Annales Henri Poincaré - Volume 9

Ann. Henri Poincaré 9 (2008), 1–33 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/010001-33, published online...

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Ann. Henri Poincaré 9 (2008), 1–33 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/010001-33, published online January 30, 2008 DOI 10.1007/s00023-007-0348-2

Annales Henri Poincar´ e

Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds Lars Andersson, Mingliang Cai, and Gregory J. Galloway Abstract. The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action.

1. Introduction Developments in string theory during the past decade, in particular the emergence of the AdS/CFT correspondence, have increased interest in the mathematical and physical properties of asymptotically hyperbolic Riemannian manifolds. Such manifolds arise naturally as spacelike hypersurfaces in asymptotically anti-de Sitter spacetimes. Asymptotically hyperbolic manifolds have a rich geometry at infinity, as exhibited by, e.g., renormalized volume and Q-curvature. The mass of an asymptotically hyperbolic manifold may, under suitable asymptotic conditions, be defined as the integral of a function defined at infinity, the so-called mass aspect function. This feature is related to the fact that, in contrast to the asymptotically Euclidean case, bounded harmonic functions on an AH manifold are not in general constant, but have nontrivial boundary values at infinity. In this paper we shall prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds. The results do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a novel deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action.

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L. Andersson, M. Cai, and G. J. Galloway

Ann. Henri Poincaré

Let (M n+1 , g) denote an (n + 1)-dimensional Riemannian manifold, and let H denote (n + 1)-dimensional hyperbolic space of curvature K = −1. As a prelude to proving positivity of mass in the asymptotically hyperbolic setting (see the discussion below), we first establish the following rigidity result. n+1

Theorem 1.1. Suppose (M n+1 , g), 2 ≤ n ≤ 6, has scalar curvature S[g] satisfying, S[g] ≥ −n(n+1), and is isometric to Hn+1 outside a compact set. Then (M n+1 , g) is globally isometric to Hn+1 . In the case that (M n+1 , g) is a spin manifold, this theorem follows from a result of Min-oo [19] (see also [2, 13]), as well as from the rigidity part of the more recently proved positive mass theorem for asymptotically hyperbolic manifolds [10, 26]. The main point of Theorem 1.1 is that it does not require a spin assumption. We note, for comparison, that there have been some other recently obtained rigidity results for hyperbolic space [4,7,21,24] that do not require a spin assumption, but these impose conditions on the Ricci curvature. The proof of Theorem 1.1 is based on the general minimal surface methodology of Schoen and Yau [22], adapted to a negative lower bound on the scalar curvature. This means, in our approach, that minimal surfaces are replaced by non-zero constant mean curvature surfaces, and the area functional is replaced by the so-called BPS brane action, as utilized by Witten and Yau [27] in their work on the AdS/CFT correspondence. From the regularity results of geometric measure theory, we require M to have dimension ≤ 7 in order to avoid the occurrence of singularities in co-dimension one minimizers of the brane action1 . In Section 2 we prove a local warped product splitting result, where the splitting takes place about a certain minimizer of the brane action. This splitting result, which extends to the case of negative lower bound on the scalar curvature previous results of Cai and Galloway [5, 6], is then used to prove Theorem 1.1. Our original motivation for proving Theorem 1.1 was to obtain a proof of positivity of mass for asymptotically hyperbolic manifolds that does not require a spin assumption. In [15], Gibbons et al. adapted Witten’s spinorial argument to prove positivity of mass in the 3 + 1 asymptotically AdS setting. More recently, Wang [26], and, under weaker asymptotic conditions, Chru´sciel and Herzlich [10] have provided precise definitions of the mass in the asymptotically hyperbolic setting and have given spinor based proofs of positivity of mass in dimensions ≥ 3. These latter positive mass results may be paraphrased as follows: Theorem 1.2. Suppose (M n+1 , g), n ≥ 2, is an asymptotically hyperbolic spin manifold with scalar curvature S ≥ −n(n + 1). Then M has mass m ≥ 0, and = 0 iff M is isometric to standard hyperbolic space Hn+1 .

1 However,

the work of Christ and Lohkamp [8, 17] offers the possibility of eliminating this dimension restriction.

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Physically, M corresponds to a maximal (mean curvature zero) spacelike hypersurface in spacetime satisfying the Einstein equations with cosmological constant Λ = −n(n + 1)/2. For then the Gauss equation and weak energy condition imply S ≥ −n(n + 1). Here we present the following version of Theorem 1.2, which does not require M to be spin. Theorem 1.3. Let (M n+1 , g), 2 ≤ n ≤ 6, be an asymptotically hyperbolic manifold with scalar curvature S[g] ≥ −n(n+ 1). Assume that the mass aspect function does not change sign, i.e., that it is either negative, zero, or positive. Then, either the mass of (M, g) is positive, or (M, g) is isometric to hyperbolic space. As noted above, the mass aspect function is a scalar function whose integral over conformal infinity determines the mass; see Section 3 for precise definitions. Our approach to proving Theorem 1.3 is inspired by Lohkamp’s variation [18] of the Schoen–Yau [23] proof of the classical positive mass theorem for asymptotically flat manifolds. Our proof makes use of Theorem 1.1, together with a deformation result, which shows roughly that if an asymptotically hyperbolic manifold with scalar curvature satisfying, S ≥ −n(n + 1), has negative mass aspect then the metric can be deformed near infinity to the hyperbolic metric, while maintaining the scalar curvature inequality. This deformation result (Theorem 3.2), along with an analysis of the case in which the mass aspect vanishes identically (Theorem 3.9), and their application to the proof of Theorem 1.3 are presented in Section 3.

2. The rigidity result The aim of this section is to give a proof of Theorem 1.1. 2.1. The brane action Let (M n+1 , g) be an (n + 1)-dimensional oriented Riemannian manifold with volume form Ω. Assume there is a globally defined form Λ such that Ω = dΛ. Let Σn be a compact orientable hypersurface in M . Then Σ is 2-sided in M ; designate one side as the “outside” and the other as the “inside”. Let ν be the outward pointing unit normal along Σ, and let Σ have the orientation induced by ν (i.e., determined by the induced volume form ω = iν Ω). Then, for any such Σ, we define the brane action B by, B(Σ) = A(Σ) − nV(Σ) , (2.1) where A(Σ) = the area of Σ, and V(Σ) = Σ Λ. If Σ bounds to the inside then, by Stokes theorem, V(Σ) = the volume of the region enclosed by Σ. Although Λ is not uniquely determined, Stokes theorem shows that, within a given homology class, B is uniquely determined up to an additive constant. We wish to consider the formulas for the first and second variation of the brane action. First, to fix notations, let A denote the second fundamental form

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of Σ; by our conventions, for each pair of tangent vectors X, Y ∈ Tp Σ, A(X, Y ) = ∇X ν, Y ,

(2.2)

where ∇ is the Levi–Civita connection of (M, g) and h = , is the induced metric on Σ. Then H = tr A is the mean curvature of Σ. variation of Σ = Σ0 , with variation Let t → Σt , − < t < , be a normal ∂ ∞ vector field V = ∂t = φν, φ ∈ C (Σ). Abusing notation slightly, set B(t) = t=0 B(Σt ). Then for first variation we have, (2.3) B (0) = (H − n)φ dA . Σ

Thus Σ is a stationary point for the brane action if and only if it has constant mean curvature H = n. Assuming Σ has mean curvature H = n, the second variation formula is given by B (0) = φL(φ) dA , (2.4) Σ

where, 1 L(φ) = − φ + (SΣ − S − |A|2 − H 2 ) φ , (2.5) 2 and where SΣ is the scalar curvature of Σ and S is the scalar curvature of M . Here L is the stability operator associated with the brane action, and is closely related to the stability operator of minimal surface theory. Using the fact that H = n, L can be re-expressed as, 1 L(φ) = − φ + (SΣ − Sn − |A0 |2 ) φ , 2

(2.6)

where Sn = S + n(n + 1) and A0 is the trace free part of A, A0 = A − h. We note that, in our applications, Sn will be nonnegative. A stationary point Σ for the brane action is said to be B-stable provided for all normal variations t → Σt of Σ, B (0) ≥ 0. For operators of the form (2.6), the following proposition is well-known. Proposition 2.1. The following conditions are equivalent. 1. Σ is B-stable. 2. λ1 ≥ 0, where λ1 is the principal eigenvalue of L. 3. There exists φ ∈ C ∞ (Σ), φ > 0, such that L(φ) ≥ 0. In particular, if λ1 ≥ 0, φ in part 3 can be chosen to be an eigenfunction. 2.2. Warped product splitting In this section we prove the local warped product splitting result alluded to in the introduction. As a precursor, we prove the following infinitesimal rigidity result.

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Proposition 2.2. Let (M n+1 , g) be an oriented Riemannian manifold with scalar curvature S satisfying, S ≥ −n(n + 1) . (2.7) n Let Σ be a compact orientable B-stable hypersurface in M which does not admit a metric of positive scalar curvature. Then the following must hold. (i) Σ is umbilic, in fact A = h, where h is the induced metric on Σ. (ii) Σ is Ricci flat and S = −n(n + 1) along Σ. Proof. By Proposition 2.1, there exists φ ∈ Σ, φ > 0, such that L(φ) ≥ 0. The 2 ˜ = φ n−2 h is then given scalar curvature S˜ of Σ in the conformally rescaled metric h by, n n − 1 |∇φ|2 S˜ = φ− n−2 −2 φ + SΣ φ + n−2 φ 2 n − 1 |∇φ|2 (2.8) = φ− n−2 2φ−1 L(φ) + Sn + |A0 |2 + n − 2 φ2 where, for the second equation, we have used (2.6) with f = φ. Since all terms in parentheses above are nonnegative, (2.8) implies that S˜ ≥ 0. If S˜ > 0 at ˜ to some point, then by well known results [16] one can conformally change h a metric of strictly positive scalar curvature, contrary to assumption. Thus S˜ vanishes identically, which implies L(φ) = 0, Sn = 0, A0 = 0 and φ is constant. Equation (2.6), with f = φ then implies that S ≡ 0. By a result of Bourguignon (see [16]), it follows that Σ carries a metric of positive scalar curvature unless it is Ricci flat. Thus conditions (i) and (ii) are satisfied. Proposition 1.1 will be used in the proof of the following local warped product splitting result. Theorem 2.3. Let (M n+1 , g) be an oriented Riemannian manifold with scalar curvature S ≥ −n(n + 1). Let Σ be a compact orientable hypersurface in M which does not admit a metric of positive scalar curvature. If Σ locally minimizes the brane action B then there is a neighborhood U of Σ such that (U, g|U ) is isometric to the warped product ((−, ) × Σ, dt2 + e2t h), where h, the induced metric on Σ, is Ricci flat. By “locally minimizes” we mean, for example, that Σ has brane action less than or equal to that of all graphs over Σ with respect to Gaussian normal coordinates. A related result has been obtained by Yau [28] in dimension three. Proof. Let H(u) denote the mean curvature of the hypersurface Σu : x → expx u(x)ν, u ∈ C ∞ (Σ), u sufficiently small. H has linearization H (0) = L, where L is the B-stability operator (2.6). But by Proposition 2.2, L reduces to − , and hence H (0) = − . We introduce the operator, ∗ ∞ ∞ ∗ H : C (Σ) × R → C (Σ) × R , H (u, k) = H(u) − k, u , (2.9) Σ

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which one easily checks has invertible linearization about (0, 0), since the kernel of H (0) contains only the constants. By the inverse function theorem, for each τ sufficiently small there exists u = uτ and k = kτ such that H(uτ ) = kτ and u dA = τ . Since u (0) ∈ ker H (0), the latter equation implies that u (0) = Σ τ const > 0. Thus for τ sufficiently small, the hypersurfaces Σuτ form a foliation of a neighborhood U of Σ by constant mean curvature hypersurfaces. Using coordinates on Σ and the normal field to the Σuτ ’s to transport these coordinates to each Σuτ , we have, up to isometry, U = (−, ) × Σ

g|U = φ2 dt2 + ht ,

(2.10)

where ht = hij (t, x)dxi dxj , φ = φ(t, x) and Σt = {t} × Σ has constant mean curvature. Since Σ locally minimizes the brane action, we have, B(0) ≤ B(t) for all t ∈ (−, ), for sufficiently small. Let H(t) denote the mean curvature of Σt . H = H(t) obeys the evolution equation, dH = L(φ) , (2.11) dt where for each t, L is the operator on Σt given in Equation (2.5). Since Σ locally minimizes the brane action, we have H(0) = n. We show H ≤ n for t ∈ [0, ). If this is not the case, there exists t0 ∈ (0, ) such that H(t0 ) > n. Moreover, t0 can be chosen so that H (t0 ) > 0. Let S˜ be the scalar curvature of Σt0 in the 2 conformally related metric ˜ h = φ n−2 ht0 . Arguing similarly as in the derivation of (2.8), Equations (2.5) and (2.11) imply, 2 n − 1 |∇φ|2 − n−2 −1 2 2 ˜ S=φ 2φ H (t0 ) + S + |A| + H + , (2.12) n − 2 φ2 where all terms are evaluated on Σt0 . The Schwartz inequality gives, |A|2 ≥ H 2 /n > n. This, together with the assumed scalar curvature inequality (2.7), implies that S + |A|2 + H 2 > 0. We conclude from (2.12) that Σt0 carries a metric of positive scalar curvature, contrary to assumption. Thus, H ≤ n on [0, ), as claimed. Now, by the formula for the first variation of the brane action, it follows that (H − n)φ dA ≤ 0 , for all t ∈ [0, ) . (2.13) B (t) = Σt

But since B achieves a minimum at t = 0, it must be that B (t) = 0 for t ∈ [0, ). Hence, the integral in (2.13) vanishes, which implies that H = n on [0, ). A similar argument shows that H = n on (−, 0], as well. Equation (2.11) then implies that L(φ) = 0 on each Σt . Hence, by Proposition 2.1, each Σt is B-stable. From Proposition 2.2, we have that At = ht , where At is the second fundamental form of Σt , and that φ only depends on t. By a simple change of t-coordinate in (2.10), we may assume without loss of generality that φ = 1. Then the condition ∂h At = ht becomes, in the coordinates (2.10), ∂tij = 2hij . Upon integration this 2t gives, hij (t, x) = e hij (0, x), which completes the proof of the theorem.

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2.3. Proof of the rigidity result In order to prove Theorem 1.1 it is convenient to work with an explicit representation of hyperbolic space Hn+1 . We start with the half-space model (H n+1 , gH ), where, H n+1 = {(y, x1 , . . . , xn ) : y > 0}, and 1 (2.14) gH = 2 dy 2 + (dx1 )2 + · · · + (dxn )2 , y and make the change of variable y = e−t , to obtain Hn+1 = (Rn+1 , g0 ), where, (2.15) g0 = dt2 + e2t (dx1 )2 + · · · + (dxn )2 . As in the statement of Theorem 1.1, let (M n+1 , g) be a Riemannian manifold with scalar curvature S[g] satisfying S[g] ≥ −n(n + 1). We assume that there are compact sets K ⊂ M , K0 ⊂ Rn+1 such that M − K is diffeomorphic to Rn+1 − K0 , and, with respect to Cartesian coordinates (t, x1 , . . . , xn ) on the complement of K, g = g0 . We want to show that (M n+1 , g) is globally isometric to Hn+1 . Since M is simply connected near infinity, it is in fact sufficient to show that M has everywhere constant curvature KM = −1. Our approach to proving this is to partially compactify (M, g) and then minimize the brane action in a suitable homology class. To partially compactify, we use the fact that the translations xi → xi + xi0 are isometries on (Rn+1 , g0 ). Choosing a > 0 sufficiently large, we can enclose the compact set K in an infinitely long rectangular box, with sides determined by the “planes”, xi = ±a, i = 1, . . . , n, −∞ < t < ∞. We can then identify points on opposite sides, xi = −a, xi = a, i = 1, . . . , n, of the box in the obvious manner to ˆ , gˆ). Note that outside the obtain an identification space which we denote by (M compact set K, ˆ = R × T n , gˆ = dt2 + e2t h , M (2.16) n ˆ , gˆ) is just a standard hyperbolic where h is a flat metric on the torus T . Thus, (M cusp outside the compact set K, with scalar curvature satisfying S[ˆ g] ≥ −n(n + 1) globally. ˆ bounded Choose b > 0 large so that K is contained in the region of M between the toroidal slices t = ±b, and fix a t-slice Σ0 = {t0 } × T n , t0 > b. Σ0 ˆ into an “inside” and an “outside”, the inside being the component separates M ˆ of M − Σ0 containing the cusp end t = −∞. We consider the brane action of hypersurfaces Σ homologous to Σ0 , B(Σ) = A(Σ) − nV (Σ) .

(2.17)

We note, as is needed to define B(Σ) unambiguously, that since Σ is homologous to Σ0 it, too, has a distinguished “inside” and “outside”, determined by the fact that both Σ0 and Σ are homologous to a t-slice far out on the cusp end. We now want to minimize the brane action B in the homology class [Σ0 ]. The basic approach is to consider a minimizing sequence Σ1 , Σ2 , . . . and use the compactness results of geometric measure theory to extract a regular limit surface. The potential difficulty with this approach is that, in principle, the surfaces

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Σ

Σ

U

ˆ Σ

K

Σ

Σ0 t ˆ Figure 1. Replacing Σ by Σ.

ˆ. Σ1 , Σ2 , . . ., or portions of them, may drift out to infinity along either end of M But, in fact, that can be avoided in the present situation, owing to the existence of natural barrier surfaces, namely the t-slices themselves. Fix t-slices Σ = {t1 } × Σ, t1 > t0 , and Σ = {t2 } × Σ, t2 < −b. We show that any minimizing sequence can be replaced by a minimizing sequence contained in the region between Σ and Σ . To this end, consider a hypersurface Σ homologous to Σ0 that extends beyond Σ into the region t < t2 . Without loss of generality we may assume Σ meets Σ transversely. Let D be the part of Σ meeting {t ≤ t2 }, and let U be the domain bounded by Σ and D. Then ∂U consists of D and a ˆ be the hypersurface homologous to Σ0 obtained from Σ by part D of Σ . Let Σ replacing D with D (see Figure 2). Since U is contained in a region where the metric (2.16) applies, and since in this region div(∂t ) = n, we apply the divergence theorem to obtain, ˆ nV(Σ) − nV(Σ) = n vol(U ) = div(∂t )dV U = ∂t , ndA + ∂t , ∂t dA D

D

ˆ . ≥ −A(D) + A(D ) = −A(Σ) + A(Σ) ˆ ≤ B(Σ). By a similar argument Rearranging this inequality gives the desired, B(Σ) the same conclusion holds if Σ extends beyond Σ into the region {t > t1 }. Thus, we can choose a minimizing sequence Σi , for the brane action within the homology class [Σ0 ] that is confined to the compact region between Σ and Σ . Since V(Σi ) ≤ V(Σ ), we are ensured that limi→∞ B(Σi ) > −∞. Then the compactness and regularity results of geometric measure theory (see, e.g., [14, 22] and

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references therein) guarantee the existence of a regular embedded hypersurface S homologous to Σ0 that achieves a minimum of the brane action on [Σ0 ]. In general S will be a sum of connected embedded surfaces, S = S1 + · · · + Sn . The next thing we wish to observe is that there is a nonzero degree map from S to the n-torus Σ0 . This map comes from the ‘almost product’ structure ˆ given in (2.10). A simple deformation of the t-lines in the vicinity of K of M ˆ → Σ0 such can be used to produce a continuous projection type map P : M that K gets sent to a single point on Σ0 under P , and such that P ◦ j = id, ˆ is inclusion. Then f = P ◦ i : S → Σ0 , where i : S → M ˆ where j : Σ0 → M is inclusion, is the desired nonzero degree map. Indeed, f induces the map on homology f∗ : Hn (S) → Hn (Σ0 ), and using that S is homologous to Σ0 , we compute, f∗ [S] = P∗ (i∗ [S]) = P∗ (j∗ [Σ0 ]) = id∗ [Σ0 ] = Σ0 = 0. Thus, by linearity of f∗ , at least one of the components of S, S1 , say, admits a nonzero degree map to the n-torus. By a result of Schoen and Yau [22], which does not require a spin assumption, S1 does not admit a metric of positive scalar curvature. (In fact it admits a metric of nonnegative scalar curvature only if it is flat). Moreover, we know that S1 minimizes the brane action in its homology class (otherwise there would exist a hypersurface homologous to Σ0 with brane action strictly less than that of S). Thus, we can apply Theorem 2.3 to conclude that a neighborhood U of S1 splits as a warped product, U = (−u0 , u0 ) × S1

gˆ|U = du2 + e2u h ,

(2.18)

where the induced metric h on S1 is flat. But since S1 in fact globally maximizes the brane action in its homology class, by standard arguments this local warped product structure can be extended to arbitrarily large u-intervals. Hence K will eventually be contained in this constructed warp product region. It now follows ˆ has constant curvature K ˆ = −1. This in turn implies that M has conthat M M stant curvature KM = −1. By previous remarks, we conclude that M is globally isometric to hyperbolic space.

3. Positivity of mass The aim of this section is to give a proof of Theorem 1.3 on the positivity of mass in the asymptotically hyperbolic setting. We shall adopt here the definition of asymptotically hyperbolic given in Wang [26]: Definition 3.1. A Riemannian manifold (M n+1 , g) is asymptotically hyperbolic provided it is conformally compact, with smooth conformal compactification ˜ , g˜), and with conformal boundary ∂ M ˜ = S n , such that the metric g on a (M n ˜ deleted neighborhood (0, T ) × S of ∂ M = {t = 0} takes the form g = sinh−2 (t)(dt2 + h) ,

(3.19)

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where h = h(t, ·) is a family of metrics on S n , depending smoothly on t ∈ [0, T ), of the form, h = h0 + tn+1 k + O(tn+2 ) , (3.20) n n where h0 is the standard metric on S , and k is a symmetric 2-tensor on S . We refer to k as the mass aspect tensor; it is the leading order measure of the deviation of the metric g from the hyperbolic metric. Its trace with respect to h0 , trh0 k, is called the mass aspect function. Up to a normalizing constant, the integral of the mass aspect function over the sphere defines the mass (or energy) of (M, g), mass = S n trh0 k. For convenience, we repeat here the statement of our positivity of mass result. Theorem 3.1. Let (M n+1 , g), 2 ≤ n ≤ 6, be an asymptotically hyperbolic manifold with scalar curvature S[g] ≥ −n(n + 1). Assume that the mass aspect function trh0 k does not change sign, i.e., that it is either negative, zero, or positive. Then, either the mass of (M, g) is positive, or (M, g) is isometric to hyperbolic space. The proof, which makes use of the rigidity result Theorem 1.1, is carried out in the next two subsections. In Subsection 3.1 we obtain the deformation result mentioned in the introduction, see Theorem 3.2 below. This, together with Theorem 1.1, implies that the mass aspect function cannot be negative, see Proposition 3.3. In Subsection 3.2, it is proved, using Theorem 1.1 again, that if the mass aspect function vanishes then (M, g) is isometric to hyperbolic space, see Theorem 3.9. These results together imply Theorem 3.1. 3.1. The deformation result Suppose (M n+1 , g) is asymptotically hyperbolic in the sense of Definition 3.1. Then by making the change of coordinate, t = arcsinh( 1r ), in (3.19) it follows that there is a relatively compact set K such that M \ K = S n × [R, ∞), R > 0, and on M \ K, g has the form, 1 dr2 + r2 h , (3.21) g= 1 + r2 where h = h(·, r) is an r-dependent family of metrics on S n of the form, 1 (3.22) h = h0 + n+1 k + σ , r where h0 is the standard metric on S n , k is the mass aspect tensor and σ = σ(·, r) is an r-dependent family of metrics on S n such that for integers , m ≥ 0, one has, |(r∂r ) ∂xm σ| ≤ C/rn+2 ,

(3.23)

for some constant C. For the proof of the deformation theorem and the positive mass theorem, it is sufficient to assume condition (3.23) for 0 ≤ , m ≤ 2. Let (M n+1 , g) be asymptotically hyperbolic, with scalar curvature satisfying, S[g] ≥ −n(n + 1). What we now prove is that if the mass aspect function of (M, g) is pointwise negative then g can be deformed on an arbitrarily small neighborhood of infinity to the hyperbolic metric, while preserving (after a change of scale) the

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scalar curvature inequality S ≥ −n(n + 1). A more precise statement is given below. Theorem 3.2. Let the metric g be given as above. Suppose that the scalar curvature of g, S[g], satisfies S[g] ≥ −n(n+1). If the mass aspect function trh0 k is pointwise negative, then for any sufficiently large R1 > R there exists a metric gˆ on M such that, g, R ≤ r ≤ R1 (3.24) gˆ = 1 2 2 ga = dr + r h0 , 9λR1 ≤ r < ∞ , r2 1+

a

where a ∈ (0, 1), and such that, n(n + 1) . (3.25) a (The constant λ > 1 depends only on the mass aspect function; see Section 3.1.1.) S[ˆ g] ≥ −

Theorems 1.1 and 3.2 may be combined to give the following result. Proposition 3.3. Let (M n+1 , g), 2 ≤ n ≤ 6, be an asymptotically hyperbolic manifold with scalar curvature satisfying, S[g] ≥ −n(n + 1). Then the mass aspect function trh0 k cannot be everywhere pointwise negative. Proof. Suppose to the contrary that the mass aspect function is strictly negative. Given any p ∈ M , choose R large enough so that p is not in the end S n × [R, ∞). Theorem 3.2, together with a rescaling of the metric, implies the existence of a metric g˜ on M such that (M, g˜) satisfies the hypotheses of Theorem 1.1. Hence, (M, g˜) is isometric to hyperbolic space Hn+1 . But, modulo the change of scale, by our construction, g˜ will differ from g only at points on the end S n × [R, ∞). It follows that (M n+1 , g) has constant negative curvature curvature in a neighborhood of p. Since p is arbitrary, (M n+1 , g) must have globally constant negative curvature, which, by the asymptotics of (M n+1 , g), must equal −1. Since M is simply connected at infinity, we conclude that (M n+1 , g) is isometric to hyperbolic space Hn . But this contradicts the assumption that the mass aspect function is negative. Proof of Theorem 3.2. We now turn to the proof of the deformation result. Introduce coordinates x = (x1 , x2 , . . . , xn ) on S n . We use the convention that for a function f = f (x, r), f (x, r) = ∂r f (x, r), and f (x, r) = ∂r2 f (x, r). Let ωij and kij be the components of h0 and k, respectively, with respect to the coordinates (x1 , x2 , . . . , xn ). Then g in (3.21) takes the form

dr2 αij g= + r2 ωij + n+1 dxi dxj 2 1+r r where αij = kij (x) +

βij (x, r) . r

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We are assuming (cf., (3.23)) αij satisfies the bounds, |(r∂r ) ∂xm αij | ≤ Λ .

(3.26)

for all integers , m, 0 ≤ , m ≤ 2. Let μ denote the the mass aspect function, μ = trh0 k = ω ij kij ; by assumption, |μ| > 0. Let μ ¯ = maxx |μ(x)|, μ = minx |μ(x)|. We shall be making estimates of geometric quantities in terms of the above defined constants. In particular, we shall use a generic constant C = C(n, R, Λ, μ ¯, μ) depending only on n, R, Λ, μ ¯, μ, which may change from line to line. We shall further use the notation O(1/rk ) for a quantity bounded by C(n, R, Λ, μ ¯ , μ)/rk . 3.1.1. Preliminary definitions. Fix R1 > R to be specified later. Set 1 n+1 μ ¯ λ= . μ Let a = a(n, μ ¯, μ, R1 ) ∈ (0, 1) be a number such that μ 1 n+1 4 μ ¯ n+1 3 < . − 1 < n 3 (4λR1 )n+1 a n 4 (3λR1 )n+1 To show such an a exists, it suffices to show that μ μ ¯ n+1 4 n+1 3 < n+1 n 3 (4λR1 ) n 4 (3λR1 )n+1 or equivalently n+1 3 4 μ ¯ λ > . 4 3 μ By our choice of λ, this is equivalent to n 4 >1 3 which is obviously true. It follows from the definition that a 1 as R1 ∞. It is straightforward to show the existence of a smooth function ψ : R → R+ such that for any R1 > R.

1 , r ≤ 7λR1 ψ= (3.27a) 0 , r ≥ 8λR1 ψ (r) ≤ 0 for all r

(3.27b)

b (3.27c) r c |ψ (r)| ≤ 2 (3.27d) r where b, c are positive constants. In the following, we consider a fixed function ψ satisfying the conditions (3.27). |ψ (r)| ≤

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In the deformation construction, we shall consider functions f : S n ×[R, ∞) → R, satisfying the following conditions. f (x, R) = 1 ,

for x ∈ S n ,

(3.28a)

and for (x, r) ∈ S n × [R, ∞) the conditions 1 ≤f ≤2 2 1 |f (x, r)| ≤ 2 r 1 |ΔS n f | ≤ n r n 1 |∇S f |2 ≤ n . r In particular, the constant function f ≡ 1 satisfies conditions (3.28). In carry out the deformation from the metric g to the metric ga , we shall metrics of the form, 1 ψαij 2 2 gf,ψ = dr + r ωij + n+1 dxi dxj . 1 + r2 f r

(3.28b) (3.28c) (3.28d) (3.28e) order to consider

(3.29)

Given ψ, the main objective is to construct an f satisfying (3.28) so that gf,ψ has the required properties. 3.1.2. Scalar curvature formulas. We need formulas for the scalar curvature S[gf,ψ ] of the metric gf,ψ . Lemma 3.4. Let f, ψ satisfy the assumptions (3.28) and (3.27). Then the metric gf,ψ has scalar curvature, 1 S[gf,ψ ] = −n(n + 1)f − nrf + n (n|μ|ψ − r|μ|ψ )f + J , (3.30) r where J is a term bounded by C(n, R, Λ, μ ¯, μ)/rn+2 , and such that J = 0 for r ≥ 8λR1 . Proof. We describe our approach to carrying out this computation. Setting, ψαij h = 1 + r2 f and gij = r2 ωij + n+1 , (3.31) r gf,ψ becomes, 1 2 dr + gij dxi dxj . h Applying the Gauss equation to an r-slice Σ = S n × {r} gives, gf,ψ =

S[gf,ψ ] = SΣ + |B|2 − H 2 + 2Ric(N, N ) ,

(3.32)

(3.33)

where SΣ , B and H are the scalar curvature, second fundamental form and mean curvature of Σ, respectively, and Ric(N, N ) is the ambient Ricci curvature in the ∂ direction of the unit normal N = h1/2 ∂r . In terms of coordinates, B and H are

14



√ given by, bij = B(∂i , ∂j ) = 12 h ∂r gij , and H = g ij bij . We then compute each term in (3.33) in turn. For the first three terms we obtain, making use of the bounds (3.26) and (3.27) 1 n(n − 1) SΣ = + O (3.34) r2 rn+3 1 1 n n + 1 |μ|ψ 1 |μ|ψ + − + O (3.35) H = h2 r 2 rn+2 2 rn+1 rn+3 n 1 |μ|ψ |μ|ψ |B|2 = Bi j Bj i = h 2 + (n + 1) n+3 − n+2 + O . (3.36) n+4 r r r r Equations (3.35) and (3.36) combine to give, n(n − 1) |μ|ψ − (n − 1)(n + 1) n+3 |B|2 − H 2 = h − r2 r |μ|ψ 1 . + (n − 1) n+2 + O r rn+4

(3.37)

∂ Applying the Raychaudhuri equation to the unit normal N = h1/2 ∂r to the level sets of r, we have, √ 1 (3.38) Ric(N, N ) = −N (H) − |B|2 − hΔΣ √ . h Making use of equations (3.35) and (3.36), we derive from (3.38), 1 n n + 1 |μ|ψ 1 1 |μ|ψ Ric(N, N ) = − h + − + O 2 r 2 rn+2 2 rn+1 rn+3 1 n(n + 1) |μ|ψ |μ|ψ 1 |μ|ψ − n + + O +h 2 rn+3 rn+2 2 rn+1 rn+4 √ 1 (3.39) − hΔΣ √ . h Equations (3.34), (3.37) and (3.39) then combine to give, n + 1 |μ|ψ 1 n(n − 1) n(n − 1) n 1 |μ|ψ S[gf,ψ ] = h − h − − − + O h r2 r2 r 2 rn+2 2 rn+1 rn+3 1 |μ|ψ |μ|ψ |μ|ψ + (n + 1) n+3 − (n + 1) n+2 + n+1 + O h r r r rn+4 √ 1 1 − 2 h Σ √ + O . (3.40) rn+3 h

Setting h = 1 + r2 f in the above, and making use of the bounds (3.28) and (3.27), one derives in a straight forward manner equation (3.30). Moreover, it is clear from the computations that all ‘big O’ terms vanish once ψ vanishes. The following Corollary gives the form of the scalar curvature which will be used in the deformation construction.

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Corollary 3.5. Let f, ψ satisfy the assumptions (3.28) and (3.27). Then, there is a nonnegative function A1 : [R, ∞) → R+ , independent of f , such that A1 ≤ C(n, R, Λ, μ ¯ , μ)

and

A1 = 0

for

r ≥ 9λR1 ,

and such that the scalar curvature S[gf,ψ ] of gf,ψ satisfies the inequality r n n+1 A1 (r) n+1 S[gf,ψ ] ≥ − n |μ|ψ − |μ|ψ f − + . r r n n r

(3.41)

(3.42)

Proof. Using the product rule, equation (3.30) may be expressed as, n 1 S[gf,ψ ] = − n (rn+1 f ) + n (n + 1)|μ|ψ − r|μ|ψ f r r 1 1 − n (n + 1)|μ|ψ − r|μ|ψ f + O r rn+2 which, by the bounds (3.28) and (3.27) simplifies to, 1 r n n+1 n+1 S[gf,ψ ] = − n |μ|ψ + |μ|ψ f + O − r . r n n rn+2

(3.43)

It follows that there exists a smooth function A : [R, ∞) → R+ , satisfying, A ≤ C(n, R, Λ, μ ¯, μ) and A = 0 for r ≥ 9λR1 , such that, r n n+1 A(r) n+1 S[gf,ψ ] ≥ − n |μ|ψ + |μ|ψ f − n+2 . − (3.44) r r n n r Now define A1 : [R, ∞) → R+ by, A1 (r) =

r n

r

9λR1

A(t) dt . t2

One easily checks that the properties (3.41) hold. Moreover, since, A1 (r) 1 A(r) =− , r n r2

(3.45)

(3.46)

inequality (3.42) follows from (3.44). Finally, it is clear from the construction that A and hence also A1 may be chosen to be independent of f . 3.1.3. Defining η: Rounding the corner. Let η1 (x, r), η2 (r) be given by η1 (x, r) = rn+1 + η2 (r) =

r

r n+1 A1 (r) |μ|ψ − |μ|ψ − , n n r

(3.47a)

n+1

, (3.47b) a where the function A1 appearing in η1 is the function A1 defined in Corollary 3.5. By (3.42), S[g1,ψ ] ≥ − rnn η1 . Further, ga has constant curvature −1/a, and hence S[ga ] = − rnn η2 . In the rest of the argument we shall be choosing R1 sufficiently large, so that the required conditions are satisfied. We shall successively increase R1 as required.

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η η1 η2 R1

3λR1

4λR1

7λR1

Figure 2. Construction of η. Lemma 3.6. There is an R1 > R, R1 = R1 (n, R, Λ, μ ¯ , μ) such that the following inequalities hold. η1 (x, r) < η2 (r) ,

R1 ≤ r ≤ 7λR1

(3.48a)

η1 (x, r) − η2 (r) > μ/20 ,

R1 ≤ r ≤ 3λR1

(3.48b)

η2 (r) − η1 (x, r) > μ ¯/10 ,

4λR1 ≤ r ≤ 7λR1 .

(3.48c)

Proof. We need to consider only the interval R1 ≤ r ≤ 7λR1 . There, ψ ≡ 1, and ψ ≡ 0. We start by proving (3.48a). By construction, we have for r ≥ R1 , using |μ| > 0 and the properties of a, after some manipulations, 1 A(r) 1 − 1 (n + 1)rn+1 − η2 − η1 > r a nr 2 μ ¯ 4 (n + 1) 1 A(r) . > − r 3 n (4λ)n+1 nr Since A(r) ≤ C(n, Λ, R, μ ¯, μ), it follows that (3.48a) holds for R1 sufficiently large. Next we prove (3.48b). Since (η1 − η2 ) < 0 for r ≥ R1 , it is sufficient to prove the inequality at r = 3λR1 . We have μ 1 n+1 (3λR1 )n+1 1− η1 (x, 3λR1 ) − η2 (3λR1 ) ≥ + a n (3λR1 )n+1 A1 (3λR1 ) − , 3λR1 which by using the properties of a and simplifying gives, 3 n+1 A1 (3λR1 ) μ− > 1− . 4 n 3λR1

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One checks that 1 − 3/4 > 1/10. Thus, in view of the fact that A1 ≤ C(n, R, Λ, μ ¯ , μ), by possibly increasing R1 we see that (3.48b) can be made to hold. We proceed in a similar fashion to prove (3.48c). We have 1 n+1 μ ¯ −1 − η2 (4λR1 ) − η1 (x, 4λR1 ) ≥ (4λR1 )n+1 a n (4λR1 )n+1 n+1 4 −1 μ ¯. ≥ 3 n We note that 4/3 − 1 > 1/10. This completes the proof of Lemma 3.6. We shall define η = η(x, r) to be a suitably increasing function of the variable r that smoothly transitions from η1 to η2 (see Figure 1). In order to make meaningful estimates, its construction shall be made fairly explicit. Its construction depends on two auxiliary functions α and β, which we now introduce. Let α : [0, ∞) → R be a function satisfying, 1. α(r) = 0 for r ≤ R1 , 2R1 ≤ r ≤ 5λR1 , and r ≥ 6λR1 . 2. α > 0 for R1 < r < 2R1 , α < 0 for 5λR1 < r < 6λR1 , and 6λR1 ∞ α(t)dt = α(t)dt = 0 .

(3.49)

−∞

R1

Consider, α(r) . η1 (x, r) − η2 (r) It follows from Lemma 3.6 and the properties of α that γ is nonnegative and bounded, 0 ≤ γ ≤ C(n, Λ, μ, μ ¯) . γ(x, r) =

Next, define m(x) by the condition 1 + m(x)

6λR1

γ(x, t) dt = 0 ,

(3.50)

R1

and let

r

γ(x, t) dt .

β(x, r) = 1 + m(x) R1

Then, β satisfies, β = 1,

for r ≤ R1

(3.51a)

β = 0,

for r ≥ 6λR1

(3.51b)

for all r .

(3.51c)

0 ≤ β ≤ 1,

Conditions (3.51a), (3.51b) are clear. For (3.51c), we consider, β = mγ. As observed above, γ ≥ 0, and hence from (3.50), m ≤ 0, so that mγ ≤ 0. Hence β is decreasing, which implies that 0 ≤ β ≤ 1.

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Now we are ready to define η. Let r β(x, t)η1 (x, t) + 1 − β(x, t) η2 (t) dt . η(x, r) = η1 (x, R1 ) + R1

Lemma 3.7. η defined as above satisfies the conditions

η1 (x, r) r ≤ R1 n+1 η(x, r) = η2 (r) = r a , r ≥ 6λR1 η (x, r) ≤ η2 (r) =

(n + 1)rn , a

Proof. For r ≤ R1 , we have β = 1, so

r

η(x, r) = η1 (x, R1 ) + R1

(3.52a)

R1 ≤ r < ∞ .

(3.52b)

η1 (x, t)dt = η1 (x, r) .

In the following calculations, which only involve derivatives and integrals with respect to r, we suppress reference to x in order to avoid clutter. For r ≥ 6λR1 , we have β = 0, which gives r 6λR1 η(r) = η1 (R1 ) + βη1 + (1 − β)η2 dt + η2 dt (3.53) R1

= η2 (r) − η2 (6λR1 ) + η1 (R1 ) +

6λR1

6λR1

R1

βη1 + (1 − β)η2 dt .

A partial integration gives 6λR1 βη1 + (1 − β)η2 dt R1

(3.54)

(3.55)

r=6λR1 − = η2 (6λR1 ) − η2 (R1 ) + β(η1 − η2 ) r=R1

6λR1

β (η1 − η2 )dt (3.56)

R1

use the properties of β and γ = η2 (6λR1 ) − η2 (R1 ) − η1 (R1 ) − η2 (R1 ) − m

6λR1

αdt

(3.57)

R1

use (3.49) = η2 (6λR1 ) − η1 (R1 ) .

(3.58)

Substituting this into the formula for η(x, r), we obtain, η(x, r) = η2 (r) ,

for

r ≥ 6λR1 .

We have now established (3.52a). Next we prove (3.52b). We have η = βη1 + (1 − β)η2

(3.59)

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and hence, by (3.48), η ≤ βη2 + (1 − β)η2 = η2 =

(3.60)

n

(n + 1)r . a

(3.61)

We shall now make use of the function η defined above to define a function f , which shall be shown to satisfy the conditions (3.28). This fact allows us to apply the result of Corollary 3.5 to estimate the scalar curvature of the deformed metric gf,ψ defined in terms of this f . 3.1.4. Defining f . We define f by, f=

A1 r η1 + Ar1

η+

,

(3.62)

which implies,

r n+1 A1 |μ|ψ − |μ|ψ f − = η. (3.63) rn+1 + n n r Here, A1 = A1 (r) is the function determined in Corollary 3.5. It is crucial to note here that A1 is independent of the particular f , as long as it satisfies the conditions (3.28). Our task is now to show that f defined as above does satisfy these conditions as long as R1 is chosen sufficiently large. This will be demonstrated in Lemma 3.8 below. It then follows from equation (3.63) above, Corollary 3.5 and (3.52b) that S[gf,ψ ] satisfies, n n(n + 1) . η ≥− rn a In addition we have, gf,ψ = g on [R, R1 ] and gf,ψ = ga on [9λR1 , ∞). Thus, subject to the following lemma, Theorem 3.2 has been proven. S[gf,ψ ] ≥ −

Lemma 3.8. Let η be as in Section 3.1.3, and let f be given in terms of η by (3.62). ¯, μ) sufficiently large, so that the inequaliThen, there is an R1 = R1 (n, R, Λ, μ ties (3.28) are valid. Proof. The condition (3.28a) is clear from the construction. Since (3.28c) implies (3.28b) we only need to verify that |f | ≤ 1/r2 and |∂x f | ≤ 1/rn , |∂x2 f | ≤ 1/rn . We begin by showing there is an R1 sufficiently large, and not smaller than the previously made choices of R1 , so that |f | ≤ 1/r2 . We have, f =1+ This gives f =

(η − η1 ) . η1 + A1 /r

(η − η1 ) (η − η1 ) − η + (A1 /r) . η1 + A1 /r (η1 + A1 /r)2 1

(3.64)

20


Since

r

η(x, r) = η1 (x, R1 ) + R1 r

η1 (x, r) = η1 (x, R1 ) +

R1

we have,

r

η − η1 =


βη1 + (1 − β)η2 dt , η1 dt

(1 − β)(η2 − η1 ) .

R1

so

(η − η1 ) = (1 − β)(η2 − η1 ) .

Hence,

|(η − η1 ) | ≤ (η2 − η1 ) =

1 − 1 rn (n + 1) + O(1/r) a

(3.65)

C . (3.66) r Here and below C = C(n, R, Λ, μ ¯, μ) is a generic constant. Recall that by construction we have C (A1 /r) ≤ 2 . r Combining the above inequalities, we have (η − η1 ) C (3.67) η1 + A1 /r ≤ rn+2 . ≤

Now since |(η − η1 ) | ≤

C r

we have |η − η1 | ≤ C for r ∈ [R1 , 9λR1 ]. This together with

(3.68)

|η1 | ≤ Crn implies that

(η − η1 ) C η + (A /r) 1 (η1 + A1 /r)2 1 ≤ rn+2 . Equations (3.67) and (3.69) imply C |f | ≤ n+2 r for r ∈ [R1 , 9λR1 ]. By choosing R1 large enough, we have 1 |f | ≤ 2 r which gives (3.28c).

(3.69)

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Next we demonstrate that by, if necessary, further increasing R1 , we can ensure that the condition 1 |∂xk f | ≤ n , k = 1, 2 , r holds, where ∂x denotes partial differentiation with respect to any one of the coordinates xi . Recalling (3.64), we have ∂x f =

η − η1 ∂x (η − η1 ) − (∂x η1 ) . η1 + A1 /r (η1 + A1 /r)2

We estimate the second term first. By (3.68), |η − η1 | ≤ C, so 1 η − η1 (η1 + A1 /r)2 ∂x η1 ≤ r2n+2 |∂x η1 | . Recalling r n+1 A1 |μ|ψ − |μ|ψ − n n r we find |∂x η1 | ≤ C. Hence the modulus of the second term in ∂x f is bounded by C/r2n+2 . Now consider the first term in ∂x f . Recall r η − η1 = (1 − β)(η2 − η1 ) dt R1 r r = (1 − β)(η2 − η1 ) + (η2 − η1 )β dt η1 = rn+1 +

R1

R1

r

= [1 − β](η2 − η1 ) +

(η2 − η1 )β dt

R1

so ∂x (η − η1 ) = −(∂x β)(η2 − η1 ) − (1 − β)∂x η1 r

∂x (η2 − η1 ) β + (η2 − η1 )∂x β dt . + R1

Now,

r

∂x β = (∂x m) R1

Recall,

α η1 − η2

6λR1

1+m R1

r

+m

∂x

R1

α η1 − η2

α =0 η1 − η2

so 1 m = − 6λR1 R1

α η1 −η2

= O(1/R1 ) .

.

22


Hence,

6λR1 R1

∂x

∂x m = − 6λR1 R1

One can see that ∂x so

α η1 − η2


α η1 −η2 2 α η1 −η2

.

= O(1) ,

∂x m = O(1/R1 ) . Hence we have, using m = O(1/R1 ), ∂x β = O(1) and therefore ∂x β(η2 − η1 ) = O(1) . Next we consider the second term in ∂x f . We have |∂x η1 | ≤ C , so (1 − β)∂x η1 = O(1) . Since m = O(1/R1 ), we have, β = m

α = O(1/R1 ) η1 − η2

but, ∂x (η2 − η1 ) = O(1) so,

∂x (η2 − η1 )β = O(1/R1 ) .

Similarly,

∂x β = O(1/R1 )

and η2 − η1 = O(1) . Hence, and therefore,

∂x (η2 − η1 ) β + (η2 − η1 )∂x β = O(1/R1 ) , r

∂x (η2 − η1 ) β + (η2 − η1 )∂x β = O(1) .

R1

Adding the three terms, we get, ∂x (η − η1 ) = O(1) . This implies that the first term in ∂x f is O(1) = O(1/rn+1 ) , η1 + A1 /r

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i.e., |∂x f | ≤

C rn+1

.

Similar arguments give, |∂x2 f | ≤

C . rn+1

By choosing R1 large enough we obtain, |∂xk f | ≤

1 , rn

k = 1, 2 .

Lemma 3.8 follows.

As discussed above, now that Lemma 3.8 is established, we have completed the proof of Theorem 3.2. 3.2. The case of vanishing mass aspect function In this subsection, we prove the following. Theorem 3.9. Let (M n+1 , g), 2 ≤ n ≤ 6, be an asymptotically hyperbolic manifold with scalar curvature satisfying, S[g] ≥ −n(n + 1). If the mass aspect function trh0 k vanishes identically, then (M, g) is isometric to hyperbolic space. We note that while Theorem 3.9 generalizes Theorem 1.1, its proof relies on it. We note also that our positivity of mass result, Theorem 3.1, follows immediately from Proposition 3.3 and Theorem 3.9. For notational convenience we set d = n + 1. Further, let capital latin indices run from 1, . . . , d − 1, let lowercase latin indices run from 1, . . . , d, and let y A be coordinates on S d−1 . Further, let (xi ) = (t, y A ) be coordinates on (0, T ) × S d−1 , and, as usual, let h0 be the standard metric on S d−1 . 3.2.1. Conformal gauge. Consider a conformally compact d-dimensional manifold ˜ = M ∪ ∂M , and (M, g) where M is the interior of a manifold with boundary M −2 suppose g is of the form g = ρ g˜, with ρ a defining function for ∂M for a metric g˜ ˜. which is smooth on M ˜ . Letting g˜ → θ2 g˜ and ρ → θρ leaves g Let θ be a positive function on M unchanged. Such a transformation can therefore be viewed as a change of conformal gauge. ˜ which in a neighborhood of ∂M can be written in Let g˜ be a metric on M the form g˜ = dt2 + h0 + td γ (3.70) where γ = γij dxi dxj is a smooth tensor field on (0, T ) × S d−1 for some T > 0, such that the restriction of γ to ∂M is a smooth tensor on S d−1 , i.e., γ ∂M = γ(0, y)AB dy A dy B . The following lemma shows that after a change of conformal gauge we may assume that g˜ is in Gauss coordinates based on ∂M .

24



Lemma 3.10. Consider the conformally compact metric g = sinh−2 (t)˜ g . There is −2 ˆ ˆ a conformal gauge change so that g takes the form g = sinh (t)g˜, where tˆ(p) = ˜ is of the form dgˆ˜ (p, ∂M ), and gˆ gˆ˜ = dtˆ2 + h0 + tˆd γˆ where γˆ = γˆ(tˆ, y)AB dy A dy B is a tˆ-dependent tensor field on S d−1 , such that γˆ ∂M = γ ∂M . In particular, g is asymptotically hyperbolic in the sense of Def inition 3.1, with mass aspect tensor k = γ ∂M . Proof. Let ρ = sinh(t). Arguing as in [2, Section 5], we shall find a function θ such that ρˆ := θρ = sinh(tˆ), where tˆ = dgˆ˜ (p, ∂M ), is the distance to the boundary in the metric gˆ ˜. This is equivalent to the condition that fˆ := arcsinh(ˆ ρ) = tˆ, with ˆ |df |ˆ = 1. A calculation as in the proof of [2, Lemma 5.3] shows that this condition g ˜

is equivalent to the equation ρ˜ g(dθ, dθ) + 2θ˜ g (dθ, dρ) = θ4 ρ + θ2 a

(3.71)

where a = ρ−1 (1 − g˜(dρ, dρ)). For g˜ of the form (3.70), we have a = −ρ + O(td ). Equation (3.71) is a system of first order partial differential equations, with characteristics transversal to ∂M = {t = 0}, and satisfies the conditions for existence of solutions with initial condition θ = 1 at ∂M , see [25, volume 5, pp. 39–40]. Hence there is a small neighborhood U of ∂M , and a solution θ to (3.71) on U . We shall need the following fact. Claim. θ = 1 + td+1 w, where w is smooth up to ∂M . The proof of the claim is straightforward and is left to the reader. Now we have that tˆ = arcsinh(θ sinh(t)) = t[1 + O(td+1 )], and hence t = tˆ 1 + O(tˆd+1 ) where the O(td+1 ) and O(tˆd+1 ) terms are smooth functions of (t, y) and (tˆ, y), respectively. It is straightforward to verify that sinh−2 (t) = sinh−2 (tˆ)[1 + tˆd+1 ] and g = sinh−2 (tˆ)gˆ ˜, with gˆ˜ = dtˆ2 + h0 + tˆd γˆ where γˆ has the property that γˆ(0, y) = γ(0, y). By construction, tˆ is the distance to ∂M , and hence the above is the form ˆ of g˜ in Gauss coordinates, based on ∂M . It follows that γˆ is a tˆ-dependent tensor on S d−1 .

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3.2.2. Conformal deformation. Assume (M, g) is asymptotically hyperbolic in the sense of Definition 3.1. Then, in slightly different notation, (M, g) has a conformal ˜ , g˜) with conformal boundary ∂ M ˜ the round sphere, and such compactification (M ˜ that near ∂ M , g has the form, g = ρ−2 g˜ (3.72) where ρ = sinh(t) and (3.73) g˜ = dt2 + h0 + td γ , d−1 where h0 is the standard metric on S and γ = γ(t, ·) is a t-dependent family of d−1 ˜ . Note that the mass aspect tensor is given by, metrics on S smooth up to ∂ M k = γAB ∂ M˜ . Let h = h0 + td γ be the metric induced on the level sets of t, and let s denote the scalar curvature defined with respect to h. Further, let Kij = 12 ∂t hij . The only nonvanishing components of K are KAB = 12 dtd−1 γAB + O(td ). denote the covariant derivative defined with respect to g˜ and let S˜ deLet ∇ note the scalar curvature of g˜. The formula for the scalar curvature of conformally related metrics gives, l ρ + ρ2 S˜ . l ρ∇ l ρ + (2d − 2)ρ∇ l∇ S = −d(d − 1)∇ Claim. S has the asymptotic form, S = −d(d − 1) + O(td+1 ) .

(3.74)

Indeed, by Taylor’s theorem, we have s = S[h0 ] + O(td ), and hence, S˜ = s − 2hAB ∂t KAB − (hAB KAB )2 + 3KAB K AB 0

0

d−1 = (d − 1)(d − 2) − d(d − 1)td−2 hAB ). 0 γAB + O(t

Further, using ρ = sinh t, l∇ l ρ = sinh(t) − g˜ij Γ tij cosh(t) ∇ d = sinh(t) + hAB 0 KAB + O(t )

d d = sinh(t) + td−1 hAB 0 γAB + O(t ) . 2 l ρ∇ l ρ = cosh2 (t). Putting this together, one finds afFinally, we note that ∇ ter a few manipulations that the terms involving the mass aspect function, μ = ˜ ), in S[g], at order td , cancel. Equation 3.74 follows. hAB 0 (γAB ∂ M By standard results [3, Theorem 1.2], there is a unique positive solution u such that limx→∞ u(x) = 1, to the Yamabe equation for prescribed scalar curvature −d(d − 1) in dimension d, −

d+2 4(d − 1) Δu + Su + d(d − 1)u d−2 = 0 . d−2

Let v = u − 1 and let Sˆ =

d−2 S + d(d − 1) . 4(d − 1)

(3.75)

26



Then the Yamabe equation takes the form ˆ = −Sˆ − F(v) −Δv + dv + Sv

(3.76)

d+2 d(d − 2) d+2 v . (1 + v) d−2 − 1 − 4 d−2 2 In particular F (v) = O(v ). A straightforward application of the maximum principle shows that since S[g] ≥ −d(d − 1), we have v ≤ 0 and hence

where

F (v) =

u ≤ 1. Linearizing the Yamabe equation around u = 1, we obtain the equation −Δ¯ u + d¯ u + Sû¯ = 0 . The indicial exponents of this equation are −1, d. It follows that the solution to the Yamabe equation is of the form u = 1+v with v = vd,1 td log t+vd td +higher order. However since by Equation (3.74), Sˆ = O(td+1 ), it follows [1] that vd,1 = 0, and in fact v is smooth up to boundary, with v = vd td + higher order . Let L = −Δ + d. Equation (3.76) takes the form Lv = f

(3.77)

with f given by ˆ − F(u − 1) . f = −Su In particular, f ≤ 0 and f = 0 except when Sˆ = 0. Let Lt be the operator defined by Lt u = − sinh2 (t)∂t2 u + (d − 2) sinh(t) cosh(t)∂t u + du . We have √ Lu = Lt u − sinh2 (t)∂t det h∂t u − sinh2 (t)Δh u , where Δh is the Laplacian on S d−1 with respect to the metric h(t, ·) = h0 + td γ. In particular Δh involves only y A -derivatives. We now introduce a function w which will be used as a supersolution, in order to control the leading order term in v. Let w = −td (1 + dt). Lemma 3.11. There exists constants t1 = t1 (d) > 0, A = A(d) > 0 such that Lt w > Atd+1 ,

for

0 < t < t∗ .

Proof. Using w = −td (1 + dt), we obtain, Lt (w) = sinh2 (t) d(d − 1)td−2 + d2 (d + 1)td−1 − (d − 2) sinh(t) cosh(t) dtd−1 + d(d + 1)td − d(td + dtd+1 ) = d(d − 1)td + d2 (d + 1)td+1 − (d − 2) dtd + d(d + 1)td+1 − [dtd + d2 td+1 ] + O(td+2 ) = d(d + 2)td+1 + O(td+2 ) , and the lemma follows.

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We have,


27

√ Lw = Lt w − sinh2 (t)∂t det h∂t w .

For metrics of the form we are considering, trh ∂t h = O(td−1 ). Hence by Lemma 3.11, we have Lw > Atd+1 − Ct2d where C = C(d, γ). This means there exists t2 > 0, t2 = t2 (d, γ) such that Lw > 0 ,

for 0 < t < t2 .

Lemma 3.12. Let (M, g) be asymptotically hyperbolic, so that (3.72) and (3.73) ˜ , with hold. Let f be a function on (M, g), and assume that f is smooth up to ∂ M d+1 fall off f = O(t ). Let L = −Δ + d, and let v be the unique solution to Lv = f ˜ and J with v = O(td ). Then v = vd td + td+1 J with vd = vd (y) smooth on ∂ M ˜ . If f ≤ 0, f = 0, then vd < 0. smooth up to ∂ M Proof. Let v¯a = aw and let v be as in (3.77). We have f ≤ 0, and hence by the ˜ . It follows that there is > 0, strong maximum principle v < 0 in the interior of M t2 > t3 > 0, so that sup v(t3 , y) < − . y∈S d−1

For each a ≥ 0, v¯a is a supersolution to L in the region 0 < t < t3 and for 0 ≤ a ≤ a∗ , we have that v¯a (t3 ) > v(t3 , y) for y ∈ S d−1 . Further, we clearly have v¯a (0) = v(0, y) = 0 for y ∈ S d−1 . It follows from the maximum principle that for small a, v¯a > v in the region 0 < t < t3 . Fix an a with this property. Since v¯a = −atd + O(td+1 ), dividing the inequality, v ≤ v¯a , by td and letting t 0 gives vd ≤ −a. We are now ready to state the following analogue of a well-known result in the asymptotically flat setting (cf., [20]). Proposition 3.13. Let (M, g) be asymptotically hyperbolic in the sense of Definition 3.1, with scalar curvature S[g] ≥ −d(d − 1), and with strict inequality somewhere. Then there exists a conformally related metric gˆ such that 1. (M, gˆ) is asymptotically hyperbolic, 2. S[ˆ g] = −d(d − 1), and 3. μ[ˆ g ] < μ[g], where μ[g], μ[ˆ g ] are the mass aspect functions of (M, g), (M, gˆ), respectively. Proof. Lemma 3.12, together with the discussion prior to Lemma 3.11, shows that the solution u to the Yamabe equation is of the form u = 1 + ud td + td+1 J

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˜ . Then, after a change of with ud = ud (y) < 0, and with J smooth up to ∂ M coordinates, gˆ = u4/(d−2) g can be brought into the form, 4 1 gˆ = sinh−2 (t) dt2 + h0 + td γ + 1+ ud h0 + td+1 z d−2 d ˜. where z = zij dxi dxj is smooth up to ∂ M By Lemma 3.10, after a change of conformal gauge, we have 4 1 gˆ = sinh−2 (tˆ) dtˆ2 + h0 + tˆd γˆ + 1+ u d h0 d−2 d ˜ and γˆ (0, y) = γ(0, y). It follows where γˆ = γˆ(tˆ, y)AB dy A dy B is smooth up to ∂ M from the above that the mass aspect functions satisfy, 4(d − 1) 1 (3.78) μ[ˆ g ] = μ[g] + 1+ ud < μ[g] , d−2 d

since ud < 0.

For the purpose of establishing Theorem 3.9, we need the following immediate consequence of Propositions 3.3 and 3.13. Corollary 3.14. Let (M, g) be as in Theorem 3.9; in particular, assume μ[g] = 0. Then g has constant scalar curvature S[g] = −d(d − 1). Proof. Suppose S[g] > −d(d − 1) somewhere. Then, by Proposition 3.13, there exists a conformally related metric gˆ such that (M, gˆ) is asymptotically hyperbolic, S[ˆ g] = −d(d − 1), and μ[ˆ g ] < μ[g] = 0. But this directly contradicts Proposition 3.3. 3.2.3. Deforming the metric. Now we will show that if g has constant scalar curvature S = −d(d − 1) and vanishing mass aspect function, then it is Einstein, Ricg = −(d − 1)g. Thus, let (M, g) be as in Theorem 3.9, and assume S[g] = −d(d − 1). Let = Ric − S g Ric d denote the traceless part of Ric. Note that since g has constant scalar curvature have vanishing divergence. Ric and Ric For the subsequent analysis, we shall need detailed information about the asymptotic behavior of Ric. Lemma 3.15. Let (M, g) be as in Theorem 3.9 (so that (3.72) and (3.73) hold, and the mass aspect vanishes). Then = − d td−2 γ + td−1 z Ric 2 where z = zij dxi dxj .

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∇ denote the Ricci tensor and covariant derivative defined with Proof. Let Ric, respect to g˜. The conformal transformation formula for Ricci curvature is jρ + ∇ l ρ˜ l ρ∇ i∇ ij + ρ−1 (d − 2)∇ l∇ l ρ˜ gij − (d − 1)ρ−2 ∇ gij (3.79) Ricij = Ric where in the right hand side, indices are raised with g˜. Note that if γ = 0, then Ric = (d − 2)h0 and the only nonvanishing terms in the formula for Ric are Rictt = (d − 1) sinh−2 (t) RicAB = (d − 1) sinh−2 (t)h0 AB . Now we consider the case with nonvanishing γ, but with vanishing mass aspect function, i.e., μ = hAB 0 γAB t=0 = 0. 1 Let Kij = 2 ∂t g˜ij . Then the nonvanishing terms in K are KAB = 12 dtd−1 γAB + d O(t ). Let h, ∇ /, ric denote the induced metric, covariant derivative and Ricci tensor on the level sets Mt of t. We use coordinates y A on these level sets, and raise and lower indices with h. Note that h = h0 + td γ, and hence, since ric involves no t-derivatives, we have by Taylor’s theorem, ric = Ric[h0 ] + O(td ) = (d − 2)h0 + O(td ) . We have from the Gauss, Codazzi, and second variation equations Rictt = −hAB ∂t KAB + KAC K C B 1 = − d(d − 1)td−2 hAB γAB + O(td−1 ) 2 which using μ = 0 gives, = O(td−1 ) , B RictA = ∇ / KBA − ∇ /A (hBC KBC )

= O(td ) , AB = ricAB − ∂t KAB + 2KAC K C B − KAB hCD KCD Ric 1 = (d − 2)h0 − d(d − 1)td−2 γAB + O(td−1 ) . 2 Thus we have = (d − 2)h0 − 1 d(d − 1)td−2 γ + O(td−1 ) . Ric 2 We next consider the remaining terms jρ + ∇ l ρ˜ l ρ∇ i∇ l∇ l ρ˜ Bij = ρ−1 (d − 2)∇ gij − (d − 1)ρ−2 ∇ gij . t Recall that since we are in a Gauss foliation, the only non-vanishing terms in Γ ij are tAB = −KAB . Γ

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We have Btt = −(d − 1) sinh−2 (t) , BtA = 0 , BAB = sinh−1 (t) (d − 2)KAB cosh(t) + sinh(t)hAB − (d − 1) sinh−2 (t) cosh2 (t)hAB , 1 = −(d − 1) sinh−2 (t)h0 AB + d(d − 2)td−2 γAB + O(td−1 ) . 2 This shows that d Ric[g] = −(d − 1)g − td−2 γ + O(td−1 ) 2 which gives the Lemma. Consider the curve,

λs = u4/(d−2) gs s

(3.80)

where, for s small, gs is the smooth curve of metrics, gs = g − sRic[g], and us is the conformal factor such that S[λs ] = −d(d − 1). Note that u0 = 1. By Lemmas 3.10, 3.15, and our earlier discussion on the asymptotic form of solutions to the Yamabe equation, λs is asymptotically hyperbolic in the sense of Definition 3.1. Let μs denote the mass aspect function of λs . s Let u ¯ = ∂u ∂s |s=0 . Then, by differentiating the Yamabe equation (3.75), with u = us and g = gs , with respect to the parameter s, we obtain the equation, 2. −Δ¯ u + d¯ u = −|Ric| By Lemma 3.12, we have u ¯ = u¯d td + O(td+1 ) , = 0. with u ¯d = u ¯d (y) < 0 if Ric 4(d − 1) 1 ∂s μs s=0 = 1+ u¯d . d−2 d Proof. Clearly, α = ∂s μs s=0 is of the form α = αRic + αu , where α = ∂s μ(gs )

Lemma 3.16.

Ric

and

s=0

αu = ∂s μ(u4/(d−2) g)s=0 . s

Let gs = sinh−2 (t)˜ gs . In order to determine the s-dependence of the mass aspect function we consider g˜s . It follows from Lemma 3.15 that d sinh2 (t)∂s g˜s = td γ + O(td+1 ) . 2 Since by assumption trh0 γ ∂M = 0, we have αRic = 0. It follows that the first order change in the mass aspect function of λs is given by αu , which clearly is determined by the first order change in the conformal factor us . The result follows.

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We are now ready to complete the proof of Theorem 3.9. Proof of Theorem 3.9. Let (M, g) be as in Theorem 3.9. Recall d = n + 1. By = 0. Corollary 3.14, S[g] = −d(d − 1). Suppose that g is not Einstein, i.e., Ric 4/(d−2) Let λs = us gs as in (3.80). As previously observed, λs is asymptotically hyperbolic with scalar curvature S[λs ] = −d(d − 1), and with mass aspect μs . By Lemma 3.16, ∂s μs |s=0 < 0, and hence for small s > 0, λs has negative mass aspect function. But in view of Proposition 3.3, μs < 0 gives a contradiction, and hence it must hold that Ric = 0. We can now apply the rigidity result of Qing [21] (see also [4, 7]) to conclude that in fact (M, g) is isometric to hyperbolic space. This concludes the proof of the positive mass theorem in the case of vanishing mass aspect function. Naturally, it would be desirable to find a way to remove the sign condition on the mass aspect from our positive mass result. Within the context of the approach taken in this paper, one possible way to accomplish this would be to extend the results of Corvino–Schoen [11, 12] and Chru´sciel–Delay [9] on initial data deformations to the asymptotically hyperbolic setting. Starting from Proposition 3.13, the aim would be to deform the time-symmetric initial data to be exactly Schwarzschild-AdS outside a compact set, without changing the scalar curvature and the sign of the mass. Starting from Schwarzschild-AdS with mass m < 0, the deformation result of Section 3.1 is then easily proved.

Acknowledgements We thank Piotr Chru´sciel for comments and discussion. This work was initiated at the Conference on Mathematical Aspects of Gravitation at the Mathematical Research Institute in Oberwolfach in 2003. We thank the Institute for its hospitality and support. This work was supported in part by NSF grants DMS-0407732 and DMS-0405906.

References [1] L. Andersson and P. T. Chru´sciel, Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary conditions”, Dissertationes Math. (Rozprawy Mat.) 355 (1996), 100. [2] L. Andersson and M. Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), no. 1, 1–27. [3] L. Andersson, P. T. Chru´sciel and H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations, Commun. Math. Phys. 149 (1992), 587–612. [4] V. Bonini, P. Miao, and J. Qing, Ricci curvature rigidity for weakly asymptotically hyperbolic manifolds, Comm. Anal. Geom. 14 (2006), no. 3, 603–612.

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[5] M. Cai, Volume minimizing hypersurfaces in manifolds of nonnegative scalar curvature, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 1–7. [6] M. Cai and G. J. Galloway, Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature, Comm. Anal. Geom. 8 (2000), no. 3, 565–573. [7] M. Cai and J. Qing, On the rigidity of AdS spacetime, preprint (2006). [8] U. Christ and J. Lohkamp, Singular minimal hypersurfaces and scalar curvature, math.DG/0609338 (2006). [9] P. T. Chru´sciel and E. Delay, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr. (N.S.) (2003), no. 94, vi+103. [10] P. T. Chru´sciel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), no. 2, 231–264. [11] J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137–189. [12] J. Corvino and R. M. Schoen, On the asymptotics for the vacuum Einstein constraint equations, J. Differential Geom. 73 (2006), no. 2, 185–217. [13] E. Delay, Analyse précisée d’équations semi-linéaires elliptiques sur l’espace hyperbolique et application a ` la courbure scalaire conforme, Bull. Soc. Math. France 125 (1997), no. 3, 345–381. [14] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. [15] G. W. Gibbons, S. W. Hawking, G. T. Horowitz, and M. J. Perry, Positive mass theorems for black holes, Comm. Math. Phys. 88 (1983), no. 3, 295–308. [16] J. L. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. (2) 101 (1975), 317–331. [17] J. Lohkamp, The higher dimensional positive mass theorem I, math.DG/0608795 (2006). [18] J. Lohkamp, Scalar curvature and hammocks, Math. Ann. 313 (1999), no. 3, 385–407. [19] M. Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989), no. 4, 527–539. ´ Murchadha and J. W. York Jr., Gravitational energy, Phys. Rev. D 10 (1974), [20] N. O no. 8, 2345–2357. [21] J. Qing, On the rigidity for conformally compact Einstein manifolds, Int. Math. Res. Not. (2003), no. 21, 1141–1153. [22] R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1–3, 159–183. [23] R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. [24] Y. Shi and G. Tian, Rigidity of asymptotically hyperbolic manifolds, Comm. Math. Phys. 259 (2005), no. 3, 545–559. [25] M. Spivak, A comprehensive introduction to differential geometry, second ed., Publish or Perish Inc., Wilmington, Del., 1979.

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[26] X. Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), no. 2, 273–299. [27] E. Witten and S.-T. Yau, Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), no. 6, 1635–1655 (2000). [28] S. T. Yau, Geometry of three manifolds and existence of black hole due to boundary effect, Adv. Theor. Math. Phys. 5 (2001), no. 4, 755–767. Lars Andersson∗ , Mingliang Cai, and Gregory J. Galloway Department of Mathematics University of Miami Coral Gables, FL 33124 USA and ∗ Max-Planck-Institut f¨ ur Gravitationsphysik Am M¨ uhlenberg 1 D-14476 Postdam Germany e-mail: [email protected] [email protected] [email protected] Communicated by Sergiu Klainerman. Submitted: March 16, 2007. Accepted: June 14, 2007.



Projecting Massive Scalar Fields to Null Infinity Claudio Dappiaggi Abstract. It is known that, in an asymptotically flat spacetime, null infinity cannot act as an initial-value surface for massive scalar fields. Exploiting tools proper of harmonic analysis on hyperboloids and global norm estimates for the wave operator, we show that it is possible to circumvent such obstruction at least in Minkowski spacetime. Hence we project norm-finite solutions of the Klein–Gordon equation of motion in data on null infinity and, eventually, we interpret them in terms of boundary free field theory.

1. Introduction In the study of classical fields over four dimensional Lorentzian curved backgrounds, Penrose conformal completion techniques have played since their introduction a pivotal role. In particular the related notion of asymptotic simplicity/flatness entails the embedding of a (four dimensional) physical spacetime (M, gμν ) as a bounded open , gμν ) being g a conformal rescaling of g. In set in an unphysical background (M this setting the image of M in M can be naturally endowed with a boundary structure usually referred to as ± , i.e., future or past null infinity. Heuristically the endpoint of all the null geodesics in (M, gμν ), ± is thus the geometrical locus where the trajectory of zero rest mass particles ends. Hence it is manifest how null conformal boundaries can be exploited as a powerful tool to study either the asymptotic properties of radiation fields associated to massless wave equations, either the scattering properties of massless fields [12]. Furthermore, from the perspective of quantum field theory over curved backgrounds, ± plays a key role in the realization of the holographic principle. The latter conjectures that the information of any field theory on a D-dimensional Lorentzian background M can be recovered by means of a suitable second field theory constructed over a codimension one submanifold Σ embedded in M . Hence,

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in asymptotically flat spacetimes it is natural to conjecture that the role of Σ is played by the null conformal boundary and this idea has been deeply investigated both at a classical and at a quantum level in [4]. To better understand the main rationale underlying the success of Penrose conformal techniques from a field theoretical perspective, let us consider a working example, namely the massless Klein–Gordon scalar field ψ conformally coupled to gravity in a globally hyperbolic and asymptotically flat four dimensional space time M . Barring a few technical assumptions, each solution of g − R6 ψ = 0 with compactly supported initial data on a Cauchy surface can be mapped into a solution of g ψ − R ψ = 0 , (1) 6 . , where ψ = Ω−1 ψ. Although ψ is strictly defined only over the image of M in M global hyperbolicity of M and uniqueness of solutions for second order hyperbolic . Accordingly PDE, allows us to extend ψ to a smooth solution for (1) over all M ± we can define the projection of ψ over the boundary simply as its restriction: . ∞ ± Ψ± = ψ| ± ∈ C ( ). It is Ψ the key ingredient to study properties of bulk physical phenomena starting from boundary data in the unphysical spacetime as exploited, to quote just a few examples, in [4, 12]. Nonetheless the situation is not heavenly as it may seem since the above construction drastically fails whenever one considers massive fields. Even in the simplest situation of the Klein–Gordon scalar field over flat four dimensional Minkowski spacetime, conformal invariance of the equation of motion is broken. Furthermore it has been argued in [9, 24] that ± cannot be used as an initial value surface for massive fields and that it is not possible to project any solution of g − m2 ψ = 0 into a smooth function over ± . This result has been established with an elegant argument in [9]: the space of sections of any vector bundle on ± which is homogeneous for the action of the Poincaré group carries only massless representations1. Hence it seems impossible to exploit the powerful means of Penrose compactification whenever we deal with solutions of partial differential equations containing a term proportional to a scale length such as the mass. In other words, since the information of the data evolving to infinity along causal timelike curves flows in to future timelike infinity i+ (a codimension 2 subthe unphysical spacetime M manifold of M hence not a proper boundary), it seems impossible to exploit null infinity as a tool to study massive fields. 1A

reader familiar with Penrose compactification techniques could argue that the relevant symmetry group on null infinity is not the Poincaré but the BMS group which is the semidirect product between the proper ortochronous component of the Lorentz group and the smooth functions over the 2-sphere thought as an Abelian group under addition. Nonetheless, since in Helffer construction the key role is played by the translational subgroup of the Poincaré group, the result can be extended also in a BMS framework remembering that it exists a four dimensional normal subgroup of the full BMS group homomorphic to T 4 [4].

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Projecting Massive Scalar Fields to Null Infinity

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The aim of this paper is to provide a way to circumvent the above obstruction at least in Minkowski background. In particular we will exploit both tools of harmonic analysis and global norm estimates for the wave equation in order to project a solution for the massive Klein–Gordon equation of motion into meaningful data over null infinity. More in detail, the outline of the analysis and hence of the paper will be the following: in the next subsection we recollect some basic details about the notion of asymptotic flatness. In Section 2, instead, we specialize to Minkowski background and we consider solutions of the massive Klein–Gordon equation of motion satisfying a finite norm condition in such a way that their Fourier transform is a square integrable function over the mass hyperboloid Hm . Exploiting a few results due to Strichartz on harmonic analysis over hyperboloids we shall introduce a unitary map between two copies of L2 (Hm ) and the space of square integrable functions over the light cone C. Furthermore such a map will also act as an intertwiner between the quasi-regular representations of the Lorentz group on L2 (Hm ) and L2 (C). Afterward we exploit global norm estimates to associate to each square integrable function over the light cone a norm finite solution for the wave equation in Minkowski spacetime. By means of Penrose compactification techniques and trace theorems, we project these functions on null infinity. Eventually, in Section 3, we show how the projected data can be interpreted in terms of a diffeomorphism invariant field theory intrinsically constructed over null infinity. 1.1. On asymptotically flat spacetimes In this section we recollect some known facts about the definition and the properties of asymptotically flat spacetimes. Although we are going to work in Minkowski background, the following summary can be useful for a twofold reason: from one side, in Section 3, we shall interpret the projection of the data from a bulk massive scalar field in terms of a field theory on future null infinity whereas, from the other side, we look at this paper as the first step to solve the same problem on a generic asymptotically flat background. Hence it could be interesting to understand where our construction relies on properties specific of Minkowski spacetime and where, on the opposite, our results could be traded to a more general scenario. In the literature there are several different notions of asymptotic flatness at (future or past) null infinity which are obviously all equivalent if the bulk spacetime is Minkowski; hence a reader familiar to any of these can skip to the next section without a second thought. We shall instead adopt the specific definition first introduced by Friedrich (see [6] and references therein from the same author) of a class of spacetimes which are flat at future null infinity and they admit future time completion at i+ . The reason for this choice lies in the realm of quantum field theory over curved backgrounds. In particular, in [4] it has been shown that it is possible to project the Weyl *-algebra of observables for a massless scalar field in Minkowski spacetime as a subsector of a suitable counterpart at null infinity

38

C. Dappiaggi


because the Lichnerowicz propagator for the wave operator is strictly supported on the light cone. On the opposite, in a generic curved background, a priori this does not hold true since the support includes a tail strictly contained in the cone and, hence, in the conformal completion language propagating at future timelike infinity. Thus, in order to recast the result of [4] in a generic scenario, Friedrich definition is the most appealing (to this avail see the analysis in [14]). In detail a four dimensional future time oriented spacetime M with a smooth metric gμν which solves the vacuum Einstein equation is called an asymptotically flat and vacuum spacetime with future time infinity i+ if it exists a second four , dimensional spacetime (M gμν ) with a preferred point i+ , a diffeomorphism λ : M → λ(M ) ⊂ M and a non negative scalar function Ω on λ(M ) such that g = Ω2 λ∗ g and the following facts hold: ) is closed and λ(M ) = J − (i+ ) \ ∂J − (i+ ; M ). Moreover ∂λ(M ) = 1. J − (i+ ; M + + + . − + + ∪ i where = ∂J (i ; M ) \ {i } is future null infinity. 2. λ(M ) is strongly causal. . 3. Ω can be extended to a smooth function on M + ν Ω(i+ ) = μ∇ 4. Ω|∂J − (i+ ;M = 0, dΩ(x) = 0 for x ∈ and dΩ(i+ ) = 0, but ∇ ) −2 gμν (i+ ). . ν Ω, it exists a strictly positive smooth function ω, defined 5. Calling nμ = gμν ∇ μ (ω 4 nμ ) = 0 on + , such that the in a neighbourhood of + and satisfying ∇ −1 μ integral curves of ω n are complete on + . From now we shall refer to λ(M ) simply as M since no confusion will arise in the manuscript due to this identification. Furthermore we point out that, with minor adaption, the above definition can be recast for spacetimes which are asymptotically flat with past time infinity i− and henceforth we shall refer only to + though the reader is warned that all our results hold identically for − . Thus let us consider any asymptotically flat spacetime as per the previous definition; the metric structure of future null infinity is not uniquely determined but it is affected by a gauge freedom in the choice of the compactification factor namely, if we rescale Ω as ωΩ with ω ∈ C ∞ (+ , R+ ), the topology and the differentiable structure of future null infinity is left unchanged. Hence the difference between the possible geometries for the conformal boundary is caught by equivalence classes of the following triplet of data (+ , na , hab ) where + stands for the . a the covariant derivative with Ω (being ∇ S2 × R topology of null infinity, na = ∇ . + respect to gab ) and hab = gab |+ . Two triplets ( , na , hab ) and (+ , na , hab ) are called equivalent if and only if it exists a gauge factor ω such that hab = ω 2 hab whereas na = ω −1 na . The set of all these equivalence classes is universal in the sense that, given any two asymptotically flat spacetimes M1 and M2 with associated triplets (+ 1 , n1a , + + + ab ) and ( , n , h ), it always exists a diffeomorphism γ ∈ Dif f ( , hab 2a 1 2 2 1 2 ) such ab that γ ∗ hab = h and γ n = n . ∗ 1a 2a 2 1

Vol. 9 (2008)


39

The set of all group elements γ ∈ Dif f (+ , + ) mapping a triplet into a gauge equivalent one2 is called the Bondi–Metzner–Sachs group (BMS). It is always possible to choose ω in such a way that on null infinity we can introduce the socalled Bondi frame (u, z, z¯) where u is the affine parameter along the null complete geodesics generating + and (z, z¯) are the complex coordinates constructed out of a stereographic projection on C from (θ, ϕ) ∈ S2 . Accordingly the BMS group is SO(3, 1) C ∞ (S2 ) acting as u −→ u = KΛ (z, z¯) u + α(z, z¯) , (2) ¯z¯ + ¯b . a . az + b , z¯ −→ Λ¯ z= , (3) z −→ Λz = cz + d c¯z¯ + d¯

a b where Λ is identified with the matrix satisfying ad − bc = 1, whereas c d KΛ (z, z¯) =

1 + |z|2 . |az + b|2 + |cz + d|2

A direct inspection of this formula shows that the BMS group is a regular semidirect product and it is much larger than the Poincaré group. In a generic scenario such a problem cannot be easily overcome though one can recognize that any element in the Abelian ideal C ∞ (S2 ) can be expanded in real spherical harmonics as l l 1 ∞ α(z, z¯) = αlm Slm (z, z¯) + αlm Slm (z, z¯) . ∀α(z, z¯) ∈ C ∞ (S2 ) . l=0 m=−l

l=2 m=−l

Here we have separated the set of first four components – known as the translational component of the BMS group – since it is homomorphic to the Abelian group T 4 . Furthermore the following proposition holds: Proposition 1.1. The subset SO(3, 1) T 4 of the BMS group made of elements (Λ, α(z, z¯)), where α(z, z¯) is any real linear combination of the first four spherical harmonics, is a BMS subgroup and if we associate to α(z, z¯) the vector

3 a00 μ √ , a1−1 , a10 , a11 , a =− 4π 3 the action of Λ ∈ SO(3, 1) on aμ is equivalent to the transformation of the 4-vector in Minkowski background under the standard Lorentz action. The proof of this theorem has been given in Propositions 3.11 and 3.12 in [4]. We wish to stress that, in a generic asymptotically flat spacetime, we cannot exploit this last statement to select a preferred Poincaré subgroup in the BMS since, acting per conjugation over the above SO(3, 1) T 4 subset with any 2 Although

at first sight we are considering a subgroup of the whole set of diffeomorphisms, one should take into account that the constraint we impose is equivalent to require that the bulk geometry is left unchanged, i.e., we are working on a fixed background.

40

C. Dappiaggi


element (I, Slm (z, z¯)) ∈ SO(3, 1) C ∞ (S2 ) with l > 1, we end up with a different though equivalent Poincaré subgroup. Nonetheless, since in this paper we are taking into account only Minkowski background, we can exploit a result due to Geroch, Ashtekar and Xanthopoulos [2, 8] namely Proposition 1.2. In any asymptotically flat spacetime (M, gμν ) it holds and a) any Killing vector ξ in M smoothly extends to a Killing vector ξ in M ˜ the restriction ξ of the latter to is tangent to null infinity; it is uniquely determined by ξ and it generates a one-parameter subgroup of the BMS. b) the map ξ → ξ˜ is injective and, if the one-parameter subgroup of the BMS generated by ξ˜ lies in C ∞ (S2 ), then it must also be a subgroup of T 4 . According to the last proposition, in a Minkowski background, the Poincaré isometries identify a preferred subgroup of the BMS group, i.e., the set 2 ¯ ¯ 0 1 z+z 2 z−z 3 |z| − 1 R = Λ, α(z, z¯) α(z, z¯) = a + a +a +a , (4) 1 + |z|2 1 + |z|2 1 + |z|2 which is homomorphic to SO(3, 1) T 4 .

2. From massive to massless scalar fields on Minkowski spacetime Let us consider four dimensional flat Minkowski spacetime (M, ημν ) and a scalar field φ : M → C satisfying the Klein–Gordon equation with squared mass m2 > 0: η φ − m2 φ = 0 .

(5)

In the most general framework we should seek for tempered distributions solutions to such PDE and their Fourier transform is a function supported on the mass 3 hyperboloids Hm (see Section IX.9 of [15]) η μν pμ pν = p20 − i=1 p2i = m2 , being pμ = (p0 , pi ) with i = 1, . . . , 3 the standard global coordinates3 on each fibre of the cotangent bundle T ∗ M canonically identified as R4 × R4 . . The mass hyperboloids can be parameterized4 with the coordinates r = | p|≡ 1 3 . . p p0 2 2 2 3 p ∈ [0, ∞), ζ = ∈ S → R and = = ±1. The value of p2 is i=1

i

| p|

|p0 |

0

p|2 , whereas the variable provides a way fixed by the defining relation p20 = m2 + | to distinguish in R4 between the upper and lower hyperboloid and we will keep track of it for the sake of generality. An interested reader can adapt the following constructions to a single hyperboloid with minor efforts. To conclude, identifying Hm with the coset O(3,1) O(3) , we can endow it with the O(3, 1) invariant measure dμ(Hm ) = 3 The

2 √ r drdζ. r 2 +m2

symbols are here adopted with respect to the standard high energy physics terminology though we do not seek at the moment any physical interpretation of the forthcoming analysis leaving it for the conclusions. 4 We wish to stress to a potential reader that we are adopting nomenclatures and symbols for coordinates as in [20] which are partly different from those in [21], where some results of this section were first derived.

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Moreover, as outlined in the introduction, the aim of this section is to associate to each solution of the Klein–Gordon equation of motion a massless scalar field. To this avail we need to take into account only a subclass of all possible solutions of (5), namely those which are finite with respect to a suitable norm. Such problem was already pointed out in [20] where the following solution was suggested: it must exists a real number α ≥ 0 and a function f (r, ζ, ) such that, being x the spatial component of xμ and · the standard Euclidean scalar product on R3 , √ 2 2 φ(xμ ) = dμ(Hm ) eirx·ζ e−i r +m t f (r, ζ, ) =±1H m

and φ2α =

α

dμ(Hm )|(r2 + m2 ) 2 f (r, ζ, )|2 < ∞ .

(6)

=±1H m

Dropping from now on all references to dμ(Hm ), we may call the Hilbert spaces of functions satisfying (6) as L2α and, out of a direct inspection of the above formula, one can realize that L20 coincides with L2 (Hm ), the space of square integrable functions over the mass hyperboloid. Furthermore, for a generic φ ∈ L2α the following chain of inequalities holds: ∞ > φ2α ≥ dμ(Hm )(r2 + m2 )α |f (r, ζ, )|2 =±1H m

= m2α+2

dμ(H1 )(r2 + 1)α |f (r , ζ, )|2

=±1H 1

> m2α+2

dμ(H1 )|f (r , ζ, )|2 ,

=±1H 1

where, in the equality, we rescaled the coordinate r to mr = r and H1 stands for the hyperboloid with m = 1. Hence f (r , ζ, ) lies in L2 (H1 ) or, equivalently, the original φ ∈ L20 ≡ L2 (Hm ). This identity shows that the following chain of inclusions holds: L2α ⊂ L2α ⊂ 2 L (Hm ) ≡ L20 for all 0 < α < α . For this reason, from now on, we will work only within the space L20 unless stated otherwise. To summarise the key point, the constraint (6) allows us a way to select only those solutions φ of (5) whose Fourier transform admits a restriction – f – on the mass hyperboloid which is square integrable with respect to the O(3, 1)-invariant measure. Furthermore we can require the O(3, 1) group to act on f with the quasiregular scalar representation, i.e., for any Λ ∈ O(3, 1) and for any pμ ∈ Hm → R4 U (Λ)f (pμ ) = f (Λ−1 pμ ) ,

f ∈ L2 (Hm )

being U unitary strongly continuous but not irreducible.

(7)

42

C. Dappiaggi


Henceforth our plan is to discuss and later to exploit the following Strichartz result: it is possible to construct an operator T from L2 (Hm ) ⊕ L2 (Hm ) into the space of square-integrable functions over the light cone with respect to the O(3, 1)invariant measure and T is also a unitary intertwiner5 between the quasi-regular O(3, 1)-representations. 2.1. From hyperboloids to light cones The analysis and the statements in this section are based upon the theorems proved in [21] even though part of the results have been independently developed also in [11] by means of integral transforms associated to horospheres. The proof of most of the following results strongly relies upon the embedding of the mass hyperboloid and of the light cone in R4 . All the analysis can be recast in terms of the intrinsic structures over these symmetric spaces and we refer to [17] for an interested reader. As a starting point we shall briefly discuss and characterize some properties of square integrable functions over the light cone. Let us quickly recall that the latter is the geometric locus p |2 = 0 \ (0, 0) , C = pμ = (p0 , pi ), η μν pμ pν = p20 − | where pμ = (p0 , p ) = (p0 , pi ) with i = 1, . . . , 3 are the same global coordinates introduced in the previous section. As for Hm we can represent C, together with an O(3, 1)-invariant measure in a more convenient coordinate system which is basically constructed with a limiting procedure (i.e., m → 0) from the counterpart . . on the mass hyperboloid. Namely, if we refer to r = | p |, ζ = |pp | ∈ S2 → R3 and . p|2 whereas = |pp00 | = ±1, the value of p20 is set by the defining relation p20 = | the measure is dμ(C) = rdrdζ. Here the two values of allow us to distinguish between the future and the past light cone and, as for the massive case, we keep track of them for the sake of completeness. The next step consists of a specific characterization for square integrable functions over the light cone with respect to dμ(C). Let us consider the set D0σ and D1σ respectively of even and odd smooth functions over C homogeneous of degree σ in the r-variable, i.e., of the form rσ g(ζ, ). Then the following proposition holds Proposition 2.1. If σ = −1 + iρ with ρ ∈ R, then D0σ and D1σ can be closed to Hilbert space Hσ0 and Hσ1 with respect to the norm dζ|g(ζ, )|2 . rσ g(ζ, )2σ = =±1

S2

Furthermore 5 We recall that, given a group G with the representations U and U on the Hilbert spaces H and H , a bounded linear map T : H → H is called an intertwiner if U (g)T = T U (g) for all g ∈ G.

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1. the quasi-regular O(3, 1) scalar representation acting on the functions over C as U (Λ)F (pμ ) = F (Λ−1 pμ ) ,

∀pμ ∈ C → R4 ∧ ∀F ∈ L2 (C) ,

is strongly continuous unitary and irreducible on both Hσ0 and Hσ1 . 2. for any F ∈ L2 (C) it exists a unique function ϕ0 in Hσ0 and ϕ1 in Hσ1 such that, calling F0 (pμ ) = 12 (F (pμ ) + F (−pμ )) and F1 (pμ ) = 12 (F (pμ ) −F (−pμ )), then ∞ dρ 2 dζ |ϕj (ρ, ζ, )| , j = 0, 1 (8) Fj (pμ )L2 (C) = 2π =±1 −∞

S2

and ∞ dρ Fj (r, ζ, ) = r−1+iρ ϕj (ρ, ζ, ) . 2π =±1

j = 0, 1 .

(9)

−∞

The image of the map F → (ϕ0 , ϕ1 ) is onto all pairs with a finite right hand side in (8). Proof. We here sketch the main details of the proof as in [21]. To start, let us notice that the norm over D0σ and D1σ is well defined since, up to the sum over , it is equivalent to the norm over L2 (S2 , d2 x) being d2 x the Lesbegue measure on S2 . The unitarity and strong continuity of the quasi-regular representation arises due to the O(3, 1)-invariance of the measure on the light cone. Hence for any F (pμ ) ∈ L2 (C) with pμ ∈ R4 satisfying η μν pμ pν = 0, it holds: 2 dμ(C)|U (Λ)F (pμ )| = dμ(C)|F (Λ−1 pμ )|2 C

C

= C

dμ(ΛC)|F (pμ )|2 =

dμ(C)|F (pμ )|2 ,

C

where, in the second equality, we performed the coordinate change pμ → Λpμ . To prove irreducibility let us note that any function f ∈ Hσj with j = 0, 1 can ∞ l be decomposed in spherical harmonics, i.e., f (r, ζ, ) = rσ g(ζ, ) = l=0 m=−l alm Ylm (ζ)k rσ where k = 0, 1 and the coefficients ajm must vanish if j = 0 and l + k is odd or if j = 1 and l + k is even. Consider now, as a special case, a function in Hσj with all but one of the coefficients alm equal to zero. We show now that the action of the quasi-regular O(3, 1) representation generates a second function with the coefficients al+1,m = 0. To this avail let us choose an element of SO(1, 1) ⊂ O(3, 1) parameterized by an angle α, apply it to f and then let us differentiate with respect to α. The resulting function f evaluated in α = 0 is f (r, ζ, ) =

(σ − l)(l + 1) σ r Yl+1 (ζ)k+1 . 1 + 2l

44

C. Dappiaggi


Since all these operations should map any irreducible subspace of Hσj into itself, the statement in point 1 of the theorem holds. To demonstrate point 2 let us associate to any F (pμ ) ≡ F (r, ζ, ) ∈ L2 (C), the functions gj (r, ζ, ) = rFj (r, ζ, ) with j = 0, 1. Hence for each j ∞ 2 dμ(C)|Fj (r, ζ, )| = dζ dr r−1 |gj (r, ζ, )|2 < ∞ , C

0

S2

∞

−1

2

which implies that 0 drr |gj (r, ζ, )| < ∞ per Fubini’s theorem. We can now apply Mellin inversion theorem to write ∞ ∞ iρ gj (r, ζ, ) = dρ r ϕj (ρ, ζ, ) = dρ eiρ ln(r) ϕj (ρ, ζ, ) . −∞

−∞

The apply Plancherel theorem to conclude ∞ us to −1 ∞ that ∞ last identity suggests 2 2 d ln(r)|g (r, ζ, )| = dρ(2π) |ϕ (ρ, ζ, )| and that ϕ (ρ, ζ, ) = j j j 0 −∞ 0 d ln(r) is ln(r) 2 e gj (r, ζ, ). Hence, upon integration over the compact S -coordinates, we recover (8) and (9). The overall construction relies only on Mellin inversion formula and the Plancherel theorem; thus the map from Fj onto ϕj exists whenever the latter is square-integrable. To conclude the analysis on the functions over a light cone, let us recall the following result still from [21]: Lemma 2.1. Whenever ρ = 0 then ρ A0 (ρ)ϕ(ζ , ) = |ζ · ζ − |−1−iρ ϕ(ζ, )dζ , π =±1 S2 ρ A1 (ρ)ϕ(ζ , ) = |ζ · ζ − |−1−iρ sgn ζ · ζ − ϕ(ζ, )dζ , π =±1

(10)

(11)

S2

are unitary operators respectively on odd and on even functions in L2 (S2 × ±1). In (10) and (11) “ · ” stands for the standard Euclidean scalar product on R3 , whereas a function f (ζ, ) ∈ L2 (S2 × ±1) if and only if |f (ζ, )|2 < ∞ . =±1

S2

We can now put together the previous lemma and Proposition 2.1 in order to represent any F ∈ L2 (C) in terms of a scalar function supported on the whole R4 as ∞ 1 2 dρ ρ dζ | p · ζ − p0 |−1+iρ ψ0 (ρ, ζ , ) F (pμ ) = 3 2π =±1−∞ S2 p · ζ − p0 ) , (12) + ψ1 (ρ, ζ , )sgn(

Vol. 9 (2008)


. being ψk (ρ, ζ , ) = to (9). Furthermore, F 22 =

π ρ Ak (ρ)ϕk (ρ, ζ , )

=±1

S2

dζ

∞

−∞

45

with k = 0, 1 and ϕk chosen according

dρ 2 ρ |ψ0 (ρ, ζ , )|2 + ψ1 (ρ, ζ , )|2 , 2π

being the norm , 2 that defined in (6) fixing α = 2. The original F is recovered simply imposing in (12) the constraint p0 = | p|. Such a property will be exploited in Proposition 2.2. Let us now move back to the square integrable functions over Hm and, to = − ∂ 22 + 3 ∂ 22 . We wish to stress that is the fix notations, let us call i=1 ∂pi ∂p0 D’Alembert wave operator in the momentum space and, in this manuscript, it has no relation with , the counterpart in the coordinate space. Instead of working with (p0 , pi ) or (m, r, ζ), we introduce as a useful tool for next proposition the new local coordinates (m, s, ζ) defined in the region | p| < . 2 . |p| 2 2 p| ∈ (0, ∞), s = |p0 | ∈ [0, 1), ζ ∈ S and = ±1 which |p0 | as m = p0 − | distinguishes between positive and negative values of p0 . The advantage of this new frame lies in the expression for the D’Alembert = − ∂ 22 − 3 ∂ + H2 , where H is the Laplacian wave operator which becomes ∂m m ∂m m on the unit hyperboloid. We stress to the reader that in the forthcoming discussion all derivatives act in a weak/distributional sense. It is standard result that H is a selfadjoint operator on f ∈ L2 (Hm ), | H f ∈ L2 (Hm ) , with a continuous negative spectrum; furthermore it commutes with the quasi = 0 for any Λ ∈ regular O(3, 1) representation, i.e., [U (Λ), H ] = [U (Λ), ] O(3, 1). The strategy is to consider the mass hyperboloid as a non characteristic initial surface for the wave equation u(m, s, ζ, ) = 0 to be solved in the region m ≥ 0. In particular the following lemma holds: Lemma 2.2. Calling B = −H − 1, then for any f, g ∈ L2 (Hm ) the function √

u(m, s, ζ, ) = m−1+i

B

√

f (s, ζ, ) + m−1−i

B

g(s, ζ, )

(13)

= 0 for m > 0 with Cauchy data satisfies u u(1, s, ζ, ) = f (s, ζ, ) + g(s, ζ, ) , iB − 2 ∂(mu) ∂m (1, s, ζ, ) = g(s, ζ, ) − f (s, ζ, ) . 1

Furthermore for all m > 0 it holds 2 f 22 + g22 = =±1

S2

1 dζ 0

(ms)2 ds (1 − s2 )2

2 − 1 ∂(mu) 2 (s, ζ, ) , |u(m, s, ζ, )| + B ∂m

where , 2 is the norm (6) with α = 2.

2

46

C. Dappiaggi


Proof. If we show that u(m, s, ζ, ) is a solution of D’Alembert wave equation then the statement on Cauchy data holds per direct substitution and the identity √ i B 2 on L (Hm ). between norms stands per unitarity of the operator m Hence let us consider any but fixed v ∈ C0∞ (R4 ) whose support does not include the origin. Dropping the dependence which is irrelevant in the proof, integration per parts grants: 4 μ μ μ d p v(p )u(p ) = d4 p u(pμ )v(p ). R4

R4

Since both the coordinates (m, r, ζ) and (m, s, ζ) cover the mass hyperboloid, a direct inspection of the transformation relation between these two frame systems (ms)2 shows that dμ(Hm ) = (1−s 2 )2 dsdζ. Hence the above formula reads

∞ dm 0

Hm

H ∂2 3 ∂ + dμ(Hm ) mu(m, s, ζ) − − v(m, s, ζ) ∂m2 m ∂m m2

∂ ∂2 dμ(Hm ) m−1 v(m, s, ζ) H − m2 − 3m u(m, s, ζ) , ∂m2 ∂m

∞

dm

= 0

Hm

which, inserting the expression for u(m, s, ζ) in the hypothesis, becomes

∞ dm 0

dμ(Hm ) m−1 v(m, s, ζ) (H + 1 + B) u(m, s, ζ) = 0 ,

Hm

. being B = −H − 1.

The choice of the initial surface as the unitary hyperboloid is pure convenience and no generality is lost in this process since it is possible to pick any Hm and none of the forthcoming results would be modified. The independence from m in the norm identity in the last lemma and the equality limm→0 dμ(Hm ) = dμ(C) suggests that we are now in position to construct a unitary intertwining operator T : L2 (C) → L2 (Hm ) ⊕ L2 (Hm ). As a matter of fact all the needed ingredients can be found in the previous lemma and in formula (12): Proposition 2.2. Given any function F ∈ L2 (C) let us consider it as a function over all R4 by means of decomposition (12) and let us split it as F = F+ + F− where + represents the contribution of the integral in the ρ-variable between 0 and infinity in (12) whereas the pedex − refers to that between minus infinity and 0. . . Then let us call as f = F+ |Hm and g = F− |Hm the restriction of F+ and F− on the mass hyperboloid obtained imposing on the coordinates pμ in (12) the constraint p20 = | p|2 + m2 . It holds either that f, g ∈ L2 (Hm ), either that the function u constructed as in Lemma 2.2 coincides with F .

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Furthermore the map T : L2 (C) −→ L2 (Hm ) ⊕ L2 (Hm ) , which associates to each function F ∈ L2 (C) the pair (f, g) above defined, is an intertwiner between the O(3, 1) representations. The demonstration is left to [21]. Remark 2.1. A consequence of the above proposition is that any f ∈ L2 (Hm ) can be represented as a function f (pμ ) defined over the whole R4 ∞ dρ μ 2 − p0 |−1+iρ ψ0 (ρ, ζ , ) f (p ) = ρ dζ | p · ζ 2π 3 =±1 0 S2 (14) + sgn p · ζ − p0 ψ1 (ρ, ζ , ) , where ψ0 (ρ, ζ , ) =

d4 p δ(η μν pμ pν − m2 )f (pμ )| p · ζ − p0 |−1−iρ ,

R4

and ψ1 (ρ, ζ , ) =

d4 p δ(η μν pμ pν − m2 )f (pμ )| p · ζ − p0 |−1−iρ sgn p · ζ − p0 .

R4

The original f ∈ L2 (Hm ) is obtained from the above representation imposing on p0 the constraint/defining relation p20 = | p|2 + m2 . Let us also pinpoint 1. although (14) is written in terms of the global coordinates pμ , we can first restrict ourselves to Hm as explained above and, then, we can switch to intrinsic coordinates (r, ζ, ). In this way the generic function f ∈ L2 (Hm ) is explicitly decomposed into a direct integral in terms of irreducible representations of O(3, 1). 2. Proposition 2.2 provides a way to explicitly construct the inverse intertwiner . T = T −1 : L2 (Hm ) ⊕ L2 (Hm ) → L2 (C). As a matter of fact starting from any two functions f, g ∈ L2 (Hm ), one can generate a solution of D’Alembert wave equation out of (13) whose restriction to the light cone is a function F ∈ L2 (C); in a few words T(f, g) = F . From our perspective this a slightly inconvenient situation since we start with a solution of (5) and, hence, with a single function f ∈ L2 (Hm ). Unfortunately the Cauchy problem, upon which (13) is based, requires two initial conditions. Hence we adopt the choice to imbed L2 (Hm ) into the diagonal component of L2 (Hm ) ⊕ L2 (Hm ), namely we fix the map i : L2 (Hm ) → L2 (Hm ) ⊕ L2 (Hm ) such that i(f ) = (f, f ). Clearly this choice is not unique and the resulting function on the light cone we will construct depends also upon the choice of i.

48

C. Dappiaggi


To summarise we have set the map T ◦ i : L2 (Hm ) → L2 (C) such that T (i(f )) = F . In order to complete our task, a last question must be answered namely if, to any element of L2 (C), it corresponds a function in Minkowski spacetime which solves the wave equation. A positive answer has been already given in [22] and6 , thus, we end up with: Proposition 2.3. If F ∈ L2 (C), then it is the restriction on the light cone of the Fourier transform of a function ψ ∈ L4 (M, d4 x) which solves the wave equation η ψ(xμ ) = 0 with Cauchy data ∂ψ ψ(0, xi ) = f1 (xi ) , 0, xi = f2 (xi ) , ∂t √ 3 ∂ 2 1 1 with K 2 f1 (xi ) and K − 2 f2 (xi ) ∈ L2 (R3 , d3 x) where K = − and = i=1 ∂x 2. i Furthermore it exists a suitable constant C such that 1 1 (15) ψ(xμ )L4 (M) ≤ C K 2 f1 (xi )L2 (R3 ) + K − 2 f2 (xi )L2 (R3 ) . Proof. The first part of the proposition is proved in Lemma 1 of Strichartz seminal paper [22]. Hence we know that ψ(xμ ) is a solution for the wave equation lying in L4 (M, d4 x) and we need only to focus on Cauchy data. In a standard Minkowski frame with coordinates xμ = (t, x) ∈ R4 we can decompose the solution for the wave equation constructed out of F as d3 p ei(p·x−t|p|) F1 (p) + ei(p·x+t|p|) F−1 (p) , ψ(t, x) = 16π 3 |p| R3

where F1 and F−1 are respectively the restriction of F to the upper and lower light cone. Taking into account the identity d3 p F1 (p) F−1 (p) − 12 ψ(t, x) = −i √ K ei(p·x−t|p|) + ei(p·x+t|p|) , 3 | p| | p| 16π 3 R

1

and evaluating this expression for t = 0, we discover that K 2 ψ(0, x) is up to a multiplicative constant complex number the sum of the Fourier transform of 1 F±1 (p) √ ; hence, being F ∈ L2 (C), per Plancherel theorem K 2 ψ(0, x) ∈ L2 (R3 , d3 x). 2| p|

Deriving now once in the time variable and exploiting the same kind of identity, we end up with d3 p 1 F1 (p) F−1 (p) ∂ψ − ei(p·x+t|p|) (t, x) = i √ K 2 ei(p·x−t|p|) . 3 ∂t | p| | p| 16π R3

6 For

an interested reader, we point out that the first successful attempt to construct global norm estimates for solutions of the D’Alembert wave equation is due to Segal in [16] though he considers only the two dimensional scenario.

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Hence, evaluating at t = 0 this expression and still exploiting the Plancherel 1 x) ∈ L2 (R3 , d3 x). theorem as in the previous case, we end up with K 2 ∂ψ ∂t (0, To conclude the demonstration it suffices to notice that the field ψ(xμ ) and the functions f1 (xi ), f2 (xi ) satisfy the hypotheses of corollary 2 in [22] where the norm estimate (15) for the homogeneous D’Alembert wave equation has been proved. 2.1.1. From bulk to null infinity. The results from the previous section can be applied to introduce a “projection” of finite-norm solutions φ for the massive Klein–Gordon equation to null infinity. In particular let us summarise that all the informations of φ can be encoded in the following triplet of data: 1. the function ψ(xμ ) which solves the massless Klein–Gordon equation of motion constructed along the lines of Proposition 2.3, 2. the quasi-regular representation U (Λ), 3. the intertwiner T : L2 (C) → L2 (Hm ) ⊕ L2 (Hm ). We stress that the inclusion of T and U as a datum to reconstruct the original massive field is essential since neither assigning U implies that it exists a unique unitary intertwiner T nor fixing T implies that it is an intertwiner for a unique pair of representations U and U . Thus the overall problem reduces to find a projection for ψ(xμ ) to null infinity. As a first step let us remember that Minkowski spacetime can be compactified in the Einstein static universe [23]. More in detail, let us consider the coordinates (u, v, θ, ϕ) being (θ, ϕ) the standard coordinates on S2 , u = t + r and v = t − r with r as radial coordinate and let us choose as conformal factor −1 Ω2 = 4 (1 + u2 )(1 + v 2 ) . (16) Hence the flat metric is rescaled to ds2 = gμν dxμ dxν =

4 (u − v)2 2 (θ, ϕ) , −dudv + dS (1 + u2 )(1 + v 2 ) 4

. with dS2 (θ, ϕ) = dθ2 + sin2 θdϕ2 . If we perform the change of variables τ = tan−1 u + tan−1 v ,

η = tan−1 u − tan−1 v ,

(17)

then we can realize the original Minkowski spacetime as the locus (−π, π) × (−π, π) × S2 ⊂ R × S3 with respect to the metric g μν dXμ dXν = −dτ 2 + dη 2 + sin2 η dS2 (θ, ϕ) , ds2 =

(18)

. Let us notice that, the closure of the image i.e., that of Einstein static universe M gμν ) is compact and that + is nothing but of Minkowski spacetime in (R × S3 , the locus τ + η = π. More importantly this new background is still globally hyperbolic and, if we . introduce ψ = Ω−1 ψ, then it is a solution of the Klein–Gordon equation g ψ − . μν R ∇μ ∇ν is the wave operator with respect to the metric gμν g =g 6 ψ = 0 where and R = 1 is the scalar curvature of Einstein static universe. Furthermore, since

50

C. Dappiaggi


, we can recast the original Cauchy surface R3 at t = 0 is mapped into τ = 0 in M the Cauchy problem in Proposition 2.3 as ⎧ μ ψ(X ) μ ⎪ ⎪ ⎨ g ψ(X ) = 6 X i ) = f1 (X i ) (19) ψ(0, ⎪ ⎪ ⎩ ∂ ψ i = f2 (X i ) , ∂τ 0, X . 12 f1 (X i ) ∈ L2 (S3 ) and K − 12 f2 (X i ) ∈ where X μ = (τ, X i ) = (τ, η, θ, ϕ) and where K the square-root of the Laplace–Beltrami operator out of the spaL2 (S3 ) being K tial component of the metric (18). Here square integrability is meant with respect to the measure dμ = sin2 η sin θdηdθdϕ. μ ) satisfies the Klein–Gordon equation with m2 = 1/6, it coincides Hence ψ(X −1 and furthermore it lies in with Ω ψ in the image of Minkowski spacetime in M 4 4 L (M, | g|d X) since ψ(xμ )4L4 = |ψ(xμ )|4 d4 x R4

= R4

μ )|4 Ω4 d4 x = |ψ(x

π π

d4 X

μ )|4 , | g ||ψ(X

−π −π S2

where in the last equality we exploited the coordinate change (17). Unfortunately, since our aim is to project ψ on null infinity, the best available tools to define a function on + are trace theorems for Sobolev spaces. In order to exploit them the set of solutions for the wave equation we are taking into account is too big and thus we need to consider only more regular solutions for the wave equation. To understand which is the less restrictive constraint we have to impose, let us gather all the needed ingredients. As a first step we point out that, being Minkowski spacetime an openset of finite volume (either with respect to Lesbegue g|dτ dηdS2 (θ, ϕ)) in Einstein static universe, then measure or with respect to | Hölder inequality grants us that Lp (M ) ⊂ Lq (M ) for all 1 ≤ q < p ≤ ∞ (see Chapter 2, Lemma 3.7 in [19]). This property can be recast at a level of first order Sobolev spaces in Lp (M ), i.e., W 1,p (M ) ⊂ W 1,q (M ) for 1 ≤ q < p ≤ ∞. As a second step we aim to exploit Proposition 4.3 in [19] according to which, if Ω is a bounded domain in RN with a three dimensional C 1 -boundary ∂Ω, then it exists a linear trace operator γ : W 1,p (Ω) → Lp (∂Ω) which is continuous and uniquely determined by the boundary value of the functions u ∈ C 1 (Ω). Furthermore the kernel of γ is W01,p (Ω), i.e., the closure of C0∞ (Ω) in W 1,p (Ω). Our scenario meets all the geometric requirements in the above hypothesis since Minkowski background is a bounded open set in Einstein static universe R × S3 which, in its turn, can be identified as an open set of R5 . Furthermore the boundary of M consists of two smooth null hypersurfaces – future and past

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null infinity – and thus, taking into account that ψ lies in L4 (M, d4 X) and hence in Lp (M, d4 X) for all 1 ≤ p ≤ 4, we can apply such proposition only to those ψ ∈ W 1,p (M ) still with 1 ≤ p ≤ 4. With this further condition set and with the inclusion relations between the Sobolev spaces as discussed before, we are entitled to introduce the map γ|+ : W 1,p (M ) → Lq (+ ) where q can be fixed to any value lower or equal to p. Here + is the locus (−π, π) × S2 and the measure on + is the Lesbegue one. Hence, being + in this reference frame an open set of R × S2 , each function on Lq (+ ) can be also read as an element in Lq (R × S2 ). This property will be exploited in the next section. Taking into account that, both from a physical point of view and for the analysis in the next section, it is better to work with Hilbert spaces on the boundary we can summarise the previous discussion as: Proposition 2.4. Assume that Minkowski spacetime is conformally embedded as , an open set – M – of Einstein static universe (M g) with g as in (18). Then, . 4 4 4 for anysolution of the wave equation ψ ∈ L (R , d x) the function ψ = Ω−1 ψ ∈ g|d4 X) – with Ω chosen as in (16) – solves (19). Furthermore, whenL4 (M, | ever ψ ∈ W 1,p (M ) with p ≤ 4, it exists a continuous projection operator γ|+ : W 1,p (M ) → Lq (+ ) where we fix q = 2 if 2 ≤ p ≤ 4 whereas q = 1 if p = 1. The image Ψ under γ|+ of ψ will be referred to as its restriction on future null infinity. Remark 2.2. This last proposition partly overlaps the scenarios envisaged in [4,12] where only solutions ψ to the D’Alembert wave equation with compactly supported initial data where taken into account. As partly discussed in the introduction, in this case, ψ ∈ C ∞ (R4 ) and accordingly also ψ ∈ C ∞ (M ) adopting the nomenclature of the previous analysis. Furthermore the uniqueness of the solution for the Cauchy problem of the Klein–Gordon equation in the Einstein static universe al coinciding with ψ if restricted lows to construct a unique function in the whole M to M . Hence, in this case, restriction to + simply means evaluation of the solution on future null infinity. Remark 2.3. We point out that the additional regularity condition (i.e., ψ ∈ W 1,p (M )) on the solutions for the D’Alembert wave equation has been set in the Einstein static universe because a direct inspection of the previous construction shows that, although, whenever f ∈ Lp (R4 , d4 x), Ω−1 f ∈ Lp (R4 , Ω4 d4 x) for p ≤ 4, this does not held true for first order Sobolev spaces. In other words, even if f ∈ W 1,p (R4 ), then, exploiting Liebinitz rule, one can realize that, due to the contribution of the derivatives of the conformal factor (16), Ω−1 f lies in Lp (R4 , Ω4 d4 x) but not necessary in W 1,p (R4 , Ω4 d4 x). Hence we have achieved our goal since all the information from the original massive field φ satisfying (5) has been projected onto null infinity in the triplet

52

C. Dappiaggi


(Ψ, U , T ) where U is the quasi-regular O(3, 1) representation acting on the massless field and T is the intertwiner constructed in the previous section. Two natural questions arise at this stage: • What about Poincaré covariance? • How can we translate at a field theoretical level the assertion that (Ψ, U , T ) contains the information of the massive scalar field? Let us answer to the first and simpler question. Up to now we have considered only the quasi-regular O(3, 1) action on the set L2 (C) or L2 (Hm ). If we want to deal instead with Poincaré covariant scalar field theories, a function φ satisfying either (5) or D’Alembert wave equation would transform in a momentum frame as μ ) = eiaμ pμ φ(Λ −1 pμ ) , (Λ, aμ )φ(p U where the hat symbol stands for the Fourier transform. This identity supplemented with the constraints η ρσ pρ pσ φ(pμ ) = m2 φ(pμ ) with m2 ≥ 0 and sgn(p0 ) > 0 is a unitary irreducible representation for the full Poincaré group [3]. In order to relate the two above points of view, beside the trivial restriction from O(3, 1) to SO(3, 1), we need only to invoke the induction-reduction theorem (c.f. Chapter 18 in [3]) according to which the quasi-regular representation U (Λ) on L2 (Hm ) is a) the SO(3, 1) representation induced from the identity representation of SO(3), b) the restriction of the scalar Poincaré representation to the Lorentz group. At the same time, if we start from U (Λ), it induces the unitary and irreducible scalar representation of the full Poincaré group. A similar reasoning and conclusion holds if we consider L2 (C) with the associated quasi-regular representation U (Λ). 2.2. Data reconstruction on null infinity In this subsection we face the last and most important question namely in which sense the information from the bulk massive field, projected on null infinity out of U , T ), can be interpreted from a classical field theory perspective. To this end (ψ, we shall exploit some recent analysis according to which it is possible to explicitly construct a diffeomorphism invariant field theory on future null infinity. Afterwards our aim will be to show how the above triplet can be interpreted in terms of such a boundary free field theory. Bearing in mind the notations and the nomenclatures of Subsection 1.1, we review some features of the construction of a Poincaré invariant field theory on + – thought as a null differentiable manifold7 – for smooth scalar fields invariant under the R subgroup of the BMS as discussed in [1, 4, 5]. For a different approach to holographic related issues which also takes into account the BMS group, a reader may wish to consult [18]. 7 More

appropriately one should claim that we are constructing a QFT on the equivalence class of triplets (+ , na , hab ) associated to the bulk Minkowski spacetime.

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We shall follow two possible roads: the first starts from a massless bulk scalar field and it imposes BMS invariance on the projection of such a field on null infinity whereas the second ignores the bulk and it constructs a scalar free field theory on + by means of the Mackey–Wigner programme, i.e., we only exploit the knowledge of the symmetry group. We stress that the full construction has been developed for a generic asymptotically flat spacetime due to the universality of the boundary structure. Hence, although both the above mentioned approaches have been fully accounted for in [4, 5], here we will only review the details adapted to the case of Minkowski bulk spacetime and, thus, Poincaré symmetry group on null infinity leaving an interested reader to the above cited manuscripts for a careful analysis. Let us thus start from the first part of this programme; in order to construct a meaningful scalar field theory on + starting from the bulk, we can focus only on smooth solutions ψ for the D’Alembert wave equation. As per Remark 2.2 such a bulk field projects to Ψ ∈ C ∞ (+ ). Then, if we wish to define a suitable representation of R acting on each Ψ, the following proposition holds [4]: Proposition 2.5. Let us take Minkowski spacetime (M, ημν ) and an associated , spacetime (M gμν ) out of Penrose compactification process (not necessarily Einstein static universe) and let us fix an arbitrary gauge factor ω. Then, for any but fixed λ ∈ R and for any but fixed g ∈ R ⊂ BM S, a R-representation is A(λ) (g) : λ is C ∞ (+ ) → C ∞ (+ ) such that the map t → A(λ) (gt )Ψ = lim+ (ωΩ) gt∗ (ψ) + smooth for every fixed bulk scalar field ψ with smooth projection Ψ on and for every but fixed one-parameter subgroup of the bulk Poincaré group. In the Bondi frame (u, z, z¯) it reads (λ) A (g)Ψ (u , z , z¯ ) = KΛ−λ (z, z¯)Ψ(u, z, z¯) , ∀g = Λ, α(z, z¯) ∈ R where the primed coordinates and KΛ (z, z¯) are defined as in (2) and (3). Since our aim is to deal with unitary and irreducible representations we have to go one step further, i.e., Proposition 2.6. Let us consider the set S(+ ) ⊂ C ∞ (+ ) of real functions Ψ such that Ψ itself and all its derivatives decay faster than any power of |u| when |u| → ∞ and uniformly in (z, z¯). Then S(+ ) can be endowed with the strongly non degenerate symplectic form ∂Ψ1 ∂Ψ2 σ(Ψ1 , Ψ2 ) = − Ψ1 Ψ2 dudS2 (z, z¯) , ∂u ∂u R×S2

and (S(+ ), σ) is invariant only under A(1) . Furthermore if we introduce the pos + of Ψ ∈ S(+ ) as itive frequency part Ψ du iEu + (E, z, z¯) = √ Ψ e Ψ(u, z, z¯) , E ∈ [0, ∞) (20) 2π R

54

C. Dappiaggi


+ . If we denote with S(+ )C the complex linear combi=Ψ + + Ψ we can write Ψ nations of these functions Ψ(E, z, z¯), then + 1. S( )C can be closed to Hilbert space H with respect to the Hermitian inner product 1, Ψ 1, Ψ 2 = −iσ(Ψ 2) . Ψ Furthermore the Hilbert space (H, , ) is unitary isomorphic to the space L2 (R × S2 , EdEdS2 (z, z¯)). 2. the representation A(1) of R on H acts as (1) (E, z, z¯) A (g)Ψ −1 −1 EKΛ (Λ−1 z, Λ−1 z¯), Λ−1 z, Λ−1 z¯ , (21) = eiEKΛ (Λ z,Λ z¯)α(z,¯z) Ψ for any g = (Λ, α(z, z¯)) ∈ R and A(1) is unitary on H. The proof of this theorem is a recollection with minor modifications of the demonstration of Proposition 2.9, 2.12 and 2.14 in [4]. Hence we refer to such paper for an interested reader. We now state a useful lemma out of this last proposition: Lemma 2.3. The projection on + of each function ψ constructed as in Proposition 2.4 can be unitary mapped into an element of (H, , ). Proof. In Proposition 2.4 we projected a function with support on the image of Minkowski spacetime in Einstein static universe to a function Ψ ∈ L2 (+ ) being + , in that specific background, (−π, π) × S2 . Since S2 is compact and (−π, π) is an open bounded set of R, Ψ can also be read as an element of L2 (R × S2 ). We stress that, switching from the Lesbegue measure in L2 (+ ) to the natural SO(3)-invariant measure on S2 for L2 (R × S2 ) is harmless. ∈ According to Plancherel theorem and to (20) the Fourier transform Ψ L2 (R × S2 , EdEdS2 ) and, hence, according to Proposition 2.6, it can be unitary mapped in (H, , ). This concludes the first part of our programme though a complete analysis would require the proof that A(1) is irreducible or how it decomposes in irreducible components. The answer to this question will be a byproduct of the Wigner– Mackey analysis that we discuss now. Such approach calls for the construction of a classical free field theory on a generic manifold only by means of the symmetry group, R ⊂ BM S in our case. Although R is homomorphic to the Poincaré group we cannot simply refer to the standard construction for a covariant field theory in Minkowski background as discussed, to quote just one example, in Chapter 21 of [3]. On the opposite we need to consider R as a subgroup of the BMS and, hence, we shall adapt the analysis in [4] to this simpler scenario. Referring to this last cited paper for further details, let us introduce the character associated to an element of N ≡ C ∞ (S2 ) as a group homomorphism

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χ : N → U (1). Since N can be endowed with a nuclear topology (see Theorem 2.1 in [5]) it can be seen as an element of the Gelfand triplet N ⊂ L2 (S2 ) ⊂ N ∗ where N ∗ is the set of real continuous linear functionals on N (endowed with the induced topology). Hence, as shown in Proposition 3.6 in [4], for any character χ it exists a distribution β ∈ N ∗ such that χ(α) = ei(β,α) ,

(22)

where (, ) stands for the pairing between N ∗ and N . Such a result can be applied also to the translational subgroup of the Poincaré 4 ∞ 2 group on + provided either that one exploits the inclusion 4T∗ ⊂ C (S ) previ4 ously discussed either that the dual space of T – namely T – is characterized in the following way [13]: if we construct the annihilator of T 4 as & 4 0 % T = β ∈ N ∗ | β, α(z, z¯) = 0, ∀α(z, z¯) ∈ T 4 , 4 ∗ is (isomorphic to) the quotient T

N∗ . (T 4 )0

Still referring to [4], the Wigner–Mackey approach for the BMS group introduces the intrinsic covariant scalar field on null infinity as a map ψ : N ∗ → H which transforms under the unitary representation D of SO(3, 1) C ∞ (S2 ) as D Λ, α(z, z¯) ϕ (β) = χβ (α)ϕ(Λ −1 β) , ∀ Λ, α(z, z¯) ∈ SO(3, 1) C ∞ (S2 ) where χβ is a character. Whenever the bulk spacetime is the Minkowski background and hence we deal with the R subgroup of the BMS, the above expression translates in ⎧ ∗ ⎨ ϕ : T4 → R (23) ⎩ D Λ, α(z, z¯) ϕ (β) = χβ (α)ϕ(Λ −1 β) ∀ Λ, α(z, z¯) ∈ R , where now β must be thought both as a distribution and as a representative for ∗ an equivalence class in the coset (TN4 )0 . Remark 2.4. It is important to point out that, in the above discussion, the real difference between a scalar field on Minkowski background and on null infinity is due to the action of the representation or more properly of the U (1) phase factor. To be more precise, Proposition 3.2 in [5] grants us that, being T 4 a subspace ∗ of a locally convex topological linear space, namely C ∞ (S2 ), the coset (TN4 )0 is a 4-dimensional space. Thus, if we introduce the set of dual spherical harmonics ∗ Ylm with l = 0, 1, m = −l, . . . , l defined as (Yl∗ m , Ylm (z, z¯)) = δll δmm , then any 4 ∗ β∈ T can be decomposed as β=

l 1 l=0 m=−l

∗ βlm Ylm .

56

C. Dappiaggi

Hence we can extract from each β the four-vector

3 β00 μ √ , β1−1 , β10 , β11 . β =− 4π 3


(24)

Moreover we define the action of Λ ∈ SO(3, 1) on a generic distribution β ∈ N ∗ as Λβ, α(z, z¯) = β, Λ−1 α(z, z¯) ∀α(z, z¯) ∈ C ∞ (S2 ) , (25) being the action of Λ on α(z, z¯) the one defined in (2) and (3). A direct inspection ∗ ∗ of Proposition 1.1 and of the isomorphism between (TN4 )0 and T 4 shows that β μ transforms as a covector and the quantity m2 = ημν β μ β ν

(26)

is SO(3, 1) invariant. Furthermore m2 is also a Casimir for the unitary and irreducible representations of the BMS group and hence also for the R subgroup. This confirms the previous statement according to which (23) differs from the counterpart in Minkowski background only in the character. The covariant scalar field (23) does not transform under an irreducible representation of the R group and, hence, in a physical language it represents only a kinematically allowed configuration. On the opposite, if we look for a genuine free field, ϕ should transform under a unitary and irreducible representation; to overcome such a discrepancy we can still exploit Wigner–Mackey theory which calls for imposing a further constraint to (23). From a more common perspective in classical field theory, this operation amounts to impose on ϕ the equations of motion written in the momenta representation; for the above scalar field it reads [4]: μν = 0, (27) η βμ βν − m2 ϕ[β] where βμ is the four vector as in (24). Two comments on (27) are in due course: 1. the equation under analysis could be recast in the more appropriate language of white noise calculus. In the general framework of BMS free field theory ϕ[β] is a functional over a distribution space which is square integrable with respect to a suitable Gaussian measure μ. Hence (27) should be recast in this scenario in terms of (multiplication) operators acting on L2 (N ∗ , dμ) and such analysis has been carried out in [5]. In ∗ paper we can avoid such this techniques exploiting the identification of T 4 with R4 which grants us that (27) acquires the standard meaning, i.e., the support of ϕ[β] is localized over the mass hyperboloid if m2 = 0 and over the light cone if m2 = 0. Most importantly the function ϕ corresponds to an element in L2 (C). 2. the equations (23) and (27) are equivalent to a function transforming under a unitary and irreducible representation of the Poincaré group induced from the SO(3) or from the SO(2) T 2 little groups depending if m = 0 or m = 0. At the same time a direct inspection of the analysis of Chapter 3

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in [4] immediately shows that the representation in (23) is nothing but the scalar BMS representation restricted to the R subgroup. Before concluding our analysis we still need the last ingredient which relates the two above constructions of a massless scalar field theory on + . Theorem 2.1. A field Ψ on + constructed as in Proposition 2.4 corresponds to a R field (23) which satisfies (27) with m = 0. Hence the representation A(1) is also irreducible on L2 (R × S2 , EdEdS2 ). Proof. We provide here a much shorter proof than that of Theorem 3.35 in [4]. satisfies (21). Let us recall that, according to Lemma 2.3, Ψ Furthermore, following the characterization of a light cone embedded in R4 . as discussed at the beginning of Section 2.1 and identifying E with r = | p|, we the Fourier transform of Ψ constructed as in ∈ L2 (C) being Ψ end up with Ψ can be read on its own as the Proposition 2.4. According to Theorem 1 in [22], Ψ satisfying D’Alembert restriction on C of the Fourier transform for a function Ψ ∈ wave equation and, hence, lying in L4 (R4 , d4 x). The Fourier transform for Ψ = 0 and the Poincaré group R still acts S (R4 ) satisfies the constraint η μν pμ pν Ψ (1) as A . To conclude the demonstration, let us now consider (23) which satisfies (27) with m = 0. Exploiting the identification between the distribution β and the covector pμ , a direct inspection shows that the scalar R representation acts on (23) as the representation A(1) . Thus each Ψ constructed in Proposition 2.4 has been mapped into a massless R scalar free field. Irreducibility of A(1) is now a consequence of Mackey construction which grants us that the scalar R (and, thus, the A(1) ) representation induced from the scalar E(2) representation is irreducible. At this stage one could require, as a natural property, that the bulk-toboundary projection operation must be compatible with the Poincaré action. In other words, we must prove that, if we act on a norm-finite solution φ of (5) with an element of SO(3, 1)↑ T 4 , this corresponds to the action of the same element in R on Ψ, the projection on + . Since the bulk massive field is related to a massless scalar field in the bulk out of an intertwiner between the quasi-regular Lorentz group representations (univocally inducing the full Poincaré group scalar representation as previously discussed), we can switch from φ to ψ, solution of the bulk D’Alembert wave equation. The following proposition holds: Proposition 2.7. Let us consider any massless scalar field ψ constructed as in Proposition 2.3 such that ψ ∈ W 1,4 (R4 ). Let us also call γ |+ : W 1,4 (R4 ) → L2 (+ ) , . the map which rescales ψ to ψ = Ω−1 ψ with Ω as in (16) and then projects ψ + on as in Proposition 2.4. Then,

58

C. Dappiaggi


1. for any g ∈ SO(3, 1)↑ T 4 , γ |+ ψ(xμ ) , γ |+ ψ(g −1 xμ ) = A1 (g ) being g the elements of R correspondent to g out of the homomorphism in (4), 2. γ |+ is, up to the homomorphism between R and SO(3, 1)↑ T 4 , an intertwiner between the Poincaré scalar representation in the bulk and the R representation A1 on the boundary. Proof. As a starting point we recall that, according to Proposition 1.2, the Poincaré algebra is generated by a set of Killing vectors for the flat metric which smoothly extend on + in a set of ten independent BMS algebra elements generating by exponential map the whole R subgroup per completeness of + . Hence we can exploit Proposition 3.4 in the second paper in [14] to conclude that the first of the two statements in the proposition is true for all smooth solutions of the D’Alembert wave equation in Minkowski spacetime. The strategy we shall now follow is to exploit this last result together with the density of C0∞ (R4 ) in the Banach space L4 (R4 ). Hence, omitting the explicit dependence on the point xμ for sake of simplicity, for any ψ ∈ W 1,4 (R4 ) seen as an element of L4 (R4 ), we can choose a sequence ψn ∈ C0∞ such that ψn → ψ as n → ∞ in the topology of L4 (R4 ). We now wish to show that we can choose each ψn in such a way that it satisfies the D’Alembert wave equation. For a generic convergent sequence, we know that, in the worst case scenario, ψn = ρn = 0 with ρn ∈ C0∞ (R4 ). For any n let us select a new function – say fn – which solves fn = ρn , fn (0, x) = 0 , (∂t fn )(0, x) = 0 . Hence we can invoke again the results from [22] to claim that, since ρn ∈ C0∞ (R4 ) 4 and, thus, ρn ∈ L 3 (R4 ), it holds that fn ∈ L4 (R4 ) ∩ C ∞ (R4 ) with fn L4 ≤ Cρn 43 . Per direct inspection, fn → 0 as n → ∞ and, if we introduce ψn = L ψn − fn , we know that, per construction, ψn = 0 and ψn → ψ in the topology of L4 (R4 ) since ψn − ψL4 ≤ ψn − ψL4 + fn L4 . Furthermore, since ψn ∈ . C ∞ (R4 ) we know that ψn = Ω−1 ψn is a smooth solution of the D’Alembert wave equation conformally coupled to gravity in the image of Minkowski in Einstein static universe. The initial data for the Cauchy problem g − R6 ψn = 0 (being g the metric (18)) are compactly supported since ψn ∈ C0∞ (R4 ) and fn vanishes on the surface t = 0. Hence, per uniqueness of the solutions for second order hyperbolic PDE, we are entitled to extend ψn to a smooth solution of the same partial differential equation on the whole Einstein static universe. Consequently we can restrict each ψn on + and, per construction of traces on Sobolev spaces (see Chapter 2 of [19]), ψn |+ = γ|+ (ψn ). Let us now consider a generic element g ∈ SO(3, 1)↑ T 4 . Calling D(g) the scalar Poincaré group representation which coincides with the Lorentz quasiregular representation U whenever g = (Λ, 0), we end up with ψ(g −1 xμ ) =

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D(g)ψ(xμ ) and, being D bounded, strongly continuous and irreducible, ' ' ψ(g −1 xμ ) − ψ (g −1 xμ )L4 = 'D(g) ψ(xμ ) − ψ (xμ ) ' 4 → 0 . n

n

Let us now consider ' ' 'γ γ |+ (ψ)' |+ D(g)ψ − A1 (g )

L2

L

' ' |+ D(g)ψn 'L2 ≤ 'γ |+ D(g)ψ − γ ' ' |+ D(g)ψ − A1 (g ) + 'γ γ |+ (ψ)' n

L2

.

Exploiting the linearity of the trace γ|+ , the first term becomes ' ' 'γ |+ D(g)(ψ − ψ ) ' 2 . n

L

Taking into account that γ|+ is also continuous, hence bounded and that rescaling g |d4 X) where M is the per Ω−1 is a unitary operator from L4 (R4 ) in L4 (M, | image of Minkowski spacetime in the Einstein static universe, we can conclude that the above norm tends to 0 as n goes to infinity. The second term can be rewritten as A1 (g ) γ |+ (ψn − ψ)L2 (+ ) = γ |+ (ψn − ψ)L2 (+ ) where, first, we exploited the result in Proposition 3.4 in the second paper in [14] and, later, the unitarity of A1 . With the same reasoning as for the first term, we can conclude that, since ψn → ψ, the above norm tends to 0 as n → ∞. Hence the first statement in the theorem is proved. The second is linear arises simply recalling that ψ(g −1 xμ ) = D(g)ψ(xμ ), the trace operator and bounded and Ω−1 is a unitary operator from L4 (R4 ) in L4 (M, | g |d4 X). We have now all the ingredients to conclude our analysis on the projection of a massive bulk scalar field: Theorem 2.2. Let us consider any norm-finite solution φ of (5) with the associated triplet (ψ, U , T ). The latter projects to a triple (Ψ, U , T ) on future null infinity which identifies two Poincaré invariant free scalar fields constructed a ` la Wigner– Mackey and solving (27) with the same mass value as φ. Proof. According to the hypothesis of the theorem we can associate to φ the triplet (ψ, U , T ) where ψ solves the D’Alembert wave equation. We can now exploit Proposition 2.4 to project ψ in a square integrable function Ψ over + : Ψ = ρ(ψ) . −1 where ψ = Ω ψ. Hence, being U and T respectively a representation and an intertwiner thus independent from coordinates, we construct on null infinity the triplet (Ψ, U , T ). The representation U (Λ) is the quasi-regular representation of the Lorentz group and it unambiguously induces (or it is the restriction of) the scalar R representation which acts on Ψ as the representation A(1) from (21). We can now exploit Theorem 2.1 according to which the pair (Ψ, A(1) ) corresponds to one R invariant field (23) which satisfies (27) with m = 0. Hence (Ψ, A(1) ) can be traded with (ϕ, D(Λ, α(z, z¯))) where (Λ, α(z, z¯)) ∈ R and D is the scalar representation in (23). Still the induction-reduction theorem for group representations (Chapter 18 in [3]) grants us that the restriction of D to SO(3, 1) is exactly U and that the quasi-regular representation unambiguously induces the scalar R representation. U , T ). The circle has Hence we have mapped the original triplet (Ψ, U , T ) in (ϕ,

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been almost closed and our last step consists of exploiting the same reasoning as in the proof of Theorem 2.1, i.e., we can read ϕ as a solution for the massless wave C . Hence we can now equation constructed out of an element of L2 (C) – say ϕ| 2 exploit our last ingredient namely the intertwiner T : L (C) → L2 (Hm ) ⊕ L2 (Hm ), C ) = (f, g). Accordingly both f and g lies in L2 (Hm ) and the Lorentz i.e., T (ϕ| group acts as [U (Λ)f ] (pμ ) = f (Λ−1 pμ ) for all Λ ∈ SO(3, 1). Still exploiting Theorem 1 in [22], we can interpret f (or g) as the restriction on the mass hyf or ϕ g – whose Fourier transform satisfies perboloid Hm of a function – say ϕ the Klein–Gordon equation of motion with mass m and it lies in Lp (R4 , d4 x) with 10/3 ≤ p ≤ 4. If we now take into account that the original field ϕ is an intrinsic R free field, we are entitled to switch from pμ to the variables β μ . To conclude we can exploit Remark 2.4 according to which a covector β μ transforming under ∗ the standard SO(3, 1) action corresponds to a distribution β ∈ T 4 ⊂ N ∗ on which Λ ∈ SO(3, 1) acts according to (25). Eventually still the induction theorem f , U ) allows us to construct from U the scalar R representation D. Hence both (ϕ and (ϕ g , U ) correspond unambiguously to a R massive scalar field as in (23) with support on the mass hyperboloid, i.e., with the same value for m2 as the original Minkowski field φ. Remark 2.5. The projection of a bulk massive scalar field into two boundary massive scalar fields is a natural byproduct of the intertwining operator. In the projection of f ∈ L2 (Hm ) to a function over the light cone, we could imbed f into the element (f, f ) of the diagonal subgroup of L2 (Hm ) ⊕ L2 (Hm ); on the opposite on the boundary we perform the inverse operation mapping a square integrable function over the light cone into L2 (Hm ) ⊕ L2 (Hm ). Hence there is no guarantee that the intertwiner identifies an element of the diagonal subgroup and we are forced to take into account two massive fields instead of a single one.

3. Conclusions In this paper we have shown that, exploiting Strichartz harmonic analysis on hyperboloids, it is possible to project the information of a norm finite massive scalar field φ in Minkowski spacetime into a triplet of data on null infinity: (Ψ, U , T ) where Ψ is the projection on + out of a trace operator of a solution for the bulk D’Alembert wave equation, U is the SO(3, 1) quasi-regular representation whereas T is a unitary intertwiner from L2 (C) to L2 (Hm ) ⊕ L2 (Hm ). The result we achieve has a twofold advantage. From one side it is coherent with Helfer result which states that the space of sections for any vector bundle over null infinity homogeneous under the action of the Poincaré group carries only the massless representation. As a matter of fact Ψ can be ultimately interpreted as a free field on the conformal boundary with m = 0. From the other side we can recover the original interpretation of massive fields exploiting the action of T and, as shown in Theorem 2.2, the original single field φ corresponds to two separate massive free fields in the R invariant theory constructed a` la Wigner–Mackey.

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Although we believe the result is rather appealing opening a wide range of possible applications, it is fair to admit that it is to a certain extent not sharp. As a matter of fact, in the whole construction we performed three arbitrary choices: the first, already discussed, refers to the imbedding of an element f ∈ L2 (Hm ) (the restriction on the mass hyperboloid of the Fourier transform of φ) into the diagonal subgroup of L2 (Hm ) ⊕ L2 (Hm ). The second and the strongest between the performed choices arises in the projection to + ; the general solution ψ of the D’Alembert wave equation we con 4 4 g|d x) but, in order to apply trace theorems, we needed structed lies in L (M, | to consider at least Sobolev spaces of first order. This restricts the range of validity of our results and it will be interesting to eliminate such constraint from our analysis. The third and less pernicious of the choices lies in the construction of the above mentioned trace operator. As a matter of fact we embedded Minkowski spacetime into an open region of Einstein static universe. Hence this amounts to select a preferred gauge factor ω according to the definitions of Section 1.1 contrary to the projection operator introduced in [4, 12] which provides a smooth function over + for any possible choice of ω. Nonetheless we feel that, fixing ω in our analysis, does not lead to a loss of generality since we can ultimately interpret our results in terms of a general field theory constructed over + without the need for such a choice of the gauge factor. To conclude we wish to discuss possible applications of our results. Our main target is an holographic interpretation of bulk field theory along the lines of [4] and the previous section was written with this goal in mind. As a matter of fact we have proved that, at least in Minkowski background, it is possible to project each solution of a massive Klein–Gordon equation of motion into a suitable counterpart at null infinity. Within this respect we must stress that our result may not be sharp since the information of the bulk massive field is encoded in a unique boundary massless scalar field which, on its turn, corresponds to two massive scalar fields on + . It is conceivable that a modification of our construction may lead to define a bulk-to-boundary projection map in such a way that the massless field on future null infinity gives rise to a single Klein–Gordon field out of the intertwiner T ; we feel this subject worth of a deep analysis in the forthcoming future. Such bulk-toboundary interplay does not represent the only possible application of our analysis and we envisage that our results could be possibly exploited also in other research fields such as, to quote an example, conformal scattering problems. Nonetheless we believe that the most interesting perspective consists of the development of a similar result in a generic globally hyperbolic and asymptotically flat spacetime. Already at a first reading of this manuscript, one can realize that the extensive use of tools proper of harmonic analysis forbids to mimic our procedure in a more generic scenario8 . Nonetheless we feel that finding a way to project the 8 It

is possible to wonder if our construction extends to higher dimensional flat backgrounds and, since both the tools proper of harmonic analysis and Strichartz estimates can be generalized to any dimension, there is no apparent obstruction. On the opposite the question if one can derive

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information for a massive scalar field on null infinity in Minkowski background is an encouraging starting point to deal with the same problem in more complicated frameworks. A positive conclusion of such a research project would also open the way to develop in a generic background (and not only in Minkowski), with tools proper of the algebraic quantization of field theory, a correspondence between the quantized bulk massive scalar theory and the boundary counterpart. We aim to address such an issue in future works.

Acknowledgements The author is grateful to V. Moretti and N. Pinamonti for fruitful discussions at early stages of this research project and to O. Maj for several useful discussions on hyperbolic partial differential equations. This work has been partly supported by GNFM-INdAM (Istituto Nazionale di Alta Matematica) under the project “Olografia e spazitempo asintoticamente piatti: un approccio rigoroso”.

References [1] G. Arcioni and C. Dappiaggi, Exploring the holographic principle in asymptotically flat spacetimes via the BMS group, Nucl. Phys. B 674 (2003), 553, [arXiv:hepth/0306142]. [2] A. Ashtekar and B. C. Xanthopoulos, Isometries compatible with asymptotic flatness at null infinity: A complete description, J. Math. Phys 19 (1978), 2216. [3] A. O. Barut, R. Raczka, Theory of group representations and applications, World Scientific 2ed (1986), [4] C. Dappiaggi, V. Moretti and N. Pinamonti, Rigorous steps towards holography in asymptotically flat spacetimes, Rev. Math. Phys. 18 (2006), 349, [arXiv:grqc/0506069]. [5] C. Dappiaggi, Free field theory at null infinity and white noise calculus: A BMS invariant dynamical system, arXiv:math-ph/0607055. [6] H. Friedrich, On static and radiative space-times, Comm. Math. Phys. 119 (1988), 51. [7] I. M. Gel’fand, M. I. Graev and N. Ya. Vilenkin, Generalized functions: Application of harmonic analysis, Vol. 4 (1966), Academic Press. [8] R. Geroch, Asymptotic structure of space-time, (1977) ed. P. Esposito and L. Witten, Plenum, New York. [9] A. D. Helfer, Null infinity does not carry massive fields, J. Math. Phys. 34 (1993), 3478. [10] S. Hollands and A. Ishibashi, Asymptotic flatness and Bondi energy in higher dimensional gravity, J. Math. Phys. 46 (2005), 022503, [arXiv:gr-qc/0304054]. a similar result for higher dimensional curved backgrounds is less adequate since conformal completion techniques ` a la Penrose may run into serious difficulties [10].

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[11] N. Limic, J. Niederle and R. Raczka, Eigenfunction expansions associated with the second-order invariant operator on hyperboloids and cones. III, J. Math. Phys. 8 (1967), 1079. [12] L. J. Mason, J.-P. Nicolas, Conformal scattering and the Goursat problem, J. Hyper. Diff. Eq. 1 (2004), 197. [13] P. J. McCarthy, The Bondi–Metzner–Sachs group in the nuclear topology, Proc. Royal Soc. London A343 (1975), 489. [14] V. Moretti, Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence, Comm. Math. Phys. 268 (2006), 727, [arXiv:gr-qc/0512049]; Quantum ground states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property, [arXiv:grqc/0610143], to appear in Comm. Math. Phys. [15] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, Self-Adjointness, (1975) Academic Press. [16] I. Segal, Space-time decay for solutions of wave equations, Adv. in Math. 22 (1976), 305. [17] W. Rossmann, Analysis on real hyperbolic spaces, J. Func. Anal. 30 (1978), 448. [18] B. Schroer, Area density of localization-entropy, Class. Quant. Grav. 23 (2006), 5227, [arXiv:hep-th/0507038]. [19] R. E. Showalter, Monotone operators in Banach spaces and partial differential equations, Mathematical Surveys and Monographs 49 (1997) AMS. [20] R. S. Strichartz, The stationary observer problem for u = M u and related equations, J. Diff. Eq. 9 (1971), 205. [21] R. S. Strichartz, Harmonic analysis on hyperboloids, J. Func. Anal. 12 (1973), 341. [22] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. Jour. 44 (1977), 705. [23] R. M. Wald, General relativity, (1984) Chicago University Press. [24] J. Winicour, Massive fields at null infinity, J. Math. Phys. 29 (1988), 2117. Claudio Dappiaggi Dipartimento di Fisica Nucleare e Teorica Universit` a di Pavia via A. Bassi 6 I-27100 Pavia Italy e-mail: [email protected] Communicated by Klaus Fredenhagen. Submitted: May 7, 2007. Accepted: July 16, 2007.



Renormalization of Orientable Non-Commutative Complex Φ63 Model Zhituo Wang and Shaolong Wan Abstract. In this paper we prove that the Grosse–Wulkenhaar type noncommutative orientable complex scalar ϕ63 theory, with two non-commutative coordinates and the third one commuting with the other two, is renormalizable to all orders in perturbation theory. Our proof relies on a multiscale analysis in x space.

1. Introduction Since the rebirth of non-commutative quantum field theory [2, 5, 27, 29], people encountered a major difficulty. A new kind of divergences appeared in noncommutative field theory [20], the UV/IR mixing. It is a kind of infrared divergence which appears after integrating the high scale variables and can’t be eliminated. It lead people to declare such theories non-renormalizable. But a real breakthrough of that deadlock came from H. Grosse and R. Wulkenhaar [11,13]. They found that the right propagator for the scalar field theory in non-commutative space should be modified to obey the Langmann–Szabo duality [18]. In a series of paper they proved that the ϕ4 scalar field theory in 4 dimensional Moyal plane, ϕ4 4 for short, is renormalizable to all orders using Polchinski’s equation [21] in the matrix base. Rigorous estimates on the propagator required by the Grosse–Wulkenhaar analysis and a more explicit multiscale analysis were provided in [23]. Then Gurau et al. gave another proof that the non-commutative ϕ4 4 is renormalizable, also with a multiscale analysis but completely in position space [15]. The corresponding parametric representation of the model was also built in [16]. Recently the model has been shown to have no Landau ghost, so that it is actually better behaved than its commutative counterpart [3,4,12] and can presumably be built non perturbatively. Apart from the ϕ4 4 theory, many other theories in non-commutative space have now also been proved to be renormalizable to all orders, such as the Gross– Neveu model in 2 dimensional Moyal plane [30], the LSZ model [19] and the ϕ3 theory in various dimensional space [8–10]. For an updated review, see [24, 25].

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In this paper we prove that the orientable non-commutative complex ϕ6 field theory, (ϕ¯ ϕ)3 for short, in 2 + 1 dimensional space, with two dimensions equipped with non-commutative Moyal product and the third one which commutes with the two others, is renormalizable to all orders of perturbation theory. In the first section we derive the propagator and establish the x-space power counting of the theory. In the second section we prove that the divergent subgraphs can be renormalized by counterterms of the form of the initial Lagrangian. Our proof, based solely on x space with multiscale analysis, follows closely the strategy of [15]. For technical reasons, we restrict ourselves here to the simpler orientable case, but we plan to study the nonorientable case or real scalar ϕ63 model as well. We are motivated by the fact that the quantum Hall effect at finite temperature should also be described by a 2+1 dimensional field theory with two anticommuting space and one commuting imaginary time coordinates [17, 22, 24, 28]. Our model is therefore a first step towards understanding how to renormalize such theories. We plan to compute in a future publication the renormalization group flow of this model, which involves three parameters λ, g and Ω, instead of two in the ϕ44 case.

2. Power counting in x-space 2.1. Model, notations 3 The simplest orientable non-commutative complex ϕ6 3 theory is defined on R equipped with the associative and non-commutative Moyal product d2 k (2.1) d2 y a x+ 12 θ·k b(x+y) eik·y . (a b)(x) = 2 (2π) The action functional is xμ ϕ) ¯ (˜ xμ ϕ) + μ20 ϕ¯ ϕ S[ϕ] = d2 x dx0 ∂μ ϕ¯ ∂ μ ϕ + ∂0 ϕ¯ ∂ 0 ϕ + Ω2 (˜ λ g + ϕ¯ ϕ ϕ¯ ϕ + ϕ¯ ϕ ϕ¯ ϕ ϕ¯ ϕ x, x0 (2.2) 2 3 where x ˜μ = 2(θ−1 )μν xν , and x = (xμ ), μ = (1, 2) are the non-commutative variables and x0 is the commutative variable, that is [xμ , xν ] = iθμν ,

[x0 , xμ ] = 0 .

Here θμν is a constant matrix and the Euclidean metric is used. Lemma 2.1. The kernel of the propagator in our (ϕ¯ ϕ)3 model is 1

(x −x )2 Ω coth(2Ωt) Ω Ω(2t)− 2 x·x − 0 4t 0 −μ20 t (x2 +x2 )+ sinh(2Ωt) 2 C(x, x ) = √ 3 , e− 2π sinh(2Ωt) 2

2

2

with x2 = x21 + x22 , x = x 1 + x 2 , x · x = x1 x1 + x2 x2 .

(2.3)

Orientable Non-Commutative Complex Φ63 Model

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Proof. The propagator of interest is expressed via the Schwinger parameter trick as: ∞ dte−tH . (2.4) H −1 = 0

Let H be H = −∂12 − ∂22 − ∂02 + Ω2 x2 + μ20

(2.5)

where μ0 is the mass of the field. The integral kernel of the operator e−tH is: 1

e

−tH

Ω(2t)− 2

(x, x ) = √ 3 e−A , 2π sinh(2Ωt) Ω cosh(2Ωt) 2 Ω (x0 −x0 )2 (x +x2 )− x · x + +μ20 t . A= 2 sinh(2Ωt) sinh(2Ωt) 4t

(2.6) (2.7)

At first we note that the kernel is correctly normalized: as Ω → 0, we have e−tH (x, x ) →

1 (4πt)

3 2

e−

|x−x |2 +|x0 −x0 |2 4t

,

(2.8)

which is the normalized heat kernel. Then we must check the equation d −tH e + He−tH = 0 . dt

(2.9)

In fact,

1 2 d −tH Ωe−A (2t)− 2 Ω2 e x + x2 =√ 3 − 2Ω coth(2Ωt) + 2 dt sinh (2Ωt) 2π sinh(2Ωt) 2 (x0 − x0 )2 1 −1 2Ω cosh(2Ωt) 2 − μ0 . − t − x·x + (2.10) 2 4t2 sinh2 (2Ωt)

Moreover, (−∂12

−

∂22 )e−tH

−∂02 e−tH

1 Ωe−A (2t)− 2 Ω2 (x2 + x2 ) =√ 3 2Ω coth(2Ωt) − sinh2 (2Ωt) 2π sinh(2Ωt) 2Ω2 coth(2Ωt) 2 2 + x·x −Ω x , (2.11) sinh Ωt 1 (x0 − x0 )2 1 Ωe−A (2t)− 2 − =√ 3 . (2.12) 4t2 2π sinh(2Ωt) 2t

It is now straightforward to verify the differential equation (2.9), which proves the lemma. Now let’s consider the interaction vertices. The non-commutative complex ϕ63 model may a priori exhibit both orientable vertices: Vo =

1 1 ϕ¯ ϕ ϕ¯ ϕ(x) + ϕ¯ ϕ ϕ¯ ϕ ϕ¯ ϕ(x) 2 3

(2.13)

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and non-orientable vertices: 1 1 Vno = ϕ¯ ϕ¯ ϕ ϕ(x) + ϕ¯ ϕ¯ ϕ¯ ϕ ϕ ϕ(x) 2 3 1 1 + ϕ¯ ϕ¯ ϕ ϕ¯ ϕ ϕ(x) + ϕ¯ ϕ¯ ϕ ϕ ϕ¯ ϕ(x) . 3 3

(2.14)

In this paper we limit ourselves to the case of action (2.2), hence to orientable vertices. In the two dimensional non-commutative space the interaction vertices (orientable or not) can be written as [6, 30]: V (x1 , x2 , x3 , x4 ) = δ(x1 − x2 + x3 − x4 )ei

1≤i<j≤4 (−1)

i+j+1

xi θ −1 xj

(2.15)

for four point vertices and V (x1 , x2 , x3 , x4 , x5 , x6 ) = δ(x1 − x2 + x3 − x4 + x5 − x6 ) ei

1≤i<j≤6 (−1)

i+j+1

xi θ −1 xj

(2.16)

for six point vertices. Here we note xθ−1 y ≡ 2θ (x1 y2 − x2 y1 ). These vertices are completed to make them local in the commutative t coordinate, that is we have to multiply them by δ(x01 − x02 )δ(x01 − x03 )δ(x01 − x04 ) or δ(x01 − x02 )δ(x01 − x03 )δ(x01 − x04 )δ(x01 − x05 )δ(x01 − x06 ) respectively. The main result of this paper is a proof in configuration space of Theorem 2.1 (BPHZ theorem for non-commutative (ϕ¯ ϕ)3 ). The theory defined by the action (2.2) is renormalizable to all orders of perturbation theory. Let G be an arbitrary connected graph. The amplitude associated with this graph is (with selfexplaining notations):

AG = dxv,i dx0v dxv,i dx0v dtl (2.17) v∈V4 ,i=1,...4

v∈V6 ,i=1,...6

l

i+j+1 xv,i θ −1 xv,j δ(xv,1 − xv,2 + xv,3 − xv,4 )eı i<j (−1) V4

i+j+1 xv,i θ −1 xv,j δ(xv,1 − xv,2 + xv,3 − xv,4 + xv,5 − xv,6 )eı i<j (−1) V6

l

×e

1

Ω(2tl )− 2

√ 3 2π sinh(2Ωtl ) −

Ω coth(2Ωtl ) Ω (x2v,i(l) +x2 )+ sinh(2Ωt xv,i(l) ·xv ,i (l) − 2 v ,i (l) l)

(x0,l −x0,l )2 −μ2 t 4tl

0 l

.

For each line l of the graph joining positions xv,i(l) and xv ,i (l) , we choose an orientation (see next section) and we define the “short” variable ul = xv,i(l) − xv ,i (l) , u0l = x0v(l) − x0v (l) and the “long” variable vl = xv,i(l) + xv ,i (l) just as the

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69

Figure 1: Orientation of a tree.

work of Gurau et al. [15]. With these notations, defining 2Ωtl = αl , the propagators in our graph can be written as: √ − 12 Ωαl dαl − Ω4 coth( α2l )u2l − Ω4 tanh( α2l )vl2 − μ2Ω20 αl − 2αΩ u0l 2 l . (2.18) e √ 3 2 2π sinh(α ) l l 2.2. Orientation and position routing We solve the δ function at every vertex by a “position routing”, following the strategy and notations of [15]. The position routing is similar to the “momentum routing” of the commutative case, but we have to take care of the cyclic invariance of the vertex. Consider a connected graph G. We choose a rooted spanning tree in G, then we start from an arbitrary orientation of a first field at the root and inductively climbing into the tree, at each vertex we follow the cyclic order to alternate entering and exiting lines. This is pictured in Figure 1. Let n = n6 + n4 be the number of vertices of the graph, with n6 of the ϕ6 and n4 of the ϕ4 type, N the number of its external fields, and L the number of internal lines of G. We have L = 3n6 + 2n4 − N/2. Every line of the spanning tree by definition has one end exiting a vertex and one end entering another. This may not be true for the loop lines, which join two “loop fields”. Among these, some exit one vertex and enter another; they are called well-oriented. But others may enter or exit at both ends. These loop lines are subsequently referred to as “clashing lines” [15]. If there are no clashing lines, the graph is called orientable. This is exactly the case in this paper, because ϕ variables can contract only to ϕ¯ ones. Choosing the ϕ variables as entering and the ϕ¯ as exiting, the form of the vertices in (2.2) ensure alternance of entering and exiting lines.

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We also define the set of “branches” associated to the rooted tree T . There are n − 1 such branches b(l), one for each of the n − 1 lines l of the tree. The full tree itself is called the root branch and noted b0 . Each branch is made of the subgraph Gb containing all the vertices “above l” in T , plus the tree lines and loop lines joining these vertices. It has also “external fields” which are the true external fields hooked to Gb , plus the loop fields in Gb for the loops with one end (or “field”) inside and one end outside Gb , plus the upper end of the tree line l itself to which b is associated. We call Xb the set of all external fields f of b. We can now describe the position routing associated to T . Here we will not limit ourselves to orientable graphs but will deal with the non-orientable graphs as well. There are n δ functions in (2.17), hence n linear equations for the 6n6 + 4n4 positions, one for each vertex. The position rooting associated to the tree T solves this system by passing to another equivalent system of n linear equations, one for each branch of the tree. This equivalent system is obtained by summing the arguments of the δ functions of the vertices in each branch. To do this we firstly fix a particular branch Gb , with its subtree Tb . In the branch sum we find a sum over all the ul short parameters of the lines l in Tb and no vl long parameters since l both enters and exits the branch. This is also true for the set Lb of welloriented loops lines with both fields in the branch. For the set Lb,+ of clashing loops lines with both fields entering the branch, the short variable disappears and the long variable remains; the same is true but with a minus sign for the set Lb,− of clashing loops lines with both fields exiting the branch. Finally we find the sum of positions of all external fields for the branch (with the signs according to entrance or exit). Obviously the Jacobian of this transformation is 1, so we simply get another equivalent set of n δ functions, one for each branch. For instance in the particular case of Figure 2, the delta function is δ ul1 + ul2 + ul3 + uL1 + uL2 + uL3 − vL4 + vL5 + X1 − X2 + X3 − X4 − X5 + X6 . (2.19) For an orientable graph, the position routing is summarized by: Lemma 2.2 (Position routing). We have, calling IG the remaining integrand in (2.17):

δ(xv,1 − xv,2 + xv,3 − xv,4 ) (2.20) AG = v4

δ(xv,1 − xv,2 + xv,3 − xv,4 + xv,5 − xv,6 ) IG xv,i , x0v,i v6

=

b

⎛ δ⎝

l∈Tb ∪Lb

ul +

⎞ ε(f )xf ⎠ IG

xv,i , x0v,i

f ∈Xb

where ε(f ) is ±1 depending on whether the field f enters or exits the branch.

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Orientable Non-Commutative Complex Φ63 Model X1

+

_

_

X2

+

X6

_

X5_

vL5

+

_ +

_

u L3 u L2

ul1 _

X4

ul2

_

uL1

_

+ +

ul3 +

vL4

+

71

+ _

+ _

+ X3

Figure 2: A branch.

Using the above equations one can at least solve all the long tree variables vl in terms of external variables, short variables and long loop variables, using the n − 1 non-root branches. There remains then the root branch δ function. If Gb is orientable, this δ function of branch b0 contains only short and external variables. Here we shouldn’t forget that each external variable can be written as linear combination of short variable and long variable. If Gb is non-orientable one can solve for an additional “clashing” long loop variable. We can summarise these observations in the following lemma just like that in [15]: Lemma 2.3. The position routing solves any long tree variable vl as a function of: • the short tree variable ul of the line l itself, • the short tree and loop variables with both ends in Gb(l) , • the short and long variables of the loop lines with one end inside Gb(l) and the other outside, • the true external variables x hooked to Gb(l) . In the orientable case the root branch δ function contains only short tree variables, short loop variables and external variables but no long variables, hence gives a linear relation among the short variables and external positions. In the non-orientable case it gives a linear relation between the long variables w of all the clashing loops in the graph some short variables u’s and all the external positions. From now on, each time we use this lemma to solve the long tree variables vl in terms of the other variables, we shall call wl rather than vl the remaining

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2n6 + n4 + 1 − N/2 independent long loop variables. Hence looking at the long variables names the reader can check whether Lemma 2.3 has been used or not. 2.3. Multiscale analysis and crude power counting In this section we follow the standard procedure of multiscale analysis [26]. First the parametric integral for the propagator is sliced in the usual way: ∞ C u, u0 , v = C 0 u, u0 , v + C i u, u0 , v ,

(2.21)

i=1

with C u, u0 , v = 0

1

√

∞

1

Ωα− 2 dα

√ 3 2 2π sinh(α)

μ2

e

2 2 α Ω α Ω 0 −Ω 4 coth( 2 )u − 4 tanh( 2 )v − 2Ω α− 2α

2

l

(u0 )

(2.22) and

√ −1 Ωα 2 dα − Ω4 coth( α2 )u2 − Ω4 tanh( α2 )v2 − μ2Ω20 α− 2αΩ (u0 )2 l . e √ 3 M −2i 2 2π sinh(α) (2.23) We have an associated decomposition of any amplitude of the theory as μ AG = AG . (2.24)

C i u, u0 , v =

M −2(i−1)

μ

Lemma 2.4. For some constants K (large) and c (small): C i (u, v) ≤ KM i e−c [M

i

u+M i u0 +M −i v]

(2.25)

(which a posteriori justifies the terminology of “long” and “short” variables). We can use the second order approximation of the hyperbolic functions near the origin to prove this lemma. Taking absolute values, hence neglecting all oscillations, leads to the following crude bound:

|AG | ≤ dul du0l dvl C il ul , u0l , vl δv , (2.26) μ

l

v

where μ is the standard assignment of an integer index il to each propagator of each internal line l of the graph G, which represents its “scale”. We will consider only amputated graphs. Therefore we have only external vertices of the graph; in the renormalization group spirit, the convenient convention is to assign all external indices of these external fields to a fictitious −1 “background” scale. To any assignment μ and scale i are associated the standard connected components Gik , k = 1, . . . , k(i) of the subgraph Gi made of all lines with scales j ≥ i. These components are partially ordered according to their inclusion relations and the (abstract) tree describing these inclusion relations is called the Gallavotti– Nicol` o tree [7, 15]; its nodes are the Gik ’s and its root is the complete graph G.


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More precisely for an arbitrary subgraph g one defines: ig (μ) = inf il (μ) , l∈g

eg (μ) =

sup

l external line of g

il (μ) .

(2.27)

The subgraph g is a Gik for a given μ if and only if ig (μ) ≥ i > eg (μ). Now we should choose the real tree T compatible with the abstract Gallavotti–Nicol` o tree to optimize the bound over spatial integrations, which means that the restriction Tki of T to any Gik must still span Gik . This is always possible (by a simple induction from leaves to root). We pick such a compatible tree T and use it both to orient the graph as in the previous section and to solve the associated branch system of δ functions according to Lemma 2.3. We obtain:

il il 0 −il M il dul du0l dvl e−c [M ul +M u +M vl ] δb |AG,μ | ≤ K n l b l

il il 0 −il ≤ K n M il dul du0l dwl e−c [M ul +M u +M vl (u,w,x)] δb0 . (2.28) l

l

Then we can find that any long variable integrated at scale i costs KM 2i . The integration over the non-commutative short variable at scale i brings KM −2i , and the commutative one brings KM −i (there is no long variable in the commutative dimension) so the integration over each tree line at scale i brings a total convergent factor KM −3i . The variables “solved” by the δ functions bring or cost nothing. For an orientable graph we should solve the n − 1 long variables vl ’s of the tree propagators in terms of the other variables, because this is the maximal number of long variables that we can solve, and they have highest possible indices because T has been chosen compatible with the Gallavotti–Nicol` o tree structure. We should study more carefully the commutative variable which is the 0th dimension of any tree line of T . While the model for the non-commutative variables is non local, it is local for the commutative variables. So we can’t integrate over all the position variables (or the equivalent line variables) but have to save one, the root (we name it xν0 ). We will use this point when we perform the renormalization where the imputed amplitude of any connected component depends only on one commutative external position xν0 . This point is also very important for the power counting of the non-orientable model as it implies the maximal number of commutative short variable we can integrate over is n − 1 not n. Finally we still have the last δb0 function (equivalent to the overall momentum conservation in the commutative case). It is optimal to use it to solve one external variable (if any) in terms of all the short variables and the external ones. Since external variables are typically smeared against unit scale test functions, this leaves power counting invariant. We now define S the set of long variables to be solved via the δ functions hence the set of n − 1 tree lines as there are only orientable graphs in our model. Gathering all the corresponding factors together with the propagators prefactors M i leads to the following bound:

|AG,μ | ≤ K n M il M −3il . (2.29) l

l∈S

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In the usual way of [26] we write

M il =

il

l i=1

l

and

M=

l∈S

M

−3il

=

M=

i,k l∈Gik

il

M −3 =

l∈S i=1

(2.30)

M −3

(2.31)

i,k

i,k

i

M l(Gk )

l∈Gik ∩S

and we must now only count the number of elements in Gik ∩ S. As remarked above Gik ∩ S = Tki , and the cardinal of Tki is n(Gik ) − 1. Using the fact that 2l(Gik ) − 6n6 (Gik ) − 4n4 = −N (Gik ) we can summarise these results in the following lemma: Lemma 2.5. The following bound holds for a connected graph of (ϕ¯ ϕ)3 model (with external arguments integrated against fixed smooth test functions):

i |AG,μ | ≤ K n M −ω(Gk ) (2.32) i,k

for some (large) constant K, with

ω(Gik )

= N (Gik )/2 + n4 − 3 .

This lemma proves the power counting for orientable graphs. But it is not yet sufficient for a renormalization theorem to all orders of perturbation. Indeed only planar graphs with a single broken face look like Moyal products when their internal indices become much higher than their external ones. So we must prove that the non-planar graphs or graphs with more than one broken face have better power counting than what Lemma 2.5 states. Vertices oscillations should be taken into account to prove that, and this is done in the next section. 2.4. Improved power counting Recall that for any non-commutative Feynman graph G we can define the genus of the graph, called g and the number of faces “broken by external legs”, called B [13, 23]. For a general graph, we have g ≥ 0 and B ≥ 1. In the previous section we established that ω(G) ≥ N/2 + n4 − 3 ,

if G orientable .

(2.33)

The subgraphs with g = 0 and B = 1 are called planar regular. We want to prove that they are the only non-vacuum graphs with ω ≤ 0. It is easy to check that planar regular subgraphs are orientable, but the converse is not true. To prove that orientable non-planar subgraphs or orientable planar subgraphs with B ≥ 2 are irrelevant requires to use a bit of the vertices oscillations to improve Lemma 2.5 and get: Lemma 2.6. For orientable subgraphs with g ≥ 1 we have ω(G) ≥ N/2 + n4 + 1 .

(2.34)


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For orientable subgraphs with g = 0 and B ≥ 2 we have ω(G) ≥ N/2 + n4 − 1 .

(2.35)

This lemma is sufficient for the purpose of this paper. It implies directly that graphs which contain only irrelevant subgraphs have finite amplitudes which are uniformly bounded by K n , using the standard method of [26] to bound the assignment sum over μ in (2.26). The rest of this subsection is essentially devoted to the proof of this Lemma. We return before solving δ functions, hence to the v variables. We will need only to compute the oscillations which are quadratic in the long variables v’s to prove (2.34) and the linear oscillations in vθ−1 x to prove (2.35). Fortunately an analog problem was solved in momentum space by Filk and Chepelev-Roiban [1,6], and adapted to position routing by Gurau et al. [15]. We just borrow from the method of [15]. As the procedures for our paper are almost the same as that for ϕ44 in [15], we reproduce the argument as concisely as possible, and we refer to [15,30] for more details. The short variables are inessential in this subsection, as the integration of them always bring about convergent terms. But it is convenient to treat on the same footing the long v and the external x variables, so we introduce a new global notation y for all these variables. Then the vertices rewrite as

i+j+1 yi θ −1 yj +yQu+uRu) δ y1 −y2 + y3 −y4 + y5 −y6 + εi ui eı( i<j (−1) (2.36) v

for some inessential signs εi and some symplectic matrices Q and R. As there are no oscillations for the commutative coordinates, there are no Filk moves for them. Since the precise oscillations in the short u variables is not important to this problem, we will note in the sequel Eu any linear combination of the u variables. Let’s consider the first Filk reduction [6], which contracts tree lines of the graph. It creates progressively generalized vertices with even number of fields. At a given induction step and for a tree line joining two such generalized vertices with respectively p and q − p + 1 fields (suppose p is even and q is odd), we assume by induction that the two vertices are δ (y1 − y2 + y3 · · · − yp + Eu ) δ (yp − yp+1 + · · · − yq + Eu ) eı(

1≤i<j≤p (−1)

i+j+1

yi θ −1 yj +

p≤i<j≤q (−1)

i+j+1

yi θ −1 yj +yQu+uRu)

. (2.37)

Using the second δ function we see that: yp = yp+1 − yp+2 + · · · + yq − Eu .

(2.38)

Substituting this expression in the first δ function we get: δ(y1 − y2 + . . . − yp+1 + . . . − yq + Eu )δ(yp − yp+1 + · · · − yq + Eu ) eı(

1≤i<j≤p (−1)

i+j+1

yi θ −1 yj +

p≤i<j≤q (−1)

i+j+1


. (2.39)

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2 4

5

3

1 6

7

Figure 3: A typical rosette.

The quadratic terms which include yp in the exponential are (taking into account that p is an even number): p−1

q

i=1

j=p+1

(−1)i+1 yi θ−1 yp +

(−1)j+1 yp θ−1 yj .

(2.40)

Using the expression (2.38) for yp we see that the second term gives only terms in yLu, as θ is antisymmetry. The first term yields: p−1

q

(−1)i+1+j+1 yi θ−1 yj =

i=1 j=p+1

q−1 p−1

(−1)i+k+1 yi θ−1 yj ,

(2.41)

i=1 k=p

which reconstitutes the crossed terms, and we have recovered the inductive form of the larger generalized vertex. After each Filk move we will have two more vertices. So by this procedure we will always treat only even vertices. We finally rewrite the product of the two vertices as: δ(y1 − y2 + · · · + yp−1 − yp+1 + · · · − yq + Eu )δ(yp − yp−1 + · · · − yq + Eu ) eı(

1≤i<j≤q (−1)

i+j+1


, (2.42)

where the exponential is written in terms of the reindexed vertex variables. In this way we can contract all lines of a spanning tree T and reduce G to a single vertex with “tadpole loops” called a “rosette graph” [1]. In this rosette to keep track of cyclicity is essential so we draw the rosette as a cycle (which is the border of the former tree) bearing loops lines on it (see Figure 3). Remark that the rosette can also be considered as a big vertex, with r = 4n6 + 2n4 + 2 fields, on which N are external fields with external variables x and 4n6 + 2n4 + 2 − N are loop fields for the corresponding 2n6 + n4 + 1 − N/2 loops. When the graph is orientable, the

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long variables yl for l in T will disappear in the rosette. Let us call z the set of remaining long loop and external variables. Then the rosette vertex factor is i+j+1 zi θ −1 zj +zQu+uRu) δ z1 − z2 + · · · − zr + Eu eı( 1≤i<j≤r (−1) . (2.43) We can go on performing inductively the first Filk move and the net effect is simply to rewrite the root branch δ function and the combination of all vertices oscillations (using the other δ functions) as the new big vertex or rosette factor (2.43). The second Filk reduction [6] further simplifies the rosette factor by erasing the loops of the rosette which do not cross any other loops or arch over external fields. Putting together all the terms in the exponential which contain zl we conclude exactly as in [6] that these long z variables completely disappear from the rosette oscillation factor, which simplifies as in [1] to δ(z1 − z2 + · · · − zr + Eu )eı(zIz+zQu+uRu)

(2.44)

where Iij is the antisymmetric “intersection matrix” of [1] (up to a different sign convention). Here Iij = +1 if oriented loop line i crosses oriented loop line j coming from its right, Iij = −1 if i crosses j coming from its left, and Iij = 0 if i and j do not cross. These formulas are also true for i external line and j loop line or the converse, provided one extends the external lines from the rosette circle radially to infinity to see their crossing with the loops. Finally when i and j are external lines one should define Iij = (−1)p+q+1 if p and q are the numbering of the lines on the rosette cycle (starting from an arbitrary origin). If a node Gik of the Gallavotti–Nicol` o tree is orientable but non-planar (g ≥ 1), there must therefore exist at least two intersecting loop lines in the rosette corresponding to this Gik , with long variables w1 and w2 . Moreover since Gik is orientable, none of the long loop variables associated with these two lines belongs to the set S of long variables eliminated by the δ constraints. Therefore, after integrating the variables in S the basic mechanism to improve the power counting of a single non planar subgraph is the following: −2i1 2 w1 −M −2i2 w22 −iw1 θ −1 w2 +w1 E1 (x,u)+w2 E2 (x,u) dw1 dw2 e−M −2i1 (w1 )2 −M −2i2 (w2 )2 +iw1 θ −1 w2 +(u,x)Q(u,x) = dw1 dw2 e−M 2i1 −2i2 M −2i1 )(w2 )2 = KM 2i1 dw2 e−(M +M = KM 2i1 ≤ K . (2.45) 1 + M −2(i1 +i2 ) In these equations we used for simplicity M −2i instead of the correct but more complicated factor (Ω/4) tanh(α/2) (see (2.18)) (of course this does not change the argument) and we performed a unitary linear change of variables w1 = w1 + 1 (x, u), w2 = w2 + 2 (x, u) to compute the oscillating w1 integral. The gain in (2.45) is M −2i1 −2i2 , which is the difference between O(1) and the normal factor M 2i1 +2i2 that would be generated by the integrals over w1 and w2 if there were not the oscillation term iw1 θ−1 w2 .

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So after the integration of the non-commutative part of the two clashing lines the gain is almost M −4i . This basic argument must then be generalized to each non-planar leaf in the Gallavotti–Nicol` o tree. Actually, in any orientable non-planar ‘primitive’ Gik node (i.e., not containing sub non-planar nodes) we can choose an arbitrary pair of crossing loop lines which will be integrated as in (2.45) using this oscillation. The corresponding improvements are independent. This leads to an improved amplitude bound:

i |AG,μ | ≤ K n M −ω(Gk ) (2.46) i,k

ω(Gik )

N (Gik )/2

where now = + n4 + 1 if Gik is orientable and non planar (i.e., g ≥ 1). This bound proves (2.34). Finally it remains to consider the case of nodes Gik which are planar orientable but with B ≥ 2. In that case there are no crossing loops in the rosette but there must be at least one loop line arching over a non trivial subset of external legs in the Gik rosette (see line 6 in Figure 3). We have then a non trivial integration over at least one external variable, called x, of at least one long loop variable called w. This “external” x variable without the oscillation improvement would be integrated with a test function of scale 1 (if it is a true external line of scale 1) or better (if it is a higher long loop variable)1 . But we get now −2i 2 −1 +2i 2 (2.47) dxdwe−M w −iwθ x+w.E1 (x ,u) = KM 2i dxe−M x = K . We find that a factor M 2i in the former bound becomes O(1) hence is improved by M −2i . So the power counting is ω(Gik ) = N (Gik )/2 − 1 + n4 . We find that the two point graphs with n4 = 0 and N (Gik ) = 2 maybe logarithmically divergent. They do not appear renormalizable at first sight. But we remark that in the orientable (ϕ¯ ϕ)3 model there will never be such subgraphs with N (Gik ) = 2 and B = 2. This is the reason we limit ourselves to this case2 . Then all graphs with B ≥ 2 are also safe. The only divergent graphs which need renormalization are the planar regular graphs.

3. Renormalization In this section we need to consider only divergent subgraphs, namely the planar two point, four point and six point subgraphs with a single external face (g = 0, B = 1, N = 2, 4, 6 for n4 = 0 ,N = 2, 4 for n4 = 1, and N = 2 for n4 = 2). We shall prove that they can be renormalized by appropriate counterterms of the form 1 Since

the loop line arches over a non trivial (i.e., neither full nor empty) subset of external legs of the rosette, the variable x cannot be the full combination of external variables in the “root” δ function. 2 We thank our referee for correcting an earlier version of this paper, which lead us to this important point.

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of the initial Lagrangian. We would like to remark that for any graph, contrary to the non-commutative variables, the commutative variables of the external points are local. So there is only one integral over the commutative variable for each vertex. 3.1. Renormalization of the six-point function Consider a 6 point subgraph which needs to be renormalized, hence is a node of the Gallavotti–Nicol` o tree. This means that there is (i, k) such that N (Gik ) = 6. The six external positions of the amputated graph are labelled x1 , x2 , x3 , x4 , x5 and x6 . We also define Q, R and S as three skew-symmetric matrices of respective sizes 6 × l(Gik ), l(Gik ) × l(Gik ) and 2[n6 (Gik ) − 1] × l(Gik ), where we recall that 2(n(G)6 − 1) is the number of loops of a 6 point graph with n6 vertices. The amplitude associated to the connected component Gik is then A(Gik )(x1 , x2 , x3 , x4 , x5 , x6 , x0ν )

= du du0 C x, u, u0 , w ∈Tki

dul du0l dwl Cl

0 ul ul , ul , wl δ x1 − x2 + x3 − x4 + x5 − x6 +

l∈Gik , l∈T ı( p i(l) and this term is irrelevant. 4 We 5 We

thank our referee for pointing out this important point. are very grateful to Prof. Rivasseau for explaining this.

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• The term AXU · U · ∇ϕ≤e (x, x0 ) is smaller as U brings M −i and there is no long loop variables. So it is irrelevant. • We can easily find that A1,R is smaller hence irrelevant. Now we consider the second order expansion in external variables in (3.13). We only have to expand f (t) to first order. We have 1 (U · ∇)2 δ(x − y) eı(URU+USW ) A2 = dxdydx0 ϕ¯≤e x, x0 ϕ≤e y, x0 2

× dul dwl Cl (ul , wl ) du du0 C u , u0 , U , W l∈Gik ,l∈T

× f (0) +

1

∈Tki

dtf (t)dt .

(3.23)

0

The first term is 1 (U · ∇)2 δ (x − y) eı(URU+USW ) A2,0 = dxdydx0 ϕ¯≤e x, x0 ϕ≤e y, x0 2

× dul dwl Cl (ul , wl ) du du0 C u , u0 , U , W . (3.24) l∈Gik ,l∈T

∈Tki

The terms with μ = ν do not survive by parity. The other ones reconstruct a counterterm proportional to the Laplacian. The power-counting of this factor A2,0 is improved (with respect to A) by a factor M −2(i−e) which makes it only logarithmically divergent, as should be for a wave-function counterterm. The second term is 1 (U · ∇)2 δ(x − y) eı(URU+USW ) A2,0 = dxdydx0 ϕ¯≤e x, x0 ϕ≤e y, x0 2

× dul dwl Cl (ul , wl ) du du0 C u , u0 , U , W l∈Gik ,l∈T

×

0

1

∈Tki

dt iXQU − 2tAXX − 2AX(W + U ) .

(3.25)

It is irrelevant as the terms in the integral bring at least a convergent factor M −(i−e) . Putting together the results of the two previous section, we have proved that the usual effective series which expresses any connected function of the theory in terms of an infinite set of effective couplings, related one to each other by a discretize flow [26], have finite coefficients to all orders. Reexpressing these effective series in terms of the renormalized couplings would reintroduce in the usual way the Zimmermann’s forests of counterterms and build the standard renormalized series. The most explicit way to check finiteness of these renormalized series in order to complete the “BPHZ theorem” is to use the “classification of forests” which distributes Zimmermann’s forests into packets such that the sum over assignments in each packet is finite [26]. This part is identical to the commutative case. Hence the proof of Theorem 2.1 is completed.


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Appendix A. The non-commutative ϕ63 model In this appendix we discuss briefly the non-orientable real scalar ϕ63 model, and its renormalizability which is questionable. The action functional, with the notations of (2.2) is now 1 Ω2 1 1 S[ϕ] = d2 x dx0 xμ ϕ) + μ20 ϕ ϕ ∂μ ϕ ∂ μ ϕ + ∂0 ϕ ∂ 0 ϕ + (˜ xμ ϕ) (˜ 2 2 2 2 λ g + ϕ ϕ ϕ ϕ + ϕ ϕ ϕ ϕ ϕ ϕ (x) . (A.1) 4 6 In the real ϕ6 model, there are only two kinds of cyclically invariant vertices, namely the ϕ4 term: V (x1 , x2 , x3 , x4 ) = δ(x1 − x2 + x3 − x4 )ei

1≤i<j≤4 (−1)

i+j+1

xi θ −1 xj

(A.2)

and the ϕ6 term: V (x1 , x2 , x3 , x4 , x5 , x6 ) = δ(x1 − x2 + x3 − x4 + x5 − x6 ) ei

1≤i<j≤6 (−1)

i+j+1

xi θ −1 xj

(A.3)

times the local factor in the time direction. Again we note xθ−1 y ≡ 2θ (x1 y2 −x2 y1 ). The discussion is almost the same as that in (ϕ¯ ϕ)3 model, with a difference in the power counting of the non-orientable graph. When several disjoint Gik subgraphs are non-orientable it is better to solve longer clashing loop variables, essentially one per disjoint non-orientable Gik , because they spare higher costs than if tree lines were chosen instead. We define S to be the set of n long variables to be solved via the δ functions. First we put in S all the n − 1 long tree variables vl . Then we scan all the connected components Gik starting from the leaves towards the root, and we add a clashing line to S each time when a new non-orientable component Gik appears. We also remove p − 1 tree lines from S so that each time p ≥ 2 non-orientable components merge into a single one. In the end we obtain a new set S of exactly n − 1 + p − (p − 1) = n long variables. So thanks to inductive use of Lemma 2.3 in each Gik , we can solve all the long variables in the set S with the branch system of δ functions associated to T plus an additional loop variable. But for the commutative dimension, there are always n − 1 short tree variables to be integrated. So for a general non-orientable graph we will earn only a convergent factor M −2i and the degree of divergence given by this crude analysis becomes ω(Gik ) = N (Gik )/2 + n4 − 1 (recall Lemma 2.5). Let us consider the improved analysis taking oscillations into account. From the analog of Lemma 2.6 we see that graphs with g = 0, n4 = 0, B = 2 and N = 2 remain dangerous. Such graphs can’t appear in the (ϕ¯ ϕ)3 model as they are non-orientable. In the ϕ6 model they can appear, are logarithmic divergent and don’t look like the initial quadratic terms in the Lagrangian. So the two point function of this (ϕ¯ ϕ)3 model seems non-renormalizable, but maybe the situation can be rescued by combining all renormalizations together, as is done, e.g., in [30]

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or maybe we can solve this problem by exploring further the vertex oscillations. The study of this problem is still in progress.

Acknowledgements The author is very grateful to V. Rivasseau for successful direction and encouragement and many useful discussions on this work. We are also very grateful to the anonymous referee for critical comments. This work is supported by NSFC Grant No.10675108.

References [1] I. Chepelev and R. Roiban, Convergence theorem for non-commutative Feynman graphs and renormalization, JHEP 03 (2001), 001, http://www.arXiv.org/abs/ hep-th/0008090hep-th/0008090. [2] A. Connes, M. R. Douglas, and A. Schwarz, Non-commutative geometry and matrix theory: Compactification on Tori, JHEP 02 (1998), 003, http://www.arXiv.org/ abs/hep-th/9711162hep-th/9711162. [3] M. Disertori and V. Rivasseau, Two and three loops beta function of non commutative Φ44 theory, Eur. Phys. J. C50 (2007) 661, http://www.arXiv.org/abs/hep-th/ 0610224hep-th/0610224. [4] M. Disertori, R. Gurau, J. Magnen and V. Rivasseau, Vanishing of beta function of non commutative Φ44 to all orders, Phys. Lett. B649 (2007), 95–102, http://www.arXiv.org/abs/hep-th/0612251hep-th/0612251. [5] M. R. Douglas and N. A. Nekrasov, Non-commutative field theory, Rev. Mod. Phys. 73 (2001), 977–1029, http://www.arXiv.org/abs/hep-th/0106048hep-th/0106048. [6] T. Filk, Divergencies in a field theory on quantum space, Phys. Lett. B376 (1996), 53–58. [7] G. Gallavotti and F. Nicol` o, Renormalization theory in four-dimensional scalar fields. i, Commun. Math. Phys. 100 (1985), 545–590. [8] H. Grosse and H. Steinacker, Renormalization of the non-commutative ϕ3 model through the Kontsevich model (2005). [9] H. Grosse and H. Steinacker, A nontrivial solvable non-commutative ϕ3 model in 4 dimensions, http://www.arXiv.org/abs/hep-th/0603052hep-th/0603052. [10] H. Grosse and H. Steinacker, Exact renormalization of a non-commutative phi**3 model in 6 dimensions, http://www.arXiv.org/abs/hep-th/hep-th/ 0607235hep-th/0607235. [11] H. Grosse and R. Wulkenhaar, Power-counting theorem for non-local matrix models and renormalization, Commun. Math. Phys. 254 (2005), no. 1, 91–127, http://www.arXiv.org/abs/hep-th/0305066hep-th/0305066. [12] H. Grosse and R. Wulkenhaar, The beta-function in duality-covariant noncommutative ϕ4 -theory, Eur. Phys. J. C35 (2004), 277–282, http://www.arXiv.org/ abs/hep-th/0402093hep-th/0402093.

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[13] H. Grosse and R. Wulkenhaar, Renormalization of ϕ4 -theory on non-commutative R4 in the matrix base, Commun. Math. Phys. 256 (2005), no. 2, 305–374, http://www.arXiv.org/abs/hep-th/0401128hep-th/0401128. [14] R. Gurau, V. Rivasseau, and F. Vignes-Tourneret, Propagators for non-commutative field theories, Ann. H. Poincaré 7 (2006), 1601–1628, http://www.arXiv.org/abs/ hep-th/0512071hep-th/0512071. [15] R. Gurau, J. Magnen, V. Rivasseau and F. Vignes-Tourneret, Renormalization of non-commutative ϕ44 field theory in x space, Commun. Math. Phys. 267 (2006), no. 2, 515–542, http://www.arXiv.org/abs/hep-th/0512271hep-th/0512271. [16] R. Gurau and V. Rivasseau, Parametric representation of non-commutative field theory, Commun. Math. Phys. 272 (2007), 811–835, http://www.arXiv.org/abs/ math-ph/0606030math- ph/0606030. [17] S. Hellerman and M. Van Raamsdonk, Quantum Hall Physics = noncommutative field theory, JHEP 0110 (2001), 039, http://www.arXiv.org/abs/ hep-th/0103179hep-th/ 0103179 [18] E. Langmann and R. J. Szabo, Duality in scalar field theory on non-commutative phase spaces, Phys. Lett. B533 (2002), 168–177, http://www.arXiv.org/abs/ hep-th/0202039hep-th/0202039. [19] E. Langmann, R. J. Szabo, and K. Zarembo, Exact solution of quantum field theory on non-commutative phase spaces, JHEP 01 (2004), 017, http://www.arXiv.org/ abs/hep-th/0308043hep-th/0308043. [20] S. Minwalla, M. Van Raamsdonk, and N. Seiberg, Noncommutative perturbative dynamics, JHEP 02 (2000), 020, http://www.arXiv.org/abs/hep-th/ 9912072hep-th/9912072. [21] J. Polchinski, Renormalization and effective Lagrangians, Nucl. Phys. B231 (1984), 269. [22] A. Polychronakos, Quantum Hall states as matrix Chern–Simons theory, JHEP 0104 (2001), 011, http://www.arXiv.org/abs/hep-th/0103013hep-th/0103013. [23] V. Rivasseau, F. Vignes-Tourneret, and R. Wulkenhaar, Renormalization of noncommutative ϕ4 -theory by multi-scale analysis, Commun. Math. Phys. 262 (2006), 565–594, (Online First) DOI: 10.1007/s00220-005-1440-4 (2005), http://www.arXiv. org/abs/hep-th/0501036hep-th/0501036. [24] V. Rivasseau and F. Vignes-Tourneret, Renormalization of non-commutative field theories, http://www.arXiv.org/abs/hep-th/0702068hep-th/0702068. [25] V. Rivasseau, Non-commutative renormalization, http://lanl.arxiv.org/abs/ 0705.0705hep-th/0705.0705. [26] V. Rivasseau, From perturbative to constructive renormalization. Princeton series in physics. Princeton Univ. Pr., Princeton, USA, 1991, 336 p. [27] N. Seiberg and E. Witten, String theory and non-commutative geometry, JHEP 09 (1999), 032, http://www.arXiv.org/abs/hep-th/9908142hep-th/9908142. [28] L. Susskind, The quantum hall fluid and non-commutative Chern–Simons theory, hep-th/0101029 [29] R. J. Szabo, Quantum field theory on non-commutative phase spaces, Phys. Rep. 378 (2003), 207, http://www.arXiv.org/abs/hep-th/0109162hep-th/0109162.

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[30] F. Vignes-Tourneret, Renormalization of the orientable non-commutative Gross– Neveu model, Ann. H. Poincaré 8 (2007), 427–474, http://www.arXiv.org/abs/ math-ph/0606069math-ph/0606069. Zhituo Wang and Shaolong Wan Institute for Theoretical Physics and Department of Modern Physics University of Science and Technology of China Hefei, 230026 P. R. China e-mail: [email protected] [email protected] Communicated by Raimar Wulkenhaar. Submitted: April 10, 2007. Accepted: July 20, 2007.



Sinai Billiards under Small External Forces II Nikolai Chernov Abstract. We study perturbations of Sinai billiards, where a small stationary force acts on the moving particle between its collisions with scatterers. In the previous work [7] we proved that the collision map preserved a unique Sinai– Ruelle–Bowen (SRB) measure that was Bernoulli and had exponential decay of correlations. Here we add several other statistical properties, including bounds on multiple correlations, the almost sure invariance principle (ASIP), the law of iterated logarithms, and a Kawasaki-type formula. We also show that the corresponding flow is Bernoulli and satisfies a central limit theorem.

1. Introduction This a continuation of our paper [7], and we use the symbols and notation of the latter for compatibility. Let B1 , . . . , Bs be open convex domains on the unit 2D torus T2 . Assume that B¯i ∩ B¯j = ∅ for i = j, and for each i the boundary ∂Bi is a C 3 smooth closed curve with nonvanishing curvature. Let a particle of unit mass move in D = T2 \ ∪i Bi according to equations q˙ = p ,

p˙ = F

(1.1)

where q = (x, y) is the position vector, p = (u, v) is the momentum (velocity) vector, and F(x, y, u, v) = (F1 , F2 ) is a stationary force (independent of time). Upon reaching the boundary ∂D = ∪i ∂Bi , the particle gets reflected elastically, according to the classical rule (1.2) p+ = p− − 2 n(q) · p− n(q) ; here q ∈ ∂D is the point of reflection, n(q) is the inward unit normal vector to ∂D, and p− , p+ are the incoming and outgoing velocity vectors, respectively. The case F = 0 corresponds to the ordinary billiard dynamics on the table D. It preserves the kinetic energy, so that one can fix it by setting p = 1. Then the phase space of the system is a compact 3D manifold Ω0 = D × S 1 . The billiard flow Φt0 on Ω0 preserves the Liouville measure μ0 (which is uniform on Ω0 ).

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In the study of billiards, one uses the collision space M0 = (q, p) ∈ Ω0 : q ∈ ∂D, p · n(q) ≥ 0 ,

(1.3)

which consists of all outgoing velocity vectors at reflection points. The first return map F0 : M0 → M0 is called the billiard map (or the collision map). The space M0 can be parameterized by (r, ϕ), where r is the arclength along ∂D and ϕ ∈ [−π/2, π/2] is the angle between p and n(q). In these coordinates, M0 = ∂D × [−π/2, π/2]. The map F0 preserves a finite smooth measure on M0 with density dν0 = const · cos ϕ dr dϕ. The flow Φt0 is a suspension flow over the base map F0 under the ceiling function τ0 (X) = min{t > 0 : Φt0 X ∈ M} (the next collision time). Billiards on tables D = T2 \ ∪i Bi as described above are known as dispersing billiards or Sinai billiards. The map F0 is ergodic, mixing [28], Bernoulli [15], and has other strong statistical properties, such as exponential decay of correlations [6, 32] and the central limit theorem [3]. The billiard flow Φt0 is also ergodic, mixing [28], and Bernoulli [15]. Under an additional assumption of finite horizon (see below) the flow Φt0 enjoys stretched exponential decay of correlations [9] and satisfies the central limit theorem [3]. Various perturbations of Sinai billiards have been studied in [1,18–21,29,30], see also a survey [22]. Most notably, when F is a small constant force with a Gaussian thermostat (see below), then one can rigorously prove a one-particle version of classical Ohm’s law and the Einstein relation [4, 5]. More recently, perturbed Sinai billiards were used in the analysis of the Galton board [12] and self-similar Lorentz channels [2, 11]. A general class of Sinai billiards with small external forces F = 0 was studied in [7] under the following assumptions: Assumption A (Additional integral). A smooth function E(q, p) is preserved by the dynamics (1.1)–(1.2). Its level surface, Ω = {E(q, p) = const} is a compact 3-D manifold such that p = 0 on Ω and for each q ∈ D and p ∈ S 1 the ray {(q, sp), s > 0} intersects the manifold Ω in one point. Under Assumption A, Ω can be parameterized by (x, y, θ), where (x, y) = q ∈ D and 0 ≤ θ < 2π is a cyclic coordinate, the angle between p and the positive x axis. The dynamics (1.1)–(1.2) restricted to Ω is a flow that we denote by Φt . In the coordinates (x, y, θ) the equations of motion (1.1) can be rewritten as x˙ = p cos θ ,

y˙ = p sin θ ,

θ˙ = ph ,

(1.4)

where p = p > 0

and

h = (−F1 sin θ + F2 cos θ)/p2 .

It is also useful to note that p˙ = F1 cos θ + F2 sin θ .

(1.5)

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Both h = h(x, y, θ) and p = p(x, y, θ) are assumed to be C 2 smooth functions on Ω, and note that 0 < pmin ≤ p ≤ pmax < ∞ . (1.6) There are two particularly interesting types of forces satisfying Assumption A. One is a potential force F = −∇U , where U = U (q) is a potential function; it preserves the total energy T = 12 p2 + U (q). The other type is isokinetic forces satisfying (F · p) = 0, they preserve the kinetic energy K = 12 p2 , so one can set p = 1 as in billiards. For example, given any force F, one can construct an isokinetic force by adding Gaussian thermostat: q˙ = p ,

p˙ = F − αp where α = (F · p)/(p · p) .

(1.7)

For a function f on Ω, let fx , fy , fθ denote its partial derivatives and f C 2 the maximum of f and its first and second partial derivatives over Ω. Put (1.8) B0 = max p−1 min , pC 2 , hC 2 . Assumption B (Smallness of the force). We assume that the force F and its first derivatives are small, i.e., max |h|, |hx |, |hy |, |hθ | ≤ δ0 . More precisely, we require that for any given B∗ > 0 there should be a small δ∗ = δ∗ (D, B∗ ) such that all our results will hold whenever B0 < B∗ and δ0 < δ∗ . We note that the smallness of δ∗ is required in [7] at several crucial steps, some of those requirements are more severe than others. Physical implications of those requirements are discussed in Remark on p. 232 in [7]. Assumption C (Finite horizon). There is an L > 0 so that every straight line of length L on the torus T2 crosses at least one obstacle Bi . Now we introduce the collision space of the system (1.1)–(1.2): M = (q, p) ∈ Ω : q ∈ ∂D, p · n(q) ≥ 0

(1.9)

and the corresponding collision map F : M → M. The space M can be parameterized by (r, ϕ) as before, in these coordinates M is identical to M0 in (1.3). Theorem 1.1 ([7]). Under Assumptions A, B, and C, the map F : M → M is a smooth hyperbolic map with singularities that has uniform expansion and contraction rates. It admits a unique SRB measure ν, which is positive on open sets, K-mixing and Bernoulli. It enjoys exponential decay of correlations (with bounds uniform in the force F) and satisfies the central limit theorem. We remark that in unperturbed Sinai billiards the hyperbolicity results from a rather obvious geometric fact that divergent families of trajectories remain divergent (under the action of the flow) and grow exponentially in size at time goes on. In our perturbed billiards, the same property holds for the so-called strongly divergent families of flow lines (whose orthogonal cross-section has curvature bounded

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below by a positive constant), see precise definition in [7, p. 208]. Similarly, strongly convergent families are defined (those remain convergent and expand in the timereversal flow).

2. Advanced statistical properties of F Here we derive further statistical properties of the collision map F by employing the coupling method recently introduced by L.-S. Young [33] and modified by D. Dolgopyat [10, Appendix A]. It is based on iterations of probability measures supported on unstable curves. First we recall a few definitions and facts following [7]. An unstable (or stable) curve γ ⊂ M is a trace of a strongly divergent (resp., convergent) family of flow lines [7, p. 213]. We may assume [7, p. 215] that the curvature of unstable and stable curves is uniformly bounded. To ensure distortion control we cut M into countably many homogeneity strips [7, p. 216] that accumulate near ∂M, thus F becomes discontinuous on the preimages of the boundaries of those strips. A curve is homogeneous if it lies in a single homogeneity strip (i.e., in one connected component of M). For any X ∈ γ denote by Jγ F n (X) the Jacobian of the map F n restricted to γ at X. If F i (γ) is a homogeneous unstable curve for all 0 ≤ i ≤ n, then we have the following distortion bound, see [7, Lemma 4.2]: | ln Jγ F n (X) − ln Jγ F n (Y )| ≤ C|F n (γ)|1/3 ,

X, Y ∈ γ

(2.1)

where |γ| denotes the length of γ, and by C we will denote various positive constants independent of the force F. Accordingly, if γ u is a homogeneous unstable manifold (called h-fiber in [7]) and ργ u is the u-SRB density on γ u , i.e., the unique probability density satisfying ργ u (X) Jγ u F −n (X) = lim , X, Y ∈ γ u , (2.2) n→∞ Jγ u F −n (Y ) ργ u (Y ) d then (2.1) implies dX ln ργ u (X) ≤ C|γ u |−2/3 ; see, e.g., [8, Section 5.6]1 . If γ1 , γ2 are unstable curves and ξ a stable h-fiber crossing each γi in a point Xi , then the Jacobian of the holonomy map h : γ1 → γ2 at X1 satisfies e−C(β+δ

1/3

)

≤ J h(X1 ) ≤ e−C(β+δ

1/3

)

(2.3)

where δ = |ξ(X1 , X2 )| is the length of the segment of ξ between X1 and X2 , and β is the angle between the tangent vectors to γ1 and γ2 at X1 and X2 , respectively. A little cruder estimate was proved in [7, Lemma 4.3], but a close examination of the proof shows that it in fact implies (2.3). Alternatively, one can prove (2.3) directly, as in [8, Theorem 5.42]. Given X, Y ∈ M, denote by s+ (X, Y ) ≥ 0 the future separation time (the first time when the images F n (X) and F n (Y ) for n ≥ 0 lie in different connected 1 The

book [8] is devoted to classical (unperturbed) billiards with smooth invariant measures, but many technical facts proven there hold in our case as well.

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components of the collision space M), and similarly let s− (X, Y ) ≥ 0 denote the past separation time (this definition takes into account that M is cut along the boundaries of the homogeneity strips, cf. [7, p. 223]). Observe that if X and Y lie on one unstable curve γ ⊂ M, then |γ(X, Y )| ≤ CΛ−s+ (X,Y ) , where Λ > 1 is the hyperbolicity constant for F , cf. [8, Eq. (5.32)]. Now (2.3) implies (see, e.g., [8, Proposition 5.48]) that for any X, Y ∈ γ1 | ln J h(X) − ln J h(Y )| ≤ Cϑs+ (X,Y ) ,

(2.4)

where ϑ = Λ−1/6 < 1. Following Young [32, p. 597], we call the property (2.4) the ‘dynamically defined H¨ older continuity’ of J h. Next we define a class of probability measures supported on unstable curves, following [8,10]. A standard pair = (γ, ν) is a homogeneous unstable curve γ ⊂ M with a probability measure P on it, whose density ρ (with respect to the Lebesgue measure on γ) satisfies | ln ρ(X) − ln ρ(Y )| ≤ Cr ϑs+ (X,Y ) .

(2.5)

Here Cr > 0 is a sufficiently large constant (independent of F). For any standard pair = (γ, ρ) and n ≥ 1 the image F n (γ) is a finite or countable union of homogeneous unstable curves (h-components) on which the density of the measure F n (P ) satisfies (2.5); hence the image of a standard pair under F n is a family of standard pairs (with a factor measure). More generally, a standard family is an arbitrary (countable or uncountable) collection G = {α } = {(γα , ρα )}, α ∈ A, of standard pairs with a probability factor measure λG on the index set A. Such a family induces a probability measure PG on the union ∪α γα (and thus on M) defined by PG (B) = Pα (B ∩ γα ) dλG (α) ∀B ⊂ M . Any standard family G is mapped by F n into another standard family Gn = F n (G), and PGn = F n (PG ). For every α ∈ A, any point X ∈ γα divides the curve γα into two pieces, and we denote by rG (X) the length of the shorter one. Now the quantity ZG = supε>0 ε−1 PG (rG < ε) reflects the ‘average’ size of curves γα in G; we only consider standard families with ZG < ∞. The growth lemma [7, Proposition 5.3] implies that ZGn ≤ C(θn ZG + 1) for all n ≥ 0 and some constant θ ∈ (0, 1), see a proof in [8, Proposition 7.17]; this estimate effectively asserts that standard families grow under F n exponentially fast. A standard pair (γ, ρ) is proper if |γ| ≥ δp , where δp > 0 is a small but fixed constant. A standard family G is proper if ZG ≤ Cp , where Cp is a large but fixed constant (chosen so that a family consisting of a single proper standard pair is proper, as a family). The image of a proper standard family under F n is proper for every n ≥ 1. A smooth foliation of M by (long enough) unstable curves gives us a proper standard family G such that PG = ν0 , the billiard invariant measure, see [8, p. 172].

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Also, there is a special standard family E consisting of (maximal) unstable hfibers γ u for the map F with the SRB densities ργ u on them and the factor measure generated by ν; in that case PE = ν, the family E is proper (due to [7, Proposition 5.6]) and obviously F -invariant. Next we present the key tool of the Young–Dolgopyat approach – the coupling lemma (for a detailed account see [8, Appendix A] and [8, Section 7.5]). Given a standard pair = (γ, ρ), we consider a ‘rectangle’ γˆ = γ × [0, 1] and equip it with ˆ with density a probability measure P ρˆ(X, t) = ρ(X) dX dt ;

(2.6)

the map F can be naturally defined on γˆ. Given a standard family G = (γα , ρα ) γα , ρˆα ) the family of the correspondwith a factor measure λG , we denote by Gˆ = (ˆ ˆ G the ing rectangles and equip it with the same factor measure λG ; we denote by P induced measure on the union ∪α γˆα . n

Lemma 2.1 (Coupling lemma). Let G = (γα , ρα ), α ∈ A, and F = (γβ , ρβ ), β ∈ B, be two proper standard families. Then there exist a bijection (called coupling map) ˆG ) = P ˆ E , and a (coupling time) Θ : ∪α γˆα → ∪β γˆβ that preserves measure; i.e., Θ(P function Υ : ∪α γˆα → N such that A. Let (X, t) ∈ γˆα , α ∈ A, and Θ(X, t) = (Y, s) ∈ γˆβ , β ∈ B. Denote m = Υ(X, t) ∈ N. Then the points F m (X) and F m (Y ) lie on the same stable h-fiber in M. B. There is a uniform exponential tail bound on the function Υ: ˆ G1 Υ > n ≤ CΥ ϑn , P (2.7) Υ for some constants CΥ > 0 and ϑΥ < 1 (independent of F, in the sense of Assumption B). A detailed (and lengthy) proof is given in [8, Chapter 7] for unperturbed Sinai billiards. It applies to our case with one little modification. While for unperturbed billiards the construction of the so called ‘magnet’ rectangle, see [8, Proposition 7.83], is relatively simple as it deals with one (billiard) map, in our case it requires a more elaborate argument, as we deal with a class of maps and need uniformity in F. In fact, an analogue of the ‘magnet rectangle’ (called rhombus) is constructed in [7, Lemma 6.5] and its necessary properties are proved in [7, Corollary 6.8]. The coupling lemma has many remarkable implications, some of them we state next. Motivated by (2.4), we say that a function f : M → R is dynamically H¨ older continuous if there are ϑf ∈ (0, 1) and Kf > 0 such that for any X and Y lying on one unstable curve s (X,Y )

|f (X) − f (Y )| ≤ Kf ϑf+

(2.8)

and for any X and Y lying on one stable curve s (X,Y )

|f (X) − f (Y )| ≤ Kf ϑf−

.

(2.9)

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We denote the space of such functions by H. It contains every piecewise H¨ older continuous function whose discontinuities coincide with those of F ±m for some m > 0. For example, the return time function τ (X) = min{t > 0 : Φt (X) ∈ M} belongs in H. Proposition 2.2 (Equidistribution). Let G be a proper standard family. For any dynamically H¨ older continuous function f ∈ H and n ≥ 0 n f ◦ F dPG − f dν ≤ Bf θfn (2.10) M

M

where Bf = 2CΥ (Kf + f ∞ ) and θf = [max{ϑΥ , ϑf }]1/2 < 1. In other words, iterations of measures on standard pairs converge to the SRB measure exponentially fast. For the proof, see [8, Theorem 7.31]. To estimate multiple correlations, let f0 , f1 , . . . , fr ∈ H and g0 , g1 , . . . , gk ∈ H be such that f ’s have identical parameters ϑf = ϑfi , Kf = Kfi , and f ∞ = fi ∞ for all 0 ≤ i ≤ r, and g’s have identical parameters ϑg = ϑgi , Kg = Kgi , and g∞ = gi ∞ for all 0 ≤ i ≤ k. Consider two products f˜ = f0 · (f1 ◦ F i−1 ) · (f2 ◦ F i−2 ) · · · (fr ◦ F i−r ) for some 0 > i−1 > · · · > i−r and g˜ = g0 · (g1 ◦ F i1 ) · (g2 ◦ F i2 ) · · · (gk ◦ F ik ) for some 0 < i1 < · · · < ik . We use a short-hand notation ν(f ) =

M

f dν.

Theorem 2.3 (Exponential bound on multiple correlations). For all n > 0 n ν f˜ · (˜ g ◦ F n ) − ν(f˜)ν(˜ g ) ≤ Bf˜,˜g θf,g (2.11) where

1/4 θf,g = max ϑΥ , ϑf , ϑg , α1 < 1,

α1 < 1 is a constant from [7, Corollary 5.4], and

Kf g∞ Kg f ∞ r k Bf˜,˜g = Cf ∞ g∞ + + f ∞ g∞ . 1 − ϑf 1 − ϑg For the proof, see [8, Theorem 7.41]. We remark that the theorem remains valid if fi ’s only satisfy (2.8) and gi ’s only satisfy (2.9). Not only the exponential bound (2.11) is novel and important itself, but the exact formulas for θf,g and Bf˜,˜g are essential in the proof of the central limit theorem and the almost sure invariance principle (ASIP), see [8, Sections 7.8–7.9]: Theorem 2.4 (Almost sure invariance principle). Let f ∈ H such that ν(f ) = 0 ,

σf2

=

∞ n=−∞

ν f · (f ◦ F n ) = 0

(2.12)

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Denote Sn = f + f ◦ F + · · · + f ◦ F n−1 and define a continuous function WN (s; X) of s ∈ [0, 1] by n S (X) n WN ;X = √ N σf N at rational points s = n/N and by linear interpolation in between. Then there is a standard Wiener process (a Brownian motion) B(s; X) on M with respect to the measure ν so that for some λ > 0 |WN (s; X) − B(s; X)| = O(N −λ )

(2.13)

for ν-almost all X ∈ M. Corollary 2.5 (Law of iterated logarithm). For ν-a.e. point X ∈ M lim sup Sn / 2nσf2 log log n = 1 . n→∞

For proofs, see [8, Sections 7.9]. We emphasize that all our constants are independent of F and the convergence is always uniform in F. We should note that our bound on the error term in (2.13) is not optimal; better bounds, e.g., O(N −1/4+ε ), can be obtained by using methods of recent works [14, 26]. A more general (vector) version of the ASIP can be derived based on the results of the manuscript [25] (not yet published). Since τ (X) is dynamically Hölder continuous, it satisfies the central limit √ theorem, i.e., (tn − nν(τ ))/ n converges to a normal distribution N (0, στ2 ); here tn = τ + τ ◦ F + · · · + τ ◦ F n−1 is the time of the nth collision. Actually, στ > 0 (this follows from the mixing property of the flow proved in the next section, as explained in [8, Remark 7.63]). Also, let nX (T ) denote the number of collisions on the trajectory Φt (X), √ 0 < t < T . Then T /nX (T ) → ν(τ ) for a.e. X ∈ M and (nX (T ) − T /ν(τ ))/ T converges to a normal distribution N (0, σ 2 ) with σ 2 = στ2 /[ν(τ )]3 ; for a proof, see [8, Sections 7.10]. Next we derive a formula specific to the case F = 0 (motivated by Kawasaki formulas in nonlinear response theory [31]). For any f ∈ H ν(f ) = lim ν0 (f ◦ F k ) n→∞

= ν0 (f ) + lim

n→∞

= ν0 (f ) + lim

n→∞

n k=1 n

ν0 (f ◦ F k ) − (f ◦ F k−1 ) ν0 (f ◦ F k )(1 − g) ,

(2.14)

k=1

where g = dF −1 ν0 /dν0 is the Jacobian of the map F with respect to the billiard invariant measure ν0 . A direct calculation gives

τ (X) p(X) exp g(X) = div Γ(X) dt p F (X) 0

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where Γ = p cos θ, p sin θ, ph is the vector field in Ω generating the flow Φt , cf. (1.4). Since ln p + phθ , div Γ = px cos θ + py sin θ + pθ h + phθ = d dt τ (X) we have g(X) = exp 0 phθ dt . Observe that g = 1 + O(δ0 ), where δ0 is a small constant in Assumption B. Also note that ν0 (g) = 1 and g(X) is a piecewise smooth function whose discontinuities coincide with those of the map F . Furthermore, ln g is dynamically H¨ older continuous with ϑln g = Λ−1/2 and Kln g = CpC 2 hC 2 being independent of F. We now use the special standard family E consisting of maximal unstable h-fibers γ u ⊂ M with SRB densities ργ u . The probability measure d˜ ν = g dν0 on M induces a conditional density ρ˜γ u on each γ u , which is proportional to gργ u , hence its logarithm is dynamically H¨ older continuous:

| ln ρ˜γ u (X) − ln ρ˜γ u (Y )| ≤ (Cr + Kln g ) ϑs+ (X,Y ) . for X, Y ∈ γ u . Of course, the density ρ˜γ u may not satisfy (2.5), but its images under F m will smooth out (due to distortion bounds [8, p. 203]) and then satisfy (2.5) for all m ≥ m0 , where m0 = m0 (pC 2 , hC 2 ). Thus, F m0 (˜ ν ) will coincide with PG for some proper G. Now standard family Proposition 2.2 implies that both ν0 (f ◦ F k ) and ν0 g · (f ◦ F k ) = ν˜(f ◦ F k ) converge to ν(f ) exponentially fast (and uniformly in F), thus the sum in (2.14) is bounded by a geometric series. This yields the desired Kawasaki formula: ν(f ) = ν0 (f ) +

∞

ν0 (f ◦ F k )(1 − g) ,

(2.15)

k=1

where the series converges exponentially fast and uniformly in F.

3. Bernoulli property of the flow Φt In this section we study the flow Φt : Ω → Ω. It is shown in [7] that Φt is a hyperbolic flow with uniform expansion and contraction rates. Its weakly unstable (and stable) manifolds are 2D surfaces in Ω that are made by families of strongly divergent (resp., convergent) flow lines. Strongly unstable and stable manifolds of the flow Φt are cross-sections (but not necessarily orthogonal!) of the corresponding families of flow lines. Clearly, Φt is a suspension flow over the base map F : M → M under a ceiling function τ . Strictly speaking, in a suspension flow the velocity must be equal to one, which can be achieved by changing the metric within Ω, but this change does not affect the existence or ergodicity of the SRB measure. Thus Theorem 1.1 easily implies the following: Corollary 3.1. The flow Φt : Ω → Ω admits a unique SRB measure μ, which is positive on open sets and ergodic.

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However, the mixing of the flow Φt requires a more elaborate argument. Loosely speaking, a hyperbolic flow is not mixing if its stable and unstable foliations are (locally) jointly integrable, i.e., for any phase point X ⊂ Ω all short chains consisting of alternating stable and unstable manifolds starting at X lie in a submanifold of codimension one transversal to the flow. For the billiard flow Φ0 , the lack of joint integrability was observed by Sinai [28], it is related to the opposite convexity of stable and unstable manifolds in Ω, see a detailed argument in [8, Section 6.11]. In our case, stable and unstable manifolds of the flow may not have opposite convexity, so we use a roundabout way to establish their nonintegrability. First we sharpen certain facts established in [7] for the map F . For any curve γ ⊂ M let mγ denote the Lebesgue measure on γ. For any point X ∈ M we denote by γ u (X) and γ s (X) the stable and unstable h-fibers through X. The point X divides the curve γ α (X), α = u, s, into two pieces, and we denote by rα (X) the length of the shorter one. Lemma 3.2. For every homogeneous unstable curve γ ⊂ M and mγ -almost every point X ∈ γ the stable h-fiber γ s (X) exists, i.e., rs (X) > 0. Moreover (3.1) mγ rs (X) < ε ≤ Cε for all ε > 0. The dual statement holds for stable curves. The existence of γ s (X) follows from two results of [7]: Eq. (6.4) and the Fact stated on p. 227. The estimate (3.1) is a local version of [7, Proposition 5.6], and for its proof see [8, Theorem 5.66]. Lemma 3.3. For every stable curve γ we have ν(∪X∈γ γ u (X)) > 0. For every unstable curve γ we have ν(∪X∈γ γ s (X)) > 0. Proof. The first statement follows from the proof of [7, Lemma 6.12]. Note that the second statement is not dual to the first one, since the SRB measure ν is not preserved under the reversal of time. The second statement follows from Lemma 3.2, the absolute continuity ([7, Lemma 4.3]), and the first statement here. Corollary 3.4 (Sinai’s fundamental theorem). Let X ∈ M and F n be continuous at X for all n > 0. Then for any ε > 0 and A > 0 there exists an open neighborhood U ⊂ M of X such that for any unstable curve γ ⊂ U mγ Y ∈ γ : rs (Y ) > A|γ| ≥ (1 − ε) |γ| . Similarly, if F n is continuous at X for all n < 0, then for any stable curve γ ⊂ U mγ Y ∈ γ : ru (Y ) > A|γ| ≥ (1 − ε) mγ |γ| . Observe that if A 1 is large, then a vast majority of points Y ∈ γ lie on stable (unstable) h-fibers which are much longer than the curve γ itself. This corollary follows from Lemma 3.2, see [8, Section 5.13]. We recall [7, p. 233] that given a stable h-fiber γ s and ε > 0 we denote by Γε (γ s ) the union of all stable h-fibers in M that are ε-close to γ s in the

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Hausdorff metric. We call γ s a density h-fiber if for every ε > 0 the set Γε (γ s ) has positive Lebesgue measure in M. (This is an analogue of Lebesgue density points of subsets of Rn .) The union of density h-fibers has full Lebesgue measure [7, Lemma 6.12]. Now we call γ s a ν-density h-fiber if for every ε > 0 the set Γε (γ s ) has positive ν-measure. Similarly, we define unstable ν-density h-fibers. Lemma 3.5. Every density h-fiber is also a ν-density h-fiber. Proof. For stable h-fibers, this follows from Lemmas 3.3 and the absolute continuity. The claim for unstable h-fibers can be proved by a construction similar to the proof of [7, Proposition 6.13]. Precisely, let γ u be a density h-fiber and ε > 0. Let γ be a stable curve crossing γ u . By reducing γ, if necessary, we can ensure that mγ (γ ∩ Γε (γ u )) > (1 − ε )mγ (γ), where ε > 0 is arbitrary small. Now we pull the entire structure back under F −n until we get mγ (γ(−n)) ≥ β˜2 mγ (γ) (here and below we use the notation of [7, Section 6]). The set F −n γ(−n) consists of stable curves that straddle the fixed rhombus R. Then it is not hard to deduce that the u set F n (RF ) ∩ Γε (γ u ) has a positive ν measure. We will only consider density h-fibers without saying that explicitly. Consider a continuous curve in M that is a finite union of segments of stable and unstable h-fibers (of course, stable and unstable h-fibers must alternate). Such curves are called Hopf chains or zigzag lines or us-paths (‘us’ stays for ‘unstablestable’). We require that at every “junction point” where two segments of h-fibers meet, those can be continued beyond the junction point. If a chain is not simple, i.e., has self-intersections, it can be shortened by the removal of extra loops, hence we will only consider simple chain. A chain is called a loop (or n-loop) if it is a simple closed curve in M (consisting of n segments of h-fibers). Chains and loops are instrumental in many proofs of ergodicity that go back to Hopf [16, 17]. Lemma 3.6 (“Zigzag lemma”). For every open connected set V ⊂ M and two curves γ1 , γ2 ⊂ V let distV (γ1 , γ2 ) denote the minimal length of smooth curves lying in V and connecting γ1 with γ2 . Then there is a zigzag line that starts on γ1 and ends on γ2 , lies entirely in V , and whose total length is ≤ C · distV (γ1 , γ2 ). This follows from Sinai’s fundamental theorem (Corollary 3.4), one just constructs a zigzag line starting at γ1 , moving in a general direction along a curve γ ⊂ V connecting γ1 with γ2 , whose length is nearly minimal, and eventually crossing γ2 ; a detailed construction of such zigzag lines is described in [8, Section 6.5]. Since stable and unstable h-fibers are uniformly transversal [7, Lemma 3.10], one can easily ensure the necessary bound on the length of the chain. Lemma 3.7. There is a global constant d0 = d0 (D) > 0 such that for every small force F there is a simple 4-loop (four is the minimal number of h-fibers in a loop), where all the h-fibers have length ≥ d0 . Proof. This follows from Corollary 3.4 and Lemma 3.5.

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We now turn to the flow Φt : Ω → Ω. Let X ∈ M and L ⊂ M a loop consisting of segments of h-fibers γ1 , . . . , γn , so that γi ∩ γi+1 = ∅ and γ1 ∩ γn = {X}. Consider Y = Φt X for some t ∈ (0, τ (X)). We now ‘lift’ the loop L from M to Ω in the following way: for each stable or unstable h-fiber γi let Γi be a stable (resp., unstable) manifold of the flow Φt that projects down onto γi and such that Γi ∩ Γi+1 = ∅ for i = 1, . . . , n − 1. (We assume that L is small enough so that the entire construction lies in the interior of Ω and avoids intersections with the boundary ∂Ω.) Now it is clear that ∪ni=1 Γi is a continuous curve in Ω starting at Y and terminating at some point Y lying on the trajectory Φt (Y ), i.e., Y = Φτ (L) (Y ) for some small τ (L). We call L a closed loop if τ (L) = 0 and an open loop otherwise. Clearly, reversing the orientation of L (i.e., traversing L in the opposite direction) simply changes the sign of τ (L). Given an orientation of L, one can easily verify that τ (L) does not depend on the choice of the initial point X ∈ L or t. In the billiard systems (where F = 0), all the loops are open, and in fact |τ (L)| equals the ν0 -measure of the domain bounded by L, see [8, Lemma 6.40]. Next we verify the existence of open loops for small forces. Lemma 3.8. For every ε > 0 there is a δ∗ = δ∗ (D, B0 , ε) > 0 such that whenever a force F satisfies Assumptions A–C with δ0 < δ∗ , then 1 − max p(x, y, θ)/ min p(x, y, θ) < ε , x,y,θ x,y,θ i.e., the function p(x, y, θ) is almost constant on Ω. Proof. Due to (1.5), p slowly changes along the trajectories of the flow, and due to (1.2) it does not change at collisions. Now the lemma follows from the uniform bound (1.8) on the derivatives of p, and the uniform bounds on correlations in Theorem 1.1. Lemma 3.9. For every ε > 0 there is a δ∗ = δ∗ (D, B0 , ε) > 0 such that whenever a force F satisfies Assumptions A–C with δ0 < δ∗ , then all the first order derivatives of the function p(x, y, θ) are less than ε. This lemma follows from Lemma 3.8 and the following elementary fact: Sublemma 3.10. Let Ω be a smooth compact manifold with boundary, f : Ω → R a C 2 function whose first and second order derivatives are uniformly bounded by a constant B0 . Then for every ε > 0 there is a δ = δ(Ω, B0 , ε) > 0 such that if |f (X) − f (Y )| < δ for all X, Y ∈ Ω, then all the first order derivatives of f are less than ε. The following lemma sharpens [7, Lemma 3.6]: Lemma 3.11. For every ε > 0 there is a δ∗ = δ∗ (D, B0 , ε) > 0 such that whenever a force F satisfies Assumptions A–C with δ0 < δ∗ , then for any strongly divergent family of trajectories on an interval (t0 , ∞) we have |αt | < ε for all t > t0 + c for some constant c > 0.

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Proof. The proof is a modification of that of Lemma 3.6 in [7]. The main difference is that now, in the equation (3.22), all the terms can be made arbitrarily small, except κ, which is still positive and bounded away from zero. Hence, the term −κ(α − pθ /p) drives α to zero whenever α is not small enough, and the other terms are not strong enough to stop this drive. We now recall the meaning of the function αt , see [7, p. 204]. Let Γu ⊂ Ω be an unstable manifold of the flow, then its trajectories {Φs (Y )}, Y ∈ Γu , s > 0, make a strongly divergent family. For every t > 0 the image Φt (Γu ) is an unstable manifold, too; its projection onto the table D is a curve that is a cross-section of the above family of flow lines. Now αt is the cotangent of the angle between that cross-section and the corresponding flow-line. If αt = 0, then the cross-section is orthogonal (and this is the case for unperturbed billiards); if αt is small, then the cross-section is almost orthogonal. The above lemma now implies that unstable manifolds of the perturbed flow Φt are good approximations to those of the billiard flow Φt0 . Due to the time reversibility, a similar property holds for stable manifolds. Now it takes a simple geometric argument to conclude the following: Corollary 3.12. Let L ⊂ M be a simple 4-loop from Lemma 3.7. If ε in Lemma 3.11 is small enough, we have τ (L) = 0, i.e., L is an open loop. Next we establish a general fact: Proposition 3.13. If there exists an open loop L ⊂ M, then the flow Φt is mixing and Bernoulli. Proof. For simplicity we assume that L is an open 4-loop (and one exists due to Corollary 3.12); it will be clear from our argument that it applies to arbitrary loops, too. The following lemma can be verified by direct inspection: Lemma 3.14. Let L1 , . . . , Lk ⊂ M be simple loops, oriented in the same way (say, all – clockwise), and bounding nonoverlapping domains V1 , . . . , Vk . Suppose that these domains are adjacent to each other so that V = V1 ∪ · · · ∪ Vk is a simply connected domain in M . Then V is bounded by a loop L and we have τ (L1 ) + · · · + τ (Lk ) = τ (L). Now, let γ1u , γ2s , γ3u , and γ4s denote the sides of the 4-loop L (which are alternating unstable and stable h-fibers). Let X1 and X2 denote the midpoints of γ1u and γ3u , respectively, and V the open ε-neighborhood of the straight line joining X1 with X2 , where ε dist(X1 , X2 ). Due to Lemma 3.6, there is a zigzag line L joining γ1u ∩ V with γ3u ∩ V , lying entirely in V , and having length |L | ≤ C · dist(X1 , X2 ). The zigzag line L divides the domain bounded by L into 2 subdomains, each bounded by a loop. Due to Lemma 3.14 one of these two loops is open, we denote it by L1 . It has four sides: two unstable sides (≈halves of γ1u and γ3u ) and two stable sides: one is γ2s or γ4s and the other is L (the latter is, of course, a zigzag line, but it stretches along a stable curve joining X1 and X2 , so we call it a stable side). Now

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we apply the same argument to construct a zigzag line joining the middle part of γ2s (or γ4s ) with the middle part of L and stretching along an unstable curve. This gives us two smaller loops, one of them will be open (again by Lemma 3.14), and we denote it by L2 . Now we repeat our construction inductively and obtain a sequence of open loops Ln , n ≥ 1, such that |Ln | → 0, hence τ (Ln ) → 0, as n → ∞ (but τ (Ln ) = 0 for every n). In other words, there are open loops with arbitrarily small values of τ (L)! Recall that all sides of our loops are density h-fibers, hence there are plenty of open loops with arbitrarily small τ values. More precisely, if we remove an arbitrary collection of h-fibers of the total ν-measure zero from M, then there still remain open loops with arbitrarily small (but non-zero) τ values. Next, it is known that, under general assumptions, a completely hyperbolic flow (which Φt is) with an ergodic SRB measure is either Bernoulli or Bernoulli times rotation. In the latter case a factor of the flow Φt is a circle rotation (and, of course, in this case the flow is not even mixing). Hence, the Bernoulli property is equivalent here to the mixing property. A general result of this sort was obtained by Ornstein and Weiss [27]. In the particular setting of certain perturbations of billiard flows, this fact was proved earlier by Kubo and Murata [21]. Assume now that the flow Φt is Bernoulli+rotation, i.e., there is a factor Ξ : Ω → S 1 so that Ξ ◦ Φt ◦ Ξ−1 is the rotation of the circle S 1 at constant speed. We call the sets Ξ−1 (p) ⊂ Ω, for p ∈ S 1 , layers. It is clear that μ-almost every stable or unstable manifold Γ ⊂ Ω lies in one layer; we call such stable and unstable manifolds typical. We call the projections of typical stable and unstable manifolds on M, along the trajectories of the flow, typical h-fibers; then ν-almost every h-fiber is typical. Let L ⊂ M be any loop consisting of typical h-fibers. Its ‘lift’ in Ω, as constructed above, will consist of typical stable and unstable manifolds of the flow Φt . Since every typical stable and unstable manifold lies in one layer, the entire lift of the loop L belongs in one layer, too. Hence, τ (L) is a multiple of the period of the rotation Ξ ◦ Φt ◦ Ξ−1 of S 1 . But we have seen that there are plenty of loops consisting of typical h-fibers with arbitrarily small non-zero τ values, thus the period of rotation must be zero. This proves Theorem 3.15. The flow Φt : Ω → Ω is mixing and Bernoulli. Lastly we present a central limit theorem for the flow Φt . Let F : Ω → R be a

τ (X) bounded function such that f (X) = 0 F (Φt X) dt is a dynamically Hölder continuous function on M (this holds, for example, when F is smooth with bounded derivatives). Note that F dμ = ν(f )/ν(τ ) . μ(F ) = Denote St (X) =

t 0

Ω

F (Φ X) dt for all t > 0 and X ∈ Ω. t

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Theorem 3.16√(Central limit theorem). There is σF ≥ 0 such that the function (St − μ(F ) t)/ t on Ω converges to a normal distribution N (0, σF2 ). This theorem follows from the central limit theorem for the map F applied to f and τ , see, e.g., [8, Theorem 7.68]. Note that we are not using the mixing property of the flow Φt . In fact, probabilistic limit theorems usually hold for suspension flows regardless of their mixing properties, as long as the base map is strongly chaotic, see, e.g., [13, 24].

t ˜ (t) = 0 p(s) ds denote the position of the moving particle As a corollary, let q

on the universal cover of the torus T2 . Then there exists a 2-vector a = Ω p dμ √ and a 2 × 2 positive definite matrix V such that (˜ q(t) − at)/ t converges to a twodimensional normal distribution N (0, V). The matrix V is close to the diffusion matrix of the unperturbed Sinai billiard, which is known to be non-singular [3], thus V cannot be singular either. In Theorem 3.16, σF2 = σf2 /ν(τ ), where σf2 is defined by the infinite series (2.12). If the flow Φt has rapidly decaying correlations, one usually can prove that ∞ 2 σF = μ (F ◦ Φt ) · F dt . (3.2) −∞

For the unperturbed billiard flow Φt0 , correlations decay at least as fast as a ‘stretched exponential’ function [9], which ensures (3.2) for that special case. But the results of [9] to dot extend to the perturbed flow Φt . By using a different approach, Melbourne [23] obtained fairly strong (‘superpolynomial’) bounds on correlations for rather general hyperbolic flows under the assumption that four periodic orbits exist whose periods satisfy a Diophantine-type condition, see (2.1) in [23]. This result implies rapid mixing, and therefore (3.2), for typical (in certain topological and measure-theoretic senses) flows Φt . It would be interesting to obtain bounds on correlations for all flows Φt covered in this paper.

Acknowledgements The author is grateful to I. Melbourne and the anonymous referee for very useful comments. The author is partially supported by NSF grant DMS-0354775.

References [1] P. R. Baldwin, Soft billiard systems, Physica D 29 (1988), 321–342. [2] F. Barra and T. Gilbert, Steady-state conduction in self-similar billiards, Phys. Rev. Lett. 98 (2007) paper 130601, 4 pp. [3] L. A. Bunimovich, Ya. G. Sinai, and N. I. Chernov, Statistical properties of twodimensional hyperbolic billiards, Russ. Math. Surv. 46 (1991), 47–106.

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[4] N. I. Chernov, G. L. Eyink, J. L. Lebowitz and Ya. G. Sinai, Steady-state electrical conduction in the periodic Lorentz gas, Comm. Math. Phys. 154 (1993), 569–601. [5] N. I. Chernov, G. L. Eyink, J. L. Lebowitz and Ya. G. Sinai, Derivation of Ohm’s law in a deterministic mechanical model, Phys. Rev. Lett. 70 (1993), 2209–2212. [6] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys. 94 (1999), 513–556. [7] N. Chernov, Sinai billiards under small external forces, Ann. H. Poincaré 2 (2001), 197–236. [8] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, AMS, Providence, RI (2006), 316 pp. [9] N. Chernov, A stretched exponential bound on time correlations for billiard flows, J. Statist. Phys., 127 (2007), 21–50. [10] N. Chernov and D. Dolgopyat, Brownian Brownian Motion – I, Memoirs AMS, to appear. [11] N. Chernov and D. Dolgopyat, Particle’s drift in self-similar billiards, manuscript. [12] N. Chernov and D. Dolgopyat, Diffusive motion and recurrence on an idealized Galton Board, Phys. Rev. Lett. 99 (2007), 030601. [13] M. Denker and W. Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function, Ergod. Th. Dynam. Syst. 4 (1984), 541–552. [14] M. Field, I. Melbourne, and A. T¨ or¨ ok, Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions, Ergod. Th. Dynam. Syst. 23 (2003), 87–110. [15] G. Gallavotti and D. S. Ornstein, Billiards and Bernoulli schemes, Comm. Math. Phys. 38 (1974), 83–101. [16] E. Hopf, Statistik der geodetischen Linien in Mannigfaltigkeiten negativer Kr¨ ummung, Ber. Verh. S¨ achs. Akad. Wiss. Leipzig 91 (1939), 261–304. [17] E. Hopf, Statistik der L¨ osungen geod¨ atischer Probleme vom unstabilen Typus, II, Math. Annalen 117 (1940), 590–608. [18] A. Knauf, Ergodic and topological properties of Coulombic periodic potentials, Commun. Math. Phys. 110 (1987), 89–112. [19] A. Kr´ amli, N. Sim´ anyi and D. Sz´ asz, Dispersing billiards without focal points on surfaces are ergodic, Commun. Math. Phys. 125 (1989), 439–457. [20] I. Kubo, Perturbed billiard systems, I., Nagoya Math. J. 61 (1976), 1–57. [21] I. Kubo and H. Murata, Perturbed billiard systems II, Berboulli properties, Nagoya Math. J. 81 (1981), 1–25. [22] C. Liverani, Interacting particles, In: Springer Encycl. Math. Sci. 101 (2000), 179– 216. [23] I. Melbourne, Rapid decay of correlations for nonuniformly hyperbolic flows, Trans. Amer. Math. Soc. 359 (2007), 2421–2441. [24] I. Melbourne and A. T¨ or¨ ok, Statistical limit theorems for suspension flows, Israel J. Math. 144 (2004) 191–209. [25] I. Melbourne and M. Nicol, A vector-valued almost sure invariance principle for hyperbolic dynamical systems, preprint.

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[26] N. Nagayama, Almost sure invariance principle for dynamical systems with stretched exponential mixing rates, Source: Hiroshima Math. J. 34 (2004), 371–411. [27] D. Ornstein and B. Weiss On the Bernoulli nature of systems with some hyperbolic structure, Ergod. Th. Dynam. Sys. 18 (1998), 441–456. [28] Ya. G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Russ. Math. Surv. 25 (1970), 137–189. [29] P. A. Vetier, Sinai billiard in potential field (construction of fibers), Coll. Math. Soc. J. Bolyai, 36 (1982), 1079–1146. [30] P. A. Vetier, Sinai billiard in potential field (absolute continuity), Proc. 3rd Pann. Symp. Math. Stat., 1983, 341–351. [31] T. Yamada and K. Kawasaki, Nonlinear effects in the shear viscosity of a critical mixture, Prog. Theor. Phys. 38 (1967), 1031–1051. [32] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147 (1998) 585–650. [33] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153– 188. Nikolai Chernov Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 USA e-mail: [email protected] Communicated by Eduard Zehnder. Submitted: May 14, 2007. Accepted: September 14, 2007.



On the Volume of Nodal Sets for Eigenfunctions of the Laplacian on the Torus Zeév Rudnick and Igor Wigman Abstract. We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4π 2 E with growing multiplicity N → ∞, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on√the eigenspaces. We show that the expected volume of the nodal set is √ const E. Our main result √ is that the variance of the volume normalized by E is bounded by O(1/ N ), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

1. Introduction The nodal set of a function on a manifold is the set of points where it vanishes. Nodal sets for eigenfunctions of the Laplacian on a smooth, compact Riemannian manifold have been studied intensively for some time now. For instance, it is known [6] that except for a subset of lower dimension, the nodal sets of eigenfunctions are smooth manifolds of codimension one in the ambient manifold. In particular one can define their hypersurface volume (in two dimensions this is the length). A conjecture of Yau is that the volume of the nodal set is bounded above and below by constant multiples of square root of the Laplace eigenvalue. Yau’s conjecture was proven for real-analytic metrics by Donnelly and Fefferman [7]. The lower bound in the case of smooth surfaces is due to Br¨ uning [4], see also [5] for planar domains. In this paper we study the volume of nodal sets for eigenfunctions of the Laplacian on the standard flat torus Td = Rd /Zd , d ≥ 2. We write the eigenvalue equation as Δf = −4π 2 Ef , where E ≥ 0 is an integer. The eigenvalues on the torus always have multiplicities, with the dimension N = N (E) of an eigenspace Z. Rudnick was supported by the Israel Science Foundation (grant No. 925/06). I. Wigman was supported by CRM analysis laboratory fellowship.

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corresponding to eigenvalue 4π 2 E being the number of integer vectors λ ∈ Zd so d that |λ|2 = E. In dimension d ≥ 5 this number grows as E → ∞ roughly as E 2 −1 , but for small values of d, particularly for d = 2, the behaviour is more erratic, and depends on the prime decomposition of E. We will consider random eigenfunctions on the torus, that is random linear combinations 1 bλ cos 2πλ, x − cλ sin 2πλ, x (1.1) f (x) = √ 2N λ∈Zd :|λ|2 =E with bλ , cλ ∼ N (0, 1) real Gaussians of zero mean and variance 1 which are independent save for the relations b−λ = bλ , c−λ = −cλ . Let E = EE be the eigenspace associated to the eigenvalue 4π 2 E (i.e., the space of functions of form (1.1)). We denote by E(•) the expected value of the quantity • in this ensemble. For instance, the expected amplitude of f is E(|f (x)|2 ) = 1. Denote by Z(f ) the volume of the nodal set of an eigenfunction (1.1). Our first result, Proposition 4.1, is that the expected value of Z is √ E(Z) = const · E for a certain constant depending only on the dimension d. This is of course consistent with the bounds of Donnelly and Fefferman [7]. √ Our main result, Theorem 6.1, is that the variance of the normalized volume Z/ E is bounded by Z 1 Var √

√ , as N → ∞ . E N (We believe that the correct upper bound for the variance is O(1/N )). Thus the √ value die out as the multiplicity N fluctuations of Z(f )/ E around its mean √ tends to ∞. Note however that Z(f )/ E is not asymptotically constant; for ind stance, if E = dm2 then for the eigenfunction f (x) = j=1 sin 2πmxj we have √ √ Z(f )/ E =√2 d while if E = m2 then for the eigenfunction f (x) = sin 2πmx1 we have Z(f )/ E = 2. Theorem 6.1 can be viewed as lending support to the expectation1 that for eigenfunctions on negatively curved manifolds, which are believed to behave similarly to random waves [2], the volumes of nodal sets, normalized by the square-root of the eigenvalue, do tend to a limiting value. See [12] for some work on the complexified nodal set of eigenfunctions in this context. Previous work in this vein is due to Bérard [1], who computed the expected surface measure of the nodal set for eigenfunctions of the Laplacian on spheres. Neuheisel [10] also worked on the sphere and gave an upper bound for the variance. Berry [3] computed the expected length of nodal lines for isotropic, monochromatic 1 We

thank Steve Zelditch for a discussion of this.

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random waves in the plane, which are eigenfunctions of the Laplacian with eigenvalue E. He found that the expected length (per unit area) is again of size about √ E and argued that the variance should be of order log E. More recently, F. Oravecz and ourselves have investigated a different characteristic of the nodal set of eigenfunctions on the torus, namely the Leray nodal measure [11], and have succeeded in obtaining the precise asymptotics of the variance of the Leray measure as N → ∞. 1.1. Plan of the paper We employ a version of the Kac–Rice formula for the volume of the nodal set, which using the Dirac delta function can be written as δ f (x) |∇f (x)|dx , Z(f ) = Td

see Section 3 for the rigorous version. To compute the expected value of Z is then a simple matter once we find that f (x) and ∂f /∂xj are independent Gaussians. This is done in Section 4. In Section 5 we derive a formula for the second moment of Z, which requires knowing the covariance structure of the 2d + 2-dimensional Gaussian vector v(x, y) = (f (x), f (y), ∇f (x), ∇f (y)). That v(x, y) is indeed a nondegenerate 2d + 2 dimensional Gaussian is verified in the appendix. As a result, we find that E(Z 2 ) = Td K(z)dz, with exp − 12 vΩ(z)−1 v T 1 dv

K(z) =

v1 v2 , d+1 2 (2π) 2d 1 − u(z) R det Ω(z) where u(z) = E(f (x)f (x+z)) is the two-point function of the ensemble, and where Ω(z) is a certain positive definite 2d × 2d matrix which enters into the covariance structure of the Gaussian vector v(x, y). In Section 6, which is the heart of the paper, we bound the variance of Z.

2. The model: Random eigenfunctions on the torus 2.1. Random eigenfunctions We consider non-constant eigenfunctions of the Laplacian on the standard flat torus Td = Rd /Zd . The solutions of the eigenvalue equation Δψ + 4π 2 Eψ = 0 ,

E = 0 ,

form a finite dimensional vector space E = EE , having as a basis the exponentials e2πiλ,x , for λ in the frequency set Λ = ΛE = {λ ∈ Zd , |λ|2 = E} . We define an ensemble of Gaussian random functions f ∈ E by 1 bλ cos 2πλ, x − cλ sin 2πλ, x f (x) = √ 2N λ∈Λ

112



with bλ , cλ ∼ N (0, 1) real Gaussians of zero mean and variance 1 which are independent save for the relations b−λ = bλ , c−λ = −cλ . Thus we can rewrite 2 f (x) = bλ cos 2πλ, x − cλ sin 2πλ, x (2.1) N λ∈Λ/±

where now only independent random variables appear. With our normalization, we have E(|f (x)|2 ) = 1 for all x ∈ Td . Definition 2.1. An eigenfunction f ∈ E is singular if ∃x ∈ Td with f (x) = 0 and ∇f (x) = 0. An eigenfunction f ∈ E is nonsingular if ∇f = 0 on the nodal set. Lemma 2.2 ([11], Lemma 2.3). The set of singular eigenfunctions has codimension at least 1 in E, and so has measure zero in E. 2.2. Properties of the frequency set The dimension N = dim E is the number of the frequencies in Λ, which is the number of ways of expressing E as a sum of d integer squares. For d ≥ 5 this grows roughly as E d/2−1 as E → ∞. For d ≤ 4 the dimension of the eigenspace need not grow with E. For instance, for d = 2, N is given in terms of the prime β decomposition of E as follows: If E = 2α j pj j k qk2γk where pj ≡ 1 mod 4 and qk ≡ 3 mod 4 are odd primes, α, βj , γk ≥ are integers, then N = 4 j (βj + 1), and otherwise E is not a sum of two squares and N = 0. √ On average (over integers which are sums of two squares) the dimension is const · log E. The frequency set Λ is invariant under the group Wd of signed permutations, consisting of coordinate permutations and sign-change of any coordinate, e.g., (λ1 , λ2 ) → (−λ1 , λ2 ) (for d = 2). In particular Λ is symmetric under λ → −λ and since 0 ∈ / Λ, we find N is even. We write Λ/± to denote representatives of the equivalence class of Λ under λ → −λ. We will need some simple properties of Λ: Lemma 2.3. For any subset O ⊂ Λ which is invariant under the group Wd , we have E 1 · δj,k . λj λk = (2.2) |O| d λ∈O

Moreover for any C ∈ R , d

1 E C, λ2 = |C|2 . |O| d

(2.3)

λ∈O

Proof. For i = j use the symmetry of O under the sign change of the i-th coordinate to change variables and deduce that the LHS of (2.2) vanishes. For i = j

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note that the sum λ∈O λ2i is independent of i since O is symmetric under permutations; hence we may average the RHS over i to find that λ∈O

λ2i =

d 1 2 1 |O|E , λi = ||λ||2 = d i=1 d d λ∈O

λ∈O

proving (2.2). To prove (2.3) we expand C, λ2 =

d

j,k=1 cj ck λj λk

and use (2.2).

Note that (2.3) implies that the frequency set Λ spans Rd . 2.3. The two point function The two-point function of the ensemble is 2 u(z) := E f (x + z)f (x) = N

cos 2πλ, z .

(2.4)

λ∈Λ/±

The two-point function clearly satisfies |u(z)| ≤ 1. We will need to know some of its basic properties, proved in [11], which we summarize as: Proposition 2.4. The two point function satisfies 1. 2. 3. 4.

There are only finitely many x ∈ Td where u(x) = ±1. points 2 The mean square of u is Td u = 1/N . The mean fourth power of u is bounded by2 Td u4 1/N . √ The kernel 1/ 1 − u2 is integrable on Td .

Part 1 follows from [11, Lemma 2.2], part 3 is [11, Proposition 7.1], and part 4 is [11, Lemma 5.3].

3. A formula for the volume of the nodal set Let χ be the indicator function of the interval [−1, 1]. We define for > 0 f (x) 1 Z (f ) := χ |∇f (x)|dx . 2 Td Lemma 3.1. Suppose that f ∈ E is non-singular. Then vol f −1 (0) = lim Z (f ) . →0

Proof. By the co-area formula [8], for f smooth and φ integrable, we have ∞ φ(x)|∇f (x)|dx = φ(x)dx ds . Td 2 Except

−∞

f −1 (s)

possibly in dimensions d = 3, 4 we have a better bound in [11] of o(1/N ), though we have no use for this finer information in this paper.

114


Taking φ(x) :=


f (x) 1 2 χ( ),

which is constant on the level sets f −1 (s) gives 1 Z (f ) = vol f −1 (s) ds . 2 −

Now if f is non-singular then s → vol(f −1 (s)) is continuous at s = 0 and so by the fundamental theorem of calculus, 1 lim vol f −1 (s) ds = vol f −1 (0) . →0 2 − Thus lim→0 Z (f ) = vol(f −1 (0)) as claimed.

Lemma 3.2. For all f ∈ E we have

√ Z (f ) ≤ 6d E .

We begin with the one variable case which we state as a separate lemma (cf. [9, Lemma 2]): Lemma 3.3. Let g(t) be a trigonometric polynomial of degree at most M . Then for all > 0 we have 1 |g (t)|dt ≤ 6M . 2 {t:|g(t)|≤} Proof. We partition the set {t : |g(t)| ≤ } ⊆ [0, 1] into a union of maximal closed intervals [ak , bk ] (with ak < bk ), disjoint except perhaps for common edges, such that on each such interval g has constant sign, that is either g ≥ 0 or g ≤ 0. If g ≥ 0 on [ak , bk ] then either g(ak ) = − or g (ak ) = 0 and ak is a local minimum for g, and g(bk ) ≤ + . If g ≤ 0 on [ak , bk ] then either g(ak ) = + or g (ak ) = 0 and ak is a local maximum for g, and g(bk ) ≥ − . If g ≥ 0 on [ak , bk ] then bk bk |g (t)|dt = g (t)dt = g(bk ) − g(ak ) ≤ 2 , ak

ak

while if g ≤ 0 on [ak , bk ] then bk |g (t)|dt = ak

bk

−g (t)dt = g(ak ) − g(bk ) ≤ 2 .

ak

Thus the total integral is bounded by the number ν of intervals [ak , bk ]: 1 |g (t)|dt ≤ ν . 2 {t:|g(t)|≤} Now the number of intervals is bounded by the number of a’s for which g(a) = ± plus the number of a’s for which g (a) = 0. Since both g and g are trigonometric polynomials of degree ≤ M , the number of such intervals is therefore 3·2M = 6M . This gives the required bound. We now prove Lemma 3.2 by reduction to the one-dimensional case.

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Proof. Since |∇f | ≤


d

∂f j=1 | ∂xj |

115

we have

d f (x) ∂f 1 χ Z (f ) ≤ ∂xj dx 2 Td j=1 and we will bound each term. Taking j = 1, we have

∂f (t, y) 1 f (x) ∂f 1 dt dy . χ ∂x1 dx = 2 Td 2 {t∈T1 :|f (t, y)|≤} ∂t y∈Td−1 d−1 In the inner integral we have a one variable polynomial g(t) = √ for each y ∈ T f (t, y) √ of degree at most E and hence by Lemma 3.3, the inner integral is at most 6 E. Summing over j introduces another factor of d.

As a consequence of the fact that for nonsingular functions we can compute the volume Z(f ) of the nodal set of f ∈ E via Lemma 3.1 and the fact that almost all f ∈ E are nonsingular (Lemma 2.2), we find: Corollary 3.4. The first and second moments of the volume Z(f ) of the nodal set of f are given by lim Z1 Z2 . E(Z) = E lim Z , E(Z 2 ) = E 1 ,2 →0

→0

4. The expected volume of the nodal set In this section we show Proposition 4.1. For d ≥ 1,

√ E(Z) = Id E

where

Id =

4π Γ d+1 d2 . d Γ 2

Proof. Since Z is uniformly bounded by Lemma 3.2, we can use the Dominated Convergence Theorem to write E(Z) = E lim Z = lim E(Z ) . →0

By Fubini’s theorem,

→0

1 f (x) χ E(Z ) = E |∇f (x)|dx 2 Td f (x) 1 χ = E K (x)dx . |∇f (x)| dx =: 2 Td Td

116



1 Now for each x ∈ Td , the function f → 2 χ( f (x) )|∇f (x)| is bounded and hence we may evaluate the integral by using the joint probability density of the variable (f (x), ∇f (x)), whose components are Gaussian of zero mean with covariance ∂f 2 (x) = 0 E f (x) = 1 , E f (x) ∂xj

and

E

∂f (x) ∂f (x) ∂xj ∂xk

=

E 2 · 4π 2 λj λk = 4π 2 · δj,k N d

(4.1)

λ∈Λ/±

by (2.2). Thus 1 K (x) = 2

χ R

a

e

b|2 db d| |b| exp − 2 8π E (2π)d/2 (4π 2 E/d)d/2 Rd a 2 1 2 1 e−a /2 da . |v | exp − |v | dv χ 2 2 d R R

−a2 /2

√ 4π 2 E =√ d · (2π)(d+1)/2

da √ 2π

Integrating over Td and taking the limit → 0 gives √ E(Z) = Id E where

1 |v | exp − |v |2 dv . 2 Rd 2 In the one-dimensional case, I1 = R |v|e−v /2 dv = 2. For d ≥ 2 , ∞ 2 1 |v | exp − |v |2 dv = vol(S d−1 ) re−r /2 rd−1 dr . 2 0 Rd 1 Id = √ d(2π)(d−1)/2

Using d

vol(S d−1 ) = gives

2π 2 , Γ d2

∞

rd e−r

2

/2

dr = 2

d−1 2

Γ

0

d+1 2

d+1 √ 1 2 d/2 Γ 2 |v | exp − |v | dv = 2(2π) 2 Γ d2 Rd

(which is consistent with the computation for d = 1). Thus 4π Γ d+1 2 Id = d Γ d2 as claimed.

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5. An integral formula for the second moment 5.1. The covariance matrix The covariance matrix Σ(x, y) of the Gaussian vector (f (x), f (y), ∇f (x), ∇f (y)) is given by A B (5.1) Σ= BT C where E (f(x)f (y)) E(f (x)∇f (x)) E(f (x)∇f (y)) E f (x)2 , B = A= E (f (y)f (x)) E f (y)2 E(f (y)∇f (x)) E(f (y)∇f (y)) E(∇f (x)T ∇f (x)) E(∇f (x)T ∇f (y)) C= . E(∇f (y)T ∇f (x)) E(∇f (y)T ∇f (y)) For generic (x, y), the covariance matrix Σ(x, y) is nonsingular (see Appendix A).

and

Lemma 5.1. The covariance matrix Σ(x, y) depends only on the difference z = x−y and is given in terms of the two-point function u by A(z) B(z) Σ(x, y) = B(z)T C(z) where

A(z) =

1 u(z) , u(z) 1

(here 0,∇u are row vectors), and

C(z) = where H =

∂2u ∂xj ∂xk

B(z) =

0 −∇u(z) 0 ∇u(z)

−H(z) 2 −H(z) 4πd E I 4π 2 E d I

is the Hessian of u.

Proof. By definition of the two point function, we have A = pute B, use

E

and hence In particular

1 u

∂f ∂ (x)f (y) = E f (x)f (y) = ∂j u(x − y) ∂xj ∂xj

E f (x)∂j f (y) = ∂j u(y − x) = −∂j u(x − y) . E f (x)∇f (x) = 0 .

Therefore

B(z) =

0 −∇u(z) 0 ∇u(z)

(where 0 denotes the d-dimensional zero row vector).

u . To com1

118



To compute C, use (4.1) to find 4π 2 E Id . E ∇f (x)T ∇f (x) = d More generally E ∂j f (x)∂k f (y) = and so

∂2 ∂2u E f (x)f (y) = − (x − y) ∂xj ∂yk ∂xj ∂xk

E ∇f (x)T ∇f (y) = −

Thus C=

∂2u (x − y) = −H(x − y) . ∂xj ∂xk j,k 4π 2 E d I

−H

−H

4π 2 E d I

as claimed. The inverse of Σ (when it exists) is given by ∗ ∗ Σ−1 = ∗ Ω−1 with Ω being the 2d × 2d matrix Ω = C − B T A−1 B . We will call Ω the reduced covariance matrix. We have det Σ = det A det Ω = (1 − u2 ) det Ω . By Lemma 5.1, we have 2 T 1 4π (E/d)I D D −H Ω= − −H 4π 2 (E/d)I 1 − u2 uDT D

(5.2) uDT D DT D

(5.3)

2

∂ u where D(z) = ∇u(z) and H = ( ∂x ) is the Hessian of u. j xk

5.2. A formula for the second moment Proposition 5.2. The second moment of Z(f ) is given by E(Z 2 ) = K(x)dx

(5.4)

Td

where 1 K(x) = √ 1 − u2 Denote K1 ,2 (x, y) :=

1 4 1 2

We have the following

R2d

exp − 21 vΩ−1 v T dv √ v1 v2 . (2π)d+1 det Ω

E

∇f (x) ∇f (y) χ

f (y) f (x) χ dμ(f ) . 1 2

(5.5)

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Lemma 5.3. For (x, y) ∈ Td × Td with u(x − y)2 = 1 E , K1 , 2 (x, y) d

1 − u2 (x − y)

(5.6)

where the implied constant depends only on the dimension d. Proof. Write f (x) = f, U (x), where U (x) is the unit vector √ 2 cos 2πλ, x, sin 2πλ, x λ∈Λ/± ∈ S N −1 , U (x) = √ N and where we identify the function f with a vector in RN via (2.1). Note that U (x), U (y) = u(x − y) is the cosine of the angle between U (x) and U (y). We have ∇f (x) = DU · f where the derivative DU is a d × N matrix. Equivalently, ∂ U (x) , 1 ≤ i ≤ d. ∇f (x) i = f, ∂xi By the triangle and Cauchy–Schwartz inequalities, d √ ∂ E f , f · U (x) ∇f (x) ≤ ∂xi i=1 by a computation of

∂U ∂xi .

Therefore

E K1 ,2 (x, y)

4 1 2

|f (x)| 2, we substitute (A.4) in (A.3) with any i = 1, j = 1 to get b1 2 ai aj = 0 a1 that is since b1 = 0, that ai aj = 0 ,

∀i = j ,

i, j = 1 .

Thus there is at most one index k = 1 with ak = 0, say k = 2, so we find that aj = 0 for j = 1, 2, and by (A.4) we therefore have bj = 0 for j = 1, 2. Thus b1 A = (a1 , a2 , 0) , B = a1 , −a2 , 0 a1 (if d = 2 this still holds, we just ignore the extra coordinates). Plugging this into (A.2) with λ so that λ1 = ±λ2 (which exists if N d 1) gives a1 (A.5) (a1 λ1 )2 − (a2 λ2 )2 = (β 2 − α2 ) b1 and replacing λ = (λ1 , λ2 , . . . ) with (λ2 , λ1 , . . . ) gives a1 (a1 λ2 )2 − (a2 λ1 )2 = (β 2 − α2 ) . (A.6) b1 Comparing (A.5) with (A.6) gives (a1 λ1 )2 − (a2 λ2 )2 = (a1 λ2 )2 − (a2 λ1 )2 that is

(λ21 − λ22 )(a21 + a22 ) = 0 . Since we chose λ2 = ±λ1 this gives a1 = a2 = 0, contradicting a1 =

0. Thus we are done with step 1.

Step 2: We take C = ±D and α = ±β in (A.1) and wish to show that for almost all z ∈ Td , the only solutions are α = 0 and C = 0. If either (α = β and C = D) or (α = −β and C = −D), then (A.1) gives e2πiz,λ = −1 which is a measure zero condition. Otherwise, assume α = β and C = −D (the other case is treated similarly). Here we have α + iC, λ = −e2πiλ,z α − iC, λ . (A.7) If α = 0 and C = 0 then there is some λ ∈ Λ so that C, λ = 0 and (A.7) forces e2πiz,λ = 1, that is z lies on one of the hyperplanes ! ∪λ∈Λ z : λ, z = 0 mod 1 , which is a measure zero condition.

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If α = 0, we replace C by − α1 C and α by 1 and drop the negative sign. Taking the real part of (A.7), we have 1 + cos 2πλ, z = C, λ sin 2πλ, z . We may assume that the sine on the RHS doesn’t vanish, since sin 2πλ, z = 0 is a measure zero condition. Therefore, we may divide to get C, λ =

1 + cos 2πλ, z = cot πλ, z . sin 2πλ, z

Now square and average the result over an orbit O ⊂ Λ of the group Wd of all permutations and sign changes of the coordinates. The LHS gives 1 E C, λ2 = |C|2 |O| d λ∈O

by (2.3), which is independent of the orbit chosen. The RHS gives 1 1 1 cot2 πλ, z = 1 + 2 |O| O sin πλ, z λ∈O

that is we find

λ∈O

1 1 E |C|2 − 1 = 2 . d |O| λ∈O sin πλ, z

Since 1/(sin πλ, z)2 is even, we get the same term for λ and −λ and so we may replace the average over O by the average over O/± where we have taken only one of λ, −λ. Thus 1 1 E |C|2 − 1 = (A.8) 2 . d |O/ ± | λ∈O/± sin πλ, z Assuming that N > |Wd | = 2d d!, we can find a different orbit O ⊂ Λ and then comparing with (A.8) gives 1 1 1 1 (A.9) 2 = 2 |O/ ± | |O / ± | λ∈O/± sin πλ, z λ∈O /± sin πλ, z that is we have eliminated the variable C. We claim that (A.9) forces the point z to lie on a measure zero subset of Td . Indeed, the functions involved are meromorphic in Cd /Zd and hence if (A.9) does not hold for all z, it can only hold on a complex submanifold of codimension (at least) one and in particular its real points will have codimension at least one in Td . But near the origin z = 0, each of the functions 1/(sin πλ, z)2 has singularities on the hyperplane z, λ = 0 and these hyperplanes are distinct for λ’s which are not collinear (here the condition d ≥ 2 comes in), as is the case for those appearing in (A.9). Thus these functions are linearly independent and so (A.9) is not valid for all z.

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References [1] P. Bérard, Volume des ensembles nodaux des fonctions propres du laplacien, ´ Bony–Sj¨ ostrand–Meyer seminar, 1984–1985, Exp. No. 14, 10 pp., Ecole Polytech., Palaiseau, 1985. [2] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), no. 12, 2083–2091. [3] M. V. Berry, Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature, J. Phys. A 35 (2002), 3025–3038. ¨ [4] J. Br¨ uning, Uber Knoten Eigenfunktionen des Laplace–Beltrami Operators, Math. Z. 158 (1978), 15–21. ¨ [5] J. Br¨ uning and D. Gromes, Uber die L¨ ange der Knotenlinien schwingender Membranen, Math. Z. 124 (1972), 79–82. [6] S. Y. Cheng, Eigenfunctions and nodal sets, Comm. Math. Helv. 51 (1976), 43–55. [7] H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), 161–183. [8] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491. [9] M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49, (1943), 314–320. Correction, ibid. 49 (1943), 938. [10] J. Neuheisel, The asymptotic distribution of nodal sets on spheres, Johns Hopkins Ph.D. thesis (2000). [11] F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus, math-ph/0609072, to appear in Annales de l’Institut Fourier 57 (2007). [12] S. Zelditch, Complex zeros of real ergodic eigenfunctions, Invent. Math. 167 (2007), 419–443. Zeév Rudnick School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel e-mail: [email protected] Igor Wigman Department of Mathematics and Statistics McGill University and Centre de Recherches Mathématiques (CRM) Université de Montréal C.P. 6128, succ. centre-ville Montréal Québec H3C 3J7 Canada e-mail: [email protected] Communicated by Jens Marklof. Submitted: June 28, 2007. Accepted: August 20, 2007.



Spectral Analysis of Magnetic Laplacians on Conformally Cusp Manifolds Sylvain Golénia and Sergiu Moroianu Abstract. We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When B is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When B is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect. In the non-gauge invariant case, we exhibit a strong Aharonov–Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limiting absorption principle. We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential.

1. Introduction Let X be a smooth manifold of dimension n, diffeomorphic outside a compact set to a cylinder (1, ∞) × M , where M is a possibly disconnected closed manifold. On X we consider asymptotically conformally cylindrical metrics, i.e., perturbations of the metric given near the border {∞} × M by: gp = y −2p (dy 2 + h) ,

y→∞

(1.1)

132

S. Golénia and S. Moroianu


where h is a metric on M and p > 0. If p = 1 and h is flat, the ends are cusps, i.e., complete hyperbolic of finite volume. For p > 1 one gets the (incomplete) metric horns. The refined properties of the essential spectrum of the Laplace–Beltrami operator Δp := d∗ d have been studied by Froese and Hislop [10] in the complete case. For the unperturbed metric (1.1), they get κ(p) = 0 , for p < 1 n−1 2 σess (Δp ) = κ(p)∞ , where (1.2) κ(1) = 2 . The singular continuous part of the spectrum is empty and the eigenvalues distinct from κ(p) are of finite multiplicity and may accumulate only at κ(p). Froese and Hislop actually show a limiting absorption principle, a stronger result, see also [8,11,12,25] for the continuation of their ideas. Their approach relies on a positive commutator technique introduced by E. Mourre in [39], see also [2] and references therein. See for instance [20, 28] for different methods. Consider more generally a conformal perturbation of the metric (1.1). Let ρ ∈ C ∞ (X, R) be such that inf y∈X (ρ(y)) > −1. Consider the same problem as above for the metric (1.3) g˜p = (1 + ρ)gp , for large y . To measure the size of the perturbation, we compare it to the lengths of geodesics. Let L ∈ C ∞ (X) be defined by 1−p y for p < 1 1−p L ≥ 1 , L(y) = , for y big enough . (1.4) ln(y) for p = 1 In [10], one essentially asks that L2 ρ, L2 dρ

and L2 Δg ρ are in L∞ (X) .

to obtain the absence of singular continuous spectrum and local finiteness of the point spectrum. On one hand, one knows from the perturbation of a Laplacian by a short-range potential V that only the speed of the decay of V is important to conserve these properties. On the other hand, in [15] and in a general setting, one shows that only the fact that ρ tends to 0 is enough to ensure the stability of the essential spectrum. Therefore, it is natural to ask whether the decay of the metric (without decay conditions on the derivatives) is enough to ensure the conservation of these properties. In this paper, we consider that ρ = ρsr + ρlr decomposes in short-range and long-range components. We ask the long-range component to be radial. We also assume that there exists ε > 0 such that L1+ε ρsr and dρsr , Δg ρsr ∈ L∞ (X) , (1.5) Lε ρlr , L1+ε dρlr and Δg ρsr ∈ L∞ (X) . Going from 2 to 1 + ε is not a significant improvement as it relies on the use of an optimal version of the Mourre theory instead of the original theory, see [2] and references therein. Nevertheless, the fact that the derivatives are asked only to be bounded and no longer to decay is a real improvement due to our method. We

Vol. 9 (2008)

Magnetic Laplacians on Cusp Manifolds

133

prove this result in Theorem 6.4. In the Mourre theory, one introduces a conjugate operator to study a given operator. The conjugate operator introduced in [10] is too rough to handle very singular perturbations. In our paper, we introduce a conjugate operator local in energy to avoid the problem. We believe that our approach could be implemented easily in the manifold settings from [5, 8, 10–12, 25, 27] to improve results on perturbations of the metric. A well-known dynamical consequence of the absence of singular continuous spectrum and of the local finiteness of the point spectrum is that for an interval J that contains no eigenvalue of the Laplacian, for all χ ∈ Cc∞ (X) and φ ∈ L2 (X), the norm χeitΔp EJ (Δp )φ tends to 0 as t tends to ±∞. In other words, if you let evolve long enough a particle which is located at scattering energy, it eventually becomes located very far on the exits of the manifold. Add now a magnetic field B with compact support and look how strongly it can interact with the particle. Classically there is no interaction as B and the particle are located far from each other. One looks for a quantum effect. The Euclidean intuition tells us that is no essential difference between the free Laplacian and the magnetic Laplacian ΔA , where A is a magnetic potential arising from a magnetic field B with compact support. They still share the spectral properties of absence of singular continuous spectrum and local finiteness of the point spectrum, although a long-range effect does occur and destroys the asymptotic completeness of the couple (Δ, ΔA ); one needs to modify the wave operators to compare the two operators, see [31]. However, we point out in this paper that the situation is dramatically different in particular on hyperbolic manifolds of finite volume, even if the magnetic field is very small in size and with compact support. We now go into definitions and describe our results. A magnetic field B is a smooth real exact 2-form on X. There exists a real 1-form A, called vector potential, satisfying dA = B. Set dA := d+iA∧ : Cc∞ (X) → Cc∞ (X, T ∗ X). The magnetic Laplacian on Cc∞ (X) is given by ΔA := d∗A dA . When the manifold is complete, ΔA is known to be essentially self-adjoint, see [46]. Given two vector potentials A and A such that A − A is exact, the two magnetic Laplacians ΔA and ΔA are unitarily equivalent, by gauge invariance. Hence when 1 (X) = 0, the spectral properties of the magnetic Laplacian do not depend on HdR the choice of the vector potential, so we may write ΔB instead of ΔA . The aim of this paper is the study of the spectrum of magnetic Laplacians on a manifold X with the metric (2.9), which includes the particular case (1.1). In this introduction we restrict the discussion to the complete case, i.e., p ≤ 1. 1 (X) = 0, and we simplify We focus first on the case of gauge invariance, i.e., HdR the presentation assuming that the boundary is connected. We classify magnetic fields. Definition 1.1. Let X be the interior of a compact manifold with boundary X. 1 (X) = 0 and that M = ∂X is connected. Let B be a magnetic Suppose that HdR field on X which extends smoothly to a 2-form on X. We say that B is trapping if 1) either B does not vanish identically on M , or

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2) B vanishes on M but defines a non-integral cohomology class [2πB] inside 2 the relative cohomology group HdR (X, M ). Otherwise, we say that B is non-trapping. This terminology is motivated by the spectral consequences a) and c) of Theorem 1.2. The definition can be generalized to the case where M is disconnected (Definition 7.1). The condition of B being trapping can be expressed in terms of any vector potential A (see 1 Definition 3.2 and Lemma 7.2). When HdR (X) = 0, the trapping condition makes sense only for vector potentials, see Section 3.2 and Theorem 1.3. Let us fix some notation. Given two Hilbert spaces H and K , we denote by B(H , K ) and K(H , K ) the bounded and compact operators acting from H to K , respectively. Given s ≥ 0, let Ls be the domain of Ls equipped with the graph norm. We set L−s := Ls∗ where the adjoint space is defined so that Ls ⊂ L2 (X, gp ) ⊂ Ls∗ , using the Riesz lemma. Given a subset I of R, let I± be the set of complex numbers x ± iy, where x ∈ I and y > 0. For simplicity, in this introduction we state our result only for the unperturbed metric (1.1). Theorem 1.2. Let 1 ≥ p > 0, gp the metric given by (1.1). Suppose that H1 (X, Z) = 0 and that M is connected. Let B be a magnetic field which extends smoothly to X. If B is trapping then: a) The spectrum of ΔB is purely discrete. b) The asymptotic of its eigenvalues is given by ⎧ n/2 ⎪ for 1/n < p , ⎨C1 λ NB,p (λ) ≈ C2 λn/2 log λ for p = 1/n , (1.6) ⎪ ⎩ 1/2p for 0 < p < 1/n C3 λ in the limit λ → ∞, where C3 is given in Theorem 4.2, and C1 =

Vol(X, gp ) Vol(S n−1 ) , n(2π)n

C2 =

Vol(M, h) Vol(S n−1 ) . 2(2π)n

(1.7)

If B c) d) e)

is non-trapping with compact support in X then The essential spectrum of ΔB is [κ(p), ∞). The singular continuous spectrum of ΔB is empty. The eigenvalues of ΔB are of finite multiplicity and can accumulate only in {κ(p)}. f) Let J a compact interval such that J ∩ ({κ(p)} ∪ σpp (H)) = ∅. Then, for all s ∈]1/2, 3/2[ and all A such that dA = B, there is c such that (ΔA − z1 )−1 − (ΔA − z2 )−1 B(Ls ,L−s ) ≤ cz1 − z2 s−1/2 , for all z1 , z2 ∈ J± .

The statements a) and b) follow from general results from [38]. This part relies on the Melrose calculus of cusp pseudodifferential operators (see, e.g., [35]) and is proved in Theorem 4.2 for the perturbed metric (2.9). We start from the basic observation that for smooth vector potentials, the magnetic Laplacian belongs to

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the cusp calculus with positive weights. For this part, we can treat the metric (2.9) which is quasi-isometric (but not necessarily asymptotically equivalent) to (1.1). Moreover, the finite multiplicity of the point spectrum (which is possibly not locally finite) in e) follows from Appendix B for this class of metrics. The statement c) follows directly from the analysis of the free case in Section 5. The perturbation of the metric is considered in Proposition 6.2 and relies on general results on stability of the essential spectrum shown in [15]. The points d), f) and e) rely on the use of an optimal version of Mourre theory, see [2]. They are developed in Theorem 6.4 for perturbations satisfying (1.5). Scattering theory under short-range perturbation of a potential and of a magnetic field is also considered. The condition of being trapping (resp. non-trapping) is discussed in Section 7.1 and is equivalent to having empty (resp. non-empty) essential spectrum in the complete case. The terminology arises from the dynamical consequences of this theorem and should not be confused with the classical terminology. Indeed, when B is trapping, the spectrum of ΔB is purely discrete and for all non-zero T φ in L2 (X), there is χ ∈ Cc∞ (X) such that 1/T 0 χeitΔB φ2 dt tends to a nonzero constant as T tends to ±∞. On the other hand, taking J as in f), for all χ ∈ Cc∞ (X) one gets that χeitΔB EJ (ΔA )φ tends to zero, when B is non-trapping and with compact support. 1 (M ) = 0 (take M = S 1 for instance), there exist some trapping magIf HdR netic fields with compact support. We construct an explicit example in Proposition 7.3. We are able to construct some examples in dimension 2 and higher than 4 but there are topological obstructions in dimension 3, see Section 7.1. As pointed out above regarding the Euclidean case, the fact that a magnetic field with compact support can turn off the essential spectrum and even a situation of limiting absorption principle is somehow unexpected and should be understood as a strong long-range effect. We discuss other interesting phenomena in Section 7.2. Consider M = S 1 and take a trapping magnetic field B with compact support and a coupling constant g ∈ R. Now remark that ΔgB is non-trapping if and only if g belongs to the discrete / cB Z and p ≥ 1/n, the spectrum of group cB Z, for a certain cB = 0. When g ∈ ΔgB is discrete and the eigenvalue asymptotics do not depend either on B or on g. It would be very interesting to know whether the asymptotics of embedded eigenvalues, or more likely of resonances, remain the same when g ∈ cB Z, (see [6] for the case g = 0). It would be also interesting to study the inverse spectral problem and ask if the magnetic field could be recovered from the knowledge of the whole spectrum, since the first term in the asymptotics of eigenvalues does not feel it. 1 (X) = 0. In quanAssume now that gauge invariance does not hold, i.e., HdR tum mechanics, it is known that the choice of a vector potential has a physical meaning. This is known as the Aharonov–Bohm effect [1]. Two choices of magnetic potential may lead to in-equivalent magnetic Laplacians. In R2 with a bounded

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obstacle, this phenomenon can be seen through a difference of wave phase arising from two non-homotopic paths that circumvent the obstacle. Some long-range effect appears, for instance in the scattering matrix like in [42–44], in an inversescattering problem [40, 48] or in the semi-classical regime [3]. See also [22] for the influence of the obstacle on the bottom of the spectrum. In all of these cases, the essential spectrum remains the same. In Section 7.3, we discuss the Aharonov–Bohm effect in our setting. In light of Theorem 1.2, one expects a drastic effect. We show that the choice of a vector potential can indeed have a significant spectral consequence. For one choice of vector potential, the essential spectrum could be empty and for another choice it could be a half-line. This phenomenon is generic for hyperbolic surfaces of finite volume, and also appears for hyperbolic 3-manifolds. We focus the presentation on magnetic fields B with compact support. We say that a smooth vector potential A (i.e., a smooth 1-form on X) is trapping if ΔA has compact resolvent, and nontrapping otherwise. By Theorems 4.2 and 6.4, for p ≤ 1, this is equivalent to Definition 3.2. It also follows that when the metric is of type (1.1), A is trapping if and only if a)–b) of Theorem 1.2 hold for ΔA , while A is non-trapping if and only if ΔA satisfies c)–f) of Theorem 1.2. Theorem 1.3. Let X be a complete oriented hyperbolic surface of finite volume and B a smooth magnetic field on the compactification X. • If X has at least 2 cusps, then for all B there exists both trapping and nontrapping vector potentials A such that B = dA. • If X has precisely 1 cusp, choose B = dA = dA where A, A are smooth vector potentials for B on X. Then B ∈ 2πZ . A is trapping ⇐⇒ A is trapping ⇐⇒ X

This follows from Corollary 8.1. More general statements are valid also in dimension 3, see Section 8. This implies on one hand that for a choice of A, as one has the points c), d) and e), a particle located at a scattering energy escapes from any compact set; on the other hand taking a trapping choice, the particle will behave like an eigenfunction and will remain bounded. It is interesting that the dimension 3 is exceptional in the Euclidean case [47] and that we are able to construct examples of such a behavior in any dimension. In the first appendix, we discuss the key notion of C 1 regularity for the Mourre theory and make it suitable to the manifold context and for our choice of conjugate operator. As pointed in [13], this is a key hypothesis in the Mourre theory in order to apply the Virial theorem and deduce the local finiteness of the embedded eigenvalues. In the second appendix, we recall that (cusp) elliptic, not necessarily fully elliptic, cusp operators have L2 eigenvalues of finite multiplicity. Finally, in the third appendix, we give a criteria of stability of the essential spectrum, by cutting a part of the space, encompassing incomplete manifolds. Some of the

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results concerning the essential spectrum and the asymptotic of eigenvalues were already present in the unpublished preprint [18].

2. Cusp geometry 2.1. Definitions This section follows closely [38], see also [35]. Let X be a smooth n-dimensional compact manifold with closed boundary M , and x : X → [0, ∞) a boundarydefining function. A cusp metric on X is a complete Riemannian metric g0 on X := X \ M which in local coordinates near the boundary takes the form g0 = a00 (x, y)

n−1 n−1

dx2 dx + a (x, y) dy + aij (x, y)dyi dyj 0j j 4 2 x x j=1 i,j=1

(2.8)

such that the matrix (aαβ ) is smooth and non-degenerate down to x = 0. For example, if a00 = 1, a0j = 0 and aij is independent of x, we get a product metric near M . If we set y = 1/x, a cusp metric is nothing but a quasi-isometric deformation of a cylindrical metric, with an asymptotic expansion for the coefficients in powers of y −1 . We will focus on the conformally cusp metric gp := x2p g0 ,

(2.9)

where p > 0. Note that (1.1) is a particular case of such metric. Let I ⊂ C ∞ (X) be the principal ideal generated by the function x. Recall [35] that a cusp vector field is a smooth vector field V on X such that dx(V ) ∈ I 2 . The space of cusp vector fields forms a Lie subalgebra c V of the Lie algebra V of smooth vector fields on X. In fact, there exists a natural vector bundle c T X over X whose space of smooth sections is c V, and a natural map c T X → T X which induces the inclusion c V → V. Let E, F → X be smooth vector bundles. The space of cusp differential operators Diff c (X, E, F ) is the space of those differential operators which in local trivializations can be written as composition of cusp vector fields and smooth bundle morphisms down to x = 0. The normal operator of P ∈ Diff c (X, E, F ) is the family of operators defined by R ξ → N (P )(ξ) := eiξ/x P e−iξ/x ∈ Diff M, E|M , F|M . |x=0

Example 2.1. N (x2 ∂x )(ξ) = iξ. Note that ker N = I · Diff c , which we denote again by I. The normal operator map is linear and multiplicative. It is also invariant under the conjugation by powers of x. Namely, if P ∈ Diff c and s ∈ C then xs P x−s ∈ Diff c and N (xs P x−s ) = N (P ). Concerning taking the (formal) adjoint, one needs to specify the volume form on the boundary.

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Lemma 2.2. Let P ∈ Diff c (X, E, F ) be a cusp operator and P ∗ its adjoint with respect to g0 . Then N (P ∗ )(ξ) is the adjoint of N (P )(ξ) with respect to the metric on E|M , F|M induced by restriction, for the volume form a0 1/2 volh0 , where the metric h0 on M is defined from rewriting g0 as in (6.41). The principal symbol of a cusp operator on X extends as a map on the cusp cotangent bundle down to x = 0. This implies that a cusp operator of positive order cannot be elliptic at x = 0 in the usual sense. A cusp operator is called cusp-elliptic if its principal symbol is invertible on c T ∗ X \ {0} down to x = 0. Definition 2.3. A cusp operator is called fully elliptic if it is cusp-elliptic and if its normal operator is invertible for all values of ξ ∈ R. An operator H ∈ x−l Diff kc (X, E, F ) is called a cusp differential operator of type (k, l). Fix a product decomposition of X near M , compatible with the boundarydefining function x. This gives a splitting of the cusp cotangent bundle on X in a neighborhood of M : c ∗ T X T ∗ M ⊕ x−2 dx . (2.10) Lemma 2.4. The de Rham differential d : C ∞ (X) → C ∞ (X, T ∗ X) restricts to a cusp differential operator d : C ∞ (X) → C ∞ (X, c T ∗ X). Its normal operator in the decomposition (2.10) is M d N (d)(ξ) = iξ where dM is the partial de Rham differential in the M factor of the product decomposition. Proof. Let ω ∈ C ∞ (X) and decompose dω according to (2.10): dx . x2 Since dM commutes with x, it follows from the definition that N (dM ) = dM . The result follows using Example 2.1. dω = dM ω + ∂x (ω)dx = dM ω + x2 ∂x (ω)

2.2. Relative de Rham cohomology Recall [4] that the cohomology of X and the relative cohomology groups of (X, M ) (with real coefficients) can be computed using smooth differential forms as follows let Λ∗ (X) denote the space of forms smooth on X down to the boundary. Let Λ∗ (X, M ) denote the subspace of those forms whose pull-back to M vanishes. These spaces form complexes for the de Rham differential (because d commutes with pull-back to M ) and their quotient is the de Rham complex of M : 0 → Λ∗ (X, M ) → Λ∗ (X) → Λ∗ (M ) → 0 . The induced long exact sequence in cohomology is just the long exact sequence of the pair (X, M ).

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2.3. Cusp de Rham cohomology Notice that d preserves the space of cusp differential forms. Indeed, since d is a derivation and using Lemma 2.4, it suffices to check this property for a set of local generators of C ∞ (X, c T ∗ X). Choose local coordinates (yj ) on M and take as generators x−2 dx and dyj , which are closed. Let cH ∗ (X) denote the cohomology of the complex of cusp differential forms ∞ (C (X, Λ∗ (c T X)), d) with respect to the de Rham differential. Proposition 2.5. cH k (X) = H k (X) ⊕ H k−1 (M )2 . Proof. The short exact sequence of de Rham complexes 0 → Λ∗ (T X) → Λ∗ (c T X) → Λ∗−1 (T M )2 → 0 , where the second map is given by Λk (c T X) ω → (x2 ∂x ω)x=0 , ∂x (x2 ∂x ω) x=0 ,

(2.11)

gives rise to a long exact sequence in cohomology. Now the composition Λ∗ (T X) → Λ∗ (c T X) → Λ∗ (T X) is a quasi-isomorphism, since de Rham cohomology can be computed either with smooth forms on X, or with smooth forms on X. Thus in cohomology the map induced from Λ∗ (T X) → Λ∗ (c T X) is injective.

3. The magnetic Laplacian 3.1. The magnetic Laplacian on a Riemannian manifold A magnetic field B on the Riemannian manifold (X, g) is an exact real-valued 2-form. A vector potential A associated to B is a 1-form such that dA = B. We form the magnetic Laplacian acting on C ∞ (X): ΔA := d∗A dA . This formula makes sense for complex-valued 1-forms A. Note that when A is real, dA is a metric connection on the trivial bundle C with the canonical metric, and ΔA is the connection Laplacian. If we alter A by adding to it a real exact form, say A = A + df , the resulting magnetic Laplacian satisfies ΔA = e−if ΔA eif 1 so it is unitarily equivalent to ΔA in L2 (X, g). Therefore if HdR (X) = 0 (for instance if π1 (X) is finite; see [4]) then ΔA depends, up to unitary equivalence, only on the magnetic field B. This property is called gauge invariance. For a more refined analysis of gauge invariance, see [19]. One usually encounters gauge invariance as a consequence of 1-connectedness (i.e., π1 = 0). But in dimensions at least 4, every finitely presented group (in particular, every finite group) can be realized as π1 of a compact manifold. Thus

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the hypothesis π1 = 0 is unnecessarily restrictive, it is enough to assume that its abelianisation is finite. While the properties of ΔA in Rn with the flat metric are quite well understood, the (absence of) essential spectrum of magnetic Laplacians on other manifolds has not been much studied so far. One exception is the case of bounded geometry, studied in [26]. However our manifolds are not of bounded geometry because the injectivity radius tends to 0 at infinity. 3.2. Magnetic fields and cohomology Recall that A is a (smooth) cusp 1-form (not to be confused with the notion of cusp form from automorphic form theory) on X if A ∈ C ∞ (X, T ∗ X) is a real-valued 1-form satisfying near ∂X dx (3.12) A = ϕ(x) 2 + θ(x) x where ϕ ∈ C ∞ (X) and θ ∈ C ∞ ([0, ε) × M, Λ1 (M )), or equivalently A is a smooth section in c T ∗ X over X. Proposition 3.1. Let B be a cusp 2-form. Suppose that B is exact on X, and its image by the map (2.11) is exact on M . Then there exists a smooth cusp 1-form A on X such that dA = B. Proof. Note that B is exact as a form on X, so dB = 0 on X. By continuity, dB = 0 on X (in the sense of cusp forms) so B defines a cusp cohomology 2-class. By hypothesis, this class maps to 0 by restriction to X. Now the pull-back of B to the level surfaces {x = ε} is closed; by continuity, the image of B through the map (2.11) is closed on M . Assuming that this image is exact, it follows from Proposition 2.5 that B is exact as a cusp form. By Lemmata 2.2 and 2.4, ΔA is a cusp differential operator of order (2p, 2). Definition 3.2. Let A be a (complex-valued) cusp vector potential. Given a connected component M0 of M , we say A is a trapping vector potential on M0 if • either the restriction ϕ0 := ϕ(0) is not constant on M0 , • or θ0 := θ(0) is not closed on M0 , 1 • or the cohomology class [θ0 |M0 ] ∈ HdR (M0 ) does not belong to the image of 1 2πH 1 (M0 ; Z) → H 1 (M0 , C) HdR (M0 ) ⊗ C

and non-trapping on M0 otherwise. We say that A is trapping if it is trapping on each connected component of M . The vector potential is said to be non-trapping if it is non-trapping on at least one connected component of M . If A is non-trapping on all connected component of M , we say that it is maximal non-trapping. Remark 3.3. The trapping notion can be expressed solely in terms of the magnetic field B = dA when H 1 (X) = 0, see Lemma 7.2.

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We comment briefly the terminology. When A is real, constant in x near M , the multiplicity of the absolutely continuous part of the spectrum of ΔA will be given by the number of connected component of M on which A is non-trapping. Hence, taking A maximal non-trapping maximizes the multiplicity of this part of the spectrum. We refer to [4] for an exposition of cohomology with integer coefficients. The trapping property is determined only by the asymptotic behavior of A. More precisely, if A is also of the form (3.12) with ϕ(0) = 0 and θ(0) = 0 then A is a (non-)trapping vector potential if and only if A + A is. The term “trapping” is motivated by dynamical consequences of Theorems 4.1 and 6.4 and has nothing to do with the classical trapping condition. This terminology is also supported by the examples given in Section 7. For a trapping vector potential, x2p ΔA is a fully-elliptic cusp operator. In turn, this implies that ΔA has empty essential spectrum so from a dynamical point of view, a particle can not diffuse, in other words it is trapped in the interior of X. Indeed, given a state φ ∈ L2 (X), there exists χ (the characteristic function T of a compact subset of X) such that 1/T 0 χeitΔA φ2 dt tends to a positive constant as T goes to infinity. On the other hand, if A is a non-trapping vector potential, then ΔA is not Fredholm between the appropriate cusp Sobolev spaces. If the metric is an exact cusp metric and complete, we show that ΔA has nonempty essential spectrum also as an unbounded operator in L2 , given by [κ(p), ∞) by Proposition 6.2. We go even further and under some condition of decay of ϕ and θ at infinity, we show that there is no singular continuous spectrum for the magnetic Laplacian and that the eigenvalues of R \ {κ(p)} are of finite multiplicity and can accumulate only in {κ(p)}. Therefore given a state φ which is not an eigenvalue of ΔA , one obtains that for all χ, χeitΔA φ tends to 0 as t → ∞. When M is connected, the class of non-trapping vector potentials is a group under addition but that of trapping vector potential is not. When M is disconnected, none of these classes is closed under addition. Directly from the definition, we get however: Remark 3.4. Let A be a maximal non-trapping vector potential and let A be a 1-form smooth up to the boundary. Then A is trapping if and only if A + A is. Let B be a smooth magnetic field on X (i.e., a 2-form) whose pull-back to M vanishes. Since B is exact, it is also closed, thus it defines a relative de Rham class as in Subsection 2.2. If this class vanishes, we claim that there exists a vector potential A for B which is maximal non-trapping. Indeed, let A ∈ Λ1 (X, M ) be any (relative) primitive of B. Then A is clearly a cusp form, the singular term φ(0) vanishes, and the pull-back of A to each boundary component vanishes by definition, in particular it defines the null 1-cohomology class. From Remark 3.4 we get

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Corollary 3.5. Let B be a cusp magnetic field. Let B be a smooth magnetic field on X which vanishes on the boundary and which defines the zero relative cohomology class in H 2 (X, M ). Then B admits (non-)trapping vector potentials if and only if B + B does. Note that when H 1 (X) = 0, a given magnetic field may admit both trapping and non-trapping vector potentials. See Theorem 1.3 and Sections 7.3 and 8.

4. The trapping case 4.1. The absence of essential spectrum In this section, given a smooth cusp 1-form, we discuss the link between its behavior at infinity and its trapping properties. Theorem 4.1. Let p > 0, gp a metric on X given by (2.9) near ∂X and A a smooth cusp 1-form given by (3.12). Then ΔA is a weighted cusp differential operator of order (2p, 2). If A is trapping then ΔA is essentially self-adjoint on Cc∞ (X), it has purely discrete spectrum and its domain is x2p H 2 (X, gp ). If p ≤ 1 then gp is complete so ΔA is essentially self-adjoint [46]. This fact remains true for a trapping A in the incomplete case, i.e., p > 1. Proof. Using Lemma 2.4, we get

dM + iθ0 N (dA )(ξ) = . i(ξ + ϕ0 )

Suppose that x2p ΔA is not fully elliptic, so there exists ξ ∈ R and 0 = u ∈ ker(N (x2p ΔA )(ξ)). By elliptic regularity, u is smooth. We replace M by one of its connected components on which u does not vanish identically, so we can suppose that M is connected. Using Lemma 2.2, by integration by parts with respect to the volume form a0 1/2 dh0 on M and the metric h0 on Λ1 (M ), we see that u ∈ ker(N (ΔA )(ξ)) implies u ∈ ker(N (dA )(ξ)). Then (ξ + ϕ0 )u = 0

and (dM + iθ0 )u = 0 ,

(4.13)

so u is a global parallel section in the trivial bundle C over M , with respect to the connection dM + iθ0 . This implies 0 = (dM )2 u = dM (−iuθ0 ) = −i(dM u) ∧ θ0 − iudM θ0 = −iudM θ0 . By uniqueness of solutions of ordinary differential equations, u is never 0, so dM θ0 = 0. Furthermore, from (4.13), we see that ϕ0 equals the constant function −ξ. It remains to prove the assertion about the cohomology class [θ0 ]. ˜ be the universal cover of M . Denote by u ˜. Let M ˜, θ˜0 the lifts of u, θ0 to M M The equation (d + iθ0 )u = 0 lifts to ˜ (dM + iθ˜0 )˜ u = 0.

(4.14)

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˜ , hence it is exact The 1-form θ˜0 is closed on the simply connected manifold M ˜ , C) = H1 (M ˜ , C) = H1 (M ˜ ; Z) ⊗ C, (by the universal coefficients formula, H 1 (M ˜ ˜ ˜ ) be and H1 (M ; Z) vanishes as it is the abelianisation of π1 (M )). Let v ∈ C ∞ (M ˜ M −iv ˜ ˜ ˜ = Ce for some constant a primitive of θ0 , i.e., d v = θ0 . Then, from (4.14), u C = 0. ˜ via deck transformaThe fundamental group π1 (M ) acts to the right on M tions. The condition that u˜ be the lift of u from M is the invariance under the action of π1 (M ), in other words u ˜(y) = u ˜ y[γ] for all closed loops γ in M . This is obviously equivalent to ˜. v y[γ] − v(y) ∈ 2πZ , ∀y ∈ M Let γ˜ be the lift of γ starting in y. Then ˜ θ˜0 = θ0 . v y[γ] − v(y) = dM v = γ ˜

γ ˜

γ

Thus the solution u ˜ is π1 (M )-invariant if and only if the cocycle θ0 evaluates to an integer multiple of 2π on each closed loop γ. These loops span H1 (M ; Z), so [θ0 ] lives in the image of H 1 (M ; Z) inside H 1 (M, C) = Hom(H1 (M ; Z), C). Therefore the solution u must be identically 0 unless ϕ0 is constant, θ0 is closed and [θ0 ] ∈ 2πH 1 (M ; Z). Conversely, if ϕ0 is constant, θ0 is closed and [θ0 ] ∈ 2πH 1 (M ; Z) then u˜ = −iv e as above is π1 (M )-invariant, so it is the lift of some u ∈ C ∞ (M ) which belongs to ker(N (x2p ΔA )(ξ)) for −ξ equal to the constant value of ϕ0 . The conclusion of the theorem is now a consequence of general properties of the cusp calculus [38, Theorem 17]. Namely, since ΔA is fully elliptic, there exists an inverse in x2p Ψ−2 c (X) (a micro-localized version of Diff c (X)) modulo compact operators. If p > 0, this pseudo-inverse is itself compact. The operators in the cusp calculus act by closure on a scale of Sobolev spaces. It follows easily that for p ≥ 0, a symmetric fully elliptic cusp operator in x2p Ψ2c (X) is essentially self-adjoint, with domain x2p Hc2 (X). Thus ΔA is self adjoint with compact inverse modulo compact operators, which shows that the spectrum is purely discrete. The cusp calculus [35] is a particular instance of Melrose’s program of microlocalizing boundary fibration structures. It is a special case of the fibered-cusp calculus [33] and can be obtained using the groupoid techniques of [29]. 4.2. Eigenvalue asymptotics for trapping magnetic Laplacians If A is a trapping vector potential, the associated magnetic Laplacian has purely discrete spectrum. In this case we can give the first term in the eigenvalue growth law.

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Theorem 4.2. Let p > 0, gp a metric on X given by (2.9) near ∂X and A ∈ C ∞ (X, T ∗ X) a complex-valued trapping vector potential in the sense of Definition 3.2. Then the eigenvalue counting function of ΔA satisfies ⎧ n/2 ⎪ for 1/n < p < ∞ , ⎨C1 λ n/2 NA,p (λ) ≈ C2 λ (4.15) log λ for p = 1/n , ⎪ ⎩ C3 λ1/2p for 0 < p < 1/n in the limit λ → ∞, where C1 =

Vol(X, gp ) Vol(S n−1 ) , n(2π)n

Vol(M, a0 1/2 h0 ) Vol(S n−1 ) C2 = . 2(2π)n

(4.16)

If moreover we assume that g0 is an exact cusp metric, then Γ 1−p 2p 1 C3 = √ ζ ΔhA0 , − 1 , 1 p 2 πΓ 2p where ΔhA0 is the magnetic Laplacian on M with potential A|M with respect to the metric h0 on M defined in Section 2. We stress that the constants C1 and C2 do not depend on the choice of A or B, but only on the metric. This fact provides some very interesting coupling constant effect, see Section 7.2. Note also that the hypotheses of the theorem are independent of the choice of the vector potential A inside the class of cusp 1-forms. Indeed, assume that A = A + dw for some w ∈ C ∞ (X) is again a cusp 1-form. Then dw must be itself a cusp form, so dx dw = x2 ∂x w 2 + dM w ∈ C ∞ (X, c T ∗ X) . x Write this as dw = ϕ dx + θ . For each x > 0 the form θx is exact. By the Hodge x x2 decomposition theorem, the space of exact forms on M is closed, so the limit θ0 is also exact. Now dw is an exact cusp form, in particular it is closed. This implies that x2 ∂x θ = dM ϕ . Setting x = 0 we deduce dM (ϕ )x=0 = 0, or equivalently ϕ|x=0 is constant. Hence the conditions from Theorem 4.2 on the vector potential are satisfied simultaneously by A and A . Proof. From Theorem 4.1, the operator x2p ΔA is fully elliptic when A is trapping. The result follows directly from [38, Theorem 17]. Let us explain the idea: the complex powers Δ−s A belong to the cusp calculus, and are of trace class for sufficiently large real part of s. The trace of the complex powers is holomorphic for such s and extends meromorphically to C with two families of simple poles, coming from the principal symbol and from the boundary. The leading pole governs eigenvalue

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asymptotics, by the Delange theorem. In case the leading pole is double (by the superposition of the two families of poles), we get the logarithmic growth law. The explicit computation of the constants (given by [38, Prop. 14 and Lemma 16]) is straightforward.

5. Analysis of the free case for non-trapping potentials We have shown in Theorems 4.1 and 4.2 that the essential spectrum of the magnetic Laplacian ΔA is empty when A is trapping (see Definition 3.2), and we have computed the asymptotics of the eigenvalues. We now consider the case of a nontrapping vector potential A. One can guess that in this last case, the essential spectrum is not empty when the metric is complete. In this section we concentrate on the unperturbed metric (5.17) with the model non-trapping potential (5.18). We take advantage of the decomposition in low- and high-energy functions from Section 5.1. The computation of the essential spectrum is based on Proposition 5.1 and on the diagonalization of the magnetic Laplacian performed in Section 5.2. In Section 5.3, we construct a local conjugate operator and state the Mourre estimate (Theorem 5.6). In Section 5.4 we introduce the classes of perturbation under which we later give a limiting absorption principle. The perturbation theory is developed in Section 6. We refer to Proposition 6.2 for the question of the essential spectrum and to Theorem 6.4 for its refined analysis under short/long range perturbations. We localize the computation on the end X := (0, ε) × M ⊂ X. We assume that on X := (0, ε) × M ⊂ X we have 2 dx gp = x2p + h (5.17) 0 , x4 Af = Cdx/x2 + θ0 ,

(5.18)

where C is a constant, θ0 is closed and independent of x, and the cohomology class [θ0 ] ∈ H 1 (M ) is an integer multiple of 2π. By a change of gauge, one may assume that C = 0. Indeed, it is enough to subtract from Af the exact 1-form d(−C/x). 5.1. The high and low energy functions decomposition Set dθ˜ := dM + iθ0 ∧. We now decompose the L2 space as follows: L2 (X ) = Hl ⊕ Hh ,

(5.19)

where Hl := K ⊗ ker(dθ˜) with K := L ((0, ε), x dx), and where Hh = ˆ ker(dθ˜)⊥ . We do not emphasize the dependence on ε for these spaces as the K⊗ properties we are studying are independent of ε. The subscripts l, h stand for low and high energy, respectively. The importance of the low energy functions space Hl is underlined by the following: 2

np−2

Proposition 5.1. The magnetic Laplacian ΔAf stabilizes the decomposition (5.19) of L2 (X ). Let ΔlAf and ΔhAf be the Friedrichs extensions of the restrictions of ΔAf

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to smooth compactly supported functions in Hl , respectively in Hh . Then ΔhAf has compact resolvent, while ΔlAf = D∗ D + c20 x2−2p ⊗ 1 , where c0 := ((2 − n)p − 1)/2, and D := x2−p ∂x − c0 x1−p acts in Kk,ε . By Proposition C.3, the essential spectrum is prescribed by the space of low energy functions. This kind of decomposition can be found in various places in the literature, see for instance [32] for applications to finite-volume negatively-curved manifolds. Remark 5.2. The dimension of the kernel of dθ˜ equals the number of connected components of M on which Af non-trapping. Indeed, take such a connected component M1 and let v be a primitive of θ0 on its universal cover. On different sheets of the cover, v changes by 2πZ so eiv is a well-defined function on M1 which spans ker(dθ˜). This decomposition is also valid in the trapping case, only that the low energy functions space is then 0. Using Proposition C.3, we obtain the emptiness of the essential spectrum for the unperturbed metric and along the way a special case of Theorem 4.1 (we do not recover the full result in this way, since the metric (2.9) can not be reached perturbatively from the metric (5.17)). Proof. We decompose the space of 1-forms as the direct sum (2.10). Recall that δM is the adjoint of dM with respect to h0 . We compute M d + iθ0 ∧ dAf = x2 ∂x (5.20) d∗Af = x−np δM − iθ0 −x2 ∂x x(n−2)p ΔAf = x−2p dθ˜∗ dθ˜ − (x2 ∂x )2 − (n − 2)px(x2 ∂x )0 . On the Riemannian manifold (M, h), dθ˜∗ dθ˜ is non-negative with discrete spectrum. Since −(x2 ∂x )2 − (n − 2)px(x2 ∂x ) = (x2 ∂x )∗ (x2 ∂x ) is non-negative, one has that ΔhAf ≥ ε−2p λ1 , where λ1 is the first non-zero eigenvalue of dθ˜∗ dθ˜. By Proposition C.3, the essential spectrum is independent of ε. By letting ε → 0 we see that it is empty; thus ΔhAf has compact resolvent. The assertion on ΔlAf is a straightforward computation. 5.2. Diagonalization of the free magnetic Laplacian In order to analyze the spectral properties of ΔAf on (X, gp ), where Af is given by (5.18) with C = 0 and gp by (5.17), we go into some “Euclidean variables”. We concentrate on the complete case, i.e., p ≤ 1. We start with (5.19) and work on K . The first unitary transformation is L2 (xnp−2 dx) → L2 (xp−2 dx)

φ → x(n−1)p/2 φ .

Then we proceed with the change of variables z := L(x), where L is given by (6.44). Therefore, K is unitarily sent into L2 ((c, ∞), dz) for a certain c. We indicate operators and spaces obtained in the new variable with a subscript 0.

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Thanks to this transformation, we are pursuing our analysis on the manifold X0 = X endowed with the Riemannian metric dr2 + h ,

r → ∞,

(5.21)

X0

on the end = [1/2, ∞) × M . The subscript r stands for radial. The magnetic Laplacian is unitarily sent into an elliptic operator of order 2 denoted by Δ0 . On Cc∞ (X0 ), it acts by Δ0 := Qp ⊗ dθ˜∗ dθ˜ + (−∂r2 + Vp ) ⊗ 1 , ˆ 2 (M, h), where on the completed tensor product L2 ([1/2, ∞), dr)⊗L 2 e2r (n−1)/2 Vp (r) = and Q (r) = for 2p/(1−p) p c0 r−2 (1−p)r

(5.22)

p=1 p < 1.

Recall that c0 is defined in Proposition 5.1 and dθ˜ = dM + iθ0 ∧. We denote also by L0 the operator of multiplication corresponding to L, given by (6.44), in the new variable r. It is bounded from below by a positive constant, equals 1 on the compact part and on the trapping ends, and equals r → r on the non-trapping ends. Let H0s := D((1 + Δ0 )s/2 ) for s > 0. By identifying H0 with H0∗ by the Riesz isomorphism, by duality, we define H0s for s < 0 with H0−s . We need the next well-known fact. Lemma 5.3. For every γ ∈ Cc∞ (X), we have that γ : H s ⊂ H s for all s ∈ R. Proof. A computation gives that there is c such that (Δ0 + i)γϕ ≤ c(Δ0 + i)ϕ, for all ϕ ∈ Cc∞ (X0 ). Since Cc∞ (X0 ) is a core for Δ0 , we get the result for s = 2. By induction we get it for s ∈ 2N. Duality and interpolation give it for s ∈ R. 5.3. The local conjugate operator In this section, we construct some conjugate operators in order to establish a Mourre estimate. By mimicking the case of the Laplacian, see [10], one may use the following localization of the generator of dilations. Let ξ ∈ C ∞ ([1/2, ∞)) such that the support of ξ is contained in [2, ∞) and that ξ(r) = r for r ≥ 3 and let χ ˜ ∈ C ∞ ([1/2, ∞)) with support in [1, ∞), which equals 1 on [2, ∞). By abuse of ˜ ⊗ 1 ∈ C ∞ (X0 ) with the same symbol. Let χ := 1 − χ ˜ . On notation, we denote χ ∞ Cc (X0 ) we set: ˜, S∞ := − i(ξ∂r + ∂r ξ) ⊗ P0 χ (5.23) where P0 is the orthogonal projection onto ker(dθ˜). The presence of P0 comes from the decomposition in low and high energies. Remark 5.4. By considering a C0 -group associated to a vector field on R like in [2, Section 4.2], one shows that −i(ξ∂r + ∂r ξ) is essentially self-adjoint on Cc∞ (R). ˜ and then it is easy to This C0 -group acts trivially away from the support of χ ˜ )) ⊗ ker(d ˜), construct another C0 -group G0 which acts like the first on L2 (supp(χ θ 2 and trivially on the rest of L (X0 ). Let S∞ be the generator of G0 . Since G0 (t)

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leaves invariant Cc∞ (X0 ) the Nelson lemma implies that S∞ is the closure of S∞ , in other words S∞ is essentially self-adjoint on Cc∞ (X0 ). This kind of approach has been used in [5] for instance. Therefore, we denote below also by S∞ the self-adjoint closure of the operator defined by (5.23) on Cc∞ (X0 ). However this operator is not suitable for very singular perturbations like that of the metric considered in this paper. To solve this problem, one should consider a conjugate operator more “local in energy”. Concerning the Mourre estimate, as it is local in energy for the Laplacian, one needs only a conjugate operator which fits well on this level of energy. Considering singular perturbation theory, the presence of differentials in (5.23) is a serious obstruction; the idea is to replace the conjugate operator with a multiplication operator in the analysis of perturbations, therefore reducing the rˆ ole of derivatives within it. The approach has been used for Dirac operators for instance in [16] to treat very singular perturbations. The case of Schr¨ odinger operators is summarized in [2, Theorem 7.6.8]. We set: ˜, ˜ ΦR (−i∂r )ξ + ξΦR (−i∂r ) ⊗ P0 χ (5.24) SR := χ where ΦR (x) := Φ(x/R) for some Φ ∈ Cc∞ (R) satisfying Φ(x) = x for all x ∈ [−1, 1]. The operator ΦR (−i∂r ) is defined on R by F −1 ΦR F , where F is the unitary Fourier transform. Let us also denote by SR the closure of this operator. Unlike (5.23), SR does not stabilize Cc∞ (X0 ) because ΦR (−i∂r ) acts like a convolution with a function with non-compact support. This subspace is sent into χ ˜ S (R), where S (R) denotes the Schwartz space. To motivate the subscript R, note that SR tends strongly in the resolvent sense to S, as R goes to infinity. We give some properties of SR . The point (2) is essential to be able to replace SR by L0 in the theory of perturbations. The point (3) is convenient to be able to express a limiting absorption principle in terms of L0 , which is very explicit. Of course these two points are false for S∞ and this explains why we can go further in the perturbation theory compared to the standard approach. Lemma 5.5. Let SR denote the closure of the unbounded operator (5.24). 1. For all R ∈ [1, ∞], the operator SR is essentially self-adjoint on Cc∞ (X). 2 ∞ 2. For R finite, L−2 0 SR : Cc (X0 ) → D(Δ0 ) extends to a bounded operator in D(Δ0 ). 3. For R finite, D(Ls0 ) ⊂ D(|SR |s ) for all s ∈ [0, 2]. Proof. The case R = ∞ is discussed in Remark 5.4, so assume that R is finite. We compare SR with L0 , defined in Section 5.2, which is essentially self-adjoint on Cc∞ (X). Noting that it stabilizes the decomposition (5.19), we write also by L0,l its restriction to Hl , which is simply the multiplication by r. On Cc∞ (X0 ), ˜ 2ΦR (−i∂r )ξL−1 + ξ, ΦR (−i∂r ) L−1 ⊗ P0 χ ˜ L0 . SR = χ 0,l 0,l ∞ and using Lemma 5.11, we get Noting that ξL−1 0,l is bounded and that ξ ∈ L ∞ SR ϕ ≤ aL0 ϕ, for all ϕ ∈ Cc (X0 ).

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On the other hand, [SR , L0 ] is equal to the bounded operator χ ˜ ˜ ΦR (−i∂r ), L0,l ξL−1 [SR , L0 ] = χ 0,l ⊗ P0 L0 ˜ L−1 ˜. ⊗ P0 χ + L0 χ 0,l ξ ΦR (−i∂r ), L0,l This gives |SR ϕ, L0 ϕ − L0 ϕ, SR ϕ| ≤ bL0 ϕ2 , for all ϕ ∈ Cc∞ (X0 ). Finally, one uses [41, Theorem X.37] to conclude that SR is essentially self-adjoint. On Cc∞ (X0 ), we have 2 2 χ −1 χ χ −1 χ ⊗ P0 . L−2 (5.25) 0 SR = 2 L0,l ξΦR (−i∂r ) ⊗ P0 + L0,l ΦR (−i∂r ), ξ 1/2

All these terms are bounded in L2 (X0 ) by Lemma 5.11 and by density. We now 2 compute Δ0 L−2 0 SR . By Lemma 5.3, it is enough to show that ΦR (−i∂r ) and [ΦR (−i∂r ), ξ] stabilize the domain Δ in L2 (R). The first one commutes with Δ. For the second one, we compute on Cc∞ (R). Since [ΦR (−i∂r ), ξ] is bounded in L2 (R), it is enough to show that the commutator [Δ, [ΦR (−i∂r ), ξ]] is also bounded in L2 (R). By Jacobi’s identity, it is equal to [ΦR (−i∂r ), [Δ, ξ]] = [ΦR (−i∂r ), 2ξ ∂r + ξ ] = 2ΦR (−i∂r )∂r ξ − 2ΦR (−i∂r )ξ + 2ξ ΦR (−i∂r ) + ξ . This is a bounded operator in L2 (R) and we get point (2). 2 −2 We now note that (5.25) is bounded in L2 (X0 ). Then, since SR L0 is also 2 2 2 ∞ bounded, we get SR ϕ ≤ cL0 ϕ for all ϕ ∈ C (X0 ). Taking a Cauchy sequence, 2 we deduce D(L20 ) ⊂ D(SR ). An argument of interpolation gives point (3). The aim of this section is the following Mourre estimate. Theorem 5.6. Let R ∈ [1, ∞]. Then eitSR H02 ⊂ H02 and Δ0 ∈ C 2 (SR , H02 , H0 ). Given an interval J inside σess (Δ0 ), there exist εR > 0 and a compact operator KR such that EJ (Δ0 )[Δ0 , iSR ]EJ (Δ0 ) ≥ 4 inf(J ) − εR EJ (Δ0 ) + KR holds in the sense of forms, and such that εR tends to 0 as R goes to infinity. Proof. The regularity assumptions follow from Lemmata 5.9 and 5.10. The left hand side of (5.30) is the commutator [Δ0 , iSR ] in the sense of forms. It extends to a bounded operator in B(H02 , H0−2 ) since Cc∞ (X0 ) is a core for Δ0 . We can then apply the spectral measure and obtain the inequality using Lemma 5.8. Compared to the method from [10, Lemma 2.3] (for the case R = ∞), we have a relatively more direct proof based on Lemma 5.8. However this has no real impact on applications of the theory. We now go in a series of lemmata to prove this theorem. Given a commutator [A, B], we denote its closure by [A, B]0 . Lemma 5.7. For R = ∞, the commutators [Δ0 , iS∞ ]0 and [[Δ0 , iS∞ ], iS∞ ]0 belong to B(H02 , H0 ). For R finite, [Δ0 , iSR ]0 and [[Δ0 , iS∞ ], iSR ]0 belong to B(H0 ). Moreover, if p = 1, all higher commutators extend to bounded operators in B(H02 , H0 ) for R = ∞ and in B(H0 ) for R < ∞.

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Proof. Let ϕ ∈ Cc∞ (X0 ) such that ϕ = ϕR ⊗ ϕM where ϕM ∈ C ∞ (M ) and ϕR ∈ Cc∞ ([1/2, ∞)). Note that P0 ϕM ∈ C ∞ (M ) by the Hodge decomposition, since d2A = 0. Applying the brackets to ϕ, by a straightforward computation, we get ˜ ⊗ P0 . [Δ0 , iS∞ ] = − 4ξ ∂r2 + 4ξ ∂r + ξ − 2Vp χ (5.26) By linearity and density, we get [Δ0 , iS∞ ]ϕ ≤ C(Δ0 + i)ϕ for all ϕ ∈ Cc∞ (X0 ). Take now ϕ ∈ Cc∞ (X0 ), considering the support of the commutator, we get [Δ0 , iS∞ ]ϕ = [Δ0 , iS∞ ]Ξϕ ≤ C (Δ0 + i)ϕ ≤ C(Δ0 + i)ϕ + [Δ0 , Ξ]ϕ [1,∞)×M = 1. Therefore, since ∈ C ∞ (X) with support in X0 such that Ξ| where Ξ ∞ Cc (X0 ) is a core for Δ0 , we conclude [Δ0 , iS∞ ]0 ∈ B(H02 , H0 ). In the same way we compute [Δ0 , iS∞ ], iS∞ = − 16(ξ )2 − 8ξξ ∂r2 + (24ξ ξ − 8ξξ )∂r ˜ ⊗ P0 + 4(ξ )2 + 4ξ − 2ξ − 4Vp χ (5.27) and get [[Δ0 , iS∞ ], iS∞ ]0 ∈ B(H02 , H0 ). For p = 1, the boundedness of higher commutators follows easily by induction (Vp = 0 in this case). We compute next the commutators of Δ0 with SR . As above we compute for ϕ = ϕr ⊗ ϕM . For brevity, we write ΦR instead of ΦR (−i∂r ). As ΦR is not a local operator, we first note that the commutator [Δ0 , SR ] ˜ sends ϕr to Cc∞ (R) (note that could be taken in the operator sense. Indeed, χ [1/2, ∞) is injected in a canonical way into R), then ΦR ξ + ξΦR sends to the ˜ sends to χ ˜ S (R) which belongs to D(Δ0 ). Schwartz space S (R) and finally χ ˜ (ΦR ξ + ξΦR )χ ˜ ] ⊗ P0 . Against ϕr ⊗ ϕM , we have: We compute [∂r2 , χ ˜ ] = Ξ[∂r2 , χ ˜ ]Ξ = Ξ[∂r2 , ΦR r + rΦR ]Ξ + Ψcomp ˜ ΦR ξ + ξΦR χ ˜ ΦR ξ + ξΦR χ [∂r2 , χ ˜ + Ψcomp , ˜ ∂r ΦR χ = 4Ξ∂r ΦR Ξ + Ψcomp = 4χ

(5.28)

where Ψcomp denotes a pseudo-differential operator with compact support such that its support in position is in the interior of X0 . For p < 1, the potential part Vp arises. We treat its first commutator: ˜ p , 2ΦR r − iΦ ]Ξ ˜ + Ψcomp ˜ ] = Ξ[V ˜ ΦR ξ + ξΦR χ [Vp , χ R ˜ 2[Vp , ΦR ]ξ − i[Vp , ΦR ] χ ˜ + Ψcomp , =χ (5.29) where ΦR = ΦR (−i∂r ). Applying Lemma 5.11, we get that [Vp , ΦR ]ξ and [Vp , ΦR ] ˜ we get [Δ0 , iSR ]ϕ ≤ are bounded in L2 (R) also. Therefore, using like above Ξ, ∞ Cϕ for all ϕ ∈ Cc (X0 ). This implies that [Δ0 , iSR ]0 ∈ B(H0 ). ˜ ΦR ξ + ξΦR χ ˜ is given For higher commutators, the n-th commutator with χ nχ n n ˜ χ ˜ by 2 ∂r ΦR + Ψcomp . Note that ∂r ΦR is a compactly supported function of ∂r , so the contribution of this term is always bounded. Consider now the second commutator of Vp . As above, since we work up to ˜ χ ˜ and Ψcomp , it is enough to show that the next commutator defined on S (R) Ξ,

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extend to bounded operators in L2 (R). We treat only the most singular part of the second commutator: [Vp , ΦR ]r, ΦR r = [Vp , ΦR ], ΦR r r + [Vp , ΦR ][r, ΦR ]r = [Vp , ΦR ], ΦR r2 + ΦR [Vp , ΦR ], r r − i[Vp , ΦR ]ΦR r = [Vp , ΦR ], ΦR r2 − ΦR [r, ΦR ], Vp r − i[Vp , ΦR ][ΦR , r] − i[Vp , ΦR ]rΦR = [Vp , ΦR ], ΦR r2 + iΦR [ΦR , Vp ]r + [Vp , ΦR ]ΦR − i[Vp , ΦR ]rΦR . These terms extend to bounded operators by Lemma 5.11.

The following lemma is the key-stone for the Mourre estimate. Lemma 5.8. For all R ∈ [1, ∞], there exists K ∈ K(H02 , H0−2 ) and NR ∈ B(H02 , H0−2 ) such that Δ0 ϕ, iSR ϕ + iSR ϕ, Δ0 ϕ = 4 ϕ, Δ0 − inf σess (Δ0 ) ϕ + ϕ, NR ϕ + ϕ, Kϕ, for all ϕ ∈

Cc∞ (X0 )

(5.30)

and such that NR B(H 2 ,H −2 ) tends to 0 as R goes to infinity. 0

0

Proof. First note that the essential spectrum of Δ0 is [Vp (∞), ∞), by Proposition 6.2. We now act in three steps. Let Ξ be like in the proof of Lemma 5.7 and ˜ and Ξ have disjoint supports, one has for all R that let ϕ ∈ Cc∞ (X0 ). Since χ [Δ0 , iSR ]Ξϕ. ϕ, [Δ0 , iSR ]ϕ = Ξϕ, For the first step, we start with R = ∞. By (5.26) and since ξ = 1 on [2, ∞), the Rellich–Kondrakov lemma gives = Ξϕ, 4 Δ0 − Vp (∞) (1 ⊗ P0 )Ξϕ [Δ0 , iS∞ ]Ξϕ + ϕ, K1 ϕ (5.31) Ξϕ, for a certain K1 ∈ K(H02 , H0−2 ). Indeed, 1 − ξ , Vp − Vp (∞), ξ , ξ , Vp belong to K(H01 , H0 ) since they tend to 0 at infinity. We consider now R finite. We add (5.28) and (5.29). We have [Δ0 , iSR ]Ξϕ = Ξϕ, 4 Δ0 − Vp (∞) − TR (1 ⊗ P0 )Ξϕ Ξϕ, + ϕ, K2 ϕ (5.32) for a certain K2 = K2 (R) ∈ K(H02 , H0−2 ) and with TR = ∂r (∂r − ΦR (∂r )). The compactness of K2 follows by noticing that L−1 ∈ K(H02 , H0 ) and that 0 L0 [Vp , iSR ]0 ∈ B(H0 , H0 ), by Lemma 5.11. We control the size of TR by showing R (1⊗P0 )Ξ that ΞT B(H02 ,H0−2 ) tends to 0 as R goes to infinity. By Lemma 5.3, one belongs to B(H 2 , L2 (R)). It remains has that Ξ stabilizes H0±2 , therefore −∂r2 Ξ 0 2 −2 to note that (−∂r + i) TR tends to 0 in norm by functional calculus, as R goes to infinity. The second step is to control the high energy functions part. Consider the Friedrichs extension of Δ0 on H0,h := L2 ([1/2, ∞)) ⊗ P0⊥ L2 (M ). We have: Δ0 P0⊥ Ξϕ = (Δ0 P0⊥ + i)−1 (Δ0 P0⊥ + i)Ξϕ, = ϕ, K1 ϕ (5.33) Δ0 P0⊥ Ξϕ Ξϕ,

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˜ ∈ B(H02 , H0,h ) and where K1 ∈ K(H02 , H0−2 ). Indeed, note first that Δ0 P0⊥ χ ⊥ −1 that (Δ0 P0 + i) ∈ K(H0,h ), since Qp in (5.22) goes to infinity. Therefore the left hand side belongs K(H02 , H0 ) and the right hand side belongs to B(H02 , H0 ). The third step is to come back on the whole manifold. It is enough to note ˜ ] ∈ K(H01 , H0 ) and to add (5.31), (5.32) with (5.33). that [Δ0 , χ We now turn to the regularity assumptions. Lemma A.2 plays a central rˆ ole. Lemma 5.9. For R ∈ [1, ∞], one has Δ0 ∈ C 1 (SR ) and eitSR D(Δ0 ) ⊂ D(Δ0 ). Proof. We start by showing that Δ0 ∈ C 1 (SR ). We check the hypothesis of Lemma A.2. Let χn (r) := χ(r/n) and D = Cc∞ (X0 ). Remark that supp(χn ) ⊂ [n, 2n] (k) and that ξ χn tends strongly to 0 on L2 (R+ ), for any k ≥ 1. By the uniform boundedness principle, this implies that supn χn D(H) is finite. Remark 5.4 and Lemma 5.5 give that D is a core for SR . Assumption (1) is obvious, assumption (2) holds since (1 − χn ) has support in [2n, ∞) and assumption (3) follows from the fact that H is elliptic, so the resolvent of Δ0 sends D into C ∞ (X0 ). The point (A.6) follows from Lemma 5.8. We now show that (A.5) is true. ˜ φ+ χn χ ˜ φ. We ˜ φ = 2χn ∂r χ Let φ ∈ C ∞ (X0 )∩D(Δ0 ). We have [Δ0 , χn ]φ = [Δ0 , χn ]χ 2 ˜ φ + (4χξn + ξ χn )∂r P0 χ ˜ φ + (2ξ χn + ξ χn )P0 χ ˜φ have iS∞ [Δ0 , χn ]φ = 2ξ χn ∂r P0 χ ˜ φ+ ˜ (2ΦR (∂r )ξ + [ξ, ΦR (∂r )])(2χn ∂r P0 χ and, for a finite R, we get iSR [Δ0 , χn ]φ = χ χn P0 χ ˜ φ). Both terms are tending to 0 because of the previous remark, Lemma 5.3 and the fact that [ξ, ΦR (∂r )] is bounded by Lemma 5.11. From that, we can apply the lemma and obtain H ∈ C 1 (SR ). By Lemma 5.7, we have that [Δ0 , iSR ]0 ∈ B(H02 , H0 ) and [13, Lemma 2] gives that eitA H02 ⊂ H02 . The invariance of the domain under the group eitSR implies that eitSR H0s ⊂ for s ∈ [−2, 2] by duality and interpolation. This allows one to define the class C k (SR , H0s , H0−s ) for s ∈ [−2, 2], for instance; we recall that a self-adjoint operator H is in this class if t → eitSR He−itSR is strongly C k from H0s to H0−s . H0s

Lemma 5.10. Let R ∈ [1, ∞]. Then Δ0 belongs to C 2 (SR , H02 , H0 ) for p ≤ 1. Proof. From Lemma 5.7, the commutators with SR extend to bounded operators from H02 to H0 . We finally give an estimation of commutator that we have used above. Lemma 5.11. Let f ∈ C 0 (R) with polynomial growth, Φj ∈ Cc∞ (R) and g ∈ C k (R) with bounded derivatives. Let k ≥ 1. Assume that supt∈R,|s−t|≤1 |f (t)g (l) (s)| < ∞, for all 1 ≤ l ≤ k. Then the operator f [Φ1 (−i∂r ), [Φ2 (−i∂r ) . . . [Φk (−i∂r ), g] . . .], defined on Cc∞ (R), extends also to a bounded operator. Proof. Take k = 1. We denote with a hat the unital Fourier transform. We get 1 − t)f (t) g(s) − g(t) ϕ(s)ds , Φ(s f Φ(−i∂r ), g ϕ (t) = 2π

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for ϕ ∈ Cc∞ (R). In order to show that the L2 norm of the left hand side is uniformly bounded by f 2 , we separate the integral in |s − t| lower and bigger than one. We (k) (t), Φ is replaced by Φ when = (−1)k Φ start with the first part. Recalling tk Φ(t) we divide by |s − t| the term with g. Now since Φ ∈ L1 , supt∈R,|s−t|≤1 |f (t)(g(s) − g(t))/(s − t)| is finite, and the convolution by a L1 function is bounded in L2 , we control this part of the integral. We turn to the part |s − t| ≥ 1. Let R ∈ N such that |f (t)| ≤ C(1 + |t|R ). We let appear the Fourier transform of Φ(R+1) . Now to conclude, note that sups,t∈R |(g(s) − g(t))/(s − t)| is finite and that t → |f (t)/(s − t)R | is in L∞ uniformly in s. For higher k, one repeats the same decomposition and let appear the terms in the l-th derivative of g by regrouping terms. 5.4. A short-range and long-range class of perturbations In the early versions of Mourre theory, one asked [[H, SR ], SR ] to be H−bounded to obtain refined results of the resolvent like the limiting absorption principle and the Hölder regularity of the resolvent. In this section, we check the optimal class of regularity C 1,1 (SR ), for R finite. This is a weak version of the two-commutators hypothesis. We refer to [2] for definition and properties. This is the optimal class of operators which give a limiting absorption principle for H in some optimal Besov spaces associated to the conjugate operator SR . The operator Δ0 belongs to C 2 (SR ) by Lemma 5.10 and therefore also to 1,1 C (SR ). We now consider perturbations of Δ0 which are also in C 1,1 (SR ). We define two classes. Consider a symmetric differential operator T : D(Δ0 ) → D(Δ0 )∗ . Take θsr ∈ ∞ Cc ((0, ∞)) not identically 0; V is said to be short-range if ∞ θsr L0 T dr < ∞ (5.34) r 1 B(D(Δ0 ),D(Δ0 )∗ ) and to be long-range if ∞ L0 ˜ [T, L0 ]θlr L0 ˜ Ξ + Ξ[T, P0 ]L0 θlr r r 1 B(D(Δ0 ),D(Δ0 )∗ ) B(D(Δ0 ),D(Δ0 )∗ ) L0 ˜ dr ˜ 0 such that L1+ε 0 T B(D(Δ0),D(Δ0 )∗ ) < ∞

(5.36)

and the second one if Lε0 [T, L0 ]B(D(Δ0 ),D(Δ0 )∗ ) + L1+ε 0 Ξ[T, P0 ]ΞB(D(Δ0 ),D(Δ0 )∗ ) + L1+ε 0 Ξ[T, ∂r ]P0 ΞB(D(Δ0 ),D(Δ0 )∗ ) < ∞ (5.37)

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The condition with P0 essentially tells that the non-radial part of T is a short-range perturbation. This is why we will ask the long-range perturbation to be radial. To show that the first class is in C 1,1 (SR ) for R finite, one use [2, Theorem 7.5.8]. The hypotheses are satisfied thanks to Lemmata 5.5 and 5.14. Concerning the second class, one shows that [T, SR ] ∈ C 0,1 (SR ) by using [2, Proposition 7.5.7] (see the proof of [2, Proposition 7.6.8] for instance). We go back to the x coordinate. For G = D(Δ1/2 ), the short and long-range perturbation of the electric and magnetic perturbations are given by: ˜ ∞ < ∞ (respecLemma 5.12. Let V ∈ L∞ (X) and A˜ ∈ L∞ (X, T ∗ X). If L1+ε A 1+ε ∗ ˜ + (iA∧) ˜ ∗ dA ) (resp. V ) tively L V ∞ < ∞) then the perturbation (dAf (iA∧) f ˜ < ∞ and L1+ε x2−p ∂x A ˜ < is short-range in B(G , G ∗ ). If A˜ is radial, Lε A 2 2 1 ∞, where these norms are in B(L (X, g), L (X, Λ , g)) (respectively V radial, and L1+ε x2−p ∂x V ∞ < ∞) then the same perturbation is long-range in B(G , G ∗ ). Proof. We deal with the magnetic perturbation. Start with the short-range. We ˜ ∗ dA )f = dA f, L1+ε (iA∧)f ˜ have L1+ε (iA∧) and on the other hand, we have f 1+ε ∗ 1+ε ∗ −ε ε ˜ ˜ ˜ dA f . This is L dAf (iA∧)f, f = [L , dAf ]L L (iA∧)f, f + L1+ε (iA∧)f, 2 ∞ bounded by f + dA f uniformly in f ∈ Cc (X). We deal now with the long-range perturbation by checking (5.37). The condition with Lε is treated as above. In the variable of the free metric gp (5.17), ∂r is given by ∂L := x2−p ∂x − (n − 1)p/2 x1−p . We extend ∂L on 1-forms by setting ∂L = x2−p ∂x − (n + 1)p/2 x1−p . Note it is symmetric on 1-forms with compact support on the cusp. First, we have on smooth functions with compact support on the cusp that: (5.38) [dAf , ∂L ]P0 = 2(1 − p)x1−p dAf + c(1 − p)x2(1−p) xp−2 dx ∧ P0 . Here, we used dAf P0 = (dx∧∂x ·)P0 . Note that xp−2 dx∧ is a bounded operator from function to 1-forms. In the following, we drop P0 and Ξ to lighten the notation. For ϕ ∈ Cc∞ (X), we have ˜ + (iA∧) ˜ ∗ dA ϕ = [∂L , dA ]L1+ε ϕ, A˜ ∧ ϕ L1+ε ϕ, d∗Af (iA∧) f f + A˜ ∧ L1+ε ϕ, [dAf , ∂L ]ϕ ˜ ∂L ] + 1 dAf L1+ε ϕ, [A∧, 1+ε ˜ ϕ, dAf ϕ . + [∂L , A∧]L Once dAf commuted with L1+ε , the two last terms are controlled by the assumption 1+ε ˜ . The two first ones are 0 for p = 1 using (5.38). When p < 1, on [∂L , A∧]L 1−p 1+ε note that x L = cLε and control the term using Lε A˜ bounded. We now describe the perturbation of the metric following the two classes. We keep the notation from Theorem 6.4. We introduce the canonical unitary transformation due to the change of measure. Set ρ to be ρsr , ρlr or ρt . Let U be the operator of multiplication by (1 + ρ)−n/4 in L2 (X, g) and V the operator of multiplication by (1 + ρ)(2−n)/4 in L2 (X, T ∗ X, g). The operator U is a unitary operator

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from L2 (X, g) onto L2 (X, g˜) and V a unitary operator from L2 (X, T ∗ X, g) onto L2 (X, T ∗ X, g˜). A be the magnetic Laplacian of vector potential Af acting in Lemma 5.13. Let Δ f 2 A U . Then, L (X, g˜). Let W0 := U −1 Δ f

1. On Cc∞ (X), W0 acts by U d∗Af V 2 dAf U . In L2 (X, g), it is essentially selfadjoint and its domain is D(ΔAf ). 2. One has (W0 − ΔAf + i)−1 − (ΔAf + i)−1 is compact. 3. For ρsr and ρt , W0 is a short-range perturbation of the magnetic Laplacian ΔAf in the space B(D(ΔAf ), D(ΔAf )∗ ). 4. For ρlr , W0 is a long-range perturbation of ΔAf in B(D(ΔAf ), D(ΔAf )∗ ). A with the help of the operator dA . Since the manifold is Proof. We write Δ f f complete, ΔAf is essentially self-adjoint on Cc∞ (X). In particular it corresponds to d˜∗Af dÃf , the Friedrichs extension. This is equal to U 2 d∗Af V 2 dAf in L2 (X, g˜). Now remark that (1+ρ)α stabilizes D(ΔAf ), D(dAf ) and D(d∗Af ), for all α ∈ R to obtain the first point. We now compare the two operators in L2 (X, g). We compute on Cc∞ (X). D := W0 − ΔAf = U −1 U 2 d∗Af V 2 dAf U − d∗Af dAf A (U − 1) + U d∗ (V 2 − 1)dA + (U − 1)ΔA . = U −1 Δ f Af f f

(5.39)

We focus on point (3). The two first terms need a justification. We start with the first term. A + [L1+ε , Δ A (U − 1) = U −1 Δ A ]L−1−ε L1+ε (U − 1) . L1+ε U −1 Δ f

f

f

Using Lemma 5.14 and the invariance of the domain under (1 + ρ)α , we obtain A + [L1+ε , Δ A ]L−ε )∗ is bounded from D(ΔA ) to L2 (X, g). Again using that (Δ f f f properties of (1 + ρ)α , for all ϕ ∈ Cc∞ (X) we get ϕ, L1+ε U −1 Δ A (U −1)ϕ f ∗ 1+ε = Δ , ΔAf ]L−1−ε U −1 ϕ, L1+ε (U −1)ϕ ≤ c(ΔAf + i)ϕ2 . Af +[L For the second term, we have L1+ε (U d∗Af (V 2 −1)dAf ) = U (d∗Af +[L1+ε , d∗Af ]L−1−ε ) L1+ε (V 2 − 1)dAf . By Lemma 5.14 and the invariance of D(dAf ) by ρα , we obtain ϕ, L1+ε U d∗Af (V 2 − 1)dAf ϕ = dAf + L−1−ε [dAf , L1+ε ] U ϕ, L1+ε (V 2 − 1)dAf ϕ ≤ c(ΔAf + i)ϕ2 , for all ϕ ∈ Cc∞ (X). To finish, use the fact that Cc∞ (X) is a core for ΔAf . We now deal with (4) by checking (5.37). The real point to check is that ˜ ˜ B(D(Δ),D(Δ)∗) is finite. We take ∂L like in the proof of Lem∂L ]P0 Ξ L1+ε Ξ[D, ma 5.12. First, [d∗ d, ∂L ]P0 = c(1 − p)x3(1−p) P0 . (5.40)

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˜ and P0 for clarity. We have: We start with the easy part of D. We drop Ξ L1+ε ϕ, (U − 1)d∗ d, ∂L ϕ = L1+ε ϕ, [U, ∂L ]d∗ dϕ + L1+ε (U − 1)ϕ, [d∗ d, ∂L ]ϕ . This is bounded by (Δ + i)ϕ2 . Indeed, the first term follows since L1+ε [U, ∂L ] is bounded in L2 (X, g). The second one is 0 for p = 1 and equals otherwise to cϕ, Lε−2 (U − 1)ϕ by (5.40). Turn now to: L1+ε ϕ, U d∗ (V 2 − 1)d, ∂L ϕ = dU L1+ε ϕ, [V 2 , ∂L ]dϕ + [U, ∂L ]L1+ε ϕ, d∗ (V 2 − 1)dϕ + (V 2 − 1)dU L1+ε ϕ, [d, ∂L ]ϕ + [d, ∂L ]U L1+ε ϕ, (V 2 − 1)dϕ . The first is controlled by commuting L1+ε with d like above and by using that L1+ε [V 2 , ∂L ] is bounded in L2 (X, Λ1 , g). For the second one, [U, ∂L ]L1+ε is bounded in L2 (X, Λ1 , g). Turn the two last ones and use (5.40), for p = 1 this is 0. Focus on the very last one for example. Now commute L1+ε with d like above. The most singular term being dU ϕ, x1−p L1+ε (V 2 − 1)dϕ. Now remember that x1−p L1+ε = cLε and use the fact that Lε (V 2 − 1) is bounded to control it. To ˜ − 1), ∂L ]. conclude, repeat the same arguments for [U −1 Δ(U We turn to point (2), W0 and ΔAf have the same domain. We take the proof of Lemma 6.3 replacing G with this domain. We then obtain a rigorous version of (6.42). Therefore, it remains to check that W0 − ΔAf ∈ K(D(ΔAf ), D(ΔAf )∗ ). This comes directly using Rellich–Kondrakov lemma and (5.39). Finally, we gather various technicalities concerning the operator L. Lemma 5.14. We have that dL is with support in (0, ε) × Mnt and dL = f (x)dx where f : (0, ε) → R such that f is 0 in a neighborhood of ε and such that f (x) = −xp−2 for x small enough. Moreover: 1. The operator L−ε d(L1+ε )∧ belongs to B(L2 (X, g), L2 (X, T ∗ X, g)) and the commutator L−ε [ΔAf , L1+ε ] with initial domain Cc∞ (X) extends to a bounded operator in B(D(ΔAf ), L2 (X, g)). 2. eitL D(ΔAf ) ⊂ D(ΔAf ) and eitL B(D(ΔAf )) ≤ c(1 + t2 ). 3. L−1−ε D(ΔAf ) ⊂ D(ΔAf ). Proof. With the diagonalization of Section 5.2, the operator ΔAf is given by (5.22). The operator L corresponds to the operator L0 of multiplication by r ⊗ 1Mnt on (c, ∞) in this variable and by 1 on the rest of the manifold. Hence, points (1) and (3) are easily obtained. Moreover eitL0 /(1 + t2 ) and its first and second derivative belong to L2 (X0 ), uniformly in t, from which (2) follows.

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6. The non-trapping case for perturbed metrics 6.1. The essential spectrum In this section, we compute the essential spectrum of a magnetic Laplacian given by a non-trapping vector potential. Unlike the trapping case, it is non-empty in the complete case. To show this, we apply perturbation techniques to the results of the previous section. We restrict ourselves to conformal perturbations of exact cusp metrics. In a fixed product decomposition of X near M we rewrite (2.8) as 2 dx + α(x) + h(x) (6.41) g0 = a x2 where a ∈ C ∞ (X), α ∈ C ∞ ([0, ε) × M, Λ1 (M )) and h ∈ C ∞ ([0, ε) × M, S 2 T M ). By [38, Lemma 6], the function a0 := a(0), the metric h0 := h(0) and the class (modulo exact forms) of the 1-form α0 := α(0), defined on M , are independent of the chosen product decomposition and of the boundary-defining function x inside the fixed cusp structure. Definition 6.1. The metric g0 is called exact if a0 = 1 and α0 is an exact 1-form. If α0 = df is exact, then by replacing x with the boundary-defining function x = x/(1 + xf ) inside the same cusp structure, we can as well assume that α0 = 0 (see [38]). It follows that g0 is quasi-isometric to a cylindrical metric near infinity. Proposition 6.2. Let (X, g˜p ) be a Riemannian manifold with a conformal exact cusp metric g˜p := (1 + ρ)gp , where g0 is exact, gp = x2p g0 and ρ ∈ L∞ (X; R) ,

inf ρ(x) > −1 ,

x∈X

ρ(x) → 0 ,

as x → 0 .

Let A be a non-trapping vector potential given by (3.12). Then • For 0 < p ≤ 1, the Friedrichs extension of ΔA has essential spectrum σess (ΔA ) = [κ(p), ∞), where κ(p) = 0 for p < 1 and κ(1) = (n − 1)/2. Moreover, if ρ is smooth, the ΔA is essential self-adjoint on Cc∞ (X). • If p > 1 and g˜p := gp is the unperturbed metric given in (1.1), then every self-adjoint extension of ΔA has empty essential spectrum. Note that for p > 1, the unperturbed metric gp given in (1.1) is essentially of metric horn type [30]. Proof. Using the Weyl theorem, Lemma 6.3 and by changing the gauge, we can suppose without loss of generality that A is of the form (5.18) with C = 0. We start with the complete case. The essential self-adjointness follows from [46]. In the exact case, gp is quasi-isometric to the metric (5.17). Using [15, Theorem 9.4] (see Theorem 9.5 for the case of the Laplacian), to compute the essential spectrum we may replace h(x) in (6.41) by the metric h0 := h(0) on M , extended to a symmetric 2-tensor constant in x near M , and we may set ρ = 0. By Proposition C.3, computing σess (ΔA ) is the same as computing σess (ΔlA ) on X

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of Proposition 5.1. By the results of Section 5.2, this is given on L2 (R+ ) by σess (−Δ + Vp ) = [Vp (∞), ∞). Let now p > 1. The metric is no longer complete and (X, gp ) is not proper; one can not apply [15, Theorem 9.4]. By Lemma B.1 and by the Krein formula, all self-adjoint extensions have the same essential spectrum. So it is enough to consider the Friedrichs extension of ΔA . We now use Propositions 5.1 and C.3. The operator D∗ D is non-negative, so the spectrum of Δ0A is contained in [ε2−2p c0 , ∞). By Proposition C.3, the essential spectrum does not depend on the choice of ε. Now we remark that p > 1 implies c0 = 0. Indeed, the equality would imply that 1/p ∈ Z, which is impossible. Thus by letting ε → 0 we conclude that the essential spectrum is empty. We have used above a general lemma about compact perturbations of magnetic Laplacians: Lemma 6.3. Let A and A in L∞ (X, T ∗ X) be two magnetic fields on a smooth Riemannian manifold (X, g) (possibly incomplete) with a measurable metric g. ∗ ∗ ∗ Suppose that A−A belongs to L∞ 0 (X, T X). Let ΔA = dA dA and ΔA = dA dA . Then (ΔA + i)−1 − (ΔA + i)−1 is compact. ∗ ∞ ∗ Here, L∞ 0 (X, T X) denotes the space of those forms of L (X, T X) which are norm limit of compactly supported forms. Note that this lemma holds without any modification for a C 1 manifold equipped with a (RM) structure, see [15, Section 9.3]. Unlike the result on the stability of the essential spectrum of the (magnetic) Laplacian from [15, Theorem 9.5], we do not ask for the completeness of the manifold. A magnetic perturbation is much less singular than a perturbation of the metric.

Proof. Note that the form domain of ΔA and of ΔA is given by G := D(d), because A and A are in L∞ (X, T ∗ X). We write with a tilde the extension of the magnetic Laplacians to B(G , G ∗ ). We aim to give a rigorous meaning to (ΔA + i)−1 − (ΔA + i)−1 = (ΔA + i)−1 (ΔA − ΔA )(ΔA + i)−1 .

(6.42)

We have (ΔA +i)−1∗ H ⊂ G . This allows one to deduce that (ΔA +i)−1 extends to a unique continuous operator G ∗ → H . We denote it for the moment by R. From R(ΔA + i)u = u for u ∈ D(ΔA ) we get, by density of D(ΔA ) in G and continuity, R(Δ A + i)u = u for u ∈ G , in particular −1 . (ΔA + i)−1 = R(Δ A + i)(ΔA + i)

Clearly, −1 (ΔA + i)−1 = (ΔA + i)−1 (ΔA + i)(ΔA + i)−1 = R(Δ . A + i)(ΔA + i)

We subtract the last two relations to get −1 (ΔA + i)−1 − (ΔA + i)−1 = R(Δ A − ΔA )(ΔA + i)

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Since R is uniquely determined as the extension of (ΔA + i)−1 to a continuous map G ∗ → H , one may keep the notation (ΔA + i)−1 for it. With this convention, the rigorous version of (6.42) that we shall use is: A − Δ A )(ΔA + i)−1 . (ΔA + i)−1 − (ΔA + i)−1 = (ΔA + i)−1 (Δ

(6.43)

∗ ∈ L∞ 0 (X, T X), 2 ∗

Since A = i(A − A ) the Rellich–Kondrakov lemma gives that A − Δ A = (A ∧)∗ dA − d∗ A ∧ ∈ A ∧ belongs to K(G , L (X, T X)). Therefore, Δ A ∗ K(G , G ). This gives the announced compactness. 6.2. The spectral and scattering theory In this section, we refine the study of the essential spectrum given in Proposition 6.2 for non-trapping vector potential. As the essential spectrum arises only in the complete case, we will suppose that p ≤ 1. We give below our main result in the study of the nature of the essential spectrum and in scattering theory under short-range perturbation. It is a consequence of the Mourre theory [39] with an improvement for the regularity of the boundary value of the resolvent, see [14] and references therein. We treat some conformal perturbation of the metric (5.17). To our knowledge, this is the weakest hypothesis of perturbation of a metric obtained so far using Mourre theory. Compared to previous approaches, we use a conjugate operator which is local in energy and therefore can be compared directly to a multiplication operator. We believe that this procedure can be implemented to all known Mourre estimates on manifolds to improve the results obtained by perturbation of the metric. We fix Af a non-trapping vector potential of the form (5.18). By a change of gauge, one can suppose that Af = θ0 which is constant in a neighborhood of M . Let Mt (resp. Mnt ) be the union of the connected components of M on which Af is trapping (resp. non-trapping). Let L be the operator of multiplication by a smooth function L ≥ 1 which measures the length of a geodesic going to infinity in the directions where θ0 is non-trapping: − ln(x) for p = 1 , L(x) = (6.44) p−1 − xp−1 for p < 1 c on (0, ε/4) × Mnt for small x, and L = 1 on the trapping part (0, ε/2) × Mnt . Given s ≥ 0, let Ls be the domain of Ls equipped with the graph norm. We set L−s := Ls∗ . Using the Riesz theorem, we obtain the scale of spaces Ls ⊂ L2 (X, g˜) ⊂ Ls∗ , with dense embeddings and where g˜ is defined in the theorem below. Given a subset I of R, let I± be the set of complex number x ± iy, where x ∈ I and y > 0. The thresholds {κ(p)} are given in Proposition 6.2. For shorthand, perturbations of short-range type (resp. trapping type) are denoted with the subscript sr (resp. t); they are supported in (0, ε) × Mnt (resp. in ((0, ε/2) × Mnt )c ). We stress that the class of “trapping type” perturbations is also of short-range nature, in the sense described in Section 5.4, even if no decay is required. This is a rather amusing phenomenon, linked to the fact that no essential

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spectrum arises from the trapping cusps. The subscript lr denotes long-range type perturbations, also with support in (0, ε) × Mnt . We ask such perturbations to be radial, i.e., independent of the variables in M . In other words, a perturbation Wlr satisfies Wlr (x, m) = Wlr (x, m ) for all m, m ∈ M . Theorem 6.4. Fix ε > 0. Let X be endowed with the metric g˜ = (1+ρsr +ρlr +ρt )g, where g = gp is given in (5.17) for some 0 −1 , ρt (x) = o(1) as x → 0 x∈X

and such that L1+ε ρsr , dAf ρsr , ΔAf ρsr , Lε ρlr , L1+ε dAf ρlr , ΔAf ρlr , dAf ρt , ΔAf ρt belong to L∞ . A be the magnetic Laplacian with a non trapping potenIn H = L2 (X, g˜), let Δ tial A = Af + Alr + Asr + At , where Af is as in (5.18), Asr , Alr and At are in C ∞ (X, T ∗ X) such that: L1+ε Asr ∞ , Lε Alr ∞ , L1+ε Lx(2−p) ∂x Alr ∞ < ∞

and

At = o(1) ,

where L denotes the Lie derivative. Let V = Vloc + Vsr + Vlr + Vt and Vlr be some A -compact and potentials, where Vloc is measurable with compact support and Δ f ∞ Vsr , Vlr and Vt are in L (X) such that: L1+ε Vsr ∞ , L1+ε dAf Vlr ∞ < ∞

and Vt = o(1) . A +A +Vlr and H = Δ A +V . Consider the magnetic Schr¨ odinger operators H0 = Δ f lr Then 1. H has no singular continuous spectrum. 2. The eigenvalues of R \ {κ(p)} have finite multiplicity and no accumulation points outside {κ(p)}. 3. Let J a compact interval such that J ∩ ({κ(p)} ∪ σpp (H)) = ∅. Then, for all s ∈ (1/2, 3/2), there exists c such that (H − z1 )−1 − (H − z2 )−1 B(Ls ,L−s ) ≤ cz1 − z2 s−1/2 , for all z1 , z2 ∈ J± . 4. Let J = R \ {κ(p)} and let E0 and E be the continuous spectral component of H0 and H, respectively. Then, the wave operators defined as the strong limit Ω± = s− lim eitH e−itH0 E0 (J ) t→±∞

exist and are complete, i.e., Ω± H = E(J )H . Remark 6.5. Any smooth 1-form A on X is a short-range perturbation of a free vector potential Af as in (5.18). Remark 6.6. If one is interested only in the free metric gp , the conclusions of A -form compact perturbations, the theorem hold for Vloc in the wider class of Δ f using [2, Theorem 7.5.4]. Similar results should hold for a smooth metric on X and Dirichlet boundary conditions, as in [5].

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Proof. We start with ΔAf in L2 (X, g). In Section 5.3, we transform it unitarily into Δ0 given by (5.22). For R finite, we construct a conjugate operator SR to Δ0 given by (5.24). Theorem 5.6 gives a Mourre estimate for Δ0 and the regularity of Δ0 compared to the self-adjoint operator SR . We go back by unitary transform into L2 (X, g). Since the dependence on R is no longer important, we denote simply by S the image of the conjugate operator SR . Therefore, we have ΔAf ∈ C 2 (S, D(ΔAf ), L2 (X, g)) and given J an open interval included in σess (H0 ), there is c > 0 and a compact operator K such that the inequality EJ (T )[T, iS]EJ (T ) ≥ cEJ (T ) + K

(6.45)

2

holds in the sense of forms in L (X, g), for T = ΔAf . A acting in L2 (X, g). By Lemma 5.13, Let W0 be the unitary conjugate of Δ f 1,1 ∗ W0 ∈ C (S, D(ΔAf ), D(ΔAf ) ) is a sum of short and long-range perturbation as described in Section 5.4. In particular, we get W0 ∈ Cu1 (S, D(ΔAf ), D(ΔAf )∗ ). By the point (2) of Lemma 5.13 and [2, Theorem 7.2.9] the inequality (6.45) holds for T (up to changing c and K). We now go into L2 (X, g˜) using U defined before Lemma 5.13. We write ˜ Therefore, Δ ˜ A belongs to the conjugate operator obtained in this way by S. f 1,1 ˜ ∗ C (S, D(ΔA˜f ), D(ΔA˜f ) ) and given J an open interval included in σess (H0 ), there is c > 0 and a compact operator K such that ˜ J (T˜) ≥ cEJ (T˜) + K EJ (T˜)[T˜, iS]E

(6.46)

˜ A . We now add the perturbation holds in the sense of forms in L2 (X, g˜) for T˜ = Δ f given by Asr , Alr , At , Vsr , Vlr , and Vt . Note that H has the same domain as H0 and that (H + i)−1 − (H0 + i)−1 is compact by Rellich–Kondrakov lemma and Lemma 6.3. By Lemma 5.12, we obtain H ∈ C 1,1 (S, D(H), D(H)∗ ). As above, the inequality 6.46 is true for T˜ = H. We now deduce the different claims of the theorem. The first comes from [2, Theorem 7.5.2]. The second ones is a consequence of the Virial theorem. For the third point first note that Ls ⊂ D(|A|s ) for s ∈ [0, 2] by Lemma 5.5 and use [14] for instance (see references therein). Finally, the last point follows from [2, Theorem 7.6.11].

7. The non-stability of the essential spectrum and of the situation of limiting absorption principle In Rn , with the flat metric, it is well-known that only the behavior of the magnetic field at infinity plays a rˆ ole in the computation of the essential spectrum. Moreover, [26, Theorem 4.1] states that the non-emptiness of the essential spectrum is preserved by the addition of a bounded magnetic field, even if it can become purely punctual. Concerning a compactly support magnetic field, the essential spectrum remains the same, see [36]. However, it is well-known that one obtains a long-range effect from it, in other words it acts on particles which have support away from it.

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In the case of Rn with a hole of some kind, this phenomena are of special physical interests and are related to the Aharonov–Bohm effect, see Section 7.3 and references therein. In contrast with the Euclidean setting, Theorem 4.2 indicates that in general the essential spectrum may vanish under compactly supported perturbations of the magnetic field. In the next sections, we discuss this effect both with and without the hypothesis of gauge invariance, and we investigate the coupling constant effect. 7.1. The case H 1 (X) = 0 In this section, we assume gauge invariance. We first characterize trapping condition in terms of the magnetic field, see Definition 3.2 for the case of a magnetic potential. We recall that if H 1 (X) = 0, given a magnetic field B the spectral properties of the magnetic Laplacian ΔA will not depend on the choice of vector potential A such that dA = B. Indeed, given A, A such that dA = dA , the operators ΔA and ΔA are unitarily equivalent by a gauge transformation. Therefore, we denote the magnetic Laplacian by ΔB and express the condition of being (non-)trapping in function of B. Let p > 0 and let X be the interior a compact manifold X endowed with the metric gp given by (2.9). For simplicity, assume that B is a smooth 2-form on X such that its restriction to X is exact. Then there exists A ∈ C ∞ (X, T ∗ X) such that B = dA (since the cohomology of the de Rham complex on X equals the singular cohomology of X, hence that of X). Let M = α∈A Mα be the decomposition of the boundary M into its connected components. Set ! A0 := α ∈ A; H 1 (Mα ; R) = 0 . For some B ⊂ A set MB = β∈B Mβ and consider the long exact cohomology sequence of the pair (X, MB ) with real coefficients: i ∂ H 1 X; R −→ H 1 (MB ; R) −→ H 2 X, MB ; R −→ H 2 X; R Since we assume that H 1 (X; R) = 0 it follows that the connecting map ∂ is injective. If B vanishes under pull-back to MB then (since it is exact on X) it defines a class in H 2 (X, MB ; R) which vanishes under the map i, so it belongs to the image of the injection ∂. We denote by [B]β the component of [B] inside ∂H 1 (Mβ ) ⊂ H 2 (X, MB ; R). Definition 7.1. Assume H 1 (X) = 0. Let B be a smooth exact 2-form on X. Denote by B the set of those α ∈ A such that B vanishes identically on Mα . The field B is called trapping if for each β ∈ B, the component [B]β ∈ ∂H 1 (Mβ ) ⊂ H 2 (X, MB ; R) is not integral, i.e., it does not live in the image of the map of multiplication by 2π 2π· H 2 X, MB ; Z −→ H 2 X, MB ; R , and non-trapping otherwise.

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This definition is consistent with Definition 1.1 when M is connected. Note that if B is trapping then B must contain the index set A0 defined above. In order to apply Theorem 6.4 and Theorem 4.2 we use the following lemma: Lemma 7.2. 1. Let A be a smooth vector potential on X such that dA = 0 in a neighbourhood of M = ∂X. Then there exists a smooth vector potential A , constant in x in a neighborhood of M , such that A = A on M and d(A − A ) = 0. 2. Assume that H 1 (X, R) vanishes. Let B be a trapping magnetic field on X. Then every vector potential for B will be trapping. 3. Assume moreover that H1 (X; Z) = 0. Let B be non-trapping such that ı∗M B = 0. Then every vector potential for B will be non-trapping. 1 (X) = 0. Recall that π1 (X) = 0 =⇒ H1 (X; Z) = 0 =⇒ HdR

Proof. 1) Let us show that one can choose A to be constant in x near the boundary, in the sense that near M it is the pull-back of a form from M under the projection π : [0, ε) × M → M for ε small enough. Indeed, A − π ∗ ı∗M A is closed on the cylinder [0, ε) × M and vanishes when pulled-back to M . Now M is a deformationretract of the above cylinder, so the map of restriction to M induces an isomorphism in cohomology and thus the cohomology class [A− π ∗ ı∗M A] ∈ H 1 ([0, ε)× M ) must be zero. Let f ∈ C ∞ (X, R) be a primitive of this form for x ≤ ε/2, then A−df is the desired constant representative. 2) Consider the commutative diagram ∂ H 1 MB ; −−−−→ H 2 X, MB ; R −−−−→ H 2 X, R " " " ⏐ ⏐ ⏐ (7.47) ⏐2π· ⏐2π· ⏐2π· ∂ H 1 MB ; Z −−−−→ H 2 X, MB ; Z −−−−→ H 2 X, Z where the horizontal maps come form the long exact sequence of the pair (X, M ) and the vertical maps are multiplication by 2π. Let B be trapping and choose a vector potential A for B, smooth on X. We claim that A is trapping. Indeed, A is not closed on the components Mα , α ∈ A \ B, while it is closed on MB . We note that ∂[A|MB ] = [dA] = [B], so ∂[A|Mβ ] = [B]β . Assume that for some β ∈ B, the class [A|Mβ ] were integral. Then using the first square from diagram (7.47), it would follow that [B]β was also integral, contradiction. 3) If B is non-trapping, there exists β ∈ B and b ∈ H 2 (X, MB , Z) with [B]β = 2πb. The image of [B]β ∈ ∂H 1 (Mβ , R) in H 2 (X, R) is zero, thus b maps to a torsion element in H 2 (X, Z). From H1 (X, Z) = 0 we see using the universal coefficients theorem 0 → Ext H1 (X, Z), Z → H 2 (X; Z) → Hom H2 (X, Z), Z → 0 that H 2 (X; Z) is torsion-free. Thus b comes from some a ∈ H 1 (MB , Z). By commutativity we have ∂(2πa) = [B]β . Since ∂ is an injection, it follows that every vector

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potential smooth on X for B will define an integral cohomology class on Mβ , thus will be non-trapping. A spectacular example is a compactly supported magnetic field which induces very strong long-range effects. If H 1 (X; Z) = 0 and B be is an exact 2-form with compact support in X, then B is maximal non-trapping (see Definition 3.2) if and only if its class in H 2 (X, M ; R) is integral. We will stress more on this aspect in the following section. We can also construct compactly supported magnetic fields for which the consequences of Theorem 4.2 hold true. We summarize this fact in the next proposition and give an explicit construction in the proof. Proposition 7.3. Let X be the interior of a compact manifold X with boundary M = ∂X, endowed with a conformally cusp metric gp . Assume that H 1 (X) = 0 and H 1 (Mj ) = 0 for every connected component Mj of the boundary. Then there exists a non-zero smooth magnetic field B with compact support such that the essential spectrum of ΔB is empty and such that for p ≥ 1/n the growth law of the eigenvalues does not depend on B and is given by (4.15). Such fields B are generic inside compactly supported magnetic fields. Proof. We construct A like in (3.12) satisfying the hypotheses of Theorem 4.2. We take ϕ0 to be constant. Let ψ ∈ C ∞ ([0, ε)) be a cut-off function such that ψ(x) = 0 for x ∈ [3ε/4, ε) and ψ(x) = 1 for x ∈ [0, ε/2). Since H 1 (Mj ) = 0, there exists a closed 1-form βj on M which is not exact. Up to multiplying βj by a real constant, 1 (Mj ) does not belong to the we can assume that the cohomology class [βj ] ∈ HdR 1 image of 2πH 1 (Mj ; Z) → H 1 (Mj ; R) HdR (Mj ). Let β denote the form on M which equals βj on Mj . Choose A to be ψ(x)β for ε > x > 0 and extend it by 0 to X. The magnetic field B = dA = ψ (x)dx ∧ β has compact support in X. By Theorem 4.2, ΔA has purely discrete spectrum with the Weyl asymptotic law eigenvalues independent of B. The relative cohomology class [B] lives in the direct sum ⊕j ∂H 1 (Mj , R) ⊂ 2 H (X, M, R). The field B is non-trapping if at least one of its components in this decomposition lives in the image of H 1 (Mj , Z). Since we assume all H 1 (Mj , R) to be nonzero, the space of non-trapping magnetic fields is a finite union of subspaces of codimension at least 1. We now show that the cohomological hypothesis about X and M can be satisfied in all dimensions greater than or equal to 2, and different from 3. In dimension 2, take X = R2 endowed with the metric (2.9). Consider, for instance, the metric r−2p (dr2 + dσ 2 ) given in polar coordinates. Here M is the circle at infinity and x = 1/r for large r. Thus b1 (X) = 0 while b1 (M ) = 0. The product of this manifold with a closed, connected, simply connected manifold Y of dimension k yields an example in dimension 2 + k with the same properties. Indeed, by the K¨ unneth formula, the first cohomology group of R2 × Y vanishes, 1 1 while H (S × Y ) H 1 (S 1 ) = Z. Clearly k cannot be 1 since the only closed manifold in dimension 1 is the circle. Thus the dimension 3 is actually exceptional.

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For orientable X of dimension 3, the assumptions H 1 (X) = 0 and H 1 (M ) = 0 cannot be simultaneously fulfilled. Indeed, we have the following long exact sequence (valid actually regardless of the dimension of X) iM δ H 1 X −→ H 1 (M ) −→ H 2 X, M . (7.48) If dim(X) = 3, the spaces H 1 (X) and H 2 (X, M ) are isomorphic by Poincaré duality, hence H 1 (X) = 0 implies H 2 (X, M ) = 0 and so (by exactness) H 1 (M ) = 0. It should be possible to build a non-orientable example in dimension 3 such that one could apply Proposition 7.3 but we were not able to construct one. We finally give an example not covered by Lemma 7.2 but such that the conclusion of Theorem 4.2 holds. We considered so far magnetic fields on X with vector potential smooth on X. One may consider also more singular magnetic fields arising from Proposition 3.1. 7.1.1. Example. Let X be any conformally cusp manifold, without any cohomological assumptions. Suppose that B = df ∧ dx/x2 where f is a function on X smooth down to the boundary M of X. Assume that f is not constant on any connected component of M . Then the essential spectrum of the magnetic operator (which is well-defined by B if H 1 (X) = 0) is empty. This follows from the fact that A := f dx/x2 is trapping. Note that the pull-back to the border of the above magnetic field is zero. 7.2. The coupling constant effect In flat Euclidean space it is shown in [24], under some technical hypotheses, that the spectrum has a limit as the coupling constant tends to infinity. In contrast, in the next example, we exhibit the creation of essential spectrum for periodic values of the coupling constant. We will focus on the properties of ΔgB for some coupling constant g ∈ R. In order to be able to exploit the two sides of this work we concentrate here on the metric (5.17). We assume that M is connected, H 1 (X, Z) = 0 and H 1 (M ) = 0. Given B a magnetic potential with compact support, gB is non-trapping if and only its class in H 2 (X, M ; R) is integral. Let GB be the discrete subgroup of those g ∈ R such that gB is non-trapping. As this subgroup is possibly {0}, we start with some exact form B which represents a nonzero cohomology class in H 2 (X, M ; Z); then by exactness of the relative cohomology long sequence, [B] lives ∂

in the image of the injection H 1 (M ; R) −→ H 2 (X, M ; R). With these restrictions, GB is a non-zero discrete subgroup of Q. Now we apply Theorem 4.2 for the trapping case and Theorem 6.4 for the non-trapping case. We obtain that 1. For g ∈ GB , the essential spectrum of ΔgB is given by [κ(p), ∞), where 1κ(p) is defined in Proposition 6.2. The spectrum of ΔgB has no singular continuous part and the eigenvalues of R \ {κ(p)} are of finite multiplicity and can accumulate only in κ(p). 2. For g ∈ / GB , the spectrum ΔgB is discrete and if p ≥ 1/n, the asymptotic of the eigenvalues depend neither on g nor on B.

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We now describe the long-range effect regarding the coupling constant. Take a state φ ∈ L2 (X) such that φ is not an eigenvalue of the free Laplacian Δ0 and is located in a energy higher than κ(p). Since the Fourier transform of an absolutely continuous measure (comparing to the Lebesgue measure) tends to 0 at infinity, we obtain, for each g ∈ GgB and for each χ operator of multiplication by the characteristic function of compact support that χeitΔgB φ → 0 as t → ∞. If one considers χ being 1 above the support of the magnetic field, then after some time the norm of φ above this zone is arbitrary small. Classically, the particle stops interacting with the magnetic field. Let us denote by φ the particle at this moment. Now, if we switch on the interaction with intensity as small as one desires one gets g ∈ / GgB and then the spectrum of ΔgB is discrete. Therefore there exists χ operator of multiplication by the characteristic function of compact T support such that 1/T 0 χ eitΔgB φ 2 dt tends to a positive constant, as T → ∞. The particle is caught by the magnetic field even thought they are far from being able to interact classically. In other words, switching on the interaction of the magnetic field with compact support has destroyed the situation of limiting absorption principle. This is a strong long-range effect. For the sake of utmost concreteness, take X = R2 endowed with the metric −2p r (dr2 +dθ2 ) in polar coordinates, for r big enough and 1 ≥ p > 0. The border M 1 is S and H 2 (X, M ; R) Z. Then for every closed 2-form B with compact support and non-zero integral, the group GB defined above is non-zero. 7.3. The case H 1 (X) = 0, the Aharonov–Bohm effect Gauge invariance does not hold in this case, so one expects some sort of Aharonov– Bohm effect [1]. Indeed, given two vectors potential arising from the same magnetic field, the associated magnetic Laplacians ΔA and ΔA might be unitarily in-equivalent. The vector potential acquires therefore a certain physical meaning in this case. In flat Rn with holes, some long-range effect appears, for instance in the scattering matrix like in [42–44, 47], in an inverse-scattering problem [40, 48] or in the semi-classical regime [3]. See also [22] for the influence of the obstacle on the bottom of the spectrum. In all the above cases the essential spectrum remains the same. In light of Proposition 7.3, one can expect a much stronger effect in our context. We now give some examples of magnetic fields with compact support such that there exists a non-trapping vector potential A, constant in x in a neighborhood of M , and a trapping vector potential A , such that dA = dA = B. To ease the presentation, we stick to the metric (5.17). For A, one applies Theorem 6.4 and obtain that the essential spectrum of ΔA is given by [κ(p), ∞), that ΔA has no singular continuous part and the eigenvalues of R \ {κ(p)} have finite multiplicity and can accumulate only to {κ(p)}. For A , one applies Theorem 4.2 to get the discreteness of the spectrum of ΔA and to obtain that the asymptotic of eigenvalues depends neither

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on A nor on B for n ≥ 1/p. In the next section, we describe how generic this situation is for hyperbolic manifolds of dimension 2 and 3. The easy step is to construct the non-trapping vector potential A constant in a neighborhood of M , more precisely we construct A to be maximal non-trapping, see Definition 3.2. Indeed, prescribe a closed 1-form θ on M defining integral cohomology 1-classes on each component of the boundary, and extend it smoothly to X, constant in x in a neighborhood on M , like in the proof of Proposition 7.3. The magnetic field B := dA has then compact support and one may apply Theorem 6.4 for ΔA . We now construct A by adding to A a closed form α, smooth on X. Since α is closed, A and A define the same magnetic field. In light of Remark 3.4, A is trapping if and only if α is. We can then apply Theorem 4.2 to ΔA . It remains to show that closed, trapping α do exist. We start with a concrete example. Example 7.4. Consider the manifold X = (S 1 )n−1 × [0, 1] with a metric gp as in (5.17) near the two boundary components. Let θi ∈ R be variables on the torus (S 1 )n−1 , so eiθi ∈ S 1 . Take the vector potential A to be 0, it is (maximal) nontrapping. Choose now α = A to be the closed form μdθ1 for some μ ∈ R. It is constant in a neighborhood of (S 1 )n−1 . The class [i∗M (A )] is an integer multiple of 2π if and only if μ ∈ Z. In other words, A is non-trapping if and only if μ ∈ R \ Z. Note that here the magnetic field B vanishes. In order to show the existence of such α in a more general setting, we assume that the first Betti number of each connected component of the boundary is nonzero. It is enough to find some closed α, smooth on X, which on each boundary component represents a non-zero cohomology class. Then, up to a multiplication by a constant, α will be trapping. When X is orientable and dim(X) is 2 or 3, one proceeds as follows. Proposition 7.5. Let X be a compact manifold with non-empty boundary M . Assume that one of the following hypotheses holds: 1. dim(X) = 2 and M is disconnected; 2. dim(X) = 2 and X is non-orientable; 3. dim(X) = 3, X is orientable and none of the connected components of M are spheres. Then there exists a closed smooth form α ∈ Λ1 (X), constant in x near the boundary, such that for all connected components Mj of M , the class [α|Mj ] ∈ H 1 (Mj ; R) is non-zero. Proof. Any cohomology class on X admits a smooth representative α up to the boundary. Moreover since α is closed, one can choose α to be constant in x near the boundary using Lemma 7.2. Thus, in cohomological terms, the proposition is equivalent to finding a class [α] ∈ H 1 (X) whose pull-back to each connected component of M is non-zero, i.e., H 1 (Mj ) iMj [α] = 0. Consider first the case dim(X) = 2. If X is non-orientable, H 2 (X, M ) = 0 so δ is the zero map. If X is oriented, the boundary components are all oriented circles,

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so H 1 (Mj ) R; the compactly supported cohomology H 2 (X, M ) is isomorphic to R via the integration map, and the boundary map δ : H 1 (M ) → H 2 (X, M ) restricted to H 1 (Mj ) is just the identity map of R under these identifications. Thus the kernel of δ is made of ν-tuples (where ν is the number $ of boundary components) (a1 , . . . , aν ) of real numbers, with the constraint aj = 0 in the orientable case. By exactness, this space is also the image of the restriction map H 1 (X) → H 1 (M ). Clearly there exist such tuples with non-zero entries, provided ν ≥ 2 in the orientable case. Thus the conclusion follows for dim(X) = 2. Assume now that dim(X) = 3. Then the maps iM and δ from the relative long exact sequence (7.48) are dual to each other under the intersection pairing on M , respectively on H 1 (X) × H 2 (X, M ): iM (α) ∧ β = α ∧ δβ . M

X

These bilinear pairings are non-degenerate by Poincaré duality; in particular since the pairing on H 1 (M ) is skew-symmetric, it defines a symplectic form. It follows easily that the subspace L := iM H 1 (X) ⊂ H 1 (M ) is a Lagrangian subspace (i.e., it is a maximal isotropic subspace for the symplectic form). Now the symplectic vector space H 1 (M ) splits into the direct sum of symplectic vector spaces H 1 (Mj ), By hypothesis, the genus gj of the oriented surface Mj is at least 1 so H 1 (Mj ) is non-zero for all j. It is clear that the projection of L on each H 1 (Mj ) must be non-zero, otherwise L would not be maximal. Hence, there exists an element of L = iM (H 1 (X)) which restrict to non-zero classes in each H 1 (Mj ), as desired. Remark 7.6. If one is interested in some coupling constant effect, it is interesting to choose the closed form α so that every [α|Mj ] are non-zero integral classes. In dimension 2, this amounts to choosing non-zero integers with zero sum. In dimension 3, as L = iM (H 1 (X)) = iM (H 1 (X, Z)) ⊗ R is spanned by integer classes, we can find an integer class in L with non-zero projection on all H 1 (Mj ). For real g, the vector potential gα is therefore non-trapping precisely for g in a discrete subgroup g0 Z for some g0 ∈ Q.

8. Application to hyperbolic manifolds We now examine in more detail how this Aharonov–Bohm effect arises in the context of hyperbolic manifolds of finite volume in dimension 2 and 3. These are conformally cusp manifold with p = 1, with unperturbed metric of the form (1.1) and such that every component Mj of the boundary is a circle when dim(X) = 2, respectively a flat torus when dim(X) = 3: indeed, outside a compact set, the metric takes the form g = dt2 + e−2t h, where t ∈ [0, ∞), and this is of the form (1.1) after the change of variables x := e−t . We denote by X the compactification of X by requiring that x := e−t be a boundary-defining

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function for “infinity”. These manifolds and their boundary components always have non-zero first Betti number. For complete hyperbolic surfaces with cusps, every smooth 2-form B on X must be exact because H 2 (X; R) is always zero for a non-closed surface. Call A a smooth primitive of B. If B vanishes at M then A is necessarily closed over M . In terms of cohomology classes, we have [B] = δR [A|M ] where δR is the connecting morphism from the sequence (7.48) with real coefficients. Notice that A can be chosen to define an integer class on M if and only if [B] is integer. Indeed, H 2 (X; Z) is also 0 for a compact surface with non-empty boundary, so if [B] is integer, it must lie in the image of δZ for the sequence (7.48) with integer coefficients. Conversely, if [A] is integer, i.e., [A] = 2π[AZ ] (see diagram (7.47) with M in the place of MB , where the horizontal maps are now surjective) then [B] = 2πδZ [AZ ] is also integer. We summarize these remarks in the following Corollary 8.1. Let B be a smooth 2-form on the compactification of a complete hyperbolic surface X with cusps, and denote by [B] ∈ H 2 (X, M, R) its relative cohomology class. • If either X is non-orientable, or X is orientable with at least two cusps, then B admits both trapping and non-trapping vector potentials. • If X is orientable with precisely one cusp, then B admits only non-trapping vector potentials if [B] is integral, while if [B] is not integral then B admits only trapping vector potentials. Proof. First note that B is closed since it is of maximal degree; it is exact since H 2 (X) = 0 for every surface with boundary; moreover its pull-back to the 1dimensional boundary also vanishes, so B defines a relative de Rham class. By Corollary 3.5, the existence of trapping and non-trapping vector potentials depends only on this class. If X is orientable and has precisely 1 cusp, then the map δ : H 1 (M ) → H 2 (X, M ) is an isomorphism both for real and for integer coefficients. Thus [B] is integer if and only if [A|M ] is integer. Since the boundary is connected, A is trapping if and only if the cohomology class of its restriction to the boundary is non-integer. If X is oriented and has at least two cusps, identify H 2 (X, M ) and each 1 map restricted to H 1 (Mj ) is the identity. H (Mj ) with Z, so that the boundary $ We can write [B] first as a sum αj of non-integer numbers, then also as a sum where at least one term is integer. Let A be a 1-form on X which restricts to closed forms of cohomology class αj on Mj = S 1 . Then B − dA represents the 0 class in H 2 (X, M ), so after adding to A a form vanishing at the boundary, we can assume that B = dA. Now when all αj are non-integers, A is trapping, while in the other case it is non-trapping as claimed. If X is non-orientable, the class [B] vanishes. It is enough to find trapping and non-trapping vector potentials for the zero magnetic field, which is done as in the orientable case.

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When dim(X) = 3, we have: Corollary 8.2. Let X be an orientable complete hyperbolic 3-manifold of finite volume. Then every magnetic field B smooth on the compactification X admits trapping vector potentials. Assume that the pull-back of B to the boundary M vanishes. If X has precisely one cusp, then there exists a rational (i.e., containing integer classes) infinite cyclic subgroup G ⊂ H 2 (X, M, R) so that B admits a non-trapping vector potential if and only if [B] ∈ G. In general, one of the following alternative statements holds: 1. Either every magnetic field smooth on X and vanishing at M admits a nontrapping vector potential, or 2. Generically, magnetic fields smooth on X and vanishing at M do not admit non-trapping vector potentials. There exists moreover q ∈ Z∗ such that if [B] is integer, then qB admits nontrapping vector potentials. Proof. For the existence of trapping vector potentials we use the closed form α from Proposition 7.5. Let A be any vector potential for B. It suffices to note that for u ∈ R, the form A + uα is another vector potential for B, which is trapping on each connected component of M except possibly for some discrete values of u. Let h denote the number of cusps of X. Both the Lagrangian subspace L ⊂ H 1 (M ) and the image space ∂H 1 (M ) ⊂ H 2 (X, M ) have dimension h. By hypothesis, the cohomology class of B on X is 0 so by exactness of (7.47), the relative cohomology class [B] lives in ∂H 1 (M ). Assume first that X has precisely one cusp. Let A be a vector potential for B (smooth on X). We can change A by adding to it any class in the line L without changing [B]. Notice that H 1 (M ) = Z2 . The line L has an integer generator (given by the image of H 1 (X, Z) → H 1 (M, Z)). Without loss of generality, we can assume that L is not the horizontal axis in Z2 . It follows that the translates of all integer points in Z2 in directions parallel to L form a discrete subgroup of Q. Thus B admits non-trapping vector potentials if and only if the cohomology class [B] inside the 1-dimensional image ∂H 1 (M ) lives inside a certain infinite cyclic discrete subgroup. In particular, if B is irrational (i.e., no positive integer multiple of B is an integral class) then B does not admit non-trapping vector potentials. In the general case, assume first that there exists a boundary component Mj so that L projects surjectively onto H 1 (Mj , R). Let A be a vector potential for an arbitrary magnetic field B which vanishes at M . Let [Aj ] ∈ H 1 (Mj , R) be such that [A]H 1 (Mj ) + [Aj ] is integer. Let [A ] ∈ L be an element whose component in H 1 (Mj ) is [Aj ]. Choose a representative A and extend it to a smooth 1-form on X, constant in x near the boundary. Then A + A is a non-trapping vector potential. From the definition of L, the form dA defines the zero class in relative cohomology, so from Corollary 3.5 we get the assertion on B.

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If the assumption on L is not fulfilled, we claim that ˆ j := L ∩ ⊕i =j H 1 (Mi ) L has dimension h − 1 for all j. Indeed, this dimension cannot be h (since the projection of L on H 1 (Mj ) is not zero) and it cannot be h − 2 (since the projection is ˆ j is a Lagrangian subspace of ⊕i =j H 1 (Mi ). not surjective). It follows easily that L ˆ Let v be a vector in L \ Lj . The component vˆj of v in ⊕i =j H 1 (Mi ) is clearly ˆ j , so by maximality it must orthogonal (with respect to the symplectic form) to L ˆ belong to Lj . Thus we may subtract this component to obtain, for each j, a nonzero element of L ∩ H 1 (Mj ). These elements may be taken integral since L has integer generators. Since L = ker ∂, it follows that the image of ∂ is the direct sum of the images ∂(H 1 (Mj )). As in the case of only one cusp, we see that B has a non-trapping potential if and only if at least one of the components of [B] in this decomposition belong to a certain cyclic subgroup containing integer classes. Set q to be the least common denominator of the generators of these subgroups for all j. If [B] is integer, it follows that every vector potential for qB must be maximal non-trapping.

Appendix A. The C 1 condition in the Mourre theory In this appendix, we give a general criterion of its own interest to check the, somehow abstract, hypothesis of regularity C 1 which is a key notion in the Virial theorem within Mourre’s theory, see [2] and [13]. Let A and H be two self-adjoint operators in a Hilbert space H . The commutator [H, iA] is defined in the sense of forms on D(A)∩D(H). Suppose that the commutator [H, iA] extends to B(D(H), D(H)∗ ) and denote by [H, iA]0 the extension. Suppose also that the following Mourre estimate holds true on an open interval I, i.e., there is a constant c > 0 and a compact operator K such that EI (H)[H, iA]0 EI (H) ≥ cEI (H) + K ,

(A.1)

where EI (H) denotes the spectral measure of H above I. Take now λ ∈ I which is not an eigenvalue of H. Set In := (λ − 1/n, λ + 1/n). Then EIn (H) tends strongly to 0 as n → ∞, so EIn (H)KEIn (H) tends in norm to 0. Hence for n big enough and for some 0 < c ≤ c, one gets the strict Mourre estimate EIn (H)[H, iA]0 EIn (H) ≥ c EIn (H) .

(A.2)

By supposing that H ∈ C 1 (A) (see below) or that eitA D(H) ⊂ D(H), the Virial theorem holds true, i.e., f, [H, iA]0 f = 0 for every eigenvector f of H. Note that f has no reason to lie in D(A) and that the expansion of the commutator [H − λ, iA] over f is formal. The Virial theorem is crucial to the study of embedded eigenvalues of H. Assuming (A.1), it implies the local finiteness of the point spectrum of H over I, i.e., that the sum of the multiplicities of the eigenvalues of H inside I is finite.

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To see this, apply (A.1) to a infinite sequence (fn )n∈N of orthonormal eigenvectors of H. Then, since fn , Kfn tends to 0 as n goes to infinity, one obtains a contradiction with the positivity of c. Assuming (A.2), the Virial theorem implies directly that H has no eigenvalue in In . We stress that the hypothesis [H, iA]0 ∈ B(D(H), D(H)∗ ) does not imply the Virial theorem. A counterexample is given in [13]. If one adds some conditions on the second order commutator of H and A, e.g., like in [7], one deduces from (A.2) a limiting absorption principle and therefore the absence of eigenvalues in In . In turn, assuming (A.1), we deduce that the set of eigenvalues of H in I is closed. It is not known whether the multiplicity of the point spectrum must be locally finite when the Virial theorem does not hold. Checking the Virial theorem, or a sufficient condition for it like the C 1 condition, is sometimes omitted in the Mourre analysis in a manifold context. To our knowledge, no result exists actually to show directly the C 1 regularity in a manifold context. On a class of exponentially growing manifolds, Bouclet [5] circumvents the problem by showing a stronger fact, i.e., the invariance of the domain. This method does not seem to work for our local conjugate operator, see Section 5.3. Besides giving an abstract criterion for the C 1 condition, we will explain under which additional condition we can recover the invariance of the domain from it. Given z ∈ ρ(H), we denote by R(z) = (H − z)−1 . For k ∈ N, we recall that H ∈ C k (A) if for one z ∈ / σ(H) (then for all z ∈ / σ(H)) the map t → e−itA R(z)eitA k is C in the strong topology. We recall a result following from Lemma 6.2.9 and Theorem 6.2.10 of [2]. Theorem A.1. Let A and H be two self-adjoint operators in the Hilbert space H . The following points are equivalent: 1. H ∈ C 1 (A). 2. For one (then for all) z ∈ / σ(H), there is a finite c such that Af, R(z)f − R(z)f, Af ≤ cf 2 , for all f ∈ D(A) . (A.3) 3.

a. There is a finite c such that for all f ∈ D(A) ∩ D(H): |Af, Hf − Hf, Af | ≤ c Hf 2 + f 2 .

(A.4)

b. For some (then for all) z ∈ / σ(H), the set {f ∈ D(A) | R(z)f ∈ D(A) and R(z)f ∈ D(A)} is a core for A. Note that in practice, condition (3.a) is usually easy to check and follows from the construction of the conjugate operator. The condition (3.b) could be more delicate. This is addressed in the next lemma, inspired by [5]. Lemma A.2. Let D be a subspace of H such that D ⊂ D(H) ∩ D(A), D is a core for A and HD ⊂ D. Let (χn )n∈N be a family of bounded operators such that 1. χn D ⊂ D, χn tends strongly to 1 as n → ∞, and supn χn D(H) < ∞. 2. Aχn f → Af , for all f ∈ D, as n → ∞. 3. There is z ∈ / σ(H), such that χn R(z)D ⊂ D and χn R(z)D ⊂ D.

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lim A[H, χn ]R(z)f = 0 .

n→∞

Finally, suppose that there is a finite c such that |Af, Hf − Hf, Af | ≤ c Hf 2 + f 2 ,

∀f ∈ D .

(A.5)

(A.6)

Then one has H ∈ C 1 (A). Note that (A.5) is well defined by expanding the commutator [H, χn ] and by using (3) and HD ⊂ D. Proof. By polarization and by applying (A.6) to χn R(z)f and to χn R(z)f , with f ∈ D, we see that there exists c < ∞ such that Aχn R(z)f, H χn R(z)f − H χn R(z)f, Aχn R(z)f ≤ c(H + i)χn R(z)f · (H + i)χn R(z)f ,

(A.7)

for all f ∈ D. By condition (1), the right-hand side is bounded by Cf 2 for some C. We expand the left hand side of (A.7) by commuting H with χn : χn R(z)f, Aχn f − Aχn f, χn R(z)f + χn R(z)f, A[H, χn ]R(z)f − A[H, χn ]R(z)f, χn R(z)f . Using (A.5), the second line vanishes as n goes to infinity. Taking in account the assumptions (1) and (2), we deduce: R(z)f, Af − Af, R(z)f ≤ Cf 2 , ∀f ∈ D . Finally, since D is a core for A, we obtain (A.3). We conclude that H ∈ C 1 (A).

The hypotheses of the lemma are easily satisfied in a manifold context with H being the Laplacian and A its conjugate operator, constructed as a localization on the ends of the generator of dilatation like for instance in [10]. Let D = Cc∞ (X) and χn a family of operators of multiplication by smooth cut-off functions with compact support. The fact that A is self-adjoint comes usually by some consideration of C0 -group associated to some vector fields and using the Nelson lemma and the invariance of D under the C0 -group give that D is a core for A, see Remark 5.4. The hypothesis (3) follows then by elliptic regularity. The only point to really check is (A.5). At this point one needs to choose more carefully the family χn . In Lemma 5.9, we show that the hypotheses of Lemma A.2 hold for the standard conjugate operator and for our local conjugate operator. The invariance of the domain is desirable in order to deal, in a more convenient way, directly with operators and no longer with resolvents. On a manifold the C0 -group eitA is not explicit and it could be delicate to deal with the domain of H directly. However, one may obtain this invariance of the domain using [13] and a C 1 (A) condition. We recall: Lemma A.3. If H ∈ C 1 (A) and [H, iA] : D(H) → H then eitA D(H) ⊂ D(H), for all t ∈ R.

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In light of this lemma, one understands better the importance of having some C 1 (A) criteria. On one hand, one can easily check the invariance of the domain. On the other hand, if the commutator belongs only to B(D(H), D(H)∗ ) and not B(D(H), H ), one may turn to another version of the Mourre theory like in [2] when H has a spectral gap or like in [17, 45] in the other case.

Appendix B. Finite multiplicity of L2 eigenvalues In order to classify all maximal symmetric extension of a given cusp-elliptic operator H (see Section 2.1 for definitions), one computes the defect indices, i.e., dim ker(H ∗ ± i). If they are equal and finite, one concludes that all maximal symmetric extensions of H are self-adjoint. By the Krein formulae, one hence obtains that the difference of the resolvent of two maximal extensions is finite rank. This implies by Weyl’s theorem that the essential spectrum is the same for all selfadjoint extensions. Moreover, by Birman’s theorem, the wave operators associated to a pair of such extensions exist and are complete. On the other hand, note that if the defect indices are not finite, one may have maximal symmetric extensions which are not self-adjoint. It is also interesting to control the multiplicity of eigenvalues embedded in the essential spectrum. In the next lemma we assume that X is a conformally cusp manifold with respect to the metric (2.9). We fix a vector bundle E over X (for instance the bundle of cups differential forms, although in this paper we only use the case where E is the trivial bundle C) endowed with a smooth metric up to ∂X = M . Lemma B.1. Let Δ, acting on Cc∞ (X, E), be a cusp-elliptic differential operator in x−2p Diff k (X, E) for some p, k > 0. Then the dimension of any L2 -eigenspace of Δ∗ is finite. Remember that if the operator Δ is bounded from below, then the defect indices are the same. This lemma guarantees that they are also finite. This point is not obvious when the manifold is not complete even if Δ is a Laplacian. Of course, this result is based on ideas that can be traced back to [34] and which are today quite standard. This lemma generalizes a result of [18]. Proof. We start by noticing that Δ can be regarded as an unbounded operator in a larger L2 space. Namely, let L2ε be the completion of Cc∞ (X, E) with respect 2ε to the volume form e− x dgp for some ε > 0. Clearly then L2ε contains L2 . A distributional solution of Δ − λ in L2 is evidently also a distributional solution of Δ − λ in L2ε . Thus the conclusion will follow by showing that Δ has in L2ε a unique closed extension with purely discrete spectrum. The strategy for this is by now clear. First we conjugate Δ through the isometry ε

L2ε → L2 , φ → e− x φ .

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ε

We get an unbounded operator e− x Δe x in L2 , which is unitarily equivalent to Δ (acting in L2ε ). Essentially from the definition, see Section 2.1, ε ε N x2p e− x Δe x (ξ) = N x2p Δ (ξ + iε) . The normal operator is a polynomial in ξ, in particular it is entire. Then by analytic Fredholm theory [34, Prop. 5.3], the set of complex values of ξ for which N (x2p Δ)(ξ) is not invertible, is discrete. Thus there exists ε > 0 such that N (x2p Δ)(ξ + iε) is invertible for all ξ ∈ R. For such ε the operator Δ in L2ε is unitarily equivalent to a fully elliptic cusp operator of order (k, 2p) in L2 . It is then a general fact about the cusp algebra [38, Theorem 17] that such an operator has a unique closed extension and admits a compact inverse modulo compact operators in L2ε . In particular, its eigenvalues have finite multiplicity. As noted above, the eigenspaces of Δ in L2 are contained in the eigenspaces of Δ in L2ε for the same eigenvalue. As a corollary, the magnetic Laplacians for the metric (2.9) and for vector potentials (3.12) which are smooth cusp 1-forms, have finite multiplicity eigenvalues.

Appendix C. Stability of the essential spectrum It is well-known that the essential spectrum of an elliptic differential operator on a complete manifold can be computed by cutting out a compact part and studying the Dirichlet extension of the remaining operator on the non-compact part (see, e.g., [9]). This result is obvious using Zhislin sequences, but the approach from loc. cit. fails in the non-complete case. For completeness, we give below a proof which has the advantage to hold in a wider context and for a wider class of operator, pseudodifferential operators for instance. We start with a general lemma. We recall that a Weyl sequence for a couple (H, λ) with H a self-adjoint operator and λ ∈ R, is a sequence ϕn ∈ D(H) such that ϕn = 1, ϕn 0 (weakly) and such that (H − λ)ϕn → 0, as n goes to infinity. It is well-known that λ ∈ σess (H) if and only if there is Weyl sequence for (H, λ). Lemma C.1. Let H be a self-adjoint operator in a Hilbert space H . Let ϕn be a Weyl sequence for the couple (H, λ). Suppose that there is a closed operator Φ in H such that: 1. ΦD(H) ⊂ D(H), 2. Φ(H + i)−1 is compact, 3. [H, Φ] is a compact operator from D(H) to H . n is a Weyl sequence for (H, λ). Then there is ϕ n ∈ D(H) such that (1 − Φ)ϕ Proof. First we note (2) implies that Φϕn goes to 0. Indeed, we have Φϕn = Φ(H + i)−1 ((H − λ)ϕn + (i + λ)ϕn 1) and the bracket goes weakly to 0. Similarly, using (3) we get that [H, Φ]ϕn → 0. Therefore we obtain (1 − Φ)ϕn ≥ 1/2 for n

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large enough. We set ϕ n := ϕn /(1 − Φ)ϕn . Note that (1 − Φ)ϕ n 0. Finally, (H − λ)(1 − Φ)ϕ n → 0 since [H, Φ]ϕn → 0. This shows that the essential spectrum is given by a “non-compact” part of the space. We now focus on Friedrichs extension. Given a dense subspace D of a Hilbert space H and a positive symmetric operator on D. Let H1 be the completion of D under the norm given by Q(ϕ)2 = Hϕ, ϕ+ ϕ2 . The domain of the Friedrichs extension of H, is given by D(HF ) = {f ∈ H1 | D g → Hg, f + g, f extends to a norm continuous function on H }. For each f ∈ D(HF ), there is a unique uf such that Hg, f + g, f = g, uf , by Riesz theorem. The Friedrichs extension of H is defined by setting HF f := uf − f . It is a self-adjoint extension of H, see [41]. Let (X, g) be a smooth Riemannian with distance d. We fix K a smooth compact sub-manifold of X of same dimension. We endow it with the induced Riemannian metric. We set X = X \ K. In the following, we embed L2 (X ) in L2 (X). We will need the next definition within the proof. Definition C.2. We say that Φ is a (smooth) cut-off function for K if Φ ∈ Cc∞ (X) and Φ|K = 1. We say that it is an ε-cut-off is supp(Φ) ⊂ B(K, ε). We are now able to give a result of stability of the essential spectrum. Proposition C.3. Let d be a differential form of order 1 on Cc∞ (X) → C ∞ (X, Λ1 ) with injective symbol away from the 0 section of the cotangent bundle. We denote by dX and dX the closure of d in L2 (X) and L2 (X ), respectively. Consider ΔX := d∗X dX and ΔX = d∗X dX , the Friedrichs extensions of the operator d∗ d, acting on Cc∞ (X) and Cc∞ (X ), respectively. One has σess (ΔX ) = σess (ΔX ). Proof. Let f ∈ L2 (B(K, ε)c ) and let Φ be a ε-cut-off for K. We first show that f ∈ D(dX ) if and only if f ∈ D(dX ). Suppose that f ∈ D(dX ), then for all η > 0, there is ϕ ∈ Cc∞ (X) such that f − ϕn + df − dϕn < η. Because of the support of f , one obtain that Φϕ < ηΦ∞ and that [d, Φ]ϕ < η[d, Φ]∞ . Therefore (1 − Φ)ϕn ∈ Cc∞ (X ) and is Cauchy in D(dX ), endow with the graph norm. By uniqueness of the limit, one obtains that f ∈ D(dX ) and that dX f = dX f . The opposite implication is obvious. Using again the ε-cut-off, one shows that g ∈ D(d∗X ) if and only if g ∈ D(d∗X ) and that d∗X g = d∗X g for g ∈ L2 (Λ1 (B(K, ε)c ). Finally, we obtain that f ∈ D(ΔX ) if and only if f ∈ D(ΔX ) and that ΔX f = ΔX f , for f ∈ L2 (B(K, ε)c ). From the definition of the Friedrichs extension and the injectivity of the 2 symbol of d, the domain of ΔX , ΔX is contained in H01 (X)∩Hloc (X), respectively 2 in H01 (X ) ∩ Hloc (X ). By taking the same Φ as above and using the Rellich– Kondrakov lemma, the hypotheses of Lemma C.1 are satisfied. Finally, we apply it to ΔX and ΔX and since the Weyl sequence is with support away from K, the first part of the proof gives us the double inclusion of the essential spectra.

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Acknowledgements We acknowledge useful discussions with Barbu Berceanu, Dan Burghelea, Jan Derezi´ nski, Vladimir Georgescu, Bernard Helffer, Andreas Knauf, Fran¸cois Nicoleau, Marius M˘ antoiu and Radu Purice. We are also grateful to the referee for helpful remarks. The authors were partially supported from the contract MERG 006375, funded by the European Commission. The second author was partially supported by the contracts 2-CEx06-11-18/2006 and CNCSIS-GR202/19.09.2006 (Romania).

References [1] Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959), no. 2, 485–491. [2] W. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -Groups, commutator methods and spectral theory of N -body Hamiltonians, Birkh¨ auser, Basel, Boston, Berlin, 1996. [3] J.-M. Bily and D. Robert, The semi-classical Van–Vleck formula. Application to the Aharonov–Bohm effect, World Scientific. Ser. Concr. Appl. Math. 1 (2001), 89–106. [4] R. Bott and L. W. Tu, Differential forms in algebraic topology, Graduate Texts Math. 82, Springer-Verlag, New York, Berlin, 1982. [5] J. M. Bouclet, Resolvent estimates for the Laplacian on asymptotically hyperbolic manifolds, Ann. H. Poincaré 7 (2006), 527–561. [6] T. Christiansen, Weyl asymptotics for the Laplacian on manifolds with asymptotically cusp ends, J. Funct. Anal. 187 (2001), 211–226. [7] H. Cycon, R. Froese, W. Kirsch and B. Simon, Schr¨ odinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. [8] S. Debièvre, P. D. Hislop and I. M. Sigal, Scattering theory for the wave equation on non-compact manifolds, Rev. Math. Phys. 4 (1992), no. 4, 575–618. [9] H. Donnelly and P. Li, Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J. 46 (1979), no. 3, 497–503. [10] R. Froese and P. D. Hislop, Spectral analysis of second-order elliptic operators on noncompact manifolds, Duke Math. J. 58 (1989), no. 1, 103–129. [11] R. Froese and P. D. Hislop and P. Perry, A Mourre estimate and related bounds for hyperbolic manifolds with cusps of non-maximal rank, J. Funct. Anal. 98 (1991), no. 2, 292–310. [12] R. Froese and P. D. Hislop and P. Perry, The Laplace operator on hyperbolic three manifolds with cusps of non-maximal rank, Invent. Math. 106 (1991), no. 2, 295–333. [13] V. Georgescu and C. Gérard, On the Virial theorem in quantum mechanics, Commun. Math. Phys. 208 (1999), no. 2, 275–281. [14] V. Georgescu, C. Gérard and J. Møller, Commutators, C0 -semigroups and resolvent estimates, J. Func. Analysis 216 (2004), no. 2, 303–361. [15] V. Georgescu and S. Golénia, Decay preserving operators and stability of the essential spectrum, preprint math.SP/0411489, to appear in Journal of Operator Theory.

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[16] V. Georgescu and M. M˘ antoiu, On the spectral theory of Dirac type Hamiltonians, J. Operator Theory 46 (2001), 289–321. [17] S. Golénia and T. Jecko, A new look at Mourre’s commutator theory, Complex Analysis Operator Theory 1 (2007), no. 3, 399–422. [18] S. Golénia and S. Moroianu, The spectrum of magnetic Schr¨ odinger operators and k-form Laplacians on conformally cusp manifolds, Unpublished preprint math.DG/0507443. [19] M. J. Gruber, Bloch theory and quantization of magnetic systems, J. Geom. Phys. 34 no. 2 (2000), 137–154. [20] L. Guillopé, Théorie spectrale de quelques variétés ` a bouts, Ann. Sci. Ecole Norm. Sup. 22 no. 4 (1989), 137–160. [21] O. Hebbar, Bohm–Aharonov effects for bounded states in the case of systems, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 4, 489–500. [22] B. Helffer, Effet d’Aharonov–Bohm sur un état borné de l’équation de Schr¨ odinger, Comm. Math. Phys. 119 (1988), no. 2, 315–329. [23] B. Helffer and A. Mohamed, Caractérisation du spectre essentiel de l’opérateur de Schr¨ odinger avec un champ magn´ etique, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 2, 95–112. [24] I. Herbst and S. Nakamura, Schr¨ odinger operators with strong magnetic fields: quasiperiodicity of spectral orbits and topology, Differential operators and spectral theory, 105–123, Amer. Math. Soc. Transl. Ser. 2, 189, Providence, RI, 1999. [25] P. D. Hislop, The geometry and spectra of hyperbolic manifolds, Proc. Indian Acad. Sci., Math. Sci. 104 (1994), no. 4, 715–776. [26] V. Kondratiev and M. Shubin, Discreteness of spectrum for the magnetic Schr¨ odinger operators, Comm. Partial Diff. Equ. 27 (2002), no. 3–4, 477–525. [27] D. Krejcir´ık, R. Tiedra de Aldecoa, The nature of the essential spectrum in curved quantum waveguides, J. Phys. A, Math. Gen. 37 (2004), no. 20, 5449–5466. [28] H. Kumura, Limiting absorption principle and absolute continuity of the Laplacian on a manifold having ends with various radial curvatures, preprint math/0606125. [29] R. Lauter and V. Nistor, On spectra of geometric operators on open manifolds and differentiable groupoids, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 45–53. [30] M. Lesch and N. Peyerimhoff, On index formulas for manifolds with metric horns, Comm. Partial Diff. Equ. 23 (1998), 649–684. [31] M. Loss and B. Thaller, Scattering of particles by long-range magnetic fields, Ann. Physics 176 (1987), no. 1, 159–180. [32] J. Lott, On the spectrum of a finite-volume negatively-curved manifold, Amer. J. Math. 123 (2001), no. 2, 185–205. [33] R. R. Mazzeo and R. B. Melrose, Pseudodifferential operators on manifolds with fibered boundaries, Asian J. Math. 2 (1998), 833–866. [34] R. B. Melrose, The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics 4, A. K. Peters, Wellesley, MA (1993). [35] R. B. Melrose and V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, preprint funct-an/9606005. [36] K. Miller and B. Simon, Quantum magnetic Hamiltonians with remarkable spectral properties, Phys. Rev. Lett. 44 (1980), 1706–1707.

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[37] S. Moroianu, K-Theory of suspended pseudo-differential operators, K-Theory 28 (2003), 167–181. [38] S. Moroianu, Weyl laws on open manifolds, to appear in Math. Ann. [39] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys. 91 (1981), 391–408. [40] F. Nicoleau, An inverse scattering problem with the Aharonov–Bohm effect, J. Math. Phys. 41 (2000), no. 8, 5223–5237. [41] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, self-adjointness, Academic Press, New York–San Francisco–London, 1975. [42] P. Roux, Scattering by a toroidal coil, J. Phys. A, Math. Gen. 36 (2003), no. 19, 5293–5304. [43] P. Roux and D. Yafaev, On the mathematical theory of the Aharonov–Bohm effect, J. Phys. A, Math. Gen. 35 (2002), no. 34, 7481–7492. [44] P. Roux and D. Yafaev, The scattering matrix for the Schr¨ odinger operator with a long-range electromagnetic potential, J. Math. Phys. 44 (2003), no. 7, 2762–2786. [45] J. Sahbani, The conjugate operator method for locally regular Hamiltonians, J. Oper. Theory 38 (1997), no. 2, 297–322. [46] M. Shubin, Essential self-adjointness for semi-bounded magnetic Schr¨ odinger operators on non-compact manifolds, J. Funct. Anal. 186 (2001), no. 1, 92–116. [47] D. Yafaev, There is no Aharonov–Bohm effect in dimension three, preprint. [48] R. Weder, The Aharonov–Bohm effect and time dependent inverse scattering theory, Inverse Probl. 18 (2002), no. 4, 1041–1056. Sylvain Golénia Mathematisches Institut der Universit¨ at Erlangen-N¨ urnberg Bismarckstr. 1 1/2 D-91054 Erlangen Germany e-mail: [email protected] Sergiu Moroianu Institutul de Matematic˘ a al Academiei Romˆ ane P.O. Box 1-764 RO-014700 Bucharest Romania and S ¸ coala Normal˘ a Superioar˘ a Bucharest Calea Grivit¸ei 21 Bucharest Romania e-mail: [email protected] Communicated by Christian Gérard. Submitted: February 6, 2007. Accepted: August 20, 2007.



On Spectral Properties of Translationally Invariant Magnetic Schrödinger Operators Dimitri Yafaev Abstract. We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr¨ odinger operator H with such a potential. In particular, we show that the spectrum of H is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions exp(−iHt)f of the time dependent Schr¨ odinger equation. It turns out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to an evolution of a one-dimensional free particle but “exits” to +∞ and −∞ in the direction of the potential might be essentially different.

1. Introduction 1.1. Translationally invariant magnetic fields B(x) = (b1 (x), b2 (x), b3 (x)), x = (x1 , x2 , x3 ), div B(x) = 0, give important examples where a non-trivial information can be obtained about spectral properties of the corresponding Schr¨ odinger operators H. We suppose for definiteness that B(x) does not depend on the x3 variable so that H commute with translations along the x3 -axis. There are two essentially different (and in some sense extreme) classes of translationally invariant magnetic fields. The first class consists of fields B(x) = (0, 0, b3 (x1 , x2 )) of constant direction. For such fields, the momentum p of a classical particle in the x3 -direction is conserved, and in the Schr¨ odinger equation the variable x3 can be separated. Thus, we arrive to a two-dimensional problem in the (x1 , x2 )-plane. Furthermore, if b3 is a function of r = (x21 + x22 )1/2 only, then we get a set of problems on the half-line r > 0 labelled by the magnetic quantum number m. The most important example of this type is a constant magnetic field b3 (r) = const (see [10]). Some class of functions b3 (r) decaying as r → ∞ was discussed in [12] (see also [2]) where new

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interesting effects were found. Another famous case is b3 (x1 , x2 ) = δ(x1 , x2 ) (δ(·) is the Dirac delta-function) studied in [1]. Scattering by an arbitrary short-range (decaying faster than |x|−2−ε , ε > 0, as |x| → ∞) magnetic field b3 (x1 , x2 ) turns out to be rather similar to this particular case (see [16]). The second class consists of fields B(x) = (b1 (x1 , x2 ), b2 (x1 , x2 ), 0) orthogonal to the x3 -axis. In this case the corresponding magnetic potential A(x), defined (up to gauge transformations) by the equation curl A(x) = B(x), can be chosen as A(x) = 0, 0, −a(x1 , x2 ) (1.1) so that it has the constant direction. In contrast to fields of the first class, now the variable x3 cannot be separated in the Schr¨ odinger equation. Nevertheless due to the invariance with respect to translations along the x3 -axis the operator (we always suppose that the charge of a particle is equal to 1) 2 H = i∇ + A(x) (1.2) can be realized, after the Fourier transform in the variable x3 , in the space L2 (R; L2 (R2 )) as the operator of multiplication by the operator-valued function 2 H(p) = −Δ + a(x1 , x2 ) + p : L2 (R2 ) → L2 (R2 ) , p ∈ R . (1.3) Moreover, if a(x1 , x2 ) = a(r), then the subspaces with fixed magnetic quantum number m ∈ Z are invariant subspaces of H(p) so that the operator H(p) reduces to the orthogonal sum over m ∈ Z of the operators Hm (p) = −

2 m2 1 d d r + 2 + a(r) + p r dr dr r

(1.4)

acting in the space H = L2 (R+ ; rdr). In this case the field is given by the equation B(x) = b(r)(− sin θ, cos θ, 0)

(1.5)

where b(r) = a (r) and θ is the polar angle. Thus, vectors B(x) are tangent to circles centered at the origin. An important example of such type is a field created by a current along an infinite straight wire (coinciding with the x3 -axis). In this case b(r) = b0 r−1 so that a(r) = b0 ln r. The Schr¨ odinger operator with such magnetic potential was studied in [15]. 1.2. In this article we consider magnetic fields (1.5) with a sufficiently arbitrary function b(r). Our goal is to study basic spectral properties of the corresponding Schr¨ odinger operator H such as the absolute continuity, location and multiplicity of the spectrum, as well as the long-time behaviour of the unitary group exp (−iHt). We emphasize that for magnetic fields considered here, the problem is genuinely three-dimensional, and actually the motion of a particle in the x3 -direction is of a particular interest. Using the cylindrical invariance of field (1.5), we can start either from translational or from rotational (around the x3 -axis) symmetries. The rotational invariance implies that the operator H is the orthogonal sum of its restrictions Hm on the

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subspaces of functions with magnetic quantum number m ∈ Z = {0, ±1, ±2, . . .}. It can be identified with the operator (we keep the same notation for this operator) 2 m2 1 d d d r + 2 + i + a(r) (1.6) Hm = − r dr dr r dx3 acting in the space H = L2 (R+ × R; rdrdx3 ). In view of the translation invariance, every operator Hm can be realized (again after the Fourier transform in the variable x3 ) in the space L2 (R; L2 (R+ ; rdr)) as the operator of multiplication by the operator-valued function Hm (p) defined by (1.4). Suppose now that b(r) does not tend to zero too fast so that a(r) → ∞ as r → ∞. Then the spectrum of each operator Hm (p) is discrete. Let λn,m (p), n ∈ N = {1, 2, . . .}, be the increasing sequence of its eigenvalues (they are simple and positive), and let ψn,m (r, p) be the corresponding sequence of its eigenfunctions. The functions λn,m (p) are known as dispersion curves of the problem. They determine the spectral properties of the operator Hm . Note that if a(r) is replaced by −a(r), then λn,m (p) is replaced by λn,m (−p), so that the case a(r) → −∞ as r → ∞ is automatically included in our considerations. Recall that, for a magnetic field B(x), the magnetic potential A(x) such that curl A(x) = B(x) is defined up to a gauge term grad ϕ(x). In particular for magnetic fields (1.5) in the class of potentials A(x) = (0, 0, −a(r)) one can always add to a(r) an arbitrary constant c. This leads to the transformations λn,m (p) → λn,m (p − c) and ψn,m (r, p) → ψn,m (r, p − c). 1.3. The precise definitions of the operators Hm and H and their decompositions into the direct integrals over the operators Hm (p) and H(p) are given in Section 2. To put it differently, we construct a complete set of eigenfunctions of the operator H. They are parametrized by the magnetic quantum number m, the momentum p in the direction of the x3 -axis and the number n of an eigenvalue λm,n (p) of the operator Hm (p). Thus, if we set ψ n,m,p (r, θ, x3 ) = eipx3 eimθ ψn,m (r, p) ,

(1.7)

ψ n,m,p = λn,m (p)ψ ψ n,m,p . Hψ

(1.8)

then In Section 3, we show that for all n ∈ N and m ∈ Z: • Under very general assumptions λn,m (p) → ∞ as p → ∞ (Proposition 3.3). • If b(r) → 0 as r → ∞, then λn,m (p) → 0 as p → −∞ (Proposition 3.5). • If b(r) admits a finite positive limit b0 as r → ∞, then λn,m (p) → (2n − 1)b0 for all m as p → −∞ (Proposition 3.6). • If b(r) → ∞ as r → ∞, then λn,m (p) → ∞ as p → −∞ (Proposition 3.6). Related results concerning the dispersion curves for Schrödinger operator with constant magnetic fields defined on unbounded domains Ω ⊂ R2 have been obtained in [5] (the case where Ω is a strip) and in [4], [7, Section 4.3] (the case where Ω is a half-plane).

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In Theorem 3.8 we formulate the main spectral results which follow from the asymptotic properties of the dispersion curves λn,m (p), p ∈ R. First, the analyticity and the asymptotics as p → ∞ of λn,m (p) imply immediately that the spectra σ(Hm ) and σ(H) of the operators Hm , m ∈ Z, and H are purely absolutely continuous. Moreover, σ(Hm ) = [Em , ∞) ,

m ∈ Z,

σ(H) = [E0 , ∞) ,

(1.9)

where Em = inf λ1,m (p) ≥ 0 . p∈R

(1.10)

Next, in the case where the magnetic field tends to 0 as r → ∞, the spectra of Hm , m ∈ Z, coincide with [0, ∞) and have infinite multiplicity. On the other hand, in the case where the magnetic field tends as r → ∞ to a positive finite limit, or to infinity, we have that Em > 0 for all m ∈ Z and each of the spectra σ(Hm ) contains infinitely many thresholds. Further, in Section 4, we obtain a convenient formula for the derivatives λn,m (p) which play the role of asymptotic group velocities. Our formula for λn,m (p) yields sufficient conditions (see Theorem 4.3) for positivity of these functions. The leading example when these conditions are met, is b(r) = b0 r−δ ,

b0 > 0 ,

δ ∈ [0, 1] ,

(1.11)

and m = 0. If δ = 1, this result remains true for all m (cf. [15]). On the contrary, if δ = 0 and m = 0, then λn,0 (p) < 0 for all n on some interval of p (lying on the negative half-axis). Similar results concerning for the dispersion curves for the Schr¨ odinger operator with constant magnetic field, defined on the half-plane with Dirichlet (resp., Neumann) boundary conditions, can be found in [4] (resp., [3] and [7, Section 4.3]). Finally, in Section 5 we discuss the long-time behaviour of a quantum particle. The time evolution of a quantum system is determined by the unitary groups exp (−iHm t), m ∈ Z, so that an analysis of its asymptotics as t → ±∞ relies on spectral properties of the operators Hm . Since these operators have discrete spectra, a quantum particle remains localized in the (x1 , x2 )-plane. Its propagation in the x3 -direction is governed by the group velocities λn,m (p). In particular, the condition λn,m (p) > 0 for all n ∈ N and p ∈ R implies that a quantum particle with the magnetic quantum number m propagates as t → +∞ in the positive direction of the x3 -axis. Let us compare these results with the long-time behaviour of a classical particle in magnetic field (1.5). Let x(t) = (x1 (t), x2 (t), x3 (t)) be its trajectory. As shown in [15], the function x3 (t) is periodic with period T determined by initial conditions. The drift of a particle x3 (T ) − x3 (0) over the period is nonnegative if b(r) ≥ 0 and b (r) ≥ 0. Morever, it is strictly positive if b (r) > 0 for all r. In the case b(r) = const it is still strictly positive if the angular momentum m of a particle is not zero. Thus, our results for functions (1.11) correspond completely to the

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classical picture if δ = 1 or δ ∈ [0, 1) and m = 0. In the case δ = 0 and m = 0 the behaviour of quantum and classical particles turn out to be qualitatively different.

2. Hamiltonians and their diagonalizations Here we give precise definitions of the Hamiltonians and discuss their reductions due to the cylindrical symmetry. 2.1. For an arbitrary magnetic potential A : R3 → R3 such that A ∈ the self-adjoint Schr¨ odinger operator (1.2) can be defined via its quadratic form 2 h[u] = |i(∇u)(x) + A(x)u(x)| dx , x = (x1 , x2 , x3 ) . (2.1) L2loc (R3 )3 ,

R3

It is easy to see that this form is closed on the set of functions u ∈ L2 (R3 ) such that ∇u ∈ L1loc (R3 )3 and i∇u + Au ∈ L2 (R3 )3 . Similarly, if a ∈ L2loc (R2 ), then the self-adjoint Schr¨ odinger operator (1.3) can be defined via its quadratic form 2 2 |(∇u)(x)| + a(x)+p |u(x)|2 dx , x = (x1 , x2 ) , p ∈ R . (2.2) h[u; p] = R2

This form is closed on the set of functions u ∈ L2 (R2 ) such that integral (2.2) is finite. Clearly, this set does not depend on the parameter p ∈ R. Let F : L2 (R3 ) → L2 (R; L2 (R2 )) be the Fourier transform with respect to x3 , i.e., 1 e−ix3 p u(x1 , x2 , x3 )dx3 . (F u)(x1 , x2 , p) = √ 2π R If A(x) is given by formula (1.1), then ∞

h[u] = h (F u)(p); p dp −∞

which implies the equation

(F Hu)(x1 , x2 ; p) = H(p)F u (x1 , x2 ; p) .

(2.3)

This equation can be regarded as a “working” definition of the operator H. 2.2. Assume now that the function a in (1.1) depends only on r, and a ∈ L2loc [0, ∞); rdr .

(2.4)

If we separate variables in the cylindrical coordinates (r, θ, x3 ) and denote by Hm ⊂ L2 (R3 ) the subspace of functions f (r, x3 )eimθ where f ∈ L2 (R+ × R; rdrdx3 ) and m ∈ Z is the magnetic quantum number, then Hm . L2 (R3 ) = m∈Z

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The subspaces Hm are invariant with respect to H so that restrictions Hm of H on Hm are related with H by formula H= Hm . (2.5) m∈Z

Every Hm can obviously be identified with the space L2 (R+ × R; rdrdx3 ) =: H; then Hm is identified with operator (1.6). Quite similarly, if Hm ⊂ L2 (R2 ) is the subspace of functions f (r)eimθ where f ∈ L2 (R+ ; rdr), then L2 (R2 ) = Hm . m∈Z

The subspaces Hm are invariant with respect to H(p) so that restrictions Hm (p) of H(p) on Hm are related with H(p) by formula H(p) = Hm (p) . (2.6) m∈Z

Every Hm can obviously be identified with the space L2 (R+ ; rdr) =: H; then Hm (p) is identified with operator (1.4). Let Fm : Hm → L2 (R; L2 (R+ ; rdr)) be the restriction of F on the subspace Hm . Then we have (cf. (2.3)) (Fm Hm f )(r; p) = Hm (p)Fm f (r; p) . (2.7) Sometimes it is more convenient to consider instead of Hm (p) the operator Lm (p) = r1/2 Hm (p)r−1/2 = −

2 d2 m2 − 1/4 + + a(r) + p 2 2 dr r

(2.8)

acting in the space L2 (R+ ) and unitarily equivalent to the operator Hm (p). It is easy to see that the operator Lm (p) corresponds to the quadratic form ∞ 2 |g (r)|2 + (m2 − 1/4)r−2 |g(r)|2 + a(r) + p |g(r)|2 dr , (2.9) lm [g; p] = 0

defined originally on C0∞ (R+ ), and then closed in L2 (R+ ). 2.3. If a(r) → ∞ as r → ∞ ,

(2.10)

then the spectrum of the operator Hm (p), p ∈ R, m ∈ Z, is discrete. Thus, it consists of the increasing sequence λn,m (p) of simple eigenvalues. Since Hm (p), p ∈ R, is a Kato analytic family of type (B) (see [9, Chapter VII, Section 4]), all the eigenvalues λn,m (p) are real analytic functions of p ∈ R. Moreover, λn,m (p) > 0 because form (2.2) is strictly positive. In view of formula (2.7) spectral analysis of the operators Hm reduces to a study of a family of functions λn,m (p), n ∈ N. Indeed, let Λn,m be the operator of multiplication by the function λn,m (p) in the space L2 (R). We denote by ψn,m (r; p)

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real normalized eigenfunctions (defined up to signs) of the operators Hm (p) and introduce an isometric mapping Ψn,m : L2 (R) → L2 (R+ × R; rdrdp) by the formula (Ψn,m w)(p) = ψn,m (r, p)w(p) . Then

L2 (R+ × R; rdrdp) =

(2.11)

RanΨn,m

n∈N

and Hm =

∗ Fm Ψn,m Λn,m Ψ∗n,m Fm .

(2.12)

n∈N

Together with (2.5), formulas (2.11) and (2.12) justify equations (1.8) for functions (1.7).

3. Dispersion curves and spectral analysis 3.1. In this subsection we consider the operators H(p) acting in the space L2 (R2 ) by formula (1.3). Under the assumption a ∈ L2loc (R2 ) they are correctly defined by their quadratic forms (2.2). If a(x) → ∞ as

|x| → ∞ ,

x = (x1 , x2 ) ,

(3.1)

then the spectrum of H(p) consists of eigenvalues λn (p), n ∈ N. We enumerate them in the increasing order with multiplicity taken into account. Our goal is to investigate the asymptotic behaviour of the eigenvalues λn (p) as p → ∞. Below we denote by C and c different positive constants whose precise values are of no importance. We use the following elementary Lemma 3.1. Let v(x) ≥ 0. For an arbitrary ε > 0, we have the inequality

1/2 2 2 v(x)|u(x)| dx ≤ C sup v (y)dy ε|∇u(x)|2 + ε−1 |u(x)|2 dx x∈R2

R2

R2

|x−y|≤ε

(3.2) provided the supremum in the right-hand side is finite. Proof. Let Πε ⊂ R2 be a square of length ε. We proceed from the estimate 1/2 4 2 −1 2 |u(x)| dx ≤C ε |∇u(x)| dx + ε |u(x)| dx Πε

Πε

Πε

which follows from the Sobolev embedding theorem by a scaling transformation. Using the Schwarz inequality, we deduce from this estimate that 1/2 v(x)|u(x)|2 dx ≤ C v 2 (x)dx |∇u(x)|2 dx + ε−1 |u(x)|2 dx . ε Πε

Πε

Πε

Πε

(3.3)

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Let us split the space R2 in the lattice of squares Πε of length ε. Applying (3.3) (n) to every Πε and summing over all n, we arrive at (3.2). In the following assertion we do not assume (3.1). Proposition 3.2. Let a ∈ L2loc (R2 ). Set a− (x) = max{−a(x), 0}, a2− (y)dy α(ε) = sup x∈R2

|x−y|≤ε

and suppose that α(ε) → 0 as ε → 0. Then we have lim inf p−2 inf σ H(p) ≥ 1 .

(3.4)

p→∞

Proof. Applying estimate (3.2) with ε = p−1 to the function v = a− , we find that |∇u|2 + (p + a)2 |u|2 dx ≥ |∇u|2 + (−2pa− + p2 )|u|2 dx 2 R2 R ≥ |∇u|2 + p2 |u|2 dx 2 R |∇u|2 + p2 |u|2 dx . − C α(p−1 ) R2

Since α(p

−1

) → 0 as p → ∞, this implies (3.4).

Proposition 3.3. Let a ∈ L2loc (R2 ) and let condition (3.1) be satisfied. Then, for all n ∈ N, we have (3.5) λn (p) = p2 1 + o(1) , p → ∞ . Proof. Under condition (3.1) the function a− has compact support so that we can use Proposition 3.2 and estimate (3.4) implies lim inf p−2 λn (p) ≥ 1 .

(3.6)

p→∞

Set G(ε) = −Δ+ (1 + ε−1)a2 (x), ε > 0. The spectrum of G(ε) is discrete; let νn (ε), n ∈ N, be the increasing sequence of its eigenvalues. By the elementary inequality (a + p)2 ≤ (1 + ε−1 )a2 + (1 + ε)p2 ,

ε > 0,

2

we have H(p) ≤ G(ε) + (1 + ε)p so that by the minimax principle λn (p) ≤ νn (ε) + (1 + ε)p2 . Therefore, for all ε > 0, lim sup p−2 λn (p) ≤ 1 + ε , p→∞

which combined with (3.6) yields (3.5).

Corollary 3.4. Suppose that the function a depends on r only. Let conditions (2.4) and (2.10) be satisfied. Then, for all n ∈ N, m ∈ Z, we have λn,m (p) = p2 1 + o(1) , p → ∞ .

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3.2. From now on we always assume that the function a depends on r only and that conditions (2.4) and (2.10) are satisfied. In this subsection we investigate the asymptotics as p → −∞ of the eigenvalues λn,m (p) of the operators Hm (p). Actually, it is more convenient to work with the operators Lm (p) acting in the space L2 (R+ ) by formula (2.8). We suppose that the function a is differentiable at least for sufficiently big r and formulate the results in terms of the function b(r) = a (r) related to the magnetic field by formula (1.5). Remark first that if k = −p > 0 is big enough, then the equation a(r) = k

(3.7)

has at least one solution. We denote by ρk the greatest solution of (3.7). Clearly, ρk → ∞ as k → ∞. Proposition 3.5. Suppose that lim b(r) = 0 .

(3.8)

r→∞

Then for each n ∈ N and m ∈ Z we have lim λn,m (−k) = 0 .

(3.9)

k→∞

Proof. Set b(r) = sup |b(x)| x≥r

and γk = b(ρk )−1/2 .

(3.10)

1

Let us fix n ∈ N. We pick a function φ1 ∈ C0∞ (R) such that supp φ1 = 0, 2n and, for n > 1, set φj (x) = φ1 x − (j − 1)/n , x ∈ R , j = 2, . . . , n . For k > 0 large enough, we put −1/2

ϕj (r; k) = γk

φj

r − ρk γk

,

r ≥ 0,

j = 1, . . . , n .

We will prove now that for quadratic form (2.9)

lim lm ϕj (k); −k = 0 .

(3.12)

k→∞

It follows from (3.11) that

0

∞

(3.11)

|ϕj (r; k)|2 dr ≤ Cγk−2

(3.13)

with C independent of k. Further, since supp ϕj (k) ⊂ [ρk , ρk + γk ], we have ∞ r−2 |ϕj (r; k)|2 dr ≤ Cρ−2 (3.14) k . 0

Similarly,

0

∞

2 a(r) − k |ϕj (r; k)|2 dr ≤ C

sup r∈(ρk ,ρk +γk )

2 a(r) − k .

(3.15)

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Using the condition a(ρk ) = k, we obtain, for r ≥ ρk , the bound r 2 2 2 a(r) − k = a(r) − a(ρk ) = b(s)ds ≤ (r − ρk )2 b2 (ρk ) ρk

where b is function (3.10). Thus, the right-hand side in (3.15) is bounded by Cγk2 b2 (ρk ). Putting together this result with inequalities (3.13), (3.14) and taking into account (3.10), we get

. lm ϕj (k); −k ≤ C b(ρk ) + ρ−2 k This yields (3.12). Let us use now that the supports of the functions ϕj (k), j = 1, . . . , n, are disjoint and set Ln (k) = span ϕ1 (k), . . . , ϕn (k) . (3.16) Then dim Ln (k) = n and according to (3.12) lm [ϕ(k); −k] → 0 as k → ∞ for all ϕ(k) ∈ Ln (k) with ϕ(k) = 1. By the mini-max principle this implies (3.9). The proof of Proposition 3.6 relies on a comparison of the operator Lm (−k) with the “model” operator d2 + b2 (k )(x − k )2 , x ∈ R , (3.17) dx2 acting in the space L2 (R). Let fj be the normalized real-valued eigenfunctions in L2 (R) of the harmonic oscillator (defined up to sign), i.e., T (k) = −

−fj (x) + x2 fj (x) = (2j − 1)fj (x) , Then

x ∈ R,

j ∈ N.

ψj (x; k) = b(k )1/4 fj b(k )1/2 (x − k )

(3.18) (3.19)

are normalized eigenfunctions of the operator T (k), that is T (k)ψj (k) = b(k )(2j − 1)ψj (k) ,

j ∈ N.

(3.20)

The proof of the following result follows the general lines of the proof of [2, Theorem 11.1]. Proposition 3.6. Suppose that a(r) is locally semibounded from above. For r > 0 large enough, we assume that the function b(r) is differentiable and that conditions b(r) > 0 ,

(3.21)

2

lim r b(r) = ∞ ,

(3.22)

r→∞

as well as lim b(r)−3 b21 (r) = 0 ,

r→∞

where

b1 (r) =

sup r/2≤x≤3r/2

|b (x)| ,

(3.23)

are satisfied. Let also lim k −2 b(ρk ) = 0 .

k→∞

(3.24)

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Then, for all n ∈ N, m ∈ Z, we have λn,m (−k) = b(k ) 2n − 1 + o(1) ,

k → ∞.

191

(3.25)

Proof. Due to the minimax principle, it suffices to show that: (i) For each n ∈ N and sufficiently large k there exists a subspace Ln (k) of L2 (R+ ) such that dim Ln (k) = n, Ln (k) ⊂ D(Lm (−k)), and for each ϕ(k) ∈ Ln (k) we have Lm (−k)ϕ(k), ϕ(k) ≤ b(k ) 2n − 1 + o(1) ϕ(k) 2 , k → ∞ . (3.26) (ii) For each n ∈ N there exists a bounded operator Rn (k) such that rank Rn (k) ≤ n − 1 (hence, R1 (k) = 0), and Lm (−k) ≥ b(k ) 2n − 1 + o(1) I + Rn (k) , k → ∞ . (3.27) We pick γk > 0 such that γk → 0 , γk k b(k )1/2 → ∞ , γk−3

−3/2

b(k )

b1 (k ) → 0

(3.28) (3.29) (3.30)

as k → ∞. Note that (3.29) is compatible with (3.28) due to (3.22), and (3.30) is compatible with (3.28) due to (3.23). Proof of (i). Let ζ ∈ C0∞ (R) be such that 0 ≤ ζ(x) ≤ 1 , ζ(x) = 1 for |x| ≤ 1/2 and supp ζ = [−1, 1]. For k large enough, set (3.31) ζ(r; k) = ζ γk b(k )1/2 (r − k ) , r ∈ R+ , and (3.32) ϕj (r; k) = ψj (r; k)ζ(r; k) , r ∈ R+ , j ∈ N , the functions ψj (r; k) being defined in (3.19). It follows from (3.29) that

supp ϕj (k) = k − γk−1 b(k )−1/2 , k + γk−1 b(k )−1/2 ⊂ [k /2, 3k /2] and, in particular, ϕj (k) ∈ D(Lm (−k)). Note that ϕj (k), ϕl (k) L2 (R+ ) = δjl − ψj (x; k)ψl (x; k) 1−ζ 2 (x; k) dx = δjl +o(1) (3.33) R

as k → ∞. Indeed, the integral here can be estimated by 2 |fj (x)fl (x)| 1 − ζ (γk x) dx ≤ |fj (x)fl (x)|dx |x|≥(2γk )−1

R

which tends to zero according to (3.28). In particular, (3.33) implies that for all n ∈ N the functions ϕ1 (k), . . . , ϕn (k) are linearly independent if k is large enough. Thus, the space Ln (k) defined by (3.16) has dimension n. Let us set n ψ(x; k) = cj ψj (x; k) , cj ∈ C , (3.34) j=1

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ϕ(r; k) = ψ(r; k)ζ(r; k) and consider Lm (−k)ϕ(k), ϕ(k). Integrating by parts, we find that −2Re ψ (k)ζ (k), ψ(k)ζ(k) − ψ(k)ζ (k), ψ(k)ζ(k) = ψ(k)ζ (k) 2 so that 2 Lm (−k)ϕ(k), ϕ(k) = Re − ψ (k) + a(r) − k ψ(k), ψ(k)ζ 2 (k) + ψ(k)ζ (k) 2 + (m2 − 1/4) r−1 ϕ(k) 2 .

(3.35)

We assume that ϕ(k) = 1 and hence according to (3.33) ψ(k) = 1 + o(1). The second and third terms in the right-hand side of (3.35) are negligible. Indeed, differentiating (3.31) and using condition (3.28), we find that (3.36) ψ(k)ζ (k) 2 = O b(k )γk2 = o b(k ) . Since r−1 ≤ 2−1 k on the support of ϕ(k), relation (3.22) implies k → ∞. r−1 ϕ(k) 2 = O(−2 k ) = o b(k ) ,

(3.37)

Further we consider the first term in the right-hand side of (3.35). It follows from (3.20) that 2 (3.38) −ψj (k) + a(r) − k ψj (k) = b(k )(2j − 1)ψj (k) + α(k)ψj (k) where the function

2 α(r; k) = a(r) − k − b2 (k )(r − k )2 .

(3.39)

Let us estimate the right-hand side. In view of the equation a(ρk ) = k, a secondorder Taylor expansion of a at k yields r b (s)(r − s)ds . a(r) = k + b(k )(r − k ) + k

Therefore,

α(r; k) = 2b(k )(r − k )

r

k

b (s)(r − s)ds +

r

k

b (s)(r − s)ds

2 ,

and hence |α(r; k)| ≤ b(k )b1 (k )|r − k |3 + 4−1 b21 (k )(r − k )4 ≤ γk−3 b(k )−1/2 b1 (k ) + 4−1 γk−4 b(k )−2 b21 (k ) , provided that |r − k | ≤ γk−1 b(k )−1/2 . In view of conditions (3.28) and (3.30), this gives us the estimate sup |α(r; k)| = o b(ρk ) (3.40) |r−k |≤γk−1 b(k )−1/2

so that

2 − ψj (k) + a(r) − k ψj (k) ζ(k) = b(k )(2j − 1)ϕj (k) + o b(ρk ) .

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Thus, using also (3.33) we obtain that 2 Re − ψ (k) + a(r) − k ψ(k), ψ(k)ζ 2 (k) = b(k )

n

(2j − 1)cj c¯l ϕj (k), ϕl (k) + o b(ρk )

j,l=1

≤ b(k )(2n − 1) + o b(ρk ) .

(3.41)

Together with (3.36) and (3.37), this implies estimate (3.26) for each ϕ(k) ∈ Ln (k). Proof of (ii). Let functions ζ ∈ C0∞ (R) and η ∈ C ∞ (R) satisfy ζ 2 (x) + η 2 (x) = 1, x ∈ R; moreover, as before, we require that 0 ≤ ζ(x) ≤ 1 , ζ(x) = 1 for |x| ≤ 1/2 and supp ζ = [−1, 1]. By analogy with (3.31) set (3.42) η(r; k) = η γk b(k )1/2 (r − k ) , r ∈ R+ . Then we have

ζ 2 (r; k) + η 2 (r; k) = 1 , r ∈ R+ . We proceed from the localization formula (known as the IMS formula – see, e.g., [2, Section 3.1]) Lm (−k) = ζ(k)Lm (−k)ζ(k) + η(k)Lm (−k)η(k) − ζ (k)2 − η (k)2 ,

where ζ(k), η(k), ζ (k) and η (k) are understood as operators of multiplication by the functions ζ(r, k), η(r, k), ζ (r, k) and η (r, k), respectively. According to (3.28) it follows from definitions (3.31) and (3.42) that max ζ (r, k)2 + η (r, k)2 = O γk2 b(k ) = o b(k ) , k → ∞ . (3.43) r∈R+

Next, we check that η(k)Lm (−k)η(k) ≥ νk b(k )η 2 (k)

(3.44)

with νk → ∞ as k → ∞. By virtue of the Hardy inequality d2 m2 − 1/4 η(k) − 2 + η(k) ≥ 0 , dr r2 it suffices to check that for

2 a(r) − k ≥ νk b(k )

−1 (+) r ≥ k + 2γk b(k )1/2 =: k

(3.45)

−1 (−) and r ≤ k − 2γk b(k )1/2 =: k . (3.46) According to (3.21) there exists r0 such that the function a(r) is increasing for r ≥ r0 . Let first r ≥ r0 . Then (±) (3.47) |a(r) − k| = |a(r) − a(k )| ≥ ± a(k ) − a(k )

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(±)

(±)

if ±(r − k ) ≥ 0 and r ≥ r0 . It follows from definition (3.46) of the numbers k that (±) k (±) a(k ) − a(k ) = b(s)ds k

−1

= ±(2γk )

1/2

b(k )

(±)

k

+ k

b(s) − b(k ) ds .

(3.48)

The absolute value of the integral in the right-hand side can be estimated by (±) k s (±) ds |b (σ)|dσ ≤ 2−1 b1 (ρk )(k − k )2 = 8−1 b1 (ρk )γk−2 b(k )−1 k k where the function b1 is defined in (3.23). By virtue of conditions (3.28) and (3.30) this expression is o(γk−1 b(k )1/2 ) as k → ∞. Therefore the absolute value of expression (3.48) is bounded from below by (3γk )−1 b(k )1/2 . Thus, for r ≥ r0 , estimate (3.45) with νk = (3γk )−2 → ∞ is a consequence of (3.47). If r ≤ r0 , we take into account that a(r) is semibounded from above so that (a(r) − k)2 ≥ 2−1 k 2 . Hence estimate (3.45) with νk = 2−1 k 2 b(k )−1 → ∞ is satisfied according to condition (3.24). Putting together definitions (2.8) and (3.17) of the operators Lm (−k) and T (k), we see that ζ(k)Lm (−k)ζ(k) = ζ(k)T (k)ζ(k) + α(k)ζ 2 (k) ,

(3.49)

where α(k) is the operator of multiplication by function (3.39). The first term in the right-hand side is bounded from below by b(k )ζ 2 (k) because b(k ) is the first eigenvalue of the operator T (k). By virtue of (3.40) the second term satisfies the estimate (3.50) α(k)ζ 2 (k) = o b(k ) . It follows that operator (3.49) is bounded from below by b(k )ζ 2 (k) − o(b(k ))I. Combining this result with (3.43) and (3.44), we get estimate (3.27) in the case n = 1. If n ≥ 2, we denote by Pn (k) the orthogonal projection onto the span of the first n − 1 eigenfunctions of the operator T (k). Then T (k)(I − Pn (k)) ≥ (2n − 1)(I − Pn (k)) and hence ζ(k)T (k)ζ(k) = ζ(k)T (k) I − Pn (k) ζ(k) + ζ(k)T (k)Pn (k)ζ(k) ≥ b(k )(2n − 1)ζ(k) I − Pn (k) ζ(k) + ζ(k)T (k)Pn (k)ζ(k) = b(k )(2n − 1)ζ 2 (k) + Rn (k) where

(3.51)

Rn (k) = ζ(k) T (k) − b(k )(2n − 1)I Pn (k)ζ(k) .

Clearly, rank Rn (k) ≤ n − 1. Putting together (3.43), (3.44) and (3.49)–(3.51), we obtain (3.27) in the case n ≥ 2.

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Example 3.7. Let b(r) = b0 r−δ , b0 > 0, δ ≤ 1, for sufficiently large r. Then b1 (r) = b0 δr−δ−1 and conditions (3.21)–(3.23) are satisfied. Moreover, ρk = c1 k ν and k 2 b(ρk ) = c2 k −1−ν where ν = (1 − δ)−1 and c1 , c2 > 0 if δ < 1. If δ = 1, then −1 2 2 ρk = exp(b−1 0 k) and k b(ρk ) = k b0 exp(−b0 k). In both cases condition (3.24) is also satisfied. Thus, Proposition 3.6 implies the following results. If δ > 0, then λn,m (p) → 0 as p → −∞ (this result follows also from Proposition 3.5). If δ = 0, then the functions λn,m (p) have finite limits b0 (2n − 1) as p → −∞. If δ < 0, then these functions tend to +∞ as p → −∞. 3.3. Let us return to the Hamiltonians Hm and H defined in Section 2. Theorem 3.8. Assume (2.4) and (2.10). (i) Then all operators Hm , m ∈ Z, and hence H are absolutely continuous and their spectra coincide with the half-axes defined by (1.9) and (1.10). (ii) If the hypotheses of Proposition 3.5 hold true, then Em = 0 for all m ∈ Z. Moreover, the multiplicities of all spectra σ(Hm ) and hence of σ(H) are infinite. (iii) Let the hypotheses of Proposition 3.6 hold true. If b(r) → ∞, then the infimum in (1.10) is attained (at a finite point) so that for all m ∈ Z Em = min λ1,m (p) > 0 . p∈R

(iv) Let the hypotheses of Proposition 3.6 hold true. If b(r) admits a finite positive limit b0 as r → ∞, then Em ∈ (0, b0 ] for all m ∈ Z. Proof. It suffices to prove only the assertions concerning the operators Hm . In view of decomposition (2.12) they reduce to corresponding statements about the operators Λn,m . These operators are absolutely continuous because the eigenvalues λn,m (p) are real analytic functions of p ∈ R which are non constants since according to Corollary 3.4 λn,m (p) → ∞ as p → ∞. Moreover, we have that σ(Λn,m ) = [En,m , ∞) where

En,m = inf λn,m (p) ≥ 0 p∈R

(3.52)

because λn,m (p) > 0 for all p ∈ R. This implies relations (1.9) with Em defined by (1.10). In case (ii) it suffices to use that according to (3.9) En,m = 0 and hence σ(Λn,m ) = [0, ∞) for all m and n. In case (iii) Proposition 3.6 implies that λn,m (p) → ∞ as p → −∞ for all n and m so that (3.53) En,m = min λn,m (p) > 0 p∈R

and hence infimum in (1.10) can be replaced by minimum. In case (iv) we use that according to (3.25) En,m ≤ (2n − 1)b0 . Moreover, En,m > 0 because λn,m (p) > 0 for all p ∈ R. For n = 1, this gives the desired result.

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Remark 3.9. According to (2.12) and (3.52) the spectrum of the operator Hm consists of the “branches” [En,m , ∞) where the points En,m are called thresholds. In cases (iii) and (iv) En,m < En+1,m (3.54) for all n ∈ N. Indeed, in case (iii) (3.54) is a consequence of the estimate λn,m (p) < λn+1,m (p) valid for all p ∈ R and of formula (3.53). In case (iv) one has to take additionally into account that the limit of λn,m (p) as p → −∞ is strictly smaller than that of λn+1,m (p). Inequality (3.54) means that there are infinitely many distinct thresholds in each of the spectra σ(Hm ), m ∈ Z, and hence in σ(H). Remark 3.10. In case (iii) the multiplicity of the spectrum of all operators Λn,m equals at least to 2 whereas in cases (ii) and (iv) it might be equal to 1.

4. Group velocities 4.1. In this subsection we obtain a formula for the derivative λn,m (p), n ∈ N, m ∈ Z, which yields sufficient conditions for the monotonicity of λn,m (p) as a function of p. Recall that the operators Hm (p), m ∈ N, p ∈ R, were defined in the space H by formula (1.4). The proof of Theorem 4.3 relies on integration by parts. To prove that nonintegral terms disappear at r = 0, we use standard bounds on ψn,m (r; p). Unfortunately, we were unable to find necessary results in the literature and therefore give their brief proofs. Let us consider the differential equation of Bessel type −r−1 (ry ) + m2 r−2 y + q(r)y = 0 ,

m = 0, 1, 2, . . . ,

(4.1)

in a neighborhood (0, r0 ) of the point r = 0. If q(r) = 0, then it has the regular and (reg) (sing) (reg) (r) = r−m for m = 0 and y0 (r) = 1 singular solutions y0 (r) = rm and y0 (sing) and y0 (r) = ln r for m = 0. Lemma 4.1. Let m = 0, and let the function rq(r) belong to the class L1 (0, r0 ). Then (4.1) has a solution y (reg) (r) satisfying the relation y (reg) (r) = rm + o(rm ) ,

r → 0.

(4.2)

For its derivative, we have the bound dy (reg) (r)/dr = O(rm−1 ) .

(4.3)

Let m = 0. Suppose that the function r ln rq(r) belongs to the class L1 (0, r0 ). Then (4.1) has a solution y (reg) (r) satisfying relation (4.2) where m = 0. For its derivative, we have the bound r (reg) (r)/dr = O |q(s)|ds . (4.4) dy 0

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Moreover, if the function r ln2 rq(r) belongs to the class L1 (0, r0 ), then (4.1) has a solution y (sing) (r) satisfying the relation y (sing) (r) = ln r + o(1) ,

r → 0.

(4.5)

In this case any bounded solution of (4.1) coincides (up to a constant factor) with the regular solution y (reg) (r). Proof. We construct the function y (reg) (r) as the solution of the Volterra integral equation (reg)

y (reg) (r) = y0

(r) r (reg) (sing) (reg) (sing) (r)y0 (s)−y0 (s)y0 (r) q(s)y (reg) (s)ds (4.6) +κm s y0 0

0 and κ0 = −1. Differentiating it twicely, we see where κm = (2m)−1 for m = that y (reg) (r) satisfies (4.1). Equation (4.6) can be solved by iterations, that is y (reg) (r) =

∞

yn(reg) (r) .

(4.7)

n=0

Hereby the nth -iteration obeys the bound |yn(reg) (r)| ≤

C n m r n!

n

r

s|q(s)|ds 0

if m = 0; if m = 0, then s|q(s)| should be replaced by s| ln s||q(s)|. This ensures the convergence of series (4.7) as well as relation (4.2). Differentiating equation (4.6) and using (4.2), we get bounds (4.3) and (4.4) on the derivative of y (reg) (r). If m = 0, we can construct the function y (sing) (r) as the solution of (4.6) (reg) (sing) where the first term, y0 (r), in the right-hand side is replaced by y0 (r), that is r

y (sing) (r) = ln r +

s ln(r/s)q(s)y (sing) (s)ds .

0

This equation can again be solved by iterations which, in particular, implies estimate (4.5). This result can be supplemented by the following Lemma 4.2. Let m = 0, and let the function rq 2 (r) belong to the class L1 (0, r0 ). Assume additionally that q = q¯. If ψ is a solution of (4.1) from the class L2 ((0, r0 ); rdr), then it coincides (up to a constant factor) with the regular solution y (reg) (r) and hence satisfies estimates (4.2) and (4.3). Proof. Let us extend the function q(r) to (r0 , ∞) by zero, and let us consider the differential operator hy = −r−1 (ry ) + m2 r−2 y + q(r)y

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in the space L2 (R+ ; rdr) on domain C0∞ (R+ ). If q = 0, we denote this operator by h0 . The operator h0 is essentially self-adjoint. To prove the same for h, it suffices to check that q 2 (r)|f (r)|2 rdr ≤ ε h0 f 2 + C f 2 , f ∈ C0∞ (R+ ) , ε < 1 . (4.8) R+

Let us use the estimate q 2 (|x|)|u(x)|2 dx ≤ |x|≤r0

q 2 (|x|)dx max2 |u(x)|2 x∈R |(Δu)(x)|2 dx + Cε−1

|x|≤r0

≤ε

R2

R2

|u(x)|2 dx ,

∀ε > 0 .

Restricting it on the subspace of functions u(x) = f (r)eimθ , we obtain estimate (4.8) which implies that h is essentially self-adjoint as well as h0 . Thus, (4.1) has at most one solution from L2 ((0, r0 ); rdr) which is necessarily proportional to y (reg) (r). Now we are in a position to obtain a formula for the derivative λn,m (p). In addition to our usual assumptions that b(r) is not too singular at r = 0, an integration-by-parts machinery requires that b(r) does not vanish too rapidly as r → 0. The precise conditions are formulated rather differently in the cases m = 0 and m = 0. We start with the first case. Theorem 4.3. Let m = 0. Suppose that b ∈ C 3 (R+ ) and b(r) > 0, r ∈ R+ . Assume (2.10) and that b(r) = O(ecr ) for some c > 0 as r → ∞. At r = 0 we suppose that b(r) = O(r−γ ) where γ < 3/2. Moreover, we assume that for some β < 2|m| − 1 b(r)−1 (k) ≤ Cr−β−k , k = 0, 1, 2, 3 , r → 0 . (4.9) Put

v(r) = r r−1 rb(r)−1 .

Then λn,m (p) = −2 −2

∞

0 −1

rb−2 (r)b (r)ψn,m (r; p)2 dr

+ 2m2

∞ 0

0

2 v (r)ψn,m (r; p)dr

∞

2 r−2 b−1 (r)ψn,m (r; p)dr ,

(4.10)

where the eigenfunctions ψn,m (r; p) of the operator Hm (p) are real and normalized, that is ψn,m = 1. Proof. Below we integrate several times by parts. In view of the equation 2 ) − m2 r−2 ψn,m + λn,m ψn,m (4.11) a(r) + p ψn,m = r−1 (rψn,m

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we can apply to the function ψn,m the results of Lemmas 4.1 and 4.2 where q(r) = (a(r) + p)2 − λn,m . Thus, Lemma 4.2 implies that ψn,m (r; p) = O(r|m| ) (r; p) = O(r|m|−1 ) as r → 0 which ensures that non-integral terms disand ψn,m appear at r = 0. To prove the same for non-integral terms corresponding to r → ∞, we use super-exponential decay of eigenfunctions ψn,m (r; p) of the operators Hm (p). This result is valid [14] (see also [6]) for all one-dimensional Schrödinger operators with discrete spectra. In view of the condition a(r) = O(ecr ), it follows from (4.11) that the derivatives ψn,m (r; p) also decay super-exponentially. Let us proceed from the formula of the first order perturbation theory (known as the Feynman–Hellman formula) 2 ∞ ∂ a(r) + p 2 ψn,m λn,m (p) = (r; p)rdr ∂p 0 2 ∞ ∂ a(r) + p 2 ψn,m = (r; p)τ (r)dr (4.12) ∂r 0 where τ (r) = rb(r)−1 . Using that a(r) = O(r1−γ ) and τ (r) = O(r1−β ), we integrate by parts and get ∞ 2 (r; p) dr . λn,m (p) = − a(r) + p ψn,m (r; p) τ (r)ψn,m (r; p) + 2τ (r)ψn,m 0

Now it follows from (4.11) that ∞ ∞ 2 2 2 λn,m (p) = −λn,m (p) τ (r)ψn,m (r; p) dr + m r−2 τ (r)ψn,m (r; p) dr 0 0 ∞ −1 rψn,m (r; p) τ (r)ψn,m (r; p) + 2τ (r)ψn,m r (r; p) dr . (4.13) − 0

2 By the condition τ (r)ψn,m (r; p) → 0 as r → 0, the first term in the right-hand side equals zero. In the second term we integrate by parts which yields ∞ ∞ 2 2 r−2 τ (r)ψn,m (r; p) dr = 2 r−3 τ (r)ψn,m (r; p)dr 0

−2

0

2 τ (r)ψn,m (r; p)

→ 0. because r In the last integral in the right-hand side of (4.13), we also integrate by parts (r; p)ψn,m (r; p)τ (r) → 0 as r → 0. Thus, we have that using that ψn,m ∞ ∞ −1 − rψn,m (r; p) τ (r)ψn,m (r; p)dr = r τ (r)ψn,m (r; p)2 dr 0 0 ∞ v(r)ψn,m (r; p)ψn,m (r; p)dr . + 0

(4.14)

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The last integral in the right-hand side equals ∞ 2 v (r)ψn,m (r; p)dr −2−1 0

2 v(r)ψn,m (r; p)

→ 0 as r → 0. Similarly, we get that ∞ ∞ −1 rψn,m (r; p) τ (r)ψn,m (r; p)dr = − r r−2 τ (r)d rψn,m (r; p)2 0 ∞0 r2 r−2 τ (r) ψn,m (r; p)2 dr =

because −2

0

τ (r)ψn,m (r; p)2

since → 0 as r → 0. Putting the results obtained together, we arrive at representation (4.10). Corollary 4.4. If b (r) ≤ 0 and r2 b(r)v (r) ≤ 4m2 for all r ≥ 0, then λn,m (p) ≥ 0 for all p ∈ R and n. If, moreover, one of these inequalities is strict on some interval, then λn,m (p) > 0. 1+δ 2 , v(r) = b−1 Corollary 4.5. If b(r) = b0 r−δ , δ ∈ [0, 1], then τ (r) = b−1 0 r 0 (δ − 1)rδ−1 and ∞ −1 λn,m (p) = 2b0 δ rδ ψn,m (r; p)2 dr 0 2 ∞ −2+δ 2 −1 −1 2 r ψn,m (r; p)dr . + b0 2m − 2 (1 − δ) (1 + δ) 0

For b0 > 0, this expression is strictly positive (so that the functions λn,m (p) are strictly increasing for all p ∈ R) for m = 0 since (1 − δ)2 (1 + δ) ≤ 1. Moreover, for δ = 1 this result is true for all m ∈ Z. In the case m = 0 we consider for simplicity only fields (1.11). Proposition 4.6. If b(r) = b0 r−δ , δ ∈ [0, 1], then ∞ −1 rδ ψn,0 (r; p)2 dr λn,0 (p) = 2b0 δ 0 ∞ 2 −1 2 2 − b−1 2 (1 − δ) (1 + δ) r−2+δ ψn,0 (r; p) − ψn,0 (0; p) dr . 0 0

If b0 > 0 and δ = 1, then

λn,0 (p)

> 0 for all p ∈ R.

Proof. Let us proceed again from formula (4.12). We use now that the function ψn,0 (|x|; p) of x ∈ R2 belongs to the Sobolev class H2loc (R2 ), and therefore ψn,0 (r; p) (r; p) = O(r1−ε ) for any ε > 0 has a finite limit as r → 0. Thus, by Lemma 4.1 ψn,0 as r → 0. These results allow us to integrate by parts as in the case m = 0. The only difference is with the second integral in the right-hand side of (4.14). Now

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2 δ−1 v(r) = b−1 and this integral equals 0 (δ − 1)r ∞ ∞ 2 2 v(r)ψn,0 (r; p)ψn,0 (r; p)dr = 2−1 v(r)d ψn,0 (r; p) − ψn,0 (0; p) 0 0 ∞ 2 −1 2 = −2 v (r) ψn,0 (r; p) − ψn,0 (0; p) dr 0

2 2 (r; p) − ψn,0 (0; p)) as r → 0. because v(r)(ψn,0

4.2. In this subsection we show that for linear potentials, that is for magnetic fields not depending on r, all eigenvalues λn,0 (p), n ∈ N, of the operator H0 (p) are not monotonous functions of p ∈ R. We follow closely the proof of the first part of Proposition 3.6. However we now use that eigenfunctions of the harmonic oscillator decay faster than any power of r−1 at infinity (actually, they decay super-exponentially). Proposition 4.7. Assume that for sufficiently large r b(r) = b0 > 0 .

(4.15)

Then, for all n ∈ N, some γn > 0 and sufficiently large k > 0, we have λn,0 (−k) ≤ (2n − 1)b0 − γn k −2 .

(4.16)

Proof. Let ζ be the same function as in the proof of the first part of Proposition 3.6. 1/2 −1 We set ρk = b−1 and define the functions ζ(r; k) and ϕj (r; k) by 0 k, γk = 2b0 k formulas (3.31) and (3.32), respectively. It suffices to check that L0 (−k)ϕ(k), ϕ(k) ≤ 2n − 1 − γn k −2 (4.17) for sufficiently large k and all normalized functions ϕ(k) from subspace (3.16). Let us proceed from formula (3.35). Since the functions ψj (x; k) decay faster than any power of |x|−1 as |x| → ∞, the term o(1) in (3.33) is actually O(k −∞ ). Similarly, estimate (3.36) can be formulated in a more precise form as ψ(k)ζ (k) 2 = O(k −∞ ) .

(4.18)

Since r ≤ 2−1 3k on the support of ϕ(k), we have that r−1 ψ(k) 2 ≥ (2/3)2 k −2 .

(4.19)

Now function (3.34) is zero if r and k are large enough. Therefore equation (3.38) yields the exact equality n Re − ψ (k) + (b0 r − k)2 ψ(k), ψ(k)ζ 2 (k) = b0 (2j − 1)cj c¯l ϕj (k), ϕl (k) j,l=1

(cf. (3.41)). Up to terms O(k −∞ ), the right-hand side here is estimated by b0 (2n−1). Together with (4.18) and (4.19), this implies estimate (4.17).

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Combining relations (3.25) and (4.16), we see that the eigenvalues λn,0 (p) tend as p → −∞ to their limits (2n−1)b0 from below. On the other hand, according to (3.5) λn,0 (p) → ∞ as p → ∞. Thus, all functions λn,0 (p) have necessarily local minima. We can obtain an additional information using the following elementary Lemma 4.8. Suppose that (4.15) is satisfied for all r > 0 and that a(r) = b0 r. Then λn,m (0) = 2b0 (2n − 1 + |m|) (4.20) for all n ∈ N and m ∈ Z. Proof. Let us consider the two-dimensional harmonic oscillator T = −Δ + b20 (x21 + x22 ). Separating the variables x1 , x2 , we see that its spectrum consists of the eigenvalues 2b0 (l1 + l2 − 1) where l1 , l2 ∈ N. It follows that the operator T has the eigenvalues 2b0 j, j ∈ N, of multiplicity j. On the other hand, separating the variables in the polar coordinates, we see that the spectrum of T consists of the eigenvalues λn,m (0) of the operators Hm (0). For the proof of (4.20) we take into account that all eigenvalues λn,m (0) are simple and that λn,m+1 (0) > λn,m (0) for all n and m ≥ 0. Clearly, the operator H0 (0) has an eigenvalue 2b0 j if and only if its multiplicity j is odd. This gives formula (4.20) for m = 0. We shall show that for every j ∈ N λ1,j−1 (0) = λ2,j−3 (0) = · · · = λ2,−j+3 (0) = λ1,−j+1 (0) = 2b0 j

(4.21)

which is equivalent to formula (4.20) for all m. Let us choose some j0 and suppose that (4.21) holds for all j ≤ j0 . Then we check it for j = j0 + 1. First we remark that if an operator Hm (0) for some m > 0 has n eigenvalues in the interval [2b0 , 2b0 (j0 + 1)], then the operator Hm−1 (0) has at least n eigenvalues in the interval [2b0 , 2b0 j0 ]. Then using (4.21) for j ≤ j0 , we see that if an operator Hm (0) has the eigenvalue 2b0 (j0 + 1), then necessarily the operator Hm−1 (0) has the eigenvalue 2b0 j0 . Therefore according to (4.21) for j = j0 , only the operators Hm (0) with m = j0 , j0 − 2, . . . , −j0 + 2, −j0 might have the eigenvalue 2b0 (j0 + 1). There are j0 + 1 of such operators and the multiplicity of this eigenvalue equals j0 + 1. Thus, all the operators Hm (0) for m = j0 , j0 − 2, . . . , −j0 + 2, −j0 and only for such m have the eigenvalue 2b0 (j0 + 1). This proves (4.21) for j0 + 1. Comparing this result with (3.25), we see that, for potentials a(r) = b0 r, lim λn,m (p) = b0 (2n − 1) < 2b0 (2n − 1 + |m|) = λn,m (0) .

p→−∞

Together with (4.16), this implies that the functions λn,0 (p) have negative local minima. Thus, we get the following Theorem 4.9. Under the hypotheses of Proposition 4.7 the eigenvalues λn,0 (p), n ∈ N, of the operator H0 (p) are not monotonous functions of p ∈ R. Moreover, if (4.15) is satisfied for all r > 0 and a(r) = b0 r, then the functions λn,0 (p) lose their monotonicity for p < 0.

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We do not know how many minima have the functions λn,0 (p). The problem of monotonicity of the eigenvalues λn,0 (p) for fields b(r) = b0 r−δ where δ ∈ (0, 1) remains also open. 4.3. In a somewhat similar situation the break down of monotonicity of group velocities was exhibited in [7]. In this paper one considers the Schr¨ odinger operator ∂ 2 ∂2 + i −bx with constant magnetic field b > 0, defined on the semiH(N ) = − ∂x 2 ∂y 2 plane (x, y) ∈ R : x > 0 with the Neumann boundary condition at x = 0. Let H (N ) (p) = −d2 /dx2 + (bx + p)2 , p ∈ R, be the self-adjoint operator in the space L2 (R+ ) corresponding to the boundary condition u (0) = 0. Then the operator H(N ) is unitarily equivalent the partial Fourier transform with respect to y, ⊕ (Nunder ) to the direct integral R H (p)dp. It is shown in [7, Section 4.3] that the lowest eigenvalue μ1 (p) of H (N ) (p) is not monotonous for p < 0. This follows from the inequality μ1 (0) > 0 proven1 in [3] and the relations lim μ1 (p) = μ1 (0) = b .

p→−∞

(4.22)

Our proof of non-monotonicity of the functions λn,0 (p) is essentially different since in contrast with (4.22) we have limp→−∞ λn,0 (p) < λn,0 (0).

5. Asymptotic time evolution 5.1. Combined with the stationary phase method, the spectral analysis of the operators H = H(a) allows us to find the asymptotics for large t of solutions u(t) = odinger equation. It follows from (1.2) exp(−iHt)u0 of the time dependent Schr¨ that exp − iH(a)t u0 = exp iH(−a)t u0 . Therefore it suffices to consider the case a(r) → +∞. Moreover, on every subspace Hm with a fixed magnetic quantum number m, the problem reduces to the asymptotics of the function u(t) = exp(−iHm t)u0 . Let us proceed from decomposition (2.12). Suppose that Fm u0 ∈ Ran Ψn,m . Then (see (2.11)) (Fm u0 )(r, p) = ψn,m (r, p)f (p) (5.1) ∗ Ψn,m e−iΛn,m t f , that is where f = Ψ∗n,m Fm u0 and u(t) = Fm ∞ eipx3 −iλn,m (p)t ψn,m (r, p)f (p)dp . (5.2) un,m (r, x3 , t) = (2π)−1/2 −∞

The analytic function λn,m (p) might have only a countable set of zeros pn,m,l with possible accumulations at ±∞ only. The function λn,m (p) is monotone on every interval (pn,m,l , pn,m,l+1 ) and takes there all values between λn,m (pn,m,l ) =: αn,m,l and λn,m (pn,m,l+1 ) =: βn,m,l . We consider the asymptotics of integral (5.2) on each of the subspaces L2 (pn,m,l , pn,m,l+1 ) separately. Let us set γ = x3 t−1 . First 1 Note

that in [3] and [7] the parameter p is chosen with the opposite sign.

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we suppose that f ∈ C0∞ (pn,m,l , pn,m,l+1 ). The stationary points of integral (5.2) are determined by the equation λn,m (p) = γ .

(5.3)

If γ ∈ (αn,m,l , βn,m,l ), it does not have solutions from the interval (pn,m,l , pn,m,l+1 ). Therefore integrating directly by parts, we find that function (5.2) decays in this region of x3 /t faster than any power of (|x3 | + |t|)−1 (and r). If γ ∈ (αn,m,l , βn,m,l ), then on the interval (pn,m,l , pn,m,l+1 ) equation (5.3) has a unique solution which we denote by νn,m,l (γ). Let us set Φn,m,l (γ) = νn,m,l (γ)γ − λn,m νn,m,l (γ) and denote by χn,m,l the characteristic function of the interval (αn,m,l , βn,m,l ). For γ from this interval, we apply the stationary phase method to integral (5.2) which yields −1/2 (±) un,m (r, x3 , t) = τn,m,l eiΦn,m,l (γ)t ψn,m r, νn,m,l (γ) λn,m νn,m,l (γ) × f νn,m,l (γ) χn,m,l (γ)|t|−1/2 + u∞ (r, x3 , t) , (±)

γ = x3 t−1 ,

t → ±∞ ,

(5.4)

where τn,m,l = e∓πi sgn(λn,m (p))/4 for p ∈ (pn,m,l , pn,m,l+1 ) and lim u∞ (·, t) = 0 .

t→±∞

(5.5)

Differentiation of (5.3 ) with respect to γ yields λn,m (p) νn,m,l (γ) νn,m,l (γ) = 1 . Using this equation and the normalization ∞ ψn,m r, νn,m,l (γ) 2 rdr = 1 , −∞

we find that the square of the norm in the space H of the first term in the right-hand side of (5.4) equals ∞ 2 |t|−1 |νn,m,l (γ)|f νn,m,l (γ) χn,m,l (γ)dx3 . −∞

Making here the changes of variables x3 = γt and p = νn,m,l (γ), we see that this expression equals the square of the norm of the function f in the space L2 (pn,m,l , pn,m,l+1 ). Therefore asymptotics (5.4) extends by continuity to all functions (5.1) with an arbitrary f from this space. Thus, we have proven Theorem 5.1. Assume (2.4) and (2.10). Let u(t) = exp(−iHm t)u0 where u0 satisfies (5.1) with f ∈ L2 (pn,m,l , pn,m,l+1 ). Then the asymptotics as t → ±∞ of this function is given by relations (5.4), (5.5).

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Of course asymptotics (5.4), (5.5) extends automatically to all f ∈ L2 (R) with compact support and to linear combinations of functions ψn,m (r, p)fn (p) over different n. By virtue of formulas (5.4), (5.5) a quantum particle in magnetic field (1.5) remains localized in the (x1 , x2 )-plane but propagates in the x3 -direction. If f ∈ L2 (pn,m,l , pn,m,l+1 ), then a particle “lives” as |t| → ∞ in the region where x3 ∈ (αn,m,l t, βn,m,l t). In particular, if λ (p) > 0 (λ (p) < 0) for p ∈ (pn,m,l , pn,m,l+1 ), then a particle propagates in the positive (negative) direction as t → +∞. Thus, according to Corollary 4.5 if b(r) = b0 r−δ , δ ∈ [0, 1], b0 > 0, then a particle with the magnetic quantum number m = 0 propagates always in the positive direction of the x3 -axis. If δ = 1, then this result remains true from all m. On the contrary, if δ = 0 and m = 0, then a particle will propagate in a negative direction for some interval of momenta p. 5.2. Theorem 5.1 implies the existence of asymptotic velocity in the x3 direction. The corresponding operator is defined by the equation (cf. (2.12)) ∗ Fm Ψn,m Λn,m Ψ∗n,m Fm , Hm = n∈N

Λn,m

where are the operators of multiplication by the functions λn,m (p). To put it differently, the operator Hm acts as multiplication by λn,m (p) in the spectral representation of the operator Hm where it acts as multiplication by the functions λn,m (p). Proposition 5.2. Assume (2.4) and (2.10). Then, for an arbitrary bounded function Q, s-lim|t|→∞ exp (iHm t)Q(x3 /t) exp (−iHm t) = Q (Hm ) (5.6) (in particular, the strong limit in the left-hand side exists). Proof. We shall check that for all u0 ∈ Hm lim Q(x3 /t) exp (−iHm t)u0 − exp (−iHm t)Q(Hm )u0 = 0

|t|→∞

(5.7)

which is equivalent to relation (5.6). Remark that if u0 satisfies (5.1), then Ψn,m Q(Hm )u0 (r, p) = ψn,m (r, p)Q λn,m (p) f (p) . (5.8) It suffices to prove (5.7) on a dense set of elements u0 such that equality (5.1) is true with f ∈ L2 (pn,m,l , pn,m,l+1 ). Applying the operator Q(x3 /t) to asymptotic relation (5.4), we see that the asymptotics of Q(x3 /t) exp (−iHm t)u0 is given again by formula (5.4) where the function f (νn,m,l (γ)) in the right-hand side is replaced by the function Q(γ)f (νn,m,l (γ)). Similarly, it follows from Theorem 5.1 and relation (5.8) that the asymptotics of exp (−iHm t)Q(Hm )u0 is given by formula (5.4) where the function f (νn,m,l (γ)) in the right-hand side is replaced by the function Q(λn,m (νn,m,l (γ)))f (νn,m,l (γ)). So for the proof of (5.7), it remains to take (5.3) into account.

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Relation (5.6) shows that Hm can naturally be interpreted as the operator of asymptotic velocity in the x3 -direction. Similar results concerning the Iwatsuka model (see [8] or [2]) have been obtained in [11]. Numerous useful discussions with Georgi Raikov as well as a financial support by the Chilean Science Foundation Fondecyt under Grant 7050263 are gratefully acknowledged.

References [1] Y. Aharonov, D. Bohm, Significance of electromagnetic potential in the quantum theory, Phys. Rev. 115 (1959), 485–491. [2] H. Cycon, R. Froese, W. Kirsch, B. Simon, Schr¨ odinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, SpringerVerlag, Berlin, Heidelberg, New York, 1987. [3] M. Dauge, B. Helffer, Eigenvalues variation. I. Neumann problem for Sturm– Liouville operators, J. Diff. Eq. 104 (1993), 243–262. [4] S. De Bièvre, J. V. Pulé, Propagating edge states for a magnetic Hamiltonian, Math. Phys. Electron. J. 5 (1999), Paper 3, 17 pp. [5] V. Ge˘ıler, M. Senatorov, The structure of the spectrum of the Schr¨ odinger operator with a magnetic field in a strip, and finite-gap potentials, Sb. Math. 188 (1997), 657–669. [6] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Moscow, Fizmatgiz, 1963 (Russian). [7] B. Helffer, Introduction to semi-classical methods for the Schr¨ odinger operator with magnetic field. Vienna version, Lecture notes of a course given at the ESI, 2006, available at http://www.math.u-psud.fr/˜helffer/syrievienne2006.pdf. [8] A. Iwatsuka, Examples of absolutely continuous Schr¨ odinger operators in magnetic fields, Publ. Res. Inst. Math. Sci. 21 (1985), no. 2, 385–401. [9] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag New York, Inc., New York 1966. [10] L. D. Landau, E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1965. [11] M. Mˇ antoiu, R. Purice, Some propagation properties of the Iwatsuka model, Comm. Math. Phys. 188 (1997), 691–708. [12] K. Miller, B. Simon, Quantum magnetic Hamiltonians with remarkable spectral properties, Phys. Rev. Lett. 44 (1980), 1706–1707. [13] M. Reed, B. Simon, Methods of modern mathematical physics, IV. Analysis of operators, Academic Press, New York, 1978. [14] I. E. Shnol’, On the behavior of eigenfunctions of the Schr¨ odinger equation, Matem. Sb. 42 (1957), 273–286 (in Russian). [15] D. Yafaev, A particle in a magnetic field of an infinite rectilinear current, Math. Phys. Anal. Geom. 6 (2003), 219–230. [16] D. Yafaev, Scattering by magnetic fields, St. Petersburg Math. J. 17 (2006), no. 5, 875–895.

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Dimitri Yafaev IRMAR Université de Rennes I Campus de Beaulieu F-35042 Rennes Cedex France e-mail: [email protected] Communicated by Christian Gérard. Submitted: June 7, 2007. Accepted: August 20, 2007.

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Ann. Henri Poincaré 9 (2008), 209–274 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020209-66, published online April 10, 2008 DOI 10.1007/s00023-008-0355-y


Tunnel Effect for Kramers–Fokker–Planck Type Operators Frédéric Hérau, Michael Hitrik, and Johannes Sjöstrand Abstract. We consider operators of Kramers–Fokker–Planck type in the semiclassical limit such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues. Résumé. On considère des opérateurs du type de Kramers–Fokker–Planck dans la limite semi-classique tels que l’exposant du maxwellien associé soit une fonction de Morse avec deux minima et un point selle. Sous des hypothèses supplémentaires convenables on établit un développement asymptotique complet de l’écart exponentiellement petit entre les deux premières valeurs propres.

1. Introduction This paper is a natural continuation of the work [14], investigating the low lying eigenvalues of the Kramers–Fokker–Planck operator γ P = y · h∂x − V (x) · h∂y + (−h∂y + y) · (h∂y + y) , x, y ∈ Rn , (1.1) 2 where γ > 0. Physically the semiclassical limit h → 0 corresponds to the low temperature limit. As explained in [14], the original motivation for that work was to give more explicit versions of some results in [13] and later in [8], giving estimates on the time of return to equilibrium, or more or less equivalently, on the gap between the first eigenvalue 0 (when the potential V tends to plus infinity sufficiently fast at infinity) and the second eigenvalue. See also [25] for further developments in that direction. The methods of those works as well as the one of Eckmann and Hairer [5] are inspired by those of hypoellipticity for H¨ ormander type operators. We will not repeat here all the motivations of [14], coming also

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from the works [4,16] and others, including recent developments in pseudospectral theory. In [14] it was assumed that V in (1.1) is a smooth real Morse function with finitely many critical points, Uj , j = 1, . . . , N , that ∂ α V (x) is bounded for all multiindices α of length |α| ≥ 2 and that |∇V (x)| ≥ 1/C for |x| ≥ C for some C > 0. Under these assumptions it was shown that the eigenvalues in any disc D(0, Ch) are of the form λj,k (h) ∼ h(μj,k + h1/Nj,k μj,k,1 + h2/Nj,k μj,k,2 + · · · ) ,

h→0

(1.2)

where the index j labels the critical points and the leading coefficients μj,k can be given explicitly in terms of the Hessian of V at the corresponding critical point. The values μj,k are confined to a sector {|arg (z − μj,0 )| ≤ θj } for some θj ∈ [0, π2 [, and λj,0 is the eigenvalue with the smallest real part of all the λj,k . (This comes formally from a harmonic oscillator approximation.) Further μj,k = μj,0 for k = 0 and the asymptotic expansion (1.2) for λj,0 contains only integer powers of h. See [14] for more details. Using this result, as well as control over the resolvent along suitable contours in the right half plane, the authors were able to give asymptotic expansions for large times of exp (−tP/h), that emphasize the role of the eigenvalues close to 0 given in (1.2). Indeed, there are no others in a certain parabolic neighborhood of the imaginary axis. Moreover, we have μj,0 ≥ 0 with equality precisely when Uj is a local minimum of the potential, and in the case of such a minimum it follows from the above results that λj,0 is actually exponentially small. In this paper we address the question of determining more precisely the size of these exponentially small eigenvalues. In the case when V has precisely one local minimum, say U1 , and tends to +∞, when x → ∞, we know that the corresponding eigenvalue λ1,0 is equal to zero (with the Maxwellian exp (−(y 2 /2 + V (x))/h) as the corresponding eigenfunction) and that this eigenvalue is separated from the other ones by a gap of size h. This means that we have return to equilibrium with a speed that is roughly 1. The situation becomes more complicated when there is more than one local minimum. We are then in the presence of a tunneling problem which is much more complicated than the corresponding ones for the semiclassical Schr¨ odinger operators since our operators are non-elliptic. In principle one should be able to follow the general approach of earlier works in the Schr¨ odinger case as [9]. However it seems that one necessarily runs into a tunneling problem where the wave functions have to be studied also in a neighborhood of some intermediate saddle points of V , and as known from [10] that can indeed be done in the Schr¨ odinger case with techniques that are very useful in a variety of problems. To carry out such an approach in the case of Kramers–Fokker–Planck would require one to accumulate the difficulties of non-resonant wells with the ones coming from the lack of ellipticity. This seems to lead to considerations of degenerate non-symmetric Finsler distances (see for instance [1, 18]). For the Witten Laplacian (see [11]) we are also in the presence of a tunneling problem with intermediate non-resonant wells and in that case one could avoid

Vol. 9 (2008) Tunnel Effect for Kramers–Fokker–Planck Type Operators

211

the detailed study near the non-resonant wells by studying directly the Witten complex as a tunneling problem between critical points of neighboring indices. More recently M. Klein, B. Helffer, and F. Nier [7] have used that approach to study the exponentially small non-vanishing eigenvalues of the Witten Laplacian. Also in [13], explicit estimates relating such small eigenvalues for the Witten Laplacian and the Kramers–Fokker–Planck operator were established. This relation with the Witten complex was strengthened further in the works of J. Tailleur, S. Tanase-Nicola, J. Kurchan [24] and J. M. Bismut [2], who showed using respectively the languages of supersymmetry and differential forms, that the Kramers–Fokker–Planck operator can be viewed as a Witten Laplacian in degree 0 associated to a certain non-semidefinite scalar product in the spaces of differential forms. See also [17] for a quick introduction to the differential form version of Bismut and [3]. In the present paper, we use this supersymmetric approach. Our main result, valid also for a class of more general operators, is that if a certain weight function φ (which in the KFP-case is the function y 2 /2 + V (x)) has precisely two local minima U± and an intermediate saddle point U0 then we can get a complete asymptotic expansion for the second eigenvalue of the corresponding Witten Laplacian (reducing to the KFP operator in the special case). The logarithm of this eigenvalue is equal to −2h−1 (min(φ(U0 ) − φ(U1 ), φ(U0 ) − φ(U−1 )) + o(1)), but actually we do have a complete asymptotic expansion. See Theorem 11.1 for a complete statement. It seems clear that this result can be somewhat generalized but a more complete result might require exponential estimates and asymptotics for eigenfunctions also far from the critical points. In our present approach we are able to get such information for the eigenfunctions of the degree 0 operator in the basin of attraction of each minimum and for the degree 1 Laplacian in a small neighborhood of the saddle point. In most of the paper we work with a scalar real second order non-elliptic operator, which is also non-selfadjoint, and we were led to reconsider some steps in [14]. The plan of the paper is the following: In Section 2 we do some very simple and elementary exponential estimates mainly designed to get the appropriate control near infinity. In Section 3 we establish the m-accretivity for our operators so that the step from a priori estimates to spectral information becomes possible. In Section 4 we study certain auxiliary weights, somehow related to escape functions in resonance theory (see [12] and a large number of more recent works) in connection with some dynamical conditions. In Section 5 we use those weights together with a machinery of Fourier integral operators with complex phase in order to get phase space a priori estimates away from the critical points. This section is perhaps technically the most complicated one, but the underlying ideas are now quite standard. Alternative methods are certainly possible and we might return to this step in future works. This section and the subsequent one are quite technical and should not be studied in detail in the first reading.

212



In Section 6 we study the conjugation of our original operator under the Fourier integral operators of the preceding section and in Section 7 we finally obtain the a priori estimates that we need. In Section 8 it is now quite easy to get detailed asymptotic results by adapting the methods of [14]. In Section 9 we show that the eigenfunction associated to λj,0 for each (nondegenerate) critical point Uj has the form a(x; h)e−φ(x)/h in a neighborhood of that point, where a(x; h) has an asymptotic expansion in integer powers of h with coefficients in C ∞ (neigh (Uj )) and φ = φj is a smooth function of the order of magnitude |x − Uj |2 . It would be very interesting to extend such descriptions further away “beyond caustics”. In Section 10 we review the supersymmetric approach of [2,24] (see also [17]) and establish various interesting links between the dynamical conditions of Section 4 and old results for non-selfadjoint operators with double characteristics [22]. This sheds additional light on some related computations in [14]. Finally in Section 11, we can put the various results together and establish the precise exponential asymptotics of the spectral gap between the first and the second eigenvalue (both real and the first one being zero.) We expect that the spectral results of the present paper will give rise to precise asymptotics for the associated heat-evolution problem in the limit of large times and we plan to treat that problem in a separate paper.

2. A priori estimates In this section we establish some simple a priori estimates which will be important in Section 6 and at other places. They illustrate the technique of gaining ellipticity by means of exponential weights that we shall later employ also in a micro-local setting. Let M denote either the space Rn , or a smooth compact n-dimensional manifold equipped with a strictly positive smooth density of integration dx. On M we consider a second order differential operator P =

n

hDxj ◦ bj,k (x) ◦ hDxk +

j,k=1

= P2 + iP1 + P0 ,

Dx j =

n 1 cj (x)h∂xj + h∂xj ◦ cj (x) + p0 (x) 2 j=1

1 ∂ , i ∂xj

(2.1)

where the coefficients bj,k , cj , p0 are assumed to be smooth and real, with bj,k = bk,j . In the manifold case, we use local coordinates such that dx = dx1 . . . dxn . To P we associate the symbol in the semiclassical sense, p(x, ξ) = p2 (x, ξ) + ip1 (x, ξ) + p0 (x) , p2 (x, ξ) =

n j,k=1

bj,k (x)ξj ξk ,

p1 (x, ξ) =

(2.2) n j=1

cj (x)ξj ,

(2.3)


213

so that pj (x, ξ) is a real-valued polynomial in ξ, positively homogeneous of degree j. (It is well-defined on T ∗ M and coincides with the Weyl symbol mod O(h2 ) locally uniformly.) We assume that p2 (x, ξ) ≥ 0 ,

p0 (x) ≥ 0 .

(2.4)

In the case M = Rn , we impose the following growth conditions at infinity: ∂xα bj,k (x) = O(1) , ∂xα cj (x) ∂xα p0 (x)

|α| ≥ 0 ,

(2.5)

= O(1) ,

|α| ≥ 1 ,

(2.6)

= O(1) ,

|α| ≥ 2 .

(2.7)

When discussing P in the Operator theoretical sense we will assume that it is the closure of P : S(M ) → S(M ) as an unbounded Operator in L2 (M ). (When M is compact, we identify the Schwartz space S(M ) with C ∞ (M ).) Let D(P ) ⊂ L2 (M ) be the domain. Lemma 2.1. We have p0 (x)|u(x)|2 dx + bj,k (x)(hDxj u)(hDxk u)dx = Re (P u|u) ,

(2.8)

j,k

for all u ∈ S(M ).

Proof. Immediate by integration by parts.

In the manifold case, we view B(x) = (bj,k (x)) as a positive semi-definite matrix Tx∗ M → Tx M and if we choose some smooth Riemannian metric on M , 1 we can view B(x) as a map Tx∗ M → Tx∗ M and define B(x) 2 similarly. (2.8) then becomes 1

1

p0 (x) 2 u 2 + B(x) 2 hDu 2 = Re (P u|u) ,

(2.9)

implying 1

1

p02 u + B 2 hDu ≤ C0 ( P u + u ) . 1 2

(2.10)

1

In particular, u ∈ D(P ) ⇒ p0 u + B 2 hDu < ∞. Using the anti-selfadjoint part iP1 we shall obtain a similar estimate where the averages of p0 along the trajectories of ν(x, ∂x ) =

n

cj (x)∂xj

(2.11)

1

will play a role. In general, if ψ(x) is a smooth real-valued function, the Operator Pψ := eψ/h ◦ P ◦ e−ψ/h

(2.12)

214



is of the same form as (2.1) with new coefficients cj , p0 and the new symbol (2.13) pψ (x, ξ) = p2 x, ξ + iψ (x) + ip1 x, ξ + iψ (x) + p0 (x) = p2 (x, ξ) + i p1 (x, ξ) + ∂ξ p2 x, ψ (x) · ξ + p0 (x) − p1 x, ψ (x) − p2 x, ψ (x) =: p2 (x, ξ) + ip1,ψ (x, ξ) + p0,ψ (x) . In this section we choose ψ very small and treat p2 (x, ψ (x)) as a perturbation. Notice that (2.14) p1 x, ψ (x) = ν(x, ∂x )ψ . Let f (t) ∈ C ∞ ([0, ∞[; [0, 3/2]) be an increasing function with f (t) = t on [0, 1], f (t) = 3/2 on [2, ∞[, f (t) ≤ t. Put f (t) = f (t/) and consider for T0 > 0 fixed, t (2.15) ψ = k f ◦ p0 ◦ exp (tν)dt , T0 where

⎧ ⎪ ⎨0 , |t| ≥ 1/2 , k(t) = t + 12 , − 12 ≤ t < 0 , ⎪ ⎩ −k(−t) , 0 < t ≤ 12 .

Then ν(ψ ) = f ◦ p0 − f ◦ p0 T0 , where f ◦ p0 T0 =

1 T0

T0 /2

−T0 /2

(2.16)

f ◦ p0 ◦ exp (tν)dt

(2.17)

is the time T0 average of f ◦ p0 along the integral curves of ν. Clearly, |ψ | ≤

3T0 T0 3 = . 4 2 8

(2.18)

From (2.6) it is easy to see that Φt (x) := exp (tν)(x) is well-defined for all t ∈ R, x ∈ M and that |∂xα Φt (x)| ≤ C(α)eC0 |α||t| ,

α ∈ Nn ,

|α| ≥ 1 .

(2.19)

In particular, |∂xα Φt (x)| ≤ Cα,T0 , |t| ≤ T0 . On the other hand, since p0 ≥ 0, 1

p0 = O(1) (by (2.7)), we know that |p0 | ≤ O(1)p02 and this quantity is O(1/2 ) in the region where 0 ≤ p0 ≤ 2. It follows that ∂ α (f ◦ p0 ) = O(1−|α|/2 ), α ∈ Nn and together with (2.19), we get ∂xα ψ = O(1−|α|/2 ) , for every fixed T0 .

∀α ∈ Nn ,

(2.20)


215

Then from (2.13), (2.14), (2.16), (2.20) and the uniform boundedness of the bj,k , we get p0,δψ = p0 (x) − δf ◦ p0 + δf ◦ p0 T0 − δ 2 p2 (x, ψ )

(2.21)

2

= p0 (x) − δf ◦ p0 + δf ◦ p0 T0 − O(δ ) , n

uniformly on R for 0 ≤ δ ≤ 1. Using the properties of f , we notice that p0 (x) − δf ◦ p0 ≥ (1 − δ)p0 , so p0,δψ ≥ (1 − δ)p0 + δf ◦ p0 T0 − O(δ 2 ) .

(2.22)

Since the coefficients of Pδψ grow at most polynomially, Lemma 2.1 can be applied and gives p0,δψ (x) − Re z |u|2 dx ≤ Re (Pδψ − z)u|u , z ∈ C , Re z ≤ , (2.23) Z where Z 1 is independent of . (For Lemma 2.1 we do not need that p0 ≥ 0.) Let μ > 0 and rewrite (2.23) as max( p0,ψ , μ)|u|2 dx ≤ Re (Pψ − z) + (μ − p0,ψ )+ u|u 1 = Re max( p0,ψ , μ)− 2 Pψ − z 1 p0,ψ , μ) 2 u . + (μ − p0,ψ )+ u| max( Here we write ψ for δψ and p0,ψ = p0,ψ − Re z. Then using Cauchy–Schwarz,

(μ − p0,ψ )+ 1 1 1 2 u , max( p p0,ψ , μ)− 2 (Pψ −z)u + , μ) max( p0,ψ , μ) 2 u ≤ max( 0,ψ μ leading to 1

1

max( p0,ψ , μ) 2 u ≤ max( p0,ψ , μ)− 2 (Pψ − z)u 1

+ 1{p0,ψ ≤μ} max( p0,ψ , μ) 2 u . 1 2

(2.24)

1 2

Notice that 1{p0,ψ ≤μ} max( p0,ψ , μ) = 1{p0,ψ ≤μ} μ . We have p0,ψ = p0 (x) − δf ◦ p0 + δf ◦ pT0 − O(δ 2 ) − Re z ≥ p0 (x) − δf ◦ p0 + δf ◦ pT0 − O(δ 2 ) − . Z Choose Z = δ −2 , so that p0,ψ ≥ p0 (x) − δf ◦ p0 + δf ◦ pT0 − O(δ 2 ) . Choose δ > 0 small enough (but independent of ) so that O(δ 2 ) ≤ δ/C0 , where we shall fix C0 sufficiently large. Then p0,ψ ≥ (1 − δ)p0 (x) + δ f ◦ p0 T0 − . C0

216


When f ◦ p0 T0 ≥

2 3 ,

we get p0,ψ ≥ (1 − δ)p0 (x) +

When f ◦ p0 T0 < Choose

2 3


we have p0,ψ ≥ (1 − δ)p0 − μ=

δ . 2 δ C0 .

δ . C0

Then p0,ψ , μ) . (1 − δ)p0 + μ ≤ p0,ψ + 2μ ≤ 3 max( δ C0

(2.25)

δ C0 ,

2 C0 .

Moreover, if p0,ψ ≤ μ, then δf ◦ p0 T0 − ≤ so f ◦ p0 T0 ≤ From (2.24) and (2.25) we then infer that 1 − 12 δ 2 δ u ≤ 3 (1 − δ)p0 + (Pψ − z)u (2.26) (1 − δ)p0 + C0 C0 √ δ u {f ◦p0 T0 ≤ C2 } , + 3 0 C0 for Re z ≤ Z . Here we can take = M h with M 1. Then by (2.18) we have |ψ/h| = |δψ /h| ≤ C(M, T0 )δ , so there is a constant C independent of h (but depending on M, δ) such that 1 ≤ eψ/h ≤ C , C From the discussion above, in particular (2.26), we get Proposition 2.2. Let P be of the form (2.1), where bj,k , cj , p0 are smooth and real and satisfy (2.2)–(2.7). Define f ◦ p0 T0 as in (2.17) with f defined after (2.14). > 0 such that Then for every C > 0, there exists C 1 (p0 + h)− 12 (P − z)u + h 12 u (2.27) (p0 + h) 2 u ≤ C {f ◦p0 T ≤Ch} , C

0

for u ∈ S, Re z ≤ Ch. Notice that (2.8) implies that 1

1

1

1

B 2 hDu 2 ≤ (p0 + h)− 2 (P − z)u (p0 + h) 2 u + C h 2 u 2 .

(2.28)

3. From injectivity to the resolvent Let P = P2 + iP1 + P0 with symbol p = p2 + ip1 + p0 be as in Section 2, so that we have (2.8) p0 (x)|u(x)|2 dx + (P2 u|u) = Re (P u|u) , u ∈ S(M ) ,


217

leading to 1 |Re z| u 2 + p02 u 2 + (P2 u|u) = Re (P − z)u|u ≤ (P − z)u u ,

Re z < 0 .

We get 1

1

1

(|Re z| + p0 ) 2 u 2 ≤ (|Re z| + p0 )− 2 (P − z)u (|Re z| + p0 ) 2 u , 1

1

(|Re z| + p0 ) 2 u ≤ (|Re z| + p0 )− 2 (P − z)u , 1

1

|Re z| 2 (|Re z| + p0 ) 2 u ≤ (P − z)u , |Re z| u ≤ (P − z)u . From this we get (P2 u|u) ≤ (P − z)u u ≤

1 (P − z)u 2 , |Re z|

and putting some of the estimates together, 1

|Re z|2 u 2 + |Re z| p02 u 2 + |Re z|(P2 u|u) ≤ 2 (P − z)u 2 .

(3.1)

By P we also denote the graph closure of P : S → S. From the estimates above we see that the range R(P − z) is closed in L2 when Re z < 0. Proposition 3.1. R(P − z) = L2 ,

Re z < 0 .

Proof. It suffices to prove that R(P − z) = L2 for some z with Re z < 0, because the a priori estimate then implies that (P −z)−1 ≤ |Re z|−1 and this fact extends by standard arguments to the whole left half plane. For the same z it suffices to show that if u ∈ L2 and (P ∗ − z)u = 0 in the sense of distributions, then u = 0. Now the formal adjoint P ∗ = P2 − iP1 + P0 has the same properties as P , so in order to simplify the notations, we may just as well prove the corresponding fact for P − z instead of P ∗ − z: There exists a z with Re z < 0 such that if u ∈ L2 and (P − z)u = 0 in the sense of distributions, then u = 0. When M = Rn , let Oph (q) denote the Weyl quantization of q(x, hξ) and put −N Λ = Oph (x, ξ) where N ≥ 2 is fixed and > 0 is small and fixed. (When M is a compact manifold, choose a Riemannian metric and put Λ = (1 − h2Δ)−N/2 .) Consider the equation (P − z)u = v ,

u, v ∈ L2 .

Then Λ (P − z)u = Λ v, so (P − z)Λ u = Λ v + [P, Λ ]u .

(3.2) 1 2

Since N ≥ 2, we can find a sequence uj ∈ S such that Λ uj → Λ u, p0 Λ uj → 1

p02 Λ u, Λ (P − z)uj → Λ v, [P, Λ ]uj → [P, Λ ]u in L2 , (P2 Λ uj |Λ uj ) →

218



(P2 Λ u|Λ u). This means that (3.1) is applicable to (3.2), and we get 1

|Re z|2 Λ u 2 + |Re z| p02 Λ u 2 + |Re z|(P2 Λ u|Λ u) ≤ 4 Λ v 2 + [P, Λ ]u 2 . (3.3) −1 and Consider [P, Λ ]u = ([P, Λ ]Λ−1 )Λ u. We study the Operator [P, Λ ]Λ n assume that M = R in order to fix the ideas. If ρ = (x, ξ), we have

∂ρα ρ−N = O(1)ρ−N −|α| , so

|α| = O(1)ρ−N ρ−|α| , ρ uniformly with respect to . From this we deduce that the symbol of [P, Λ ] is h 2 i {p, Λ } + O0 (h Λ ), using the same letters for Operators and their symbols (except for P, Pj where we already introduced a distinction by using lower case letters for the symbols), and using the notation O0 (h2 Λ ) for a symbol q satisfying ∂ρα q = Oα (h2 Λ ), uniformly in . Here {p, Λ } = O0 Λ (ρ)ρ , ∂ρα ρ−N = O(1)ρ−N

is so the symbol of [Λ , P ]Λ−1 h {p, Λ } + O0 (h2 ) . i Λ Recall that p(x, ξ) = p0 (x) + ip1 (x, ξ) + p2 (x, ξ). We get {p0 , Λ } x = O0 = O0 (1) , Λ ρ {p1 , Λ } x ξ = O0 + O0 = O0 (1) , Λ ρ ρ {p2 , Λ } = Q + O0 (1) , Λ 1 ∇ξ Λ Q=− · ∇x p2 = O0 · ∇x p2 (ρ) . Λ ρ From these computations, we retain that [P, Λ ]Λ−1 w ≤ O(h) w + O(h) Qw . The symbol Q is real and ∂ξα Q = O0 (ξ2−|α| /ρ), so Qw 2 = (Q2 w|w) , where Q ◦ Q has the symbol Q2 + O0 (h2 ). Since p2 ≥ 0, ∂x2 p2 = O(ξ 2 ) we know that 1

|∂x p2 | ≤ O(1)|ξ|p2 (x, ξ) 2 ,

(3.4)


219

and we conclude that on the symbol level Q2 ≤ Cp2 . Hence by the semi-classical Fefferman–Phong inequality for operators with symbols 2 (see for example Subsection 7.2 in [14] for a short in the H¨ ormander class S1,0 review of the Weyl–H¨ ormander calculus): Qw 2 ≤ C(P2 w|w) + O(h2 ) w 2 . Using this in (3.4), we get

2 2 2 . [P, Λ ]Λ−1 w ≤ Ch (P2 w|w) + w

we use this in (3.3) with [P, Λ ]u =

[P, Λ ]Λ−1 Λ u

(3.5)

and get

1

|Re z| p02 Λ u 2 + (|Re z|2 − Ch2 ) Λ u 2 + (|Re z| − Ch2 )(P2 Λ u|Λ u) ≤ 4 Λ (P − z)u 2 . (3.6)

√ So if Re z < − Ch and (P − z)u = 0, u ∈ L2 , we have u = 0.

Corollary 3.2. The maximal closed extension Pmax of P (with domain given by {u ∈ L2 ; P u ∈ L2 }) coincides with the graph closure (the minimal closed extension), already introduced. √ Proof. Let z be fixed with Re z < − Ch. Let u, v ∈ L2 with (Pmax − z)u = v. Denote by Pmin the graph closure. Since R(Pmin −z) = L2 , there exists u ∈ D(Pmin ) u = v. Hence (P − z)(u − u ) = 0 in the sense of distributions. such that (Pmin − z) We saw in the proof of the proposition that this implies that u − u = 0. This result can be extended to the following auxiliary problem: Let R− : CN → L2 , R+ : L2 → CN be bounded Operators and assume that P − z R− : D(P ) × CN → L2 × CN , P(z) = R+ 0 is injective with u + |u− | ≤ C(h)( v + |v+ |) ,

z ∈ Ω(h) ,

(3.7)

u v P(z) = . (3.8) u− v+ Here we assume that Ω(h) is open and connected, intersecting the resolvent set ρ(P ) of P . From (3.7), we see that P(z) is injective and has closed range for every z ∈ Ω(h). When z is also in the resolvent set of P , we can rewrite (3.8) as 1 (P − z)−1 R− u (P − z)−1 v = , (3.9) v+ R+ 0 u− whenever

and the matrix appearing here is injective. On the other hand it is a finite rank perturbation of the identity in L2 × CN and the injectivity implies the bijectivity.

220



It follows that P(z) is bijective for z ∈ Ω(h) ∩ ρ(P ). Combining this with (3.7) and recalling that Ω is connected we see by a standard argument that P(z) is bijective for all z ∈ Ω and that P(z)−1 ≤ C(h).

4. Geometric preparations In this section we construct a special bounded weight that we shall implement in Section 5 with the help of Fourier integral operators with complex phase. Let P be the Operator introduced in the beginning of Section 2, satisfying (2.1)–(2.7). The symbol p(x, ξ) = p0 (x) +

p2 (x, ξ) ξ2

is non-negative and satisfies α ∂x , ξ∂ξ p = O(1) ,

|α| ≥ 2 ,

locally uniformly with respect to x. It follows that α 1 p 2 ) , |α| = 1 . ∂x , ξ∂ξ p = O(

(4.1)

(4.2) (4.3)

We now introduce a critical set associated to p. Hypothesis 4.1. Assume The set x ∈ M ; p0 (x) = 0, ν(x, ∂x ) = 0 is finite = {x1 , . . . , xN } .

(4.4)

Let ρj = (xj , 0) and put C = {ρ1 , . . . , ρN } .

(4.5)

Notice that p1 , p0 , p2 , p vanish to second order at each ρj . Our weight will be of the form t (4.6) ψ = − k p ◦ exp (tHp1 )dt , 0 < 1 , T0 where p will be specified below and k is the same function as in Section 2. Notice that p T0 − p , (4.7) Hp1 ψ = where we now write in general, T0 /2 1 qT0 = q ◦ exp (tHp1 )dt . T0 −T0 /2 Let g(t) ∈ C ∞ ([0, +∞[; [0, 1]) be a smooth decreasing function with 1, 0 ≤ t ≤ 1, g(t) = −1 t , t ≥ 2.

(4.8)

Notice that g (k) (t) = O(t−1−k ). In a domain 0 ≤ |ρ − ρj | < 1/C, C 1, we put |ρ − ρj |2 p (ρ) = g p , (4.9)


221

√ (so that p (ρ) = p(ρ) for |ρ − ρj | ≤ ). It is easy to show the symbol estimates ∂ α p = O (4.10) , 0 ≤ |ρ − ρj | < 1/C . ( + |ρ − ρj |2 )|α|/2 Away from any fixed neighborhood of C, we simply get ∂ α p = O() . Let χj ∈ C0∞ (R2n ) be equal to 1 near ρj and have its support close to C. To define p further out from C, we put N |ρ − ρj |2 p (ρ) = χj (ρ)g (4.11) p + 1 − χj (ρ) p , 1 so that (4.10) remains valid in a neighborhood of C, while ∂xα ∂ξβ p = O ξ−|β| = O ξ−|β|

(4.12)

outside any such region where |x| is bounded. We also have ∂xα ∂ξβ (ξ · ∂ξ p ) = O ξ−2−|β|

(4.13)

in the same region. In fact, (4.13) follows if we write for |ξ| 1: p2 1 p2 1 p2 ξ · ∂ξ p = ξ · ∂ξ 2 = ξ · ∂ξ ξ · ∂ 1 − = − . ξ ξ |ξ|2 ξ2 |ξ|2 ξ2 p when dist (ρ, C) ≥ 1/C and in particular From (4.11) we see that p (ρ) = for ρ = (x, ξ) with |x − xj | ≥ 1/C, ∀j. Further away from {x1 , . . . , xN } we want to make p (x, ξ) independent of ξ, so we replace p = p0 (x) + p2 (x, ξ)/ξ2 in (4.11) by χ(x) p2 (x, ξ) , (4.14) pnew (x, ξ) = p0 (x) + ξ2 where χ ∈ C0∞ (M ; [0, 1]) is equal to 1 near {x1 , . . . , xN }. Even further out (in the case when M = Rn ) we want to avoid problems caused by p0 being large, so when |x| 1 we want to replace p = p0 there by f (p0 ), where f (t) = f (t/) is the function introduced in Section 2. Thus with a new cutoff function χnew ∈ C0∞ (M ; [0, 1]) being equal to 1 in a large neighborhood of supp χ, we get the final choice of p : N |ρ − ρj |2 p (ρ) = χnew (x) χj (ρ)g p + 1 − χj (ρ) pnew 1 + 1 − χnew (x) f (p0 ) . (4.15) Notice that by construction p ≤ p ≤ p0 + p2 . We also get

(4.16)

222



Proposition 4.2. We have (4.10) near C and (4.12), (4.13) in any closed subset of T ∗ M disjoint from C and supp (1 − χnew ) × Rn , where χnew is the wider cutoff in (4.15). Over supp (1 − χnew ) (where p only depends on x) we have ∂xα p = O(1−|α|/2 ). From the definition (4.6), we see that ψ satisfies the same estimates as p and will only depend on x for x outside a neighborhood πx (C) (that we can choose as small as we like) and that the region where we only have ∂xα ψ (x) = O(1−|α|/2 ) can be any neighborhood of infinity. This follows from the fact that our various symbol estimates (as well as the ξ-independence) are conserved by the flow of Hp1 , as for (4.13), we here also use that Hp1 commutes with dilations in ξ. It also follows from the construction that √ when |ρ − ρj | ≤ , (4.17) p (ρ) = p(ρ) , √ p (ρ) p , when ≤ |ρ − ρj | ≤ 1/C , (4.18) |ρ − ρj |2 p (ρ) pnew , when dist (ρ, C) ≥ 1/C , |ρx | ≤ C , (4.19) p (ρ) = f (p0 ) ,

. |ρx | ≥ C

when

(4.20)

We introduce the following dynamical conditions where T0 > 0 is fixed: Hypothesis 4.3. 1 |ρ − ρj |2 , C 1 In any set |x| ≤ C , dist (ρ, C) ≥ , C

pT0 ≥ Near each ρj we have

we have pT0 (ρ) ≥

(4.21)

1 , C(C) > 0. C(C)

(4.22)

When M = Rn we also need a modified dynamical assumption ∀ neighborhood U of πx C, and ∀ x ∈ Rn \ U , ∃C > 0 , 1 T0 T0 1 meas t∈ − , ; p0 exp tν(x) ≥ ≥ . (4.23) 2 2 C C When M is compact, we just assume (4.21), (4.22), where it is understood that the estimate in (4.22) should hold for all ρ ∈ T ∗ M of distance ≥ 1/C from C. Notice that in the region dist (x, πx (C)) ≥ 1/C, |x| ≤ C, (4.22) is equivalent (up to a change of C(C)) to p0 T0 ,ν (x) ≥ where qT0 ,ν =

1 T0

1 , C(C)

T0 /2

−T0 /2

C(C) > 0,

q ◦ exp (tν)dt .

(4.24)


223

This follows from the fact that πx ◦ exp tHp1 = exp (tν) ◦ πx , where πx ((x, ξ)) := x and the fact that {ξ = 0} is invariant under the Hp1 flow. Actually, if we only assume (4.21), then we see that p2 T0 (xj , ξ) ξ 2 , and if we also assume (4.24), we recover (4.22), first over a neighborhood of each xj and then again by (4.24) over any bounded set in M . Assuming the dynamical conditions (4.21), (4.22), ((4.24), (4.23)), we see that √ p T0 dist (ρ, C)2 , dist (ρ, C) ≤ , (4.25) √ p T0 , dist (ρ, C) > . (4.26) From the estimates on ψ in the various regions that we mentioned after Proposition 4.2 we shall often only retain that ∂xα ∂ξβ ψ (x, ξ) = O 1−|α+β|/2 ξ−|β| , (4.27) and we write this for short as

. ψ = O() Similarly, we have in view of (4.13), and (4.10) also valid for ψ , that ξ−2 . ξ · ∂ξ ψ = O

(4.28)

1

From [20] let us recall that if f is a C function locally defined on complexified phase space, then at every point where ∂f = 0, we have Re σ Im σ f = HRe H f = HIm f ,

Re σ Im σ if = J H f = H−Im H f = HRe f ,

(4.29)

∂f ∂ ∂f ∂ Hf = − ∂ξj ∂xj ∂xj ∂ξj is the complex (1,0) Hamilton field of f with respect to the complex symplectic f = Hf + Hf is the real vector field which acts as form σ = dξj ∧ dxj and H Im σ Hf on holomorphic functions. HIm f denotes the real Hamilton field of Im f with respect to the real symplectic form Im σ, and similarly for the other Hamilton fields appearing in (4.29). As usual, J denotes the action on tangent vectors induced by multiplication by i. (When M is compact we may assume without loss of generality that M is real-analytic.) Assume first that M = Rn and put Λ0 = T ∗ Rn = R2n , Λs = ρ + isHψ (ρ); ρ ∈ Λ0 , 0 ≤ s 1 . (4.30) where

If we extend ψ to be a function on the complexified phase space C2n , by setting ψ (ρ) = ψ (Re ρ) ,

(4.31)

then we have the equivalent description of Λs as σ Λs = exp (sHψIm )(Λ0 ) .

(4.32)

It follows that Λs is an I-Lagrangian manifold, i.e., a manifold which is Lagrangian for Im σ.

224



Now we would like to parametrize Λs by means of a canonical transformation κ(s) : Λ0 → Λs for the symplectic form Re σ (implying that Λs like Λ0 is an IRmanifold, i.e., a manifold which is Lagrangian for Im σ and symplectic for the ρ) be the unique function which restriction of Re σ). For 0 ≤ s 1, let ψ,s = ψ(s, is affine in Im ρ and satisfies ∂ ρ ψ,s = 0 on Λs ,

ψ,s = ψ

on Λs .

(4.33)

∂ Recalling that ψ (ρ) = ψ (Re ρ), we get with the notation (ψ )ξ = ∂Re ξ ψ , (ψ )x = ∂ ∂Re x ψ to indicate derivatives in the real directions, ψ,s (x, ξ) = ψ (x, ξ) + iax Re (x, ξ) · Im x − s(ψ )ξ + iaξ Re (x, ξ) · Im ξ + s(ψ )x , (4.34)

where

ax (ψ )x 1 − is F ψ Re (x, ξ) = (ψ )ξ aξ t

and

Fψ =

(ψ )ξx −(ψ )xx

(ψ )ξξ −(ψ )xξ

(4.35)

(4.36)

is the fundamental matrix of ψ . It also follows from the construction that 12 ) , aξ = O 12 ξ−1 . (4.37) ax = O( Im σ Since ψ,s = ψ on Λs , we know that HRe ψ

,s

σ − HψIm is tangent to Λs for

every s. We can therefore define κ(s) : Λ0 → Λs , by d Im σ κ(s)(ρ) = HRe ,s κ(s)(ρ) , ψ ds Im σ The second relation in (4.29) implies that HRe ψ

,s

ρ ∈ Λ0 . Re σ = H−Im ψ

,s

(4.38) on Λs (given that

∂ ψ,s = 0 there) and hence κ(s) is symplectic for Re σ: κ(s) (Re σ) = Re σ. Notice that (4.38) again shows that Λs is an I-Lagrangian manifold. A priori, κ(s)(ρ) is well-defined only for s ≥ 0 small enough depending on ρ, but we shall next derive symbol estimates for ψ,s and κ(s) that will imply that κ(s)(ρ) is indeed well-defined for s small enough independently of ρ. Assume that we work near a point (x0 , rξ0 ), x0 ∈ Rn , ξ0 ∈ S n−1 , r ≥ 0 and replace (x, ξ) by where ( x, ξ), √ √ x = x0 + x , ξ = rξ0 + r ξ. (4.39) ∗

x, ξ), so that Define ψ = ψ,x0 ,r,ξ0 by ψ (x, ξ) = ψ( √ √ x, ξ) = 1 ψ x0 + ψ( x, rξ0 + r ξ , ≤ Const. ∂xα ∂ β ψ = O(1) , ∀α, β , for |( x, ξ)| ξ

(4.40) (4.41)


In the new coordinates Λs takes the form . = s H Re ( x, ξ) Im ( x, ξ) ψ r

225

(4.42)

The scaling and the construction of ψ,s commute so if we put √ √ 1 x, rξ0 + r ξ , ψ,s = ψ,s x0 + then (4.41) still holds. In particular ψ,s satisfies the same estimates on Λs as ψ in (4.27), now with ∂x , ∂ξ replaced by (∂Re x , ∂Im x ), (∂Re ξ , ∂Im ξ ). The transformation κ(s) can be scaled similarly and the scaling commutes with (4.38) up to a factor r: If we put √ √ κ(s)(ρ) = x(s), ξ(s) = x0 + x(s), rξ0 + r ξ(s) , then from (4.38), (4.29) we get i ∂ξψ,s x (s), ξ(s) , r = − i ∂x ψ ∂s ξ(s) (s), ξ(s) . ,s x r √ √ We conclude that with (x(0), ξ(0)) = (y, η) = (x0 + y , rξ0 + r η ), −k k α β = O r ∂s ∂y ∂η ( x, ξ) , (s) = ∂s x

(4.43)

(4.44)

and hence for (x, ξ) = κ(s)(y, η):

1 |α+β| ∂sk ∂yα ∂ηβ x = O 2 − 2 η−k−|β| , 1 |α+β| ∂sk ∂yα ∂ηβ ξ = O 2 − 2 η1−k−|β| ,

(4.45)

when k + |α + β| ≥ 1. Notice that the right hand sides in (4.43) reduce to ir−1 ∂ξψ and −ir−1 ∂x ψ when s = 0, where the derivatives are taken in the real directions. The flow in (4.45) is therefore tangent to the one in (4.32) at s = 0 and we get κ(s)(y, η) = (y, η) + isHψ (y, η) + (z, ζ) , 1 |α+β| ∂yα ∂ηβ z = O 2 − 2 s2 η−2−|β| , 1 |α+β| ∂yα ∂ηβ ζ = O 2 − 2 s2 η−1−|β| . From (4.45) and the subsequent remark, we have for κ(δ)(x, ξ) = κ(δ, x, ξ). 1 1 1 2 2 κ(δ, x, ξ) = (x, ξ) + iδHψ (x, ξ) + δ O ,O . (4.46) 2 ξ ξ Recalling how κ(δ) was constructed we conclude that + iδHψ ( κ(δ, x, ξ) = ( x, ξ) x, ξ) ,

(4.47)

is real and ( 1 2 ), O( 1 )). Put αδ (x, ξ) = ( x, ξ). where ( x, ξ) x, ξ)−(x, ξ) = δ 2 (O( ξ ξ 1 2

226



The essential part of the discussion above took part near the points of C. In that region the discussion is the same in the case when M is compact.

5. Quantization of weights We will follow [12, 23] with one modification; instead of analyticity we will use that our weights are “moderate” allowing us to use almost holomorphic extensions. Another minor difference is that we shall not use FBI-transforms explicitly, but rather rely on certain Fourier integral operators with complex phase. We will assume that M = Rn for simplicity, but as in the preceding section the essential part of the work will take place near C and here there is no difference between the case M = Rn and the case when M is compact. As a first step towards introducing some Fourier integral operators we shall study the function h(y, η) on T ∗ Rn , given by κ(δ)∗ (ξ · dx) − η · dy = dh . Recall here that

d κ(s)ρ = Hiψ,s κ(s)ρ , ds

ρ ∈ Λ0 = T ∗ Rn ,

(5.1)

(5.2)

where ψ,s is given by (4.34), satisfying (4.33). Of course, (5.2) remains unchanged if we replace ψ,s by an almost holomorphic extension from Λs = Λsψ . Now using Cartan’s formula, we get d κ(s)∗ (ξ · dx) = κ(s)∗ LiHψ,s (ξ · dx) ds = iκ(s)∗ Hψ,s d(ξ · dx) + d(Hψ,s ξ · dx) ∂ ψ,s ∗ = iκ(s) Hψ,s σ + d ξ · ∂ξ ∂ ψ,s ∗ = iκ(s) d ξ · − dψ,s ∂ξ ∂ ∗ = idκ(s) ξ · ψ,s − ψ,s . ∂ξ Thus, we can take δ h=i (ξ · ∂∂∂ξ ψ,s − ψ,s ) ◦ κ(s)ds . (5.3) 0

,s ∂ψ ∂ξ

,s ∂ψ ∂Re ξ

On Λs we have = and we recall that ψ (x, ξ) = ψ (Re (x, ξ)). Using (4.34), we get on Λs : ∂ ∂ ψ (x, ξ) − is ax Re (x, ξ) |(ψ )ξξ ξ ξ · ψ,s = ξ · ∂ξ ∂Re ξ + is aξ Re (x, ξ) |(ψ )xξ ξ , (5.4)


227

where again we use the notation ( · )ξ , ( · )xξ etc to indicate derivatives in the real directions. We now estimate the terms in the right hand side of (5.4). On Λs we have Im ξ = s(ψ )ξ , so ξ·

∂ ψ = Re ξ · (ψ )ξ − is(ψ )x · (ψ )ξ . ∂Re ξ

Hence by (4.27), (4.28), ξ· Next, look at

∂ ξ−1 . ψ = O ∂Re ξ

ax |(ψ )ξξ ξ = ax |(ψ )ξξ Re ξ − is ax |(ψ )ξξ (ψ )x .

−2 ) by (4.27), (4.37). The first term is equal to The last term is O(ξ ∂ (ψ )ξ · Re ξ − (ψ )ξ · ax . ax |(ψ )ξξ Re ξ = ax · ∂Re ξ From (4.28), (4.27), (4.37) this is O(ξ−1 ), so Re ξ−1 . ax |(ψ )ξξ ξ = O

Similarly,

(5.5)

(5.6)

aξ |(ψ )xξ ξ = aξ |(ψ )xξ Re ξ − is aξ |(ψ )xξ (ψ )x Re ξ−2 = aξ |(ψ )xξ Re ξ + O ∂ψ ∂ Re ξ−2 , · Re ξ + O = aξ · ∂Re x ∂Re ξ

so

Re ξ−2 . aξ |(ψ )xξ ξ = O

(5.7)

Returning to (5.4), we get ξ·

∂ Re ξ−1 ψ,s = O on Λs . ∂ξ

(5.8)

Also recall that ψ,s = ψ on Λs , so ξ·

∂ ψ,s − ψ,s = O() ∂ξ

on Λs .

Combining this with (4.45), we get ∂ , ξ · ψ,s − ψ,s ◦ κ(s) = O() ∂ξ

(5.9)

(5.10)

and finally from (5.3) we obtain Lemma 5.1.

h = O(δ) .

(5.11)

228



Following [12] we shall now quantize κ(δ) by means of a Fourier integral operator i Au(x; h) = e h φ(x,y,α) a(x, y, α; h)χ(x, y, αx )u(y)dydα , (5.12) (y,α)∈Rn ×T ∗ M

3n n 2 +m, 2 +k

with a ∈ S , and where χ is a standard cutoff to a neighborhood of the diagonal: |x − αx |, |y − αx | < 1/C, equal to 1 on a smaller neighborhood of the same type. Here we take with β = κ(δ, α) = κ(δ)(α): φ(x, y, α) = (x − βx ) · βξ + (αx − y) · αξ + ψ(x − βx , y − αx , α) + h(α) ,

(5.13)

where on the real domain,

ψ(x − βx , y − αx , α) = O αξ (x − βx )2 + (y − αx )2 ,

(5.14)

in the symbol sense:

∂xk ∂y ∂αmx ∂αp ξ ψ(x, y, α) = O αξ 1−|p| (|x| + |y|)(2−|k+|)+ ,

and we take almost holomorphic extensions satisfying the same estimates in the 3n n 3n n complex domain. Further, a ∈ S 2 +m, 2 +k means that a = O(h− 2 −m αξ 2 +k ) 3n n in the symbol sense, ∂x,y ∂αp x ∂αq ξ a(x, y, α; h) = O(h− 2 −m αξ 2 +k−|q| ). We also assume that (5.15) Im ψ αξ (x − βx )2 + (y − αx )2 . Viewing h as a function on the graph of κ(δ), we have dh = βξ · dβx −αξ · dαx , so for x = κ(δ, α)x , y = αx , (5.16) we have dα φ = 0. Moreover, in a neighborhood of that set, we have with ψ = ψ(x, y, α), dα φ = x − κ(α)x · dα κ(α)ξ + (αx − y) · dαξ (5.17) ∂ψ ∂ψ ∂ψ − x − κ(α)x , y − αx , α · dα κ(α)x − · dαx + · dα . ∂x ∂y ∂α So φ is a non-degenerate phase function in the sense of H¨ ormander (a part from the homogeneity condition in the fiber variables) with a critical set (5.16), the associated canonical transformation is κ = κ(δ). Similarly, to κ(δ)−1 we can associate i Bv(x; h) = e h φ(x,y,γ)b(x, y, γ; h)χ(x, y, γ)v(y)dydγ , (5.18) (y,γ)∈Rn ×T ∗ M

b∈S

3n n 2 +m, 2 +k

, m, k ∈ R, where with β = κ(δ)(γ):

y, γ) = (x − γx ) · γξ + (βx − y) · βξ + ψ(x − γx , y − βx , γ) − h(γ) φ(x,

(5.19)

and ψ satisfies (5.14). Again this is a non-degenerate phase and the critical set is given by x = γx , y = βx , β = κ(δ)(γ) , (5.20)


229

and the associated canonical transformation is κ(δ)−1 . For x, y ∈ Rn , |x − αx |, |y − βx | ≤ 1/C, we have with β = κ(α) = κ(δ, α): Im φ(x, y, α) = (x − Re βx ) · Im βξ − Im βx · Re βξ + Im ψ(x − βx , y − αx , β, α) + Im h(α) . Here

21 Re αξ −1 , Im βx = δ O

12 , Im βξ = δ O δ ∂ψ Im βx · Re βξ = δRe βξ · =O , ∂Re βξ Re βξ 2

and Im h(α) = O(δ) , |x − Re βx |2 C 1 2 2 |y − αx | − C|Im βx | , Im ψ(x − βx , y − αx , α) ≥ αξ + C so αξ (|x − Re βx |2 + |y − αx |2 ) C δ 2 αξ 1 −C − C|x − Re βx |δ 2 − Cδ , αξ 2 αξ . Im φ ≥ (|x − Re βx |2 + |y − αx |2 ) − Cδ 2C Recalling that = Ah, A 1, we get i αξ φ(x,y,α) 2 2 h (|x − Re βx | + |y − αx | ) + CδA . | ≤ exp − |e 2Ch Im φ(x, y, α) ≥

(5.21)

(5.22)

It follows that A, B are well-defined Operators: S → A similar estimate holds for φ. S with semi-norm estimates that are uniform in powers of h. Also for every s ∈ R, there is an s ∈ R such that A, B : H s → H s with norms bounded by some power of h. Moreover, our Operators are independent of the choice of almost holomorphic extensions of the phase and amplitude modulo Operators whose integral kernels are O(h∞ ) with all their derivatives and supported in a domain of the form |x − y| ≤ O(1). Proposition 5.2. If m, k, m, k = 0, then A, B = O(exp O(1)δA) : L2 (Rn ) → 2 n L (R ) The proof will be given later. −m−m Proposition 5.3. We have BA = Oph (c), where c = O(h αξ k+k )

Proof. We have i BAu(x) = e h (φ(x,z,γ)+φ(z,y,α))b(x, z, γ; h)a(z, y, α; h)u(y)dydαdzdγ ,

230



where the cutoffs χ, χ have been incorporated in a, b. Here z, γ) + φ(z, y, α) = (x − γx ) · γξ + κ(γ)x − z · κ(γ)ξ φ(x, + z − κ(α)x · κ(α)ξ + (αx − y) · αξ + ψ x − γx , z − κ(γ)x , γ + ψ z − κ(α)x , y − αx , α + h(α) − h(γ) .

(5.23)

The contribution to the distribution kernel from a region with |γξ −αξ |/αξ ≥ C −1 is O(h∞ ), as can be seen by integration by parts with respect to z. More precisely the distribution kernel of BA is |γξ − αξ | i (BA)(x, y) = e h (φ+φ) baχ dαdzdγ + R(x, y; h) , αξ where χ ∈ C0∞ ([0, ∞[) is equal to 1 near 0, and ∂xk ∂y R = O(h∞ ). Here (5.22) and i

the analogous estimate for e h φ are essential of course. we get a localization in Next using (5.21) and the similar estimate for Im φ, |γx − αx |, leading to i (γξ − αξ )2 (φ+φ) 2 h (BA)(x, y) = e baχ (γx − αx ) + dαdzdγ αξ 2 y; h) , + R(x, (5.24) = O(h∞ ). Here χ ∈ C ∞ ([0, 0 [), χ = 1 near 0, and 0 can be any where ∂xk ∂y R 0 fixed number. In the integral (5.24), we may assume that |x−γx |, |y−αx |, |z−κ(α)x | < 1/C, where C is as large as we like, since the integral in the complementary region is exponentially small. We now want to eliminate integration variables by means of the method of stationary phase, and we start by carrying out the z-integration, so we first look for the critical point of z, γ) + φ(z, y, α) , z → φ(x, (5.25) where φ, φ also denote almost holomorphic extensions. Let Fδ (x, γ, y, α) denote the corresponding critical value. In order to understand this function, we first treat γ = κ(γ), α = κ(α) as independent variables (writing κ(α) instead of κ(δ, α) for short). Let γx − z) · γξ + (z − α x ) · α ξ + (αx − y) · αξ G(x, γ, γ, y, α, α ) = vcz (x − γx ) · γξ + ( ! − γx , z − γ + ψ(x x , γ) + ψ(z − α x , y − αx , α) . Here the critical point z = zc (x, γ, γ , y, α, α ) satisfies α ξ − γ ξ , (x − γx )2 , (y − αx )2 zc = α x + O α x − γ x , γξ in the natural symbol sense. Notice that γ ; αx , α, γ ) = 0 . G(γx , γ,


231

Moreover, dG = γξ · dx − αξ · dy + γ ξ · d γx − γξ · dγx − ( αξ · d αx − αξ · dαx ) ξ γ ξ − α x − α x , + O γξ x − γx , y − αx , γ γξ in the same symbol sense, with the convention that the remainder term is expressed as linear combination of the “normalized” forms dx, dy, dαx , α x , dγx , γ x , γξ −1 dαξ , . . . . Now, Fδ = G"" + h(α) − h(γ) , =κ(γ), γ α =κ(α)

and we get dFδ = γξ · dx − αξ · dy ξ γ ξ − α + O γξ x − γx , y − αx , γ x − α x , . " γξ " γ =κ(γ)

(5.26)

α =κ(α)

x = αx + O Moreover, Fδ (γx , γ; γx , γ) = 0. From (4.45), we know that α 1/2 −1 1/2 ξ = αξ + O(δ ) when α = κ(α), so (δ αξ ), α δ γξ − αξ γx − α x = γx − αx + O γx − αx , γξ γξ and similarly for Hence,

γξ − αξ αξ .

γξ − αξ dFδ = γξ · dx − αξ · dy + O(γξ ) x − γx , y − αx , γx − αx , γξ γ − α ξ ξ + O(δ) γx − αx , , γξ

(5.27)

and integrating this, we get Fδ (x, γ; y, α) = γξ · (x − γx ) − αξ · (y − αx ) 2 γξ − αξ x − γx , y − αx , γx − αx , , (5.28) + O(γξ ) + O(δ) γξ where the loss of 1/2 for each differentiation appears in the variables α, γ only. When δ = 0, we have on the real domain 2 γξ − αξ 2 2 2 , (5.29) Im Fδ αξ (x − γx ) + (y − αx ) + (γx − αx ) + γξ and in view of (5.28) this persists for 0 ≤ δ 1.

232



When applying stationary phase to (5.24) we also have to make a deformation of the integration contour in order to pass through the critical point zc . Here we recall from [19] and (5.21) and its analogue for φ that 1 Cδ 1 Im Fδ ≥ (Im zc )2 − , αξ C αξ so the error from ∂ z of the almost holomorphic extension, appearing in Stokes’ formula, is O((δ + Im Fδ )/αξ )∞ ). Since = O(h), we conclude that i (BA)(x, y) = e h Fδ (x,γ,y,α)d(x, γ, y, α; h)χ (γx − αx )2 +

γξ − αξ αξ

2 , dαdγ + R

has the same properties as R in (5.24) and where R |+p| n n k 2 α k+k+ 2 −|p| − 2 . ∂x,y ∂γx ,αx ∂γpξ ,αξ d = O h−m−m−3n+ ξ

(5.30)

(5.31)

We now compute the Weyl symbol c of BA by means of the formula w w −iwξ/h c(x, ξ; h) = (BA) x + , x − e dw . (5.32) 2 2 in (5.30) is O(h∞ ξ−∞ ) with all its derivatives. The The contribution from R contribution from the integral in (5.30) is i w w w w e h (Fδ (x+ 2 ,γ,x− 2 ,α)−w · ξ) d x + , γ, x − , α; h χ (γx − αx )2 2 2 2 γξ − αξ + dwdαdγ . αξ The contribution from a region {|ξ − αξ | ≥ C1 αξ } is O(h∞ ξ−∞ ) with all its derivatives and the remaining region can be treated with the method√of stationary γx , . . . as in phase by working in the dilated tilde variables given by γx = x0 + the addendum below. The proposition follows. Addendum. Stationary phase with O-symbols. Assume φ ∈ C ∞ (Rn ) ,

|φ (x)| |x| ,

Im φ ≥ 0 ,

, φ = x · O(1)

φ(0) = 0 ,

det φ (0) = 0 uniformly in .

Let a = O(1). We shall establish a stationary phase development for i −n 2 I(h) = h e h φ(x) a(x)dx in powers of h = h/. (Assume h 1.)


233

1

Put x = 2 x ,

− n2 i h e h φ(x) a( x)d x, I(h) =

(5.33)

1 1 x), φ( x) = φ( 12 x ) = φ( ), a( x) = a( 2 x ). where φ( 2 x 1 a = O(1), and Then a = O(1) in the symbol sense; ∂ α " " " 1 √ " " |φ ( x)| = " √ φ ( x)"" | x) = x · O(1) x| , φ (

and φ (0) = φ (0). The contribution to I(h) from | x| ≥ 1 is O( h∞ ) by repeated integrations by parts, and the contribution from | x| < 1 can be handled in the usual way since φ = O(1) here. Thus I(h) ∼

∞ 0

hj , cj

π (2π)n/2 c0 = # ei 4 sgn φ (0) . det φ (0)

(5.34)

It follows from the proof that the proposition remains valid if we relax the symbol condition in y and only assume 3n n 1 (5.35) ∂x ∂αq x ,y ∂αp ξ a = O h− 2 −m αξ 2 +k−|p| − 2 (|q|+|p|) , i.e., we also allow for a loss of 1/2 for each y-derivation. Similarly for B (cf. (5.18)) we can content ourselves with 3n n 1 (5.36) ∂y ∂γqx ,x ∂γpξ b = O h− 2 −m αξ 2 +k−|p| − 2 (|q|+|p|) . Moreover, if b and a are elliptic, then c is elliptic. We get by standard arguments, Proposition 5.4. Let A be an elliptic Fourier integral operator of order (m, k) with symbol as in (5.35). Then there exists an elliptic Fourier integral operator B of order (−m, −k) with symbol as in (5.36), such that BA = 1 + R ,

(5.37)

h )∞ ξ−∞ ). In parwhere R is 1-negligible in the sense that its symbol R is O(( −1 ticular A has the left inverse (1 + R) B when /h 1. We notice that when δ = 0, then A, B are elliptic pseudodifferential operators and hence (1+R)−1 B is also a left inverse. (By the Beals lemma we also know that h )∞ ).) For (1 + R)−1 is an h-pseudodifferential operator with symbol 1 + O(( ξ general small δ, R(Aδ ) is closed. Using suitable deformations of A we can produce a continuous deformation of closed subspaces in L2 from L2 to R(Aδ ). All the deformed subspaces then have to be equal to L2 and R(Aδ ) = L2 , so (1 + Rδ )−1 Bδ is also a left inverse of Aδ .

234



We next turn to Egorov’s theorem and start with some preparations. Recall that by (4.45) δ 21 αξ −1 , κ(α)ξ = αξ + O δ 12 . κ(α)x = αx + O (5.38) It follows that

δ dα κ(α)ξ = dαξ + O(δ)dα dαξ x +O αξ δ δ dα κ(α)x = dαx + O dαξ . dαx + O αξ αξ 2

Substituting this into (5.17), we get dα φ = x − κ(α)x − (y − αx ) · dαξ ∂ψ ∂ψ + x − κ(α)x , y − αx , αξ · dαx − ∂x ∂y δ + x − κ(α)x , y − αx · O(δ)dαx + O dαξ αξ ∂ψ x − κ(α)x , y − αx , α · dα , + ∂α which we can write ∂ψ ∂ψ ∂ψ /αξ ∂αx φ/αξ ∂αx − ∂x + ∂y = ∂ψ ∂αξ φ (x − κ(α)x ) − (y − αx ) + ∂α ξ δ x − κ(α)x +O . y − αx αξ

(5.39)

(5.40)

(5.41)

1 ν Here we write a(x, y, α) = O(m) if ∂x,y ∂αμx ∂αρ ξ a = O(m− 2 (|μ|+|ρ|) αξ −|ρ| ). The differential of this vector at a point of the critical set is given by the matrix δ + ψyx ) −αξ −1 (ψxy + ψyy ) −αξ −1 (ψxx +O . (5.42) 1 −1 αξ + ψyx + ψxy + If ttxy is in the kernel of the first term, we get tx = ty and (ψxx ψyy )tx = 0. Here we recognize the Hessian of z → ψ(z, z, α) which is invertible because of the assumption on the imaginary part. More precisely, the matrix (5.42) has a uniformly bounded inverse. From (5.41), we get ∂αx φ/αξ x − κ(α)x , = Mδ (x, y, α) y − αx ∂αξ φ

|x − κ(α)x | , where Mδ (x, y, α) = M0 (x, y, α) + O k ∂ M0 = O(αξ −|| ). and ∂x,y,α x αξ

|y − αx | ≤ 1/O(1) , (5.43) δ αξ

,

(5.44)


235

ξ m Lemma 5.5. Let Φ(x, y, α) = O(α ) in the sense that 1 p − 2 (||+|p|) m−|p| . ∂xk ∂y,α ∂ Φ = O α ξ x αξ

ξ n ) be elliptic. Then Similarly, let a = O(α i e h φ Φ(x, y, α)a(x, y, α; h)χ(x, y, α)u(y)dydα (5.45) i = e h φ Φ κ(α)x , αx , α a(x, y, α; h)χ(x, y, α)u(y)dydα i h αξ m−1 + e h φO a(x, y, α; h) χ(x, y, α)u(y)dydα , where χ is a similar cutoff. Proof. We have

x − κ(α)x −1 m Φ(x, y, α) = Φ κ(α)x , αx , α + O 2 αξ y − αx −1 2 αξ m−1 − 12 αξ m = Φ κ(α)x , αx , α + O · ∂αx φ + O · ∂αξ φ .

The contribution from the remainders to the left hand side in (5.45) is therefore ∂ i φ − 1 h e h · O 2 αξ m−1 aχudydα ∂αx ∂ i φ − 1 e h · O 2 αξ m aχudydα , +h ∂αξ and it suffices to integrate by parts.

Actually, we shall not use the lemma directly, only its proof. The next result is closely related and could probably be obtained from Lemma 5.5. We will give a different proof however. Lemma 5.6. Under the same assumptions as in the preceding lemma, we have i e h φ Φaχu(y)dydα = AQ , (5.46) where Q is an h-pseudodifferential operator with symbol h ξm−1 Q = Φ κ(x, ξ)x , x, x, ξ + O . Here i Au(x) = e h φ aχu(y)dydα .

(5.47)

(5.48)

Proof. In order to harmonize with Proposition 5.3, we may change the assumption 3n n ξ m+ 2 h− 2 ) and assume that a is elliptic in this class. Here O on a to a = O(α 1/2 for each differentiation in α, y. Let B be of the form (5.18) indicates a loss of 3n n ξ −m+ 2 h− 2 ) where the 1/2 loss is now for each differentiation with b = O(α

236



in (x, α). We also assume that b is elliptic. Then from Proposition 5.3 and the is elliptic. remark following its proof, we know that BA = Oph (c), where c = O(1) Recalling that the proof was by stationary phase, we see that if A is the Operator = Oph ( c), where given by the left hand side of (5.46), then B A h m−1 ξ c = Φ κ(x, ξ), x, x, ξ c + O . = BAQ, with Q as in the lemma. By pseudodifferential calculus, we get B A Moreover, B is invertible. Remark 5.7. Later on, we shall meet the special situation when Φ = Φ(α) = ξ m1 )P (α) =: RP , P (α) = O(ξm2 ). In this case we have O(α ξm1 #P + O √h ξm1 +m2 −1 . Q=O (5.49) In fact (anticipating on a part of the proof of Proposition 5.9), let i Bu = e h φ Raχu(y)dydα . Then (with Op = Oph when nothing else is specified) i t B ◦ Op(P )u = P (y, hDy )(e h φ Raχ)u(y)dydα h2 αξ m1 +m2 −2 e hi φ u(y)dydα . P (y, −φy )Raχ + O = As we shall see in the proof of Proposition 5.9, i i P (y, −φy )Raχe h φ u(y)dydα = e h φ Φaχu(y)dydα h i φ m +m −1 1 2 + e h O √ αξ u(y)dydα . Applying Lemma 5.6 to B and the various remainder terms, we get (5.49). Proposition 5.8. Let As be a Fourier integral operator quantizing κ(s) as in Proposition 5.3. Also, assume that a = as is elliptic and depends smoothly on s in the sense that n h−m− 3n 2 α k+ 2 ∂sk as = O , ξ indicates a loss of 1/2 for differentiations in α, y. Then where O hDs As + iAs ψˇs = 0 , 0 ≤ s ≤ δ0 , 0 < δ0 1 ,

(5.50)

where Ds = i−1 ∂s and ψˇs is an h-pseudodifferential operator with symbol h + hξ−1 + ψ,s ◦ κ(s) . ψˇs = O

(5.51)

The term “h” in the remainder can be dropped if as is independent of s.


Proof. Assume for simplicity that as is independent of s. Then, i hDs As u(x) = e h φ a∂s φ udydα . Recalling that,

237

(5.52)

φs (x, y, α) = x − κ(s, α)x · κ(s, α)ξ + (αx − y) · αξ + ψ x − κ(s, α)x , y − αx , α + hs (α) ,

we get

∂s φs (x, y, α) = x − κ(s, α)x · ∂s κ(s, α)ξ − ∂s κ(s, α)x · κ(s, α)ξ (5.53) − (∂x ψ) x − κ(s, α)x , y − αx , α · ∂s κ(s, α)x + ∂s hs (α) .

The restriction to the critical set (5.16) is (∂s φs ) κ(s, α)x , αx , α = (∂s hs )(α) − ∂s κ(s, α)x · κ(s, α)ξ ∂ ψ,s · ξ ◦ κ(s) = (∂s hs )(α) − i ∂ξ = −iψ,s ◦ κ(s)(α)

(5.54)

, = O() where we used (5.3). More generally, from (5.53) we get 12 ) · x − κ(s, α)x , y − αx . ∂s φs = −iψ,s ◦ κ(s)(α) + O(

(5.55)

As in the proof of Lemma (5.5) we can make integrations by parts and see that the contribution from the remainder to (5.52) becomes i n h−m− 3n 2 α k+ 2 hα −1 e h φO u(y)dydα . ξ ξ Combining this with (5.55) and Lemma 5.6, we get the proposition.

We can now prove Proposition 5.2. Proof. We only consider A and we may assume that A = A(δ) where A(s) is a smooth family as in the preceding proposition. The result for B will be the same since B is like the adjoint of A. From (5.50) we get h∂s A∗ = ψˇ∗ A∗ . s

If u ∈ S, then

s

s

A∗s u

∈ S and we get h∂s A∗ u 2 = (ψˇ∗ A∗ u|A∗ u) + (A∗ u|ψˇ∗ A∗ u) , s

s

s

s

s

s

s

so h∂s A∗s u 2 ≤ 2 ψˇs∗ A∗s u 2 . But ψˇs∗ = O(), so A∗s u ≤ eO()|s|/h u = eO(1)A|s| u ( = Ah) .

238



Then A∗δ ≤ eO(1)Aδ and the proposition follows since Aδ and A∗δ have the same norm. Let P = O(ξM ) in the symbol sense: ∂xα ∂ξβ P = O(ξM−|β| ). We study n m+ 3n 2 ,k+ 2 (so P A√ s − As P where As is as in Proposition 5.8, ∂s as bounded in S no -loss in the symbol for simplicity). Working with the Weyl quantization we know that i i h∞ αξ −∞ P (x, hDx ; h) e h φ(x,y,α) a(x, y, α; h) = e h φ b + O i i t h∞ αξ −∞ , P (y, hDy ; h) e h φ(x,y,α) a(x, y, α; h) = e h φ c + O where

n h− 3n 2 −m α k+ 2 +M h2 α −2 b ≡ P x, φx (x, y, α); h a(x, y, α; h) + O , ξ ξ 3n n h− 2 −m αξ k+ 2 +M h2 αξ −2 , c ≡ P y, −φy (x, y, α); h a(x, y, α; h) + O

refers to √-loss only with respect to differentiations in α. (Recall and the O the general fact that the Weyl symbol of e−iφ/h ◦ pw (x, hDx ) ◦ eiφ/h is equal to p(x, ξ + φx ) + O(h2 ).) s + As ◦ Op (O(h 2 αξ −2 )), where We conclude that P As − As P = A i s u = A e h φ P x, φx (x, y, α); h − P (y, −φy ; h) au(y)dydα . (5.56) On the critical set (5.16), we have

P (x, φx ; h) − P (y, −φy ; h) = P κ(s)(α) − P (α) ,

and more generally,

P (x, φx ) − P (y, −φy ) = P κ(s)(α) − P (α) s 21 αξ M−1 + x − κ(s, α)x · O αξ M + O s 21 αξ M−1 . + (y − αx ) · O αξ M + O

Using (5.43), (5.44), we get

P (x, φx ) − P (y, −φy ) = P κ(s, α) − P (α) (5.57) 1 s 2 αξ M−1 . + φαξ , αξ −1 φαx · O αξ M + O The contribution from P (κ(s)(α)) − P (α) in (5.56) is of the form s √h αξ M−2 As ◦ Op P κ(s)(α) − P (α) + O

M−2 in general, and the remainder estimate improves to O(shα ) if P (κ(s, α)) − ξ M−1 ). By integration by parts in α, we see that the contribution P (α) = O(sα


239

from the two remainder terms in (5.57) is 1 i 1 φ M M−1 h 2 (idem) a u(y)dydα h e ∂αξ O αξ + O s αξ + ∂αx αξ i sαξ M−2 au(y)dydα . =h e h φ O αξ M−1 + O In conclusion, Proposition 5.9. We have P As = As Ps ,

Ps = P + Qs ,

where

h M−2 hαξ M−1 Qs = P κ(s)(α) − P (α) + O s 1 αξ +O 2 12 αξ M−1 ). in the general case, when P (κ(s)(α)) − P (α) = O(s M−1 In the special case when P (κ(s, α)) − P (α) = O(sα ) the first remainξ M−2 ) so in that case, der term improves to O(shαξ hαξ M−1 . Qs (α) = P κ(s, α) − P (α) + O In order to treat certain conjugations, we need a more precise description of Qs in the general case in the last proposition. The proof above shows that i hξM−1 , (5.58) As Qs u = e h φ P κ(s, α) −P (α) au(y)dydα+As Op O and we need to take a closer look at the oscillatory integral. By Taylor’s formula and (4.45), √ 1 s2 αξ M−2 . P κ(α) − P (α) = O(s Pα x (α) + O ) · Pα ξ (α), αξ √), then When passing to Weyl composition of symbols, we notice that if r = O(s 1 1 shαξ M−2 . Pα x = r · Pα ξ , Pα x + O r# Pα ξ , αξ αξ Lemma 5.6 and Remark 5.7 then show that on the symbol level 2 √ M−1 M−2 1 P . + O s ξ + O(s )# Pξ , Qs = O hξ ξ x

(5.59)

Here the first term to the right is too large; we would like to have hξM−2 , so we take a closer look at Qs using Proposition 5.8, where we now add the assumption m = k = 0,

A0 = 1 ,

as is independent of s .

Then as noticed, (5.51) improves to ψˇs = ψs ◦ κ(s) + O

h ξ

. = O()

(5.60)

(5.61)

240



We get on the symbol level, writing [A, B] = A#B − B#A: h∂s Qs = [Ps , ψˇs ] = [P, ψˇs ] + [Qs , ψˇs ] hξM−1 h s2 ξM−2 h = [P, ψˇs ] + O +O ξ ξ √ √ h 1 ξM−1 √ h s # Pξ , Px + O(s )#O +O ξ ξ ξ 2 M−2 2 M−3 ˇ = [P, ψs ] + O h ξ + O s hξ √ s h 1 shξM−2 . P +O +O # Pξ , ξ ξ x On the other hand,

3 h ξM h [P, ψˇs ] = {P, ψˇs } + O i ξ3 3/2 h P − 12 h3 ξM−3 +O = ih ψs ◦ κ(s), P + h Pξ , x · O 1/2 ξ ξ 3 h Px h ξM−3 , = ih ψs ◦ κ(s), P + hO , # P + O ξ ξ ξ1/2

so h∂s Qs = ih ψs ◦ κ(s), P + O

h2 s1/2 h + ξ 1/2 ξ 2 M−2 M−2 h ξ + shξ +O .

P # Pξ , x ξ

Since κ(s) is obtained from integrating iHψs , we obtain, using also that Q0 = 0, 2√ −1/2 P δ + δh Qδ = P ◦ κ(δ) − P + O # Pξ , x ξ ξ ξ M−2 2 M−2 (5.62) + δ ξ + O hδξ under the assumption (5.60). (Recall that here M is the order of P .)

6. The conjugated pseudodifferential operator Let P be the Operator introduced in the beginning of Section 2 satisfying the assumptions (2.1)–(2.7). To start with, we assume Hypothesis 4.1 and define C, ψ as in (4.5), (4.6). Later, we shall also use the dynamical Hypothesis 4.3 (implying (4.24)). −2 The function ψ satisfies ψ = O(), ξ · ∂ξ ψ = O(ξ ) in the sense of (4.27), (4.28), but we also know that these bounds improve considerably away from C. Let A = Aδ be an elliptic Fourier integral operator of order 0,0 quantizing κ(δ) as in Section 5, and let Bδ be a corresponding Operator quantizing κ(δ)−1 .


241

Then we can apply Proposition 5.3 to see that BA = Oph (c), where c = O(1) is elliptic. Now restrict the attention to the intermediate region, where ψ = ψ(x), ψ(x) ∈ C ∞ . Using that = Ah with A large but fixed, we find here that e−δψ /h ◦ A = Oph (c) where c = O(1) in the standard symbol sense, ∂xα ∂ξβ c = O(ξ−|β| ) and c is elliptic in this class. Let c = O(1) be an elliptic symbol which is equal to 1 in the interior region and which changes from 1 to c when going outwards in the intermediate c)−1 . (Here we assume that A0 = 1, to avoid region. We now replace A by A ◦ Oph ( a topological difficulty.) Then we have achieved that Au = eδψ /h u for u supported in the exterior region including a shell where ψ = ψ, ψ = ψ(x) ∈ C ∞ as above and the region further out where ψ = ψ (x) = O(). Next, we consider P δ defined as in Proposition 5.9 by P Aδ = Aδ P δ . Recall that P = P2 + iP1 + P0 and write Pj Aδ = Aδ Pjδ , P δ = P + Qδ , Pjδ = Pj + Qδj . For Pj , j = 0, 2, we apply (5.62) and get on the symbol level over any bounded set in x-space √ 1 δ 2 δh− 2 ∂x pj δ + # ∂ξ pj , Qj = pj ◦ κ(δ) − pj + O ξ ξ ξ hδξj−2 + δ 2 ξj−2 +O √ ∂x pj (hδ + δ 2 )ξj−2 . (6.1) )# ∂ξ pj , = O(δ +O ξ For p1 we need the more precise information about ψ given in (4.10), (4.12), (4.13), satisfied the fact that ψ (ρ) is independent of in √ also by ψ , as well as √ |ρ − ρj | ≤ (actually in |ρ − ρj | ≤ /C for some C > 0, but we can always dilate in ). From these estimates it follows that ∂x p1 , ∂ξ p1 ⊗ ∂x ψ , ξ∂ξ ψ = O() (6.2) ξ over a neighborhood of the set where ψ also depends on ξ. It also follows that p1 κ(δ)(x, ξ) − p1 (x, ξ) = O(δ) . (6.3) From Proposition 5.9, we deduce that Qδ1 = p1 κ(δ)(x, ξ) − p1 (x, ξ) + O(h) .

(6.4)

By construction of κ(δ), especially (4.47), we see that 2) . p1 κ(δ)(x, ξ) − p1 (x, ξ) = iδHψ p1 (x, ξ) + O(δ

(6.5)

In fact, this is quite obvious in the intermediate and far out regions, and near √ a ρ. point ρj ∈ C, we choose canonical coordinates so that ρj = (0, 0) and put ρ = Then if ψ (ρ) = ψ ( ρ), we get Hψ = Hψ , where the tilde on the H indicates that we take the Hamilton field with respect to the ρ-variables. The manifold

242



(Re ρ) while p1 = O( Λδ : Im ρ = δHψ (Re ρ) now becomes Im ρ = δ H ρ2 ) and ψ since ψ satisfies (4.10), 1 α ∂ρ ψ = O . ρ|α| ( Then κ(δ)(ρ) becomes κ(δ)( ρ) = ρ+iδ H ρ)+O(δ 2 ρ−2 ), so p1 ( κ(δ) ρ) = p1 (ρ)+ (p1 ) + O(δ 2 ), leading to (6.5). iδ H ψ It follows that

ψ

2 + h) = O(δ Qδ1 = −iδHp1 ψ + O(δ + h) .

(6.6)

Away from any neighborhood C we have the improved estimates Qδ1 = −iδHp1 ψ + O(δ 2 + h) in the usual symbol sense, as long as we stay away from the outer region where and where we can apply the analysis of ψ = ψ (x) only satisfies ψ = O(), Section 2, so (6.7) Qδ1 = −iδHp1 ψ = −iδν(ψ ) there.

7. Estimates for the conjugated pseudodifferential operator and localization of the spectrum Let P δ be the conjugated Operator of Section 6. We shall study lower bounds for Re (P δ u|u) = Re (P0δ + P2δ )u|u − Im (P1δ u|u) , u ∈ S(M ) . (7.1) Using (6.1), we get Re (P2δ u|u)

√ ∂x p2 2 2 ∂ξ p2 , ≥ (P2 u|u) − Cδ u Op u − C(hδ + δ ) u ξ 2 1 ∂x p2 2 2 Op ∂ ≥ (P2 u|u) − p , u ξ 2 − C2 (hδ + δ ) u , C1 ξ

where C1 can be chosen arbitrarily large and C2 depends on C1 . Here 2 1 1 ∂x p2 p2 − ≥ 0, ∂ξ p2 , 2 C1 ξ if C1 is large enough, so 2 2 p p 1 ∂ 1 ∂ x 2 x 2 Op u|u (P2 u|u) − u ∂ξ p2 , = Op p2 − C1 ∂ξ p2 , ξ C1 ξ + O(h2 ) u 2 1 ≥ (P2 u|u) − O(h2 ) u 2 , 2


243

where the estimate follows from the Fefferman–Phong inequality in the semiclassical setting (see [14]). It follows that 1 (P2 u|u) − C(h2 + hδ + δ 2 ) u 2 . 2 For P0 we have the same conclusion, Re (P2δ u|u) ≥

(7.2)

1 (P0 u|u) − C(h2 + hδ + δ 2 ) u 2 , (7.3) 2 since the same analysis applies for x in a bounded region and further out, we just have P0δ = P0 ≥ 0. Combining this with (6.6), we get 1 Re (P δ u|u) ≥ δ Op (Hp1 ψ )u|u + (P0 + P2 )u|u − C(h + hδ + δ 2 ) u 2 . (7.4) 2 Here we recall that Hp1 ψ = p T0 − p by (4.7), that p T0 satisfies (4.25), (4.26) and that p ≤ p ≤ p0 + p2 by (4.16). Write, Re (P0δ u|u) ≥

1 1 δHp1 ψ + (p0 + p2 ) = δ p T0 − δ p + (p0 + p2 ) 2 2 p − p ) + δ(p0 + p2 − p) = δ p T0 + δ( 1 − δ (p0 + p2 ) . + 2

(7.5)

Here we want a lower bound for Op ( p − p ). This is quite straight forward away from C, so we concentrate on a neighborhood of a point ρj ∈ C. Assume ρj = 0 for simplicity. Then near 0 we have by (4.15) 2 2 2 |ρ| |ρ| p = g p , p − p = 1 − g p = 1 − (ρ) p , where

(ρ) =

|ρ|2

=O

+ |ρ|2

,

and we may assume that g has been chosen so that 1 − g = (1 − )2 with smooth. Here (1 − )# p#(1 − ) = p − # p − p# + # p# 2 = 1 − (ρ) p + O(h) . From this we conclude that Op ( p − p )u|u ≥ −Ch u 2 . Similarly,

Op (p0 + p2 − p)u|u ≥ −Ch2 u 2 ,

by the Fefferman–Phong inequality or by a direct argument.

(7.6) (7.7)

244



√ Let 0 ≤ k = O() be equal to in C + B(0, ) and have its support in C + √ B(0, 2). Let K = Op (k ). By (4.25), (4.26) we have p T0 min (, dist ( · , C)2 ), so . k + p T0 = O() (7.8) Hence, p T0 u|u u 2 . (7.9) K + Op Combining (7.4)–(7.9), we get Proposition 7.1. We have δ Re (P δ + K )u|u ≥ u 2 + C

1 − δ (P2 + P0 )u|u − Ch u 2 , 2 u ∈ S(M ) . (7.10)

Here we recall that δ > 0 should be small enough, = Ah with A arbitrarily large and fixed, and C in (7.10) is independent of δ, A while h is small enough depending on these two parameters. From (7.10) we get the a priori estimate δAh − Ch − Re z u ≤ (P δ + K − z)u , u ∈ S(M ) , (7.11) C when Re z < δAh/C − Ch. From Section 3 we know that P has no spectrum in the open left half-plane. We shall next prove Proposition 7.2. For every constant B > 0 there is a constant D > 0 such that P has no spectrum in {z ∈ C; Re z < Bh, |Im z| > Dh} (7.12) when h > 0 is small enough. Moreover (P − z)−1 = OB (h−1 ) for z in the set (7.12). Proof. Choose δ > 0 small, then A large enough, so that δAh C − Ch − Re z ≥ /Const. when Re z < Bh. Then (7.11) gives u ≤ (P δ + K − z)u , u ∈ S(M ) . (7.13) C0 Take z in the set (7.12). When Re z < 0, we already know that z ∈ σ(P ), so we may assume that 0 ≤ Re z < Bh. √ and supported in C + B(0, 2). Now recall that the symbol of K is O() = O(Ah) On that set we have pδ = O() and hence + |z| , when |Im z| > Dh , C0 and D is large enough, assuming still that 0 ≤ Re z < Bh. It follows that we can find E = O(/( + |z|)) such that |pδ − z| >

K = E ◦ (P δ − z) + F ,

F = O(h) ,


245

where E, F also denote the corresponding h-pseudodifferential operators. In particular, (7.14) K u ≤ O(1) (P δ − z)u + O(h) u , where the Os are uniform in . Combining this with (7.13) with h, we get u ≤ O(1) (P δ − z)u ,

u ∈ S(M ) , δ

(7.15) A−1 δ P Aδ ,

for 0 ≤ Re z < Bh, 0 ≤ Im z < Dh. Now recall that P = 2 2 2 Aδ , A−1 δ : S → S, L → L and have L norm OA (1). Then (7.15) gives h u ≤ OB (1) (P − z)u ,

u ∈ S(M ) ,

where (7.16)

for z in the set (7.12). From Section 3 we then know that z ∈ σ(P ) and that (P − z)−1 ≤ OB (1)/h.

8. Asymptotics of eigenvalues Let ρj ∈ C and let Fρj be the matrix of the linearization of Hp at ρj (the so called fundamental matrix of p at the doubly characteristic point ρj ). Thanks to the fact that the quadratic approximation of pδ = p ◦ κ(δ) at ρj is elliptic on Tρj (Λδ ) and takes its values in a closed angle contained in the union of {0} and the open right half plane, we know from [22] that the eigenvalues of Fρj are of the form ±λj,k , 1 ≤ k ≤ n, when repeated with their multiplicity, with Im λj,k > 0. Let (p, ρj ) = 1 tr λj,k . (8.1) i k

In our case the subprincipal symbol of P at ρj is zero and will not enter into the description of the eigenvalues. Put q(x, ξ) = −p(x, iξ) = p2 + p1 − p0 . Let Fq , Fp be the fundamental matrices of q, p at one of the critical points ρj ∈ C. Since 1 ∂ ∂ − px (x, η) · Hq (x, ξ) = pξ (x, η) · , with η = iξ , i ∂x ∂η we see that Fq and 1i Fp have the same eigenvalues; ± 1i λk , k = 1, . . . , n (j being fixed) where Re ( 1i λk ) > 0. Now q is real-valued and we can apply the stable manifold theorem as in [9] (and at many other places) to see that the Hq -flow has a stable outgoing manifold Λ+ passing through ρj such that Tρj ΛC + is spanned by the generalized eigenvectors corresponding to + 1i λk , k = 1, . . . , n. We also know that Λ+ is a Lagrangian manifold and that q vanishes on Λ+ . Lemma 8.1. Assume for simplicity that ρj = (0, 0). Then Tρj Λ+ is transversal both to {x = 0} and to {ξ = 0}.

246



Proof. Since we consider the linearized situation we may assume right away that q is a quadratic form (the second order Taylor polynomial at ρj ), so that pj are quadratic forms as well. The dynamical condition (4.21) implies that p0 + p2 T0 > 0

as a quadratic form .

(8.2)

Let L = Λ+ ∩ {x = 0}. Since p0 = 0, p1 = 0 on {x = 0}, we know that p2 = 0 on L and hence Hq = Hp1 on L. Now Hq is tangent to Λ+ and Hp1 is tangent to {x = 0}, so Hq = Hp1 is tangent to L. Thus L is an Hq - and Hp1 invariant subspace on which p2 + p0 = 0, and since p2 + p0 T0 > 0 away from 0, we necessarily have L = 0. The proof of the fact that Λ+ ∩ {ξ = 0} = 0 is the same after permuting the roles of p0 and p2 . It follows from the lemma that Λ+ : ξ = φ+ (x) ,

x ∈ neigh (0) ,

(8.3)

where φ+ ∈ C ∞ (neigh (0); M ), φ+ (0) = 0, φ+ (0) = 0, det φ+ (0) = 0. Let Λ− be the stable incoming Hq -invariant manifold such that Tρj ΛC − is spanned by the generalized eigenvectors of Fq corresponding to − 1i λk , 1 ≤ k ≤ n. The lemma is valid also for Λ− and (8.3) has an obvious analogue for Λ− where we let φ− denote the corresponding generating function. Proposition 8.2. We have φ+ (0) > 0, φ− (0) < 0. Proof. Again we can consider the linearized quadratic case. If we make a smooth deformation of q, then Λ+ , Λ− , φ+ (0), φ− (0) vary smoothly with the deformation parameter and det φ± (0) = 0, provided of course that we maintain the condition (8.2). Consider the deformation from q = q0 to to ξ 2 − x2 = q1 : qt (x, ξ) = (1 − t)q(x, ξ) + t(ξ 2 − x2 ) = pt2 + pt1 − pt0 , with p2 (ξ) = (1 − t)p2 (ξ) + tξ 2 ,

pt0 (x) = (1 − t)p0 (x) + tx2 .

pt2 and pt0 are positive definite for t > 0, so (8.2) is maintained. For t = 1 we have φ1± (x) = ±x2 /2 so ±φ± (0) is positive definite. Since the signatures of φ± (0) are independent of t, we get the lemma. From Section 6 we recall that the conjugated operator Pδ = Pδ, has the symbol √ ∂x pj + δ 2 ) O(δ p + δHp1 ψ + )# ∂ξ pj , + O(h ξ j=0,2 and that we have the a priori estimate (7.10) expressing that the real part of P δ √ √ δ is ≥ δ C − Ch outside C + B(0, ). In the set C + B(0, ) the symbol P is independent of modulo O(( h )∞ ) and is of the form Pδ ∼ pδ + hr1 + h2 r2 + · · · , where |pδ | dist ( · , C)2 , Re pδ dist ( · , C)2 . In the following we shall assume for


247

a while (in order to simplify the notations) that C is reduced to a single point ρ0 = (0, 0). We fix B > 0 and restrict the spectral parameter z to the disc D(0, Bh). We shall take = Ah with A B sufficiently large. Assume that M = Rn for simplicity. Following basically [14] we recall the construction of a well-posed Grushin problem, first for Pδ − z and then for Pδ − z. Let Λ = (x2 + (hDx )2 )1/2 so that Pδ is equipped with the natural domain D(Pδ ) = {u ∈ L2 ; Λ2 u ∈ L2 (Rn )}. In Section 11 of [14] the authors constructed operators R− : CN0 → L2 ,

R+ : L2 → CN0

(8.4)

of the form R− u− =

N0

u− (j)ehj (x) ,

R+ u(j) = u|fjh (x) ,

(8.5)

j=1

with the following properties: x h −n 4 ej (x) = h ej √ , h ej (x) = pj (x)eiΦ0 (x) ,

x = h fj √ , h fj (x) = qj (x)eiΨ0 (x) ,

fjh (x)

−n 4

(8.6) (8.7)

where pj , qj are polynomials and Φ0 , Ψ0 are quadratic forms with Im Φ0 , Im Ψ0 > 0, Φ0 = φ+ (0). If δ > 0 is small and fixed, A sufficiently large, the problem δ (Pδ − z)u + R− u− = v ,

δ R+ u = v+ ,

(8.8)

for v ∈ L2 (Rn ), v+ ∈ CN0 has a unique solution u ∈ D(Pδ ), u− ∈ CN0 , where δ δ = R+ Aδ , R− = A−1 R+ δ R− , Moreover, for the solution, we have the a priori estimate Λ2 u + |u− | ≤ C( v + h|v+ |) .

(8.9)

Notice here that (h−1/2 Λ)N R− = O(1) : CN0 → L2 , for every N ∈ R and similarly δ, δ δ for R+ , R± . From this, it follows that R− = R− depends weakly on in the sense A ≥ A, then that if = Ah, 1 δ, δ, R− − R− L(CN0 ,L2 ) = O , A δ . and similarly for R+ We shall derive an a priori estimate for the problem δ (Pδ − z)u + R− u− = v ,

δ R+ u = v+ ,

(8.10)

when u ∈ S. Let χ ∈ C0∞ (B(0, 2)) be equal to one on B(0, 1), and put χ√ (x, ξ) = χ(−1/2 (x, ξ)). We use the same notation for the corresponding h-quantization. We ∞ ) on supp (χ√ ). From the first equation may assume that Pδ − Pδ = O((h/)

248



in (8.10), we get ⎧ δ √ √ √ ⎪ ⎨(Pδ − z)χ u + R− u− = χ v + χ (Pδ − Pδ )u +[Pδ , χ√ ]u + (1 − χ√ )R− u− , ⎪ ⎩ δ √ δ R+ χ u = v+ − R+ (1 − χ√ )u . Here

(8.11)

hh χ√ #(Pδ − Pδ ) = O hh [Pδ , χ√ ] = O 1

1

δ δ u− = (1 − χ√ )(h− 2 Λ)−N (h− 2 Λ)N R− u− (1 − χ√ )R− N h 2 =O |u− | , 1

(8.12)

1

δ δ h|R+ (1 − χ√ )u| = h|R+ (h− 2 Λ)N (h− 2 Λ)−N (1 − χ√ )u| N h 2 =O h u .

(8.13)

Thus, applying the a priori estimate (8.9) to (8.11), we get h h 2 √ √ Λ χ u + |u− | ≤ C χ v + O h u + O |u− | + h|v+ | . (8.14) We next look for an a priori estimate for (1 − χ√ )u. Apply 1 − χ√ to (8.10): (Pδ − z)(1 − χ√ )u = (1 − χ√ )v + [Pδ , χ√ ]u − (1 − χ√ )R− u− . As before, [Pδ

, χ√

(8.15)

h √ √ : L2 → L2 , ] = [Pδ , χ ] + [Pδ − Pδ , χ ] = O h

and using also (8.12), we get from (2.12), (8.15) that (1 − χ√ )u ≤ (1 − χ√ )v + K (1 − χ√ )u C h h h u + O +O |u− | ,

(8.16)

√ ). Since where we can take K (x, ξ) = χ( 3(x,ξ) N h √ K #(1 − χ ) = O : L2 → L2 ,

we get

h h √ √ (1 − χ )u ≤ (1 − χ )v + O h u + O |u− | . C

(8.17)


249

Let L = O(min(h + dist (x, ξ; 0, 0)2 , )) be an elliptic symbol in the class defined by the right hand side, so that Λ2 χ√ u + (1 − χ√ )u L u . Then summing (8.14), (8.17), we can absorb the various remainder terms to the right, and obtain 1 ( L u + |u− |) ≤ v + h|v+ | , (8.18) C when (8.10) holds. δ δ , R− have been defined from P , R+ , R− by conjugation Now recall that Pδ , R+ with Aδ , and use also that 1 h u ≤ L u ≤ CAh u , C to see that if u, v ∈ S(M ) and (P − z)u + R− u− = v R+ u = v+

,

z ∈ D(0, Bh) ,

(8.19)

then h u + |u− | ≤ C( v + h|v+ |) .

(8.20)

From the discussion after the proof of Corollary 3.2, we conclude that P − z R− (8.21) : D(P ) × CN0 → L2 × CN0 P(z) = R+ 0 is bijective with a bounded inverse E(z) E+ (z) E= : L2 × CN0 → D(P ) × CN0 , E− (z) E−+ (z) for z ∈ D(0, Bh), and (8.20) shows that 1 E(z) = O , h E+ (z) = O(1) ,

E− (z) = O(1) ,

(8.22)

(8.23)

E−+ (z) = O(h) .

In Section 11 of [14] the authors studied the action of P(z) on spaces of functions of the form (a(x; h)eiΦ0 (x)/h , u− ), where a is a symbol, and deduced that E−+ (z; h) has an asymptotic expansion in half powers of h with a certain additional structure. From that was obtained the asymptotic expansion of the zeros of det E−+ , i.e., of the eigenvalues of P in D(0, Bh). That discussion goes through without any changes in the present situation, so we get the asymptotics for the eigenvalues in any disc D(0, Bh), when h → 0.

250



Theorem 8.3. We make the assumptions (2.1)–(2.7), (4.4), (4.21), (4.22), and recall the definition of C in (4.5). Let B > 0. Then there exists h0 > 0 such that for 0 < h ≤ h0 , the spectrum of P in D(0, Bh) is discrete and the eigenvalues are of the form λj,k (h) ∼ h μj,k + h1/Nj,k μj,k,1 + h2/Nj,k μj,k,2 + · · · , (8.24) where the μj,k are all the numbers in D(0, B) of the form n

μj,k =

1 1 νj,k, λj, + tr (p, ρj ) , i 2

with

νj,k, ∈ N ,

(8.25)

=1

for some j ∈ {1, . . . , N }, N = #C. (Possibly after changing B, we may assume that |μj,k | = B, ∀j, k.) Recall here that ±λj are the eigenvalues of Fp . This description also takes into account the multiplicities in the natural way. If the coefficients νj,k, in (8.25) are unique, then Nj,k = 1 and we have only integer powers of h in the asymptotic expansion (8.24). Theorem 8.4. We make the same assumptions as in Theorem 8.3. For every B, C > 0 there is a constant D > 0 such that h D , for z ∈ D(0, Bh) with dist z, σ(P ) ≥ . (8.26) (z − P )−1 ≤ h C The last result follows from the formula (z − P )−1 = −E(z) + E+ (z)E−+ (z)−1 E− (z) , −1 (8.23) and the fact that E−+ (z) = O(h−1 ) when (8.26) holds. Still with j = j0 fixed, let n 1 1 μ= ν λ + tr (p, j0 ) , ν ∈ N i 2

(8.27)

=1

be a value as in (8.25) and assume that μ is simple in the sense that (ν1 , . . . , νn ) ∈ Nn is uniquely determined by μ. In particular, every λ for which ν = 0 is a simple eigenvalue of Fp . Then as in [9] (see also Chapter 3 in [21]) we can construct λ(h) ∼ h(μ + hμ1 + h2 μ2 + · · · )

(8.28)

with uniquely determined coefficients μ1 , μ2 , . . . and

(8.29) a(x; h) ∼ a0 (x) + ha1 (x) + · · · in C ∞ neigh (xj0 ) , (m−2j)+ where aj (x) = O(|x − xj0 | ), m = ν and a0 has a non-vanishing Taylor polynomial of order m, such that P − λ(h) a(x; h)e−φ+ (x)/h = O(h∞ )e−φ+ (x)/h (8.30) in a neighborhood of xj0 . Actually any neighborhood Ω ⊂⊂ Rn will do, provided that 1) φ+ is well-defined in a neighborhood of Ω. 2) Hq | = 0 on Ω \ {xj0 }. Λ+


251

3) Ω is star-shaped with respect to the point xj0 and the integral curves of the vector field ν+ := (πx )∗ (Hq | ), where πx ((x, ξ)) = x. Λ+

We also know that λ(h) is equal mod O(h∞ ) to the corresponding value in (8.24). As in [9] we notice that if γ ⊂ D(0, B) is a closed h-independent contour avoiding all the values μj,k in (8.25), and 1 (z − P )−1 dz (8.31) πhγ = 2πi hγ the corresponding spectral projection, then, using also Theorem 8.4, πhγ (χae−φ+ /h ) − χae−φ+ /h L2 = O(h∞ )

(8.32)

if χ ∈ C0∞ (Ω) is equal to one near xj0 . It follows that χae−φ+ /h is a linear combination of generalized eigenfunctions of P with eigenvalues inside hγ up to an error O(h∞ ) in L2 -norm.

9. Exponentially weighted estimates In this section we keep the general assumptions on P and assume for simplicity that C is reduced to a single point: C = {(0, 0)} .

(9.1)

∞

If ψ ∈ C (neigh (0, M ); R), we have eψ/h ◦ P ◦ e−ψ/h = Pψ ,

(9.2)

pψ (x, ξ) = p2 (x, ξ) − q x, ψ (x) + i(∂ξ q) x, ψ (x) · ξ ,

(9.3)

with the symbol (cf. (2.13)) where we recall that

q(x, ξ) = p2 (x, ξ) + p1 (x, ξ) − p0 (x) .

(9.4)

Notice that ξ → q(x, ξ) is a convex function for every x. Let φ = φ+ (x) ∈ C ∞ (neigh (0; R)) be the function introduced in Section 8 so that Λ+ = Λφ is the stable outgoing manifold through (0, 0) for the Hq -flow. Recall that by Proposition 8.2 φ (0) > 0 . (9.5) We have the eikonal equation (9.6) q x, φ (x) = 0 , so pφ (x, ξ) = p2 (x, ξ) + i(∂ξ q)(x, φ (x)) · ξ. The vector field (∂ξ q)(x, φ (x)) · ∂x is the x-space projection of Hq | , so its linearization at x = 0 has all its eigenvalues Λφ

with real part > 0. Consequently (as we shall see in more detail in the proof of Lemma 10.1 below), there exists G ∈ C ∞ (neigh (0, M ); R) such that (∂ξ q) x, φ (x) · ∂x G x2 , G(x) x2 . (9.7)

252



Let ΩG (r) = {x ∈ neigh (0); G(x) ≤ r} for 0 < r 1. Outside the set ΩG (C0 ), we put ψ = φ − g(G) , (9.8) for a suitable function g and for 0 < 1. (Eventually will be proportional to h.) Using (9.6), we get q x, ψ (x) = −g (G)(∂ξ q) x, φ (x) · ∂x G(x) + 2 g (G)2 p2 x, G (x) . (9.9) Choose g(G) = ln G for G ≥ C0 , so that g (G) = 1/G. Then 1 p2 x, G (x) q x, ψ (x) = − (∂ξ q) x, φ (x) · ∂x G + . G(x) G(x) G(x) Here 1 (∂ξ q)(x, φ ) · ∂x G 1 , G We conclude that , q x, ψ (x) ≤ − C0

(9.10)

p2 (x, G ) = O(1) . G

x ∈ neigh (0, M ) \ ΩG (C0 ) ,

(9.11)

if C0 > 0 is large enough. Outside a small fixed neighborhood of 0 we want to flatten out the weight. Let fδ (t) = δf ( δt ) be the function introduced in Section 2. For some small and fixed δ0 > 0, we put = fδ0 φ − g(G) (9.12) ψ = fδ0 (ψ) which is also well-defined as the constant 3δ0 /2 for large x. From (9.11), the fact that q(x, 0) ≤ 0 and the convexity of q, we get , q x, ψ (x) ≤ − fδ0 (ψ) (9.13) C where we keep in mind that 0 ≤ fδ0 ≤ 1. We extend the definition of ψ to a full neighborhood of x = 0, by putting g(G) = ln(C0 ) +

1 (G − C0 ) , C0

for

0 ≤ G ≤ C0 .

(9.14)

Then in ΩG (C0 ), we have ψ = ψ = φ − G /C0 , so q(x, ψ ) = O() by (9.9). ψ is small, the conjugated Operator In the exterior region where ψ = fδ0 (ψ) P = Pψ is close to the unperturbed Operator P and we can apply the method of Section 2. Write p1 + p0 p = p2 + i for the symbol of P, so that by (9.3), we have p1 = (∂ξ q) x, ψ (x) · ξ , p0 = −q(x, ψ ) ,

(9.15)

(9.16)


253

to be compared with p1 = (∂ξ q(x, 0)) · ξ, p0 = −q(x, 0). Let ν(x, ∂x ) = cj (x)∂xj , cj (x)ξj and define ψ as in (2.15), now for P instead of P and with where p1 = an additional cut-off: t ψ = 1 − χ(x) k ν )dt . (9.17) f ◦ p0 ◦ exp (t T0 Here χ ∈ C0∞ (M ) is equal to 1 near x = 0 and has its support in a small neighborhood of that point. Then for Pδψ = eδψ /h ◦ P ◦ e−δψ /h we get (cf. (2.22)), p0 + δf ◦ p0 T0 − O(δ 2 ) outside supp χ, pδψ ≥ (1 − δ)

(9.18)

where the time average in the second term to the right is taken along the trajectories of ν. In the region where χ = 1, we get Pδψ = P of course, and in the intermediate region, supp (∇χ), we have p0,δψ = p0 + O(δ) .

(9.19)

If δ1 > 0 is small enough, we know that fδ0 ψ(x) , ≤ δ1 ⇒ f ◦ p0 T0 (x) ≥ C1 for some constant C1 > 0. In fact, the ν and ν trajectories through a given point ≤ δ stay close for some fixed time > 0 , so the conditions (4.24), with fδ0 (ψ(x)) (4.23) imply that the ν-trajectory will encounter points with p0 ≥ 1/Const during a non-trivial interval of time. Then ≤ δ1 , we have p0,δψ ≥ (1 − δ) p0 + Cδ1 (and we • In the region where fδ0 (ψ(x)) recall that p0 ≥ 0). • In the region where fδ0 (ψ(x)) > δ1 , and G(x) ≥ C0 , we have p0,δψ = δ1 p0 + O(δ) ≥ C + O(δ), by (9.13). • In ΩG (C0 ), we have p0,δψ = p0 = O(). Choosing first δ1 > 0 small enough, then δ > 0 small enough, we conclude that ≥ C outside ΩG (C0 ) , p0,δψ (9.20) = O() in ΩG (C0 ) . Now Pδψ = eψ /h ◦ P ◦ e−ψ /h = Pψ , where ψ = ψ + δ ψ ,

(9.21)

and where we recall that ψ also depends on . Combining Lemma 2.1 for Pψ with (9.20), we get − Re z |u|2 dx + (P2 u)udx M\ΩG (C0 ) C M ≤ Re (Pψ − z)u|u + O() + Re z |u|2 dx . (9.22) ΩG (C0 )

254



If Re z = O(h), we choose = Ah with A so large that C − Re z ≥ h and deduce that h u 2 + (P2 u|u) ≤ Re (Pψ − z)u|u + O(h) u 2ΩG (C0 ) , (9.23) leading to the a priori estimate h u ≤ (Pψ − z)u + O(h) u ΩG (C0 ) .

(9.24)

Re-injecting this estimate in (9.23), we get 1

h2 u 2 + h B 2 hDu 2 ≤ (1 + α) (Pψ − z)u 2 + Oα (h2 ) u 2ΩG (C0 ) ,

(9.25)

for every fixed α > 0. Here (Pψ − z)u = eψ /h (P − z)v, u = eψ /h v, so from (9.25) we get the a priori estimate for the original Operator 1

1

h eψ /h v + h 2 B 2 hD(eψ /h v) ≤ O(1) eψ /h (P − z)v + O(h) eψ /h v ΩG (C0 ) ,

(9.26)

uniformly, for |Re z| ≤ Ch provided that = Ah for A large enough depending on C. Now let λ(h) = λ1,k (h) be an eigenvalue of P as in (8.24), (8.28) and assume that μ is given by (8.27) and is simple, as explained after that equation. Then λ(h) is a simple eigenvalue of P and is the only eigenvalue in some disc D(λ(h), h/C0 ). Let uWKB (x; h) be the approximate solution given in (8.29), (8.30) and let u = πhγ (χuBKW ) be the corresponding exact eigenfunction, where γ = ∂D(μ, 2C1 0 ). Theorem 9.1.

a) Outside any h-independent neighborhood of 0, we have 1

u, B 2 hDu = O(e−1/(Ch) ) in L2 -norm. b) There exists a neighborhood Ω of 0, where u(x; h) = (a + r)e−φ+ (x)/h , r L2 (Ω) ,

(9.27)

1

B 2 hDr L2 (Ω) = O(h∞ ) .

Proof. Apply (9.26) with v = u, z = λ(h), = Ah, A 1, to get 1

1

h eψ /h u + h 2 B 2 hD(eψ /h u) ≤ O(h−N ) , ψ /h

N = N (A) ,

−N

= O(h ) in ΩG (C0 ). Here, ψ = fδ0 (φ − g(G)) + where we also used that e O(δ) is larger than a positive constant outside any fixed neighborhood of 0, so u = O(e−1/(Ch) ) in L2 -norm there. Moreover, since eψ /h u = O(h−N ), we have 1

1

1

1

1

O(h−N ) ≥ h 2 B 2 hD(eψ /h u) ≥ h 2 eψ /h B 2 hDu − O(h 2 ) |∇ψ |eψ /h u . 1

Here ∇ψ = O(1) (as we shall see more in detail below), so eψ /h B 2 hDu = 1 1 1 O(h−N − 2 ), so B 2 hDu = O(e− Ch ) in L2 -norm away from any given fixed neighborhood of 0. The proof of a) is complete. To prove b), we apply (9.26) to v = u−χuWKB, z = λ(h) with = Ah, A 1. Since u − χuWKB = O(h∞ ) by (8.32), and eψ /h (P − z)χuWKB = O(h∞ ) if we


255

arrange so that ψ < φ on supp ∇χ, we conclude that eψ /h (u−χuWKB ) ΩG (C0 ) = O(h∞ ) and hence 1 1 h eψ /h (u − χuWKB ) + h 2 B 2 hD eψ /h (u − χuWKB ) ≤ O(h∞ ) . (9.28) In a small neighborhood of 0, we have φ

eψ /h = e h −

g(G) h

φ

= e−Ag(G) e h ,

where by (9.14) e−Ag(G) = O(1)−A e

−Ag(G)

−A

= G(x)

for G(x) ≤ C0 , for G(x) ≥ C0 .

It then follows from (9.28), that u = (a + r)e−φ+ /h in a neighborhood Ω of 0, with r L2 (Ω) = O(h∞ ). 1 To get the corresponding bound on B 2 hDr, we just have to proceed as in the proof of a) and use that ∇ψ = ∇φ − g (G)∇G, where 1

Thus ∇ψ O(h∞ ).

g (G)∇G = O(− 2 ) for G(x) < C0 , 1 ∇G = O(− 2 ) for G(x) ≥ C0 . g (G)∇G = G √ 1 = ∇φ + O( ) = O(1) and we conclude that B 2 hDr L2 (Ω) =

Remark 9.2. If we drop the assumption (9.1) and allow N−1 more points ρ2 , . . . , ρN in C, then Theorem 9.1 is still valid, provided that all μj,k in 8.24 with j ≥ 2 are different from the value μ, associated to ρ1 = (0, 0).

10. Supersymmetric approach The Witten approach has been independently extended to the case of non-elliptic Operators like the Kramers–Fokker–Planck Operator in [24] (in supersymmetric language) and in [2] (in terms of differential forms). See also [17]. We start by a quick review of that in the semiclassical case, then we establish some basic facts about the principal and subprincipal symbols, especially at the critical points of the given weight function. 10.1. Generalities Let A(x) : Tx∗ M → Tx M ,

x∈M,

(10.1)

be an invertible map depending smoothly on x ∈ M . Then we have the real nondegenerate bilinear form u|vA(x) = ∧k A(x)(u)|v , u, v ∈ ∧k Tx∗ M . (10.2)

256



If a : ∧k Tx∗ M → ∧ Tx∗ M is a linear map, we define the “adjoint” aA,∗ : ∧ Tx∗ M → ∧k Tx∗ M by (10.3) au|vA(x) = u|aA,∗ vA(x) . (In the complexified case, we use the sesquilinear scalar product (u|v)A = u|vA and define aA,∗ the same way.) If ω is a one form and u and v are k − 1 and k forms respectively, we get at x: ω ∧ u|vA = Aω ∧ (∧k−1 A)u|v = ∧k−1 Au|(Aω) v = u|(Aω) v A , (10.4) so (ω ∧ )A,∗ = (Aω) ,

(10.5)

where denotes the usual Operator of contraction. Let μ(dx) be a locally finite measure with a smooth positive density. When M = Rn , μ will be the Lebesgue measure. Sometimes we also use the symbol μ for the corresponding density. If u, v are smooth k forms with supp u ∩ supp v compact, we define u(x)|v(x) A(x) μ(dx) , (u|v)A = u(x)|v(x) A(x) μ(dx) u|vA = and denote by aA,∗ the formal adjoint of an Operator a : C0∞ (M ; ∧k T ∗ M ) → D (M ; ∧ T ∗ M ). If we fix some local coordinates x1 , . . . , xn and write μ(dx) = μ(x)dx (by slight abuse of notation), we can consider ∂xj : C0∞ (M ; ∧k T ∗ M ) → C0∞ (M ; ∧k T ∗ M ) , acting coefficient-wise, and a straightforward computation shows that −1 (−∂xj ) ◦ μ t(∧k A) (∂xj )A,∗ = μ t(∧k A) ∂x μ = −∂xj − j − t(∧k A)−1t ∂xj (∧k A) . μ

(10.6)

We only retain that (h∂xj )A,∗ = −h∂xj + O(h) ,

(10.7)

where O(h) stands for multiplication by a smooth matrix, which is O(h) with all its derivatives, uniformly on all of M when M = Rn , and which is = 0 when A(x), μ(x) are constant. Let φ ∈ C ∞ (M ; R) and introduce the Witten (de Rham) complex dφ = e−φ/h ◦ hd ◦ eφ/h = hd + (dφ)∧ : C0∞ (M ; ∧k T ∗ M ) → C0∞ (M ; ∧k+1 T ∗ M ) ,

(10.8)

with d2φ = 0. In local coordinates (always the canonical ones when M = Rn ) we have dφ =

n 1

(h∂xj + ∂xj φ) ◦ dx∧ j ,

(10.9)


257

where h∂xj + ∂xj φ acts coefficient-wise and commutes with dx∧ j , so dA,∗ = φ

n

− h∂xk + ∂xk φ + O(h) ◦ A(dxk ) ,

(10.10)

1

where from now on, O(h) and O(h2 ) will have the same meaning as after (10.7). The corresponding Witten–Hodge Laplacian is given by A,∗ −ΔA = dA,∗ φ dφ + dφ dφ .

Since

(10.11)

2 2 A,∗ = 0, (dA,∗ φ ) = (dφ )

we also have

2 −ΔA = (dφ + dA,∗ φ ) ,

and −ΔA conserves the degree of differential forms. Choose local coordinates x1 , . . . , xn (to be the standard ones when M = Rn ) and write A(dxk ) = Aj,k (x)∂xj , Aj,k (x) = A(dxk )|dxj . j

+ ∂xk φ. Notice that [Zj , dx∧ Let Zj = h∂xj + ∂xj φ, ZkA,∗ = h∂xA,∗ k ] = 0, so k ! ZkA,∗ , A(dxj ) = 0 . A,∗ Writing dφ = n1 Zj ◦ dx∧ = n1 ZkA,∗ ◦ A(dxk ) , we get j , dφ A,∗ ∧ A,∗ Zk ◦ A(dxk ) dx∧ . −ΔA = j Zj + Zj dxj A(dxk ) Zk j,k

Here, we use the general identity ν ∧ μ + μ ν ∧ = ν, μ1 on the first term in the parenthesis to get −ΔA = I + II + III , A,∗ Zk Aj,k Zj , I= j,k

II = −

ZkA,∗ dx∧ j A(dxk ) Zj

j,k

III =

A,∗ dx∧ j Zj Zk A(dxk ) ,

j,k

where Aj,k = dxj , A(dxk ). We have ! Zj , A(dxk ) = h (∂xj A)(dxk ) . t

Using the identity (U A,∗ ) A,∗ = U (see Subsection 10.4), we see that % $ t A,∗ [ZkA,∗ , dx∧ = tA(dxj ) , Zk = −h (∂xk tA)(dxj ) , j]

258


hence


t −1 ∧ ∂xk tA(dxj ) . [ZkA,∗ , dx∧ j ] = −h A

Using these commutator relations, we move dx∧ j to the left and A(dxk ) to the right in II and combining with III, we finally obtain that A,∗ A,∗ −ΔA = Zk Aj,k Zj + dx∧ (10.12) j [Zj , Zk ]A(dxk ) j,k

+h

j,k

−1 ∧ t ZkA,∗ dx∧ +h A (∂xk tA)dxj A(dxk ) Zj j (∂xj A)(dxk )

j,k

j,k

−1 +h (tA (∂xk tA)dxj )∧ (∂xj A)dxk . 2

j,k

Modulo O(h)(h∂x + ∂x φ) + O(h)(−h∂x + ∂x φ) + O(h2 ), we get −ΔA ≡

(−h∂xk + ∂xk φ)Aj,k (x)(h∂xj + ∂xj φ)

(10.13)

j,k

+

2h∂xj ∂xk φ ◦ dx∧ j A(dxk ) ,

j,k

where the error terms vanish when A(x) and μ(x) are constant (for the chosen coordinates). Now write A(x) = B(x) + C(x) ,

t

B(x) = B(x) ,

t

C(x) = −C(x) .

(10.14)

Then (10.13) gives −ΔA ≡

(−h∂xk + ∂xk φ)Bj,k (x)(h∂xj + ∂xj φ)

(10.15)

j,k

+

(∂xk φ)Cj,k h∂xj + h∂xj ◦ Cj,k ◦ (∂xk φ) − h∂xk (Cj,k )h∂xj j,k

+

j,k

2h∂xj ∂xk φ ◦

dx∧ j A(dxk )

.

j,k

Again, this becomes an equality when A, μ are constant. Note that the last term vanishes on 0-forms, i.e., on scalar functions. To recover the Kramers–Fokker– Planck Operator (cf. [24]), replace n by 2n, put M = R2n x,y , 1 A= 2

0 1 , −1 γ


259

and let μ = dxdy be the Lebesgue measure. Then (10.15) is an equality and we get for 0-forms: n γ (0) (−h∂yj + ∂yj φ)(h∂yj + ∂yj φ) (10.16) −ΔA = 2 j=1 +

=

! 1 (∂yj φ)h∂xj − (∂xj φ) h∂yj + h∂xj ◦ ∂yj φ − h∂yj ◦ ∂xj φ 2 j

n γ (−h∂yj + ∂yj φ)(h∂yj + ∂yj φ) + hHφ , 2 j=1

where Hφ =

(∂yk φ ∂xk − ∂xk φ ∂yk )

is the Hamilton field of φ with respect to the standard symplectic form dyj ∧dxj . If we choose 1 (10.17) φ(x, y) = y 2 + V (x) , 2 we get the Kramers–Fokker–Planck Operator γ (0) −ΔA = y · h∂x − V (x) · h∂y + (−h∂y + y) · (h∂y + y) . (10.18) 2 10.2. The principal symbol of the Hodge Laplacian The principal symbol of −ΔA in the sense of h-differential operators is scalar and given by Aj,k (−iξk + ∂xk φ)(iξj + ∂xj φ) (10.19) p(x, ξ) = j,k

=

Bj,k (ξj ξk + ∂xj φ ∂xk φ) + 2i

j,k

Cj,k ∂xk φ ξj .

j,k

The corresponding real symbol q(x, ξ) = −p(x, iξ) is given by q(x, ξ) = Aj,k (ξk + ∂xk φ)(ξj − ∂xj φ) j,k

=

Bj,k (ξj ξk − ∂xj φ ∂xk φ) + 2

j,k

(10.20)

Cj,k ∂xk φ ξj .

j,k

It vanishes on the two Lagrangian manifolds Λ±φ . We define ν± = Hq | .

(10.21)

Using x1 , . . . , xn as coordinates on Λ±φ , we get ν+ = 2 Aj,k ∂xk φ ∂xj = 2A(x) φ (x) · ∂x

(10.22)

Λ±φ

j,k

ν− = −2

j,k

Aj,k ∂xj φ ∂xk = −2 tA(x) φ (x) · ∂x .

(10.23)

260



(Even more radically, we could say that ν+ = 2A(x)(dφ(x)), where A(x) is viewed as a map Tx∗ M → Tx M , and similarly for ν− .) Let x0 be a non-degenerate critical point of φ, so that Λφ and Λ−φ intersect transversally at (x0 , 0). The spectrum of the linearization Fq of Hq at (x0 , 0) is equal to the union of the spectra of the linearizations 0 0 = 2A(x0 )φ (x0 )x · ∂x and ν− = − 2tA(x0 )φ (x0 )x · ∂x (10.24) ν+ of ν+ and ν− respectively at x0 . Thus we are interested in the eigenvalues of the matrices Aφ , tAφ , where we write A = A(x0 ), and φ = φ (x0 ) for short. Here, −1 we notice that tAφ = φ t(Aφ )φ has the same eigenvalues as Aφ and similarly t φ A, φ A are isospectral to Aφ . Thus The eigenvalues of Fq are given by ± 2λj , where λ1 , . . . , λn are the eigenvalues of Aφ . (10.25) From questions about hypoellipticity (see [22]) we would like to know when all the eigenvalues of Fp avoid the real axis, or equivalently, when all the eigenvalues of Fq (the linearization of Hq at (x0 , 0)) avoid the imaginary axis. We assume from now on that B(x) ≥ 0 ,

x∈M.

(10.26)

Then, if φ0 (x) = 12 φ x · x is the Hessian quadratic form of φ at x0 , we have 0 ν+ (φ0 ) = 2Bφ x, φ x ≥ 0 .

(10.27)

Lemma 10.1. Let μ(x, ∂x ) = M x · ∂x be a real linear vector field on Rn . Let n± ∈ N, n+ + n− = n. Then the following two statements are equivalent: (A) M has n+ eigenvalues with real part > 0 and n− eigenvalues with real part < 0. (B) There exists a quadratic form G : Rn → R of signature (n+ , n− ) and a constant C > 0, such that μ(x, ∂x )(G) ≥

1 2 |x| , C

x ∈ Rn .

Proof. Assume first that (n+ , n− ) = (n, 0). If (A) holds, we know that etM x 2 ≥

1 t/C e x 2 , C

t ≥ 0,

for some constant C > 0, and we can put (by a classical argument) 1 T tM 2 G(x) = GT (x) = e x dt , T 1 . T 0

(10.28)


261

Then

1 T 1 T d tM 2 tM 2 e x dt M x · ∂x e x dt = M x · ∂x GT (x) = T 0 T 0 dt 1 1 1 T /C 1 TM 2 2 e x 2 , = ( e x − x ) ≥ − 1 x 2 ≥ T T C 2T

if T is large enough. Thus we get (B). Conversely, if (B) holds, we have with a new constant C > 0, that μ(x, ∂x )G ≥ and hence

1 G, C

1 d G(etM x) ≥ G(etM x) , dt C

so

G(etM x) ≥ et/C G(x) , tM

t ≥ 0.

−1 t/C

Thus e x ≥ C e x for some new positive constant C and we conclude that the eigenvalues of M all have positive real parts. Now consider the case of general (n+ , n− ) and assume first that (A) holds. Then we have the M -invariant decomposition, Rn = L+ ⊕ L+ where dim (L± ) = n± and σ(M| L ) belongs to the open right half plane in the + case and to the ± open left half plane in the − case. Hence we have positive definite quadratic forms G± on L± such that 1 ±M x · ∂x (G± ) ≥ |x|2 , x ∈ L± . C Then G = G+ ⊕ (−G− ) (defined in the obvious way) has the required properties in (B). + be an n+ -dimensional subspace Conversely, assume that (B) holds. Let L on which G is positive definite. By (10.28), we have for all x ∈ Rn , μ(x, ∂x )G(x) ≥

1 G(x) , C

C > 0,

+ , we get so if x ∈ L

G(etM x) ≥ et/C G(x) , t ≥ 0 , and hence with a new constant C > 0, 1 |etM x| ≥ et/C |x| , t ≥ 0 . C − ) = n− and G is negative definite on L − , we get Similarly, if dim (L 1 |t|/C − , e |x| , x ∈ L C Now we have the M -invariant decomposition |etM x| ≥

Rn = L + ⊕ L 0 ⊕ L − ,

t ≤ 0.

262



C C where LC + , L0 , L− are the sums of generalized eigenspaces of M corresponding to the eigenvalues with real parts > 0, = 0 and < 0 respectively. We see that + ∩ (L0 ⊕ L− ) = 0, so necessarily, L

+ ≤ dim L+ . n+ = dim L Similarly, − ≤ dim L− , n− = dim L so dim L+ = n+ , dim L− = n− , dim L0 = 0, and (A) follows.

If B > 0, then (10.27) with strict inequality for x = 0, and Lemma 10.1 imply 0 has n± eigenvalues with ± real part > 0, where (n+ , n− ) is the signature of that ν± φ (x0 ). It follows in that case that Fp has no real eigenvalues. This last conclusion also follows from [22]. Indeed, in that case the quadratic approximation p0 (x, ξ) of p at (x0 , 0) is elliptic in the sense that |p0 (x, ξ)| |x|2 + |ξ|2 and takes its values in an angle − π2 + ≤ arg p0 ≤ π2 − for some > 0. Now return to the general case, when we only assume (10.26) and φ = φ (x0 ) is non-degenerate of signature (n+ , n− ). We next make some remarks about the quadratic approximation p0 of p at (x0 , 0). 0 Proposition 10.2. a) Assume that the matrix Aφ of ν+ has m± eigenvalues with ± real part > 0, m+ + m− = n. Then there exists a real quadratic form G(x, ξ) on R2n such that Re p0 (x, ξ) + iHG (x, ξ) ≥ |(x, ξ)|2 , (x, ξ) ∈ R2n , 0 < 1 . (10.29) C

b) Conversely, assume that there exists a quadratic form G such that (10.29) holds. Then Aφ has n± eigenvalues with ± real part > 0, where (n+ , n− ) is the signature of φ (0). Recall that the condition (4.21) implies the existence of G as in a) of the proposition. (The converse is not true however. It is easy to find examples of purely imaginary quadratic forms p0 for which there exist G as in a) of the proposition.) , −G ). Proof. a) Choose G(x, ξ) of the form G(x) + G(ξ), so that HG (x, ξ) = (G x ξ 0 Recall that p is the quadratic approximation of p at (x0 , 0), obtained from (10.19) by freezing Aj,k at x0 and replacing ∂x φ by φ x = (φ0 ) (x), φ = φ (x0 ). We get Bj,k (ξj ξk + ∂xj φ0 ∂xk φ0 ) Re p0 (x, ξ) = j,k

⎛ ⎞ k ξj ⎠ + O 2 |(x, ξ)|2 , + 2 ⎝ Cj,k (∂xk φ0 )∂xj G − Cj,k (φ ∂ξ G) j,k

j,k


263

so it suffices to have 1 2 −1 |x| on φ N (B) , (10.30) C ≥ 1 |ξ|2 on N (B) , φ Cξ · ∂ξ G (10.31) C where we also used the antisymmetry of C in the last equation. This would follow from 1 ≥ 1 |ξ|2 , x, ξ ∈ Rn , Aφ x · ∂x G ≥ |x|2 , φ Aξ · ∂ξ G C C it suffices to apply Lemma 10.1 to and in order to find such functions G and G, the isospectral matrices Aφ and φ A. b) Let G be as in (10.29). The quadratic form p0 (x, ξ) = p0 exp iHG (x, ξ) = p0 (x, ξ) + iHG p0 + O(2 ) Cφ x · ∂x G ≥

is elliptic on the real phase space and takes its values in an angle π

π

ei[− 2 + C , 2 − C ] [0, +∞[ , so we know from [22] that Fp0 has no real eigenvalues. On the other hand p0 and p0 are related by a canonical transformation, so Fp0 and Fp0 are isospectral. Hence Aφ has m± eigenvalues with ± real part > 0, where m+ +m− = n. To see that m± = n± , we just replace B by B+δ1, 0 < δ 1, to reduce ourselves to the elliptic case, and apply the observation after the proof of Lemma 10.1. 10.3. The subprincipal symbol We next look at the subprincipal term in (10.13). Write Aν,k ∂xν , A(dxk ) = Aν,k ∂x ν , A(dxk ) = ν

ν

so the second sum in (10.13) becomes 2h φj,k Aν,k ◦ dx∧ (φ ◦ tA)j,ν ◦ dx∧ j ∂xν = 2h j ∂xν j,ν

j,k,ν

which simplifies further to 2h

(φ ◦ tA)(dxj )∧ ∂x j .

(10.32)

j

Now we restrict the attention to a non-degenerate critical point x0 of φ and we shall compute the subprincipal symbol of −ΔA at the corresponding doubly characteristic point (x0 , 0). At that point φ ◦ tA : Tx∗0 M → Tx∗0 M is invariantly defined and it is easy to check that (10.32) is also invariantly defined: we get the same quantity if we replace dx1 , . . . , dxn , ∂x1 , . . . , ∂xn , by ω1 , . . . , ωn , ω1∗ , . . . , ωn∗ , where ω1 , . . . , ωn is any basis in the complexified cotangent space and ω1∗ , . . . , ωn∗ is the dual basis of tangent vectors for the natural bilinear pairing.

264



Assume that the equivalent conditions of Proposition 10.2 hold and denote the corresponding eigenvalues (that are also the eigenvalues of φ ◦tA) by λ1 , . . . , λn with Re λj > 0 for 1 ≤ j ≤ n+ and with Re λj < 0 for n+ + 1 ≤ j ≤ n = n+ + n− . The eigenvalues of Fp are then ±2iλj (in view of (10.25) and the isospectrality of Fp and iFq reviewed prior to Lemma 8.1), so Fp := 1 tr i

μ∈σ(Fp ) Im μ>0

μ=

n+

2λj −

1

n

2λj .

(10.33)

n+ +1

The subprincipal symbol of the first term in (10.13) (at (x0 , 0)) is equal to 1 Aj,k {−iξk + ∂xk φ, iξj + ∂xj φ} = − Aj,k φj,k 2i j,k

j,k

= −tr (Aφ ) = −

n

λj .

(10.34)

1

The eigenvalues of j (φ ◦ tA)(dxj )∧ ∂xj on the space of m-forms are easily calculated, if we replace dx1 , . . . , dxn by a basis of eigenvectors ω1 , . . . , ωn of φtA, so that (φ ◦ tA)(ωj ) = λj ωj , and ∂xj by the corresponding dual basis vectors ωj∗ . (Here we assume to start with that there are no Jordan blocks. This can be achieved by an arbitrarily small perturbation of A, and we can extend the end result of our calculation to the general case by continuity.) We get (φ ◦ tA)(dxj )∧ ∂x j = λj ωj∧ ωj∗ . (10.35) j

j

A basis of eigenforms of this Operator is given by ωj1 ∧ · · · ∧ ωjm , 1 ≤ j1 < j2 < · · · < jm ≤ n and the corresponding eigenvalues are λj1 + · · · + λjm . Let SP be the subprincipal symbol of (10.13) at (x0 , 0). Then the eigenvalues of 1 tr Fp + SP , acting on m forms 2 are ⎞ ⎛ n+ n n n λj − λj − λj + 2(λj1 + · · · + λjm ) = 2 ⎝λj1 + · · · + λjm − λj ⎠ , 1

n+ +1

1

n+ +1

1 ≤ j1 < · · · < jm ≤ n . (10.36) We conclude that if m = n− , then all the eigenvalues have a real part > 0 and if m = n− , then precisely one eigenvalue is equal to 0, while the others have positive real part.


265

10.4. A symmetry for adjoints Our last remark in this section concerns symmetry relations for the A, ∗ adjoints. If D : L2 (Ω; ∧k T ∗ Ω) → L2 (Ω; ∧j T ∗ Ω), then a simple calculation shows that −1 ∗ j t D ∧ ( A) , DA,∗ = ∧k (tA) where D∗ denotes the adjoint with respect to the measure μ. We also have (u|v)A = (v|u)tA . Playing with these relations we see that t

D = (DA,∗ ) A,∗ . This can be applied to −ΔA and we get t

(−ΔA ) A,∗ = −ΔtA .

11. The double well case In this section we assume that M = Rn ,

A = A(x)

is independent of x, and invertible . ∞

(11.1) n

We decompose A as in (10.14) and assume (10.26). Let φ ∈ C (R ; R) be such that ∂xα φ(x) = O(1) , ∂xα (B∂x φ, ∂x φ) = O(1) , |α| ≥ 2 . (11.2) (q)

Consider P (q) = −ΔA which according to (10.15) becomes P (q) = hDxj Bj,k hDxk + (∂xj φ)Bj,k (∂xk φ) − htr (Bφ ) j,k

j,k

(∂xk φ)Cj,k h∂xj + h∂xj ◦ Cj,k (∂xk φ) + j,k

+ 2h

(∂xj ∂xk φ)dx∧ j A(dxk ) .

(11.3)

j,k

Apart from the third and the last terms which are O(h) with all their derivatives, this is of the form (2.1) with bj,k = Bj,k , cj (x) = k Cj,k ∂xk φ, p0 (x) = B∂x φ, ∂x φ. We define p2 , p1 , p0 as in Section 2 and see that (2.4)–(2.7) hold. Assume that φ is a Morse function with critical points x1 , . . . , xN ∈ Rn ,

|φ (x)| ≥ 1/C ,

|x| ≥ C . ∗

n

(11.4) (11.5)

Let C = {ρj ; j = 1, . . . , N } where ρj = (xj , 0) ∈ T R . Then ρj are critical points with critical value 0 for p2 , p1 , p0 . Since A is invertible, N (B) ∩ N (C) = 0, and hence (4.4) holds. We adopt the dynamical assumptions (4.21), (4.22) (or equivalently (4.21), (4.24)) and (4.23). Then we can apply the results of Sections 8, 9 to P (q) since neither the presence of the bounded subprincipal symbol in (11.3) nor

266



the non-scalar nature of the Operators has any serious influence. In the preceding section we saw that we are in the case when the conditions of Proposition 10.2 are fulfilled. The only change in Theorem 8.3 is that the μj,k in (8.24) are of the form μj,k =

n 1 1 νj,k, λ + tr Fp (ρj ) + γj,k , i 2

(11.6)

l=1

where γj,k is any eigenvalue of the subprincipal symbol SP (q) at (xj , 0). From the calculations in Subsection 10.3 we notice that the μj,k will be confined to a sector {0} ∪{|arg z| < π/2− 1/C} around [0, +∞[ and it is precisely when xj is of index q (i.e., when the Hessian of φ at xj has precisely q negative eigenvalues) that one of the μj,k may be equal to 0. We now add more specific conditions for the double well case. Assume that φ has precisely three critical points, two local minima U±1 , and a “saddle point” U0 of index one . (11.7) Then φ(x) → +∞ with φ(x) ≥ C1 |x|, for |x| ≥ C. Put Sj = φ(U0 ) − φ(Uj ), j = ±1, so that Sj > 0. The set φ−1 (] − ∞, φ(U0 )[) has precisely two connected components Dj , j = ±1, determined by the condition (0) Uj ∈ Dj . Under these assumptions we know that P (0) = −ΔA has precisely two eigenvalues μ0 , μ1 = o(h) spanning a corresponding 2-dimensional spectral subspace E (0) . Actually one of these two eigenvalues, say μ0 , is equal to 0 with e−φ/h as the corresponding eigenfunction and since a truncation of this function can be used as a quasimode near each of U±1 we also know that μ1 = O(h∞ ) (1) (k) (cf. (11.13)). Moreover, −ΔA has precisely one eigenvalue μ 1 = o(h) and −ΔA has no eigenvalues = o(h) for k ≥ 2. Since our Operators are real we know that 1 are real. From the the spectra are symmetric around the real axis, hence μ0 , μ1 , μ intertwining relations (1)

(0)

−ΔA dφ = dφ (−ΔA ) ,

(0)

(1)

−ΔA dA,∗ = −dA,∗ φ φ ΔA ,

we then also know that μ 1 = μ1 . In fact, when B > 0 it follows from the ellipticity and the estimates in Section 2 that all eigenforms and generalized eigenforms corresponding to an eigen(0) value in D(0, Ch) belong to S(Rn ), so if μ1 = 0 and (−ΔA − μ1 )u = 0, u ∈ L2 , (1) then u ∈ S(Rn ) and 0 = dφ u ∈ S is an eigenform for −ΔA with the same eigen(0) value. A priori we cannot exclude that μ1 = 0 and that −ΔA u = Const e−φ/h . (1) Then again, 0 = dφ u ∈ S is a corresponding eigenvector of −ΔA . In the general case, we let 0 < B → B =: B0 when 0. (Take for instance B = B + I.) (q) On a small circle D(0, h/C) we know that (P − z)−1 = O( h1 ) uniformly for (q) (q) 0 ≤ 1. If (P0 − z)u = v, u, v ∈ S (for = 0), then (P − z)u = v + r , (q) (q) r → 0, so (P − z)−1 v = u − (P − z)−1 r → u, → 0. Since (P (q) − z)(S) is (q) dense in L2 we conclude that (P − z)−1 → (P (q) − z)−1 strongly for |z| = h/C.


267

We have then the corresponding fact for the finite rank spectral projections and (q) their compositions with P : 1 1 −1 (z − P ) dz and z(z − P )−1 dz . 2πi |z|=h/C 2πi |z|=h/C These are finite rank Operators and converge therefore in norm. It follows that σ(P(q) ) ∩ D(0, h/C) → σ(P (q) ) ∩ D(0, h/C) ,

when → 0 ,

(11.8)

so we get μ 1 = μ1 also in the general case. Let χj ∈ C0∞ (Dj ) be equal to 1 on Dj ∩ φ−1 (] − ∞, φ(U0 ) − 0 ]) for 0 > 0 fixed but arbitrarily small. Consider 1

fj = h−n/4 cj (h)e− h (φ(x)−φ(Uj )) χj (x) ,

j = ±1 ,

(11.9)

where cj ∼ cj,0 + hcj,1 + · · · > 0 is a normalization constant with cj,0 > 0, such that fj = 1 . (11.10) We also have 1

1

P (0) (fj ) = [P (0) , χj ](cj e− h (φ(x)−φ(Uj )) ) = O(h−N0 e− h (Sj −0 ) ) , for some N0 > 0. If (0)

Π

1 = 2πi

(z − P

(0) −1

)

h γ = ∂D 0, C

dz ,

γ (0)

(11.11)

(11.12)

is the spectral projection of P onto E (0) we know from Theorem 8.4 that Π(0) = O(1). It follows from (11.11) that 1

ej := Π(0) fj = fj + O(h−N1 e− h (Sj −0 ) ) In fact, we write (11.11) as P

(0)

in L2 .

(11.13)

fj = rj ,

(z − P (0) )(fj ) = zfj − rj , 1 (z − P (0) )−1 fj = fj + (z − P (0) )−1 z −1 rj z and integrate, using the bounds on the resolvent provided by Theorem 8.4. From (11.13) we see that 1

ej 2 = 1 + O(h−N2 e− h (Sj −0 ) ) , (e1 |e−1 ) = O(h−N2 e

1 −h (Smin −0 )

),

(11.14) (11.15)

where Smin = min(S−1 , S1 ) . (11.16) (1) Let E be the one-dimensional eigenspace of P corresponding to μ1 . From an easy extension of Theorem 9.1 to the non-scalar case with the presence of other non-resonant wells (U±1 ) as in Remark 9.2, we know that E (1) is generated by an eigenform 1 n (11.17) e0 (x; h) = χ0 (x)e− h φ+ (x) h− 4 a0 (x; h) + O(e−S0 /h ) , (1)

268



where χ0 ∈ C0∞ (neigh (U0 )) is equal to one near U0 , S0 > 0, a0 (x; h) ∼

∞

a0,k (x)hk

0

is a symbol as in Theorem 9.1 with a0,0 (U0 ) = 0, and φ+ ∈ C ∞ (neigh (U0 ); [0, ∞[) satisfies φ+ (x) |x − U0 |2 (11.18) and solves the eikonal equation

q x, φ+ (x) = 0 ,

with q = p2 + p1 − p0 , p2 = B(x)ξ, ξ , p1 (x, ξ) = 2 C(x)φ (x), ξ ,

(11.19) p0 (x) = B(x)φ (x), φ (x) .

Λφ+ is the stable outgoing manifold through (U0 , 0) for the Hq -flow and recall that φ+ (U0 ) > 0 by Proposition 8.2. (Similarly we have a stable incoming manifold Λφ− .) Let k± be the number of eigenvalues of the linearization of Hq | Λφ

at that point with ± real part > 0, so that k+ + k− = n. Let K+ , K− ⊂ Λφ be the corresponding stable outgoing and incoming submanifolds of dimension k+ and k− respectively. Then K+ ⊂ Λφ+ , K− ⊂ Λφ− and φ − φ(U0 ) − φ± vanishes to the second order on πx (K± ). Since φ (U0 ) has signature (n − 1, 1), we conclude that dim K+ = n − 1, dim K− = 1. (This also follows from Proposition 10.2.) It is also clear that Λφ , Λφ± intersect cleanly along K± , so we get 2 φ+ − φ − φ(U0 ) dist x, πx (K+ ) , (11.20) 2 φ − φ(U0 ) − φ− dist x, πx (K− ) . t

We next make some remarks about the adjoint Operator −ΔtA = (−ΔA ) A,∗ (cf. Subsection 10.4). The principal symbol is p2 − ip1 + p0 = p(x, −ξ) = pˇ(x, ξ) and the corresponding real “q”-symbol is qˇ(x, ξ) = q(x, −ξ). Since our dynamical conditions are invariant under a change of sign of the Hp1 -direction, all our assumptions are equally valid for −ΔtA . This also holds for the geometric discussion above, so if Λφ∗+ , Λφ∗− denote the outgoing and incoming Hqˇ-invariant Lagrangian ∗ ⊂ Λφ the outgoing/incoming manifolds for manifolds through (U0 , 0) and K± ∗ ∗ Hqˇ| (noting that qˇ = 0 on Λφ ), then dim K+ = n − 1, dim K− = 1 and Λφ

∗ 2 φ∗+ − φ − φ(U0 ) dist x, πx (K+ ) , ∗ 2 ) . φ − φ(U0 ) − φ∗− dist x, πx (K−

(11.21)

In view of the general relation J∗ (Hq ) = −Hqˇ ,

where

J : (x, ξ) → (x, −ξ) ,

(11.22)

we see that Λφ∗− = J(Λφ+ ), Λφ∗+ = J(Λφ− ), or more simply φ∗− = −φ+ ,

φ∗+ = −φ− ,

(11.23)


269

giving in particular from (11.20), (11.21), 2 φ − φ(U0 ) + φ∗+ dist x, πx (K− ) , ∗ 2 ) . φ − φ(U0 ) + φ+ dist x, πx (K−

(11.24)

(0) (0) Let μ∗0 = 0, μ∗1 be the two eigenvalues of P∗ := −ΔtA that are o(h) and (0) (0) let Π∗ be the spectral projection onto the corresponding spectral subspace E∗ . (0) (0) (1) Then e∗j = Π∗ fj , j = ±1, span E∗ , and satisfy (11.13). Similarly for P∗ = (1) −ΔtA we have the generating eigenform 1

∗

n

e∗0 (x; h) = χ0 (x)e− h φ+ (x) h− 4 a∗0 (x; h) + O(e−S0 /h )

in L2

(11.25)

(1)

for the one dimensional eigenspace E∗ corresponding to μ∗1 . Now, using that our eigenvalues and Operators are real, we know by duality that μ∗1 = μ1 , (11.26) (0)

(1)

and that (E∗ , E (0) ) and (E∗ , E (1) ) are dual pairs for the scalar products (u|v)L2 (q) and (u|v)A respectively. In fact, ((z − P∗ )−1 )A,∗ = (z − P (q) )−1 . From Subsection 10.3 we know that a0,0 (U0 ) is an eigenvector corresponding to the negative eigenvalue of φ ◦ tA at U0 , and a∗0,0 (U0 ) is an eigenvector corresponding to the negative eigenvalue of φ ◦ A. Since (φ ◦ A)A,∗ = φ ◦ tA, we know that the two eigenvalues are equal and that the A-product of the two eigenvectors is = 0; ∗ (11.27) a0,0 (U0 )|a0,0 (U0 ) A = 0 . It follows that (e∗0 |e0 )A 1 and after renormalization of e∗0 we may assume that (e∗0 |e0 )A = 1 .

(11.28)

Similarly, using (11.10) 1

(e∗j |ek ) = δj,k + O(e− Ch ) ,

j, k = ±1 . (11.29) Let λ−1 λ1 be the matrix of dφ : E (0) → E (1) with respect to the bases e−1 , e1 and (e0 ). (Strictly speaking, we approximate our operators by elliptic ones as in (11.8) and pass to the limit.) Let ∗ λ−1 λ∗1 be the matrix of dA,∗ for the same bases. The eigenvalue μ1 can be viewed as the φ (0) → E (0) or equivalently as the scalar dφ dA,∗ : second eigenvalue of dA,∗ φ dφ : E φ (1) (1) (2) has no eigenvalue = o(h)). Either way, we get E → E (using also that P μ1 = λ∗−1 λ−1 + λ∗1 λ1 .

(11.30)

270



We get λk = (e∗0 |dφ ek )A , ∗ λj

where

= (gj |dA,∗ φ e0 )A ,

k = ±1 ,

(11.31)

j = ±1 ,

(11.32)

1 g1 = e∗−1 e∗1 1 + O(e− Ch ) (11.33) (0) is the base in E∗ that is dual to e−1 e1 . Here the complex conjugate signs are superfluous since we work with real Operators, eigenvalues and functions. Let χ ∈ C0∞ (neigh (U0 ); [0, 1]) be equal to 1 near U0 . Using that dφ e−1 = λ−1 e0 , we get, dropping the bars from now on, g−1

λ−1 = (e∗0 |dφ e−1 )A =

(e∗0 |χdφ e−1 )A

(11.34)

+ λ−1 e∗0 |(1 − χ)e0 A 1

= (e∗0 |χdφ e−1 )A + O(e− Ch )λ−1 . Here

t (e∗0 |χdφ e−1 )A = e∗0 |[χ, dφ ]e−1 A + (dφA,∗ e∗0 |χe−1 )A . t

(1)

(0)

Now the matrix of dφA,∗ : E∗ → E∗ (0)

(1)

(11.35)

with respect to the dual bases is the adjoint

t

dφA,∗ e∗0

= λ−1 g−1 + λ1 g1 , and expressing gj as of the one of dφ : E → E , so linear combinations of the e∗j by means of (11.33) and using (11.13) for the e∗±1 we see that the last term in (11.24) is of the form t

1

1

(dφA,∗ e∗0 |χe−1 )A = O(e− Ch )λ−1 + O(e− Ch )λ1 . Thus we have obtained 1 1 1 + O(e− Ch ) λ−1 + O(e− Ch )λ1 = e∗0 |[χ, dφ ]e−1 A = −h e∗0 |(dχ) ∧ e−1 A ,

(11.36)

(11.37)

and we shall study the last expression. The contribution from the remainder 1 1 in (11.13) is O(h−N2 )exp h1 (−S−1 + 0 − C1 ) = O(1)e− h (S−1 + 2C ) if we choose 0 small enough. A similar estimate holds for the contribution from the remainder term in (11.25). As we shall see, the contribution from the leading terms in (11.13), (11.25) will be larger. It is equal to 1 ∗ n χ−1 (x) A(x)a∗0 (x; h)|dχ(x) e− h (φ+ (x)+φ(x)−φ(U−1 )) dx . (11.38) −c−1 (h)h1− 2 Here by (11.24),

φ∗+ (x) + φ(x) − φ(U−1 ) = φ∗+ + φ(x) − φ(U0 ) + S−1

2 S−1 + dist x, πx (K− ) , (11.39) 1

so we expect (11.38) to behave like some power of h times e− h S−1 with the main contribution coming from a neighborhood of supp (dχ) ∩ supp (χ−1 ) ∩ πx (K− ). Now φ(x) − φ(U0 ) −|x − U0 |2 on πx (K− ) while χ−1 has its support in D−1 and


271

equals 1 in the subset of D−1 where φ(x) − φ(U0 ) ≤ −0 with 0 arbitrarily small. Because of the presence of dχ which has its support in an annular region around U0 , we see that χ−1 (x) = 1 in neigh (D−1 ∩ supp (dχ) ∩ πx (K− )), so we can forget about χ−1 in (11.38) and just integrate over neigh (supp (dχ) ∩ πx (K−1 ), D−1 ). Let us look at A(x)a∗0,0 (x; h)|dχ(x1 ) at a point x1 ∈ supp (dχ)∩πx (K− ). At U0 we know that a∗0,0 is an eigenvector of φ ◦ A associated to the negative eigenvalue, so Aa∗0,0 is a corresponding eigenvector for Aφ which is the linearization of 1 . Hence Aa∗0,0 at U0 is tangent to πx (K− ). This will remain approximately 2 Hq | Λφ

true at x1 since the latter point is close to U0 . Choosing χ to be a “circular” standard cut-off, we see that Aa∗0,0 , dχ is non-vanishing of constant sign at every point x1 ∈ πx (K− ) ∩ D−1 where dχ = 0. By stationary phase it is now clear that the integral (11.38) is equal to 1

1

h 2 −1 (h)e− h S−1 ,

−1 ∼ −1,0 + h−1,1 + · · · ,

−1,0 = 0 .

(11.40)

Returning to (11.37) and modifying −1 by an exponentially small term, we get 1 1 1 1 (11.41) 1 + O(e− Ch ) λ−1 + O(e− Ch )λ1 = h 2 −1 (h)e− h S−1 . Similarly, 1 1 1 1 O(e− Ch )λ−1 + 1 + O(e− Ch ) λ1 = h 2 1 (h)e− h S1 , 1 ∼ 1,0 + h1,1 + · · · ,

1,0 = 0 . (11.42)

Inverting the system, we get h 12 −1 (h)e− h1 S−1 1 λ−1 − Ch ) = 1 + O(e . 1 1 λ1 h 2 1 (h)e− h S1 Now turn to λ∗j in (11.32). In view of (11.33), we have ∗ α−1 1 λ−1 − Ch ) = 1 + O(e , λ∗1 α1

(11.43)

(11.44)

where αj = (dφ e∗j |e0 )A = (e0 |dφ e∗j )tA which can be identified with the expression (11.31) after replacing A by tA and making the corresponding substitutions, e∗0 → e0 , ej → e∗j . Hence we have the analogue of (11.43), ∗ α−1 h 21 ∗−1 (h)e− h1 S−1 1 1 λ−1 − Ch − Ch ) ) = 1 + O(e = 1 + O(e , 1 1 λ∗1 α1 h 2 ∗1 (h)e− h S1 ∗j (h) ∼ ∗j,0 + h∗j,1 + · · · ,

∗j,0 = 0 . (11.45)

We finally claim that j,0 ∗j,0 > 0. Indeed, this number is real and different from zero and if we deform our matrices to reach the selfadjoint case (with A > 0) we see that we have a positive sign. Combining this with (11.30) we get the main result of this work:

272



(0)

Theorem 11.1. Let P = −ΔA where we assume (11.1), (11.2), (11.7). We also assume that P satisfies the additional dynamical conditions (4.21), (4.22) (or equivalently (4.21), (4.24)) and (4.23). Then for C > 0 large enough, P has precisely 2 eigenvalues, 0 and μ1 in the disc D(0, h/C) when h > 0 is small enough. Here μ1 is real and of the form (11.46) μ1 = h a1 (h)e−2S1 /h + a−1 (h)e−2S−1 /h , where aj (h) are real, aj (h) ∼ aj,0 + aj,1 h + · · · , aj,0 > 0, Sj = φ(U0 ) − φ(Uj ).

Acknowledgements We are grateful to B. Helffer who pointed out the work [24] to us, and for pointing out that an earlier version of our condition (11.2) was not directly applicable to the Kramers–Fokker–Planck operator. The research of the second author is supported in part by the National Science Foundation under grant DMS–0304970 and by an Alfred P. Sloan Research Fellowship. He is happy to acknowledge the hospitality of ´ Ecole Polytechnique and Université de Reims, where part of this work was done.

References [1] D. Bao, S.-S. Chern, Z. Shen, An introduction to Riemann–Finsler geometry, Graduate texts in Mathematics, 200. Springer-Verlag, New York, 2000. [2] J. M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc. 18 (2005), 379–476. [3] J. M. Bismut, G. Lebeau, The hypoelliptic Laplacian and Ray–Singer metrics, preprint (2006). [4] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker–Planck equation, Comm. Pure Appl. Math., 54 (1) (2001), 1–42. [5] J.-P. Eckmann, M. Hairer, Spectral properties of hypoelliptic operators, Comm. Math. Phys. 235 (2) (2003), 233–253. [6] M. I. Freidlin, A. D. Wentzell, Random perturbations of dynamical systems, SpringerVerlag, New York, 1984. [7] B. Helffer, M. Klein, F. Nier, Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp. 26 (2004), 41–85. [8] B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for Fokker–Planck operators and Witten Laplacians, Lecture Notes in Mathematics, 1862. Springer-Verlag, New York, 2005. [9] B. Helffer, J. Sj¨ ostrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations 9 (4) (1984), 337–408, Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation, Ann. Inst. H. Poincaré Phys. Théor. 42 (2) (1985), 127–212. [10] B. Helffer, J. Sj¨ ostrand, Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells, Math. Nachr. 124 (1985), 263–313.


273

[11] B. Helffer, J. Sj¨ ostrand, Puits multiples en mécanique semi-classique. IV. Etude du complexe de Witten, Comm. Partial Differential Equations 10 (3) (1985), 245–340. [12] B. Helffer, J. Sj¨ ostrand, Résonances en limite semi-classique, Bull. de la S.M.F., Mémoire 24/25, Suppl. du Tome 114 (3) (1986). [13] F. Hérau, F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker– Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2) (2004), 151–218. [14] F. Hérau, J. Sj¨ ostrand, C. Stolk, Semiclassical analysis for the Kramers–Fokker– Planck equation, Comm. Partial Differential Equations 30 (4–6) (2005), 689–760. [15] M. Hitrik, Boundary spectral behavior for semiclassical operators in dimension one, Int. Math. Res. Not. 64 (2004), 3417–3438. [16] V. N. Kolokoltsov, Semiclassical analysis for diffusions and stochastic processes, Lecture Notes in Mathematics, 1724, Springer-Verlag, Berlin, 2000. [17] G. Lebeau, Le bismutien, Séminaire équations aux dérivées partielles, Ecole Polytechnique 2004–05, I.1–I.15. [18] Y. Li, L. Nirenberg, The distance function to the boundary, Finsler’s geometry and the singular set of viscosity solutions of some Hamilton–Jacobi equations, Comm. Pure Appl. Math, 58 (2005), 85–146. [19] A. Melin, J. Sj¨ ostrand, Fourier integral operators with complex-valued phase functions, Springer Lect. Notes in Math., 459. [20] A. Melin, J. Sj¨ ostrand, Determinants of pseudodifferential operators and complex deformations of phase space, Meth. Appl. Analysis 9 (2002), 177–238. [21] J. Sj¨ ostrand, M. Dimassi, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Series 269, Cambridge University Press 1999. [22] J. Sj¨ ostrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. f. Mat. 12 (1) (1974), 85–130. [23] J. Sj¨ ostrand, Density of resonances for strictly convex obstacles, Can. J. Math. 48 (2) (1996), 397–447. [24] J. Tailleur, S. Tanase-Nicola, J. Kurchan, Kramers equation and supersymmetry, J. Stat. Phys. 122 (4) (2006), 557–595. (preprint: arxiv.org/abs/cond-mat/0503545). [25] C. Villani, Hypocoercivity, preprint, 2006 (arxiv.org/abs/math.AP/0609050).

Frédéric Hérau Laboratoire de Mathématiques UMR 6056–CNRS Université de Reims Moulin de la Housse B.P. 1039 F-51687 Reims Cedex 2 France e-mail: [email protected]

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Michael Hitrik Department of Mathematics University of California Los Angeles, CA 90095–1555 USA e-mail: [email protected] Johannes Sj¨ ostrand CMLS UMR7640–CNRS Ecole Polytechnique F-91120 Palaiseau Cedex France e-mail: [email protected] Communicated by Christian Gérard. Submitted: April 3, 2007. Accepted: October 4, 2007.


Ann. Henri Poincaré 9 (2008), 275–327 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020275-53, published online April 10, 2008 DOI 10.1007/s00023-008-0356-x


Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems Petr Girg and Peter Takáˇc Abstract. The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem −Δp u = λ|u|p−2 u + h x, u(x); λ in Ω ; (P) u = 0 on ∂Ω . Here, Ω is a bounded domain in RN (N ≥ 1), Δp u = div(|∇u|p−2 ∇u) denotes the Dirichlet p-Laplacian on W01,p (Ω), 1 < p < ∞, and λ ∈ R is a spectral parameter. Let μ1 denote the first (smallest) eigenvalue of −Δp . Under some natural hypotheses on the perturbation function h : Ω × R × R → R, we show that the trivial solution (0, μ1 ) ∈ E = W01,p (Ω) × R is a bifurcation point for problem (P) and, moreover, there are two distinct continua, Zμ+1 and Zμ−1 , consisting of nontrivial solutions (u, λ) ∈ E to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ1 ). The continua Zμ+1 and Zμ−1 are either both unbounded in E, or else their intersection Zμ+1 ∩ Zμ−1 contains also a point other than (0, μ1 ). For the semilinear problem (P) (i.e., for p = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union Zμ+1 ∩ Zμ−1 looks like (for p > 2) in an interesting particular case. Our proofs are based on very precise, local asymptotic analysis for λ near μ1 (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer’s original work. def

1. Introduction This work is concerned with bifurcations of continua of “positive” and “negative” solutions to quasilinear elliptic problems of the following type: −Δp u = λ|u|p−2 u + h x, u(x); λ in Ω ; (1.1) u = 0 on ∂Ω .

276

P. Girg and P. Tak´ aˇc


Here, Ω denotes a bounded domain in RN (N ≥ 1), Δp stands for the Dirichlet def

p-Laplacian defined by Δp u = div(|∇u|p−2 ∇u) for 1 < p < ∞, λ (λ ∈ R) serves as a bifurcation parameter, and h : Ω × R × R → R is a Carathéodory function with h(x, · ; · ) continuous for a.e. x ∈ Ω. When considering bifurcations from a trivial solution, we naturally assume also h(x, 0; λ) = 0 and h(x, u; λ)/|u|p−1 → 0 as u → 0, pointwise for a.e. x ∈ Ω and uniformly for every λ ∈ R. A trivial solution def of (1.1) is any pair (0, λ) ∈ E = W01,p (Ω) × R. In analogy with classical results of Dancer [10] for the semilinear case p = 2, our main goal is to show the existence of two distinct continua of nontrivial (weak) solutions to problem (1.1), “positive” and “negative” ones, that bifurcate from the set of trivial solutions at the point (0, μ1 ) in the positive and negative directions ϕ1 and −ϕ1 , respectively (see Lemma 3.6). As usual, μ1 denotes the first (smallest) eigenvalue of −Δp which is known to be simple with a positive eigenfunction ϕ1 ∈ W01,p (Ω). Under a continuum in a Banach space we mean a closed connected set which contains at least two distinct points. Similarly to bifurcations from zero we treat also bifurcations from infinity under the condition h(x, u; λ)/|u|p−1 → 0 as |u| → ∞, pointwise for a.e. x ∈ Ω and uniformly for every λ ∈ R. To be more specific about our present results, let us begin by considering the semilinear case p = 2 first: The classical global bifurcation result of Rabinowitz [31, Theorem 1.3] exhibits a continuum of nontrivial solutions to problem (1.1) which emanates from the set of trivial solutions at the bifurcation point (0, μ1 ). Furthermore, Dancer’s result [10, Theorem 2] guarantees the bifurcation of two continua of “positive” and “negative” solutions to problem (1.1) in the directions ±ϕ1 . Indeed, in a sufficiently small neighborhood of (0, μ1 ) these continua contain only solutions (u, λ) ∈ E of problem (1.1) satisfying u = τ (ϕ1 + v ) where τ ∈ R and v /ϕ1 L∞ (Ω) → 0 as τ → 0. Hence, u > 0 in Ω (u < 0 in Ω, respectively) if and only if τ > 0 (τ < 0), provided |τ | > 0 is small enough. Now let us consider the quasilinear case p = 2. The analogue of Rabinowitz’ result [31, Theorem 1.3] for problem (1.1) has been obtained in del Pino and Man´ asevich [30] with a continuum of nontrivial solutions bifurcating from the point (0, μ1 ) and having the same properties as in the case p = 2. In the work reported here we obtain the corresponding analogue (Theorem 3.7 below) of Dancer’s result [10, Theorem 2] for 1 < p < ∞. We treat problems with a more general (p − 1)-homogeneous part than just (1.1) treated in [30]. Similarly to a bifurcation from zero at (0, μ1 ) sketched above, under a bifurcation from infinity at (+∞, μ1 ) ((−∞, μ1 ), respectively) we mean a continuum of solutions (u, λ) ∈ E of problem (1.1) satisfying u = t−1 (ϕ1 + v ) where 0 = t ∈ R and v /ϕ1 L∞ (Ω) → 0 as t → 0. Again, u > 0 in Ω (u < 0 in Ω, respectively) if and only if t > 0 (t < 0), provided |t| > 0 is small enough. In an analogy with the case p = 2, we use the fact that μ1 is a simple eigenvalue of −Δp with a positive eigenfunction ϕ1 in an essential way. Under a rather restrictive hypothesis, this extension of Dancer’s result has already been stated in Dr´ abek [14, Theorem 14.20, p. 191] without proof. His hypothesis [14,

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Eq. (14.43), p. 191] has been verified in Dr´ abek et al. [16, Theorem 4.1] for the special case h(x, u; λ) ≡ f (x) independent from u and λ, for bifurcations from infinity at λ = μ1 . Extending a new asymptotic technique developed recently in Dr´ abek et al. [16, Theorem 4.1] and Tak´ aˇc [34, Section 5] and [35, Section 6], we are able to verify Dr´ abek’s hypothesis [14, Eq. (14.43), p. 191] and thus extend Dancer’s result to a broader class of quasilinear elliptic operators of second order. Last but not least, for the radially symmetric problem (1.1) in a ball Ω ⊂ RN , a local bifurcation result of Crandall–Rabinowitz-type [9, Theorem 1.7, p. 325] has been obtained in Garc´ıa–Meli´ an and Sabina de Lis [21, Theorem 2, p. 30]. We remark that Crandall–Rabinowitz’ result guarantees only local existence of a (smooth) bifurcation curve of nontrivial solutions together with their uniqueness, whereas Dancer’s result guarantees global existence of “positive” and “negative” bifurcation continua of nontrivial solutions without uniqueness. Since Crandall– Rabinowitz’ result is concerned only with bifurcations from simple eigenvalues, one can clearly determine the “positive” and “negative” parts of the (smooth unique local) bifurcation curve of nontrivial solutions. A direct consequence of our extension of Dancer’s result is the following dichotomy for the simplified bifurcation problem −Δp u = λ|u|p−2 u + f (x) in Ω ; (1.2) u = 0 on ∂Ω , where f ∈ L∞ (Ω), f ≡ 0 in Ω: There exist two continua Zμ+1 and Zμ−1 (⊂ E) of solutions (u, λ) to problem (1.2) bifurcating from (+∞, μ1 ) and (−∞, μ1 ), respectively, such that either (i) Zμ+1 ∩Zμ−1 = ∅, i.e., Zμ+1 ∪Zμ−1 is a continuum connecting large positive with large negative solutions of types (+∞, μ1 ) and (−∞, μ1 ), respectively, or else (ii) the intersections of both Zμ±1 with the set {(u, λ) ∈ E : |λ − μ1 | > δ} are unbounded (in E) for every δ > 0 small enough. This extension also fills the gap left open in several results on global bifurcations abek et al. [16, Section 5] and in from (±∞, μ1 ) for problem (1.1) obtained in Dr´ Dr´ abek, Girg, and Tak´ aˇc [15, Section 3] as well. This work is organized as follows. For the sake of clarity of our presentation we always begin by treating the case of bifurcation from the trivial solution at (0, μ1 ) in all details and then reduce our treatment of bifurcation from infinity at (±∞, μ1 ) to highlighting the necessary changes. In the next section (Section 2) we introduce basic notations, state our hypotheses, and deduce a few simple consequences. Our main results are stated in Section 3: bifurcations from zero in Section 3.1 (Proposition 3.5 and Theorem 3.7) and bifurcations from infinity in Section 3.2 (Proposition 3.8 and Theorem 3.10). We begin Section 4 by showing the simplicity of the first eigenvalue μ1 for the quasilinear eigenvalue problem (2.7) in Section 4.1 (Remark 4.1). In particular, we

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generalize also the well-known inequality of D´ıaz and Saa which they have established only for Δp ; see Remark 4.4. In Section 4.2 we adapt some results from Arcoya and G´ amez [6, Lemma 24, p. 1905] (proved there only for Δp and bifurcation from infinity) on the asymptotic analysis of local bifurcation from zero to our setting in problem (2.1); see Proposition 4.5. Analogous results for local bifurcations from infinity are established in Section 4.3 (Proposition 4.10). Section 5 contains the complete proofs of our main results, Proposition 3.5 and Theorem 3.7. In these proofs we employ the topological degree due to Browder and Petryshyn [7] and Skrypnik [33] which we describe in Section 5.1. The corresponding results for bifurcations from infinity, Proposition 3.8 and Theorem 3.10, respectively, are derived from Proposition 3.5 and Theorem 3.7 by applying the standard transformation u → v = u/u2W 1,p (Ω) for u ∈ W01,p (Ω) \ {0}; see Remark 3.9. 0

Section 6 features an interesting example of problem (1.1) with Ω = (0, πp ) ⊂ R1 an interval and 2 < p < ∞, cf. (6.1). The nonhomogeneous perturbation function h(x, u; λ) is chosen in such a way that the continuum Zμ+1 ∪ Zμ−1 (⊂ E) oscillates through the hyperplane {(u, λ) ∈ E : λ = μ1 } in E while approaching the bifurcation point (+∞, μ1 ) or (−∞, μ1 ). In other words, λ−μ1 oscillates about zero as uW 1,p (Ω) → ∞. Rather involved asymptotic formulas from [16, Theorem 4.1] 0 are required to handle these oscillations. Finally, we collect some auxiliary results in Appendices A, B, and C. An a priori boundedness result in L∞ (Ω) (due to Anane [4, Théorème A.1, p. 96]) is stated in Appendix A. Some useful consequences thereof (for bifurcations from zero and infinity) follow in Appendix B. Our treatment of Example 6.1 is based on a ramification of Erdélyi’s asymptotic formula [17, Theorem on p. 52] which we establish in Appendix C.

2. Preliminaries 2.1. Notation We set R+ = [0, ∞) and N = {1, 2, 3, . . . }. The closure, interior, and boundary of a set S ⊂ RN are denoted by S, int(S), and ∂S, respectively, and the characteristic def function of S by χS : RN → {0, 1}. We write |S|N = RN χS (x) dx if S is also Lebesgue measurable. Let Ω be a bounded domain in RN (N ≥ 1). Given older space of an integer k ≥ 0 and 0 ≤ α ≤ 1, we denote by C k,α (Ω) the H¨ all k-times continuously differentiable functions u : Ω → R whose all (classical) partial derivatives of order ≤ k possess a continuous extension up to the boundary and are α-Hölder continuous on Ω. The norm uC k,α(Ω) in C k,α (Ω) is defined in a natural way. As usual, we abbreviate C k (Ω) ≡ C k,0 (Ω). The linear subspace of C k (Ω) consisting of all C k functions u : Ω → R with compact support is k denoted by Cck (Ω); we set Cc∞ (Ω) = ∩∞ k=0 Cc (Ω). Given 1 ≤ p ≤ ∞, we denote p by L (Ω) the Lebesgue space of all (equivalence classes of) Lebesgue measurable functions u : Ω → R with the standard norm. Finally, for an integer k ≥ 1,

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we denote by W k,p (Ω) the Sobolev space of all functions u ∈ Lp (Ω) whose all (distributional) partial derivatives of order ≤ k also belong to Lp (Ω). Again, the norm uk,p ≡ uW k,p (Ω) in W k,p (Ω) is defined in a natural way. The closure in W k,p (Ω) of the set of all C k functions u : Ω → R with compact support is denoted by W0k,p (Ω). We refer to Adams and Fournier [1] or Kufner, John, and Fuˇc´ık [24] for details about these and other similar function spaces. All Banach and Hilbert spaces used in this article are real. The Euclidean inner product in RN is denoted by · , · . We work with def the standard inner product in L2 (Ω) defined by u, vL2 (Ω) = Ω uv dx for u, v ∈ L2 (Ω). The orthogonal complement in L2 (Ω) of a set M ⊂ L2 (Ω) is denoted by 2 M⊥,L , 2 def M⊥,L = u ∈ L2 (Ω) : u, vL2 (Ω) = 0 for all v ∈ M . The inner product · , · L2 (Ω) induces the canonical duality between the space of test functions D(Ω) ≡ Cc∞ (Ω) and the space of distributions D (Ω). More generally, if X is a Banach space, D(Ω) ⊂ X ⊂ D (Ω), such that the embedding D(Ω) → X is dense and continuous, we denote by · , · X the duality between X and its dual space X induced by the canonical duality between D(Ω) and D (Ω). Since D(Ω) is reflexive, also the embedding X → D (Ω) is dense and continuous. If no confusion may arise, we often leave out the index X in · , · X . In particular, the inner product · , · L2 (Ω) induces the duality · , · Lp (Ω) between the Lebesgue spaces Lp (Ω) and Lp (Ω), where 1 ≤ p < ∞ and 1 < p ≤ ∞ with 1/p + 1/p = 1, and the duality · , · W 1,p (Ω) between the Sobolev space W01,p (Ω) and its dual

0

space W −1,p (Ω), as well. We use analogous notation also for the duality between the Cartesian products [Lp (Ω)]N and [Lp (Ω)]N . 2.2. Structural hypotheses Let us consider the following more general version of problem (1.1), namely, − div a(x, ∇u) = λ B(x) |u|p−2 u + h x, u(x); λ in Ω ; u=0

on ∂Ω .

(2.1)

In the sequel we always assume that the domain Ω satisfies the following regularity hypothesis: Hypothesis (Ω). If N ≥ 2 then Ω is a bounded domain in RN whose boundary ∂Ω is a compact manifold of class C 1,α for some α ∈ (0, 1), and Ω satisfies also the interior sphere condition at every point of ∂Ω. If N = 1 then Ω is a bounded open interval in R1 . It is clear that for N ≥ 2, hypothesis (Ω) is satisfied if Ω ⊂ RN is a bounded domain with C 2 boundary. We always assume that the function A of (x, ξ) ∈ Ω × RN and its partial ∂A N gradient ∂ξ A ≡ ∂ξ with respect to ξ ∈ RN satisfy the following structural i i=1 def 1 p

hypothesis, upon the substitution a(x, ξ) =

∂ξ A(x, ξ) with ai =

1 ∂A p ∂ξi :

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Hypothesis (A). A : Ω × RN → R+ verifies the positive p-homogeneity hypothesis A(x, tξ) = |t|p A(x, ξ)

for all t ∈ R

(2.2)

and for all (x, ξ) ∈ Ω × RN . Furthermore, we assume that A ∈ C 1 (Ω × RN ), and its ∂A partial gradient ∂ξ A : Ω × RN → RN satisfies p1 ∂ξ = ai ∈ C 1 (Ω × (RN \ {0})) for i all i = 1, 2, . . . , N , together with the following ellipticity and growth conditions: There exist some constants γ, Γ ∈ (0, ∞) such that N ∂ai (x, ξ) · ηi ηj ≥ γ · |ξ|p−2 · |η|2 , ∂ξ j i,j=1

N ∂ai p−2 , ∂ξj (x, ξ) ≤ Γ · |ξ| i,j=1 N ∂ai p−1 , ∂xj (x, ξ) ≤ Γ · |ξ|

(2.3)

(2.4)

(2.5)

i,j=1

for all x ∈ Ω, all ξ ∈ RN \ {0}, and all η ∈ RN . It is evident that it suffices to require inequalities (2.3), (2.4), and (2.5) for |ξ| = 1 only; the general case ξ ∈ RN \ {0} follows from the positive p-homogeneity hypothesis (2.2). ∂A (x, 0) = 0 for all x ∈ Ω and Hypothesis (2.2) forces A(x, 0) = 0 and ∂ξ i i = 1, 2, . . . , N . It follows that A(x, · ) is strictly convex and satisfies Γ γ |ξ|p ≤ A(x, ξ) ≤ |ξ|p p−1 p−1

for all ξ ∈ RN .

(2.6)

These inequalities are a direct consequence of Taylor’s formula combined with (2.3) and (2.4), which yields

γ Γ |ξ|p ≤ A(x, ξ) − A(x, 0) − ∂ξ A(x, 0), ξ ≤ |ξ|p p−1 p−1 for all (x, ξ) ∈ Ω × RN . The weight function B is assumed to satisfy Hypothesis (B). B : Ω → R+ belongs to L∞ (Ω) and does not vanish identically (almost everywhere) in Ω, i.e., B ≡ 0 in Ω. Now consider the (p − 1)-homogeneous nonlinear eigenvalue problem − div a(x, ∇u) = λ B(x) |u|p−2 u in Ω ; u=0

on ∂Ω ,

with an eigenvalue λ ∈ R and an eigenfunction u ∈ W01,p (Ω) \ {0}.

(2.7)

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Remark 2.1. The first (smallest) eigenvalue μ1 for problem (2.7) is given by the Rayleigh quotient

μ1 = inf A(x, ∇u) dx : u ∈ W01,p (Ω) with B(x) |u|p dx = 1 . (2.8) Ω

Ω

W01,p (Ω)

p

Since the Sobolev embedding

→ L (Ω) is compact, the infimum above is attained and satisfies 0 < μ1 < ∞. It is well-known that μ1 is a simple eigenvalue for problem (2.7) with the associated eigenfunction ϕ1 normalized by ϕ1 > 0 in Ω and Ω B(x)ϕp1 dx = 1; see Takáˇc, Tello and Ulm [37, Theorem 2.6, p. 80]. The special case of the positive Dirichlet p-Laplacian A = −Δp is due to Anane [3, Théorème 1, p. 727] and in a more general domain Ω to Lindqvist [26, Theorem 1.3, p. 157]. Moreover, it is shown in Anane [3, Théorème 2, p. 727] or Anane and Tsouli [5, Prop. 2, p. 5] that μ1 is an isolated eigenvalue of A and the next eigenvalue μ2 > μ1 has a variational characterization. An interested reader can easily derive this fact from the proof of the anti-maximum principle in Tak´ aˇc [36, proof of Theorem 4.4, Eq. (4.13) on p. 408]. 2.3. Hypotheses on the nonhomogeneous perturbation Finally, we assume that h satisfies hypothesis (H0 ) (for bifurcations from zero) or hypothesis (H∞ ) (for bifurcations from infinity) stated below: Hypothesis (H0 ). h : Ω × R × R → R is a Carathéodory function, i.e., h( · , u; λ) : Ω → R is Lebesgue measurable for each pair (u, λ) ∈ R2 and h(x, · ; · ) : R×R → R is continuous for almost every x ∈ Ω. Furthermore, we assume that there exists a constant C ∈ (0, ∞) such that h(x, u; λ) ≤ C |u|p−1 (2.9) for all a.e. x ∈ Ω and all (u, λ) ∈ R × R, and h(x, u; λ)/|u|p−1 → 0 as

u→0

(2.10)

uniformly for a.e. x ∈ Ω and uniformly in λ from bounded intervals in R. Hypothesis (Hn0 ). We say that a sequence of functions hn : Ω × R × R → R, n ∈ N, satisfy hypothesis (Hn0 ) if functions hn satisfy (H0 ) for each n ∈ N and the bounds (2.9) and convergence in (2.10) are uniform in n ∈ N. Hypothesis (H∞ ). h : Ω × R × R → R is a Carathéodory function. Furthermore, there exists a constant C ∈ (0, ∞) such that h(x, u; λ) ≤ C 1 + |u|p−1 (2.11) for all a.e. x ∈ Ω and all (u, λ) ∈ R × R, and h(x, u; λ)/|u|p−1 → 0

as |u| → ∞

uniformly for a.e. x ∈ Ω and in λ from bounded intervals in R.

(2.12)

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As we work with the zero Dirichlet boundary conditions, sometimes it will be necessary to assume inequality (2.9) also for bifurcations from infinity; this inequality is stronger than (2.11). These hypotheses are satisfied by the following “canonical” examples of h(x, u; λ) = g(x, u) + f (x) for (x, u, λ) ∈ Ω × R × R: • Condition (2.10) holds if f ≡ 0 in Ω and g takes either of the forms c(x) |u|q(x)−2 u ; (2.13) g(x, u) = c(x) |u|q(x)−1 , for |u| ≤ 1, where c, q ∈ L∞ (Ω) and q satisfies ess inf q > p. Ω

• Condition (2.12) holds if f ∈ L∞ (Ω) and g takes either of the forms (2.13) for |u| ≥ 1, where c, q ∈ L∞ (Ω) and q satisfies q(x) > 1 for a.e. x ∈ Ω together with ess sup q < p. Ω

The following simple lemma is a very useful consequence of hypothesis (H0 ) or (H∞ ) in our functional formulation of problem (2.1). Lemma 2.2. Let u ∈ L∞ (Ω) be arbitrary, u ≡ 0 in Ω. (a) If hypothesis (H0 ) is satisfied, then ∞ h x, u(x); λ /up−1 L∞(Ω) → 0 as uL (Ω) → 0 holds pointwise for a.e. x ∈ Ω and uniformly for every λ ∈ R. (b) If hypothesis (H∞ ) is satisfied, then ∞ h x, u(x); λ /up−1 L∞ (Ω) → 0 as uL (Ω) → ∞

(2.14)

(2.15)

holds pointwise for a.e. x ∈ Ω and uniformly for every λ ∈ R. The proof is given in the Appendix, Section B.1. In order to obtain easy-to-verify a priori estimates for the “positive” and “negative” (nontrivial) branches of solutions to the bifurcation problem (2.1), we impose the following additional hypothesis (H0 ) (for bifurcations from zero) or (H∞ ) (for bifurcations from infinity) on the perturbation function h where we assume that h is independent from λ: Hypothesis (H0 ). h(x, u(x); λ) ≡ h(x, u(x)) and there exist a constant C0 ∈ (0, ∞) and functions f0+ , f0− ∈ L∞ (Ω), f0± ≡ 0 in Ω, and g0 : R → R, g0 continuous with g0 (τ ) = 0 for τ ∈ R \ {0}, g0 differentiable in (−δ, δ) \ {0} for some δ > 0, such that u for a.e. x ∈ Ω and all u ∈ R ; h(x, u) ≤ C0 g0 (2.16) ϕ1 (x) h(x, u) → f0± (x) g0 ϕ1u(x)

pointwise for a.e. x ∈ Ω

τ g0 (τ ) < ∞. Γ0 = sup 0δ g∞ (τ )

(2.21)

Some remarks on these hypotheses are in order. Remark 2.3. Rewriting (2.17) as u p−1 h(x, u) ϕ1 (x) f0± (x) −→ · |u|p−1 ϕ1 (x)p−1 g0 ϕ1u(x)

for a.e. x ∈ Ω

as u → 0±

we observe that (2.10) forces g0 (τ )/|τ |p−1 → 0 as τ → 0. Similarly, combining (2.20) with (2.12) we arrive at g∞ (τ )/|τ |p−1 → 0 as τ → ±∞ . Remark 2.4. Condition (2.18) guarantees g τ (1 + θ) 0 − 1 ≤ 4 Γ0 |θ| g0 (τ )

(2.22)

def

for all τ, θ ∈ R such that 0 < |τ | ≤ 1/2δ and |θ| ≤ θ0 = min{1/2, 4 1Γ0 }. This can be seen as follows. Let 0 < |τ | ≤ 1/2δ and |θ| ≤ θ0 . From

1 g0 τ (1 + s θ) ds g0 τ (1 + θ) − g0 (τ ) = τ θ 0

we obtain

g0 τ (1 + θ) − g0 (τ ) ≤ |τ | |θ| · sup g0 τ (1 + s θ) . 0≤s≤1

Now we apply (2.18) and |θ| ≤ 1/2 to conclude that g0 τ (1+θ) −g0 (τ ) ≤ 2 Γ0 |θ| · sup g0 τ (1 + sθ) 0≤s≤1

≤ 2 Γ0 |θ| ·

sup g0 τ (1+sθ) −g0 (τ ) + g0 (τ ) . (2.23)

0≤s≤1

Since θ ∈ R is arbitrary with |θ| ≤ θ0 , (2.23) yields 1 · sup g0 τ (1 + sθ) − g0 (τ ) ≤ 2 Γ0 |θ| g0 (τ ) 2 0≤s≤1 from which (2.22) follows.

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Remark 2.5. Condition (2.21) guarantees g τ (1 + θ) ∞ − 1 ≤ 4 Γ∞ |θ| g∞ (τ )

(2.24)

def

for all τ, θ ∈ R such that |τ | ≥ 2δ and |θ| ≤ θ0 = min{1/2, 4 Γ1∞ }. The proof of (2.24) is analogous to that of (2.22).

3. Main results def

Set E = W01,p (Ω) × R. Under a solution of problem (2.1) (in the weak sense) we understand a pair (u, λ) ∈ E that satisfies the integral identity

a(x, ∇u), ∇φ dx = λ B(x) |u|p−2 u φ dx + h x, u(x); λ φ dx (3.1) Ω

for all φ ∈

Ω

W01,p (Ω).

Ω

The last equation is equivalent to the operator equation A(u) = λ B(u) + H(u; λ)

(3.2)

with all terms valued in the dual space X = W −1,p (Ω) of X = W01,p (Ω) and the operators A, B, H( · ; λ) : X → X defined as follows, for all u, φ ∈ X and λ ∈ R:

a(x, ∇u), ∇φ dx ; (3.3) A(u), φ X =

Ω

B(u), φ X = B(x) |u|p−2 u φ dx ; (3.4) Ω

H(u; λ), φ X = h(x, u; λ) φ(x) dx . (3.5) Ω

Owing to our conditions (2.2) through (2.5), the operator A : X → X is continuous, coercive, and strictly monotone. The operator B˜ : X → X can be extended to a continuous operator B : Lp (Ω) → (Lp (Ω)) = Lp (Ω) in a unique way. Consequently, B decomposed as B˜

B : X → Lp (Ω) −→ Lp (Ω) → X is compact by Rellich’s theorem. Finally, given λ ∈ R, also the operator H( · ; λ) : ˜ · ; λ) : Lp (Ω) → Lp (Ω) X → X can be extended to a continuous operator H( in a unique way. Again, H : X × R → X is compact by Rellich’s theorem. Furthermore, given any F ∈ X , the mapping u → A(u) − F equals the Fréchet derivative of the energy functional

def 1 J0 (u) = A(x, ∇u) dx − F, uX , u ∈ X = W01,p (Ω) , (3.6) p Ω which is coercive and strictly convex on X. Thus, the equation A(u) = F has exactly one solution u ∈ X, i.e., A is invertible with the inverse mapping A−1 : X → X which is continuous because X is a uniformly convex space (cf. Takáˇc,

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Tello, and Ulm [37, Prop. 4.3, p. 87]). Consequently, the operator λB + H( · ; λ) on the right-hand side of equation (3.2) may be viewed as a compact perturbation of the invertible operator A : X → X . This setting will enable us to apply the Browder–Petryshyn degree theory (Browder and Petryshyn [7] or Skrypnik [33]) to the mapping def

Φλ (u) ≡ Φ(u, λ) = A(u) − λB(u) − H(u; λ) ,

(u, λ) ∈ E = X × R .

(3.7)

In terms of Φ(u, λ), equation (3.1) reads Φ(u, λ) = 0. Now consider the (p − 1)homogeneous nonlinear eigenvalue problem (2.7), i.e., A(u)−λB(u) = 0 for (u, λ) ∈ E; recall that E = X × R. Definition 3.1. Let μ0 ∈ R. We say that (0, μ0 ) ∈ E is a bifurcation point (from zero) for problem (2.1) if there exists a sequence of pairs {(un , λn )}∞ n=1 ⊂ E such that equation (3.1) holds with (u, λ) = (un , λn ) for all n = 1, 2, . . . , and (un , λn ) → (0, μ0 ) in E as n → ∞. Proposition 3.2. Let (0, μ0 ) ∈ E be a bifurcation point (from zero) for problem (2.1). Then μ0 is an eigenvalue for the nonlinear eigenvalue problem (2.7), i.e., A(u) − μ0 B(u) = 0 with some u ∈ X \ {0}. The proof follows a standard pattern for this kind of result (the necessary condition for bifurcation via compactness). The reader is referred to the monograph by Fuˇc´ık et al. [20, Proof of Theorem II.3.2, pp. 61–62] for details. Definition 3.3. Let μ0 ∈ R. We say that (∞, μ0 ) is an (asymptotic) bifurcation point from infinity for problem (2.1) if there exists a sequence of pairs {(un , λn )}∞ n=1 ⊂ E such that equation (3.1) holds with (u, λ) = (un , λn ) for all n = 1, 2, . . . , and (un X , λn ) → (∞, μ0 ) as n → ∞. For u ∈ X, u = 0, set v = u/u2X . Then (3.2) is equivalent to 2(p−1) A(v) − λ B(v) = vX H v/v2X ; λ , and so the term

def

G(v; λ) =

2(p−1)

vX

H v/v2X ; λ if

v = 0 ;

0 if

v = 0,

for λ ∈ R, represents a compact perturbation “of higher order” in the variable v in the equation A(v) − λ B(v) = G(v; λ) . (3.8) It follows immediately from this transformation that the pair (∞, μ0 ) is a bifurcation point from infinity for (3.2) if and only if (0, μ0 ) is a bifurcation point zero for (3.8). For C ⊂ X × R we define (the set) C to be the closure in X × R of the set of all pairs (v, μ) ∈ X × R such that v = 0 and (v/v2X , μ) ∈ C. Using this transformation, a necessary condition for bifurcations from infinity easily follows from Proposition 3.2.

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Proposition 3.4. Let (∞, μ0 ) be a bifurcation point from infinity for problem (2.1). Then μ0 is an eigenvalue for the nonlinear eigenvalue problem (2.7), i.e., A(u) − μ0 B(u) = 0 with some u ∈ X \ {0}. In our treatment below we will prove a number of bifurcation results only for bifurcations from zero and leave obvious adjustments for bifurcations from infinity to the interested reader. More precisely, the counterparts for bifurcations from infinity corresponding to bifurcations from zero are obtained by the standard transformation described in Definition 3.3 above. We will make necessary comments in case when this procedure is not straightforward. 3.1. Bifurcations from zero – main results The closure of the set of all nontrivial solutions of problem (2.1) in E = W01,p (Ω)×R will be denoted by S, i.e., E def . S = (u, λ) ∈ E : Φ(u, λ) = 0, u = 0 Our first result concerns global bifurcation of solutions (u, λ) ∈ S from zero at the point (0, μ1 ) ∈ E. Proposition 3.5. Let μ1 be defined by formula (2.8) and assume that h satisfies hypothesis (H0 ). Then the pair (0, μ1 ) is a bifurcation point (from zero) for problem (2.7). Moreover, there exists a maximal closed set C ⊂ S (in the ordering by set inclusion), such that C is connected in E and has the following properties: (i) there exists a sequence {(un , λn )}∞ n=1 ⊂ C such that (un , λn ) → (0, μ1 ) in E; (ii) either C is unbounded, or else there exist another eigenvalue μ0 of A(u) − λB(u) = 0 such that μ0 > μ1 and another sequence {(un , λn )}∞ n=1 ⊂ C satisfying (un , λn ) → (0, μ0 ) in E. The proof of this result is postponed till Section 5.2. Our main result below provides more details about the bifurcation from Proposition 3.5. In order to formulate and prove this result, it is convenient to introduce Dancer’s notation [10]. Given any μ ∈ R and 0 < s < ∞, we consider an open neighborhood of (0, μ) in E defined by def E s (μ) = (u, λ) ∈ E : uW 1,p (Ω) + |λ − μ| < s . 0

Next, we define a functional ∈ W −1,p (Ω) by

def φ ϕ1 dx for all φ ∈ W01,p (Ω) . (φ) = ϕ1 −2 L2 (Ω) Ω

Thus, (ϕ1 ) = 1. Notice that, using the reflexivity of W01,p (Ω), we may identify −1/p we define with the function ϕ1 −2 L2 (Ω) ϕ1 . Finally, for any 0 < η < μ1 def Kη = (u, λ) ∈ E : |(u)| > η uW 1,p (Ω) 0

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and decompose it in a disjoint union Kη = Kη+ ∪ Kη− of the sets def Kη± = (u, λ) ∈ E : ±(u) > η uW 1,p (Ω) . 0

287

(3.9)

In particular, both Kην (ν = ±) are convex cones, Kη− = −Kη+ , and νtϕ1 ∈ Kην for every number t > 0. The symbol −ν will denote the sign opposite to ν. For a precise formulation of our main result, we also need the following lemma which is a variant of Rabinowitz’ result [31, Lemma 1.24] adapted to the quasilinear problem (2.1) with λ in a neighborhood of the first eigenvalue μ1 . Here we introduce the (technical) constant 1/p γ def η0 = · ϕ1 −1 (3.10) L∞ (Ω) > 0 , (p − 1) (μ1 + 1)BL∞ (Ω) + C |Ω|N where γ > 0 and C > 0 are the constants from inequalities (2.6) and (2.9), respectively. Lemma 3.6. For every η ∈ (0, η0 ) there exists a number S, 0 < S ≤ 1, such that S \ (0, μ1 ) ∩ E S (μ1 ) ⊂ Kη . Moreover, if (u, λ) ∈ (S \ {(0, μ1 )}) ∩ E S (μ1 ) then u = τ (ϕ1 + v ), where τ = (u) ∈ R and v ∈ C 1,β (Ω) (0 < β < β) satisfy |τ | > η uW 1,p (Ω) and (v ) = 0 0 together with |λ − μ1 | → 0 and v C 1,β (Ω) → 0 as τ → 0. The proof of this result is postponed till Section 5.2. Let S > 0 be the constant from Lemma 3.6. For 0 < ε ≤ S and ν = ± we define Dμν 1 ,ε to be the component of {(0, μ1 )} ∪ (S ∩ Eε ∩ Kην ) containing −ν (0, μ1 ), and Zμν1 ,ε to be the component of Zμν1 \ Dμ1 ,ε containing ν(0, μ1 ). Finally, ν we define Zμ1 to be the closure of 0 μ1 and there is a sequence {(un , λn )}∞ n=1 ⊂ C satisfying (un X , λn ) → (∞, μ0 ). Remark 3.9. The assumption (3.11) implies that (3.2) cannot have a trivial solution (u, λ) = (0, λ) in E and, therefore, C contains no sequence of pairs (uk , λk ) ˆ) in E for some μ ˆ ∈ R. Hence, the statement of Proposition 3.8 with (uk , λk ) → (0, μ follows directly from Proposition 3.5 using the transformation u → v = u/u2X . μ± ,ε as the component Let S be as in Lemma 3.6. For 0 < ε ≤ S we define D 1 of {(0, μ1 )} ∪ (S ∩ Eε ∩ Kη± ) containing (0, μ1 ), and Zμ±1 ,ε as the component of ± μ∓1 ,ε containing (0, μ1 ). Finally, we define Z± def Zμ1 \ D μ1 = 0 0 the convexity of Jλ is known to be lost, see Fleckinger et al. [18, Example 2, p. 148] for 1 0 almost everywhere in Ω, one may substitute v = up and investigate the functional

def (0) A x, ∇(v 1/p ) dx − λ B(x) v dx (4.2) Kλ (v) = p · Jλ (v 1/p ) = Ω

Ω

instead, which is defined on the set • def V + = v : Ω → (0, ∞) : v 1/p ∈ W01,p (Ω) . The second summand in (4.2) being linear in the variable v, it suffices to focus on the convexity of the first one,

• def A x, ∇(v 1/p ) dx , v ∈ V + . (4.3) K(v) = Ω

Notice that K is positively homogeneous, i.e., K(tv) = t · K(v)

for any number

t > 0.

4.1. Convexity on the cone of positive functions We would like to point out for future references that all results that are stated throughout this paragraph remain valid for any bounded domain Ω ⊂ RN ; the smoothness of its boundary ∂Ω required in hypothesis (Ω) is not necessary here. •

It is shown in Tak´ aˇc, Tello, and Ulm [37, Lemma 2.4, p. 79] that V + is a •

•

convex cone, i.e., v0 , v1 ∈ V + =⇒ α0 v0 + α1 v1 ∈ V + for all α0 , α1 ∈ (0, ∞), and •

•

the functional K : V + → R is ray-strictly convex , i.e., for all v0 , v1 ∈ V + and θ ∈ (0, 1) we have K (1 − θ)v0 + θv1 ≤ (1 − θ) K(v0 ) + θ K(v1 ) where equality may hold only if v0 and v1 are colinear, i.e., v1 = αv0 for some α ∈ (0, ∞). For the special case A(x, ξ) = |ξ|p , (x, ξ) ∈ Ω × RN , this lemma is due to D´ıaz and Saa [12]. It has had a number of important consequences in the past; some of them are surveyed in Takáˇc [36, Sect. 3]. However, several of these important

290



consequences can be obtained in a “parallel” way we adopt here, using supporting hyperplanes to the epigraph of K. In particular, also this method yields the simplicity of the first (smallest) eigenvalue μ1 for the Euler equation corresponding to the energy functional Jλ on W01,p (Ω); recall that μ1 is given by formula (2.8). We begin with the notion of the subdifferential ∂K(v0 ) of the functional •

•

K : V + → R which we define as follows: Given v0 ∈ V + , we introduce the weighted Sobolev space def −(p−1)/p ∈ W01,p (Ω) D(v0 ) = φ : Ω → R : φ v0 endowed with the natural norm def −(p−1)/p φD(v0 ) = φ v0

W01,p (Ω)

p 1/p −(p−1)/p = . ∇ φ v0 Ω

For φ ∈ D(v0 ) we set

def 1/p −(p−1)/p ∂K(v0 ), φ = a x, ∇(v0 ) , ∇ φ v0 dx .

(4.4)

Ω

Combining our hypotheses on a(x, ξ) = 1/p ∂ξ A(x, ξ) with the H¨ older inequality, we conclude that the last integral is absolutely convergent and ∂K(v0 ) is a bounded linear functional on D(v0 ). •

An easy calculation reveals that v ∈ D(v0 ) whenever v ∈ V + satisfies v/v0 ∈ L∞ (Ω). •

Remark 4.1. It is easy to see that v0 ∈ V + ∩ C 0 (Ω) implies ∂K(v0 ) ∈ D (Ω). More precisely, ∂K(v0 ) belongs to the dual space of the Fréchet space Cc1 (Ω) and thus to D (Ω). Employing formal integration by parts, we may write in D (Ω) −(p−1)/p 1/p · div a x, ∇(v0 ) . (4.5) ∂K(v0 ) = − v0 In addition, the expression ∂K(v0 ), φ gives the directional derivative of the functional K at v0 in direction φ ∈ Cc1 (Ω). The conclusion of the lemma below is very close to being equivalent to the •

claim that K : V + → R is ray-strictly convex. Its consequences are similar; see e.g., Picone’s identity for the p-Laplacian used in Allegretto and Huang [2, Theorem 2.1, p. 821]. •

Lemma 4.2. For any pair v0 , v ∈ V + with v ∈ D(v0 ) we have

K(v) ≥ ∂K(v0 ), v .

(4.6)

Equality holds if and only if the functions v0 and v are colinear, i.e., v = αv0 •

for some α ∈ (0, ∞). In particular, we may take any pair v0 , v ∈ V + satisfying v/v0 ∈ L∞ (Ω).

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Proof. The function A(x, · ) being strictly convex on RN , by the ellipticity condition (2.3) (hypothesis (A)), we observe that the inequality

A(x, ξ) − A(x, ξ0 ) > ∂ξ A(x, ξ0 ), ξ − ξ0 holds for all ξ, ξ0 ∈ RN with ξ = ξ0 . Using the identities a(x, ξ0 ) = 1/p ∂ξ A(x, ξ0 ) and A(x, ξ0 ) = a(x, ξ0 ), ξ0 , we can rewrite it equivalently as p−1 ξ0 A(x, ξ) > p a(x, ξ0 ), ξ − (4.7) for ξ = ξ0 . p Now let v, v0 ∈ (0, ∞) and ξ, ξ0 ∈ RN . Substituting the fractions ξ/v and ξ0 /v0 for ξ and ξ0 , respectively, from (4.7) we derive p − 1 ξ0 ξ ξ ξ0 A x, , − ≥ p a x, v v0 v p v0 where equality holds if and only if ξ/v = ξ0 /v0 . Finally, we apply the positive p-homogeneity hypothesis (2.2) to get ! ξ ξ0 p−1 v 1 A x, (p−1)/p ≥ (p−1)/p a x, (p−1)/p , ξ − ξ0 . (4.8) p v0 pv v0 p v0 Equality holds if and only if ξ/v = ξ0 /v0 . To conclude our proof, we use the identities ∇(v 1/p ) = p−1 v −(p−1)/p ∇v and v 1 p−1 v ∇ (p−1)/p = (p−1)/p ∇v − ∇v0 p v0 v0 v0 •

for v0 , v ∈ V + . If also v ∈ D(v0 ), we may take ξ = ∇v and ξ0 = ∇v0 a.e. in Ω, substitute them into inequality (4.8), and then integrate the result over Ω, thus arriving at

1/p −(p−1)/p 1/p A x, ∇(v ) dx ≥ a x, ∇(v0 ) , ∇ v v0 dx . (4.9) Ω

Ω

Equality holds if and only if v −1 ∇v = v0−1 ∇v0 a.e. in Ω. The latter equality is equivalent to v/v0 ≡ const in Ω. Clearly, (4.6) and (4.9) are the same. The lemma is proved. Remark 4.3. Lemma 4.2 implies immediately that the first eigenvalue μ1 given by the Rayleigh quotient (2.8) must be simple; cf. Tak´ aˇc, Tello, and Ulm [37, proof of Theorem 2.6, p. 81]. Indeed, since the Sobolev embedding W01,p (Ω) → Lp (Ω) is compact by Rellich’s theorem, the infimum in (2.8) is attained and satisfies 0 < μ1 < ∞. Now write u = u+ − u− where u+ = max{u, 0} and u− = max{−u, 0}, respectively, denote the positive and negative parts of a real-valued function u ∈ W01,p (Ω). We have u± ∈ W01,p (Ω), see Gilbarg and Trudinger [22, Theorem 7.8, p. 153]. More precisely, also ∇u+ = ∇u almost everywhere in Ω+ = {x ∈ Ω : u(x) > 0} and ∇u+ = 0 almost everywhere in Ω \ Ω+ . The corresponding result holds for u−

292



and −u as well. It follows from the proof of Theorem 2.6 (p. 81) in [37] that if a minimizer u ∈ W01,p (Ω) for μ1in (2.8) changes sign in the set {x ∈ Ω : B(x) > 0}, then also both functions u+ / Ω B(x)(u+ )p dx and u− / Ω B(x)(u− )p dx are minimizers for μ1 . We apply the strong maximum principle (due to Tolksdorf [38, Prop. 3.2.1 and 3.2.2, p. 801] and V´ azquez [40, Theorem 5, p. 200]) to conclude 1,p that any minimizer u ∈ W0 (Ω) for μ1 is either almost everywhere positive or else almost everywhere negative in Ω; we may assume u > 0 a.e. in Ω. Hence, K(up ) = μ1 . Denote by ϕ1 any such minimizer for μ1 (with ϕ1 > 0 a.e. in Ω). From (4.5) we get ∂K(ϕp1 ) = μ1 B( · ) in Ω. We conclude that equality must hold in K(up ) ≥ ∂K(ϕp1 ), up , which forces u/ϕ1 ≡ const in Ω, by Lemma 4.2. Furthermore, if the boundary ∂Ω is of class C 1,α for some α ∈ (0, 1), then ϕ1 ∈ C 1,β (Ω) for some β ∈ (0, α), by Proposition A.1. Finally, if Ω satisfies also an interior sphere condition at a point x0 ∈ ∂Ω, then (∂ϕ1 /∂ν)(x0 ) < 0, by the Hopf maximum principle (see [38, Prop. 3.2.1 and 3.2.2, p. 801] or [40, Theorem 5, p. 200]). We will need these facts throughout the rest of this work. Remark 4.4. Lemma 4.2 implies also the monotonicity of the subdifferential ∂K •

in the following sense: If u, v ∈ V + satisfy u/v, v/u ∈ L∞ (Ω), then one has

∂K(u) − ∂K(v), u − v ≥ 0 (4.10) where equality holds if and only if u and v are colinear. Indeed, using inequality (4.6) we get

∂K(u) − ∂K(v), u = K(u) − ∂K(v), u ≥ 0 ,

∂K(v) − ∂K(u), v = K(v) − ∂K(u), v ≥ 0 . We obtain (4.10) by adding these two inequalities. Notice that (4.10) is the well-known inequality of D´ıaz and Saa established in [12] for the special case A(x, ξ) = |ξ|p , (x, ξ) ∈ Ω × RN : Let u0 , u1 ∈ W01,p (Ω) be such that u0 > 0 and u1 > 0 in Ω and both u0 /u1 and u1 /u0 are in L∞ (Ω). Then we have

div a(x, ∇u1 ) div a(x, ∇u0 ) + (4.11) − (up0 − up1 ) dx ≥ 0 p−1 up−1 u Ω 0 1 where equality holds if and only if v1 /v0 ≡ const in Ω. 4.2. Bifurcations from zero for λ near μ1 Now, assuming hypothesis (H0 ), let us consider problem (2.1) again. By Lemma B.2 in Appendix B.2, for any λ ∈ R sufficiently close to μ1 , every weak solution u ∈ W01,p (Ω) of problem (2.1) with a sufficiently small norm uW 1,p (Ω) takes the 0 form u = t(ϕ1 + v ) where t ∈ R \ {0} and v ∈ C 1 (Ω) , (4.12) and v satisfies v , ϕ1 = 0 together with |v | ≤ 1/2ϕ1 in Ω. Moreover, one has the asymptotic formulas λ → μ1 and v C 1 (Ω) → 0 as t → 0 (t = 0). In particular, we work only with such solutions u ∈ C 1 (Ω) of problem (2.1) that are

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293

either positive or negative throughout Ω. More precisely, we have t−1 u ≥ 1/2ϕ1 in Ω. We need a more accurate estimate on the difference λ − μ1 as t → 0 which we derive from Lemma 4.2 next. Proposition 4.5. Let hypothesis (H0 ) be satisfied. Then, for any t ∈ R \ {0} with |t| small enough, we have −1 p−2 −1 p − |t| t B (ϕ1 + v ) dx h x, t(ϕ1 + v ); λ (ϕ1 + v ) dx ≤ λ − μ1

Ω

−1 ≤ − |t|p−2 t

Ω

−(p−1) v ϕ1 dx , 1+ ϕ1

h x, t(ϕ1 + v ); λ

Ω

(4.13)

where v /ϕ1 L∞ (Ω) → 0 as t → 0. Proof. We treat the case t > 0 only. The case t < 0 is analogous; one has to replace u and h(x, u; λ) by −u and −h(x, u; λ), respectively. The lower bound on λ − μ1 is obtained as follows. The Euler equation for the minimizers in formula (2.8) for μ1 combined with the normalization of ϕ1 yield

∂K(ϕp1 ), φ = μ1 B φ dx for every φ ∈ D(ϕ1 ) . (4.14) Ω

Multiplying equation (2.1) by u and integrating over Ω we get

K(up ) = λ B up dx + h(x, u; λ) u dx . Ω

(4.15)

Ω

From inequality (4.6) we obtain K(up ) ≥ ∂K(ϕp1 ), up . Combining this inequality with (4.14) and (4.15) we arrive at

λ B up dx + h(x, u; λ) u dx ≥ μ1 B up dx . Ω

Ω

Finally, using (4.12), we get

(λ − μ1 ) B (ϕ1 + v )p dx Ω

≥ −t

−(p−1)

Ω

h x, t(ϕ1 + v ); λ (ϕ1 + v ) dx (4.16) Ω

for t > 0 small enough. Now we estimate λ − μ1 from above. From equation (2.1) we deduce

p B φ dx + h(x, u; λ) u−(p−1) φ dx (4.17) ∂K(u ), φ = λ Ω

Ω

for every φ ∈ D(ϕ1 ). Formula (2.8) for μ1 yields

p K(ϕ1 ) = μ1 B ϕp1 dx = μ1 . Ω

(4.18)

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From inequality (4.6) we obtain K(ϕp1 ) ≥ ∂K(up ), ϕp1 . We combine this inequality with (4.17) and (4.18) to get

p (λ − μ1 ) B ϕ1 dx + h(x, u; λ) u−(p−1) ϕp1 dx ≤ 0 . Ω

Ω

Finally, using (4.12), we arrive at

B ϕp1 dx λ − μ1 = (λ − μ1 ) Ω

≤ − t−(p−1)

h x, t(ϕ1 + v ); λ Ω

−(p−1) v ϕ1 dx 1+ ϕ1

for t > 0 small enough. Hence, (4.13) is a combination of inequalities (4.16) and (4.19).

(4.19)

Our hypothesis (H0 ) on h(x, u; λ) implies the following asymptotic behavior of the integrals

h x, t(ϕ1 + φ); λ (ϕ1 + φ) dx (4.20) Ω

and

−(p−1) φ ϕ1 dx (4.21) 1+ ϕ1 Ω from (4.13), for |t| → 0, t = 0, and φ/ϕ1 L∞ (Ω) small enough (φ ∈ C 1 (Ω)). Given a number 0 < η ≤ 1/2, for each t ∈ R we define the expressions def h x, t(ϕ1 + φ); λ (ϕ1 + φ) dx ; (4.22) Θ(1) (t) = sup η

h x, t(ϕ1 + φ); λ

|φ|≤ηϕ1 |λ−μ1 |≤1

def Θ(2) η (t) =

sup |φ|≤ηϕ1 |λ−μ1 |≤1

Ω

−(p−1) φ 1+ ϕ1 dx , h x, t(ϕ1 + φ); λ Ω ϕ1

(4.23)

where both suprema are taken over all functions φ ∈ C 1 (Ω) that satisfy |φ| ≤ ηϕ1 in Ω. Clearly, we have 3 Θ(1) Θη (t) ; (4.24) η (t) ≤ (1 + η) Θη (t) ≤ 2 −(p−1) Θη (t) ≤ 2p−1 Θη (t) , (4.25) Θ(2) η (t) ≤ (1 − η) where we have denoted def

Θη (t) =

sup Ω

|s|≤η |λ−μ1 |≤1

h x, t(1 + s)ϕ1 ; λ · ϕ1 dx .

(4.26)

Lemma 4.6. Let hypothesis (H0 ) be satisfied. Then, given any 0 < η ≤ 1/2, we have Θη (t)/|t|p−1 → 0 as |t| → 0. In particular, also p−1 →0 Θ(1) η (t)/|t|

and

p−1 Θ(2) →0 η (t)/|t|

as

|t| → 0 .

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Proof. We estimate

Θη (t) ≤ Θ1/2 (t) =

sup Ω

where sup |s|≤1/2 |λ−μ1 |≤1

|s|≤1/2 |λ−μ1 |≤1

h x, t(1 + s)ϕ1 ; λ · ϕ1 dx

p−1 3 p−1 ϕ1 h x, t(1 + s)ϕ1 ; λ ≤ C|t| 2

295

(4.27)

for a.e. x ∈ Ω ,

by (2.9). It follows that |t|−(p−1) ·

sup |s|≤1/2 |λ−μ1 |≤1

p−1 3 ϕ1 h x, t(1 + s)ϕ1 ; λ ≤ C 2

for a.e. x ∈ Ω and 0 < |t| ≤ 1. Now we can apply the Lebesgue dominated convergence theorem to the integral in (4.27) to obtain Θη (t)/|t|p−1 → 0 as |t| → 0, by (2.10). The remaining two claims now follow from inequalities (4.24) and (4.25), respectively. Given any function u ∈ Lp (Ω), for every n = 1, 2, . . . we replace the reaction function h(x, u) by the expression " # def hn u( · ); λ (x) = h x, u(x); λ + Rn (|t|)(|t|) B(x) ϕ1 (x)p−1 , x ∈ Ω , (4.28) where t = ϕ1 −2 L2 (Ω) Ω u ϕ1 dx, and : R+ → (0, +∞) and Rn : R+ → R+ are continuous functions with the following properties: (i) For every 0 < r ≤ 1 we have max{1, p−1} 1+η (r) > 2 1−η

sup 0 0 (4.30) 2 Ω

296



and

hn x, t(ϕ1 + φ); λ

Ω

−(p−1) φ ϕ1 dx 1+ ϕ1 1 ≥ (1 + η)−(p−1) |t|p−1 (|t|) > 0 . (4.31) 2

Proof. Recall that 0 < |t| ≤ 1/(2n) implies Rn (|t|) = |t|p−1 . We combine (4.24) with (4.29) to get

hn x, t(ϕ1 + φ); λ (ϕ1 + φ) dx Ω

≥ −Θ(1) (t) + R (|t|) (|t|) B ϕp−1 (ϕ1 + φ) dx n η 1 Ω

≥ − (1 + η) Θη (t) + (1 − η) |t|p−1 (|t|) ≥

1 (1 − η) |t|p−1 (|t|) > 0 . 2

Similarly, a combination of (4.25) with (4.29) yields −(p−1)

φ hn x, t(ϕ1 + φ); λ ϕ1 dx 1+ ϕ1 Ω −(p−1)

φ p (2) B ϕ1 1 + dx ≥ −Θη (t) + Rn (|t|) (|t|) ϕ1 Ω ≥ − (1 − η)−(p−1) Θη (t) + (1 + η)−(p−1) |t|p−1 (|t|) 1 ≥ (1 + η)−(p−1) |t|p−1 (|t|) > 0 . 2

The lemma is proved.

We finish this section by inserting the integrals from Lemma 4.7 into Proposition 4.5. Thus, the boundary value problem they relate to reads − div a(x, ∇u) = λ B(x) |u|p−2 u + hn (u; λ) (x) in Ω ; (4.32) u = 0 on ∂Ω , where [hn (u; λ)](x) has been defined in (4.28). In analogy with (4.12), we decompose a solution of (4.32) with uL∞ (Ω) sufficiently small as u = t(ϕ1 + v ) where

t ∈ R \ {0} and v ∈ C 1 (Ω) ,

(4.33)

and v satisfies v , ϕ1 = 0 together with |v | ≤ 1/2ϕ1 in Ω. We insert estimates (4.30) and (4.31) into (4.13) in order to obtain the following asymptotic formulas for λ − μ1 . Again, we work only with such solutions u ∈ C 1 (Ω) of problem (4.32) that are either positive or negative throughout Ω.

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Corollary 4.8. Let hypothesis (H0 ) be satisfied. Then, for any t ∈ R \ {0} with |t| small enough, we have 1 (1 + η)−p (1 − η) (−t) > 0 if t < 0 ; 2 1 λ − μ1 ≤ − (1 + η)−(p−1) (t) < 0 if t > 0 2 λ − μ1 ≥

(4.34) (4.35)

together with vn /ϕ1 L∞ (Ω) → 0 as t → 0. In the following lemma, Θ(i) with i = 1, 2 are given by

def h x, t(ϕ1 + v ) (ϕ1 + v ) dx Θ(1) (t) =

(4.36)

Ω

and Θ

(2)

def

h x, t(ϕ1 + v )

(t) =

Ω

−(p−1) v ϕ1 dx , 1+ ϕ1

(4.37)

respectively. Lemma 4.9. Let hypotheses (H0 ) and (H0 ) be satisfied. Then we have

Θ(i) (t) → f0± ϕ1 dx as t → 0± , g0 (t) Ω

(4.38)

and, in particular, Θ(i) (t) →0 |t|p−1

as

|t| → 0 ,

|t| > 0 .

Proof. For t ∈ R \ {0} we compute

h x, t(ϕ1 + v ) Θ(1) (t) = (ϕ1 + v ) dx g0 (t) g0 (t) Ω

v t 1 + g h x, t(ϕ1 + v ) 0 ϕ 1 dx = · (ϕ1 + v ) · v g0 (t) Ω g0 t 1 + ϕ 1

→ f0± ϕ1 dx

(4.39)

(4.40)

Ω

by the Lebesgue dominated convergence theorem which makes use of (2.16) and (2.17) combined with Remark 2.4 and v /ϕ1 L∞ (Ω) → 0 as |t| → 0. The case of Θ(2) is analogous. 4.3. Bifurcations from infinity for λ near μ1 Given a Banach space X (X = W01,p (Ω) in our case), we apply a standard method to transform bifurcations from infinity to bifurcations from zero using the transfordef mation · : u → u = u−2 X u which maps bijectively {u ∈ X : 1/r < uX < ∞} onto { u ∈ X : 0 < uX < r}, for 0 < r < ∞ small enough.

298



More precisely, in our case we take r > 0 small enough and thus obtain u = τ (ϕ1 + v ) with |τ | → ∞ and v C 1 (Ω) → 0 as r → 0. This yields u = t(ϕ1 + v ) with −1 ϕ1 −2 as r → 0 . t = τ u2X = τ −1 ϕ1 + v −2 X =τ X + o(r) This means that it suffices to replace the scalar τ ∈ R\{0} in u = τ (ϕ1 +v ), |τ | → ∞ as r → 0, by t ∈ R \ {0} from u = t(ϕ1 + v ), t → 0 as r → 0. Consequently, we can easily reformulate all our results from the previous section (Section 4.2) as follows. Proposition 4.10. Let hypothesis (H∞ ) be satisfied. Then, for any τ ∈ R \ {0} with |τ | large enough, we have −1 p−2 p −|τ | τ B (ϕ1 + v ) dx h x, τ (ϕ1 + v ); λ (ϕ1 + v ) dx Ω

≤ λ − μ1

Ω

(4.41)

−(p−1) v p−2 h x, τ (ϕ1 + v ); λ 1+ ϕ1 dx , ≤ −|τ | τ ϕ1 Ω

where v /ϕ1 L∞ (Ω) → 0 as |τ | → ∞. Our hypothesis (H∞ ) on h(x, u; λ) implies the following asymptotic behavior of the integrals (4.20) and (4.21) from (4.41), for |τ | large enough (τ ∈ R) and φ/ϕ1 L∞ (Ω) small enough (φ ∈ C 1 (Ω)). Lemma 4.11. Let hypothesis (H∞ ) be satisfied. Then, given any 0 < η ≤ 1/2, we have Θη (τ )/|τ |p−1 → 0 as |τ | → ∞. In particular, also p−1 Θ(1) →0 η (τ )/|τ |

and

p−1 Θ(2) →0 η (τ )/|τ |

as

|τ | → ∞ .

Given any function u ∈ Lp (Ω), for every n = 1, 2, . . . next we replace the reaction function h(x, u; λ) by the expression " # def hn u( · ); λ (x) = h x, u(x); λ

+ Rn (|τ |) (|τ |) B(x) ϕ1 (x)p−1 ,

x ∈ Ω,

(4.42)

where τ = ϕ1 −2 L2 (Ω) Ω u ϕ1 dx, and : R+ → (0, ∞) and Rn : R+ → R+ are continuous functions with the following properties: (i) For every 1 ≤ r < ∞ we have max{1, p−1} 1+η Θη (τ ) (r) > 2 sup p−1 1−η |τ |≥r |τ |

(4.43)

and (r) → 0 as r → +∞. (ii) We require Rn (r) = 0 if 0 ≤ r ≤ n, Rn (r) is monotone increasing for n ≤ r ≤ 2n, and Rn (r) = rp−1 if 2n ≤ r < ∞.

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The following lemma guarantees that the integrals in (4.41), with [hn (u; λ)](x) in place of h(x, u; λ), are positive. Lemma 4.12. Assume that hypothesis (H∞ ) is satisfied. Let 0 < η ≤ 1/2 and n ∈ N be arbitrary. Then for every u ∈ C 1 (Ω) such that u = τ (ϕ1 + φ), where |τ | ≥ 2n, Ω φ ϕ1 dx = 0, and φ/ϕ1 L∞ (Ω) ≤ η, we have

1 hn x, τ (ϕ1 + φ); λ (ϕ1 + φ) dx ≥ (1 − η) |τ |p−1 |τ | > 0 (4.44) 2 Ω and

hn x, τ (ϕ1 + φ); λ

1+

Ω

≥

φ ϕ1

−(p−1) ϕ1 dx

1 (1 + η)−(p−1) |τ |p−1 (|τ |) > 0 . 2

(4.45)

We finish this section by inserting the integrals from Lemma 4.12 into Proposition 4.10. Thus, the boundary value problem they relate to reads − div a(x, ∇u) = λ B(x) |u|p−2 u + hn (u; λ) (x) in Ω ; (4.46) u = 0 on ∂Ω , where [hn (u; λ)](x) has been defined in (4.42). In analogy with (4.12), we decompose a solution of (4.46) with uL∞ (Ω) sufficiently large as u = τ (ϕ1 + v ) where

τ ∈ R \ {0} and v ∈ C 1 (Ω) ,

(4.47)

and v satisfies v , ϕ1 = 0 together with |v | ≤ 1/2ϕ1 in Ω. We insert estimates (4.44) and (4.45) into (4.41) in order to obtain the following asymptotic formulas for λ − μ1 . Again, we work only with such solutions u ∈ C 1 (Ω) of problem (4.46) that are either positive or negative throughout Ω. Corollary 4.13. Let hypothesis (H∞ ) be satisfied. Then, for any τ ∈ R \ {0} with |τ | large enough, we have 1 (1 + η)−p (1 − η) (−τ ) > 0 if τ < 0 ; 2 1 λ − μ1 ≤ − (1 + η)−(p−1) (τ ) < 0 if τ > 0 , 2 λ − μ1 ≥

(4.48) (4.49)

together with v /ϕ1 L∞ (Ω) → 0 as τ → ∞. In the following lemma, Θ(i) with i = 1, 2 are given by (4.36) and (4.37), respectively. Lemma 4.14. Let hypotheses (H∞ ) and (H∞ ) be satisfied. Then we have

Θ(i) (τ ) → f±∞ ϕ1 dx as τ → ±∞ , g∞ (τ ) Ω

(4.50)

300



and, in particular, Θ(i) (τ ) →0 |τ |p−1

as

|τ | → ∞ .

Proof. For τ ∈ R \ {0} we compute

h x, τ (ϕ1 + v ) Θ(1) (τ ) = (ϕ1 + v ) dx g∞ (τ ) g∞ (τ ) Ω

g∞ τ 1 + vϕ1 h x, τ (ϕ1 + v ) dx = · (ϕ1 + v ) · v g∞ (τ ) Ω g∞ τ 1 + ϕ 1

→ f± ϕ1 dx

(4.51)

(4.52)

Ω

by the Lebesgue dominated convergence theorem which makes use of (2.19) and (2.20) combined with Remark 2.5 and v /ϕ1 L∞ (Ω) → 0 as |τ | → ∞.

5. Global bifurcation results This section is divided into three paragraphs. The first one is devoted to preliminary results concerning Browder–Petryshyn and Skrypnik degree for perturbations of monotone operators. The definition and basic properties thereof can be found, e.g., in [42, Chapter 36, pp. 1002–1007]. Then we present proofs of global bifurcation results of Rabinowitz type, i.e., Propositions 3.5 and 3.8 in the second paragraph. Our main results, Theorems 3.7 and 3.10, which are global bifurcation results of Dancer’s type, are proved in the third paragraph. 5.1. The Browder–Petryshyn and Skrypnik degree Definition 5.1. Let us consider an operator T : X → X where X is a real separable reflexive Banach space. The operator T is said to satisfy condition α(X) if for an arbitrary sequence {un }∞ n=1 ⊂ X the relations

(5.1) un u0 weakly in X and lim sup T (un ), un − u0 X ≤ 0 n→∞

imply un → u0 strongly in X. def

It is proved in [14, Chapter 5, p. 188] that in the special case a(x, v) = def

|v|(p−2) v and B(x) = 1 for all x ∈ Ω and all v ∈ RN the operator T = A − λB satisfies condition α(X) from [33] (which is nothing else but condition (S+ ) from [7]) and so its (Browder–Petryshyn) degree can be defined. The following well-known inequalities, (Ai ) A(u) − A(v), u − vX ≥ γ(uX + vX )p−2 u − v2X for all 1 0; (Aii ) A(u) − A(v), u − vX ≥ γu − vpX for all 2 0,

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cf. [8, Thm. 3; p. 736], play the crucial role in the verification of condition α(X) for T = A. We prove only (Ai ); the proof of (Aii ) is analogous and well-known. So let 1 < p < 2. From (2.3) we deduce

N 1 ∂ai x, u (5.2) a(x, u) − a x, v · (u − v) dx = 0 ∂ξj Ω i,j=1 + s(v − u) ds (ui − vi )(uj − vj ) dx

1

≥γ Ω

≥γ Ω

u + s(v − u) p−2 ds |u − v|2 dx

0

max u + s(v − u)

p−2 |u − v|2 dx .

0≤s≤1

Next, we use the classical Hölder inequality with the exponents 2/p and older inequality that p/2 + 2−p 2 = 1) to get the “reversed” H¨ 2/p (p−2)/p

f (x)g(x) dx ≥ f (x)p/2 dx g(x)p/(p−2) dx Ω

Ω

2 2−p

(note

(5.3)

Ω

for any measurable functions f, g : Ω → R, f ≥ 0 and g > 0 a.e. in Ω, such that f g ∈ L1 (Ω) and 1/g ∈ Lp/(2−p) (Ω); hence, f p/2 ∈ L1 (Ω). Now take any vector-valued functions u, v ∈ [Lp (Ω)]N , uLp (Ω) + vLp (Ω) > 0. Inserting p−2 def def 2 f = |u − v| and g = max u + s(v − u) 0≤s≤1

into (5.3) we arrive at 2/p

p−2 2 p max u |u − v| ≥ |u − v| dx max u + s(v − v) 0≤s≤1

Ω

Ω

p (p−2)/p + s(v − u) ≥ u − v 2Lp (Ω)

Ω

0≤s≤1

p |u| + |v|

(5.4) (p−2)/p

Ω

p−2 = u − v2Lp (Ω) |u| + |v|Lp (Ω) p−2 ≥ u − v2Lp (Ω) uLp(Ω) +vLp (Ω) . Combining inequalities (5.2) and (5.4) we conclude that

p−2 a(x, u) − a(x, v) (u − v) dx ≥ γ uLp(Ω) + vLp (Ω) u − v2Lp (Ω) Ω

which proves inequality (Ai ). def

By means of (Ai ) and (Aii ), condition α(X) for T = A can be verified.

302



Proposition 5.2. The operator A : X → X defined by identity (3.3) satisfies condition α(X). Proof. Assume that A and {un }∞ n=1 ⊂ X satisfy (5.1), i.e., un u0 weakly in X and lim supn→∞ (A(un ), un − u0 )X ≤ 0. We combine the last inequality with limn→∞ A(u0 ), un − u0 X = 0 to obtain

lim sup A(un ) − A(u0 ), un − u0 X ≤ 0 . (5.5) n→∞

Now we distinguish between the cases 1 < p < 2 and p ≥ 2. Let 1 < p < 2. From (Ai ) we find p−2

A(un ) − A(u0 ), un − u0 X ≥ γ un X + u0 X un − u0 2X

(5.6)

for n = 1, 2, 3 . . . . Since every weakly convergent sequence in X is also bounded, we have (un X +u0 X )p−2 ≥ C > 0 for some constant C. We apply (5.5) to (5.6) to get un − u0 X → 0 as n → ∞ which verifies condition α(X). For p ≥ 2 we use inequality (Aii ) in place of (Ai ) to get the same conclusion. A standard method for proving a discontinuity in the Browder–Petryshyn degree (cf. [14, Chapter 5, Thm. 14.18, p. 189] for the p-Laplacian) is based on the variational structure of the p-homogeneous part A−λB of the operator A−λB−H. For the sake of completeness we begin by proving the following result. Proposition 5.3. For all r > 0 and all 0 < δ < μ2 − μ1 we have Deg A − (μ1 ± δ)B; Br (0), 0 = ∓1 .

(5.7)

Proof. Given R > 0 fixed, we define ψ : R+ → R as follows: ψ(t) = 0 for 0 ≤ t ≤ R, ψ(t) = δ/R (t − R)2 for R < t < 2R, and ψ(t) = 2δ(t − 2R) + δR for 2R ≤ t < ∞. Clearly, ψ is continuously differentiable, monotone increasing, and convex on R+ , with 0 ≤ ψ (t) ≤ 2δ for every t ∈ R+ . Now consider the functional Fλ : X → R defined by 1

λ

def 1

A(u), u X − B(u), u X + ψ B(u), u X , u ∈ X . Fλ (u) = p p p Every critical point u0 ∈ X of Fλ is a solution of the operator equation $ % 1

Fλ (u) = A(u) − λ − ψ B(u) = 0 in X . B(u), u X p

(5.8)

Of course, u = 0 ∈ X is a solution. Assuming −∞ < λ ≤ μ1 + δ (< μ2 ) we have 1

B(u), u X ≤ λ ≤ μ1 + δ (< μ2 ) . (λ − 2δ ≤) λ − ψ p Therefore, if u0 ∈ X \ {0} is a nonzero solution of (5.8), we must have 1

B(u0 ), u0 X = μ1 λ−ψ p

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and u0 = αϕ1 for some constant where α satisfies λ−ψ (|α|p /p) = μ1 . α ∈ R\{0}, p (Recall that B(ϕ1 ), ϕ1 X = Ω B(x)ϕ1 dx = 1.) Since ψ (t) = 0 for 0 ≤ t ≤ R, ψ (t) = 2δ R (t − R) for R ≤ t ≤ 2R, and ψ (t) = 2δ for 2R ≤ t < ∞, the last equation for α ∈ R possesses a nonzero solution α = ±αλ (αλ > 0) if and only if λ ∈ [μ1 , μ1 + 2δ]. If λ = μ1 , we may take any α ∈ R with 0 < |α|p /p ≤ R. If μ1 < λ ≤ μ1 + δ, we determine α ∈ R from p 2δ |α|p − R = λ − μ1 ∈ (0, δ] ; ψ |α| /p = (5.9) R p p

hence, R < |α|p ≤ 3/2R (< 2R). Let μ1 − δ ≤ λ < μ1 . The only critical point of Fλ is the zero function u = 0 ∈ X; it is the global minimizer for Fλ . We apply Skrypnik [33, Thm. 1.5.1, p. 42] to conclude that (5.10) Deg Fλ ; Br (0), 0 = 1 for all r > 0 . More precisely, this claim is proved in [33, Thm. 1.5.1, p. 42] for r > 0 small enough only. Arbitrary r > 0 is then allowed by the fact that u = 0 ∈ X is the only critical point of Fλ . Now let us consider the case μ1 < λ ≤ μ1 + δ. The functional Fλ is coercive on X, owing to the following inequalities which hold for all u ∈ X such that 1/pB(u), uX ≥ 2R: 1

1

λ

Fλ (u) = A(u), u X − B(u), u X + 2δ B(u), u X − R p p p 1

λ − 2δ

= A(u), u X − B(u), u X − 2δR p p 1

μ1 − δ

B(u), u X − 2δ ≥ A(u), u X − p p μ1 − δ

1 1− ≥ A(u), u X − 2δR p μ1 1

=δ A(u), u X − 2R −→ +∞ pμ1 as uX → ∞. Recall that αλ ∈ (0, ∞) is uniquely determined by equation (5.9) αp and satisfies R < pλ ≤ 3/2R (< 2R). It is easy to see that Fλ (±αl ϕ1 ) < 0 = Fλ (0). Since Fλ has no other critical points than 0 ∈ X and ±αλ ϕ1 , both ±αλ ϕ1 must be the global minimizers for Fλ . By [33] again, we find (5.11) Deg Fλ ; B (±αλ ϕ1 ), 0 = 1 for every > 0 small enough. We assume also < 1/2αμ1 +δ ϕ1 X . Set rδ = 2αμ1 +δ ϕ1 X ; hence, μ1 < λ ≤ μ1 + δ implies αλ ≤ αμ1 +δ and therefore rδ > αλ ϕ1 X + . Thus, by an argument with a homotopy connecting Fμ 1 −δ with Fμ 1 +δ we get Deg Fμ 1 +δ ; Br (0), 0 = Deg Fμ 1 −δ ; Br (0), 0 = 1 for every r > rδ . (5.12)

304



Now, since u = 0 ∈ X is an isolated solution to (5.8), the degree Deg[Fμ 1 +δ ; B (0), 0] is well defined for every > 0 small enough; we assume < 1/2αμ1 +δ ϕ1 X . From the additivity property of the degree we deduce Deg Fμ 1 +δ ; B (αμ1 +δ ϕ1 ), 0 + Deg Fμ 1 +δ ; B (−αμ1 +δ ϕ1 ), 0 + Deg Fμ 1 +δ ; B (0), 0 = Deg Fμ 1 +δ ; Br (0), 0 = Deg Fμ 1 −δ ; Br (0), 0 = 1 for every r > rδ . Finally, we apply (5.11) to get Deg[Fμ 1 +δ ; B (0), 0] = −1. Since $ % 1

A(u) − (μ1 + δ)B(u) = A(u) − (μ1 + δ) + ψ B(u), u B(u) = Fμ 1 +δ p holds whenever B(u), uX < R, we find Deg A − (μ1 + δ)B; B (0), 0 = Deg Fμ 1 +δ ; B (0), 0 = −1 provided > 0 from above is taken so small that also B(u), uX ≤ μ11 A(u), uX < R holds for every u ∈ B (0) ⊂ X. Since u = 0 ∈ X is the only solution to the operator equation A(u) − (μ1 + δ)B(u) = 0 in X , we arrive at Deg A − (μ1 + δ)B; Br (0), 0 = −1 for every r > 0 . By analogous arguments, we infer from (5.10) that Deg A − (μ1 − δ)B; Br (0), 0 = 1 for every r > 0 .

The proof is now complete.

5.2. Proof of a Rabinowitz-type bifurcation theorem As in the semilinear case, in order to prove Theorem 3.7, we begin with the proof of Proposition 3.5 (which is a Rabinowitz-type bifurcation theorem). Proof of Proposition 3.5. We have 0 < μ1 < μ2 by Remark 2.1. We begin the proof by showing that for every 0 < δ < μ2 − μ1 there exists R > 0 such that Deg Φμ1 ±δ ; Br (0), 0 = ∓1 whenever 0 < r < R . (5.13) By Proposition 3.2, for each λ ∈ (−∞, μ2 )\ {μ1 }, u = 0 ∈ X is an isolated solution of Φλ (u) = 0. Thus one can find R > 0 small enough, such that Deg[Φμ1 ±δ ; Br (0), 0] remains constant with respect to r ∈ (0, R). We will show later that there exists R ∈ (0, R) such that A(u) − (μ1 ± δ)B(u) − αH(u; μ1 ± δ) = 0 . holds for all u ∈ ∂BR (0) and α ∈ [0, 1]. Therefore, the homotopy A(u) − (μ1 ± δ)B(u) − αH(u; μ1 ± δ) ,

α ∈ [0, 1] ,

(5.14)

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connecting Φμ1 ±δ with A − (μ1 ± δ)B, is admissible with respect to BR (0) and 0. Consequently, we have Deg Φμ1 ±δ ; Br (0), 0 = Deg Φμ1 ±δ ; BR (0), 0 = Deg A − (μ1 ± δ)B; BR (0), 0 = ∓1 , by Lemma 5.3. Using Deg[Φμ1 ±δ ; Br (0), 0] = ∓1 we can proceed step by step as in the original proof of Rabinowitz [31, Theorem 1.3, pp. 490–491], cf. also Dr´ abek [14, Theorem 14.9, pp. 178–183]. It remains to prove (5.14). Suppose the contrary, i.e., for all R ∈ (0, R) there exist u ∈ ∂BR (0) and α ∈ [0, 1] such that A(u) − (μ1 ± δ)B(u) − αH(u; μ1 ± δ) = 0 .

(5.15)

∞ sequence {rn }∞ n=1 ⊂ (0, R), rn → 0, together with {un }n=1 ⊂ ∞ and {αn }n=1 ⊂ [0, 1], such that (5.15) holds with un and

Thus, we can find a X, un ∈ ∂Brn (0), αn in place of u and α, respectively. Since un ∈ ∂Brn (0), we have un X → 0. Observe that the functions αn h( · , · ; · ) : Ω × R × R → R satisfy (Hn0 ) because h : Ω × R × R → R satisfies (H0 ) and αn ∈ [0, 1]. We apply Lemma B.2 with λn = μ1 ± δ (< μ2 ) to equation (5.15) to conclude that λn → μ1 as n → ∞, which is absurd (δ > 0). This concludes the proof.

Now we continue by giving the proof of Lemma 3.6 which is another ingredient in the proof of Theorem 3.7. Proof of Lemma 3.6. Suppose that for some η ∈ (0, η0 ) such a number 0 < S ≤ 1 does not exist. Then we can find a decreasing sequence 0 < Sn ≤ 1 with Sn 0 and another sequence (un , λn ) ∈ (S \ {(0, μ1 )}) ∩ E Sn (μ1 ) such that |(un )| ≤ η un W 1,p (Ω) for each n = 1, 2, . . . . Notice that owing to (un , λn ) = (0, μ1 ) we 0 must have un ≡ 0 in Ω for all n ≥ 1 large enough, because μ1 is an isolated eigenvalue of the (p − 1)-homogeneous operator A, as shown in Anane [3, Théorème 2, p. 727] or Anane and Tsouli [5, Prop. 2, p. 5]. Discarding a finite number of members of this sequence if necessary, we may assume un ≡ 0 in Ω for all n ≥ 1. Since (un , λn ) ∈ S and Sn 0 as n → ∞, we have also un L∞ (Ω) → 0 and un C 1,β (Ω) → 0, by Lemma B.2 (Appendix B). This lemma shows also that the def

normalized sequence wn = un /unL∞ (Ω) , with wn L∞ (Ω) = 1, is the union of ∞ two disjoint subsequences {wn }∞ n=1 and {wn }n=1 , one of them possibly empty, such that, if nonempty, they satisfy wn → ϕ1 /ϕ1 L∞ (Ω) and/or wn → −ϕ1 /ϕ1 L∞ (Ω) in C 1,β (Ω) as n → ∞. Consequently, we get (wn ) → ϕ1 /ϕ1 L∞ (Ω) = ϕ1 −1∞ L (Ω) as n → ∞ . Furthermore, by our assumption we have (wn ) =

un W 1,p (Ω) |(un )| 0 ≤η un L∞ (Ω) un L∞ (Ω)

for each n = 1, 2, . . . .

306



Combining the last two facts we arrive at ϕ1 −1 L∞ (Ω) ≤ η · lim inf n→∞

un W 1,p (Ω) 0

un L∞ (Ω)

.

(5.16)

To find an upper bound for the fraction above, we first take (u, λ) = (un , λn ) and φ = un in equation (2.1) which yields

p A(x, ∇un ) dx = λn B(x) |un | dx + h x, un (x); λn un (x) dx . Ω

Ω

Ω

Now we apply inequalities (2.6), λn ≤ μ1 + Sn ≤ μ1 + 1, and (2.9) to get

γ |∇un |p dx ≤ A(x, ∇un ) dx p−1 Ω Ω

p ≤ (μ1 + 1) B(x) |un | dx + C |un |p dx Ω Ω

≤ (μ1 + 1)BL∞ (Ω) + C |un |p dx Ω ≤ (μ1 + 1)BL∞ (Ω) + C |Ω|N un pL∞ (Ω) . Consequently, we have un W 1,p (Ω) ≤ c0 un L∞ (Ω) where 0

1/p (p − 1) (μ1 + 1)BL∞ (Ω) + C |Ω|N def c0 = > 0. γ Combining the last inequality with (5.16) we obtain ϕ1 −1 L∞ (Ω) ≤ η c0 or, equivalently, η ≥ η0 with η0 > 0 defined by (3.10). But this contradicts our choice of η ∈ (0, η0 ). The lemma is proved. Let us consider the sequence of boundary value problems − div a(x, ∇u) = λ B(x) |u|p−2 u + hn (x, u; λ) in Ω ; u=0

on ∂Ω ,

(5.17)

where the sequence of functions hn : Ω × R × R → R satisfies condition (Hn0 ). With

Hn (u; λ), φ X = hn (x, u; λ) ϕ dx satisfied for all u, φ ∈ X , Ω

the operator formulation of (5.17) reads as follows, A(u) − λB(u) − Hn (u; λ) = 0

in X .

For each n ∈ N, we also define E def . (u, λ) ∈ E : A(u) − λB(u) − Hn (u; λ) = 0, u = 0 Sn = The following lemma is a version of Lemma 3.6 for (5.17) which is uniform in n ∈ N.

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Lemma 5.4. Let hn : Ω × R × R → R satisfy condition (Hn0 ). For every η ∈ (0, η0 ) there exists a number S, 0 < S ≤ 1, such that Sn \ (0, μ1 ) ∩ E S (μ1 ) ⊂ Kη for all n ∈ N . Moreover, if (u, λ) ∈ (Sn \ {(0, μ1 )}) ∩ E S (μ1 ) then u = τ (ϕ1 + v ), where τ = (u) ∈ R and v ∈ C 1,β (Ω) (0 < β < β) satisfy |τ | > η uW 1,p (Ω) and (v ) = 0 0 together with |λ − μ1 | → 0 and v C 1,β (Ω) → 0 as τ → 0. The proof of Lemma 5.4 is almost identical with the proof of Lemma 3.6. The ∞ only difference is that we consider (um , λm ) ∈ Snm (here {Snm }∞ m=1 ⊂ {Sn }n=1 ). Then we continue literally as in the proof of Lemma 3.6 because all estimates are uniform with respect to n ∈ N due to (Hn0 ). 5.3. Proof of a Dancer-type bifurcation theorem As in the semilinear case in Dancer [10, Theorem 2, p. 1071], our proof of Theorem 3.7 is based on the following three lemmas. Lemma 5.5. Suppose δ1 , δ2 > 0 are such that 0 < δ1 + δ2 < S and Φλ (u) = 0 if uX = δ1 and |λ − μ1 | ≤ δ2 . If 0 < σ < δ2 and β > 0 is sufficiently small, β ≡ β(σ), then uX < β together with Φμ1 ±σ (u) = 0 imply u = 0 and, moreover, Deg Φμ1 +σ ; W ν , 0 − Deg Φμ1 −σ ; W ν , 0 = 1 where Wν =

def

u ∈ X : (u, λ) ∈ Kην and β < uX < δ1 ,

ν = ±.

−1/p

Recall that η is arbitrary with 0 < η < μ1 , and Kην ⊂ X × R has been ν defined in (3.9). In the definition W ⊂ X above we can use any λ ∈ R. Proof. We follow Dancer [10, Proof of Lemma 1, p. 1071]. We define ⎧ H(u; λ) if Ω uϕ1 dx ≤ −ηuX ; ⎪ ⎪ ⎨ def uϕ1 dx Ω H− (u; λ) = − ηuX < Ω uϕ1 dx ≤ 0 ; η u X H(u; λ) if ⎪ ⎪ ⎩ − H(−u; λ) if Ω uϕ1 dx > 0 , and

− (5.18) Φ− λ (u) = A(u) − λB(u) + H (u; λ) . − The mapping Φλ : X → X is odd. Since the (p − 1)-homogenous part of Φ− λ is the same as that of Φλ , also Φ− satisfies condition α(X). λ By our hypothesis, the equation Φμ1 +σ (u) = 0 has no solution on ∂Eδ1 , ∂Eβ , or in Eδ1 \ W + ∪ W − ∪ Eβ , see Lemma 3.6. It follows that − − − Deg Φ− μ1 +σ ; Eδ1 , 0 = Deg Φμ1 +σ ; Eβ , 0 + Deg Φμ1 +σ ; W , 0 + + Deg Φ− μ1 +σ ; W , 0 def

308



which is

− − − + Deg Φ− μ1 +σ ; W , 0 + Deg Φμ1 +σ ; W , 0 = Deg Φμ1 +σ , Eδ1 , 0 − Deg Φ− μ1 +σ , Eβ , 0 .

By the oddness of Φ− μ1 +σ : X → X and the definition of the Browder–Petryshyn degree, we find that − + − Deg Φ− μ1 +σ , W , 0 = Deg Φμ1 +σ , W , 0

and so

− − − 2 · Deg Φ− μ1 +σ , W , 0 = Deg Φμ1 +σ , Eδ1 , 0 − Deg Φμ1 +σ , Eβ , 0 .

(5.19)

Analogously, − − − 2 · Deg Φ− μ1 −σ , W , 0 = Deg Φμ1 −σ , Eδ1 , 0 − Deg Φμ1 −σ , Eβ , 0 .

(5.20)

As in the proof of Proposition 3.5 (in Section 5.2) one can show that − (5.21) Deg Φ− μ1 −σ ; Eβ , 0 = −1 and Deg Φμ1 +σ ; Eβ , 0 = 1 . Subtracting (5.20) from (5.19) and using (5.21), we arrive at − − − ; W , 0 − Deg Φ ; W , 0 = Deg Φ− 2 · Deg Φ− μ1 +σ μ1 −σ μ1 −σ ; Eβ , 0 − Deg Φ− μ1 +σ ; Eβ , 0 = 1 − (−1) = 2 . Φ− μ1 −σ (u)

−

Since = Φμ1 −σ (u) for all u ∈ W and all λ ∈ R, due to the definition of − Φλ , we must have − Deg Φμ1 +σ ; W − , 0 − Deg Φμ1 −σ ; W − , 0 = Deg Φ− μ1 +σ ; W , 0 − − Deg Φ− μ1 −σ ; W , 0 = 1 . If uX = δ1 and |λ − μ1 | ≤ δ2 , we have Φλ (u) = 0 by our assumptions. Consequently, for σ ∈ (0, δ2 ) the homotopy Φ− λ (where μ1 −σ ≤ λ ≤ μ1 +σ) is admissible on Eδ1 whence − − Deg Φ− μ1 −σ ; Eδ1 , 0 = Deg Φλ ; Eδ1 , 0 = Deg Φμ1 +σ ; Eδ1 , 0 holds for all λ ∈ [μ1 − σ, μ1 + σ].

Remark 5.6. It is worthwhile to note that in proving a Dancer-type bifurcation result for an elliptic boundary value problem we have to consider an abstract functional differential equation; observe that the definition of H− contains uX and Ω uϕ1 dx. For 0 < ε < S we define Tμ−1 ,ε to be the component of Zμ1 \ (E ε (μ1 ) ∩ Kη+ ) containing (0, μ1 ). Lemma 5.7. If 0 < ε < S, zero is an isolated solution of Φμ1 (u) = 0, and Tμ−1 ,ε is bounded in E, then ∂E ε (μ1 ) ∩ Kη+ ∩ Tμ−1 ,ε = ∅ .

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The proof of Lemma 5.7 would almost literally copy that of its semilinear counterpart in Dancer [10, Proof of Lemma 2, p. 1072], and therefore is omitted. In Lemmas 5.5 and 5.7 we have assumed that zero is an isolated solution of Φμ1 (u) = 0. This is the case when, e.g., the function h(x, u; λ) satisfies not only (H0 ) but also (H0 ) and, moreover, Ω f0± ϕ1 dx = 0. (Recall that functions f0± are defined in (H0 ).) Under these assumptions it follows from the asymptotic estimates (4.13) in Proposition 4.5 that there exists c > 0 such that, for any sequence {(un , λn )}∞ n=1 ⊂ X ×R of solutions to Φλn (un ) = 0 satisfying un X → 0 and λn → μ1 , one must have λn = μ1 . Thus Φμ1 (u) = 0 provided 0 < uX < c, and so u = 0 is an isolated solution of Φμ1 (u) = 0. In the following lemma we drop the assumption that zero is an isolated solution to Φμ1 (u) = 0. This is possible with help from an approximation scheme based on the results from Lemma 4.7. Lemma 5.8. The statement of Lemma 5.7 holds without the assumption that zero is an isolated solution of Φμ1 (u) = 0. Proof. We proceed as in the proof of [10, Lemma 3, p. 1072] by considering a (n) sequence of boundary value problems Φλ (u) = 0, where (n)

def

Φλ (u) = A(u) − λB(u) − Hn (u; λ) ,

(5.22)

with Hn (u; λ) being defined by (3.5) with [hn (u)](x) defined by (4.28) in place of h(x, u; λ). Note that, by Lemma 4.7, the corresponding integrals with [hn (u)](x) defined by (4.28) in place of h(x, u; λ) in (4.13) are positive for each n ∈ N. (n) Thus, for all n ∈ N, zero is an isolated solution of Φμ1 (u) = 0 and, consequently, Lemma 5.7 applies. By Lemma 5.4, let us first fix η ∈ (0, η0 ) and then choose S, 0 < S ≤ 0, such that Sn \ (0, μ1 ) ∩ E S (μ1 ) ⊂ Kη for all n ∈ N . Now let 0 < ε < S and assume that Tμ−1 ,ε is bounded in E. Let Tn be a component of Sn \(E ε (μ1 )∩Kε+ ) containing (0, μ1 ). Suppose that the conclusion of our lemma is false. This means that ∂E ε (μ1 ) ∩ Kη+ ∩ Tμ−1 ,ε = ∅ . Recall that, by the definition of Tμ−1 ,ε , this set is connected and satisfies E ε (μ1 ) ∩ Kη+ ∩ Tμ−1 ,ε = ∅ . In addition, as Tμ−1 ,ε is assumed to be bounded, we can find R > 0 such that Tμ−1 ,ε ⊂ E R (μ1 ). We combine these facts with a classical topological result from Whyburn [41, Chap. I, Statement (9.3), p. 12], to conclude that S ∩ E R (μ1 ) \ E ε (μ1 ) ∩ Kη+ = k1 ∪ k2 where k1 , k2 are compact sets in E, such that k1 ∩ k2 = ∅, Tμ−1 ,ε ⊂ k1 , and S ∩ ∂E R (μ1 ) ∪ S ∩ ∂E ε (μ1 ) ∩ Kη+ ⊂ k2 .

310



Consequently, there exists a bounded open set U in E such that k1 ⊂ U and k2 ∩ U = ∅. The properties of k1 and k2 entail (0, μ1 ) ∈ U , ∂U ∩ S ⊂ E ε (μ1 ) ∩ Kη+ , and (5.23) ∂E ε (μ1 ) ∩ Kη+ ∩ U = ∅ . Recalling our definition of Tn as the component of Sn \ (E ε (μ1 ) ∩ Kε+ ) con(n) taining (0, μ1 ), we can apply Lemma 5.7 to the mapping Φλ in place of Φλ , thus obtaining ∂E ε (μ1 ) ∩ Kη+ ∩ Tn = ∅ for each n ∈ N . (5.24) Note that our asymptotic estimates (4.13), (4.30), and (4.31) guarantee the crucial (n) assumption of Lemma 5.7, namely, that zero is an isolated solution of Φμ1 (u) = 0 for each n ∈ N. Combining the facts (5.23) and (5.24), we arrive at ∂U ∩ Tn = ∅. So let us choose (un , λn ) ∈ ∂U ∩ Tn for each n ∈ N. Since U is bounded in E, we may pass ∗ to a subsequence {unk , λnk }∞ k=1 that converges weakly in E, that is, unk u 1,p ∗ weakly in W0 (Ω) and λnk → λ in R as k → ∞. Consequently, unk → u∗ (n ) strongly in Lp (Ω), by Rellich’s theorem. Since Φλnk (unk ) = 0 for each k ∈ N, with k

(n )

Φλnk defined by (5.22), we conclude that k

A(unk ) → λ∗ B(u∗ ) + H(u∗ ; λ∗ )

in Lp (Ω)

as k → ∞ .

This implies unk → u∗ strongly in W01,p (Ω) as k → ∞. (Note that A−1 : X → X is continuous due to (Ai ) or (Aii ).) The embeddings W01,p (Ω) → Lp (Ω) and Lp (Ω) → W −1,p (Ω) being compact, we have also A(unk ) → A(u∗ ) strongly in W −1,p (Ω). It follows that Φλ∗ (u∗ ) ≡ A(u∗ ) − λ∗ B(u∗ ) − H(u∗ ; λ∗ ) = 0 . ∗

(5.25)

∗

The boundary ∂U being closed, we conclude that (u , λ ) ∈ ∂U . Moreover, by (5.25), we have also (u∗ , λ∗ ) ∈ S ∩ E R (μ1 ) \ E ε (μ1 ) ∩ Kη+ ⊂ k2 . However, this contradicts ∂U ∩ k2 = ∅.

Proof of Theorem 3.7. With Lemma 5.8 in hand, the proof of Theorem 3.7 follows the same pattern as in Dancer [10, Proof of Theorem 2, p. 1073]. Therefore, we omit the details.

6. Parameter oscillations about μ1 This section provides an example of oscillations of the parameter λ around μ1 for solutions (u, λ) ∈ E, where uW 1,p (Ω) → ∞ and λ → μ1 . We need some subtle 0 regularity properties of ϕ1 to perform some computations in the following example.

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Unfortunately, these properties are known only for p > 2 in the one-dimensional case to which we restrict ourselves. In the one-dimensional case it is convenient to work with Ω = (0, πp ), where

1 1 2π def πp = 2 ds = . p 1/p p sin πp 0 (1 − s ) In the following example, we will use an asymptotic formula from [16] that takes πp pa more readable form if the positive eigenfunction ϕ1 is normalized by 0 ϕ1 dx = 1. For this asymptotic formula we also need the weighted Sobolev space Dϕ1 defined to be the completion of W01,p (0, πp ) with respect to the norm def π uDϕ1 = ( 0 p |ϕ1 (x)|p−2 |u (x)|2 dx)1/2 . We have the embedding Dϕ1 → C β [0, πp ] 1 where β = p−1 ∈ (0, 1) (see [34, Lemma 4.5] or [35, Lemma 4.4]). π Example 6.1. Take p > 2, 0 ≤ α < p−2, and f ∈ L∞ (0, πp ) with 0 p f ϕ1 dx = 0 and f ≡ 0. Let us consider the boundary value problem − |u |p−2 u − λ|u|p−2 u + |u|α sin(u) = f + aϕ1 in (0, πp ) (6.1) u(0) = u(πp ) = 0 . Note that Theorem 3.10 applies to this problem. Below we will show that if either def of the conditions a = 0 or α > 1/p = 1 − (1/p) (p = p/(p − 1)) is satisfied, then there exist two continua Zμ±1 ⊂ S as in Theorem 3.10 and such that Zμ+1 and Zμ−1 exhibit the following additional “oscillation” phenomenon (with λ oscillating about μ1 ). We write down this phenomenon for Zμ+1 only; for Zμ−1 it is analogous: ∞ (OC) There exist a number δ > 0 and two sequences {βn }∞ n=1 , {γn }n=1 ⊂ R, 0 < βn < γn < βn+1 < γn+1 for all n ∈ N, with βn , γn → +∞ as n → ∞, and such that for all (u, λ) ∈ Zμ+1 ∪ Zμ−1 with |λ − μ1 | < δ we have π (i) 0 p uϕ1 dx = βn =⇒ λ > μ1 ; π (ii) 0 p uϕ1 dx = γn =⇒ λ < μ1 .

As a consequence, the set Zμ+1 being connected, for every n ∈ N large enough π there exists (un , μ1 ) ∈ Zμ+1 such that βn < 0 p un ϕ1 dx < γn ; see Figure 1. Clearly, πp 0 un ϕ1 dx → +∞ as n → ∞. π Notice that, in this example, 0 p un ϕ1 dx → +∞ as n → ∞ forces un W 1,p (Ω) → ∞. This means that Dr´ abek’s hypothesis [14, Eq. (14.43), p. 191] 0 for bifurcations from infinity at (±∞, μ1 ) is violated for the boundary value problem (6.1). This example relates to results from Dancer [11] obtained for the semilinear case p = 2 and for α = 0. Proof of the statement (OC ) in Example 6.1. Our proof is based on asymptotic estimate proved in [16] and stationary phase argument, see, e.g., [17].

312



Let f = f + aϕ1 We take t ∈ R \ {0} with |t| small enough. Multiplying equation (6.1) by tα and denoting w = tα/(p−1) u, we arrive at ⎧ p−2 p−2 ⎪ w = tα f + aϕ1 ⎪ ⎨ − |w | w − λ|w| − |t|α(p−α−1)/(p−1) |w|α sin t−α/(p−1) w ; (6.2) ⎪ ⎪ ⎩ w(0) = w(π ) = 0 p

The right-hand side satisfies the assumptions of Theorem 4.1 from [16] and we thus obtain that large solutions (u, λ) ∈ Zμ+1 satisfy u = t−1 (ϕ1 + v ) with t → 0+ and

πp " # tα aϕ1 + |ϕ1 + v |α sin t−1 (ϕ1 + v ) ϕ1 dx λ − μ1 = −tp−1−α 0 2(p−1−α) + (p − 2)t (6.3) Q0 (V , V ) + o |t|2(p−1−α) where V ∈ Dϕ1 is the limit t−p+α+1 v → V in Dϕ1 as t → 0+. Thanks to the generalized Riemann–Lebesgue lemma [32, Prop. 2.1], we have ∗ |ϕ1 + v |α sin t−1 (ϕ1 + v ) 0 weakly∗ in L∞ (0, 1). Hence, the right hand-side of equation (6.2) converges weakly∗ in L∞ (0, 1) to a function f ∗ given by f ∗ = 0 if α > 0, and f ∗ = f if α = 0 (in which is case a = 0 by our hypothesis). The limit function V ∈ Dϕ1 is the solution of the linearization of problem (6.2) at (u, λ) = (ϕ1 , μ1 ), that is, p−2

πp d dϕ1 dV 1 p−2 p−1 ∗ ∗ −μ ϕ V = −ϕ (f ϕ )dx in (0, 1) ; f 1 1 1 1 dx dx dx p−1 0

V (0) = 0, V (1) = 0 ; 1

V ϕ1 dx = 0 ;

(6.4)

0

see [16, Thm. 4.1, pp. 445–446]. Owing to 0 ≤ α < p − 2, we find that t−1 v → 0 in Dϕ1 as t → 0+. This fact will later allow us in Corollary C.2 to use a “stationary phase argument” to our problem and prove that

πp α |ϕ1 + v |α sin t−1 (ϕ1 + v ) ϕ1 dx = −K ϕ1 (πp /2) ϕ1 (πp /2) 0 π · sin − + t−1 ϕ1 (πp /2) 2p −1/p ·t (6.5) + o t−1/p

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holds as t → 0+. Here 1/p+1/p = 1 and K > 0 is independent of t. Inserting (6.5) in the asymptotic estimate (6.3) and dividing it by tp−1−α+1/p we get π p−1−α+1/p α −1 (λ − μ1 )/t = − K |ϕ1 (πp /2)| ϕ1 (πp /2) · sin − + t ϕ1 πp /2 2p

πp + o(1) − tα−1/p a ϕ21 dx 0 p−1−α−1/p + (p − 2)t Q0 (V , V ) + o |t|p−1−α−1/p ) . An easy calculation shows that p − 1 − α − 1/p =

(p − 1)2 − αp >0 p

thanks to 0 < α 1/p we find that

(λ − μ1 )/tp−1−α+1/p = −K|ϕ1 (πp /2)|α ϕ1 (πp /2) π −1 · sin − + t ϕ1 (πp /2) + o(1) 2p as t → 0+. Then the desired numbers βn and γn are, for instance,

πp ϕ1 (x)2 dx/ϕ1 (πp /2) βn = 3/2 + 1/(2p ) + 2n + 2n0 π 0

and

γn = 1/2 + 1/(2p ) + 2n + 2n0 π

πp

ϕ1 (x)2 dx/ϕ1 (πp /2)

0

with n0 ∈ N large enough. This completes the proof.

Appendices Appendix A. A priori regularity results Here we state the main regularity result for a weak solution u ∈ W01,p (Ω) of the Dirichlet boundary value problem − div a(x, ∇u) = h x, u(x) in Ω ; u = 0 on ∂Ω . (A.1) This a priori regularity is used throughout the entire article.

314



c βn+5 βn+4

βn+3

βn+2

βn+1

βn

γn+5

γn+4

γn+3 γn+2

γn+1

γn

Zμ+1

μ1

λ

Figure 1. Bifurcations from infinity of solutions to (6.1): a sketch def π of the set Zμ+1 for large positive solutions; here c = 0 p uϕ1 dx. Proposition A.1. Let 1 < p < ∞ and let hypotheses (A) and (H) be satisfied. Assume that u ∈ W01,p (Ω) is a weak solution of problem (A.1). Then u ∈ C 1,β (Ω) where β ∈ (0, 1) is a constant independent from u. If, in addition, ∂Ω is a compact manifold of class C 1,α for some α ∈ (0, 1), then β ∈ (0, α) can be chosen such that u ∈ C 1,β (Ω). Moreover, β is again independent from u, and uC 1,β (Ω) ≤ C where C > 0 is some constant depending solely upon Ω, A, h, N , p, and the norm uLp0 (Ω) with p if p < N ; p∗ = NN−p p0 = 2p if p ≥ N . Notice that, owing to the Sobolev embedding W01,p (Ω) → Lp0 (Ω), we have also uC 1,β (Ω) ≤ C , where the constant C depends solely upon Ω, A, h, N , p, and the norm uW 1,p (Ω) . Similarly, one obtains uC 1,β (Ω) ≤ C as well, where 0 the constant C depends solely upon Ω, A, h, N , p, and the norm uL∞ (Ω) . These two consequences of Proposition A.1 will be used quite often in the sequel.

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Proposition A.1 is, in fact, a combination of the following two lemmas, in which we keep our hypotheses and notation from the proposition: Lemma A.2. Let g : Ω × R → R be a Carathéodory function such that g( · , s) ∈ L1loc (Ω) for every s ∈ R, and the following inequality holds with some constants a > 0 and b ≥ 0: s · g(x, s) ≤ a|s|p + b|s|

for all

s∈R

Assume that u ∈ W01,p (Ω) satisfies

a(x, ∇u), ∇φ dx = g x, u(x) φ dx Ω

and a.e.

for all

Ω

x ∈ Ω.

φ ∈ Cc∞ (Ω) .

Then u ∈ L∞ (Ω) and there exists a constant c > 0 such that uL∞ (Ω) ≤ c, where c depends solely upon a, b, N , p, and uLp0 (Ω) . This is a special case of a more general result shown in Anane’s thesis [4, Théorème A.1, p. 96]. Although his proof is carried out only for a(x, ξ) ≡

1 ∂ξ A(x, ξ) = |ξ|p−2 ξ , p

(x, ξ) ∈ Ω × RN ,

(A.2)

one can rewrite it directly for our more general case. Lemma A.3. Assume that u ∈ W01,p (Ω) is a weak solution of problem (A.1) such that u ∈ L∞ (Ω). Then u ∈ C 1,β (Ω) where β ∈ (0, 1) is a constant independent from u. If, in addition, ∂Ω is a compact manifold of class C 1,α for some α ∈ (0, 1), then β ∈ (0, α) can be chosen such that u ∈ C 1,β (Ω). Moreover, β is again independent from u, and uC 1,β (Ω) ≤ C where C > 0 is some constant depending solely upon Ω, A, h, N , p, and the norm uL∞(Ω) . The first statement of this lemma, interior regularity in C 1,β (Ω), was established independently by DiBenedetto [13, Theorem 2, p. 829] and Tolksdorf [39, Theorem 1, p. 127]. The second statement, regularity near the boundary, is due to Lieberman [25, Theorem 1, p. 1203]. The constant β depends solely upon α, N and p. We keep the meaning of the constants α and β throughout the entire article and denote by β an arbitrary, but fixed number such that 0 < β < β < α < 1. Last but not least, Lieberman’s regularity results have been shown for the Neumann boundary conditions as well. While Anane’s proof of Lemma A.2 is based on the special form of a(x, ξ) ≡ 1/p ∂ξ A(x, ξ) with the positively p-homogeneous potential A(x, · ) satisfying also N hypothesis (2.2), Lemma A.3 is valid with any vector field a ≡ (ai )N i=1 : Ω × R → N R satisfying ai ∈ C 0 (Ω × RN ) ∩ C 1 Ω × RN \ {0} (i = 1, 2, . . . , N ) together with the ellipticity and growth conditions (2.3), (2.4) and (2.5).

316



Appendix B. Bifurcations from zero or infinity Let us consider a sequence of nontrivial solutions {(un , λn )}∞ n=1 ⊂ S of problem (2.1), i.e., for each n = 1, 2, . . . the integral identity

p−2 a(x, ∇un ), ∇φ dx = λn B |un | un φ dx + h(x, un ; λn ) φ dx (B.1) Ω

holds for all φ ∈

Ω

W01,p (Ω).

Ω

We assume

−∞ < λn ≤ μ2 − δ

(n = 1, 2, . . . )

(B.2)

where δ ∈ (0, μ2 − μ1 ) is a constant and μ2 stands for the second eigenvalue of the quasilinear operator A : W01,p (Ω) → W −1,p (Ω) : u → − div a · , ∇u( · ) . A variational characterization of μ2 by a minimax formula is due to Anane [4, Remarques 2.2, pp. 15–16]. It is shown in Anane and Tsouli [5, Prop. 2, p. 5] that there is no eigenvalue in the open interval (μ1 , μ2 ). Although the last two claims have been proved only for the case of the positive Dirichlet p-Laplacian −Δp , that is, for 1 a(x, ξ) ≡ ∂ξ A(x, ξ) = |ξ|p−2 ξ , (x, ξ) ∈ Ω × RN , p their proofs carry over directly to our more general case. Now we need to distinguish between solutions (un , λn ) with un having arbitrarily small or arbitrarily large norm (bifurcations from zero or infinity, respectively). More precisely, we will show that for this purpose any of the three norms un W 1,p (Ω) , un L∞ (Ω) , or un C 1,β (Ω) can be employed. Recall that 0 < β < 0 α < 1 are constants from Proposition A.1. To verify this claim, we need to employ Lemma 2.2, the proof of which is given next. B.1. Proof of Lemma 2.2 To verify (a), we first notice that h(x, u(x); λ) = 0 if u(x) = 0, and estimate h x, u(x); λ h x, u(x); λ |u(x)| p−1 h x, u(x); λ = ≤ (B.3) uL∞(Ω) |u(x)|p−1 u(x) p−1 up−1 L∞ (Ω) if u(x) = 0. From uL∞ (Ω) → 0 we get u(x) → 0 uniformly for a.e. x ∈ Ω. Finally, using (2.10) we arrive at (2.14). ∞ To prove (b), let us take a sequence {un }∞ n=1 ⊂ L (Ω) with un L∞ (Ω) → ∞ as n → ∞. We split the domain Ω = An ∪ Bn where 1 An = x ∈ Ω : un (x) ≤ un L2 ∞ (Ω) , 1 Bn = x ∈ Ω : un (x) > un L2 ∞ (Ω) .

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For x ∈ An we infer from inequality (2.11) that h x, un (x); λ h x, un (x); λ 1 + un (x) p−1 = p−1 · un p−1 un p−1 1 + un (x) L∞ (Ω) L∞ (Ω) (p−1)/2

≤C

1 + un L∞ (Ω) un p−1 L∞ (Ω)

For x ∈ Bn we infer from (B.3) that h x, un (x); λ un p−1 L∞ (Ω)

≤

.

(B.4)

h x, un (x); λ |un (x)|p−1

.

(B.5)

Now let χAn and χBn denote the characteristic functions of the sets An and Bn , respectively. Combining inequalities (B.4) and (B.5) we arrive at |h(x, un (x); λ)| un p−1 L∞ (Ω)

(p−1)/2

≤C

1 + un L∞ (Ω) un p−1 L∞ (Ω)

χAn (x)

|h(x, un (x); λ)| χBn (x) (B.6) |un (x)|p−1 for x ∈ Ω. The first summand clearly tends to zero as n → ∞, whereas the second one tends to zero pointwise for a.e. x ∈ Ω and uniformly for every λ ∈ R, by (2.12). This proves (2.15). +

Lemma 2.2 has the following important corollary. Corollary B.1. Let 1 ≤ q < ∞. In both alternatives, (a) and (b), of Lemma 2.2 we have h · , u( · ); λ q /up−1 (B.7) L∞(Ω) → 0 L (Ω) as uL∞ (Ω) → 0 or uL∞(Ω) → ∞, respectively, uniformly for every λ ∈ R. Proof. In the situation of alt. (a), we can combine inequality (2.9) with the Lebesgue dominated convergence theorem to obtain (B.7) as uL∞ (Ω) → 0. The same argument applies to alt. (b), of course, with (2.9) replaced by (2.11). B.2. A priori results – bifurcations from zero Lemma B.2. Let {(un , λn )}∞ n=1 ⊂ S be as specified above. Then the following three statements are equivalent, as n → ∞: (i) un W 1,p (Ω) → 0; 0 (ii) un L∞ (Ω) → 0; (iii) un C 1,β (Ω) → 0. def

Moreover, in all three cases we have λn → μ1 and the sequence wn = un / ∞ un L∞ (Ω) is the union of two disjoint subsequences {wn }∞ n=1 and {wn }n=1 , one of them possibly empty, such that, if nonempty, they satisfy wn → ϕ1 /ϕ1 L∞ (Ω)

318



and/or wn → −ϕ1 /ϕ1 L∞ (Ω) in C 1,β (Ω) as n → ∞. Here, β ∈ (0, β) is arbitrary. Proof. Clearly, (iii) implies (i) and (ii). We will prove (i) =⇒ (ii) and (ii) =⇒ (iii). def

(i) =⇒ (ii): The function wn = un /un W 1,p (Ω) satisfies wn W 1,p (Ω) = 1 0 0 together with

a(x, ∇wn ), ∇φ dx = gn x, wn (x) φ dx (B.8) for all φ ∈

Ω 1,p W0 (Ω),

Ω

by equation (B.1), where we have abbreviated h x, sun W 1,p (Ω) ; λn def p−2 0 s+ , (x, s) ∈ Ω × R . gn (x, s) = λn B(x) |s| un p−1 1,p (Ω) W 0

Our hypotheses 0 ≤ B ∈ L∞ (Ω) and (2.9) guarantee s · gn (x, s) ≤ a|s|p + b|s| for all s ∈ R

and a.e. x ∈ Ω ,

where a = (μ2 − δ)BL∞ (Ω) + C > 0 and b = 0 . We may apply Lemma A.2 to (B.8) to conclude that wn ∈ L∞ (Ω) and there exists a constant c > 0 such that wn L∞ (Ω) ≤ c, where c is independent from n = 1, 2, . . . . Consequently, we have un L∞ (Ω) ≤ c un W 1,p (Ω) for n = 1, 2, . . . , 0 which proves (i) =⇒ (ii). def

(ii) =⇒ (iii): This time we take wn = un /un L∞ (Ω) which satisfies wn L∞ (Ω) = 1 together with

B(x) |wn |p−2 wn φ dx a(x, ∇wn ), ∇φ dx = λn Ω Ω

h x, wn (x)un L∞ (Ω) ; λn + φ dx . (B.9) un p−1 Ω L∞ (Ω) for all φ ∈ W01,p (Ω), by equation (B.1). We claim: lim inf n→∞ λn ≥ μ1 . On the contrary, suppose that there is a subsequence of {(un , λn )}∞ n=1 , denoted again in the same way, such that for some δ ∈ (0, μ1 ) and for each n = 1, 2, . . . we have λn ≤ μ1 − δ . Taking φ = wn in equation (B.9) we obtain

A(x, ∇wn ) dx = λn B(x) |wn |p dx Ω Ω

h x, wn (x)un L∞ (Ω) ; λn + wn (x) dx . un p−1 Ω L∞ (Ω) Now we use λn ≤ μ1 − δ and the variational characterization of μ1 from (2.8) to get

h x, wn (x)un L∞ (Ω) ; λn δ A(x, ∇wn ) dx ≤ |wn (x)| dx . (B.10) μ1 Ω un p−1 Ω L∞ (Ω)

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Next, we apply Corollary B.1, alt. (a), and wn L∞ (Ω) = 1 to the right-hand side of equation (B.10) to conclude that Ω A(x, ∇wn ) dx → 0 as n → ∞. But this def

means wn W 1,p (Ω) → 0 as n → ∞, by inequality (2.6). Finally, let us define zn = 0 wn /wn W 1,p (Ω) . This function satisfies zn W 1,p (Ω) = 1 together with 0 0

a(x, ∇zn ), ∇φ dx = λn B(x) |zn |p−2 zn φ dx Ω Ω

h x, νn zn (x); λn φ dx + νnp−1 Ω def

for all φ ∈ W01,p (Ω), by equation (B.9), where νn = wn W 1,p (Ω) un L∞ (Ω) → 0 0 as n → ∞. The same arguments we have used above in the proof of (i) =⇒ (ii) now reveal that also wn L∞ (Ω) → 0 as n → ∞, a contradiction with wn L∞ (Ω) = 1 for n = 1, 2, . . . . Therefore, our claim lim inf n→∞ λn ≥ μ1 must be valid. Taking n ∈ N large enough and using (B.2), we may assume 0 ≤ λn ≤ μ2 − δ for every n = 1, 2, . . . . Thus, applying 0 ≤ B ∈ L∞ (Ω) and (2.9) we observe that the function h x, wn (x)un L∞ (Ω) ; λn def p−2 , x ∈ Ω, fn (x) = λn B(x) |wn | wn + un p−1 L∞ (Ω) on the right-hand side of equation (B.9) is uniformly bounded a constant, fn (x) ≤ M = (μ2 − δ)BL∞ (Ω) + C > 0 , x ∈ Ω . Now we may apply Lemma A.3 to (B.9) to conclude that wn ∈ C 1,β (Ω) and there exists a constant c > 0 such that wn C 1,β (Ω) ≤ c , where c is independent from n = 1, 2, . . . . Consequently, we have un C 1,β (Ω) ≤ c un L∞ (Ω) for n = 1, 2, . . . , which proves (ii) =⇒ (iii). To complete the proof, we will derive λn → μ1 from our proof of (ii) =⇒ (iii). Let us fix any β ∈ (0, β). The embedding C 1,β (Ω) → C 1,β (Ω) being compact by Arzel` a–Ascoli’s theorem, the sequence {wn }∞ n=1 contains a subsequence that converges in C 1,β (Ω) to some w; we denote it again by wn → w. Notice that wL∞ (Ω) = 1. Extracting yet another convergent subsequence from {λn }∞ n=1 we may assume also λn → λ∗ . We let n → ∞ in equation (B.9) and use Corollary B.1, alt. (a), to conclude that w ∈ C 1,β (Ω) must satisfy

∗ Y a(x, ∇w), ∇φ dx = λ B(x) |w|p−2 w φ dx (B.11) Ω

W01,p (Ω).

Ω ∗

for all φ ∈ Since 0 ≤ λ ≤ μ2 − δ and μ1 is the only eigenvalue of the operator A in the open interval (−∞, μ2 ), we must have λ∗ = μ1 . In addition, μ1 being a simple eigenvalue, we have w = κϕ1 in Ω where κ ∈ R satisfies |κ| · ϕ1 L∞ (Ω) = 1. The sequence {λn }∞ n=1 ⊂ [0, μ2 − δ] being bounded and the cluster point ∗ λ = μ1 unique, we conclude that λn → μ1 (n → ∞) holds not only for a

320



suitable subsequence of {λn }∞ n=1 , but also for the entire original sequence as well. Finally, the original sequence {wn }∞ n=1 can have at most two cluster points, ±ϕ1 /ϕ1 L∞ (Ω) . The lemma is proved. B.3. A priori results – bifurcations from infinity Lemma B.3. Let {(un , λn )}∞ n=1 ⊂ S be as specified above. Then the following two statements are equivalent, as n → ∞: (i) un W 1,p (Ω) → ∞; 0 (ii) un L∞ (Ω) → ∞. Moreover, if (B ) 0 < b0 ≤ B(x) ≤ b1 < ∞ for a.e. x ∈ Ω (b0 , b1 – constants), def

then in both cases we have λn → μ1 and the sequence wn = un /un L∞ (Ω) is the ∞ union of two disjoint subsequences {wn }∞ n=1 and {wn }n=1 , one of them possibly empty, such that, if nonempty, they satisfy wn → ϕ1 /ϕ1 L∞ (Ω) and/or wn → −ϕ1 /ϕ1 L∞ (Ω) in C 1,β (Ω) as n → ∞. Here, β ∈ (0, β) is arbitrary. Finally, either of the statements (i) and (ii) is equivalent to (as n → ∞) (iii) un C 1,β (Ω) → ∞ provided that, in addition to (B ), either of the following two conditions is satisfied: (I) −∞ < Λ ≤ λn ≤ μ2 − δ for all n = 1, 2, . . . (Λ, δ – constants, δ > 0); (II) inequality (2.9) holds. Notice that (all or some of) (B), (B.2), and (2.11), respectively, have been replaced by stronger hypotheses (B ), (I), and (2.9). Proof. It is obvious that either of (i) and (ii) implies (iii). (ii) =⇒ (i) is proved by contradiction using similar arguments as in the proof of (i) =⇒ (ii) in Lemma B.2 above. (i) =⇒ (ii) is proved by contradiction, as well. Taking φ = un in equation (B.1) we obtain

A(x, ∇un ) dx = λn B(x) |un |p dx + h x, un (x); λn un (x) dx. (B.12) Ω

Ω

Ω

Now we apply inequalities (2.6), λn ≤ μ2 − δ, and (2.11) to get


|un |p + |un | dx . ≤ (μ2 − δ) B(x) |un |p dx + C Ω

{un }∞ n=1

Ω

Consequently, if contains a subsequence bounded in L∞ (Ω), then the same subsequence must be bounded also in W01,p (Ω), thus contradicting (i). To verify the remaining claims, we assume (B ) throughout the rest of the proof.

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Knowing (i) ⇐⇒ (ii) already, let us assume (i). We begin by establishing λn > −2C/b0

for all n ≥ n0 ,

(B.13)

where C > 0 is the constant from (2.11) and n0 ∈ N is taken large enough. Contrary to (B.13), suppose that a subsequence of {λn }∞ n=1 , denoted identically, satisfies λn ≤ −2C/b0 (< 0) for every n ≥ 1. We apply inequalities (2.6), (2.11), and (B ) to equation (B.12) to estimate


p = λn B(x) |un | dx + h x, un (x); λn un (x) dx Ω

Ω

p ≤ λn b0 |un | dx + C |un |p + |un | dx Ω Ω

p ≤ (λn b0 + 2C) |un | dx + C |Ω|N ≤ C |Ω|N . Ω

This is a contradiction to (i), so (B.13) must be valid. The convergence λn → μ1 , def

together with all the remaining claims for wn = un /un L∞ (Ω) , is now derived in the same way as in the proof of Lemma B.2. Finally, assume that (iii) and (B ) hold together with (I) or (II). First, it turns out that condition (II) implies (I); this can be deduced from equation (B.12), using (II): If λn ≤ 0 and un ≡ 0 in Ω, then in equation (B.12) we can estimate

p A(x, ∇un ) dx − λn b0 |un | dx ≤ C |un |p dx . Ω

Ω

Ω

In particular, we must have −λn b0 ≤ C. Thus, condition (I) holds with Λ = −C/b0 . Assuming now condition (I), we may apply Lemma A.3 to equation (B.1) ∞ to conclude that if {un }∞ n=1 contains a subsequence bounded in L (Ω), then the same subsequence is bounded also in C 1,β (Ω), which contradicts (iii). Hence, (iii) =⇒ (ii).

Appendix C. Stationary phase argument We restrict ourselves to 2 < p < ∞ throughout this section. Lemma C.1 (A generalization of Erdélyi [17, Theorem on p. 52]). Let g : [0, πp /2]× R+ → C be continuous with g( · , τ ) → gˆ uniformly on [0, πp /2] as τ → +∞. (Hence, also gˆ : [0, πp /2] → C is continuous.) Assume that the functions g( · , τ ) : [0, πp /2] → C are absolutely equicontinuous, that is, (AEC ) for every ε > 0 there is δ > 0 such that

∂g (x, τ ) dx < ε dx < δ =⇒ ∂x M

M

holds for every Lebesgue-measurable set M ⊂ [0, πp /2].

322



Finally, let h : [0, πp /2] → R be continuously differentiable with h (x) = (πp /2 − x)σ−1 h1 (x) ,

(C.1)

where σ > 1 is a constant and h1 ∈ C 1 [0, πp /2] is a strictly positive function. Then there exists a constant K ≡ K(σ, h1 (πp /2)) > 0, independent from τ , such that

πp /2 π g(x, τ ) · eiτ h(x)dx = −K gˆ(πp /2) · ei[− 2σ +τ h(πp/2)] · τ −1/σ + o τ −1/σ 0

as τ → +∞. Proof. We proceed as in [17, proof of the theorem on p. 52]. We take λ = μ = 1, hypothesis α = 0, β = πp /2, = 1, and σ > 1. In [17] a stronger smoothness ∂g is imposed on g( · , τ ), namely, that g( · , τ ) ∈ C 1 [0, πp ] and ∂x (x, τ ) ≤ C ≡ const. < +∞ for all (x, τ ) ∈ [0, πp /2] × R+ . However, this hypothesis is essentially used only in an estimate contained in the second displayed formula from the bottom of page 55. Nevertheless, it is easy to see that this estimate holds also under the weaker hypothesis (AEC). Recall that we deal only with the special case λ = μ = 1 in that estimate; hence, (AEC) is sufficient. The remaining parts of the proof are identical with [17]. Corollary C.2. Let u = t−1 (ϕ1 +v ) be a solution of (6.1) with t > 0 small enough. Then the asymptotic formula

πp α |ϕ1 + v |α sin t−1 (ϕ1 + v ) ϕ1 dx = −K ϕ1 (πp /2) ϕ1 (πp /2) 0 π · sin − + t−1 ϕ1 (πp /2) 2p −1/p ·t + o t−1/p holds as t → 0+. Proof. This claim is derived from Proposition C.1 as follows. One splits the integral π /2 π π as 0 p . . . dx = 0 p . . . dx + πpp/2 . . . dx and applies Proposition C.1 to both integrals on the right. We treat only the first integral in detail; the second one can be treated analogously. Observe that

πp /2 α sin t−1 ϕ1 (x) + v (x) ϕ1 (x) + v (x) ϕ1 (x)dx 0 + * πp /2 α it−1 ϕ1 (x) it−1 v (x) e e ϕ1 (x) + v (x) ϕ1 (x)dx . = m 0

We set τ = t−1 , g(x, τ ) = |ϕ1 (x) + v (x)|α · eiτ v (x) · ϕ1 (x), and h(x) = ϕ1 (x) for x ∈ [0, πp /2] and τ > 0. Recall that ϕ1 (x) > 0 for every x ∈ (0, πp /2), by Remark 2.1. It is obvious that α g(x, τ ) → gˆ(x) = ϕ1 (x) ϕ1 (x) = ϕ1 (x)α+1

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uniformly for x ∈ [0, πp /2] as τ → +∞. This follows from τ p−1−α · v → V in Dϕ1 as τ → +∞ combined with p − 1 − α > 1 and the embedding Dϕ1 → C β [0, πp ] 1 ∈ (0, 1) (see [34, Lemma 4.5] or [35, Lemma 4.4]). The first where β = p−1 eigenfunction ϕ1 of the p-Laplacian on the interval (0, πp ) can be expressed by means of a special function sinp and a constant πp defined, e.g., in [27, 28]; we set π ϕ1 (x) = κ sinp (x) where κ = 1/ 0 p (sinp (x))p dx. We set cosp (x) = sinp (x). As a consequence of formulas (30)–(32) on page 332 in [27], we obtain cosp (x) p−2 cosp (x) = πp − x (p − 1) 1 − J(x) 2 def

for 0 ≤ x ≤ πp /2, cf. estimate (33) on page 332 in [27], where we have introduced π /2 p if 0 ≤ x < πp /2 ; | sinp (t)|p−2 cosp (t) πpt−x def x /2−x dt J(x) = if x = πp /2 . 0 Taking into account cosp (x) > 0 on [0, πp /2), we find 1−J(x) > 0 for 0 ≤ x ≤ πp /2 and thus p −1 π p −1 p −x cosp (x) = (p − 1)p −1 1 − J(x) . 2 This implies also J(x) → J(πp /2) = 0 as x → (πp /2)−. Hence, the function h(x) = sinp x satisfies assumption (C.1), where σ = p and h1 (x) = (p − 1)p −1 [1 − J(x)]p −1 is continuously differentiable for 0 ≤ x ≤ πp /2. Indeed, substituting def

φ(t) =

1 p−2 p−1 | sinp (t)|

J(x) =

πp /2

φ (t)

x

and

sinp (t) for 0 ≤ t ≤ πp , we observe that both t−x 1 dt = φ(πp /2) − πp /2 − x πp /2 − x

πp /2

φ(t) dt

(C.2)

x

πp /2

πp /2 1 1 φ (t)(t − x) dt − φ (t) dt (πp /2 − x)2 x πp /2 − x x φ(πp /2) − φ(x) 1 J(x) − (C.3) = πp /2 − x πp /2 − x

J (x) =

are continuous functions of x ∈ [0, πp /2). Moreover, we compute lim

x→(πp /2)−

J(x) = φ(πp /2) − φ(πp /2) = 0 = J(πp /2)

and lim

x→(πp /2)−

J(πp /2) − J(x) φ(πp /2) − φ(x) − lim πp /2 − x x→(πp /2)− πp /2 − x = −J (πp /2) − φ (πp /2) = −J (πp /2) ,

J (x) = −

lim

x→(πp /2)−

324



where J (πp /2) = −φ (πp /2)/2 = 0, by the following calculation: J (πp /2) =

J(πp /2) − J(x) x→(πp /2)− πp /2 − x

=−

lim

lim

x→(πp /2)−

φ(πp /2) −

1 πp /2−x

πp /2 x

φ(t)dt

πp /2 − x

=−

1 x→(πp /2)− (πp /2 − x)2

=−

1 x→(πp /2)− (πp /2 − x)2

(πp /2 − x)φ(πp /2) −

lim

πp /2

lim

x

πp /2

φ(t)dt

x

φ(πp /2) − φ(t) dt

πp /2 πp /2 1 =− lim φ (σ)dσdt x→(πp /2)− (πp /2 − x)2 x t

1 =− lim φ (σ)dσdt = −φ (πp /2)/2 x→(πp /2)− (πp /2 − x)2 x≤t≤σ≤πp /2 as φ is continuous on [0, πp ]. This completes the proof.

Acknowledgements ˇ cka Example 6.1 is motivated by numerical experiments performed by Jan Cepiˇ from University of West Bohemia, Plzeˇ n. Petr Girg was supported by a one year fellowship (for the academic year 2003–2004) from the Alexander von Humboldt Foundation during his stay at the University of Rostock and his research is supported in part by Ministry of Education, Youth and Sports of the Czech Republic through the Research Plan VZ MSM4977751301. This work was supported in part also by the German Academic Exchange Service (DAAD, Germany) within the exchange program “Acciones Integradas” with Spain, and by the Federal Ministry for Education and Research (BMBF, Germany) through its International Office, Grant No. CZE-01/004.

References [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd Ed., Academic Press, New York, Oxford, 2003. [2] W. Allegretto and Y. X. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal., 32 (7) (1998), 819–830. [3] A. Anane, Simplicité et isolation de la première valeur propre du p-Laplacien avec poids, Comptes Rendus Acad. Sc. Paris, Série I, 305 (1987), 725–728. [4] A. Anane, Etude des valeurs propres et de la résonance pour l’opérateur p-Laplacien, Thèse de doctorat, Université Libre de Bruxelles, 1988, Brussels.

Vol. 9 (2008)


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[5] A. Anane and N. Tsouli, On the second eigenvalue of the p-Laplacian, in Nonlinear Partial Differential Equations, Proc. Conf. Fès, Morocco, May 9–14, 1994, pp. 1–9, A. Benkirane and J. P. Gossez; eds., Pitman Research Notes in Mathematics Series Vol. 343. Longman Ltd., Essex, U.K., 1996. [6] D. Arcoya and J. L. G´ amez, Bifurcation theory and related problems: Anti-maximum principle and resonance, Comm. P.D.E. 26 (9&10) (2001), 1879–1911. [7] F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Functional Anal. 3 (1969), 217–245. [8] Y. Cheng, NOTE: H¨ older continuity of the inverse of p-Laplacian, J. Math. Anal. Appl. 221 (1998), 734–748. [9] M. G. Crandall; P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. [10] E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J. 23 (11) (1974), 1069–1076. [11] E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. 131 (1982), 167–185. [12] J. I. D´ıaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, Comptes Rendus Acad. Sc. Paris, Série I 305 (1987), 521–524. [13] E. DiBenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (8) (1983), 827–850. [14] P. Dr´ abek, Solvability and Bifurcations of Nonlinear Equations, Pitman Research Notes in Mathematics Series, Vol. 264. Longman Scientific & Technical, U.K., 1992. [15] P. Dr´ abek, P. Girg, and P. Tak´ aˇc, Bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity, J. Differential Equations 204 (2) (2004), 265–291. [16] P. Dr´ abek, P. Girg, P. Tak´ aˇc, and M. Ulm, The Fredholm alternative for the pLaplacian: Bifurcation from infinity, existence and multiplicity of solutions, Indiana Univ. Math. J. 53 (2) (2004), 433–482. [17] A. Erdélyi, Asymptotic Expansions, Dover, New York, 1956. [18] J. Fleckinger, J. Hern´ andez, P. Tak´ aˇc, and F. de Thélin, Uniqueness and positivity for solutions of equations with the p-Laplacian, in G. Caristi and E. Mitidieri; eds., Proceedings of the Conference on Reaction-Diffusion Equations, 1995, Trieste, Italy. Lecture Notes in Pure and Applied Math. Vol. 194, pp. 141–155. Marcel Dekker, New York and Basel, 1998. [19] S. Fuˇc´ık, Solvability of Nonlinear Equations and Boundary Value Problems. D. Reidel Publ. Co., Dordrecht, Holland, 1980. [20] S. Fuˇc´ık, J. Neˇcas, J. Souˇcek, and V. Souˇcek, Spectral Analysis of Nonlinear Operators, Lecture Notes in Mathematics Vol. 346. Springer-Verlag, New York, Berlin, Heidelberg, 1973. [21] J. Garc´ıa-Meli´ an and J. Sabina de Lis, A local bifurcation theorem for degenerate elliptic equations with radial symmetry, J. Differential Equations 179 (2002), 27–43.

326



[22] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, Berlin, Heidelberg, 1977. [23] A. Kufner, Weighted Sobolev Spaces, in Teubner-Texte zur Mathematik, Vol. 31. Teubner-Verlag, Leipzig, 1980. [24] A. Kufner, O. John, and S. Fuˇc´ık, Function Spaces, Academia, Prague, 1977. [25] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (11) (1988), 1203–1219. [26] P. Lindqvist, On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0, Proc. Amer. Math. Soc. 109 (1) (1990), 157–164. [27] R. F. Man´ asevich and P. Tak´ aˇc, On the Fredholm alternative for the p-Laplacian in one dimension, Proc. London Math. Soc. 84 (2002), 324–342. [28] M. A. del Pino, P. Dr´ abek, and R. F. Man´ asevich, The Fredholm alternative at the first eigenvalue for the one-dimensional p-Laplacian, J. Differential Equations 151 (1999), 386–419. [29] M. A. del Pino, M. Elgueta, and R. F. Man´ asevich, A homotopic deformation along p of a Leray–Schauder degree result and existence for (|u |p−2 u ) + f (t, u) = 0, u(0) = u(T ) = 0, p > 1, J. Differential Equations 80 (1) (1989), 1–13. [30] M. A. del Pino and R. F. Man´ asevich, Global bifurcation from the eigenvalues of the p-Laplacian, J. Differential Equations 92 (1991), 226–251. [31] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Anal. 7 (1971), 487–513. [32] S. Solimini, On the solvability of some elliptic partial differential equations with the linear part at resonance, J. Math. Anal. Appl. 117 (1986), 138–152. [33] I. V. Skrypnik, Nonlinear Elliptic Boundary Value Problems” (in Russian), Naukovaja Dumka, Kyiev, 1973. English Translation: in Teubner-Texte zur Mathematik, Vol. 91. Teubner-Verlag, Leipzig, 1986. [34] P. Tak´ aˇc, On the Fredholm alternative for the p-Laplacian at the first eigenvalue, Indiana Univ. Math. J. 51 (1) (2002), 187–237. [35] P. Tak´ aˇc, On the number and structure of solutions for a Fredholm alternative with the p-Laplacian, J. Differential Equations 185 (2002), 306–347. [36] P. Tak´ aˇc, Nonlinear spectral problems for degenerate elliptic operators, in M. Chipot and P. Quittner; eds., Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 1, pp. 385–489. Elsevier Science B.V., Amsterdam, The Netherlands, 2004. [37] P. Tak´ aˇc, L. Tello, and M. Ulm, Variational problems with a p-homogeneous energy, Positivity 6 (1) (2001), 75–94. [38] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. P.D.E. 8 (7) (1983), 773–817. [39] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150. [40] J. L. V´ azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202. [41] G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, N.J., 1964.

Vol. 9 (2008)


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[42] E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, Berlin, Heidelberg, New York, 1990. Petr Girg Department of Mathematics Faculty of Applied Sciences University of West Bohemia P.O. Box 314 CZ-30614 Plzeˇ n Czech Republic e-mail: [email protected] Peter Tak´ aˇc Institut f¨ ur Mathematik Universit¨ at Rostock Universit¨ atsplatz 1 D-18055 Rostock Germany e-mail: [email protected] Communicated by Rafael D. Benguria. Submitted: July 28, 2007. Accepted: November 8, 2007.

Ann. Henri Poincaré 9 (2008), 329–346 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020329–18, published online April 10, 2008 DOI 10.1007/s00023-008-0357-9


Principe d’Incertitude et Positivité des Opérateurs à Trace ; Applications aux Opérateurs Densité Maurice de Gosson et Franz Luef Résumé. Nous discutons une condition nécessaire (mais non suffisante en général) pour qu’un opérateur ` a trace auto-adjoint soit positif. Ceci nous permet d’énoncer une relation entre l’opérateur de densité et la notion de cellule quantique symplectique introduite dans un travail précédent. Nous appliquons également le principe d’incertitude de Hardy ` a l’étude des majorations de la distribution de Wigner par des gaussiennes ; ceci nous permet de retrouver très simplement le fait que la transformée de Wigner d’une fonction de carré intégrable ne peut être ` a support compact. Abstract. We discuss a necessary (but generally not sufficient) condition for a self-adjoint trace-class operator to be positive. This allows us to state a relation between density operators and the notion of symplectic quantum cell introduced in a previous work. We also apply Hardy’s uncertainty principle to Gaussian estimates for the Wigner distribution. This allows us to recover in a very simple way the fact that the Wigner transform of a square integrable function cannot have compact support.

1. Introduction Dans un article précédent [5] l’un d’entre nous (MdG) a étudié de manière précise la relation entre la positivité des transformées de Wigner et de Husimi et la notion √ de cellule quantique (une cellule quantique est l’image d’une boule de rayon dans R2n par un automorphisme symplectique de R2n ; est la constante de Planck divisée par 2π). Dans cet article ces résultats sont reformulés en termes Les auteurs de cet article ont ét´ e financés par le projet Eucetifa MEXT-CT-2004-517154 de l’Union Europ´ eenne.

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de la notion topologique de capacité symplectique, qui permet d’énoncer le principe d’incertitude de la mécanique quantique de manière concise et invariante par symplectomorphismes. Dans cet article-ci, qui constitue une suite naturelle de [5], nous étendons cette étude au cas plus général des opérateurs densité, qui sont des opérateurs positifs a` trace unité. Outre leur intérêt intrinsèque, ces opérateurs jouent un rˆ ole important en mécanique quantique o` u ils représentent les »états mixtes« mélanges statistiques d’états bien déterminés (les »états purs«). Nous prouverons, entre autres, les propriétés suivantes : • Si Σρ est la matrice des covariances d’un opérateur densité ρ alors l’ellipso¨ıde de Wigner associée 1 −1 Σρ z, z ≤ 1 WΣρ : 2 doit contenir une cellule quantique (de manière équivalente, la capacité symplectique de cette ellipso¨ıde est au moins égale a` 1/2h) ; si la distribution de Wigner de ρ est une gaussienne, alors cette condition est également suffisante. • Nous donnons une application originale du principe d’incertitude de Hardy en montrant que si, réciproquement, la distribution de Wigner de ρ satisfait une majoration −1 1 ρ(z) ≤ Ce− 2 Σ z,z avec C > 0, Σ > 0 alors l’ellipso¨ıde 1 −1 Σ z, z ≤ 1 WΣ : 2 doit contenir une cellule quantique. (Une conséquence de cette propriété est la propriété connue que le support d’une distribution de Wigner n’est jamais compact). Notations. La forme symplectique standard sur R2n ≡ Rn × Rn est définie par σ(z, z ) = p, x − p , x si = (x, p), z = (x , p ) ( · , · désignera toujours le produit scalaire euclidien sur les espaces Rm ; la norme associée sera notée | · |). On notera par Sp(n) le groupe symplectique de (R2n , σ) : S ∈ Sp(n) si et seulement si S est un automorphisme de R2n tel que σ(Sz, Sz ) = σ(z, z ) pour tous z, z . Soit 0 I J= ; −I 0 on a alors σ(z, z ) = Jz, z et S ∈ Sp(n) si et seulement si S T JS = SJS T = J. On notera Mp(n) le groupe métaplectique de Sp(n) : c’est la représentation unitaire dans L2 (Rn ) du revêtement a` deux feuillets de Sp(n) ; la projection π : Mp(n) −→ Sp(n) est notée S −→ S. La transformation de Fourier unitaire F est définie par n/2 1 i F ψ(p) = e− p,x ψ(x)dx 2π n R pour ψ ∈ L2 (Rn ).

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2. Rappels et préliminaires Nous donnons dans cette section une rapide revue des principales propriétés des opérateurs a` trace dont nous aurons besoin ; nous rappelons aussi un résultat ancien (et malheureusement, semble-t-il, peu connu) de Kastler [11] sur la relation entre la positivité des opérateurs a` trace et la notion de fonction de type -positif (voir aussi Loupias et Miracle-Sole [14, 15]). 2.1. Opérateurs à trace Soit un espace de Hilbert séparable H et L(H) l’algèbre normée des opérateurs continus H −→ H. Un opérateur A ∈ L(H) est de Hilbert–Schmidt si pour une (et donc toutes) base othonormée (ψj ) de H on a j ||Aψj ||2H < ∞. Les opérateurs de Hilbert–Schmidt forment une sous-algèbre LHS (H) de L(H). Lorsque H = L2 (Rn ) les opérateurs de Hilbert–Schmidt sont précisément ceux dont le noyau est de carré intégrable : KA (x, y)φ(y)dy , KA ∈ L2 (Rn × Rn ) ; Aφ(x) = Rn

en particulier un tel opérateur est compact. Définissant le symbole a de A ∈ LHS (L2 (Rn )) par la formule 1 1 − i p,y (1) a(z) = e KA x + y, x − y dy 2 2 Rn c’est-à-dire

KA (x, y) =

1 2π

n

i

Rn

e p,x−y a

1 (x + y), p dp 2

on a ||a||L2 = (2π)n/2 ||KA ||L2 . De manière équivalente, pour ψ ∈ L2 (R2 ), n 1 aσ (z0 )T(z0 )ψ(x)dz0 Aψ(x) = 2π R2n o` u aσ est la transformée de Fourier symplectique de a : n 1 i aσ (z) = F a(−Jz) = e− σ(z,z ) a(z )dz . 2π 2n R

(2)

(3)

(4)

et, avec z0 = (x0 , p0 ), i 1 T(z0 )ψ(x) = e (p0 ,x− 2 p0 ,x0 ) ψ(x − x0 )

(5)

´ (T(z0 ) est l’opérateur de Heisenberg–Weyl usuel). Evidemment ||aσ ||L2 = ||a||L2 . En résumé : On a A ∈ LHS (L2 (Rn )) si et seulement si l’une des trois conditions équivalentes est satisfaite : (i) KA ∈ L2 (R2n ) ,

(ii) a ∈ L2 (R2n ) ,

(iii) aσ ∈ L2 (R2n ) .

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Soit un opérateur positif A ∈ L(H). On définit la trace de A par la formule Tr(A) = (Aψj , ψj )H i

o` u (ψj ) est une base orthonormée de H (voir [22, 23]). On a Tr(A) ∈ R+ ou bien Tr(A) = ∞. Dans les deux cas, Tr(A) ne dépend pas du choix de la base orthonormée (ψj ) (voir Reed et Simon [22]). On dit que A est »à trace« si on a Tr(|A|) < ∞. Les opérateurs a` trace forment une sous-algèbre LTr (H) de L(H) et l’on a Tr(A + B) = Tr(A) + Tr(B), Tr(λA) = λ Tr(A) pour λ ∈ C, ainsi que Tr(AB) = Tr(BA). Les opérateurs a` trace sont en particulier des opérateurs compacts. Une propriété caractéristique est : On a A ∈ LTr (H) si et seulement si A = BC ∗ avec B, C ∈ LHS (H) ; si A est en outre positif il existe B ∈ LHS (H) tel que A = BB ∗ (ou B ∗ B). Supposons de nouveau H = L2 (Rn ). Alors A est un opérateur 1 a`n trace sur ∗cσ o` bσ u L (R ) si et seulement si il existe bσ , cσ ∈ L2 (Rn ) tels que aσ = 2π ∗ est la »convolution gauche«, définie par i (bσ ∗cσ )(z) = e− 2 σ(z,z ) bσ (z )cσ (z − z )dz . (6) 2

n

R2n

En termes des symboles a, b, c cette formule s’écrit : 1 2n 1 1 i a(z) = 4π e 2 σ(u,v) b z + u c z − v dudv . 2 2 Rn ×Rn Notons que l’on a :

Tr(A) = ainsi que

Tr(A) =

1 2π

1 2π

(7)

n R2n

bσ (z)cσ (−z)dz = aσ (0)

n R2n

b(z)c(z)dz =

1 2π

(8)

n a(z)dz

(9)

(voir Grossmann et al. [7] et Loupias et Miracle-Sole [14,15] ; voir aussi la pertinente discussion de Reed et Simon [22] sur la validité des »formules de trace« en général). 2.2. Fonctions de type -positif Soit f une fonction sur R2n , a` valeurs complexes. Suivant Kastler [11] nous dirons que f est de type -positif si pour tout entier m ≥ 1 et tout (z1 , . . . , zm ) ∈ (R2n )m la matrice F = (Fjk (zj , zk ))1≤j,k≤m avec i

Fjk (zj , zk ) = e− 2 σ(zj ,zk ) f (zj − zk ) est telle que F ≥ 0. On montre que toute fonction continue de type -positif est de carré intégrable (Loupias et Miracle-Sole [14], Th. 4). L’intérêt de cette notion provient du résultat suivant :

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Proposition 1. Soit A un opérateur a ` trace sur H = L2 (Rn ). On a A ≥ 0 si et seulement si la transformée de Fourier symplectique aσ du symbole a de A est de type -positif. Cette propriété est en un sens l’analogue »quantique« du théorème bien connu de Bochner (voir Katznelson [12], p. 137) qui dit qu’une fonction définie sur R est la transformée de Fourier–Stieltjes d’une mesure positive si et seulement si elle est continue et de type positif. La démonstration de la Proposition 1 se trouve dans Kastler [11] (voir aussi Loupias et Miracle-Sole [14]). Notons que Narcowich et O’Connell en donnent dans [21] une justification heuristique (mais qui peut facilement être adaptée de manière à en faire une démonstration rigoureuse). Nous aurons besoin du résultat suivant dans notre étude de la matrice des covariances : Lemme 1. Si f : R2n −→ C est de classe C 2 au voisinage de 0 et de type -positif alors on a −2f (0) + iJ ≥ 0 (10) o` u f (0) est la matrice Hessienne de f en 0. Démonstration. (Cf. le Lemme 2.1 dans Narcowich [20]). Pour (λ1 , . . . , λm ) ∈ Cm et ε ∈ R posons R(ε) =

m

iε2 λj λk e− 2 σ(zj ,zk ) f ε(zj − zk ) .

j,k=1

Si f est de

type -positif alors R(ε) ≥ 0 pour tout ε ; choisissons les λj de fa¸con a ce que j λj = 0 ; alors R(0) = 0 et R (0) ≥ 0. Un calcul un peu long, mais ` élémentaire, montre que

R (0) = Z T − 2f (0) + i−1 J Z

avec Z = j λj zj ∈ C2n . Les λj , zj étant arbitraires on a donc −2f (0)+ i−1 J ≥ 0, ce qui démontre le lemme. 2.3. Les opérateurs densité ` partir de maintenant nous supposons que A est un opérateur a` trace positif A (donc auto-adjoint) sur l’espace de Hilbert H = L2 (Rn ). En particulier, puisque A est compact, il admet, vu la théorie de Riesz–Schauder [22], la décomposition spectrale λj Pj (11) A= j

o` u les λj > 0 sont les valeurs propres de A et les Pj les projections orthogonales de H sur les espaces propres Hj correspondants (qui sont de dimension finie : dim Hj < ∞). La trace de A est donnée par la formule λj dim Hj . Tr(A) = j

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Supposons en particulier que Tr(A) = 1. Suivant l’usage de la mécanique quantique nous appellerons alors A un opérateur densité et nous le noterons ρ. La fonction ρ = (2π)−n a est, par définition, la distribution de Wigner de ρ. Notons que l’on a, vu la formule (8), ρ(z)dz = 1 . (12) Tr( ρ) = R2n

Avec ces notations nous avons : 1 i (x + y), p ψ(y)dydp ρψ(x) = e p,x−y ρ 2 Rn ×Rn ainsi que

ρψ(x) =

R2n

ρσ (z0 )T(z0 )ψ(x)dz0

(13)

(14)

o` u ρσ est la transformée de Fourier symplectique de ρ : n i 1 ρσ (z) = e− σ(z,z ) ρ(z )dz . 2π 2n R Appliquant la formule de décomposition spectrale (11) à ρ on peut écrire ρ = αj ρj avec αj = 1 , αj > 0 j

j

o` u ρj est la projection orthogonale sur l’espace propre Hj . Soit (ψjk )k une base orthonormée de Hj = ker( ρj ) ; pour tout φ ∈ L2 (Rn ) on a (φ, ψjk )L2 ψjk ρj (φ) = k

donc le noyau de ρj est Kj = ψjk ⊗ ψjk ; son symbole aj est par conséquent donné par la formule 1 1 − i p,y aj (z) = e ψjk x + y ψjk x − y dy . 2 2 Rn k

Rappelant que la transformée (ou : distribution) de Wigner de ψ ∈ L2 (Rn ) est définie par n i 1 1 1 W ψ(z) = (15) e− p,y ψ x + y ψ x − y dy 2π 2 2 Rn on a donc le résultat suivant : pour que l’opérateur ρ défini par (13) soit un opérateur densité, il faut et il suffit qu’il existe des réels αj tels que αj ≥ 0,

2 n j αj = 1 et des fonctions ψj ∈ L (R ), ||ψj ||L2 = 1 tels que l’on ait ρ= αj W ψj ; (16) j

la fonction ρ est donc une somme convexe (en général infinie) de transformées de Wigner (15) de fonctions de carré intégrable orthonormales.

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Remarque 1. Si ρ est un opérateur densité alors 0 ≤ Tr( ρ2 ) ≤ 1 ; le nombre 2 μ( ρ) = Tr( ρ ) s’appelle la »pureté« de l’opérateur ρ, et on a μ( ρ) = 1 si et seulement si ρ = W ψ pour une fonction ψ ∈ L2 (Rn ).

3. Opérateur densité et principe d’incertitude Dans ce paragraphe nous énon¸cons et prouvons une condition nécessaire, mais non suffisante, pour qu’un opérateur a` trace soit positif. Pour les opérateurs densité cette condition s’interprète comme une conséquence du principe d’incertitude. Cette propriété peut s’énoncer de manière lapidaire dans le langage de la mécanique quantique en disant que »le principe d’incertitude est néccessaire mais non suffisant pour assurer la quantification des états«. ´ 3.1. Enonc´ e et démonstration du résultat principal ` A partir de maintenant nous supposerons toujours que ρ est un opérateur densité sur L2 (Rn ) dont la distribution de Wigner ρ satisfait la condition suivante :

La fonction z −→ 1 + |z|2 ρ(z) est dans L1 (R2n ) (17) (donc en particulier ρ ∈ L1 (R2n )). Cette condition garantira l’existence de la matrice des covariances ; notons d’ores et déjà qu’elle implique que la transformée de Fourier symplectique ρσ est de classe C2 . La condition Tr( ρ) = 1 s’écrivant ρ(z)dz = 1 les notations suggèrent (comme elles sont d’ailleurs sensées le faire !) que ρ pourrait jouer le rˆ ole d’une densité de probabilité sur R2n ; ceci n’est pas toutefois le cas en général car ρ peut prendre des valeurs négatives (voir néanmoins le paragraphe 4 ci-dessous). Ceci ne nous empêchera pourtant pas de définir, par analogie avec la mécanique statistique classique, les valeurs moyennes associées à ρ (et, en particulier, la matrice des covariances). Soit A un opérateur auto-adjoint sur L2 (Rn ), et considérons le composé ρA = ρ ◦ A. Par définition Aρ = Tr( ρA)

(18)

est la valeur moyenne de A par rapport a` ρ. Nous conviendrons d’étendre cette définition au cas o` u A est un opérateur essentiellement auto-adjoint, défini (au moins) sur l’espace de Schwartz S(Rn ) des fonctions rapidement décroissantes, ainsi que leurs dérivées de tous les ordres. Interprétant A comme un opérateur de Weyl, la formule ci-dessus s’écrit ρ(z)a(z)dz (19) Aρ = R2n

o` u a est le symbole de A. Si A et B sont (essentiellement) auto-adjoints on définit leur covariance par la formule Cov(A, B)ρ =

1 AB + BAρ − Aρ Bρ . 2

(20)

336



Choisissons en particulier A l’opérateur Xj de multiplication par la coordonnée xj et pour B l’opérateur Pj = −i∂xj . Par définition la matrice des covariances de l’opérateur densité ρ est la matrice réelle symétrique ΣXX,ρ ΣXP,ρ Σρ = ΣP X,ρ ΣP P,ρ o` u ΣXX,ρ , ΣXP,ρ = ΣTP X,ρ , et ΣP P sont les matrices de dimension n × n définies par :

ΣXX,ρ = Cov(Xj , Xk )ρ 1≤j,k≤n

ΣXP,ρ = Cov(Xj , Pk )ρ 1≤j,k≤n

ΣP P,ρ = Cov(Pj , Pk )ρ 1≤j,k≤n . Notons que puisque les symboles des opérateurs Xj Xk , Pj Pk , et 1/2(Pj Xk +Xk Pj ) sont, respectivement xj xk , pj pk , et pj xk on trouve, utilisant (19), que

Cov(Xj , Xk )ρ = xj − xj ρ xk − xk ρ ρ(z)dz 2n R

Cov(Pj , Pk )ρ = pj − pj ρ pk − pk ρ ρ(z)dz 2n R

Cov(Xj , Pk )ρ = xj − xj ρ pk − pk ρ ρ(z)dz R2n

Remarque 2. L’existence des covariances et les formules ci-dessus résulte immédiatement de la condition (17) sur ρ. Posant, conformément a` l’usage de la mécanique quantique (Messiah [17]), (ΔXj )2ρ = Cov(Xj , Xj )ρ ,

(ΔPj )2ρ = Cov(Pj , Pj )ρ

on a le résultat que voici ; il généralise la remarque suivant la démonstration de la Proposition 4 dans notre précédent article [5] : Proposition 2. Soit un opérateur densité ρ. (i) La matrice des covariances Σρ associée satisfait 1 (21) Σρ + iJ ≥ 0 2 et cette condition est équivalente aux inégalités de Heisenberg–Robertson–Schr¨ odinger

2 1 (22) (ΔXj )2ρ(ΔPj )2ρ ≥ Cov(Xj , Pj )ρ + 2 4 (j = 1, . . . , n) et (ΔXj )2ψ (ΔPk )2ψ ≥ 0 si j = k. Démonstration. (i) La matrice Σρ + 1/2iJ est hermitienne : cela résulte immédiatement de la symétrie de Σρ et du fait que J T = −J. Remarquons maintenant

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que Σρ = Σ u ρ0 est la fonction définie par ρ0 (z) = ρ(z + zρ ) avec zρ = ρ0 o` (xρ , pρ ) : on a zρ0 = 0 et donc Cov(Xj , Xk )ρ0 = xj xk ρ0 (z)dz = Cov(Xj , Xk )ρ ; R2n

de même Cov(Xj , Pk )ρ0 = Cov(Xj , Pk )ρ , Cov(Pj , Pk )ρ0 = Cov(Pj , Pk )ρ . Il suffit donc de prouver la proposition pour l’opérateur ρ0 (qui est évidemment aussi de densité). Déterminons la matrice Hessienne ρ0,σ (0). Un calcul direct montre que l’on a ΣXP,ρ0 −ΣP P,ρ0 2 ρ0,σ (0) = (2π)−n ΣP X,ρ0 −ΣXX,ρ0 c’est-à-dire 2 ρ0,σ (0)

=

1 2π

n JΣρ J .

(23)

Puisque ρσ = (2π)−n aσ la positivité de l’opérateur ρ implique, vu la Proposition 1 et le Lemme 1, que M = −2−1 JΣρ J + iJ ≥ 0 ; la condition M ≥ 0 étant équivalente a` J T M J ≥ 0 on a bien (21). (ii) L’équivalence des conditions (21) et (22) se vérifie aisément en notant que Σρ + 1/2iJ ≥ 0 est semi-définie positive si et seulement si tous ses mineurs principaux le sont (voir de Gosson [5]). Remarque 3. On vérifie facilement que la matrice Σρ est semi-définie positive. Il se trouve qu’en fait Σρ est même définie positive : ceci est prouvé dans [20]. Notons que les relations (22) impliquent les inégalités de Heisenberg usuelles (ΔXj )ρ (ΔPj )ρ ≥ 1/2 ; comme nous l’avons expliqué dans [5, 6] ces inégalités ne sont qu’une forme faible du principe d’incertitude de la mécanique quantique. Le résultat suivant montre d’ailleurs pourquoi on a tout intérêt à utiliser la formulation complète (21) de ce principe : L’opérateur composé SρS−1 est Proposition 3. Soit S ∈ Mp(n) et S = π(S). l’opérateur densité correspondant ` a la distribution de Wigner ρ ◦ S −1 et sa matrice de covariance est (24) ΣS ρS −1 = SΣρS T . Démonstration. On a SρS−1 ≥ 0 ainsi que

ρ) = 1 Tr SρS−1 = Tr( donc SρS−1 est bien un opérateur densité. On rappelle que si l’opérateur A est associé à a par (1) alors SρS−1 est associé à a ◦ S −1 ; un calcul immédiat montre qu’en outre (a ◦ S −1 )σ = aσ ◦ S −1 , donc la distribution de Wigner de SρS−1 est la fonction ρ◦S −1 . La formule (24) résulte très simplement de la relation (23) utilisée

338



lors de la démonstration de la proposition précédente : posons f (z) = 2 ρσ (S −1 z) ; on a n

1 (S −1 )T JΣρJS −1 . f (0) = (S −1 )T 2 ρσ (0) S −1 = 2π Vu les relations (S −1 )T J = JS et JS −1 = S T J (S est symplectique) on a donc n 1 f (0) = JSΣρ S T J 2π et on termine la démonstation par le même argument que dans la Proposition 2. 3.2. Matrice des covariances et cellules symplectiques Dans [5] nous avons défini les notions de »cellule quantique« et d’ »ellipso¨ıde admissible«. Une cellule√quantique est l’image par une transformation symplectique affine de la boule B( ) = {|x|2 + |p|2 ≤ } de l’espace symplectique standard (R2n , σ) ; un ellipso¨ıde admissible est un ellipso¨ıde de R2n contenant une cellule quantique. Une propriété caractéristique est la suivante : un ellipso¨ıde BM est admissible si et seulement si sa section par n’importe quel plan affine passant par son centre, et parallèle a ` ` un plan symplectique de (R2n , σ), a une aire au moins égale a 1/2h = π. (Il suffit en fait de se limiter, pour la condition suffisante, aux plans de coordonnées conjuguées xj , pj ). Il est avantageux – ne serait-ce que par souci d’économie de notations ! – d’exprimer la définition ci-dessus dans le langage de la topologie symplectique, en utilisant la notion de capacité symplectique (voir [9] pour une définition et étude détaillée de cette notion ; nous en donnons une revue dans [5]). On rappelle qu’une capacité symplectique c sur l’espace symplectique (R2n , σ) associe à tout sous-ensemble Ω de R2n un nombre c(Ω) ∈ [0, +∞] tel que : • c(Ω) ≤ c(Ω ) si Ω ⊂ Ω ; • c(f (Ω)) = c(Ω) pour tout symplectomorphisme f de (R2n , σ) ; • c(Zj (r)) = c(B(r)) = πr2 o` u Zj (r) = {z ∈ R2n : x2j + p2j ≤ r2 } et B(r) = {z ∈ R2n : |z| ≤ r}. L’existence de capacités symplectiques peut s’établir en invoquant par exemple le théorème de Gromov sur l’impossibilté d’envoyer la boule B(R) dans un cylindre Zj (r) au moyen de symplectomorphismes si r < R. Pour d’autres constructions et une étude précise de la notion de capacité syplectique, voir le traˆıté [9] de Hofer et Zehnder. Il existe une infinité de capacités symplectiques différentes sur (R2n , σ). Elles co¨ıncident toutefois sur les ellipso¨ıdes de R2n . Nous parlerons donc de la capacité symplectique d’un ellipso¨ıde. Cette capacité se calcule comme suit : soit M une matrice symétrique réelle et positive. Posons BM = z : M z, z ≤ .

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Les valeurs propres de JM ont la forme ±iλj avec λj > 0 car JM est équivalente u a la matrice antisymétrique M 1/2 JM 1/2 ; la suite Specσ (M ) = (λ1 , . . . , λn ) o` ` λ1 ≥ · · · ≥ λn est appelée le spectre symplectique de M et on a la formule c(BM ) =

π . λ1

(25)

L’un d’entre nous (MdG) a prouvé dans [5], Proposition 2, que : Un ellipso¨ıde BM est admissible si et seulement si c(BM ) ≥ 1/2h. Si c(BM ) = 1/2h alors BM contient une cellule quantique unique. Soit Σ = 1/2M −1 ; suivant la terminologie de Littlejohn [13] nous appellerons

1 −1 Σ z, z ≤ 1 z: 2 l’ellipso¨ıde de Wigner associée à Σ. Avec cette terminolgie et ces notations la Proposition 2 s’énonce concisément de manière suivante : WΣ =

Proposition 4. Si ρ est un opérateur densité, alors c(BM ) ≥ 1/2h avec M = ere équivalente : l’ellipso¨ıde de Wigner BM = WΣρ est admis1/2Σ−1 ρ . De mani` sible. Démonstration. Elle est la même, mutatis mutandis, que celle de la Proposition 4(ii) dans de Gosson [5]. Notons que l’existence de Σ−1 ρ est garantie par la Remarque suivant l’énoncé de la Proposition 2. En calcul de Weyl le symbole a d’un opérateur A borné est réel si et seulement si A est auto-adjoint (c’est d’ailleurs cette propriété qui en fait un outil de choix en mécanique quantique car elle assure que l’opérateur associé à une »observable classique« est essentiellement auto-adjoint). Toutefois, la condition a ≥ 0 n’est ni suffisante ni nécessaire pour assurer la positivité de l’opérateur A. Cette particularité est en fait une manifestation du principe d’incertitude (voir par exemple Fefferman et Phong [3]). On pourrait de prime abord penser que la condition c(BM ) ≥ 1/2h serait suffisante pour garantir la positivité de l’opérateur A. Il n’en est toutefois rien, comme Narcowich et O’Connell [21] l’ont montré sur l’exemple suivant : prenons pour symbole a la transformée de Fourier symplectique de la fonction 1 2 1 2 −(α2 x4 +β 2 p4 ) , α, β > 0 (26) aσ (x, p) = 1 − αx − 2 βp e 2 (on suppose ici n = 1). On vérifie sans difficulté que a est réel, que l’opérateur A correspondant est a` trace (avec Tr(A) = 1), et que la matrice des covariances est α 0 Σ= . 0 β La condition Σ + 1/2iJ ≥ 0 est donc équivalente a` αβ ≥ 2 /4. Toutefois, même dans ce cas nous n’avons pas A ≥ 0 : quel que soit le choix des paramètres α, β > 0

340


on trouve


p4 a(x, p)dxdp = −24α2 < 0

ce qui n’est pas possible si A ≥ 0.

4. Majorations par des Gaussiennes Le cas o` u ρ est une gaussienne est d’un intérêt tout particulier, non seulement parce qu’il est relativement facile à traiter mathématiquement, mais peut-être surtout parce que c’est celui qui est le plus utile dans les applications, notamment a` l’optique quantique dans le contexte des »états mixtes gaussiens« (voir par exemple R. Simon et al. [24, 25]). On rappelle la propriété suivante de la transformée de Wigner n 1 1 − i p,y e ψ x + y ψ(x − 12 y)dy W ψ(z) = 2π 2 Rn d’une fonction ψ ∈ L2 (Rn ) telle que W ψ ∈ L1 (R2n ) : W ψ(x, p)dp = |ψ(x)|2 et W ψ(x, p)dx = |F ψ(p)|2 . Rn

(27)

Rn

4.1. La gaussienne caractéristique d’un ellipso¨ıde Soit BM : M z, z ≤ une ellpiso¨ıde quelconque dans R2n . Convenons d’appeler »gaussienne caractéristique« de BM la fonction ρM : R2n −→ R définie par : n −1 1 1 det(M −1 )e− M z,z . (28) ρM (z) = π Si M = I (l’identité) alors ρI est la transformée de Wigner de la gaussienne φ0 définie par 2 1 (29) φ0 (x) = (π)−n/4 e− 2 |x| (c’est »l’état cohérent standard« de la mécanique quantique [13]). Plus généralement, si BM est une cellule quantique, alors ρM est la transformée de Wigner d’une Gaussienne normalisée (vouir Littlejohn [13] et de Gosson [5], Proposition 3). Rappelant la formule élémentaire e−Kz,z dz = π n (det K)−1/2 si K = K T > 0 (30) R2n

on vérifie immédiatement que

R2n

ρM (z)dz = 1

(31)

donc ρM est a priori un bon candidat pour définir un opérateur densité via les formules (13) ou (14). Il se trouve – et ceci est une conséquence du principe d’incertitude – que ce n’est le cas que si la fonction ρM n’est pas trop »concentrée« autour de l’origine dans R2n . Le résultat suivant précise cette idée en énoncant un critère nécessaire et suffisant :

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Proposition 5. La fonction ρM définie par (28) est la distribution de Wigner d’un opérateur densité ρM si et seulement si c(BM ) ≥ 1/2h (c’est-` a-dire si et seulement si l’ellpiso¨ıde BM : M z, z ≤ est admissible). Démonstration. Elle est analogue à celle de la Proposition 4 dans de Gosson [5]. Un calcul sans difficulté montre que la matrice de covariance associée à ρM est Σ = 1/2M −1. Si ρM est un opérateur densité on doit donc avoir, vu la Proposition 4, u D est la matrice diagonale M −1 + iJ ≥ 0. Soit S ∈ Sp(n) telle que M = S T DS o` Λ 0 (32) D= , Λ = diag(λ1 , . . . , λn ) 0 Λ o` u (λ1 , . . . , λn ) est le spectre symplectique de M (forme normale de Williamson [26] ; de Gosson [5] en donne une revue) ; puisque S T JS = J la condition ome caractéristique de M −1 + iJ ≥ 0 est équivalente a` D−1 + iJ ≥ 0. Le polynˆ D−1 + iJ est le produit des n facteurs −2 Fj (t) = t2 − 2λ−1 j t + λj − 1 ,

1≤j≤n

dont les zéros tj = ±1 + λ−1 sont positifs ou nuls si et seulement si λj ≤ 1, j c’est-à-dire si et seulement si c(BM ) ≥ 1/2h. Notons que lorsque la condition c(BM ) ≥ 1/2h est satisfaite, la pureté de ρM (voir la Remarque 1) est donnée par ρ2M ) = (det M )−1 . μ( ρM ) = Tr( En effet, vu la formule (9) et tenant compte du fait que le symbole aM de ρM est (2π)n ρM on a n 1 a2M (z)dz = (2π)n ρ2M (z)dz . μ( ρM ) = 2π Calculant l’intégrale au moyen de la formule (30) on a bien μ( ρM ) = (det M )−1 . 4.2. Applications du principe d’incertitude de Hardy Rappelons le théorème suivant de Hardy [8] : supposons que la fonction f ∈ L2 (Rn ) et sa transformée de Fourier F f satisfont b 2 a 2 (33) f (x) = O e− 2 |x| , F f (p) = O e− 2 |p| pour |x|, |p| → ∞, (a, b > 0). Alors : • Si ab > 1, alors f = 0 ; a

2

• Si ab = 1, il existe C ∈ C telle que f (x) = Ce− 2 |x| ; • Si ab < 1, l’ensemble des fonctions satisfaisant (33) n’est pas vide (il contient les fonctions propres de la transformation de Fourier F , qui s’expriment en termes des fonctions de Hermite usuelles).

342



Le théorème de Hardy a pour conséquence immédiate la propriété suivante de la transformation de Wigner : soit ψ ∈ L2 (Rn ) ; supposons qu’il existe une 2 1 constante C > 0 telle que |W ψ(z)| ≤ Ce− |z| . Alors ψ est proportionnelle a` la 2 1 gaussienne ψ0 (x) = (π)−n/4 e− 2 |x| . En effet, intégrant successivement en p et x cette inégalité, il vient, compte tenu des formules (27) : 2 1 2 2 − |ψ(x)| = W (ψ)(z)dp ≤ CX e |x| 2 1 2 − |F ψ(p)|2 = W (ψ)(z)dp ≤ CX e |p| 1

2

1

2

o` u CX , CP > 0. On a donc a` la fois |ψ(x)| ≤ CX e− 2 |x| et |F ψ(p)| ≤ CP e− 2 |p| et il suffit d’appliquer le théorème de Hardy. Nous allons montrer, en utilisant cette idée somme toute très simple, un résultat analogue pour l’opérateur de densité. On rappelle la formule de covariance symplectique pour la transformée de Wigner : si S ∈ Sp(n) alors (z) W ψ(S −1 z) = W Sψ (34) o` u S est l’un quelconque des deux éléments du groupe métaplectique Mp(n) couvrant S. Proposition 6. Soit ρ un opérateur densité et ρ sa distribution de Wigner. 1

(i) S’il existe C > 0 et M > 0 tels que ρ(z) ≤ Ce− Mz,z alors l’ellipso¨ıde BM : M z, z ≤ est admissible : c(BM ) ≥ 1/2h. √ (ii) Si BM est une cellule quantique S(B( )) alors ρ est proportionnelle a ` = S et φ0 est la gaussienne (29). 0 ) o` u π(S) W (Sφ Démonstration. (i) Soit S ∈ Sp(n) telle que M = S T DS, D comme dans (32). On a 1 ρ(S −1 z) ≤ Ce− Dz,z donc, rempla¸cant ρ par S−1 ρS et tenant compte du fait que la capacité symplectique est invariante par symplectomorphismes, on se ramène au cas diagonal ´ M = D. Ecrivons maintenant ρ= αj W ψj o` u αj > 0, on a

j

j

αj = 1, et ||ψj ||L2 = 1. Intégrant ρ successivement en p et en x

ρ(z)dp =

αj

j

ρ(z)dx =

j

W ψj (z)dp =

αj |ψj (x)|2

j

αj

W ψj (z)dx =

j

αj |F ψj (p)|2 ;

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puisque Dz, z = Λx, x + Λp, p on a d’autre part les inégalités 1 1 ρ(z)dp ≤ C e− Dz,z dp = C e− Λx,x 1 1 ρ(z)dx ≤ C e− Dz,z dx = C e− Λp,p avec C > 0. Il existe donc, pour chaque indice j, une constante Cj > 0 telle que 1

|ψj (x)| ≤ Cj e− 2 Λx,x ,

1

|F ψj (p)| ≤ Cj e− 2 Λp,p ;

(35)

on rappelle que Λ = diag(λ1 , . . . , λn ) ,

λ1 ≥ · · · ≥ λn .

(36)

Posons maintenant ψ1j (x1 ) = ψj (x1 , 0, . . . , 0) et notons F1 la transformation de Fourier en la première variable. Vu la première inégalité (35) on a λ1

2

|ψ1j (x1 )| ≤ Cj e− 2 x1 .

(37)

Par définition de F on a aussi n/2 i 1 F ψj (p)dp2 · · · dpn = e− p · x ψj (x)dxdp2 · · · dpn ; 2π c’est-à-dire, utilisant la formule d’inversion de Fourier, (n−1)/2 F ψj (p)dp2 · · · dpn = (2π) F1 ψ1j (p1 ) . On a donc, vu la seconde inégalité (35), (n−1)/2

n 2 1 1 |F1 ψ1j (p1 )| ≤ Cj e− 2 j=1 λj pj dp2 · · · dpn 2π c’est-à-dire λ1

2

|F1 ψ1j (p1 )| ≤ Cj e− 2 p1

(38)

pour une constante Cj > 0. Vu (36) et le théorème de Hardy on doit avoir λ21 ≤ 1, c’est-à-dire c(BM ) ≥ 1/2h vu la formule (25). 2 1 (ii) Soit S ∈ Sp(n) tel que BM : S T Sz, z ≤ . On a ρ(S −1 z) ≤ Ce− |z| et par le même raisonnement que dans la démonstration de (i) il existe Cj > 0 tel que 1 1 |x|2 |p|2 j (x)| ≤ C e− 2 j (p)| ≤ C e− 2 |Sψ , |F Sψ . j

j

j = Aj φ0 , et Vu le théorème de Hardy il existe une constante Aj ∈ C telle que Sψ donc ρ= αj Aj W S−1 φ0 = CW S−1 φ0 . j

344



Dans [4] Folland et Sitaram ont conjecturé que la transformée de Wigner d’une fonction ψ ∈ L2 (Rn ) ne peut être a` support compact. Dans [10] Janssen a démontré cette conjecture. La Proposition 6 ci-dessus permet de retrouver immédiatement ce résultat, et de l’étendre aux distributions de Wigner des opérateurs de densité. L’idée est simple : on montre que si la distribution de Wi1 gner est à support compact alors on peut trouver M avec ρ(z) ≤ Ce− Mz,z et ´ c(BM ) < 1/2h, ce qui falsifie la proposition ci-dessus. Enon¸ cons et prouvons ceci de manière précise : Corollaire. Soit ρ un opérateur densité. Le support de sa distribution de Wigner ρ n’est pas compact. Démonstration. Supposons que Supp(ρ) soit contenu dans la boule B(R) et que 2 λ ρ(z) ≤ CMax pour tout z. Pour tout λ > 0 on a ρ(z) ≤ Cλ e− |z| avec Cλ = λ 2 1 CMax e R . Choisissons λ > 1 et M = λI. On a donc ρ(z) ≤ Cλ e− Mz,z et c(BM ) < 1/2h d’o` u une contradiction vu (i) dans la Proposition 6.

5. Conclusions. . . et Perspectives On sait depuis longtemps que les questions de positivité pour les opérateurs de Weyl sont intimement associées au principe d’incertitude ; nous avons donné dans cet Article un énoncé précis et invariant par symplectomorphimes de ce principe. Ceci nous a permis d’analyser la positivité des opérateurs a` trace en termes de la notion de capacité symplectique de l’ellipso¨ıde de Wigner. Il serait évidemment très utile d’énoncer et de prouver une condition suffisante générale ; a` notre connaissance une telle condition n’a pas été énoncée clairement (et encore moins prouvée !) jusqu’` a présent (voir cependant les remarques et conjectures dans le §III de l’article de Narcowich et O’Connell [21]). Remarquons enfin que dans [1] Bonami et al. généralisent une variante du théorème de Hardy (due a` Beurling) ; en particulier ils montrent que si ψ ∈ L2 (Rn ) est telle que 2 2 a b e 2 |pj | e 2 |xj | |F ψ(p)| |ψ(x)|

N dx < ∞ ,

N dp < ∞ 1 + |xj |2 1 + |pj |2 (N un entier ≥ 0) pour tous j = 1, . . . , n, alors ψ = P φ0 , P un polynˆ ome. Il serait certainement intéressant d’appliquer leurs résutats dans le sens de la Proposition 6. Un dernier point qui mériterait sans doute d’être étudié. Nous avons vu (Proposition 2) que la condition Σρ +1/2iJ ≥ 0 est nécessaire pour que l’opérateur ρ soit de densité ; vu la même proposition, cette condition est équivalente aux inégalités

2 1 (ΔXj )2ρ (ΔPj )2ρ ≥ Cov(Xj , Pj )ρ + 2 . 4 Ces dernières restent évidemment vraies si l’on remplace par une constante < , mais l’inverse n’est pas vrai. En d’autres termes, pour un opérateur ρ

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donné la condition Σρ + 1/2iJ ≥ 0 peut être violée si l’on augmente la valeur de (que la positivité de ρ puisse a` priori dépendre de est au demeurant clair vu la Proposition 35). Quelles sont les classes d’opérateurs qui restent positifs quelle que soit la valeur de , et quelle est l’interprétation de ces opérateurs en mécanique quantique ? Un début de réponse se trouve dans les travaux de Narcowich [18, 19] et de Bröckner et Werner [2], mais on manque de résultats complets dans le cas non gaussien, qui est loin d’être trivial.

Références [1] A. Bonami, B. Demange et P. Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transform, Rev. Mat. Iberoamericana 19 (1) (2003) 23–55. [2] T. Br¨ ockner et F. Werner, Mixed states with positive Wigner fuctions, J. Math. Phys 36 (1) (1995), 62–75. [3] C. Fefferman et D. H. Phong, The uncertainty principle and sharp G˚ arding inequalities, Comm. Pure Appl. Math. 75 (1981) 285–331. [4] G. B. Folland et A. Sitaram, The uncertainty principle : A mathematical survey, Journ. Fourier Anal. Appl. 3 (3) (1997), 207–238. [5] M. de Gosson, Cellules quantiques symplectiques et fonctions de Husimi–Wigner, Bull. Sci. Math. 129 (2005), 211–226. [6] M. de Gosson, Uncertainty principle, phase space ellipsoids and Weyl calculus, Operator Theory : Advances and applications. Vol. 164 (2006), 121–132, Birkh¨ auser Verlag Basel. [7] A. Grossmann, G. Loupias, et E. M. Stein, An algebra of pseudo-differential operators and quantum mechanics in phase space, Ann. Inst. Fourier, Grenoble 18 (2) (1968), 343–368. [8] G. H. Hardy, A theorem concerning Fourier transforms, J. London. Math. Soc. 8 (1933), 227–231. [9] H. Hofer et E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkh¨ auser Advanced Texts, Birkh¨ auser Verlag, 1994. [10] A. J. E. M. Janssen, Proof of a conjecture on the supports of Wigner distributions, Journ. Fourier Anal. Appl. 4 (6) (1998), 723–726. [11] D. Kastler, The C ∗ -algebras of a free boson field, Commun. math. Phys. 1 (1965), 14–48. [12] Y. Katznelson, An Introduction to Harmonic Analysis, Dover, New York (1976). [13] R. G. Littlejohn, The semiclassical evolution of wave packets, Physics Reports 138 (4–5) (1986), 193–291. [14] G. Loupias et S. Miracle-Sole, C ∗ -Algèbres des systèmes canoniques, I, Commun. math. Phys. 2 (1966), 31–48. [15] G. Loupias et S. Miracle-Sole, C ∗ -Algèbres des systèmes canoniques, II Ann. Inst. Henri Poincaré 6 (1) (1967), 39–58.

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[16] O. V. Man’ko, V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, et F. Zaccaria, Does the uncertainty relation determine the quantum state ?, Phys. Lett. A 357 (2006), 255–260. [17] A. Messiah, Mécanique Quantique. Dunod, Paris, 1961, 1995 (Vol. 1) [traduction anglaise : Quantum Mechanics, North–Holland, 1991]. [18] F. J. Narcowich, Conditions for the convolution of two Wigner distributions to be itself a Wigner distribution, J. Math. Phys. 29 (9) (1988), 2036–2041. [19] F. J. Narcowich, Distributions of -positive type and applications, J. Math. Phys. 30 (11) (1989), 2565–2573. [20] F. J. Narcowich, Geometry and uncertainty, J. Math. Phys. 31 (2) (1990). [21] F. J. Narcowich et R. F. O’Connell, Necessary and sufficient conditions for a phasespace function to be a Wigner distribution, Phys. Rev. A 34 (1) (1986), 1–6. [22] M. Reed et B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York, 1972. [23] B. Simon, Trace Ideals and Their Applications, Cambridge University Press, 1979. [24] R. Simon, E. C. G. Sudarshan et N. Mukunda, Gaussian–Wigner distributions in quantum mechanics and optics, Phys. Rev. A 36 (8) (1987), 3868–3880. [25] R. Simon, N. Mukunda et B. Dutta, Quantum noise matrix for multimode systems : U(n)-invariance, squeezing and normal forms, Phys. Rev. A 49 (1994), 1567–1583. [26] J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math. 58 (1963), 141–163. Maurice de Gosson et Franz Luef Universit¨ at Wien Nu HAG, Mathematische Fakult¨ at Nordbergstrasse 15 A-1090 Wien Austria e-mail: [email protected] [email protected] Communicated by Jean Bellissard. Submitted: February 19, 2007. Accepted: October 3, 2007.

Ann. Henri Poincaré 9 (2008), 347–372 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020347-26, published online April 10, 2008 DOI 10.1007/s00023-008-0358-8


A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners along a Hypersurface Vincent Bonini and Jie Qing Abstract. In this paper we take an approach similar to that in [13] to establish a positive mass theorem for spin asymptotically hyperbolic manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a solution to a perturbed eigenfunction equation to obtain an asymptotic expansion of the solution in the right order. This allows us to understand the change of the mass aspect of a conformal change of asymptotically hyperbolic metrics.

1. Introduction In this paper we study the change of mass aspect for asymptotically hyperbolic manifolds under a conformal change of metric and establish a positive mass theorem for a class of asymptotically hyperbolic manifolds admitting corners along a hypersurface. This work follows an approach similar to that in [13]. The dimensions of all manifolds concerned in this paper are greater than 2. Positive mass theorems for asymptotically hyperbolic manifolds have been studied in many works, notably in [3,6,14,21]. A Riemannian manifold (M, g) with corners along a hypersurface Σ is a manifold that is separated by an embedded hypersurface Σ ⊂ M such that each individual part is a smooth Riemannian manifold and the metric g is continuous across the hypersurface Σ. An asymptotically hyperbolic manifold with corners along a hypersurface is a Riemannian manifold with corners along a hypersurface with one part compact and the other part asymptotically hyperbolic. The issue at hand is to investigate the validity of a positive mass theorem for asymptotically hyperbolic manifolds with corners along a hypersurface if each part satisfies the The first named author supported by MSRI Postdoctoral Fellowship. The second named author supported partially by NSF grant DMS 0402294 .

348

V. Bonini and J. Qing


scalar curvature condition. A good motivation given in [13] to initiate the study of such question is to use the Ricatti equation ∂H , (1.1) R = RΣ − |A|2 + H 2 − 2 ∂n which allows one to consider the scalar curvature in distributional sense across the hypersurface. It also turns out to relate to a notion of quasi-local mass in relativity (cf. [4, 13, 19, 20]. It is desirable to have a non-negative quantity associated with a compact domain Ω of an asymptotically hyperbolic manifold M , which is zero if and only if Ω can be isometrically embedded into the hyperbolic space and converges to the total mass when Ω exhausts M . Analogous to the suggestion for the asymptotically flat setting in [4], a natural candidate for such a quantity is given by taking the infimum of the total mass over the class of all asymptotically hyperbolic manifolds in which Ω can be isometrically embedded and to which positive mass theorem can apply. For more details readers are referred to [4, 13, 19, 20]. In case of an asymptotically hyperbolic manifold with corners along a hypersurface we will call the compact part the inside and the non-compact part the outside. We will denote the mean curvature of the hypersurface with respect to the inside metric in the outgoing direction by H− and the mean curvature of the hypersurface with respective to the outside metric in the direction inward to the outside by H+ . Our main theorem is as follows: Theorem 1.1. Suppose that (M n , g) is a spin asymptotically hyperbolic manifold of dimension n ≥ 3 with corners along a hypersurface. And suppose that the scalar curvature of both the inside and outside metrics are greater than or equal to −n(n − 1) and that H− (x) ≥ H+ (x) for each x on the hypersurface. Then, if in a coordinate system at the infinity, ρn −2 2 n+1 h + O(ρ g = sinh ρ dρ + g0 + ) , n then Trg0 h(x)dvolg0 (x) ≥ xTrg0 h(x)dvolg0 (x) . (1.2) n−1 n−1 S

S

In [21] the vanishing of the mass is proved to imply the asymptotically hyperbolic manifold is isometric to the hyperbolic space. However, we did not find it is a straightforward consequence to have the same conclusion in our context nor did Miao in [13] in the context of asymptotically flat manifolds. We will give an affirmative answer to this question in a forthcoming paper. We would like to point out though it is easy to see that the scalar curvature should be the constant as the hyperbolic space. We adopt an approach from [13] to smooth the corners, then conformally deform the metric so that the scalar curvature is greater than or equal to −n(n−1) and then apply the positive mass theorem in [21]. Instead of solving an equation

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349

which is a perturbation of Laplace equation as in [13, 18] for asymptotically flat case, we realize, with our experience in [5,16], that we should consider an equation which is a perturbation of the eigenfunction equation −Δv + nv = 0

(1.3)

on an asymptotically hyperbolic manifold, where n ∂2 Δ= ∂x2i i=1 on Rn in our notation in this paper. We also learned that in fact in each case the operator is simply the linearization of the Yamabe equation at the constant scalar curvature one. One of the consequences of this consideration gives hope that v decays in the right order to allow us to estimate the change of mass aspect after a conformal change of metric while another is the following key observation. Lemma 1.2. Suppose that (M n , g) is a Riemannian manifold and v is a positive smooth solution to the linear equation − − n−2 n−2 (1.4) −Δv + nv − R + n(n − 1) v = R + n(n − 1) . 4(n − 1) 4(n − 1) 4

Then the scalar curvature of the metric gv = (1 + v) n−2 g satisfies Rgv ≥ −n(n − 1) .

(1.5)

To find a solution v to (1.4) we use the analysis of weighted function spaces and uniformly degenerate elliptic equations, which are well developed in, for example, [1, 2, 8–12]. The positivity of the solution v to (1.4) follows from a clever use of a generalized maximum principle in [15]. We have noticed that the existence of the expansion of the solution v was studied in [2, 12]. But we need the explicit formula to estimate the change of mass aspects here. We followed the approach taken in [18] which used an integral representation to obtain an asymptotic expansion. To obtain an integral representation we used an explicit formula for the fundamental solution to the eigenfunction equation in the hyperbolic space cn (1.6) θ cosh dH (x, y) , GH (x, y) = n−2 2 sinh dH (x, y) cosh dH (x, y) where dH (x, y) is the hyperbolic distance between x and y in hyperbolic space H n , 1 , cn = (n − 2)vol(S n−1 ) ∞

i 1 n θ(s) = 1+ (1.7) 1− s−2i+2 θ0 2j + n − 1 i=2 j=2 and θ0 = 1 +

∞

i 1− i=2 j=2

n 2j + n − 1

.

(1.8)

350



For more detailed account on the above generalized eigenfunctions please see [2,12]. Thus Lemma 1.3. Suppose that (M n , g) is an asymptotically hyperbolic manifold, Mc is a compact set in M and r0 is a large number. Let x = ψ(p) : M \ Mc → Rn \ Br0 (0) , be a coordinate at the infinity in which ρn −2 2 n+1 g = sinh ρ dρ + g0 + h + O(ρ ) , n where sinh ρ = |x|−1 . Suppose that v ∈ Cδ2,α (M ) with δ > 0 solves the equation −Δv + nv + f v = w , with

f ∈ Cκ0,α (M ) and w ∈ Cη2,α (M ) , for some κ > 2 and η > n + 1. Then

x v(x) = A |x|−n + O |x|−(n+1) |x|

(1.9)

for some function A on S n−1 . x Note that the function A( |x| ) in the above lemma in our proof will be given as a sum of several integrals which later allow us to estimate the size of change of the mass aspects, please see Lemma 6.5 in this note. The paper is organized as follows: Section 2 is devoted to establishing an isomorphism theorem for a class of uniformly degenerate operators based on work in [10]. In Section 3 we introduce a linear equation whose solution gives a conformal factor for a metric with the scalar curvature greater than or equal to −n(n − 1). In Section 4 we derive an explicit formula for the fundamental solutions to the eigenfunction equation on hyperbolic space H n . In Section 5 we use the standard fundamental solution to construct an approximate fundamental solution on an asymptotically hyperbolic manifold. This gives us an integral representation of a solution to the eigenfunction equation and the desired asymptotic expansion. In Section 6 we prove our main theorem by calculating the mass aspect of the deformed metric and applying the positive mass theorem in [21].

2. Analytic preliminaries In this section we discuss some preliminaries of the analysis on weakly asymptoti¯ n be a smooth compact n-dimensional manifold cally hyperbolic manifolds. Let M n ¯ with boundary ∂M and M be its interior. A nonnegative smooth function ρ on M is said to be a defining function for ∂M if ρ>0 ρ=0

in M on ∂M

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and dρ never vanishes on ∂M . For any non-negative integer m and any 0 ≤ β < 1, a smooth Riemannian metric g on M is then said to be conformally compact of class C m,β if for any defining function ρ for ∂M , the conformal metric g¯ = ρ2 g ¯ . The metric g¯ restricted to T (∂M ) induces a extends as a C m,β metric on M metric gˆ := g¯|T (∂M) on ∂M which rescales upon change in defining function and therefore defines a conformal structure [ˆ g] on ∂M called the conformal infinity of (M, g). When m + β ≥ 2, a straightforward computation as in [11] shows that the sectional curvatures of g approach −|dρ|2g¯ at ∂M . As in [5], we define weakly asymptotically hyperbolic manifolds as follows: Definition 2.1. A connected complete Riemannian manifold (M n , g) is said to be weakly asymptotically hyperbolic of class C m,β if g is conformally compact of class C m,β with m + β ≥ 2 and |dρ|2g¯ = 1 on ∂M for a defining function ρ. We will use the definitions of weighted function spaces from the papers of Lee [9, 10] (see also [1, 8]. Let (M n , g) be a weakly asymptotically hyperbolic manifold and let ρ be a defining function. The weighted H¨ older spaces are defined, for δ ∈ R, (2.1) Cδk,α (M ) := ρδ C k,α (M ) = ρδ u : u ∈ C k,α (M ) with the norm uC k,α (M) := ρ−δ uC k,α (M) . δ

The weighted Sobolev spaces are defined, for δ ∈ R, Wδk,p (M ) := ρδ W k,p (M ) = ρδ u : u ∈ W k,p (M )

(2.2)

with the norm uW k,p := ρ−δ uW k,p (M) . δ

We recall the following weighted Sobolev embedding theorem from [10]. Lemma (Sobolev embedding). Let (M n , g) be weakly asymptotically hyperbolic manifold of class C m,β and U ⊂ M an open subset. For 1 < p, q < ∞, 0 < α < 1, δ ∈ R, 1 ≤ k ≤ m, and k + α ≤ m + β, the inclusions n n Wδk,q (U ) → Wδj,p (U ) for k − ≥ j − (2.3) q p and

Wδk,p (U ) → Cδj,α (U )

for

k−

n ≥j+α p

(2.4)

are continuous. The readers are referred to [10] (see also [1,8,9] for a more complete discussion of properties of the weighted H¨ older and Sobolev spaces on weakly asymptotically hyperbolic manifolds. Our goal in this section is to derive an isomorphism result from [8, 10], particularly Theorem C in [10], for the operator −Δ + n + f . We first state a simpler version of Theorem C in [10].

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Lemma 2.2. Suppose that (M n , g) is a weakly asymptotically hyperbolic manifold of class C m,β . Let k + 1 + α ≤ m + β and f ∈ Cγ0,α for some γ > 0. Then −Δ + n + f : Cδ2,α (M ) → Cδ0,α (M ) is a zero index Fredholm operator whenever δ ∈ (0, n). The possible kernel is the L2 -kernel of −Δ + n + f . Then we derive an isomorphism result by asking that −Δ + n + f is a perturbation of −Δ + n with the negative part of f small in integral sense. We will denote f = f+ − f− where f + = max{f, 0} and f − = − min{f, 0}. Proposition 2.3. Suppose that (M n , g) is a weakly asymptotically hyperbolic manifold of class C m,β . Let 4 ≤ m + β and f ∈ Cγ0,α for some γ > 0. Then there is a positive number 0 such that, if n2 n |f − | 2 dvol ≤ 0 , (2.5) M

then

−Δ + n + f : Cδ2,α (M ) → Cδ0,α (M ) is an isomorphism when δ ∈ (0, n).

(2.6)

Proof. Suppose that v is a function in the L2 -kernel of the operator −Δ + n + f . Due to some standard weighted L2 estimates (cf. Lemma 4.8 in [10], for instance) we know that v ∈ W 2,2 (M ) and solves the equation −Δv + nv + f v = 0 .

(2.7)

Let ρ be a geodesic defining function for the weakly asymptotically hyperbolic manifold (M n , g). For > 0 let M = p ∈ M : 0 < ρ(p) < . Multiplying (1) by v and integrating by parts over M \M we see 0= −vΔv + f v 2 + nv 2 M\M ∂v dσ . = (|∇v|2 + nv 2 ) + f v2 + v ∂ n M\M M\M {ρ= } Now v ∈ W 2,2 (M ) so for a fixed small number 1 > 0 1 ds = |v||∇v|dσ |v||∇v| < ∞ . s 0 ρ=s M\M1 Therefore, there is a sequence of i → 0 such that |v||∇v|dσ → 0 , ρ= i

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which implies

M

|∇v|2 + nv 2 = −

Then, by H¨ older inequality, |∇v|2 + nv 2 ≤ M

M

f − v2 ≤

353

f v2 .

M

n

(f − ) 2

n2

2n

1− n2

v n−2

M

.

M

Next we apply the Sobolev embedding theorem and obtain n2 2 2 − n 2 |∇v| + nv ≤ C |∇v|2 + v 2 , (f ) M

M

(2.8)

M

where C here is the Sobolev constant, which is independent of v. Thus, for 1

0 = , 2C we may conclude that v = 0. So the proposition follows from Lemma 2.2.

3. Conformal deformations In this section we discuss the conformal deformation of the scalar curvature on an asymptotically hyperbolic manifold (M n , g). This idea comes from the work in [18] where the analogous situation was treated in the context of asymptotically flat manifolds. Lemma 3.1. Suppose that v is a positive solution to the following equation − − n−2 n−2 R + n(n − 1) v = R + n(n − 1) −Δv + nv − 4(n − 1) 4(n − 1) on a manifold (M n , g). Then 4 R (1 + v) n−2 g ≥ −n(n − 1) . Proof. Let u = 1 + v. Then −Δu +

n−2 n(n − 2) n−2 Ru = −Δv + R + n(n − 1) u − u 4(n − 1) 4(n − 1) 4 − n−2 ≥ −Δv + nv − R + n(n − 1) v 4(n − 1) − n−2 R + n(n − 1) − 4(n − 1) n(n − 2) (1 + v) − nv − 4 4 v 1 + n−2 n+2 (n − 2) 1+v n−2 n(n − 1) =− u . 4 4(n − 1) (1 + v) n−2

(3.1)

354



Hence to prove the lemma is to show that 4 4 1 4 − ≤ (1 + v) n−2 . (3.2) 1+ n−2 n−21+v We differentiate the two sides with respect to v and compare 4 4 4 (1 + v)−2 < (1 + v) n−2 −1 . n−2 n−2 Therefore, by the fact that the two sides are the same when v = 0, the lemma follows. The rest of this section is devoted to solving for a positive solution to the equation (−Δ + n + f )v = h (3.3) n on an asymptotically hyperbolic manifold (M , g) with the function f suitably small in an integral sense. By the isomorphism proposition in the previous section we know, for δ ∈ (0, n) and each h ∈ Cδ0,α (M ), there is a unique solution v ∈ Cδ2,α (M ) to the equation (3.3). Hence what really need to do is to show that v > 0 in M . For simplicity we will denote − n−2 R + n(n − 1) ≤ 0 . f =− 4(n − 1) Proposition 3.2. Suppose that (M n , g) is a weakly asymptotically hyperbolic manifold of class C m,β with m + β ≥ 4. Let 0 be the small positive number in Proposition 2.3 in the previous section and α ∈ (0, 1). Suppose that f ∈ Cδ0,α (M ) for some δ ∈ (0, n) and that n2 n |f | 2 ≤ 0 . (3.4) M

Then there is a positive solution v ∈ Cδ2,α (M ) to the equation −Δv + nv + f v = −f .

(3.5)

Proof. We first prove that v has to be nonnegative in M . Assume otherwise that v is negative somewhere in M so that v− = min v(p) : p ∈ M < 0 . Let us consider instead the function u = v + v0 for a small positive number v0 < min{1, − v2− }. Then −Δu + nu + f u = −f (1 − v0 ) + nv0 > 0 in M and min{u(p) : p ∈ M } < 0. Since v ∈ Cδ2,α (M ) for δ > 0, for a geodesic defining function ρ, we may assume that u > 0 on ∂(M \ Mτ ) = p ∈ M : ρ(p) = τ provided that τ > 0 is sufficiently small. Now we are going to apply the generalized maximum principle in Section 2.5 in [15] to the function u on the manifold M \Mτ . According the generalized maximum principle what we need is to verify that the

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first eigenvalue of the operator −Δ + n + f on the domain M \ Mτ for some τ < τ with Dirichlet boundary condition is positive. Therefore, for any φ ∈ Cc∞ (M \Mτ ), we consider the ratio (|∇φ|2 + nφ2 + f φ2 ) M φ2 M n2 n 1 1 ≥ (|∇φ|2 + φ2 ) − C |f | 2 (|∇φ|2 + φ2 ) ≥ . 2 2 φ M M M M Thus the first eigenvalue of the operator −Δ + n + f on the domain M \ Mτ with the Dirichlet boundary condition is always positive. We may apply Theorem 10 in Section 2.5 of the book [15] to the function u/φ, where φ is the positive first eigenfunction over M \ Mτ , to obtain a contradiction. Therefore v is nonnegative in M . To show that v is in fact positive in M , for each τ > 0, we apply the Hopf strong maximum principle to the function v/φ on the domain M \ Mτ , where φ is the positive first eigenfunction over M \ Mτ for any 0 < τ < τ . Thus the proof is complete.

4. The fundamental solutions on the hyperbolic space The materials in this section are well known and readers are refered to [1, 2, 10, 12] for more detailed account on the references. But for the convenience of the readers we will present a construction briefly. Let us first recall the definition of the hyperbolic space as a hyperboloid in the Minkowski space-time. The Minkowski space-time is Rn+1 equipped with the Minkowski metric −dt2 + |dx|2 for (t, x) ∈ Rn+1 . The upper hyperboloid is the submanifold (4.1) H n = (t, x) ∈ Rn+1 : −t2 + |x|2 = −1, t > 0 . Hence

(d|x|)2 n 2 (H , gH ) = R , + |x| gS n−1 , 1 + |x|2 n

(4.2)

where gS n−1 is the standard metric on the unit round (n − 1)-sphere. We want to find the solution to the equation −ΔH n G0 (x) + nG0 (x) = δ0 (x) ,

(4.3)

which defines the Green’s function in x centered at the origin of the differential operator −Δ + n on hyperbolic space H n . We first compute, for r = |x|, (−ΔH n + n)r−n+2 t−k = −(k − 2)(k + n − 1)r−n+2 t−k + k(k + 1)r−n+2 t−k−2 .

356



We then observe inductively that, for even number k (−ΔH n + n) −n+2 t−2 + r

2 · 3 −4 2 · 3 · 4 · 5 · · · (k − 1) −k t + ···+ t 2(n + 3) (2(n + 3) · · · (k − 2)(k + n − 1) 2 · 3 · 4 · 5 · · · k · (k + 1) = r−n+2 t−k−2 . (2(n + 3) · · · (k − 2)(k + n − 1)

Therefore we consider the function ∞

i ˜ = 1+ θ(t) 1− i=2 j=2

n 2j + n − 1

1 t2i−2

.

(4.4)

Notice that the infinite series θ˜ is obviously convergent when t > 1. In fact, when t = 1, taking the logarithm of the general term we see i

n−1 n n log 1− ≤ − log i + + c(n) 2j + n − 1 2 2 j=2 for some dimensional constant c(n). Thus the infinite series ∞

i n ˜ θ(1) = 1 + 1− 2j + n − 1 i=2 j=2

(4.5)

converges for all n ≥ 3. We set θ(t) =

˜ θ(t) ˜ θ(1)

(4.6)

and easily conclude that Lemma 4.1. Let G0 (x) =

1 θ(t) . (n − 2)vol(S n−1 ) rn−2 t2

(4.7)

Then −ΔH n G0 (x) + nG0 (x) = δ0 (x) n

on hyperbolic space H . To write the fundamental solution at any point in the hyperbolic space we want to express hyperbolic translation in the hyperboloid model of hyperbolic space H n . Recall that the changes of coordinates between the ball model and hyperboloid model of the hyperbolic space are x=

2 x ¯, 1 − |¯ x|2

and x ¯=

t=

1 x. 1+t

1 + |¯ x|2 , 1 − |¯ x|2

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357

Also recall that hyperbolic translation by ¯b in the ball model is given in [17] by x · ¯b + 1 ¯ 1 − |¯b|2 |¯ x|2 + 2¯ x) = x ¯ + b, (4.8) τ¯b (¯ 2 2 2 2 ¯ ¯ ¯ |¯ x| |b| + 2¯ x·b + 1 |¯ x| |b| + 2¯ x · ¯b + 1 where tx = 1 + |x|2 and tb = 1 + |b|2 . Therefore we have x·b b 1 + tb

(4.9)

|Tb (x)| = sinh dH (x, −b) .

(4.10)

cosh dH (x, b) = tx tb − x · b .

(4.11)

GH (x, y) = Gy (x) = G0 T−y (x) .

(4.12)

Tb (x) = x + tx b + with One key fact here is that Thus and explicitly GH (x, y) =

sinh

n−2

cn θ cosh dH (x, y) 2 dH (x, y) cosh dH (x, y)

where cn =

1 . (n − 2)vol(S n−1 )

(4.13)

(4.14)

5. Asymptotic behavior So far, for a weakly asymptotically hyperbolic manifold (M n , g) with n − n2 − 0,α R + n(n − 1) R + n(n − 1) ∈ Cδ and ≤ 02 , M

4

we have obtained a conformal deformation gv = (1 + v) n−2 g such that R[gv ] ≥ −n(n − 1) and

0 < v ∈ Cδ2,α (M ) , provided that δ ∈ (0, n). Unfortunately the decay rate of v just misses the decay rate on which the mass aspect of an asymptotically hyperbolic manifold is defined. We will use the Green’s function we constructed in the pervious section to obtain an expansion at the infinity of the solution v to the equation − − n−2 n−2 R + n(n − 1) v = R + n(n − 1) . −Δv + nv − 4(n − 1) 4(n − 1) We follow the idea used in [18] to write an integral representation of the solution v with the help of the approximate Green’s function GH (x, y) on the asymptotically hyperbolic manifold M . Let us start with a definition of asymptotically hyperbolic manifolds, which should be compared with the definition of weakly asymptotically hyperbolic manifolds given in Section 2. Since we will adopt the definition of mass

358



aspect and mass for asymptotically hyperbolic manifolds from the work [21] we use his definition for asymptotically hyperbolic manifolds. Definition 5.1. (M n , g) is said to be an asymptotically hyperbolic manifold if (M n , g) is a weakly asymptotically hyperbolic manifold with the standard round sphere (S n−1 , [g0 ]) as its conformal infinity, and, for a geodesic defining function ρ, in the conformally compact coordinates at the infinity, 1 n −2 2 n+1 g = sinh ρ dρ + g0 + ρ h + O(ρ ) , (5.1) n where h is symmetric two tensors on S n−1 at each point. In the light of the above definition, we set up a conformally compact coordinate at the infinity associated with a defining function ρ as follows. Let ψ : M \ Mc → Rn \ Br0 (0) , for some compact subset Mc ⊂ M , such that gH =

(d|x|)2 + |x|2 g0 = sinh−2 ρ(dρ2 + g0 ) 1 + |x|2

(5.2)

1 for |x| > r0 and sinh ρ = |x| . We construct an approximate Green’s function of an asymptotically hyperbolic manifold (M n , g). At each point y ∈ Rn \ Br0 (0), we consider the hyperbolic space H n in the coordinate so that 1 dr2 + ry2 (x)g0 = (˜ gH (x) = gH )ij (x)dxi dxj , 1 + ry2 (x)

where ry (x) =

Aij (y)xi xj .

(5.3)

This coordinate can be made into the standard coordinate by the linear transformation B : Rn → Rn such that B 2 = A. More importantly we need to ask (˜ gH )ij (y) = gij (y) .

(5.4)

A simple calculation yields Aik xk Ajl xl . 1 + Akl xk xl

(5.5)

gik (y)yk gjl (y)yl . 1 − gkl (y)yk yl

(5.6)

(˜ gH )ij (x) = Aij − Hence Aij (y) = gij (y) + Therefore, since gij (x) = δij −

xi xj ˜ ij (x) + O |x|−n h 1 + |x|2

(5.7)

and ˜ ij (x)xj = 0 , h

(5.8)

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we have Aij (y) = gij (y) +

yi yj ˜ ij (y) . = δij + O |y|−n h 2 1 + |y|

359

(5.9)

gH )ij (x)dxi dxj Let d˜H (x, y) be the hyperbolic distance function in the metric (˜ and let θ cosh d˜H (x, y) 1 Gy (x) = . (5.10) (n − 2)vol(S n−1 ) sinhn−2 d˜H (x, y) cosh2 d˜H (x, y) In the geodesic ball B1 (y) in the metric g we calculate gij (x) = (˜ gH )ij (x) + d˜H (x, y)O |y|−n and

1 det gg ij ∂j ∂j det g = ΔH + d˜H (x, y)O |y|−n ΔH + (˜ gH )ij ∂i O(|y|−n )d˜H (x, y) ∂j .

Δg = √

(5.11)

(5.12)

Thus, for any x ∈ B1 (y) and x = y,

Ψy (x) = −Δg Gy (x) + nGy (x) = O |y|−n O d˜H (x, y)−n+1 ,

(5.13)

as |x − y| → 0 and |y| → ∞. On the other hand, outside the geodesic ball B1 (y), we simply need ˜ ij (x) + O |y|−n ξij (x, y) , gH )ij (x) + O |x|−n h gij (x) = (˜ as |x| → ∞ and |y| → ∞, which follows from some calculations, where ˜ ˜ ˜ ˜ ij (y) − hik (y)xk xj + hjk xk xi + xi xj hkl xk xl . ξij (x, y) = h 2 2 1 + |x| 1 + |x| 1 + |x|2 Therefore ˜ kl xk xl ˜ ij xj h h xi ξij xj = − 1 + |x|2 1 + |x|2 1 + |x|2 and (˜ gH )ij = δij + xi xj + O |y|−n ξij . This implies ˜ ij + higher order terms . gH )ij + O |y|−n ξij + O |x|−n h g ij = (˜ Here we use the facts that (δik + xi xk )ξkl (δlj + xl xj ) = ξij and ˜ kl (δlj + xl xj ) = h ˜ ij . (δik + xi xk )h Therefore, outside the geodesic ball B1 (y), Δg = Δg˜H + O |y|−n + O |x|−n Δg˜H

+ (˜ gH )ij ∂i O(|y|−n ) + O(|x|−n ) ∂j .

360



One last calculation we need is an estimate for Ψy (x) outside the geodesic ball B1 (y). We compute Aik (y)xk 1 ∂i d˜H (x, y) = ty − Aik (y)yk , tx sinh d˜H (x, y) 2θ(cosh s) θ (cosh s) (n − 2)θ(cosh s) G (s) = cn − − + , sinhn−1 s cosh s sinhn−3 s cosh3 s sinhn−3 s cosh2 s and

coshn d˜H (y, x)G d˜H (y, x) → −ncn ,

as d˜H (y, x) → ∞. Thus, outside the geodesic ball B1 (y), Ψy (x) = −Δg Gy (x) + nGy (x) = O |x|−n + O |y|−n O

1 n ˜ cosh dH (x, y)

.

(5.14)

Lemma 5.2. Suppose that (M n , g) is an asymptotically hyperbolic manifold. Then −ΔGy (x) + nGy (x) = δy (x) + Ψy (x)

(5.15)

where Ψy (x) satisfies the estimates (5.13) and (5.14). As a consequence we have the following integral representation. Proposition 5.3. Suppose that (M n , g) is an asymptotically hyperbolic manifold and that ψ : M \ Mc → Rn \ Br0 (0) is a conformally compact coordinate associated with a defining function ρ in which ρn g = sinh−2 ρ dρ2 + g0 + h + O ρn+1 . n Suppose that v ∈ Cδ2,α (M ) solves the equation −Δv + nv + f v = w ∈ Cδ0,α (M ) , where f ∈ Cδ0,α (M ) and δ ∈ (0, n). Then, for each x ∈ Rn \ Br0 (0), v(y)Ψx (y)dvolg (y) v(x) = −

Rn \Br0 (0)

+ −

Rn \Br0 (0)

∂Br0 (0)

+ ∂Br0 (0)

w(y) − f (y)v(y) Gx (y)dvolg (y)

∂Gx (y)v(y)dσg (y) ∂n ∂v (y)Gx (y)dσg (y) . ∂n

(5.16)

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361

Proof. We use the density property (cf. [10]) of the the space Cδ2,α (M ) to have a sequence of functions vn ∈ Cc∞ (M ) such that vn → v Then from (5.16) we have, for vn , vn (x) = −

Rn \Br0 (0)

+ Rn \Br0 (0)

−

∂Br0 (0)

+

∂Br0 (0)

in

Cδ2,α (M ) .

vn (y)Ψx (y)dvolg (y) (−Δvn + nvn )Gx (y)dvolg (y)

∂Gx (y)vn (y)dσg (y) ∂n

(5.17)

∂vn (y)Gx (y)dσg (y) . ∂n

Hence, by taking the limit, we obtain (5.16) for v.

Now we are ready to state and prove our main result of this section. Theorem 5.4. Suppose that (M n , g) is an asymptotically hyperbolic manifold and that ψ : M \ Mc → Rn \ Br0 (0) is a conformally compact coordinate associated with a defining function ρ in which ρn h + O(ρn+1 ) . g = sinh−2 ρ dρ2 + g0 + n Suppose that v ∈ Cδ2,α (M ) with δ > 0 solves the equation −Δv + nv + f v = w with f ∈ Cκ0,α (M )

and

w ∈ Cη2,α (M )

for some κ > 2 and η > n + 1. Then, for each x ∈ Rn \ Br0 (0),

x v(x) = A |x|−n + O |x|−(n+1) . |x|

(5.18)

Remark 5.5. We would like to point out that the expansion (5.18) is a simple consequence of the work in [2, 12]. But we need some explicit expression of the coefficient A in (5.18) to prove Theorem 6.3 and Lemma 6.5 in the following section, which we did not find that it is easier to extract it from [2, 12] than to obtain it in the way presented here. The explicit expression of A will be obtained in the course of the following proof of Theorem 5.5 based on the integral representation of the solution v in (5.16).

362



Proof of Theorem 5.4. We are going to study the asymptotic behavior of v(x) term by term in (5.16). We treat the easy ones first. First we consider ∂v n (y)Gx (y)dσ(y) , |x| ∂Br0 (0) ∂n as |x| → ∞ and y ∈ ∂Br0 (0). Now |x|n Gx (y) =

cn coshn−2 d˜H (y, x) |x|n θ cosh d˜H (y, x) , n−2 cosh d˜H (y, x) sinh d˜H (y, x) n

where cosh d˜H (y, x) = tx ty − Aij (x)xi y j and

Aij (x) = δij + O |x|−n .

Hence |x|n Gx (y) =

1 tx |x| ty

n

i

x − Aij (x) |x| yj

and

cn coshn−2 d˜H (y, x) θ cosh d˜H (y, x) ∈ C 1 (M ) n−2 ˜ sinh dH (y, x)

−n x x (y) = c lim λ Gλ |x| ·y . n ty − λ→∞ |x| n

Therefore n

|x| and

A1

x |x|

∂v (y)Gx (y)dσ(y) ∈ C 1 (M ) ∂n

∂Br0 (0)

∂v x (y)dσ(y) (y)Gλ |x| ∂n ∂Br0 (0) −n ∂v x (y) ty − ·y dσ(y) . = cn |x| ∂Br0 (0) ∂n n

= lim λ λ→∞

Next we consider n

|x|

∂Br0 (0)

(5.19)

∂Gx (y)v(y)dσ(y) , ∂n

as |x| → ∞ and y ∈ ∂Br0 (0). We compute |x|n

∂ d˜H (y, x) ∂Gx (y) = |x|n ρ(y)cn G d˜H (y, x) ∂n ∂r n

= |x| ρ(y)G where

tx

gij y i y j |y|ty

− gij xi sinh d˜H (y, x)

yj |y|

,

2θ(cosh s) θ (cosh s) (n − 2)θ(cosh s) G (s) = cn − − + sinhn−1 s cosh s sinhn−3 s cosh3 s sinhn−3 s cosh2 s

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and

363

coshn d˜H (y, x)G d˜H (y, x) → −ncn

as d˜H (y, x) → ∞. Therefore x ∂Gλ |x| x (y)v(y)dσ(y) A2 = lim λn λ→∞ |x| ∂n ∂Br0 (0) −n |y| − x · y x ty |x||y| = −ncn ·y dσ(y) . v(y) ty − x |x| ty − |x| · y ∂Br0 (0) For the term n

|x|

we know, for any given y

(5.20)

(h − f v)Gx (y)dvolg (y) ,

Rn \Br0 (0) ∈ Rn \ Br0 (0),

−n x x (y) = c · y lim λn Gλ |x| − . t n y λ→∞ |x|

We observe that x · y = ty − |y| cos φ ≥ (1 − cos φ)|y| , |x| x , we easily see that for where φ is the angle between x and y. Fixing a direction |x| any 0 > 0 x (y)dvol (y) lim λn (h − f v)Gλ |x| g ty −

λ→∞

{y∈Rn \Br0 (0): cos φ≤1− 0 }

= {y∈Rn \Br0 (0):

−n x ·y (h − f v) ty − dvolg (y) . |x| cos φ≤1− 0 }

On the other hand, when cos φ > 1 − 0 , it suffices to verify the claim ∞ rn−1 (h − f v)(ty − r cos φ)−n dσ0 dr < ∞ . ty r0 {cos φ>1− 0 }

(5.21)

Here we need to use the fact that η > n. We simply notice that 1 + sin2 φ|y|2 . ty + |y| cos φ

ty − |y| cos φ = Hence {cos φ>1− 0 }

−n ty − |y| cos φ dσ

0

1

S n−2 0

0

0

+

S n−2

−n n−2 ty − |y| cos φ φ dσdφ −n n−2 ty − |y| cos φ φ dσdφ −n n−2 ty − |y| cos φ φ dσdφ

1 S n−2 n n−1 |y| 1 + |y|−n −n−1 1

|y|

364



for 1 = |y|−1 < 0 . Therefore −n rn−1 (h − f v) ty − r cos φ dσ0 = O r−ι+n−1 , ty {cos φ>1− 0 } where ι = min{η, n + 12 δ} > n, which implies our claim (5.21). Thus x n x (y)dvol (y) A0 (h − f v)Gλ |x| = lim λ g λ→∞ |x| Rn \Br0 (0) −n x = ·y (h − f v) ty − dvolg (y) . |x| Rn \Br0 (0)

(5.22)

A similar argument yields the next order when we have κ > 2 and η > n + 1. For the last term |x|n v(y)Ψx (y)dvolg (y) , Rn \Br0 (0)

we need to use the estimates about the correction term Ψx (y) in (5.13) and (5.14). We first look at 1 |x|n v(y)Ψx (y)dvolg (y) |x|n v(y)Ψx (y) sinhn−1 rdσdr S n−1

0

B1 (x)

|x|n

0

1

|y|−n+ |x|−n r−n+1 rn−1 dr

for any small positive number . Clearly lim |x|n v(y)Ψx (y)dvolg (y) = 0 |x|→∞

(5.23)

B1 (x)

since |y| ≥ c|x| for y ∈ B1 (x) and |x| → ∞. Next we look at n |x| v(y)Ψx (y)dvolg (y) . (Rn \Br0 (0))\B1 (x)

In the light of (5.14) and (5.23), using the argument we used to treat last term to obtain (5.21) and (5.22), we have x x (y)dvol (y) A−1 v(y)Ψλ |x| = lim λn g λ→∞ |x| Rn \Br0 (0) (5.24) n x = lim λ v(y)Ψλ |x| (y)dvolg (y) . λ→∞

(Rn \Br0 (0))\B1 (x)

We have thus proven the theorem with x x x x x A = A−1 + A0 + A1 + A2 . |x| |x| |x| |x| |x|

(5.25)

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6. Proof of the main theorem In this section we prove the main theorem. We first recall a positive mass theorem for asymptotically hyperbolic manifolds from [21]. Readers are referred to [6] for more elaborated and complete discussions of positive mass theorems for asymptotically hyperbolic manifolds. Recall that, on an asymptotically hyperbolic manifold (M n , g) as defined in Definition 5.1, we have a coordinate at the infinity such that ρn h + O ρn+1 . g = sinh−2 ρ dρ2 + g0 + (6.1) n In [21] it was proven that Theorem 6.1 (Xiaodong Wang). Suppose that (M n , g) is a spin asymptotically hyperbolic manifold and that Rg ≥ −n(n − 1). Then (6.2) Trg0 h(x)dvolg0 (x) ≥ Trg0 h(x)xdvolg0 (x) . S n−1

S n−1

Moreover the equality holds if and only if (M n , g) is isometric to the standard hyperbolic space H n . We adopt the idea from [13] to deal with asymptotically hyperbolic manifolds with corners along a hypersurface. Definition 6.2. A Riemannian manifold (M n , g) is said to have corners along a hypersurface Σ if there is a smooth embedded hypersurface Σ ⊂ M such that M \ Σ = M− M+ and the inside (M− , g− ) = (M− , g) is a smooth compact Riemannian manifold with a boundary Σ and the outside (M+ , g+ ) = (M+ , g) is a smooth Riemannian manifold with a boundary Σ. Moreover g− and g+ agree on the boundary Σ, that is, g continuous across the hypersurface Σ ⊂ M . We will consider the outward mean curvature H− of the hypersurface Σ in (M− , g− ) and the inward mean curvature H+ of the hypersurface Σ in (M+ , g+ ). Near the hypersurface Σ we may use Gauss coordinates, that is, for some ν0 > 0, a point p within distance ν0 from the hypersurface Σ is labeled by a point x on the hypersurface Σ and the signed distance d = dist(p, Σ) to the hypersurface Σ. We now recall the smoothing operation given in Proposition 3.1 in [13] to have C 2 metrics on M approximating g. Proposition 6.3 (Pengzi Miao). Suppose that (M, g) is a manifold with corners along a hypersurface Σ. Then there is a family of C 2 metrics gν , for ν ∈ (0, ν0 ), on M such that gν uniformly converges to g on M and gν = g outside Σ × (− 12 ν, 12 ν). Furthermore, the scalar curvature Rν of the metric gν satisfies ν2 ν when d ∈ 100 ,2 Rν (p) = O(1)in 100 100 (6.3) ν2 Rν (p) = O(1) + 2(H− − H+ ) ν 2 φ ν 2 when d ≤ 100 , where O(1) stands for terms bounded independent of ν and φ(t) ∈ Cc∞ (−1, 1) is a standard mollifier.

366



Our next goal is to conformally deform the metric gν so that the scalar curvature is greater than or equal to −n(n − 1) so that the positive mass theorem in [21] applies. The reason that gν admits such conformal deformation relies on the fact that n n − 2 n−2 Rν + n(n − 1) dvolgν ≤ 02 4(n − 1) M whenever ν is sufficiently small and H− − H+ ≥ 0. Thus we are ready to state and prove our main theorem. Theorem 6.4. Suppose that (M, g) is a spin Riemannian manifold with corners along a hypersurface Σ and that the outside is an asymptotically hyperbolic manifold and the inside is compact. Suppose that the scalar curvature of both the inside and outside metrics are greater than or equal to −n(n − 1) and that H− (x) ≥ H+ (x) for each x on the hypersurface. Then, if in a coordinate system at the infinity, ρn −2 2 n+1 g = sinh ρ dρ + g0 + h + O(ρ ) , n then Trg0 h(x)dvolg0 (x) ≥ Trg0 h(x)xdvolg0 (x) . n−1 n−1 S

S

Proof. We first use the smoothing operation given in [13] as stated in the above proposition. For each small ν < ν0 , we then solve the equation −Δgν v + nv + fν v = −fν

(6.4)

on M for

− n−2 Rν + n(n − 1) . 4(n − 1) According to Proposition 6.2 above n fν2 dvolgν ≤ C(g)ν , fν = −

M

where C(g) depends only on the metric g. For sufficiently small ν we apply Proposition 3.2 in Section 3 to obtain a positive solution vν to the above equation (6.4). Then we consider the new metric 4

g˜ν = (1 + vν ) n−2 gν . ˜ ν of In the light of Lemma 3.1 in Section 3 we know that the scalar curvature R the new metric g˜ν is greater than or equal to −n(n − 1). To finish the proof we need to establish the following two lemmas. Lemma 6.5. Suppose that (M n , g) is an asymptotically hyperbolic manifold and in a coordinate at the infinity associated with a geodesic defining function r ρn −2 2 n+1 g = sinh ρ dρ + g0 + h + O(ρ ) , n

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where r= And suppose that

367

cosh ρ − 1 . sinh ρ

x ρn + O(ρn+1 ) |x| is a positive function on M . Then there is a geodesic defining function r˜ for g˜ = 4 (1 + v) n−2 g such that ρñ ˜ g˜ = sinh−2 ρ˜ d˜ h + O(˜ ρn+1 ) , ρ2 + g 0 + n v=A

where r˜ =

cosh ρ˜ − 1 sinh ρ˜

and ˜ = 4(n + 1) A h n−2

x |x|

g0 + h .

(6.5)

Proof. First we recall that the geodesic defining function of the metric g is a defining function s such that |ds|s2 g = 1 near the infinity. We refer the readers to Lemma 2.1 in [7] for the existence and uniqueness of the geodesic defining function associated with each boundary metric in the conformal infinity. We start with a geodesic defining function r for g. Then for each θ ∈ S n−1 , let r˜ = ew r and w(θ, 0) = 0 . By the definition, w satisfies

4 ∂w 1 4 2 + r|dw|2r2 g = (1 + v) n−2 − 1 = Arn−1 + O(rn ) . ∂r r n−2 By an inductive argument we obtain

(6.6)

∂kw (θ, 0) = 0 ∂rk for k ≤ n − 1 and

∂nw 2 A(θ) . (θ, 0) = (n − 1)! n ∂r n−2

(6.7)

Hence w(θ, r) =

2 A(θ)rn + O(rn+1 ) . n(n − 2)

This gives r˜(θ, r) = r +

2 A(θ)rn+1 + O(rn+2 ) . n(n − 2)

(6.8)

368



By the construction of the coordinate associated with a geodesic defining function, ∂ we need to compare the integral curves of the vector field ∂r and ∂∂r˜ . We know 2(n + 1) n 2 ∂A d˜ r = 1+ Ar dr + rn+1 dθi + O(rn+1 ) , n(n − 2) n(n − 2) ∂θi which implies 4 2(n + 1) n ∂ ∂ − n−2 1+ = (1 + v) Ar ∂˜ r n(n − 2) ∂r n+2 ij ∂ 4 ∂A 2 − n−2 n+1 (6.9) + (1 + v) +O r r gr n(n − 2) ∂θj ∂θi 2(n − 1) n ∂ ∂ − Ar + O rn+1 . = ∂r n(n − 2) ∂r Therefore ˜ r) = θ + O rn+1 . (6.10) θ(θ, Thus 4 ∂ ∂ ∂ ∂ 2 2 n−2 sinh ρ˜ g˜ , g , = sinh ρ˜ (1 + v) + O rn+1 . ˜ ˜ ∂θ ∂θ ∂ θi ∂ θj i j In the light of the fact that

sinh ρ˜ 2 = r˜ = r 1 + Arn + O(rn+1 ) 1 + cosh ρ˜ n(n − 2) sinh ρ 2 = Arn + O(rn+1 ) 1+ 1 + cosh ρ n(n − 2)

we have sinh2 ρ˜ = sinh2 ρ

1 + cosh ρ˜ 1 + cosh ρ

2 1+

4 Arn + O rn+1 , n(n − 2)

where 1 + cosh ρ˜ cosh ρ˜ − cosh ρ =1+ 1 + cosh ρ 1 + cosh ρ = 1 + O(r)(˜ ρ − ρ) = 1 + O(r) tanh−1 r˜ − tanh−1 r = 1 + O rn+1 . Finally, we arrive at

4(n + 1) n ρñ ˜ ρn h + O(˜ ρn+1 ) = g0 + ρ A(θ)g0 + h(θ) + O(ρn+1 ) , (6.11) n n−2 n which gives ˜ = 4(n + 1) A x g0 + h h (6.12) n−2 |x| So the calculation is completed. g0 +

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The next lemma is an estimate of the perturbation of mass aspect x Aν ( |x| )g0 in terms of the small number ν as ν → 0 when v = vν .

4(n+1) n−2

Lemma 6.6. Suppose that (M, g) is a complete Riemannian manifold with corners along a hypersurface and that the outside is an asymptotically hyperbolic manifold. Suppose that the scalar curvature of both the inside and outside metrics are greater than or equal to −n(n − 1) and that H− (x) ≥ H+ (x) for each x on the hypersurface. Let gν be constructed as in Proposition 6.2. Then there is a unique positive solution vν ∈ Cδ2,α (M ) to the equation − − n−2 n−2 Rν + n(n − 1) v = Rν + n(n − 1) , −Δgν v + nv − 4(n − 1) 4(n − 1) when ν is sufficiently small. Moreover, in a coordinate at the infinity associated with a geodesic defining function r, x vν = Aν rn + O(rn+1 ) |x| and

1 Aν x ≤ Cν n+1 , |x|

(6.13)

where C is independent of ν. Proof. By Proposition 6.2 we have − n−2 Rν + n(n − 1) ≤ C 4(n − 1) with compact support inside ∂Ω × [− ν2 , ν2 ], where C is independent of ν. Hence n − 2 n−2 Rν + n(n − 1) dvolgν ≤ Cν . 4(n − 1) M Therefore, by Proposition 3.2 and Theorem 5.5, there is exists the unique positive solution to the equation − − n−2 n−2 Rν + n(n − 1) v = Rν + n(n − 1) , −Δgν v + nv − 4(n − 1) 4(n − 1) when ν is sufficiently small and in a coordinate at the infinity associated with a geodesic defining function r, x vν = Aν rn + O(rn+1 ) , |x| x ) is given in (5.25). where A( |x| First of all, since n−2 − 4(n − 1) Rν + n(n − 1)

1

Wγ0,n+1 (M)

≤ Cν n+1

(6.14)

370



for any γ, we know by an isomorphism theorem similar to Proposition 2.3 (cf. Theorem C in [10]), that 1

vν Wγ2,n+1 (M) ≤ Cν n+1 for any γ
0 ,

λ → ∞,

(2.25)

which defines Jas as a homogeneous of degree −3 asymptote of J, supported inside the future and past lightcones. Moreover, in all physical scattering situations there is (2.26) x ∧ Jas (x) = 0 , which reflects the fact that Jas is only due to asymptotically free matter carrying electric charge (no magnetic charges). Accordingly, we shall admit as the space of test fields a class of conserved currents of the type carried by charged matter, moving freely at early and late times, but having compact support in spacelike directions. As we shall see, this is sufficient to result in the appearance of infrared degrees of freedom in the algebra. This should not come as a surprise if one recalls classical analogues: an infrared singular field, however low-energetic, induces a finite phase change of the wave function of a quantum particle, and also causes an adiabatic shift of the trajectory of a classical charged particle [13,14]. In both cases the size of the effect depends on

380

A. Herdegen


the infrared characteristic of the electromagnetic field, which shows that physically typical currents are able to “test” those aspects of the field. We supplement, however, this extension of the test space with the following restriction. The test currents space will slightly differ for the “in” and “out” cases. In the former only the past asymptotics, and in the latter the future asymptotics admitted by (2.25) may be different from zero. This seems rational from the point of view of the regions in which those fields are “tested”. On the other hand, as we shall see, each of these classes of currents produces the same class of radiation fields (and the same as that obtained without these restrictions on asymptotics), so in each case the role of W (J1 ) is the same. We note that the restrictions automatically imply that the test currents are globally charge-free (but may carry nonzero charges in different asymptotic directions). With the test space of currents thus extended, we shall have to be more cautious with the symplectic form (2.7). As we shall see, this form is not completely gauge-independent any more on the enlarged space. Therefore from now on we put for Ai in this form radiation potentials obtained according to (2.4). The integrand in (2.7) will be thus specified, and absolutely integrable. On the other hand, the integrand in the double integral of (2.8) will not be absolutely integrable in general, so this form will not be used. After this specification the symplectic form will become unambiguous for charge-free test currents. We now add charged particles. We assume that the fields interact only weakly, but we want to construct for this situation a closed algebra. The only remnant of the interaction which we take into account is the fact that free charged particles carry their Coulomb fields. Thus the quantum variables will now be interpreted as: ψ(χ1 ) – free charged field carrying its Coulomb field, (2.27) A(J1 ) – total electromagnetic field. For A and ψ separately we retain previous commutation relations, but the above interpretation implies that these variables should not be assumed to commute with each other, one should expect a relation of the intuitive form A ψ = ψ [A+ Coulomb field carried by ψ]. Recall once more that A(J1 ) is loosely {J1 , J}. Moreover, with the use of Fourier-transformed fields 1 χ 1 (p) = (2.28) χ1 (x)eip · x dx (2π)2

dp, with ψ(p)

we have ψ(χ1 ) = χ 1 (p)ψ(p) describing a particle with charge −e moving with the momentum p. Therefore we postulate the relation

A(J1 )ψ(p) = ψ(p) A(J1 ) − {J1 , Jp/m } , (2.29) where Jv is the current connected with the particle with charge e moving freely with four-velocity v; remember that ψ is supported on the mass hyperboloid. This current is non-radiating, but on the other hand it differs from test currents of the “in” and “out” space by having non-vanishing both asymptotes.

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Infrared Problem and Spatially Local Observables

381

A priori, one has a potential difficulty in the relation (2.29): a particle with fixed momentum is completely delocalized, so there is an ambiguity in the current Jp/m . It turns out, however, that taking for Jv the current of a point particle moving along any straight line parallel to v one obtains the same value of {J1 , Jv }, depending only on the long-range tail of the potential produced by J1 . We rewrite (2.29) in the form W (J1 )ψ(χ1 ) = ψ(SJ1 χ1 )W (J1 ) ,

(2.30)

where

−i{J1 ,Jp/m } S χ 1 (p) for p2 = m2 . (2.31) J1 χ1 (p) = e There are two important points to be made. Recall that the test spinors for the Dirac field were assumed smooth and compactly supported. This turns out to be inconsistent with the above relation – if χ1 is compactly supported, then SJ1 χ1 is not. Therefore, similarly as in the electromagnetic case, we have to extend the test function space. We shall find that it is possible to choose these fields as compactly supported in spacelike directions and decaying polynomially in the timelike directions; the degree of the decay may be chosen arbitrarily high without changing the element ψ(χ1 ). The second point concerns gauge invariance. Consider element W (J1 ) with J1 producing pure gauge potential. As stated above, the symplectic form is unambiguously defined for the currents in one of the test classes (“in” or “out”), so this element commutes with electromagnetic field. However, it does not commute with the Dirac field and (2.30) gives in that case

W (J1 )ψ(χ1 ) = eiΛ1 e ψ(χ1 )W (J1 ) ,

A1 pure gauge ,

(2.32)

where the scalar Λ1 is determined by the infrared characteristic of the Lorentz potential A1 produced by J1 . Thus for such J1 the element W (J1 ) should be interpreted as exp[−iΛ1 Q], with Q – the total charge observable. The quantization of charge in units of e means that Λ1 e should be interpreted as a phase variable, which we shall take into account below. The elements W (J1 ) and ψ(χ1 ) satisfying relations (2.17), (2.24) and (2.30) form our algebra. In the next section we give precise meaning to test fields of these generating elements. Section 4 gives then a precise formulation of the algebra.

3. Test functions spaces The geometry of the spacetime is given by the affine Minkowski space M. If a reference point O is chosen, then each point P in M is represented by a vector x in the associated Minkowski vector space M according to P = O + x. We mostly keep O fixed and use this representation, but we also remember to control the independence of structures from O. If a Minkowski basis (e0 , . . . , e3 ) in M is chosen, then we denote x = xi ei . We also then use the standard multi-index notation xα = (x0 )α0 . . . (x3 )α3 , |α| = α0 + · · · + α3 , Dβ = ∂0β0 . . . ∂3β3 , where ∂i = ∂/∂xi . We associate with the chosen Minkowski basis a Euclidean metric with unit matrix

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in that basis, and denote |x| the norm of x in that metric. For a tensor or spinor C ∞ (M) field φ we introduce for each κ ≥ 0 and l = 0, 1, . . . a seminorm κ (3.1)

φ κ, l = sup 1 + |x| |Dβ φj (x)| , where supremum is taken over x ∈ M , all β such that |β| = l, and j running over component indexes of the field in the chosen basis. For fixed l the seminorms form an increasing net over κ ≥ 0. For fixed κ and l seminorms . κ,l and . κ,l associated with two reference systems (O, (ei )) and (O , (ei )) are equivalent. For fixed O this follows from the equivalence of norms |x| and |x| and from the linearity of components transformations, while for the translation O = O + a from the estimate (1 + |x − a|)κ ≤ const(κ, a)(1 + |x|)κ . If one denotes φa (x) = φ(x − a) then it follows (3.2)

φa κ, l ≤ const φ κ, l . Seminorms (3.1) are used in this section to construct the spaces Jas and K which will supply test functions for elements W (J) and ψ(χ) respectively. We also equip the space Jas with the natural topology, although it will not be used in this paper. Space K , being a subspace of the Hilbert space of the scalar product (2.23), inherits its topology. The reference for inductive limit spaces are the books [15] and [16]. 3.1. Spaces Sκ+ Consider the space C ∞ (M) of fields of a given geometric type – not to burden notation this type will be kept implicit. For each κ > 0 we define the subspace Sκ = φ ∈ C ∞ | φ κ+l, l < ∞, l = 0, 1, . . . . (3.3) With the topology determined by the family of seminorms which define them, these spaces are locally convex, Fréchet spaces, independent of the choice of a reference system (O, (ei )). We denote the topology of Sκ by Tκ . For each κ the net of spaces Sκ+ , ∈ (0, 1), is decreasing, so the union Sκ+ (3.4) Sκ+ = 0 Tκ+ > Tκ ,

(3.5)

where the sign > means that the topology to the left is stronger than the topology induced by the one to the right.

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κ 3.2. Spaces Sκ+ Consider the homogeneity operator H = x · ∂ and denote Hκ = H + κ id for κ > 0. If Hκ f (x) = 0 then f is homogeneous of degree −κ, so Hκ is injective on C ∞ . 3 One has |Dβ Hκ φ(x)| ≤ i=0 |xi ||∂i Dβ φ(x)| + (κ + |β|)|Dβ φ(x)|, so for φ ∈ Sκ+ there is

Hκ φ κ++l, l ≤ const φ κ++l+1, l+1 + φ κ++l, l . (3.6)

Conversely, let now ψ ∈ Sκ+ and define 1 φ(x) = uκ−1 ψ(ux)du .

(3.7)

0

Using the bounds on Dβ ψ(x) one finds that differentiation may be pulled under the integral sign 1 Dβ φ(x) = uκ+|β|−1 [Dβ ψ](ux) du , (3.8) 0

and then it is easily seen that Hκ φ = ψ. The estimation of the integral gives −κ−|β| |Dβ φ(x)| ≤ const ψ κ++|β|,|β| 1 + |x| , (3.9) so we find

φ κ+l, l ≤ const Hκ φ κ++l, l . Furthermore, define

φas (x) =

∞

uκ−1 ψ(ux)du ,

(3.10) (3.11)

0

which is homogeneous of degree −κ and C ∞ outside x = 0. Then ∞ Dβ [φ − φas ](x) = − uκ+|β|−1 [Dβ ψ](ux) du .

(3.12)

Estimating the integral one finds that for |x| ≥ 1 we have |x|κ++|β| Dβ [φ − φas ](x) ≤ const Hκ φ κ++|β|, |β| .

(3.13)

1

It follows that φas (x) = lim Rκ φ(Rx) ,

(3.14)

R→∞

and this asymptote is independent of the choice of the central point O. With the use of the estimates (3.10) and (3.13) one finds that in the special case of vanishing asymptote we have a bound stronger than (3.10):

φ κ++l, l ≤ const Hκ φ κ++l, l

iff

φas = 0 .

(3.15)

The estimates (3.6) and (3.10) imply that Hκ Sκ+ ⊂ Sκ+ ⊂ Hκ Sκ , κ = Hκ−1 (Sκ+ ), and Sκ+ onto so Hκ−1 maps Sκ+ bijectively onto Sκ+ κ κ = Sκ+ . Sκ+ 0 Tκκ+ .

(3.21)

For each φ in this space there is: supp φas ⊆ V+ ∪ V− . Finally, we note that the derivative ∂i maps continuously Sκ → Sκ+1 and κ+1 κ Sκ+

→ Sκ+1+ , while multiplication by xi maps continuously Sκ+1 → Sκ and κ+1 κ Sκ+1+

→ Sκ+ . Then similar continuous connections also take place between pairs κ κ κ of spaces of the type Sκ+ , Sκ+ , Sκ+ (C) and s.l.(Sκ+ ). 3.4. Spaces Jin and Jout 3 Choose now κ = 3 and consider the space s.l.(S3+ ) of vector fields. We denote by Jin (Jout ) the subspace of fields J which satisfy the following additional conditions: ∂ ·J = 0,

x ∧ Jas = 0 ,

supp Jas ⊆ V− (resp. V+ ) ,

(3.22)

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(compare (2.25) and (2.26)). Let Jλ be any net in Jin (Jout ), J an element of 3 s.l.(S3+ ), and Jλ → J. The mapping J → ∂ · J is continuous (between suitable spaces – see the end of the last subsection) so the first condition is conserved under the limit and ∂ · J = 0. As s.l.(Tκκ+ ) is stronger than the topology induced by Tκ it is easy to see that the support properties of (Jλ )as are conserved under the limit, so J satisfies the third condition. Finally, using the continuity of the mapping J → x ∧ J and again the conservation of support properties one finds that J satisfies the second condition. Thus Jin and Jout are closed subspaces of 3 ). We shall write Jas for Jin or Jout , and we shall also set Jin = Jas in s.l.(S3+ Jin , and Jout = Jas in Jout . We denote by Jas (C) the subspace of Jas consisting of currents supported in C. Let s ∈ R and l be a future-pointing lightlike vector and choose a region C. It is shown by a straightforward calculation that for κ > 2 δ(s − x · l) const(C) κ dx ≤ (3.23) κ−3 , C |x| + 1 |s| + 1 where δ(.) is the Dirac delta function. If vectors l are scaled to l0 = 1 then the bounding constant in the above relation is l-independent. Therefore for J ∈ Jas the integral V (s, l) = J(x)δ(s − x · l) dx (3.24) is absolutely convergent, the function V (s, l) is homogeneous of degree −1: V (μs, μl) = μ−1 V (s, l) for μ > 0, and if l’s are scaled to l0 = 1 then it is bounded. We denote Lab = la ∂/∂lb − lb ∂/∂la , Xab = xa ∂/∂xb − xb ∂/∂xa and observe that (Lab + Xab )δ(s − x · l) = 0 ,

(s∂s + x · ∂ + 1)δ(s − x · l) = 0 .

(3.25)

Using these identities we find that V (s, l) is infinitely differentiable (outside the vertex of the cone; operators Lab incorporate all intrinsic derivatives in the cone), and we have (dot denotes the derivative ∂/∂s ) ˙ sLa1 b1 . . . Lan bn Vc (s, l) = δ(s − x · l)Xa1 b1 . . . Xan bn H3 Jc (x) dx . (3.26) 3 Let J be in Jas ∩ S3+ (C). Then estimating the integrand by

|Xa1 b1 . . . Xan bn H3 Jc (x)| ≤ const

n

H3 J 3++l, l (1 + |x|)−3−

(3.27)

l=0

and using (3.23) we obtain n −1− La1 b1 . . . Lan bn V˙ (s, l) ≤ const(C)

J 33++l, l 1 + |s| l=0

(3.28)

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for l’s scaled to l0 = 1. The limits V (±∞, l) are determined by Jas and one finds V (−∞, l) = Jin (x)δ(x · l + 1) dx , V (+∞, l) = 0 for J ∈ Jin , (3.29) V (−∞, l) = 0 , V (+∞, l) = Jout (x)δ(x · l − 1) dx for J ∈ Jout . In addition the following identities are satisfied L[ab Vc] (±∞, l) = 0 ,

l · V (s, l) = 0 .

(3.30)

The first of them is the consequence of the second condition in (3.22). To prove the second one we observe that ∂s (s l · V˙ (s, l)) = δ(s − x · l)H4 ∂ · J(x) dx = 0 by (3.26) and conservation of J, and then the result follows by (3.29). For currents with non-vanishing both future and past asymptotes Jas the rhs of the second equation in (3.30) is the total charge. The radiation potential produced by the current J ∈ Jas is completely determined by V˙ (s, l) according to the formula [9] 1 V˙ (x · l, l) d2 l , A(x) = − (3.31) 2π which follows from the representation D(x) = −(1/8π 2 ) δ (x · l) d2 l. Here d2 l is the invariant measure on the set of null directions: we remind the reader that if f (l) is homogeneous of degree −2 then the integral f (l) d2 l = f (1, l) dΩ(l) , (3.32) where dΩ(l) is the solid angle measure in the direction of the unit 3-vector l, is independent of the choice of Minkowski basis, and satisfies (3.33) Lab f (l) d2 l = 0 . Using (3.31) to express A1 and A2 in the symplectic form (2.7) we find that the integrand in that form is absolutely integrable and one obtains 1 (3.34) {J1 , J2 } = V˙ 1 · V2 − V˙ 2 · V1 (s, l) ds d2 l , 4π so Jas becomes a symplectic space. Moreover, using (3.28) and the fact that one of the asymptotic limits V (±∞, l) vanishes, one easily obtains the estimate {J1 , J2 } ≤ const(C) J1 3

J2 3 . (3.35) 3+, 0

3+, 0

Using properties of Vi it is easy to find the kernel of the symplectic form: Ker{. , .} = {J ∈ Jas | l ∧ V = 0} .

(3.36)

If Jv is a current of a point particle carrying charge e and moving freely along any world-line parallel to the four-velocity v = p/m then it is easy to find that p · ΔV (l) 2 e d l, (3.37) {J, Jp/m } = 4π p·l

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where ΔV (l) = V (+∞, l) − V (−∞, l) = ∓V (∓∞, l)

(3.38)

in the “in” and “out” case respectively. By the substitution p → w = p + iq the rhs of (3.37) may be analytically extended to the domain {w | p2 > 0 or q 2 > 0}; we denote this extension FJ (w). It is shown in Appendix A that there exist continuous limit functions on M : FJ± (p) = limλ0 FJ (p ± λiq), where q is a timelike futurepointing vector, with FJ± (p) = FJ (p) for p2 > 0. The electromagnetic potential represented by (3.31) is infrared singular (has a spacelike tail of the decay rate of the Coulomb field) if ΔV (l) = 0, and this function characterizes this singularity completely. We observe that our form of the symplectic structure remains well-defined for those fields. Also, note that (3.37) vanishes for infrared-regular fields, so W (J) for such currents commutes with charged particle field (see (2.30), (2.31) and below). 3.5. Space K For the space C ∞ (M) of fields of given geometric type we define subspaces κ+n n β Sκ = {φ ∈ C ∞ | sup 1 + |x| | D φj (x)| < ∞ , x

∀n = 0, 1, . . . , ∀β , ∀j}

(3.39)

(independent of the choice of reference system). We also introduce subspaces Sκ (C) of functions with support in C and the algebraic inductive limit space s.l.(Sκ ) = Sκ (C) . (3.40) C

Consider the space of 4-spinor fields of the type K = s.l.(S5 ); we shall also write K (C) = S5 (C). The Fourier transforms of fields in that space (after fixing the origin) are among continuous functions vanishing faster then polynomially at infinity. Two test fields χ1 and χ2 are in one class producing the same Dirac field 2 to the hyperboloid according to (2.19) if, and only if, the restrictions of χ 1 and χ p2 = m2 are equal. For χ ∈ K we shall denote by [χ] the class of fields producing the same Dirac field as χ, and by [K ] the quotient space of these classes; also, [K (C)] will denote the set of classes [χ] with χ ∈ K (C). Let χ ∈ S5 (C) and k > 5. Then we have χ = χk + (m2 + )χk , k−5 − χ ∈ Sk (C) , χk = m2

k−6 − − 1 χk = 2 1 + + ···+ χ ∈ S5 (C) . m m2 m2

(3.41)

Therefore for each χ ∈ K the class [χ] contains for each k ≥ 5 a field χk ∈ Sk with the same support properties as χ.

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For each J ∈ Jas we define a linear operator SJ in [K ] as follows. For χ ∈ S5 (C) with C = C+ ∪ C− we find any χk ∈ [χ] ∩ Sk (C) with k ≥ 10 and split χk = χk+ + χk− , χk± ∈ Sk (C± ), which is possible, as shown in Appendix B. We put SJ [χ] = [χ ] ,

−iFJ− (p) where χ (p) = e−iFJ+ (p) χ χ k+ (p) + e k− (p) ,

(3.42)

and the functions FJ± were defined at the end of the last subsection. The results of Appendix B guarantee that this is a correct definition, i.e., the function χ is in S5 (C) and the class [χ ] is independent of the choice of χk and its split into χk± . Moreover, it is easy to convince oneself that (3.43) SJ1 SJ2 = SJ1 +J2 , S0 = id , SJ K (C) = K (C) , so SJ is bijective for each J and the mapping S• : J → SJ is a homomorphism of the additive group of currents into the group of automorphisms of [K ] and of each of the spaces [K (C)]. The kernel of this homomorphism is the subgroup of currents given by (3.44) Ker S• = J ∈ Jas | {J, Jv } = 2kπ, k = 0, ±1, . . . . The mappings are unitary with respect to the scalar product (2.23): if

[χi ] = SJ [χi ] then

χ1 , χ2 = χ1 , χ2 .

(3.45)

4. Asymptotic algebras of fields The structures of the last section allow now for a rigourous formulation of our ideas in the algebraic form. 4.1. ∗ -algebras Bas We define the ∗ -algebras of fields. For each J ∈ Jas we assume an element of the algebra Was (J), and for each χ ∈ K an element ψas (χ). We also assume a unit element E and impose the algebraic relations Was (J)∗ = Was (−J) , Was (0) = E ,

i Was (J1 )Was (J2 ) = exp − {J1 , J2 } Was (J1 + J2 ) , 2 ψas (χ1 ), ψas (χ2 ) + = 0 , ψas (χ1 ), ψas (χ2 )∗ + = χ1 , χ2 , Was (J)ψas (χ) = ψas (χ )Was (J) ,

(4.1)

where [χ ] = SJ [χ] .

We want to identify those elements Was (J) which generate the same relations. The following is a subgroup of the additive group of currents Jas : 0 = Ker{. , .} ∩ Ker S• Jas (4.2) 2 = J ∈ Jas | V (s, l) = lα(s, l), (e/4π) Δα(l) d l = 2kπ, k = 0, ±1, . . . ,

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where Δα(l) = α(+∞, l) − α(−∞, l) = ∓α(∓∞, l) in the “in” and “out” case re0 spectively. We shall denote by [Jas ] the quotient group Jas /Jas , and set Was (J1 ) = Was (J2 ) if [J2 ] = [J1 ]. Similarly we identify ψas (χ1 ) = ψas (χ2 ) if [χ1 ] = [χ2 ]. After these identifications we shall call the ∗ -algebra generated by the relations (4.1) the field ∗ -algebra Bas . For each C the elements Was (J) and ψas (χ) with J ∈ Jas (C) and χ ∈ K (C) generate a subalgebra, denoted Bas (C), and we have Bas = ∪C Bas (C). This construction can also be characterized as follows. The relations (4.1) generate a ∗ -algebra. The elements A Was (J) − E , J ∈ [0], and Aψas (χ), χ ∈ [0], where A goes over all elements of the algebra, generate a two-sided ideal of the algebra. The quotient of the algebra through this ideal is the ∗ -algebra Bas , as=in or out. + The elements ψas (χ) generate a subalgebra Bas of the CAR type, and elements − Was (J) – a subalgebra Bas of the CCR type. Each element of Bas may be brought k + to the form i=1 Ci Was (Ji ), where Ci ∈ Bas and with currents Ji such that [Ji ] = [Jj ] for i = j. The last relation in (4.1) may be used to define a group of + automorphisms of Bas : βJ (C) = Was (J)CWas (−J) , βJ1 βJ2 = βJ1 +J2 ,

β0 = id .

+ C ∈ Bas ,

(4.3)

The universal covering group P of the Poincaré group has a representation in the automorphism group of algebra Bas . After choosing the origin in M each element in P is represented by (a, A), a ∈ M , A ∈ SL(2, C), and the respective automorphism αa,A is given by the standard formulas αa,A Was (J) = Was (Ta,A J) , [Ta,A J](x) = Λ(A)J Λ(A)−1 (x − a) , (4.4) αa,A ψas (χ) = ψas (Ra,A χ) , [Ra,A χ](x) = S(A)χ Λ(A)−1 (x − a) , where Λ(A) and S(A) are elements of the vector and 4-spinor representations of SL(2, C) respectively. In the above construction of the algebras Bas we identified elements labelled by test functions falling into a common equivalence class. The net effect of these identifications is that the elements ψ(χ) could be labelled by the restriction of the Fourier transform of χ to the hyperboloid p2 = m2 , and the elements W (J) – by 0 the corresponding classes of V ’s up to addition of an element of Jas . If formulated in this way, the algebra Bout is a subalgebra of the algebra B of [9], Eqs. (3.40–43). The use of the present test spaces adds to the elements of the algebra the spacetime localization properties. In the following (sub)sections we use results of [9], although with some modifications indicated below. The change from Bout to Bin is almost trivial in all what follows and we shall not separate these two cases. To be precise, there is one minor difference in the use of V ’s in [9] and the present paper. Let us concentrate on the “out” case. Here we have V (−∞, l) = 0 in that case, while in [9] we used variable V out (s, l) for which V out (+∞, l) = 0. The relation between the two variables is V out (s, l) = V (s, l) − V (+∞, l). It is easy to check that the value of the symplectic form remains unchanged under this

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transformation, and also ΔV out (l) = ΔV (l), so SJ is unaffected. The convention of V is more naturally connected with the outgoing test current, while the convention of V out is connected with the description of the future null asymptotics of the radiation potential (3.31): for a lightlike future-pointing vector k one has lim RA(x + Rk) = V out (x · k, k) .

(4.5)

R→∞

Similar connections, with past interchanged with future, take place in the “in” case. 4.2. Selection of representations We consider ∗ -representations π of the algebra Bas by bounded operators in a Hilbert space H (all representations considered will be ∗ -representations and we suppress this qualification in the sequel). We are interested only in regular representations – those for which all one-parameter groups λ → π Was (λJ) are strongly continuous; the Weyl exponentiation is a technical device, which should be invertible on the level of representations having physical significance. Moreover, we further restrict attention to the translationally covariant representations with positive energy. This means that there exists a representation of the translation group by unitary operators U (a) in H which implement the automorphisms αa ≡ αa,1 , i.e., π(αa (B)) = U (a)π(B)U ∗ (a) for each B ∈ Bas , and whose spectrum is contained in V+ . + Let πF be the standard positive energy Fock representation of Bas on the Hilbert space HF , with the Fock vacuum vector denoted ΩF , and πr be a regular, − translationally covariant positive energy representation of Bas on Hr . Define the following operators π(A) on the space H = HF ⊗ Hr by + , π(C) = πF (C) ⊗ idr , C ∈ Bas π Was (J) [πF (B)ΩF ⊗ ϕ] = πF (βJ B)ΩF ⊗ πr Was (J) ϕ ,

+ B ∈ Bas .

(4.6)

Then π extends to a regular, translationally covariant positive energy representation of Bas . Conversely, if π is a representation of Bas with these properties then up to a unitary equivalence it has the form given by (4.6). The theorem formulated in the last paragraph results from a slight modification of the Theorem 4.4 of [9], taken over here by the remarks of the two last paragraphs of the preceding subsection. The modification is twofold. First, the theorem is formulated in terms of a C ∗ -algebra F generated by B, but in the proofs of this theorem and its lemmas one can replace F by B. Second, we need to replace B by Bas . This however poses no problem, one only has to use the fact that our present test space K is dense in the Hilbert space of the scalar product (2.23). We now restrict the choice of representations still further. We demand that the elements π(Was (J)) are gauge-invariant when acting on the subspace ΩF ⊗ Hr . The physical motivation for that seems plausible enough – the electromagnetic field alone should be gauge-invariant. Let [[Jas ]] be the quotient space Jas / Ker{. , .} with elements denoted by [[J]]. The assumed gauge-invariance means that πr (Was (J1 )) = πr (Was (J2 )) if [[J1 ]] = [[J2 ]]. This is equivalent to the assumption

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− that πr is a representation of the algebra [[Bas ]] (in which such elements Was (J1 ) and Was (J2 ) have been identified).

4.3. C ∗ field algebras F as We can now equip the algebra Bas with a C ∗ -norm defined by A = π(A) , where π is any representation in the class defined by the above assumptions. This norm is independent of the choice of the particular representation π. The uniqueness follows in three steps. First, the representation πF extends uniquely to the Fock representation of the full (unique) CAR Dirac fields algebra – this is because fields [χ], χ ∈ K , are dense in the Hilbert space defined by the product (2.23) (as, in particular, compactly supported χ’s are). Second, as the symplectic form is − ]] generates the unique C ∗ Weyl algebra. nondegenerate on [[Jas ]], the algebra [[Bas Each representation πr in the assumed class extends to a representation of this Weyl algebra. Third, these extended representations πF and πr are faithful, so the + − ) ⊗alg πr (Bas ) is independent of the choice of πr . This is operator norm on πF (Bas sufficient to conclude the claimed uniqueness. We shall call the C ∗ -algebra generated by Bas equipped with the norm intro. duced above the C ∗ asymptotic field algebra, Fas = Bas . Each representation π of Bas in the assumed class extends to a faithful representation of Fas . We also .

introduce algebras Fas (C) = Bas (C) . We note that the present construction of the algebras Fas is more restrictive than the one leading from B to F in [9]. There the construction of F was based on all possible Hilbert space representations of B. However, now I think that this is both more involved and unjustified. Some of the elements of this larger algebra could be brought to zero in representations having physical interpretation. 4.4. Examples of the representations π r Let ρ be a real smooth function on M of compact support, such that ρ(x)dx = 1. For each J ∈ Jas (C) we denote (4.7) Jρ = J − ρ ∗ Jas , (ρ ∗ Jas )(x) = ρ(x − y)Jas (y) dy . Modifying slightly the steps of Appendix C one finds that ρ ∗ Jas , Jρ ∈ Jas (C). The asymptote of ρ ∗ Jas is equal to Jas , therefore the asymptote of Jρ vanishes, Jρas = 0. Moreover, taking into account the support property of Jas one finds V (s, l) − Vρ (s, l) = δ(s − x · l)(ρ ∗ Jas )(x)dx = ρ(z) δ(s − z · l − y · l)Jas (y) dy dz = θ ∓ (s − z · l) ρ(z)dz δ(y · l ± 1)Jas (y)dy , where the upper and lower signs refer to the “in” and “out” case respectively and the last step results from rescaling s − z · l in the inner integral to ∓1 respectively.

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If we introduce


H(s, l) =

sgn(s − z · l)ρ(z)dz ,

with

lim H(s, l) = ±1 ,

s→±∞

(4.8)

then

1 1 ∓ H(s, l) V (∓∞, l) . 2 Using this split in (3.34) it is easy to show that V (s, l) = Vρ (s, l) +

{J1 , J2 } = {J1ρ , J2ρ } + {J1 , ρ ∗ J2as } − {J2 , ρ ∗ J1as } .

(4.9)

(4.10)

Currents Jiρ produce infrared-regular fields and the first term on the rhs coincides with the standard symplectic form for such fields, and expressed in terms of V ’s reads 1 V˙ 1ρ · V2ρ − V˙ 2ρ · V1ρ (s, l) ds d2 l , (4.11) {J1ρ , J2ρ } = 4π The rest of the rhs of (4.10) is another symplectic form which we now transform. By a straightforward calculation we have 1 V˙ 1 H(s, l)ds · ΔV2 (l) d2 l , (4.12) {J1 , ρ ∗ J2as } = 4π with ΔV (l) defined in (3.38). We recall from [9] that for vector fields f (l) homogeneous of degree −1 and orthogonal to l the scalar product (f, g)0 = − f (l) · g(l) d2 l defines a Hilbert space H0 of equivalence classes, and that the fields satisfying in addition L ∧ f = 0 (cf. Eq. (3.30)) span its subspace HIR . (All fields differing by fields of the form lα(l) fall into one class. To simplify notation we suppress the square brackets which were used in [9] to distinguish a class [f ] 1 from the field f .) Following the notation of [9] we denote p(V˙ ) = 2π ΔV . Also, we ˙ and again following [9] write rh (V˙ ) for the orthogonal projection in set h = π H, H0 onto HIR of 12 V˙ H(s, l)ds. Now we can write {J1 , ρ ∗ J2as } − {J2 , ρ ∗ J1as } = p(V˙ 1 ) ⊕ rh (V˙ 1 ), p(V˙ 2 ) ⊕ rh (V˙ 2 ) IR , (4.13) where {g1 ⊕ k1 , g2 ⊕ k2 }IR = (g1 , k2 )HIR − (g2 , k1 )HIR

(4.14)

is a nondegenerate symplectic form on the space HIR ⊕ HIR . The additive split of the symplectic form into the parts (4.11) and (4.13) ˆ (V ) generate the standard Weyl algesuggests the following construction. Let W bra over the space of infrared-regular fields with the symplectic form (4.11), and w(g ⊕ k) generate the Weyl algebra over the symplectic space given by (4.14). Choose representations πreg and πsing of these two algebras. Then the formula ˆ (Vρ ) ⊗ πsing w p(V˙ ) ⊕ rh (V˙ ) πr W (J) = πreg W (4.15) − defines a representation of the algebra Bas . If πreg and πsing are cyclic, determined by the GNS construction from the states ωreg and ωsing respectively, then πr is acyclic representation determined by the state ωr (W (J)) =

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ˆ (Vρ ))ωsing (w(p(V˙ ) ⊕ rh (V˙ ))). In particular, we take for ωreg the standard ωreg (W ˆ (Vρ )) = exp[− 1 F (V˙ ρ , V˙ ρ )], where vacuum state, ωreg (W 2 dω 2 d l V˙ 1 (ω, l) · V˙ 2 (ω, l) F (V˙ 1 , V˙ 2 ) = − ω ω≥0 1 (4.16) = log(s − τ − i0) V˙ 1 (s, l) · V˙ 2 (τ, l) ds dτ d2 l , (2π)2 1 V˙ (ω, l) = (4.17) eiωs V˙ (s, l) ds . 2π Let B be a positive, trace-class operator in HIR such that B 1/2 HIR contains the ∞ subspace CIR of all smooth fields in HIR . Then the formula

1 ωsing w(g ⊕ k) = exp − s(g ⊕ k, g ⊕ k) , (4.18) 4 1 s(g1 ⊕ k1 , g2 ⊕ k2 ) = B −1/2 g1 , B −1/2 g2 HIR + 2 B 1/2 k1 , B 1/2 k2 HIR , (4.19) 2 defines a quasi-free state. The resulting representation πr satisfies all our selection HIR

∞ = HIR , then πr is irreducible [9]. conditions. If, in addition, B −1/2 CIR Representations thus obtained seem to depend on the choice of ρ. However, this dependence is spurious: it was shown in [9] that (with fixed B) they are all unitarily equivalent. In fact, one can construct them in a version which does not need this auxiliary function. States ωr with different ρ’s are then realized by different vector states in the representation space. The spectrum of energy-momentum covers in each representation from the given class the whole future lightcone, and is purely continuous [9]. Thus there is no vacuum state, but the energy content can be arbitrarily close to zero. In fact, one finds that in our formulation the translational invariance is in contradiction with the regularity of infrared-singular Weyl operators. However, for infrared-regular test fields the states of the form discussed above can approximate (weakly) the vacuum. Indeed, for these test currents J there is Jas = 0, so Jρ = J and the state ωreg is the vacuum. Next, we have to consider the infrared part ωsing . As p(V˙ ) = 0, we have ωsing (ω(0 ⊕ rh (V˙ ))) = exp[− B 1/2 rh (V˙ ) 2IR /2]. Take now a family of states with ρλ (x) = ρ(x + λt), where t is any timelike vector. It is then easy to show that for λ → ±∞ the corresponding Hλ tends point-wise to ∓1, and then rhλ (V˙ ) tends in norm to zero. As the operator B is bounded, the singular part tends to 1, which ends the proof.

5. Scattering In standard treatments of scattering the ingoing and outgoing fields are different representations of one asymptotic field algebra Fas (say, of the free scalar field) unitarily connected by the scattering operator: πout (A) = S ∗ πin (A)S, A ∈ Fas . The “in” and “out” fields are (expected to be) obtainable from the actual field

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variables of the theory by some limiting process. We have to explain in what way we expect this picture should be extended to accommodate two asymptotic algebras Fin and Fout . 5.1. Canonical isomorphism γ ι : Fin → Fout We still conjecture that the asymptotic fields are outcomes of some limiting process. The gap between Fin and Fout will be bridged by showing that there exists a canonical isomorphism γι : Fin → Fout . This automorphism will be interpreted to describe the “no interaction” situation, in the sense that the formula πout (γι A) = πin (A), A ∈ Fin , gives the “out” field in terms of the “in” field in that case. In general case we expect the relation πout (γι A) = S ∗ πin (A)S ,

A ∈ Fin ,

(5.1)

with S playing the role of the scattering operator. To construct the automorphism γι we first define the group isomorphism ι : [Jin ] → [Jout ]. To this end we first define relation R ⊂ [Jin ] × [Jout ] as follows: ([J1 ], [J2 ]) ∈ R iff the current J1 − J2 radiates no electromagnetic field and (e/4π) [ΔV1 (l) − ΔV2 (l)] d2 l = 2kπ. Recalling the definition of [Jas ] as given after (4.2) it is easy to see that this is an unambiguous definition independent of the choice of Ji in the respective classes, and that the relation is one to one. Moreover, if the pairs ([J1 ], [J2 ]) and ([J1 ], [J2 ]) satisfy the relation, then also does the pair ([J1 ] + [J1 ], [J2 ] + [J2 ]). We show in Appendix C that for each J1 there exists J2 such that ([J1 ], [J2 ]) satisfy the relation (and conversely, for each J2 there is a respective J1 ). With this result we can set ι[J1 ] = [J2 ] iff ([J1 ], [J2 ]) ∈ R, and conclude that this defines a group isomorphism. With the results of Appendix C it is also easily shown that ι is a symplectic mapping, and also that {J1 , Jv } = {J2 , Jv } for [J2 ] = ι[J1 ]. We now set (5.2) γι ψin (χ) = ψout (χ) , γι Win (J1 ) = Wout (J2 ) , [J2 ] = ι[J1 ] . If πin is the representation of Bin in the assumed class (see (4.6) and the accompanying discussion) then πout defined by πout (ψout (χ)) = πin (ψin (χ)), πout (Wout (J2 )) = πin (Win (J1 )), [J2 ] = ι[J1 ], defines a representation of Bout in the same class. Therefore γι extends to a topological isomorphism γι : Fin → Fout . 5.2. Radiation by external current How to construct a scattering theory in the given language in the general case of full quantum theory is an open question. Here we consider only electromagnetic field scattered by classical external current. Let Jext be a current satisfying the conditions of spaces Jas , except that its asymptote has both an incoming and outgoing parts (has support in V+ ∪ V− ); this is a classical conserved current typical of charged matter. We denote Vext (s, l) = Jext (x)δ(s − x · l)dx. This current produces the Lorenz radiation potential Aext in accordance with (2.4). We need test currents producing the same radiation as

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Jext but belonging to Jin or Jout . Let Jext,in and Jext,out denote the incoming and outgoing asymptote of Jext respectively. We denote further + (x) = Jext,out (x) + Jext,out (−x) , Jext,out + (x) = Jext,in (x) + Jext,in (−x) , Jext,in

+ Jext,1 = Jext − ρ ∗ Jext,out , + Jext,2 = Jext − ρ ∗ Jext,in .

(5.3)

Then proceeding as in Appendix C one shows that Jext,1 ∈ Jin , Jext,2 ∈ Jout and they both produce radiation potential Aext ; the classes [Jext,1 ] and [Jext,2 ] are independent of the choice of ρ and [Jext,2 ] = ι[Jext,1 ]. Denoting Vext,i (s, l) = Jext,i δ(s − x · l)dx (i = 1, 2) we have Vext,1 (s, l) = Vext (s, l) − Vext (+∞, l), Vext,2 (s, l) = Vext (s, l) − Vext (−∞, l). The process of scattering of electromagnetic field by the external classical current should have the effect of adding the radiation field of the current. Therefore, remembering the interpretation of W (J) we expect πout Wout (J2 ) = πin Win (J1 ) e−i{J2 ,Jext,2 } = πin Win (J1 ) e−i{J1 ,Jext,1 } , (5.4) where [J2 ] = ι[J1 ]. Using the commutation relations it is easy to find that the operator S = πin Win (Jext,1 ) = πout Wout (Jext,2 ) (5.5) satisfies the condition (5.1) and turns to identity when there is no radiation. There is no “infrared catastrophe” difficulty in this formulation. Formula (5.5) cannot be directly applied when Jext is a current of a point charge, (5.6) Jpoint (x) = e z(τ ˙ )δ 4 x − z(τ ) dτ . However, if πin (and πout ) is of the form discussed in Section 4.4 it can be extended to this case. One easily finds that z˙ τ (s, l) , Vpoint (s, l) = e z˙ τ (s, l) · l (5.7) ˙ − u, l)Vpoint (u, l) du , Vpointρ (s, l) = H(s where τ (s, l) is the solution of z(τ ) · l = s. For a smooth trajectory z(τ ) with sufˆ (Vpointρ ) is well defined ficiently fast achieved asymptotic velocities the element W and the formula (5.5) may be extended with the use of the rhs of (4.15).

6. Summary This paper introduces the notion of spatially local observables and fields in electrodynamics; we have developed the test functions machinery needed for that. We have shown that the enlarged algebra of fundamental asymptotic fields naturally includes the infrared degrees of freedom, which care for the implementation of Gauss’ law at the algebraic level. The algebra proved to be a reformulation of

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the algebra postulated before by the author, but now we gained control over the spacetime localization of fields. The perturbational construction of the standard electrodynamics is based on the uncoupled free fields algebra. We expect that the scattering theory based in similar way on the algebra discussed in this article could throw new light on the infrared and charge problems of the standard theory. Here, as a first step towards this task, we have shown how to apply our formalism to the simple case of the radiation produced by external classical current. The description is free from “infrared catastrophe”.

Appendices Appendix A. Functions F± (p) Let Va (l) be a smooth vector function on the future lightcone, homogeneous of degree −1 and such that l · V (l) = 0. If t is any unit timelike future-pointing vector and V (l) is continued to a neighbourhood of the cone with the preservation of its properties then Lab [ta V b (l)/t · l] = ∂ · V (l). Therefore the rhs is independent of the extension in the assumed class and ∂ · V (l) d2 l = 0 . (A.1) Let now F (w) be defined for w = p + iq, p, q ∈ M , q 2 > 0, by w · V (l) 2 F (w) = d l. (A.2) w·l This is a homogeneous of degree 0, analytical function on its domain. Choosing any unit timelike future-pointing vector t and using property (A.1) we rewrite F as t · V (l) 2 w·l 2 d l+ d l. (A.3) F (w) = − ∂ · V (l) log t·l t·l This is used to show that F (w) is bounded on its domain. Due to homogeneity it is sufficient to consider two cases: |q| ≤ |p| = 1/2, and |p| ≤ |q| = 1/2. In the first case the first integral is bounded by const [| log(p0 − p · l)2 | + 2π]dΩ(l) ≤ const; the second case is treated similarly. It is easy to see that there exist limit functions on M \ {0} F± (p) = lim F (p ± iλt) λ0

π |p · l| ∓ i sgn(p · l) d2 l = − ∂ · V (l) log t·l 2 t · V (l) 2 d l = F∓ (−p) , + t·l

(A.4)

which are independent of the choice of t (specified as above), homogeneous and equal to F (p) for p ∈ V± . We show below that these limits are achieved uniformly

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on each set separated from zero, F± are continuous outside p = 0 and C ∞ outside p2 = 0, and for each multiindex α functions (p2 )|α| Dα F± (p) have continuous extensions to M . Proof. If g(l) is any smooth function on the cone, homogeneous of degree −2 and such that g(l) d2 l = 0, then for q 2 > 0 we have

∂ wb w·l 2 w·l 2 d d l l = g(l) log Lba g(l) log ∂wa t·l w2 t·l (A.5) l a tb − l b ta 2 d l . + g(l) t·l This is shown by transferring Lba on the rhs by parts and observing that [Lba + Wba ] log(w · l/t · l) = (la tb − lb ta )/t · l , where Wba = wb ∂/∂wa − wa ∂/∂wb . Applying (A.5) inductively to (A.3) one finds that for q 2 > 0 w·l 2 d l + Q(w) , Qi (w) hi (l) log (w2 )|α| Dα F (w) = t·l i where Qi and Q are polynomials, homogeneous of degree |α|, and hi are smooth functions, homogeneous of degree −2 and such that hi (l)d2 l = 0. To end the proof it is now sufficient to show that uniform limits (A.4) exist on each set sepa rated from zero for each function of the form G(w) = h(l) log(w · l/t · l) d2 l, with 2 h(l)d l = 0. We have (λ > 0)

2 p·l p·l 1 (p ± iλt) · l π = log + λ2 ∓ i arctan log ±i , t·l 2 t·l λt·l 2 but the last term falls out of the integral. Denote

π |p · l| G± (p) = h(l) log ∓ i sgn(p · l) d2 l . t·l 2 If we denote k = p/λ then G(p ± iλt) − G± (p) = G(k ± it) − G± (k)

2 t·l ≤ const log 1 + k·l

|k · l| + π − 2 arctan d2 l t·l

0 const |k |+|k| 1 = log 1 + 2 + π − 2 arctan |u| du u |k| |k0 |−| k| log 1 + |k 0 | + |k| 1 ≤ const log 1 + + . |k 0 | + |k| |k 0 | + |k|

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where the spherical coordinates have been used. To get this result we first estimate the integrand by const log(1 + 1/|u|) and then consider the cases (a) |k 0 | ≥ 2|k| and (b) |k 0 | ≤ 2|k| separately. In case (a) the integral is bounded by const × log[1 + 1/(|k 0 | − |k|)], which is sufficient as here |k 0 | − |k| ≥ (|k 0 | + |k|)/3. In case (b) we extend the integration limits to ±(|k 0 | + |k|), calculate the integral explicitly and observe that here |k| ≥ (|k 0 | + |k|)/3, which yields the final result. For |p0 | + | p| ≥ r ≥ λ we now have |G(p ± iλt) − G± (p)| ≤ const

1 + log[1 + r/λ] → 0 for r/λ

which ends the proof.

λ → 0,

Appendix B. Transformation E F Let F and F± be defined as in Appendix A and let χ ∈ Sk (C± ) for a given k ≥ 10. F χ by Define linear mappings χ → E± F ˆ . (B.1) E ± χ(p) = exp − iF± (p) χ(p) F We show here that there is E± χ ∈ Sk−5 (C± ). Furthermore, if χ ∈ Sk (C) and C = C+ ∪ C− then it is possible to separate χ = χ1+ + χ1− , χ1± ∈ Sk (C± ), F F and define E1F χ = E+ χ1+ + E− χ1− ∈ Sk−5 (C); subscript 1 indicates the choice of (in general non-unique) separation. If another separation is indexed by 2 then F χ(p) = E F χ(p) for p2 > 0. E 1

2

Proof. If χ ∈ Sk (k ≥ 5) then one shows by induction with respect to |α| that χ(p) ˆ is C ∞ outside p2 = 0 and for |α| ≤ k − 5 + n the functions (p2 )n Dα χ(p) ˆ have continuous extensions to M , vanishing faster than polynomially at infinity. Using this fact and the properties of F± shown in Appendix A it is easy to show that the ˆ It follows same remains valid when χ(p) ˆ is replaced by χˆ (p) = exp[−iF± (p)]χ(p). α n β that supx |x D χ (x)| < ∞ whenever |α| ≤ (k − 5) + n. This is sufficient to F χ ∈ Sk−5 . conclude that E± F χ ∈ Sk−5 . We can choose the reference Let χ ∈ Sk (C± ), k ≥ 10, so E± point at the vertex of the cone, and then the support of χ is in V± . Then the Fourier transform χ(w) ˆ exists as an analytical function of the complex variable ˆ + iq)| < const(1 + |p|)−n for each n for w ∈ M + iV± , satisfies the bound |χ(p and gives limλ0 χ(p ˆ + iλq) = χ(p). ˆ Using the properties of F (w) one finds that also exp[−iF (w)]χ(w) ˆ is analytical in the same domain, satisfies similar bounds ˆ + iλq) = exp[−iF± (p)]χ(p). and gives in the limit limλ0 exp[−iF (p + iλq)]χ(p By the theorem connecting cone-like support properties of a distribution with the analyticity properties of its transform (see [17], Thm.IX.16) this is sufficient to F conclude that E± χ ∈ Sk−5 (C± ). Let now χ ∈ Sk (C), C = C− ∪C+ . As C− ∩C+ = ∅ one can choose the reference system such that C± = ∓Re0 + V± , R > 0 (if the origin is identified with the zero

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vector). Let f be a real smooth function of a real variable, such that f (s) = −1 for s < −1/3, f (s) = 1 for s > 1/3. We define functions ρ± on M as follows: ⎧ " # 1 Rx0 ⎪ ⎪ for x ∈ C− ∩ C+ ⎨ 2 1 ± f R2 −| x|2 (B.2) ρ± (x) = 1 on the rest of C± ⎪ ⎪ ⎩0 outside C± . It is easy to show that ρ± ∈ C ∞ (C) and the functions χ± = ρ± χ are in Sk (C± ) respectively (use the fact that χ and its derivatives vanish at the boundary of C faster than any power of the Euclidean distance from that boundary). As the sum ρ+ + ρ− is the characteristic function of the set C we have χ = χ+ + χ− and this separation satisfies conditions stated at the beginning. If χ = χi+ + χi− , i = 1, 2, F χ(p) − E F χ(p) = are any two separations satisfying these conditions, then E 2

1

ˆ2+ (p) − χ ˆ1+ (p)), which vanishes for p2 > 0. (exp[−iF+ (p)] − exp[−iF− (p)])(χ

Appendix C. Non-radiating currents Let J1 ∈ Jin (C), C = C− ∪ C+ , with the asymptote J1as . Choose a real smooth function ρ on M with support in C− ∩ C+ , such that ρ(y)dy = 1, and define + + + , (ρ ∗ J1as )(x) = ρ(x − y)J1as (y) dy , J2 = J1 − ρ ∗ J1as + J1as (y) = J1as (y) + J1as (−y) .

(C.1)

Then we have J2 ∈ Jout (C), J2 − J1 produces no radiation potential and J2as (x) = −J1as (−x). In terms of V ’s we have V2 (s, l) = V1 (s, l) − V1 (−∞, l) and ΔV2 (l) = V2 (+∞, l) = −V1 (−∞, l) = ΔV1 (l). Proof. It is easy to show that the asymptote J1as (x) is divergence-free outside x = 0 (as a limit of a conserved current). But it is also a distribution and by smearing it with a test function one also shows that J1as is a conserved distributional current. + is a smooth, conserved current. The support Then it is immediate that ρ ∗ J1as + , and consequently also properties of ρ and J1 imply that the support of ρ ∗ J1as the support of J2 , are contained in C. Let C− ∩ C+ be contained in |x| ≤ R. Then for |x| ≥ 2R we have β D H3 (ρ ∗ J + )(x) = ρ(z) z · ∂Dβ J + (x − z)dz ≤ const |x|−|β|−4 . 1as 1as + 3 , J2 ∈ S3+ (C) together with J1 . The asymptote This guarantees that ρ ∗ J1as + + + 3 )(λx) = J1as (x) (use the assumption of ρ ∗ J1as is easily found: limλ→∞ λ (ρ ∗ J1as on the integral of ρ). Thus J2as (x) = −J1as (−x) and J2 ∈ Jout (C).

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Finally, we calculate + )(x)dx V1 (s, l) − V2 (s, l) = δ(s − x · l)(ρ ∗ J1as + = lim ρ(z) δ(s − z · l − y · l)J1as (y)dy dz . λ0 |s−z · l|≥λ + + But now using the fact that |ξ|3 J1as (ξy) = J1as (y) for all ξ = 0 one can scale s − z · l to −1, and then one finds + V1 (s, l) − V2 (s, l) = δ(y · l + 1)J1as (y)dy = δ(y · l + 1)J1as (y)dy = V1 (−∞, l) ,

which ends the proof.

References [1] J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons, Springer, Berlin, 1976. [2] G. Morchio and F. Strocchi, Infrared problem, Higgs phenomenon and long range interactions, in Fundamental Problems in Gauge Field Theory, eds. G. Velo and A. S. Wightman, Plenum Press, New York, 1986. [3] R. Haag, Local Quantum Physics, 2nd ed., Springer, Berlin, 1996. [4] O. Steinmann, Perturbative Quantum Electrodynamics and Axiomatic Field Theory, Springer, Berlin, 2000. [5] D. Buchholz, Commun. Math. Phys. 85 (1982), 49. [6] D. Buchholz, M. Porrmann and U. Stein, Phys. Lett. B 267 (1991), 377. [7] P. P. Kulish and L. D. Faddeev, Teor. Matem. Fiz. 4 (1970), 153; [Theor. Math. Phys. 4 (1971), 745]. [8] D. Zwanziger, Phys. Rev. D 14 (1976), 2570. [9] A. Herdegen, J. Math. Phys. 39 (1998), 1788. [10] A. Herdegen, Acta Phys. Pol. B 36 (2005), 35. [11] J. L. Gervais and D. Zwanziger, Phys. Lett. B 94 (1980), 389. [12] A. Staruszkiewicz, Ann. Phys. (NY) 190 (1989), 354. [13] A. Staruszkiewicz, Acta Phys. Pol. B 12 (1981), 327. [14] A. Herdegen, J. Math. Phys. 36 (1995), 4044. [15] G. K¨ othe, Topological Vector Spaces, vol. I, Springer, New York, 1969. [16] A. P. Robertson and W. Robertson, Topological Vector Spaces, 2nd ed., Cambridge Univ. Press, Cambridge, 1973. [17] M. Reed and B. Simon, Methods of Mathematical Physics, vol. II., Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.

Vol. 9 (2008)


Andrzej Herdegen Institute of Physics Jagiellonian University Reymonta 4 PL-Cracow 30–059 Poland e-mail: [email protected] Communicated by Klaus Fredenhagen. Submitted: May 14, 2007. Accepted: October 2, 2007.

401

Ann. Henri Poincaré 9 (2008), 403–424 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020403-22, published online April 10, 2008 DOI 10.1007/s00023-008-0360-1


Constructive φ4 Field Theory without Tears Jacques Magnen and Vincent Rivasseau Abstract. We propose to treat the φ4 Euclidean theory constructively in a simpler way. Our method, based on a new kind of “loop vertex expansion”, no longer requires the painful intermediate tool of cluster and Mayer expansions.

1. Introduction Constructive field theory builds functions whose Taylor expansion is perturbative field theory [15, 24]. Any formal power series being asymptotic to infinitely many smooth functions, perturbative field theory alone does not give any well defined mathematical recipe to compute to arbitrary accuracy any physical number, so in a deep sense it is no theory at all. In field theory “thermodynamic” or infinite volume quantities are expressed by connected functions. One main advantage of perturbative field theory is that connected functions are simply the sum of the connected Feynman graphs. But the expansion diverges because there are too many such graphs. However to know connectedness does not require the full knowledge of a Feynman graph (with all its loop structure) but only the (classical) notion of a spanning tree in it. This remark is at the core of the developments of constructive field theory, such as cluster expansions, summarized in the constructive golden rule: “Thou shall not know most of the loops, or thou shall diverge!” Some time ago Fermionic constructive theory was quite radically simplified. It was realized that it is possible to rearrange perturbation theory order by order by grouping together pieces of Feynman graphs which share a common tree [1, 22]. This is made easily with the help of a universal combinatoric so-called forest formula [2,5] which once and for all essentially solves the problem that a graph can have many spanning trees. Indeed it splits any amplitude of any connected graph in a certain number of pieces and attributes them in a “democratic” and “positivity preserving” way between all its spanning trees. Of course the possibility for such a rearrangement to lead to convergent resummation of Fermionic perturbation

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theory ultimately stems from the Pauli principle which is responsible for analyticity of that expansion in the coupling constant. Using this formalism Fermionic theory can now be manipulated at the constructive level almost as easily as at the “perturbative level to all orders”. It leads to powerful mathematical physics theorems such as for instance those about the behavior of interacting Fermions in 2 dimensions [8, 11, 25], and to more explicit constructions [9] of just renormalizable Fermionic field theories such as the Gross– Neveu model in two dimensions first built in [13, 14]. But Bosonic constructive theory remained awfully difficult. To compute the thermodynamic functions, until today one needed to introduce two different expansions one of top of the other. The first one, based on a discretization of space into a lattice of cubes which breaks the natural rotation invariance of the theory, is called a cluster expansion. The result is a dilute lattice gas of clusters but with a remaining hardcore interaction. Then a second expansion called Mayer expansion removes the hardcore interaction. The same tree formula is used twice once for the cluster and once for the Mayer expansion1 , the breaking of rotation invariance to compute rotation invariant quantities seems ad hoc and the generalization of this technique to many renormalization group steps is considered so difficult that despite courageous attempts towards a better, more explicit formalization [4,6], it remains until now confined to a small circle of experts. The Bosonic constructive theory cannot be simply rearranged in a convergent series order by order as in the Fermionic case, because all graphs at a given order have the same sign. Perturbation theory has zero convergence radius for bosons. The oscillation which allows resummation (but only, e.g., in the Borel sense) of the perturbation theory must take place between infinite families of graphs of different orders. To explicitly identify such families and rearrange the perturbation theory accordingly seemed until now very difficult. The cluster and Mayer expansion perform this task but in a very complicated and indirect way. In this paper we at last identify such infinite families of graphs. They give rise to an explicit convergent expansion for the connected functions of Bosonic φ4 theory, without any lattice and cluster or Mayer expansion. In fact we stumbled upon this new method by trying to adapt former cluster expansions to large matrix φ4 models in order to extend constructive methods to non-commutative field theory (see [26] for a recent review). The matrix version is described in a separate publication [27]. Hopefully it should allow a non-perturbative construction of the φ4 theory on Moyal space R4 , whose renormalizable version was pioneered by Grosse and Wulkenhaar [16].

2. The example of the pressure of φ4 We take as first example the construction of the pressure of φ44 in a renormalization group (RG) slice. The goal is, e.g., to prove its Borel summability in the coupling 1 It

is possible to combine both expansions into a single one [3], but the result cannot be considered a true simplification.

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constant uniformly in the slice index, without using any lattice (breaking Euclidean invariance) nor any cluster or Mayer expansion. The propagator in a RG slice j is, e.g., M −2j+2 2 2 j Cj (x, y) = e−αm e−(x−y) /4α α−2 dα ≤ KM 2j e−cM |x−y| (1) M −2j

where M is a constant defining the size of the RG slices, and K and c from now on are generic names for inessential constants, respectively large and small. We could also use compact support cutoffs in momentum space to define the RG slices. 4 4 Consider a local interaction λ φ (x)d x = λTrφ4 where the trace means spatial integration. For the moment assume the coupling λ to be real positive and small. We decompose the φ4 functional integral according to an intermediate field as: 4

dμCj (φ)e−λTrφ =

1

dν(σ)e− 2 Tr log(1+iH)

(2)

where dν is the ultralocal measure on σ with covariance δ(x − y), and H = 1/2 23/2 λ1/2 Dj σDj is an Hermitian operator, with Dj = Cj . The pressure is known to be the Borel sum of all the connected vacuum graphs with a particular root vertex fixed at the origin. We want to prove this through a new method. We define the loop vertex 2 V = − 12 Tr log(1 + iH). This loop vertex can be pictured as in the left hand side of Figure 1. The trace means integration over a “root” x0 . Cyclic invariance means that this root can be moved everywhere over the loop. It is convenient to also introduce an arrow, by convention always turning counterclockwise for a +iH convention, and anti-clockwise for a complex conjugate loop vertex V¯ = − 12 Tr log(1 − iH). n We then expand the exponential as n Vn! . To compute the connected graphs we give a (fictitious) index v, v = 1, . . . , n to all the σ fields of a given loop vertex Vv . This means that we consider n different copies σv of σ with a degenerate Gaussian measure dν({σv }) whose covariance is σv σv ν = δ(x−y). The functional integral over dν(σ) is equal to the functional integral over dν({σv }). We apply then the forest formula of [2] to test connexions between the loop vertices from 1 to n. (The lines of this forest, which join loop vertices correspond to former φ4 vertices.) The logarithm of the partition function log Z(Λ) at finite volume Λ is given by this formula restricted to trees (like in the Fermionic case [1]), and spatial Z(Λ) is integration restricted to Λ. The pressure or infinite volume limit of log|Λ| given by the same rooted tree formula but with one particular position fixed at the origin, for instance the position associated to a particular root line 0 . More precisely: 2 To

avoid any confusion with the former φ4 vertices we shall not omit the word loop.

406


11 00 00 11 00 11


1 0 0 1 0 1

x0

A loop vertex

x0

A tree on loop vertices

Figure 1. Loop vertices and a tree on them. Theorem 2.1.

1

∞ log Z(Λ) 1 = lim dw GT (σ, x0 )|x0 =0 Λ→R4 |Λ| n! 0 n=1 T ∈T d4 x d4 y dνT {σv }, {w} GT (σ, x0 ) = ∈T

δ(x − y ) ∈T

δ

δ

δσv() (x ) δσv () (y )

(3)

Vv ,

(4)

v

where • each line of the tree joins two different vertices Vv() and Vv () at point x and y , which are identified through the function δ(x − y ) (since the covariance of σ is ultralocal), • the sum is over rooted trees over n vertices, which have therefore n − 1 lines, with root 0 , • the normalized Gaussian measure dνT ({σv }, {w}) over the vector field σv has covariance σv , σv = δ(x − y)wT v, v , {w} where wT (v, v , {w}) is 1 if v = v , and the infimum of the w for running over the unique path from v to v in T if v = v . This measure is well-defined because the matrix wT is positive. Proof (sketched ). This is the outcome of the universal tree formula of [2] in this case. To explicit further this formula, consider a loop vertex Vv of coordination kv in the tree, and let us compute more explicitly the outcome of the kv derivatives

kv δ i=1 δσ(xi ) acting on 1 V = − T r log(1 + iH) 2 which created this loop vertex.

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Consider the operator 1 Dj . (5) 1 + iH Calling x1 the root position for the loop vertex Vv , that is the unique position from which a path goes to the root of T , the loop vertex factor Vv after action of the derivatives is k kv v √ δ 1 Vv = (−i λ)kv Cj (σ, xτ (i) , xτ (i+1) ) (6) δσ(xi ) 2 τ i=1 i=1 Cj (σ) = Dj

where the sum is over all permutations τ of [2, . . . , k], completed by τ (1) = τ (k + 1) = 1. To check it, we need only to move by cyclicity the local root of each loop nearest to the global root in the tree. This global root point is chosen for simplicity in formulas above at a particular root line 0 , but in fact it could be fixed anywhere in an arbitrarily chosen “root loop”, as shown on the right hand side of Figure 1 (with all loops oriented counterclockwise). There is another representation of the same object. A tree on connecting loops such as the one shown in the right hand side of Figure 1 can also be drawn as a set of dotted lines dividing in a planar way a single loop as in Figure 2. Each dotted line carries a δ(x − y ) function which identifies pairs of points on the border of the loop joined by the dotted line, and is equipped with a coupling constant, because it corresponds to an old φ4 vertex. This second picture is obtained by turning around the tree. The pressure corresponds to the sum over such planar partitions of a single big loop with an arbitrary root point fixed at the origin. The corresponding interpolated measure dν can be described also very simply in this picture. There is now a σv field copy for every domain v inside the big loop, a w parameter for each dotted line, and the covariance of two σv and σv fields is the ordinary δ function covariance multiplied by a weakening parameter which is the infimum of the w parameters of the dotted lines one has to cross to go from v to v . The counterclockwise orientation of the big loop corresponds to the +iH convention. In this new picture we see indeed many loops. . . but the golden rule is not violated. In this new representation it simply translates into “Thou shall see only planar (or genus-bounded ) structures. . . ” (Recall that genus-bounded graphs are not many and don’t make perturbation theory diverge.) Let us prove now that the right hand side of formula (3) is convergent as series in n. Theorem 2.2. The series (3) is absolutely convergent for λ small enough, and the sum is bounded by K|λ|M 4j , where K is some constant. Proof. To bound the integrals over all positions except the root, we proceed by induction along the tree, starting from the leaves and working towards the root.

408



1 0 0 1

Figure 2. The big loop representation. For n = 1, that is for the empty tree corresponding to a single loop vertex, we remark that H at σ = 0 being 0, hence log(1 + iH)|σ=0 = log 1 = 0, we can perturb a first σ out of the T r log(1 + iH) and Wick contract it. In this way we get a contribution with one vertex and two loops, similar to the contribution at n = 2 where the tree has exactly one line (except that there is no w parameter. . . ). In this way our induction can really start from n = 2. The first step of the induction is a bound on a single loop or “leaf” uniform in the position x of the root of that leaf: Lemma 2.1. There exists K such that for any x and any v |Cj (σv )(x, x)| ≤ KM 2j

∀σv .

(7)

Since iH is anti-hermitian we have (1 + iH)−1 ≤ 1. It is obvious from (1) that Cj ≤ KM −2j , hence Dj ≤ KM −j . We have Cj (σv ) (x, x) = dydzDj (x, y)A(y, z)Dj (z, x) = f, Af (8) for f = Dj (x, .) and A = (1 + iH)−1 . The norm of the operator A is bounded by 1. Since f 2 ≤ KM 2j , the result follows. Remark that by combining the single coupling constant with two bounds M 2j of this type for the two leaves, we get the theorem for n = 2 (and also for n = 1 by the remark above). We now consider a subgraph of the tree, obtained by cutting a particular branch B at some place above the root, containing n ≥ 1 loop vertices. This branch has a root at a point called x. We assume as our induction hypothesis a

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bound similar to (7) for the amplitude IB (x, x) of that branch. It reads: |IB (x, x)| ≤ |λK|n−1 M 2j

∀σ .

(9)

Then we prove the same bound for a branch with n + 1 loop vertices which can be seen as a root loop with root x0 with k branches B1 , . . . , Bk (of order n1 , . . . nk ) inserted at x1 , . . . xk along the loop. We now use the induction hypothesis on these branches and the fact that the bound (9) is uniform in x to deduce that the multiplication operator M IBi in x space (with diagonal coefficients IBi (x, x)) is bounded in norm by |λK|ni −1 M 2j . This is the essential point. But the root loop corresponds to f, A1 Dj M IB1 Dj A2 Dj M IB1 Dj . . . Dj M IBk Dj Ak+1 f . Inserting the uniform norm bounds on the operators along the loop (and taking into account the correct number of coupling constants) proves the bounds at order n + 1 for that bigger branch. Remark however that when we complete the tree, the last root is common to two branches. Multiplying the two corresponding M 2j factors gives a M 4j global n independent factor, as should be the case for vacuum graphs in the φ4 theory in a single RG slice. To conclude the proof of the theorem the reader may worry about the combinatoric of trees and the w integrals. But we can integrate the previous bound over the complicated measure dνT and over the {w } parameters. But since our bound is independent of σ v , since the measure dν(σ) is normalized, and since each w runs from 0 to 1, this does not change the result. Finally by Cayley’s theorem the sum over trees costs (kn!v −1)! . The n! cancels v with the 1/n! of (3) and the 1/(kv − 1)! factors compensate for the sums over

permutations τ in (6), which have exactly v (kv − 1)! elements. It remains a geometric series bounded by 12 M 4j (λK)n−1 hence convergent for small λ, and the sum is bounded by K.M 4j .

3. Uniform Borel summability Rotating to complex λ and Taylor expanding out a fixed number of φ4 vertices proves Borel summability in λ uniformly in j. Definition. A family fj of functions is called Borel summable in λ uniformly in j if • Each fj is analytic in a disk DR = {λ|Re λ−1 > 1/R}; • Each fj admits an asymptotic power series k aj,k λk (its Taylor series at the origin) hence: fj (λ) =

r−1

aj,k λk + Rj,r (λ)

(10)

k=0

such that the bound |Rr,j (λ)| ≤ Aj ρr r!|λ|r

(11)

410



holds uniformly in r and λ ∈ DR , for some constant ρ ≥ 0 independent of j and constants Aj ≥ 0 which may depend on j. k Then every fj is Borel summable [29], i.e., the power series k aj,k tk! converges for |t| < 1/ρ, it defines a function Bj (t) which has an analytic continuation in the j independent strip Sρ = {t| dist (t, R+ ) < 1/ρ}. Each such function satisfies the bound t |Bj (t)| ≤ Bj e R for t ∈ R+ (12) for some constants Bj ≥ 0 which may depend on j. Finally each fj is represented by the following absolutely convergent integral: 1 ∞ −t fj (λ) = e λ Bj (t)dt for λ ∈ CR . (13) λ 0 Theorem 3.1. The series for the pressure is uniformly Borel summable with respect to the slice index. Proof. It is easy to obtain uniform analyticity for Re λ > 0 and |λ| small enough, a region which obviously contains a disk DR . Indeed all one has to do is to reproduce the previous argument but adding that for H Hermitian, the operator (1+ieiθ H)−1 √ is bounded √ by 2 for |θ| ≤ π/4. Indeed if π/4 ≤ Argz ≤ 3π/4, we have |(1 + iz)−1 | ≤ 2. Then the uniform bounds (11) follow from expanding the product of resolvents in (6) up to order r − 2(n− 1) in λ by an explicit Taylor formula with integral remainder followed by explicit Wick contractions. The sum over the contractions leads to the ρr r! factor in (11).

4. Connected functions and their decay To obtain the connected functions with external legs we need to add resolvents to the initial loop vertices. A resolvent is an operator Cj (σr , x, y). The connected functions S c (x1 , . . . , x2p ) are obtained from the normalized functions by the standard procedure. We have the analog of formula (3) for these connected functions: Theorem 4.1.

1

∞ 1 4 4 dw d x d y S (x1 , . . . , x2p ) = n! 0 π n=0 T ∈T

δ δ dνT ({σv }, {σr }, {w}) δ(x − y ) δσv() (x ) δσv () (y ) c

∈T

v

where

Vv

p r=1

Cj (σr , xπ(r,1) , xπ(r,2) ) , (14)

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• the sum over π runs over the pairings of the 2p external variables into pairs (xπ(r,1) , xπ(r,2) ), r = 1, . . . , p, • each line of the tree joins two different loop vertices or resolvents Vv() and Vv () at point x and y , which are identified through the function δ(x − y ) because the covariance of σ is ultralocal, • the sum is over trees joining the n + p loop vertices and resolvents, which have therefore n + p − 1 lines, • the measure dνT ({σv }, {σr }, {w}) over the {σ} fields has covariance σα , σα = δ(x − y)wT α, α , {w} where wT (α, α , {w}) is 1 if α = α (where α, α ∈ {v}, {r}), and the infimum of the w for running over the unique path from α to α in T if α = α . This measure is well-defined because the matrix wT is positive. Now we want to prove not only convergence of this expansion but also scaled tree decay between external arguments: Theorem 4.2. The series (14) is absolutely convergent for λ small enough, its sum is uniformly Borel summable in λ and we have: |S c (z1 , . . . , z2p )| ≤ (2p)!K p |λ|p−1 M 2pj e−cM

j

d(z1 ,...,z2p )

(15)

where d(z1 , . . . , z2p ) is the length of the shortest tree which connects all the points z1 , . . . , zp . The proof of convergence (and of uniform Borel summability) is similar to the one for the pressure. We shall provide only a sketch of this proof and in particular we do not take care of listing all different constants K that occur in the induction below. These constants K do not build up into a problem for the proof because each can be paired with a fractional power of a different coupling constant. The tree decay (15) is well known and standard to establish through the traditional cluster and Mayer expansion. It is due to the existence of a tree of Cj propagators between external points in any connected function. In the present expansion, this tree is hidden in the resolvents and loop vertices, so that an expansion on these resolvents (and loop vertices) is necessary in one form or another to prove (15). It does not seem to follow from bounds on operator norms only: the integral over the σ field has to be bounded more carefully. The standard procedure to keep resolvent expansions convergent is a so-called large/small field expansion on σ. In the region where σ is small the resolvent expansion converges. In the large field region there are small probabilistic factors coming from the dνT measure. This is further sketched in Subsection 5.2. However the large/small field expansion again requires a discretization of space into a lattice: a battery of large/small field tests is performed, on the average of the field σ over each cube of the lattice. We prefer to provide a new and different proof of (15). It relies on a single resolvent step followed by integration by parts, to establish a Fredholm inequality on the modulus square of the 2p point function. From this Fredholm inequality the desired decay follows easily. The rest of this

412



y

x 11 00 00 11

11 00 00 11

Figure 3. Three resolvents with two branching subtrees.

x

11 00 00 11

1 0

y

Figure 4. The half-circle representation of Figure 3. section is devoted to the proof of (15) in the simplest case p = 1. The most general case is sketched in Subsection 5.1. The two point function S c is simply called S(x, y) from now on, and for p = 1 (15) reduces to |S(x, y)| ≤ KM 2j e−cM

j

|x−y|

.

(16)

We work with n, T and {w} fixed in (14). We use the resolvent as root for T , from which grow q subtrees T1 , . . . , Tq . In more pictorial terms, (14) represents a chain of resolvents from x to y separated by insertions of q subtrees. Figure 3 is therefore the analog of Figure 1 in this context3 . A representation similar to the big loop of Figure 2 pictures the decorated resolvent as a half-circle going from x to y, together with a set of planar dotted lines for the vertices. The +i convention again corresponds to a particular orientation. For reason which should become clear below, we picture the planar dotted lines all on the same side of the x-y line, hence inside the half-disk. To each such drawing, or graph G, there is an associated Gaussian measure dνG which is the one from which the drawing came as a tree. Hence it has a field copy associated to each planar region of the picture, a weakening parameter w associated to each dotted line, and the covariance between the σ fields of different regions is given by the infimum over the parameters of the dotted lines that one has to cross to join these two regions. 3A

similar figure is a starting point for the 1PI expansion of the self-energy in [8, 25].

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There is also for each such G an amplitude. Let us write simply dνG for the 1 normalized integral 0 ∈G dw dνG ({σ}, {w}). If the graph has n dotted lines hence 2n + 1 resolvents from x to y, its amplitude is 2n+1 n 4 d x Cj (σi , xi−1 , xi ) (17) dνG AG (x, y) = λ i=1

∈G

where the product over runs over the dotted lines and the product over i runs over the resolvents along the half-circle, with x0 = x and x2n+1 = y. In (17) σi is the field copy of the region just before point xi and the 2n positions x1 , . . . , x2n are equal in pairs to the n corresponding x ’s according to the pairings of the dotted lines. We shall prove Lemma 4.1. There exists some constant K such that for λ small enough sup

|AG (x, y)| ≤ (|λ|K)n/2 M 2j e−cM

j

|x−y|

.

(18)

G,n(G)=n

From this lemma (16) obviously follows. Indeed the remaining sum over Cayley trees costs at most K n n!, which is compensated by the 1/n! in (14). In the language of planar graphs the planar dotted lines cost only K n . Hence the sum over n converges for λ small enough because of the |λ|n/2 factor in (18). Remark that this factor |λ|n/2 is not optimal; |λ|n is expected; but it is convenient to use half of the coupling constants for auxiliary sums below. We apply a Schwarz inequality to |AG (x, y)|2 , relatively to the normalized measure dνG : |AG (x, y)|2 ≤ AG∪G¯ (x, y) , 2n 4 4 AG∪G¯ (x, y) = |λ| d x d x ¯ dνG

(19)

∈G 2n+1

Cj (σi , xi−1 , xi )C¯j (σi , x¯i−1 , x¯i )

(20)

i=1

with hopefully straightforward notations. The quantity on the right hand side is now pointwise positive for any σ. It can be considered as the amplitude AG∪G¯ (x, y) associated to a mirror graph ¯ Such a mirror graph is represented by a full disk, with x and y diametrally G ∪ G. opposite, and no dotted line crossing the corresponding diameter. The upper halfcircle represents the complex conjugate of the lower part. Hence the upper half-disk is exactly the mirror of the lower half-disk, with orientation reversed, see Figure 5. The Gaussian measure associated to such a mirror graph remains that of G, hence it has a single weakening w parameter for each dotted line and its mirror line, and it has a single copy of a σ field for each pair made of a region of the disk and its mirror region. Let’s call such a pair a “mirror region”. The covariance between two fields belonging to two mirror regions is again the infimum of the w

414


x


0y 1 0 1

1 0 0 1

11 00

¯ for the graph G of Figure 4. Figure 5. The mirror graph G ∪ G parameters crossed from one region to the other, but, e.g., staying entirely in the lower half-disk (or the upper half-disk). We shall now perform a single resolvent expansion step and integration by parts, together with a bound which reproduces an amplitude similar to AG∪G¯ . The problem is that the category of mirror graphs is not exactly stable in this operation; this bound generates other graphs with “vertical” dotted lines between the lower and upper half of the circle. To prove our bound inductively we need therefore to generalize slightly the class of mirror graphs and their associated ¯ ∪ V , called generalized Gaussian measures to a larger category of graphs G ∪ G mirror graphs or GM graphs and pictured in Figure 6. They are identical to mirror graphs except that they can have in addition a certain set V of “vertical” dotted lines between the lower and upper half of the circle, again without any crossing. There is a corresponding measure dνG,V with similar rules; there is a single w parameter for each pair of dotted line and its mirror, in particular there is a w parameter for each vertical line. Again the covariance between two fields belonging to two mirror regions is the infimum of the w parameters crossed from one mirror region to the other, staying entirely in, e.g., the lower half-disk. The upper halfpart is still the complex conjugate of the lower half-part. The order of a GM graph is again the total number L = 2n + |V | of dotted lines and its amplitude is given by a pointwise positive integral similar to (20): L 4 4 (x, y) = |λ| d x d x ¯ dy dνG∪V AG∪G∪V ¯ ∈G

∈V

2n+|V |+1

i=1

Cj (σi , zi−1 , zi )C¯j (σi , z¯i−1 , z¯i ) , (21)


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x00 00 11 11

415

11 00 11 y 00

Figure 6. The generalized mirror graphs. where the z’s and z¯’s are either x ’s, x ¯ ’s or y ’s according to the graph. (x, y) of a GM graph so that AG∪G∪V (x, y) = Defining the integrand I ¯ ¯ G∪ G∪V dνG∪V IG∪G∪V (x, y), we have: ¯ Lemma 4.2. For any GM graph we have, uniformly in σ, x and y: IG∪G∪V (x, y) ≤ (K|λ|)L M 4j . ¯

(22)

Indeed the quantity IG∪G∪V (x, y) is exactly the same as the amplitude of ¯ a pressure graph but with two fixed points and some propagators replaced by complex conjugates, hence the proof through the norm estimates of Lemma 2.1 is almost identical to the one of Theorem 2.2. We now write the resolvent step which results in an integral Fredholm inequality for the supremum of the amplitudes of any generalized mirror graph. Let us define the quantity ΓL (x, y) =

sup GM graphs G,V | L(G)=L

|λ|−L/2 AG∪G∪V (x, y) . ¯

(23)

We shall prove by induction on L: Lemma 4.3. There exists some constant K such that for λ small enough 4j −cM j |x−y| 1/2 −cM j |x−z| + |λ| ΓL (z, y) . dze ΓL (x, y) ≤ KM e From that lemma indeed obviously follows

(24)

416



Lemma 4.4. There exists some constant K such that for λ small enough ΓL (x, y) ≤ KM 4j e−cM

j

|x−y|

.

(25)

Indeed iterating the integral Fredholm equation (24) leads obviously to (25). Taking (21) and (23) into account to reinstall the λL/2 factor, considering the equation L = 2n + V and taking a square root because of (19), Lemma 4.1 is then nothing but Lemma 4.4 for the particular case V = 0. The rest of this section is therefore devoted to the proof of Lemma 4.3, by a simple induction on L. If L = 0, Γ0 (x, y) = dνCj (σ, x, y, )C¯j (σ, x, y, ). Expanding the Cj (σ, x, y) propagator, we get √ Γ0 (x, y) = dν Cj (x, y) − i λ dzCj (x, z)σ(z)Cj (σ, z, y) C¯j (σ, x, y) . (26) For the first term | dνCj (x, y)C¯j (σ, x, y)|, we simply use bounds (1) and (22) in the case L = 0. For the second term we Wick contract the σ field (i.e., integrate δ by parts over σ). There are two subcases: the Wick contraction δσ hits either Cj (σ, z, y) or C¯j (σ, x, y). We then apply the inequality |ABC| ≤

A (M 2j |B|2 + M −2j |C|2 ) , 2

(27)

which is valid for any positive A. In the first subcase we take A = dzCj (x, z), B = Cj (σ, z, y) and C = Cj (σ, z, z)C¯j (σ, x, y), hence write dzCj (x, z)Cj (σ, z, z)Cj (σ, z, y)C¯j (σ, x, y) Cj (x, z) 2j M |Cj (σ, z, y)|2 + M −2j |Cj (σ, z, z)C¯j (σ, x, y)|2 (28) ≤ dz 2 and in the second subcase we write similarly dzCj (x, z)Cj (σ, z, y)C¯j (σ, x, z)C¯j (σ, z, y) Cj (x, z) 2j M |Cj (σ, z, y)|2 + M −2j |C¯j (σ, x, z)C¯j (σ, z, y)|2 . (29) ≤ dz 2 Using the uniform bound (22) on the “trapped loop” |Cj (σ, z, z)|2 or |C¯j (σ, x, z)|2 in the C term we obtain 4j −cM j |x−y| + |λ|K Γ0 (x, y) Γ0 (x, y) ≤ KM e 4j −cM j |x−z| +M Γ0 (z, y) (30) dze so that (24) hence Lemmas 4.3 and 4.4 hold for L = 0.


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We now assume that (24), hence also (25), is true up to order L and we want to prove (24) at order L + 1. Consider a GM graph of order L + 1. If V ≥ 1 we can decompose it as a convolution of smaller GM graphs: AG∪G∪V (x, y) = λ dy1 AG1 ∪G¯ 1 (x, y1 )AG2 ∪G¯ 2 ∪V2 (y1 , y) (31) ¯ with total orders L1 for G1 and L2 for G2 , V2 = V − {1} strictly smaller than L + 1. Applying the induction hypothesis (25) to these smaller GM graphs we get directly that sup

¯ G,V |L(G∪G∪V )=L+1,V >0

|λ|−(L+1)/2 AG∪G∪V (x, y) ≤ KM 4j e−cM ¯

j

|x−y|

.

(32)

Hence we have now only to prove (24) for mirror graphs with V = ∅. Consider now such a mirror graph G. Because of the |λ|−L/2 in (23), we should remember that we have only a remaining factor |λ|L/2 to use for our bounds on ΓL . Starting at x we simply √ expand the first resolvent propagator Cj (σ, x, x1 ) as Cj (x, x1 ) − dzCj (x, z)i λσ(z)Cj (σ, z, x1 ). For the first term we call xi1 the point to which x1 is linked by a dotted line and apply a Schwarz inequality of the (27) type, with: A = dx1 Cj (x, x1 ) , (33) dxi Cj (σ, xi−1 , xi ) , B= C=

i1 +1≤i≤2n

2≤i≤i1 −1

dxi

i1 +1≤i≤2n+1

2≤i≤i1

Cj (σ, xi−1 , xi )

2n i=1

d¯ xi

C¯j (σ, x¯i−1 , x ¯i ) .

1≤i≤2n+1

It leads, using again the norm bounds of type (22) on the “trapped loop” in the first part of C, to a bound j |λ|1/2 K ΓL (x, y) + M 4j dx1 e−cM |x−x1 | Γr (x1 , y) (34) for some r < L. Applying the induction hypothesis concludes to the bound (24). Finally for the second term we Wick contract again the σ field. There are δ again two subcases: the Wick contraction δσ hits either a Cj or a C¯j . Let us call i the number of half-lines, either on the upper or on the lower circles, which ¯i1 , . . . x ¯ik the positions of the are inside the Wick contraction, and xi1 , . . . xik or x dotted lines crossed by the Wick contraction. We have now two additional difficulties compared to the L = 0 case: • we have to sum over where the Wick contraction hits, hence sum over i (because the Wick contraction creates a loop, hence potentially dangerous combinatoric). The solution is that the norm bound on the “trapped loop” in the C term of (27) erases more and more coupling constants as the loop gets longer: this easily pays for choosing the Wick contraction.

418



S(x,y)

x11 00 11 00 Cj

11 00 00 y 11

z

TL(z,z)

R(z,y)

Xi 1

Xi 1

z

Figure 7. The Wick contraction. • the dotted lines crossed by the Wick contraction should be kept in the A term in inequality (27). In other words they become vertical lines at the next step, even if no vertical line was present in the initial graph. This is why we had to extend our induction to the category of GM graphs. This extension is what solves this difficulty. We decompose the amplitude of the graph in the first subcase of Figure 7 as ¯ y) (35) dzdxi1 , . . . dxik Cj (x, z)T Lxi1 ,...xik (z, z)Rxi1 ,...xik (z, y)S(x, i

with hopefully straightforward notations, and we apply the Schwarz inequality (27), with: A = |λ|i/8 dzdxi1 , . . . dxik Cj (x, z) , i

B = Rxi1 ,...xik (z, y) , ¯ y) . C = |λ|−i/8 T Lxi1 ,...xik (z, z)S(x,

(36)

Now the first remark is that i|λ|i/8 is bounded by K for small λ so we need only to find a uniform bound at fixed i. The A|B|2 is a convolution of an explicit propagator bounded by (1) with a new GM graph (with vertical lines which are the crossed lines at xi1 , . . . xik ) either identical to G or shorter. If it is shorter we apply the induction hypothesis. If it

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x11 0

x 112 00 00 11

x31 0

x 114 00 00 11

0 1

0 1

419

Figure 8. A connected 4 point function. is not shorter we obtain a convolution equation term like in the right hand side of (24). The A|C|2 contains a trapped loop T L with i vertices. Each half-vertex of the trapped loop has only |λ|1/8 because of the |λ|−i/8 factor in (36). The trapped loop is again of the GM nature with vertical lines which are the crossed lines at xi1 , . . . xik . But we can still apply the bound (22) to this trapped loop. Therefore the bound on the sum of the A|B|2 and A|C|2 is again of the type (34). δ hits a C¯j , is exFinally the second subcase where the Wick contraction δσ actly similar, except that the “almost trapped loop” is now something of the type T¯L(x, z) rather than T L(z, z), but the bound (22) also covers this case, so that everything goes through. Collecting the bounds (34) in every case completes the proof of Lemmas 4.3 and 4.4 for ΓL+1 . This concludes the proof of Lemmas 4.3 and 4.4 for all L.

5. Further topics 5.1. Higher functions The analysis of 2p point functions is similar to that of the previous section. The general 2p point function S c (x1 , . . . , x2p ) defined by (14) contains p resolvents of the Cj (σ) type and a certain number of loop vertices joining or decorating them. Turning around the tree we can still identify the drawing as a set of decorated resolvents joined by local vertices or dotted lines as in Figures 8 and 9, which are the analogs of Figures 3 and 4. This is because any chain of loop vertices joining resolvents can be “absorbed” into decorations of one of these resolvents. The factor 2p! in (15) can be understood as a first factor 2p!! to choose the pairing of the points in p resolvents and an other p! for the choice of the tree of connecting loop vertices between them. We can again bound each term of the initial expansion by a “mirror” term pointwise positive in σ with p disks as shown in Figure 10. A Lemma similar to Lemma 4.1 is again proved by a bound on generalized mirror graphs such as Figure 10 but with additional vertical lines between the p disks. This bound is proved inductively by a single resolvent step followed by a Fredholm bound similar to Lemmas 4.3 and 4.4. Verifications are left to the reader.

420


x 11 2 00

x1 0 1 1 0


x3

x4

00 11

0 1

1 0

1 0

Figure 9. The “half-disk” representation of that connected 4 point function.

00 11 00x 2 11 00 11

0 x1 1 0 1 0 1 0 1

x3 1 0 0 1

1 0 0x 4 1 0 1 0 1

1 0 0 1

1 0

1 0

Figure 10. The mirror representation of the same connected 4 point function. 5.2. Large/small field expansion To prove the tree decay of the 2p-point connected functions as external arguments are pulled apart, it is possible to replace the Fredholm inequality of the previous section by a so-called large/small field expansion. It still relies on a resolvent expansion, but integration by parts is replaced by a probabilistic analysis over σ. We recall only the main idea, as this expansion is explained in detail in [4,20] but also in a very large number of other publications. A lattice D of cubes of side M −j is introduced and the expansion is 4j 2 4j 2 M |λ| σ (x)dx + 1 − χ M |λ| σ (x)dx (37) 1= χ Δ∈D

Δ

Δ

where χ is a compact support function (eventually smooth). The small field region S is the union of all the cubes for which the χ factor has been chosen. The complement, called the large field region L, is decomposed as

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the union of connected pieces Lk . Every large field region has a small probabilistic factor for each of its cube using, e.g., some standard Tchebycheff inequality. The field is decomposed according to its localization as σ = σS + k σLk . Then the resolvent Cj (σ, x, y) is simply bounded in norm if x and y belong to the same Lk region because the decay is provided by the probabilistic factor associated to Lk . The σS piece is expanded according to resolvent formulas such as √ (38) Cj (σS , x, y) = Cj (x, y) − i λ dzCj (x, z)σS (z)Cj (σS , z, y) , which can be pushed to infinity because the σS field is not integrated with the Gaussian measure but bounded with the help of the small field conditions. Then inside each connected large field region Lk the resolvent Cj (σLk , x, y) is simply bounded in norm. The decay is provided by the probabilistic factor associated to Lk . Between different connected large field regions, the decay is provided by the small field resolvent expansion. However one advantage of the loop expansion presented in this paper is to avoid the need of any lattice of cubes for cluster/Mayer expansions. If possible, it seems better to us to avoid reintroducing the lattice of cubes for the small/large field analysis. 5.3. Multiscale expansions The result presented in this paper for a single scale Bosonic model should be extended to a multiscale analysis. This means that every loop-vertex or resolvent should carry a scale index j which represents the lowest scale which appears in that loop or resolvent. Then we know that the forest formula used in this paper should be replaced by a so-called “jungle” formula [2] which is nothing but a multiforest formula in which links are built preferentially between loop vertices and resolvents of highest possible index. This jungle formula is to be completed with a “vertical” expansion which tests whether connected contributions of higher scales have less or more than four external lower legs. A renormalization expansion then extracts the local parts of such two and four point contributions and hides them into effective couplings. This would provide a new completely explicit Bosonic renormalization-group-resummed expansion, which in contrast with [4] would avoid any cluster and Mayer expansion. The expansion could be completed by auxiliary resolvent expansions, either with integration by parts in the manner of Section 4 or with a small/large field analysis as in Subsection 5.2 above. This is necessary to establish scaled spatial decay, which in turn is crucial to prove that the renormalized two and four point contributions are small. But these new auxiliary expansions shall be used only to prove the desired bounds, not to define the expansion itself. 5.4. Vector models The method presented here is especially suited to the treatment of large N vector models. Indeed we can decompose a vector φ4 interaction with an intermediate

422



scalar field so that the flow of vector indices occur within the loop-vertices. Every loop vertex simply carries therefore a global N factor where N is the number of colors. Hence we expect that the loop expansion presented here is the right tool to glue different regimes of the renormalization group governed respectively, e.g., in the ultraviolet regime by a small coupling expansion and in the infrared by a “non-perturbative” large N expansion of vector type. This gluing problem occurs in many different physical contexts, from mass generation of the two-dimensional Gross–Neveu [20] or non-linear σ-model [21] to the BCS theory of supraconductivity [12]. These gluing problems have been considered until now too complicated in practice for a rigorous constructive analysis. 5.5. Matrix models and φ4 4 The loop expansion is also suited for the treatment of large N matrix models and was in fact found for this reason [27]. Our first goal is to apply it to the full construction of non-commutative φ4 4 [16], either in the so-called matrix base [17, 28] or in direct space [18]. One needs again to develop for that purpose the multiscale version of the expansion and the resolvent bounds analogs to Section 4 or Subsection 5.2 above. Indeed neither the matrix propagator nor the Mehler x space propagator are diagonal (except at the very special ultraviolet fixed point where the matrix propagator of φ4 4 becomes diagonal). Ultimately we hope better understanding of non commutative models of the matrix or quasi-matrix type should be useful in many areas of physics, from physics beyond the standard model [7, 10] to more down to earth physics such as quark confinement [19] or the quantum Hall effect [23].

References [1] A. Abdesselam and V. Rivasseau, Explicit Fermionic cluster expansion, Lett. Math. Phys. 44, 77–88 (1998), arXiv:cond-mat/9712055. [2] A. Abdesselam and V. Rivasseau, Trees, forests and jungles: A botanical garden for cluster expansions, in Constructive Physics, ed. by V. Rivasseau, Lecture Notes in Physics 446, Springer Verlag, 1995, arXiv:hep-th/9409094. [3] A. Abdesselam, J. Magnen and V. Rivasseau, Bosonic monocluster expansion, Commun. Math. Phys. 229 (2002), 183, arXiv:math-ph/0002053. [4] A. Abdesselam and V. Rivasseau, An explicit large versus small field multiscale cluster expansion, Rev. Math. Phys. 9 (1997), 123, arXiv:hep-th/9605094. [5] D. Brydges and T. Kennedy, Mayer expansions and the Hamilton–Jacobi equation, Journal of Statistical Physics, 48, 19 (1987). [6] D. Brydges, Weak perturbations of massless Gaussian measures, in Constructive Physics, LNP 446, Springer 1995. [7] A. H. Chamseddine, A. Connes and M. Marcolli, Gravity and the standard model with neutrino mixing, arXiv:hep-th/0610241v1, and references therein.

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[8] M. Disertori and V. Rivasseau, Interacting Fermi liquid in two dimensions at finite temperature, Part I: Convergent attributions, Commun. Math. Phys. 215, 251 (2000); Part II: Renormalization, in two dimensions at finite temperature, Part I: Convergent attributions, Commun. Math. Phys. 215, 291 (2000). [9] M. Disertori and V. Rivasseau, Continuous constructive Fermionic renormalization, Annales Henri Poincaré, 1 (2000), 1, arXiv:hep-th/9802145. [10] M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. 73 (2001), 977–1029, arXiv:hep-th/0106048. [11] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Commun. Math. Phys. 247 (2004): A two dimensional Fermi liquid. Part 1: Overview, 1–47; Part 2: Convergence, 49–111; Part 3: The Fermi surface, 113–177; Particle–Hole Ladders, 179–194; Convergence of perturbation expansions in Fermionic models. Part 1: Nonperturbative bounds, 195–242; Part 2: Overlapping loops, 243–319. [12] J. Feldman, J. Magnen, V. Rivasseau and E. Trubowitz, An intrinsic 1/N expansion for many Fermion systems, Europhysics Letters 24 (1993), 437–442. [13] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, A renormalizable field theory: The massive Gross–Neveu model in two dimensions, Commun. Math. Phys. 103 (1986), 67. [14] K. Gawedzki and A. Kupiainen, Gross–Neveu model through convergent perturbation expansions, Commun. Math. Phys. 102 (1985), 1. [15] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd edition, Springer, (1987). [16] H. Grosse and R. Wulkenhaar, Renormalization of φ4 -theory on noncommutative R4 in the matrix base, Commun. Math. Phys. 256 (2005), 305–374, arXiv:hepth/0401128. [17] H. Grosse and R. Wulkenhaar, Power-counting theorem for non-local matrix models and renormalization, Commun. Math. Phys. 254 (2005), 91–127, arXiv:hepth/0305066. [18] R. Gurau, J. Magnen, V. Rivasseau and F. Vignes-Tourneret, Renormalization of non-commutative φ44 field theory in x space, Commun. Math. Phys. 267 (2006), 515–542, arXiv:hep-th/0512271. [19] G. ’t Hooft, A planar diagram theory for strong interactions, Nuclear Physics B 72 (1974), 461. [20] C. Kopper, J. Magnen and V. Rivasseau, Mass generation in the large N Gross– Neveu-Model, Commun. Math. Phys. 169 (1995), 121–180. [21] C. Kopper, Mass generation in the large N nonlinear σ-model, Commun. in Math. Phys. 202 (1999), 89–126. [22] A. Lesniewski, Effective action for the Yukawa2 quantum field theory, Commun. Math. Phys. 108, 437 (1987). [23] A. Polychronakos, Noncommutative fluids, Poincaré Seminar 2007, to appear in “Quantum Spaces”, Birkha¨ user Verlag, arXiv:hep-th/0706.1095. [24] V. Rivasseau, From Perturbative to Constructive Renormalization, Princeton University Press (1991).

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[25] V. Rivasseau, The two dimensional Hubbard model at half-filling: I. Convergent contributions, Journ. Stat. Phys. Vol 106 (2002), 693–722; S. Afchain, J. Magnen and V. Rivasseau, Renormalization of the 2-point function of the Hubbard model at halffilling, Ann. Henri Poincaré 6 (2005), 399 ; The Hubbard model at half-filling, part III: The lower bound on the self-energy, Ann. Henri Poincaré 6 (2005), 449. [26] V. Rivasseau, Noncommutative renormalization, Poincaré Seminar 2007, to appear in “Quantum Spaces”, Birkha¨ user Verlag, arXiv.org/0705.0705. [27] V. Rivasseau, Constructive matrix theory, arXiv:hep-th/0706.1224. [28] V. Rivasseau, F. Vignes-Tourneret and R. Wulkenhaar, Renormalization of noncommutative φ4 -theory by multi-scale analysis, Commun. Math. Phys. 262 (2006), 565–594, arXiv:hep-th/0501036. [29] A. Sokal, An improvement of Watson’s theorem on Borel summability, Journ. Math. Phys, 21 (1980), 261–263. Jacques Magnen Centre de Physique Théorique CNRS & Ecole Polytechnique F-91128 Palaiseau Cedex France e-mail: [email protected] Vincent Rivasseau Laboratoire de Physique Théorique CNRS & Université Paris XI F-91405 Orsay Cedex France e-mail: [email protected] Communicated by Joel Feldman. Submitted: June 19, 2007. Accepted: December 6, 2007.

Ann. Henri Poincaré 9 (2008), 425–455 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/030425-31, published online May 5, 2008 DOI 10.1007/s00023-008-0361-0


Absolute Quantum Energy Inequalities in Curved Spacetime Christopher J. Fewster and Calvin J. Smith Dedicated to Klaus Fredenhagen on the occasion of his 60th birthday.

Abstract. Quantum Energy Inequalities (QEIs) are results which limit the extent to which the smeared renormalized energy density of the quantum field can be negative, when averaged along a timelike curve or over a more general timelike submanifold in spacetime. On globally hyperbolic spacetimes the minimally-coupled massive quantum Klein–Gordon field is known to obey a ‘difference’ QEI that depends on a reference state chosen arbitrarily from the class of Hadamard states. In many spacetimes of interest this bound cannot be evaluated explicitly. In this paper we obtain the first ‘absolute’ QEI for the minimally-coupled massive quantum Klein–Gordon field on four dimensional globally hyperbolic spacetimes; that is, a bound which depends only on the local geometry. The argument is an adaptation of that used to prove the difference QEI and utilizes the Sobolev wave-front set to give a complete characterization of the singularities of the Hadamard series. Moreover, the bound is explicit and can be formulated covariantly under additional (general) conditions. We also generalise our results to incorporate adiabatic states.

1. Introduction The classical minimally coupled scalar field, like most matter models studied in classical general relativity, obeys the weak energy condition (WEC). That is, the stress energy tensor Tab obeys the inequality Tab v a v b ≥ 0 for all timelike vector fields v a , which entails that observers encounter only non-negative energy densities. However, it has been known since 1965 that no Wightman quantum field theory can obey the weak energy condition [8], (see [16, 17] for simple arguments as to why this is true). Moreover, under many circumstances there is no lower bound to the energy densities available in quantum field theory (QFT). This surprising feature of QFT has often been used to support proposals for exotic spacetimes, such

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as warp drive and traversable worm holes, which require WEC-violating matter distributions. In addition, the validity of the second law of thermodynamics is called into question by WEC violations in QFT [26]. With these concerns in mind, substantial effort has been directed to understand the magnitude and extent of negative energy densities permitted by QFT, starting with the work of Ford in 1978 [26]. Given the failure of pointwise energy inequalities, attention has been focused on averages of the stress tensor along timelike worldlines or over spacetime regions. In many QFT models such averages turn out to obey Quantum Energy Inequalities (QEIs); that is, their expectation values are bounded from below as the state varies within the class of physically reasonable states [9–11, 14, 26–28, 41]. Since their inception QEIs have been applied to a variety of physical problems; they form the basis of the arguments constraining exotic spacetimes such as the warp drive [2,28] and traversable wormholes [19,27]. In their most common form, QEIs place lower bounds on the expectation value of the averaged stress energy tensor relative to that obtained in a reference state. For this reason they are called difference QEIs. To give a specific example [10], consider the minimally coupled scalar field of mass μ ≥ 0 in a globally hyperbolic spacetime (M, g) of dimension m ≥ 2, and let γ : R → M be a smooth timelike curve (not necessarily a geodesic) with velocity v a . Then for any real-valued f ∈ C0∞ (R) the QEI ren ren γ(t) ≥ −BD dt f 2 (t) v a v b Tab ω γ(t) − v a v b Tab (1) ω0 R

holds for all Hadamard states ω, where the bound ∞

∧ dξ split BD = f ⊗ f ϑ∗ v a v b Tab (−ξ, ξ) ω0 π 0

(2)

depends only on f , γ, and the reference state ω0 (which may be any Hadamard state); note that it does not depend on the state of interest ω. Here the hat denotes the Fourier transform given, in our conventions, by f(ξ) = Rn dn x f (x)eiξ · x . The split quantity v a v b Tab ω0 (x, x ) is the (unrenormalized) point split energy density defined, in a neighbourhood of γ, by

3 1 a 1 2 a b split b e ∇a ⊗ eα ∇b + μ ½ ⊗ ½ Λω0 (x, x ) (3) v v Tab ω0 (x, x ) = 2 α=0 α 2 where {eaα }α=0,1,2,3 is a tetrad field satisfying ea0 |γ = v a (see Section 3 of [10] for a more detailed discussion) and Λω0 is the two point function of the state ω0 . split Finally, ϑ∗ v a v b Tab ω0 denotes the pull-back split split ϑ∗ v a v b Tab γ(τ ), γ(τ ) (τ, τ ) = v a v b Tab (4) ω

ω0

which may be defined rigorously as a distribution on R2 using the techniques of microlocal analysis, which also guarantee that the bound (2) is finite. Similar bounds also hold for the free spin-1/2 and spin-1 field in comparable generality

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and rigour [5,12–14]. Recently, quantum energy inequalities have also been proven for free spin-3/2 fields in Minkowski spacetime [37, 49]. As already mentioned, one application of the QEIs is to place constraints on exotic spacetimes. However, the bound given above, while valid in any globally hyperbolic curved spacetime, depends crucially on the choice of a reference state. Although Hadamard states exist on any globally hyperbolic spacetimes [29, 30], closed form expressions for two point functions are known only in very special circumstances. For instance, no such expression is available for any Hadamard state on the warp drive spacetime. Typically these problems have been avoided by heuristic appeals to the equivalence principle to justify the use of Minkowski spacetime QEIs on sufficiently small scales. To date, this approach, while physically reasonable, lacks full mathematical justification and control over the scales on which it is valid. It would clearly be preferable to employ a lower bound which did not require the specification of a reference state and placed constraints directly on ren ω . Bounds of this type, known as absolute QEIs, have been established in flat Tab spacetimes [24] but the only curved spacetime absolute QEI is that of Flanagan [25] (see also [15, 47]) which applies to massless free fields in two-dimensional globally hyperbolic spacetimes. This approach relies on the conformal invariance of the theory and does not generalise to higher dimensions or non-zero mass. However, it does provide the basis for a QEI on arbitrary positive energy conformal field theories in Minkowski spacetime, including interacting examples [18]. In this paper we present the first absolute QEI applicable to the scalar field of mass μ ≥ 0 in four-dimensions, by refining and modifying the argument presented in [10]. For averaging along timelike worldline γ, our result takes the form ren dt f 2 (t) v a v b Tab (t) ≥ −BA , (5) ω R

where

BA =

R+

∧ dξ (−ξ, ξ) + “local curvature terms” f ⊗ f ϑ∗ T splitH π

(6)

is constructed in the same fashion as v a v b T split and T split H ab ω0 but with (essentially) the first few terms of the Hadamard series replacing the reference two-point function. At the technical level, we invoke a refined version of microlocal analysis which keeps track of the order of singularities. The structure of this paper is as follows. In Section 2 we review the algebraic formulation of quantum field theory in curved spacetimes, review two (equivalent) formulations of the Hadamard condition and give a detailed analysis of the singularity structure of the Hadamard series in terms of Sobolev wave-front sets. Section 3 contains our main result, Theorem 3.1, which is then used to give a number of examples of absolute QEIs. Although our bound depends on a choice of coordinates, we describe how the dependence can be eliminated by restricting the choice of smearing tensor, thus providing a covariant formulation of our bounds.

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2. Quantum field theory in curved spacetime 2.1. The algebra of observables and the first definition of Hadamard states We shall employ the algebraic framework for describing the scalar quantum field in a classical curved four-dimensional spacetime (M, g). Here M is a four-dimensional smooth manifold (assumed Hausdorff, paracompact and without boundary) with a Lorentz metric gab of signature (+ − −−). Furthermore, we require (M, g) to be globally hyperbolic, that is M contains a Cauchy surface. Where index notation is used, Latin indices will run over the range 0, 1, 2, 3 unless explicitly stated otherwise, while Greek characters will denote frame indices and also run over 0, 1, 2, 3 unless explicitly stated otherwise. We employ units in which c = = 1. The minimally coupled scalar field φ obeys the Klein–Gordon equation (∇2 + μ2 )φ = 0, where ∇2 = g ab ∇a ∇b and μ ≥ 0 is the mass of the field quanta. Global hyperbolicity entails the existence of unique global advanced (E − ) and retarded (E + ) Green functions E ± : C0∞ (M) → C ∞ (M) for the Klein–Gordon equation obeying (7) (∇2 + μ2 )E ± f = E ± (∇2 + μ2 )f = f , and (8) supp E ± f ⊂ J ± (supp f ) ∞ ± for all f ∈ C0 (M), where J (S) denote the causal future (+) and past (−) of a set S. One may use the set of smooth functions having compact support in M, C0∞ (M), to label a set of abstract objects {φ(f ) | f ∈ C0∞ (M)} which generate a free unital ∗-algebra A over C. The algebra of smeared fields A(M, g) is defined to be the quotient of A by the following relations: i) Hermiticity, φ(f )∗ = φ(f ) ∀f ∈ C0∞ (M); ii) Linearity, φ(αf + βf ) = αφ(f ) + βφ(f ) ∀α, β ∈ C and ∀f, f ∈ C0∞ (M); iii) Field equation, φ((∇2 + μ2 )f ) = 0 ∀f ∈ C0∞ (M); iv) Canonical commutation relations, [φ(f ), φ(f )] = iE(f, f )½ ∀f , f ∈ C0∞ (M). Here, E = E − − E + is the advanced-minus-retarded Green’s function for the Klein–Gordon operator and by E(f, f ) we mean E(f, f ) = dvol(x) f (x)(Ef )(x) . (9) M

It is relation (iv) that quantizes the field theory. In this framework, a state is a linear functional ω on A(M, g) which is normalized so that ω(½) = 1 and is positive in the sense that ω(A∗ A) ≥ 0 for all A ∈ A(M, g). The two point function associated with the state ω is a bilinear map Λω : C0∞ (M) ⊗ C0∞ (M) → C given by Λω (f, f ) = ω(φ(f )φ(f )). We will only consider states for which Λω is a distribution, i.e., Λω ∈ D (M × M). It is clear from (iv) that the antisymmetric part of Λω , 1 i Λω (f, f ) − Λω (f , f ) = E(f, f ) , (10) 2 2 is state independent.

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As already mentioned, we will largely be concerned with Hadamard states. There are two equivalent formulations of the Hadamard condition, both of which will be used in the sequel. The original definition, given in a precise form by Kay & Wald [39], involves a local series expansion of the two point function Λω associated with a state ω and is based upon Hadmard’s work on the fundamental solution for hyperbolic operators. In order to give the precise formulation of the Hadamard series construction we first introduce some geometrical structures, following [39,42,44]. We denote by X ⊂ M × M the set X = (x, x ) ∈ M × M | x, x causally related and J + (x) ∩ J − (x ) & J − (x) ∩ J + (x ) are contained

within a convex normal neighbourhood . (11)

For each (x, x ) ∈ X let Ux,x be any convex normal neighbourhood containing J + (x) ∩ J − (x ) and J − (x) ∩ J + (x ). Then (cf. Lemma 3.1 in [42]) X = (x,x )∈X Ux,x ×Ux,x is an open neighbourhood of X in M×M on which the signed1 squared geodesic separation of points σ is well-defined and smooth, and on which the Hadamard construction (to be described shortly) can be carried out. Any open neighbourhood of X defined in this way will be called a regular domain. For each k = 0, 1, 2, . . . , we may define a distribution Hk ∈ D (X) by ⎧ 1 k j 1 ⎨ Δ 2 (x, x ) σ+ (x, x ) σ (x, x ) + Hk (x, x ) = 2 vj (x, x ) 2(j+1) ln 4π ⎩ σ+ (x, x ) j=0 2 ⎫ k σ j (x, x ) ⎬ + , (12) wj (x, x ) 2(j+1) ⎭ j=0

where we have introduced a length scale to make σ/ 2 dimensionless2 and the coefficient functions Δ, vj and wj will be explained below. We also set H−1 = Δ1/2 /(4π 2 σ+ ). By F (σ+ ), for some function F , we mean the distributional limit F (σ+ ) = lim F (σ ) , →0+

(13)

where σ (x, x ) = σ(x, x )+2i (t(x)−t(x ))+ 2 and t is a time function; that is, ∇a t is a normalized future directed timelike vector field on X. We shall occasionally use the notation t = t(x) and t = t(x ). The function Δ ∈ C ∞ (X) is the van 1 We

adopt the convention that σ(x, x ) > 0 if x, x are spacelike separated, σ(x, x ) < 0 if x, x are timelike separated and σ(x, x ) = 0 if they are null separated. In Minkowski spacetime, (R4 , η), for example σ(x, x ) = −ηab (x − x )a (x − x )b . 2 A different choice of length scale can be absorbed into a redefinition of the local curvature terms Cab appearing in the renormalization of the stress-energy tensor; see Section 2.2.

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Vleck–Morette determinant bi-scalar and is given by det − ∇a ⊗ ∇b σ(x, x ) Δ(x, x ) = − . −g(x) −g(x )


(14)

The functions vj and wj are found by fixing x and applying (∇2 + μ2 ) ⊗ ½ to Hk 2 and equating all the coefficients of 1/σ+ , 1/σ+ , ln σ+ etc to zero. This determines a system of equations (known as the Hadamard recursion relations, given in appendix A) which can be solved uniquely (in X) for the vj series. The wj series is specified once the value of w0 is fixed; we adopt Wald’s prescription that w0 = 0 [48]. We remark that the k → ∞ limit of the right-hand side of (12) has a nonzero radius of convergence in analytic spacetimes, but not in general [32]. Let N be a causal normal neighbourhood of a Cauchy surface C [39]; that is, C is a Cauchy surface for N and every double-cone J + (x) ∩ J − (y) with x, y ∈ N is contained in a convex normal neighbourhood of (M, g) (see Lemma 2.2 in [39] for the existence of causal normal neighbourhoods). We may further choose an open neighbourhood X∗ of the set of pairs of causally related points in N × N whose closure is contained in X ∩ (N × N ) and a cut-off function χ : N × N → [0, 1] so that (15) χ|X∗ = 1 and χ|(N ×N )\X = 0 . See Lemma 3.3 in [42] for the existence of X∗ and χ with these properties. Given the above, a state ω on A(M, g) is said to be Hadamard if for each k ∈ N there exists a Fk ∈ C k (N × N ) such that Λω = χHk + Fk

(16)

in N × N . We remark that this definition can be shown to be independent of the choices of C, N , t, χ, X and X∗ [39, 44]. In the special case in which M is a convex normal neighbourhood, we note that M would be a causal normal neighbourhood of any of its Cauchy surfaces and we could take X∗ = X = M × M and χ ≡ 1, so (16) becomes Λω = Hk + Fk and holds on the whole of M × M. In the general case, it is easy to see (e.g., using the microlocal characterization of the Hadamard condition) that if ω is Hadamard then so is its restriction to any open globally hyperbolic subset of M, considered as a spacetime in its own right. Thus Λω − Hk is C k for all k on any set of the form U × U where U is a globally hyperbolic convex normal neighbourhood. As every k point x ∈ M has such a neighbourhood Ux we may conclude that Λω − Hk is C for all k in an open neighbourhood of the diagonal of the form x∈M Ux × Ux ; we will refer to any such open neighbourhood as an ultra-regular domain. Note. The need to introduce the notion of an ultra-regular domain only came to light as the final version of this paper was prepared for publication, and after [45] had gone to press. One of us (CJS) would like to warn the reader that some results in [45] hold on ultra-regular domains as opposed to the stated regular domain; with this modification, the results of [45] are unchanged.

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2.2. Renormalization of the stress tensor The Hadamard series construction forms the basis for the renormalization of the stress-energy tensor in curved spacetimes, to which we now turn. The classical stress tensor of the real scalar field 1 1 (17) Tab (x) = ∇a ⊗ ∇b − gab g cd ∇c ⊗ ∇d + μ2 gab ½ ⊗ ½ ϕ ⊗ ϕ (x, x) 2 2 must be renormalized in QFT owing to the divergent behaviour of the two point function. Define the point-split stress-energy operator (which should not be confused with the stress-energy tensor itself) by 1 1 split Tab = ∇a ⊗ ∇b − gab g cd ∇c ⊗ ∇d + μ2 gab ½ ⊗ ½ 2 2

(18)

near the diagonal in M × M, where gab (x, x ) is the parallel propagator3. If ω is ren a Hadamard state we may define Tab ω (x) at any point x ∈ M by the following procedure: a) note that Λω − Hk ∈ C 2 (X) for k ≥ 2 and any ultra-regular domain X, so split Λω − Hk is defined and continuous near the diagonal in M × M; Tab b) define fin split gb b (x, x )Tab (19) Tab ω (x) = lim (Λω − Hk )(x, x ) x →x

for k ≥ 2; fin c) make finite corrections to Tab ω in order to obtain a conserved tensor ren Tab ω (x) with the correct properties in Minkowksi space. fin ω is not covariantly conserved Step (c) is needed because the tensor Tab and cannot be considered as an appropriate stress-energy tensor (it could not be inserted on the right hand side of the Einstein equations, for example). However, it fin ω is of the form ∇b Q where Q is a local quantity, determined turns out that ∇a Tab fin ω we therefore obtain up to a constant [48]; subtracting Qgab by hand from Tab a conserved quantity. The undetermined constant in Q is fixed by the requirement that in Minkowski spacetime the vacuum expectation value vanishes. If we require ren ren ω − Tab ω0 should be given by that the difference Tab ren ren split gb b (x, x )Tab (20) Tab ω − Tab ω0 = lim Λω − Λω0 (x, x ) x →x

then any remaining finite renormalization must take the form of a stateindependent conserved local curvature term Cab that vanishes in Minkowski space, and the finite renormalized expection value of the quantum stress energy is given by fin ren (x) − Q(x)gab (x) + Cab (x) . (21) Tab ω (x) = Tab ω 3 Thus,

if vb ∈ Tx M, gab (x, x )vb is its parallel transport to Tx M along the unique geodesic joining x to x, which is well-defined sufficiently close to the diagonal in M × M.

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We take the view that the tensor Cab is a necessary part of the specification of a given species of scalar field, alongside the mass and curvature coupling. Given sufficient experimental accuracy Cab should, in principle, be measurable. The renormalization prescription we have outlined above is vulnerable to the ren criticism that the Qgab term needed to restore conservation of Tab ω is only found to be a local curvature term a posteriori. Moretti [40] has shown that this problem can be circumvented by an alternative construction of the quantum stress tensor. The basic idea is to modify the classical stress energy tensor Tab by adding a term of the form αϕ(∇2 + μ2 )ϕ for constant α. While this addition does not affect the classical physics, it has a non-trivial quantization. A judicious choice of α ensures that the quantized stress energy tensor is conserved a priori (see Theorem 2.1 of [40]) and agrees with the usual quantization up to conserved local curvature terms. Although Moretti’s approach is certainly elegant, it turns out that the usual quantization is better adapted to the derivation of QEIs. In particular, split in a symmetric form our argument relies crucially on being able to write Tab which is not possible with a term of the form ½ ⊗ (∇2 + μ2 ). A similar problem arises for the non-minimally coupled scalar field. In this case one must smear the stress-energy tensor even to obtain an inequality on the classical field [21], which necessitates a more complicated analysis at the quantum level [22]. 2.3. The wave-front set and second definition of Hadamard state The above discussion shows that Hadamard states are characterized by their singularity structure. For this reason the techniques of microlocal analysis, which focus attention on singular behaviour, are ideally suited to this theory. This realization has led to a number of important developments in the theory of quantum fields in curved backgrounds, following initial work of Radzikowski [42], particularly in regard to renormalization [3, 33, 34]. In addition, the theory of (smooth) wave-front sets is a key tool in the proof of general difference QEIs [6, 10, 12, 14] in general globally hyperbolic spacetimes. Our absolute QEIs require the finer control on singularities of distributions afforded by the Sobolev wave-front set. In this subsection we briefly review the definition of the smooth and Sobolev wave-front sets and explain how they may be used to give a purely microlocal definition of the Hadamard condition, as first identified by Radzikowksi [42]. In addition, we will state a result of Junker and Schrohe [38] on the Sobolev wave-front set of the two-point functions of Hadamard states. This will form the basis of our analysis of the Sobolev wave-front sets of individual terms in the Hadamard series in the next subsection. To begin, let u ∈ D (Rm ) be any distribution. We say that u is smooth O ⊂ Rm of x and a smooth function at x if there exists an open neighbourhood ∞ m ϕ ∈ C (O) such that u(f ) = Rm d x ϕ(x)f (x) for all test functions f ∈ C0∞ (O). The singular support, singsupp u, of a distribution u ∈ D (Rm ) is the complement in Rm of the set of all points at which u is smooth. In particular, a distribution is smooth if and only if its singular support is empty.

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While the singular support tells us ‘where’ a distribution u fails to be smooth, Fourier transforms of localizations of u contain additional information. A covector ζ ∈ Rm \ {0} is a direction of rapid decay for u at x if there exists a conic neighbourhood Γ ⊂ Rm \ {0} of ζ and a localizer χ ∈ C0∞ (Rm ) which does not vanish at x such that (1 + |ξ|)N | χu(ξ)| −→ 0

as ξ → ∞ in

Γ,

∀N ∈ N .

(22)

The set of singular directions of u ∈ D (Rm ) at x, Σx (u), is the complement in Rm \ {0} of the set of directions of rapid decay of u at x. The wave-front set of u assembles this information in a convenient way (see Section 8.1 of [35] for more detail). Definition. The (smooth) wave-front set W F (u) of a distribution u ∈ D (Rm ) is ! W F (u) = (x, ξ) ∈ Rm × Rm \ {0} | ξ ∈ Σx (u) (23) As an example, it is easy to verify that the Dirac δ and Heaviside θ distributions have the following wave-front sets. W F (δ) = W F (θ) = (0, ξ) | ξ ∈ R \ {0} . (24) The wave-front set is a closed cone in Rm × (Rm \ {0}), whose elements transform as covectors under coordinate transformations (see Theorem 8.2.4 in [35]). Accordingly, the definition of wave-front set may be extended to distributions on a smooth manifold M in the following way: We say (x, ξ) ∈ W F (u) ⊂ T ∗ M \ {0} if and only if there exists a chart neighbourhood (κ, U) of x such that the corresponding coordinate expression of (x, ξ) belongs to W F (u ◦ κ−1 ) ⊂ Rm × (Rm \ {0}), where m is the dimension of M. The Sobolev wave-front set provides greater structure on the information in the wave-front set. Recall that the Sobolev space H s (Rm ), s ∈ R, is the set of all tempered distributions u on Rm such that s dm ξ 1 + |ξ|2 | u(ξ)|2 < ∞ . (25) Rm

We summarise some relevant properties of Sobolev spaces for convenience. Proposition 2.1. The Sobolev spaces H s (Rm ) have the following properties: i) If s > k + m/2 then H s (Rm ) ⊂ C k (Rm ) for k ∈ N; ii) H s (Rm ) ⊂ H s (Rm ) ∀s ≥ s ; iii) H s (Rm ) is closed under multiplication by smooth functions. Associated with the scale of Sobolev spaces, there is a refined notion of the wave-front set. Just as W F (u) informs us where a distribution u fails to be smooth, the Sobolev wave-front set W F s (u) contains information about where in phase space the distribution fails to be H s .

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Definition. The distribution u ∈ D (Rm ) is said to be microlocally H s at (x, ξ) ∈ Rm ×(Rm \{0}) if there exists a conic neighbourhood Γ of ξ and a smooth function ϕ ∈ C0∞ (Rm ), ϕ(x) = 0, such that s dm ζ 1 + |ζ|2 |[ϕu]∧ (ζ)|2 < ∞ . (26) Γ

The Sobolev wave-front set W F s (u) of a distribution u ∈ D (Rm ) is the complement, in T ∗ Rm \ {0}, of the set of all pairs (x, ξ) at which u is microlocally H s . It is easy to verify, for example, that " (0, ξ) | x ∈ R \ {0} s ≥ −1/2 W F s+1 (θ) = W F s (δ) = ∅ s < −1/2

(27)

and we see how this refines the information in (24). Like the wave-front set, W F s (u) is a closed cone in T ∗ Rn \ {0}. Furthermore, part (ii) of Proposition 2.1 entails that W F s (u) ⊂ W F s (u) ⊂ W F (u) for all s ≤ s . In fact, one may show that W F (u) = s∈R W F s (u). Additionally, if ϕ ∈ C0∞ (Rn ) does not vanish in a neighbourhood of x then (x, ξ) ∈ W F s (u) if and only if (x, ξ) ∈ W F s (ϕu); we also have W F s (u + w) ⊂ W F s (u) ∪ W F s (w). One may show that W F s (u) can be characterized in a coordinate-independent way as a subset of the cotangent bundle which then permits the definition to be extended to distributions on a manifold by referring back to (any choice of) local coordinates (see, e.g., remark (i) follows (M) ing Proposition B.3 in [38]). We shall occasionally use the notation u ∈ Hloc s if W F (u) = ∅ for a distribution u ∈ D (M) (see also the remarks following Definition 8.2.5 of [36]). In [42], Radzikowski proved the remarkable result that the definition of a Hadamard state in terms of the series construction previously given is equivalent to a condition on the wave-front set of the two-point function. Namely, the wavefront set is required to lie in a particular subset of the bicharacteristic set of the Klein–Gordon operator, which we now define. Denote by R = {(x, ξ) ∈ T ∗ M | g ab (x)ξa ξb = 0 , ξ = 0} the set of nonzero null covectors over M. Since (M, g) is time orientable we may decompose R into two disjoint sets R± defined by R± = {(x, ξ) ∈ R | ±ξ 0} where by ξ 0 (ξ ∈ Tx∗ M) we mean that ξa is in the dual of the future light cone at x. We define the notation (x, ξ) ∼ (x , ξ ) to mean that there exists a null geodesic γ : [0, 1] → M such that γ(0) = x, γ(1) = x and ξa = γ˙ b (0)gab (x), ξa = γ˙ b (1)gab (x ). In the instance where x = x , (x, ξ) ∼ (x, ξ ) shall mean that ξ = ξ is null. Then, the set (28) C = (x, ξ; x , ξ ) ∈ R × R | (x, ξ) ∼ (x , ξ ) is the bicharacteristic relation for the Klein–Gordon operator. We also define the related sets C +− = (x, ξ; x , −ξ ) ∈ C | ξ 0 (29) and

C −+ = (x, −ξ; x , ξ ) ∈ C | ξ 0 .

(30)

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We may now state the relevant portion of Radzikowski’s equivalence theorem [42]: Theorem 2.2. Let (M, g) be a four-dimensional globally hyperbolic spacetime and suppose Λ ∈ D (M × M) satisfies the Klein–Gordon equation and has antisymmetric part iE/2 modulo smooth functions. Choose a Cauchy hypersurface C, a causal normal neighbourhood N of C and a time function t. Then, the following two conditions are equivalent: i) Λ has the Hadamard series structure given by (16) on N × N , ii) W F (Λ) = C +− . As a consequence of this equivalence theorem we may now adopt the condition that W F (Λω ) = C +− as the second definition of the state ω, on A(M, g), being Hadamard. Junker and Schrohe [38] applied the theory of Sobolev wave-front sets to study both Hadamard states and the larger class of adiabatic states (see Section 3.2.3). In particular, Lemma 5.2 of [38] gives the Sobolev singularity structure of the two point function of Hadamard states. Theorem 2.3. Let ω be a Hadamard state on A(M, g) where M is a smooth fourdimensional globally hyperbolic spacetime. Then, the two point function, Λω ∈ D (M × M), associated to ω has the following Sobolev wave-front set: " +− C s ≥ −1/2 (31) W F s (Λω ) = ∅ s < −1/2 . 2.4. Sobolev microlocal analysis of the Hadamard series We will now employ Theorem 2.3 to study the Sobolev wave-front sets of the individual terms in the Hadamard series, working within an ultra-regular domain X on which the distributions 1/σ+ and σ j ln σ+ featuring in (12) may be defined. (The length scale will be suppressed from now on.) We shall establish the following statement: " +− C s ≥ −1/2 s+j+1 j s WF (σ ln σ+ ) ⊂ W F (1/σ+ ) = (32) ∅ s < −1/2 . In order to do this we observe that the terms appearing in the Hadamard series are (loosely) related to one another via differentiation. If P is any partial differential operator of order r on a smooth manifold M, i.e., in local coordinates pα (x)(−i∂)α (33) P = |α|≤r

where α is a multi-index and pα are smooth functions, then the principal symbol, pr (x, ξ), of P is pr (x, ξ) = pα (x)ξ α . (34) |α|=r

436



The characteristic set of P , Char P , is the set of (x, ξ) ∈ T ∗ M \ {0} at which the principal symbol vanishes. Corollaries 8.4.9–10 of [36] encapsulate the effect of partial differential operators on the Sobolev wave-front set of a distribution: Lemma 2.4. Let M be a smooth manifold. For u ∈ D (M) and any partial differential operator P of order r with smooth coefficients then W F s (P u) ⊂ W F s+r (u) and W F s+r (u) ⊂ W F s (P u) ∪ Char P . Lemma 2.4 enables us to quantify our earlier observation about the relationship between 1/σ+ and σ j ln σ+ : Proposition 2.5. Within an ultra-regular domain we have W F s+1+j (σ j ln σ+ ) ⊂ W F s (1/σ+ )

∀s ∈ R

and

∀j ∈ {0} ∪ N .

(35)

Proof. We employ induction on j. If v ∈ C ∞ (T M) is a smooth vector field, then (v · ∇ ⊗ ½) ln σ+ = [(v · ∇ ⊗ ½)σ]/σ+ . Hence, by Lemma 2.4, we have W F s+1 (ln σ+ ) ⊂ W F s (1/σ+ ) ∪ Char (v · ∇ ⊗ ½) . As v is arbitrary,

⎛

⎞

%

W F s+1 (ln σ+ ) ⊂ W F s (1/σ+ ) ∪ ⎝

(36)

Char (v · ∇ ⊗ ½)⎠

(37)

v∈C ∞ (T M)

and since

Char (v · ∇ ⊗ ½) = (x, ξ; x , ξ ) ∈ T ∗ X \ 0 | v(x) · ξ = 0

(38)

it is clear that the intersection is empty and the statement holds for j = 0. Now suppose it holds for some j ∈ {0} ∪ N. The identity ( ) (39) (v · ∇ ⊗ ½)σ j+1 ln σ+ = (v · ∇ ⊗ ½)σ (j + 1)σ j ln σ+ + σ j and the inductive hypothesis give W F s+2+j (σ j+1 ln σ+ ) ⊂ W F s+1+j (σ j ln σ+ ) ∪ Char (v · ∇ ⊗ ½)

(40)

and taking the intersection over all v ∈ C ∞ (T M) as before, we establish the result for j + 1 and hence all j ∈ {0} ∪ N by induction. We next prove the intuitively reasonable result that Λω is as singular as the leading term in the Hadamard series. Proposition 2.6. Let ω be a Hadamard state. Then, within any ultra-regular domain X, we have W F s (Λω ) = W F s (1/σ+ )

∀s ∈ R .

(41)

Proof. Recall that for every k ∈ N there exists a Fk ∈ C (X) such that Λω = Hk + Fk . Hence, as W F s (σ j ln σ+ ) ⊂ W F s+j+1 (σ j ln σ+ ) ⊂ W F s (1/σ+ ), k

W F s (Λω ) ⊂ W F s (Δ1/2 /σ+ ) ∪ W F s (Fk ) .

(42)

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We remark that it is known (from, say, [42]) that Δ does not vanish where x, x are null related4 and as such W F s (Λω ) ⊂ W F s (1/σ+ ) ∪ W F s (Fk ). Moreover, given any particular s we can always find a k sufficiently large such that W F s (Fk ) = ∅ and it remains to prove W F s (1/σ+ ) ⊂ W F s (Λω ). Let (x, ξ; x , ξ ) ∈ W F s (1/σ+ ) such that W F s− (1/σ+ ) = ∅ for any > 0. Hence, by proposition 2.5, W F s (σ j ln σ+ ) ⊂ W F s−j−1 (1/σ+ ) = ∅ for j ≥ 0 and we have Hk − s (X). Therefore, (x, ξ; x , ξ ) ∈ W F s (Hk ) and by the nesting Δ1/2 /4π 2 σ+ ∈ Hloc property (x, ξ; x , ξ ) ∈ W F s (Hk ) for all s ≥ s. As a consequence of Theorem 2.3 we now have the Sobolev wave-front sets of the constituent distributions in the Hadamard series. Corollary 2.7. The distributions 1/σ+ , σ j ln σ+ ∈ D (X), where X is an ultraregular domain, have the following Sobolev wave-front sets: " +− C s ≥ −1/2 s (43) W F (1/σ+ ) = ∅ s < −1/2 " +− C s ≥ j + 1/2 W F s (σ j ln σ+ ) ⊂ (44) ∅ s < j + 1/2 . In consequence, we also have, for arbitrary j ≥ −1, " +− C s ≥ −1/2 s W F (Hj ) ⊂ ∅ s < −1/2 and

" W F (Hj+j − Hj ) ⊂ s

C +− ∅

s ≥ j + 3/2 s < j + 3/2

(45)

(46)

for j > 0. Proof. As we have already established, 1/σ+ possesses the lowest order singularity which Lemma 2.6 states is precisely that of Λω . The remaining results follow from Proposition 2.5. The remainder of this section is devoted to calculating the Sobolev wavefront sets of the advanced-minus-retarded fundamental solution E and a quantity k ∈ D (X) (X an ultra-regular domain) defined by H k (x, x ) = 1 Hk (x, x ) + Hk (x , x) + iE(x, x ) , H (47) 2 which plays an important role in our main result Theorem 3.1. As iE is the antisymmetric part of Λω , for all Hadamard ω, Theorem 2.3 implies that " +− −+ C C s ≥ −1/2 s W F (iE) ⊂ (48) ∅ s < −1/2 . 4 That

is, Δ1/2 restricted to singsupp 1/σ+ is non-vanishing.

438



k Proposition 2.8. Within an ultra-regular domain, the Sobolev wave-front set of H satisfies ⎧ +− −+ C s ≥ k + 3/2 ⎨ C k ) ⊂ s ∈ [−1/2, k + 3/2) C +− (49) W F s (H ⎩ ∅ s < −1/2 . Proof. Suppose first that s < k + 3/2. It follows from the covariant commutation relations that, within an ultra-regular domain X, iE(x, x ) = Hk+2 (x, x ) − Hk+2 (x , x) modulo C k+2 (X). Hence there exists F ∈ C k+2 (X) such that k − Hk ](x, x ) = [Hk+2 − Hk ](x, x ) − [Hk+2 − Hk ](x , x) + F (x, x ). [H

(50)

s As k + 2 ≥ s, we have C k+2 (X) ⊂ Hloc (X), so all three terms on the right-hand s k = Hk modulo H s (X) side will belong to Hloc (X), using (46) as well. Thus H loc for all s < k + 3/2, which establishes (49) for s in this range. For s ≥ k + 3/2 k , the wave-front set of Hk (and its the result follows from the definition (47) of H behaviour under interchange of the arguments x and x ) together with the rule for wave-front sets of sums of distributions.

Finally, we end this discussion of the microlocal properties with the following k which follows directly from the proof result concerning the singularities of Λω − H of Corollary 2.7. Proposition 2.9. Within an ultra-regular domain, the Sobolev wave-front set of k is given by Λω − H " +− −+ C C s ≥ k + 3/2 k ) ⊂ W F s (Λω − H (51) ∅ s < k + 3/2 . 2.5. Restriction results and a point-splitting lemma In addition to the results of the previous subsection, our main result will make use of three additional technical results. The first, Beals’ restriction theorem, enables us to restrict Λω , Hk and their derivatives to certain submanifolds of M × M. The second, taken from [10], shows that positive type is preserved under such restrictions, while the third result is a technical tool that enables us to write integrals over the diagonal on product manifolds in terms of their ‘point-split’ Fourier transforms. Our QEI results will encompass averages of the stress-energy tensor smeared over timelike submanifolds, e.g., timelike curves or hyperplanes, as well as averages over spacetime volumes. For these purposes, it is necessary to understand how restricting distributions (such as Λω , Hk and their derivatives) to a submanifold alter the Sobolev wave-front set. A theorem due to Beals (see Lemma 11.6.1 of [36]) tells us that, for suitably well behaved restrictions, the Sobolev order of the wave-front set is reduced by an amount proportional to the codimension of the restriction, while its elements are transformed according to the associated pull-back mapping. We will state a specialization of Beals’ result to the case we will need, in which we restrict from a product manifold M × M to a submanifold Σ × Σ, where Σ is

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a submanifold of M. Writing the embedding of Σ in M as a map ι : Σ → M and defining ϑ = ι ⊗ ι : Σ × Σ → M × M, the restriction of u ∈ D (M × M) to Σ × Σ may also be regarded as the formation of a pull-back ϑ∗ u. Beals’ result hinges on the relationship between the Sobolev wave-front set of u and the conormal bundle N ∗ Σ of Σ defined by ! (52) N ∗ Σ = ι(x), ξ ∈ T ∗ M; x ∈ Σ, ι∗ (ξ) = 0 . Theorem 2.10 (Beals’ restriction theorem). Let u ∈ D (M × M) and ϑ ∈ C ∞ (Σ × Σ, M × M) be defined as above, and suppose M and Σ have dimensions m and n respectively. If N ∗ Σ × N ∗ Σ ∩ W F s (u) = ∅ for some s > m − n then ϑ∗ u is a well defined distribution in D (Σ × Σ). Moreover, W F s−(m−n) (ϑ∗ u) ⊂ ϑ∗ W F s (u) ∗

where the set ϑ W F (u) is defined to be ϑ∗ W F s (u) = t, ι∗ (ξ); t , ι∗ (ξ ) ∈ (T ∗ Σ × T ∗ Σ) | ! ι(t), ξ; ι(t ), ξ ∈ W F s (u) .

(53)

s

(54)

The next result, Theorem 2.2 of [10], asserts that the positive type condition is preserved under the restrictions carried out by Beals’ theorem. Lemma 2.11. If, in addition to the hypotheses of Theorem 2.10, u ∈ D (M × M) is of positive type, then ϑ∗ u is of positive type on Σ × Σ. Finally, we present a point-splitting identity for distributions of sufficient regularity. Beginning in Rn × Rn , we have the following. Lemma 2.12. For all u ∈ C0 (Rn × Rn ), we have the identity dn ξ −|ξ|2 dn t u(t, t) = lim+ e u (−ξ, ξ) . n →0 Rn Rn (2π)

(55)

In particular, this holds if u ∈ H s (Rn × Rn ) ∩ E (Rn × Rn ) for s > n, by virtue of the Sobolev embedding of H s (Rn × Rn ) in C(Rn × Rn ). Proof. By definition of the Fourier transform, we have 2 dn ξ −|ξ|2 dn ξ e u (−ξ, ξ) = dn τ dn τ e−|ξ| −iξ · (τ −τ ) u(τ, τ ) n n Rn (2π) Rn (2π) Rn ×Rn (56) As the integrand is absolutely integrable on R3n , Fubini’s theorem permits us to reorder the integrations and perform the ξ integral first, thus obtaining dn ξ −|ξ|2 e u (−ξ, ξ) = dn τ dn τ ϕ (τ − τ )u(τ, τ ) n Rn (2π) Rn ×Rn = dn t dn t ϕ (t )u(t + t /2, t − t /2) Rn ×Rn n = d t ϕ (t ) dn t u(t + t /2, t − t /2) (57) Rn

Rn

440



2

where ϕ (t) = (4π )−n/2 e−|t| /(4) . We have made the change of variables t = (τ + τ )/2, t = τ − τ (for which the Jacobian is unity) and reordered integrals using Fubini’s theorem again. As u is continuous and compactly supported, the inner integral exists for each t and defines a continuous compactly supported function. The limit → 0+ exists and yields the value of this function at t = 0 because ϕ is an approximate identity. This is the required result. Note that if ξ → u (−ξ, ξ) is absolutely integrable on Rn then the dominated convergence theorem permits us to dispense with the limiting procedure on the right-hand side.5 For our application, we will need a straightforward generalization of the above to distributions on manifolds. Lemma 2.13. Let (Σ, h) be a n-dimensional pseudo-Riemannian manifold and u ∈ E (Σ × Σ) ∩ H s (Σ × Σ) for s > n. Suppose the support of u is contained within U × U where U is a single coordinate chart of Σ with associated coordinate map κ : U → Rn . Then dn ξ −|ξ|2 U (−ξ, ξ) (58) dvol(x) u(x, x) = lim e →0+ Rn (2π)n Σ where U : κ(U) × κ(U) → C is defined by U (x, x ) = |hκ |1/4 ⊗ |hκ |1/4 uκ (x, x ) , where uκ = u ◦ (κ chart κ.

−1

⊗κ

−1

) and hκ is the determinant of the metric in coordinate

Proof. We have dvol(x)u(x, x) = Σ

(59)

Rn

d x |hκ (x)| n

1/2

uκ (x, x) =

dn x U (x, x) .

(60)

Rn

As hκ is a positive smooth function bounded away from zero and therefore has smooth fractional powers, we may apply Lemma 2.12 to the function U to obtain the desired result.

3. An absolute quantum inequality 3.1. Main result We now come to the statement and proof of our main result. Let Σ be any ndimensional timelike submanifold of (M, g) for 1 ≤ n ≤ 4, that is, h = ι∗ g is a Lorentzian metric on Σ, where ι : Σ → M embeds the submanifold Σ in M. We also equip Σ with the time orientation induced from M, so that non-zero future-directed causal covectors on (M, g) pull back to non-zero future-directed causal covectors on (Σ, h). In our conventions a positive definite metric on a onedimensional manifold is regarded as Lorentzian. As Σ is timelike, its tangent space 5 Our

hypotheses are strong enough to guarantee that u is absolutely integrable on Rn × Rn , and hence ξ → u (−ξ + η, ξ + η) is absolutely integrable a.e. in η by Fubini’s theorem. However, a simple proof of integrability for η = 0 was not forthcoming.

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T Σ can be annihilated only by covectors which can annihilate at least one nonzero timelike vector; in particular, all covectors in the conormal bundle N ∗ Σ are spacelike. Our aim is to obtain lower bounds, as ω varies among Hadamard states, on quantities of the form dvol(x) f 2 (x) Q ⊗ Q(Λω − H2 ) (x, x) (61) Σ

where Q = q ∇a + b is a partial differential operator with smooth real-valued coefficients q a and b defined on a neighbourhood of Σ and f ∈ C0∞ (Σ) is real valued. Note that Λω − H2 is C 2 so the coincidence limit is well defined. For simplicity it is convenient to assume in addition that Σ may be covered by a single coordinate chart with certain properties. a

Definition. A small sampling domain is an n-dimensional timelike submanifold Σ of (M, g) such that (i) Σ is contained in a globally hyperbolic convex normal neighbourhood in M; (ii) Σ may be covered by a single hyperbolic coordinate chart, i.e., a coordinate system x0 , . . . , xn−1 on Σ with ∂/∂x0 future-pointing and timelike, and for which there exists a constant c > 0 such that all causal covectors ua on Σ obey * +n−1 + c|u0 | ≥ , u2j (62) j=0

(i.e., the coordinate speed of light is bounded from above). A sufficient condition for the existence of a maximum coordinate speed of light is that h00 > ε and | det(hij )n−1 i,j=1 | > ε for some ε > 0. It is easy to verify (e.g. by using suitable normal coordinates) that every point of a general timelike submanifold Σ has a neighbourhood (in Σ) which is a small sampling domain. Thus any integral over a compact subset of a timelike submanifold may be decomposed into finitely many integrals over small sampling domains by a partition of unity. Suppose then, that Σ is a small sampling domain in (M, g) with hyperbolic chart {xa }a=0,...,n−1 . We may express these coordinates by a map κ : Σ → Rn , κ(p) = (x0 (p), . . . , xn−1 (p)) and write Σκ = κ(Σ). Any function F on Σ determines a function Fκ = F ◦ κ−1 on Σκ ; in particular, we have a smooth map ικ : Σκ → M. The significance of κ being hyperbolic is that the bundle R+ of (non-zero) future pointing null covectors on (M, g) pulls back under ικ so that ι∗κ R+ ⊂ Σκ × Γ

(63)

where Γ ⊂ Rn is the set of all ua with u0 > 0 and satisfying (62), which means that Γ is a proper subset of the upper half-space R+ × Rn−1 of Rn (here, we regard R+ = [0, ∞)). We now state our main result:

442



Theorem 3.1. Let Σ be a small sampling domain of dimension n in (M, g) with hyperbolic coordinate map κ and suppose Q is a partial differential operator of order at most one with smooth real-valued coefficients in a neighbourhood of Σ. Set k = max{n + 3, 5}. For any real-valued f ∈ C0∞ (Σ) and any Hadamard state ω we have the inequality dvol(x) f 2 (x) Q ⊗ Q(Λω − H2 ) (x, x) ≥ −B > −∞ (64) Σ

where B=2

R+ ×Rn−1

∧ dn ξ 1/4 1/4 ∗ |h | f ⊗ |h | f ϑ (−ξ, ξ) , Q ⊗ Q H κ κ κ κ k κ (2π)n

(65)

hκ is the determinant of the matrix κ∗ h and ϑ : Σ × Σ → M × M is the map ϑ(x, x ) = (ι ⊗ ι)(x, x ). Remarks. This bound depends nontrivially on the coordinates (and on any partition of unity used to reduce a general timelike submanifold into small sampling domains). In Section 3.3 we will discuss some classes of QEI averages which, in a sense, determine a natural choice of coordinates. Note that although H2 is suf k for ficient to renormalize the left-hand side, the bound is given in terms of H k ≥ 5. Similar results hold when Q has higher order, for suitably modified values of k; we have restricted attention to the cases relevant to QEIs. Our strategy will be to mimic the proof of the general worldline quantum energy inequality presented in [10] but with Sobolev wave-front sets, as opposed to the smooth wave-front sets used in that paper. Proof of Theorem 3.1. We will break the proof into three parts. Part one will establish that dn ξ −|ξ|2 2 dvol(x) f (x)Q ⊗ Q Λω − H2 (x, x) = 2 lim e →0+ R+ ×Rn−1 (2π)n Σ ∧ k × |hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ Q Λω − H (−ξ, ξ) (66) for the given values of k. Part two contains a positivity result which enables us to discard the state dependent contribution to the right hand side of (66) to obtain the inequality dn ξ −|ξ|2 2 dvol(x) f (x)Q ⊗ Q Λω − H2 (x, x) ≥ −2 lim+ e n →0 Σ R+ ×Rn−1 (2π) ∧ k × |hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ QH (−ξ, ξ) (67) Then, in part three, we show that the right-hand side of this expression is finite and equal to the required lower bound −B.

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k coincide on the Part one: We begin by observing that Λω − H2 and Λω − H diagonal in Σ × Σ, so we may write dvol(x) f 2 (x) Q ⊗ Q(Λω − H2 ) (x, x) Σ k ) (x, x) . (68) = dvol(x) f 2 (x) ϑ∗ Q ⊗ Q(Λω − H Σ

k is symmetric and more regular than The latter form has the merit that Λω − H Λω − H2 . We have also written in the restriction map ϑ∗ explicitly, anticipating later steps in the proof. By hypothesis on k, we may choose s ∈ (6, k + 3/2), and Proposition 2.9 tells us that within an ultra-regular domain X ⊂ M × M s k ∈ Hloc Λω − H (X) .

As Q ⊗ Q is at most second order, Lemma 2.4 entails that k ∈ H s−2 (X) . Q ⊗ Q Λω − H loc

(69)

(70)

k )) is therefore empty and s−2 > 4−n, As the wave-front set W F s−2 (Q⊗Q(Λω − H Beals’ restriction theorem, Theorem 2.10, yields k ∈ H n+s−6 (Σ × Σ) , ϑ∗ Q ⊗ Q Λω − H (71) loc so the point-splitting identity, Lemma 2.13, may be applied to give dn ξ −|ξ|2 k (x, x) = lim dvol(x) f 2 (x) ϑ∗ Q ⊗ Q Λω − H e →0+ Rn (2π)n Σ ∧ k × |hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ Q Λω − H (−ξ, ξ) . (72) k is symmetric and C k , the integrand of (72) is invariant under Then, as Λω − H ξ → −ξ, so we may replace the integration over Rn with that over R+ ×Rn−1 at the expense of a factor of two, thus obtaining (66). Note that we are now integrating over those ξ with ξ0 ≥ 0. This particular half-space of Rn is chosen because it contains the cone Γ defined after (63). k exist Part two: We now assert that the pull-backs of (Q ⊗ Q)Λω and (Q ⊗ Q)H separately, which enables us to split the integrand on the right-hand (side of (66) into two parts. Moreover, we will show that the first of these, namely |hκ |1/4 fκ ⊗ )∧ |hκ |1/4 fκ ϑ∗κ Q ⊗ QΛω (−ξ, ξ), is nonnegative for all ξ ∈ Rn . The existence of the required pull-backs follows because we have already observed that non-zero covectors in N ∗ Σ must be spacelike. As the covectors in k are null, it follows that there is no the wave-front set of (Q ⊗ Q)Λω and (Q ⊗ Q)H intersection between their wave-front sets (at any Sobolev order) and N ∗ Σ × N ∗ Σ. Therefore the pull-backs exist. )∧ ( To establish that |hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ QΛω (−ξ, ξ) ≥ 0, we define a one parameter family of functions fκξ (x) = |hκ (x)|1/4 fκ (x)eiξ · x then, as fκ ∈

444



C0∞ (Rn ), [|hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ QΛω ] ∈ E (Σκ × Σκ ) where the Fourier transform is defined by u (ξ, ξ ) = u(eiξ · , eiξ · ) ∀u ∈ E (Rn × Rn ). Therefore,

∧ |hκ |1/4 fκ ⊗|hκ |1/4 fκ ϑ∗κ Q ⊗ QΛω (−ξ, ξ)

= |hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ QΛω (e−iξ · , eiξ · ) (73)

(74) = ϑ∗κ Q ⊗ QΛω (fκξ , fκξ ) where we have exploited the fact that f is real valued. It is clear that if u is a distribution of positive type then Q ⊗ Qu is also of positive type because Q has real coefficients. Accordingly, Lemma 2.11 establishes that ϑ∗ Q ⊗ QΛω is a distribution of positive type and the assertion is justified. Hence, we obtain the inequality (67) provided that the limit on the right-hand side exists and is finite, which is the remaining step in the proof. Note that the integral converges for each

> 0 because the Fourier transform of any compactly supported distribution is polynomially bounded. Part three: Our aim is to show that

∧ k (−ξ, ξ) I(ξ) := |hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ QH is absolutely integrable on the integration region R+ × Rn−1 , for then we may conclude that the limit on the right-hand side of (67) exists by dominated convergence and equals −B (which is thereby finite). To do this, we introduce an arbitrary Hadamard state ω0 and use the Hadamard series definition of Hadamard states k = Λω0 + Fk for some Fk ∈ C k (X). We consider the contributions of to write H these terms to I(ξ) in turn. First, the results of Radzikowski and Beals entail that W F (ϑ∗κ Q ⊗ QΛω0 ) ⊂ ϑ∗κ W F (Q ⊗ QΛω0 ) ⊂ ϑ∗κ C +− ⊂ ϑ∗κ R+ × R− , (75) where R± are the bundles of future- and past-directed null covectors defined earlier. Thus, we have W F (ϑ∗κ Q ⊗ QΛω0 ) ⊂ ι∗κ R+ × ι∗κ R− ⊂ (Σκ × Γ) × Σκ × (−Γ) (76) using equation (63) and its obvious analogue for R− . By Proposition 8.1.3 in [35], it follows that the Fourier transform of localizations of ϑ∗κ Q ⊗ QΛω0 is of rapid decay outside the cone Γ × (−Γ); in particular we have rapid decay in the cone (−R+ × Rn−1 ) × (R+ × Rn−1 ) (here we have used the assumption that Γ is a proper subset of R+ × Rn−1 because κ is hyperbolic). 1 1 Accordingly we find that [|hκ | 4 fκ ⊗ |hκ | 4 fκ ϑ∗κ Q ⊗ QΛω0 ]∧ (−ξ, ξ) is rapidly decaying in the integration region R+ × Rn−1 and is therefore absolutely integrable there. It remains to show that the Fk dependent contribution to I(ξ) is also absolutely integrable. As Fk ∈ C k (X), we have (Q ⊗ Q)Fk ∈ C k−2 (X). Hence there is

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a constant c such that -

∧ - |hκ |1/4 fκ ⊗ |hκ |1/4 fκ ϑ∗κ Q ⊗ QFk (ξ, ξ )- ≤

(1 +

|ξ|2

445

c + |ξ |2 )(k−2)/2

(77)

for all (ξ, ξ ) ∈ Rn × Rn because fκ is compactly supported. As (k − 2) > n for the values of k given in the hypotheses, it follows in particular that left-hand side is absolutely integrable on R+ × Rn−1 . Accordingly, we have shown that I ∈ L1 (R+ × Rn−1 ), and the dominated convergence argument mentioned above completes the proof. Two points should noted about the foregoing proof. First, the state ω0 was introduced purely as a convenient way of showing that our bound is finite; the bound itself does not depend on any reference state. Second, in the difference QEIs studied in [10] the Gaussian cut-off was not necessary, because the pointsplitting lemma was applied to a smooth compactly supported function. Moreover, k was taken by the two-point function of a reference state Λω0 and the the place of H fact that W F (Λω0 ) = C +− was used to show that the integrand decays rapidly in the integration region. This line of argument was not available to us here, k (in contrast to Hk ) has a portion of its wave-front set lying in C −+ because H (see Proposition 2.8). In the next subsection, we will show how Theorem 3.1 may be used to obtain QEI bounds, by appropriate choices of the operator Q. 3.2. Examples 3.2.1. Worldvolume absolute quantum null energy inequality. Our first example is a quantum null energy inequality (QNEI), that is, a lower bound on quanren ω , where F ab = na nb and na is a smooth, tities of the form M dvol F ab Tab compactly supported null vector field on (M, g) that is future-directed where it is nonzero. We will show how Theorem 3.1 allows us to obtain an absolute QEI ren ω . To do this, we suppose that F ab is supported within an on M dvol F ab Tab open subset Σ that is a four-dimensional small sampling domain in (M, g) with hyperbolic chart κ. Noting that a ab ren b n Λ (x, x ) + Cab F ab (x) , (78) F Tab ω (x) = lim ∇ ⊗ n ∇ − H a b ω 2 x →x

we apply Theorem 3.1 with Q = n · ∇ and f ∈ C0∞ (Σ) chosen to be real-valued and to equal unity on the support of F ab . This yields the absolute QNEI: ren dvol F ab Tab ω M . /∧ d4 ξ 1/4 1/4 (n · ∇ ⊗ n · ∇) H ≥ −2 | ⊗ |g | (−ξ, ξ) |g κ κ 7 4 κ R+ ×R3 (2π) dvol F ab Cab (79) + M

for all Hadamard states ω.

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Clearly the right-hand side of this inequality depends explicitly on the choice of coordinates κ. In Section 3.3 we will explain how this problem may be removed by restricting the class of sampling tensors F ab in such a way that there is a canonical class of coordinate systems, all of which give the same lower bound. 3.2.2. Worldline absolute quantum weak energy inequality. Our second example applies Theorem 3.1 to the energy density sampled along a smooth timelike worldline γ. This was the situation studied in [10], where a difference QEI was obtained. We assume γ is given in a proper time parameterization as a smooth function γ : I → R, where I is a possibly unbounded open interval of R and denote the four-velocity of the curve by u = γ. ˙ The curve forms a small sampling domain, with the proper time parameterization as a hyperbolic coordinate system, provided that the track of γ can be contained in a globally hyperbolic convex normal neighbourhood in M. The classical energy density of a field ϕ(x) along γ may be written in the form (80) ua ub Tab (x) = T split (ϕ ⊗ ϕ) (x, x) where the point split energy density operator is defined within a suitable neighbourhood U of γ by T split =

3 1 a 1 e ∇a ⊗ ebα ∇b + μ2 ½ ⊗ ½ , 2 α=0 α 2

(81)

and {eaα }α=0,1,2,3 is any smooth tetrad defined in a neighbourhood of γ such that ea0 = ua on γ. This operator may be used to define the renormalized energy density in the usual fashion. Given any real-valued f ∈ C0∞ (I), we may apply Theorem 3.1 in turn to the operators Qα = eα · ∇ to obtain the absolute quantum weak energy inequality ren dτ f 2 (τ )ua ub Tab ω γ(τ ) R

∧ dξ 5 (−ξ, ξ) |hκ |1/4 f ⊗ |hκ |1/4 f ϑ∗ T split H ≥− + π R + dτ f 2 (τ ) Q + ua ub Cab |γ(τ ) (82) R

where ϑ : (τ, τ ) → (γ(τ ), γ(τ )). For the purposes of comparison with existing QEI results, let us consider this bound for the massless Klein–Gordon field in Minkowski spacetime (R4 , η) for a worldline along the time axis. In this case, the bound simplifies because the full Hadamard series is given by the leading term; that is, Λω (x, x ) − 1/(4π 2 σ+ (x, x )) is smooth and symmetric for any Hadamard state ω. Of course, σ+ is globally defined in Minkowski space. Now consider the example above, applied to the case

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where γ is an inertial curve, def ren γ(τ ) ≥ −B dτ f 2 (τ ) ua ub Tab = − ω

∧ dξ f ⊗ f ϑ∗ T splitH−1 (−ξ, ξ) R R+ π (83) −1 . Then, where we have denoted H−1 = 1/(4π 2 σ+ ) = H dξ 1 3 lim dt dt f (t)f (t ) e−iξ(t−t ) (84) B= 2 4 + 2π R+ π →0 R×R (t − t − i ) 3 dξ 1 = lim dt F (−t) e−iξt (85) 2 + 2π R+ π →0 R (t − i )4 where F (t) = R dt f (t − t)f (t ) has Fourier transform F (ξ) = |fˆ(ξ)|2 . Thus dξ 1 dζ|fˆ(ξ + ζ)|2 ζ 3 (86) B= 4π 2 R+ π R+ η 1 = dη dζ |fˆ(η)|2 ζ 3 (87) 4π 2 R+ 0 where we have utilized the fact that the Fourier transform of 1/(t − i0+ )4 is πξ 3 θ(ξ)/3 [31] and changed variables to η = ξ + ζ. Hence, a b ren 1 2 dt f (t) u u Tab ω γ(t) ≥ − dη |fˆ(η)|2 η 4 (88) 3 16π + R R

which is the same as the QEI for the massless field in Minkowski spacetime obtained in [9]. This example is of particular importance as on small length scales one expects the massive quantum field in a curved background to behave like its massless counterpart in flat spacetime. We expect that the same should hold for the quantum inequalities, i.e., on small length scales the dominant contribution to the bound arises from the 1/σ+ contribution to the Hadamard series. This will be investigated in a future work. 3.2.3. QEIs for adiabatic states. Finally, we show how our analysis of the Hadamard series using the Sobolev wave-front set allows us to establish QEIs for adiabatic states. We refer the reader to [38] for a detailed study of adiabatic states and further references. Adiabatic states, like Hadamard states, are defined in terms of their singular structure: following [38],6 a state ω on A(M, g) is an adiabatic state of order N if its associated two point function Λω satisfies W F s (Λω ) ⊂ C +− for all s < N + 3/2. From this definition, we see that any Hadamard state is adiabatic to all orders. In what follows the following lemma, taken from [38], is essential: 6 In

[38] the definition of adiabatic states was given, as for Hadamard states, only for quasi-free states, so the present usage is a slight extension.

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Lemma 3.2. Let ω be a Hadamard state and ω be an adiabatic state of order N , with associated two point functions Λω , Λω respectively, on A(M, g). Then W F s (Λω − Λω ) = ∅

(89)

for all s < N + 3/2. An immediate corollary is that (89) also holds for all s < N + 3/2 if ω and ω are any two adiabatic states of order N . By the Sobolev embedding theorem (Proposition 2.1 (i)), differences of this type will be in C 2 (M × M) provided N > 9/2, thus permitting the construction of a normal ordered stress-energy tensor. Similarly, if ω is adiabatic of order N > 9/2 and k ≥ 2, a difference of the form Λω − Hk will be twice continuously differentiable on an ultra-regular domain, permitting the computation of the renormalized stress-energy tensor. It is straightforward to modify the proof of Theorem 3.1 to obtain the following. Theorem 3.3. (a) Theorem 3.1 continues to hold (with the same lower bound) under the weaker hypothesis that ω is an adiabatic state of order N > 9/2. (b) Using the assumptions and notation of Theorem 3.1, except that ω is assumed only to be an adiabatic state of order N > 9/2, there is a difference inequality dvol(x)f 2 (x) Q ⊗ Q Λω − Λω (x, x) Σ

∧ dn ξ 1/4 1/4 ∗ ≥ −2 |h | f ⊗ |h | f ϑ Q ⊗ QΛ (−ξ, ξ) , (90) κ κ κ κ ω κ n R+ ×Rn−1 (2π) for any reference state ω which is adiabatic of order N > n + 11/2. Proof. We sketch the main points only. For (a), note that the hypotheses on N and k entail that we may choose s ∈ (6, min{N, k} + 3/2). Part one of the proof k ∈ H s . To see this, of Theorem 3.1 will continue to hold provided that Λω − H loc we introduce an arbitrary Hadamard state ω0 and note that k ⊂ W F s (Λω − Λω0 ) ∪ W F s Λω0 − H k = ∅ (91) W F s Λω − H using Lemma 3.2 together with the fact that s < N + 3/2, and Proposition 2.9 together with s < k + 3/2. Part two of the proof holds because Λω is a bisolution to the Klein–Gordon equation, so all covectors in its wave-front set are null, from which it follows that the required pull-back exists. The third part is identical to the original argument. For (b), the hypotheses on N and N permit us to choose s ∈ (6, min{N, N } + 3/2), and we have W F s (Λω − Λω ) = ∅ by the remark following Lemma 3.2. Part one of the argument then goes through, as does the k ). The remaining issue is to check that the second part (with Λω replacing H bound is finite. Introducing a reference Hadamard state ω0 as before, we note s that Λω − Λω0 ∈ Hloc (M × M) for some s ∈ (n + 7, N + 3/2), and hence n+3 (M × M) (this is the reason for the constraint N > n + 11/2). Λ ω − Λ ω0 ∈ C k . This is sufficient for Part three to apply to Λω in place of H

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3.3. Covariance The QEIs obtained from Theorem 3.1 are not covariant in full generality because they depend non-trivially on the coordinates used, and, in some cases, on a choice of tetrad near Σ. However, covariance may be rescued if we can restrict the freedom to choose coordinates and the tetrad in a covariant fashion so that the bound is independent of any residual choice. This strategy was successfully employed in [20] for worldline difference QEIs, and the same techniques would also apply to our worldline bounds; here we show how this may be accomplished for worldvolume averages (timelike submanifolds of other dimensions could be handled in an analogous fashion, but we will not do this here for brevity). Consider the quantum null energy inequality studied in Section 3.2.1. We will show that if the sampling tensor F ab picks out a preferred smooth timelike curve γ in a covariant way and we employ a system of Fermi normal coordinates near γ, then residual choices in our construction cannot affect the bound. With these restrictions, our absolute QEI would be locally covariant in the sense of [20]; see also [23] for a more abstract discussion of these ideas in the formulation of locally covariant quantum field theory developed by Brunetti, Fredenhagen & Verch [4] in terms of category theory. The requirement that F ab should pick out a unique timelike curve may be addressed in various ways. For example, if we restrict to sampling tensors for which there exists a (necessarily unique) pair of points x, x ∈ M such that the support of the sampling tensor obeys supp F ab = J − (x) ∩ J + (x )

(92)

and is contained within a convex normal neighbourhood then the unique timelike geodesic between x and x may be used as our choice of γ. From now on we assume that the sampling tensor does indeed select a preferred timelike curve, and that γ is given in a proper time parameterization. As already mentioned, we will restrict our coordinate system to belong to the class of Fermi normal coordinates about γ. For completeness, we briefly summarise the salient features of Fermi–Walker transport and Fermi normal coordinates, mainly following Chapters 1 §4 and 2 §10 of [46]. Recall that a vector field ξ defined on γ is said to be Fermi–Walker transported along it if DF W ξ = 0, where (93) DF W ξ a = (γ˙ · ∇)ξ a − gbc γ˙ c αa − αc γ˙ a ξ b and αa = γ˙ · ∇γ˙ a . Since α · γ˙ = 0 and γ˙ 2 = 1 it is easy to see that DF W γ˙ = 0; hence, the velocity vector is preserved under Fermi–Walker transport. Moreover, it is possible to show that Fermi–Walker transport of two vectors along γ preserves their inner-product. Therefore, a tetrad remains an orthogonal frame along γ under Fermi–Walker transport. If γ is a timelike geodesic, then Fermi–Walker and parallel transport coincide. The construction of Fermi normal coordinates near γ proceeds as follows. Let y lie on γ and construct an oriented and time-oriented orthonormal frame {eaα }α=0,1,2,3 at y with ea0 = γ˙ a |y . Fermi–Walker transport yields a tetrad along

450



the whole of γ. In a convex normal neighbourhood U of γ each point x ∈ U will be joined to γ by a unique spacelike geodesic segment c which is orthogonal to γ and which meets it at some γ(t). Assuming that c is parameterized by proper length, the Fermi normal coordinates xa of x are x0 = t ;

xi = sc˙ · ei |γ(t) ,

(94)

where s is the proper length of c. This construction has two important features. First, the metric takes the Minkowski form in these coordinates everywhere on γ. By continuity, this guarantees that the Fermi normal coordinates form a hyperbolic chart in a neighbourhood of γ. Second, the only freedom in the construction is the choice of origin on γ (which amounts to the freedom to add a constant to x0 ) and the choice of the spatial tetrad vectors ei (i = 1, 2, 3) at y, which are determined only up to a rotation. Owing to the angle-preserving nature of Fermi–Walker transport, any two coordinate systems obtained by the construction are therefore globally related 0 i by x = x0 + λ, x = S ij xj for constant scalar λ and constant rotation matrix S ∈ SO(3). If that the sampling tensor is supported within the neighbourhood of γ in which the Fermi normal coordinates are hyperbolic, it is easy to see that the absolute QEI (79) is independent of the particular system of Fermi normal coordinates chosen. The key point is that the Jacobian determinant for a change of coordinates between two Fermi normal coordinate systems is identically unity by the remarks given above. (Note also that F ab may be written uniquely as F ab = na nb under the constraint that na is future-pointing and null where it is nonzero.) For more general QEIs one also needs to construct a tetrad throughout the support of the sampling tensor. This may be done by taking the tetrad formed along γ and propagating it by parallel transport along spacelike geodesics which meet γ orthogonally.

4. Conclusion We have given the first explicit absolute quantum energy inequalities for the massive minimally coupled Klein–Gordon field in arbitrary four-dimensional globally hyperbolic backgrounds, by refining the argument of [10] to make use of the theory of the Sobolev wave-front set, and analysing microlocal properties of the components of the Hadamard series. The lower bounds are given in terms of partial sums of the Hadamard series, which are computed locally. Previously explicit absolute quantum energy inequalities were known only for the massless field in two dimensions [25] (a similar argument could be used to extend this to general positive energy unitary conformal field theories, based on [18]). Although the bounds make use of coordinate systems, we have shown that by restricting the class of sampling tensors, there are circumstances in which the bound is covariant. Absolute QEIs may also be found for higher spin fields. In the case of the Dirac field, which will be reported elsewhere [45], one adapts the difference QEI

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obtained in [6] in a similar fashion to the way in which [10] has been adapted here. Moreover, it is expected that one should be able to employ our method to prove an absolute QEI for the spin-1 vector bosons, using the formulation of the Hadamard condition for the Maxwell and Proca fields given in [14]. One possibility which is opened up by our work is to obtain control over the size of spacetime region in which the absolute QEI bound can be approximated to a good degree by the QEIs obtained in Minkowski space for massless fields. This would potentially result in very simple bounds of wide applicability, and is the subject of ongoing work.

Appendix A. Hadamard recursion relations In this appendix we briefly summarise the method for generating the coefficient functions, {vj }j=0,...,k and {wj }j=0,...,k , featuring in (12). These are obtained as the coefficients in the formal power series solution ⎧ ∞ j 1 ⎨ Δ1/2 (x, x ) σ+ (x, x ) σ (x, x ) H(x, x ) = 2 vj (x, x ) 2(j+1) ln + 4π ⎩ σ+ (x, x ) 2 j=0 ⎫ ∞ σ j (x, x ) ⎬ + wj (x, x ) 2(j+1) . (95) ⎭ j=0

to

2 (∇ + μ2 ) ⊗ ½ Hk (x, x ) = 0

subject to

w0 = 0 .

(96)

The series does not actually converge except in analytic spacetimes, which is why one makes use of the partial sums Hk . The recursion relations for the vj for the massive field in a curved background are: 0 = 2 (∇2 + μ2 )Δ1/2 + 2∇v0 · ∇σ + 4v0 + v0 ∇2 σ

(97)

0 = (∇ + μ )vj + 2(j + 1)∇vj+1 · ∇σ 2

2

2

− 4j(j + 1)vj+1 + (j + 1)vj+1 ∇2 σ

(98)

where j ∈ {0} ∪ N. In a regular domain X the system of differential equations uniquely determines the series of vj ’s. The wj series is specified once the value of w0 is fixed; we have adopted Wald’s prescription that w0 = 0 [48] and with this boundary condition the recursion relations are: 0 = 2∇w1 · ∇σ + w1 ∇2 σ + 2∇v1 · ∇σ − 4v1 + v1 ∇2 σ

(99)

0 = 2 (∇2 + μ2 )wk + 2(k + 1)∇wk+1 · ∇σ − 4k(k + 1)wk+1 + (k + 1)wk+1 ∇2 σ + 2∇vk+1 · ∇σ − 4(2k + 1)vk+1 + vk+1 ∇2 σ

(100)

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where k ∈ N. The system of (97)–(100) is known as the Hadamard recursion relations; these relations for the massless field may be found in [1, 7] where the dependency on a choice of length scale is suppressed.

Acknowledgements The authors would like to thank S. P. Dawson and L. Osterbrink for their many helpful comments on the manuscript, and J. Schlemmer and R. Verch for pointing out some typographical errors in the Appendix.

References [1] S. L. Adler, J. Lieberman and Y. Ng, Regularization of the stress energy tensor for vector and scalar particles propagating in a general background metric, Ann. Phys. 106 (1976), 279–321. [2] M. Alcubierre, The warp drive: Hyper-fast travel within general relativity, Class. Quantum Grav. 11 (1994), L73–L77. [3] R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208 (2000), 623–661. [4] R. Brunetti, K. Fredenhagen and R. Verch, The generally covariant locality principle – A new paradigm for local quantum field theory, Commun. Math. Phys. 237 (2003), 31–68. [5] S. P. Dawson, A quantum weak energy inequality for the Dirac field in twodimensional flat spacetime, Class. Quantum Grav. 23 (2006), 287–293. [6] S. P. Dawson and C. J. Fewster, An explicit quantum weak energy inequality for Dirac fields in curved spacetimes, Class. Quantum Grav. 23 (2006), 6659–6681. [7] B. S. DeWitt and R. W. Brehme, Radiation damping in a gravitational field, Ann. Phys. 9 (1960), 220–259. [8] H. Epstein, A. Jaffe and V. Glaser, Nonpositivity in the energy density in quantised field theories, Nuovo Cimento 36 (1965), 1016–1022. [9] C. J. Fewster and S. P. Eveson, Bounds on negative energy densities in flat spacetime, Phys. Rev. D 58 (1998), 084010. [10] C. J. Fewster, A general worldline quantum inequality, Class. Quantum Grav. 17 (2000), 1897–1911. [11] C. J. Fewster and E. Teo, Bounds on negative energy densities in static space-times, Phys. Rev D 59 (1999), 104016. [12] C. J. Fewster and R. Verch, A quantum weak energy inequality for Dirac fields in curved spacetime, Commun. Math. Phys. 225 (2002), 331–359. [13] C. J. Fewster and B. Mistry, Quantum weak energy inequalities for the Dirac field in flat spacetime, Phys. Rev. D 68 (2003), 105010. [14] C. J. Fewster and M. J. Pfenning, A weak quantum energy inequality for spin-one fields in curved spacetime, J. Math. Phys. 44 (2003), 4480–4513.

Vol. 9 (2008)


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[15] C. J. Fewster, Quantum energy inequalities in two dimensions, Phys. Rev. D 70 (2002), 127501. [16] C. J. Fewster, Energy inequalities in quantum field theory, in XIVth International Congress on Mathematical Physics, ed. J. C. Zambrini (World Scientific, Singapore, 2005). See math-ph/0501073 for an expanded and updated version. [17] C. J. Fewster, Quantum energy inequalities and stability conditions in quantum field theory, in Rigorous Quantum Field Theory: A Festschrift for Jacques Bros, A. Boutet de Monvel, D. Buchholz, D. Iagolnitzer, U. Moschella (Eds.) Progress in Mathematics, Vol. 251. (Birkh¨ auser, Boston, 2007), math-ph/0502002. [18] C. J. Fewster and S. Hollands, Quantum energy inequalities in two-dimensional conformal field theory, Rev. Math. Phys. 17 (2005), 577–612. [19] C. J. Fewster and T. A. Roman, On wormholes with arbitrarily small quantities of exotic matter, Phys. Rev. D 72 (2005), 044023. [20] C. J. Fewster and M. J. Pfenning, Quantum energy inequalities and local covariance I: Globally hyperbolic spacetimes, J. Math. Phys. 47 (2006), 082303. [21] C. J. Fewster and L. W. Osterbrink, Averaged inequalities for the non-minimally coupled classical scalar field, Phys. Rev. D 74 (2006), 044021. [22] C. J. Fewster and L. W. Osterbrink, Quantum energy inequalities for the nonminimally coupled scalar field, (2007), preprint arXiv:0708.2450, to appear in J. Phys. A: Math. Theor. . [23] C. J. Fewster, Quantum energy inequalities and local covariance II: Categorical formulation, Gen. Rel. and Grav. 39 (2007), 1855–1890. ´ E. ´ Flanagan, Quantum inequalities in two-dimensional Minkowski spacetime, [24] E. Phys. Rev. D 56 (1997), 4922–4926. ´ E. ´ Flanagan, Quantum inequalities in two-dimensional curved spacetimes, Phys. [25] E. Rev. D 66 (2002), 104007. [26] L. H. Ford, Quantum coherence effects and the second law of thermodynamics, Proc. R. Soc. Lond. A. 364 (1978), 227–236. [27] L. H. Ford and T. A. Roman, Quantum field theory constrains traversable wormhole geometries, Phys. Rev. D 53 (1996), 5496–5507. [28] L. H. Ford and M. J. Pfenning, The unphysical nature of “warp drive”, Class. Quantum Grav. 14 (1997), 1743–1751. [29] S. A. Fulling, M. Sweeny and R. M. Wald, Singularity structure of the two-point function in quantum field theory in curved spacetime, Commun. Math. Phys. 65 (1978), 257–264. [30] S. A. Fulling, F. J. Narcowich and R. M. Wald, Singularity structure of the twopoint function in quantum field theory in curved spacetime II, Ann. Phys. (N.Y.) 136 (1981), 243–272. [31] I. M. Gel’fand and G. E. Shilov, Generalised functions, Academic Press, New York and London (1964). [32] P. G¨ unther, Huygen’s principle and hyperbolic equations, Academic Press Inc, New York (1988). [33] S. Hollands and R. M. Wald, Local wick polynomials and time ordered products of quantum fields in curved spacetime, Commun. Math. Phys. 223 (2001), 289–326.

454



[34] S. Hollands and R. M. Wald, Existence of local covariant time ordered products of quantum fields in curved spacetime, Commun. Math. Phys. 231 (2002), 309–345. [35] L. H¨ ormander, The analysis of linear partial differential operators I, second edition, Springer-Verlag, New York (1989). [36] L. H¨ ormander, Lectures on nonlinear hyperbolic differential equations, Springer, New York (1996). [37] B. Hu, Y. Ling and H. Zhang, Quantum inequalities for massless spin-3/2 field in Minkowski spacetime, Phys. Rev. D 73 (2006), 045015. [38] W. Junker and E. Schrohe, Adiabatic vacuum states on general spacetime manifolds: Defintion, construction, and physical properties, Annales Poincare Phys. Theor. 3 (2002), 1113–1182. [39] B. S. Kay and R. M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon, Phys. Rep. 207 (1991), 49–136. [40] V. Moretti, Comments on the stress-energy tensor operator in curved spacetime, Commun. Math. Phys. 232 (2003), 189–221. [41] M. Pfenning M., DPhil Thesis Quantum inequality restrictions on negative energy densities on curved spacetimes, (1998), pre-print gr-qc/9805037. [42] M. Radzikowski M., Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553. [43] M. Reed M. and B. Simon, Methods of modern mathematical physics, Vol II, Fourier analysis and self adjointness, Academic Press, New York (1975). [44] H. Sahlmann and R. Verch, Microlocal spectral condition and Hadamard form for vector-valued quantum fields in curved spacetime, Rev. Math. Phys. 13 (2001), 1203– 1246. [45] C. J. Smith, An absolute quantum energy inequality for the Dirac field in curved spacetime, Class. Quantum Grav. 24 (2007), 4733–4750. [46] J. Synge, Relativity: The general theory, North Holland, Amsterdam (1960). [47] D. N. Vollick, Quantum inequalities in curved two dimensional spacetimes, Phys. Rev. D 61 (2000), 084022. [48] R. M. Wald, Trace anomaly of a conformally invariant quantum field in curved spacetime, Phys. Rev. D 17 (1978), 1477–1484. [49] H. Yu and P. Wu, Quantum inequalities for the free Rarita–Schwinger fields in flat spacetime, Phys. Rev. D 69 (2004), 064008.

Christopher J. Fewster Department of Mathematics University of York Heslington, York, YO10 5DD United Kingdom e-mail: [email protected]

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Calvin J. Smith7 School of Mathematical Sciences University College Dublin Belfield, Dublin 4 Ireland e-mail: [email protected] Communicated by Klaus Fredenhagen. Submitted: February 27, 2007. Accepted: November 22, 2007.

7 Address

from 1 January 2007.

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Ann. Henri Poincaré 9 (2008), 457–508 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/030457-52, published online May 8, 2008 DOI 10.1007/s00023-008-0364-x


Bethe–Sommerfeld Conjecture Leonid Parnovski Dedicated to the memory of B. M. Levitan

Abstract. We consider Schr¨ odinger operator −Δ + V in Rd (d ≥ 2) with smooth periodic potential V and prove that there are only finitely many gaps in its spectrum.

1. Introduction This paper is devoted to proving the Bethe–Sommerfeld conjecture which states that number of gaps in the spectrum of a Schr¨ odinger operator −Δ + V (x) ,

x ∈ Rd

(1.1)

with a periodic potential V is finite whenever d ≥ 2. We prove the conjecture for smooth potentials in all dimensions greater than one and for arbitrary lattices of periods. The conjecture so far was proved by V. Popov and M. Skriganov [9] (see also [11]) in dimension 2, by M. Skriganov [12, 13] in dimension 3, and by B. Helffer and A. Mohamed [3] in dimension 4; M. Skriganov [12] has also shown the conjecture to hold in arbitrary dimension under the assumption that the lattice of periods is rational. In the case d = 3 the conjecture was proved in [5] for non-smooth or even singular potentials (admitting Coulomb and even stronger singularities). An interesting approach to proving the conjecture was presented by O. A. Veliev in [15]. There is a number of problems closely related to the Bethe–Sommerfeld conjecture on which extensive work has been done; the relevant publications include, but are by no means restricted to, [2, 5] (and references therein), [7, 8]. Methods used to tackle these problems range from number theory [7, 8, 12, 13] to microlocal analysis in [3] and perturbation theory in [5, 14] and [15]. The approach used in the present paper consists, mostly, of perturbation theoretical arguments with a bit of geometry and geometrical combinatorics thrown in at the end.

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There are certain parallels between the approach of our paper and the approach used in [15]. In particular, there are several important intermediate results in our paper and in [15] which look rather similar to each other. Examples of such similarities are: precise asymptotic formulae for eigenvalues in the non-resonance regions and some, although not very precise, formulae in the resonance regions; proving that the eigenvalue is simple when we move the dual parameter ξ along a certain short interval, and, finally, the use of geometrical combinatorics. However, here the similarities end, because the detailed methods used on each step are completely different. For example, paper [15] makes a heavy use of the asymptotic formulae for the eigenfunctions, whereas in our paper they are not needed. On the other hand, we prove that each eigenvalue close to λ is described by exactly one asymptotic formula (i.e., the mapping f constructed in our paper is a bijection in a certain sense), and this plays an essential role in our proof, but in [15] this property is not required at all. In [15] a very important role is played by the isoenergetic surface, whereas we don’t need it. This list can be continued, but it is probably better to stop here and state once again: the methods of [15] and our paper are different, despite the similarity of some intermediate results. It is also worthwhile to mention that asymptotic expressions for eigenfunctions as well as asymptotic formulae for isoenergetic surfaces were obtained by Yu. Karpeshina (see for example [5]). In many of the papers mentioned above, proving the conjecture in special cases comes together with obtaining lower bounds for either of the functions describing the band structure of the spectrum: the multiplicity of overlapping m(λ) and the overlapping function ζ(λ) (we will give a definition of these functions in the next section). For example, in dimensions d = 2, 3, 4 it has been proved in [2, 12, 13], and [8] that for large λ we have m(λ) λ and ζ(λ) λ

d−1 4

3−d 4

;

(1.2)

however, these estimates do not seem likely to hold in high dimensions. The estimates of the present paper are rather weaker, but they hold in all dimensions. Unfortunately, our approach does not allow to say anything stronger than m(λ) ≥ 1 for large λ (this inequality is equivalent to the finiteness of the number of spectral gaps). However, it is possible to give a nontrivial lower bound for the overlapping function: we will show that in all dimensions for sufficiently large λ ζ(λ) λ

1−d 2

.

(1.3)

The rest of the introduction is devoted to the informal discussion of the proof. Since the proof of the main Theorem 2.1 is rather complicated and technically involved, the major ideas are outlined here. After an affine change of coordinates, we can re-write our operator (1.1) as H = H0 + V (x) ,

H0 = DGD ,

(1.4)

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with the potential V being smooth and periodic with the lattice of periods of V equal (2πZ)d (D = i∇ and G = F2 is d × d positive matrix, where the matrix F is also assumed to be positive). Without loss of generality, we assume that the average of the potential V over the cell [0, 2π]d is zero (otherwise we simply subtract this average from the potential, which will just shift the spectrum of the problem). Let us fix a sufficiently large value of spectral parameter λ = ρ2 ; we will prove that λ is inside the spectrum of H. The first step of the proof, as usual, consists in performing the Floquet–Bloch decomposition to our operator (1.4): H= H(k)dk , (1.5) ⊕

where H(k) = H0 + V (x) is the family of ‘twisted’ operators with the same symbol as H acting in L2 (T d ) where T d := {x ∈ Rd , |xj | ≤ π, 1 ≤ j ≤ d}. The domain the boundary conditions D(k) consists of functions f ∈ H 2 (T d ) satisfying of H(k) ∂f ∂f f xj =π = ei2πkj f xj =−π , ∂xj xj =π = ei2πkj ∂x . These auxiliary operators xj =−π j are labelled by the quasi-momentum k ∈ Rd /Zd ; see [10] for more details about this decomposition. The next step is to assume that the potential V is a finite trigonometric polynomial whose Fourier coefficients Vˆ (m) vanish when |m| > R. The justification of the fact that it is enough to prove the conjecture in this case is not too difficult once we keep careful control of the dependence of all the estimates on R. The main part of the argument consists of finding an asymptotic formula for all sufficiently large eigenvalues of all operators H(k), with an arbitrarily small power of the energy in the remainder estimate. In order to be able to write such a formula, however, we have to abandon the traditional way of labeling eigenvalues of each H(k) in the non-decreasing order. Instead, we will label eigenvalues by means of the integer vectors n ∈ Zd . Consider, for example, the unperturbed operator H0 (k). Its eigenfunctions and eigenvalues are i(n+k)x e n∈Zd and

|F(n + k)|2

n∈Zd

(1.6)

correspondingly. However, despite our precise knowledge of eigenvalues, it is extremely difficult to write them in increasing order or, indeed, even to derive the one-term asymptotic formula for the j-th eigenvalue with the precise remainder estimate. It is rather convenient to introduce one parameter which takes care of both the quasi-momentum k and the integer vector n which labels eigenvalues in (1.6). We denote ξ := n + k (notation indicates that ξ can be thought of as being a dual variable) so that n = [ξ] and k = {ξ} (integer and fractional parts, respectively). Then we can reformulate formula (1.6) for the unperturbed eigenvalues as follows: there is a mapping f : Rd → R, given by the formula f (ξ) = |F(n + k)|2 such that for each k the restriction of f to {ξ ∈ Rd : {ξ} = k} is a bijection onto the set of all eigenvalues of H(k) (counting multiplicities). We want to give an analogue of

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this (trivial) statement in the general case. Let us define the spherical layer A := ξ ∈ Rd , |Fξ|2 − λ ≤ 40v (v is the L∞ -norm of V ). Let N ∈ N be a fixed number. We will construct two mappings f, g : A → R which satisfy the following properties: (I) for each k the first mapping f is an injection from the set of all ξ with fractional part equal to k into the spectrum of H(k) (counting multiplicities). Each eigenvalue of H(k) inside J := [λ−20v, λ+20v] has a pre-image ξ ∈ A with {ξ} = k. The perturbation inequality |f (ξ) − |Fξ|2 | ≤ 2v holds for all ξ ∈ A. (II) for ξ ∈ A satisfying |Fξ|2 ∈ J we have: |f (ξ) − g(ξ)| < ρ−N ; (III) one can divide the domain of g in two parts: A = B ∪ D (both B and D are intersections of A with some cones centered at the origin – at least modulo very small sets) such that g(ξ) is given by an explicit formula when ξ ∈ B, we have some control over g(ξ) when ξ ∈ D, and the ratio of volumes of B and D goes to infinity when ρ → ∞. The set B, called the non-resonance set, contains, among others, all points ξ ∈ A which satisfy the inequality | ξ, Gθ| ≥ ρ1/3 |Fθ|

(1.7)

for all non-zero integer vectors θ with |θ| RN . The precise formula for g will imply, in particular, that when ξ ∈ B we have g(ξ) = |Fξ|2 + G(ξ) with all partial derivatives of G being O(ρ− ) for some > 0. When ξ belongs to the resonance set D, we can give good estimates only of the partial derivative of g along one direction; this direction has a small angle with the direction of ξ. The behaviour of g along all other directions is much worse. Indeed, by considering potentials V which allow to perform the separation of variables, one can see that the function g can not, in general, be made even continuous in the resonance set. However, we still have some (although rather weak) control over the behaviour of g along all directions inside the resonance set; see Lemma 7.11 for the precise formulation of these properties. One should mention that asymptotic formulae of non-resonance eigenvalues (i.e., the function g(ξ) for ξ ∈ B in our notation) and some resonance eigenvalues were obtained before in certain cases, using completely different methods, by O. A. Veliev, [14] and [15] and Y. E. Karpeshina (see [4, 5] and references therein). However, as has been already mentioned, there are certain distinctions between the settings of [15] and [5] and the settings of our paper. Because of this, and in order to make our paper self-contained, it seems sensible to include an independent proof of the asymptotic formula for eigenvalues. Before describing how to construct these mappings, we explain first how to prove the Bethe–Sommerfeld conjecture using them. Put δ = ρ−N . For each η ∈ Rd of unit length we denote Iη the interval consisting of points ξ = tη, t > 0 satisfying g(ξ) ∈ [λ − δ, λ + δ]; we will consider only vectors η for which Iη ⊂ B. Suppose we have found an interval Iη on which the mapping f is continuous. Then

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property (II) above together with the intermediate value theorem would imply that there is a point ξ(λ) ∈ Iη satisfying f (ξ(λ)) = λ, which would mean that λ is in the spectrum of H. Thus, if we assume that λ belongs to the spectral gap, this would imply that the mapping f is not continuous on each of the intervals Iη . A simple argument shows that in this case for each point ξ ∈ B with |g(ξ) − λ| ≤ δ there exists another point ξ 1 = ξ with ξ − ξ 1 ∈ Zd and |g(ξ 1 ) − λ| δ. The existence of such a point ξ 1 (which we call a conjugate point to ξ) is a crucial part of the proof; it seems that similar arguments based on the existence of conjugate points could be helpful in analogous problems. Afterwards, a geometrical combinatorics 1−d would do) some (moreover, argument shows that for sufficiently small δ (δ ρ −1 most) of the points ξ ∈ B ∩ g [λ − δ, λ + δ] have no conjugate points; the important part in the proof is played by the fact that the surface g −1 (λ) ∩ B has positive curvature in each direction. Now let us discuss how to construct mappings f and g with properties described above. This is done in several steps. First, we prove Lemma 3.2 which states that under certain conditions it is possible instead of studying eigenvalues of the operator H = H0 + V , to study eigenvalues of the operator P j HP j , (1.8) j j

where P are spectral projections of H0 ; the error of this approximation is small. This result can be applied to the operators H(k) from the direct integral (1.5). We want therefore to study the spectrum of the (direct) sum (1.8) where P j are projections ‘localized’ in some domains of the ξ-space. The geometrical structure of these projections will depend on whether the localization happens inside or outside the resonance regions. The case of a projection P j ‘localized’ around a point ξ ∈ B is relatively simple: the rank of such projections does not depend on ρ or the ‘localization point’ ξ. Thus, in this case we will need to compute the eigenvalue of the finite matrix P j H(k)P j . This can be done by computing the characteristic polynomial of this matrix and then using the iterative process based on the Banach fixed point theorem to find the root of this characteristic polynomial. It is much more difficult to construct projections P j corresponding to the points ξ located inside the resonance set D. The form of projections will depend on, loosely speaking, how many linearly independent integer vectors θ for which (1.7) is not satisfied are there. The construction of such projections is the most technically difficult part of the paper. Once these projections are constructed, it turns out that the eigenvalues of P j HP j with large ρ can be easily expressed in terms of the eigenvalues of the operator pencil rA + B where A and B are fixed and r ∼ ρ is a large parameter. The rest is a relatively simple perturbation theory. The approach used in this paper can be applied to various related problems. For example, it seems possible to obtain several new terms of the asymptotics of the integrated density of states using these methods. It might even be possible to obtain the complete asymptotic formula; however, this would require much more careful analysis of the mapping g in the resonance set. As an immediate

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‘free’ corollary of our results we obtain the Theorem 7.17 which seems to be new. Loosely speaking, it states that there are no ‘simultaneous clusters’ of eigenvalues of all operators H(k). The approach of this paper works almost without changes for the polyharmonic operators (−Δ)l + V with a smooth periodic potential V . Another possible field of applications of the results of this paper is studying the structure of the (complex) Bloch and Fermi varieties. The rest of the paper is constructed as follows: Section 2 has all necessary preliminaries; also, in this section for the convenience of the reader we, taking into account the size of the paper, give references to the definitions of all major objects in the paper. Section 3 proves the abstract result allowing to reduce computation of the spectrum of H = H0 + V to the computation of the spectrum of j P j HP j , P j being the spectral projections of H0 . Section 4 proves various estimates of angles between lattice points which are needed to keep track on the dependence of all results on R – the size of the support of the potential. In Section 5 we apply the abstract lemma from Section 3 to our case and perform the reduction of H to the sum of simpler operators. In Section 6 we compute the eigenvalues of these simpler operators corresponding to the non-resonance set; we also give the formula for g(ξ) when ξ ∈ B. Section 7 is devoted to the study of the properties of these simpler operators and the mapping g restricted to the resonance set D. Finally, in Section 8 we prove the Bethe–Sommerfeld conjecture. When this manuscript was ready, I have learned that another article of Veliev [16] was published recently.

2. Preliminaries We study the Schr¨ odinger operator H = H0 + V (x) ,

H0 = DGD ,

(2.1)

with the potential V being infinitely smooth and periodic with the lattice of periods equal (2πZ)d . Here, D = i∇, and G = F2 is d × d positive matrix; F is also taken to be positive. Throughout the paper we use the following notation. If A is a bounded below self-adjoint operator with compact resolvent, then we denote by {μj (A)} (j = 1, 2, . . . ) the set of eigenvalues of A written in non-decreasing order, counting multiplicities. As we have already mentioned, the spectrum of H is the union over k ∈ Rd /Zd of the spectra of the operators H(k), the domain of each H(k) is D(k) and H(k) := DGD + V (x). By H := L2 (T d ) we denote the Hilbert space in which all the operators H(k) act. We also denote by H0 (k) the operator DGD with the domain D(k). Let λj (k) = μj (H(k)) be the jth eigenvalue of H(k). Then it is well-known (see, for example, [10]) that each function λj ( · ) is continuous and piecewise smooth. Denote by j the image of λj ( · ). Then j is called the jth

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spectral band. We also define, for each λ ∈ R, the following functions: m(λ) = #{j : λ ∈ j } is the multiplicity of overlapping (# denotes the number of elements in a set). The overlapping function ζ(λ) is defined as the maximal number t such that the symmetric interval [λ − t, λ + t] is entirely contained in one of the bands j : ζ(λ) = max max t ∈ R : [λ − t, λ + t] ⊂ j . j

Finally,

N (λ) = [0,1]d

# j : λj (k) < λ dk

(2.2)

is the integrated density of states of the operator (1.4). For technical reasons sometimes it will be convenient to assume that the dimension d is at least 3 (in the 2-dimensional case the conjecture has been proved already, so this assumption does not restrict generality). The main result of the paper is the following: Theorem 2.1. Let d ≥ 3. Then all sufficiently large points λ = ρ2 are inside the spectrum of H. Moreover, there exists a positive constant Z such that for large enough ρ the whole interval [ρ2 − Zρ1−d , ρ2 + Zρ1−d ] lies inside some spectral band.

Without loss of generality we always assume that [0,2π]d V (x)dx = 0. Abusing the notation slightly, we will denote by V both the potential itself and the operator of multiplication by V . By B(R) we denote a ball of radius R centered at the origin. By C or c we denote positive constants, depending only on d, G, and norms of the potential in various Sobolev spaces H s . In Section 5 we will introduce parameters p, qj and M ; constants are allowed to depend on the values of these parameters as well. The exact value of constants can be different each time they occur in the text, possibly even each time they occur in the same formula. On the other hand, the constants which are labelled (like C1 , c3 , etc) have their values being fixed throughout the text. Whenever we use O, o, , , or notation, the constants involved will also depend on d, G, M , and norms of the potential; the same is also the case when we use the expression ‘sufficiently large’. Given two positive functions f and g, we say that f g, or g f , or g = O(f ) if the ratio fg is bounded. We say f g if f g and f g. By λ = ρ2 we denote a point on the spectral axis. We will always assume that λ is sufficiently large. We also denote by v the L∞ -norm of the potential V , and J := [λ − 20v, λ + 20v]. Finally, (2.3) A := ξ ∈ Rd , |Fξ|2 − λ ≤ 40v and

A1 := ξ ∈ Rd , |Fξ|2 − λ ≤ 20v .

(2.4) Notice that the definition of A obviously implies that if ξ ∈ A, then |Fξ| − ρ ρ−1 .

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Given several vectors η1 , . . . , η n ∈ Rd , we denote by R(η1 , . . . , η n ) the linear subspace spanned by these vectors, and by Z(η 1 , . . . , η n ) the lattice generated by them (i.e., the set of all linear combinations of ηj with integer coefficients; we will use this notation only when these vectors are linearly independent). We denote by M(η 1 , . . . , ηn ) the d × n matrix whose jth column equals η j . Given any lattice Γ, we denote by |Γ| the volume of the cell of Γ, so that if Γ = Z(η1 , . . . , η d ), then |Γ| is the absolute value of the determinant of M(η1 , . . . , η d ). We also denote, for any linear space V ⊂ Rd , B(V; R) := V ∩ B(R). For any non-zero vector ξ ∈ Rd we ξ . Any vector ξ ∈ Rd can be uniquely decomposed as ξ = n + k denote n(ξ) := |Fξ| d with n ∈ Z and k ∈ [0, 1)d . We call n = [ξ] the integer part of ξ and k = {ξ} the fractional part of ξ. Whenever P is a projection and A is an arbitrary operator acting in a Hilbert space H, the expression P AP means, slightly abusing the notation, the operator P AP : P H → P H. Throughout the paper we use the following convention: vectors are denoted by bold lowercase letters; matrices by bold uppercase letters; sets (subsets of Rd ) by calligraphic uppercase letters; linear subspaces by gothic uppercase letters. By vol(C) we denote the Lebesgue measure of the set C. If Cj ⊂ Rd , j = 1, 2 are two subsets of Rd , their sum is defined in the usual way: C1 + C2 = ξ ∈ Rd : ξ = ξ1 + ξ 2 , ξ j ∈ Cj . Finally, for the benefit of the reader we will list here either the definitions of the major objects introduced later in the paper or references to the formulas in which they are defined. f, g : A → R are mappings satisfying properties listed in Theorem 7.13 (if the Fourier transform of V has compact support) and in Corollary 7.15 for general potentials. The sets Θj and Θj are defined in (5.5). The projections P(k) (C) are defined immediately before Lemma 5.14. V(n), ξ V , ξ ⊥ V , and Θ(V) are defined at the beginning of Subsection 7.1. The sets Ξ(V) and Ξj (V) (j = 0, . . . , 3) are defined in formulas (5.7)–(5.11); the sets Υj (ξ), Υ(ξ), Υ(ξ 1 ; ξ2 ), and Υ(ξ; U ) are defined by formulas (7.1), (7.2), (7.11), and (7.15) correspondingly. The numbers p and qn are defined in (5.15), K = ρp and Ln = ρqn . The projection P (ξ) and the operator H (ξ) are defined in (7.3) and (7.4) correspondingly. The sets B and D are defined in (5.13) and (5.12). r(ξ) and ξV are defined by formula (7.8). Operators A and B are defined by (7.30) and (7.31). Finally, the sets A(δ), B(δ), and D(δ) are defined before Lemma 8.1.

3. Reduction to invariant subspaces: General result The key tool in finding a good approximation of the eigenvalues of H(k) will be the following two lemmas. Lemma 3.1. Let H0 , V and A be self-adjoint operators such that H0 is bounded below and has compact resolvent, and V and A are bounded. Put H = H0 + V and

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ˆ = H0 + V + A and denote by μl = μl (H) and μ ˆ the sets of eigenvalues H ˆl = μl (H) of these operators. Let {Pj } (j = 0, . . . , n) be a collection of orthogonal projections commuting with H0 such that Pj = I, Pj V Pk = 0 for |j − k| > 1, and A = Pn A. Let l be a fixed number. Denote by aj the distance from μl to the spectrum of Pj H0 Pj Assume thatfor j ≥ 1 we have aj > 4a, where a := ||V || + ||A||. Then n |ˆ μl − μl | ≤ 22n a2n+1 j=1 (aj − 2a)−2 . Proof. Let Ht = H + tA, 0 ≤ t ≤ 1 and let μ(t) = μl (H(t)) be the corresponding family of eigenvalues. We also choose the family of corresponding normalized eigenfunctions φ(t) = φl (t). We will skip writing the index l in the rest of the proof. Elementary perturbation theory (see, e.g., [6]) implies that μ(t) is piecewise differentiable and dμ(t) = Aφ(t), φ(t) . (3.1) dt Let Φj = Φj (t) := Pj φ(t), and let Vkj := Pk V Pj (so that Vjk = 0 if |j − k| > 1). Then the eigenvalue equation for φ(t) can be written in the following way: H0 Φ0 + V0 0 Φ0 + V0 1 Φ1 = μ(t)Φ0 H0 Φj + Vj j−1 Φj−1 + Vj j Φj + Vj j+1 Φj+1 = μ(t)Φj ,

1≤j 1 and Pjm V Q = 0 if j < jm . Let v := ||V || and let us fix an interval J = [λ1 , λ2 ] on the spectral axis which satisfies the following properties: spectra of the operators QH0 Q and Pjk H0 Pjk , j ≥ 1 lie outside J; moreover, the distance from the spectrum of QH0 Q to J is greater than 6v and the distance from the spectrum of Pjk H0 Pjk (j ≥ 1) to J, which we denote by akj , is greater than 16v. Denote by μp ≤ · · · ≤ μq all eigenvalues of H = H0 + V which are inside J. Then the corresponding eigenvalues μ ˜p , . . . , μ ˜q of the operator ˜ := H P m HP m + QH0 Q m

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P m HP m , and they satisfy ⎡ ⎤ jm −2 ⎦ |˜ μr − μr | ≤ max ⎣(6v)2jm +1 ; (am j − 6v)

are eigenvalues of

m

m

j=1

˜ are outside the interval [λ1 + 2v, λ2 − 2v]. all other eigenvalues of H Proof. Assumptions of the lemma imply that m m m m ˜ + Q+ H=H Pjm V Q + Pjm − Pjm V Pjm . m

m

m

m

m m ˜ − 2v(Q + ˜ Therefore, H m Pjm ) ≤ H ≤ H + 2v(Q + m Pjm ), and the elementary perturbation theory implies that for all l m m ˜ − 2v Q + ˜ + 2v Q + μl H ≤ μl (H) ≤ μl H . (3.10) P P jm

jm

m

m

˜ ± 2v(Q + P m ) split into the sum of invariant operators The operators H m jm QH0 Q ± 2vQ and P m HP m ± 2vPjm (m = 0, . . . , n). The spectrum of operators m QH0 Q ± 2vQ is outside [λ1 − 4v, λ2 + 4v] due to the assumptions of the lemma. Therefore, since the shift of an eigenvalue is at most the norm of the perturbation, m ˜ ± 2v(Q + for p ≤ l ≤ q, μl (H m Pjm )) is an eigenvalue of one of the operators m m m P HP ±2vPjm . If we now apply Lemma 3.1 to each of the operators P m HP m ± 2vPjm with A := ±2vPjm and a = 3v, we will obtain m m ) − μk (P m HP m )| |μk (P m HP m ± 2vPjm m ≤ 62jm +1 v 2jm +1 max m

⎡ ≤ max ⎣(6v)2jm +1 m

jm j=1

jm

−2 (am j − 6v)

⎤

(3.11)

−2 ⎦ =: τ , (am j − 6v)

j=1

provided μk (P HP ) ∈ [λ1 − 4v, λ2 + 4v]. Let us now define the bijection F ˜ ˜ mapping the set of all eigenvalues of H to the set of all eigenvalues of {μl (H + 2v(Q + m Pjm ))} (counting multiplicities) in the following way. Suppose, μ is an m m m ˜ eigenvalue of H. Then either μ = μk (QH0 Q), or μ = μk (P HP ) for some k, m. We define F (μ) := μk (QH0 Q + 2vQ) in the former case, and F (μ) := μk (P m HP m + 2vPjm ) in the latter case. Then the mapping F satisfies the folm lowing properties: |F (μ) − μ| ≤ 2v ; (3.12) m

m

moreover, if μ ∈ [λ1 − 4v, λ2 + 4v], then |F (μ) − μ| ≤ τ

(3.13)

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(this follows from (3.11)). A little thought shows that this implies m ˜ ˜ Pjm − μl (H) ≤ τ μl H + 2v Q +

(3.14)

m

for p ≤ l ≤ q. Indeed, suppose that (3.14) is not satisfied for some l, say m ˜ + 2v Q + ˜ >τ; μl H P − μl (H) jm

(3.15)

m

˜ +2v(Q+ P m )) > λ1 −2v. Then the pigeonhole in particular, this implies μl (H m jm ˜ k ≤ l to principle shows that F maps at least one of the eigenvalues μk (H), m ˜ + 2v(Q + ˜ μt (H m Pjm )) with t ≥ l. If μk (H) < λ1 − 4v, this contradicts (3.12), ˜ and if μk (H) ≥ λ1 − 4v, this contradicts (3.13). These contradictions prove (3.14). Similarly, we prove that m ˜ − 2v Q + ˜ ≤τ. (3.16) Pjm − μl H μl H m

Estimates (3.14) and (3.16) together with (3.10) prove the lemma.

Corollary 3.3. If all conditions of Lemma 3.2 are satisfied, there exists an injec tion G defined on a set of eigenvalues of the operator m P m HP m (all eigenvalues are counted according to their multiplicities) and mapping them to a subset of the set of eigenvalues of H (again considered counting multiplicities) such that: (i) all eigenvalues of H inside J have a pre-image, (ii) If μj ∈ [λ1 + 2v, λ2 − 2v] is an eigenvalue of m P m HP m , then ⎡ ⎤ jm −2 ⎦ |G(μj ) − μj | ≤ max ⎣(6v)2jm +1 , (am j − 6v) m

j=1

and (iii) G(μj ( m P m HP m )) = μj+l (H), where l is the number of eigenvalues of QH0 Q which are smaller than λ1 . Proof. Statements (i) and (ii) follow immediately from Lemma 3.2, and to prove (iii) we just notice that if μj ( m P m HP m ) ∈ J, then m m m m P HP P HP = μj+l QH0 Q + . μj m

m

4. Lattice points In this section, we prove various auxiliary estimates of angles between integer vectors.

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Lemma 4.1. Let η 1 , . . . , η n ∈ Zd be linearly independent. Let Γ = Z(η 1 , . . . , η n ) and suppose that ν 1 , . . . , ν n−1 ∈ Γ ∩ B(R). Then there exists a vector θ ∈ Γ, θ = 0 orthogonal to all ν j ’s, such that |θ| ≤ 2n |Γ|

n−1

|ν j |

(4.1)

j=1

and, therefore, |θ| ≤ 2n |Γ|Rn−1 . Proof. For r > 1 let Ar ⊂ R(η 1 , . . . , ηn ) be the set Ar = ξ ∈ R(η 1 , . . . , η n ) : | ξ, ν j | < 1, j = 1, 2, . . . , n − 1, & |ξ| < r . This set is obviously convex and symmetric about the origin. Moreover, vol(Ar ) > r

n−1

|ν j |−1 .

j=1

By Minkowski’s convex body theorem (see, e.g., [1], §III.2.2 Theorem II), under the condition vol(Ar ) > |Γ|2n the set Ar contains at least two non-zero points n−1 ±θ ∈ Γ. The above condition is satisfied if r j=1 |ν j |−1 ≥ 2d |Γ|, that is if r ≥ 2d |Γ| n−1 j=1 |ν j |. Since ν j ’s and θ are integer vectors, the condition | θ, ν j | < 1 is equivalent to θ, ν j = 0. This implies the required result. Lemma 4.2. Let θ 1 , . . . , θ n , μ ∈ Zd ∩ B(R) be linearly independent. Then the angle between μ and R(θ1 , . . . , θn ) is R−n−1 . Proof. Suppose this angle is smaller than R−n−1 . Then the lattice Γ = Z(θ1 , . . . , θn , μ) has |Γ| ≤ 1. Lemma 4.1 then implies that there exists a vector θ ∈ Γ, θ ⊥ θj , |θ| Rn . Then, since θ and μ are non-orthogonal integer vectors, we have: | μ, θ| ≥ 1, and sin of the angle between μ and R(θ1 , . . . , θn ), which equals cos of the angle between μ and θ, is bounded below by |θ|−1 |μ|−1 R−n−1 . Corollary 4.3. Let θ1 , . . . , θn , μ ∈ Zd ∩ B(R) be linearly independent. Then the angle between Fμ and R(Fθ1 , . . . , Fθn ) is R−n−1 . Proof. This is equivalent to saying that for each ξ ∈ R(θ1 , . . . , θn ) the distance between F(n(μ)) and Fξ is larger than cR−n−1 . But the distance between F(n(μ)) and Fξ is not greater than the largest eigenvalue of F times the distance between n(μ) and ξ. Now the statement follows from Lemma 4.2. It is possible to generalize Lemma 4.2 a bit: if we talk about distance from a vector to a linear sub-space instead of the angle between a vector and a subspace, we can drop the assumption that |μ| ≤ R: Lemma 4.4. Let θ 1 , . . . , θ n ∈ Zd ∩ B(R) and μ ∈ Zd be linearly independent. Then the distance between μ and R(θ 1 , . . . , θ n ) is R−n .

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1 ,...,θ n )| Proof. The distance between μ and R(θ1 , . . . , θn ) equals |Z(μ,θ |Z(θ 1 ,...,θn )| . The square of the denominator of this fraction is the determinant of the n × n matrix A with Ajk := θ j , θk = O(R2 ), so the denominator is O(Rn ). Similarly, the square of the numerator is the determinant of (n + 1) × (n + 1) non-singular matrix with integer entries. Therefore, the absolute value of the numerator is at least 1. This proves our statement.

The following result is a generalization of Lemma 4.1 and the proof is similar: Lemma 4.5. Let Γ be as above and let ν 1 , . . . , ν m ∈ Γ ∩ B(R) (m < n) Then there exist linearly independent vectors θ 1 , . . . , θ n−m ∈ Γ such that each θl is orthogonal to each ν j and n−m m |θ l | |Γ| |ν j | ≤ |Γ|Rm . (4.2) j=1

l=1

Proof. Applying Lemma 4.1 n − m times, we see that the set of vectors from Γ which are orthogonal to ν j form a lattice Γn−m of dimension n − m. Let θ j (j = 1, . . . , n − m) be successive minimal vectors of Γn−m . That means that θ1 is the smallest nonzero vector in Γn−m ; θ2 ∈ Γn−m is the smallest vector linearly independent of θ 1 ; θ3 ∈ Γn−m is the smallest vector linearly independent of θ1 , θ2 , etc. For r > |θ 1 | let Ar ⊂ R(η 1 , . . . , ηn ) be the set Ar = ξ ∈ R(η 1 , . . . , η n ) : | ξ, ν j | < 1, j = 1, 2, . . . , m, & |ξ| < r . This set is obviously convex and symmetric about the origin. Moreover, m |ν j |−1 . vol(Ar ) rn−m

(4.3)

j=1

Applying again Minkowski’s convex body theorem, we find that the set Ar contains at least m −n −1 −1 n−m |ν j |−1 (4.4) N = 2 |Γ| vol(Ar ) |Γ| r j=1

pairs of points ±μk ∈ Γ, k = 1, . . . , N . Obviously, each μk is orthogonal to each ν j . Suppose, r < |θn−m |. Then, obviously, |θp | ≤ r < |θ p+1 | for some p ≤ n − m − 1. The dimension of R(μ1 , . . . , μN ) is then ≤ p, and each μk is a linear combination of θ1 , . . . , θp with integer coefficients. Denote Γp := Z(θ1 , . . . , θp ). Minkowski’s second theorem (see, e.g., [1], §VIII.2, Theorem I) shows that p

|θl | |Γp | .

(4.5)

l=1

A simple packing argument shows that N |Γp | is smaller than the volume of the ball of radius (p + 1)r in R(θ 1 , . . . , θ p ), i.e., N |Γp | rp . Estimate (4.5) implies N

p l=1

|θ l | rp .

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Therefore, if the condition N

p

|θl | > Crp

(4.6)

l=1

is satisfied, where C is sufficiently large, this implies that r ≥ |θp+1 |. Estimate (4.4) shows that if r > C|Γ|

m

|ν j |

j=1

n−m−1

|θl |−1 ,

(4.7)

l=1

then condition (4.6) with p = n − m − 1 will be satisfied and this would guarantee that r > |θn−m |. In other words, if r is greater than the RHS of (4.7), then r > |θ n−m |. This implies |θn−m | |Γ|

m j=1

which finishes the proof.

|ν j |

n−m−1

|θl |−1 ,

l=1

Let ν 1 , . . . , ν n ∈ Rd (n ≤ d). We denote by M = M(ν 1 , . . . , ν n ) the d × n matrix whose jth column equals ν j . We also denote ν 1 ∧ · · · ∧ ν n 2 := det(M∗ M) (4.8) (M∗ M is obviously non-negative, and so is the determinant). The reason for the notation is that we can think of ν 1 ∧ · · · ∧ ν n 2 as being the Hilbert–Schmidt norm of the tensor ν 1 ∧ · · · ∧ ν n . Lemma 4.6. Let ν 1 , . . . , ν n , μ1 , . . . , μm ∈ Rd . Let V1 = R(ν 1 , . . . , ν n ) and V2 = R(μ1 , . . . , μm ). Let α be the angle between V1 and V2 . Then the following inequality holds: ν 1 ∧ · · · ∧ ν n ∧ μ1 ∧ · · · ∧ μm 2 . (4.9) sin α ≥ ν 1 ∧ · · · ∧ ν n 2 μ1 ∧ · · · ∧ μm 2 Proof. If we multiply matrix M from the right by a non-singular n × n matrix B, the expression (4.8) is multiplied by det B. This observation shows that elementary transformations of the set of vectors ν (i.e., multiplying ν j by a non-zero scalar, adding ν j to ν k , etc) do not change both sides of (4.9); the same is the case for elementary transformations of the vectors μ. Thus, we may assume that vectors ν form an orthonormal basis of V1 , vectors μ form an orthonormal basis of V2 , and the angle between ν 1 and μ1 equals α. Notice that now the denominator of the RHS of (4.9) equals 1. Next, we notice that an orthogonal change of coordinates results in multiplying M from the left by a d×d orthogonal matrix and thus doesn’t change (4.8) and the RHS of (4.9); the LHS of (4.9) is obviously invariant under an orthogonal change of coordinates as well. Assume, without loss of generality, that n ≥ m. Then, applying an orthogonal change of coordinates, we can make our vectors to have the following form: ν j = ej (j = 1, . . . , n, where ej are standard

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basis vectors), μj = pj ej + qj en+j , (pj , qj ≥ 0, p2j + qj2 = 1). Elementary geometry implies cos α = p1 , and so sin α = q1 . Computing the determinant, we obtain: n ν 1 ∧ · · · ∧ ν n ∧ μ1 ∧ · · · ∧ μm 2 = qj ≤ q1 . j=1

The lemma is proved.

Lemma 4.7. Let ν 1 , . . . , ν n , μ1 , . . . , μm ∈ Zd ∩ B(R) be linearly independent. Let R(ν 1 , . . . , ν n ) and V2 = R(μ1 , . . . , μm ). Then the angle between V1 and V2 V1 = n m is j=1 |ν j |−1 l=1 |μl |−1 ≥ R−n−m . Remark 4.8. It is not difficult to see that the power −n − m in Lemma 4.7 is optimal. Proof. We use the inequality (4.9) and notice that the numerator of the RHS is a square root of an integer number (since all vectors involved are integer) and is non-zero (since the vectors are linearly independent). Therefore, the numerator is m n at least 1. The denominator is, obviously, j=1 |ν j | l=1 |μl |. This finishes the proof. Using the same argument we have used while proving Corollary 4.3, we can prove the following Corollary 4.9. Let ν 1 , . . . , ν n , μ1 , . . . , μm ∈ Zd ∩B(R) be linearly independent. Let V1 = R(Fν 1 , . . . , Fν n ) and V2 = R(Fμ1 , . . . , Fμm ). Then the angle between V1 and V2 is R−n−m . Lemma 4.10. Let ν 1 , . . . , ν n ∈ Zd ∩ B(R) and μ1 , . . . , μm ∈ Zd ∩ B(R) be two sets. We assume that each set consists of linearly independent vectors (but the union of two sets is not necessary linearly independent). Let V1 = R(ν 1 , . . . , ν n ) and V2 = R(μ1 , . . . , μm ). Suppose, dim (V1 ∩ V2 ) = l. Then there are l integer linearly independent vectors θ1 , . . . , θl ∈ Zd ∩ V1 ∩ V2 such that |θj | Rm+n−l+1 . Moreover, the angle between orthogonal complements to (V1 ∩ V2 ) in V1 and V2 is bounded below by CR−α , α = α(n, m, l) = n + m + 2l(m + n − l + 1). Proof. Denote M = M(ν 1 , . . . , ν n , −μ1 . . . , −μm ). The rank of M equals k := m + n − l. Without loss of generality we can assume that the top left k × k minor of this matrix is non-zero (otherwise we just change the order of the vectors −μj or the order of the coordinates xj ). In order to find the basis of the intersection V1 ∩ V2 we have to solve the system of equations Mt = 0 .

n

(4.10) m

Indeed, if t = (t1 , . . . , tn+m ) is a solution of (4.10), then p=1 tp ν p = q=1 tn+q μq ∈ V1 ∩ V2 . Now the simple linear algebra tells us that the basis of solutions of (4.10) is formed by the vectors of the form (s1 , . . . , sk , 1, 0, . . . , 0), (t1 , . . . , tk , 0, 1, . . . , 0), . . . , (τ1 , . . . , τk , 0, . . . , 0, 1). Using Cramer’s rule, we find that each of the numbers sj , tj , τj , etc is a ratio of two determinants, each of them T

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an integer number Rk ; moreover, the denominator is the same for all of the numbers sj , tj , etc. After multiplication by the denominator, we obtain an integer basis of solutions of (4.10) with entries Rk . For any such solution t the following estimate holds: | np=1 tp ν p | Rk+1 . This proves the first statement of lemma. To prove the second statement, we first use Lemma 4.5 to construct integer bases {η1 , . . . , η n−l } and {ξ1 , . . . , ξ m−l } of the orthogonal complements to V1 ∩ V2 in V1 and V2 correspondingly with properties n−l

|η j | Rn+l(m+n−l+1)

j=1

and m−l

|ξ j | Rm+l(m+n−l+1) .

j=1

Now Lemma 4.7 produces the required estimate. This finishes the proof.

Using the same argument we have used while proving Corollary 4.3, we can prove the following Corollary 4.11. Let ν 1 , . . . , ν n , μ1 , . . . , μm ∈ Zd ∩B(R) be two linearly independent families of vectors. Let V1 = R(Fν 1 , . . . , Fν n ) and V2 = R(Fμ1 , . . . , Fμm ). Then the angle between orthogonal complements to V1 ∩V2 in V1 and V2 is R−α(n,m,l) .

5. Reduction to invariant subspaces Let λ = ρ2 be a large real number. In this section, we use Lemma 3.2 to construct ˜ the family of operators H(k) the spectrum of which (or at least the part of the spectrum near λ) is close to the spectrum of H(k). Consider the truncated potential V (x) = Vˆ (m)em (x) , (5.1) m∈B(R)∩Zd

where em (x) :=

1 ei m,x , (2π)d/2

and

m ∈ Zd

Vˆ (m) = [0,2π]d

V (x)e−m (x)

(5.2)

are the Fourier coefficients of V . R is a large parameter the precise value of which will be chosen later; at the moment we just state that R ∼ ργ with γ > 0 being small. Throughout the text, we will prove various statements which will hold under conditions of the type R < ργj . After each statement of this type, we will always assume, without possibly specifically mentioning, that these conditions are always satisfied in what follows; at the end, we will choose γ = min γj .

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Since V is smooth, for each n we have sup |V (x) − V (x)| < Cn R−n .

(5.3)

x∈Rd

This implies that if we denote H (k) := H0 (k) + V with the domain D(k), the following estimate holds for all n: μj H(k) − μj H (k) < Cn R−n . (5.4) Throughout this and the next two sections, we will work with the truncated operators H (k). These sections will be devoted to the construction of mappings f, g with properties specified in the introduction. Let M ∈ N be a fixed number. For each natural j we denote Θj := Zd ∩ B(jR) ,

Θ0 := {0} ,

Θj := Θj \ {0} .

(5.5)

Let V ⊂ Rd be a linear subspace of dimension n and r > 0. We say that V is an integer r-subspace if V = R(θ1 , . . . , θn ) and each θj is an integer vector with length smaller than r. The set of all integer r-subspaces of dimension n will be denoted by V(r, n). We mostly will be dealing with V(6M R, n); for brevity we will denote V(n) := V(6M R, n). If ξ ∈ Rd and V ∈ V(n), we denote ξ V and ξ ⊥ V vectors such that ξ = ξV + ξ⊥ V,

ξV ∈ V ,

Gξ ⊥ V ⊥ V.

(5.6)

If V ∈ V(n), we put Θ(V) := Θ6M ∩V, Θ (V) := Θ(V)\{0}. By p, qn (n = 1, . . . , d) we denote positive constants smaller than 1/3; the precise value of these constants will be specified later; we also denote K = ρp and Ln = ρqn . Let V ∈ V(n). We denote (5.7) Ξ0 (V) := ξ ∈ A, |ξ V | < Ln , Ξ1 (V) := Ξ0 (V) + V ∩ A , (5.8) d Ξ2 (V) := Ξ1 (V) \ ∪m=n+1 ∪W∈V(m): V⊂W Ξ1 (W) , (5.9) Ξ3 (V) := Ξ2 (V) + B(V, K) ,

(5.10)

and finally, Ξ(V) := Ξ3 (V) + ΘM .

(5.11)

These objects (especially Ξ3 (V) and Ξ(V)) play a crucial role in what follows; the pictures of them are shown in Figures 1–4 in the case d = 2 (here, the integer subspaces V are 1-dimensional, so V = R(θ) with θ ∈ Θ ; we have called Ξj (θ) := Ξj (R(θ))). It may seem that the definition of these objects is overcomplicated; for example, one may be tempted to define Ξ3 (V) by Figure 5. This definition is indeed simpler and it would work in the 2-dimensional case; however, if we try to extend this definition to higher dimensions, we would find out that Lemma 5.12 no longer holds. One more remark concerning the definitions of the sets Ξ is that it is very difficult to make a mental picture of them in high dimensions (even when d = 3).

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θ

475

Ξ (θ) 0

Figure 1. The set Ξ0 (θ) in the 2-dimensional case.

θ

Ξ (θ) 1

Figure 2. The set Ξ1 (θ) = Ξ2 (θ) in the 2-dimensional case. A good approach to working with these sets is to do it on a purely formal level, without trying to imagine how they look like. We also put D := ∪dm=1 ∪W∈V(m) Ξ1 (W) (5.12) and B := A \ D .

(5.13)

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L. Parnovski

θ


Ξ (θ) 3

Figure 3. The set Ξ3 (θ) in the 2-dimensional case.

θ

Ξ(θ)

Figure 4. The set Ξ(θ) in the 2-dimensional case; here, Θ = {(0, 0), (±1, 0), (0, ±1)} consists of five elements. We will often call the set D the resonance region and the set B the non-resonance region. Note that the definitions (5.7)–(5.11) make sense for the subspace U0 := {0} ∈ V(0). In particular, we have Ξ0 (U0 ) = Ξ1 (U0 ) = A, Ξ2 (U0 ) = Ξ3 (U0 ) = B, and (5.14) Ξ(U0 ) = B + ΘM .

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θ

Figure 5. Bad definition of the set Ξ3 (θ) in the 2-dimensional case. Let us now formulate several properties of the sets Ξj . In what follows, we always assume that ρ and R are sufficiently large. We also assume that Ln = ρqn with qn+1 ≥ qn + 3p for all n, qd ≤ 1/3, and K = ρp with q1 ≥ 3p > 0. We also put q0 = 0 so that L0 = 1. From now on, we fix the values p and qn satisfying these conditions; say, we put qn = 3np ,

p = (9d)−1 .

(5.15)

Finally, we assume that M > 2 and that ρp > R2β , where β is the maximal possible value the exponent α(n, m, l) from Lemma 4.10 can attain. Lemma 5.1. Ξ0 (Rd ) = ∅. Proof. This statement is obvious since if V = Rd , then for each ξ we have ξ = ξ V ; therefore one cannot have a point ξ ∈ A with |ξ V | < Ld ≤ ρ1/3 . Lemma 5.2. Let V ∈ V(n), 0 ≤ n < d, and ξ ∈ Ξ1 (V). Then |ξ V | < 2Ln . Proof. The condition ξ ∈ Ξ1 (V) means that ξ ∈ A and there exists ξ ∈ A, |ξ V | < Ln such that ξ − ξ ∈ V. These conditions imply ||Fξ V |2 − |Fξ V |2 | = ||Fξ|2 − |Fξ |2 | 1 . Now the statement is obvious.

Corollary 5.3. If ξ ∈ Ξ(V), then |ξ V | Ln . Lemma 5.4. Suppose, V1 ∈ V(n1 ) and V2 ∈ V(n2 ) are two subspaces such that neither of them is contained in the other one. Let ξ j ∈ Ξ2 (Vj ). Then |ξ 1 −ξ2 | > L1 .

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Proof. The conditions of lemma imply |(ξ j )Vj | Lnj , j = 1, 2. Let W = V1 + V2 , U = V1 ∩ V2 , dim U = l. Then W is an integer 6M R-subspace, say W ∈ V(m). Also, conditions of lemma imply that W = Vj , so m > nj . Suppose, the statement of lemma does not hold, i.e., |ξ 1 − ξ 2 | ≤ L1 . Then |(ξ 1 − ξ 2 )V2 | L1 and thus |(ξ 1 )V2 | Ln2 . By Corollary 4.11, the angle between FV1 and FV2 is at least CR−α(n1 ,n2 ,l) . Since the projections of ξ 1 onto V1 and V2 are smaller than Ln1 and 2Ln2 respectively, it is a simple geometry to deduce that |(ξ 1 )W | (Ln1 + Ln2 )Rα(n1 ,n2 ,l) . Due to the conditions stated before Lemma 5.1, this implies |(ξ 1 )W | < Lm . Therefore, ξ 1 ∈ Ξ1 (W). Now definition (5.9) implies that ξ1 ∈ Ξ2 (V1 ), which contradicts our assumptions. Thus, |ξ 1 − ξ 2 | > L1 . Corollary 5.5. Suppose, V1 ∈ V(n1 ) and V2 ∈ V(n2 ) are two subspaces such that neither of them is contained in the other one. Let ξ j ∈ Ξ(Vj ). Then |ξ1 −ξ2 | L1 . 2 Lemma 5.6. Let V ∈ V(n) and ξ ∈ Ξ3 (V). Then ||Fξ|2 −ρ2 | KLn and ||Fξ ⊥ V| − 2 2 ρ | Ln .

Proof. The assumption of lemma imply that there exists η ∈ Ξ2 (V) such that ξ − η ∈ V and |ξ − η| < K. Lemma 5.2 implies |ξ V | Ln , and thus ||Fξ|2 − |Fη|2 | = ||Fξ V |2 − |FηV |2 | KLn . The first statement now follows from the fact that η ∈ A. Now we compute: 2 ⊥ 2 2 2 2 ρ2 − Fξ⊥ V | = |Fξ| + O(KLn ) − |Fξ V | = |Fξ V | + O(KLn ) = O Ln by Corollary 5.3.

Factorizing the LHS’s of the estimates from this lemma, we immediately obtain the following Corollary 5.7. Let V ∈ V(n) and ξ ∈ Ξ3 (V). Then ||Fξ| − ρ| ρp+qn −1 and 2qn −1 . ||Fξ ⊥ V | − ρ| ρ Lemma 5.8. Let V ∈ V(n) and ξ ∈ Ξ3 (V). Suppose, for some η ∈ A we have ξ − η ∈ V. Then η ∈ Ξ2 (V). Proof. Definition (5.8) implies that η ∈ Ξ1 (V). Therefore, in order to prove our lemma, we need to show that for any W ∈ V(m) (m > n), V ⊂ W, we have η ∈ Ξ1 (W). Suppose, this is not the case and η ∈ Ξ1 (W). Then the fact that ξ ∈ Ξ3 (V) means that there exists a vector ξ˜ ∈ Ξ2 (V) with ξ − ξ˜ ∈ V. But then η − ξ˜ ∈ V ⊂ W. Therefore, ξ˜ ∈ Ξ1 (W). This contradicts the assumption ξ˜ ∈ Ξ2 (V). The lemma is proved. Lemma 5.9. Let V ∈ V(n) and ξ ∈ Ξ3 (V). Suppose, for some η ∈ V we have α := ξ + η ∈ Ξ3 (V). Then ||Fα|2 − ρ2 | K 2 . ˜ be the point which satisfies the following conditions: α ˜ − α ∈ V, Proof. Let α ˜ ∈ A, and the vector αV is a non-negative multiple of α ˜ V (a simple geometrical α argument shows that such a point always exists). Then Lemma 5.8 implies that

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˜ ∈ Ξ2 (V). Therefore, since α ∈ Ξ3 (V), we have ||Fα ˜ V | − |FαV || = |Fα ˜V − α FαV | K. Moreover, ˜ 2 | = ||FαV |2 − |Fα ˜ V |2 | ||Fα|2 − |Fα|

2 ˜ V || ||FαV | + |Fα ˜ V || ≥ |FαV | − |Fα ˜ V| K 2 . = ||FαV | − |Fα

˜ 2 − ρ2 | 1. This finishes the proof, since ||Fα|

Lemma 5.10. Let V ∈ V(n) and ξ ∈ Ξ3 (V). Suppose, θ ∈ Θ6M , θ ∈ V. Denote η := ξ + θ. Then ||Fη|2 − ρ2 | K 2 Ln . Proof. Let W be the linear span of V and θ, and let U := R(θ) be the onedimensional subspace. Assume first that |ξ U | ≤ K 2 Ln . Then, since |ξ V | ≤ Ln , the geometrical argument similar to the one used in the proof of Lemma 5.4 implies that |ξ W | < Ln+1 /2 (recall that the assumption we have made on the exponents p and qn imply ˜ < K. that Ln+1 ≥ K 3 Ln ). Since ξ ∈ Ξ3 (V), there exists a vector ξ˜ ∈ Ξ2 (V), |ξ− ξ| Therefore, |ξ˜W | ≤ |ξW | + |(ξ˜ − ξ)W | < Ln+1 , which implies ξ˜ ∈ Ξ1 (W). This contradicts the condition ξ˜ ∈ Ξ2 (V). Therefore, we must have |ξU | > K 2 Ln . This implies ||Fη|2 − |Fξ|2 | = ||F(ξ U + θ)|2 − |Fξ U |2 | K 2 Ln . Now it remains to notice that Lemma 5.6 implies that ||Fξ|2 − ρ2 | KLn . This finishes the proof. Corollary 5.11. Let V ∈ V(n) and ξ ∈ Ξ3 (V). Suppose, θ ∈ Θ6M and η = ξ + θ ∈ Ξ3 (V). Then ||Fη|2 − ρ2 | K 2 . Proof. If θ ∈ V, then the statement follows from Lemma 5.9, and if θ ∈ V, the statement follows from Lemma 5.10. Lemma 5.12. For each two different integer subspaces Vj ∈ V(nj ), j = 1, 2, 0 ≤ nj < d we have (Ξ(V1 ) + Θ1 ) ∩ (Ξ(V2 ) + Θ1 ) = ∅. Proof. Suppose, ξ ∈ (Ξ(V1 ) + Θ1 ) ∩ (Ξ(V2 ) + Θ1 ). Then Corollary 5.5 implies that one of the subspaces Vj is inside the other, say V1 ⊂ V2 . Moreover, there exist two points, ξ1 ∈ Ξ3 (V1 ) and ξ2 ∈ Ξ3 (V2 ) such that θj := ξj − ξ ∈ ΘM+1 . Then θ := ξ 1 − ξ 2 = θ1 − θ2 ∈ Θ3M . There are two possibilities: either θ ∈ V2 , or θ ∈ V2 . Assume first that θ ∈ V2 . Since ξ j ∈ Ξ3 (Vj ), there exist points ξ˜j ∈ Ξ2 (Vj ) such that ξ˜j − ξ j ∈ Vj , |ξ˜j − ξ j | < K. But then ξ˜1 − ξ˜2 ∈ V2 . Since ξ˜2 ∈ Ξ2 (V2 ) ⊂ Ξ1 (V2 ), according to definition (5.8) this means that ξ˜1 ∈ Ξ1 (V2 ). Now definition (5.9) implies ξ˜1 ∈ Ξ2 (V1 ) which contradicts our assumption. Assume now θ ∈ V2 . Then Lemma 5.10 implies ||Fξ 1 |2 − ρ2 | = ||F(ξ 2 + θ)|2 − ρ2 | K 2 Ln2 . However, this contradicts the inequality ||Fξ 1 |2 − ρ2 | KLn1 which was established in Lemma 5.6.

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Corollary 5.13. Each point ξ ∈ A belongs to precisely one of the sets Ξ(V). Proof. Indeed, definitions (5.7)–(5.14) imply that each point ξ ∈ A belongs to at least one of the sets Ξ(V). The rest follows from Lemma 5.12. Let us introduce more notation. Let C ⊂ Rd be a measurable set. We denote by P (C) the orthogonal projection in H = L2 ([0, 2π]d ) onto the subspace spanned by the exponentials eξ (x), ξ ∈ C, {ξ} = k. (k)

Lemma 5.14. For arbitrary set C ⊂ Rd and arbitrary k we have: V P(k) (C) = P(k) (C + Θ1 )V P(k) (C) .

(5.16)

Proof. This follows from the obvious observation that if ξ = m + k ∈ C and |n| ≤ R, then ξ + n ∈ C + Θ1 . We are going to apply Lemma 3.2 and now we will specify what are the projections Pjl . The construction will be the same for all values of quasi-momenta, so often we will skip kfrom the superscripts. For each V ∈ V(n), n = 0, 1, . . . , d − 1 we put P (V) := P(k) Ξ(V) . We also define Pj (V) := P(k) (Ξ3 (V)+Θj )\(Ξ3 (V)+Θj−1 ) , j = 1, . . . , M , P0 (V) = P(k) (Ξ3 (V). We also denote Q := I − V P (V) (the sum is over all integer 6M R-subspaces of dimension n = 0, 1, . . . , d − 1). Now we apply Lemma 3.2 with the set of projections being {P (V)}, J := [λ− 20v, λ+ 20v], and H0 = H0 (k). Let us check that all the conditions of Lemma 3.2 are satisfied assuming, as before, that all the conditions before Lemma 5.1 are fulfilled. Indeed, Lemmas 5.12 and 5.14 imply that P (V1 )P (V2 ) = 0 and P (V1 )V P (V2 ) = 0 for different subsets V1 and V2 (in particular, Q is also a projection). Properties M P (V) = j=0 Pj (V), Pj (V)V Pl (V) = 0 for |j − l| > 1 and Pj (V)V Q = 0 for j < M follow from the construction of the projections Pj (V) and Lemma 5.14. Since A ⊂ ∪V Ξ3 (V), the distance between the spectrum of QH0 Q and J is greater than 6v. Corollary 5.11 implies that the distances between the spectra of Pj (V)H0 Pj (V), j = 1, . . . , M and J are K 2 . All these remarks imply that we can apply Lemma 3.2 (or rather Corollary 3.3) and, instead of studying eigen ˜ := V P (V)H (k)P (V); the values inside J of H (k), study eigenvalues of H(k) distance between any eigenvalue of H (k) lying inside J and the corresponding ˜ eigenvalue of H(k) is ρ−4Mp . To be more precise, we do the following. Assume ξ = n + k ∈ A. Then ξ ∈ Ξ(V) for some uniquely defined V ∈ V(n). In the following sections, we will define a mapping g˜ : ξ → μτ (ξ) (P (V)H (k)P (V)), where τ = τ (ξ) is a function with values in N. The mapping g˜ will be an injection and any eigenvalue of P (V)H (k)P (V) inside J will have a pre-image under g˜. Then, g˜(ξ) is also an eigenvalue of V P (V)H (k)P (V) + QH (k)Q, say P (V)H (k)P (V) + QH (k)Q . g˜(ξ) = μτ1 (ξ) V

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Then Lemma 3.2 implies that |˜ g(ξ) − μτ1 (ξ) (H (k))| ρ−4Mp . We then define g(ξ) − f (ξ)| ρ−4Mp . In order to construct the f (ξ) := μτ1 (ξ) (H(k)) so that |˜ mapping g, we compute g˜ (or at least obtain an asymptotic formula for it) and then, roughly speaking, throw away terms which are sufficiently small for our purposes. In the next two sections, we discuss how to obtain an asymptotic formula for g˜ when ρ is large. We will consider separately the case ξ ∈ Ξ2 (U0 ) = B (recall that U0 = {0} ∈ V(0) and we have called B the non-resonance region) and the case of ξ lying inside the resonance region D. We start by looking at the case ξ ∈ B.

6. Computation of the eigenvalues outside resonance layers First of all, we notice that Lemma 5.10 implies that the operator P (U0 )H (k)P (U0 ) splits into the direct sum of operators. Namely, P(k) (ξ + ΘM )H (k)P(k) (ξ + ΘM ) , (6.1) P (U0 )H (k)P (U0 ) = the sum being over all ξ ∈ B with {ξ} = k. We denote by g˜(ξ) the eigenvalue of P(k) (ξ + ΘM )H (k)P(k) (ξ + ΘM ) which lies within the distance v from |Fξ|2 (Lemma 5.10 implies that this eigenvalue is unique). Our next task is to compute g˜(ξ). In this section we will prove the following lemma: Lemma 6.1. Let R < ρpd

−1

/2

. Then the following asymptotic formula holds:

g˜(ξ) ∼ |Fξ|2 +

∞

An1 ,...,nr ξ, Gη 1 −n1 · · · ξ, Gηr −nr

(6.2)

r=1 η 1 ,...,η r ∈ΘM n1 +···+nr ≥2

in the sense that for each m ∈ N we have g˜(ξ) − |Fξ|2 −

m

An1 ,...,nr ξ, Gη1 −n1 · · · ξ, Gη r −nr

r=1 η 1 ,...,η r ∈ΘM 2≤n1 +···+nr ≤m

= O ρ−(m+1)p , (6.3)

−1

uniformly over R < ρpd /2 . Here, An1 ,...,np is a polynomial of the Fourier coefficients Vˆ (η j ) and Vˆ (η j − η l ) of the potential. Proof. Let us denote a(η) = |F(ξ + η)|2 .

(6.4)

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The matrix of P (ξ + ΘM )H (k)P (ξ + ΘM ) has the ⎛ a(0) Vˆ (η 1 ) Vˆ (η 2 ) ... ⎜ˆ ˆ ⎜ V (η 1 ) a(η 1 ) V (η 2 − η 1 ) . . . ⎜ ⎜ Vˆ (η ) Vˆ (η − η ) a(η 2 ) ... ⎜ 2 2 1 ⎜ . . . .. ⎜ .. .. .. . ⎜ ⎜ˆ ⎜V (η n ) Vˆ (η n − η 1 ) Vˆ (η n − η 2 ) . . . ⎝ .. .. .. . . . ...

following form:

⎞ ... ⎟ Vˆ (η n − η 1 ) . . .⎟ ⎟ Vˆ (η n − η 2 ) . . .⎟ ⎟ ⎟ .. . . . .⎟ ⎟ ⎟ a(η n ) . . .⎟ ⎠ .. .. . . Vˆ (η n )

(6.5)

The diagonal elements of this matrix equal |F(ξ + η)|2 (with η running over ΘM ) and off-diagonal elements are Fourier coefficients of the potential (and are thus bounded). Let L be the number of columns of this matrix; obviously, L Rd . Let us compute the characteristic polynomial p(μ) of (6.5). The definition of the determinant implies ⎞ ⎛ L a(η) − μ ⎠ + Jm (μ) , (6.6) p(μ) = ⎝ η∈ΘM

m=2

where Jm consists of products of exactly (L − m) diagonal terms of (6.5) and m off-diagonal terms. Put Jm = Jm + Jm , where Jm (resp. Jm ) consists of all terms, not containing (resp. containing) (a(0) − μ). Then we can re-write (6.6) as ⎛ ⎞ ! (6.7) p(μ) = ⎝ a(η) − μ ⎠ a(0) − μ + I(μ) , where I(μ) :=

L−1

η∈ΘM

m=1 Im (μ)

+

L

˜

m=2 Im (μ) with Jm+1

Im :=

η∈ΘM

and I˜m :=

η∈ΘM

a(η) − μ

Jm . a(η) − μ

We can easily compute the first several terms: |Vˆ (η)|2 , I1 (μ) := − a(η) − μ η∈ΘM

I2 (μ) := −

η,η ∈ΘM ,η=η

I˜2 (μ) := − a(0) − μ

(6.8)

2 Vˆ (η)Vˆ (η − η )Vˆ (η ) , a(η) − μ a(η ) − μ

η,η ∈ΘM ,η=η

|Vˆ (η − η )|2 . a(η) − μ a(η ) − μ

(6.9)

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Overall, Im is the sum of O(Rdn ) terms of the form Wm (η 1 , . . . , η n ) , a(η 1 ) − μ . . . a(η n ) − μ

(6.10)

and I˜m is the sum of O(Rdn ) terms of the form ˜ m (η 1 , . . . , η n ) W . a(0) − μ a(η 1 ) − μ . . . a(η n ) − μ

(6.11)

˜ m (η 1 , . . . , η n ) are some polynomials of Vˆ (η j ) and Here, Wm (η 1 , . . . , η n ) and W ˆ V (η j − η l ). On the interval [a(0)−v, a(0)+v] the equation p(μ) = 0 has a unique solution, which we have called g˜(ξ); this is the solution of the equation a(0) − μ + I(μ) = 0. After denoting F (μ) := a(0)+I(μ), this equation becomes equivalent to μ = F (μ). Throughout the rest of the section we will assume that μ ∈ [a(0) − v, a(0) + v]. Then, since ξ ∈ B, Lemma 5.10 guarantees that |a(η) − a(0)| ρ2p for η ∈ ΘM . This implies In (μ) = O(Rdn ρ−2np ) = O(ρ−np ); similarly, Iñ (μ) = O(ρ−np ). d d ˜ In (μ) are O(ρ−np ) as well. In (μ) and dμ Computing the derivatives, we see that dμ Slightly more careful analysis shows that in fact I1 = O(Rd ρ−4p ) = O(ρ−2p ) and d d −4p ) = O(ρ−2p ). Indeed, we have: dμ I1 = O(R ρ I1 (μ) = −

η∈ΘM

|Vˆ (η)|2 a(η) − μ

" # 1 1 1 ˆ 2 + =− |V (η)| 2 a(η) − μ a(−η) − μ η∈ΘM a(η) + a(−η) − 2μ 1 ˆ 2 |V (η)| =− 2 a(η) − μ a(−η) − μ η∈ΘM a(0) + |Fη|2 − μ 2 ˆ , |V (η)| =− a(η) − μ a(−η) − μ η∈Θ M

and it remains to notice that

η∈ΘM

(6.12)

|Vˆ (η)|2 is bounded by the square of the −1

L2 -norm of V . These estimates show that when R < ρpd /2 , we have I(μ) = d d ˜ F (μ) = dμ I(μ) = O(ρ−2p ). We will find λ(ξ) using a sequence of O(ρ−2p ) and dμ approximations. We define a sequence μk in the following way: μ0 = a(0), μk+1 = g (ξ)| = |F (μk )−F (˜ g (ξ))| = |μk −˜ g (ξ)|O(ρ−2p ) F (μk ) = a(0)+I(μk ). Since |μk+1 −˜ and |μ0 − g˜(ξ)| = O(1), we have: (6.13) |μk − g˜(ξ)| = O ρ−2kp . Therefore, we will prove the lemma if we show that for all k ≥ 1 the approximation μk enjoys the same asymptotic behaviour (6.2), at least up to an error O(ρ−kp ).

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This computation is straightforward. For example, we have μ1 = |Fξ|2 + I1 |Fξ|2 + I2 |Fξ|2 + I˜2 |Fξ|2 + O ρ−3p , and, using (6.12), we obtain: |V (η)|2 I1 a(0) = − η∈ΘM

|Fη|2 a(η) − a(0) a(−η) − a(0)

|Fη|2 = |V (η)| 2 Fξ, Fη + |Fη|2 2 Fξ, Fη − |Fη|2 η∈ΘM " # |Fη|2 2 = |V (η)| 4 Fξ, Fη2 − |Fη|4 η∈ΘM " −1 # 4 |Fη|2 Fξ, Fη−2 = |V (η)|2 1 − 4−1 |Fη|4 Fξ, Fη−2

2

(6.14)

η∈ΘM

=

|V (η)|2

η∈ΘM

=

∞

4−n |Fη|4n−2 Fξ, Fη−2n

n=1

|V (η)|2

η∈ΘM

∞

4−n |Fη|4n−2 ξ, Gη−2n .

n=1

Computations of I2 (a(0)) are similar (and, obviously, I˜2 (a(0)) = 0), only now the result will have terms which involve inner products of ξ with two different η’s. Thus, An1 ,n2 ξ, Gη 1 −n1 ξ, Gη2 −n2 + O(ρ−3p ) , (6.15) μ1 = |Fξ|2 + η 1 ,η 2 ∈ΘM n1 ,n2

the sum being over all n1 , n2 with n1 + n2 ≥ 2 (in fact, we can take the sum over n1 + n2 = 2, since other terms will be O(ρ−3p )). Using induction, it is easy to prove now that μk = |Fξ|2 +

k+1

r=1

η 1 ,...,η r ∈ΘM

n1 ,...,nr

An1 ,...,nr ξ, Gη1 −n1 · · · ξ, Gη r −nr

(6.16)

+ O ρ−(k+2)p , r the sum being over 2 ≤ j=1 nj ≤ k + 1; An1 ,...,np is a polynomial of {Vˆ (η j )} and {Vˆ (η j − ηl )}. Indeed, if μk satisfies (6.16), then a calculation similar to (6.14) 1 can be decomposed as a sum of shows that for each η ∈ ΘM the fraction a(η)−μ k products of negative powers of ξ, Gηj . Therefore, all functions In (μk ) (and, thus,

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I(μk )) admit similar decomposition. This implies that the next approximation μk+1 = |Fξ|2 + I(μk ) also satisfies (6.16). Estimate (6.13) now shows that the asymptotic formula (6.3) holds. We now define g(ξ) as the finite part of the RHS of the expansion (6.2), namely g(ξ) = |Fξ|2 +

4M

An1 ,...,nr ξ, Gη 1 −n1 . . . ξ, Gηr −nr . (6.17)

r=1 η 1 ,...,η r ∈ΘM n1 +···+nr ≥2

Lemma 6.2. We have: |g(ξ) − g˜(ξ)| ρ−4Mp . Proof. This follows from Lemma 6.1.

(6.18)

7. Computation of the eigenvalues inside resonance layers Now let us fix V ∈ V(n), 1 ≤ n ≤ d − 1, and try to study the eigenvalues of P (V)H (k)P (V). Let ξ = n + k ∈ Ξ3 (V). We denote ! j = 0, 1, 2, 3 , (7.1) Υj = Υj (ξ) := ξ + (V ∩ Zd ) ∩ Ξj (V) , Υ = Υ(ξ) := Υ3 (ξ) + ΘM , P (ξ) := P(k) Υ(ξ) ,

(7.2) (7.3)

H (ξ) := P (ξ)H (k)P (ξ) ,

(7.4)

H0 (ξ) := P (ξ)H0 (k)P (ξ) ,

(7.5)

and Vξ := P (ξ)V P (ξ) .

(7.6)

Out of all sets denoted by the letter Υ, we will mostly use Υ3 (ξ) and Υ(ξ); see Figures 6 and 7 for an illustration of these sets when d = 2. Let us establish some simple properties of these sets. Lemma 7.1. Suppose, η ∈ Υ(ξ) \ Υ3 (ξ). Then ||Fη|2 − λ| K 2 (in particular, η ∈ A). Proof. The assumptions of the lemma imply that η = ξ˜ + θ with ξ˜ ∈ Υ3 (ξ) and θ ∈ ΘM . If θ ∈ V, the statement follows from Lemma 5.10. Assume θ ∈ V. Then η ∈ Ξ3 (V) (otherwise we had η ∈ Υ3 (ξ)). Now the statement follows from Lemma 5.9. Lemma 7.2. We have Υ3 (ξ) ⊂ Ξ3 (V) and Υ(ξ) ⊂ Ξ(V). If η ∈ Υ(ξ), then η −ξ ∈ Zd . If for some ξ 1 , ξ2 ∈ Ξ3 (V) we have Υ(ξ 1 ) ∩ Υ(ξ 2 ) = ∅, then Υ(ξ 1 ) = Υ(ξ 2 ).

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θ

ξ

Figure 6. The sets Ξ3 (θ) and Υ3 (ξ).

ϒ (ξ)

θ

ξ

Figure 7. The sets Ξ3 (θ) and Υ(ξ); here, Θ = {(0, 0), (±1, 0), (0, ±1)} consists of five elements. Proof. The first three statements follow immediately from the definitions. Assume Υ(ξ 1 ) ∩ Υ(ξ 2 ) = ∅, say η ∈ Υ(ξ 1 ) ∩ Υ(ξ 2 ). Then η = ξ˜j + θ j , j = 1, 2, with ξ˜j ∈ Ξ3 (V) and θj ∈ ΘM . Then ξ˜1 = ξ˜2 + (θ 2 − θ 1 ). Since θ2 − θ1 ∈ Θ2M ,

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Lemmas 5.10 and 5.6 imply that θ 2 − θ1 ∈ V. Therefore, ξ 2 − ξ1 ∈ (V ∩ Zd ), so Υ(ξ 1 ) = Υ(ξ 2 ). Lemma 7.2 implies that the operator P (V)H (k)P (V) splits into the direct sum: P (V)H (k)P (V) = H (ξ) , (7.7) the sum being over all classes of equivalence of ξ ∈ Ξ3 (V) with {ξ} = k. Two vectors ξ 1 and ξ 2 are equivalent if Υ(ξ 1 ) = Υ(ξ 2 ). Remark 7.3. The programme formulated at the end of Section 5 requires to put into correspondence to each point ξ ∈ Ξ3 (V) a number g˜(ξ) which is an eigenvalue of P (V)H (k)P (V). It is natural to choose g˜(ξ) to be an eigenvalue of H (ξ), say g˜(ξ) = μj (H (ξ)), where j = j(ξ) is some natural number, and the mapping j : Υ(ξ) → N is (at least) an injection. There are certain technical problems with defining the function j. The first problem is that the sets Υ(ξ 1 ) and Υ(ξ 2 ) can have different number of elements for different ξ 1 , ξ 2 ∈ Ξ3 (V) (as Figure 8 illustrates), and the mapping j obviously has to take care of this fact. The second problem is that the mapping j cannot possibly be continuous (otherwise, since it takes only natural values, it would be constant and therefore not an injection), so g˜ as well cannot be continuous. Finally, we want g˜(ξ) not to change too much when we change ξ a little. We cannot exactly achieve this (since, as we mentioned above, g˜ must be discontinuous), but we can achieve some weaker version of this (see Lemma 7.11 for the precise statement).

ϒ3(ξ1) ϒ3(ξ2)

θ

ξ1

ξ

2

Figure 8. The sets Υ3 (ξ 1 ) (dots) and Υ3 (ξ 2 ) (crosses) have different number of elements.

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L. Parnovski


Hence, we will study operators H (k) for each ξ = m + k ∈ Ξ(V) with ⊥ {ξ} = k. Recall that we have denoted by ξ V and ξ ⊥ V vectors such that ξ = ξ V +ξ V , ξ V ∈ V, Gξ⊥ V ⊥ V. Let us also define r = r(ξ) := |Fξ ⊥ V| ,

ξV := ξ ⊥ V /r ,

(7.8)

|FξV |

so that nates on

= 1. We can think of the triple (r, ξ V , ξ V ) as the cylindrical coordiΞ(V). Corollary 5.3 implies that |ξV | ρqn ; Corollary 5.7 implies |r − ρ| = O ρ2qn −1 = O ρ−1/3 ,

(7.9)

since qn ≤ 1/3; in particular, we have r > 0. The current objective is to express the asymptotic behaviour of eigenvalues of H (ξ) inside J in terms of r. In order to do this, we want to compare the eigenvalues of H (ξ 1 ) and H (ξ 2 ) when ξ 1 , ξ 2 ∈ Ξ(V) are two points which are close to each other. Since the operators H (ξ 1 ) and H (ξ 2 ) act in different Hilbert spaces P (ξ j )H, we first need to map these Hilbert spaces onto each other. A natural idea is to employ the mapping Fξ1 ,ξ2 : P (ξ 1 )H → P (ξ 2 )H defined in the following way: Fξ1 ,ξ2 (eη ) = eη+ξ2 −ξ1 .

(7.10)

This mapping is ‘almost’ an isometry, except for the fact that it is not well-defined, i.e., it could happen for example that η ∈ Υ(ξ 1 ), but (η + ξ 2 − ξ 1 ) ∈ Υ(ξ 2 ) (Figure 8 illustrates how this can happen). In order to avoid this, we will extend the sets Υ(ξ). We do this in the following way. First, for ξ1 , ξ2 ∈ Ξ2 (V) we define (7.11) Υ(ξ 1 ; ξ2 ) := Υ(ξ 1 ) ∪ Υ(ξ 2 ) − ξ 2 + ξ 1 and, similarly,

Υ3 (ξ 1 ; ξ 2 ) := Υ3 (ξ 1 ) ∪ Υ3 (ξ 2 ) − ξ 2 + ξ 1 (the set Υ3 (ξ 2 ; ξ1 ) is shown on Figure 9). We also define P (ξ 1 ; ξ2 ) := P(k1 ) Υ(ξ 1 ; ξ 2 ) ,

(7.12)

where k1 := {ξ 1 }; H (ξ 1 ; ξ 2 ) := P (ξ 1 ; ξ 2 )H (ξ 1 )P (ξ 1 ; ξ2 ) ,

(7.13)

H0 (ξ 1 ; ξ 2 ) := P (ξ 1 ; ξ 2 )H0 (ξ 1 )P (ξ 1 ; ξ 2 ) .

(7.14)

and

Suppose also that ξ ∈ Ξ2 (V) and let U ⊂ Ξ2 (V) be a set containing ξ of diameter ρ−1 . Denote Υ(ξ; U ) := ∪η∈U Υ(ξ; η) ,

(7.15)

Υ3 (ξ; U ) := ∪η∈U Υ3 (ξ; η) , P (ξ; U ) = P(k) Υ(ξ; U ) , H (ξ; U ) := P (ξ; U )H (ξ)P (ξ; U ) ,

(7.16)

H0 (ξ; U ) := P (ξ; U )H0 (ξ)P (ξ; U ) . Notice that Υ(ξ 1 ; ξ2 ) = Υ(ξ 1 ; {ξ1 , ξ2 }).

(7.17)

and

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ϒ (ξ ) 3

1

ϒ (ξ ; ξ ) 3

θ

ξ1

2

1

ξ2

Figure 9. Now the sets Υ3 (ξ 1 ; ξ 2 ) = Υ3 (ξ 1 ) (dots) and Υ3 (ξ 2 ; ξ1 ) (stars) have the same number of elements. Now if we define the mapping Fξ1 ,ξ2 : P (ξ 1 ; ξ2 )H → P (ξ 2 ; ξ 1 )H by formula (7.10), this mapping will be a bijection and an isometry, since obviously Υ(ξ 2 ; ξ 1 ) = Υ(ξ 1 ; ξ 2 ) + ξ 2 − ξ 1 . Similarly, if U is any set containing ξ 1 and ξ 2 , then the mapping Fξ1 ;U : P (ξ 1 ; U )H → P (ξ 2 ; U )H defined by (7.10) will be a bijection and an isometry, since Υ(ξ 2 ; U ) = Υ(ξ 1 ; U ) + ξ2 − ξ 1 . Note also that if η ∈ Υ(ξ; U ), then η − ξ ∈ Zd . The problem, of course, is that in general the spectra of H (ξ 1 ) and H (ξ 1 ; ξ2 ) (or H (ξ 1 ; U )) can be quite far from each other. However, we can give sufficient conditions which guarantee that the spectra of H (ξ 1 ) and H (ξ 1 ; ξ 2 ) (or rather the parts of the spectra lying inside J) are within a small distance (of order O(ρ−4Mp )) from each other. The following statement is a straightforward corollary of Lemma 3.2. Lemma 7.4. a) Let ξ 1 , ξ 2 ∈ Ξ2 (V) ⊂ A satisfy |ξ 1 − ξ 2 | < ρ−1 . Then there exists a bijection G = Gξ1 ,ξ2 defined on a subset of the set of all eigenvalues of H (ξ 1 ) and mapping them to a subset of the set of all eigenvalues of H (ξ 1 ; ξ2 ) (eigenvalues in both sets are counted including multiplicities) satisfying the following properties: (i) all eigenvalues of H (ξ 1 ) (resp. H (ξ 1 ; ξ 2 )) inside J are in the domain (resp. range) of Gξ1 ,ξ2 ;

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(ii) for any eigenvalue μj (H (ξ 1 )) ∈ J (and thus in the domain of Gξ1 ,ξ2 ) we have: ! (7.18) μj H (ξ 1 ) − G μj H (ξ 1 ) ρ−4Mp . b) Suppose ξ ∈ U ⊂ Ξ2 (V) and the diameter of U is ρ−1 . Then there exists a bijection G = Gξ,U defined on a subset of the set of all eigenvalues of H (ξ) and mapping them to a subset of the set of all eigenvalues of H (ξ; U ) (eigenvalues in both sets are counted including multiplicities) satisfying the following properties: (i) all eigenvalues of H (ξ) (resp. H (ξ; U )) inside J are in the domain (resp. range) of Gξ,U ; (ii) for any eigenvalue μj (H (ξ)) ∈ J (and thus in the domain of Gξ,U ) we have: ! (7.19) μj H (ξ) − G μj H (ξ) ρ−4Mp . Proof. Let us prove part a) of this lemma; part b) is proved analogously. Suppose, ξ ∈ (Υ(ξ 1 ; ξ2 ) \ Υ(ξ 1 )). Let us prove that then ||Fξ|2 − ρ2 K 2 .

(7.20)

Then definiIndeed, we obviously have ξ˜ := ξ + ξ 2 − ξ 1 ∈ Υ(ξ 2 ) ⊂ Ξ(V). ˜ +a+θ, η ∈ Ξ2 (V)∩(ξ 2 +V) , tions (5.10), (5.11), (7.1) and (7.2) imply that ξ˜ = η a ∈ B(V, K), θ ∈ ΘM . If θ ∈ V, (7.20) follows from Lemma 5.10 and the inequality ˜ 2| 1 , ||Fξ|2 − |Fξ| which in turn follows from the conditions of lemma. Suppose θ ∈ V. Then ξ −ξ 1 = a + θ + (η − ξ2 ) ∈ V and, since ξ ∈ Υ(ξ 1 ), we have ξ ∈ Υ3 (ξ 1 ), which in turn implies ξ ∈ Ξ3 (V). Now (7.20) follows from Lemma 5.9. Inequality (7.20) shows that as before we can apply Lemma 3.2, or rather its Corollary 3.3. This time, we apply this lemma with H = H (ξ 1 ; ξ 2 ), H0 = 0 (k1 ) H0 (ξ 1 ; ξ 2 ), n = 0, P 0 = (Υ3 (ξ 1 )), P (ξ1 ), Q = P (ξ 1 ; ξ 2 ) − P (ξ 1), P0 = P 0 (k1 ) (Υ3 (ξ 1 )+Θj )\(Υ3 (ξ 1 )+Θj−1 ) P (ξ 1 ; ξ 2 ), j = 1, . . . , M . Pj (V) := P (ξ 1 ; ξ 2 )P The fulfillment of all conditions of Lemma 3.2 follows from (7.20) and Lemma 5.14. Now the statement of lemma immediately follows from Corollary 3.3. Remark 7.5. Part (iii) of Corollary 3.3 shows that the bijection Gξ1 ,ξ2 is given by the following formula. Let l = l(ξ1 , ξ2 ) be the number of points η ∈ Υ(ξ 1 ; ξ2 ) \ Υ(ξ 1 )

(7.21)

with |Fη|2 < λ (notice that if η satisfies (7.21), then |Fη| ∈ J). Then Gξ1 ,ξ2 (μj (H (ξ 1 ))) = μj+l (H (ξ 1 ; ξ 2 )). Similarly, if l = l(ξ, U ) is the number of points η ∈ Υ(ξ; U ) \ Υ(ξ) with |Fη|2 < λ, then Gξ,U (μj (H (ξ))) = μj+l (H (ξ; U )).

(7.22)

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The next lemma shows that the eigenvalues of H (ξ; U ) do not change much if we increase U ; this lemma is an immediate corollary of Lemma 7.4. Lemma 7.6. Let ξ ∈ U1 ⊂ U2 ⊂ Ξ2 (V) and let the diameter of U2 be ρ−1 . Denote by l = l(ξ; U1 , U2 ) the number of points η ∈ Υ(ξ; U2 ) \ Υ(ξ; U1 )

(7.23)

with |Fη|2 < λ. Then: a) for any eigenvalue μj (H (ξ; U1 )) ∈ J we have: μj H (ξ; U1 ) − μj+l H (ξ; U2 ) ρ−4Mp ;

(7.24)

b) the number l(ξ; U1 , U2 ) does not depend on ξ, i.e., if ξ 1 , ξ 2 ∈ U1 , then l(ξ1 ; U1 , U2 ) = l(ξ 2 ; U1 , U2 ). Proof. Part a) of lemma follows from Lemma 7.4 and Remark 7.5, since l(ξ; U1 , U2 ) = l2 − l1 , lj := l(ξ; Uj ), and we have μj H (ξ) − μj+lj H (ξ; Uj ) ρ−4Mp , j = 1, 2 . (7.25) Let us prove part b). Suppose, η 1 ∈ Υ(ξ 1 ; U2 ) \ Υ(ξ 1 ; U1 ). Then, in the same way as we have proved (7.20), we can show that ||Fη 1 |2 − λ| K 2 . Denote η 2 := η 1 + (ξ2 − ξ1 ). The definitions of the sets Υ imply that η 2 ∈ Υ(ξ 2 ; U2 ) \ Υ(ξ 2 ; U1 ). Since |ξ2 − ξ 1 | ρ−1 , we have ||Fη 2 |2 − |Fη 1 |2 | 1. Therefore, the inequality |Fη 1 |2 < λ is satisfied if and only if the inequality |Fη 2 |2 < λ is satisfied. This proves that l(ξ 1 ; U1 , U2 ) = l(ξ2 ; U1 , U2 ). As we have already mentioned, if ξ, η ∈ U , we have Υ(η; U ) = Υ(ξ; U ) + (η − ξ), which implies that the mapping Fξ,η : P (ξ; U )H → P (η; U )H defined by (7.10) is an isometry. Thus, by considering the sets Υ(ξ; U ) instead of Υ(ξ) we have overcome the first difficulty mentioned in Remark 7.3. Now we will try to face the other problems mentioned there. Let η 0 = 0, η1 , . . . , η p be the complete system of representatives of ΘM modulo V (we assume of course that ηj ∈ ΘM ). That means that each vector θ ∈ ΘM has a unique representation θ = η j + a, a ∈ V. Denote Ψj = Ψj (ξ) := ξ + η j + (V ∩ Zd ) ∩ Υ(ξ). Then $ Υ(ξ) = Ψj , (7.26) j

and this is a disjoint union (on Figure 7, the set Ψ0 is the middle column of dots, and Ψ1 and Ψ2 are the left and right columns). Let us compute diagonal elements of H (ξ). Let η ∈ Υ(ξ). Then η can be uniquely decomposed as η = ξ + μ + ηj

(7.27)

492

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with μ ∈ V∩Zd . Recall that H (ξ) = H0 (ξ)+Vξ and H0 (ξ)eη = |Fη|2 eη whenever η ∈ Υ(ξ). Since Fξ V ⊥ Fμ and FξV ⊥ Fξ V , we have: |Fη|2 = |F(ξ + η j + μ)|2 = |F(rξ V + ξV + ηj + μ)|2 2 = r2 + 2 FξV , Fη j r + F (ξ + η j )V + μ .

(7.28)

This simple computation implies that H (ξ) = r2 I + rA + B . A(ξ V , ξ V , r)

and B = B(ξ) = Here, A = A(ξ) = operators acting in P (ξ)H in the following way: A=2

p

(7.29) B(ξ V , ξV , r)

are self-adjoint

Fξ V , Fηj P(k) (Ψj ) ;

j=0

in other words,

% & Aeη = 2 Fξ V , Fη j eη = 2 Fξ V , F(η − ξ) eη ,

(7.30)

and

2 (7.31) Beη = F (ξ + ηj )V + μ eη + Vξ eη = |Fη V |2 + Vξ eη for all η ∈ Ψj (ξ) with η j and μ being defined by (7.27). These definitions imply that ker A = P(k) (Ψ0 )H. Notice that A(ξ) R < ρ1/3

(7.32)

and

(7.33) B(ξ) L2n < ρ2/3 due to our assumptions made before Lemma 5.1; see also Corollary 5.3. The dependence of the operator pencil H = r2 I + rA + B on r is two-fold: together with the obvious quadratic dependence, the coefficients A and B depend on r as well. However, as we will show in Lemma 7.7, the second type of dependence is rather weak. Put D(ξ) := r(ξ)A(ξ) + B(ξ) . By {νj (ξ)} we denote the eigenvalues of D(ξ). Then according to (7.29) the eigenvalues of H (ξ) are equal to λj (ξ) = r2 (ξ) + νj (ξ) .

(7.34)

If ξ 1 , ξ 2 ∈ Ξ3 (V), then we can define the operator A(ξ 1 ; ξ 2 ) as the operator defined by (7.30) with the domain P (ξ 1 ; ξ2 )H. Similarly, if U is a set containing ξ of diameter ρ−1 , then we define the operator A(ξ; U ) as the operator defined by (7.30) with the domain P (ξ; U )H. In the same way, we can define B(ξ 1 ; ξ 2 ), B(ξ; U ) (they are defined by means of (7.31)), D(ξ 1 ; ξ2 ) = r(ξ 1 )A(ξ 1 ; ξ2 ) + B(ξ 1 ; ξ 2 ), and D(ξ; U ). We also denote by νj (ξ 1 ; ξ 2 ) the eigenvalues of D(ξ 1 ; ξ 2 ) and by λj (ξ 1 ; ξ 2 ) = r2 (ξ 1 ) + νj (ξ 1 ; ξ 2 ) the eigenvalues of H (ξ 1 ; ξ 2 ); νj (ξ; U ) and λj (ξ; U ) are defined analogously.

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Let us now study how the eigenvalues change under the change of r. Lemma 7.7. Let ξ ∈ Ξ3 (V), r = r(ξ). Let U be a set of diameter ρ−1 containing ξ. Let t be a real number with |t − r| ρ−1 and a = a(t) ∈ Ξ2 (V) be a unique point satisfying (a)V = (ξ)V , (a)V = (ξ)V , and r(a) = t (thus, when we vary t, the path a(t) is a straight interval which goes along the F-perpendicular dropped from the point ξ onto V). Suppose, a ∈ U . Let νj (t) (resp. λj (t)) denote the eigenvalues of D(a(t); U ) (resp. H (a(t); U )). Then dνj (t) = O ρ1/3 (7.35) dt and dλj (t) = 2t + O ρ1/3 . (7.36) dt Proof. Let t1 , t2 be real numbers satisfying |tj − r| ρ−1 and a1 = a(t1 ), a2 = a(t2 ) be the corresponding points inside Ξ3 (V) ∩ U . First of all, we notice that the mapping Fa1 ,a2 defined by (7.10) is an isometry which maps P (a1 ; U )H onto P (a2 ; U )H. Moreover, the definitions of the operators A and B imply that A(a1 ; U ) = Fa2 ,a1 A(a2 ; U )Fa1 ,a2 ; similarly, B(a1 ; U ) = Fa2 ,a1 B(a2 ; U )Fa1 ,a2 . These unitary equivalences show that the eigenvalues νj (t) are in fact the eigenvalues of the linear operator pencil tA + B, with A and B being any of the operators A(a; U ) and B(a; U ) with a satisfying (a)V = (ξ)V and (a)V = (ξ)V ; it does not matter which particular point a we have chosen, since all corresponding operators are unitarily equivalent. For example, we can choose A = A(ξ; U ) and B = B(ξ; U ). Now an elementary perturbation theory shows that dνj = Auj , uj , (7.37) dt where uj is the eigenvector of D corresponding to the eigenvalue νj . The estimate (7.32) shows that dνj /dt = O(ρ1/3 ). This proves (7.35). The estimate (7.36) follows from this and the identity λj (t) = t2 + νj (t). Using similar perturbative argument, we can study how the eigenvalues change when we change the other variables, namely, ξV and ξ V . Lemma 7.8. Let ξ ∈ Ξ3 (V), and let a ∈ Ξ3 (V) be the point satisfying r(a) = r(ξ), |a − ξ| ρ−1 . Suppose, ξ, a ∈ U . Then |λj (a; U ) − λj (ξ; U )| = |νj (a; U ) − νj (ξ; U )| |a − ξ|ρ1/3 .

(7.38)

Proof. Formula (7.34) and the condition r(a) = r(ξ) imply that λj (a; U )−λj (ξ; U ) = νj (a; U ) − νj (ξ; U ). Moreover, definitions (7.30) and (7.31) imply that ||A(ξ; U ) − Fa,ξ A(a; U )Fξ,a || |ξ V − aV |R |a − ξ|ρ−2/3

(7.39)

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and ||B(ξ; U ) − Fξ,a B(a; U )Fa,ξ || |ξ V − aV |(|ξ V | + Ln ) |a − ξ|ρ1/3 .

(7.40)

Indeed, let us check for example (7.39). Suppose, η ∈ Υ (say, η ∈ Ψj ). Then we have: A(ξ; U )η = 2 Fξ V , Fη j eη and Fa,ξ AFξ,a η = 2 FaV , Fη j eη . Since |aV − ξ V | |a − ξ|ρ−1 and |η j | R , we have (7.39). The estimate (7.40) can be proved analogously. Therefore, since r(ξ) ∼ ρ, we have ||D(ξ; U ) − Fξ,a D(a; U )Fa,ξ || |a − ξ|ρ1/3 . Since the spectra of D(a; U ) and Fξ,a D(a; U )Fa,ξ coinside, this implies |νj (ξ; U ) − νj (a; U )| |a − ξ|ρ1/3 , which finishes the proof.

Let us summarize the information about the spectra of H (ξ) we have obtained so far. Recall that A1 is a slightly ‘slimmed down’ version of A; it consists of all points ξ with |Fξ|2 ∈ J. Lemma 7.9. Let ξ1 , ξ2 ∈ U ⊂ Ξ2 (V) ∩ A1 with the diameter of U being ρ−1 . Assume that μj (H (ξ 1 ; U )) ∈ J. Then μj H (ξ 1 ; U ) − μj H (ξ 2 ; U ) ρ|ξ 1 − ξ2 | + ρ−4Mp . If we assume, moreover, that (ξ 1 )V = (ξ 2 )V and (ξ 1 )V = (ξ 2 )V , then μj H (ξ 1 ; U ) − μj H (ξ 2 ; U ) = 2ρ + O(ρ1/3 ) r(ξ 1 ) − r(ξ 2 ) + O(ρ−4Mp ) . Finally, if (ξ 1 )V = (ξ 2 )V , (ξ 1 )V = (ξ 2 )V , and U contains the interval I joining ξ1 and ξ 2 , then μj H (ξ 1 ; U ) − μj H (ξ2 ; U ) = 2ρ + O(ρ1/3 ) r(ξ 1 ) − r(ξ 2 ) . Proof. The last statement follows directly from Lemma 7.7. Assume now that (ξ 1 )V = (ξ 2 )V and (ξ 1 )V = (ξ 2 )V . Denote U1 := U , U2 := U ∪ I, where I is the interval joining ξ 1 and ξ2 , and l = l(ξ1 ; U1 , U2 ) = l(ξ 2 ; U1 , U2 ) (the last equality follows from Lemma 7.6). Then Lemma 7.6 implies that μj H (ξ m ; U1 ) − μj+l H (ξ m ; U2 ) ρ−4Mp , m = 1, 2 . Now the statement follows from Lemma 7.7. If r(ξ 1 ) = r(ξ 2 ), the statement follows in a similar way from Lemmas 7.6 and 7.8. In the general case, we join ξ 1 and ξ2 by a path consisting of intervals falling into either of the two cases above.

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Now we will ‘globalize’ the local mappings constructed so far, in other words, we will define the function j : Υ(ξ) → N mentioned in the Remark 7.3. Let ξ ∈ Ξ(V) and {ξ} = k. Then the set of eigenvalues {μj (H0 (ξ))} of the unperturbed operator H0 (ξ) coincides with the set {|Fη|2 , η ∈ Υ(ξ)}. Let us label all numbers {|Fη|2 , η ∈ Υ(ξ)} in the increasing order; if there are two different vec˜ ∈ Υ(ξ) with |Fη|2 = |F˜ tors η, η η |2 , we label them in the lexicographic order of ˜ if either η1 < η˜1 , or η1 = η˜1 and η2 < η˜2 , their coordinates (i.e., we put η before η etc.) Then to each point η ∈ Υ(ξ) we have put into correspondence a natural number j = j(η) such that (7.41) |Fη|2 = μj H0 (ξ) . Next we define

g˜(η) := μj(η) H (ξ) .

This mapping is well-defined and satisfies the following obvious property: |˜ g (η) − |Fη|2 | ≤ v (recall that v = ||V ||∞ ). The problem with the mapping g˜ defined in this way is that we cannot apply Lemma 7.7 to it, since Lemma 7.7 treats not the eigenvalues of H (ξ), but the eigenvalues of H (ξ; U ) with the set U containing certain intervals perpendicular to V. Thus, we need to introduce a different definition which takes care of Lemma 7.7 and at the same time is reasonably canonical. Let ξ ∈ Ξ2 (V). Denote X = X(ξ) := η ∈ Ξ2 (V) : ηV = ξ V , η V = ξV . Simple geometry implies that X(ξ) is an interval of length ρ−1 . Similarly to our actions when we were defining g˜, we notice that the set of eigenvalues {μj (H0 (ξ; X(ξ)))} coincides with the set {|Fη|2 , η ∈ Υ(ξ; X)}. Let us label all numbers {|Fη|2 , η ∈ Υ(ξ; X)} in the increasing order; if there are two different vectors η 1 , η 2 ∈ Υ(ξ; X) with |Fη 1 |2 = |Fη 2 |2 , we label them in the lexicographic order of their coordinates. Then to the point ξ we have put into correspondence a natural number i = i(ξ) such that |Fξ|2 = μi H0 (ξ; X) . (7.42) Next we define g(ξ) := μi(ξ) (H (ξ; X)). This mapping is well-defined and satisfies the property |g(ξ) − |Fξ|2 | ≤ v. Lemma 7.10. Let ξ ∈ Ξ2 (V) ∩ A1 . Then the following properties are satisfied: (i) |g(ξ) − g˜(ξ)| ρ−4Mp ; (ii) g(ξ) = r2 + s, where r = r(ξ) and s = s(ξ) = s(ξ V , ξV , r) is a function which ∂s smoothly depends on r with ∂r = O(ρ1/3 ). Proof. Let us prove the first statement. First, we notice that the difference i(ξ) − j(ξ) is equal to the number of points η ∈ (Υ(ξ; X(ξ)) \ Υ(ξ) satisfying |Fη|2 < λ. Now the statement follows from Lemma 7.4 and Remark 7.5.

496

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Let us now prove the second statement. Suppose, ξ 1 ∈ X(ξ) and η ∈ Υ(ξ; X). Then η 1 := η + (ξ 1 − ξ) ∈ Υ(ξ 1 ; X) . Note that (ξ)V = (ξ 1 )V and therefore (η)V = (η 1 )V . Let us assume that |Fξ| ≥ |Fη|

(7.43)

and prove that this implies |Fξ 1 | ≥ |Fη 1 |. Indeed, there are two possible cases: (i) ξ − η ∈ V. Then, since ξ 1 − η 1 = ξ − η ∈ V, we have: |Fξ 1 |2 − |Fη 1 |2 = |F(ξ 1 )V |2 − |F(η 1 )V |2 = |F(ξ)V |2 − |F(η)V |2 = |Fξ|2 − |Fη|2 ≥ 0 . (ii) ξ − η ∈ V. Then, in the same way we have proved estimate (7.20), using Lemma 5.10 we can show that ||Fη|2 − λ| K 2 . But since ξ ∈ Ξ2 (V) ⊂ A, we have ||Fξ|2 − λ| 1. Thus, (7.43) implies λ − |Fη|2 K 2 . Since |η 1 − η| = |ξ 1 − ξ| ρ−1 , we have λ − |Fη 1 |2 K 2 . Since ξ 1 ∈ A, this implies |Fξ 1 | ≥ |Fη 1 |. Thus, we have proved that the inequality |Fξ 1 | ≥ |Fη 1 | is equivalent to |Fξ| ≥ |Fη|. This implies that i(ξ 1 ) = i(ξ), where i is the function defined by (7.42). Now the second statement of lemma follows from Lemma 7.7. This lemma shows that the mapping g behaves in a nice way as a function of r. Unfortunately, the dependence on other variables is not quite so nice. In fact, this mapping is not continuous, even modulo O(ρ−4Mp ), because the functions i(ξ) are not continuous; moreover, a little thought shows that we cannot, in general, define the mapping g to have all properties formulated in the introduction and be continuous at the same time. Indeed, if the function i = i(ξ) were continuous, it would necessary have been a constant. Thus, the function i has discontinuities, and the function g may have discontinuities at the same points as i. However, Lemmas 7.4, 7.7, and 7.8 show that for each small neighbourhood U in the space of quasi-momenta we can find a family of representatives of the functions g which is ‘almost’ smooth. Namely, the following statement holds: Lemma 7.11. Let I = [a, b] ⊂ Ξ2 (V) ∩ A1 be a straight interval of length L := |b − a| ρ−1 . Then there exists an integer vector n such that |g(b + n) − g(a)| Lρ + ρ−4Mp . Moreover, suppose in addition that there exists an integer vector m = 0 such that the interval I + m is entirely inside Ξ2 (V) ∩ A1 . Then there exist two different integer vectors n1 and n2 such that |g(b + n1 ) − g(a)| Lρ + ρ−4Mp and |g(b + n2 ) − g(a + m)| Lρ + ρ−4Mp . Proof. Lemmas 7.4 and 7.10 show that g(a) = λl (a; I) + O(ρ−4Mp ) for some integer l. Lemma 7.9 now implies that |λl (a; I) − λl (b; I)| Lρ + ρ−4Mp .

(7.44)

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Once again using Lemma 7.4, we deduce that λl (b; I) = g(η) + O(ρ−4Mp ) for some η ∈ Υ(b; I); in particular, we have η = b + n for some integer vector n. This proves the first statement. Let us prove the second statement. Conditions of lemma imply that g(a) = λj (a; I)+O(ρ−4Mp ) and g(a+m) = λl (a+m; I +m)+O(ρ−4Mp ) for some integers j, l. Moreover, if a + m ∈ Υ(a; I) (so that Υ(a; I) = Υ(a + m; I + m)), then, since m = 0 we have j = l . (7.45) Lemma 7.9 now implies that together with (7.44) we have |λl (a + m; I + m) − λl (b + m; I + m)| Lρ + ρ−4Mp .

(7.46) −4Mp

Once again using Lemma 7.4, we deduce that λj (b; I) = g(η1 ) + O(ρ ) and λl (b + m; I + m) = g(η2 ) + O(ρ−4Mp ) for different points η 1 ∈ Υ(b; I) and η 2 ∈ Υ(b + m; I + m) (the points η 1 and η 2 are different because of (7.45)). In particular, these inclusions imply η 1 − b ∈ Zd and η2 − b ∈ m + Zd = Zd . This proves the second statement. Thus, we have proved the following lemma, which is the main result of this section: Lemma 7.12. Let V ∈ V(n). Then there are two mappings g˜, g : Ξ2 (V) → R which satisfy the following properties: (i) g˜(ξ) is an eigenvalue of P (V)H (k)P (V) with {ξ} = k. All eigenvalues of P (V)H (k)P (V) inside J are in the image of g˜. (ii) If ξ ∈ A1 , then |˜ g (ξ) − g(ξ)| ≤ Cρ−4Mp and |g(ξ) − |Fξ|2 | ≤ 2v. ∂s 2 1/3 (iii) g(ξ) = r + s(ξ) with r := |Fξ ⊥ ). V | and ∂r = O(ρ Proof. The only statement which has not been checked so far is that |g(ξ)−|Fξ|2 | ≤ 2v. This follows immediately from the second statement of this lemma together with the inequality |˜ g (ξ) − |Fξ|2 | ≤ v and |˜ g(ξ) − g(ξ)| ≤ Cρ−4Mp . Now, we can put together the results of the previous sections. Theorem 7.13. Suppose, R is sufficiently large, all conditions before Lemma 5.1 −1 are satisfied, and R < ρpd /2 . Then there are two mappings f, g : A → R which satisfy the following properties: (i) f (ξ) is an eigenvalue of H (k) with {ξ} = k; |f (ξ) − |Fξ|2 | ≤ 2v. f is an injection (if we count all eigenvalues with multiplicities) and all eigenvalues of H (k) inside J are in the image of f . (ii) If ξ ∈ A1 , then |f (ξ) − g(ξ)| ≤ Cρ−4Mp . 'd−1 (iii) ' We can decompose the domain of g into the disjoint union: A = B ∪ n=1 V∈V(6MR,n) Ξ2 (V). For any ξ ∈ B g(ξ) = |Fξ|2 +

2M

j=1

η 1 ,...,η j ∈ΘM

2≤n1 +···+nj ≤2M

Cn1 ,...,nj ξ, Gη 1 −n1 · · · ξ, Gη j −nj . (7.47)

498

L. Parnovski


For any ξ ∈ Ξ2 (V) g(ξ) = r2 (ξ) + s(ξ) , with r(ξ) =

|Fξ ⊥ V |,

s(ξ) =

∂s s(r, ξ V , ξ V ), ∂r

= O(ρ

(7.48) 1/3

).

Proof. We have described the construction of the mapping f at the end of Section 5. Mapping g is constructed in Sections 6 and 7. Let us formulate an important property of the mapping g, which is a global version of Lemma 7.11. Lemma 7.14. Let I = [a, b] ⊂ A1 be a straight interval of length L := |b−a| ρ−1 . Then there exists an integer vector n such that |g(b + n) − g(a)| Lρ + ρ−4Mp+d . Moreover, suppose in addition that there exists an integer vector m = 0 such that the interval I + m is entirely inside A1 . Then there exist two different integer vectors n1 and n2 such that |g(b + n1 ) − g(a)| Lρ + ρ−4Mp+d and |g(b + n2 ) − g(a + m)| Lρ + ρ−4Mp+d . Proof. Let us parametrise the interval I so that I = ξ(t), t ∈ [tmin , tmax ] , a = ξ(tmin ), b = ξ(tmax ). Let us prove the first statement. If the interval ξ(t) lies entirely inside B, then the statement is obvious since the length of the gradient of g inside B is ρ, so we can take n = 0. If the interval ξ(t) lies entirely inside Ξ2 (V) for V ∈ V(n), the statement has been proved in Lemma 7.11. Consider the general case. Denote by yj (k) := μj (H (k)) the j-th eigenvalue of H (k). Then the definition of the mapping f implies that if yj (k) ∈ J, then yj (k) = f (n + k) for some integer vector n; the opposite is also true, namely if f (n + k) ∈ J, then f (n + k) = yj (k) for some j. Notice also that for each j the function yj is continuous. Now let us return to the study of the behaviour of the function g(ξ(t)). Suppose for definiteness that ξ(tmin ) ∈ B. Then, as we mentioned in the beginning of proof, since the gradient of g has length ρ, we have |g(ξ(t)) − g(ξ(tmin ))| |ξ(t) − ξ(tmin )|ρ as soon as ξ(t) stays inside B. Suppose that t1 is the point at which ξ(t) crosses the boundary of B. Then g ξ(t1 − 0) − g ξ(tmin ) |ξ(t) − ξ(tmin )|ρ . (7.49) According to the relationship between the mapping f and functions yj stated above, there exists an index j such that f (ξ(t1 − 0)) = yj ({ξ(t1 − 0)}) (recall that if ξ = n + k, then we call k = {ξ} the fractional part of ξ). Since yj is continuous function, yj ({ξ(t1 − 0)}) = yj ({ξ(t1 + 0)}). Using the relationship between the mapping f and functions yj again, we deduce that there exists an integer vector n1 such that yj ({ξ(t1 + 0)}) = f (ξ(t1 + 0) + n1 ). Property (ii) of

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Theorem 7.13 implies that f (ξ(t1 − 0)) = g(ξ(t1 − 0)) + O(ρ−4Mp ) and, similarly, f (ξ(t1 + 0) + n1 ) = g(ξ(t1 + 0) + n1 ) + O(ρ−4Mp ). All these estimates imply g ξ(t1 + 0) + n1 = g ξ(t1 − 0) + O ρ−4Mp . (7.50) Since ξ(t1 +0)+n1 ∈ A1 , we have either ξ(t1 +0)+n1 ∈ Ξ2 (V) or ξ(t1 +0)+n1 ∈ B. Assume the former. Let t2 > t1 be the smallest value of t at which ξ(t)+ n1 crosses the boundary of Ξ2 (V). Then Lemma 7.11 implies that thee exists an integer vector n2 such that g ξ(t2 − 0) + n2 − g ξ(t1 + 0) + n1 |ξ(t2 ) − ξ(t1 )|ρ + O ρ−4Mp . (7.51) Now repeating the argument we have already used at the moment t1 , we deduce that there exists an integer n3 such that (7.52) g ξ(t2 + 0) + n3 = g ξ(t2 − 0) + n2 + O ρ−4Mp . Now we repeat the process and increase t beginning from t2 until we hit another piece of boundary of some Ξ2 (W) at t = t3 , etc. The shift of the function g at each of the points tj of hitting the boundary is O(ρ−4Mp ). The number of such points is ρd , since for each fixed integer vector m the number of intersections of the interval (ξ(t) + m) (t ∈ [tmin , tmax ]) with the boundaries of all sets Ξ2 (V) is finite, and the number of possible integer vectors m allowed here is ρd (obviously, the length of each of these integer vectors is ρ). Now formulas (7.49)–(7.52) lead to the desired result. The proof of the second statement is similar and can be derived from the proof of the first statement in the same way as the proof of the second part of Lemma 7.11 follows from the proof of the first part of that lemma. Now it remains to extend the above results to the ‘full’ operator H(k). Corollary 7.15. For each natural N there exist mappings f, g : A → R which satisfy the following properties: (i) f (ξ) is an eigenvalue of H(k) with {ξ} = k; |f (ξ) − |Fξ|2 | ≤ 2v. f is an injection (if we count all eigenvalues with multiplicities) and all eigenvalues of H(k) inside J are in the image of f . (ii) If ξ ∈ A1 , then |f (ξ) − g(ξ)| ≤ ρ−N . ' (iii) ' We can decompose the domain of g into the disjoint union: A = B ∪ d−1 n=1 V∈V(n) Ξ2 (V). For any ξ ∈ Bρ g(ξ) = |Fξ|2 +

2M

j=1

η 1 ,...,η j ∈ΘM

2≤n1 +···+nj ≤2M

Cn1 ,...,nj ξ, Gη 1 −n1 · · · ξ, Gη j −nj

(7.53)

with M = [(N + d)(4p)−1 ] + 1. For any ξ ∈ (Ξ2 (V) ∩ A1 ) g(ξ) = r2 + s(ξ) , with r :=

|Fξ ⊥ V |,

s(ξ) =

s(r, ξV , ξ V )

and

∂s ∂r

(7.54) = O(ρ

1/3

).

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(iv) Let I = [a, b] ⊂ A1 be a straight interval of length L := |b − a| ρ−1 . Then there exists an integer vector n such that |g(b + n) − g(a)| Lρ + ρ−N . Moreover, suppose m = 0 is a given integer vector such that the interval I +m is entirely inside A1 . Then there exist two different integer vectors n1 and n2 such that |g(b+n1 )−g(a)| Lρ+ρ−N and |g(b+n2 )−g(a+m)| Lρ+ρ−N . Proof. We use Theorem 7.13 for the operator H (k) with M = [(N + d)(4p)−1 ]+ 1. Estimate (5.4) implies that |μj (H(k)) − μj (H (k))| < ρ−N −1 , so that all the required properties are fulfilled. Remark 7.16. The function f is not necessarily continuous. Before we continue with the proof of the Bethe–Sommerfeld conjecture, let us formulate a theorem which immediately follows from our results, just to illustrate their usefulness. Recall that by N (λ) we have denoted the integrated density of states of the operator (1.4) defined in (2.2). Theorem 7.17. For each natural n we have the following estimate: N (λ + λ−n ) − N (λ − λ−n ) = O(λd/2−n−1 ). Proof. We use Corollary 7.15 with N = 2n + 1. Then ! N (λ + λ−n ) − N (λ − λ−n ) = vol f −1 [λ − λ−n , λ + λ−n ] ! ≤ vol g −1 [λ − 2λ−n , λ + 2λ−n ] = O λd/2−n−1 ,

(7.55)

the last equality being an easy geometric exercise (which will anyway be established in the next section). Remark 7.18. As it was pointed out to the author by Yu. Karpeshina, it seems possible that using the results of this paper (including the results from the next section) one can prove the following lower bound: N (λ + ε) − N (λ) ελ(d−2)/2 , uniformly over ε < 1 as λ → ∞ (in particular, ε does not have to be a negative power of λ). We will not prove this estimate in our paper though.

8. Proof of the Bethe–Sommerfeld conjecture Throughout this section we keep the notation from the previous section. Without specific mentioning, we always assume that ρ is sufficiently large; the precise value of the power N will be chosen later. In what follows, it will be convenient to consider a slightly slimmed down resonance set. Namely, we introduce the set ˜ := ξ ∈ A1 : |ξ | > ρ1/2 , ∀U ∈ V(1) . B U

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˜ consists of all points ξ ∈ A1 the F-projections of which to all In other words, B ˜ ⊂ B. We also denote vectors η ∈ Θ6M has F-length larger than ρ1/2 . Obviously, B ˜ ˜ D := A1 \ B. Now we will study various properties of mappings f and g. We begin with the function g. For each positive δ ≤ v denote A(δ), B(δ), and D(δ) to be intersections of ˜ and D ˜ correspondingly. The following is a simple g −1 ([ρ2 − δ, ρ2 + δ]) with A1 , B, geometry: Lemma 8.1. The following estimates hold: vol A(δ) ρd−2 δ , vol B(δ) ρd−2 δ , and

vol D(δ) ρ(3d−7)/3 δ .

(8.1) (8.2) (8.3)

Proof. Let ξ = rξ ∈ B, |Fξ | = 1. Then the definition of g implies that ∂g ρ ∂r

(8.4)

uniformly over ξ . Therefore, for each fixed ξ the intersection of g −1 ([ρ2 −δ, ρ2 +δ]) with the set {rξ , r > 0} is an interval of length δρ−1 . Integrating over ξ , we obtain (8.2). Estimate (8.3) is obtained in a similar way, only for ξ ∈ Ξ(V) we put r := |Fξ ⊥ V |. Then the estimate (8.4) is still valid. Let η ∈ ΘM . Then (8.4) implies that the set of all points ξ ∈ A(δ) such that the F-projection of ξ onto η has F-length smaller than ρ1/2 has volume O(ρ(2d−5)/2 δ). Since the number of elements in ΘM is O(Rd ) = O(ρp/2 ), we have vol D(δ) ρ(2d−5+p)/2 δ ρ(3d−7)/3 δ , since p < 1/3. Finally, (8.1) is the sum of (8.2) and (8.3).

Remark 8.2. Putting δ = 2λ−n in (8.1), we establish the last equality in (7.55). The next estimate is more subtle. Lemma 8.3. Let d ≥ 3. Then for large enough ρ and δ < ρ−1 the following estimate holds uniformly over a ∈ Rd with |a| > 1: ! (8.5) vol B(δ) ∩ B(δ) + a δ 2 ρd−3 + δρ−d . If d = 2, similar estimate holds with δ 3/2 + δρ−2 in the RHS. Proof. After making the substitution ν = Fξ, the function g in new coordinates will have the form h(ν) = |ν|2 + G(ν), with G(ν) = O |ν|−1/2 (8.6)

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and

∂G ≤ C2 |ν|−1 (8.7) ∂νj ˜ these estimates follow from Lemma 6.1. We for all j = 1, . . . , d, provided ν ∈ FB; need to estimate the volume of the set ( X = ν ∈ F(B) ∩ (F(B) + Fa) , h(ν) ∈ [ρ2 − δ, ρ2 + δ], ) h(ν − Fa) ∈ [ρ2 − δ, ρ2 + δ] . (8.8)

Indeed, we have X = F(B(δ) ∩ (B(δ) + a)), so the volume of X equals det F times the volume of the set B(δ) ∩ (B(δ) + a). Denote b := Fa. First, we will estimate the 2-dimensional area of the intersection of X with arbitrary 2-dimensional plane containing the origin and vector b; the volume of X then can be obtained using the integration in cylindrical coordinates. So, let V be any 2-dimensional plane containing the origin and b, and let us estimate the area of XV := V ∩ X. Let us introduce cartesian coordinates in V so that ν ∈ V has coordinates (ν1 , ν2 ) with ν1 going along b, and ν2 being orthogonal to b. For any ν ∈ XV estimate (8.6) implies h(ν) = ν12 + ν22 + O ρ−1/2 , and so

2 2δ ≥ |h(ν) − h(ν − b)| = |ν12 − |b| − ν1 | + O ρ−1/2 . This implies that |b| 2|b| < ν1 < 3 3 when ρ is sufficiently large, and therefore

(8.9)

∂h(ν) |b| (8.10) ∂ν1 whenever ν ∈ XV . Thus, for any fixed t ∈ R, the intersection of the line ν2 = t with XV is an interval of length |b|−1 δ. Let us cut XV into two parts: XV = X1V ∪ X2V with X1V := {ν ∈ XV , |ν2 | ≤ −1 2C2 ρ }, X2V = XV \X1V , and estimate the volumes of these sets (C2 is the constant from (8.7)). A simple geometrical argument shows that if X1V is nonempty, then |b| ρ. This, together with the remark after (8.10), implies that the area of X1V is ρ−2 δ. Now we define the ‘rotated’ set X1 which consists of the points from X which belong to X1V for some V. Computing the volume of this set using integration in the cylindrical coordinates, we obtain vol(X1 ) ρ−d δ . Now consider X2V . Let us decompose X2V = X2V ∪ X2V , where X2V = ν ∈ X2V : ν2 > 0 and

X2V = ν ∈ X2V : ν2 < 0 .

(8.11)

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ν2

t

ν

l

r

ν

ν

b

ν

ν

1

b Figure 10. The set X2V (the area bounded by four arcs). 2

Notice that for any ν ∈ XV , formula (8.7) implies ∂h(ν) ν2 . ∂ν2

(8.12)

Let ν l = (ν1l , ν2l ) be the point in the closure of X2V with the smallest value of the first coordinate: ν1l ≤ ν1 for any ν = (ν1 , ν2 ) ∈ X2V . Analogously, we define ν r to be the point in the closure of X2V with the largest first coordinate, ν t the point with the largest second coordinate, and ν b the point with the smallest second coordinate (see Figure 10 for an illustration). Note that ν t ρ. Let us prove that ν1r − ν1l δ . (8.13) Indeed, suppose first that ν2r ≥ ν2l . Let ν rl := (ν1r , ν2l ). Then, since h is an increasing function of ν2 when ν2 > 2C2 ρ−1 , we have h(ν rl ) ≤ h(ν r ) ≤ ρ2 + δ. Therefore, h(ν rl ) − h(ν l ) ≤ 2δ. Estimate (8.10) then implies (8.13). Suppose now that ν2r ≤ ν2l . Let ν lr := (ν1l , ν2r ). Then h(ν lr −b) ≤ h(ν l −b) ≤ 2 ρ + δ. Therefore, h(ν lr − b) − h(ν r − b) ≤ 2δ. Now, (8.9) and (8.10) imply (8.13). Thus, we have estimated the width of X2V . Let us estimate its hight (i.e., t ν2 − ν2b ). Let us assume that ν1t ≥ ν1b ; otherwise, we use the same trick as in the previous paragraph and consider h( · − b) instead of h. Let ν bt := (ν1b , ν2t ). Then

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h(ν bt ) ≤ h(ν t ) ≤ ρ2 + δ. Therefore, h(ν bt ) − h(ν b ) ≤ 2δ. Now, (8.12) implies ν2t ν2t t 2 b 2 ∂h b ν2 − ν2 = 2 ν1 , ν2 dν2 ≤ 2δ . ν2 dν2 (8.14) ν2b ν2b ∂ν2 Therefore, we have the following estimate for the hight of X2V : δ ν2t − ν2b t . ν2 + ν2b

(8.15)

Now, we can estimate the volume of X2 := X\X1 using estimates (8.13) and (8.15). The cylindrical integration produces the following: d−3 δ 2 t d−2 vol X2 t ≤ δ 2 ν2t ≤ δ 2 ρd−3 . (8.16) ν2 b ν2 + ν2 Equations (8.11) and (8.16) imply (8.5). If d = 2, we have to notice that (8.14) implies ν2t − ν2b δ 1/2 and then use (8.11) and (8.13). As was mentioned already, the function f is not necessarily continuous. We now give a sufficient condition for its continuity. Recall that v is the L∞ -norm of the potential V . Lemma 8.4. Let ξ ∈ B(v) be a point of discontinuity of f . Then there is a non-zero vector n ∈ Zd such that |g(ξ + n) − g(ξ)| ≤ 2ρ−N .

(8.17)

Proof. If ξ = m + k ∈ B(v) is a point of discontinuity of a bounded function f , there exist two sequences {ξj } and {ξ˜j } which both converge to ξ, such that the ˜ limits λ(ξ) := lim f (ξ j ) and λ(ξ) := lim f (ξ˜j ) exist and are different. Since the points f (ξ j ) are eigenvalues of H({ξj }), the limit λ is an eigenvalue of H(k) (it is well-known that the spectrum of H(k) is continuously dependent on k). The same ˜ is also an eigenvalue of H(k). Since λ = λ, ˜ at most one of argument implies that λ ˜ ˜ these points can be equal to f (ξ). Say, λ = f (ξ). But since λ is inside J, it must ˜ = f (ξ), ˜ {ξ} ˜ = {ξ}. Thus, ξ˜ = ξ + n with n ∈ Zd . belong to the image of f , say λ ˜ Since the function g is continuous in B, lim g(ξ˜j ) = g(ξ), and so ˜ = lim |g(ξ˜ ) − f (ξ˜ )| ≤ ρ−N . |g(ξ) − λ| j j ˜ − λ| ˜ = |g(ξ) ˜ − f (ξ)| ˜ ≤ ρ−N . The last two inequalities But we also have |g(ξ) imply (8.17). Corollary 8.5. There is a constant C3 with the following properties. Let I := ξ(t) : t ∈ [tmin , tmax ] ⊂ B(v) . be a straight interval of length L < ρ−1 δ. Suppose that there is a point t0 ∈ [tmin , tmax ] with the property that for each non-zero n ∈ Zd g(ξ(t0 ) + n) is either outside the interval

g ξ(t0 ) − C3 ρ−N − C3 ρL, g ξ(t0 ) + C3 ρ−N + C3 ρL or not defined. Then f (ξ(t)) is a continuous function of t.

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Proof. Suppose not. Then previous lemma implies that there is a point t1 ∈ [tmin , tmax ] and a non-zero vector n ∈ Zd such that |g(ξ(t1 )+n)−g(ξ(t1 ))| ≤ 2ρ−N . Since |ξ(t1 ) − ξ(t0 )| ≤ |ξ(tmax ) − ξ(tmin )| ≤ L, it follows that (I + n) ⊂ A1 , and now Lemma 7.14 implies that for two different integer vectors m1 and m2 we have |g(ξ(t0 )+m1 )−g(ξ(t1 )+n)| ρL+ρ−N and |g(ξ(t0 )+m2 )−g(ξ(t1 ))| ρL+ρ−N . Since ξ(t) ∈ B for all t and the length of the gradient of g is ρ in B, we also have |g(ξ(t1 ))−g(ξ(t0 ))| ρL. Thus, we have |g(ξ(t0 )+mj )−g(ξ(t0 ))| ≤ Cρ−N +CρL (j = 1, 2). Since at least one of vectors mj is non-zero, this contradicts the assumption of the corollary. Now we are ready to prove the Bethe–Sommerfeld conjecture. Since in the two-dimensional case it has been proved, we will assume that d ≥ 3. Theorem 8.6. Let d ≥ 3. Then all sufficiently large points λ = ρ2 are inside the spectrum of H. Moreover, there exists a positive constant c4 such that for large enough ρ the whole interval [ρ2 − c4 ρ1−d , ρ2 + c4 ρ1−d ] lies inside some spectral band. Proof. Put N = d in the Corollary 7.15. Also put δ = c4 ρ1−d (the precise value of c4 will be chosen later). For each unit vector η ∈ Rd we denote Iη to be the intersection of {rη, r > 0} with A(δ). We will consider only vectors η for ˜ As was mentioned in the proof of Lemma 8.1, the length L of any which Iη ⊂ B. interval Iη satisfies L δρ−1 . Let us prove that f is continuous on at least one ˜ Suppose this is not the case. Then Corollary 8.5 tells us of the intervals Iη ⊂ B. that for each point ξ ∈ B(δ) there is a non-zero integer vector n such that |g(ξ + n) − g(ξ)| ≤ C3 ρ−d + ρL ρ−d + δ . (8.18) Since |g(ξ) − ρ2 | ≤ δ, this implies |g(ξ + n) − ρ2 | ≤ C5 (ρ−d + δ) =: δ1 , and thus ξ + n ∈ A(δ1 ); notice that C5 > 1 and so δ1 > δ. Therefore, each point ξ ∈ B(δ) also belongs to the set (A(δ1 ) − n) for a non-zero integer n; obviously, |n| ρ. In other words, $ $ $ A(δ1 ) − n = B(δ1 ) − n ∪ D(δ1 ) − n . (8.19) B(δ) ⊂ n=0

n∈Zd ∩B(Cρ),n=0

n=0

To proceed further, we need more notation. Denote D0 (δ1 ) to be the set of all points ν from D(δ1 ) for which there is no non-zero n ∈ Zd satisfying ν − n ∈ B(δ); D1 (δ1 ) to be the set of all points ν from D(δ1 ) for which there is a unique non-zero n ∈ Zd satisfying ν − n ∈ B(δ); and D2 (δ1 ) to be the rest of the points from D(δ1 ) (i.e., D2 (δ1 ) consists of all points ν from D(δ1 ) for which there exist at least two different non-zero vectors n1 , n2 ∈ Zd satisfying ν − nj ∈ B(δ)). Then a little thought shows that we can replace D(δ1 ) by D1 (δ1 ) in the RHS of (8.19). Indeed, this is shown in the following lemma. Lemma 8.7. The following formulae hold: ⎛ ⎞ * $ B(δ) ⎝ D0 (δ1 ) − n ⎠ = ∅ , n=0

(8.20)

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and

⎛ ⎝

$


⎞ $ D2 (δ1 ) − n ⎠ ⊂ B(δ1 ) − n .

n=0

(8.21)

n=0

Proof. The first formula is an immediate corollary of the definition of D0 (δ1 ). Let us prove the second formula. Suppose, ν ∈ D2 (δ1 ). Then there exist two integer vectors, n1 and n2 such that ν − nj ∈ B(δ). Let m be an integer vector. Then m is different from either n1 or n2 , say m = n1 . Since δ1 ≥ δ, this implies: $ ν − m = ν − n1 − (m − n1 ) ∈ B(δ) − (m − n1 ) ⊂ B(δ1 ) − n . n=0

This finishes the proof of the lemma. This lemma shows that we can re-write (8.19) as $ $ $ B(δ) ⊂ B(δ1 ) − n D1 (δ1 ) − n . n=0

This, obviously, implies !$ $ ! $ B(δ1 ) − n ∩ B(δ) D1 (δ1 ) − n ∩ B(δ) . B(δ) = n=0

(8.22)

n=0

(8.23)

n=0

Now let us compare volumes of the sets in both sides of (8.23). The volume of the LHS we already know from (8.2): it is ρd−2 δ. The definition of the set D1 implies that ! $ vol D1 (δ1 ) − n ∩ B(δ) (8.24) n (3d−7)/3 (3d−7)/3 −d ≤ vol D1 (δ1 ) ρ δ1 ρ (ρ + δ) . Finally, Lemma 8.3, inequality δ < δ1 and the fact that the union in (8.23) consists of no more than Cρd terms imply ! $ B(δ1 ) − n ∩ B(δ) ρd (δ12 ρd−3 + δ1 ρ−d ) vol n

ρd (ρ−d + δ)2 ρd−3 + (ρ−d + δ)ρ−d

(8.25)

δ 2 ρ2d−3 + δρd−3 + ρ−3 . Putting all these inequalities together, we get

! ρd−2 δ < C6 δ 2 ρ2d−3 + δρ(3d−7)/3 + ρ−7/3 .

It is time to recall that δ = c4 ρ1−d . Plugging this into (8.26), we obtain ! c4 ρ−1 < C6 c24 ρ−1 + c4 ρ−4/3 + ρ−7/3 .

(8.26)

(8.27)

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Now, if we choose c4 to be small enough (i.e., c4 < C6−1 ), the inequality (8.27) will not be satisfied for sufficiently large ρ. Thus, our assumption that function f is discontinuous on every interval Iη ⊂ B(δ) leads to a contradiction (provided we have chosen small enough c4 ). Therefore, there is an interval Iη ⊂ B(δ) on which f is continuous. Since the value of f on one end of this interval is ≤ ρ2 − c4 ρ1−d , and the value on the other end is ≥ ρ2 + c4 ρ1−d , the point ρ2 must be in the range of f . The first part of the theorem is proved. In order to prove the second part of the theorem, we notice that the interval Iη which we found satisfies the following condition: for each point ξ ∈ Iη and each non-zero integer vector n such that ξ + n ∈ A1 we have |g(ξ + n) − g(ξ)| > 2ρ−N . This implies f (ξ + n) − f (ξ) = 0. Therefore, f (ξ) is a simple eigenvalue of H({ξ}) for each ξ ∈ Iη . This implies that the interval [ρ2 − c4 ρ1−d , ρ2 + c4 ρ1−d ] is inside the spectral band. The theorem is proved.

Acknowledgements First and foremost, I am deeply grateful to Alex Sobolev. I was introduced to periodic problems by working jointly with him, and our numerous conversations and discussions resulted in much better understanding of this subject by me (and, I do hope, by him as well). He has read the preliminary version of this manuscript and made essential comments. Thanks also go to Keith Ball who made several important suggestions which have substantially simplified proofs of the statements from Section 4. I am also immensely grateful to Gerassimos Barbatis, Yulia Karpeshina, Michael Levitin, and Roman Shterenberg for reading the preliminary version of this manuscript and making very useful comments and also for helping me to prepare the final version of this text.

References [1] J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin, 1959. [2] B. E. J. Dahlberg and E. Trubowitz, A remark on two dimensional periodic potentials, Comment. Math. Helvetici 57 (1982), 130–134. [3] B. Helffer and A. Mohamed, Asymptotics of the density of states for the Schr¨ odinger operator with periodic electric potential, Duke Math. J. 92 (1998), 1–60. [4] Y. E. Karpeshina, Perturbation series for the Schr¨ odinger operator with a periodic potential near planes of diffraction, Comm. Anal. Geom. 4 (1996), no. 3, 339–413. [5] Y. E. Karpeshina, Perturbation Theory for the Schr¨ odinger Operator with a Periodic Potential, Lecture Notes in Math. Vol. 1663, Springer Berlin 1997. [6] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980. [7] L. Parnovski and A. V. Sobolev, Bethe–Sommerfeld Conjecture for Polyharmonic Operators, Duke Math. J., 2001.

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[8] L. Parnovski and A. V. Sobolev, Perturbation Theory and the Bethe–Sommerfeld Conjecture, Annals H. Poincaré, 2001. [9] V. N. Popov and M. Skriganov, A remark on the spectral structure of the two dimensional Schr¨ odinger operator with a periodic potential, Zap. Nauchn. Sem. LOMI AN SSSR 109 (1981), 131–133 (Russian). [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Academic Press, New York, 1975. [11] M. Skriganov, Proof of the Bethe–Sommerfeld conjecture in dimension two, Soviet Math. Dokl. 20 (1979), 1, 89–90. [12] M. Skriganov, Geometrical and Arithmetical Methods in the Spectral Theory of the Multi-dimensional Periodic Operators, Proc. Steklov Math. Inst. Vol. 171, 1984. [13] M. Skriganov, The spectrum band structure of the three-dimensional Schr¨ odinger operator with periodic potential, Inv. Math. 80 (1985), 107–121. [14] O. A. Veliev, Asymptotic formulas for the eigenvalues of the periodic Schr¨ odinger operator and the Bethe–Sommerfeld conjecture, Functional Anal. Appl. 21 (1987), no. 2, 87–100. [15] O. A. Veliev, On the spectrum of multidimensional periodic operators, Theory of Functions, functional analysis and their applications, Kharkov University 49, (1988), 17–34 (in Russian). [16] O. A. Veliev, Perturbation theory for the periodic multidimensional Schr¨ odinger operator and the Bethe–Sommerfeld Conjecture, Int. J. Contemp. Math. Sci. 2 (2007), no. 2, 19–87. Leonid Parnovski Department of Mathematics University College London Gower Street London, WC1E 6BT United Kingdom e-mail: [email protected] Communicated by Christian Gérard. Submitted: September 5, 2007. Accepted: November 12, 2007.



Absence of Quantum States Corresponding to Unstable Classical Channels Ira Herbst and Erik Skibsted Abstract. We develop a general theory of absence of quantum states corresponding to unstable classical scattering channels. We treat in detail Hamiltonians arising from symbols of degree zero in x and outline a generalization in an Appendix.

1. Introduction and results The purpose of this paper is to show in a class of models that there are no quantum states corresponding to unstable classical channels. A principal example treated in detail is the following: Consider a real-valued potential V on Rn , n ≥ 2, which is smooth outside zero and homogeneous of degree zero. Suppose that the restriction of V to the unit sphere S n−1 is a Morse function. We prove that there are odinger equation i∂t φ = (−2−1 Δ + V )φ which asympno L2 -solutions to the Schr¨ totically in time are concentrated near local maxima or saddle points of V|S n−1 . Consequently all states concentrate asymptotically in time in arbitrarily small open cones containing the local minima, cf. [15] and [18]. In the bulk of the paper we consider the following general situation: Suppose h(x, ξ) is a real classical Hamiltonian in C ∞ ((Rn \ {0}) × Rn ), n ≥ 2, satisfying x · ∇x h(x, ξ) = 0

(1.1)

in a neighborhood of a point (ω0 , ξ0 ) ∈ S × R . Suppose in addition that this neighborhood is conic in the x-variable and that the orbit (0, ∞) t → (x(t), ξ(t)) = (tk0 ω0 , ξ0 ) with k0 > 0 is a solution to Hamilton’s equations n−1

dx = ∇ξ h(x, ξ) , dt

n

dξ = −∇x h(x, ξ) , dt

E. Skibsted is (partially) supported by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation.

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or equivalently, ∇x h(ω0 , ξ0 ) = 0 ,

∇ξ h(ω0 , ξ0 ) = k0 ω0 .

(1.2)

We consider situations in which for each energy E near E0 = h(ω0 , ξ0 ) there is a (typically unique) (ω(E), ξ(E)) ∈ S n−1 × Rn near (ω0 , ξ0 ) depending smoothly on E such that the above structure persists, namely h ω(E), ξ(E) = E , (1.3) ∇x h ω(E), ξ(E) = 0 , (1.4) ∇ξ h ω(E), ξ(E) = k(E)ω(E) . (1.5) Although we shall not elaborate here, we remark that one may easily derive a criterion for (1.3)–(1.5) using the implicit function theorem. Let us restrict attention to the constant energy surface h(x, ξ) = E and to values of (ˆ x, ξ, E) close to (ω(E), ξ(E), E0 ). (Here and henceforth x ˆ = |x|−1 x.) Introduce a change of variables x = xn ω(E) + u , ξ = ξ(E) + η + μω(E) ; (1.6) u · ω(E) = η · ω(E) = 0 . This amounts to considering coordinates (u, xn , η, μ) ∈ Rn−1 × R × Rn−1 × R. We can solve the equation h(ω(E) + u, ξ(E) + η + μω(E)) = E for μ using the implicit function theorem, because ∂μ h ω(E), ξ(E) + μω(E) |μ=0 = k(E) > 0 for E near E0 . We obtain μ = −g(u, η, E) where g is smooth in a neighborhood of (0, 0, E0 ) and g(0, 0, E0 ) = 0. After introducing the “new time” τ = ln xn (t) = ln (x(t) · ω(E)) Hamilton’s equations reduce to dη du = ∇η g(u, η, E) , = −∇u g(u, η, E) . (1.7) dτ dτ (See [3, p. 243].) After linearization of these equations around the fixed point (u, η) = (0, 0) we obtain with w = (u, η) dw 0 I I 0 = B(E)w ; B(E) = A(E) − , −I 0 0 0 dτ (1.8) g gu,η A(E) = u,u . gη,u gη,η u+

Here the real symmetric matrix A(E) of second order derivatives is evaluated at (0, 0, E). We assume all eigenvalues of B(E) have nonzero real part (the hyperbolic case). These eigenvalues are easily proved to come in quadruples, λ, −1 − λ, and their complex conjugates (if λ is not real). If all eigenvalues of B(E) have negative real part then this corresponds to a stable channel. We prefer the word channel because in the case considered xn (t) grows linearly in time. If at least one of the eigenvalues of B(E) has a positive real part then the usual stable/unstable manifold theorem shows that there are always classical orbits (on the stable manifold)

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for which (ˆ x(t), ξ(t)) → (ω(E), ξ(E)) for t → ∞ (throughout this paper we use the convention t → ∞ to mean t → +∞ ). In this situation the question is, do there exist quantum states whose propagation is governed by a self-adjoint quantization H of h(x, ξ) on L2 (Rn ) (possibly with the singularity at x = 0 removed) which exhibit this behavior? With a mild further requirement (see (1.10) below), we will answer this question in the negative. To be precise, let us first fix a (small) neighborhood U0 ⊆ (Rn \ {0}) × Rn of (k(E0 )ω0 , ξ0 ). Then we consider a small open neighborhood I0 of E0 and states of the form ψ = f (H)ψ with f ∈ C0∞ (I0 ) such that: For all g1 , g2 ∈ C0∞ (Rn ) g1 (t−1 x) − g1 k(H)ω(H) 1I0 (H) ψ(t) → 0 for t → ∞ , g2 (p) − g2 ξ(H) 1I0 (H) ψ(t) → 0 for t → ∞ ;

ψ(t) = e−itH ψ , while 1

∞

(1.9)

p = −i∇x ,

2 t−1 aw (t−1 x, p)ψ(t) dt < ∞ for all a ∈ C0∞ U0 \ γ(I0 ) ; γ(I0 ) = k(E)ω(E), ξ(E) | E ∈ I0 .

(1.10)

(Here aw signifies Weyl quantization, and 1I0 is the characteristic function of I0 .) Notice that by (1.9), at least intuitively, for all such symbols a w −1 a (t x, p)ψ(t) → 0 for t → ∞ , (1.11) so that (1.10) appears as a weak additional assumption (or as part of our definition of a quantum channel). See the beginning of Section 3 where (1.11) is proved from (1.9) and assumptions about the pseudodifferential nature of H (conditions (H1)–(H3)). On the other hand, (1.11) is also a consequence of (1.10) as may be shown by a subsequence argument (cf. the proof of (8.22)). The states ψ obeying the above conditions (with fixed I0 ) form a subspace whose closure, say H0 , is H-reducing. We show the following (main) result. Theorem 1.1. Suppose B(E0 ) has an eigenvalue with a positive real part. Then under a certain assumption concerning possible resonances (and other technical conditions, see (H1)–(H8) in Section 2) there exists a sufficiently small open neighborhood I0 of E0 such that H0 = {0} . (1.12) There is the following slightly more general result not involving (1.9). Theorem 1.2. Under the conditions of Theorem 1.1 there exists a sufficiently small open neighborhood I0 of E0 such that if a state ψ(t) = e−itH f (H)ψ with f ∈ C0∞ (I0 ) obeys (1.10), then in fact the pointwise decay (1.11) holds for all a ∈ C0∞ (U0 ).

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A symbol satisfying the conditions (1.4) and (1.5) was studied by Guillemin and Schaeffer [12]. In their paper the roles of x and ξ are reversed and their h is homogeneous of degree one in ξ. There is only one half-line of points in question rather than a one parameter family of half-lines (their critical set of points is at zero energy). Under the condition of no resonances they obtain a conjugation of H to a simpler normal form from which they draw conclusions about propagation of singularities for an equation of the form Hψ = φ. x) To see what Theorem 1.1 means in the model where h(x, ξ) = 2−1 ξ 2 + V (ˆ with V a Morse function on S n−1 we recall from [15]: The spectrum of H = 2−1 p2 + V (ˆ x) is purely absolutely continuous and

Pl , (1.13) I= ωl ∈Cr

where Pl are H-reducing orthogonal projections defined as follows: Pick any family {χl |ωl ∈ Cr } of smooth functions on S n−1 with χk (ωl ) = δkl (the Kronecker symbol); here Cr is the finite set of non-degenerate critical points in S n−1 for V . Then Pl = s − lim eitH χl (ˆ x)e−itH , t→∞

see [15] and [1]. Furthermore in [15] the existence of an asymptotic momentum p+ was proved and its relationship to the above projections was shown. (There was the restriction in [15] to n ≥ 3 but this is easily removed using the Mourre estimate [1, Theorem C.1].) We notice that (1.13) has an analog in Classical Mechanics: Any classical orbit (except for the exceptional ones that collapse at the origin) obeys |x| → ∞ with x ˆ → ωl for some ωl ∈ Cr . Obviously the collection (1.3)–(1.5) corresponds in the potential model ex actly to Cr : (ω(E), ξ(E)) = (ωl , 2(E − V (ωl ))ωl ) with ωl ∈ Cr . The assumption that the real part of one eigenvalue is positive corresponds to ωl being either a local maximum or a saddle point of V . Moreover we have the identification (1.14) H0 = Ran Pl 1I0 (H) . Whence, upon varying I0 , Theorem 1.1 yields the following for the potential model. Theorem 1.3. Suppose ωl ∈ Cr is the location of a local maximum or a saddle point of V . Then (1.15) Pl = 0 . Of course we will need to verify (1.14) in order to use Theorem 1.1 and this involves verifying (1.9) and (1.10) for ψ ∈ RanP satisfying ψ = f (H)ψ, f ∈ C0∞ (I0 ) (see Section 8). A detailed analysis of the large time asymptotic behavior of states in the range of the projections Pl which correspond to local minima was accomplished recently in [18]. In particular for any local minimum, Pl = 0. Moreover in this case we have (1.14) for the analogous space of that in Theorem 1.1. One may

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easily include in Theorem 1.3 a short-range perturbation V1 = O(|x|−1−δ ), δ > 0, ∂xα V1 = O(|x|−2 ), |α| = 2, to the Hamiltonian H, see Remarks 8.3 (1). The results Theorems 1.1 and 1.2 are much more general than Theorem 1.3. In particular, as a further application, we apply them to a problem of a charged quantum particle in two dimensions subject to an electromagnetic vector potential which is asymptotically homogeneous of degree zero in x, see [6]. (For another magnetic field problem in this class, see [5].) Here the magnetic field which is given asymptotically by B0 (x) = b(θ)/r (in polar coordinates) can be zero on certain rays. If qb (θ0 ) < 0 at a zero θ0 of b (here q denotes the charge of the particle), the orbit with magnetic field B0 for which r → ∞ and whose x-space projection is the ray θ = θ0 has both stable and unstable manifolds associated with it. Using Theorems 1.1 and 1.2 it is shown in [6] that quantum orbits corresponding to the classical stable manifold do not exist. Let us also give a simple example from Riemannian geometry. Example 1.4. Consider the symbol h on (R2 \ {0}) × R2 −1 2 1 2 1 h = h(x, ξ) = 1 + ax22 |x|−2 ξ1 + ξ2 ; a > 0 . (1.16) 2 2 √ The family (1, 0; 2E, 0), E > 0, consists of points obeying (1.3), (1.4) and √ (1.5). For the linearized reduced flow (1.8) we find the eigenvalues − 12 (−1 ± 1 + 4a), and we conclude that the fixed points are saddle points. If a is irrational there are no resonances of any order (see Section 2 for definition), whence we may infer from Theorem 1.1 that there is no quantum channel associated to the family of fixed points in this case. Using the absence of low order resonances condition (2.6) we may in fact obtain this conclusion for a = 34 , 2; see Remark 2.1 for a further discussion. We have tacitly assumed that the symbol (1.16) is suitably regularized at x = 0 (for the quantization). Our proof of Theorem 1.1 consists of three steps: I) Assuming ψ(t) = e−itH ψ does localize in phase space as t → ∞ in the region |u| + |η| ≤ for any > 0 in the sense of (1.8) and (1.9), we prove a stronger localization. Namely, for some small positive δ, the probability (assuming here that ψ is normalized) that ψ(t) is localized in the region |u| + |η| ≥ t−δ goes to zero as t → ∞. See Section 4. II) Using I) and an iteration scheme, we construct an observable Γ which decreases “rapidly” to zero. This iteration scheme is based on one used by Poincaré (see [2, pp. 177–180]) to obtain a change of coordinates which linearizes (1.7). The fact that if one eigenvalue of B(E) has a positive real part then another has real part < −1 is relevant here. Our observable Γ is in first approximation roughly a quantization of a component of w in (1.8) which decays as exp (λτ ) with Re λ < −1. See Section 5. III) Using Mourre theory we prove an uncertainty principle lemma for two self-adjoint operators P and Q satisfying i[P, Q] ≥ cI, c > 0, and some technical conditions. A consequence of this lemma is that if 0 ≤ δ1 < δ2 and g1 and g2 are

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two bounded compactly supported functions then lim g1 (t−δ1 Q)g2 (tδ2 P ) = 0 . t→∞

If ψ is normalized this bound implies that the localizations of I) and II) are incompatible. See Sections 6 and 7. The basic theme of our paper may be phrased as absence of certain quantum mechanical states which are present in the corresponding classical model. Notice that given any critical point ωl ∈ Cr (restricting for convenience the discussion to the potential model) there are indeed classical orbits with |x| → ∞ and x ˆ → ωl ; in particular this is the case for any given local maximum or saddle point. Intuitively, Theorem 1.1 is true because the associated classical orbits occur for only a “rare” set of initial conditions as fixed by the stable manifold theorem. Alternatively, for some components of (ˆ x, ξ) the convergence to (ωl , ξ + ) is “too fast” thus being incompatible with the uncertainty principle in Quantum Mechanics. These two different explanations are actually connected. For another example of this theme we refer to [11, 24] and [25]. We addressed the problem of Theorem 1.3 in a previous work, [16], where we proved (1.15) at local maxima but only had a partial result for saddle points (using a different time-dependent method). Also in the case of homogeneous potentials similar and related results were obtained in [13] and [14] by stationary methods. The present paper is an expanded version of the preprint [17]. This paper is organized as follows: In Section 2 we elaborate on all technical conditions needed for Theorem 1.1 and give a more detailed outline of its proof, cf. the steps I)–III) indicated above. In Section 3 we have collected a few technical preliminaries. In Section 4 we prove the t−δ -localization, cf. step I), while the localization of Γ is given in Section 5. Finally, Section 6 is devoted to the Mourre theory for this observable. We complete the proof of Theorem 1.1 in Section 7 (the proof of Theorem 1.2 is omitted since it follows the same pattern) and give a few missing details of the proof of Theorem 1.3 in Section 8. In Appendix A we study possible generalizations of the homogeneity condition (1.1).

2. Technical conditions and outline of proof We fix (ω0 , ξ0 ) ∈ S n−1 × Rn and a small open neighborhood I0 of E0 = h(rω0 , ξ0 ) as in Section 1. We shall elaborate on conditions for the real-valued symbol h(x, ξ), see (H1)–(H8) below. For convenience we remove a possible singularity at x = 0 caused by the imposed (local) homogeneity assumption of Section 1. This may be done as follows. Let N0 be as small open neighborhood of (ω0 , ξ0 ). We shall now and henceforth assume that for some r0 > 0

ˆ, ξ) in C0 := (x, ξ) | (ˆ x, ξ) ∈ N0 , |x| > r0 , h(x, ξ) = h(r0 x (H1) h ∈ C ∞ (Rn × Rn ) .

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Notice that this modification intuitively is irrelevant for the issue of Theorem 1.1 (which concerns states propagating linearly in time in configuration space). We assume that for some r, l ≥ 0 1/2 h ∈ S ξ r x l , g0 ; g0 = x −2 dx2 + dξ 2 , x = (1 + |x|2 ) , (H2) and that

H = hw (x, p) is essentially self-adjoint on C0∞ (Rn ) . (See Section 3 for notation.)

(H3)

Remark. There is some freedom in choosing a global condition like (H2). For −2δ 2δ example it suffices to have (H2) with g0 replaced by x 1 dx2 + x 2 dξ 2 with 0 ≤ δ2 < δ1 ≤ 1 . We assume (H4) (1.3)–(1.5) for E ∈ I0 . We define ωn (E) = ω(E), and shrinking I0 if necessary we pick smooth functions ω1 (E), . . . , ωn−1 (E) ∈ S n−1 such that ω1 (E), . . . , ωn (E) are mutually orthogonal. We define, cf. (1.6), xj = x · ωj (E) for j ≤ n, uj = xj /xn and ηj = (ξ − ξ(E)) · ωj (E) for j ≤ n − 1 and μ = (ξ − ξ(E)) · ωn (E). Let w = (u, η) = (u1 , . . . , un−1 , η1 , . . . , ηn−1 ). As for the matrix B(E) of (1.8) in these coordinates we need the condition: The real part of each eigenvalue of B(E) is (H5) nonzero for E ∈ I0 . Let us order the eigenvalues as β1s (E), . . . , βns s (E), β1u (E), . . . , βnuu (E) where Re (βjs (E)) < 0 (βjs (E) are the stable ones) and Re (βju (E)) > 0 ( βju (E) are the unstable ones). Let β(E) refer to the C2n−2 -vector of eigenvalues (β1s (E), . . . , βnuu (E)) counted with multiplicity. We are interested in the case nu = nu (E) ≥ 1 . s

(H6)

u

Let V (E) and V (E) be the sum of the generalized eigenspaces of B(E) corresponding to stable and unstable eigenvalues, respectively. Then we have the decomposition C2n−2 = V s (E) ⊕ V u (E) . Using basis vectors respecting this structure we can find a smooth M2n−2 (C)valued function T (E) such that T (E)−1 B(E)T (E) = diag B s (E), B u (E) . (2.1) We may assume the following at E = E0 : Corresponding to the decomposition into generalized eigenspaces C2n−2 = V s ⊕ V u = V1s ⊕ · · · ⊕ Vnss ⊕ V1u ⊕ · · · ⊕ Vnuu , T (E0 )−1 B(E0 )T (E0 ) = diag (B1s , . . . , Bnuu ) ,

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where for all entries Nj# := Bj# − βj# (E0 )Idim (V # ) is strictly lower triangular. j

Given any > 0 we may assume (by rescaling the basis vectors) that Nj# ≤ .

(2.2) s

u

We introduce a vector of new variables γ = (γ , γ ) = (γ1 , . . . , γ2n−2 ) −1 γ = γ w(E), E = T (E) w(E) , (2.3) where γ s and γ u are the vectors of coordinates of the part of w(E) in V s (E) and V u (E), respectively. We shall make the assumption (using “tr” to denote transposed): tr

There exists a smooth eigenvector v(E) of B(E) in E ∈ I0 , such that Re (λ(E)) < −1 for the corresponding eigenvalue λ(E).

(H7)

See Remark 2.3 below for an alternative condition. The ordering of the eigenvalues may be chosen such that β1s (E) = λ(E) .

(2.4) −1

It may also be assumed that v(E) is the first row of T (E) . Clearly by (2.4) β1s (E) is smooth for E ∈ I0 . We call E0 a resonance of order m ∈ {2, 3, . . . } for an eigenvalue βj# (E0 ) if for some α = (α1 , . . . α2n−2 ) ∈ (N∪{0})2n−2 with |α| = m, βj# (E0 ) = β(E0 ) · α .

(2.5)

E0 is not a resonance of order ≤ m0 for β1s (E0 ) .

(H8)

We assume that

Here m0 may be extracted from the bulk of the paper; the condition 1 + Re β1s (E0 ) 1 + Re βns s (E0 ) ,..., m0 > max 4, − Re β1s (E0 ) − Re βns s (E0 )

(2.6)

suffices. Remark 2.1. Typically the set of resonances of all orders will be dense in I0 . The theorem proved with (H8) does not exclude cases where there are low order resonances as long as they constitute a discrete set. This is used in the proof of Theorem 1.3 in Section 8. For the exceptional values a = 34 and a = 2 of Example 1.4 there are resonances of order 5 and 4, respectively. For these values of a all positive energies are resonances, and consequently our theorem is not applicable. We shall build a (classical) observable Γ from the first coordinate γ1 = γ1 (w(E), E) = v(E) · w(E) of γ s = γ s (w(E), E) 2 (2.7) Γ = γ1 w(E), E + O γ w(E), E .

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In the study of an analogous quantum observable we consider in detail the case where for some 1 ≤ l ≤ n − 1 ∂ηl γ1 (w, E0 )|w=0 = 0 .

(2.8)

We notice that if (2.8) is not true then for some 1 ≤ l ≤ n − 1 ∂ul γ1 (w, E0 )|w=0 = 0 .

(2.9)

The construction of the quantum Γ in the case of (2.8) and an elaboration of its decay properties will be given in Section 5. A Mourre estimate is given in Section 6, and we complete the proof of Theorem 1.1 in this case in Section 7. We refer the reader to Remarks 5.3, 6.4 and 7.2 for the modifications needed for showing Theorem 1.1 in the case of (2.9). 2.1. Outline of proof of Theorem 1.1 Consider a classical orbit with (ˆ x(t), ξ(t)) → (ω(E), ξ(E)) for t → ∞ (and E near E0 ). How do we prove the bound |u| + |η| ≤ Ct−δ for some positive δ? We consider the observables q s = |γ s |2 ,

q u = |γ u |2 ,

q− = qu − qs ,

q + = q u + q s = |γ|2 .

Using (1.7) and (2.1) we compute ∂μ h s d γ= B (E)γ s , B u (E)γ u + O(q + ) . dt xn

(2.10)

(2.11)

For > 0 small enough in (2.2) the equation (2.11) leads to d − d d q = 2 Re γ u , γ u − 2 Re γ s , γ s ≥ δ − t−1 q + u s dt dt dt n n C C

(2.12)

for some positive δ − (which may be chosen independent of E close enough to E0 ) and for all t ≥ t− (with t− large enough). In particular q − is increasing and hence q− ≤ 0 ;

t ≥ t− .

(2.13)

Using (2.11), (2.13) and the Cauchy–Schwarz inequality we compute d s s d s q = 2 Re γ , γ ≤ −2δ s t−1 q s (2.14) dt dt Cns for some positive δ s and all t ≥ ts . Integrating (2.14) yields s

q s ≤ C s t−2δ ,

t ≥ ts .

(2.15) s −2δ s

Finally from (2.13) and (2.15) we conclude that q ≤ 2C t +

that

and therefore

(2.16) |γ| ≤ Ct−δ ; δ ≤ δ s . This classical proof will be the basis for our quantum arguments in Section 4 which constitute step I) of the proof of Theorem 1.1.

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Remarks 2.2. 1. We may choose the positive δ in (2.16) as close to the (optimal) exponent min (Re (−β1s (E0 )), . . . , Re (−βns s (E0 ))) as we wish (provided E is taken close enough to E0 ). 2. Although not needed, one may easily prove using similar differential inequal2 ities that indeed q u = O((q s ) ) in complete agreement with the stable manifold theorem. Classical Γ. To implement step II) of the proof, we shall for each m ∈ {1, . . . , m0 } construct a γ (m) of the form (2.7) such that d (m) ∂μ h s (m) γ = β1 γ + O(|γ|m+1 ) ; β1s = β1s (E) . (2.17) dt xn Specifically we shall require γ (1) = γ1 ,

and γ (m) = γ1 +

cα γ α ;

m ≥ 2,

(2.18)

2≤|α|≤m α

2n−2 with γ α = γ1α1 · · · γ2n−2 . (It will follow from the construction below that the coefficients cα = cα (E) will be smooth; this will be important for “quantizing” the symbol.) We proceed inductively. Clearly by (2.11) wehave (2.17) for m = 1. Now suppose we have constructed a function γ (m−1) = |α|≤m−1 cα γ α obeying ⎛ ⎞

d (m−1) ∂μ h s ⎝ (m−1) γ = β γ + dα γ α + O(|γ|m+1 )⎠ , dt xn 1

|α|=m

then we add to γ (m−1) a function of the form

|α|=m cα γ

α

and we need to solve

d

∂μ h s

cα γ α = β (cα − dα )γ α + O(|γ|m+1 ) . dt xn 1 |α|=m

(2.19)

|α|=m

For that we compute the derivative using again (2.11). Let us denote by tr tr βij the ij’th entry of the matrix diag(B s (E) , B u (E) ). Then (2.19) reduces to solving

˜ i +ej α ˜ i βij cα˜ γ α−e = β1s (cα − dα )γ α , (2.20) i,j |α|=m ˜

|α|=m

which in turn reduces to solving the system of algebraic equations

(αi + 1 − δij )βij cα+ei −ej = β1s (cα − dα ) ; |α| = m .

(2.21)

i,j

Here ei and ej denote canonical basis vectors in R2n−2 and δij is the Kronecker symbol.

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Clearly (2.21) amounts to showing that β1s is not an eigenvalue of the linear ˜ on Cn˜ with map B (m + 2n − 3)! 2n−2 n ˜ = # α ∈ N∪{0} | |α| = m = (2n − 3)!m! given by ⎛ ⎞

˜ Cn˜ c = (cα ) → Bc =⎝ (αi + 1 − δij )βij cα+ei −ej ⎠ ∈ Cn˜ . α

α

i,j

α

Since βij = βij (E) depends continuously on E ∈ I0 we only need to show that ˜ 0 ) − β1s (E0 )I is invertible . B(E (2.22) By the condition (H8) indeed (2.22) holds since m ≤ m0 and the spectrum

˜ 0 ) = β(E0 ) · α| |α| = m . σ B(E tr

tr

The latter is obvious if diag (B s (E0 ) , B u (E0 ) ) is diagonal. In general this may be seen by a perturbation argument, see [22, p. 37]. Finally we define Γ = γ (m0 ) . If we have m0 so large that δ(m0 + 1) > −β1s (E) where δ is given as in (2.16) ∂ h we infer by integrating (2.17) (since limt→∞ t xμn = 1) that s (2.23) Γ = γ1 + O(|γ|2 ) = O tβ1 (E)+ ; > 0 . Remark 2.3. We could have used a different observable constructed by a similar iteration using as γ (1) a component of γ corresponding to an eigenvector with eigenvalue λ(E) having Re (λ(E)) > 0. We would again need smoothness of the eigenvector and a non-resonance condition for λ(E0 ), cf. (H7) and (H8). The analogous observable γ (m) decreases as t−δ(m+1) with no upper bound on m (assuming E0 is not a resonance of any order). But as we will see below, the correspondence between classical and quantum behavior is not so precise as to allow a similar statement in Quantum Mechanics. Thus it does not much matter which of these observables is used. Quantum Γ. To get a statement like (2.23) in Quantum Mechanics we need to quantize the classical symbol γ (m) = γ (m) (x, ξ). We choose a quantization that takes into account localizations of the states ψ = f (H)ψ obeying (1.9) and (1.10). We fix m = m0 depending on an analogue of the classical bound (2.16), cf. the classical case discussed above. Without going into details, in the case of (2.8) this operator takes the form Γ = Γ(t) = p − ξ(E0 ) · ωl (E0 ) + B1 (t) ; B1 (t) bounded . We want B1 (t) to be bounded to facilitate our uncertainty principle argument (see Section 6). The fact that this works even though the classical Γ does not have this form rests on the localizations of ψ. Strictly speaking, to get the above

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expression we first make the modification of the classical Γ of dividing by the constant cl = ∂ηl γ1 (w, E0 )|w=0 and then taking the real part; we shall not discuss the case of (2.9) here. We show the following analog of (2.23): Given σ > 0 we have for some Γ of this form the strong localization 1[tσ−1 ,∞) (|Γ|)e−itH ψ → 0 for t → ∞ . (2.24) We notice that (2.24) is a weaker bound than (2.23); to control various commutators we need to have σ positive. On the other hand it may appear somewhat surprising that such localization result can be proved at all for σ < 2−1 . According to folklore wisdom there is usually a strong connection for pseudodifferential operators between the functional calculus and the pseudodifferential calculus, see for example [8, Appendix D]. In our case one might think that (2.24) is equivalent to a statement like −itH e−itH ψ ≈ aw ψ t (x, p)e

for t → ∞ ,

(m0 ) Re (c−1 )) l γ

where the symbol at = h(t for suitable h ∈ C0∞ (R) and γ (m0 ) given by the classical symbol (possibly modified by cut-offs) discussed above. However for σ < 2−1 such symbols at do not fit into any standard (parameterdependent) pseudodifferential calculus which by the uncertainty principle essen tially would require the uniform bounds ∂ξβ ∂xα at = O tδ2 |β|−δ1 |α| with δ2 < δ1 . As a consequence we shall base our proof of (2.24) on a functional calculus approach. Using a differential equality related to (2.17) we can indeed bound certain quantum errors in a calculus even for σ < 2−1 . It is important that we can take σ small; see below. Somewhat related problems were studied in [10] and [5]. 1−σ

Remarks. 1. There is a subtle point suppressed in the above discussion which is very important technically. Although the t−δ -localization proved in step I) is needed to construct the quantum Γ and prove (2.24), its full force cannot be used for this purpose. The reason is that an effective use of the operator calculus limits the strength of this localization (this is basically the uncertainty principle again). Thus using a strong t−δ -localization results in a weaker localization for Γ. The full force of the t−δ -localization is only exploited at the very end of the proof of Theorem 1.1 in Section 7. 2. Another technical point not discussed here is the use of a certain hierarchy of localizations in the construction of Γ (and A¯ below) necessary because of the variation of (ω(E), ξ(E)) with E. The fact that our procedure here actually works may look almost miraculous at first glance (see (6.5) and (6.6)). Implementing the uncertainty principle. The last step in our proof of Theorem 1.1 is the decisive one; here Quantum Mechanics enters crucially. We show that a localization similar to the classical bound (2.16) and (2.24) are incompatible unless ψ = 0. First fix δ > 0 in agreement with (2.16). More precisely we need the localization ¯ −itH ψ → 0 for t → ∞ , (2.25) e−itH ψ ≈ h2 (A)e

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for some h2 ∈ C0∞ (R) and some operator of the form A¯ = tδ−1 xl + B2 (t) ; B2 (t) = O tδ , xl = x · ωl (E0 ) . ¯ = t1−δ Γ. Then fix any σ ∈ (0, δ) and introduce with Γ as in (2.24) the operator H We prove a global Mourre estimate ¯ A] ¯ ≥ 2−1 I . i[H, (2.26) Abstract Mourre theory and (2.26) lead to the bound ¯ ≤ Ct(σ−δ)/2 , ¯ 1 (tδ−σ H) h2 (A)h

(2.27)

valid for all h1 , h2 ∈ C0∞ (R). Finally picking localization functions in agreement with (2.25) and (2.24) we conclude from (2.27) that ¯ −itH ψ → 0 for t → ∞ , ¯ 1 (tδ−σ H)e e−itH ψ ≈ h2 (A)h completing the proof.

3. Preliminaries We use the notation Ψ(m, g) for the space of operators given by quantizing symbols in the symbol class S(m, g) as defined by [19, (18.4.6)]. For the weight functions m and metrics g relevant for this paper it does not matter here whether “quantize” refers to Weyl or Kohn–Nirenberg quantization. For a ∈ S(m, g) we use the notation aw (x, p) to denote the Weyl quantization of a. We refer the reader to [8, Appendix D] and [19, Chapter 18] for a detailed account of the calculus of pseudodifferential operators. We shall deal with various kinds of parameterdependent symbols. In one case the parameter is time t ≥ 1 and for that we introduce the following shorthand notation. Definition 3.1. A family {at |t ≥ 1} of symbols in S(m, g) is said to be uniform in S(m, g) if for all semi-norms || · ||k on S(m, g) (cf. [19, (18.4.6)]) supt at k < ∞. In this case we write at ∈ Sunif (m, g) and aw t (x, p) ∈ Ψunif (m, g). Given this uniformity property various bounds from the calculus of pseudodifferential operators are uniform in the parameter (by continuity properties of the calculus). We shall also deal with parameter-dependent metrics. Specifically we shall consider for 0 ≤ δ2 < δ1 ≤ 1 and t ≥ 1 gt = gtδ1 ,δ2 = t−2δ1 dx2 + t2δ2 dξ 2 .

(3.1)

Similarly to Definition 3.1 we shall write (for given l ∈ R), at ∈ Sunif (tl , gt ) l and aw t (x, p) ∈ Ψunif (t , gt ) meaning that for all (time-dependent) semi-norms supt at t,k < ∞. Also in this case various bounds from the calculus of pseudodifferential operators will be uniform in the parameter. Some extensions of this idea will be used without further comment.

522



One may verify that (1.11) follows from (1.9) by applying a partition of unity to the f of any state ψ = f (H)ψ of (1.9) to decompose it as f = fi and by noticing that (1.9) remains valid for the sharper localized states ψ → ψi = fi (H)ψ. (Notice that if supp (fi ) is located near Ei this leads to t−1 x ≈ k(Ei )ω(Ei ) and p ≈ ξ(Ei ) along ψi (t).) The latter follows readily upon commutation and applying Lemma 3.2 stated below. The same argument shows that indeed H0 is H-reducing. (This property may also be verified without appealing to Lemma 3.2.) ˜1 ∈ C0∞ (Rn ) such that g1 = 1 in a (small) neighPick non-negative g1 , g˜1 , g˜ borhood of k(E0 )ω0 , g˜1 = 1 in a neighborhood of supp (g1 ) and g˜˜1 = 1 in a neighborhood of supp (˜ g1 ). Similarly, pick non-negative g2 , g˜2 , g˜˜2 ∈ C0∞ (Rn ) such that g2 = 1 in a neighborhood of ξ0 , g˜2 = 1 in a neighborhood of supp (g2 ) and g2 ). We suppose supp (g˜˜1 ) × supp (g˜˜2 ) ⊆ U0 g˜ ˜2 = 1 in a neighborhood of supp (˜ (with U0 given as in (1.10)), and in fact that the supports are so small that for some t0 ≥ 1 the symbol g1 (t−1 x)˜ g2 (ξ) ht (x, ξ) := h(x, ξ)˜ ˆ, ξ)˜ g1 (t−1 x)˜ g2 (ξ) ; = h(r0 x

(3.2)

t ≥ t0 ,

cf. (H1). By the assumption (H2) we then have ht ∈ Sunif (1, g0 ) ∩ Sunif (1, gt1,0 ) .

(3.3)

Lemma 3.2. For all f ∈ C0∞ (R) the family 1,0 f hw t (x, p) ∈ Ψunif (1, g0 ) ∩ Ψunif (1, gt ) and

(3.4)

(x, p) − f (H) g1 (t−1 x)g2 (p) f hw = O(t−∞ ) . t

(3.5)

This lemma facilitates the transition between the functional calculus and the pseudo-differential operator calculus, both of which are used in this paper. Proof. As for (3.4) we may proceed as in the proofs of [8, Propositions D.4.7 and D.11.2]. (One verifies the Beals criterion using the representation (3.10) given below and the calculus of pseudodifferential operators.) w w For (3.5) we let B = hw t (x, p) and G = h (x, p) − ht (x, p). By (3.10) 1 −1 −1 ∂¯f˜ (z)(B − z) G(H − z) dudv . (x, p) − f (H) = (3.6) f hw t π C For any large m ∈ N we may decompose −1

(B − z)

G=

m

k=1

−k

adkB (G)(B − z)

−1

+ (B − z)

−m

adm B (G)(B − z)

,

(3.7)

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yielding (by the calculus) −1

g1 (t−1 x)g2 (p)(B − z)

G=

m

−k

Rk (B − z)

k=1 −m + g1 (t−1 x)g2 (p)(B − z)−1 adm ; B (G)(B − z)

(3.8)

Rk = O(t−∞ ) . l−m l−m , g0 ) and therefore adm ), whence By (H2), adm B (G) ∈ Ψunif ( x B (G) = O(t −1 (3.9) g1 (t−1 x)g2 (p)(B − z) G ≤ Ctl−m | Im z|−(m+1) uniformly in z ∈ supp f˜ . Clearly (3.5) follows from (3.6) and (3.9).

Remark 3.3. The statements of Lemma 3.2 extend to any smooth function f dk m−k with dλ ) (for fixed m ∈ R); in particular Lemma 3.2 holds for k f (λ) = O(λ f (λ) = λ. Definition 3.4. Let F+ denote the largest set of F = F+ ∈ C ∞ (R), that √ such √ 0 ≤ F ≤ 1, F ≥ 0, F ∈ C0∞ (( 12 , 34 )), F ( 12 ) = 0, F ( 34 ) = 1 and 1 − F , F , √ F ∈ C ∞ , which is stable under the maps F → F m and F → 1 − (1 − F )m ; m ∈ N. Let F− denote the set of functions F− = 1 − F+ where F+ ∈ F+ . We shall in Section 5 use a modification of the abstract calculus [7, Lemma A.3 (b)], see also [8, Appendix C], [10, Appendix] or [21]. ¯ and B are self-adjoint operators on a complex Hilbert Lemma 3.5. Suppose H space H , and that {B(t) | t > t0 } is a family of self-adjoint operators on H with the ¯ is bounded, that the commutacommon domain D(B(t)) = D(B). Suppose that H ¯ tor form i[H, B(t)] defined on D(B) is a symmetric operator with same (operator) −1 is continuously domain D(B) and that the B(H)-valued function B(t)(B − i) differentiable. Then (A) For any given F ∈ C0∞ (R) we let F˜ ∈ C0∞ (C) denote an almost analytic extension. In particular −1 1 ∂¯F˜ (z) B(t) − z dudv , z = u + iv . (3.10) F B(t) = π C The B(H)-valued function F (B(t)) is continuously differentiable, and introd ¯ · ], the form ducing the Heisenberg derivative D = dt + i[H, d ¯ F B(t) F B(t) + i H, dt is given by the bounded operator −1 −1 1 ∂¯F˜ (z) B(t) − z DB(t) B(t) − z DF B(t) = − dudv . (3.11) π C

524


Ann. Henri Poincaré 1

In particular if DB(t) is bounded then for any > 0 (with z = (1 + |z|2 ) 2 ) DF B(t) ≤ C sup z +2 | Im z|−2 |(∂¯F˜ )(z)| ||DB(t)|| . (3.12) z∈C

(B) Suppose in addition that we can split DB(t) = D(t) + Dr (t), where D(t) and Dr (t) are symmetric operators on D(B) and that the form ik adkB(t) (D(t)) =

i[ik−1 adk−1 B(t) (D(t)), B(t)] for k = 1 defined on D(B) is a symmetric operator on D(B); ad0B(t) (D(t)) = D(t). (No assumption is made for the form when k = 2.) Then the contribution from D(t) to (3.11) can be written as −1 1 −1 ∂¯F˜ (z) B(t) − z − D(t)(B(t) − z) dudv π C 1 (3.13) = F B(t) D(t) + D(t)F B(t) + R1 (t) ; 2 −2 1 R1 (t) = ∂¯F˜ (z) B(t) − z 2π C −2 · ad2B(t) D(t) B(t) − z dudv . For all f ∈ C0∞ (R) 1 2 f B(t) D(t) + D(t)f 2 B(t) 2 = f B(t) D(t)f B(t) + R2 (t) ; −1 −1 1 R2 (t) = 2 ∂¯f˜ (z2 ) ∂¯f˜ (z1 ) B(t) − z2 B(t) − z1 2π C C −1 −1 B(t) − z2 ad2B(t) D(t) B(t) − z1 du1 dv1 du2 dv2 .

(3.14)

(C) Suppose in addition to previous assumptions that for all t > t0 the form i[D(t), B(t)] extends from D(B) to a bounded self-adjoint operator. Similarly suppose the operator Dr (t) extends to a bounded self-adjoint operator. Then for all F ∈ F+ the B(H)-valued function F (B(t))(B − i)−1 is continuously differentiable, and there is an almost analytic extension with (∂¯F˜ )(z) ≤ Ck z −1−k | Im z|k ; k ∈ N , (3.15) yielding the representation 1 1 DF B(t) = F 2 B(t) D(t)F 2 B(t) + R1 (t) + R2 (t) + R3 (t) , (3.16) √ where R1 (t) is given by (3.13), R2 (t) by (3.14) with f = F and R3 (t) is the contribution from Dr (t) to (3.11). Remarks. 1. The left hand side of (3.16) is initially defined as a form on D(B) while the terms on the right hand side are bounded operators. We shall use the stated representation formulas for bounding these operators in an application in the

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proof of Proposition 5.1; this will be in the spirit of (3.12) although somewhat more sophisticated. ¯ is bounded; 2. There are versions of Lemma 3.5 without the assumption that H they are not needed in this paper.

4. t−δ -localization Let ψ = f (H)ψ be any state obeying (1.9) and (1.10) with f supported in a very small neighborhood of E0 (in agreement with the smallness of the neighborhood I0 of Theorem 1.1). Let g1 , g˜1 , g2 , g˜2 ∈ C0∞ (Rn ) be given as in (3.2) and (3.5). In particular we have g1 k(E)ω(E) f (E) = f (E) , g2 ξ(E) f (E) = f (E) . Consider for t, κ ≥ 1 symbols

a = at,κ (x, ξ) = F+ κq − (x, ξ) g˜1 (t−1 x)˜ g2 (ξ) , −

(4.1) −

where F+ is given as in Definition 3.4 and q is built from the q of (2.10) by ˆ, ξ) cf. (3.2), writing q − = q − (w(E), E) and substituting for E the symbol h(r0 x ˆ, ξ) , h(r0 xˆ, ξ) . (4.2) q = q − w h(r0 x We shall consider κ ∈ [1, tν ] with ν > 0. To have a good calculus for the symbol a we need ν < 1/2. Notice that at,κ ∈ Sunif (1, gt1−ν,ν ) ,

(4.3) 2ν−1

. and that the “Planck constant” for this symbol class is h = t Denoting by · t the expectation in the state ψ(t) = e−itH ψ we have the following localization. Lemma 4.1. For all ν ∈ (0, 2/5) w at,tν (x, p) t → 0

for

t → ∞.

(4.4)

This lemma is a quantum version of (2.13). Proof. We shall use a scheme of proof from [7]. Let ∗

L1 (t) = g1 (t−1 x)g2 (p) .

At,κ = L1 (t) aw t,κ (x, p)L1 (t) ;

(4.5)

From (1.11) and the calculus of pseudodifferential operators we immediately conclude that for fixed κ As,κ s → 0 for yielding

− At,κ t =

t

∞

s → ∞,

DAs,κ s ds ,

(4.6)

d + i[H, · ]. We shall show that where D refers to the Heisenberg derivative D = ds the expectation of DAs,κ is essentially positive (in agreement with (2.12)). Up to

526



d terms O(s−∞ ) we may replace D by Ds = ds + i[hw s (x, p), · ], cf. Remark 3.3. First we notice that −1 x)g2 (p) ≥ −Cs5ν−3 , (4.7) g2 (p)g1 (s−1 x) Ds aw s,κ (x, p) g1 (s

where C > 0 is independent of κ ∈ [1, tν ]. This bound follows from the calculus. The classical Poisson bracket contributes by a positive symbol when differentiating q(x, ξ). The Fefferman–Phong inequality (see [19, Theorem 18.6.8 and Lemma 18.6.10]) for this term yields the 2 lower bound O(sν−1 (s2ν−1 ) ) = O(s5ν−3 ). Hence (uniformly in κ) DAs,κ ≥ {T + T ∗ } − Cs5ν−3 ;

−1 x)g2 (p) . T = g2 (p)g1 (s−1 x)aw s,κ (x, p)Ds g1 (s

For the contribution from the first term on the right hand side we invoke (1.10) after symmetrizing. We conclude that ∞ DAs,κ s ds ≥ o(t0 ) − Ct5ν−2 uniformly in κ ∈ [1, tν ] . (4.8) t

Pick κ = tν . By combining (4.6), (4.8), and the Fefferman–Phong inequality, we infer that At,tν t → 0

for t → ∞ ,

and therefore (4.4). +

s

u

ˆ, ξ) Let q , q and q be given as in (2.10) upon substituting the symbol h(r0 x for E, cf. the use of q − above. We introduce the symbols a1t = tν−1 q − (x, ξ)F+ tν q − (x, ξ) g˜1 (t−1 x)˜ g2 (ξ) , g2 (ξ) . a2t = tν−1 q + (x, ξ)F+ tν q − (x, ξ) g˜1 (t−1 x)˜ We get the following integral estimate from the above proof employing the uniform boundedness of the family of “propagation observables” At,tν , cf. a standard argument of scattering theory see for example [7, Lemma A.1 (b)]. Lemma 4.2. In the state ψ1 (t) = L1 (t)ψ(t) ∞ 1 w (a ) (x, p) + (a2 )w (x, p) dt < ∞ . t t t t 1

Proof. We substitute κ = tν in the construction (4.5). Then up to integrable terms the left hand side of (4.7) (with s = t) is given by cw t (x, p) with

2 2 ct (x, ξ) = g2 (ξ) g1 (t−1 x) νtν−1 q − (x, ξ) + tν h(x, ξ), q − (x, ξ) F+ tν q − (x, ξ) , where { · , · } signifies Poisson bracket. We have the bounds for some C > 0 and all large enough t 2 2 C −1 ct (x, ξ) ≤ g2 (ξ) g1 (t−1 x) a1t (x, ξ) + a2t (x, ξ) ≤ Cct (x, ξ) , from which we readily get the lemma by the Fefferman–Phong inequality.

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527

Remark 4.3. We shall not directly use Lemma 4.2. However the proof will be important. In particular we shall need the non-negativity of the above symbol ct . Let for t, κ ≥ 1 and 0 < 2δ < min (ν, 2δ s ) with ν < 2/5 and δ s as in (2.14) (this number may be taken independent of E close to E0 , cf. Remarks 2.2 (1)), bt,κ (x, ξ)

= F+ κ−1 t2δ q s (x, ξ) F− tν q − (x, ξ) g˜1 (t−1 x)˜ g2 (ξ) ∈ Sunif (1, gt1−ν,ν ) .

Lemma 4.4. For all > 0

bw t,t (x, p) t → 0

for

t → ∞.

(4.9)

Proof. We shall use another scheme of proof from [7]. Let Bt,κ = L1 (t)∗ bw t,κ (x, p)L1 (t) , cf. (4.5), and write for any (large) t0

Bt,κ t = Bt0 ,κ t0 +

t

t0

DBs,κ s ds .

(4.10)

(4.11)

To show that the left hand side of (4.11) vanishes as t → ∞ (with κ = t ) we look at the integrand on the right hand side: As in the proof of Lemma 4.1 we may replace D by Ds up to a term rs,κ such that t rs,κ ds → 0 uniformly in κ ≥ 1 and t ≥ t0 as t0 → ∞ . t0

Using (1.10) and Remark 4.3 we may estimate the integrand up to terms of this type as w · · · ≤ L1 (s)∗ (b1s,κ ) (x, p)L1 (s) s , where

b1s,κ (x, ξ) = κ−1 s2δ 2δs−1 q s (x, ξ) + h(x, ξ), q s (x, ξ) cs,κ (x, ξ) ; g2 (ξ) . cs,κ (x, ξ) = F+ κ−1 s2δ q s (x, ξ) F− sν q − (x, ξ) g˜1 (s−1 x)˜

We compute, cf. (2.14), that for all large s and a large constant C > 0 −Cs2δ−ν−1 − Cs2δ−1 q s (x, ξ)cs,κ (x, ξ) ≤ b1s,κ (x, ξ) ≤ Cs2δ−ν−1 − C −1 s2δ−1 q s (x, ξ)cs,κ (x, ξ) , from which we conclude that lim sup

t

sup

t0 →∞ κ≥1,t≥t0

t0

DBs,κ s ds ≤ 0 .

(4.12)

As for the first term on the right hand side of (4.11), obviously for fixed t0 Bt0 ,κ t0 → 0 for

κ → ∞.

(4.13)

528



Combining (4.12) and (4.13) we conclude (by first fixing t0 ) that lim sup Bt,t t ≤ 0 , t→∞

whence we infer (4.9).

Next we “absorb” the of Lemma 4.4 into the δ and introduce the symbols bt (x, ξ) = F− t2δ q s (x, ξ) F− tν q − (x, ξ) g˜1 (t−1 x)˜ g2 (ξ) , (4.14) ν − 2δ s 1 −1 g2 (ξ) , bt (x, ξ) = −t F− t q (x, ξ) F− t q (x, ξ) g˜1 (t−1 x)˜ where 0 < 2δ < min (ν, 2δ s ) with ν < 2/5 and δ s as in (2.14). Clearly bt (x, ξ) ∈ Sunif (1, gt1−ν

,ν

) ⊆ Sunif (1, gt1−ν,ν ) ;

ν = ν − δ .

We have the following integral estimate. Lemma 4.5. In the state ψ1 (t) = L1 (t)ψ(t) ∞ 1 w (bt ) (x, p) dt < ∞ . t

(4.15)

1

Proof. We use the proofs of Lemmas 4.2 and 4.4. Notice that to leading order “the derivative” of the symbol F+ t2δ q s (x, ξ) F− tν q − (x, ξ) g˜1 (t−1 x)˜ g2 (ξ) is indeed non-positive, and that F+ = −F− .

By combining Lemmas 4.1 and 4.4 we conclude the following localization result. Proposition 4.6. For any state ψ = f (H)ψ obeying (1.9) and (1.10) with f ∈ C0∞ (I0 ) where I0 is a sufficiently small neighborhood of E0 ||ψ(t) − bw t (x, p)ψ(t)|| → 0

for

t → ∞.

(4.16)

Using the symbol bt (x, ξ) we can bound powers of γ, cf. (2.17). If we define γ = γ(x, ξ) as in (2.3) upon substituting E by the symbols h(r0 x ˆ, ξ) we may consider the symbol 2n−2 . (4.17) γtα (x, ξ) := γ α (x, ξ)bt (x, ξ) ; α ∈ N ∪ {0} We have the bounds ||(γtα ) (x, p)|| = O(t−δ|α| ) . w

(4.18)

Proposition 4.6 and the accompanying (4.18) give the t−δ -localization of step I) of the proof of Theorem 1.1. We will also need the integral estimate of Lemma 4.5 as well as Remark 4.3 in the proof that Γ is well localized in the state ψ(t) (see the proof of Proposition 5.1).

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5. Γ and its localization With the assumption (2.8) we define operators G and Γ as follows: The right hand side of (2.18) with m = m0 is of the form

cα γ α , γ (m0 ) = γ1 + 2≤|α|≤m0

with cα as well as γ1 and γ substitute

α

depending smoothly of E. As done in (4.17) we

E = h(r0 x ˆ, ξ) (5.1) −1 and multiply suitably by the factors g˜˜1 (t x) and g˜˜1 (ξ) as introduced in Section 3 (with small supports). Precisely we pick l ≤ n − 1 such that (2.8) holds and write γ1 = cl ξ − ξ(E0 ) · ωl (E0 ) + rE (x, ξ) ; cl = ∂ηl γ1 (w, E0 )|w=0 . Then we define the operator G = Gt = γtw (x, p) by the symbol γt (x, ξ) = γ 1 (x, ξ) + γt2 (x, ξ) ; γ 1 (x, ξ) = ξ − ξ(E0 ) · ωl (E0 ) , ⎛ γt2 (x, ξ)

−1 ⎝

= (cl )

rE (x, ξ) +

⎞

(5.2)

cα γ (x, ξ)⎠˜ g˜1 (t−1 x)g˜˜2 (ξ) . α

2≤|α|≤m0

For the second term the substitution (5.1) is used. Let Γ = Γt = Re (G). w Clearly the quantization of this second term B1 (t) = (γt2 ) (x, p) is bounded. We shall assume that (5.3) δ(m0 + 1) ≥ 1 , where δ < 2−1 min (ν, 2δs ) is given as in Proposition 4.6. Our proof that Γ is well localized in the state e−itH ψ (see Corollary 5.2) rests on the quantum analog of the differential equation (2.17) and the t−δ -localizations proved in Section 4. In addition we will need integral estimates to bound terms which arise when these “t−δ -localizations” are differentiated (see the proof of Proposition 5.1). We shall use the operator L1 (t) given in (4.5). Let us introduce the notation L2 (t) = bw t (x, p) for the quantization of the first symbol of (4.14). Let us also introduce the “bigger” localization operator w L3 (t) = (˜bt ) (x, p) ; ˜bt (x, ξ) = F− 2−1 t2δ q s (x, ξ) F− 2−1 tν q − (x, ξ) g˜1 (t−1 x)˜ g2 (ξ) .

Notice that also ˜bt (x, ξ) ∈ Sunif (1, g 1−ν t

,ν

);

ν = ν − δ ,

and that indeed for example I − L3 (t) L2 (t)L1 (t) = O(t−∞ ) .

(5.4)

530



We obtain from (2.17), (5.3) and bounds like (4.18) that L3 i[H, G]L3 = −L3 t˜−1 GL3 + O(t−2 ) , ˜−1

where t is omitted in the notation and t −

(5.5)

is the Weyl quantization of the symbol

∂μ h(x, ξ) β1s h(x, ξ) g˜˜1 (t−1 x)g˜˜2 (ξ) . x · ω h(x, ξ)

We may assume that the supports of g˜˜1 and g˜˜1 are so small that Re (t˜−1 ) ≥ t−1 Re g˜˜1 (t−1 x)g˜˜2 (p) + O(t−2 ) . ∗

(5.6)

∗

Next introduce P = Pt = GG + G G where G = Gt is given as above. Using the calculus we compute (with some patience) L3 i[H, P ]L3 = 2 Re L3 i[H, G]L3 G∗ + G∗ L3 i[H, G]L3 + Re L3 i[H, G][G∗ , L3 ] − [G∗ , L3 ]i[H, G]L3 + Re L3 i[H, G∗ ][G, L3 ] − [G, L3 ]i[H, G∗ ]L3 2ν −3 = 2 Re L3 i[H, G]L3 G∗ + G∗ L3 i[H, G]L3 + cw , t (x, p) + O t where ct (x, ξ) = ct = 2 Re

˜bt , ˜bt , γt {h, γt } ∈ Sunif t3ν −3 , g 1−ν ,ν . t

Applying (5.5) to the first two terms on the right hand side and symmetrizing yields

L3 i[H, P ]L3 = −L3 P Re (t˜−1 ) + h.c. L3 + Re GO(t−2 ) + G∗ O(t−2 ) + O(t2ν −3 ) . (5.7) (Here and henceforth the notation h.c. refers to hermitian conjugate, viz. S+h.c. = S + S ∗ .) Notice that the contribution from cw t (x, p) disappears and that we use P Re (t˜−1 ) + h.c. = 2G Re (t˜−1 )G∗ + 2G∗ Re (t˜−1 )G + O(t−3 ) .

(5.8)

We have the following localization result. Proposition 5.1. Let ψ, ν and δ be given as in Proposition 4.6 and suppose (5.3). Then for all σ ∈ (ν , 1 − ν ), ν = ν − δ, and with P = Pt = GG∗ + G∗ G where G = Gt is given as above F+ (t2−2σ P )ψ(t) → 0 for t → ∞ . (5.9) Proof. We shall use the scheme of the proof of Lemma 4.4. Consider with κ = t

for a small > 0 the observable ∗ 2 A(t, κ) = L1 (t) F+ B(t) L2 (t) F+ B(t) L1 (t) ; ¯G ¯∗ + G ¯, G ¯ = G(t; ¯ κ) = κ−1 t1−σ Gt . ¯∗G B(t) = B(t; κ) = G

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531

As before we first compute the Heisenberg derivative treating κ as a parameter and split (with Lj = Lj (t)) DA(t, κ) = T1 (t, κ) + T2 (t, κ) + T3 (t, κ) ; T1 = L∗1 F+ B(t) L22 DF+ B(t) L1 + h.c. , T2 = L∗1 F+ B(t) (DL22 )F+ B(t) L1 , T3 = L∗1 F+ B(t) L22 F+ B(t) DL1 + h.c. The analog of (4.11) is A(t, κ) t = A(t0 , κ) t +

0

We shall prove that

t

lim sup sup

t0 →∞ t≥t0

t0

t

t0

T1 (s, κ) + T2 (s, κ) + T3 (s, κ) s ds .

Ti (s, κ) s ds ≤ 0 ;

i = 1, 2, 3 .

(5.10)

(5.11)

To do this we may replace D by the modified Heisenberg derivative d ¯ ·]; H ¯ = L3 HL3 , L3 = L3 (t) , D3 = + i[H, dt cf. (5.4) and arguments below for (5.17). With this modification we first look at the most interesting bound (5.11) with i = 1. We use (3.16) to write 1 1 D3 F+ B(t) = F+2 B(t) D(t)F+2 B(t) + R1 (t) + R2 (t) + R3 (t) ;

2 − 2σ B(t) − L3 B(t) Re (t˜−1 ) + h.c. L3 . (5.12) D(t) = t Notice that here R3 (t) is given by the integral representation (3.11) of Lemma 3.5 in terms of the bounded operator Dr (t) = D3 B(t) − D(t) which by (5.7) is of the form

d Dr (t) = κ−2 t2−2σ P + κ−2 t2−2σ L3 Hi[L3 , P ] + h.c. dt (5.13) + κ−2 t2−2σ Re GO(t−2 ) + Re G∗ O(t−2 ) + O(t2ν −3 ) . First we examine the contribution from the expectation of the term

2 · · · L2 (s) R1 (s) + R2 (s) L1 (s) + h.c. of the integrand of (5.11) (after substituting (5.12)). We may write, omitting here and henceforth the argument s,

i[D, B] = −i L3 B Re (t˜−1 ) + h.c. L3 , B

= − L3 B Re (t˜−1 ) + h.c. i[L3 , B] + h.c. (5.14) −1 − L3 B Re i[t˜ , B] + h.c. L3 .

532



Substituted into the representation formulas (3.13) and (3.14) of Lemma 3.5 the first term to the right can be shown to contribute by terms of the form κ−2 O(s−∞ ) (using the factors of L1 and L2 and commutation), however the bound κ−1 O(sν −1−σ ) suffices. Here and henceforth O(s−˜ ) refers to a term bounded by Cs−˜ uniformly in t (recall that B contains a factor κ−2 = t−2 ). To demonstrate this weaker bound we compute ¯ ∗ i[L3 , G] + h.c. , ¯ ∗ + κ−1 s1−σ G i[L3 , B] = κ−1 s1−σ i[L3 , G]G i[L3 , G] = O(sν

−1

).

Since the middle factor Re (t˜−1 ) = O(s−1 ) we get the bound κ−1 O(s1−σ )O sν −2 = κ−1 O sν −1−σ . ¯ G ¯ ∗ and B may be considered as bounded in combination with the We used that G, resolvents of B; explicitly we exploited the uniform bounds (after commutation) −1

¯ − z) ||G(B

−1

||(B − z)

|| ,

−1

¯ ∗ (B − z) ||G −1

|| ≤ C| Im z|

,

|| ≤ C

1/2

z , | Im z| −1

||B(B − z)

(5.15)

2 z || ≤ . | Im z|

Similarly, since ¯ ∗ + κ−1 O(s−1−σ )G ¯ + h.c. Re i[t˜−1 , B] = κ−1 O(s−1−σ )G

(5.16)

the second term to the right in (5.14) contributes by a term of the form κ−1 O(s−1−σ ). Using the representation for R3 = R3 (s) and commutation we claim the bound · · · L22 R3 L1 + h.c. = κ−1 O(s−1 ) + κ−1 O(s−1−σ ) + κ−2 O(s2ν −2

−1−2σ

−∞

).

(5.17)

The contributions from the first two terms of (5.13) are κ O(s ) and therefore in particular κ−1 O(s−1 ). Let us elaborate on this weaker bound for the first term: Write −2 2−2σ d −1 1−σ ¯ d ∗ ∗ d ¯ G G +G P =κ s G + h.c. , κ s ds ds ds and compute the time-derivative of the symbol g˜˜1 (s−1 x) that defines the timedependence of the symbol of G d ˜ −1 g˜1 (s x) = −s−2 x · (∇g˜˜1 )(s−1 x) . ds The contribution from this expression is treated by using the factor g1 (s−1 x) of L1 . First we may insert the j’th power of F = g˜1 (s−1 x) next to a factor L1 . Then we place one factor of F next to any of the factors of the time-derivative of G by commuting through the resolvent of B, and repeat successively this procedure for the “errors” given in terms of intermediary commutators. At each step a factor of κ−1 sν −σ = O(sν −σ ) will be gained. (In fact for the first term of (5.13) treated

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here we have the stronger estimate O(s−σ ).) This means that if we put σ = σ − ν then h = s−σ will be an “effective Planck constant”. Notice that ! i (B − z)−1 , F

−1 ¯ ¯ ∗ O(sν −1 ) + h.c. (B − z)−1 . GO(sν −1 ) + G = κ−1 s1−σ (B − z) Repeated commutation through such an expression by factors of F provides even ¯ − z)−1 tually the power hj = s−σ j . Again the finite numbers of factors like G(B ¯ ∗ (B − z)−1 may be estimated by (5.15) before integrating with respect to and G the z-variable. We choose j so large that σ (j + 1) ≥ 1. The contribution to (5.17) from the second term of (5.13) may be treated very similarly. Clearly the last term of (5.13) contributes by terms of the form of the last two terms to the right in (5.17). Next we move the factors of L2 next to those of L1 (and other commutation) for the contribution to (5.11) from the first term to the right in (5.12) yielding, as a conclusion, that ˘ D(s)ψ˘ + κ−1 O(s−1 ) + O(sν −2 ) ; T1 (s, κ) s ≤ ψ, (5.18) 1 ψ˘ = (F+2 ) 2 B(s) L2 (s)L1 (s)ψ(s) . 1

1

Notice that commutation of D(s) with the factors of L2 (s), F+2 (B(s)) and (F+2 ) 2 (B(s)) (when symmetrizing) involves the calculus of Lemma 3.5 and the effective Planck constant h = s−σ in a similar fashion as above. For the first term on the right hand side of (5.18) we infer from (5.6) and (5.8) that ˘ D(s)ψ˘ ≤ C1 κ−2 s−1−2σ + C2 s−2 . ψ, (5.19) By combining (5.18) and (5.19) we finally conclude (5.11) for i = 1. As for (5.11) for i = 2 we use Remark 4.3, the integral estimate of Lemma 4.5 and the factors of L1 . Notice that the leading (classical) term from differentiating the symbol bt may be written as a sum of three terms: The contribution from “differentiating” the factor F− (tν q − (x, ξ)) is non-positive, cf. Remark 4.3. The contribution from “differentiating” the first factor F− (t2δ q s (x, ξ)) may after a symmetrization be treated by Lemma 4.5. The commutation through the factors of F+ (B(s)) (when symmetrizing) involves the calculus of Lemma 3.5 in a similar fashion as above. Finally the contribution from “differentiating” the last two factors are integrable due to the factors of L1 . We omit further details. As for (5.11) for i = 3 we use the integral estimate (1.10) and commutation. We omit the details. We conclude (5.11), and therefore by Proposition 4.6 the bound (5.9) first with σ replaced by σ + and then (since is arbitrary) by any σ as specified in the proposition.

534



Corollary 5.2. Under the conditions of Proposition 5.1 and with Γ = Γt = Re (G) F+ (t1−σ |Γ|)ψ(t) → 0 for t → ∞ . (5.20) Proof. Let σ ∈ (2ν , 1) be given. Fix σ1 ∈ (2ν , σ). By Proposition 5.1 it suffices to show that F+ (t1−σ |Γ|)F− (t2−2σ1 P ) = O(tσ1 −σ ) . Clearly by the spectral theorem this estimate follows from 1−σ t 1 ΓF− (t2−2σ1 P ) ≤ 1 , which in turn follows from substituting Γ = 2−1 (G + G∗ ) and then estimating 1−σ t 1 ΓF− (t2−2σ1 P ) ≤ 2−1 t1−σ1 GF− ( · ) + 2−1 t1−σ1 G∗ F− ( · ) 1/2 ≤ 2−1 t2−2σ1 F− ( · )G∗ GF− ( · ) 1/2 + 2−1 t2−2σ1 F− ( · )GG∗ F− ( · ) 1/2 ≤ F− ( · )t2−2σ1 P F− ( · ) ≤ 1. Remark 5.3. In the case of (2.9) we define Γ as follows: We pick l ≤ n − 1 such that (2.9) holds and write x · ωl (E0 ) + rt,E (x, ξ) ; γ1 = cl x ñ cl = ∂ul γ1 (w, E0 )|w=0 , x ñ = tk(E0 ) . The operator G = Gt = γtw (x, p) is given by the symbol (using the substitution (5.1)) γt (x, ξ) = γt1 (x, ξ) + γt2 (x, ξ) ; =t

(5.21)

−1

x · ωl (E0 ) , ⎛ k(E0 ) ⎝ γt2 (x, ξ) = rt,E (x, ξ) + cl

γt1 (x, ξ)

⎞ cα γ α (x, ξ)⎠˜ g˜1 (t−1 x)g˜˜2 (ξ) ,

2≤|α|≤m0

cf. (5.2). One proves Proposition 5.1 with this G in the same way as before. Let Γ = Re (G). We have (5.20) for this Γ.

6. Mourre theory for Γ The goal of this section is to show that Γ (modified by a constant) and a certain conjugate operator which we introduce below satisfy a version of the uncertainty principle. We accomplish this using Mourre theory. The abstract version of the uncertainty principle we shall need is the following. ¯ and A¯ are two self-adjoint operators on the same Hilbert Lemma 6.1. Suppose H space such that

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¯ ∩ D(A) ¯ is dense in D(H). ¯ 1. D(H) ¯ isA ¯ ¯ 2. sup|s| 0 . i[H, ¯ A], ¯ A] ¯ extends to a bounded operator B satisfying 4. The form i[i[H, B ≤ C1 < ∞ . Then there exists C2 = C(c1 , C1 ) > 0 such that for all h ∈ C0∞ (R) (with 1/2 A¯ := (1 + A¯2 ) ) −1 A¯ ¯ A¯ −1 ≤ C2 hL1 . h(H) (6.1) In particular, for all h1 , h2 ∈ C0∞ (R), δ2 > δ1 ≥ 0 and t ≥ 1 ¯ 2 (tδ2 H) ¯ ≤ C3 t(δ1 −δ2 )/2 ; h1 (t−δ1 A)h

(6.2)

C3 = C2 h2 L2 sup | x h1 (x)| . Proof. We readily obtain by keeping track of constants in the method of [20] that for some positive constant C depending only on c1 and C1 −1 ¯ − z)−1 A¯ −1 ≤ C ; Im z = 0 . A¯ (H (6.3) " −1 ¯ = π −1 lim ↓0 h(λ) Im H ¯ − λ − i dλ and then usRepresenting h(H) ing (6.3) we conclude (6.1). ¯ → tδ1 H, ¯ and with h(x) = As for (6.2) we use (6.1) with A¯ → t−δ1 A¯ and H δ2 −δ1 2 |h2 (t x)| . Notice that (3) and (4) hold with the same constants for this replacement. To apply Lemma 6.1 we shall need a specific construction of Γ given in terms of a hierarchy of sharp localizations in our observables (see (6.5) and (6.6)). We are forced to use such hierarchy due to the energy variation of (ω(E), ξ(E)). Let Γ be as in Section 5 (assuming first (2.8)). The m0 of (5.2) is here considered as arbitrary (but fixed); the condition (5.3) (needed before for dynamical statements) is not imposed. We introduce for 0 < δ¯ ≤ 1 the operators ¯ = t1−δ¯Γ , A¯ = a H ¯w (6.4) t (x, p) ; ¯ a ¯t (x, ξ) = tδ−1 x · ωl (E0 ) + x · ωl h(x, ξ) − ωl (E0 ) g˜˜1 (t−1 x)g˜˜2 (ξ) . We shall need a specific construction of the functions g˜˜1 and g˜˜2 in the definitions (6.4) in terms of a small parameter > 0: The factor g˜ ˜1 (t−1 x) is the product of the n functions F− −3 |t−1 x · ωj (E0 )| ; j = 1, . . . , n − 1 , (6.5) F− −2 |t−1 x · ωn (E0 ) − k(E0 )| .

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The factor g˜ ˜2 (ξ) is the product of the n functions F− −2 | ξ − ξ(E0 ) · ωl (E0 )| , F− −3 | ξ − ξ(E0 ) · ωj (E0 )| ; j = 1, . . . , n − 1, F− −4 | ξ − ξ(E0 ) · ωn (E0 )| .

j = l ,

(6.6)

Now, indeed for we may apply Lemma 6.1 to the example introduced by (6.4). Lemma 6.2. There exists 0 > 0 such that for all positive ≤ 0 there exists t0 ≥ 1 ¯ =H ¯ t, and such that for all t ≥ t0 the conditions of Lemma 6.1 are fulfilled for H ¯ ¯ A = At, with constants independent of t ≥ t0 . Proof. We shall verify Lemma 6.1 (3) and (4) only (Lemma 6.1 (1) and (2) follow readily from the calculus of pseudodifferential operators). As for (3) we claim that for all small enough ¯ A] ¯ ≥ 2−1 ; t ≥ t0 = t0 () . (6.7) i[H, To see this we notice that clearly the first term in (5.2) and the first term of the symbol a ¯ contribute by ! w ¯ ¯ i t1−δ (γ 1 ) (x, p), tδ−1 x · ωl (E0 ) = 1 , so it remains to estimate w 1−δ¯ Re (γt2 ) (x, p), A¯ ≤ 4−1 ; i t

t ≥ t0 ,

(6.8)

and

1−δ¯ 1 w ! ¯ i t (γ ) (x, p), A¯ − tδ−1 x · ωl (E0 ) ≤ 4−1 ; t ≥ t0 . (6.9) Let us denote by at (x, ξ) the Weyl symbol of the operator in (6.8) or the one in (6.9). We have in both cases that at ∈ Sunif (1, gt1,0 ), so it suffices to show (cf. [19, Theorem 18.6.3] and the proof of [8, Proposition D.5.1]) that sup

x,ξ∈Rn ,t≥t0

|at (x, ξ)| ≤ ν0 ,

(6.10)

where ν0 is a (universal) small positive constant associated for example to the L2 -boundedness result [19, Theorem 18.6.3]. For (6.10) we note the uniform bounds h(x, ξ) − E0 = O(4 ) , t∂xj h(x, ξ) = O(2 ) , ∂ξj h(x, ξ) = O(2 ) for γj (x, ξ) = O(2 ) ,

j ≤ n−1,

∂ξn h(x, ξ) = O(0 ) ,

t∂x γj (x, ξ) = O(0 ) ,

∂ξ γj (x, ξ) = O(0 ) ,

on the support of the function g˜˜1 (t−1 x)g˜˜2 (ξ) given by (6.5) and (6.6). Here we used (1.4) and (1.5), and the notation xj = x · ωj (E0 ) ,

ξj = ξ · ωj (E0 ) .

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By estimating the leading term of the symbol using these bounds we may show (with some patience) that sup

x,ξ∈Rn ,t≥t0

|at (x, ξ)| ≤ C ,

(6.11)

from which (6.10) and (therefore) (6.7) follow. As for (4) we have the bound ! ¯ ¯ A], ¯ A¯ = O(tδ−1 i i[H, ) = O(1) .

(6.12)

As an immediate consequence of Lemmas 6.1 and 6.2 we have. Corollary 6.3. Suppose h1 , h2 ∈ C0∞ (R) and 0 ≤ σ < δ¯ ≤ 1. Then there exists 0 > 0 such that for all positive ≤ 0 there exists C > 0 such that for all t ≥ 1 ¯ ¯ ¯ ≤ Ct(σ−δ)/2 ¯ 2 (tδ−σ H) . (6.13) h1 (A)h Remark 6.4. In the case of (2.9) we introduce (with Γ as in Remark 5.3) ¯ = t1−δ¯Γ , A¯ = a H ¯w t (x, p) ; ¯ a ¯t (x, ξ) = tδ ξ − ξ(E0 ) · ωl (E0 ) + b(x, ξ)g˜˜1 (t−1 x)g˜˜2 (ξ) , b(x, ξ) = ξ − ξ h(x, ξ) · ωl h(x, ξ) − ξ − ξ(E0 ) · ωl (E0 ) . Here the factor g˜ ˜1 (t−1 x) is the product of the n functions F− −2 |t−1 x · ωl (E0 )| , F− −3 |t−1 x · ωj (E0 )| ; j = 1, . . . , n − 1 , F− −2 |t−1 x · ωn (E0 ) − k(E0 )| , while the factor g˜ ˜2 (ξ) is the product of F− −3 | ξ − ξ(E0 ) · ωj (E0 )| ; F− −4 | ξ − ξ(E0 ) · ωn (E0 )| .

(6.14)

j = l ,

j = 1, . . . , n − 1 ,

One verifies (6.13) under the same conditions as in Corollary 6.3 along the same line as before.

7. Proof of Theorem 1.3 The proof of Theorem 1.1 is based on Proposition 4.6, and Corollaries 5.2 and 6.3 (with the assumption (2.8)); we show that the t−δ -localization and the strong localization of Γ are incompatible with the uncertainty principle as expressed in Corollary 6.3. We recall the assumptions of Proposition 4.6: 0 < 2δ < min (ν, 2δ s ) with ν < 2/5 and δ s as in (2.14).

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Lemma 7.1. With A¯ = A¯t given in terms of any (small) > 0 and of δ¯ = δ (with δ as above) by either (6.4) (in the case of (2.8)) or (6.14) (in the case of (2.9)) ¯ lim ||F+ (|A|)ψ(t)|| = 0,

(7.1)

t→∞

where ψ = f (H)ψ is given as in Proposition 4.6 (with the support of f being sufficiently small possibly depending on ). Proof. We fix δ1 such that 2δ < 2δ1 < min (ν, 2δ s ). Let bt (x, ξ) be given by (4.14) in terms of δ1 and ν. By Proposition 4.6 it suffices to show that ¯ w ||F+ (|A|)b t (x, p)|| → 0 for

t → ∞,

and therefore in turn ¯ w (x, p)|| = O(tδ−δ1 ) . ||Ab t ¯ w For the latter bound one easily checks that the symbol of Ab t (x, p) belongs to Sunif (tδ−δ1 , gt1−ν

,ν

);

ν = ν − δ1 .

Now, we first fix δ as above and conclude from Lemma 7.1 that ¯ → 0 for t → ∞ , ||ψ(t) − F− (|A|)ψ(t)||

(7.2)

where ψ = f (H)ψ is given as in Proposition 4.6. This holds for f ∈ C0∞ (I0 ); I0 = I0 (). Next we fix any σ ∈ (0, δ) in agreement with Corollary 5.2 which means that ||F+ (|t1−σ Γ|)ψ(t)|| → 0

for t → ∞ .

(7.3)

Here the input of δ in Proposition 5.1 say δ1 (needed to fix the m0 in the definition of the Γ of Corollary 5.2) is different; we need to have σ > ν , ν = ν1 − δ1 , for which δ1 < δ is needed. The construction of this Γ depends on the same as above, cf. Section 6. Combining (7.2) and (7.3) leads to ¯ − (|t1−σ Γ|)ψ(t)|| → 0 ||ψ(t) − F− (|A|)F

for t → ∞ .

(7.4)

By combining Corollary 6.3 and (7.4) we conclude (by finally fixing > 0 sufficiently small) that ||ψ(t)|| → 0 for

t → ∞,

(7.5)

and therefore that ψ = 0 proving Theorem 1.1. Remark 7.2. With the assumption (2.9) we proceed similarly using Remarks 5.3 and 6.4, and Lemma 7.1.

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8. Proof of Theorem 1.3 We shall here elaborate on the derivation of Theorem 1.3 from our general result Theorem 1.1. First we remove the singularity at x = 0 by defining h(x, ξ) = 2−1 ξ 2 + V˜ (x) ;

V˜ (x) = F+ (|x|)V (ˆ x) ,

where (as before) V is a Morse function on S n−1 . (See Remarks 8.3 for extensions.) In this case clearly the hypotheses (H1)–(H3) of Section 2 are satisfied, and (H4) ∈ Cr and energy E > V (ωj ) upon putting ω(E) = ωl , holds for any critical point ωl ξ(E) = k(E)ωl and k(E) = 2(E − V (ωl )). For (1.7) we put # g(u, η, E) = 2 E − V (ωl ) − 2E − η 2 − 2V (ωl + u) , yielding (1.8) with A(E) = k(E)

−1

V (2) (ωl ) 0

0 . I

⊥

We may choose an orthonormal basis in {ωl } ⊆ Rn for which V (2) (ωl ) is diagonal, say V (2) (ωl ) = diag (q1 , . . . , qn−1 ). The eigenvalues of B(E) take the form # 1 1 βj+ (E) = − + 1 − 2qj / E − V (ωl ) or 2 2 (8.1) # 1 1 − βj (E) = − − 1 − 2qj / E − V (ωl ) 2 2 √ √ say with ζ := i −ζ if ζ < 0. Clearly the hypothesis (H5) is the non-degeneracy condition, qj = 0 for all j, while hypothesis (H6) amounts to qj < 0 for some j, i.e., ωl is a local maximum or a saddle point of V . As for (H7) one easily checks that there exists a smooth basis of eigenvectors of B(E)tr for E − V (ωl ) ∈ (0, ∞) \ {2q1 , . . . , 2qn−1 }. Elementary analyticity arguments show that given any m ∈ {2, 3, . . . } the set of resonances of order m for any of the eigenvalues of B(E) is discrete in (V (ωl ), ∞). In conclusion, the hypotheses (H1)–(H8) are satisfied for any local maximum or saddle point ωl of a Morse function V for E0 ∈ (V (ωl ), ∞) \ D where D is discrete in (V (ωl ), ∞). Due to the possible existence of bound states we change the definition of Pl to be Pl = s − lim eitH χl (ˆ x)e−itH Eac (H) , t→∞

where Eac (H) is the orthogonal projection onto the absolutely continuous subspace of H, see [15] and [1, Theorem C.1]. This gives (1.13) with the left hand side replaced by Eac (H).

540



Now, to get (1.15) it suffices by Theorem 1.1 to verify (1.14) for any E0 ∈ (V (ωl ), ∞). Invoking the discreteness of the set of eigenvalues of H on the complement of the set of critical values of V , cf. [1, Theorem C.1], one may easily conclude (1.14) from the following statement: Consider any open set I0 ⊆ (V (ωl ), ∞) such that I0 ∩ σpp (H) ∪ V (Cr ) = ∅ . Let H0 be the closure of the subspace of states ψ = f (H)ψ, f ∈ C0∞ (I0 ), obeying (1.9) and (1.10). Then for all ψ = Pl f (H)ψ where f ∈ C0∞ (I0 ) ψ ∈ H0 .

(8.2)

We shall verify (8.2) by showing that indeed ψ = Pl f (H)ψ obeys (1.9) and (1.10). We shall proceed a little more generally than needed in that we here assume that the U0 of (1.10) is given by U0 = U = C˜ × Rn ;

C˜ = x ∈ Rn \ {0}|ˆ x ∈ C , C = {ω ∈ S n−1 | |ω − ωl | < } , where > 0 is taken so small that C ∩ Cr = {ωl }. Pick f˜ ∈ C0∞ (I0 ) such that 0 ≤ f˜ ≤ 1 and f˜ = 1 in a neighborhood of supp (f ). Let r ∈ C ∞ (Rn ) be given in terms of any F+ ∈ F+ by |x| 1 r(x) = F+ (s)ds + F− (s)ds . (8.3) 0

0

(Notice that r(x) = |x| for |x| ≥ 1.) Let 1 p|| = (∇r · p + h.c.) , p˜|| = f˜(H)p|| f˜(H) . 2 ∞ Lemma 8.1. Let χl ∈ C0 (C ) be given with 0 ≤ χl ≤ 1 and χl = 1 in a neighborhood of ωl , and g˜2 ∈ C0∞ (R) by g˜2 (s) = f˜ 2−1 s2 + V (ωl ) 1(0,∞) (s) . Let real-valued g1− , g1+ ∈ C0∞ (R) be given with − c− ˜− ; c− + < c + = sup supp (g1 ) , + ˜+ ; c+ c+ − > c − = inf supp (g1 ) ,

c˜− = inf supp (˜ g2 ) , c˜+ = sup supp (˜ g2 ) .

Let F+ ∈ F+ , F− ∈ F− and $ C > 2 2 sup supp (f ) − min (V ) . Then, in the state ψ(t) = e−itH Pl f (H)ψ ∞ r−1−δ t dt < ∞ ; δ > 0 , −∞ ∞ | p · r(2) p t |dt < ∞ , −∞

(8.4) (8.5)

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−∞ ∞ −∞ ∞

1

1

∞

∞

1 ∞

1

∞


541

r|∇V˜ |2 t dt < ∞ ,

(8.6)

−1 2 1 χ ˜l r 2 (η + u2 )r− 2 χ ˜l t dt < ∞ ;

χ ˜l = χl (ˆ x)F+ (r) ,

(8.7)

−t−1 F− (C −1 t−1 r) t dt < ∞ , t−1 ||g(˜ p|| )F− (C −1 t−1 r)ψ(t)||2 dt < ∞ ;

(8.8) g ∈ C0∞ ((−∞, 0)) ,

g¯ = g , (8.9)

2 t−1 1 − g˜2 (˜ p|| ) F− (C −1 t−1 r)χ ˜l ψ(t) dt < ∞ ,

(8.10)

t−1 ||B − (t)ψ(t)||2 dt < ∞ ;

B − (t) = g1− (t−1 r)˜ g2 (˜ p|| ) ,

(8.11)

t−1 ||B + (t)ψ(t)||2 dt < ∞ ;

B + (t) = g1+ (t−1 r)˜ g2 (˜ p|| ) .

(8.12)

1

Proof. For (8.4), (8.5) and (8.6) we refer to [15] and [1, Theorem C.1]. The bound (8.7) follows from those estimates by Taylor expansion. As for (8.8) we consider the “propagation observable” Φ(t) = f (H)F− (C −1 t−1 r)f (H) . We may bound its Heisenberg derivative as DΦ(t) ≥ −t−1 f (H)F− (C −1 t−1 r)f (H) + O(t−2 ) ;

> 0.

As for (8.9) we consider the observable p|| )f˜(H) . Φ(t) = f˜(H)g(˜ p|| )t−1 rF− (C −1 t−1 r)g(˜ We write its Heisenberg derivative as DΦ(t) = T1 + T2 + T3 ; T1 = f˜(H) Dg(˜ p|| ) t−1 rF− (C −1 t−1 r)g(˜ p|| )f˜(H) + h.c. , p|| )f˜(H) + h.c. , p|| )t−1 r DF− (C −1 t−1 r) g(˜ T2 = 2−1 f˜(H)g(˜ T3 = 2−1 f˜(H)g(˜ p|| )f˜(H) + h.c. , p|| ) D(t−1 r) F− (C −1 t−1 r)g(˜ and notice the identities Dp|| = p · r(2) p + O(r−3 ) .

Dr = p|| ,

(8.13)

Using (8.4), (8.5), the second identity of (8.13) and (3.11) we readily obtain after symmetrization that ∞ | T1 t |dt < ∞ . (8.14) 1

As for the the term T2 we use the first identity of (8.13) and (8.8) to derive ∞ | T2 t |dt < ∞ . (8.15) 1

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For the term T3 we compute using the first identity of (8.13) and (3.11) T3 = Re t−1 f˜(H)g(˜ p|| )f˜(H) + O(t−2 ) p|| )(p|| − t−1 r)F− (C −1 t−1 r)g(˜ p|| )F− (C −1 t−1 r)g(˜ p|| )f˜(H) + O(t−2 ) ; > 0 . (8.16) ≤ −t−1 f˜(H)g(˜ We conclude (8.9) from (8.14), (8.15) and (8.16). The bound (8.10) follows from elementary energy bounds, Taylor expansion and the previous estimates. (For this we need (8.9) to deal with the “region” where p2|| energetically has the right size, but p|| < 0.) As for (8.11) we consider s Φ(t) = f˜(H)˜ g2 (˜ p|| )F (t−1 r)˜ g2 (˜ p|| )f˜(H) ; F (s ) = g1− (s)2 ds . −∞

We write its Heisenberg derivative as DΦ(t) = T1 + T2 ; T1 = f˜(H) D˜ g2 (˜ p|| ) F (t−1 r)˜ g2 (˜ p|| )f˜(H) + h.c. , T2 = f˜(H)˜ g2 (˜ p|| ) DF (t−1 r) g˜2 (˜ p|| )f˜(H) . Using (8.4), (8.5), the second identity of (8.13) and (3.11) as for (8.9) we obtain that ∞ | T1 t |dt < ∞ . (8.17) 1

As for the the term T2 we compute using the first identity of (8.13) and (3.11) ∗ T2 = t−1 f˜(H)B − (t) (p|| − t−1 r)B − (t)f˜(H) + O(t−2 ) 2 − −2 ˜ p|| ) − c− ) ≥ t−1 B − (t)∗ p˜|| 1[˜c− ,∞) (˜ + f (H) B (t) + O(t ∗

≥ t−1 B − (t) B − (t) + O(t−2 ) ;

= c˜− −

c− +

(8.18)

.

Clearly (8.11) follows by combining (8.17) and (8.18). As for (8.12) we may proceed similarly using Φ(t) = f˜(H)˜ g2 (˜ p|| )F (t−1 r)˜ g2 (˜ p|| )f˜(H) ; F (s ) =

s

−∞

g1+ (s)2 ds .

Corollary 8.2. Let ψ, χl ∈ C0∞ (C ) and g˜2 be given as in Lemma 8.1. Let g1 ∈ C0∞ (R) be given such that 0 ≤ g1 ≤ 1 and g1 = 1 in an open interval containing supp (˜ g2 ). Then ||ψ(t) − g1 (t−1 r)˜ g2 (˜ p|| )χl (ˆ x)f˜(H)ψ(t)|| → 0 Proof. From the very definition of ψ we have x)f˜(H)ψ(t)|| → 0 ||ψ(t) − χl (ˆ

for

t → ∞.

(8.19)

for t → ∞ .

Next, from [15, Theorems 4.10 and 4.12] we learn that ||ψ(t) − g˜2 (˜ p|| )χl (ˆ x)f˜(H)ψ(t)|| → 0

for t → ∞ .

(8.20)

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Whence to show (8.19) it suffices to verify that

g1 (t−1 r) − g1 (˜ p|| ) g˜2 (˜ p|| )f˜(H)ψ(t) → 0

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for t → ∞ ,

which in turn is reduced (by a standard density argument using that the energy bounds the momentum) to verifying that for all constants C large enough

F− (C −1 t−1 r) g1 (t−1 r) − g1 (˜ p|| ) g˜2 (˜ p|| )f˜(H)ψ(t) → 0 for t → ∞ . (8.21) For (8.21) we consider the observable 2

ΦC (t) = f˜(H)˜ g2 (˜ p|| )F− (C −1 t−1 r)(˜ p|| − t−1 r) F− (C −1 t−1 r)˜ g2 (˜ p|| )f˜(H) . Using Lemma 8.1 as well as the proof of this lemma we easily show that ∞ ∞ d dt, Φ | (t) t−1 ΦC (t) t dt < ∞ , C t dt 1 1 from which we conclude that along some sequence tk → ∞ indeed ΦC (tk ) tk → 0, and then in turn that (8.22) ΦC (t) t → 0 . We easily obtain (8.21) using (8.22), (3.10) and commutation.

Now, one may easily verify (8.2) for ψ = Pl f (H)ψ as follows: We introduce a partition f = fi of sharply localized fi ‘s and for each of these a “slightly larger” f˜i . Using these functions and the states ψi = Pl fi (H)ψ as input in Corollary 8.2 the bounds (1.9) follow from the conclusion of the corollary and [15, Theorems 4.10 and 4.12]. As for (1.10) we may use the same partition and then conclude the result from Lemma 8.1 (applied with f˜ replaced by f˜i ). Remarks 8.3. 1. Using the Mourre estimate [1, Theorem C.1] one may easily include a shortrange perturbation V1 = O(|x|−1−δ ), δ > 0, ∂xα V1 = O(|x|−2 ), |α| = 2, to the Hamiltonian H. In particular Theorem 1.3 holds for the strictly homogeneous case as discussed in Section 1. 2. The non-degeneracy condition at ωl is important for the method of proof presented in this paper. However it is not important that the set of critical points Cr is finite; it suffices that ωl is an isolated non-degenerate critical point and that V (Cr ) is countable. 3. At a local maximum we proved a somewhat better result in [16] (by a different method): A larger class of perturbations was included and we imposed a somewhat weaker condition than the non-degeneracy condition. The method of [16] yielded only a limited result at saddle points. Although there are indications that this method of proof might be extended to included Theorem 1.3 (by using a certain complicated iteration scheme) the proof presented in this paper is probably much simpler.

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4. The components of the γ of (2.3) may be taken of the form # γj = ηj + 2 E − V (ωl ) βj# (E)uj , where βj# (E) is given by one of the expressions of (8.1). In particular both of the conditions (2.8) and (2.9) are satisfied in the potential case. 5. We applied the Sternberg linearization procedure in [18] to the equations (1.7) in the case of a local minimum. In this case the union of all resonances (of all orders and for all eigenvalues) is discrete on (V (ωl ), ∞). One needs to exclude this set of resonances to construct a smooth Sternberg diffeomorphism, see for example [22, Theorem 9]. The construction of the symbol γ (m) in (2.18) may be viewed as a rudiment of this procedure. However, the union of all resonances at a local maximum or a saddle point ωl is dense in (V (ωl ), ∞), and for that reason the smooth Sternberg diffeomorphism (defined at nonresonance energies) would not be suited for quantization. Although not elaborated, one may essentially view γ (m) as being constructed by a C m Sternberg diffeomorphism.

Appendix A. A generalization of the homogeneity condition In this appendix we shall discuss possible generalizations of the homogeneity condition (1.1). We elaborate on the structure of the classical mechanics of our models. A possible formulation of the quantum problem will be proposed although not justified in general. It will be discussed for various examples. The homogeneity condition is best understood as the invariance of the Hamiltonian under the flow generated by the vector field v(x, ξ) = xj ∂/∂xj , or infinitesimally vh(x, ξ) = 0 . (A.1) Our goal is thus to find invariance conditions (A.1) which will a. reduce the dimension of phase space by two giving an autonomous dynamical system in dimension 2n − 2 (usually not Hamiltonian) b. give a natural framework for discussing stability of orbits which do not lie in a compact set. It will turn out that stability is not measured using any preexisting metric in the phase space but rather using bundles of orbits of the vector field v surrounding a given orbit of the Hamiltonian vector field, vh . The particular vector field v(x, ξ) = xj ∂xj does not generate a symplectic flow but does satisfy a crucial property. Namely Lv ω = ω where Lv is the Lie derivative in direction v and ω is the symplectic form. It will turn out (see Lemma A.1) that a geometric condition such as this, although more restrictive than necessary, will guarantee that v is a suitable vector field. We will require v to satisfy certain conditions relative to vh , where vh is a Hamiltonian vector field on a symplectic manifold (M, ω) with Hamiltonian h:

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1. In a neighborhood U0 of a point x0 ∈ M , the local flow φvt ( · ) generated by v exists for all t ∈ (−, ∞) for some > 0 and there exists a surface S ⊂ U0 containing x0 , transverse to v, and a diffeomorphism σ : B → S, where B is a ball in R2n−1 centered at 0, such that the map B × (−, ∞) (w, t) → φvt σ(w) is a diffeomorphism onto its image, K0 ⊇ U0 . We also assume v and vh are parallel (and nonzero) along the positive orbit of v originating at x0 (identified as 0 ∈ B). 2. There are smooth functions β and γ such that [v, vh ] = βvh + γv

in K0 .

3. vh = 0 in K0 . Condition (1) allows us to assume (after a change of coordinates) that K0 = B × (−, ∞), x0 = (0, 0), and v = (0, . . . , 0, 1) in K0 . With the notation x⊥ = (x1 , . . . , x2n−1 ) for x ∈ R2n , condition (2) implies (vh )⊥ (x) = k(x)(vh )⊥ (x⊥ , 0) where k(x) = exp 0 β ◦ φvs (x⊥ , 0)ds so that introducing the new time variable τ with dτ /dt = k(x(t)) the first 2n − 1 of Hamilton’s equations become " x2n

dx⊥ = (vh )⊥ (x⊥ , 0) . dτ As long as dh(x0 ) = 0, using condition (3) we can eliminate one more variable using energy conservation, h(x) = h(x⊥ , 0) = E. For example if ∂h/∂x2n−1 = 0 we obtain x2n−1 = g(w, E) with w = (x1 , . . . , x2n−2 ). Here we assume (w, E) is in a neighborhood of (0, E0 ), E0 = h(x0 ) = h(0). We obtain dw = f (w, E) , (A.2) dτ where f (w, E) = ((vh )1 (w, g(w, E), 0), . . . , (vh )2n−2 (w, g(w, E), 0)). The orbit of vh along v corresponds to w = 0, E = E0 (in which case f (0, E0 ) = 0). If det(∂fi / ∂wj (0, E0 )) = 0 there will be a smooth family of fixed points of (A.2), w = w(E), in a neighborhood of E0 (with w(E0 ) = 0). This situation is analogous to the case v(x, ξ) = xj ∂xj discussed in Section 1 and we can define stability of orbits in M in terms of the stability of the fixed points w(E). In practice one might want to place the fixed point of (A.2) at the origin by an affine change of variables, cf. Section 1. In any case one may check that for the model studied in Section 1 indeed the systems (1.7) and (A.2) are smoothly equivalent systems (up to a conformal factor). Notice that in this case we may choose S ⊂ S n−1 × Rn , for example. If a proof of absence of channels is contemplated along the lines carried out in this paper, it is necessary that low order resonances do not occur at more than a discrete set of energies. In particular, the equations (A.2) should not have a Hamiltonian structure.

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The only place where the Hamiltonian nature of the equations appeared above was where we used conservation of energy. To bring in the symplectic form ω we introduce a more geometric condition which turns out to imply condition (2) above (see Remark A.2 for an interpretation): Lemma A.1. Fix an open set U ⊆ M . a. Suppose Lv ω = αω in U for some α ∈ C ∞ (U ). Suppose in addition that vh = 0 in U . Then [v, vh ] = −αvh in U . b. Suppose v is nonzero in U and for any smooth function h on U satisfying vh = 0 in a neighborhood of a point of U , v satisfies [v, vh ] = −αvh in this neighborhood. Then Lv ω = αω in U . Proof. We shall use the general relations dh(w) = ω(vh , w), [Lv , iw ] = i[v,w] and [Lw , d] = 0. Here iw represents interior product with w (see for example [4, p. 84] or [3, p. 198]). For (a) we compute in U i[v,vh ] ω = [Lv , ivh ]ω = Lv dh − ivh αω = dLv h − iαvh ω = i−αvh ω . Since ω is non-degenerate we conclude (a). As for (b) we use the same computation to conclude that ivh (−Lv ω + αω) = 0 in open subsets where vh = 0. Since v is nonzero there are sufficiently many choices of h to conclude from this that indeed Lv ω = αω. Remark A.2. By integrating the condition of Lemma A.1 (a), Lv ω = αω, we obtain t v ∗ v α ◦ φs ds ω . (A.3) (φt ) ω = exp 0

In particular if Lv ω = αω holds in M and φvt is a global flow we see that the diffeomorphisms φvt preserve the family of Lagrangian manifolds. Conversely one may readily prove that if φvt is a global flow and the diffeomorphisms φvt preserve the family of Lagrangian manifolds, then indeed Lv ω = αω for some smooth α. We give two simple examples. Example A.3. Consider the symbol h on R2 × R2 , suitably regularized at singularities, −1 2 1 h = h(x, ξ) = x2 − aξ22 ξ ; a > 0. 2 Let v(x, ξ) = 12 (xj ∂xj + ξj ∂ξj ). Then the vector field v and the Hamiltonian vector field vh fulfill the √ conditions (1)–(3) along the positive orbit of v originating at (1 + 2E)−1/2 (1, 0; 2E, 0), E > 0. Here we take the S in condition (1) to be

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a subset of the unit-sphere S 3 . Notice also that (φvt )∗ ω = exp(t)ω, and therefore Lv ω = ω. After linearizing the reduced flow (A.2) we find the eigenvalues √ √ − 2E 1 ± 1 + 4Ea , and we conclude that the family of fixed points consists of saddle points. Resonances (of any fixed order) are discrete in (0, ∞). Example A.4. Consider the symbol h on (R2 \ {0}) × R2 κ/2 2 1 h = h(x, ξ) = x21 + bx22 ξ ; b > 0 , κ < 2 , κ(b − 1) < 0 . 2 We introduce s = 2/(2 − κ) and v = (sxj ∂xj + (1 − s)ξj ∂ξj ). The vector field v and the Hamiltonian vector field √ vh fulfill the conditions (1)–(3) along the positive orbit of v originating at (1, 0; 2E, 0), E > 0. Here we take S ⊂ {(x, ξ)| x1 = 1}. We notice that the condition κ < 2 assures that the x-component of the flow φvt grows as t → ∞; whence there is no conflict with a regularization at x = 0. (The fact that for κ ∈ (0, 2) the ξ-component decays is irrelevant.) We find the eigenvalues for the linearized reduced flow to be given by 2 − κ√ 2E 1 ± 1 − 8κ(b − 1)(2 − κ)−2 . − 4 Since by assumption κ(b − 1) < 0 we conclude that the family of fixed points consists of saddle points. For a “generic” set of parameters b and κ there are no resonances (of any order). We shall propose a formulation of the quantum problem corresponding to the classical framework discussed above, and then relate it to Examples A.3 and A.4. Let us strengthen the above conditions (1)-(3) as follows: We assume that = ∞ in (1) so that K0 is two-sided invariant under the flow φvτ , and furthermore that the condition Lv ω = αω of Lemma A.1 (a) holds in U = K0 (implying (2) with β = −α and γ = 0). Suppose also that α > 0. Under these conditions we may write φvτ (t,E0 ) (x0 ) = φvt h (x0 ) ; τ (t,E0 ) dτ (t, E0 ) v = exp − α ◦ φs (x0 )ds k(E0 ) , dt 0 vh (x0 ) = k(E0 )v(x0 ) ,

τ (0, E0 ) = 0 .

Notice that any maximal solution to this differential equation is defined at least on a positive directed half-line (i.e., τ (t, E0 ) exists for all large t’s). Denoting by x(E) ∈ S the fixed points for neighboring energies E ≈ E0 we have similar identities for the positive common orbits originating at x0 → x(E). Whence we may look at localization of states in quantum mechanics in terms of Weyl quantization of symbols of the form a(φv−τ (t,h) ) where a ∈ C0∞ (U0 ). Notice that for the model studied in the bulk of this paper this procedure is a slight modification of the one

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used in (1.10) and (1.11). In fact in this case we may take S ⊂ S n−1 × Rn and compute in terms of the function k = k(E) of (1.5) τ = ln tk(E) + 1 yielding

φv−τ (t,h) (x, ξ) = x/ tk(h) + 1 , ξ ;

h = h(x, ξ) .

We need in this setting to replace γ(I0 ) → γ(I0 ) = x(E) = ω(E), ξ(E) | E ∈ I0 . There is also a way to interpret the first factor t−1 of (1.10): Using (A.3) we may compute the Poisson bracket −τ (t,h) v v α ◦ φs ( · )ds {h, a} φv−τ (t,h) ( · ) , h, a φ−τ (t,h) ( · ) = exp 0

which indicates that the first factor to the right is a “Planck constant” (this interpretation is supported by the requirement α > 0). Effectively it is equal to t−1 for this example. Whence a possible reformulation of the integral condition (1.10) (suited for generalization) is ∞ 2 ∞ bw (A.4) t (x, p)ψ(t) dt < ∞ for all a ∈ C0 U0 \ γ(I0 ) ; 1 v at (x, ξ) = a φ−τ (t,h) (x, ξ) , −τ (t,h) −1 v bt (x, ξ) = exp 2 α ◦ φs (x, ξ)ds at (x, ξ) , 0

γ(I0 ) = x(E) | E ∈ I0 ,

ψ(t) = e−itH f (H)ψ ,

f ∈ C0∞ (I0 ) .

The analogous statement of Theorem 1.2 in general would read: For all a ∈ C0∞ (U0 ) and all localized states ψ(t) = e−itH f (H)ψ, f ∈ C0∞ (I0 ), obeying (A.4) with I0 E0 small enough w at (x, ξ)ψ(t) → 0 for t → ∞ . (A.5) Now, for Examples A.3 and A.4 we may compute 1/2 √ −1 φv−τ (t,h) (x, ξ) = t0 /(t + t0 ) (x, ξ) ; t0 = 2 2h(1 + 2h) , and φv−τ (t,h) (x, ξ) =

t √ +1 s 2h

−s

x,

t √ +1 s 2h

s−1 ξ ;

(A.6)

s = 2/(2 − κ) , (A.7)

respectively. We may use the effective Planck constant t−1 like for the other example. In conclusion, the somewhat complicated looking quantum condition (A.4) reduces to simple explicit requirements. Similarly (A.5) reads in these cases 1/2 w t0 (h)/t (x, p) ψ(t) → 0 for t → ∞ (A.8) a

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√ √ 1−s s−1 s w s 2h t−s x, s 2h t p ψ(t) → 0 for a

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t → ∞,

(A.9)

respectively. We remark that (A.4), (A.6) (or (A.7)) and (A.8) (or (A.9)) apply literally for Example A.3 (or Example A.4); the conclusion (A.8) (or (A.9)) for the states considered may be reached using Theorem 1.2 after a symplectic change of variables and invoking symplectic covariance: Example A.5. Consider a smooth symbol h on (Rn \ {0})2 obeying one of the homogeneity properties 1) h(λx, λξ) = h(x, ξ) ;

for all λ > 0 ,

or 2) for some κ2 = 0 and some κ1 = κ2 h(λ1 x, λ2 ξ) = λκ1 1 λκ2 2 h(x, ξ) ;

for all λ1 , λ2 > 0 .

For 2) the change of variables x = |y|s yˆ = |y|s−1 y, where s = κ2 /(κ2 − κ1 ), induces a symplectic map on (Rn \ {0})2 . The Hamiltonian in the corresponding new variables, denoted again by x and ξ, reads ˜ ξ) = h x x, ξ ˆ x . h(x, ˆ, ξ + (s−1 − 1) ˆ The same change of variables with s = 12 leads for 1) to a Hamiltonian of the same ˜ Up to other form. In particular (1.1) holds (in both cases) for the new symbol h. conditions we may therefore apply Theorem 1.2. Clearly Examples A.3 and A.4 are concrete examples. To stress the symplectic covariance let us note that indeed v := (sxj ∂xj + (1 − s)ξj ∂ξj ) → v˜ := xj ∂xj . We give yet another example from Riemannian geometry. Example A.6. Consider the symbol h on (R2 \ {0}) × R2 1 −1 2 g ξ , 2 where the conformal (inverse) metric factor is specified in polar coordinates x = (r cos θ, r sin θ) as g −1 = ef ; f = f (θ − c ln r). We assume f is a given smooth nonconstant 2π-periodic function and that c > 0. We introduce v = (x1 − cx2 )∂x1 + (cx1 + x2 )∂x2 − cξ2 ∂ξ1 + cξ1 ∂ξ2 . Computations show that v and the Hamiltonian vector field vh fulfill the conditions (1)–(3) along the positive orbit of v originating at (r0 , 0; ρ0 , cρ0 ); here ρ0 = 2E(1 + c2 )−1 e−f0 where f0 = f (θ0 ) is given in terms of any r0 > 0 satisfying the equation h = h(x, ξ) =

−f (θ0 ) = 2c(1 + c2 )−1 ;

θ0 = −c ln r0 ,

(A.10)

and E = h > 0 is arbitrary. (Notice that there are at least two solutions to (A.10) for all small as well as for all large values of c.) The x-space part of the orbit (a geodesic) is the logarithmic spiral given by the equation θ − c ln r = θ0 . We take

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S ⊂ {(x, ξ)| x2 = 0} and compute the eigenvalues for the linearized reduced flow to be given by # 1 −ρ0 1 ± 1 − 2(1 + c2 )2 f0 ; f0 = f (θ0 ) . (A.11) 2 For f0 < 0 the family of fixed points consists of saddles. There are no resonances for “generic” values of c, and we also notice that taking c → 0 in (A.10) and (A.11) yields the formulas for the corresponding homogeneous model (here the equations are considered to be equations in c and θ0 ). Finally, using the new angle θ˜ = θ − c ln r one may again conjugate to a homogeneous model. More precisely the relevant symplectic change of variables is induced (expressed here in terms of rectangular coordinates) by the map x → x ˜= (x1 g1 + x2 g2 , x2 g1 − x1 g2 ), where g1 = cos(c ln |x|) and g2 = sin(c ln |x|). One may ˜ given by check that v → v˜ := xj ∂xj , and that h → h

˜ = 1 ef (θ) (c sin θ + cos θ)ξ1 + (sin θ − c cos θ)ξ2 2 + {− sin θξ1 + cos θξ2 }2 ; h 2 we changed notation back to the old one, x = (r cos θ, r sin θ) for position and ξ for momentum. Remark A.7. Although we shall not elaborate, due to the general nature of the method used in the bulk of this paper the method should be generalizable to apply to the quantum problem for Examples A.3, A.4 and A.6 (without changing variables). We believe it would apply to the quantum problem for a variety of other examples of the classical theory. However we have not pursued the outlined general scheme for two reasons: 1) There are additional complications related to the pseudodifferential calculus, cf. [19, Section 18]. The treatment of these complications is somewhat cumbersome and does not add new insight to the problem. 2) The condition (A.4) has a certain global flavor in our opinion, whence it does not entirely stand alone. For instance its verification in the context of proving asymptotic completeness, cf. [6, 15, 18] and Section 8, relies on global information on the dynamics. To illustrate this point further let us look at Example A.4 in the case κ < 0 and b > 1. For the classical problem any orbit x(t) going to infinity will roughly follow either the x1 -axis or the x2 -axis. As a first step of proving asymptotic completeness in Quantum Mechanics (for the regularized Hamiltonian) one may derive estimates for states in the continuous subspace with roughly the same content, in particular the bound (A.4). Due to the eigenvalue calculation of Example A.4 only the x2 -axis is “stable” for the classical orbits. The corresponding statement in Quantum Mechanics given by (A.9) then leads to the preliminary information for asymptotic completeness, x1 /|x|ψ(t) → 0 for t → ∞. Although the dynamics of Example A.6 in general is more complicated than Example A.4 we remark that the attractive spirals (cf. the eigenvalue calculation (A.11)) similarly define non-trivial quantum channels. One can show in some cases, for example if f (θ) + 2c(1 + c2 )−1 ≤ 0 on an interval of length (1 + c2 )π/2, that those channels are the only occurring ones.

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References [1] S. Agmon, J. Cruz, I. Herbst, Generalized Fourier transform for Schr¨ odinger operators with potentials of order zero, J. F. A. 167 (1999), 345–369. [2] V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983. [3] V. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer-Verlag, New York, 1989. [4] M. P. do Carmo, Riemannian Geometry, Birkh¨ auser, Boston, 1992. [5] H. Cornean, I. Herbst, E. Skibsted, Spiraling attractors and quantum dynamics for a class of long-range magnetic fields, J. Funct. Anal. 247 (2007), 1–94. [6] H. Cornean, I. Herbst, E. Skibsted, Classical and quantum dynamics for 2Delectromagnetic potentials asymptotically homogeneous of degree zero, preprint arxiv.org/abs/math-ph/0703089. [7] J. Derezi´ nski, Asymptotic completeness for N -particle long-range quantum systems, Ann. Math. 138 (1993), 427–476. [8] J. Derezi´ nski, C. Gérard, Scattering theory of classical and quantum N -particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. [9] G. B. Folland, Harmonic analysis in phase space, Annals of Mathematical studies 122, Princeton University Press, Princeton, 1989. [10] C. Gérard, Sharp propagation estimates for N -particle systems, Duke Math. J. 67 no. 3 (1992), 483–515. [11] C. Gérard, Asymptotic completeness for 3-particles systems, Invent. Math. 114 (1993), 333–397. [12] V. Guillemin, D. Schaeffer, On a certain class of Fuchsian partial differential equations, Duke Math. J. 44 (1977), 157–199. [13] A. Hassell, R. Melrose, A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Advances in Math. 181 (2004), 1–87. [14] A. Hassell, R. Melrose, A. Vasy, Scattering for symbolic potentials of order zero and microlocal propagation near radial points, preprint 2005. [15] I. Herbst, Spectral and scattering theory for Schr¨ odinger operators with potentials independent of |x|, Amer. J. Math. 113 no. 3 (1991), 509–565. [16] I. Herbst, E. Skibsted, Quantum scattering for homogeneous of degree zero potentials: Absence of channels at local maxima and saddle points, MaPhySto preprint no. 24 August 1999, unpublished manuscript. [17] I. Herbst, E. Skibsted, Absence of channels of quantum states corresponding to unstable classical channels: Homogeneous potentials of degree zero, MaPhySto preprint no. 23 September 2003, unpublished manuscript. [18] I. Herbst, E. Skibsted, Quantum scattering for potentials independent of |x|: Asymptotic completeness for high and low energies, Comm. PDE. 29 no. 3–4 (2004), 547–610. [19] L. H¨ ormander, The Analysis of Partial Differential Operators III, Springer-Verlag, Berlin, 1985. ´ Mourre, Absence of singular continuous spectrum for certain self-adjoint opera[20] E. tors, Commun. Math. Phys. 91 (1981), 391–408.

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[21] J. S. Møller, An abstract radiation condition and applications to N -body systems, Rev. Math. Phys. 12 no. 5 (2000), 767–803. [22] E. Nelson, Topics in dynamics I, Flows, Princeton U. Press and U. Tokyo Press, Princeton, 1969. [23] M. Reed, B. Simon, Fourier analysis, self-adjointness. Methods of modern mathematical physics II, Academic Press, New York, 1975. [24] E. Skibsted, Long-range scattering of three-body quantum systems, Asymptotic completeness, Invent. Math. 151 (2003), 65–99. [25] E. Skibsted, Long-range scattering of three-body quantum systems, II, Ann. Henri Poincaré 4 (2003), 1–25. Ira Herbst Department of Mathematics University of Virginia Charlottesville, VA 22903 USA e-mail: [email protected] Erik Skibsted Institut for Matematiske Fag Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark e-mail: [email protected] Communicated by Claude-Alain Pillet. Submitted: September 18, 2007. Accepted: January 14, 2008.



Quasi-static Limits in Nonrelativistic Quantum Electrodynamics Lucattilio Tenuta Abstract. We consider a system of N nonrelativistic particles of spin 1/2 interacting with the quantized Maxwell field (mass zero and spin one) in the limit when the particles have a small velocity. Two ways to implement the limit are considered: c → ∞ with the velocity v of the particles fixed, the case for which rigorous results have already been discussed in the literature, and v → 0 with c fixed. The second case can be rephrased as the limit of heavy particles, mj → ε−2 mj , observed over a long time, t → ε−1 t, ε → 0+ , with kinetic energy Ekin = O(1). Focusing on the second approach we construct subspaces which are invariant for the dynamics up to terms of order ε log(ε−1 ) and describe effective dynamics, for the particles only, inside them. At the lowest order the particles interact through Coulomb potentials. At the second one, ε2 , the mass gets a correction of electromagnetic origin and a velocity dependent interaction, the Darwin term, appears. Moreover, we calculate the radiated piece of the wave function, i.e., the piece which leaks out of the almost invariant subspaces and calculate the corresponding radiated energy.

1. Introduction A system of nonrelativistic particles of spin 1/2 interacting with the quantized radiation field is described by the so-called Pauli–Fierz Hamiltonian, or “nonrelativistic quantum electrodynamics”. The model is thought to have an extremely wide range of validity, apart from phenomena connected to gravitational forces and from other ones typical of high-energy physics like pair creation, whose description requires the use of full relativistic QED. This belief is mainly based on the analysis of some formal limit cases, which can be accurately studied both from a theoretical and an experimental point of view. Indeed, the interaction between charged particles is usually described by instantaneous pair potentials of Coulomb-type, without introducing the field as

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dynamical variable. This is known to be a good approximation if the particles move sufficiently slowly. One aim of this paper is a mathematically rigorous justification of this fact, i.e., the derivation of the Schr¨ odinger equation with Coulomb potentials, and the second order velocity dependent corrections to them, starting from nonrelativistic quantum electrodynamics. In addition, a formula is provided for the wave function of the radiated photons and the corresponding radiated energy, which is the quantum equivalent of the Larmor formula of classical electrodynamics. In more detail the model considered is given, excluding the addition of the electronic spin, by the canonical quantization of a system of N classical charges interacting through the Maxwell field. A sharp ultraviolet cutoff is introduced assuming that each charge has a charge distribution given by j (x) = ej ϕ(x) ,

x ∈ R3 ,

(1)

−3/2

for |k| ≤ Λ, 0 otherwise (note that where the form factor satisfies ϕ(k) ˆ = (2π) there is no infrared cutoff ). The classical equations of motion are given by 1 ∂ B(x, t) = −∇ × E(x, t) , c t N (2) q˙j (t) 1 ∂t E(x, t) = ∇ × B(x, t) − , ej ϕ x − qj (t) c c j=1 with the constraints ∇ · E(x, t) =

N

ej ϕ x − qj (t) ,

∇ · B(x, t) = 0 ,

(3)

j=1

and the Newton equations for the particles, q˙l (t) ml q¨l (t) = el Eϕ ql (t), t + × Bϕ ql (t), t , c

l = 1, . . . N ,

(4)

where Eϕ (x, t) := (E ∗x ϕ)(x, t) and analogously for Bϕ . The canonical quantization of this system in the Coulomb gauge is described, e.g., in [23, Chapter 13]. The Hilbert space of the pure states is given by H := Hp ⊗ F .

(5)

The space for the particles, Hp , is defined by1 Hp := L2 (R3 × Z2 )⊗N ,

(6)

where R is the configuration space of a single particle and Z2 represents its spin. 3

1 The

formalism presented holds also in the case when all the particles are equal and their Hilbert space is given by the subspace of totally antisymmetric wave functions. In this case, the dipole radiation given in (29) is zero. We consider therefore the general case of different particles.

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The state space for a single photon is L2 (R3 ×Z2 ), where R3 is the momentum space of the photon and Z2 represents its two independent physical helicities. The photon Fock space is therefore M 2 3 F := ⊕∞ M=0 ⊗(s) L (R × Z2 ) ,

(7)

0 2 3 where ⊗M (s) denotes the M -symmetric tensor product and ⊗(s) L (R × Z2 ) := C. We denote by ΩF the vector (1, 0, . . .), called the Fock vacuum. The dynamics of the system are generated by the Hamiltonian 2 N 1 1 c H := + Vϕ coul (x) + cHf , (8) σj · −i∇ − √ ej Aϕ (xj ) 2mj c j=1

where all the operators appearing are independent of c and we use units in which = 1. σj is a vector whose components are the Pauli matrices of the jth particle, Aϕ (xj ) denotes the quantized transverse vector potential in the Coulomb gauge, Vϕ coul is the smeared Coulomb potential and Hf the free field Hamiltonian. The reader who is not familiar with the notation is advised to look at Section 2, where the model is described in more detail. To implement practically the idea that the particles move “slowly”, a standard procedure, applied also in classical electrodynamics (see, e.g., [14, 17]), is to take the limit c → ∞2 . Since c is a quantity with a dimension, one should actually say that |v|/c → 0, where v is a typical velocity of the particles. This can be achieved in two ways, fixing v and letting c → ∞ or fixing c and letting v → 0. In the classical case this is reflected in the fact that the limit c → ∞ is equivalent, up to a rescaling of time, to the limit of heavy particles, as one can easily verify replacing in equations (2)–(4) ml with ε−2 ml , t with ε−1 t, and looking at the limit ε → 0. We will show that in the quantum case the two procedures are non equivalent anymore, a fact that can be intuitively explained by the presence of an additional scale given by . In this paper we concentrate on the limit of heavy particles observed over a long time. An additional aim is to point out similarities and differences between the two limits in the quantum context and to compare the results we get for the 2 In

the classical case, a more refined and precise analysis is carried through in [15,16]. The authors consider, loosely speaking, initial conditions which represent free particles moving together with the field they generate (“dressed” particles or charge solitons), with a velocity of order O(ε1/2 ) with respect to the speed of light. Assuming that the particles are at time t = 0 far apart (relative distance of order O(ε−1 )) and rescaling suitably the dynamical variables, they show that the particles remain at a relative distance of order O(ε−1 ) for long times (of order O(ε−3/2 )) and on this time scale their motion is governed by effective dynamics. The possibility to implement an analogous limit in the quantum case is unclear, because there is no obvious quantum counterpart to the classical charge solitons. The Pauli–Fierz Hamiltonian without infrared cutoff has indeed no ground state in the Fock space for fixed total momentum different from zero [4, 10]. We stick therefore to the more pragmatic choice c → ∞. The states which we define through the dressing transformation Uε in equation (27) should be considered approximate dressed states valid for small velocities of the particles.

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Pauli–Fierz model with the ones valid for the Nelson model, where the particles and the photons are spinless, [24]. We recall briefly in the next subsection some known results about the case c → ∞ and then illustrate in more detail the limit ε → 0. 1.1. The limit c → ∞ This case, as observed by Spohn [23], has the form of a weak coupling limit, fact which was already noted for the Nelson model by Davies [6], who formulated also a general scheme to analyze the limit dynamics in the weak coupling case [5] (an extended notion of weak coupling limit for Pauli–Fierz systems in which the Hilbert space of the particles is finite dimensional has been examined in [7]). Davies looks loosely speaking at the limit λ → 0 for the time evolution generated by an Hamiltonian of the form λ−2 (H0 + λHint ) , which corresponds physically to a weak interaction, whose effect is however observed over the long time scale λ−2 . The Hamiltonian H c assumes a similar form if we consider it on the long time scale defined by c2 . Putting λ := c−3/2 we get indeed c2 H c = λ−2 Hf + λ2/3 Hp + λh1 + λ4/3 h4/3 := λ−2 Hλ ,

(9)

where Hp := −

h1 :=

h4/3 :=

N 1 Δj + Vϕ coul , 2m j j=1

(10)

N ej ej i∇xj · Aϕ (xj ) − σj · Bϕ (xj ) , m 2m j j j=1

(11)

N e2j : Aϕ (xj )2 : , 2m j j=1

(12)

where Bϕ := ∇ × Aϕ and we normal order the quadratic term. Applying Davies scheme one gets in the end [23, Theorem 20.5] N

Theorem 1. Let ψ ∈ H 1 (R3N , C2 ), then

c 2 2 lim e−iH c t − e−iHdarw c t ψ ⊗ H = 0 , c→∞

(13)

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where Hdarw := Hp + c−2 Vdarw + c−2 Vspin , N 2 ej el |ϕ(k)| ˆ dk eik · xj i∇xj · (1 − κ ⊗ κ)i∇xl e−ik · xl , Vdarw := − m j m l R3 2|k|2 j,l=1

Vspin := −

N j,l=1

ej el σj · σl 12ml mj

R3

2 ik · (xj −xl ) dk |ϕ(k)| ˆ e ,

(14) (15)

(16)

where (κ ⊗ κ)ij := κi κj and κ := k/|k|. Vdarw gives rise to a correction of electromagnetic origin to the mass of the particles and to a velocity dependent potential, called the Darwin term. It already appears in classical electrodynamics when the dynamics of the particles are expanded up to terms of order (v/c)2 (see, e.g., [14] or [17]). For the convenience of the reader and to ease the comparison with the results for the limit of heavy masses we give a formal derivation of this theorem in Appendix A. We note here that the method employed in the weak coupling case forces one to consider as initial condition for the field just the Fock vacuum, which contains no photons at all. There is therefore no analogy with the physical picture that every particle should be described by a “dressed state”, loosely speaking the particle itself dragging with it a cloud of “virtual” photons. 1.2. The limit m → ∞ The situation is different in the case mj → ∞, which is more conveniently studied adopting units where c = 1. Replacing mj by ε−2 mj we get then the Hamiltonian N 1 2 ej ej H := pˆj + Vϕ coul + Hf − ε pˆj · Aϕ (xj ) − ε2 σj · Bϕ (xj )+ 2m m 2m j j j j=1 ε

(17)

e2j + ε2 : Aϕ (xj )2 : , 2mj where we have indicated with pˆj the ε-momentum of the jth particle pˆj := −iε∇xj .

(18)

As already pointed out talking about the classical case, the dynamics have to be observed over times of order O(ε−1 ). This is necessary in order to see non trivial effects, because we consider initial states with bounded kinetic energy. Since the particles have a mass of order O(ε−2 ) this means that their velocity is in the original time scale of order O(ε). To analyze the limit ε → 0 we construct a unitary dressing transformation Uε : H → H , which allows us to define dressed states for small velocities of the particles and to introduce a clear notion of real and virtual photons. More precisely, in the new representation defined by Uε , the vacuum sector Hp ⊗ ΩF corresponds

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to states of dressed particles without real photons, while in the original Hilbert space a state with M real photons is a linear combination of states of the form Uε−1 ψ ⊗ a(f1 )∗ · · · a(fM )∗ ΩF , where ψ ∈ Hp , f1 , . . . , fM ∈ L2 (R3 × Z2 ) . The projector on the subspace corresponding to dressed particles with M real photons is therefore ε PM := Uε∗ (1p ⊗ QM )Uε ,

where QM denotes the projector on the M -particles subspace of the Fock space. ε H are approximately invariant In short, we will show that the subspaces PM ε for the dynamics defined by H on times of order O(ε−1 ). Moreover, on this time scale we will give effective dynamics for states inside such a subspace, with an error of order O(ε2 log(ε−1 )). The effective dynamics contain the Darwin correction described in (15), but no spin dependent term. One can get an idea of why this happens comparing the expression of H ε with that of Hλ , equation (17) and (9). In H ε the spin dependent term is of second order, while in Hλ is of the first one. In the limit ε → 0 the analogue of Vspin would be of order O(ε4 ), therefore it does not appear in an expansion of the time evolution till second order. Finally we compute the leading order part of the state which makes a transition between P0ε and P1ε , which corresponds to the emission of one real photon. The corresponding radiated energy is given by a quantum analogue of the Larmor formula. The procedure to construct the unitary Uε is explained in detail in [24] for the Nelson model. The technique used is based on space-adiabatic perturbation theory [25], a method which allows to expand the dynamics generated by a pseudodifferential operator with an ε-dependent semiclassical symbol. The main difficulty in all models concerning the interaction of particles with a quantized field of zero mass is that, because of soft photons, the principal symbol of the Hamiltonian has no spectral gap, which is a condition required to apply the methods of [25]. In the case of H ε we have indeed h0 (p, q) :=

N 1 2 pj + Vϕ coul (q) + Hf , 2m j j=1

(p, q) ∈ R3N × R3N .

(19)

For every fixed (p, q) this is an operator on F and has a ground state given by ΩF , at the threshold of the continuous spectrum. The corresponding eigenvalue E0 (p, q) =

N 1 2 pj + Vϕ coul (q) 2m j j=1

(20)

is the symbol of an Hamiltonian acting just on Hp and describing the particles interacting through the smeared Coulomb potential. The trouble connected to the absence of the spectral gap is solved by introducing an effective gap, considering the Hamiltonian H ε,σ where the form factor ϕˆ (see equation (1)) is replaced by ϕˆσ (k) := (2π)−3/2 for σ < |k| < Λ, 0 otherwise.

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Proposition (see Proposition 3). Suppose that the cutoff σ is a function of ε, σ = σ(ε), such that σ(ε) < ε2 , then e−itH where

ε

/ε

− e−itH

ε,σ

/ε

L(H0 ,H ) ≤ C|t|σ(ε)1/2

H0 := D H0ε )

(21)

(22)

is the domain of the free Hamiltonian H0ε :=

N 1 2 pˆj + Hf 2m j j=1

(23)

with the corresponding graph norm. Fixing σ, e.g., as a sufficiently high power of ε we can then replace the original dynamics with infrared cutoff ones. For H ε,σ it is possible to build a dressing operator Uε,σ which can be expanded in a series of powers of ε with σ-dependent coefficients which are at most logarithmically divergent. Using it we define the dressed Hamiltonian ε,σ ∗ Hdres := Uε,σ H ε,σ Uε,σ

(24)

which can be expanded in a series of powers of ε in L(H0 , H ), with coefficients which are also at most logarithmically divergent in σ. The different coefficients in the expansion correspond to different physical effects which can be now clearly separated according to their order of magnitude in ε. The first result we find, as we already mentioned above, is that the dressed M -photons subspaces are approximately invariant for the dynamics: Theorem (see Corollary 2). Given a χ ∈ C0∞ (R) and a function σ(ε) such that ε−2 σ(ε)1/2 → 0 , ε log σ(ε)−1 → 0 , ε → 0+ , (25) then

√

−iH ε t ε

ε

e ε,P M + 1|t|ε log σ(ε)−1 , M χ(H ) L(H ) = O

(26)

where ε ∗ PM := Uε,σ(ε) (1p ⊗ QM )Uε,σ(ε) .

(27)

The adiabatic decoupling which guarantees the invariance of the subspaces holds uniformly only on states in which the particles have a uniformly bounded kinetic energy. For this reason we introduce a cutoff function on the total energy χ, which gives rise automatically to a bounded kinetic energy for the slow particles. In the following we assume that the function σ(ε) has been fixed so that (25) is satisfied. One can then approximate the dynamics of the particles inside each almost invariant subspace.

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Theorem (see Theorem 6). Let S be a bounded observable for the particles, S ∈ ε L(Hp ), and ω ∈ I1 (PM χ(H ε )H ) a density matrix for a mixed dressed state with M free photons, whose time evolution is defined by ω(t) := e−itH

ε

/ε

ωeitH

ε

/ε

.

We have then ε ε TrH (S ⊗ 1F )ω(t) = TrHp Se−itHeff TrF (ω)eitHeff + O ε3/2 |t| (1 − δM0 ) + O ε2 log σ(ε)−1 |t| + |t|2 , where δM0 = 1, when M = 0, 0 otherwise, and ε Heff :=

N 1 2 pˆj + Vϕ coul 2m j j=1

− ε2

N 2 ej el |ϕ(k)| ˆ dk eik · xj pˆj · (1 − κ ⊗ κ)ˆ pl e−ik · xl 2 m j m l R3 2|k|

(28)

l,j=1

=

N 1 2 pˆj + Vϕ coul + ε2 Vdarw . 2m j j=1

ε Remark 1. Even though the subspaces PM depend on the choice of the infrared cutoff, the effective Hamiltonian is infrared regular and therefore independent of σ. Moreover, as we briefly mentioned above, it contains the corrections to the mass of the particles and the Darwin term, but no spin dependent term (compare with Theorem 1). This topic is further discussed in the proof of Theorem 5 and in Remark 4. ε are only approximately invariant, there is a piece Since the subspaces PM of the wave function which “leaks out” in the orthogonal complement. This correspond physically to the emission or absorption of free photons. For a system starting in the dressed vacuum the leading order of the wave function of the emitted photon is given in the next theorem.

Theorem (see Corollary 3). Up to terms of order O(ε2 log(σ(ε)−1 )(|t| + |t|2 )), the radiated piece for a system starting in the dressed vacuum (M = 0) is given by t

ε

Ψrad (t) := (1 − P0ε )e−i ε H P0ε χ(H ε )Ψ ˆσ(ε) (k) ˆ ε iε ϕ ∼ eλ (k) = −e−ith0 √ 2 |k|3/2 t ¨ · ds ei(s−t)|k|/ε OpW ε D(s; x, p) ψ(x) , 0

(29)

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where eλ (k) is the polarization vector of a photon with helicity λ, N 1 2 pˆ + Vϕ coul + Hf , 2mj j j=1 ˆ ε )Ψ ∈ Hp , ψ(x) := ΩF , χ(h 0 F

ˆ ε := h 0

D(s; x, p) :=

N j=1

ej cl x (s; x, p) , mj j

(30) (31) (32)

3N × OpW ε denotes the Weyl quantization acting on a suitable symbol space on R 3N cl R and xj is the solution to the classical equations of motion cl ¨cl mj x j (s; x, p) = −∇xj Vϕ coul x (s; x, p) ,

xcl j (0; x, p) = xj ,

−1 x˙ cl j (0; x, p) = pj mj ,

j = 1, . . . , N .

Remark 2. As explained in detail in Remark 5, generically the norm of the radiated piece is bounded below by O(ε log(εσ(ε)−1 )), which means that the subspace P0ε is near optimal, i.e., the transitions are at least of order O(ε log(εσ(ε)−1 )). Note that, like in classical electrodynamics, when all the particles are equal, the leading order of the radiated piece vanishes, because D is then proportional to the position of the center of mass, whose acceleration is zero. Remark 3. Even though the radiated wave function has no limit when ε → 0, / L2 (R3 ), the corresponding radiated energy has a limit. because ϕ(k)|k|−3/2 ∈ Defining Erad (t) := Ψrad (t), Hf Ψrad (t) , (33) we get to the leading order (see Remark 6) d ε3 2 ¨ . Prad (t) := Erad (t) ∼ = 2 ψ, OpW ε |D(t)| ψ dt 3π Hp

(34)

In the case of the Nelson model analogous results are proved in [24], which contains also a detailed discussion of the adiabatic framework. The form of the effective dynamics is equal, the only difference, as one can expect, is in the radiated piece, which contains here explicitly the helicity of the photon. Another difference is that the principal symbol of the Pauli–Fierz Hamiltonian, defined in (19), is diagonal with respect to the Fock projectors QM , while for the Nelson Hamiltonian one needs a dressing transformation already at the leading order. This makes the analysis of the Pauli–Fierz case somewhat less technical. The effective dynamics for M = 0 (dressed vacuum) was calculated by Spohn [23, Section 20.2] in the case when the photon has a small mass, mph > 0, which introduces a gap in the principal symbol of the Hamiltonian. He however states that these effective dynamics are identical with the ones calculated for the case c → ∞, while we have already remarked that the spin dependent term cannot be present when ε → 0. In the case mph > 0, moreover, the transitions between

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the almost invariant subspaces become smaller than any power of ε and therefore it is not known how to get an explicit expression for them. The Pauli–Fierz Hamiltonian has also been extensively studied to get informations about its spectral and scattering structure. Not pretending to be exhaustive, we refer the reader interested to these aspects to [3, 8, 11, 12] and references therein. In Section 2 we complete the description of the model and discuss the approximation of the original dynamics through infrared cutoff ones. In Section 3 the construction of the dressing operator U is discussed, and applied in Section 4 to the study of the dressed Hamiltonian. The main results on the effective dynamics and the radiated piece are contained in Section 5. Finally, Appendix A contains a sketch of the proof of Theorem 1, for the case c → ∞.

2. Preliminary facts In this section we elaborate on the definition of the Pauli–Fierz model and discuss some preliminary facts like the self-adjointness of the Hamiltonian and the approximation of the original dynamics through infrared cutoff ones. 2.1. Fock space and field operator (The proofs of the statements we claim can be found in [21, Section X.7]. We denote by Ffin the subspace of the Fock space, defined in (7), for which Ψ(M) = 0 for all but finitely many M . Given f ∈ L2 (R3 × Z2 ), one defines on Ffin the annihilation operator by 2 √ (M) a(f )Ψ (k1 , λ1 ; . . . ; kM , λM ) := M + 1 dk f (k, λ)∗ 3 (35) λ=1 R · Ψ(M+1) (k, λ; k1 , λ1 . . . , kM , λM ) . The adjoint of a(f ) is called the creation operator, and its domain contains Ffin . On this subspace they satisfy the canonical commutation relations

a(f ), a(g)∗ = f, gL2 (R3 ×Z2 ) , (36)

a(f ), a(g) = 0 , a(f )∗ , a(g)∗ = 0 . Since the commutator between a(f ) and a(f )∗ is bounded, it follows that a(f ) can be extended to a closed operator on the same domain of a(f )∗ . On this domain one defines the Segal field operator 1 (37) Φ(f ) := √ a(f ) + a(f )∗ 2 which is essentially self-adjoint on Ffin . Moreover, Ffin is a set of analytic vectors for Φ(f ). From the canonical commutation relations it follows that

(38) Φ(f ), Φ(g) = i f , gL2 (R3 ×Z2 ) .

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Given a self-adjoint multiplication operator by the function ω on the domain D(ω) ⊂ L2 (R3 ), we define Fω,fin := L ΩF , a(f1 )∗ · · · a(fM )∗ ΩF : M ∈ N, fj ∈ D(ω) ⊗ C2 , j = 1, . . . , M , (39) where L means “finite linear combinations of”. On Fω,fin we define the second quantization of ω, dΓ(ω), by

dΓ(ω)Ψ

(M)

(k1 , λ1 ; . . . ; kM , λM ) :=

M

ω(kj )Ψ(M) (k1 , λ1 ; . . . ; kM , λM ) ,

j=1

dΓ(ω)ΩF := 0 , which is essentially self-adjoint. In particular, the free field Hamiltonian Hf acts as M (M) (k1 , λ1 ; . . . ; kM , λM ) = |kj |Ψ(M) (k1 , λ1 ; . . . ; kM , λM ) , (Hf Ψ) j=1

Hf ΩF = 0 , and is self-adjoint on its maximal domain. From the previous definitions, given f ∈ D(ω)⊗C2 , one gets the commutation properties

dΓ(ω), a(f )∗ = a(ωf )∗ ,

dΓ(ω), a(f ) = −a(ωf ) , (40)

dΓ(ω), iΦ(f ) = Φ(iωf ) . 2.2. The Pauli–Fierz model Using the Segal field operator one can write the quantized vector potential and the magnetic field appearing in (8) as Aϕ (x) = Φ(vx ) ,

(41)

eλ (k) ϕ(k) ˆ , vx (k, λ) := f (k, λ)e−ik · x , f (k, λ) := |k|1/2 Bϕ (x) = ∇x × Aϕ (x) = −Φ(ik × vx ) ,

(42) (43)

where eλ (k), λ = 1, 2, are, for simplicity, real photon polarization vectors satisfying eλ (k) · eμ (k) = δλμ ,

k · eλ (k) = 0 .

The smeared Coulomb potential is given by N 2 |ϕ(k)| ˆ 1 Vϕ coul (x) = ej el dk eik · (xj −xl ) . 2 2 |k| R3

(44)

(45)

j,l=1

Analogous expressions hold for the infrared cutoff Hamiltonian H ε,σ , where the form factor ϕˆ is replaced by ϕˆσ .

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To separate more clearly the terms of different order in the Hamiltonian H ε , equation (17), it is useful to write it as Hε =

2

ˆε εi h i

(46)

i=0

where ˆ ε := h 0

N 1 2 pˆj + Vϕ coul + Hf , 2m j j=1

N ej pˆj · Φ(vxj ), m j j=1 N 2 e e j j ˆ ε := h σj · Φ(ik × vxj ) + : Φ(vxj )2 : . 2 2m 2m j j j=1

ˆ ε := − h 1

(47)

(48)

ˆ ε is of order O(1) when applied to functions of bounded kinetic Each of the h i ˆ ε,σ . energy. The coefficients for H ε,σ will be denoted by h i As proved by Hiroshima [13] using functional integral techniques the Hamiltonian H ε (and analogously H c ) is self-adjoint on H0 for every value of the masses, charges and number of particles. Since however we study the limit ε → 0 (respectively c → ∞) it is enough for our purposes to show this using Kato theorem, like, e.g., in [3]. Even though the proof is well known, we repeat it because we need to show that the graph norms which appear are equivalent uniformly in ε and σ. Moreover, the estimates which appear in the proof will be useful in Propositions 3 and Lemma 1. Given f ∈ L2 (R3 × Z2 ), we define 1/2 . (49) f ω := f |k|−1/2 2L2 (R3 ×Z2 ) + f 2L2(R3 ×Z2 ) One has then the basic estimate Proposition 1. a (f1 ) · · · a (fn )(Hf + 1)−n/2 L(F ) ≤ Cn f1 ω · · · fn ω ,

(50)

where a (f ) can be a(f ) or a∗ (f ). Proposition 2. Both Hamiltonians H ε and H ε,σ are self-adjoint on H0 . Moreover the graph norms they define are equivalent to the one defined by H0ε uniformly in ε and σ. The same holds for the graph norm defined by (H ε )1/2 and (H ε,σ )1/2 . Proof. (We give the proof for H ε,σ , the one for H ε is the same.) The regularized Coulomb potential is a bounded function, therefore for it the statement is trivial. We choose a vector Ψ in a core of H0ε made up of smooth functions with compact support both in x and k.

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For the term of order ε we get then

N N

e e j j

−ε

pˆj · Aϕσ (xj )Ψ = −ε Φ(vxj ,σ ) · pˆj Ψ

m m

j=1 j

j=1 j

H

H

≤ε

N 3 |ej | j=1

≤ε

N j=1

mj

Φ(vxαj ,σ )ˆ pα j Ψ

α=1

3 |ej | Φ(vxαj ,σ )(Hf + 1)−1/2 L(H ) mj α=1

· (Hf + 1)1/2 pˆα j ΨH ≤ Cε

N |ej | j=1

mj

vxj ,σ ω ΨH0 ,

so for ε sufficiently small this term is Kato small with respect to the free Hamiltonian, with a constant uniformly bounded in ε and σ. An analogous estimate holds for the term with the magnetic field. For the remaining one we have

N 2

N 2 e2j

e j 2 2 2

ε : Aϕσ (xj ) : Ψ = ε : Φ(vxj ,σ ) : Ψ

j=1 2mj

j=1 2mj

H

H

N e2j 2 ≤ Cε vxj ,σ 2ω (Hf + 1)ΨH , 2m j j=1

which completes the proof. Proposition 3. If σ(ε) < ε2 then

−itH ε /ε ε,σ

e − e−itH /ε L(H

0 ,H

)

≤ C|t|σ 1/2 .

(51)

Proof. From the previous proposition we know that both Hamiltonians are selfadjoint on H0 , so, given Ψ ∈ H0 , we can apply Duhamel formula to get

−itH ε /ε

ε,σ ε 1 t

e − e−itH /ε Ψ H ≤ ds H ε − H ε,σ e−isH /ε Ψ H . ε 0 Putting Ψs := e−isH

ε

/ε

Ψ, the difference of the two Hamiltonians is

(H ε − H ε,σ )Ψs = (Vϕ coul − Vϕσ coul )Ψs −ε

N ej pˆj · Φ 1(0,σ) (k)vxj Ψs mj j=1

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− ε2

N ej σj · Φ − ik × 1(0,σ) (k)vxj Ψs 2mj j=1

+ ε2

N e2j Aϕ (xj )2 − Aϕσ (xj )2 Ψs . 2mj j=1

(52)

Using the explicit expression (45), the term with the Coulomb potential gives N 1 dk |Vϕ coul (x) − Vϕσ coul (x)| ≤ |ej el | ||ϕ| ˆ 2 − |ϕˆσ |2 | 2 |k|2 j,l=1

=

N 1 |ej el |σ 12π 2 j,l=1

⇒ Vϕ coul − Vϕσ coul L(H ) = O(σ) .

(53)

For the term of order ε, proceeding as in the proof of Proposition 2 we get that

N N

e |ej | j

−ε ˜ p ˆ · Φ 1 (k)v ≤ Cε 1(0,σ) (k)vxj ,σ Ψs H0 . Ψ j xj s

(0,σ)

mj

j=1 mj j=1 H

From the same proposition it follows that the graph norm associated to H0ε and the one associated to H ε are equivalent uniformly in ε and σ, therefore

N N

e |ej |

j

−ε

1(0,σ) (k)vxj ,σ ΨH0 p ˆ · Φ 1 (k)v ≤ Cε Ψ j x s (0,σ) j

mj (54)

j=1 mj j=1 H

= O(εσ 1/2 )ΨH0 . The same reasoning holds for the term containing the spin, which has however a |k| more, which gives in the end

N

2 ej

= O(ε2 σ 3/2 )ΨH0 .

−ε Ψ σ · Φ − ik × 1 (k)v (55) j xj s

(0,σ)

2m j

j=1 H

Concerning the last term we have ε2

N N e2j e2j Aϕ (xj )2 − Aϕσ (xj )2 Ψs = ε2 Φ(vxj )2 − Φ(vxj ,σ )2 Ψs 2m 2m j j j=1 j=1 N 2 e2j Φ(vxj ,σ ) + Φ(1(0,σ) (k)vxj ) 2mj j=1 − Φ(vxj ,σ )2 Ψs

= ε2

= ε2

N e2j Φ(vxj ,σ )Φ 1(0,σ) (k)vxj 2mj j=1

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+ Φ 1(0,σ) (k)vxj Φ(vxj ,σ ) 2 Ψs . + Φ 1(0,σ) (k)vxj Using again the basic estimate in Proposition 1 we get, for example,

Φ(vxj ,σ )Φ 1(0,σ) (k)vxj Ψs ≤ Cvxj ,σ ω 1(0,σ) (k)vxj Ψs H0 = O(σ 1/2 )ΨH0 ,

(56)

by the same reasoning we used for the terms of order ε. Lemma 1. Given a function χ ∈ C0∞ (R) and assuming σ(ε) < ε2 , then χ(H ε ) − χ(H ε,σ )L(H ) ≤ Cεσ 1/2

(57)

Proof. Using the Hellfer–Sj¨ ostrand formula (see, e.g., [9, Chapter 8]), given a selfadjoint operator A, we can write 1 ¯ a (z)(A − z)−1 , z := x + iy , χ(A) = dxdy ∂χ (58) π R2 where χa ∈ C0∞ (C) is an almost analytic extension of χ, which satisfies the properties ¯ a | ≤ D ¯ |z|N¯ , ¯ ∈ N ∃ DN¯ : |∂χ ∀N N χa|R = χ . (For the explicit construction of such a χa see [9]). Applied to our case (58) yields

1 ¯ a (z) (H ε − z)−1 − (H ε,σ − z)−1 . χ(H ε ) − χ(H ε,σ ) = dxdy ∂χ π R2 Since both Hamiltonians are self-adjoint on H0 we have (H ε − z)−1 − (H ε,σ − z)−1 = (H ε,σ − z)−1 (H ε,σ − H ε )(H ε − z)−1 ,

(59)

and hence χ(H ε ) − χ(H ε,σ )L(H ) 1 ¯ a (z)|(H ε,σ − z)−1 L(H ) (H ε,σ − H ε )(H ε − z)−1 L(H ) . ≤ dxdy |∂χ π R2 In addition we have that (H ε,σ − z)−1 L(H ) ≤

C . |z|

(60)

This follows because H0 is dense in the domain of H ε=0,σ=0 , and for every Ψ ∈ H0 H ε,σ Ψ → H ε=0,σ=0 Ψ as

(ε, σ) → (0, 0) .

According to Theorem VIII.25 [20], this implies that (H ε,σ − z)−1 Ψ → (H ε=0,σ=0 − z)−1 Ψ ,

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therefore |z|(H ε,σ −z)−1 Ψ is bounded for every Ψ and the uniform boundedness principle gives (60). For the second norm we find that for z ∈ supp χa (H ε,σ − H ε )(H ε − z)−1 L(H ) ≤ (H ε,σ − H ε )L(H0 ,H ) · (H ε − z)−1 L(H ,H0 ) C (H ε,σ − H ε )L(H0 ,H ) . ≤ |z| The right-hand side was already estimated in the previous proposition ((52) and the following calculations), the only difference being that here we have to multiply the final result by ε.

3. Construction of the dressing operator In this section we construct the unitary dressing operator Uε,σ . We apply to the Pauli–Fierz Hamiltonian the general procedure explained in some detail in [24] and, for the case with spectral gap, in [25]. All the calculations expounded in Section 3.1 are formal, and they serve as a guide for the rigorous definition of Uε,σ given in Section 3.2. 3.1. The formal procedure The main idea is to build an approximate projector, π ˆ (1) , which satisfies formally

ˆ (1) = O(ε2 ) , π ˆ (1) , H ε = O(ε2 ) . (ˆ π (1) )2 − π ε

Integrating over time the second equation one gets in a loose sense that [e−itH /ε , π ˆ (1) ] = O(ε|t|). The projector π ˆ (1) is found using an iterative procedure, which assumes that one can expand it in powers of ε, π ˆ (1) = π ˆ0 + εˆ π1 , where the coefficient π ˆ0 is a known input and must commute with the coefficient of order zero in the expansion of the Hamiltonian H ε , see equation (46). As it turns out, the procedure does not work directly for H ε , but only for the infrared cutoff Hamiltonian H ε,σ . An obvious choice for π ˆ0 is π ˆ 0 = QM , ˆ ε,σ , QM ] [h 0

which satisfies = 0. Proceeding now in the same way as described in [24] we get a formal expression for the first order almost projection given by (1)

π ˆM := QM + εˆ π1M , ⎡ ⎤ N iv (λ, k) e xj ,σ j ⎦. π ˆ1M := ⎣QM , i pˆj · Φ m |k| j j=1

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For brevity, we put from now on ivxj ,σ (λ, k) Φj,σ := Φ . |k|

569

(61)

It is clear from equation (61) that QM π ˆ1M QM = (1 − QM )ˆ π1M (1 − QM ) = 0 , so (1)

(1)

(ˆ πM )2 − π ˆM = O(ε2 ) . (1)

π ˆM is also almost invariant for the total dynamics, in the sense that

M ε,σ

M ε,σ

(1) ε,σ ˆ ε,σ + ε π ˆ = QM , h ˆ1 , h + ε QM , ˆhε,σ + ε2 π ˆ1 , ˆh1 π ˆM , H 0 0 1

M ε,σ ˆ ε,σ + ε3 π ˆ + ε2 QM , h ˆ1 , h = O ε2 log(σ −1 ) . 2 2 To justify this claim we note that

N N ej ej ε,σ ε,σ ˆ ε,σ + QM , ˆ ˆ π ˆ1M , h = i p ˆ − h · [Q , Φ ], h pˆj j M j,σ 0 1 0 m m j j j=1 j=1

N

ej pˆ2 · QM , Φ(vxj ,σ ) = i pˆj · [QM , Φj,σ ], l mj 2 j,l=1

+i

N j=1

ej pˆj · [QM , Φj,σ ], Vϕσ coul mj

N ej +i pˆj · [QM , Φj,σ ], Hf m j j=1

−

N

ej pˆj · QM , Φ(vxj ,σ ) = O ε log(σ −1 ) mj j=1

+i

N

ej pˆj · QM , [Φj,σ , Hf ] m j j=1

N

ej pˆj · QM , Φ(vxj ,σ ) mj j=1 = O ε log(σ −1 ) .

−

To analyze in a simple way the restriction of the dynamics to the subspace (1) defined by π ˆM one builds an almost unitary U (1) , which maps the almost projections to a reference projection up to terms of order O(ε2 ). Using the formal expression we get for U (1) , we will define in next section a true unitary operator (1) which will allow us to construct a rigorous version of the almost projections π ˆM .

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A natural choice for the reference projections, linked to the physics of the system, is to choose them equal to the QM s. We assume then that also U (1) can be expanded in powers of ε, U (1) := 1 + εU1 , with the condition U1 + U1∗ = O(ε). This ensures that U (1) U (1)∗ = O(ε2 ) ,

U (1)∗ U (1) = O(ε2 ) .

To determine U1 we impose that U (1) intertwines the almost invariant projections with the reference projections QM up to terms of order O(ε2 ): !

(1)

U (1) π ˆM U (1)∗ = QM + O(ε2 ) . The left-hand side gives (1)

ˆM U (1)∗ = (1 + εU1 )(QM + εˆ π1M )(1 − εU1 ) + O(ε2 ) U (1) π = QM + ε [U1 , QM ] + π ˆ1M + O(ε2 ) ⎤⎞ ⎛ ⎡ N ej = QM + ε ⎝[U1 , QM ] − ⎣i Φj,σ · pˆj , QM ⎦⎠ + O(ε2 ) , m j j=1 so we can choose U (1) = 1 + iε

N ej Φj,σ · pˆj . m j j=1

(62)

3.2. Rigorous definition To get a well-defined unitary operator from the formal expression for U (1) we first cutoff the number of photons in the field operator Φj,σ , replacing it by ΦLj,σ := Q≤L Φj,σ Q≤L ,

(63)

where L is fixed, but otherwise arbitrary. We introduce then a cutoff in the total energy, to cope with the unboundedness of the momentum of the electrons pˆj . This reflects the fact that the adiabatic approximation holds uniformly only on states where the kinetic energy of the slow particles in uniformly bounded. More precisely, given a function χ ∈ C0∞ (R), we define (1)

N N ej L ej L Φj,σ · pˆj − iε 1 − χ(H ε,σ ) Φj,σ · pˆj 1 − χ(H ε,σ ) mj mj j=1 j=1 L L = 1 + εχ(H ε,σ )U1,σ + ε 1 − χ(H ε,σ ) U1,σ χ(H ε,σ ) , (64)

UL,χ := 1 + iε

where we have defined L := i U1,σ

N ej L Φj,σ · pˆj . m j j=1

(65)

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571

L∗ L Note that it follows from the Coulomb gauge condition that U1,σ = −U1,σ ,

so

L (1) ∗ L UL,χ = 1 − εχ(H ε,σ )U1,σ − ε 1 − χ(H ε,σ ) U1,σ χ(H ε,σ ) .

(66)

Lemma 2. 1. For each x ∈ R3 , ΦLj,σ (x) ∈ L(F ), and ΦLj,σ (x)∗ = ΦLj,σ (x). Moreover, ΦLj,σ : R3 → L(F ) , x → ΦLj,σ (x) , ∈ Cb∞ R3 , L(F ) ,

(67)

and, for σ small enough,

√ ΦLj,σ L(H ) ≤ C L + 1 log(σ −1 ) .

(68)

Given α ∈ N3 with |α| > 0, it holds instead √ ∂xα ΦLj,σ = ∂xα ΦLj,0 + O σ |α| L + 1 L(H ) ,

(69)

where ∂xα ΦLj,0 := (∂xα ΦLj,σ )|σ=0 (70) is a well-defined bounded operator on H . 2. The statements of point 1 (except for the self-adjointness of ΦLj,σ (x)) remain true if F is replaced by D(Hf ). Corollary 1. The fibered operators ∂xα ΦLj,σ belong to L(H ) ∩ L(H0 ) ∀α ∈ N3 . Proof. The proof of both statements follows from the facts that

√

Q≤L Φ gx ( · ) Q≤L

≤ 21/2 L + 1 sup gx( · )L2 (R3 ×Z L(H )

and that

x∈R3

2)

−ik · x ∂xα vx (k, λ)|k|−1 = (−i)|α| |k|−3/2+|α| eλ (k)ϕ(k)e ˆ .

(1)

Lemma 3. The operator UL,χ is closable and its closure, which we denote by the same symbol, belongs to L(H ) ∩ L(H0 ). Moreover

(1)

−1 ) ,

U

(71) L,χ L(K ) ≤ C 1 + ε log(σ (1) ∗

where K = H or H0 . The same holds for UL,χ . Proof. The operator χ(H ε,σ )ΦLj,σ · pˆj is defined on D(ˆ pj ) and, since χ(H ε,σ )ΦLj,σ is a bounded operator, we have ∗ χ(H ε,σ )ΦLj,σ · pˆj = pˆj · ΦLj,σ χ(H ε,σ ) which is clearly bounded. This shows that χ(H ε,σ )ΦLj,σ · pˆj is closable and its closure belongs to L(H ). The same reasoning can be applied to the operator H ε,σ ΦLj,σ · pˆj χ(H ε,σ ) (1)

which shows that UL,χ is also in L(H0 ).

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The estimate on the norm follows now from the estimate on the norm of ΦLj,σ given in Lemma 2. Theorem 2. Assume that σ = σ(ε) satisfies the condition ε log σ(ε)−1 → 0 , ε → 0+ , then the operator

(72)

−1/2 (1) (1) ∗ (1) U := UL,χ UL,χ UL,χ

(73)

is well-defined and unitary, for ε small enough. Both U and U ∗ belong to L(H ) ∩ L(H0 ), with the property that U L(H0 ) ,

U ∗ L(H0 ) ≤ C ,

(74)

where C is independent of ε and σ. Moreover we can expand them in powers of ε and the corresponding series converges both in L(H ) and L(H0 ). Proof. It follows from equations (64) and (66) that, defining L L + 1 − χ(H ε,σ ) U1,σ χ(H ε,σ ) , Tσ := χ(H ε,σ )U1,σ we have then (1)

UL,χ = 1 + εTσ(ε) ,

Tσ∗ = −Tσ ,

Tσ L(K ) ≤ C

(75)

log(σ −1 ) ,

where K = H or H0 . From this expression we get immediately that (1) ∗

(1)

2 , UL,χ UL,χ = 1 − ε2 Tσ(ε) 2 so, choosing ε small enough, we have that ε2 Tσ(ε) L(K ) < 1, therefore the square root is well-defined, and can be expressed through a convergent power series in L(K ): ∞ −1/2 (2j − 1)!! 2j 2j 2 2 ε Tσ(ε) . 1 − ε Tσ(ε) = (76) (2j)!! j=0

From standard calculations it follows in the end that U is unitary on H .

4. The dressed Hamiltonian We define the dressed Hamiltonian as the unitary transform of H ε,σ , ε,σ := U H ε,σ U ∗ . Hdres

(77)

ε,σ is self-adjoint on H0 , and using the Since U is a bijection on H0 , Hdres ε,σ expansion of U on L(H0 ), we can expand Hdres in L(H0 , H ).

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Theorem 3. Assume that σ = σ(ε) satisfies conditions (25). The expansion of the dressed Hamiltonian up to the second order is then given by ε,σ ˆ dres + εh ˆ dres + ε2 h ˆ dres + O ε3 (log σ −1 )3/2 =h , (78) Hdres 0 1 2 L(H0 ,H ) where ˆ dres = h ˆε = h 0 0

N 1 2 pˆ + Vϕ coul + Hf , 2mj j j=1

ˆ dres is given in equation (81) and h ˆ dres in equation (84). h 1 2 Proof. Applying equation (76), we get that U = 1 + εT +

ε2 2 T + O(ε3 T 3 ) , 3 2

therefore ε,σ Hdres = (1 + εT + ε2 T 2 /2)H ε,σ (1 − εT + ε2 T 2 /2) + O ε3 (log σ −1 )3/2 L(H ,H ) 0

ε (75) = H ε,σ + ε[T, H ε,σ ] + [T, T, H ε,σ ] + O ε3 (log σ −1 )3/2 L(H0 ,H ) = 2

ˆ ε,σ + “(1 − χ) · · · χ” ˆ ε,σ + ε h ˆ ε,σ + χ(H ε,σ ) U L , h =h 1,σ 0 1 0

ˆ ε,σ + 1 T, χ(H ε,σ )[U L , h ˆ ε,σ ] ˆ ε,σ + χ(H ε,σ ) U L , h + ε2 h 1,σ 1,σ 2 1 0 2 3 + “(1 − χ) · · · χ” + O ε (log σ −1 )3/2 L(H0 ,H ) ˆ dres + εh ˆ dres + ε2 ˆ =: h + O ε3 (log σ −1 )3/2 L(H ,H ) hdres 0 1 2 2

0

where “(1 − χ) · · · χ” indicates that for every term containing χ · · · we have to add a corresponding term containing (1 − χ) · · · χ, as in equation (75). Using equation (53) to eliminate the σ, we get immediately ˆ dres = h ˆε . h 0 0

(79)

The commutator in the term of order ε gives: N

L ej L ˆ ε,σ = −Q≤L ˆhε,σ Q≤L + ε U1,σ , h Φ · ∇xj Vϕσ coul 0 1 mj j,σ j=1

+ε

N j,l=1

3 The

(80)

ej ∇x (ΦL · pˆj ) · pˆl + O(ε2 ) , ml mj l j,σ

expansion of U till the second order coincides with that of eεT , however eεT is not the correct dressing transformation to every order. Following the formal procedure sketched in Section 3.1 one can construct a second order expression for U , which depends however also in general ˆ ε,σ and has not a simple exponential form. on the second order coefficient of the Hamiltonian, h 2

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therefore, taking into account equation (54), ˆ ε Q≤L 1 − χ(H ε,σ ) ˆ dres = 1 − χ(H ε,σ ) Q≤L h h 1

1

ˆ ε Q>L + Q>L h ˆ ε Q≤L + Q>L h ˆ ε Q>L . + Q≤L h 1 1 1

(81)

We will show below that this term vanishes in the effective dynamics. Concerning the terms of second order we have

N ej el L L ˆ ε,σ Φj,σ · pˆj , Φ(vxl ,σ ) · pˆl , h1 = −i U1,σ mj ml j,l=1

N

ej el β = −i Q≤L Φα ˆα ˆβj j,σ , Φ(vxl ,σ ) Q≤L p jp mj ml

(82)

j,l=1

+ RL−1 + O(ε)L(H0 ,H ) , where RL−1 is a term which vanishes on the range of Qj when j < L − 1. Using equation (38) to calculate the commutator of the two field operator we get in the end −i

N ej el α Φj,σ , Φ(vxβl ,σ ) pˆα ˆβj jp mj ml

j,l=1

=i

% α & N vxj ,σ β ej el , vxl ,σ · pˆα ˆβj jp mj ml |k| 2 3 2 L (R ,dk)⊗C

j,l=1

N 2 ej el |ϕ(k)| ˆ =i dk eik · (xj −xl ) pˆj · (1 − κ ⊗ κ)ˆ pl m j m l R3 |k|2 j,l=1

N 2 ej el |ϕ(k)| ˆ =i dk eik · xj pˆj · (1 − κ ⊗ κ)ˆ pl e−ik · xl . m j m l R3 |k|2

(83)

j,l=1

The remaining term of second order gives, (80) L ˆ ε,σ ˆ ε,σ Q≤L + O(ε) T, χ(H ε,σ )[U1,σ , h0 ] = − T, χ(H ε,σ )Q≤L h 1

L (75) ˆ ε,σ Q≤L − χ2 Q≤L U L , ˆhε,σ Q≤L = −χ U1,σ , χ Q≤L h 1,σ 1 1 ˆ ε,σ Q≤L ]U L . − χ[χ, Q≤L h 1,σ 1

Putting the calculations above together, and using equations (55) and (56), we get in the end ˆ ˆε + hdres =h 2 2 +

N ej L Φj,σ · ∇xj Vϕ coul m j j=1

N j,l=1

ej ∇x (ΦL · pˆj ) · pˆl ml mj l j,σ

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575

L

ˆ ε,σ − 1 χ2 Q≤L U L , h ˆ ε,σ Q≤L + (1 − χ) U L , h ˆ ε,σ χ + χ U1,σ ,h 1,σ 1,σ 1 1 1 2 L 1 1 L ε,σ ˆ ˆ ε,σ Q≤L , − χ χ, Q≤L h1 Q≤L U1,σ − χ U1,σ , χ Q≤L h (84) 1 2 2 L ˆ ε,σ , h1 ] is given in equations (82) and (83). We will show where the commutator [U1,σ below that only few of the terms written above contribute to the effective dynamics, giving the correction to the mass of the electrons and the Darwin term.

5. The effective dynamics We quote without proof a number of lemmas, which, with minor modifications, are identical to the ones proved in [24]. Lemma 4 (see Corollary 4 [24]). Given a function χ ∈ C0∞ (R) and a σ > 0, we have ivxj ,σ (λ, k) ˆε) a QM χ(h 0 |k| ˆ ε )a ivxj ,σ (x, λ) QM χ(h ˆ ε ) + O0 (ε∞ ) , (85) = QM−1 ξ(h 0 0 |k| where ξ ∈ C0∞ (R), ξχ = χ and cξ = 2dχ + E∞ , where dχ := 2cχ + E∞ + min{cχ , Λ} , cχ := sup{|k| : k ∈ supp χ} , E∞ := sup |Vϕ coul (x)| , x∈R3N

and we can choose sup{|k| : k ∈ supp ξ} arbitrarily close to cξ . An analogous statement holds for the creation operator. Lemma 5. Assume that σ satisfies conditions (25), then 1. Given a function χ ˜ ∈ C0∞ (R), we have ˆ ε ) = εRε , ˜h χ(H ˜ ε,σ ) − χ( 0 χ

(86)

where Rεχ ∈ L(H , H0 ), Rεχ L(H ,H0 ) = O(1) and ˆ ε ) = (QM+1 + QM−1 )ξ(h ˆ ε )Rε QM χ( ˆ ε ) + O(ε)L(H ,H ) , ˜h ˜h Rεχ QM χ( 0 0 χ 0 0

(87)

where ξ has the properties described in Lemma 4. 2. Moreover, we have that ε,σ ˆ ε ) = O(ε)L(H ,H ) , χ(H ˜ dres ) − χ( ˜h 0 0

(88)

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and that

ε,σ ˆ ε )χ(H ˜ ε,σ ) + O ε2 log(σ −1 ) QM χ(H ˜ dres ) = QM χ( ˜h 0 ˜ dres

L(H ,H0 )

,

(89)

˜ ˜˜ = χ ˜˜ = χ, ˜˜ M < L − 1. where χ ˜ is any C0∞ (R) function such that χ ˜χ ˜ and χχ Proof. We give the proof just for point 1. The proof of point 2 is analogous and can be found also in [24, Lemma 7]. Proceeding as in Lemma 1, we apply Hellfer–Sj¨ ostrand formula and we get

ˆε) = 1 ¯ a (z) (H ε,σ − z)−1 − (h ˆ ε − z)−1 . χ(H ε,σ ) − χ(h dxdy ∂χ 0 0 π R2 Since both Hamiltonians are self-adjoint on the same domain we get, iterating equation (59), ˆ ε − z)−1 = −ε(h ˆ ε − z)−1 ˆhε,σ (h ˆ ε − z)−1 + O(ε2 |z|−3 )L(H ,H ) . (H ε,σ − z)−1 − (h 0 0 0 1 0 ˆ ε,σ using LemThe statement now follows from the explicit expression of h 1 ma 4. Theorem 4 (Zero order approximation to the time evolution). √ ε,σ t ˆε t ε,σ ˜ dres )L(H ) = O M + 1|t|ε log σ(ε)−1 , (90) (e−iHdres ε − e−ih0 ε )QM χ(H √ ε,σ t ˆε t ε,σ QM (e−iHdres ε − e−ih0 ε )χ(H ˜ dres )L(H ) = O M + 1|t|ε log σ(ε)−1 , (91) ˜ = χ. ˜ for every χ ˜ ∈ C0∞ (R) such that χχ ε,σ ˆ ε ), since the difference, being Proof. Using Lemma 5 we replace χ(Hdres ) with χ(h 0 of order O(ε), is smaller than the error we want to prove. Both Hamiltonians are self-adjoint on H0 , therefore, applying Duhamel formula, we get: −itH ε,σ /ε ˆε ˆε) dres − e−ith0 /ε QM χ( ˜h e 0 t ε,σ i ε,σ ˆ ε )e−ishˆ ε0 /ε QM χ( ˆε) ds ei(s−t)Hdres /ε (Hdres −h ˜h =− 0 0 ε 0 t √ ε,σ ˆ ε ) + O ε M + 1 log(σ −1 ) = −i ds ei(s−t)Hdres /ε h1,χ e−ish0 /ε QM χ( ˜h 0 L(H ) 0 t √ ε,σ ˆ ε )e−ish0 /ε + O ε M + 1 log(σ −1 ) ˆ dres QM χ( = −i ds ei(s−t)Hdres /ε h ˜h . 1 0 L(H ) 0

ˆ dres , equation (81), and obPutting in the previous equation the expression of h 1 ˆ ε ), we get that serving that we can replace, again√by Lemma5, χ(H ε,σ ) with χ(h 0 −1 the right-hand side is of order O( M + 1|t|ε log(σ )).

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577

For the second estimate we apply again Duhamel formula, inverting the position of the unitaries, ε,σ ˆε ε,σ ˜ dres ) QM e−itHdres /ε − e−ith0 /ε χ(H t ε,σ i ˆε ε,σ ε,σ /ε ˆ ε )e−isHdres ds QM ei(s−t)h0 /ε (Hdres −h χ(H ˜ dres ) =− 0 ε 0 t √ ε,σ ˆε ε,σ ˆ dres χ(H ds ei(s−t)h0 /ε QM h ˜ dres )e−isHdres /ε +O ε M + 1 log(σ −1 ) , = −i 1 0

ε,σ ˆ ε ), our claim is proved. ) and χ(H ε,σ ) by χ(h so, replacing χ(Hdres 0

ε Corollary 2. The dressed projectors PM are almost invariant with respect to the original dynamics, i.e., √

−iH ε t ε

ε

e ε,P χ(H ) = O M + 1|t|ε log σ(ε)−1 . M L(H )

Proof. From the previous theorem it follows that √

−iH ε,σ t

ε,σ

e dres ε , Q ) = O M + 1|t|ε log σ(ε)−1 . χ(H M dres L(H ) ε,σ , equation (77), we deduce therefore that From the definition of Hdres √

−iH ε,σ t ε

ε,σ

−1

e ε,P χ(H , ) = O M + 1|t|ε log σ(ε) M L(H )

but

−iH ε t ε

−iH ε t

−iH ε,σ t ε ε,σ t ε ε ε ε,P ε − e−iH ε ,Pε ε,P e e M χ(H ) = M χ(H ) + e M χ(H ) ε,σ t ε χ(H ε,σ ) − χ(H ε,σ ) + e−iH ε , PM = O σ(ε)1/2 |t| L(H ) + O εσ(ε)1/2 L(H ) √ + O M + 1|t|ε log σ(ε)−1 , L(H )

where we have used equations (51) and (57), the fact that χ(H ε ) ∈ L(H , H0 ) with norm uniformly bounded in ε and equation (74). Lemma 6. The truncated dressed Hamiltonian (2) ˆ ε + εh ˆ dres + ε2 h ˆ dres Hdres := h 0 1 2

(92)

is self-adjoint on H0 for ε small enough. Proof. The proof follows from a symmetric version of the Kato theorem [21, Theorem X.13].

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Theorem 5 (First order approximation to the time evolution). Given a function χ ˜ ∈ C0∞ (R), ε,σ

(2)

ε,σ ε,σ e−itHdres /ε QM χ(H ˜ dres ) = e−itHD /ε QM χ(H ˜ dres ) t ˆε ˆε ε,σ − iε ds ei(s−t)h0 /ε h2,OD e−ish0 /ε QM χ(H ˜ dres ) 0 + O ε3/2 |t| L(H ) (1 − δM0 ) (93) + O ε2 log(σ −1 )(|t| + |t|2 ) L(H ) , (2)

where δM0 = 1 when M = 0, 0 otherwise, [HD , QM ] = 0 ∀ M , (2)

HD :=

N 1 2 pˆj + Vϕ coul + Hf 2m j j=1

N 2 |ϕ(k)| ˆ ej el −ε dk eik · xj pˆj · (1 − κ ⊗ κ)ˆ pl e−ik · xl m j m l R3 2|k|2 2

l,j=1

=

ˆ hε0

+ ε2 Vdarw .

(94)

The off-diagonal Hamiltonian is defined by h2,OD

N ej := Φj,σ · ∇xj Vϕ coul . m j j=1

(95)

Remark 4. The spin term, as mentioned in the introduction, does not appear in the effective dynamics, even though it is apparently of order O(ε2 ). This is due to the fact that it is off diagonal with respect to the decomposition of the Hilbert space associated to the QM s and that the coupling function in the magnetic field, equation (43), goes to zero like |k|1/2 when |k| → 0+ . This implies that the term is actually smaller than O(ε2 ), as explained in the proof.

Proof. We split the proof into three parts. In the first one, we show that equa˜ (2) given by tion (93) is true with a diagonal Hamiltonian H D ˜ (2) := H D

N N e2j 1 2 pˆj + Vϕ coul + Hf + ε2 a(vxj )∗ a(vxj ) 2m 2m j j j=1 j=1

− ε2

N 2 ej el |ϕ(k)| ˆ dk eik · xj pˆj · (1 − κ ⊗ κ)ˆ pl e−ik · xl , 2 m j m l R3 2|k|

l,j=1

(96)

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579

and an off-diagonal one ˜ h2,OD defined by ˜ h2,OD :=

N N ej ej Φj,σ · ∇xj Vϕ coul + ∇xl (Φj,σ · pˆj ) · pˆl m m j l mj j=1 j,l=1

+

N j=1

N e2j ej a(vxj )2 + a(vxj )∗ 2 ] . σj · Φ(ik × vxj ) + 2mj 2m j j=1

(97)

In the second part we prove that if one neglects the term ε2

N e2j a(vxj )∗ a(vxj ) 2m j j=1

(2)

˜ , one gets an error of order O(ε3/2 |t|) in the time evolution. Note that this in H D term is exactly zero if the initial state for the field is the Fock vacuum. ˜ 2,OD with h2,OD . In the third part, we prove analogously that we can replace h (2) ˜ ˜ More specifically, the terms which we neglect in H D and h2,OD give rise to higher order contributions to the time evolution, although their norm in L(H0 , H ) is not small. This is caused by the fact that they are strongly oscillating in |k|, so that their behavior is determined by the value of the density of states in a neighborhood of k = 0. For all these terms, the density however vanishes for k = 0, uniformly in σ, and this implies that they are of lower order with respect to the leading pieces (2) whose density is constant (for the terms in HD ) or diverges logarithmically in σ (for the terms in h2,OD ). We elaborate on this last observation in a corollary to this theorem. (2) ε,σ by Hdres . We start showing that we can, up to the desired error, replace Hdres (2) ε,σ By Lemma 6, Hdres is self-adjoint on H0 like Hdres , therefore we can apply the Duhamel formula and use Theorem 3 to get ε,σ (2) (2) ε,σ ε,σ i t (2) −itHdres /ε −itHdres /ε −e =− ds ei(s−t)Hdres /ε Hdres − Hdres e−isHdres /ε e ε 0 3/2 . = O ε2 log(σ −1 ) ε,σ ˆ ε )χ(H ˜ ε,σ ). Moreover, using Lemma 5, we can replace QM χ(H ˜ dres ) by QM χ( ˜h 0 ˜ dres (2) ˜ Since the diagonal Hamiltonian H D is also self-adjoint on H0 for ε sufficiently small (the proof can be given along the same lines of Lemma 6), we apply again Duhamel formula, −itH (2) /ε ˜ (2) ˆ ε )χ(H ˜ ε,σ ) dres − e−itHD /ε QM χ( ˜h e 0 ˜ dres t (2) (2) i (2) ˜ (2) e−isH˜ D /ε QM χ( ˆ ε )χ(H ˜ ε,σ ) ds ei(s−t)Hdres /ε Hdres − H ˜h =− 0 ˜ D dres ε 0

580

L. Tenuta

t

= −i 0

(2)

˜ (2) /ε

ˆ dres e−isHD ds ei(s−t)Hdres /ε h 1

− iε 0

t

(2)


ˆ ε )χ(H ˜ ε,σ ) QM χ( ˜h 0 ˜ dres (2)

ˆ dres − h2,D )e−isH˜ D ds ei(s−t)Hdres /ε (h 2

/ε

ˆ ε )χ(H ˜ ε,σ ) . QM χ( ˜h 0 ˜ dres

To analyze the first term, we remark that, proceeding as in Lemma 5, one can prove that ˆ ε ) − χ( ˜ (2) ) = O ε2 log(σ −1 ) χ( ˜h ˜H , 0 D L(H ) so

˜ (2) /ε

e−itHD

ˆ ε ) = O ε2 log(σ −1 ) , χ( ˜h , 0 L(H )

therefore, t (2) (2) ˆ ε )χ(H ˆ dres e−isH˜ D /ε QM χ( ˜ ε,σ ) −i ds ei(s−t)Hdres /ε h ˜h 1 0 ˜ dres 0 t (2) (2) ˆ ε )e−isH˜ D /ε χ(H ˜˜ ε,σ ) ds ei(s−t)Hdres /ε ˆhdres QM χ( ˜h = −i 1 0 dres 0 2 + O ε |t| log(σ −1 ) L(H ) . From equation (81) it follows ˆ dres QM χ( ˆ ε ) = 1 − χ(H ε,σ ) Q≤L h ˆ ε ) = O(ε2 )L(H ) , ˆ ε Q≤L 1 − χ(H ε,σ ) QM χ( h ˜h ˜h 1 0 1 0 using Lemma 5 twice and Lemma 4. Concerning the second one, applying once again the Duhamel formula, we have t (2) (2) ˆ dres − h2,D )e−isH˜ D /ε QM χ( ˆ ε )χ(H ˜ ε,σ ) ds ei(s−t)Hdres /ε (h ˜h −iε 2 0 ˜ dres 0 t (2) ˆ dres − h2,D )e−ishˆ ε0 /ε QM χ( ˆ ε )χ(H ˜ ε,σ ) ds ei(s−t)Hdres /ε (h ˜h = −iε 2 0 ˜ dres 0 2 2 + O ε |t| log(σ −1 ) L(H ) , (2) ˆ dres − h2,D )QM χ( ˆ ε ). Following a procedure so we have to look at ei(s−t)Hdres /ε (h ˜h 2 0 ˆ dres , already employed several times, we first observe that, in the expression for h 2 ε,σ ˆ ε ). equation (84), we can replace, making an error of order O(ε), χ(H ) with χ(h 0 Applying Lemma 4, we see that the last three terms in (84) vanish, and L ˆ ε,σ that the terms containing [U1,σ , h1 ] combine to give the Darwin term and the ˜ 2,OD . Finally, we apply expression of the effective mass. What remains is exactly h (2) again Duhamel formula to approximate the time evolution generated by Hdres , getting (2)

ˆ dres − h2,D )QM χ( ˆ ε ) = ei(s−t)h0 /ε ˜h2,OD QM χ( ˆε) ˜ h ˜ h ei(s−t)Hdres /ε (h 2 0 0 −1 + O ε|t| log(σ ) .

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˜ (2) with H (2) , up to an error We proceed now to show that we can replace H D D of order O(ε3/2 |t|)L(H ) . ε,σ ˜ dres )Ψ0 , we Applying repeatedly Duhamel formula, and putting Ψ := QM χ(H get t N −itH˜ (2) /ε (2) e2j ˆε ˆε −itHD /ε D e Ψ = −iε −e ds ei(s−t)h0 /ε a(vxj )∗ a(vxj )e−ish0 /ε Ψ 2m j 0 j=1 2 2 + O ε |t| L(H ) . To streamline the presentation, we assume that M = 1, the calculations for M > 1 are basically the same, but more cumbersome. The integral gives therefore ˆε

e− ε th0 f (k1 , λ1 ) i

2

dk f (k, λ)∗

R3

λ=1

ˆε

·

2 λ=1

t

t

ds ei ε (|k1 |−|k|) ei ε hp eixj · (k−k1 ) e−i ε hp Ψ(k, λ) s ˆε

s

s ˆε

0

= e− ε th0 f (k1 , λ1 ) i

R3

dk

f (k, λ)∗ [1 + i(|k1 | − |k|)ε−1 ] 1 + i |k1 | − |k| ε−1

ds ei ε (|k1 |−|k|) ei ε hp eixj · (k−k1 ) e−i ε hp Ψ(k, λ) . s ˆε

s

s ˆε

(98)

0

Integrating by parts we get −1 t ˆε ˆε i |k1 | − |k| ε ds eis(|k1 |−|k|)/ε eishp eixj · (k−k1 ) e−ishp Ψ(k, λ) 0

ˆε ˆε = eit(|k1 |−|k|)/ε eithp eixj · (k−k1 ) e−ithp Ψ − eixj · (k−k1 ) Ψ i t ˆ ε ˆ ε ixj · (k−k1 ) −ish ˆε pΨ , − ds eis(|k1 |−|k|)/ε eishp h e p, e ε 0

where the commutator is of order O(ε) when applied to functions of bounded kinetic energy, so that the right-hand side is uniformly bounded in ε. We have now to put this expression back in (98) and estimate the single terms. We show how to do this for the first one, the others being entirely analogous. We ignore the unitary on the left, which does not change the norm, so we have to consider f (k1 , λ1 )

2 λ=1

dk

R3

· 0

t

f (k, λ)∗ · 1 + i |k1 | − |k| ε−1

ds eis(|k1 |−|k|)/ε eishp eixj · (k−k1 ) e−ishp Ψ(k, λ) . ˆε

ˆε

582

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Using twice the Cauchy–Schwarz inequality we get 2 · · ·2H = dx dk1 |· · ·|2 R3N

σ1 ,...,σN ,λ1 =1

2

≤ |t|

R3

dx

dk1 |f (k1 , λ1 )|2

σ1 ,...,σN ,λ1 =1

·

2

t

dk

dk

λ=1

|f (k, λ)|2 2 1 + |k1 | − |k| ε−2

' '2 ˆε ' ˆε ' ds 'eishp eixj · (k−k1 ) e−ishp Ψ(k, λ)'

0

λ=1

= |t|2 Ψ2H

2

dk1 dk

λ1 ,λ=1

≤ Cε |t| 4

2

2

Ψ2H

Λ/ε

|f (k, λ)f (k1 , λ1 )|2 2 1 + |k1 | − |k| ε−2

dk1 0

Λ/ε

dk 0

k1 k = O ε|t|2 Ψ2H . 1 + (k1 − k)2

We proceed now to examine the last three terms in (97) to show that they can be neglected. The second and the third term are a sum of terms of the form Φ g(λ, k)e−ik · xj Tˆ , where the function g |k|α , |k| → 0+ , with α = −1/2 or +1/2, and Tˆ is a Pauli matrix, or the product of two momentum operators. ε,σ ˜ dres )Ψ0 , For a term of this form we get then, putting again Ψ := QM χ(H t ˆε ˆε ε,σ iε ds ei(s−t)h0 /ε Φ g(λ, k)e−ik · xj Tˆ e−ish0 /ε QM χ(H ˜ dres )Ψ0 0 t ε ˆε ˆε = i√ ds ei(s−t)h0 /ε a(ge−ik · xj ) + a(ge−ik · xj )∗ Tˆ e−ish0 /ε Ψ . 2 0 Expression of this type have already been estimated in [24, Theorem 4]. For convenience of the reader, we give the proof for the annihilation part, referring to [24] for the creation part, which is entirely analogous. The annihilation part gives t 2 ε ˆε √ ˆε ˆε ds eishp /ε dk g(k, λ)∗ eik · xj Tˆ e−is|k|/ε e−ishp /ε · i √ e−ith0 /ε M 3 2 0 λ=1 R · Ψ(x, σ; k, λ, k1 , λ1 , . . . , kM , λM ) 2 ε −ithˆ ε0 /ε √ g(k, λ)∗ √ =i e 1 − i|k|ε−1 · M dk −1 1 − i|k|ε 3 2 λ=1 R t ˆε ˆε ds e−is|k|/ε eishp /ε eik · xj Tˆ e−ishp /ε Ψ(x, σ; k, λ, . . .) . · 0

(99)

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583

Integrating by parts we get t ˆε ˆε ds e−is|k|/ε eishp /ε eik · xj Tˆ e−ishp /ε Ψ(x, σ; k, λ, . . .) −i|k|ε−1 0

ˆε ˆε = e−it|k|/ε eithp /ε eik · xj Tˆ e−ithp /ε Ψ − eik · xj Tˆ Ψ i t ˆ ε ˆ ε ik · xj ˆ −ish ˆ ε /ε p T e − ds e−is|k|/ε eishp /ε h Ψ, p, e ε 0

where

ˆ ε , eik · x Tˆ = h p

1 ik · xj 2εe k · pˆj + ε2 |k|2 eik · xj Tˆ + eik · xj Vϕ coul , Tˆ . 2mj

The commutator on the right-hand side is zero if Tˆ is a Pauli matrix, or it is a term of order ε times pˆ, if Tˆ is the product of two momentum operators. Therefore it has the same form as the first part of the right-hand side and can be treated in the same way. We have now to put the result of the integration by parts back in equation (99) and estimate what comes out. We show how to do this for the first term, all the other ones can be treated in the same way. Ignoring the unitary on the left and the constants, we consider then t 2 g(k, λ)∗ ˆε ˆε ε dk ds e−is|k|/ε eishp /ε eik · xj Tˆ e−ishp /ε Ψ(x, σ; k, λ, . . .) . −1 1 − i|k|ε 0 R3 λ=1

Using the Cauchy–Schwarz inequality we get 2 |g(k, λ)|2 2 2 |· · ·| ≤ ε dk 1 + |k|2 ε−2 3 λ=1 R t 2 ˆε ˆε · |t| dk ds |eishp /ε eik · xj Tˆ e−ishp /ε Ψ|2 , R3

λ=1

so · · ·2H

≤ε

2

2 λ=1

0

|g(k, λ)|2 dk |t| 1 + |k|2 ε−2 R3

0

t

ˆε ds Tê−ishp /ε Ψ2H .

The left integral gives Λ |g(k, λ)|2 |k|2(1−α) dk ≤ C d|k| = O ε2 log(Λε−1 ) , 2 −2 2 −2 1 + |k| ε 1 + |k| ε 0 R3 when α = 1/2, or −1/2. The same analysis can be carried out for the remaining term N e2j a(vxj )2 + a(vxj )∗ 2 ] . 2m j j=1

584

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The proof is identical to the one given in [24, Theorem 4] for a similar term appearing in the case of Nelson model, and depends as above on the fact that vxj (λ, k) |k|−1/2 , |k| → 0+ . Corollary 3. At the leading order, the radiated piece (i.e., the piece of the wave function which makes a transition between the almost invariant subspaces) for a system starting in the Fock vacuum, Ψ0 (x) = ψ(x)ΩF , ψ(x) ∈ Hp , is given by ⎛ ⎞ t N ϕ ˆ (k) ε ej cl ˆ iε σ(ε) ⎝ eλ (k) · ds ei(s−t)|k|/ε OpW x ¨j (s; x, p)⎠ψ(x) , (100) −e−ith0 √ ε m 2 |k|3/2 j 0 j=1 where xcl j is the solution to the classical equations of motion cl mj x ¨cl j (s; x, p) = −∇xj Vϕ coul x (s; x, p) , xcl j (0; x, p) = xj , x˙ cl j (0; x, p)

=

(101)

pj m−1 j ,

j = 1, . . . , N .

This coincides with the leading order of the radiated piece corresponding to the original Hamiltonian H ε , for a system starting in the dressed vacuum, Uε∗ ΩF . Proof. Applying equation (93) for the case M = 0 we get at the leading order t ε,σ ˆε ˆε −itHdres /ε Q⊥ e ψ(x)Ω = −iε ds ei(s−t)h0 /ε h2,OD e−ish0 /ε ψ(x)ΩF F 0 0

ε ej = −√ 2 j=1 mj N

t

ˆε

ds ei(s−t)hp /ε ei(s−t)|k|/ε ∇xj Vϕ coul

0

ivxj ,σ (k, λ) −ishˆ ε /ε p e ψ(x) |k| iε ϕˆσ(ε) (k) eλ (k) = −√ 2 |k|3/2 t N ej ˆε ds ei(s−t)|k|/ε ei(s−t)hp /ε (e−ik · xj − 1) · mj 0 j=1 ·

iε ϕˆσ(ε) (k) ˆε eλ (k) · ∇xj Vϕ coul e−ishp /ε ψ(x) − √ 2 |k|3/2 t N ej ˆε ˆε ds ei(s−t)|k|/ε ei(s−t)hp /ε ∇xj Vϕ coul e−ishp /ε · m j 0 j=1 · ψ(x) iε ϕˆσ(ε) (k) = −√ eλ (k) 2 |k|3/2

Vol. 9 (2008)


·

585

t N ej ˆε ds ei(s−t)|k|/ε ei(s−t)hp /ε (e−ik · xj − 1) m j 0 j=1

ˆσ(ε) (k) ˆε ˆ ε iε ϕ eλ (k) · ∇xj Vϕ coul e−ishp /ε ψ(x) − e−ith0 √ 2 |k|3/2 ⎛ ⎞ t N ej cl ⎝ · ds ei(s−t)|k|/ε OpW x ¨ (s; x, p)⎠ ψ(x) ε mj j 0 j=1 + O(ε2 |t|)L(H ) ψL2 (R3n ) , ˆε

ˆε

where we have used Egorov’s theorem to approximate eishp /ε ∇xj Vϕ coul e−ishp /ε (see, e.g., [22]). To end the proof of the first statement we have to show that the norm of the first term is small. For the jth term in the sum we get ' e2j |ϕˆσ(ε) (k)|2 ' t ˆε ' · · ·2H ≤ ε2 2 ds eis|k|/ε eishp /ε (eik · xj − 1) dx dk ' 3 m |k| 0

j spin

'2 ' ˆ ε /ε −ish p ψ(x)'' · ∇xj Vϕ coul (x)e

' e2j |ϕˆσ(ε) (εk)|2 '' t ˆε =ε 2 ds eis|k| eishp /ε (eiεk · xj − 1) dx dk ' 3 mj spin |k| 0 '2 ' ˆε · ∇xj Vϕ coul (x)e−ishp /ε ψ(x)'' 2

≤ ε2 |t|

e2j m2j

dk

|ϕˆσ(ε) (εk)|2 |k|3

− 1)∇xj Vϕ coul (x)e ≤ Cε4 |t|

ˆ ε /ε −ish p

Λ/ε

1 |k| t

d|k| σ(ε)/ε

= Cε4 |t| log

Λ σ(ε)

t

ds (eiεk · xj

0

2

ψ(x)

Hp

t

2 ˆε

ds xj ∇xj Vϕ coul (x)e−ishp /ε ψ(x)

0

2 ˆε

ds xj ∇xj Vϕ coul (x)e−ishp /ε ψ(x)

Hp

0

Hp

.

For the second statement we have ε

ε

ε,σ

(1 − P0ε )e− ε tH P0ε χ(H ε )Ψ = (1 − P0ε )(e− ε tH − e− ε tH )P0ε χ(H ε )Ψ

ε i + (1 − P0ε )e− ε tH P0ε χ(H ε ) − χ(H ε,σ ) Ψ i

i

i

ε,σ

+ (1 − P0ε )e− ε tH P0ε χ(H ε,σ )Ψ ε,σ −itHdres /ε = O σ(ε)1/2 |t|ΨH + U ∗ Q⊥ ψ(x)ΩF U 0e ε,σ −itHdres /ε ψ(x)ΩF + O ε2 log σ(ε)−1 ΨH = Q⊥ 0e + O σ(ε)|t|ΨH , i

586

L. Tenuta

with


ˆ ε )Ψ ∈ Hp . ψ(x) := ΩF , χ(h 0 F

Remark 5. Denoting by Tt (k) := OpW ε

⎛ ⎞ t N ej cl ⎝ ds ei(s−t)k x¨j (s; x, p)⎠ , m j 0 j=1

the operator acting on Hp which appears in (100), the norm squared of the leading part of the radiated piece is |ϕˆσ(ε) (k)|2 ε2 |eλ (k) · Tt |k|/ε ψ(x)|2 dx dk 3 2 spin |k| λ=1,2

|ϕˆσ(ε) (|k|)|2

4

Tt |k|/ε ψ 2 = πε2 d|k| Hp ⊗C3 3 |k| Λε−1

ε2 1

Tt |k| ψ 2 = d|k| Hp ⊗C3 2 6π σ(ε)ε−1 |k| a

ε2 1

Tt |k| ψ 2 ≥ d|k| , Hp ⊗C3 6π 2 σ(ε)ε−1 |k| where a > 0 can be chosen arbitrarily small. The symbol of Tt (k) is an ε-independent function, which for k = 0 is different from the null function t N N ej cl ej cl x˙ j (t; x, p) − pj , ds x ¨j (s; x, p) = m m j j 0 j=1 j=1 so Tt (0) is different from the zero operator. We expect therefore that for a generic state ψ inf Tt (|k|)ψHp ⊗C3 > 0 .

0 0, Σ0 ∩ [0, a[= [0, a[ and Σ0 ∩ [−δ, 0[= ∅. We now consider a random Schrödinger operator of the form Hλ,w = H0 + λVw = H0 + λ wγ V ( · − γ) . (1.1) γ∈Γ

where we assume the following: (H.0.1). The single site potential V is a compactly supported function. (H.0.2). The random variables (wγ )γ∈Γ are independent identically distributed, bounded, non-negative and their essential support contains zero and is not reduced to this single point. (H.0.3). λ is a positive coupling constant.

596

F. Ghribi


Assumption (H.0) ensures that, for any realization of w, Hλ,w is self-adjoint on H2 (Rd ). By (H.0.2), Hλ,w is ergodic. The classical theory of ergodic random operators tells us that, w-almost surely, the spectrum of Hλ,w does not depend on the realization of w (see [3, 18]). Let Σλ = σ(Hλ,w ) be the almost sure spectrum of Hλ,w . It follows from the standard characterization of Σλ using admissible periodic operators (see [4,18]) that Σλ contains Σ0 and is a union of intervals. The connected components of R\Σλ will be the gaps of Σλ . The edges of the gaps are the points in Σλ ∩ R\Σλ . In [16], Najar studied the spectrum minimum for this model. For λ sufficiently small, the gap [−δ, 0[ in the spectrum of the unperturbed periodic Schr¨ odinger operator H0 stays open when the random magnetic perturbation turned on. For λ small, let Eλ be the spectral band edge of Σλ closest to 0. Then, for some δ > 0, we have Σλ ∩ [−δ , Eλ [= ∅. Now, we define the main object of our study, the integrated density of states. It is one of the simplest but quite important characteristics of the ranl,D dom Schr¨ odinger operators. For Hλ,w defined as above and l > 0, we define Hλ,w to be the Dirichlet restriction of Hλ,w to the cube Cl centered in zero of side l,D length l. Hλ,w has only discrete spectrum and is bounded from below. For A ⊂ R,

l,D we denote by νl (A) the number of eigenvalues (counting multiplicity) of Hλ,w in the set A. νl ( · ) is a point measure with mass m at eigenvalues of multiplicity m. For E ∈ R, we define l Nλ,w (E) =

νl (] − ∞, E]) , Vol(Cl )

(1.2)

where Vol(Cl ) denotes the volume of Cl . l We show that, w-almost surely, Nλ,w (E) has a limit when l → +∞ (see [3,18]). This limit is independent of the realization of w. It is the integrated density of states of Hλ,w . We denote it by Nλ (E). Physically, it can be interpreted as the number of states per unit of volume below energy E for a system governed by Hλ,w . Our mean question we are interested in this paper: How does the integrated density of states Nλ (E) behave near the fluctuation band edge Eλ ? The most remarkable phenomena exhibited by the integrated density of states for many random models near fluctuation band edge is the so-called Lifshits tail. This asymptotic behavior of the integrated density of states at fluctuation band edge was discovered by I. M. Lifshits [11, 12] in the early 60’s. I. M. Lifshits produced a heuristic showing that, at the bottom of the spectrum of a random Schr¨ odinger operator, the density of states decays exponentially fast. Mathematical work on Lifshits tails appeared much later. It has been studied first at the bottom of the spectrum. This problem has been widely studied for the nonmagnetic case. Among the models the most studied, we cite the Poisson model, the Anderson model and acoustic operators.

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597

The Poisson model with nonnegative single site potential is the random model for which Lifshits tails are best understood. The results were not obtained following the original Lifshits heuristic but using Wiener integrals and the Donsker– Varadham technique to estimate the Laplace transform of N (E) (see [17, 19]). This point of view is directly related to the study of Brownian motion among Poissonian obstacles. One has log N (E) ∼ CE − 2 , d

(1.3)

for E ↓ 0. Here 0 is the bottom of the spectrum. Proofs of Lifshits tails at the bottom of the spectrum of the density of states for the continuous Anderson model in the case of sign definite single site potential following the original heuristic of Lifshits were developed by many authors in 80’s (see [5,6,14,22]). They don’t obtain an asymptotic as precise as (1.3) but only the weaker version log log N (E) d =− . (1.4) lim + log(E − E ) 2 E→E0 0 Lifshits tails were also predicted at all fluctuational edges of the spectrum (see [13]). Much fewer results are known. In dimension 1, in [15], Mezincescu proved that log log N (E) − N (E0 ) d lim =− (1.5) E→E0 log(E − E0 ) 2 holds at all band edges. In dimension larger than 1, in [7, 8] and [9] Klopp proved that (1.5) occurs if and only if n(E), the density of states of a well chosen underlying periodic Schr¨ odinger operator satisfies log n(E) − n(E0 ) d lim =− . (1.6) E→E0 log(E − E0 ) 2 For random Schr¨ odinger operators with definite sign single-site potentials, the idea behind Lifshits tail is that the random potential hoist the eigenvalues near fluctuation band edge, so that the integrated density of states there decays exponentially fast. This is due the monotonic variation of the eigenvalues of Hw,Λ (the operator Hλ,w restricted to the cube Λ) in vicinity of the fluctuation band edge with respect to the random variables (wγ )γ∈Γ . Then, the questions concerning Lifshits tails are tightly related to the monotonicity of the random perturbation with respect the random variables. However, in our case, the random perturbation of H0 is given by λVw = λ wγ V ( · − γ) . (1.7) γ∈Γ

It is not monotonic with respect to a variation of w. If we change one random variable wγ with others fixed, some eigenvalues of Hλ,w,Λ go up but others go down. Then, the first question that we will focus in this paper is: what are the realizations that contribute most to the existence of the spectrum in the vicinity of Eλ ?

598

F. Ghribi


To overcome this difficulty, we use the idea developed in [2] concerning the same problem for random magnetic Schr¨ odinger operator. We require some further assumptions on the single site potential V and on the nature of the spectrum of H0 near 0. Under these hypotheses, we prove for weak disorder regime, the realizations that contribute most to the existence of the spectrum near Eλ are the regions of realizations of potential that vanishes. Then, we prove that the integrated density of states Nλ (E) has the following behavior near the band edge Eλ , for all λ sufficiently small, log | log Nλ (E) − Nλ (Eλ ) | d =− . (1.8) lim + log(E − E ) 2 E→Eλ λ This result says that the integrated density of states decays exponentially fast in the vicinity of Eλ . This can be used to prove band-edge localization for Hλ,w .

2. The main assumptions and results To describe our main assumptions, we need to recall some preliminary considerations facts on the periodic Schr¨ odinger operators. Basic references are [21] and [20]. 2.1. Some preliminary considerations on periodic Schr¨ odinger operators 2.1.1. The Floquet decomposition. As VΓ is Γ-periodic, we know that, for any γ ∈ Γ, τγ ◦ H0 ◦ τγ∗ = τγ ◦ H0 ◦ τ−γ = H0 , where τγ : L2 (Rd ) → L2 (Rd ) denotes the translation by γ operator, i.e., for ϕ ∈ L2 (Rd ) and x ∈ Rd , (τγ ϕ)(x) = ϕ(x − γ). For θ ∈ Rd and u ∈ S(Rd ), the Schwartz space of rapidly decreasing functions, we define eiγ · θ u(x − γ) . (U u)(θ, x) = γ∈Γ

U can be extended as a unitary isometry from L2 (Rd ) to H. Its inverse is given by the formula, 1 for v ∈ H , (U ∗ v)(x) = v(θ, x)dθ . Vol(T) Td One sees that, for 1 ≤ j ≤ d, [∂j , U ] = 0. (Here ∂j denotes the jth partial derivative.) Moreover, as VΓ is periodic, [VΓ , U ] = 0. Hence, H0 admits the Floquet decomposition ⊕ H0 (θ)dθ , (2.9) U H0 U ∗ = Td

where H0 (θ) is the differential operator H0 acting on Hθ = {u ∈ L2loc (Rd ), ∀γ ∈ Γ, τγ u = e−iγ · θ u}. H0 (θ) is self-adjoint and elliptic.

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As H0 is elliptic, we know that H0 (θ) has a compact resolvent; hence its spectrum is discrete. Let us denote its eigenvalues, called Floquet eigenvalues of H0 , by E0 (θ) ≤ E1 (θ) ≤ · · · ≤ En (θ) ≤ · · · . The functions (θ → En (θ))n∈N are Lipschitz-continuous in the variable θ; they are even analytic in θ when they are simple. Moreover, Weyl’s law tells us that En (θ) → +∞ as n → +∞ (uniformly in θ) .

(2.10)

Its spectrum Σ0 is purely absolutely continuous and is given by Σ0 = ∪n∈N En (Td ). So, the spectrum of H0 is the union of closed intervals called bands of the spectrum; the connected components of R\Σ0 are called the gaps of the spectrum of H0 . Definition 2.1. Let E0 ∈ Σ. We say that E0 is simple if the set {p ∈ N; ∃θ ∈ Td , Ep (θ) = E0 } is reduced to a single integer. 2.1.2. The density of states. We define n(E), the integrated density of states of H0 by (1.2). Following [21], for E ∈ R, we compute 1 n(E) = dθ . (2.11) (2π)d θ∈Td ;En (θ)≤E n∈N

It is a continuous, positive, and increasing function. It is constant in the gaps of the spectrum of H0 , and its growth points are exactly the points of Σ0 . 2.1.3. Wannier basis. Let E ⊂ L2 (Rd ) be a closed subspace Γ-translation-invariant; that is, if ΠE is the orthogonal projector on E, for γ ∈ Γ, one has ΠE = τγ ΠE τγ∗ . The Floquet decomposition of ΠE tells us that ⊕ E ΠE (θ)dθ , Π = Td E

E

E

where Π (θ) is Π acting on Hθ . Π (θ) is an orthogonal projector acting on L2 (C0 ), where C0 = {x ∈ Rd ; ∀1 ≤ j ≤ d, − 12 < xj ≤ 12 } is the fundamental cell of Γ. The family of projectors (ΠE (θ))θ∈Td is continuous in θ; hence it is of constant rank. For θ ∈ Td , we can find an orthogonal system of vectors (u(θ, · ))n∈N (where N ⊂ N is a set of indices independent of θ) that spans the range of ΠE (θ). So, transporting (un (θ, · ))n∈N over to L2 (Rd ) using U ∗ , we see that there exists an orthogonal system of vectors (ˆ un,0 )n∈N such that if, for γ ∈ Γ, we set u ˆn,γ = τγ (ˆ un,0 ), then (ˆ un,γ )n∈N ;γ∈Γ is an orthogonal basis for E. We call it a Wannier basis of E. The vectors (ˆ un,0 )n∈N are the Wannier generators for E. For a more details, we refer to [7]. We say that E ⊂ H2 (Rd ), a Γ-translation-invariant subspace, is of finite energy if ΠE H0 ΠE is a bounded operator. If so, E has only finitely many Wannier generators.

600

F. Ghribi


2.2. The main assumptions Let us now state our main assumptions. On the underlying periodic Schr¨ odinger operator H0 = −Δ + VΓ , we assume (H.1) For some a > 0 and δ > 0, Σ0 ∩ [0, a[= [0, a[ and Σ0 ∩ [−δ, 0[= ∅. We see that 0 is a lower edge of a band of H0 . (H.2) 0 is a simple. Let En0 (θ), the unique Floquet eigenvalue of H0 that vanishes at some point θ ∈ Td . Define the set Z = {θ ∈ Td ; En0 (θ) = 0}. Then, there exists η > 0 such that: for n < n0 , for all θ ∈ Td , En (θ) < −δ; for n > n0 , for all θ ∈ Td , En (θ) > η. This implies that, for θ in some neighborhood of Z, the Floquet eigenvalue En0 (θ) is simple, hence the function θ → En0 (θ) is real analytic in some neighborhood of Z. Remark 2.2. It is proved in [10] that, generically, the band edges of periodic Schr¨ odinger operators are simple. (H.3) The density of states n of the unperturbed periodic Schr¨ odinger operator H0 has non-degenerate behavior at 0; that is, log n( ) − n(0) d = . (2.12) lim log 2 →0+ Remark 2.3. The conditions (H.1), (H.2) and (H.3) are known to hold at the bottom of the spectrum of H0 in any dimension, see [1, 20]. Let θ0 ∈ Z. In [7], Klopp proves that if (H.1) and (H.3) hold, then, there exists a constant c > 1 and Vθ0 , neighborhood of θ0 such that, for all θ ∈ Vθ0 , 1 ξθ0 (θ) ≤ En0 (θ) ≤ cξθ0 (θ) , (2.13) c d where ξθ0 (θ) = i=1 (θi − θi0 )2 . Hence, θ0 is isolated and there are only finitely many of θ0 (as Td is compact). Let Zn0 = {θ1 , θ2 , . . . , θm }. We need a more precise assumption on the single site potential V ; that is, (H.4) The matrix defined by

M = V ϕn0 (θi , · ), ϕn0 (θj , · ) , (2.14) 1≤i,j≤m

is positive-definite or negative-definite. Here, ϕn0 (θ0 , · ) is the Floquet eigenvector of H0 associated to En0 (θ0 ) and · , · denotes the scalar product. Remark 2.4. The assumption (H.2) made on the simplicity of 0 is not really necessary. Here, it is assumed to simplify the proof. We also need to know more about the random variables (wγ )γ∈Γ ; that is, we assume that, the i.i.d random variables (wγ )γ∈Γ are bounded, non negative

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and their essential support contains zero and is not reduced to this single point. Moreover, we assume that the random variables (wγ )γ∈Γ satisfy (H.5) log log P{w0 ≤ } = 0. lim sup | log | →0+ Let us formulate the main results of this paper. 2.3. The results We want to study the behavior of the integrated density of states Nλ (E) in the vicinity of Eλ . Therefore, we need to know more information about the band edge Eλ . Under assumptions (H.0) and (H.1), Σλ contains some interval of the form [0, a](a > 0) (consult [4]). Then, Eλ ≤ 0. Theorem 2.5. Let Hλ,w be a random Schr¨ odinger operator described by (1.1), and assume that (H.0), (H.1), (H.2), (H.3), (H.4) and (H.5) hold. Then, there exists λ0 > 0 such that for all λ ∈ [0, λ0 ], we have log log Nλ ( ) − Nλ (0) d =− . (2.15) lim log 2 →0+ Theorem 2.5 says that for λ small, the integrated density of states Nλ decays exponentially fast at the band edge 0. The behavior of the integrated density of states Nλ (E) at fluctuation boundary 0 given by (2.15) is called a Lifshits tail and constitutes an important feature of disordered models. It very roughly means that −d

Nλ (E) − Nλ (0) has an exponential decay of the form e−E 2 , when E tend to 0 from above, that is, for E inside Σλ . Mathematically speaking, they can be used to formalize the intuition, namely that the spectrum near fluctuation boundary 0 is due to rare events. Proposition 2.6. We assume that the assumptions (H.0), (H.1), (H.2), (H.3) hold. Furthermore, we assume that the matrix M defined by (2.14) is positive-definite. Then, there exists λ0 > 0 such that for all λ ∈ [0, λ0 ], Eλ = 0. Proposition 2.6 says that, under the additional assumptions (H.2), (H.3) and (H.4), for λ sufficiently small, zero stays the edge of a gap for Σλ when the random perturbation turned on. Lemma 2.7. Assume that the assumptions (H.0), (H.1), (H.2) and (H.3) hold. Furthermore, If the matrix M defined by (2.14) is definite negative, then, for λ sufficiently small, Eλ = inf En0 (λ, θ) , θ∈Td

where En0 (λ, · ) is the n0 -th Floquet eigenvalue of the Γ-periodic realization Hλ,w+ given by Hλ,w+ = H0 + λw+ τγ V . γ∈Γ

602

F. Ghribi


In this paper, we will prove the Theorem 2.5 when the matrix M defined by (2.14) is definite-positive. The result when the matrix M is definite-negative can be proven in same way. We just have to replace the single site potential V by −V and w by −w. In this paper, we follow the point of view of Klopp. We use widely the techniques and ideas developed in [7]. We refer to it for further details. However, in our case, we suffer from the non-monotonicity of the random perturbation with respect to the random variables (wγ )γ∈Γ . The first difficulty one has to overcome is to know what are the realizations of Hλ,w that contribute most to the presence of the spectrum to the vicinity of Eλ . To surmount this difficulty, we use the idea developed in [2] concerning the same problem for the random magnetic Schr¨ odinger operator.

3. A reduction procedure We decompose our random Hamiltonian on different translation-invariant subspaces. The obtained random operators are the reference operators. Let us first describe the decomposition. We denote by Π0 (θ), Π− (θ) and Π+ (θ) the orthogonal projections in Hθ on the vector spaces spanned by, respectively, ϕn0 (θ, · ), (ϕj (θ, · ))jn0 . (We recall that the (ϕj (θ, · ))j≥1 are the Floquet eigenvectors for H0 ). Obviously, these projectors are mutually orthogonal and their sum is the identity for any θ ∈ Td . Define Πα = U ∗ Πα (θ)U , where α ∈ {−, 0, +}. Πα is an orthogonal projector 2 on L (Rd ) and, for γ ∈ Γ, we have τγ∗ Πα τγ = Πα . It is clear that Π− , Π0 , and Π+ are mutually orthogonal and that Π− + Π0 + Π+ = IdL2 (Rd ) . For α ∈ {−, 0, +}, we set Eα = Πα (L2 (Rd )). These spaces are translation-invariant. Moreover, E− and E0 are of finite energy. The first tool needed for the reduction will be Theorem 3.1. Under the assumptions (H.0), (H.1), (H.2), (H.3), (H.4) and (H.5), there exists λ0 > 0, C1 = C(λ0 ), C2 = C2 (λ0 ) > 0 and α = α(λ0 ) such that for λ ∈ [0, λ0 ] and for E ∈ [0, α], we have Nλ,0 (C2 · E) ≤ Nλ (E) − Nλ (0− ) ≤ Nλ,0 (C1 · E) ,

(3.16)

where Nλ,0 is the integrated density of states of the random operator Π0 Hλ,w Π0 . Before proving Theorem 3.1, we need to prove the following theorem. Theorem 3.2. We assume that the assumptions (H.0), (H.1), (H.2), (H.3) et (H.4) hold. Then, there exists λ0 > 0 and m = m(λ0 ) ∈ ]0, 1[ such that, for λ ∈ [0, λ0 ], we have (3.17) K1,λ,w,n ≤ Hλ,w,n ≤ K2,λ,w,n , where K1,λ,w,n = (1 + m)Π− Hλ,w,n Π− ⊕ (1 − m)Π0 Hλ,w,n Π0 ⊕ (1 − m)Π+ Hλ,w,n Π+

(3.18)

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and K2,λ,w,n = (1 − m)Π− Hλ,w,n Π− ⊕ (1 + m)Π0 Hλ,w,n Π0 ⊕ (1 + m)Π+ Hλ,w,n Π+ . (3.19) To prove Theorem 3.2, we need to study the reference operator Π0 Hλ,w Π0 . Theorem 3.2 is proved in Appendix. 3.1. The study of Π0 Hλ,w Π0 In this section, we give a more explicit form for the reference operator ΠE0 Hλ,w ΠE0 . For u ∈ L2 (T∗ ), we define (3.20) Pϕn0 (u) = U ∗ u(θ)ϕn0 (θ, · ) . The mapping Pϕn0 : L2 (T∗ ) → E0 defines a unitary equivalence. For v ∈ E0 , its inverse is given by

Pϕ∗n0 (v) = (U v)(θ, · ), ϕn0 (θ, · ) , eiγ · θ v(x−γ). Hence, ΠE0 H ΠE0 is unitarily equivalent where (U v)(θ, x) = λ,w

γ∈Γ

to the operator hλ,w = Pϕ∗n0 Π0 Hλ,w Π0 Pϕn0 acting on L2 (T∗ ) and defined by hλ,w = h + λvw , where h is the multiplication by En0 (θ), vw is the operator with the kernel

vw (θ, θ ) = Vw ϕn0 (θ, · ), ϕn0 (θ , · ) (3.21) Definition 3.3. Let v ∈ L2 (T∗ , Hθ2 ). We define the three norms

v 21,H,∞ = sup v( · , θ), Hθ v( · , θ) L2 (C0 ) + v( · , θ) 2L2 (C0 ) , θ∈T∗

v 22,H,∞ = sup Hθ v( · , θ) 2L2 (C0 ) + v( · , θ) 2L2 (C0 ) , θ∈T∗

v 22,∞ = sup v( · , θ) 2L2 (C0 ) . θ∈T∗

Definition 3.4. Let v ∈ L2 (T∗ , Hθ2 ) such that v 22,∞ < ∞. We define the operator Pv : L2 (T∗ ) → L2 (Rd ) by ∀u ∈ L2 (T∗ ) , Pv (u) (x) = v(x, θ)u(θ)dθ . T∗

Definition 3.5. Let (χk )1≤k≤m be a C0∞ -partition of unity on T∗ such that for k ∈ {1, . . . , m}, 0 ≤ χk ≤ 1, χk ≡ 1 in some neighborhood of θk and satisfying ∀θ ∈ T∗ ,

1 ≤ χ2k (θ) ≤ 1 . m m

k=1

We define the mapping S : L2 (Td ) → L2 (T∗ ) ⊗ Cm by S(u) = (χk u)1≤k≤m ,

if u ∈ L2 (T∗ )

(3.22)

604

F. Ghribi


The adjoint of S, S ∗ : L2 (T∗ ) ⊗ Cm → L2 (Td ) is defined by S ∗ (u) = χk uk , if u = (uk )1≤k≤m ∈ L2 (T∗ ) ⊗ Cm . 1≤k≤m

One easily checks that that S is one to one.

1 mI

≤ S ∗ ◦ S ≤ I (here, I is the identity of L2 (T∗ )), so

Definition 3.6. We define for 1 ≤ k ≤ m and for (θ, x) ∈ Td × Rd , the functions ϕñ0 ,k and δϕn0 ,k by: ϕñ0 ,k (θ, x) = ϕn0 (θk , x)ei(θ−θk ) · x and

1 δϕn0 ,k (θ, x) = ϕn0 (θ, x) − ϕñ0 ,k (θ, x) , ξk (θ) d where ξk (θ) = j=1 (θj − θk,j )2 .

Remark 3.7. Obviously, for 1 ≤ k ≤ m, ϕñ0 ,k ∈ L2 (Td , Hθ2 ) and ϕñ0 ,k 2,∞ = 1. On the other hand, as the Floquet eigenvalue En0 (θ) is simple in a some neighborhood of θk , the Floquet eigenvector ϕn0 (θ, · ) is analytic in some neighborhood of θk . Hence, δϕn0 ,k is well defined. Furthermore, δϕn0 ,k 1,H0 ,∞ , δϕn0 ,k 2,H0 ,∞ et δϕn0 ,k 2,∞ are finite. Remark 3.8. We remark that, for 1 ≤ k ≤ m and u ∈ L2 (Td ) ξk u . Pϕn0 (u) = Pϕñ0 ,k (u) + Pδϕn0

(3.23)

Proposition 3.9. Assume that (H.0), (H.1), (H.2) and (H.3) hold. Then, there exists a constant c > 1 such that for u ∈ L2 (T∗ ), we have the following: 1 hu, uL2 (Td ) ≥ ξk χk u, χk u . (3.24) c 1≤k≤m

where (χk )1≤k≤m is the partition of unity on T∗ defined above. Proposition 3.9 is an immediate consequence of (2.13). 3.1.1. Lower bound of vw . Proposition 3.10. Under the assumptions (H.0), (H.1), (H.2), (H.3) and (H.4), there exists λ0 > 0 and constants K1 , v1 > 0 such that for λ ∈ [0, λ0 ] and for u ∈ L2 (Td ), we have 2 v1 wγ (χ ξk χk uj , χk uj . vw u, uL2 (Td ) ≥ k uj )(γ) − K1 4 1≤k≤m

γ∈Γ

1≤k≤m

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Proof of Proposition 3.10. We estimate vw u, uL2 (Td ) for u ∈ L2 (Td ). By definition,

vw u, uL2 (Td ) = Vw Pϕn0 (u), Pϕn0 (u) L2 (Rd ) . Using the partition of unity (χk )1≤k≤m on Td and the linearity of the operator Pϕn0 , we can write Pϕn0 (u) = Pϕn0 (χk u) . (3.25) 1≤k≤m

Furthermore, using the relation (3.23), we obtain

Vw Pϕñ0 ,k (χk u), Pϕñ0 ,k (χk u) L2 (Rd ) vw u, uL2 (Td ) = 1≤k,k ≤m

+

Vw Pδϕn0 ,k

1≤k,k ≤m

+2

ξk χk u , Pδϕn0 ,k ξk χk u

Vw Pϕñ0 ,k (χk u), Pδϕn0 ,k ξk χk u

1≤k,k ≤m

L2 (Rd )

L2 (Rd )

.

By the assertion (5.59) of Lemma 5.1, we have

Vw Pϕñ0 ,k (χk u), Pϕñ0 ,k (χk u) L2 (Rd ) 1≤k,k ≤m

− wγ mϕn0 ,θk ,θk (x)dx (χ k u)(γ)(χk u)(γ) d R 1≤k,k ≤m γ∈Γ 2 1 ≤ cα wγ ξk χk u, χk u (χk u)(γ) + c 1 + α γ∈Γ 2 wγ + cα (χk u)(γ)

γ∈Γ

1 +c 1+ (3.26) ξk χk u, χk u . α √ √ We bound |Vw Pδϕn0 ,k ( ξk χk u), Pδϕn0 ,k ( ξk χk u)L2 (Rd ) | for 1 ≤ k, k ≤ m. By Cauchy–Schwarz and the assertions (5.58) and (5.56) of Lemma 5.1, ξk χk u , Pδϕn0 ,k ξk χk u Vw Pδϕn0 ,k L2 (Rd ) 2 ≤ 2−1 Vw Pδϕn0 ,k ξk χk u L2 (Rd ) + 2−1 Pδϕn0 ,k ξk χk u 2L2 (Rd ) ≤ 2−1 cδϕn0 ,k 22,H0 ,∞ ξk χk u, χk u 2 + 2−1 δϕn0 ,k ∞ ξk χk u, χk u .

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Thus, we deduce that there exist a constant c18 > 0 such that ξk χk u , Pδϕn0 ,k ξk χk u Vw Pδϕn0 ,k 2 d L (R )

1≤k≤m

≤ c18

ξk χk u, χk u . (3.27)

1≤k≤m

√ We estimate Vw Pϕñ0 ,k (χk u), Pδϕn0 ,k ( ξk χk u)L2 (Rd ) for 1 ≤ k, k ≤ m. Using again the inequality of Cauchy–Schwarz, we obtain, ξk χk u Vw Pϕñ0 ,k (χk u), Pδϕn0 ,k L2 (Rd ) 2 1 2 ≤ αVw Pϕñ0 ,k (χk u)L2 (Rd ) + ξk χk u . (3.28) Pδϕn0 ,k 4α L2 (Rd ) By the assertions (5.59) and (5.56) of Lemma 5.1, there exist two constants c19 , c20 > 0 such that ξk χk u Vw Pϕñ0 ,k (χk u), Pδϕn0 ,k 2 d 1≤k,k ≤m

L (R )

1 ≤ c19 1 + α

+ c20 α

ξk χk u, χk u

(3.29)

1≤k≤m

2 wγ | (χ k u)(γ) | .

γ∈Γ

1≤k≤m

We addition the relations (3.26), (3.27) and (3.29), we deduce that there exist λ0 > 0 and a constant c21 > 0 such that for λ ∈ [0, λ0 ] and for u ∈ L2 (Td ), vw u, uL2 (Td ) − wγ mϕn0 ,θk ,θk (x)dx (χk u)(γ)(χk u)(γ) 1≤k,k ≤m

Rd

γ∈Γ

1 ≤ c21 1 + α + c21 α

1≤k≤m

1≤k≤m

By assumption (H.4), the matrix

M = V ϕn0 (θk , · ), ϕn0 (θk , · )

ξk χk u, χk u

2 wγ | (χ k u)(γ) | .

(3.30)

γ∈Γ

1≤k,k ≤m

= Rd

mϕ0,n0 ,θk ,θk (x)dx

1≤k,k ≤m

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is definite positive. Then, there exist a constant v1 > 0 such that for u ∈ L2 (Td ), 2 (χk u)(γ) v1 wγ 1≤k≤m γ∈Γ

≤

wγ

Rd

1≤k,k ≤m γ∈Γ

mϕn0 ,θk ,θk (x)dx (χ k u)(γ)(χk u)(γ) (3.31)

Finally, combining the inequalities (3.31) and (3.30), we obtain the following lower bound of vw : 2 wγ (χ vw u, uL2 (Td ) ≥ (v1 − c21 α) k uj )(γ) 1≤k≤m

1 − c21 1 + α

γ∈Γ

ξk χk uj , χk uj .

1≤k≤m

Pick α sufficiently small, we deduce that there exist a constant K1 > 0 such that for u ∈ L2 (Td ), we have 2 v1 vw u, uL2 (Td ) ≥ wγ (χ ξk χk uj , χk uj . k uj )(γ) − K1 2 1≤k≤m

γ∈Γ

1≤k≤m

This ends the proof of Proposition 3.10. 3.1.2. Lower bound of hλ,w .

Theorem 3.11. Assume that the assumptions (H.0), (H.1), (H.2), (H.3) and (H.4) hold. Then, there exist λ0 > 0 and a constant C > 1 such that for λ ∈ [0, λ0 ] and for u ∈ L2 (Td ), we have 1

hλ,w Su, Su L2 (Td )⊗Cm ≤ hλ,w u, uL2 (Td ) , where the operator h1λ,w acting on L2 (Td ) ⊗ Cm is defined by ⎞ ⎛ 1 0 ··· 0 h1,λ,w ⎟ ⎜ 0 h12,λ,w · · · 0 ⎟ ⎜ h1λ,w = ⎜ ⎟ .. .. .. .. ⎠ ⎝ . . . . 1 0 0 · · · hm,λ,w with for 1 ≤ k ≤ m,

⎛ h1k,λ,w =

1 ⎝ ξk + λ C

⎞ wγ πγ ⎠ ,

γ∈Γ

and πγ is the orthogonal projector on the vector eiγ · θ in L2 (Td ).

608

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Proof of Theorem 3.11. It results from Propositions 3.9 and 3.10 the following lower bound of hλ,w : hλ,w u, u ≥ ρ1 (λ) ξk χk u, χk uL2 (Td ) 1≤k≤m

+ λρ2 (λ)

2 (χk u)(γ) wγ

1≤k≤m γ∈Γ

where ρ1 (λ) = 1c −K1 λ and ρ2 (λ) = v21 −(K2 +K3 )λ. Obviously, ρ1 (λ) (respectively ρ2 (λ)) converges to 1c (respectively v21 ) as λ → 0. Thus, there exist λ0 > 0 and a constant C > 1 such that for λ ∈ [0, λ0 ] and for u ∈ L2 (Td ), we have ⎞ ⎛ 2 ⎟ 1 ⎜ (χk u)(γ) ⎠ . hλ,w u, u ≥ ⎝ ξk χk u, χk uL2 (Td ) + λ wγ C 1≤k≤m

1≤k≤m

γ∈Γ

This ends the proof of Proposition 3.11.

Remark 3.12. Obviously, the random bounded operator Π0 Hλ,w Π0 is positive for λ sufficiently small. The positivity of the operator Π0 Hλ,w Π0 and Theorem 3.1 imply that 0 stay the edge band of Σλ . By Theorem 3.11 and Theorem 3.1, we notice that near 0, the spectrum is entirely due to sufficiently large fluctuations of the random potential. It is a fluctuation boundary. Fluctuation boundaries were first identified by I. Lifshits. They are a particular feature of random ergodic Schr¨ odinger operators. The spectral values near 0, inside Σλ exist because of large cubes in which all random variables wγ are near 0. By the independence of the random variables, such cubes occur somewhere in space almost surely. But if we restrict our attention to a fixed large cube, it is almost impossible that on a large cube all the random variables take value near 0. Then, the energies near the fluctuation boundary 0 are rarely hit by the eigenvalues of the local Hamiltonians Hλ,w,Λ . This suggests that the energies are very scarce near the band edge 0. 3.2. Periodic approximations odinger operator: Let n ∈ N∗ , and define the following periodic Schr¨ wγ V (x − γ − β) , Hλ,w,n = −Δ + λ γ∈Cn ∩Γ

(3.32)

β∈(2n+1)Γ

where Cn = {x ∈ Rd ; for j = 1, . . . , d, −n − 12 < xj ≤ n + 12 }. Then, for any w ∈ Supp(dP) and n ∈ N∗ , Hλ,w,n is a (2n + 1)Γ-periodic, essentially self-adjoint Schr¨ odinger operator. It is an H0 -bounded perturbation of H0 with relative bound zero. we notice that the boundedness assumption on the (wγ )γ∈Γ implies that the family (Hλ,w,n )w,n is uniformly H0 -bounded. Hλ,w,n also denotes its self-adjoint extension.

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According to [20] or [21], Hλ,w,n admits an integrated density of states denoted by Nλ,w,n ; it is given by (2.11), the torus T∗ being replaced by the torus T∗n = Rd /((2n + 1)Γ)∗ . In [7], it was proved that for any E ∈ R, a continuity point of Nλ , lim E Nλ,w,n (E) = Nλ (E) . n→+∞

and

lim E Nλ,w,n,0 (E) = Nλ,0 (E) ,

n→+∞

where Nλ,w,n,0 (respectively Nλ0 ) is the density of states of the random operator Π0 Hλ,w,n Π0 (respectively Π0 Hλ,w,n Π0 ). Then, to get Theorem 3.1, it is sufficient to prove the following lemma. Lemma 3.13. Under the assumptions (H.0), (H.1), (H.2), (H.3), (H.4) and (H.5), there exists λ0 > 0, C1 = C(λ0 ), C2 = C2 (λ0 ) > 0 and α = α(λ0 ) such that for all λ ∈ [0, λ0 ] and for E ∈ [0, α], Nλ,w,n,0 (C2 · E) ≤ Nλ,w,n (E) − Nλ,w,n (0− ) ≤ Nλ,w,n,0 (C1 · E) ,

(3.33)

where Nλ,w,n,0 is the integrated density of states of the (2n + 1)Γ-periodic random operator Π0 Hλ,w,n Π0 . Proof of Lemma 3.13. To obtain the estimate (3.33), we will use the Floquet theory for the periodic operator Hλ,w,n . By E− (resp. E+ ), we denote the subspace un,γ )0≤n Π+ . (3.35) 2 The projector Π0 is Γ-periodic, hence it is (2n + 1) · Γ-periodic. So we may Floquet decompose it jointly with Hλ,w,n . More precisely ⊕ Π− = Π−,n (θ)dθ . Td n

For θ ∈ Tdn , Π−,n (θ) is an orthogonal projector. The same statements hold for Π+,n (θ), Π0,n (θ). We define Hλ,w,n,− (θ) = Π−,n (θ)Hλ,w,n (θ)Π−,n (θ) , and in an analogous way Hλ,w,n,+ (θ), Hλ,w,n,0 (θ).

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By Theorem 3.2, for λ sufficiently small, w ∈ Supp(dP) and n ∈ N, K1,λ,w,n ≤ Hλ,w,n ≤ K2,λ,w,n , where K1,λ,w,n (respectively K2,λ,w,n) defined by (3.18)(respectively (3.19)). For w ∈ Ω and n ∈ N∗ , the operators Hλ,w,n , K1,λ,w,n and K2,λ,w,n are (2n + 1)Γr-periodic. Then, they admit the Floquet decomposition. ⊕ ⊕ Hλ,w,n = Hλ,w,n (θ)dθ , K1,λ,w,n = K1,λ,w,n (θ)dθ , Tn

Tn

and

K2,λ,w,n =

⊕

Tn

K2,λ,w,n (θ)dθ ,

where for θ ∈ Tdn , Hλ,w,n (θ) = Πn (θ)Hλ,w,n Πn (θ), K1,λ,w,n (θ) = (1 + m)Hλ,w,n,− (θ) ⊕ (1 − m)Hλ,w,n,0 (θ) ⊕ (1 − m)Hλ,w,n,+ (θ) and K2,λ,w,n (θ) = (1 − m)Hλ,w,n,− (θ) ⊕ (1 + m)Hλ,w,n,0 (θ) ⊕ (1 + m)Hλ,w,n,+ (θ) . Their integrated densities of states are defined by 1 Nλ,w,n (E) = ϑ Hλ,w,n (θ), E dθ , d (2π) Tdn 1 ϑ K1,λ,w,n (θ), E dθ N1,λ,w,n (E) = d (2π) Tdn and N2,λ,w,n (E) =

1 (2π)d

Td n

ϑ K2,λ,w,n (θ), E dθ

where for an operator with discrete spectrum D and E ∈ R, we denote by ϑ(D, E) the number of eigenvalues of D less than E. Clearly, for all E ∈ R and θ ∈ Tdn , ϑ K2,λ,w,n (θ), E ≤ ϑ Hλ,w,n (θ), E ≤ ϑ K1,λ,w,n (θ), E . (3.36) Using the fact that ϑ(A ⊕ B ⊕ C, E) ≥ ϑ(A, E) + ϑ(B, E) + ϑ(C, E), we get ϑ K2,λ,w,n (θ), E ≥ +ϑ Hλ,w,n,− (θ), (1 − m)−1 · E + ϑ Hλ,w,n,0 (θ), (1 + m)−1 · E + ϑ Hλ,w,n,+ (θ), (1 + m)−1 · E . Let δ = 2−1 (1 − m)δ and α = 2−1 (1 − m)b0 . By (3.34) and (3.35), for all λ ∈ [0, λ0 ], θ ∈ Tdn and E ∈ [−δ , α], ϑ Hλ,w,n,− (θ), (1 − m)−1 · E = (2n + 1)d J− and

ϑ Hλ,w,n,+ (θ), (1 + m)−1 · E = 0 .

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Then, for all λ ∈ [0, λ0 ], θ ∈ Tdn and E ∈ [−δ , α], ϑ Hλ,w,n,0 (θ), C2 · E ≤ ϑ K2,λ,w,n (θ), E − (2n + 1)d J− where C2 = (1 + m)−1 . Let θ ∈ Tdn . We decompose Hn,θ in the following way: + , Hn,θ = Hn,θ,− ⊕ Hn,θ,λ + where Hn,θ,− = Πn,− (θ)H and Hn,θ = Π+ n (θ)H = (Πn,0 (θ) ⊕ Πn,+ (θ))H. We denote (E1,j (λ, w, n, θ))j≥1 , the eigenvalues of K1,λ,w,n (θ). By Mini Max principle,

E1,j (λ, w, n, θ) =

sup

E⊂Hn,θ , dim E=j−1

inf

ϕ⊥E,ϕ∈H2n,θ ,ϕL2 (Cn ) =1

K1,λ,w,n (θ)ϕ, ϕ .

(3.37)

We notice that dim Hn,θ,− = (2n + 1)d J− where J− = {1, . . . , n0 − 1}. If j = (2n + 1)d J− + i, we can choose in the variational formula (3.37) a vectorial space E = Hn,θ,− . Thus, we obtain the following inequality: + (λ, w, n, θ) E1,(2n+1)d J− +i (λ, w, n, θ) ≥ E1,i + + where E1,i (λ, w, n, θ) are the eigenvalues of Π+ n (θ)K1,λ,w,n (θ)Πn (θ). Then, we have + ϑ K1,λ,w,n(θ), E − (2n + 1)J− ≤ ϑ Π+ n (θ)K1,λ,w,n (θ)Πn (θ), E = ϑ Hλ,w,n,0 (θ) ⊕ Hλ,w,n,+ (θ), C1 · E

where C1 = (1 − m)−1 . By (3.35), for E ≤ α, ϑ Hλ,w,n,0 (θ) ⊕ Hλ,w,n,+ (θ), (1 − m)−1 · E = ϑ Hλ,w,n,0 (θ), C1 · E . Finally, we have for all E ∈ [−δ , α], ϑ Hλ,w,n,0 (θ), C2 · E ≤ ϑ Hλ,w,n (θ), E − (2n + 1)d J− ≤ ϑ Hλ,w,n,0 (θ), C1 · E .

(3.38)

Indeed, to get the estimate (3.33), one only needs to integrate (3.38) with respect to θ.

4. Lifshits tail The upper and lower bound are proved separately. 4.1. The upper bound Theorem 4.1. Under the assumptions (H.0), (H.1), (H.2), (H.3), (H.4) and (H.5), there exists λ0 > 0 such that for λ ∈ [0, λ0 ], log log Nλ ( ) − Nλ (0) d ≤− . (4.39) lim sup + log

2 →0

612

F. Ghribi


Proof of Theorem 4.1. By Theorem 3.1, to upper bound Nλ (E) for E close to Eλ , we just need to upper bound Nλ,0 (E) for E close to Eλ . This presents two advantages. First, Π0 Hλ,w Π0 is a discrete operator that, is via a Wannier basis of E0 unitarily equivalent to a random operator acting on l2 (Γ). Moreover, for Hλ,w , Eλ is an interior edge of the spectrum. But, it is the bottom of the spectrum of Π0 Hλ,w Π0 . In this case, more techniques are available. It was proved at the time of the study of Π0 Hλ,w Π0 that, under our assumptions, there exists λ0 > 0 and a constant C > 1 such that for λ ∈ [0, λ0 ] and for u ∈ L2 (Td ), we have

C h1λ,w Su, Su L2 (Td )⊗Cm ≤ hλ,w u, uL2 (Td ) , where h1λ,w , acting on L2 (Td ) ⊗ Cm is defined by ⎞ ⎛ 1 0 ··· 0 h1,λ,w ⎟ ⎜ 0 0 h12,λ,w · · · ⎟ ⎜ h1λ,w = ⎜ ⎟ .. .. . .. .. ⎠ ⎝ . . . 0 0 · · · h1m,λ,w with h1k,λ,w = ξk + λ γ∈Γ wγ πγ . We denote by Nλ1 (E) and Nλ,0 (E), the integrated densities of h1λ,w and hλ,w . Then, for E close to 0, Nλ,0 (E) ≤ Nλ1 C −1 · E . We note that, using the discrete Fourier transform, the operator h1k,λ,w is 1,a unitarily equivalent to the discrete Anderson model Hk,λ,w acting on l2 (Γ) and defined by: for u ∈ l2 (Γ), 1,a u(γ) − u(β) + λwγ u(γ) . Hk,λ,w u (γ) = |β−γ|=1

We deduce then that for E close to 0, we have Nλ,0 (E) ≤ Nλ1,a C −1 · E . 1,a where Nλ1,a is the integrated density of states of the discrete Anderson model Hλ,w defined by 1,a 1,a Hλ,w = ⊕1≤k≤m Hk,λ,w .

To end the proof of Lemma 4.1, we just need to use the well known results concerning the Lifshits tails for discrete Anderson model (see [6]). 4.2. The lower bound Theorem 4.2. Under the assumptions (H.0), (H.1), (H.3) and (H.5), there exists λ0 > 0 such that for λ ∈ [0, λ0 ], we have log log Nλ ( ) − Nλ (0) d lim inf ≥− . (4.40) log 2 →0+

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Proof of Theorem 4.2. We recall that for λ sufficiently small, Σλ has a spectral gap below 0 of length at least δ . Thus, for < δ , we have Nλ ( ) − Nλ (0) = Nλ ( ) − Nλ (− ) . To prove Theorem 4.2, we will lower bound Nλ ( ) − Nλ (− ). For N large enough, D we restrict Hλ,w to the cube ΛN and show that, with a good probability, Hλ,w,Λ N has many eigenvalues in the interval [− , ]. This is done by explicitly constructing orthogonal family of approximate eigenfunctions associated with energies in [− , ] and the operator Hλ,w,ΛN . These functions will be constructed from an eigenvector of H0 associated to 0. Locating this eigenvector in θ near θ0 ∈ Z, we obtain an approximate eigenfunction of Hλ,w,Λ ˜ N . Then, we locate this eigenfunction in x in several disjoints boxes, we get a great orthogonal family of eigenfunctions. Let θ0 ∈ Z, be a point where En0 (θ) vanishes. To simplify the notation, we assume that θ0 = 0. By assumption (H.3), near θ0 , the behavior of En0 (θ) is nondegenerate. Then, by (2.13), there exists a constant c > 1 such that H0 (θ)ϕn0 ( · , θ) 2 ≤ c|θ|2 L (C0 )

where ϕn0 ( · , θ) denotes the Floquet eigenvector of H0 associated to En0 (θ). Let 0 < ζ < 1 be a small constant. Let χ ∈ C0∞ (R) be such that χ ≥ 0, supported in [ ζ2 , ζ] and [ ζ ,ζ] χ(t)2 dt = 2. 2 For > 0, we define d

u (θ) = − 4

d

1

χ( − 2 θj ) ∈ L2 (Td ) .

j=1

We also define f ( · , θ) = u (θ)ϕn0 ( · , θ). Using the definition of ϕn0 ( · , θ), one checks that d f 2H ≥ χ2 (t)dt = 2d (4.41) j=1

and that

H0 f 2H

1 = V ol(Td ) c2 ≤ V ol(Td ) ≤

2 2

c V ol(Td )

Td

[ ζ2 ,ζ]

H0 (θ)ϕn0 ( · , θ)2 2 u (θ)2 dθ L (C0 )

−d 2

Td

[ ζ2 ,ζ]

|θ| · 4

d

1 χ2 − 2 θj dθ

j=1

| θ |4 ·

d

χ2 (θj )dθ

j=1

2 , (4.42) ≤ 16 if we pick ζ sufficiently small (independent of ). Now, for β ∈ Γ and α > 0, we define f,β ( · , θ) = e−iβ · θ f ( · , θ) and fα,,β ( · , θ) = e−iβ · θ ΠΛα () f ( · , θ) ,

614

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where Λα ( ) is the cube defined by ! 1 Λα ( ) = γ ∈ Γ; | γj |≤ −( 2 +α) , ∀1 ≤ j ≤ d and ΠΛα () is the orthogonal projection on Λα ( ). We set f,β fα,,β U (x) = f,β (x, θ) dθ and U (x) = Td

Td

fα,,β (x, θ) dθ .

For N large enough and β, w well chosen, U fα,,β (x) is an approximate eigenfunction associated to an eigenvalue in [− , ] and the operator Hλ,w,ΛN . Let us first notice that U fα,,β and U f,β are close to each other. More precisely, for n ∈ N, there exists a constant Cn > 0 such that, for β ∈ Γ, we have (4.43) vol(Td ) · U fα,,β − U f,β 2 = fα,,β − f,β 2H ≤ Cn αn . Indeed fα,,β − f,β 2H is obviously independent of β, so let us take β = 0. Then, to estimate the norm of fα,,β − f,β , we only need to estimate the γ’s Fourier coefficient of f for γ outside of Λα ( ). Using the smoothness of the function θ → ϕn0 (x, θ) and the non-stationary phase, we obtain that for n ∈ N and for some constant Cn (x) > 0,

e

−iγ · θ

Td

d u (θ)ϕn0 (x, θ)dθ = 4

1

e

−i 2 γ · θ

[ ζ2 ,ζ]d

d j=1

χ(θj )ϕn0 (x, θ)dθ

≤ Cn (x) − 2 | γ |−n . n

1 2

(4.44)

Notice that, as ∂θk ϕn0 (x, · ) ∞ ∈ L2loc,unif (Rd ), the constant Cn (x) will be in L2loc,unif (Rd ). If we sum (4.44) over γ outside of Λα ( ), we get (4.43). Notice that, combining (4.43) with (4.41), we get that, for small enough, U fα,,β ≥ 1. Now, we look for a condition on w for which we have Hλ,w,ΛN U fα,,β 2 ≤ 2 . Notice that Hλ,w,ΛN U fα,,β 2 ≤ Hλ,w U fα,,β 2 , ≤ 2 H0 U fα,,β 2 +2λ2 Vw U fα,,β 2 .

(4.45)

The inequality (4.42) gives the bound of the first member of (4.45). It just remains to us to control the second term Vw U fα,,β 2 . As the random variables (wγ )γ∈Γ are bounded and V is a compactly supported function, Vw is bounded uniformly in w. Using the boundedness of Vw and

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inequality (4.43), we get 2

2 + Vw2 (x) ϕn0 (θ, x)u (θ)eiβ · θ dθ dx 32 Rd Td 2

V (x + η − γ) · V (x + η − γ ) + ≤ wγ wγ 32 C0 η,γ,γ ∈Γ 2 −i(η−β) · θ ϕn0 (θ, x)u (θ)e dθ dx ·

Vw U fα,,β 2 ≤

Td

⎛ ⎞2

2 ⎝ + wγ V L∞ (Cη−γ ) ⎠ ≤ 32 η∈Γ γ∈Γ 2 −i(η−β) · θ · ϕn0 (θ, x)u (θ)e dθ dx . C0

(4.46)

Td

For sufficiently small, the support of V is in the cube Λα ( ), thus wγ V L∞ (Cη−γ ) = wγ V L∞ (Cη−γ ) . γ∈Γ

γ∈η+Λα ()

Using again the boundedness of the random variables (wγ )γ∈Γ , the fact that V is compactly supported and the inequality (4.44), we get that for all n ∈ N, there exists cn > 0 such that ⎛

⎝

η∈β+Λα ()

⎞2

wγ V L∞ (Cη−γ ) ⎠

γ∈η+Λα ()

· C0

Td

2 ϕn0 (θ, x)u (θ)e−i(η−β) · θ dθ dx ≤ cn αn . (4.47)

On the other hand, there exists a constant c > 0 such that ⎛

⎝

η∈β+Λα ()

· C0

⎞2 wγ V L∞ (Cη−γ ) ⎠

γ∈η+Λα ()

Td

ϕn0 (θ, x)u (θ)e

−i(η−β) · θ

" 2 dθ dx ≤ c

#2 sup

wγ

. (4.48)

γ∈β+2Λα ()

Combining the inequalities (4.46), (4.47) and (4.48), we obtain that for small enough and some c > 1, we have " #2

2 fα,,β 2 +c sup ≤ wγ . (4.49) Vw U 16 γ∈β+2Λα ()

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F. Ghribi


Hence, using (4.42), (4.45) and (4.49), we get that for some K > 0, " #2

2 fα,,β 2 +K sup Hλ,w,ΛN U ≤ wγ . 2 γ∈β+2Λα ()

(4.50)

Now, for N ∈ N very large, we may divide ΛN into N ( ) disjoints sets of size 2Λα ( ). More precisely, we may write N () ∪j=1 βj + Λα ( ) ⊂ ΛN , where

βj + 2Λα ( ) ∩ βj + 2Λα ( ) = ∅ , Notice that, for some a constant C > 0, we have N ( )

(2N )d 1

−d( 2 +α)

≥

for βj = βj .

(N )d . C

(4.51)

(4.52) f

Equation (4.51) implies that, for βj = βj , the functions U fα,,βj and U α,,βj are orthogonal. We denote the counting function of the eigenvalues of Hλ,w,N below E by Θλ,ΛN (E). Hence, by , we get that ! E Θλ,ΛN ( ) − Θλ,ΛN (− ) = E eigenvalues of Hλ,w,N in [− , ] ! ≥ E 1 ≤ j ≤ N ( ); Hλ,w,N U fα,,βj ≤ ⎛ ⎞ N () ≥ E⎝ Bj (w)⎠ , (4.53) j=1

$

where

1 si supγ∈β+2Λα () wγ ≤ √2K 0 sinon . The (Bj )1≤j≤N () are independent, identically distributed, Bernoulli random variables. So, by (4.52) and (4.53), we get that, for some C > 0, ! 1 E eigenvalues of Hλ,w,N in [− , ] Nλ,ΛN ( ) − Nλ,ΛN (− ) = d (2N + 1) 1 N ( ) P(Bj = 1) ≥ d P Bj = 1 . ≥ d (2N + 1) C Hence, taking the limit N → +∞, we get that, for > 0 small enough, 1 Nλ ( ) − Nλ (− ) ≥ d P(Bj = 1) . (4.54) C It just remains to estimate P(Bj = 1). If, for any γ ∈ βj + 2Λα ( ), we have , then Bj (w) = 1. As the random variables are independent identically wγ ≤ √2K distributed, one has " $ % # 2Λα ()

w0 ≤ √ . P Bj = 1 ≥ P 2K

Bj (w) =

Vol. 9 (2008)


617

Hence, taking the double logarithm of (4.54), using assumption (H.5) and the fact 1 that Λα ( ) = −d( 2 +α) , we get that log log Nλ ( ) − Nλ (0) d lim inf ≥ − − dα . (4.55) log 2 →0+ Now, we may choose α > 0 as small as we like. This ends the proof of Theorem 4.2.

5. Appendix Lemma 5.1. Let θ1 and θ2 two points in Td , ψ a function in L2 (Td , Hθ2 ) and α a strictly positive real number. Designate by ψ1 , ψ2 , ξ1 , ξ2 and mψ,θ1 ,θ2 the functions defined by ψ1 (θ, x) = ei(θ−θ1 ) · x ψ(θ1 , x), ψ2 (θ, x) = ei(θ−θ2 ) · x ψ(θ2 , x), ξ1 (θ) = d d 2 2 j=1 (θj −θ1,j ) , ξ2 (θ) = j=1 (θj −θ1,j ) and mψ,θ1 ,θ2 (x) = V (x)ψ(θ1 , x)ψ(θ2 , x). There exists a constant c > 0 such that for u, v ∈ L2 (Td ), we have Pψ (u) 2L2 (Rd ) ≤ ψ 22,∞ · u 2L2 (Td ) ,

Vw Pψ (v), Pψ (v) ≤ c ψ 21,H0 ,∞ · v 2L2 (Td ) Vw Pψ (v)2 2 d L (R )

Vw Pψ1 (v)2 2 d L (R )

(5.56)

(5.57)

≤ c ψ 22,H0 ,∞ · v 2L2 (Td ) Vw Pψ1 (u), Pψ2 (v) (5.58) − mψ,θ1 ,θ2 (x) dx wγ u ˆ(γ) · vˆ(γ) d R γ∈Γ 2 2 1 ≤ cα wγ uˆ(γ) + cα wγ vˆ(γ) + c 1 + ξ2 v, v α γ∈Γ γ∈Γ 1 +c 1+ (5.59) ξ1 u, u , α 2 ˆ(γ) + cξ1 u, uL2 (Td ) . ≤c wγ u (5.60) γ∈Γ

Proof of Lemma 5.1. Vw is H0 -bounded uniformly in w. Then, there exist c1 , c2 > 0 (independents of w) such that for u ∈ L2 (Td ), $ % 2

H0 Pψ (u), Pψ (u) + Pψ (u) Vw Pψ (u), Pψ (u) ≤ c1 ≤ c1 ψ 21,H0 ,∞ · u 2L2 (Td ) and

Vw Pψ (v)2 2 d ≤ c2 H0 Pψ (u)2 + Pψ (u)2 L (R ) ≤ c2 ψ 22,H0 ,∞ · u 2L2 (Td ) .

This ends the proof of the assertions (5.56) and (5.58).

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We estimate now Vw Pψ1 (u), Pψ2 (v)L2 (Rd ) for u, v ∈ L2 (Td ).

Vw Pψ1 (u), Pψ2 (v) = Vw (x)ψ(θ1 , x) ψ2 (θ2 , x) Rd

· Φθ1 (u) (x) Φθ2 (v) (x)dx , (5.61) where for i ∈ {1, 2}, u ∈ L2 (Td ) and x ∈ Rd , [Φθi (u)](x) = Td ei(θ−θi ) · x u(θ)dθ. By definition of Vw ,

wγ V (x)ψ(θ1 , x) ψ(θ2 , x) Φθ1 ,γ (u) (x) Vw Pψ1 (u), Pψ2 (v) L2 (Rd ) = Rd

γ∈Γ

· Φθ1 ,γ (v) (x) dx where for i ∈ {1, 2}, γ ∈ Γ and u ∈ L2 (Td ), Φθi ,γ (u) (x) = ei(θ−θi ) · x eiγ · θ u(θ) dθ . Td

We recall that θi is the unique zero of ξi (θ) on Td . Furthermore, it is nondegenerate. Thus, the function gi (θ, x) is well defined on Td × Rd . Moreover, there exists a constant ri > 0 such that for all (θ, x) ∈ Td × Rd , we have gi (θ, x) ≤ ri 1+ | x | . (5.62) We remark that, for i ∈ {1, 2}, γ ∈ Γ, u ∈ L2 (Td ) and x ∈ Rd , Φθi ,γ (u) (x) = uˆ(γ) + gi (θ, x)eiγ · θ ξi (θ)u(θ)dθ .

(5.63)

Using the relation (5.63), we obtain that

Vw Pψ1 (u), Pψ2 (v) L2 (Rd ) = vβ,w,ψ,θ1 ,θ2 u, v ,

(5.64)

Td

0≤β≤3

where v0,w,ψ,θ1 ,θ2 u, v =

wγ

γ∈Γ

v1,w,ψ,θ1 ,θ2 u, v =

wγ

γ∈Γ

Rd

Rd

mψ,θ1 ,θ2 (x) dx u ˆ(γ) · vˆ(γ) ,

mψ,θ1 ,θ2 (x) ·

Td

g1 (θ, x) eiγ · θ

g2 (θ , x) eiγ · θ ξ2 (θ ) v(θ ) dθ dx , Td v2,w,ψ,θ1 ,θ2 u, v = wγ uˆ(γ) mψ,θ1 ,θ2 (x) ·

γ∈Γ

·

Td

(5.65) ξ1 (θ) u(θ) dθ (5.66)

Rd

g2 (θ , x) eiγ · θ

ξ2 (θ ) v(θ ) dθ dx

(5.67)

Vol. 9 (2008)


and v3,w,ψ,θ1 ,θ2 u, v =

wγ vˆ(γ)

γ∈Γ

·

619

Rd

mψ,θ1 ,θ2 (x)

g1 (θ, x) eiγ · θ

Td

ξ1 (θ) u(θ) dθ dx .

(5.68)

ˆ(γ) · vˆ(γ). We estimate Vw Pψ1 (u), Pψ2 (v) − Rd mψ,θ1 ,θ2 (x) dx γ∈Γ wγ u

mψ,θ1 ,θ2 (x) dx wγ u ˆ(γ) · vˆ(γ) Vw Pψ1 (u), Pψ2 (v) − Rd γ∈Γ vβ,w,ψ,θ1 ,θ2 u, vL2 (Td ) . ≤ 1≤β≤3

We estimate | v1,w,ψ,θ1 ,θ2 u, v |. Using the fact that the random variables (wγ )γ∈Γ are bounded, Cauchy–Schwarz inequality, then, Parseval identity and finally the inequality (5.62), we obtain that v1,w,ψ,θ1 ,θ2 u, v ≤ c5 ξ1 u, uL2 (Td ) + c6 ξ2 v, vL2 (Td ) , (5.69) −1 + 2 2 where c5 = 2 w r1 Rd | mψ,θ1 ,θ2 (x) | (1+ | x |) dx and mψ,θ1 ,θ2 (x) 1+ | x | 2 dx . c6 = 2−1 w+ r22 Rd

Estimate v2,w,ψ,θ1 ,θ2 u, v. Let α > 0. By Cauchy–Schwarz inequality, 2 mψ,θ1 ,θ2 (x) dx v2,w,ψ,θ1 ,θ2 u, v ≤ α wγ uˆ(γ) Rd

1 + wγ 4α γ∈Γ

R

mψ,θ1 ,θ2 (x) d

Td

γ∈Γ

g2 (θ, x)e

iγ · θ

2 ξ2 (θ)v(θ)dθ dx .

Using the fact that the random variables (wγ )γ∈Γ are bounded, Cauchy–Schwarz inequality, Parseval identity and finally the inequality (5.62), we obtain that 2 v2,w,ψ1 ,ψ2 u, v ≤ α mψ,θ1 ,θ2 (x) dx wγ u ˆ(γ) Rd

γ∈Γ

c6 ξ2 v, vL2 (Td ) . + 2α

(5.70)

Doing the same computation for | v3,w,ψ1 ,ψ2 u, v |, we get the same estimate 2 v3,w,ψ1 ,ψ2 u, vL2 (Td ) ≤ α mψ,θ1 ,θ2 (x) dx wγ vˆ(γ) Rd

c5 ξ1 u, uL2 (Td ) . + 2α

γ∈Γ

(5.71)

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F. Ghribi


Finally, the inequalities (5.69), (5.70) and (5.71) imply that

mψ,θ1 ,θ2 (x) dx wγ uˆ(γ) · vˆ(γ) Vw Pψ1 (u), Pψ2 (v) − Rd γ∈Γ 2 c7 ≤ αc8 + c5 ξ1 u, u + α(2c3 + c8 ) wγ uˆ(γ) + wγ | vˆ(γ) |2 2α γ∈Γ γ∈Γ c6 ξ2 v, v, + 2c4 + c6 + 2α where c8 = [ Rd | mψ,θ1 ,θ2 (x) | dx]. This ends the proof of the assertion (5.59). We estimate now Vw Pψ1 (u) 2L2 (Rd ) . By definition of Vw , Vw Pψ1 (u)2 = Vw (x)ψ(θ1 , x)2 Φθ1 (u) (x)2 dx Rd 2 = wγ wβ Vγ (x)Vβ (x)ψ(θ1 , x)ψ(θ1 , x) · Φθ1 (u) (x) dx Rd

γ,β∈Γ

=

wγ wβ

γ,β∈Γ

≤

2 V (x)Vβ−γ (x)ψ(θ1 , x)ψ(θ1 , x) · Φθ1 ,γ (u) (x)

Rd

wγ wη+γ

Rd

γ,η∈Γ

V (x)Vη (x) · ψ(θ1 , x)2 Φθ1 ,γ (u) (x)2 dx

≤ c1,w,ψ1 [u] + c2,w,ψ1 [u] , where c1,w,ψ1 [u] = 2

2 ˆ(γ) wγ wη+γ u

γ,η∈Γ

and c2,w,ψ1 [u] = 2

wγ wη+γ

γ,η∈Γ

·

Td

e

iγ · θ

Rd

Rd

V (x)Vη (x)ψ(θ1 , x)2 dx

V (x)Vη (x) · ψ(θ1 , x)2

2 g1 (θ, x) ξ1 (θ)u(θ) dθ dx .

As the random variables (wγ )γ∈Γ are bounded, we have 2 c1,w,ψ1 [u] ≤ c9 ˆ(γ) , wγ u γ∈Γ

(5.72)

where c9 = 2w+ η∈Γ Rd | V (x)Vη (x) | · | ψ(θ1 , x) |2 dx < ∞ as the single site potential V is compactly supported. Using the fact that the random variables (wγ )γ∈Γ are bounded, Cauchy– Schwarz inequality, then, Parseval identity and finally the inequality (5.62), we obtain that c2,w,ψ1 [u] ≤ c10 ξ1 u, uL2 (Td ) , (5.73) + 2 2 2 2 where c10 = (2w ) r1 η∈Γ Rd | V (x)Vη (x) | · | ψ(θ1 , x) | (1+ | x |) dx.

Vol. 9 (2008)


Finally, by (5.72) and (5.73), we get 2 Vw Pψ1 (u)2 2 d ≤ c9 wγ u ˆ(γ) + c10 ξ1 u, uL2 (Td ) . L (R )

621

(5.74)

γ∈Γ

This ends the proof of the assertion (5.60). Proof of Lemma 3.2. We define the operator Dλ,w by Πi Hλ,w Πi . Dλ,w := Hλ,w − i∈{−,0,+}

Let ϕ ∈ H . We notice that, 2

Dλ,w ϕ, ϕ = Hλ,w ϕ, ϕ − Π− Hλ,w Π− ϕ, ϕ + Π0 Hλ,w Π0 ϕ, ϕ + Π+ Hλ,w Π+ ϕ, ϕ = 2λ Π− Vw Π+ ϕ, ϕ + 2λ Π− Vw Π0 ϕ, ϕ + 2λ Π+ Vw Π0 ϕ, ϕ .

(5.75)

To get Theorem 3.2, we just need to prove that there exists λ0 > 0 et m = m(λ0 ) ∈]0, 1[ such that for λ ∈ [0, λ0 ] and ϕ ∈ H2 , we have Dλ,w ϕ, ϕ ≤ m − Π− Hλ0 ,w Π− ϕ, ϕ + Π0 Hλ,w Π0 ϕ, ϕ + Π+ Hλ,w Π+ ϕ, ϕ . (5.76) Let us now estimate | Dλ,w ϕ, ϕ |. Dλ,w ϕ, ϕ ≤ 2λΠ− Vw Π0 ϕ, ϕ | +2λ | Π+ Vw Π0 ϕ, ϕ + 2λΠ+ Vw Π− ϕ, ϕ . We control first | Π− Vw Π0 ϕ, ϕ |. Using Cauchy–Schwarz inequality,

Π− Vw Π0 ϕ, ϕ ≤ Vw Π0 ϕ · Π− ϕ ≤ 1 λ2η Π0 Vw2 Π0 ϕ, ϕ 4 (5.77) + λ−2η Π− ϕ 2 where η is a real number in ] 12 , 1[. Doing the same computation for | Π+ Vw Π0 ϕ, ϕ |, we get

Π+ Vw Π0 ϕ, ϕ ≤ 1 λ2η Π0 V 2 Π0 ϕ, ϕ + λ−2η Π+ ϕ 2 . w 4 Finally, for | Π+ Vw Π− ϕ, ϕ |, we obtain Π+ Vw Π− ϕ, ϕ ≤ 1 Vw Π− ϕ 2 + 1 Π+ ϕ 2 . 2 2 We addition the contributions (5.77), (5.78) and (5.79), we obtain that Dλ,w ϕ, ϕ ≤ dλ,w,− [ϕ] + dλ,w,0 [ϕ] + dλ,w,+ [ϕ] ,

(5.78)

(5.79)

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F. Ghribi


where dλ,w,− [ϕ] = λ Vw Π− ϕ 2 +2λ1−2η Π− ϕ 2 ,

dλ,w,0 [ϕ] = λ1+2η Π0 Vw2 Π0 ϕ, ϕ and dλ,w,+ [ϕ] = (λ + 2λ1−2η ) Π+ ϕ 2 . In order to complete the proof, it suffices to prove that there exists λ0 > 0 and m = m(λ0 ) ∈]0, 1[ such that, for λ ∈ [0, λ0 ] and for ϕ ∈ H, we have dλ,w,− [ϕ] ≤ −mΠ− Hλ,w Π− ϕ, ϕ ,

(5.80)

dλ,w,0 [ϕ] ≤ mΠ0 Hλ,w Π0 ϕ, ϕ

(5.81)

and dλ,w,+ [ϕ] ≤ mΠ+ Hλ,w Π+ ϕ, ϕ . (5.82) First, we prove the assertion (5.80). By the definition of Π− , we have Π− H0 Π− ≤ −δΠ− .

(5.83)

This implies that the operator Π− H0 Π− , acting on Π− (H2 ) is invertible. Furthermore, we know that the operator Vw is H0 -bounded uniformly on w. Then, there exists two constants d1 > 0 (independent of w) such that, for ϕ ∈ H2 , Π− Vw Π− ϕ, ϕ ≤ −d1 Π− H0 Π− ϕ, ϕ . (5.84) Combining (5.84) and (5.83), we obtain that for ϕ ∈ H2 , Π− Hλ,w Π− ϕ, ϕ ≤ −(1 − d1 λ)δ Π− ϕ 2 .

(5.85)

As Π− is of finite energy and Vw is uniformly relatively H0 -bounded, Vw Π− is a bounded family of operators. Then, there exists d2 > 0 (independent on w) such that for ϕ ∈ H, we have Vw Π− ϕ ≤ d2 Π− ϕ .

(5.86)

By relations (5.85) and (5.86), we get the following inequality dλ,w,− [ϕ] ≤ −m1 (λ)Π− Hλ,w Π− ϕ, ϕ + 2λ1−2η ] · (1 − d1 λ)−1 δ −1 . where m1 (λ) = We prove now the assertion (5.81). By definition of Π+ , [d22 λ

Π+ H0 Π+ ≥ b0 Π+ ,

(5.87)

where b0 = inf{En0 +1 (θ), θ ∈ T } > 0. This implies that Π+ H0 Π+ , acting on Π+ (H2 ) is invertible. We recall that the operators Vw is H0 -bounded uniformly in w. Then, there exists a constant e1 > 0 (independent of w) such that for ϕ ∈ H2 , Π+ Vw Π+ ϕ, ϕ ≤ e1 Π+ H0 Π+ ϕ, ϕ . (5.88) d

Combining the relations (5.88) and (5.87), we obtain that for ϕ ∈ H2 , Π+ Hλ,w Π+ ϕ, ϕ ≥ (1 − e1 λ)Π+ H0 Π− ϕ, ϕ ≥ (1 − e1 λ)b0 Π+ ϕ 2 . 2−2η

Then, we obtain the assertion (5.81) where m2 (λ) = (λ + 2λ

(5.89) (5.90) ) · ((1 − e1 λ)b0 )−1 .

Vol. 9 (2008)


623

Finally, we prove the assertion (5.82). Identifying L2 (Td ) to E0 = Π0 (L2 (Rd )), we need just to prove for λ sufficiently small hλ,w u, uL2 (Td ) ≤ fw u, uL2 (Td )

for u ∈ L2 (Td )

where fw = Pϕ∗n0 Π0 Vw2 Π0 Pϕn0 . We calculate fw u, uL2 (Td ) for u ∈ L2 (Td ). Using the partition of unity (χk )1≤k≤m , linearity of Pϕn0 and Cauchy–Schwarz inequality, we obtain Vw Pϕñ (χk u)2 2 d fw u, uL2 (Td ) ≤ 2m 0

1≤k≤m

+ 2m

L (R )

2 ξk χk u 2 Vw Pδϕn0

L (Rd )

.

1≤k≤m

By assertions (5.59) and (5.58) of Lemma 5.1, there exists a constant K > 0 such that for u ∈ L2 (Td ), 2 wγ ξk χk u, χk u . (χk u)(γ) + K fw u, uL2 (Td ) ≤ K 1≤k≤m

γ∈Γ

1≤k≤m

Thus, for u ∈ L2 (Td ), λ2η fw u, uL2 (T2 ) ≤ m3 (λ)hλ,w u, uL2 (Td ) , where m3 (λ) = CK λ2η−1 . Let m(λ) = sup1≤j≤3 mj (λ). As η is a real number chosen in ] 12 , 1[, Clearly, m(λ) converges to 0 as λ → 0. This ends the proof of Theorem 3.24.

References [1] M. Sh. Birman, Perturbations of periodic Schr¨ odinger operators. Lectures given at the Mittag–Leffler Insitute during the program “Spectral Problems in Mathematical Physics”, 1992. [2] F. Ghribi, Internal Lifshits tails for random magnetic Schr¨ odinger operators, Journal of Functional Analysis 208 (2007), 387–427. [3] W. Kirsch, Random Schr¨ odinger Operators: A Course in Schr¨ odinger Operators, (Sonderborg, 1989), ed. A. Jensen and H. Holden, Lecture Notes in Phys. 345, Springer-Verlag. Berlin, 1989, 264–370. [4] W. Kirsch and F. Martinelli, On the spectrum of Schr¨ odinger operators with a random potential, Comm. Math. Phys. 85 (1982), 329–350. [5] W. Kirsch and F. Martinelli, Large deviations and Lifshitz singularities of the integrated density of states of random Hamiltonians, Comm, Math. Phys. 89 (1983), 27–40. [6] W. Kirsch and B. Simon, Lifshits tails for the Anderson model, J. Statist. Phys. 38 (1985), 65–76. [7] F. Klopp, Internal Lifshits tails for random perturbations of periodic Schr¨ odinger operators, Duke Math. J. 98 (1999), 335–396.

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[8] F. Klopp, Erratum to the paper “Internal Lifshits tails for random perturbations of periodic Schr¨ odinger operators”, Duke Math, J. 98 (1999), 335–396. [9] F. Klopp, Weak disorder localisation and Lifshits tails, Comm, Math. Phys. 232 (2002), 125–155. [10] F. Klopp and J. Ralston, Endpoints of the spectrum of periodic operators are generically simple, Methods Appl. Anal. 7 (2000), 459–464. [11] I. M. Lifshitz, Structure of the energy spectrum of impurity bands in disordered solid solutions, Soviet Phys. JETP 17 (1963), 1159–1170. [12] I. M. Lifshitz, Energy spectrum structure and quantum states of disordered condensed systems, Sov. Phys. Usp. 7 (1965) 549–573. [13] I. M. Lifshits, S. A. Gredskul and L. A. Pastur, Introduction to the Theory of Disorederd Systems, Willey, New York, 1988. [14] G. Mezincescu, Lifshitz singularities for periodic operators plus random potentials, Journal of Statistical Physics 49 (1987), 1081–1090. [15] G. Mezincescu, Lifshitz singularities for one dimensional Schr¨ odinger operators. Comm. Math. Phys 158 (1993), 315–325. [16] H. Najar, The spectrum minimum for random Schr¨ odinger operators with indefinite sign potentials, J. Math. Phys. 47 (2006), no. 1, 013515, 13 pp. [17] S. Nakao, On the spectral distribution of the Schr¨ odinger operator with random potential, Japan J. Math. (N.S.) 3 (1977), 111–139. [18] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Grundlehren Math. Wiss. 297, Springer-Verlag, Berlin, 1992. [19] L. Pastur, Behaviour of some Wiener integrals as t → +∞ and the density of states of Schr¨ odinger equation with a random potential, Teo. Math. Fiz. (Russian) 32 (1977), 88–95. [20] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of operators, Academic Press, New York, 1978. [21] J. Sj¨ ostrand, Microlocal analysis for periodic magnetic Schr¨ odinger equation and related questions, in Microlocal Analysis and Applications, Lecture Notes in Math. 1495, Springer-Verlag, Berlin, 1991. [22] P. Stollmann, Caught by Disorder. Bound States in Random Media, Birkh¨ auser, Boston. MA, 2001. Fatma Ghribi Département de Mathématiques Site de Saint-Martin Université Cergy-Pontoise 2, avenue Adolphe Chauvin F-95302 Cergy-Pontoise Cedex France e-mail: [email protected] Communicated by Vincent Rivasseau. Submitted: April 17, 2007. Accepted: December 13, 2007.



Reflections on Refraction (Geometrical Optics) Michel Mendès France and Ahmed Sebbar a Jacques Peyri` ` ere

Abstract. Consider an optical system made of an unknown number N of layers of homogeneous transparent plates with different unknown refraction indices. Observing beams of monochromatic light through the system, find the number N of plates together with their respective indices and their thicknesses. The mathematical analysis of the problem involves the so-called Hadamard quotient of two power series. We shall also discuss fractal optical systems consisting of infinitely many infinitely thin plates. If the index of refraction varies in an erratic way there may be multiple refraction. These systems could be called “refractals”. We conclude the paper with independent considerations on a general system consisting of one plate with continuous varying index n(x, y) ≥ 1. To determine the function (x, y) → n(x, y) seems to be a difficult problem. Our contribution to solving it is thus very modest.

1. Determining the index of refraction Fascinated by the simplicity of geometrical optics and by the Snell–Descartes law “sin i = n sin r”, we ask the natural question which may be well known to specialists. Consider a non-crystalline transparent plate whose unknown index of refraction n(x, y) ≥ 1 depends on the point (x, y) within the plate. We ignore partial reflection and absorption. Analyzing the outcoming beams in terms of monochromatic incident beams should enable one to recover the function (x, y) → n(x, y). We shall discuss briefly the general problem using geometry at the very end of our work. The main part of the paper is concerned with the much simpler case where the optical system consists of N transparent layers of respective thicknesses a1 , a2 , . . . , aN and with respective indices of refraction n1 , n2 , . . . , nN . Our approach is analytical.

626

M. Mendès France and A. Sebbar


y

θ M O

Α

θ

x

Figure 1. Ray through plates. The system is represented in the (x, y) plane so that the front face of the plate is the y-axis and the opposite face is the vertical line x=

N

a = d .

=1

An incident ray hits the plate at the origin O at angle θ ∈ [0, π/2[, penetrates within the system and meets the opposite side at M whose coordinates are x = d, y = φ(θ). Does the knowledge of θ → φ(θ) determine the number of layers, their indices and their thicknesses? In Sections 6 and 7 we study a somewhat limit case. There we are given the refraction indices and we ask what can be said about φ(θ). We shall discover that for certain distribution of indices the outcoming ray may well be multiple. Of course this multiple refraction has nothing to do with the well known double refraction of calcite.

2. The mathematical formulation Let θk be the angle of the ray as it penetrates the k th plate. The Snell–Descartes law asserts that sin θ = n1 sin θ1 = n2 sin θ2 = . . . Let M0 = O, M1 , M2 , . . . be the points where the ray hits the plates 1, 2, 3, . . . The ordinate of these points are y0 = 0 ,

y1 = a1 tan θ1 , . . . , yk = yk−1 + ak tan θk , . . .

Vol. 9 (2008)

Reflections on Refraction (Geometrical Optics)

θk Mk−1 θk−1

Mk

627

θk+1

θk

Figure 2. Indices. Since sin θk sin θ tan θk = = 2 1 − sin θk n2k − sin2 θ we have φ(θ) = yN =

N =1

a sin θ . n2 − sin2 θ

More generally, should the index vary continuously then ∞ sin θ dσ(x) φ(θ) = 0 n2 (x) − sin2 θ

(1)

(2)

where σ is a bounded non-decreasing function. We shall come back to this aspect in Sections 6 and 7. The above representation is actually general since it encompasses also the discrete case where σ is then a step function and a δ x , a > 0 dσ =

where δx is the Dirac measure at x . The question raised in Section 1 reduces to the problem of expressing N, n1 , n2 , . . . , nN , a1 , a2 , . . . , aN in terms of φ(θ). For a given φ, if these values do exist they are not unique since permuting the layers does not change φ (provided all the indices are finite). Therefore in this case, with no loss of generality we can always assume 1 ≤ n1 < n2 < · · · < nN and agree that layers of same index are put together to form a single plate.

628



3. A functional equation Put ξ = sin θ so that 0 ≤ ξ < 1. Define Ψ(sin θ) = φ(θ). Equation (1) reads − 12 N ξ ξ2 a . Ψ(ξ) = 1− 2 n n =1

The problem now is to find N, a > 0 and the increasing sequence 1 ≤ n1 < n2 < · · · < nN in terms of Ψ. This is of course quite trivial. Indeed n1 is the smallest singularity of the odd analytic function 2k+1 Ψ(ξ) = Σ∞ k=0 f2k+1 ξ

so 1 1 = lim |f2k+1 | 2k+1 . k→∞ n1

Then

Ψ(ξ) n21 − ξ 2 . ξ→n1 ξ The couple (n2 , a2 ) is obtained in a similar way by considering the function Ψ1 a1 = lim

a1 ξ Ψ1 (ξ) = Ψ(ξ) − 2 . n1 − ξ 2 At step l Ψl−1 = Ψ(ξ) −

l−1 j=1

aj ξ n2j − ξ 2

determines nl and al . At some point, ΨN ≡ 0 and the algorithm ends. It may be interesting to observe that the above problem falls in a general scheme, namely to discuss the equation N ξ F (ξ) = al H (3) nl l=1

where F and H are given functions and where N, al , nl are unknown. In our context we assume that H and therefore F are odd functions holomorphic in a neighbourhood of the origin: f2k+1 ξ 2k+1 , H(ξ) = h2k+1 ξ 2k+1 . F (ξ) = k≥o

k≥o

We shall suppose that for all k, h2k+1 = 0. The previous case corresponds to 1 2 − 12 k −2 = (−1) H(ξ) = ξ(1 − ξ ) ξ 2k+1 k k≥0

and F = Ψ.

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Define the Hadamard quotient of the power series F and H by f2k+1 z 2k+1 . F/H(z) = h2k+1 k≥0

Theorem 1. Under the above assumptions, the following two equalities are equivalent N ξ F (ξ) = al H nl l=1

and F/H(z) =

N

z . n2l − z 2

al n l

l=1

Proof. The first equality reads

f2k+1 ξ

2k+1

=

k≥o

N

al

l=1

h2k+1

k≥o

ξ nl

2k+1 .

Therefore

f2k+1 z 2k+1 h2k+1

k≥0

N

al 2k+1 n l=1 l N N z 2k+1 z = al = al n l 2 . nl nl − z 2

f2k+1 = h2k+1

l=1

k≥0

(4)

l=1

Corollary 1. Given Ψ, the number N of plates of the optical system is the number of positive poles of the rational function 1

Ψ/z(1 − z 2 )− 2 =

N

al n l

l=1

n2l

z . − z2

The positive poles n1 , n2 , . . . nN are the respective indices of refraction and the respective thicknesses of the plates are given by 1 al = − residue at nl . 2nl 1 Corollary 2. The only optical system such that for all k, f2k+1 ∈ −k2 Z is the trivial one, i.e., its index of refraction is uniformly equal to 1. Then necessarily 1 f2k+1 = (−1)k −k2 a where a is a positive integer. Proof. According to the hypothesis 1

Ψ/z(1 − z 2 )− 2 =

e2k+1 z 2k+1

k

where e2k+1 ∈ Z. The radius of convergence is then surely ≤ 1 (it cannot vanish). Since it is R = n1 ≥ 1, the radius equals 1. A classical theorem of Polya’s [5]

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asserts that a power series with integral coefficients whose radius of convergence is 1, is either a rational function or a transcendental function where |z| = 1 is a natural boundary. Since a n z 1 Ψ/z(1 − z 2 )− 2 = n2 − z 2

∞ or maybe 0 n(x)zdσ(x) n2 (x)−z 2 , then necessarily 1

Ψ/z(1 − z 2 )− 2 =

a1 z 1 − z2

so that a1 ξ Ψ(ξ) = . 1 − ξ2

4. An algebraic statement Given the function φ(θ), we know that the optical system is determined given that the index of refraction increases. What if we only know some derivatives of φ at the origin? Theorem 2. Suppose there exists no nontrivial polynomial R ∈ Z[X0 , X1 , . . . , X2N ] of degree at most (2N + 1)! such that R φ (0), φ (0), . . . , φ(4N +1) (0) = 0 . Then the optical system consists of at least N + 1 plates. Proof. The derivatives of φ are rational linear combinations of those of Ψ so that the theorem can be stated with the coefficients f2k+1 of the Taylor expansion of Ψ Ψ(ξ) =

2N

f2k+1 ξ 2k+1 + O(ξ 4N +3 ) .

k=0

Equation (4) reads N

f2k+1 1 (−1)k −k2 for k = 0, 1, . . . . In order to simplify notations in the following discussions put al 2k+1 n l=1 l

al = αl , nl

=

1 = νl , n2l

f2k+1 1 = gk . (−1)k −k2

Consider the system of 2N + 1 equations where the αl and νl could be thought of as 2N unknowns and where the gk , k = 0, 1, . . . , 2N are given N l=1

αl νlk = gk ,

0 ≤ k ≤ 2N .

(5)

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The number 2N +1 of equations exceeds the number 2N of unknown and therefore unless the parameters gk are linked the system should have no solutions. And indeed a classical result as found for example in O. Perron’s book [4] (Algebra I, p. 129, satz 57) states: Given M + 1 polynomials in M variables of respective degrees m1 , m2 , . . . , mM+1 with a common zero, these polynomials are neces M+1 sarily linked by an algebraic polynomial of degree at most j=1 mj . Therefore if the system (5) is solvable then necessarily there exist a nontrivial polynomial R of degree (2N + 1)! such that R (g0 , g1 , . . . , g2N ) = 0 . If no such polynomial exists then (5) has no solution and the optical system must contain at least N + 1 plates. Theorem 2 is a partial answer to a question of Ken Falconner’s who asked what can be said about an optical system where φ(θ) is not precisely known. Our result shows that if we know that the M first nonzero derivatives of Ψ are not linked by an algebraic relation of degree (2N + 1)! then necessarily the number of plates N is larger than M 2 .

5. A remarkable example with infinitely many layers Suppose Ψ(ξ) =

(−1)k

k≥0

1 −2 ζ(2k + 1 + α)ξ 2k+1 k

where α > 0 is a given parameter and where ζ is the Riemann function. We then know that a n z 1 Ψ/z(1 − z 2 )− 2 = = ζ(2k + 1 + α)z 2k+1 2 2 n − z 1≤

=

0≤k

z 2k+1

1≤

0≤k

1 2k+1+α

1 z = . α 2 − z 2 1≤

Equating both sides, we get n = and a = −α . If α > 1 the total thickness of the optical system is finite. As increases to infinity, n tends to infinity. At the limit, the luminous rays are parallel to the x-axis within the system. This is somehow a paradoxical situation since the outcoming rays keep the same horizontal direction contrary to the finite case where the outcoming ray is parallel to the incoming ray. Paradoxical as this may seem, we should not be too surprised. Infinity often creates strange situations. If for example the horizontal ray hits the system from the right hand side, the outcoming ray on the left of the system is completely

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undetermined and we could agree to say that it diffuses. Of course we are here far from the real world where the highest known indices seem not to exceed 4. Suppose now α = 1. The optical system is infinitely thick. For all θ < π/2, φ(θ) stays finite since 1 sin θ 0 and dσ = xβ dx, β < −1, and compute Ψ or its coefficients f2k+1 . As we just noted −1 ∞ e−α(2k+1)x xβ dx f2k+1 = (−1)k 2 k 0 −1 ∞ du = (−1)k 2 e−u uβ β+1 k 0 α(2k + 1) −1 Γ β+1 k 2 = (−1) β+1 k α(2k + 1) Γ β + 1 Γ k + 12 1 = √ β+1 . k! (2k + 1)β+1 πα And therefore

Γ β + 1 Γ k + 12 ξ 2k+1 Ψ(ξ) = √ β+1 . πα k! (2k + 1)β+1 k≥0

In particular, for β = 0 2 − 12

Ψ/z(1 − z ) so that

= 0

∞

1+z zeαx dx 1 log = e2αx − z 2 2α 1−z

1 Γ k + 12 ξ 2k+1 Ψ(ξ) = √ k! 2k + 1 πα k≥0

1+ξ 1 1 log

ξ(1 − ξ 2 )− 2 = 2α 1−ξ where represents the Hadamard product.

7. Refractals Consider once again the case of a unique plate with varying index x → n(x). As already mentioned ∞ sin θ dσ(x) φ(θ) = 0 n2 (x) − sin2 θ provided the function n(.) is σ-integrable.

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If however it is not, it is bounded from below and from above by two measurable functions. We can then consider both the lower and the upper integrals ∞ ∞ sin θ dσ(x) sin θ dσ(x) and φ∗ (θ) = . φ∗ (θ) = 0 0 n2 (x) − sin2 θ n2 (x) − sin2 θ If φ∗ (θ) < φ∗ (θ) we interpret this as multirefraction: the incoming θ-ray emerges as a beam of width (φ∗ (θ) − φ∗ (θ)) cos θ. We propose to call such an optical system a refractal. Refractals do not diffract light nor they diffuse, disperse nor scater light. The phenomenon is entirely different but unfortunately no experiment will ever demonstrate it! The real world ignores non-measurable functions which as we mathematicians know, depend on the Axiom of choice! Here we are far away from the trends of today’s philosophy according to which there is no room for metaphysics within physics. Refractals are an instance of what could be called “abstract physics” which describes a non-existing world.

8. The average of φ(θ) The average of

∞

φ(θ) = 0

sin θ dσ(x) n2 (x) − sin2 θ

with respect to the incident angle θ ∈ [0, π/2[ and with respect to the thickness of the system is ∞ π 2 ∞ 2 sin θ dθdσ(x) dσ(x) M(φ) = π 0 0 0 n2 (x) − sin2 θ which is easily seen to be M(φ) =

1 π

∞

log 0

n(x) + 1 dσ(x) n(x) − 1

∞

dσ(x) . 0

If the index of refraction n(x) is constant from one layer to the other, then 2 n+1 n+1 1 1 = log M(φ) = log . π n−1 2π n−1 2 The ratio ( n−1 n+1 ) is a well known quantity called the reflectivity R of the surface of the plate (see for example M. Born, E. Wolf; Principles of Optics, page 42 [1]). Then R = exp − 2πM(φ) .

The right hand side of the formula makes sense even for non-constant indices. Our formula could then be considered as an average measure of the reflectivity.

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Here is a simple application. Given the measure dσ, minimize the harmonic average n ˆ ∞ ∞ dσ(x) 1 dσ(x) = 1 = , n ˆ n(x) 0 0 knowing that the mean M(φ) is equal to a known value M . The problem is to find the extrema of the above integral given the constraint ∞ n(x) + 1 dσ(x) = πM . log n(x) − 1 0 Following Lagrange we introduce the parameter λ: d 1 n+1 + λ log =0 dn n n−1 1 1 1 − =0 − 2 +λ n n+1 n−1 and therefore 1 1 − 2λ which shows that the index is constant. Then n+1 = πM log n−1 n2 (x) =

−1

and finally n ˆ = n = (tanh(πM/2)) . The system consists of one plate with constant index, and this indeed minimizes n ˆ.

9. Blind spots Consider a plate P of thickness d where now the index of refraction n(x, y) ≥ 1 varies continuously with the point p = (x, y) ∈ P . We actually assume that the map p → n(p) is continuously differentiable on P . As before, the front face ∂P0 is identified with the axis x = 0 and the back face ∂Pd with x = d. To each A ∈ ∂P0 we associate the gradient of n at A denoted grad n(A). If none of the incident rays at A reach ∂Pd , A is said to be a blind point or blind spot. We shall show that if there exists such a point then the plate P cannot be too thin. Before stating our theorem we define |||grad log n||| = sup ||grad log n(p)|| . p∈P

Theorem 3. If A is a blind point then the thickness d of the plate satisfies the inequality ∂n(A) 1 ∂y 1− . d> |||grad log n||| ||grad n(A)||

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If all the spots of ∂P0 are blind then P is opaque even though sup n(p) may be finite. No rays from either side of P manage to reach the opposite side , and then ∂n(A) 1 ∂y 1 − inf . d> A∈∂P ||grad n(A)|| |||grad log n||| In this formula, ∂P = ∂P0 ∪ ∂Pd . The second hand side of the inequality in Theorem 3 is invariant if the index n is multiplied by a positive scalar. In other words, if the plate P is plunged into a medium of constant index ν and if for that new optical system A is blind , then the inequality holds. It can be shown that the inequality is sharp for ν = 0, since then all incident rays at A are reflected following the tangent grad n(A). In terms of physics this is of course nonsense since ν = 0 corresponds to a medium where the speed of light is infinitely larger than in vacuum.

10. Proof of Theorem 3 The proof requires two steps: Step 1: It is well known that if the index of refraction n varies continuously from one point to another then the luminous ray Γ satisfy the partial differential equation d dp grad n(p) = n(p) , p∈Γ ds ds where s is the arclength of Γ (see for example W. Pauli [3], p. 5 and p. 6). A simple and classical consequence is that if R(p) is the radius of curvature of Γ at p, then R(p) = ||grad log n(p)||−1 and therefore R(p) ≥ |||grad log n|||−1 . Step 2: Consider the incident ray at A which is colinear with grad n(A). As it penetrates P it is not immediately deflected since it coincides with the normal to the curve n(x, y) = n(A). Both the angles of incidence and refraction at A are equal to the acute angle ω between grad n(A) and the x−axis . Therefore ∂n(A) ∂x ω = arccos . ||grad n(A)|| Consider two circles tangent at A to grad n(A), of radius R = |||grad log n|||−1 . According to Step 1, the refracted luminous curve Γ cannot penetrate neither of the two discs if d ≤ d0 where d0 is defined on the figure. In other terms, if d ≤ d0 the luminous ray Γ necessarily reaches ∂Pd . Therefore, if A is a blind spot, d > d0 .

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637

y

d0

•

grad n(A) A

ω

x

•

•

Figure 3. Definition of d0 . To conclude the proof of the theorem we are left to compute d0 . Easily seen, d0 = R(1 − sin ω). Now ∂n(A) ∂x sin ω = sin arccos ||grad n(A)|| 2 12 ∂n(A) = 1− ||grad n(A)||−1 ∂x ∂n(A) ||grad n(A)||−1 . = ∂y

11. Appendix The functional Equation (3) met in Section 3 is quite general. Here is an example in Number Theory where the motivations are obviously entirely different and where F and H are no longer continuous. Put H(ξ) = 1ξ [ξ] ([.] represent the integer part). Define ∞ a H(ξ/) F (ξ) = =1

where the unknown real sequence a verifies lim inf a > −∞. A. E. Ingham establishes the subtle Tauberian theorem according to which ∞ lim F (ξ) = λ ⇒ a = λ ξ→∞

(see [2]).

=1

638



Acknowledgements Many thanks to Jean Creignou and particularly to Alain Yger.

References [1] M. Born, E. Wolf, Principles of Optics, Pergamon Press, sixth edition reprinted 1989. [2] A. E. Ingham, Some Tauberian theorems connected with prime numbers, Jour. London Math. Soc. 20 (1945), 171–180. [3] W. Pauli, Optics and the Theory of Electrons, Pauli Lectures on Physics 2, Dover Publ. 2000. [4] O. Perron, Algebra I, Die Grundlagen , Walter de Gruyter, Berlin 1951. ¨ [5] G. P´ olya, Uber Potenzreihen mit ganzzahligen Koeffizienten, Math. Annalen 77 (1916), 497–518. Michel Mendès France and Ahmed Sebbar Institut de Mathématiques de Bordeaux UMR 5251 Université Bordeaux I F-33405 Talence Cedex France e-mail: [email protected] [email protected] Communicated by Vincent Rivasseau. Submitted: December 20, 2007. Accepted: February 20, 2008.



Singular Yamabe Metrics and Initial Data with Exactly Kottler–Schwarzschild–de Sitter Ends Piotr T. Chru´sciel and Daniel Pollack Abstract. We construct large families of initial data sets for the vacuum Einstein equations with positive cosmological constant which contain exactly Delaunay ends; these are non-trivial initial data sets which coincide with those for the Kottler–Schwarzschild–de Sitter metrics in regions of infinite extent. From the purely Riemannian geometric point of view, this produces complete, constant positive scalar curvature metrics with exact Delaunay ends which are not globally Delaunay. The ends can be used to construct new compact initial data sets via gluing constructions. The construction provided applies to more general situations where the asymptotic geometry may have non-spherical cross-sections consisting of Einstein metrics with positive scalar curvature.

1. Introduction There exists very strong evidence suggesting that we live in a world with strictly positive cosmological constant Λ [38, 42]. This leads to a need for a better understanding of the space of solutions of Einstein equations with Λ > 0. The most general method available for constructing such solutions proceeds by solving a Cauchy problem [2, 10, 21]. In view of the general relativistic constraint equations this, subsequently, requires understanding the corresponding collection of initial data sets. In particular one is led to the question of boundary conditions satisfied by the fields. When Λ vanishes a natural set of boundary conditions arises from the obvious model solution – the Minkowski space-time. A tempting further restriction is then the requirement of a well defined and finite total mass, leading to a well understood set of asymptotic boundary conditions [1, 5, 11, 34]. When Λ > 0 the question of asymptotic conditions seems to be much less clear cut. One wants to consider a class of space-times which includes all solutions of physical interest. Until there is overwhelming evidence to the contrary, “physical interest” should carry a notion of “non-singular”. The simplest possibility, widely adopted, is to assume that the Cauchy surface S is a compact manifold without boundary.

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P. T. Chru´sciel and D. Pollack


However, an appealing more general way of ensuring regularity of the initial data is to suppose that (S , g) is a complete Riemannian manifold. One would then like to understand the space of solutions of those general relativistic constraint equations with (S , g) – complete. An interesting class of asymptotic models for such initial data has already been explored in the mathematical literature, the time symmetric initial data provided by the Delaunay 1 metrics [19, 20, 41]. These describe the family of complete rotationally symmetric, conformally flat metrics with constant positive scalar curvature, and are in fact well known to general relativists as the time-symmetric slices of the Kottler–Schwarzschild–de Sitter solutions [22, 25] (however the connection between these two subjects has apparently not been previously noted). In the Riemannian geometric context, the Delaunay metrics form the local asymptotic model for isolated singularities of locally conformally flat constant positive scalar curvature metrics [8, 9, 24, 36] (in dimensions n ≤ 5 this also holds in the non-conformally flat setting [26]). The known results concerning the existence of complete constant positive scalar curvature metrics with asymptotically Delaunay ends [7, 24, 27, 30–32, 36, 37, 40] may thus be reinterpreted, via their space-time development, as the existence of space-times satisfying the Einstein field equations with a positive cosmological constant which have asymptotically Kottler– Schwarzschild–de Sitter ends. The object of this work is to point out that every constant positive scalar curvature (CPSC) asymptotically Delaunay metric is naturally accompanied by a CPSC metric with an exactly Delaunay end, and moreover these metrics may be chosen to coincide away from the end in question. Such metrics are of interest in general relativity for at least four reasons: 1. They provide, via their maximal development, a large class of space-times satisfying the Einstein field equations with a positive cosmological constant with exactly controlled geometry in the asymptotic regions; in fact the spacetime development is explicitly known in the domain of dependence of the Delaunay regions. 2. They demonstrate that the special horizon behavior, with alternating cosmological and event horizons, which is exhibited by the Kottler–Schwarzschild– de Sitter space-time, occurs in large classes of non-stationary solutions. 3. Any two metrics which carry exactly Delaunay ends with identical mass (Delaunay) parameters may be glued together using obvious identifications on the ends. (A more difficult end-to-end asymptotic gluing theorem of this sort was established by Ratzkin [37], however with exactly Delaunay ends this construction is effortless.) Thus Delaunay ends can easily be used as bridges to create wormholes, or to make connected sums of initial data sets. Wormhole constructions are already known to be possible by completely different techniques [16] in the setting of a non-positive cosmological constant. Here 1 These are also often called Fowler solutions; see §2.3 for further remarks on the history and choice of terminology used here.

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we provide such a construction for positive cosmological constants, with the added bonus of an explicit knowledge of the space-time development in the domain of dependence of the middle part of the connecting neck, which may be of arbitrary (quantized by multiples of the period of the exact Delaunay metric) length. 4. The asymptotically Delaunay metrics are uniquely characterised by a simple geometric criterion [24, 26], see Section 2.3 below. A natural setting for our considerations is provided by the generalised Kottler metrics and generalised Delaunay metrics, as described in Sections 2.2 and 2.4 below. Our gluing construction applies in this more general setting. In an accompanying paper [14], by one of us (PTC) and Erwann Delay, analogous constructions are carried out with a negative cosmological constant. With hindsight, within the family of Kottler metrics with Λ ∈ R, the gluing in the current setting is the easiest, while that in [14] is the most difficult. This is due to the fact that for Λ > 0, as considered here, one deals with one linearised operator with a one-dimensional kernel; in the case Λ = 0 the kernel is (n + 1)-dimensional; while for Λ < 0 one needs to deal with a one-parameter family of operators with (n + 1)-dimensional kernels.

2. Kottler–Schwarzschild–de Sitter space and metrics of constant positive scalar curvature with asymptotically Delaunay ends In this section we review some results concerning the Kottler–Schwarzschild–de Sitter space and CPSC metrics which are asymptotically Delaunay. In order to fix notations and conventions we start with some standard facts. Recall that initial data for the Einstein field equations with a cosmological constant Λ on an n-dimensional manifold M consist of a pair (g, K) consisting of a Riemannian metric g on M and a symmetric 2-tensor K satisfying the vacuum constraint equations (2.1) R(g) − 2Λ + |K|2g − (trg K)2 = 0 Di (K ij − trg Kg ij ) = 0

(2.2)

where R(g) is the scalar curvature (Ricci scalar) of the metric g. If one considers time-symmetric initial data, for which K ≡ 0, then these equations reduce to the requirement that g has constant scalar curvature R(g) = 2Λ. Here we restrict to the corresponds case where Λ is positive, and note that the normalization Λ = n(n−1) 2 to R(g) = n(n − 1), the scalar curvature of the standard sphere of radius one in Rn+1 . 2.1. Kottler–Schwarzschild–de Sitter metrics The Kottler–Schwarzschild–de Sitter space-time [25] metric in n + 1 dimensions, with cosmological constant Λ > 0 and mass m ∈ R may be written as ◦

ds2 = −V dt2 + V −1 dr2 + r2 h ,

where

V = V (r) = 1 −

2m r2 − 2, n−2 r

(2.3)

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where > 0 is related to the cosmological constant Λ by the formula 2Λ = ◦ n(n − 1)/2 , while h denotes the standard metric on the unit (n − 1)-sphere in Rn . To avoid a singularity lying at finite distance on the level sets of t we will assume m > 0. Equation (2.3) provides then a spacetime metric satisfying the Einstein equations with cosmological constant Λ > 0 and with well behaved spacelike hypersurfaces when one restricts the coordinate r to an interval (rb , rc ) on which V (r) is positive; such an interval exists if and only if

2 (n − 1)(n − 2)

n−2

Λn−2 m2 n2 < 1 .

(2.4)

When n = 3 this corresponds to the condition that 9m2 Λ < 1, and the case of equality is referred to as the extreme Kottler–Schwarzschild–de Sitter spacetime (for which the coordinate expression (2.3) is no longer valid). In the limit where Λ tends to zero with m held constant, the space-time metric approaches the Schwarzschild metric with mass m, and in the limit where m goes to zero with Λ held constant the metric tends to that of the de Sitter space-time with cosmological constant Λ. The breakdown of the coordinate description above at the horizons r = rb and r = rc can be handled by taking extensions [4, 22]: In fact, the Kottler– Schwarzschild–de Sitter metric admits an analytic extension (analogous to the Kruskal extension of the Schwarzschild metric) as an r-periodic metric on (t, r, θ) ∈ R × R × Sn−1 . This is most easily seen via the associated conformal CarterPenrose diagrams [22]. The time-symmetric slice t = 0 of the (extended) Kottler– Schwarzschild–de Sitter metrics are thus a one-parameter family (parameterized by their mass m) of periodic, spherically symmetric, metrics on R × Sn−1 with constant positive scalar curvature R = 2Λ. Finally note that, due to the spherical symmetry, each of these metrics is conformally flat. 2.1.1. Extreme limit. It is of some interest to enquire what happens when m → m ˚ , where m ˚ denotes the values at which equality is achieved in (2.4). In this limit rb and rc coalesce to a single value which we will denote by ˚ r. From the space-time point of view the situation is the following: recall that the CarterPenrose diagram for the maximally extended KSdS space-times with 0 < m < m ˚ is built out of diamond shaped regions corresponding to rb < r < rc , where the Killing vector ∂t is time-like, and of triangle shaped (either upright, or upside˚ the down) regions where ∂t is spacelike [22]. After passing to the limit m → m diamond-shaped regions disappear, and the resulting diagram consists of a string of triangles. The Killing vector ∂t is then spacelike everywhere, except on the degenerate horizons ˚ r = rb = rc . On the level sets of t a rather different analysis applies, this is discussed in Section 2.3.1.

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2.2. Generalised Kottler metrics All the results discussed in Section 2.1 remain valid if m = 0 and if the metric ˚ h in (2.3) is an Einstein metric on an (n−1)-dimensional manifold N n−1 with scalar curvature equal to (n − 1)(n − 2) [6]. We will refer to such metrics as generalised ◦ Kottler metrics. Note that m = 0 requires (N n−1 , h) to be the unit round metric if one does not want r = 0 to be a singularity at finite distance along the level sets of t. 2.3. Delaunay metrics The Delaunay metrics, in dimension n ≥ 3, may be defined as the (two parameter) family of metrics ◦ (2.5) g = u4/(n−2) (dy 2 + h) , ◦

where h is the unit round metric on Sn−1 , which are spherically symmetric and have constant scalar curvature R(g) = n(n − 1). Thus the functions u = u(y) > 0 must satisfy the ODE n+2 n(n − 2) n−2 (n − 2)2 u+ u = 0. (2.6) 4 4 The two parameters correspond respectively to a minimum value ε for u, with n−2 n−2 4 (2.7) 0 ≤ ε ≤ ε¯ = n

u −

(ε is called the Delaunay parameter or neck size) and a translation parameter along the cylinder. A straightforward ODE analysis (see [32]) shows that all the positive solutions are periodic. The degenerate solution with ε = 0 corresponds to the round metric on a sphere from which two antipodal points have been removed. The solution with ε = ε¯ corresponds to the rescaling of the cylindrical metric so that the scalar curvature has the desired value. Note that the Delaunay ODE was first studied by Fowler [19, 20], however the name used here and elsewhere in the literature is inspired from the analogy with the Delaunay surfaces: the complete, periodic CMC surfaces of revolution in R3 [18]. As is well known, the analogy between the “conformally flat metrics of constant positive scalar curvature” and “complete embedded CMC surfaces of in R3 ” goes far beyond this correspondence (see, e.g., [30]). Regarding the Delaunay metrics as singular solutions of the Yamabe equation on (Sn , g0 ) one has a number of uniqueness results. Among these are the facts that no solution with a single singular point exists, and that any solution with exactly two isolated singular points must be conformally equivalent to a Delaunay metric. These results can be proved by a generalization of the classical Alexandrov reflection argument (the method of moving planes), see [23]. The first general existence result for complete conformally flat metrics of constant scalar curvature with asymptotically Delaunay ends is due to Schoen [40]. Of immediate interest to us is the fact that conformally flat metrics, with constant positive scalar curvature, and with an isolated singularity of the conformal

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factor are necessarily asymptotic to a Delaunay metric [24]; in fact, in dimensions n = 3, 4, 5 the conformal flatness condition is not needed [26]. Specifically, with respect to spherical coordinates about an isolated singularity of the conformal factor, there is a half-Delaunay metric which g converges to, exponentially fast in r, along with all of its derivatives. This fact is used in [27, 31, 32, 36, 37] where complete, constant scalar curvature metrics, conformal to the round metric on S \ {p1 , . . . , pk } were studied and constructed. (This is one instance of the more general “singular Yamabe problem”.) By the uniqueness of solutions to ODEs, or otherwise, we have: Proposition 2.1. The time symmetric initial data sets for Kottler–Schwarzschild– , are precisely the Delaunay de Sitter space in spatial dimension n with Λ = n(n−1) 2 metrics with constant positive scalar curvature R = n(n − 1). This correspondence continues to hold for any choice of positive cosmological constant Λ provided that one homothetically rescales the Delaunay metrics so that R = 2Λ. Comparing (2.3) and (2.5) we find 2 dy = V −1/2 , (2.8) r = u n−2 , r dr which allows us to determine y as a function of r on any interval of r’s on which V has no zeros. 2.3.1. Extreme limit. Let m ˚ and ˚ r be as in Section 2.1.1 and suppose that 0 < m<m ˚ , denote by r∗ ∈ (rb , rc ) the value at which the maximum value V∗ of V is attained, shifting y by a constant we can assume that the corresponding value r and V∗ → 0 as y∗ = y(r∗ ) of the y coordinate in (2.6) is zero. We have r∗ → ˚ m→m ˚ , and it clearly follows from (2.8) that the correspondence y ↔ r breaks down in the limit. This singular behavior with respect to the r coordinate is of course resolved by the coordinate y of (2.5). A somewhat more explicit way of seeing this is to replace r by a new coordinate w through the formula r = r∗ + V∗ w √ √ which scales up the interval r ∈ (rb , rc ) to w ∈ ((rb − r∗ )/ V∗ , (rc − r∗ )/ V∗ ). Equations (2.8) become √ 2 V∗ dy √ , r∗ + V∗ w = u n−2 , (2.9) = √ dw (r∗ + V∗ w) V which are regular in the limit m → m ˚ . In the new coordinates we have V 1 2 ∗ h= h −→ dw2 + r∗2˚ h as m → m ˚, dr + r2˚ dw2 + (r∗ + V∗ w)2˚ V V with the limit being uniform over compact sets of the w coordinate. This shows in which sense the space sections of the KSdS metrics approach a cylindrical geometry in the extreme limit. It should, however, be borne in mind that the space-time picture of Section 2.1.1 is rather different.

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2.4. Generalised Delaunay metrics Similarly to Section 2.2, the analysis presented at the beginning of Section 2.3 remains valid when the parameter is positive and if the metric ˚ h in (2.5) is an Einstein metric on an (n − 1)-dimensional manifold N n−1 with scalar curvature equal to (n − 1)(n − 2) . We refer to the resulting metrics as generalised Delaunay ◦ metrics. As before, = 0 requires (N n−1 , h ) to be the unit round sphere if one wants to avoid a singularity at the set u(y) = 0. 2.5. Complete metrics with constant positive scalar curvature and asymptotically Delaunay ends Conformal gluing constructions for constant scalar curvature metrics g˜ = u4/(n−2) g with R(˜ g ) = R(g) = n(n − 1) have given rise to a wide variety of such metrics with asymptotically Delaunay ends. The linearisation of this equation about a solution leads to the operator Lg = Δg + n. The key nondegeneracy assumption of any conformal gluing construction is that Lg is surjective when acting on appropriately defined function spaces. The Delaunay metrics themselves are nondegenerate in this sense [32], moreover the solutions constructed by Mazzeo-Pacard on S \ {p1 , . . . , pk } [27] are non-degenerate. On the other hand, the standard metric on the n-sphere, (S, g0 ), is degenerate due to the fact that the restrictions of the linear functions in Rn+1 span an (n + 1)-dimensional co-kernel of Lg0 . In addition to the original construction of Schoen [40], the constructions of [31] and [37] use non-degenerate solutions as building blocks to produce new non-degenerate solutions. All of the constructions alluded to above are in the setting where the metrics are locally conformally flat everywhere. This is clearly not necessary. A general conformal gluing theorem was established by Byde [7]: Theorem 2.2 (Byde [7]). Let (M, g) be a compact Riemannian manifold, possibly with boundary, of constant scalar curvature n(n−1), which is non-degenerate in the sense described above, and let x0 ∈ int(M) be a point in a neighborhood of which g is conformally flat. Then there is a constant ρ0 and a one parameter family of complete metrics gρ on M \ {x0 } defined for ρ ∈ (0, ρ0 ), conformal to g, with constant scalar curvature n(n − 1). Moreover, each gρ is asymptotically Delaunay and gρ → g uniformly on compact sets in M \ {x0 } as ρ → 0. This result is exactly analogous to results for constant mean curvature surfaces established in [28, 29]. Byde goes further and shows how one can also glue asymptotically Delaunay ends onto non-compact, non-degenerate solutions (though without the uniform convergence to the original metric away from the gluing locus). Note that, in light of Proposition 2.1, all of these results, and others, on the existence of CPSC metrics with asymptotically Delaunay ends have an immediate reinterpretation, after considering the maximal development of the initial data set, as statements regarding the existence of space-times with asymptotically Kottler– Schwarzschild–de Sitter ends.

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3. Perturbation to exactly Delaunay, or generalised Delaunay ends The gluing construction of Corvino-Schoen [17, 39] (compare [13]) generalises to the positive cosmological constant setting as follows: Theorem 3.1. Let N n−1 be compact, let (M, g) satisfy R(g) = n(n − 1), and suppose that M contains an end E ≈ [0, ∞) × N n−1 on which g is asymptotic to a n−2 4 , together with derivageneralised Delaunay metric ˚ g =˚ g ε , with 0 < < ( n−2 n ) tives up to order four. Then for every δ > 0 there is an ε satisfying |ε − ε | < δ and a metric g with R(g ) = n(n − 1), which differs from g only far away on E, and which is a generalised Delaunay metric with Delaunay parameter ε on the complement E of a compact subset of E. Remark 3.2. By taking the maximal, globally hyperbolic, space-time development of the time-symmetric initial data set (M, g ), we obtain a solution of the vacuum Einstein equations with cosmological constant Λ = n(n−1)/2 such that the metric on the domain of dependence of the end E is isometric to a subset of the Kottler– Schwarzschild–de Sitter space-time. Proof. Let g asymptote to a generalised Delaunay metric ˚ gε on R × N n−1 . We can write ˚ g ε as ◦

˚ g = dx2 + e2f (x) h , ◦

(3.1)

where h is an Einstein metric on N n−1 , normalised as described above. Consider a connected component of the set on which V > 0, where V√ is the function appearing in (2.3) for the metric ˚ g ε . It follows from (2.3) that V is the normal component of the Killing vector ∂t on the level sets of t. From the general results in [33] it follows that any such function, for a static space-time, √ solves equation (3.2) below. It further follows from the analysis in [12] that V ˚, by can be smoothly continued to a real-analytic function on M , which we call N ˚ and V = ±N ˚2 changing signs across the zero level sets of V . Furthermore, both N are functions of x only in the representation (3.1) of ˚ g. Let T = T (ε) be the period of f , and let Ωi = [iT + σ, iT + T + σ] × N n−1 , where σ will be chosen below. Let (R × N n−1 ,˚ gε ) be a generalised Delaunay metric with parameter ε near ε, |ε − ε | < δ. Let gε be a metric on Ωi obtained by interpolating between g and ˚ g ε using any i-independent cut-off function smoothly varying from zero to one. The cut-off should be supported away from the end-points of the interval [i, i + T ]. To achieve constant scalar curvature we will, first, correct the metric gε to a new metric g˜ε using the operator L of [13], as restricted to time-symmetric data, so that Y ≡ 0 there. The correction will be of the order of the perturbation introduced, namely O(|ε − ε |). This will, however, not quite solve the problem because the operator L at g = ˚ gε has a cokernel, which consists of functions

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solving the “static KIDs equation”: Di Dj N = N Rij + Δg N gij .

(3.2)

We thus have to understand the space of solutions of (3.2): Lemma 3.3. Let g = ˚ g be a generalised Delaunay metric as in (3.1). n−1 ˚ 1. If (N , h) is the round sphere and if m = 0, then the space of solutions ˚ together with functions of the of (3.2) is (n + 1)-dimensional, spanned by N f i i form e α , where f is as in (3.1) and α is the restriction of the Euclidean coordinate xi to Sn−1 under the standard embedding Sn−1 → Rn . ˚. 2. Otherwise all solutions of (3.2) are proportional to N ◦

Proof. Let v A be coordinates on the level sets of x; we have ΓxAB = −f hAB , A A ˚A ˚A Γxxj = 0, ΓA xB = f δB , and ΓBC = ΓBC , where the ΓBC ’s are the Christoffel ◦ ˚ depends only upon x, and satisfies (3.2), we immediately symbols of h. Since N find RxA = 0 away from the zero-set of V ; by continuity this holds everywhere (this conclusion could also have been reached directly from the warped product structure of ˚ g). But then (3.2) gives 0 = Dx DA N = ef ∂x (e−f ∂A N ) , hence ˚(v A ) + M ˚ (x) , N (x, v A ) = ef (x) P (3.3) A ˚=P ˚(v ), M ˚ =M ˚ (x). for some functions P Set ˚, α := N/N ˚. Since both N and N ˚ satthen α is smooth away from the zero-level sets of N isfy (3.2) one finds that α is a solution of the equation ˚ + Dj αDi N ˚ = 0. ˚Di Dj α + Di αDj N N

(3.4)

˚ = 0 we obtain From DA N DA DB α = 0 . −ΓxAB

(3.5)

◦

Let λAB = = f hAB be the second fundamental form of the level sets of x, (3.5) can be rewritten as ◦

D˚A D˚B α + αx λAB ≡ D˚A D˚B α + αx f hAB = 0 ,

(3.6)

=:ϕ

◦

where D˚ is the covariant derivative operator of the metric h, and αx = ∂x α. Applying D˚B to (3.6) and commuting derivatives one obtains (recall that the ◦ ◦ Ricci tensor of h equals (n − 2)h) D˚A D˚B D˚B α + (n − 2)α − ϕ = 0 .

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Contracting A with B in (3.6) we find ϕ = D˚B D˚B α/(n − 1), which allows us to conclude that there exists a constant C such that D˚B D˚B (α + C) = −(n − 1)(α + C) . (3.7) ◦

Suppose, first, that (N n−1 , h) is the unit round sphere. Equation (3.7) shows that solutions of (3.6) are linear combinations of the constant function α0 = 1 and of the functions αi = xi |Sn−1 , where xi is a canonical coordinate in Rn , with Sn−1 being embedded in Rn in the obvious way. Hence, there exist functions λμ (x), μ = 0, . . . , n, such that (3.8) α(x, v A ) = λμ (x)αμ (v A ) . f ˚−1˚ λi , without however But then (3.3) implies λi (x) = e N λi for some constants ˚ imposing any constraints on λ0 (x) which remains undetermined so far. ◦ On the other-hand, if (N n−1 , h) is not the unit round sphere, then by a theorem of Obata [35] the function α does not depend upon v A , so that (3.8) again holds with αi ≡ 0 and α0 = 1. Inserting (3.8) into (3.4) with ij = xx one finds

2 −1 f (x)˚ i A ˚ ˚ ∂x N (x)∂x λ0 (x) + N (x)e λi α (v ) = 0. (3.9) In order to analyse this equation, it is useful to compare (2.3) with (3.1) to conclude that 1 1 dr = ± = ± . (3.10) ˚ dx r2 N 1 − r2m n−2 − 2 ˚2 ∂x = ±N ˚3 ∂r and We then have N

⎡ ⎤

r ˚ ˚2 ∂x λ0 + N ˚−1 ef ˚ ˚3 ∂r ⎣λ0 + ±N λi αi (v A ) = N λi αi (v A )⎦ 2 1 − r2 − r2m n−2 mn ˚ ˚3 ∂r λ0 + 1 − λi αi (v A ) . =N rn−2 So (3.9) will hold if and only if this is a function which depends at most upon v A . Hence λ0 is a constant and ˚ λi = 0 unless m = 0, in which case the ˚ λi ’s are arbitrary, as desired.

Returning to the proof of Theorem 3.1, the metric g˜ε is obtained by solving the equation R(˜ gε ) − n(n − 1) Ωi ∈ (Im L)⊥ using the implicit function theorem, compare [13, Theorem 5.9]; this can be done on Ωi for all i large enough. As already mentioned, the perturbation introduced is O(|ε − ε |). In view of Lemma 3.3, the obstruction to solving the problem is thus the vanishing of ˚ R(˜ N gε ) − R(˚ gε ) dμ˚ (3.11) g, Ωi

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˚ is the static KID associated with the generalised Delaunay metric ˚ where N gε . We need the following identity, from [15]: ˚(Rg − Rb ) = ∂i Ui (N ˚) + det g (ρ + Q) , det g N (3.12) where

i[k j]l ˚g j]k ejk , ˚) := 2 det g N ˚g g D˚j gkl + D[i N Ui (N ˚ − Δb N ˚bij g ik g j ek , ˚Ric(b)ij + D˚i D˚j N ρ := − N ˚(g ij − bij + g ik g j ek )Ric(b)ij + Q . Q := N

(3.13) (3.14) (3.15)

Brackets over a symbol denote anti-symmetrisation, with an appropriate numerical factor (1/2 in the case of two indices). Here Q denotes an expression which is bilinear in e ≡ eij dxi dxj := (gij − bij )dxi dxj , and in D˚k eij , where D˚ denotes now the covariant derivative operator of the met˚, dN ˚ and HessN ˚, with coefficients which are constants in an ON ric b, linear in N frame for b. The idea behind this calculation is to collect all terms in Rg that contain second derivatives of the metric in ∂i Ui ; in what remains one collects in ρ the terms which are linear in eij , while the remaining terms are collected in Q; one should note that the first term at the right-hand-side of (3.15) does indeed not contain any terms linear in eij when Taylor expanded at gij = bij . Note that ρ ˚. So the integrand is quadratic in eij , up to vanishes when b = ˚ g ε by choice of N terms O(|ε − ε |)eij , and up to the divergence which produces a boundary term Ui dSi . ∂Ωi

For our next lemma it is convenient to write two generalised Delaunay metrics g and b as ◦ ◦ dr2 dr2 g = 2 + r2 h , b = + r2 h . (3.16) 2 ˚ N N We claim: Lemma 3.4. Let g and b be two generalised Delaunay metrics with mass parameters ˚(r) = 0 and N (r) = 0. If {r} × N n−1 is positively m and m0 . Let r be such that N oriented, then ˚N −1 (m − m0 ) . Ui dSi = 2ωn−1 (n − 1)N (3.17) {r}×N n−1

where ωn−1 is the volume of N n−1 . Proof. Let us denote by Γijk the Christoffel symbols of the metric b. We have ◦ ˚ N ˚, Γr = −rN ˚2 hAB , ΓA = ˚ ˚A Γrrr = −∂r N/ ΓA AB BC BC (where, as before, the ΓBC ’s are

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◦

−1 A the Christoffel symbols of the metric h), ΓA δB , while the remaining Γ’s rB = r −2 −2 2 ˚ vanish. It holds that e = (N − N )dr , from which one easily finds ◦ r −1 ˚ U = 2(n − 1)N N (m − m0 ) det h , (3.18)

and the result follows by integration.

We are ready to show that one can choose ε – equivalently m – so that the obstruction vanishes. So we consider the integral (3.11). We wish to use (3.12) with g = g˜ε . Note that the integration in (3.11) is taken with respect to the measure dμ˚ g , while (3.12) involves dμg . The difference between the two volume integrals comes thus with a prefactor O(|ε−ε |), and produces an error term which is O((ε − ε )2 ): ˚ R(˜ N gε ) − R(˚ g ε ) dμ˚ g Ωi ˚ R(˜ N gε ) − R(˚ = g ε ) dμg + O (ε − ε )2 Ω i = Ui dSi − Ui dSi + O (ε − ε )2 . (3.19) {i+T }×N n−1

{i}×N n−1

We now choose σ in the definition of Ωi so that the number ˚|x=σ λ := 4ωn−1 (n − 1)N does not vanish; note that λ equals, up to O(|ε − ε |), the number in front of (m − m0 ) in (3.17). Since g approaches ˚ g together with its first derivatives as i goes to infinity, the first integral in the last line of (3.19) is o(1), where o(1) tends to zero as i tends to infinity. By Lemma 3.4 the second integral in the last line ˚ ) = O(|ε − ε |), where m is the mass parameter of γε of (3.19) equals λ(m − m while m ˚ is that of ˚ g . We infer that ˚ R(˜ N gε ) − R(˚ g ε ) dμ˚ ˚ ) + o(1) + O (ε − ε )2 . g = λ(m − m Ωi

Clearly this can be made positive or negative when i is large enough by choosing ε appropriately; by continuity there exists an ε which makes the integral vanish, and the result is proved.

4. Concluding remarks Our work leads naturally to the following questions: 1. In light of the results in [26] one should expect that, at least for n = 3, 4, 5, a construction similar to Byde’s [7] could be carried out without any assumption of conformal flatness in a neighborhood of the omitted point (which forms the end of the resulting complete metric). Moreover, with or without the conformally flat condition, it should be straightforward to iterate Byde’s construction to produce any number of asymptotically Delaunay ends. If one

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could add asymptotically Delaunay ends at any chosen set of points in any positive constant scalar curvature manifold, then our analysis here could then be used to replace them with exactly Delaunay ends. (Furthermore, all of this should be doable via a local deformation of the metric near points without static KIDs, using the techniques of [3, 13, 16].) Alternatively, can one generically deform a constant positive scalar curvature metric, keeping the scalar curvature fixed, to a metric which is conformally flat near a set of prescribed points? Proposition 4.1 of [41] could perhaps be used as an intermediate step here. 2. In Byde’s construction, or in a variation thereof as just suggested, can one ensure that the range of masses of the resulting Delaunay ends covers an interval of the form (0, ), for some > 0? This is a natural condition which has appeared elsewhere as a necessary hypothesis (see, e.g., [37]). Now, it is clear that the masses of our exactly Delaunay ends are continuous functions of the initial mass. Given any two points with the associated families of exactly Delaunay ends, one could then always adjust the masses to be the same, ensuring that the ends can be glued together. 3. We did not carry out the gluing in situations when the metric g approaches h along the asymptotic end; such metrics arise a cylindrical metric dy 2 + ˚ in black-hole space-times with degenerate horizons. This deserves further attention. 4. It would be of interest to extend the current gluings to general relativistic initial data with non-vanishing extrinsic curvature.

Acknowledgements D. Pollack would like to thank Mihalis Dafermos for first raising the question of whether space-times with Kottler–Schwarzschild–de Sitter horizon behavior exist more generally, and Frank Pacard for a number of illuminating discussions.

References [1] R. Bartnik, The mass of an asymptotically flat manifold, Commun. Pure Appl. Math. 39 (1986), 661–693. [2] R. Bartnik and J. Isenberg, The constraint equations, The Einstein equations and the large scale behavior of gravitational fields, Birkh¨ auser, Basel, 2004, pp. 1–38. MR2098912 (2005j:83007). [3] R. Beig, P. T. Chru´sciel, and R. Schoen, KIDs are non-generic, Ann. H. Poincaré 6 (2005), 155–194, arXiv:gr-qc/0403042. MR2121280 (2005m:83013). [4] R. Beig and J. M. Heinzle, CMC-slicings of Kottler–Schwarzschild-de Sitter cosmologies, Commun. Math. Phys. 260 (2005), 673–709. MR2183962 (2006k:83029). ´ Murchadha, The Poincaré group as the symmetry group of canon[5] R. Beig and N. O ical general relativity, Ann. Phys. 174 (1987), 463–498.

652



[6] D. Birmingham, Topological black holes in anti-de Sitter space, Class. Quantum Grav. 16 (1999), 1197–1205, arXiv:hep-th/9808032. [7] A. Byde, Gluing theorems for constant scalar curvature manifolds, Indiana Univ. Math. Jour. 52 (2003), 1147–1199. MR2010322 (2004h:53049). [8] L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math. 42 (1989), 271–297. MR982351 (90c:35075). [9] C. C. Chen and C. S. Lin, Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent, Duke Math. Jour. 78 (1995), 315–334. MR1333503 (96d:35035). [10] Y. Choquet-Bruhat and J. York, The Cauchy problem, General Relativity (A. Held, ed.), Plenum Press, New York, 1980, pp. 99–172. [11] P. T. Chru´sciel, A remark on the positive energy theorem, Class. Quantum Grav. 33 (1986), L115–L121. [12] P. T. Chru´sciel, On analyticity of static vacuum metrics at non-degenerate horizons, Acta Phys. Pol. B36 (2005), 17–26, arXiv:gr-qc/0402087. [13] P. T. Chru´sciel and E. Delay, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. de France. 94 (2003), vi+103, arXiv:gr-qc/0301073v2. MR2031583 (2005f:83008). [14] P. T. Chru´sciel and E. Delay, Gluing constructions for asymptotically hyperbolic manifolds with constant scalar curvature, (2007), arXiv:0710.xxx [gr-qc]. [15] P. T. Chru´sciel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), 231–264, arXiv:dg-ga/0110035. MR2038048 (2005d:53052). [16] P. T. Chru´sciel, J. Isenberg, and D. Pollack, Initial data engineering, Commun. Math. Phys. 257 (2005), 29–42, arXiv:gr-qc/0403066. [17] J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys. 214 (2000), 137–189. MR1794269 (2002b:53050). [18] C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, Jour. de Math. 6 (1841), 309–320. [19] R. H. Fowler, The form near infinity of real continuous solutions of a certain differential equation of the second order, Quart. Jour. Pure Appl. Math. 45 (1914), 289–349. [20] R. H. Fowler, Further studies of Emden’s and similar differential equations, Quart. Jour. Math., Oxf. Ser. 2 (1931), 259–288. [21] H. Friedrich and A. Rendall, The Cauchy problem for the Einstein equations, Einstein’s field equations and their physical implications, Lecture Notes in Phys., vol. 540, Springer, Berlin, 2000, pp. 127–223. MR1765130 (2001m:83009). [22] G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D15 (1977), 2738–2751. [23] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243. MR544879 (80h:35043).

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[24] N. Korevaar, R. Mazzeo, F. Pacard, and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math. 135 (1999), 233– 272. MR1666838 (2001a:35055). ¨ [25] F. Kottler, Uber die physikalischen Grundlagen der Einsteinschen Gravitationstheorie, Annalen der Physik 56 (1918), 401–462. [26] F. C. Marques, Isolated singularities of solutions of the Yamabe equation, Calc. of Var. (2007), doi:10.1007/s00526–007–0144–3. [27] R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities, Duke Math. J. 99 (1999), 353–418. MR1712628 (2000g:53035). [28] R. Mazzeo, F. Pacard, and D. Pollack, Connected sums of constant mean curvature surfaces in Euclidean 3 space, J. Reine Angew. Math. 536 (2001), 115–165. MR1837428 (2002d:53020). [29] R. Mazzeo, F. Pacard, and D. Pollack, The conformal theory of Alexandrov embedded constant mean curvature surfaces in R3 , Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 525–559. MR2167275 (2006g:53009). [30] R. Mazzeo and D. Pollack, Gluing and moduli for noncompact geometric problems, Geometric theory of singular phenomena in partial differential equations (Cortona, 1995), Sympos. Math., XXXVIII, Cambridge Univ. Press, Cambridge, 1998, pp. 17– 51. MR1702086 (2000i:53058). [31] R. Mazzeo, D. Pollack, and K. Uhlenbeck, Connected sum constructions for constant scalar curvature metrics, Topol. Methods Nonlinear Anal. 6 (1995), 207–233. MR1399537 (97e:53076). [32] R. Mazzeo, D. Pollack, and K. Uhlenbeck, Moduli spaces of singular Yamabe metrics, Jour. Amer. Math. Soc. 9 (1996), 303–344. MR1356375 (96f:53055). [33] V. Moncrief, Space-time symmetries and linearization stability of the Einstein equations. II, Jour. Math. Phys. 17 (1976), 1893–1902. ´ Murchadha, Total energy momentum in general relativity, Jour. Math. Phys. [34] N. O 27 (1986), 2111–2128. [35] M. Obata, Certain conditions for a Riemannian manifold to be iosometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340. MR0142086 (25 #5479). [36] D. Pollack, Compactness results for complete metrics of constant positive scalar curvature on subdomains of S n , Indiana Univ. Math. Jour. 42 (1993), 1441–1456. MR1266101 (95c:53052). [37] J. Ratzkin, An end to end gluing construction for metrics of constant positive scalar curvature, Indiana Univ. Math. Jour. 52 (2003), 703–726. MR1986894 (2004m:53066). [38] A. G. Riess et al., New Hubble Space Telescope discoveries of type Ia Supernovae at z > 1: Narrowing constraints on the early behavior of dark energy, Astroph. Jour. 659 (2007), 98–121, arXiv:astro-ph/0611572. [39] R. M. Schoen, Vacuum spacetimes which are identically Schwarzschild near spatial infinity, talk given at the Santa Barbara Conference on Strong Gravitational Fields, June 22–26, 1999, http://doug-pc.itp.ucsb.edu/online/gravity_c99/schoen/.

654



[40] R. M. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Commun. Pure Appl. Math. 41 (1988), 317– 392. MR929283 (89e:58119). [41] R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987), Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120– 154. MR994021 (90g:58023). [42] W. M. Wood-Vasey et al., Observational constraints on the nature of the dark energy: First cosmological results from the essence supernova survey, (2007), arXiv:astroph/0701041. Piotr T. Chru´sciel LMPT Fédération Denis Poisson Tours France Mathematical Institute and Hertford College Oxford United Kingdom e-mail: [email protected] Daniel Pollack University of Washington Department of Mathematics Box 354350 Seattle, WA 98195-4350 USA e-mail: [email protected] Communicated by Sergiu Klainerman. Submitted: October 19, 2007. Accepted: February 11, 2008.

Ann. Henri Poincaré 9 (2008), 655–683 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/040655-29, published online May 26, 2008 DOI 10.1007/s00023-008-0363-y


Dimensional Regularization and Renormalization of Non-Commutative Quantum Field Theory Razvan Gur˘au and Adrian Tanas˘a Abstract. Using the recently introduced parametric representation of noncommutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized Φ4 4 model on the Moyal space.

1. Introduction and motivation Non-commutative geometry (see [6]) is one of the most appealing frameworks for the quantification of gravitation. Quantum field theory (QFT) on these type of spaces, called non-commutative quantum field theory (NCQFT) – for a general review see [12, 36] – is now one of the most appealing candidates for new physics beyond the Standard Model. Also, NCQFT arises as the effective limit of some string theoretical models [7, 34]. Moreover, NCQFT is well suited to the description of the physics in background fields and with non-local interactions, like for example the fractional quantum Hall effect [25, 30, 35]. However, naive NCQFT suffers from a new type of non renormalizable divergences, known as the ultraviolet (UV)/ infrared (IR) mixing. The simplest example of this kind of divergences is given by the nonplanar tadpole: it is UV convergent, but inserting it an arbitrary number of times in a loop gives rise to IR divergences. Interest in NCQFT has been recently revived with the introduction of the Grosse–Wulkenhaar scalar Φ4 4 model, in which the UV/IR mixing is cured: the model is renormalizable at all orders in perturbation theory [17, 18]. The idea of Grosse–Wulkenhaar was to modify the kinetic part of the action in order to satisfy the Langmann–Szabo duality [28] (which relates the infrared and ultraviolet regions). We refer to this modified theory as the vulcanized Φ4 4 model.

656

R. Gur˘ au and A. Tanas˘ a


A general proof, using position space and multiscale analysis, has then been given in [21], and the parametric representation of this model was computed in [23]. Furthermore, it was recently proved that that the vulcanized Φ4 4 is better behaved than the commutative φ44 model: it does not have a Landau ghost [10, 11, 20]. In commutative QFT dimensional renormalization is the only scheme which respects the symmetries of gauge theories see [2, 3, 5]. It also is the appropriate setup for the Connes–Kreimer Hopf algebra approach to renormalization (see [8, 9, 27] for the case of commutative QFT). A second class of renormalizable NCQFT exists. These models, called covariant, are characterized by a propagator which decays in position space as x−y tends to infinity (like the Grosse–Wulkenhaar propagator of (2.8)) but it oscillates when x + y goes to infinity, rather than decaying. In this class of NCQFT models enters the non-commutative Gross–Neveu model and the Langmann–Szabo–Zarembo model [29]. The non-commutative orientable Gross–Neveu model was proven to be renormalizable at all orders in perturbation theory [38]. The parametric representation was extended to this class of models [33]. For a general review of recent developments in the field of renormalizable NCQFT see [32]. The parametric representation introduced in [23] is the starting point for the dimensional regularization and renormalization performed in this paper. Our proof follows that of the commutative Φ4 4 model, as presented in [2, 3]. This paper is organized as follows. Section 2 is a summary of the parametric representation of the vulcanized Φ4 4 model. The non-commutative equivalent HUG and HVG of the Symanzik polynomials UG and VG are recalled. In Section 3 we prove the existence in the polynomial HUG of some further leading terms in the ultraviolet (UV) regime. This is an improvement of the results of [23], needed to correctly identify the meromorphic structure of the Feynman amplitudes. In Section 4 we prove the factorization properties of the Feynman amplitudes. These properties are needed in order to prove that the pole extraction is equivalent to adding counterterms of the form of the initial lagrangean. This factorization is essential for the definition of a coproduct Δ necessary for the implementation of a Hopf algebra structure in NCQFT [37]. Section 5 uses the results of the previous sections to perform the dimensional regularization, prove the counterterm structure for NCQFT and complete the dimensional renormalization program. Section 6 is devoted to some conclusion and perspectives.

2. The non-commutative model In this section we give a brief overview of the Grosse–Wulkenhaar Φ4 model. Our notations and conventions as well as some notions of diagrammatics and the results of the parametric representation follow [23]. To define the Moyal space of dimension D, we introduce the deformed Moyal product on RD so that [xμ , xν ] = iΘμν ,

(2.1)

Vol. 9 (2008) Dimensional Regularization and Renormalization of NCQFT

where the matrix Θ is

⎛

0 ⎜ −θ ⎜ ⎜ Θ=⎜ ⎜ ⎝

657

⎞

θ 0

0 ..

. 0 −θ

0

⎟ ⎟ ⎟ ⎟. ⎟ θ ⎠ 0

(2.2)

The associative Moyal product of two functions f and g on the Moyal space writes

dD k D 1 (f g)(x) = Θ · k g(x + y)eık · y d y f x + (2π)D 2 −1 1 = D (2.3) dD ydD z f (x + y)g(x + z)e−2ıyΘ z . π | det Θ| The Euclidian action introduced in [17] is

1 Ω2 1 2 4 μ μ ∂μ φ ∂ φ + (˜ xμ φ) (˜ x φ) + m φ φ + φ φ φ φ , (2.4) S= d x 2 2 2 where x˜μ = 2(Θ−1 )μν xν .

(2.5)

The propagator of this model is the inverse of the operator ˜2 . −Δ + Ω2 x

(2.6)

The results we establish here hold for orientable models (in the sense of Subsection 2.1). This corresponds to a Grosse–Wulkenhaar model of a complex scalar field

1 ¯ μ Ω2 4 μ ¯ ¯ ¯ ∂μ φ ∂ φ + (˜ xμ φ) (˜ (2.7) x φ) + φ φ φ φ . S= d x 2 2 ˜ = 2Ω/θ, the kernel of the propagator is (Lemma 3.1 of [24]) Introducing Ω ∞ ˜ ˜ ˜ 2 2 Ωdα Ω α − Ω coth( α 2 )(x−y) − 4 tanh( 2 )(x+y) . C(x, y) = (2.8) D/2 e 4 0 2π sinh(α) Using (2.3) the interaction term in (2.7) leads to the following vertex contribution in position space (see [21]) δ(x1 − x2 + x3 − x4 )e2i

1≤i<j≤4 (−1)

i+j+1

xi Θ−1 xj

.

(2.9)

with x1 , . . . , x4 the 4-vectors of the positions of the 4 fields incident to the vertex. To any such vertex V one associates a hypermomentum pV using the relation dpV pV σ(x1 −x2 +x3 −x4 ) δ(x1 − x2 + x3 − x4 ) = e . (2.10) (2π)4

658



Figure 1. The first Filk move: the line 2 is reduced and the 2 vertices merge. 2.1. Some diagrammatics for NCQFT; orientability In this subsection we introduce some useful conventions and definitions, some of them used in [21] and [24] but also some new ones. Let a graph G with n(G) vertices, L(G) internal lines and F (G) faces. The Euler characteristic of the graph is 2 − 2g(G) = n(G) − L(G) + F (G) ,

(2.11)

where g(G) ∈ N is the genus of the graph. Graphs divide in two categories, planar graph with g(G) = 0, and non-planar graphs with g(G) > 0. Let also B(G) denote the number of faces broken by external lines and N (G) be number of external points of the graph. The “orientable” form (2.9) of the vertex contribution of our model allows us to associate “+” sign to a corner φ¯ and a “−” sign to a corner φ of the vertex. These signs alternate when turning around a vertex. As the propagator allways relates a φ¯ to a φ, the action in (2.7) has orientable lines, that is any internal line joins a “−” corner to a “+” corner 1 . Consider a spanning tree T in G. In has n − 1 lines and the remaining L − (n − 1) lines form the set L of loop lines. Amongst the vertices V one chooses a special one V¯G , the root of the tree. One associates to any vertex V the unique tree line which hooks to V and goes towards the root. We introduce now some topological operations on the graph which allow one to reexpress the oscillating factors coming from the vertices of the graph G. Let a tree line in the graph = (i, j) and its endpoints i and j. Suppose it connects to the root vertex V¯G at i and to another vertex V at j. In Figure 1, 2 is the tree line, y4 is i and x1 is j. The first Filk move, inspired by [13], consists in removing such a line from the graph and gluing the two vertices together respecting the ordering. Thus the point i on the root vertex is replaced by the neighbors of j on V . This is represented in Figure 1 where the new root vertex is y2 , y3 , x4 , x1 , x2 , y1 . 1 The

orientability of our theory allows us to simplify the proofs. It should however be possible, although tedious, to follow the same procedure for the non orientable model.


659

Figure 2. Two rosettes obtained by contracting a tree via the first Filk. Note that the number of faces or the genus of the graph do not change under this operation. A technical point to be noted here is that one must chose the field j on the vertex V to be either the first (if the line enters V ) or the last (if the line exits V ) in the ordering of V . Of course this is allways possible by the use of the δ functions in the vertex contribution. Iterating this operation for the n − 1 tree lines, one obtains a single final vertex with all the loop lines hooked to it – a rosette (see Figure 2). The rosette contains all the topological information of the graph. If no two lines cross (on the left in Figure 2) the graph is planar. If on the contrary we have at least a crossing (on the right in Figure 2) the graph is non planar (for details see [23]). For a nonplanar graph we define a nice crossing in a rosette as a pair of lines such that the end point of the first is the successor in the rosette of the starting point of the other. A genus line of a graph is a loop line which is part of a nice crossing on the rosette (lines 2 and 4 on the right of Figure 2). In the sequel we are interested in performing this operation in a way adapted to the scales introduced by the Hepp sectors: we perform the first Filk move only for a subgraph S (we iterate it only for a tree in S). Thus, the subgraph S will be shrunk to its corresponding rosette inside the graph G. If S is not primitively divergent we have a convergent sum over its associated Hepp parameter. We will prove later that S is primitively divergent if and only if g(S) = 0, B(S) = 1, N (S) = 2, 4. For primitively divergent subgraphs the first Filk move above shrinks S to a Moyal vertex inside the graph G. For example, consider the graph G of Figure 3 and its divergent sunshine subgraph S given by the set of lines 4 , 5 and 6 . Under the first Filk move for the subgraph S, G will have a rosette vertex insertion like in Figure 4, Denote G − S graph G with its subgraph S erased (see Figure 5). It becomes the graph G/S with a Moyal vertex like in Figure 6.

660



Figure 3. A graph containing a primitive divergent subgraph given by the lines 4 , 5 and 6 .

Figure 4. The graph G with S reduced to a rosette.

Figure 5. The graph G − S obtained by erasing the lines and vertices of the primitive divergent subgraph S. In the commutative case, this operation corresponds to the shrinking of S to a point: it represents the “Moyality” (instead of locality) of the theory. 2.2. Parametric representation for NCQFT In this subsection we recall the definitions and results obtained in [23] for the parametric representations of the model defined by (2.7). First let us recall that, when considering the parametric representation for commutative QFT, one has translation invariance in position space. As a consequence of this invariance, the


661

Figure 6. The graph G/S obtained by shrinking to a Moyal vertex the sunshine primitive divergent subgraph S. first polynomial vanishes when integrating over all internal positions. Therefore, one has to integrate over all internal positions (which correspond to vertices) save one, which is thus marked. However, the polynomial is a still a canonical object, i.e., it does not depend of the choice of this particular vertex. As stated in [23], in the non-commutative case translation invariance is lost (because of non-locality). Therefore, one can integrate over all internal positions and hypermomenta. However, in order to be able to recover the commutative limit, we also mark a particular vertex V¯ ; we do not integrate on its associate hypermomenta pV¯ . This particular vertex is the root vertex. Because there is no translation invariance, the polynomial does depend on on the choice of the root; however the leading ultraviolet terms do not. We define the (L×4)-dimensional incidence matrix εV for each of the vertices V . Since the graph is orientable (in the sense defined in Subsection 2.1 above) we can choose εVi = (−1)i+1 , if the line hooks to the vertex V at corner i.

(2.12)

Let also V ηi = |εVi | ,

V = 1, . . . , n ,

= 1, . . . , L and i = 1, . . . , 4 .

(2.13)

From (2.12) and (2.13) one has V ηi = (−1)i+1 εi .

We introduce with the “short” u and “long” v variables by 1 V V v = √ ηi xi , 2 V i 1 V V u = √ εi xi . 2 V i Conversely, one has

1 V xVi = √ ηi v + εVi u . 2

(2.14)

(2.15)

(2.16)

662



From the propagator 2.8 and vertices contributions 2.9 one is able to write the amplitude AG,V¯ of the graph G (with the marked root V¯ ) in terms of the non-commutative polynomials HUG,V¯ and HVG,V¯ as (see [23] for details) AG,V¯ (xe , pV¯ ) =

L

˜ Ω D

2 2 −1

L 1 0 =1

dt (1 −

D t2 ) 2 −1

e−

HV ¯) ¯ (t ,xe ,pv G,V HU ¯ (t) G,V D

HUG,V¯ (t) 2

,

(2.17)

with xe the external positions of the graph and t = tanh

α , 2

= 1, . . . , L .

(2.18)

where α are the parameters associated by (2.8) to the propagators of the graph. In [23] it was proved that HU and HV are polynomials in the set of variables t. The first polynomial is given by (see again [23]) 1

HUG,V¯ = (detQ) D

L

t ,

(2.19)

=1

where Q = A ⊗ 1D − B ⊗ σ ,

(2.20)

with A a diagonal matrix and B an antisymmetric matrix. The matrix A writes ⎛ ⎞ S 0 0 A = ⎝ 0 T 0⎠ , (2.21) 0 0 0 where S and resp. T are the two diagonal L by L matrices with diagonal elements c = coth( α2 ) = 1/t , and resp. t . The last (n − 1) lines and columns are have 0 entries. The antisymmetric part B is

sE C , (2.22) B= −C t 0 with s= and CV =

1 2 = , ˜ Ω θΩ

4

(−1)i+1 Vi

4i=1 , i+1 V ηi i=1 (−1)

uu E E= E vu

E uv . E vv

(2.23)

(2.24)

(2.25)


663

The blocks of the matrix E are vv E, =

4 V

uu E, =

uv E, =

i,j=1

4 V

(−1)i+j+1 ω(i, j) Vi V j ,

i,j=1

4 V

V V (−1)i+j+1 ω(i, j)ηi η j ,

(−1)i+j+1 ω(i, j) Vi ηV j .

(2.26)

i,j=1

The symbol ω(i, j) takes the values ω(i, j) = 1 if i < j, ω(i, j) = −1 is j < i and ω(i, j) = 0 if i = j. In (2.24) of the matrix C we have rescaled by s the hypermomenta pV . For further reference we introduce the integer entries matrix:

E C B = . (2.27) −C t 0 In [23] it was proven that detQ = (det M )D ,

(2.28)

M = A+B.

(2.29)

where Thus (2.19) becomes: HUG,V¯ = det M

L

t .

(2.30)

=1

Let I and resp. J be two subsets of {1, . . . , L}, of cardinal |I| and |J|. Let kI,J = |I| + |J| − L − F + 1 ,

(2.31)

and nIJ = Pf(BIˆJˆ), the Pffafian of the matrix B with deleted lines and columns I among the first L indices (corresponding to short variables u) and J among the next L indices (corresponding to long variables v). The specific form 2.21 allows one to write the polynomial HU as a sum of positive terms: s2g−kI,J n2I,J t t . (2.32) HUG,V¯ (t) = I,J

∈I

∈J

In [23], some non-zero terms were identified. They correspond to subsets I = {1, . . . , L} and J admissible, that is • J contains a tree T˜ in the dual graph, • the complement of J contains a tree T in the direct graph. Amongst this terms, the leading UV terms, (i.e., terms with the smallest global degree in the t variables) are given by choosing J minimal, that is • J = T˜ , tree in the dual graph,

664



Figure 7. The bubble graph.

Figure 8. The sunshine graph. • the complement of J is the union of a tree T in the direct graph and 2g genus lines. However, the list of these leading terms, as already remarked in [23], is not exhaustive. In the next section we complete this list with further terms, necessary for the sequel 2 . We end this section with the explicit example of the bubble and the sunshine graph (Figure 7 and Figure 8). For the bubble graph one has (2.33) HUG,V¯ = (1 + 4s2 ) t1 + t2 + t21 t2 + t1 t22 . For the sunshine graph one has HUG,V¯ = t1 t2 + t1 t3 + t2 t3 + t21 t2 t3 + t1 t22 t3 + t1 t2 t23 (1 + 4s2 )2 + 16s2 t22 + t21 t23 .

(2.34)

For further reference we also give the polynomial of the graph of Figure 6 HUG,V¯ = (1 + 4s2 )(t1 + t2 + t3 + t1 t2 t3 ) 1 + t2 t3 + t1 (t2 + t3 ) . (2.35) The polynomial HV is more involved. One has ˜ HVG,V¯ Ω xe = HUG,V¯ 2 2 For

pV¯ P Q

−1

P

t

xe pV¯

,

the purposes of [23], the existence of some non-zero leading terms was sufficient.

(2.36)


665

where P is some matrix coupling the external positions xe and the root hypermomenta pV¯ with the short u and long v variables and the rest of the hypermomenta pV (V = V¯ ). Explicit expressions can be found in [23].

3. Further leading terms in the first polynomial HU To procede with the dimensional regularization one needs first to correctly isolate the divergent subgraphs in different Hepp sectors. Let a subgraph S in the graph G. If S is nonplanar the leading terms in HUG suffice to prove that S is convergent in every Hepp sector as will be explained in Section 5. This is not the case if S is planar with more that one broken face. Let ˜ be the line of G which breaks an internal face of S. If ˜ is a genus line in G (see Subsection 2.1), one could still use only the leading terms in HUG to prove that S is convergent. But one still needs to prove that S is convergent if the line ˜ is not a genus line in G. This is for instance the case of the sunshine graph: in the Hepp sector t1 < t3 < t2 one must prove that the subgraph formed by the lines l1 and l3 is convergent. This is true due to the term 16s2 t22 in (2.34). Note that the variable t2 is associated to the line which breaks the internal face of the subgraph 1 , 3 . One needs to prove that for arbitrary G and S we have such terms. If we reduce S to a rosette there exists a loop line 2 ∈ S which either crosses ˜ or encompasses it. This line separates the two broken faces of S. Definition 3.1. Let J0 a subset of the internal lines of the graph G. J0 is called pseudo-admissible if: • its complement is the union of tree T in G and 2 , • neither ˜ nor 2 belong to T , ˜ This implies |I| = L(G) − 1 and |J| = Let I0 = { 1 . . . L } − ˜ ≡ I − . F (G)− 2 + 2g(G). For the sunshine graph (see Figure 8) I = { 1 , 3 } and J = { 2 }. One has the theorem Theorem 3.1. In the sum 2.32 the term associated to I0 and J0 above is n2I0 ,J0 = 4 , = 16 ,

if ˜ is a genus line, in G if ˜ is not a genus line in G.

Proof. The proof is similar to the one concerning the leading terms of HU given in Lemma III.1 of [23], being however more involved. The matrix whose determinant we must compute is obtained from B by deleting the lines and columns corresponding to the subsets I0 and J0 (as explained in the previous section). The matrix BIˆ ,Jˆ has 0

0

• a line and column corresponding to u˜, the short variable of ˜

666



• n lines and columns corresponding to the v variables of the n − 1 tree lines of T and the supplementary line 2 • n − 1 lines and columns associated to the hypermomenta. We represent the determinant of this matrix by the Grassmann integral: ¯ (3.1) det BIˆ ,Jˆ = dψ¯u˜ dψu˜ dψ¯v dψ v dψ¯p dψ p e−ψB ψ . 0

0

The quadratic form in the Grassman variables in the above integral is: V v − ψ¯u˜ (−1)i+1 ω(i, j)εVi ˜ εj ψ V

−

V

+

V

−

l;i,j

j+1 u ψ˜ ψ¯v ω(i, j)εVi εVj ˜ (−1)

pV − ψ¯u˜ (−1)i+1 εVi ˜ψ

V

+

l;i,j

V

V

u ψ¯pV (−1)i+1 εVi ˜ ψ˜

ψ¯v ω(i, j)εVi εV j ψv

, ;i,j

ψ¯v εVi ψ pV −

V

;i

ψ¯pV εVi ψv .

(3.2)

;i

We implement the first Filk move as a Grasmann change of variales. At each step we reduce a tree line 1 = (i, j) connecting the root vertex V¯G to a normal vertex V and gluing the two vertices. This is achieved by performing a change of variables for the hypermomenta and reinterpreting the quadratic form in the new variables as corresponding to a new vertex V¯G : the quadratic form essentially reproduces itself under the change of variables! Take 1 = (i, j) a line connecting the “root” vertex V¯G to a vertex V . We make the change of variables ¯ −ω(i, k)εVGk + ω(j, k)εV k ψv ψ pV = χpV + =1

+

k

¯pV + ψ¯pV = χ

¯G V −ω(i, k)εVk (−1)k+1 ψu˜ + ω(j, k)ε ˜ ˜ k

=1

+

k

k

¯ −ω(i, k)εVGk + ω(j, k)εV k ψ¯v

k

¯G V −ω(i, k)εVk (−1)k+1 ψ¯u˜ , ˜ ˜ + ω(j, k)εk

(3.3)

where the first sum is performed on the internal lines of G (note that because of the presence of the incidences matrices ε this sum reduces to a sum on the lines hooked to the two vertices V¯G and V ). The corners i and resp. j are the corners where the tree line 1 hooks to the vertex V¯G and resp. V . At each step, let us consider the coupling between the variables associated to the line 1 and to the hypermomentum pV and the rest of the variables. Using (3.2)


667

one has

− ψ¯u˜

p

V V v G VG (−1)p+1 ω(p, i)εVp ˜ ε1 j ψ1 ˜ ε1 i + ω(p, j)εp

V V G VG ω(i, p)εVp ψu˜ (−1)p+1 − ψ¯v1 ε + ω(j, p)ε ε ˜ ˜ 1 i p 1 j

+

V ;p

p

pV − ψ¯u˜ (−1)p+1 εVp ˜ ψ

V ;p

u ψ¯pV (−1)p+1 εVp ˜ ψ˜

¯ ¯ ω(i, k)εV1Gi εVGk + ω(j, k)εV1 j εV k ψv

ψ¯v1

−

=1 ;k

−

=1 ;k

¯ ¯ ψ¯v ω(k, i)εVGk εV1Gi + ω(k, j)εV k εV1 j ψv1

+ ψ¯v1 εV1 j ψ pV +

ψ¯v εV k ψ pV

=1 ;k

¯pV

−ψ

εV1 j ψv

−

ψ¯pV εV k ψv ,

(3.4)

=1 ;k

which rewrites as V V G VG − ψ¯u˜ (−1)p+1 ω(p, i)εVp ˜ ε 1 j ˜ ε1 i + ω(p, j)εp p

+

=

1 ;k

− ψ¯v1 +

ψ¯v

V ;p

+

¯ ¯ ω(i, k)εV1Gi εVGk

+

pV − ψ¯u˜ (−1)p+1 εVp ˜ ψ

=

+ ω(k, j)εV k εV1 j + ψ¯pV εV1 j ψv1

p

¯ ¯ ω(k, i)εVGk εV1Gi

V V u p+1 G VG ω(i, p)εVp ˜ ε1 j ψ˜ (−1) ˜ ε1 i + ω(j, p)εp

=1 ;k

+

1 ;k

ψ¯v εV k ψ pV −

ω(j, k)εV1 j εV k

=

V ;p

ψv

−

εV1 j ψ pV

u ψ¯pV (−1)p+1 εVp ˜ ψ˜

ψ¯pV εV k ψv .

(3.5)

1 ;k

Performing now in (3.4) the change of variable (3.3) for the hypermomentum pV of the vertex V associated to the tree line 1 of G and taking into account that ε1 i = −ε1 j the first two lines of 3.5 are simply −χ ¯pV εV1 j ψv1 + ψ¯v1 εV1 j χpV .

(3.6)

668



As ψv1 and ψ¯v1 do not appear anymore in the rest of the terms we are forced to pair them with χ ¯pV and χpV . The rest of the terms in the quadratic form are: ¯ ψ¯u˜ (−1)p+1 εV˜ −ω(i, k)εVG + ω(j, k)εV ψ v +

V ;p

=1

k

¯G V + −ω(i, k)εVk (−1)k+1 ψu˜ ˜ ˜ + ω(j, k)εk k

+

ψ¯v εV p

=1 ;p

=1

k

−

=1

V ;p

¯ −ω(i, k)εVGk + ω(j, k)εV k ψv

k

¯G V −ω(i, k)εVk (−1)k+1 ψu˜ + + ω(j, k)ε ˜ ˜ k

k

k

p


k

¯ V u G −ω(i, k)εVk (−1)k+1 ψ¯u˜ (−1)p+1 εVp + ˜ ˜ ψ˜ ˜ + ω(j, k)εk k

−

=1


k

¯G V −ω(i, k)εVk (−1)k+1 ψ¯u˜ + ˜ ˜ + ω(j, k)εk k

εV p ψv .

(3.7)

=1 ;p

We analyse the different terms in the above equation. The term ψ¯u˜ ψu˜ is: ¯G V p+1 V k+1 εVp (−1) + ω(j, k)ε − ω(i, k)ε ˜ ˜ (−1) ˜ k k p,k

−

p,k

¯G V k+1 V p+1 − ω(i, k)εVk εp = 0. ˜ (−1) ˜ (−1) ˜ + ω(j, k)εk

(3.8)

The term in ψ¯u˜ ψv is given by: ¯ −ω(i, k)εVGk + ω(j, k)εV k ψv ψ¯u˜ (−1)p+1 εVp + ˜ p

−

p

=1

¯G −ω(i, p)εVp ˜

k

(−1)p+1 ψ¯u˜ + ω(j, p)εVp εV k ψv . ˜

(3.9)

=1 ;k

Setting j to be either the first or the last halfline on the vertex V we see that the last terms in the two lines above cancel eachother. The first two terms hold: ¯ ¯G V V (3.10) ψ¯u˜ ψv (−1)p+1 − ω(i, k)εVGk εVp ˜ + ω(i, p)εp ˜ ε k . p =1

k


This can be rewritten in the form: ¯ V ¯ V G G − ψ¯u˜ ψv (−1)p+1 ω(p, k)εp ˜ ε k , p =1

669

(3.11)

k

where V¯G is a new root vertex in which the vertex V has been glued to the vertex V¯G and the halflines on the vertex V have been inserted on the new vertex at the place of the halfline i. The coupling between ψ v ’s with ψ v ’ is 4 ¯ + εVG ω(i, p) + εV ω(j, k ) ψ v − ψ¯v εV k

−

=

−

1

k

p

=1 ,k

p

k =1

¯ εVGp ω(i, p) +

p

4

εV k ω(j, k) ψ¯v

k=1

=1

εVk ψv .

(3.12)

;k

Again, as j is either the first or the last halfline on the vertex V , the last two terms in the two lines of (3.12) cancel each other. The rest of the terms give exactly the contacts between the ψ¯v and ψ v on a new vertex V¯G obtained by gluing V on V¯G This is the first Filk move on the line 1 and its associated vertex V . One iterates now this mechanism for the rest of the tree lines of G. Hence we reduce the graph G to a rosette (see Subsection 2.1). The quadratic form writes finally as ¯G V ¯G ¯G V ¯G ψ¯u˜ ψv (−1)p+1 ω(p, k)εVp ψ¯v ψu˜ (−1)p+1 ω(k, p)εVp − ˜ ε k − ˜ ε k ;p,k

−

;p,k

¯ ¯ ψ¯v εVjG εVGk ω(j, k)ψv

.

(3.13)

, ;j,k

The sum concerns only the rosette vertex V¯G . Therefore the last line is 0, as by the first Filk move we have exhausted all the tree lines of T . As ˜ breaks the face separated from the external face by 2 we have k1 < p < k2 . By a direct inspection of the terms above, one obtains the requested result.

4. Factorization of the Feynman amplitudes Take now S to be a primitively divergent subgraph. We now prove that the Feynman amplitude AG factorizes into two parts, one corresponding to the primitive divergent subgraph S and the other to the graph G/S (defined in Section 2.1). This is needed in order to prove that divergencies are cured by Moyal counterterms 3 . In Section 5 we will prove that only the planar (g = 0), one broken face B = 1, N = 2 or N = 4 external legs subgraphs are primitively divergent. We now deal only with such subgraphs. 3 It

is also the property needed for the definition of a coproduct Δ for a Connes–Kreimer Hopf algebra structure (see [8, 9, 27]). Details of this construction are given elsewhere [37].

670



4.1. Factorization of the polynomial HU We denote all the leading terms in the first polynomial associated to a graph S by HUSl . If we rescale all the t parameters corresponding to a subgraph S by ρ2 , HUG becomes a polynomial in ρ. We denote the terms of minimal degree in ρ l(ρ) in this polynomial by HUG . It is easy to see that for the subgraph S we have l(ρ) HUS = ρ2[L(S)−n(S)+1] HUSl |ρ=1 . We have the following theorem. Theorem 4.1. Under the rescaling tα → ρ2 tα

(4.1)

of the parameter corresponding to a divergent subgraph S of any Feynman graph G, the following factorization property holds l(ρ)

l(ρ)

HUG,V¯ = HUS,V¯S HUG/S,V¯ .

(4.2)

Proof. In the matrix M defined in (2.29) (corresponding to the graph G) we can rearrange the lines and columns so that we place the matrix MS (corresponding to the subgraph S) into the upper left corner. We place the line (and resp. the column) associated to the hypermomentum of the root vertex of S to be the last line (and resp. column) of M (without loss of generality we consider that the root of the subgraph S is not the root of G). M takes the form ⎛ uS uS S S S S S G−S S G−S S G−S ⎞ Eu v Cu p Eu u Eu v Cu p E S S S S S S S G−S S G−S S G−S ⎟ ⎜ Ev u Ev v Cv p Ev u Ev v Cv p ⎟ ⎜ S S S G−S S G−S ⎟ ⎜ pS uS Cp v 0 Cp u Cp v 0 ⎟ ⎜ C ⎜ uG−S uS G−S S G−S S G−S G−S G−S G−S G−S G−S ⎟ , u v u p u u u v u p ⎟ ⎜E E C E E C ⎜ G−S S G−S S G−S S G−S G−S G−S G−S G−S G−S ⎟ u v p u v p ⎠ ⎝E v Ev Cv Ev Ev Cv G−S

Cp

uS

G−S

Cp

vS

G−S

Cp

0

uG−S

G−S

Cp

v G−S

0

(4.3) S S where we have denoted by E u u a coupling between two short variables corresponding to internal lines of S etc.. I. We first write the determinant of the matrix above under the form of a Grassmannian integral ¯ det M = dψ¯u dψ u dψ¯v dψ v dψ¯p dψ p e−ψMψ . (4.4) Denote a generic line of the subgraph S by S and a generig line of the subgraph G − S by G−S . We perform a Grassmann change of variables of Jacobian 1. The value of the integral (4.4) does not change under this change of variables. We will prove that the following properties hold for the different terms in the Grassmanian quadratic form S

S

S

G−S

= diag(tl ) , E v u = 0, E v v G−S G−S G/S G/S G/S G/S u u u u uG−S v G−S =E , E = E u v E G−S G−S G/S G/S v E v = E v v .

E v

S G−S

v

=0 (4.5)


671

The new matrix of the quadratic form M will now be ⎛ uS uS S S S S S G−S S G−S E u v Cu p E u u E u v E S S S S ⎜ E vS uS E v v Cv p 0 0 ⎜ S S S S ⎜ p u p v pS uG−S pS v G−S C 0 C C ⎜ C ⎜ uG−S uS uG−S pS uG/S uG/S uG/S v G/S ⎜E 0 C E E ⎜ G−S S G−S S G/S G/S G/S G/S u p ⎝E v 0 Cv E v u E v v G−S S G−S S G−S G−S G−S G−S u v u v Cp Cp 0 Cp Cp

S G−S ⎞ Cu p S G−S ⎟ Cv p ⎟ ⎟ 0 ⎟ . uG−S pG−S ⎟ ⎟ C G−S G−S ⎟ p ⎠ Cv 0 (4.6) The first part of the proof follows that of Section 3, where we replace ψu (4.7) ψ¯u˜ → ψ¯u . ψu˜ →

∈G−S

∈G−S

Thus the appropriate change of variables is now ¯ ψ pV = χpV + −ω(i, k)εVSk + ω(j, k)εV ;k ψv =1 ;k

¯ −ω(i, k)εVSk + ω(j, k)εV k (−1)k+1 ψu + ;k ∈G−S

¯pV + ψ¯pV = χ

¯ −ω(i, k)εVSk + ω(j, k)εV k ψ¯v

=1 ;k

¯ −ω(i, k)εVSk + ω(j, k)εV k (−1)k+1 ψ¯u . +

(4.8)

;k ∈G−S

We emphasize that this Grassmann change of variables can be viewed as forming appropriate linear combinations of lines and columns. As we only use lines and columns associated to hypermomenta pS , this manipulations can not change the value of the determinant in the upper left corner: it will always correspond precisely to the first polynomial of the subgraph S. The relevant terms in the quadratic form are those of (3.4) and (3.5), with the substitutions (4.7). Again, after the change of variables (4.8) the only surviving contacts of ψv1 and ψ¯v1 are given by (3.6). Finally, the remaining terms are given by (3.7) with the substitutions (4.7). One needs again to analyse the different terms in this equation. The quadratic term in ψ¯u ψ u is: ¯ ¯ ψ¯u ψu (−1)k+p+1 ω(i, k)εV p εVSk − ω(j, k)εV p εV k , ∈G−S

¯ − ω(i, k)εVSk εV p + ω(j, k)εV k εV p . (4.9)

As j is either the first or the last halfline on the vertex V , we see that the last terms in the two lines above cancel. The remaining two terms give the contacts

672



amongst ψ¯u and ψ u on the rosette new vertex V¯S , obtained by gluing V¯S and V respecting the ordering. The contacts between ψ¯u and ψ v become ¯ −ω(i, k)εVGk + ω(j, k)εV k ψv + ψ¯u (−1)p+1 εV p

p

p

=1

k

¯ −ω(i, p)εVGp + ω(j, p)εV p (−1)p+1 ψ¯u − εV k ψv .

(4.10)

=1 ;k

Again that the last terms in the two lines above cancel. Rearanging the rest as before we end recover again the terms corresponding to a rosette. Finally for the ψ¯v and ψ v contacts (3.12) goes through. We iterate the change of variables only for a tree in the subgraph S, hence we reduce the subgraph S to a rosette (see Subsection 2.1). As the quadratic form reproduced itself, the root vertex V¯S is now a Moyal vertex, with either 2 or 4 external legs. Let r be an external half line of the subgraph S. As S is planar one broken face any line lS = (p, q) will either have r < p, q or p, q < r. Thus S G−S ¯ ¯S = ω(p, r)εVSSp εVG−S (4.11) E v v r = 0. l,p,r

The reader can check by similar computations that (4.5) holds. II. To obtain in the lower right corner the matrix corresponding to the graph G/S, we just have to add the lines and columns of the hypermomenta corresponding to the vertices of S to the ones corresponding to the root of S. Furthermore, S G−S (which had only one non-trivial by performing this operation, the block C v p column, the column corresponding to the hypermomentum pV¯S ) becomes identically 0. Forgetting the primes, the matrix in the quadratic becomes: ⎛ uS uS S S S S S G−S S G−S S G−S ⎞ E Eu v Cu p Eu u Eu v Cu p S S S S ⎜ Ev u ⎟ t δ v S v S Cv p 0 0 0 ⎜ ⎟ S S S G−S S G−S ⎜ C pS uS ⎟ Cp v 0 Cp u Cp v 0 ⎜ ⎟ ⎜ uG−S uS G−S S G/S G/S G/S G/S G/S G/S ⎟ . (4.12) p ⎜E 0 Cu Eu u Eu v Cu p ⎟ ⎜ G−S S G−S S G/S G/S G/S G/S G/S G/S ⎟ u p ⎝E v 0 Cv Ev u Ev v Cv p ⎠ G−S

Cp

uS

0

G/S

Cp

0

uG/S

G/S G/S

Cp

v

0

This is equivalent to the Grassmannian change of variables: S

S

S

S

χp = χp + ψ pV¯S , ¯p + ψ¯pV¯S . χ ¯p = χ

(4.13)

III. We finally proceed with the rescaling with ρ2 of all the parameter tα coresponding to the divergent subgraph S (see 4.1). Recall that these parameters S S are present as t1α on the diagonal of the block E u u and as tα on the diagonal of the block E v

S S

v

.


673

We factorize ρ1 on the first L(S) lines of columns of M (corresponding to the uS variables) and ρ on the next L(S) lines and columns (coresponding to the v S variables). We also factorize ρ1 on the n(S) − 1 lines and columns corresponding to the hypermomenta pS . This is given by the Grassmann change of variables 1 ψ u = ψ u ρ ψ v = ρψ v 1 χp = χp . (4.14) ρ ⎛

In the new variables the matrix of the quadratic form is

cl δuu + ρ2 E uu ⎜ E vu ⎜ ⎜ ρ2 C pu ⎜ ⎜ ρE uu ⎜ ⎜ ⎝ ρE vu ρC pu

E uv tl δvv C pv 0 0 0

ρ2 C up C vp 0 ρC up ρC vp 0

ρE uu 0 ρC pu G/S

ρE uv 0 ρC pv

G/S

E u u G/S G/S E v u G/S G/S C p u

G/S G/S

E u v G/S G/S E v v G/S G/S C p v

ρC up 0 0

⎞

⎟ ⎟ ⎟ ⎟ uG/S pG/S ⎟ . C ⎟ G/S G/S ⎟ C v p ⎠ 0 (4.15)

The determinant of the original matrix is obtained by multiplying the overall factor ρ−2n(S)+2 (coming from the Jacobian of the change of variables) with the determinant of the matrix (4.15). To obtain HUG , according to (2.30) we must multiply this determinant by a product over all lines of t . The determinant upper left corner, coresponding to the subgraph S, multiplied by the appropriate product of t as in (2.30) and by the Jacobian factor holds l(ρ) the complete polynomial HUS . At leading order in ρ it is HUS . The determinant of the lower right corner, multiplied by its corresponding product of t holds the complete polynomial HUG/S and no factor ρ. At the leading order in ρ the off diagonal blocks become 0. Therefore we have l(ρ)

HUG

l(ρ)

= HUS HUG/S .

(4.16)

Let us illustrate all this with the example of the graph of Figure 3, where the primitive divergent subgraph is taken to be the sunshine graph of lines 4 , 5 and 6 . A direct computation showed that, under the rescaling t4 → ρ 2 t4 ,

t5 → ρ 2 t 5 ,

t6 → ρ 2 t 6 ,

(4.17)

the leading terms in ρ of the polynomial HUG factorize as ρ4 (1 + 4s2 )t4 (t5 + t6 ) + t5 t6 + 8s2 (2t5 + t6 + 2s2 t6 ) × (1 + 4s2 )(t1 + t2 + t3 + t1 t2 t3 ) 1 + t2 t3 + t1 (t2 + t3 ) . (4.18)

674



The first line of this formula corresponds to the leading terms under the rescaling with ρ of HUS , while the second line is nothing but the polynomial HUG/S of (2.35). 4.2. The exponential part of the Feynman amplitude In order to perform the appropriate subtractions we need to check the factorization also at the level of the second polynomial. Throughout this subsection we suppose that S is a completely internal subgraph, that is none of its external points is an external point of G. The general case is treated by the same methods with only slight modifications. We have the following lemma. Proposition 4.2. Under the rescaling tα → ρ2 tα of all the lines of the subgraph S we have: HVG/S HVG = . (4.19) HUG ρ=0 HUG/S HVG is given by the inverse matrix Q−1 due to (2.36) which in Proof. The ratio HU G turn is given by M −1 (see [23] for the exact relation). Thus any property which holds for M −1 will also hold for Q−1 . We write the matrix elements of the inverse of M with the help of Grassmann variables ¯ ¯ ¯ −ψMψ dψdψψ i ψj e −1 (M )ij = . (4.20) ¯ −ψMψ ¯ dψdψe As S is a completely internal subgraph we only must analyse the inverse matrix entries ¯ −ψMψ ¯ ¯ dψdψψ G−S ψG−S e −1 (M )G−S,G−S = , (4.21) ¯ −ψMψ ¯ dψdψe

the only ones which intervene in the quadratic form due to the matrix P in (2.36). None of the changes of variables of the previous section involve any Grassmann variable associated with the G − S sector. We conclude that (M −1 )G−S,G−S = (M −1 )G−S,G−S ,

(4.22)

with M in (4.12). After the rescaling with ρ, the matrix M becomes (4.15). At leading order we set ρ to zero so that M becomes block diagonal. Consequently −1 −1 = MG/S,G/S . MG−S,G−S

(4.23)

4.3. The two point function The results proven above must be refined further for the two point function. The reason is that, as explained in Section 5, the two point functions have two singularities so that one needs also to analyse subleading behaviour. In the sequel we replace QG by MG , QG/S by MG/S , etc., the difference between the Q’s and the M ’s being inessential.


675

When integrating over the internal variables of G we start by integrating over the variables associated to S first. All the variables u, v and p appearing in the sequel belong then to G/S. The amplitude of the graph G will then write, after rescaling of the parameters of the subgraph S and having perform the first Filk move AG =

[dt dρ]

[dudvdp]G/S ρ2L(S)−1

e

−

u v

p

+ρ2 δM ) (MG/S D

u

v p

,

(4.24)

HUS2 (ρ)

where [dtdρ] is a short hand notation for the measure of integration on the Schwinger parameters, to be developed further in the next section. Here [dudvdp]G/S is the measure of integration for the internal variables of G/S, and M is given in (4.15). We explicitate δM as ⎞⎛ ⎞−1 ⎛ G−S S G−S S S S u p 0 Cu 0 cl δuS uS E u v E u G−S S ⎟ ⎜ S S S S⎟ ⎜ G−S uS p δM = ⎝E v 0 Cv tl δvS vS C v p ⎠ ⎠ ⎝ E v u G−S S S S u C p 0 0 0 Cp v 0 ⎞ ⎛ uS uG−S uS v G−S uS pG−S E C E ⎠. ⎝ (4.25) 0 0 0 pS uG−S pS v G−S C 0 C The Taylor development in ρ of the exponential gives u

v G/S − u v p MG/S p [dt ] [dudvdp] e u 1 − ρ2 u v p δM vp dρρ2L(S)−1 . D HUS2 (ρ)

(4.26)

The first term in the integral over ρ above corresponds to a (quadratic) mass divergence. The second term (logarithmically divergent) coresponds to the insertion of some operator which we now compute. The interaction is real. This means that we should symmetrize our amplitudes over complex conjugation of all vertices. For instance, at on loop one should allways symmetrize the left and right tadpoles [10, 11, 20] 4 . Consequently, the inverse matrix in (4.25) is actually a sum over the two possible choices of orientation of vertices. We must also sum over all possible choices of signs for the entries in the contact matrices in (4.25) as a similar symmetrization must be performed for the hypermomenta. 4 Following

[21] one can prove using this argument that if terms like x∂ do not appear in the initial Lagrangean for the complex orientable model, they will not be generated by radiative corrections, which not proven there.

676

R. Gur˘ au and A. Tanas˘ a S

Ann. Henri Poincaré G−S

G−S

As ψ¯u couples only to the linear combination ψv +εVi ψu in the initial S G−S S G−S u u V = εlG−S i E u v . matrix as well as in the change of variables, we have E Due to the sums over choices of signs, the only non zero entries in δM are δMu S uS , δMv S vS , δMv S uS , δMu S vS and δMp V¯ pV¯ . S S We denote the two external lines of S by 1 and 2 . A tedious but straightforeward computation holds ⎛ ⎞ u ¯ u v p δM ⎝v ⎠ = A1 (εV1Si u1 + v1 )2 + A2 (εV2 i u2 + v2 )2 + B1 p2V¯S . (4.27) p The first two terms are an insertion of the operator Ωx2 whereas the last is the insertion of an operator −Δ + x2 , being of the form of the initial Lagrangean. Take the graph G/S with the insertion of this operator at S, and with the adition of eventual mass subdivergencies (due to the first term). We denote it by an operator OS action on the graph G/S. We can then sum up the results of this section in the formula −

HVG (ρ)

−

HVG/S

e HUG (ρ) 1 e HUG/S 2 ≈ (1 + ρ O ) , S l(ρ) D/2 HUG (ρ)D/2 [HUS ]D/2 HUG/S

(4.28)

where by ≈ we mean the divergent part.

5. Dimensional regularization and renormalization of NCQFT In this section we proceed to the dimensional regularization and renormalization of NCQFT. We detail the meromorphic structure and give the form of the subtraction operator. Dimensional regularization and meromorphic structure of Feyman amplitudes for this model was also established in [22]. However, for consistency reasons we will give here an independent proof of this results. However, as the proof of convergence of the renormalized integral for this model is identical with that for the commutative Φ44 (up to substituting the Φ4 4 commutative subtraction operator with our subtraction operator) we will not detail it here. 5.1. Meromorphic structure of NCQFT In this subsection we prove the meromorphic structure of a Feynman amplitude A. We follow here the approach of [2]. We express the amplitude by (2.17) HVG,V ¯ (t ,xe ,pv ¯) L L − 1 HUG,¯ ˜ v (t) Ω e 2 D −1 dt (1 − t ) 2 . (5.1) AG,V¯ (xe , pV¯ , D) = D HUG,V¯ (t)D/2 2 2 −1 0 =1

We restrict our analysis to connected non-vacuum graphs. As in the commutative case we extend this expression to the entire complex plane. Take a Hepp sector σ


677

defined as 0 ≤ t 1 ≤ · · · ≤ tL ,

(5.2)

and perform the change of variables t =

L

x2j ,

= 1, . . . , L .

(5.3)

j=

We denote by Gi the subgraph composed by the lines t1 to ti . As before, we denote L(Gi ) = i the number of lines of Gi , g(Gi ) its genus, F (Gi ) its number of faces, etc.. The amplitude is ⎛ ⎞2 ⎞ D2 −1 ⎛ L L L 1 ˜ Ω ⎜ ⎟ 2 dxi AG,V¯ = ⎝1 − ⎝ xj ⎠ ⎠ 2(D−4)/2 0 i=1 j=i L i=1

2 ¯ (x ) (x2 )

HV

2L(Gi )−1 e

xi

− HUG,V

¯ G,V

.

HUG,V¯ (x2 )

(5.4)

In the above equation we factor out in HUG,V¯ the monomial with the smallest degree in each variable xi ⎛ ⎞2 ⎞ D2 −1 ⎛ L L L 1 ˜ Ω ⎜ ⎟ AG,V¯ (xe , pv¯ ) = dx ⎝1 − ⎝ x2j ⎠ ⎠ D 22 0 =1 j= HV

2L(Gi )−1−Db (Gi ) xi

e

¯

− HUG,V

¯ G,V

D . asb + F (x2 ) 2

(5.5)

The last term in the above equation is always bounded by a constant. Divergences can arise only in the region xi close to zero (it is known that this theory does not have an infrared problem, even at zero mass). The integer b (Gi ) is given by the topology of Gi . It is ⎧ ⎪ ⎨≤ L(Gi ) − n(Gi ) − 1 − 2g(Gi ) if g(Gi ) > 0 b (Gi ) = ≤ L(Gi ) − n(Gi ) if g(Gi ) = 0 and B(Gi ) > 1 . (5.6) ⎪ ⎩ = L(Gi ) − n(Gi ) − 1 if g(Gi ) = 0 and B(Gi ) = 1 To prove the first and the third line, let I = {1 . . . L}. We will exhibit a J admissible in G with the right scaling in the xi . Let a tree in Gi , T (Gi ) and complete it to a tree T (G) in the graph G. Furthermore, let 2g(Gi ) genus lines in Gi . Let J the complement of this set. It obviousely contains a tree in the dual graph of G being therefore admissible and has the right scaling in xi .

678



For the second line one must take I = {1 . . . L} − ˜ and J pseudo-admissible as in Section 3, choosing again the tree T (G) contained in the complement of J to be a subtree in the component Gi . We see that b (Gi ) is at most L(Gi ) − n(Gi ) + 1 and that the maximum is achieved if and only if g(Gi ) = 0 and B(Gi ) = 1. The convergence in the UV regime (xi → 0) is ensured if 2L(Gi ) − Db (Gi ) > 0 , i = 1 . . . L . (5.7) 2L(Gi ) − Db (Gi ) > 2L(Gi ) − D L(Gi ) − n(Gi ) + 1 ,

As

(5.8)

we always have convergence provided D < 2 ≤

2L(Gi ) 4n(Gi ) − N (Gi ) ≤ . n(Gi ) − N (Gi )/2 + 1 L(Gi ) − n(Gi ) + 1

(5.9)

where N (Gi ) is the number of external points of Gi 5 . Thus AG,V¯ (D) is analytic in the strip (5.10) Dσ = {D | 0 < D < 2} . We extend now A as a function of D for 2 ≤ D ≤ 4. We claim that if • g(Gi ) > 0 • g(Gi ) = 0 and B(Gi ) > 1 • N (Gi ) > 4, the strip of analyticity can be immediately extended up to Dσ = {D | 0 < D < 4 + εG } .

(5.11)

for some small positive number εG depending on the graph. Indeed, for the first two cases we have b (Gi ) ≤ L(Gi ) − n(Gi ) so that the integral over xi converges for 2L(Gi ) 4n(Gi ) − N (Gi ) = , (5.12) D ≤ 4 < n(Gi ) − N (Gi )/2 L(Gi ) − n(Gi ) whereas in the third case, as N (Gi ) > 4 the integral over xi converges for D ≤ 4
0. If A1 , A2 , . . . , Am ⊂ R are disjoint bounded Borel sets, then the random vector ω,e ξk (A1 ), ξkω,e (A2 ), . . . , ξkω,e (Am ) , has a multinomial distribution

P ξkω,e (A1 ) = r1 , ξkω,e (A2 ) = r2 , . . . , ξkω,e (Am ) = rm = where qk,s = P

k! rm+1 q r1 q r2 · · · qk,m+1 , r1 !r2 ! . . . rm+1 ! k,1 k,2

kω(1) − e ∈ As =

rs = 0, . . . , k ,

γ(t)dt ,

e+k−1 As

lim kqk,s = γ(e)L(As ) ,

rs = k ,

s=1

s = 1, . . . , m + 1 ,

m and Am+1 = R\( s=1 As ). Continuity of γ at e yields that k→∞

m+1

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and hence m

Pλs ({rs }) , lim P ξkω,e (A1 ) = r1 , ξkω,e (A2 ) = r2 , . . . , ξkω,e (Am ) = rm =

k→∞

s=1

with λs = γ(e)L(As ). Hence the random variables ξkω,e (As ), s asymptotically independent and have Poisson distributions Pλs .

= 1, . . . , m, are

In nontrivial situations, the operator H0 = 0 introduces statistical dependence to the eigenvalues of Hkω l2 (Bk ) and therefore the analysis of the rescaled measure ξkω,e is more involved. If this dependence is not too big in a suitable sense, then Minami’s method allows to show that ξkω,e (As ), s = 1, . . . , m are still asymptotically independent Poisson random variables. In the next subsection, we discuss a general limit theorem needed for Minami’s method. 2.2. The Poisson point process and Grigelionis’ limit theorem Although ξkω,e as well as the other measures of interest to us are on R, we discuss, for sake of clarity, the general situation of random point measures on a metric space S. We equip S with the Borel σ-algebra BS , i.e. the σ-algebra generated by open sets. We denote by M the set of all nonnegative Borel measures μ on (S, BS ) such that μ(A) < ∞ for every bounded Borel set A ⊂ S. A measure μ ∈ M is called a point measure if μ can be written in the form μ= δ(xj ) , xj ∈ S , j∈J

where J is a countable index set. We denote by Mp the set of all point measures on (S, BS ). A point process on S is map ω → μω from some probability space (Ω, F , P) to Mp such that for every bounded Borel set A ⊂ S, the map ω → μω (A) is measurable. If μω is a point process, then the map ν(B) = Eμω (B) ,

B ∈ BS ,

defines a measure on (S, BS ). The measure ν is called the intensity measure of the point process μω . Definition 2.1. Let ν ∈ M. A Poisson point process on S with intensity ν is a point process ξ ω with the following properties: 1. for every bounded Borel set A ⊂ S, the random variable ξ ω (A) has a Poisson distribution with parameter ν(A). 2. given disjoint bounded Borel sets A1 , A2 , . . . , Am in S, the random variables ξ ω (A1 ), ξ ω (A2 ), . . . , ξ ω (Am ) are independent. It can be shown [17] that given any ν ∈ M, there exists a Poisson process on S with intensity ν, constructed on a suitable probability space. The Poisson point process is an idealized model of noninteraction and the point process ξkω,e in the study of eigenvalue statistics never exactly verifies Conditions (1) and (2) of Definition 2.1.

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Definition 2.2. A sequence ξkω of point processes on S, defined on the same probability space, is said to converge to a Poisson point process on S with intensity ν ∈ M if for any given disjoint bounded Borel sets A1 , A2 , . . . , Am in S, we have m

lim P ξkω (A1 ) = r1 , ξkω (A2 ) = r2 , . . . , ξkω (Am ) = rm = Pν(As ) ({rs }) , (2.1)

k→∞

s=1

for all r1 , r2 , . . . , rm ∈ N. Hence, in the previous subsection, the sequence of point processes ξkω,e on R converges to a Poisson process on R with intensity γ(e)L. In general, it can be difficult to verify the Condition (2.1) directly and it is more convenient to verify an equivalent condition in terms of the characteristic functions, namely lim Eei

k→∞

m

ω s=1 ts ξk (As )

=

m

exp ν(As )(eits − 1) ,

(2.2)

s=1

for all t1 , t2 , . . . , tm ∈ R. Both (2.1) and (2.2) are equivalent to the usual definition of convergence in law for random vectors in Nm . The basic limit theorem guaranteeing the convergence of a sequence of point processes to a Poisson point processes is due to Griegelionis [13]. Originally formulated for step processes on R, Grigelionis’ theorem remains valid in more general settings and in our case it translates to: Theorem 2.3 (Grigelionis, 1963). Let (nk )k≥1 be a natural subsequence, let for each ω ω ω , ξk,2 , . . . , ξk,n be independent point processes on S and let k ≥ 1, ξk,1 k ξkω =

nk

ω ξk,j .

j=1

Let ν ∈ M and assume that for every bounded Borel set A ⊂ S, we have

ω (1) lim max P ξk,j (A) ≥ 1 = 0 , (2)

k→∞ 1≤j≤nk nk

lim

k→∞

ω P ξk,j (A) ≥ 1 = ν(A) ,

j=1

and (3)

lim

k→∞

nk

ω P ξk,j (A) ≥ 2 = 0 . j=1

Then ξkω converges to a Poisson point process on S with intensity ν. Theorem 2.3 is well known and can be found in the literature e.g. [9, 16] as a corollary of more general results on point processes. For completeness, we include a self-contained proof here, following the original arguments of [13].

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m m Proof. m We use themstandard notation abm= s=1 as bs , for a, b ∈ R and m|α| = s=1 αs for α ∈ N . We denote by {es }s=1 the standard basis vectors of R . Let A1 , A2 , . . . , Am be given disjoint bounded Borel sets in S. Let Xkω be the random vector Xkω = ξkω (A1 ), ξkω (A2 ), . . . , ξkω (Am ) , and let φk : Rm → C be the corresponding characteristic function ω

φk (t) = EeitXk ,

t ∈ Rm .

According to (2.2), we have to show that for all t ∈ Rm , m

lim φk (t) =

k→∞

We set

exp ν(As )(eits − 1) .

(2.3)

s=1

ω ω ω ω = ξk,j (A1 ), ξk,j (A2 ), . . . , ξk,j (Am ) , Xk,j ω

φk,j (t) = EeitXk,j , and A=

t ∈ Rm , m

As .

s=1

By assumption (1), there is a k0 such that for k ≥ k0 ,

ω (A) ≥ 1 < 1/4 . max P ξk,j 1≤j≤nk

Hence for k ≥ k0 and 1 ≤ j ≤ nk ,

ω iαt

ω ω ≤2 P X = α (e − 1) P Xk,j = α = 2P ξk,j (A) ≥ 1 < 1/2 , k,j |α|≥1 |α|≥1 and we can write φk,j (t) = 1 +

ω P Xk,j = α (eiαt − 1)

|α|≥1

⎛ = exp ⎝

⎞

ω iαt P Xk,j = α (e − 1) + Ek,j ⎠ ,

(2.4)

|α|≥1

where

⎛ Ek,j = f ⎝

⎞

ω iαt P Xk,j = α (e − 1)⎠ ,

|α|≥1

and f (z) = log(1 + z) − z. The function f is analytic in the open disk {|z| < 1} and 2 |f (z)| ≤ C |z| for |z| < 1/2 , (2.5)

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where 0 < C < ∞ is a numerical constant. Next, we write

ω ω P Xk,j = α (eiαt − 1) = P Xk,j = α (eiαt − 1) + Fk,j |α|≥1

|α|=1

= =

m

ω P Xk,j = es (eits − 1) + Fk,j s=1 m

(2.6)

ω P ξk,j (As ) = 1 (eits − 1) + Gk,j + Fk,j ,

s=1

where Fk,j =

ω P Xk,j = α (eiαt − 1) ,

|α|≥2

and Gk,j =

m

ω ω P{Xk,j = es } − P ξk,j (As ) = 1 (eits − 1) .

s=1

Hence,

m

ω φk,j (t) = exp P ξk,j (As ) = 1 (eits − 1) + Hk,j

,

s=1

where Hk,j = Ek,j + Fk,j + Gk,j . We then have, by independence, that φk (t) =

nk j=1

φk,j (t) ⎛

= exp ⎝

m s=1

⎛

⎞ ⎞ nk nk

ω ⎝ P ξk,j (As ) = 1 ⎠ (eits − 1) + Hk,j ⎠ j=1

(2.7)

j=1

The assumptions (2) and (3) imply that lim

k→∞

nk

ω P ξk,j (As ) = 1 = ν(As ) .

(2.8)

j=1

We claim that lim

k→∞

nk

Hk,j = 0 .

(2.9)

j=1

If (2.9) holds, then (2.8), (2.9) and (2.7) together yield the desired conclusion (2.3) and we are done. We now prove (2.9). We have

ω (A) ≥ 2 , (2.10) |Fk,j | ≤ 2P ξk,j

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and the bound (2.5) yields ⎞2 ⎛

2 ω ω P Xk,j = α ⎠ = 4C P ξk,j (A) ≥ 1 . |Ek,j | ≤ C ⎝2

(2.11)

|α|≥1

To estimate |Gk,j |, note that ω

ω Xk,j = es ⊂ ξk,j (As ) = 1 , and

ω ω ω ξk,j (As ) = 1 \ Xk,j = es ⊂ ξk,j (A) ≥ 2 .

Hence

ω |Gk,j | ≤ 2mP ξk,j (A) ≥ 2 .

(2.12)

We now combine the bounds (2.11), (2.10) and (2.12) to get nk nk

ω ≤ (2m + 2) H P ξk,j (A) ≥ 2 k,j j=1 j=1 nk

ω

ω + 4C max P ξk,j (A) ≥ 1 P ξk,j (A) ≥ 1 . 1≤j≤nk

j=1

The assumptions (1), (2) and (3) imply that the right hand side of last inequality converges to zero as k → ∞, completing the proof. 2.3. Corollaries of Grigelionis’ limit theorem For the point processes ξ ω on S = R arising in the study of eigenvalue statistics, it more natural to obtain information about the Poisson integrals is sometimes −1 ω Im(t − z) dξ (t), Imz > 0, rather than about the events {ξ ω (A) ≥ 1} and R ω {ξ (A) ≥ 2}. In this subsection, we replace the Conditions (2) and (3) of Theorem 2.3 by sufficient conditions in terms of the Poisson integrals. We refer the reader to [15] for the general theory of Poisson integrals and their applications to spectral theory. For a positive Borel measure μ on S and a Borel function f : S → [0, ∞), we set f (t)f (t )dμ(t)dμ(t ) . I(μ, f ) = t=t

If μ = j δ(tj ) is a point measure on S and f (t) = 1A (t) is the indicator function of a bounded Borel set A ⊂ S, then we have I(μ, 1A ) = 1A (ti )1A (tj ) = μ(A) μ(A) − 1 , i=j

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and therefore I(μ, 1A ) = 0 ⇔ μ(A) ≥ 2. If ξ ω is a point process on S, then

P ξ ω (A) ≥ l = (l − 1)P ξ ω (A) = l l≥2

l≥2

≤

l(l − 1)P ξ ω (A) = l

l≥2

= EI(ξ ω , 1A ) . Since

P ξ ω (A) ≥ 1 = Eξ ω (A) − P ξ ω (A) ≥ l , l≥2

we conclude that the conditions

(2 )

lim

k→∞

and

(3 )

lim

k→∞

nk

ω Eξk,j (A) = ν(A) ,

j=1 nk

ω EI(ξk,j , 1A ) = 0 ,

j=1

together imply Conditions (2) and (3) of Theorem The next step is to replace, 2.3. ω ω in (2’) and (3’), the quantity Eξk,j (A) by E f dξk,j for f in a sufficiently rich family F of functions. ω ω ω , ξk,2 , . . . , ξk,n be point processes on S and Theorem 2.4. For each k ≥ 1, let ξk,1 k n k av ω let ξk = j=1 Eξk,j . Let ν ∈ M. Suppose that there is a measure μ ∈ M s.t. that ν and (ξkav )k≥1 are absolutely continuous with respect to μ, with uniformly bounded densities, i.e. there is a constant 0 < C < ∞ such that for all bounded Borel sets A ⊂ S, ν(A) ≤ Cμ(A) ,

and ξkav (A) ≤ Cμ(A) ,

k ≥ 1.

Suppose that F ⊂ L1 (S, μ) is a family of functions such that finite linear combinations of functions in F are dense in L1 (S, μ) and such that for every bounded Borel set A ⊂ S, there exists f ∈ F with f ≥ 1A . Suppose that for all f ∈ F , we have (2 )

lim

k→∞

and (3 )

lim

k→∞

f dξkav = nk

f dν ,

ω EI(ξk,j , f) = 0 .

j=1

Then (2’) and (3’) hold for all bounded Borel sets A ⊂ S.

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Proof.Let A be a bounded Borel set. Let ε > 0. There isa finite linearcombination gdν − ν(A) < Cε and g = c f , f ∈ F , with |g − 1A | dμ i i i i < ε. Then gdξ av − ξ av (A) < Cε. Since limk→∞ gdξ av = gdν, we have k k k ν(A) − 2Cε ≤ lim inf ξkav (A) ≤ lim sup ξkav (A) ≤ ν(A) + 2Cε , k→∞

k→∞

and (2’) is obtained after letting ε ↓ 0. Now let f ∈ F be such that f ≥ 1A . Since, ω ω I(ξk,j , 1A ) ≤ I(ξk,j , f ), (3’) follows from (3”). The special case when S = R, μ = L is the Lebesgue measure on R, ν = λL for a λ > 0 and F is the family of functions Im(t − z)−1 Imz>0 yields ω ω , ξk,2 , Theorem 2.5. Let (nk )k≥1 be a natural subsequence, let for each k ≥ 1, ξk,1 ω . . . , ξk,nk be independent point processes on R and let

ξkω =

nk

ω ξk,j .

j=1

We make the following four hypotheses: (H0): there is a constant 0 < C < ∞ such that for all k ≥ 1 and every bounded Borel set A ⊂ R, nk ω Eξk,j (A) ≤ CL(A) . j=1

(H1): for every bounded Borel set A ⊂ R,

ω lim max P ξk,j (A) ≥ 1 = 0 . k→∞ 1≤j≤nk

(H2): there is a constant 0 < λ < ∞ such that for Imz > 0, lim

k→∞

nk j=1

E

R

ω Im(t − z)−1 dξk,j (t) = πλ .

(H3): for Imz > 0, nk lim E

k→∞

j=1

t=t

ω ω Im(t − z)−1 Im(t − z)−1 dξk,j (t)dξk,j (t ) = 0 .

Then ξkω converges to a Poisson point process on R with intensity λL. Theorem 2.5 is implicitly derived in [25] and is suitable for applications to eigenvalue statistics of general random discrete Schrödinger operators.

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3. Poisson statistics of eigenvalues in the hierarchical Anderson model 3.1. Definition of the model and its basic properties In this subsection, we review the definition and the basic properties of the hierarchical Anderson model. For additional information, we refer the reader to [21– 23, 27, 28]. Theorems 3.1 and 3.2 collect, for reference purposes, the main known results on the hierarchical Anderson model and are stated without proof. We consider the set X = {0, 1, 2, . . . }. Given an integer n ≥ 2, X has a metric space structure with the distance d : X × X → [0, ∞)

d(x, y) = min r : q(x, nr ) = q(y, nr ) , where q(x, nr ) denotes the quotient of the division of x by nr . The (closed) ball with center x and radius r is denoted by

B(x, r) = y ∈ X : d(x, y) ≤ r . The main property of d is that two balls of the same radius are either disjoint or identical, and that each B(x, r + 1) is a disjoint union of n balls of radius r. For x ∈ X, the unit vector δx ∈ l2 (X) denotes the Kronecker delta function at x: δx (x) = 1 and δx (y) = 0 for y = x. For each integer r ≥ 1, we set Er : l2 (X) → l2 (X), ψ(y) . (Er ψ)(x) = n−r d(y,x)≤r

Thus Er is the orthogonal projection onto the subspace of l2 (X) consisting of functions that are constant on every ball of radius r. The hierarchical Laplacian is then defined by the formula Δ=

∞

pr Er ,

r=1

∞ where (pr )r≥1 is a given sequence such that pr > 0 and r=1 pr = 1. We assume that C2 C1 ≤ pr ≤ r , r ρ ρ for some fixed constants ρ > 1, C1 > 0, C2 > 0. The number d = d(n, ρ) = 2

log n , log ρ

(3.1)

is called the spectral dimension of Δ. The following theorem [21, 28] summarizes some of the spectral features of Δ. Theorem 3.1. Δ is a bounded self-adjoint operator on l2 (X) and its spectrum consists of infinitely degenerate isolated eigenvalues λ0 = 0 ,

λ1 = p1 ,

λ2 = p1 + p2 ,

λ3 = p1 + p2 + p3 , . . .

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and of their accumulation point λ∞ = 1, which is not an eigenvalue. For each x ∈ X, δx |Δδy = 1 , y∈X

and hence Δ generates a random walk on X.The random walk is recurrent when d ≤ 2 and transient when d > 2. The hierarchical Anderson model is the random discrete Schr¨ odinger operator Hω = Δ + Vω , as in the framework of the introduction, with H0 = Δ. If the set {ω(x) : x ∈ X} is unbounded, then Vω and Hω are unbounded self-adjoint operators with the domain 2 2 Dω = ψ : |ψ(x)| 1 + |ω(x)| < ∞ . x∈X

Theorem 3.2. Hω has the following generic spectral properties. (1) [22, 23] If the support of γ is connected, supp(γ) = [a, b], then for P-a.e. ω ∈ Ω, the spectrum of Hω is given by ∞ Σ= [λr + a, λr + b] . r=0

(2) [22] If the model has a spectral dimension d < 4 then, for P-a.e. ω ∈ Ω, the spectrum of Hω is dense pure-point in Σ. (3) [28] For any spectral dimension d < ∞, the same conclusion as in (2) holds provided the random variables ω(x) have a Cauchy distribution, i.e. the density γ(t) is of the special form: v 1 , (3.2) γ(t) = π (u − t)2 + v 2 for some u ∈ R, v > 0. 3.2. The density of states We denote by C0 (R) the space of continuous functions f : R → C vanishing at infinity, i.e. lim|t|→∞ |f (t)| = 0. If (νk )k≥1 and ν are Borel probability measures on R, we say that νk converges to ν in the weak-* topology if for every f ∈ C0 (R), lim f (t)dνk (t) = f (t)dν(t) . k→∞

The finite volume approximations to Hω are defined as follows. We fix x0 ∈ X and we consider the increasing sequence of balls Bk = B(x0 , k) k ≥ 0 . Each Bk has then size |Bk | = nk . We define Hkω to be the truncated operator Hkω =

k s=1

ps Es + Vω .

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Note that the subspace

l2 (Bk ) = ψ ∈ l2 (X) : ψ(x) = 0 for x ∈ / Bk ,

is invariant for Hkω . The normalized eigenvalue counting measure μω k is then given by (1.1). The averaged spectral measure for Hω is the unique Borel probability measure μav on R defined by f (t)dμav (t) = E δx0 |f (Hω )δx0 , f ∈ C0 (R) . (3.3) By symmetry, f (t)dμav (t) = Eδx |f (Hω )δx for all x ∈ X. The content of the following theorem is that the averaged spectral measure μav is naturally interpreted as the density of states for Hω . av Theorem 3.3. For P-a.e. ω ∈ Ω, μω in the weak-* topology as k → ∞, i.e. k → μ there is a set Ω ∈ F with P(Ω) = 1 such that for all ω ∈ Ω and f ∈ C0 (R) we have ω lim f (t)dμk (t) = f (t)dμav (t) . k→∞

ω + We start the proof of Theorem 3.3 with resolvent bounds. Since Hrω = Hr−1 pr Er , the resolvent identity yields ω ω − z)−1 − (Hrω − z)−1 = pr (Hr−1 − z)−1 Er (Hrω − z)−1 , (Hr−1

for z ∈ C\R. Therefore: ! ω ! !(Hr−1 − z)−1 − (Hrω − z)−1 ! ≤ |Imz|−2 pr ,

z ∈ C\R .

(3.4)

Iterating (3.4) yields for r < k, k ! ω ! !(H − z)−1 − (H ω − z)−1 ! ≤ |Imz|−2 ps , r k

z ∈ C\R .

(3.5)

z ∈ C\R .

(3.6)

s=r+1

and letting k → ∞, ∞ ! ω ! !(H − z)−1 − (Hω − z)−1 ! ≤ |Imz|−2 ps , r s=r+1

Proposition 3.4. For every z ∈ C\R there is a set Ωz ∈ F, with P(Ωz ) = 1 and such that for all ω ∈ Ωz , the difference −1 ω Dk,ω = (t − z) dμk (t) − (t − z)−1 dμav (t) , converges to 0 as k → ∞. Proof. Let ε > 0 be given. We take r = r(ε, z) big enough so that −2

|Imz|

∞ s=r+1

ps < ε/2 .

(3.7)

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Then for r < k, −1

Dk,ω = |Bk |

δx |(Hkω − z)−1 δx − E δx0 |(Hω − z)−1 δx0

x∈Bk

δx | (Hkω − z)−1 − (Hrω − z)−1 δx

−1

|Bk |

=


x∈Bk

+

−1

|Bk |

ω −1 −1 δx |(Hr − z) δx − E δx0 |(Hω − z) δx0

(3.8)

x∈Bk

= Ik,ω + IIk,ω . The bounds (3.5) and (3.7) yield |Ik,ω | < ε/2. We proceed with estimating |IIk,ω |. Note that Bk is a disjoint union of nk−r balls of radius r, Bk =

k−r n

Bk,j ,

j=1

and therefore 2

l (Bk ) =

k−r n"

l2 (Bk,j ) .

j=1

Since each subspace l2 (Bk,j ) is invariant for Hrω , we can write −1

|Bk |

δx |(Hrω

−1

− z)

δx =

x∈Bk

1 nk−r

k−r n

n−r

j=1

δx |(Hrω − z)−1 δx , x∈Bk,j

and recognize that the right hand side is an average of nk−r i.i.d. bounded random variables. Hence, Kolmogorov’s strong law of large numbers yields that there is a set Ωz,ε ∈ F with P(Ωz,ε ) = 1 and such that for all ω ∈ Ωz,ε , −1 lim |Bk | E δx |(Hrω − z)−1 δx , δx |(Hrω − z)−1 δx = n−r (3.9) k→∞

x∈Bk

x∈B

where B is some fixed ball of radius r. The bounds (3.5) and (3.7) yield δx |(Hrω − z)−1 δx − δx |(Hω − z)−1 δx < ε/2 , which combined with (3.9) yields lim sup |IIk,ω | < ε/2 . k→∞

Hence#for ω ∈ Ωz,ε , lim supk→∞ |Dk,ω | < ε, and the statement follows after taking ∞ Ωz = m=1 Ωz,1/m . Theorem 3.3 is a consequence of Proposition 3.4 and a density argument. Let G be a countable dense set in C\R. Since any function f ∈ C0 (R) can be uniformly approximated by finite linear combinations of the functions t → (t − # z)−1 , with z ranging through G, Theorem 3.3 follows after taking Ω = z∈G Ωz .

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Remarks on Theorem 3.3. There is no restriction on the spectral dimension d. Also, the theorem and the above proof remain valid without the assumption that the random variables ω(x) have a density γ. 3.3. Fine eigenvalue statistics For our study of fine eigenvalue statistics, we need the following two well-known general estimates for random discrete Schr¨ odinger operators. For both estimates, the density γ plays a fundamental role. Lemma 3.5 (Wegner estimate [31]). Let M0 be any self-adjoint operator on l2 (X) and let Mω = M0 + Vω . Then for every bounded Borel measurable function h : R → [0, ∞) and x ∈ X, E δx |h(Mω )δx ≤ γ ∞ h(t)dt . (3.10) Hence, if νω is the spectral measure for δx and Mω and ν av = Eνω is the corresponding averaged measure, then ν av is absolutely continuous with respect to Lebesgue measure, dν av (t) = υ(t)dt , and

υ ∞ ≤ γ ∞ . Lemma 3.6 (Minami’s estimate [4, 12, 25]). Let M0 be any self-adjoint operator on l2 (X) and let Mω = M0 + Vω . Then for every x, y ∈ X and Imz > 0 δx |Im(Mω − z)−1 δx δx |Im(Mω − z)−1 δy 2 ≤ π 2 γ ∞ . (3.11) E det δy |Im(Mω − z)−1 δx δy |Im(Mω − z)−1 δy Wegner estimate yields that μav is absolutely continuous with respect to Lebesgue measure, dμav (t) = η(t)dt , and

η ∞ ≤ γ ∞ . If e ∈ and ε > 0 are given, then in view of Theorem 3.3 we expect the number of eigenvalues of Hkω l2 (Bk ) in the interval (e − ε, e + ε),

# i : eω,k ∈ (e − ε, e + ε) , i to have typical size of order |Bk | μav (e − ε, e + ε) for large k. The precise statistical behavior of the eigenvalues eω,k near e is captured by the rescaled measure ξkω,e j given by (1.2). We make the following regularity assumption on e: for Imz > 0, lim Im(t − e − εz)−1 η(t)dt = πη(e) . (3.12) ε↓0

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For example, if η is continuous at e, then (3.12) holds. However, it is in general a difficult problem to establish the continuity of η for random discrete Schr¨ odinger operators. In the case of the Cauchy random potential (3.2), η is known to be analytic [24]. If the Fourier transform of γ(t) decays exponentially, then it is possible [7] to prove analyticity of η after increasing the disorder, i.e. replacing Vω with σVω for a sufficiently large σ. When continuity of η is not available, one appeals to a classical theorem in harmonic analysis (see for example [20]), due to Fatou, guaranteeing that (3.12) holds for Lebesgue almost all e ∈ R. We now state our main result. Theorem 3.7. Assume that the hierarchical model has a spectral dimension d < 1. Assume that η(e) > 0 and that e verifies the regularity Condition (3.12). Then ξkω,e converges to a Poisson point process on R with intensity η(e)L. Remarks on Theorem 3.7. Our theorem is the analogue of Minami’s result for the localized Anderson model on Zd (see the appendix). The proof of Poisson statistics for the hierarchical model is technically simpler than the corresponding proofs for the Anderson model on Zd , because of the low spectral dimension assumption and because of the high degree of self-similarity of the hierarchical model. The rest of the section is devoted to the proof of Theorem 3.7. The main idea is to approximate Hkω with Hrω for r < k, as in the proof of Theorem 3.3. This time we choose r to depend on k, r = rk , such that rk = c, (3.13) lim k→∞ k where d < c < 1. (3.14) Let eω,k ≤ eω,k ≤ · · · ≤ eω,k 1 2 |Bk | , denote the eigenvalues of Hrωk l2 (Bk ) and let |Bk |

ξkω,e =

δ |Bk | (eω,k − e) , i i=1

be the corresponding rescaled measure near e. Since Bk is a disjoint union of nk−rk balls of radius rk , Bk =

nk−r k

Bk,j

j=1

we have the corresponding direct sum decomposition Hrωk l2 (Bk ) =

k−rk n"

j=1

Hrωk l2 (Bk,j ) .

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Therefore the point process ξkω,e is the sum of nk−rk independent point processes, ξkω,e

=

k−rk n

ω,e , ξk,j

j=1

where

r

ω,e = ξk,j

n k δ |Bk | (eω,k,j − e) , l l=1

eω,k,j , l

l = 1, . . . , n are the eigenvalues of Hrωk l2 (Bk,j ). The proof of Theorem 3.7 is organized as follows. We first establish that the point processes ξkω,e and ξkω,e are asymptotically close in the following sense:

and

rk

Proposition 3.8. For every f ∈ L1 (R, dt), ω,e ω,e f dξk − f dξk = 0 . lim E k→∞

(3.15)

Corollary 3.9. Let A1 , A2 , . . . , Am be given disjoint bounded Borel sets in R. Let Xkω and Xkω be the random vectors Xkω = ξkω,e (A1 ), ξkω,e (A2 ), . . . , ξkω,e (Am ) , Xkω = ξkω,e (A1 ), ξkω,e (A2 ), . . . , ξkω,e (Am ) . and let φk , φk : Rm → C be the corresponding characteristic functions ω

φk (t) = EeitXk , Then for all t ∈ Rm ,

ω

φk (t) = EeitXk ,

t ∈ Rm .

lim φk (t) − φk (t) = 0 .

k→∞

Then we establish Proposition 3.10. The point process ξkω,e converges to a Poisson point process on R with intensity η(e)L. Proposition 3.10 and Corollary 3.9 together imply Theorem 3.7. The Wegner estimate plays a crucial role in the proofof Propositions 3.8 and 3.10. For ω every Borel set A ⊂ R, we have ξkω,e (A) = x∈Bk δx |f (Hk )δx , where f (t) = 1A (|Bk | (t − e)). Wegner estimate (3.10) yields that for all x ∈ Bk , −1 ω (3.16) E δx |f (Hk )δx ≤ γ ∞ f (t)dt = γ ∞ |Bk | L(A). Summing (3.16) over all x ∈ Bk yields Eξkω,e (A) ≤ γ ∞ L(A) .

(3.17)

Eξkω,e (A) ≤ γ ∞ L(A) .

(3.18)

Similarly

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Proof of Proposition 3.8. Step 1: We first prove (3.15) for the family of functions gz (t) = Im(t − z)−1 ,

Imz > 0 .

Setting −1

zk = e + |Bk |

z,

(3.19)

we have $ % −1 ω,e δx (Hrωk − zk )−1 − (Hkω − zk )−1 δx . gz dξk − gz dξkω,e = |Bk | Im x∈Bk

Hence

2k ∞ n −2 gz dξ ω,e − gz dξ ω,e ≤ |Imzk |−2 ps = const |Imz| k k ρr k s=rk +1

The formulas (3.13) and (3.14) imply that for large enough k, ρrk ≥ ρc k where d < c < c < 1. Therefore 2 k n n2k ≤ , r k ρ ρc and

n2 ρc

< 1 because of the formula (3.1). This proves (3.15). Step 2: To prove (3.15) for general f ∈ L1 (R, dt), note that span {gz , Imz > 0} is dense in L1 (R, dt). Hence given ε > 0, there is a finite linear combination g(t) =

p

aj Im(t − z (j) )−1 ,

Imz (j) > 0 ,

j=1

with

R

|f (t) − g(t)| dt ≤ ε .

The triangle inequality ω,e ω,e E f dξk − f dξk (t) ≤ E |f − g| dξkω,e ω,e + E gdξk − gdξk + E |g − f | dξkω,e , together with Step 1 and the bounds (3.17) and (3.18) imply ω,e ω,e lim sup E f dξk − f dξk (t) ≤ 2 γ ∞ ε , k→∞

and (3.15) follows after letting ε ↓ 0.

ω,e Proof of Proposition 3.10. I suffices to show that ξkω,e and the ξk,j verify the four hypotheses of Theorem 2.5.

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(H0): holds because of the bound (3.18). (H1): we need to to establish that for every bounded Borel set A ⊂ R, ω,e lim max P ξk,j (A) ≥ 1 = 0 . k→∞ 1≤j≤nk−rk

703

(3.20)

Proof. Chebyshev’s inequality and the bound (3.16) yield ω,e ω,e P ξk,j (A) ≥ 1 ≤ Eξk,j (A) ≤

|Bk,j |

γ ∞ L(A) |Bk |

= nrk −k γ ∞ L(A) ,

and (3.20) follows. (H2): We need to establish that for all Imz > 0, lim E Im(t − z)−1 dξkω,e (t) = πη(e) . k→∞

Proof. We have −1 δx |(Hrωk − zk )−1 δx E Im(t − z)−1 dξkω,e (t) = |Bk | EIm x∈Bk −1

= |Bk |

EIm

$ % δx (Hrωk − zk )−1 − (H ω −zk )−1 δx

x∈Bk

+ EIm δx0 |(H ω − zk )−1 δx0 = Ik,ω + IIk,ω . Now IIk,ω → πη(e) by 3.12 and Ik,ω → 0, as in the proof of Proposition 3.8.

(H3): We need to establish that for every function gz (t) = Im(t − z)−1 , Imz > 0, lim

k→∞

k−rk n

ω,e EI(ξk,j , gz ) = 0 .

(3.21)

j=1

Proof. We have, ⎞2 2 ω,e |Bk | I(ξk,j δx |Im(Hrωk − zk )−1 δx ⎠ , gz ) = ⎝ ⎛

x∈Bk,j

−

$ 2 % δx Im(Hrωk − zk )−1 δx

x∈Bk,j

=

x,y∈Bk,j

δ |Im(Hrωk −zk )−1 δx δx |Im(Hrωk −zk )−1 δy det x . δy |Im(Hrωk −zk )−1 δx δy |Im(Hrωk −zk )−1 δy

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Using Minami’s estimate (3.11) we get the bounds ω,e |Bk |2 EI(ξk,j , gz ) ≤ π 2 γ 2∞ |Bk,j |2 ,

and hence k−rk n

ω,e EI(ξk,j , gz ) ≤ π 2 γ ∞ n−rk , 2

j=1

which yields (3.21).

Appendix A. Minami’s proof of Poisson statistics for the localized Anderson model on Zd ω For a rectangle B ⊂ Zd , we denote by HB the restriction of Hω to l2 (B) with ω Dirichlet boundary conditions: i.e. δx |HB δy = δx |Hω δy if both x, y ∈ B, and ω δx |HB δy = 0 otherwise. For k ≥ 1, let Bk be the rectangle {x ∈ Zd : maxi=1,...,d ω |xi | ≤ k}, and let Hkω = HB . As before, eω,k ≤ eω,k ≤ · · · ≤ eω,k 1 2 |Bk | , are the k ω 2 ω eigenvalues of Hk l (Bk ), μk is the corresponding normalized counting measure given by (1.1) and ξkω,e is the rescaled measure near e given by (1.2). We refer the reader to [18] for a discussion of the regime where both space and energy are rescaled. The averaged spectral measure for Hω is given by (3.3) and the Wegner estimate yields that μav has a bounded density η(t) with respect to L. A basic result for the Anderson model is that for P-a.e. ω ∈ Ω, the spectrum of Hω is av equal to [−2d, 2d] + supp(γ) = supp(μav ) and μω in the weak-* k converges to μ topology as k → ∞ [6, 8, 29].

Theorem A.1 (Minami, 1996). Assume that there are constants 0 < C < ∞, 0 < D < ∞ and 0 < s < 1 such that s ω − z)−1 δy ≤ Ce−D|x−y| , x, y, ∈ Zd , (A.1) E δx |(HB for all z with e1 < Rez < e2 , Imz = 0 and for all rectangles B ⊂ Zd . Assume that e ∈ (e1 , e2 ) verifies the regularity Condition (3.12) and that η(e) > 0. Then ξkω,e converges to a Poisson point process on R with intensity η(e)L. We refer the reader to [14] for a discussion of the set of e for which η(e) > 0. Condition (A.1) is called fractional-moments localization. It implies that within (e1 , e2 ), for P-a.e. ω ∈ Ω the spectrum of Hω , if any, is pure-point with exponentially decaying eigenfunctions [1, 2]. For d = 1, Condition (A.1) holds for all energy intervals (e1 , e2 ) [25]. In dimensions d ≥ 2, condition (A.1) is obtained by either moving the energy interval (e1 , e2 ) to ±∞ or by increasing the disorder. The two main techniques for proving that are the multiscale analysis [10, 11] and the Aizenman–Molchanov theory [1].

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Proof of Theorem A.1. We fix α ∈ (0, 1) and for each k, we make a partition Bk =

nk

Bk,j ,

j=1

where Bk,j are disjoint rectangles with side ∼ (2k)α . Hence nk ∼ k d(1−α) . Let ω , l = 1, . . . , |Bk,j | denote the eigenvalues of HB l2 (Bk,j ) and let eω,k,j l k,j |Bk,j | ω,e ξk,j

=

δ |Bk | (eω,k,j − e) , l l=1

ξkω,e =

nk

ω,e . ξk,j

j=1

ξkω,e

ω,e Hence the point process is the sum of nk independent point processes ξk,j . As in Section 3, Theorem A.1 follows from the following two propositions.

Proposition A.2. For every f ∈ L1 (R, dt), lim E f dξkω,e − f dξkω,e = 0 . k→∞

(A.2)

Proposition A.3. The point process ξkω,e converges to a Poisson point process on R with intensity η(e)L. Proof of Theorem A.2. As in the proof of Proposition 3.8, it is enough to prove (A.2) for the family of functions gz (t) = Im(t − z)−1 , We set

−1

zk = e + |Bk | Then

Imz > 0 . z.

(A.3)

gz dξkω,e −

gz dξkω,e −1

= |Bk |

Im

nk % $ ω −1 ω −1 δx (HB δx . − z ) − (H − z ) k k k k,j j=1 x∈Bk,j

Let vk = β ln k, where β > 0 is a fixed big enough constant to be specified later. We set

int(Bk,j ) = x ∈ Bk,j : dist(x, ∂Bk,j ) ≥ vk , and

wall(Bk,j ) = x ∈ Bk,j : dist(x, ∂Bk,j ) < vk . Then ω,e ω,e E gz dξk − gz dξk ≤ E |Ik,ω | + E |IIk,ω | ,

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where −1

Ik,ω = |Bk |

Im

nk


& ' ω −1 ω −1 δx , − z ) − (H − z ) δx (HB k k k k,j

j=1 x∈wall(Bk,j ) −1

IIk,ω = |Bk |

Im

nk

& ' ω −1 ω −1 δx . − z ) − (H − z ) δx (HB k k k k,j

j=1 x∈int(Bk,j )

The Wegner estimate (3.18) yields that −1

E |Ik,ω | ≤ 2π γ ∞ |Bk |

nk

|wall(Bk,j )| ,

j=1

and the right hand side converges to zero as k → ∞. To estimate E |IIk,ω |, we use the resolvent identity & ' ω −1 ω −1 δx − z ) − (H − z ) δx (HB k k k k,j ω − zk )−1 δy δy |(Hkω − zk )−1 δx , = δx |(HB k,j (y,y )

where the sum is over all pairs (y, y ), with y ∈ ∂Bk,j , y ∈ / Bk,j and |y − y | = 1. Hence, E |IIk,ω | −1

≤ |Bk |

nk

ω −1 ω −1 δ E δx |(HB − z ) δ |(H − z ) δ k y y k x . k k,j

j=1 x∈int(Bk,j ) (y,y )

(A.4) For k large enough so that e1 < Rezk < e2 , we use the main assumption (A.1) together with the bound ω −1 ω −1 −2 δ − z ) δ |(H − z ) δ = (|Bk | /Imz)2 , δx |(HB k y y k x ≤ (Imzk ) k k,j to obtain ω −1 ω −1 δ − z ) δ |(H − z ) δ E δx |(HB k y y k x k k,j s/2 ω ≤ (|Bk | /Imz)2(1−s/2) E δx |(HB − zk )−1 δy δy |(Hkω − zk )−1 δx k,j s 1/2 ω −1 ≤ (|Bk | /Imz)2(1−s/2) E δx |(HB − z ) δ k y k,j s 1/2 E δy |(Hkω − zk )−1 δx ≤ (|Bk | /Imz)2(1−s/2) Ce−Dvk .

(A.5)

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Since, in (A.4), there are O(k α(d−1) ) pairs (y, y ) for each Bk,j , the bounds (A.4) and (A.5) yield 2(1−s/2) −Dvk

E |IIk,ω | ≤ O(k α(d−1) |Bk | = O(k

e

α(d−1)+2d(1−s/2) −Dβ ln k

e

) )

Hence, if we choose β > D−1 (α(d − 1) + 2d(1 − s/2)), then E |IIk,ω | → 0 as k → ∞. Proof of Proposition A.3. As in the proof of Propositon 3.10, it suffices to show ω,e verify the four hypotheses of Theorem 2.5. The proof of that ξkω,e and the ξk,j (H0), (H1) and (H3) is the same as in Propositon 3.10. It remains to show that (H2) holds, i.e. for Imz > 0, lim E gz dξkω,e = πη(e) . (A.6) k→∞

The argument of the proof of Proposition A.2, with Hkω replaced by Hω , yields that ω,e av gz dξk − gz dμ = 0, (A.7) lim E k→∞

and then (A.6) follows from (A.7) and (3.12).

Acknowledgements We are grateful to Vojkan Jaksic for suggesting this research project. We benefited from discussions with Michael Aizenman, Vojkan Jaksic, Rowan Killip, Stas Molchanov and Mihai Stoiciu. This work was supported by FQRNT, ISM and McGill Majors grants.

References [1] M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Comm. Math. Phys. 157 (1993), 245–278. [2] M. Aizenman, J. Schenker, R. Friedrich, and D. Hundertmark, Finite-volume fractional moment criteria for Anderson localization, Comm. Math. Phys. 224 (2001), 219–253. [3] M. Aizenman and S. Warzel, The canopy graph and level statistics for random operators on trees, Mathematical Physics, Analysis and Geometry 9, no. 4, (2006). [4] J. V. Bellissard, P. D. Hislop, and G. Stolz, Correlations estimates in the lattice Anderson model, to appear in J. Statist. Phys. [5] A. Bovier, The density of states in the Anderson model at weak disorder: A renormalization group analysis of the hierarchical model, J. Statist. Phys. 59 (1990), no. 3–4, 745–779.

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[6] R. Carmona and J. Lacroix, Spectral Theory of Random Schr¨ odinger Operators, Birkh¨ auser Boston, Inc., Boston, MA, 1990. [7] F. Constantinescu, J. Fr¨ ohlich and T. Spencer, Analyticity of the density of states and replica method for random Schr¨ odinger operators on a lattice, J. Statist. Phys. 34 (1984), no. 3–4, 571–596. [8] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics. Springer Study Edition. Springer-Verlag, Berlin, 1987. [9] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics. Springer-Verlag, New York, 1988. [10] H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124 (1989), 285–299. [11] J. Fr¨ ohlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151–184. [12] G. M. Graf and A. Vaghi, A Remark on the estimate of a determinant by Minami, Lett. Math. Phys. 79 (2007), no. 1, 17–22. [13] B. Grigelionis, On the convergence of sums of random step processes to a Poisson process, Theory of Probability and its Applications 8 (1963), 177–182. [14] P. D. Hislop and P. M¨ uller, A lower bound for the density of states of the lattice Anderson model, http://front.math.ucdavis.edu/0705.1707. [15] V. Jaksic, Topics in spectral theory. Open Quantum Systems I. The Hamiltonian Approach, Lecture Notes in Mathematics 1880 (2006), 235–312, Springer. [16] O. Kallenberg, Random Measures, Academic Press. 4th. ed. 1986. [17] J. F. C. Kingman, Poisson Processes, Oxford Studies in Probability, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. [18] R. Killip and F. Nagano, Eigenfunction statistics in the localized Anderson model, Ann. Henri Poincaré 8 (2007), 27–36. [19] R. Killip and M. Stoiciu, Eigenvalue statistics for CMV matrices: From Poisson to clock via circular beta ensembles, arXiv preprint - math-ph/0608002. [20] P. Koosis, Introduction to Hp Spaces, Second edition. Cambridge Tracts in Mathematics, 115. Cambridge University Press, Cambridge, 1998. [21] E. Kritchevski, Spectral localization in the hierarchical Anderson model, Proc. Amer. Math. Soc. 135 (2007), 1431–1440. [22] E. Kritchevski, Hierarchical Anderson model, CRM Proceedings and Lecture Notes 42 (2007), 309–322. [23] E. Kritchevski, PhD Thesis, McGill University. In preparation. [24] P. Lloyd, Exactly solvable model of electronic states in a three-dimensional disordered Hamiltonian: Non-existence of localized states, J. Phys. C: Solid State Phys. 2 no. 10 (Oct. 1969), 1717–1725. [25] N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tightbinding model, Comm. Math. Phys. 177 (1996), 709–725. [26] S. Molchanov, The local structure of the spectrum of the one-dimensional Schr¨ odinger operator, Comm. Math. Phys. 78 (1981), 429–446.

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[27] S. Molchanov, Lectures on random media. Lectures on probability theory (SaintFlour, 1992), 242–411, Lecture Notes in Math., 1581, Springer, Berlin, 1994. [28] S. Molchanov, Hierarchical random matrices and operators. Application to Anderson model. Multidimensional statistical analysis and theory of random matrices, (Bowling Green, OH, 1996), 179–194, VSP, Utrecht, 1996. [29] L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 297. Springer-Verlag, Berlin, 1992. [30] M. Stoiciu, Poisson statistics for eigenvalues: From random Schr¨ odinger operators to random CMV matrices, CRM Proceedings and Lecture Notes 42 (2007), 465–475. [31] F. Wegner, Bounds on the density of states in disordered systems, Z. Phys. B. 44 (1981), 9–15. Evgenij Kritchevski Department of Mathematics and Statistics McGill University 805 Sherbrooke Street West Montreal QC, H3A 2K6 Canada e-mail: [email protected] Communicated by Claude-Alain Pillet. Submitted: October 8, 2007. Accepted: December 17, 2007.

Ann. Henri Poincaré 9 (2008), 711–742 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/040711-32, published online June 6, 2008 DOI 10.1007/s00023-008-0370-z


Hartree–Fock Theory for Pseudorelativistic Atoms Anna Dall’Acqua, Thomas Østergaard Sørensen, and Edgardo Stockmeyer Abstract. We study the Hartree–Fock model for pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudorelativistic operator (|p|c)2 + (mc2 )2 − mc2 . We prove the existence of a Hartree–Fock minimizer, and prove regularity away from the nucleus and pointwise exponential decay of the corresponding orbitals.

1. Introduction and results We consider a model for an atom with N electrons and nuclear charge Z, where the kinetic energy of the electrons is described by the expression (|p|c)2 + (mc2 )2 − mc2 . This model takes into account some (kinematic) relativistic effects; in units where = e = m = 1, the Hamiltonian becomes N 1 Z −2 −2 −4 −α Δj + α − α − H = Hrel (N, Z, α) = + |x | |x − xj | j i j=1 1≤i<j≤N

=

N j=1

α−1 T (−i∇j ) − V (xj ) +

1≤i<j≤N

1 , |xi − xj |

(1)

with T (p) = E(p) − α−1 = |p|2 + α−2 − α−1 and V (x) = Zα/|x|. Here, α is Sommerfeld’s fine structure constant; physically, α 1/137.036. The operator H acts on a dense subspace of the N -particle Hilbert space 2 3 q HF = ∧N i=1 L (R ; C ) of antisymmetric functions, where q is the number of spin states. It is bounded from below on this subspace (more details below). The (quantum) ground state energy is the infimum of the spectrum of H considered as an operator acting on HF : E QM (N, Z, α) := inf σHF (H) = inf q(Ψ, Ψ) | Ψ ∈ Q(H), Ψ, Ψ = 1 ,

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where q is the quadratic form defined by H, and Q the corresponding form domain N (see below); , is the scalar product in HF ⊂ L2 (R3N ; Cq ). In the Hartree–Fock approximation, instead of minimizing the functional q in the entire N -particle space HF , one restricts to wavefunctions Ψ which are pure wedge products, also called Slater determinants:

N 1 det ui (xj , σj ) i,j=1 , Ψ(x1 , σ1 ; x2 , σ2 ; . . . ; xN , σN ) = √ (2) N! 2 3 q with {ui }N i=1 orthonormal in L (R ; C ) (called orbitals). Notice that this way, Ψ ∈ HF and Ψ L2 (R3N ;CqN ) = 1. The Hartree–Fock ground state energy is the infimum of the quadratic form q defined by H over such Slater determinants: (3) E HF (N, Z, α) := inf q(Ψ, Ψ) | Ψ Slater determinant . For the non-relativistic Hamiltonian, N Z 1 Hcl (N, Z) = − Δj − + 2 |xj | j=1

1≤i<j≤N

1 , |xi − xj |

(4)

the mathematical theory of this approximation has been much studied, the groundbreaking work being that of Lieb and Simon [19]; see also [21] for work on excited states. For a comprehensive discussion of Hartree–Fock (and other) approximations in quantum chemistry, and an extensive literature list, we refer to [16]. The aim of the present paper is to study the Hartree–Fock approximation for the pseudorelativistic operator H in (1). (See Remark 2, below, for a discussion of other relativistic models.) We turn to the precise description of the problem. The one-particle operator h0 = T (−i∇)− V (x) is bounded from below (by α−1 [(1 − (πZα/2)2 )1/2 − 1]) if and only if Zα ≤ 2/π (see [13], [15, 5.33 p. 307], and [33]; we shall have nothing further to say on the critical case Zα = 2/π). More precisely, if Zα < 1/2, then V is a small operator perturbation of T . In fact [13, Theorem 2.1 c)], |x|−1 (T (−i∇) + 1)−1 B(L2 (R3 )) = 2. As a consequence, h0 is selfadjoint with D(h0 ) = H 1 (R3 ; Cq ) when Zα < 1/2. It is essentially selfadjoint on C0∞ (R3 ; Cq ) when Zα ≤ 1/2. If, on the other hand, 1/2 ≤ Zα < 2/π, then V is only a small form perturbation of T : Indeed [15, 5.33 p. 307], π |f (x)|2 dx ≤ |p||fˆ(p)|2 dp for f ∈ H 1/2 (R3 ) , (5) |x| 2 3 3 R R where fˆ denotes the Fourier transform of f . Hence, the quadratic form v given by v[u, v] := (V 1/2 u, V 1/2 v)

for u, v ∈ H 1/2 (R3 ; Cq )

(6)

(multiplication by V 1/2 in each component) is well defined (for all values of Zα). Here, ( , ) denotes the scalar product in L2 (R3 ; Cq ). Let e be the quadratic form with domain H 1/2 (R3 ; Cq ) given by

e[u, v] := E(p)1/2 u, E(p)1/2 v for u, v ∈ H 1/2 (R3 ; Cq ) . (7)

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By abuse of notation, we write E(p) for the (strictly positive) operator E(−i∇) = √ −Δ + α−2 . Then, using (5) and that |p| ≤ E(p), v[u, u] ≤ Zα

π e[u, u] for u ∈ H 1/2 (R3 ; Cq ) \ {0} . 2

(8)

Hence, by the KLMN theorem [24, Theorem X.17], if Zα < 2/π there exists a unique self-adjoint operator h0 whose quadratic form domain is H 1/2 (R3 ; Cq ) such that (with t = e − α−1 ) (u, h0 v) = t[u, v] − v[u, v] for

u, v ∈ H 1/2 (R3 ; Cq ) ,

(9)

and h0 is bounded below by − α−1 . Moreover, if Zα < 2/π then the spectrum of h0 is discrete in [−α−1 , 0) and absolutely continuous in [0, ∞) [13, Theorems 2.2 and 2.3]. As for the N -particle operator in (1), when Zα < 2/π, (5) implies that the quadratic form q(Ψ, Φ) =

N

E(pj )1/2 Ψ, E(pj )1/2 Φ − α−1 Ψ, Φ − V (xj )1/2 Ψ, V (xj )1/2 Φ

j=1

+

|xi − xj |−1/2 Ψ, |xi − xj |−1/2 Φ ,

Ψ, Φ ∈

1≤i<j≤N

N

H 1/2 (R3 ; Cq ) ,

i=1

is well-defined, closed, and bounded from below. The operator H can then be defined as the corresponding (unique) self-adjoint operator. It satisfies N

H 1 (R3 ; Cq ) ⊂ D(H) ⊂ Q(H) =

i=1

N

H 1/2 (R3 ; Cq ) ,

i=1

q(Ψ, Φ) = Ψ, HΦ ,

Φ ∈ D(H) ,

Ψ ∈ Q(H) .

1 3 q For Zα < 1/2, D(H) = ∧N i=1 H (R ; C ). All this follows from (the statements and proofs of) [24, Theorem X.17] and [27, Theorem VIII.15]. See [20] for further references on H. We shall not have anything further to say on H in this paper, however, but will only study the Hartree–Fock problem mentioned above. We now discuss this in more detail. It is convenient to use the one-to-one correspondence between Slater determinants and projections onto finite dimensional subspaces of L2 (R3 ; Cq ). Indeed, if Ψ 1/2 is given by (2) with {ui }N (R3 ; Cq ), orthonormal in L2 (R3 ; Cq ), and γ is i=1 ⊂ H the projection onto the subspace spanned by u1 , . . . , uN , then the kernel of γ is given by

γ(x, σ; y, τ ) =

N j=1

uj (x, σ)uj (y, τ ) .

(10)

714



Let ργ ∈ L1 (R3 ) denote the 1-particle density associated to γ given by ργ (x) =

q

γ(x, σ; x, σ) =

σ=1

q N

|uj (x, σ)|2 .

σ=1 j=1

Then the energy expectation of Ψ depends only on γ, more precisely, q(Ψ, Ψ) = Ψ, HΨ = E HF (γ) , where E HF is the Hartree–Fock energy functional defined by E HF (γ) = α−1 Tr E(p)γ − α−1 Tr[γ] − Tr[V γ] + D(γ) − Ex(γ) .

(11)

Here, N Tr E(p)γ := e[uj , uj ] ,

Tr[V γ] :=

j=1

N

v[uj , uj ] = Zα

j=1

D(γ) is the direct Coulomb energy, 1 ργ (x)ργ (y) D(γ) = dx dy , 2 R3 R3 |x − y|

R3

ργ (x) dx , |x|

(12)

and Ex(γ) is the exchange Coulomb energy, q 1 |γ(x, σ; y, τ )|2 Ex(γ) = dx dy . 2 σ,τ =1 R3 R3 |x − y| This way,

E HF (N, Z, α) = inf E HF (γ) | γ ∈ P , (13) 2 3 q 2 3 q P = γ : L (R ; C ) → L (R ; C ) | γ projection onto span{u1 , . . . , uN }, ui ∈ H 1/2 (R3 ; Cq ), (ui , uj ) = δi,j . (Notice that if one of the orbitals ui of γ is not in H 1/2 (R3 ; Cq ), then E HF (γ) = +∞ (since Zα < 2/π).) We now extend the definition of the Hartree–Fock energy functional E HF , in order to turn the minimization problem (13) (that is, (3)) into a convex problem. A density matrix γ : L2 (R3 ; Cq ) → L2 (R3 ; Cq ) is a self-adjoint trace class operator that satisfies the operator inequality 0 ≤ γ ≤ Id. A density matrix γ has the integral kernel λj uj (x, σ)uj (y, τ ) , (14) γ(x, σ; y, τ ) = j

where λj , uj are the eigenvalues and corresponding eigenfunctions of γ. We choose the uj ’s to be orthonormal in L2 (R3 ; Cq ). As before, let ργ ∈ L1 (R3 ) denote the 1-particle density associated to γ given by q 2 ργ (x) = λj |uj (x, σ)| . (15) σ=1

j

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Define

A := γ density matrix Tr E(p)γ < +∞ ,

(16)

where, by definition, for γ written as in (14), Tr E(p)γ := λj e[uj , uj ] .

(17)

j

Notice that if γ ∈ A then all the terms in E HF (γ) (see (11)) are finite. Indeed, for γ ∈ A and written as in (14), ργ (x) dx (18) λj v[uj , uj ] = Zα Tr[V γ] := |x| 3 R j is finite, due to (8). In particular, uj ∈ H 1/2 (R3 ; Cq ) ⊂ L3 (R3 ; Cq ) ,

(19)

the last inclusion by Sobolev’s inequality [18, Theorem 8.4]. On the other hand, if γ ∈ A then ργ ∈ L1 (R3 ) ∩ L4/3 (R3 ) .

(20)

This follows from Daubechies’ inequality, see [6, pp. 519–520]. By H¨ older’s inequality, ργ ∈ L6/5 (R3 ). The Hardy–Littlewood–Sobolev inequality [18, Theorem 4.3] then implies that D(γ) (see (12)) is finite. Finally, Ex(γ) ≤ D(γ), since D(γ) − Ex(γ) =

q |ui (x, σ)uj (y, τ ) − uj (x, σ)ui (y, τ )|2 1 dxdy ≥ 0 . λi λj 2 i,j |x − y| 3 R3 σ,τ =1 R

Therefore, E HF defined by (11) extends to γ ∈ A. This way, with h0 defined as in (9), Tr[h0 γ] = Tr E(p)γ − α−1 Tr[γ] − Tr[V γ] , and so E HF (γ) = α−1 Tr[h0 γ] + D(γ) − Ex(γ) , γ ∈ A . Consider γ ∈ A and define, with ργ as in (15), ργ (y) dy . Rγ (x) := R3 |x − y|

(21)

(22)

We have that Rγ ∈ L∞ (R3 ) ∩ L3 (R3 ) .

(23)

∞

This follows from (8) (for L ), and (20) and the weak Young inequality [18, p. 107] (for L3 ). Next, define the operator Kγ with integral kernel Kγ (x, σ; y, τ ) :=

γ(x, σ; y, τ ) . |x − y|

The operator Kγ is Hilbert–Schmidt; we prove this fact in Lemma 2 below.

(24)

716



Note that, using (14) and the Cauchy–Schwarz inequality, (u, Rγ u) ≥ (u, Kγ u) (multiplication by Rγ is in each component). Denote by bγ the (non-negative) quadratic form given by bγ [u, v] := α(u, Rγ v) − α(u, Kγ v) Then, using (u, Kγ u) ≥ 0 and (8), q 0 ≤ bγ [u, u] ≤ α(u, Rγ u) = α σ=1

R3

R3

for u, v ∈ H 1/2 (R3 ; Cq ) .

2 ργ (y)|u(x, σ)|2 dx dy ≤ α Tr[γ] e[u, u] . |x − y| π

Therefore (by the statements and proofs of [24, Theorem X.17] and [27, Theorem VIII.15]), there exists a unique self-adjoint operator hγ (called the Hartree– Fock operator associated to γ), which is bounded below (by −α−1 ), with quadratic form domain H 1/2 (R3 ; Cq ) and such that (u, hγ v) = t[u, v] − v[u, v] + bγ [u, v] for

u, v ∈ H 1/2 (R3 ; Cq ) .

(25)

The operator hγ has infinitely many eigenvalues in [−α−1 , 0) (when N < Z), and σess (hγ ) = [0, ∞); both of these facts will be proved in Lemma 2 below. The main result of this paper is the following theorem. Theorem 1. Let Zα < 2/π, and let N ≥ 2 be a positive integer such that N < Z +1. Then there exists an N -dimensional projection γ HF = γ HF (N, Z, α) minimizing the Hartree–Fock energy functional E HF given by (11), that is, E HF (N, Z, α) in (13) (and therefore, in (3)) is attained. In fact, E HF (γ HF ) = E HF (N, Z, α) = inf E HF (γ) γ ∈ A, γ 2 = γ, Tr[γ] = N = inf E HF (γ) γ ∈ A, Tr[γ] = N = inf E HF (γ) γ ∈ A, Tr[γ] ≤ N . (26) Moreover, one can write γ HF (x, σ; y, τ ) =

N

ϕi (x, σ)ϕi (y, τ ) ,

(27)

i=1

with ϕi ∈ H 1/2 (R3 ; Cq ), i = 1, . . . , N , orthonormal, such that the Hartree–Fock orbitals {ϕi }N i=1 satisfy: (i) With hγ HF as defined in (25), hγ HF ϕi = εi ϕi ,

i = 1, . . . , N ,

(28)

with 0 > εN ≥ · · · ≥ ε1 > − α−1 the N lowest eigenvalues of hγ HF . (ii) For i = 1, . . . , N ,

(29) ϕi ∈ C ∞ R3 \ {0}; Cq . (iii) For all R > 0 and β < νεN := −εN (2α−1 + εN ), there exists C = C(R, β) > 0 such that for i = 1, . . . , N , |ϕi (x)| ≤ C e−β|x|

for

|x| ≥ R .

(30)

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717

Remark 1. (i) In fact, we prove that (29) holds for any eigenfunction ϕ of hγ HF , and (30) for those corresponding to negative eigenvalues ε. More precisely, if hγ HF ϕ = εϕ for some ε ∈ [εN , 0), then (30) holds for ϕ for all β < νε := −ε(2α−1 + ε) for some C = C(R, β) > 0. (ii) Note that, in general, eigenfunctions of hγ HF can be unbounded at x = 0; therefore (29) and (30) can only be expected to hold away from the origin. (iii) Both the regularity and the exponential decay above are similar to the results in the non-relativistic case (i.e., for the operator in (4); see [19]). However, the proof of Theorem 1 is considerably more complicated due to, on one hand, the non-locality of the kinetic energy operator E(p), and, on the other hand, the fact that the Hartree–Fock operator hγ HF is only given as a form sum for Zα ∈ [1/2, 2/π). (iv) We show the existence of the Hartree–Fock minimizer by solving the minimization problem on the set of density matrices. This method was introduced in [30]. The same method was used in [5] in the Dirac–Fock case (see Remark 2 below). (v) Notice that nothing is known on the question of uniqueness of the minimizer of the Hartree–Fock functional defined on density matrices (up to the trivial invariance properties of the Hartree–Fock energy functional) [16]. The Hartree–Fock functional is not convex. This is a major difference compared to the reduced (restricted) Hartree–Fock theory. The reduced Hartree–Fock functional has no exchange term and so the uniqueness of the minimizer is assured by the convexity of the functional; see [30]. (vi) For any eigenfunction of the Hartree–Fock operator that is orthogonal to the Hartree–Fock orbitals ϕ1 , . . . , ϕN the corresponding eigenvalue ε satisfies ε > εi , i = 1, . . . , N . In other words, there are no unfilled shells. This follows from the result in [4] since the only crucial assumption is that the two-body interaction is repulsive (i.e., positive definite). The particular choice of the one-particle operator does not play any role. (vii) As mentioned earlier, we have to assume that Zα < 2/π; the reason is that our proof that Tr[E(p)γn ] is uniformly bounded for a minimizing sequence {γn }n∈N does not work in the critical case Zα = 2/π. (viii) As will be clear from the proofs, the statements of Theorem 1 (appropriately modified) also hold for molecules. More explicitely, for a molecule with K nuclei of charges Z1 , . . . , ZK , fixed at R1 , . . . , RK ∈ R3 , replace v in (6) by v[u, v] :=

K k=1

1/2

(Vk

1/2

u, Vk

v)

for u, v ∈ H 1/2 (R3 ; Cq ) ,

(31)

K with Vk (x) = Zk α/|x − Rk |, Zk α < 2/π. Then, for N < 1 + k=1 Zk , there exists a Hartree–Fock minimizer, and the corresponding Hartree–Fock orbitals have the regularity and decay properties as stated in Theorem 1, away from each nucleus.

718



Remark 2. In our model the kinetic energy of the (relativistic) electrons is given by a non-local operator. Another choice would be to consider the Dirac operator: D0 = α · (−i∇) + βα−1 with α = (α1 , α2 , α3 ), β the Dirac matrices (α is still the fine structure constant); see [32]. The Dirac operator is local but it has a negative continuous spectrum which is not bounded from below. The analogue of the Hartree–Fock approximation in this model is called the Dirac–Fock model. Esteban and Séré [7] proved that the Dirac–Fock functional has infinitely many critical points, giving rise to infinitely many solutions to the Dirac–Fock equations; see also [23]. In this model the rigorous definition of a ground state is a delicate problem since the energy functional is not bounded from below; see [8], [9]. Nevertheless, there are Hartree–Fock-type models, coming from the Dirac operator, that do have a minimizer. We refer to [5], [11], and [12], and the references therein, for the description of these models.

2. Proof of Theorem 1 For simplicity of notation, we give the proof of Theorem 1 only in the spinless case. It will be obvious that the proof also works in the general case. 2.1. Existence of the Hartree–Fock minimizer The proof of the existence of an N -dimensional projection γ HF minimizing E HF , the equalities in (26), and that the corresponding Hartree–Fock orbitals {ϕi }N i=1 solve the Hartree–Fock equations (28), will be a consequence of the following two lemmas. Lemma 1. Let Zα < 2/π and N ∈ N. Then HF E≤ (N, Z, α) := inf E HF (γ) γ ∈ A, Tr[γ] ≤ N is attained. Lemma 2. Let γ ∈ A. Then the operator Kγ , defined by (24), is Hilbert–Schmidt. If Zα < 2/π then the operator hγ , defined in (25), satisfies σess (hγ ) = [0, ∞). If furthermore Tr[γ] < Z, then hγ has infinitely many eigenvalues in [−α−1 , 0). Before proving these two lemmas, we use them to prove the parts of Theorem 1 mentioned above. Proof. For computational reasons we first state and prove a lemma in the spirit of [3, Lemma 1]. Lemma 3. Let γ ∈ A, u1 , u2 ∈ H 1/2 (R3 ), and let 1 , 2 ∈ R be such that γ˜ given by γ˜ (x, y) := γ(x, y) + γu (x, y) , γu (x, y) := γu1 ,u2 (x, y) = 1 u1 (x)u1 (y) + 2 u2 (x)u2 (y) is again an element of A.

(32) (33)

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Then we have that E HF (˜ γ ) = E HF (γ) + α−1 1 (u1 , hγ u1 ) + α−1 2 (u2 , hγ u2 ) + 1 2 Ru , where hγ is given in (25), and |u1 (x)u2 (y) − u2 (x)u1 (y)|2 1 dxdy . Ru := Ru1 ,u2 = 2 R3 R3 |x − y| Proof of Lemma 3. We have that E

HF

(˜ γ) = E

HF

(γ) + α

−

R3

1 2

R3

R3

(35)

ργ (x)ργu (y) dxdy |x − y| 1 γ(x, y)γu (x, y) ργu (x)ργu (y) dxdy + dxdy |x − y| 2 R3 R3 |x − y| −1

−

(34)

R3

Tr[h0 γu ] +

R3

R3

γu (x, y)γu (x, y) dxdy |x − y|

= E HF (γ) + α−1 1 (u1 , hγ u1 ) + α−1 2 (u2 , hγ u2 ) (36) 1 ργu (x)ργu (y) γu (x, y)γu (x, y) 1 dxdy − dxdy . + 2 R3 R3 |x − y| 2 R3 R3 |x − y| Using (33), that ργu (x) = 1 |u1 (x)|2 + 2 |u2 (x)|2 , and (35), we obtain (34).

By Lemma 1 a minimizer γ HF ∈ A, with Tr[γ HF ] ≤ N, exists. We may write λk ϕk (x)ϕk (y) , (37) γ HF (x, y) = k 1/2

with 1 ≥ λ1 ≥ · · · ≥ 0 and {ϕk }k ⊂ H (R3 ) an orthonormal (in L2 (R3 )) system (it might be finite). Extend {ϕk }k to an orthonormal basis {ϕk }k ∪ {u }∈N for L2 (R3 ), with u ∈ H 1/2 (R3 ). Let K + 1 be the first index such that λK+1 < 1. Fix j ∈ {1, . . . , K}, choose u ∈ {ϕk }k≥K+1 ∪ {u }∈N , and consider, for to be chosen,

1 ϕj (x) + u(x) ϕj (y) + u(y) . λk ϕk (x)ϕk (y) + γ(j) (x, y) := 2 1 + m k=j

(j)

(j)

Choosing m ≥ 1 assures that Tr[γ ] ≤ N . Then 0 ≤ γ ≤ Id for || small enough (j) (depending on u). Since γ HF minimizes E HF , and γ0 = γ HF , d HF (j) (E ) γ 0= = α−1 (ϕj , hγ HF u) + α−1 (u, hγ HF ϕj ) . d =0

Repeating the computation for iu we get that (u, hγ HF ϕj ) = 0, from which it follows that hγ HF maps span{ϕ1 , . . . , ϕK } into itself. Diagonalising the restriction of hγ HF to span{ϕ1 , . . . , ϕK }, we can choose ϕ1 , . . . , ϕK to be eigenfunctions of hγ HF with eigenvalues εn1 , . . . , εnK , nj ∈ N (numbering the eigenvalues of hγ HF in increasing order, −α−1 < ε1 ≤ ε2 ≤ · · ·). Since λ1 = · · · = λK = 1, this does not change (37).

720



To show that, for j > K, ϕj is also an eigenfunction of hγ HF (corresponding to an eigenvalue εnj ) one repeats the argument above, with u ∈ {ϕk }k=1,...,K,j ∪ {u }∈N , and

λj ϕj (x) + u(x) ϕj (y) + u(y) . λk ϕk (x)ϕk (y) + γ(j) (x, y) = 1 + m2 k=j

Moreover, the eigenvalues εnk (of hγ HF ) corresponding to the eigenfunctions ϕk are non-positive. In fact, if εnk > 0, then we could lower the energy: Define γ˜ (x, y) = γ HF (x, y) − λk ϕk (x)ϕk (y), then, using Lemma 3, we get that E HF (˜ γ) = E HF (γ HF ) − α−1 λk εnk < E HF (γ HF ). It remains to show that Tr[γ HF ] = N , that γ HF is a projection, and that the N {ϕj }j=1 are eigenfunctions corresponding to the lowest (negative) eigenvalues of hγ HF (that is, to ε1 ≤ ε2 ≤ · · · ≤ εN < 0). Consider first the case N < Z. Assume, for contradiction, that Tr[γ HF ] < N . Let K ∈ N be the multiplicity of the eigenvalue 1 in (37). Since (by Lemma 2), for N < Z, hγ HF has infinitely many eigenvalues in [−α−1 , 0) we can find a (normalized) eigenfunction u, corresponding to a negative eigenvalue of hγ HF , and orthogonal to ϕ1 , . . . , ϕK . Let > 0 be sufficiently small that γ(x, y) := γ HF (x, y) + u(x)u(y) defines a density matrix satisfying Tr[γ] ≤ N . By Lemma 3 (with u1 = u, 1 = and 2 = 0) we get that E HF (γ) = E HF (γ HF ) + α−1 (u, hγ HF u) < E HF (γ HF ) , HF

(38)

HF

leading to a contradiction. Hence, Tr[γ ] = N . That γ is a projection follows from Lieb’s Variational Principle (see [17]) which we prove for completeness. If this is not the case, there exist indices p, q such that 0 < λp , λq < 1. Consider γ˜ (x, y) := γ HF (x, y) + ϕq (x)ϕq (y) − ϕp (x)ϕp (y) with such that 0 ≤ γ˜ ≤ Id. Choose > 0 if εnq ≤ εnp and < 0 otherwise. By Lemma 3, we get that E HF (˜ γ ) < E HF (γ HF ). Consider now the case Z ≤ N < Z + 1 (and N ≥ 2), so that N − 1 < Z. Let HF γN −1 denote the density matrix where inf E HF (γ) γ ∈ A, Tr[γ] ≤ N − 1 HF HF is attained. By the above, Tr[γN −1 ] = N − 1 and γN −1 is a projection, so its integral kernel is given by HF γN −1 (x, y) =

N −1

φi (x)φi (y) ,

i=1

where the φi ’s are eigenfunctions of hγNHF−1 . We first prove that inf E HF (γ) γ ∈ A, Tr[γ] ≤ N

(39)

HF is not attained at the density matrix γN ˜ with −1 by constructing a density matrix γ HF HF HF γ ) < E (γN −1 ). Indeed, since hγNHF−1 has infinitely many Tr[˜ γ ] ≤ N such that E (˜

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721

strictly negative eigenvalues (by Lemma 2; N − 1 < Z) there exists a (normalized) eigenfunction u of hγNHF−1 corresponding to a negative eigenvalue, and orthogonal to span{φ1 , . . . , φN −1 }. Let γ˜ be defined by HF γ˜(x, y) = γN −1 (x, y) + u(x)u(y) .

Then Tr[˜ γ ] = N and, by a computation like in (38), HF −1 HF E HF (˜ γ ) = E HF (γN (u, hγNHF−1 u) < E HF (γN −1 ) + α −1 ) .

Hence, inf E HF (γ) γ ∈ A, Tr[γ] ≤ N < inf E HF (γ) γ ∈ A, Tr[γ] ≤ N − 1 .

(40)

Let γN be a density matrix where (39) is attained (the existence of such a minimizer follows, as before, from Lemma 1). By the above it follows that N − 1 < Tr[γN ] ≤ N . We now show that there exists a minimizer γ HF with Tr[γ HF ] = N . The integral kernel of γN is given by λj ϕj (x)ϕj (y) , γN (x, y) = j

where 1 ≥ λ1 ≥ · · · ≥ 0 and the ϕj ’s are (orthonormal) eigenfunctions of hγN . If γ ] ≤ N and E HF (˜ γ) ≤ Tr[γN ] < N we can define a new density matrix γ˜ with Tr[˜ HF E (γN ). Indeed, if Tr[γN ] < N (and bigger than N − 1) then there exists a (first) j0 such that 0 < λj0 < 1. We define γ˜ with integral kernel γ˜ (x, y) = γN (x, y) + rϕj0 (x)ϕj0 (y) ,

(41)

with r = min{1 − λj0 , N − Tr[γN ]} > 0. Recall that hγN ϕj = εnj ϕj , εnj ≤ 0, for all j. By Lemma 3 we have that E HF (˜ γ ) = E HF (γN ) + α−1 rεnj0 . γ ) < E HF (γN ). On the other hand, if εnj0 = 0, If εnj0 < 0, it follows that E HF (˜ HF HF then E (˜ γ ) = E (γN ), and Tr[γN ] < Tr[˜ γ ] ≤ N . Either Tr[˜ γ ] = N , in which case we let γ HF := γ˜ , and, as above, we are done. Or, we repeat all of the above argument on j0 γ˜ (x, y) = ϕj (x)ϕj (y) + λj ϕj (x)ϕj (y) . j=1

j>j0

Since the trace stays bounded by N , this procedure has to stop eventually. Hence, with γ HF the resulting density matrix, Tr[γ HF ] = N and by Lieb’s Variational Principle it follows (as above) that γ HF is a projection. Finally, let {ϕj } be the eigenfunctions of hγ HF , now numbered corresponding to the eigenvalues ε1 ≤ ε2 ≤ · · · , where ε1 is the lowest eigenvalue of hγ HF . We know that, for some j1 , . . . , jN ∈ N, γ HF (x, y) =

N k=1

ϕjk (x)ϕjk (y) .

722



Suppose for contradiction that {εj1 , . . . , εjN } = {ε1 , . . . , εN }. Then there exists a k ∈ {1, . . . , N } with εjk > εk . For δ ∈ (0, 1) define γ˜ (x, y) = γ HF (x, y) + δϕk (x)ϕk (y) − δϕjk (x)ϕjk (y) . By Lemma 3, E HF (˜ γ ) = E HF (γ HF ) + δα−1 (εk − εjk ) − δ 2 Rϕj ,ϕjk < E HF (γ HF ) , where the last inequality follows by choosing δ small enough. It remains to prove that ε1 , . . . , εN are strictly negative. For N < Z this follows directly from Lemma 2. In the case Z ≤ N < Z +1, assume, for contradiction, that εN = 0; then the density matrix γ˜ (x, y) := γ HF (x, y) − ϕN (x)ϕN (y) satisfies E HF (˜ γ ) = E HF (γ HF ) (by Lemma 3) and Tr[˜ γ ] = N − 1. This is a contradiction to (40). This finishes the proof of the first part of Theorem 1. It remains to prove Lemma 1 and Lemma 2. Proof of Lemma 1. We minimize on density matrices following the method in [30]. In the pseudorelativistic context one faces the problem that the Coulomb potential is not relatively compact with respect to the kinetic energy. This problem has been addressed in [5] and we follow the idea therein. HF The quantity E≤ (N, Z, α) is finite since for any density matrix γ, with Tr[γ] ≤ N , E HF (γ) ≥ α−1 Tr E(p)γ − α−1 N − Tr[V γ] ≥ − α−2 N . Here we used that D(γ) − Ex(γ) ≥ 0, and (8) (see also (17) and (18)). HF Let {γn }∞ n=1 be a minimizing sequence for E≤ (N, Z, α), more precisely, γn ∈ HF A (with A as defined in (16)), Tr[γn ] ≤ N , and E HF (γn ) ≤ E≤ (N, Z, α) + 1/n. The sequence Tr[E(p)γn ] is uniformly bounded. Indeed, for every n ∈ N, using (8), E HF (N, Z, α) + 1 ≥ E HF (γn ) ≥ α−1 Tr E(p)γn − α−1 N − Tr[V γn ] π Tr E(p)γn − α−2 N . ≥ α−1 1 − Zα 2 The claim follows since Zα < 2/π. It is this argument that prevents us from proving Theorem 1 for the critical case Zα = 2/π. Define γñ := E(p)1/2 γn E(p)1/2 . Then, by the above, {˜ γn }n∈N is a sequence of Hilbert–Schmidt operators with uniformly bounded Hilbert–Schmidt norm. Hence, by Banach–Alaoglu’s theorem, there exist a subsequence, which we denote again by γñ , and a Hilbert–Schmidt operator γ˜(∞) , such that for every Hilbert–Schmidt operator W , Tr[W γñ ] → Tr[W γ˜(∞) ] , n → ∞ .

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Let γ(∞) := E(p)−1/2 γ˜(∞) E(p)−1/2 . We are going to show that γ(∞) is a minimizer of E HF (in fact, of αE HF , which is equivalent). We first prove that γ(∞) ∈ A, then that E HF is weak lower semicontinuous on A. Let {ψk }k∈N be a basis of L2 (R3 ) with ψk ∈ H 1/2 (R3 ). Then, for all k ∈ N ,

lim (ψk , γn ψk ) = lim ψk , E(p)−1/2 γñ E(p)−1/2 ψk n→∞

n→∞

= (ψk , γ(∞) ψk ) . From this follows, by Fatou’s lemma, that

(ψk , γn ψk ) = lim inf Tr[γn ] ≤ N , Tr[γ(∞) ] = ψk , γ(∞) ψk ≤ lim inf n→∞

k

and

n→∞

k

Tr E(p)1/2 γ(∞) E(p)1/2 ≤ lim inf Tr E(p)1/2 γn E(p)1/2 < ∞ . n→∞

Since also 0 ≤ γ(∞) ≤ Id we see that γ(∞) ∈ A. To reach the claim it remains to show the weak lower semicontinuity of the functional E HF . As mentioned in the introduction, the spectrum of the one-particle operator h0 , defined in (9), is discrete in [−α−1 , 0) and purely absolutely continuous in [0, ∞). Let Λ− (α) denote the projection on the pure point spectrum of h0 and Λ+ (α) := Id − Λ− (α). We write αE HF (γn ) = T1 (γn ) + T2 (γn ) + αT3 (γn ) , with

T1 (γn ) = Tr Λ+ (α)h0 Λ+ (α)γn ,

(42)

T2 (γn ) = Tr Λ− (α)h0 Λ− (α)γn ,

T3 (γn ) = D(γn ) − Ex(γn ) . We consider these three terms separately. For the first term in (42), fix (as above) a basis {ψk }k∈N of L2 (R3 ), with {ψk }k∈N ⊂ H 1/2 (R3 ). Defining

1/2 ψk , fk := Λ+ (α)h0 Λ+ (α) we have that

1/2

1/2 γn Λ+ (α)h0 Λ+ (α) T1 (γn ) = Tr Λ+ (α)h0 Λ+ (α)

E(p)−1/2 fk , γñ E(p)−1/2 fk . (fk , γn fk ) = = k

Since the projection

k

Hk := E(p)−1/2 fk E(p)−1/2 fk

is a non-negative Hilbert–Schmidt operator, we find, by Fatou’s lemma, that Tr[Hk γñ ] ≥ Tr[Hk γ˜(∞) ] = T1 (γ(∞) ) . lim inf T1 (γn ) = lim inf n→∞

n→∞

k

k

As for the second term in (42), we have limn→∞ T2 (γn ) = T2 (γ(∞) ) since the operator Λ− (α)h0 Λ− (α) is Hilbert–Schmidt; see Lemma 7 in Appendix A.

724



Finally, for the last term in (42), following the reasoning in [5, pp.142–143] (here we need that N ∈ N), we get that lim inf T3 (γn ) ≥ T3 (γ(∞) ) . n→∞

This finishes the proof of Lemma 1.

Proof of Lemma 2. In order to prove that Kγ is Hilbert–Schmidt it is enough to prove that its integral kernel belongs to L2 (R6 ). We have that (see (24) and (14)) |γ(x, y)|2 |Kγ (x, y)|2 dxdy = dxdy (43) 2 R6 R6 |x − y| uk (x)uj (x)uk (y)uj (y) λj λk dxdy =: λj λk Ij,k . = |x − y|2 R6 j,k

j,k

The last integral can be estimated using the Hardy–Littlewood–Sobolev, H¨ older, and Sobolev inequalities (in that order), to get Ij,k ≤ uk uj 23/2 ≤ uk 23 uj 23 ≤ C uk 2H 1/2 uj 2H 1/2 . Inserting (44) in (43) we obtain (since γ ∈ A) R6

|Kγ (x, y)|2 dxdy ≤ C

j,k

λj λk uk 2H 1/2 uj 2H 1/2

2 = C Tr E(p)γ < ∞.

(44)

⎛ =C⎝ λj uj 2

H 1/2

⎞2 ⎠

j

To prove the statement on the essential spectrum, define ˜hγ := hγ + αKγ . Since Kγ is Hilbert–Schmidt, and σess (h0 ) = [0, ∞) (see the introduction), it is ˜ γ + η)−1 − (h0 + η)−1 is compact for some η > 0 large enough to prove that (h ˜ γ ) ⊂ D(Rγ ), we have that enough [25, Theorem XIII.14]. Since D(h0 ) = D(h ˜ γ + η)−1 αRγ (h0 + η)−1 . ˜ γ + η)−1 − (h0 + η)−1 = − (h (45) (h From Tiktopoulos’ formula (see [29, (II.8), Section II.3]), it follows that

−1/2

−1/2

−1/2

−1/2 −1 1− T (p)+η T (p)+η V T (p)+η . (h0 +η)−1 = T (p)+η (46) Since, by (5), (T (p) + η)−1/2 V 1/2 < 1 for Zα < 2/π and η > α−1 , the right side of (46) is well defined. Inserting (46) in (45) one sees that it suffices to prove that Rγ (T (p) + η)−1/2 is compact. That this is indeed the case follows by using [26, Theorem XI.20] together with the observation that, for ε > 0 and η > α−1 , Rγ and (T (p) + η)−1/2 (as a function of p) belong to the space L6+ε (R3 ) (for Rγ , see (23)). Finally, we show that if Tr[γ] = N < Z then hγ has infinitely many eigenvalues in [−α−1 , 0). By the min-max principle [25, Theorem XIII.1] and since σess (hγ ) = [0, ∞), it is sufficient to show that for every n ∈ N we can find n orthogonal functions u1 , . . . , un in L2 (R3 ) such that (ui , hγ ui ) < 0 for i = 1, . . . , n.

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Let n ∈ N. Fix δ := 1 − N/Z and let h0,δ be the unique self-adjoint operator whose quadratic form domain is H 1/2 (R3 ) such that (u, h0,δ v) = t[u, v] − δ v[u, v] for

u, v ∈ H 1/2 (R3 ) .

By [13, Theorems 2.2 and 2.3], σess (h0,δ ) = [0, ∞). Moreover, h0,δ has infinitely many eigenvalues in [−α−1 , 0). This follows by the min-max principle and the inequality h0,δ ≤ α/2(−Δ) − δZα/|x|. Hence, we can find u1 , . . . , un spherically symmetric and orthonormal such that (ui , h0,δ ui ) < 0 for i = 1, . . . , n. Then, by the positivity of Kγ , by Newton’s theorem [18, p. 249], and since Tr[γ] = N we get, for i = 1, . . . , n, that (ui , hγ ui ) ≤ t[ui , ui ] − v[ui , ui ] + α(ui , Rγ ui ) N v[ui , ui ] = (ui , h0,δ ui ) < 0 . ≤ t[ui , ui ] − v[ui , ui ] + Z The claim follows.

2.2. Regularity of the Hartree–Fock orbitals Here we prove that any eigenfunction of hγ HF is in C ∞ (R3 \ {0}). Proof. Let ϕ be a solution of hγ HF ϕ = εϕ for some ε ∈ R. Then ϕ belongs to the domain of the operator and in particular to H 1/2 (R3 ; Cq ). We are going to prove that ϕ ∈ H k (Ω) for all bounded smooth Ω ⊂ R3 \ {0} and all k ∈ N. The claim will then follow from the Sobolev imbedding theorem [2, Theorem 4.12]. We will use results on pseudodifferential operators; see Appendix B. We briefly summarize these here. 1) For all k, ∈ R, E(p) maps H k (R3 ) to H k− (R3 ). 2) For all k, ∈ R, and any χ ∈ C0∞ (R3 ), the commutator [χ, E(p) ] maps H k (R3 ) to H k−+1 (R3 ). 3) For all k, , m ∈ R and χ1 , χ2 ∈ C0∞ (R) with supp χ1 ∩supp χ2 = ∅, χ1 E(p) χ2 maps H k (R3 ) to H m (R3 ). Such an operator is called ‘smoothing’. Fix Ω a bounded smooth subset of R3 \ {0}. We proceed by induction on k ∈ N. Assume that ϕ ∈ H k (Ω) for some k ≥ 0, i.e., χϕ ∈ H k (R3 ) for all χ ∈ C0∞ (Ω). Notice that H k (R3 ) = D(E(p)k ). Since χϕ ∈ H k+1 (R3 ) is equivalent to χϕ ∈ D(E(p)k+1 ), and D(E(p)k+1 ) = D((E(p)k+1 )∗ ), it is sufficient to prove that χϕ ∈ D((E(p)k+1 )∗ ), or equivalently, that there exists v ∈ L2 (R3 ) such that

χϕ, E(p)k+1 f = (v, f ) for all f ∈ H k+1 (R3 ) . Let f ∈ H k+1 (R3 ). Then

χϕ, E(p)k+1 f = e ϕ, E(p)−1 χE(p)k+1 f

= (ε + α−1 ) ϕ, E(p)−1 χE(p)k+1 f + v ϕ, E(p)−1 χE(p)k+1 f

− bγ HF ϕ, E(p)−1 χE(p)k+1 f , (47)

726



where we use that hγ HF ϕ = εϕ. We study the terms in (47) separately. In the following, χ ˜ denotes a function in C0∞ (Ω) with χ ˜ ≡ 1 on supp χ. For the first term in (47) we find that

ϕ, E(p)−1 χE(p)k+1 f = χE(p)−1 ϕ, E(p)k+1 f = χ, E(p)−1 ϕ, E(p)k+1 f

(48) + E(p)−1 χϕ, E(p)k+1 f . Since χϕ ∈ H k (R3 ) by the induction hypothesis, we have that E(p)−1 χϕ ∈ H k+1 (R3 ) and hence there exists w1 ∈ L2 (R3 ) such that

E(p)−1 χϕ, E(p)k+1 f = (w1 , f ) . It remains to study the first term in (48). We have that ˜ E(p)k+1 f χ, E(p)−1 ϕ, E(p)k+1 f = χ, E(p)−1 χϕ, ˜ E(p)k+1 f . + χ, E(p)−1 (1 − χ)ϕ, Since χϕ ˜ ∈ H k (R3 ) by the induction hypothesis, it follows from Proposition 2 that [χ, E(p)−1 ]χϕ ˜ belongs to H k+2 (R3 ). On the other hand since the supports of χ and χ ˜ are disjoint the operator [χ, E(p)−1 ](1 − χ) ˜ is a smoothing operator. Hence there exists a w2 ∈ L2 (R3 ) such that χ, E(p)−1 ϕ, E(p)k+1 f = (w2 , f ) . As for the second term in (47), we find, with χ ˜ as before,

−1 k+1 −1 v ϕ, E(p) χE(p) f = ϕ, V E(p) χE(p)k+1 f

= χϕ, ˜ V E(p)−1 χE(p)k+1 f

+ (1 − χ)ϕ, ˜ V E(p)−1 χE(p)k+1 f .

(49)

Since χ ˜ has support away from zero, V χϕ ˜ ∈ H k (R3 ) and hence there exists w3 ∈ 2 3 L (R ) such that

χϕ, ˜ V E(p)−1 χE(p)k+1 f = (w3 , f ) . For the second term in (49) we proceed via an approximation. Let {ϕn }∞ n=1 ⊂ C0∞ (R3 ) such that ϕn → ϕ, n → ∞, in L2 (R3 ). Since (1 − χ)V ˜ E(p)−1 χE(p)k+1 f belongs to L2 (R3 ), we have that

ϕ, (1 − χ)V ˜ E(p)−1 χE(p)k+1 f = lim ϕn , (1 − χ)V ˜ E(p)−1 χE(p)k+1 f . n→+∞

For each n ∈ N, V (1 − χ ˜)ϕn ∈ H (R ) for all m, since ϕn ∈ C0∞ (R3 ), and V maps ˜ n ∈ L2 (R3 ), H k (R3 ) into H k−1 (R3 ) for all k. Therefore, E(p)k+1 χE(p)−1 V (1−χ)ϕ and so

˜ E(p)−1 χE(p)k+1 f = E(p)k+1 χE(p)−1 V (1− χ)ϕ ˜ n, f ϕn , (1− χ)V

−1 = E(p)k+1 χE(p)−1 (1− χ)E(p)E(p) ˜ V ϕn , f . m

3

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Here E(p)−1 V is bounded by (8), and χE(p)−1 (1 − χ) ˜ is a smoothing operator by the choice of the supports of χ and χ. ˜ It then follows that {E(p)k+1 χE(p)−1 (1 − −1 χ)E(p)E(p) ˜ V ϕn }n∈N is a uniformly bounded sequence in L2 (R3 ) and hence there exists w4 ∈ L2 (R3 ) such that

˜ E(p)−1 χE(p)k+1 f = (w4 , f ) . lim ϕn , (1 − χ)V n→+∞

For the third term in (47), we have to separate the cases k = 0 and k ≥ 1. Let k = 0. The terms Rγ HF ϕ and Kγ HF ϕ belong to L2 (R3 ), since Rγ HF ∈ L∞ (R3 ) (see (23)) and Kγ HF is Hilbert–Schmidt (see Lemma 2), and therefore

bγ HF ϕ, E(p)−1 χE(p)f = α E(p)χE(p)−1 (Rγ HF − Kγ HF )ϕ, f . Assume now k ≥ 1. With χ ˜ as before,

˜ γ HF − Kγ HF )ϕ, E(p)−1 χE(p)k+1 f (50) bγ HF ϕ, E(p)−1 χE(p)k+1 f = α χ(R

−1 k+1 χE(p) f . + α (1 − χ)(R ˜ γ HF − Kγ HF )ϕ, E(p) By the induction hypothesis and Lemma 6 (see Appendix A) we have that χR ˜ γ HF ϕ and χK ˜ γ HF ϕ belong to H k (R3 ). Therefore there exists w5 ∈ L2 (R3 ) such that

χ(R ˜ γ HF − Kγ HF )ϕ, E(p)−1 χE(p)k+1 f = (w5 , f ) . For the second term in (50) we find, since Rγ HF ϕ, Kγ HF ϕ ∈ L2 (R3 ), that

−1 (1 − χ)(R ˜ χE(p)k+1 f γ HF − Kγ HF )ϕ, E(p)

k+1 ˜ f , = χE(p)−1 (1 − χ)(R γ HF − Kγ HF )ϕ, E(p) and the result follows since χE(p)−1 (1 − χ) ˜ is a smoothing operator.

2.3. Exponential decay of the Hartree–Fock orbitals The pointwise exponential decay (30) will be a consequence of Proposition 1 and Lemma 4 below. Proposition 1. Let γ HF be a Hartree–Fock minimizer, let hγ HF be the corresponding Hartree–Fock operator as defined in (25), and let {ϕi }N i=1 be the Hartree–Fock orbitals, such that hγ HF ϕi = εi ϕi ,

i = 1, . . . , N ,

with 0 > εN ≥ · · · ≥ ε1 > − α−1 the N lowest eigenvalues of hγ HF . (i) Let νεN := −εN (2α−1 + εN ). Then ϕi ∈ D(eβ| · | ) for every β < νεN and i ∈ {1, . . . , N }. (ii) Assume hγ HF ϕ = εϕ for some ε ∈ [εN , 0), and let νε := −ε(2α−1 + ε). Then ϕ ∈ D(eβ| · | ) for every β < νε .

728



Lemma 4. Let E < 0 and νE := | − E(2α−1 + E)| = |α−2 − (E + α−1 )2 |. √ Then the operator T (−i∇) − E = −Δ + α−2 − α−1 − E is invertible and the integral kernel of its inverse is given by α−1 K1 (α−1 |x − y|) (E + α−1 )e−νE |x−y| + 4π|x − y| 2π 2 |x − y| −1 −1 −νE | · | K1 (α | · |) e 2 α ∗ ) 2 + (α−2 − νE (x − y) , (51) 2π |·| 4π| · |

(T − E)−1 (x, y) = GE (x − y) =

where K1 is a modified Bessel function of the second kind [1]. Moreover, 0 ≤ GE (x) ≤ Cα,E

e−νE |x| α−1 K1 (α−1 |x|) + , 4π|x| 2π 2 |x|

eβ| · | GE ∈ Lq (R3 )

for all

β < νE

and

(52) q ∈ [1, 3/2) .

(53)

Proof of Lemma 4. The formula (51) for the kernel of (T − E)−1 can be found in [22, eq. (35)]. The estimate (52) is a consequence of the bound K1 (α−1 | · |) e−νE | · | e−νE |x| ∗ (x) ≤ Cα,E . |·| 4π| · | 4π|x| This estimate, on the other hand, follows from Newton’s theorem (see e.g. [18]), K1 (α−1 |x − y|) e−νE |y| K1 (α−1 |x − y|) eνE |x−y| dy ≤ e−νE |x| dy |x − y| 4π|y| |x − y| 4π|y| R3 R3 e−νE |x| K1 (α−1 |z|) νE |z| ≤ e dz . 4π|x| R3 |z| The last integral is finite since νE < α−1 , using the following properties of K1 (see [10, 8.446, 8.451.6]): 1 for all t > 0 , (54) K1 (t) ≤ |t| and for every r > 0 there exists cr such that e−t K1 (t) ≤ cr √ for all t ≥ r . t The estimate (53) is a consequence of (52), (54), and (55).

(55)

Before proving Proposition 1, we apply it, and Lemma 4, to prove the pointwise exponential decay, i.e., the estimate in (30). Proof of Theorem 1 (iii). Fix i ∈ {1, . . . , N }. If Zα < 1/2 we can rewrite the Hartree–Fock equation (28) as

Zα ϕi − αRγ HF ϕi + αKγ HF ϕi . −Δ + α−2 − α−1 ϕi = εi ϕi + (56) |x|

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The idea of the proof is to study the elliptic regularity of the corresponding parametrix. By Lemma 4 we find that Zα ϕi −αRγ HF ϕi + αKγ HF ϕi (y)dy . ϕi (x) = (T −εN )−1 (x, y) (εi −εN )ϕi + |·| R3 In the case 1/2 ≤ Zα < 2/π, on the other hand, the operator of which we are studying the eigenfunctions cannot be written as a sum of operators acting on L2 (R3 ) and hence we cannot write directly the equation (28) as in (56). However, since the eigenfunctions are smooth away from the origin we are able to write a pointwise equation for a localized version of ϕi . In fact, let χ ∈ C ∞ (R3 ) be such that 0 ≤ χ ≤ 1 and 1 if |x| ≥ 1 , χ(x) = 0 if |x| ≤ 1/2 , and let, for R > 0, χR (x) = χ(x/R). We will derive an equation (similar to (56)) for T (−i∇)(χR ϕi ). Indeed, for every u ∈ H 1/2 (R3 ) we have that

u, hγ HF (χR ϕi ) = e(u, χR ϕi ) − α−1 (u, χR ϕi ) − v(u, χR ϕi ) + bγ HF (u, χR ϕi ) = (χR u, hγ HF ϕi ) + e(u, χR ϕi ) − e(χR u, ϕi ) + bγ HF (u, χR ϕi ) − bγ HF (χR u, ϕi ) . Note that

e(u, χR ϕi ) − e(χR u, ϕi ) = u, E(p), χR ϕi ,

where [E(p), χR ] is a bounded operator in L2 (R3 ) (see Appendix B), and bγ HF (u, χR ϕi ) − bγ HF (χR u, ϕi ) = (u, Kϕi ) , with K the bounded operator on L2 (R3 ) given by the kernel K(x, y) = α

N

ϕj (x)ϕj (y)

j=1

χR (x) − χR (y) . |x − y|

(57)

Therefore there exists w ∈ L2 (R3 ) such that e(u, χR ϕi ) = (εi + α−1 )(u, χR ϕi ) + v(u, χR ϕi ) − bγ HF (u, χR ϕi ) + u, E(p), χR ϕi + (u, Kϕi ) = (u, w) . Hence χR ϕi ∈ H 1 (R3 ) and we can write the pointwise equation Zα χR ϕi − αRγ HF χR ϕi ( −Δ + α−2 − α−1 )χR ϕi = εi χR ϕi + |x| + αKγ HF (χR ϕi ) + E(p), χR ϕi + Kϕi .

(58)

This is the substitute for (56) in the case 1/2 ≤ Zα < 2/π; if Zα < 1/2, the proof below simplifies somewhat, using (56) directly.

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By Lemma 4, (58) implies that Zα χR ϕi − αRγ HF χR ϕi + αKγ HF (χR ϕi ) χR (x)ϕi (x) = (T − εN )−1 (x, y) |·| R3 + (εi − εN )χR ϕi + E(p), χR ϕi + Kϕi (y) dy . (59) We will first show that, for all R > 0 and β < νεN , χR ϕi eβ| · | ∈ Lp (R3 ) + L∞ (R3 )

for

p ∈ [2, 6) ,

(60)

and then, by a bootstrap argument, that χR ϕi eβ| · | ∈ L∞ (R3 ), which is the claim of Theorem 1 (iii). We multiply (59) by χR/2 (x)eβ|x| . Using that |(Zα/|y|)χR (y)| ≤ (Zα)/R for all y ∈ R3 , (23), (24), and (57) (recall (27), that ϕj ∈ H 1/2 (R3 ), and (5)) we get, for some constant C = CR,α > 0, that χR (x)ϕi (x)eβ|x| ⎡ ⎤ N (T − εN )−1 (x, y) ⎣|ϕi (y)| + |ϕj (y)|⎦ dy ≤ CχR/2 (x)eβ|x| R3

j=1

β|x| −1 + χR/2 (x)e 3 (T − εN ) (x, y) E(p), χR ϕi (y) dy .

(61)

R

We will show that the first term on the right side of (61) belongs to Lp (R3 ) for p ∈ [2, 6), and that the second belongs to L∞ (R3 ). This will prove (60). The first term on the right side of (61) is a sum of terms of the form hf (x) := χR/2 (x)eβ|x| (T − εN )−1 (x, y) |f (y)| dy , (62) R3

with f such that, by Proposition 1, f eβ| · | ∈ L2 (R3 ). By Lemma 4 we have, using e|x|−|y| ≤ e|x−y|, that |hf (x)| ≤ C eβ|x−y|GεN (x − y)eβ|y| |f (y)| dy . R3

From Young’s inequality it follows that hf ∈ Lp (R3 ) for all p ∈ [2, 6), since β < νεN , so (by Proposition 1) f eβ| · | ∈ L2 (R3 ) and (by Lemma 4) eβ| · | GεN ∈ Lq (R3 ) for all q ∈ [1, 3/2). We now prove that the second term on the right side of (61) is in L∞ (R3 ). This follows from Young’s inequality once we have proved that eβ| · | E(p), χR ϕi ∈ Lp (R3 ) for p ∈ [2, ∞) , (63)

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since e

−1 3 (T − εN ) (x, y) E(p), χR ϕi (y) dy R ≤ eβ|x−y|GεN (x − y)eβ|y| E(p), χR ϕi (y) dy ,

β|x|

R3

and eβ| · | GεN ∈ Lq (R3 ) for q ∈ [1, 3/2). To prove (60) it therefore remains to prove (63). To do so, we consider a new localization function. Let η ∈ C0∞ (R3 ) be such that 0 ≤ η ≤ 1 and # 1 if R/4 ≤ |x| ≤ 3R/2 η(x) = 0 if |x| ≤ R/8 or |x| ≥ 2R , and consider the following splitting

eβ| · | E(p), χR ϕi = eβ| · | η E(p), χR (ηϕi ) + eβ| · | η E(p), χR (1 − η)ϕi + eβ| · | (1 − η) E(p), χR (ηϕi ) (64) + eβ| · | (1 − η) E(p), χR (1 − η)ϕi . Since ηϕi ∈ H k (R3 ) for all k ∈ N (as proved earlier), [E(p), χR ](ηϕi ) belongs to H k (R3 ) for all k ∈ N. Hence, since η has compact support away from x = 0, the first term on the right side of (64) is in Lp (R3 ) for p ∈ [1, ∞] by Sobolev’s imbedding theorem (the term is smooth). For the second term in (64) we proceed by duality: We will prove that

ψ(x) := eβ| · | η E(p), χR (1 − η)ϕi (x) defines a bounded linear functional on Lq (R3 ) for any q ∈ (1, 2]. It then follows that ψ ∈ Lp (R3 ) for all p ∈ [2, ∞). Note that [18, 7.12 Theorem (iv)]

g,[ −Δ + α−2 − α−1 ]g |g(x) − g(y)|2 α−2 = K2 (α−1 |x − y|) dxdy 4π 2 R3 R3 |x − y|2

for

g ∈ S(R3 ) , (65)

where K2 is a modified Bessel function of the second kind (in fact, K2 (t) = d −1 −t dt [t K1 (t)]), satisfying [1] K2 (t) ≤ Ct−1 e−t

for

t ≥ 1.

(66)

732



Let f ∈ C0∞ (R3 ). Using (65) and polarization, we have that f (x)ψ(x) dx R3

= f, eβ| · | η E(p), χR (1 − η)ϕi α−2 χR (x) − χR (y) = K2 (α−1 |x − y|) 4π 2 |x − y|2 |x−y|≥R/4

× f (x)eβ|x| η(x) 1 − η(y) ϕi (y) − f (y)eβ|y| η(y) 1 − η(x) ϕi (x) dxdy , by the properties of χ and η. Hence, f (x)ψ(x) dx 3 R ≤ CR |f (x)|eβ|x−y| K2 (α−1 |x − y|)eβ|y| |ϕi (y)| dxdy , |x−y|≥R/4 ≤ CR |f (x)|eβ|x−y|K2 (α−1 |x − y|)χR/4 (|x − y|)eβ|y| |ϕi (y)| dxdy . (67) Note that, since β < νεN < α−1 , (66) implies that eβ| · | K2 (α−1 | · |)χR/4 is in Lr (R3 ) for all r ≥ 1. Since (by Proposition 1) eβ| · | ϕi ∈ L2 (R3 ), Young’s inequality therefore gives that β| · |

e K2 (α−1 | · |)χR/4 ∗ (eβ| · | |ϕi |) ∈ Ls (R3 ) for all s ∈ [2, ∞) . This, (67), and H¨ older’s inequality (with 1/q + 1/s = 1) imply that, for all f ∈ C0∞ (R3 ) and all q ∈ (1, 2] $ $

≤ CR $ eβ| · | K2 (α−1 | · |)χR/4 ∗ (eβ| · | |ϕi |)$ f q . f (x)ψ(x) dx s R3

By density of C0∞ (R3 ) in Lq (R3 ), it follows that ψ defines a bounded linear functional on Lq (R3 ) for any q ∈ (1, 2], and therefore, that ψ ∈ Lp (R3 ) for all p ∈ [2, ∞). Proceeding similarly one shows that the two remaining terms in (64) are also in Lp (R3 ) for all p ∈ [2, ∞). This finishes the proof of (63), and therefore of (60). Finally we prove that χR ϕi eβ| · | ∈ L∞ (R3 ). We start again from (61). We already know that the second term is in L∞ (R3 ). The first term is a sum of terms of the form (see also (62)) β|x| hf (x) = χR/2 (x)e (T − εN )−1 (x, y)|f (y)| dy , R3

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with f ∈ L2 (R3 ) and χR/4 eβ| · | f ∈ Lp (R3 ) + L∞ (R3 ) for p ∈ [2, 6) by what just proved, replacing R by R/4 in (60). We find that hf (x) ≤ χR/2 (x) eβ|x−y|(T − εN )−1 (x, y)eβ|y| χR/4 (y)|f (y)| dy R3 + χR/2 (x) eβ|x−y|(T − εN )−1 (x, y)eβ|y| (1 − χR/4 )(y)|f (y)| dy , R3

and, again by Young’s inequality, we see that both terms are in L∞ (R3 ). Notice that in the second integrand |x − y| > R/4. This finishes the proof of Theorem 1 (iii). It therefore remains to prove Proposition 1. Proof of Proposition 1. We start by proving (i). It will be convenient to write the Hartree–Fock equations hγ HF ϕi = εi ϕi , i = 1, . . . , N , (see (28)) as a system. Let t be the quadratic form with domain [H 1/2 (R)]N defined by t(u, v) =

N

t(ui , vi )

for all

N u, v ∈ H 1/2 (R3 ) ,

i=1

where ui denotes the i-th component of u ∈ [H 1/2 (R3 )]N and t is the quadratic form defined in (7). Similarly we define the quadratic forms v, rγ and kγ , all with domain [H 1/2 (R3 )]N , by v(u, v) =

N i=1

v(ui , vi ) ,

rγ (u, v) = α

N

(ui , Rγ vi ) ,

kγ (u, v) = αu, Kγ v ,

i=1

with v defined in (6), Rγ defined in (22), and Kγ the N × N -matrix given by ϕi (y)ϕj (y) dy . (Kγ )i,j = |x − y| 3 R The effect of writing the Hartree–Fock equations as a system is that Kγ is a (nondiagonal) multiplication operator. This idea was already used in [19]. Note that (Kγ )i,j ∈ L3 (R3 ) ∩ L∞ (R3 ); the argument is the same as for (22). Let finally E be the N × N matrix defined by (E)i,j = −εi δi,j . We then define the quadratic form q by q(u, v) = t(u, v) − v(u, v) + rγ (u, v) − kγ (u, v) + u, Ev .

(68)

One sees that the quadratic form domain of q is [H 1/2 (R3 )]N , that q is closed (since t is closed), and that there exists a unique selfadjoint operator H with D(H) ⊂ [H 1/2 (R3 )]N such that N u, Hv = q(u, v) for all u ∈ H 1/2 (R3 ) , v ∈ D(H) . Notice that the vector Φ = (ϕ1 , . . . , ϕN ) satisfies HΦ = 0. Let W (κ), κ ∈ C3 , denote the multiplication operator from a subset of 2 [L (R3 )]N to [L2 (R3 )]N given by f (x) → eiκ · x f (x). Instead of proving directly

734



the claim of the proposition, we are going to prove the following statement, which implies the proposition:

(69) Φ ∈ D W (κ) for Im(κ) R3 < νεN , where Φ = (ϕ1 , . . . , ϕN ). Here, κ = Re(κ) + iIm(κ) with Re(κ), Im(κ) ∈ R3 . We know that W (κ)Φ is well defined on [L2 (R3 )]N for κ ∈ R3 and we need to show that it has a continuation into the ‘strip’ ΣνεN , where Σt := κ ∈ C3 | Im(κ) R3 < t . We shall also need Σα−1 ; note that Σα−1 ⊃ ΣνεN . The idea is to use O’Connor’s Lemma (see Lemma 5 below). Starting from the quadratic form q defined in (68) we define the following family of quadratic forms on [H 1/2 (R3 )]N :

q(κ)(u, u) := q W (−κ)u, W (−κ)u , depending on the real parameter κ ∈ R3 . From the definition, q(κ)(u, u) = t(κ)(u, u) − v(u, u) + rγ (u, u) − kγ (u, u) + u, Eu , where t(κ)(u, u) =

⎛

N i=1

⎝α−2 + R3

3

⎞1/2 (pj − κj )2 ⎠

|ˆ ui (p)|2 dp − α−1 u, u .

(70)

j=1

One sees that q(κ) extends to a family of sectorial forms with angle θ < π4 , and that q(κ) is holomorphic in the strip Σα−1 (indeed, Im(κ) R3 < α−1 is needed to assure that the complex number under the square root in (70) has non-negative real part for all p ∈ R3 ). Moreover, q(κ) is closed. Indeed, it is sufficient to prove that the real part of q(κ) is closed, which will follow from

(71) v(u, u) + rγ (u, u) + kγ (u, u) + u, Eu ≤ b Re t(κ) (u, u) + Ku, u , with b < 1, K > 0 and Re(t(κ)) closed. We now prove (71). We already know that rγ (u, u) + kγ (u, u) + u, Eu ≤ K u, u

for

K > 0 .

(72)

By (8) we find N π |p| |ˆ ui (p)|2 dp 2 i=1 R3 & % N π ≤ (Zα) R |ˆ ui (p)|2 dp + |p| |ˆ ui (p)|2 dp . 2 i=1 |p|≤R |p|≥R

v(u, u) ≤ (Zα)

(73)

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735

Let δ > 0 be such that Zα π2 (1 − δ)−1 < 1. Since 1/2 N 3 −2

2 α + Re t(κ) (u, u) = (p − κ ) cos θ(p, κ) |ˆ ui (p)|2 dp j j 3 i=1 R j=1 − α−1 u, u , with

α−2 +

2 cos θ(p, κ) − 1 = 2

3

− Re(κj ))2 − (Im(κj ))2 , 3 |α−2 + j=1 (pj − κj )2 | j=1 (pj

there exists R > 0 such that cos(θ(p, κ)) ≥ (1 − δ) for |p| > R. Hence we find that 1/2 N 3 −2

2 α + Re t(κ) (u, u) ≥ (1−δ) (p −κ ) |ˆ ui (p)|2 dp − α−1 u, u j j i=1 |p|>R j=1 N ≥ (1−δ) (|p|−C)| u ˆ i (p)|2 dp−α−1 u, u , (74) i=1

|p|>R

with C > Re(κ) R3 . The estimate in (71) follows combining (72) with (73) and (74). The fact that Re(t(κ)) is closed follows from N N

1 √ (|p| − C) |ˆ ui (p)|2 dp ≤ Re t(κ) (u, u) ≤ (|p| + C)|ˆ ui (p)|2 dp , 2 i=1 i=1 with C ≥ 2α−1 + Re(κ). Hence, q(κ) is an analytic family of forms of type (a) [15, p. 395]. The associated family H(κ) of sectorial operators is a holomorphic family of operators of type (B) and has domain in a subset of [H 1/2 (R3 )]N . We are interested now in locating the essential spectrum of H(κ). Since Kγ is a Hilbert–Schmidt operator, the essential spectrum of H(κ) coincides with the essential spectrum of the operator associated to t(κ)(u, u) − v(u, u) + α rγ (u, u) + u, Eu . Notice that the operator associated to this quadratic form is diagonal. Proceed) = [0, ∞) (Lemma 2), one sees that σess (H(κ)) ⊂ ing as in the proof of σess (hγ σess (T (κ) − εN ) with T (κ) := α−2 + 3j=1 (pj − κj )2 − α−1 . Hence we find that

σess H(κ) ⊂ z ∈ C Re(z) ≥ α−2 − Im(κ) 2R3 − α−1 − εN . Hence 0, eigenvalue of H(0), remains disjoint from the essential spectrum of H(κ) for all κ ∈ ΣνεN (recall that ΣνεN ⊂ Σα−1 ) . Since H(κ) is an analytic family of type (B) [25, p. 20] in Σνε , 0 is an eigenvalue of H(0) and moreover, 0 remains disjoint from the essential spectrum of H(κ), it

736



follows that 0 is an eigenvalue in the pure point spectrum of H(κ) for all κ ∈ ΣνεN (reasoning as in [25, page 187]). Let P(κ) be the projection onto the eigenspace corresponding to the eigenvalue 0 of the operator H(κ). Then P(κ) is an analytic function in ΣνεN and for κ ∈ ΣνεN and κ0 ∈ R we have P(κ + κ0 ) = W (κ0 )P(κ)W (−κ0 ) . Here we used that W (−κ0 ) is a unitary operator. The result of the lemma follows ˜ (θ) := eiθκ · x with κ ∈ R3 , κ R3 = νεN , and by applying Lemma 5 below to W ˜ (θ) = W (θκ) and that the projection θ ∈ {z ∈ C | |Im(z)| < 1}. Notice that W ˜ (θ0 )~ ˜ (−θ0 ) for θ0 ∈ R. ~ P(θ) := P(θκ) is analytic and satisfies ~ P(θ + θ0 ) = W P(θ)W This finishes the proof of (i). To prove (ii), we can work directly with the Hartree–Fock equation, since, from (i), the function Kγ HF ϕ is exponentially decaying. Therefore, let q[u, v] = (u, hγ HF v) − ε(u, v)

for

u, v ∈ H 1/2 (R3 ) ,

(75)

and note that, by assumption, 0 is an eigenvalue for the corresponding operator (ϕ is an eigenfunction). Define, for κ ∈ R3 , q(κ)[u, v] = q W (−κ)u, W (−κ)v = t(κ)[u, v] − v[u, v] + bγ HF (κ)[u, v] − ε(u, v) ,

(76)

with W (κ) and t(κ) as before (but now on H 1/2 (R3 )), see (70), and

bγ HF (κ)[u, v] = α(u, Rγ HF v) − α u, Kγ HF (κ)v ,

(77)

where Kγ HF (κ)(x, y) =

N ϕj (x)eiκx e−iκy ϕj (y)

|x − y|

j=1

.

(78)

Using (i) of the proposition (exponential decay of the Hartree–Fock orbitals {ϕj }N j=1 ) one now proves that (78) extends to a holomorphic family of Hilbert– Schmidt operators in ΣνεN . One can now repeat the reasoning in the proof of (i) to obtain the stated exponential decay of ϕ. Lemma 5 ([25, p. 196]). Let W (κ) = eiκA be a one-parameter unitary group (in particular, A is self-adjoint) and let D be a connected region in C with 0 ∈ D. Suppose that a projection-valued analytic function P (κ) is given on D with P (0) of finite rank and so that W (κ0 )P (κ)W (κ0 )−1 = P (κ + κ0 )

for

κ0 ∈ R

and

κ, κ + κ0 ∈ D .

Let ψ ∈ Ran(P (0)). Then the function ψ(κ) = W (κ)ψ has an analytic continuation from D ∩ R to D.

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Appendix A. Some useful lemmata Lemma 6. Let Ω be an open subset of R3 \ {0} with smooth boundary and let f1 , f2 ∈ H k (Ω) for some k ≥ 1. Then the function f1 (y)f2 (y) F (x) := dy |x − y| 3 R belongs to C k (Ω) if k ≥ 2, while if k = 1, it belongs to W 1,p (Ω) for all p ≥ 1, and hence to C(Ω). Proof. We are going to prove the following equivalent statement. If k ≥ 2, χF ∈ C k (R3 ) for all χ ∈ C0∞ (Ω), while if k = 1, χF ∈ W 1,p (R3 ) for all p ≥ 1 and χ ∈ C0∞ (Ω). Fix χ ∈ C0∞ (Ω) and take χ ˜ ∈ C0∞ (Ω) verifying χ ˜ ≡ 1 on supp χ and such that there is a strictly positive distance between supp χ and supp (1 − χ). ˜ We write χF (x) = χF1 (x) + χF2 (x) with

f1 (y)f2 (y) χ(y)f ˜ 1 (y)f2 (y) dy and F2 (x) = dy . F1 (x) = 1 − χ(y) ˜ |x − y| |x − y| R3 R3 The term χF2 is clearly in C ∞ (R3 ). For the other term we use Young’s inequality: if f ∈ Lp (R3 ) and g ∈ Lq (R3 ) then 1 1 1 (79)

f ∗ g r ≤ C f p g q with 1 + = + . r p q Moreover, if 1/p + 1/q = 1 then f ∗ g is continuous (see [31, Lemma 2.1]). Let α ∈ N30 with |α| ≤ k. Then 1 α β1 β2 (80) D (χf |D (χF1 )(x)| ≤ |D χ(x)| ˜ 1 f2 )(y)dy . |x − y| 3 β1 +β2 =α, β1 ,β2 ∈N30

R

If f1 , f2 ∈ H k (Ω), k ≥ 2, then Dβ2 (χf ˜ 1 f2 ) ∈ L5/3 (R3 ) for all β2 as in (80). From (79), (80) and χ/| ˜ · | ∈ L5/2 (R3 ) it follows that Dα (χF1 ) is continuous and, since α is arbitrary, that χF ∈ C k (R3 ). ˜ 1 f2 ) ∈ L3/2 (R3 ) and from (79) we get (only) that If f1 , f2 ∈ H 1 (Ω) then ∂(χf p 3 ∂(χF ) ∈ L (R ) for all p ≥ 1. It then follows that F ∈ W 1,p (Ω) for all p ≥ 1 and therefore (by the Sobolev imbedding theorem) F ∈ C(Ω). Lemma 7. Let, for Zα < 2/π, h0 be the self-adjoint operator defined in (9), and let Λ− (α) be the projection onto the pure point spectrum of h0 . Then the operator Λ− (α)h0 Λ− (α) is Hilbert–Schmidt. Proof. Let > 0 be such that Zα(1 + ) ≤ 2/π(1 − ). We are going to prove that there exists a constant M = M () such that ' ( 1 C h0 ≥ P −Δ − P, (81) M + 2α−1 |·|

738



with C = Zα(M + 2α−1 )(1 + 1/) and P = χ[0,M] (T (p)). The claim will then follow from (81) since ) *2

2 1 C Tr [h0 ]− ≤ Tr −Δ − < ∞. (M + 2α−1 )2 |·| − The last inequality follows since the eigenvalues of −Δ−C/| · | are −C 2 /4n2 , n ∈ N, with multiplicity n2 . We now prove (81). For > 0 and any projection P (with P ⊥ = 1 − P ), we have that Zα Zα ⊥ P − P⊥ P h0 = P h0 P + P ⊥ h0 P ⊥ − P |·| |·| ' ' ( ( 1 Zα Zα ≥ P h0 − (82) P + P ⊥ h0 − P⊥ . |·| |·| By a direct computation one sees that there exists a constant M = M () such that 1 T (p) ≥ M implies T (p) ≥ (1 − )|p| and T (p) ≤ M implies T (p) ≥ M+2α −1 (−Δ). Hence, with this choice of M and P = χ[0,M] (T (p)), (82) implies that √ 1 Zα ⊥ −1 Zα ⊥ h0 ≥ P (−Δ) − (1 + ) P + P (1 − ) −Δ − (1 + ) P . M + 2α−1 |·| |·|

The inequality (81) follows directly by the choice of .

Appendix B. Pseudodifferential operators In this appendix we collect facts needed from the calculus of pseudodifferential operators (ψdo’s) (for references, see e.g. [14] or [28]). Define the standard (H¨ ormander) symbol class S μ (Rn ), μ ∈ R, to be the set ∞ n of functions a ∈ C (Rx × Rnξ ) satisfying α β ∂x ∂ a(x, ξ) ≤ Cα,β (1 + |ξ|2 )(μ−|β|)/2 for all (x, ξ) ∈ Rnx × Rnξ . (83) ξ

Here, α, β ∈ Nn and |α| = α1 + · · · + αn . Furthermore, S μ (Rn ) ⊂ S μ (Rn ) for μ ≤ μ . We denote S ∞ (Rn ) = ∪μ∈R S μ (Rn ) and S −∞ (Rn ) = ∩μ∈R S μ (Rn ). Finally, note that ab ∈ S μ1 +μ2 (Rn ), ∂xα ∂ξβ a ∈ S μ1 −|β| (Rn ) when a ∈ S μ1 (Rn ), b ∈ S μ2 (Rn ). A symbol a ∈ S μ (Rn ) defines a linear operator A = Op(a) ∈: Ψμ (‘pseudodifferential operator of order μ’) by Op(a)u (x) = (2π)−n eix · ξ a(x, ξ)ˆ u(ξ) dξ , (84) Rn

where uˆ is the Fourier-transform of u. The operator A is well-defined on the space S(Rn ) of Schwartz-functions; it extends by duality to S (Rn ), the space of tempered distributions. Note that for aα (x)ξ α (85) a(x, ξ) = 0≤|α|≤μ

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739

(with aα smooth and with all derivatives bounded, i.e., aα ∈ B(Rn )), A = Op(a) ∈ Ψμ is the partial differential operator given by Op(a)u (x) = aα (x)Dα u(x) . (86) 0≤|α|≤μ

Note also that, with a = a(x) and b = b(ξ), Op(a)u (x) = a(x)u(x) and

Op(b)u (ξ) = b(ξ)ˆ u(ξ) .

If a ∈ S μ (Rn ), then Op(a), defined this way, maps H k (Rn ) continuously into H k−μ (Rn ) for all k ∈ R. Here, H k (Rn ) is the Sobolev-space of order k, consisting of u ∈ S (Rn ) for which

u 2H k (Rn ) := |ˆ u(ξ)|2 (1 + |ξ|2 )k dξ (87) Rn

k

is finite; this defines the norm on H (Rn ). We denote + , H k (Rn ) , H −∞ (Rn ) = H k (Rn ) . H ∞ (Rn ) = k∈R 0

k∈R

In particular, symbols in S (R ) define bounded operators on L2 (Rn ) = H 0 (Rn ). Furthermore, operators defined by symbols in S −∞ (Rn ) maps any H k (Rn ) into H ∞ (Rn ); such operators are called ‘smoothing’. We need to compose ψdo’s. There exists a composition # of symbols, n

# : S μ1 (Rn ) × S μ2 (Rn ) → S μ1 +μ2 (Rn ) (a, b) → a#b , such that Op(a)Op(b) = Op(a#b). It is given by 1 (a#b)(x, ξ) = e−iy · ξ a(x, ξ − η)b(x − y, η) dydη . (2π)n Rn ×Rn Here, the integral is to be understood as an oscillating integral. The symbol a#b has the expansion i−|α| a#b ∼ (∂xα a)(∂ξα b) . α! α Here, ‘∼’ means that for all j ∈ N, i−|α| (∂xα a)(∂ξα b) ∈ S μ1 +μ2 −j (Rn ) a#b − α!

(88) (89)

(90)

(91)

(92)

|α|<j

(recall that associative.

(∂xα a)(∂ξα b)

∈ S μ1 +μ2 −|α| ). One easily sees that the composition is

Proposition 2. If a ∈ S m1 (Rn ), b ∈ S m2 (Rn ) then the symbol associated to [Op(a), Op(b)] belongs to S m1 +m2 −1 (Rn ).

740



In particular, if φ1 , φ2 ∈ B ∞ (Rn ) (the smooth functions with bounded derivatives) with supp φ1 ∩supp φ2 = ∅ and a ∈ S μ (Rn ), a(x, ξ) = a(ξ), then φ1 #a#φ2 ∼ 0, and so, with A := Op(a), φ1 Aφ2 = Op(φ1 )Op(a)Op(φ2 ) is smoothing.

Acknowledgements The authors wish to thank Heinz Siedentop for useful discussions. Support from the EU IHP network Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems, contract no. HPRN-CT-2002-00277, and from the Danish Natural Science Research Council, under the grant Mathematical Physics and Partial Differential Equations, is gratefully acknowledged. TØS wishes to thank the Department of Mathematics, LMU Munich, for its hospitality in the spring of 2007.

References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, §9.6: Modified Bessel Functions I and K., New York: Dover, 1972. [2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, New YorkLondon, 2003, Pure and Applied Mathematics, Vol. 140. [3] V. Bach, Error Bound for the Hartree–Fock Energy of Atoms and Molecules, Comm. Math. Phys. 147 (1992), no. 3, 527–548. [4] V. Bach, E. H. Lieb, M. Loss, and J. P. Solovej, There Are No Unfilled Shells in Unrestricted Hartree–Fock Theory, Phys. Rev. Lett. 72 (1994), no. 19, 2981–2983. [5] J.-M. Barbaroux, W. Farkas, B. Helffer, and H. Siedentop, On the Hartree–Fock Equations of the Electron-Positron Field, Comm. Math. Phys. 255 (2005), no. 1, 131–159. [6] I. Daubechies, An Uncertainty Principle for Fermions with Generalized Kinetic Energy, Comm. Math. Phys. 90 (1983), no. 4, 511–520. [7] M. J. Esteban and E. Séré, Solutions of the Dirac–Fock Equations for Atoms and Molecules, Comm. Math. Phys. 203 (1999), no. 3, 499–530. [8] M. J. Esteban and E. Séré, Nonrelativistic Limit of the Dirac–Fock Equations, Ann. Henri Poincaré 2 (2001), no. 5, 941–961. [9] M. J. Esteban and E. Séré, A Max-Min Principle for the Ground State of the Dirac– Fock Functional, Mathematical Results in Quantum Mechanics (Taxco, 2001), Contemp. Math., vol. 307, Amer. Math. Soc., Providence, RI, 2002, pp. 135–141. [10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1980, Corrected and enlarged edition edited by A. Jeffrey, Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tse˘ıtlin], Translated from the Russian.

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[11] C. Hainzl, M. Lewin, and E. Séré, Existence of a Stable Polarized Vacuum in the Bogoliubov–Dirac–Fock Approximation, Comm. Math. Phys. 257 (2005), no. 3, 515– 562. [12] C. Hainzl, M. Lewin, and E. Séré, Self-consistent Solution for the Polarized Vacuum in a No-photon QED Model, J. Phys. A 38 (2005), no. 20, 4483–4499. [13] I. W. Herbst, Spectral Theory of the Operator (p2 + m2 )1/2 − Ze2 /r, Comm. Math. Phys. 53 (1977), no. 3, 285–294. [14] L. H¨ ormander, The Analysis of Linear Partial Differential Operators III, Pseudodifferential Operators, Classics in Mathematics, Springer, Berlin, 2007, Reprint of the 1994 edition. [15] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. [16] C. Le Bris and P.-L. Lions, From Atoms to Crystals: A Mathematical Journey, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 3, 291–363 (electronic). [17] E. H. Lieb, Variational Principle for Many-Fermion Systems, Phys. Rev. Lett. 46 (1981), no. 7, 457–459. [18] E. H. Lieb and M. Loss, Analysis, second ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. [19] E. H. Lieb and B. Simon, The Hartree–Fock Theory for Coulomb Systems, Comm. Math. Phys. 53 (1977), no. 3, 185–194. [20] E. H. Lieb and H.-T. Yau, The Stability and Instability of Relativistic Matter, Comm. Math. Phys. 118 (1988), no. 2, 177–213. [21] P.-L. Lions, Solutions of Hartree–Fock Equations for Coulomb Systems, Comm. Math. Phys. 109 (1987), no. 1, 33–97. [22] T. Østergaard Sørensen and E. Stockmeyer, On the Convergence of Eigenfunctions to Threshold Energy States, Proc. Roy. Soc. Edinburgh Sect. A 138A (2008), 169–187. [23] E. Paturel, Solutions of the Dirac–Fock Equations without Projector, Ann. Henri Poincaré 1 (2000), no. 6, 1123–1157. [24] M. Reed and B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975. [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. [26] M. Reed and B. Simon, Methods of Modern Mathematical Physics III. Scattering Theory, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979. [27] M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis, second ed., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. [28] X. Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. [29] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, N. J., 1971, Princeton Series in Physics. [30] J. P. Solovej, Proof of the Ionization Conjecture in a Reduced Hartree–Fock Model, Invent. Math. 104 (1991), no. 2, 291–311.

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[31] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin Heidelberg, 2007. [32] B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. [33] R. A. Weder, Spectral Analysis of Pseudodifferential Operators, J. Functional Analysis 20 (1975), no. 4, 319–337. Anna Dall’Acqua Institut f¨ ur Analysis und Numerik Fakult¨ at f¨ ur Mathematik Otto-von-Guericke Universit¨ at Postfach 4120 D-39016 Magdeburg Germany e-mail: [email protected] Thomas Østergaard Sørensen Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G DK-9220 Aalborg East Denmark e-mail: [email protected] Edgardo Stockmeyer Mathematisches Institut Universit¨ at M¨ unchen Theresienstraße 39 D-80333 Munich Germany e-mail: [email protected] Communicated by Rafael D. Benguria. Submitted: August 1, 2007. Accepted: November 8, 2007.

Ann. Henri Poincaré 9 (2008), 743–773 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/040743-31, published online May 30, 2008 DOI 10.1007/s00023-008-0371-y


Resonances of the Confined Hydrogen Atom and the Lamb–Dicke Effect in Non-Relativistic QED Jérémy Faupin Abstract. We study a model describing a system of one dynamical nucleus and one electron confined by their center of mass and interacting with the quantized electromagnetic field. We impose an ultraviolet cutoff and assume that the fine-structure constant is sufficiently small. Using a renormalization group method (based on [3, 4]), we prove that the unperturbed eigenvalues turn into resonances when the nucleus and the electron are coupled to the radiation field. This analysis is related to the Lamb–Dicke effect.

1. Introduction and statements of results 1.1. Introduction In this paper, we study a model of a hydrogen atom (or, more generally, a hydrogeno¨ıd ion) confined by its center of mass. This model is used in theoretical physics to explain the Lamb–Dicke effect (see [7]); our purpose is to present, in the framework of non-relativistic quantum electrodynamics, a mathematically rigorous aspect of this phenomenon. Let us begin with describing it briefly. First, consider a hydrogen atom whose nucleus is treated as static in quantum mechanics. The Schrödinger operator associated with it can be written as −Δ/2m1 + V , where m1 is the mass of the electron and V is the Coulomb potential. If the electron is initially in some excited state of energy Ei , it can fall into a state of lower energy Ef by spontaneous emission of a photon of energy |k| = Ei − Ef .

(1)

Here k is the momentum of the emitted photon and we have set = c = 1, where = h/2π, h is the Planck constant and c is the velocity of light. Next, consider a more realistic model where the nucleus is treated as dynamical. The internal energy of the atom is associated to the Schr¨ odinger operator

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−Δ/2μ + V , where μ is the reduced mass, μ = m1 m2 /(m1 + m2 ). We denote by M = m1 + m2 the total mass of the system. Assume that the atom is initially in some excited state of internal energy (m2 /M )Ei . If the center of mass motion of the atomic system is free, then, as the internal state falls into a state of lower energy (m2 /M )Ef , the energy of the emitted photon is modified as follows: |k| =

m2 k·P k2 (Ei − Ef ) − + . M 2M M

(2)

This identity comes from the conservation of both the energy and the momentum. Here P denotes the momentum of the center of mass before the emission process. The term k 2 /2M corresponds to the recoil energy, whereas the term k · P/M is due to the Doppler effect. Suppose finally that the center of mass of the atom is confined. Then the energy associated with its motion is quantized, as well as the internal energy. If ei and ef denote the energies associated with the center of mass motion respectively before and after the emission process, we have |k| =

m2 (Ei − Ef ) + ei − ef . M

(3)

In the Lamb–Dicke regime (in particular, for a sufficiently “strong” confinement), one can see that the most likely transitions are the ones such that ei = ef . In other words, in the scattering spectrum of the physical system, the most intense rays correspond exactly to the internal transitions of energies (m2 /M )(Ei − Ef ). These rays are, in addition, neither shifted by recoil effect nor broadened by Doppler effect. The suppression of both the recoil shift and the Doppler effect is then called the Lamb–Dicke effect. This effect seems to have been discussed for the first time by R. H. Dicke in [9] where the author studies the reduction of the Doppler width of the light emitted by the molecules of a dense gas. It is assumed that the effect of collisions between the molecules of the gas is to confine their centers of mass. Then an emitter whose center of mass is trapped in a 1-dimensional square potential is considered. Let us also mention that the Lamb–Dicke effect is frequently used in theoretical physics, for instance to study the cooling of atoms or ions by lasers (see, e.g., [16, 17]). The idea of sideband laser cooling for an atom or an ion is indeed as follows: with the notations above, consider an ion in its ground state of energy (m2 /M )E0 . Let ω0 = (m2 /M )(E1 − E0 ), and let γ be the radiative line width of an excited state of energy (m2 /M )E1 . Assume that the ion is confined by a harmonic well of vibration frequency ω; we suppose that ω γ, which insures in particular that the rays centered at the energies ω0 ± (n + 1/2)ω are well-resolved. Assume furthermore that the external energy is initially equal to ei = (i + 1/2)ω. The aim, to cool the ion, is to decrease this energy associated with the center of mass motion. To this end, a laser beam is tuned to a frequency ωL = ω0 − ω. The ion then “jumps” to a state of energy E1 + ei−1 , and is next likely to “fall”

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into a state of energy E0 + ei−1 , according to the Lamb–Dicke effect. This process is repeated, so that cooling proceeds, in principle, until the lowest vibrational energy e0 is reached. In [6, 7], the Lamb–Dicke effect is explained with the help of a Pauli–Fierz Hamiltonian describing a hydrogen atom confined by its center of mass. We shall consider the same model in this paper. Mathematically, we shall have to face some difficulties in the spectral study of the Hamiltonian, because the electron and the nucleus themselves are not confined: they only interact with each other through the Coulomb potential. On the other hand, the confinement only exists in the direction of the center of mass. Thus one can imagine some states where the nucleus and the electron are localized very far from each other, and where, yet, the energy associated with the center of mass motion is low. In [1], we obtained the existence of a ground state for the same model, for all values of the fine-structure constant. Now, as explained above, when instead of fixing the nucleus, it is only assumed that the center of mass of the atom or ion is confined, new intense rays appear in the scattering spectrum of the physical system. Thus, some resonances depending on the confining potential, with a very small imaginary part, should appear in the spectrum of the Hamiltonian. The aim of our paper is to prove this, assuming here that the fine-structure constant is sufficiently small. In the standard model of non-relativistic quantum electrodynamics, the Hamiltonian that describes the system we consider is written as 1 2 pj − qj A(xj ) + Hf + U (R) + V (r) . (4) HUV = 2mj j=1,2 It acts on the Hilbert space L2 (R6 ) ⊗ Fs , where Fs is the symmetric Fock space over L2 (R3 × Z2 ) for the transversal photons. The spins of the electron and of the nucleus are not taken into account. Here xj , qj , pj = −i∇j , mj denote respectively the position, the charge, the momentum and the mass of the particle j (the electron or the nucleus). The position of the center of mass R and the internal variable r are defined by m1 x1 + m2 x2 , r := x1 − x2 . (5) R := m1 + m2 Moreover, A is the quantized electromagnetic vector potential in the Coulomb gauge, Hf is the free photon energy field, U is a confining potential acting on the center of mass of the atomic system, and V denotes the Coulomb potential. An ultraviolet cutoff at a scale Λ is imposed on A, for some arbitrary but finite Λ > 0. Recall that the units have been chosen such that = c = 1. We shall consider throughout the paper, for the sake of simplicity, a harmonic potential U (R) = β 2 R2 + c0 , for some β > 0 and c0 ∈ R. To address the problem of the existence of resonances, we follow the strategy developed in [3–5] and [2], based on a renormalization group analysis. In order to improve the infrared behavior of the interaction, as in [3], we shall transform HUV

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through the usual Pauli–Fierz transformation. As a consequence, the interaction part in the transformed Hamiltonian can be treated as a perturbation growing in x1 , x2 . We shall use, to control the latter, the confining potential U (R) together with a spatial cutoff χ(r) restricting the electron position to bounded distances V be the Hamiltonian obtained after the Pauli– from the nucleus position. Let H U Fierz transformation and the spatial regularization. The main result of this paper is Theorem 1 below which gives the existence of V as q1 , q2 are = 0. Technically, the proof is significantly different resonances for H U from the one in [3,4]. Indeed, since a “part” of the x1 , x2 -behavior in the interaction will be controlled with the help of the confining potential U (R), we shall have to use complex dilatations of both the photons and the nucleus-electron positions. Therefore we shall have to deal with non self-adjoint operators and non-orthogonal projections. We shall prove in Proposition 4 that the family of complex dilated Hamiltonians is analytic of type (A) in a neighborhood of the origin, which is not straightforward because of the presence of the confining potential. Besides, in the first step of the renormalization procedure, using U (R) instead of introducing a cutoff χ(R) in the interaction requires estimates different from [3, 4]. The main V can ones will be stated in Lemmas 9, 10 and 14. Our aim is to prove that H U be seen as a good starting point for the renormalization group analysis, referring next to [2] for the renormalization procedure itself. Thus we shall only reproduce the new aspects of the proofs, and refer otherwise to [3–5] or [2]. Note in addition that we will have to require a hypothesis related to the Fermi Golden Rule similar to the ones used in [3, 5]; we will show in the appendix how to verify that it is satisfied. The paper is organized as follows: in the remaining part of this section, we define our assumptions on the model we will work with, and we state our results. V associated with In Section 2, we study the Pauli–Fierz Hamiltonians HUV and H U the model of the confined hydrogen atom. We prove in particular that θ → HUV (θ) is analytic of type (A), which, by [5], leads to the absolute continuity of σ(HUV ) on an interval (Theorem 2). Next, in Section 3, we show how the renormalization group method developed in [3,4] and [2] can be applied to our model. This gives the V (Theorem 1). Finally as mentioned above, we verify existence of resonances for H U in the appendix that a hypothesis related to the Fermi Golden Rule is satisfied. 1.2. Assumptions on the model We shall work in this paper with more general operators than the Hamiltonians V describing the model of the confined hydrogen atom. The operators HUV and H U we consider are supposed to act on L2 (R6 ) ⊗ Fs and are defined as Hg = H0 + Wg = Hat ⊗ 1 + 1 ⊗ Hf + Wg .

(6)

Here H0 = Hat ⊗1+1⊗Hf denotes the unperturbed Hamiltonian. We assume that Hat is a Schr¨ odinger operator on L2 (R6 ), such that its ground state energy λ0,0 is strictly less than its ionization threshold Σ := inf σess (Hat ), with at least one

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isolated eigenvalue located between λ0,0 and Σ. We denote by (λj,0 ) the increasing sequence of eigenvalues of Hat located below Σ. The operator Hf is defined by Hf = |k|a∗ (k)a(k)dk . (7) R3

Here we have used the notations R3 = R3 × Z2 ,

n R3n = R3 ,

and, for k = (k, λ) ∈ R3 and a# = a∗ or a, dk = dk , R3

λ=1,2

R3

a# (k) = a# λ (k) .

(8)

(9)

The usual creation operator a∗λ (k) and annihilation operator aλ (k) obey the following Canonical Commutation Rules (in the sense of operator-valued distributions): aλ (k), a∗λ (k ) = δλλ δ(k − k ) , (10) aλ (k), aλ (k ) = a∗λ (k), a∗λ (k ) = 0 . The interaction Wg is written as

with

W1 := W2 :=

R3

R6

Wg := gW1 + g 2 W2 ,

(11)

G1,0 (k) ⊗ a∗ (k) + G0,1 (k) ⊗ a(k) dk ,

(12)

G2,0 (k, k ) ⊗ a∗ (k)a∗ (k ) + G0,2 (k, k ) ⊗ a(k)a(k )

+ G1,1 (k, k ) ⊗ a∗ (k)a(k ) dkdk .

(13)

We assume that, for all k, k , Gm,n (k, k ) defines an operator on L2 (R6 ). We will sometimes use the notations W1,0 := G1,0 (k) ⊗ a∗ (k)dk, and similarly for the other operators Wm,n , m + n ∈ {1, 2}. As in [5], we shall dilate both the positions xj of the two particles (the electron and the nucleus) and the momenta k of the photons in the following way: xj → eθ xj ,

k → e−θ k .

(14)

We denote by Hat (θ), Wg (θ), Gm,n (θ) and Hg (θ) the families of operators obtained respectively from Hat , Wg , Gm,n and Hg through this complex scaling. We shall ¯ In addition we set assume throughout the paper that Gm,n (θ)∗ = Gn,m (θ). ⎧ There exists θ0 > 0 such that for all θ in the disc D(0, θ0 ), ⎪ ⎨ [Hat (θ) − Hat ][Hat + i]−1 ≤ b(θ0 ), where b(θ0 ) → 0 as θ0 → 0. (Han ) ⎪ ⎩ Moreover, the map θ → Hg (θ) is analytic of type (A) on D(0, θ0 ).

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As for the interaction, we require the following hypotheses: ⎧ There exist a non-negative function J−1/2 (k) and θ0 > 0 such ⎪ ⎪ ⎪ ⎪ that: ⎪ ⎪ ⎪ ⎪ ⎪ (i) sup Gm,n (k; θ)|Hat + i|−1/2 ≤ J−1/2 (k) for m + n = 1, ⎪ ⎪ ⎪ |θ|≤θ ⎪ 0 ⎪ ⎪ ⎨ (ii) sup Gm,n (k, k ; θ) ≤ J−1/2 (k)J−1/2 (k ) for m + n = 2, (H−1/2 ) |θ|≤θ0 ⎪ ⎪ ⎪ −1 ⎪ (iii) J−1/2 (k)2 dk < ∞. 3 1 + |k| ⎪ R ⎪ ⎪ ⎪ ⎪ Moreover, the maps θ → Gm,n (k, k ; θ) are bounded analytic on ⎪ ⎪ ⎪ ⎪ D(0, θ0 ) with respect to the norms given in (i) and (ii) ⎪ ⎪ ⎩ respectively. A similar assumption is required in [3,4]. Here we need another assumption related to the confinement of the center of mass imposed in our model: ⎧ There exist a non-negative function J1/2 (k) and θ0 > 0 such that: ⎪ ⎪ ⎪ ⎪ ⎪ (i) sup Gm,n (k; θ)|Hat + i|−1 ≤ J1/2 (k) for m + n = 1, ⎪ ⎪ ⎪ |θ|≤θ0 ⎪ ⎪ ⎪ ⎪ ⎪ (ii) sup Gm,n (k, k ; θ)|Hat + i|−1 ≤ J1/2 (k)J1/2 (k ), ⎪ ⎪ ⎨ |θ|≤θ0 (H1/2 ) sup Gm,n (k, k ; θ)|Hat + i|−1/2 ≤ J1/2 (k)J−1/2 (k ), ⎪ ⎪ |θ|≤θ0 ⎪ ⎪ ⎪ for m + n = 2, where J−1/2 is defined in Hypothesis ⎪ ⎪ ⎪ ⎪ (H−1/2 ), ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ (iii) sup |k| 2 (1−μ) J1/2 (k) < ∞ for some μ > 0. ⎩ k∈R3

Note that in Hypothesis (H1/2 ), it is implicitly assumed that (H−1/2 ) is satisfied and that for m+n = 2, Gm,n is symmetric under the permutation of the variables k and k . 1.3. Statement of results Our main result is Theorem 1 below which provides the existence of resonances for Hg under Hypotheses (Han ), (H−1/2 ) and (H1/2 ). More precisely, considering an eigenvalue λj,0 of H0 , we shall prove that there exist resonances λj,g of Hg , such that λj,g → λj,0 . (15) g→0

We follow the strategy of [3, 4] and use the smooth Feshbach map defined in [2]. The proof is not straightforward since, as explained above, we have to modify carefully Hypotheses 2 and 3 stated in [3, 4] in such a way that, on one hand, our new hypotheses are well adapted to our model, and on the other hand, they are still sufficient to perform a renormalization group analysis. This is reflected in our choice of Hypotheses (H−1/2 ) and (H1/2 ) above. We shall prove:

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Theorem 1. Let λj,0 be a non-degenerate eigenvalue of H0 such that λ0,0 < λj,0 < Σ. Let the coupling parameter g > 0 be sufficiently small. Assume that Hypotheses (Han ), (H−1/2 ) and (H1/2 ) hold for some θ0 > 0 sufficiently small. Let δj be the distance from λj,0 to the rest of the spectrum of Hat . Pick θ = η + iν in the disc D(0, θ0 ) and ρ0 > 0 sufficiently small, such that ρ0 ≤ (δj sin ν)/2 < 1. Then the spectrum of Hg (θ) in Dρ0 /2 := D(λj,0 , ρ0 /2) is located as follows: (16) σ Hg (θ) ∩ Dρ0 /2 ⊂ λj,g (θ) + Kl,n (θ) , where λj,g (θ) is a non-degenerate eigenvalue of Hg (θ), and where Kl,n (θ) is a complex domain defined, for some τ > 1 and 0 < C < 1, by: (17) Kl,n (θ) := λj,g (θ) + e−iν a + b, 0 ≤ a ≤ 1, |b| ≤ Caτ . In particular, λj,g (θ) is independent of θ. Thus, as stated at the beginning of this subsection, provided that Hypothesis (HΓj ) (see Subsection 3.1) related to the Fermi golden rule holds, the unperturbed eigenvalue λj,0 turns into a resonance when the nucleus and the electron are coupled to the photons. If Hypothesis (H1/2 ) is not satisfied, following [5], we still have: Theorem 2. Let g > 0 sufficiently small. Assume that Hypotheses (Hat ) and (H−1/2 ) hold for some θ0 > 0 sufficiently small. Then, provided that Hypotheses (HΓj ) (see Subsection 3.1) hold for all non-perturbed eigenvalues λj,0 located below the ionization threshold Σ of Hat , the spectrum of Hg is absolutely continuous on [Eg , Σ]\V, where Eg denotes the ground state energy of Hg and V is a neighborhood of order O(g 2 ) of {Eg } ∪ {Σ}. The proof of this result follows from the existence of a Feshbach operator associated with Hg (θ); this can be obtained from Lemmas 9 and 10 instead of using the method of [5]. Finally, in the next section, we shall prove that Hypotheses (Han ), (H−1/2 ) and (H1/2 ) are well-adapted to the model of the confined hydrogen atom, that is 1. The initial Hamiltonian HUV fulfills Hypotheses (Han ) and (H−1/2 ). V fulfills Hypotheses (Han ), (H−1/2 ) and 2. The regularized Hamiltonian H U (H1/2 ).

2. The model of the confined hydrogen atom 2.1. Definition of the Hamiltonian HUV Recall that the Hamiltonian HUV we want to study is written as 1 (pj − qj Aj )2 + Hf + U + V . HUV := 2m j j=1,2

(18)

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J. Faupin

E 0 +e0 E 0 +e1 E 0 +e2

E 1 +e0 E 1 +e1 E 1 +e2


e0

e1

e2

Figure 1. Spectrum of the atomic Hamiltonian Hat . It acts on the Hilbert space L2 (R6 ) ⊗ Fs L2 (R6 ; Fs ), where Fs denotes the symmetric Fock space of transversally polarized photons over L2 (R3 ), that is Fs := Fs L2 (R3 ) = C ⊕ Sn ⊗nk=1 L2 (R3 ) . (19) n≥1

Here Sn denotes the symmetrization of the n components in the tensor product ⊗nk=1 L2 (R3 ). The vector potential Aj in the Coulomb gauge is defined by ⊕ Aj := A(xj )dX , (20) R6

with X = (x1 , x2 ) and 1 χ Λ (k) ελ (k) a∗λ (k)e−ik · x + aλ (k)eik · x dk . A(x) := 2π 3 |k| λ=1,2 R

(21)

Here χ Λ denotes an ultraviolet cutoff function. Some analyticity of χ Λ is required, so that we choose: 2 4 8 2 χ Λ (k) = e−k /(Z q Λ ) . (22) Besides, in (21), ε1 (k) and ε2 (k) denote polarization vectors that are perpendicular to each other and to k. The free field energy operator Hf is defined in (7). We write the Coulomb potential V as C (23) V (r) := −Zq 2 , |r| where C is a positive constant. We set p1 p p2 P := p1 + p2 , := − . (24) μ m1 m2 The atomic Hamiltonian is then given in this case by 2 2 p2j P p +V ⊗1+1⊗ +U . (25) +U +V Hat = 2mj 2μ 2M j=1,2 We denote by (Ej )j≥0 the non-decreasing sequence of eigenvalues of p2 /2μ + V and by (ej )j≥0 the non-decreasing sequence of eigenvalues of P 2 /2M + U . The spectrum of Hat is pictured in Figure 1. As in [3], we proceed to a change of units in order to exhibit the perturbative character of the problem. More precisely, we consider the unitary operator U1 that dilates the electron and nucleus positions, and the photons momenta, through (xj , k) → (xj /Zq 2 , Z 2 q 4 k). This leads to 1 1 j (Zq 2 · ) 2 + Hf + V + U . pj − qj Zq 2 A U1 HUV U1∗ = (26) 2 4 Z q 2mj j=1,2

Vol. 9 (2008)


E 0 +e0 E 0 +e1 E 0 +e2

E 1 +e0 E 1 +e1 E 1 +e2

e0

e1

751

e2

Figure 2. Spectrum of the unperturbed Hamiltonian H0 . The vector potential A(x) denotes A(x) where χ Λ is replaced by χ Λ (Z 2 q 4 · ). More2 4 (R) := (1/Z q )U (R/Zq 2 ). We redefine over we have set V (r) := −C/|r| and U (R). Thus the new HamiltonΛ (Z 2 q 4 k), V (r) := V (r) and U (R) := U χ Λ (k) := χ V ian, still denoted by HU , that we have to consider, is HUV =

2 1 pj − qj Zq 2 Aj (Zq 2 · ) + Hf + U + V . 2mj j=1,2

(27)

Setting g := (q 2 Λ)3/2 , we can write HUV as HUV := H0 + Wg + Λ0 , where H0 is defined by H0 :=

j=1,2

p2j + U + V + Hf = Hat ⊗ 1 + 1 ⊗ Hf . 2mj

(28)

Since the spectrum of Hf consists of the simple eigenvalue 0, and the half-axis ]0; ∞[ as absolutely continuous spectrum, we obtain σ(H0 ) as pictured in Figure 2. the interaction Wg as in (11)–(13), where the operators Gm,n := We write j j=1,2 Gm,n are given by G11,0 (k) = G10,1 (k)∗ =

χ Λ (k) −iZq2 k · x1 iZ e ελ (k) · ∇x1 , 3/2 2m1 Λ 2π |k|

G21,0 (k) = G20,1 (k)∗ =

Λ (k) −iZq2 k · x2 −iZ 2 χ e ελ (k) · ∇x2 , 2m2 Λ3/2 2π |k|

(29)

G12,0 (k, k ) = G10,2 (k, k )∗ =

2 2 Λ (k) χΛ (k ) Z2 χ ελ (k) · ελ (k )e−iZq k · x1 e−iZq k · x1 , 3 2m1 Λ 4π 2 |k||k |

G22,0 (k, k ) = G20,2 (k, k )∗ =

2 2 Λ (k) χΛ (k ) Z4 χ ελ (k) · ελ (k )e−iZq k · x2 e−iZq k · x2 , 3 2 2m2 Λ 4π |k||k |

G11,1 (k, k ) =

2 2 Λ (k) χΛ (k ) Z2 χ ελ (k) · ελ (k )e−iZq k · x1 eiZq k · x1 , 2m1 Λ3 2π 2 |k||k |

G21,1 (k, k ) =

2 2 Λ (k) χ (k ) Z4 χ Λ ελ (k) · ελ (k )e−iZq k · x2 eiZq k · x2 . 3 2 2m2 Λ 2π |k||k |

(30)

(31)

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Finally the real number Λ0 is defined by Λ0 := j=1,2 Λj0 where χ Λ (k) χ Λ (k) Z2 Z4 2 dk , Λ dk . Λ10 = = 0 2 3 2 3 4π m1 Λ R3 |k| 4π m2 Λ R3 |k|

(32)

Henceforth we remove this constant from the interaction by redefining H0 := H0 + Λ0 , Wg := Wg − Λ0 . 2.2. Analyticity of HUV (θ) In this paper, in the same way as in [5], we scale both the electron-nucleus positions and the photons momenta through xj → eθ xj ,

k → e−θ k .

(33)

The scaling parameter θ is assumed to lie in a disc D(0, θ0 ) ⊂ C. For real θ, the transformations (33) determine a unitary operator Uθ such that: 2 p p2 + V Uθ∗ = e−2θ + e−θ V , ∗ Uθ 2μ 2μ 2 P P2 (34) + U Uθ∗ = e−2θ + U (eθ · ) , ∗ Uθ 2M 2M ∗ Uθ Hf Uθ∗ = e−θ Hf . Recall that U (R) = β 2 R2 + c0 . For θ ∈ D(0, θ0 ), we define the quadratic forms qVθ θ and qU on H1 (R3 ) and H1 (R3 ) ∩ Q(U + ) respectively by e−2θ (pφ, pψ) − e−θ (V − )1/2 φ, (V − )1/2 ψ , 2μ e−2θ θ (P φ, P ψ) + e2θ β 2 (Rφ, Rψ) + c0 (φ, ψ) . (φ, ψ) := qU 2M

qVθ (φ, ψ) :=

(35) (36)

Lemma 3. Let θ0 be sufficiently small. For θ in D(0, θ0 ), let H0 (θ) be the operator associated with the strictly m-sectorial quadratic form θ θ (Φ, Ψ) = qVθ (Φ, Ψ) + qU (Φ, Ψ) + e−θ (Hf Φ, Hf Ψ) , qH 0 1/2

1/2

(37)

on Q(p21 + p22 ) ∩ Q(U + ) ∩ Q(Hf ). Then θ → H0 (θ) is analytic of type (A) on D(0, θ0 ) and σ H0 (θ) = σ(e−2θ p2 /2μ + e−θ V ) + σ e−2θ P 2 /2M + U (eθ · ) + σ(e−θ Hf ) , (38) where e−2θ p2 /2μ + e−θ V and e−2θ P 2 /2M + U (eθ · ) are the operators associated θ . respectively with the strictly m-sectorial quadratic forms qVθ and qU Proof. Fix θ in D(0, θ0 ) for a sufficiently small θ0 . It is easy to see that the quaθ dratic forms qVθ and qU defined in (35)–(36) are strictly m-sectorial. Applying θ is also strictly m-sectorial and Ichinose’s lemma (see [14]), we obtain that qH 0

Vol. 9 (2008) E 0 +e0 E 0 +e1 E 0 +e2

Resonances of the Confined Hydrogen Atom E 1 +e0 E 1 +e1 E 1 +e2

e e 00000000000 11111111111 00000 11111 000000 111111 00000 11111 00000 11111 000000 111111 00000 11111 00000 11111 000000 111111 00000 11111 00000 11111 000000 111111 00000 11111 00000 11111 000000 111111 00000 11111

e0 −Imθ

753 1

2

Figure 3. Spectrum of the complex dilated unperturbed Hamiltonian H0 (θ). that (38) is satisfied. It remains to prove analyticity of type (A). Let Φ ∈ D(H0 ). For all Ψ ∈ D(H0 (θ)∗ ), we have θ H0 (θ)∗ Ψ, Φ ≤ |(Ψ, H0 Φ)| + qH (Ψ, Φ) − qH0 (Ψ, Φ) 0 (39) ≤ b(θ0 ) Ψ p2 Φ + V Φ + P 2 Φ + R2 Φ + Hf Φ , where b(θ0 ) is such that b(θ0 ) → 0 as θ0 → 0. It follows that D(H0 ) ⊂ D(H0 (θ)). Then, from [H0 − H0 (θ)][H0 + i]−1 ≤ b(θ0 ), we get 1

H0 Φ ≤

H0 (θ)Φ + b(θ0 ) Φ , (40) 1 − b(θ0 ) for all Φ ∈ D(H0 ), provided that b(θ0 ) < 1. This implies that H0 (θ) is closed on D(H0 ) and hence that D(H0 ) = D(H0 (θ)). The analyticity of the map θ → H0 (θ)Φ for Φ ∈ D(H0 ) is then straightforward. According to (38), the spectrum of H0 (θ) is given by Figure 3. Now, the operator Uθ Wg Uθ∗ is Wg where the operator-valued functions Gjm,n are replaced by Gjm,n (θ); we get Gjm,n (θ) by adding a factor e−2θ and replacing Λ (e−θ k) in Gjm,n . For instance χ Λ (k) by χ G11,0 (k; θ) = G10,1 (k; θ)∗ = e−2θ

iZ χ Λ (e−θ k) −iZq2 k · x1 e ελ (k) · ∇x1 . 3/2 2m1 Λ 2π |k|

(41)

The operator Wg (θ) is well-defined on D(H0 ). Proposition 4. Let θ0 and g be sufficiently small. For θ in D(0, θ0 ), let HUV (θ) := H0 (θ) + Wg (θ) on D(H0 ). Then HUV (θ) fulfills Hypotheses (Han ) and (H−1/2 ), with 2 2 J−1/2 (k) ≤ Cste |k|−1/2 e−k /Λ . (42) Proof. The fact that [Hat (θ) − Hat ][Hat + i]−1 ≤ b(θ0 ), for some b(θ0 ) such that b(θ0 ) → 0 as θ0 → 0, follows in the same way as in the proof of Lemma 3. As in [5, Lemma 1.1], one can see that Wg (θ)[H0 + i]−1 ≤ Cg, for some positive constant C. This implies that HUV (θ) is closed on D(H0 ) for g and θ0 sufficiently small. Since θ → Wg (θ)[H0 + i]−1 is analytic on D(0, θ0 ) by our choice (22) of the ultraviolet cutoff, the analyticity of type (A) of θ → HUV (θ) follows from Lemma 3. Finally, using the definition of Gm,n (θ) given in (29)–(31) and (41), one can straightforwardly prove that HUV (θ) fulfills Hypothesis (H−1/2 ) with J−1/2 given by (42).

754

J. Faupin


V 2.3. The regularized Hamiltonian H U We proved in Proposition 4 that HUV (θ) fulfills Hypothesis (H−1/2 ). This will be sufficient to obtain the absolute continuity of σ(HUV ) on an interval between λ0,0 and Σ, but this is not sufficient to apply the renormalization group method of [3,4]. To face this problem, we begin with performing the Power–Zienau–Woolley transformation (sometimes called the Pauli–Fierz transformation) on HUV . More precisely, setting X = (x1 , x2 ), we define a unitary operator T by T =

⊕

R6

T (X)dX

with

T (X) = e−i

j=1,2

qj Zq2 xj · A(0)

.

(43)

Then bλ (k, X) := T (X)aλ (k)T ∗ (X) = aλ (k) − iwλ (k, X), with Λ (k) 1 χ ελ (k) · qj Zq 2 xj . 1/2 2π |k| j=1,2

wλ (k, X) =

(44)

V , which is unitary equivalent to H V , is defined by The Hamiltonian H U U UV := T HUV T ∗ = H

j=1,2

j = with A

⊕ R6

1 j (Zq 2 · ) 2 + H f + U + V , pj − qj Zq 2 A 2mj

j (X)dX, H f = A

j (X) = A(xj ) − A(0) , A

⊕ R6

(45)

f (X)dX, and H

f (X) = H

λ=1,2

R3

|k|b∗λ (k, X)bλ (k, X)dk .

(46)

Expanding (45), we can write UV := H 2 , 0 + W g := H at + Hf + W g := H at + Hf + g W 1 + g 2 W H

(47)

with 2 2 χ Λ (k)2 at = Hat + g 2 Z H ελ (k) · r dk 3 2 Λ 4π 3 λ=1,2 R Z Z2 χ Λ (k)2 1 2 2 2 2 sin (Zq k · x /2) + sin (Zq k · x /2) dk , + 2g 2 3 1 2 Λ R3 π 2 |k| 2m1 2m2 (48)

and 1 = W

R3

1,0 (k) ⊗ a∗ (k) + G 0,1 (k) ⊗ a(k) dk . G

(49)

Vol. 9 (2008)


755

m,n are defined for m + n = 1 by The “new” coupling operators G 1,0 (k) = G 0,1 (k)∗ G =

χ Λ (k) −iZq2 k · x1 iZ e − 1 ελ (k) · ∇x1 2m1 Λ3/2 2π |k| −

iZ 2 χ Λ (k) −iZq2 k · x2 − 1 ελ (k) · ∇x2 e 3/2 2m2 Λ 2π |k|

−

iZ |k|1/2 χ Λ (k) ελ (k) · r . 3/2 2π Λ

(50)

2 is W2 where the terms Note that we have set r := x1 − Zx2 . Moreover, W ±iZq2 k · xj ±iZq2 k · xj e are replaced by e − 1 in Gm,n , for m + n = 2. We define 0 (θ), W g (θ) and G m,n (θ) by means of the complex scaling at (θ), H the operators H operator Uθ associated with the dilatations (33). Then, in the same way as for HUV (θ), one can prove: V (θ) := Proposition 5. Let θ0 and g be sufficiently small. For θ in D(0, θ0 ), let H U 0 (θ) + W g (θ) on D(H 0 ). Then H V (θ) fulfills Hypotheses (Han ), (H−1/2 ) and H U (H1/2 ), with 2 2 Cste |k|1/2 e−k /Λ . J1/2 (k) ≤ (51) g The factor of order g −1 in the bound (51) is due to the last term in (50) which at + i|1/2 /g. This appears to be a problem, is relatively bounded with respect to |H 2 g := g W 1 + g 2 W because we require that all the terms of the perturbation W are small compared to the unperturbed Hamiltonian H0 := Hat + Hf , when g is small. To avoid this difficulty, in a way similar to what is done in [3], we consider g , which restricts the a simplified model where a spatial cutoff is imposed on W position of the electron to bounded distances from the position of the nucleus. g;reg where g by W More precisely, we replace W g;reg := χr0 (r)W g . W

(52)

, where r0 Here χr0 is a cutoff that, for concreteness, we choose as χr0 (r) := e−r V is some arbitrary positive real number. Then we define the Hamiltonian H U;reg by 2

/r02

V H U;reg := H0 + Wg;reg ,

(53)

and one can verify that: V (θ) := Proposition 6. Let θ0 and g be sufficiently small. For θ in D(0, θ0 ), let H U;reg 0 (θ) + W g;reg (θ) on D(H 0 ). Then H V (θ) fulfills Hypotheses (Han ), (H−1/2 ) H U;reg

and (H1/2 ), with J−1/2 (k) := Cste |k|−1/2 e−k

2

/Λ2

,

J1/2 (k) := Cste |k|1/2 e−k

2

/Λ2

.

(54)

756

J. Faupin ~ E 0 +e0

~ E 0+e1

~ E 1+e0

~ E 1+e1

Ann. Henri Poincaré ~ e 0 +Ec0

~ e 1 +Ec0

~ e 2 +Ec0

at for Z = 1. Figure 4. Spectrum of the new atomic Hamiltonian H g := W g;reg and H V := H V . Henceforth, we redefine W U U;reg To conclude with this subsection, we describe the spectrum of the “new” at . Assume that we are dealing with the hydrogen atom, atomic Hamiltonian H that is Z = 1. Then (48) implies 2 2 at = p + V + Cg 2 r2 + P + U H 2μ 2M (55) 2 1 1 χ (k) 1 Λ 2 2 2 2 2 + 2g 3 sin (q k.x1 /2) + sin (q k.x2 /2) dk , Λ R3 π 2 |k| 2m1 2m2

where C is positive. One can see that the spectrum of p2 /2μ+V +Cg 2 r2 is discrete: l , and the continuous the eigenvalues El of p2 /2μ + V are slightly shifted to E 2 spectrum of p /2μ+ V turns into a non-decreasing sequence of discrete eigenvalues c (g), l ≥ 1, such that E l c lc (g) − E l−1 E (g) → 0 . 2

2 2

g→0 2

Thus the spectrum of p /2μ + V + Cg r + P /2M + U is purely discrete, and at : its eigenvalues are E l + en and E c + en , slightly perturbed the same holds for H l c + en the eigenvalues l + en and E by the last term in (55). We still denote by E l at as described in Figure 4. at . Hence we obtain the spectrum of H of H Note that, if Z > 1, we can not use the same argument to state that the essential spectrum of p2 /2μ + V turns into discrete spectrum. However, we still have that the eigenvalues El + en of Hat are slightly shifted, remaining eigenvalues at . Since we only study such eigenvalues in the sequel (we shall not study the of H c + en ), the case Z > 1 can be treated in the same way as the case behavior of E l Z = 1.

3. Absolute continuity of the spectrum and existence of resonances We proved in the previous section that the Pauli–Fierz Hamiltonian associated with the model of the confined hydrogen atom, HUV , fulfills Hypotheses (Han ) V , fulfills Hypotheses (Han ), and (H−1/2 ), whereas the regularized Hamiltonian, H U (H−1/2 ) and (H1/2 ). From these results, considering operators Hg of the form introduced in Subsection 1.2, we shall obtain in this section the absolute continuity V . of σ(HUV ) on an interval, and the existence of resonances for H U 3.1. Preliminaries. Absolute continuity of the spectrum Recall from Subsection 1.2 that Hg = H0 + Wg = Hat ⊗ 1 + 1 ⊗ Hf + Wg . Consider an eigenvalue λj,0 of Hat strictly located between the ground state energy λ0,0 and

Vol. 9 (2008)


757

the ionization threshold Σ. We set δj := dist(λj,0 , σ(Hat )\{λj,0 }). The projection Pat,j (θ) onto the eigenspace of Hat (θ) corresponding to λj,0 , and P¯at,j (θ), are defined by dz i , P¯at,j (θ) := 1 − Pat,j (θ) . (56) Pat,j (θ) := δj 2π |z−λj,0 |= 2 Hat (θ) − z As in [5], setting Pat,j := Pat,j (0), P¯at,j := P¯at,j (0), we define the matrices od Pat,j G0,1 (k)P¯at,j [Hat − λj,0 + |k| − i0]−1 P¯at,j G1,0 (k)Pat,j dk , (57) Zj := 3 R dk . (58) Pat,j G0,1 (k)Pat,j G1,0 (k)Pat,j Zjd := 3 |k| R For θ ∈ D(0, θ0 ), we set Zjod (θ) := Uθ Zjod Uθ−1 , Zjd (θ) := Uθ Zjd Uθ−1 , Zj (θ) := Zjd (θ) − Zjod (θ) . (59) Let Γj := min σ(Im(Zjod )) ; then for the needs of the proof, we have to require the following hypothesis, related to the Fermi golden rule: (HΓj ) Γj > 0 . We shall verify in the appendix that Hypothesis (HΓ1 ) is indeed satisfied in the case of HUV , for suitably chosen parameters. Now, to simplify, we assume moreover that: (HΓ j ) Γj > 0 and Γj is a simple eigenvalue of Im(Zjod ) . Let φj,0 be a normalized eigenvector associated with the eigenvalue Γj of Im(Zjod ); we define (60) Δj := Re (φj,0 , Zj φj,0 ) . Let Sj , Rj (ε, C) denote the following sets: Sj := λj,0 + g 2 (Δj − iΓj ) − iQj , Rj (ε, C) := Sj + e−θ R+ + D(0, Cg 2+ε ) ,

(61)

where ε is a small positive constant and C ∈ R+ . The set Qj := {z ∈ C| − μ ≤ arg(z) ≤ μ} (for some 0 < μ < π/2) is supposed (by Hypothesis (HΓ j )) to be such that (62) (φ, Zj φ), φ = 1 ⊂ Δj − iΓj − iQj . Finally, the set Aj (ε) is defined by 1 2−2ε Aj (ε) := z ∈ C, Re(z) ∈ Ij , |Im(z)| ≤ g sin ν , 2

(63)

where Ij is the interval ]λj,0 − δj /2; λj,0 + δj /2[. The following result is proved as in [5] (see also Proposition 11 below for the proof of the existence of a Feshbach operator associated with Hg (θ)):

758

J. Faupin

Ann. Henri Poincaré Aj

λj,0

Ij

Sj

1111 0000 0000 1111 0000 1111 0000 1111

Rj

Figure 5. The resolvent set of Hg (θ): Aj \Rj ⊂ ρ(Hg (θ)) (see Theorem 7).

Theorem 7. Let g > 0 sufficiently small. Assume that Hypotheses (Han ) and (H−1/2 ) hold for a sufficiently small θ0 . Let 0 < ε < 1/3 and fix θ = iν in D(0, θ0 ) with ν > 0. Assume that (HΓ j ) is fulfilled. Then there exists a positive constant C such that Aj (ε)\Rj (ε, C) ⊂ ρ Hg (θ) , (64) where ρ(HUV (θ)) denotes the resolvent set of Hg (θ) (see Figure 5). This implies that the spectrum of Hg is absolutely continuous in Ij . As a corollary of Theorem 7, we obtain Theorem 2, which implies by Proposition 4: Corollary 8. Let g > 0 sufficiently small. Let EUV be the ground state energy of HUV . Assume that Hypotheses (HΓj ) hold for all non-perturbed eigenvalues λj,0 located below e0 . Then σ(HUV ) is absolutely continuous on [EUV , e0 ]\V, where V is a neighborhood of order O(g 2 ) of {EUV } ∪ {e0 }. 3.2. The smooth Feshbach map applied to Hg (θ) Henceforth, to simplify, we suppose that the eigenvalue λj,0 of Hat is non-degenerate and located at 0. The complex parameter θ is fixed to θ := iν for some ν > 0. For any ρ0 > 0 such that ρ0 ≤ (δj sin ν)/2 < 1, we define the functions of Hf , χρ0 (Hf ) and χ ¯ρ0 (Hf ), by π χρ0 (Hf ) := sin Θ(Hf /ρ0 ) , ! 2 π χ ¯ρ0 (Hf ) := 1 − χ2ρ0 (Hf ) = cos Θ(Hf /ρ0 ) , (65) 2 where Θ ∈ C∞ 0 ([0, ∞[; [0, 1]) is such that Θ = 1 on [0, 3/4[ and Θ = 0 on [1, ∞[. Next we use (56) and (65) to define P (θ) := Pat,j (θ) ⊗ χρ0 (Hf ) ,

P¯ (θ) := Pat,j (θ) ⊗ χ ¯ρ0 (Hf ) + P¯at,j (θ) ⊗ 1 . (66)

Note that (66) implies P (θ)2 + P¯ (θ)2 = 1. A key point in the proof of Theorem 7 is to get the existence of the Feshbach operator (or smooth Feshbach operator [2])

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759

FP (θ) (Hg (θ) − z, H0 (θ) − z) defined by FP (θ) Hg (θ) − z, H0 (θ) − z := H0 (θ) − z + P (θ)Wg (θ)P (θ) − P (θ)Wg (θ)P¯ (θ) (H0 (θ) − z) −1 + P¯ (θ)Wg (θ)P¯ (θ) P¯ (θ)Wg (θ)P (θ) .

(67)

where z is a spectral parameter belonging to Dρ0 /2 := {z ∈ C, |z| ≤ ρ0 /2}. Here we give two lemmas that lead to the existence of FP (θ) (Hg (θ) − z, H0 (θ) − z). Lemma 9. Assume that Hypothesis (H−1/2 ) holds for some θ0 > 0 sufficiently small. Then there exists a positive constant C such that for all θ ∈ D(0, θ0 ), ρ > 0, and m, m , n, n ∈ N, m + n ∈ {1, 2}, m + m + n + n = 2: m n |Hat + i|− 2 [Hf + ρ]− m2 Wm,n (θ)[Hf + ρ]− n2 |Hat + i|− 2 ≤ C . (68) Proof. (68) follows from Hypothesis (H−1/2 ) and well-known “Pull-through formulas”; we refer for instance to [5] for details. See also Lemma 14 below. Lemma 10. Assume that Hypothesis (Han ) holds for some θ0 > 0 sufficiently small. Then there exists a positive constant C such that for all θ = iν ∈ D(0, θ0 ), 0 < ρ0 < (δj sin ν)/2, z ∈ Dρ0 /2 , and m, m , n, n ∈ N, m + m + n + n = 2: −1 m m |Hat + i| 2 [Hf + ρ0 ] 2 P¯ (θ) (H0 (θ) − z)1Ran(P¯ (θ)) n n C − m +n P¯ (θ)[Hf + ρ0 ] 2 |Hat + i| 2 ≤ ρ0 2 . (69) δj sin ν Proof. Let us prove (69) for m = n = 1. The other cases could be obtained similarly. First consider the contribution of P¯at,j (θ) ⊗ 1 in P¯ (θ). We write P¯at,j (θ) as j−1 P¯at,j (θ) = Pat,l (θ) + Pat,>j (θ) , (70) j

l=0

where Pat,>j (θ) := 1− i=0 Pat,l (θ). Hypothesis (Han ) implies that [Hat +i]Pat,l (θ) is bounded for any l ≤ j. Hence by the Spectral Theorem, j−1 −1 1/2 |Hat + i|1/2 Pat,l (θ) ⊗ 1 H0 (θ) − z 1Ran(P¯ (θ)) [Hf + ρ0 ] l=0

≤

C . (71) δj sin ν

To estimate the contribution of Pat,>j (θ), note that Pat,>j (θ) ⊗ 1 H0 (θ) − z = (Pat,>j ⊗ 1)[Hat + e−θ Hf − z] (72) + Pat,>j (θ) ⊗ 1 Hat (θ) − Hat " # + Pat,>j (θ) − Pat,>j ⊗ 1 [Hat + e−θ Hf − z] ,

760

J. Faupin


where Pat,>j := Pat,>j (0). Since [(Pat,>j ⊗ 1)[Hat + e−θ Hf − z]]−1 ≤ Cste δj−1 , a Neumann series expansion, Hypothesis (Han ) and the Spectral Theorem yield −1 C 1/2 |Hat + i|1/2 Pat,>j (θ) ⊗ 1 H0 (θ) − z 1Ran(P¯ (θ)) [Hf + ρ0 ] ≤ . (73) δj Finally, using that [Hat +i]Pat,j (θ) is bounded together with the Spectral Theorem, the contribution of Pat,j (θ) ⊗ χ ¯ρ0 (Hf ) is estimated as −1 1/2 |Hat + i|1/2 Pat,j (θ) ⊗ χ ¯ρ0 (Hf ) H0 (θ) − z 1Ran(P¯ (θ)) [Hf + ρ0 ] ≤

C −1/2 ρ . (74) δj sin ν 0

Hence the proof is complete if m = n = 1. The other cases are similar.

From Lemmas 9 and 10, we obtain the existence and isospectrality of the Feshbach operator (67) in the following sense: Proposition 11. Let g > 0 sufficiently small. Assume that Hypotheses (Han ) and (H−1/2 ) hold for some θ0 > 0 sufficiently small. Let 0 < ρ0 < (δj sin ν)/2. Then, for all z ∈ Dρ0 /2 , the Feshbach operator (67) is well-defined. Moreover, z is an eigenvalue of Hg (θ) with multiplicity m if and only if 0 is an eigenvalue of FP (θ) (Hg (θ) − z, H0 (θ) − z)Ran(P (θ)) with the same multiplicity. Proof. Expanding the resolvent in the rhs of (67) into a Neumann series yields −1 H0 (θ) − z 1Ran(P¯ (θ)) + P¯ (θ)Wg (θ)P¯ (θ) −1 " −1 #n = H0 (θ) − z 1Ran(P¯ (θ)) . −P¯ (θ)Wg (θ)P¯ (θ) H0 (θ) − z 1Ran(P¯ (θ)) n≥0

(75) −1/2

sufficiently small, which can The rhs of the last equation is well-defined for gρ0 be seen by using the estimates of Lemmas 9 and 10. One can prove similarly that FP (θ) (Hg (θ) − z, H0 (θ) − z) is well-defined and satisfies all the assumptions of [11]. The isospectrality then follows by [11]. Using (75), our aim in the next subsection will be to prove that the Feshbach operator (67) can be identified with an element of a suitably chosen Banach space constructed in a way similar to the one in [2]. The main property of this Banach space that we shall require is that the operator coming from the interaction is irrelevant under renormalization.

Vol. 9 (2008)


761

3.3. A Banach space of Hamiltonians For the convenience of the reader, we describe precisely the Banach space of Hamiltonians constructed here in a way similar to [2]. We set: # # # W≥0 := C ⊕ T ⊕ W≥1 := C ⊕ T ⊕ WM,N , (76) M+N ≥1

where

$ T :=

% f ∈ C1 ([0, 1]), f (0) = 0, f T := sup |f (γ)| < ∞ ,

(77)

γ∈[0,1]

and

# N WM,N := fM,N : [0, 1] × B M 1 × B 1 → C such that: , ∗ fM,N ( · ; k(M) ; k˜(N ) ) ∈ C1 ([0, 1]) for every (k (M) ; k˜(N ) ) ∈ B M+N 1 ∗ fM,N (γ; k(M) ; k˜(N ) ) is symmetric w.r.t. k(M) and k˜ (N ) , ∗ fM,N # := fM,N + ∂γ fM,N < ∞ .

(78)

Here B 1 = B1 × {1, 2} where B1 is the unit ball in R3 . Moreover ∂γ fM,N is the partial derivative of fM,N with respect to the first variable, and

fM,N :=

sup +N [0,1]×B M 1

M N & & |ki |−1/2 |k˜j |−1/2 . fM,N (γ; k (M) ; k˜(N ) ) i=1

(79)

j=1

Note that we have used the notations k (M) := (k 1 , . . . , k M ) ∈ R3M ,

k˜(N ) := (k˜1 , . . . , kÑ ) ∈ R3N .

(80)

Note also that, in [2], a L2 -norm is used instead of the L∞ -norm considered in (79). # Next, the space W≥1 := {w := (wM,N )M+N ≥1 } is equipped with the norm := ζ −(M+N ) wM,N # , (81)

w # ζ,1 M+N ≥1

where 0 < ζ < 1 is a parameter that we will precise below. Defining # W0,0 := f0,0 ∈ C1 ([0, 1]), f0,0 # := f0,0 ∞ + ∂γ f0,0 ∞ < ∞ ,

(82)

one can see that there is a natural isomorphism between the Banach spaces C ⊕ T # # and W0,0 , so that we identify C ⊕ T and W0,0 . Thus, we can write an element of # # W≥0 as w := (wM,N )M+N ≥0 , and W≥0 is equipped with the norm := ζ −(M+N ) wM,N # . (83)

w # ζ M+N ≥0 # with an operator on the Hilbert space Now, we want to identify an element of W≥0

Hred := 1Hf 0 and ρ0 > 0 sufficiently small. Assume that Hypotheses (Han ), (H−1/2 ) and (H1/2 ) hold for some θ0 sufficiently small. Fix ρ and ζ as in Theorem 15. Then, for all θ = iν ∈ D(0, θ0 ), where ν > 0 is chosen such that ρ0 ≤ (δj sin ν)/2 < 1, the spectrum of Hg (θ) is located as follows: σ Hg (θ) ∩ Dρ0 /2 ⊂ E(∞) + K(∞) . (133) −1 −1 −1 ◦ Z(1) ◦ · · · ◦ Z(n) (0) is a simple eigenvalue of Hg (θ), Here, E(∞) := limn→∞ Z(0) and K(∞) is a complex domain defined by K(∞) := e−iν a + b, 0 ≤ a ≤ 1, |b| ≤ Caτ , (134)

where τ > 1 and where C is a positive constant which can be chosen strictly less than 1 provided that g is sufficiently small. Assuming moreover that (HΓj ) is fulfilled, this implies that E(∞) is a resonance for Hg .

Appendix A. Proof of a hypothesis related to the Fermi Golden Rule In this appendix, we prove that, for suitably chosen parameters, the assumption (HΓ1 ) (see Subsection 3.1) related to the Fermi Golden Rule is satisfied in the case of the confined hydrogen atom HUV . Recall that {El } denotes the sequence of the eigenvalues of p2 /2μ+ V and that {en } denotes the sequence of the eigenvalues of P 2 /2M + U , where U (R) = β 2 R2 + c0 . Note that e1 − e0 = 2/M β, whereas E1 − E0 = O(1). Thus, for β sufficiently small, E0 + e1 is the first eigenvalue (with multiplicity 3) above the bottom of the spectrum E0 + e0 of Hat . We have to show that (135) Γ1 = min σ Im(Z1od ) is strictly positive, where Z1od is the matrix −1 od Pat,1 G0,1 (k)P¯at,1 Hat − (E0 + e1 ) + |k| − i0 P¯at,1 G1,0 (k)Pat,1 dk . Z1 = R3

(136) Here Pat,1 denotes the projection onto the eigenspace of Hat associated with E0 + e1 , and P¯at,1 = 1 − Pat,1 . We choose explicit polarization vectors: (k2 , −k1 , 0) ε1 (k) := 2 , k1 + k22

ε2 (k) :=

k ∧ ε1 (k) , |k|

(137)

where k = (k1 , k2 , k3 ). Recall that q 2 = g 2/3 /Λ. Proposition 17. Let 0 < q 2 β 5/2 1. Then (HΓ1 ) is satisfied, with Γ1 = Cβ 5/2 + O(q 2 ) , where C is a positive constant.

(138)

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Proof. Let Ψ0 be a non-degenerate ground state of p2 /2μ + V . Let φ0 and φ1 be first two normalized eigenstates associated with the 1-dimensional harmonic oscillator d2 /dx2 + β 2 x2 : .1/4 -√ √ 3/4 √ √ 2M β 2M β − 2Mβx2 /2 − 2M βx2 /2 e , φ1 (x) = xe . (139) φ0 (x) = π π 1/4 Then we can write the eigenstates Φ0 and Φj1 (for j = 1, 2, 3) associated respectively with the eigenvalues e0 and e1 of P 2 /2M + U as ⎛ ⎞ & & φ0 (Rj ) , Φj1 (R) = ⎝ φ0 (Ri )⎠ φ1 (Rj ) , (140) Φ0 (R) = j=1,2,3

i=1,2,3,i=j

with R = (R1 , R2 , R3 ). By (136), we have to compute for i, j ∈ {1, 2, 3}: od i Im(Z1 ) i,j = Ψ0 (r)Φ1 (R)G0,1 (k)Ψ0 (r)Φ0 (R)drdR R3

R6

×C Ψ0 (r)Φj1 (R)G0,1 (k)Ψ0 (r)Φ0 (R)drdR R6 × δ |k| − (e1 − e0 ) dk ,

(141)

where C denotes the complex conjugation in C. Recall that G0,1 (k) =

χ Λ (k) iZq2 k · x1 −iZ e ελ (k) · ∇x1 3/2 2m1 Λ 2π |k| +

χ Λ (k) iZq2 k · x2 iZ 2 e ελ (k) · ∇x2 . 2m2 Λ3/2 2π |k|

(142)

2 We denote by A the matrix obtained from Im(Z1od ) by replacing eiZq k · xj by 1 in the definition of G0,1 (k). Let us show that A is diagonal and positive. In the spherical coordinates defined for ρ ≥ 0, 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ π by

k1 = ρ cos θ sin ϕ ,

k2 = ρ sin θ sin ϕ ,

k3 = ρ cos ϕ ,

(143)

the polarization vectors (137) become ε1 (k) = (sin θ, − cos θ, 0) ,

ε2 (k) = (cos θ cos ϕ, sin θ cos ϕ, − sin ϕ) .

Let us compute, for instance, A1,2 : we have β2χ Λ ( 2/M β) √ A1,2 = C0 β 2π π φ1 , (sin θ)φ0 × φ1 , (− cos θ)φ0 sin ϕdϕdθ ×

0 2π

0 π

+ 0

0

φ1 , (cos θ cos ϕ)φ0 × φ1 , (sin θ cos ϕ)φ0 sin ϕdϕdθ ,

(144)

(145)

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J. Faupin


where C0 is a positive constant and where φ0 := dφ0 /dx. Integrating over θ yields A1,2 = 0. The same would hold for Ai,j with i = j, as follows by integrating over θ or ϕ. Now, let us compute, for instance, the diagonal coefficient A1,1 : we have β2χ Λ ( 2/M β) √ A1,1 = C1 β 2π π 2 φ1 , (sin θ)φ0 sin ϕdϕdθ ×

0 2π

+ 0

0 π

2 φ1 , (cos θ cos ϕ)φ0 sin ϕdϕdθ ,

(146)

0

where C1 a positive constant. The second term in the previous sum vanishes, whereas the first one gives: 2π π 2 φ1 , (sin θ)φ0 sin ϕdϕdθ = c1 (φ1 , φ0 )2 = c1 β , (147) 0

where c1 and

c1

0

are positive constants. Thus A1,1 = C1 χ Λ 2/M β β 5/2 ,

(148)

where C1 is a positive constant. Likewise one could show that Λ 2/M β β 5/2 , A3,3 = C3 χ Λ 2/M β β 5/2 , A2,2 = C2 χ

(149)

where C2 and C3 are positive constants. To finish the proof, we have to check that Im(Z1od )−A is small compared to A. But the matrix Im(Z1od )−A is Im(Z1od ) where, 2 in at least one of the two operators G0,1 (k, 1), G0,1 (k, 2), the terms eiZq k · xj are 2 replaced by eiZq k · xj − 1. Hence, using that iZq2 k · xj − 1 ≤ Zq 2 |k||xj | , (150) e one can see that for i, j ∈ {1, 2, 3}: Im(Z1od ) − A i,j ≤ C(β)q 2 ,

(151)

where C(β) is a bounded positive constant depending on β. Provided that q 2 is sufficiently small, this implies that Γ1 > 0 and the proof is complete. Remark 18. Let j be such that E1 +e0 is the jth eigenvalue of Hat . Using arguments similar to the ones of the proof of Proposition 17, under the same assumptions, one could show that Γj > 0, with Γj β provided that β is chosen sufficiently small. Thus, we see that the first eigenvalues E0 + en (with n small) lead to resonances that should be much closer to the real axis than the other ones coming from the eigenvalues of the form El + en with l ≥ 1. In some sense, this is related to the Lamb–Dicke effect mentioned in our introduction.

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References [1] L. Amour and J. Faupin, The confined hydrogeno¨ıd ion in non-relativistic quantum electrodynamics, Cubo Math. J. 9(2) (2007), 103–137. [2] V. Bach, T. Chen, J. Fr¨ ohlich and I. M. Sigal, Smooth Feshbach map and operatortheoretic renormalization group methods, J. Funct. Anal. 203 (2002), 44–92. [3] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. in Math. 137 (1998), 299–395. [4] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Renormalization group analysis of spectral problems in quantum field theory, Adv. in Math. 137 (1998), 205–298. [5] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation fields, Comm. Math. Phys. 207(2) (1999), 249–290. [6] C. Cohen-Tannoudji, Cours au collège de France 2003–2004, www.phys.ens/cours/ college-de-france. [7] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Processus d’interaction entre photons et atomes (Edition du CNRS, 2001). [8] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons et atomes, (Edition du CNRS, 2001). [9] R. H. Dicke, The effect of collisions upon the Doppler width of spectral lines, Phys. Rev. 89(2) (1953), 472–473. [10] A. Fehli, Analyse spectrale de l’hamiltonien associ´ e` a un atome non relativiste couplé a un champ électromagnétique quantifié, Thèse de l’université Paris 13, (2002). ` [11] M. Griesemer and D. Hasler, On the smooth Feshbach–Schur map, Preprint mp-arc, 07-102, (2007). [12] S. Gustafson and I. M. Sigal, Mathematical concepts of quantum mechanics, (Springer-Verlag, 2003). [13] T. Kato, Perturbation theory for linear operators, (Springer-Verlag, 1966). [14] M. Reed and B. Simon, Methods of modern mathematical physics, vol. IV, Analysis of operators, (Academic Press, New York, 1978). [15] H. Spohn, Dynamics of charged particles and their radiation field, (Cambridge University press, 2004). [16] T. Ta¨ıeb, R. Dum, J. I. Cirac, P. Marte and P. Zoller, Cooling of atoms in laserinduced potential wells, Phys. Rev. A 49 (1994), 4876–4887. [17] D. J. Wineland and W. M. Itano, Laser cooling of atoms, Phys. Rev. A 20(4) (1979), 1521–1540. Jérémy Faupin Laboratoire de Mathématiques EDPPM, UMR-CNRS 6056, Université de Reims Moulin de la Housse – BP 1039 F-51687 Reims Cedex 2, France e-mail: [email protected] Communicated by Vincent Rivasseau. Submitted: October 19, 2007. Accepted: January 22, 2008.

Ann. Henri Poincaré 9 (2008), 775–815 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/040775-41, published online June 2, 2008 DOI 10.1007/s00023-008-0372-x


Semiclassical Resolvent Estimates for Schr¨ odinger Operators with Coulomb Singularities Fran¸cois Castella, Thierry Jecko, and Andreas Knauf Abstract. Consider the Schr¨ odinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h−1 ) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.

1. Introduction In the late eighties and the beginning of the nineties, many semiclassical results were obtained in stationary scattering theory. In this setting, the long time evolution of a system is studied via the resolvent, which appears in representation formulae for the main scattering objects. One can distinguish two complementary domains: on the one hand semiclassical results concerning scattering objects at non-trapping energies (when resonances are negligible), and on the other hand studies of resonances and of their influence on scattering objects. We refer to [15, 20, 29, 39, 47] and also to [46] for an overview of the subject. These results often show a Bohr correspondence principle for the scattering states. Many studies treat (non-relativistic) molecular systems described by a (many body) Schr¨ odinger operator. From a physical point of view, it is natural to let the potential admit Coulomb singularities in that context. In the spectral analysis of the operator, these singularities do not produce difficulties in dimension ≥ 3, thanks to Hardy’s inequality (cf. (2.9)). In the semiclassical regime however, little is known when Coulomb singularities occur. We point out the propagation results in [13, 27, 32]. In the above

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mentioned domains of stationary scattering theory, we do not know of any semiclassical result, except that of [29,53]. We think that the main obstacle stems from the difficulty to develop a semiclassical version of Mourre’s theory (cf. [14, 40, 47]) in this situation. This task is performed in [29] when all singularities are repulsive, a situation where the associated classical Hamilton flow is complete. Recently semiclassical resolvent estimates (and further interesting results) were obtained by Wang in [53] but in a non optimal framework (see comments below). When attractive singularities occur, the classical flow is not complete anymore, while it can be regularized (cf. [13, 27, 32]). The aim of this article is to contribute to the development of such a semiclassical analysis of molecular scattering. In [22,23], the author faced similar difficulties in the study of a matricial Schr¨ odinger operator. He adapted in [24,25] an alternative approach, previously used in [5]. We here follow the same approach, combined with ideas from [6, 13, 32, 52], in order to extend, in the case of potentials with arbitrary Coulomb singularities, a result established in [29, 47]. We now introduce some notation and present the main results of this paper. 1.1. The Schr¨ odinger operators Let d ∈ N := {0, 1, 2, . . .} with d ≥ 2. For x ∈ Rd , we denote by |x| the usual norm of x and we set x := (1 + |x|2 )1/2 . We denote by Δx the Laplacian in Rd . We consider a long-range potential V which is smooth except at N Coulomb singularities (N ∈ N∗ ) located at the sites sj where j ∈ {1, 2, . . . , N }. Let ˆ := Rd \ S , M

with S := {sj ; 1 ≤ j ≤ N }

ˆ ; R) such and R0 := max{|sj | + 1; 1 ≤ j ≤ N }. Technically, we take V ∈ C ∞ (M that ∃ρ > 0 ; ∀α ∈ Nd , ∀x ∈ Rd , |x| > R0 , |∂xα V (x)| = Oα x−ρ−|α| . (1.1) Furthermore, we assume that for all j ∈ {1, 2, . . . , N }, we can find smooth functions fj , Wj in C0∞ (Rd ; R) such that fj (sj ) = 0 and, near sj , V (x) =

fj (x) + Wj (x) . |x − sj |

(1.2)

If fj (sj ) < 0 (resp. fj (sj ) > 0), we say that sj is an attractive (resp. a repulsive) Coulomb singularity. Let N ≥ 0 be the number of attractive singularities. We may assume that they are labelled by {1, 2, . . . , N }. Given some h∗ ∈]0; 1[, we introduce a semiclassical parameter h ∈]0; h∗ ]. The semiclassical Schrödinger operator is given by P (h) := −h2 Δx + V , acting in L2 (Rd ). Under the previous assumptions, it is well known that P (h) is self-adjoint (see [8, 26, 43]). When d ≥ 3, this fact follows from Hardy’s inequality (cf. (2.9)) and from Kato’s theorem on relative boundedness. The domain of P (h) then is the Sobolev space H2 (Rd ), i.e. the domain of the Laplacian. When d = 2, selfadjointness follows when considering the quadratic form associated with P (h) and

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777

using Kato’s theorem on relative boundedness for forms: the form is seen to be closable and bounded below, and the associated self-adjoint operator is P (h) [7,33]. The situation is rather different for d = 1 (see Section I.1 in [43]), which is the reason why we exclude this dimension here. 1.2. The function spaces and main notation For z belonging to the resolvent set ρ(P (h)) of P (h), we set R(z; h) := (P (h)−z)−1 . We are interested in the size of the resolvent R(z; h) as a bounded operator from some space S into its dual S∗ , i.e. as an element of the space L(S; S∗ ). We denote by · S,S∗ the usual operator norm on L(S; S∗ ). If S = L2 (Rd ), we also use the notation · in place of · S,S∗ . The relevant spaces S are introduced below. If a is a measurable subset of Rd , we denote by · a (resp. · , · a ) the usual norm (resp. the right linear scalar product) of L2 (a) (and we skip the subscript a if a = Rd ). For s ∈ R, we denote by L2s the weighted L2 -space of measurable functions f such that x → xs f (x) belongs to L2 (Rd ). Its dual space is identified with L2−s . For j ∈ Z, we set cj := {x ∈ Rd ; 2j−1 < |x| ≤ 2j }

and c = {x ∈ Rd ; |x| ≤ 1} .

(1.3)

˙ be the space of functions f locally in Let B (resp. its homogeneous version B) 2 d 2 d L (R ) (resp. L (R \ {0})) such that ⎛ ⎞ ∞

f B := f c + (1.4) 2j/2 f cj ⎝resp. f B˙ := 2j/2 f cj ⎠ j=1

j∈Z

is finite. Its dual B∗ (resp. B˙ ∗ ) is equipped with

−j/2 −j/2

f cj

f cj .

f B∗ := max f c ; sup 2 resp. f B˙ ∗ := sup 2 j≥1

j∈Z

(1.5) One can easily check that the embeddings L2s ⊂ B ⊂ L21/2 , for any s > 1/2, and ˙ ˙ are all continuous. Notice that, for S = L2 , B, and B, B ⊂ B, s ∀f ∈ S∗ , ∀g ∈ S , f¯g ∈ L1 and f , g ≤ f S∗ · g S . (1.6) For z ∈ ρ(P (h)), R(z; h) can be viewed as a bounded operator from L2s to L2−s , for s ≥ 0, and from B to B∗ , being a bounded operator on L2 (Rd ). When d ≥ 3, one can show using Hardy’s inequality (2.9) that, for z ∈ ρ(P (h)), R(z; h) can even be viewed as a bounded operator from B˙ to B˙ ∗ (cf. [54]), a stronger result. Let I be a compact interval included in ]0; +∞[ and d ≥ 3. By [12], we know that I contains no eigenvalue of P (h). By Mourre’s commutator theory (cf. [3,40]), we also know that for fixed h, R( · ; h) S,S∗ is bounded on {z ∈ C; z ∈ I, z = 0} whenever S = L2s (s > 1/2) or S = B. Adapting an argument by [54], the above ˙ Summarizing, for s > 1/2 and any given norm is even seen bounded when S = B.

778



h > 0, the following chain of inequalities holds true sup R(z; h) L2s ,L2−s ≤ sup R(z; h) B,B∗ ≤ sup R(z; h) B, ˙ B ˙∗ < ∞.

z∈I

z=0

z∈I

z=0

(1.7)

z∈I

z=0

1.3. The non-trapping condition We now estimate the terms involved in (1.7) as h → 0. When V = 0, it is known that sup R(z; h) S,S∗ = O(1/h) , (1.8) z∈I

z=0

whenever S = L2s (s > 1/2), or S = B. Our aim is to characterize those potential V for which (1.8) holds true with S = B. If V ∈ C ∞ (Rd , R) and satisfies (1.1), then a characterization of those V ’s such that (1.8) holds true is well known, at least in the case S = L2s (s > 1/2) or S = B, as we now describe. Let T ∗ Rd (x, ξ) → p(x, ξ) := |ξ|2 + V (x) be the symbol of P (h). Since the potential V is bounded below, the speed |ξ| is bounded above on p−1 (λ), for all energies λ . Thus the particle cannot escape to infinity in finite time and p defines a complete smooth Hamilton flow (φt )t∈R on T ∗ Rd . The symbol p is said non-trapping at the energy λ whenever ∀(x, ξ) ∈ p−1 (λ) ,

lim |φt (x, ξ)| = +∞ and

t→−∞

lim |φt (x, ξ)| = +∞ . (1.9)

t→+∞

In many cases it is easy to show trapping by topological criteria, see [34]. Let S = L2s with s > 1/2, or S = B. Then (1.8) holds true if and only if any energy λ ∈ I is non-trapping for p (cf. [14, 24, 39, 47, 50, 52]). This statement has been extended to the homogeneous space S = B˙ (d ≥ 3) by [6], for V ’s of class C 2 only. First note that such a characterization is a Bohr correspondence principle: in the limit h → 0, a qualitative property of the classical flow (the non-trapping condition) is connected to a propagation property of the quantum evolution operator U (t; h) = exp(−ih−1 tP (h)). Indeed the propagation estimate (3.11) turns out to be equivalent to the above estimate (1.8). Second, it is also useful to develop a semiclassical, stationary scattering theory (the case S = L2s actually suffices). If the non-trapping condition is true, one expects to deduce from (1.8) bounds on several scattering objects (as is done when V ∈ C ∞ (Rd ), cf. [46, 47]). If trapping occurs, one expects that the resonant phenomena have a leading order influence on the scattering objects (cf. [15, 46]). Of course, these two motivations are still present if Coulomb singularities are allowed. When only repulsive Coulomb singularities occur, it was proved in [29] that the non-trapping condition implies that (1.8) is true with S = L2s (s > 1/2). If at least one attractive Coulomb singularity is present, the flow is not complete anymore and the previous non-trapping condition does not even make sense. However, it is known that one can “regularize the flow” (see [32] and references therein),

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and it turns out the regularization is easier to deal with in dimension d = 2 and d = 3. In the present paper, we choose to focus on the case d = 3, which is the physically important situation. Our study is devoted to generalizing the previous characterization, in a case where the potential admits arbitrary Coulomb singularities. Note that we do expect our results extend to the case d = 2. Let d = 3 and assume that S contains an attractive singularity. Let (x, ξ) ∈ ˆ = T ∗ (R3 \ S). As we shall see in Subsection 2.2, there exists some at most T ∗M countable subset coll(x, ξ) ⊂ R and a smooth function φ( · ; x, ξ) : R \ coll(x, ξ) −→ ˆ such that φ( · ; x, ξ) solves the Hamilton equations generated by the symbol p T ∗M of P (h) with initial value (x, ξ) (see (2.15) and (2.14) below). Furthermore, for all t ∈ R \ coll(x, ξ), p(φ(t; x, ξ)) = p(x, ξ). The function φ replaces the usual flow. It is thus natural to say that p is non-trapping at energy λ whenever ∀(x, ξ) ∈ p−1 (λ) ,

lim |πx φ(t; x, ξ)| = +∞ and

t→−∞

lim |πx φ(t; x, ξ)| = +∞ ,

t→+∞

(1.10) ˆ. where πx φ(t; x, ξ) denotes the configuration or base component of φ(t; x, ξ) ∈ T ∗ M 1.4. Survey In view of (1.7) and (1.10), we can now state our main result. Theorem 1.1. Let V be a potential satisfying the assumptions (1.1) and (1.2). If there are no attractive singularities (N = 0), let d ≥ 3 else let d = 3. Let I0 be an open interval included in ]0; +∞[. The following properties are equivalent. 1. For all λ ∈ I0 , p is non-trapping at energy λ. 2. For any compact interval I ⊂ I0 , there exists C > 0 such that, for h ∈]0; h∗ ], sup R(z; h) B,B∗ ≤ C h−1 .

(1.11)

z∈I

z=0

In [53], the point 2 of Theorem 1.1 is derived from a virial-like assumption, which is stronger than the non-trapping condition. It is assumed there that only one singularity occurs and that (1.2) holds true for a constant fj . The statement “1 =⇒ 2” of Theorem 1.1 is proved in [29] when N = 0. Theorem 1.1 provides the converse. More importantly, it extends the result to the delicate case N > 0. To complete the picture given by Theorem 1.1, we study in Section 5 the non-trapping condition. In the case of a single Coulomb singularity, we show that it is always satisfied when the energy λ is large enough, as in the case of a smooth potential (see Remark 5.5). The classically forbidden region in configuration space then is a point (for attracting Coulomb potential), or it is diffeomorphic to a ball (in the repelling case). Conversely Proposition 5.1 says that – irrespective of the number of singularities and the energy – only for the case of a single point or ball trapping does not need to occur. In particular, Corollary 5.2 states that trapping always occurs for two or more singularities at large enough energies. We point out that our proof of Theorem 1.1 gives some additional insight about the case when the non-trapping condition fails at some energy λ > 0 (cf.

780



Proposition 4.9). In such a situation, “semiclassical trapping” occurs, as described by (4.1) and (4.2). Notice that a resonance phenomenon (cf. [20, 35]) is a particular case of the quasi-resonance phenomenon defined in [16], the latter being a particular case of our “semiclassical trapping” criterion. Propositions 4.5 and 4.9 show that the “semiclassical trapping” is microlocalized near “trapped trajectories” (see (4.10) for a precise definition). It would be interesting to check whether a (quasi-)resonance phenomenon is related to our “semiclassical trapping” (cf. Remark 4.6). A traditional study of the resonances “created” by a bounded trajectory (see [16] and references therein) would also be of interest. We do hope that the present paper may help to overcome the difficulties due to the singularities. While the proof of “2 =⇒ 1” in Theorem 1.1 follows the strategy developed by [52] for smooth potentials, we use a rather different argument compared to [14, 29,47] when showing “1 =⇒ 2”. In these papers, a semiclassical version of Mourre’s commutator theory is used (cf. [3, 40]), and the Besov-like space B is replaced by the weaker L2s (s > 1/2). An alternative approach is given in [5] for compactly supported perturbations of the Laplacian, using a contradiction argument due to G. Lebeau in [36]. This method was adapted in [24] to include long-range, smooth perturbations, the study still being carried out in the space L2s (s > 1/2). This technique was further developed in [6] to tackle the estimates in the optimal ˙ by combining and adapting an original estimate derived homogeneous space B, in [42]. Note that both works [42] and [6] only require C 1 resp. C 2 smoothness on the potential. Note also that the extension of Theorem 1.1 to the homogeneous estimate in B˙ still is open. Now, the contradiction argument of [6, 24] is a key ingredient of the present study. Concerning the treatment of the singularities, we stress that our study uses many results from [13], the propagation results being here crucial. The main features we need on the regularization of the classical flow are provided by [13, 32]. Our main new contributions are given in Proposition 3.6 and in Section 4.3. Finally, we give some nonrelativistic, physical situations for which our result applies. In both examples below, we may add to the operator a smooth exterior potential satisfying (1.1). Example 1.2. The behaviour of a particle with charge e0 in the presence of fixed, pointlike ions, with nonzero charges z1 , . . . , zN , is governed by the operator (here d = 3) N e0 zj 2 . (1.12) P1 (h) := −h Δx + |x − sj | j=1 The hydrogen atom corresponds to N = 1, z1 > 0, and e0 < 0. Clearly (1.1) and (1.2) hold true. If charges have different sign, the model has attractive and repulsive singularities. Example 1.3. Consider a molecule with N nuclei having positive charges z1 , . . . , zN , binding K > 0 electrons with charge −1. We assume the nuclei are fixed (Born–Oppenheimer idealization) and we neglect electron-electron repulsion.

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The behaviour of each electron is then governed by P1 (h) in (1.12). Let h0 > 0 be fixed. Let ψk be the normalized wavefunction of P1 (h0 ) of electron number k. Let ρk = |ψk |2 be its charge density. Consider another, much heavier particle with charge e0 . Its scattering by the molecule can be described by P (h) where ⎛ ⎞ N K zj ρk (q) V (x) := e0 ⎝ + dq . Wk (x)⎠ , with Wk (x) := − |x − s | |q − x| d j R j=1 k=1

(1.13) As we show in Section 6, it turns out that the ψk ’s are “nice enough” to make Wk well defined, smooth away from the singularities s1 , . . . , sN , and to make Wk satisfy (1.1). Though (1.2) does not hold, we show the proof of our result applies in this case.

2. Preliminaries We shall often use well known facts concerning h-pseudodifferential calculus, functional calculus, and semiclassical measures in the sequel. For sake of completeness, we recall here the main results we need, referring to [9, 17, 18, 21, 27, 37, 38, 41, 45] for further details. Since our Schr¨ odinger operator has Coulomb singularities, it does not define a pseudodifferential operator yet. For this reason, we also explain here how we can use pseudodifferential calculus “away from the singularities”: the required results are essentially contained in [13]; notice however that we do not need the results in the appendix of [29], which are, by the way, not known if an attractive Coulomb singularity is present. Last, we also recall basic results on the regularization of the Hamilton flow when an attractive singularity is present, referring to [13, 27, 32] for details. 2.1. Symbolic calculus with singularities Let d ∈ N∗ . For (r, m) ∈ R2 , we consider the vector space (space of symbols)

Σr;m := a ∈ C ∞ (T ∗ Rd ) ; ∀γ = (γx , γξ ) ∈ N2d , ∃Cγ > 0 ; sup x−r+|γx | ξ−m+|γξ | (∂ γ a)(x, ξ) ≤ Cγ . (2.1) (x,ξ)∈T ∗ Rd

If r, m ≤ 0, then Σr;m is contained in the vector space of bounded symbols, which are smooth functions a : T ∗ Rd −→ C such that γ (∂ a)(x, ξ) ≤ Cγ . ∀γ ∈ N2d , ∃Cγ > 0 ; sup (2.2) (x,ξ)∈T ∗ Rd

For a larger class of symbols a, one can define the Weyl h-quantization of a, denoted ∞ d by aw h . It acts on u ∈ C0 (R ) as follows (cf. [9, 38, 41, 45]). w −d eiξ · (x−y)/h a (x + y)/2, ξ u(y) dy dξ . (2.3) ah u (x) = (2πh) Rd

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2 d If a is a bounded symbol, then aw h extends to a bounded operator on L (R ), uniformly with respect to h, by Calder´ on–Vaillancourt’s theorem (cf. [9,38,45]). We shall also use the following functional calculus of Helffer–Sj¨ ostrand, which can be found in [9,38]. Given θ ∈ C0∞ (R), one can construct an almost analytic extension θC ∈ C0∞ (C) (with ∂θC (z) = O((z)∞ )). Let H be a self-adjoint operator in some Hilbert space. The bounded operator θ(H), defined by the functional calculus of self-adjoint operators, can be written as −1 θ(H) = ∂θC (z) · (z − H)−1 dL2 (z) . (2.4) π C

where L2 denotes the Lebesgue mass on C. Let us now recall some well known facts about semiclassical measures, which can be found in [17, 18, 27, 37]. Let (un )n be a bounded sequence in L2 (Rd ). Up to extracting a subsequence, we may assume that it is pure, i.e. it has a unique semiclassical measure μ. By definition μ is a finite, nonnegative Radon measure on the cotangent space T ∗ Rd . Furthermore, there exists a sequence hn → 0 such that, for any a ∈ C0∞ (T ∗ Rd ), w a(x, ξ) μ(dx dξ) =: μ(a) . (2.5) lim un , ahn un = n→∞

T ∗ Rd

One may relate the total mass of μ to the L2 -norm of the un ’s (see [18], or [27,37]), through the following Proposition 2.1 ([18]). Let (un )n be a pure bounded sequence in L2 (Rd ) such that |un (x)|2 dx = 0 , (2.6) lim lim sup R→+∞ n→∞ |x|≥R lim lim sup |Fun (x)|2 dξ = 0 , (2.7) R→+∞ n→∞

|ξ|≥R/hn

where F un denotes the Fourier transform of un . Then the sequence ( un 2 )n converges to the total mass μ(T ∗ Rd ) of its semiclassical measure μ.

Proof. See the proof of Proposition 1.6 in [18].

Besides, transformation of the semiclassical measure upon composition of the un ’s with a diffeomorphism is described in the Proposition 2.2 ([18]). Let Φ : U −→ V be a C ∞ diffeomorphism between two open subsets of Rp (p ≥ 1). Let Φc : T ∗ U −→ T ∗ V be the symplectomorphism −1 η . (2.8) (y, η) → Φ(y) ; Φ (y)T Here Φ (y)T denotes the transpose of Φ (y). Given a ∈ C0∞ (T ∗ V ), let b ∈ C0∞ (T ∗ U ) be defined by b = a ◦ Φc . Then, for every compact subset K of V , lim sup (aw u) ◦ Φ − bw (u ◦ Φ) = 0 . h→0

u≤1

supp u⊂K

h

h

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Let K be a compact subset of V and (un )n be a pure bounded sequence in L2 (V ) such that, for all n, supp un ⊂ K. Denote by μ its semiclassical measure. Then the sequence (un ◦ Φ)n is bounded in L2 (U ), its semiclassical measure μ ˜ is given (μ), and μ(a) = μ ˜ (b). by |DetΦ |−1 Φ−1 c

Proof. See the proof of Lemma 1.10 in [18].

We now focus on the treatment of Coulomb singularities in dimension d ≥ 3, in combination with the h-pseudodifferential framework. To begin with, let us recall Hardy’s inequality. 4 4 |f (x)|2 ∞ d dx ≤

∇x f 2 = 2

h∇x f 2 , ∀f ∈ C0 (R ) , 2 2 |x| (d − 2) h (d − 2)2 d R (2.9) where the last bound is relevant in the present, semiclassical regime. We next discuss how one can use h-pseudodifferential calculus “away from ˆ = Rd \ S. Let χ ∈ C ∞ (Rd ) with χ = 1 near the the singularities”. Recall that M 0 set S of all singularities . Define the (truncated) h-pseudodifferential operator Its symbol

Pχ (h) := −h2 Δx + (1 − χ)V .

(2.10)

T ∗ Rd (x, ξ) → pχ (x, ξ) = |ξ|2 + 1 − χ(x) V (x)

(2.11)

belongs to Σ0;2 (cf. (2.1)). The following lemma is essentially proved in [13]. Lemma 2.3. Let d ≥ 3. Let χ ∈ C0∞ (Rd ) with χ = 1 near S and θ ∈ C0∞ (R). Let Pχ (h) be given by (2.10). Let T > 0, k, k ∈ R, r, m ∈ R, and [−T ; T ] t → a(t) ∈ Σr;m be a continuous function such that, for all t ∈ [−T ; T ], a(t) = 0 near supp χ. Then, in C 0 ([−T ; T ]; L(L2k ; L2k )), w P (h) − Pχ (h) a( · ) h = O(h2 ) , (2.12) w a( · ) h = O(h2 ) . (2.13) and, if m ≤ 2 , θ P (h) − θ Pχ (h) Proof. Let r, m, k, k ∈ R. For a ∈ Σr;m and f ∈ S(Rd ), the Schwartz space on Rd , k 2 −1 k · (−h2 Δ + 1)χ · k aw · k P (h) − Pχ (h) aw h · f = V (−h Δ + 1) h· f , where V (−h2 Δ+1)−1 ∈ L(L2 ; L2 ) has norm O(1/h2 ) by (2.9). Now, if a is replaced by a continuous map t → a(t) with a(t) = 0 near supp χ for all t, then, for all N ∈ N, N k (−h2 Δ + 1)χ · k a(∗)w h· = O h in C 0 ([−T ; T ]; L(L2 ; L2 )), by the usual h-pseudodifferential calculus. This yields (2.12). On the other hand it is known that, for all k ∈ R, the resolvents (P (h)+i)−1 and (Pχ (h)+ i)−1 are bounded from L2k to L2k (see [44], Sect. XIII.8), and, by (2.9), there exists some α(k) ≥ 0 such that P (h) + i −1 2 2 = O h−α(k) , Pχ (h) + i −1 2 2 = O h0 . L(L ;L ) L(L ;L ) k

k

k

k

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Besides, there is a χ1 ∈ C ∞ (Rd ) with χχ1 = 0, χ1 = 1 at infinity, and χ1 a(t) = a(t) k N 0 2 2 for all t. Hence, for all N ∈ N, (1−χ1 )a(∗)w h · = O(h ) in C ([−T ; T ]; L(L ; L )). We may now adapt the arguments in the proof of Lemma 3.1 in [13] to get (2.13). 2.2. Extension of the flow Here we explain how the usual flow can be extended when attractive singularities occur (more details are given in [27, 32]). Let d = 3. We still denote by p the smooth function defined by p : Pˆ −→ R ,

ˆ. where Pˆ := T ∗ M

(x, ξ) → |ξ|2 + V (x) ∗

(2.14)

Let πx (resp. πξ ) be the projection T R −→ R defined by πx (x, ξ) := x (resp. πξ (x, ξ) := ξ). As for any smooth dynamical system, the hamiltonian initial value problem, d

dX (t) = ∇ξ p X(t); Ξ(t) , dt X(0); Ξ(0) = x∗ = (x, ξ) ∈ Pˆ

d

dΞ (t) = −∇x p X(t); Ξ(t) , dt (2.15)

ˆ → Pˆ with has a unique maximal solution φ : D ˆ = (t, x∗ ) ∈ R × Pˆ ; t ∈ ]T − (x∗ ), T + (x∗ )[ , D where the functions T ± : Pˆ → R satisfy T − < 0 < T + and are lower resp. upper semi-continuous with respect to the natural topology on the extended line ˆ ⊆ R × Pˆ is open. R := {−∞} ∪ R ∪ {+∞}. In particular, the set D If no attractive singularity is present (i.e. N = 0 in the notation of Paraˆ = R× Pˆ . Otherwise a maximal solution can fall on an attractive graph 1.1), then D singularity s at finite time T + (x∗ ) > 0. Such a time is called a collision time. In that case, it turns out that, setting ⎧ ∅ if T − (x∗ ) = −∞, T + (x∗ ) = ∞ ⎪ ⎪ ⎪ ⎨{T + (x∗ )} if T − (x∗ ) = −∞, T + (x∗ ) < ∞ coll(x∗ ) := ⎪ {T − (x∗ )} if T − (x∗ ) > −∞, T + (x∗ ) = ∞ ⎪ ⎪ + ∗ ⎩ + ∗ − ∗ {T (x )} + Z T (x ) − T (x ) if T − (x∗ ) > −∞, T + (x∗ ) < ∞ and D := (t, x∗ ) ∈ R × Pˆ ; t ∈ coll(x∗ ) , the map φ can be uniquely extended to a smooth map D → Pˆ , still denoted by φ. Even more, when T + (x∗ ) < ∞, backscattering occurs, that is, for 0 < t < T + (x∗ ) − T − (x∗ ), we have πx φ T + (x∗ ) + t; x∗ = πx φ T + (x∗ ) − t; x∗ , (2.16) πξ φ T + (x∗ ) + t; x∗ = −πξ φ T + (x∗ ) − t; x∗ ,

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and one may set πx φ(T + (x∗ ); x∗ ) = s. We mention that the momentum πξ φ( · ; x∗ ) however blows up at T + (x∗ ), in the following sense: lim

t→T+ (x∗ )

|πξ φ(t; x∗ )| = ∞ , while

v :=

πξ φ(t; x∗ ) πξ φ(t; x∗ ) = − lim tT+ (x∗ ) |πξ φ(t; x∗ )| t T+ (x∗ ) |πξ φ(t; x∗ )| lim

exists .

For any x∗ ∈ Pˆ , we obtain in this way a configuration trajectory (πx φ(t; x∗ ))t∈R , which has a countable set coll(x∗ ) of collision times t0 for which lim πx φ(t; x∗ ) ∈ {sj , 1 ≤ j ≤ N } and

t→t0

lim |πξ φ(t; x∗ )| = ∞ .

t→t0

(2.17)

Although φ is not a complete flow on Pˆ , the broken trajectory (φ(t; x∗ ))t∈R\coll(x∗ ) is a solution of (2.15) on R \ coll(x∗ ). Its values lie in the energy shell p−1 (p(x∗ )). Note that no collision with the repulsive singularities can occur. For t ∈ R, it is convenient to introduce φt : Dt → Pˆ defined by and φt (x∗ ) := φ(t; x∗ ) . (2.18) Dt := x∗ ∈ Pˆ ; t ∈ coll(x∗ ) Note further that the Hamiltonian system (Pˆ , ω0 , p) with canonical symplectic form ω0 can be uniquely extended to a smooth Hamiltonian system with a complete flow (see Section 5). An important feature to analyse the pseudo-flow φ is the Kustaanheimo– Stiefel transformation (KS-transform for short). We briefly describe it here and refer to [13, 27, 32, 49], for further details. For z = (z0 , z1 , z2 , z3 )T ∈ R4 , let ⎞ ⎛ z0 −z1 −z2 z3 Λ(z) = ⎝ z1 z0 −z3 −z2 ⎠ . z2 z3 z0 z1 Let K : R4 −→ R3 be defined by ⎞ ⎛ 2 z0 − z12 − z22 + z33 K(z) := Λ(z) · z = ⎝ 2z0 z1 − 2z2 z3 ⎠ . For all z ∈ R4 , 2z0 z2 + 2z1 z3

|K(z)| = |z|2 .

(2.19) We call it the Hopf map. See the appendix for more information. Let R3± := {(x1 , x2 , x3 ) ∈ R3 ; ±x1 > 0} and z ∈ R4 . It turns out (see [13]) √ that, if x := K(z) ∈ R3+ , A+ (z) := 2(x1 + |z|)−1/2 (z0 + iz3 ) ∈ S 1 and, if √ x := K(z) ∈ R3− , A− (z) := 2(−x1 + |z|)−1/2 (z1 + iz2 ) ∈ S 1 . Furthermore, one can explicitly construct smooth maps J± : R3± × S 1 −→ R4 such that, locally, (K, A± ) ◦ J± = Id in R3± × S 1

and

J± ◦ (K, A± ) = Id in J± (R3± × S 1 ) . (2.20)

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For z = J± (x; θ), for x ∈ R3± and θ ∈ S 1 , we have dz = C|x|−1 dx dθ for some constant C > 0. In particular, there exists C > 0 such that, for all f, g : R3 −→ C measurable, −1 |x| · |f (x)g(x)| dx = C |f ◦ K(z) g ◦ K(z)| dz . (2.21) R3

R4

It is useful to consider the following extension to phase space. For z∗ = (z; ζ) ∈ T ∗ R4 , we set as usual πz z∗ = z and πζ z∗ = ζ. If (x; ξ) ∈ T ∗ (R3 \ {0}), let z ∈ R4 such that x = K(z) = Λ(z) · z (z is not unique). Then, we define ⎛ ⎞ ⎛ ⎞ z1 z2 z0 ξ1 ⎜ ⎟ −z z z 1 0 3 ⎟⎝ ξ2 ⎠ , ζ := 2Λ(z)T ξ = 2 ⎜ (2.22) ⎝ −z2 −z3 z0 ⎠ ξ3 z3 −z2 z1 which is a solution of the equation 2|x|ξ = Λ(z)ζ. The KS-transform is defined by

1 K∗ : T ∗ R4 \ {0} −→ T ∗ R3 \ {0} , K∗ (z; ζ) = Λ(z) · z ; Λ(z) · ζ . 2|z|2 (2.23) Assume that an attractive singularity sits at 0. Recall that, by (1.2), V (x) = f (x)/|x| + W (x) on Ω \ {0} , where Ω := {x ∈ R3 ; |x| < r} for some r > 0, with ˜ := K−1 (Ω). Let x∗ = (x0 ; ξ0 ) ∈ Pˆ be such that the f, W ∈ C0∞ (Rd ; R). Let Ω 0 ∗ ∗ first collision of (πx φ(t; x0))t∈R takes place at 0 at Let T0 be the time t+ (x0 ) > 0. ∗ connected component of t ∈ R; πx φ(t; x0 ) ∈ Ω containing t+ (x∗0 ). Let z0 ∈ R4 be such that x0 = K(z0 ) and let ζ0 be the ζ given by (2.22) with (z; ξ) = (z0 ; ξ0 ). For t∗ = (t; τ ) ∈ T ∗ R, z∗ = (z; ζ) ∈ T ∗ R4 , let p˜(t∗ ; z∗ ) := |ζ|2 + f ◦ K(z) + |z|2 (W ◦ K(z) − τ ). Since p˜ is smooth on T ∗ R × T ∗ R4 = T ∗ (Rt × R4z ), independent of t, and since its Hamilton vector field at point (t, τ ; z, ζ) is given by (−|z|2 , 0; 2ζ, 2τ z) outside a compact region in (z, ζ), there exists a unique maximal solution R s → t(s); τ (s); z(s); ζ(s) = (t∗ (s); z∗ (s)) to the Hamilton equations associated with p˜

(dz/ds)(s) = ∇ζ p˜ (t∗ (s); z∗ (s)) (dζ/ds)(s) = −∇z p˜ (t∗ (s); z∗ (s)) , (dτ /ds)(s) = −∇t p˜ (t∗ (s); z∗ (s)) (dt/ds)(s) = ∇τ p˜ (t∗ (s); z∗ (s)) (2.24) with initial condition (t∗ (0); z∗ (0)) = (t∗1 ; z∗1 ). We denote it by ˜ t∗ ; z∗ ) := t∗ (s; t∗ ; z∗ ); z∗ (s; t∗ ; z∗ ) φ(s; 1 1 1 1 1 1 = t(s; t∗1 ; z∗1 ); τ (s; t∗1 ; z∗1 ); z(s; t∗1 ; z∗1 ); ζ(s; t∗1 ; z∗1 ) . Let (t∗0 ; z∗0 ) = (0; p(x∗0 ); z0 ; ζ0 ). It turns out that, for all t1 ∈ T0 , there exists a unique s ∈ R such that t1 = t(s; t∗0 ; z∗0 ). Furthermore, if t1 = t+ (x∗0 ), z(s; t∗0 ; z∗0 ) = 0 and (2.25) φ(t1 ; x∗0 ) = φ t(s; t∗0 ; z∗0 ); x∗0 = K∗ z∗ (s; t∗0 ; z∗0 ) .

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3. Towards the non-trapping condition The aim of this section is to prove the implication “2 =⇒ 1” of Theorem 1.1. We thus assume that 2 holds true and we want to show that p is non-trapping at energy λ, for all λ ∈ I0 . Let λ0 be such an energy. We can find a compact interval I ⊂ I0 such that λ0 belongs to the interior of I. By assumption, (1.11) holds true for I. This implies, by (1.7), that (1.8) holds true for S = L2s , for any s > 1/2. As in [25], we follow the strategy in [52]. We translate the bound on the resolvent into a bound on a time integral of the associated propagator U (t; h) := exp − ih−1 tP (h) . (3.1) If an attractive singularity is present (and d = 3), we need some information on the time-dependent microlocalization of U (t; h)uh , for some family (uh )h of L2 (R3 ) functions. Most of it is already available in [13]. We also need some well-known facts on the classical flow, which we borrow from [32]. In Subsection 3.1, we shall recall results from [13, 32] and extend them a little bit. Then we proceed with the announced proof in Subsection 3.2. In the repulsive case (N = 0 and d ≥ 3), we show in Subsection 3.3 that Wang’s proof may be carried over with minor changes. 3.1. Coherent states evolution In this subsection we are interested in the case where an attractive singularity occurs (i.e. N > 0) but the results hold true for N = 0. Better results in the latter case are given in Subsection 3.3. Proposition 3.6 is the main result of the subsection. Before considering the time evolution of coherent states, we recall some basic facts on the classical dynamics, in particular on the dilation function a0 : T ∗ Rd −→ R ,

a0 (x, ξ) := x · ξ .

Lemma 3.1. Consider a dimension d ≥ 2 and energies λ > 0. 1. Then for some R1 = R1 (λ) ≥ R0 and all x∗0 := (x0 , ξ0 ) ∈ p−1 (]λ/2; ∞[) |x0 | ≥ R1 =⇒ {p, a0 }(x∗0 ) ≥ λ/2 ,

lim inf |πx φ(t; x∗0 )| t→±∞

> R1 =⇒

lim |πx φ(t; x∗0 )| t→±∞

and

= +∞ .

(3.2) (3.3)

2. For any T, R > 0, there is some R2 > R1 such that, for all x∗0 = (x0 , ξ0 ) ∈ p−1 (]λ/2; 2λ[) |x0 | > R2 =⇒ |πx φ(t; x∗0 )| > R for all t ∈ [−T ; T ] . (3.4) Proof. We shortly recall the standard arguments. Thanks to the decay properties (1.1) of V , {p, a0 }(x∗0 ) = 2 p(x∗0 ) − V (x0 ) − x0 , ∇V (x0 ) ≥ λ/2 for large |x0 |, implying (3.2). As the dilation function a0 is the time derivative of the phase space function |x|2 /2, composed with φ, the second time derivative of the

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latter function is eventually bounded below by λ/2 > 0, if the l.h.s. of (3.3) is satisfied. Thus t → |πx φ(t; x∗0 )|2 goes to infinity, showing (3.3). Let V0 = inf |x|≥R0 V (x). Relation (3.4) follows, since the speed is bounded by |ξ0 | ≤ (4λ− 2V0 )1/2 < ∞. h

For h ∈]0; h∗ ] the dilation operator Eh on L2 (Rd ), given by Eh (f )(x) := f (h−1/2 x), is unitary, as are the Weyl operators w(x∗0 ; h) := exp ih−1/2 (x0 · x − ξ0 · Dx ) for x∗0 := (x0 , ξ0 ) ∈ T ∗ Rd ,

−d/4

(cf. [21], p. 151, [11]). The coherent states operators, microlocalized at x∗0 , are c(x∗0 ; h) := Eh · w(x∗0 ; h) .

(3.5)

A direct computation gives that

1/2 w Eh∗ aw ; h−1/2 ) 1 , h Eh = a(h w ∗ 1/2 c(x∗0 ; h)∗ aw ; ξ0 + h−1/2 ) 1 , h c(x0 ; h) = a(x0 + h

where b(; ) denotes the symbol (x; ξ) → b(x; ξ). It is known (cf. [51]) that ∀a ∈ Σ0,0 ,

∀f ∈ S(Rd ) ,

∗ ∗ c(x∗0 ; h)∗ aw h c(x0 ; h)f = a(x0 )f + O(h) ,

(3.6)

where S(Rd ) denotes the Schwartz space on Rd . Let uh be the function given by uh := c(x∗0 ; h) π −d/4 exp −| · |2 /2 . (3.7) Then (uh )h is a family of L2 (Rd )-normalized coherent states microlocalized at x∗0 . We collect properties of the family (U ( · ; h)uh )h of the propagated states. In the remainder part of Subsection 3.1 we consider initial conditions in phase space x∗0 := (x0 , ξ0 ) ∈ Pˆ with energy λ := p(x∗0 ) > 0 and the associated coherent states (uh )h microlocalized at x∗0 . In [13] the following energy localization of (uh )h is obtained. We give a short proof using Lemma 2.3. Lemma 3.2 ([13]). Let d ≥ 3 and θ ∈ C0∞ (R) such that θ = 1 near λ. Then, in L2 (Rd ), (1 − θ(P (h)))uh = O(h). Proof. Let χ, χ ˜ ∈ C0∞ (Rd ) with χ, χ ˜ = 1 near S, χ, χ ˜ = 0 near x0 = πx x∗0 , and 2 d χχ ˜ = χ. From (3.7), we see that χu ˜ h = O(h) in L (R ). By Lemma 2.3, 1−θ P (h) uh = 1−θ P (h) (1−χ)u ˜ h +O(h) = 1−θ Pχ (h) (1−χ)u ˜ h +O(h) , in L2 (Rd ), where Pχ (h) is as in (2.10). Besides, thanks to (3.6) and using (2.11), 1 − θ P (h) uh = 1 − θ Pχ (h) uh + O(h) = 1 − θ pχ (x∗0 ) uh + O(h) = 0 + O(h) . From [13] we pick the following localization away from singularities.

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Lemma 3.3 ([13]). Let d = 3. Let K be a compact subset of R such that K ∩ coll(x∗0 ) = ∅ (cf. (2.17)). If σ ∈ C0∞ (R3 ) has small enough a support near the set S of singularities, then K t → σU (t; h)uh is of order O(h) in C 0 (K; L2 (R3 )). Proof. See the proof of Theorem 1, p. 25 in [13].

A careful inspection of the result in [13] on the frequency set shows that even after a collision, we have the following localization along our broken trajectories. Lemma 3.4. Let d = 3. Let K be a compact subset of R such that K ∩ coll(x∗0 ) = ∅ (cf. (2.17)). Let > 0 and K t → a(t; ∗) ∈ C0∞ (Pˆ ) be continuous functions such that a(t; x, ξ) = 0 if |x − πx φ(t; x∗0 )| ≤ . Then (a( · ; ∗))w h U ( · ; h)uh = O(h) in C 0 (K; L2 (R3 )). Proof. See the proof of Theorem 1, p. 25 in [13].

We also need to complete Lemma 3.4 with a bound on (U ( · ; h)uh )h near infinity in position space and, since the singularities are far away, we can assume d ≥ 3. This is the purpose of the following Lemma 3.5. Let d ≥ 3. Let T > 0 and R := max(R0 ; 1 + |x0 |}. Let R2 > R1 large enough such that (3.4) holds true. Let R3 > R2 + 1 and κ ∈ C ∞ (Rd ; R) such that supp κ ⊂ {y ∈ Rd ; |y| > R2 + 1} and κ = 1 on {y ∈ Rd ; |y| > R3 }. Then κU ( · ; h)uh = O(h) in C 0 [−T ; T ]; L2(R3 ) . Proof. The proof is based on an Egorov type estimate which is valid although P (h) is not a pseudodifferential operator. • Let τ ∈ C0∞ (Rd ) such that τ = 1 on {y ∈ Rd ; |y| ≤ R0 } and τ = 0 near the set

−1 t (x, ξ); |x| > R2 ]λ/2; 2λ[ ∩ φ p πx . t∈[−T ;T ]

This is well-defined by (2.18), (3.4), and the choice of R. Let pτ be defined as in (2.11). Let θ ∈ C0∞ (R) with supp θ ⊂]λ/2; 2λ[ such that θ = 1 near λ. Set a : T ∗ Rd −→ C , a(x, ξ) = κ(x) θ pτ (x, ξ) . (3.8) Thanks to (3.4), [−T ; T ] t → a ◦ φt is a Σ0;0 -valued, C 1 -function. Therefore, by Calder´ on–Vaillancourt (a ◦ φt )w h is h-uniformly bounded, and for t ∈ [−T ; T ], strongly in H2 (Rd ), t w U(t; h)∗ aw (3.9) h U (t; h) − a ◦ φ h t d = U (s; h)∗ (a ◦ φt−s )w h U (s; h) ds ds 0

t i ∗ t−s w t−s w U (s; h) = U (s; h) ds . P (h), (a ◦ φ )h + (d/ds)a ◦ φ h h 0

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The support properties of a and the choice of τ ensure that, for all r ∈ [−T ; T ], (d/dr)a ◦ φr = {p, a ◦ φr } = {pτ , a ◦ φr }. Thus, (3.9) equals

t i t−s w P (h), (a ◦ φt−s )w − {p U (s; h)∗ , a ◦ φ } U (s; h) ds . τ h h h 0 By Lemma 2.3, (3.9) equals

t i ∗ t−s w t−s w Pτ (h), (a ◦ φ )h − {pτ , a ◦ φ } h U (s; h) ds + hBh (t) U (s; h) h 0 where [−T ; T ] t → Bh (t) ∈ L(L2 (Rd )) is bounded, uniformly with respect to h. By the usual pseudodifferential calculus, 2 d i r w [−T ; T ] r → Pτ (h), (a ◦ φr )w h − {pτ , a ◦ φ } h ∈ L L (R ) h is O(h) in C 0 [−T ; T ]; L(L2(Rd )) . Thus, so is (3.9). t t w • Since a2◦ φ vanishes near x0 , for t ∈ [−T ; Tt ],w t → (a ◦ φ )h uh is O(h) in 0 C [−T ; T ]; L , by (3.6). Thus so is U ( · ; h) (a ◦ φ )h uh . • Using the previous points, the Lemmata 3.2 and 2.3, the fact that θ(P (h)) and U (t, h) commute, and the usual pseudodifferential calculus, κU (t; h)uh = κU (t; h)θ P (h) uh + O(h) = κθ P (h) U (t; h)uh + O(h) w = κθ Pτ (h) U (t; h)uh + O(h) = κθ(pτ ) h U (t; h)uh + O(h) w = U (t; h) κθ(pτ ) ◦ φt h uh + O(h) = O(h) . (3.10) From these lemmata, we can deduce the following information on the time evolution of the coherent states uh . Proposition 3.6. Let N > 0 and d = 3. Let K be a compact subset of R such that K ∩ coll(x∗0 ) = ∅ (cf. (2.17)). Let τ ∈ C0∞ (R3 ) with τ = 1 near 0. For t ∈ R and x ∈ R3 , set τt (x) := τ (x − πx φ(t; x∗0 )). Take the support of τ small enough such that, for all t ∈ K, supp (τt ) ∩ S = ∅. Then, for any a ∈ Σ0;0 and any t ∈ K, w aw h U (t; h)uh = (τt a)h U (t; h)uh + e(t) ,

where K t → e(t) is O(h) in C 0 (K; L2 (Rd )). Proof. Let T > 0 such that K ⊂ [−T ; T ]. Let κ0 , κ1 ∈ C ∞ (R3 ; R) such that κ0 + κ1 = 1 and κ := κ1 satisfies the assumptions of Lemma 3.5. Then, by Lemma 3.5, w aw h U (t; h)uh = ah κ0 U (t; h)uh + O(h) ,

in C 0 (K) := C 0 (K; L2 (Rd )). Now let σ0 ∈ C0∞ (R3 ; R) such that σ0 = 1 near sj for any 1 ≤ j ≤ N , and, for all t ∈ K, supp σ0 ∩ supp τt = ∅. Upon possibly decreasing the support of σ0 , we may apply Lemma 3.3. This yields w aw h U (t; h)uh = ah κ0 (1 − σ0 ) U (t; h)uh + O(h) ,

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in C 0 (K). Let σ ∈ C0∞ (R3 ; R) such that σ = 1 near each singularity sj and σσ0 = σ. For an energy cutoff θ as in Lemma 3.2, we obtain, as in the proof of Lemma 3.5 (see (3.10)), w aw h κ0 (1 − σ0 ) U (t; h)uh = ah κ0 (1 − σ0 ) θ Pσ (h) U (t; h)uh + O(h) , in C 0 (K), since 1−σ0 is localized away from the singularities. By pseudodifferential calculus, w aw h U (t; h)uh = bh U (t; h)uh + O(h) , in C 0 (K), where b := θ(pσ ) (1 − σ0 ) κ0 a ∈ C0∞ (Pˆ ). Applying Lemma 3.4 to a(t) = (1 − τt )b, w w aw h U (t; h)uh = (τt b)h U (t; h)uh + O(h) = (1 − σ0 ) κ0 τt a h U (t; h)uh + O(h) , in C 0 (K). Since τt (1 − σ0 ) κ0 = τt , for all t ∈ K, we obtain the desired result.

3.2. Necessity of the non-trapping condition Assuming N > 0 and d = 3, we want to show that (1.11) implies the non-trapping condition, yielding the proof of “2 =⇒ 1”. The proof below actually works if N = 0, but a more straightforward and easier proof is provided in Subsection 3.3. In view of (1.7), we assume (1.8) for S = L2s with s > 1/2. This means that, for any θ ∈ C0∞ (I0 ; R), · −s θ(P (h)) is Kato smooth with respect to P (h) (by Theorem XIII.30 in [44]). This can be formulated in the following way (cf. Theorem XIII.25 in [44]). There exists Cs > 0 such that for any θ ∈ C0∞ (I0 ; R), −s 2 d · U (t; h) θ P (h) u2 dt ≤ Cs · u 2 . (3.11) ∀u ∈ L (R ) , R

uniformly in h ∈]0; h∗ ]. Take λ ∈ I0 and a function θ ∈ C0∞ (I0 ; R) such that θ = 1 near λ. Let x∗0 := (x0 , ξ0 ) ∈ p−1 (λ) and consider the coherent states uh given by (3.7). Let (tj )j∈J , with J ⊂ N∗ , be the set of collision times of the broken trajectory (φ(t; x∗0 ))t∈R (cf. (2.17)). Eq. (3.11) implies that, for all T > 0 and all h ∈]0; h∗ ], −s · U (t; h) θ P (h) uh 2 dt ≤ Cs . [−T ;T ]

We know from Subsection 2.2 that the collision times in coll(x∗0 ) have positive minimal distance T + (x∗0 ) − T − (x∗0 ). Thus we can choose > 0 smaller than one fourth of that distance, and define for T > 0 the compact sets K(T ) := t ∈ [−T, T ]; dist t, coll(x∗0 ) ≥ . Notice that the length of K(T ) goes to infinity when T → ∞, while −s · U (t; h) θ P (h) uh 2 dt ≤ Cs . K(T )

By energy localization of the coherent state (Lemma 3.2) and Pythagoras’ theorem,

· −s U (t; h) uh 2 dt + OT (h) ≤ 2Cs , (3.12) K(T )

792



where OT (h) is a T -dependent O(h). We apply Proposition 3.6 for the bounded symbol (x, ξ) → a(x, ξ) = x−s and the compact K(T ) introduced above. This yields

τt · −s U (t; h) uh 2 dt + OT (h) ≤ 2Cs . K(T )

We can require that the support of the function τ is so small that, for all t ∈ K(T ), supp (τt ) ∩ S = ∅ and τt2 · −2s ≥ (1/2)τt2 πx φ(t; x∗0 )−2s . Therefore, −2s πx φ(t; x∗0 ) U (t; h)uh , τt2 U (t; h)uh dt + OT (h) ≤ 4Cs . K(T )

Now, we apply Proposition 3.6 again for the bounded symbol (x, ξ) → a(x, ξ) = 1, yielding −2s πx φ(t; x∗0 ) dt + OT (h) ≤ 4Cs , (3.13) K(T )

since the uh are normalized. Letting h tend to 0, we obtain, for all T > 0, −2s πx φ(t; x∗0 ) dt ≤ 4Cs . (3.14) K(T )

Assume semi-boundedness of the trajectory, that is, for some t0 ∈ R, πx φ(t; x∗0 ) , ±t ≥ t0 ⊂ {y ∈ R3 ; |y| ≤ R1 } ,

(3.15)

then, by (3.14), 4Cs is larger than R1−2s times the length of K(T ) \ {t ∈ R; ±t < t0 } = [−T ; T ] \ t ∈ R; ±t < t0 and dist t, coll(x∗0 ) < . This is a contradiction since the latter tends to ∞ as T → ∞. Thus (3.15) is false and we can apply (3.3), yielding the non-trapping condition (1.10). 3.3. The repulsive case Here we consider the case where any singularity is repulsive (i.e. N = 0) and d ≥ 3. We want to show that (1.11) implies the non-trapping condition. Thanks to Proposition 3.7 below, we show that Wang’s proof can be followed in the present case, yielding a much simpler proof than the one in Subsection 3.2. First of all, we show that an important ingredient in Wang’s proof is available, namely the following weak version of Egorov’s theorem. Proposition 3.7. Let N = 0 and d ≥ 3. Let T > 0 and a ∈ Σ0;0 . Let θ, γ ∈ C0∞ (R) such that γθ = θ. Then [−T ; T ] t → γ(p)(a ◦ φt ) is a Σ0;0 -valued, C 1 -function. Furthermore, there exists C > 0, depending on θ and a, such that, for any > 0, for any t ∈ [−T ; T ], w U (t; h)∗ aw γ(p)(a ◦ φt ) h + r(t) θ P (h) , h U (t; h)θ P (h) = where [−T ; T ] t → r(t) is bounded by C + O ,T (h) in C 0 [−T ; T ]; L(L2(Rd )) .

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Proof. Let > 0. Since the singularities are repulsive, there exists some σ0 ∈ C0∞ (Rd , [0, 1]) which equals 1 near each singularity, such that, 2 V ≥ 1 on the support of σ0 and (σ0 ◦ πx )(γ ◦ p) = 0. Thus, for g ∈ L2 (Rd ) and f = θ(P (h))U (t; h)g,

σ0 f 2 ≤ f , σ02 2 V f + 2 σ0 f , −h2 Δ σ0 f ≤ 2 σ02 f , P (h)f + 2 σ0 f , [−h2 Δ, σ0 ]f ≤ Cθ2 2 f 2 + 2 σ0 f , [−h2 Δ, σ0 ]f , where Cθ depends only on θ. Since [−h2 Δ, σ0 ]θ(P (h)) = O (h) in L(L2 (Rd )), σ0 θ P (h) U (t; h) ≤ Cθ + O (h) (3.16) 0 2 d ∞ d in C [−T ; T ]; L(L (R )) . Let σ ∈ C0 (R ) with σ = 1 near each singularity such that σσ0 = σ. Using (3.16), Lemma 2.3, and pseudodifferential calculus, w ∗ U (t; h)∗ aw h U (t; h)θ P (h) = U (t; h) a(1 − σ0 ) h γ Pσ (h) U (t; h)θ P (h) + r1 (t) w = U (t; h)∗ a(1 − σ0 )γ(p) h U (t; h)θ P (h) + r2 (t) , where the rj are bounded by C + O (h) in C 0 [−T ; T ]; L(L2(Rd )) . By the choice of σ0 , a(1 − σ0 )γ(p) = aγ(p) =: aγ . Furthermore, for all t ∈ [−T ; T ], aγ ◦ φt = γ(p)(a ◦ φt ) and (d/dt)aγ ◦ φt = {p, aγ ◦ φt } = {pσ , aγ ◦ φt }. This allows us to follow the arguments in the proof of Lemma 3.5 showing that (3.9) with a = aγ is O ,T (h) in C 0 [−T ; T ]; L(L2 (Rd )) . Let λ ∈ I0 . As in Subsection 3.2, (1.11) implies the existence of some constant Cs > 0 such that (3.11) holds true, for θ ∈ C0∞ (I0 ; R) with θ(λ) = 1. Since no collision occurs, we choose K(T ) = [−T ; T ], take a : (x, ξ) → x−2s , and write (3.12) as U (t; h)uh , aw (3.17) h U (t; h)uh dt + OT (h) ≤ 2Cs . [−T ;T ]

By Lemma 3.2, Proposition 3.7 with = Cs , and (3.6), " −2s ! t w U (t; h)u , γ(p)(a◦φ ) u +b2 (t) +b1 (t) = πx φ(t; x∗0 ) U (t; h)uh , aw = u h h h h h where the bj are bounded by CCs + O(h) in C 0 ([−T, T ]). This yields (3.13), with bound 4Cs replaced by (2 + C)Cs , and the non-trapping condition as in Subsection 3.2.

4. Semiclassical trapping This section is devoted to the proof of the implication “1 =⇒ 2” of Theorem 1.1. We assume the non-trapping condition true on I0 and we want to prove the bound (1.11), for any compact interval I ⊂ I0 . Here we follow the strategy in [5,24]. We assume that the bound (1.11) is false, for some I. This means precisely that the following situation occurs, which we call “semiclassical trapping”. There exist a sequence (fn )n of nonzero functions of H2 (Rd ), a sequence (hn )n ∈]0; h0 ]N tending

794



to zero, and a sequence (zn )n ∈ CN with (zn ) → λ ∈ I and (zn )/hn → r ≥ 0, such that

fn B∗ = 1 and P (hn ) − zn fn B = o(hn ) . (4.1) ∗ As in [6], we shall see that “the (fn )n has no B -mass at infinity” (see Proposition 4.2 below). This yields the existence of some large R1 > 0, of a sequence (gn )n of nonzero functions of H2 (Rd ), of a sequence (hn )n ∈]0; h0 ]N tending to zero, and of a sequence (λn )n ∈ RN with λn → λ ∈ I, such that supp gn ⊂ {x ∈ R3 ; |x| ≤ R1 } ,

gn = 1 ,

and P (hn ) − λn gn = o(hn ) . (4.2)

Possibly after extraction of a subsequence, we may assume that the sequence (gn )n has a unique semi-classical measure μ, satisfying (2.5) with un replaced by gn (see Lemma 4.3). Now, we look for a contradiction with the non-trapping condition. While, in the regular case, it is quite easy to show the invariance of μ under the flow generated by p, this is not clear in the present situation. We shall show the invariance for repulsive singularities in Subsection 4.2. In Subsection 4.3 however, we only show a weaker form of invariance, if there is an attractive singularity. This Subsection 4.3 contains the main novelty of the paper. The other steps of the strategy are essentially the same as in [24], as explained in Subsection 4.1. If the reader is only interested in the bound (1.11) with B replaced by some L2s (s > 1/2), we propose a simpler proof in Subsection 4.4. 4.1. Main lines of the proof In this subsection, we give the main steps leading to the contradiction between the “semiclassical trapping” and the non-trapping condition. Here we focus on the steps which are essentially proved as in [24]. Lemma 4.1. The sequence fn 2 (zn )/hn n goes to 0 and limn→∞ (zn )/hn = 0. Proof. We write fn 2 (zn ) = fn , (P (hn )−zn )fn , which is o(hn ) by (1.6) and (4.1). This gives the first result. Now, assume that r > 0. Since fn 2 ((zn )/hn ) goes to 0, fn must go to 0, while fn ≥ fn B∗ = 1. This is a contradiction. Using (1.1), we show as in [6] the following localization in position space. Proposition 4.2. There exists R0 > R0 such that limn→∞ 1I{| · |>R0 } fn B∗ = 0. 2 2 Proof. Let a ∈ Σ0;0 . It is known that (aw h )h∈]0;h∗ ] is uniformly bounded in L(Ls ; Ls ) for any s ∈ R. Even more, using a partition of unity adapted to the decom# position Rd = c ∪ (∪j≥1 cj ) from (1.3), say 1 = τ (x) + j≥1 τj (x), and writ# ing, for any u ∈ B ∗ , the identity u = τ u + j≥1 τj u, standard pseudodifferential calculus and almost orthogonality properties allow to easily establish that ∗ ∗ (aw h )h∈]0;h∗ ] is uniformly bounded in L(B ; B ) (see [6] for a complete proof). Now, −1 w let αn := fn , ihn [P (hn ), ahn ]fn . Expanding the commutator, using (1.6), (4.1) and Lemma 4.1, we observe that αn → 0. For any s > 1/2, (fn )n is bounded in

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L2−s , since L2s ⊂ B. Now, we assume that a vanishes near the set S of all singularities. We can find χ ∈ C0∞ (Rd ; R) such that aχ = 0 and χ = 1 near the singularities. By Lemma 2.3 with k = k = −s, " ! w αn = fn , ih−1 n Pχ (hn ), ahn fn + O(hn ) . Let θ ∈ C0∞ (R; R) with θ = 1 near I and θ˜ := 1 − θ. Since zn → λ ∈ I, ˜ (hn ))(P (hn ) − zn )−1 )n is uniformly bounded. Thus there exists C > 0 ( θ(P such that

−j/2 ˜ ˜ ˜ θ P (hn ) fn B ∗ ≤ max θ P (hn ) fn c ; sup 2 θ P (hn ) fn cj j≥1 ≤ C P (hn ) − zn fn = o(hn ) , (4.3) since (4.1) implies that (P (hn ) − zn )fn = o(hn ). Using further that, for w s ∈]1/2; 1], · s ih−1 n [Pχ (hn ), ahn ] is uniformly bounded, ! " w αn = fn , ih−1 n Pχ (hn ), ahn θ P (hn ) fn + O(hn ) . w Since ih−1 n [Pχ (hn ), ahn ] is a h-pseudodifferential operator, we may apply Lemma 2.3 with k = k = −s, yielding ! " w P θ P αn = fn , ih−1 (h ), a (h ) χ n χ n fn + O(hn ) . n hn

Using similar arguments again, we arrive at ! " w αn = θ Pχ (hn ) fn , ih−1 n Pχ (hn ), ahn θ Pχ (hn ) fn + O(hn ) .

(4.4)

Now we specify the symbol a more carefully. By [6] (see Proposition 8 and the second step of the proof of Proposition 7 therein), we can find c > 0 and a function χ1 ∈ C0∞ (Rd ) such that, for all β = (βj ) ∈ 1 with |β| 1 = 1, there exists a symbol a ∈ Σ0;0 satisfying the following properties. The function χ1 = 1 on a large enough neighbourhood of 0 and of the support of χ. The semi-norms of a in Σ0;0 are bounded independently of β and, uniformly with respect to β, 2 −j (1 − χ1 )θ Pχ (hn ) fn cj + o(1) . βj 2 αn ≥ c · j By the above arguments, αn → 0, uniformly in β. This implies that 2 sup 2−j (1 − χ1 )θ Pχ (hn ) fn cj and therefore j

sup 2−j/2 (1 − χ1 )θ Pχ (hn ) fn cj j

tend to 0. In other words, (1 − χ1 )θ(Pχ (hn ))fn B ∗ → 0. Since B ⊂ L21/2−

continuously, for any > 0, we derive from Lemma 2.3 that (1 − χ1 ) θ Pχ (hn ) − θ P (hn ) fn ∗ → 0 , B

796



yielding (1 − χ1 )fn B ∗ → 0, thanks to (4.3). Now the desired result follows for R0 large enough such that |x| ≥ R0 =⇒ χ1 (x) = 0. Lemma 4.3. Let R1 > R0 . There exist a sequence (gn )n of nonzero functions of H2 (Rd ), bounded in L2 (Rd ) and having a unique semiclassical measure μ, a sequence (hn )n ∈]0; h∗ ]N tending to zero, and a sequence (λn )n ∈ RN with λn → λ ∈ I, such that (4.2) holds true. Proof. Let τ, κ ∈ C0∞ (Rd ; R) be such that supp τ, supp κ ⊂ {x ∈ Rd ; |x| ≤ R1 }, κ = 1 on {x ∈ Rd ; |x| ≤ R0 }, and τ κ = κ. The sequence (τ fn )n is bounded in L2 (Rd ). Possibly after extraction of a subsequence, we may assume that it has a unique semiclassical measure μ. We shall show that (4.5) supp μ ⊂ (x, ξ) ∈ T ∗ Rd ; |x| ≤ R0 , supp μ ∩ T ∗ (Rd \ S) ⊂ p−1 (λ) .

(4.6)

By Proposition 4.2, 1I{| · |>R0 } τ fn goes to 0. Using (2.5), this implies (4.5). Now let a ∈ C0∞ (T ∗ Rd ) be such that a = 0 near p−1 (λ) ∪ S. Since ( · −1 fn )n is bounded by (4.1), w τ fn , aw hn τ fn = τ fn , (τ a)hn fn + O(hn ) ! " = τ fn , (τ a)w (4.7) hn θ P (hn ) fn + O(hn ) " ! −1 ˜ + τ fn , (τ a)w (P (hn ) − zn )fn , hn θ P (hn ) P (hn ) − zn where θ ∈ C0∞ (R; R) with θ = 1 near λ, such that θ(p)a = 0, and θ˜ = 1 − θ. By (4.1), (P (hn ) − zn )fn = o(hn ) and the last term in (4.7) is a o(hn ). We can find χ ∈ C0∞ (Rd ; R) such that aχ = 0 and χ = 1 near the singularities. By Lemma 2.3, we recover ! " w τ fn , aw τ f = τ f , (τ a) θ P (h ) n n χ n fn + O(hn ) = O(hn ) hn hn since aθ(pχ ) = 0. By (2.5), this yields (4.6). The symbol of [−h2n Δ, κ] belongs to Σ−∞,1 and is supported in {(x, ξ) ∈ T ∗ Rd ; R0 < |x| < R1 }. Let τ˜ ∈ C0∞ (Rd ) such that τ˜ = 1 on supp ∇κ and supp τ˜ ⊂ {x ∈ Rd ; R0 < |x| < R1 }. Then −1 Pχ (hn ) + i τ˜fn τ fn = [−h2n Δ, κ] Pχ (hn ) + i [−h2n Δ, κ]fn = [−h2n Δ, κ]˜ −1 = [−h2n Δ, κ] Pχ (hn ) + i [−h2n Δ, τ˜]fn −1 + [−h2n Δ, κ] Pχ (hn ) + i τ˜ P (hn ) − zn fn −1 + [−h2n Δ, κ] Pχ (hn ) + i (i + zn )˜ τ fn =: r1 + r2 + r3 . Standard pseudodifferential calculus together with Proposition 4.2 provide r1 = o(h2n ), r2 = o(h2n ), and r3 = o(hn ) in L2 (Rd ). Thus, setting gn := κfn , P (hn ) − zn gn = κ P (hn ) − zn fn + o(hn ) = o(hn )

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in L2 (Rd ). By Proposition 4.2 and (4.1), gn → c, with c > 0, and (zn )gn = o(hn ) in L2 (Rd ), by Lemma 4.1. Setting λn := (zn ), we obtain (P (hn ) − λn )gn = o(hn ). Using (2.5) and the previous arguments, μ is the unique semi classical measure of (gn )n . We now collect properties of the gn and their semiclassical measure μ, defined in Lemma 4.3. Lemma 4.4. Let a ∈ C0∞ (T ∗ Rd ) such that a = 0 near the set S of all singularities. 1. Then μ({p, a}) = 0 (“μ is invariant under the flow”). 2. If a = 0 near p−1 (λ) or near {(x, ξ) ∈ T ∗ Rd ; |x| ≤ R0 } then μ(a) = 0. 3. Let τ ∈ C ∞ (Rd ) such that τ = 0 near S. Then the sequence ( τ ihn ∇gn )n is bounded. Proof. 1) Let χ ∈ C0∞ (Rd ; R) such that aχ = 0 and χ = 1 near the set S of all singularities. In particular, {p, a} = {pχ , a}. By Lemma 2.3, " ! " ! w −1 w an := gn , ih−1 (4.8) n P (hn ), ahn gn = gn , ihn Pχ (hn ), ahn gn + O(hn ) = gn , ({pχ , a})w (4.9) hn gn + O(hn ) . By (2.5), the r.h.s. of (4.9) goes to μ({pχ , a}), as n → ∞. As in [24], we replace P (hn ) by P (hn ) − λn in the commutator on the l.h.s. of (4.8) and expand the commutator. Using (4.2), we show that an = o(1), as n → ∞, yielding μ({p, a}) = 0. 2) The second assertion was established in the proof of Lemma 4.3. ˆ . Since supp gn ⊂ {|x| ≤ R }, 3) Let τ ∈ C ∞ (Rd ) with support in M 1 ! " 2 τ gn , h2n Δx gn ≤ τ 2 gn , P (hn ) − λ gn + O(n0 ) , where O(n0 ) means O(1) as n → ∞. Thus ihn ∇x gn , τ 2 ihn ∇x gn ≤ 2hn (∇x τ )gn , τ ihn ∇x gn + O(n0 )

τ ihn ∇x gn 2 ≤ O(hn ) · τ ihn ∇x gn + O(n0 ) , yielding the boundedness of ( τ ihn ∇gn )n . We introduce B± (λ) := x∗ ∈ p−1 (λ); 0 ≤ ±t → πx φ(t; x∗ ) is bounded

(4.10)

and B(λ) := B+ (λ) ∩ B− (λ). By (3.3), the non-trapping condition (1.10) exactly means that B+ (λ) and B− (λ) are empty. Proposition 4.5. Let d ≥ 3 if N = 0 else let d = 3. The measure μ is nonzero. If N = 0, μ vanishes near the (repulsive) singularities, is invariant under the complete flow t → φt , and supp μ ⊂ B(λ). If N > 0, then, outside the attractive singularities, μ is supported in B(λ) that is supp μ ∩ T ∗ (R3 \ S) ⊂ B(λ) . (4.11)

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Proof. For the case of purely repulsive singularities (i.e. N = 0) the proof is given in Subsection 4.2. The other case appears in Subsection 4.3. Remark 4.6. If (1.11) is really false, one expects that the fn are “close to some resonant state”. Proposition 4.5 and Proposition 4.9 below roughly say that this resonant state should be microlocalized on trajectories in B(λ). However, it does not give any information above the attractive singularities. If the potential V is smooth (i.e. N = N = 0), the arguments used in [24] actually prove Proposition 4.5 in this case. Lemma 4.7. Let d ≥ 3 if N = 0 else let d = 3. If p is non-trapping at energy λ (cf. (1.10)) then μ = 0. Proof. Let N = 0. By Proposition 4.5, supp μ ⊂ B(λ), which is empty by the nontrapping condition. Thus μ = 0. The other case is treated in Subsection 4.3. Now Proposition 4.5 and Lemma 4.7 produce the desired contradiction. 4.2. Repulsive singularities We show Proposition 4.5 for the case N = 0, d ≥ 3, by first showing a decay estimate for the Fourier transform of the gn ’s. Since we only have repulsive singularities, there exists some positive c such that

N gn , (1/| · − sj |) gn gn , (−h2n Δx ) gn + j=1

! " ≤ c gn , P (hn ) − λ gn + O(n0 ) . (4.12)

By Lemma 4.3, (P (hn ) − λ)gn → 0 and the r.h.s of (4.12) is bounded. Now, we show that μ = 0. Let χ ∈ C0∞ (Rd ; R) such that 0 ≤ χ ≤ 1 and χ = 1 near 0. Let us denote by F g the Fourier transform of g. Setting χR (ξ) = χ(ξ/R), for R > 0 and ξ ∈ Rd , we observe that $ % (1 − χR )(hn · ) O(n0 ) 2 gn , (−h2n Δ)gn . |h · | F g F gn , ≤ n n 2 2 |hn · | R The bracket on the r.h.s is bounded uniformly w.r.t. R. Thus lim lim sup F gn , (1 − χR )(hn · )F gn = 0 . R

(4.13)

n

Recall that, for all n, supp gn ⊂ {|x| ≤ R1 } (cf. Lemma 4.3). By Proposition 2.1, this implies that gn 2 → μ(1I), yielding μ = 0. Now let τ ∈ C0∞ (R; R+ ) be supported on a neighborhood of the singularities such that τ = 1 near them. Since V − λ is large and positive near the singularities, we can choose the support of τ such that, (4.14)

τ gn 2 ≤ τ gn , (V − λ)τ gn .

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Thus τ gn , (−h2n Δx ) τ gn + τ gn 2 ≤ 2 τ gn , (P − λ)τ gn = o(1) ,

799

(4.15)

using Lemma 4.4. In particular, τ gn 2 → μ(τ 2 ) (cf. Proposition 2.1) and τ gn → 0. Thus μ is supported away from the (repulsive) singularities. By Lemma 4.4, we conclude that μ is invariant under the flow (φt )t∈R . If the trajectory t → πx φt (x, ξ) goes to infinity when ±t → +∞, then the invariance of μ under the flow implies that μ vanishes on this trajectory. This shows that supp μ ⊂ B(λ) and finishes the proof of Proposition 4.5 in the case N = 0 and d ≥ 3. 4.3. The general case in dimension 3 In this subsection, we assume that N > 0 and d = 3 and we give successively the proofs of Proposition 4.5 and Lemma 4.7 (at the end of the subsection). In view of (4.13) and of Proposition 2.1, we want to show that gn , (−h2n Δx )gn is bounded to get μ = 0. We also need a kind of invariance of μ under the pseudo-flow φt (cf. (2.18)). To realize this programme, we want to use the KS-transform (2.23) to lift the property (4.2) in R4 , locally near each attractive singularity. Let (τj )0≤j≤N ∈ (C0∞ (Rd ; R+ ))N +1 be such that #N 2 d • j=0 τj = 1 near {x ∈ R ; |x| ≤ R1 }, • for 1 ≤ j ≤ N , τj = 1 near sj and is supported away from the other singularities, • τ0 = 1 near the set of repulsive singularities and is supported away from the other singularities. There exists c > 0 such that

N τ0 gn , (1/|x − sj |) τ0 gn ≤ c τ0 gn , (V − λ)τ0 gn .

(4.16)

j=N +1

Thus

N τ0 gn , (−h2n Δx ) τ0 gn + τ0 gn , (1/|x − sj |) τ0 gn j=N +1

≤ (1 + c) τ0 gn , (P − λ)τ0 gn + O(n0 ) = O(n0 ) . (4.17) Here we used the fact that τ0 gn , (P −λ)τ0 gn → 0, by Lemma 4.3 and Lemma 4.4. Let 1 ≤ j ≤ N . For the same reason, τj gn , (P − λ)τj gn → 0. Thus, since (V − fj /| · − sj |)τj is bounded, τj gn , (−h2n Δx ) τj gn + τj gn , (fj /| · − sj |) τj gn = O(n0 ) . (4.18) We introduce the KS-transformation (cf. (2.23)) which is adapted to the singularity at sj : x = Kj (zj ) := sj + K(zj ) (cf. (2.19)) and, for x = sj , (x, ξ) = Kj∗ (z; ζ) := (sj , 0) + K∗ (z; ζ) .

(4.19)

800



For all n, let gñ,j := gn ◦ Kj . Let χj ∈ C0∞ (R3 ) such that χj τj = χj , χj = 1 near sj , and χj τk = 0, for k = j. Denote by χ ˜j the function χj ◦Kj . For λ ∈ R, 4 we introduce the differential operator in Rzj P˜j (h; λ ) := −h2 Δzj + (τj V )◦Kj − λ | · |2 , (4.20) which can be seen as the Weyl h-quantization of the symbol T ∗ R4 (zj , ζj ) → p˜j,λ (zj , ζj ) := |ζj |2 + (τj V ) K(zj ) − λ · |zj |2 .

(4.21)

˜j , Notice that p˜j,λ ∈ Σ2;2 . We can write, for zj ∈ supp χ |zj |2 (τj V ) Kj (zj ) − λ = fj (sj ) + fj Kj (zj ) − fj Kj (0) + |zj |2 Wj Kj (zj ) − λ ˜ j,λ (zj ) . =: fj (sj ) + W

(4.22)

˜ j,λ is a quadratic perturbation of the constant fj (sj ), vanishing at sj , and So W ˜ j,λ . P˜j (h; λ ) = −h2 Δzj + fj (sj ) + W

(4.23)

Lemma 4.8. Let 1 ≤ j ≤ N . The sequence (˜ gn,j )n = (gn◦Kj )n is bounded in L2 (R4 ). Up to subsequence, we may assume that it has a unique semiclassical measure μ ˜j . Besides, ∀n ∈ N , supp gñ,j ⊂ zj ; |zj | ≤ (R0 + R1 )1/2 , (4.24) and χ ˜j P˜j (hn ; λn )˜ gn,j = o(hn ) in L2 (R4 ) . (4.25) Let φ˜sj := φ˜j (s; · ) be the Hamiltonian flow associated to (t, λ , z, ζ) → p˜j,λ (z, ζ) by (2.24). Let ˜b ∈ C0∞ (T ∗ R4 ) and Tb := {s > 0; ∀t ∈ [0; s] , (˜b ◦ φ˜tj )(1 − χ ˜j ) = 0}. Then, for s ∈ Tb , (4.26) ˜j ˜b ◦ φ˜sj . μ ˜j (˜b) = μ Proof. • Eq. (4.24) follows from the scaling |K(z)| = |z|2 of the Hopf map (see (2.19)) and the estimate (4.2) for the support of gn . • Since (−h2n Δx + V − λn )gn = o(hn ) and gn = O(n0 ) in L2 (R3 ), we use (2.19), (2.21), and the arguments of Proposition 2.1 in [13] to get ˜j P˜j (hn ; λn )˜ gn,j = o(hn ) and | · |˜ gn,j = O(n0 ) in L2 (R4 ) . | · |−1 χ

(4.27)

This yields (4.25). • Now, we show that χ ˜j gñ,j = O(n0 ) in L2 (R4 ). Together with (4.27), this then will imply the desired boundedness of (˜ gn,j )n in L2 (R4 ). Thanks to (2.19), (2.21), and to Part 3 of Lemma 4.4, (4.28)

1Isupp ∇χ˜j hn ∇zj gñ,j = O 1Isupp ∇χj hn ∇x gn = O(n0 ) , 0

1Isupp ∇χ˜j gñ,j = O 1Isupp ∇χj gn = O(n ) . (4.29)

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801

Let An,j := (zj · hn ∇zj + hn ∇zj · zj )/(2i) and " ! ˜ an,j := gñ,j , h−1 ˜j An,j χ ˜j gñ,j . n Pj (hn ; λn ), iχ Expanding the commutator and using (4.27), we see, on one hand, that |an,j | ≤ o(n0 ) · hn 1Isupp ∇χ˜j gñ,j + χ ˜j ihn ∇zj gñ,j + O(n0 ) ˜j ihn ∇zj gñ,j + o(n0 ) , ≤ o(n0 ) · χ

(4.30)

thanks to (4.29). On the other hand, writing 2iAn,j = 2zj · hn ∇zj + 4hn , " ! " ! 2 an,j = gñ,j , 2 −h2n Δzj , χ ˜2j gñ,j − 2 (zj · hn ∇zj )χ ˜j gñ,j , h−1 ˜j gñ,j n −hn Δzj , χ " ! ˜j (hn ; λn ), zj · hn ∇zj χ + χ ˜j gñ,j , h−1 P ˜ g ˜ j n,j . n By (4.28) and (4.29), " ! ˜ ˜j gñ,j = O(n0 ) . ˜j gñ,j , h−1 an,j − χ n Pj (hn ; λn ), iAn,j χ 2 ˜ ˜ As a differential operator, h−1 n [Pj (hn ; λn ), iAn,j ] = 2(−hn Δzj ) − zj · ∇zj Wj,λn (zj ) (cf. (4.23)) and, by (4.24), there exists some cj > 0 such that, for all n and for all ˜ j,λn (zj )| ≤ cj |zj |2 . By (4.27), zj ∈ supp gñ,j , |zj · ∇zj W an,j − χ ˜j gñ,j , −2h2n Δzj χ ˜j gñ,j = O(n0 ) .

This, together with (4.30), implies that 0≤ χ ˜j gñ,j , −2h2n Δzj χ ˜j gñ,j ≤ o(n0 ) · χ ˜j ihn ∇zj gñ,j + O(n0 ) . Writing

(4.31)

˜j ihn ∇zj gñ,j 2 + h2n (∇zj χ χ ˜j gñ,j , −h2n Δzj χ ˜j gñ,j = χ ˜j )˜ gn,j 2 + 2hn (∇zj χ ˜j )˜ gn,j , χ ˜j ihn ∇zj gñ,j

and using again (4.28) and (4.29), we arrive at ˜j ihn ∇zj gñ,j + O(n0 ) .

χ ˜j ihn ∇zj gñ,j 2 ≤ o(n0 ) · χ This yields

χ ˜j ihn ∇zj gñ,j = O(n0 ) and Now

χ ˜j gñ,j , −h2n Δzj χ ˜j gñ,j = O(n0 ) .

(4.32)

˜j gñ,j , χ χ ˜j gñ,j , P˜j (hn ; λn )χ ˜j gñ,j = χ ˜j P˜j (hn ; λn )˜ gn,j ˜j ]˜ gn,j + χ ˜j gñ,j , [−h2n Δzj , χ

and is bounded by (4.27), (4.28), and (4.29). Thus ˜ j,λn χ ˜j gñ,j + f (sj ) χ ˜j gñ,j 2 + χ ˜j gñ,j , W ˜j gñ,j χ ˜j gñ,j , −h2n Δzj χ = O(n0 ) . (4.33) In (4.33), the first and third terms are O(n0 ), by (4.32) and by (4.27) respectively. Since fj (sj ) = 0, we conclude that (χ ˜j gñ,j )n is bounded in L2 (R4 ).

802



• We now show the invariance (4.26). It suffices to show that, for all λ ∈ R and all ˜b ∈ C0∞ (T ∗ R4 ) such that ˜b(1 − χ ˜j ) = 0, μ ˜j ({˜ pj,λ , ˜b}) = 0. Take such a ˜b w ˜ and λ ∈ R. Since bhn is uniformly bounded, " ! w 0 ˜ ˜ χ ˜ g ˜ (h ; λ ), b gñ,j , ih−1 P j j n n n hn n,j = o(n ) , by expanding the commutator, using (4.25), and using the boundedness in L2 (R4 ) of (˜ gn,j )n . Now we compute the leading term of the commutator and arrive at o(n0 ) = gñ,j , {χ ˜j p˜j,λn , ˜b}w ñ,j + O(hn ) = gñ,j , {χ ˜j p˜j,λ , ˜b}w ñ,j + o(n0 ) hn g hn g = gñ,j , {˜ pj,λ , ˜b}w gñ,j + o(n0 ) , hn

˜j ({˜ pj,λ , ˜b}) = 0. As in the proof of (4.6), since χ ˜j = 1 on the support of ˜b. Thus μ −1 we see that supp μ ˜ j ⊂ (˜ pj,λ ) (0). Since the last two components of φ˜sj ( · , λ, · , · ) actually form the flow generated by p˜j,λ , we obtain (4.26). Proof of Proposition 4.5. Let 1 ≤ j ≤ N . The boundedness of the sequence (χ ˜j gñ,j )n in L2 (R4 ) precisely means that (χj gn , (1/| · − sj |)χj gn )n is bounded (cf. (2.21)) and so is also (τj gn , (1/| · − sj |)τj gn )n . By (4.18), this implies that (τj gn , −h2n Δx τj gn )n is bounded. By the IMS localization formula (cf. Chapter 3.1 of [8]), gn , −h2n Δx gn =

N j=0

τj gn , −h2n Δx τj gn − h2n

N

(∇x τj )gn 2 = O(n0 ) , (4.34)

j=0

thanks to (4.17). As in Subsection 4.2, we can derive (4.13) and prove that μ = 0. Consider a trajectory (φ(t; x∗0 ))t∈coll(x∗0 )) such that πx φ(t; x∗0 ) goes to infinity as t → ±∞. If it does hit a singularity then πx φ(t; x∗0 ) must come from infinity, hit the singularity and then go back to infinity (coll(x∗0 ) contains one point). Since μ vanishes on some {x∗ ∈ T ∗ R3 ; |x| ≥ C}, μ vanishes near the tail(s) of (φ(t; x∗0 ))t∈coll(x∗0 )) which is (are) inside this set. By invariance (cf. Lemma 4.4), μ vanishes near each φ(t; x∗0 ), for t ∈ coll(x∗0 ). This proves (4.11). ˜j ) = 0 and τ˜ = 0 Proof of Lemma 4.7. Let 1 ≤ j ≤ N and τ˜ ∈ C0∞ (R4 ) with τ˜(1−χ near zj = 0. Then |˜ τ |2 μ ˜j is the semiclassical measure of (˜ τ gñ,j )n (see [18]). We τ gñ,j 2 . may assume that τ = τ˜ ◦ Jj,+ is well defined. By (2.21), τ gn | · |−1/2 2 = ˜ By (2.19), τ1 := τ | · |−1/2 is smooth. Thus τ1 gn , (P − λ)τ1 gn → 0, by Lemma 4.3 and Lemma 4.4. This yields the bound (4.34) and Eq. (4.13) with gn replaced by τ1 gn . By Proposition 2.1, τ1 gn 2 → |τ1 |2 μ(1I). But the latter is zero since, by Proposition 4.5 and the non-trapping assumption, μ may only have mass above ˜j μ ˜j may only the attractive singularities. Thus lim ˜ τ gñ,j = 0. This implies that χ have mass above zj = 0. Now let τ ∈ C0∞ (R3 ) supported near sj and inside the set χ−1 j (1), and set pj,λ )−1 (0) ∩ τ˜ = τ ◦ Kj . Let ϕ˜ ∈ C0∞ (T ∗ R4 ) such that ϕ˜ = 1 on a neighborhood of (˜ 4 (supp χ ˜j × R ). Let r ∈ R. For n large enough, the well defined symbols τ˜(1 −

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803

ϕ)(˜ ˜ pj,λn )−1 belong to Σr,−2 and form a bounded sequence in this set. Writing τ˜(1− ϕ) ˜ = τ˜(1 − ϕ)(˜ ˜ pj,λn )−1 · χ ˜j p˜j,λn and using pseudodifferential calculus and (4.25), w w τ˜(1−ϕ) ˜ hn gñ,j = τ˜(1−ϕ)(˜ ˜ pj,λn )−1 hn χ ˜j P˜j (hn ; λn )˜ gn,j +O(hn ) = O(hn ) (4.35) in L2 (R4 ). Notice that if (0, ζ) ∈ (˜ pj,λ )−1 (0) then |ζ|2 = −f (sj ) = 0 and ζ = 0. Now, using (2.19), we can choose the support of τ small enough around z = 0 ˜ ; · ) ⊂ χ such that, for some s0 > 0, supp (˜ τ ϕ) ˜ ◦ φ(s ˜−1 j (1), for 0 ≤ s ≤ s0 , and 4 ∗ 4 ˜ ; · ) = 0 near {0} × R ⊂ T R . Using (2.21), (4.35), and (2.5) applied (˜ τ ϕ) ˜ ◦ φ(s ˜j , to gñ,j and μ

τ gn 2 = ˜ τ gñ,j | · | 2 = ˜ τ ϕ˜ ˜gn,j | · | 2 + O(hn ) = μ ˜j (˜ τ 2 ϕ˜2 | · |2 ) + o(n0 ) . ˜ ; · )) = 0, by the choice of s . Thereτ 2 ϕ˜2 | · |2 ) = μ ˜j ((˜ τ 2 ϕ˜2 | · |2 )◦φ(s By (4.26), μ ˜j (˜ 0 0 2 fore lim τ gn = 0, yielding μ = 0 near sj . Thus μ = 0. Actually, if trapping occurs, we have the following stronger result on the measure μ. Proposition 4.9. Let N > 0 and d = 3. If x∗ ∈ supp μ ∩ T ∗ (R3 \ S) and t ∈ coll(x∗ ) then φ(t; x∗ ) ∈ supp μ. Proof. Let 1 ≤ j ≤ N . Let x∗0 := (x0 , ξ0 ) ∈ p−1 (λ) such that χj = 1 near x0 . By the properties of the KS-transform (4.19) (cf. (2.22)), there exists z∗0 = (z0 , ζ0 ) ∈ T ∗ R4 such that x∗0 = Kj∗ (z∗0 ). Let t∗0 = (0, p(x∗0 )) = (0, λ). We consider the trajectory {πx φt (x∗0 ), t ∈ R} and assume that it hits the singularity sj at time t0 . Let t > t0 such that χj (πx φ(t ; x∗0 )) = 1. There exists some s ∈ R such that t = tj (s ; t∗0 , z∗0 ) (cf. (2.25)). Here tj (s; t∗ , z∗ ) is the first component of the flow φ˜j (s; t∗ , z∗ ) given by (2.25) with p˜ replaced by (4.21). Let τ0 ∈ C0∞ (R3 ) such that χj = 1 near supp τ0 , τ0 = 1 near x0 , and τ0 = 0 near sj . The semiclassical measure μ1 of the sequence (τ0 gn )n , viewed as a bounded sequence in L2 (R3 × S 1 ), is μ ⊗ 1 ⊗ δ0 on T ∗ R3 × T ∗ S 1 . Let ψ ∈ C0∞ (R) such that ψ = 1 near 0 and K0 ⊂⊂ R3 be a vicinity of ξ0 . Let a ∈ C0∞ (T ∗ R3 ) such that τ0 = 1 near πx supp a and πξ supp a ⊂ K0 . For (x∗ ; θ∗ ) := (x; ξ; θ; σ) ∈ T ∗ R3 × T ∗ S 1 , set a1 (x∗ ; θ∗ ) = ψ(σ)a(x∗ ). Let ψ1 + ψ2 = 1 be a smooth partition of unity on S 1 . Notice that μ(a) = τ0 μ(a) = τ0 μ1 (a1 ) =

2

τ0 μ1 (a1 ψk ) .

(4.36)

k=1

For each k ∈ {1; 2}, we may apply Proposition 2.2 with un = τ0 gn ψk ∈ L2 (R3 ×S 1 ) and Φ = (Kj , Aj,+ ), since (Kj , Aj,+ ) is a local diffeomorphism near supp τ0 × supp ψk by (2.20). Thus ((τ0 ψk ) ◦ (Kj , Aj,+ ))˜ μj (˜bk ) = τ0 μ1 (a1 ψk ), where ˜bk = (a1 ψk ) ◦ (Kj , Aj,+ )c , since ((τ0 ψk ) ◦ (Kj , Aj,+ ))˜ μj is the semiclassical measure of ((τ0 gn ψk ) ◦ (Kj , Aj,+ ))n . Now we can choose K0 and supp τ0 small enough such that, for all k ∈ {1; 2}, ˜bk ◦ φ˜sj = 0 near {0} × R4 and (1 − χ ˜j )˜bk ◦ φ˜tj = 0, for 0 ≤ t ≤ s . Thus (4.26) holds true with s = s and ˜b = ˜bk . Let τ˜k ∈ C0∞ (R4 ) such that χ ˜j = 1 near supp τ˜k , τ˜k = 1 near πz supp ˜bk ◦ φ˜sj , and τ˜k = 0 near

804



zj = 0. We may assume that Jj,+ is a local diffeomorphism with local inverse (Kj , Aj,+ ) near πz supp ˜bk ◦ φ˜sj (cf. (2.20)). Thus we can apply Proposition 2.2 ˜j (˜bk ◦ φ˜sj ) = τk μ1 (as ,k ), with un = τ˜k gñ,j ∈ L2 (R4 ) and Φ = Jj,+ . This yields τ˜k μ where τk = τ˜k ◦ Jj,+ and as ,k = ˜bk ◦ φ˜sj ◦ (Jj,+ )c . Now we see that, if μ is zero near φ(t ; x∗0 ), then we can choose K0 and supp τ0 small enough such that τk μ1 (as ,k ) = 0, for k ∈ {1; 2}. By (4.36), this implies that μ(a) = 0, for a with small enough support near x∗0 . Since we can reverse the time direction, we get the desired result. 4.4. A simpler proof for weighted L2 estimates In Subsections 4.1, 4.2, and 4.3, we proved that the non-trapping condition implies the Besov estimate (1.11). By (1.7), the latter implies the existence of some C > 0 such that, for all s > 1/2, sup R(z; h) L2s ,L2−s ≤ C · h−1 ,

(4.37)

z∈I

z=0 2

a weighted L estimate. This derivation of (4.37) from the non-trapping condition uses Proposition 4.2, the proof of which is based on arguments borrowed from [6]. The latter are rather involved since, in [6], the potential is assumed to be C 2 only. In particular, a special pseudodifferential calculus, adapted to this low regularity, is used there. Since our potential here is C ∞ outside the singularities, we want to give a simpler proof of the following, slightly weaker result. Proposition 4.10. Under the assumptions of Theorem 1.1, we assume that p is non-trapping at each energy λ ∈ I0 . Then, for any compact interval I ⊂ I0 and any s > 1/2, there exists Cs > 0 such that (4.37) holds true with C = Cs . Proof. Let d ≥ 3. We can follow the arguments in Subsections 4.1, 4.2, and 4.3, if Proposition 4.2 is replaced by lim 1I{| · |>R0 } fn L2 = 0 . (4.38) ∃R0 > R0 ; n→∞ −s Indeed, for functions localized in {x ∈ Rd ; |x| ≤ R0 }, the norms · B and · L2s are equivalent and so are the norms · B ∗ and · L2−s . So we are left with the proof of (4.38). We follow the proof of Proposition 4.2 and arrive at (4.4). Now, by [24], we can find c > 0, a function χ1 ∈ C0∞ (Rd ), and a symbol a ∈ Σ0;0 satisfying the following properties. The function χ1 = 1 on a large enough neighbourhood of 0 and of the support of χ and 2 αn ≥ c · (1 − χ1 )θ Pχ (hn ) fn L2 + o(1) . −s

Following again the proof of Proposition 4.2, we get (4.38).

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805

5. On the validity of the non-trapping condition The aim of this section is to provide examples both of validity and of invalidity of the non-trapping condition (1.10). As we shall see in Corollary 5.2 below, the non-trapping property is seldom fulfilled if there is some singularity (N > 0), even at positive energies. This is in strong contrast to the smooth case, for which p is always non-trapping at large enough positive energies. To study the non-trapping condition (1.10) when an attractive singularity is present (and d = 3), we need to review the regularization of the Hamilton flow ˆ = R3 \ S. of p, described in Section 2, in a more sophisticated way. Recall that M #3 ∗ 3 Let ω0 be the natural symplectic two-form on T R given by j=1 dxi ∧ dξi and also its restriction to Pˆ . It is well known (see [32], Thm. 5.1) that there exists an extension (M, ω, m) of the Hamiltonian system (Pˆ , ω0 , p), where as a set the six-dimensional smooth manifold M equals M := Pˆ ∪

N

(R × S 2 ) .

i=1

Here the ith copy R × S 2 parameterizes energy and direction of the particle colliding with the attractive singularity si . Using the symplectic form ω on M , the Hamiltonian function m ∈ C ∞ (M ) generates a smooth complete flow Φ : R × M −→ M ,

(t; x∗ ) → Φ(t; x∗ ) =: Φt (x∗ ) .

(5.1)

A collision time for x∗ ∈ M is a time t0 such that Φ(t0 ; x∗ ) ∈ Pˆ . If t is not a collision time for x∗ ∈ Pˆ then Φ(t; x∗ ) = φ(t; x∗ ), defined just before (2.17). Proposition 5.1. Consider for d = 2 or 3 a regular value λ > 0 of V . If the set Hλ := x ∈ Rd ; V (x) ≥ λ or x ∈ S is not homeomorphic to a d-dimensional ball or a point, then p is trapping at energy λ, i.e. (1.10) is false. ˜ λ ∪{s ˙ 1 , . . . , sN } with Proof. We write Hλ as the disjoint union H ˜ λ := x ∈ Rd ; V (x) ≥ λ or x ∈ {sN +1 , . . . , sN } . H ˜ λ is a d-dimensional manifold with boundary, since by assumption λ is a Then H regular value of V . It is compact since by assumption lim|x|→∞ V (x) = 0 but λ > 0. ˜ λ is a neighbourhood of the repulsive singularities sN +1 , . . . , sN , Notice that H but there exist neighbourhoods of the attractive singularities s1 , . . . , sN that are ˜ λ . In the presence of repulsive singularities H ˜ λ is nonempty. In any disjoint from H case, Hλ is a nonempty compact set. We denote by Int(Hλ ) the interior of Hλ . Now we assume that Hλ is not homeomorphic to a d-dimensional ball nor to a point, and we construct a periodic orbit, thus proving trapping. We discern two cases.

806



First case: Hλ has two or more connected components. Here the idea is to construct a periodic orbit (using curve shortening), whose projection on configuration space is a curve connecting two components of Hλ . Let gEuclid denotes the euclidean metric on Rd . We now use the Jacobi metric gˆλ , defined on Rd \ Hλ by (5.2) gˆλ (q) := λ − V (q) gEuclid . It is known (see e.g. [28] and [4]) that for regular curves c : [0, 1] → Rd \ Int(Hλ ) with c(1) = si (i ≤ N ) the length t& gˆλ c(s) c(s), L(c) := lim ˙ c(s) ˙ ds t1

0

is finite. By compactness of Hλ the number of connected components of Hλ is finite. Denoting them by Hλ;1 , . . . , Hλ; , for 1 ≤ i < j ≤ Dλ (i, j) :=

inf

c:c(0)∈Hλ;i , c(1)∈Hλ;j

L(c) > 0 ,

that is, the different components have positive geodesic distances. Taking R large enough, we can ensure that these mutual distances are smaller than the corresponding geodesic distance of the Hλ;i to the region {x ∈ Rd ; |x| ≥ R}. The (standard) approach is to consider the negative gradient flow of the energy functional 1 E(c) := ˙ c(s) ˙ ds , with c(0) ∈ Hλ;i0 and c(1) ∈ Hλ;i1 gˆλ c(s) c(s), 0

in order to approximate geodesic segments, which are then critical points of E with respect to these boundary conditions. Due to the degeneracy of the Jacobi metric (5.2) at ∂(Rd \Hλ ) still no Palais– Smale condition is satisfied for E, that is, a vanishing gradient of E at c does not ensure that c is a geodesic (see Klingenberg [30], Chapter 2.4 for a discussion of the Palais–Smale condition). However, as λ is assumed to be a regular value of V , the regularization technique devised by Seifert in [48] and later by Gluck and Ziller in [19] can be applied to yield a geodesic segment of length equal to Dλ (i0 , i1 ) = mini<j Dλ (i, j) > 0, with c(0) ∈ Hλ;i0 and c(1) ∈ Hλ;i1 . We denote the restriction of the flow Φt to m−1 (λ) by Φtλ . Away from the end points, and up to time parameterization, the geodesic segment in the Jacobi metric corresponds to a segment of a Φtλ -solution curve. See [1], Thm. 3.7.7 for a proof. This segment is part of a periodic orbit, whose period is twice the time needed to parametrize the segment: • If V (c(ik )) = λ, then (by our regularity assumption for the value λ) ∇V (c(ik )) = 0. Furthermore the geodesic segment at this point has a normalized tangent lim c(s) − c(ik ) −1 c(s) − c(ik ) sik

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which is parallel to ∇V (c(ik )) (see [19], Sect. 6). Thus the solution curve can be continued by time reversal (cf. (2.16)) c(ik + s) := c(ik − s)

(s > 0) .

(5.3)

• Similarly, if instead c(ik ) ∈ {s1 , . . . , sN }, that is, c(t) converges to an attracting singularity, then, time reversal (5.3) again continues the geodesic segment c and thus the Φtλ -solution curve as well. In both cases we thus constructed a periodic Φλ -orbit. Second case: Hλ has only one component, which however is not homeomorphic to a d-dimensional ball nor a point. Thus it is a connected compact d-dimensional submanifold of Rd with boundary not homeomorphic to Sd−1 . • If Rd \ Int(Hλ ) contains a compact connected component, then this arises as the projection on configuration space of a connected component of the regularized energy surface m−1 (λ). This flow-invariant component is compact too, and thus consists of trapped orbits. • If, however Rd \Int(Hλ ) does not contain a compact component, it necessarily is connected since d ≥ 2 and Hλ is compact. In this situation, the boundary ∂Hλ consists of one component, which is not homeomorphic to Sd−1 . In this situation Corollary 3.3 of [31] ensures the existence of a periodic so– called brake orbit, that is a trapped orbit in the terminology of our paper (although [31] treats smooth potentials, in the case at hand all singularities of our potential are repelling. Thus the dynamics at energy λ is unaffected by the singularities.). A converse of Proposition 5.1 does not hold true in general. That is, there are potentials like Yukawa’s potential V (x) = −e−|x| /|x| for which Hλ consists only of one point but still there are trapped orbits for small λ > 0, see [28]. Yet Proposition 5.1 gives us the Corollary 5.2. Consider for d = 2 or 3 a regular value λ > 0 of V . If N > 1 or if N = 1 and N > N , then p is trapping at energy λ. If N ≥ 2 then p is trapping at energy λ, for λ large enough. Proof. In all cases, Hλ has several connected components. Thus Proposition 5.1 gives the result. However one can find non-trapping situations as in Examples 5.3 and 5.4 below. # Example 5.3. Let N ∈ N∗ . Let V be defined on Rd \ S by V (x) = N j=1 fj /|x − sj | with fj > 0, for any 1 ≤ j ≤ N . It satisfies (1.2) with N = 0. For 0 < λ < # −1 , p is non-trapping at energy λ. ( N j=1 |sj |/fj )

808



Proof. Recall that a0 (x, ξ) = x · ξ. For (x, ξ) ∈ p−1 (λ),

N x − sj x − sj 2 {p, a0 }(x, ξ) = 2|ξ| + fj x · = |ξ| + λ + f j sj · 3 |x − s | |x − sj |3 j j=1 j=1 ⎞ ⎛ N N |sj |λ2 |s | j ⎠ ≥λ− = λ ⎝1 − λ > 0. f f j j j=1 j=1 2

N

Here we used that 0 < fj /|x−sj | ≤ λ, for (x, ξ) ∈ p−1 (λ). Now standard arguments yields the result (see the proof of Lemma 3.1, for instance). Example 5.4. Let λ, c, ρ > 0 and W ∈ C ∞ (Rd ; R) such that ∀α ∈ Nd ,

∃Cα > 0 ;

∀x ∈ Rd ,

|∂xα W (x)| ≤ Cα x−ρ−|α| .

Let V ∈ C ∞ (Rd \{0}; R) defined by V (x) = −c/|x|+W (x). Depending on c and λ, one can find small enough (Cα )|α|≤1 ’s such that p is non-trapping at energy λ. Proof. The function p˜ defined just before (2.24) takes the following form: p˜(t∗ ; z∗ ) = p˜(t, τ ; z, ζ) = |ζ|2 − c + |z|2 (W ◦ K(z) − τ ). Let b0 : T ∗ R × T ∗ Rd −→ R be defined by b0 (t∗ ; z∗ ) = ζ · z. Then {˜ p , b0 }(t∗ ; z∗ ) = 2˜ p(t∗ ; z∗ ) + 2c + |z|2 4τ − 4W ◦ K(z) − z · ∇z (W ◦ K)(z) . (5.4) Thanks to (2.19), we can choose the (Cα )|α|≤1 ’s small enough such that, for τ = λ > 0, the last term in (5.4) is everywhere non-negative. Thus, on p˜−1 (]−c/2; c/2[), {˜ p , b0 } ≥ c. This implies that, for any solution s → (t(s), λ; z(s), ζ(s)) of (2.24) leaving in p˜−1 (0), the function s → |z(s)|2 is strictly convex. It must go to infinity in both time s directions. By (2.25), this implies that any broken trajectory (φ(t; x∗ ))t∈R\coll(x∗ ) with p(x∗ ) = λ goes to infinity in both time t directions. Remark 5.5. By inspection of (5.4) we see that, for a potential of the form V (x) = f (x) |x| + W (x) with f (0) < 0 and meeting (1.1), no trapping occurs for high enough energies.

6. Scattering by a molecular potential We now show that our analysis can be applied to Example 1.3. # −1 ˆ = The potential x → of P1 (h0 ) is smooth on M j e0 zj |x − sj | 3 R \{s1 , . . . , sN } and satisfies (1.1). By local elliptic regularity (see [43], Thm. ˆ. IX.26), the electronic eigenfunctions ψk ∈ L2 (R3 ) of P1 (h0 ) are smooth on M 3 Furthermore, they are continuous on R (see [8], Thm. 2.4) and the corresponding eigenvalues Ek are negative by [12] (see also [8], Thm. 4.19). Using [2] outside the ball B := {x ∈ R3 ; |x| ≤ R0 } (cf. (1.1)), one can show that the ψk ’s decay exponentially. This means, for any k, that there exists ck , Ck > 0 such that x ∈ B =⇒ |ψk (x)| ≤ Ck e−ck |x| .

(6.1)

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By (1.1), the result in [2] can be applied to the derivatives of the ψk outside B. Thus (6.1) holds true for these derivatives with possibly different constants ck , Ck . For any j ∈ {1, . . . , N }, it turns out that ψkj : R4 z → ψk (sj + K(z)), with K(z) defined in (2.19), is smooth near z = 0. Indeed, we can show as in [13] (see also the proof of (4.25) in Lemma 4.8) that the equation P1 (h0 )ψk = Ek ψk can be lifted to a Schr¨ odinger equation in R4 with smooth potential solved by the function ψkj . Again, the elliptic regularity gives the desired result. Therefore the charge densities ˆ and continuous on Rd . The ρk and their derivatives ρk := |ψk |2 are smooth on M satisfy (6.1). For any j ∈ {1, . . . , N }, ρkj : R4 z → ρk (sj + K(z)) is smooth near z = 0. This allows us to obtain the following properties for the Wk . ˆ , continuous Proposition 6.1. Let k ∈ {1, . . . , K}. The potential Wk is smooth on M 3 on R , and satisfies (1.1). For any j ∈ {1, . . . , N }, the function Wkj : R4 z → Wk (sj + K(z)), with K(z) defined in (2.19), is smooth near z = 0. Proof. Since | · |−1 ∈ L1 (R3 ) + L∞ (R3 ), ρk ∈ L1 (R3 ), and ρk is continuous, Wk is ˆ , and consider a well defined and continuous on R3 . Let j ∈ {1, . . . , N }, y ∈ M #N 3 ∞ partition of unity in R of the form j=0 χj = 1 with χj ∈ C0 and χj = 1 near sj , for j ≥ 1, and χ0 = 1 near y. Denoting by ∗ the convolution product, we can write near y, for any k and any multiindex α ∈ Nd ,

Dxα Wk

=

Dxα (ρk

−1

∗ |·|

)=

N

(ρk χj ) ∗ Dxα | · |−1 + Dxα (ρk χ0 ) ∗ | · |−1

(6.2)

j=1

(as distributions). This defines a continuous function near y. Using the exponential decay of the functions ρk , we can show that Wk satisfies (1.1). Let j ∈ {1, . . . , N }. We want to show that the function R4 z → (ρk ∗ −1 | · | )(sj + K(z)) is a constant times the function R4 z → (ρkj ∗| · |−2 )(z). Notice that, for an f ∈ C(R4 ) ∩ L1 (R4 ), f ∗ | · |−2 is a well defined continuous function since | · |−2 ∈ L1 (R4 ) + L∞ (R4 ). Now, it is convenient to view R4 as the quaternion space H and to use the representation of K on this space (see the appendix). In particular, one can use formula (3) from [13], saying that for x := K(Y ), Y ∈ H, |Y |2 dY = c · dx dθ for some constant c > 0 (dθ is uniquely defined by (2.20), compare also with the group action (A.4)). Then, using Lemma 6.2 below, we get ρk (sj + K(Y )) 2 −2 |Y | dY (ρkj ∗ | · | )(Z) = |Y − Z|2 4 R ρk (sj + x) = c· dx dθ |Y (x, θ) − Z|2 3 1 R ×S ρk (sj + x) = c · dx |x − K(Z)| 3 R ρk (sj + x) dx = c · |s + x − sj − K(Z)| 3 j R = c (ρk ∗ | · |−1 ) sj + K(Z) ,

810



for Z ∈ H, and c > 0. Now, since ρkj is smooth near 0 and | · |−1 is smooth away from 0, we can use a formula similar to (6.2) to show that ρkj ∗ | · |−2 is smooth near 0. Lemma 6.2. For X, Z ∈ H with K(Z) = K(X) 2π | exp(I1 θ)Z − X|−2 dθ = 2π · |K(Z) − K(X)|−1 . 0

Proof. Assuming the condition, both sides are well-defined. Then, using the definition of the real part of a quaternion (see appendix) 2π | exp(I1 θ)Z − X|−2 dθ 0

2π

|Z| + |X| − 2Re 2

=

0

0

2π

2π

= =

2

cos(θ) + I1 sin(θ) ZX ∗

−1 dθ

−1 |Z|2 + |X|2 − 2 Re(ZX ∗ ) cos(θ) + Re(I1 ZX ∗ ) sin(θ) dθ |Z|2 + |X|2 − 2

& −1 2 2 Re(ZX ∗ ) + Re(I1 ZX ∗ ) cos(ψ) dψ

0

2 2 −1/2 = 2π · (|Z|2 + |X|2 )2 − 4 Re(ZX ∗ ) − 4 Re(I1 ZX ∗ ) = 2π · |Z ∗ I1 Z − X ∗ I1 X|−1 = 2π · |K(Z) − K(X)|−1 , the last two equations being due to (A.5) and (A.3).

Now we are able to explain why the proof of our results can be adapted to treat the potential V defined in (1.13). In the proof of the necessity of the non-trapping condition in Section 3, the results away from the singularities work since V satisfies (1.1). Since the Wkj are smooth near 0, the results in [13] (see Lemmata 3.3 and 3.4) are still valid. Since each Wk is bounded, it is small compared to a repulsive potential +| · −sj |−1 near the corresponding repulsive singularity sj . So Section 3.3 is also valid. In the proof of the converse in Section 4, the results away from the singularities hold true since (1.1) is still valid. The fact that the Wk is small compared to the size of a singular potential ±| · − sj |−1 near the corresponding singularity sj explains why Section 4.2 works and also the validity of (4.18). The fact that the Wkj are smooth near 0, ensures that Lemma 4.8 still works.

Appendix A. The Hopf map We use the following notation for the quaternion algebra over R:

w1 −w2 w ,w ∈ C ∼ H := = R4 w ¯2 w ¯1 1 2

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with matrix multiplication, and 1 (I0 , I1 , I2 , I3 ) := 0

811

basis

0 i 0 0 1 0 −i , , , . 1 0 −i −1 0 −i 0

The direct sum decomposition H = R · 1l ⊕ ImH with ImH := {Z ∈ H | Z 2 = λ · 1l with λ ≤ 0} = SpanR (I1 , I2 , I3 ) into real and imaginary space is orthogonal w.r.t. the inner product H × H → R,

X, Y :=

1 tr(XY ∗ ) , 2 1

¯ t being the conjugation. The norm |X| := X, X 2 is multiplicative: X → X ∗ := X |XY | = |X| |Y | (X, Y ∈ H) . The real part of a quaternion equals Re(X) := 12 tr(X). See, e.g., [10] for more information on H. The Hopf map equals

w1 w ¯1 − w2 w ¯2 −2w ¯1 w2 ∗ K : H → ImH , K(Z) := Z I1 Z = i −2w1 w ¯2 w2 w ¯2 − w1 w ¯1

(A.3)

which is a surjection R4 → R3 whose preimages are the orbits of the isometric group action α0 : S 1 → Aut(H) , α0 (θ)(Z) := exp(θI1 )Z . (A.4) This action is free on H \ {0}. We call K the Hopf map, since its restriction to S 3 is the Hopf fibration S 3 −→ S 2 with fibre S 1 . Writing w1 := z0 + iz3 , w2 := z2 + iz1 we get formula (2.19) in the basis (I1 , I2 , I3 ) of ImH. Finally we prove the formula ' 2 2 (A.5) |Z ∗ I1 Z − X ∗ I1 X| = (|Z|2 + |X|2 )2 − 4 Re(ZX ∗ ) + Re(I1 ZX ∗ ) used in Section 6. Notice that, for all A, B ∈ H, Re(Ik A∗ ) = −Re(Ik A), Re(A∗ ) = Re(A), Re(A∗ A) = |A|2 , and Re(AB) = Re(A)Re(B) −

3

Re(Ik A)Re(Ik B) .

(A.6)

k=1

Setting A := I1 ZX ∗ and B := I1 XZ ∗ in (A.6), we get 2 2 Re (I1 ZX ∗ )(I1 XZ ∗ ) = − Re(XZ ∗ ) − Re(I1 XZ ∗ ) 2 2 + Re(I2 XZ ∗ ) + Re(I3 XZ ∗ ) . #3 2 Similarly it follows from (A.6) that |A|2 = k=0 (Re(Ik A)) , so that for ∗ A := ZX , 2 2 2 2 |Z|2 |X|2 = |A|2 = Re(ZX ∗ ) + Re(I1 ZX ∗ ) + Re(I2 XZ ∗ ) + Re(I3 XZ ∗ ) .

812



So |Z ∗ I1 Z − X ∗ I1 X|2 = (Z ∗ I1 Z − X ∗ I1 X)(−Z ∗ I1 Z + X ∗ I1 X) = Z ∗ I1 (−|Z|2 )I1 Z + X ∗ I1 (−|X|2 )I1 X + (Z ∗ I1 ZX ∗ I1 X) + (Z ∗ I1 ZX ∗ I1 X)∗ = |Z|4 + |X|4 + 2Re (I1 ZX ∗ )(I1 XZ ∗ ) 2 2 = |Z|4 + |X|4 + 2 − Re(XZ ∗ ) − Re(I1 XZ ∗ ) 2 2 + Re(I2 XZ ∗ ) + Re(I3 XZ ∗ ) 2 2 . = (|Z|2 + |X|2 )2 − 4 Re(ZX ∗ ) + Re(I1 ZX ∗ ) This proves the claim.

Acknowledgements The authors would like to thank the anonymous referee for numerous valuable remarks and suggestions, that helped to improve the clarity of this text, as well as for pointing out reference [53].

References [1] R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd edition, Benjamin, 1978. [2] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schr¨ odinger operators, Mathematical Notes, 29. Princeton, New Jersey: Princeton University Press, (1982). [3] W. O. Amrein, A. Boutet de Monvel, I. V. Georgescu, C0 -groups, commutator methods and spectral theory of N -body hamiltonians, Birkh¨ auser, 1996. [4] S. V. Bolotin, P. Negrini, Regularization and topological entropy for the spatial ncenter problem, Ergodic Theory and Dynamical Systems 21 (2001), 383–399. [5] N. Burq, Semiclassical estimates for the resolvent in non trapping geometries, Int. Math. Res. Notices 2002, no. 5 (2002), 221–241. [6] F. Castella, Th. Jecko, Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and C 2 potential, J. Diff. Eq. 228, N.2 (2006), 440–485. [7] K. Chantelau, Coulombic potentials in the semi-classical limit, Lett. Math. Phys. 19 (1990), 285–298. [8] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schr¨ odinger operators with applications to quantum mechanics and global geometry, Springer, 1987. [9] J. Derezi´ nski, C. Gérard, Scattering theory of classical and quantum N-particle systems, Springer, 1997. [10] H. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert, J. Ewing, Numbers, Springer, 1996.

Vol. 9 (2008)


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[11] G. B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, Princeton University Press, 1989. [12] R. G. Froese, I. Herbst, Exponential bounds and absence of positive eigenvalues for N -body Schr¨ odinger operators, Comm. Math. Phys. 87 (1982), 429–447. [13] C. Gérard, A. Knauf, Collisions for the quantum Coulomb Hamiltonian, Comm. Math. Phys. 143 (1991), 17–26. [14] C. Gérard, A. Martinez, Principe d’absorption limite pour des opérateurs de Schr¨ odinger ` a longue portée, C.R. Acad. Sci. 306 (1988), 121–123. [15] C. Gérard, A. Martinez, Semiclassical asymptotics for the spectral function of longrange Schr¨ odinger operators, J. of Funct. Anal. 85, no. 1 (1989), 226–254. [16] C. Gérard, I. M. Sigal, Space-time picture of semiclassical resonances, Comm. Math. Phys. 145 (1992), 281–328. [17] P. Gérard, Mesures semi-classiques et ondes de Bloch, Séminaire Equations aux ´ Dérivées Partielles, exp 6 (1990–1991), Ecole Polytechnique, Palaiseau. [18] P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71, no. 2 (1993), 559–607. [19] H. Gluck, E. Ziller, Periodic motions of conservative systems, In: Seminar on minimal submanifolds. Ed.: E. Bombieri. Ann. of math. Studies 103, 65–98, Princeton University Press, 1983. [20] B. Helffer, J. Sj¨ ostrand, Résonances en limite semi-classique, Bull. Soc. Math. France 114 (1986), nos. 24–25. [21] L. H¨ ormander, The analysis of linear partial differential operators III. Springer, 1985. [22] Th. Jecko, Estimations de la résolvante pour une mol´ ecule diatomique dans l’approximation de Born–Oppenheimer, Comm. Math. Phys. 195 (1998), 585–612. [23] Th. Jecko, Semiclassical resolvent estimates for Schr¨ odinger matrix operators with eigenvalues crossing, Math. Nachr. 257 (2003), 36–54. [24] Th. Jecko, From classical to semiclassical non-trapping behaviour, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 545–548. [25] Th. Jecko, Non-trapping condition for semiclassical Schr¨ odinger operators with matrix-valued potentials, Math. Phys. Electronic Journal, 11 (2005), no. 2. [26] T. Kato, Perturbation theory for linear operators, Springer, 1995. [27] S. Keraani, Wigner measures dynamics in a Coulomb potential, J. Math. Phys. 46, no. 6 (2005), 063512. [28] M. Klein, A. Knauf, Classical Planar Scattering by Coulombic Potentials, LNP m13, Springer, 1992. [29] M. Klein, A. Martinez, X. P. Wang, On the Born–Oppenheimer approximation of diatomic wave operators. II. Singular potentials, J. Math. Phys. 38 (3) (1997), 1337– 1397. [30] W. P. Klingenberg, Riemannian geometry, De Gruyter, 1995. [31] A. Knauf, Qualitative aspects of classical potential scattering, Regular and Chaotic Dynamics, 4 (1999), no. 1, 1–20. [32] A. Knauf, The n-centre problem of celestial mechanics for large energies, J. Eur. Math. Soc. 4 (2002), 1–114.

814



[33] A. Knauf, Coulombic periodic potentials: The quantum case, Annals of Physics 191 (1989), 205–240. [34] A. Knauf, M. Krapf, The non-trapping degree of scattering, arXiv:0706.3124 (2007). [35] A. Lahmar-Benbernou, A. Martinez, On Helffer–Sj¨ ostrand’s theory of resonances, Int. Math. Res. Not. 13 (2002), 697–717. ´ [36] G. Lebeau, Equations des ondes amorties, In A. Boutet de Monvel and V. Marchenko, editors, Algebraic and geometric methods in mathematical physics, p. 93–109, Kluwer Academic, The Netherlands, 1996. [37] P-L. Lions, T. Paul, Sur les mesures de Wigner, Revista Mat. Iberoamericana, 9 (1993), 553–618. [38] A. Martinez, An introduction to semiclassical and microlocal analysis, Universitext Springer, 2002. [39] A. Martinez, Resonance free domains for non globally analytic potentials, Ann. Henri Poincaré 4 (2002), 739–756. [40] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Commun. in Math. Phys. 78 (1981), 391–408. ´ [41] F. Nier, A semiclassical picture of quantum scattering, Ann. Sci. Ecole Norm. Sup. (4) 29, no. 2 (1996), 149–183. [42] B. Perthame, L. Vega, Morrey–Campanato estimates for Helmholtz equation, J. Funct. Anal. Vol. 164, N. 2 (1999), 340–355. [43] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II : Fourier Analysis, Self-adjointness, Academic Press, 1979. [44] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. IV: Analysis of operators, Academic Press, 1978. [45] D. Robert, Autour de l’approximation semi-classique, Birkh¨ auser, 1987. [46] D. Robert, Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results, Helv. Phys. Acta 71 1998, 44–116. [47] D. Robert, H. Tamura, Semiclassical estimates for resolvents and asymptotics for total cross-section. Ann. IHP 46, 415–442 (1987). [48] H. Seifert, Periodische Bewegungen mechanischer Systeme. Math. Zeitschrift 51, 197– 216 (1948). [49] E. L. Stiefel, G. Scheifele, Linear and regular celestial mechanics. Grundlehren der Mathematischen Wissenschaften, Band 174, Berlin, Heidelberg, New York: Springer, 1975. [50] A. Vasy, M. Zworski, Semiclassical estimates in asymptotically euclidean scattering. Comm. Math. Phys. 212, no. 1, 205–217 (2000). ´ [51] X. P. Wang, Etude semi-classique d’observables quantiques. Ana. Fac. Sci. Toulouse, Math. (5) 7, 101–135 (1985). [52] X. P. Wang, Semiclassical resolvent estimates for N -body Schr¨ odinger operators. J. Funct. Anal. 97, 466–483 (1991). [53] X. P. Wang, Microlocal estimates of the Schr¨ odinger equation in semi-classical limit. Preprint available from www.math.sciences.univ-nantes.fr/~wang, 2006. [54] X. P. Wang, P. Zhang, High frequency limit of the Helmholtz equation with variable refraction index. J. Funct. Anal. 230, 116–168 (2006).

Vol. 9 (2008)


Fran¸cois Castella IRMAR & IRISA – Université de Rennes 1 Campus de Beaulieu F-35042 Rennes Cedex France e-mail: [email protected] Thierry Jecko IRMAR – Université de Rennes 1 Campus de Beaulieu F-35042 Rennes Cedex France e-mail: [email protected] Andreas Knauf Mathematisches Institut Universit¨ at Erlangen-N¨ urnberg Bismarckstr. 1 1/2 D-91054 Erlangen Germany e-mail: [email protected] Communicated by Christian Gérard. Submitted: March 19, 2007. Accepted: March 3, 2008.

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Ann. Henri Poincaré 9 (2008), 817–834 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/040817-18, published online June 6, 2008 DOI 10.1007/s00023-008-0373-9


Eigenvalue Inequalities in Terms of Schatten Norm Bounds on Differences of Semigroups, and Application to Schrödinger Operators Michael Demuth and Guy Katriel∗ Abstract. We develop a new method for obtaining bounds on the negative eigenvalues of self-adjoint operators B in terms of a Schatten norm of the difference of the semigroups generated by A and B, where A is an operator with non-negative spectrum. Our method is based on the application of the Jensen identity of complex function theory to a suitably constructed holomorphic function, whose zeros are in one-to-one correspondence with the negative eigenvalues of B. Applying our abstract results, together with bounds on Schatten norms of semigroup differences obtained by Demuth and Van Casteren, to Schr¨ odinger operators, we obtain inequalities on moments of the sequence of negative eigenvalues, which are different from the Lieb–Thirring inequalities.

1. Introduction Let A be a self-adjoint operator on a complex Hilbert space, whose spectrum is non-negative. If B is another self-adjoint operator, such that the difference Dt = e−tB − e−tA of the semigroups corresponding to A, B belongs to a Schatten ideal (trace class or Hilbert–Schmidt class), we will prove inequalities which provide bounds from above on the negative eigenvalues of B, in terms of Schatten norms of Dt . The usefulness of such results follows from the fact that for concrete operators, for example when B is a Schr¨ odinger operator B = −Δ + V , and A is the free Schr¨ odinger operator A = −Δ, it is known that, under appropriate conditions on the potential V , Dt belongs to a Schatten ideal, and explicit bounds on the Schatten norm of Dt are available [3]. Indeed such results are important in the study of the absolutely continuous spectrum of the perturbed operator B. The theorems proven here show that these bounds on the Schatten norms of Dt can also be used in the study of the discrete spectrum of B. ∗ Partially

supported by the Minerva Foundation (Germany).

818

M. Demuth and G. Katriel


The method used to prove our results is based on constructing a holomorphic function whose zeros are in one-to-one correspondence with the negative eigenvalues of B, and using complex function theory to bound these zeros. Specifically we will use the Jensen identity (see, e.g., [6], p. 307): Lemma 1. Let Ωr be an open disk centered at 0 and with radius r. Let h : U → C be a holomorphic function on the open set U , where Ωr ⊂ U , and assume h(0) = 1. Then ⎞ ⎛ 2π r r n(u) 1 ⎠= du , log |h(reiθ )| dθ = log ⎝ 2π 0 |z| u 0 ¯ z∈Ωr ,h(z)=0

¯ u. where n(u) (0 ≤ u ≤ r) denotes the number of zeros of h in Ω In Section 2 we prove general theorems which give bounds on the moments (sums of powers) of the sequence of negative eigenvalues of an operator B in terms of the trace norm of the semigroup difference. In Section 3 we prove analogous bounds in terms of the Hilbert–Schmidt norm of the semigroup difference. In Section 4 we apply the theorems of Section 3 to derive inequalities for the negative eigenvalues of Schr¨ odinger operators under some conditions on the potential, which are different from the well-known Lieb–Thirring inequalities.

2. Eigenvalue inequalities in terms of trace-norm bounds on semigroup differences In this section we will prove results under the assumption that A,B are selfadjoint operators, with the spectrum of A non-negative, and such that the difference of semigroups Dt = e−tB − e−tA is of trace class. This implies that the negative spectrum of B, which we denote by σ − (B) = σ(B) ∩ (−∞, 0) , consists only of eigenvalues, which can accumulate only at 0 (of course compactness of Dt is sufficient for this property). We shall denote by N (−s) the number of eigenvalues λ of B which satisfy λ < −s. We begin by proving identities expressing the moments of the negative eigenvalues of the operator B in terms of an integral. It should be noted that the identities hold also in the case that one side is infinite – which implies that the other side is infinite too. Theorem 1. Let A,B be self-adjoint in a complex Hilbert space H, with σ(A) ⊂ [0, ∞). Assume that D = e−B − e−A is of trace class. Then, for any γ > 1, we have 2π

γ(γ − 1) 1 1 γ γ−2 | log(r)| |λ| = log |Det I − F (reiθ ) | dθdr , (1) 2π 0 r 0 − λ∈σ (B)

where F (z) is the operator-valued function defined by F (z) = z[I − ze−A ]−1 D ,

(2)

Vol. 9 (2008)

Semigroup Differences and Eigenvalues

819

and for γ = 1 we have 2π

1 |λ| = lim log |Det I − F (reiθ ) | dθ . r→1 2π 0 −

(3)

λ∈σ (B)

Proof. We have, for all z ∈ C I − ze−B = I − ze−A − zD ,

(4)

and if also |z| < 1, so that ze−A < 1 then I − ze−A is invertible, so that F (z) given by (2) is well defined, and we can write (4) as [I − ze−A ]−1 [I − ze−B ] = I − F (z) . Thus we have the following equivalence for |z| < 1: 1 log(z) ∈ σ(B) ⇔ ∈ σ(e−B ) ⇔ 1 ∈ σ F (z) , z so that

σ − (B) = log(z) | |z| < 1, 1 ∈ σ F (z) .

(5)

Since we assume D is of trace class, then so is F (z). We note also that F (0) = 0 .

(6)

Since F (z) is a trace class operator, the determinant h(z) = Det I − F (z) is well defined, and we have that h is holomorphic in the unit disk and h(z) = 0 ⇔ 1 ∈ σ F (z) ⇔ log(z) ∈ σ − (B) . Thus

σ − (B) = so that, for all s > 0,

log(z) | |z| < 1, h(z) = 0

,

N (−s) = n(e−s ) , (7) where n(r) denotes the number of zeros of h in Ωr = {z | |z| < r}. By (6) we have h(0) = Det I − F (0) = Det(I) = 1 .

Applying the Jensen identity, Lemma 1, we have, for any 0 < r < 1, 2π r 1 n(u) du , log |h(reiθ )| dθ = 2π 0 u 0

(8)

and making the substitution u = e−s in the integral on the right-hand side of (8) and using (7) we get 2π ∞ 1 iθ log |h(re )| dθ = N (−s)ds . (9) 2π 0 log( 1r ) We now recall the well-known identity

|λ|γ = γ λ∈σ− (B)

0

∞

sγ−1 N (−s)ds .

(10)

820


Taking γ = 1, (10) becomes

|λ| =

λ∈σ− (B)


∞

N (−s)ds .

0

Taking the limit r → 1 in (9), we have 2π ∞ 1 iθ lim log |h(re )| dθ = N (−s)ds . r→1 2π 0 0

(11)

(12)

From (11) and (12), we conclude

1 r→1 2π

|λ| = lim

λ∈σ− (B)

0

2π

log |h(reiθ )| dθ ,

so that we have (3). We now assume that γ > 1. Multiplying (9) by 1r | log(r)|γ−2 and integrating over r ∈ [0, 1], we obtain 1 ∞ 1 2π 1 1 1 | log(r)|γ−2 log |h(reiθ )| dθdr = | log(r)|γ−2 N (−s)dsdr 2π 0 0 r 0 r log( 1r ) 1 ∞ 1 | log(r)|γ−2 drds N (−s) = 0 e−s r ∞ 1 N (−s)sγ−1 ds , = γ−1 0 which, together with (10), implies

γ(γ − 1) 1 2π 1 γ | log(r)|γ−2 log |h(reiθ )| dθdr , |λ| = 2π r 0 0 − λ∈σ (B)

so that we have (1).

By bounding the function h of Theorem 1 from above, we obtain bounds on the moments of the negative eigenvalues. Theorem 2. Let A,B be self-adjoint in a complex Hilbert space H, with σ(A) ⊂ [0, ∞). Assume that D = e−B − e−A is of trace class. Then for any γ > 1, 2π

γ(γ − 1) 1 |λ|γ ≤ | log(r)|γ−2 [I − reiθ e−A ]−1 Dtr dθdr , (13) 2π 0 0 − λ∈σ (B)

and for γ = 1 we have

λ∈σ− (B)

|λ| ≤ lim sup r→1

1 2π

0

2π

[I − reiθ e−A ]−1 Dtr dθ .

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Proof. We recall the general inequality for trace class operators T (see, e.g., [7]) |Det(I − T )| ≤ eT tr ,

(14)

which gives log |Det I − F (reiθ ) | ≤ F (reiθ )tr = r[I − reiθ e−A ]−1 Dtr . Substituting this inequality into (1), (3), we obtain the results.

Bounding the integral on the right-hand side of (13), we get the following theorem. Although we shall later prove a stronger result, Theorem 4, it is useful to present Theorem 3, whose proof is more straightforward, and for which the coefficient in the inequalities can be evaluated explicitly, in terms of Euler’s Γfunction and Riemann’s ζ-function. Theorem 3. Let A,B be self-adjoint in a complex Hilbert space H, with σ(A) ⊂ [0, ∞). Assume that, for some t > 0, Dt = e−tB − e−tA is of trace class. Then, for any γ > 2, we have the inequality

1 |λ|γ ≤ Γ(γ + 1)ζ(γ − 1) γ Dt tr , (15) t − λ∈σ (B)

and the right-hand side is finite. Proof. We note first that it suffices to prove (15) for t = 1, that is, setting D = D1 = e−B − e−A , to prove

|λ|γ ≤ Γ(γ + 1)ζ(γ − 1)Dtr , (16) λ∈σ− (B)

since (15) follows from (16) by replacing A, B by tA, tB. Since σ(A) ⊂ [0, ∞), we have e−A ≤ 1, so that, for |z| < 1, 1 , [I − ze−A ]−1 ≤ 1 − |z| hence F (reiθ )tr = r[I − reiθ e−A ]−1 Dtr ≤ r[I − reiθ e−A ]−1 Dtr r . ≤ Dtr 1−r From the inequality (13) of Theorem 2 we thus have 2π

γ(γ − 1) 1 |λ|γ ≤ | log(r)|γ−2 [I − reiθ e−A ]−1 Dtr dθdr 2π 0 0 λ∈σ− (B) 1 1 dr | log(r)|γ−2 ≤ γ(γ − 1)Dtr 1 − r 0 ∞ γ−2 x dx = γ(γ − 1)Dtr x−1 e 0 = Γ(γ + 1)ζ(γ − 1)Dtr (17) so we have (16).

822



A more refined estimate on the integral in (13) yields the following theorem, which is stronger than Theorem 3. We note that this theorem is valid for γ > 1, rather than γ > 2 as in Theorem 3. The value of the constant Ctr (γ) is given in the proof of the theorem, in terms of some integrals. Theorem 4. Let A,B be self-adjoint in a complex Hilbert space H, with σ(A) ⊂ [0, ∞). Assume that, for some t > 0, Dt = e−tB − e−tA is of trace class. Then, for any γ > 1, we have the inequality

1 |λ|γ ≤ Ctr (γ) γ Dt tr , (18) t − λ∈σ (B)

where Ctr (γ) is a finite constant depending only on γ. Proof. As noted in the proof of Theorem 3, it suffices to prove (18) for t = 1, that is, setting D = e−B − e−A , to prove

|λ|γ ≤ Ctr (γ)Dtr . (19) λ∈σ− (B)

Since σ(e

−A

) ⊂ [0, 1], we have

|z|[I − ze−A ]−1 = [z −1 I − e−A ]−1 1 ≤ minu∈[0,1] |z −1 − u| ⎧ 1 ⎪ Re(z −1 ) ≥ 1 ⎨ |z−1 −1| , 1 −1 ) 1,

|λ|γ λ∈σ− (B)

γ(γ − 1) ≤ 2π ≤

1

| log(r)|

γ−2

0

0

2π

[I − reiθ e−A ]−1 Dtr dθdr

arccos(r) 1 γ(γ − 1) 1 Dtr | log(r)|γ−2 dθdr 2 − 2r cos(θ) + 1 π r 0 0 1 1 π π2 1 γ−2 γ−2 + dθdr + | log(r)| | log(r)| dθdr . π 0 0 arccos(r) | sin(θ)| 2

(21)

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823

We estimate the integrals in (21) from above: making the substitution s = 1 , we have y = cos(θ) c1 (γ) =

1

| log(r)|

0

∞

= 1

≤

arccos(r)

γ−2

∞

1

= 1

∞

0 γ−2

(log(s)) √ s s2 + 1

(log(s))γ−2 √ s s2 + 1

s

1

1 r,

1 dθdr 2 r − 2r cos(θ) + 1

1 1 1 √ dyds 2−1 2s y y y − s2 +1

s

1 1 √ dyds 2s y −1 1 y − s2 +1 √ ( s(s + 1) + s2 + 1)2 (log(s))γ−2 √ ds log s−1 s s2 + 1

and since, for any > 0, the integrand in the last integral is O((s − 1)γ−2− ) as s → 1+ and O(s−2 ) as s → ∞, this integral is finite whenever γ > 1, so c1 (γ) is finite. 1 π2 1 γ−2 dθdr | log(r)| c2 (γ) = 0 arccos(r) | sin(θ)| 1 1 1+r = | log(r)|γ−2 log dr , 2 0 1−r and since, for any > 0, the integrand in the last integral is O(r1− ) as r → 0 and O((1 − r)γ−2− ) as r → 1, this integral is finite whenever γ > 1, so c2 (γ) is finite. Finally, we have 1 π π ∞ γ−2 −x π c3 (γ) = | log(r)|γ−2 dθdr = x e dx = Γ(γ − 1) . π 2 0 2 0 2 From (21) we thus obtain, for γ > 1,

1 |λ|γ ≤ γ(γ − 1) c1 (γ) + c2 (γ) + c3 (γ) Dtr . π − λ∈σ (B)

so that we have (19), with Ctr (γ) =

1 γ(γ − 1) c1 (γ) + c2 (γ) + c3 (γ) . π

One could ask what is the best constant Ctr (γ) in inequality (18), that is, given γ > 1, what is the smallest number Ctr (γ) for which (18) will hold for any pair of selfadjoint operators with σ(A) ⊂ [0, ∞). We do not know how to answer this question, but we can give a simple lower bound for the possible values of Ctr (γ). We recall that the Lambert W-function is defined on [−e−1 , ∞) as the inverse of the function f (x) = xex .

824



Proposition 1. If γ > 1 and Ctr (γ) is a constant for which Theorem 4 holds, then γ−1 . (22) Ctr (γ) ≥ −W (−γe−γ ) γ + W (−γe−γ ) Proof. If the inequality (18) holds then in particular it must hold when A and B are 1 × 1 matrices. Thus let A = 0, B = −b, (b > 0), t = 1. Then the moment of the negative eigenvalues of B of order γ is simply bγ , and D1 = e−B −e−A = eb −1. Thus inequality (18) becomes in this case bγ ≤ Ctr (γ)(eb − 1) . Since this must hold for all b > 0, we have Ctr (γ) ≥ sup b>0

bγ . eb − 1

(23)

By differentiating the function of b on the right-hand side of (23) we find its maximum on [0, ∞) to be given by the expression on the right-hand of (22). As an example, we take γ = 2. From (22) we obtain Ctr (2) ≥ 0.647.. Theorem 4 gives (evaluating the integrals numerically) Ctr (2) ≤ 2.5.. Using the above argument one can see that for γ < 1, Theorem 4 cannot be true. Indeed, if γ < 1, then the expression on the right-hand side of (23) goes to +∞ as b → 0, so that the supremum is infinite. We remark that the inequalities for the moments of eigenvalues derived here imply inequalities for the number of eigenvalues less than a given negative number −s (s > 0), which we denote by N (−s). Indeed since

|λ|γ ≥ |λ|γ ≥ sγ = sγ N (−s) , λ∈σ− (B)

λ∈σ(B)∩(−∞,−s)

λ∈σ(B)∩(−∞,−s)

we have, from (18), assuming that Dt is trace-class for all t > 0,

Ctr (γ) 1 N (−s) ≤ γ |λ|γ ≤ inf Dt tr . t>0,γ>1 s (st)γ λ∈σ(B)∩(−∞,0)

3. Eigenvalue inequalities in terms of Hilbert–Schmidt norm bounds on semigroup differences In this section we prove theorems analogous to those in the previous section, for the case in which the semigroup difference is Hilbert–Schmidt rather than trace class. The proofs are similar, the difference being that we have to get around the fact that the determinant is not defined for a general Hilbert–Schmidt perturbation of the identity. In the applications to Schr¨ odinger operators, it is easier to verify that the semigroup difference is Hilbert–Schmidt than to verify that it is trace class, so the theorems of this section will be used in these applications, to be presented in Section 4.

Vol. 9 (2008)


825

The following theorem is the Hilbert–Schmidt analog of Theorem 1. It should be noted, however, that unlike in Theorem 1, here we have only inequalities rather than identities. Theorem 5. Let A,B be self-adjoint in a complex Hilbert space H, with σ(A) ⊂ [0, ∞). Assume that D = e−B − e−A is Hilbert–Schmidt. Then we have, for any γ > 1, 2π

γ(γ − 1) 1 1 γ γ−2 iθ 2 | log(r)| |λ| ≤ log Det I − F (re ) dθdr . 2π 0 r 0 − λ∈σ (B)

(24) where F (z) is the operator-valued function defined by F (z) = z[I − ze−A ]−1 D , and for γ = 1

λ∈σ− (B)

1 r→1 2π

|λ| ≤ lim

2π 0

2 log Det I − F (reiθ ) dθ .

(25)

Proof. Like in the proof of Theorem 1, we have

σ − (B) = log(z) | |z| < 1, 1 ∈ σ F (z) . Since we assume D is Hilbert–Schmidt, then so is F (z), and this implies that (F (z))2 is trace class, so we can define the holomorphic function 2 , h(z) = Det I − F (z) and we have

2 1 ∈ σ F (z) ⇒ 1 ∈ σ F (z) ⇔ h(z) = 0 ,

and thus

σ − (B) ⊂ log(z) | |z| < 1, h(z) = 0 . (26) Since (26) is an inclusion rather than an equality as in (5), (7) is replaced by the inequality N (−s) ≤ n(e−s ) , Since F (0) = 0 we have h(0) = 1. Applying the Jensen identity, as in the proof of Theorem 1, we get the results. The next theorem is the Hilbert–Schmidt analog of Theorem 2. Theorem 6. Let A,B be self-adjoint in a complex Hilbert space H, with σ(A) ⊂ [0, ∞). Assume that D = e−B − e−A is Hilbert–Schmidt. Then, for any γ > 1, we have the inequality 2π

γ(γ − 1) 1 |λ|γ ≤ r| log(r)|γ−2 [I − reiθ e−A ]−1 D2HS dθdr . (27) 2π 0 0 − λ∈σ (B)

826



and for γ = 1 we have 2π

1 |λ| ≤ lim sup [I − reiθ e−A ]−1 D2HS dθ . 2π r→1 0 − λ∈σ (B)

Proof. Using (14), we have 2 2 log Det I − F (z) ≤ F (z)

tr

,

(28)

and since, for any Hilbert–Schmidt operator T we have T 2 tr ≤ T 2HS , we get 2 F (z) tr ≤ F (z)2HS . (29) From (28) and (29), together with (24), (25), we obtain the results.

The following theorem is the Hilbert–Schmidt analog of Theorem 4. Theorem 7. Let A,B be self-adjoint in a complex Hilbert space H, with σ(A) ⊂ [0, ∞). Assume that, for some t > 0, Dt = e−tB − e−tA is Hilbert–Schmidt. Then, for every γ > 2, we have the inequality

1 |λ|γ ≤ CHS (γ) γ Dt 2HS , (30) t − λ∈σ (B)

where CHS (γ) is a finite constant depending only on γ. Proof. We first note that it suffices to prove (30) with t = 1, that is, setting D = D1 = e−B − e−A , to prove

|λ|γ ≤ CHS (γ)D2HS , (31) λ∈σ− (B)

since (30) follows from (31) by replacing A, B by tA, tB. Using the inequality (20), we have [I − eiθ e−A ]−1 D2HS ≤ [I − eiθ e−A ]−1 2 D2HS ⎧ 1 cos(θ) ≥ r ⎨ r2 −2r cos(θ)+1 2 1 0 < cos(θ) < r ≤ DHS 2 (sin(θ)) ⎩ 1 cos(θ) ≤ 0 Therefore from inequality (27) of Theorem 6

|λ|γ λ∈σ− (B)

γ(γ − 1) ≤ 2π

1

2π

r| log(r)| [I − reiθ e−A ]−1 D2HS dθdr 0 0 arccos(r) 1 r 2 γ(γ − 1) ≤ Dt HS dθdr | log(r)|γ−2 2 − 2r cos(θ) + 1 π r 0 0 1 1 π π2 1 γ−2 γ−2 + r| log(r)| dθdr + r| log(r)| dθdr . 2 π 0 0 arccos(r) (sin(θ)) 2 γ−2

Vol. 9 (2008)


827

To verify that the above integrals are indeed finite for γ > 2, we estimate from above: arccos(r) 1 r γ−2 c4 (γ) = | log(r)| dθdr 2 − 2r cos(θ) + 1 r 0 0 ! 1 2r 1+r dr , = | log(r)|γ−2 arctan 2 1 − r 1−r 0 and since, for any > 0 the integrand is O(r1− ) as r → 0, and O((1 − r)γ−3 ) as r → 1, the integral is finite when γ > 2. 1 π2 1 γ−2 c5 (γ) = r| log(r)| dθdr 2 (sin(θ)) 0 arccos(r) = 0

1

r2 | log(r)|γ−2 √ dr , 1 − r2 5

and since, for any > 0, the integrand is O(r2− ) as r → 0, and O((1 − r)γ− 2 ) as r → 1, the integral is finite when γ > 32 . Finally, 1 π π ∞ −2x γ−2 c6 (γ) = r| log(r)|γ−2 dθdr = e x dx = π2−γ Γ(γ − 1) , π 2 0 0 2 finite for any γ > 1. From (27) we thus have, for γ > 2,

1 |λ|γ ≤ γ(γ − 1) c4 (γ) + c5 (γ) + c6 (γ) D2HS . π − λ∈σ (B)

so that (31) holds, with CHS (γ) =

1 π γ(γ

− 1)[c4 (γ) + c5 (γ) + c6 (γ)].

An argument involving one-dimensional operators, like in the end of the previous section, shows that Theorem 7 is not true if γ < 2.

4. Application to Schr¨ odinger operators We now apply our general results to the study of the discrete spectrum of Schr¨ odinger operators −Δ + V . Recall that the potential V : Rd → R is said to belong to the class K(Rd ) if t lim sup (eηΔ |V |)(x)dη = 0 . t→0 x∈Rd

0

V is said to belong to class K (R ) if χQ V ∈ K(Rd ) for any ball Q ⊂ Rd , where χQ denotes the characteristic function of Q. V is said to be a Kato potential if V− = min(V, 0) ∈ K(Rd ) and V+ = max(V, 0) ∈ K loc (Rd ). By the min-max principle, the eigenvalues of −Δ + V− are smaller then or equal to the corresponding eigenvalues of −Δ + V , and therefore we have

|λ|γ ≤ |λ|γ , (32) loc

λ∈σ− (−Δ+V )

d

λ∈σ− (−Δ+V− )

828



so that to bound the left-hand side of (32) it suffices to bound the right-hand side. We shall therefore take A = H0 = −Δ, B = H0 + V− , so that Dt = e−t(H0 +V− ) − e−tH0 . We quote the following bounds for the Hilbert–Schmidt norm of Dt ( [3], Theorem 5.7) Lemma 2. Assuming V− ∈ K(Rd ), we have 2 Dt HS ≤ 2t e−2t(H0 +V− ) (x, x)|V− (x)|dx . Rd

Lemma 3. Assuming V− ∈ K(Rd ), we have Dt 2HS ≤ t2 e−2t(H0 +V− ) (x, x)|V− (x)|2 dx . Rd

We also quote the following inequality (see [3], p. 66, in the proof of Theorem 2.9): Lemma 4. Assuming V− ∈ K(Rd ), we have 1 1 e−t(H0 +V− ) (x, y) ≤ e−t(H0 +2V− ) L2 1 ,L∞ e−tH0 (x, y) 2 .

Since e−tH0 (x, x) =

1 d

(4πt) 2

Dt 2HS ≤ Dt 2HS ≤

, Lemmas 2, 3 and 4 imply

2t (8πt) t2 (8πt)

1

d 4

e−2t(H0 +2V− ) L2 1 ,L∞ V− L1 ,

(33)

1

d 4

e−2t(H0 +2V− ) L2 1 ,L∞ V− 2L2 .

(34)

From (33) and Theorem 7 we have Theorem 8. Let V be a Kato potential, and assume also V− ∈ L1 (Rd ). We have the following inequality for any γ > 2, 1

λ∈σ− (−Δ+V

|λ|γ ≤ )

2CHS (γ) d

(8π) 4

e−2t(H0 +2V− ) L2 1 ,L∞

V− L1 inf

tγ+ 4 −1 d

t>0

.

Similarly, from (34) and Theorem 7 we have Theorem 9. Let V be a Kato potential, and assume also V− ∈ L2 (Rd ). We have the following inequality for any γ > 2,

λ∈σ− (−Δ+V )

1

|λ| ≤ γ

CHS (γ) d

(8π) 4

V− 2L2 inf t>0

e−2t(H0 +2V− ) L2 1 ,L∞ tγ+ 4 −2 d

.

Vol. 9 (2008)


829

In order to make the bounds given by Theorems 8, 9 more explicit we are going to bound e−2t(H0 +2V− ) L1 ,L∞ in terms of the quantity (c > 0) β(c) = (c − Δ)−1 V− L∞ .

(35)

We note that (see, e.g., [2], Lemma 4.2.4) V− ∈ K(R ) implies that d

lim β(c) = 0 .

(36)

c→∞

From [3], Proposition 2.2, we have Lemma 5. Assume V is a Kato potential. Then, for any c > 0 for which β(c) < 1, we have ect . e−t(H0 +V− ) L∞ ,L∞ ≤ 1 − β(c) Lemma 6. Let V be a Kato potential. If c > 0 is such that 4β(c) < 1 , then e−2t(H0 +2V− ) L1 ,L∞ ≤

(37)

ect . (4πt) 1 − 4β(c) 1

d 2

Proof. We have (as in [3], proof of Theorem 2.9): e−2t(H0 +2V− ) L1 ,L∞ ≤ e−t(H0 +2V− ) L1 ,L2 e−t(H0 +2V− ) L2 ,L∞ = e−t(H0 +2V− ) 2L2 ,L∞ ≤ e−t(H0 +4V− ) L∞ ,L∞ e−tH0 L1 ,L∞ 1 −t(H0 +4V− ) = L∞ ,L∞ . d e (4πt) 2

Using Lemma 5, we get the result. Using Lemma 6, Theorem 8 implies, for c satisfying (37),

λ∈σ− (−Δ+V )

2 4 +1 d

|λ|γ ≤

(8π)

d 2

CHS (γ)

t

1

1

e 2 ct

γ+ d 2 −1

[1 − 4β(c)] 2

1

V− L1 .

(38)

We can now minimize the expression on the right-hand side of (38) over t. Since 1 γ+ d2 −1 e 2 ct ec min = d t>0 tγ+ 2 −1 2γ + d − 2 we obtain Theorem 10. Let V be a Kato potential, and assume also V− ∈ L1 (Rd ). If c > 0 is such that 4β(c) < 1, then, for any γ > 2, γ+ d2 −1 d

ec 1 2 4 +1 γ |λ| ≤ 1 V− L1 . d CHS (γ) 2γ + d − 2 2 2 [1 − 4β(c)] (8π) − λ∈σ (−Δ+V ) Similarly, from Theorem 9 we obtain

830



Theorem 11. Let V be a Kato potential, and assume also V− ∈ L2 (Rd ). If c > 0 is such that 4β(c) < 1, then, for any γ > 2, γ+ d2 −2 d

ec 1 24 2 |λ|γ ≤ C (γ) HS 1 V− L2 . d 2γ + d − 4 2 2 [1 − 4β(c)] (8π) λ∈σ− (−Δ+V ) We note that (36) assures us that there always exists c > 0 with 4β(c) < 1, so that Theorems 10, 11 apply. The dependence on V− in Theorems 10, 11 is both through its L1 -norm and through the quantity β(c). The quantity β(c) can be written more explicitly by using the integral representation of (c − Δ)−1 , 1 d−2 (c − Δ)−1 V− (x) = c 2 G c 2 (x − y) V− (y)dy , Rd

where G(x) =

1 (2π)

d 2

K d −1 (|x|) 2

1 |x| 2 −1 d

,

in which K d −1 is the modified Bessel function of the third kind (see, e.g., [1]). 2 Thus 1 d−2 β(c) = c 2 sup G c 2 (x − y) |V− (y)|dy x∈Rd Rd 1 1 G(x − y)|V− (c− 2 y)|dy . (39) = sup c x∈Rd Rd We now introduce an apparently new norm on potentials, which is natural in this context, in terms of which we can derive some useful inequalities from Theorems 10, 11. For α > 0, we say that a measurable function W : Rd → R belongs to K α (Rd ) if W K α < ∞, where W K α = sup cα (c − Δ)−1 |W |L∞ c>0 1 α−1 = sup c G(x − y)|W (c− 2 y)|dy . x∈Rd ,c>0

(40)

Rd

K α (Rd ) is a normed space with the above norm, and we have K α (Rd ) ⊂ K(Rd ) for all α > 0. By the definition of the K α -norm and by (39) we have, when V− ∈ K α (Rd ) β(c) ≤ V− K α c−α , ∀c > 0 . (41) To see that K α (Rd ) is a sufficiently large class of functions, we note that Lemma 7. If d ≥ 3 and p > have, for all W ∈ Lp (Rd ),

d 2

then Lp (Rd ) ⊂ K α (Rd ), where α = 1 −

W K α ≤ Cd,p W Lp ,

d 2p ,

and we (42)

Vol. 9 (2008)


where

Cd,p =

Rd

|G(x)|

p p−1

831

p−1 p dx .

(43)

Proof. Using Hölder’s inequality we have d 1 G(x − y)|W (c− 2 y)|dy ≤ Cd,p c 2p W Lp = Cd,p c1−α W Lp , Rd

which, using (40), implies (42). We note that the fact that Cd,p is finite follows p d from the condition p > d2 , which implies p−1 < d−2 . Another fact, which shows that K α (Rd ) contains functions which are not in any Lp (Rd ) is Lemma 8. If W is measurable and |W (x)| ≤

A , |x|η

∀x ∈ Rd ,

where η ∈ (0, 2), then W ∈ K 2−η (Rd ), and d−η 1 1 η Γ W K 2−η ≤ d η+1 Γ 1 − A 2 2 π2 2 Proof. We have η 1 G(x − y)|W (c− 2 y)|dy ≤ Ac 2 G(x − y)|y|−η dy Rd Rd η 2 ≤ Ac G(y)|y|−η dy Rd d−η 1 1 η Γ = d η+1 Γ 1 − Acη−1 , 2 2 π2 2 where the second inequality follows from the fact that both G(x) and |x|−η are radially symmetric functions which are decreasing in |x|, so that their convolution is maximized at the origin. We now derive eigenvalue inequalities using the norms V− K α . From Theorem 10 and (41) we have γ+ d2 −1 ec d +1

2γ+d−2 24 |λ|γ ≤ CHS (γ) (44) 1 V− L1 . d (8π) 2 [1 − 4V− K α c−α ] 2 λ∈σ− (−Δ+V ) We now wish to minimize the right-hand side of (44) with respect to c. We compute 1

cγ+ 2 −1 d

min

1

c>(4V− K α ) α

1

[1 − 4V− K α c−α ] 2

2δ (2δ + 1)δ+ 2 = V− δK α , δδ

832


where δ=

1 α


d γ+ −1 . 2

Thus from (44) we get Theorem 12. Let V be a Kato potential, and assume also V− ∈ L1 (Rd ) ∩ K α (Rd ), where α > 0. Then, for any γ > 2,

|λ|γ ≤ κV− L1 V− δK α , λ∈σ− (−Δ+V )

where the constants are given by 1 d δ = δd,α,γ = γ+ −1 , α 2 κ = κd,α,γ = CHS (γ)

d 1 2 4 +1 2δ (2δ + 1)δ+ 2 e δα . d δδ 2δα (8π) 2

Similarly, using Theorem 11 we obtain Theorem 13. Let V be a Kato potential, and assume also V− ∈ L2 (Rd ) ∩ K α (Rd ), where α > 0. Then, for any γ > 2,

|λ|γ ≤ κV− 2L2 V− δK α , λ∈σ− (−Δ+V )

where the constants are given by 1 d δ = δd,p,γ = γ+ −2 , α 2

1 2δ (2δ + 1)δ+ 2 e δα = CHS (γ) . d δδ 2δα (8π) 2 d

κ = κd,α,γ

24

We particularize to the case in which d ≥ 3, V− ∈ Lp (Rd ), p > Lemma 7, Theorem 12, 13 imply

d 2.

Using

Corollary 1. Assume d ≥ 3. Let V be a Kato potential, and assume also V− ∈ L1 (Rd ) ∩ Lp (Rd ), where p > d2 . Then, for any γ > 2,

|λ|γ ≤ κV− L1 V− δLp , (45) λ∈σ− (−Δ+V )

where the constants are given by δ = δd,p,γ =

γ+

d 2

1−

−1 d 2p

,

(46) d

κ = κd,p,γ

1

2 4 +1 (2δ + 1)δ+ 2 = CHS (γ)(2Cd,p )δ d δδ (8π) 2

with Cd,p given by (43).

e 2δ(1 −

d δ(1− 2p )

d 2p )

,

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833

Corollary 2. Assume d ≥ 3. Let V be a Kato potential, and assume also V− ∈ L2 (Rd ) ∩ Lp (Rd ), where p > d2 . Then, for any γ > 2,

|λ|γ ≤ κV− 2L2 V− δLp , (47) λ∈σ− (−Δ+V )

where the constants are given by δ = δd,p,γ =

γ+

d 2

1−

−2 d 2p

, 1

d

κ = κd,p,γ

(2δ + 1)δ+ 2 = CHS (γ)(2Cd,p )δ d δδ (8π) 2 24

e 2δ(1 −

d δ(1− 2p )

d 2p )

,

with Cd,p given by (43). It is interesting to compare the inequalities given by Corollaries 1, 2 with a different bound on the moments of eigenvalues, given by the Lieb–Thirring inequalities [4, 5]. These state that

γ+ d |λ|γ ≤ Cd,γ V− γ+2 d , (48) L

λ∈σ− (−Δ+V )

2

holds for any γ ≥ 0 when d ≥ 3, for any γ > 0 when d = 2, and for any γ ≥ 12 when d = 1. Let us compare the bounds given by the inequalities when both of them are valid. The following argument shows that our inequality (45) and the Lieb–Thirring inequality are independent, in the sense that neither of them is stronger than the other: fixing γ > 2, p > d2 , if we take some potential W ∈ L1 (Rd ) ∩ Lp (Rd ) ∩ d Lγ+ 2 (Rd ), and define the family Vμ (μ > 0) by d

d

Vμ (x) = μ γ+ 2 W (μx) then, for any r > 0, d

Vμ− Lr = μ γ+ 2

d

−d r

W− Lr ,

hence Vμ−

γ+ d 2

d Lγ+ 2

= W−

γ+ d 2

d

Lγ+ 2

,

2dδ

Vμ− L1 Vμ− δLp = μ− (2γ+d)p W− L1 W− δLp , where δ is defined by (46). Thus the right-hand side of the inequality (45) is arbitrarily small for μ large and arbitrarily large for μ small, while the righthand side of (48) does not depend on μ, so that our inequalities are sometimes weaker and sometimes stronger than the Lieb–Thirring inequalities – depending on the potential V . In particular (45) is better than the bound given by the Lieb– Thirring inequality when μ is large. A similar conclusion holds with respect to the inequality (47) of Corollary 2.

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Acknowledgements We are grateful to M. Hansmann for his critical reading of the manuscript and helpful comments.

References [1] N. Aronszajn & K. T. Smith, Theory of Bessel potentials. I., Ann. Inst. Fourier 11 (1961), 385–475. [2] M. Demuth & M. Krishna, Determining Spectra in Quantum Theory, Birkh¨ auser (Boston), 2005. [3] M. Demuth & J. A. Van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration Approach, Birkh¨ auser (Basel), 2000. [4] A. Laptev & T. Weidl, Recent results on Lieb–Thirring inequalities, Journées ´ Equations aux dérivées partielles (2000), 1–14. [5] E. H. Lieb & W. Thirring, Inequalities for the moments of eigenvalues of the Schr¨ odinger Hamiltonian and their relation to Sobolev inequalities, Studies in Math. Phys., Essays in honor of Valentine Bargmann, Princeton, 269–303 (1976). [6] W. Rudin, Real and Complex Analysis, McGraw-Hill (New-York), 1987. [7] B. Simon, Trace Ideals and their Applications, London Math. Soc. Lecture Notes, 1979. Michael Demuth and Guy Katriel Institute of Mathematics Technical University of Clausthal D-38678 Clausthal-Zellerfeld Germany e-mail: [email protected] [email protected] Communicated by Christian Gérard. Submitted: September 4, 2007. Accepted: December 11, 2007.

Ann. Henri Poincaré 9 (2008), 835–880 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/050835-46, published online August 7, 2008 DOI 10.1007/s00023-008-0374-8


KAM Theory for Equilibrium States in 1-D Statistical Mechanics Models Rafael de la Llave Abstract. We extend the Lagrangian proof of KAM for twist mappings [34,51] to show persistence of quasi-periodic equilibrium solutions in 1-D statistical mechanics models. The interactions in the models considered here do not need to be of finite range but they have to decrease sufficiently fast with the distance (a high enough power suffices). In general, these models do not admit an interpretation as a dynamical system. Even when they do, the Hamiltonian description may be very singular or the number of degrees of freedom may be very large, so that the Hamiltonian KAM theory does not apply. We formulate the main result in an “a-posteriori” way. We show that if we are given a quasi-periodic function which solves the equilibrium equation with sufficient accuracy, which has a Diophantine frequency and which satisfies some non-degeneracy conditions, then, there is a true solution of the equilibrium equation which is close to the approximate solution. As an immediate consequence, we deduce that quasi-periodic solutions of the equilibrium equation with one Diophantine frequency persist under small modifications of the model. The main result can also be used to validate numerical calculations or perturbative expansions. Indeed, the method of proof lends itself to very efficient numerical implementations. We also show that some perturbative expansions (Lindstedt series) can be computed to all orders and that they converge.

1. Introduction The goal of this paper is to present a KAM theory for equilibrium solutions 1 dimensional models in statistical mechanics with long range interactions. We refer to Section 2 for a fuller description of the models and the physical interpretation of the solutions. We anticipate that these models appear naturally both in solid state physics and in dynamical systems. In the solid state interpretation, the models we consider can be thought of as describing a sequence of particles. The state of each particle is given by a real

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variable. Each particle interacts with all the other particles (but the interaction decreases with the distance). We allow many body interactions and interactions of unbounded range, but we require that the interactions are translation invariance and that they satisfy some periodicity condition (this periodicity is implied, for example if the variables are spin variables or phases, but it happens in other models). We are interested in equilibrium configurations. That is, solutions in which the net forces on each of the particles vanish. More precisely, we will seek quasi-periodic equilibrium solutions. Our main result, Theorem 1 is a KAM theorem, which, following [10, 11, 26, 41, 50, 51, 54] and many others, we will formulate in an “a posteriori” way. We will show that if there is an approximate solution of the equilibrium equations, which has an internal frequency with Diophantine properties, and which satisfies some non-degeneracy equations, then, we can conclude the existence of an exact solution with the same frequency, which is also close to the approximate solution we started with. As pointed in the above references, this formulation of the KAM results has several advantages. It immediately implies the usual formulation for quasiintegrable systems (take as approximate solution in the quasi-integrable system the exact solutions of the integrable system). A less immediate advantage, pointed out in [41, 54] is that an a-posteriori result for analytic problems immediately implies a result for finitely differentiable problems as well as bootstrap of regularity for sufficiently smooth solutions. We will not consider this result here since, in spite of being standard, requires to introduce notation and consider the dependence of several estimates of approximation theory on the dimension (HL is a function of L + 1 variables). For applications it is important to note that this a-posteriori formulations can be used to justify approximate solutions produced by non-rigorous methods such as numerical calculations or formal expansions. In relation to this, we show in Section 8.2 that the method of proof can be translated into very efficient numerical algorithms: The number of operations required by a Newton step for a discretization in N terms is O(N ln(N )). More importantly, the storage required is O(N ). Implementations of these algorithms have been done in [7]. The proof of Theorem 1 we present is based on the proof of [34]. It is based on the Lagrangian formalism. It is worth remarking that the results in [34,51] included assumptions that make the Lagrangian description equivalent to a Hamiltonian description. In contrast, several models to which we apply the results here do not have have a Hamiltonian description or even a formulation as a dynamical systems. Of course, the PDE models considered in [43] have a Lagrangian description but not a Hamiltonian one. Conversely, there are Hamiltonian versions of the KAM theorem [13] which do not have a Lagrangian counterpart. Even for models that admit both descriptions (For example, finite range models), the Hamiltonian description may have a singular dependence on parameters while the Lagrangian description is completely regular.

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This paper is organized as follows: In Section 2 we present the models that are going to be treated. In Section 2.1, we review models in the literature which are particular cases. In Section 3 we introduce definitions and collect some easy preliminary results (these preliminary results could be omitted in a first reading and used only as reference). In particular, in Section 3.1.1, we reformulate the equilibrium equations for quasi-periodic functions. This equation will be the centerpiece of our analysis. In Section 3.2 we introduce norms that we will use and collect some elementary estimates. In Section 4 we formulate the main result. In Section 5 we indicate the iterative step. We omit estimates and several details (such as verifying that the compositions indicated make sense). In Section 6 we go over the steps indicated, provide estimates and check that all the compositions indicated can be done. The fact that the step can be iterated and convergence is established in Section 6.3. In Section 7 we generalize an argument of [46] (more modern expositions appear in [16, 24]) to show that, under some mild conditions, for quasi-integrable systems, it is possible to show that there are perturbative expansions defined to all orders. Furthermore, by an argument of [42], these series converge. Somewhat surprisingly, the natural conditions we find for the existence of Lindstedt series to all order are slightly different from the conditions for convergence. In Section 8, we include a different approach for some cases based in normal hyperbolicity. We also argue that the method of proof leads to very efficient numerical algorithms. The implementation of this algorithms and results of the explorations is developed in [7]. 1.0.1. Relation with variational results. We point out that there is a variational theory for quasi-periodic solutions of models of the type we consider. For nextneighbor interactions, existence of these solutions, was one of the first results in Aubry–Mather theory [1, 35]. Generalizations to longer range interactions (and to higher dimensions) have been established in [8, 30, 32, 33]. As usual, the variational theory does not need Diophantine properties on the frequency and has very moderate regularity assumptions on the interaction. On the other hand, the variational theory requires convexity assumptions which are not needed in the KAM theory. The KAM theory also contains higher regularity in the conclusions. In the case that the model satisfies the convexity assumptions needed by the variational theory, one can use a Hilbert-functional argument to show that the KAM solutions considered here minimizers. Of course, there are models without convexity for which the KAM argument applies. For these models there could be KAM solutions but no minimizers.

2. Description of the models considered We will consider one dimensional chains of interacting particles. This is a very standard area in Mathematical Physics. See, for example [37]. As we will see,

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there are also models that have appeared motivated by Dynamical Systems or Numerical Analysis. We will assume that the state of each site is described by a real local variable. Hence, the configuration of a system is described by x ≡ {xi }i∈Z . We will find more convenient to think of x as a function x : Z → R. The particles at different sites interact with all the other ones. This interaction is described, as standard in Statistical Mechanics [47] by a formal functional S(x), which is obtained by assigning a energy to every finite subset of Z. We will assume that the interaction is invariant under translation. Hence, we will only consider models of the form HL (xk , . . . , xk+L ) . (1) S {xn }n∈Z = k∈Z L∈N

We will find it convenient to denote the arguments of HL starting from 0 so that we can write ∂x∂k+j HL (xk , . . . , xk+L ) = ∂j HL (xk , . . . , xk+L ). This will simplify the notations and subsequent calculations. In the case that HL ≡ 0 when L > R, we say that the interaction is of finite range R. The sum in (1) is not meant to converge, even in the case of finite range interactions. We will seek configurations (i.e. sequences {xi }i∈Z ) which are solutions of the Euler–Lagrange equations of the formal functional S. ∂xi S(x) = 0 i ∈ Z . For models of the form (1), the equations (2) are just 0= ∂j HL (xk , . . . , xk+j , . . . , xk+L ) ∀i ∈ Z . L

(2)

(3)

k+j=i j=0,...,L

In contrast with (1), which is not meant to converge, we will make assumptions on the decay of the interactions with the distance which imply that the R. H. S. of (3) converges uniformly, and, indeed, with some uniformity and some derivatives. We will postpone the description of the precise conditions till Section 4, when we have developed some notations about convergence. Here, we just note that, when the interactions are finite range, for a fixed k the RHS of (3) is just a finite sum. Our study will be based exclusively on the equations (3) and the functional S will not play any role except for motivation. We will use very strongly that the equations (3) have a variational structure coming from a translation invariant functional. Another assumption that we will make on the models is that they are invariant under a change of phase. We will assume that HL (x0 , . . . , xL ) = HL (x0 + 1, . . . , xL + 1) .

(4)

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In the case that the variables xi have the interpretation of a phase (e.g. when xi are spins and the model is a model of ferromagnetism or when the xi are the phases of an oscillator) it is natural to assume that HL (x0 , . . . , xL ) = HL (x0 + j0 , . . . , xL + jL ) for any j0 , . . . , jL ∈ Z. This, of course, clearly implies (4). In the dynamical systems interpretation, assumption (4) appears naturally as a periodicity of the phase space. The final assumptions that we will make to get our results, are non-degeneracy assumptions on the approximate solutions. We emphasize that these non-degeneracy assumptions are not just assumptions on the model, but they need to be verified for the approximate solutions. The non-degeneracy assumption means that, in a neighborhood of the solution considered, a mean field model is ferromagnetic or antiferromagnetic. In the dynamical systems interpretation, the averaged system has twist. Of course, if the original system has some global ferromagnetism, then the assumption is easy to verify but, even in the case that the system does not have global ferromagnetism (or twist) assumptions, it could be possible to find approximate solutions which satisfy our assumption. We will illustrate them in examples in the next section before formulating them in general. 2.1. Examples and different motivations The models of the form (1) generalize at the same time, several models that have been considered in the literature under different physical or mathematical motivations. In this section, we will present some of these models. We hope that this can be used as a motivation for some of the assumptions of our main results, which have appeared under different names. 2.1.1. Frenkel Kontorova model. The most famous example of models of the form (1) is, perhaps, the Frenkel–Kontorova model 1 (xi+1 − xi − a)2 + V (xi ) (5) S(x) = 2 i λ where V (t + 1) = V (t). For example, V (t) = 2π cos(2π). This model is a particular case of (1) taking H0 (t) = V (t), H1 (x, y) = 12 (y − x − a)2 and HL ≡ 0 for L ≥ 2. The Euler–Lagrange equations for this model are

xi+1 + xi−1 − 2xi + V (xi ) = 0 .

(6)

The model (5) was introduced in [19] as a rather crude microscopic theory of plasticity due to dislocations. More modern studies [9] realized that dislocations have a long range effect and that it is more realistic to consider interactions of many sites. The model (5) has been considered, e.g. in [1] as a model of deposition of material over a periodic 1-dimensional substratum. The xi denote the position of

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the i particle and the term 12 (xi+1 − xi − a)2 models the energy of interaction of neighboring particles, and the term V (xi ) models the energy of interaction with the substratum. Again, the consideration of longer range interactions is natural. One can consider (5) as an approximation to the XY model of ferromagnetism, which we will discuss in the next section. Some recent surveys of results on these models are [5, 6]. We refer to them for catalogues of the solution studied. 2.1.2. The Heisenberg XY model. In this interpretation, xi are angles that describe the orientation of the spin si = (cos(2πxi ), sin(2πxi )) The Heisenberg models contains two effects: interaction with an external field λ B = ( 2π , 0) and exchange interactions between nearest neighbors. In the Heisenberg model, the form of the interaction is H0 (xi ) = si · B = H1 (xi , xi+1 ) = si · si+1

λ cos(2πxi ) 2π = cos 2π(xi+1 − xi ) .

The equilibrium equations of the XY model are 2π sin 2π(xi+1 − xi ) − 2π sin 2π(xi − xi−1 ) − λ sin(2πxi ) = 0 .

(7)

(8)

When |xi+1 −xi | is small, so that cos(2π(xi+1 −xi )) ≈ 1− 12 (2π)2 (xi+1 −xi )2 , we recover the model (5). We note that since the exchange interaction is a Coulomb interaction, it is more realistic to consider also longer range effects such as dipole interactions [38]. We note for future reference that ∂1 ∂2 H1FK (xi , xi+1 ) = −1 < 0

∂1 ∂2 H1Heis (xi , xi+1 ) = −(2π)2 cos 2π(xi − xi+1 ) . In the first case, we have a definite sign for the mixed derivatives but in the second case we do not. This is an important motivation for our formulation of the non-degeneracy condition on the solutions. The ground states of the classical model without any external field but with boundary conditions have been studied in [20]. The quasi-periodic solutions we study are spin waves. For a very complete bibliographic treatment of models in 1-dimension we refer to [37]. 2.2. Twist mappings It is well known in Hamiltonian mechanics that orbits of a twist map of the annulus A ≡ T×R can be identified with critical points of the functional given by the formal sum S({xn }) = S(xn , xn+1 ) (9) n∈Z

where S is the so-called generating function of the map.

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The standard assumptions for the generating functions of the twist mappings of the annulus are: ∂x ∂y S(x, y) ≤ C < 0

(10)

S(x + 1, y + 1) = S(x, y) .

(11)

The condition (10) is called the twist condition in dynamical systems, it corresponds to the ferromagnetism condition considered. The condition (11) corresponds to the fact that the configuration space is a circle and the phase space is an annulus. See (13). The equilibrium equations in this case are just ∂1 S(xn , xn+1 ) + ∂2 S(xn−1 , xn ) = 0 .

(12)

Setting yn = ∂1 S(xn , xn+1 ) we have that the fact that xn satisfies (12) is equivalent to saying {(xn , yn )}n∈Z is an orbit of the mapping T defined by y = ∂x S(x, x ˜) (˜ x, y˜) = T (x, y) ⇔ (13) ˜) . y˜ = −∂x˜ S(x, x The reason why the equations in (13) define a map is that, because of assumption (10), given x, y we can use the first equation to determine a unique x ˜, then, evaluate the second equation to compute y˜. It is also easy to see that the mapping T is symplectic. We recall that the very deep Aubry–Mather theory [3, 36, 39], uses (10) to establish a the existence of a great variety of orbits of these mappings. On the other hand, we note that for models such as the XY model (7) as noted the twist condition fails and indeed, the interpretation of the equilibrium equations of the XY model as dynamical systems is rather problematic. Indeed, in order to λ sin(2πxi )| ≤ 1. be able to isolate xi+1 from (8) we need | sin(2π(xi − xi−1 )) + 2π Even when this condition is satisfied, xi+1 is not defined uniquely mod 1. 2.2.1. Monotone recurrences. The action principle (9) can be generalized naturally, see [2, 53] to action principles that involve more variables. S(xn , . . . , xn+L ) (14) S({xn }) = n∈Z

where, the analogues of (10), (11) that: ∂i ∂j S ≤ C < 0 ,

i = j

(15)

S(x0 + 1, . . . , xL + 1) = S(x0 , . . . , xL ) .

(16)

In [53], it is shown that, under (15), (16), the equilibrium equations of (14) allow to determine xn+L+1 as a function of xn+L , . . . xn and, by a procedure similar to that in (13), define a symplectic map in TL+1 × RL+1 . In [2] it is shown that

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one can develop many properties of the Aubry–Mather theory (existence of quasiperiodic orbits, shadowing properties, etc.). The existence of minimax orbits was taken up in [30, 32, 33]. Again, we emphasize that the results of this paper require only something weaker than (10). For our purposes, the fact that the system describes a recurrence will not play a role. 2.2.2. A specific toy model. Some interesting examples that can serve to fix ideas are Aj 1 (xn+j − xn − bj )2 (xn+1 − xn − a)2 + S({xn }) = 2 n 2 (17) j≥2 + λV (xn ) with Aj decaying sufficiently fast. A physical interpretation of these models is either models of dislocations with an influence kernel [9]. It is also low amplitude model for XY models incorporating long range interactions. The equilibrium equations for (17) are: Aj · (xn+j − 2xn + xn−j ) + λV (xn ) = 0 . (18) (xn+1 − 2xn + xn−1 ) + j≥2

Note that when λ = 0, this model is integrable. In Section 7.1 we will study the existence of Lindstedt series for equations generalizing (18). When all the A are different from zero, equilibrium equations do not admit an interpretation as a dynamical system. When just a finite number of A are different from zero, (say AR = 0 and Ai = 0 for i > R), if x satisfies the equilibrium equation, then xn+R = −xn−R + 2xn −

R−1 1 Aj · (xn+j − 2xn + xn−j ) + λV (xn ) . AR j=1

Recurrences of this type are extensively studied in [14]. A further interpretation of (18) is as as a multistep discretization of the differential equation x + V (xn ) = 0 obtained by setting g xn = x(nλ−1/2 ). With some appropriate choice of the coefficients Aj , the equations (18) can become discretizations of very high order in the step. If we carry out to higher order the Taylor expansions in λ of xn+j +xn−j −2xn , multiply by Aj and add them, for j = 2, . . . M , the conditions that the coefficients for the even derivatives higher than the second vanish are just 1 + 24 A2 + 34 A3 + · · · M 4 AM = 0 1 + 26 A2 + 36 A3 + · · · M 6 AM = 0 ··· 1 + 2M −1 A2 + 3M −1 A3 + · · · M M −1 AM = 0 .

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The above set of equations can be solved explicitly since they are the well known Vandermonde determinants. Hence, by choosing Aj = −

j

(1 − k −2 )−1 1 − (j + 1)2 k −2 ,

k=2

for j ≤ M , in (18), we obtain systems that approximate the solution of the ODE (and which are, therefore integrable) up to order λM/2 . Of course, as usual with multistep methods, the discretization does not only give very accurate solutions to the ODE, but also some spurious solutions. We note that the quasi-periodic solutions obtained applying the results in this paper are indeed close to solutions of the ODE.

3. Preliminaries In this section, we collect some standard definitions and prove some elementary properties of the concepts. In Section 3.1 we introduce some standard concepts in the theory of quasiperiodic functions and see how can they be adapted to our case. We also collect some elementary results on the equations. The main result of Section 3.1 is a reformulation of the equilibrium equations for quasi-periodic equilibria. This reformulation is an extension of the ideas of [44, 45]. In Section 3.2, we collect several standard definitions of spaces of analytic functions. These are extremely standard, but we need them to formulate the smallness of the residual. We also collect some standard results (Cauchy estimates etc.) Except for the definitions, this section is not needed in the statement of the main results. 3.1. Plane-like configurations, hull functions, Percival variational principle We will be interested in equilibrium configurations {xn }n∈Z ⊂ R that can be written as: xn = h(nω) (19) where ω ∈ R and h : R → R satisfies h(x + e) = h(x) + e ∀e ∈ Z .

(20)

In solid state physics the function h is often referred as “hull” function of the configuration. In dynamical systems, the function h gives a semi-conjugacy between the dynamics and a rotation on a torus. KAM theory always looks for solutions of the form (19). The main goal of these section is to reformulate the equilibrium equations in terms of the hull function. Notice that because of the periodicity assumption (20) h can be considered a map from the torus T = R/Z to itself. In our applications, we will assume that h is a diffeomorphism of the torus.

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We will use the notation h(θ) = θ + u(θ) and often work with the u function which is periodic. An obvious consequence of the form (19) is that |xi − ωi| ≤ u C 0 . Hence, the configurations that can be represented by hull functions are almost linear (usually called “plane-like” in homogenization theory). Our assumption that h is a diffeomorphism of the circle implies that u C 0 ≤ 1. Remark 1. The property that a configuration is given by a hull function is equivalent to satisfying the so-called “Birkhoff property” or “non-intersection property” or “self-conforming property”, which was introduced in [1, 35]. This property is very important in variational calculus. 3.1.1. Equilibrium equations for hull functions. We note that the Euler–Lagrange equations (3), that express that the configuration is in equilibrium, evaluated on a configuration described by a hull function h = Id +u are just ∂j HL h(θ + kω), . . . , h θ + (k + j)ω , . . . , h θ + (k + L)ω 0= L k+j=i;j=0,...,L

=

∂j HL θ + kω + u(θ + kω), . . . , θ + (k + j)ω + u θ + (k + j)ω , . . . ,

L k+j=i

θ + (k + L)ω + u θ + (k + L)ω .

(21)

If ω is irrational, (21) are satisfied if and only if E[u](θ) defined below vanishes identically Eω [u](θ) ≡

L

∂j HL h(θ − jω), . . . , h(θ), . . . , h θ + (L − j)ω

L j=0

≡

L

∂j HL θ − jω + u(θ − jω), . . . , θ + u(θ), . . . ,

L j=0

θ + (L − j)ω + u θ + (L − j)ω .

(22)

Note that under the periodicity assumptions for HL and for u, Eω [u](θ) is a periodic function of θ. When ω is understood – in most of this paper we will be considering ω a fixed Diophantine number – we will suppress the ω from the notation. Remark 2. We also remark that E[u](θ) is the Euler–Lagrange equation associated to the functional P ω defined on periodic functions by 1 1 HL θ + u(θ), . . . , θ + Lω + u(θ + Lω) . P[u] = (23) L 0 L

This is an extension of the functional introduced in [44, 45] for computational purposes in the case of nearest neighbor interactions. This functional was

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used in [35] to establish existence of quasi-periodic solutions which were given by monotone – but possibly non-smooth – hull functions h. 3.1.2. A symmetry of the equilibrium equations. If h(θ) = θ + u(θ) is the hull function for a configuration x, then for k, ∈ Z ˜ h(θ) = θ + kω + u(θ + kω) +

(24)

is the hull function for the configuration obtained shifting the argument in x by k and adding an integer to it. Since the interaction is invariant under translation (see (1)) and by addition of integers to a configuration (see (4)), it is clear that if the configuration xn = h(nω) is an equilibrium, the configuration whose hull ˜ given in (24) is also an equilibrium. If ω is irrational, then, kω is function is h dense in the torus. Therefore, if u is a continuous solution of the equilibrium equation (21), so is (25) uσ = u(θ + σ) + σ for any choice of σ ∈ T. The fact that (25) is a solution of (21) for all σ can be also checked by substituting directly. It is valid for all ω, including rational ones. In summary, in general, the quasi-periodic solution of the equilibrium equations are not unique and indeed appear in one parameter families. This corresponds to the choice of the origin of time in the of the torus parameterization, which is a symmetry of the problem. The diffeomorphism of the torus hσ corresponding to uσ is just the translation of the origin in h. That is h(θ + σ) = hσ (θ). The fact that the solutions come in families will play an important role in the study of the equations of equilibrium. Using (25), we see that given one u, we can find a unique σ such that uσ has zero average. An important consequence of the symmetry of the problem under translation and changes of phases is the identity: E[uσ ](θ) = E[u](θ + σ) .

(26)

This identity will play an important role in in Section 5.3. Actually, the symmetry under changes of the origin of the phases is true not only for the equilibrium equations but also for the variational principle (23). We have (27) P[uσ ] = P[u] . This identity will also play a role in Section 5.3. Given the importance of the symmetry under shifts in the formulas, it is important for the analysis that the norms we use are also invariant under shifts. The norms we introduce in Section 3.2, are indeed invariant under shifts in the parameterization. The non-degeneracy conditions we will consider in Theorem 1 are also invariant under changes of the origin of the phase. Hence, we can assume without loss of generality that the approximate solutions we consider are such that the average of E[u] is zero.

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3.2. Some families of norms in analytic periodic functions In this section, we study the spaces and the norms that we use in these spaces. We will denote by

Tρ = z ∈ C/Z | | Im z| ≤ ρ . We denote by Aρ the Banach space of functions from Tρ taking values in CL (when L is understood from the context we will omit it from the notation) such that • They are real for a real argument. • They are holomorphic in the interior of Tρ and continuous on Tρ . We consider Aρ endowed with the norm u ρ = sup |u(z)| .

(28)

z∈Td,ρ

This norm makes Aρ a Banach space. Since the set of L that we will consider will be unbounded, it is important to specify that the norm we will use in CL is the supremum of the coordinates. Of course, for finite L all the norms in CL are equivalent, but the constants given the equivalence could be unbounded as L grows, so that one needs to pay attention to the choice of norms. We have not optimized the choice of norms in CL . So that it is quite possible that other choices could lead to sharper results. This will also become important when we choose norms in the space of interactions in Section 3.3. It is clear that if ⊗ is a bilinear operation of norm 1 (e.g an inner product, multiplication or matrices with their operator norms), we have: u ⊗ v ρ ≤ u ρ · v ρ .

(29)

We also recall that we have Cauchy estimates for derivatives and for the Fourier coefficients in terms of the family of norms (28). We will write the Fourier series of a function u ∈ Aρ as uk e2πikθ . u(θ) = k∈Z

Proposition 1. Let u ∈ Aρ,L , with ρ > 0, L ∈ N, then Dθj u ∈ Aρ ,L for every 0 < ρ < ρ. Moreover, Dθj u ρ ≤ C (ρ − ρ )−j u ρ |uk | ≤ e−2πρ|k| u ρ . The proof of this proposition can be found in any book in complex analysis. In Section 3.3 we will present in detail similar standard proofs in other more complicated cases to make sure that we obtain the dependence in L. The following lemma estimates the composition of functions and shows it is differentiable.

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Lemma 1. Let u ∈ Aρ,L . Let U ⊂ CL be such that dist CL − U, u(Tρ ) ≥ δ > 0 . Let f : U → C be an analytic function f (z) L∞ (U ) ≤ M . Let η ∈ Aρ,L . Then, we have: a) If ||η − u||ρ < δ, then f ◦ η ∈ Aρ . Moreover, |f ◦ η|ρ ≤ M . b) If ||u − η||ρ ≤ δ/2. Then, the mapping f˜ : Aρ → Aρ defined by f˜[γ] = f ◦ (η + γ) is an analytic mapping from the set γ ρ < δ/2 to Aρ . Moreover, we have the following explicit formula for the derivative of the operator Df˜ and bounds for the reminder c) (30) (Df˜)[η]γ = f ◦ η · γ . If ϕ ρ < δ/2, f˜[η + ϕ] − f˜[η] − Df˜[η]ϕ ρ ≤ 2M δ −2 ϕ 2ρ f ◦ (η + ϕ) − f ◦ η ρ

(31)

≤ 2M ϕ ρ . The proof of Lemma 1 is an straightforward and standard application of the Taylor theorem with uniform estimates and the Cauchy estimates for the derivatives. We leave it for the reader. Remark 3. We emphasize that the Df˜ in (31) refers to the derivative of the operator f˜ acting on a space of functions. It is interesting to compare this with the derivative with respect to the varid able θ of the function f˜[η](θ) = f ◦ η(θ). We have dθ f [η](θ) = f ◦ η(θ). d ˜ Even if Df and dθ f [η](θ) are conceptually very different, they have the same formula. This will play an important role later leading to interesting cancellations. 3.3. Properties of the interactions We will assume that the interactions HL are defined on a complex set and analytic there. We will need that the functions HL are analytic in a complex domain large enough to allow their evaluation on the interactions we are considering. The goal of this section is to define norms in the interactions that measure their sizes so that we can state precisely the results. For the spaces that we will consider, it will be quite important to consider not only the sizes but also the domains. Even if we will not consider this in this paper, we hope that this will allow to extend the results to finite differentiable interactions using the techniques of [40, 41, 54] which characterize finite differentiable functions by their approximation properties. Thinking about this further extension, the result we will present will pay special attention to the dependence of the smallness conditions and the change required in the conclusions with respect to the domain of the function.

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In the applications to physical problems, the assumption that the interactions are analytic in a large domain will be often satisfied. Since our result will be formulated for an approximate solution, it is natural to consider domains for the interaction which are defined in a neighborhood of the range of of the approximate solution. Given u ∈ Aρ , we consider

DL,u,δ = (z0 , . . . , zL ) ⊂ (C)L+1 (32) ∃ θ ∈ Td,ρ , |zi − h(θ + iω)| ≤ δ, i = 0, . . . , L . As usual, we suppress the dependence in ω from the notation unless it can cause confusion. Again, we recall that the norms we use in CL are the supremum norms. Since our configurations will be such that they map real values into real values, in some applications it suffices to consider the simpler domains

˜ L,δ = (z0 , . . . , zL ) ⊂ (C)L+1

| Im(zi )| ≤ δ . D (33) Clearly, ˜ L,u +δ . DL,u,δ ⊂ D ρ

(34)

Since L will be unbounded, we will need to estimate the dependence in L of several standard results such as Cauchy estimates and the like. Again, we repeat that we have not optimized the in the chose of norms. With the choice of supremum norm in CL+1 , we have

sup u(θ), u(θ+ω), . . . , u(θ + Lω) θ∈Tρ (35) − u ˜(θ), u ˜(θ + ω), . . . , u ˜(θ + Lω) ≤ ||u − u ˜||ρ . Therefore, we have

HL u(θ), u(θ+ω), . . . , u(θ + Lω) − HL u ˜(θ), u ˜(θ + ω), . . . , u ˜(θ + Lω)

(36)

≤ ||DHL ||L∞ ||u − u ˜||ρ . On the other hand, we note that the estimate of the norm of a derivative in terms of the partial derivatives does have a dependence on L. |DHL | ≤ (L + 1) max |∂xj HL | . j=0,...L

The Cauchy bounds may also have a dependence in L. In Lemma 2, we state the version of Cauchy estimates which we will use (even if we do not know if it is optimal).

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˜ and dist(Ω, CL+1 − Ω) ˜ ≥ δ we have: Lemma 2. If Ω ⊂ Ω ||DHL ||Ω ≤ C(L + 1)δ −1 ||HL ||Ω˜ .

(37)

Proof. Given z ∈ Ω we can find circles γi centered in zi with radius δ such that γ = (γ0 , . . . , γL ) ⊂ Ω. Cauchy formula gives: HL (w) 1 . dw · · · dwL HL (z) = 0 L+1 (2πi) (w0 − z0 ) · · · · (wL − zL ) γ0 γL Therefore, given a direction η, Dη HL , the directional derivative is: 1 Dη HL (z) = dw · · · dwL HL (w) 0 (2πi)L+1 γ0 γL η0 + ··· 2 (w0 − z0 ) · (w1 − z1 ) · · · · (wL − zL )

ηL . (w0 − z0 ) · (w1 − z1 ) · · · · (wL − zL )2

Using that

(38)

1 ≤ 2π |w − zi | i γ i 1 dwi ≤ 2πδ −1 |w − zi |2 i γi dwi

we obtain the claimed result.

˜ Corresponding to the domains DL,u,δ , D(L, δ) we consider the spaces HL,u,δ , HL,δ consisting of functions analytic in the interior and continuous in the whole domain. We endow these spaces with the supremum norm, which makes them Banach spaces. HL L,u,δ = sup |HL (z)| z∈DL,u,δ

HL L,δ = sup |HL (z)| .

(39)

˜ L,δ z∈D

By (34), we have HL,u,δ ⊂ HL,uρ +δ and HL L,u,δ ≤ HL L,uρ +δ .

4. Statement of the main result Following standard practice in KAM theory, we will denote by C numbers that depend only on on combinatorial factors but are independent of the size of the domains considered, the Diophantine constants κ or the size of the error assumed. In our case, we will also require that they are independent of L, the range of the interactions. The meaning of this constants can change from one formula to the next.

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The main result of this paper is the following Theorem 1. Note that, as indicated, above, we have formulated it in an “a posteriori” way. We will assume that, we are given an approximate solution of the invariance equation, give some finite list of conditions to verify (some version of the twist and some measure of the smoothness of the function). Then, if the solution satisfies the equilibrium equation up to an error which is sufficiently small with respect to the other conditions, then we conclude that there is one solution close to the given one. Furthermore, the solution is unique in some neighborhood. Theorem 1. Let H be a translation invariant interaction as in (1) satisfying the periodicity condition (4). Let ω ∈ R. Let h = Id +u, with u ∈ Aρ , T u = 0 be an analytic diffeomorphism of T. Assume: H1) ω is Diophantine, i.e., for some κ > 0, τ > 0 |pω − q| ≥ κ|q|−τ

∀p ∈ Z − {0} ,

q ∈ Z.

(40)

H2) The interactions HL ∈ HL,u,δ for some δ > 0. Denote ML = max( Di HL L,u,δ ) , i = 0, 1, 2, 3 ML L4 α=C L≥2

where C is a combinatorial constant that will be made explicit during the proof. H3) Assume that the inverses indicated below exist and have the indicated bounds. H3.1) (∂0 ∂1 H1 )−1 u(θ), u(θ + ω) ≤ T . ρ H3.2) Define

C0,1,1 (θ) = ∂0 ∂1 H1 u(θ), u(θ + ω) h (θ)h (θ + ω) .

(See (57) later for a justification of including the subindices in C) Assume that:

−1

−1 C

≤ U .

T 0,1,1

H4) The following bounds measure the non-degeneracy and the accuracy with which the approximate solution solves the problem. H4.1) Id +u ρ ≤ N+ , (Id +u )−1 ρ ≤ N− . H4.2) E[u] ρ ≤ ε. H5) Assume furthermore that the above upper bounds satisfy the following relations: H5.1) T α < 1/2, U T α < 1/4 H5.2) u ρ + ρ ≤ 12 δ

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H5.3) ε ≤ ε∗ (N+ , N− , τ, α, T, U, δ)κ4 ρ4τ +A where ε∗ > 0 is function which we will make explicit along the proof. The function ε∗ makes quantitative the relation between the smallness conditions and the nondegeneracy conditions. A ∈ R+ is a number which will also be made explicit along the proof. Then, there exists a periodic function u∗ ∈ Aρ/2 such that and

˜ω [u∗ ] = 0 E

(41)

∗

u = 0. Moreover

u − u∗ ρ/2 ≤ Cκ−2 ρ−2τ −A T E[u] ρ . (42) ∗ The function u is the only function in a ball of radius centered at u of radius Cκ−2 ρ−2τ −A T E[u] ρ satisfying (41) and the normalization. u∗ = 0. Hypotheses H1) just indicates that the frequency we are considering is Diophantine. The hypothesis H2) just indicates that the interaction can be defined comfortably in the range of u and indicates some constant. Condition H3) is the twist condition. We note that we only evaluate the twist condition in a neighborhood of the approximate solution. this allows us to deal with systems that do not have global convexity properties. Condition H4) is a measure of the smoothess of the approximate function. (We measure the size of h and h−1 in a smooth norm.) The crucial condition is H5) which states that the approximate solution u solves the equilibrium equation very accurately. The accuracy required depends on properties of the equation (a very degenerate equation will require higher accuracy). The fact that the quality of the test function enters is natural since we can always get the equation to be satisfied by extremely complicated functions. The expression of H5) is easy to implement in numerical calculation. It is just the composition of several rather simple conditions. A non-rigorous implementation can give some idea of whether the calculation is reliable and help weed out spurious solutions. More details are given in [7]. Note also that the dependence of the smallness conditions on the domain of analyticity ρ is a power and also that the effect of the analyticity domain in the correction (42) is also a power of ρ. It is well known to experts that such a result can be used to prove a finitely differentiable result. We hope to come back to this problem. We also note that such a result implies that there is Lipschitz dependence on the Diophantine frequencies. (Just take the solution for a frequency as an approximate solution for a nearby frequency). Indeed, by combining a result such as the one above with the Lindstedt series, one gets that the dependence on the frequencies is smooth in the sense of Whitney. See [31, 52]. Theorem 1 implies the result of persistence of solutions for quasi-integrable systems. In our case, the integrable systems are the linear systems: 1 HL0 (x0 , . . . , xL ) = AL |x0 − xL |2 . (43) 2

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For systems of the form (43), given ω ∈ R, xn = ωn is a solution, which corresponds to u = 0. If we consider a system HL = HL0 + μFL . If the perturbation satisfies F L,δ L2 < ∞, for |μ| sufficiently small, we can consider u = 0 as an approximate solution of the system. Note that, to verify the hypothesis of Theorem 1, ε – the error – is bounded by μ and the non-degeneracy conditions remain uniformly bounded as μ approaches zero. Hence, we can obtain the existence of solutions with corresponding frequencies. A more detailed discussion of results for quasi-integrable systems happens in Section 7.1.

5. Description of the proof of Theorem 1 The proof is based on an iterative procedure very similar to that of [34, 43, 51]. Given a function u analytic and bounded in a domain Dρ , so that E[u] ρ is sufficiently small compared with other properties of the function and with ρ /ρ, the iterative procedure constructs another function u ˜ defined on Dρ and such that 2 E[˜ u] ρ ≤ CT A(τ, N+ , N− , T, U )κ−2 (ρ − ρ )−4τ −2 E[u] ρ (44) where A is a very easy to write algebraic function. As it is well known in KAM theory, the estimates (44) imply that the iterative procedure can be repeated indefinitely and the resulting sequence converges to a function satisfying all the claims of Theorem 1. We will present details later. The procedure to produce the improved solution we will use here is very similar to that of the papers [34, 43, 51]. Nevertheless, we have to deal with the fact that the interactions are not nearest neighbor. The method of the previous papers depended on some identities obtained in [25,43] by combining derivatives of the approximate solution, and using the variational structure and the symmetry of the problem. These identities allow us to reduce the problem to a constant coefficients problem. As it turns out, similar identities hold in our case. and we can obtain a similar factorization. Once this factorization is obtained, we can treat the non-nearest neighbor interactions perturbatively. 5.1. Heuristic description of the iterative step The iterative step is very similar to the iterative step in the previous papers [34, 43, 51]. The step is based on a modification of the Newton method which makes the linearized equation used in the Newton method readily solvable but does not change the quadratic convergence. See (54). In this section we will describe the iterative procedure somewhat formally. We will specify the manipulations to be carried out with the functions but ignore questions of domains, definition and convergence of series involved. These questions will be addressed in Section 6 where we will develop estimates for the objects considered. These estimates will allow us to verify that the algorithm can indeed be well defined and that the steps indicated formally can be carried out (i.e. the

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compositions can be defined since the domains match). The estimates will also verify that the step improves the solution in the sense that they satisfy (44). Undoubtedly some experts will be able to carry out the detailed estimates in Section 6 without reading it. In this presentation, we have followed the notation of [34, 43, 51] as much as possible. In Sections 5.2, 5.3, 5.4, 5.5 we will motivate and specify an iterative procedure that given a sufficiently approximate solution produces a better one. This procedure will be rigorously analyzed in the subsequent sections. In Section 6 we will present estimates that make precise the fact that the iterative procedure improves the approximation (albeit in a smaller domain). In Section 6.3 we will show that the procedure can be iterated indefinitely, and that converges to a solution. Given the estimates in Section 6, the estimates in Section 6.3 are very similar to those in [55]. In Section 6.4 we will show that the solution is unique in a ball. 5.2. A Newton step The iterative procedure we will use will be a modification of the Newton procedure. The important thing for us that it leads to estimates as in (44). The step will be similar to the step in [34] – except in the perturbative treatment of the non-nearest neighbor interactions –. We have tried to follow the notation of [34] as much as possible. As motivation, and to introduce the notation, we will start by discussing the standard Newton method. Given that we have an approximate solution u, a step of the Newton method consists in setting u ˆ=u+v where v is obtained by solving E [u]v = −E[u]

(45)

where E denotes the derivative of the functional E with respect to its argument. Proceeding formally for the moment, (we will present precise estimates in Section 6) we compute (the computation can readily be justified using Lemma 1) that:

L L E [u]v (θ) = ∂j ∂i HL θ − jω + u(θ − jω), . . . , θ + u(θ), L j=0 i=0

. . . θ + (L − j)ω, u θ + (L − j)ω v θ + (i − j)ω .

(46)

We introduce the notation h(θ) = θ + u(θ) h(i) (θ) = θ + iω + u(θ + iω) γL (θ) = h(θ), h(1) (θ), . . . , h(L) (θ) (i) γL (θ) = h(i) (θ), h(1+i) (θ), . . . , h(L+i) (θ) . (0)

In particular, h(0) (θ) = h(θ), γL (θ) = γL (θ).

(47)

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Using the notation above, the equation (46) can be written more concisely as

L L (j) E [u]v (θ) = ∂j ∂i HL γL (θ) v (i−j) (θ) .

(48)

L j=0 i=0

5.3. Two important geometric identities An identity for the operator E [u] introduced in (46) which will be extremely important later is d E[u](θ) = E [u]h (θ) . (49) dθ The identity (49) can be verified by a direct calculation, taking derivatives with respect to θ in (22) and comparing with E [u]h in (46). More conceptually, we note that the identity (49) is just the derivative with respect to σ of the equation (26) E[˜ uσ ](θ) = E[u](θ + σ) evaluated at σ = 0. We recall that the equation (26) expressed the invariance of the problem under changes of the origin of the internal phase. If we take derivatives with respect to σ in (26) and evaluate for σ = 0, taking into account that

d uσ σ=0 = Id +u = h dσ we obtain (49). Remark 4. It has been remarked in [4] that the introduction of the correction term E [u]h is similar to the use of Ward identities in quantum field theory. Recall that the gist of Ward identities is that taking derivatives of solutions with respect to a symmetry of the theory, we obtain an identity that can be used to absorb terms in the perturbation theory or in the renormalization group. In our case, the symmetry of the theory is the covariance under the choice of the origin of the phase expressed by (26). Another interpretation from the point of view of differential equations can be found in [25, 43]. There other possible identities obtained through similar ideas. For example, [55] points out that other useful identities in the Hamiltonian formalism are related to the invariance of the equations under canonical changes of variables (The group structure). Of course, since our equations (or the partial differential equations of [43]) do not admit a transformation theory, the two phenomena seem different. Nevertheless, the method of proof in [51] uses symmetries of the equation – for flows – that allow one to reduce the Newton method to the case of the identity. I am very indebted to A. Gonz´ alez-Enriquez who explained me many of these symmetries. Another identity that will play an important role is: Lemma 3. With the notations above we have h (θ)E[u](θ) dθ = 0 . T

(50)

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Proof. By changing variables in each term of the sum below, we obtain: L dθ h (θ)∂j HL h(−j) (θ) . . . h(L−j) (θ) j=0

T

=

L j=0

=

T

dθ T

dθ ∂j HL h(θ) . . . hL (θ) h(j (θ) d HL h(θ) . . . hL (θ) dθ

=0 where we have used the periodicity properties of h (20) and the periodicity properties of HL (11). Recalling the expression of E[u] given in (22), the desired result is obtained adding the previous calculation over L. More conceptually, we note that the equation (50) is related to the variational structure and the invariance under translation of the problem. We recall that, as mentioned before, the equilibrium configurations are critical points of the functional P introduced in (23). The variational principle (23) is invariant under changes of the origins of the phase and in (25) we had P [uσ ] = P [u]. A simple calculation shows that:

d

P [uσ ] σ=0 = h (θ)E[u](θ) dθ dσ and therefore, (50) is a consequence of the invariance under translation of the variational principle. 5.4. The quasi-Newton method Unfortunately, the equation (45) is hard to solve in our case since it involves difference equations with non-constant coefficients. The trick that works in our case is very similar to the one that was used in [25,43,51] and specially in [34]. Namely, our step consists in solving the following equation, which is a modification of (45), the equation suggested by the Newton method. (51) h (θ) E [u]v (θ) − v(θ) E [u]h (θ) = −h (θ)E(θ) . The equation (51) is just (45) multiplied by h and added the extra term v(E [u]h ). We will show that indeed, the equation (51) can be solved by reducing it to constant coefficient equations plus an elementary perturbative argument. We will also show that the iterative step, leads to quadratic estimates of the form (44). The reason why the added extra term is small is that, because of (49) we can write d (52) v(θ) E [u]h (θ) = v(θ) E[u](θ) . dθ

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So that, when we show in Section 6 that v ρ is estimated by E ρ we will have that the extra term is controlled by E 2ρ . The role of the added extra term will be to make the RHS of equation (51) be factorizable. This same phenomenon happened in [43, 51]. 5.5. Solution of the equation of the quasi-Newton method The goal of this section is to specify the steps of an algorithm that can be used to solve the equation (51). Once we have specified how to break down (51) into auxiliary problems, we will present estimates for them in Section 6. Following the previously mentioned references, we introduce a new variable w by (53) v(θ) = h (θ)w(θ) . Notice that the nondegeneracy assumption H4.a of Theorem 1 implies that h is boundedly invertible, so that, the unknowns v and w are equivalent. Substituting (53) by (48) into (51) and using the notation in (47), we obtain that the equation to be solved for the step of the modified Newton method is: L L

(−j)

∂j ∂i HL ◦ γL

(θ)h (θ)h(i−j) (θ)w(i−j) (θ)

L j=0 i=0

−

L L

(−j)


(θ)h(i−j) (θ)w(θ)h (θ)

(54)

L j=0 i=0

= −h (θ)E[u](θ) . We will analyze separately the terms that appear in the RHS of (54). We first fix L and then, we consider the terms that correspond to a certain i, j. Our goal is to show that the equations can be factored into simpler equations. We first note that, when i = j, the terms in the RHS of (54) cancel. The two terms in the first sum cancel the two terms in the second sum. When i = j we observe that we have four terms involving the mixed deriva(j) tives ∂j ∂i HL ◦ γL , namely, (−j)


(−i)

+ ∂i ∂j HL ◦ γL

(θ)h (θ)h(j−i) (θ)w(j−i) (θ)

(−j)

− ∂j ∂i HL ◦ γL

(−i)

− ∂i ∂j HL ◦ γL

(θ)h (θ)h(i−j) (θ)w(i−j) (θ) (θ)h(i−j) (θ)h (θ)w(θ)

(55)

(θ)h(j−i) (θ)h (θ)w(θ) .

We rearrange (55) as

(θ)h (θ)h(i−j) (θ) w(i−j) (θ) − w(θ) (−i) − ∂i ∂j HL ◦ γL h (θ)h(j−i) (θ) w(θ) − w(j−i) (θ) . (−j)


(56)

An observation that is important for us is that the second term in (56) is just the first term shifted (j − i)ω.

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We introduce the notation [Δ w](θ) ≡ w(θ + ω) − w(θ) ≡ w( ) (θ) − w(θ) Ci,j,L ≡ ∂j ∂i HL ◦

(57)

γL−j (θ) h (θ)h(i−j) (θ) .

With the notations (57) above, the four terms in (56) corresponding to a fixed i, j, i = j can be written: (58) −Δj−i Ci,j,L (Δi−j w) . Therefore, the equation (54) can be written as: L

Δj−i Ci,j,L Δi−j w = h E[u] .

(59)

L j,i=0 j>i

Remark 5. As a curiosity, we note that some of the identities between the terms in (55) can be considered physically as a consequence of the action-reaction principle. Note that the interpretation of each of the terms is that of the force on one body from another. They are clearly related to the fact that the equations come from a variational principle. The other identities used come from the invariance under shifting of the origin of the internal phase, which in turn is a consequence of the invariance under translations of the model and the invariance under global changes of the phase by an integer. The basic idea we will use is that, under the hypothesis i) of Theorem 1, the equation (59) can be treated as a perturbation of the term corresponding to L = 1 corresponding to nearest neighbor interactions. To accomplish this perturbative treatment, it will be important to study conditions for invertibility of the operators Δ . A precise formulation of this study will be carried out in Section 6.1. For the moment, we will just perform formal manipulations to motivate and specify the procedure to be followed in Procedure 1. Later we will provide estimates that show that the procedure can be implemented and give bounds on the results. The first observation is that the operator Δ1 is diagonal on Fourier series. Hence, if we consider the equation for w given η

Δ w = η

(60)

2πikθ

, provided that η0 = 0, we can find (at least formal) solutions when η = ηk e of (60) by setting, for k = 0: (61) w = η /(e2πi k · ω − 1) k

k

(recall that we are assuming that ω is Diophantine, hence, in particular, · ω ∈ Z =⇒ k = 0) These solutions η are unique up additive constants. We will denote by Δ−1 1 the operator that given η produces the w with zero average. This makes into a linear operator.

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These operators appear very often in KAM theory and have been extensively studied. In Section 6.1 we state the results of [48, 49], which are optimal with respect to the loss of differentiability. Hence, we can define the operators −1 L±

= Δ±1 Δ

acting on all the functions in Aρ and the operators −1 R±

= Δ Δ±1

defined for all functions with zero average. ± Note that all the operators L±

and R have as range the set of functions with zero average. ± ± Remark 6. The operators R±

L are not identical since R can only be defined on functions with integral zero. When applied to functions with integral zero, the operators R, L agree. In other words, the only difference among them is that R±

has a domain which is a codimension 1 space while the domain of L±

is the whole space.

The key observation that allows to treat the equation (54) as a perturbation of the nearest neighbor case is the fact that, in spite of the fact that Δ−1 ±1 are unbounded operators, we have: Δ−1 ±1 Δ ρ ≤ ||

(62)

Δ Δ−1 ±1 ρ ≤ || .

The (elementary) proof of this key result is postponed till Lemma 6. Coming back to the solution of the equation (54), we observe that separating explicitly the nearest neighbor terms, we have that (59) can be written as h E[u] = Δ1 C0,1,1 Δ−1 w + Δj−i Ci,j,L Δi−j w ⎡ = Δ1 ⎣C0,1,1 +

L≥2 i>j

⎤

(63)

−1 ⎦ Δ−1 1 Δj−i Ci,j,L Δi−j Δ−1 Δ−1 w .

L≥2 j>i

By the twist assumption property H3, the operator C0,1,1 is boundedly invertible from the spaces Aρ (with zero integral) to itself. Similarly, we observe that the operator G is defined on the functions Aρ with zero average. This is acceptable for our applications because the operator G in (63) is only applied to Δ−1 w, which has zero average. We also note that the operator G always produces functions with zero average. All the terms in the sum defining G in (64) have in the left an operator L− i−j which produces functions with zero average.

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The decay properties on the interaction which we assumed in Theorem 1 show that −1 Δ−1 (64) G≡ 1 Δj−i Ci,j,L Δi−j Δ−1 L≥2 j>i

has a small norm from A to Aρ for all 0 < ρ ≤ ρ. Indeed we have using the estimates assumed in Theorem 1 and the bounds −1 for ||Δ−1 −1 Δi−j ||ρ ||Δj−i Δ1 ||ρ obtained in Lemma 6 ML |i − j|2 G ρ ≤ ρ

L≥2

≤C

0≤i≤j≤L

(65)

ML L4

L≥2

= α. Hence, under the assumptions i) of Theorem 1, the usual Neumann series shows that the operator C0,1,1 + G is boundedly invertible from Aρ to Aρ . Moreover, we have (C0,1,1 + G)−1 − (C0,1,1 )−1 ρ ≤ 1/(1 − T α) (C0,1,1 )−1 G ρ ≤ 2T α . The equation for T ∈ R can be written as: −1 (C0,1,1 + G)−1 − (C0,1,1 )−1 T = (C0,1,1 + G)−1 ϕ . C0,1,1 T + T

T

T

Under the assumptions H4.a , we see that the second term in the left hand side of the above equation can be treated as a perturbation of the first term. Therefore, |T | ≤ U/(1 − 2U T α) ϕ ρ ≤ 2U ϕ ρ . Then, w can be obtained by solving the equation Δ1 w = ϕ + T . and, we obtain v by recalling that, by definition, v = h w. In summary, the procedure to solve (63) is to follow the following steps Procedure 1.

1. Observe that, by Lemma 3, we have T h E[u] = 0. 2. Find a function ϕ (normalized so that ϕ = 0) solving the equation Δ1 ϕ = h E[u] .

(66)

Therefore, for any constant T Δ1 (ϕ + T ) = h E[u] . The equation (66) is the standard small divisors equation, which has been studied in e.g. [48].

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3. Choose T in such a way that (C0,1,1 + G)−1 (ϕ + T ) = 0 .


(67)

To accomplish that, we will show that, under the non-degeneracy assumptions, the linear operator on T defined by the LHS of (67) is is invertible. This follows from assumption H3) because the operator G is invertible. We emphasize that T ∈ R and that this step is dividing by a number, which is trivial if the number is non-zero. 4. Obtain w by solving Δ−1 w = (C0,1,1 + G)−1 (ϕ + T ) . 5. We set u ˜(θ) = u(θ) + h (θ)w(θ) as our improved solution. Remark 7. We note that the analysis above also allows to conclude uniqueness of the solution of the equation (51). Note that the factorization of the equation into steps is just an identity. Each of the steps provides with unique solutions. As we will see, uniqueness of the solution of this linearized problem will translate into uniqueness of the solution of the full non-linear problem. In turn, this will be useful for the study of problems with finite regularity.

6. Estimates for the iterative step The goal of this section is to provide precise estimates for the iterative step described in Section 5.1. The precise statement of the result of this section is the following Lemma 4. This lemma establishes that the procedure indicated in Section 5.1 can be carried out if there are some smallness assumptions (that guarantee that compositions can be defined). The most crucial results is the estimates (71) that establish that the error after the procedure is quadratically small with respect to the original error. Finally, since the final estimates of the step depend on the constants that measure the non-degeneracy properties of the solution, we estimate how these nondegeneracy properties change. This will be important to show that the procedure can be iterated indefinitely using (70). This is very standard in KAM theory. Remark 8. For further applications (e.g. Section 7.3) it is important to remark that the size of the change of u produced in the iterative step depends only on the size of the non-degeneracy conditions. In particular, if we have families of problems and families of approximate solutions so that they satisfy the hypothesis of the step uniformly, then, we can carry the step for all of them and then, the result has uniform estimates.

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Lemma 4. Under the assumptions of Theorem 1. Let 0 < ρ = ρ − σ < ρ, ρ” = ρ − σ/2. Assume that κ−2 (ρ − ρ )−2τ (N+ )2 E[u] ρ ≤ δ/4 .

(68)

Then, the procedure indicated above can be carried out to produce a solution v ∈ Aρ of (51) satisfying v = 0. We have the following estimates: w ρ ≤ Cκ−1 σ −τ ϕ ρ ≤ Cκ−2 σ −2τ N+ ε .

(69)

˜ = Id +h ˜ to be the improved approximate solutions Denote by u ˜ = u + v,h obtained applying the procedure, we have: ˜ u − u ρ ≤ Cκ−2 (ρ − ρ )−2τ (N+ )2 E[u] ρ ˜ − h ρ ≤ Cκ−2 (ρ − ρ )−2τ −1 (N+ )2 E[u] ρ . h

(70)

We also have have that E[˜ u] is well defined and, moreover, the solution solves the problem more accurately in the sense that: 2 4 E[˜ u] ρ ≤ C (N+ ) N− + N+ ML L · (71) L · κ−4 (ρ − ρ )−4τ −1 ||E[u]||2ρ . As a consequence of (70), we have the following estimates for the constants that measure the non-degeneracy. That is, the constants that enter in hypothesis H3). We use the same notation as in Theorem 1 but use the˜to indicate that they are evaluated on the function u ˜. We define Δ = εκ−2 (ρ − ρ )−2τ (N+ )2 Δ = εκ2 (ρ − ρ )−2τ −1 (N+ )2 the RHS of the equations (70). We have: T˜ ≤ T + Cδ −1 Δ ˜ ≤ U + Cδ −1 ΔK(U, T, Δ) U ˜+ ≤ N+ + CΔ N ˜+ ≤ N− + CΔ K(N+ , N+ , Δ ) N

(72)

where K are very simple algebraic functions that will be made explicitly in the proof. Keep in mind that, as it is standard in KAM theory, when we repeat the procedure, we will show that ε will be much smaller than (ρ − ρ )−2τ −1 , hence the estimate for the RHS in (71) is indeed much smaller. Also, the Δ, Δ in (72)

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will be very small. Hence the intuition of the step is that it reduces the domain slightly, worsens slightly the non-degeneracy assumptions, but the error is drastically reduced. The losses of domain will decrease rapidly enough that there is some domain left. Similarly, the non-degeneracy conditions will remain bounded. The quadratic convergence of the error to zero overcomes the other problems. This is perhaps the most important principle in the KAM method. See the subsequent sections for details. In the rest of this section, we will prove Lemma 4. The most subtle part of the proof is the estimates in points (2) and (4) of the algorithm, but these estimates are classical estimates for small divisors. Rather optimal versions were obtained in [48, 49]. The rest of the steps are more elementary in nature. As indicated before, the step (3) is just a perturbative argument based on Neumann series. From this, we get (70). Then, using Taylor’s theorem for the operator, we get (71). Then, given (70), the estimates (72) are just based on the Neumann series (this is the reason why we get the functions K). 6.1. Estimates for equations involving small divisors The following result is proved in [48]. Lemma 5. Assume that ω ∈ R satisfies (40). Then, given any function η ∈ Aρ,L satisfying η = 0 there is one and only one function ϕ ∈ L2 (T, CL ) satisfying Δ±1 ϕ = η ϕ = 0. (73) Moreover, ϕ ∈ Aρ ,L for all ρ < ρ and ϕ ρ ≤ Cκ−1 (ρ − ρ )−τ η ρ .

(74)

The constant C is independent of L. Since we are using the supremum norm in CL , it is clear that to prove this result, we can just reduce to studying the equation for each of the components. Each component can be estimated by the norm of the function. Hence, it is clear that the final constant will be independent of the dimension of the image. We refer to [48,49] for the proof of a more general result (e.g. it studies similar equations for functions defined over Td . In this paper, we only need d = 1. ) To prove a weaker result with a worse exponent in ρ − ρ it suffices to notice that the estimates (40) provide an upper bound for the multipliers in (61). Using the Cauchy estimates that estimate |ηk | in terms of ||η||ρ , one obtains the same result but with bounds Cκ−1 (ρ − ρ )−τ −1 η ρ which would also be enough for our case. The more subtle estimates of [48] use also that the estimates (40) cannot be saturated for k that are very close. Another version of the estimates can be found in [28] and the revised version of [27].

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We denote by Δ−1 ±1 the mapping that to η associates ϕ solving (73). Lemma 6. For every 0 < ρ we have: Δ Δ−1 ±1 η ρ ≤ || η ρ

(75)

Δ−1 ±1 Δ η ρ ≤ || η ρ . Proof. Denoting by ϕ = Δ−1 1 η, we have for > 0, Δ Δ−1 1 η(θ) = Δ ϕ(θ) = ϕ(θ + ω) − ϕ(θ) = ϕ(θ + ω) − ϕ θ + ( − 1)ω + ϕ θ + ( − 1)ω − ϕ θ + ( − 2)ω

(76)

+ ···+ + ϕ(θ + ω) − ϕ(θ) = η θ + ( − 1)ω + · · · + η(θ) . If < 0, we have Δ Δ−1 1 η(θ) = Δ ϕ(θ) = ϕ(θ + ω) − ϕ(θ) = ϕ(θ + ω) − ϕ θ + ( + 1)ω + ϕ θ + ( + 1)ω − ϕ θ + ( + 2)ω

(77)

+ ···+ + ϕ(θ − ω) − ϕ(θ) = −η(θ − ω) − · · · − η(θ − ω) . Hence, clearly in both cases, Δ Δ−1 1 ρ ≤ ||. All the other cases are proved in the same way or can be deduced from the present ones. Remark 9. Note that the result here does not depend on the fact that ω is a Diophantine. The fact that ω is Diophantine is used to define Δ−1 ±1 as an operator on the whole space. Note however that, for any irrational ω, the operator Δ Δ−1 ±1 can be defined for trigonometric polynomials and we have the identities (76),(77). Hence, the operator Δ Δ−1 ±1 is bounded in the space of trigonometric polynomials, which makes it uniquely defined and bounded in the whole space of analytic functions. Remark 10. The diagonal elements of Δ Δ−1 1 are 1 + e2πikω + (e2πikω )2 + · · · + (e2πikω ) −1 e2πiωk ω − 1 = e2πikω − 1 −1 − (e2πikω ) −1 − (e2πikω ) −1 − · · · − (e2πikω )

>1 < −1 .

(78)

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The identity (78) is just the elementary sum of the geometric series in the left. Of course, the identity for the diagonal factors can also be obtained from the identity (76). Remark 11. Note that the identity (78) shows that Δ1 Δ−1 2 is not bounded since the diagonal terms of this operator are 1 e2πikω − 1 = 2πikω . (79) 2πik2ω e −1 e +1 Note that the denominator becomes arbitrarily close to zero as k ranges over the integers. Hence, it is not possible to use a simple perturbative argument to treat the general case as a perturbation of the interactions at length 2. We think that it would be quite interesting to study if there is a KAM method that can deal with this problem. Some indications that there indeed may be invariant tori are: Numerical experiments see [7]; Formal series expansions exist to all orders (see Section 7.1). The variational methods in [8, 30, 32, 33] produce – may be less regular – solutions for systems with convexity. 6.2. Estimates for the step in Lemma 4 In this section we prove estimates (69), (70) for the sizes of the changes produced by the iterative step. The proof is rather straightforward. We just follow the steps of Procedure 1 but we take care of ensuring that all the steps are well defined and give estimates for them. Since we will have to loose domain repeatedly we will introduce auxiliary numbers ρ = ρ − (ρ − ρ )/2, We will denote ρ − ρ = σ. Hence, ρ − ρ = 2−1 σ, ρ − ρ = 4−1 σ, so that we can estimate from below the distance between two of these by Cσ. In step 1) we estimate h E[u] ρ ≤ N+ ε using the Banach algebra property. Then, by Lemma 5, we have ϕ ρ ≤ Cκ−1 σ −τ N+ ε. −1 ρ ≤ T δ −2 as an operator from Aρ The assumption H3 implies that C0,1,1 to itself. The assumption implies that G ≤ α also as an operator from Aρ to itself. Since, by assumption H3, T α < 1, C0,1,1 + G is invertible in Aρ . Moreover, −1 (C0,1,1 + G)−1 − C0,1,1 ≤ α(1 − αT )

≤ 1/2T . Using the fact that C0,1,1 is a multiplication operator, we write the equation for T ∈ R as −1 −1 (C0,1,1 + G)−1 − C0,1,1 T = − (C0,1,1 + G)−1 ϕ . C0,1,1 T + (80) T

T

T

−1 By assumption H3, ( C0,1,1 ) is an invertible matrix and its inverse has norm −1 T is less than U . We have shown that the operator T → (C0,1,1 + G)−1 − C0,1,1 has norm less than 1/2T . By the assumptions U · T ≤ 1/2, we can treat (80)

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as a perturbation of the first term. Since the RHS of (80) has norm less than 1/2T ϕ ρ , we have that |T | ≤ 2U (1/2)T ϕ ρ 1 ≤ ϕ ρ . 2 Hence T + ϕ ρ ≤ 32 ϕ ρ . Applying Lemma 5 we obtain (69). Therefore, u − u ρ ≤ h w ρ ˜ u − u ρ ≤ ˜ ≤ Cκ−2 σ −2τ (N+ )2 ε . Applying Cauchy inequalities, we obtain ˜ − h ρ ≤ (ρ − ρ )−1 ˜ h u − u ρ ≤ Cκ−2 σ −2τ −1 (N+ )2 ε .

(81)

(82)

The equations (81), (82) give us (70) for the solution of (51). We note that the solution consists on applying identities to factor the equation (51) into several different steps. We have shown that the solution is in Aρ for every ρ < ρ. Nevertheless it is important to realize that the solution is unique among the solutions in Aρˆ for any 0 < ρˆ. Hence, the solution of (51) is unique among the solutions in Aρˆ for any 0 < ρˆ. This establishes the uniqueness claim in Lemma 4. 6.2.1. Proof of (71). The proof consists in showing that u ˜ is still in the domain of the functional E. Then, we just add and subtract appropriate terms and estimate what remains using the Taylor estimates. Because of (70) and the assumption (68) we obtain that ||u − u ˜||ρ ≤ δ/4

(83)

where δ denotes – see the assumptions in Theorem 1 – the distance of the range of u to the boundary of the domain of the interaction. Therefore, we have E[u + v] = h−1 h E[u] + h E [u]v + E[u + v] − E[u] − E [u]v = h−1 vE[u]h + E[u + v] − E[u] − E [u]v (84) d −1 v E[u] + E[u + v] − E[u] − E [u]v . =h dθ The first identity is just adding and subtracting. The second equation uses that v solves equation (51) and the third identity is just (49). Using the Cauchy inequality, the Banach algebra property and the estimates for v obtained in (81), we have: −1 d ≤ Cκ−2 σ −2τ −1 (N+ )2 N− ε2 . (h ) E[u] (85) v dθ ρ

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The equation (83) tells us that we can apply the estimates in Lemma 1 to each of the terms defining the error E and we obtain: 1 ML L v 2ρ E[u + v] − E[u] − E [u]v ρ ≤ 2 L (86) −4 −4τ 4 2 ≤C ML L κ σ (N+ ) ε . L

Adding (85) and (86) and using the obvious estimates σ −4τ ≤ σ −4τ −1 we obtain (71). 6.2.2. Proof of the estimates for the change in the induction hypothesis in Lemma 4. We use the notation introduced in (47) and denoting by γ˜L the one corresponding to u ˜ instead of u. Given the estimates (70) and (68), we have (83) that tells us that we can apply Lemma 1. We first observe that dist γ˜L (Tρ ), C − Domain(HL ) ≥ dist γL (Tρ ), C − Domain(HL ) − ||γL − γ˜L ||ρ . Since ||γL − γ˜L ||ρ = ||u − u ˜||ρ , applying (70), we see that the new function u ˜ satisfies assumption H2 with δ˜ = δ − εκ−2 (ρ − ρ )−2τ (N+ )2 E[u] ρ . If we do that, we see that the ML do not need to be changed because they are the supremum of functions over an smaller set. We also note that, by Cauchy estimates, we have that, by the mean value theorem,

∂0 ∂1 H1 u( · ), u( · + ω) − ∂0 ∂1 H1 u ˜( · ), u ˜( · + ω)

≤ 2||u − u ˜||ρ ρ

= 2M1 Δ . Using that ∂0 ∂1 H1 (u(θ), u(θ + ω)) is invertible for all θ, we obtain, using the u( · ), u ˜( · + ω)) and we Neumann series that if Δ is small enough, so is ∂0 ∂1 H1 (˜ get the bounds claimed in (72). Adding and subtracting, we also get ||C0,1,1 −C˜0,1,1 ||ρ

˜( · ), u ˜( · + ω) h ( · )h ( · + ω)

ρ ≤

∂0 ∂1 H1 u( · ), u( · + ω) ∂0 ∂1 H1 u

˜ ( · ) h ( · + ω)

+

∂0 ∂1 H1 u( · ), u( · + ω) h ( · ) − h ρ

˜

+ ∂0 ∂1 H1 u( · ), u( · + ω) h ( · ) h ( · + ω) − h ( · + ω)

ρ 2 ≤ 2M1 ΔN+ + M1 Δ N+ M1 Δ N+ 2 = Δ 2M1 (N+ + N+ ) .

From this, we deduce immediately the claim in (72) using the Neumann series.

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6.3. Iteration of the inductive step and convergence to a solution The rest is very standard and can be done by invoking some of the implicit function theorems in the literature. For example, [31, 52, 54, 55]. See [27] for an exposition of different methods. However, since we have used some non-standard spaces it is perhaps clearer to run over the argument giving the convergence. We hope that this will make this paper more self-contained. We consider a system which satisfies the hypothesis of Theorem 1. We label with a subindex n all the elements corresponding to the n iterative step. We start with a function defined in a domain parametrized by ρ0 . We choose a sequence of parameters ρn = ρn−1 − ρ0 2−n−1 . We try the iterative step so that the n iterative step starts with a function un defined in a domain ρn−1 and ends up with a function un+1 defined in a domain of radius ρn+1 . We note that assuming that we can take J steps to which we can apply Lemma 4 and that, in all the steps, the non-degeneracy conditions are bounded uniformly, we have: −J(2τ +A) εJ ≤ κ−2 ρ−η K(N+ , N− , T, T˜)ε2J−1 0 2 −η(1+2) −(J+2(J−1))η 1+2 22 ≤ κ−2−2 · 2 ρ 2 K ε

J−2

0

≤

J −η(1+2+···+2J ) κ−2−2 · 2−···−2 · 2 ρ0

2−(J+2(J−1)+···+2

J−1

≤ (κ−4 ρ−2η 2B K 2 )2 0

(87)

· 1)η K 1+2+ · +2 ε2 0 J

J

J

∞ where B = j=0 j2−j . We see that under the assumptions in Theorem 1, the term in parenthesis in the RHS of (87) is smaller than 1. Indeed, by making ε0 small enough, we can make it as small as desired. 6.4. Uniqueness of the solution The proof of uniqueness is based on uniqueness of the solution of (51). In the language of [54], the uniqueness shows that there is an approximate inverse. We give the easy details. We assume that, besides u, there was another solution u ˆ. We note that, since they are solutions, applying Lemma 1 0 = E[ˆ u] − E[u] = E [u](ˆ u − u) + R(u, u ˆ − u) u − u||2ρ . with ||R||ρ ≤ M ||ˆ Now, denoting as before h = Id +u and recalling that d E [u]h (θ) = E[u] = 0 dθ we can write the equation (88) as: h (θ) E [u](ˆ u − u) (θ) − (ˆ u − u)(θ) E [u]h (θ) = −h R .

(88)

(89)

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The left hand side of (89) is the equation we studied in Section 5.4 and the subsequent sections. Notingthat the factorization (59) of the LHS of (89) achieved in Section 5.5 shows that h R = 0. Using the uniqueness statements for the solution, we conclude that for any 0 < ρ < ρ, ˆ||2ρ ||u − u ˆ||ρ ≤ Cκ−2 (ρ − ρ )−η K||u − u

(90)

where K is an algebraic expression which depends on the non-degeneracy constants. If we apply repeatedly the argument above choosing a sequence of domains ρn = ρn−1 − 2−n ρ0 , we obtain proceeding as in (87) 2n ||u − u ˆ || ||u − u ˆ||ρ ≤ C 2 κ−4 K 2 2B ρ−2η ρ 0

(91)

with B the same fixed number before. ||u − u ˆ||ρ < 1, we conclude, by taking the limit as n → ∞ If C 2 κ−4 K 2 2B ρ−2η 0 in (91) that ||u − u ˆ||ρ0 /2 = 0 which is the desired conclusion about uniqueness of the solution.

7. Lindstedt series for quasiperiodic solutions in extended systems If our model has a small parameter μ measuring the distance to integrable, it is natural to try to solve the equilibrium equations (21) as formal power series in μ. These formal power series have been considered for Hamiltonian systems and are called the Lindstedt series. In this section, we want to discuss the existence of these series to all orders (see Lemma 7) and the convergence in some cases (see Lemma 2). 7.1. Existence of Lindstedt series to all orders In the case of Hamiltonian systems, the existence of Lindstedt series to all orders is proved in [46]. The proof presented here is similar to the proof in [16,27] for the twist mapping case. Lemma 7. Consider the model given by 1 HL (x0 , . . . , xL ) = AL (x0 − xL )2 + μHL1 (x0 , . . . , xL ; μ) 2 where all the HL1 are analytic in all the variables. Assume that H1 The frequency ω satisfies

2AL cos(2πkω) − 1 ≥ κ|k|−τ ∀k ∈ Zd − {0} .

L

(92)

(93)

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H2 The interactions HL satisfy for some ρ > 0 and for all μ of sufficiently small modulus ||HL1 ||ρ ≤ R L

|AL | ≤ R

(94)

L

where A is uniform in μ. ∞ Then, there exist uμ (θ) = 0 μi ui (θ) a formal power series in μ with coefficients analytic in θ solving the equation (21) in the sense of formal power series. The solution is unique if we impose the normalization condition dθ u(θ; μ) = 0 . (95) T

Moreover, in the case that the interactions are finite range and trigonometric polynomials, the result above can be improved in two ways: • We do not need the assumption (93). It suffices that

2AL cos(2πkω) − 1 = 0 ∀k ∈ Z .

L

• The un are trigonometric polynomials of degree less than Kn. It is interesting to compare the conditions of Theorem 1 and Lemma 7. Of course, one important difference is that Theorem 1 is not formulated with reference to an integrable model, whereas Lemma 7 is. Nevertheless, for the common ground of nearly integrable systems, they present differences. In particular, the Diophantine conditions are different. Note that the non-degeneracy conditions of Theorem 1 are implicit in the choice of the unperturbed model. We also note that the condition on ω in Theorem 1 is the standard Diophantine condition whereas the Diophantine condition in Lemma 7 is the condition (93). The later condition depends on the coefficients of the integrable part of the interaction. We will explore the relation between the Diophantine coefficients of both results later. In Lemma 8 we will find some conditions on the coefficients AL that ensure that the sets of ω satisfying the conditions of Lemma 7 are of full measure. Nevertheless the conditions (93) are stronger than those in Theorem 1. The most interesting difference among the conditions is that Lemma 7 requires much weaker decay properties in the interaction than those in the main KAM result, Theorem 1. Of course, one can wonder whether the Lindstedt series whose existence is asserted in Lemma 7 converge or not. In Lemma 2 we will present a result that asserts the convergence of the series provided that the model satisfies the hypothesis of Theorem 1 as well as the hypothesis of Lemma 7. The argument is an indirect one based on the KAM theorem following an argument of [42].

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Since some of the hypothesis of Lemma 7 are weaker than those of Theorem 1 we think it would be interesting to see if there is a direct proof of convergence of the Lindstedt series, specially if such a proof can deal with decay conditions such as those considered by the KAM approach Theorem 1. See [12,15,21,22] for proofs of some standard KAM theorems using the method of compensations. Proof. In our case, the equation (21) reads 0=

∞

AL u(θ + Lω) + u(θ − Lω) − 2u(θ)

L=1

+μ

∂j HL θ + kω + u(θ + kω; μ), . . . ,

L k+j=i

(96)

θ + (k + j)ω + u θ + (k + j)ω; μ , . . . , θ + (k + L)ω + u θ + (k + L)ω; μ ; μ .

We denote [Γu](θ) =

∞

AL u(θ + Lω) + u(θ − Lω) − 2u(θ) .

L=1

We write equation (96) as Γuμ = N (uμ ) where N (u) is defined as the other terms in (96). Proposition 2. In the conditions (94), given a function

u : Tρ˜ × {μ ∈ C |μ| < β} → C such that • For each fixed θ, u(θ; μ) is polynomial in μ. • u(θ; 0) = 0. Then, the sums

HL γμL (θ)

L

converge uniformly in θ ∈ T × {μ ∈ C||μ| < a1 for some a1 > 0. ρ˜

The proof of Proposition 2 is just the observation that for small enough μ then u(θ, μ) is in the domain of definition of the all the H L and, therefore the sup norm the HL controls the composition. By the assumption (93), we see that equations for ϕ given η with η(θ) satisfying Γϕ = η

(97)

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have an analytic solution if and only if η = 0. Moreover, the solution is unique up to an additive constant. In particular, it is unique if we add the normalization ϕ = 0. The solution of (97) is given, in Fourier coefficients by 2AL cos(2πkω) − 1 . ϕk = ηk / L

Hence, if η is an analytic function and ω satisfies the inequalities (93), we obtain that ϕ is analytic. If η is a trigonometric polynomial, provided that the RHS of (93) does not vanish, we obtain that there is trigonometric polynomial ϕ solving the equations (97). Equating the terms independent of μ in (96), we obtain Γu0 = 0 . Which shows that u0 is a constant. If we impose the normalization (95) Equating terms of order μn , n ≥ 1 on both sides of (96) we obtain Γun = ηn (u0 , u1 , . . . , un−1 )

(98)

where ηn is a polynomial whose coefficients depend on the derivatives of HL up to order n − 1. Note that, by the assumption H2, using Proposition 2, we conclude that if u0 , . . . , un−1 are analytic functions on some strip so is ηn , therefore, we can formulate the equations for un+1 . Our next task is that the equations can be solved and that the solution un+1 is analytic in Tρ for any ρ < ρ˜. Note that if the interactions are finite range and all of the terms are trigonometric polynomials, it follows by induction that ηn is a trigonometric polynomial and that therefore, the un is a trigonometric polynomial. Using the discussion of the solutions (97), we see that if we know u0 , . . . , un−1 we can find un provided that ηn = 0 and that this un is unique under the normalization (95), which implies un = 0. Hence, Lemma 7 is proved once we show that we have dθ ηn u0 (θ), u1 (θ), . . . , un−1 (θ) = 0 . (99) T

To establish this, we now assume that we have u≤n (θ) = fying (96) up to errors of order μn+1 . We have, therefore that Γu[≤n] − N (u[≤n] ) = μn+1 ηn + μn+2 Sn . By Lemma 3, we have

n i=0

μi ui (θ) satis-

(100)

(h≤n ) Γu[≤n] − N (u[≤n] ) n+1 =μ ηn (h[≤n] ) + O(μn+2 ) .

0=

(101)

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Taking into account that (h[≤n] ) = Id +O(μ), we obtain that the RHS of (101) is μn+1 ηn + O(μn+2 ) therefore, we obtain, equating the terms of μn+1 in (101), we obtain ηn = 0 as desired. 7.2. On the Diophantine conditions (93) The conditions (93) are different from the standard Diophantine conditions. In this section, we study these conditions. Our first result is a result on the abundance of numbers satisfying (93). Definition 1. We say that a function F of period 1 is non degenerate when 1. It has a finite number of zeros in [0, 1). 2. All the zeros are of finite order. That is, if F (z0 ) = 0, there exists N ∈ Z such that F (N ) (z0 ) = 0. Lemma 8. With the notations of Lemma 7. Assume that the function F (z) = 2AL cos(2πz) − 1 L=1

is a non-degenerate function in the sense of Definition 1. Then, the set of ω satisfying (93) is of full measure. Note that that the simplest case of the Lemma above is the case when F (z) is a C 2 Morse function. This condition is generic among the set of coefficients |Aj | ≤ Cj −(3+δ) endowed with the product topology. When the interaction is finite range, only a finite number of AL are zero, so that the function F (z) is a trigonometric polynomial which satisfies the assumption of Lemma 8. As a corollary of Lemma 8, we obtain that, under the conditions of the Lemma – in particular, for finite range interactions – there is a full measure set of ω satisfying both (40) and (93). On the other hand, for any Diophantine ω, it is possible to choose AL so that the denominators in (93) become zero (or grow very fast). Note that the functions F (z) always satisfy F (0) = 0. The conditions (93) are, roughly, that the values of ωk − n are not too close to the zeros of F . Hence, in particular, all the numbers that satisfy condition (93) are Diophantine. On the other hand, it is not hard to produce numbers that are Diophantine but do not satisfy condition (93). Proof. By the assumption on F , since F has a finite number of critical points, it has a finite number of intervals of monotonicity. Therefore, given an interval I, F −1 (I) is the union of a number of intervals smaller than the number of intervals of monotonicity of F . We have bounds for the length |F −1 (I)| ≤ A|I|α . Therefore, the sets

Cκ,τ,k = ω||F (k · ω)| ≤ κ|k|−τ

are the union of a finite number of regions bounded by planes.

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If Br ⊂ R is a ball of radius r we can estimate the measure |Cκ,τ,k ∩ Br | ≤ Cκα k −(τ +1)α . We observe that if (τ + 1)α > 1, we have | ∪k∈Z−{0} Cκ,τ,k ∩ Br | ≤ Cκα . This shows that, if we take κ small enough, we can make the measure of the sets where the inequality (93) fails as small as desired. Note that, as a consequence of the proof, if we take κ ≤ Cr1/α , we can ensure that there is a point satisfying (93) in any ball of radius α. 7.3. Convergence of the Lindstedt series The argument is based in an argument due to [42], but it is simpler in this case. The basic idea is that we can apply a rapidly convergent quadratic procedure in a space of functions which are analytic in μ. We note that an interaction of the form (92), if we take μ = 0, the system satisfies the non-degeneracy assumptions of Theorem 1. Moreover, taking μ small enough, we obtain that the smallness assumptions are satisfied. Of course, to apply Theorem 1 we need that the interactions decrease fast enough. Theorem 2. Consider a system of the form (92) satisfying the hypothesis of Lemma 7 Assume furthermore that for small μ, the system satisfies uniformly the hypothesis of Theorem 1 and that the frequency ω is Diophantine. Then, the Lindstedt series produced in Lemma 7 converges in a neighborhood of zero. Note that this theorem requires both the Diophantine conditions of the Lemma 7 and the rapid decrease conditions for the interaction on Theorem 1. We think that it would be quite interesting to study whether there is a proof of convergence of the Lindstedt series without using fast decay properties of the interaction. This would lead to a proof of existence of smooth quasi-periodic solutions in situations not covered by Theorem 1. Proof. Following [42], we consider a space of functions which are jointly analytic in μ, θ and consider the original problem of finding solutions of (3) as a functional analysis problem in the space of analytic functions of the variables μ, θ. We claim that there is Theorem analogous to Theorem 1 for functions depending analytically on extra parameters. We just indicate the argument. Some more details on how to lift KAM theorems from functions to families of functions can be found in [29, 31, 52]. The paper [29] lifts the main theorem from [54] to spaces of families obtain KAM theorems for some very degenerate system. We consider spaces of functions Ar,ρ whose domain is the set |μ| ≤ r, θ ∈ Tρ . We will assume that the functions are continuous in the whole domain and analytic in the interior. We endow these spaces with the norm of the supremum.

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Our goal is to show an analogue of Theorem 1 for families depending on parameters. That is, we will assume that we have families of interactions depending analytically on parameters and satisfying the non-degeneracy assumption uniformly for the parameters. Then, if we have a analytic family that satisfies the equilibrium equations with enough accuracy, then there is another family which satisfies the equilibrium equation exactly. Moreover we also have the same estimates for the distance between the approximate solution and the true one. We note that the Procedure 1 can be implemented in spaces of functions depending on the parameter μ. Provided that the estimates assumed in Theorem 1 hold uniformly, we observe that for each value of the parameter μ we obtain uniform estimates of the same form as those we had before. We can also check that if the family u(θ; μ) depends analytically in the parameter μ, then the improved solution also depends analytically in μ. This is obvious from the fact that the procedure to find u ˜ consists only in composing the function with the interaction terms, applying some elementary perturbation argument and then, solving some small divisors whose coefficients do not depend on μ. Therefore, we conclude that the new solution belongs to Ar,ρ˜ and that the new error satisfies the same estimates. The rest of the argument for convergence does not need any change. As was emphasized in Theorem 1, we obtain that the size of ε which is allowed in the argument, only depends on the size of the non-degeneracy conditions. Once we have that the result Theorem 1 can be adapted to families, we see that we can take as initial guess just u(θ, μ) = 0. Then, we obtain that there is a solution which is analytic. By the uniqueness of the Lindstedt series, we obtain that the Lindstedt series coincides with the Taylor expansion of the function.

8. Some final remarks 8.1. A normal hyperbolicity approach for quasi-integrable systems with finite range In this section we study the normal hyperbolicity properties of the quasi-integrable systems whose interaction terms are in (92). We note that the equilibrium equations are written explicitly in (18). In order to be able to use a a dynamical description, we will assume that the interaction has finite range. That is, we will assume that Aj = 0 for j ≥ J. For μ = 0, the equilibrium equations are linear. They are: (xn+1 + xn−1 − 2xn ) + A1 (xn+2 + xn−2 − 2xn ) + · · · + AJ−1 (xn+J + xn−J − 2xn ) = 0 . We can write (102) as a dynamical system in R2J . Denoting xn = (xn+J , xn+(J−1)) , . . . , xn−J+1 , xn−J ) .

(102)

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The equilibrium equations can be written as a linear system xn+1 = Mxn where

Mx =

1 (xn+1 + xn−1 − 2xn ) AJ−1 A1 + (xn+2 + xn−2 − 2xn ) AJ−1 + ··· AJ−2 + (xn+(J−2) + xn−(J−2) − 2xn ) AJ−1 − xn−J + 2xn , xn+J−1 , xN +J−2 , . . . , xn−J+1 .

(103)

It is not hard to compute the characteristic equation for the operator M. It suffices to try solutions of the form xn = λn in (102). The characteristic equation is: 0 = λJ−1 (λ − 1)2 + A1 λJ−2 (λ2 − 1)2 + · · · + AJ−1 (λJ − 1)2 = (λ − 1)2 λJ−1 + A1 λJ−2 (λ + 1) + · · · + AJ−1 (λJ−1 + λJ−2 + · · · λ1 ) .

(104)

We note that the characteristic polynomial P (λ) in (104) satisfies P (1/λ) = P (λ)λ2J . One implication is that if λ0 is a root of the characteristic polynomial, then 1/λ0 is also a root. From the factorization in (104), we conclude has always a double root 1. Note also that the A coefficients are factors of the highest degree terms in the characteristic equation. We note that for the case A = 0 it has 0 as a root of multiplicity J −1. We conclude that if all the A’s are small there will be J −1 roots (counted with multiplicity) near zero. Because of the symmetry of the polynomial, we conclude that their inverses will also be roots. Hence, for AJ > 0 and all the roots small, there will be two eigenvalues exactly 1, J − 1 eigenvalues of small modulus and J − 1 of large modulus. The space corresponding to the eigenvalues 1 will be invariant and will be normally hyperbolic. If we now add a non-integrable perturbation of the system, applying the theory of persistence of normally hyperbolic manifolds [17, 18, 23], we conclude that there will be a two dimensional invariant manifold. By the results of [53], the perturbed evolution system is symplectic. Hence, the system restricted to the invariant manifold will preserve the restricted form. The restricted form will be closed and, since in the unperturbed case, it is nondegenerate, the restricted form in the quasi-integrable case, will be symplectic.

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Therefore, the quasi-periodic solutions in the quasi-integrable case are quasi-periodic solutions corresponding to a one degree of freedom hamiltonian system. Of course, this approach does not give a clue of what can happen for systems far from integrable or for systems with interactions of infinite range. Since center manifolds could fail to be even C ∞ it is not clear that this approach can produce straightforwardly the analytic results presented here. On the other hand, it seems that this normal hyperbolicity approach can produce results in the case that the xi are multidimensional variables. 8.2. Numerical implementation Note that Procedure 1 is an algorithm that can be readily implemented as a numerical algorithm. An efficient way of carrying out the computation is to keep the function u discretized as a Fourier series. Denote by N the number of Fourier coefficients kept. The step 1 is diagonal in Fourier series so, it has a cost O(N ). The operator G that is needed for step 2 is just a combination of translation and evaluation of derivatives. We note that, in Fourier series, translating or differentiating Fourier series is just an O(N ) operation. If we use the Fast Fourier Transform algorithm (henceforth FFT), we can compute the products needed in O(N log(N )). Again, the computation of C0,1,1 + G)−1 is diagonal on real space, so that if we use the FFT (cost O(N log(N ))), then the computation is just O(N ). The computation of T is just division by a number. Actually, for the next step, the only thing that we need is to project (C0,1,1 + G)−1 (ϕ), which in Fourier series can be accomplished by setting to zero the zeroth order coefficient. Then, the computation of w is O(N ) in Fourier coefficients. The computation of u ˜ is O(N ) in real space (some of the computations before give us a computation of h , which is diagonal in Fourier space anyway). In summary, the steps of Procedure 1 are diagonal either in real space or in Fourier space. These diagonal operations cost O(N ) operations. The FFT needed to switch from real space to Fourier representation have a cost of O(N log(N )). Therefore, Procedure 1 can be implemented in O(N log(N )) operations. Preliminary implementations in [7] indicate that indeed it is possible to implement this algorithm quite efficiently. We also note that the existence of a variational principle (23) can also be taken advantage using minimization methods. In the case that the system is ferromagnetic, it is not too hard to show that all the invariant circles are minimizers [36]. Hence, in the case that the system is ferromagnetic, one can combine the good global properties of the variational method till one gets close enough to a solution so that one can use the Newton method. For the case of twist mappings this is the method that was used in [44, 45].

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Acknowledgements I thank R. Calleja, A. Gonz´ alez-Enr´ıquez and J. Vano for discussions on the subject. A. Gonz´ alez-Enr´ıquez and R. Calleja gave a very careful reading to a preliminary version of the manuscript, which improved the presentation and removed several mistakes. The work of the author has been supported by NSF grants.

References [1] S. Aubry and P. Y. Le Daeron, The discrete Frenkel–Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D 8 (3) (1983), 381–422. [2] S. B. Angenent, Monotone recurrence relations, their Birkhoff orbits and topological entropy, Ergodic Theory Dynam. Systems 10 (1) (1990), 15–41. [3] V. Bangert, Mather sets for twist maps and geodesics on tori, In Dynamics reported, Vol. 1, pages 1–56. Teubner, Stuttgart, 1988. [4] J. Bricmont, K. Gaw¸edzki and A. Kupiainen, KAM theorem and quantum field theory, Comm. Math. Phys. 201 (3) (1999), 699–727. [5] O. M. Braun and Y. S. Kivshar, Nonlinear dynamics of the Frenkel–Kontorova model, Phys. Rep. 306 (1–2) (1998), 108. [6] O. M. Braun and Y. S. Kivshar, The Frenkel–Kontorova model, Texts and Monographs in Physics. Springer-Verlag, Berlin, 2004. ISBN 3-540-40771-5. Concepts, methods, and applications. [7] R. Calleja and R. de la Llave, Computation of quasiperiodic equilibrium states in 1-d statistical mechanics models, (2008), Manuscript. [8] A. Candel and R. de la Llave, On the Aubry–Mather theory in statistical mechanics, Comm. Math. Phys. 192 (3) (1998), 649–669. [9] P. Cardaliaguet, F. Da Lio, N. Forcadel and R. Monneau, Dislocation dynamics: A non-local moving boundary, In Free boundary problems, volume 154 of Internat. Ser. Numer. Math., pages 125–135. Birkh¨ auser, Basel, 2007. [10] A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computerassisted applications, In Dynamics Reported, pages 60–129. Springer, Berlin, 1995. [11] A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc. 187 (878) (2007), viii+134. [12] L. Chierchia and C. Falcolini, A direct proof of a theorem by Kolmogorov in Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 21 (4) (1994), 541–593. [13] A. Delshams and R. de la Llave, KAM theory and a partial justification of Greene’s criterion for nontwist maps, SIAM J. Math. Anal. 31 (6) (2000), 1235–1269 (electronic). [14] S. Elaydi, An introduction to difference equations, Undergraduate Texts in Mathematics. Springer, New York, third edition, 2005. ISBN 0-387-23059-9. [15] L. H. Eliasson, Absolutely convergent series expansions for quasi periodic motions, Math. Phys. Electron. J. 2 (1996), Paper 4, 33 pp. (electronic).

878

R. de la Llave


[16] C. Falcolini and R. de la Llave, A rigorous partial justification of Greene’s criterion, J. Statist. Phys. 67 (3–4) (1992), 609–643. [17] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/1972), 193–226. [18] N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1973/74), 1109–1137. [19] J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning, Acad. Sci. U.S.S.R. J. Phys. 1 (1939), 137–149. [20] J. Fr¨ ohlich and C.-E. Pfister, Spin waves, vortices, and the structure of equilibrium states in the classical XY model, Comm. Math. Phys. 89 (3) (1983), 303–327. [21] G. Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review. Rev. Math. Phys. 6 (3) (1994), 343–411. [22] G. Gallavotti and G. Gentile, Majorant series convergence for twistless KAM tori, Ergodic Theory Dynam. Systems 15 (5) (1995), 857–869. [23] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Springer-Verlag, Berlin, 1977. Lecture Notes in Mathematics, Vol. 583. ` Jorba, R. de la Llave and M. Zou, Lindstedt series for lower-dimensional tori. [24] A. In Hamiltonian Systems with Three or More Degrees of Freedom (S’Agar´ o, 1995), pages 151–167. Kluwer Acad. Publ., Dordrecht, 1999. [25] S. M. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch. 46 (1983), 99–123. English translation: Trans. Moscow Math. Soc., Issue 2 (1984), 101–126. ` Jorba and J. Villanueva, KAM theory without action[26] R. de la Llave, A. Gonz´ alez, A. angle variables, Nonlinearity 18 (2) (2005), 855–895. [27] R. de la Llave, A tutorial on KAM theory, In Smooth ergodic theory and its applications (Seattle, WA, 1999), pages 175–292. Amer. Math. Soc., Providence, RI, 2001. Revised version available from ftp.ma.utexas.edu/pub/papers/llave/tutorial. pdf. [28] R. de la Llave, Improved estimates on measure occupied and the regularity of invariant tori in hamiltonian systems by zehnder’s method, Preprint (2005). [29] R. de la Llave and R. Obaya, Decomposition theorems for groups of diffeomorphisms in the sphere, Trans. Amer. Math. Soc. 352 (3) (2000), 1005–1020. [30] R. de la Llave and E. Valdinoci, Ground states and critical points for Aubry–Mather theory in statistical mechanics, MP ARC # 06-279. [31] R. de la Llave and J. Vano, A Whitney–Zehnder implicit function theorem, Manuscript (2000). [32] R. de la Llave and E. Valdinoci, Critical points inside the gaps of ground state laminations for some models in statistical mechanics, J. Stat. Phys. 129 (1) (2007), 81–119. [33] R. de la Llave and E. Valdinoci, Ground states and critical points for generalized Frenkel–Kontorova models in Zd , Nonlinearity 20 (10) (2007), 2409–2424.

Vol. 9 (2008)


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[34] M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, In Smooth ergodic theory and its applications (Seattle, WA, 1999), volume 69 of Proc. Sympos. Pure Math., pages 733–746. Amer. Math. Soc., Providence, RI, 2001. [35] J. N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology 21 (4) (1982), 457–467. [36] J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems. In Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), pages 92–186. Springer, Berlin, 1994. [37] D. C. Mattis, The many-body problem. An encyclopedia of exactly solved models in one dimension, Singapore: World Scientific, 1993. [38] D. C. Mattis, The Theory of Magnetism Made Simple: An Introduction to Physical Concepts and to Some Useful Mathematical Methods. Singapore: World Scientific., 2006. ISBN 9812386718. [39] J. D. Meiss, Symplectic maps, variational principles, and transport, Rev. Modern Phys. 64 (3) (1992), 795–848. [40] J. Moser, A rapidly convergent iteration method and non-linear differential equations. II, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 499–535. [41] J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 265–315. [42] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. [43] J. Moser, A stability theorem for minimal foliations on a torus, Ergodic Theory Dynamical Systems 8∗ (Charles Conley Memorial Issue) (1988), 251–281. [44] I. C. Percival, Variational principles for the invariant toroids of classical dynamics, J. Phys. A 7 (1974), 794–802. [45] I. C. Percival, A variational principle for invariant tori of fixed frequency, J. Phys. A 12 (3) (1979), L57–L60. [46] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, volume 1, 2, 3, Gauthier-Villars, Paris, 1892–1899. [47] D. Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York, Amsterdam, 1969. [48] H. R¨ ussmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. In Dynamical Systems, Theory and Applications (Battelle Rencontres, Seattle, Wash., 1974), pages 598–624. Lecture Notes in Phys., Vol. 38. Springer, Berlin, 1975. [49] H. R¨ ussmann, Note on sums containing small divisors, Comm. Pure Appl. Math. 29 (6) (1976), 755–758. [50] D. A. Salamon, The Kolmogorov–Arnold–Moser theorem, Math. Phys. Electron. J. 10 paper 3, 37 pp. (electronic), (2004). [51] D. Salamon and E. Zehnder, KAM theory in configuration space, Comment. Math. Helv. 64 (1) (1989), 84–132. [52] J. A. Vano, A Nash–Moser Implicit Function Theorem with Whitney Regularity and Applications, Ph.D. thesis, Univ. of Texas at Austin, 2002. MP ARC #02-276.

880

R. de la Llave


[53] A. P. Veselov, Integrable mappings, Uspekhi Mat. Nauk 46 (5(281)) (1991), 3–45, 190. [54] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math. 28 (1975), 91–140. [55] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math. 29 (1) (1976), 49–111. Rafael de la Llave Department of Mathematics University of Texas 1 University Station C1200 Austin, TX 78712-0257 USA e-mail: [email protected] Communicated by Jean Bellissard. Submitted: January 16, 2008. Accepted: March 3, 2008.

Ann. Henri Poincaré 9 (2008), 881–926 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/050881-46, published online July 9, 2008 DOI 10.1007/s00023-008-0375-7


Fibrés de Green et Régularité des Graphes C 0-Lagrangiens Invariants par un Flot de Tonelli Marie-Claude Arnaud Abstract. In this article, we prove different results concerning the regularity of the C 0 -Lagrangian invariant graphs of the Tonelli flows. For example : • in dimension 2 and in the autonomous generic case, we prove that such a graph is in fact C 1 on some set with (Lebesgue) full measure; • under certain dynamical additional hypothesis, we prove that these graphs are C 1 . Résumé. Dans cet article, on démontre différents résultats concernant la régularité des graphes C 0 -lagrangiens invariants par des flots de Tonelli. Par exemple : • en dimension 2, dans le cas autonome et générique, on montre que ces graphes sont de classe C 1 sur un ensemble de mesure (de Lebesque) pleine ; • sous certaines hypothèses concernant la dynamique restreinte, on montre que ces graphes sont de classe C 1 .

1. Introduction 1.1. Préambule Dans cet article, nous nous intéressons à la régularité des graphes C 0 lagrangiens invariants par un flot de Tonelli, ou, ce qui revient au même, à la régularité des solutions continˆ ument différentiables de l’équation de Hamilton– Jacobi associée à un hamiltonien de Tonelli. Il est bien connu que de telles solutions sont de classe C 1,1 (i.e. que le graphe 0 C -lagrangien correspondant est lipschitz). Ce résultat est montré par A. Fathi dans [10] (nous renvoyons a` la section 4 pour plus de détails). Avant lui, et dans un cadre un peu différent, le même résultat avait été obtenu par :

882

M.-C. Arnaud


• G. D. Birkhoff dans le cas des applications de l’anneau déviant la verticale (voir par exemple [16]) ; • M. Herman dans le cas des difféomorphismes symplectiques du fibré cotangent T ∗ Tn du tore qui admettent une fonction génératrice globale (voir [15]). Citons aussi les beaux résultats de J. Mather concernant les mesures minimisantes, résultats contenus dans l’article [21] et qui ont des liens certains avec ceux que nous venons de citer. On se demande alors si on peut dire plus sur la régularité de ces solutions, sous diverses hypothèses : dynamiques, de dimension . . . Avant de parler des résultats que nous obtenons dans ce sens, nous allons rappeler quelques définitions et notations. 1.2. Notations et définitions Dans toute la suite, M sera une variété compacte et connexe munie d’une métrique riemannienne. Un point du fibré tangent T M sera noté (x, v) avec x ∈ M et v vecteur tangent en x. La projection π : T M → M s’écrit alors (x, v) → x. Un point du fibré cotangent T ∗ M sera noté (x, p) avec p ∈ Tx∗ M et π ∗ : T ∗ M → M désignera la projection canonique (x, p) → x. On s’intéresse alors à un lagrangien L : T M × T → R qui est de classe au moins C 2 et : • uniformément superlinéaire : uniformément en (x, t) ∈ M × T, on a : limv→+∞ L(x,v;t) = +∞ ; v • strictement convexe : pour tout (x, v; t) ∈ T M × T, positive ;

∂2L ∂v 2 (x, v; t)

est définie

• complet. Un tel lagrangien sera appelé un “lagrangien de Tonelli”. On peut associer à un tel lagrangien l’application de Legendre L = LL : eomorphisme T M ×T → T ∗ M ×T définie par : L(x, v; t) = ∂L ∂v (x, v; t) qui est un diff´ fibré de classe C 1 et un hamiltonien H : T ∗ M × T → R défini par : H(x, p; t) = p(L−1 (x, p; t)) − L(L−1 (x, p; t)). Le hamiltonien H est alors uniformément superlinéaire, strictement convexe dans la fibre, de classe C 2 et complet (un tel hamiltonien sera dit de Tonelli). Tout comme a` un lagrangien de Tonelli on peut associer un hamiltonien de Tonelli, a` chaque hamiltonien de Tonelli on peut associer un L ) le flot d’Euler–Lagrange associé à L lagrangien de Tonelli. On notera alors (ft,s H L −1 et (Φt,s ) le flot hamiltonien associé à H ; on a alors : ΦH . t,s = L ◦ ft,s ◦ L 0 Suivant [15], on appellera “graphe C -lagrangien” le graphe d’une 1-forme λ : M → T ∗ M qui est continue et fermée au sens des distributions. Remarquons a ce sujet que tout au long de l’article nous parlerons de “graphe”, comme c’est ` l’usage, alors qu’il serait sans doute plus correct de parler de section du fibré cotangent. Les résultats que nous obtenons sont alors les suivants :

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1.3. Résultats Théorème 1. 1 Soit M une variété de dimension 2, L : T M → R un lagrangien indépendant du temps satisfaisant les hypothèses de Tonelli, H : T ∗ M → R le hamiltonien qui lui est associé dont on suppose que toutes les singularités sont non dégénérées. Soit G un graphe C 0 -lagrangien invariant par le flot hamiltonien de H. Soit λ la 1-forme fermée (au sens des distributions) dont G est le graphe. Alors, il existe un Gδ dense D de mesure pleine de M tel qu’en tout point de D, λ est différentiable et Dλ est continue. Signalons qu’“avoir toutes ses singularités non dégénérées” est une condition générique pour les hamiltoniens de classe C 2 (et même de classe C k avec k ≥ 2). Nous renvoyons le lecteur a` la sous-section 4.2.2 pour la définition précise. Ce théorème nous dit donc que pour les hamiltoniens génériques des fibrés cotangents des surfaces, les graphes C 0 -lagrangiens invariants par le flot hamiltonien sont plus réguliers que simplement lipschitz, puisqu’ils sont de classe C 1 sur un ensemble de (Lebesgue) mesure pleine. Nous donnons en fin de sous-section 4.2.2 des exemples qui montrent pourquoi notre démonstration de ce résultat n’est pas valable en dimension plus grande. Mais nous ne connaissons pas de contre-exemple au résultat en dimension plus grande, et il serait intéressant de pouvoir en exhiber. Le cas des hamiltoniens dépendant du temps sur T ∗ T est similaire, et a déjà été traité dans [1]. Nous nous contentons de le rappeler : Théorème ([1]). Soit L : T T1 × T1 → R un lagrangien satisfaisant les hypothèses de Tonelli, H : T ∗ T1 × T1 → R le hamiltonien qui lui est associé et G un graphe continu invariant par le temps 1 du flot hamiltonien de H. On suppose que ce graphe invariant est le graphe de λ. Alors il existe un Gδ dense D de mesure pleine de T tel qu’en tout point de D, λ est différentiable et Dλ est continue. Donnons maintenant les résultats que nous obtenons en faisant des hypothèses sur la dynamique : Théorème 2. 2 Soit L : T Tn × T1 → R un lagrangien satisfaisant les hypothèses de Tonelli, H : T ∗ Tn × T1 → R le hamiltonien qui lui est associé et G un graphe C 0 -lagrangien invariant par le temps 1 du flot hamiltonien de H. On suppose que le temps 1 du flot hamiltonien de H restreint a ` G est bi-lipschitz conjugué a ` une rotation. Alors le graphe G est de classe C 1 . Dans le cas de la dimension 1, on obtient un résultat plus précis : 1 Une

version plus précise de ce r´ esultat se trouve en sous-section 4.2.2 sous le nom de proposition 4.18. 2 Ce r´ esultat se retrouve dans le corps de l’article sous le nom de corollaire 4.13, en sous-section 4.2.

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Proposition 1.1. 3 Soit L : T T1 ×T1 → R un lagrangien satisfaisant les hypothèses de Tonelli, H : T ∗ T1 × T1 → R le hamiltonien qui lui est associé et G un graphe continu invariant par le temps 1 du flot hamiltonien de H. On suppose que le temps 1 du flot hamiltonien de H restreint a ` G est bi-lipschitz conjugué a ` un difféomorphisme du cercle de classe C 2 de nombre de rotation irrationnel. Alors le graphe G est de classe C 1 . Enfin, le dernier résultat que nous obtenons concerne les hamiltoniens C 0 intégrables. Brièvement, un hamiltonien H : T ∗ M × T → R est dit C 0 intégrable s’il existe une partition P de T ∗ M en graphes C 0 -lagrangiens invariants par le temps 1 du flot hamiltonien telle que de plus l’application qui a` un élément de P associe sa classe de cohomologie est surjective sur H 1 (M, R). Théorème 3. Soit H : T ∗ M × T → R un hamiltonien de Tonelli C 0 -intégrable et Λ1 ⊂ Λ(M ) tel que {Gλ ; λ ∈ Λ1 } soit une partition de T ∗ M en graphes C 0 lagrangiens invariants par le temps 1 du flot hamiltonien (φH t ) de H. Alors, il existe un Gδ dense G(H) de Λ1 dont tout élément est de classe C 1 . Remarquons que nous ne connaissons pas d’exemple de hamiltonien C 0 intégrable qui ne soit pas C 1 -intégrable . . . Aussi, il serait intéressant soit d’en donner un exemple, soit d’améliorer le résultat du théorème précédent en montrant qu’un hamiltonien C 0 -intégrable est C 1 -intégrable. 1.4. Arguments-clefs des démonstrations Rappelons quelle est l’idée géométrique qui permet de montrer des inégalités à priori sur les graphes lagrangiens lipschitziens invariants par un flot de Tonelli (voir [16] en dimension quelconque, ou [15] pour le cas de l’anneau) : il est usuel d’utiliser l’image par le flot de Tonelli linéaire (i.e. la différentielle du flot de Tonelli hamiltonien) de la verticale V (x, p) = ker Dπ ∗ (x, p). Il se trouve que cette verticale est un plan lagrangien, et qu’il existe une manière classique de comparer entre eux différents plans lagrangiens transverses à un plan lagrangien donné 4 . Pour montrer des inégalités à priori, on “coince” alors le tangent au graphe invariant (aux points de différentiabilité du graphe lipschitz, qui est un ensemble de mesure pleine) entre deux images (l’une en temps positif, l’autre en temps négatif) de la verticale. Une idée extrêment naturelle est alors d’utiliser toutes les images des verticales (i.e. pour des temps quelconques) pour coincer ce plan tangent au graphe, puis de passer à la limite. On trouve alors deux fibrés lagrangiens au dessus des points de différentiabilité du graphe, qui s’appellent les fibrés de Green : ceci est détaillé en sous-section 3.2. Ces fibrés furent introduits par Leon W. Green dans [13] dans le cas du flot géodésique d’une métrique riemannienne, puis généralisés au cas des métriques finsleriennes par Patrick Foulon dans [12], au cas des hamiltoniens autonomes optiques par Gonzalo Contreras et Renato Iturriaga dans [9] et enfin au cas des applications symplectiques de T ∗ Td déviant la verticale par Misha Bialy 3 Ce

r´ esultat se retrouve dans le corps de l’article sous le nom de corollaire 4.14, en sous-section 4.2. est détaill´ e en sous-section 3.1

4 Ceci

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et Robert S. Mackay dans [8]. Ils ont été utilisés pour montrer des résultats de rigidité, des résultats sur l’hyperbolicité, mais à notre connaissance l’usage que nous en faisons ici, pour encadrer des plans tangents, et même, ainsi que nous le verrons lors de la sous-section 4.1, des cônes tangents (c’est une généralisation, empruntée à l’optimisation non lisse, de la notion de plan tangent), est nouvelle. Aux points o` u les deux fibrés de Green co¨ıncident, on arrive alors a` montrer que le graphe admet continument un plan tangent, i.e. un résultat de régularité C 1 . Un étude fine des fibrés de Green permet alors, sous des hypothèses variées (de dimension, dynamiques. . . ), de montrer qu’ils co¨ıncident en de nombreux points (voir le critère dynamique des sous-sections 3.5 et 3.6). Enfin, signalons que les arguments utilisés sont des arguments classiques des dynamiques lagrangiennes et hamiltonniennes : notion de point conjugué, de plan lagrangien, de solution de l’équation de Hamilton–Jacobi. . . sauf en ce qui concerne la démonstration du théorème 3, en sous-section 4.2.3, dont les arguments font de plus appel aux récentes théories K.A.M. faible développées par A. Fathi et P. Bernard (voir [11] et [6]) et aux théories concernant les mesures minimisantes développées par J. Mather (voir [21]). 1.5. Plan de l’article Tout le début de l’article sert a` étudier des fibrés (non forcément continus) lagrangiens ; c’est dans ce but qu’en section 2, on définit une notion de semi-continuité sur les champs de formes quadratiques, après avoir défini une relation d’ordre sur l’ensemble de ces champs, relation qui permettra en sous-section 3.1 de comparer entre eux des plans lagrangiens et de parler de fibrés lagrangiens semi-continus. Dans le reste de la section 3, on construit les fibrés de Green le long des orbites sans points conjugués, puis on montre différents résultats concernant ces fibrés : • on montre qu’ils sont semi-continus ; • on explique qu’ils servent a` encadrer les fibrés lagrangiens invariants par le flot hamiltonien qui sont transverses a` la verticale (sous-section 3.4) ; • on explique comment la dynamique du flot linéarisé (et plus spécifiquement la détermination de vecteurs dont l’orbite ne sort pas de tout compact) permet de trouver des vecteurs des fibrés de Green : ce résultat s’appelle le critère d’appartenance aux fibrés de Green (section 3.5) ; • en passant au quotient par la direction du champ de vecteurs hamiltonien, on construit, dans le cas autonome, des fibrés de Green “réduits”. En sous-section 4.1, on introduit une notion de différentielle généralisée (empruntée à l’optimisation non lisse) et de cˆ one tangent a` un graphe lipschitz. On explique alors pourquoi ces cˆ ones tangents sont eux-aussi encadrés par les fibrés de Green. Enfin, en sous-section 4.2, on démontre tous les résultats annoncés précédemment qui concernent la régularité des graphes C 0 -lagrangiens invariants.

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2. Semi-continuité des champs de formes quadratiques Nous intéressant à des fibrés lagrangiens transverses a` la verticale, nous serons amenés à étudier des champs de formes quadratiques (en effet, un plan lagrangien peut être vu, en coordonnées symplectiques, comme le graphe d’une matrice symétrique). Notation 1. Si E est un R-espace vectoriel de dimension finie, Q(E) est l’espace vectoriel des formes quadratiques sur E et Q+ (E) est le cône des formes quadratiques définies positives sur E. Définition 2.1. Soit E un espace vectoriel de dimension finie et q1 , q2 deux formes quadratiques sur E. On écrit : q1 ≺ q2 si : ∀v ∈ E, q1 (v) ≤ q2 (v). On écrit : q1 q2 si q2 − q1 est définie positive. Une suite (qn )n∈N est croissante (resp. strictement croissante) si : ∀n ∈ N, qn ≺ qn+1 (resp. qn qn+1 ). Remarques 2.2. Si (qn )n∈N est une suite croissante d’éléments de Q(E), on a : ∀n ≤ n , qn ≺ qn . La relation ≺ est transitive ; c’est même une relation d’ordre (partiel) sur Q(E). Définition 2.3. Soit π : F → B un fibré vectoriel topologique dont la fibre Fx = π −1 (x) est de dimension constante finie (par souci de concision, désormais, on dira juste “fibré vectoriel”). On appelle champ de formes quadratiques sur F une application q définie sur la base B telle que : ∀x ∈ B, q(x) ∈ Q(Fx ). Remarques 2.4. En d’autres termes, si Π : Q(F ) → B désigne le fibré vectoriel des formes quadratiques dans la fibre de F , un champ de formes quadratiques sur F est une section de Π. On note Q(F ) l’ensemble des champs de formes quadratiques de F . Notation 2. On note alors Qc (F ) l’ensemble des champs continus de formes quadratiques sur F . De plus, Q+ (F ) désignera l’ensemble des éléments de Q(F ) dont la restriction a chaque fibre est définie positive. ` Nous introduisons une notion de semi-continuité analogue a` celle concernant les fonctions a` valeurs réelles : Définition 2.5. Soit q un champ de formes quadratiques sur le fibré vectoriel π : F → B. Ce champ q est semi-continu supérieurement (resp. semi-continu inférieurement) si pour tout champ de formes quadratiques continu f : B → Q(F ) de F , l’ensemble {x ∈ B; q(x) f (x)} est ouvert dans B (resp. {x ∈ B; q(x) f (x)} est ouvert dans B). Remarque 2.6. Remarquons que q est semi-continue supérieurement si et seulement si −q est semi-continue inférieurement. En utilisant la remarque suivante :

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Remarque 2.7. Si E est un espace vectoriel de dimension finie et si q0 est une forme quadratique sur E, une base de voisinages ouverts de q0 dans Q(E) est u Uε = {q ∈ Q(E); q0 − ε q q0 + ε}. (Uε )ε∈Q+ (E) o` On montre : Proposition 2.8. On suppose que la base B du fibré vectoriel π : F → B est un espace normal (par exemple un espace métrique ou un espace compact). Soit q un champ de formes quadratiques sur le fibré vectoriel π : F → B. Alors, q est continue si et seulement si elle est semi-continue supérieurement et inférieurement. Démonstration de la proposition 2.8. Un champ de formes quadratiques qui est continu est toujours semi-continu inférieurement et supérieurement (pour cette implication, on n’a pas besoin de supposer que B soit normal). Supposons maintenant que q soit un champ de formes quadratiques qui est a` la fois semi-continu inférieurement et supérieurement. Soit x ∈ B ; il existe alors un voisinage ouvert U de x dans B et un homéomorphisme H : U ×Fx → V de U ×Fx dans l’ouvert V = π −1 (U ) de F tel que : ∀(y, v) ∈ U ×Fx , π◦H(y, v) = y. Comme B est normal, on peut a` l’aide du théorème de Tietze–Urysohn construire une fonction continue η : B → [0, 1] qui vaut 1 sur un voisinage de x et qui est nulle en dehors de U . Si maintenant σ est une forme quadratique définie positive sur Fx , on définit q− , q+ ∈ Qc (F ) comme suit : pour tout y ∈ U : ∀v ∈ Fy , q− (y)(v) = η(y)(q(x) − u p2 : U × F x → F x σ)(p2 ◦ H −1 (v)) (et q+ (y)(v) = η(y)(q(x) + σ)(p2 ◦ H −1 (v))) o` est la projection (y, v) → v et pour y ∈ / U : q− (y)(v) = q+ (y)(v) = 0. Comme q est semi-continue supérieurement et inférieurement, l’ensemble : {y ∈ B; q− (y) q(y) q+ (y)} est un ouvert qui contient x. Ceci joint a` la remarque 2.7 nous permet de conclure. De même : Proposition 2.9. Soit π : F → B un fibré vectoriel et soit E une partie de R (totalement) ordonnée par ≤ ou ≥ (on note son ordre, qui est donc ≤ ou ≥) et (qt )t∈E une famille croissante de champs de formes quadratiques sur F , c-` a-d : ∀t, t ∈ E, t t ⇒ qt ≺ qt . On suppose que chaque qt est semicontinue supérieurement et que : ∀x ∈ B, ∀v ∈ Fx , inf t∈E qt (x)(v) est fini. On note : q(x)(v) = inf t∈E qt (x)(v) ; q est alors un champ de formes quadratiques qui est semi-continu supérieurement. Remarque 2.10. On montre de même qu’un supremum de champs de formes quadratiques semi-continues inférieurement est semi-continu inférieurement. Démonstration de la proposition 2.9. Avec les notations de la proposition, q est un champ de formes quadratiques. Il reste a` montrer qu’il est semi-continu supérieurement. Soit f un champ continu de formes quadratiques sur F . Soit x ∈ B. Les propositions suivantes sont alors équivalentes :

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• q(x) f (x) ; • ∀v ∈ Fx \{0}, inf t∈E qt (x)(v) < f (x)(v) ; • ∀v ∈ Fx \{0}, ∃t ∈ E, qt (x)(v) < f (x)(v). Montrons alors qu’on peut choisir le t de l’assertion précédente indépendamment de v ; notons T = inf E et (Tn )n∈N une suite décroissante (pour ) d’éléments de E qui a pour limite T (T peut être infini). On déduit du fait que (qt ) est croissante que : ∀v ∈ Fx , q(x)(v) = limn→∞ qTn (x)(v). La suite (qTn (x))n∈N est donc une suite décroissante d’éléments de Q(Fx ) qui converge simplement en décroissant vers q(x) ; elle y converge donc uniformément sur les compacts (l’espace vectoriel Fx est de dimension finie), ce qui nous dit bien que t est indépendant de v ; on peut alors poursuivre la suite d’assertions équivalentes : • ∃t ∈ E, ∀v ∈ Fx \{0}, qt (x)(v) < f (x)(v) ; • ∃t ∈ E, qt (x) f (x) ; • x ∈ t∈E {y ∈ B; qt (x) f (x)}. Cette dernière condition est bien ouverte car chaque qt est semi-continu supérieurement. Donnons maintenant un résultat qui est une version adaptée à notre cas du théorème de Dini ; pour l’énoncer, on a besoin d’une définition : Définition 2.11. Si π : F → B est un fibré vectoriel dont la base B est compacte et si (qn )n∈N une suite de champs de formes quadratiques sur F , on dira que (qn ) converge uniformément vers le champ de formes quadratiques q sur F si pour tout champ de formes quadratiques ε ∈ Qc (F ) sur F qui est défini positif sur chaque fibre, il existe un rang N ∈ N tel que : ∀n ≥ N ,

∀x ∈ B ,

q(x) − ε(x) qn (x) q(x) + ε(x) .

Il existe bien entendu une version de cette définition et de la proposition qui suit pour les familles (non forcément indexées par N) de champs de formes quadratiques. Proposition 2.12. Soit π : F → B un fibré vectoriel dont la base est compacte et soit (qn )n∈N une suite décroissante de champs de formes quadratiques sur F telle que chaque qn est semi-continue supérieurement. On suppose de plus que pour chaque x ∈ B, pour chaque v ∈ Fx , la quantité q(x)(v) = inf n∈N qn (x)(v) est finie : q est alors un champ de formes quadratiques sur F qui est semi-continu supérieurement. Si de plus q est continu, on a convergence uniforme de (qn ) vers q. Démonstration de la proposition 2.12. Elle s’inspire de celle du théorème classique de Dini. Fixons un champ continu de formes quadratiques ε sur F qui est défini positif sur chaque fibre et définissons pour chaque n ∈ N : On = x ∈ B; qn (x) − q(x) ε(x) = x ∈ B; qn (x) ε(x) + q(x) .

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Chaque qn étant semi-continue supérieurement et q + ε étant continue, chaque On est ouvert. De plus, les On forment un recouvrement du compact B. La suite On étant de plus croissante (car (qn ) est décroissante), on en déduit l’existence d’un N ∈ N tel que B = ON , et donc par la décroissance de la suite (qn ) : ∀n ≥ N ,

∀x ∈ B ,

q(x) ≺ qn (x) q(x) + ε(x) .

Une autre proposition nous sera utile : Proposition 2.13. On suppose que la base B de F est normale. Une somme finie de champs de formes quadratiques semi-continus supérieurement (resp. inférieurement) est semi-continue supérieurement (resp. inférieurement). Remarque 2.14. Il est immédiat qu’un multiple par un réel strictement positif d’un champ de formes quadratiques semi-continu supérieurement est semi-continu supérieurement. Démonstration de la proposition 2.13. Soient q1 , q2 deux champs de formes quadratiques semi-continus supérieurement sur le fibré vectoriel π : F → B, soit f un champ de formes quadratiques continu sur F et soit x ∈ B tel que : q1 (x)+q2 (x) f (x). On cherche un voisinage V de x dans B tel que : ∀y ∈ V, q1 (y)+q2 (y) f (y). On utilise alors deux formes quadratiques r1 et r2 définies sur Fx telles que : q1 (x) r1 , q2 (x) r2 et r1 +r2 f (x). En utilisant la définition de fibré vectoriel et une fonction plateau sur B qui vaut 1 en x et 0 en dehors d’un petit voisinage de x (on utilise ici le théorème de Tietze–Urysohn et donc le fait que B est normal, comme dans la proposition 2.8), on prolonge r1 et r2 en des champs de formes quadratiques R1 et R2 continus. Alors, l’ouvert {y ∈ B; q1 (y) R1 (y)} ∩ {y ∈ B; q2 (y) R2 (y)} ⊂ {y ∈ B; R1 (y) + R2 (y) f (y)} répond a` la question. Avec notre notion de semi-continuité, on obtient un autre résultat, lui aussi similaire a` ceux qui concernent les fonctions semi-continues a` valeurs réelles : Proposition 2.15. Soit π : F → B un fibré vectoriel dont la base B est normale tel que Q+ (F ) ∩ Qc (F ) = ∅. Soient q− et q+ deux champs de formes quadratiques de F qui vérifient les hypothèses suivantes : • q+ est semi-continue supérieurement ; • q− est semi-continue inférieurement ; • q − ≺ q+ . Alors, si G = {x ∈ B; q− (x) = q+ (x)}, G est un Gδ de B. Si de plus B est normal,et si q ∈ Q(F ) est tel que q− ≺ q ≺ q+ , alors q est continue en tout point de G ; par exemple, q− et q+ sont continues en tout point de G. Remarque 2.16. On peut se demander si l’hypothèse Q+ (F ) ∩ Qc (F ) = ∅ n’est pas redondante. Sur le fibré tangent d’une variété (le cas qui nous intéresse), elle est automatique (c’est l’existence d’une métrique riemannienne). Plus généralement, si on suppose que B est à la fois paracompact et normal, on a l’existence d’un

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recouvrement localement fini de B par des ouverts (Ui ) tel que chaque π −1 (Ui ) soit un fibré trivial, donc admette un qi ∈ Q+ (π −1 (Ui )) ∩ Qc (π −1 (Ui )). On utilise alors une partition del’unité subordonnée à (Ui ) ; αi : B → [0, 1] avec suppαi ⊂ Ui pour construire q = αi qi ∈ Q+ (F ) ∩ Qc (F ). D´ emonstration de la proposition 2.15. Soit ε ∈1 Q+ (F ) ∩ Qc (F ). On a : G = u Gn = {x ∈ B; q+ (x) − q− (x) n ε(x)} ; comme q+ − q− est semin∈N∗ Gn o` continue supérieurement (par la proposition 2.13), chaque Gn est ouvert et G est donc bien un Gδ . Soit maintenant x ∈ G. On adapte la démonstration de la proposition 2.8 a` notre cas (en fait la proposition 2.8 est un cas particulier de ce que nous sommes en train de montrer). Etant donné ε ∈ Q+ (Fx ), on peut comme dans la pro et q+ de formes quadratiques position 2.8 construire deux champs continus q− sur F telles que : q(x) − ε = q− (x) q(x) q+ (x) = q(x) + ε. Comme q+ est semi-continue supérieurement et q− est semi-continue inférieurement, l’ensemble : (y) q− (y) ≺ q+ (y) q+ (y)} est un ouvert qui contient x et qui {y ∈ B; q− est contenu dans {y ∈ B; q− (y) q(y) q+ (y)} puisque q− ≺ q ≺ q+ . Aussi, {y ∈ B; q− (y) q(y) q+ (y)} est un voisinage de x. Ceci joint a` la remarque 2.7 nous permet de conclure. Remarque 2.17. En fait, on a dans la proposition précédente montré un résultat d’équicontinuité : tous les q tels que q− ≺ q ≺ q+ forment une famille qui est équicontinue sur G.

3. Les fibrés de Green Ainsi que nous l’avions annoncé au début de la section précédente, nous allons utiliser les champs de formes quadratiques pour représenter des fibrés lagrangiens transverses à la “verticale”. 3.1. Comparaison des sous-espaces lagrangiens à l’aide de formes quadratiques Il est très classique d’utiliser des matrices symétriques pour représenter des sousespaces vectoriels lagrangiens ; la nouveauté de ce que nous faisons est que nous n’avons pas besoin de fixer de sous-espace lagrangien “horizontal” pour définir nos formes quadratiques. Il semble en effet raisonnable que si on s’est fixé une “verticale”, on n’ait pas besoin de dire ce qu’est une “horizontale” pour comparer deux sous-espaces lagrangiens transverses à la verticale. Définition 3.1. Soit (E, Ω) un espace vectoriel symplectique de dimension finie et V un sous-espace lagrangien fixé de E. On note p : E → E/V la projection canonique. Si L1 et L2 sont deux sous-espaces lagrangiens de E transverses à V , on définit une forme quadratique Q(L1 , L2 ) sur E/V par : ∀w ∈ E/V , Q(L1 , L2 )(w) = Ω (p|L1 )−1 (w), (p|L2 )−1 (w) . Cette forme quadratique s’appelle alors hauteur de L2 au dessus de L1 (relativement à V ).

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Si on se fixe V (que nous appellerons plus tard la “verticale”) et L1 (l’“horizontale”), la donnée de cette hauteur détermine L2 : Proposition 3.2. Soit (E, Ω) un espace vectoriel symplectique de dimension finie et V un sous-espace lagrangien fixé de E. Notons LV (E) l’ensemble des sousespaces lagrangiens de E qui sont transverses a ` V et fixons L ∈ LV (E). Alors, ` L ∈ LV (E) associe Q(L, L ) est l’application Q(L, .) : LV (E) → Q(E/V ) qui a une bijection. −1 De plus, on a : L ∩ L = p−1 |L (ker Q(L, L )) = p|L (ker Q(L, L )). Remarque 3.3. 1. Si on munit les deux ensembles de leurs topologies usuelles, cette application est un homéomorphisme ; 2. de la dernière phrase de cet énoncé, on déduit que L est tranverse à L si et seulement si Q(L, L ) n’est pas dégénérée. Démonstration de la proposition 3.2. Fixons V et L comme dans l’énoncé, et considérons q ∈ Q(E/V ). On note alors = (p|L )−1 : c’est un isomorphisme de E/V sur L. On cherche alors tous les L ∈ LV (E) tels que Q(L, L ) = q. On commence par remarquer que si L ∈ LV (E), il s’écrit de fa¸con unique comme { (x) + v( (x)); x ∈ E/V } o` u v : L → V est une application linéaire telle que : ∀x, y ∈ E/V, Ω( (x) + v( (x)), (y) + v( (y))) = 0. De plus, comme L,L et V sont lagrangiens :

∀x, y ∈ E/V , 0 = Ω (x) + v (x) , (y) + v (y)

= Ω (x), v (y) − Ω (y), v (x) donc la forme bilinéaire symétrique associée à Q(L, L ) est : ϕ(L, L )(x, y) = Ω( (x), v( (y))) = Ω( (y), v( (x))). L’application étant un isomorphisme de E/V sur L, si on définit ψ(L, L ) par ψ(L, L ) = ϕ(L, L ) ◦ (p|L , p|L ), alors ψ(L, L ) est une forme bilinéaire symétrique et la correspondance est un isomorphisme de l’ensemble des formes bilinéaires symétriques sur E/V sur l’ensemble des formes bilinéaires symétriques sur L. On a de plus : ∀x, y ∈ L, ψ(L, L )(x, y) = Ω(x, v(y)). Or, Ω étant symplectique et V et L des espaces lagrangiens transverses, l’application : Ω∗ : V → L∗ qui a` x ∈ V associe Ω(., x) ∈ L∗ est un isomorphisme. On en déduit que pour toute forme bilinéaire symétrique ψ de L, il existe une unique application linéaire v : L → V , définie par v(x) = (Ω∗ )−1 (ψ(., x)) telle que : ∀y ∈ L, ψ(y, x) = Ω(y, v(x)). Du fait que ψ est symétrique, on déduit : ∀x, y ∈ L, Ω(x, v(y)) = Ω(y, v(x)), donc que l’ensemble {x + v(x); x ∈ L} est un sous-espace lagrangien de E, ce qui permet de conclure que Q(L, .) est bien une bijection. De plus, il vient alors immédiatement que L ∩ L = { (x); x ∈ E/V, v( (x)) = 0} = { (x)+v( (x)); x ∈ E/V, v( (x)) = 0}. Comme la forme bilinéaire symétrique associée à Q(L, L ) est : ϕ(L, L )(x, y) = Ω( (x), v( (y))) = Ω( (y), v( (x))) ; le fait que ker ϕ(L, L ) = p(ker v) en découle immédiatement.

892

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Proposition 3.4. Soit (E, Ω) un espace vectoriel symplectique de dimension finie et V un sous-espace lagrangien fixé de E. On a : • ∀L1 , L2 ∈ LV (E), Q(L1 , L2 ) = −Q(L2 , L1 ) ; • ∀L1 , L2 , L3 ∈ LV (E), Q(L1 , L2 ) + Q(L2 , L3 ) = Q(L1 , L3 ). Démonstration de la proposition 3.4. Le premier point résulte de l’antisymétrie de la forme symplectique Ω. Montrons le second point. Soient donc L1 , L2 , L3 ∈ LV (E). Etant donné x ∈ E/V , convenons de noter : −1 (x) = 2 (x) ; • p|L2 −1 (x) = 2 (x) + v3 (x) ; on a alors : v3 (x) ∈ V ; • p|L3 −1 (x) = 2 (x) + v1 (x) ; on a alors : v1 (x) ∈ V ; • p|L1 et calculons :

Q(L1 , L2 )(x) + Q(L2 , L3 )(x) = Ω 2 (x) + v1 (x), 2 (x) − Ω 2 (x) + v3 (x), 2 (x) = Ω v1 (x) − v3 (x), 2 (x) = Ω v1 (x) + 2 (x), v3 (x) + 2 (x) = Q(L1 , L3 )(x) .

Définition 3.5. Soit (E, Ω) un espace vectoriel symplectique de dimension finie et V un sous-espace lagrangien fixé de E. Si L, L ∈ LV (E), on dira que L est au dessus (resp. strictement au dessus) de L si 0 ≺ Q(L, L ) (resp. 0 Q(L, L )). Il nous arrivera d’écrire L ≺ L (resp. L L ). Remarquons que “être au dessus” est une relation d’ordre (partiel) sur LV (E) (la transitivité se déduit du deuxième point de 3.4). On peut alors parler de famille croissante d’éléments de LV (E). On peut aussi bien entendu parler de fibré semicontinus de sous-espaces lagrangiens transverses à la verticale. 3.2. Construction des fibrés de Green pour les orbites sans points conjugués Nous allons maintenant utiliser a` la fois la représentation des sous-espaces lagrangiens et les résultats que nous avons démontrés en section 2 concernant les champs de formes quadratiques pour construire sur T ∗ M les fibrés de Green au dessus des orbites minimisantes d’un lagrangien de Tonelli. Ces fibrés furent introduits par Leon W. Green dans [13] dans le cas du flot géodésique d’une métrique riemannienne, puis généralisés au cas des métriques finsleriennes par Patrick Foulon dans [12], au cas des hamiltoniens autonomes optiques par Gonzalo Contreras et Renato Iturriaga dans [9] et enfin au cas des applications symplectiques de T ∗ Td déviant la verticale par Misha Bialy et Robert S. Mackay dans [8]. La référence [18] donne une démonstration très courte de l’existence de ces fibrés dans le cas riemannien et passe en revue différentes utilisations de ces fibrés. Dans toutes ces constructions, le choix d’un fibré lagrangien “horizontal” est fait. Même si plus tard il nous arrivera de fixer un tel fibré pour montrer certains résultat, nous allons voir que nous n’avons en fait pas besoin de

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cela pour construire les fibrés de Green : la représentation a` l’aide des champs de formes quadratiques suffit. Supposons désormais que L : T M × T → R soit un lagrangien de Tonelli et que H : T ∗ M × T → R soit le hamiltonien qui lui est associé. Le flot de H est noté (φtt ), et on notera : φt0 = φt . Définissons le fibré lagrangien vertical : si π ∗ : T ∗ M → M désigne la projection usuelle, si x ∈ T ∗ M , la verticale en x, notée V (x) est :

V (x) = ker Dπ ∗ (x) = Tx π ∗−1 π ∗ (x) . Chaque verticale V (x) est alors un sous-espace lagrangien de Tx (T ∗ M ). Etant donné t ∈ R et x ∈ T ∗ M , on définit : Gt (x) = Dφ0−t (φ−t (x))(V (φ−t (x)). On rappelle : Définition 3.6. L’orbite de x ∈ T ∗ M est sans point conjugué si :

∀t, t ∈ R , t = t ⇒ Dφtt φt (x) V φt (x) ∩ V φt (x) = {0} . On a alors : Proposition 3.7. Soit x ∈ T ∗ M sans point conjugué. On a alors : 1. pour tout t ∈ R∗ , le sous espace lagrangien Gt (x) de Tx (T ∗ M ) est transverse a la verticale V (x) ; on s’intéressera alors aux hauteurs relativement a ` ` V (x) (voir définition 3.1) ; 2. pour tous t, t ∈ R tels que 0 < t < t , on a : Q(Gt (x), Gt (x)) 0 : la hauteur ` V (x) est définie négative i.e : de Gt (x) au dessus de Gt (x) relativement a Gt (x) Gt (x) ; 3. pour tous t, t ∈ R tels que 0 > t > t, on a : Q(Gt (x), Gt (x)) 0 : la ` V (x) est définie positive hauteur de Gt (x) au dessus de Gt (x) relativement a i.e : Gt (x) Gt (x) ; 4. pour tous t, t ∈ R tels que t < 0 < t , on a : Q(Gt (x), Gt (x)) 0 : la ` V (x) est définie positive hauteur de Gt (x) au dessus de Gt (x) relativement a i.e : Gt (x) Gt (x). Corollaire 3.8. Sous les mêmes hypothèses, G− (x) = limt→−∞ Gt (x) et G+ (x) = limt→+∞ Gt (x) sont deux sous-espaces lagrangiens de Tx (T ∗ M ) tels que : • Q(G+ (x), G− (x)) ≺ 0 : suivant la définition 3.5, G+ (x) est au dessus de G− (x) i.e ; G− (x) ≺ G+ (x) ; • ∀k ∈ Z, Dφk (x)G− (x) = G− (φk (x)) et Dφk (x)G+ (x) = G+ (φk (x)) et si de plus le hamiltonien est indépendant du temps : ∀t ∈ R, Dφt (x)G− (x) = G− (φt (x)) et Dφt (x)G+ (x) = G+ (φt (x)). G− et G+ sont alors les fibrés de Green en x.

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Démonstration de la proposition 3.7 et du corollaire 3.8. Nous allons successivement démontrer la proposition et son corollaire. Le 1. découle du fait que l’orbite est sans point conjugué. Toujours du fait que l’orbite est sans point conjugué, on déduit que : ∀t = t , Gt (x) ∩ Gt (x) = {0}. Cela implique que Q(Gt (x), Gt (x)) est non dégénérée. L’application (t, t ) → Q(Gt (x), Gt (x)) étant continue (l` a o` u elle est définie, c’est-à-dire pour t et t non nuls), on en déduit que sur chacun des ensembles connexes suivants : {(t, t ); 0 < t < t }, {(t, t ); 0 > t > t }, {(t, t ); t < 0 < t }, Q(Gt (x), Gt (x)) est de signature constante. Il suffit donc pour prouver les points 2, 3, 4, de trouver la signature de Q(Gt (x), Gt (x)) en un point des ensembles considérés. Pour cela, on écrit les équations de Hamilton linéarisées dans une carte au voisinage de x ; on choisit des coordonnées (x1 , . . . , xn ) sur U ⊂ M que l’on complète avec les coordonn´ duales de T ∗ M : le point de coorn ees 1 n 1 n k k données (x , . . . , x , p , . . . p ) est k=1 p dx ; pour t petit, on note Mt (y) = at ct la matrice de Dφt (y) dans ces coordonnées. Cette matrice est symplecbt dt tique (puisque φt et les coordonnées le sont) et on déduit des équations de Hamilton linéarisées que : ct = tHpp (y, 0) + ot→0 (t) et dt = 1 + ot→0 (1) et at = 1 + ot→0 (1), ces équivalents étant uniformes sur un voisinage compact de x. Aussi, si t = 0 est assez petit, Dφt (y)(V (y)) est le graphe de la matrice symétrique dt (ct )−1 ∼t→0 −1 1 qui est définie positive pour t > 0, définie négative pour t < 0. t (Hpp (y, 0)) Notons Z(y) le sous-espace lagrangien horizontal dans ces coordonnées. On a donc −1 (en identifiant la forme quadratique et sa Q(Z(y), Gt (y)) ∼t→0 1t (Hpp (y, 0)) matrice dans la base de Ty (T ∗ M )/V (y) obtenue en passant au quotient la base donnée pas les coordonnées dans l’espace horizontal) ce qui donne pour t = −t ou t = 2t ` a l’aide du deuxième point de la proposition 3.4 : −1 1 1 Hpp (y, 0) − . Q Gt (y), Gt (y) ∼(t,t )→(0,0) t t Ceci permet de déterminer les signatures cherchées. Montrons maintenant le corollaire. Il suffit de remarquer que d’après la proposition 3.7, (Gt (x))t>0 et (Gt (x))t0 est minorée et (Gt (x))t 0 tel que, pour chaque t ∈ [−ε, ε], Gt (x) ne rencontre pas l’horizontale en x (cette horizontale est définie dans les coordonnées utilisées). On a vu lors de la démonstration de la proposition 3.7 que pour t > 0 assez petit, Gt (x) est le graphe (dans la carte (T ∗ Ui , ϕ∗i )) de la matrice symétrique −1 définie positive St (x) ∼ 1t (Hpp (y, 0)) . Comme pour t ∈]0, ε], Gt (x) ne rencontre pas l’horizontale, la signature de St (x) ne peut changer et la matrice est définie positive. Comme t → Gt (x) est continue et vaut V (x) en t = 0, on en déduit que pour tout C > 0, il existe η tel que pour chaque t ∈]0, η], C1 St (x) : pour des temps positifs assez petits, les images de la verticale sont au dessus d’un plan lagrangien fixé. En particulier, pour t ∈]0, η], Gt (x) est strictement au dessus de F (0). On a déjà montré que pour chaque temps, Gt (x) est transverse à F (0), donc la hauteur QV (x) (F (0), Gt (x)) ne dégènère pas pour t ∈ R∗+ et reste donc définie positive, i.e. F (0) Gt (x). On conclut en passant a` la limite. 3.5. Un critère dynamique d’appartenance aux fibrés de Green Le critère que nous allons donner est démontré dans le cas autonome mais d’une fa¸con très calculatoire dans [9] et dans le cas des applications symplectiques de l’anneau déviant la verticale dans [1]. C’est a` la suite d’une discussion avec Sylvain Crovisier concernant l’interprétation géométrique de ce critère en dimension 1 que m’est venu l’idée la démonstration qui va suivre. Proposition 3.12. Soit x ∈ T ∗ M un point dont l’orbite est relativement compacte et sans point conjugué et v ∈ Tx (T ∗ M ). Alors : • si v ∈ / G− (x), alors limn→+∞ D(π ∗ ◦ φn )(x)v = +∞ ; • si v ∈ / G+ (x), alors limn→+∞ D(π ∗ ◦ φ−n )(x)v = +∞. Démonstration de la proposition 3.12. Nous allons juste montrer la première affirmation, l’autre étant similaire. L’orbite de x étant relativement compacte, il en est de même de sa projection sur M , contenue dans un compact que l’on peut recouvrir par un nombre fini de carte (Ui , ϕi )0≤i≤N . On note alors : ∀x ∈ Ui , ϕi (x) = (x1i , . . . , xni ) ; pour ees duales de obtenir des coordonnées sur T ∗ Ui , on complète avec les coordonn´ n T ∗ M : le point de coordonnées (x1i , . . . , xni , p1i , . . . pni ) est k=1 pki dxki ; on note alors (T ∗ Ui , ϕ∗i ) cette carte duale. Etant donné t ∈ R, il existe alors i = i(t) tel que π ∗ ◦ φt (x) ∈ Ui . On note alors Mt la matrice de Dφt (x) exprimée dans les coordonnées symplectiques (T ∗ Ui(0) , ϕ∗i(0) ) et (T ∗ Ui(t) , ϕ∗i(t) ). Cette matrice est de la forme : Mt = actt dbtt . Comme de plus G− est lagrangien et transverse a` la verticale, il existe pour tout t ∈ R une unique matrice symétrique S − (t) telle que G− (φt (x)) est le graphe de S − (t). De même, on a vu que chaque Dφ1 (φt−1 (x))V (φt−1 (x)) et chaque Dφ−1 (φt+1 (x))V (φt+1 (x)) est un sous-espace lagrangien transverse à la verticale, et est donc le graphe d’une matrice symétrique S1+ (t), S1− (t).

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On a supposé que l’orbite de x est relativement compacte, d’adhérence compacte notée Γ. Comme l’ensemble des points dont l’orbite est sans point conjugué est un fermé, on peut en fait définir tous ces fibrés sur Γ tout entier. Or, les deux ument fibrés y ∈ Γ → Dφ1 (y)V (y) et y ∈ Γ → Dφ−1 (y)V (y) dépendent continˆ de y et sont transverses à la verticale ; comme on a un nombre fini de cartes, on en déduit que leurs matrices lues en cartes forment une partie bornée et comme : S1− S − S1+ , les {S − (t); t ∈ R} forment une partie relativement compacte. 1 0 Considérons les matrices symplectiques : Pt = S − (t) 1 ; elles sont donc uni 1 0 formément bornées et Pt−1 = −S − (t) 1 est aussi uniformément bornée. On utilise alors ces matrices pour faire un changement de base symplectique dans la fibre de T (T ∗ M ). Plus précisément, si les coordonnées initiales dans TΦt (x) (T ∗ M ) étaient ξ (ξi , ρi ), les nouvelles coordonnées sont Pt−1 ρξ = ρ−S − (t)ξ , i.e. on choisit des coordonnées dans lesquelles le fibré de Green G− devient l’“horizontale”. Comme la matrice Pt et son inverse sont uniformément bornées, les convergences des fibrés se voient aussi bien dans ces nouvelles coordonnées que dans les anciennes. Désormais, on travaillera donc dans ces bases on gardera la même notation Mt . Comme G− aet t bt est invariant, on a alors : Mt = 0 dt . Pour n ≥ 1, notons Sn− la matrice symétrique dont G−n (x) est le graphe et la matrice symétrique dont Gn (φn (x)) est le graphe. Alors : dn = Sn+ bn . De t + t plus, comme Mn est symplectique, on a : Mn−1 = bn0Sn −t abn donc : an = −bn Sn− . n −bn Sn− bn d’o` u finalement : Mn = . 0 S+ b

Sn+

n

n

Comme le fibré de Green G− correspond dans ces coordonnées à l’horizontale, on a : limn→+∞ Sn− = 0. De plus, avec les mêmes notations que précédemment, mais exprimées dans les nouvelles coordonnées : S1− (n) Sn+ ≺ S1+ (n) : le fibré Gn est encadré par des images de la verticale dont on a vu qu’elles sont uniformément bornées. A cause de l’inégalité entre les deux fibrés de Green, comme l’un deux est l’horizontale, on a même : 0 Sn+ ≺ S1+ (n). Ainsi, (Sn+ )n≥1 est une famille uniformément bornée de matrices symétriques définie positives. Il existe donc une constante C > 0 telle que : ∀v ∈ R2d , 0 ≤ t vSn+ v ≤ Cv2 . Or, comme Mn et symplectique, on a : Sn− (t bn )Sn+ bn = −1 donc (Sn− )−1 = −t bn Sn+ bn . On en déduit : ∀v ∈ R2d ,

−t v(Sn− )−1 v = t v t bn Sn+ bn v ≤ Cbn v2 .

Comme limn→+∞ Sn− = 0, on en déduit que pour tout v = 0, la suite (bn v)n≥1 tend vers +∞ et même que pour tout A ≥ 0, il existe un rang a` partir duquel, pour tout v : bn v ≥ Av (*). En d’autres termes, on a montré que si p− : T (T ∗ M ) → T (T ∗ M ) désigne la projection sur G− parallèlement à la verticale, alors pour tout vecteur v ∈ V (x)\{0} : limn→+∞ p− ◦ Dφn (x)v = +∞. Soit maintenant v ∈ Tx (T ∗ M )\G− (x) de coordonnées (v1 , v2 ) ; alors : p− ◦ Dφn (x)v a pour coordonnées bn (v2 − Sn− v1 ). / G− (x) et donc Or, la suite (v2 − Sn− v1 )n≥1 converge vers v2 = 0 puisque v ∈

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par (*) la suite (bn (v2 − Sn− v1 ))n≥1 converge vers +∞. On a finalement montré que pour tout v ∈ Tx M \G− (x), on a : limn→+∞ p− ◦ Dφn (x)v = +∞. On remarque que si maintenant on revient dans les coordonnées initiales (l’“horizontale” n’est plus G− ), on peut exprimer : p− (v1 , v2 ) = (v1 , S− v1 ) et Dπ ∗ (v1 , v2 ) = v1 , donc si limn→+∞ p− ◦ Dφn (x)v = +∞, comme S− est uniformément bornée, on a aussi : limn→+∞ Dπ ∗ ◦ Dφn (x)v = +∞. Exemples. Nous expliquons quelles conséquences simples on peut tirer du critère dynamique, dans le cas autonome ou/et dans le cas partiellement hyperbolique. Dans le cas autonome, toutes ces conséquences étaient déjà données dans [9], nous en rappelons l’argument pour être complets : 1. Dans le cas d’une partie invariante K ⊂ T ∗ M relativement compacte et formée d’orbites sans point conjugué telle que l’application tangente au temps 1 du flot hamiltonien restreinte a` TK (T ∗ M ) est hyperbolique, G− (x) est le tangent au feuilletage stable et G+ (x) le tangent au feuilletage instable : en effet, les orbites des vecteurs des fibrés stables sont bornées en temps positif, alors que les orbites des vecteurs des fibrés instables sont bornées en temps négatifs, et on peut donc utiliser le critère dynamique pour conclure. 2. dans le cas d’un hamiltonien indépendant du temps, si on suppose qu’une orbite non critique et sans point conjugué est relativement compacte, le champ de vecteurs hamiltonien a une orbite bornée sous l’application tangente au flot. On a donc dans ce cas : RXH (x) ⊂ G− (x) ∩ G+ (x) . Comme G− (x) et G+ (x) sont lagrangiens, on en déduit qu’ils sont dans l’orthogonal pour la forme symplectique, au champ de vecteurs hamiltonien, c’est-à-dire dans le fibré tangent a` l’hypersurface d’énergie. 3. supposons maintenant que K soit une partie relativement compacte, invariante et sans point conjugué telle que la dynamique sur TK M soit partiellement hyperbolique (voir par exemple [7] pour une définition) ; on note E s ⊕ E c ⊕ E u la décomposition correspondante, avec E u fibré instable, E s fibré stable et E c fibré centre. Alors : ∀x ∈ K, E s (x) ⊂ G− (x) ⊂ E s (x) ⊕ E c (x) et

E s (x) ⊂ G− (x) ⊂ E s (x) ⊕ E c (x) .

Pour la première inclusion, la démonstration est exactement la même que dans le cas hyperbolique. Comme le flot est symplectique, le sous-espace E s est alors isotrope et son orthogonal pour la forme symplectique est (E s )⊥ = E c ⊕ E s (voir [7] pour ces propriétés des fibrés symplectiques). Comme G− s ⊥ c s est lagrangien et contient E s , on en déduit : G− = G⊥ − ⊂ (E ) = E ⊕ E , soit l’inclusion voulue.

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3.6. Les fibrés de Green réduits dans le cas autonome Dans le cas o` u le hamiltonien est indépendant du temps, on sait que les hypersurfaces d’énergie sont invariantes. Fixons donc une telle hypersurface E et intéressons-nous à une partie F de E qui est invariante par le flot hamiltonien et qui ne contient pas de point x en lequel le champ hamiltonien est vertical : / V (x). ∀x ∈ F, XH (x) ∈ En chaque x ∈ F, la 2-forme ω(x)|Tx E est alors dégénérée de noyau RXH (x) si XH désigne le champ de vecteurs hamiltonien. On définit alors au dessus de F le fibré symplectique F qui se déduit de TF E en effectuant une réduction symplectique, c’est-à-dire qu’on on passe au quotient par ker ω|TF E : ∀x ∈ F , F (x) = Tx E/RXH (x) et ∀u, v ∈ Tx E , Ω p(u), p(v) = ω(u, v) o` u p : TF E → F désigne la projection canonique. Dans ce fibré symplectique F , on peut définir une verticale : v(x) = p(V (x) ∩ Tx E) qui définit une sous-espace lagrangien de F (x) (car on a supposé que XH (x) n’est pas vertical). On peut dans ce nouveau fibré symplectique utiliser les notions définies en section 3.1 et comparer deux-à-deux les sous-espaces lagrangiens de F (x), parler de leur hauteur relativement a` v(x). Alors, si 1 , 2 sont deux sous-espaces lagrangiens de F (x) qui sont transverses a` v(x), L1 = p−1 ( 1 ) et L2 = p−1 ( 2 ) sont deux sous-espaces lagrangiens de Tx E (donc Tx (T ∗ M )) qui sont transverses à V (x). La hauteur de 2 au dessus de 1 a même signature que la hauteur de L2 au dessus de L1 ; simplement, la nullité (dimension du noyau) augmente d’une unité quand on passe de F (x) a` Tx (T ∗ M ). Pour faire la comparaison dans l’autre sens, c’est-` a-dire partir de deux sousespaces lagrangiens L1 et L2 de Tx (T ∗ M ) qui sont transverses a` V (x) et comparer 1 = p(L1 ∩ Tx E) et 2 = p(L2 ∩ Tx E), il faut faire attention a` plusieurs choses ; on s’intéresse au cas o` u ni L1 , ni L2 ne sont inclus dans Tx E ; on doit alors vérifier : 1. pour que les Li ∩ Tx E se projettent par p en des sous-espaces lagrangiens de / Li ; F (x), il faut s’assurer que XH (x) ∈ 2. si on suppose que L1 et L2 sont transverses, il n’est pas sˆ ur que 1 = p(L1 ∩ Tx E) et 2 = p(L2 ∩ Tx E) soient transverses ; ceci n’est vrai que si : L2 ∩ Tx E ∩ (L1 ⊕ RXH (x)) = {0}. Au dessus de F, le fibré en droite RXH est invariant par (Dφt ). On peut donc aussi passer au quotient Dφt , qui définit un cocycle symplectique (Mt ) sur le fibré F . Supposons maintenant que l’orbite de x ∈ F soit sans point conjugué. On avait défini en section 3.2 les Gt (x). Comme on a supposé que sur F, et donc le long de l’orbite de x, le champ de vecteurs n’est pas sur la verticale, on a : / Gt (x). On en déduit que chaque gt (x) = p(Gt (x) ∩ Tx E) est un ∀t ∈ R, XH (x) ∈ sous-espace lagrangien de F (x). De plus, on déduit de la définition de Mt et Gt que : gt (x) = Mt (φ−t x)v(φ−t x). Pour essayer de faire dans F (x) une démonstration analogue a` celle faite dans Tx (T ∗ M ) en section 3.5, on a besoin de vérifier que les gt (x) sont transverses, ce

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qui revient a` montrer que pour chaque t > 0, gt (x) est transverse à v(x). Montrons donc : Lemme 3.13. Soit x ∈ F sans point conjugué ; alors, pour chaque t > 0, gt (x) est transverse a ` v(x). Démonstration du lemme 3.13. Introduisons une notation : g˜t (x) = (Gt (x) ∩ Tx E) ⊕ RXH (x) et V(x) = V (x) ∩ Tx E. Montrer que gt (x) est transverse à v(x) revient a montrer que g˜t (x) ∩ V(x) = {0}. ` Soit donc v ∈ V(φ−t x)\{0} tel que Dφt (φ−t x).v = w + λXH (x) avec w ∈ V(x). Comme v = 0, on a : λ = 0. Aussi, quitte a` diviser v par λ, on peut supposer que λ = 1. Nous pouvons nous placer dans les coordonnées introduites dans la démonstration de la proposition 3.12. L’horizontale est alors le fibré de Green G− et −bt St− bt avec les mêmes notations que la matrice de Dφt (φ−t x) est : Nt = 0 St+ bt dans la démonstration de la proposition 3.12. Faisant un abus de notation, nous identifierons les vecteurs avec leurs coordonnées. On a donc : v = (0, v0 ) et XH (x) = (h1 , 0) puisque XH (x) ∈ G− (x) ; on a de plus : h1 = 0. Du fait que Dφt (φ−t x).v = w + XH (x) avec w = (0, w1 ) ∈ V (x), on déduit alors aisément que bt v0 = h1 et que donc Dφt (φ−t x)v = (h1 , St+ h1 ). Donc : ω Dφt (φ−t x)v, XH (x) = ω (h1 , St+ h1 ), (h1 , 0) = −t h1 St+ h1 = 0 . Ceci contredit le fait que v ∈ V(φ−t x) ⊂ Tφ−t x E = XH (φ−t x)⊥ω .

De ce lemme on déduit que non seulement les gt (x) sont transverses à la verticale v(x), mais aussi qu’ils sont deux a` deux transverses. On peut donc parler de la hauteur de gt (x) au dessus se gt (x) (relativement à v(x)), et on sait que ces hauteurs sont non dégénérées. De fa¸con analogue a` la proposition 3.7, on peut même préciser : Proposition 3.14. Soit x ∈ F sans point conjugué. On a alors : 1. pour tout t ∈ R∗ , le sous espace lagrangien gt (x) de F (x) est transverse a ` la verticale v(x) ; on s’intéressera alors aux hauteurs relativement a ` v(x) (voir définition 3.1) ; 2. pour tous t, t ∈ R tels que 0 < t < t , on a : Q(gt (x), gt (x)) 0 : la hauteur ` v(x) est définie négative i.e : de gt (x) au dessus de gt (x) relativement a gt (x) gt (x) ; 3. pour tous t, t ∈ R tels que 0 > t > t, on a : Q(gt (x), gt (x)) 0 : la ` v(x) est définie positive hauteur de gt (x) au dessus de gt (x) relativement a i.e : gt (x) gt (x) ; 4. pour tous t, t ∈ R tels que t < 0 < t , on a : Q(gt (x), gt (x)) 0 : la ` v(x) est définie positive hauteur de gt (x) au dessus de gt (x) relativement a i.e : gt (x) gt (x).

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Démonstration de la proposition 3.14. Le premier point découle du lemme 3.13. Comme les trois autres points se démontrent de fa¸con similaire, nous allons seulement montrer l’un d’entre eux. Supposons donc par exemple que 0 < t < t . Comme dans la démonstration de la proposition 3.7, les gt étant transverses deux à deux, il suffit de montrer le résultat pour un choix de t et t , que nous allons prendre très petits. Cela nous permet de raisonner dans une seule carte en x ∈ F. Comme le champ de vecteurs sur F n’est pas vertical, on peut construire, au voisinage U ⊂ T ∗ U de x, un hamiltonien K qui vaut 1 sur U ∩ E et qui, dans les coordonnées sur U déjà utilisées dans la démonstration du lemme, est homogène de degré 2 dans la fibre, ce qui signifie : on choisit un morceau de graphe lagrangien L au dessus de U qui est proche de x et inclus dans {y; H(y) < H(x)} puis on considère pour K l’unique fonction définie sur U telle que : • K|E∩U = 1 ; • ∀y = (y1 , y2 ) ∈ L ∩ U, ∀v ∈ Ty1 M, w = (y1 , y2 + v) ∈ U et z = (y1 , y2 + μv) ∈ U ⇒ K(z) = μ2 K(w). Le hamiltonien K est alors aussi de classe C 2 et convexe dans la fibre (voir par exemple [2] ou [3]). De plus, comme E = {K = 1} = {H = H(x)} ∩ U, le flot hamiltonien de K restreint à E se déduit de celui de H à l’aide d’une reparamétrisation. Comparer gt (x) et gt (x) pour 0 < t < t petits revient alors au même que l’on parle du flot hamiltonien de K ou de H (en effet, on peut avoir les Gt ∩ E qui sont différents pour H et K, par contre les gt (x) co¨ıncident). On peut donc supposer que H vérifie les même hypothèses que K. Mais alors, de l’homogénéité dans la fibre, on déduit qu’il existe en chaque y ∈ E une droite verticale D(y) ⊂ V (y) telle qu’au dessus de E, le fibré en plans symplectiques P (y) = D(y) ⊕ RXH (y) est continu et invariant par le flot symplectique (local) de H (voir par exemple [2]). Le fibré orthogonal a` P pour la forme symplectique est alors un fibré continu en sous-espaces symplectiques G de dimension 2(d − 1) qui sont inclus dans Ty E et transverses au champ de vecteurs. Or, si y ∈ E, V(y) = V (y) ∩ Ty E est à la fois orthogonal a` D(y) (car V (y) est lagrangien) et a` RXH (y) (comme tout vecteur de Tx E), donc inclus dans G(y). Donc pour t = 0 petit, les Gt (x) ∩ Tx E = Dφt (φ−t x)V(φ−t x) sont des sous-espaces lagrangiens de G(x) transverses à V(x). + Nous avons maintenant les outils pour estimer le signe de Q(gt (x), gt (x)). Supposons que w ∈ Gt (x) ∩ Tx E et w ∈ Gt (x) ∩ Tx E soient tels que p(w) = p(w ). On cherche alors le signe de ω(w, w ). Dire que p(w) = p(w ) s’écrit : w − w ∈ / G(x), RXH (x)+V(x) ; mais comme de plus w, w ∈ G(x), V(x) ⊂ G(x) et XH (x) ∈ on a en fait : w − w ∈ V(x) ⊂ V (x). On déduit alors de la proposition 3.7 (qui dit en particulier que QV (x) (Gt (x), Gt (x)) 0) que ω(w, w ) ≤ 0, soit le résultat cherché. Remarque 3.15. La construction du hamiltonien K que nous venons de faire est un analogue local d’une construction globale bien connue en mécanique classique :

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celle de la métrique de Jacobi. Son avantage est l’existence, pour le nouveau flot, d’un fibré symplectique invariant G tel que la restriction du nouveau flot linéaire a G est conjuguée au flot linéaire initial restreint a` l’hypersurface et quotienté par ` le champ de vecteurs. De fa¸con analogue a` ce que nous avions fait en sections 3.2 et 3.4 (les démonstrations sont similaires), on obtient : Proposition 3.16. Soit x ∈ F sans point conjugué. Alors : 1. g− (x) = limt→−∞ gt (x) et g+ (x) = limt→+∞ gt (x) sont deux sous-espaces lagrangiens de F (x) tels que : • Q(g+ (x), g− (x)) ≺ 0 : suivant la définition 3.5, g+ (x) est au dessus de g− (x) i.e ; g− (x) ≺ g+ (x) ; • ∀t ∈ R, Mt (x)g− (x) = g− (φt (x)) et Mt (x)g+ (x) = g+ (φt (x)). g− et g+ sont alors les fibrés de Green réduits en x. 2. Soit K une partie de F invariante par (φt ) et dont tous les éléments sont sans points conjugués. Alors sur K, l’application g+ est semi-continue supérieurement et g− est semi-continue inférieurement. Aussi, G = {x ∈ K; g− (x) = g+ (x)} est un Gδ de K, et en tout point de G, g+ et g− sont continues. Si de plus g est un sous-fibré lagrangien de F au dessus de K tel que g− ≺ g ≺ g+ , g est continu en tout point de G. 3. Soit g un fibré vectoriel au dessus de Γ = {φt (x); t ∈ R} tel que : • pour chaque t ∈ R, g(φt (x)) est un sous-espace lagrangien de F (φt (x)) transverse a ` la verticale v(φt (x)) ; • pour chaque t ∈ R, g(φt (x)) = Mt (x)(g(x)) Alors : ∀t ∈ R ,

g− φt (x) ≺ g φt (x) ≺ g+ φt (x) .

On a même une version du critère dynamique dans ce cas : Proposition 3.17. Soit H : T ∗ M → R un hamiltonien de Tonelli. Soit E une hypersurface de niveau de H et F une partie de E invariante par le flot hamiltonien, sans point conjugué, compacte et sur laquelle l’angle du champ de vecteurs hamiltonien avec la verticale est uniformément minoré par une constante strictement positive quand il est défini (aux point o` u le champ de vecteurs ne s’annnule pas). On note px : T E → T E/RXH (x) la projection canonique. Soit x ∈ F et v ∈ Tx (E). Alors : • si v ∈ / G− (x), alors limt→+∞ pφt (x) (Dφt (x)v) = +∞ ; • si v ∈ / G+ (x), alors limt→+∞ pφ−t (x) (Dφ−t (x)v) = +∞. On a ainsi un analogue de la proposition 3.12 en passant au quotient par le champ de vecteurs dans la surface d’énergie.

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Démonstration de la proposition 3.17. On va s’inspirer de la démonstration de la proposition 3.12 : on a le même ordre sur les gt que celui qu’on avait sur le Gt ; la seule difficulté est l’argument qui permet de dire que les matrices de changement de bases symplectiques sont uniformément bornées : si on est exactement en un zéro du champ de vecteurs, on n’a pas de problème car en ce point notre énoncé est exactement le même que celui de la proposition 3.12 ; le problème est quand on raisonne sur l’ensemble F0 = {x ∈ F; XH (x) = 0}, qui n’est pas forcément compact. Pour résoudre de problème, on est amené à éclater les singularités. On commence par définir : P = (x, Δ); x ∈ F et Δ droite vectorielle de Tx E . Comme F est compacte, P est aussi compacte. De plus, P est invariante par le flot (Dφt ). Ensuite, on introduit :

R = x, R.XH (x) ; x ∈ F0 ⊂ P ¯ qui est l’adhérence de R qui est aussi invariante par le flot (Dφt ). Alors, C = R dans P est une partie compacte de P invariante par (Dφt ). Contrairement a` R, C n’est pas une section de P au dessus d’une partie de F, puisqu’au dessus d’un zéro de XH il peut exister plusieurs éléments. On définit alors le fibré vectoriel T au dessus de C par : (x, Δ, v ∈ Δ⊥ω ) ∈ T → (x, Δ). La fibre de T au dessus de (x, Δ) ∈ C est donc Δ⊥ω ; en un point régulier (tel que XH (x) = 0), c’est Tx E. Comme Δ ⊂ Δ⊥ω et même Δ = ker ω|Δ⊥ω , on peut passer T au quotient par son sous-fibré T0 défini par : (x, Δ, v) ∈ T0 ⇔ v ∈ Δ, et on obtient ainsi un fibré vectoriel T ∗ qui est symplectique. Au dessus des (x, D) avec x régulier, cela revient, comme on l’a déjà expliqué précédemment, a quotienter Tx E par RXH (x). On notera : p : T → T ∗ ce passage au quotient. ` On peut alors définir sur T le flot “éclaté” de (Dφt ) par : Ft : (x, Δ, v) ∈ T → φt x, Dφt (x)Δ, Dφt (x)v . le sous-fibré T0 de T est alors invariant par Ft , et on peut donc passer (Ft ) au quotient de manière à définir le cocycle symplectique Ft∗ : T ∗ → T ∗ au dessus de C. A cause de l’hypothèse faite sur F (l’angle du champ de vecteurs hamiltonien avec la verticale est uniformément minoré par une constante strictement positive quand il est défini), pour chaque (x, Δ) ∈ C, on a : Δ ∩ V (x) = {0}. On peut alors reprendre le raisonnement que nous avions fait au début de cette sous-section au voisinage des points réguliers en lesquels le champs hamiltonien n’est pas vertical, mais cette fois au dessus de tous les points de C : si (x, Δ) ∈ C, v(x, Δ) = p(V (x) ∩ ∗ de T ∗ au dessus de (x, Δ). On Δ⊥ω ) est un sous-espace lagrangien de la fibre T(x,Δ) définit alors gt (x, Δ) par : {(φt x, Dφt (x)Δ)} × gt (φt x, Dφt (x)Δ) = Ft∗ ({(x, Δ)} × v(x, Δ)) ; les gt (x, Δ) sont alors disposés comme l’étaient les Gt (x) (on l’a montré en proposition 3.14 pour x point régulier, on l’obtient aux autres points par passage a la limite) , ce qui permet de définir par passage à la limite (de manière analogue ` a G− et G+ ) les sous-espaces lagrangiens g− (x, Δ) et g+ (x, Δ) de T ∗ (x, Δ). Cette `

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fois, comme on travaille sur un compact, on peut finir la démonstration comme celle de la proposition 3.12, et cela donne la conclusion voulue en regardant juste ce qui se passe aux points réguliers.

4. Liens entre les fibrés de Green et les graphes C 0 lagangiens invariants On rappelle qu’une sous-variété de classe C 1 de T ∗ M est dite lagrangienne si elle a même dimension que M et si la restriction de la forme symplectique a` son fibré tangent est identiquement nulle. On sait alors qu’une telle sous-variété est le graphe d’une fonction de classe C 1 si et seulement si il existe une 1-forme fermée (de classe C 1 ) dont elle est le graphe (voir par exemple [23]). Qu’appelle-t-on alors un graphe C 0 -lagrangien ? De fa¸con naturelle et suivant [15] (I.8.13) : Définition 4.1. On appelle graphe C 0 -lagrangien dans T ∗ M le graphe d’une 1forme λ (continue) qui est fermée au sens des distributions. Remarque 4.2. De manière équivalente, un graphe C 0 lagrangien est le graphe d’une section s : M → T ∗ M du cotangent telle qu’il existe une 1-forme fermée c de classe C ∞ de M et une fonction u : M → R de classe C 1 telles que s = c + du. Le premier théorème de Birkhoff (voir par exemple [16]) affirme que si M = T1 et si L : T ∗ T × T → R est un Lagrangien de Tonelli, tout graphe invariant continu par le temps 1 du flot d’Euler–Lagrange de L est en fait le graphe d’une application lipschitzienne. Signalons aussi qu’il existe en dimension 1 des résultats qui montrent qu’en fait ces graphes invariants sont plus réguliers que simplement lipschitz (voir [1]). En dimension supérieure, la même question se pose pour les graphes C 0 lagrangiens de T ∗ M . Nous renvoyons le lecteur a` l’excellent texte de Michel Herman [15] dans lequel il est expliqué à l’aide de contre-exemples pourquoi il faut imposer aux graphes considérés d’être lagrangiens, pourquoi il faut une “torsion” définie . . . pour espérer des résultats analogue et dans lequel est donnée la démonstration d’un résultat analogue pour les difféomorphismes qui admettent une fonction génératrice globale. En dimension supérieure, le premier théorème de Birkhoff est aussi vrai. Expliquons plus en détail d’o` u cela vient. Tout d’abord, remarquons que quitte a utiliser un difféomorphisme symplectique de la forme (x, p) → (x, p + c(x)) et ` changer le hamiltonien H en H1 (x, p, t) = H(x, p+c(x), t), on peut supposer qu’on s’intéresse à des graphes exacts symplectiques (i.e. λ = du0 est exacte). Or, si le graphe de du0 (avec u0 ∈ C 1 (M, R)) est invariant par le temps 1 du flot hamiltonien (φt ) du hamiltonien de Tonelli H, il existe u ∈ C 1 (M × R, R) qui est une solution de l’équation de Hamilton–Jacobi : ∂u (x, t) + H dx u(x, t), t = 0 (H − J) ∂t

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et vérifie : • ∀x ∈ M, u(x, 0) = u0 (x) ; • ∀x ∈ M, dx u(x, 1) = dx u(x, 0). Ce résultat est classique. Par souci de complétude, nous en donnons une démonstration dans l’appendice, démonstration qui nécessite d’être familier avec la théorie K.A.M. faible. Dans [10], A. Fathi montre que toute solution de classe C 1 de l’équation de Hamilton–Jacobi est en fait de classe C 1,1 . Ainsi, il généralise à toute dimension le premier théorème de Birkhoff : tout graphe C 0 lagrangien invariant par le temps 1 d’un flot hamiltonien de Tonelli est donc le graphe d’une application lipschitzienne. C’est pourquoi il n’est pas plus restrictif de parler de graphe Lipschitz invariant que de graphe C 0 lagrangien invariant, et dans la suite tous nos graphes seront supposés lipschitziens. De plus, toutes les orbites partant d’un tel graphe invariant sont globalement minimisantes (voir par exemple [10]), donc sans point conjugué. Comme nous les utiliserons par la suite, mettons ces résultats en valeur : Résultats. Tout graphe C 0 -lagrangien invariant par un flot de Tonelli dans le cas autonome, par le temps 1 d’un flot de Tonelli dans le cas dépendant du temps, est lipschitzien. De plus, l’orbite de tout point d’un tel graphe est globalement minimisante au sens de la section 3.3. Par souci de complétude concernant les résultats sur les graphes lipschitz, signalons aussi les résultats obtenus par J. Mather dans [21], résultats qui concernent les supports des mesures minimisantes. Un autre résultat nous sera, puisque nous utiliserons dans certaines démonstrations le fait que l’espace tangent a` un graphe lipschitz lagrangien en un point de différentiabilité est dans le tangent a` une hypersurface d’énergie : Résultat. Tout graphe lipschitz lagrangien invariant par le flot d’un hamiltonien de Tonelli autonome est contenu dans une hypersurface d’énergie. La justification de ce résultat est élémentaire : soit λ la 1-forme fermée qui définit le graphe invariant Gλ nous intéressant et x un point de différentiabilité de λ. De l’invariance par le flot de XH on déduit que XH (λ(x)) ∈ Tλ(x) Gλ . Aussi, le sousespace lagrangien Tλ(x) Gλ est dans l’orthogonal de XH (λ(x)) pour la forme symplectique, c’est-à-dire dans le noyau de DH(λ(x)). On en déduit qu’en (Lebesgue) presque tout point de M (ceux o` u λ est différentiable), on a : D(H ◦ λ)(x) = 0. Le résultat en découle. 4.1. Inégalités entre les fibrés de Green et les différentielles généralisées des graphes lagrangiens invariants Dans cette section, on suppose que u : M × R → R est une solution de l’équation de Hamilton–Jacobi telle que dx u(., 0) = dx u(., 1), et on définit : u0 = u(., 0). Le graphe N de λ = du0 est lipschitzien et est invariant par le flot hamiltonien au

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temps 1. Nous convenons de noter pour chaque t ∈ R Nt = φt (N ) ⊂ T ∗ M (donc ∗ = pour n ∈ Z, Nn = N ), N t × {t} ⊂ T M × R, N1 = t∈R N t∈[0,1] Nt × {t} ⊂ T ∗ M × R et N = t∈R Nt × {t¯} = t∈[0,1] Nt × {t¯} ⊂ T ∗ M × T. Remarquons de Nt est le graphe de dut , o` u ut = u(., t) et que donc Nt est une sous-variété Lipschitz exacte lagrangienne de T ∗ M . 4.1.1. Etude aux points de différentiabilité. Par le théorème de Rademacher, λ est donc différentiable Lebesgue presque partout. Nous allons commencer par montrer un résultat concernant la différentielle en ces points de différentiabilité. Proposition 4.3. Soit x ∈ M un point de différentiabilité de du0 . Alors, l’espace tangent en (x, λ(x)) ` a N est un sous-espace lagrangien noté Tx N qui vérifie : G− (x) ≺ Tx N ≺ G+ (x). Démonstration de la proposition 4.3. Soit x un point de différentiabilité de λ. Comme Nt = φt (N ), en tout point φt (x) de l’orbite de x il existe un tangent a Nt , donc comme dut est lipschitzienne, c’est un point de différentiabilité de ` dut (on ne pourrait avoir un espace tangent qui ne soit pas transverse a` la ver˜ et y est un point de différentiabilité de dut , Ty Nt est ticale). Donc si (y, t) ∈ N un sous-espace lagrangien transverse a` la verticale V (y). Aussi, pour tout τ ∈ R, Dφt+τ (y)(Ty Nt ) = Tφt+τ (y) Nt+τ est transverse à Dφt+τ (y)V (y). t t t Si x ∈ N , on définit alors : F (t) = Tφt (x) Nt . Ce fibré vérifie toutes les hypothèses de la proposition 3.11, donc aussi sa conclusion. 4.1.2. Notions de vecteurs tangents généralisés et de différentielle généralisée. Aux points de différentiabilité de u0 , nous avons réussi à “coincer” le sous-espace tangent entre les deux fibrés de Green. Nous aimerions obtenir un résultat similaire aux points o` u la fonction n’est pas différentiable. Dans ce cas, nous somme amenés a introduire une notion de “vecteur tangent généralisé à N en x”. ` Définition 4.4. Soit U un ouvert de Rd et h : U → Rn un plongement topologique. Soit x ∈ U . Un vecteur w ∈ Rn est dit vecteur tangent généralisé à h en x s’il existe une suite (xk ) de points de U tendant vers x, une suite (tk ) de réels strictement positifs tendant vers 0 telles que : 1 h(xk ) − h(x) . w = lim k→∞ tk On note alors TxG h l’ensemble de ces vecteurs. C’est un cône appelé cône tangent a h. ` Bien entendu, dans le cas o` u h est différentiable en x de différentielle en x injective, ce cône n’est rien d’autre que l’image de Dh(x).On montre alors facilement : Proposition 4.5. Si ψ : (Rn , h(x)) → (Rn , ψ ◦ h(x)) est un difféomorphisme local et ϕ : (Rd , ϕ−1 (x)) → (Rd , x) est un homéomorphisme local, alors : TϕG−1 (x) (ψ ◦ h ◦ ϕ) = Dψ(h(x))TxG h.

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Cette proposition nous permet d’étendre la notion de vecteur tangent généralisé à des plongement topologiques de sous-variétés dans des variétés. Si maintenant on s’intéresse au graphe N de λ avec λ lipschitzienne, on conviendra de noter TxG N = TxG (λ). On dira que TxG N est l’espace tangent généralisé à N ; un de ses éléments sera un “vecteur tangent généralisé a ` N en x”. De la proposition 4.5, on déduit que si ψ est un difféomorphisme de T ∗ M qui envoie le graphe N (λ) de λ sur G G N (μ) = Dψ(λ(q))Tλ(q) N (λ). En parle graphe N (μ) de μ, alors : ∀q ∈ M, Tψ(λ(q)) ticulier, le “fibré tangent généralisé” à un graphe invariant par le difféomorphisme ψ est invariant par Dψ. Proposition 4.6. Soit N le graphe lipschitz de λ : M → T ∗ M . Soit x ∈ M en aN lequel : TxG λ est un plan P . Alors λ est différentiable en x et le plan tangent ` en λ(x) est P . Démonstration de la proposition 4.6. Adoptons les hypothèses de l’énoncé. Comme λ est lipschitzienne, le plan P est transverse à la verticale V (λ(x)), et est donc le graphe d’une application linéaire L : Tx M → Tλ(x) (T ∗ M ). Supposons que λ ne soit pas différentiable en x ; passant en coordonnées pour pouvoir soustraire des points, on trouve une suite (xk ) de points différents de x qui converge vers x et un ε > 0 tels que : ∀k ∈ N , i.e : ∀k ∈ N ,

λ(xk ) − λ(x) − L(xk − x) ≥ εxk − x

xk − x 1 xk − x λ(xk ) − λ(x) − L xk − x ≥ ε

quitte a` extraire une sous-suite, on peut : • supposer que la suite xxkk −x −x converge vers un vecteur w de norme 1 ; • comme λ est lipschitzienne supposer que la suite xk1−x (λ(xk ) − λ(x)) converge vers W ∈ TxG λ ⊂ Tλ(x) (T ∗ M ). Remarquons qu’alors Dπ ∗ (λ(x))W = w. On a alors : W − Lw ≥ ε, donc W ∈ TxG λ\P = ∅, donc on obtient une contradiction. Nous introduisons maintenant la notion de différentielle généralisée d’une application lipschitzienne, puis expliquerons dans le cadre qui nous intéresse comment relier les deux notions. Cette notion de différentielle généralisée est couramment utilisée en optimisation non lisse. Définition 4.7. Soient M , N deux variétés et λ : M → N une application lipschitzienne. On sait alors que l’ensemble D des points de M en lesquels λ est différentiable est une partie dense de M (par le théorème de Rademacher). Pour chaque x ∈ M , on appelle différentielle généralisée de λ en x et on note DG λ(x) l’enveloppe convexe de l’ensemble des valeurs d’adhérences des suites de la forme (Dλ(xk )) avec xk ∈ D et limk→∞ xk = x.

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Par le théorème de Carathéodory, la différentielle généralisée de λ en x est automatiquement une partie compacte (et non vide) de L(Tx M, Tλ(x) N ) puisque c’est l’enveloppe convexe d’une partie compacte (compacte car λ est lipschitzienne) en dimension finie. Remarquons que même en un point de différentiabilité de λ, sa différentielle généralisée peut contenir plus qu’un seul élément. Par contre : Proposition 4.8. Si λ : M → N est lipschitzienne, si pour un x ∈ M DG λ(x0 ) ne contient qu’un élément noté L, alors λ est différentiable en x et Dλ(x) = L. Démonstration de la proposition 4.8. On se place en carte (ce qui permet de soustraire) et on suppose que le résultat soit faux. Il existe alors ε > 0 et une suite (xk ) tendant vers x telle que : ∀n ,

λ(xn ) − λ(x) − L(x − xn ) ≥ ε. x − xn

On choisit alors α > 0 tel que : ∀y ∈ B(x, 2α) ∩ D, L − Dλ(y) < 4ε . De plus, on note R = L + ε. A partir d’un certain rang N , les xn sont tous dans B(x, α). On utilise alors le fait que λ est lipschitzienne pour dire que D est de mesure pleine, puis Fubini x−xn n pour trouver pour chaque n ≥ N un yn ∈ B(x, α2 ) tel que xynn−x −yn − x−xn < λ(x)−λ(yn ) ε 4R , xn −yn

≤

ε 4,

x − xn = yn − xn et pour presque tout t ∈ [0, 1],

n ) txn + (1 − t)yn ∈ D. On majore alors pour n ≥ N la quantité λ(xn )−λ(x)−L(x−x x−xn par : 1

y −x λ(x) − λ(yn ) n n Dλ tyn + (1 − t)xn − L dt + x − xn yn − xn 0 yn − xn x − xn + L. − yn − xn x − xn

qui est majoré par

3ε 4 ,

ce qui contredit le choix de ε.

On en déduit immédiatement : Corollaire 4.9. Soit λ : M → N lipschitzienne et x ∈ M . On a équivalence de : • DG λ(x) est un singleton ; • λ est différentiable en x et x est un point de continuité de Dλ. Dans le cas o` u DG λ(x) ne contient qu’un élément, on en déduit que λ est différentiable en x et donc que si de plus Dλ(x) est injective, alors TxG λ, cône tangent a` λ, est un plan. En fait, il existe une relation entre le cˆ one tangent et la différentielle généralisée, bien connue des gens qui font de l’optimisation non lisse : Proposition 4.10. Soit λ : M → N une application bi-lipschitzienne et x ∈ M . Alors : TxG λ ⊂ Lv; v ∈ Tx M, L ∈ DG λ(x) . Nous en donnons rapidement une démonstration :


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Démonstration de la proposition 4.10. On raisonne en carte. Soit v ∈ TxG λ. Il existe alors une suite (tk ) de réels strictement positifs tendant vers 0 et une suite (xk ) de points de M convergent vers x tels que : 1 v = lim λ(xn ) − λ(x) . n→∞ tn Comme λ est bilipschitz, il existe c > 0 tel que : ∀x, y, λ(x) − λ(y) ≥ cx − y. On en déduit que la suite de terme général ( xntn−x ) est bornée. Comme dans la démonstration précédente, on choisit yn proche de x tel que 1 1 λ(y n ) − λ(x) ≤ n , xn − yn = xn − x et pour presque tout t ∈ [0, 1], λ tn est différentiable en txn + (1 − t)yn . Le vecteur v est alors la limite de la suite de terme général : 1 1 1 λ(xn ) − λ(yn ) = Un = Dλ tyn + (1 − t)xn (yn − xn )dt . tn tn 0 n Quitte a` extraire une sous-suite, on peut supposer que la suite ( xxnn −y −yn ) converge vers w. On a alors : xn − yn 1 Un = Dλ tyn + (1 − t)xn w + o(1) dt . tn 0 n On a vu que la suite ( xnt−y ) = ( xntn−x ) est bornée, donc quitte a` extraire une n sous-suite on peut supposer qu’elle converge vers ∈ R+ . Etant donné ε > 0, on peut trouver α > 0 tel que pour tout y ∈ B(x, 2α), il existe Ly ∈ DG λ(x) tel que Ly − Dλ(y) < ε ; on peut même choisir Ly de fa¸con mesurable en fonction de y. Aussi, pour n assez grand : 1 Un − Ltyn +(1−t)xn dt + o(1) w + o(1) ≤ + o(1) εw + o(1) .

0

On a : L = voulue.

1 0

Ltyn +(1−t)xn dt ∈ DG λ(x) donc v = L( .w) est de la forme

4.1.3. Encadrement de la différentielle généralisée et des vecteurs tangents généralisés. Revenons au cas qui nous intéresse, c’est-à-dire a` l’étude des graphes lipschitz lagrangiens invariants, graphes de du. Dans ce cas, l’application du est un plongement qui est bi-lipschitz, et on est donc dans le cadre de la proposition 4.10, et si x ∈ M , tout vecteur tangent généralisé au graphe N de du est dans l’image d’une différentielle généralisée de du en x. a dérivée lipschitzienne telle que Proposition 4.11. Soit u : M → R de classe C 1 ` le graphe N de du est invariant par (φ1 ). Alors : ∀x ∈ M , ∀L ∈ DG (du)(x) , G− λ(x) ≺ L(Tx M ) ≺ G+ λ(x) . Cette proposition découle immédiatement de la proposition 4.3 : il suffit de passer à la limite en utilisant la semi-continuité de G− et G+ .

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Ainsi, en quelque sorte, les “sous-espaces tangents généralisés” en x à N , que l’on trouve en prenant les limites des sous-espaces tangents aux points de différentiabilité de du, sont “coincés” entre les deux fibrés de Green, au sens de l’ordre entre les sous-espaces lagrangiens que nous avions défini précédemment. En ce qui concerne le “cône tangent” en x à N , que nous avions noté TxG N , on peut ` N en x, il existe un donc dire : pour tout v ∈ TxG N vecteur tangent généralisé a sous-espace lagrangien L ⊂ Tx (T ∗ M ) tel que v ∈ L et G− (x) ≺ L ≺ G+ (x). Remarquons que pour un tel v ∈ TxG N , on a alors par définition de la relation d’ordre entre les sous-espaces lagrangiens : ω(x)(v, (Dπ ∗ (x)|G+ (x) )−1 (Dπ ∗ (x)v)) ≥ 0 et ω(x)(v, (Dπ ∗ (x)|G− (x) )−1 (Dπ ∗ (x)v)) ≤ 0, i.e. plus simplement en coordonnées, si G+ (x) est le graphe de S+ et G− (x) celui de S− , si v = (v1 , v2 ) : t

v1 S+ v1 − t v1 .v2 ≥ 0 et

t

v1 .v2 − t v1 S− v1 ≥ 0 .

On peut alors se demander si la vérification de ces deux dernières conditions pour un vecteur v ∈ Tx (T ∗ M ) suffisent pour obtenir l’existence de L ⊂ Tx (T ∗ M ) sousespace lagrangien tel que v ∈ L et G− (x) ≺ L ≺ G+ (x). La réponse est non, comme le lecteur peut s’en convaincre avec l’exemple suivant : 1 0 3 0 ; S+ = ; v = (1, 0, 2, 3) . S− = 0 1 0 3 Dans ce cas, on cherche L comme graphe de S = 23 d3 telle que S− ≺ S ≺ S+ ; 1 −3 eterminant négatif or, dans ce cas forcément d ∈ [1, 3] et S+ − S = −3 3−d a un d´ et est donc indéfinie. Par contre, bien entendu, si M est de dimension 1, la condition signifie juste qu’une pente est comprise entre deux valeurs. 4.2. Résultats de régularité des graphes C 0 lagrangiens invariants 4.2.1. Lien entre la dynamique sur un graphe invariant et la régularité sur ce graphe. Proposition 4.12. Soit L : T M × T1 → R un lagrangien satisfaisant les hypothèses de Tonelli, H : T ∗ M × T1 → R le hamiltonien qui lui est associé et G un graphe C 0 -lagrangien invariant par le temps 1 du flot hamiltonien de H. On suppose que : ∀x ∈ G ,

G+ (x) = G− (x) .

Alors le graphe G est de classe C et : 1

∀x ∈ G ,

Tx G = G+ (x) = G− (x) .

Démonstration de la proposition 4.12. Soit λ l’application dont G est le graphe. De la proposition 4.11, on déduit que : ∀x ∈ M , ∀L ∈ DG (λ)(x) , G− λ(x) ≺ L(Tx M ) ≺ G+ λ(x) ce qui implique qu’en tout point x de M , DG (λ)(x) est un singleton et le corollaire 4.9 implique alors qu’en chaque x ∈ M , λ est différentiable et Dλ est continue.

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Corollaire 4.13. Soit L : T Tn × T1 → R un lagrangien satisfaisant les hypothèses de Tonelli, H : T ∗ Tn × T1 → R le hamiltonien qui lui est associé et G un graphe C 0 -lagrangien invariant par le temps 1 du flot hamiltonien de H. On suppose que le temps 1 du flot hamiltonien de H restreint a ` G est bi-lipschitz conjugué a ` une rotation. Alors le graphe G est de classe C 1 . Démonstration du corollaire 4.13. Notons F le temps 1 du flot hamiltonien de H et f sa restriction à G. Il existe alors une rotation R : Tn → Tn (définie par R(θ) = θ+α) et un homéomorphisme bilipschitz h : G → Tn tel que f = h−1 ◦R◦h. Soit alors C une constante de Lipschitz commune à h et h−1 . Comme R est un isométrie, on a alors : ∀x, y ∈ G , d F n (x), F n (y) ≤ C 2 d(x, y) ; u G est le graphe de λ). Il existe donc une suite soit alors x ∈ M et v ∈ TxG λ (o` (xk ) de points de M tendant vers x, une suite (tk ) de réels strictement positifs tendant vers 0 telles que : 1 λ(xk ) − λ(x) w = lim k→∞ tk (bien entendu, la soustraction précédente n’a pas de sens si on ne se place pas en cartes). On a alors (toujours avec la convention qu’on travaille en cartes pour définir la soustraction de points) : 1 f n λ(xk ) − f n λ(x) ≤ C 2 1 λ(xk ) − λ(x) ∀k, n ∈ N , ∀x, y ∈ M , tk tk donc en passant a` la limite : n Df λ(x) w ≤ C 2 w . Aussi, par le critère dynamique d’appartenance aux fibrés de Green (proposition 3.17), w ∈ G− (λ(x)). Donc : TxG λ ⊂ G− (λ(x)). Du fait que G est un graphe lipschitz, on déduit aisément qu’en chaque point x on a : Dπ ∗ (x)TxG λ = Tx M , ace à la proposition 4.6, on en déduit donc forcément : TxG λ = G− (λ(x)). Grˆ que λ est différentiable en x et que Tλ(x) G = G− (λ(x)) ; de fa¸con analogue, Tλ(x) G = G+ (λ(x)) et donc finalement G− (λ(x)) = G+ (λ(x)), ce qui permet de conclure par la proposition 4.12. Corollaire 4.14. Soit L : T T1 × T1 → R un lagrangien satisfaisant les hypothèses de Tonelli, H : T ∗ T1 × T1 → R le hamiltonien qui lui est associé et G un graphe continu invariant par le temps 1 du flot hamiltonien de H. On suppose que le temps 1 du flot hamiltonien de H restreint a ` G est bi-lipschitz conjugué a ` un difféomorphisme du cercle de classe C 2 de nombre de rotation irrationnel. Alors le graphe G est de classe C 1 . Démonstration du corollaire 4.14. On utilise le résultat VII-1-9 contenu dans [14] : étant donné un difféomorphisme g du cercle de classe C 2 et de nombre de rotation α irrationnel, si les qn désignent les dénominateurs des réduites de α, il existe

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une constante c > 0 telle que : ∀x ∈ T1 , 1c ≤ g qn (x) ≤ c. Ceci implique que les applications g qn , g −qn sont toutes c-lipschitzienne. Si maintenant f|G est bi-lipschitz conjuguée à une telle application g, il existe alors une constante C > 0 telle que qn −qn les f|G , f|G sont toutes C-lipschitziennes. On conclut alors de manière analogue a ce qui a été faite dans la démonstration du corollaire 4.13. ` 4.2.2. Le cas des petites dimensions. Dans le cas des lagrangiens indépendants du temps à deux degrés de liberté, un phénomène intéressant se produit : les graphes continus invariants sans point critique du hamiltonien sont en fait plus réguliers que simplement Lipschitz : ils sont en quelque sorte “de classe C 1 ” sur un ensemble de mesure pleine, i.e : dans le cas autonome à deux degrés de liberté, les solutions correspondantes de classe C 1 de l’équation de Hamilton–Jacobi sont “de classe C 2 ” sur un ensemble de mesure pleine. Proposition 4.15. Soit M une variété de dimension 2, L : T M → R un lagrangien indépendant du temps satisfaisant les hypothèses de Tonelli, H : T ∗ M → R le hamiltonien qui lui est associé et G un graphe C 0 -lagrangien invariant par le flot hamiltonien de H qui ne contient pas de point critique. Soit λ la 1-forme fermée (au sens des distributions) dont G est le graphe. Alors, il existe un Gδ dense D de mesure pleine de M tel qu’en tout point de D, la différentielle généralisée de λ soit un singleton. En particulier, en chaque point de D, λ est différentiable et Dλ est continue. Rappelons le résultat correspondant dans le cas non autonome : Proposition 4.16. Soit L : T T1 × T1 → R un lagrangien satisfaisant les hypothèses de Tonelli, H : T ∗ T1 × T1 → R le hamiltonien qui lui est associé et G un graphe continu invariant par le temps 1 du flot hamiltonien de H. On suppose que ce graphe invariant est le graphe de λ. Alors il existe un Gδ dense D de mesure pleine de T tel qu’en tout point de D, la différentielle généralisée de λ soit un singleton. En particulier, en chaque point de D, λ est différentiable et Dλ est continue. Avec des hypothèses légèrement différentes, ce dernier résultat est démontré dans [1]. La démonstration dans ce cas en est une copie, nous ne la donnons pas ici. Nous allons nous contenter de démontrer la proposition 4.15 : Démonstration de la proposition 4.15. Prenons les hypothèses de la proposition. On a vu qu’on peut, quitte a` changer le hamiltonien, supposer que le graphe invariant G est exact : λ = du0 avec u0 ∈ C 1 (M, R). De plus, on a vu qu’alors, λ est automatiquement lipschitzienne et que toute orbite issue de G est globalement minimisante donc sans point conjugué. Ceci permet de définir les fibrés de Green G− et G+ en tout point de G, ainsi que les fibrés de Green réduits g− et g+ en ces mêmes points. La proposition 3.9 nous dit que D = {x ∈ M ; G− (x, λ(x)) = G+ (x, λ(x))} est un Gδ de M . D’après la proposition 4.11, on sait alors que : ∀x ∈ M ,

∀L ∈ DG (du0 )(x) ,

G− (x) ≺ L(Tx M ) ≺ G+ (x)

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ce qui implique qu’en tout point x de D, DG (du0 )(x) est un singleton et le corollaire 4.9 implique alors qu’en chaque x ∈ D, λ est différentiable et Dλ est continue. Il nous faut alors, pour conclure, montrer que D est de mesure pleine. Soit donc μ une mesure de Lebesque sur M , définie a` l’aide d’une métrique riemanienne. On a : μ(M ) = M dμ. Notons (ϕt ) le flot (φt|G ) projeté sur M . Comme chaque (ϕt ) est un homéomorphisme bi-lipschitzien, on a en faisant un changement de variable : μ(M ) = E | det(Dϕt )|dμ (cette différentielle Dϕt est définie presque partout, sur un ensemble noté E, qui est invariant par le flot (ϕt )) ; tout point de E est un point de différentiabilité de λ, o` u det est pris dans des bases de volume 1. Aussi, pour toute suite T = (tn ) de réels, on trouve en utilisant le lemme de Fatou : +∞ > μ(M ) ≥ lim inf | det(Dϕtn )|dμ E n→∞

donc en presque tout x ∈ M , on a : lim inf n→∞ | det(Dϕtn (x))| < +∞ ; on note alors D(T ) (qui est une partie de E) l’ensemble de ces points, qui est donc de mesure pleine. Quand la suite (tn ) est bornée, ceci est une évidence en tout point de x. Par contre, quand la suite (tn ) tend vers +∞ (resp. −∞), on déduit du fait que x ∈ D(T ) que G− (λ(x)) = Tλ(x) G (resp. G+ (λ(x)) = Tλ(x) G). Expliquons pourquoi. On aura ainsi montré que D est de mesure pleine et fini la démonstration. Supposons donc par exemple que lim inf n→∞ | det(Dϕtn (x))| < +∞ avec limn→∞ tn = +∞. Le long de l’orbite de x, λ est différentiable et la projection ∗ restreinte au graphe est différentiable, de différentielle uniformément sur M π|G bornée, et d’inverse différentiable et de différentielle uniformément bornée (ceci car le graphe est lipschitzien). Aussi, si on choisit de remonter les bases de volume 1 de Tx M en des bases de volume 1 de Tλ(x) G par cette application (cela revient a dire quelle forme volume on utilise le long de l’orbite de λ(x)), on en déduit ` que : lim inf n→∞ | det(Dφtn (λ(x))|T G )| < +∞. Comme le champ de vecteurs XH n’a pas de singularité sur le compact G, il existe une constante C > 1 telle que : ∀y, z ∈ M ,

XH (λ(z)) 1 ≤ ≤C. C XH (λ(y))

Aussi : Dφtn (x)XH (λ(x)) = XH (φtn λ(x)) ≥ C1 XH (λ(x)). Si maintenant on regarde l’action Mtn de Dφtn sur la droite T G/RXH , du fait que suivant XH (λ(x)) les Dφtn ne contractent pas trop et du fait que lim inf n→∞ | det(Dφtn (λ(x))|T G )| < +∞, on déduit : < +∞ . lim inf Mtn λ(x) n→∞

|Tλ(x) G/RXH λ(x))

A l’aide de la proposition 3.17, on en déduit que Tλ(x) G/RXH (λ(x)) ⊂ g− (λ(x)), donc que Tλ(x) G ⊂ G− (λ(x)), donc que Tλ(x) G(λ(x)) = G− (λ(x)). Question. Que se passe-t-il quand le graphe invariant contient des singularités ?

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Nous n’avons pu démontrer un résultat analogue a` la proposition 4.15 que dans le cas o` u les singularités sont toutes non dégénérées. Dans le cas général, le problème reste ouvert. Définition 4.17. Soit x0 une singularité du hamiltonien H (c-à-d XH (x0 ) = 0) ; on dit qu’elle est non dégénérée si DXH (x0 ) a toutes ses valeurs propres deux à deux distinctes. Rigoureusement, cette définition est fausse puisque DXH (x0 ) n’a pas même espace de départ et d’arrivée, mais il existe une fa¸con canonique et standard d’identifier ces deux espaces. Pour un hamiltonien de classe C k avec k ∈ [2, ∞], la propriété d’avoir toutes ses singularités non dégénérées est générique. Pour un tel hamiltonien, les singularités sont alors isolées. Comme de plus elles forment un ensemble fermé, tout compact de T ∗ M n’en contient qu’un nombre fini. Proposition 4.18. Soit M une variété de dimension 2, L : T M → R un lagrangien indépendant du temps satisfaisant les hypothèses de Tonelli, H : T ∗ M → R le hamiltonien qui lui est associé dont on suppose que toutes les singularités sont non dégénérées. Soit G un graphe C 0 -lagrangien invariant par le flot hamiltonien de H. Soit λ la 1-forme fermée (au sens des distributions) dont G est le graphe. Alors, il existe un Gδ dense D de mesure pleine de M tel qu’en tout point de D, la différentielle généralisée de λ soit un singleton. En particulier, en chaque point de D, λ est différentiable et Dλ est de classe C 1 . Bien entendu, le théorème 1 est une conséquence immédiate de cette proposition. Démonstration de la proposition 4.18. On prend les hypothèses de l’énoncé. Si x0 est un point critique qui appartient a` G, alors il est forcément hyperbolique. Supposons en effet que ce ne soit pas le cas. Comme x0 est une singularité non dégénérée, chaque Dφt (x0 ) est diagonalisable (sur C). Supposons que DXH (x0 ) ait une valeur imaginaire pure iλ. Il existe alors un plan symplectique P ⊂ Tx0 (T ∗ M ) invariant par (Dφt (x0 )) tel que les valeurs propres de chaque Dφt (x0 )|P soient e±itλ . Alors, pour chaque v ∈ P , l’ensemble {Dφt (x0 )v; t ∈ R} est borné et donc P ⊂ G− (x0 ) ∩ G+ (x0 ). Mais comme P est symplectique, il ne peut pas être inclus dans un plan lagrangien, d’o` u une contradiction. La situation est donc la suivante : l’ensemble S = {s1 , . . . , sn } des points critiques contenus dans G est fini, et chacun de ses points est hyperbolique. n La réunion des variétés stables des si intersectées avec G, W = i=1 W s (si ) ∩ G, est alors mesurable. Il en est de même de R = G\W . On va alors montrer que pour presque tout point de R et presque tout point de W , on a : Tx G = G− (x). On en déduira qu’en presque tout point de G, Tx G = G− (x) et de fa¸con symétrique u le résultat cherché. Tx G = G+ (x) = G− (x), d’o`

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Cas o` u x ∈ R : On construit pour chaque i ∈ {1, . . . , n} et chaque k ∈ N un petit disque Dk (i) ⊂ G ouvert contenant si dont la frontière est un lacet lipschitzien. On suppose les Dk (i) pour i ∈ {1, . . . , n} deux a` deux disjoints et que : ecroissante i }. On peut aussi supposer chaque suite (Dk (i))k∈N d´ k∈N Dk (i) = {s et poser : Dk = 1≤i≤n Dk (i) et Rk = G\Dk . On note alors f = φ1 le temps 1 du flot hamiltonien restreint a` G et Fkm l’application (non définie partout) qui a` un point x de R associe le mième point de l’orbite de x sous f à appartenir a` Rk . Soit Ek l’intersection des ensembles de définition des Fkm pour m ∈ N∗ avec Rk , alors k∈N Ek = R puisque si x ∈ R, il existe un point y ∈ ω(x) de l’ensemble ω-limite de x qui n’est pas dans S, donc dans l’un des Rk . On va alors montrer que pour chaque k, pour presque tout x ∈ Ek , on a : Tx G = G− (x), ce qui donnera le résultat cherché en presque tout point de R. Déj` a, on se limite aux points de Ek en lesquels G admet un plan tangent (ceci est vrai presque partoutcar on a une variété lipschitzienne), et qui ne sont pas sur m∈N f −m (∂Dk ) = m∈N φ−m (∂Dk ) ; on note Ek leur ensemble ; a` cause de l’hypothèse faite lors de la construction de Dk (i), Ek est de mesure pleine dans Ek . Soit alors x ∈ Ek . Pour chaque m ∈ N, il existe nm = nm (x) ≥ m tel que Fkm (x) = f nm (x) = φnm (x) ∈ Rk . Vu les hypothèses faites dans la définition de Ek , pour y assez proche de x, on a nm (y) = nm (x) = nm et : Fkm (y) = f nm (y) = φnm (y), et donc Fkm est différentiable en x, de différentielle : DFkm (x) = Dφnm (x)|Tx G . On a alors, si μ désigne la mesure de Lebesgue sur G : m m (Fk )∗ (dμ) = | det(DFkm )|dμ μ(Rk ) ≥ μ Fk (Ek ) = Ek

Ek

=

Ek

| det(Dφnm |T G )|dμ .

Donc en utilisant le lemme de Fatou, en presque tout point de Ek : lim inf det Dφn (x) (x)|T G < +∞ . m→∞

m

x

De plus, vu la définition de Rk , il existe C ≥ 1 tel que : ∀y, z ∈ Rk ,

XH (z) 1 ≤ ≤C. C XH (y)

Comme dans la démonstration de la proposition 4.16, de ceci et du fait que : lim inf det Dφn (x) (x)|T G < +∞ m→∞

m

x

on déduit que si on passe au quotient p : T G → T G/RXH (x) par XH , on obtient : lim inf p ◦ Dφnm (x)|Tx G < +∞ . m→∞

A l’aide de la proposition 3.17, on en déduit que Tλ(x) G/RXH (x) ⊂ g− (x), donc que Tx G ⊂ G− (x), donc que Tx G(x) = G− (x). Cas o` u x ∈ W : déjà, W1 = W \S est de mesure pleine dans W . On appellera Wloc l’intersection de la réunion des variétés locales stables des si avec G. Soit alors

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x ∈ W1 ; il existe alors T > 0 tel que x ∈ φ−T (Wloc ). On dira alors que x est simple s’il existe un voisinage Ux de x tel que les seuls points de Ux contenus dans φ−T (Wloc ) sont ceux qui sont sur l’orbite de x : en d’autres termes, l’orbite de x est “isolée” dans la variété locale stable de S. L’ensemble des points simples de W est alors un ensemble de mesure nulle. Aussi, on travaillera désormais dans W , ensemble des points de W1 qui ne sont pas simples et en lesquels G admet un espace tangent. Cet ensemble est de mesure pleine dans W . Si maintenant x ∈ W , alors s (si )) ; du fait que x n’est pas simple, on déduit pour un i on a : x ∈ φ−T (Wloc s alors que : Tx G = Tx W (si ) ; comme : G− (x) = Tx W s (si ), on en déduit le résultat cherché. Soulignons que le phénomène de co¨ıncidence presque partout des deux fibrés de Green est un phénomène typique aux basses dimensions : nous donnons parmi les exemples qui vont suivre un exemple de graphe invariant de dimension 3 sur lequel le flot hamiltonien est hyperbolique, donc les fibrés de Green transverses. Dans ce cas donc, sans hypothèse supplémentaire sur la dynamique, on ne saurait espérer utiliser les fibrés de Green pour montrer une régularité supérieure a` Lipschitz pour les graphes invariants. Exemple de graphe C 0 lagrangien invariant qui n’est pas de classe C 1 . En fait, on ne connait pas d’exemple de tel graphe qui soit très “sauvage”. L’exemple le plus simple est celui formé par la séparatrice du pendule simple rigide, qui donne un graphe partout C 1 sauf en un point. Bien entendu, en faisant un produit direct de tels pendules, on a un exemple en dimension plus grande. Il existe dans [15] des exemples analogues pour les difféomorphismes de l’anneau déviant la verticale. Exemple de graphe invariant pour un système à trois degrés de liberté tel que les fibrés de Green sont transverses dans la surface d’énergie en tout point du graphe. Considérons un flot géodésique hyperbolique défini sur le fibré tangent unitaire C = T 1 S d’une surface S à courbure négative. Soit X le champ de vecteurs associé à la métrique riemannienne considérée : X : C → T C est de classe C ∞ . u . est une métrique On de´finit alors L : T C → R par L(x, v) = v − X(x)2 o` quelconque de C. Alors, L est un lagrangien de Tonelli, et le graphe dans T C de X est formé de courbes minimisantes, donc est une variété invariante par le flot d’Euler–Lagrange. Si H : T ∗ C → désigne le hamiltonien de Tonelli associé a L, son flot (φt ) laisse invariant la section nulle N de T ∗ C (qui est un graphe ` C ∞ lagrangien) et la restriction du flot a` cette section nulle est Anosov. De ceci et du caractère symplectique du flot hamiltonien, on déduit que si E désigne la surface d’énergie qui contient la section nulle, alors la restriction de (Dφt ) a` T E|N est partiellement hyperbolique, avec un fibré central de dimension 1 dirigé par le champ de vecteurs hamiltonien. Aussi, les fibrés de Green, qui co¨ıncident avec les variétés stables et instables, sont transverses dans la surface d’énergie en tout point de N , et on n’a donc pas, comme dans le cas des systèmes à deux degrés de libertés, co¨ıncidence des deux fibrés presque partout.

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Aussi, dans ce cas, la méthode développée pour démontrer la proposition 4.15 ne peut s’appliquer. Bien sˆ ur, ceci ne prouve pas que le résultat de cette proposition soit faux en plus de degrés de liberté, et il serait intéressant de donner un contreexemple dans ce cas. 4.2.3. Le cas intégrable. Dans cette partie, nous allons devoir de faire appel a` la théorie de Mather–Ma˜ né–Fathi développée dans [20, 21] et [11]. Notation 3. Etant donné une variété M , Λ(M ) désignera l’ensemble des 1-formes fermées continues de M . Définition 4.19. Un hamiltonien de Tonelli H : T ∗ M × T → R est C 0 -intégrable s’il existe une partie Λ1 ⊂ Λ(M ) telle que : 1. l’application λ ∈ Λ1 → [λ] ∈ H 1 (M ) est surjective ; 2. pour chaque λ ∈ Λ1 , le graphe Gλ de λ est invariant par le flot hamiltonien (φH t ) de H dans le cas autonome, par le temps 1 de ce flot dans le cas dépendant du temps ; 3. l’ensemble {Gλ ; λ ∈ Λ1 } est une partition de T ∗ M . Exemple. Si on munit le tore Tn de sa métrique plate, le hamiltonien obtenu est C 0 intégrable ; de plus, les tores invariants correspondants sont tous dans ce cas de classe C ∞ (et non seulement continus). Nous allons commencer par nous limiter au cas autonome, afin en particulier de pouvoir utiliser les résultats de [4] et [11] sans trop de difficulté. Ensuite, nous nous intéresserons au cas dépendant du temps, un peu plus délicat. Remarque 4.20. 1. Si H : T ∗ M → R est C 0 -intégrable et autonome, nous verrons un peu plus loin qu’il existe une action de T1 sur M qui est sans point fixe. Ainsi, M ne peut être n’importe quelle variété (sa caractéristique d’Euler est nulle, mais on a même mieux) ; si par exemple M est une surface, c’est forcément un tore. 2. Avec ces hypothèses, l’application λ ∈ Λ1 → [λ] ∈ H 1 (M ) est injective. En effet, si elle n’est pas injective, il existe λ, μ ∈ Λ1 tels que λ = μ et [λ] = [μ]. Or, deux graphes de même classe de cohomologie s’intersectent ; on a donc : Gλ ∩ Gμ = ∅ et Gλ = Gμ , ce qui contredit le fait que {Gν ; ν ∈ Λ1 } est une partition de T ∗ M ; 3. Avec ces hypothèses, Λ1 est exactement l’ensemble des graphes C 0 invariants H par (φH t ). En effet, si Gλ est invariant par (φt ) pour un certain λ ∈ Λ(M ), soit μ ∈ Λ1 tel que [μ] = [λ]. Si λ = μ, il existe η ∈ Λ1 tel que η = μ et Gη ∩Gλ = ∅. né Soit alors (x, p) ∈ Gη ∩Gλ . Alors comme Gλ est inclus dans l’ensemble de Ma˜ dual N ∗ ([λ]) (voir [4] pour les notations/définitions), l’ensemble W ω-limite de (x, p) est dans l’ensemble d’Aubry dual A∗ ([λ]) = A∗ ([μ]) ⊂ Gμ ; comme de plus Gη est fermé et invariant par (φH t ), on a aussi : W ⊂ Gη donc finalement W ⊂ Gη ∩ Gμ, ce qui contredit Gμ ∩ Gη = ∅.

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Théorème 4. Soit H : T ∗ M → R un hamiltonien de Tonelli C 0 -intégrable et Λ1 ⊂ Λ(M ) tel que {Gλ ; λ ∈ Λ1 } soit une partition de T ∗ M en graphes C 0 lagrangiens invariants par le flot hamiltonien (φH t ) de H. Alors, il existe un Gδ dense G(H) de Λ1 dont tout élément est de classe C 1 . L’ingrédient essentiel pour prouver ce théorème est : Proposition 4.21. Soit H : T ∗ M → R un hamiltonien de Tonelli C 0 -intégrable et Λ1 ⊂ Λ(M ) tel que {Gλ ; λ ∈ Λ1 } soit une partition de T ∗ M en graphes C 0 lagrangiens invariants par le flot hamiltonien (φH t ) de H. Alors, il existe une partie dense D de Λ1 tel que pour chaque λ ∈ D, Gλ est rempli par des orbites périodiques pour (φH eme période et deux a ` deux t ) de mˆ homotopes. Démonstration de la proposition 4.21. Nous allons adopter un angle d’attaque déjà développé dans [4]. On considère H et Λ1 comme dans l’énoncé. Etant donné T > 0, on considère une orbite (φH t (x, p))t∈R telle que : ∗ H π (φT (x, p)) = x. Alors, il existe un λ ∈ Λ1 tel que (x, p) ∈ Gλ . Comme Gλ est un graphe invariant par le flot, on a forcément : φT (x, p) = (x, p) et donc (x, p) est périodique. En d’autres termes et avec la terminologie développée dans [4], tout point radialement transformé est périodique. Fixons une classe d’homotopie de lacet Γ dans M et une période T > 0. Pour chaque x ∈ M , il existe p ∈ Tx∗ M tel que (π ∗ ◦ φH t (x, p))t∈[0,T ] minimise l’action lagrangienne parmi les chemins γ : [0, T ] → M absolument continus de Γ eriodique (grˆ ace qui joignent x ` a x. Dans ce cas, (φH t (x, p))t∈R est une orbite T p´ au paragraphe précédent) incluse dans un certain graphe Gλ avec λ ∈ Λ1 . Or, né dual N ∗ ([λ]) (voir [11]) de la classe de Gλ est contenu dans l’ensemble de Ma˜ cohomologie [λ] de λ. Donc la mesure μ supportée par l’orbite périodique de (x, p) a son support inclus dans l’ensemble de Mather dual M∗ ([λ]), puisque c’est une mesure invariante de support inclus dans l’ensemble de Ma˜ né dual correspondant. De plus, la donnée de T et Γ détermine complètement le nombre de rotation ρ de cette mesure (voir [21] pour la définition) et cette mesure μ est aussi une mesure minimisante a` nombre de rotation ρ fixé. Si maintenant on fait varier x sans changer Γ et T , on trouve une autre mesure minimisante μ qui a même nombre de rotation ρ ; elle est donc elle aussi minimisante parmi les mesures invariantes ayant ρ comme nombre de rotation. Elle est donc aussi minimisante pour la classe de cohomologie [λ], comme l’était μ (ceci découle des résultats de J. Mather explicités dans la section 2 de [21]). Elle est donc de support dans l’ensemble de Mather dual M∗ ([λ]). Or, M∗ ([λ]) est inclus dans l’ensemble d’Aubry dual A∗ ([λ]), qui est inclus dans Gλ (voir [11]). Donc le support de μ est inclus dans le même graphe Gλ que μ. Aussi, Gλ est rempli par des orbites périodiques de même période et deux a` deux homotopes. Il reste à montrer que les λ que l’on trouve ainsi forment une partie dense de Λ1 . Commen¸cons par montrer :

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Lemme 4.22. Sous ces hypothèses, si on fixe x ∈ M , l’application λ ∈ Λ1 → λ(x) ∈ Tx∗ M est un homéomorphisme. Démonstration du lemme 4.22. Il s’agit d’une application continue et bijective. Si maintenant on fixe un compact K de Tx∗ M , la réunion des niveaux d’énergie qui rencontrent K est un compact K1 de T ∗ M . Il alors existe alors une constante C > 0 telle que tout graphe C 0 lagrangien inclus dans K1 est C-lipschitzien. Ceci découle des résultats contenus dans [11]. Un argument alternatif est de dire que les fibrés de Green sont uniformément “loin” de la verticale sur le compact sans point conjugué K1 (sans point conjugué car tout point est sur un graphe C 0 lagrangien invariant), a` cause des propriétés des fonctions semi-continues sur les compacts, donc les différentielles généralisées des graphes C 0 lagrangiens invariants contenus dans K1 sont uniformément minorées et majorées. Grâce au théorème d’Ascoli, on en déduit alors que {λ ∈ Λ1 , λ(x) ∈ K} est relativement compact dans Λ(M ). Montrons qu’en fait il est compact. Soit (λn )n∈N une suite d’éléments de Λ1 telle que λn (x) ∈ K qui converge vers λ∞ ∈ Λ(M ). On veut montrer que eme de Gλ∞ . Ceci λ∞ ∈ Λ1 . Chaque Gλn étant invariant par (φH t ), il en est de mˆ joint au point 3. de la remarque 4.20 permet de conclure que λ∞ ∈ Λ1 . L’application considérée étant propre, c’est un homéomorphisme. On a vu précédemment qu’à chaque T > 0 et chaque classe d’homotopie de lacets h ∈ π1 (M ), on peut associer une classe de cohomologie λ(T, h) ∈ Λ1 telle que Gλ(T,h) est rempli par des orbites périodiques de période T et de classe d’homotopie h. Nous voulons montrer que D = {λ(T, h); T > 0 et h ∈ π1 (M )} est dense dans Λ1 . Pour cela, on fixe λ0 ∈ Λ1 . Alors, l’ensemble d’Aubry dual A∗ ([λ0 ]) est une partie non vide incluse dans Gλ0 . Choisissons un (x0 , p0 ) = (x0 , λ0 (x0 )) ∈ A∗ ([λ0 ]). A chaque T > 0, on associe (xT , pT ) : [0, T ] → T ∗ M orbite sous le flot hamilT T tout chemin γ : [0, T ] → M tonien (φH t ) telle que x (0) = x (T ) = x0 et pour T absolument continu et de mêmes extrémités que xT , 0 (L − λ0 )(xT (t), x˙ T (t))dt ≤ T (L − λ0 )(γ(t), γ(t))dt. ˙ On a alors : limT →+∞ (xT (0), pT (0)) = (x0 , p0 ) (voir par 0 exemple [4]). Si maintenant hT désigne la classe d’homotopie de xT , on a a` fortiori : (xT (0), pT (0)) ∈ G(λ(T, hT )), i.e : (xT (0), pT (0)) = (x0 , λ(T, hT )(x0 )). Aussi : limT →+∞ λ(T, hT )(x0 ) = λ0 (x0 ). Par le lemme 4.22, ceci implique que : lim λ(T, hT ) = λ0

T →+∞

avec chaque λ(T, hT ) dans D, donc prouve bien que D est dense dans Λ1 .

Démonstration du théorème 4. On reprend les mêmes hypothèses et notations que dans l’énoncé du théorème. Rappelons qu’alors dans ce cas toute orbite du flot hamiltonien (φH e. On peut donc définir en chaque (x, p) ∈ t ) est sans point conjugu´ T ∗ M les deux fibrés de Green G− (x, p) et G+ (x, p). On a alors vu que : E = M . On reprend alors les {(x, p) ∈ T ∗ M ; G− (x, p) = G+ (x, p)} est un Gδ de T ∗ notations de la proposition 4.21 ; alors, l’ensemble D = λ∈D G(λ) est une partie dense dans T ∗ M . On va alors montrer que D ⊂ E. On en déduira que E est un Gδ dense de T ∗ M .

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Pour cela, considérons λ = λ(T, h) ∈ D. Alors : ∀x ∈ M , ∀N ∈ Z , φH N T x, λ(x) = x, λ(x) . Aussi, si x ∈ M et si w ∈ TxG λ, on a : ∀k ∈ Z, DφH kT (x, λ(x))w = w. De ceci et G G ⊂ G+ (λ(x)) ∩ du critère dynamique (proposition 3.17), on déduit que : Tλ(x) G− (λ(x)), et on conclut comme dans la fin de la démonstration du corollaire 4.13 que Gλ ⊂ E. ∗ Ayant maintenant montré que E est un Gδ dense de T M qui contient D = eresse à : Λ2 = {λ ∈ Λ1 ; Gλ ⊂ E}. Puisque D ⊂ Λ2 , Λ2 est λ∈D Gλ , on s’int´ dense dans Λ1 . Remarquons alors que : F : (x, λ) ∈ M × Λ1 → λ(x) ∈ T ∗ M est un homéomorphisme : c’est une conséquence assez immédiate du lemme 4.22. (E) est un Gδ dense de M × Λ1 , Gδ qui contient M × D. Ecrivons : Aussi, F −1 u chaque Un est un ouvert dense de M × Λ1 . Comme chaque F −1 (E) = n∈N Un o` ensemble M × {λ} est compact, l’ensemble Vn = {λ ∈ Λ1 , M × {λ} ⊂ Un } est ouvert etvérifie M × Vn ⊂ Un ; comme il contient D, il est dense dans Λ1 . Aussi, G(H) = n∈N Vn est un Gδ dense de Λ1 tel que M × G(H) ⊂ F −1 (E) i.e. tel que pour tout λ ∈ G(H) et tout x ∈ M , G− (x, λ(x)) = G+ (x, λ(x)). Par la proposition 4.12, ceci implique que chaque λ ∈ G(H) est de classe C 1 . Le théorème 4 a son analogue dans le cas dépendant du temps : Théorème 5. Soit H : T ∗ M × T → R un hamiltonien de Tonelli C 0 -intégrable et Λ1 ⊂ Λ(M ) tel que {Gλ ; λ ∈ Λ1 } soit une partition de T ∗ M en graphes C 0 lagrangiens invariants par le temps 1 du flot hamiltonien (φH t ) de H. Alors, il existe un Gδ dense G(H) de Λ1 dont tout élément est de classe C 1 . Remarque 4.23. L’argument du troisième point de la remarque 4.20 reste valable dans le cas non autonome, et donc même dans ce cas, Λ1 est exactement l’ensemble des graphes C 0 invariants par le temps 1 de (φH t ). On va aussi montrer un analogue de la proposition 4.21 : Proposition 4.24. Soit H : T ∗ M ×T → R un hamiltonien de Tonelli C 0 -intégrable et Λ1 ⊂ Λ(M ) tel que {Gλ ; λ ∈ Λ1 } soit une partition de T ∗ M en graphes C 0 lagrangiens invariants par le temps 1 du flot hamiltonien (φH t ) de H. Alors, il existe une partie dense D de Λ1 tel que pour chaque λ ∈ D, Gλ est rempli par des orbites périodiques pour (φH eme période et deux a ` deux 1 ) de mˆ homotopes. La différence entre la démonstration de cette proposition et celle de la proposition 4.21 vient du fait qu’il ne nous a pas été possible de démontrer un analogue du lemme 4.22 dans le cas dépendant du temps : dans le cas non autonome, sans hypothèse supplémentaire sur H, hormis le cas dans le cas du fibré cotangent du cercle (ce résultat est dˆ u a` Birkhoff), il ne nous a pas été possible de montrer que, étant donné un compact K fixé, l’ensemble des graphes C 0 -lagrangiens qui rencontrent K est borné en C 0 -topologie.

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Démonstration de la proposition 4.24. En raisonnant de fa¸con similaire à ce qui est fait dans la démonstration de la proposition 4.21, on peut associer a` chaque couple (N, h) ∈ N∗ × π1 (M ) un λ(N, h) ∈ Λ1 tel que Gλ(N,h) est rempli par des orbites périodiques de période N qui sont toutes homotopes a` h. Nous adoptons alors la terminologie développée dans [5] et [6] (sauf que par rapport a` ces articles, pour ne pas surcharger, nous avons retiré les “ ˜ ” et que, comme c’est fait dans le cas autonome dans [4], nous sommes passé via l’application de Legendre au fibré cotangent). Ainsi, si c ∈ H 1 (M, R) est une classe de cohomologie : • G(c) est l’ensemble des points de T ∗ M dont l’orbite est minimisante pour L−λ o` u λ désigne n’importe quelle 1-forme fermée de classe de cohomologie c ; • N (c) est l’ensemble de Ma˜ né dual associé à la classe c ; • A(c) est l’ensemble d’Aubry dual associé à la classe c. Rappelons qu’alors : A(c) ⊂ N (c) ⊂ G(c). De plus, si (x, p) ∈ G(c), alors l’ensemble ω-limite de (x, p) est dans A(c), et même : si une orbite est c-minimisante sur [0, +∞[, alors son ensemble ω-limite est dans A(c) (ceci est démontré dans [5] par exemple). Nous allons alors démontrer : Lemme 4.25. On reprend les hypothèses de la proposition. Alors, pour tout λ ∈ Λ1 , on a : Gλ = A([λ]) = N ([λ]) = G([λ]) . Démonstration du lemme 4.25. D’après [6], on a : A([λ]) ⊂ Gλ ⊂ N ([λ]) ⊂ G([λ]). Soit maintenant (x, p) ∈ G([λ]). Il existe alors ν ∈ Λ1 tel que (x, p) ∈ Gν . Alors : • comme (x, p) ∈ Gν et comme Gν est invariant par le temps 1 du flot hamiltonien, on a : ω(x, p) ⊂ Gν ; • comme (x, p) ∈ G([λ]), on a : ω(x, p) ⊂ A([λ]) ⊂ Gλ . Donc λ = ν et on a : G([λ]) = N ([λ]) = Gλ . Déterminons maintenant les solutions KAM faibles associées à la classe de cohomologie [λ], o` u plutˆ ot, suivant la terminologie de [6], les pseudo-graphes qui sont des points fixes du semi-groupe de Lax–Oleinik associé à la classe de cohomologie [λ]. Comme l’ensemble ω-limite d’un point d’un tel pseudo-graphe est dans A([λ]), forcément ce pseudographe est inclus dans Gλ (puisque sinon son ensemble ω-limite serait dans un A([ν]) avec ν ∈ Λ1 et ν = λ). Donc ce pseudo-graphe est Gλ . Aussi, avec les notations de [6] : A([λ]) = I(Gλ ) = Gλ .

Lemme 4.26. Soit λ ∈ Λ1 et ε > 0. Il existe N0 ∈ N tel que : pour tout N ≥ N0 , pour tout x ∈ M , si γ : [0, N ] → M absolument continue minimise l’action de L − λ entre x et x, alors : γ(0) ˙ − ∂H ∂r (x, λ(x)) < ε.

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Démonstration du lemme 4.26. Si ce lemme est faux, on trouve une suite (Nk ) tendant vers +∞, une suite de chemins absolument continus minimisant γk : [0, Nk ] → M tels que γk (0) = γk (Nk ) = xk et tels que : ∂H ≥ε (∗) γ ˙ x (0) − , λ(x ), 0 ∀k ∈ N , k k k ∂r le lemme de compacité à priori (voir par exemple [5], lemme 2.3) nous permet d’affirmer que la suite (γ˙ k (0)) est bornée, donc quitte a` extraire une sous-suite on peut supposer que : lim γk (0) = x∞

k→∞

et

lim γ˙ k (0) = v∞ .

k→∞

Soit (x∞ , p∞ ) l’image de (x∞ , v∞ ) par l’application de Legendre. Etant limite d’orbites minimisantes (voir par exemple [5]), l’orbite de (x∞ , p∞ ) est minimisante sur [0, +∞[. Son ensemble ω-limite est donc dans A([λ]) = Gλ ; on a déjà vu que cela implique que (x∞ , p∞ ) est dans Gλ ; aussi : v∞ − ∂H ∂r (x∞ , λ(x∞ ), 0) = 0 puisque p∞ = λ(x∞ ). Ceci contredit (∗). Comme dans la démonstration de la proposition 4.21, nous allons prendre pour D : D = λ(N, h); N ∈ N∗ , h ∈ Π1 (M ) . Pour montrer que D est dense dans Λ1 , fixons λ ∈ Λ1 et ε > 0. Grˆ ace au lemme 4.26, on trouve un entier N > 0 tel que, pour tout x ∈ M , toute courbe γ : [0, N ] → M minimisante pour L − λ joignant x à x vérifie que : γ(0) ˙ − ∂H (γ(0), λ(γ(0), 0) ≤ ε. Remarquons que d’apr` e s ce qu’on a fait pr´ e c´ e demment, ∂r en fait γ est la projection d’une orbite N périodique du flot hamiltonien. De plus, par des propriétés classiques sur les minima des fonctions continues, l’ensemble E des (x, p) ∈ T ∗ M tels que (t ∈ [0, N ] → π ∗ ◦ φt (x, p)) est minimisant pour L − λ parmi les courbes joignant x ` a x est un compact. On peut donc choisir (x0 , p0 ) ∈ E N d (π ∗ ◦ φt (x, p)), t)dt. Si h désigne la classe qui minimise la quantité 0 (L − λ)( dt d’homotopie de (φt (x0 , p0 ))t∈[0,N ] , alors (x0 , p0 ) ∈ Gλ(N,h) . De plus, tout point (x, p) de Gλ(N,h) = N ([λ(N, h)]) a son orbite qui est N -périodique et dans la classe d’homotopie h ; aussi la mesure équidistribuée sur cette orbite est (L − λ(N, h)) minimisante, donc a même (L − λ(N, h))-action que la mesure équidistribuée sur l’orbite de (x0 , p0 ) : N d ∗ π ◦ φt (x, p) , t dt L − λ(N, h) dt 0 N d ∗ L − λ(N, h) π ◦ φt (x0 , p0 ) , t dt . = dt 0 Comme ces deux orbites sont dans la même classe d’homotopie, on en déduit que : N N d ∗ d ∗ π ◦ φt (x, p) , t dt = π ◦ φt (x0 , p0 ) , t dt . (L − λ) (L − λ) dt dt 0 0

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Vu comment on a choisi (x0 , p0 ) dans E, cela implique que (x, p) ∈ E, i.e : G(λ(N, h)) ⊂ E, donc λ(N, h) est proche de λ. Démonstration du théorème 3. On commence par remarquer que Λ1 est une partie fermée de Λ(M ) muni de la topologie de la convergence uniforme, donc un espace de Baire. En effet, Λ1 est aussi l’ensemble des graphes C 0 -lagrangiens invariants par le temps 1 du flot hamiltonien, ce qui est une condition fermée. On en déduit que Λ1 × M est aussi un espace de Baire. De plus, l’application ((λ, x) ∈ Λ1 × M → (x, λ(x)) étant continue, l’ensemble E = {(λ, x) ∈ Λ1 × M ; G− (x, λ(x)) = G+ (x, λ(x))} est un Gδ qui contient D×M , donc un Gδ dense de Λ1 ×M . Un argument similaire a` celui utilisé dans la démonstration du théorème 4 nous permet de conclure que {λ ∈ Λ1 ; ∀x ∈ M, G− (x, λ(x)) = G+ (x, λ(x))} est un Gδ dense de Λ1 .

5. Appendice Comme annoncé en début de section 4, nous allons donner une démonstration du classique résultat : Si le graphe de du0 (avec u0 ∈ C 1 (M, R)) est invariant par le temps 1 du flot hamiltonien (φt ) du hamiltonien de Tonelli H, il existe u ∈ C 1 (M × R, R) qui est une solution de l’équation de Hamilton–Jacobi : ∂u (x, t) + H dx u(x, t), t = 0 (H − J) ∂t et vérifie : • ∀x ∈ M, u(x, 0) = u0 (x) ; • ∀x ∈ M, dx u(x, 1) = dx u(x, 0). La démonstration que nous donnons nécessite d’être familier avec la théorie K.A.M. faible. Nous utilisons dans cet appendice des notions qui sont développées dans [6]. La notion fondamentale utilisée dans [6] est la notion de fonction semiconcave. Nous n’en rappelons pas la définition ici (disons que localement, i.e. en carte, c’est la somme d’une fonction concave et d’une fonction de classe C 2 (et même C ∞ )). En plus de savoir que certaines fonctions classiques sont semiconcaves, ce qui va nous servir est la propriété suivante des fonctions semiconcaves : soit u une fonction semi-concave et v une fonction de classe C 1 . Soit x0 un point en lequel un minimum local de u + v est atteint. Alors u est différentiable en x0 et du(x0 ) = −dv(x0 ). Expliquons maintenant quelles sont les fonctionnelles classiques qui sont semiconcaves. Etant donné s > t deux réels et x et y deux points de M , suivant [6], on convient de noter Σ(t, x; s, y) l’ensemble des courbes absolument continues γ : [t, s] → M telles que γ(t) = x et γ(s) = y. On sait alors définir : s L σ, γ(σ), γ(σ) ˙ dσ ; A(t, x; s, y) = min γ∈Σ(t,x;s,y)

t

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l’ensemble des courbes en lequel le minimum est atteint est alors noté Σm (t, x; s, y) et est compact pour la topologie de la convergence uniforme. Il est alors montré dans [6] que chaque (x, y) → A(t, x; s, y) est semi-concave et que les trois propriétés suivantes sont équivalentes : 1. l’ensemble Σm (t, x; s, y) est réduit a` un point ; 2. la fonction A(t, .; s, y) est différentiable en x ; 3. la fonction A(t, x; s, .) est différentiable en y. De plus, en un tel point, si on note Σm (t, x; s, y) = {γ} on a : ∂L ∂A ∂A ∂L t, x, γ(t) ˙ , s, y, γ(s) ˙ (t, x; s, y), (t, x; s, y) . − = ∂v ∂v ∂x ∂y Pour démontrer le résultat annoncé, considérons u0 ∈ C 1 (M, R) telle que le graphe de du0 est invariant par le flot hamiltonien (φt ) du hamiltonien de Tonelli H. Posons alors : L(x, y) = A(0, x; 1, y), puis K(x, y) = u0 (x) + L(x, y) − u0 (y). Comme u0 est de classe C 1 et comme L est semi-concave, chaque point x en lequel le minimum de K(., y) est atteint est un point de différentiabilité de L par rapport a` x, donc par le résultat rappelé auparavant un point de différentiabilité de L par rapport a` y et il existe une seule courbe minimisante γ dans Σm (0, x; y, 1). De plus : ∂L ∂L ∂L ∂L (x, y) = 0, x, γ(0) ˙ et (x, y) = 0, y, γ(1) ˙ . du0 (x) = − ∂x ∂v ∂y ∂v Aussi, (γ, γ) ˙ est l’orbite du flot d’Euler–Lagrange qui correspond a` l’orbite (φt (x, du0 (x)))t∈[0,1] du flot hamiltonien. Le fait que γ soit unique implique que si y = y , si on leur associe comme précédemment les chemins γ et γ , alors : ∀t ∈ [0, 1], γ(t) = γ (t). On en déduit que pour chaque t ∈ [0, 1], l’image par φt du graphe de du0 est un graphe. Du fait que (φt ) est un flot hamiltonien, on déduit que ce graphe est C 0 lagrangien (et même exact lagrangien), graphe de dut . il n’est pas difficile d’en déduire l’existence d’une solution de classe C 1 de l’équation de Hamilton–Jacobi, solution de la forme : u(x, t) = ut (x) + c(t).

Remerciements Cet article n’aurait pas existé sans les enrichissants exposés concernant les tores invariants des difféomorphismes symplectiques donnés par Michel Herman a` son séminaire à Paris 7. Sans une remarque limpide de Sylvain Crovisier concernant l’interprétation géométrique en dimension 1 du critère dynamique d’appartenance aux fibrés de Green, la démonstration de ce critère aurait été beaucoup plus lourde, et je l’en remercie. Je remercie aussi Jean-Christophe Yoccoz pour m’avoir indiqué certains résultats concernant les difféomorphismes du cercle et du tore. Enfin, merci aux organisateurs des Atelier sur les aspects mathématiques de la mécanique céleste qui ont eu lieu a` l’ Institut Henri Poincaré en décembre 2007 de m’avoir donné l’occasion de faire un mini-cours sur les notions développées dans cet article. Merci aussi au referee de cet article pour ses remarques et corrections.

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Références [1] M.-C. Arnaud, Three results on the regularity of the curves that are invariant by an ´ exact symplectic twist map, ` a paraˆıtre aux Publ. Math. Inst. Hautes Eudes Sci. [2] M.-C. Arnaud, Type des orbites p´ eriodiques des flots associ´ es a ` des lagrangiens optiques homogènes, Bull. Braz. Math. Soc. (N.S.) 37 no. 2 (2006), 153–190. [3] M.-C. Arnaud, Hyperbolic periodic orbits and Mather sets in certain symmetric cases, Ergodic Theory Dynam. Systems 26 no. 4 (2006), 939–959. [4] M.-C. Arnaud, The tiered Aubry set for autonomous Lagrangian functions, ` a paraˆıtre aux Ann. Inst. Fourier (Grenoble). [5] P. Bernard Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble) 52 no. 5 (2002), 1533–1568. [6] P. Bernard The dynamics of pseudographs in convex Hamltonian systems, ` a paraˆıtre a J. Amer. Math. Soc. ` [7] J. Bochi & M. Viana Lyapunov exponents : how frequently are dynamical systems hyperbolic ? Modern dynamical systems and applications, 271–297, Cambridge Univ. Press, Cambridge, (2004). [8] M. L Bialy & R. S. MacKay, Symplectic twist maps without conjugate points, Israel J. Math. 141 (2004), 235–247. [9] G. Contreras & R. Iturriaga, Convex Hamiltonians without conjugate points, Ergodic Theory Dynam. Systems 19 no. 4 (1999), 901–952. [10] A. Fathi, Regularity of C 1 solutions of the Hamilton–Jacobi equation, Ann. Fac. Sci. Toulouse Math. (6) 12 no. 4 (2003), 479–516. [11] A. Fathi, Weak KAM theorems in Lagrangian dynamics, livre en préparation. [12] P. Foulon, Estimation de l’entropie des syst` emes lagrangiens sans points conjugués, Ann. Inst. H. Poincaré Phys. Théor. 57 no. 2 (1992), 117–146. [13] L. W. Green, A theorem of E. Hopf, Michigan Math. J. 5 (1958), 31–34. [14] M. Herman, Sur la conjugaison diff´ erentiable des difféomorphismes du cercle a ` des ´ rotations, Inst. Hautes Etudes Sci. Publ. Math. 49 (1979), 5–233. [15] M.-R. Herman, Inégalités “a priori” pour des tores lagrangiens invariants par des ´ difféomorphismes symplectiques. , vol. I, Inst. Hautes Etudes Sci. Publ. Math. 70 (1989), 47–101. [16] M.-R. Herman, Sur les courbes invariantes par les difféomorphismes de l’anneau, Astérisque 103–104, (1983). [17] M. Herman, Sur les courbes invariantes par les difféomorphismes de l’anneau, Astérisque 144, (1986). [18] R. Iturriaga, A geometric proof of the existence of the Green bundles, Proc. Amer. Math. Soc. 130 no. 8 (2002), 2311–2312. [19] P. Libermann & Charles-Michel Marle, Géométrie symplectique, bases théoriques de la mécanique. Tome I. Publications Mathématiques de l’Université Paris VII , 21. Université de Paris VII, U.E.R. de Math´ ematiques, Paris, 176 pp, (1986).

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[20] R. Mané, Lagrangian flows : the dynamics of globally minimizing orbits, Int. Pitman Res. Notes Math. Ser. 362 (1996), 120–131. [21] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), 169–207. [22] J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble) 43 no. 5 (1993), 1349–1386. [23] A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), 329–346. [24] J.-C. Yoccoz, Travaux de Herman sur les tores invariants, Séminaire Bourbaki 784, Astérisque 206 (1992), 311–344. Marie-Claude Arnaud Université d’Avignon et des Pays de Vaucluse Laboratoire d’Analyse non linéaire et Géométrie (EA 2151) F-84018 Avignon France e-mail: [email protected] Communicated by Viviane Baladi. Submitted: July 23, 2007. Accepted: February 14, 2008.



Finite Energy Scattering for the Lorentz–Maxwell Equation Pierre Germain Abstract. In the case where the charge of the particle is small compared to its mass, we describe the asymptotics of the Lorentz–Maxwell equation (Abraham model) for any finite-energy data. As time goes to infinity, we prove that the speed of the particle converges to a certain limit, whereas the electromagnetic field can be decomposed into a soliton plus a free solution of the Maxwell equation. It is the first instance of a scattering result for general finite energy data in a field-particle equation.

1. Introduction 1.1. Presentation of the equation The Abraham model The Abraham model describes the interaction of a charged particle with the field that it generates. It consists of a coupling of the Lorentz equation (which governs the movement of the particle), and the Maxwell equation (which gives the evolution of the field). In the Abraham model, the particle has a fixed, spherically symmetric, charge distribution. This feature is not relativistically invariant, but we will use the relativistic version of the Lorentz equation. We provide the system with initial data and consider the following Cauchy problem. mp(t) ˙ = eE ρ q(t) + eq(t) ˙ × B ρ q(t) ˙ E(x, t) = curl B − eρ x − q(t) q(t) ˙

with

p=

q˙ 1 − q˙2

(1a) (1b)

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˙ B(x, t) = − curl E(x, t)

(1c)

div B(x, t) = 0

(1d)

div E(x, t) = eρ x − q(t)

(1e)

(q, q, ˙ E, B)|t=0 = (q0 , q˙0 , E0 , B0 ) .

(1f)

The above system is set in the three-dimensional space: (t, x) ∈ R×R3 . We denote q the position of the particle, p its momentum, and ρ its charge distribution, which is such that ρ ∈ C0∞ , ρ ≥ 0 , ρ = ρ(|x|) ρ = 1. R3

We also denote, for a function f fρ = f ∗ ρ . The total charge of the particle is given by e, its mass by m, and we take all other physical constants (including the speed of light) to be one. The electric and magnetic field are denoted E and B. The initial data are assumed to satisfy the constraints div B0 (x) = 0

div E0 (x) = eρ(x − q0 ) .

We can then forget about the conditions on div E and div B (equations (1d) and (1e)) in the above system : if they hold true for the data, this is propagated by the flow given by equations (1a)–(1c). This model was introduced by Abraham [1] in 1903 in order to describe the dynamics of the electron. It has recently been intensively studied by mathematicians. Our general reference will be the textbook of Spohn [9], in which the interested reader can find further bibliographical indications. We will be concerned about the asymptotics of the Abraham model. Another important problem, that will not be considered here, is to understand the point particle limit, that is the limit when the support of ρ shrinks to a point. Conserved quantities The Abraham model actually derives from a Lagrangian, see [9]. The conservation of the Hamiltonian can also be expressed as the conservation of 2 m 1 E(q, ˙ E, B) = E + B2 . + 2 R3 1 − q˙2 Another conserved quantity is the generalized momentum Π = mp + E×B. R3

Scaling invariance The system (1a)–(1f) is invariant by the following scaling transformation E −→ λ3/2 E(λ · , λ · ) B −→ λ3/2 B(λ · , λ · )

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q −→ λ−1 q(λ · ) q˙ −→ q(λ ˙ ·) ρ −→ λ5/2 ρ(λ · ) . In particular, q, ˙ and the L2 norm of E and B, hence the energy E, are left invariant by this scaling. In other words, the energy is at the scaling of the equation, the Lorentz–Maxwell system is therefore critical. Solitons We will call solitons the solutions of the system (1a)–(1e) which travel at constant speed, that is of the form q = vt E = eEv (x − vt) B = eBv (x − vt) . Thus, the fields Ev , Bv solve the elliptic problem ⎧ ρ ⎨ Ev (0) + v × Bvρ (0) = 0 −v · ∇Ev = curl Bv − ρv ⎩ −v · ∇Bv = − curl Ev .

(2)

For any speed v less than 1 in norm there exists only one solution up to space translation (see Spohn [9]), and it is given by Ev (x) = −∇φv (x) + v v · ∇φv (x) Bv (x) = −v × ∇φv (x) , (3) where φv (x) = ρ(x) ∗

1 . 2 4π (1 − v )x2 + (v · x)2

(4)

Related models for field-particle interaction We would like to review here some of the models considered in the literature. It is physically more relevant to allow for a spin of the particle, this possibility is considered in [4]. One can modify the nature of the field interacting with the particle: instead of the electromagnetic field, one can consider a scalar field [8], a Schr¨ odinger field [6] or a Klein–Gordon field [5]. The qualitative behaviour of all these systems seems to remain essentially the same. Another possibility is to consider the non-relativistic Lorentz equation [2], that is to take as a new definition of the momentum in (1a) simply p = q. ˙ However, the Maxwell equation remains relativistic and as a consequence there are no solitons that propagate at a speed v larger than 1. One can also consider a fully relativistic model by letting the shape of the particle be relativistically invariant: this is the Lorentz model. It is presented in [9], but not so many results seem to be available about it.

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1.2. Large time behavior of solutions We would like to discuss here known results on large time behavior of solutions. Orbital stability and scattering It is shown in [3] that the Lorentz–Maxwell system exhibits orbital stability, that is to say if data are close to a soliton, the solution remains close to it. This relies on the following property: for a given generalized momentum Π, there exists v such that the energy is minimized by the solutions (x0 +vt, v, eEv (t), eBv (t)). Once orbital stability is proved, the next question is: as t → ∞, is it possible to decompose the solution into a free electromagnetic field plus a soliton? More specifically this would mean: there exists x0 , v ∈ R and EL , BL free solutions of the Maxwell equation such that ⎧ ⎨ q˙ −→ v E −→ Ev (x − q(t)) + EL (t) as t → ∞ , (5) ⎩ B −→ Bv (x − q(t)) + BL (t) (we leave for the moment unprecise the meaning of the above convergences). If (5) holds, we say that the solution scatters. We will see in the following that the solutions scatters under appropriate conditions. This means that the Lorentz–Maxwell equation exhibits the behaviour which is expected for general dispersive field equations: for large time, the solution can be decomposed into solitons and a free solution, that move away from one another. Proving such a large time behaviour is very difficult though for nonlinear field equations. The Lorentz–Maxwell equation is much easier to study, since it involves no nonlinear interaction of the electromagnetic field with itself. This is one of the great mathematical interests of the Lorentz–Maxwell equation: it provides a tractable model which reproduces some of the features of much more complex situations. We have seen that orbital stability holds regardless of the “constants” of the problem, that is m, e and ρ. Let us now discuss under which conditions the solutions scatters. Small charge If the quotient

e2 m

is small, and if the initial fields satisfy

|B0 (x)| , |E0 (x)| ≤

C |x|3/2+

and

|∇B0 (x)| , |∇E0 (x)| ≤

C |x|5/2+

,

(6)

it is shown in [9] that scattering occurs. The condition on the moduli of B0 , E0 almost includes L2 fields, but the condition on their gradients is very strong. Our aim in this paper will be to remove these two conditions and prove a scattering result for fields that are merely of finite energy, that is L2 .


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Wiener condition Another hypothesis under which the solution scatters is the so-called Wiener condition, which requires that ∀ξ ,

ρ (ξ) = 0 .

Scattering under this Wiener condition and condition (6) is proved in [3]. The beautiful idea, introduced in [7], is to make use of the following physical fact: a particle which accelerates radiates energy to infinity. Since this amount of radiated energy is bounded, we get a bound on q¨ that enables one to conclude. Periodic solutions? It has been speculated since the introduction of the Abraham model that periodic in time solutions exist, apparently without any rigorous proof till now, see [9]. The conditions that we have reviewed under which the solution scatters show that this can happen only for a large charge and with a ρ that violates the Wiener condition.

2. Main result 2.1. Statement Theorem 2.1. If

e2 m

is small enough, then for any data of finite energy, that is E(q˙0 , E0 , B0 ) < ∞ ,

the solution of (1a)–(1f) scatters. More precisely, there exists v∞ ∈ R3 (of norm less than 1) and EL , BL finite-energy solutions of the free Maxwell equation such that ⎧ q(t) ˙ −→ v∞ ⎪ ⎨ L2 E(t) − Eq(t) as t → ∞ , ˙ ( · − q(t)) − EL (t) −→ 0 . ⎪ ⎩ L2 B(t) − Bq(t) ˙ ( · − q(t)) − BL (t) −→ 0 To our knowledge, this theorem is the first instance of a scattering result for a field-particle interaction equation with general finite energy data; it is also the first instance of such a result where global convergence of the fields is proved. 2.2. Reduction of the problem Consider data as in the above theorem. Using conservation of energy, the global existence of a solution is not hard to prove (we refer to [9]); conservation of energy also implies that the speed of the particle is bounded away from 1: |q(t)| ˙ ≤ 1 − ,

> 0.

(7)

From now on, we fix initial data (q0 , q˙0 , E0 , B0 ) of finite energy. The associated solution (q, q, ˙ E, B) satisfies (7). Our strategy will be the following: we reduce matters to a linear equation in p, ˙ whose coefficients however depend on q, q. ˙ Solving this equation, we obtain the convergence of q˙ (or p); convergence of the fields follows.

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The first step is to introduce the modified fields (following Spohn [9]) ¯ t) = E(x, t) − eEq(t) x − q(t) E(x, ˙ ¯ t) = B(x, t) − eBq(t) B(x, x − q(t) . ˙ These new fields somehow measure the distance from the solution to the soliton. In these new coordinates, the system (1a)–(1f) becomes ¯ ρ q(t) ¯ ρ q(t) + eq(t) ˙ ×B mp(t) ˙ = eE

with

p=

¯˙ ¯ t) − e¨ E(x, t) = curl B(x, q (t)∇v Eq(t) x − q(t) ˙ ¯˙ ¯ t) − e¨ B(x, t) = − curl E(x, q (t)∇ B x − q(t) v

q˙ 1 − q˙2

(8a) (8b) (8c)

q(t) ˙

¯ B) ¯ |t=0 = (q0 , q˙0 , E ¯0 , B ¯0 ) . (q, q, ˙ E,

(8d)

UE 0 curl the semi group generated by ∂t − − curl , the Let us now denote U (t) = U 0 B equations (8b)–8c can be rewritten in an integral form as

¯ E(t) ¯ B(t)

setting

t ¯0 E ∇v Eq(s) (x − q(s)) ˙ = U (t) U (t − s)¨ q (s) ds +e ¯0 B (x − q(s)) ∇v Bq(s) ˙ 0 t ¯L (t) E = U (t − s)¨ q (s)S s, x − q(s) ds , + e ¯L (t) B 0

¯L (t) E ¯L (t) B

= U (t)

¯0 E ¯0 B

S(s, x) =

(x) ∇v Eq(s) ˙ (x) ∇v Bq(s) ˙

(9)

.

E¯ (t) So B¯L (t) is simply a solution of the free Maxwell equation, whereas S(s) L only consists of derivatives of the solitons. Inserting the above equality in (8a), we obtain ¯ ρ q(t) + eq(t) ¯ ρ q(t) mp(t) ˙ = eE ˙ ×B L L t

2 UE (t − s)¨ +e q (s)S ρ (s) q(t) − q(s) ds 0 t

UB (t − s)¨ ˙ × q (s)S ρ (s) q(t) − q(s) ds . + e2 q(t) 0

Finally, the equalities q˙ =

p 1 + p2

def = F (p)

q¨ = F (p)p˙


give

933

¯ ρ q(t) + eq(t) ¯ ρ q(t) mp(t) ˙ = eE ˙ ×B L L t 2 ρ +e ˙ (s) q(t) − q(s) ds UE (t − s)F p(s) p(s)S 0 t ρ + e2 q(t) UB (t − s)F p(s) p(s)S ˙ ˙ × (s) q(t) − q(s) ds . 0

If one considers any function but p˙ as given, we have thus reduced matters to a system which is linear. This will be even more clear setting ¯ ρ q(t) α(t) = eE L ¯ ρ q(t) β(t) = eB L t (10) UE (t − s)F p(s) f (s)S ρ (s) q(t) − q(s) ds A : f → 0 t UB (t − s)F p(s) f (s)S ρ (s) q(t) − q(s) ds . + q(t) ˙ × 0

so that our problem becomes mp(t) ˙ = α(t) + q(t) ˙ × β(t) + e2 (Ap)(t) ˙ .

(11)

We consider α, β, A and q˙ as ‘fixed’ functions (even though they actually depend on the solution of (1a)–(1f)); as stressed above, p˙ solves a linear equation. 2.3. Outline of the proof We give here the outline of the proof, which will essentially consist on estimates on α, β and A. The actual proof of Propositions 2.1–2.5 will be found in Sections 3–5. As a first and easy step, we shall prove that Proposition 2.1. If (7) holds, the functions α(t) and β(t) belong to L2 ([0, ∞)). Proposition 2.2. If (7) holds, the operator A is bounded on L2 ([0, ∞)). If

e2 m

is small enough, we thus get p˙ = (m Id −e2 A)−1 (α + q˙ × β)

∈ L2 ([0, ∞)) .

Notice that this gives at once that q¨ ∈ L2 ([0, ∞)). This also implies a better decay of α and β, as well as better boundedness properties for A, as appears in the two next propositions: Proposition 2.3. If (7) holds and p˙ ∈ L2 ([0, ∞)), the functions α and β belong to L1 ([0, ∞)) + L2 ∩ ∂L2 ([0, ∞)). (We denote ∂L2 ([0, ∞)) for functions f such that f = g, ˙ with g ∈ L2 , g(0) = 0)

934

P. Germain


It is easy to see (noticing that q¨ is bounded by the equation) that the last proposition implies that α + q˙ × β ∈ L1 + L2 ∩ ∂L2 . The next proposition will give us the boundedness of A on this space. Proposition 2.4. If (7) holds and p˙ ∈ L2 ([0, ∞)), then A is bounded on L1 ([0, ∞))+ L2 ∩ ∂L2 ([0, ∞)). As a conclusion, p˙ = (m Id −e2 A)−1 (α + q˙ × β)

∈ L1 + L2 ∩ ∂L2 ([0, ∞)) .

Now we observe that a function which belongs to L2 together with its derivative has to go to zero. So integrating p, ˙ we get p(t) −→ p∞

as

t → ∞,

q(t) ˙ −→ q˙∞

as

t → ∞.

which implies (12)

In order to complete the proof of the theorem, it remains to prove that the fields converge as stated. But since the particle is the only source of the fields, the information that we have gathered on p˙ will enable us to conclude. Proposition 2.5. If p˙ ∈ L1 + L2 ∩ ∂L2 , there exists a finite energy solution of the free Maxwell equation (EL , BL ) such that L2 E(t) − Eq(t) ˙ ( · − q(t)) − EL (t) −→ 0 as t → ∞ , . L2 B(t) − Bq(t) ˙ ( · − q(t)) − BL (t) −→ 0

3. Estimates on α and β: Proofs of Propositions 2.1 and 2.3 3.1. The semi group U (s)

0 curl . Taking the Fourier Recall the semi-group U (s) is generated by ∂t − − curl 0 transform (that we denote ), it is not hard to see that, denoting E0 (t), B0 (t) = U (t)(E0 , B0 ) ,

one has 0 (ξ) 0 (ξ, t) = cos(t|ξ|)E 0 (ξ) + i sin(t|ξ|) ξ × B E |ξ| 0 (ξ) . 0 (ξ, t) = cos(t|ξ|)B 0 (ξ) − i sin(t|ξ|) ξ × E B |ξ|


935

It is well-known that the inverse Fourier transform of sin(t|ξ|) (which is the funda|ξ| 1 mental solution of the scalar wave equation) is given by 4πt δ(|x| − t). We deduce from this and the above equality that 1 1 E0 (y) dy + E0 (x, t) = 4πt2 |y−x|=t 4πt |y−x|=t 1 + curl B0 (y) dy 4πt |y−x|=t 1 1 B0 (x, t) = B0 (y) dy + 4πt2 |y−x|=t 4πt |y−x|=t 1 − curl E0 (y) dy 4πt |y−x|=t

y−x · ∇E0 dy |y − x|

y−x · ∇B0 dy |y − x|

(13)

(the above integrals over surfaces are understood with respect to the standard surface measure).

3.2. A change of variable Combining the above formula and (10), that is the definition of α and β, we see that in order to prove Propositions 2.1 and 2.3, we have to study surface integrals of functions which are L2 in space. More specifically, α and β can be written as linear combinations of functions of the type 1 g1 (t) = 2 t

f ρ (z) dz |z−q(t)|=t

z − q(t) 1 · ∇f ρ (z) dz t |z−q(t)|=t |z − q(t)| 1 g3 (t) = curl f ρ (z) dz , t |z−q(t)|=t g2 (t) =

(14)

where f is an L2 function. Instead of proving estimates on α and β, we will prove estimates on g1 and g2 , the case of g3 being very similar. Since our aim will be to integrate in time, it appears, considering the definition of the gi , that the following change of variable is natural: φ:

R3 y = sω

→ →

R3 z = q(s) + sω ,

where s ∈ R+ , ω ∈ S2 , in other words, (s, ω) are the polar coordinates of y.

(15)

936

P. Germain


Assuming that ω = e1 (the first vector of an orthonormal basis (e1 , e2 , e3 )), we get ∂s ∂s 1 ∂s = = 3 =0 1 1 2 ∂z 1 + q˙ ∂z ∂z ∂ω 1 = 0 for any i ∂z i ∂ω 2 ∂ω 2 1 q˙2 1 ∂ω 2 =− = =0 1 1 2 ∂z s 1 + q˙ ∂z s ∂z 3 (16) ∂ω 3 ∂ω 3 1 q˙3 ∂ω 3 1 =− =0 = ∂z 1 s 1 + q˙1 ∂z 2 ∂z 3 s ⎛ 1 ⎞ q˙2 q˙3 ∂y ⎝ 1+q˙1 − 1+q˙1 − 1+q˙1 ⎠ = . 0 1 0 ∂z 0 0 1 This implies in particular that ∂y 1 = . ∂z 1 + q(s) ˙ , ω It is at this point that (7) plays a crucial role: due to this inequality, φ is a global diffeomorphism. 3.3. Proof of Proposition 2.1 As we saw above, we reduce matters to studying g1 and g2 . Thus to prove Proposition 2.1, it suffices to prove that t → g1 (t) and t → g2 (t) belong to L2 . We will prove this for g2 – it being easier for g1 . By definition of g2 , 2 ∞ z − q(t) 1 2 ρ · ∇f (z) dz g2 L2 ([0,∞)) = dt t |z−q(t)|=t |z − q(t)| 0 ∞ |∇f ρ (z)|2 dz dt by H¨ older’s inequality ≤C 0 |z−q(t)|=t ρ ∇f φ(y) 2 dy where φ is defined in (15) =C 3 R dz changing coordinates =C |∇f ρ (z)|2 1 +

q(s) ˙ , ω 3 R ≤ C∇f ρ 22

using (7) .

3.4. Proof of Proposition 2.3 As in the previous subsection, we prove estimates on g1 and g2 instead of α and β. Thus our aim here will be to prove that g1 and g2 belong to L1 + L2 ∩ ∂L2 . The result for g1 is easily obtained: a straightforward estimate as in the previous subsection yields: g1 ∈ L1 .


937

Now we would like to prove that g2 belongs to L1 + L2 ∩ ∂L2 . The idea is to integrate g2 in time, and try and write the result as the integral of an L1 function, plus an L2 function. Differentiating this equality, we will get the desired result. Let us introduce the following notation: Bt is the Euclidean ball of radius t, and nt the exterior normal to φ(∂Bt ). By definition of g2 , T T 1 z − q(t) · ∇f ρ (z) dz dt g2 (t) dt = t |z − q(t)| 0 0 |z−q(t)|=t y dz proceeding as in Section 3.3 · ∇f ρ (z) = 2 |y| 1 +

q(s) ˙ , ω φ(BT ) y dz = nT (z) · 2 f ρ (z) |y| 1 +

q(s) ˙ , ω φ(∂BT ) y 1 ρ − f (z)∇z · dz |y|2 1 + q(s) ˙ , ω φ(BT ) y dz nT (z) · 2 f ρ (z) = |y| 1 + q(s) ˙ , ω φ(∂BT ) y 1 1 + q(s) ˙ , ω dy − f ρ φ(y) · ∇z 2 |y| 1 + q(s) ˙ , ω BT def = K(T ) − L(y) dy . BT

(in the above, z, y, s, and ω are functions of y or z, given by (15)). Differentiating in time, ˙ g2 (t) = K(t) − L(y) dy . ∂Bt

This is the decomposition of g2 that we were looking for. Indeed, we have the following Claim 3.1. With K and L defined above, one has (i) t → ∂Bt L ∈ L1 (ii) K˙ ∈ L2 ∩ ∂L2 . Proof of the claim. (i) First, using (16), we see that ρ y 1 |L(y)| ≤ C f φ(y) ∇z 2 |y| 1 + q(s) ˙ , ω 1 |¨ q (|y|)| ≤ C f ρ φ(y) + . |y|2 |y| But f ρ ◦ φ ∈ L2 ∩ L∞ (R3 ), and, since q¨ ∈ L2 ([0, ∞)), 1 |¨ q (|y|)| y → + ∈ L1 + L2 (R3 ) . |y|2 |y| Thus L ∈ L1 (R3 ), which implies that t → ∂Bt L ∈ L1 ([0, ∞)).

(17)

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P. Germain


(ii) The definition of K ensures as in Section 3.3 that it belongs to L2 ([0, ∞)), and that K(0) = 0. Finally, K˙ can be written as ˙ K(t) = g2 (t) + L(y) dy , ∂Bt

where g2 ∈ L ; to complete the proof it suffices to show that t → ∂Bt L(y) dy belongs to L2 . Using H¨ older’s inequality, we get 2 ∞ dt ≤ C L(y) dy |y|2 L(y)2 dy 3 0 ∂Bt R 2 1 ≤C f ρ φ(y) q¨(y)2 + 2 dy by (17) |y| R3 −∞; the constant may depend on the curve γ and the weighting function h, but is required to be independent of the choice of state ω ∈ L. In principle this concept makes sense for arbitrary (C 1 and causal) curves γ, in practice one however usually restricts the class of curves to timelike (or lightlike) geodesics. Such a limiting case of a QEI is the following: If γ is a complete (lightlike or null) geodesic, then one says that a set of states L fulfills the averaged null energy

948

J. Schlemmer and R. Verch


condition (ANEC) if lim inf λ→0+

dt ≥ 0 h2 (λt)γ˙ a (t)γ˙ b (t) Tab γ(t) ω

holds for all states ω ∈ L. Conditions of such form (and related conditions, see (39)), if valid for all complete null geodesics, allow conclusions about focussing of null geodesics for solutions to the semiclassical Einstein equations similar to that resulting from a pointwise null energy condition [5, 32, 36, 42]. (See also the beginning of Section 4.) Thus, the ANEC is a key property for deriving singularity theorems for solutions to the semiclassical Einstein equations. Quantum energy inequalities have been investigated extensively for quantum fields subject to linear field equations in the recent years, and there is now a wealth of results in this regard. We refer to the reviews by Fewster and by Roman [13, 33] for representative lists of references. Important to mention, however, is the fact that for many linear fields, like the minimally coupled scalar field, the Dirac field and the electromagnetic field, it could be shown that the set of Hadamard states fulfills a QEI with respect to timelike curves γ in generic globally hyperbolic spacetimes [12, 15, 19]. Hadamard states are regarded as physical states in quantum field theory in curved spacetime, and expectation values of the stress-energy tensor at any given spacetime point are well-defined for these states (up to finite renormalization ambiguities), cf. [41] for discussion. There is also an intimate relation between QEIs, the Hadamard condition and thermodynamic properties of linear quantum fields [20]. It has been shown that QEIs put strong limitations on the possibility of solutions to the semiclassical Einstein’s equations to allow exotic spacetime scenarios such as wormholes or warpdrive [17, 22, 31]. It is also worth mentioning two other recent results. First, it has been shown that the non-minimally coupled linear scalar field on any spacetime violates QEIs for the class of Hadamard states; nevertheless, the class of Hadamard states fulfills in this case weaker bounds, called “relative QEIs”, cf. [14] for results and discussion. Secondly, one is interested in lower bounds q(γ, h) which depend (apart from renormalization constants entering the definition of expectation value of the stress energy tensor) only on the underlying spacetime geometry in a local and covariant manner, and one also aims at making this dependence as explicit as possible. Considerable progress on this issue, for the case of the minimally coupled linear scalar field on globally hyperbolic spacetimes, has been achieved in [18]. In the present article, we will derive QEI-like bounds on sets of LTE-states of the non-minimally coupled linear scalar field φ(x) on generic globally hyperbolic spacetimes. More precisely, we consider LTE states ω whose thermal function ϑω (x) = : φ2 : (x)ω is bounded by some constant T20 (corresponding to a maximal squared temperature) and we will show that there are upper and lower bounds for the averaged energy density ∞

η(τ )v a v b Tab γ(τ ) dτ , −∞

ω

Vol. 9 (2008)

Local Thermal Equilibrium States and QEIs

949

averaged against a C 2 -weighting function η ≥ 0 with compact support along any causal geodesic γ with affine parameter τ and tangent v a = γ˙ a . The lower bound depends only on T20 , the geodesic γ and η, while the upper bound depends additionally on local tetrads entering into the definition of LTE states. The lower bound is therefore state-independent within each set of LTE states ω with a fixed maximal value of ϑω . The bounds depend on the spacetime geometry in a local covariant manner which, together with their dependence on T20 , we will make explicit. This result holds for all values of curvature coupling ξ in the field equation (4), and upon averaging along causal geodesic, not only those which are timelike. Hence, the result is not immediate from know quantum energy inequalities for Hadamard states, as these are violated in general for non-minimally coupled fields [14], and upon averaging along null geodesics [16]. Furthermore, we will show that the ANEC holds for LTE states ω of the quantized linear scalar field with curvature couplings 0 ≤ ξ ≤ 1/4, provided that the growth of the thermal function ϑω along the null geodesics γ fulfills certain bounds. Despite the fact that we have to assume that the LTE states we consider are Hadamard states – in order to have a well-defined, local covariant expression of expected stress-energy for these states – our derivation of QEIs and ANEC makes no further use of the Hadamard property but uses only properties of LTE states. Therefore, one may expect that, in principle, similar results could be derived for LTE states of interacting quantum fields. This prospect can actually be seen as one of our motivations in view of the fact that quantum energy inequalities seem to be very difficult to obtain (if valid at all) for very general sets of states in interacting quantum field theory, and that, on the other hand, one may argue that only special classes of states are of physical interest. We will come back to this point in Section 6. This article is organized as follows. We will discuss the concept of LTE states, as far as needed for our purposes, in Section 2. In Section 3 we derive upper and lower bounds for the geodesically averaged expectation values of energy density for LTE states. The validity of ANEC for certain LTE states will be studied in Section 4. In Section 5 we indicate that the results of Sections 3 and 4 hold also for a more general notion of LTE states. We conclude with discussion and outlook in Section 6.

2. Local thermal equilibrium states The system under investigation in the present article is the non-minimally coupled linear scalar field on globally hyperbolic spacetimes. A globally hyperbolic spacetime will be denoted by a pair (M, g) where M is the spacetime manifold (assumed to be C ∞ ) and g is the Lorentzian metric. We will consider the case of spacetime dimension equal to 4 with metric signature (+ − −−), but most of our considerations can be readily generalized, with appropriate modifications, to arbitrary spacetime dimensions. We recall that global hyperbolicity means that

950



the spacetime is time-orientable and possesses Cauchy surfaces [2, 40]. Our conventions for curvature quantities, like in [14], are those of Birrell and Davies, i.e. [-,-,-] in the classification scheme of Misner, Thorne and Wheeler. The classical linear scalar field ϕ on a globally hyperbolic spacetime (M, g) obeys the field equation (∇μ ∇μ + ξR + m2 )ϕ = 0

(4)

where ∇ is the covariant derivative of g and R is the scalar curvature corresponding to g; the constants ξ ≥ 0 and m ≥ 0 are the curvature coupling and the mass parameters, respectively. The case ξ = 0 corresponds to minimal coupling. The quantization of the system proceeds as follows. Owing to global hyperbolicity, there are (for each fixed ξ and m) two uniquely determined linear maps E ± : C0∞ (M, R) → C ∞ (M, R) so that E ± (∇μ ∇μ + ξR + m2 )f = f = (∇μ ∇μ + ξR + m2 )E ± f

(5)

holds for all f ∈ C0∞ (M, R), and additionally, supp(E ± f ) ⊂ J ± (supp(f )), where J ± (G) is the causal future/past set of G ⊂ M [2]. These are called the advanced/retarded fundamental solutions of the wave-operator (∇μ ∇μ + ξR + m2 ), and with their help one can construct the real bilinear form f1 (x)(E − f2 − E + f2 )(x) dvol(x) (6) E (f1 , f2 ) = M

C0∞ (M, R)

which turns out to be antisymmetric. Note that E is uniquely deon termined by (M, g), ξ and m. Fixing ξ and m, one can now define the complex ∗-algebra A(M, g) = A((M, g), ξ, m) with unit element 1 as being generated by a family of objects φ(f ), f ∈ C0∞ (M, R) which are required to fulfil the following relations: (a) f → φ(f ) is real-linear, (b) φ(f )∗ = φ(f ), (c) φ((∇μ ∇μ + ξR + m2 )f ) = 0, (d) [φ(f1 ), φ(f2 )] = iE (f1 , f2 )1. Here [A, B] = AB − BA denotes the commutator. Since the generators φ(f ) of A(M, g) obey, according to (d), the canonical commutation relations in a covariant manner, one has thus obtained a quantization of the system in an abstract form. The hermitean elements in A(M, g) correspond to observables of the quantized system, but they do not contain all observables that one may wish to consider, so that the algebra A(M, g) will have to be enlarged to include those additional observables as well. We will come back to this point. For the moment, a state ω of the quantized linear scalar field is, by definition, a linear functional ω : A(M, g) → C, A → ω(A) ≡ Aω , with the additional property that ω is positive, meaning ω(A∗ A) ≥ 0 for all A ∈ A(M, g), and also with the property that ω is normalized, i.e. ω(1) = 1. Now it is known from examples that not every state according to this definition corresponds to a physically reasonable configuration of the system and that selection criteria for physical states are needed. In the case of the linear

Vol. 9 (2008)


951

fields on curved spacetime, the best candidates for physical states are quasifree Hadamard states, and most other physical states can be derived from those [38]. We will very briefly summarize the concept of a quasifree Hadamard state. (For a more in-depth discussion, see [28].) For any state ω on A, the n-point functions are the maps f1 ⊗ · · · ⊗ fn → Wnω (f1 , . . . , fn ) = ω φ(f1 ) · · · φ(fn ) . (7) Clearly, each state on A(M, g) is determined by all the n-point functions. A quasifree state ω on A(M, g) is a state which is entirely determined by its twopoint function, by requiring that the truncated n-point functions vanish [6]. This 2 ω can also be expressed as ω(eitφ(f ) ) = e−t W2 (f,f )/2 , to be interpreted as a sequence of relations in the sense of formal power series in t. A (quasifree) state ω is called Hadamard state if its two-point function is of Hadamard form. This is the case, in turn, if for any geodesic convex neighbourhood N of any given point xo , and any time function t on the underlying spacetime M , one can find a sequence Hkω ∈ C k (N × N, C) such that for all f1 , f2 ∈ C0∞ (N, R) one has W2ω (f1 , f2 ) = lim

ε→0+

1 4π 2

N ×N

Gk,ε (x, x ) + Hk (x, x ) f1 (x)f2 (x ) dvol(x) dvol(x ) ,

(8)

where Gk,ε (x, x ) =

U (x, x ) σ(x, x ) + 2i(t(x) − t(x ))ε + ε2 + Vk (x, x )ln σ(x, x ) + 2i t(x) − t(x ) ε + ε2 .

(9)

Here,

Vk (x, x ) =

k

Uj (x, x )σ(x, x )j ,

(10)

j=0

σ(x, x ) is the squared geodesic distance from x to x1 , and U and Uj are smooth functions on N × N determined by the Hadamard recursion relations. Thus, the term Gk,ε is, for each k, determined by the local spacetime geometry and the parameters ξ and m of the scalar field equation. For later use, we define the distribution 1 (11) Gk (f1 , f2 ) = lim Gk,ε (x, x ) f1 (x)f2 (x ) dvol(x) dvol(x ) ε→0+ 4π 2 for test-functions f1 , f2 supported in a geodesic convex neighbourhood N . It is worth noting that the existence of very many quasifree Hadamard states (spanning an infinite dimensional space) has been established for the linear scalar [18], we choose σ positive for x and x spacelike related and negative for x and x timelike related, so that e.g. on Minkowski spacetime we have σ(x, x ) = −gab (x − x )a (x − x )b 1 Following

952



field on all globally hyperbolic spacetimes. Moreover, in stationary, globally hyperbolic spacetimes, the canonical ground state as well as the thermal equilibrium states (KMS states) are known to be quasifree Hadamard states [34]. Let us now turn to the concept of local thermal equilibrium states introduced in [9] and further investigated in [8,10]. This will be done first for the case that the underlying spacetime is just Minkowski spacetime. Our discussion here is limited to the linear scalar field, but as explained in [9], the discussion can be generalized to include general quantum field theories. For the quantized linear scalar field on Minkowski spacetime, there is in each Lorentz frame a unique quasifree thermal equilibrium state at given temperature. Actually, for fixed temperature this state depends only on the time-direction of the Lorentz frame. Let e0 be such a time-direction, i.e. a timelike, future-pointing unit vector on Minkowski spacetime, and let e1 , e2 , e3 be a set of spacelike unit vectors so that e = (e0 , e1 , e2 , e3 ) forms an orthonormal tetrad on Minkowski spacetime. When choosing coordinates (x0 , x1 , x2 , x3 ) on Minkowski spacetime such that the coordinate axes are aligned with the tetrad, the two-point function W2βe of the unique quasifree thermal equilibrium (KMS) state ω βe at inverse temperature β > 0 2 with respect to the Lorentz frame defined by e is given by μ d4 p , (12) W2βe (x, x ) = e−i(x−x ) pμ (p0 )δ(pμ pμ − m2 ) (2π)3 (1 − e−βp0 ) to be interpreted in the sense of distributions, where (p0 ) is the sign function of p0 . The uniqueness implies that every intensive thermal property (e.g. pressure, density etc) can be expressed as a function of the timelike vector βe0 . The passage from these global equilibrium states to states of local thermodynamic equilibrium now uses spaces Sx of observables located at spacetime points x. Mathematically this implies that these observables are no longer defined as operators but only as quadratic forms (their products are usually not defined). In physical terms, the observables in Sx should model idealized limits of measurements of intensive thermal properties of states in smaller and smaller spacetime regions. This culminates in the requirement that, for s(x) ∈ Sx , the functions Φs (x) = ω βe s(x) have to be independent of x and non-constant as functions of β. Furthermore, for many s(x) ∈ Sx one can identify the thermal quantity to which Φs , as a function of β, actually corresponds. As an example, one calculates that for s(x) =: φ2 : (x) (Wick-square) one obtains for the massless case 1 k 2 T2 Φ:φ2 : = ω βe : φ2 : (x) = = B 2 12β 12 which leads to the identification of this observable as a “scalar thermometer”, giving the square of the local temperature T times some fixed constant. Choosing appropriate units to measure T, this constant can be set equal to 1, justifying 2β

= 1/(kB T) where kB is Boltzmann’s constant and T is absolute temperature

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953

the notation T2 for the expectation value of the Wick-square in thermal equilibrium states already alluded to above. Actually this seems quite similar to what one would do to construct a thermometer in the laboratory: Take some (small) device which, when exposed to a situation known to be in equilibrium at some temperature T, gives a reading which is a simple function of T. In the investigations of local thermal equilibrium for linear scalar fields φ(x) on Minkowski spacetime carried out in previous articles [8–10], the spaces Sx are chosen as the linear spaces generated by elements s(x) = ðμ : φ2 : (x), referred to as the balanced derivatives of the Wick-squared field : φ2 : (x). 3 Here, μ = (μ1 , . . . , μn ) ∈ Nn is a multi-index of arbitrary length n, and the balanced derivatives are defined by (13) ðμ : φ2 : (x) = lim ∂μ φ(x + ζ)φ(x − ζ) − ω vac φ(x + ζ)φ(x − ζ) 1 ζ→0

where ∂μ = ∂ζ μ1 · · · ∂ζ μn , and where ω vac is the vacuum state. The limit is taken along spacelike directions ζ, so that φ(x + ζ)φ(x − ζ) is well defined as a quadratic form, and the limit defines an operator-valued distribution after smearing in x with test-functions. For multi-index length equal to 0, the balanced derivative equals just the Wick-square : φ2 : (x). For linear fields on Minkowski spacetime, this definition of the Wick-square coincides with the usual normal ordering prescription. Owing to the translation invariance of the KMS-states ω βe one can easily check that the thermal functions Φðμ :φ2 : = ω βe (ðμ : φ2 : (x)) are independent of x. Following [8, 9], a state of the linear scalar field on Minkowski spacetime is said to be locally in thermal equilibrium at a spacetime point x if it looks like a global thermal equilibrium state ω βe as far as the expectation values of elements in Sx are concerned. The following definition, taken from [9], expresses this more formally. Definition 2.1. A state ω of the quantized linear scalar field φ(x) on Minkowski spacetime is called Sx -thermal at the spacetime point x if there are an orthonormal tetrad e with e0 timelike and future-pointing, and β > 0, such that (14) ω s(x) = ω βe s(x) holds for all s(x) ∈ Sx , where Sx is spanned by ðμ : φ2 : (x) as μ ranges over all multi-indices. If O is some open set of spacetime points, a state ω of the quantized linear scalar field on Minkowski spacetime is called SO -thermal if (14) holds and ω(s(x)) varies continuously with x, for s(x) ∈ Sx and x ∈ O. That means, ω is Sx -thermal at each x ∈ O, where β and e in (14) may vary with x. For SO -thermal states, the expectation value Φs ω (x) of an extensive thermal quantity Φs at x, whose local measurement is modelled by s(x) ∈ Sx , is then given as Φs ω (x) = ω(s(x)). This leads for SO -thermal states to an assignment of thermal quantities to each x ∈ O whose values in general vary with x, and this 3 In

the references [8, 9], the notation ðμ is used, but we prefer to view μ as a co-tensor index

954



assignment is consistent in the sense that relations among the thermal quantities (like equations of state) also hold at each point. In the case of s(x) =: φ2 : (x), one obtains in this way for an SO -thermal state ω an assignment of the expected squared temperature to each spacetime point x ∈ O. The concept of states which are locally in thermal equilibrium has also been generalized in [9] to allow mixtures of global thermal equilibrium states on the right hand side of (14). We will summarize this generalized concept in Section 5. In attempting to extend the concept of local thermal equilibrium states to quantum fields in curved spacetime, one faces a couple of difficulties which are, of course, connected to the occurrence of curvature and the related lack of global vacuum states and global equilibrium states. Primarily, these difficulties are: (i) The definition (13) of balanced derivatives ðμ : φ2 : (x) uses the Minkowski vacuum state ω vac as preferred vacuum state. (ii) Moreover, the definition (13) uses the affine space structure of Minkowski spacetime. (iii) Definition 2.1 uses global thermal equilibrium states ω βe on Minkowski spacetime, for which there is no counterpart on generic curved spacetimes. Thus, there is no verbatim translation of the concept of local thermal equilibrium states given in Definition 2.1. It is clear that problems (i) and (ii) concern the definition of balanced derivatives of a Wick-squared quantum field in curved spacetime. We will soon turn to that problem. Assuming that the definition of balanced derivatives in curved spacetime is settled, a proposal was made in [10] to surpass problem (iii). The idea is to define that a state ω of the quantized linear scalar field φ on a curved spacetime (M, g) is Sx -thermal at a point x in M if ω(s(x)) = ωoβe (so (xo )) holds for all s(x) ∈ Sx . Here, ωoβe is a thermal equilibrium state of the free scalar field φo (with same parameters as φ) on Minkowski spacetime Mo , xo is a point in Mo , and so (xo ) is the flat space counterpart of s(x). To explain what this latter phrase means precisely can be seen as part of the definition of balanced derivatives in curved spacetime, but certainly one would require that so (xo ) corresponds to a balanced derivative of the Wick-square of φo if s(x) corresponds to a balanced derivative of the Wick-square of φ. In the approach of [10], the requirement of local thermality on ω is thus not implemented by comparing expectations values of pointlike thermal observables with the corresponding expectation values in a global thermal state (as such states need not exist), but with the “flat space version” of a thermal equilibrium situation for the quantum field. The motivation for this approach is that Sx -thermality is a pointwise property which should not be affected by curvature; this, in turn, rests largely on the equivalence principle. For a linear scalar quantum field φ on a curved spacetime it is simple enough to know what its flat space counterpart φo should be. However, as pointed out in [10], one may invoke the concept of a local covariant quantum field theory [7,26] to know this also for more general types of quantum fields. The concept of local covariance affects also the elements s(x) ∈ Sx ; in our situation where we start

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from a linear scalar field φ in curved spacetime – which is known to have the structure of a local covariant quantum field – the balanced derivatives of Wicksquares of φ should be defined in such a way that they are also local covariant quantum fields. Let us thus discuss our proposal for the generalization of the concept of balanced derivatives in a curved spacetime. Our discussion is greatly facilitated by the circumstance that for the purpose of deriving QEIs for LTE states we need only focus on balanced derivatives up to second order, corresponding to a multi-index length of μ not greater than two. Accordingly, we will define the LTE property on curved spacetime only with balanced derivatives of the Wick-squared field up to second order, see below. We proceed in two steps. First, we shall consider the generalization of expressions like ∂ζ μ ∂ζ ν f (x + ζ, x − ζ)|ζ=0 for C 2 -functions f from Minkowski spacetime to curved spacetime. This discussion is entirely of differential geometric nature. In a second step, we have to give a generalization of the quantity f (x + ζ, x − ζ) = ω φ(x + ζ)φ(x − ζ) − ω vac φ(x + ζ)φ(x − ζ) for Hadamard states ω on curved spacetime where there is no counterpart of ω vac . In doing this we have to ensure, as mentioned, that the resulting balanced derivatives of the Wick-ordered linear scalar field give rise to local covariant quantum fields. Turning to the first step, let (M, g) be a spacetime and suppose that N is a geodesically convex neighbourhood of some point x in M . The exponential map at x will be denoted by expx . A fairly obvious generalization of the first balanced derivative of a function f ∈ C 2 (N × N ) arises by requiring

d

v a ða f (x) = f expx (λv), expx (−λv)

dλ λ=0 for all spacelike vectors v = v a ∈ Tx M lying in exp−1 x (N ). By linearity, this ∗ r r determines a co-vector ða f (x) in Tx M . We define T (s s )(M × M ) as the bundle over M × M whose fibre at (y, y ) ∈ M × M is given by Ty (rs )M ⊗ Ty (rs )M , where Ty (rs )M coincides with the space of r-fold contravariant and s-fold covariant tensors at y. If (y, y ) → V (y, y ) is any C 1 section in T (rs sr )(M × M ), then we denote by ∇a V the covariant derivative with respect to the y-entry and by ∇a V the covariant derivative with respect to the y -entry. Furthermore, we denote by V x = V (x, x) the coincidence value of V (y, y ) for y = x = y . With these conventions, one has ða f (x) = ∇a f x −∇a f x , and if f is C 2 , this defines a C 1 co-vector field as x varies. The second order balanced derivative ðab f (x) can then be defined as follows. ∇a f (y, y ) is a y -dependent co-vector at y, and vice versa for ∇a f (y, y ). Let v = v a be a (spacelike) vector in Tx M which lies in exp−1 x (N ), so that ηv : λ → expx (λv) (−1 ≤ λ ≤ 1) is the geodesic determined by v at x. Corre∗ ∗ spondingly, we can define the map of parallel transport Pv,λ : Texp (λv) M → Tx M x

956



of co-vectors from expλv = ηv (λ) to x = ηv (0) along the geodesic ηv . A geometrically natural definition of the second order balanced derivative ðab f (x) of f at x is then obtained by demanding that

d

a b wb Pv,λ ∇b f expx (λv), expx (−λv) v w ðab f (x) = dλ λ=0

d

wb P−v,λ ∇b f expx (λv), expx (−λv) − dλ λ=0 holds for all (spacelike) vectors v, w ∈ Tx M with v ∈ exp−1 x (N ). Using the properties of the parallel transport, it follows that M x → ðab f (x) is a continuous (if f is C 2 ) (02 )-tensor field on M , and ðab f (x) = ∇a ∇b f x −∇a ∇b f x

(15)

− ∇a ∇b f x +∇a ∇b f x . Note here that the covariant derivatives on the right hand side act on y for unprimed indices and on y for primed indices, and primed and unprimed tensor indices are identified at the coincidence point y = x = x . Looking at the final result obtained in (15) one notices that this is exactly what one would have obtained by replacing ordinary by covariant derivatives in the prescription for Minkowski spacetime. This is an indication that at the level of the second order balanced derivatives ordering problems for multiple (covariant) derivatives do not yet arise (as is to be expected for a torsion-free connection). Such ambiguities can however be expected when defining higher order balanced derivatives. Other possible modifications of the balanced derivatives include the addition of suitable geometric terms like gab R, but such terms will be dealt with in the transition from the classical to the quantum expressions anyway and it is the assumption of general covariance for the resulting quantum fields which then imposes restrictions on the possibility of such terms. Turning to the second step, suppose that (M, g) is a globally hyperbolic spacetime, and that N is a geodesic convex neighbourhood of some point x ∈ M . Then define the distributions (k ≥ 2) i 1 Gk (f1 , f2 ) = Gk (f1 , f2 ) + Gk (f2 , f1 ) + E (f1 , f2 ) , 2 2

f1 , f2 ∈ C0∞ (N, R) ,

where Gk and E are defined above for the quantized linear scalar field φ on (M, g). Next, let ω be a quasifree Hadamard state of φ on (M, g); then define the point-split renormalized two-point function obtained by subtracting the symmetrized Hadamard parametrix (SHP) Gk from the two-point function: SHP (f1 , f2 ) = W2ω (f1 , f2 ) − Gk (f1 , f2 ) , Wω,k

f1 , f2 ∈ C0∞ (N, R) .

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SHP SHP SHP A first observation is that Wω,k is symmetric, Wω,k (f1 , f2 ) = Wω,k (f2 , f2 ). FurSHP thermore, for k ≥ 2, Wω,k is given SHP Wω,k (f1 , f2 ) =

N ×N

SHP Wω,k (y, y )f1 (y)f2 (y )dvol(y)dvol(y ) ,

f1 , f2 ∈ C0∞ (N, R) ,

SHP with Wω,k ∈ C 2 (N × N ) for k ≥ 2, which follows from the definition of Hadamard form (see also the arguments in [18]). Consequently, one may now define for any quasifree Hadamard state ω the expectation value of the SHP-Wick square of φ and the corresponding second balanced derivatives in the following way:

Definition 2.2.

SHP x , ω : φ2 :SHP (x) = Wω,k 2 SHP ω ðab : φ :SHP (x) = ðab Wω,k x

(16) (17)

for x ∈ M , with k ≥ 2. We add a few observations to this definition. (α) One can likewise define the first balanced derivative of ω(: φ2 :SHP (x)), but SHP is symmetric, its first balanced derivative vanishes. This is similar since Wω,k to the property of balanced derivatives of the Wick-square of the quantized linear scalar field φo on Minkowski spacetime, which can be traced back to the symmetry of ωo φo (y)φo (y ) − ω vac φo (y)φo (y ) with respect to y and y , for each quasifree Hadamard state ωo of φo . This provides motivation why we define the Wick-ordering by subtraction of the symmetrized Hadamard parametrix. SHP SHP (y, y ) and ∇a ∇b Wω,k (y, y ) depend on the time-function t (β) Actually, Wω,k entering into the definition of Gk, , but for k ≥ 2, this dependence vanishes in the SHP SHP x and ∇a ∇b Wω,k x are indepencoincidence limit y = x = y . Similarly, Wω,k dent of k for k ≥ 2. (γ) One purpose of using the point-split renormalization by subtraction of the symmetrized Hadamard parametrix Gk is that the latter is a locally constructed geometric quantity which is state independent, so that : φ2 : SHP and ðab : φ2 : SHP become local, covariant fields. If one applies this technique to the linear scalar field φo on Minkowski spacetime, one finds that : φo 2 : SHP and ðμν : φo 2 : SHP deviate for m > 0 from the usual flat-space definitions of : φo 2 : and ðμν : φo 2 :, described above, by constants. This deviation actually reflects the fact that the conventional normal-ordering procedure on Minkowski spacetime does not give rise to covariant Wick powers. This is related to the fact that the argument of the logarithm in the definition of the Hadamard parametrix, equation (9), can be chosen as lσ2 where l is a length-scale. Our definition of the Gk can be seen as the choice l = 1 and using

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the expression for W2ω − Gk (k ≥ 2), one can calculate that, for each Hadamard state ωo of φo , one has (18) ωo : φo 2 : SHP (xo ) = ωo : φo 2 : (xo ) + c0,m , 2 2 (19) ωo ðμν : φo : SHP (xo ) = ωo ðμν : φo : (xo ) + c2,m ημν , vac

at all points in Minkowski spacetime (details in Appendix B, see also [26] and [29] for a discussions of the relations between local covariance and the appearance of l). Here, m is the mass parameter of the linear scalar field, and ημν is the Minkowski metric. The constants c0,m and c2,m vanish for m = 0; for m > 0, they are given by e2γ m2 m2 c0,m = ln −1 , (20) (4π)2 4 e2γ m2 m4 5 , (21) ln c2,m = − − (4π)2 4 2 where γ denotes the Euler–Mascheroni constant. This needs to be taken into account in the definition of thermal equilibrium states below. We can now define the concept of a local thermal equilibrium state on a globally hyperbolic curved spacetime (M, g). Let e = (e0 , e1 , e2 , e3 ) be an orthonormal tetrad at x ∈ M , with e0 timelike and future-pointing. Then e induces an identification of Tx M with Minkowski spacetime Mo , whereupon e is identified with a basis of Mo , again with e0 timelike and future-pointing in Minkowski spacetime. This identification is used in the following definition. Definition 2.3. Let ω be a state with two-point function of Hadamard form for the quantized linear scalar field φ on a globally hyperbolic spacetime (M, g). Then let φo denote the quantized linear scalar field, with the same parameters as φ, on Minkowski spacetime Mo . (2) (a) We say that ω is Sx -thermal at a point x ∈ M if, with some orthonormal tetrad e = (e0 , e1 , e2 , e3 ) at x such that e0 is timelike and future-pointing, there is a thermal equilibrium state ωoβe of φo so that – upon identification of e with a basis tetrad of Mo – the equalities (22) ω : φ2 : SHP (x) = ωoβe : φo 2 : SHP (xo ) βe 2 = ωo : φo : (xo ) + c0,m , a b 2 (23) v w ω ðab : φ : SHP (x) = v μ wν ωoβe ðμν : φo 2 : SHP (xo ) = v μ wν ωoβe ðμν : φo 2 : (xo ) + c2,m v μ wν ημν hold for all (spacelike) vectors v, w ∈ Tx M with coordinates v μ eμ = v, wν eν = w, for some xo ∈ Mo . (By translation-invariance of ωoβe , the particular choice of xo is irrelevant.) (2) (b) Let N be a subset of M . We say that ω is SN -thermal if ω(: φ2 : SHP (x)) and ω(ðab : φ2 : SHP (x)) are continuous in x ∈ N and if, for each x ∈ N , ω is (2) Sx -thermal at x.

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(2)

The definition of Sx -thermal states demands the coincidence of expectation values of the SHP Wick square of φ and its balanced derivatives up to second order with the thermal equilibrium situation in flat spacetime. This amounts to saying (2) that Sx consists of linear combinations of the unit operator 1 and of the quadratic forms : φ2 : SHP (x) and ðab : φ2 : SHP (x) whose evaluations (i.e. expectation values) on states ω are given by (16) and (17). Thus, for the linear scalar field on Minkowski (2) (2) spacetime, Sx is a small subset of Sx , and thus an Sx -thermal state fulfills less constraints on its thermal properties than an Sx -thermal state. We shall not follow (2) (2) up that matter at this point. Our definition of Sx -thermal states (or SO -thermal states) turns out to be sufficient to derive quantum energy inequalities. (2) Given an Sx -thermal state ω, we shall now use the abbreviations ϑω (x) = ω : φ2 : SHP (x) , (24) 1 2 (25) εω ab (x) = − ω ðab : φ : SHP (x) . 4 Then we have Lemma 2.4. Let φ be a linear scalar field on (M, g), with mass parameter m, and (2) let ω be an Sx -thermal state (x ∈ M ), satisfying (22) and (23) for some β > 0 and an orthonormal tetrad e = (e0 , e1 , e2 , e3 ) at x. Then the following statements hold. (a) ϑω (x) = β12 χ0,m (β) + c0,m , with ∞ ρ2 dρ 1 √ . χ0,m (β) = 2π 2 0 (e ρ2 +β 2 m2 − 1) ρ2 + β 2 m2 a 2 (b) εω a (x) = m χ0,m (β) − c2,m (c) Suppose that v is a lightlike vector at x, va v a = 0, or a timelike vector at x with unit proper length, va v a = 1, and set v 0 = (e0 )a v a . Then one has the bound

ζ(4)

c2,m 6(v 0 )2 (v 0 )2 a c2,m a b ≥ v − v v v ε (x) ≥ χ2,m (β) − va v a a ab 2 4 4 π β 4 β 4

where 1 χ2,m (β) = 2π 2

∞

0

ρ2 ρ2 + β 2 m2 √ 2 2 2 dρ e ρ +β m − 1

(26)

(27)

(ζ(4) is the value of the ζ-function at 4.) Proof. The proof is based on the fact that, with respect to coordinates induced by the basis tetrad e, 1 1 ωoβe : φo 2 : (xo ) = d3 p , 3 βp 0 (2π) R3 (e − 1)p0 pμ pν 1 1 2 βe d3 p , − ðμν ωo : φo : (xo ) = 3 βp 4 (2π) R3 (e 0 − 1)p0

960



where (pμ )μ=0,...,3 = (p0 , p) and p0 = |p|2 + m2 in the integrals. The stated relations then basically result from transforming the integrals into spherical polar coordinates. For the upper bound in (26) notice that the integrand is given by 2 √ 2 2 β |p| +m 0 2 2 2 2 −1 |p| + m v |p| + m − v · p e which is bounded above by ⎛ |p| ⎝v 0 −

v·p |p|2 + m2

⎞2 ⎠

(eβ|p| − 1) .

Upon integration over p, this can be bounded by the integrand 2(v 0 )2 |p|/(eβ|p| −1), using that (v 0 )2 − |v|2 = 1 in the timelike case and (v 0 )2 − |v|2 = 0 in the lightlike case. Now we introduce a set of states whose local temperature is bounded above by some fixed value. Definition 2.5. Let β > 0, x ∈ M . Then we define Lβ (x) as the set of all Sx thermal states ω of the linear scalar field on (M, g) so that (2)

ϑω (x) ≤

1 χ0,m (β ) + c0,m . (β )2

(28)

(2)

If N ⊂ M , we define Lβ (N ) as the set of all SN -thermal states of the linear scalar field on (M, g) so that (28) is fulfilled for all x ∈ N . In other words, ω is in Lβ (x) if the relations (22) and (23) are fulfilled for 1/β < 1/β . Now let N be an open subset of M , and let γ : [τ0 , τ1 ] → N , τ → γ(τ ) be a geodesic with affine parameter τ , and denote by v a = γ˙ a the tangent vector field of γ. By the geodesic equation, it holds that d2 (v a v b ∇a ∇b ϑω ) γ(τ ) = 2 ϑω γ(τ ) . dτ Consequently, we obtain for ω ∈ Lβ (N ) and η ∈ C02 ((τ0 , τ1 )),

η(τ )(v a v b ∇a ∇b ϑω ) γ(τ ) dτ = η (τ )ϑω γ(τ ) dτ

1

≤ ||η ||L1

2 χ0,m (β ) + c0,m

. (β ) Here, η is the second derivative of η.

(29)

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3. Quantum energy inequalities Let (M, g) be a globally hyperbolic spacetime, and let ϕ be the classical linear scalar field on (M, g) with mass parameter m ≥ 0 and conformal coupling parameter ξ. If ϕ is a field configuration, i.e. a smooth solution to the field equation (4), (ϕ) then the corresponding classical stress-energy tensor is a (02 ) co-tensor field Tab given by 1 (ϕ) Tab (x) = ∇a ϕ(x) ∇b ϕ(x) + gab (x) m2 ϕ2 (x) − (∇c ϕ)(∇c ϕ)(x) 2 + ξ gab (x)∇c ∇c − ∇a ∇b − Gab (x) ϕ2 (x) , x ∈ M , where Gab = Rab − 12 gab R is the Einstein tensor. Now let φ be the quantized linear scalar field on (M, g), corresponding to the choice of parameters m and ξ. The definition of the renormalized expectation value of products and derivatives for the quantized linear scalar field φ in a state ω having two-point function of Hadamard form proceeds, similarly to what was done in the previous chapter, by point-splitting and subtraction of the SHP (see [39,41]). To this end, we define: SHP x ω : φ∇a φ : SHP (x) = ∇a Wω,k SHP ω : φ∇a ∇b φ : SHP (x) = ∇a ∇b Wω,k x SHP ω : (∇a φ)(∇b φ) :SHP (x) = ∇a ∇b Wω,k x with k ≥ 2, x ∈ M . (Note again that a and a are identified upon taking the coincidence limit y = x = y on the right hand side of each equation.) Owing to SHP (k ≥ 2), one can easily check that the following Leibniz the symmetry of Wω,k rule is fulfilled for SHP Wick-products involving derivatives: ω ∇a : φ2 : SHP (x) = 2ω : φ∇a φ : SHP (x) , ω ∇a : φ∇b φ : SHP (x) = ω : (∇a φ)(∇b φ) : SHP (x) + ω : φ∇a ∇b φ : SHP (x) . The renormalized expectation value of stress-energy is then obtained via replacing the classical expressions ϕ2 (x), (∇a ϕ(x))(∇b ϕ(x)), and so on, by ω(: φ2 :SHP (x)), ω(: (∇a φ)(∇b φ) : SHP (x)), etc. Using also the Leibniz rule for SHP Wick products, this leads to SHP 1 ω Tab (x) = ω − : φ∇a ∇b φ : SHP (x) + ∇a ∇b : φ2 : SHP (x) 4 1 + − ξ ω ∇a ∇b : φ2 : SHP (x) − gab (x)∇c ∇c : φ2 : SHP (x) 4 1 + gab (x)ω : φ∇c ∇c φ : SHP (x) + m2 : φ2 : SHP (x) 2 − ξGab (x)ω : φ2 : SHP (x)

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This expression, however, has the defect of a non-vanishing divergence. The way to cope with this problem, following Wald [39, 41], is like this: It can be shown SHP (x)) = ∇b Q(x)4 where (apart from a free constant which can be set that ∇a ω(Tab to a preferred value depending on the mass parameter m) Q is a function which is determined by the local geometry of (M, g); in particular, Q is independent of SHP (x)) to the state ω. One may therefore subtract the term Q(x)gab (x) from ω(Tab make the resulting quantity have vanishing divergence. There remains an ambiguity in that one may still add other (02 ) co-tensor fields Cab which are determined by the local geometry of (M, g) and have vanishing divergence. We take here the same view as put forward in [18], namely that the specification of Cab is a further datum of the underlying quantum field φ on (M, g), in addition to the parameters m and ξ. An alternative, elegant method has been proposed by Moretti [29], which nevertheless we won’t follow here mainly because we would like to maintain close contact to other works on quantum energy inequalities. This understood, we finally define the renormalized expectation value of the stress energy tensor in some state ω (with two-point function of Hadamard form) of the linear scalar field φ on (M, g) as SHP ren (x) = ω Tab (x) − Q(x)gab (x) + Cab (x) , x ∈ M . (30) ω Tab Again note that Q and Cab are state-independent and constructed locally out of the spacetime metric g = gab . Let us next observe that, for each state ω of φ with two-point function of Hadamard form, and with R = Raa denoting the scalar curvature, F (x) = ω : φ(∇a ∇a + m2 + ξR)φ : SHP (x) = ω : φ(∇a ∇a φ) : SHP (x) + m2 + ξR(x) ω : φ2 : SHP (x) is a continuous function of x ∈ M , independent of the state ω, entirely determined by the local geometry of (M, g) and the parameters m and ξ of φ. To see this, note that SHP x . ω : φ(∇a ∇a + m2 + ξR)φ :SHP (x) = (∇a ∇a + m2 + ξR)Wω,k SHP SHP (y, y ) is the integral kernel of Wω,k = W2ω − Gk (k ≥ 2), On the other hand, Wω,k ω b 2 a SHP and since W2 (f, (∇ ∇b +m +ξR)h) = 0, it follows that (∇ ∇a +m2 +ξR)Wω,k SHP is independent of ω as the ω-dependent part of Wω,k is annihilated by the wave SHP x operator (∇a ∇a + m2 + ξR). In consequence, F (x) = (∇a ∇a + m2 + ξR)Wω,k is state-independent, continuous in x, and actually it is determined by the local geometry of (M, g) since so is Gk (by the Hadamard recursion relations). An explicit calculation using this relations gives the relation [29]

F = 3Q 4 it should however be noted, that this Q differs from the one given by Wald by a numerical factor, which is due to the fact that we define the Wick squares using a symmetric Hadamard Parametrix which does not fulfill the wave equation in any of its two arguments, whereas Wald uses an asymmetric one which fulfills the wave equation in the first argument

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ren Using the Leibniz rule, we can now rewrite the expression for ω(Tab ) as follows: ren 1 (x) = ω − : φ∇a ∇b φ : SHP (x) + ∇a ∇b : φ2 : SHP (x) (31) ω Tab 4 1 − ξ ω ∇a ∇b : φ2 : SHP (x) + 4 1 c c 2 gab + (4ξ − 1) ω − : φ∇ ∇c φ : SHP (x) + ∇ ∇c : φ : SHP (x) 4 1 + (1 − 4ξ)(m2 + ξR) − ξR gab − ξGab ω : φ2 : SHP (x) 2 5 + 12ξ − Q(x)gab (x) + Cab (x) 2

By (15) and once more the Leibniz rule, ab can also be expressed as 1 2 ab = ω − : φ∇a ∇b φ : SHP (x) + ∇a ∇b : φ : SHP (x) 4 (2)

Thus, if ω is an Sx -thermal state of φ, we obtain ren ωc (x) = εω (32) ω Tab ab (x) + (4ξ − 1)gab (x)εc (x) 1 + − ξ ∇a ∇b ϑω (x) + gab (x)ψ(x) − ξRab (x) ϑω (x) 4 + (12ξ − 5/2)Q(x)gab (x) + Cab (x) , where we use the abbreviation

ψ(x) = (1 − 4ξ) m2 + ξR(x) .

(33) a b

ren With this expression, we are now in the position to derive bounds on v v ω(Tab ) for lightlike or timelike vectors v. We will treat lower bounds first.

Theorem 3.1. Let φ be the quantized linear scalar field on (M, g), with parameters m, ξ and Cab , and let ω be a state of φ having two-point function of Hadamard form. (a) Suppose that ξ = 1/4, and let v be a lightlike vector at x ∈ M , or a timelike vector at x with va v a = 1. If ω is in Lβ (x), β > 0, then ren (x) ≥ q(x, v; β ) (34) v a v b ω Tab where

1 1 q(x, v; β ) = − v a v b Rab (x)

2 χ0,m (β ) + c0,m

4 β 3 1 + Q(x) − c2,m va v a + v a v b Cab . 2 4

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(b) Let ξ be arbitrary, let N ⊂ M , and let γ : [τ0 , τ1 ] → N be an affinely parametrized lightlike geodesic defined on a finite interval, with tangent vector field v a = γ˙ a . Suppose that η is in C02 ((τ0 , τ1 )) with η ≥ 0. If ω ∈ Lβ (N ), there holds the bound ren γ(τ ) dτ ≥ q0 (γ, η; β ) . (35) η(τ )v a v b ω Tab Here, writing

R[γ] = max γ˙ a (τ )γ˙ b (τ )Rab γ(τ ) , τ ∈[τ0 ,τ1 ]

and defining C[γ] analogously, the bounding constant is given by

1

q0 (γ, η; β ) = − |ξ|R[γ]

2 χ0,m (β ) + c0,m

+ C[γ] ||η||L1 β

1

1

− − ξ · 2 χ0,m (β ) + c0,m

· ||η ||L1 4 β (c) Let ξ be arbitrary, let N ⊂ M , and let γ : [τ0 , τ1 ] → N be an affinely parametrized timelike geodesic with tangent vector field v a = γ˙ a , so that v a va = 1. Assume that η is in C02 ((τ0 , τ1 )) with η ≥ 0. If ω ∈ Lβ (N ), there holds the bound ren γ(τ ) dτ ≥ q1 (γ, η; β ) , (36) η(τ )v a v b ω Tab where, using the notation ψ[γ] = maxτ ∈[τ0 ,τ1 ] |ψ(γ(τ ))|, and defining Q[γ] similarly, the bounding constant is given by

1

1

q1 (γ, η; β ) = − − ξ · 2 χ0,m (β ) + c0,m

· ||η ||L1 4 β

1

− (ψ[γ] + |ξ|R[γ] ) · 2 χ0,m (β ) + c0,m

· ||η||L1 β |c2,m | 5 − |12ξ − 2 |Q[γ] + |4ξ − 1||c2,m | + C[γ] + · ||η||L1 . 4 Proof. The proof of the statement consists just of inserting the estimates of Lemma 2.4 and discarding manifestly positive terms, in combination with estimate (29) for the average of the second derivatives of ϑω along the geodesic. The term involving second derivatives of ϑω doesn’t occur for ξ = 1/4, which makes it possible to give a pointwise lower bound in this case. The central assertion of Theorem 3.1 is that the lower bound of the energy density averaged along a causal geodesic depends only on the temperatures an LTE state attains on the geodesic, and is otherwise state-independent. The bound worsenes (shifts towards the left on the real axis) as the temperature increases, i.e. with increasing 1/β . This is related to the question of the sharpness of the obtained bounds. The point to notice here is that they were obtained by bounding the term ab v a v b from below by the temperature independent term −v a va |c2 |/4. However,

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ab v a v b grows with temperature as can be seen from (26) and the growth is (for high temperatures) with the fourth power of temperature. As the ϑω dependent term, which is responsible for the worsening of the bounds, grows (asymptotically) with the square of the temperature, it will be compensated for sufficiently high temperatures by the dropped term. By a more careful investigation one could therefore hope to obtain a lower bound where the temperature dependence is replaced by a dependence on the spacetime geometry and γ. Finally it should also be noted that the bounds are local covariant. For upper bounds on the averaged energy density of LTE states, an additional state-dependence shows up: The bounds depend also on the tetrad e ap(2) pearing in the condition of Sx -thermality, Definition 2.3. In this sense, the lower bounds on the averaged energy densities of LTE states are stronger than the upper bounds. This is similar to what holds for averages of energy densities for arbitrary Hadamard states of the linear scalar field [14]. Let x ∈ M , and let e = (e0 , . . . , e3 ) be an orthonormal tetrad at x with e0 timelike and future-pointing. We define Lβ (x, e) as the set of all states ω in (2) Lβ (x) where the Sx -thermality conditions (22) and (23) hold with respect to the given tetrad. Similarly, let N be a subset of M , and let N x → e(x), e(x) = (e0 (x), . . . , e3 (x)) be a C 0 field of orthonormal tetrads over N , with e0 (x) timelike and future-pointing for all x. Then we define Lβ (N, e) as the set of all (2) states ω in Lβ (N ) such that, for each x ∈ N , ω satisfies the Sx -thermality conditions (22) and (23) with respect to e = e(x). With these conventions, we obtain the following upper bounds on (averaged) energy densities. Theorem 3.2. Let φ be the quantized linear scalar field on (M, g), with parameters m, ξ and Cab , and let ω be a state of φ having two-point function of Hadamard form. (a) Suppose that ξ = 1/4, let v be a lightlike vector at x ∈ M , or a timelike vector at x with va v a = 1, If ω is in Lβ (x, e), β > 0, then ren p(v, x; β , e) ≥ v a v b ω Tab (x) (37) where

6(v 0 )2 1 p(v, x; β , e) =ζ(4) 2 4 + v a v b Rab 2 π β + q(x, v; β )

1

β 2 χ0,m (β ) + c0,m

with v 0 = va (e0 )a . (b) Let ξ be arbitrary, N ⊂ M , and let γ : [τ0 , τ1 ] → N be an affinely parametrized lightlike geodesic defined on a finite interval, with tangent vector field v a = γ˙ a . Suppose that η is in C02 ((τ0 , τ1 )) with η ≥ 0. If ω is in Lβ (N, e), then ren p0 (γ, η; β , e) ≥ η(τ )v a v b ω Tab γ(τ ) dτ (38)

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where p0 (γ, η; β , e) =

6ζ(4) 0 2 v ||η||L1 + |q0 (γ, β ; η)| π 2 (β )2 [γ]

0 = maxτ ∈[τ0 ,τ1 ] γ˙ a (τ )ea0 (γ(τ )). with v[γ]

4. Averaged null energy condition (ANEC) (2)

In this section we derive the averaged null energy condition (ANEC) for SN thermal states of the quantized linear scalar field φ on a globally hyperbolic spacetime (M, g). The ANEC on a state ω of φ demands that τ+ ren γ(τ ) dτ ≥ 0 (39) v a v b ω Tab lim inf τ± →±∞

τ−

for all complete lightlike geodesics γ in M with affine parameter τ and tangent v a = γ˙ a . If this condition holds, and if (M, g) together with φ and ω are a solution to the semiclassical Einstein equation in the form ren Gab (x) = 8πω Tab (x) , x ∈ M , (40) then this implies that

τ+

lim inf

τ± →±∞

v a v b Gab γ(τ ) dτ ≥ 0

(41)

τ−

for all complete lightlike geodesics γ. (We address the issue for the semiclassical Einstein equations with an additional contribution by a classical stress-energy tensor below.) It has been shown that this weaker form of the usual pointwise null energy condition, which demands that a b Gab (x) ≥ 0 for all lightlike vectors a at each x ∈ M , is still sufficient to reach the same conclusions with respect to singularity theorems as obtained from the pointwise null energy condition, i.e. that congruences of geodesics will focus with expansion diverging to −∞ at finite affine geodesic parameter [25]. The validity of (39) is therefore of importance for the properties of the spacetime structure of solutions to the semiclassical Einstein equations. It has been argued in [42] that condition (39) may be replaced by the following condition: ∞ ren γ(τ ) dτ ≥ 0 (42) ηλ (τ )v a v b ω Tab lim inf λ→0

−∞

for any η ∈ η ≥ 0, with η(0) > 0 and ηλ (τ ) = η(λτ ) for λ > 0. More precisely, in [42] it has been shown that (42) and (40) imply that the expansion of a congruence of lightlike geodesics around γ becomes singular along γ (in the sense of diverging to −∞ at a finite value of the affine parameter) unless it vanishes identically on γ. (In [42] this argument is given for half-line geodesics, but it carries over to the case at hand as will be shown in our Appendix A.) C02 (R),

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(2)

Now let ω ∈ SN , and let γ be a complete lightlike geodesic in N ⊂ M with affine parameter τ and tangent v a = γ˙ a . Then, from (32), 1 a b ren a b ω − ξ v a v b ∇a ∇b ϑω − ξv a v b Gab ϑω + v a v b Cab (43) v v ω(Tab ) = v v εab + 4 holds along γ. Therefore, positivity properties of the (integrated) energy density ren v a v b ω(Tab ) depend also on the behaviour of Gab and Cab . The sign of the term involving Gab is not known. To circumvent this difficulty, we assume that the underlying spacetime (M, g) together with φ and ω are solutions to the semiclassical Einstein equations (40), since it is this situation in which the ANEC is applied to deduce (41) and the ensueing statements about focussing of lightlike geodesics. Supposing that (M, g) together with φ and ω are solutions to the semiclassical (2) Einstein equations, and also that ω is an SN -thermal state, we obtain upon combination of (40) and (43) the equation 1 a b ω a b ω ω − ξ ∇a ∇b ϑ v v Gab (1 + 8πξϑ ) − 8πCab = 8πv v εab + (44) 4 on N . In order to draw further conclusions, one must specify Cab . We recall that Cab is a datum of the linear quantum field φ, a priori only restricted by the ren be a local covariant quantum field and divergence-free, thus requirement that Tab Cab should be locally constructed from the spacetime metric. Following Wald [41], one can make the assumption that Cab have canonical dimension, which leads to the form δ δ (45) Cab = Agab + BGab + Γ ab S1 (g) + D ab S2 (g) δg δg where S1 (g) = M R2 dvolg , S2 (g) = M Rab Rab dvolg , and δ/δg ab means functional differentiation with respect to the metric, with constants A, B, Γ, D as remaining renormalization ambiguity for the quantum field φ (see [41] for additional discussion). For the rest of our discussion, we will simplify matters by assuming Γ, D = 0. Making these assumptions and observing that hence, v a v b Cab = Bv a v b Gab for all lightlike vectors v a , (44) assumes on N the form 1 ω − ξ ∇ v a v b Gab 1 + 8π(ξϑω − B) = 8πv a v b εω + ∇ ϑ . (46) a b ab 4 The constant B is still free, and one may now try to choose B in such a way that (46) entails the ANEC for all lightlike geodesics in N ⊂ M and an as large (2) as possible class of SN -thermal states ω. We will show that this is possible with different conditions on B for the cases ξ = 1/4, 0 < ξ < 1/4, ξ = 0. Theorem 4.1. Let (M, g) be a globally hyperbolic spacetime, let φ be the quantized linear scalar field on (M, g), with parameters m, ξ, Cab , where Cab = Agab + BRab , with real constants A, B.

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Ann. Henri Poincaré (2)

Suppose further that ω is a quasifree Hadamard state for φ, that ω ∈ SN for N ⊂ M , and that (M, g) together with φ and ω provides a solution to the semiclassical Einstein equation (40). Let γ be a complete lightlike geodesic in N with affine parameter τ and tangent v a = γ˙ a , and let η ∈ C02 (R), η ≥ 0. Then ∞ ren lim γ(τ ) dτ ≥ 0 (47) η(λτ )v a v b ω Tab λ→0

−∞

holds if any of the following groups of conditions is assumed: 1.) ξ = 1/4, B < 1 + 2πc0,m . In this case one even has ren v a v b ω Tab (x) ≥ 0 pointwise for all x ∈ M and all lightlike vectors v a at x. 2.) 0 < ξ < 1/4, B ≤ ξc0,m + 1/(8π), λ ln ϑω γ(τ /λ) → 0 as λ → 0 for almost all τ , r

λ ln ϑω γ(τ /λ) dτ < k < ∞ for small λ and all

(48) s < r ∈ R.

(49)

s

3.) ξ = 0, B < 1/8π, λϑω γ(τ /λ) → 0 as λ → 0 r λϑω γ(τ /λ) dτ < K < ∞

for almost all for small

λ

τ, and all

(50) s < r ∈ R.

(51)

s

Remark. (a) If, instead of (40), the semiclassical Einstein equations are assumed to hold in the form ren class Gab (x) = 8π Tab (x) + ω Tab (x) class for classical, macroscopic matter distribution, with a stress-energy tensor Tab and if it is assumed that this stress-energy tensor fulfills the pointwise null energy class (x) ≥ 0 for all lightlike vectors a at each point x ∈ M , then the condition a b Tab class ren ren statements of the theorem remain valid with Tab + ω(Tab ) in place of ω(Tab ). ω (b) Conditions (48) and (49) say, roughly speaking, that ϑ (γ(τ )) should not grow (1−) for |τ | → ∞, while (50) and (51) say that ϑω (γ(τ )) should not faster than e|τ | grow faster than |τ |1− as |τ | → ∞. Since ϑω (γ(τ )) = (β(γ(τ ))−2 χ0,m (β(γ(τ ))) + c0,m and since ∞ ρ 1 χ0,m (β) → dρ for β → 0 , 2 2π 0 eρ

this means that the growth of the temperature 1/β(γ(τ )) at γ(τ ) appearing in (1−)/2 (2) Definition 2.3 of Sγ(τ ) -thermality ω should not exceed e|τ | and |τ |(1−)/2 as |τ | → ∞, respectively.

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Proof of Theorem 4.1. 1.) If ξ = 1/4, then (46) assumes the form v a v b Gab 1 + 8π(ϑω /4 − B) = 8πv a v b εω ab .

969

(52)

If B < 1 + 2πc0,m , then the factor 1 + 8π(ϑ /4 − B) is strictly positive, as is the right hand side of (52). This equality holds pointwise at all x ∈ M and for all lightlike vectors v a , thus proving, in combination with the assumed property (40), the statement of the theorem. 2.) For 0 < ξ < 1/4, B = ξc0,m + 1/(8π) − ξc, where c ≥ 0, (46) takes the form a b ω (53) v a v b Gab 8πξ(ϑω − c0,m + c) = 8πv a v b εω ab + 8π(1/4 − ξ)v v ∇a ∇b ϑ . ω

Observing that v a v b ∇a ∇b c0,m = 0, the last equation is turned into v a v b Gab =

(1/4 − ξ)v a v b ∇a ∇b (ϑω − c0,m ) v a v b εω ab + − c0,m + c) ξ(ϑω − c0,m + c)

ξ(ϑω

(54)

where it was used that ϑω − c0,m + c > 0. The first term on the right hand side of (54) is positive. Upon integration against a non-negative C02 weighting function η along the geodesic γ we obtain, using the abbreviation u(τ ) = ϑω γ(τ ) − c0,m , the inequality

1/4 − ξ η(τ )(v a v b Gab ) γ(τ ) dτ ≥ ξ

η(τ )

u (τ ) dτ . u(τ ) + c

By partial integration, 2 u (τ ) u (τ ) η(τ ) dτ = η(τ ) dτ + ln(u(τ ) + c)η (τ ) dτ . u(τ ) + c u(τ ) + c Thus, since the first integral on the right hand side is non-negative, (1/4−ξ)/ξ > 0 for the ξ considered and using the monotonicity of the logarithm together with c ≥ 0, 1/4 − ξ λ ln u(τ /λ) η (τ ) dτ , η(λτ )(v a v b Gab ) γ(τ ) dτ ≥ ξ and owing to assumptions (48) and (49), the expression on the right hand side converges to 0 as λ → 0. Equation (47) is then again implied by the assumed property (40). 3.) If ξ = 0, equation (46) turns into 1 a b ω (55) v a v b Gab (1 − 8πB) = 8πv a v b εω ab + v v ∇a ∇b ϑ , 4 and by the condition on B, the factor 1 − 8πB is strictly positive. Observing 2 again positivity of 8πv a v b εω ab , upon integration against a non-negative C0 weighting function η along γ one obtains 1 a b λu(τ /λ)η (τ ) dτ η(λτ )v v Gab γ(τ ) dτ ≥ 4(1 − 8πB)

970



and the right hand side converges to 0 as λ → 0 by assumptions (50) and (51). Again (47) is deduced from the assumed validity of (40).

5. Generalized local thermal equilibrium states (2)

The notion of LTE states in [9], and the related definition of Sx -thermal states, is actually more general than the definition given in Section 2. In [9] the possibility was considered that an LTE state ω coincides on Sx -observables not necessarily with a thermal equilibrium state at sharp temperature in a certain Lorentz frame, but with a mixture of such states. In our setting, where we work with the linear scalar field, this corresponds to a modification of Definition 2.3 as follows. As a consequence of eqn.(12), ωoβe , the quasifree thermal equilibrium state with respect to the Minkowski tetrad e = (e0 , e1 , e2 , e3 ) at inverse temperature β, depends only on β = βe0 . This quantity completely parametrizes ωoβe , so we write ωoβ in place of ωoβe . The vectors β take values in V + , the set of future-directed timelike vectors in Minkowski spacetime. Let (M, g) be a globally hyperbolic spacetime, let Vx+ ⊂ Tx M be the set of future-directed timelike vectors at x ∈ M , and let ρx be a Borel measure on Vx+ supported on a compact subset Bx ⊂ Vx+ , with Bx dρx (β) = 1. Then we say that a Hadamard state ω of the linear scalar field φ on (M, g) is a generalized (2) Sx -thermal state if 2 ωoβ : φo 2 : (xo ) dρx (β) + c0,m , ω : φ : SHP (x) = Bx a b 2 μ ν ωoβ ðμν : φo 2 : (xo ) dρ(β) + c2,m v μ wν ημν v w ω ðab : φ : SHP (x) = v w Bx

holds for all (spacelike) vectors v, w ∈ Tx M for some xo ∈ Mo . Making further the assumption that F → M Bx F (x, β)dρx (β)dvol(x), F ∈ C0∞ (T M, C) is a distribution (on the manifold T M ), such that x → Bx F (x, β)dρx (β) is C 2 , one (2)

(2)

can define generalized SN -thermal states in analogy to the definition of SN thermal states in Section 2. With these conventions and assumptions, the results of Theorems 3.1, 3.2 (2) and 4.1 extend to generalized SN -thermal states, under identical assumptions, except that the bounds have to be corrected for the ρx -integrations. It should be obvious how this is to be done.

6. Discussion and outlook We have generalized the concept of local thermal equilibrium states of [9], or rather, (2) the concept of Sx -thermal states, to the quantized linear scalar field models on generic globally hyperbolic spacetimes, and have shown that one can derive certain quantum energy inequalities for such states. The lower bounds appearing in the quantum energy inequalities of local thermal equilibrium states depend only

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on the local temperature of the states, i.e. thermal function ϑω , corresponding to the expectation value of the Wick-square in local thermal equilibrium states. The upper bounds, instead, depend also on the local frames with respect to which (2) Sx -thermality is defined. In this sense, the lower bounds are stronger (have less dependence on the states) than the upper quantum energy inequality bounds. This is a feature also found for quantum energy inequalities of general Hadamard states of the linear scalar fields, and has led to the proposal to consider ‘relative quantum energy inequalities’ as a more general variant of quantum energy inequalities which has the potential to be valid also in interacting quantum field theories [18]. Moreover, the quantum energy inequalities for thermal equilibrium states are local covariant. A major purpose of quantum energy inequalities, especially in local covariant form, is to provide information about the structure of spacetime geometries appearing as solutions to the semiclassical Einstein equations. Quite generally, they serve as stability conditions on quantum matter, and ensure that correspondingly the (semiclassical) gravitational interaction is attractive, at least when averaged over sufficiently extended spacetime regions. The averaged null energy condition which we proved for certain values of the curvature coupling ξ and certain values of the renormalization constants is of a similar nature. One may also take the requirement that the ANEC should be fulfilled for suitable thermal equilibrium states as a constraining condition on the largely free choice of renormalization constants for the stress-energy tensor. Certainly a demand in this spirit leads to further relations between the renormalization constants, the parameters fixing the field model, and possibly geometrical quantities, and for this reason it is attractive to further study quantum energy inequalities and ANEC in the context of solutions to the semiclassical Einstein equations. Furthermore it might well be that the condition of local thermality has to be strengthened by fixing some renormalization freedom; in the setup chosen here this would mean imposing requirements on Cab but as the concept of local thermality is defined with reference to the balanced derivatives these requirements on Cab could then also be shifted to modified balanced derivatives. One important issue we haven’t addressed at all so far is the existence of (2) local thermal equilibrium states, or at least SN states for subsets N in spacetime. We have simply assumed that there are such local thermal equilibrium states to which our results apply. The question if there are local thermal equilibrium states in generic spacetimes is an interesting and difficult problem, for which we can’t offer, as yet, any route to its solution. However, the existence of LTE states for the massless and the massive Klein–Gordon fields on (parts of) Minkowski spacetime has been established, with an interesting relation to situations resembling a big bang scenario [8, 27]. The question if local thermal equilibrium states exist is a first step towards the question how generic they are. One is inclined to think that within certain time- and energy scales, local thermal equilibrium states should be the archetypical physical states in the sense that, if one is asked to randomly pick a state in

972



the physical state space (of a quantum field theory), then the result would be a local thermal equilibrium state with overwhelming likelihood. At least this is expected for interacting quantum field theories since interaction tends to equilibrate subsystems (or degrees of freedom) of a large system. If this turned out to be true, and if the temperature distribution of such states turned out to allow ANEC results similar to those of Theorem 4.1, then one would be led to conclude that (under general additional assumptions) the occurrence of singularities in solutions to the semiclassical Einstein equations is a generic feature. It would be of utmost interest to investigate this circle of questions further particularly in scenarios of early cosmology.

Appendix A. A. We will present a result on real-valued solutions θ(t) of the differential equation θ (t) + μθ(t)2 = −f (t) ,

t ∈ R,

(56)

where μ > 0 and f ∈ C 1 (R, R), with initial condition θ(0) = θ0 .

(57)

It follows from the Picard–Lindel¨ of Theorem that there is an open interval (a, b) containing 0, which may be finite, semi-finite or infinite (i.e. coinciding with R), such that this interval is the domain of the unique, inextensible C 1 solution θ of (56) satisfying the initial condition. In this case, we call θ the maximal solution of (56) defined by the initial condition, and refer to (a, b) as the maximal domain. The following statement is a variation on a similar result in [42], and it uses a very similar argument, the main difference being that the assumption (58) here is slightly different from that in [42], where the integral is taken over a semi-axis. Note also that our parameter λ corresponds to 1/λ in the notation of [42]. Theorem A.1. Suppose that f ∈ C 1 (R, R) has the property ∞ lim inf f (t)η(λt) dt ≥ 0 λ→0

(58)

−∞

for the function η(t) = (1 − t2 )4 for |t| < 1, η(t) = 0 for |t| ≥ 1. Then either the maximal domain of θ coincides with all of the real axis and θ(t) = 0 for all t ∈ R, or the maximal domain (a, b) of θ is a finite or semifinite interval. In this case, θ(t) → ∓∞ for t approaching the finite boundary at the right/left side of the maximal domain (in the finite case this holds with the respective sign for both boundaries). In particular, this is the case if θ(t0 ) = 0 for some t0 in the maximal domain of θ. Proof. Consider the auxiliary differential equation u (t) +

f (t) u(t) = 0 . μ

(59)

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973

For the initial values u(0) = 1, u (0) = θ0 and the given f this linear differential equation has by the Picard–Lindel¨ of Theorem a unique, global solution u ∈ C 2 (R, R). Furthermore, this solution is nonzero in some neighbourhood of 0. For points from this neighbourhood, one can then rewrite (59) as d dt

u (t) u(t)

+

u (t) u(t)

2 =−

f (t) u(t)

ˆ ≡ u (μt) fulfills equation (56). Furthermore, θˆ also satiswhich implies that θ(t) u(μt) fies the initial condition (57) and by the uniqueness part in the Picard–Lindel¨ of Theorem it therefore agrees with θ. This however implies that the only way in which θ can fail to be C 1 at a boundary point c = a or c = b of a semi-finite interval is a zero of u at μc. At this zero u has to differ from zero, otherwise u as a C 2 -solution to (59) with initial conditions u(μc) = 0, u (μc) = limx→μc u (x) = 0 would be identically zero in contradiction to the initial values for u at 0. By conti (μt) nuity, u is therefore nonzero in a neighbourhood of μc, and by (56), θ(t) = uu(μt) approaches the value −∞ for t → c, t < c (right boundary point) or the value +∞ for t → c, t > c (left boundary point). For proving that θ diverges at the boundary (boundaries) of a semi-finite interval it is therefore sufficient to show that θ cannot be continued as a C 1 function beyond this boundary. With the definition of η as above, and provided that the maximal domain of θ coincides with all of R, one has for 0 < λ < 1,

∞

−∞

θ (t)η(λt) dt = −

∞

θ(t)λη (λt) dt

−∞ 1/λ

= 8λ −1/λ

≥ −8λ

3 θ(t)(λt) 1 − (λt)2 dt

1/λ −1/λ

2 |θ(t)| 1 − (λt)2 dt

owing to the fact that both |λt| and |(1 − (λt)2 )| are bounded by 1 on the domain of integration. Combining this with (56) and (58) leads to lim sup −8λ λ→0

1/λ −1/λ

2 |θ(t)| 1 − (λt)2 dt + μ

1/λ

−1/λ

4 θ(t)2 1 − (λt)2 dt ≤ 0 . (60)

Using also the Cauchy–Schwarz inequality

1/λ

−1/λ

2 |θ(t)| 1 − (λt)2 dt ≤

1/λ

4 θ(t) 1 − (λt)2 dt

1/2

−/λ

1/2

1/λ

2

1 dt −1/λ

,

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the estimate (60) can be replaced by 1/2 1/λ 8√ 2 2 4 lim sup − 2λ θ(t) 1 − (λt) dt μ λ→0 −1/λ 1/λ 4 + θ(t)2 1 − (λt)2 dt ≤ 0 ,

(61)

−1/λ

∞ which shows that −∞ θ(t)2 dt = 0 upon using Levi’s theorem. Since θ is C 1 , this implies that θ(t) = 0 for all t. We have therefore shown that the assumption of θ being C 1 on all of R implies θ(t) = 0 for all t ∈ R; if on the other hand θ is C 1 only on a maximal finite or semi-finite interval, then by the statement in the first paragraph of the proof, it will diverge at the finite boundaries of this interval in the indicated way. B. Here we will calculate the constants c0,m , c2,m that arise when defining the Wick-square and the second balanced derivative on Minkowski spacetime using the covariant point-split renormalization. A similar calculation can also be found in the Appendix B of [29], the different conventions adapted here however lead to small changes in some of the formulas appearing. The Hadamard recursion-relations satisfied by the functions Uj in (10) read with our sign-conventions: −2(∇κ σ)∇κ U0 − (4 + ∇κ ∇κ σ)U0 = (∇κ ∇κ + m2 + ξR)U (∇κ ∇κ + m2 + ξR)Uj . j+1 For Minkowski spacetime, U is identically one, ∇κ ∇κ σ = −8 and the unique solutions of the resulting recursion relations −2(∇κ σ)∇κ Uj+1 + (4j − ∇κ ∇κ σ)Uj+1 =

4(x − x )κ ∇κ U0 + 4U0 = m2 4(x − x )κ ∇κ Uj+1 + 4(2 + j)Uj+1 =

(m2 + ∇κ ∇κ )Uj j+1

that remain bounded for x → x are easily calculated (e.g. using the method of characteristics) as 2 j+1 m 1 Uj = . j!(j + 1)! 4 For non-lightlike x − x where Gk, is a regular distribution (the corresponding function being obtained as the pointwise limit → 0) we have with the abbreviation (x − x )2 := ηab (x − x )a (x − x )b : 2 2 2 (x − x ) 1 m 1 −m ln − (x − x )2 1 + + G1,0 (x, x ) = 4π 2 −(x − x )2 4 8 (it will be seen in the course of the calculation, that G1,0 is actually sufficient to calculate the second balanced derivative, one does not need G2,0 ). The two-point

Vol. 9 (2008)

function W2ω by [3]

Local Thermal Equilibrium States and QEIs vac

975

of the Minkowski vacuum state ω vac for spacelike (x − x ) is given

K1 m −(x − x )2 m W2ω (x, x ) = 4π 2 −(x − x )2 and using the asymptotic expansion of the modified Bessel function K1 for small arguments, the terms up to the order (x − x )2 of the two-point function are given by −m2 (x − x )2 1 m2 1 m2 ω vac 2 ln (x − x W2 (x, x ) = + ) 1 − 4π 2 −(x − x )2 4 4 8 m2 −m2 (x − x )2 + (2γ − 1) + (2γ − 5/2) 4 8 vac

(here and in the following, x − x is now assumed to be spacelike). The difference vac W2ω (x + ζ, x − ζ) − G1,0 (x + ζ, x − ζ) to the order required for the calculation of ω ∞ (: φ2 : SHP (x)) and ω ∞ (ðμν : φ2 : SHP (x)) is then m2 m2 ω vac ln W2 (x + ζ, x − ζ) − G1,0 (x + ζ, x − ζ) = + 2γ − 1 (4π)2 4 2 m −m2 ζ 2 + ln + 2γ −5/2 . 4 2 With this expression one calculates ω

∞

ω ∞ ðμν

e2γ m2 m2 : φ : SHP (x) = ln − 1 =: c0,m (4π)2 4 e2γ m2 m4 5 2 ημν =: c2,m ημν ln : φ : SHP (x) = − − (4π)2 4 2 2

and from this one reads of the equations (18) and (19).

Acknowledgements The authors would like to thank D. Buchholz for discussions on local thermal equilibrium states. J. Schlemmer gratefully acknowledges financial support by the International Max Planck Research School (IMPRS).

References [1] M. Alcubierre, The Warp Drive: Hyper-Fast Travel within General Relativity, Class. Quant. Grav. 11 (1994), L73, gr-qc/0009013.

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[2] C. B¨ ar, N. Ginoux and F. Pf¨ affle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Z¨ urich (2007). [3] N. N. Bogolubov, A. A. Logunov, A. I. Oksak and I. T. Todorov, General Principles of Quantum Field Theory, vol. 10 of Mathematical Physics and Applied Mathematics, Kluwer Academic Publishers Group, Dordrecht (1990). [4] M. Bordag, U. Mohideen and V. M. Mostepanenko, New Developments in the Casimir Effect, Phys. Rept. 353 (2001), 1. [5] A. Borde, Geodesic Focusing, Energy Conditions and Singularities, Class. Quant. Grav. 4 (1987), 343. [6] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, Texts and Monographs in Physics, Springer-Verlag, Berlin, second edn. (1997). [7] R. Brunetti, K. Fredenhagen and R. Verch, The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Physics, Commun. Math. Phys. 237 (2003), 31, math-ph/0112041. [8] D. Buchholz, On Hot Bangs and the Arrow of Time in Relativistic Quantum Field Theory, Commun. Math. Phys. 237 (2003), 271, hep-th/0301115. [9] D. Buchholz, I. Ojima and H. Roos, Thermodynamic Properties of Non-Equilibrium States in Quantum Field Theory, Annals Phys. 297 (2002), 219, hep-ph/0105051. [10] D. Buchholz and J. Schlemmer, Local Temperature in Curved Spacetime, Class. Quant. Grav. 24 (2007), F25, gr-qc/0608133. [11] H. Epstein, V. Glaser and A. Jaffe, Nonpositivity of the Energy Density in Quantized Field Theories, Nuovo Cimento 36 (1965), 1016. [12] C. J. Fewster, A General Worldline Quantum Inequality, Class. Quant. Grav. 17 (2000), 1897, gr-qc/9910060. [13] C. J. Fewster, Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory, (2005), math-ph/0502002. [14] C. J. Fewster and L. W. Osterbrink, Quantum Energy Inequalities for the NonMinimally Coupled Scalar Field, J. Phys. A41 (2008), 025402, arXiv:0708.2450 [gr-qc]. [15] C. J. Fewster and M. J. Pfenning, A Quantum Weak Energy Inequality for Spin-One Fields in Curved Spacetime, J. Math. Phys. 44 (2003), 4480, gr-qc/0303106. [16] C. J. Fewster and T. A. Roman, Null Energy Conditions in Quantum Field Theory, Phys. Rev. D67 (2003), 044003, gr-qc/0209036. [17] C. J. Fewster and T. A. Roman, On Wormholes with Arbitrarily Small Quantities of Exotic Matter, Phys. Rev. D72 (2005), 044023, gr-qc/0507013. [18] C. J. Fewster and C. J. Smith, Absolute Quantum Energy Inequalities in Curved Spacetime, (2007), gr-qc/0702056. [19] C. J. Fewster and R. Verch, A Quantum Weak Energy Inequality for Dirac Fields in Curved Spacetime, Commun. Math. Phys. 225 (2002), 331, math-ph/0105027. [20] C. J. Fewster and R. Verch, Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition, Commun. Math. Phys. 240 (2003), 329, math-ph/0203010.

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[21] L. H. Ford, Quantum Coherence Effects and the Second Law of Thermodynamics, Proc. R. Soc. Lond. A 364 (1978), 227. [22] L. H. Ford and T. A. Roman, Quantum Field Theory Constrains Traversable Wormhole Geometries, Phys. Rev. D53 (1996), 5496, gr-qc/9510071. [23] S. A. Fulling, Nonuniqueness of Canonical Field Quantization in Riemannian SpaceTime, Phys. Rev. D7 (1973), 2850. [24] D. Guido and R. Longo, A Converse Hawking – Unruh Effect and dS(2)/CFT Correspondence, Ann. Henri Poinc. 4 (2003), 1169. [25] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press, London (1973). [26] S. Hollands and R. M. Wald, Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime, Commun. Math. Phys. 223 (2001), 289, gr-qc/0103074. [27] R. H¨ ubener, Lokale Gleichgewichtszust¨ ande massiver Bosonen, Diplomarbeit, University of G¨ ottingen (2005). [28] B. S. Kay and R. M. Wald, Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Spacetimes with a Bifurcate Killing Horizon, Phys. Rept. 207 (1991), 49. [29] V. Moretti, Comments on the Stress-Energy Tensor Operator in Curved Spacetime, Commun. Math. Phys. 232 (2003), 189, gr-qc/0109048. [30] M. S. Morris, K. S. Thorne and U. Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition, Phys. Rev. Lett. 61 (1988), 1446. [31] M. J. Pfenning and L. H. Ford, The Unphysical Nature of ‘Warp Drive’, Class. Quant. Grav. 14 (1997), 1743, gr-qc/9702026. [32] T. A. Roman, On the ‘Averaged Weak Energy Condition’ and Penrose’s Singularity Theorem, Phys. Rev. D37 (1988), 546. [33] T. A. Roman, Some Thoughts on Energy Conditions and Wormholes, in M. Novello, S. Perez Bergliaffa, R. Ruffini, eds., Proceedings of the MG10 Meeting, p. 1909, gr-qc/0409090. [34] H. Sahlmann and R. Verch, Passivity and microlocal spectrum condition, Commun. Math. Phys. 214 (2000), 705, math-ph/0002021. [35] F. Serry, D. Walliser and G. J. Maclay, The Role of the Casimir Effect in the Static Deflection and Stiction of Membrane Strips in Microelectromechanical Systems (MEMS), J. Appl. Phys. 84 (1998), 2501. [36] F. J. Tipler, Energy Conditions and Spacetime Singularities, Phys. Rev. D17 (1978), 2521. [37] W. G. Unruh, Notes on Black Hole Evaporation, Phys. Rev. D14 (1976), 870. [38] R. Verch, Local Definiteness, Primarity and Quasiequivalence of Quasifree Hadamard Quantum States in Curved Space-Time, Commun. Math. Phys. 160 (1994), 507. [39] R. M. Wald, Trace Anomaly of a Conformally Invariant Quantum Field in Curved Space-Time, Phys. Rev. D17 (1978), 1477. [40] R. M. Wald, General Relativity, University of Chicago Press, Chicago, IL (1984).

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[41] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, Chicago Lectures in Physics, University of Chicago Press, Chicago, IL (1994). [42] R. M. Wald and U. Yurtsever, General Proof of the Averaged Null Energy Condition for a Massless Scalar Field in Two-dimensional Curved Space-Time, Phys. Rev. D44 (1991), 403. Jan Schlemmer Max-Planck-Inst. f. Mathematics in the Sciences Inselstr. 22 D-04103 Leipzig Germany and Institut f. Theoretische Physik Universit¨ at Leipzig Postfach 100 920 D-04009 Leipzig Germany e-mail: [email protected] Rainer Verch Institut f. Theoretische Physik Universit¨ at Leipzig Postfach 100 920 D-04009 Leipzig Germany e-mail: [email protected] Communicated by Klaus Fredenhagen. Submitted: February 27, 2008. Accepted: May 5, 2008.



Large Deviations for Quantum Markov Semigroups on the 2 × 2-Matrix Algebra Henri Comman Abstract. Let (T∗t ) be a predual quantum Markov semigroup acting on the full 2 × 2-matrix algebra and having an absorbing pure state. We prove that for any initial state ω, the net of orthogonal measures representing the net of states (T∗t (ω)) satisfies a large deviation principle in the pure state space, with a rate function given in terms of the generator, and which does not depend on ω. This implies that (T∗t (ω)) is faithful for all t large enough. Examples arising in weak coupling limit are studied.

1. Introduction The integral representation of states on a unital separable C ∗ -algebra establishes that each state is the barycentre of a measure concentrated on the set of pure states P [15]. There are in general various such representing measures, a class of which is the so called orthogonal measures. In the case of the algebra of compact operators on some separable complex Hilbert space H, when the state ∞ ω is given by the positive trace-one operator ρ, to each diagonal form ρ = i=1 ai |ei ei | ∞ is associated the orthogonal measure μ = i=1 ai δω|ei ei | , where ω|ei ei | is the pure state given by the projection |ei ei | (note that such a measure is uniquely determined by ρ if and only if all eigenvalues are simple). In this paper we study large deviations for nets of orthogonal measures, and in particular when these nets are given by a quantum Markov semigroup acting on the full 2 × 2-matrix algebra M2 . More precisely, let (Tt ) be such a semigroup having an absorbing state ω∞ (i.e., in physical terminology, (Tt ) converges to the equilibrium), and let (T∗t ) denotes its predual semigroup. For each initial state ω, we consider the net of states (T∗t (ω)). Our main result establishes that when ω∞ is pure, the net of orthogonal measures representing (T∗t (ω)) satisfies a large deviation principle in P with powers (1/t); the rate function takes the values {0, η − a, +∞} where a, η are parameters given by the generator of (T∗t ), and in particular it does not depend on ω (Theorem 4). This gives an exponential rate of

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H. Comman


“purification” of the state T∗t (ω) in terms of the generator, in the large deviation sense (i.e. the rate with which the mass assigned to sets not containing the limit state vanishes). This rate is given by the eigenvalues of the operator J ∗ (|e1 e1 |), where e1 is the unit vector determining ω∞ , and J a completely positive operator on M2 appearing in the generator. As a consequence, we obtain an exponential rate of convergence of the semigroup on projections (Corollary 1); this result can be interpreted as a noncommutative large deviation principle as defined by the author in previous works, and the so-called “rate operator” is exactly J ∗ (|e1 e1 |) (see Remark 1). The proof rests essentially on two operator-theoretic ingredients: the first one is the well-known form of the generator of quantum Markov semigroups acting on all bounded operators on H, and having a pure stationary state; the second one is a general result that we prove for these semigroups, when they act on M2 and admit an absorbing state. It establishes that T∗t (ω) is faithful for all states ω = ω∞ and all t large enough (Theorem 3); this property will be, in turn, recovered as a consequence of large deviations. By a compactness argument combined with large deviations techniques, this result allows us to reduce the proof of the general initial state case to the one given by 12 I, where I is the identity. Although we are mainly interested by orthogonal measures arising from dynamics as above, we begin in Section 2 by considering a general family of such measures (H infinite dimensional), for which we give sufficient conditions to have large deviations (Proposition 1). This requires recent results in large deviation theory, and in particular a notion of exponential τ -smoothness, weaker than the usual exponential tightness [6]. We then specialize to the case where H is N -dimensional and the net of states is converging (Proposition 2). The problem of large deviations for orthogonal measures given by the evolution of quantum Markov semigroups is posed even in absence of convergence; this contrasts with the usual approach where it is the distance to some limit state which is measured. As a motivation to study in this way the asymptotic behavior we can mention some models of information dynamics, where algorithms are represented by the semigroups, the input by the initial state and the output by the limit state, which is typically pure [4]; the complexity of the state under the evolution is represented by its support. Indeed, our large deviations describe the rate with which this support decreases. In fact, the method used here for the two-dimensional case give some indications about possible extensions to higher dimensions. Since the main tools for the proofs are the representation of the generator given by Theorem 1 and large deviations techniques, which both are valid in higher dimensions, we could reasonably expect that similar results hold at least in finite dimensions when there exists an absorbing pure state (the infinite dimensional case is more delicate because of the non-compactness of the pure state space). A crucial argument in the proof uses the fact that the operator y in (1) is diagonalizable, which is a particular feature of dimension two; this suggests that in dimension N , some extra conditions on the generator may have to be add. Note that Proposition 2 shows

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981

that a strict N -dimensional analogue of the large deviation principle proved here would imply the convergence of eigenvectors. It is likely that such a large deviation principle implies its noncommutative counterpart, namely an exponential rate of convergence on projections (see Remark 1); in other words, part (b) of Corollary 1 should admit a generalization. Since any noncommutative large deviation principle admits a unique rate operator [5, Proposition 5.2], a natural question arises: Is this rate operator still given by J ∗ (|e1 e1 |)? Or coming back to the classical setting: Are the eigenvalues of J ∗ (|e1 e1 |) still correspond to the finite values of the rate function? On the other hand, the only hypothesis in Theorem 1 being the existence of a stationary pure state, a similar study can be made in more general situations when there is no absorbing state. For instance, when there is a set S∞ of pure states such that for each initial state ω, T∗t (ω) converges to some element of S∞ ; we should then obtain various rate functions indexed by the elements of S∞ . This kind of semigroups belongs to the class of the so-called “generic” semigroups, which arise in an extended version of the weak coupling limit [1]. A class of such semigroups admitting an absorbing pure state (i.e. S∞ contains only one element) is studied in Section 4. 1.1. Notations and background material 1.1.1. Quantum Markov semigroups. Let H be a complex separable Hilbert space, and let K(H) be the set of compact operators acting on H. Let P be the pure state space of K(H) provided with the weak∗ topology, and note that P is completely regular Hausdorff, and compact when H is finite dimensional. For each x ∈ K(H), we denote by x ˆ the map defined on P by x ˆ(ω) = ω(x), and note that x ˆ is continuous. The full N × N -matrix algebra is denoted by MN . By convention, for any N self-adjoint ρ ∈ MN , the expression ρ = i=1 ai |ei ei | has to be considered as a formal sum (i.e., ai can be zero), which means that the set {ei : 1 ≤ i ≤ N } is an orthonormal basis diagonalizing ρ, where each ei is an eigenvector corresponding to the eigenvalue ai . given by the positive trace-one operator ρ. When ρ Let ωρ denotes the state ∞ admits a diagonal form ρ = i=1 ai |ei ei |, we will consider the measure μ = ∞ a δ , where the sum has to be understood in the sense of the weak i ω |ei ei | i=1 topology for Borel measures on P . It is easy to see that μ is an orthogonal measure representing ωρ , in the sense of the theory of integral representation of states [15]. Clearly, when H has dimension 2, each state is represented by a unique orthogonal measure. We shall use the following lemma whose proof is straightforward. Lemma 1. For any net (ωt ) of states on MN and any state ω on MN , the following statements are equivalent. (i) lim ωt = ω; (ii) ω (resp. ωt ) is represented by an orthogonal measure μ (resp. μt ) such that lim μt = μ; (iii) lim μt = μ for all orthogonal measures μt and μ representing ωt and ω, respectively.

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By a quantum Markov semigroup acting on M2 , we mean a w∗ -continuous one-parameter semigroup (Tt )t≥0 of completely positive linear maps on M2 preserving the identity I. We denote by (T∗t ) the predual semigroup, and by (T∗t ) the associated semigroup acting on M2 , obtained by identifying ωρ and ρ. In other words, (T∗t ) is a strongly continuous semigroup of completely positive contractions on M2 defined by the relation ωT∗t (ρ) = T∗t (ωρ ) for all states ωρ and extended by linearity. A state ω is stationary if T∗t (ω) = ω for all t ≥ 0, and a state ω∞ is said to be absorbing if lim T∗t (ω) = ω∞ for all states ω; note that an absorbing state is stationary. In the following theorem, we collect various results that will be used in the sequel. They appear in [8], and are valid more generally for uniformly continuous quantum Markov semigroups acting on the algebra of all bounded operators on H infinite dimensional. Theorem 1. Let (Tt ) be a quantum Markov semigroup on M2 having a stationary pure state given by the projection |ee|. Then there exist y, z1 , z2 in M2 such that the following hold. (a) The generator L˜∗ of (T˜∗t ) has the form L˜∗ (ρ) = yρ + ρy ∗ + J (ρ) , (1) 2 where J is defined on M2 by J (ρ) = i=1 zi ρzi∗ . (b) ye = y ∗ e = z1 e = z2 e = 0. (c) y is the generator of a one-parameter semigroup of contractions (Ct )t≥0 on H, and the semigroup (St )t≥0 on M2 defined by St (ρ) = Ct ρCt∗ , satisfies for all t ≥ 0, (2) ∀ρ ≥ 0 , 0 ≤ St (ρ) ≤ T˜∗t (ρ) ∀ρ ∈ M2 ,

and

T˜∗t = St +

t

T˜∗t−s J Ss ds .

(3)

0

1.1.2. Large deviations. We recall now some large deviations results for a net (μt )t≥0 of Borel probability measures on a completely regular Hausdorff topological space X. For each [−∞, +∞[-valued Borel measurable function h on X, we put 1/t 1/t μt (eth ) = ( X eth(x) μt (dx))1/t , and define Λ(h) = log lim μt (eth ) when the limit exists. By definition, (μt ) satisfies a large deviation principle with powers (1/t) if there exists a [0, +∞]-valued lower semi-continuous function J on X such that lim sup μt (F ) ≤ sup e−J(x)

for all closed

sup e−J(x) ≤ lim inf μt (G)

for all open G ⊂ X ;

1/t

x∈F

F ⊂X

and 1/t

x∈G

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983

1/t

J is called the rate function for (μt ), and for each Borel set A ⊂ X such that 1/t supx∈Int(A) e−J(x) = supx∈A e−J(x) , the limit lim μt (A) exists and satisfies lim μt (A) = sup e−J(x) 1/t

(4)

x∈A

(such a set A is called a J-continuity set). The following notions have been introduced in [6]. Definition 1. The net (μt ) is exponentially τ -smooth if for all open covers {Gi : i ∈ I} of X and for all ε > 0, there exists a finite set {Gi1 , . . . , GiN } ⊂ {Gi : i ∈ I} such that ⎛ ⎞ 1/t Gij ⎠ < ε . lim sup μt ⎝X\ 1≤j≤N

Definition 2. A class A of [−∞, +∞[-valued continuous functions on X is said to be approximating if for each x ∈ X, each open set G containing x, each real s > 0, and each real r > 0, A contains some function h satisfying e−r 1{x} ≤ eh ≤ 1G ∨ e−s . Under exponential τ -smoothness, the existence of Λ( · ) on some approximating class A is sufficient to get large deviations, with a rate function which can be expressed in terms of A. However, with slight extra conditions on A, this expression is substantially simplified [6, Corollary 2 and 4]. This variant is stated in the following theorem, and will be used in the proof of Proposition 1. Theorem 2. Let A be an approximating class of bounded above functions such that for each x ∈ X, each open set G containing x, and each real s > 0, A contains some function h satisfying 1{x} ≤ eh ≤ 1G ∨ e−s . If (μt ) is exponentially τ -smooth and if Λ(h) exists for all h ∈ A, then (μt ) satisfies a large deviation principle with powers (1/t) and rate function − Λ(h) f or all x ∈ X . J(x) = sup h∈A,h(x)=0

2. Large deviations for orthogonal measures In this section, we first give sufficient conditions for a net of orthogonal measures to satisfy a large deviation principle (Proposition 1). Next, we specialize to the case where H is finite dimensional and the net of states is converging (Proposition 2). Proposition 1. Let (ωt )t≥0 be a net ∞of states on K(H), where each ωt is represented by the orthogonal measure μt = i=0 ai,t δω|ei,t ei,t | , and assume that the following conditions hold:

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∞ (i) limN lim sup( i=N ai,t )1/t = 0. (ii) The net (ω|ei,t ei,t | )t≥0 converges in P , for all i ∈ N. 1/t

(iii) lim ai,t exists for all i ∈ N. Then, (μt ) satisfies a large deviation principle with powers (1/t) and rate function − Λ(h) for all ω ∈ P , (5) J(ω) = sup h∈A,h(ω)=0

where A = ω∈P Aω with Aω the set of all finite infima of elements in {−|ˆ x− x ˆ(ω)| : x ∈ K(H)}. Proof. By (ii), for each i ∈ N there exists ei ∈ H such that lim ω|ei,t ei,t | = ω|ei ei | . Let G0 be an open cover of P , and let for each i ∈ N some Gi ∈ G0 containing ω|ei ei | . By (i), for each ε > 0 there exists Nε ∈ N such that ∞ lim sup( i=Nε +1 ai,t )1/t < ε. Since for each t ≥ 0, ε μt (P \∪N i=0 Gi ) =

Nε

∞

ε ai,t δω|ei,t ei,t | (P \∪N i=0 Gi )+

i=0

ε ai,t δω|ei,t ei,t | (P \∪N i=0 Gi ) ,

i=Nε +1

with Gi containing ω|ei,t ei,t | for all i ∈ {0, . . . , Nε } and all t large enough, we get 1/t

ε lim sup μt (P \ ∪N i=0 Gi ) < ε .

This shows that (μt ) is exponentially τ -smooth since G0 is arbitrary. For each ω ∈ P , each open set G ⊂ P containing ω, and each s > 0, by definition of the w∗ -topology, there exists a finite set Kω,G,s ⊂ K(H) such that − supx∈K

1{ω} ≤ e

ω,G,s

|ˆ x−ˆ x(ω)|

≤ 1G ∨ e−s ,

hence A is an approximating class for P satisfying the hypothesis of Theorem 2. For any ω ∈ P and any finite subset K ⊂ K(H), there is some N ∈ N such that N 1/t 1/t −t supx∈K |ˆ x−ˆ x(ω)| lim sup μt (e ) = lim sup ai,t e−t supx∈K |ei,t ,xei,t −ω(x)| = sup 1≤i≤N

i=1 1/t lim ai,t e− supx∈K |ei,t ,xei,t −ω(x)|

(6)

= lim μt (e−t supx∈K |ˆx−ˆx(ω)| ) , 1/t

where the first equality follows from (i) and the second from (ii) and (iii). Since 1/t ω and K are arbitrary, lim μt (eth ) exists and so Λ(h) exists for all h ∈ A. By Theorem 2, (μt ) satisfies a large deviation principle with powers (1/t) and rate function (5). Lemma 2. Let (ρt )t≥0 be a net of hermitian matrices in MN converging in norm N to a hermitian matrix ρ = i=1 ai |ei ei | with a1 > · · · > aN . Then for each t large N enough ρt admits a diagonal form ρt = i=1 ai,t |ei,t ei,t | such that lim ai,t = ai and lim |ei,t ei,t | = |ei ei | (in norm) for all i ∈ {1, . . . , N }.

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Proof. Let mi,t denote the multiplicity of ai,t (1 ≤ i ≤ N ). Since ρt converges in trace-norm, the assertion concerning the eigenvalues follows from the well-known N inequality i=1 |ai − ai,t | ≤ ||ρ − ρt ||1 where || · ||1 denotes the trace norm [14]. Let −1 ε < 12 minN i=1 {|ai −ai+1 |}. Since (ρt ) is uniformly converging, it converges in norm ρt ρ ρ converges uniformly to E]a = E{a resolvent sense, so that E]a i −ε,ai +ε[ i −ε,ai +ε[ i} for each i ∈ {1, . . . , N }. For each ε ≤ ε/4 we have ρ ρ ρt E{a = E]a = lim E]a 1} 1 −ε,a1 +ε[ 1 −ε,a1 +ε[ t

N 1 ρt E]a j,t −ε ,aj,t +ε [ m j,t j=1

= lim

N 1 ρt E]a 1 −ε,a1 +ε[∩]aj,t −ε ,aj,t +ε [ m j,t j=1

= lim

m1,t 1 ρt E m1,t j=1 ]a1 −ε,a1 +ε[∩]aj,t −ε ,aj,t +ε [

= lim

m1,t 1 ρt ρt ρt E . = lim E]a = lim E{a 1,t −ε ,a1,t +ε [ 1,t } t t m1,t j=1 ]aj,t −ε ,aj,t +ε [

t

t

t

ρt ρ = E{a for all i ∈ {2, . . . , N }, which proves the Similarly we get limt E{a i,t } i} lemma.

Part (a) of the following proposition shows that when (ωρt ) converges to some state ωρ∞ , and under some extra condition on eigenvectors, large deviations for a suitable representing net of orthogonal measures are determined by the asymptotic behavior of the eigenvalues of ρt . The interesting case occurs when ωρ∞ is not faithful, otherwise the rate function (7) is trivial since r = N ; it gives then the rate with which the support of ρt gets smaller. Note that by Lemma 2 the hypotheses are always satisfied in dimension 2 when ωρ∞ = 12 I, and in particular when ω∞ is pure. Although this will not be used in the sequel, it is worth noticing that (assuming lim ωρt = ρ∞ ) a large deviation principle with rate function (7) implies the convergence of some eigenvectors, as establishes (b). Proposition 2. Let (ωρt )t≥0 be a net of states on MN , and assume that (ωρt ) w∗ converges to some state ωρ∞ . Let a1,t ≥ · · · ≥ aN,t be the eigenvalues of ρt , and a1 ≥ · · · ≥ ar be the non-zero eigenvalues of ρ∞ counted with multiplicity (ai = 0 for i > r), and consider the following property (Pi ) for any i ∈ {1, . . . , N }. (Pi ) For each t large enough ai,t (resp. ai ) admits an eigenvector ei,t (resp. ei ) such that lim ω|ei,t ei,t | = ω|ei ei | . Then, (a) If (Pi ) holds for all i ∈ {1, . . . , N }, then the associated net (μt ) of orthogonal measures satisfies a large deviation principle with powers (1/t) if and only if lim 1t log ai,t exists for all i ∈ {r + 1, . . . , N }. In this case, the rate function

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is

⎧ ⎨ 0 − lim 1t log ai,t J(ω|ee| ) = ⎩ +∞

if |ee| ∈ {|ei ei | : 1 ≤ i ≤ r} if |ee| = |ei ei |, r + 1 ≤ i ≤ N elsewhere .

(7)

In particular, for each i ∈ {1, . . . , N } and for each open set G ⊂ P containing ω|ei ei | such that G ∩ {ω|ej ej | : 1 ≤ j ≤ N, j = i} = ∅, lim 1t log μt (G) exists and satisfies 1 1 lim log μt (G) = lim log ai,t . t t (b) Conversely, if (ωρt ) is represented by a net of orthogonal measures (μt ) satisfying a large deviation principle with rate function (7) (where ρ∞ = N i=1 ai |ei ei |), then (Pi ) holds for all i where J(ω|ei ei | ) < +∞. Proof. Assume that (Pi ) holds for all i ∈ {1, . . . , N }. The convergence of states implies lim ||ρt − ρ∞ || = 0 so that lim ai,t = ai for all i ∈ {1, . . . , N }, and in 1/t 1/t particular lim ai,t = 1 when 1 ≤ i ≤ r. Assume that lim ai,t exists for all i ∈ {r + 1, . . . , N }. All the hypotheses of Proposition 1 hold, and the large deviations follow for (μt ), with rate function given by (5). Let ω|ee| ∈ P . For each h ∈ A with h(ω|ee| ) = 0 there exist ω ∈ P and a finite set K ⊂ MN such that h = inf x∈K {−|ˆ x−x ˆ(ω )|} and ω|ee| (x) = ω (x) for all x ∈ K. We put hK,ω = h, and 1 ci = lim t log ai,t for all i ∈ {1, . . . , N }. By (6) we have Λ(hK,ω ) = sup ci − sup lim |ei,t , xei,t − ω (x)| 1≤i≤N

= sup 1≤i≤N

x∈K

ci − sup |ei , xei − e, xe| ,

(8)

x∈K

so that Λ(h) = 0 for all h ∈ A when |ee| ∈ {|ei ei | : 1 ≤ i ≤ r} (since in this case ci = 0), hence J(ω|ee| ) = 0 by (5). Assume now that |ee| ∈ {|ei ei | : 1 ≤ i ≤ N }. Let x ∈ MN such that ei , xei = e, xe for all i ∈ {1, . . . , N }, and put δ = inf 1≤i≤N |ei , xei − e, xe|. For all M > 0 there exists rM > 0 such that inf |ei , rM xei − e, rM xe| > M , and since by (8) −Λ(h{rM x},ω ) ≥ inf |ei , rM xei − e, rM xe| , 1≤i≤N

we get by letting M → +∞, +∞ = sup − Λ(h{rM x},ω ) ≤ M

sup

− Λ(h) ,

h∈A,h(ω|ee| )=0

that is J(ω|ee| ) = +∞. Assume now that |ee| = |ei ei | for some i ∈ {r + 1, . . . , N }. By the extended version of Varadhan’ theorem for [−∞, +∞[valued bounded above functions (see Corollary 3.2 of [7]), large deviations im1/t t 1/t t ply that lim μt ( x ) exists for all positive x ∈ MN , and satisfies lim μt ( x) = −J(ω) supω∈P x (ω)e . Taking x = |ei ei | yields

Vol. 9 (2008)


1/t

lim μt

t e | |e = i i

sup

987

|u, ei |2 e−J(ω|uu| ) ,

u∈H,u=1

and by the preceding cases we see that the only possible non-zero value of the map |uu| → |u, ei |2 e−J(ω|uu| ) is obtained at the point |ei ei |, so that t 1/t lim μt |e = e−J(ω|ei ei | ) . i ei | Since limei , ej,t = 0 for all j = i and limei , ei,t = 1, it follows that t 1/t 1/t 1/t lim μt e | |e = max lim aj,t |ei , ej,t |2 = lim ai,t i i 1≤j≤N

hence J(ω|ei ei | ) = − lim log ai,t . We have proved the “if” part of the first assertion of (a), and the second assertion. If (μt ) satisfies a large deviation principle t 1/t with powers (1/t), then lim μt (|e i ei | ) exists and t 1/t 1/t 1/t lim μt (9) |ei ei | = max lim sup aj,t |ei , ej,t |2 = lim ai,t 1 t

1≤j≤N

for all i ∈ {r + 1, . . . , N }; this proves the “only” part of the first assertion of (a). The last assertion follows from (4) since (7) implies that any open set G ⊂ P containing |ei ei | with G ∩ {|ej ej | : 1 ≤ j ≤ N, j = i} = ∅, is a J-continuity set. The proof of (a) is complete. Assume that the hypotheses of (b) hold. The extended version of Varadhan’s theorem together with (7) yield for each i ∈ {1, . . . , N }, t 1/t 1/t lim μt |ei ei | = max lim sup aj,t |ei , ej,t |2 1≤j≤N (10) 1/t |u, ei |2 e−J(ω|uu| ) = lim ai,t . = sup u∈H,u=1

1/t

Let s be the greatest integer less than N such that lim ai,t > 0 for all i ≤ s, 1/t

1/t

and note that s ≥ r ≥ 1. By (10) we have lim a1,t = lim aj,t |e1 , ej,t |2 for some j ∈ {1, . . . , N }. If j = 1, then clearly lim |e1 , e1,t | = 1 and the conclusion holds. If s = 1 the proof is complete so let us assume that s > 1. Assume that j > 1. Since 1/t 1/t 1/t 1/t lim a1,t ≥ lim aj,t we have necessarily lim a1,t = lim aj,t and lim |e1 , ej,t | = 1. With the new labeling of the eigenvalues of ρt obtained interchanging (1, t) and (j, t), the conclusion holds for the first eigenvector. Suppose that the result holds for all i − 1 < s, and assume that 1/t

1/t

lim ai,t = lim aj,t |ei , ej,t |2 for some j ∈ {1, . . . , N }. Since lim |ej,t ej,t | = |ej ej | for all j ≤ i − 1 and 1/t lim ai,t > 0, it follows that j ≥ i, and the proof is the same as for the case i = 1. We conclude with a finite recurrence (note that at each step the new labeling 1/t 1/t preserves the inequalities lim ai,t ≥ lim aj,t when j ≥ i).

988

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Proposition 3. Let (ωt )t≥0 be a net of states on MN . If (ωt ) is represented by a net of orthogonal measures which satisfies a large deviation principle with a rate function vanishing at a unique point ω|ee| , then lim ωt = ω|ee| . Proof. Let μt = 1≤i≤N ai,t δω|ei,t ei,t | be the orthogonal measure representing ωt as above, let ε > 0, let x ∈ MN with ||x|| = 1, and let G be the open neighborhood of ω|ee| defined by {ω ∈ P : |ω(x) − ω|ee| (x)| < ε}. Then, x) = ai,t ω|ei,t ei,t | (x) μt (ˆ 1≤i≤N,ω|ei,t ei,t | ∈G

+

(11) ai,t ω|ei,t ei,t | (x)

1≤i≤N,ω|ei,t ei,t | ∈G

and note that

ai,t ω|ei,t ei,t | (x) ≤ μt (P \G) .

(12)


We have

ω|ee| (x)


≤

ai,t − ε

ai,t


ai,t ω|ei,t ei,t | (x)


≤

ai,t (ω|ee| (x) + ε)


≤ ω|ee| (x) + ε . Let ε → 0 and get ω|ee| (x)

ai,t


≤

ai,t ω|ei,t ei,t | (x) ≤ ω|ee| (x) .

(13)


The large deviations and the hypothesis on the rate function imply that lim μt (P \G) = 0 (exponentially fast) hence lim ai,t = lim μt (G) = 1 , t

and (13) yields lim t


t

ai,t ω|ei,t ei,t | (x) = ω|ee| (x) .

(14)


x) = ω|ee| (x), which proves the proposition by Then (11), (12), (14) give lim μt (ˆ Lemma 1.

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989

3. The case of states arising from quantum Markov semigroups on M2 In this section, we first prove a general property of quantum Markov semigroups on M2 having an absorbing state (Theorem 3). Looking the state space of M2 as the unit ball in R3 , it says that when the absorbing state ω∞ is on the sphere (i.e., pure), T∗t (ω) approaches ω∞ remaining inside the open unit ball, for all states ω = ω∞ . This property is crucial for the proof of the large deviations result (Theorem 4). Theorem 3. Let (Tt ) be a quantum Markov semigroup acting on M2 , and having an absorbing state ω∞ . Then, for each state ω = ω∞ , T∗t (ω) is faithful for all t large enough. Proof. Clearly, the conclusion holds when ω∞ is faithful, so that let us assume that ω∞ is pure given by some projection |e1 e1 |, and let e2 be a unit vector orthogonal to e1 . First assume that ω is a pure state given by a unit vector e = α1 e1 +α2 e2 with α2 = 0, and suppose the conclusion does not hold. There exists a sequence (tn ) such that T˜∗tn (|ee|) is a rank-one projection for all n ∈ N. Let (St ), J , y, z1 , z2 as in Theorem 1, and note that ye1 = y ∗ e1 = 0 implies that y is normal and diagonalizes in the basis {e1 , e2 }, hence ye2 = γe2 for some γ ∈ C. Put −η = γ + γ, and note that −η < 0 since −η is the least eigenvalue of y + y ∗ . For each n ∈ N, let un be a unit vector such that un , T˜∗tn (|ee|)un = 0, and get by (2) and (3) 2 tn un , T˜∗tn −s (|zi esy ezi esy e|)un ds = 0 , 0

i=1

and so for i ∈ {1, 2} and for each s ∈ [0, tn ] un , T˜∗tn −s (|zi esy ezi esy e|)un = 0 . Since |esy eesy e| = |α1 |2 |e1 e1 | + |α2 |2 e−sη |e2 e2 | + α1 α2 esγ |e1 e2 | + α2 α1 esγ |e2 e1 | we get for i ∈ {1, 2} |zi esy ezi esy e| = |α2 |2 e−sη |zi e2 zi e2 | hence ∀n ∈ N ,

∀s ∈ [0, tn ]

un , T˜∗s |zi e2 zi e2 | un = 0 .

Put zi e2 = α1,i e1 + α2,i e2 for i ∈ {1, 2} and get T˜∗s |zi e2 zi e2 | = |α1,i |2 |e1 e1 | + |α2,i |2 T˜∗s |e2 e2 | + α2,i α1,i T˜∗s |e1 e2 | + α1,i α2,i T˜∗s |e2 e1 | .

(15)

(16)

990

H. Comman


By (3) we have 2 T˜∗s |e2 e2 | = e−sη |e2 e2 | +

=e

|e2 e2 | +

T˜∗s−r zi |erγ e2 erγ e2 |zi∗ dr

0

i=1

−sη

s

2 s

−rη

e

T˜∗s−r |zi e2 zi e2 | dr .

(17)

0

i=1

Then, 2 T˜∗s |e1 e2 | = Ss |e1 e2 | +

T˜∗s−r zi |ery e1 ery e2 |zi∗ dr

s

T˜∗s−r |zi e1 zi erγ e2 |

0

i=1

= Ss |e1 e2 | +

s

2

(18)

0

i=1

= Ss |e1 e2 | = esγ |e1 e2 | . In the same way we get

T˜∗s |e2 e1 | = esγ |e2 e1 | .

(19)

Then (16)–(19) give for each n ∈ N, un , T˜∗s |zi e2 zi e2 | un = |α1,i |2 |e1 , un |2 + +2Re α1,i α2,i esγ e1 , un un , e2 (20) + |α2,i |2 un , T˜∗s |e2 e2 | un 2 2 sγ = |α1,i | |e1 , un | + 2Re α1,i α2,i e e1 , un un , e2 + |α2,i |2 e−sη (|e2 , un |2 2 s + |α2,i |2 e−rη un , T˜∗s−r |zj e2 zj e2 | un dr . j=1

0

By (15) and (20) we obtain for each n ∈ N and each s ∈ [0, tn ] |α1,i |2 |e1 ,un |2 +2Re α1,i α2,i esγ e1 ,un un , e2 +|α2,i |2 e−sη |e2 , un |2 = 0 . (21) Taking the limit when n → +∞ in (21) with s = tn yields lim |α1,i |2 |e1 , un |2 = 0 , n

(22)

and (15) with s = 0 gives ∀n ∈ N ,

α1,i e1 , un + α2,i e2 , un = 0 .

(23)

First assume α2,i = 0. If α1,i = 0, then |un un | = |e2 e2 | for all n ∈ N by (23), and (2) implies ∀n ∈ N , e2 , Stn |ee| e2 = etn γ e2 , |ee|etn γ e2 = |α2 |2 e−tn η = 0 which gives the contradiction since α2 = 0. It follows that α1,i = 0, and so zi = 0. Assume now α2,i = 0. It is easy to see that (22) and (23) imply lime2 , un = 0 and

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991

α1,i = 0, hence |un un | = |e1 e1 | for all n ∈ N, which gives the contradiction since lime1 , T˜∗tn (|ee|)e1 = 1. We obtain finally that (22) and (23) imply z1 = z2 = 0, that is T˜∗t = St for all t ≥ 0. Since min σ(T˜∗tn (|ee|)) = 0 and tr(T˜∗tn (|ee|)) = 1, we have |α1 |2 |u, e1 |2 + |α2 |2 e−tn η |u, e2 |2 sup 1 = Stn |ee| = u∈H,||u||=1

+ 2Re α1 α2 etn γ e1 , uu, e2 , and letting tn → +∞ it follows that 1=

sup

|α1 |2 |u, e1 |2 ,

u∈H,||u||=1

which implies |α1 | = 1 and the contradiction since α2 = 0. The theorem is proved when ω is pure. Assume now that ω is given by some strictly positive operator ρ, ∗ i.e. cI ≤ ρ ≤ I for some c > 0. Since St (cI) = cet(y+y ) , (2) gives ce−tη = min σ St (cI) ≤ min σ T˜∗t (ρ) , which shows that T∗t (ω) is faithful for all t ≥ 0.

The following theorem is our large deviation result. It shows that when (Tt ) has an absorbing pure state ω∞ , and for any initial state ωρ = ω∞ , the net (T∗t (ωρ )) converges exponentially fast (in the large deviation sense expressed by (24)) toward ω∞ with rate η − a, where η, a are parameters given by the generator. Note that (24) implies a2,t,ρ > 0 for all t large enough, so that Theorem 4 contains Theorem 3. Theorem 4. Let (Tt ) be a quantum Markov semigroup on M2 , and having an absorbing pure state ω|e1 e1 | . Let y, z1 , z2 be the parameters of the generator L˜∗ as in Theorem 1, and let e2 be a unit vector orthogonal to e1 . Then for each state ωρ = ω|e1 e1 | , the net (μt,ρ ) of orthogonal measures representing (T∗t (ωρ )) satisfies a large deviation principle with powers (1/t) and rate function ⎧ if |ee| = |e1 e1 | ⎨ 0 η−a if |ee| = |e2 e2 | J(ω|ee| ) = ⎩ +∞ elsewhere , where a = |z1 e2 , e2 |2 + |z2 e2 , e2 |2 and η is the greatest eigenvalue of −(y + y ∗ ); moreover, η − a > 0. In particular, the rate function does not depend on the initial state ωρ , and for each open sets G ⊂ P containing |e2 e2 | such that |e1 e1 | ∈ G, lim 1t log μt,ρ (G) exists and satisfies 1 1 log μt,ρ (G) = lim log a2,t,ρ = a − η , t t is the least eigenvalue of T˜∗t (ρ). lim

where a2,t,ρ

(24)

992

H. Comman


Proof. Let ωρ = ω|e1 e1 | be a state, and let T˜∗t (ρ) have the diagonal form T˜∗t (ρ) = 2 i=1 ai,t,ρ |ei,t,ρ ei,t,ρ | as in Lemma 2. By Proposition 2 we only have to check 1/t that lim a2,t,ρ exists and equals ea−η . Since T˜∗t (I) = T˜∗t |e1 e1 | + T˜∗t |e2 e2 | = |e1 e1 | + T˜∗t |e2 e2 | , (25) by (17) we have (26) min σ T˜∗t (I) ≤ e2 , T˜∗t (I)e2 = e2 , T˜∗t |e2 e2 | e2 2 t = e−tη + e−sη e2 , T˜∗t−s |zi e2 zi e2 | e2 i=1

0

and by using the first equality of (20) (with e2 in place of un and t − s in place of s), the last above equality becomes t e2 , T˜∗t |e2 e2 | e2 = e−tη + a e−sη e2 , T˜∗t−s |e2 e2 | e2 0

with a = |α2,1 | + |α2,2 | . By putting u(t) = e2 , T˜∗t (|e2 e2 |)e2 for all t ≥ 0, and applying the Laplace transform, it is easy to see that the equation t −tη +a e−sη u(t − s)ds u(t) = e 2

2

0

has the unique solution ∀t ≥ 0 , u(t) = et(a−η) , with a − η < 0 (since lime2 , T˜∗t (|e2 e2 |)e2 = 0). It follows from (26) that 1/t 1/t lim sup min σ T˜∗t (I) ≤ lim e2 , T˜∗t |e2 e2 | e2 = ea−η < 1 .

(27)

(28)

For each t ≥ 0 let ut be a unit vector such that min σ T˜∗t (I) = ut , T˜∗t (I)ut . By (25) we have

min σ T˜∗t (I) = |ut , e1 |2 + ut , T˜∗t |e2 e2 | ut .

Let (utj ) be a subnet of (ut ) such that 1/t 1/tj lim inf min σ T˜∗t (I) = lim min σ T˜∗tj (I) and get 1/t lim inf min σ T˜∗t (I)

1/tj = lim sup |utj , e1 |2/tj ∨ lim sup utj , T˜∗tj |e2 e2 | utj .

(29)

Let (utk ) be a subnet of (utj ). Then |utk utk | has a subnet |utl utl | converging to some projection |uu|. If |uu| = |e2 e2 |, then lim inf min σ(T˜∗t (I))1/t = 1, which contradicts (28). Therefore, |uu| = |e2 e2 | and since the subnet |utk utk |

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993

is arbitrary, |ut ut | converges to |e2 e2 |. Put ut = b1,t e1 + b2,t e2 for all t ≥ 0, and get utj , T˜∗tj |e2 e2 | utj = |b1,tj |2 e1 , T˜∗tj |e2 e2 | e1 + |b2,tj |2 e2 , T˜∗tj |e2 e2 | e2 (30) + b1,tj b2,tj e1 , T˜∗tj |e2 e2 | e2 + b2,tj b1,tj e2 , T˜∗tj |e2 e2 | e1 . Then (17) combined with (16), (18), (19) yield t ∀t ≥ 0 , e1 , T˜∗t |e2 e2 | e2 = a e−sη e1 , T˜∗t−s |e2 e2 | e2 ds , 0

and by an application of Laplace transform we get as unique solution e1 , T˜∗t |e2 e2 | e2 = 0 for all t ≥ 0 .

(31)

Similar calculations yield e2 , T˜∗t |e2 e2 | e1 = 0 for all t ≥ 0 .

(32)

It follows from (27) and (30) that utj , T˜∗tj |e2 e2 | utj ≥ |b2,tj |2 etj (a−η) . Since lim |utj utj | = |e2 e2 |, we have lim |b2,tj | = 1 and by (29) 1/t 1/tj lim inf min σ T˜∗t (I) ≥ lim sup utj , T˜∗tj |e2 e2 | utj ≥ ea−η .

(33)

Then (28) and (33) yield 1/t 1/t lim min σ T˜∗t (I) = lim e2 , T˜∗t |e2 e2 | e2 = e(a−η) .

(34)

If ρ is strictly positive, then cI ≤ ρ ≤ I and ! ! ! c min σ T˜∗t (I) ≤ min σ T˜∗t (ρ) ≤ min σ T˜∗t (I) for some c > 0, hence !1/t 1/t 1/t lim a2,t,ρ = lim min σ T˜∗t (ρ) = lim min σ T˜∗t (I) = ea−η .

(35)

Assume now that ρ = |ee| for some unit vector e = α1 e1 + α2 e2 with α2 = 0. Let 1/t 1/t (a2,tjj,ρ ) be a subsequence of (a2,t,ρ ), and consider the corresponding subsequence (μtj ,ρ ) of (μt,ρ ). Note that by Theorem 3, a2,tj ,ρ > 0 for all j large enough. By a well-known compactness result in large deviation theory (see Lemma 4.1.23 of [10], or Corollary 5 of [6] for a general version), (μtj ,ρ ) has a subsequence (μtjk ,ρ )

994

H. Comman

Ann. Henri Poincaré 1/tj

satisfying a large deviation principle, so that lim a2,tj k,ρ exists by Proposition 2. k

1/tj

Put e−l = lim a2,tj k,ρ , and get for each k ∈ N, k

− l = lim

log min σ T˜∗tjk −tj

1

+tj

(ρ)

tjk − tjk + tjk ! tjk − tjk 1 = lim · log min σ T˜∗tjk −tj T˜∗tj (ρ) k k k tjk − tj + tj tjk − tj k k k ! 1 log min σ T˜∗tjk −tj T˜∗tj (ρ) . lim k k k tjk − tj k k

k

k

Since limk (tjk − tjk ) = +∞, the sequence (T˜∗tjk −tj (T˜∗tj (ρ)))k∈N,tjk ≥tj is a k k k subsequence of (T˜∗t (T˜∗tj (ρ))t>0 . Since T˜∗tj (ρ) is strictly positive for k large k k 1/t = ea−η hence enough, (35) (with T˜∗t (ρ) in place of ρ) implies lim a 2,t,T˜∗tj

j k

−l = lim = lim

tjk tjk

k

(ρ)

! 1 log min σ T˜∗tjk −tj T˜∗tj (ρ) k k − tjk 1 log a2,tj −tj ,T˜∗t (ρ) = a − η . j k k − tjk k 1/t

It follows that each subsequence of (a2,t,ρ ) has a subsequence converging to ea−η , 1/t

hence lim a2,t,ρ = ea−η .

The following result shows that the large deviation principle as well as the exponential rate of convergence on projections are given by the eigenvalues of J ∗ (|e1 e1 |). Corollary 1. Let (Tt ) be a quantum Markov semigroup on M2 having an absorbing pure state ω|e1 e1 | , let e2 be a unit vector orthogonal to e1 , and let J be the operator on M2 appearing in the generator (1). Then the following conclusions hold. (a) J ∗ (|e1 e1 |) = (η − a)|e2 e2 | with η, a as in Theorem 4; (b) For each state ω = ω|e1 e1 | and each projection p ∈ M2 \{0} we have " 1 a−η if p = |e2 e2 | lim log ω Tt (p) = 0 otherwise . t Proof. Differentiating (27), (31), (32) and taking the value at t = 0 yields respectively e2 , L˜∗ (|e2 e2 |)e2 = a − η, e1 , L˜∗ (|e2 e2 |)e2 = 0, e2 , L˜∗ (|e2 e2 |)e1 = 0. By (27) and the preservation of the trace we have e1 , T∗t (|e2 e2 |)e1 = 1−et(a−η) hence e1 , L˜∗ (|e2 e2 |)e1 = η − a. Then (a) follows by noting that J (I) = J |e2 e2 | = L˜∗ |e2 e2 | + η|e2 e2 | . Clearly the conclusion of (b) holds when p is two dimensional, so let us assume that p = |ee| for some unit vector e. Since ω|e1 e1 | is absorbing we have ! lim ω Tt |ee| = |e, e1 |2

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995

for all states ω. When |ee| = |e2 e2 | the above limit is strictly positive hence lim ω(Tt (|ee|))1/t = 1 and the conclusion holds. Assume that |ee| = |e2 e2 | and write ω = ωρ . We have min σ T˜∗t (I) ≤ e2 , T˜∗t (I)e2 = e2 , T˜∗t |e2 e2 | e2 hence

1/t 1/t lim min σ T˜∗t (I) = lim e2 , T˜∗t (I)e2 = ea−η by (34), and finally 1/t !1/t lim ωρ Tt |e2 e2 | = lim e2 , T˜∗t (ρ)e2 = ea−η for all ρ strictly positive since in this case cI ≤ ρ ≤ I for some c > 0. Assume now that ρ = |f f | for some unit vector f , and let h be a unit vector orthogonal to f . We have 1/t lim e2 , T˜∗t (I)e2 " # 1/t 1/t ˜ ˜ = max lim sup e2 , T∗t |f f | e2 , lim sup e2 , T∗t |hh| e2 = ea−η hence

!1/t 1/t ≤ lim inf e2 , T˜∗t |f f | e2 lim inf min σ T˜∗t |f f | 1/t ≤ lim sup e2 , T˜∗t |f f | e2 ≤ ea−η ,

and since lim min σ(T˜∗t (|f f |))1/t = ea−η by (24) we get lime2 , T˜∗t (|f f |)e2 1/t = ea−η . The existence of an absorbing pure state can be seen as some uniform (with respect to the initial state) large deviation principle, as establishes the following corollary (the implication (ii) ⇒ (i) is a direct consequence of Proposition 3). Corollary 2. For any quantum Markov semigroup (Tt ) on M2 the following statements are equivalent. (i) (Tt ) admits an absorbing pure state; (ii) There exists a function J on the pure state space vanishing at a unique point such that for each state ω distinct from this point, the net of orthogonal measures representing (T∗t (ω)) satisfies a large deviation principle with powers (1/t) and rate function J. When this holds the absorbing state is the point where J vanishes. Remark 1. In [5] we defined a noncommutative large deviation principle for any net of states on any C ∗ -algebra A, where all the basic ingredients of the classical theory are replaced by their noncommutative counterparts, using the framework of noncommutative topology. Namely, open (resp. closed) sets are replaced by open (resp. closed) projections living in A∗∗ , and the rate function J by a rate operator

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(more exactly, in order to avoid possibly infinite-valued operator we use instead the bounded upper semi-continuous operator as the counterpart of exp −J, belonging also to A∗∗ ). Since A = A∗∗ when A is finite dimensional, all self-adjoint operators in A are continuous, in particular all projections are clopen. In this simple case, by definition, a net (ωt ) of states is said to satisfy a noncommutative large deviation principle with governing operator z if z = 0 for all projections p ∈ A , lim ωt (p)1/t = sup λ ∈ σ(z) : pE{λ} z where σ(z) and E{λ} denotes respectively the spectrum of z and the eigenspace corresponding to the eigenvalue λ (and inf ∅ = +∞ by convention). The r.h.s. of the above equality can be written in the symbolic form “supp e−z ” since it is the exact noncommutative version of supY e−J for Y open or closed [5, Theorem 4.2]. It follows that part (b) of Theorem 2 amounts to say that for each state ω = ω|e1 e1 | the net of states (T∗t (ω))t≥0 satisfies a noncommutative large deviation principle with governing operator exp −J ∗ (|e1 e1 |).

Remark 2. There are situations not covered by Theorem 4 for which large deviations, when they hold, are trivial. There are those where there is a faithful absorbing state ω∞ ; we distinguish two cases. (a) ω∞ = 12 I. By Lemma 2 the hypothesis of Proposition 2 (a) hold, hence (taking r = 2) the associated net of orthogonal measures satisfies a large deviation principle with a rate function vanishing at the two points given by the eigenvectors of ω∞ , and infinite-valued elsewhere. (b) When ω∞ = 12 I the large deviation is equivalent to the convergence of the one-dimensional projections given by eigenvectors, in which case the rate function has the same form as above. The “if” part of this assertion as well as the form of the rate function follow from Proposition 2. Conversely, if a large deviation principle holds for some net of representing measures, then necessarily the rate function vanishes on some point, say ω|e1 e1 | . Varadhan’s theorem implies t 1/t 1/t lim μt e | |e = max lim sup aj,t |e1 , ej,t |2 1 1 1≤j≤2

=

sup

|u, e1 |2 e−J(ω|uu| ) = 1 .

u∈H,u=1 1/t

1/t

Since lim a1,t = lim a2,t = 1 we have lim |e1 , ej,t | = 1 for some j (say j = 1) so that lim |e1,t e1,t | = |e1 e1 |. Since lim

2

1 aj,t |e1 , ej,t |2 = ω∞ |e1 e1 | = 2 j=1

with lim a1,t = lim a2,t = 12 we conclude that lim |e1 , e2,t | = 0 hence lim |e2,t e2,t | = |e2 e2 | for some unit vector e2 orthogonal to e1 .

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4. Examples In this section, we study a class of quantum Markov semigroups on M2 arising in a special instance of the weak coupling limit [3, 9, 12, 13]. We show that each of these semigroups has an absorbing state, so that large deviations follow from Theorem 4 when this state is pure. Before to describe the model we first recall briefly how works the weak coupling limit theory in this particular case [2, 11]. 4.1. Weak coupling limit and squeezed-vacuum state A quantum system with underlying Hilbert space H0 is coupled with the bosonic reservoir on some Hilbert space H1 (we shall assume H1 = L2 (Rd ) to simplify) where some reference state φ on the associated CCR algebra is given. We will consider a special case where φ is a so-called squeezed-vacuum state, which can be defined as follows. For each pair of reals r, s let Tr,s be the operator on H1 defined by ∀f ∈ H1 , Tr,s (f ) = (cosh r)f − exp(−2is)(sinh r)f . Then Tr,s is real linear, invertible, and so induces a unique ∗-automorphism of the CCR algebra, defined on the Weyl operators by W (f ) → W (Tr,s f ). The vacuum state is transformed by the above automorphism into the state φr,s defined by 1 2 ∀f ∈ H1 , φr,s W (f ) = exp − ||Tr,s f || . 2 By means of the GNS representation of the CCR algebra with state φr,s , we obtain for all f ∈ H1 a strongly continuous unitary group (Wr,s (f ))t∈R ; let Br,s (f ) denote its infinitesimal generator. By definition such a φr,s is a squeezed-vacuum state if the following conditions hold. • φr,s (Br,s (f )) = 0 for all f ∈ H1 ; • There exists f ∈ H1 such that φr,s |Br,s (f )|2 = φr,s |Br,s (if )|2 and

$ 1 %! φr,s |Br,s (f )|2 · φr,s |Br,s (if )|2 = φr,s Br,s (f ), Br,s (if ) . 4 In the sequel we assume that φ is such a squeezed vacuum state, and the corresponding Br,s is simply denoted by B; we also fix some g ∈ H1 \{0} and some ω0 > 0. The evolution of the composite system is given by the Hamiltonian H λ = H0 ⊗ 1 + 1 ⊗ H1 + λV , where H0 is the Hamiltonian of the system, H1 is the Hamiltonian of the free evolution of the reservoir (i.e. the second quantization of the one-particle Hamiltonian), λ the coupling constant, and 1 1 † B(g) − iB(ig) − D ⊗ B(g) + iB(ig) , V =i D⊗ 2 2

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where D is a bounded operator on H0 satisfying exp(itH0 )D exp(−itH0 ) = exp(−iω0 t)D and for each u in a dense subset of H0 ∞ |u, Dn u| n=1

[n/2]!

< ∞.

We put Utλ = exp(itH 0 ) exp(−itH λ ) , (t ∈ R) . We moreover assume that the function t → exp(−iω0 t)f, St h is integrable on R for all f, h in a dense linear subspace of the domain of Q, where (St ) is the unitary one-particle free evolution, and Q is a real linear operator on H1 satisfying ∀f ∈ H1 , f, Qf = φ |B(f )|2 . λ Then, Ut/λ 2 converges (in some appropriate sense) to some unitary transformation U (t) (solution of a quantum stochastic differential equation) which induces on the algebras B(H0 ) of all bounded operators on H0 a quantum Markov semigroup (Tt ). More precisely, for each normal state ω on B(H0 ) and each x ∈ B(H0 ) we have λ † λ lim (ω ⊗ φ)Ut/λ (36) 2 (x ⊗ 1)Ut/λ2 = ω Tt (x) . λ→0

The generator of (Tt ) is obtained in terms of D and some parameters depending on φ, g, ω0 . 4.2. The model We consider a two-level system so that H0 = C2 , and we choose D = ( 01 00 ). In this case, the generator L˜∗ associated to the predual semigroup arising as in (36) has Lindblad form ν+η † (D Dρ − 2DρD† + ρD† D) ∀ρ ∈ M2 , L˜∗ (ρ) = iξ[DD† − D† D, ρ] − 2 ν − (DD† ρ − 2D† ρD + ρDD† ) + ζDρD + ζD† ρD† , 2 where ζ ∈ C and η, ν, ξ are reals satisfying η > 0, ν ≥ 0 and |ζ|2 ≤ ν(ν + η) .

(37)

These parameters depend on the (implicitly fixed) choice of φ, g, ω0 , as described in 4.1; in particular, +∞ exp(−iω0 t)g, St g , (38) η= −∞

Since our large deviation results will occur when ν = 0 (equality which will be expressed in terms of η) and the other ones will not play any role, we do not give them here and refer to [11]. In order to show that each of these semigroups has an absorbing state ω∞ , 1 0 we will use the Pauli matrices I = ( 10 01 ), σ1 = ( 01 10 ), σ2 = ( 0i −i 0 ), σ3 = ( 0 −1 ). Recall that any self-adjoint operator ρ ∈ M2 can be written in a unique way as

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ρ = 12 (tr ρI + u(ρ) · σ) where u(ρ) = (u1 (ρ), u2 (ρ), u3 (ρ)) with ui (ρ) = tr ρσi for 3 i ∈ {1, 2, 3}, σ = (σ1 , σ2 , σ3 ), and u(ρ) · σ denotes the product i=1 ui (ρ)σi . The diagonal form of ρ is given by ρ=

1 1 1 + ||u(ρ)|| p1,ρ + 1 − ||u(ρ)|| p2,ρ , 2 2

u(ρ) u(ρ) where p1,ρ = 12 (I + ||u(ρ)|| · σ) and p2,ρ = 12 (I − ||u(ρ)|| · σ) are the projections on the one-dimensional eigenspaces. Note that for each real a and each self-adjoint operator x ∈ M2 , tr 12 (I + u(ρ) · σ)(aI + u(x) · σ) = a + u(ρ) · u(x).

Lemma 3. For each positive trace-one operator ρ ∈ M2 we put ρt = T˜∗t (ρ) for all t ≥ 0. η η ) − 2ν+η . (a) u3 (ρt ) = e−(2ν+η)t (u3 (ρ) + 2ν+η 2 2 (b) If |ζ| −4ξ > 0, then there exist constants a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 such that – u1 (ρt ) = em1 t (a1 u1 (ρ) + b1 u2 (ρ)) + em2 t (a2 u1 (ρ) + b2 u2 (ρ)). m2 t +& d2 u2 (ρ)). – u2 (ρt ) = em1 t (c1 u1 (ρ) & + d1 u2 (ρ)) + e (c2 u1 (ρ) η where m1 = −(ν + 2 ) + |ζ|2 − 4ξ 2 , m2 = −(ν + η2 ) − |ζ|2 − 4ξ 2 . (c) If |ζ|2 −4ξ 2 < 0, then there exist constants a& 1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 such that – u1 (ρt ) = emt ((a1 u1 (ρ) + b1& u2 (ρ)) cos t 4ξ 2 − |ζ|2 ) + (a2 u1 (ρ) + b2 u2 (ρ)) sin t 4ξ 2 − |ζ|2& ) mt – u2 (ρt ) = e ((c1 u1 (ρ) + d1& u2 (ρ)) cos t 4ξ 2 − |ζ|2 ) + (c2 u1 (ρ) + d2 u2 (ρ)) sin t 4ξ 2 − |ζ|2 ) where m = −(ν + η2 ). (d) If |ζ|2 −4ξ 2 = 0, then there exist constants a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 such that – u1 (ρt ) = e2mt ((a1 u1 (ρ) + b1 u2 (ρ)) + t(a2 u1 (ρ) + b2 u2 (ρ))). – u2 (ρt ) = e2mt ((c1 u1 (ρ) + d1 u2 (ρ)) + t(c2 u1 (ρ) + d2 u2 (ρ))). where m = −(ν + η2 ).

Proof. Direct calculations yield (i) (ii) (iii) (iv)

L˜∗ (I) = −ησ3 L˜∗ (σ1 ) = −(ν + η2 − ζ)σ1 + (2ξ − ζ)σ2 , L˜∗ (σ2 ) = −(2ξ + ζ)σ1 − (ν + η2 + ζ)σ2 , L˜∗ (σ3 ) = −(2ν + η)σ3 . ˜

Differentiating T˜∗t = etL∗ yields differential equations • • • •

dT˜∗t dt

= L˜∗ ◦ T˜∗t , hence the following system of

dT˜∗t (I) = −η T˜∗t (σ3 ) dt dT˜∗t (σ1 ) = −(ν + η2 − ζ)T˜∗t (σ1 ) + (2ξ dt ˜ dT∗t (σ2 ) = −(2ξ + ζ)T˜∗t (σ1 ) − (ν + η2 dt ˜ dT∗t (σ3 ) = −(2ν + η)T˜∗t (σ3 ) dt

− ζ)T˜∗t (σ2 ) + ζ)T˜∗t (σ2 )

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with initial conditions T˜∗0 (I) = I, T˜∗0 (σ1 ) = σ1 , T˜∗0 (σ2 ) = σ2 , T˜∗0 (σ3 ) = σ3 . We then obtain η (e−(2ν+η)t − 1)σ3 . T∗t (I) = I + 2ν + η T∗t (σ3 ) = e−(2ν+η)t σ3 , which gives (a). Put x(t) = T˜∗t (σ1 ), y(t) = T˜∗t (σ2 ). Clearly, x(t) has only nonzero components x1 (t) on σ1 and x2 (t) on σ2 ; similarly, y(t) = (y1 (t), y2 (t)). The characteristic equation associated to the matrix given by the system " x (t) = − ν + η2 − ζ x(t) + (2ξ − ζ)y(t) (39) y (t) = −(2ξ + ζ)x(t) − ν + η2 + ζ y(t) with initial conditions x(0) = (σ1 , 0), y(0) = (0, σ2 ), is on each component X 2 + (2ν + η)X +

η2 + ν 2 + νη + 4ξ 2 − |ζ|2 2

(40)

and so Δ = 4(|ζ|2 − 4ξ 2 ). Assume |ζ|2 − 4ξ 2 > 0. The solutions of (40) being m1 = −(ν + η2 ) + & & 2 |ζ| − 4ξ 2 and m2 = −(ν + η2 ) − |ζ|2 − 4ξ 2 , the general solution of (39) is " x1 (t) = a1 em1 t + a2 em2 t , x2 (t) = c1 em1 t + c2 em2 t y1 (t) = b1 em1 t + b2 em2 t , y2 (t) = d1 em1 t + d2 em2 t for suitable constants a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 . It follows that T˜∗t (σ1 ) = (a1 em1 t + a2 em2 t )σ1 + (c1 em1 t + c2 em2 t )σ2 , T˜∗t (σ2 ) = (b1 em1 t + b2 em2 t )σ1 + (d1 em1 t + d2 em2 t )σ2 , which gives (b). & Assume |ζ|2 − 4ξ 2 < 0. Then m1 = −(ν + η2 ) + i 4ξ 2 − |ζ|2 , m2 = −(ν + & η 2 2 2 ) − i 4ξ − |ζ| , and the general solution of (39) is ' & & x1 (t) = emt (a1 cos t 4ξ 2 − |ζ|2 + a2 sin t 4ξ 2 − |ζ|2 ) & & y1 (t) = emt (b1 cos 4ξ 2 − |ζ|2 + b2 sin 4ξ 2 − |ζ|2 ) ' & & x2 (t) = emt (c1 cos t 4ξ 2 − |ζ|2 + c2 sin t 4ξ 2 − |ζ|2 ) & & y2 (t) = emt (d1 cos 4ξ 2 − |ζ|2 + d2 sin 4ξ 2 − |ζ|2 ) where m = −(ν + η2 ), and a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 are suitable constants. We then obtain & & T˜∗t (σ1 ) = emt a1 cos t 4ξ 2 − |ζ|2 + a2 sin t 4ξ 2 − |ζ|2 σ1 & & + emt c1 cos t 4ξ 2 − |ζ|2 + c2 sin t 4ξ 2 − |ζ|2 σ2 . & & T˜∗t (σ2 ) = emt b1 cos t 4ξ 2 − |ζ|2 + b2 sin t 4ξ 2 − |ζ|2 σ1 & & + emt d1 cos t 4ξ 2 − |ζ|2 + d2 sin t 4ξ 2 − |ζ|2 σ2 , and (c) follows easily.

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If |ζ|2 − 4ξ 2 = 0, then the unique solution of (40) is 2m with m = −(ν + η2 ), and the general solution of (39) is " x1 (t) = e2mt (a1 + a2 t), x2 (t) = emt (c1 + c2 t) y1 (t) = e2mt (b1 + b2 t), y2 (t) = emt (d1 + d2 t) for suitable constants a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 , so that T˜∗t (σ1 ) = e2mt (a1 + a2 t)σ1 + e2mt (c1 + c2 t)σ2 T˜∗t (σ2 ) = e2mt (b1 + b2 t)σ1 + e2mt (d1 + d2 t)σ2 ,

which imply (d). Proposition 4. The state ω∞ given by the operator ρ∞ = 12 (I − sorbing for the semigroup (Tt )t≥0 .

η (2ν+η) σ3 )

is ab-

Proof. Let ωρ be a state and put ρt = T˜∗t (ρ) for all t ≥ 0. The hypotheses η > 0, |ζ|2 ≤ ν(ν + η), ν ≥ 0 imply (with the notations of Lemma 3) m1 m2 = η2 2 2 2 2 + ν + νη + 4ξ − |ζ| > 0, and so m1 < 0, m2 < 0, m < 0. It follows that −η lim u1 (ρt ) = 0 = lim u2 (ρt ) and lim u3 (ρt ) = (2ν+η) , hence for any ρ initial, and any positive x = aI + u(x) · σ in M2 , we have η u3 (x) = trρ∞ x . lim trρt x = lim a + u(ρt ) · u(x) = a − 2ν + η The above proposition shows that ω∞ is pure if and only if ν = 0. As it is easily seen from the expressions given in [11], we have +∞ ! 1 exp(−iω0 t) g, St Q(g) + ig, St Q(ig) . (41) ν = 0 ⇐⇒ η = 2 −∞ Note that this holds in particular when Q = I (i.e. φ is the vacuum state). We have ρ∞ = ( 00 10 )(= |e1 e1 |), and since ζ = 0 by (37), the Lindblad form of the generator becomes η ρ ∈ M2 , L˜∗ (ρ) = iξ[DD† − D† D, ρ] − (D† Dρ − 2DρD† + ρD† D) , 2 or equivalently as in Theorem 1, ∀ρ ∈ M2 ,

L˜∗ (ρ) = yρ + ρy ∗ + zρz ∗ ,

η

0 with y = ( − 2 0−iξ iξ ) and z = ( √0η 00 ); in particular J ∗ (|e1 e1 |) = η|e2 e2 |. The following large deviation result follows from Theorem 4 and Corollary1. It shows that the exponential asymptotic behavior of (T∗t ) is controlled by the parameter (38); moreover, it does not depend on the choice of the squeezed-vacuum state φ, provided that φ satisfies the condition of (41).

Proposition 5. If ν = 0, then for each initial state ω = ωρ∞ the following conclusions hold.

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(a) The net of orthogonal measures representing (T∗t (ω))t≥0 satisfies a large deviation principle with powers (1/t) and rate function ⎧ if |ee| = ρ∞ ⎨ 0 η if |ee| = I − ρ∞ J(ω|ee| ) = ⎩ +∞ otherwise . (b) For each projection p ∈ M2 \{0} we have " 1 −η lim log ω Tt (p) = 0 t

if p = I − ρ∞ otherwise .

Remark 3. The explicit expressions of T∗t (ω) given by Lemma 3 (c) allow a direct proof of Proposition 5 (a). Indeed, easy calculations yield η u1 (ρt ) = e− 2 t u1 (ρ) cos(2ξt) − u2 (ρ) sin(2ξt) , η u2 (ρt ) = e− 2 t u1 (ρ) sin(2ξt) + u2 (ρ) cos(2ξt) , so that

2 ! 1 − ||u(ρt )||2 = −e−2ηt 1 + u3 (ρ) − e−ηt u1 (ρ)2 + u2 (ρ)2 − 2 u3 (ρ) + 1 .

If u1 (ρ)2 + u2 (ρ)2 − 2(u3 (ρ) + 1) = 0, then necessarily u3 (ρ) = −1, u1 (ρ) = u2 (ρ) = 0, and ρ = ρ∞ , which is excluded. It follows that u1 (ρ)2 + u2 (ρ)2 − 2(u3 (ρ) + 1) < 0 and lim(1 − ||u(ρt )||2 )1/t = e−η . Since lim(1 + ||u(ρt )||)1/t = 1 and 1/t 1/t 1/t = lim sup 1 − ||u(ρt )|| lim 1 + ||u(ρt )|| , e−η = lim 1 − ||u(ρt )||2 we get lim(1 − ||u(ρt )||)1/t = e−η . Since ρt = 12 (1 + ||u(ρt )||)p1,ρt + 12 (1 − ||u(ρt )||) p2,ρt the conclusion follows from Proposition 2.

Acknowledgements The author wishes to thank S. Attal for many stimulating discussions, as well as for the support and the warm hospitality he enjoyed during a visit at the Institut Jordan in 2006. This work has been supported by FONDECYT grant No. 7070061.

References [1] L. Accardi, F. Fagnola and R. Hachicha, Generic q-Markov semigroups and speed of convergence of q-algorithms, Infinite Dim. Anal. Quant. Prob. and Related Topics 9 (2006), 567–594. [2] L. Accardi and Y. G. Lu, Squeezing noises as weak coupling limit of a Hamiltonian system, Rep. Math. Phys. 29 (1991), 227–256. [3] L. Accardi, A. Frigerio and Y. G. Lu, The weak coupling limit as a quantum functional central limit, Comm. Math. Phys. 131 (1990), 537–570. [4] V. P. Belavkin and M. Ohya, Entanglement, quantum entropy, mutual information, R. Soc. Lond. Proc. Ser. A 458 (2002), 209–231.

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[5] H. Comman, Upper regularization for extended self-adjoint operators, J. Operator Theory 55 (2006), 91–116. [6] H. Comman, Functional approach of large deviations in general spaces, J. Theoretical Prob. 18 (2005), No. 1, 187–207. [7] H. Comman, Criteria for large deviations, Trans. Amer. Math. Soc. 355 (2003), 2905–2923. [8] E. B. Davies, Generators of dynamical semigroups, J. Funct. Analysis 34 (1979), 421–432. [9] E. B. Davies, Markovian master equation, Commun. Math. Phys. 39 (1974), 91–110. [10] A. Dembo and O. Zeitouni, Large deviations techniques and applications, Second Edition, Springer, New-York, 1998. [11] F. Fagnola, R. Rebolledo and C. Saavedra, Reduction of noise by sqeezed vacuum, Stochastic Analysis and Mathematical Physics 96 (R. Rebolledo, ed.), World Scientific, 1996. [12] K. O. Friedrich, On the perturbation of continuous spectrum, Comm. Math. Phys. 1 (1948), 361–406. [13] L. van Hove, Quantum mechanical perturbations giving rise to a statistical transport equation, Physica 21 (1955), 617–640. [14] R. T. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. Math. 186 (1967), 138–171. [15] M. Takesaki, Theory of operator algebras, Springer, 1979. Henri Comman Department of Mathematics University of Santiago de Chile Bernardo O’Higgins 3363 Santiago Chile e-mail: [email protected] Communicated by Claude-Alain Pillet. Submitted: January 9, 2008. Accepted: April 15, 2008.



On the Lifetime of Quasi-Stationary States in Non-Relativistic QED David Hasler, Ira Herbst, and Matthias Huber Abstract. We consider resonances in the Pauli–Fierz model of non-relativistic QED. We use and slightly modify the analysis developed by Bach, Fr¨ ohlich, and Sigal [3, 4] to obtain an upper and lower bound on the lifetime of quasistationary states.

1. Introduction and main result Spectral properties of models of non-relativistic QED were investigated by Bach, Fr¨ ohlich, Sigal, and Soffer [1,3–5] and by many others. Bach, Fr¨ ohlich, and Sigal [4] proved, among other things, an upper bound on the lifetime of quasi-stationary states. We show an upper and lower bound on the lifetime of quasi-stationary states. We heavily rely on the analysis developed in [3, 4], but choose a different contour of integration and make use of an additional cancellation of terms. Moreover, we neither require a non-degeneracy assumption nor a spectral cutoff. However, we do not provide time dependent estimates on the remainder term and there are no photons in our quasi-stationary state. We comment in some depth on the differences between our approach and the work [4] in Remark 4 in Section 2 after the proof of the main theorem. Estimates similar to ours were obtained before by different authors for other models, see e.g. [16, 17]. In order to be self-contained, we give all the necessary definitions for the model considered. For details, we refer the reader to [4]. We consider an atom in interaction with the second quantized electromagnetic field. The Hilbert space of the system is given by H := Hel ⊗ F , where Hel := AN L2 [(R3 × Z2 )]N

1006

D. Hasler, I. Herbst, and M. Huber


is the Hilbert space of N electrons with spin, and where F :=

∞

SN L2 [(R3 × Z2 )]N

N =0

is the Fock space (with vacuum Ω) of the quantized electromagnetic field, allowing two transverse polarizations of the photon. AN and SN are the projections onto the subspaces of functions anti-symmetric and symmetric, respectively, under a permutation of variables. Strictly speaking, we would have to take the physical units into account in the definition of these spaces. However, we refrain from doing so in order not to complicate the notation. The operator ⎤ ⎡ N 2 2 −Z 1 e ⎦ ⎣ Δ3N + + := − Hel 2m 4π0 j=1 |xj | |xi − xj | 1≤i<j≤N

describes the electrons, and the operator for the total system is Hg :=

N

2 1 : σj · − i∇xj − eAκ (xj ) : +Hf 2m j=1 ⎡ N 2 e ⎣ −Z + + 4π0 j=1 |xj |

1≤i<j≤N

⎤ 1 ⎦, |xi − xj |

where −eZ is the charge of the nucleus, e < 0 the charge of the electron, is Planck’s constant, 0 is the permittivity of the vacuum, m the mass of the electron, σj is the vector of Pauli matrices for the jth electron, and : · · · : denotes normal ordering. The kinetic energy of the photons is Hf := c dk|k|a∗ μ (k)aμ (k) , μ=1,2

k∈R3

where the a∗ μ (k) and aμ (k) are the usual creation and annihilation operators. The second quantized electromagnetic field is Aκ (x) := Aκ (x)+ + Aκ (x)− , where

Aκ (x)+ := ε (k)e−ik · x a∗ dkκ (|k|) μ (k) . 3 μ 2 c|k|(2π) 0 μ=1,2

and Aκ (x)−

:=

μ=1,2

dkκ (|k|)

ε (k)eik · x aμ (k) . 20 c|k|(2π)3 μ

εμ (k),

μ = 1, 2, are the polarization vectors of the photon, depending only Here on the direction of k, and c is the velocity of light. Let us note that we use SI units here; for details about these operators, we refer the reader to [9, 10]. We set ) (Bohr radius), ζ := a20 and ξ −1 := 2α a0 := α−1 ( mc a0 . Moreover, κ (r) := κ(rξ) is a cutoff function depending on the fine structure constant α =

e2 4π0 c .

κ is a

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function, which is positive on [0, ∞), satisfies κ(r) → 1 as r → 0, and has an analytic continuation to a cone around the positive real axis which is bounded 4 and decays faster than any inverse polynomial, e.g., κ(r) := e−r . Following [3], we scale the operator with the transformation xj → ζxj and k → ξ −1 k. We denote the corresponding unitary transformation by U . After this transformation the electron positions are measured in units of 12 a0 , photon wave vectors in units of 2α α2 mc2 . The creation a0 , and energies in units of 4Ry, where the Rydberg is Ry := 2 and annihilation operators transform as U aμ (k)U −1 = ξ 3/2 aμ (ξk) , Moreover, we set

−1 U a∗ = ξ 3/2 a∗μ (ξk) . μ (k)U

εμ (k) := εμ (ξ −1 k) ,

μ = 1, 2 .

Accordingly, we obtain

U Hg U −1 = 2α2 (mc2 )Hg , with Hg := H0 + Wg and H0 := Hel ⊗ 1f + 1el ⊗ Hf , where Hel := −Δ3N +

N −Z j=1

|xj |

Here Hf :=

μ=1,2

k∈R3

+

1≤i<j≤N

1 . |xi − xj |

dk|k|a∗μ (k)aμ (k)

and the interaction is given by Wg :=

N 3/2 2α Aκ (αxj ) · (−i∇xj ) + α3 : A2κ (αxj ) : j=1

+ α5/2 σj · (∇ × Aκ )(αxj ) ,

where the second quantized electromagnetic field is Aκ (x) := Aκ (x)+ + Aκ (x)− with dkκ(|k|) εμ (k)e−ik · x a∗μ (k) . Aκ (x)+ := 2 |k| 4π μ=1,2 and Aκ (x)− :=

dkκ(|k|) εμ (k)eik · x aμ (k) . 2 |k| 4π μ=1,2

As in [4], we set g := α3/2 . Henceforth, we let the coupling constant g := α > 0 be the perturbation parameter. We assume that the spectrum of Hel has the structure σ(Hel ) = {E0 , E1 , . . .} ∪ [Σ, ∞) , where Σ := inf σess (Hel ) and E0 < E1 < · · · are (at least two) eigenvalues (possibly) accumulating at Σ. In the following, we will look at one (fixed) eigenvalue Ej of Hel with j ≥ 1. For 0 < < 1/3 we set ρ0 := g 2−2 , and A(δ, ) := 3/2

1008



[Ej − δ/2, Ej + δ/2] + i[−g 2− , ∞), where δ := dist(Ej , σ(Hel ) \ {Ej }) > 0. We define the operators Hel (θ) := Uel (θ)Hel Uel (θ)−1 ,

Hg (θ) := U(θ)Hg U(θ)−1 , Wg (θ) := U(θ)Wg U(θ)−1

(1)

for real θ, where U(θ) is the unitary group associated to the generator of dilations. It is defined in such a way that the space coordinates of the electrons are dilated as xj → eθ xj and the momentum coordinates of the photons as k → e−θ k. It can be shown [4, Corollary 1.3, Corollary 1.4] that the operators defined in equation (1) are analytic families for |θ| ≤ θ0 for some θ0 > 0. We introduce the convention θ := iϑ with ϑ > 0. Moreover, Uel (θ) is the above dilation acting on the electronic space only. We define (with r > 0 small enough) Pel,i (θ) := −(2πi)−1 |Ei −z|=r (Hel (θ) − z)−1 dz to be the projection onto the eigenspace corresponding to the eigenvalue Ei of Hel (θ) and set P el,i (θ) := 1 − Pel,i (θ). Furthermore, we define P (θ) := Pel,j (θ) ⊗ χHf ≤ρ0 and P (θ) := 1 − P (θ). We abbreviate Pel,i := Pel,i (0). Note that if we consider operators of the form P AP , where A is a closed operator and P a projection with Dom A ⊂ Ran P , then our notation does not distinguish between the operators P AP and P AP |Ran P . It will be clear from the context, how the symbol P AP is to be understood. Following [4], we make crucial use of the Feshbach operator

FP (θ) Hg (θ) − z := P (θ) Hg (θ) − z P (θ) −1

P (θ)Wg (θ)P (θ) . (2) − P (θ)Wg (θ)P (θ) P (θ) Hg (θ) − z P (θ) For the convenience of the reader, we summarize its most important properties including its existence in Appendix A. For details, we refer the reader to [3, Section IV] and [4]. It was shown in [3, 4] that the Feshbach operator can be approximated in a sense to be shown using the operators (0) od ˜ Zj (α) := lim dkPel,j w0,1 (k, μ) ↓0

μ=1,2

k∈R3

× P el,j [P el,j Hel − Ej + |k| − i]−1 P el,j w1,0 (k, μ)Pel,j (0)

and Z˜jd (α) :=

μ=1,2

k∈R3

dk (0) (0) Pel,j w0,1 (k, μ)Pel,j w1,0 (k, μ)Pel,j . |k|

(θ)

(3) (4)

(θ)

Here the coupling functions w0,1 (k, μ) and w1,0 (k, μ) will be needed later with θ = 0. Denoting the momentum of the jth electron by pj , they are (θ)

¯ (θ)

∗

w0,1 (k, μ) := w1,0 (k, μ) :=

N j=1

(θ) 2e−θ G(θ) xj (k, μ) · pj + σj · Bxj (k, μ) ,

(5)

Vol. 9 (2008)


where

e−θ κ(e−θ |k|) iαk · x e μ (k) 4π 2 |k|

(6)

αe−2θ κ(e−θ |k|) iαk · x k × μ (k) . e 2 i 4π |k|

(7)

G(θ) x (k, μ) := and Bx(θ) (k, μ) :=

1009

We set −1 ˜ ˜ Z(α, θ) := Uel (θ)Z(α)U , el (θ) ˜ ˜ 0) . Z(θ) := Z(0, θ) , and Z := Z(0,

˜ Z(α) := Zjd (α) + Zjod (α) ,

(8)

We consider the Feshbach operator FP (θ) (Hg (θ) − z) as an operator on Ran P (θ). ˜ ˜ Similarly, we consider Z(α) := Zjd (α) + Zjod (α) and Z(α, θ) as operators on Ran Pel,j and Ran Pel,j (θ) respectively. We are now able to formulate our main result. It will be proven in Section 2. Theorem 1. Let 0 < < 1/3 and g small enough. Let φ1 and φ2 be normalized eigenvectors of Hel with eigenvalue Ej and ψi := φi ⊗ Ω. Assume moreover that 1 the imaginary part Im Z := 2i (Z − Z ∗ ) of Z is strictly positive on Ran Pel,j . Then, in terms of a dimensionless time parameter s ≥ 0, ψ1 , e−isHg ψ2 = φ1 , e−is(Ej −g

2

Z)

φ2 + b(g, s) ,

where |b(g, s)| ≤ Cg for some C ≥ 0.

The theorem has the following immediate corollary: Corollary 2. Under the assumptions of Theorem 1, if 0 < τ := g 2 s is kept fixed, if φ := φ1 = φ2 is an eigenvector of Z with eigenvalue Γ, and if ψ := φ ⊗ Ω, then lim | ψ, e−isHg ψ | = e−τ Im Γ . g↓0

We close the introductory section with the following remarks: Remark 1. The theorem can be rewritten in terms of the original operators: Let φ1 and φ2 be normalized eigenvectors of Hel with eigenvalue 2α2 mc2 Ej and −1

ψi := φi ⊗ Ω. Then ψ1 , e−it 2 2 −it 2α mc

φ1 , e Here

(Ej −g Z) 2

Hg

ψ2 = φ1 , e−it

2α2 mc2

φ2 + O(α3/2 ), where φi ⊗ Ω

(Ej −g 2 Z ) φ2 = U [φi ⊗ Ω].

+ O(α3/2 ) =

2 κ (|k|)2 Z := Pel,j μ (k) · p P el,j dk lim 4 3 2 2 8α m c ↓0 μ=1,2 k∈R3 4π |k|

× [P el,j Hel − 2α2 mc2 Ej + c|k| − i]−1 P el,j μ (k) · p Pel,j dk κ (|k|)2 Pel,j μ (k) · p Pel,j μ (k) · p Pel,j . (9) + 2 3 c|k| 4π |k| μ=1,2 k∈R

1010



Pel,i is the projection onto the eigenspace of Hel belonging to the eigenvalue n 2 2 2α mc Ei , pj := −i∇xj and p := j=1 pj .

˜ Remark 2. Note that the matrix Z(α) depends on the fine structure constant α, since the coupling functions defined in equations (5), (6), and (7) do. Due to the ˜ exponential decay of the eigenfunctions of the electronic operator, Z(α) can be developed in a power series in α = g 2/3 . The zero order term corresponds to electric dipole (E1) transitions, the higher order terms to magnetic dipole transitions as well as to higher order electric and magnetic transitions. We have for some C > 0 ˜ g 2 Z(α, θ) − Z(θ) ≤ Cg 2+2/3 .

(10)

It is easy to see that the imaginary part of Z is (see also [3, Formula (IV.19)]) j−1 Im Z = π dω(Ei − Ej )2 i=0 μ=1,2

×

|ω|=1

4κ(Ej − Ei )2 Pel,j μ (ω) · p Pel,i μ (ω) · p Pel,j 2 4π (Ej − Ei )

8 (Ej − Ei )κ(Ej − Ei )2 Pel,j pPel,i pPel,j . (11) 3 i=0 In the last step we used the relationships μ=1,2 (μ (ω))m (μ (ω))n = δm,n − ωm ωn 4πδm,n and dωωm ωn = , where δm,n is the Kronecker symbol. Moreover, p := 3 N p and the expression Pel,j pPel,i pPel,j indicates a Euclidean inner product. j=1 j N Analogously, we set x := j=1 xj . Using the commutation relation j−1

=

[x, Hel ] = 2ip we find

2 (Ej − Ei )3 κ(Ej − Ei )2 Pel,j xPel,i xPel,j . 3 i=0 j−1

Im Z =

(12)

We analyze equation (12) for the case of a hydrogen atom in Appendix B. We show there that Im Z is indeed strictly positive unless j = 1. If j = 1, Im Z has a zero eigenvalue, since the 2s state of hydrogen cannot decay via electric dipole transitions. However, the 2p states can decay via an electric dipole transition. It would be interesting to prove time decay estimates also in the latter case. Note that the transition rate is proportional to g 2 α2 ∝ α5 , in accordance with physics textbooks (see e.g. [6, Section 59]). Remark 3. The eigenvectors of Hel are analytic vectors for the generator of dilations, and therefore Uel (θ) : ker(Hel (0) − Ej ) → ker(Hel (θ) − Ej ) is a (bounded and bounded invertible) mapping between finite dimensional vector spaces. (The latter is true for |Im θ| < π/2.) This implies that the matrices Z(0) and Z(θ) are similar. In particular, the bounded operators [−g 2 Z(0)⊗1f +e−θ 1el ⊗Hf ]|Ran P (0)

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and [−g 2 Z(θ) ⊗ 1f + e−θ 1el ⊗ Hf ]|Ran P (θ) are similar (cf. [4, Section 3]). This fact will be used in the proof of Theorem 1 and in Section 3.

2. Proof of the main result In this section we prove our main result. The technical estimates needed in the proof are collected in a series of lemmas and deferred to Section 3. For the proof we need the operator (see [3, Formula (IV.67)]) (θ) dkP (θ) w0,1 (k, μ) ⊗ 1f Q(θ) (z) := μ=1,2

k∈R3

(θ) P (θ)(|k|) w1,0 (k, μ) ⊗ 1f P (θ) , × −iϑ Hel (θ) + e (Hf + |k|) − z

(13)

defined on Ran P (θ) and for z ∈ A(δ, ). Here we used the definition P (θ)(|k|) := (θ) P el,j (θ) ⊗ 1f + Pel,j (θ) ⊗ χHf +|k|≥ρ0 . Moreover, we need the the operator Q0 (z) (θ)

on Ran Pel,j (θ), defined by [Q0 (z)φ]⊗Ω := Q(θ) (z)[φ⊗Ω] for all φ ∈ Ran Pel,j (θ). It is defined by the formula (θ)

Q0 (z) =

Pel,j (θ) (θ) w1,0 (k, μ) Pel,j (θ) −iϑ |k| − z E + e 3 j μ=1,2 k∈R P el,j (θ) (θ) (θ) dkPel,j (θ)w0,1 (k, μ) + w1,0 (k, μ) Pel,j (θ) . −iϑ |k| − z H (θ) + e 3 el k∈R μ=1,2 (θ) dkχ|k|≥ρ0 Pel,j (θ)w0,1 (k, μ)

(14) We remark that both operators are analytic for z ∈ A(δ, ). This follows from the fact that the resolvents in their definitions can be bounded uniformly in z ∈ A(δ, ). (θ) (See the proof of Lemma 3 for a proof in the case of Q0 (z). The proof for Q(θ) (z) is similar and uses additionally the spectral theorem for Hf .) Note that by assumption there exists a constant c > 0 such that Im Z ≥ c. Since Z is bounded, there are constants a, b > 0 such that NumRan Z is localized as NumRan Z ⊂ A(c, a, b), where A(c, a, b) := ic + [−a, a] + i[0, b] (see Figure 2). We set ν := min{ϑ, arctan(c/(2a))}. Finally for w ∈ C and r > 0 we define D(w, r) := {z ∈ C||z − w| < r}, and for A ⊂ C we set D(A, r) := {z ∈ C| dist(z, A) < r}. The notation [z, w] denotes either the line segment between z ∈ C and w ∈ C or a linear contour from z ∈ C to w ∈ C. Accordingly, [z1 , w1 ] + [z2 , w2 ] is to be understood either as the sum of the sets [z1 , w1 ] ⊂ C and [z2 , w2 ] ⊂ C or as a generalized contour. Proof of Theorem 1. First, we show that we can introduce a spectral cutoff with an error of O(g): We choose a function F ∈ C0∞ ((Ej − δ/2, Ej + δ/2)) with 0 ≤

1012



F (x) ≤ 1 for all x ∈ [Ej −δ/2, Ej +δ/2] and F (x) = 1 for all x ∈ [Ej −δ/4, Ej +δ/4]. By the almost analytic functional calculus [11, 14] 1 ∂ F˜ (z) (Hg − z)−1 F (Hg ) = dxdy π ∂ z¯ 1 1 ∂ F˜ (z) ∂ F˜ (z) (H0 − z)−1 − (Hg − z)−1 Wg (H0 − z)−1 = dxdy dxdy π ∂ z¯ π ∂ z¯ ˜ where F˜ ∈ C0∞ (C) is an almost analytic extension of F (x) with | ∂ F∂ (z) z¯ | = O(|Im z|2 ). Since ∂ F˜ (z) isHg 1 dxdy e ψ1 , (H0 − z)−1 ψ2 = eisHg ψ1 , ψ2 π ∂ z¯ and 1 ∂ F˜ (z) isHg −1 −1 dxdy e (Hg − z) Wg (H0 − z) ψ2 ψ1 , π ∂ z¯ ∂ F˜ (z) 1 ≤ dxdyψ1 (Hg − z)−1 |(Ej − z)−1 |Wg ψ2 ≤ Cg , ∂ z¯ π

we find that ψ1 , e−isHg ψ2 = ψ1 , e−isHg F (Hg )ψ2 + O(g). Analogous to [15] and [4], we can write 1 −isHg ψ1 , e lim dλe−iλs F (λ) f (0, λ − i) − f (0, λ + i) F (Hg )ψ2 = − 2πi ↓0 1 dλe−iλs F (λ) f (θ, λ) − f (θ, λ) , =− 2πi 1 ψ2 (θ) with ψi (θ) := φi (θ) ⊗ Ω and φi (θ) := where f (θ, λ) := ψ1 (θ), Hg (θ)−λ Uel (θ)φi . We used Stone’s theorem in the first step. In the second step we used the analyticity of Hg (θ) and the fact that Hg (θ) has no spectrum in the interval [Ej − δ/2, Ej + δ/2] (see [4, Theorem 3.2] and also Corollary 8 below). 1 Noting that ψ1 (θ), Hg (θ)−λ ψ2 (θ) = ψ1 (θ), FP (θ) (Hg (θ) − λ)−1 ψ2 (θ) (see [3, Formula (IV).14] and also Lemma A.6) and using the resolvent equation, we obtain

−1 f (θ, λ) = ψ1 (θ), FP (θ) Hg (θ) − λ ψ2 (θ) −1 (θ) = φ1 (θ), Ej − λ − g 2 Q0 (λ) φ2 (θ)

−1 (θ) − ψ1 (θ), Ej − λ − g 2 Q0 (λ) ⊗ 1f

× FP (θ) Hg (θ) − λ − Ej − λ + e−θ 1el ⊗ Hf − g 2 Q(θ) (λ) P (θ) !

−1 ψ2 (θ) =: f˜(θ, λ) + B(θ, λ) , (15) × FP (θ) Hg (θ) − λ

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Figure 1. The integration contour. where f˜(θ, λ) is the first term in the sum. The strategy is now to move the contour for the first term in order to pick up a pole contribution (see Figure 1), and to estimate the second term on the real axis: dλe−iλs F (λ) f (θ, λ) − f (θ, λ) −iλs = dλe F (λ) B(θ, λ) − B(θ, λ) + dze−izs F (z) f˜(θ, z) − f˜(θ, z) C1 +C5 −izs ˜ ˜ f (θ, z) − f (θ, z) − dze dze−izs f˜(θ, z) − f˜(θ, z) , + C2 +C3 +C4

C0

where we set C := C1 + C2 + C3 + C4 + C5 , with C1 := [Ej − δ/2, Ej − δ/4], C2 := [Ej −δ/4, Ej −δ/4−ig 2− /2], C3 := [Ej −δ/4−ig 2− /2, Ej +δ/4−ig 2− /2], C4 := [Ej + δ/4 − ig 2− /2, Ej + δ/4] and C5 := [Ej + δ/4, Ej + δ/2]. C0 is a suitable contour to pick up the pole contribution from f˜(θ, z). The analyticity properties required for this process will be discussed below. Note that the contour C cannot (θ) simply be moved down much further, since Q0 (z) may have singularities outside of A(δ, ). Estimates on the real axis: We divide the integration interval [Ej − δ/2, Ej + δ/2] into two parts: On [Ej −δ/2, Ej +δ/2]\(Ej −ρ0 /2, Ej +ρ0 /2) we use Lemma A.7 2+ and Lemma 7 to obtain |B(θ, λ)| ≤ C · (sin ϑ)2 (|λ−Eg j |−Cg2 / sin ϑ)2 . Since (sin ϑ)−2

∞ g 2−2 2

dλ = (sin ϑ)−2 g −2 (λ − Cg 2 / sin ϑ)2 = (sin ϑ)−2 g −2 = O(g −2+2 ) ,

∞ g −2 2

dλ (λ − C/ sin ϑ)2

1 g −2 /2 − C/ sin ϑ

1014



we see that the error term for this region is of the order g 3 . On (Ej − ρ0 /2, Ej + g 2+ ρ0 /2) we estimate using Lemma A.7 and Lemma 6: |B(θ, λ)| ≤ Cν −2 · (Ej −λ) 2 +c2 g 4 . g 2+ Since dλ (Ej −λ)2 +c2 g4 is easily seen to be of order g , and the same estimates hold ¯ λ), the estimate on the real axis is proven. for B(θ, Estimates on the contour C: We estimate the integral C |e−isz ||f˜(θ, z) − f˜(θ, z)||dz|. Note that f˜(θ, z) =

1 φ1 (θ), φ2 (θ) Ej − z 1 1 (θ) 2 Q (z) φ2 (θ) . + g φ1 (θ), (θ) Ej − z 0 Ej − z − g 2 Q0 (z)

(16)

Thus, the zero order terms of f (θ, z) and f (θ, z) cancel each other, and it suffices to (θ) show that the higher order terms are at least of order g . Since Q0 (z) is uniformly bounded in z ∈ A(δ, ) by Lemma 3, we estimate using Corollary 4 (see Figure 3) 1 (θ) Q (λ − ig 2− ) g φ1 (θ), Ej − (λ − ig 2− ) 0 2

φ2 (θ) × (θ) 2− 2 2− Ej − (λ − ig ) − g Q0 (λ − ig ) 1

≤C·

(Ej −

g2 . + (g 2− )2

λ)2

Thus the integral along C3 of the above expression is easily seen to be of order g . The integral over the remaining contour is of order g 2 , since dist(z, Ej ) can be estimated independently of g along this part of the contour. The integral of f˜(θ, z) can be estimated in the same way. (θ) Estimates on the pole term. Since Q0 (z) is uniformly bounded for z ∈ A(δ, ) by Lemma 3, the function f˜(θ, z) has no poles in A(δ, ) \ D(Ej , ρ0 /2). It follows (θ) by Lemma 6 that Ej − z − g 2 Q0 (z) is bounded invertible if z ∈ D(Ej , ρ0 /2) \ 2 2+ (Ej − g A(c, a, b) + D(0, C1 · g )) ⊂ D(Ej , ρ0 /2) \ [NumRan(Ej − g 2 Z(0)) + D(0, C1 · g 2+ )], i.e., all poles of f˜(θ, z) are in the set W := Ej − g 2 A(c, a, b) + D(0, C1 · g 2+ ). Moreover, by Lemma 6 we have the estimate (Ej − z − g 2 Q0 (z))−1 ≤ C dist(z, NumRan(Ej − g 2 Z))−1 for some C > 0 if z ∈ D(Ej , ρ0 /2) \ W . In order to estimate the pole terms, we choose a contour C0 around W such that the length of the contour and its distance to W are of order g 2 . A possible choice is C0 = [Ej + g 2 (−(a + c/2) − ic/2), Ej + g 2 ((a + c/2) − ic/2)] + [Ej + g 2 ((a + c/2) − ic/2), Ej +g 2 ((a+c/2)−i(b+3c/2))]+[Ej +g 2 ((a+c/2)−i(b+3c/2)), Ej +g 2 (−(a+ c/2)−i(b+3c/2))]+[Ej +g 2 (−(a+c/2)−i(b+3c/2)), Ej +g 2 (−(a+c/2)−i(c/2))]. (θ)

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We now use the expansion

−1 (θ) φ1 (θ), Ej − z − g 2 Q0 (z) φ2 (θ)

−1 φ2 (θ) = φ1 (θ), Ej − z − g 2 Z(θ)

−1 (θ)

−1 (θ) Q0 (z) − Z(θ) Ej − z − g 2 Z(θ) φ2 (θ) . + g 2 φ1 (θ), Ej − z − g 2 Q0 (z) The integral over the first term gives the claimed leading term, the second term is of order g by Corollary A.9 and Lemma 6. ¯ the function f˜(θ, z) Since by the above considerations (with θ replaced by θ) has no poles in the lower half-plane, there is no pole contribution from this function (see also Remark 5 in Appendix A.2). Remark 4. We compare our proof and our time decay estimate to the one obtained by Bach, Fr¨ ohlich, and Sigal [4] in some detail here: These authors choose a trial −2 state ψj := F (Hg )e−g Hf φj ⊗ η, where η describes a photon cloud, and estimate its time decay behaviour by " # 2 2+ | ψj , e−isHg ψj | ≤ Bη Ce−s(g Γj −Cg ) + CL s−L g 4 , with some constants Bη , C, CL > 0 of unknown size and Γj := inf σ(Im Z) is assumed to be non-degenerate. Thus they show that the survival probability becomes smaller than any δ > 0 if one chooses s ≥ s0 for some s0 = O(g −2 ). We choose a trial state ψj := φj ⊗ Ω and obtain the estimate ψ1 , e−isHg ψ2 = φ1 , e−is(Ej −g

2

Z)

φ2 + b(g, s) ,

with |b(g, t)| ≤ g and 0 < < 1/3. This estimate shows that for times smaller than some s0 = O(g −2 ln g), the exponential term dominates and thus we obtain a fairly precise description of the amplitude for these times. Note that we neither require a spectral cutoff in our trial state nor a non-degeneracy assumption on Γj . On a technical level, we make use of the analysis of Bach, Fr¨ ohlich, and Sigal [3, 4], but in order to prove a lower bound we have to extend it in various directions. Since the matrix Z(θ) does not seem to approximate the Feshbach operator FP (θ) (Hg (θ) − z) globally on A(δ, ) with an error smaller than O(g 2 ) (see the remark after Lemma 6), we have to replace Z(θ) by the z-dependent approximation Q(θ) (z) defined in equation (13), which appears as an intermediate object in [3]. As a consequence, we have to split the set A(δ, ) into two different regions, which depend on g. In the region close to Ej we use a variant of the numerical range estimate obtained by Bach, Fr¨ ohlich, and Sigal (cf. Lemma 6). Away from Ej , we use a different estimate on the numerical range (cf. Lemma 7). Note that the choice of integration contours is somewhat subtle: Since the operator Q(θ) (z) may have singularities for z ∈ / A(δ, ), we cannot move the integration contours outside of A(δ, ). On the other hand, choosing the contours too close to the numerical range of the Feshbach operator FP (θ) (Hg (θ)−z) would result

1016



in estimates diverging as g → 0. Thus our strategy is as follows: Contrary to [4], we 1 use the identity ψ1 (θ), Hg (θ)−λ ψ2 (θ) = ψ1 (θ), FP (θ) (Hg (θ) − λ)−1 ψ2 (θ) before moving any contours. Then, in equation (15), we separate the leading order term (θ) f˜(θ, λ) := φ1 (θ), [Ej −λ−g 2 Q0 (λ)]−1 φ2 (θ) from the term ψ1 (θ), FP (θ) (Hg (θ)− ¯ λ) close to λ)−1 ψ2 (θ) . Instead of moving the remainder terms B(θ, λ) and B(θ, −s(g 2 Γj −Cg 2+ ) the singularities and bounding them from above by Ce with some C > 0 of unknown size, we estimate them on the real axis and bound them from above by Cg uniformly in s. Since Q(θ) (z) leaves the photon subspaces invariant, we can move the integra¯ λ) across the singularities. When doing tion contour for the terms f˜(θ, λ) and f˜(θ, this for f˜(θ, λ), we pick up a pole contribution which gives rise to the exponential time decay. ¯ z) on the moved contour, we extract once In order to estimate f˜(θ, z) and f˜(θ, ¯ z) are identical and more a leading term. Since the leading terms of f˜(θ, z) and f˜(θ, appear with different signs (see equation (16)), these terms cancel, since, unlike [4], ¯ z). we integrate over the same contour for both f˜(θ, z) and f˜(θ,

3. Technical lemmas As in [4], we need estimates on the numerical range and on the norm of the inverse of various operators. We make use of numerous results shown by Bach, Fr¨ ohlich, and Sigal [4], which are summarized in Appendix A. We use the following definitions from [4]: For η > 0 such that Ej + δ/2 < Σ − η we define Pdisc (θ) := i:Ei ≤Σ−η Pi (θ) and P disc (θ) := 1 − Pdisc (θ). (θ)

Since the operator valued function Q0 is relevant for the location of the pole term in the time decay estimates, we need certain properties: (θ)

Lemma 3. Let ϑ sufficiently small and g /sinϑ ≤ 1/2. Then Q0 (z) is uniformly bounded for z ∈ A(δ, ). Proof. The proof follows [3, Chapter IV], using, however, the following estimates: For the first summand in equation (14), we use the estimate |e−iϑ |k| + Ej − z| ≥ |Im (e−iϑ |k| + Ej − z)| ≥ | sin ϑ|k| − g 2− | ≥ | sin ϑ| · ||k| − ρ0 /2| ≥ 1/2 sin ϑ|k|, for |k| ≥ ρ0 . For the second summand in (14), observe that for all Ei with i = j we have |Ei + e−θ |k| − z| ≥ sin ϑδ/2 − g 2− ≥ 1/4δ sin ϑ and that by Lemma A.1 $ $

2 $ Hel (θ) − (z − e−iϑ |k|) −1 P disc (θ)$ ≤ . Σ − η − Re z + cos ϑ|k| This has the following immediate corollary: Corollary 4. There exists a constant C > 0 such that for all z ∈ A(δ, )

(θ) NumRan Ej − g 2 Q0 (z) ⊂ D(Ej , C · g 2 ) . We use the following lemma to estimate the inverse of the Feshbach operator:

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Lemma 5. Suppose A is a bounded operator on a Banach space and let A be similar to A, i.e., there exists a bounded, bounded invertible operator G such that A = GAG−1 . Moreover, let B be a another bounded operator. Then for any q > 1 and for all z ∈ / D(NumRan(A), q · BG · G−1 ) the following estimate holds: $ $

$(A + B − z)−1 $ ≤ G · G−1 q · dist z, NumRan(A) −1 . q−1 In particular, σ(A + B) ⊂ D(NumRan(A), q · BG · G−1 ). Proof. First, observe that for all z ∈ / NumRan(A) by similarity

−1 (A −z)−1 ≤ G · G−1 · (A−z)−1 ≤ G · G−1 · dist z, NumRan(A) . By a series expansion we obtain for z ∈ / D(NumRan(A), q · B · G · G−1 ) (A + B − z)−1 = (A − z)−1

∞

− B(A − z)−1

n

.

n=0

Taking the norm of both sides implies the claim.

Following [4], we control the Feshbach operator FP (θ) (Hg (θ) − z) for z ∈ D(Ej , ρ0 /2) as follows (see Figure 2): Lemma 6. Let 0 < ϑ < θ0 and 0 < g ϑ small enough. Then the following statements hold: a) There are constants C1 , C2 > 0 such that FP (θ) (Hg (θ) − z) is bounded invertible for all z ∈ D(Ej , ρ0 /2) \ D(NumRan(Ej − g 2 Z(0) ⊗ 1f + e−θ 1el ⊗ Hf )|Ran P (0) , C1 · g 2+ ), and for λ ∈ [Ej − ρ0 /2, Ej + ρ0 /2] the estimate $

$ $FP (θ) Hg (θ) − λ −1 $ ≤

sin ν

C2 (Ej − λ)2 + cg 4

holds, where c and ν were defined in Section 2 after Equation (14). The same holds for (Ej − z − g 2 Q(θ) (z) ⊗ 1f + e−θ 1el ⊗ Hf )|Ran P (θ) . b) There is a constant C > 0 such that for all z ∈ C \ NumRan(Ej − g 2 Z(0))|Ran Pel,j (0) the operator (Ej − z − g 2 Z(θ))|Ran Pel,j (θ) is bounded invertible and fulfills the estimate $ −1 $

$ $ $ $ Ej − z − g 2 Z(θ) |Ran Pel,j (θ) ≤

C . dist(z, NumRan(Ej − g 2 Z(0))|Ran Pel,j (0) )

(17)

There are constants C1 , C2 > 0 such that for all z ∈ D(Ej , ρ0 /2) \ D(NumRan(Ej − g 2 Z(0))|Ran Pel,j (0) , C1 · g 2+ ) the operator (Ej − z − g 2

1018



Figure 2. The numerical ranges of the operators Ej − g 2 Z(0) and NumRan(Ej − g 2 Z(0) ⊗ 1f + e−θ 1el ⊗ Hf )|Ran P (0) . (θ)

Q0 (z))|Ran Pel,j (θ) is bounded invertible and fulfills $ −1 $

$ $ (θ) $ Ej − z − g 2 Q0 (z) |Ran Pel,j (θ) $ ≤

C2 . dist(z, NumRan(Ej − g 2 Z(0))|Ran Pel,j (0) )

(18)

Proof. By similarity (cf. Remark 3) we obtain immediately for some C3 > 0 that $ −1 $

$ $ $ Ej − z − g 2 Z(θ) ⊗ 1f + e−θ 1el ⊗ Hf |Ran P (θ) $ #−1 "

≤ C1 · dist z, NumRan Ej − g 2 Z(0) ⊗ 1f + e−θ 1el ⊗ Hf |Ran P (0) . By Lemma A.7, Corollary A.9, and Lemma 5 there are constants C1 , C2 > 0 such that $

$ $FP (θ) Hg (θ) − z −1 $ #−1 "

≤ C2 dist z, NumRan Ej − g 2 Z(0) ⊗ 1f + e−θ 1el ⊗ Hf |Ran P (0) (19) follows for z ∈ / D(NumRan(Ej − g 2 Z(0) ⊗ 1f + e−θ 1el ⊗ Hf )|Ran P (0) ), C1 · g 2+ ). It follows that NumRan[(−g 2 Z(0) ⊗ 1f + e−θ 1el ⊗ Hf )|Ran P (0) ] ⊂ −g 2 A(c, a, b) + e−θ [0, ρ0 ] (see Figure 2). By geometrical considerations, we see that this set is contained in the conical region −i 2c g 2 − i{reiφ | − (ν − π2 ) ≤ φ ≤ ν − π2 , r ∈ [0, ∞)}. This, in turn, implies the claim. The claims in b) follow by the same reasoning. We do not see that the estimate of Lemma 6 a) is true for λ ∈ [Ej − δ/2, Ej + δ/2] \ (Ej − ρ0 /2, Ej + ρ0 /2) as used in [4, Proof of Theorem 3.5] (see also the

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Figure 3. Global localization of the numerical range of Ej + (θ) g 2 Q0 (z) and of the Feshbach operator FP (θ) (Hg (θ) − z) + z. remark after Lemma A.8). Thus we bound FP (θ) (Hg (θ) − λ)−1 differently in that region in the next lemma. Lemma 7. Let 0 < ϑ < θ0 and 0 < g ϑ small enough. Then FP (θ) (Hg (θ) − z) is bounded invertible for all z ∈ A(δ, ) \ D(Ej , ρ0 /2) and there is a constant C > 0 such that for g small enough and z ∈ A(δ, ) \ D(Ej , ρ0 /2) the numerical range of FP (θ) (Hg (θ) − z) is localized as " #

NumRan FP (θ) Hg (θ) − z + z ⊂ D Ej + e−θ [0, ρ0 ], Cg 2 . In particular, for λ ∈ [Ej − δ/2, Ej + δ/2] \ (Ej − ρ0 /2, Ej + ρ0 /2) the estimate $

$ 1 $FP (θ) Hg (θ) − λ −1 $ ≤ sin ϑ|λ − Ej | − Cg 2 holds. Analogous statements hold with FP (θ) (Hg (θ) − z) replaced by Ej − z − (θ) g 2 Q0 (z). Proof of Lemma 7. We have by Lemma A.7 that P (θ)Wg (θ)P (θ) = O(g 2+ ). Therefore, it suffices to show that $ $ −1

$ $ P (θ)Wg (θ)P (θ)$ = O(g 2 ) . $P (θ)Wg (θ)P (θ) P (θ) Hg (θ) − z P (θ) Following [4, Proof of Lemma 3.14], we use a Neumann expansion: −1

P (θ) P (θ) Hg (θ) − z P (θ) P (θ) =

∞

−1

P (θ) P (θ) H0 (θ) − z P (θ) P (θ)

n=0

n −1

P (θ) . × − P (θ)Wg (θ)P (θ) P (θ) H0 (θ) − z P (θ)

(20)

1020



This expansion is valid if z ∈ A(δ, ) with Im z ≥ C for some C > 0 (independent of g.) We define Bθ (ρ) := Hel (θ) ⊗ 1f − Ej + e−θ (1el ⊗ Hf + ρ) as in [4]. The right handside of equation (20) is equal to ∞ n=0

−1

|Bθ (ρ0 )|−1/2 |Bθ (ρ0 )|1/2 P (θ) P (θ) H0 (θ) − z P (θ) P (θ)|Bθ¯(ρ0 )|1/2

× − |Bθ¯(ρ0 )|−1/2 Wg (θ)|Bθ (ρ0 )|−1/2 |Bθ (ρ0 )|1/2 P (θ) n −1

P (θ)|Bθ¯(ρ0 )|1/2 |Bθ¯(ρ0 )|−1/2 =: R(z) . × P (θ) H0 (θ) − z P (θ)

(21)

for all z ∈ A(δ, ) with Im z ≥ C. By Lemma A.4 and Corollary A.3 the series in equation (21) converges uniformly for z ∈ A(δ, ) and is thus a holomorphic function of z ∈ A(δ, ). Thus −1

P (θ) P (θ) Hg (θ) − z P (θ) P (θ) = R(z) for all z ∈ A(δ, ) by holomorphic continuation and for all z ∈ A(δ, ) −1

P (θ)Wg (θ)P (θ) P (θ) Hg (θ) − z P (θ) P (θ)Wg (θ)P (θ) = P (θ)|Bθ¯(ρ0 )|1/2 |Bθ¯(ρ0 )|−1/2 Wg (θ)R(z)Wg (θ)|Bθ (ρ0 )|−1/2 |Bθ (ρ0 )|1/2 P (θ) . (22) Note that |Bθ (ρ0 )|P (θ) = Bθ (ρ0 )P (θ) and P (θ)|Bθ¯(ρ0 )| = P (θ)Bθ (ρ0 ). 1/2 Thus, using P (θ)|Bθ¯(ρ0 )|1/2 ≤ P (θ)|Bθ¯(ρ0 )| · |Bθ¯(ρ0 )|−1/2 = O(ρ0 ), and counting the powers of ρ0 in (22), the first claim follows. The estimate on the (θ) inverse follows by geometrical considerations. The claim on Ej − z − g 2 Q0 (z) follows from Lemma 3. Note that due to the appearance of the interaction Wg (θ) on both sides of the resolvent [P (θ)(Hg (θ) − z)P (θ)]−1 and due to the projections P (θ), the divergence of the resolvent for ρ0 → 0 is completely eliminated (see also the remark after Lemma A.6). We use the following corollary instead of [4, Theorem 3.2]. Corollary 8. Let 0 < ϑ < θ0 and 0 < g ϑ small enough. Then " #

A(δ, ) \ Ej − D g 2 A(c, a, b), C · g 2+ + e−θ [0, ρ0 ] ⊂ ρ Hg (θ) for some C > 0. In particular, the interval [Ej − δ/2, Ej + δ/2] is contained in the resolvent set ρ(Hg (θ)). Proof. By Lemma 7, FP (θ) (Hg (θ) − z) is bounded invertible for all z ∈ A(δ, ) \ D(Ej , ρ0 /2). By Lemma 6, it is bounded invertible for all z ∈ D(Ej , ρ0 /2) \ D(NumRan(Ej −g 2 Z(0)⊗1f +e−θ 1el ⊗Hf )|Ran P (0) , C1 · g 2+ ). Lemma A.6 implies the claim.

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Appendix A. Estimates taken from Bach, Fr¨ ohlich, and Sigal [4] In the appendix, we quote some important technical lemmas from [4], which we frequently use. We do not give their proofs, since they are very lengthy. However, for the orientation of the reader, we describe the essential points of the proofs in words. A.1. Existence of the Feshbach operator First we need certain relative bounds on the interaction and bounds on the resolvent. Lemma A.1 ([4], Lemma 3.8). Let z ∈ C with Re z < Σ − η. Then, for |θ|(1 + (Σ − η − Re z)−1 ) sufficiently small, Hel (θ) − z is invertible on Ran P¯disc (θ) and $ $

$ P¯disc (θ)Hel (θ)P¯disc (θ) − z −1 P¯disc (θ)$ ≤ 2(Σ − η − Re z)−1 . This lemma is proved by using that the estimate holds for θ = 0 with constant one (instead of two) and using that Hel (0) − Hel (θ) is relatively Hel (0) bounded. We remind the reader that as in [4] we define Bθ (ρ) := Hel (θ)⊗1f −Ej +e−θ (1el ⊗ Hf + ρ). Note that A(δ, ) ⊂ ρ(P (θ)H0 (θ)). Lemma A.2 ([4], Lemma 3.11). There exists a constant C > 0 such that for 0 < −1/2 ϑ < θ0 , for all g with 0 ≤ gρ0 ≤ 1/3 and 0 < ρ0 ≤ (δ/3) sin ϑ, and for all z ∈ A(δ, ) $ $ $ $ $Bθ (ρ0 ) P (θ) $ ≤ C . (23) $ H0 (θ) − z $ ϑ The proof of Lemma A.2 is based on Lemma A.1, the fact that Hel (θ) restricted to Pdisc (θ) is similar to a self-adjoint operator, and various other estimates on the resolvent of Hel (θ) as well as the application of the spectral theorem for Hf . The following corollary was used in [2]: Corollary A.3. There exists a constant C > 0 such that for 0 < ϑ < θ0 , all g with −1/2 0 ≤ gρ0 ≤ 1/3 and 0 < ρ0 ≤ (δ/3) sin ϑ, and for all z ∈ A(δ, ) $ $ $ $ $|Bθ (ρ0 )|1/2 P (θ) |Bθ¯(ρ0 )|1/2 $ ≤ C . $ $ H0 (θ) − z ϑ (θ) Proof. By taking adjoints in equation (23), we find that H0P(θ)−z |Bθ¯(ρ0 )| ≤ C ϑ. The claim follows by complex interpolation.

Lemma A.4 ([4], Lemma 3.13). There is a constant C > 0 such that for 0 < ϑ < θ0 sufficiently small, θ1 , θ2 ∈ {±iϑ} and for all ρ > 0 |Bθ1 (ρ)|−1/2 Wg (θ)|Bθ2 (ρ)|−1/2 ≤ g

C (1 + ρ−1/2 ) . ϑ

(24)

1022


Ann. Henri Poincaré 1/2

The proof of Lemma A.4 uses that Aκ (x)− ψ ≤ CHf ψ and Aκ (x)+ ψ ≤ 1/2

C(Hf + 1)1/2 ψ for some 0 < C and all ψ in the domain of Hf , that Hel (0) − Hel (θ) is relatively Hel (0) bounded, and some other estimates. The term proportional to ρ−1/2 is due to the +1 in the bound for the creation operator Aκ (x)+ and to the appearance of a similar constant in the estimate for the relative boundedness of Hel (0) − Hel (θ). Note that the symmetric form of the estimate (24) is essential. Estimates on Wg (θ)|Bθ2 (ρ)|−1 lead to worse behavior as ρ → 0. Lemma A.5 ([4], Lemma 3.14). There is a C > 0 such that for ϑ ∈ (0, θ0 ), ρ0 < −1/2 (δ/3) sin ϑ, 0 < gρ0 ϑ2 , and for all z ∈ A(δ, ) the operator P (θ)Hg P (θ) − z is invertible on Ran P (θ) and fulfills $ $ $ P (θ)Hg (θ)P (θ) − z −1 P (θ)$ ≤ C . ϑρ0 The proof of Lemma A.5 uses Corollary A.3, Lemma A.4 and a Neumann series expansion. Lemma A.6 ([4], Lemma 3.15). Assume that ϑ ∈ (0, θ0 ). Let ρ0 < (δ/3) sin ϑ and −1/2 0 < gρ0 ϑ2 . Then for all z ∈ A(δ, ) the Feshbach operator FP (θ) (Hg (θ) − z) defined in equation (2) exists. If z ∈ A(δ, ), then Hg (θ) − z is bounded invertible if and only if the Feshbach operator FP (θ) (Hg (θ) − z) is bounded invertible, and the equation

−1

−1 Hg (θ) − z = P (θ) − P (θ) P (θ)Hg (θ)P (θ) − z P (θ)Wg (θ)P (θ)

−1

−1 P (θ) − P (θ)Wg (θ)P (θ) P (θ)Hg (θ)P (θ) − z P (θ) × FP (θ) Hg (θ) − z −1 P (θ) (25) + P (θ) P (θ)Hg (θ)P (θ) − z holds, where the left side exists if and only if the right side exists. Moreover, there is a constant C > 0, independent of g and θ, such that for all z ∈ A(δ, ) $ $

$ P (θ)Hg (θ)P (θ) − z −1 P (θ)Wg (θ)P (θ)$ ≤ Cg (26) 1/2 ϑρ0 and

$

$ $P (θ)Wg (θ)P (θ) P (θ)Hg (θ)P (θ) − z −1 $ ≤ Cg . 1/2 ϑρ0

(27)

Equations (26) and (27) are proved similarly as Lemma A.5. Together with Lemma A.5 they imply the existence of the Feshbach operator and the validity of Equation (25) (see [3, Theorem IV.1]). Note that the operator Wg (θ) in Formulas (26) and (27) reduces the divergence as ρ0 → 0 in comparison to Lemma A.5. A.2. Approximations of the Feshbach operator The following lemma gives an approximation of the Feshbach operator globally for all z ∈ A(δ, ) (see [4], Lemma 3.16, estimates on Rem0 through Rem3 ):

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Lemma A.7. Let 0 < < 1/3 and 0 < ϑ < θ0 . Then there is a constant C ≥ 0 such that for all g > 0 sufficiently small with ρ0 < (δ/3) sin ϑ, and for all z ∈ A(δ, ) $ $

$ $ $ FP (θ) (Hg (θ) − z) − Ej − z + e−θ 1el ⊗ Hf − g 2 Q(θ) (z) P (θ)$ ≤ Cg 2+ . Moreover P (θ)Wg (θ)P (θ) = O(g 2+ ). The lengthy and technical proof of Lemma A.7 is based on a Neumann series expansion, estimates similar to Lemma A.4, and the pull-through formula. ˜ θ) (see [4, For z sufficiently close to Ej , Q(θ) (z) can be approximated by Z(α, Lemma 3.16, Estimates on Rem4 and Rem5 ]). Lemma A.8. Let 0 < < 1/3 and 0 < ϑ < θ0 . Then there is a constant C ≥ 0 such that for all g > 0 sufficiently small with ρ0 < (δ/3) sin ϑ, and for all z ∈ D(Ej , ρ0 /2) ˜ θ) ≤ Cg 2+ . g 2 Q(θ) (z) − Z(α, The proof requires some additional estimates to eliminate the z-dependence of Q(θ) (z). However, we not see that Lemma A.8 holds for all z ∈ A(δ, ), which seems to be used in [4]. Lemma A.8 and Equation (10) imply Corollary A.9. Under the assumptions of Lemma A.8 g 2 Q(θ) (z) − Z(θ) ≤ Cg 2+ . ¯ Remark 5. Note that in order to approximate the Feshbach operator FP (θ) ¯ (Hg (θ)− z) for θ = iϑ with ϑ > 0, the −i in definition (3) has to be replaced by +i. In particular, when considering the spectral analysis of this operator, the localization of the numerical range and of the spectrum have to be reflected about the real axis.

Appendix B. The hydrogen atom In this section we discuss the applicability of the presented method to the hydrogen atom. In particular, we show that Im Z is strictly positive unless j = 1. For compatibility with physics literature, we number the eigenvalues of the hydrogen atom according to the principal quantum number n = i + 1. We denote the corresponding eigenvalues by En , i.e., En = Ei for all i ≥ 0. We will ignore the (trivial) ˜ 0) in this appendix. spin dependence of Z = Z(0, B.1. The hydrogen eigenfunctions We define the associated Laguerre polynomials (see [6, Formula (3.5)]) for λ, μ ∈ N0 with 0 ≤ μ ≤ λ by ( % &μ ' % &λ −r λ

d d μ r e r Lλ (r) := e dr dr

1024



and set (see [6, Formula (3.16)]) " #l 1 (n − l − 1)!1/2 2l+1 3/2 −r/(2n) r (2/n) e Ln+l (r/n) . (28) Rn,l (r) := − √ n 8 (n + l)!3/2 (2n)1/2 Note that the Hamiltonian in [6] has an additional factor of 1/2 in front of the Laplacian, so that the radial functions and certain other quantities have to be adapted accordingly. We would like to warn the reader that there are different conventions for the indices of the associated Laguerre functions. For n ∈ N and l, m ∈ Z with 0 ≤ l ≤ n − 1 and −l ≤ m ≤ l the normalized eigenfunctions to the eigenvalue En are un,l,m (r, θ, φ) := Rn,l (r)Yl,m (θ, φ) ,

(29)

where the Yl,m are spherical harmonics (see [6, Section 1]) and we introduced polar coordinates by x = r sin θ cos φ y = r sin θ sin φ z = r cos θ with 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. Note that in this appendix x, y, and z denote the cartesian coordinates of the electron, contrary to the main part of the paper, where x and z have different meanings. Moreover, note that the eigenvalues En are n2 -fold degenerate. B.2. Selection rules for dipole transitions In this subsection, we give some important results from [6]. We define ∞ n ,l Rn,l := drr3 Rn ,l (r)Rn,l (r) .

(30)

0

These integrals have been evaluated by Gordon [13] (see also [6, Section 63]). Below, we need (see [6, Formula (63.4)])

215 n9 (n − 2)2n−6 n,0 | = 2· . (31) |R2,1 3(n + 2)2n+6 For the dipole moments (un ,l ,m , zun,l,m ) one finds (see [6, Formula (60.11)]) for all n, n ∈ N0 that (un ,l ,m , zun,l,m ) = 0

unless l = l ± 1 and

m = m .

Moreover, we will need the relation (see [6, Formula (60.7)]) ) 1 n ,0 R (un ,0,0 , zu2,1,0 ) = . 3 2,1 The selection rules given in [6, Formula (60.11)] imply immediately (un ,l ,m , xun,l,m ) = (un ,l ,m , yun,l,m ) = 0

unless l = l ± 1 and m = m ± 1.

(32)

(33)

(34)

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B.3. The imaginary part of Z In this subsection, we show that the method presented in this paper applies to the hydrogen atom, except for the case n = 2. Theorem B.1. Fix n ∈ N and consider 2 (Ej − Ei )3 κ(Ej − Ei )2 3 i=0 j−1

Im Z =

× [Pel,j xPel,i xPel,j + Pel,j yPel,i yPel,j + Pel,j zPel,i zPel,j ] for j = n − 1 as in equation (12). Then for all l, m, l , m ∈ N0 with 0 ≤ l ≤ n − 1, −l ≤ m ≤ l, 0 ≤ l ≤ n − 1, and −l ≤ m ≤ l (un,l ,m , Im Zun,l,m ) = 0 unless l = l and m = m , and for all l, m ∈ N0 with 0 ≤ l ≤ n − 1, −l ≤ m ≤ l (un,l,m , Im Zun,l,m ) > 0 unless n = 2. In particular, Im Z is positive, unless n = 2. Proof. Off-diagonal matrix elements: Since Im Z is invariant under rotations, it is diagonal in the basis {un,l,m | 0 ≤ l ≤ n − 1, −l ≤ m ≤ l}. This can also be verified using the explicit formulas for the dipole matrix elements in [6, Section 63] . Note that the matrices Pel,j xPel,i xPel,j , Pel,j yPel,i yPel,j , and Pel,j zPel,i zPel,j are not diagonal separately. We would like to mention that also the real part is diagonal in the basis {un,l,m | 0 ≤ l ≤ n − 1, −l ≤ m ≤ l}. Diagonal matrix elements: Let us first remark that the matrix element

u2,0,0 , [Pel,1 xPel,0 xPel,1 + Pel,1 yPel,0 yPel,1 + Pel,1 zPel,0 zPel,1 ]u2,0,0 vanishes by the selection rules (34) and (32). Suppose now that n ≥ 3. We have to prove that there is an i < j = n − 1 such that for all φ ∈ Ran Pel,j Pel,i pυ φ2 > 0 . υ=x,y,z

Since Im Z is diagonal in the basis {un,m,l |l = 0 . . . n− 1, m = −l, . . . , l}, it suffices to show Pel,i pυ un,l,m 2 > 0 υ=x,y,z

for all 0 ≤ l ≤ n − 1 and −l ≤ m ≤ l. For the case l = 0, m = 0 it follows from equations (33) and (31) that the transition (n, 0, 0) → (2, 1, 0) is an allowed n,0,0 electric dipole transition, since z2,1,0 > 0. Consequently (un,0,0 , Im Zun,0,0 ) > 0. Thus, it suffices to consider the case l > 0. The proof is by contradiction. Assume that υ=x,y,z Pel,i pυ un,l,m 2 = 0 for all i < j = n − 1 and some l, m. This would imply that for υ = x, y, z (pυ un,l,m , Hel pυ un,l,m ) ≥ Ej (pυ un,l,m , pυ un,l,m ) .

1026



For l > 0, it is easy to see by Equation (28) that pυ un,l,m ∈ Dom(Hel ) and, using partial integration and the fact that un,l,m (0) = 0, we see that (pυ un,l,m , Hel pυ un,l,m ) = Ej (pυ un,l,m , pυ un,l,m ) . υ=x,y,z

υ=x,y,z

Thus, we conclude by the variational principle that Hel pυ un,l,m = Ej pυ un,l,m . However, Ej pυ un,l,m = Hel pυ un,l,m = Ej pυ un,l,m + [Hel , pυ ]un,l,m for υ = x, y, z and

x r3 2 2 2 with r = x + y + z , so that we arrive at a contradiction. [Hel , px ] = −i

B.4. Numerical illustration In this subsection we give explicit numerical values for the matrix Im Z for the case n = 3 setting the cutoff function κ identically equal to one. Using Maple and the explicit form of the eigenfunctions in equation (29), we calculate the matrices Pel,0 xPel,2 and Pel,1 xPel,2 as well as the corresponding matrices for the coordinates y and z, where Pel,0 is the projection onto the groundstate, Pel,1 the projection onto the eigenspace belonging to E2 , and Pel,2 the projection onto the eigenspace belonging to E3 . With these matrices, we calculate Im Z according to equation (12). The numerical values for other principal quantum numbers could be calculated in the same way. The matrix Im Z (and also Z) is diagonal in the basis {u3,l,m | 0 ≤ l ≤ 2, −l ≤ m ≤ l}. The diagonal elements depend only on l, but not on m. We find 192 738423 , (u3,1,m , Im Zu3,1,m ) = 250000000 for −1 ≤ m ≤ 1, (u3,0,0 , Im Zu3,0,0 ) = 1953125 49152 and (u3,2,m , Im Zu3,2,m ) = 48828125 for −2 ≤ m ≤ 2. Let us remark that the eigen−1 values of 2 · (2α5 mc2 /)Im Z are precisely the inverse lifetimes τn,l,m of the corresponding eigenstates of the hydrogen atom. The additional factor two is due to the fact that lifetimes are defined via survival probabilities and not via survival amplitudes. Inserting α = 7.29735 · 10−3 , m = 9.10939 · 10−31 kg, c = 2.99792 · 108 m/s and = 1.05457 · 10−34 Js we find τ3,0,0 = 1.58303 · 10−7 s, τ3,1,m = 5.26860 · 10−9 s for −1 ≤ m ≤ 1, and τ3,2,m = 1.54593 · 10−8 s for −2 ≤ m ≤ 2. Experimental values for these lifetimes are not very precise. We quote a value of τ3,1,m = (5.5 ± 0.2) × 10−9 s given in [8]. [7] find a value of τ3,1,m = (5.58 ± 0.13) × 10−9 s and [12] find τ3,1,m = (5.41 ± 0.18) × 10−9 s. Notice that the experimental values are in reasonable agreement with the calculated value.

Acknowledgements M. Huber wishes to thank I. Herbst for the hospitality of the Mathematics Department of the University of Virginia and V. Bach for interesting conversations. He

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acknowledges support by the Deutsche Forschungsgemeinschaft (DFG), grant no. SI 348/12-2. His stay at the University of Virginia was supported by a “Doktorandenstipendium” from the German Academic Exchange Service (DAAD), which he gratefully acknowledges.

References [1] V. Bach, J. Fr¨ ohlich, and I. M. Sigal, Mathematical theory of nonrelativistic matter and radiation, Lett. Math. Phys. 34 (3) (1995), 183–201. [2] V. Bach, J. Fr¨ ohlich, and I. M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Preprint mp arc 98-728 (1998). [3] V. Bach, J. Fr¨ ohlich, and I. M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (2) (1998), 299–395. [4] V. Bach, J. Fr¨ ohlich, and I. M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Comm. Math. Phys. 207 (2) (1999), 249–290. [5] V. Bach, J. Fr¨ ohlich, I. M. Sigal, and A. Soffer, Positive commutators and the spectrum of Pauli–Fierz Hamiltonian of atoms and molecules, Comm. Math. Phys. 207 (3) (1999), 557–587. [6] H. A. Bethe and E. E. Salpeter, Quantum mechanics of one- and two-electron atoms, Berlin, G¨ ottingen, Heidelberg: Springer-Verlag, 1957. [7] W. S. Bickel and A. S. Goodman, Mean lives of the 2p and 3p levels in atomic hydrogen, Physical Review 148 (2) (1970), 1–4. [8] E. L. Chupp, L. W. Dotchin, and D. J. Pegg, Radiative mean-life measurements of some atomic-hydrogen excited states using beam-foil excitation, Physical Review 175 (1) (1968). [9] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions – Basic Processes and Applications. John Wiley and Sons, Inc., 1992. [10] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons & Atoms. WILEYVCH Verlag GmbH& Co. KGaA, 2004. [11] E. B. Davies, The functional calculus, J. London Math. Soc. (2) 52 (1) (1995), 166– 176. [12] R. C. Etherton, L. M. Beyer, W. E. Maddox, and L. B. Bridwell, Lifetimes of 3p, 4p, and 5p states in atomic hydrogen, Physical Review A 2 (6) (1970), 2177–2179. [13] W. Gordon, Zur Berechnung der Matrizen beim Wasserstoffatom, Annalen d. Physik 5 (2) (1929), 1031–1056. ´ [14] B. Helffer and J. Sj¨ ostrand, Equation de Schr¨ odinger avec champ magnétique et équation de Harper, In Schr¨ odinger operators (Sønderborg, 1988), volume 345 of Lecture Notes in Phys., pages 118–197. Springer, Berlin, 1989. [15] W. Hunziker, Resonances, metastable states and exponential decay laws in perturbation theory, Comm. Math. Phys. 132 (1) (1990), 177–188. [16] V. Jakˇsić and C.-A. Pillet, On a model for quantum friction. I. Fermi’s golden rule and dynamics at zero temperature, Ann. Inst. H. Poincaré Phys. Théor. 62 (1) (1995), 47–68.

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[17] Ch. King, Resonant decay of a two state atom interacting with a massless nonrelativistic quantised scalar field, Comm. Math. Phys. 165 (3) (1994), 569–594. David Hasler and Ira Herbst Department of Mathematics P.O. Box 400137 University of Virginia Charlottesville, VA 22904-4137 USA e-mail: [email protected] [email protected] Matthias Huber Mathematisches Institut Ludwig-Maximilians-Universit¨ at M¨ unchen Theresienstraße 39 D-80333 M¨ unchen Germany e-mail: [email protected] Communicated by Vincent Rivasseau. Submitted: September 30, 2007. Accepted: March 28, 2008.

Ann. Henri Poincaré 9 (2008), 1029–1067 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/061029-39, published online September 26, 2008 DOI 10.1007/s00023-008-0385-5


Asymptotic Behavior of Spherically Symmetric Marginally Trapped Tubes Catherine Williams Abstract. We give conditions on a general stress-energy tensor Tαβ in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay per se is not required for this asymptotic result, and in this general setting, such decay only implies that the marginally trapped tube has finite length with respect to the induced metric. We do, however, impose a smallness condition (B1) which one may obtain in practice by imposing decay on the Tvv component of the stress-energy tensor along the event horizon. We give two applications of the theorem to self-gravitating Higgs field spacetimes, one using weak Price law decay, the other certain strong smallness and monotonicity assumptions.

1. Introduction Black holes have long been one of the most-studied features of general relativity. The global nature of their mathematical definition, however, makes them somewhat inaccessible to physical considerations (e.g. black hole mechanics) or numerical simulation. In recent years, considerable work has gone into developing more tractable quasi-local notions to capture black hole behavior. In particular, a program initiated by Hayward and modified and refined by the work of Ashtekar and Krishnan [4, 14] proposes certain hypersurfaces to model the surfaces of dynamical and equilibrium black holes, called dynamical and isolated horizons respectively; such horizons are examples of a more general class of hypersurfaces known as marginally trapped tubes. The geometry of such hypersurfaces has been the subject of a flurry of investigation recently, and nice existence, uniqueness, and compactness results for marginally trapped tubes have been established [1–3]. Major open questions remain, however. One area for further exploration is the relationship between marginally trapped tubes and the event horizons of black holes.

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Numerical simulations appear to indicate that marginally trapped tubes that form during gravitational collapse or in the collision of two black holes should become achronal and asymptotically approach the event horizon [4, 5, 16], but this prediction has not been proven except in two specific matter models: that of a scalar field in a (possibly) charged spacetime [7, 9], and that of collisionless matter [13]. In this paper, we give conditions on a general stress-energy tensor Tαβ in a spherically symmetric black hole spacetime which are sufficient to guarantee both that the black hole will contain a (spherically symmetric) marginally trapped tube and that that marginally trapped tube will be achronal, connected, and asymptotic to the event horizon. We then derive some additional results pertaining to the affine lengths of both the black hole event horizon and the marginally trapped tube and show how our main result can be applied for Vaidya and Higgs field matter models. A spherically symmetric spacetime admits an SO(3)-action by isometries, so it is both natural and convenient to formulate and prove these results at the level of the 1+1 Lorentzian manifold obtained by taking a quotient by this action. In particular, we restrict ourselves to a characteristic rectangle in Minkowski 2-space with conformal metric and past boundary data constrained in such a way that the rectangle could indeed lie inside the quotient of a spherically symmetric black hole spacetime, with one of its edges coinciding with the event horizon. In order to make this regime both generic and physical, we assume that our spacetime is a globally hyperbolic open subset of this characteristic rectangle and contains its two past edges; this would indeed be the case if it were the maximal future development of some initial data for the metric and stress-energy tensor prescribed along these hypersurfaces. We use no explicit evolution equations for Tαβ . Instead we assume that Tαβ satisfies the dominant energy condition throughout the spacetime, and in addition we require that the spacetime satisfy a nontrivial extension principle, one which arises in the evolutionary setting for many ‘physically reasonable’ matter models. Our conditions then take the form of four inequalities which must hold near a point which we call future timelike infinity and denote by i+ . The inequalities relate components of the stress-energy tensor to the conformal factor and radial function for the metric. It is worth mentioning that the conditions we impose on Tαβ do not directly include or imply ‘Price’s law’. (Originally formulated as an estimate of the decay of radiation tails of massless scalar fields in the exterior of a black hole [15], the appellation ‘Price’s law’ is now widely used to refer to inverse power decay of any black hole “hair” along the event horizon itself.) In [9], which addressed the double characteristic initial value problem for the Einstein–Maxwell-scalar field equations, Dafermos showed that imposing a weak version of Price law decay on data along an outgoing characteristic yields a maximal future development which does indeed contain an achronal marginally trapped tube asymptotic to the event horizon. Consequently, one might have expected such decay to be central for obtaining the same result in the general setting. In this paper, however, we show that the analogous decay of Tvv (v an outgoing null coordinate) is only a priori related to the length of the marginally trapped tube, not its terminus. The conditions we

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use instead to control the tube’s asymptotic behavior entail only smallness and integrability of certain quantities. However, it appears that some sort of decay is always necessary in order to retrieve our conditions in practice. Indeed, in the self-gravitating Higgs field setting, our conditions follow rather naturally from the assumption of weak Pricelaw-like decay on the derivatives of the scalar field and the potential (Theorem 3), exactly analogously to Dafermos’ result for Einstein–Maxwell-scalar fields. On the other hand, in Theorem 4, we are able to derive these conditions without making use of an explicit decay rate, instead using only smallness and monotonicity, and indeed one can construct examples which satisfy our conditions but violate even the weak version of Price’s law. Still, the specific monotonicity assumptions are themselves quite strong and do imply decay, if not that which is specifically called Price’s law. The paper proceeds as follows: in Section 2, we present the setting for the main theorem, including all the assumptions necessary to insure that our characteristic rectangle represents the correct portion of a black hole and precise statements of the energy condition and extension principle to be used. In Section 3, we present a weak version of the first of the conditions used in the main result, show in Proposition 3 that it is sufficient by itself to guarantee achronality of the marginally trapped tube, and then use that result to establish Proposition 4, a key ingredient for the proof of the main theorem. In Section 4, we state the remaining three conditions and prove the main result, Theorem 1, that a marginally trapped tube must form, have i+ as a limit point, and be achronal and connected near i+ . Theorem 2 shows how Price law decay implies that the marginally trapped tube has finite length. In Section 5, we give some applications of Theorem 1: first we discuss how Theorems 1 and 2 apply to the (ingoing) Vaidya spacetime, then we turn to self-gravitating Higgs field spacetimes, presenting two different ways in which Theorem 1 can be applied. In Theorem 3, we postulate an explicit inverse power decay rate in order to extract the hypotheses of Theorem 1, whereas in Theorem 4 we make only certain smallness and monotonicity assumptions (which are nonetheless quite strong).

2. Background assumptions 2.1. Spherical symmetry & the initial value problem A spacetime (M, g) is said to be spherically symmetric if the Lie group SO(3) acts on it by isometries with orbits which are either fixed points or spacelike 2spheres. If we assume that the quotient Q = M/SO(3) is a manifold with (possibly empty) boundary Γ corresponding to the points fixed by the SO(3)-action, that Q inherits a 1+1-dimensional Lorentzian structure, and that its topology is such that we may conformally embed it into Minkowski space (R2 , η), then the image of this conformal embedding retains the important causal and asymptotic features of the original spacetime. In particular, such features of (M, g) as black holes, event

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horizons, and marginally trapped tubes are preserved and may be studied at this quotient level. Specifically, we fix double null coordinates (u, v) on R2 , such that the Minkowski metric η takes the form η = −du dv and such that (R2 , η) is time oriented in the usual way, with u and v both increasing toward the future. (In our Penrose diagrams, we will always depict the positive u- and v-axes at 135◦ and 45◦ from the usual positive x-axis, respectively.) With respect to a conformal embedding, the metric on a 1+1 Lorentzian quotient manifold Q as above then takes the form −Ω2 du dv, where Ω = Ω(u, v) is a smooth positive function on Q. Suppressing pullback notation, the original metric g may be expressed g = −Ω2 du dv + r2 gS 2 , 2

2

2

(1) 2

where gS 2 = dθ +sin θ dϕ is the standard metric on S , and the radial function r is a smooth nonnegative function on Q such that r(q) = 0 if and only if q ∈ Γ. Choosing units so that the Einstein field equations take the form 1 Rαβ − Rgαβ = 2Tαβ , 2 by direct computation we find that the field equations for the metric (1) on M yield the following system of pointwise equations on Q: ∂u (Ω−2 ∂u r) = −rΩ−2 Tuu

(2)

∂v (Ω−2 ∂v r) = −rΩ−2 Tvv

(3)

2

−2

(Tuv ∂u r − Tuu ∂v r)

(4)

2

−2

(Tuv ∂v r − Tvv ∂u r) ,

(5)

∂u m = 2r Ω ∂v m = 2r Ω

where Tuu , Tuv , and Tvv are component functions of the stress-energy tensor Tαβ on M and r m = m(u, v) = (1 + 4Ω−2 ∂u r ∂v r) (6) 2 is the Hawking mass. Note that the null constraints (2) and (3) are just Raychaudhuri’s equation applied to each of the two null directions in Q. The study of spherically symmetric 3+1-dimensional spacetimes is thus essentially equivalent to the study of conformal metrics paired with radial functions on subsets of (R2 , η), and the relative simplicity of the latter recommends it as a starting point. Without a priori knowledge about an “upstairs” spacetime (M, g) or the embedding Q → R2 , it is natural to begin with a generalized initial value problem for the system (2)–(5). First, for any values u, v > 0, we use K(u, v) to denote the characteristic rectangle given by K(u, v) = [0, u] × [v, ∞) . Next, suppose we choose some values u0 , v0 > 0, fix the specific rectangle K(u0 , v0 ), and define initial hypersurfaces Cin = [0, u0 ] × {v0 } and Cout = {u0 } × [v0 , ∞). In a specific matter model, we would prescribe initial data for the metric and the

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i+

Cout

Cin

Figure 1. The characteristic rectangle K(u0 , v0 ) contains the spacetime G(u0 , v0 ). In applications to specific matter models, G(u0 , v0 ) would be the maximal future development of initial data prescribed on Cin ∪ Cout . stress-energy tensor along Cin and Cout in such a way that the four equations (2)– (5) were satisfied, then use these equations coupled to the evolution equation(s) associated with the matter model to establish the existence of a maximal future development of the initial data. In our situation, however, we are working with a general stress-energy tensor with no evolution equations of its own beyond those imposed by the Bianchi identity, divg T = 0, so we will instead assume only that we have a globally hyperbolic (relatively) open spacetime G(u0 , v0 ) ⊂ K(u0 , v0 ) such that Cin ∪ Cout ⊂ G(u0 , v0 ). For simplicity we also assume that our rectangle does not intersect the center of symmetry along its two past edges, i.e. r > 0 on Cin ∪ Cout ; in practice one could always shrink the rectangle so as to insure this. See Figure 1 for a representative Penrose diagram. 2.2. Black hole & energy assumptions In this section, we make a number of assumptions to which we will refer later, namely in the statement and proof of our main result. These assumptions are the basic requirements that the spacetime and stress-energy tensor must satisfy in order to be physically reasonable and relevant to black hole spacetimes. We label them here with upper-case Roman numerals for convenience. First, on physical grounds, we want the “upstairs” stress-energy tensor Tαβ to satisfy the dominant energy condition. In the 1+1-setting, this condition yields the following pointwise inequalities at the quotient level: I

Tuu ≥ 0 ,

Tuv ≥ 0 ,

and

Tvv ≥ 0 .

Second, because we are not working with a specific global existence result in an evolutionary setting, we explicitly require that the spacetime G(u0 , v0 ) obtained in the previous section be a past subset of K(u0 , v0 ), i.e. II J − G(u0 , v0 ) ⊂ G(u0 , v0 ) .

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Next, we assume that along Cout the functions r and m satisfy r ≤ r+ ,

III and

0 ≤ m ≤ m+ ,

IV

where the constants r+ , m+ < ∞ are chosen to be the respective suprema of r and m along Cout . Since our aim is to say something about the interiors of spherically symmetric black holes, we want to choose the data on the initial hypersurface Cout of our characteristic rectangle K(u0 , v0 ) to insure that they would agree with that we would find along (the quotient of) the event horizon in a general spherically symmetric black hole spacetime – assumption III provides that correspondence. In particular, the boundedness of r along Cout is precisely the requirement that Cout must satisfy in order to lie inside a black hole in (the quotient of) an asymptotically flat spacetime, at least provided the black hole has bounded surface area (or equivalently, bounded ‘entropy’). Indeed, in the general spherically symmetric setting described in [11], whether r is bounded or tends to infinity along outgoing null rays is precisely what distinguishes the black hole region from the domain of outer communications, respectively. Since Cout is necessarily defined for arbitrarily large values of v, while outgoing null rays past the event horizon need not be, it is natural to interpret Cout as lying along the event horizon itself. For this reason we will often refer to its terminal point (0, ∞) – which is not in the spacetime, strictly speaking – as i+ , future timelike infinity. The first inequality of IV is physically natural, since it just requires that the quasi-local mass m be nonnegative, while the second inequality is actually slightly redundant: given equation (6), the boundedness of m along Cout follows immediately from the fact that it is nonnegative and that r is bounded. Indeed, we must have the relation m+ ≤ 12 r+ . For the next two of our assumptions, we must introduce notions of trappedness in our spacetime. Each point (u, v) of G(u0 , v0 ) represents a 2-sphere of radius r = r(u, v) in the original manifold M , and the two future null directions orthogonal to this sphere are precisely ∂u and ∂v . We designate u as the “ingoing” direction and v the “outgoing” direction and use θ− and θ+ to denote the expansions in the directions ∂u and ∂v , respectively. The induced Riemannian metric on this twosphere is of course just hab = r2 (gS 2 )ab , and a straightforward calculation shows that θ− = 2(∂u r)r−1 and θ+ = 2(∂v r)r−1 . Since r is strictly positive away from the center of symmetry Γ, the signs of θ+ and θ− are exactly those of ∂v r and ∂u r, respectively. This fact motivates the definitions of the following three subsets of interest: the regular region R = (u, v) ∈ G(u0 , v0 ) : ∂v r > 0 and ∂u r < 0 , the trapped region T = (u, v) ∈ G(u0 , v0 ) : ∂v r < 0 and ∂u r < 0 ,

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i+ T A

Cout

R Cin

Figure 2. The spacetime G(u0 , v0 ) comprises the regular and trapped regions, R and T , and a marginally trapped tube A, shown here as a dotted curve. Note that the marginally trapped tube shown is not achronal, but it does comply with Proposition 2. and the marginally trapped tube, A = (u, v) ∈ G(u0 , v0 ) : ∂v r = 0 and ∂u r < 0 . Note that A is in fact a hypersurface of G(u0 , v0 ) provided that 0 is a regular value of ∂v r. It is then called a tube because “upstairs” it is foliated by 2-spheres. For us, an anti-trapped surface is one for which ∂u r ≥ 0. (Here we are adding the marginal case ∂u r = 0 into the usual definition in order to avoid introducing unnecessary terminology.) We impose the assumption that no anti-trapped surfaces are present initially: V

∂u r < 0 along

Cout .

This assumption is motivated primarily by the following result due to Christodoulou (see [8, 11]): Proposition 1. If ∂u r < 0 along Cout , then G(u0 , v0 ) = R ∪ T ∪ A – that is, anti-trapped surfaces cannot evolve if none are present initially. See Figure 2 for a representative Penrose diagram. Proof. Let (u, v) be any point in G(u0 , v0 ). Then integrating Raychaudhuri’s equation (2) along the ingoing null ray to the past of (u, v), we obtain u −2 −2 (Ω ∂u r)(u, v) = (Ω ∂u r)(0, v) − r Ω−2 Tuu (U, v) dU . 0

Since we have assumed that ∂u r < 0 along Cout and that Tuu ≥ 0 by assumption I, the righthand side of this equation is strictly negative, and hence so is the lefthand side. Thus assumption V guarantees that a spacelike marginally trapped tube in G(u0 , v0 ) is indeed a dynamical horizon as defined by Ashtekar and Krishnan [4], since their definition requires both that θ+ = 0 and θ− < 0 along A.

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One further consequence of the dominant energy condition (assumption I), the Einstein equations (2)–(5), and our definitions of A and T is the following proposition due to Christodoulou, which will be of considerable use later on: Proposition 2. If (u, v) ∈ T ∪ A, then (u, v ∗ ) ∈ T ∪ A for all v ∗ > v. Similarly, if (u, v) ∈ T , then (u, v ∗ ) ∈ T for all v ∗ > v. Proof. Integrating Raychaudhuri’s equation (3) along the null ray to the future of (u, v) yields v∗ −2 ∗ −2 r Ω−2 Tvv (u, V ) dV (Ω ∂v r)(u, v ) = (Ω ∂v r)(u, v) − v ∗

for v > v. Since Tvv ≥ 0 everywhere by assumption I, the righthand side of this equation will be nonpositive if (Ω−2 ∂v r)(u, v) ≤ 0, and strictly negative if (Ω−2 ∂v r)(u, v) < 0. Since Ω > 0 everywhere, both statements of the proposition now follow immediately. In a black hole spacetime, the trapped region T must be contained inside the black hole. Since we would like Cout to represent an event horizon, we must therefore require that ∂v r ≥ 0 along Cout . Combining this inequality with Proposition 2, we see that if A intersects Cout at a single point, then the two must in fact coincide to the future of that point. This is indeed the case in the Schwarzschild and Reisner–Nordstr¨ om spacetimes, in which the black hole coincides exactly with the trapped region T . However, we are really only interested in the cases in which the marginally trapped tube A does not coincide with the event horizon, so we will instead assume 0 < ∂v r

VI

along

Cout .

It is now clear that the values r+ and m+ specified in assumptions III and IV are in fact the asymptotic values of r and m along Cout , respectively (the monotonicity of m follows from assumption I and equation (5)). Finally, we will need to assume that G(u0 , v0 ) satisfies the extension principle formulated in [11]. This principle holds for self-gravitating Higgs fields and selfgravitating collisionless matter [10, 12], and it is expected to hold for a number of other physically reasonable models [11]. Regarding set closures as being taken with respect to the topology of K(u0 , v0 ), the extension principle may be formulated as follows: VII

If p ∈ R , and q ∈ R ∩ I − (p) such that J − (p) ∩ J + (q)\{p} ⊂ R ∪ A ,

111111 000000 ?

R∪A

p

q

p

then p ∈ R ∪ A.

R∪A

q

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Note that the formulation of the extension principle given in [11] specifically excludes the possibility that the point p ∈ Γ, where Γ denotes the center of symmetry of G(u0 , v0 ), i.e. the set of points at which r = 0. However, in our situation, Proposition 2 and our assumption that r > 0 on Cin ∪ Cout together imply that r > 0 everywhere in R, so our formulation is slightly simpler.

3. Achronality & connectedness The proof of our main result, Theorem 1, which appears in Section 4.1, relies on two more general propositions, both of which are of interest in their own right. These propositions require a weak version of one of the four conditions appearing in Theorem 1, a condition which we now state separately: 1 A Tuv Ω−2 < 2 . 4r In practice, we will only require that condition A hold in some small subset of our spacetime G(u0 , v0 ). The expression Tuv Ω−2 takes a particularly simple form in many matter models. For a perfect fluid of pressure P and energy density ρ, it is the quantity 14 (ρ − P ). For a self-gravitating Higgs field φ with potential V (φ), it is 12 V (φ). And for an Einstein–Maxwell massless scalar field of charge e, it is 1 2 −4 . 4e r Proposition 3. Suppose (G(u0 , v0 ), −Ω2 du dv) is a spacetime obtained as in Section 2.1 with radial function r, and suppose it satisfies assumptions I–VII of Section 2.2. If A is nonempty and condition A holds in A, then each of its connected components is achronal with no ingoing null segments. Remark. In [6], Booth et al. give a necessary and sufficient condition for a general marginally trapped tube (not necessarily spherically symmetric) to be achronal; that condition is precisely A in our setting. The proof of Proposition 3 essentially duplicates their reasoning, although it is formulated somewhat differently. Proof. To begin, we must establish that A is in fact a hypersurface in G(u0 , v0 ). Since A is defined as a level set, this is equivalent to showing that 0 is a regular value of ∂v r, i.e. that the differential D(∂v r) is non-degenerate at points where 2 2 r and ∂vv r, it suffices to show that ∂v r = 0. Since D(∂v r) has components ∂uv 2 ∂uv r < 0 along A. Rearranging equation (4) and then combining with equations (2) and (6) yields 2r2 Ω−2 Tuv ∂u r = 2r2 Ω−2 Tuu ∂v r + ∂u m 1 2 r)Ω−2 ∂u r , = (∂u r) + 2(∂u r)2 (∂v r)Ω−2 + 2r(∂uv 2 2 r, we obtain and solving for ∂uv 1 2 ∂uv r = − Ω2 r−2 α , 2

(7)

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where α is given by α = m − 2r3 Ω−2 Tuv .

(8)

Since r and Ω are strictly positive in G(u0 , v0 ), it is enough to show that α > 0 along A. Using condition A and the fact that ∂v r = 0 on A, we have α = m − 2r3 Ω−2 Tuv r = − 2r3 Ω−2 Tuv 2 r3 1 −2 = − 4 Ω T uv 2 r2 > 0, 2 so ∂uv r < 0 along A as desired. Now, we have A = ∅, and since we now know it is a 1-dimensional submanifold of the spacetime, we can parameterize any connected component of it by a curve γ(t) = (u(t), v(t)). Since ∂v r ≡ 0 along A, at points on A we have

γ(∂ ˙ v r) = 0 =

du 2 dv 2 (∂uv r) + (∂ r) . dt dt vv

(9)

By the result of Proposition 2, we know that we can describe all but the outgoing null segments of A in terms of a function v(u), defined on some (possibly disconnected) subset of [0, u0 ]. From equation (9) we see that its slope is given by dv dv/dt ∂2 r = = − uv 2 r du du/dt ∂vv

(10)

2 at points where ∂vv r = 0. dv ≤0 Showing that A is achronal thus amounts to showing that this slope du wherever it is defined, since the points at which the slope is not defined correspond to points on outgoing null segments. In fact, we will show that, where it is defined, dv du < 0, thereby excluding the possibility of ingoing null segments. Expanding the lefthand side of (3), we see that along A, 2 r = −r Ω−2 Tvv , Ω−2 ∂vv

or rather, 2 r = −rTvv . ∂vv

(11)

Substituting equations (11) and (7) into equation (10) then yields dv Ω2 α =− 3 . du 2r Tvv Since r, Ω, and α are all positive along A and Tvv is nonnegative by assumption I, dv < 0 at points along A at the dominant energy condition, we conclude that du which Tvv > 0, which is exactly what was needed.

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For technical reasons, in the next proposition we use K(u0 , v0 ) to denote the “compactification” of our initial rectangle K(u0 , v0 ), that is, K(u0 , v0 ) = [0, u0 ] × [v0 , ∞]. Set closures are taken with respect to K rather than K so as to include points at infinity (the Cauchy horizon). We also want to confine ourselves to regions of the spacetime in which r is close to r+ , so for any δ > 0, let W(δ) = (u, v) ∈ G(u0 , v0 ) : r(u, v) ≥ r+ − δ . Proposition 4. Suppose (G(u0 , v0 ), −Ω2 du dv) is a spacetime obtained as in Section 2.1 with radial function r, suppose it satisfies assumptions I–VII of Section 2.2, and suppose there exists δ > 0 such that for W = W(δ), condition A is satisfied in A ∩ W. If G(u0 , v0 ) does not contain a marginally trapped tube which is asymptotic to the event horizon, then W ∩ R contains a rectangle K(u1 , v1 ) for some u1 ∈ (0, u0 ], v1 ∈ [v0 , ∞). Proof. The proposition is an immediate consequence of the following two lemmas: Lemma 1. Under the hypotheses of Proposition 4, if A ∩ W = ∅, then A ∩ W is connected and terminates either at i+ (in which case it is asymptotic to the event horizon) or along the Cauchy horizon (0, u0 ]×{∞} ⊂ K(u0 , v0 ). In the latter case, W ∩ R contains a rectangle K(u1 , v1 ) for some u1 ∈ (0, u0 ], v1 ∈ [v0 , ∞). Lemma 2. Under the hypotheses of Proposition 4, if A ∩ W = ∅, then W ∩ R contains a rectangle K(u1 , v1 ) for some u1 ∈ (0, u0 ], v1 ∈ [v0 , ∞). Remark. The existence of the rectangle in W ∩R is what is required for the proof of the main result, but the first statement of Lemma 1 establishing the connectedness of A is of independent interest. As we will see in the proof of the lemma, that result hinges on the extension principle (VII) and the achronality of A. Proof of Lemma 1. First we lay some groundwork. Let S denote any connected component of ∂W ∩ G(u0 , v0 ), that is, a connected component of the level set {r = r+ − δ}. Since ∂u r is strictly negative by Proposition 1, the differential Dr is nondegenerate in all of G(u0 , v0 ), and thus S is a smooth curve segment whose endpoints lie on ∂G(u0 , v0 ). Parameterizing S by a curve γ(t) = (u(t), v(t)), we compute that du dv + (∂v r) . (12) 0 = γ(r) ˙ = (∂u r) dt dt dv Now, ∂v r > 0 in R and ∂u r < 0 everywhere, so du dt and dt must have the same sign in R, which in turn implies that du dv

γ, ˙ γ˙ = −Ω2 du dv (γ, ˙ γ) ˙ = −Ω2 v(˜ for any q˜ ∈ S0 , q˜ = q. Now, unless p1 lies on S0 , p1 must have v-coordinate greater than v(q), i.e. v∗ ≥ v(q) – otherwise, the fact that r(p1 ) = r+ − δ and r ≡ r+ − δ along S0 would contradict assumptions V and VI, the strict monotonicity of r along ingoing null rays and along Cout . Furthermore, p1 ∈ ∂W implies that p1 ∈ W,

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Cauchy horizon

11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 ?

q

S0

p0

W

i+

p1

A0

Cout

Cin

Figure 3. A priori, the curves S0 and A0 used in the proof of Lemma 1 may be situated as shown. Their endpoints q, p0 and p1 may lie in the spacetime G(u0 , v0 ) or in its boundary, G(u0 , v0 ) \ G(u0 , v0 ). The dashed curves and diagonal lines indicate boundaries and regions which also may or may not be part of the spacetime. Lemma 1 shows that the point p1 must in fact lie along the Cauchy horizon.

which in turn yields that if v∗ ≥ v(q), then either p1 = q or u∗ < u(q) since J + (q) \ {q} ∩ W = ∅. Thus the only possibilities remaining are that either p1 ∈ S0 , or u∗ < u(q) and v∗ ≥ v(q). If u∗ < u(q) and v∗ ≥ v(q), then since p1 ∈ A, the outgoing null ray to the past of p1 must lie in R ∪ A by Proposition 2, but it must also contain some point q˜ ∈ int(W). Then r(˜ q ) > r+ − δ, but r(p1 ) = r+ − δ and ∂v r ≥ 0 in R ∪ A, a contradiction. For the case p1 ∈ S0 , first note that since D(∂v r) is nondegenerate at p1 (see the proof of Proposition 3), it must be the case that the curve A leaves W at p1 as v increases. Together with the facts that A is achronal, S0 is causal, and p1 was chosen to be the endpoint of A0 with the largest v-coordinate, this implies that p1 = q. We are now in a position to derive the contradiction for this case. First note that both curves A and ∂W must extend smoothly through p1 = q. As noted in the preceding paragraph, by definition of A0 and p1 , the curve A leaves W at p1 as v increases. On the other hand, by definition of S0 and the characterization of its causal behavior given above, the curve ∂W must leave R ∪ A at q as v increases, passing into T ; in particular, it must become spacelike for v > v(q). And since ∂W is leaving R ∪ A, this spacelike curve-continuation past q = p1 must lie to the future of that of A at least locally. On the other hand, since A is leaving W at p1 = q, its continuation must lie to the future of that of ∂W, locally. But the two cannot coincide past p1 = q, by the choices of both p1 and q, so we have arrived at a contradiction.

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p1 = (u∗ , v∗ ) (˜ u, v∗ )

A0

Cout u=u ˜

v = v∗ v = v˜ v = v∗ −

Figure 4. The dashed line indicates the boundary of the spacetime, G(u0 , v0 ) \ G(u0 , v0 ). The points p1 = (u∗ , v∗ ) and (˜ u, v∗ ) lie in this boundary, not in G(u0 , v0 ) itself. The dark-shaded rectanu, v∗ ) ∩ J + (0, v˜) \ {(˜ u, v∗ )} to which we apply gle is the set J − (˜ the extension principle (assumption VII) and derive a contradiction.

Thus p1 cannot lie in G(u0 , v0 ), so we must have p1 ∈ G(u0 , v0 ) \ G(u0 , v0 ). If v∗ < ∞, then consider the ingoing null ray to the past of p1 , [0, u∗ ] × {v∗ }. Since the point (0, v∗ ) ∈ G(u0 , v0 ) and G(u0 , v0 ) is open, there must exist some u, v∗ ) ∈ / G(u0 , v0 ). Since p1 is a limit point of A but smallest u ˜ ∈ (0, u∗ ] such that (˜ is not in the spacetime, we must have p0 = p1 , and so since A0 is achronal with no ingoing segments, we can parameterize a portion of A0 in a neighborhood of p1 by (u(v), v), v ∈ (v∗ − , v∗ ], some > 0. For each value v ∈ (v∗ − , v∗ ), the ingoing null ray to the past of the point (u(v), v) ∈ A ∩ W must be contained in W ∩ R – the ray must lie in W since ∂u r < 0 along it, and it must lie in R since ∂u ∂v r < 0 in A ∩ W. So in particular, if we choose some v˜ ∈ (v∗ − , v∗ ), then u, v∗ ) ∩ J + (0, v˜) \ {(˜ u, v∗ )} ⊂ R ∪ A, and hence by the extension principle, we J − (˜ must have (˜ u, v∗ ) ∈ R ∪ A as well, a contradiction. (See Figure 4 for a Penrose diagram of this situation.) Thus we conclude that v∗ = ∞, and we are done; this is what we wanted to show. Now we have shown that an arbitrary connected component of A ∩ W must terminate along the Cauchy horizon [0, u0 ] × {∞}. To see that this component is unique, i.e. A ∩ W is connected, recall that since condition A holds in A ∩ W, we have ∂u (∂v r) < 0 there; see the proof of Proposition 3. This monotonicity in turn implies that any future-directed ingoing null ray intersecting A∩W must pass from R into T , and so every such ingoing null ray can intersect at most component of A ∩ W (note that if the ray leaves W, then it cannot reenter it since ∂u r < 0). Therefore, since every connected component of A ∩ W must exist for arbitrarily large v, there can be at most one component. The last statement of the lemma follows immediately. Proof of Lemma 2. If A ∩ W = ∅, then W ⊂ R, so in fact W ∩ R = W. Fix a reference point (u1 , v1 ) ∈ int(W) ∩ R such that u1 > 0. Note that since ∂u r < 0,

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the past-directed ingoing null ray behind (u1 , v1 ) must also be in int(W), that is, [0, u1 ] × {v1 } ⊂ int(W). If K(u1 , v1 ) is not wholly contained in W, then there exists some q ∈ (∂W) ∩ K(u1 , v1 ) = ∅, where the boundary ∂W is taken here with respect to K(u0 , v0 ) (as opposed to G(u0 , v0 )). In particular, q cannot lie on [0, u1 ] × {v1 } by choice of (u1 , v1 ), nor can it lie elsewhere in W ∩ K(u1 , v1 ), since that would imply that r(q) = r+ − δ, violating the monotonicity of r in R (∂v r > 0) and the fact that r > r+ − δ on [0, u1 ] × {v1 }. Hence q ∈ G(u0 , v0 ) \ G(u0 , v0 ). Fix v∗ to be the smallest value in (v1 , ∞) such that [0, u1 ]×{v∗ }∩(G(u0 , v0 )\ G(u0 , v0 )) = ∅, and set u∗ to be the smallest value in (0, u1 ] such that (u∗ , v∗ ) ∈ G(u0 , v0 ) \ G(u0 , v0 ). Then by construction, the rectangle J − (u∗ , v∗ ) ∩ J + (0, v1 ) \ {(u∗ , v∗ )} ⊂ R, and so since r is bounded below by r+ − δ near (u∗ , v∗ ), the extension principle implies that (u∗ , v∗ ) ∈ G(u0 , v0 ), a contradiction. Thus we must in fact have (∂W) ∩ K(u1 , v1 ) = ∅, which implies that K(u1 , v1 ) ⊂ W ∩ R.

4. Main results 4.1. Asymptotic behavior We are now ready to state and prove our main result characterizing the asymptotic behavior of certain marginally trapped tubes. Afterward we describe how an estimate obtained in the proof of theorem implies that the event horizon of the given spacetime must be future geodesically complete. We then present a second theorem relating the lengths of such tubes to Price law decay. Theorem 1. Suppose (G(u0 , v0 ), −Ω2 du dv) is a spacetime obtained as in Section 2.1 with radial function r, and suppose it satisfies the assumptions I–VII of Section 2.2. Define W(δ) = (u, v) ∈ G(u0 , v0 ) : r(u, v) ≥ r+ − δ and assume that there exist a constant 0 < c0 < constants 0 <
0,

− c0 and v ≥ v0 , and some small δ > 0 such that for

W = W(δ) the following conditions hold: A

Tuv Ω−2 ≤ c0 in W ∩ R;

B1 Tuu /(∂u r)2 ≤ c1 in W ∩ R; B2 ∂v (Ω−2 Tuv )(u, · ) ∈ L1 ([v0 , ∞)) for all u ∈ [0, u0 ], and v ∂ (Ω−2 Tuv )(u, v˜) d˜ v < for all (u, v) ∈ W ∩ R with v ≥ v ; v v C

(−∂u r)Ω−2 ≤ c2 along Cout ∩ W.

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i+ T

R

A Cout

Figure 5. Theorem 1 says that the spacetime must contain a (small) characteristic rectangle whose Penrose diagram looks like this – in particular, the marginally trapped tube A is achronal and terminates at i+ .

Then the spacetime G(u0 , v0 ) contains a marginally trapped tube A which is asymptotic to the event horizon, i.e. for every small u > 0, there exists some v > v0 such that (u, v) ∈ A. Furthermore, for large v, A is connected and achronal with no ingoing null segments. See Figure 5 for a representative Penrose diagram. Remark. The physical meaning of condition A , which is somewhat stronger than condition A, is readily apparent from Proposition 3: it controls the causal behavior of the marginally trapped tube, if one exists. Conditions B1 and B2 have no obvious general physical meaning. (But note that B2 is automatically satisfied if ∂v (Ω−2 Tuv ) ≤ 0 in W ∩ R.) Condition C determines a gauge along the event horizon; it may alternately be expressed as saying that the quantities (1 − 2m r ) and ∂v r approach zero at proportional rates as v tends to infinity along Cout , which in turn implies that m → 12 r along Cout , i.e. that 2m+ = r+ . See also the remarks concerning conditions A and C following the proof of the theorem. These four conditions, as well as the proof of the theorem, were obtained by extrapolating portions of the bootstrap argument in Section 7 of [9] for Einstein– Maxwell scalar fields. Conditions A , B2, and C are all satisfied for sufficiently small δ in any Einstein–Maxwell-scalar field black hole, provided e < 2m+ . (The particular choice of the upper bound for c0 in A is analogous to the condition in [9] that the black hole not be “extremal in the limit,” i.e. e < 2m+ .) Condition B1 also holds in all the spacetimes considered in [9], but there the proof hinges on the Price law decay imposed on Tvv ; one integrates the scalar field equation by parts and uses the polynomial decay of Tvv in the v-direction to obtain the bound on Tuu /(∂u r)2 . (In fact one obtains something stronger than B1 this way, that Tuu /(∂u r)2 decays polynomially with v.)

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Proof. Our first step is to shrink δ in order to align with the choice of . For this, consider the quantity 3 r+ − δ ∗ 1

δ∗ ∗ c1 r + δ ∗ 1 − 2 − c c e (2c0 + 3M ) , Λ( ∗ , δ ∗ ) := 2 0 − 2 r+ 4r+ r+

where M :=

v

sup (u,v)∈G(u0 ,v0 )

|∂v (Ω−2 Tuv )(u, v˜)| d˜ v.

v0

Clearly Λ is positive for ∗ and δ ∗ sufficiently small and Λ

1 2 4r+

−c0 as ∗ , δ ∗ 0.

Then since < 4r12 −c0 , there exist 1 , δ1 > 0 such that Λ( 1 , δ1 ) > . Without loss + of generality, we may assume that δ1 ≤ δ, and henceforth we restrict our attention to the (possibly) smaller region W(δ1 ), using W to denote it rather than W(δ). We make use of 1 in what follows. Now, from condition A it follows immediately that condition A holds in W ∩ R. This in turn implies that condition A holds in A ∩ W. (To see that A \ ∂R = ∅, observe that a point p ∈ A \ ∂R would have to have a neighborhood lying entirely in T ∪ A; hence by Proposition 2, any connected component of A \ ∂R must be an outgoing null ray to the past of p and must intersect either ∂R or Cin . The former case contradicts the achronality of A at points where condition A holds (Proposition 3), and the latter case we can exclude, without any loss of generality, by suitably shrinking u0 .) Thus by Proposition 4, either W ∩ R contains a rectangle K(u1 , v1 ) for some u1 ∈ (0, u0 ], v1 ∈ [v0 , ∞), or the spacetime contains a marginally trapped tube A which is asymptotic to the event horizon. We will show that the existence of the rectangle in the former case leads to a contradiction and thus conclude that the latter statement is true. Furthermore, given how Proposition 4 was proved, i.e. via Lemma 1, we will then know that in fact it is A ∩ W which is asymptotic to the event horizon, so in particular A must be achronal with no ingoing null segments (by Proposition 3) and connected (by Lemma 1), proving the theorem. For the remainder of the proof, we restrict our attention to the region K(u1 , v1 ). (We may assume without loss of generality that v1 ≥ v .) Define 1 . Now, using equation (2) and condiκ = − 14 Ω2 (∂u r)−1 . Then −Ω−2 ∂u r = 4κ tion B1, we have ∂u log(−Ω−2 ∂u r) = −r(∂u r)−1 Tuu ≤ −c1 r(∂u r) , so integrating along an ingoing null ray, we have u κ(0, v) log ∂u log(−Ω−2 ∂u r)(˜ u, v)d˜ u = κ(u, v) 0 u ≤− c1 r(∂u r)(˜ u, v)d˜ u 0

c1 2 =− r (u, v) − r2 (0, v) 2

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c1 2 (r+ − δ1 )2 − r+ 2 ≤ c1 r+ δ1 ,

≤− which yields

κ(u, v) ≥ κ(0, v)e−c1 r+ δ1 . Then since condition C implies that κ(0, v) ≥ 4c12 for all v ≥ v1 , we have a lower bound 1 −c1 r+ δ1 κ ≥ κ0 := e >0 4c2 in all of K(u1 , v1 ). Next, since r(0, v) → r+ as v → ∞, ∂v r cannot have a positive lower bound along Cout ; thus there must exist some V ≥ v1 such that ∂v r(0, V ) < 1 . By continuity, there is a neighborhood of (0, V ) in G(u0 , v0 ) in which this inequality holds, and in particular, there exists some 0 , we have 3 r+ − δ1 1 δ1 c1 r+ δ1 − c0 , (2c0 + 3M ) + < 2 1 − 2 1 c2 e r+ r+ 4r+ or equivalently, setting r0 = r+ − δ1 , 2 3 (2c0 + 3M ) + 2r+ < 2δ1 r+

r03 2

1 2 r+

1−

1 2κ0

− 4c0 .

We can thus fix constants α0 and α1 such that 1 r3 1 2 3 2δ1 r+ (2c0 + 3M ) + 2r+ < α0 < 0 1 − − 4c 0 2 2 r+ 2κ0

(14)

and 2 3 (2c0 + 3M ) − 2r+ > 0. α1 = α0 − 2δ1 r+ As in the proof of Proposition 3, define a function α on G(u0 , v0 ):

(15)

α(u, v) = m − 2r3 Ω−2 Tuv (u, v) . Using our lower bound for κ, (13), (14), and condition A , we see that α > α0 on [0, U ] × {V }: α = m − 2r3 Ω−2 Tuv r 1 + 4 Ω−2 ∂u r ∂v r − 2r3 Ω−2 Tuv = 2 r3 1 1 −2 ∂v r − 4 Ω Tuv = 1− 2 r2 2κ 1 r03 1 ≥ 1− − 4c0 2 2 r+ 2κ0 > α0 .

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Our goal now is to deduce that α > α1 in K(U, V ), using B2. First we compute: ∂v α = ∂v (m − 2r3 Ω−2 Tuv ) = ∂v m − ∂v (2r3 Ω−2 Tuv ) = 2r2 Ω−2 (Tuv ∂v r − Tvv ∂u r) − 2r3 ∂v (Ω−2 Tuv ) − 6r2 (∂v r)Ω−2 Tuv = −4r2 Ω−2 Tuv ∂v r − 2r2 Ω−2 Tvv ∂u r − 2r3 ∂v (Ω−2 Tuv ). ≥ −4r2 Ω−2 Tuv ∂v r − 2r3 ∂v (Ω−2 Tuv ) ,

(16)

where the last inequality follows from assumptions I and V. The next step is to integrate (16) along an outgoing null ray {u} × [V, v), but first let us consider the two summands on the right hand side separately. First, v v 4 −4r2 Ω−2 Tuv ∂v r = − (∂v r3 )(Ω−2 Tuv ) 3 V Vv 4 − c0 (∂v r3 ) ≥ 3 V 4 3 > − c0 r+ − (r+ − δ1 )3 3 2 > −4c0 r+ δ1 . For the second summand of (16), we use the following notation: given a function f , f + = max{f, 0} and f − = max{−f, 0}, so that f = f + − f − . Then: v v

+

− 2r3 ∂v (Ω−2 Tuv ) = 2r3 ∂v (Ω−2 Tuv ) − 2r3 ∂v (Ω−2 Tuv ) V Vv

+

− 3 ∂v (Ω−2 Tuv ) − 2(r+ − δ1 )3 ∂v (Ω−2 Tuv ) ≤ 2r+ Vv 3 ∂v (Ω−2 Tuv ) = 2r+ V v

− 2 2(3r+ δ1 − 3r+ δ12 + δ13 ) ∂v (Ω−2 Tuv ) + V v

− 3 2 ∂v (Ω−2 Tuv ) ≤ 2r+ + 6r+ δ1 Vv 3 2 ≤ 2r+ + 6r+ δ1 |∂v (Ω−2 Tuv )| V 3 2 ≤ 2r+ + 6r+ δ1 M .

Integrating (16) now yields 2 3 2 δ1 − (2r+ + 6r+ δ1 M ) α(u, v) > α(u, V ) − 4c0 r+ 3 2 − 2r+ δ1 (2c0 + 3M ) > α0 − 2r+ = α1 .

Thus we conclude that α > α1 in all of K(U, V ).

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Finally, recall from equation (7) that 1 2 r = − Ω2 r−2 α , ∂uv 2 or, using the definition of κ, 2 ∂uv r = 2κr−2 (∂u r)α .

Rearranging and applying our bounds in our region of interest, we have ∂v log(−∂u r) = 2κr−2 α −2 > 2κ0 r+ α1 ,

and so integrating along an outgoing ray yields −2 ∂u r(u, v) > e2κ0 r+ α1 (v−V ) , ∂u r(u, V )

and hence

−2

−∂u r(u, v) > −∂u r(u, V ) e2κ0 r+ Assume ∂u r(u, V ) ≤ −b0 < 0 for all 0 ≤ u ≤ U , let

α1 (v−V )

.

(17)

−2 b1 = 2κ0 r+ α1 ,

and set

b2 = b0 e−b1 V ;

then −∂u r(u, v) > b2 eb1 v and so integrating along an ingoing null ray, we get r(0, v) − r(u, v) > b2 eb1 v u , i.e. r(u, v) < r(0, v) − b2 eb1 v u . But for any u > 0, the right-hand side tends to −∞ as v → ∞, while the left-hand side is positive. Thus we have arrived at a contradiction, so no such rectangle K(U, V ) can be contained in R, and the statement of the theorem follows. Remark. Given the last estimate obtained in the proof of Theorem 1, it is essentially immediate that the event horizon of the black hole is future geodesically complete, i.e. Cout has infinite affine length, whenever the hypotheses of Theorem 1 are satisfied. (Note, however, that this result does not require all of the hypotheses of Theorem 1; the argument below could be refined such that assumptions C and A (restricted to W ∩ Cout ) are sufficient.) Suppose s is an affine parameter for Cout = {0} × [v0 , ∞) = {(0, v(s))} which ∂ increases to the future. Then the vector field X = dv ds ∂v satisfies ∇X X = 0, which in this setting becomes dv dv dv 2 ∂v (log Ω ) = 0 , ∂v + ds ds ds

Vol. 9 (2008)


or equivalently, since

dv ds

> 0,

2 dv

∂v log Ω

ds

1049

= 0.

Integrating, we have dv = a0 Ω−2 ds for some a0 > 0, and so now taking s as a function of the outgoing null coordinate v, we have v

Ω2 (0, v)dv ,

s(v) = a1 + a2 v0

some a1 , a2 ∈ R, a2 > 0. Thus s has infinite range if and only if Ω ∈ / L2 ([v0 , ∞)). Now, one of the hypotheses of Theorem 1 was that (−∂u r)Ω−2 ≤ c2 along Cout ∩ W for some constant c2 > 0 (condition C), and in the proof of the theorem, we found that for some U > 0, V ≥ v0 and some b1 , b2 > 0, −∂u r(u, v) > b2 eb1 v for all 0 ≤ u ≤ U , V ≤ v (equation (17)). Putting these inequalities together and evaluating along Cout , we have b2 eb1 v < −∂u r(0, v) ≤ c2 Ω2 (0, v) for all V ≤ v, so in fact Ω ∈ / L2 ([v0 , ∞)) and Cout is future complete.

5. Immediate applications 5.1. Price law decay & length Theorem 2. Suppose (G(u0 , v0 ), −Ω2 du dv) is a spacetime obtained as in Section 2.1 with radial function r, and suppose it satisfies assumptions I–VII of Section 2.2. Suppose A0 is a connected component of A along which condition A is satisfied, and suppose that in addition, Tvv ≤ c3 v −2−

along

A0

for some c3 ≥ 0, > 0. Then A0 has finite length with respect to the induced metric. Remark. The rate of decay of Tvv given corresponds to that of Price’s law; cf. [9,15]. Theorem 2 applies in particular to the case of the marginally trapped tube A ∩ W obtained in Theorem 1, but it does not require that the tube terminate at i+ in order to be valid. In the context of Theorem 1, this decay rate gives a direct measure of how quickly the tube approaches the event horizon. Proof. Since A0 is connected and achronal with no ingoing null segments, we may parameterize it by its v coordinate, i.e. γ(v) = (u(v), v) ∈ A0 . If the domain of

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the v coordinate is bounded, then the result is trivially true, so suppose v has domain [V, ∞) for some V ≥ v0 . Then we have du 2 2 |γ(v)| ˙ = γ(v), ˙ γ(v) ˙ = −Ω . dv Using the relation γ(∂ ˙ v r) = 0, we readily compute that ∂2 r 2r3 Tvv du = − vv =− 2 , 2 dv ∂uv r Ω α so

2r3 Tvv . α As inthe proof of Proposition 3, we compute that α > α0 along A, so setting 3 α−1 , we have b3 = 2r+ 0 ∞

∞ ∞ 12 |γ(v)|dv ˙ ≤ b3 Tvv γ(v) dv ≤ b3 c3 v −1−/2 dv < ∞ , 2 |γ(v)| ˙ =

V

V

V

i.e., the length of A0 is finite.

5.2. Vaidya spacetimes Perhaps the simplest example at hand when one is working with dynamical horizons is that of the (ingoing) Vaidya spacetime, the spherically symmetric solution to Einstein’s equations with an ingoing null fluid as source. It is widely accepted that the Vaidya marginally trapped tube is asymptotic to the event horizon, but the literature seems to be lacking an analytical proof of this behavior for an arbitrary mass function, so it is worth seeing how our results apply to this case. Recall that the ingoing Vaidya metric is given in terms of the ingoing Eddington–Finkelstein coordinates (v, r, θ, ϕ) by 2M (v) g =− 1− dv 2 + 2dvdr + r2 (dθ2 + sin2 θdϕ2 ) , r with stress-energy tensor M˙ (v) 2 dv , r2 where and M (v) is any smooth function of v. One can show directly that the marginally trapped tube is the hypersurface at which r = 2M (v); it is spacelike where M˙ (v) > 0, null where M˙ (v) = 0, and timelike where M˙ (v) < 0. By inspection T satisfies the dominant energy condition, assumption I, if and only if M (v) is nondecreasing. We restrict our attention to a characteristic rectangle in which M is strictly positive and indicate how the remaining assumptions II–VII are satisfied: the metric is regular everywhere in the rectangle except at the singularity at r = 0 (albeit not in the coordinates given above), so it follows that assumption VII is satisfied, and furthermore, that singularity is evidently spacelike, so II is satisfied as well. The inner expansion of each round 2-sphere is (some positive multiple of) − 2r , i.e. it is strictly negative, so assumption V holds. T =

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Finally, assuming M (v) < M0 for some limiting value M0 < ∞, assumptions III, IV, and VI are all satisfied as well (the strict inequality is what yields VI). Now, in order to check the hypotheses of Theorem 1, it seems we ought to convert from Eddington–Finkelstein to double-null coordinates. Unfortunately, to make such a conversion explicitly is impossible in general; see [17]. However, we can still compute the relevant quantities from first principles. Following the treatment given in [17], we find that in double-null coordinates (u, v, θ, ϕ) with v scaled such that its domain is [v0 , ∞), the only nonzero component of Tαβ is Tvv . Thus conditions A , B1, and B2 are all trivially satisfied. We can also deduce that the term −(∂u r)Ω−2 = 12 everywhere, so C is satisfied as well. Thus we conclude from Theorem 1 that the marginally trapped tube is asymptotic to the event horizon. Furthermore, as one might expect, Theorem 2 tells us that the length of the tube depends on the rate of decay of M˙ (v) as v → ∞. In particular, if M˙ (v) = O(v −2− ) for some > 0, or more generally, if (M˙ (v))1/2 ∈ L1 ([V, ∞)) for some V , then the tube has finite length. 5.3. Self-gravitating Higgs fields We next consider a self-gravitating Higgs field with non-zero potential. This matter model consists of a scalar function φ on the spacetime and a potential function V (φ) such that φ = g αβ φ;αβ = V (φ) . (18) The stress-energy tensor then takes the form 1 Tαβ = φ;α φ;β − φ;γ φ;γ + V (φ) gαβ . (19) 2 In our spherically symmetric setting, φ = φ(u, v), so the evolution equation (18) becomes 2 V (φ) = −4Ω−2 ∂uv φ + ∂u φ (∂v log r) + ∂v φ (∂u log r) (20) and in double-null coordinates, (19) yields Tuu = (∂u φ)2 Tvv = (∂v φ)2 and

1 2 Ω V (φ) . 2 Note that the dominant energy condition (I) is satisfied if and only if V (φ) is nonnegative. The extension principle (VII) is known to hold for self-gravitating Higgs fields if V ≥ −C for any finite C [10]. We give two applications of Theorem 1 to self-gravitating Higgs field black hole spacetimes. In Theorem 3, we assume that the scalar field and the potential both satisfy weak Price-law-like decay conditions on the event horizon, namely that |∂v φ| and |V (φ)| ∈ O(v −p ) for some constant p > 12 . For Theorem 4, we make only certain monotonicity and smallness assumptions, including that V is convex. In both cases, we extract the hypotheses of Theorem 1 and conclude that the Tuv =

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respective black holes each contain a marginally trapped tube which is asymptotic to the event horizon, achronal with no ingoing null segments, and connected for large v. The advantage of Theorem 4 over Theorem 3 is that it does not require an explicit rate of decay. However, the assumptions which we must make in the absence of such a decay rate are highly nontrivial. In particular, in the case of a Klein– Gordon potential of mass m, i.e. V (φ) = 12 m2 φ2 , the hypotheses of Theorem 4 are satisfied only if φ decays exponentially along the event horizon. In the case of an exponential potential V (φ) = cekφ , the hypotheses cannot even be satisfied simultaneously. By contrast, Theorem 3 can be applied in both such settings. On the other hand, for any potential of the form V (φ) = cφk+2 , k > 0, the hypotheses 1 of Theorem 4 may be readily satisfied; in this case they imply that φ ∈ O(v − k ) 1 and V (φ) ∈ O(v −1− k ) along the event horizon but make no a priori restriction on |∂v φ|. Indeed, it is possible to construct an admissible φ along the event horizon in this setting such that lim supv→∞ |∂v φ|v p = ∞ for any p > 0, which then implies that Price law decay per se does not hold. For both of the following theorems, we assume we have initial data r, Ω, φ for a self-gravitating Higgs field along the null hypersurfaces Cin ∪Cout = [0, u0 ]×{v0 }∪ {0} × [v0 , ∞) with nonnegative potential function V ∈ C 2 (R), and we suppose the data satisfy assumptions III-VI, namely: r ≤ r+ and 0 ≤ m ≤ m+ along Cin ∪ Cout , and ∂u r < 0, ∂v r > 0 along Cout . Theorem 3. Fix a constant p > 12 and a function η(v) > 0 such that η(v) decreases monotonically to 0 as v tends to infinity. Suppose the second derivative of the potential V is bounded, i.e. there exists a constant B such that |V (x)| ≤ B on the interval (φ0 − δ0 , φ1 + δ0 ) for some δ0 > 0, where φ0 and φ1 are the (possibly infinite) inf and sup of φ along Cout , respectively. If along Cout the initial data satisfy ∂v r < η(v) , 1 |∂v φ| < b1 v −p , 2 1 |V (φ)| < b2 v −p , 2 1 −2 c3 ≤ −(∂u r)Ω ≤ c2 , 2

(21)

and lim inf V (φ) < v→∞

1 2 , 4r+

(22)

for some positive constants b1 , b2 , c2 , and c3 , then the result of Theorem 1 holds for maximal development of these initial data.

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Remark. Note that if the constant p > 1, then ∂v φ is integrable along the event horizon, which in turn implies that the domain of φ is compact and hence that V is a priori bounded on the relevant domain. Thus the hypothesis that V be bounded is only necessary for 12 < p ≤ 1. Theorem 4. Suppose there exist positive constants c0 , c1 , c2 , c3 , and c4 such that c0 < along Cin

∂u φ ∂u r

1 2 ; 4r+

(23)

2 < c1 ;

(24)

and along Cout c3 ≤ −(∂u r)Ω−2 ≤

1 c2 2

(25)

and V (φ) < c4 |∂v φ| .

(26)

Suppose also that the potential V satisfies 0 ≤ V (x) ≤ B

(27)

on the interval (φ0 − δ0 , φ1 ) for some δ0 > 0, where φ0 and φ1 are the (possibly infinite) inf and sup of φ along Cout , respectively, and B is a constant satisfying −2 . B < r+ If along Cout the initial data satisfy ∂v r < 1 V (φ) < 2 1 |∂v φ| < 2 for sufficiently small , , and > 0, as well as ∂v φ < 0 ∂u φ < 0 √ −1 V (φ) − 4 c1 c2 (∂v log r) − 4c3 r+ (∂v φ) > 0 ,

and either V (φ) ≤ 0 or |φ0 | < ∞ , then the result of Theorem 1 holds for maximal development of these initial data.

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Remark. Stated more precisely, the requirement that the constants , , and be sufficiently small is the following: −2 −2 < r+ −B, 2 + 8c2 r+ 1 < min − 2c , 2c 0 0 , 2 2r+

and

√ 2 c1 c3 (1 − 2r+ − 4c2 ) . < c2 (c4 r+ + 4c3 )

The existence of constants c0 and c1 satisfying (23) and (24) is not restrictive, but that of constants c2 and c3 in (25) is. Inequalities (43) and (44) together imply that the scalar field φ has a timelike gradient. Note that if V (φ) ≤ 0 along Cout , then (26) is trivially true for any c4 . Proof of Theorem 3. Let G(u0 , v0 ) denote the maximal development of the given initial data. By (22), we may choose a positive constant c0 such that 1 lim inf V φ(0, v) < c0 < 2 . v→∞ 4r+ Thus there exist some small 0 < < 2c0 and a sequence {v k } → ∞ such that V (φ(0, v k )) < c0 − 12 for all k. Choose > 0 such that 1 1 < min c0 − , 2 − c0 . 2 4r+ Since v k → ∞, we can find K sufficiently large that for v ≥ v K , b1 b2 1−2p v < , 2p − 1 2 2c2 η(v) < r+ , and

2c2 r+ p

log v v

2 − 2c2 η(v) . < r+

Set v1 = v K and note that by construction V (φ(0, v1 )) < c0 − 12 . Next, set b0 = η(v1 ), let b2 2c2 r+ 2b1 b3 = 2 + 2 − 2c b 4c3 r+ r+ 2 0 −p 2c2 r+ p log v1 −p · v1 + 1 − , 2 − 2c b r+ v1 2 0 and fix b3 such that

∂u φ p (0, v1 ) v1 , b3 . b3 > max 2 ∂u r

(28)

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Now, continuity at the point (0, v1 ) and our initial conditions along Cout imply that there exists u1 > 0 sufficiently small that ∂v r(u, v1 ) < b0 , 1 |∂v φ(u, v1 )| < b1 v1−p , 2 V φ(u, v1 ) < 1 b2 v −p , 1 2 1 ∂u φ −p ∂u r (u, v1 ) < 2 b3 v1 , and

1 V φ(u, v1 ) < c0 − 2 for all u ∈ [0, u1 ]. Set Cin = [0, u1 ] × {v1 } and Cout = {0} × [v1 , ∞). Henceforth we will consider the subregion of G(u0 , v0 ) given by G(u1 , v1 ) := K(u1 , v1 ) ∩ G(u0 , v0 ) , ∪ Cout . i.e. the maximal development of the induced initial data on Cin Finally, we choose −1 2b0 b3 + 2b1 b2 δ0 v1p v12p r+ b1 b2 , + , , , 0 < δ < min , 2 2 4c3 r+ 2Bb3 b3 2b23 r+

and as usual, define

W = (u, v) ∈ G(u1 , v1 ) | r(u, v) ≥ r+ − δ .

Define a region V as the set of points (u, v) ∈ G(u1 , v1 ) such that the following seven inequalities hold for all (˜ u, v˜) ∈ J − (u, v) ∩ G(u1 , v1 ): ∂v r < b0 |∂v φ| < b1 v

(29) −p −p

|V (φ)| < b2 v ∂u φ −p ∂u r < b3 v −(∂u r)Ω−2 < c2 V (φ) < 2c0 − ∂v r > 0 .

(30) (31) (32) (33) (34) (35)

Note that (35) implies that V ⊂ R, so W ∩ V ⊂ W ∩ R. Since we can easily extract the hypotheses of Theorem 1 from these inequalities, our goal is to prove that in fact W ∩ R ⊂ V. We accomplish this by means of a bootstrap argument showing that W ∩ R = W ∩ V. Before proceeding, however, let us show that the hypotheses of Theorem 1 are satisfied.

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First we check that conditions A –C hold in V. Note that V is a past set by ⊂ V. Thus for (u, v) ∈ V, definition, and our choices of u1 and v1 imply that Cin we have v v b1 b2 1−2p ∂v V φ(u, v˜) d˜ v v ≤ b1 b2 v˜−2p d˜ v< < . (36) 2p − 1 1 v1 v1 Since < 4r12 − c0 , (36) implies that B2 is satisfied, and since < c0 − 12 , it also + implies that 1 1 1 1 1 V φ(u, v) < V φ(u, v1 ) + < c0 − < c0 < 2 , 2 2 2 2 4r+ so A is satisfied as well. Condition B1 follows immediately from (32), and C follows from the hypothesis (21). We must also verify that assumptions I–VII hold in G(u1 , v1 ). The requirement that V be nonnegative implies I, while II follows by construction, since ∪ Cout . Assumptions III G(u1 , v1 ) is the maximal development of initial data on Cin and IV hold on Cin ∪ Cout by monotonicities of r and m in R, respectively, while V and VI were among the hypotheses of the theorem. Finally, assumption VII holds by [10] since V is nonnegative. We now turn to the bootstrap argument, which we carry out as follows: we first retrieve (strict) inequalities (32) and (34) in V, where the set closure is taken with respect to G(u1 , v1 ), i.e. V = V ∩ G(u1 , v1 ). Since inequalities (29)– by hypothesis, a continuity argument then (31), (33) and (35) hold along Cout . Thus implies that Cout ⊂ V, i.e. that both (32) and (34) hold along all of Cout + W ∩ V = ∅, since W must contain a neighborhood of i = (0, ∞). We then retrieve inequalities (29)–(31) and (33) in W ∩ V and conclude, again by continuity, that in fact W ∩ V = W ∩ R. It will again be convenient to use the quantity κ introduced in the proof of Theorem 1, ∂v r 1 . (37) κ = − Ω2 (∂u r)−1 = 4 1 − 2m r Equation (2) implies that ∂u κ ≤ 0, and combining this fact with (21), we have κ ≤ 4c13 in all of G(u1 , v1 ). The bootstrap inequality (33) implies that κ ≥ 4c12 in all of V as well, so by (29), 2m (38) 1− (u, v) = (∂v r)κ−1 (u, v) ≤ 4c2 b0 r in V. Also, note that combining equations (2) and (4) (or alternately (3) and (5)) yields 1 2 −1 2m 2 2 ∂uv r = Ω r 2r V (φ) + 1 − −1 . (39) 4 r

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Let us now retrieve inequality (32) in V. First we observe that (20) may be rearranged as 1 2 φ = − Ω2 V (φ) − (∂u φ)(∂v log r) − (∂v φ)(∂u log r) . (40) ∂uv 4 Set r1 = r(0, v1 ) and note that ∂v r > 0 implies that r ≥ r1 on all of Cout . We compute: 2 ∂u φ ∂u φ ∂uv r ∂2 φ ∂v = uv − ∂u r ∂u r ∂u r ∂u r 1 = − Ω2 V (φ) − (∂u φ)(∂v log r) − (∂v φ)(∂u log r) (∂u r)−1 4 ∂u φ 1 2 −1 2m Ω r − 2r2 V (φ) − 1 − − 1 (∂u r)−1 ∂u r 4 r ∂u φ ∂v φ = κV (φ) − (∂v log r) − ∂u r r ∂u φ 2m − κr−1 1 − 2r2 V (φ) − 1 − ∂u r r φ ∂ v = κV (φ) − r ∂u φ 2m − κr−1 1 − 2r2 V (φ) − 1 − + (∂v log r) . ∂u r r Let A := κr

−1

2m 2 1 − 2r V (φ) − 1 − + (∂v log r) , r

so that we may write ∂u φ ∂u φ ∂v φ ∂v = −A + κV (φ) − . ∂u r ∂u r r Using (34), (38) and the fact that c0
+ =: a0 . 4c2 r1 2c2 r1 The constant a0 is positive by our choice of b0 . Also, (30) and (31) imply that κV (φ) − ∂v φ ≤ b2 + b1 r−1 v −p =: a1 v −p . 1 r 4c3

A≥

Then for (u, v) ∈ V, integrating (41) along the outgoing null ray {u} × [v1 , v] yields ∂u φ ∂u φ − v A(u,¯ v ) d¯ v (42) (u, v) = e v1 (u, v1 ) ∂u r ∂u r v v ∂v φ e− v˜ A(u,¯v) d¯v κV (φ) − + (u, v˜) d˜ v, r v1

1058

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so


v ∂u φ (u, v) ≤ e− v1 A(u,¯v) d¯v ∂u φ (u, v1 ) ∂u r ∂u r v v ∂v φ (u, v˜) d˜ v e− v˜ A(u,¯v) d¯v κV (φ) − + r v1 v v 1 − v a d¯ v ≤ b3 v1−p e v1 0 + a1 v˜−p e− v˜ a0 d¯v d˜ v 2 v1 v 1 v˜−p ea0 v˜ d˜ v. ≤ b3 v1−p ea0 (v1 −v) + a1 e−a0 v 2 v1

Integrating the second term by parts, we have: v −p a0 v1 −a0 v −p a0 v ˜ −a0 v −p a0 v v˜ e d˜ v = a1 e e − a−1 a−1 a1 e 0 v 0 v1 e v1

v

+

a−1 v −p−1 ea0 v˜ d˜ v 0 p˜

v1

≤

−p a1 a−1 0 v

+

−a0 v pa1 a−1 0 e

v

v˜−p−1 ea0 v˜ d˜ v.

v1

Furthermore, we estimate that v v v˜−p−1 ea0 v˜ d˜ v= v1

v− ap 0

≤ ea0 v

v˜−p−1 ea0 v˜ d˜ v+

v− ap log v 0

v˜−p−1 ea0 v˜ d˜ v

v1

log v v

v˜−p−1 d˜ v+

v− ap log v 0

v− ap log v 0 + ea0 v−p log v v˜−p−1 d˜ v v1 (v − ap0 log v)−p −v −p a0 v + =e + p p −(v − ap0 log v)−p v1 −p a0 v −p + +e v p p −p p −1 −1 a0 v −p −p ≤p e v −1 + 1 − v log v + v1 . a0 Putting it all back together yields ∂u φ 1 −p a0 (v1 −v) −p −a0 v + a1 a−1 + (pa1 a−1 ) 0 v 0 e ∂u r (u, v) ≤ 2 b3 v1 e −p p −1 −1 a0 v −p −p −1 + 1 − v log v + v1 · p e v a0

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−p 1 p −1 −p a0 (v1 −v) −1 −p −p ≤ b3 v1 e + a1 a0 v + v1 1 − v log v 2 a0 1 1 p −p a0 (v1 −v) b3 v v1 e + b3 v −p < 2 2 < b3 v −p , where in the second to last line we have used the monotonicity of v −1 log v together with the definition of b3 and the fact that r1 < r+ , and in the last line we have used the fact that v p e−a0 v decreases monotonically for v > pa−1 0 , a lower bound which is guaranteed by (28) and our choice of v1 (assuming without loss of generality that v1 ≥ e). Thus we have retrieved (32) in V. For (34), we compute as in (36) that for (u, v) ∈ V, v 1 ∂v V (φ) (u, v˜) d˜ v < c0 − + < 2c0 − , V φ(u, v) ≤ V φ(u, v1 ) + 2 v1 where the last inequality follows from our choice of . Thus we have retrieved (34) in V. As discussed above, we can now conclude that Cout ⊂ V and hence that W ∩ V = ∅. We now turn to improving inequalities (29)–(31) and (33) in W ∩ V. For (29), note that equation (39) and inequalities (34) and (29) imply that 1 2 −1 2 Ω r 2r+ (2c0 − ) + 4c2 b0 − 1 4 1 2 ≤ Ω2 r−1 4c2 b0 − 2r+ 4 ≤0

2 ∂uv r≤

in V. Since ∂v r(0, v) < η(v) ≤ η(v1 ) = b0 for all v ≥ v1 , this yields (29). Next we turn to (30). Rearranging (40), we have ∂u φ ∂v φ 2 ∂uv φ = (∂u r) κV (φ) − (∂v log r) − , ∂u r r so for (u, v) ∈ W ∩ V, using inequalities (29)–(32) and the fact that r ≥ r+ − δ implies b2 b0 b3 + b 1 2 |∂uv φ(u, v)| ≤ − ∂u r(u, v) + v −p . 4c3 r+ − δ Thus

|∂v φ(u, v)| ≤ |∂v φ(0, v)| +

u

2 |∂uv φ(˜ u, v)| d˜ u u b2 b0 b3 + b 1 −p ≤ |∂v φ(0, v)| − + ∂u r(˜ u, v) d˜ u v 4c3 r+ − δ 0 0

1060

C. Williams

0 ,

∂v r <

V (φ) <

|∂v φ| < √ |∂u φ| < c1 |∂u r| −2

−(∂u r)Ω

(45) (46) (47) (48) (49) (50)

< c2

(51)

∂v r > 0 .

(52)

Clearly V0 is open in G(u0 , v0 ). Consequently, our assumptions on the initial data imply that V0 must contain some neighborhood of (0, v0 ) in G(u0 , v0 ), so by shrinking u0 as necessary, we may in fact assume that they all hold along Cin . Since all of the inequalities except (50) are known to hold on Cout , our first step will be to retrieve (50) in V0 , where the set closure is taken relative to G(u0 , v0 ), i.e. V0 = V0 ∩ G(u0 , v0 ). Then by a continuity argument, we can conclude that Cin ∪ Cout ⊂ V0 . As in the proof of Theorem 3, recall that quantity κ is given by (∂v r) 1 . κ = − Ω2 (∂u r)−1 = 4 1 − 2m r From (51), we have κ ≥ 4c12 in V0 , and since ∂u κ ≤ 0 by equation (2), (25) implies κ ≤ 4c13 in all of G(u0 , v0 ). Also, from (47) we have that 2m (53) 1− = (∂v r)κ−1 ≤ 4c2 r in V0 . Let r0 = r(0, v0 ) and observe that ∂v r > 0 implies that r ≥ r0 on all of Cout . Now, equation (42) may be derived exactly as in the proof of Theorem 3, namely ∂u φ ∂u φ − vv A(u,˜ v ) d˜ v 0 (u, v) = e (u, v0 ) ∂u r ∂u r v ∂v φ − vv A(u,˜ v ) d˜ v e κV (φ) − + (u, v ) dv , r v0 where A := κr

−1

2m 2 1 − 2r V (φ) − 1 − + (∂v log r) . r

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C. Williams


Using the above bounds, (48), and (52), in V0 we estimate A≥

2 1 − 2r+ − 4c2 =: a0 . 4c2 r0

The constant a0 is positive by our choices of and . Also, (49) and (45) imply that c4 ∂v φ 1 ≤ κV (φ) − + =: a1 . r 4c3 r0 Then applying these bounds and using (24), for (u, v) ∈ V0 we have v √ ∂u φ e−a0 (v−v ) dv (u, v) < c1 e−a0 (v−v0 ) + a1 ∂u r v0

√ −1 −a0 (v−v0 ) = c1 e + a1 a0 1 − e−a0 (v−v0 ) √ + a1 a−1 c1 − a1 a−1 = e−a0 (v−v0 ) 0 0 . √ Our choices of , and imply that c1 − a1 a−1 0 > 0, so for v ≥ v0 , we have √ √ ∂u φ + a1 a−1 c1 − a1 a−1 c1 . (u, v) < 0 0 = ∂u r Thus (50) holds in all of V0 , so in particular, Cout ⊂ V0 . Our next step is to choose a suitably small δ > 0 to use in defining W: we let

c r −1 4 + δ0 r+ 1 − 4c +1 3 , r δ < min √ , √ 1 − 2 , + c1 8 c1 c3 2c1 r+ and set W = W(δ) = {(u, v) ∈ G(u0 , v0 ) | r(u, v) ≥ r+ − δ}. Now, r r+ along Cout , so there must exist some v1 ≥ v0 such that W contains an open neighborhood of the ray {0} × [v1 , ∞). Also, since V0 and W each contain some neighborhood of the point (0, v1 ), we can find 0 < u1 ≤ u0 such that [0, u1 ] × {v1 } ⊂ V0 ∩ W. Set = [0, u1 ] × {v1 } and Cout = {0} × [v1 , ∞). Henceforth we restrict our attention Cin to the subset G(u1 , v1 ) = K(u1 , v1 ) ∩ G(u0 , v0 ), that is, the maximal development ∪ Cout . of the induced data on Cin The proof now proceeds by a bootstrap argument. Let V be the set of all u, v˜) ∈ V0 ∩ W for all (˜ u, v˜) ∈ J − (u, v) ∩ points (u, v) ∈ G(u1 , v1 ) such that (˜ ⊂ V. We will G(u1 , v1 ). Clearly V ⊂ W ∩ R. By construction, we have Cin ∪ Cout retrieve inequalities (43)–(51) in V and consequently conclude that V = W ∩ R ∩ G(u1 , v1 ). At that point we can easily extract the hypotheses of Theorem 1. We proceed through the ten inequalities in order, beginning with (43) and (44). Rearranging equation (20) and applying (50) and our bounds for κ yields 1 2 ∂u φ ∂v φ 2 −1 (∂v log r)κ − ∂uv φ = − Ω V (φ) − 4 ∂u r rκ √ 1 −1 ≤ − Ω2 V (φ) − 4 c1 c2 (∂v log r) − 4c3 r+ (∂v φ) , 4

Vol. 9 (2008)


1063

so (46) now implies that 2 φ≤0 ∂uv Cout

(54) Cin ,

in V. Thus since (43) and (44) hold along and respectively, they must hold in V as well. Now we turn to (45) and we compute that 2 2 V (φ) φ) φ) V (φ)(∂uv V (φ)(∂u φ)(∂v φ) − V (φ)(∂uv ≥− . ∂u = 2 ∂v φ (∂v φ) (∂v φ)2 Suppose (u, v) ∈ V. If V (φ(u, v)) ≤ 0, then clearly (45) holds at (u, v), since |∂v φ| > 0. If V (φ(u, v)) > 0, let u∗ be the smallest value such that V (φ) ≥ 0 along [u∗ , u) × {v} and integrate the above inequality, noting that the righthand side is positive along this ray by (54). If u∗ = 0, then by our hypotheses on Cout , we have V (φ(u∗ , v))(∂v φ(u∗ , v))−1 > −c4 . On the other hand, if u∗ > 0, then by choice of u∗ , V (φ(u∗ , v))(∂v φ(u∗ , v))−1 = 0. Thus in either case we have V (φ) V (φ) (u, v) ≥ (u∗ , v) > −c4 , ∂v φ ∂v φ thereby obtaining (45) in V. Next, before proceeding to (46), let us first show that our initial bounds for V (φ) continue to hold in V. On one hand, integrating (50) yields √ √ φ(0, v) − φ(u, v) ≤ c1 r(0, v) − r(u, v) < c1 δ < δ0 . On the other hand, (44) implies φ(u, v) ≤ φ(0, v) . Thus for (u, v) ∈ V we have φ(u, v) ∈ (φ0 − δ0 , φ1 ), and hence 0 ≤ V (φ(u, v)) ≤ B . 2 r in V. Recalling equaFor (46), we first derive an upper estimate for ∂uv tion (39), we have 1 2m 2 ∂uv r = Ω2 r−1 2r2 V (φ) + 1 − −1 4 r 2

−1 2r+ + 4c2 − 1 ≤ −κ(∂u r)r

∂u r 2 1 − 2r+ − 4c2 , ≤ 4c2 r+

where we have used (48), (53), and κ ≥ a2 :=

1 4c2 .

Setting

1 2 1 − 2r+ − 4c2 > 0 , 4c2 r+

we thus have 2 r ≤ a2 (∂u r) . ∂uv

(55)

1064

C. Williams


Consequently, differentiating the lefthand side of (46) and using inequalities (50), (44), (54), (47), and (55) yields √ −1 (∂v φ) ∂u V (φ) − 4 c1 c2 (∂v log r) − 4c3 r+ √ −1 2 2 = V (φ)(∂u φ) − 4 c1 c2 (∂uv log r) − 4c3 r+ (∂uv φ) √ √ √ 2 r + 4 c1 c2 r−2 (∂u r)(∂v r) ≥ c1 B(∂u r) − 4 c1 c2 r−1 ∂uv √ √ √ −1 −2 ≥ c1 B(∂u r) − 4 c1 c2 a2 r+ (∂u r) + 4 c1 c2 r+ (∂u r) √ −1 −2 = c1 (∂u r) B − 4c2 a2 r+ + 4c2 r+ √ −2 −2 = c1 (∂u r) B − r+ + 2 + 8c2 r+ > 0, where the last line follows from the choices of and . Thus we have retrieved (46) in V. Inequality (47) follows immediately from (55), since the latter implies that 2 ∂uv r < 0. For (48), observe that from (46), we have √ −1 −1 V (φ) ≥ 4 c1 c2 (∂v log r) + 4c3 r+ (∂v φ) ≥ 4c3 r+ (∂v φ) . Multiplying through by ∂u φ and using (44), (50), and (49) yields −1 |∂v φ||∂u φ| ∂u V (φ) ≤ 4c3 r+ √ −1 ≤ −4 c1 c3 r+ (∂u r) . Integrating and using the assumption that V (φ) < 12 on Cout then gives √ 1 −1 + 4 c1 c3 r+ δ 2 <

V (φ)(u, v) ≤

by our choice of δ. Next we turn to (49). Using equation (20) and inequalities (43)–(45), we have 2 ∂u log |∂v φ| = (∂uv φ)(∂v φ)−1 1 = − Ω2 V (φ)(∂v φ)−1 − (∂u φ)(∂v φ)−1 (∂v log r) − (∂u log r) 4 ≤ −c4 κ(∂u r) − ∂u log r c4 r+ ≤ −∂u log r +1 , 4c3

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so integrating yields |∂v φ(u, v)| ≤ |∂v φ(0, v)| 1 2 < ,
0, for some scalars ck . Then lim ck Tr(e−sAk ) = ck dim ker Ak . s→+∞

k

k

Proof. Clearly lims→+∞ Tr(e−sA ) = dim ker A if e−sA is trace class and A is nonnegative. The finite trace condition implies that Ak bounded below. Let Ak = − + − −sA− k ) is a linear A+ k ck Tr(e k + Ak with Ak ≥ 0 and Ak < 0. Then the function combination of exponential of the form eas with a > 0, and vanishes at infinity, thus it must be identically zero. It follows that + ck Tr(e−sAk ) = ck Tr(e−sAk ) −→ ck dim ker A+ ck dim ker Ak . k = k

s=∞

k

k

k

Lemma 29. Let A be a Fermi modular net. If λ is supersymmetric then ˜ Sρ,ν null(ν, c/24) Trs (e−2πtHλ ) = 2

(18)

ν∈R

where null(ν, h) ≡ dim ker(L0,ν − h). Proof. The left hand side of (17) is constant by McKean–Singer lemma, so we have ˜ that ν∈R Sρ,ν Tr(e−2πsL0,ν ) is a constant function of s > 0. Thus (18) holds by the sublemma. With λ as in the above lemma, by eq. (16) we have ˜

Trs (e−2πtHλ ) = ind(Qλ+ ) .

Vol. 9 (2008)

Structure and Classification of Superconformal Nets

Therefore by Lemma 29 we have ind(Qλ+ ) = 2

1095

Sρ,ν null(ν, c/24)

ν∈R

then, writing Rehren definition of the S matrix, we have d(ρ) ind(Qλ+ ) = 2 √ K(ρ, ν)d(ν) null(ν, c/24) μAb ν∈R

where μAb is the μ-index of Ab . By [38] we have μAb = 4μA therefore: Theorem 30. Let A be a Fermi conformal net as above and λ a supersymmetric irreducible representation of A. Then d(ρ) ind(Qλ+ ) = √ K(ρ, ν)d(ν) null(ν, c/24) μA ν∈R

where ρ is one of the two irreducible components of λb and μA is the μ-index of A. In the above formula the Fredholm index of the supercharge operator Qλ+ is expressed by a formula involving the Jones index of the Ramond representations whose lowest eigenvalue c/24 modulo integers. Corollary 31. If ind(Qλ ) = 0 there exists a σ-Fermi sector ν such that c/24 is an eigenvalue of L0,ν . Corollary 32. Suppose that, in Theorem 30, the Ramond sector ρ has lowest eigenvalue c/24 and no other Ramond sector has lowest eigenvalue c/24 modulo integers. Then d(ρ)2 1 K(ρ, ρ) = . Sρ,ρ = √ μAb 2 Proof. By Lemma 29 we have: ind(Qλ+ ) = 2Sρ,ρ null(ρ, c/24) . On the other hand by formula (16) ind(Qλ+ ) = Tr(ΓEc/24 ) = null(ρ, c/24) − null(ρ , c/24) = null(ρ, c/24) because Hλ = L0,ρ ⊕ L0,ρ and c/24 is not in the spectrum of L0,ρ , so we get our formula.

6. Super-Virasoro algebra and super-Virasoro nets We now focus on model analysis and shall consider the most basic superconformal nets, namely the ones associated with the super-Virasoro algebra.

1096

S. Carpi, Y. Kawahigashi, and R. Longo


6.1. Super-Virasoro algebra The super-Virasoro algebra governs the superconformal invariance [20,24]. It plays in the supersymmetric context the same rôle that the Virasoro algebra plays in the local conformal case. Strictly speaking, there are two super-Virasoro algebras. They are the superLie algebras generated by even elements Ln , n ∈ Z, odd elements Gr , and a central even element c, satisfying the relations c [Lm , Ln ] = (m − n)Lm+n + (m3 − m)δm+n,0 (19) 12 m − r Gm+r [Lm , Gr ] = 2 c 1 2 [Gr , Gs ] = 2Lr+s + r − δr+s,0 . 3 4 Here the brackets denote the super-commutator. In the Neveu–Schwarz case r ∈ Z + 1/2, while in the Ramond case r ∈ Z. We shall sometime use the term superVirasoro algebra to indicate either the Neveu–Schwarz algebra or the Ramond algebra. The point is that, although the Neveu–Schwarz algebra and the Ramond algebra are not isomorphic graded Lie algebras, they are representations of a same object, in a sense that we shall later see (they have the same “isomorphic completion”). By definition, the Neveu–Schwarz algebra and the Ramond algebra are both extensions of the Virasoro algebra. The super-Virasoro algebra is equipped with the involution L∗n = L−n , G∗r = G−r , c∗ = c and we will be only interested in unitary representations on a Hilbert space, i.e. representations preserving the involution. Note that unitary representations have automatically positive energy, namely L0 ≥ 0. Indeed we have 1 1 G 12 G∗1 + G∗1 G 12 ≥ 0 (NS case) L0 = [G 12 , G− 12 ] = 2 2 2 2 1 2 L0 = [G0 , G0 ] = G0 + c/24 ≥ c/24 (R case) . 2 The unitary, lowest weight representations of the super-Virasoro algebra, namely the unitary representations of the super-Virasoro algebra on a Hilbert space H with a cyclic vector ξ ∈ H satisfying L0 ξ = hξ ,

Ln ξ = 0 ,

n > 0,

Gr ξ = 0 ,

r > 0,

are studied in [20,24]; in the NS case they are irreducible and uniquely determined by the values of c and h. In the Ramond case one has to further specify the action of G0 on the lowest energy subspace. It turns out that for a possible value of c and h there two inequivalent irreducible lowest weight representations (but for the case c = h/24 when the representation is unique and graded). Note also that for c = h/24 the direct sum of the two inequivalent irreducible representations

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has a cyclic lowest energy vector and is the unique Ramond (c, h) lowest weight representation where grading is implemented. The possible values are either c ≥ 3/2, h ≥ 0 (h ≥ c/24 in the Ramond case) or 3 8 c= 1− , m = 2, 3, . . . (20) 2 m(m + 2) and h = hp,q (c) ≡

[(m + 2)p − mq]2 − 4 ε + 8m(m + 2) 8

where p = 1, 2, . . . , m−1, q = 1, 2, . . . , m+1 and p−q is even or odd corresponding to the Neveu–Schwarz case (ε = 0) or Ramond case (ε = 1/2). Note that the Neveu–Schwarz algebra has a vacuum representation, namely a irreducible representation with 0 as eigenvalue of L0 , the Ramond algebra has no vacuum representation. 6.2. Stress-energy tensor Let c be an admissible value as above with Ln (n ∈ Z), Gr , the operators corresponding to a Neveu–Schwarz (r ∈ Z + 12 ) or Ramond (r ∈ Z) representation. The Bose and Fermi stress-energy tensors are defined by z −n−2 Ln (21) TB (z) = n

1 −r−3/2 TF (z) = z Gr 2 r

(22)

namely

TB (z) = n∈Z z −n−2 Ln Neveu–Schwarz case: TF (z) = 12 m∈Z z −m−2 Gm+ 12

TB (z) = n z −n−2 Ln Ramond case: √ TF (z) = 12 m∈Z z −m−2 zGm .

(23) (24)

Let’s now make a formal calculation for the (anti-)commutation relations of the Fermi stress energy tensor TF . We want to show that, setting w ≡ z2 /z1 , we have 1 −1 3 −3 − 32 c 2 TF (z1 ), TF (z2 ) = z1 TB (z1 )δ(w) + z1 w (25) w δ (w) + δ(w) 2 12 4 both in the Neveu–Schwarz and in the Ramond case.

1098



Setting k = r = s ∈ Z we have, say in the Ramond case, 1 −r−3/2 −s−3/2 [Gr , Gs ]z1 z2 TF (z1 ), TF (z2 ) = 4 r,s c 1 1 −r−3/2 −s−3/2 −r−3/2 r−3/2 = Lr+s z1 z2 + z2 r2 − z1 2 r,s 12 4 r c 1 1 −r−3/2 −k+r−3/2 −3/2 −3/2 = Lk z1 z2 + z1 z2 r2 − wr 2 12 4 r r,k c 1 3 1 1 = z1−1 Lk z2−k−2 w 2 wr + z1−3 w− 2 r2 − wr 2 12 4 r r k c 1 3 1 −1 1 −3 − 2 r 2 2 = z1 TB (z2 )w w + z1 w r − wr 2 12 4 r r 1 3 c 1 −1 1 −3 − 2 2 2 = z1 TB (z2 )w δ(w) + z1 w w δ (w) + w − δ(w) 2 12 4 3 c 1 3 = z1−1 TB (z1 )δ(w) + z1−3 w− 2 w2 δ (w) + δ(w) . 2 12 4 As

r∈Z

1 r − 4 2

r

w =

r∈Z+ 12

1 r − 4 2

3 wr = w2 δ (w) + δ(w) 4

√ the above calculation (by using equalities as δ(w) w = δ(w) and similar ones) shows that the commutation relations for the Fermi stress energy tensor TF and, analogously, the commutation relations for TB and TF are indeed the same in the Neveu–Schwarz case and in the Ramond case, namely they are representations of the same (anti-)commutation relations. This is basic reason to view the Neveu– Schwarz and Ramond algebras as different types of representations of a unique algebra. However the above calculation is only formal. To give it a rigorous meaning, and have convergent series, we have to smear the stress energy tensor with a smooth test function with support in an interval. We then arrive naturally to consider the net of von Neumann algebras of operators localised in intervals. In the case of central charge c < 3/2 we shall see that Neveu–Schwarz and Ramond representations correspond to DHR representations and general solitons of the associated super-Virasoro net. 6.3. Super-Virasoro nets We give here the definition of the super-Virasoro nets for all the allowed values of the central charge. We follow the strategy adopted in [8] for the case of Virasoro nets, cf. also [12, 40]. An alternative construction in the case of the discrete series (c < 3/2) is outlined in Section 6.4.

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Let λ be a unitary positive energy representation of the super-Virasoro algebra on a Hilbert space Hλ . The corresponding conformal Hamiltonian L0 is a self-adjoint operator on Hλ and we denote by Hλ∞ the dense subspace of smooth vectors for L0 , namely the subspace of vectors belonging to the domain of Ln0 for all positive integers n. The operators Ln , n ∈ Z satisfy the linear energy bounds 3

Ln v ≤ M (1 + |n| 2 )(1 + L0 )v ,

v ∈ Hλ∞ ,

(26)

for a suitable constant M > 0 depending on the central charge c, cf. [8, 13]. Moreover from the relations [G−r , Gr ] = 2L0 + 3c (r2 − 14 ), we find the energy bounds 1 c 12 Gr v ≤ 2 + r2 (1 + L0 ) 2 v , 3

v ∈ Hλ∞ ,

(27)

where r ∈ Z + 1/2 (resp. r ∈ Z) if λ is a Neveu–Schwarz (resp. Ramond) representation. We now consider the vacuum representation of the super-Virasoro algebra with central charge c and denote by H the corresponding Hilbert space and by Ω the vacuum vector, namely the unique (up to a phase) unit vector such that L0 Ω = 0. Let f be a smooth function on S 1 . It follows from the linear energy bounds in Eq. (26) and the fact that the Fourier coefficients π dθ , n ∈ Z, (28) fˆn = f (eiθ )e−inθ 2π −π are rapidly decreasing, that the smeared Bose stress-energy tensor TB (f ) = fˆn Ln

(29)

n∈Z

is a well defined operator with invariant domain H∞ . Moreover, for f real, TB (f ) is essentially self-adjoint on H∞ (cf. [8]) and we shall denote again TB (f ) its self-adjoint closure. Now let f be a smooth function on S 1 whose support does not contain −1. Then also the coefficients π 1 dθ , r ∈Z+ , (30) f (eiθ )e−irθ fˆr = 2π 2 −π are rapidly decreasing and it follows from the energy bounds in Eq. (27) that the corresponding smeared Fermi stress-energy tensor 1 ˆ TF (f ) = (31) fr Gr 2 1 r∈Z+ 2

is also a well defined operator with invariant domain H∞ . Again, for f real, TF (f ) is essentially self-adjoint on H∞ (cf. [8]) and we denote its self-adjoint closure by the same symbol.

1100



As in Section 3.1 we identify R with S 1 \{−1} and consider the family IR of nonempty, bounded, open intervals of R as a subset of the family I of intervals of S 1 . We define a net SVirc of von Neumann algebras on R by SVirc (I) ≡ eiTB (f ) , eiTF (f ) : f ∈ C ∞ (S 1 ) real, suppf ⊂ I , I ∈ IR . (32) Isotony is clear from the definition and we have to show that the net is graded local and covariant. We first consider graded locality. The spectrum of L0 is contained in Z/2 and hence the unitary operator Γ = ei2πL0 is an involution such that ΓΩ = Ω. It is straightforward to check that if f1 , f2 are smooth functions on S 1 and the support of f2 does not contain −1 then ΓTB (f1 ) = TB (f1 )Γ and ΓTF (f1 ) = −TF (f1 )Γ. Hence, γ = AdΓ is a Z2 -grading on the net SVirc , namely ΓSVirc (I)Γ∗ = SVirc (I), for all I ∈ IR . Now let I1 , I2 ∈ IR be disjoint intervals and let f1 , f2 be real smooth functions on S 1 with support in I1 , I2 respectively. Then the operators TB (f1 ) and ∞ TF (f1 ) commute with ZTB (f2 )Z ∗ and ZTF (f2 )Z ∗ , Z = 1−iΓ 1−i , on H , cf. the ∞ ∞ (anti-)commutation relations in Section 6.2 (note that ZH = H ). Using the energy bounds in Eq. (26) and Eq. (27) and the fact that Z commutes with L0 one can apply the argument in [8, Sect. 2] to show that eiTB (f1 ) and eiTF (f1 ) commute with ZeiTB (f2 ) Z ∗ and ZeiTF (f2 ) Z ∗ . It follows that the net SVirc is graded local, namely (33) SVirc (I1 ) ⊂ ZSVirc (I2 ) Z ∗ whenever I1 , I2 are disjoint interval in IR . We now discuss the covariance. The crucial fact here is that the representation of the Virasoro algebra on H integrates to a strongly continuous unitary projective positive-energy representation of Diff (∞) (S 1 ) on H by [25, 53] which factors through Diff (2) (S 1 ) because ei4πL0 = 1. Hence there is a strongly continuous projective unitary representation U of Diff (2) (S 1 ) on H such that, for all real f ∈ C ∞ (S 1 ) and all x ∈ B(H), ∗ (34) U exp(2) (tf ) xU exp(2) (tf ) = eitTB (f ) xe−itTB (f ) , where, exp(2) (tf ) denotes the lift to Diff (2) (S 1 ) of the one-parameter subgroup d exp(tf ) of Diff(S 1 ) generated by the (real) smooth vector field f (eiθ ) dθ . Moreover, (2) (2) 1 if θ → r (θ) is the lift to Diff (S ) of the one-parameter subgroup of rotations in Diff(S 1 ) we have (35) U r(2) (θ) = eiθL0 , for all θ in R. The following properties of U follow rather straightforwardly. (1) The restriction of U to the subgroup M¨ ob(2) ⊂ Diff (2) (S 1 ) is (after multiplication by a phase factor) a unique strongly continuous unitary representation which we again denote by U . If exp(2) (tf ) ∈ M¨ ob(2) for all t ∈ R, this unitary representation satisfies (36) U exp(2) (tf ) = eitTB (f ) , U exp(2) (tf ) Ω = Ω , for all t ∈ R.

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(2) If the support of the real smooth function f is contained in I ∈ IR then U (g)eiTB (f ) U (g)∗ ∈ SVirc (gI) ˙

(37)

for all g ∈ Diff (S ) such that gI ˙ ∈ IR . (3) For all g ∈ Diff (2) (S 1 ) we have (2)

1

U (g)ΓU (g)∗ = Γ

(38)

for all g ∈ Diff (2) (S 1 ). Note that property (3) follows from the fact that Diff (2) (S 1 ) is connected and Γ2 = 1. We now consider the covariance properties of the Fermi stress-energy tensor TF . From the commutation relations in equations (19) we find, 1 i TB (f1 ), TF (f2 ) v = TF f f2 − f2 f1 v , v ∈ H∞ (39) 2 1 where f1 , f2 are real smooth functions on S 1 , suppf2 ⊂ I for some I ∈ IR and, d f (eiθ ). For any g ∈ Diff(S 1 ) for any f ∈ C ∞ (S 1 ), f is defined by f (eiθ ) = dθ consider the function Xg : S1 → R defined by d log(geiθ ) . (40) dθ Since g is a diffeomorphism of S 1 preserving the orientation then Xg (z) > 0 for all z ∈ S 1 . Moreover Xg ∈ C ∞ (S 1 ). Another straightforward consequence of the definition is that Xg1 g2 (z) = Xg1 (g2 z)Xg2 (z) . (41) 1 As a consequence the family of continuous linear operators β(g), g ∈ Diff(S ), on the Fréchet space C ∞ (S 1 ) defined by 1 β(g)f (z) = Xg (g −1 z) 2 f (g −1 z) (42) Xg (eiθ ) = −i

gives a strongly continuous representation of Diff(S 1 ) leaving the real subspace of real functions invariant. Moreover if f1 , f2 ∈ C ∞ (S 1 ) are real then vector valued function t → β(exp(tf1 ))f2 is differentiable in C ∞ (S 1 ) and d 1 β exp(tf1 ) f2 |t=0 = f1 f2 − f1 f2 . (43) dt 2 Now let suppf2 be a subset of some interval I ∈ IR and let LI ⊂ R be the connected component of 0 in R of the open set {t ∈ R : exp(tf1 )I ∈ IR }. Then, for any v ∈ H∞ the function LI t → TF (β(exp(tf1 ))f2 )v is differentiable in H and it follows from Eq. (39) and Eq. (43) that d TF β exp(tf1 ) f2 v|t=0 = i TB (f1 ), TF (f2 ) v . (44) dt We now specialize to the case of Möbius transformations i.e. we assume that exp(tf1 ) ∈ M¨ ob for all t ∈ R. The map LI t → v(t) ∈ H given by v(t) = TF β exp(tf1 ) f2 U exp(2) (tf1 ) v (45)

1102



is well defined because, U (exp(2) (tf1 ))H∞ = H∞ for all t ∈ R. Note also that v(t) ∈ H∞ for all t ∈ LI . Now using Eq. (44) and the energy bounds in Eq. (27) it can be shown that v(t) is differentiable (in the strong topology of H) and that it satisfy the following differential equation on H

It follows that

d v(t) = iTB (f1 )v(t) . dt

(46)

v(t) = U exp(2) (tf1 ) TF (f2 )v

(47)

∞

and since v ∈ H was arbitrary we get, for all t ∈ LI , the following equality of self-adjoint operators (48) U exp(2) (tf1 ) TF (f2 )U exp(2) (−tf1 ) = TF β exp(tf1 ) f2 . (2)

Now, if we denote by UI the connected component of the identity in M¨ ob(2) of the open set {g ∈ M¨ ob(2) : gI ∈ IR } it follows that U (g)TF (f )U (g)∗ = TF β(g)f ˙ , (49) (2)

for any real smooth function on S 1 with suppf ⊂ I and all g ∈ UI . From Eq. (49) and Eq. (37) we have ˙ , U (g)SVirc (I)U (g)∗ = SVirc (gI)

I ∈ IR ,

(2)

g ∈ UI .

(50)

Hence SVirc extends to a Möbius covariant net on S 1 satisfying graded locality, see Section 3.2. Note that we have not yet shown that the vacuum vector Ω is cyclic and hence we still don’t know if the net satisfy all the requirements of Property 3 in the definition of M¨ obius covariant Fermi nets on S 1 given in Section 2.1. We shall however prove the cyclicity of the vacuum as a part of the following theorem. Theorem 33. SVirc is an irreducible Fermi conformal net on S 1 for any of the allowed values of the central charge c. Proof. Since Ω is the unique (up to a phase) unit vector in the kernel of L0 , we only have to show that Ω is cyclic and that the strongly continuous positiveenergy projective representation U of Diff (2) (S 1 ) defined above makes the net diffeomorphism covariant in the appropriate sense. We first show that Ω is cyclic for the net. Let K ⊂ H be the closure of I∈I SVirc (I)Ω. We have to show that K = H. Clearly U (g)K = K for all g ∈ M¨ ob(2) . It follows that if j ∈ Z/2 , Pj is the orthogonal projection of H onto the kernel of L0 − j1 then Pj K ⊂ K ∩ H∞ . Now let r ∈ Z + 1/2. Since the smooth functions on S 1 whose support does not contain the point −1 is dense in L2 (S 1 ) we can find an interval I ∈ IR and a real smooth function f with suppf ⊂ I such that fˆr = 0. Since TF (f )Pj K ⊂ K we find Gr Pj K =

1 Pj−r TF (f )Pj K ⊂ K . fˆr

(51)

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A similar argument applies to the operators Ln , n ∈ Z and hence the linear span L of the subspaces Pj K, j ∈ Z/2 is invariant for the representation of the superVirasoro algebra. Since Ω ∈ L is cyclic for the latter representation it follows that L is dense in H. Hence K = H because L ⊂ K. Hence SVirc is an irreducible M¨ obius covariant Fermi net on S 1 . To show that SVirc is diffeomorphism covariant we first observe that by [40, Sect. V.2] for any I ∈ I the group generated by diffeomorphisms of the form exp(f ) with suppf ⊂ I is dense in Diff I (S 1 ). It follows that the group generated (2) by elements of the form exp(2) (f ) with suppf ⊂ I is dense in Diff I (S 1 ). Hence, for (2) any I ∈ I and any g ∈ Diff I (S 1 ), U (g) ∈ SVirc (I) and, by graded locality, U (g) ∈ SVir(I ) because U (g) commutes with Γ. Now, an adaptation of the argument in the proof of [12, Proposition 3.7] shows that SVirc is diffeomorphism covariant and the proof is complete. 6.4. The discrete series of super-Virasoro nets We shall now use the construction in [24] to study SVirc with c < 3/2 an admissible value. First consider three real free Fermi fields in the NS representation. They ˆ ˆ ⊗F ˆ where = F ⊗F define a graded-local net on S 1 . This net coincides with F ⊗3 F is the net generated by a single real free Fermi field in the NS representation ˆ denotes the graded tensor product. The net ASU(2) embeds as a (cf. [2]) and ⊗ 2 ˆ ⊗3 subnet of F . Actually, from the discussion in [24, page 115] we have ASU(2)2 = (F ⊗3 )b . ˆ

ˆ Now consider the conformal net FN (on the Hilbert space HN ) given by F ⊗3 ⊗ ASU(2)N , N positive integer. Consider the representation of the super-Virasoro algebra on HN with central charge 3 8 cN = 1− , (52) 2 (N + 2)(N + 4)

constructed in [24, Sect. 3] (coset construction). Then the corresponding stress energy-tensors TB and TF generate a family of von Neumann algebras on HN as in eq. (32). Using the energy bounds in Eq. (26) and Eq. (27) it can be shown that this family defines a Fermi subnet of FN as in eq. (32) which can be identified with the super-Virasoro net SVircN . In this way we obtain all the super-Virasoro nets corresponding to the discrete series. Using [24] we can identify these super-Virasoro nets as coset subnets. From the embedding ASU(2)2 ⊗ ASU(2)N ⊂ FN

(53)

ASU(2)N +2 ⊂ FN .

(54)

we have the embedding

1104



It follows from Eq. (3.13) and the claim at the end of page 114 in [24, Sect. 3] that SVircN is contained in the coset (ASU(2)N +2 )c = (ASU(2)N +2 ) ∩ FN .

(55)

Moreover it follows from the branching rules in [24, Eq. 4.15] that these nets coincide (cf. [35]) namely SVircN = (ASU(2)N +2 )c .

(56)

As a consequence the Bose subnet SVir0cN ≡ (SVircN )b of the super-Virasoro net SVircN is equal to the coset (57) (ASU(2)N +2 ) ∩ ASU(2)2 ⊗ ASU(2)N and hence, by [61, Corollary 3.4] and [48, Theorem 24] SVir0cN is completely rational, see also [48, Corollary 28]. Now we look at representations. We denote (N S) and (R) the Neveu–Schwarz and Ramond representations for three Fermion fields respectively. In (N S) the lowest energy eigenspace is one-dimensional (“nondegenerate vacuum”), whilst in (R) it is two-dimensional (“2-fold degenerate vacuum”).5 It is almost obvious that (N S) corresponds to the vacuum representation ˆ and, arguing as in the proof of [2, Lemma 4.3], it can be shown πN S of F ⊗3 that (R) corresponds to a general soliton πR of the latter net. Clearly (N S) and (R) restrict to positive-energy representations of SU(2)2 . We denote by π(N,l) the representation of ASU(2)N with spin l. At level N the possible values of the spin are those satisfying 0 ≤ 2l ≤ N . Then the following identities hold (see [24, page 116]): πN S |ASU(2)2 = π(2,0) ⊕ π(2,1) πR |ASU(2)2 = π(2, 12 ) .

(58) (59)

Note that the restriction of (R) remains irreducible because the grading automorphism is not unitarily implemented, cf. Proposition 22. Denote by (cN , hp,q )N S , resp. (cN , hp,q )R , a NS, resp. R, irreducible representation of super-Virasoro algebra with central charge cN and lowest energy 2

hp,q =

[(N + 4)p − (N + 2)q] − 4 , 8(N + 2)(N + 4)

resp. 2

1 [(N + 4)p − (N + 2)q] − 4 + , 8(N + 2)(N + 4) 16 where p = 1, 2, . . . , N + 1, q = 1, 2, . . . N + 3 and p − q is even in the NS case and odd in the R case. hp,q =

5 Different

Ramond representations could be defined corresponding to different choices of the corresponding representation of the Dirac algebra of the 0-modes on the subspace of lowest energy vectors, cf. page 113 and page 115 of [24].

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As already mentioned, in the NS case, for every value of the central charge, the lowest energy hp,q completely determines the (equivalence class of) the representation. In contrast for a given values of the central charge and of the lowest energy hp,q there are two Ramond representations one with cN Ψh G0 Ψhp,q = hp,q − 24 p,q and the other with

cN Ψh , G0 Ψhp,q = − hp,q − 24 p,q where Ψhp,q is the lowest energy vector. These two representations are connected by the automorphism Gr → −Gr and become equivalent when restricted to the even (Bose) subalgebra. Accordingly (cN , hp,q )R denotes indifferently these two N . representations which are clearly inequivalent when hp,q = c24 For a given N the equality hp,q = hp ,q when p − q and p − q are both even or odd hold if and only if p = N + 2 − p and q = N + 4 − q. Note also that it may happen that hp,q = hp ,q when p−q is even and p −q is odd. For example, if N = 2 then h2,2 = h1,2 = 1/16. Accordingly there are values of N for which a given value of the lowest energy corresponds to three distinct irreducible representations of super-Virasoro algebra: one NS representation and two R representations. From [24, Section 4] we can conclude that there exist DHR representations S , p − q even, and general solitons πhRp,q , p − q odd, of SVircN (associated to the πhNp,q representations (cN , hp,q )N S , resp. (cN , hp,q )R of the of super-Virasoro algebra) such that S πN S ⊗ π(N, 12 [p−1]) |ASU(2)N +2 ⊗SVircN = π(N +2, 12 [q−1]) ⊗ πhNp,q , (60) q

1 ≤ q ≤ N + 3, p − q even, and πR ⊗ π(N, 12 [p−1]) |ASU(2)N +2 ⊗SVircN = π(N +2, 12 [q−1]) ⊗ πhRp,q ,

(61)

q

1 ≤ q ≤ N + 3, p − q odd. 0 S R NS R We now denote ρN hp,q , resp. ρhp,q , the restriction of πhp,q , resp πhp,q , to SVircN . NS In the representation space of πhp,q the grading is always unitarily implemented and hence we have the direct sum N S+ N S− S ρN hp,q = ρhp,q ⊕ ρhp,q

of two (inequivalent) irreducible representations corresponding to the eigenspaces with eigenvalues 1 and -1 of the grading operator respectively. In contrast in the case of πhRp,q the grading automorphism is unitarily implemented only if hp,q = cN /24. This happens if and only if N is even and p = (N + 2)/2, q = (N + 4)/2. In this case π R cN is a supersymmetric general 24 representation of the Fermi conformal net SVircN (with supercharge Q = G0 ).

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Moreover we have the decomposition into irreducible (inequivalent) subrepresentations R− ρRcN = ρR+ cN ⊕ ρ cN . 24

24

24

In the remaining cases ρR hp,q is irreducible. Restricting Eq. (60) and Eq. (61) to the Bose elements and using Equations (58), (59) we get S+ π(2,0) ⊗ π(N, 12 [p−1]) |ASU(2) π(N +2, 12 [q−1]) ⊗ ρN (62) ⊗SVir0c = hp,q , N +2

N

q

π(2,1) ⊗ π(N, 12 [p−1]) |ASU(2)

N +2

1 ≤ q ≤ N + 3, p − q even, and π(2, 12 ) ⊗ π(N, 12 [p−1]) |ASU(2)

=

⊗SVir0c

N +2

N

⊗SVir0c

N

S− π(N +2, 12 [q−1]) ⊗ ρN hp,q ,

(63)

q

=

π(N +2, 12 [q−1]) ⊗ ρR hp,q ,

(64)

q

1 ≤ q ≤ N + 3, p − q odd. Now, recalling the identification of SVir0cN as a coset in Eq. (57), it follows from [61, Corollary 3.2] that every irreducible DHR representation of this net S+ N S− R is equivalent to one of those considered before, namely ρN hp,q , ρhp,q and ρhp,q R− (hp,q = cN /24), ρR+ cN and ρ cN . 24

24

Remark. One may wonder whether there are supersymmetric representations of S although there is no obvious supercharge operator in this case (G0 is the form πhNp,q S missing in NS representations). This is not the case as we now show. Set λ ≡ πhNp,q and let Hλ be the corresponding conformal Hamiltonian. Suppose that Qλ is a selfadjoint operator such that Q2λ = Hλ − a for some a ∈ R. Then hp,q ≥ a. Moreover, by the McKean–Singer formula Trs (e−t(Hλ −a) does not depend on t > 0 so dim ker(Hλ −) is even if = a. But this is impossible; in fact in an irreducible NS representation both ker(Hλ − hp,q ) and ker(Hλ − hp,q − 1/2) are one-dimensional if hp,q > 0 (spanned by Ψhp,q and by G−1/2 Ψhp,q ) respectively. If hp,q = 0, ker(Hλ − hp,q ) is again one-dimensional, G−1/2 Ψhp,q = 0, yet G−3/2 Ψhp,q = 0 and spans ker(Hλ − hp,q − 3/2). 6.5. Modularity of local super-Virasoro nets We state here explicitly the modularity of the Bose subnet super-Virasoro nets for c < 3/2. In this case the Bose super-Virasoro net can be obtained as the coset (57). Then the Rehren S and T matrices as been computed by Xu in [62, Sect. 2.2] (see also Section 7 below). These matrices agree with those in [23] and [21] giving modular transformations of specialised characters. Accordingly we have the following. Theorem 34. For a positive even integer N then SVir0cN is a modular conformal net.

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7. Classification of superconformal nets in the discrete series By a superconformal net (of von Neumann algebras on S 1 ) we shall mean a Fermi net on S 1 that contains a super-Virasoro net as irreducible subnet. If the central 8 charge c of a superconformal net is less than 3/2, it is of the form c = 32 1− m(m+2) for some m = 3, 4, 5, . . . [20]. We classify all such superconformal nets. 7.1. Outline of classification As above, we denote the super Virasoro net with central charge c and its Bosonic 0 part by SVirc and SVir We are interested in the case c < 3/2. In this c , respectively. 3 8 for some m = 3, 4, 5, . . . , and we have already case, we have c = 2 1 − m(m+2) seen that in this case the local conformal net SVir0c is realised as a coset net for the inclusion SU (2)m ⊂ SU (2)m−2 ⊗ SU (2)2 . This net is completely rational in the sense of [38] by [61]. The DHR sectors of the local conformal net SVir0c is described as follows by [62, Section 2.2]. Label the DHR sectors of the local conformal nets SU (2)m , SU (2)m−2 , and SU (2)2 by k = 0, 1, . . . , m, j = 0, 1, . . . , m − 2, and l = 0, 1, 2, respectively. Then we consider the triples (j, k, l) with j − k + l being even. For l = 0, 2, we have identification (j, k, l) ↔ (m − 2 − j, m − k, 2 − l) , thus it is enough to consider the triples (j, k, 0) with j − k being even. Each such triple labels an irreducible DHR sector of the coset net SVir0c . For the case l = 1, we also have identification (j, k, l) ↔ (m − 2 − j, m − k, 2 − l) , but if we have a fixed point for this symmetry, that is, if m is even, then the fixed point (2/m − 1, m/2, 1) splits into two pieces, (2/m − 1, m/2, 1)+ and (2/m − 1, m/2, 1)− . All of these triples, with this identification and splitting, label all the irreducible DHR sectors of the coset net SVir0c . The sectors with l = 0 and l = 1 are called Neveu–Schwarz and Ramond sectors, respectively. The conformal spin of the sector (j, k, l) is given by πi j(j + 2) k(k + 2) l(l + 2) − + exp . 2 m m+2 4 (This also works for the case (j, k, l) = (2/m − 1, m/2, 1).) For example, if m = 3, we have six irreducible DHR sectors and they are labeled with triples (0, 0, 0), (0, 2, 0), (1, 1, 0), (1, 3, 0), (0, 3, 1), (1, 2, 1). (This local conformal net is equal to the Virasoro net with c = 7/10.) For m = 4, we have 13 irreducible DHR sectors and they are labeled with (0, 0, 0), (0, 2, 0), (0, 4, 0), (1, 1, 0), (1, 3, 0), (2, 0, 0), (2, 2, 0), (2, 4, 0), (0, 3, 1), (1, 4, 1), (2, 3, 1), (1, 2, 1)+ , (1, 2, 1)− , where the two labels (1, 2, 1)+ , (1, 2, 1)− arise from the fixed point (1, 2, 1) of the symmetry of order 2. For all m, the irreducible DHR sector (m − 2, m, 0) has a dimension 1 and a spin −1. The superconformal net SVirc arises as a non-local extension of SVir0c as a crossed product by Z2 using identity and this sector.

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Let A be any superconformal net on the circle with c < 3/2. Then let B be its Bosonic part. By a similar argument to that in [35, Proposition 3.5], we know that the local conformal net B is an irreducible extension of the local conformal net SVir0c , where c is the central charge of A. By the strategy in [35] based on [4], we know that the dual canonical endomorphism θ of an extension is of the form θ = λ Z0,λ λ, where Z is the modular invariant arising from the extension as in [4]. Cappelli [9] gave a list of type I modular invariants and conjectured that it is a complete list. From his list, it is easy to guess that the dual canonical endomorphisms we use for obtaining extensions are those listed in Table 1. (Cappelli also considered type II modular invariants, but they do not correspond to local extensions, so we ignore them here.) We will prove that each of the dual canonical endomorphisms in Table 1 gives a local extension of SVir0c in a unique way and that an arbitrary such local extension of SVir0c gives one of the dual canonical endomorphisms in Table 1. 7.2. Study of type I modular invariants We study type I modular invariants for the coset nets for the inclusions SU (2)m ⊂ SU (2)m−2 ⊗ SU (2)2 . First we recall the S and T matrices for SU (2)m . For j, k = 0, 1, 2, . . . , m, we have the following. (j + 1)(k + 1) 2 (m) sin π , Sjk = m+2 m+2 1 πi (j + 1)2 (m) − Tjk = δjk exp . 2 m+2 2 For odd m, we have no problem arising from a fixed point of the order two symmetry, and in this case, the modular invariants have been already classified by Gannon–Walton [23], which shows that the identity matrix is the only modular invariant. So we have no non-trivial extensions in these cases. So we now deal with the case of even m in the rest of this section and put m = 2m0 . In this case, the S-matrix of the modular tensor category of the Table 1. List of candidates of the dual canonical endomorphisms.

(1) (2) (3) (4) (5) (6) (7) (8) (9)

m any m m = 4m m = 4m + 2 m = 10 m = 12 m = 28 m = 30 m = 10 m = 12

θ θ θ θ θ θ θ θ θ θ

= (0, 0, 0) = (0, 0, 0) ⊕ (0, m, 0) = (0, 0, 0) ⊕ (m − 2, 0, 0) = (0, 0, 0) ⊕ (0, 6, 0) = (0, 0, 0) ⊕ (6, 0, 0) = (0, 0, 0) ⊕ (0, 10, 0) ⊕ (0, 18, 0) ⊕ (0, 28, 0) = (0, 0, 0) ⊕ (10, 0, 0) ⊕ (18, 0, 0) ⊕ (28, 0, 0) = (0, 0, 0) ⊕ (0, 6, 0) ⊕ (8, 6, 0) ⊕ (8, 6, 0) = (0, 0, 0) ⊕ (6, 0, 0) ⊕ (0, 12, 0) ⊕ (6, 12, 0)

Label (Am−1 , Am+1 ) (A4m −1 , D2m +2 ) (D2m +2 , A4m +3 ) (A9 , E6 ) (E6 , A13 ) (A27 , E8 ) (E8 , A31 ) (D6 , E6 ) (E6 , D8 )

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irreducible DHR-sectors of the coset net is already not so easy to obtain, and it has been computed by Xu [62]. As in the previous section, we label the irreducible DHR sectors of the coset net for the inclusion SU (2)m ⊂ SU (2)m−2 ⊗SU (2)2 with triples (j, k, l) with j−k+ l being even, except for the case (m/2 − 1, m/2, 1). For this fixed point of the order two symmetry, we use labels ϕ1 = (m/2 − 1, m/2, 1)+ , ϕ2 = (m/2 − 1, m/2, 1)− to denote the two irreducible DHR sectors. We also use the symbols S (m−2) , S (m) , S (2) and S for the S-matrices for the nets SU (2)m−2 , SU (2)m , SU (2)2 and the coset for SU (2)m ⊂ SU (2)m−2 ⊗ SU (2)2 , respectively. Then Xu’s computation for the matrix S in [62] gives the following, where n = 1, 2. (m−2)

1. S(j,k,l),(j ,k ,l ) = 2Sjj

Skk Sll for (j, k, l), (j , k , l ) = ϕ1,2 . (m)

(2)

(m−2)

(m)

(2)

2. S(j,k,l),(j ,k ,l ),ϕn = Sϕn ,(j,k,l),(j ,k ,l ) = Sj,m/2−1 Sk,m/2 Sl1 for (j, k, l) = ϕ1,2 . (m−2)

(m)

(2)

3. Sϕn ,ϕn = δnn + (Sm/2−1,m/2−1 Sm/2,m/2 S11 − 1)/2. The T -matrix of the coset is described as follows more easily. (m−2)

1. T(j,k,l),(j ,k ,l ) = Tjj

Tkk Tll for (j, k, l), (j , k , l ) = ϕ1,2 . (m)

(2)

(m−2)

(m)

(2)

2. T(j,k,l),ϕn = Tϕn ,(j,k,l) = Tj,m/2−1 Tk,m/2 Tl1 for (j, k, l) = ϕ1,2 . (m−2)

(m)

(2)

3. Tϕn ,ϕn = δnn Tm/2−1,m/2−1 Tm/2,m/2 T11 . Suppose we have a modular invariant Z for the coset for SU (2)m ⊂ SU (2)m−2 ⊗ SU (2)2 . We define a new matrix Z˜(j,k,l),(j ,k ,l ) where the triples (j, k, l) and (j , k , l ) satisfy j, j ∈ {0, 1, . . . , m − 2}, k, k ∈ {0, 1, . . . , m}, l, l ∈ {0, 1, 2}. Note that we have no identification or splitting for the triples (j, k, l) and (j , k , l ) here. 1. If j − k + l, j − k + l ∈ 2Z and (j, k, l), (j , k , l ) = (m/2 − 1, m/2, 1), then we set Z˜(j,k,l),(j ,k ,l ) = Z(j,k,l),(j ,k ,l ) . 2. If j − k + l ∈ 2Z and (j, k, l) = (m/2 − 1, m/2, 1), then we set Z˜(j,k,l),(m/2−1,m/2,1) = Z(j,k,l),ϕ1 + Z(j,k,l),ϕ2 . 3. If j − k + l ∈ 2Z and (j , k , l ) = (m/2 − 1, m/2, 1), then we set Z˜(m/2−1,m/2,1),j k l = Zϕ1 ,(j ,k ,l ) + Zϕ2 ,j k l . 4. Z˜(m/2−1,m/2,1),(m/2−1,m/2,1) = Zϕ1 ,ϕ1 + Zϕ1 ,ϕ2 + Zϕ2 ,ϕ1 + Zϕ2 ,ϕ2 . 5. If j − k + l ∈ / 2Z or j − k + l ∈ / 2Z, then we set Z˜(j,k,l),(j ,k ,l ) = 0. This construction is analogous to the one of Lcw in [23, page 178], but now unlike in [23], our map Z → Z˜ may not be injective because of the definition of Z˜ involving the row/column labeled with (m/2 − 1, m/2, 1). For triples (j, k, l) and (j , k , l ) satisfying j, j ∈ {0, 1, . . . , m − 2}, k, k ∈ {0, 1, . . . , m}, l, l ∈ {0, 1, 2}, we set as follows. (We do not impose the conditions j − k + l, j − k + l ∈ 2Z here.) (m−2) (m) (2) 1. S˜(j,k,l),(j ,k ,l ) = Sjj Skk Sll . (m−2) (m) (2) 2. T˜(j,k,l),(j ,k ,l ) = Tjj Tkk Tll .

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We also write ϕ = (m/2 − 1, m/2, 1) and set I = (j, k, 0) | j + k ∈ 2Z ∪ (j, k, 1) | j + k ∈ / 2Z, j + k < m − 1 , which gives representatives of {(j, k, l) | j − k + l ∈ 2Z} {ϕ} under the order 2 symmetry. We now have the following proposition. Proposition 35. If the matrix Z is a modular invariant for S, T , then the matrix ˜ T˜. Z˜ is a modular invariant for S, Proof. It is clear that the entries of the matrix Z˜ are nonnegative integers and ˜ we have Z˜(0,0,0),(0,0,0) = 1. We thus have to prove Z˜ S˜ = S˜Z˜ and Z˜ T˜ = T˜Z. The latter is clear from the definition, so we have to prove the former identity at (j, k, l), (j , k , l ). We deal with seven cases separately. Case (1). j − k + l ∈ / 2Z, j − k + l ∈ / 2Z. The identity is trivial because the both hand sides are zero. / 2Z. The right hand side is obviously Case (2). j − k + l ∈ 2Z, j − k + l ∈ ˜ and the following three identities now prove that the zero by the definition of Z, left hand side is also zero. S˜ϕ,(j ,k ,l ) = 0 , Z˜(j,k,l),(j ,k ,l ) = Z˜(j,k,l),(m−2−j ,m−k ,2−l ) , S˜(j ,k ,l ),(j ,k ,l ) = −S˜(m−2−j ,m−k ,2−l ),(j ,k ,l ) . Note that the third identity holds because j − k + l ∈ / 2Z. Case (3). j − k + l ∈ / 2Z, j − k + l ∈ 2Z. This case can be proved in the same way as above. Case (4). j − k + l ∈ 2Z, j − k + l ∈ 2Z, (j, k, l) = ϕ, (j , k , l ) = ϕ. The identity ZS = SZ gives the following identity. 2Z(j,k,l),(j ,k ,l ) S˜(j ,k ,l ),(j ,k ,l ) (j ,k ,l )∈I

+ Z(j,k,l),ϕ1 S˜ϕ1 ,(j ,k ,l ) + Z(j,k,l),ϕ2 S˜ϕ2 ,(j ,k ,l ) = 2S(j,k,l),(j ,k ,l ) Z(j ,k ,l ),(j ,k ,l ) (j ,k ,l )∈I

+ S˜(j,k,l),ϕ Zϕ1 ,(j ,k ,l ) + S˜(j,k,l),ϕ Zϕ2 ,(j ,k ,l ) . Since we now have S˜(j ,k ,l ),(j ,k ,l ) = S˜(m−2−j ,m−k ,2−l ),(j ,k ,l ) , S˜(j,k,l),(j ,k ,l ) = S˜(j,k,l),(m−2−j ,m−k ,2−l ) ˜ (j,k,l),(j ,k ,l ) = (Z˜ Z) ˜ (j,k,l),(j ,k ,l ) . we conclude (Z˜ S)

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Case (5). j − k + l ∈ 2Z, (j, k, l) = ϕ, (j , k , l ) = ϕ. The identity ZS = SZ gives the following identity for n = 1, 2. Z(j,k,l),(j ,k ,l ) S˜(j ,k ,l ),ϕ + Z(j,k,l),ϕ1 Sϕ1 ,ϕn + Z(j,k,l),ϕ2 Sϕ2 ,ϕn (j ,k ,l )∈I

=

2S˜(j,k,l),(j ,k ,l ) Z(j ,k ,l ),ϕn

(j ,k ,l )∈I

+ S˜(j,k,l),ϕ Zϕ1 ,ϕn + S˜(j,k,l),ϕ Zϕ2 ,ϕn . Adding these two equalities for n = 1, 2, we obtain the following identity, ˜ (j,k,l),ϕ = (Z˜ Z) ˜ (j,k,l),ϕ . which gives (Z˜ S) Z(j,k,l),(j ,k ,l ) S˜(j ,k ,l ),ϕ + (Z(j,k,l),ϕ + Z(j,k,l),ϕ )S˜ϕ,ϕ 1

(j ,k ,l )∈I

=

2

2S˜(j,k,l),(j ,k ,l ) (Z(j ,k ,l ),ϕ1 + Z(j ,k ,l ),ϕ2 )

(j ,k ,l )∈I

+ S˜(j,k,l),ϕ (Zϕ1 ,ϕ1 + Zϕ1 ,ϕ2 + Zϕ2 ,ϕ1 + Zϕ2 ,ϕ2 ) . Case (6). (j, k, l) = ϕ, j −k +l ∈ 2Z, (j , k , l ) = ϕ. This case can be proved in the same way as above. Case (7). (j, k, l) = (j , k , l ) = ϕ. The identity ZS = SZ gives the following identity for n = 1, 2, n = 1, 2. Zϕn ,(j ,k ,l ) S˜(j ,k ,l ),ϕ + Zϕn ,ϕ1 Sϕ1 ,ϕn + Zϕn ,ϕ2 Sϕ2 ,ϕn (j ,k ,l )∈I

=

S˜ϕ,(j ,k ,l ) Z(j ,k ,l ),ϕn

(j ,k ,l )∈I

+ Sϕn ,ϕ1 Zϕ1 ,ϕn + Sϕn ,ϕ2 Zϕ2 ,ϕn . Adding these four identities for n, n = 1, 2, we obtain the following identity, which ˜ ϕ,ϕ = (Z˜ Z) ˜ ϕ,ϕ . gives (Z˜ S) 2(Zϕ1 ,(j ,k ,l ) + Zϕ2 ,(j ,k ,l ) )S˜(j ,k ,l ),ϕ (j ,k ,l )∈I

+ (Zϕ1 ,ϕ1 + Zϕ1 ,ϕ2 + Zϕ2 ,ϕ1 + Zϕ2 ,ϕ2 )S˜ϕ,ϕ = 2S˜ϕ,(j ,k ,l ) (Z(j ,k ,l ),ϕ1 + Z(j ,k ,l ),ϕ2 ) (j ,k ,l )∈I

+ S˜ϕ,ϕ (Zϕ1 ,ϕ1 + Zϕ1 ,ϕ2 + Zϕ2 ,ϕ1 + Zϕ2 ,ϕ2 ) .

We now recall Gannon’s parity rule [22, page 696]. For n and j, we set ε2n (j) as follows. 1. If j ≡ 0, n mod 2n, then ε2n (j) = 0. 2. If j ≡ 1, 2, . . . , n − 1 mod 2n, then ε2n (j) = 1. 3. If j ≡ n + 1, n + 2, . . . , 2n − 1 mod 2n, then ε2n (j) = −1.

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Set

L = n | n, 8m(m + 2) = 1 .


Then the parity rule is the following. If we have a modular invariant Z˜ for S˜ and T˜ with Z˜000,jkl = 0, then the parity rule says that we have ε2m (n)ε2m+4 (n)ε8 (n) = ε2m n(j + 1) ε2m+4 n(k + 1) ε8 n(l + 1) for all n ∈ L. Now suppose m0 = m/2 is odd. For any n1 with (n1 , 4m0 ) = 1, choose p ∈ {1, −1} so that n1 ≡ p mod 4. Then it is easy to see that there exists n ∈ L satisfying n ≡ n1 mod 4m0 , n ≡ p mod 4(m0 + 1), n ≡ p mod 8, since we have (m0 , m0 + 1) = 1. Suppose Z˜000,jkl = 0. Then the parity rule says ε2m (n1 )ε2m+4 (p)ε8 (p) = ε2m n1 (j + 1) ε2m+4 p(k + 1) ε8 p(l + 1) , which gives ε2m (n1 ) = ε2m (n1 (j + 1)) for all n1 with (n1 , 2m) = 1. If m0 = 3, 5, 15, then we have j = 0, m − 2 by [22]. Using the order two symmetry (j, k, l) ↔ ˜ we conclude that our dual canonical (m−2−j, m−k, 2−l) and the definition of Z, endomorphism θ has a decomposition within {(0, k, l) | −k +l ∈ 2Z}. Let T1 be the tensor category of the representations of SU (2)m with the opposite braiding and T2 be the tensor category of the representations of SU (2)2 with the usual braiding. If θ gives a local extension for the tensor category generated by {(0, k, l) | −k+l ∈ 2Z}, then it also gives a local extension for the tensor category T1 ×T2 by the description of the braided tensor category having the simple objects {(0, k, l) | −k + l ∈ 2Z} by [60, Section 4.3], [64, Proposition 2.3.1, Proof of Theorem B]. (Note that we have the notion of a local extension as a local Q-system in [45] for an abstract tensor category.) Then this extension for T1 × T2 gives a type I modular invariant for the S- and T -matrices arising from T1 × T2 . Then Gannon’s classification of modular invariants for SU (2)m ⊗ SU (2)2 in [22, Section 7] shows that our θ must be one of Table 1. (Here we have an opposite braiding for T1 , but this does not matter since we can consider a modular invariant Zk l,kl instead of Zkl,k l as in [23, page 178].) Next suppose m0 = m/2 is even. For any n2 with (n2 , 4(m0 + 1)) = 1, choose p ∈ {1, −1} so that n2 ≡ p mod 4. Then it is easy to see that there exists n ∈ L satisfying n ≡ p mod 4m0 , n ≡ n2 mod 4(m0 + 1), n ≡ p mod 8, since we have (m0 , m0 + 1) = 1. Suppose Z˜(0,0,0),(j,k,l) = 0. Then the parity rule says ε2m (p)ε2m+4 (n2 )ε8 (p) = ε2m p(j + 1) ε2m+4 n2 (k + 1) ε8 p(l + 1) , which gives ε2m+4 (n2 ) = ε2m+4 (n2 (k + 1)) for all n2 with (n2 , 4(m0 + 1)) = 1. If m0 = 2, 4, 14, then we have k = 0, m by [22]. Using the order two symmetry (j, k, l) ↔ (m − 2 − j, m − k, 2 − l) and the definition of Z˜ again, we conclude that our dual canonical endomorphism θ has a decomposition within {(j, 0, l) | j + l ∈ 2Z}. If θ gives a local extension for the tensor category generated by {(0, k, l) | −k + l ∈ 2Z}, then it also gives a local extension for the tensor category of the representations SU (2)m−2 ⊗SU (2)2 by the description of the braided tensor

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category having the simple objects {(j, 0, l) | j + l ∈ 2Z} by [60, Section 4.3], [65, Proposition 2.3.1]. Then this extension gives a type I modular invariant for the S- and T -matrices arising from SU (2)m−2 ⊗ SU (2)2 , so Gannon’s classification of modular invariants for SU (2)m−2 ⊗ SU (2)2 in [22, Section 7] again shows that our θ must be one of Table 1. We now deal with the remaining exceptional cases m = 4, 6, 8, 10, 28, 30. If Z˜(0,0,0),(j,k,l) = 0 and j − k + l ∈ 2Z, then the conformal spin of (j, k.l) is 1. This condition already gives the restriction that the dual canonical endomorphisms θ has a decomposition within (j, k, 0) | j + l ∈ 2Z for m = 4, 6, 8, 10, 28. Again Gannon’s classification of modular invariants in [22] shows that our θ must be one of Table 1. The final remaining case is m = 30. In this case, the only sectors (j, k, l) with l = 1 and j − k + l ∈ 2Z having conformal spins 1 are (3, 12, 1) and (13, 18, 1). The parity rules with n = 13, 7 exclude these two sectors from θ, respectively. Thus again the dual canonical endomorphism θ has a decomposition within (j, k, 0) | j + l ∈ 2Z and Gannon’s classification in [22] shows that our θ must be one of Table 1. We have thus proved that if we have a local extension of SVir0c , then its dual canonical endomorphism must be one of those listed in Table 1. 7.3. Classification of the Fermi extensions of the super-Virasoro net, c < 3/2 We now show that each endomorphism in Table 1 uniquely produces a superconformal net extending SVirc . (1) We have the label (Am−1 , Am+1 ). The identity matrix obviously gives the coset net for SU (2)m ⊂ SU (2)m−2 ⊗ SU (2)2 itself. This has a unique Fermionic extension. (2) Now we have the label (A4m −1 , D2m +2 ) with m = 4m . The irreducible DHR sectors (0, 0, 0), (0, 4m , 0) give the group Z2 . The crossed product of the coset net by this group gives a local extension. The irreducible DHR sector (4m −2, 0, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the crossed product construction. This α-induced sector gives a unique Fermionic extension through a crossed product by Z2 . (3) Now we have the label (D2m +2 , A4m +3 ) with m = 4m + 2. The irreducible DHR sectors (0, 0, 0), (4m , 0, 0) give the group Z2 . The crossed product of the coset net by this group gives a local extension. The irreducible DHR sector (0, 4m + 2, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the crossed product construction. This again gives a unique Fermionic extension in a similar way to case (2).

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(4) Now we have the label (A9 , E6 ) for m = 10. The DHR sector (0, 0, 0) ⊕ (0, 6, 0) gives a local Q-system as in [35, Subsection 4.2] by the description of the braided tensor category having the simple objects {(0, k, 0) | k ∈ 2Z} as in [60, Section 4.3], [64, Proposition 2.3.1, Proof of Theorem B]. Uniqueness also follows as in [35, Remark 2.5], using the cohomology vanishing result in [36]. (This is also a mirror extension in the sense of [65], so locality of the extension also follows from [65].) The irreducible DHR sector (0, 10, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the local extension. This gives a unique Fermionic extension as above. (5) Now we have the label (E6 , A13 ) for m = 12. The irreducible DHR sectors (0, 0, 0), (6, 0, 0) give a local Q-system as in (4) by the description of the braided tensor category having the simple objects {(j, 0, 0) | j ∈ 2Z} as in [60, Section 4.3], [64, Proposition 2.3.1]. Uniqueness also follows as above. (This is also a coset model as in [35].) The irreducible DHR sector (10, 0, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the local extension. This gives a unique Fermionic extension as above. (6) Now we have the label (A27 , E8 ) for m = 28. The irreducible DHR sectors (0, 0, 0), (0, 10, 0), (0, 18, 0), (0, 28, 0) give a local Q-system as in case (4). Uniqueness also follows as above, using the cohomology vanishing result in [36]. (This is again also a mirror extension in the sense of [65].) The irreducible DHR sector (26, 0, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the local extension. This gives a unique Fermionic extension as above. (7) Now we have the label (E8 , A31 ) for m = 30. The irreducible DHR sectors (0, 0, 0), (10, 0, 0), (18, 0, 0), (28, 0, 0) give a local Q-system as in (5). Uniqueness also follows as above. (This is again also a coset model as in [35].) The irreducible DHR sector (0, 30, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the local extension. This gives a unique Fermionic extension as above. (8) Now we have the label (D6 , E6 ) for m = 10. The irreducible DHR sectors (0, 0, 0), (0, 6, 0), (8, 0, 0), (8, 6, 0) give a local Q-system, since this is a further index 2 extension of the local extension given in the above (4). Uniqueness holds for this second extension. The irreducible DHR sector (0, 10, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the local extension. This gives a unique Fermionic extension as above. (9) Now we have the label (E6 , D8 ) for m = 12. The irreducible DHR sectors (0, 0, 0), (6, 0, 0), (0, 12, 0), (6, 12, 0) give a local Q-system, since this is a further index 2 extension of the local extension given in the above (5). Uniqueness holds for this second extension. The irreducible DHR sector (10, 0, 0) has spin −1, and this sector still gives an irreducible DHR sector of index 1 and spin −1 after the α-induction applied to the local extension. This gives a unique Fermionic extension as above.

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All the above give the following classification theorem. Theorem 36. The following gives a complete list of superconformal nets with c < 3/2 and the dual canonical endomorphisms for the Bosonic part extending SVir0c in each case is given as in Table 1. 8 , labeled with(Am−1 , Am+1 ). 1. The super Virasoro net with c = 32 1 − m(m+2) 2. Index 2 extensions of the above (1), labeled with (A4m −1 , D2m +2 ), (D2m +2 , A4m +3 ). 3. Six exceptionals labeled with (A9 , E6 ), (E6 , A13 ), (A27 , E8 ), (E8 , A31 ), (D6 , E6 ), (E6 , D8 ).

8. Outlook A potential development of this work concerns the relation with the Noncommutative Geometrical framework of A. Connes [14]. The supersymmetric sector with lowest weight c/24 in Section 6.4 gives rise in a natural way to an infinite dimensional spectral triple whose Chern character is given by the JLO cyclic cocycle [32], but for a domain problem verification (see also [7] where an analogous domain problem is settled). According to the lines indicated in [44], we expect the computation of this cocycle to be of interest in relation to a QFT index theorem.

Appendix A. A.1. Topological covers and covering symmetries If X is a connected manifold, we denote by DIFF(X) the group of diffeomorphisms of X and by Diff(X) the connected component of the identity of DIFF(X) 6 . Let X be compact, connected manifold; then DIFF(X) is an infinite dimensional Lie group modeled on the locally convex topological vector space Vect(X). ˜ the universal covering of X and by p : X ˜ → X the covering map. The Denote by X ˜ by deck transformations: p(γx) = p(x) fundamental group Γ ≡ π1 (X) acts on X ˜ and γ ∈ Γ. for x ∈ X ˜ the subgroup of Diff(X) ˜ of diffeomorphisms commuting Denote by Diff Γ (X) Γ ˜ with the Γ-action. Given g ∈ Diff (X) there exists a unique P (g) ∈ Diff(X) P (g)(x) ≡ p(g˜ x) , x ∈ X ˜ with x ˜ any point in X s.t. p(˜ x) = x. The map P : g → P (g) is clearly a continuous ˜ into Diff(X). It is not difficult to check that P is homomorphism of Diff Γ (X) ˜ by surjective. Indeed any g ∈ DIFF(X) has a lift to an element of DIFFΓ (X) ˜ thus P extends to a surjective continuous homomorphism the uniqueness of X, ˜ → DIFF(X), that thus maps the connected component of the identity DIFFΓ (X) to the connected component of the identity. More directly, if g ∈ Diff(X) is in a 6 If X is oriented, in general Diff(X) is smaller than the orientation preserving subgroup of DIFF(X), see [49]

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sufficiently small neighbourhood U of the identity, one can lift g ∈ U to an element ˜ because p is a local diffeomorphism; as Diff(X) is generated by U, all of Diff Γ (X) elements of Diff(X) have such a lift. Denote by Diff(X) the universal central cover of Diff(X). Then Diff(X) is an infinite dimensional Lie group with center Z ≡ π1 (Diff(X)) and we denote by q the quotient map q : Diff(X) → Diff(X) whose kernel is Z. If {gt }t∈[0,1] is a loop in Diff(X) with g0 = g1 = ι and x0 ∈ X, then {gt (x0 )}t∈[0,1] is a loop in X with g0 (x0 ) = g1 (x0 ) = x0 . Homotopic loops in Diff(X) clearly go to homotopic loops in X and we thus obtain a homomorphism (65) Φ0 : π1 Diff(X) → π1 (X) . This homomorphism is certainly not surjective (π1 (Diff(S 1 )) is always abelian, π1 (X) may be not). Proposition 37. There exists a unique continuous homomorphism Φ : Diff(X) → Γ ˜ Diff (X) such that the following diagram commutes: Φ / Diff Γ (X) ˜ Diff(X) JJ s s JJ q P ss JJ ss JJ s s J% yss Diff(X)

Φ restricts to a homomorphism Φ0 : π1 Diff(X) (= Z) → center of π1 (X)

(66)

that coincides with the one given above in eq. (65). However Φ0 is not injective in Thus π1 (X) is a quotient of π1 (Diff(X)). general. For example π1 (S 2 ) = {ι} while π1 Diff(S 2 ) = π1 Diff(SO(3)) = Z2 [49] . We end this appendix with the following proposition that has been used in the paper. (2)

Proposition 38. The group Diff (2) (S 1 ) is algebraically generated by Diff I (S 1 ) as I varies in I (2) . Proof. As I varies in I, Diff I (S 1 ) algebraically generates a normal subgroup of Diff(S 1 ), hence all Diff(S 1 ) because this is a simple group [49]. With rot the rotation one-parameter subgroup of Diff(S 1 ), we denote by rot(2) its lift to Diff (2) (S 1 ). Let H be the subgroup of Diff (2) (S 1 ) algebraically generated by the (2) Diff I (S 1 )’s and z ≡ rot(2) (2π) the generator of the center of Diff (2) (S 1 ) ( Z2 ). With q : Diff (2) (S 1 ) → Diff(S 1 ) the quotient map, we then have q(H) = Diff(S 1 ) and Diff (2) (S 1 ) = H ∪ Hz. If z ∈ H we then have Diff (2) (S 1 ) = H as desired. So we may assume z ∈ / H. Then q|H is one-to-one, namely an isomorphism between H and Diff(S 1 ). Consider now the one-parameter subgroup of H given by rot (θ) ≡

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q −1 (rot(θ)). Clearly q(rot (θ)) = rot(θ) = q(rot(2) (θ)), so rot(2) (θ) = rot (θ)z m where m = 0 or m = 1 depending on θ. So rot(2) (2θ) = rot (2θ) for all θ, thus rot(2) and rot coincide, in particular z = rot (2π) = 1 which is a contradiction. A.2. McKean–Singer formula Let Γ be a selfadjoint unitary on a Hilbert space H, thus H = H+ ⊕ H− where H± is the ±-eigenspace of Γ. A linear operator Q on H is even if it commutes with Γ, namely QH± ⊂ H± , and odd if ΓQΓ−1 = −Q, namely 0 Q− Q= Q+ 0 where Q± : H± → H∓ . Clearly Q∗+ = Q− if Q is selfadjoint. Note that Q2 is even if Q is odd. We shall denote by Trs = Tr(Γ · ) the supertrace on B(H). Clearly the supertrace depends on Γ, but the Hilbert spaces in the following will be equipped with a grading unitary on which the definition of Trs will be naturally based. Note that if T ∈ B(H) then Trs (T ) is well-defined and finite iff T is a trace class operator, i.e. Tr(|T |) < ∞. Indeed |ΓT | = |T |. We recall the following well known lemma. Lemma 39 (McKean–Singer formula). Let Q be a selfadjoint odd linear operator on 2 2 H. If e−tQ is trace class for some t0 > 0 then Trs (e−tQ ) is an integer independent of t > t0 and indeed Trs (e−tQ ) = ind(Q+ ) 2

where ind(Q+ ) ≡ dim ker(Q+ ) − dim ker(Q∗+ ) is the Fredholm index of Q+ . Proof. If t > t0 both traces Tr(e−tQ ) and Tr(Q2 e−tQ ) are convergent and we have 2 2 d Trs (e−tQ ) = − Trs (Q2 e−tQ ) . dt 2

2

But Trs (Q2 e−tQ ) = 0 because 2

Trs (Q2 e−tQ ) = Tr(ΓQ2 e−tQ ) = − Tr(QΓQe−tQ ) 2

2

2

= − Tr(ΓQe−tQ Q) = − Tr(ΓQ2 e−tQ ) = − Trs (Q2 e−tQ ) , 2

2

2

thus Trs (e−tQ ) is constant. Therefore 2

Trs (e−tQ ) = lim Trs (e−tQ ) = Tr(ΓE) 2

2

t→+∞

where E is the projection onto ker(Q2 ). But Q2 = Q∗+ Q+ ⊕ Q∗− Q− in the graded decomposition of H, thus ker(Q2 ) = ker(Q+ ) ⊕ ker(Q− ) and so Tr(ΓE) = dim ker(Q+ ) − dim ker(Q− ) = ind(Q+ ) .

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Acknowledgements The authors thank F. Xu for useful conversations. Our collaboration was carried on in different occasions: R. Longo wishes to thank V. G. Kac for the invitation at the Sch¨ odinger Institute summer school in June 2005; S. Carpi and R. Longo thank Y. Kawahigashi for the invitations at the University of Tokyo in November 2005 and December 2006 respectively.

References [1] B. Bakalov, N. Nikolov, K.-H. Rehren & I. Todorov, Unitary positive-energy representations of scalar bilocal quantum fields, Commun. Math. Phys. 271 (2007) 223–246. [2] J. B¨ ockenhauer, Localized endomorphisms of th chiral Ising model, Commun. Math. Phys. 177 (1996), 265–304. [3] J. B¨ ockenhauer & D. E. Evans, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, Commun. Math. Phys. 213 (2000), 267–289. [4] J. B¨ ockenhauer, D. E. Evans & Y. Kawahigashi, On α-induction, chiral projectors and modular invariants for subfactors, Commun. Math. Phys. 208 (1999) 429–487. [5] R. Brunetti, D. Guido & R. Longo, Modular structure and duality in conformal quantum field theory, Commun. Math. Phys. 156 (1993) 201–219. [6] D. Buchholz, C. D’Antoni & R. Longo, Nuclearity and thermal states in conformal field theory, Commun. Math. Phys. 270 (2007) 267–293. [7] D. Buchholz & H. Grundling, Algebraic supersymmetry: A case study, Commun. Math. Phys. 272 (2007), 699–750. [8] D. Buchholz & H. Schulz-Mirbach, Haag duality in conformal quantum field theory, Rev. Math. Phys. 2 (1990), 105–125. [9] A. Cappelli, Modular invariant partition functions of superconformal theories, Phys. Lett. B 185 (1987) 82–88. [10] A. Cappelli, C. Itzykson & J.-B. Zuber, The A-D-E classification of minimal and (1) A1 conformal invariant theories, Commun. Math. Phys. 113 (1987) 1–26. [11] S. Carpi, Classification of subsystems for the Haag–Kastler nets generated by c = 1 chiral current algebras, Lett. Math. Phys. 47, 353–364 (1999). [12] S. Carpi, On the representation theory of Virasoro nets, Commun. Math. Phys. 244 261–284 (2004). [13] S. Carpi & M. Weiner, On the uniqueness of diffeomorphism symmetry in conformal field theory, Commun. Math. Phys. 258, 203–221 (2005). [14] A. Connes, “Noncommutative Geometry” Academic Press (1994). [15] C. D’Antoni, K. Fredenhagen & S. K¨ oster, Implementation of conformal covariance by diffeomorphism symmetry, Lett. Math. Phys. 67, 239–247 (2004). [16] C. D’Antoni, R. Longo & F. Radulescu: Conformal nets, maximal temperature and models from free probability, J. Operator Theory 45, 195–208 (2001). [17] P. Di Francesco, P. Mathieu & D. Sénéchal, “Conformal Field Theory”, Berlin, Heidelberg, New York: Springer Verlag 1996.

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[18] S. Doplicher, R. Haag & J. E. Roberts, Local observables and particle statistics. I & II, Commun. Math. Phys. 23, 199–230 (1971) & 35, 49–85 (1974). [19] D. E. Evans & Y. Kawahigashi, “Quantum Symmetries on Operator Algebras”, Oxford University Press, 1998. [20] D. Friedan, Z. Qiu & S. Shenker, Superconformal invariance in two dimensions and the tricritical Ising model, Phys. Lett. 151B, 37 (1985). [21] J. Fuchs, B. Schellekens & C. Schweigert, The resolution of field identification fixed points in diagonal coset theories, Nucl. Phys. B 461 371 (1996). [22] T. Gannon, Towards a classification of su(2)⊕· · ·⊕su(2) modular invariant partition functions, J. Math. Phys. 36 675–706 (1995). [23] T. Gannon & M. A. Walton, On the classification of diagonal coset modular invariants, Commun. Math. Phys. 173 175–197 (1995). [24] P. Goddard, A. Kent & D. Olive, Unitary representations of the Virasoro and superVirasoro algebras, Commun. Math. Phys. 103, 105–119 (1986). [25] R. Goodman & N. R. Wallach, Projective unitary positive-energy representations of Diff(S 1 ), J. Funct. Anal. 63, 299–321 (1985). [26] D. Guido & R. Longo, Relativistic invariance and charge conjugation in quantum field theory, Commun. Math. Phys. 148, 521–555 (1992). [27] D. Guido & R. Longo, The conformal spin and statistics theorem, Commun. Math. Phys. 181, 11 (1996). [28] D. Guido & R. Longo, A converse Hawking–Unruh effect and dS 2 /CF T correspondence, Ann. H. Poincaré 4, 1169–1218 (2003). [29] R. Haag, “Local Quantum Physics”, Springer-Verlag (1996). [30] M. Henkel, “Conformal Invariance and Critical Phenomena”, Texts and monographs in Physics, Springer-Verlag, Berlin (1999). [31] Y.-Z. Huang, Vertex operator algebras, the Verlinde conjecture, and modular tensor categories, Proc. Natl. Acad. Sci. USA 102, 5352–5356 (2005). [32] A. Jaffe, A. Lesniewski & K. Osterwalder, Quantum K-theory I. The Chern character, Commun. Math. Phys. 118 1–14 (1988). [33] V. F. R. Jones, Index for subfactors, Invent. Math. 72 1–25 (1983). [34] V. G. Kac & A. K. Raina, “Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras”, World Scientific, Singapore, 1987. [35] Y. Kawahigashi & R. Longo, Classification of local conformal nets. Case c < 1, Ann. of Math. 160 (2004), 493–522. [36] Y. Kawahigashi & R. Longo, Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories, Commun. Math. Phys. 244 (2004) 63–97. [37] Y. Kawahigashi & R. Longo, Noncommutative spectral invariants and black hole entropy, Commun. Math. Phys. 257, 193–225 (2005). [38] Y. Kawahigashi, R. Longo & M. M¨ uger, Multi-interval subfactors and modularity of representations in conformal field theory, Commun. Math. Phys. 219, 631–669 (2001). [39] Y. Kawahigashi, R. Longo, U. Pennig, U & K.-H. Rehren, The classification of nonlocal chiral CFT with c < 1, Commun. Math. Phys. 271, 375–385 (2007).

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[40] T. Loke, “Operator Algebras and Conformal Field Theory of the Discrete Series Representation of Diff + (S 1 )”. PhD Thesis, University of Cambridge, 1994. [41] R. Longo, Index of subfactors and statistics of quantum fields. I, Commun. Math. Phys. 126 (1989) 217–247 [42] R. Longo, Index of subfactors and statistics of quantum fields. II, Commun. Math. Phys. 130, 285–309 (1990). [43] R. Longo, An analogue of the Kac–Wakimoto formula and black hole conditional entropy, Commun. Math. Phys. 186, 451–479 (1997). [44] R. Longo, Notes for a quantum index theorem, Commun. Math. Phys. 222 (2001) 45–96. [45] R. Longo & K. H. Rehren, Nets of subfactors, Rev. Math. Phys. 4, 567–597 (1995). [46] R. Longo & K. H. Rehren, Local fields in boundary CFT, Rev. Math. Phys. 16, 909– 960 (2004). [47] R. Longo & F. Xu, Topological sectors and a dichotomy in conformal field theory, Commun. Math. Phys. 251, 321–364 (2004). [48] R. Longo, Conformal subnets and intermediate subfactors, Commun. Math. Phys. 237 (2003), 7–30. [49] J. W. Milnor, Remarks on infinite dimensional Lie groups, in: “Relativity, Groups, and Topology II, Les Houches Sessions”, de Witt, Stora, ed., vol XL, 1983, North Holland, Amsterdam, 1984. [50] K.-H. Rehren, Braid group statistics and their superselection rules, in “The Algebraic Theory of Superselection Sectors”, D. Kastler ed., World Scientific 1990. [51] K.-H. Rehren, Space-time fields and exchange fields, Commun. Math. Phys. 132 (1990), 461–483. [52] J. E. Roberts, More lectures on algebraic quantum field theory, in: “Noncommutative Geometry”, S. Doplicher and R. Longo eds., Lecture Notes in Math. 1831, p. 263–342, Springer-Verlag, Berlin, 2004. [53] V. Toledano Laredo, Integrating unitary representations of infinite-dimensional Lie groups, J. Funct. Anal. 161 (1999), 478–508. [54] M. Wakimoto, “Infinite-dimensional Lie algebras”, Providence, RI: AMS 1999. [55] A. Wassermann, Operator algebras and conformal field theory III: Fusion of positive energy representations of SU(N) using bounded operators, Invent. Math. 133 (1998) 467–538. [56] M. Weiner, Conformal covariance and positivity of energy in charged sectors, Commun. Math. Phys. 265, 493–506 (2006). [57] M. Weiner, “Conformal Covariance and Related Properties of Chiral QFT”, Ph.D. Thesis, Univ. of Rome “Tor Vergata” 2005, arXiv:math/0703336. [58] J. Wess & J. Bagger, “Supersymmetry and Supergravity”. Second edition. Princeton University Press, Princeton, NJ, 1992. [59] F. Xu, Algebraic orbifold conformal field theory, in “Mathematical Physics in Mathematics and Physics”, R. Longo ed., Fields Institute Communications Vol. 30, Amer. Math. Soc., Providence RI 2001. [60] F. Xu, New braided endomorphisms from conformal inclusions, Commun. Math. Phys. 192 (1998) 347–403.

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[61] F. Xu, Algebraic coset conformal field theories I, Commun. Math. Phys. 211 (2000) 1–44. [62] F. Xu, Algebraic coset conformal field theories II, Publ. RIMS, Kyoto Univ. 35 (1999) 795–824. [63] F. Xu, On a conjecture of Kac–Wakimoto, Publ. RIMS, Kyoto Univ. 37 (2001) 165– 190. [64] F. Xu, 3-manifolds invariants from cosets, J. Knot Theory Ramif. 14 (2005) 21–90. [65] F. Xu, Mirror extensions of local nets, Commun. Math. Phys. 270 (2007) 835–847. Sebastiano Carpi Dipartimento di Scienze Universit` a di Chieti-Pescara “G. d’Annunzio” Viale Pindaro, 42 I-65127 Pescara Italy e-mail: [email protected] Yasuyuki Kawahigashi Department of Mathematical Sciences University of Tokyo Komaba, Tokyo, 153-8914 Japan e-mail: [email protected] Roberto Longo Dipartimento di Matematica Universit` a di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1 I-00133 Roma Italy e-mail: [email protected] Communicated by Klaus Fredenhagen. Submitted: March 3, 2008. Accepted: May 5, 2008.



A Cox Process Involved in the Bose–Einstein Condensation Nathalie Eisenbaum Abstract. The point process corresponding to the configurations of bosons in standard conditions is a Cox process driven by the square norm of a centered Gaussian process. This point process is infinitely divisible. We point out the fact that this property is preserved by the Bose–Einstein condensation phenomenon and show that the obtained point process after such a condensation occured, is still a Cox process but driven by the square norm of a shifted Gaussian process, the shift depending on the density of the particles. This law provides an illustration of a “super”- Isomorphism Theorem existing above the usual Isomorphism Theorem of Dynkin available for Gaussian processes.

1. Introduction The point processes corresponding to the spatial configurations of fermions and bosons in standard conditions have been clearly identified (see Macchi [13, 14]) and are usually respectively named fermion point processes and boson point processes. Shirai and Takahashi [16] have given an unified presentation of these two classes of processes by introducing the definition of alpha-permanental (or alphadeterminantal) random point processes. Indeed they have established the existence of random point processes whose Laplace transforms are equal to the power (− α1 ) of a Fredholm determinant. When α = −1 one obtains a determinantal (or a fermion) point process, when α = 1 it is a permanental (or a boson) point process. These results allow to check the property of infinite divisibility of some alphapermanental point processes for α > 0, and in particular of the point process associated to an infinite collection of bosons in an infinite box with a density lower than the Bose–Einstein critical density. We remind that a point process N is infinitely divisible if for every integer k there exists k independent identically distributed point processes N1,k , N2,k , . . . , Nk,k such that (law)

N = N1,k + N2,k + · · · + Nk,k .

(1.1)

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Note that a fermion point process can not be infinitely divisible. One can have an intuition of that fact thanks to the “anti-bunching” property of the fermions: two fermions can not be closer than the “correlation length”, a given distance whose existence has been assumed theoretically before being recently put in evidence experimentally (see [10] Jeltes et al.). This property makes impossible the realization of (1.1) even for k = 2 since two points belonging respectively to the support of N1,2 and N2,2 can not be closer than the correlation length. The independency of N1,2 and N2,2 can not afford that. More generally for α < 0, alpha-permanental random point processes can not be infinitely divisible. Coming back to the bosons, one may ask whether a Bose–Einstein condensation would preserve this infinite divisibility property. The answer is affirmative and based on a recent paper of Tamura and Ito [19] who have obtained in a new way the law of the configurations of the particles of an ideal Bosonian gas containing particles in a Bose–Einstein condensation state. We shall analyze in Section 2 their result and show why infinite divisibility is the key to understand the factorization of the Cox process involved in the Bose–Einstein condensation. In Section 3, we show that this factorization provides an illustration of a “super”- Isomorphism Theorem existing above the usual Isomorphism Theorem of Dynkin [3].

2. An infinitely divisible Cox process Shirai and Takahashi [16] have extended the notions of boson and fermion point processes by introducing the following distributions denoted by μα,K . The corresponding random point processes are sometimes called alpha-permanental point processes. In the definition below, E is a locally compact Hausdorff space with a countable basis, λ is a nonnegative Radon measure on E, and Q is the space of nonnegative integer-valued Radon measures on E. An operator K on L2 (E, λ) is locally bounded if for every compact subset of E, A, the operator PA KPA is bounded (PA denotes the projection from L2 (E, λ) to L2 (A, λ)). Definition 2.1. For K a locally bounded integral operator on L2 (E, λ) and α a fixed number, the distribution μα,K on Q satisfies, when it exists μα,K (dξ)exp − ξ, f = Det(I + αKφ )−1/α (2.1) Q

for every nonnegative measurable function f with compact support on E, Kφ stands for the trace class operator defined by Kφ (x, y) = φ(x)K(x, y) φ(y) and

φ(x) = 1 − exp − f (x) .

The function Det denotes the Fredholm determinant.

Vol. 9 (2008) A Cox Process Involved in the Bose–Einstein Condensation

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For a different presentation of these distributions we refer the reader to the paper of Hough et al. [9] When α = 1, μ1,K is the distribution of the configurations of a Bosonian gas. Shirai and Takahashi have established sufficient conditions, on the operator K, for the existence of the distribution μα,K . In particular for K, a locally bounded integral operator on L2 (E, λ) and α a fixed positive number, they have shown that if (α, K) satisfies (B) : the kernel function of the operator Jα = K(I + αK)−1 is nonnegative then μα,K exists and is infinitely divisible. Note that μα,K is infinitely divisible iff μnα,K/n exists for every n ∈ N∗ . 1 In the special case E = Rd , α = 1 and J1 with kernel J1 (x, y) = (4πβ) d/2 2 −1 exp(−|x − y| /4β) (K is such that J1 = K(I + K) ), the distribution μ1,K can be obtained as the limit of the distributions of the positions in Rd of N identical particles following the Bose–Einstein statistics in a finite box. More precisely, one starts from the following random point measure μ(L,N ) which describes the location of an ideal Bosonian gas, composed of N particles in a volume V = [−L/2, L/2]d with d ≥ 1, at a given temperature T μ(L,N ) (dξ)e−ξ,f VN ⎛ ⎞ N =C exp ⎝− f (xj )⎠ per GL (xi , xj ) 1≤i,j≤N dx1 . . . dxN VN

j=1

where the constant C is equal to V N per(GL (xi , xj ))1≤i,j≤N dx1 . . . dxN , GL denotes the operator exp(βΔL ) with β = 1/T and ΔL is the Laplacian under the periodic condition in L2 (V ). As N and V are tending to ∞ with N/V → ρ, μ(L,N ) converges to a limit 2 dx e−β|x| depending on ρ. Indeed, denoting by ρc the critical density Rd (2π) 2 which d −β|x| 1−e is finite for d > 2, we have – if ρ < ρc , then μ(L,N ) converges to μ1,Kρ , where Kρ = (ρ)J1 (I − (ρ)J1 )−1 and (ρ) is a positive constant depending on ρ. This last result provides a justification to the fact that μ1,Kρ is the distribution of the configurations of an ideal Bosonian gas. The next result is more illuminating; indeed, in the case d > 2 and – if ρ ≥ ρc , then μ(L,N ) converges to a random point process with a distribution ζ given by ζ(dξ)e−ξ,f Q

1 − e−f , (I + Kφ )−1 1 − e−f = Det(I + Kφ )−1 exp − (ρ − ρc ) (2.2) where K = J1 (I − J1 )−1 .

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The physical explanation of this split convergence, actually a phase transition, is due to the fact that when the density of the gas becomes higher than ρc , a certain proportion of the particles tends to lower the density by reaching the lowest level of energy. This phenomenon, called the Bose–Einstein condensation, predicted by Einstein in 1925, is intensively studied today especially since this phenomenon has been experimentally obtained (for d = 3 of course) in 1995 by a team at JILA. It is interesting to see that the Bose–Einstein condensation phenomenon provides an illustration in the case d = 3 of a mathematical physics result available for any dimension d greater than 3. These results have been established by many authors. In particular they are consequences of the works of Bratteli and Robinson [2] (see Theorem 5.2.32 Chap. 5 p. 69) and of Fichtner and Freudenberg [7]. The way Tamura and Ito have obtained these results in [18] and [19], deserves a special attention because they need neither quantum field theories nor the theory of states on the operator algebras, but mostly an integral formula due to Vere-Jones [21]. Further, Tamura and Ito have actually done more than (2.2). In [19] their proof is based on the following theorem. Theorem A. Let K be a locally bounded symmetric integral operator on L2 (E, λ) such that (1, K) satisfies condition (B) and J1 (x, y)λ(dy) ≤ 1 λ(dx) a.e. (2.3) E

Then for every r > 0, there exists a unique random measure with distribution ζr on Q such that for every non-negative measurable function f on E

−ξ,f −1 −f −f ζr (dξ)e = exp − r 1 − e , (I + Kφ ) 1−e (2.4) Q

where (., .) denotes the inner product of L2 (E, λ). Tamura and Ito’s result generates several natural questions: • In (2.2) the distribution of the configurations of the particles is, by Theorem A, the convolution of two distributions: μ1,K ∗ζρ−ρc . It is tempting to imagine that μ1,K corresponds to the fraction of the particles with level of energy greater than 0 and that ζρ−ρc corresponds to the particles that did “coalesce” (i.e. without kinetic energy or similarly in a quantic state equal to 1.) Indeed (ρ) is a continuous function of ρ on (0, ρc ] that takes the value 1 at ρc . Hence the distribution of the configurations of particles with density ρc has the distribution μ1,K . The problem is to know whether the configurations of particles with 0 kinetic energy are independent of the configurations of moving particles? In what follows we shall answer this question. • The assumption of condition (B) in Theorem A, makes μ1,K infinitely divisible. Since ζr = (ζr/n )∗n , ζr is infinitely divisible as well. Consequently the distribution ζ given by (2.2) is also infinitely divisible. Moreover, thanks to Theorem A, the distribution ζ exists for any K such that (1, K) satisfies


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condition (B) and (2.3). We therefore obtain a family of infinitely divisible distributions – who are they? Their characterization is the theme of Theorem 2.3. • Besides, in their paper [16] (Theorem 6.12) Shirai and Takahashi have obtained a factorization involving μα,K for (α, K) satisfying condition (B) (see (2.13) below). In the case α = 1, is this factorization connected to (2.2)? We will show that the answer is affirmative and that they are both direct consequences of the infinite divisibility of μα,K . To analyze further the results of Tamura and Ito, we will use the notion of a Cox process. Definition 2.2. A Cox process is a Poisson point process with a random intensity σ on the space of Radon measures on E. Its distribution Πσ satisfies therefore 1 − e−f (x) σ(dx) , Πσ (dξ)exp − ξ, f = IE exp − Q

E

for every nonnegative measurable function f with compact support on E. We shall work mostly with Cox processes with random intensity ψ(x)λ(dx) where (ψ(x), x ∈ E) is a positive process such that IE(ψ(x)) is a locally bounded function of x. Such a Cox process is said to be driven by (ψ, λ). We denote its distribution by Πψ,λ or Πψ when there is no ambiguity about the measure λ. If ψ is equal to 12 η 2 with η real valued centered Gaussian process with covariance (K(x, y), x, y ∈ E) then −f (x) Π(ψ,λ) (dξ)exp − ξ, f = IE exp − (1 − e )ψ(x)λ(dx) Q

E

for every positive function f with compact support. Using the Dominated Convergence Theorem one shows then that Π(ψ,λ) = μ 12 ,K . Note that a priori the couple ( 12 , K) does not satisfy condition (B) of Shirai and Takahashi. Remark 2.2.1. The infinite divisibility of a Cox process with distribution Πψ is not equivalent to the infinite divisibility of the process ψ. Of course, the infinite divisibility of ψ implies the infinite divisibility of Πψ , but the converse is not true. This fact has been stated in 1975 by Kallenberg [11] (Ex. 8.6, p. 58 Chap. 8 – see also Shanbhag and Westcott (1977) [15]). Condition (B) of Shirai and Takahashi allows to put in evidence examples of squared Gaussian processes η 2 which are not infinitely divisible although the Cox process with distribution Πη2 is infinitely divisible. For this purpose, consider the example of the ideal Bose gas, where 1 2 J1 (x, y) = (4πβ) d/2 exp(−|x − y| /4β). By condition(B), μ1,K is infinitely divisible. Further note that, μ1,K = Π 12 η2 ,λ ∗ Π 12 η2 ,λ , where (ηx , x ∈ Rd ) is a centered Gaussian process with covariance (K(x, y), x, y ∈ Rd ) and λ is the Lebesgue measure on Rd . Hence Π 12 η2 ,λ is infinitely divisible. Using now Bapat’s characterization of Gaussian processes with infinitely divisible square [1], we can easily choose

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x, y, z ∈ Rd such that (ηx2 , ηy2 , ηz2 ) is not infinitely divisible. Indeed, if yi zi < 0 for every 1 ≤ i ≤ d then |y − z|2 > |y − x|2 + |z − x|2 for x in Rd with |x| small enough. The matrix (K(a, b), a, b ∈ {x, y, z}) has only positive coefficients and (K −1 (a, b), a, b ∈ {x, y, z}) has at least one off-diagonal positive coefficient. Consequently this last matrix can not be an M-matrix. In Lemma 2.5, we shall characterize the infinitely divisible Πψ where ψ is a nonnegative process. To state the main result of this section, Theorem 2.3, we need the following notation. Let (ηx , x ∈ E) be a centered Gaussian process with covariance (K(x, y), x, y ∈ E) and a a point not in E. Extend the process η to E ∪ {a} by setting ηa = 0 and K(a, a) = K(x, a) = K(a, x) = 0 for all x ∈ E. Let (ψx , x ∈ E) be a centered Gaussian process with covariance (K(x, y) + 1, x, y ∈ E). One can similarly extend it to E ∪ {a} in the obvious manner, namely, E(ψx ψa ) = 1 for all x in E. For a measure λ on E and > 0, define λ = λ + δa where δa is the Dirac measure with mass at a. For a positive random process (φx , x ∈ E ∪ {a})on E ∪ {a}, we denote by Πφ,λ the distribution of a Cox process with random intensity φx λ (dx) on E ∪ {a}. Note that for the process η above, we have: Πη2 ,λ = Πη2 ,λ . Without ambiguity Q will denote the space of nonnegative integer-valued Radon measures on E ∪ {a}. With these notation, we are now ready to state Theorem 2.3. Its proof is deferred to the end of this section. Theorem 2.3. Let (ηx , x ∈ E) be a centered Gaussian process with covariance (K(x, y), x, y ∈ E). Let (ψx , x ∈ E) be a centered Gaussian process with covariance (K(x, y) + 1, x, y ∈ E). Assume that the distribution Π 12 η2 ,λ is infinitely divisible, then the following five points are equivalent. (i) The distribution Π 12 (η+c)2 ,λ is infinitely divisible for every constant c in R. (ii) The distribution Π 12 ψ2 ,λ is infinitely divisible for every > 0. (iii) For every r > 0 there exists a random measure with distribution νr on Q such that Π 12 (η+r)2 ,λ = Π 12 η2 ,λ ∗ νr . (2.5) Moreover the distribution νr satisfies 1 2 −ξ,f −1 −f −f 1 − e , (I + Kφ ) 1−e νr (dξ)e = exp − r , 2 Q

(2.6)

where the inner product is taken with respect to the measure λ. (iv) For every > 0 and r > 0, there exists a random measure with distribution νr on Q satisfying (2.5) and (2.6) but for the measure λ instead of the measure λ. (v) The distribution Π 12 (η+c)2 ,λ is infinitely divisible for every constant c in R and every > 0. Remark 2.3.2. Note that the distribution νr of (iii) is not necessarily the distribution of a Cox process. In Section 3, we will see that, this makes precisely the


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difference between the infinite divisibility of Πφ and the infinite divisibility of φ. We shall see also that this remark extends from squared Gaussian processes to nonnegative processes. We are now in position to analyze further the results of Tamura and Ito. Recall that in the case ρ > ρc , the case when a Bose–Einstein condensation occurs, the obtained limit ζ is equal to μ1,K ∗ ζρ−ρc , where ζr is defined by Theorem A. First we note that by Theorem 2.3, the existence of ζρ−ρc for every ρ > ρc is equivalent to the infinite divisibility of μ1,K+1 . To check directly this last property we can, for example, verify that (1, K + 1) satisfies condition (B). Indeed we have the following general result which does not require that K be symmetric. Proposition 2.4. Let K be an integral operator on L2 (E, λ) such that (1, K) satisfies condition (B) and J1 (y, x)λ(dy) ≤ 1 λ(dx) a.e. (2.7) E

and

J1 (x, y)λ(dy) ≤ 1

λ(dx) a.e.

(2.8)

E

Then (1, K + 1) satisfies condition (B). Proof of Proposition 2.4. Set J 1 = (K + 1)(I + K + 1)−1 . We have to show that J 1 has a nonnegative kernel. We denote by 11 the integral operator on L1 (E, λ) with the kernel identically equal to 1. We then have I + K + 11 = (I − J1 )−1 + 11 = −1 (I + 11(I − J1 ))(I − J1 )−1 , which leads to J 1 = (K + 11)(I − J1 ) (I + 11(I − J1 )) . 2 −1 Let f be an nonnegative element of L (E, λ). We set g = (I + 11(I − J1 )) f and similarly f = g + 11(I − J1 )g. Note that 11(I − J1 )g is a constant function that we denote by c(g). We claim that c(g) ≥ 0. Indeed, c(g) = 11(I − J1 )(f − c(g)) and by (2.7) the integral operator 11(I − J1 ) has a positive kernel. Hence if c(g) < 0 then f − c(g) is a positive function and therefore so is c(g) = 11(I − J1 )(f − c(g)), which leads to a contradiction. Thus, c(g) ≥ 0. We now have: J 1 f = (K+11)(I−J1 )g = J1 g+11(I−J1 )g = J1 f −c(g) +c(g) = J1 f +(I−J1 )c(g) . By (2.8) (I − J1 )c(g) ≥ 0 and thanks to (B), J1 f is nonnegative. Consequently J 1 f is non negative as well. Theorem 2.3 and Proposition 2.4 prove Theorem A of Tamura and Ito. Restricting our attention to the case of ideal Bosonian particles, we see that, with the notations of Theorem 2.3, (2.2) becomes (2.9) ζ = μ1,K ∗ ν√2(ρ−ρ ) . c

Can we interpret ν√2(ρ−ρ

c)

as the law of the configurations of the particles with 0

kinetic energy and density ρ−ρc ? These particles are at temperature T = 1/β and the distribution ζ depends on T . Now imagine that we can lower the temperature T

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to 0, we then have ρc → 0 and for any positive function f with compact support, we easily obtain ζ(dξ)e−ξ,f −→ exp −ρ (1 − e−f (x) )dx . Rd

Q

The obtained limit is the distribution of a Poisson point process with uniform intensity ρdx on Rd . But at temperature T = 0, all the particles are at 0 level of kinetic energy. Hence this limit Πρ,dx is the distribution of the configurations of particles, with density ρ, at 0 level of energy and temperature 0. This has been already established (differently) by Goldin et al. [8]. Now remember that once the 0 state of kinetic energy is reached, the particles don’t move anymore, hence the law of their configurations should not vary when the temperature goes down. But obviously ν√2(ρ−ρ ) is different from Π(ρ−ρc ),dx . Consequently the answer to c the above question is negative. This implies that the presence of particles in the Bose–Einstein condensation state has an influence on the configurations of the still moving particles. Moreover, (2.9) can be rewritten as ζ = Π 12 η2 ∗ ν√ρ−ρc ∗ Π 12 η2 ∗ ν√ρ−ρc which leads to ζ = Πψ

(2.10)

√ √ (law) with (ψx , x ∈ E) = ( 12 (ηx + ρ − ρc )2 + 12 (˜ ηx + ρ − ρc )2 , x ∈ E) and η and η˜ two independent centered Gaussian processes with covariance (K(x, y), x, y ∈ Rd ). Under this writing it appears that ζ is the distribution of a Cox process. Similarly (2.9) leads to ζ = Π 1 (η+√2(ρ−ρ ))2 ∗ Π 12 η2 . (2.11) 2

c

Under this last form, one can provide a physical interpretation in terms of fields (instead of particles). We thank Yvan Castin from Laboratoire Kaestler–Brossel for the following explanation. The Bosonic field (φ(x), x ∈ Rd ) satisfies φ(x) = φ0 +φe (x), where φ0 is a (spatially) uniform field corresponding to the condensated particles and (φe (x), x ∈ Rd ) is the field corresponding to the excited particles. η ). Besides φ0 is taken This last field φe is a complex Gaussian field: φe = √12 (η + i˜ √ to be the constant ρ − ρc . The real component of φe can interfere with φ0 and provides the part Π 1 (η+√2(ρ−ρ ))2 of ζ, while the imaginary component of φe does c 2 not interfere with φ0 and its contribution to ζ is the same as for the gas without condensation Π 12 η2 . To prove Theorem 2.3 we will use the following characterization of the infinite divisible random measure. According to Theorem 11.2 (Chap. 11, p. 79) in Kallenberg’s book [11], a random measure with distribution ζ is infinitely divisible iff for almost every x,w.r.t. IE(ζ), there exists a random measure with distribution μx


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on Q such that ζ x = ζ ∗ μx

(2.12)

where (ζ x , x ∈ E) denotes the Palm measures of ζ. In the special case of a couple (α, K) satisfying condition (B), we hence obtain the existence of μx such that μxα,K = μα,K ∗ μx

(2.13)

which is precisely the factorization obtained by Shirai and Takahashi. Note that it is really an immediate consequence of the infinite divisibility of μα,K . We are going to make use of Kallenberg’s Theorem (2.12) to characterize the infinitely divisible Πψ . Lemma 2.5. Let Πψ be the distribution of a Cox process directed by a positive process (ψx , x ∈ E) with respect to λ. Assume that IE(ψx ) is a locally bounded function of x. For each b in E such that IE(ψb ) > 0, denote by (ψ (b) (x), x ∈ E) ψ(b) ; .]. The Palm measure at b of Πψ , denoted by Πbψ , the process ψ under IE[ IE(ψ(b)) admits the following factorization for IE(ψx )λ(dx) almost every b Πbψ = Πψ(b) ∗ δb where δb is the Dirac point mass at b. The distribution Πψ is infinitely divisible iff for almost every b w.r.t. IE(ψx )λ(dx), there exists a random measure with distribution μb such that Πψ(b) ∗ δb = Πψ ∗ μb . Proof of Lemma 2.5. For every nonnegative function f on E, we have: Πψ (dξ)e−ξ,f = IE exp − (1 − e−f (x) )ψ(x)λ(dx) . Q

(2.14)

E

Call X the Cox process with distribution Πψ , then X admits a first moment measure M on B(E) defined by IE(X(A)) = M (A) = IE( A ψ(x)λ(dx)), for every ˜x A ∈ B(E). Let (Πxψ , x ∈ E) be the family of Palm measures of Πψ , we define Π ψ x x x ˜ ∗ δx . This means that (Π ˜ , x ∈ E) satisfies by Πψ = Π ψ ψ ˜ xψ (dξ)u(ξ + δx , x) . Π Πψ (dξ) ξ(dx)u(ξ, x) = M (dx) Q

E

E

Q

As a consequence of this disintegration formula, we have for any f and g nonnegative functions on E with support in a compact set A d ˜ x (dξ)e−ξ+δx ,f M (dx) Π Πψ (dξ)e−ξ,f +tg |t=0 = g(x) − ψ dt Q E Q

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which thanks to (2.14) leads to

−f (x)

g(x)e E

−f (y) λ(dx)IE ψ(x)exp − (1 − e )ψ(y)λ(dy) R ˜ x (dξ)e−ξ,f e−f (x) IE ψ(x) λ(dx) . Π = g(x) ψ E

Q

Consequently IE[ψ(x)]λ(dx) a.e. ψ(x) x −ξ,f −f (y) ˜ Πψ (dξ)e exp − (1 − e = IE )ψ(y)λ(dy) , IE(ψ(x)) Q R and ˜ x = Πψ(x) . Π ψ Lemma 2.5 now follows from (2.12), the infinite divisibility of Πψ , and the definition ˜x. of Π ψ Proof of Theorem 2.3. Let N be a standard Gaussian variable independent of η. Then η + N is a centered Gaussian process with covariance K + 1 and we may take ψ = η + N . (ii) ⇒ (iii) Assume that Π 12 (η+N )2 ,λ is infinitely divisible for every > 0. Denote by Πx , x ∈ E ∪ {a} the Palm measures of Π 12 (η+N )2 ,λ . According to Lemma 2.5 there exists IE(ηx + N )2 λ (dx) almost every x, a random measure with distribution μx on Q such that Πx = Π 12 (η+N )2 ,λ ∗ μx . Since λ ({a}) = > 0, we have Πa = Π 12 (η+N )2 ,λ ∗ μa . We also have ˜ a ∗ δa Πa = Π ˜ a is the law of a Cox process with intensity where Π with respect to λ . Consequently, we obtain

1 2 (η

˜ a ∗ δa = Π 1 (η+N )2 ,λ ∗ μa . Π 2

+ N )2 under IE(N 2 , . ) (2.15)

For a fixed positive constant r > 0, the finite-dimensional Laplace transforms of the process 12 (η + r)2 are given by n 1 1 2 t 2 −1/2 −1 IE exp − αi (ηxi + r) exp − r 1 (I + αK) α1 = |I + αK| 2 i=1 2


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for every x1 , x2 , . . . , xn in E ∪{a} where 1 is the n-dimensional column vector of 1’s and 1t is its transpose. Consequently for every nonnegative function f on E ∪ {a} Π 12 (η+r)2 ,λ (dξ)e−ξ,f Q 1 1 2 −f (y) 2 = IE exp − (1 − e )ηy λ (dy) exp − r F (f, K) (2.16) 2 E∪{a} 2 √ √ with F (f, K) = ( 1 − e−f , (I + Kφ )−1 1 − e−f ), where the inner product is with respect to λ . Note that

F (f, η) = 1 − e−f (x) (I + Kφ )−1 1 − e−f (x)λ(dx) + (1 − e−f (a) ) . E

(2.17) We obtain Π 12 (η+N )2 ,λ (dξ)e−ξ,f Q 1 1 2 −f (y) 2 = IE exp − (1 − e )ηy λ (dy) IE exp − N F (f, K) 2 E∪{a} 2 (2.18) and similarly ˜ a (dξ)e−ξ,f Π Q 1 1 2 −f (y) 2 2 = IE exp − (1 − e )ηy λ (dy) IE N exp − N F (f, K) . 2 E∪{a} 2 (2.19) Now, making use of (2.18) and (2.19), equation (2.15) gives μa (dξ)e−ξ,f Q

−1 1 1 = IE N 2 exp − N 2 F (f, K) IE exp − N 2 F (f, K) e−f (a) 2 2

which thanks to elementary computations on the standard Gaussian law leads to −1 −f (a) μa (dξ)e−ξ,f = 1 + F (f, K) e = IE[e−F (f,K)T ] e−f (a) Q

where T is an exponential variable with parameter 1 independent of η. Multiplying then each member of the above equation by Q Π 12 η2 (dξ)eξ,f , we obtain μa (dξ)e−ξ,f Π 12 η2 (dξ)e−ξ,f = Π 12 η2 (dξ)e−ξ,f IE[e−F (f,K)T ] e−f (a) Q

Q

Q

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which implies μa ∗ Π 12 η2 = Π 1 (η+√2T )2 ,λ ∗ δa . 2

Now note that Π 12 η2 does not charge configurations including the site a, hence there exists a distribution μ ã such that μa = μ ã ∗ δa and we finally obtain μ ã ∗ Π 12 η2 = Π 1 (η+√2T )2 ,λ . 2

Denote by Xa , Yη2 and Y(η+√2T )2 the random measures corresponding respectively to the distributions μ ã , Π 12 η2 and Π 1 (η+√2T )2 ,λ . We then have 2

(law)

Xa + Yη2 = Y(η+√2T )2 . (law)

In particular, since Yη2 ({a}) = 0, we have Xa ({a}) = Y(η+√2T )2 ({a}). It follows that (law) Xa + Yη2 , Xa ({a}) = Y(η+√2T )2 , Y(η+√2T )2 ({a}) . Now, thanks to (2.17), we know that Y(η+√2T )2 ({a}) = N T where (Nt , t ≥ 0) is a Poisson process independent of (Y(η+√2T )2 , T ). Similarly Xa ({a}) = N T a |E

where Ta is an exponential variable with parameter 1, and (Nt , t ≥ 0) is a Poisson process independent of ((Xa + Yη2 )|E , Ta ). Moreover, since Xa ({a}) is independent of Yη2 , we may take Ta independent of Yη2 . Hence (law) = (Y(η+√2T )2 , N T ) (Xa + Yη2 )|E , N T a |E

which implies that for every nonnegative measurable function f with compact support on E and every λ > 0 IE[e−(Xa

+ Yη2 )|E ,f

−Y(η+√2T )2

e−λNTa ] = IE[e

|E

,f

e−λNT ]

which leads to IE[e−(Xa

+ Yη2 )|E ,f

−λ

e−(1−e

) Ta

−Y(η+√2T )2

] = IE[e

|E

,f

−λ

e−(1−e

) T

].

Since the above is true for every > 0, we obtain (law)

(Xa|E + Yη2 , Ta ) = (Y(η+√2T )2

|E

,T)

which leads to (law)

(Xa|E |Ta = r) + Yη2 = Y(η+√2r)2

|E

for almost every r > 0. In terms of distribution, this means that there exists a random measure on E with distribution νr satisfying νr ∗ Π 12 η2 = Π 12 (η+r)2 ,λ . With the above equation and (2.16) for λ instead of λ , we obtain 1 νr (dξ)e−ξ,f = exp − r2 1 − e−f , (I + Kφ )−1 1 − e−f 2 Q


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where the inner product is with respect to the measure λ. We use now the result contained in Exercise 5.1 p. 33 Chap. 3 in Kallenberg’s book [11], to check that for any sequence (rn , n ∈ N) of rational numbers converging to a given r, the sequence (νrn ) converges to a limit distribution satisfying both (2.5) and (2.6) for the measure λ. Hence (iii) is established for every r > 0. Since for every real r, (law)

(η + r)2 = (η − r)2 , (iii) is obtained for every real r. (iii) ⇒ (i) By assumption Π 12 η2 ,λ is infinitely divisible. Since, νr = (νr/√n )∗n , νr is infinitely divisible. Hence Π 12 (η+r)2 ,λ is infinitely divisible for every r in R. (i) ⇒ (v) Assume that Π 12 (η+r)2 ,λ is infinitely divisible for every constant r. We have Π 12 (η+r)2 ,λ (dξ)e−ξ,f Q 1 −f (y) 2 = IE exp (1 − e )(ηy + r) λ (dy) 2 E∪{a} 1 1 −f (y) 2 −f (a) 2 (1 − e )(ηy + r) λ(dy) exp − (1 − e ) r = IE exp 2 E 2 hence Π 12 (η+r)2 ,λ = Π 12 (η+r)2 ,λ ∗ Π 12 r2 , δa . As the convolution of two infinitely divisible distributions, Π(η+r)2 ,λ is infinitely divisible too for every > 0. (v) ⇒ (iv) We keep the notation of the proof of “(ii) ⇒ (iii)”. Π 12 (η+r)2 ,λ (dξ)e−ξ,f Q 1 1 2 −f (y) 2 = IE exp − (1 − e )ηy λ(dy) exp − r F (f, K) . 2 E 2 For every integer n and every r, [ Q Π 12 (η+r)2 ,λ (dξ)e−ξ,f ]1/n is still a Laplace √ transform of a random measure on Q. This is true in particular for r n, but Q

Π 12 (η+r√n)2 ,λ (dξ)e−ξ,f

1/n

1/n 1 1 2 −f (y) 2 = IE exp − (1 − e )ηy λ(dy) exp − r F (f, K) . 2 E 2

Letting n tend to ∞ we obtain, using Kallenberg’s result (Exercise 5.1 p. 33 Chap. 3 in [11]) that there exists a limiting distribution with Laplace transform exp{− 12 r2 F (f, K)}. (iv) ⇒ (ii) We start from Π 12 (η+r)2 ,λ = Π 12 η2 ,λ ∗ νr .

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Integrating the above equation with respect to IP (N ∈ dr) (recall that N is a standard Gaussian random variable) we obtain Π 12 (η+N )2 ,λ = Π 12 η2 ,λ ∗ νN where νN denotes the distribution satisfying for every positive function f on E∪{a} 1 2 −ξ,f νN (dξ)e = exp − r F (f, K) IP (N ∈ dr) , 2 Q R where F (f, K) is defined by (2.16). Since N 2 is an infinitely divisible variable, for every integer n, there exists an i.i.d. sequence (Z1 , Z2 , . . . , Zn ) of positive variables (law)

such that N 2 = Z1 + Z2 + · · · + Zn . Hence we have 1 νN (dξ)e−ξ,f = IE exp − N 2 F (f, K) 2 Q n 1 = IE exp − Z1 F (f, K) 2 n −ξ,f √ = ν Z1 (dξ)e . Q

It follows that νN is infinitely divisible and thus, so is Π 12 (η+N )2 ,λ ( ie Π 12 ψ2 ,λ ).

3. A super-Isomorphism Theorem The characterization of Gaussian processes (ηx , x ∈ E) such that (ηx2 , x ∈ E) is infinitely divisible is an old question that has been first raised up by Paul Lévy [12]. Several answers have been given since (see [5] for an extended bibliography of the subject). For example, in Remark 2.2.1 we have used Bapat’s criteria [1] for centered Gaussian processes. In [4], we have characterized the Gaussian processes (ηx , x ∈ E) such that ( 12 (ηx + r)2 , x ∈ E) is infinitely divisible for every real constant r: a Gaussian process has such a property iff its covariance is equal to the Green function of a recurrent Markov process X killed at the first hitting time of a given value a. The Markov process X is independent of η. This condition translates into the following factorization result. For every real r 1 (law) 1 2 2 x (ηx + r) , x ∈ E = η + Lτr , x ∈ E (3.1) 2 2 x where (Lxt , x ∈ E, t ≥ 0) is the local time process of X and τr = inf{s ≥ 0 : Las > 12 r2 }. This identity is a so-called “Isomorphism Theorem” which has been established in [6]. It is a variant of the first Isomorphism Theorem established by Dynkin [3]. who drew inspiration from Symanzik [17]. With the notation of Theorem 2.3, (3.1) is equivalent to Π 12 (η+r)2 ,λ = Π 12 η2 ,λ ∗ ΠLτr ,λ for every λ element of Q.


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One can hence formulate the exact difference between the infinite divisibility of (η+r)2 for every r and the infinite divisibility of Π 12 (η+r)2 ,λ for every r: (η+r)2 is infinitely divisible for every r iff for every λ of Q, Π 12 (η+r)2 ,λ is infinitely divisible for every r and νr is a Cox process with respect to λ . The identity generated by the infinite divisibility property of Π 12 (η+r)2 ,λ is an extension of the above Isomorphism Theorem to point processes. It can hence be considered as a “super”- Isomorphism Theorem. We are going now to establish a lemma that will enlarge even more the point of view on the Isomorphism Theorems. Lemma 3.1. Let (ψx , x ∈ E) be a positive process. For every a such that IE(ψa ) > 0, (a) denote by (ψx , x ∈ E) the process having the law of (ψx , x ∈ E) under the 1 IE(ψa , .). Then, ψ is infinitely divisible if and only if for every a probability IE(ψ a) (a)

such that IE(ψa ) > 0, there exists a process (lx , x ∈ E) independent of ψ such that (law) (3.2) ψ (a) = ψ + l(a) . Consider a centered Gaussian process η with a covariance equal to the Green function of transient Markov process. Then according Dynkin’s Isomorphism Theorem [3], (3.2) holds for ψ = η 2 . Hence Dynkin’s Isomorphism Theorem can be seen as a characterization of the infinite divisibility property of η 2 . Lemma 3.1 connects every infinitely divisible positive process (ψx , x ∈ E) to a family of pro(a) cesses ((lx , x ∈ E), a ∈ E). The identity (3.4) below relates the different l(a) as a varies. Similarly to Dynkin’s Isomorphism Theorem or to (3.1), Lemma 3.1 relates path properties of ψ to path properties of l(a) . For example, one immediately ob(a) tains that the continuity of ψ implies the continuity of (lx , x ∈ E) for every a. Lemma 3.1 can hence be seen as an “Isomorphism Theorem”. More generally, we can regard Lemma 2.5 as a super-Isomorphism Theorem characterizing the infinite divisibility of a given Cox process. When the considered Cox process has an infinitely divisible intensity ψ, the corresponding super Isomorphism Theorem is just the “super” identity existing above (3.2). Proof of Lemma 3.1. If ψ is infinitely divisible then for every x = (x1 , x2 , . . . , xn ) ∈ E n , there exists νx a Levy measure on Rn such that Rn 1 ∧ |y|νx (dy) < ∞ and +

for every α = (α1 , α2 , . . . , αn ) in Rn+ −

IE(e

n

i=1 αi ψxi

) = exp −

Rn +

−(α,y)

(1 − e

)νx (dy)

(3.3)

n where (α, y) = i=1 αi yi . We hence have n ψ(x1 ) − ni=1 αi ψx y1 i e e−(α,y) νx (dy) = IE(e− i=1 αi ψxi ) IE n IE(ψ(x1 )) I E(ψ(x )) 1 R+ from which it follows that there exists a process l(x1 ) independent of ψ such that ψ (x1 ) = ψ + l(x1 ) .

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Conversely, assume that for every a there exists a process l(a) satisfying (3.2). By computing the law of ψ underIE[ψa ψb , .], applying the above formula twice, we see that for every couple (a, b) of E, we must have (a) (3.4) ca IE lb F (lx(a) , x ∈ E) = cb IE la(b) F (lx(b) , x ∈ E) where cx = IE(ψx ) for every x in E. To lighten the writing, we set x1 = a. We also have ∂ − n n i=1 αi ψxi ) (a) ∂α1 IE(e n = −IE(e− i=1 αi lxi )IE(ψx1 ) − i=1 αi ψxi IE(e ) and hence IE(e−

n i=1

−

= IE(e

αi ψxi

n

)

i=1 αi ψxi

)|α1 =0 exp −IE(ψx1 )IE

1 − e−α1 lx1

(a)

(a)

−

e

n

(a) i=2 αi lx i

. (3.5)

lx1

We now use (3.5) and an induction argument to end our proof. For n = 1 it ∞ −α1 y1 )νx (dy1 ) where follows immediately from (3.5) that IE(e−α1 ψx1 ) = e− 0 (1−e IE(ψx1 ) (a) νx (dy1 ) = y1 IP (lx1 ∈ dy1 ). Assume now that the law of (ψx1 , ψx2 , . . . , ψxn−1 ) is given by n−1 − n−1 α ψ − α y i x i i i=1 i=1 i ) = exp IE(e − (1 − e )νx (dy) , [0,∞)n−1

with νx (dy) =

cx1 y1

(x1 )

IP (lx

∈ dy). (xn )

c

By (3.4) νx (dy) = R+ yxnn IP (lx from x1 , x2 , . . . , xn−1 . Using this in (3.5) we obtain IE(e−

n i=1

αi ψxi

= exp −

(x )

∈ dy, lxnn ∈ dyn ), for every xn distinct

) −

(1 − e

n i=2

αi yi

)νx (dy)

[0,∞)n−1

exp − = exp −

−

(e

n

i=2 αi yi

−

−e

[0,∞)n

(1 − e− [0,∞)n

n i=1

αi yi

)

n

i=1 αi yi

cx1 (x1 ) ) IP (lx ∈ dy1 dy2 . . . dyn ) y1

cx1 IP (lx(x1 ) ∈ dy1 dy2 . . . dyn ) . y1

We mention that Theorem 2.3 is easily extendable from squared Gaussian processes to permanental processes which are real valued processes characterized by the fact that any joint moment of such a process is equal to a permanent. These processes have been properly defined by Vere-Jones [20].


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Acknowledgements We thank Yvan Castin for interesting discussions.

References [1] R. B. Bapat, Infinite divisibility of multivariate gamma distribution and M-matrices, Sankhya 51 (1989), 73–78. [2] O. Bratteli and W. D. Robinson, Operator Algebras and Quantum-Statistical Mechanics II, Springer-Verlag, New York/Berlin 1979, 1981. [3] E. B. Dynkin, Local times and quantum fields, Seminar on Stochastic Processes, Birkhauser (1983), 64–84. [4] N. Eisenbaum, A connection beetween Gaussian processes and Markov processes, Electronic Journal of Probability 10, 6 (2005), 202–215. [5] N. Eisenbaum and H. Kaspi, A characterization of the infinitely divisible squared Gaussian processes, Annals of Probability 34, 2 (2006), 728–742. [6] N. Eisenbaum, H. Kaspi, M. B. Marcus, J. Rosen and Z. Shi, A Ray–Knight theorem for symmetric Markov processes, Annals of Probability 28, 4 (2000),1781–1796. [7] K. H. Fichtner and W. Freudenberg, Point processes and the position distribution of infinite boson systems, J. Statist. Phys. 47 (1987), 959–978. [8] G. A. Goldin, J. Grodnik, R. T. Powers and D. H. Sharp, Nonrelativistic current algabra in the N/V limit, J. Math. Phys. 15 (1974), 88–100. [9] J. B. Hough, M. Krishnapur, Y. Peres and B. Vir´ ag, Determinantal processes and independence, Probab. Surv. 3 (2006), 206–229. [10] T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect and C. I. Westbrook, Comparison of the Hanbury Brown–Twiss effect for bosons and fermions, Nature 445 (2007), 402–405. [11] O. Kallenberg, Random measures, Akademie-Verlag, Berlin (1975). [12] P. Lévy, The arithmatical character of the Wishart distribution, Proc. Camb. Phil. Soc. 44 (1948), 295–297. [13] O. Macch, The fermion process, a model of stochastic point process with repulsive points. Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians (Tech. Univ. Prague, Prague, 1974), Vol. A, pp. 391–398. [14] O. Macchi, The coincidence approach to stochastic point processes, Adv. Appl. probab. 7 (1975), 83–122. [15] D. N. Shanbhag and M. Westcott, A note on infinitely divisible point processes, J. Roy. Statist. Soc. Ser. B 39, 3 (1977), 331–332. [16] T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes, Journal of Functional Analysis 205 (2003), 414–463. [17] K. Symanzik, Euclidean quantum field theory, Local Quantum Theory, Academic Press, ed: R. Jost, New York/London (1969).

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[18] H. Tamura and K. R. Ito, A canonical ensemble approach to the Fermion/Boson random point proceses and its applications, Commun. Math. Phys. 263 (2006), 353– 380. [19] H. Tamura and K. R. Ito, A random point field related to Bose–Einstein condensation, J. Funct. Anal. 243 1 (2007), 207–231. 82B10 (60K40). [20] D. Vere-Jones, Alpha-permaments and their applications to multivariate gamma, negative binomial and ordinary binomial distributions, New Zealand Journal of Maths. 26 (1997), 125–149. [21] D. Vere-Jones, A generalization of permanents and determinants, Linear Algebra Appl. 111 (1988), 119–124. Nathalie Eisenbaum Laboratoire de probabilités et modèles aléatoires UMR 7599 – CNRS Universités Paris 6 et 7 4, Place Jussieu Case 188 F-75252 Paris Cedex 05 France Communicated by Vincent Rivasseau. Submitted: February 8, 2008. Accepted: March 5, 2008.



Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries Peter D. Hislop1 and Eric Soccorsi2 Abstract. Devices exhibiting the integer quantum Hall effect can be modeled by one-particle Schr¨ odinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to bounded or unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barriers create edge currents. This is the second of two papers in which we review recent progress and prove explicit lower bounds on the edge currents associated with one- and two-edge geometries. In this paper, we study various unbounded and bounded, two-edge geometries with soft and hard confining potentials. These two-edge geometries describe the electron confined to unbounded regions in the plane, such as a strip, or to bounded regions, such as a finite length cylinder. We prove that the edge currents are stable under various perturbations, provided they are suitably small relative to the magnetic field strength, including perturbations by random potentials. The existence of, and the estimates on, the edge currents are independent of the spectral type of the operator.

1. Introduction This is the second of two papers (the first is [13]) dealing with lower bound estimates on edge currents associated with quantum Hall devices. The integer quantum Hall effect (IQHE) refers to the quantization of the Hall conductivity in integer multiples of e2 /h. The IQHE is observed in planar quantum devices at zero temperature and can be described by a Fermi gas of noninteracting electrons. This 1

Supported in part by NSF grant DMS-0503784. also Centre de Physique Théorique, Unit´ e Mixte de Recherche 6207 du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l’Université du Sud Toulon-Var-Laboratoire affilié ` a la FRUMAM, F-13288 Marseille Cedex 9, France.

2

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P. D. Hislop and E. Soccorsi


simplification reduces the study of the dynamics to the one-electron approximation. Typically, experimental devices consist of finitely-extended, planar samples subject to a constant perpendicular magnetic field B. An applied electric field in the x-direction induces a current in the y-direction, the Hall current, and the Hall conductivity σxy is observed to be quantized. Furthermore, the Hall conductivity is a function of the electron Fermi energy, or, equivalently, the electron filling factor, and plateaus of the Hall conductivity are observed as the filling factor is increased. It is now accepted that the occurrence of the plateaus is due to the existence of localized states near the Landau levels that are created by the random distribution of impurities in the sample. We refer to [2] and references mentioned there for a more detailed discussion. Since the earliest theoretical discussions, the existence of edge currents has played a major role in the explanation of the quantum Hall effect. To describe the two-edge geometries dealt with in this paper, we first recall the theory for the plane. The Landau Hamiltonian HL describes a particle constrained to R2 , and moving in a constant, transverse magnetic field with strength B ≥ 0. Let px = −i∂x and py = −i∂y be the two momentum operators in the absence of a magnetic field. The operator HL is defined on the dense domain C0∞ (R2 ) ⊂ L2 (R2 ) by HL = (−i∇−A)2 = p2x +(py −Bx)2 , in the Landau gauge for which the vector potential is A(x, y) = B(0, x). This operator extends to a selfadjoint operator with point spectrum given by {En (B) = (2n+1)B | n = 0, 1, 2, . . .}, and each eigenvalue is infinitely degenerate. As in [13], we define the edge current carried by the state ψ as the expectation of the y-component of the velocity operator Vy ≡ (py − Bx) in the state ψ, that is, Jy (ψ) ≡ |ψ, Vy ψ|. We will consider states ψ with energy concentration between two successive Landau levels En (B) and En+1 (B). As we discuss, edge currents may be localized near the left or the right edge, depending on the choice of ψ, and this determines the sign of the matrix element ψ, Vy ψ as minus or plus, respectively. 1.1. Edge currents in two-edge geometries: The main results Our main results in this paper are lower bound estimates on Jy (ψ) for certain states ψ and for a variety of two-edge geometries and confining potentials. We also show that the currents are localized near the edges of the devices. The currents are stable under a suitable class of perturbations. Roughly speaking, we consider states ψ in the range of the spectral projector E(Δ) for H with Δ a small interval between two successive Landau levels. For such states, we prove that Jy (ψ) ≥ CB 1/2 ψ2 , for a positive constant C > 0 depending on various parameters of the model. Concerning the types of confining potentials, our models can be described as follows. 1. Two-edge, unbounded geometries: We study the strip case for which the electron is constrained to the region −/2 ≤ x ≤ /2, a strip of width > 0,

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Edge Currents for Quantum Hall Systems

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by the sharp confining potential V0 (x) = V0 χ{|x|>/2} (x) ,

V0 > 0 ,

(1.1)

where χI denotes the characteristic function of the set I. The main results are Theorems 2.1 and 2.2. 2. Two-edge, unbounded geometries: We extend these results to the parabolic channel model (Section 3.1), to various soft confining potentials given by polynomials (Lemma 3.1), and to Dirichlet boundary conditions (Theorem 3.1). 3. Two-edge, bounded geometries: We study models on a cylinder CD = R × [−D/2, D/2], for D > 0, for which the electron is confined to the bounded region [−/2, /2] × [−D/2, D/2] by a sharp confining potential (1.1). The main results are Theorems 4.1 and 4.2. In addition, we study the spectral properties of various models using commutator methods (see, for example, Proposition 1.1). In all cases, the unperturbed Hamiltonian has the form H0 = HL + V0 , acting on the Hilbert space L2 (R2 ). This is a nonnegative, selfadjoint operator. Our strategy is to analyze the unperturbed operator via the partial Fourier transform in the y-variable. We write fˆ(x, k) for the partial Fourier transform of the function f (x, y). For the case of unbounded geometry, we have k ∈ R, whereas for the case of bounded geometry, the allowable k values are discrete. In either case, this decomposition reduces the problem to a study of the fibered operators of the form h0 (k) = p2x + (k − Bx)2 + V0 (x), acting on L2 (R). Since the effective, nonnegative, potential (k − Bx)2 + V0 (x) is unbounded as x → ±∞, the resolvent of h0 (k) is compact and the spectrum is discrete. We denote the eigenvalues of h0 (k) by ωj (k), with corresponding normalized real eigenfunctions ϕj (x; k), so that h0 (k)ϕj (x; k) = ωj (k)ϕj (x; k) ,

ϕj ( · ; k) = 1 .

(1.2)

As in [4] and [13], the properties of the curves k ∈ R → ωj (k) play an important role in the proofs. These curves are called the dispersion curves for the unperturbed Hamiltonian H0 . The importance of the properties of the dispersion curves comes from an application of the Feynman–Hellmann formula. To illustrate this, let us first consider the two-edge geometry of a strip with the sharp confining potential. We note that unlike the case of one-edge geometries, the dispersion curves are no longer monotonic in k, see Figure 1 in Section 2.1. For simplicity, we consider in this introduction a closed interval Δ0 ⊂ (B, 3B) and a normalized wave function ψ satisfying ψ = E0 (Δ0 )ψ, where E0 (Δ0 ) denotes the spectral projection of H0 associated with Δ0 . Such a function admits a decomposition of the form 1 eiky β0 (k)ϕ0 (x; k) dk , ψ(x, y) = √ 2π ω0−1 (Δ0 ) ˆ · ; k), ϕ0 ( · ; k) . (1.3) with β0 (k) ≡ ψ(

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The matrix element of the velocity operator Vy in such a state is ψ, Vy ψ = dx dk|β0 (k)|2 (k − Bx)ϕ0 (x; k)2 . R

ω0−1 (Δ0 )

From (1.2) and the Feynman–Hellmann Theorem, we find that ω0 (k) = 2 dx (k − Bx)ϕ0 (x; k)2 ,

(1.4)

R

so that we get

1 |β0 (k)|2 ω0 (k) dk . (1.5) Jy (ψ) = 2 R It follows from (1.5) that in order to obtain a lower bound on Jy (ψ) we need to bound the derivative ω0 (k) from below for k ∈ ω0−1 (Δ0 ). The next step of the proof involves relating the derivative ω0 (k) to the trace of the eigenfunction ϕ0 (x; k) on the boundary of the strip. Taking into account the eigenvalue equation (1.2), and the fact that ∂k (k − Bx)2 = −B −1 ∂x (k − Bx)2 , we integrate by parts in (1.4), and find that V0 ϕ0 (/2; k)2 − ϕ0 (−/2; k)2 . (1.6) ω0 (k) = B Consequently, we are left with the task of estimating the trace of the eigenfunction along the two boundary components at x = ±/2. The key point that allows us to distinguish these two traces is the following. The dispersion curves are symmetric about k = 0 if V0 (x) is an even function. Consequently, if a wave function ψ satisfies ψ = E0 (Δ0 )ψ, we have to study the decomposition of ψ in k-space according to the decomposition ω0−1 (Δ0 ) = ω0−1 (Δ0 )− ∪ ω0−1 (Δ0 )+ , where ω0−1 (Δ0 )± ≡ ω0−1 (Δ0 ) ∩ R± . These two components correspond to currents propagating in opposite directions along the left and right edges of the band, respectively. To construct a left-edge current, we construct states ψ so that the coefficients β0 (k) in (1.3) satisfy supp β0 (k) ⊂ ω0−1 (Δ0 )− . Such a state is spatially concentrated near the left edge x = −/2. Hence, the contribution to the left-edge current coming from ϕ0 (/2; k) will be exponentially small since the domain x ≈ /2 is in the classically forbidden region for energies ω0 (k), for k ∈ ω0−1 (Δ0 )− . Consequently, the contribution to the integral in (1.5) will be exponentially small. Thus, we prove that if ψ = E0 (Δ0 )ψ is spectrally concentrated in the set ω0−1 (Δ0 )− , then the matrix element ψ, Vy ψ is negative, and the corresponding left-edge current Jy (ψ) is bounded from below by a constant times B 1/2 ψ2 . Much of our technical work, therefore, is devoted to obtaining lower bounds on quantities of the form V0 ϕ0 (±/2; k)2 for such left-edge current states. We also mention that similar results hold for the right-edge current. Of course, in the unperturbed case with a symmetric confining potential V0 , we expect that the net current across any line y = C is zero. We will prove this in Proposition 2.2 below. For the unperturbed Hamiltonian H0 , the edge current estimate is valid for time-dependent states. That is, if ψt = e−itH0 ψ0 , and ψ0 carries an edge current

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so that Jy (ψ0 ) ≥ C0 B 1/2 ψ0 2 , we then have that Jy (ψt ) ≥ C0 B 1/2 ψ0 2 . The situation is different for the perturbed Hamiltonian H1 because the decomposition ˜ + E0 (Δ ˜ c )ψ (see Theorem 2.2) is not invariant under the time of ψ as ψ = E0 (Δ)ψ evolution generated by H1 . We expect that the perturbation V1 mixes left and right edge currents. We proved in [4] that the edge current carried by ψt has a 2 definite sign and remains O(B 1/2 ) for times up to O(eC0 B ). 1.2. Edge conductance for two-edge geometries The quantization of the Hall conductance σH for the planar models with no confining potential was proved in [2] for lattice models and in [1,17] for models on R2 , under various assumptions. For one-edge geometries, one can define an edge conductance corresponding to the edge currents. For b > a ∈ R, a switch function g[a,b] is a nonnegative, differentiable, decreasing function with g[a,b] (s) = 1 for s < a, and g[a,b] (s) = 0, for s > b so that supp g[a,b] ⊂ [a, b]. Let χ(x, y) ≡ χ2 (y) be an x-translation invariant switch function with [a, b] = [−1/2, 1/2]. For the half-plane model on R+ × R with edge at x = 0, the edge conductance associated with an (H)i[H, χ2 ]). It can be energy interval [a, b] ⊂ R is defined by σe ([a, b]) = −Tr(g[a,b] shown that this is well-defined and independent of the choice of switch functions. One may also define a bulk Hall conductance σH for the half-space models using the usual Kubo formula. It is this quantity that is quantized, even for the halfplane model. The main result of research on σe is the equality σe = σH [3, 6, 22] showing the quantization of the edge conductance. One may also define edge conductance for the two-edge geometries in the same manner. A simple calculation based on the direct integral decomposition (2.1) and the Feynman–Hellmann formula (1.4) proves that σe = 0 for the free Hamiltonian H0 with left and right confining potentials provided the interval Δ lies below V0 . What is also true is that σe = 0 for the perturbed problem H1 = H0 + V1 for a wide family of potentials V1 . This is proved in [3]. The explanation is that even though the perturbation mixes left- and right-edge currents (see, for example, the calculation in [4]), the total conductance remains unchanged as the left- and right-edge currents, which are probably not well-localized, cancel. 1.3. Spectral properties of two-edge geometries It follows from the direct integral decomposition of H0 presented in Section 2.1 that the spectrum of H0 is contained in the half-line [B, ∞) and is purely absolutely continuous. Unlike the one-edge, half-plane case, the spectral type of the perturbed two-edge operator H1 = H0 + V1 is generally unknown. A Mourre estimate [5, 20] can be proved for H0 as in the one-edge case, but it is more complicated due to the fact that the dispersion curves (1.6) are not monotonic. We sketch this construction and refer to the archived version [14] for the details. Let Uα = eiαpy , for py = −i∂y and α ∈ R, be the unitary translation group in the y-direction defined by (Uα g)(y) = g(y + α). We define the selfadjoint operator Sα = 2i (Uα y−yU−α ). The operator Sα commutes with px and V0 . As Vy = py −Bx,

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it is easy to check that [Vy , Sα ] =

1 (Uα − U−α ) = i sin(αpy ) , 2

(1.7)

so that (1.8) i[H0 , Sα ] = −2 sin(αpy )Vy , as a quadratic form on D(H0 ) ∩ Dy . A nice property of Sα is that the double commutator with H0 vanishes. Consequently, a positive commutator will imply absolutely continuous spectrum (cf. [5]) in the range of the corresponding spectral projector. Proposition 1.1. Let n ∈ N and Δn be defined by (2.7). Then there are two constants n = n (a, c) > 0 and τn = τn (a, c) > 0 depending only on n, a and c, such that for all B2 ≥ n , all V0 ≥ En+1 (B), any subinterval Δ of Δn such that |Δ| < δn , where δn = δn (B, , V0 ) is as in Lemma 2.1, and all α > 0 sufficiently small we have −iE0 (Δ)[H0 , Sα ]E0 (Δ) ≥ (Cn /2)(a − 1)2 (3 − c)3 B 1/2 E0 (Δ) ,

(1.9)

where Cn > 0, which is defined in Lemma 2.2, depends only on n. As an application of the commutator method, we prove that the spectrum of H1 is purely absolutely continuous if 1) V1 (x, y) is periodic with respect to y with sufficiently small period, or 2) V1 (x, y) is such that |yV1 (x, y)| is bounded. These results are similar to those of Exner, Joye, and Kovarik [7] who used the commutator method to study perturbations of the parabolic channel model (see Section 3). Following the idea developed by Macris, Martin and Pulé in [19] for the half-plane geometry, we can actually prove H1 = H0 + V1 has purely absolutely continuous spectrum for the two-edge geometry if the perturbation V1 is bounded and integrable in R2 . The proof relies on the perturbation of semigroups and we refer the reader to the archived version [14] for the details. We point out that for the more general class of perturbations V1 treated in Sections 4 and 5 of [13], such as random potentials, we do not know the spectral type of the operator H1 . However, we still know that there are states carrying nontrivial edge currents. The existence of edge currents is not tied to the spectral properties of H1 . Indeed, the cylinder geometry model (see Section 4 and [8, 9]) shows that the full Hamiltonian may have only pure point spectrum, yet there are nontrivial edge currents. Hence, the existence of edge currents is not directly tied to the existence of continuous spectrum. We will discuss this in more detail in Section 4. 1.4. Contents This paper is organized as follows. Section 2 is devoted to the estimation of edge currents with a sharp confining potential. In Section 3, we mention the case of soft confining potentials, the parabolic channel model, and the strip with Dirichlet boundary conditions. In Section 4, we address cylinder geometries models and

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prove the existence of edge currents for Hamiltonians with pure point spectrum in this framework. The appendix in Section 5 presents basic properties of the dispersion curves and the precise exponential decay of eigenfunctions needed in the proofs.

2. Edge currents for two-edge geometries Many quantum devices can be modeled by a confining potential forcing the electrons into a strip of infinite extent in one direction. The dynamics of electrons in an infinite-strip with a transverse magnetic field B > 0 are different from the half-plane cases treated in [13]. We study an electron in a strip of width > 0 in the x-direction, and unbounded in the y-direction. As stated in the theorems, our main results hold for B2 sufficiently large. This condition insures that the edge states are well-defined and localized near each edge. We first consider confining potential V0 (x) that are step functions. We refer to these as sharp confining potentials. We will consider other confining potentials in Section 3. After some basic analysis of these models that is independent of the precise form of the confining potential, we study edge currents for sharp confining potentials. 2.1. Basic analysis of two-edge geometries As in [13], we study the existence of edge currents for a general confining potential V0 (x). We obtain lower bounds on the appropriately localized velocity along the y-direction Vy . The strip geometry is a two-edge geometry. Thus, we expect that there is a current associated with each edge. Classically, these currents propagate along the edges in opposite directions. For the unperturbed system, one expects that the net current flow across the line y = C, for any C ∈ R, to be zero, and we prove this in Proposition 2.2. Once a perturbation V1 is added, this may no longer be true, and the persistence of edge currents may depend upon a relationship between B and . We continue to use the same notation as in [13]. We write H0 = HL + V0 for the unperturbed operator with an even, two-edge confining potential V0 . Since we have translational invariance in the y-direction, this operator admits a direct sum decomposition ⊕ dkh0 (k) , with h0 (k) = p2x +(k −Bx)2 +V0 (x) , on L2 (R) . (2.1) H0 = R

We explicitly treat the case of the sharp confining potential (1.1) in this section, and comment on other confining potentials in Section 3. We first prove that the total edge current carried by certain wave number symmetric states of finite energy vanishes. For this, it is essential that the confining potential be an even function. We consider states of finite energy ψ, with ψ ∈ E0 (Δn )L2 (R2 ), for an interval Δn ⊂ (En (B), En+1 (B)), for any n ≥ 0. The partial Fourier transform ψˆ of ψ in

1148


Ann. Henri Poincaré ω ( k) 2

ω ( k) 1

5B

ω ( k)

Δ

0

3B

ν

Ο

B k

ωΟ (Δ)− −1

0

(Δ) −

−1 1

ω

(Δ)+

−1 1

ω

ωΟ (Δ)+ −1

Figure 1. Dispersion curves of H0 and various spectral intervals. the y-variable can be expressed in terms of the eigenfunctions ϕj (x; k) as ˆ k) = ψ(x,

n

χω−1 (Δn ) (k)βj (k)ϕj (x; k) , j

(2.2)

j=0

or equivalently as

n 1 eiky χω−1 (Δn ) (k)βj (k)ϕj (x; k) dk , ψ(x, y) = √ j 2π j=0 R

(2.3)

ˆ · ; k), ϕj ( · ; k), and the normalization condition where the coefficients βj (k) ≡ ψ( is n 2 |βj (k)|2 dk . (2.4) ψL2 (R2 ) = j=0

ωj−1 (Δn )

We recall that the properties of the dispersion curves ωj (k) result in the disjoint decomposition ωj−1 (Δn ) = ωj−1 (Δn )− ∪ ωj−1 (Δn )+ with ωj−1 (Δn )± ≡ ωj−1 (Δn ) ∩ R± , see Figure 1. Lemma 2.1. Let n ∈ N and Δn ⊂ (En (B), En+1 (B)). Then there is δn = δn (B, , V0 ) > 0 such that if |Δn | < δn we have ωj−1 (Δn ) = ∅ ,

j ≥ n + 1,

(2.5)

and, if n ≥ 1, ωj−1 (Δn ) ∩ ωl−1 (Δn ) = ∅ ,

j = l ,

j, l = 0, 1, . . . , n .

(2.6)

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Proof. First (2.5) is evident since ωn+1 (k) ≥ En+1 (B) for all k ∈ R. Next set δn = δn (B, , V0 ) = min0≤j≤n−1 inf k∈R (ωj+1 (k) − ωj (k)). Due to Lemma 5.2, part 2, we have δn > 0. For all l, j = 0, 1, . . . , n and k ∈ ωj−1 (Δn ) ∩ ωl−1 (Δn ), we have |ωl (k) − ωj (k)| ≤ |Δn |, which leads to a contradiction if |Δn | < δn and j = l. Henceforth ωj−1 (Δn ) ∩ ωl−1 (Δn ) = ∅ if j = l. It is clear from the fact the potential in h0 (k) is centered at x0 = k/B that the wave function ψ may be more localized near one edge or another depending upon the properties of the weights βj (k). For example, if the βj (k) are supported only by negative wave numbers k, then the wave function and edge current are essentially localized in the region of width B −1/2 near the left edge −/2. Proposition 2.1. Let n ∈ N and Δn be given by Δn ≡ (2n + a)B, (2n + c)B , for

1 < a < c < 3.

(2.7)

Let ψ ∈ E0 (Δn )L2 (R2 ), as in (2.3), be a finite energy state such that βj (k) = 0 ,

k ∈ ωj−1 (Δn )+ ,

j = 0, 1, . . . , n .

There are two constants αn = αn (a) > 0, depending only on n and a, and θn = θn (a, c) > 0, depending only on n, a and c, such that, defining a narrow strip near −/2, I(δ± ) = [−/2 − δ− , −/2 + αn B −1/2 + δ+ ], then for all V0 ≥ En+1 (B), and for all B2 ≥ θn , we have 1/2 1/2 1/2 |ψ(x, y)|2 dxdy ≥ (1 − 8e−(αn /4)B δ+ − e−(3−c) B δ− )ψ2 . I(δ± )×R

Proof. In light of (2.2) and the Parseval’s Theorem we have

n 1 2 2 2 ψ (x, y)dxdy = βj (k) ϕj (x; k) dx dk ; 2π j=0 ωj−1 (Δn ) I(δ± )×R I(δ± ) Hence the result follows from this, (2.4) and Lemma 5.7.

Such wave function should carry a net left-edge current. We will prove this in Theorem 2.1. We will first prove that if the Fourier transform of a wave function is symmetrically distributed (in the sense of (2.9)), with respect to the Fourier variable k, then it carries no net edge current: The left-edge current cancels the right-edge current. Proposition 2.2. Let n ∈ N and Δn ⊂ (En (B), En+1 (B)) be small enough so Lemma 2.1 holds true. Let ψ ∈ E0 (Δn )L2 (R2 ), as in (2.2), be a finite energy state. Then, the current carried by such a state has the following expression: n 1 βj (k)|2 − |βj (−k) 2 ωj (k)dk . ψ, Vy ψ = (2.8) 2 j=0 ωj−1 (Δn )−

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If ψ is such that |βj (k)| = |βj (−k)| ,

j = 0, 1, · · · , n ,

(2.9)

then the current carried by ψ vanishes: Jy (ψ) = ψ, Vy ψ = 0. Proof. The velocity Vy = py − Bx has a Fourier transform that we write as Vˆy = Vˆy (k) = k − Bx. Using the Fourier decomposition (2.2), the matrix element of the velocity operator Vy is n χω−1 (Δn ) (k)χω−1 (Δn ) (k)β j (k)βl (k) ϕj ( · ; k), Vˆy (k)ϕl ( · ; k) dk . Jy (ψ) = j,l=0

R

j

l

(2.10) As a consequence of the result of Lemma 2.1, the cross-terms in (2.10) vanish, at least for |Δn | sufficiently small, giving n ψ, Vy ψ = χω−1 (Δn ) (k)|βj (k)|2 ϕj ( · ; k), Vˆy (k)ϕj ( · ; k) dk =

j=0 R n 0 j=0

−∞

j

χω−1 (Δn ) (k) |βj (k)|2 ϕj ( · ; k), Vˆy (k)ϕj ( · ; k) j

+ |βj (−k)|2 ϕj ( · ; −k), Vˆy (−k)ϕj ( · ; −k) dk ,

(2.11)

where we used the fact, proved in Lemma 5.1 in the appendix, that the dispersion curves are even functions of k, that is, ωj (k) = ωj (−k). We also note that the Hamiltonian h0 (k) commutes with the operation P that implements (x, k) → (−x, −k). The simplicity of the eigenfunctions then implies (this is shown in Lemma 5.1) that P ϕj = θj ϕj with θj = ±1. Hence the last term in the right hand side of (2.11), ϕj ( · ; −k), Vˆy (−k)ϕj ( · ; −k) becomes 2 ϕj (x; −k) (−k − Bx)dx = ϕj (−x; −k)2 (−k + Bx)dx R R = − ϕj (x; k)2 (k − Bx)dx R = − ϕj ( · ; k), Vˆy (k)ϕj ( · ; k) , and the result follows from this, (2.11) and the Feynman–Hellmann formula ωj (k) = 2 ϕj ( · ; k), (k − Bx)ϕj ( · ; k) . (2.12) It can be seen from Lemma 5.1 that a state ψ defined by (2.3) and symmetric about the origin (i.e. ψ(−x, −y) = ψ(x, y) for (x, y) ∈ R2 ) is characterized by Fourier coefficients βj , j = 0, 1, . . . , n, satisfying the condition βj (−k) = θj βj (k) ,

k ∈ ωj−1 (Δn )− ,

j = 0, 1, . . . , n .

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Consequently, symmetric states about the origin are among states satisfying (2.9), though a state satisfying (2.9) is not necessarily symmetric about the origin. Moreover, it should be noticed that states satisfying the condition (2.9) are not necessarily symmetric about the y-axis either, since the condition ψ(−x, y) = ψ(x, y) for (x, y) ∈ R2 is equivalent to βj (k) = 0 if θj = −1 ,

k ∈ ωj−1 (Δn )− ,

j = 0, 1, . . . , n .

2.2. Estimation of the edge current for a strip We turn now to the estimation of the left-edge current for a strip of width > 0 carried by appropriately chosen states ψ. That is, for all n ∈ N we want to obtain a lower bound on the matrix element of the localized velocity operator (2.8), carried by a state ψ ∈ E0 (Δn )L2 (R2 ) associated to the energy interval Δn ⊂ (En (B), En+1 (B)). Much of the technical work in this paper is devoted to bounding −ωj (k), j = 0, 1, . . . , n, from below, uniformly for k in ωj−1 (Δn )− . Lemma 2.2. Let n ∈ N and Δn be given by (2.7). Then, there are two constants βn = βn (a) > 0, depending only on n and a, and Cn > 0, depending only on n, such that we have k ∈ ωj−1 (Δn )− ,

−ωj (k) ≥ Cn (a − 1)2 (3 − c)3 B 1/2 ,

j = 0, 1, . . . , n ,

(2.13)

provided B2 ≥ βn and V0 ≥ En+1 (B). The proof of Lemma 2.2 being rather technical, it is postponed to Section 2.3. In light of (2.8) and Lemma 2.2, let us see now the current carried by a state ψ, whose coefficients βj (k), j = 0, 1, · · · , n, are mostly supported on the set of negative wave numbers k, is of size B 1/2 . More precisely, if Δ ⊂ (En (B), En+1 (B)), we consider states of finite energy ψ ∈ E0 (Δ), whose Fourier coefficients βj (k), defined after (2.3), satisfy the condition |βj (k)|2 ≥ (1 + γ 2 )|βj (−k)|2 ,

k ∈ ωj−1 (Δ)− ,

j = 0, 1, · · · , n ,

(2.14)

for some γ > 0. If γ goes to infinity we find that βj (−k) = 0, for k ∈ ωj−1 (Δ)− and j = 0, 1, · · · , n, whence ψ is localized in a strip of width O(B −1/2 ) along the left edge x = −/2 according to Proposition 2.1. Analogously we may expect that all states satisfying (2.14) for some γ > 0 are mostly supported in the left side of the strip [−/2, /2] × R. Theorem 2.1. Let n ∈ N and Δn be given by (2.7). Let βn = βn (a) and Cn be defined as in Lemma 2.2. Then for all B2 ≥ βn and V0 ≥ En+1 (B), there is a constant δn = δn (B, , V0 ) > 0 such that for all intervals Δ ⊂ Δn with size |Δ| < δn and all states ψ ∈ E0 (Δ)L2 (R2 ) satisfying the condition (2.14) for the interval Δ, we have Jy (ψ) = −ψ, Vy ψ ≥

γ2 Cn (a − 1)2 (3 − c)3 B 1/2 ψ2 . 2 + γ2

(2.15)

1152



Proof. Due to (2.14) we have |βj (k)|2 − |βj (−k)|2 ≥ γ 2 /(1 + γ 2 )|βj (k)|2 for all j = 0, 1, . . . , n and k ∈ ωj−1 (Δ)− , whence n 1 + γ2 |βj (k)|2 dk ≥ ψ2 , (2.16) 2 −1 2 + γ j=0 ωj (Δ)− from the normalization condition (2.4). Further, the size of Δ being sufficiently small so Lemma 2.1 holds true, the total current carried by the state ψ is n 1 (|βj (k)|2 − |βj (−k)|2 )ωj (k)dk , ψ, Vy ψ = 2 j=0 ωj−1 (Δ)− so the result follows from this, (2.16) and Lemma 2.2 since ωj−1 (Δ)− ⊂ ωj−1 (Δn )− for j = 0, 1, . . . , n. 2.3. Estimation of the derivative of the dispersion curves We prove Lemma 2.2 in this section. For all n ∈ N, Δn defined by (2.7) and for j = 0, 1, . . . , n, it is clear from the definition of ωj−1 (Δn )− that sup0≤j≤n ωj−1 (Δn )− ≤ 0. Actually, Lemma 5.4 tells us this supremum can be bounded by any number greater than (−B)/2 upon taking B2 sufficiently large. This means that the region x ≥ 0 is in the classically forbidden zone for energies ωj (k), k ∈ ωj−1 (Δn )− , at least in the intense magnetic field regime. This is because the parabolic part of the effective potential Wj (x; k) ≡ (k − Bx)2 + V0 (x) − ωj (k) ,

(2.17)

is centered at the coordinate k/B. As a consequence, the eigenfunctions ϕj (.; k) of h0 (k) are exponentially decaying in the region x ≥ 0 for all k ∈ ωj−1 (Δn )− . This is not true in the region x ≤ 0, and ϕj (/2; k)2 is also expected to be small relative to ϕj (−/2; k)2 . Since V0 ϕj (/2; k)2 − ϕj (−/2; k)2 , B by arguing as in the derivation of (1.6), we then get that ωj (k) =

(2.18)

V0 ϕj (−/2; k)2 . B This remark is made precise below. Namely we know from Lemma 5.6 of the appendix that upon choosing B2 ≥ ζn we have ωj (k) ≈ −

V0 ϕj (/2; k)2 ≤ (νn (B2 )−1/2 )B 3/2 ,

k ∈ ωj−1 (Δn )− ,

j = 0, 1, . . . , n , (2.19)

with both constants ζn > 0 and νn > 0 depending only on n and a. Hence the remaining term V0 ϕj (/2; k)2 is uniformly bounded by a constant times B−1 . We turn now to computing a lower bound on the main term V0 ϕj (−/2; k)2 . We shall show that it is of size B 3/2 . This will require several steps.

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Step 1: Harmonic oscillator eigenfunction comparison The proof of Lemma 2.2 in [13] is based on the properties of the eigenfunctions ψm (.; k) of the harmonic oscillator hL (k) = p2x + (Bx − k)2 . It applies without change to the case of the strip geometry. Let Pn denote the projection on the eigenspace spanned by the first n eigenfunctions ψm (.; k) of hL (k), Pn ϕj (x; k) ≡

n

(j) αm (k)ψm (x; k) ,

with

(j) αm (k) ≡ ϕj (.; k), ψm (.; k) . (2.20)

m=0

We recall that the explicit expression of ψm (x; k) is 1/4 2 B 1 Hm B 1/2 (x − k/B) e(−B/2)(x−k/B) , ψm (x; k) = √ m 2 m! π

(2.21)

where Hm denotes the mth Hermite polynomial as in [13]. It follows from this method that for all k ∈ ωj−1 (Δn )− , we have the lower bound ϕj (.; k), V0 Pn ϕj (.; k) ≥

1 ωj (k)−En (B) En+1 (B)−ωj (k) . (2.22) 2(n + 1)B

The strategy consists in computing an upper bound on |ϕj (.; k), V0 Pn ϕj (.; k)|, involving the trace V02 ϕj (−/2; k)2 . To do this, we use the expansion of Pn ϕj (.; k) as in (2.20), in ϕj (.; k), V0 Pn ϕj (.; k), and obtain: n (j) ϕj (.; k), V0 Pn ϕj (.; k) ≤ V0 |αm (k)| |ϕj (x; k)||ψm (x; k)|dx . |x|≥/2

m=0

(2.23) The set |x| > /2 is the classically forbidden region for electrons with energy less than V0 , so 0 ≤ ϕj (x; k) ≤ ϕj (±/2; k)e∓(V0 −ωj (k))

1/2

(x∓/2)

,

±x ≥ /2 ,

(2.24)

according to Proposition 8.3 in [13]. Substituting the corresponding exponentially decreasing term for ϕj (.; k) in (2.23), we have ϕj (.; k), V0 Pn ϕj (.; k) n (j) (j) (j) ≤ V0 |αm (k)| (Im,− )ϕj (−/2; k) + (Im,+ )ϕj (/2; k) , (2.25) m=0

where (j) Im,±

≡

±x≥/2

|ψm (x; k)|e∓(V0 −ωj (k))

1/2

(x∓/2)

dx .

(2.26)

Step 2: Estimate on the trace of the wave function (j) In view of bounding the integrals Im,± we first define the constant Hm ≡ sup Hm (u)e−u

2

u∈R

/2

.

(2.27)

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Then we substitute the following obvious consequence of (2.21) and (2.27) 1/4 B H √ m , (2.28) |ψm (x; k)| ≤ π 2m m! for |ψm (x; k)| in (2.26), and get: 1/4 B H 1 (j) √ m Im,± ≤ . (2.29) π 2m m! (V0 − ωj (k))1/2 Now combining (2.25) with (2.29), we obtain ϕj (., k), V0 Pn ϕj (., k) 1/4 n V0 B Hm (j) √ ≤ |αm (k)| ϕj (−/2; k) + ϕj (/2; k) . 1/2 π (V0 − ωj (k)) 2m m! m=0 (2.30) Let us define the constant H(n) by H(n)

⎞1/2 H2 m ⎠ ≡⎝ . 2m m! ⎛

(2.31)

m≤n

By applying the Cauchy–Schwarz inequality to the sum in (2.30), and using the normalization condition n (j) |αm (k)|2 = Pn ϕj ( · ; k)2 ≤ 1 , m=0

we obtain: ϕj (., k), V0 Pn ϕj (., k) V0 ≤ (V0 − ωj (k))1/2

B π

1/4

H(n) ϕj (−/2; k) + ϕj (/2; k) .

This, together with (2.19) and (2.22), yield 1/2

V0 ϕj (−/2; k) ≥ fn (B2 )B 3/4 , where π 1/4 fn (B ) = 2(n + 1)H(n) 2

ωj (k) 1− V0

1/2

(2.32)

(a − 1)(3 − c) − νn1/2 (B2 )−1/4 ,

provided B2 ≥ ζn . Further for all V0 ≥ En+1 (B) we have 1−

3−c ωj (k) , ≥ V0 2n + 3

uniformly in k ∈ ωj−1 (Δn )− , so fn (B2 ) can be made greater than (a−1)(3−c)3/2 / (2(n + 1)(2n + 3)1/2 H(n) ) by taking B2 sufficiently large, and (2.13) then follows from (2.32), (2.18) and (2.19).

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2.4. Perturbation of edge currents We now consider the perturbation of edge currents by adding a bounded impurity potential V1 (x, y) to H0 . As in Section 2.3 of [13] for unbounded one-edge geometries, we prove that the lower bound on the edge currents is stable with respect to these perturbations provided V1 = V1 ∞ is sufficiently small. Theorem 2.2. Let n ∈ N and Δn be as in (2.7). Let βn be defined as in Lemma 2.2, B2 ≥ βn and V0 ≥ En+1 (B). We consider two subintervals of Δn , all with the ˜ ⊂ Δn with |Δ|, |Δ| ˜ < δn , where δn = δn (B, , V0 ) > same center point Em , Δ ⊂ Δ 0 is as in Lemma 2.1. Let V1 be a bounded potential and let E0 ( · ) and E( · ) be the spectral families for H0 , and H1 = H0 + V1 , respectively. We decompose ˜ ˜ c )ψ. Let φ have ψ ∈ E(Δ)L2 (R2 ) as ψ = φ + ξ, where φ ≡ E0 (Δ)ψ and ξ ≡ E0 (Δ an expansion as in (2.3) with coefficients βj (k) satisfying the condition (2.14) for ˜ Then we have the interval Δ. γ2 2 3 Cn (a − 1) (3 − c) − Fn B 1/2 ψ2 , (2.33) Jy (ψ) = −ψ, Vy ψ ≥ 2 + γ2 ˜ −1 ), where Cn > 0 is the constant defined in Lemma 2.2. With ζ ≡ ((|Δ|+2V1 )|Δ| ˜ is given by the constant Fn ≡ Fn (B, V1 , |Δ|, |Δ|) 3/2 2 γ 2 ζ 3/2 1/2 −1 1/2 2 3 + Cn (a − 1) (3 − c) 2 2n + c + V1 B . Fn = ζ ˜ 2 + γ2 |Δ| (2.34) Further, for a fixed level n, if |Δ| and V1 are sufficiently small compared with ˜ there is a constant Cñ > 0 independent of B such that |Δ|, Jy (ψ) = −ψ, Vy ψ ≥ Cñ B 1/2 ψ2 .

(2.35)

Proof. Decomposing ψ = φ + ξ as in the theorem, and recalling that ψ = 1, we find that −ψ, Vy ψ ≥ −φ, Vy φ − 2Vy ξL2 (R2 ) ψ , (2.36) by the Cauchy–Schwarz inequality. The result then follows from Theorem 2.1 provided we have a good bound on ξ and on Vy ξ. As in Section 2.3 in [13], we write (2.37) ξ2 = ψ, ξ = (H0 − Em )ψ, (H0 − Em )−1 ξ ≤ (H1 − Em − V1 )ψ(H0 − Em )−1 ξ .

(2.38)

From the definitions and spectral localization of the states, we have two estimates |Δ| (H1 − Em − V1 )ψ ≤ + V1 ψ , 2 and ˜ c )ξ = (H0 − Em )−1 ξ ≤ dist−1 (Em , Δ

2 ˜ |Δ|

ξ .

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Using these to bound the right side of (2.37), we obtain the upper bound |Δ| + 2V1 ψ . (2.39) ξ ≤ ˜ |Δ| Further we notice that ξ, H0 ξ = ψ, H1 ξ − ψ, V1 ξ, so we obtain Vy ξ2 ≤ ξ, H0 ξ ≤ (2n + c)B + V1 ξ ψ .

(2.40)

The lower bound on the main term in (2.36) follows from the estimate (2.15): ⎛ ⎞ n γ2 −φ, Vy φ ≥ Cn (a − 1)2 (3 − c)3 B 1/2 ⎝ |βj (k)|2 dk ⎠ −1 ˜ 2 + γ2 ω ( Δ ) n j j=0 ≥ since

γ2 Cn (a − 1)2 (3 − c)3 B 1/2 (ψ2 − ξ2 ) , 2 + γ2 n j=0

˜ ωj−1 (Δ)

(2.41)

|βj (k)|2 dk = φ2 = ψ2 − ξ2 .

Combining this lower bound (2.41), with the estimate on ξ in (2.39), and Vy ξ in (2.40), we find (2.33) with the constant (2.34). This completes the proof. If the distance from the midpoint Em of Δ to Δcn is not smaller than δn ˜ with size of the order of δn . In this case any state we may choose an interval Δ ψ = E(Δ)ψ satisfying the assumptions of Theorem 2.2 carries a current of size B 1/2 provided (V1 + |Δ|)/δn is small enough. If δn is of size O(B), this indicates that the edge current survives in presence of perturbations V1 sufficiently small compared with B.

3. Extensions: The parabolic channel model and more general confining potentials The estimation of edge currents can be generalized to the case of various confining potentials like polynomial confining potentials p V0 (x) = B 1+(p/2) |x| − (/2) χ{|x|>/2} (x) , p > 1 . (3.1) We refer to these as soft confining potentials. We also discuss the straight parabolic channel model studied by Exner, Joye and Kovarik in [7]. In this case, the confining potential is defined by V0 (x) = V02 x2 , V0 > 0 . This model is completely solvable and the estimation of the edge currents rather straightforward. We conclude this section by estimating the edge currents for Dirichlet boundary conditions. This can be considered the limiting case of V0 → ∞ for the sharp confining potential (1.1). In both cases V0 is a function of x alone so the direct sum decomposition (2.1) remains valid. The fibered operators h0 (k), k ∈ R, have compact resolvents since

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limx→+∞ ((k − Bx)2 + V0 (x)) = +∞. As above, we denote the eigenvalues of h0 (k) by ωj (k), j ∈ N. In light of the proof of Theorem 2.1, we remark that it is enough to give an estimation of ωj (k), for k ∈ ωj−1 (Δn ) where Δn is as in (2.7). The stability of the edge currents under perturbations then follows as in Section 2.4. We state some of the results without proof and refer the reader to the archived version [14] for a complete discussion of soft confining potentials. 3.1. Parabolic channel model For this model, the electron is confined to a parabolic channel of infinite extent in 2 the y-direction. For any E > 0, the plane R is divided into a classically allowed region given by |x| < E/V0 , and the complementary classically forbidden region. The reduced, unperturbed Hamiltonian is given by h0 (k) = p2x + (k − Bx)2 + V02 x2 2 2 V0 B 2 = px + B0 x − k + k2 , B0 B0 for the modified field strength B0 = (B 2 + V02 )1/2 . Since this is simply a shifted harmonic oscillator Hamiltonian, it is completely solvable. The dispersion curves are parabolas with equation ωj (k) = (2j + 1)B0 + (V0 /B0 )2 k 2 so the set ωj−1 (Δn ) for the interval Δn = [(2n + a)B0 , (2n + c)B0 ], 1 < a < c < 3, is explicitly known: (n) (n) ωj−1 (Δn )− = − kj (c), −kj (a) , (n)

kj (x) ≡

3/2 1/2 B0 2(n − j) + x − 1 , V0

x = a, c .

From this and (−ωj (k)) = −2(V0 /B0 )2 k ≥ 2(V0 /B0 )2 kj (a) then follows that

1/2 V0 −ωj (k) ≥ 2 2(n − j) + a − 1 , k ∈ ωj−1 (Δn )− . 1/2 B0 (n)

3.2. Soft confining potentials Soft confining potentials include the polynomial model (3.1). We assume that the confining potential V0 is an even function, so this is the case for the dispersion curves too. This follows from the arguments of Lemma 5.1. The corresponding fibered operators h0 (k) depend analytically on k, so the dispersion curves functions are differentiable (see [15] or [21]). Although technically more complicated, their derivative can be estimated following the ideas in this paper (see [14] for the details). As a result, we obtain the following lower bound. Lemma 3.1. Let n ∈ N, Δn be the same as in (2.7), and (for the sake of simplicity) V0 = (2n + c)B. Then, there are two constants ζn = ζn (a) ≥ 0, depending only on n and a, and Cn,p = Cn,p (a, c) > 0, depending on n, a, c and p, such that we have −ωj (k) ≥ Cn,p (a − 1)2 (3 − c)2 B 1/2 , provided B2 ≥ ζn .

k ∈ ωj−1 (Δn )− ,

j = 0, 1, . . . , n ,

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D

ω2 ( k ) 5B D

ω1 ( k )

3B D

ω0 ( k ) B k 0

Figure 2. Dispersion curves of HD 0. 3.3. Dirichlet boundary conditions We denote the Landau Hamiltonian HL (B) on the space L2 ([−/2, /2] × R) with Dirichlet boundary conditions along x = (±/2) by H0D . This unperturbed operator admits a direct integral decomposition with respect to the y-variable. We D denote by hD 0 (k) the corresponding fibered operator with eigenvalues ωj (k) and D eigenfunctions ϕj (x; k). Because of the Dirichlet boundary conditions, the dispersion curves satisfy limk→±∞ ωjD (k) = ∞ as shown in Figure 2. The eigenfunctions ϕD j (x; k) provide an eigenfunction expansion of any state, as in (2.3), and we denote the coefficients of this expansion by βjD (k). Many properties of the dispersion curves ωjD (k) can be derived from [12] and [18], such as D ωj+1 (k) − ωjD (k) > 0 , k ∈ R . (3.2) Since the dispersion curves are nonintersecting and continuous functions of R, (3.2) implies D δj,l (K) ≡ inf |ωlD (k) − ωjD (k)| > 0 , j = l , (3.3) k∈K

for all compact subsets K of R. The perturbed operator is denoted by HD ≡ H0D + V1 , on the same Hilbert space. We let E0D ( · ) and ED ( · ) denote the spectral families of H0D and HD , respectively. Theorem 3.1. Consider the operators H0D and HD = H0D + V1 , on H ≡ L2 ([−/ 2, /2] × R), with Dirichlet boundary conditions along x = ±/2, where V1 (x, y) is

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bounded. Let n ∈ N and Δn , defined by (2.7), be sufficiently small so that there ˜ containing Δn , such that is a larger interval Δ ˜ ∩ (ω D )−1 (Δ) ˜ = ∅, (ωjD )−1 (Δ) l

0 ≤ j = l ≤ n .

Then for B2 sufficiently large (depending on a, c and n), any state ψ ∈ ED (Δn )H with coefficients satisfying the condition (2.14) carries an edge current satisfying the lower bound (2.35) provided |Δn | and V1 are sufficiently small compared ˜ with |Δ|. We prove this theorem through a perturbation argument comparing H0D with the Hamiltonian with the sharp confining potential of height V0 , H0 = HL (B)+V0 , in the large V0 regime. We begin with an estimate of the traces of the eigenfunctions ϕj (x; k) of h0 (k) on the lines x = ±/2. Lemma 3.2. Let n ∈ N, Δn be given by (2.7). Let V0 ≥ En+1 (B) and B2 ≥ θn , where θn = θn (a, c) is as in Lemma 5.8 and depends only on n, a and c. Then there is a constant rn = rn (a, c, B2 ) > 0 depending only on n, a, c and B2 such that for j = 0, 1, . . . , n, we have 0 ≤ ϕj (±/2; k) ≤

4π(j + rn ) 1/2

3/2 V0

,

k ∈ ωj−1 (Δn ) ∪ (ωjD )−1 (Δn ) .

(3.4)

Proof. 1. For any l ∈ N it follows from the eigenvalue equation (1.2) that 2 (Bx − k)2 + V0 (x) ϕl (x; k)2 dx = ωl (k) , k ∈ R . ϕl (x; k) +

(3.5)

From this and the Feynman–Hellmann formula, it follows that 1/2 ωl (k) = 2 (k − Bx)ϕl (x; k)2 dx ≤ 2 ωl (k) ,

(3.6)

R

R

R

and consequently ωl (k)1/2 ≤ ωl (0)1/2 + |k| ,

k ∈ R,

l ∈ N,

(3.7)

by integrating (3.6) over [0, |k|]. Since ωl (k) ≤ ωlD (k) ,

k ∈ R,

l ∈ N,

from the Min–Max Theorem, (3.7) then yields 1/2 ωl (k)1/2 ≤ ωlD (0) + |k| , k ∈ R ,

(3.8)

l ∈ N.

(3.9)

2 D Further, the quadratic part B 2 x2 in hD 0 (0) being bounded by (B/2) , ωl (0) is easily seen (see [21]) to be bounded as 2 2 2πl B D ωl (0) ≤ + , l ∈ N. (3.10) 2

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Moreover, taking into account (3.8), we deduce from Lemma 5.8 there are two constants τn and θn , depending only on n, a and c, such that for j = 0, 1, . . . , n, B + τn B 1/2 , k ∈ ωj−1 (Δn ) ∪ (ωjD )−1 (Δn ) , (3.11) 2 provided B2 ≥ θn and V0 ≥ En+1 (B). 2. Let ρ ∈ C 3 (R) be a bounded real-valued function and A denote the selfadjoint operator ρ(x)px + px ρ(x) in L2 (R), with domain H 1 (R). Any function ϕ in the domain of h0 (k) belonging to H 1 (R), [A, h0 (k)]ϕ, ϕ can be defined as h0 (k)ϕ, Aϕ − Aϕ, h0 (k)ϕ, and we find that A, h0 (k) ϕ, ϕ = 4ρ ϕ , ϕ − 4B ρ(Bx − k)ϕ, ϕ − ρ ϕ, ϕ − 2V0 ρ(/2)ϕ(/2)2 − ρ(−/2)ϕ(−/2)2 . (3.12) |k| ≤

In the particular case where ϕ is an eigenfunction ϕl (.; k) of h0 (k), the scalar product −i[A, h0 (k)]ϕ, ϕ vanishes according to the Virial Theorem, so (3.12) yields 2V0 ρ(/2)ϕl (/2; k)2 − ρ(−/2)ϕl (−/2; k)2 = 4 ρ ϕl (.; k), ϕl (.; k) − 4B ρ(Bx − k)ϕl (.; k), ϕl (.; k) − ρ ϕl (.; k), ϕl (.; k) . (3.13) Let ρ be such that ρ(x) = ±1 for ±x ≥ /2 and 0 ≤ ρ(i) (x) ≤ (2/)i , for |x| ≤ /2 and i = 1, 2, 3. It follows from (3.13) that 1 2 2 ϕl (x; k)2 dx V0 ϕl (/2; k) + ϕl (−/2; k) ≤ (4/) −1

1/2

1

(Bx − k) ϕl (x; k) dx 2

+ 2B −1

2

1

+ (4/3 ) −1

ϕl (x; k)2 dx .

Restricting to l = j = 0, 1, . . . , n, the result now follows from this, (3.5), and (3.9)– (3.11). The traces estimate (3.4) is a key ingredient in proving the local convergence of the dispersion curves to those for the Dirichlet problem. Lemma 3.3. The dispersion curves ωj (k), j ∈ N, are monotonic increasing functions of V0 . Further, for n ∈ N fixed and Δn as in (2.7), for V0 ≥ En+1 (B) and for B2 ≥ θn as in Lemma 3.2, there is a constant sn (a, c, B2 ) > 0 independent of V0 , such that 0 ≤ ωjD (k) − ωj (k) ≤ sn

B 3/4 1/2 V0

,

k ∈ ωj−1 (Δn ) ,

j = 0,

...,n.

(3.14)

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Proof. The Hamiltonians h0 (k) being analytic operators in the parameter V0 , we have ∂ωj (k) = ϕj (x; k)2 dx > 0 , (3.15) ∂V0 |x|≥/2 by the Feynman–Hellmann Theorem, which shows that the dispersion curves are monotone increasing with respect to V0 . Furthermore, the rate of increase in (3.15) slows as V0 → ∞. This follows from the pointwise upper bound on ϕj (x, k) restricted to |x| ≤ /2. In particular, from (2.24) and the trace estimate (3.4) of Lemma 3.2, we have −/2 1/2 ∂ωj (k) ≤ ϕj (−/2; k)2 e2(V0 −ωj (k)) (x+/2) dx 0≤ ∂V0 −∞ +∞ 1/2 + ϕj (/2; k)2 e−2(V0 −ωj (k)) (x−/2) dx ≤ sn

B V0

/2

3/2 ,

(3.16)

for B2 ≥ θn and V0 ≥ En+1 (B), the constant sn > 0 depending only on n, a, c and B2 . To prove the rate of convergence (3.14), we use the eigenvalue equation (1.2) and take the inner product in (−/2, /2) with the Dirichlet eigenfunction ϕD l . After integration by parts, and an application of the eigenvalue equation for ϕD l , one obtains, D ωl (k) − ωj (k) ϕD l ( · ; k), ϕj ( · ; k) D = (ϕD l ) (/2; k)ϕj (/2; k) − (ϕl ) (−/2; k)ϕj (−/2; k) .

(3.17)

The estimate (3.4) in Lemma 3.2 implies that the right side of (3.17) vanishes as V0 → ∞, that is |ωlD (k) − ωj (k)| ϕD l ( · ; k), ϕj ( · ; k) D −3/2 l + rn ≤ |(ϕD ) 1/2 . (3.18) l ) (−/2; k)| + |(ϕl ) (/2; k)| (4π V0 We next show that |ϕD j ( · ; k), ϕj ( · ; k)| is uniformly bounded from below as V0 → ∞, proving the convergence of the eigenvalues. To show this, let χi and χe denote the characteristic functions onto the interval lines (−/2, /2) and (−∞, −/2]∪[/ 2, +∞), respectively. We first note that ϕj ( · ; k)2 = 1 = χi ϕj ( · ; k)2 + χe ϕj ( · ; k)2 , and the upper bound on the eigenfunction ϕj outside the strip (−/2, /2) (2.24), together with (3.4), imply that χe ϕj ( · ; k) ≤ O(V0−1 ) , so that

−3/4

χi ϕj ( · ; k) ≥ 1 − O(V0

),

(3.19)

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as V0 → ∞ and k ∈ ωj−1 (Δn ). Now, for l = j, it follows from (3.3) and the monotonicity of the dispersion curves in V0 that −1 D ωj (Δn ) > 0 . |ωlD (k) − ωj (k)| ≥ |ωlD (k) − ωjD (k)| ≥ δl,j So it follows from this and from (3.18) that for l = j D ϕl ( · ; k), ϕj ( · ; k) → 0 , as V0 → ∞ . If, in addition, the matrix element ϕD j ( · ; k), ϕj ( · ; k) also vanished as V0 → ∞, this would contradict (3.19) as the family {ϕD l ( · ; k)} is an orthonormal basis. It follows that this matrix element must be bounded from below uniformly in V0 as V0 → ∞. Consequently, the dispersion curves must converge as V0 → ∞ with the specified rate. In light of the estimates (3.4) and (3.14) we argue as in the proof of Lemma 3.3 in [13] and obtain the convergence of the projection Pj (k), for the eigenvalue ωj (k) of h0 (k), to the projector P0D (k), for the eigenvalue ωjD (k) of hD 0 (k), when V0 tends 2 to infinity, with B fixed and sufficiently large. Lemma 3.4. Let n ∈ N and Δn be given by (2.7). Let Pj (k), respectively PjD (k), for j = 0, 1, . . . , n, be the projection onto the one-dimensional subspace of h0 (k), D respectively hD 0 (k), corresponding to the eigenvalue ωj (k), respectively ωj (k). Let 2 V0 ≥ En+1 (B) and B ≥ θn as in Lemmas 3.2 and 3.3. Then, there exists a finite constant tn = tn (a, c, B2 ) > 0, such that for all j = 0, 1, . . . , n, we have Pj (k) − PjD (k) ≤

tn 1/2

V0

,

k ∈ (ωjD )−1 (Δn ) ∪ ωj−1 (Δn ) .

(3.20)

Taking into account estimates (3.3) and (3.20), Theorem 2.2, and the fact that ϕD j ( · ; k), ϕj ( · ; k) ≥ D0 > 0, as V0 → ∞, the proof of Theorem 3.1 can now be obtained by mimicking the proof of Theorem 3.1 in [13].

4. Edge currents for bounded, two-edge, cylindrical geometry We now address the case of a quantum device with bounded cylindrical geometry. The charged particle is assumed to be moving on the cylinder CD of circumference D > 0 and confined along the cylinder axis by two confining potentials separated by the distance > 0. The infinite cylinder CD is given by CD = R × J = {(x, y) | x ∈ R, y ∈ J}, where J = [−D/2, D/2] is an interval with length D, with the points y = −D/2 and y = D/2 identified. The Landau Hamiltonian on L2 (CD ) is given by HL = p2x +(py −Bx)2 initially defined on functions in C0∞ (CD ) satisfying y-periodic boundary conditions ϕ(x, −D/2) = ϕ(x, D/2)

and

∂y ϕ(x, −D/2) = ∂y ϕ(x, D/2) .

(4.1)

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This operator has a unique selfadjoint extension. The quantum particle is confined in the x-direction to the strip [−/2, /2] by a sharp confining potential V0 (1.1). We will show in Section 4.1 that H0 has pure point spectrum. We shall prove that suitable states ϕ = E0 (Δn )ϕ, Δn ⊂ (En (B), En+1 (B)), carry an edge current Jy (ϕ) of size B 1/2 localized along the edges at x = ±/2. Furthermore, this current survives in the presence of sufficiently small perturbations. Thus, the existence of edge currents does not require that the corresponding Hamiltonian have continuous spectrum. This result extends some of the results of Ferrari and Macris [8–11] to more general cylindrical regions. Ferrari and Macris extensively investigated this model in the particular case when = D. They considered the infinite cylinder C , (α) of diameter , with two confining potentials V0 , for α = l, r, separated by a (α) (α) distance . Let H0 = HL + V0 , α = l, r, be the Hamiltonian with the left, respectively, right confining potential only. They next considered the random Hamiltonian Hω = H0 + Vω where Vω is an Anderson-type random potential. Under a (α) rather technical assumption on the spectra σ(H0 ), they proved that with large probability the spectrum of the random Hamiltonian Hω = H0 + Vω in the energy interval (B + Vω ∞ , 3B − Vω ∞ ) consists of eigenvalues close to those in (l) (r) σ(H0 ) ∪ σ(H0 ). They proved that the edge current carried by an associated (α) eigenstate ϕj is of size D (with opposite signs depending on whether α = l or r). Their analysis extends to the case where is at least of size log D. The estimates on the edge currents given in this section are obtained without any constraints on the size of and D. They hold for general wave packets with energy concentrated between two consecutive Landau levels. We remark that the edge currents exist for this model despite the fact that the Hamiltonian does not have a continuous spectral component. 4.1. The spectra of HL and H0 Let us define the Fourier transform F : L2 (CD ) → l2 (Z; L2 (R)) as Fϕ(x) = (ϕˆp (x))p∈Z , where e−ikp y 2π p, p ∈ Z. (4.2) ϕˆp (x) = ϕ(x, y) √ dy and kp = D D J ⊕ Due to the periodic boundary conditions (4.1), we have FHL F ∗ = p∈Z hL (kp ), where hL (k) = p2x + (k − Bx)2 in L2 (R). It follows that the spectrum of HL is the set ∪p∈Z σ(hL (kp )) and is pure point with σ(HL ) = (2N + 1)B, each eigenvalue having infinite multiplicity. We turn now to describing the spectrum of H0 = HL + V0 . Because the confining potential V0 is a function of x alone, the operator H0 = HL + V0 also admits a direct sum decomposition with fibered operators h0 (kp ) = hL (kp ) + V0 . This operator has a compact resolvent. The eigenvalues of h0 (k) are denoted ωm (k), m ∈ N, with the corresponding normalized eigenfunction ϕm (x; k). We (p) define functions Φm (x, y) ≡ D−1/2 ϕm (x; kp )eikp y , m ∈ N, p ∈ Z. They satisfy

1164

P. D. Hislop and E. Soccorsi (p)

(p)


(p)

H0 Φm = ωm (kp )Φm and the set {Φm , m ∈ N, p ∈ Z} is an orthonormal basis of L2 (CD ). This provides a spectral representation of H0 : (p) H0 = ωm (kp )|Φ(p) m Φm | . m≥0 p∈Z

As a consequence H0 has pure point spectrum: σ(H0 ) = {ωm (kp ), m ≥ 0, p ∈ Z}. Despite the fact that each eigenvalue ωm (kp ), (m, p) ∈ N×Z, has finite multiplicity, it is not guaranteed that the spectrum of H0 is discrete. As |p| goes to infinity, each ωm (kp ) approaches Em (B) + V0 by Lemma 5.2, part 1, so the eigenvalues lying in a neighborhood of Em (B) + V0 may not be isolated. 4.2. Edge currents: The unperturbed case Let Δn for n ≥ 0, be defined by (2.7) and consider an energy localized state ϕ = E0 (Δn )ϕ. We want to estimate the current carried by ϕ along the edges of the free sample CD at x = ±/2. It turns out (see below the estimate (4.7) of the current carried by a wave packet) that this current is the weighted sum of the (p) currents carried by all the eigenstates Φm , (m, p) ∈ N×Z, such that ωm (kp ) ∈ Δn . (p) We therefore start by estimating the current carried by such an eigenstate Φm , for appropriate indices m ∈ N and p ∈ Z− . In a second step, we extend this estimate to the case of the wave packet ϕ. (p)

4.2.1. Current carried by an eigenstate. We consider an eigenfunction Φm of H0 (p) for some (m, p) in N×Z− satisfying ωm (kp ) ∈ Δn . The current carried by Φm along (p) (p) the left edge of the cylinder CD is defined as the expectation Φm , Vy Φm of the velocity operator Vy = py − Bx in the y-direction. By arguing as in Section 2.1, (p) (p) we find that Φm , Vy Φm = ωm (kp ), so we may deduce from Lemma 2.2 the following lower bound. Since y is not a global coordinate, the velocity operator Vy is not expressible as a commutator. If it were a commutator with H0 , the current of an eigenstate would be zero by the Virial Theorem. Proposition 4.1. Let Δn be defined by (2.7). Then, for any (m, p) ∈ N × Z− with ωm (kp ) ∈ Δn , we have (p) (p) 2 3 1/2 Jy (Φ(p) , m ) = −Φm , Vy Φm ≥ Cn (a − 1) (3 − c) B

provided B2 ≥ βn and V0 ≥ En+1 (B) where βn > 0 and Cn > 0 are as in Lemma 2.2. 4.2.2. Current carried by a wave packet. We turn now to estimating the current carried along CD by a the state ϕ = E0 (Δ)ϕ, where Δ is a subinterval of Δn . We assume as in Proposition 4.1 that B2 ≥ βn and V0 ≥ En+1 (B), and suppose that |Δ| < δn (B, , V0 ) so Lemma 2.1 holds true. The state ϕ decomposes in the (p) orthonormal basis {Φm , m ∈ N, p ∈ Z} as (p) (p) (p) ϕ(x, y) = βm Φm (x, y) , where βm = ϕ, Φ(p) (4.3) m . p∈Z

0≤m≤n ωm (kp )∈Δn

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Since V0 ≥ En+1 (B), there are only a finite number of indices p involved in the sum (4.3). Indeed, we know from Lemma 5.2, part 1, that lim|k|→+∞ ω0 (k) = E0 (B) + V0 , with E0 (B) + V0 > (2n + c)B, thus there is p∗n = p∗n (B, , V0 , Δ) ∈ N such that ω0 (kp∗n ) ∈ Δ and

ω0 (kp ) ∈ / Δ for all |p| > p∗n .

(4.4)

Since ωn (k) > ω0 (k) for all n ≥ 1 and k ∈ R, we have ωn (kp ) ∈ / Δ for |p| > p∗n , so (4.3) finally reduces to (p) (p) βm Φm (x, y) . (4.5) ϕ(x, y) = |p|≤p∗ n

0≤m≤n ωm (kp )∈Δ

The current carried by ϕ along the left edge of the cylinder has the following expression: (p ) (p) (p ) Jy (ϕ) = ϕ, Vy ϕ = βm βm Φ(p) (4.6) m , vy Φm . |p|,|p |≤p∗ n 0≤m,m ≤n ωm (kp )∈Δ ωm (kp )∈Δ

(p )

The cross terms Φm , Vy Φm in (4.6) vanish for p = p as follows from the identity (p) (p ) FΦm (x) = (δ(s−p)ϕm (x; kp ))s∈Z , a similar expression for F(Vy Φm )(x), and the −1 −1 (Δ) ∩ ωm unitarity of F. Furthermore, since |Δ| < δn , we have ωm (Δ) = ∅ for all m = m by Lemma 2.1. Consequently, we obtain (p) 2 (p) |βm | Φ(p) (4.7) Jy (ϕ) = ϕ, Vy ϕ = m , Vy Φm , (p)

|p|≤p∗ n

0≤m≤n ωm (kp )∈Δ (p)

which shows that the current carried by ϕ is the |βm |2 -weighted sum of the current (p) carried by the eigenstates Φm with energy ωm (kp ) in Δ. Now by combining (4.7) with Proposition 4.1 and mimicking the proof of Theorem 2.1 we obtain a lower bound on the edge current carried by a suitably constructed wave packet. Theorem 4.1. Let n ∈ N, Δn , B2 , V0 , and Δ be as in Theorem 2.1. Let ϕ = E0 (Δ)ϕ and p∗n be the smallest integer satisfying (4.4), so ϕ has expansion as (p) in (4.5). Assume there is a constant γ > 0 such that the coefficients βm defined by (4.3) satisfy (−p) 2 (p) 2 | ≥ (1 + γ 2 )|βm | , |βm

(4.8)

for all m = 0, 1, . . . , n and p = 0, 1, . . . , p∗n such that ωm (kp ) ∈ Δ. Then we have Jy (ϕ) = −ϕ, Vy ϕ ≥

γ2 2 3 Cn (a − 1) (3 − c) B 1/2 ϕ2 , 2 + γ2

the constant Cn > 0 being as in Lemma 2.2.

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4.3. Perturbation theory As in Section 2.4 for the strip geometries we now consider the perturbation of the edge currents by adding a bounded impurity potential V1 (x, y) to H0 , and show (using Theorem 4.1 and arguing in the same way as in the proof of Theorem 2.2) that the lower bound on the edge currents is stable with respect to these perturbations provided V1 is not too large compared with the constant δn defined in Lemma 2.1. ˜ be as in Theorem 2.2. Let V1 (x, y) be a Theorem 4.2. Let n, Δn , B2 , V0 , Δ, Δ bounded potential and let E(Δ) denote the spectral projection for H1 = H0 + V1 ˜ ˜ c )ψ, so that and the interval Δ. Let ψ = E(Δn )ψ. Let φ ≡ E0 (Δ)ψ and ξ ≡ E0 (Δ (p) ψ = φ+ξ. Let φ have an expansion as in (4.5) with coefficients βm satisfying (4.8). ˜ the conclusion of Then if |Δ| and V1 are sufficiently small compared with |Δ| ˜ Theorem 2.2 holds true: there is a constant Cn > 0 independent of B such that Jy (ψ) = −ψ, Vy ψ ≥ Cñ B 1/2 ψ2 .

5. Appendix: Basic properties of the eigenvalues and eigenfunctions The resolvent of the operator h0 (k) = h(k) + V0 , k ∈ R, is compact since the effective potential (Bx − k)2 + V0 (x) is unbounded as |x| → ∞. Consequently, the spectrum of h0 (k) is discrete with only ∞ as an accumulation point. We write the eigenvalues of h0 (k) in increasing order and denote them by ωj (k), j ≥ 0. The normalized eigenfunction associated to ωj (k) is ϕj (x; k). We recall from Proposition 7.2 in [13] that the eigenvalues ωj (k), j ≥ 0, are simple for all k ∈ R. In this Appendix we collect the main properties of the eigenvalues and eigenfunctions of the operator h0 (k) for an even confining potential V0 . 5.1. Symmetry properties Lemma 5.1. Let V0 be an even confining potential. For all j ∈ N, the eigenvalue ωj (k) is an even function of k, and there is θj ∈ {−1, 1} such that the corresponding eigenfunction ϕj (x; k) satisfies ϕj (−x; −k) = θj ϕj (x; k), x ∈ R, k ∈ R. Proof. Let j ∈ N. The operator P that implements x → −x satisfies P {dom h0 (k)} = dom h0 (−k) and P h0 (k) = h0 (−k)P . Consequently, ωj (k) is an eigenvalue of h0 (−k), say ωmk (−k). Since the eigenvalues are continuous in k and simple, we must have ωj (k) = ωm (−k), for some m. This implies that ωj (0) = ωm (0) so we get m = j from the simplicity of the eigenvalues. The eigenvalue equation and the evenness of ωj imply the equation h0 (k)ϕj (−x; −k) = ωj (k)ϕj (−x; −k). Due to the uniqueness of the real-valued eigenfunction ϕj (.; k), since the eigenvalues

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are simple, together with the normalization condition ϕj (.; ±k) = 1, there is θj (k) ∈ {−1, 1} such that ϕj (−x; −k) = θj (k)ϕj (x; k) ,

x ∈ R,

k ∈ R.

(5.1)

It is easy to check that θj (k) is continuous in k and therefore a constant.

5.2. Properties of the dispersion curves We now describe the asymptotic behavior of the dispersion curves and show that the dispersion curves remain separated. The remaining results in this appendix are proved for the sharp confining potential (1.1). Lemma 5.2. For any j ∈ N, we have: 1. lim|k|→+∞ ωj (k) = Ej (B) + V0 2. inf k∈R (ωj+1 (k) − ωj (k)) > 0. Proof. 1. In light of Lemma 5.1, it is enough to show the result for k > 0. First, we deduce from the obvious operator inequality hL (k) ≤ h0 (k) ≤ hL (k) + V0 and the Min–Max Theorem that 0 ≤ Ej (B) ≤ ωj (k) ≤ Ej (B) + V0 .

(5.2)

Let ε ∈ (0, 1) and ϕ be a normalized function in the domain of h0 (k). We have hL (k)ϕ, ϕ ≥ (1 − ε) hL (k)ϕ, ϕ + ε (Bx − k)2 |ϕ(x)|2 dx , |x|≤/2

whence where Rε ≡

h0 (k)ϕ, ϕ ≥ (1 − ε) hL (k)ϕ, ϕ + V0 − ε + Rε ,

|x|≤/2

(5.3)

(ε(Bx − k) − V0 )|ϕ(x)| dx. Since ε(Bx − k) − V0 ≥ 0 on 2

2

2

[−/2, /2] for all k ≥ kε ≡ (B)/2 + (V0 /ε)1/2 , (5.3) implies h0 (k)ϕ, ϕ ≥ (1 − ε) hL (k)ϕ, ϕ + V0 − ε , k ≥ kε . Let Mj denote a j-dimensional submanifold of dom h0 (k), j = 0, 1, 2, · · · , n. It follows from the above inequality and the Max-Min Principle that h0 (k)ϕ, ϕ ωj (k) ≥ min ϕ∈M⊥ j , ϕ=1

≥

min

ϕ∈M⊥ j ,

(1 − ε) h(k)ϕ, ϕ + V0 − ε ,

ϕ=1

so we obtain ωj (k) ≥ (1 − ε)Ej (B) + V0 − ε , ∀k ≥ kε , by taking the max over the Mj ’s. Now the result follows from this and (5.2). 2. Let us suppose that inf k∈R (ωj+1 (k) − ωj (k)) = 0 for some j ∈ N. This implies there is a sequence (km )m≥1 of real numbers, such that 0 ≤ ωj+1 (km ) − ωj (km )
0, independent of k, B, V0 , , and , such that 2 2 ωj (k) − Ej (B) ≤ ηj (B2 2 )−3/4 + 2(B2 2 )−1/4 e−(B )/64 , 0≤ (5.5) B for all k ∈ [−(B/2)(1 − ), (B/2)(1 − )] provided we have B2 2 ≥ 1. Proof. The left inequality being obvious it is enough to prove the right one. Let θ be a real-valued, even, and twice continuously differentiable function on R, such that 1 if x ∈ − /2(1 − /2), 0 θ (x) = 0 if x ∈ (−∞, −/2] . It is evident that θ ψj (.; k) (where ψj (x; k) still denotes the j th normalized eigenfunction of hL (k)) belongs to the domain of h0 (k). Moreover, since the supports of V0 and θ are disjoint, we have h0 (k) − Ej (B) θ (x)ψj (x; k) = h0 (k), θ ψj (x; k) = −θ ψj (x; k) + 2iθ ψj (x; k) , which leads to h0 (k) − Ej (B) θ ψj (.; k) ≤ θ ψj (.; k) + 2θ ψj (x; k) .

(5.6)

Using that k/B ∈ [−/2(1 − ), 0], together with the explicit expression (2.21) of ψj (.; k), and the vanishing of θ outside [−/2, −/2(1 − /2)] ∪ [/2(1 − /2), /2], we see there are two constants αj and βj independent of k, B, V0 , , and , such that 2 2 θ ψj (.; k) ≤ αj B(B2 2 )−3/4 e−(B )/64 θ px ψj (x; k) ≤ βj B(B2 2 )−1/4 e−(B

2 2

)/64

.

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1169

This, combined with (5.6), implies h0 (k) − Ej (B) θ ψj (.; k) ≤ δj B (B2 2 )−3/4 2 2 + 2(B2 2 )−1/4 e−(B )/64 ,

(5.7)

where δj = max(αj , βj ). Moreover, as ψj (.; k) = 1, it follows from (2.21) that B 1/2 (/2(1−/2)−k/B) 2 1 θ ψj (.; k)2 ≥ j √ Hj (y)2 e−y dy 2 j! π B 1/2 (−/2(1−/2)−k/B) (B 1/2 )/4 2 1 ≥ j √ Hj (y)2 e−y dy ≡ ζj > 0 , 2 j! π −(B 1/2 )/4 by taking, say, B2 2 ≥ 1. Upon setting ηj = δj /ζj , this lower bound and (5.7) prove the result. The main consequence of Lemma 5.3 is the following result on the inverse image of energy intervals under the dispersion curves. Lemma 5.4. Let n ∈ N, the interval Δn be as in (2.7), and ∈ (0, 1). Then there is a constant γn (a) > 0 depending only on n and a such that B (1 − ) , 2 uniformly in V0 ≥ 0, provided B2 2 ≥ γn (a). sup ωj−1 (Δn )− < −

j = 0, 1, . . . , n ,

(5.8)

Proof. It suffices to apply Lemma 5.3. Let ηñ ≡ maxj=0,...,n ηj and consider the right side of (5.5) with ηñ replacing ηj . There is a constant γn (a) > 0 so that if B2 2 > γn (a), then this expression is smaller than B −1 (inf Δn − En (B)) = a − 1. This implies that ωj (k) < inf Δn , for all k ∈ [−(B/2)(1 − ), (B/2)(1 − )] and j = 0, . . . , n, and the result follows. 5.4. Estimation of the eigenfunctions in the classically forbidden region We prove precise pointwise exponential decay estimates on eigenfunctions of h0 (k) for certain values of k and x. Let n ∈ N be fixed, the interval Δn be as in Lemma (2.7), and consider j = 0, 1, . . . , n. We prove in Lemma 5.5 that for k ∈ ωj−1 (Δn )− , j = 0, 1, . . . , n, the eigenfunctions ϕj (x; k), are exponentially decreasing functions in the domain x > −/2 provided B2 is taken sufficiently large. This is the main tool for the proof of 1) Lemma 5.6, a technical result used in the estimation of ωj (k) given in Lemma 2.2, and 2) Lemma 5.7, which states the localization properties of the eigenfunctions in view of Proposition 2.1. Finally, Lemma 5.8, that is particularly useful in Section 3, is derived from Lemmas 5.4 and 5.7. Lemma 5.5. Let n ∈ N, Δn be as in (2.7), ε ∈ (0, 1) and set xε = −(/2)(1 − ε). Then there is a constant μn = μn (a) > 0 depending only on n and a such that ϕj (x; k)2 ≤ (Bε)e−

Bε 8 (x−xε )

,

x ≥ xε , k ∈ ωj−1 (Δn )− ,

uniformly in V0 ≥ 0, provided B2 2 ≥ μn .

j = 0, 1, . . . , n ,

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Proof. Set x ˜ε = −(/2)(1−ε/2). We notice that it suffices to take B2 large enough so the effective potential Wj (x; k) defined by (2.17) is positive for k ∈ ωj−1 (Δn )− in the region x ≥ x ˜ε . This follows from Lemma 5.4 with = ε/4. Indeed, in this case, (5.8) implies Bx − k ≥

Bε , 8

k ∈ ωj−1 (Δn )− ,

x≥x ˜ε ,

so by using the fact that Wj (x; k) ≥ (Bx − k)2 − En+1 (B), we get that 2 Bε , x≥x ˜ε , k ∈ ωj−1 (Δn )− , Wj (x; k) ≥ 16

(5.9)

provided B2 ε2 ≥ max(γn (a), 162 (2n + 3)), where γn (a) is defined in Lemma 5.4. As a consequence, the H 1 (R)-solution ϕj (.; k) to the differential equation ϕ (x) = ˜ε . Namely for all k ∈ Wj (x; k)ϕ(x) is exponentially decaying in the region x ≥ x ωj−1 (Δn )− , we have 0 ≤ ϕj (t; k) ≤ ϕj (s; k)e−

Bε 8 (t−s)

,

x ˜ε ≤ s ≤ t ,

(5.10)

from Proposition 8.3 in [13]. By combining (5.10) for t = xε with (5.9), we find that ϕj (xε ; k)2 e−

Bε 8 s

≤ ϕj (s; k)2 e−

Bε 8 xε

x ˜ε ≤ s ≤ xε ,

,

k ∈ ωj−1 (Δn )− .

(5.11)

Integrating (5.11) with respect to s over (˜ xε , xε ), and using the normalization condition ϕj (.; k) = 1, we find the bound ϕj (xε ; k)2 ≤

(Bε)/8 e

Bε xε ) 8 (xε −˜

−1

k ∈ ωj−1 (Δn )− .

,

(5.12)

˜ε = (ε)/4, we may take B2 ε2 sufficiently large so (5.12) implies Since xε − x ϕj (xε ; k)2 ≤ (Bε)e−

Bε xε ) 8 (xε −˜

,

k ∈ ωj−1 (Δn )− .

The result follows from this, (5.9) and (5.10). We now give two main consequences of Lemma 5.5. Lemma 5.6. Let n ∈ N and Δn as in (2.7). For s ∈ {+, −} and t > 0, we set −1/2 t if s = + gs (t) = 1 if s = −.

Then there are two constants ζn = ζn (a) > 0 and νn = νn (a) > 0, depending on n and a, such that for all B2 ≥ ζn we have V0 ϕj (±/2; k)2 ≤ νn g± (B2 )B 3/2 , uniformly in V0 ≥ 0.

k ∈ ωj−1 (Δn )− ,

j = 0, 1, . . . , n ,

(5.13)

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1171

Proof. 1. Let j be 0, 1, . . . , n. We start by proving there are two constants αj > 0 and βj > 0 independent of B, , V0 and k such that we have +∞ 2 0≤ (Bx − k)ϕj (x; k)2 dx ≤ αj Be−βj B , k ∈ ωj−1 (Δn )− , (5.14) 0

provided B2 is sufficiently large. The left inequality in (5.14) being evident, it is enough to show the right one. By using the fact that ωj (k) ≤ En+1 (B) and arguing as in the proof of Lemma 5.5, we deduce from Lemma 5.4 there is a constant υn (a) > 0 depending only on n and a such that the potential Wj (t; k) defined by (2.17) satisfies Wj (t; k) ≥ 0

and

0 ≤ Bt − k ≤ 2Wj (t; k)1/2 ,

t ≥ 0,

k ∈ ωj−1 (Δn )− ,

upon taking B2 ≥ υn (a). Now (5.14) follows immediately from this, the exponentially decaying behavior of ϕj (.; k) in R+ , 0 ≤ ϕj (x; k) ≤ ϕj (0; k)e−

x 0

Wj (t;k)1/2 dt

,

x ≥ 0,

k ∈ ωj−1 (Δn )− ,

as stated in Proposition 8.2 in [13], and Lemma 5.5. 2. We write (3.13) for a bounded real-valued function ρ ∈ C 3 (R) such that ρ(x) = 0 if x ≤ 0, and ρ(x) = 1 if x ≥ /2, and find /2 /2 ϕj (x; k)2 dx + 4ρ ∞ ϕj (x; k)2 dx 2V0 ϕj (/2; k)2 ≤ ρ ∞ 0 0 +∞ (Bx − k)ϕj (x; k)2 dx . (5.15) + 4Bρ∞ 0

The first term in the right hand side of (5.15) is bounded by a constant times −3 . Due to the energy equation (3.5), the second one is bounded by a constant times B−1 . Summing up (5.14) and (5.15) we then get that 2 V0 ϕj (/2; k)2 ≤ cB−1 1 + 4(2n + 3) + αj B2 e−βj B , for some c > 0 independent of B, , V0 and j, whence V0 ϕj (/2; k)2 ≤ νn (B2 )−1/2 B 3/2 , upon taking B2 sufficiently large. 3. The lemma now follows from (2.18), (3.6) and (5.16).

(5.16)

Lemma 5.7. Let αn = (μn + 1)1/2 and xn = −/2 + αn B −1/2 , where μn is defined in Lemma 5.5. Then for all B > 0, > 0, V0 > 0 and Δx ≥ 0, we have +∞ 1/2 ϕj (x; k)2 dx ≤ 8e−(αn /4)B Δx , k ∈ ωj−1 (Δn )− , j = 0, 1, . . . , n . xn +Δx

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Furthermore there are two constants θn = θn (a, c) > 0 and ϑn = ϑn (a, c) > 0 depending only on n, a, and c such that for all Δx ≥ 0 we have

−/2−Δx

−∞

ϕj (x; k)2 dx ≤ ϑn e−2(3−c)

1/2

B 1/2 Δx

,

k ∈ ωj−1 (Δn )− ,

j = 0, 1, . . . , n ,

provided B2 ≥ θn and V0 ≥ En+1 (B). Proof. Set ε = 2αn /(B 1/2 ) so B2 ε2 > μn and ϕj (x; k)2 ≤ 2αn B 1/2 e−(αn /4)B

1/2

x

,

x ≥ xn ,

k ∈ ωj−1 (Δn )− ,

j = 0, 1, . . . , n , (5.17) according to Lemma 5.5. The first part of the result is obtained by integrating (5.17) over [xn + Δx, +∞). To prove the second part, we make V0 ≥ En+1 (B) so that the effective potential (2.17) satisfies Wj (x; k) ≥ (3−c)B, for all x < −/2, and subsequently, ϕj (x; k)2 ≤ ϕj (−/2; k)2 e2(3−c)

1/2

B 1/2 (x+/2)

, x ≤ −/2 ,

k ∈ ωj−1 (Δn )− ,

by Proposition 8.3 in [13], since the effective potential Wj (x; k) ≥ (3 − c)B for all x ≤ −/2. Now the result immediately follows from this and Lemma 5.6. Another immediate consequence of Lemmas 5.4 and 5.7 is the following result. Lemma 5.8. Let n ∈ N and the interval Δn be as in (2.7). Then there is a constant κn = κn (a, c) > 0 depending only on n, a and c such that k + B ≤ κn B 1/2 , k ∈ ω −1 (Δn )− , j = 0, 1, . . . , n , (5.18) j 2 provided V0 ≥ En+1 (B) and B2 ≥ θn , where θn is as in Lemma 5.7. Proof. 1. Let k ∈ ωj−1 (Δn )− , j = 0, 1, . . . , n, be of the form k = −(B/2)(1 + ) for some > 0. Taking Δx = ()/4 in Lemma 5.7, and assuming that B2 ≥ θn and V0 ≥ En+1 (B), we get that −(/2)(1+(/2)) 1/2 1/2 e−((3−c) /2)B ϕj (x; k)2 dx ≤ . (5.19) (3 − c)1/2 −∞ Further, the normalization condition ϕj (.; k) = 1 implies

+∞

−(/2)(1+(/2))

(Bx − k)2 ϕj (x; k)2 dx ≥

B 4

2

1−

−(/2)(1+(/2))

−∞

2

ϕj (x; k) dx ,

(5.20)

Vol. 9 (2008)


1173

since x − k/B ≥ ()/4 for all x ≥ −(/2)(1 + (/2)). As a consequence, by combining (5.19)–(5.20) with the basic estimate ωj (k) ≥ (Bx − k)2 ϕj (x; k)2 dx , R

we have B 2 2 2 ωj (k) ≥ 16

e−((3−c) /2)B 1− (3 − c)1/2 1/2

1/2

.

Moreover, as k ∈ ωj−1 (Δn )− , we have ωj (k) ≤ (2n + c)B, whence

1/2 1/2 B2 2 e−((3−c) /2)B 2n + c ≥ , 1− 16 (3 − c)1/2

(5.21)

(5.22)

according to (5.21). The right hand side of (5.22) is an unbounded increasing function of B2 2 depending only on c. The left hand side of (5.22) depends only on n and c. We thus have B2 2 ≤ ξn (c) , for some constant ξn (c) > 0 depending only on n and c. Therefore B − ξn (c)1/2 B 1/2 , (5.23) 2 since k = −((B)/2)(1 + ) ∈ ωj−1 (Δn )− . 2. For the rest of the proof we recall from Lemma 5.4 that B + γn (a)1/2 B 1/2 , sup ωj−1 (Δn )− ≤ − (5.24) 2 where γn (a) > 0 depends only on n and a. The result then follows from (5.23)– (5.24) by setting κn (a, c) = max(ξn (c)1/2 , γn (a)1/2 ). inf ωj−1 (Δn )− ≥ −

Acknowledgements We thank J.-M. Combes for many discussions on edge currents and their role in the IQHE. We thank E. Mourre for discussions on the commutator method described in Section 1.3. We also thank F. Germinet, G.-M. Graf, and H. SchulzBaldes for fruitful discussions. Some of this work was done when ES was visiting the Mathematics Department at the University of Kentucky and he thanks the Department for its support.

References [1] Y. Avron, R. Seiler, B. Simon, Charge deficiency, charge transport, and comparison of dimensions, Commun. Math. Phys. 159 (1994), 399–422. [2] J. Bellissard, A. van Elst, H. Schulz-Baldes, The non-commutative geometry of the quantum Hall effect, J. Math. Phys., 35 (1994), 5373–5451.

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[3] J.-M. Combes, F. Germinet, Edge and impurity effects on quantization of Hall currents, Commun. Math. Phys. 256 (2005), 159–180. [4] J.-M. Combes, P. D. Hislop, E. Soccorsi, Edge states for quantum Hall Hamiltonians, Contemporary Mathematics, 307 (2002), 69–81. [5] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schr¨ odinger Operators, SpringerVerlag, Berlin (1987). [6] P. Elbau, G.-M. Graf, Equality of bulk and edge conductance revisited, Commun. Math. Phys. 229 (2002), 415–432. [7] P. Exner, A. Joye, H. Kovarik, Magnetic transport in a straight parabolic channel, J. Phys. A 34 (2001), 9733–9752. [8] C. Ferrari, N. Macris, Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems, J. Phys. A 35 (2002), 6339–6358. [9] C. Ferrari, N. Macris, Spectral properties of finite quantum Hall systems, Operator algebras and mathematical physics (Constancta, 2001), 115–122, Theta, Bucharest, 2003. [10] C. Ferrari, N. Macris, Extended energy levels for macroscopic Hall systems, math-ph. 02-255. [11] C. Ferrari, N. Macris, Extended edge states in finite Hall systems, J. Math. Phys. 44 (2003), 3734–3751. [12] V. A. Geiler, M. M. Senatorov, Structure of the spectrum of the Schr¨ odinger operator with magnetic field in a strip and infinite-gap potentials, Sbornik: Mathematics 188:5 (1997), 21–32. [13] P. D. Hislop, E. Soccorsi, Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries, Reviews in Math. Phys. 20 no. 1 (2008), 71–115. [14] P. D. Hislop, E. Soccorsi, Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries, http://arXiv.org/math-ph/0702093, archived. [15] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, 132 Springer-Verlag New York, Inc., New York 1966. [16] J. Kellendonk, H. Schulz-Baldes, Quantization of edge currents for continuous magnetic operators, J. Funct. Anal. 209 (2004), 388–413. [17] H. Kunz, The quantum Hall effect for electrons in a random potential, Commun. Math. Phys. 112 (1987), 121–145. [18] B. M. Levitan, I. S. Sargsyan, Sturm–Liouville and Dirac operators, Kluwer, Dordrecht, 1991. [19] N. Macris, P. A. Martin, J. V. Pulé, On Edge States in semi-infinite Quantum Hall Systems, J. Phys. A: Math. and General, 32 no. 10 (1999), 1985–1996. [20] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys. 78 (1981), 519–567. [21] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. IV: Analysis of Operators. Academic Press, 1978. [22] H. Schulz-Baldes, J. Kellendonk, T. Richter, Simultaneous quantization of edge and bulk Hall conductivity, J. Phys. A: Math. Gen. 33 (2002), L27–L32.

Vol. 9 (2008)


Peter D. Hislop Department of Mathematics University of Kentucky Lexington, KY 40506–0027 USA e-mail: [email protected] Eric Soccorsi Université de la Méditerranée Luminy, Case 907 13288 Marseille France e-mail: [email protected] Communicated by Claude-Alain Pillet. Submitted: February 2, 2007. Accepted: May 27, 2008.

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Ann. Henri Poincar´ e 9 (2008), 1177–1215 c 2008 Birkh¨

auser Verlag Basel/Switzerland 1424-0637/061177-39, published online September 22, 2008 DOI 10.1007/s00023-008-0383-7


Agmon-Type Estimates for a Class of Difference Operators Markus Klein and Elke Rosenberger Abstract. We analyze a general class of self-adjoint difference operators Hε = Tε + Vε on `2 ((εZ)d ), where Vε is a one-well potential and ε is a small parameter. We construct a Finslerian distance d induced by Hε and show that short integral curves are geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by the Finsler distance to the well. This is analog to semiclassical Agmon estimates for Schr¨ odinger operators.

1. Introduction The central topic of this paper is the investigation of a rather general class of families of self-adjoint difference operators Hε on the Hilbert space `2 ((εZ)d ), as the small parameter ε > 0 tends to zero. The operator Hε is given by X Hε = (Tε + Vε ) , where Tε = aγ τγ , (1.1) γ∈(εZ)d

(τγ u)(x) = u(x + γ)

and

(aγ u)(x) := aγ (x, ε)u(x)

for x, γ ∈ (εZ)d

(1.2)

where Vε is a multiplication operator, which in leading order is given by V0 ∈ C ∞ (Rd ). We remark that the limit ε → 0 is analog to the semiclassical limit ~ → 0 for the Schr¨ odinger operator −~2 ∆ + V . This paper is the first in a series of papers; the aim is to develop an analytic approach to the semiclassical eigenvalue problem and tunneling for Hε which is comparable in detail and precision to the well known analysis for the Schrödinger operator (see Simon [24, 25] and Helffer–Sj¨ ostrand [17]). Our motivation comes from stochastic problems (see Bovier–Eckhoff–Gayrard–Klein [8, 9]). A large class of discrete Markov chains analyzed in [9] with probabilistic techniques falls into the framework of difference operators treated in this article.

1178

M. Klein and E. Rosenberger


We recall that sharp semiclassical Agmon estimates describing the exponential decay of eigenfunctions of appropriate Dirichlet realizations of the Schrödinger operator are crucial to analyze tunneling for the Schrödinger operator. We further recall that the original work of Agmon on the decay of eigenfunctions for second order differential operators is not in the semiclassical limit. It treats the limit |x| → ∞ (in a non bounded domain of Rn ). Agmon realized in [3] that for a large class of such operators the exponential rate at which eigenfunctions decay is given by the geodesic distance in the Agmon metric. This is the Riemannian metric from Jacobi’s theorem in classical mechanics: For a Hamilton function whose kinetic energy is a positive definite quadratic form in the momenta, the projection to configuration space of an integral curve of the Hamiltonian vector field is a geodesic in the Agmon (Jacobi) metric. This paper contains analog results for the class of operators Hε , including a generalization of Jacobi’s theorem. It is essential that we consider these operators as semiclassical quantizations of suitable Hamilton functions and investigate the relation of these Hamilton functions to Finsler geometry. In this generality our results are new. We recall, however, that various examples extending the original framework of the semiclassical analysis in the work of Simon [24] and Helffer– Sj¨ ostrand [17] have been analyzed: The operator cos hDx + cos x in Harpers equation (see e.g. Helffer–Sj¨ ostrand [18]), the Schrödinger operator with magnetic field (Helffer–Mohamed [15]), the Dirac and Klein–Gordon operator (see e.g. Helffer– Parisse [16], Servat [23]) and the Kac operator (Helffer [14]). If Td := Rd /(2π)Zd denotes the d-dimensional torus and b ∈ C ∞ d Rd × Td × (0, 1] , a pseudo-differential operator OpTε (b) : K (εZ)d −→ K0 (εZ)d is defined by X Z

d

OpTε (b) v(x) := (2π)−d

y∈(εZ)d

i

e ε (y−x)ξ b(x, ξ; ε)v(y) dξ ,

(1.3)

[−π,π]d

where K (εZ)d := u : (εZ)d → C | u has compact support (1.4) d d and K0 (εZ)d := {f P : (εZ) → C } is dual to K (εZ) by use of the scalar product hu , vi`2 := x u ¯(x)v(x). We remark that under certain assumptions on the aγ defining Tε in (1.1), d one has Tε = OpTε (t(., .; ε)), where t ∈ C ∞ Rd × Td × (0, 1] is given by t(x, ξ, ε) =

X γ∈(εZ)

i aγ (x, ε) exp − γ · ξ . ε d

(1.5)

Here t(x, ξ; ε) is considered as a function on R2d × (0, 1], which is 2π-periodic with respect to ξ.

Vol. 9 (2008) Agmon-Type Estimates for a Class of Difference Operators (0)

(1)

1179

(2)

Furthermore, assuming that aγ (x, ε) = aγ (x) + εaγ (x) + Rγ (x, ε), where (2) Rγ (x, ε) = O(ε2 ) uniformly with respect to x and γ, we can write t(x, ξ; ε) = t0 (x, ξ) + ε t1 (x, ξ) + t2 (x, ξ; ε) , X − εi γξ tj (x, ξ) := a(j) , j = 0, 1 γ (x)e

with

(1.6)

γ∈(εZ)d

t2 (x, ξ; ε) :=

X

i

Rγ(2) (x, ε)e− ε γξ .

γ∈(εZ)d

Thus, in leading order the symbol of Hε is h0 = t0 + V0 . In its original form, neither Jacobi’s theorem applies to h0 (x, ξ) nor Agmon estimates to Hε . Our analysis is motivated by the remark in Agmon’s book [3] to develop part of the theory of the Agmon metric in the more general context of Finsler geometry. It turns out that the Hamilton function −h0 (x, iξ) (this transformation is analog to the procedure in the case of the Schr¨ odinger operator) in a natural way induces a Finsler metric and an associated Finsler distance d on Rd . This allows to formulate and prove a generalization of Jacobi’s theorem (which might be some kind of lesser known folk wisdom in mathematical physics, which, however, we were unable to find in the literature) and prove an analog of the semiclassical Agmon estimates for Hε . We remark that Finsler distances have been used for higher order elliptic differential operators in the analysis of decay of resolvent kernels and/or heat kernels, see Tintarev [26] and Barbatis [6, 7].1 However, these papers do not develop a generalization of Jacobi’s theorem, which turns out to be crucial in our semiclassical analysis. We will now state our assumptions on Hε and formulate our results more precisely. Hypothesis 1.1. (a) The coefficients aγ (x, ε) in (1.1) are functions a : (εZ)d × Rd × (0, 1] → R ,

(γ, x, ε) 7→ aγ (x, ε) ,

(1.7)

satisfying the following conditions: (i) They have an expansion (1) (2) aγ (x, ε) = a(0) γ (x) + ε aγ (x) + Rγ (x, ε) , (i)

(j)

(1.8)

(j)

where aγ ∈ C ∞ (Rd ) and |aγ (x) − aγ (x + h)| = O(|h|) for j = 0, 1 (2) uniformly with respect to γ ∈ (εZ)d and x ∈ Rd . Furthermore Rγ ∈ ∞ d d C (R × (0, 1]) for all γ ∈ (εZ) . P (0) (0) (ii) γ aγ = 0 and aγ ≤ 0 for γ 6= 0 (iii) aγ (x, ε) = a−γ (x + γ, ε) for x ∈ Rd , γ ∈ (εZ)d 1 M. K.

thanks S. Agmon for the reference to [26], where prior to the publication of Agmon’s book a Finslerian approach was used to obtain estimates on the kernel of the resolvent and the decay of the heat kernel for higher order elliptic operators, following ideas of Agmon.

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(iv) For any c > 0 there exists C > 0 such that for j = 0, 1 uniformly with respect to x ∈ (εZ)d and ε

c|.|

c|.|

ε (j)

≤ C and e ε R.(2) (x) ≤ Cε2 (1.9)

e a. (x) 2 d 2 d `γ ((εZ) )

`γ ((εZ) )

(0) (εZ)d | aγ (x)

(v) span{γ ∈ < 0} = Rd for all x ∈ Rd . (b) (i) The potential energy Vε is the restriction to (εZ)d of a function Vbε ∈ C ∞ (Rd , R), which has an expansion Vbε (x) =

N X

εl Vl (x) + RN +1 (x; ε) ,

l=0

where V` ∈ C (R ), RN +1 ∈ C ∞ (Rd × (0, ε0 ]) for some ε0 > 0 and for any compact set K ⊂ Rd there exists a constant CK such that supx∈K |RN +1 (x; ε)| ≤ CK εN +1 . (ii) There exist constants R, C > 0 such that Vε (x) > C for all |x| ≥ R and ε ∈ (0, ε0 ]. (iii) V0 (x) has exactly one non-degenerate minimum at x0 = 0 with the value V0 (0) = 0. ∞

d

The following lemma couples the assumptions on the coefficients aγ given in Hypothesis 1.1 with properties of the symbol t and the kinetic energy Tε . Lemma 1.2. Assume Hypothesis 1.1 and let t and tj , j = 0, 1, 2 be defined in (1.5) and (1.6) respectively. Then: (a) t ∈ C ∞ (Rd × Td × (0, 1]) and the estimate supx,ξ |∂xα ∂ξβ t(x, ξ; ε)| ≤ Cα,β holds for all α, β ∈ Nd uniformly with respect to ε. Furthermore t0 and t1 are bounded and supx,ξ |t2 (x, ξ; ε)| = O(ε2 ). (b) The 2π-periodic function Rd 3 ξ 7→ t0 (x, ξ) is even and has an analytic continuation to Cd . (c) At ξ = 0, for fixed x ∈ Rd the function t0 defined in (1.6) has an expansion

t0 (x, ξ) = ξ , B(x)ξ + O |ξ|4 as |ξ| → 0 , (1.10) where B : Rd → M(d × d, R) is positive definite and symmetric. (d) The operator Tε defined in (1.1) is symmetric, bounded (uniformly in ε) d and hu , Tε ui`2 ≥ −Cεkuk2 for some C > 0. Furthermore Tε = OpTε (t) (see (1.3)). Remark 1.3. Tε being symmetric boils down to aγ being real and condition (a)(iii) of Hypothesis 1.1. In the probabilistic context, which is our main motivation, the latter is a standard reversibility condition while the former is automatic for a Markov chain (see Section 4). Since Tε is bounded, Hε = Tε + Vε defined in (1.1) possesses a self adjoint realization on the maximal domain of Vε . Abusing notation, we shall denote this realization also by Hε and its domain by D(Hε ) ⊂ `2 (εZ)d . The associated symbol is denoted by h(x, ξ; ε). Clearly, Hε commutes with complex conjugation.

Vol. 9 (2008) Agmon-Type Estimates for a Class of Difference Operators

1181

We will use the notation a ˜ : Zd × Rd 3 (η, x) 7→ a ˜η (x) := a(0) εη (x) ∈ R

(1.11)

and set ˜ 0 (x, ξ) := −h0 (x, iξ) = t˜0 (x, ξ) − V0 (x) : R2d → R , h where by Lemma 1.2 (b) X t˜0 (x, ξ) := −t0 (x, iξ) = − a ˜η (x) cosh (η · ξ) .

(1.12) (1.13)

η∈Zd

˜ 0 for fixed energy E We shall now describe, how Hamilton functions such as h introduce a Finsler geometry in configuration space. Hypothesis 1.4. Let M be a d-dimensional smooth manifold. Let h ∈ C ∞ (T ∗ M, R) be hyperregular and even and strictly convex in each fibre. Furthermore, let h(., 0) f := M \ {h(x, 0) ≥ E}. Denoting the be bounded from above. For E ∈ R set M fibre derivative of h by DF h, we associate to h the energy function Eh (x, v) := −1 h ◦ (DF h) (x, v) on T M . The notion of fibre derivative and hyperregular are standard (see Abraham– Marsden [2]). For convenience of the reader, they are repeated in Definition 2.7. Now Theorem 2.11 states that assuming Hypothesis 1.4 `h,E (x, v) := (DF h)

−1

(x, v˜) · v ,

(1.14)

f. The most where v˜ is chosen such that Eh (x, v˜) = E, is a Finsler function on M important property of `h,E is the homogeneity `h,E (x, λv) = |λ|`h,E (x, v) , λ ∈ R , p which is analog to the homogeneity of |v| = g(v, v) in the case of a Riemannian metric. This is essential to define a curve length associated to `h,E as described in Definition 2.3 by Z b s`h,E (γ) := `h,E γ(t), γ(t) ˙ dt . a

A Finsler geodesic is then a curve γ on M , for which s`h,E is extremal (see Definition 2.4). The following theorem establishes the connection between geodesics with respect to the Finsler function `h,E for a given hyperregular Hamilton function h and the integral curves of the associated Hamiltonian vector field Xh . It amplifies the Maupertuis principle in classical mechanics. f satisfy Hypothesis 1.4. Let `h := `h,E be as defined Theorem 1.5. Let h, E and M in (1.14) (see Theorem 2.11 for details). f be a base integral curve of the Hamiltonian vector field (a) Let γ0 : [a, b] → M Xh with energy E (i.e. Eh (γ0 (t), γ˙ 0 (t)) = E for all t ∈ [a, b]). Then γ0 is a f with respect to `h . geodesic on M

1182



f with respect to `h with energy Eh (γ0 , γ˙ 0 ) (b) Conversely, if γ0 is a geodesic on M = E, then γ0 is a base integral curve of Xh . ˜ 0 introduced in (1.12) actually satisfies HypotheThe Hamilton function h sis 1.4 with respect to the energy E = 0 (see Corollary 2.15) and thus induces a Finsler function ` := `h˜ 0 ,0 and a Finsler distance defined by Z 1 d` (x0 , x1 ) = inf ` γ(t), γ(t) ˙ dt , (1.15) γ∈Γ0,1 (x0 ,x1 )

0

where Γ0,1 (x0 , x1 ) denotes the set of regular curves γ with γ(0) = x0 and γ(1) = x1 . Theorem 1.6. There exists a neighborhood Ω of 0 such that d0 (x) := d` (0, x), with d` defined in (1.15), fulfills the generalized eikonal equation ˜ 0 x, ∇d0 (x) = 0 , x ∈ Ω . h Furthermore d0 (x) −

X

ϕk (x) = O(|x|N +1 )

as

x → 0,

(1.16)

1≤k≤N

where each ϕk is an homogeneous polynomial of degree k + 2. In addition d` is locally Lipschitz continuous, i.e. |d` (x, y)| ≤ C |x − y| ,

x, y ∈ Rd ,

(1.17)

where C is locally uniform in x and y. The eikonal inequality ˜ 0 x, ∇d0 (x) ≤ 0 h

(1.18)

holds almost everywhere in Rd . To analyze eigenfunctions concentrated at the potential minimum x0 = 0, we introduce a Dirichlet operator HεΣ as follows. Definition 1.7. For Σ ⊂ Rd we set Σε := Σ∩(εZ)d . Any function u ∈ `2 (Σε ) can by zero extension, i.e. via u(x) = 0 for x ∈ / Σε , be embedded in `2 ((εZ)d ). If we denote this embedding by iΣε , we can define the space `2Σε := iΣε `2 (Σε ) ⊂ `2 ((εZ)d ) and the Dirichlet operator HεΣ := 1Σε Hε |`2Σ

ε

: `2Σε → `2Σε

(1.19)

with domain D(HεΣ ) = {u ∈ `2Σε | Vε u ∈ `2Σε }. We now formulate our estimates of weighted `2 -norms of eigenfunctions of the Dirichlet operator HεΣ . We will show that they decay exponentially at a rate controlled by the Finsler distance d0 (x). Theorem 1.6 is crucial to prove these estimates.


1183

Theorem 1.8. Let Σ ⊂ Rd be a bounded open region including the point 0 such that d0 ∈ C 2 (Σ), where d0 (x) := d` (0, x) is defined by (1.15). Let E ∈ [0, εR0 ] for R0 fixed, assume Hypothesis 1.1 and let HεΣ denote the Dirichlet operator introduced in (1.19). Then there exist constants ε0 , B, C > 0 such that for all ε ∈ (0, ε0 ] and real u ∈ `2Σε

−B

d0 d0

e ε u

1+

ε

`2

" ≤C ε

−1

# −B

d0 d0

Σ e ε Hε − E u + kuk`2 .

1+

2 ε

(1.20)

`

In particular, let u ∈ `2Σε be a normalized eigenfunction of HεΣ with respect to the eigenvalue E ∈ [0, εR0 ]. Then there exist constants B, C > 0 such that for all ε ∈ (0, ε0 ]

−B

d0 d0

e ε u ≤ C . (1.21)

1+

2 ε `

Remark 1.9. The estimate (1.20) is the analog of the sharp semiclassical Agmon estimate in Helffer–Sj¨ ostrand [17] for the Schrödinger operator. We emphasize that our generalization in the context of general Finsler geometry is a result in the semiclassical limit, under the crucial hypothesis that both the kinetic and the potential energy have a non-degenerate minimum at ξ = 0, x = 0. In particular, we have nothing to report for an analog of the original Agmon estimate if ε = 1 and |x| → ∞. The plan of the paper is as follows. Section 2 is devoted to the construction and properties of a Finsler function associated to a hyperregular Hamilton function. In particular, in Subsection 2.1 we introduce the general notion of a Finsler manifold, the associated curve length and Finsler geodesics. In Subsection 2.2 we construct the absolute homogeneous Finsler function `h,E with respect to an hyperregular Hamilton function h and a fixed energy E. In particular, we prove Theorem 2.11. The proof of Theorem 1.5 is given in Subsection 2.3. In Subsection 2.4 we prove Lemma 1.2 and we show that we can apply the results derived up to this point to the Hamilton function ˜ 0 defined in (1.12). Subsection 2.5 contains the proof of Theorem 1.6. h In Section 3 we show the exponential decay of the eigenfunctions of the low lying spectrum of Hε with a rate controlled by the Finsler distance constructed in Section 2. In particular, in Subsection 3.1 we show three basic lemmata and in Subsection 3.2 we prove Theorem 1.8. In Section 4 we describe how a certain class of Markov chains fits into the framework of our hypotheses.

1184



2. Finsler distance associated to Hε 2.1. Definition and properties of Finsler manifold and Finsler metric We introduce the general notion of a Finsler manifold and Finsler distance (for detailed description of Finsler manifolds we refer e.g. to Bao–Chern–Shen [5], Abate– Patrizio [1]). For a manifold M , π : T M → M denotes the tangent bundle with fibre Tx M = π −1 (x). We denote an element of T M by (x, v) where x ∈ M and v ∈ Tx M . Analogously, (x, ξ) with ξ ∈ Tx∗ M denotes a point in the cotangent bundle π∗ : T ∗ M → M . The canonical pairing between an element v ∈ Tx M and ξ ∈ Tx∗ M is written as v · ξ. Definition 2.1. Let M be a d-dimensional C ∞ -manifold and T M \ {0} := {(x, v) ∈ T M | v 6= 0} the slit tangent bundle. (a) A (Lagrange)-function F : T M → [0, ∞) is called a Finsler function on M , if: 1) F is of class C ∞ (T M \ {0}). 2) F (x, λv) = λF (x, v) for λ > 0, i.e., F is positive homogeneous of order 1 in each fibre. 3) g(x, v) := Dv2 12 F 2 |(x,v) is positive definite as a bilinear form on Tx M for all (x, v) ∈ T M \ {0}. (b) A Finsler function F is said to be absolute homogeneous, if 4) F (x, λv) = |λ|F (x, v) for all λ ∈ R, (c) A manifold together with a Finsler function, (M, F ), is called a Finsler manifold. Pd Remark 2.2. (a) By Euler’s Theorem, 2) implies i,j=1 gij (x, v)vi vj = F 2 (x, v) and thus 3) implies F (x, v) > 0 for v 6= 0 (for details see Bao–Chern–Shen [5]). (b) In Agmon [3], the definition of a Finsler function is slightly more general: F is only required to be continuous and F (x, .) to be positive and convex. In this paper, we use the definition of Bao–Chern–Shen [5], since the natural class of Finsler functions which we construct turn out to be Finsler in this more narrow sense. Thus we can use results of Bao–Chern–Shen [5]. A Finsler function induces a curve length on M as follows. A curve γ : [a, b] → M on M is called regular, if it is C 2 and γ(t) ˙ 6= 0 for all t ∈ [a, b]. We introduce the Banach manifold n o Γa,b (x1 , x2 ) := γ ∈ C 2 [a, b], M | γ is regular and γ(a) = x1 , γ(b) = x2 . (2.1) Definition 2.3. For any Finsler function F on M , the curve length sF : Γa,b (x1 , x2 ) → R associated to F is defined as Z b sF (γ) := F γ(t), γ(t) ˙ dt . a


1185

For any δ > 0, a regular variation of γ ∈ Γa,b (x1 , x2 ) is a C 2 -map γδ : [a, b] × (−δ, δ) → M , such that γδ (t, 0) = γ(t) for all t ∈ [a, b] and γδ (., u) is regular for each u ∈ (−δ, δ). Each C 2 -map γδ : (−δ, δ) → Γa,b (x1 , x2 ) with γδ (0) = γ can be considered as a regular variation of γ with fixed endpoints (i.e. with γδ (a, u) = x1 and γδ (b, u) = x2 for all u ∈ (−δ, δ)). Therefore the tangent space of Γa,b (x1 , x2 ) at a point η is given by Tη Γa,b (x1 , x2 ) = {∂u ηδ |u=0 | ηδ is a regular variation of η with fixed endpoints} , (2.2) where ∂u ηδ |u=0 is considered as a vector field along η, i.e., as a function ∂u ηδ |u=0 : [a, b] → T M such that ∂u ηδ |u=0 (t) ∈ Tη(t) M . Since the variation ηδ has fixed endpoints, it follows that ∂u ηδ |u=0 (a) = ∂u ηδ |u=0 (b) = 0. Definition 2.4. γ ∈ Γa,b (x1 , x2 ) is called a geodesic with respect to the Finsler function F (or a Finsler geodesic), if dsF |γ = 0. Definition 2.5. Let (M, F ) denote a Finsler manifold. (a) The Finsler distance dF (x1 , x2 ) : M × M → [0, ∞] between the points x1 and x2 is defined by dF (x1 , x2 ) :=

inf γ∈Γ0,1 (x1 ,x2 )

sF (γ) .

If Γ0,1 (x1 , x2 ) is empty, the distance is defined to be infinity. (b) A geodesic γ between two points x1 and x2 is called minimal, if sF (γ) = d(x1 , x2 ). It follows easily from the definitions of a Finsler function F and the associated Finsler distance dF that dF (x1 , x2 ) ≥ 0, where equality holds if and only if x1 = x2 . Furthermore the triangle inequality dF (x1 , x3 ) ≤ dF (x1 , x2 ) + dF (x2 , x3 ) holds. If in addition the Finsler function F is absolute homogeneous, then dF (x1 , x2 ) = dF (x2 , x1 ). Thus for an absolute homogeneous Finsler function, (M, dF ) is a metric space. Definition 2.6. We denote by SM := T M/ ∼S the sphere bundle, where (x, v) ∼S (y, w) ,

if x = y

and v = λw

for any

λ > 0.

Let πs : T M → SM denote the projection πs (x, v) = [x, v]. 2.2. The Finsler function of a hyperregular Hamilton function To define a Finsler distance for which an analog of Jacobi’s Theorem holds, we briefly introduce the notion of fibre derivatives, hyperconvexity and hyperregularity of h. It is shown in Proposition 2.9 that hyperconvexity of h is a sufficient condition for hyperregularity. Definition 2.7. (a) Let M be a manifold and f ∈ C ∞ (T ∗ M, R). Then for fx := f |Tx∗ M the map DF f : T ∗ M → T M defined by DF f (x, ξ) := Dfx (ξ) is called the fibre derivative of f .

1186



(b) Analogously, the fibre derivative of a function g ∈ C ∞ (T M, R) is defined as DF g : T M → T ∗ M , DF g(x, v) := Dgx (v). (c) A function f : SM → T M is called strictly fibre preserving, if f ([x, u]) ∈ [x, u], where [x, u] denotes the equivalence class with respect to ∼S . (d) A smooth function h : T ∗ M → R (or L : T M → R) is said to be hyperregular, if its fibre derivative DF h : T ∗ M → T M (or DF L : T M → T ∗ M ) is a diffeomorphism. For a hyperregular function h ∈ C ∞ (T ∗ M ), we sometimes use the notation ξh (x, v) := (DF h)

−1

(x, v)

and vh (x, ξ) := DF h(x, ξ) .

(2.3)

Definition 2.8. For a normed vector space V we call a function L ∈ C 2 (V, R) hyperconvex, if there exists a constant α > 0 such that D2 L|v0 (v, v) ≥ αkvk2

for all

v0 , v ∈ V .

We recall that a strictly convex function L ∈ C 2 (V, R) has the properties L(v1 ) − L(v2 ) ≥ DL(v2 )(v1 − v2 ) 2

D L|v0 [v, v] > 0 DL(v1 ) − DL(v2 ) (v1 − v2 ) > 0 .

(2.4) (2.5) (2.6)

Proposition 2.9. If a real valued function h ∈ C ∞ (T ∗ M ) is hyperconvex in each fibre Tx∗ M , it is hyperregular. Proof. By definition, DF h is fibre preserving, thus in the coordinates on T M and T ∗ M induced from local coordinates on M at x0 , its derivative DDF h|(x0 ,ξ0 ) is given by the 2d × 2d-matrix 1 0 , (2.7) ∗ M where M is the matrix representation of Dξ2 h|(x0 ,ξ0 ) = D DF h|Tx∗ M . Since h was assumed to be hyperconvex in each fibre, M is positive definite and thus it follows from (2.7) that DF h is a local diffeomorphism. We claim that Dhx : Tx∗ M → Tx M is bijective for all x ∈ M . Since DF h is fibre preserving, this shows that DF h is a global diffeomorphism and finishes the proof. Thus we fix any x ∈ M and analyze Dhx . Since hx is strictly convex for each x ∈ M , by (2.6) (ξ − η) Dhx (ξ) − Dhx (η) > 0 , ξ, η ∈ Tx∗ M , η 6= ξ holds and thus DF h(x, .) = Dhx is injective. To show the surjectivity, we claim that, for any v0 ∈ Tx M , the initial value problem d ˙ , ξ(0) = 0 . v0 = Dhx ξ(t) = D2 hx ξ(t) · ξ(t) (2.8) dt


1187

has a solution ξ(t) for all t ∈ [0, 1]. Then Z 1 ˙ dt = Dhx ξ(1) − Dhx ξ(0) = Dhx ξ(1) v0 = D2 hx ξ(t) · ξ(t) 0

and thus Dhx is surjective. −1 Since h is hyperconvex, the inverse D2 hx |ξ(t) exists, thus (2.8) can be rewritten as ˙ = D2 hx |ξ(t) −1 · v0 , ξ(0) = 0 . ξ(t) (2.9) Thus (2.9) is of the form ξ˙ = F (ξ), where F is locally Lipschitz. Therefore for any v0 ∈ Tx M , (2.9) has a solution, which either exists for all t ≥ 0 or becomes infinite for a finite value of t. In order to exclude that the curve ξ reaches infinity for some t < 1, we need the hyperconvexity of h. We choose a norm k.kTx∗ M on Tx∗ M and denote by k.kTx M the norm on Tx M , which is induced by duality. Since for fixed η ∈ Tx∗ M = Tξ (Tx∗ M ) the second derivative D2 hx |ξ (η) can be regarded as an element of Tx M , it follows by the hyperconvexity of h that there exists a constant α > 0 such that for all ξ ∈ Tx∗ M kD2 hx |ξ (η)kTx M = sup

µ∈Tx∗ M

|D2 hx |ξ (η, µ)| |D2 hx |ξ (η, η)| ≥ ≥ αkηkTx∗ M , kµkTx∗ M kηkTx∗ M η ∈ Tξ (Tx∗ M )

and therefore

−1

kvkTx M = D2 hx |ξ D2 hx |ξ (v)

Tx M

−1

≥ α D2 hx |ξ (v)

Tx∗ M

(2.10)

,

v ∈ Tx M . (2.11) (2.9) together with (2.11) yields

−1

2

˙ ∗ kξ(t)k = D h | (v )

Tx M x ξ(t) 0

Tx∗ M

≤

1 kv0 k . α

Therefore the curve ξ(t) exists for all t ∈ [0, 1] and DF h(ξ(1)) = v0 .

(2.12) 2

For any hyperregular Hamilton function h ∈ C ∞ (T ∗ M ), we define the energy function Eh on T M by −1 Eh (x, v) := h ◦ (DF h) (x, v) = h x, ξh (x, v) (2.13) and the action Ah : T M → R ,

Ah (x, v) := (DF h)

−1

(x, v) · v = ξh (x, v) · v .

(2.14)

Then the Lagrange function Lh : T M → R

defined by

Lh (x, v) = Ah (x, v) − Eh (x, v)

(2.15)

(the Legendre transform of h) is hyperregular on T M and DF Lh (x, v) = (DF h)

−1

(x, v)

(2.16)

1188



(by Theorem 3.6.9 in [2], the hyperregular Lagrange functions on T M and the hyperregular Hamilton functions on T ∗ M are in bijection). In particular, by (2.14) and (2.16), Ah (x, v) = DF Lh (x, v) · v .

(2.17)

Definition 2.10. For a smooth manifold M , a C ∞ -function h : T ∗ M → R and E ∈ R, we define the singular set Sh (E) by Sh (E) := x ∈ M | h(x, 0) ≥ E . f := M \ Sh (E) is again a Since h(., 0) is continuous, Sh (E) is closed. Thus M smooth manifold. For h hyperregular, we shall now introduce an associated Finsler function f. on M Theorem 2.11. Let M and h ∈ C ∞ (T ∗ M ) satisfy Hypothesis 1.4 and let E, Sh (E) f be as described in Definition 2.10. and M f → TM f, i) Then there exists a strictly fibre preserving C ∞ -function τE : S M which is uniquely determined by the condition −1

h ◦ (DF h)

◦ τE = E .

(2.18)

f → TM f and let `h,E : T M f → R be defined by ii) Let τ˜E := τE ◦ πS : T M −1

`h,E (x, v) := (DF h)

◦ τ˜E (x, v) · v .

f. Then `h,E is an absolute homogeneous Finsler function on M f iii) For any regular curve γ : [a, b] → M , there exists a unique C 1 -function λ : [a, b] → R+ such that τ˜E γ(t), γ(t) ˙ = γ(t), λ(t)γ(t) ˙ . (2.19) f = M \ S(E), we call (M, `h,E ) a Remark 2.12. (a) Since `h,E is defined on M Finsler manifold with singularities. f to M by setting `h,E (x, v) = 0 for (b) If we continuously extend `h,E from M x ∈ S(E), the associated distance d` is well defined on all of M . Nevertheless contrary to the case of a Finsler manifold without singularities (as described for example in Bao–Chern–Shen [5]), the geodesic curves with respect to `h,E may have kinks at the “singular points”, which are the connected components of Sh (E). (c) Geometrically, the function τ˜E projects an element (x, v) of the tangent f to an element (x, λv) in the (2d − 1)-dimensional submanifold bundle T M −1 E = Eh (E).


1189

(d) Schematically the functions occurring in Theorem 2.11 are illustrated in the following diagram.

''' ' ' '' ' ' ' f w E × T Mf T M [][ [ [ u [[ `h,E

x

'') ''' wh

x

(πS ,1)

−1

Ru

f (E) × Tx M

τE ×1

f × Tx M f Sx M

w (˜v, v)

(˜ τE ,1)

v

(DF h)−1 ×1

w (ξ (˜v), v) h

(e) With the notation (2.3), `h,E (x, v) can be written as `h,E (x, v) = ξh (x, v˜) · v

where

(x, v˜) = τ˜E (x, v) ∈ E .

(2.20)

To prove Theorem 2.11, we need the following lemma. f and u ∈ Tx M f \ {0}. Then Lemma 2.13. In the setting of Theorem 2.11 fix x ∈ M for Eh : T M → R defined by (2.13), the function Eu : [0, ∞) → R , is strictly increasing with limλ→∞ Eu (λ) = ∞.

d dλ Eu

Eu (λ) := Eh (x, λu)

> 0 for λ > 0. Furthermore Eu (0) ≤ E and

Proof of Lemma 2.13. Since hx is even, Dhx (0) = 0, thus vh (x, 0) = 0, ξh (x, 0) = 0 and Eu (0) = Eh (x, 0) = h(x, 0) < E . (2.21) To show that Eu is strictly increasing, we will analyze the derivative of Eu for λ > 0. By definition DF h(x, ξ) = Dhx (ξ), thus dEu −1 |λ = Dhx |(Dhx )−1 (λu) · D (Dhx ) |λu (u) . dλ

(2.22)

Dhx |(Dhx )−1 (λu) = λu

(2.23)

We notice that and for Lh defined in (2.15) it follows from (2.16) that −1

D (Dhx )

|λu (u) = D (DLh,x ) |λu (u) .

(2.24)

1190



Inserting (2.24) and (2.23) in (2.22) yields dEu |λ = λu · D (DLh,x ) |λu (u) = λ D2 Lh,x |λu (u, u) , (2.25) dλ where we identify linear maps from Tx M to Tx∗ M with bilinear forms on Tx M . If h is strictly convex in each fibre, the same is true for Lh . Therefore by (2.25) the first derivative of Eu is strictly positive for λ ∈ (0, ∞) and thus Eu is strictly increasing. The fact that limλ→∞ Eu (λ) = ∞ can be seen as follows. From the strict convexity of h and since h(x, ξ) ≥ h(x, 0) for all ξ ∈ T ∗ M , it follows that lim|ξ|→∞ h(x, ξ) = ∞. Since h is hyperregular, DF h(x, .) = Dhx : Tx M → Tx∗ M is a global diffeomorphism. Thus for any norm k.kTx M on Tx M and the induced norm k.kTx∗ M on Tx∗ M , we have kDhx (vn )kTx∗ M → ∞ for any sequence (vn ) in Tx M satisfying kvn kTx M → ∞ (any global diffeomorphism is proper). Thus 2 lim Eu (λ) = lim h x, ξh (x, λu) = lim h(x, ξ) = ∞ . λ→∞

kξk→∞

λ→∞

f, each Proof of Theorem 2.11. i) From Lemma 2.13 it follows that for fixed x ∈ M −1 f f ray [x, u] ∈ S M intersects the Ex := Eh (E) ∩ Tx M in exactly one point v, i.e. f such that Eh (x, v) = E. for each ray [x, u] there is exactly one point (x, v) ∈ Tx M Thus [x, u] 7→ (x, v) defines a map τE , which is uniquely determined by (2.18). Clearly τE is strictly fibre preserving. To analyze the regularity of τE , we will use the Implicit Function Theorem. f \ {0} → S M f × R+ , such We remark that there exists an isomorphism ψ : T M −1 f \ {0}. Setting E bh := Eh ◦ ψ −1 and τbE := that ψ ({[x, u]} × R+ ) = [x, u] ⊂ T M f, for some λ : S M f → R+ and therefore ψ ◦ τE , we get τbE (s) = (s, λ(s)), s ∈ S M b d E h bh (s, λ(s)) = E. By Lemma 2.13, f E dλ (s0 , λ0 ) > 0 for all (s0 , λ0 ) ∈ S M × R+ . Smoothness of λ and thus of τE now follows from the Implicit Function Theorem. iii) By i), for t ∈ [a, b] fixed, there is a unique λ(t) ∈ R+ with (2.19). Since τE is smooth (as composition of smooth maps), t 7→ (γ(t), λ(t)γ(t)) ˙ is a C 1 -curve. 1 Thus λ ∈ C ([a, b], R). f → R is a Finsler function on M f, we check the ii) To show that `h,E : T M defining properties. f \ {0}) follows from the fact that h is hyper1) The regularity `h,E ∈ C ∞ (T M regular and the function τ˜E is C ∞ . 2) To show `h,E (x, λv) = λ`h,E (x, v), (λ > 0), we notice that by construction −1 τ˜E (x, λv) = τ˜E (x, v) for any λ > 0. Thus (DF h) ◦ τ˜E is homogeneous of order zero in each fibre. Since ξ · v is bilinear, it follows that −1

`h,E (x, λv) = (DF h)

◦ τ˜E (x, λv) · λv = λ`h,E (x, v) .

3) We start showing that `h,E (x, v) > 0, (v 6= 0). To this end we define ah = Ah ◦ DF h : T ∗ M → R ,

ah (x, ξ) = ξ · DF h(x, ξ) .


1191

Since h was assumed to be strictly convex in each fibre, one obtains from (2.6) (ξ − η) · DF h(x, ξ) − DF h(x, η) > 0 , ξ, η ∈ Tx∗ M , η 6= ξ . Therefore choosing ξ = −η and using that h is even in each fibre (thus DF h is odd) yields 2ξ · DF h(x, ξ) − DF h(x, −ξ) = 4ξ · DF h(x, ξ) = 4ah (x, ξ) > 0 , for

ξ 6= 0 . (2.26)

Since h is even and strictly convex, it takes its absolute minimum at ξ = 0 and thus DF h(x, 0) = 0. Since furthermore DF h is a global diffeomorphism, −1 we get (DF h) (x, v) 6= 0 for v 6= 0. By (2.26) −1 Ah (x, v) = ah x, (DF h) (x, v) > 0 , v 6= 0 . (2.27) Setting τ˜E (x, v) =: (x, v˜), it follows from the fact that τE is strictly fibre preserving that there exists a λ > 0 such that v = λ˜ v . Thus by (2.27) for v 6= 0 −1 `h,E (x, v) = (DF h) (x, v˜) · λ˜ v = λAh (x, v˜) > 0 . (2.28) To show that the matrix g is positive definite, we set `x (v) := `h,E (x, v) : f → R. Then we have to show that g(x,v) (w, w) = D2 ( 1 `2x )|v (w, w) > 0 Tx M 2 f \ {0}. for all v, w ∈ Tx M We first remark that g(x,v) (w, w) = T1 (w) + T2 (w)

where

2

T1 (w) = `x (v)D `x |v (w, w)

(2.29)

and T2 (w) = D`x |v (w)

2

.

By the definition of `x and (2.16) it follows that −1 D`x |v (w) = D2 Lx |τ˜E (v) D˜ τE |v (w), v + (Dhx ) ◦ τ˜E (v) · w .

(2.30)

To analyze D˜ τE , we use that by Lemma 2.13 the function Eh,x is strictly increasing in each fibre. Therefore, analogue to the proof of i), there exists a smooth function µx : Tx M → (0, ∞) such that τ˜E (v) = µx (v)v and thus D˜ τE |v (w) = v · Dµ|v (w) + µ(v)w .

(2.31)

Since E = Eh,x (˜ τE )(v) = Eh,x ◦ Gx v, µ(v)

where

Gx v, µ(v) := µ(v)v

we have Dv (Eh,x ◦ Gx )(v, µ(v)) = 0, leading to −1 Dµ|v (w) = − D2 (Eh,x ◦ Gx )|(v,µ(v)) D1 (Eh,x ◦ Gx )|(v,µ(v)) (w) .

(2.32)

Since by the definition of Eh in (2.13) and again by (2.16) we have DEh,x |v (w) = Dhx |(Dhx )−1 (v) · D(Dhx )−1 |v (w) = D2 L|v (v, w) ,

(2.33)

it follows that D2 (Eh,x ◦ Gx )(v, µ) = DEh,x |µv (v) = µD2 L|v (v, v) .

(2.34)

1192



By (2.33) it follows at once that D1 (Eh,x ◦ Gx )|(v,µ(v)) (w) = DEh,x |µ(v)v D1 Gx |(µ(v),v) (w) = µ2 (v)D2 Lx |µ(v)v (v, w) .

(2.35)

Since L is strictly convex, the symmetric bilinear form D2 Lx |µ(v)v (., .) is positive definite and therefore defines a scalar product hv, wiL := D2 Lx |µ(v)v (v, w)

with associated norm

kvkL .

(2.36)

Inserting (2.35) and (2.34) in (2.32) and using (2.36) gives Dµ|v (w) = −

µ(v) hv, wiL . kvk2L

(2.37)

Thus inserting (2.37) in (2.31) and the resulting term in (2.30) yields µ(v) −1 D`x |v (w) = −v hv, wi + µ(v)w ,v + (Dhx ) |µ(v)v · w L kvk2L L = (Dhx )

−1

|µ(v)v · w .

(2.38)

Using (2.38) and (2.16) we get

D2 `x |v (w, w) = (vDµ|v (w) + µ(v)w) , w L and inserting (2.32) gives (hv, wiL )2 D2 `x |v (w, w) = µ(v) hw, wiL − ≥ 0, kvk2L

(2.39)

where the last estimate follows from the Cauchy–Schwarz Inequality. Since T2 is quadratic, (2.39) together with (2.28) gives 1 2 f. D2 `x |v (w, w) ≥ 0 , v, w ∈ Tx M 2 f and v ∈ Tx M f\{0}. Assuming To prove the strict positivity, we now fix x ∈ M g(x,v) (w, w) = 0

f, for w ∈ Tx M

(2.40)

we have to show that w = 0. By (2.29) it follows from (2.40) that T1 (w) = 0 and T2 (w) = 0. We have already seen in (2.28) that `h,E (x, v) > 0 for v 6= 0, thus T1 (w) = 0 implies g(x,v) (w, w) = 0, leading by (2.39) and the Cauchy– Schwarz-inequality to w = ηv

for some

η ∈ R.

Inserting this in T2 , the homogeneity of `x shows 0 = D`x |v (ηv) = ηD`x |v (v) = η`x (v) and thus by the positivity of `x we get η = 0 and thus w = 0.


1193

4) It remains to show that `h,E is absolute homogeneous of order one. Since h −1 is even in each fibre, DF h and (DF h) are odd. Thus for (x, v) ∈ Ex −1

h ◦ (DF h)

(x, −v) = h ◦ (DF h)

−1

(x, v) = E

(2.41)

and (2.41) yields (x, v) ∈ E

=⇒

(x, −v) ∈ E .

(2.42)

f and some λ > 0, Since τE is strictly fibre preserving, we have for (x, v) ∈ Tx M using (2.42) τ˜E (−v) = λ(−v) = −˜ τE (v) . (2.43) −1

By the fact that (DF h) `h,E (−v) = (DF h)

−1

is odd and (2.43) we can conclude that

◦ τ˜E (−v) · (−v) = (DF h)

−1

◦ τ˜E (v) · v = `h,E (v) . (2.44)

By (2.44), `h,E is even in each fibre and thus for any λ ∈ R 2

`h,E (x, λv) = `h,E (x, |λ|v) = |λ|`h,E (x, v) .

2.3. Proof of Theorem 1.4 Step 4 of our proof is adapted from Abraham–Marsden [2] and uses the Maupertuis principle (at least implicitly). Step 1: We will show that Γ x1 , x2 , [a, b], E :=

(γ, α) | α : [a, b] → R

is

C 2,

dα > 0, dt

γ ∈ Γ0,α(b) (x1 , x2 ) such that Eh γ α(t) , γ˙ α(t) = E for all

α(a) = 0 ,

t ∈ [a, b]

, (2.45)

is a Banach manifold, where Γa,b (x1 , x2 ) was introduced in (2.1). Γ(x1 , x2 , [a, b], E) is the set of all pairs (γ, α), where γ is a regular curve on f M joining the points x1 and x2 and α is a change of parameter, ensuring that the d curve (γ ◦ α, γ˙ ◦ α) ∈ T M (which is not equal to the lifted curve (γ ◦ α, dt (γ ◦ α))) −1 lies on the energy shell E = Eh (E). Set A := {α : [a, b] → R | dα dt > 0 and α(a) = 0} and denote by Γ0,∞ the f. Then Γ0,∞ × A is a Banach manifold. space of all regular curves γ : [0, ∞) → M 1 We consider the C -mapping f×M f , (γ, α) 7→ γ ◦ α(a), γ ◦ α(b) . g : Γ0,∞ × A → M f×M f is a regular value of g and Then (x1 , x2 ) ∈ M n o Γ [a, b], x1 , x2 := g −1 (x1 , x2 ) = (γ, α) ∈ Γ0,∞ ×A | γ α(a) = x1 , γ α(b) = x2

1194



is a submanifold of Γ0,∞ × A. This follows from the fact that the Inverse Function Theorem holds in Banach manifolds (see Hamilton [13]). We introduce ˜h : C 1 [a, b], T M f → C 1 [a, b], R , E ˜h (η)(t) := Eh η(t) E and for n o f f | γ ∈ Γa,b (x1 , x2 ), k ∈ C 1 [a, b], R+ ΓTa,bM (x1 , x2 ) := (γ, k γ) ˙ ∈ C 1 [a, b], T M we set f Φ : Γ [a, b], x1 , x2 → ΓTa,bM (x1 , x2 ) ,

(γ, α) 7→ (γ ◦ α, γ˙ ◦ α) .

Then Φ is a diffeomorphism. In fact, by a straightforward it is bijective, R t calculation −1 with inverse Φ−1 (η, k η) ˙ = (η ◦ α−1 , α), where α(t) = a (k(s)) ds. Identifying E with the constant function E(t) = E, we obtain Γ(x1 , x2 , ˜h ◦ Φ. To show that E is a regular value of f it is [a, b], E) = f −1 (E) for f := E ˜h , i.e., that for each v ∈ C 1 ([a, b], R) sufficient to show that it is a regular value of E 1 (considered as a vector field along E ∈ C ([a, b], R)), there is a vector field X along ˜ −1 (E) with dE ˜h |η X = v. Note that η(t) = (x(t), v(t)) with v(t) 6= 0. η∈E h Since DEh (x, v) 6= 0 for v 6= 0, there is a covering of [a, b] by open intervals Ij , j ∈ J and vector fields Xj along η|Ij with DEh |η(t) Xj (t) = v(t) for all t ∈ Ij . Choosing a partition of unity (χj ) subordinate to (Ij ), we set P ˜h |η X = v. Thus E is a regular value of f and X(t) = j∈J χj (t)Xj (t). Then dE Γ([a, b], x1 , x2 , E) is a Banach manifold. Step 2: We construct a diffeomorphism bE : Γa,b (x1 , x2 ) → Γ(x1 , x2 , [a, b], E). By Theorem 2.11, there exists for any η ∈ Γa,b (x1 , x2 ) a unique C 1 -function λ : [a, b] → R+ such that Eh η(t), λ(t)η(t) ˙ =E. Set Z

t

1 f, ds and γ = η ◦ α−1 : 0, α(b) → M λ(s) a then α : [a, b] → R with α˙ > 0. From η(t) ˙ = γ(α(t)) ˙ · α(t) ˙ it follows that Eh γ α(t) , γ˙ α(t) = Eh η(t), λ(t)η(t) ˙ =E, α(t) :=

(2.46)

(2.47)

i.e. (γ, α) ∈ Γ(x1 , x2 , [a, b], E). We can conclude that there is a bijection between Banach manifolds given by bE : Γa,b (x1 , x2 ) → Γ x1 , x2 , [a, b], E bE (η) = (η ◦ α−1 , α) with Z t −1 α(t) := λ(s) ds for τ˜E (η, η) ˙ = (η, λη) ˙ . (2.48) a


1195

On the other hand, if we start with (γ, α) ∈ Γ(x1 , x2 , [a, b], E), then Eh (γ(s), γ(s)) ˙ f = E with s = α(t). Setting η := γ ◦ α : [a, b] → M it follows from γ(s) ˙ = −1 η(t) ˙ (α(t)) ˙ that −1 −1 Eh η(t), α(t) ˙ η(t) ˙ = E and thus λ(t) = α(t) ˙ . (2.49) −1 Thus the inverse function b−1 E is given by bE (γ, α) = γ ◦ α.

Step 3: We show that the critical points of the length functional s`h defined in Definition 2.3 (i.e., the geodesics of `h ) are in bijection with the critical points of the action integral Z α(b) I : Γ x1 , x2 , [a, b], E → R , I(γ, α) := Ah γ(s), γ(s) ˙ ds , (2.50) α(a)

where Ah denotes the action with respect to h defined in (2.14). Setting s = α(t) and using (2.3) gives Z α(b) Z b Ah γ(s), γ(s) ˙ ds = ξh γ α(t) , γ˙ α(t) · γ˙ α(t) α(t) ˙ dt . (2.51) α(a)

a

Setting η(t) = γ(α(t)) and using (2.49) and the definition of `h and s`h , we obtain from (2.50) and (2.51) Z b Z b −1 I(γ, α) = ξh η(t), η(t) ˙ α(t) ˙ · η(t) ˙ dt = ξh ◦ τ˜E η(t), η(t) ˙ · η(t) ˙ dt a

Z =

a b

`h,E

η(t), η(t) ˙ dt = s`h (η) .

a

Since bE (γ ◦ α) = (γ, α), it follows that s`h = I ◦ bE

ds`h |η = dI|bE (η) ◦ dbE |η .

(2.52)

dI|bE (η) = 0 η ∈ Γa,b (x1 , x2 ) .

(2.53)

and thus

Since bE is a diffeomorphism, we get ds`h |η = 0

⇐⇒

Step 4: We show (a). We set γ0 (a) = x1 , γ0 (b) = x2 . If γ0 is a base integral curve of the Hamiltonian vector field Xh with Eh (γ0 (t), γ˙ 0 (t)) = E for all t ∈ [a, b], then bE (γ0 ) = (γ0 , 1), where 1 : [a, b] → [a, b] is defined by 1(t) = t. Thus by (2.53) it remains to show that dI|(γ0 ,1) = 0 for any base integral curve γ0 ∈ Γa,b (x1 , x2 ) of the Hamiltonian vector field Xh with energy E. The tangent space of Γ(x1 , x2 , [a, b], E) at a point (γ, α) can be described by use of

1196



variations as n T(γ,α) Γ x1 , x2 , [a, b], E = ∂u (γ, α)δ |u=0 | (γ, α)δ : (−δ, δ) → Γ x1 , x2 , [a, b], E o is C 2 with (γ, α)δ (0) = (γ, α) . (2.54) We start analyzing dI|(γ,α) . We use the notation (γ( . ), α( . ))δ (u) =: (γδ ( . , u), αδ ( . , u)). From (2.54) and (2.45) it follows that αδ (a, u) = 0. Furthermore γδ (0, u) = x1 and γδ (αδ (b, u), u) = x2 for all u ∈ (−δ, δ). This leads to d d γδ (0, u) = 0 = γδ αδ (b, u), u . (2.55) du du Since Ah = Lh +Eh by the definition (2.15) of the Lagrange function Lh , it follows from the definition (2.45) of Γ(x1 , x2 , [a, b], E) that Ah γδ αδ (t, u), u , γ˙ δ αδ (t, u), u = Lh γδ αδ (t, u), u , γ˙ δ αδ (t, u), u + E , (2.56) thus the definition (2.50) of I and (2.56) yield dI|(γ,α) ∂u (γ, α)δ |u=0 = ∂u I((γδ , αδ ))|u=0 Z αδ (b,u) d = Lh γδ (s, u), γ˙ δ (s, u) + E ds du αδ (a,u)

. u=0

(2.57) We get using γδ (t, 0) = γ(t) and αδ (t, 0) = α(t) Z αδ (b,u) d Lh γδ (s, u), γ˙ δ (s, u) + E ds du αδ (a,u) u=0 " #b = Lh γ α(t) , γ˙ α(t) + E · ∂u αδ |u=0 (t) a

Z + 0

α(b)

d Lh γδ (s, u), γ˙ δ (s, u) ds . (2.58) du u=0

For the integrand on the right hand side of (2.58) we get d Lh γδ (s, u), γ˙ δ (s, u) = Dγ Lh γ(s), γ(s) ˙ · ∂u γδ |u=0 (s) du u=0 + Dγ˙ Lh γ(s), γ(s) ˙ ∂u γ˙ δ |u=0 (s) .

(2.59)

Since ∂u γ˙ δ |u=0 (s) = ∂s ∂u γδ |u=0 (s), integration by parts for the second summand on the right hand side of (2.59) gives Z α(b) h iα(b) d Lh γδ (s, u), γ˙ δ (s, u) ds = Dγ˙ Lh γ(s), γ(s) ˙ · ∂u γδ (s, u)|u=0 0 α(a) du u=0 Z α(b) d − Dγ Lh γ(s), γ(s) ˙ + Dγ˙ Lh γ(s), γ(s) ˙ · ∂u γδ |u=0 (s) ds . (2.60) ds 0


1197

It follows from (2.55) that ∂u γδ α(t), u |u=0 = −γ˙ α(t) ∂u αδ |u=0 (t) ,

t = a, b .

(2.61)

Since by (2.17) we have Ah (γ, γ) ˙ = Dγ˙ Lh (γ, γ) ˙ · γ, ˙ we get by (2.56) b − Dγ˙ Lh γ α(t) , γ˙ α(t) · γ˙ α(t) ∂u αδ |u=0 (t) a

" =−

Lh

#b γ α(t) , γ˙ α(t) + E · ∂u αδ |u=0 (t) . (2.62) a

Using (2.61) and (2.62), the boundary terms on the right hand side of (2.58) and (2.60) cancel. Combining (2.57), (2.58) and (2.60) yields dI|(γ,α) ∂u (γ, α)δ |u=0 Z α(b) d = Dγ Lh γ(s), γ(s) ˙ − Dγ˙ Lh γ(s), γ(s) ˙ · ∂u γδ |u=0 (s) ds . (2.63) ds 0 For (γ, α) = (γ0 , 1), the integrand is zero, since the integral curve (γ0 , γ˙ 0 ) of Xh solves Lagranges equation and thus dI|(γ0 ,1) = 0 . Step 5: We show (b). If γ0 is a Finslerian geodesic with energy E, then bE (γ0 ) = (γ0 , 1) and by (2.53) the integral (2.63) is zero for each tangent vector ∂u γ0,δ |u=0 . By standard arguments it follows that (γ0 , γ˙ 0 ) solves Lagranges equation. Thus γ0 is a base integral curve of Xh . 2 2.4. Application to Hε We start with Proof of Lemma 1.2. (a): These estimates and the regularity follow at once by Hypothesis 1.1, (a), (i) and (iii). (b): By standard Fourier theory, t0 is even with respect to ξ, i.e. t0 (x, ξ) = t0 (x, −ξ), if and only if for all η ∈ Zd a ˜η (x) = a ˜−η (x) .

(2.64) d

To show (2.64) we use that by Hypothesis 1.1, (a), (iv) for all η ∈ Z , x ∈ Rd and ε ∈ (0, 1] (1)

(2)

(2) a ˜η (x)+ε a(1) ˜−η (x+εη)+ε a−εη (x+εη)+R−εη (x+εη, ε) . (2.65) εη (x)+Rεη (x, ε) = a

By Hypothesis 1.1, (a), (i) we have a ˜−η (x + εη) = a ˜−η (x) + Oη (ε) .

(2.66)

1198



Combining (2.65) and (2.66) leads to (1)

|˜ aη (x) − a ˜−η (x)| ≤ ε |a−εη (x + εη) − a(1) εη (x)| + Oη (ε) (2)

(2) + |R−εη (x + εη, ε) − Rεη (x, ε)|

≤ εCx,η + ε2 Cx,η .

(2.67)

Since the left hand side of (2.67) is independent of ε, it is equal to zero and (2.64) follows. The analytic continuation of t0 follows at once from Hypothesis 1.1,(a),(v), (0) since aγ (x) are the Fourier-coefficients of t0 (x, ξ). P (c): By (a), t0 (x, ξ) = a ˜ (x) cos(η · ξ), thus its Taylor-expansion at η∈Zd η ξ = 0 yields by Hypothesis 1.1(a)(ii) X

1 a ˜η (x) 1 − (η · ξ)2 + O |ξ|4 = ξ , B(x)ξ + O |ξ|4 , (2.68) 2 d η∈Z

where the symmetric d × d-matrix B is given by 1X − a ˜η (x)ην ηµ = Bνµ (x) for µ, ν ∈ {1, . . . , d} , 2 η

x ∈ Rd .

(2.69)

P (0) Since hξ , B(x)ξi = − 2ε12 γ aγ (x)(ξ · γ)2 , by Hypothesis 1.1, (a), (iii) and (vi) the matrix B is positive definite. d i (d): First we mention that by a short calculation OpTε (e− ε γ · ξ ) = τγ . This P d implies Tε = γ aγ τγ = OpTε (t) as operator on u ∈ K((εZ)d ) for t given in (1.5). Boundedness: For u ∈ `2 ((εZ)d ), by the Cauchy–Schwarz inequality the l2 -norm of Tε u can be estimated as  2 X X  kTε uk2`2 ≤ |aγ (x, ε)u(x + γ)| x∈(εZ)d

γ∈(εZ)d

 ≤

X

X 

x∈(εZ)d

|aγ (x, ε)|2

γ∈(εZ)d

|γ| ε

d+1

 12 

 12 X |γ| −(d+1) × |u(x + γ)|2  . ε d 

(2.70)

γ∈(εZ)

By (1.9), the first factor on the right hand side of (2.70) is bounded uniformly in x. Thus X X kTε uk2`2 ≤ C |η|−(d+1) |u(x + εη)|2 ≤ Ckuk2`2 . η∈Zd

x∈(εZ)d

Thus Tε is a bounded operator on `2 ((εZ)d ).


1199

Symmetry: Since Tε is bounded, it is symmetric if and only if for any x, γ ∈ (εZ)d hTε δx , δx+γ i`2 = hδx , Tε δx+γ i`2 ,

(2.71)

where δx (y) := δxy . Since the left hand side of (2.71) is equal to a−γ (x + γ, ε) and the right hand side equals aγ (x, ε), the statement follows by Hypothesis 1.1(a)(iv). Boundedness from below: For u ∈ K((εZ)d ), we write hu , Tε ui`2 = A[u] + B[u] where    X  (0) X A[u] := a0 (x)|u(x)|2 + a(0) u(x) γ (x)u(x + γ)¯   γ6=0 x∈(εZ)d X (2) B[u] := ε a(1) ¯(x)u(x + γ) . γ (x) + Rγ (x, ε) u

(2.72)

x,γ∈(εZ)d

Then by the exponential decay of a(1) and R(2) with respect to γ (Hypothesis 1.1(a)(v)) X (2) |B[u]| ≤ u(x)|2 + |u(x + γ)|2 ≤ εCkuk2 . ε a(1) γ (x) + Rγ (x, ε) |¯ x,γ∈(εZ)d

(2.73) By Hypothesis 1.1(iii) we have XX A[u] = a(0) u(x) − |u(x)|2 γ (x) u(x + γ)¯

(2.74)

x γ6=0

 1 X (0) = aγ (x) u(x + γ)¯ u(x) − |u(x)|2  2 x γ6=0

+

X

(0)

a−˜γ (˜ x + γ˜ ) u(˜ x)¯ u(˜ x + γ˜ ) − |u(˜ x + γ˜ )|2

x ˜ γ ˜ 6=0

=−

    

1 X (0) 2 a (x) |u(x) − u(x + γ)| ≥ 0 , 2 x γ γ6=0

where for the second step we used the symmetry of Tε and the substitution x ˜ = x+ γ and γ˜ = −γ and the last estimate follows from Hypothesis 1.1(a)(iii). Inserting (2.73) and (2.74) in (2.72) gives the stated result. 2 Definition 2.1 and Theorem 2.11 allow to define a metric adapted to the Hamilton operator Hε as follows. ˜ 0 : R2d ∼ Proposition 2.14. The Hamilton function h = T ∗ Rd → R defined in (1.12) is hyperconvex in each fibre.

1200



Proof. We have to show that there exists a constant α > 0 such that D E ˜ 0 (x, ξ)v ≥ αkvk2 for all x, ξ, v ∈ Rd . v , Dξ2 h

(2.75)

˜ 0 and a(0) For simplicity of notation, we will skip the x-dependence of h γ . We have for a ˜ defined in (1.11) D E X ˜ 0 (ξ)v = − v , Dξ2 h a ˜η (γ · v)2 cosh(γ · ξ) , ξ, v ∈ Rd . (2.76) η∈Zd

By Hypothesis 1.1,(vi), we can choose a basis {η 1 , . . . , η d }, η j ∈ Zd of Rd with a ˜ηi < 0. Since by Hypothesis 1.1(a)(iv) each summand in (2.76) has positive sign D

d E X ˜ 0 (ξ)v ≥ − v , Dξ2 h a ˜ηk (η k · v)2 cosh(η k · ξ) ,

ξ, v ∈ Rd .

(2.77)

k=1

We have C = mink −˜ aηk > 0, thus (2.77) yields D

E

˜ 0 (ξ)v ≥ C v , Dξ2 h

d X

! k

2

(η · v) = hv , M vi ≥ 0 ,

for M =

k=1

C

X

ηik ηjk

.

k

˜0 The sum can take the value 0 only if v = 0 since {η k } is a basis of Rd . Thus h is hyperconvex (the lowest eigenvalue of M gives the lower bound for its second derivative). 2 Proposition 2.14 leads by Proposition 2.9 and Lemma 1.2 to the following corollary. ˜ 0 : R2d → R defined in (1.12) is hyperCorollary 2.15. The Hamilton function h regular and even and strictly convex in each fibre. ˜0 = In the setting of Theorem 2.11, we choose M = Rd , E = 0 and h = h t˜0 − V0 . Recall that by Hypothesis 1.1, the set S(0) of singular points with respect to the energy E = 0 is given by S(0) = {0}. ˜ 0 given in (1.12), we Definition 2.16. For the hyperregular Hamilton function h define ( f := Rd \ {0}, v ∈ Tx M f `˜ (x, v) , x ∈ M `(x, v) := h0 ,0 (2.78) 0 x = 0. The associated Finsler metric d` : Rd × Rd → [0, ∞) is given by Z 1 d` (x0 , x1 ) = inf ` γ(t), γ(t) ˙ dt . γ0,1 ∈Γ(x0 ,x1 )

(2.79)

0

We notice that it follows from the Definition of τ˜0 that limx→0 τ˜0 (x, v) = (0, 0). Thus ` : R2d → R defined in (2.78) is continuous.


1201

2.5. Proof of Theorem 1.5 In order to prove Theorem 1.6, we notice that if d` is locally Lipschitz continuous, it is differentiable almost everywhere in both arguments (Rademacher Theorem). Step 1: We prove (1.17). By the triangle inequality and the definition of d` (x, y), we have for any v ∈ Rd with |v| = 1 and δ > 0 Z 1 d0 (x + δv) − d0 (x) ≤ d` (x, x + δv) ≤ ` γ0 (t), γ˙ 0 (t) dt , (2.80) 0

where γ0 (t) = x + tδv. For this special curve we get by the homogeneity of ` Z 1 ` γ0 (t), γ˙ 0 (t) dt ≤ sup `(x + tδv, δv) = δ sup `(x + tδv, v) , (2.81) t∈[0,1]

0

t∈[0,1]

where by a slight abuse of notation v is considered as an element of Tx+tδv Rd . Thus (2.80) together with (2.81) proves (1.17). Step 2: We prove (1.18). By (2.80) and (2.81) we have for any v ∈ Rd with |v| = 1 almost everywhere in x ∈ Rd d0 (x + δv) − d0 (x) ∇d0 (x) · v = ∂v d0 (x) = lim δ→0 δ ≤ lim sup `(x + tδv, v) = `(x, v) . (2.82) δ→0 t∈[0,1]

Note that ∇d0 (x) can be considered as an element of Tx∗ Rd . Since both sides in (2.82) are positive homogeneous of order one with respect to v, we can extend the inequality to all v ∈ Rd . Using (2.20), the Finsler function ` can be written as `(x, v) = ξh˜ 0 (x, v˜) · v, where v is considered as an element of Tx M and will be written as (x, v). It follows from (2.82) that ξh˜ 0 (x, v˜) − ∇d0 (x) · v ≥ 0 , (x, v) ∈ T M a.e. on M . (2.83) ˜ 0 (x, ξ) is differentiable, real valued and convex in each fibre, by (2.4) the Since h inequality ˜ 0 (x, ξ) ≥ h ˜ 0 (x, η) + Dη h ˜ 0 (x, η) · (ξ − η) h d holds for all x, ξ, η ∈ R . Thus by setting ξ = ξh˜ 0 (x, v˜) and η = ∇d0 (x), we get for all (x, v) ∈ T M the estimate ˜ 0 x, ξ˜ (x, v˜) ≥ h ˜ 0 x, ∇d0 (x) +Dξ h ˜ 0 x, ∇d0 (x) · ξ˜ (x, v˜)−∇d0 (x) , (2.84) h h0

h0

where (x, v˜) ∈ E is associated to (x, v). The left hand side of (2.84) is by definition ˜ 0 (x, ∇d0 (x)) in equation (2.84) yields of v˜ equal to zero. Choosing (x, v) = Dξ h ˜ 0 x, ∇d0 (x) + v · ξ˜ (x, v˜) − ∇d0 (x) . 0≥h h0 Using (2.83), this proves (1.18).

1202



Step 3: We prove (1.16) (the eikonal equality): We consider the generalized eikonal equation ˜ 0 x, ∇ϕ(x) = t˜0 x, ∇ϕ(x) − V0 (x) = 0 . h 2

(2.85) 3

Choose coordinates such that t0 (x0 , ξ) = |ξ| + O |ξ| and V0 (x) = 2 2 3 ∞ ν=1 λν xν + O |x| . It is proven in [22] that there exists a unique positive C function ϕ defined in a neighborhood Ω of 0, solving (2.85) such that ϕ has an expansion as asymptotic series Pd

ϕ(x) ∼

d X λν ν=1

2

x2ν +

X

ϕk (x) ,

x ∈ Ω,

(2.86)

k≥1

where each ϕk is an homogeneous polynomial of order k + 2. In particular, denote by Ft the flow of the Hamiltonian vector field Xh˜ 0 . Then the Local Stable Manifold Theorem [2] tells us that there is an open neighborhood N of (0, 0) such that the two submanifolds Λ± X˜ , (0, 0) := (x, ξ) ∈ T ∗ Rd Ft (x, ξ) → (0, 0) for t → ∓∞ (2.87) h0

exist and are unique in N . They are called stable (Λ− ) and unstable (Λ+ ) manifold of Xh˜ 0 of the critical point (0, 0). Moreover they are of dimension d and contained ˜ −1 (0). It is shown in [22] that Λ± are Lagrangian manifolds in T ∗ Rd and in h 0 that the outgoing manifold can be parametrized as Λ+ = {(x, ∇ϕ(x)) | x ∈ Ω}. Thus for a given x ∈ Ω there exists an integral curve γ b0 := (γ0 , ∇ϕ(γ0 )) ⊂ Λ+ of the Hamiltonian vector field Xh˜ 0 , parametrized by [−∞, 0] such that γ b0 (0) = (x, ∇ϕ(x)) and limt→−∞ γ b0 (t) = (0, 0). Since γ b0 is an integral curve of Xh˜ 0 , it follows from Hamilton’s equations that ˜ 0 γ0 , ∇ϕ(γ0 ) (γ0 , γ˙ 0 ) = DF h and therefore

˜0 DF h

−1

(γ0 , γ˙ 0 ) = γ0 , ∇ϕ(γ0 ) .

Thus

−1 d ˜0 ϕ ◦ γ0 = ∇ϕ|γ0 · γ˙ 0 = DF h (γ0 , γ˙ 0 ) · γ˙ 0 . (2.88) dt Since γ b0 is an integral curve, (γ0 (t), γ˙ 0 (t)) ∈ E for all t. Therefore τ˜0 (γ0 , γ˙ 0 ) = (γ0 , γ˙ 0 ) and it follows at once from (2.88) and the definition of ` that d ϕ ◦ γ0 = `(γ0 , γ˙0 ) . (2.89) dt The point x = 0 is a singular point of the Finsler manifold (Rd , `), thus the base integral curve γ0 : [−∞, 0] → Ω 3 0 of Xh˜ 0 is not a regular curve on a Finsler manifold in the sense of Definition 2.3. To avoid this difficulty, we restrict the curve γ0 to [−T, 0] and set yT := γ0 (−T ). Then by (2.89) Z 0 ϕ(x) − ϕ(yT ) = ` γ0 (t), γ˙ 0 (t) dt . (2.90) −T


1203

By Proposition 1.5 the base integral curve γ0 of Xh˜ 0 is a geodesic with respect to the associated Finsler function `. It is a basic theorem in Finsler Geometry (see Abate–Patrizio [1], Theorem 1.6.6), that geodesics, which are short enough, actually minimize the curve length among all C ∞ -curves (or C 2 -curves) with the same endpoints. Thus the length of any short geodesic joining x and y is for |x − y| sufficiently small equal to the Finsler distance d` (x, y) and ϕ(x) − ϕ(yT ) = d` (yT , x) .

(2.91)

Since yT → 0, T → ∞ and d` and ϕ are continuous in x = 0, we get ϕ(x) = d0 (x)

x∈Ω 2

for |x| sufficiently small.

3. Weighted estimates for Dirichlet eigenfunctions 3.1. Preliminary results Lemma 3.1. Assume Hypothesis 1.1 and, for Σ ⊂ Rd , let HεΣ denote the Dirichlet operator introduced in Definition 1.7. Let ϕ : Σ → R be Lipschitz and constant outside some bounded set. Then for any real valued v ∈ D(HεΣ )

D E ϕ ϕ ϕ e ε HεΣ e− ε v , v `2 = Vε + Vε,Σ v, v `2 2 1X X 1 − aγ (x, ε) cosh ϕ(x) − ϕ(x + γ) v(x) − v(x + γ) , 2 ε 0 x∈Σ γ∈Σε (x)

where Σ0ε (x) := {γ ∈ (εZ)d | x + γ ∈ Σ} and ϕ Vε,Σ (x) :=

X γ∈Σ0ε (x)

aγ (x, ε) cosh

1 ϕ(x) − ϕ(x + γ) ε

,

(3.1)

where the sum on the right hand side converges. Proof. By use of the symmetry of Tε (Lemma 1.2) and since v and ϕ are assumed ϕ to be real valued and e± ε v ∈ D(HεΣ ), we have

ϕ ϕ e ε 1Σε Tε 1Σε e− ε v , v `2

ϕ ϕ ϕ ϕ 1

= 1Σε Tε 1Σε e− ε v , e ε v `2 + e− ε v , 1Σε Tε 1Σε e ε v `2 2 1 1 1 X = aγ (x, ε) e ε (ϕ(x)−ϕ(x+γ)) + e− ε (ϕ(x)−ϕ(x+γ)) v(x + γ)v(x) . 2 x,x+γ∈Σ

1204



Writing v(x + γ)v(x) = v 2 (x) − 12 2v 2 (x) − 2v(x + γ)v(x) yields by the definition ϕ of Vε,Σ

ϕ ϕ e ε Tε e− ε v , v `2 D E 1 X 1 ϕ = Vε,Σ v , v 2 − aγ (x, ε) cosh ϕ(x) − ϕ(x + γ) 2 ε ` x,x+γ∈Σ × 2v 2 (x) − 2v(x)v(x + γ) . (3.2) By Hypothesis 1.1 we have aγ (x, ε) = a−γ (x+γ, ε). Thus by use of the substitutions x0 = x + γ and γ 0 = −γ together with the fact that cosh is even X 1 aγ (x, ε) cosh ϕ(x) − ϕ(x + γ) v 2 (x) ε x,x+γ∈Σ X 1 = a−γ 0 (x0 + γ 0 , ε) cosh ϕ(x0 + γ 0 ) − ϕ(x0 ) v 2 (x0 + γ 0 ) ε x0 ,x0 +γ 0 ∈Σ X 2 0 1 0 0 0 0 0 = aγ (x , ε) cosh ϕ(x ) − ϕ(x + γ ) v (x + γ 0 ) . (3.3) ε 0 0 0 x ,x +γ ∈Σ

Inserting (3.3) into (3.2) gives

ϕ ϕ e ε 1Σε Tε 1Σε e− ε v , v `2 D E 1 X ϕ = Vε,Σ v, v − 2 `2

aγ (x, ε) cosh

x,x+γ∈Σ

1 ϕ(x) − ϕ(x + γ) ε

× v 2 (x) − 2v(x)v(x + γ) + v 2 (x + γ) . ϕ

Since Vε commutes with e− ε , the stated equality follows. The convergence of the series in (3.1) follows from the decay of aγ (x, ε) with respect to γ (Hypothesis 1.1(v)) together with the assumptions on ϕ and the mean value theorem. 2 Lemma 3.1 leads to the following norm estimate, which will be used later on to prove Theorem 1.8. Lemma 3.2. Assume Hypothesis 1.1 and, for Σ ⊂ Rd , let HεΣ denote the Dirichlet operator introduced in Definition 1.7. For E ≥ 0 fixed, let F± : Σ → [0, ∞) be a pair of functions such that F (x) := F+ (x) + F− (x) > 0 and ϕ F+2 (x) − F−2 (x) = Vbε (x) + Vε,Σ (x) − E , ϕ Vε,Σ (x)

x ∈ Σ,

(3.4)

where is given in (3.1). Then for v ∈ D(HεΣ ) real-valued with F v ∈ `2Σε and ϕ : Σ → R Lipschitz and constant outside some bounded set, we have for some C>0

2

1 ϕ ϕ

+ 8kF− vk22 + Cεkvk2 . ε (H Σ − E)e− ε kF vk2`2 ≤ 4 e v (3.5) ε `

F

2 `


1205

Proof. First observe that kF vk2`2 ≤ 2 kF+ vk2`2 + kF− vk2`2 = 2 kF+ vk2`2 − kF− vk2`2 + 4kF− vk2`2 . (3.6) By (3.4) one has D E ϕ kF+ vk2`2 − kF− vk2`2 = (Vbε + Vε,Σ − E)v , v 2 .

(3.7)

`

Hypothesis 1.1(a)(iii) and (v) yields by straightforward calculation 2 1 X 1 − aγ (x, ε) cosh ϕ(x) − ϕ(x + γ) v(x) − v(x + γ) ≥ −Cεkvk2 , 2 ε x,x+γ∈Σ

(3.8) since |ϕ(x + γ) − ϕ(x)| ≤ |γ| supy∈K |Dϕ(y)| for some compact set K ⊂ Σ. Thus it follows from Lemma 3.1 and (3.8) that D E

ϕ ϕ ϕ (3.9) (Vbε + Vε,Σ − E)v , v 2 − Cεkvk2 ≤ e ε (HεΣ − E)e− ε v , v `2 . `

(3.7) and (3.9) yield by use of the Cauchy–Schwarz inequality 2 kF+ vk2`2 −kF− vk2`2 − Cεkvk2

ϕ ϕ ≤ 2 e ε (HεΣ − E)e− ε v , v `2

√ 1 ϕ Σ

1 ϕ −

≤ 2 2 e ε (Hε − E)e ε v

2 √2 kF vk`2 F `

2

1

ϕ 1 Σ −ϕ 2 ε ε ≤ v2 v

F e (Hε − E)e

2 + 2 kF vk`2 . `

(3.10)

Inserting (3.10) into (3.6) we get

1

2 ϕ 1 2 Σ −ϕ 2 2 2

ε ε kF vk`2 ≤ 2 e (Hε − E)e v

2 + 2 kF vk`2 + 4kF− vk`2 + Cεkvk . F ` 2

This proves (3.5).

Lemma 3.3. Let Σ ⊂ Rd be an open bounded region including the point 0 and such that d0 ∈ C 2 (Σ), where d0 (x) := d` (0, x) is defined in (2.79). Let χ ∈ C ∞ (R+ , [0, 1]) such that χ(r) = 0 for r ≤ 12 and χ(r) = 1 for r ≥ 1. In addition we assume that 0 ≤ χ0 (r) ≤ log2 2 . For B > 0 we define g : Σ → [0, 1] by 0 d (x) g(x) := χ , x∈Σ (3.11) Bε and set 0 Bε B Bε 2d (x) 0 Φ(x) := d (x) − log − g(x) log , x ∈ Σ. (3.12) 2 2 2 Bε Then there exists a constant C > 0 such that for all ε ∈ (0, ε0 ] |∂ν ∂µ Φ(x)| ≤ C .

(3.13)

1206



Furthermore for any B > 0 there is C 0 > 0 such that − B2 − B2 d0 (x) 1 Φ(x) d0 (x) d0 (x) d0 (x) 0 e ε 1+ ≤e ε ≤e ε C 1+ . C0 ε ε

(3.14)

Proof. We write for simplicity d(x) := d0 (x). First we notice that there exists a C > 0 such that for α ∈ Nd , |α| ≤ 2 α

|∂ α g(x)| ≤ Cε− 2 ,

x ∈ Σ.

(3.15)

Here one uses that by (1.16) d(x) = O(|x|2 ) and ∇d(x) = O(|x|) as |x| → 0, thus √ √ x = O( ε) and ∇d(x) = O( ε) on supp ∇g ⊂ {x ∈ Σ | Bε 2 ≤ d(x) ≤ Bε}. By the definition (3.12) of Φ we have Bε 2d(x) ∂ν ∂µ Φ(x) = ∂ν ∂µ d(x) − ∂ν ∂µ g(x) log = A1 + A2 + A3 (3.16) 2 Bε with A1 := ∂ν ∂µ d(x) Bε 2d(x) Bε A2 := − (∂ν ∂µ g) (x) log + (∂ν g)(x) (∂µ d)(x) 2 Bε 2d(x) Bε +(∂µ g)(x) (∂ν d)(x) 2d(x) Bε (∂ν d)(x)(∂µ d)(x) A3 := g(x) + (∂ν ∂µ d)(x) . 2d(x) d(x) Since Σ is bounded, all derivatives of d are at least bounded by a constant independent of ε, thus A1 is bounded. Each summand in A2 includes a derivative of g and is therefore supported in the region Bε < Bε. Thus 1 < 2d(x) 2 < d(x)√ Bε < 2 and from (1.16), it follows as above that ∂ν d(x) = O( ε). By (3.15) A2 is bounded. To estimate A3 , we introduce a constant δ > 0 such that {x ∈ Σ | d(x) < δ} ⊂ Ω and δ ≥ ε0 B and analyze the regions d(x) < δ and d(x) ≥ δ separately. Case 1: d(x) < δ: By Theorem 1.6, we have ∂ν d(x) = O(|x|) and ∂ν ∂µ d(x) = O(1). Thus there exists a constant M > 0 such that X (∂ν d)(x)(∂µ d)(x) + |(∂ν ∂µ d)(x)| < M for δ small enough . d(x) ν,µ Since in addition for d(x) > Bε 2 (on the support of g), the term by 1, A3 is bounded by a constant independent of ε.

Bε 2d

is bounded

Case 2: d(x) ≥ δ: 1 We use that the derivatives of d are bounded on Σ and that d(x) ≤ 1δ . Combining Case 1 and 2 we get the boundedness of A3 and thus (3.13).


1207

To see (3.14), we first note that by definition − B2 (g(x)−1) − B2 g(x) Φ(x) d(x) B d(x) ε ε e =e . (3.17) 2 ε We notice that for any y ≥ 0 and for any B > 0 there exists C˜ > 0 such that y

1 y χ( B ) ≤ ≤ C˜ . 1+y C˜ Setting y =

d(x) ε ,

2

this leads to (3.14).

3.2. Proof of Theorem 1.7 We partly follow the ideas in the proof of Proposition 5.5 in Helffer–Sjöstrand [17]. Let X 1 Σ (0) t0 (x, ξ) := aγ (x) cos γ · ξ , (x, ξ) ∈ Σ × Td , (3.18) ε 0 γ∈Σε (x)

where

Σ0ε (x)

:= {γ ∈ (εZ)d | x + γ ∈ Σ}. We notice that t0 (x, iξ) ≤ tΣ 0 (x, iξ) ,

(0) aγ

(3.19) 0

since ≤ 0 for γ 6= 0. In the following we write for simplicity d(x) := d (x). By Theorem 1.6, for any B > 0 we may choose εB > 0 such that for all ε < εB V0 (x) + t0 x, i∇d(x) = 0 , x ∈ Σ ∩ d−1 ([0, Bε)) , (3.20) By (3.11) and (3.12) Bε d(x) 1 0 d(x) 2d(x) ∇Φ(x) = ∇d(x) 1 − χ − χ log . 2d(x) Bε 2 Bε Bε

(3.21)

Step 1: We shall show that there is C0 > 0 independent of B such that ( 0, x ∈ Σ ∩ d−1 ([0, Bε]) Σ V0 (x) + t0 (x, i∇Φ) ≥ B (3.22) x ∈ Σ ∩ d−1 ([Bε, ∞)) C0 ε , Case 1: d(x) ≤ Bε 2 Since χ(x) = χ0 (x) = 0 and the eikonal equation (2.85) holds, we get V0 (x) + t0 x, i∇Φ(x) = V0 (x) + t0 x, i∇d(x) = 0 , Bε −1 x∈Σ∩d 0, . (3.23) 2 which by (3.19) leads at once to the first estimate in (3.22) in Case 1. Case 2: d(x) ≥ Bε Since χ0 (x) = 0 in this region, we have by (3.21) Bε ∇Φ(x) = ∇d(x) 1 − . 2d(x)

(3.24)

1208



By Lemma 2.14, t0 (x, iξ) = −t˜0 (x, ξ) is concave with respect to ξ, therefore t0 x, λiξ + (1 − λ)iη ≥ λt0 (x, iξ) + (1 − λ)t0 (x, iη) for

0 ≤ λ ≤ 1,

ξ, η ∈ Rd . (3.25)

Bε Bε We have 0 ≤ (1 − 2d(x) ) ≤ 1. Thus choosing λ = (1 − 2d(x) ) and η = 0 in (3.25) d and using t0 (x, 0) = 0 for all x ∈ (εZ) , by (3.24) we get the estimate Bε V0 (x) + t0 x, i∇Φ(x) ≥ V0 (x) + 1 − t0 x, i∇d(x) 2d(x) Bε ≥ V0 (x) 1 − 1 − 2d(x) Bε = V0 (x) , (3.26) 2d(x)

where for the second estimate we used that by Theorem 1.6 the eikonal inequality t0 (x, i∇d(x)) ≥ −V0 (x) holds. It follows from Theorem 1.6 and Hypothesis (1.1)(b) respectively that d(x) = O(|x|2 ) and V0 (x) = O(|x|2 ) for |x| → 0. Since the region Σ was assumed to be bounded, it thus follows that there exists a constant C0 > 0 such that V0 (x) C0−1 ≤ ≤ C0 , x ∈ Σ ∩ d−1 ([Bε, ∞)) . (3.27) 2d(x) Combining (3.19), (3.26) and (3.27), we finally get the second estimate in (3.22). Case 3: Bε 2 < d(x) < Bε We define Bε d(x) f1 (x) := χ 2d(x) Bε

and f2 (x) :=

1 0 χ 2

d(x) Bε

log

2d(x) Bε

,

such that by (3.21) ∇Φ(x) = ∇d(x) 1 − f1 (x) − f2 (x) .

(3.28)

Since 1 < 2d(x) Bε < 2, f1 and f2 are non-negative and therefore 1−f1 (x)−f2 (x) ≤ 1. In addition it follows that 0 ≤ f1 (x) ≤ 1 and by the assumption χ0 (r) ≤ log2 2 we get 0 ≤ f2 (x) ≤ 1. Therefore 0 ≤ f1 (x) + f2 (x) ≤ 2 and thus the estimate |1 − f1 (x) − f2 (x)| ≤ 1

(3.29)

holds. Setting λ(x) := 1 − f1 (x) − f2 (x) it follows from (3.28) and (3.29) that ∇Φ(x) = λ(x)∇d(x)

with

|λ(x)| ≤ 1

x ∈ Rd .

(3.30)

Thus again from (3.25) (with η = 0 and ξ = ∇d(x)) together with (3.30), (3.19) and the fact that t0 is even with respect to iξ it follows that V0 (x) + tΣ 0 x, i∇Φ(x) ≥ V0 (x) + |λ(x)|t0 x, i∇d(x) ≥ V0 1 − |λ(x)| , (3.31) where for the second step we used (3.20). Since |λ(x)| ≤ 1 and V0 ≥ 0, (3.31) gives the first estimate in (3.22) in Case 3.


Step 2: We shall show ( −C5 ε Vbε (x) + V Φ (x) ≥ B C0 − C5 ε

for x ∈ Σ ∩ d−1 ([0, Bε]) for x ∈ Σ ∩ d−1 ([Bε, ∞)) .

Φ for some C5 > 0 independent of B, where V Φ := Vε,Σ is defined in (3.1). We write Vbε (x) + V Φ (x) = Vbε (x) − V0 (x) + V Φ (x) − tΣ x, i∇Φ(x) 0 + V0 (x) + tΣ 0 x, i∇Φ(x)

1209

(3.32)

(3.33)

and give estimates for the differences in the first two brackets on the right hand side. By Hypothesis 1.1 and since Σ is bounded, there exists a constant C1 > 0 such that Vbε (x) − V0 (x) ≥ −C1 ε , x ∈ Σ . (3.34) We shall show that Φ V (x) − tΣ ≤ εC4 . 0 x, i∇Φ(x)

(3.35)

Then inserting (3.35), (3.34) and (3.22) in (3.33) proves (3.32). Setting (see (3.1)) X 1 Φ (0) V0 (x) := aγ (x) cosh Φ(x) − Φ(x + γ) , ε 0 γ∈Σ (x)

we write Φ Φ Φ Σ V Φ (x) − tΣ 0 x, i∇Φ(x) = V (x) − V0 (x) + V0 (x) − t0 x, i∇Φ(x) =: D1 (x) + D2 (x) and analyze the two summands on the right hand side separately. Since Φ is Lipschitz and constant outside of some bounded set, it follows from Hypothesis 1.1(a) (as in the proof of (3.8)) that for some C˜ > 0 X 1 (2) ˜ . |D1 (x)| = ε a(1) Φ(x) − Φ(x + γ) ≤ Cε γ (x) + Rγ (x, ε) cosh ε γ∈Σ0 (x) (3.36) uniformly with respect to x. We have for x ∈ Σ X 1 1 (0) |D2 (x)| ≤ |aγ (x)| cosh Φ(x) − Φ(x + γ) − cosh γ∇Φ(x) . ε ε γ∈Σ0ε (x)

(3.37)

1210



By the mean value theorem for cosh z with z0 = 1ε γ∇Φ(x) and z1 = 1ε (Φ(x) − Φ(x + γ)), we get from | sinh x| ≤ e|x| cosh 1 Φ(x) − Φ(x + γ) − cosh 1 γ∇Φ(x) ε ε n o 1 {(Φ(x)−Φ(x+γ))t+γ∇Φ(x)(1−t)}| 1 | ε ≤ sup e ε Φ(x) − Φ(x + γ) + γ∇Φ(x) . t∈[0,1] (3.38) By (1.17) and the definition (3.12) of Φ there exist constants c1 , c2 > 0 such that |Φ(x)−Φ(x+γ)| ≤ c1 |γ| and |γ∇Φ(x)| ≤ c2 |γ| ,

x ∈ Σ,

γ ∈ Σ0ε (x) . (3.39)

(3.39) gives a constant D > 0 such that 1

D

e| ε {(Φ(x)−Φ(x+γ))t+γ∇Φ(x)(1−t)}| ≤ e ε |γ| .

(3.40)

By second order Taylor-expansion d 1 1 X Φ(x) − Φ(x + γ) + γ∇Φ(x) ≤ sup |γν γµ ∂ν ∂µ Φ(x + tγ)| . ε t∈[0,1] ε ν,µ=1

(3.41)

Inserting (3.13) into (3.41) shows that there exists a constant C3 > 0 independent of the choice of B such that for all ε ∈ (0, ε0 ] C3 2 1 Φ(x) − Φ(x + γ) + γ∇Φ(x) ≤ |γ| . (3.42) ε ε By (1.9), inserting (3.40) and (3.42) in (3.37) we get for any A > 0 with η = γε X Φ V0 (x) − tΣ e−A|η| eD|η| C3 ε|η|2 0 (x, −i∇Φ) ≤ η 0 ε ∈Σε (x)

≤ε

X

0

e−|η|D C3 |η|2 ≤ εC4 ,

η∈Zd

where A − D = D0 > 0. This together with (3.36) gives (3.35). Step 3: We prove (1.20) by use of Lemma 3.2. Choosing B ≥ C0 (1 + R0 + C5 ), we have B − C5 ε − E ≥ ε , E ∈ [0, εR0 ] . C0

(3.43)

Let Ω− := x ∈ Σ | Vbε (x) + V Φ (x) − E < 0

and

Ω+ := Σ \ Ω− ,

(3.44)

then from (3.43) it follows that Ω− ⊂ {d(x) < εB} and by (3.32) |Vbε (x) + V Φ (x)| ≤ ε max{C5 , R0 }

for all

x ∈ Ω− .

We define the functions F± : Σ → [0, ∞) by q F+ (x) := ε1{d(x) 0 , F− = O( ε) and F+2 −F−2 = Vbε +V Φ −E . (3.48) Φ

Lemma 3.2 yields with the choice v = e ε u

Φ 2

1 Φ 2 Φ 2

ε

Σ

ε ε u + 8 e + Cεkuk2 .

F e u 2 ≤ 4 e Hε − E u

F −

2 F ` `2 `

(3.49)

By (3.14) and (3.48)

2 − B2

Φ 2

d d

ε

e ε u

F e u 2 ≥ Cε 1 +

2 ε `

(3.50)

`

and

2

− B2

1 Φ 2 d d

Σ

e ε HεΣ − E u ≤ Cε−1 e ε Hε − E u .

1+

F

2

2 ε `

(3.51)

`

Since Ω− ⊂ {d(x) < Bε} it follows from the definition of F− that d(x) ε ≤ C on the support of F− . Therefore by (3.14) and (3.48) there exists a constant C > 0 such that

Φ 2

2 (3.52)

F− e ε u ≤ Cε kuk`2 . `2

˜ := B Inserting (3.50), (3.51) and (3.52) in equation (3.49) yields with B 2

2

2 −B˜ −B˜

d d d d

2 ˜ Cε e ε u ≤ ε−1 1 + e ε HεΣ − E u + ε kuk`2 .

1+

2

2 ε ε `

`

This proves (1.20). Step 4: We prove (1.21). If u is an eigenfunction of HεΣ with eigenvalue E, then the first summand on the right hand side of (1.20) vanishes. The normalization of u leads therefore to (1.21). 2

4. Application to Markov chains An example of self adjoint difference operators as analyzed above are generators of certain Markov chains and jump processes. For the sake of the reader we briefly recall some relevant facts on Markov chains. A Markov chain on (εZ)d is described by means of a “transition matrix” Pε : (εZ)d × (εZ)d → [0, 1]. Pε is a stochastic matrix, i.e., X Pε (x, y) = 1 , x ∈ (εZ)d . (4.1) y∈(εZ)d

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We assume that Pε satisfies a detailed balance condition, i.e., µε (x)Pε (x, y) = µε (y)Pε (y, x)

(4.2)

with respect to some family {µε }ε∈(0,ε0 ] of probability measures on (εZ)d . Then (1 − Pε ) defines a self adjoint (diffusion) operator on `2 (εZ)d , µε via X (1 − Pε )u(x) = u(x) − Pε (x, y)u(y) . y∈(εZ)d

In fact Pε is a bounded operator on `2 ((εZ)d , µε ) with kPε k = 1. To see this, we first notice that by (4.1)   ! X X 2 2   |Pε u(x)| ≤ Pε (x, y) Pε (x, y)|u(y)| γ

y∈(εZ)d

=

X

2

Pε (x, y)|u(y)| .

γ

This yields by (4.2), the Fubini-Theorem and again (4.1) X X X kPε uk2`2 ((εZ)d ,µε ) = µε (x)|Pε u(x)|2 ≤ µε (x) Pε (x, y)|u(y)|2 x

x∈(εZ)d

y

! =

X X y

Pε (y, x) µε (y)|u(y)|2

x

= kuk2`2 ((εZ)d ,µε ) , thus kPε k ≤ 1. Since the constant function u(x) = 1 belongs to `2 ((εZ)d , µε ) and fulfills kPε uk`2 ((εZ)d ,µε ) = kuk`2 ((εZ)d ,µε ) , this proves that kPε k = 1 The symmetry of Pε follows from the reversibility condition (4.2), since for u, v ∈ `2 ((εZ)d , µε ) X X hu , Pε vi`2 ((εZ)d ,µε ) = µε (x)u(x) Pε (x, y)v(y) x∈(εZ)d

=

XX y

y∈(εZ)d

µε (y)Pε (y, x)u(x)v(y) = hPε u , vi`2 ((εZ)d ,µε ) .

x

Conjugation with respect to the measure µε induces a bounded self adjoint oper1 −1 ator Hε := µε2 (1 − Pε )µε 2 on `2 (εZ)d , whose restriction to K((εZ)d ) is given by X 1 −1 Hε u(x) = u(x)−µε2 (x) Pε (x, x+γ)µε 2 (x+γ)u(x+γ) , u ∈ K (εZ)d . (4.3) γ

Note that K((εZ)d ) is dense in `2 ((εZ)d ) and Hε is linear continuous and is therefore completely determined by (4.3).


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1 −1 Proposition 4.1. The operator Hε := µε2 (1 − Pε )µε 2 on `2 (εZ)d is of the form (1.1) and fulfills Hypothesis 1.1 (a)(iii). If the coefficients aγ have an expansion (1.8), they also fulfill (ii).

Proof. Setting Tε (x) :=

X

1

−1

µε2 (x)Pε (x, x + γ)µε 2 (x + γ)(1 − τγ )

(4.4)

1 1 − −1 µε2 (x)Pε (x, x + γ) µε 2 (x) − µε 2 (x + γ)

(4.5)

γ6=0

Vε (x) :=

X γ6=0

we have the standard form Hε = Tε + Vε , where Vε is a potential energy (a multiplication operator) and Tε is of the form described in (1.1) with X 1 −1 a0 (x, ε) = µε2 (x)Pε (x, x + γ)µε 2 (x + γ) ≥ 0 (4.6) γ6=0 1

−1

aγ (x, ε) = −µε2 (x)Pε (x, x + γ)µε 2 (x + γ) ,

γ 6= 0 .

(4.7)

Since Pε (x, y) and µε (x) are non-negative numbers, it follows at once that aγ (x, ε) P ≤ 0 for all γ 6= 0 and that aγ (x, ε) = 0. Thus under the assumption (1.8) it follows that Hypothesis 1.1 (a)(ii) holds. The detailed balance condition for Pε ensures the symmetry of Hε and thus of Tε . By (2.71), this leads to aγ (x, ε) = a−γ (x + γ, ε) (Hypothesis 1.1(a)(iii)). 2 Remark 4.2. The other conditions given in Hypothesis 1.1 lead to analog conditions on the transition matrix Pε and the reversible measure µε . For example, condition (a)(iv) on the exponential decay of aγ with respect to γ, must be reflected by the fact that Pε (x, y) is assumed to be exponential small for |x − y| large. Furthermore in order to fulfill (a)(i), the measure µε should be slowly varying. Condition (a)(v) is a kind of ergodicity condition, which guarantees that jumps in each direction are possible. It follows at once from (4.4) and (4.5) that for a general probabilistic operator, the potential energy can be written in terms of the kinetic energy and the measure µε as −1

1

Vε (x) = −µε 2 (x) Tε µε2 (x) .

(4.8)

The assumptions on V0 given in Hypothesis 1.1 are conditions on the pair (µε , Pε ). The class of Markov chains satisfying these conditions is more general than the class of Markov chains analyzed in Bovier–Eckhoff–Gayrard–Klein [9], if the Markov chain acts on (εZ)d .

References [1] M. Abate, G. Patrizio, Finsler Metrics – A Global Approach, LNM 1591, Springer, 1994. [2] R. Abraham, J. E. Marsden, Foundations of Mechanics, 2.ed., The Benjamin / Cummings Pub.Comp., 1978.

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[3] S. Agmon, Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schr¨ odinger Operators, Mathematical Notes 29, Princeton University Press, 1982. [4] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2.ed., Springer-Verlag, 1989. [5] D. Bao, S.-S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, GTM 200, Springer, 2000. [6] G. Barbatis, Sharp heat Kernel bounds and Finsler-type metrics, Quart. J. Math. Oxford 2, 49 (1998), 261–277. [7] G. Barbatis, Explicit estimates on the fundamental solution of higher-order parabolic equations with measurable coefficients, Journal of Diff. Equations 174 (2001), 442– 463. [8] A. Bovier, M. Eckhoff, V. Gayrard, M. Klein, Metastability in stochastic dynamics of disordered mean-field models, Probab. Theory Relat. Fields 119 (2001), 99–161. [9] A. Bovier, M. Eckhoff, V. Gayrard, M. Klein, Metastability and low lying spectra in reversible Markov chains, Comm. Math. Phys. 228 (2002), 219–255. [10] M. Dimassi, J. Sj¨ ostrand, Spectral Asymptotics in the Semi- Classical Limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999. [11] L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, 1992. [12] M. Giaquinta, S. Hildebrandt, Calculus of Variations 2, GMW 311, Springer, 1996. [13] R. S. Hamilton, The Inverse Function Theorem of Nash and Moser, Bulletin of the American National Society, Vol. 7, Number 1, 1982. [14] B. Helffer, Décroissance exponentielle des fonctions propres pour l’opérateur de Kac: Le cas de la dimension > 1, Operator calculus and spectral theory (Lambrecht, 1991), 99–115, Oper. Theory Adv. Appl. 57, Birkh¨ auser, Basel, 1992. [15] B. Helffer, A. Mohamed, Semiclassical analysis for the ground state energy of a Schr¨ odinger operator with magnetic wells, J. Funct. Anal. 138 (1996), no. 1, p. 40– 81. [16] B. Helffer, B. Parisse, Comparaison entre la décroissance de fonctions propres pour les opérateurs de Dirac et de Klein–Gordon. Application a ` l’étude de l’effet tunnel, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 2, 147–187. [17] B. Helffer, J. Sj¨ ostrand, Multiple wells in the semi-classical limit I, Comm. in P.D.E. 9 (1984), 337–408. [18] B. Helffer, J. Sj¨ ostrand, Analyse semi-classique pour l’équation de Harper (avec application a ` l’équation de Schr¨ odinger avec champ magnétique), Mém. Soc. Math. France (N.S.) 34 (1988), 1–113. [19] W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics 1 (1982). [20] S. Lang, Differential and Riemannian Manifold, 3.ed., Springer, 1995. [21] C. Mantegazza, A. C. Mennucci, Hamilton–Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Opt. 47, 1 (2003), 1–25. [22] E. Rosenberger, Asymptotic Spectral Analyis and Tunnelling for a class of Difference Operators, Thesis, http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-7393.


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[23] E. Servat, Le splitting pour l’opérateur de Klein–Gordon: Une approche heuristique et numérique, Canad. J. Math. 59 (2007), no. 2, 393–417. [24] B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: Asymptotic expansions, Ann Inst. H. Poincare Phys. Theor. 38 (1983), 295–308. [25] B. Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. 120 (1984), 89–118. [26] K. Tintarev, Short time asymptotics for fundamental solutions of higher order parabolic equations, Comm. in Part. Diff. Equ. 7 (4) (1982), 371–391. Markus Klein and Elke Rosenberger Universit¨ at Potsdam Institut f¨ ur Mathematik Am Neuen Palais 10 D-14469 Potsdam Germany e-mail: [email protected] [email protected] Communicated by Christian Gérard. Submitted: February 23, 2008. Accepted: May 23, 2008.



Asymptotic Behavior of Bifurcation Branch of Positive Solutions for Semilinear Sturm–Liouville Problems Tetsutaro Shibata Abstract. We consider the nonlinear eigenvalue problem −u (t) + f u(t) = λu(t) , u(t) > 0 , t ∈ I := (0, 1) , u(0) = u(1) = 0 , where f (u) = up + h(u) (p > 1) and λ > 0 is a parameter. Typical example of h(u) is h(u) = ±uq with 1 < q < (p+1)/2. We establish the precise asymptotic formula for Lm -bifurcation branch λ = λm (α) of positive solutions as α → ∞, where α > 0 is the Lm -norm of the positive solution associated with λ 1.

1. Introduction We consider the following nonlinear eigenvalue problem −u (t) + f u(t) = λu(t) , t ∈ I := (0, 1) , u(t) > 0 ,

t∈I,

(1.1) (1.2)

u(0) = u(1) = 0 ,

(1.3)

p

where f (u) = u +h(u) (p > 1) and λ > 0 is a parameter. We assume the following conditions (A.1)–(A.3) on h. (A.1) h(u) is C 1 for u ≥ 0. (A.2) h(0) = h (0) = 0. Furthermore, for u 1, |h(u)/u| is increasing for u 1 and tends to infinity as u → ∞. (A.3) There exist constants 0 < δ1 1 and K > 0 such that for u 1 |h (u)| ≤ u(p−1)/2−δ1 ,

|h (u)u| ≤ K|h(u)| .

(1.4) (1.5)

Typical example of h(u) is h(u) = ±u and h(u) = u log(u+1) (1 < q < (p+1)/2). We know from [5] that if f (u) satisfies (A.1)–(A.3), then there exists a constant ¯ α0 ≥ 0 such that for α > α0 , a unique solution (λ, u) = (λm (α), uα ) ∈ R+ × C 2 (I) q

q

1218

T. Shibata


of (1.1)–(1.3) with uα m = α exists, where · m denotes usual Lm -norm (m ≥ 1). λm (α) (α > α0 ) is an unbounded increasing curve of class C 1 and is called the Lm -bifurcation branch of positive solutions to (1.1)–(1.3). The purpose of this paper is to establish the precise asymptotic formula for λm (α) as α → ∞ to clarify the global behavior of the bifurcation branch of positive solutions. Before stating our results, let us recall the mathematical, physical and biological background of the equation (1.1). Since (1.1)–(1.3) is regarded as an typical model of bifurcation problem and a nonlinear eigenvalue problem, it has been studied intensively by many authors from the viewpoint of L∞ -framework and L2 framework, especially, when f (u) = up (u ≥ 0, p > 1). We refer to [1, 3, 5–8, 10–12] for the works and the related topics in these directions. We emphasize that (1.1)–(1.3) describes many physical phenomena. We introduce here some of the well known equations. When f (u) = ±u3 , (1.1) is called the classical Duffing type equation of nonlinear vibration theory. When f (u) = u3 , it is 1D Ginzburg–Landau equation which appears in the analysis of superconductivity. It is also the stationary equation of so-called Chafee–Infante equation, which is a typical example of bistable type reaction-diffusion equation. In this paper, we treat the case where the reaction term is very strong. Furthermore, this nonlinearity f (u) = u3 appears in the study of simple pendulum equation, since sin u = u − u3 /6 + · · · when u is sufficiently small. We refer to [2, 4, 9, 15, 16] and the references therein. A further important background of (1.1)–(1.3) is that, it is a model equation of population density for some species when f (u) = u2 . Here, λ > 0 is regarded as the reciprocal number of its diffusion rate. From this biological background, it seems that one of the most important problems to study is the asymptotic behavior of λ1 (α) as α → ∞, which stands for the relation between the reciprocal number of its diffusion rate and the total number of population. The leading term of λm (α) as α → ∞ is easily obtained as follows. It is known that for t ∈ I, as α → ∞, uα (t) → 1. (1.6) λm (α)1/(p−1) This implies that as α → ∞, (1.7) λm (α) = αp−1 1 + o(1) . Since (1.1)–(1.3) is essentially an eigenvalue problem, L2 -theory is worked well, and the L2 -bifurcation branch λ2 (α) with the most typical nonlinear term f (u) = up was considered in [12]. Theorem 1.1 ([12]). Consider (1.1)–(1.3) with h(u) = 0. Let an arbitrary N ∈ N be fixed. Then as α → ∞ λ2 (α) = αp−1 + C1 α(p−1)/2 +

N n=0

an α−n(p−1)/2 + o(α−N (p−1)/2 ) ,

(1.8)

Vol. 9 (2008)

Bifurcation Branch of Positive Solutions

where

1

C1 := (p − 1)

0

1 − s2 1−

s2

− 2(1 − sp+1 )/(p + 1)

1219

ds

(1.9)

and aj (0 ≤ j ≤ N ) is a constant determined by C1 , a0 , a1 , . . . , aj−1 . Theorem 1.1 was proved by using the fine relation between λ2 (α) and the critical value associated with uα , which only holds for the case f (u) = up . Recently, by using a very straightforward calculation, (1.8) has been improved to λm (α), where m ≥ 1. Let p−1 C(m) , m 1 1 − sm C(m) := 2 ds . 1 − s2 − 2(1 − sp+1 )/(p + 1) 0 C2 =

Theorem 1.2 ([14]). Consider (1.1)–(1.3) with h(u) = 0. Let 1 ≤ m < ∞ be fixed. Further, let an arbitrary positive integer N be fixed. Then as α → ∞ λm (α) = αp−1 + C2 α(p−1)/2 +

N

bk αk(1−p)/2 + o(αN (1−p)/2 ) ,

(1.10)

k=0

where b0 =

p−1 C(m)2 , 2m

b1 =

(p − 1)(p − 1 − 2m)(p − 1 − 4m) C(m)3 24m3

and {bj }N j=2 is a constant determined by C(m), b0 , b1 , . . . , bj−1 . However, even if we focus our attention on the typical case f (u) = up ± uq and m = 2, it seems that a few results such as (1.8) and (1.10) have been obtained for λm (α) as α → ∞. Theorem 1.3 ([13]). Assume that |h (u)| ≤ Cuq−1 for u 1, where 1 < q < (3p + 1)/4. Then as α → ∞ λ2 (α) = αp−1 +

h(α) + C1 α(p−1)/2 + o(α(p−1)/2 ) . α

(1.11)

Theorem 1.3 was proved by using the modified method of that used in [12]. We note that unfortunately, if h(u) = ±uq with 1 < q < (p + 1)/2, then we do not obtain any information from (1.11) about the effect of h(u) on the asymptotic behavior of λ2 (α) as α → ∞, since the term h(α)/α is included in the remainder term o(α(p−1)/2 ). In this paper, we focus our attention on the nonlinear term f (u) = up + h(u) which includes the case f (u) = up ± uq with 1 < q < (p + 1)/2 and find the precise asymptotic formula for λm (α) as α → ∞ with the remainder estimate. Now we state our main results.

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T. Shibata


Theorem 1.4. Assume that h(u) satisfies (A.1)–(A.3). Then as α → ∞ p−1 h(α) p − 1 C(m)α(p−1)/2 + + C(m)2 λm (α) = αp−1 + m α 2m h(α) +O . α(p+1)/2

(1.12)

Corollary 1.5. Let h(u) = ±uq with 1 < q < (p + 1)/2. Then as α → ∞ p−1 p−1 C(m)α(p−1)/2 ± αq−1 + C(m)2 m 2m + O(α(2q−p−1)/2 ) .

λm (α) = αp−1 +

(1.13)

Remark 1.6. (i) The remainder estimate in (1.13) seems to be optimal, since the order (2q − p − 1)/2 comes from q − 1 − (p − 1)/2. However, to obtain the exact fifth term of λm (α), we must calculate a quite difficult definite integral which seems almost impossible to find. Therefore, the formula (1.12) seems to be best possible asymptotic formula for λm (α) at this stage. (ii) It follows from (1.4) that |h(u)| ≤ Cu(p+1)/2−δ1 ,

(u 1) ,

(1.14)

which will be used later. We prove Theorem 1.4 by quite an elementary and straightforward method which has been developed in [14]. We compare uα m with uα ∞ directly. However, more delicate and complicated observation than that in [14] is necessary here. So we carry out our calculation very carefully.

2. Proof of Theorem 1.4

u Let 1 ≤ m < ∞ be fixed. We set H(u) = 0 h(s)ds. In this section, we use the ¯ be the solution of (1.1)–(1.3) for given following notation. Let (λ, uλ ) ∈ R+ ×C 2 (I) λ 1. Therefore, α = uλ m . We write λ = λm (α) for simplicity. Since α 1 and λ 1 are equivalent, we use the notation λ 1 in this section. Finally, C denotes various positive constants independent of λ 1. It is well known that uλ (t) = uλ (1 − t) , 0 ≤ t ≤ 1 , 1 uλ = max uλ (t) = uλ ∞ , 0≤t≤1 2 1 uλ (t) > 0 , 0 ≤ t < . 2 We know that there exists a constant δ0 > 0 such that for λ 1, √ h( uλ ∞ ) f ( uλ ∞ ) + λ1 = uλ p−1 + O(λe− (p−1−δ0 )λ/2 ) . λ= ∞ + uλ ∞ uλ ∞

(2.1) (2.2) (2.3)

(2.4)

Vol. 9 (2008)


1221

Note that λ1 > 0. For λ 1 and 0 ≤ s ≤ 1, we set 2 uλ p−1 ∞ (1 − sp+1 ) p+1 λ 2 − H( uλ ∞ ) − H( uλ ∞ s) , 2 λ uλ ∞ 2 Sλ (s) := 1 − s2 − (1 − sp+1 ), p+1 1 (1 − sm )(Sλ (s) − Tλ (s)) Uλ := 2 ds . Sλ (s) Tλ (s)( Sλ (s) + Tλ (s)) 0 Tλ (s) = 1 − s2 −

(2.5)

(2.6) (2.7)

Lemma 2.1. As λ → ∞ m uλ m ∞ − uλ m =

uλ m √ ∞ C(m) + Uλ . λ

(2.8)

Proof. It follows from (1.1) that for 0 ≤ t ≤ 1

1 d 1 2 1 p+1 2 u (t) − uλ (t) − H uλ (t) + λuλ (t) = 0 . dt 2 λ p+1 2 By this and (2.2), for 0 ≤ t ≤ 1, we obtain 1 1 2 1 u (t) − uλ (t)p+1 − H uλ (t) + λuλ (t)2 = constant 2 λ p+1 2 1 1 q+1 2 uλ p+1 =− ∞ − H( uλ ∞ ) + λ uλ ∞ . p+1 2

(2.9)

We set 2 p+1 ( uλ p+1 Mλ (θ) := λ( uλ 2∞ − θ2 ) − ) ∞ −θ p+1 − 2 H( uλ ∞ ) − H(θ) , 2 p+1 uλ p+1 ) Qλ (s) := λ uλ 2∞ (1 − s2 ) − ∞ (1 − s p+1 − 2 H( uλ ∞ ) − H( uλ ∞ s) . By (2.3) and (2.9), for 0 ≤ t ≤ 1/2, uλ (t) =

Mλ uλ (t) .

(2.10)

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T. Shibata


By (2.1), (2.10), putting θ = uλ (t) and s = θ/ uλ ∞ , we obtain 1/2 uλ (t) m m uλ m dt − u = 2 uλ m λ m ∞ ∞ − uλ (t) Mλ (uλ (t)) 0 uλ ∞ 1 m =2 dθ ( uλ m ∞ − θ ) M 0 λ (θ) 1 − sm uλ m 1 ds =2 √ ∞ λ Qλ (s)/(λ uλ 2∞ ) 0 1 1 − sm uλ m ∞ ds =2 √ λ Tλ (s) 0

1 1 − sm uλ m ∞ 2 ds + Uλ = √ λ Sλ (s) 0 uλ m = √ ∞ C(m) + Uλ . λ Thus, the proof is complete. Lemma 2.2. For λ 1

h(α) |Uλ | ≤ C p . α

(2.11)

Proof. The proof is divided into two steps. Step 1. We first prove

h( uα ∞ ) . |Uλ | ≤ C uα p∞

(2.12)

Let an arbitrary 0 < 1 be fixed. By (2.4), (2.5) and (2.6), we have Xλ (s) := Sλ (s) − Tλ (s) 2 p+1 ( uλ p−1 ) = ∞ − λ)(1 − s (p + 1)λ 2 + H( uλ ∞ ) − H( uλ ∞ s) 2 λ uλ ∞ h( uλ ∞ ) −2 = + λ1 (1 − sp+1 ) (p + 1)λ uλ ∞ 2 H( uλ ∞ ) − H( uλ ∞ s) , + λ uλ 2∞ 2 h( uλ ∞ ) 2 Xλ (s) = + λ1 sp − h( uλ ∞ s) , λ uλ ∞ λ uλ ∞ 2p h( uλ ∞ ) 2 + λ1 sp−1 − h ( uλ ∞ s) . Xλ (s) = λ uλ ∞ λ

(2.13)

(2.14) (2.15)

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Then by (A.2), (1.5) and Taylor expansion, for λ 1 and 1 − ≤ s ≤ 1 h( uλ ∞ ) 2λ1 (1 − s)2 . (1 − s) + C |Xλ (s)| ≤ λ λ uλ ∞

1223

(2.16)

By the same argument as that to obtain (2.16), we see that there exists a constant 0 < δ2 1 such that for λ 1 and 1 − ≤ s ≤ 1, Sλ (s) ≥ (p − 1 − δ2 )(1 − s)2 , Tλ (s) ≥

2λ1 (1 − s) + (p − 1 − δ2 )(1 − s)2 . λ

(2.17) (2.18)

We set Uλ = U1,λ + U2,λ 1− (1 − sm )(Sλ (s) − Tλ (s)) ds := 2 Tλ (s) Sλ (s)( Tλ (s) + Sλ (s)) 0 1 (1 − sm )(Sλ (s) − Tλ (s)) ds . +2 Tλ (s) Sλ (s)( Tλ (s) + Sλ (s)) 1− √ Recall that a + b ≥ 2 ab for a, b ≥ 0. We apply it to (2.17) to obtain λ1 (1 − s)3 . Tλ (s) ≥ C λ By this, (1.4), (2.4), (2.16), (2.17), (2.18) and (2.19) 1 (1 − sm )|Xλ (s)| |U2,λ | ≤ 2 ds 1− Tλ (s) Sλ (s) 1 1 h( uλ ∞ ) (2λ /λ)(1 − s) 1 ds + C ≤C λ uλ ∞ ds (λ1 /λ)(1 − s)3 1− 1−

h( uλ ∞ ) λ1 h( uλ ∞ ) . ≤C ≤C + λ λ uλ ∞ λ uλ ∞

(2.19)

(2.20)

Next, by (A.2), (2.4), (2.13) and mean value theorem, for λ 1 and 0 ≤ s ≤ 1 − , h( uλ ∞ ) 2 + |Xλ (s)| ≤ C |h(ηs uλ ∞ )| uλ ∞ (1 − s) (2.21) λ uλ ∞ λ uλ 2∞ h( uλ ∞ ) . ≤ C λ uλ ∞ Here, 0 < η < 1. Since it is clear that Sλ (s) ≥ C, Tλ (s) ≥ C for 0 ≤ s ≤ 1 − and λ 1, by this and (2.21), we obtain 1− h( uλ ∞ ) h( uλ ∞ ) (2.22) |U1,λ | ≤ C λ uλ ∞ ds ≤ C λ uλ ∞ . 0 By this, (2.4), (2.19) and (2.20), we obtain (2.12).

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Step 2. By (1.14) and (2.11), |Uλ | → 0 as λ → ∞. By (2.8), for λ 1, we have 1/m 1 α = uλ ∞ 1 − √ (C(m) + Uλ ) (2.23) λ 1 = uλ ∞ 1 − √ C(m) + o(1) . m λ By this, mean value theorem, (1.4), (1.5) and (A.2), for 0 < β < 1 |h( uλ ∞ ) − h(α)| = h βα + (1 − β) uλ ∞ ( uλ ∞ − α) h(βα + (1 − β) uλ ∞ ) ( uλ ∞ − α) ≤C βα + (1 − β) uλ ∞ h( uλ ∞ ) uλ ∞ √1 ≤C uλ ∞ λ (1−p)/2 . ≤ C|h( uλ ∞ )|α

(2.24)

This implies that for λ 1, h(α) = h( uλ ∞ ) + h(α) − h( uλ ∞ ) = h( uλ ∞ ) 1 + O(α(1−p)/2 ) . By this, we obtain

h( uλ ∞ ) = h(α) 1 + O(α(1−p)/2 ) .

By this and (1.7), for λ 1, we obtain h( uλ ∞ ) |h(α)| λ uλ ∞ ≤ C αp . By this and (2.12), we obtain (2.11). Thus, the proof is complete.

(2.25)

(2.26)

Now we prove Theorem 1.4. Proof of Theorem 1.4. By (1.7), (1.14), (2.4), Lemmas 2.1 and 2.2, and Taylor expansion, for λ 1, λ = uλ p−1 ∞ +

h( uλ ∞ ) + λ1 uλ ∞

−(p−1)/m h( uλ ∞ ) 1 =α + + λ1 1 − √ C(m) + Uλ uλ ∞ λ p−1 1 √ C(m) + Uλ 1 + o(1) + o(α(p−1)/2 ) = αp−1 1 + m λ p−1 p−1 C(m)α(p−1)/2 + o(α(p−1)/2 ) . + =α m p−1

(2.27)

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By this, (1.14), Lemmas 2.1 and 2.2 and Taylor expansion, for λ 1,

uλ p−1 ∞

−(p−1)/m 1 = 1 − √ C(m) + Uλ λ 1 p − 1 √ C(m) + Uλ = αp−1 1 + m λ (p + m − 1)(p − 1) (C(m) + Uλ )2 −3/2 + + O(λ ) 2m2 λ p − 1 (p−1)/2 = αp−1 + α C(m) + Uλ m −1/2 p−1 × 1+ C(m)α(1−p)/2 1 + o(1) m (p + m − 1)(p − 1) + C(m)2 + o(1) 2m2 p − 1 (p−1)/2 α C(m) + Uλ = αp−1 + m p−1 (1−p)/2 C(m)α 1 + o(1) × 1− 2m (p + m − 1)(p − 1) + C(m)2 + o(1) 2m2 p−1 p−1 C(m)α(p−1)/2 + C(m)2 + o(1) . = αp−1 + m 2m uλ p−1 m

(2.28)

By (2.27) and (2.28), for λ 1, we obtain λ = αp−1 +

p−1 h(α) C(m)α(p−1)/2 + 1 + o(1) . m α

(2.29)

By this, (A.2) and Taylor expansion, for λ 1, we obtain −1/2 1 h(α) p−1 (1−p)/2 √ = α(1−p)/2 1 + C(m)α + p 1 + o(1) m α λ h(α) 1 p−1 (1−p)/2 (1−p)/2 C(m)α + p 1 + o(1) =α 1− 2 m α 2 3 (p − 1) C(m)2 α1−p 1 + o(1) + 8 m2 h(α) p−1 (1−p)/2 (1−p)/2 C(m)α =α +O 1− , 2m αp h(α) 1 p − 1 (1−p)/2 = α1−p 1 − α +O . λ m αp

(2.30)

(2.31)

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Substitute (2.30) and (2.31) in the second line of (2.28). Then by (A.2) and Lemma 2.2, for λ 1, we obtain p − 1 (p−1)/2 p−1 α C(m) + Uλ (2.32) + uλ p−1 ∞ =α m h(α) p−1 C(m)α(1−p)/2 + O × 1− 2m αp (p + m − 1)(p − 1) + C(m)2 2 2m h(α) p−1 C(m)α(1−p)/2 + O × 1− m αp + O(α(1−p)/2 ) p−1 p−1 C(m)α(p−1)/2 + C(m)2 = αp−1 + m 2m h(α) +O . α(p+1)/2 Finally, by (2.23) and (2.25), for λ 1, h( uλ ∞ ) h(α)(1 + O(α(1−p)/2 )) = uλ ∞ α(1 + α(1−p)/2 C(m)(1 + o(1))/m) h(α) 1 + O(α(1−p)/2 ) = α h(α) h(α) +O = . α α(p+1)/2

(2.33)

By (2.27), (2.32) and (2.33), we obtain Theorem 1.4. Thus, the proof is complete.

References [1] R. Benguria and M. Cristina Depassier, Upper and lower bounds for eigenvalues of nonlinear elliptic equations. I. The lowest eigenvalue, J. Math. Phys. 24 (1983), 501–503. [2] R. Benguria and M. Cristina Depassier, Variational calculation of the period of nonlinear oscillators, J. Statist. Phys. 116 (2004), 923–931. [3] H. Berestycki, Le nombre de solutions de certains probl` emes semi-linéaires elliptiques, Journal of Functional Analysis 40 (1981), 1–29. [4] N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 17–37. [5] R. Chiappinelli, Remarks on bifurcation for elliptic operators with odd nonlinearity, Israel Journal of Mathematics 65 (1989), 285–292. [6] R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term, Nonlinear Analysis 13 (1989), 871–878.

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[7] J. M. Fraile, J. L´ opez-G´ omez and J. C. Sabina de Lis, On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems, Journal of Differential Equations 123 (1995), 180–212. [8] M. Holzmann and H. Kielh¨ ofer, Uniqueness of global positive solution branches of nonlinear elliptic problems, Mathematische Annalen 300 (1994), 221–241. [9] S. Jimbo and Y. Morita, Ginzburg–Landau equation with magnetic effect in a thin domain, Calc. Var. Partial Differential Equations 15 (2002), 325–352. [10] P. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis 8 (1971), 321–340. [11] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis 7 (1971), 487–513. [12] T. Shibata, Precise spectral asymptotics for nonlinear Sturm–Liouville problems, Journal of Differential Equations 180 (2002), 374–394. [13] T. Shibata, Three-term spectral asymptotics for nonlinear Sturm–Liouville problems, NoDEA: Nonlinear Differential Equations and Applications 9 (2002), 239–254. [14] T. Shibata, Global behavior of the branch of positive solutions to a logistic equation of population dynamics, to appear. [15] T. Shibata, Layer structures for the solutions to the perturbed simple pendulum problems, Journal of Mathematical Analysis and Applications 315 (2006), 725–739. [16] T. Wakasa, Exact eigenvalues and eigenfunctions associated with linearization for Chafee–Infante problem, Funkcial. Ekvac. 49 (2006), 321–336. Tetsutaro Shibata Department of Applied Mathematics Graduate School of Engineering Hiroshima University Higashi-Hiroshima, 739-8527 Japan e-mail: [email protected] Communicated by Rafael D. Benguria. Submitted: September 27, 2007. Accepted: May 28, 2008.

Ann. Henri Poincaré 9 (2008), 1229–1273 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/071229-45, published online October 17, 2008 DOI 10.1007/s00023-008-0387-3


A Functional Integral Representation for Many Boson Systems I: The Partition Function Tadeusz Balaban, Joel Feldman, Horst Knörrer, and Eugene Trubowitz Abstract. We derive a functional integral representation for the partition function of a many Boson system for which the configuration space consists of finitely many points.

1. Introduction We are developing a set of tools and techniques for analyzing the large distance/infrared behaviour of a system of identical bosons, as the temperature tends to zero. The total energy of the many boson systems considered in this paper has two sources. First, each particle in the system has a kinetic energy. We shall denote the 1 Δ, corresponding quantum mechanical observable by h. The most common is − 2m but, in this paper, we allow any positive operator. Second, the particles interact with each other through a two–body potential, v(x, y). For stability, v is required to be repulsive. We assume that the system is in thermodynamic equilibrium and that expectations of observables are given by the grand canonical ensemble at 1 > 0 and chemical potential μ. temperature T = kβ Functional integrals are an important source of intuition about the behaviour of quantum mechanical systems. They are also an important rigorous technical tool in the analysis of, for example, Euclidean quantum field theories. In this paper and its companion [2], we derive rigorous functional integral representations for the partition function and thermodynamic correlation functions of a many boson system. There are many possible applications of our functional integral representations. However, we are motivated by the following potential specific application.

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One may speculate (in agreement with the standard picture of condensed matter physics) that, at temperature zero and infinite volume, a weakly coupled, three dimensional many boson system will undergo a phase transition at some critical chemical potential μ∗ . ◦ For μ < μ∗ , the system is in a massive phase. That is, all correlation functions decay exponentially fast at large separation. The expected value of the field φ(x) = a† (x), where a† (x) is the particle creation operator at x, is zero. ◦ For μ > μ∗ , number symmetry is broken. In this phase, correlation functions fail to decay exponentially due to the presence of extended, collective excitations (the massless Goldstone bosons). The expected value of the field is nonzero. The presence of such anomalous nonzero amplitudes is used as a general criterion for a condensed quantum fluid. The intuition behind this phase transition is easily obtained by using a formal coherent state functional integral [6, (2.66)] to express the grand canonical partition function as dφ∗τ (x) dφτ (x) A(φ∗ ,φ) e (1.1) Z = ··· 2πi x∈IR3 φβ = φ0 0≤τ ≤β where

∗

A(φ , φ) =

β

dτ 0

and ∗

K(α , φ) =

3

d x IR3

φ∗τ (x)

∂ φτ (x) − ∂τ

∗

dτ K φ∗τ , φτ

β

0

dxdy α(x) h(x, y)φ(y) − μ dx α(x)∗ φ(x) 1 + dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) . 2

Here h(x, y) is the kernel of the kinetic energy operator. In the mean field approximation, that is, when φτ (x) is independent of τ and x, the action A(φ∗ , φ) is minus the integral over τ and x of the 1 vˆ(0)|φ|4 − μ|φ|2 2 where vˆ(0) = dy v(x, y). We have assumed that v(x, y) is translation invariant and that h annihilates constants. The minimum of the naive effective potential is ◦ nondegenerate at the point φ = 0 when μ < 0 and μ when μ > 0. ◦ degenerate along the circle |φ| = vˆ(0) “naive effective potential” =

It is therefore reasonable that an attempt to rigorously justify the phase transition in the chemical potential discussed above would begin with the derivation of rigorous functional integral representations of the thermodynamic correlation functions in which the effective potential appears explicitly. We do so in this paper and the companion paper [2].

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It is common practice in condensed matter physics, to discretize space, because the overall energy scale is low. On physical grounds, this does not affect the long range behaviour of the system. For this paper, space is an arbitrary, but fixed, finite set X, that we may imagine is a subset of a lattice. The second quantized Hamiltonian H and the number operator N act on the infinite dimensional Fock space ∞ n (C|X| /Sn ) . F= n=0

The long distance behavior of the system is revealed in the thermodynamic limits of grand canonical correlation functions, such as derivatives of the partition function Z = Tr F e−β(H−μN ) . To implement the thermodynamic limit, one would take the usual family of finite spaces XL = {x ∈ ZZ3 | |xi | < L , i = 1, 2, 3} and send L to infinity. We shall not do so in this paper. Our first result (Theorem 3.13), stated somewhat informally, is the representation β ∗ dμR(p) (φ∗τ , φτ ) eF ( p , φ ,φ) (1.2) Tr e−β(H−μN ) = lim p→∞

τ ∈Tp

for the finite volume grand canonical partition function. Similar representations for general correlation functions are derived in [2]. Here, for each natural number p, the discrete time interval Tp is given by

β Tp = τ = q |q = 1 , . . . , p . p For each point (x, τ ) in the discrete space–time X × Tp , we have introduced the complex variable φ(τ, x) = φτ (x). For each r > 0, the measure dφ∗ (x) ∧ dφ(x) χr |φ(x)| dμr (φ∗ , φ) = 2πı x∈X

where, χr is the characteristic function of the closed interval [0, r]. The sequence R(p) > 0 in (1.2) tends to infinity at an appropriate rate as p → ∞. The “action” F(ε, φ∗ , φ), with ε = βp , is given by ∗ ∗ ε dτ dx φτ (x)(∂ φτ )(x) − dτ dx φ∗τ (x)(hφτ )(x) F(ε, φ , φ) = + μ dτ dx φτ (x)∗ φτ (x) 1 dτ dxdy φτ (x)∗ φτ (x) v(x, y) φτ (y)∗ φτ (y) − 2

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where

dτ dx ψ(τ, x) = ε


ψ(τ, x)

τ ∈ Tp x∈X

dτ dxdy ψ(τ, x, y) = ε

ψ(τ, x, y)

τ ∈ Tp x∈X y∈X

and the difference operator ∂ ε acts by ∂ ε φ(τ, x) = ε−1 φ(τ + ε, x) − φ(τ, x) . In (1.2), fields φτ with τ ∈ / Tp are determined by the periodicity condition φτ = φτ −β . It is easy to check that the representation (1.2) generates the usual formal graphical perturbation series. In the physics literature, coherent states1 | φ , φ ∈ CX , the formal resolution of the identity ∗ dφ (x)dφ(x) −|φτ (x)|2 e 1l = |φφ| (1.3) 2πı x∈X

and the formal trace formula ∗ dφ (x)dφ(x) −|φτ (x)|2 e φ |B | φ Tr B = 2πı

(1.4)

x∈X

are used to justify (1.1) as follows. Formally, Tr e−β(H−μN ) β

β

β

= lim Tr e− p (H−μN ) 1le− p (H−μN ) 1l · · · 1le− p (H−μN ) p→∞ ∗ dφτ (x)dφτ (x) −|φτ (x)|2 e = lim p→∞ 2πı x∈X τ ∈Tp \{β}

⎡

× Tr ⎣

⎤ −β p (H−μN )

e

| φτ φτ |⎦ e− p (H−μN ) β

τ ∈Tp \{β}

β dφ∗ (x) dφτ (x) τ −|φτ (x)|2 φτ e− p (H−μN ) φτ + β . e = lim p p→∞ 2πı x∈X τ ∈Tp

τ ∈Tp

(1.5)

are using coherent so that α | γ = e states normalized the convention that dx f (x) = x∈X f (x).

1 We

dy α(y) γ(y) .

We systematically use

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Then, one argues [6, (2.59)] that

α e−ε (H−μN ) φ = α .. e−ε (H−μN ) .. φ + O(ε2 ) (1.6)

= exp dy α(y)∗ φ(y) − εK(α∗ , φ) + O(ε2 )

where : · : denotes Wick ordering. If ε = βp , one observes that there are O(p) error terms of order O(ε2 ) = O( p12 ). “Off the cuff”, the error terms do not contribute when p → ∞ and consequently, Tr e−β(H−μN ) = lim

p→∞

×

dφ∗ (x) dφτ (x) τ 2πı

x∈X τ ∈Tp dy φτ (y)∗ [φτ +β/p (y)−φτ (y)] −

e

β ∗ p K(φτ ,φτ +β/p )

.

τ ∈Tp

It is the purpose of this paper, and its companion, to make this all precise. In §II, we review the basic formalism of bosonic quantum statistical mechanics and the formalism of coherent states in the context of a finite configuration space X. In particular, we prove, in Theorem 2.26, a rigorous version of the formal resolution of the identity (1.3) and we prove, in Proposition 2.28, that the trace formula (1.4) applies rigorously to a certain class of operators. In this way, we obtain a rigorous variant of (1.5). See Theorem 3.1. It is by no means clear that dropping, “off the cuff”, the O(p) error terms of order O(ε2 ) is justified, because the error terms are unbounded functions of the fields φτ . We circumvent this part the formal argument by directly constructing of the logarithm F (ε, α∗ , φ) = ln α e−εK φ , at least for α and φ not too large. See Proposition 3.6. To this end, we derive and then solve an evolution equation in ε for F (ε, α∗ , φ). It follows that F (ε, α∗ , φ) =

dx α(x)∗ φ(x) − εK(α∗ , φ) + O(ε2 ) .

X

We then show, in Theorem 3.13, that the “matrix element” α e−ε (H−μN ) φ ∗ ∗ can be replaced by e dy α(y) φ(y) − εK(α ,φ) in the formula for Tr e−β(H−μN ) of Theorem 3.1, provided the integration radius R(p) of (1.2) is chosen appropriately. In the physics literature, one simply “evaluates” the limit lim

p→∞

τ ∈Tp

β

∗

dμR(p) (φ∗τ , φτ ) eF ( p , φ

,φ)

=

···

φ β = φ0

dφ∗ (x) dφτ (x) ∗ τ eAX (φ ,φ) 2πi x∈X

0≤τ ≤β

1234

where

T. Balaban et al.


∂ ∗ ∗ AX (φ , φ) = dτ dx φτ (x) φτ (x) − φτ (x)(h − μ)φτ (x) ∂τ 0 X β 1 − dτ dxdy φτ (x)∗ φτ (y)∗ v(x, y) φτ (x)φτ (y) . 2 0 2 X ∗

β

The first impulse of a mathematical physicist determined to ascribe a rigorous meaning to this formal functional integral representation for the partition function, would be to construct a “complex Gaussian measure” dμC , with covariance −1 ∂ + (h − μ) C= − ∂τ out of the formal measure

β dφ∗τ (x) dφτ (x) ∂ ∗ ∗ . exp dτ dx φτ (x) φτ (x) − φτ (x)(h − μ)φτ (x) ∂τ 2πi 0 X x∈X 0≤τ ≤β

Normally, one starts by defining the integral of any polynomial in the complex fields φ∗τ (x), φτ (x), τ ∈ [0, β), x ∈ X, against dμC as cumulants of “matrix elements” of the covariance C. Then one constructs the characteristic function of dμC , as a limit of integrals of polynomials, and the corresponding measure. However, the explicit calculations in Appendix A, modelled on those of Cameron [3], show that the purely imaginary term β ∂ dτ dx φ∗τ (x) φτ (x) ∂τ 0 X in the exponential generates oscillations that are so severe that there is no complex Gaussian measure. To work with the ultraviolet limit β ∗ dμR(p) (φ∗τ , φτ ) eF ( p , φ ,φ) lim p→∞

τ ∈Tp

to, for example, construct the thermodynamic limit and justify the phase transition in the chemical potential, one must exploit the cancellations arising from the oscillations generated by the purely imaginary term in the action. One methodology for the explicit control of cancellations of this kind, when the coupling constant λ = vˆ(0) is small, is known as “multiscale analysis”. In the present case “scales” refer to blocks of frequencies in space x and inverse temperature τ . There are infinitely many scales. Cancellations are implemented at each scale. Typically, the total contribution of “large fields”, for example field configurations with φτ (x) or appropriate derivatives large, is smaller than any power of λ, reminiscent of large deviations in probability theory. This can be proven without attention to cancellations. On the other hand, oscillations are fully exploited in

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the complementary “small field regions” by high dimensional steepest descent calculations around the complex critical points of the effective actions. In the end the functional integral becomes an infinite sum over small and large field regions. The physicists formal functional integral is morally “the dominant term”.

2. Finite systems of bosons In this section we carefully review the basic formalism of bosonic quantum statistical mechanics and introduce the notation that we will systematically use. Fock space Fix a finite set X. Definition 2.1. (i) Let n ∈ IN ∪ {0}. The action π·f of a permutation π ∈ Sn on f in L2 X n is given by π·f (x1 , . . . , xn ) = f (xπ−1 (1) , . . . , xπ−1 (n) ) . The bosonic n-particle space Bn (X) = f ∈ L2 X n π·f = f for all π ∈ Sn is the n+|X|−1 dimensional complex Hilbert space of all symmetric funcn tions on X n with inner product dx1 . . . dxn f (x1 , . . . , xn )g(x1 , . . . , xn ) . f, gBn = Xn

In particular, B0 (X) = C and B1 (X) = L2 (X). (ii) The bosonic Fock space B(X) over X is the orthogonal direct sum B(X) = n≥0 Bn (X). It is an infinite dimensional complex Hilbert space. The inner product between f and g in B(X) is

f , gB = fn , gn Bn . n≥0

Definition 2.2. (i) Let h be a (single particle) operator on L2 (X), with kernel h(x, y). Assume that h(y, x) = h(x, y)∗ so that h is self–adjoint. The corresponding independent particle operator on Bn (X) is H0 (h, n, X) =

n

h(i) .

i=1 (i)

The superscript on h , i = 1, . . . , n, indicates that the single particle operator h acts on the variable xi appearing in a function g(x1 , . . . , xn ). That is, (i) dxi h(xi , xi ) g(x1 , . . . , xi , . . . , xn ) . h g (x1 , . . . , xn ) = X

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By convention, H0 (h, 0, X) = 0. The kernel of H0 (h, n, X) is H0 (h, n, X)(x1 , . . . xn , y1 , . . . yn ) =

n

δx1 (y1 ) . . . δxi−1 (yi−1 ) h(xi , yi ) δxi+1 (yi+1 ) . . . δxn (yn ) .

i=1

x=y is the delta function on X concentrated at Here δx (y) = δx,y = 10 if if x =y the point x. The second quantization of h is the direct sum H0 (h, X) = H0 (h, n, X) n≥0

! ! ! H0 (h, n, X) fn !2 < ∞ . acting on the domain f ∈ B(X) n≥0 Bn (ii) Let v(x, y) be a real valued (two body) potential on X 2 satisfying v(x, y) = v(y, x) for all x, y ∈ X. Multiplication by v determines a (two particle) operator on B2 (X). Its kernel is v(x1 , x2 ) δx1 (y1 )δx2 (y2 ). The corresponding 2 particle interaction operator on Bn (X) is

V(v, n, X) = v (i1 ,i2 ) .

i1 0. If v(x, y) is the kernel of a strictly positive definite operator acting on L2 (X), then v is repulsive with λ0 (v) at least as large as the smallest eigenvalue of the operator. If v(x, y) ≥ 0 for all x, y ∈ X and v(x, x) > 0 for all x ∈ X, then v is repulsive with λ0 (v) ≥ minx∈X v(x, x). Proposition 2.17. Let h be a single particle operator and v(x1 , x2 ) be a real, symmetric, pair potential. Assume that the self adjoint operator v acting on L2 (X) with kernel v(x, y) is strictly positive definite. Then −

1 v02 |X| ≤ 8λ0 2

λ0

N 1 − v0 N ≤ V(v, X) ≤ (ΛN − λ0 )N |X| 2 λ0 N ≤ H0 (h, X) ≤ Λ N

on the domain D(V ) = D(N 2 ). Here, λ0 = λ0 (v), Λ is the largest eigenvalues of the operator v, λ0 and Λ are the smallest and largest eigenvalues of the operator h and v0 = maxx∈X v(x, x). The leftmost bound on V is called the ‘linear lower bound’. Proof. We have )

*

δY ,

dx1 dx2 n(x1 )v(x1 , x2 )n(x2 ) δY

dx1 dx2 μY (x1 )v(x1 , x2 )μY (x2 )

=

X2

X2

≥ λ0

dx μ2Y (x) X

2 λ0 dx μY (x) |X| X λ0 2 n = |X| ≥

where Schwarz’s inequality was used in the third line. Similarly, ) δY , X2

* dx1 dx2 n(x1 )v(x1 , x2 )n(x2 ) δY

≤ Λ n2 .

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Let u = Y ∈X n /Sn ϕY δY be an arbitrary element of Bn (X). We have ) * u, dx1 dx2 n(x1 )v(x1 , x2 )n(x2 ) u X2 ) *

∗ = ϕY1 ϕY2 δY1 , dx1 dx2 n(x1 )v(x1 , x2 )n(x2 ) δY2 Y1 ,Y2 ∈X n /Sn

=

|ϕY |2

Y ∈X n /Sn

≥ =

λ0 2 n |X|

X2

dx1 dx2 μY (x1 )v(x1 , x2 )μY (x2 ) X2

|ϕY |2

Y ∈X n /Sn

λ0 2 n u2Bn . |X|

In the same way, ) u,

* dx1 dx2 n(x1 )v(x1 , x2 )n(x2 ) u

X2

≤ Λ n2 u2Bn .

Since a ≤ v(x, x) ≤ b, a similar, but simpler, argument gives ) * 2 dx v(x, x)n(x) u ≤ v0 n u2Bn . λ0 n uBn ≤ u , X

The bound on H0 (h, X) follows directly from Definition 2.2.i.

Coherent states We now review the formalism of coherent states [6, §1.5]. Definition 2.18. (i) The family | zδx , z ∈ C, of coherent states concentrated at x ∈ X is given by

1

1 ⊗n z n ψ † (x)n 1 = z n δx s | zδx = n! n! n≥0

n≥0

where 1 is the ‘vacuum’ 1 ∈ C = B0 (X). That | zδx ∈ B(X) is a consequence of Proposition 2.22, below. (ii) If φ(y) ∈ L2 (X), the coherent state | φ ∈ B(X) is + dy φ(y)ψ † (y) |φ = | φ(y)δ = e 1. y s y∈X

Lemma 2.19. For all φ in L2 (X),

|φ = n≥0

Here, φ(Y ) =

( y∈X

φ(Y ) cY δY .

Y ∈ X n /Sn

φμY (y) (y) for each Y in the quotient space X n /Sn .

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Proof. ⎛ ⎞ 1 +

+ 1 ny n ⊗ n ny s ⎝ ⎠ φ φ (y) δ = (y) δ ⊗s |φ = y s s y n! n ! y ny ≥0 y∈X

=

n≥0

= =

y∈X

n≥0

χ ⎝n =

ny ≥0 y∈X

n≥0

Y ∈ X n /Sn

n≥0

Y ∈ X n /Sn

y∈X

⎛

⎞⎛ ny ⎠ ⎝

y∈X

⎛ ⎝

y∈X

⎞

1 + ny φny (y)⎠ δ ⊗s s y ny !

y∈X

y∈X

⎞ φμY (y) (y)⎠ c2Y

y∈X

+

μ

Y δ ⊗s s y y∈X

(y)

φ(Y ) cY δY .

Coherent states have been defined to give Proposition 2.20. For all x ∈ X and φ ∈ L2 (X), ψ(x) | φ = φ(x) | φ ψ † (x) | φ =

∂ |φ . ∂φ(x)

Convention 2.21. For any φ in L2 (X) and any state f in the Fock space B(X), abusing the inner between the coherent state | φ and f is written product notation, as φ f . That is, φ f = | φ , f B Similarly, the inner product between the coherent states | φ and φ is written φ | φ . ! ! Proposition 2.22. For all α and γ in L2 (X), we have !P (m) | α !

Bm

αγ = e

=

α m √ m!

and

dy α(y) γ(y)

where P (m) is the orthogonal projection from B(X) onto the m particle subspace Bm (X). Lemma 2.23. For any single particle operator h and any φ in L2 (X), e−τ H0 (h,X) | φ = e−τ h φ . Proof. By Proposition 2.14, d −τ H0 e |φ = − dτ

dx1 dx2 ψ † (x1 ) h(x1 , x2 ) ψ(x2 ) e−τ H0 | φ

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and, by Definition 2.18.ii, −τ h † d −τ h ∂ e e dy (e φ)(y) ψ (y) 1 φ = dτ ∂τ −τ h φ)(y) ψ † (y) = dy − he−τ h φ (y) ψ † (y) e dy (e 1 = − dy ψ † (y) h e−τ h φ (y) e−τ h φ = − dx1 dx2 ψ † (x1 ) h(x1 , x2 ) ψ(x2 ) e−τ h φ . d −τ H0 d −τ h e e | φ and dτ φ satisfy the same first order ordinary differential As dτ equation and initial condition, they coincide. Lemma 2.24. Suppose both f and N 2|X| f belong to the Fock space B. Then, ⎛ ⎞ ! 2! 1 φf ≤ ⎝ ⎠ 2|X| e 12 φ !(N + 4|X|)2|X| f ! . 4 B 1 + |φ(y)| y∈X

Proof. Fix P in X n /Sn . Then, φ(P ) φ f = CY φ(P ) φ(Y ) δY , f B Y

=

)

CP Y

Y

=

CP Y

Y

=

CP Y

* CY φ(P Y ) δY , f CP Y B 21 01 3 μP Y (x)! 2 φ(P Y ) δY , f μY (x)! x∈X B 0 3 μP (x) 12 n(x) + k δY , f φ(P Y )

Y

=

Y

0 CP Y φ(P Y )

x∈X

δY ,

k=1

1 n(x) + k 2 f

μP (x)

x∈X

k=1

3

It follows from Schwarz’s inequality and Parseval’s identity that & ' 12 ! ! P (x)

! μ 1 ! CP Y φ(P Y )2 n(x) + k 2 φ(P ) φ f ≤ ! !x∈X k=1 Y ! & ' 12 ! ! ! μ P (x)

! 2 12 ! ! ! CY φ(Y ) n(x) + k ≤ f ! ! ! !x∈X k=1 Y B 2! 1 n ! φ ! ! 2 2 (N + n) f B . ≤e

B

. B

! ! ! f! ! !

B

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Adding,


! 2! 1 φf 1 + |φ(y)|4 ≤ 2|X| e 2 φ !(N + 4|X|)2|X| f !B .

y∈X

An approximate resolution of the identity In the physics literature, the formal resolution of the identity ∗ 2 dφ (x)dφ(x) 1l = e− dy |φ(y)| | φ φ | 2πı x∈X

is often used. See, for example [6, (1.123)]. However, for each φ ∈ L2 (T ), the opera − dy |φ(y)|2 tor norm of e | φ φ | is exactly one. So the nature of the convergence of the integral in the above formal resolution of the identity is not clear. We now investigate the convergence more carefully. Definition 2.25. For each r > 0, the measure dμr (φ∗ , φ) on L2 (X) is given by dφ∗ (x)dφ(x) dμr (φ∗ , φ) = χr |φ(x)| 2πı x∈X

where χr is the characteristic function of the interval [0, r]. The measure dμ(φ∗ , φ) on L2 (X) is given by dφ∗ (x)dφ(x) ∗ dμ(φ , φ) = . 2πı x∈X

Theorem 2.26. For each r > 0, let Ir be the operator that acts on f in B(X) by 2 Ir f = dμr (φ∗ , φ) e− φ | φ φ f . (a) For all n ≥ 0 and all Y ∈ X n /Sn , Ir δY = λr (Y ) δY where λr (Y ) =

Γr (μY (x)) μY (x)!

x∈X

with

Γr (s) =

r2

dt e−t ts ,

for all

s > −1 .

0

In particular, 0 ≤ λr (Y ) ≤ 1. (b) Ir commutes with N . (c) The operator norm of Ir is bounded by one for all r and Ir converges strongly to the identity operator as r → ∞. (d) For all n and r, the operator norm ! ! ! 1l − Ir Pn ! ≤ |X| 2n+1 e−r2 /2 . Here, Pn is the orthogonal projection from B(X) onto the direct sum Bm (X). 0≤m≤n

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(e) Suppose that N 2|X| f belongs to the Fock space B. Then, 2 f = dμ(φ∗ , φ) e− φ | φ φ f with the integral converging absolutely. Proof. (a,b) To verify (a), let Y ∈ X n /Sn and observe that 2 Ir δY = dμr (φ∗ , φ) e− dy |φ(y)| | φ φ δY

2 = δY cY cY dμr (φ∗ , φ) e− dy |φ(y)| φ(Y ) φ∗ (Y ) . n≥0 Y ∈X n /Sn

We have 2 dμr (φ∗ , φ) e− dy |φ(y)| φ∗ (Y ) φ(Y ) dφ∗ (x)dφ(x) 2 = χr |φ(x)| e−|φ(x)| φ∗ (x)μY (x) φ(x)μY (x) 2πı x∈X r 2π dθ −ρ2 μY (x)+μY ıθ(μY (x)−μY (x)) = ρe ρ dρ (x) e π 0 0 x∈X r −ρ2 2μY (x) = dρ 2ρ e ρ δμY (x),μY (x) x∈X

= δY,Y

0

dt e−t tμY (x)

0

x∈X

= δY,Y

r

2

Γr μY (x) .

x∈X

Consequently,

Ir δY = δY c2Y

x∈X

Γr (μY (x)) . Γr μY (x) = δY μY (x)! x∈X

Thus Ir is a diagonal operator in the orthonormal basis {δY }. Each diagonal entry 1 2 r2 1 −t μY (x) λr (Y ) = dt e t μY (x)! 0 x∈X

is between 0 and 1. Since {δY } is a basis of eigenvectors for both Ir and N , they commute, which proves parts (a) and (b). (c) Each λr (Y ) =

x∈X

1

1 μY (x)!

0

r2

dt e−t tμY (x)

2 (2.2)

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approaches 1 in the limit r → ∞. By the Lebesgue dominated convergence theorem, if f = n Y ∈X n /Sn fY δY is any vector in B(X), then

!2 ! 2 lim !f − Ir f ! = lim 1 − λr (Y ) |fY |2 = 0 . r→∞

r→∞

n Y ∈X n /Sn

The operator norm of Ir is bounded by 1 for all r > 0, because all of the eigenvalues of Ir are between 0 and 1. This completes the proof of part (c). (d) We bound ! ! $ % ! 1l − Ir Pn ! = max 1 − λr (Y ) . |Y |≤n

Fix any Y with |Y | ≤ n and set, for each x ∈ X, ∞ 1 βx = dt e−t tμY (x) . μY (x)! r2 By (2.2) 1 − λr (Y ) = 1 −

(1 − βx ) ≤

x∈X

x∈X

βx ≤ |X| max βx . x∈X

The claim now follows from μY (x) ∞ 2 t 2μY (x) ∞ βx = dt e−t ≤ 2μY (x) dt e−t/2 = 2μY (x)+1 e−r /2 μY (x)! r2 2 r2 ≤ 2n+1 e−r

2

/2

(e) By definition Ir f =

.

dμ(φ∗ , φ)

2 χr |φ(x)| e− φ | φ φ f .

x∈X

By part (c) the left hand side converges to f as r → ∞. By Lemma 2.24, the norm of the integrand of the right hand side is bounded by ⎛ ⎞ ! ! 1 ⎝ ⎠ 2|X| !(N + 4|X|)2|X| f ! 4 B 1 + |φ(y)| y∈X

which is integrable with respect to dμ(φ∗ , φ). Hence, as r → ∞, the right 2 hand side converges to dμ(φ∗ , φ) e− φ | φ φ f . The trace formula Another commonly used formal property of coherent states is the trace formula (1.4). We now develop a rigorous, but limited, version of this formula that is adequate for our purposes. Remark 2.27. We use the approximate identity Ir to prove a “cutoff” trace for any bounded operator that computes with N . By (2.2), for each fixed Y ∈ X n /Sn ,

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limr→∞ λr (Y ) = 1. On the other hand, for each fixed r, there is a constant Cr,X = r2 |X| such that 1 n max λr (Y ) ≤ Cr,X . n! Y ∈X n /Sn 1 n Thus the operator norm of Ir , restricted to Bn (X), is bounded by n! Cr,X . The n dimension of Bn (X) is bounded by |X| . Therefore, for any bounded operator B that commutes with N , BIr is trace class and

Tr BIr = lim Tr BIr Pn . n→∞

Proposition 2.28. (a) Let B be a bounded operator on B(X) that commutes with N . Then, for all r > 0, BIr is trace class and 2 Tr BIr = dμr (φ∗ , φ) e− dy |φ(y)| φ | B | φ . (b) Let B be any operator on B(X) that commutes with N and obeys P (n) BBn ≤ const (1 + n)−2|X| . Then B is trace class and Tr B = dμ(φ∗ , φ) e−

dy |φ(y)|2

φ |B | φ .

Proof. (a) We have

Tr BIr Pn =

deg Y ≤ n

=

δY | BIr | δY

dμr (φ∗ , φ) e−

deg Y ≤ n

= =


dy |φ(y)|2

dy |φ(y)|2

δ Y | B | φ φ δY

φ δY δY | B | φ

deg Y ≤ n


dy |φ(y)|

2

φ | Pn B | φ .

2 Since B is a bounded operator, e− dy |φ(y)| φ | Pn B | φ is bounded uniformly in n and φ and the dominated convergence theorem provides the limit of the right hand side as n → ∞. By Remark 2.27, the left hand side converges to Tr BIr as n → ∞. (b) As in part (a), but using part (e) of Theorem 2.26, 2 Tr BPn = dμ(φ∗ , φ) e− dy |φ(y)| φ | Pn B | φ .

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By Lemma 2.24, ⎛ φ | Pn B | φ ≤ ⎝

⎞ ! ! 2 1 ! 2|X| ⎠ 2|X| e 12 φ ! (N + 4|X|) P B | φ ! ! n 1 + |φ(y)|4 B y∈X ⎛ ⎞ ! ! 2 1 2|X| ! ⎠ 2|X| ! + 4|X|) ≤ e φ ⎝ B !(N !. 1 + |φ(y)|4 y∈X

By the Lebesgue dominated convergence theorem 2 lim dμ(φ∗ , φ) e− dy |φ(y)| φ | Pn B | φ n→∞ = dμ(φ∗ , φ) e−

dy |φ(y)|2

φ |B | φ .

|X|−1

(n+|X|−1) and the operator Since the dimension of Bn (X) is (n+|X|−1)! n!(|X|−1)! ≤ (|X|−1)! norm of the restriction of B to Bn (X) is bounded by a constant times (1 + n)−2|X| , B is trace class and

lim Tr BPn = Tr B .

n→∞

3. An integral representation of the partition function Let h be a single particle operator on X and v(x1 , x2 ) a real, symmetric, pair potential which is repulsive in the sense of Definition 2.16. For the rest of this paper, except where otherwise stated, we write K = K(h, v, X, μ) = H0 (h, X) + V(v, X) − μN . Recall that H0 (h, X) and V(v, X) were defined in Definition 2.2. The first step in the formal derivation of the functional integral representation (1.1) is the application of the resolution of the identity (1.3) and the trace formula (1.4) to give the intermediate representation (1.5). Theorem 3.1, below, is a rigorous version of (1.5). Theorem 3.1. Suppose that the sequence R(p) obeys lim p e− 2 R(p) = 0 . 2

1

p→∞

Then, Tr e−βK = lim

p→∞

4 τ ∈Tp

dμR(p) (φ∗τ , φτ ) e−

where Tp = τ = qε q = 1, . . . , p φ0 = φpε = φβ .

dy |φτ (y)|2

5

φτ −ε e−ε K φτ

τ ∈Tp

and we use the conventions that ε =

β p

and

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In Proposition 3.2, below, we prove that the grand canonical partition function Tr e−βK is well–defined. Then we prove Lemma 3.4, which provides a rigorous multiple insertion of the approximate resolution of the identity in our context. The proof of Theorem 3.1 then follows Remark 3.5. Proposition 3.2. For any β > 0, the trace Tr e−β(H0 (h,X)+zV(v,X)−μN ) on the Fock space B(X) is a holomorphic function of (z, μ) on 0 × C.

z ∈ C Re z >

Proof. Suppressing h, v and X,

Tr e−β(H0 +zV−μN ) = Trn e−β(H0 +zV−μN ) = eβμn Trn e−β(H0 +zV) n≥0

(3.1)

n≥0

where Trn denotes the trace on the n particle space Bn . When restricted to Bn , the Hamiltonian H0 + zV is an operator on a finite dimensional vector space. Therefore, each term Trn e−β(H0 +zV) is an entire function of (z, μ) on C2 . For each n ≥ 0, ! βμn ! e Trn e−β(H0 +zV) ≤ eβ|Re μ|n dim Bn (X) !e−β(H0 +zV) !Bn where · Bn is the operator norm on Bn (X). By the Trotter product formula, !6 β 7m ! ! −β(H +zV) ! β ! − m (H0 +Re zV) −ı m Im zV 0 !e ! = lim ! e e ! ! Bn m→∞ Bn ! − β (H +Re zV) !m ! −ı β Im zV !m 0 ! ! ! ! m m e ≤ lim e Bn Bn m→∞ ! − β (H +Re zV) !m 0 ! ! = lim e m . B m→∞

n

By Proposition 2.17, λ0 ! − β (H +Re zV) !m 1 n2 −v0 n)] !e m 0 ! ≤ e−β[λ0 n+ 2 Re z( |X| Bn with the λ0 = λ0 (v) of Definition 2.16. Since dim Bn (X) ≤ |X|n , the sum in (3.1) is absolutely convergent, uniformly for (z, μ) in compact subsets of z ∈ C Re z > 0 × C. This gives the desired analyticity. Remark 3.3. Observe that Tr e−β(H0 (h,X)+zV(v,X)−μN ) = ∞ when z = 0 and μ is strictly bigger than the smallest eigenvalue of h. This indicates that the “free” limit z 0 is extremely singular. Lemma 3.4. Let β > 0 and let K be any self adjoint operator on B(X) that commutes with N and obeys N K≥a −ν N |X|

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for some constants a > 0 and ν. Also, let R be any map from IN to (0, ∞) such that 2 1 lim p e− 2 R(p) = 0 . p→∞

Then,

7p−1 β 6 β Tr e−βK = lim Tr e− p K IR(p) e− p K p→∞

p exponentials e

−

β K p

8 9: ; −β K −β K −β K p p p = lim Tr e IR(p) e IR(p) · · · IR(p) e p→∞

and

6 β 7p−1 β φ e−βK φ = lim φ e− p K IR(p) e− p K φ p→∞

for all φ, φ ∈ L2 (X). Furthermore, for each 0 < η < 1, there is a constant Cη , depending on η, β, a and ν, but independent of φ, φ ∈ L2 (X), p ∈ IN and X such that 6 β 7p−1 β η 2 2 e− p K φ ≤ e 2 ( φ + φ ) eCη |X| . φ e− p K Ir Proof. Introduce the local notation β e− p K if i is odd Ai = if i is even IR(p) so that

6

β

e− p K IR(p)

7p−1

β

e− p K =

β e− p K Bi = 1l

2p−1 i=1

Ai

if i is odd if i is even

and e−βK =

2p−1

Bi .

i=1

For any n ∈ IN, &2p−1 ' −βK Ai − e Tr i=1 &2p−1 ' 2p−1 ≤ Tr Ai − e−βK Pn + Tr Ai (1l − Pn ) + Tr e−βK (1l − Pn ) i=1 i=1 &2p−1 ' 2p−1 (3.2) = Tr Ai − Bi Pn i=1 i=1 2p−1 + Tr (3.3) Ai (1l − Pn ) + Tr e−βK (1l − Pn ) . i=1 ! ! Consider the first line, (3.2). Since !IR(p) ! ≤ 1, by part (c) of Theorem 2.26, and a a K≥ (N − ν|X|)N ≥ − |X|ν 2 ≡ −K0 |X| 4

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Bosonic Integral I

we have

β e p K0 Ai , Bi ≤ 1

1251

if i is odd if i is even .

Since Aq − Bq = IR(p) − 1l for i = 2, 4, . . . , 2p − 2 and is zero otherwise, we have, for all n , p ∈ IN , ! !&2p−1 ' ! 2p−1 ! !q−1 ! 2p−1 2p−1 ! !

! ! ! ! ! Ai − Bi Pn ! ≤ Ai Aq − Bq Bi Pn ! ! ! ! ! ! ! q = 1 ! i=1 i=1 i=1 i=q+1 ! ! ≤ (p − 1)e K0 β ! IR(p) − 1l Pn ! ≤ pe K0 β |X| 2n+1 e−R(p)

2

/2

.

Consequently, 66 β 7p−1 β 7 2 e− p K − e−βK Pn ≤ pe K0 β |X| 2n+1 e−R(p) /2 Tr Pn Tr e− p K IR(p) ≤ pe K0 β |X| 2n+1 e−R(p)

2

/2

(n + |X|)! . n!|X|!

Hence, for any fixed n ∈ IN, 66 β 7p−1 β 7 e− p K − e−βK Pn = 0 . lim Tr e− p K IR(p) p→∞

m Now we consider the second line, (3.3). For all m ≥ 1, K B ≥ a |X| −ν m m

and

! !2p−1 ! ! ! ! m ! ! ! ! Ai ! , !e−βK ! ≤ e− βa( |X| −ν) m ! ! Bm ! Bm

(3.4)

i=1

and it follows that 2p−1 m (m + |X| − 1)! . Ai Bm , Tr e−βK Bm ≤ e− βa( |X| −ν) m Tr m!(|X| − 1)! i=1

n If we impose the stronger condition m ≥ n with |X| ≥ 2ν , the last inequality becomes 2p−1 2 a (m + |X| − 1)! Ai B , Tr e−βK B ≤ e− 2 |X| m β Tr m m m!(|X| − 1)! i=1

and (m + |X| − 1)! (m + |X| − 1)|X|−1 (c m)|X|−1 ≤ ≤ m!(|X| − 1)! (|X| − 1)! (|X| − 1)!

(3.5)

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where, c =

1 2ν


+ 1 . Now, we have, if n ≥ 2|X|ν, 2p−1 Ai (1l − Pn ) , Tr e−βK (1l − Pn ) Tr i=1

≤

m>n ∞

≤

e− 2|X| m a

2

a

a − 4|X| n2 β

≤e

≤ Ce

β

dt e− 2|X| t

n

−

2

a 4|X|

∞

(c m)|X|−1 (|X| − 1)! β

(c t)|X|−1 (|X| − 1)!

dt e− 4|X| t

2

a

0 n2 β

β

(3.6) (c t)|X|−1 (|X| − 1)!

for some positive constant C, depending only on β, a, ν and |X|. The first claim now follows from the observation that this converges to zero as n → ∞, uniformly in R(p) and p. In fact this proves convergence in trace norm and hence convergence in operator norm and also weak convergence, so that this also proves the second claim. Finally, we prove the bound. By (3.4) and Proposition 2.22, ∞ 6 β 7p−1 β m m φm − βa( |X| −ν) m φ √ e √ e− p K φ ≤ φ e− p K IR m! m! m=0 ≤

∞

m tm − βa( |X| −ν) m e m! m=0

φ2 + φ 2 . Observe that, for any γ > 0, 2 2 m aβ a ν+γ − νm − γm = |X| − β(ν + γ)2 |X| aβ m− |X| |X| 2 4 aβ ≥ − (ν + γ)2 |X| . 4

where t =

Thus

1 2

∞ 6 β 7p−1 β m2 tm −aβγm −aβ( |X| −νm−γm) e e− p K φ ≤ e φ e− p K IR m! m=0

≤

∞

tm −aβγm aβ (ν+γ)2 |X| e e4 m! m=0

= ee

−aβγ

t

e

aβ 2 4 (ν+γ) |X|

.

It now suffices to choose γ so that η = e−aβγ and then set Cη =

aβ 4 (ν

+ γ)2 .

Remark 3.5. If R(p) ≥ c | ln p| , then limp→∞ pe−R(p) = 0. Also if R(p) ≥ 2 1 c | ln p| 2 with c > 1, then limp→∞ pe−R(p) = 0. 1 2 +ε

2

Vol. 9 (2008)

Bosonic Integral I

1253

Proof of Theorem 3.1. By (3.6), the strong convergence of Ir to 1l, and Proposition 2.28.a Tr e−βK = lim Tr e−βK IR(p) p→∞ = lim dμR(p) (φ∗0 , φ0 ) e−

dy |φ0 (y)|2

p→∞

φ0 e−βK φ0 .

It follows from Lemma 3.4 and the dominated convergence theorem that 4 5 2 dμR(p) (φ∗τ , φτ ) e− dy |φτ (y)| φτ −ε e−ε K φτ Tr e−βK = lim p→∞

τ ∈Tp

τ ∈Tp

−ε K φ The logarithm of α e Theorem 3.1 is a rigorous version of the intermediate representation (1.5). As discussed it now remains to show that one may replace the −εin introduction, ∗ ∗ α e K φ by e dy α(y) φ(y) − εK(α ,φ) in the formula for Tr e−βK of Theorem 3.1. In Theorem 3.13, below, we show that this is indeed the case, provided R(p) is chosen appropriately. To prepare for that, we explicitly find the logarithm F (ε, α∗ , φ) = ln α e−εK φ at least for α and φ not too large, and show that ∗ F (ε, α , φ) = dx α(x)∗ φ(x) − εK(α∗ , φ) + O(ε2 ) . as desired.

X

This expression is the same, to order ε, as dxdy α(x)∗ j(ε; x, y)φ(y) F (ε, α∗ , φ) = X2 1 dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) + O(ε2 ) − ε 2 X2 provided j(ε) = 1l − ε(h − μ) + O(ε2 ). For application of renormalization group methods, the latter form is more convenient. So we show it too. We typically use the supremum norm |φ|X = max |φ(x)| x∈X

to measure the size of the field φ and the norm h1,∞ = max dy |h(x, y)| x∈X

X

to measure the size of (symmetric) integral operators on L2 (X). Proposition 3.6. For each ε > 0, there is an analytic function F (ε, α∗ , φ) such that −εK φ = eF (ε,α∗ ,φ) α e %−1/2 $ on the domain |α|X , |φ|X < Cε where Cε = 8eε( h 1,∞ +μ+v0 ) εv1,∞ with v0 = maxx∈X v(x, x).

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Let cj > 0 and j(ε; x, y) be the kernel of an operator obeying ! ! !j(ε) − e−ε(h−μ) ! ≤ cj ε2 . 1,∞ Define the function F1 (ε, α∗ , φ) by ∗ dxdy α(x)∗ j(ε; x, y)φ(y) F (ε, α , φ) = 2 X ε − dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) + F1 (ε, α∗ , φ) . 2 X2 For every ε0 > 0 there is a constant const (depending only on ε0 , h1,∞ , v, cj and μ) such that for all 0 < ε < ε0 |F1 (ε, α∗ , φ)| ≤ const ε2 (R2 + v21,∞ R6 ) |X| for all |α|X , |φ|X ≤ R ≤ 12 Cε . An immediate consequence is Corollary 3.7. We use the notation of Proposition 3.6. Define the function F0 (ε, α∗ , φ) by F (ε, α∗ , φ) = dx α(x)∗ φ(x) − εK(α∗ , φ) + F0 (ε, α∗ , φ) X

where K(α∗ , φ) =

dxdy α(x)∗ h(x, y)φ(y) − μ dx α(x)∗ φ(x) 2 X X 1 ∗ ∗ dxdy α(x) α(y) v(x, y) φ(x)φ(y) . + 2 X2

For every ε0 > 0 there is a constant const (depending only on ε0 , h1,∞ , v and μ) such that for all 0 < ε < ε0 |F0 (ε, α∗ , φ)| ≤ const ε2 (R2 + v21,∞ R6 ) |X| for all |α|X , |φ|X ≤ R ≤ 12 Cε . We now prove a number of lemmas leading up to the proof of Proposition 3.6, following Lemma 3.11. Lemma 3.8. Let ε > 0. There exists a function F (ε, α∗ , φ), analytic in α∗ and φ in a neighbourhood of the origin, such that −εK φ = eF (ε,α∗ ,φ) . α e F satisfies the differential equation ∂ ∂ F ∂ F 1 ∗ ∂ F = −K α , dxdy α(x)∗ α(y)∗ v(x, y) F− ∂ε ∂α∗ 2 X ∂α(x)∗ ∂α(y)∗ with the initial condition ∗ dx α(x)∗ φ(x) . F (0, α , φ) = X

Vol. 9 (2008)

Here,

Bosonic Integral I

∂ K α , ∂α∗ ∗

1255

∂ ∂ dxdy α(x) h(x, y) −μ dx α(x)∗ = ∗ ∂α(y) ∂α(x)∗ X X ∂ 1 ∂ + dxdy α(x)∗ α(y)∗ v(x, y) . 2 X ∂α(x)∗ ∂α(y)∗

∗

Proof. Set λ0 = λ0 (v) as in Definition 2.16. Since, by Propositions 2.22 and 2.17, ∞ −εK αm −εK φm φ ≤ α e √ √ e Bm m! m! m=0

≤

∞ m

v m αm − ε( λ20 |X| +λ0 − 20 −μ) m φ √ e √ m! m! m=0 ∞

λ v m 1 +λ0 − 20 −μ) m 2 2 m − ε( 20 |X| α + φ e 2m m! m=0 the Taylor series expansion of α e−εK φ converges for all α, φ ∈ L2 (X) so that ∗ α e−εK φ is an entire function of α∗ and φ. Since α φ = e α (x)φ(x) dx = 0, the matrix element has the representation −εK φ = eF (ε,α∗ ,φ) α e

≤

in a neighbourhood of 0, with F (ε, α∗ , φ) is analytic in α∗ , φ. We have ∗

eF (ε,α

,φ) ∂

∂ε

∂ F (ε,α∗ ,φ) ∂ −εK φ = − α Ke−εK φ e α e = ∂ε) ∂ε = − α dxdy ψ † (x) h(x, y) ψ(y) − μ dx ψ † (x)ψ(x) X2 X * 1 dxdy ψ † (x)ψ † (y) v(x, y) ψ(x)ψ(y) e−εK φ . + 2 X2

F =

By Proposition 2.20, the first term * ) † −εK α dxdy ψ (x) h(x, y) ψ(y) e φ X2 = dxdy h(x, y) α ψ † (x)ψ(y)e−εK φ 2 X = dxdy α(x)∗ h(x, y) α ψ(y)e−εK φ 2 X −εK ∂ φ = α e dxdy α(x)∗ h(x, y) ∗ ∂α(y) X2 ∗ ∂ = dxdy α(x)∗ h(x, y) eF (ε,α ,φ) . ∗ ∂α(y) 2 X

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T. Balaban et al.


Treating the other two terms similarly, ∗ ∂ ∂ ∂ dxdy α(x)∗ h(x, y) − μ dx α(x)∗ eF (ε,α ,φ) F = − ∂ε ∂α(y)∗ ∂α(x)∗ X2 X ∗ ∂ 1 ∂ + dxdy α(x)∗ α(y)∗ v(x, y) eF (ε,α ,φ) . ∗ ∗ 2 X2 ∂α(x) ∂α(y)

The differential equation for F follows. Lemma 3.9. The function F (ε, α∗ , φ) of Lemma 3.8 has an expansion ∗

F (ε, α , φ) =

∞

n=1

˜ dn y ˜ enμ Fn (ε, x ˜, y ˜) dn x

X 2n

n

α(xi )∗ φ(yi )

i=1

where

˜ = (x1 , . . . , xn ) x ˜ = (y1 , . . . , yn ) y

in powers of the fields α∗ and φ. Furthermore F1 (ε, x1 , y1 ) = e−εh (x1 , y1 ) n−1

1 ε ˜ e−(ε−τ )Hn (˜ ˜) ˜, y ˜) = − x, x dτ dn x Sx˜ ,˜y Fn (ε, x 2 0 Xn m=1 ×

m n

˜ [≤m] , y ˜ [≤m] ) Fn−m (τ, x ˜ [>m] , y ˜ [>m] ) v(xj , xk )Fm (τ, x

j=1 k=m+1

where Sx˜ ,˜y denotes independent symmetrization in the x variables and the y variables, ˜) = Hn (˜ x, x

n

k=1

h(xk , xk )

δx (x ) +

1≤≤n =k

n 1 v(xj , xk ) δx (x ) 2 j,k=1 1≤≤n

j=k

and ˜ = (x1 , . . . , xn ) x

˜ [≤m] = (x1 , . . . , xm ) x

and

˜ [>m] = (xm+1 , . . . , xn ) . x

Proof. Expand F in the power series ∞ n

˜ enεμ Fn (ε, x ˜ , φ) α(xi )∗ F (ε, α∗ , φ) = dn x n=1 ∗

i=1

˜ , φ) that are symmetric in α with coefficients Fn (ε, x under permutation of the xk ’s. The constant term is zero because α e−εK φ α=φ=0 = 1. Each ˜ , φ) has degree n in φ because the fact that e−εK preserves particle number Fn (ε, x implies that F (ε, e−iθ α∗ , eiθ φ) = F (ε, α∗ , φ) for all real θ. Observe that 1 ∂ ∂ ∗ ˜ , φ) = enεμ Fn (ε, x · · · F (ε, α , φ) n! ∂α(x1 )∗ ∂α(xn )∗ α=0

Vol. 9 (2008)

and

Bosonic Integral I

1257

1 ∂ ∂ ∂ ∗ ∗ ··· dxdy α(x) h(x, y) F (ε, α , φ) ∗ ∗ ∗ n! ∂α(x1 ) ∂α(xn ) ∂α(y) X2 α=0 n

1 ∂ ∂ = dy h(xk , y) F (ε, α∗ , φ) n! X ∂α(y)∗ ∂α(x )∗ α=0 k=1 =k n

1 = dx h(xk , xk )enεμ Fn (ε, x1 , . . . , xk−1 , xk , xk+1 , . . . , xn , φ) n! X k k=1

and 1 ∂ ∂ ··· G(α∗ )H(α∗ ) ∗ n! ∂α(x1 ) ∂α(xn )∗ 1 21 2 n m n

∂ 1 ∂ 1 ∗ ∗ = S G(α ) H(α ) m! ∂α(x )∗ (n − m)! ∂α(x )∗ m=0 =1

=m+1

where S denotes the symmetrization operator in the variables xk . Thus Lemma 3.8 gives the system of equations ∂ F1 (ε, x1 , φ) = − dx1 h(x1 , x1 )F1 (ε, x1 , φ) and for n > 1 ∂ε n−1 m n ∂ 1 ˜ Hn (˜ ˜ , φ) = − dn x ˜ )Fn (ε, x ˜ , φ) − Fn (ε, x x, x S v(xj , xk ) ∂ε 2 m=1 j=1 k=m+1

× Fm (ε, x1 , . . . , xm , φ)Fn−m (ε, xm+1 , . . . , xn , φ)

(3.7)

with the initial condition F1 (0, x1 , φ) = φ(x1 ) ,

˜ , φ) = 0 for Fn (0, x

n > 1.

The “integral” form of these equations is F1 (ε, x1 , φ) = e−εh φ (x1 ) n−1 m n

1 ε ˜) ˜ , φ) = − x, x dτ d˜ x e−(ε−τ )Hn (˜ S v(xj , xk ) Fn (ε, x 2 0 m=1 j=1 k=m+1 Fm (τ, x1 , . . . , xm , φ) Fn−m (τ, xm+1 , . . . , xn , φ) .

˜ , φ) is of degree n in φ also follows by induction We remark that the fact that Fn (ε, x on n from these equations. Writing n ˜ , φ) = d˜ ˜, y ˜) Fn (ε, x y Fn (ε, x φ(yi ) i=1

˜, y ˜ )’s symmetric under permutations of the yk ’s for each n ≥ 1, with the Fn (ε, x too, defines the functions of the lemma.

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We now estimate the norm of the operator e−εHn of Lemma 3.9, acting on functions F : X n × X n → C. The space of functions is equipped with the norm |F (x1 , . . . , x2n )| dxi . F 1,∞ = max max 1≤j≤2n xj ∈X

X 2n−1

1≤i≤2n i=j

Lemma 3.10. Write ˜ ) = hn (˜ ˜ ) + Vn (˜ ˜) Hn (˜ x, x x, x x, x where ˜) = hn (˜ x, x

n

h(xk , xk )

k=1

˜) = x, x Vn (˜

δx (x )

1≤≤n =k

n 1 v(xj , xk ) δx (x ) . 2 j,k=1 1≤≤n

j=k

For all F : X n × X n → C, we have ! ! ! ! (a) !e−εhn F !1,∞ ≤ enε h 1,∞ !F !1,∞ ! ! ! ! 1 (b) !e−εVn F !1,∞ ≤ e 2 nεv0 !F !1,∞ where v0 = maxx∈X v(x, x) ! ! ! ! 1 (c) !e−εHn F !1,∞ ≤ enε( h 1,∞ + 2 v0 ) !F !1,∞ ! ! ! ! (d) !Hn F !1,∞ ≤ n h1,∞ + 12 (n − 1)|v|X !F !1,∞ . Proof. (a) The kernel of e−εhn is ˜) = x, x e−εhn (˜

n

e−εh (xk , xk ) .

k=1

So we may view e−εhn F as n factors

n−k factors factors n 8k−1 9: 8 9: ; 8 n factors 9: ; ; 8 9: ; −εh −εh −εh 1l ⊗ · · · ⊗ 1l ⊗e ⊗ ··· ⊗ e ⊗ 1l ⊗ · · · ⊗ 1l = ⊗ 1l ⊗ · · · ⊗ 1l e k=1

acting of F , viewed as an element of L2 (X 2n ). The bounds sup dx dz1 . . . dzm L(x, x ) G(x , z1 , . . . , zm ) x dz1 . . . dzm G(x , z1 , . . . , zm ) = sup dx L(x, x ) x ! ! ≤ L1,∞ !G!1,∞

Vol. 9 (2008)

and

sup zm

Bosonic Integral I

1259

dz1 . . . dzm−1 L(x, x ) G(x , z1 , . . . , zm ) = sup dx dz1 . . . dzm−1 dx L(x, x ) G(x , z1 , . . . , zm ) zm ! ! ≤ L1,∞ !G!1,∞

dx dx

imply that factors n−k factors ! 8k−1 9: ; 8 9: ; ! ! ! ! 1l ⊗ · · · ⊗ 1l ⊗L ⊗ 1l ⊗ · · · ⊗ 1l F !

1,∞

! ! ≤ L1,∞ !F !1,∞ .

(3.8)

Part (a) now follows by repeated application of (3.8) in conjunction with ∞ 1 n n ε h (x, x ) dx e−εh (x , x) = dx e−εh (x, x ) ≤ dx n! n=0 ≤

∞

1 n ε hn1,∞ = eε h 1,∞ . n! n=0

(3.9)

(b) Since v(x, y) is the kernel of a positive definite operator n

j,k=1 j=k

so that

v(xj , xk ) =

n

v(xj , xk ) −

j,k=1

n

v(xj , xj ) ≥ −v0 n

(3.10)

j=1

−εV 1 e n F (˜ ˜ ) . ˜ ) ≤ e 2 nεv0 F (˜ x, y x, y

(c) follows from the Trotter product formula ε p ε e−εHn F = lim e− p hn e− p Vn F p→∞

and repeated application of parts (a) and (b). (d) By (3.8), ! ! ! ! !hn F ! ≤ nh1,∞ !F !1,∞ . 1,∞ Since

n v(xj , xk ) ≤ n(n − 1)|v|X j,k=1 j=k

we also have

! ! ! ! 1 !Vn F ! ≤ n(n − 1)|v|X !F !1,∞ 1,∞ 2 and the claim follows.

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˜, y ˜ ) of Lemma 3.9 obey Lemma 3.11. The functions Fn (ε, x n−1 nεK1 1 e Fn (ε, · , · )1,∞ ≤ 8εv1,∞ n3 v0 where K1 = h1,∞ + 2 . Furthermore there is a constant const , depending only on h1,∞ and v1,∞ , such that $ % 1 F2 (ε, x1 , x2 , y1 , y2 ) = − εv(x1 , x2 ) δx1 (y1 )δx2 (y2 ) + δx1 (y2 )δx2 (y1 ) 4 + F2+ (ε, x1 , x2 , y1 , y2 ) with F2+ (ε, · , · )1,∞ ≤ const ε2 v1,∞ e2εK1 . Proof. We first prove the bound on Fn (ε, · , · )1,∞ by induction on n. The case n = 1 follows immediately from (3.9). So assume that the bound has been proven for all m < n. In general, if F (x1 , . . . , xn+m ) = dy1 dy2 G(x1 , . . . , xn , y1 ) ω(y1 , y2 ) H(y2 , xn+1 , . . . , xn+m ) then F 1,∞ ≤ G1,∞ ω1,∞ H1,∞ . This is proven by repeated application of dx2 dy f (x1 , y)g(y, x2 ) ≤ dy |f (x1 , y)| dx2 |g(y, x2 )| ≤ dy |f (x1 , y)| g1,∞ ≤ f 1,∞ g1,∞ . Hence, by the inductive hypothesis, Lemma 3.9 and part (c) of Lemma 3.10, ! n−1 m n

−(ε−τ )Hn 1! ε ˜ ˜, y ˜ )1,∞ ≤ ! Fn (ε, x e (˜ x , x dτ d˜ x ) S ˜ ,˜ x y 2! 0 m=1 j=1 k=m+1 ! ! ˜ [≤m] , y ˜ [≤m] ) Fn−m (τ, x ˜ [>m] , y ˜ [>m] )! v(xj , xk )Fm (τ, x ! ≤

1 2

1,∞

ε

dτ e(ε−τ )K1 n 0

m n−1

n

v1,∞

m=1 j=1 k=m+1

× Fm (τ, · )1,∞ Fn−m (τ, · )1,∞ m n−1

n−2 ε 1 nεK1 8v1,∞ ≤ e dτ τ n−2 2 0 m=1 j=1 n

k=m+1

v1,∞

1 1 m3 (n − m)3

Vol. 9 (2008)

Bosonic Integral I

1261

1 n−1 1 n−1 1 1 nεK1 e 8εv1,∞ 16 n − 1 m=1 m2 (n − m)2 n−1 1 8 1 nεK1 e 8εv1,∞ ≤ 16 n − 1 n2 n−1 nεK1 1 ≤ 8εv1,∞ e . n3 =

The equation for n = 2 in Lemma 3.9 yields 1 ε F2 (ε, x1 , x2 , y1 , y2 ) = − dτ dx1 dx2 e−(ε−τ )H2 (x1 , x2 , x1 , x2 ) v(x1 , x2 ) 2 0 % 1 $ −τ h e (x1 , y1 ) e−τ h (x2 , y2 ) + e−τ h (x1 , y2 ) e−τ h (x2 , y1 ) 2 $ % 1 = − εv(x1 , x2 ) δx1 (y1 )δx2 (y2 ) + δx1 (y2 )δx2 (y1 ) 4 + F2+ (ε, x1 , x2 , y1 , y2 ) where the second order Taylor remainder ε ∂2 dτ (ε − τ ) 2 F2 (τ , x1 , x2 , y1 , y2 ) . F2+ (ε, x1 , x2 , y1 , y2 ) = ∂ε 0 By (3.7) ∂ F2 (ε, x1 , x2 , y1 , y2 ) = − ∂ε

(3.11)

dx1 dx2 H2 (x1 , x2 , x1 , x2 )F2 (ε, x1 , x2 , y1 , y2 )

1 − Sx,y v(x1 , x2 ) F1 (ε, x1 , y1 ) F1 (ε, x2 , y2 ) 2 and hence ∂2 F2 (ε, x1 , x2 , y1 , y2 ) = − ∂ε2

dx1 dx2 H2 (x1 , x2 , x1 , x2 )

− Sx,y v(x1 , x2 ) F1 (ε, x1 , y1 )

∂ F2 (ε, x1 , x2 , y1 , y2 ) ∂ε

∂ F1 (ε, x2 , y2 ) . ∂ε

Using part (d) of Lemma 3.10 and the bounds on F1 and F2 already proven, we have ! ! !∂ ! ! F2 (ε, · , · )! ! ∂ε ! 1,∞

! ! !2 ! 1 ≤ !H2 F2 (ε, · , · )!1,∞ + v1,∞ !F1 (ε, · , · )!1,∞ 2 ! !2 ! ! 1 1 ≤ 2 h1,∞ + |v|X !F2 (ε, · , · )!1,∞ + v1,∞ !F1 (ε, · , · )!1,∞ 2 2 1 1 ≤ v1,∞ 1 + 4ε h1,∞ + |v|X e2εK1 . 2 2

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Using the bound ! ! !∂ ! ! ! ! F1 (ε, · , · )! = !he−εh !1,∞ ≤ h1,∞ eεK1 ! ∂ε ! 1,∞ we similarly get ! 2 ! !∂ ! ! ! F (ε, · , · ) ! ∂ε2 2 ! 1,∞ ! ! ! ∂ ! ! ! ≤ !H2 F2 (ε, · , · )! + v1,∞ !F1 (ε, · , ! ∂ε 1,∞ 1 ≤ h1,∞ + |v|X v1,∞ 1 + 4ε h1,∞ + 2

! ! !∂ ! ! ! ! · ) 1,∞ ! F1 (ε, · , · )! ! ∂ε 1,∞ 1 |v|X e2εK1 2

+ v1,∞ h1,∞ e2εK1 1 2 2 1 1 ≤ v1,∞ 2h1,∞ + |v|X + 4εv1,∞ h1,∞ + |v|X e2εK1 2 2 ≤ const v1,∞ e2εK1 . Hence, by (3.11) ! ! !F2+ (ε, · , · )! ≤ const ε2 v1,∞ e2εK1 1,∞

as desired.

˜ for (x1 , . . . , xn ) and y ˜ for Proof of Proposition 3.6. We routinely write x (y1 , . . . , yn ). The value of n should be clear from the context. By Lemma 3.11, ∞

n=3

n nεμ ∗ ˜, y ˜) α(xi ) φ(yi ) d˜ xd˜ y e Fn (ε, x i=1

≤

∞

! ! |X| enεμ !Fn (ε, · , · )!1,∞ |α|nX |φ|nX

n=3

≤ |X|

∞

8εv1,∞

n−1

enε(K1 +μ) |α|nX |φ|nX .

(3.12)

n=3

∗ This gives us the desired domain of analyticity. Since α e−εK φ and eF (ε,α ,φ) are both analytic on this domain and since they agree for all sufficiently small α, φ, they coincide on the full domain.

Vol. 9 (2008)

Bosonic Integral I

Set F1 (ε, α∗ , φ) =

$ % dxdy eεμ F1 (ε, x, y) − j(ε; x, y) α(x)∗ φ(y)

2 $ 2εμ % ˜d y ˜ e ˜, y ˜) d x − 1 F2 (ε, x α(xi )∗ φ(yi ) 2

+

2

i=1

˜ d2 y ˜ F2+ (ε, x ˜, y ˜) d2 x

+ +

1263

2

α(xi )∗ φ(yi )

i=1

∞

n

n

nεμ

˜d y ˜e d x

˜, y ˜) Fn (ε, x

n=3

so that F (ε, α∗ , φ) =

n

α(xi )∗ φ(yi )

i=1

dxdy α(x)∗ j(ε; x, y)φ(y) ε dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) + F1 (ε, α∗ , φ) . − 2 X2 X2

If |α|X , |φ|X ≤ R, then, by (3.12), ∞ n ˜, y ˜) ˜ dn y ˜ enεμ Fn (ε, x α(xi )∗ φ(yi ) dn x n=3

i=1

≤ |X|

∞

8εv1,∞

n−1

enε(K1 +μ) R2n

n=3

64 |X|ε2 v21,∞ e3ε(K1 +μ) R6 ≤ 1 − 8εv1,∞ eε(K1 +μ) R2 ≤ 128 ε2 R6 |X| v21,∞ e3ε(K1 +μ) . Similarly, by Lemma 3.11, 2 ! ! 2 2 ∗ ˜d y ˜ F2+ (ε, x ˜, y ˜) α(xi ) φ(yi ) ≤ |X| !F2+ (ε, · , · )!1,∞ R4 d x i=1

≤ const ε2 R4 |X| v1,∞ e2ε(K1 +|μ|) and

2 % $ ˜, y ˜) ˜ d2 y ˜ e2εμ − 1 F2 (ε, x α(xi )∗ φ(yi ) d2 x i=1 ! ! ≤ 2ε|μ| |X| !F2 (ε, · , · )! R4 e2ε|μ| 1,∞

≤ 2|μ|ε R |X| v1,∞ e 2

4

2ε(K1 +|μ|)

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and

$ % dxdy eεμ F1 (ε, x, y) − j(ε; x, y) α(x)∗ φ(y) ! −ε(h−μ) ! ≤ |X| !e − j(ε)!1,∞ R2 ≤ cj ε2 R2 |X| .

Since

1 R4 v1,∞ = R R3 v1,∞ ≤ R2 + v21,∞ R6 2 the desired bound on F1 follows. −εK φ can have Example 3.12. Here is a simple example shows that α e −εKthat φ need not be defined for all α, φ ∈ zeroes so that the logarithm of α e L2 (X). If the finite set X, which plays the role of space here, consists of just a single point, then each n-particle space Bn with n ≥ 1 has dimension exactly one. So any operator that commutes with the number operator must be a function of the number operator. In particular, K = H0 + V − μN , which is of degree two in the annihilation operators and of degree two in the creation operators, is a polynomial in N of degree two. As a simple example, we take K = N 2 − N . Then ∞ −ε(N 2 −N ) (α∗ φ)n −εn(n−1) φ = e . αe n! n=0 Set f (z) =

∞

z n −εn(n−1) e . n! n=0

Observe that f fulfills the functional equation f (z) = f (e−2ε z) since f (e−2ε z) =

∞

z n −2εn −εn(n−1) e e n! n=0

=

∞ ∞

z n −εn(n+1) d z n+1 −εn(n+1) e e = n! dz n=0 (n + 1)! n=0

=

∞ d z n −εn(n−1) d e f (z) . = dz n=1 n! dz

We claim that f necessarily has a zero on the negative real axis, somewhere between 1 0 and − 1−e1−2ε = − 2ε+O(ε 2) . Proof. Set κ = e−2ε < 1, so that f (z) = f (κz). By inspection, f (x) > 0 for all x ≥ 0. Now assume that f has no zero on the negative real axis. Then h(x) = − log f (−x)

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is well defined for all real x and fulfills the equation h (x) =

f (−κx) f (−x) = f (−x) f (−x)

= eh(x)−h(κ x) . In particular h (x) > 0 for all x ∈ IR and h is monotonically increasing. The second derivative h (x) = h (x) − κh (κ x) eh(x)−h(κ x) 2 = eh(x)−h(κ x) − κ eh(κx)−h(κ x) eh(x)−h(κ x) . As κ < 1 and h(0) = 0, we have h (0) = 1 − κ > 0. Next we show that h (x) > 0 for all x ≥ 0. If this were not the case we would have a smallest positive zero x0 of h . Then h would be monotonically increasing in [0, x0 ]. By the formula for h above 2 eh(x0 )−h(κ x0 ) − κ eh(κx0 )−h(κ x0 ) = 0 so that h(x0 ) − h(κ x0 ) < h(κx0 ) − h(κ2 x0 ) . By the mean value theorem there exist ξ1 ∈ [κ x0 , x0 ] and ξ2 ∈ [κ2 x0 , κ x0 ] such that h(x0 ) − h(κ x0 ) = h (ξ1 ) (1 − κ) x0 h(κx0 ) − h(κ2 x0 ) = h (ξ2 ) (1 − κ) κ x0 . Then h (ξ1 ) (1 − κ) x0 < h (ξ2 ) (1 − κ) κ x0 and h (ξ1 ) < h (ξ2 ) κ . As ξ2 ≤ ξ1 ≤ x0 and κ < 1 this contradicts the monotonicity of h on [0, x0 ]. Since h(0) = 0 and h (x) > 0 for all x ≥ 0 we have h(κx) ≤ κh(x) for all x ≥ 0 and therefore h(x) − h(κx) ≥ (1 − κ) h(x). In particular h (x) = eh(x)−h(κ x) ≥ e(1−κ) h(x) ⇒ e−(1−κ) h(x) h (x) ≥ 1 ⇒ −

1 d −(1−κ) h(x) e ≥ 1. 1 − κ dx

Integrating both sides and using the initial condition h(0) = 0 gives −

% 1 $ −(1−κ) h(x) e − 1 ≥ x ⇒ e−(1−κ) h(x) ≤ 1 − (1 − κ)x 1−κ

for all x ≥ 0. Thus h(x) must have a pole at some 0 < x < 1 have a zero at some − 1−κ < x < 0.

1 1−κ

and f (x) must

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A functional integral representation of Tr e−βK Let h be a single particle operator on X and v(x1 , x2 ) a real, symmetric, pair potential which is repulsive in the sense of Definition 2.16. One of our precise formulations of the standard physics representation (1.1) for the partition function Tr e−β K , where K = H0 (h, X)+V(v, X)−μN is the second quantized Hamiltonian of a boson gas, is Theorem 3.13. Suppose that the sequence R(p) obeys lim p e− 2 R(p) = 0 2

1

p→∞

Then Tr e−β K = lim

p→∞

1

and

4 dμR(p) (φ∗τ , φτ ) e−

R(p) < p 24|X| .

∗ dy [φ∗ τ (y)−φτ −ε (y)]φτ (y)

∗

e−εK(φτ −ε ,φτ )

5

τ ∈Tp

with the conventions that ε = in Corollary 3.7.

and φ0 = φpε . The function K(α∗ , φ) was defined

β p

This theorem will be proven after Example 3.17. During the course of the proof, we will modify the factors φτ −ε e−ε K φτ of the integrand in Theorem 3.1, using the representation of these factors in Corollary 3.7. In Proposition 3.16, below, we develop some machinery to assist in proving that such modifications do not change the limit. Definition 3.14. Let r > 0. Define, for I : C2|X| → C, the seminorm Ir =

sup α,φ∈CX |α|X ,|φ|X ≤r

|I(α, φ)| .

The “r-product” of I, J : C2|X| → C, with Ir , J r < ∞ is defined to be (I ∗r J )(α, γ) = I(α, φ)J (φ, γ) dμr (φ∗ , φ) which is just the convolution with respect to the measure dμr . The q th power with respect to this product is denoted q factors

I

∗r q

8 9: ; = I ∗r I ∗r · · · ∗r I .

Example 3.15. For each ε > 0, set Iε (α, φ) = e− 2 α 1

2

− 12 φ 2 F (ε,α∗ ,φ)

e

= e− 2 α 1

2

− 12 φ 2

α e−εK φ .

Theorem 3.1 states that, for R(p) obeying limp→∞ pe− 2 R(p) = 0, −βK = lim dμr (φ∗ , φ) Iε∗r p (φ, φ) r=R(p) . Tr e 1

p→∞

2

ε=β/p

The operator K is bounded below. Suppose that K ≥ −K0 1l. Then −εK α e φ ≤ αeεK0 φ = e 12 α 2 + 12 φ 2 eεK0

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implies that Iε r ≤ eεK0 for all r > 0. Furthermore Iε∗r q (α, φ) = e− 2 α 1

2

− 12 φ 2

q−1 −εK α e−εK Ir e φ .

Thus, by part (c) of Theorem 2.26, Iε∗r q r ≤ eqεK0 for all r > 0. Proposition 3.16. Let K0 , ε, ζ > 0 and 0 < κ < 1 and r, Cβ ≥ 1 obey 3|X| 1−κ Cβ πr2 ζ ≤ ε. Let I, I˜ : C2|X| → C obey I ∗r q r ≤ eqεK0

˜ r≤ζ I − I Then, for all q ∈ IN with q ≤

for all

q ∈ IN .

Cβ ε ,

I˜ ∗r q r ≤ eqε(K0 +ζ

κ

)

I˜ ∗r q − I ∗r q r ≤ ζ κ eqε(K0 +ζ ) κ dμr (φ∗ , φ) I˜ ∗r q (φ, φ) − I ∗r q (φ, φ) ≤ ζ κ eqε(K0 +ζ ) . κ

Proof. For notational convenience, we suppress the subscript r on ∗r . We first prove, by induction on q, that (3.13) I˜ ∗q r ≤ (A + B)q I˜ ∗q − I ∗q r ≤ qB(A + B)q−1 2|X| ζ. The case q = 1 is obvious. So assume that where A = eεK0 and B = πr2 these bounds hold when q is replaced by < q. Then I˜ ∗q =

q−1

I ∗ ∗ (I˜ − I) ∗ I˜ ∗q−−1 + I ∗q .

=0

Since

|X| dμr (φ∗ , φ) ≤ πr2

we have q−1

! ! ! ∗q ! |X| |X| ! ∗q−−1 ! !˜ ! + !I ∗q ! !˜ I I !r ≤ I ∗ r πr2 I˜ − Ir πr2 r r =0

≤

q−1

=0

A B (A + B)q−−1 + Aq

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and


q−1

! ∗q ! ∗q ! !˜ I −I r ≤ A B (A + B)q−−1 . =0

Then (3.13) follows from (A + B)q =

q−1

A (A + B)q− −

=0

=

q−1

q−1

A+1 (A + B)q−−1 + Aq

=0

A B(A + B)q−−1 + Aq

=0

and

1

d (A + tB)q ≤ qB(A + B)q−1 . dt 0 To complete the proof, we observe that 6 2|X| 2|X| 7 ζ ≤ eεK0 1 + πr2 ζ ≤ eεK0 1 + εζ κ A + B = eεK0 + πr2 (A + B) − A = q

≤ eε(K0 +ζ and

q

κ

dt

)

|X| Cβ 2 3|X| πr ≤ ζ ≤ ζκ . qB, qB πr2 ε

Example 3.17. Let Iε (α, φ) = e− 2 α 1

2

− 12 φ 2 F (ε,α∗ ,φ)

e

= e− 2 α 1

2

− 12 φ 2

α e−εK φ

be as in Example 3.15 and set 2 2 ∗ ∗ 1 1 I˜ε (α, φ) = e− 2 α − 2 φ eF (ε,α ,φ)−F0 (ε,α ,φ) 2 2 ∗ 1 1 = e− 2 α − 2 φ α e−εK φ e−F0 (ε,α ,φ)

where F0 was defined in Corollary 3.7. Observe that

1 1 I˜ε (α, φ) = exp − α2 − φ2 + dx α∗ (x)φ(x) − εK(α∗ , φ) . 2 2 $ %−1/2 Let r satisfy r ≤ 12 8eε( h 1,∞ +μ+v0 ) εv1,∞ . Then, by Corollary 3.7, |F0 (ε, α∗ , φ)| ≤ const ε2 r2 + v21,∞ r6 |X| for all |α|X , |φ|X ≤ r . Consequently (assuming that r > 1 and allowing the constant to depend on v1,∞ too) 1 2 1 2 $ % ∗ − 2 α − 2 φ α e−εK φ 1 − e−F0 (ε,α ,φ) sup Iε − I˜ε r = e |α|X ,|φ|X ≤r

r |X|

2 6

≤ eεK0 const ε2 r6 |X| econst ε

.

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Proof of Theorem 3.13. We apply Proposition 3.16 with I = Iε , I˜ = I˜ε , as in 1 and Cβ = β. If ε Examples 3.15 and 3.17, ζ = ε3/2 , r = R(p), p = βε , κ = 12 is sufficiently small, the three hypotheses of the Proposition are satisfied because then 6 24 3|X| 1−κ 3 1 33 β ζ = βπ 3|X| R(p)6|X| ε 2 (1− 12 ) ≤ βπ 3|X| ε 24 ≤ ε Cβ πr2 ε and, by Example 3.17, ˜ r ≤ const ε2 I − I

1 4|X| 1 2 β 4|X| |X|eεK0 econst |X|ε (β/ε) ≤ ε3/2 = ζ ε

and, by Example 3.15,

I ∗r q r ≤ eqεK0 . By the last conclusion of Proposition 3.16, ∗p Iεr (α, α) − Iε∗r p (α, α) r=R(p) = 0 . lim dμr (α∗ , α) ˜ p→∞

ε=β/p

Recall, from Example 3.15, that Tr e−βK = lim

p→∞

so that Tr e−βK = lim

p→∞

= lim

p→∞

dμr (φ∗ , φ) Iε∗r p (φ, φ) r=R(p) ε=β/p

dμr (φ∗ , φ) I˜ε∗r p (φ, φ) r=R(p)

4

ε=β/p

dμR(p) (φ∗τ , φτ ) e− 2 φτ −ε 1

τ ∈Tp

−εK(φ∗ τ −ε ,φτ )

×e

= lim

p→∞

− 12 φτ 2 + dy φ∗ τ −ε (y)φτ (y)

τ ∈Tp

5 ∗ × e−εK(φτ −ε ,φτ ) 4 2 dμR(p) (φ∗τ , φτ )e− φτ + = lim p→∞

2

4

dy φ∗ τ −ε (y)φτ (y)

5

dμR(p) (φ∗τ , φτ ) e−

∗ dy [φ∗ τ (y)−φτ −ε (y)]φτ (y)

τ ∈Tp

5 ∗ × e−εK(φτ −ε ,φτ ) .

Appendix A. A Cameron style model computation We consider the formal infinite dimensional complex Gaussian measure 1 dφ∗τ dφτ 1 dτ φ∗τ (∂+μ)φτ e0 N 2πi 0≤τ 10p/8 . lim Np =

e−μ

Proof. Think of Tp as the discrete torus p1 ZZ/ZZ. Denote by L2 (Tp ) the p complex dimensional Hilbert space of functions on Tp with the usual inner product ψ, φ = ∗ 2 τ ∈Tp ψτ φτ . Define the operator ∂p on L (Tp ) by ∂p φ τ = p(φτ +1/p − φτ ) . Set, for each 0 ≤ n ≤ p − 1

en (τ ) = e2nπı τ Then en (τ ) n=0,...,p−1 is an (orthogonal) basis for L2 (Tp ) and each en (τ ) is an eigenvector for any translation invariant operator on L2 (Tp ). Since n ∂p en (τ ) = p e2π p ı − 1 en (τ )

the eigenvalues of ∂p are

n λn,p = p e2π p ı − 1 ,

n = 0, 1, . . . , p − 1 .

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The 2p real dimensional Gaussian integral −1 p−1 dφ∗τ dφτ p1 (φ,(∂p +μ)φ) 1 e = . − (λn,p + μ) Np = 2πi p n=0 τ ∈Tp

We have lim

p→∞

p−1 n=0

p−1 1 μ 2π n ı p − (λn,p + μ) = lim 1− −e p→∞ p p n=0 p μ = lim 1 − − 1 = e−μ − 1 p→∞ p

since p−1

2πn z − eı p = z p − 1 .

n=0

On the other hand, assuming that p is divisible by 8,

dνp (φ∗ , φ) =

−1 p−1 dφ∗τ dφτ p1 (φ,(∂p +μ)φ) 1 = e − Re (λn,p + μ) 2πi p n=0 τ ∈Tp

and p−1 n=0

p−1 n 1 − Re (λn,p + μ) = Cp − cos 2π p p n=0 where 2 n Cp − cos 2π p n=1

Cp = 1 −

μ p

p 2 −1

= (Cp − 1)(Cp + 1)

since 2 n 2 2 + 1) Cp − cos 2π p n=1

p−n n cos 2π = cos 2π p p

p 4 −1

= (Cp −

1)Cp2 (Cp

p/2 − n n cos 2π = − cos 2π p p 2 p8 2 2 −1 1 n n = (Cp − 1)Cp2 (Cp + 1) Cp2 − Cp2 − cos2 2π Cp2 − sin2 2π 2 p p n=1 p/4 − n n since cos 2π = sin 2π . p p since

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Now 2 2 2 n n n n 2 2 2 2 2 4 2 2 Cp − cos 2π = Cp − Cp + sin 2π Cp − sin 2π cos 2π p p p p 2 2 μ μ 1 n = Cp2 −2 + 2 + sin2 4π p p 4 p 1 ≤ 10 2 if p is large enough, since limp→∞ Cp = 1 and limp→∞ − 2 μp + μp2 = 0. If p is large enough, we also have 2 2 1 μ 1 1 . (Cp − 1)Cp2 (Cp + 1) Cp2 − = − Cp2 (Cp + 1) Cp2 − < 2 p 2 10

References [1] T. Balaban, A low temperature expansion and “spin wave picture” for classical N vector models, in Constructive Physics, V. Rivasseau, editor, Springer, 1995. [2] T. Balaban, J. Feldman, H. Kn¨ orrer and E. Trubowitz, A functional integral representation for many Boson systems. II: Correlation functions, Annales Henri Poincaré 9 (2008), 1275–1307. [3] R. Cameron, The Ilstow and Feynman integrals, J. Analyse Math. 10 (1962/63), 287–361. [4] J. Fr¨ ohlich, B. Simon and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50 (1976), 79–95. [5] I. M. Gel’fand and N. Ya. Vilenkin, Generalized Functions. Applications of Harmonic Analysis, Vol. 4 , Acad. Press (1968). [6] J. W. Negele and H. Orland, Quantum Many–Particle Systems, Addison–Wesley (1988). Tadeusz Balaban Department of Mathematics Rutgers The State University of New Jersey 110 Frelinghuysen Rd Piscataway, NJ 08854-8019 USA e-mail: [email protected] Joel Feldman Department of Mathematics University of British Columbia Vancouver, B.C. V6T 1Z2 Canada e-mail: [email protected]

Vol. 9 (2008)

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Horst Kn¨ orrer and Eugene Trubowitz Mathematik ETH-Zentrum CH-8092 Z¨ urich Switzerland e-mail: [email protected] [email protected] Communicated by Vincent Rivasseau. Submitted: August 20, 2007. Accepted: July 1, 2008.

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A Functional Integral Representation for Many Boson Systems II: Correlation Functions Tadeusz Balaban, Joel Feldman, Horst Knörrer, and Eugene Trubowitz Abstract. We derive functional integral representations for the partition function and correlation functions of many Boson systems for which the configuration space consists of finitely many points.

1. Introduction We are developing a set of tools and techniques for analyzing the large distance/infrared behaviour of a gas of bosons as the temperature tends to zero. In [2], we developed functional integral representations for the partition function of a many–boson system on a finite configuration space X with a repulsive two particle potential v(x, y). Let H be the Hamiltonian, N the number operator, β the inverse temperature and μ the chemical potential. The main result, Theorem 3.13, of [2] is Tr e−β(H−μN ) dμR(p) (φ∗τ , φτ ) e− = lim p→∞

∗ dy [φ∗ τ (y)−φτ −ε (y)]φτ (y)

τ ∈Tp

∗

e−εK(φτ −ε ,φτ )

(1.1)

with the conventions1 that ε = βp , φ0 = φβ and Tp = τ = q βp q = 1 , . . . , p . The “classical” H − μN is K(α∗ , φ) = dxdy α(x)∗ h(x, y)φ(y) − μ dx α(x)∗ φ(x) 1 + dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) 2 1 We

also use the convention that

dx =

x∈X .

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where h is the single particle operator. For each r > 0, dφ∗ (x) ∧ dφ(x)

∗ χr |φ(x)| dμr ( φ , φ) = 2πı x∈X

where χr is the characteristic function of the closed interval [0, r]. In [2, Theorem 3.13], we need the hypothesis that the integration radius R(p) obeys lim p e− 2 R(p) = 0 1

p→∞

2

and

1

R(p) < p 24|X| .

(1.2)

In [2], we outlined our motivation for deriving the function integral representation (1.1). We wish to use functional integrals as a starting point for analyzing the long distance behaviour of a many boson system. Such an analysis begins by directly extracting detailed properties of the ultraviolet limit p → ∞ from the finite dimensional integrals in (1.1). These detailed properties would, in turn, provide a suitable starting point for an analysis of the thermodynamic limit and the temperature zero limit. The restrictions (1.2) on the domain of integration in (1.1) are not well suited for such a program. This is particularly obvious for the |X| dependent second condition in (1.2). In Theorem 2.2, we prove a representation for the partition function, similar to (1.1), but with functional integrals that are better suited to this program. The choice of integration domain in Theorem 2.2 is motivated by the following considerations. For two particle potentials that are repulsive in the sense that dxdy ρ(x)v(x, y)ρ(y) dx ρ(x)2 = 1, ρ(x) ≥ 0 λ0 = λ0 (v) = inf for all x ∈ X > 0 (1.3) the real part of the exponent of the integrand of (1.1) is, roughly speaking, dominated by 1 2 ε 2 2 dx φτ (x) − φτ −ε (x) + dx dy φτ (x) v(x, y) φτ (y) − 2 2 τ ∈Tp 2 4 1 ≤− max φτ (x) − φτ −ε (x) + ελ0 max φτ (x) . x∈X x∈X 2 τ ∈Tp

Contributions to the integral of (1.1) of of the domain coming from the part 1 will integration where, for some τ and x, φτ (x)−φτ −ε (x) 1 or φτ (x) √ 4 ελ0 be extremely small. Consequently, we ought to restrict the domain of integration to be something like

1 φτ (x) τ ∈Tp φτ (x) − φτ −ε (x) ≤ p0 (ε), φτ (x) ≤ √ p0 (ε), 4 ελ 0 x∈X τ ∈ Tp , x ∈ X for some function, p0 (ε), that grows slowly as ε → 0.

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To study the long distance behaviour of a many boson system, one needs to study correlation functions. By definition, an n-point correlation function at inverse temperature β is (up to a sign) an expression of the form n Tr e−β(H−μN ) T j=1 ψ (†) (βj , xj ) . Tr e−β(H−μN ) Here ψ (†) refers to either ψ or ψ † and ψ (†) (τ, x) = e(H−μN )τ ψ (†) (x)e−(H−μN )τ . n The time–ordering operator T orders the product j=1 ψ (†) (βj , xj ) with smaller times to the right. In the case of equal times, ψ † ’s are placed to the right of ψ’s. Theorem 2.2 and (1.1) give functional integral representations for the denominator. The “times” βj appearing in the numerator need not be rational multiples of β. Therefore in the functional integral representations for the functions correlation we replace the set Tp of allowed times by a partition P = τ 0 ≤ ≤ p of the interval [0, β] that contains the points β1 , β2 , . . . , βn . The analogs for correlation functions of (1.1) and Theorem2.2 are Theorems 3.5 and 3.7, respectively.

2. Another integral representation of the partition function Let h be a single particle operator on X and v(x1 , x2 ) a real, symmetric, pair potential which is repulsive in the sense of (1.3). Throughout this section, except where otherwise noted, we write K = K(h, v, X, μ) = H0 (h, X) + V(v, X) − μN where, as in [2, Proposition 2.14], H0 (h, X) is the second quantized free Hamiltonian with single particle operator h, V(v, X) is the second quantized interaction and N is the number operator. In this section, we prove a variant of the functional integral representation of [2, Theorem 3.13] that is better adapted to a rigorous renormalization group analysis. Recall, from [2, Theorem 3.1], that 2 −βK dμR(p) (φ∗τ , φτ ) e− dy |φτ (y)| φτ −ε e−ε K φτ = lim Tr e p→∞

τ ∈Tp

τ ∈Tp β p

with the conventions that ε = and φ0 = φpε = φβ . Further recall, from [2, Proposition 3.6], that −εK φ = eF (ε,α∗ ,φ) α e and, from [2, Lemma 3.9], that ∞ n ∗ ˜ dn y ˜ enμ Fn (ε, x ˜, y ˜) F (ε, α , φ) = dn x α(xi )∗ φ(yi ) n=1

X 2n

i=1

where

˜ = (x1 , . . . , xn ) x ˜ = (y1 , . . . , yn ) . y

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In [2, Theorem 3.13], we approximated eεμ F1 (x, y), which is the kernel of the operator e−ε(h−μ) , by 1l − ε(h − μ). Now, more generally, we approximate it by j(ε, x, y) where we only assume that there is a constant cj such that j(ε) − e−ε(h−μ) ≤ cj ε2 (2.1) 1,∞

2

where, as in [2], for any operator A on L (X) with kernel A(x, y), the norm

A 1,∞ = max max dy |A(x, y)|, max dx |A(x, y)| . x∈X

y∈X

For fields, we use the norms 1/2

α = |α(x)|2

and |α|X = max |α(x)| . x∈X

x∈X

In [2, Theorem 3.13], the domain of integration restricted each field φτ X ≤ 1 R(p) < p 24|X| . Now we relax that condition to φτ X ≤ Rε , with Rε satisfying Hypothesis 2.1, below. In addition, the new domain of integration will restrict each “time derivative” φτ +ε − φτ X ≤ p0 (ε) with p0 satisfying Hypothesis 2.1. Let Rε > 0 and p0 (ε) ≥ ln 1ε be decreasing functions of ε defined for all 0 < ε ≤ 1. Assume that √ 1 p0 (ε) and Rε ≥ √ lim ε Rε = 0 . 4 ε→0 ε Theorem 2.2. Let Rε and p0 (ε) obey Hypothesis 2.1 and j(ε) obey (2.1). Let β > 0. Then, with the conventions that ε = βp and φ0 = φpε , ∗ dμRε (φ∗τ , φτ ) ζε (φτ −ε , φτ ) eA(ε,φτ −ε ,φτ ) Tr e−β K = lim p→∞

where

τ ∈Tp

1 1 2 dxdy α(x)∗ j(ε; x, y)φ(y) − φ 2 A(ε, α , φ) = − α + 2 2 2 X ε − dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) 2 X2 Tp = τ = qε q = 1, . . . , p ∗

and ζε (α, φ) is the characteristic function of |α − φ|X ≤ p0 (ε). The proof of Theorem 2.2, which comes at the end of this section, is similar in spirit to that of [2, Theorem 3.13], but uses, in place of [2, Example 3.15]. Example 2.3. For each ε > 0, set Iε (α, φ) = e− 2 α 1

2

− 12 φ2 F (ε,α∗ ,φ)

e

= e− 2 α 1

2

− 12 φ2

α e−εK φ

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and use ∗ε to denote the convolution (I ∗r J )(α, γ) = I(α, φ)J (φ, γ) dμr (φ∗ , φ) of [2, Definition 3.15], with r = Rε . Then 2 2 1 1 Iε∗ε q (α, φ) = e− 2 α − 2 φ α e−εK IRε e−εK IRε . . . IRε e−εK φ with q factors of e−εK . Also set

ζε (α, φ) − 1 −1/2 , in [2, where F1 was defined, for |α|X , |φ|X < 8eε(h1,∞ +μ+v0 ) ε v 1,∞ Proposition 3.6]. Here v0 = maxx∈X |v(x, x)|. δIε (α, φ) = e− 2 α 1

2

− 12 φ2 F (ε,α∗ ,φ)

e

∗

e−F1 (ε,α

,φ)

The principal difference between the proofs of Theorem 2.2 and [2, Theorem 3.13] is that is in the latter we simply bounded each integral by the supremum of its integrand multiplied by its volume of integration while in the former we use a field dependent, integrable, bound on the integrand. This demands relatively fine bounds on Iε∗ε q (α, φ) and δIε (α, φ), which we prove in the next subsection. Bounds on Iε∗ε q (α, φ) and δIε (α, φ) Set λ0 = λ0 (v) as in (1.3). By [2, Proposition 2.7], N 1 λ0 N 2 K ≥ λ0 N − μN + − v0 N = − νN (2.2) λ0 2 |X| 2 |X| where ν = λ20 max 0, v0 + μ − λ0 . Here λ0 is the smallest eigenvalue of h. Lemma 2.4. The functionals Iε (α, φ) and δIε (α, φ) of Example 2.3 obey the following. (a) For any γ > 0 and q ∈ IN, ∗q Iεε (α, φ) ≤ c1 e− 12 min{1,qελ0 γ}t where t= (b) For any q ∈ IN,

1

α 2 + φ 2 ) 2

∗q Iεε (α, φ) ≤ c2 where t=

1

α 2 + φ 2 ) 2

c1 = eqελ0 (ν+γ)

2

|X|

t 1 −c3 qεt2 e + e− 8 qελ0

c2 = 65e(1+qελ0 ν

2

)|X|

.

c3 =

λ0 . 40|X|

(c) Let β > 0 and assume that q ∈ IN and ε > 0 are such that 0 < qε ≤ β. If Rε is large enough (depending only on ν and |X|), then there is a constant const (depending only on |X|, β, λ0 and ν) such that λ 2 2 1 1 1 1 2 ∗ε q − 0 R4 qε Iε (α, φ) − e− 2 α − 2 φ α e−qεK φ ≤ const e− 4 Rε + e 54|X| ε . ε

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Proof. (a) Recalling that P(n) denotes projection onto Bn (X), ∗q Iεε (α, φ) ≤ e− 2 α e− 2 φ 2

1

1

2

n=0

× P(n) φ ≤ e− 2 α e− 2 φ 2

1

∞ (n) (n) −εK P α P e IRε e−εK IRε . . . IRε e−εK

1

2

∞ n n2

α n − 12 qελ0 ( |X| −νn) φ

√ e √ . n! n! n=0

Observe that n2 1 − νn − 2γn = |X| |X| Using that α φ ≤

1 2

2 1 ν + 2γ |X| − (ν + 2γ)2 |X| ≥ −(ν + γ)2 |X| . (2.3) n− 2 4

α 2 + φ 2 = t,

∞ n ∗q n2 t −qελ0 γn − 12 qελ0 ( |X| −νn−2γn) Iεε (α, φ) ≤ e−t e e n! n=0

≤ e−t

∞ n t −qελ0 γn qελ0 (ν+γ)2 |X| e e n! n=0 −qελ0 γ )t

= e−(1−e

eqελ0 (ν+γ)

2

|X|

.

If qελ0 γ ≤ 1, then, by the alternating series test e−qελ0 γ ≤ 1−qελ0 γ + 12 (qελ0 γ)2 ≤ 1 − 12 qελ0 γ, which implies that ∗q Iεε (α, φ) ≤ e− 12 qελ0 γt eqελ0 (ν+γ)2 |X| . If qελ0 γ ≥ 1, then ∗q Iεε (α, φ) ≤ e− 12 t eqελ0 (ν+γ)2 |X| (b) As

2 1 n2 1 − νn = n − ν|X| − ν 2 |X| ≥ −ν 2 |X| 2|X| 2|X| 2

Vol. 9 (2008)

Bosonic Integral II

1281 n2

we have, since IRε ≤ 1 and P(n) e−εK ≤ e−ελ0 ( |X| −νn) , ∗q Iεε (α, φ) ≤ e− 2 α e− 2 φ 1

2

1

2

n=0

× P(n) φ ≤ e− 2 α e− 2 φ 1

2

1

∞ (n) (n) −εK P α P e IRε e−εK IRε . . . IRε e−εK

2

∞ n n2

α n − 12 qελ0 ( |X| −νn) φ

√ e √ n! n! n=0

∞ n 1 n2 0 2 t − qελ e 4|X| n e− 2 qελ0 ( 2|X| −νn) n! n=0

≤ e−t

∞ n 2 0 2 t − qελ e 4|X| n eqελ0 ν |X| . n! n=0 2 2 If t ≤ e, it suffices to use the bound Iε∗ε q (α, φ) ≤ e−t et eqελ0 ν |X| = eqελ0 ν |X| 2 e t since 65e1 e− 8 ≥ 1 implies that c2 e− 8 ≥ eqελ0 ν |X| for all t ≤ e. So we may assume qελ0 that t > e. Similarly, if |X| ≥ 1, we may use the bound

≤ e−t

∞ n ∞ ∗q t − 1 n2 qελ0 ν 2 |X| (t/e)n − 1 n2 +n qελ0 ν 2 |X| Iεε (α, φ) ≤ e−t e 4 e e 4 ≤ e−t e n! n! n=0 n=0

≤ e−t

∞ 2 1 (t/e)n 1+qελ0 ν 2 |X| e = e1+qελ0 ν |X| e−(1− e )t . n! n=0

0 So we may also assume that qελ |X| ≤ 1. For any m > 0 (not necessarily integer) ∗q Iεε (α, φ) tn − qελ0 n2 tn − qελ0 n2 2 2 ≤ e−t e 4|X| eqελ0 ν |X| + e−t e 4|X| eqελ0 ν |X| n! n!

n≤4m|X|

≤ eqελ0 ν

2

= eqελ0 ν

2

n≥4m|X|

|X| −t

e (4m|X| + 1) sup n

|X| −t

e (4m|X| + 1) sup n

∞ n 0 2 tn − qελ t −qελ0 mn qελ0 ν 2 |X| e 4|X| n + e−t e e n! n! n=0 −qελ0 m 0 2 tn − qελ )t qελ0 ν 2 |X| e 4|X| n + e−(1−e e . n!

0 Choose the m specified in Lemma 2.5 below with ε replaced by qελ 4|X| . Applying that lemma gives ∗q Iεε (α, φ) ≤ eqελ0 ν 2 |X| e−t (4m|X| + 1)2e(m+t)/2 + e−(1−e−qελ0 m )t eqελ0 ν 2 |X|

where m is the unique solution to qελ0 1 m + ln m + = ln t 2|X| 2m

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with m ≥ 1. Since ln m ≤ ln t −

qελ0 qελ0 m ⇐⇒ m ≤ te− 2|X| m 2|X|

we have qελ0 − m ∗q Iεε (α, φ) ≤ 2eqελ0 ν 2 |X| (4m|X| + 1)e−(1−e 2|X| )t/2 + e−(1−e−qελ0 m )t eqελ0 ν 2 |X| . We treat the two terms, T 1 = 2eqελ0 ν

2

|X|

−qελ0 m )t

T 2 = e−(1−e

−

(4m|X| + 1)e−(1−e eqελ0 ν

2

qελ0 m 2|X| )t/2

|X|

separately. Case 1: First term,

qελ0 m 2|X|

≥1

qελ0

In this case 1 − e− 2|X| m ≥ 1 − e−1 ≥ 12 and, since m < t, T 1 ≤ 2eqελ0 ν 2 |X| (4m|X| + 1)e−t/4 ≤ 2eqελ0 ν 2 |X| (4t|X| + 1)e−t/4 2 t qελ0 ν 2 |X| ≤ 64e |X| 1 + e−t/4 ≤ 64e(1+qελ0 ν )|X| e−t/8 . 8 Case 2: First term,

qελ0 m 2|X|

≤1

qελ0

−α 0m ≥ α2 for all 0 ≤ α ≤ 1. Hence In this case 1 − e− 2|X| m ≥ qελ 4|X| since 1 − e T 1 ≤ 2eqελ0 ν 2 |X| (4m|X| + 1)e−qελ0 mt/(8|X|) .

Since ln m = ln t − we have

1 3 t qελ0 m − ≥ ln t − ⇒ m≥ 2|X| 2m 2 5

qελ0 2 t T 1 ≤ 2eqελ0 ν 2 |X| (4m|X| + 1)e− 40|X| qελ0 2 2 |X|2 ≤ 2eqελ0 ν |X| 8 + 1 e− 40|X| t qελ0 qελ0 2 |X|2 qελ0 ν 2 |X| 16 + |X| e− 40|X| t ≤ 2e qελ0 2 qελ0 2 2 1 − 40|X| t ≤ 32e(1+qελ0 ν )|X| e . qελ0

Case 3: Second term, qελ0 m ≥ 1 In this case 1 − e−qελ0 m ≥ 1 − e−1 ≥ 12 and T 2 ≤ e−t/2 eqελ0 ν 2 |X| .

Vol. 9 (2008)

Bosonic Integral II

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Case 4: Second term, qελ0 m ≤ 1 Now 1 − e−qελ0 m ≥ 12 qελ0 m so that qελ0 2 2 1 1 − 40|X| t qελ0 t2 qελ0 ν 2 |X| T 2 ≤ e− 12 qελ0 mt eqελ0 ν 2 |X| ≤ e− 10 e ≤ eqελ0 ν |X| e qελ0

since we again have m ≥ 5t , as in Case 2. (c) Introduce the local notation −εK −εK e e if i is odd Ai = Bi = if i is even IR ε 1l so that

e−εK IRε

q−1

e−εK =

2q−1

Ai

and

if i is odd if i is even

e−qεK =

i=1

2q−1

Bi .

i=1

For any n ∈ IN, q−1 e−εK − e−qεK φ α e−εK IRε !2q−1 # " 2q−1 ≤ α (2.4) Ai − Bi Pn φ i=1 i=1 2q−1 # + α (2.5) Ai (1l − Pn ) φ + α e−qεK (1l − Pn ) φ . i=1 Consider the first line, (2.4). Observe that IR ≤ 1, by part (c) of [2, Theorem 2.26] and K ≥ − 12 λ0 ν 2 |X|, by (2.3) with γ = 0. Hence $ 2 eελ0 ν |X| if i is odd

Ai , Bi ≤ 1 if i is even . Since A − B = IRε − 1l for = 2, 4, . . . , for all n , q ∈ IN , !2q−1 " 2q−1 2q−1 Ai − Bi Pn ≤ i=1

i=1

2q − 2 and is zero otherwise, we have,

−1

2q−1 Ai A − B Bi Pn i=1 =1 i=+1

2 ≤ (q − 1)eqε λ0 ν |X| IRε − 1l Pn

≤ qeqε λ0 ν

2

|X|

|X| 2n+1 e−Rε /2 2

by part (d) of [2, Theorem 2.26]. Consequently, q−1 2 2 1 1 e− 2 α − 2 φ α e−ε K IRε e−ε K − e−qε K Pn φ ≤ qeqε λ0 ν

2

|X|

|X| 2n+1 e−Rε /2 . (2.6) 2

1284

and

T. Balaban et al.


Now consider the second line, (2.5). For all m ≥ 1, K B

m

≥ 12 λ0

m |X|

−ν m

2q−1 1 m Ai , e−qεK ≤ e− 2 qε λ0 ( |X| −ν) m Bm Bm i=1

and it follows that 2q−1 # 1 m 2 2 1 1 Ai Bm φ , α e−qεK Bm φ ≤ e− 2 qε λ0 ( |X| −ν) m e 2 α + 2 φ . α i=1

n If we impose the stronger condition m ≥ n with |X| ≥ 2ν, the last inequality becomes 2q−1 # λ0 2 2 2 1 1 Ai B φ , α e−qεK B φ ≤ e− 4 |X| m qε e 2 α + 2 φ . α m m i=1

Now, we have, if n is large enough (depending only on ν and |X|) 2q−1 # 2 2 1 1 e− 2 α − 2 φ α Ai (1l − Pn ) φ , i=1 ∞ − λ0 m2 qε λ0 2 ≤ e 4 |X| ≤ ds e− 4 |X| s qε m>n λ0 − 6 |X|

≤e %

% =

e− 2 α 1

2

λ0

ds e− 12 |X| s

n qε 0

6|X| e λ0 qε

n2 qε

2

∞

ds e− 2 s

1 2

(2.7)

0

α e−qεK (1l − Pn ) φ .

− 12 φ2

1

2

(so that 2n ≤ e 4 Rε ) and adding (2.6) and twice (2.7)

gives 2 2 1 1 ∗ε q Iε (α, φ) − e− 2 α − 2 φ α e−qεK φ qε λ0 ν 2 |X|

≤ 2qe Lemma 2.5. Let 0 < ε ≤

qε

% 0 n2 qε 1 √ 0 n2 qε 6|X| − 6λ|X| |X| − 6λ|X| e e 2π ≤ 4 . λ0 qε 2 λ0 qε

The same estimate applies to

1 2 3 Rε

∞

λ0 − 6 |X|

=

Choosing n =

n

2

1 4

−R2ε /4

|X| e

% +8

and t ≥ e. Then sup e−εn

2

n≥1

tn ≤ 2e(m+t)/2 n!

0 |X| − 54λ|X| R4ε qε e . λ0 qε

Vol. 9 (2008)

Bosonic Integral II

1285

where m is the unique solution to 2εm + ln m +

1 = ln t 2m

with m ≥ 1. Proof. Recall that Stirling’s formula [1, 6.1.38] states that for each real n > 0, there is a 0 < θ < 1 such that √ 1 θ n! = 2πnn+ 2 e−n+ 12n . In particular, for n ≥ 1, √ √ 1 1 2πnn+ 2 e−n ≤ n! ≤ 2πnn+ 2 e−n+1 . Hence −εn2

e

1 1 n −εn2 tn ≤√ 1 e e n! 2π n 2

n 2 1 t 1 = √ en−εn +n ln t−n ln n− 2 ln n . n 2π

(2.8)

Observe that, for n ≥ 1, d 1 1 2 n − εn + n ln t − n ln n − ln n = 1 − 2εn + ln t − ln n − 1 − dn 2 2n 1 . = ln t − 2εn − ln n − 2n 1 = 2ε + n1 − 2n1 2 > 0 for all n ≥ 1 and ln t ≥ 2ε + 12 , the Since ddn 2εn + ln n + 2n equation 1 = ln t 2εm + ln m + 2m has a unique solution m ≥ 1. This solution obeys m t 1 2 2εm2 −1/2 2εm + m ln m + = m ln t ⇒ e =e . 2 m The derivative of the last exponent in (2.8) is positive for n < m and negative for n > m. Hence the last exponent of (2.8) takes its maximum value at n = m and m m/2 n 2 t 1 1 m −εm2 t e1/4 1 m t √ ≤√ e e = e e−εn n! m m 2π m1/2 2π m1/2 1/2 1/4 m m e t 1 m/2 e =√ e 1/4 mm+1/2 2π m 1/2 e1/4 1 m/2 √ tm √ ≤ e e 2π m! 2π m1/4 m 1/2 t ≤ 2em/2 m! ≤ 2e(m+t)/2 .

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For the rest of this section, except where otherwise specified, all constants may depend on |X|, v, h 1,∞ , cj , β and μ. They may not depend on ε or p. Lemma 2.6. Let δIε (α, φ) be as in Example 2.3. There are constants const and CR √R , such that, for all sufficiently small ε and all |α|X , |φ|X ≤ C ε

& '

2 4 4 δIε (α, φ) ≤ const ε2 1+|α|6X +|φ|6X e− 14 α−φ − 18 ελ0 α4 +φ4 +e− 14 p0 (ε)2 . Proof. By [2, Corollary 3.6], 1 1 Re − α 2 − φ 2 + F (ε, α∗ , φ) 2 2 1 1 = Re − α 2 − φ 2 + dx α(x)∗ φ(x) − εK(α∗ , φ) + F0 (ε, α∗ , φ) 2 2 X 1 = − α − φ 2 + Re − ε dxdy α(x)∗ h(x, y)φ(y) + εμ dx α(x)∗ φ(x) 2 X2 X ε dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) + F0 (ε, α∗ , φ) . − 2 X2 For any α, φ ∈ L2 (X), we have εμ Re

1 εμ α 2 + φ 2 2 √ √ 4μ λ0 λ0 4μ 1 |α(x)|2 + √ |φ(x)|2 = ε dx √ 2 λ0 4 λ0 4 2

1 8μ |X| ε + ελ0 α 44 + φ 44 ≤ λ0 64

dx α(x)∗ φ(x) ≤

and

dxdy α(x)∗ h(x, y)φ(y) = −ε dxdy α(x)∗ h(x, y)α(y) + ε Re dxdy α(x)∗ h(x, y)(α − φ)(y)

−ε Re

≤ ε h α

α − φ

1 1 ≤ ε3/2 h α 2 + ε1/2 h α − φ 2 2 2

h

h 3/2 1 ε λ0 α 44 + ε1/2 h α − φ 2 ≤ |X| ε3/2 + λ0 16 2

Vol. 9 (2008)

and −

ε Re 2

Bosonic Integral II

dxdyα(x)∗ α(y)∗ v(x, y) φ(x)φ(y) 1 dxdy α(x)∗ α(x) v(x, y) α(y)∗ α(y) =− ε 4 1 dxdy φ(x)∗ φ(x) v(x, y) φ(y)∗ φ(y) − ε 4 1 dxdy α(x)∗ (α−φ)(x) v(x, y) α(y)∗ φ(y) + ε Re 4 + α∗ (x)α(x)v(x, y)α∗ (y)(α−φ)(y) 1 dxdy (α − φ)∗(x)φ(x) v(x, y) α(y)∗ φ(y) − ε Re 4 + φ∗ (x)φ(x)v(x, y)(α − φ)∗ (y)φ(y)

1 1 ≤ − ελ0 α∗ α 2 + φ∗ φ 2 + ε v α∗ (α − φ) α∗ φ

4 4

+ α∗ (α − φ) α∗ α + φ∗ (α − φ) α∗ φ + φ∗ (α − φ) φ∗ φ

1 ≤ − ελ0 α 44 + φ 44 4

1 + ε v α − φ |α|X α∗ φ + α∗ α + |φ|X α∗ φ + φ∗ φ

4

1 ≤ − ελ0 α 44 + φ 44 4 CR √ ε v α − φ α∗ φ + α∗ α + α∗ φ + φ∗ φ

+ 4

1 ≤ − ελ0 α 44 + φ 44 4 CR √ ε v α − φ 2 α 4 φ 4 + α 24 + φ 24 + 4

1 ≤ − ελ0 α 44 + φ 44 4 CR √ CR √ ε v α − φ

α 24 + ε v α − φ

φ 24 + 2 2 2

v

1 ≤ − ελ0 α 44 + φ 44 + 2CR √

α − φ 2 4 λ0

1 + ελ0 α 44 + φ 44 32 2

v

7 4 4

α − φ 2 = − ελ0 α 4 + φ 4 + 2CR √ 32 λ0

1287

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T. Balaban et al.


and, by [2, Corollary 3.6],

|F0 (α∗ , φ)| ≤ c0 ε2 |X| |α|2X + |φ|2X + v 21,∞ |α|6X + v 21,∞ |φ|6X

1 1 2 4 4 2 ε v 21,∞ |X| α 44 + φ 44 . ≤ c0 ε |X| 1 + α 4 + φ 4 + c0 CR 2 2 v 1 2 ≤ 13 and c0 CR

v 21,∞ |X| ≤ 96 λ0 , All together, if ε is small enough and 2CR √ λ0 then there is a constant const such that

1 1 1 3 Re − α 2 − φ 2 + F (ε, α∗ , φ) ≤ − α−φ 2 − ελ0 α 44 + φ 44 +const ε 2 2 4 16

and

2 3 ελ0 α44 +φ44 +const ε Iε (α, φ) ≤ e− 14 α−φ − 16 .

(2.9)

Similarly, by [2, Proposition 3.6],

|F1 (α∗ , φ)| ≤ c0 ε2 |X| |α|2X + |φ|2X + v 21,∞ |α|6X + v 21,∞ |φ|6X

1 1 2 ε v 21,∞ |X| α 44 + φ 44 ≤ c0 ε2 |X| 1 + α 44 + φ 44 + c0 CR 2 2 1 2

v 21,∞ |X| ≤ 32 λ0 and ε is sufficiently small, so that, if c0 CR

1 −F1 (ε,α∗ ,φ) ελ α44 +φ44 e ≤ const e 16 0

1 ελ0 α4 +φ4 −F1 (ε,α∗ ,φ) 2 6 6 4 4 − 1 ≤ const ε 1 + |α|X + |φ|X e 16 e

and

δIε (α, φ) ≤ eRe [− 12 α2 − 12 φ2 +F (ε,α∗ ,φ)] e−F1 (ε,α∗ ,φ) − 1 ∗ + e−F1 (ε,α ,φ) ζε (α, φ) − 1 &

' ≤ const ε2 1 + |α|6X + |φ|6X + ζε (α, φ) − 1

− 14 α−φ2 − 18 ελ0 α44 +φ44 e

which yields the desired bound.

Lemma 2.7. Let β > 0 and assume that q ∈ IN and ε > 0 are such that 0 < qε ≤ β. Let Iε (α, φ) be as in Example 2.3 and Rε obey Hypothesis 2.1. Then there are constants a1 , a2 and a3 such that ∗q

Iεε (α, φ) ≤ a1 e−a2 α−φ2 + e−a3 p0 (ε)2 . Proof. Let c2 = 65e(1+βλ0 ν

2

)|X|

,

c3 =

λ0 40|X|

Vol. 9 (2008)

Bosonic Integral II

If either α of φ is larger than ∗q Iεε (α, φ) ≤ c2

CR √ qε

t 1 −c3 qεt2 e + e− 8 λ0 qε

where

c4 = c2

then, t ≥ ≤

2 4 2 c C2 + 1 λ0 CR 3 R

2 CR 2qε

1289

and, by part (b) of Lemma 2.4,

2 2 1 t 2c2 te− 2 c3 CR t +c2 e− 8 ≤ c4 e−c5 α−φ 2 λ0 CR

c5 = min

1 1 2 , c3 CR . 32 16

Here we used that

α − φ 2 ≤ 2 α 2 + 2 φ 2 = 4t . On the other hand if both α and φ are smaller than replaced by qε, and part (c) of Lemma 2.4,

CR √ qε ,

then by (2.9), with ε

λ0 ∗q R4ε qε Iεε (α, φ) ≤ e− 14 α−φ2 +const qε + const 1 e− 14 R2ε + e− 54|X| ε ' λ0 4 2 1 1 & − 1 p0 (ε)2 e 4 ≤ e− 4 α−φ +const qε + const + e− 54|X| p0 (ε) ε − 1 α−φ2

−c6 p0 (ε)2 4 . +e ≤ c7 e

In both cases,

with

∗q

Iεε (α, φ) ≤ a1 e−a2 α−φ2 + e−a3 p0 (ε)2 a1 = max{c4 , c7 ,

a2 = c5 ,

a3 = c6 .

Proof of Theorem 2.2 Lemma 2.8. Under the notation and hypotheses of Theorem 2.2 there are constants C and κ > 0 such that ∗q Iεε (α, φ) δIε (φ, γ)e−κQεγ dμR (φ∗ , φ)dμR (γ ∗ , γ) ε

ε

& ' √ −κ min{1,(Q+q+1)ε}α −κp0 (ε)2 ≤C e +e ε, min

for all 0 < ε < 1, 0 ≤ q ≤

β ε

and Q ≥ 0.

Proof. By Lemmas 2.4, 2.6 and 2.7 and the bounds

|γ|6X ≤ 26 |φ|6X + |γ − φ|6X e−δα ≤ eδ e−δα 2

e−δα4 ≤ eδ|X| e−δα ≤ eδ(|X|+1) e−δα 4

2

1 √ 5/2 q ε

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T. Balaban et al.


(for all δ > 0) we have ∗q Iεε (α, φ)

1 −κqεφ44 −κqεα −2κα−φ −2κp0 (ε)2 −κφ ˜ + e ≤ Ce e min 1, e +e qε δIε (φ, γ) ' &

4 2 ≤ C˜ ε2 1 + |φ|6X + |γ − φ|6X e−κεφ e−3κφ−γ e−κεφ4 + e−3κp0 (ε)

˜ −κεφ e−2κφ−γ + e−2κp0 (ε)2 ≤ Ce

4 ε2 1 + |φ|6X e−κεφ4 + |γ − φ|6X e−κφ−γ . (2.10)

For the last inequality of (2.10), we used that, for all |φ|X ≤ Rε , √

4 2 e−κεφ ε2 1+|φ|6X e−κεφ4 +|γ−φ|6X e−κφ−γ ≥ e−κεRε |X| ε2 ≥ const e−κp0 (ε) . First use (twice) that, for Qε ≤ 1 (if Qε > 1, replace Qε by 1), qκε α + κ α − φ + κQε φ ≥ qκε α + κQε α − φ + κQε φ

≥ κ(Q + q)ε α

(2.11)

to prove that 2 2 e−κqεα e−2κα−φ + e−2κp0 (ε) e−κεφ e−2κφ−γ + e−2κp0 (ε) e−κQεγ 2 2 ≤ e−κqεα e−2κα−φ + e−2κp0 (ε) e−κ(Q+1)εφ e−κφ−γ + e−2κp0 (ε) ≤ e−κ min{1,(Q+q+1)ε}α e−κα−φ e−κφ−γ + 3e−2κp0 (ε) . 2

Next combine

3/2 −κqεφ4 1 −κqεφ44 1 −κφ + e 4 e |φ|6X e κqε φ 44 ≤ 3 qε (κqε) 2 qε

6 1 + 6 κ φ e−κφ κ 1 ≤ Cκ 1 + (qε)5/2 and

3/2 −κεφ4 1 4 κε φ 44 e 3/2 (κε) 1 ≤ Cκ 3/2 ε

|φ|6X e−κεφ4 ≤ 4

and |γ − φ|6X e−κφ−γ ≤ Cκ

(2.12)

Vol. 9 (2008)

Bosonic Integral II

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to give

1+

4 |φ|6X e−κεφ4

+ |γ −

1 −κqεφ44 −κφ min 1, e +e qε 1 1 ≤ Cκ 3 + min , . (2.13) ε3/2 (qε)5/2

φ|6X e−κφ−γ

By (2.10), (2.12) and (2.13) ∗q √ Iεε (α, φ)δIε (φ, γ)e−κQεγ ≤ C˜ 2 Cκ 3ε2 + min ε,

1 √ 5/2

q ε −κ min{1,(Q+q+1)ε}α −κα−φ −κφ−γ 2 e e + 3e−2κp0 (ε) . e

Hence ∗q Iεε (α, φ)δIε (φ, γ)e−κQεγ dμRε (φ∗ , φ)dμRε (γ ∗ , γ) √ 1 ≤ C˜ 2 Cκ 3ε2 + min ε, 5/2 √ q ε

2|X| −2κp0 (ε)2 2 −κ min{1,(Q+q+1)ε}α D e + 3 πR2ε e with

D=

For q ≤

β ε

we have ε2 ≤

β 5/2 √ q 5/2 ε

e−κγ dμ(γ ∗ , γ) .

and the bound follows.

Proof of Theorem 2.2. Expand (Iε + δIε )∗ε p − Iε∗ε p =

p

r=1

q1 ,...,qr+1 ≥0 q1 +···+qr+1 =p−r

Iε∗ε q1 ∗ δIε ∗ Iε∗ε q2 ∗ δIε ∗ · · · ∗ Iε∗ε qr ∗ δIε ∗ Iε∗ε qr+1 .

Hence p ∗ε p ∗ε p ∗ ≤ (I (α, α) dμ + δI ) − I (α , α) ε ε R ε ε r=1 ∗q ∗ dμRε (α , α) sup Iεε 1 ∗δIε ∗· · ·∗Iε∗ε qr ∗δIε ∗Iε∗ε qr+1 (α, γ) . q1 ,...,qr+1 ≥0 q1 +···+qr+1 =p−r

|γ|X ≤Rε

(2.14)

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We now prove by backwards induction that, for each r ≥ s ≥ 0, sup Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qr ∗ δIε ∗ Iε∗ε qr+1 (α, γ) |γ|X ≤Rε

≤ (3C)r−s+1

r

$ √ min ε,

1 5/2 √

(

q ε =s+1 & ' 2 dμRε (γ ∗ , γ) Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qs ∗ δIε (α, γ) e−κ min{ps ε,1}γ + e−κp0 (ε) (2.15)

with ps = qs+1 + · · · + qr+1 + (r − s) and C the constant of Lemma 2.8. When s = 0, Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qs ∗ δIε (α, γ) is the kernel of the identity operator. Consider the initial case, s = r. By (2.10), sup Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qr ∗ δIε ∗ Iε∗ε qr+1 (α, γ) |γ|X ≤Rε

≤

∗

dμRε (α , α )

sup Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qr ∗ δIε (α, α ) Iε∗ε qr+1 (α , γ)

|γ|X ≤Rε

≤C

∗ dμRε (α , α ) Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qr ∗ δIε (α, α )e−κqr+1 εα

which provides the induction hypothesis for s = r. Now assume that the induction hypothesis holds for s. Observe that ' & 2 dμRε (γ ∗ , γ) Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qs ∗ δIε (α, γ) e−κ min{ps ε,1}γ + e−κp0 (ε) ' & 2 ∗ ≤ dμRε (α , α )dμRε (φ∗ , φ)dμRε (γ ∗ , γ) e−κ min{ps ε,1}γ + e−κp0 (ε) ∗ q

Iεε 1 ∗ δIε ∗ · · · ∗ Iε∗ε qs−1 ∗ δIε (α, α ) Iε∗ε qs (α , φ) δIε (φ, γ) . (2.16) By Lemma 2.8, with q = qs , ' & 2 dμRε (φ∗ , φ)dμRε (γ ∗ , γ) e−κ min{ps ε,1}γ + e−κp0 (ε) Iε∗ε qs (α , φ) δIε (φ, γ) 2 ≤ C e−κ min{(ps +qs +1)ε,1}α +e−κp0 (ε) ( $ √ 1 −κp0 (ε)2 −κ min{(qs +1)ε,1}α −κp0 (ε)2 e +e +e ε, 5/2 √ min ε qs ( $ & ' √ 2 1 ≤ 3C e−κ min{ps−1 ε,1}α + e−κp0 (ε) min . (2.17) ε, 5/2 √ qs ε

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Here we used Lemma 2.8 with Qε = min{ps ε, 1} for the first term in the curly bracket and with Q = 0 for the second term in the curly bracket. Inserting this result into (2.16) and then applying the inductive hypothesis (2.15) yields (2.15) with s replaced by s − 1 and γ replaced by α . In particular, when s = 1, inserting (2.17) into the inductive hypothesis (2.15) yields sup Iε∗ε q1 ∗ δIε ∗ · · · ∗ Iε∗ε qr ∗ δIε ∗ Iε∗ε qr+1 (α, γ) |γ|X ≤Rε

≤ (3C)r+1

r

$ √ min ε,

1 5/2 √

q

=1

(

& ' 2 e−κ min{pε,1}α + e−κp0 (ε) .

Applying (2.18) to (2.14), it follows that ∗ε p ∗ε p ∗ (I (α, α) dμ + δI ) − I (α , α) ε ε Rε ε $ p r √ ≤ (3C)r+1 min ε, r=1

q1 ,...,qr+1 ≥0 q1 +···+qr+1 =p−r

=1

1 5/2 √ q ε

(

' & 2 e−κ min{pε,1}α + e−κp0 (ε) ( $ ∞ r √ 1 r+1 ≤ const (3C) min ε, 5/2 √ q ε r=1 q1 ,...,qr ≥0 =1 ⎡ ⎤r ∞ √ 1 ⎣3C ≤ const min ε, 5/2 √ ⎦ . q ε r=1 q≥0 Since

min

dμRε (α∗ , α)

√ ε,

q≥0

q

1 √ 5/2

ε

≤

0≤q≤

√

ε+

1 ε2/5

≤ const

!√

q≥

ε

ε2/5

1 ε2/5

1 +√ ε

(2.19)

1 √ q 5/2 ε 1

−3/2 "

ε2/5

1

≤ const ε 10 we get that

(2.18)

ε

(2.20)

1 ∗ε p ∗ε p ∗ (Iε + δIε ) − Iε (α, α) dμRε (α , α) = O ε 10

and the theorem follows from −β K

Tr e

= lim

p→∞

which was proven in [2, Theorem 3.1].

Iε∗ε p (α, α) dμRε (α∗ , α)

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3. Correlation functions By definition, an n-point correlation function at inverse temperature β is an expression of the form n Tr e−βK T j=1 ψ (†) (βj , xj ) . Tr e−βK Here ψ (†) refers to either ψ or ψ † and ψ (†) (τ, x) = eKτ ψ (†) (x)e−Kτ . n The time–ordering operator T orders the product j=1 ψ (†) (βj , xj ) with smaller times to the right. In the case of equal times, ψ † ’s are placed to the right of ψ’s. We already have functional integral representations for the denominator, which is just the partition function. In this section, we outline the analogous construction of functional integral representations for the numerator. Recall that a partition P of the interval [0, β] is a finite set of points τ , 0 ≤ ≤ p, that obeys 0 = τ0 ≤ τ1 ≤ · · · ≤ τp−1 ≤ τp = β .

(3.1)

We shall only consider partitions all of whose subintervals τ − τ−1 are of roughly the same size. We denote by p = p(P ) the number of intervals in the partition P and set ε = ε(P ) = βp . For the rest of this section we fix β > 0, n ∈ IN and 0 = β0 ≤ β1 ≤ β2 ≤ · · · ≤ βn+1 = β. Then n Tr e−βK T j=1 ψ (†) (βj , xj ) Tr e−βK ψ (†) (xn )e−(βn −βn−1 )K ψ (†) (xn−1 ) · · · e−(β2 −β1 )K ψ (†) (x1 )e−β1 K Tr e = . Tr e−βK Definition 3.1. (a) A (β0 , . . . , βn+1 )-partition is a partition P = τ 0 ≤ ≤ p of the interval [β0 , βn+1 ] (i) that contains the points β1 , β2 , . . . , βn and for which (ii) 12 ε(P ) ≤ τ − τ−1 ≤ 2ε(P ) for all 1 ≤ ≤ p. (b) We denote by P = P(β0 , . . . , βn+1 ) the set of all (β0 , . . . , βn+1 )-partitions. When we say that lim f (P ) = F −(β−βn )K

p→∞

we mean that for every η > 0 there is an N ∈ IN such that |F − f (P )| < η for all P ∈ P(β0 , . . . , βn+1 ) with p(P ) ≥ N . The analog of [2, Theorem 3.1] is Theorem 3.2. Let R(P, ) > 0, for each P ∈ P and 1 ≤ ≤ p(P ), and assume that

p(P )

lim

p→∞

=1

e− 2 R(P,) = 0 . 1

2

(3.2)

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Then, Tr e−βK T

n

ψ (†) (βj , xj )

j=1

p(P ) = lim dμR(P,) (φ∗τ , φτ ) e− p→∞

dy |φτ (y)|2

φτ−1 e−(τ −τ−1 ) K φτ

=1 n

φβj (xj )(∗)

j=1

with the convention that φ0 = φβ . √ Example 3.3. Let C > 2. Any R(P, )’s that obey R(P, ) ≥ C ln p(P ) satisfy the hypothesis of Theorem 3.2, because

p(P )

p(P )

e− 2 R(P,) ≤ 2

1

=1

p(P )− 2 C ≤ p(P )1− 2 C . 2

1

2

1

=1

Remark 3.4. In fact Theorem 3.2 does not use condition (ii) of Definition 3.1.a. It suffices to require (3.2). For example, any R(P, )’s that obey % 1 R(P, ) ≥ C ln τ − τ−1 √ where P = {0 = τ0 , τ1 , . . . , τp−1 ≤ τp = β} and C > 2 work, as long as the mesh P = max1≤≤p(P ) (τ − τ−1 ) tends to zero, because

p(P )

p(P )

e− 2 R(P,) ≤ 2

1

=1

1

2

1

(τ − τ−1 ) 2 C ≤ P 2 C

2

−1

β.

=1

Proof of Theorem 3.2. We may assume, without loss of generality, that the numn ber ψ † ’s is the same as the number of ψ’s so that the operator j=1 ψ (†) (xj )

e−(βj+1 −βj )K commutes with the number operator. Otherwise, both sides are zero. (To see that the right hand side vanishes, use invariance under φτ → φτ eiθ .) So, by the definition of Ir (given in the statement of [2, Theorem 2.26]), [2, Proposition 2.20] and [2, Proposition 2.28] (we’ll prove boundedness of the appropriate operator shortly), the integral on the right hand side can be written

p(P )

Tr

+ e−(τ −τ−1 ) K Ψ− IR(P,) Ψ

=1

where the product is ordered with smaller indices on the right, $ n ψ(xj ) if βj = τ and ψ (†) (xj ) = ψ(xj ) − Ψ = 1l otherwise j=1

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and

$ ψ † (xj ) Ψ+ = 1l j=1 n


if βj = τ and ψ (†) (xj ) = ψ † (xj ) otherwise .

Replacing all the IR(P,) ’s by 1l gives the trace on the left hand side. on the m particle space Bm (X) Recall that P (m) is the orthogonal projection . and that Pm is the orthogonal projection on ≤m B (X). Since K and Ir preserve particle number, ψ † increases it by one and ψ decreases it by one, there are, for each m ∈ IN ∪ {0}, integers m − n ≤ m ≤ m + n, 1 ≤ ≤ p(P ), such that

p(P )

p(P )

P (m)

+ e−(τ −τ−1 ) K Ψ− IR(P,) Ψ =

=1

+ P (m ) e−(τ −τ−1 ) K Ψ− IR(P,) Ψ .

=1

Recall from (2.2) that, if and τ ≥ 0, then

m |X|

≥ 2ν (the constant ν was defined just after (2.2))

λ0 (m) −τ K m2 τ ≤ e− 4|X| P e

(3.3)

†

By [2, Lemma 2.13], the local density operator ψ (x)ψ(x), when restricted to the m particle space Bm , has eigenvalues running over the integers from 0 to m. As a consequence (m) † √ √ P ψ(x)P (m) ≤ m and ψ (x) ≤ m . (3.4) Hence if each J , 1 ≤ ≤ p(P ), is either IR(P,) or 1l, we have ) λ0 (m) p(P 2 n + −(τ −τ−1 ) K − P e Ψ J Ψ ≤ e− 4|X| (m−n) β (m + n) 2 =1 assuming that

m−n |X|

≥ 2ν. Pick any γ > 0 with 2γ
0, m ˜ large ˜2 < 2ε and then choosing p large enough that the remaining enough that 2Ce−γ m sum is smaller than 2ε . The analog of [2, Theorem 3.13] is Theorem 3.5. Let max

1≤≤p(P )

R(P, ) ≤

1 ε(P )

1 6(n+3)(|X|+1)

for each P ∈ P and assume that

p(P )

lim

p→∞

=1

e− 2 R(P,) = 0 . 1

2

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Then, Tr e−βK T

n

ψ (†) (βj , xj )

j=1

= lim

p→∞

p(P )

−

dμR(P,) (φ∗τ , φτ ) e

∗ dy [φ∗ τ (y)−φτ

−1

(y)]φτ (y)

=1 −(τ −τ−1 )K(φ∗ τ

e

−1

,φτ )

n

φβj (xj )(∗)

j=1 ∗

with the convention that φ0 = φβ . Recall that K(α , φ) was defined in [2, Corollary 3.7]. Proof. We give the proof for the case that 0 = β0 < β1 < β2 < · · · < βn < β = βn+1 . The proofs for the other cases require only very minor changes. Let, as in [2, Examples 3.15 and 3.17], 2 2 ∗ 2 2 1 1 1 1 Iε (α, φ) = e− 2 α − 2 φ eF (ε,α ,φ) = e− 2 α − 2 φ α e−εK φ 2 2 ∗ ∗ 1 1 I˜ε (α, φ) = e− 2 α − 2 φ eF (ε,α ,φ)−F0 (ε,α ,φ) 2 2 ∗ 1 1 = e− 2 α − 2 φ α e−εK φ e−F0 (ε,α ,φ)

where F0 was defined in [2, Corollary 3.7]. Recall that 1 1 2 2 ∗ ∗ ˜ Iε (α, φ) = exp − α − φ + dx α (x)φ(x) − εK(α , φ) . 2 2 For any partition P = {0 = τ0 < τ1 < · · · < τp = β} ∈ P, set, for 1 ≤ ≤ m ≤ p(P ),

I,m (φ, φ ) = Iε ∗R(P,) Iε+1 ∗R(P,+1) · · · ∗R(P,m−1) Iεm (φ, φ )

˜,m (φ, φ ) = I˜ε ∗R(P,) I˜ε ∗R(P,+1) · · · ∗R(P,m−1) I˜ε (φ, φ ) I

m

+1

where ε = τ − τ−1 . The convolution ∗r was introduced in [2, Definition 3.14]. By Theorem 3.2, if βj = τj for 1 ≤ j ≤ n, T r e−βK T

n

ψ (†) (βj , xj )

j=1

= lim

p→∞

n

j=0

dμR(P,j ) (φ∗βj , φβj )

n j=0

Ij +1,j+1 (φβj , φβj+1 )

n

φβj (xj )(∗)

(3.7)

j=1

where 0 = β0 = 0, n+1 = p(P ), φβn+1 = φ0 and R(P, 0) = R(P, p(P )). On the other hand, the right hand side of the claim of the current theorem is n n n ˜ +1, (φβ , φβ ) dμR(P,j ) (φ∗βj , φβj ) φβj (xj )(∗) . (3.8) lim I j j+1 j j+1 p→∞

j=0

j=0

j=1

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We apply Proposition 3.6.b, below with I replaced by Iε , I˜ replaced by 3/2 1 ˜ Iε , ζ = ε , r = R(P, ), κ = 12 , p = p(P ) and Cβ = β. If p(P ) is sufficiently large, the hypotheses of the Proposition are satisfied because then 13 3 1

1 (1− 12 ) 2 (n+3)(|X|+1) 1−κ (n+3)(|X|+1) Cβ π max r ζ ≤ βπ ε2 ε(P ) 25

≤ 2βπ (n+3)(|X|+1) ε24 ≤ ε 1 ε(P )

since

≤

2 ε ,

and, by [2, Example 3.17] with r =

Iε − I˜ε r−1 ,r ≤ eε K0 const ε2

2 ε

1 4|X|

1 2 24|X|

ε

≥ r−1 , r ,

|X| econst |X|ε (2/ε ) 2

1 4|X|

3/2

≤ ε

= ζ

and, as [2, Example 3.15] (just replace the q − 1 appearances of Ir by Ir , . . . , Irm−1 and the q appearances of e−εK by e−ε K , . . . , e−εm K ),

Iε ∗r Iε+1 ∗r+1 · · · ∗rm−1 Iεm r−1 ,rm ≤ e(ε +···+εm )K0 . The theorem now follows by Proposition 3.6.b, (3.7) and (3.8).

Proposition 3.6. Let K0 > 0, 0 < κ < 1, Cβ ≥ 1 and p ∈ IN. Let r0 , . . . , rp ≥ 1, ε1 , . . . , εp > 0 and ζ1 , . . . , ζp > 0 and assume that ε1 + · · · + εp ≤ Cβ . For each 1 ≤ ≤ p, let I , I˜ : C2|X| → C. Define, for each 1 ≤ ≤ m ≤ p, ˜,m = I˜ ∗r I˜+1 ∗r · · · ∗r I˜m I m−1 +1

I,m = I ∗r I+1 ∗r+1 · · · ∗rm−1 Im and assume that

I − I˜ r−1 ,r ≤ ζ

I,m r−1 ,rm ≤ e(ε +···+εm )K0

where

I r,r = (a) If

sup

φ,φ ∈C |φ|X ≤r, |φ |X ≤r

I(φ, φ )

X

2 |X| 1−κ 2 |X| Cβ πr−1 πr ζ ≤ ε

for

= 1, . . . , p

then, setting ζ = max1≤≤p ζ , ˜1,p r ,r ≤ e(ε1 +···+εp )(K0 +ζ

I 0 p

κ

)

˜1,p − I1,p r ,r ≤ ζ κ e(ε1 +···+εp )(K0 +ζ

I 0 p

κ

)

.

(b) Let n ∈ IN and 0 ≤ 1 ≤ · · · ≤ n ≤ p. If r1 , . . . , rp ≤ r with

(n+3)(|X|+1) 1−κ ζ ≤ ε for = 1, . . . , p Cβ πr2

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then p−1 p n dμr (φ∗ , φ ) I (φ−1 , φ ) φj (xj )(∗) =0

j=1

=1

−

p−1

dμr (φ∗ , φ )

=0

p

n

I˜ (φ−1 , φ )

(∗)

φj (xj )

j=1

=1

n n n = dμrj (φ∗j , φj ) Ij +1,j+1 (φj , φj+1 ) φj (xj )(∗) j=0

−

n

j=0

dμrj (φ∗j , φj )

j=0

≤ ζ κ e(ε1 +···+εp )(K0 +ζ

n

j=1

˜ +1, (φ , φ ) I j j+1 j j+1

j=0 κ

n

(∗)

φj (xj )

j=1

)

where 0 = 0, n+1 = p and, as usual, φp = φ0 .

Proof. The proof is very similar to the proof of [2, Proposition 3.16]. The analog of Theorem 2.2 is

Theorem 3.7. Let Rε and p0 (ε) obey Hypothesis 2.1 and j(ε) obey (2.1). Let β > 0. Set R(P, ) = Rε(P ) , for each partition P = 0 = τ0 < τ1 < · · · < τp = β ∈ P β and 1 ≤ ≤ p(P ). Then, with the conventions that ε = ε(P ) = p(P ) , p = p(P ), ε = τ − τ−1 and φ0 = φβ , Tr e−βK T

n

ψ (†) (βj , xj )

j=1

= lim

p→∞

where

p

A(ε ,φ∗ τ

dμRε (φ∗τ , φτ ) ζε (φτ−1 , φτ ) e

−1

,φτ )

=1

n

φβj (xj )(∗)

j=1

1 1 2 dxdy α(x)∗ j(ε ; x, y)φ(y) − φ 2 A(ε , α , φ) = − α + 2 2 2 X ε − dxdy α(x)∗ α(y)∗ v(x, y) φ(x)φ(y) 2 2 X ∗

and ζε (α, φ) is the characteristic function of |α − φ|X ≤ p0 (ε). We prove this theorem following the proof of Lemma 3.10, below. Until we start the proof of Theorem 3.7, we fix a partition P = 0 = τ0 < τ1 < · · · < τp = β ∈ P and set ε = βp , ε = τ − τ−1 and r = Rε . We let Iε and δIε be defined

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as in Example 2.3 and write I for Iε and Iˆ for Iε + δIε . Thus 2 2 ∗ 2 2 1 1 1 1 I (α, φ) = e− 2 α − 2 φ eF (ε ,α ,φ) = e− 2 α − 2 φ α e−ε K φ 2 2 ∗ ∗ 1 1 Iˆ (α, φ) = e− 2 α − 2 φ eF (ε ,α ,φ)−F1 (ε ,α ,φ) ζε (α, φ) 2 2 ∗ ∗ 1 1 δI (α, φ) = e− 2 α − 2 φ eF (ε ,α ,φ) e−F1 (ε ,α ,φ) ζε (α, φ) − 1

where F1 was defined in [2, Proposition 3.6]. Recall that 1 1 − α 2 − φ 2 + F (ε , α∗ , φ) − F1 (ε , α∗ , φ) = A(ε , α∗ , φ) . 2 2 We also introduce analogs of I and Iˆ that contain the appropriate correlation n fields from j=1 φβj (xj )(∗) . C (α, φ) = I (α, φ)Φ (φ)

Cˆ (α, φ) = Iˆ (α, φ)Φ (φ) δC (α, φ) = δI (α, φ)Φ (φ) (3.9)

where

⎧ n ⎪ ⎨φβj (xj ) Φ (φ) = φ (x )∗ ⎪ βj j j=1 ⎩1

if βj = τ and ψ (†) (xj ) = ψ(xj ) if βj = τ and ψ (†) (xj ) = ψ † (xj ) otherwise .

The various convolutions are

I,m (φ, φ ) = I ∗r I+1 ∗r · · · ∗r Im (φ, φ )

C,m (φ, φ ) = C ∗r C+1 ∗r · · · ∗r Cm (φ, φ )

ˆ ,m (φ, φ ) = Cˆ ∗r Cˆ+1 ∗r · · · ∗r Cˆm (φ, φ ) . C

We have proven, in Theorem 3.2, that the left hand side in Theorem 3.7 is lim C1,p (α, α) dμRε (α∗ , α) . p→∞

On the other hand, the right hand side in Theorem 3.7 is ˆ 1,p (α, α) dμR (α∗ , α) . lim C ε

p→∞

Lemma 3.8. Let 1 ≤ ≤ m ≤ p and write ε¯ = ε + · · · + εm . Then (a) For any γ > 0,

I,m (α, φ) ≤ c1 e− 12 min{1,¯ελ0 γ}t

where t= (b) We have

1

α 2 + φ 2 ) 2

I,m (α, φ) ≤ c2

c1 = eε¯λ0 (ν+γ)

2

|X|

t 1 −c3 ε¯t2 e + e− 8 ε¯λ0

.

(3.10)

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where t=

1

α 2 + φ 2 ) 2

c2 = 65e(1+¯ελ0 ν

2

)|X|

c3 =

λ0 . 40|X|

(c) Let β > 0 and assume that 0 < ε + · · · + εm ≤ β. If r is large enough (depending only on ν and |X|), then there is a constant const (depending only on |X|, β, λ0 and ν) such that 2 2 1 1 I,m (α, φ) − e− 2 α − 2 φ α e−¯εK φ λ0 1 − 54|X| r 4 ε¯ − 14 r 2 √ ≤ const (m − )e + . e ε¯ The proof of this lemma is virtually the same as the proof of its analog, Lemma 2.4. For the rest of this section, except where otherwise specified, all constants may depend on |X|, v, h 1,∞ , cj , β, μ and n. They may not depend on the partition P and, in particular, on ε = ε(P ) or p = p(P ). Lemma 3.9. Let I,m (α, φ) be as in (3.10). There are constants a1 , a2 and a3 such that

I,m (α, φ) ≤ a1 e−a2 α−φ2 + e−a3 p0 (ε)2 for all 1 ≤ ≤ m ≤ p. The proof of this lemma is virtually identical to that of its analog, Lemma 2.7. Lemma 3.10. Under the notation and hypotheses of Theorem 3.7 there are constants C and κ > 0 such that the following holds. Let 0 < ε < 1 and 1 ≤ ≤ m ≤ p and set q = m − + 1. Write |γ|+ = max{1, |γ|X }. (a) Denote by n the total number of φβj (xj )(∗) ’s in C,m , as defined in (3.9) and (3.10). Then & ' 2 sup C,m (α, φ) ≤ C |α|n+ e−κ min{1,qε}α + e−κp0 (ε) . φ

(b) Denote by n is the total number of φβj (xj )(∗) ’s in C,m δCm+1 , as defined in (3.9) and (3.10). Then

˜ −κQεγ C,m (α, φ) δCm+1 (φ, γ) |γ|n+ e dμr (φ∗ , φ)dμr (γ ∗ , γ) & ' √ n ˜ +n −κ min{1,(Q+q+1)ε}α −κp0 (ε)2 e min ≤ C |α|+ ε, +e for all n ˜ + n ≤ n and Q ≥ 0.

1 √ 5/2 q ε

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Proof. We start by observing, just as in (2.10), that

˜ −2κ¯εα e−3κα−φ + e−3κp0 (ε)2 I,m (α, φ) ≤ Ce 1 −κ¯εφ44 −κφ + e min 1, e ε¯

˜ −κεφ e−3κφ−γ +e−3κp0 (ε)2 δIm+1 (φ, γ) ≤ Ce

4 ε2 1+|φ|6X e−κεφ4 +|γ − φ|6X e−κφ−γ .

(3.11)

(a) From the definitions (3.9) and (3.10), we have C,m (α, φ) ≤

k j=1

dμr (φ∗j , φj ) I,1 (φ0 , φ1 ) |φ1 |X I1 +1,2 (φ1 , φ2 ) |φ2 |X · · · · · · Ik +1,m (φk , φk+1 ) |Φm (φk+1 )|X .

(3.12)

Here φ0 = α and φk+1 = φ. If k = 0, then k = − 1. The n of the statement of the lemma is $ k if Φm (φk+1 ) = 1 n = k + 1 if Φm (φk+1 ) = 1 . Insert the first bound of (3.11) into (3.12). Set q1 = 1 − + 1, q2 = 2 − 1 , . . ., qk = k − k−1 and qk+1 = m − k . Also set ε¯1 = τ1 − τ−1 , ε¯2 = τ2 − τ1 , . . ., ε¯k+1 = τm − τk . By the second condition in part (a) of Definition 3.1, each ε¯i ≥ 12 qi ε. Also q1 + · · · + qk+1 = q. When inserting the first bound of (3.11) 4 into (3.12), discard all factors min 1, ε¯1j e−κ¯εj φj 4 + e−κφj . To this point, the right hand side of (3.12) is bounded by a constant times k+1 ' & 2 e−κqi εφi−1 e−3κφi−1 −φi + e−3κp0 (ε) i=1 k i=1

|φi |X |Φm (φk+1 )|X

k

dμr (φ∗j , φj ) .

j=1

k

We now deal with the factors i=1 |φi |X and |Φm (φk+1 )|X . Use that, for any field |φ|X ≤ r, a > 0 and 0 < b < 1 −aα−φ 2 e + e−ap0 (ε) |φ|+ ≤ e−aα−φ |α|+ + e−aα−φ |α − φ|X + e−ap0 (ε) |φ|+ 2

≤ e−aα−φ |α|+ + α − φ e−aα−φ + re−ap0 (ε) 2 ≤ Ca,b e−abα−φ + e−abp0 (ε) |α|+ (3.13) k to “move” each of the n ≤ n fields in i=1 |φi |X |Φm (φk+1 )|X to |φ0 |+ = |α|+ . n We may choose b so that 3κb ≥ 2κ. Consequently, to this point, the right hand 2

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side of (3.12) is bounded by an (n-dependent) constant times |α|n+

k+1 k ' & 2 e−κqi εφi−1 e−2κφi−1 −φi + e−2κp0 (ε) dμr (φ∗j , φj ) i=1

j=1

k k+1 n k+1 −2κp0 (ε)2 −κqi εφi−1 −2κφi−1 −φi ≤ |α|+ e e + e dμr (φ∗j , φj ) . 2 i=1

j=1

Now use that, for Qε ≤ 1 (if Qε > 1, replace Qε by 1), κqi ε φ + κ φ − γ + κQε γ ≥ κqi ε φ + κQε φ − γ + κQε γ

≥ κ(Q + qi )ε φ

to prove that e−κqi εφ e−2κφ−γ e−κQεγ ≤ e−κ min{1,(Q+qi )ε}φ e−κφ−γ .

(3.14)

Applying this k times we have that the right hand side of (3.12) is bounded by a constant times k k+1 n k+1 −2κp0 (ε)2 −κ min{1,Σi qi ε}α −κφi−1 −φi e +e e dμr (φ∗j , φj ) 2 |α|+

i=1

j=1

≤ |α|n+ 2k+1 (πr2 )k|X| e−2κp0 (ε) + e−κ min{1,Σi qi ε}α Dk 2

with D= As

e−κγ dμ(γ ∗ , γ) .

k+1

i=1 qi = q, the bound follows. (b) The proof is similar to that of Lemma 2.8.

Proof 3.7. We need to show that, in the notation of (3.10), the integral of Theorem ˆ C1,p − C1,p (α, α) dμRε (α∗ , α) converges to zero as p = βε → ∞. Recall that Cˆ = C + δC and expand ˆ 1,p − C1,p = C

p

C1,q1 −1 ∗r δCq1 ∗r Cq1 +1,q2 −1 ∗r δCq2 ∗r · · ·

ρ=1 1≤q1 1 smaller than the minimum expansion along unstable vectors, see Section 2.4.2. To formulate our main results, let us recall what is ment by exponential decay of correlations and the central limit theorem. Definition 1.5. Consider a Riemannian manifold M as a phase space, with a dynamics T and a T -invariant probability measure µ. We say that the dynamical system (M, T, µ) has exponential decay of correlations (EDC), if for every f, g : M → R α-H¨ older-continuous pair of functions there exist constants C < ∞ and a(α) > 0 such that for every n ∈ N Z Z Z n n f (x)g(T x)dµ(x) − f (x)dµ(x) g(T x)dµ(x) ≤ C(f, g)e−an . M

M

M

Definition 1.6. Consider a Riemannian manifold M as a phase space, with a dynamics T and a T -invariant probability measure µ. We say that the dynamical system (M, T, µ) satisfies the Central LimitR Theorem (CLT), if for every Höldercontinuous function f : M → R such that M f dµ = 0, there is some σ ≥ 0 such that n−1 1 X D √ f ◦ T i −→ N (0, σ) . n i=0 That is, suitably normalized Birkhoff sums converge in the distribution to the normal law if the initial point is chosen according to the measure µ. Theorem 1.7. Consider a multi-dimensional dispersing billiard map that satisfies Assumptions 1.1, 1.2 and 1.3. Then the dynamics enjoys exponential decay of correlations and the central limit theorem holds. Remark 1.8. It is an interesting question how the constant C(f, g) of Definition 1.5 depends on the observables f and g. As we rely on [9] and [20], which work in a symbolic setting (Young towers), C(f, g) is not directly determined by the functions f and g themselves, but their symbolic representations. Nonetheless it is still true that C(f, g) ≤ C(α)||f ||C α ||g||C α where ||f ||C α denotes the α-Hölder norm of f . The formal proof of this theorem will be given at the end of the paper, in Section 4.6, when all the necessary ingredients are at hand. However, we describe the structure of the proof and give an outline of the paper here.

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As already mentioned in the introduction, we prove Theorem 1.7 by verifying the assumptions of a theorem from [9] that guarantees exponential decay of correlations. To make our exposition more self-contained, we present Chernov’s conditions separately in Appendix A. The rest of the paper, which provides the verification of these assumptions, is organized as follows. In Section 2 we collect the most important prerequisites on multidimensional dispersing billiards. In particular, we recall that certain conditions from Appendix A: Conditions A.1, A.2, A.4, A.5 and A.6 have already been proven for the studied billiards in [5]. In addition, we formulate and prove some further properties of similar flavour, to be applied in the later sections, which – to our knowledge – have not been considered before. The verification of Condition A.7 is the main novelty of our paper. To achieve this we are led to use a Riemannian structure different from the traditional “Euclidean” metric on the billiard phase space. However, the properties discussed in Section 2, in particular, conditions A.1, A.2, A.4, A.5 and A.6 are originally proved for the “Euclidean” metric. Thus it is to be verified that these properties remain valid with the use of the new Riemannian structure. This is the content of Section 3, which is of differential geometric nature. Condition A.7 on the “growth properties of unstable manifolds” is proved in Section 4. As already mentioned, Condition A.3, more precisely, the ergodicity of the map (and its iterates) is the only condition that we do not prove in the present paper. For the proof of ergodicity we refer to [4]. Some explanation is given in Section 4.5. This completes the proof of Theorem 1.7. In addition to Appendix A, we have also included Appendix C which contains some simple lemmas of geometric measure theory.

2. Preliminaries In this section we repeat notions and statements from [5] and [9]. These will be referred to in several parts of the discussion. Although some of the referred statements are only found in the text of [5] and are not highlighted as theorems, we do not repeat the proofs here. Instead, we give the precise references within the paper. Our aim with listing these statements is to collect all facts about billiards that are used from earlier works in one place, and keep the paper otherwise selfcontained. Notation 2.1. Throughout the paper we will use the following conventions: Positive and finite global constants whose value is unimportant, will be denoted by c or C. So e.g. f < C means that the function f is bounded from above. The letters c and C may denote different values in different equations. On the other hand, C1 , C2 , . . . will denote global constants whose values are the same throughout the paper.

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We say that two nonzero functions f and g on the phase space are equivalent, if c ≤ fg ≤ C. In this case we will use the notation f ∼ g. 2.1. Chernov’s conditions In his paper [9] Chernov has proven a theorem that guarantees exponential decay of correlations, provided that we can check that the dynamical system satisfies certain conditions about hyperbolicity, regularity of unstable manifolds and the dynamics on them, and about the growth of unstable manifolds. A dynamical system that can be handled in this way should consist of • A phase space M , which is the set of possible states of the system. • A dynamics T which is an M → M map, or at least a map defined on a large subset of M . • A measure µ on M which is T -invariant. ¯ ) is a Riemannian manifold, • A Riemannian structure on M , so that M (or M possibly with boundary. The level of smoothness of maps and subsets of this Riemannian manifold is an important issue. Instead of just referring to the work [9], we will list the conditions and repeat the statement in the Appendix. This is done mainly to make our paper easier to read, but also to point out two minor details where we use modified versions of Chernov’s conditions. Both modifications allow the proof in [9] to remain unchanged. 2.2. The dynamical system In accordance with Chernov’s conditions, we will now describe our choice of phase space, dynamics and measure. However, we postpone our choice of the Riemannian structure until Section 3, since this is a key point of our proof, and certainly doesn’t belong to the Preliminaries section. 2.2.1. The Poincaré section phase space and the dynamics. First we describe the flow phase space M. This consists of all possible positions of the particle, equipped with the possible unit velocities: M = (q, v) | q ∈ Q, v ∈ Rd , kvk = 1 . We could identify phase points with opposite velocities and the same configuration point on ∂Q, but this is not important for our purposes. We will only use the flow phase space at those occasions, when it is important to view the Poincaré section phase space as a submanifold of M. ˜ . Our phase So now we describe the usual Poincaré section phase space M ˜ space M will be a subset of this. In M , we only consider collision moments. Since kinetic energy is preserved in the system, we also fix the speed to be 1. We choose to describe the motion of the particle at a collision time by recording its velocity just after the collision (we use the ‘outgoing’ Poincaré section). So a possible state of the particle is described by giving a boundary point ˜ = q ∈ ∂Q and a unit velocity v ∈ S d−1 , which is often written roughly as M d−1 ∂Q × S+ , where the + indicates that only velocity vectors pointing inward Q

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˜ really doesn’t have a product are allowed. However, this is misleading, because M structure. So we better write (still roughly) n o

˜ = (q, v) | q ∈ ∂Q, v ∈ Rd , kvk = 1, v, n(q) ≥ 0 . M Notation 2.2. n(q) denotes the (unit) normal vector of ∂Q at q pointing inward Q. ˜ →M ˜ gives the state of the particle at the next collision as a function T :M of the present state. When we apply the theorem of [9], we will apply it to some higher iterate T n0 of this dynamics. n0 will be given later. Notation 2.3. If some quantity (e.g. q or n) is related to some phase point x, then we will often denote the corresponding quantity related to T x by the same letter and an index 1 (e.g. q1 , n1 ). 2.2.2. The invariant measure and the Euclidean metric. The natural Riemannian ˜ is described in detail in many works including [5], and we don’t restructure on M ˜ is viewed as a semisphere-bundle peat those details here. We only mention that M over ∂Q, and the different velocity semispheres (fibres) at nearby configuration points are identified using the parallel transport from the natural Riemannian structure of ∂Q as a submanifold of Td . This results in the definition of a C 2 ˜ and a local product structure. Since we need to handle differential atlas on M aspects of the dynamics, it is crucial to understand how M is embedded in M. Now the natural Riemannian structure is locally a product of the natural Riemannian structures on ∂Q and S d−1 . We will call this natural structure the “Euclidean structure”. This is a little misleading, since this is really Euclidean only if d = 2, while it is not flat in higher dimensions. However, it still resembles a Euclidean structure in the sense that the induced norm (which is called the “Euclidean norm” in [5]) has the form k(dq, dv)k2e = dq 2 + dv 2 , where dq ∈ T 1 is a tangent vector of the configuration space ∂Q and dv ∈ J is a tangent vector of d−1 ˜ becomes a C 1 the velocity hemisphere S+ . With this Riemannian structure, M Riemannian manifold with boundary, and the boundary is n o

˜ = (q, v) | q ∈ ∂Q, v ∈ Rd , kvk = 1, v, n(q) = 0 , ∂M the set of tangential collisions. ˜ , ϕ(x) will denote the angle of the velocity and Notation 2.4. For x = (q, v) ∈ M the normal vector of the scatterer: ϕ(x) = ^(n(q), v), hn(q), vi = cos ϕ(x). In two dimensions one often treats ϕ as a signed quantity, but for us, 0 ≤ ϕ ≤ π2 . With this notation, n o ˜ = (q, v) ∈ M ˜ | ϕ(x) = π . ∂M 2 ˜ The natural T -invariant measure µ on M is defined in terms of the natural Riemannian structure: let µ be absolutely continuous with respect to the 1T

and J will be introduced in Section 2.3.

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induced measure (which we will call the Lebesgue measure), and let the density be const cos ϕ(x) where const is a normalizing constant so that µ is a probability. This can vaguely be written as dµ = const cos ϕ dq dv ,

(2.1)

since the Lebesgue measure is locally a product of the natural (surface volume) measures on ∂Q and S d−1 . Remark 2.5. Note that our choice to use “outgoing” velocities in the Poincaré map phase space (which is the usual choice) is just a matter of notation. We could as well identify every incoming velocity vector with the corresponding outgoing one, and view the outgoing velocity as a representative of the equivalence class. Accordingly, there is no asymmetry in the definition of the Euclidean metric: the Euclidean metric of the inverse dynamics is the same. As a consequence, replacing velocities with their opposites (up to identification of incoming/outgoing) leaves the measure invariant. This fact will be reflected by the formulas (2.6), (2.7), (2.16) and (2.17). 2.2.3. Singularities and the phase space we use . T is not continuous in the points ˜ , which we call the primary singularity set. So, to satisfy of S = T −1 S 0 = T −1 ∂ M Condition A.1 we have to exclude S from the domain of T . Moreover, the derivative of T blows up near S, which causes Condition A.5 to fail, unless we declare certain ˜ (which are close) to be separated by “artificial boundaries”. This is points of M done in the way usual in billiard theory. We partition the original phase space into infinitely many homogeneity layers: 1 π 1 ˜ Ik = x ∈ M | 2 < − ϕ(x) < for k = k0 + 1, k0 + 2, . . . and k 2 (k − 1)2 ˜ | 1 < π − ϕ(x) . Ik0 = x ∈ M (2.2) k02 2 Here the integer constant k0 is arbitrary, and will be chosen later. The boundary of this partitioned phase space is ∞ [ Γ0 = Γ0k k=k0

where

π 1 ˜ = x ∈ M | − ϕ(x) = 2 , k = k0 , k0 + 1, . . . . 2 k Correspondingly, the countably many manifolds in the set Γ0k

Γ = T −1 Γ0

(2.3)

are the so called secondary singularities. Now we can define the phase space we will use: ∞ [ ˜ \ (S 0 ∪ Γ0 ) = Ik ∪ M =M 0 k=k0 +1

Ik ,

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where the components Ik are meant to have disjoint closures – as if they were moved apart from each other. This makes M non-compact, but many compactness arguments remain valid if we temporarily forget about artificial boundaries – ˜ . On the other hand, regularity properties may be easier, that is, we look at M specifically distortion bounds depend on secondary singularities. 2.3. Hyperbolicity, cones and fronts In billiard theory, several basic constructions and concepts are based on the notion of a local orthogonal manifold, which - for simplicity - we will call front. A front W is defined in the flow phase space M rather than in the Poincaré section. Take a smooth 1-codimensional submanifold E of the flow configuration space Q, and add the unit normal vector v(r) of this submanifold at every point r as a velocity, continuously. Consequently, at every point the velocity points to the same side of the submanifold E. Then n o W = r, v(r) |r ∈ E ⊂ M , where v : E → Sd−1 is continuous (smooth) and v ⊥ E at every point of E. The derivative of this function v, called B plays a crucial role: dv = Bdr for tangent vectors (dr, dv) of the front. B acts on the tangent plane Tr E of E, and takes its values in the tangent plane J = Tv(r) Sd−1 of the velocity sphere. These are both naturally embedded in the configuration space Q, and can be identified through this embedding. So we just write B : J → J . B is nothing else than the curvature operator – or second fundamental form – of the submanifold E of Q, which we will abbreviate as s.f.f. Clearly, this is different from the curvature of W as a submanifold of M. Obviously, B is symmetric. Notice that fronts remain fronts during time evolution - at least locally, and apart from some singularity lines. When we talk about a front, we sometimes think of it as the part of the (flow) phase space just described (for example, when we talk about time evolution under the flow), but sometimes just as the submanifold E (for example, when we talk about the tangent space or the curvature of the front). This should cause no confusion. In the rest of this section we list technical details about the evolution of fronts and the construction of invariant cone fields required by the Hyperbolicity condition (Condition A.2). Later we will only use these details in two places: • in Section 2.5.2 to understand the nature of anisotropic expansion of unstable vectors • in Sections 3.2 and 3.3 where we introduce a new Riemannian structure and show that T is uniformly hyperbolic with respect to this Riemannian structure in the sense of Condition A.2. 2.3.1. Evolution of fronts. During free propagation (that is, from one collision to the other) a tangent vector (dr+ , dv + ) of the post-collision front evolves into the

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tangent vector (dr1− , dv1− ) of the pre-collisional front at the next collision given by the formulas dr1− = dr+ + τ dv + ,

(2.4)

dv1−

(2.5)

= dv

+

where τ is the length of the free run between the two collisions. For this formula – and the next one – to make sense, we need to identify the tangent planes of the front at different moments of time. Let T = Tr ∂Q be the tangent plane of the scatterer at a collision point r. Just like J , T is viewed together with its natural embedding into Q. The identification of different J ’s is done in the usual way (cf. [15, 19]): • by translation parallel to v from one collision to the other, • by reflection with respect to T (or, equivalently, by projection parallel to n) from pre-collision to post-collision moments. At a moment of collision a tangent vector of a front changes non-continuously (the front is “scattered”): a tangent vector (dr− , dv − ) of the pre-collision front evolves into the tangent vector (dr+ , dv + ) of the post-collision front given by dr := dr+ = dr− , +

−

(2.6) ∗

dv = dv + 2hn, viV KV dr

(2.7)

where • V : J → T is the projection parallel to v: V dv = dv − hdv,ni hv,ni v ∈ T for dv ∈ J , • V ∗ : T → J (the adjoint of V ) is the projection parallel to n: V ∗ dq = dq − hdq,vi hn,vi n ∈ J for dq ∈ T , • K : T → T is the s.f.f. of the scatterer at the collision point, • hn, vi = cos ϕ, where ϕ ∈ [0, π2 ] is the collision angle. From these we can get the evolution of the second fundamental form: B + = B − + 2 cos ϕV ∗ KV

(2.8)

(equation (2.3) from [5]), where • B − : J → J is the s.f.f. just before collision, which also describes tangent vectors (dr− , dv − ) of the pre-collision front through dv − = B − dr− .

(2.9)

• B + : J → J is the s.f.f. just after collision, which also describes tangent vectors (dr+ , dv + ) of the pre-collision front through dv + = B + dr+ .

(2.10)

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2.3.2. Unstable and stable cone field. In Condition A.2, uniform hyperbolicity is formulated in terms of invariant cone fields. The construction of these cone fields is done in a standard way (see e.g. [5], Section 4.3): cones consist of tangent vectors of appropriate fronts. More precisely: since fronts are subsets of the flow phase space, cones consist not of tangent vectors of fronts, but rather the traces of these vectors on T M . The fronts defining the unstable cone field satisfy c < B+ , c 0 depending only on c1 and C1 (and the geometry of the billiard table).

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2.3.5. Expansion along unstable manifolds. The rather technical formulas of this subsection will only be used in Section 2.5.2 to prove Lemma 2.10 and its corollary. The reader is encouraged to skip these details for the first reading. Let W be a “u-manifold”, that is, a d − 1 = dim2 M dimensional submanifold of M , which has all of its tangent vectors in the unstable cone (at all of its points). e Denote by JW (x) the Jacobian of T restricted to W at x. Then we know −1 e JW (x) ∼ det( V1 ) ∼ cos(ϕ1 ) , (2.18) which is equation (4.15) from [5]. Let us now consider a further restriction of DT onto a subspace R ⊂ Tx W of the tangent plane of this u-manifold. For this we know that det( DT |R ) ∼ det(V1 |R0 )

(2.19)

(V1−1

˜ to where R = ◦ π1 ◦ DT )(R), and π is the natural projection of Tx M T = Tx ∂Q: π((dq, dw)) = dq. This is equation (4.16) from [5]. 0

2.4. Known regularity properties 2.4.1. Geometric regularity properties in the Euclidean structure. The first two regularity properties listed here are obvious, the others are proven in [5]. The phase space is always equipped with the Euclidean structure. 1. T is piecevise H¨ older continuous – i.e. it is Hölder continuous on the finitely ˜ \S, but of course also on the countably many compomany components of M ˜ nents of M \ (S ∪ Γ). Actually, the Hölder exponent is 12 , but for simplicity we p will use that ρ(T x, T y) ≤ 3 ρ(x, y) whenever ρ(x, y) is small enough, and x and y are in the same component of continuity. 2. The expansion of T is bounded when not acting near singularities. In particular, there exists a δ > 0 such that kDTx k ≤ 1δ whenever ρ(x, S) > δ. 3. Uniform transversality: the angle between vectors of the stable and unstable cones Cxs and Cxu is uniformly bounded away from zero. 4. Uniform alignment: the angle of any unstable manifold with any (one-step) singularity manifold in S or Γ is uniformly bounded away from zero. 5. Chernov’s regularity conditions: (a) Uniform hyperbolicity: We know that Condition A.2 is satisfied by some iterate T n of the dynamics. We will not make use of this fact, instead, we will prove the stronger statement of one-step uniform hyperbolicity for another Riemannian structure. (b) Uniform curvature bounds: Condition A.4 is satisfied. (c) Uniform distortion bounds: Condition A.5 is satisfied. (d) Uniform absolute continuity: Condition A.6 is satisfied. Remark 2.6. To avoid confusion we mention that Condition A.6 is not explicitly stated in this form in [5]. However, the relevant statement, Theorem 5.9 of [5] is known to imply the absolute continuity property of Condition A.6, based on a classical argument by Anosov and Sinai from [1]. Furthermore, we may allow for a

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Figure 1. A piece of the phase space cut by singularities. little more flexibility: the manifolds W1 and W2 that appear in Condition A.6 may be arbitrary u-manifolds (manifolds with tangent planes in the unstable cone). Actually, it is this slightly generalized form of the absolute continuity property that we apply in [4]. See also [11] on further details about different formulations of absolute continuity. 2.4.2. Structure and complexity of the singularity set. In this section we discuss singularities of higher iterates of the dynamics T . We introduce the notation S n = T −n S 0

n = 1, 2, . . . , −1, −2, . . .

for the “n-step singularity set”. So S 0 = ∂M is the “0-step singularity set”, and the singularity set of T is S = S 1 . The set of points where T n is singular is n [ S (n) := S n for n ≥ 1 i=1

S (n) :=

−1 [

Sn

for n ≤ −1 .

i=n

The n-step secondary singularity set Γn and the secondary singularity set of T , Γ(n) can be defined analogously. However, in this section we discuss the structure of the primary singularity set S (n) of T n . That is, the secondary singularities in (2.3) are not considered. An important feature of the singularity set of billiards is the so-called “continuation property”. This means that the primary singularities are one-codimensional submanifolds that can only terminate on each other, or on the boundary of M . More precisely, S is a finite union of one-codimensional compact submanifolds of M with boundary, and every boundary point is either an inner point of some other component, or it is on ∂M . See Figure 1. The consequence of this continuation property is that singularities do cut the phase space: if a small open subset U of M is intersected by a singularity manifold, n

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then it is indeed cut into two components. These are not necessarily connected components: U itself may be non-connected already, but even a connected set may well be cut into many pieces by a single 1-codimensional plane. So by “component” we mean those points of U which can be connected by a continuous curve in M which is disjoint from the entire (primary) singularity set S. ˜ \ S (n) consists of finitely many (open) Definition 2.7. For every n, the set M ˜ (n),i , where i is from some finite connected components, let us denote these by M ˜ (n),i -s ˜ index set. Now for any set U ⊂ M , we denote by Kn,U the number of M that are intersected by U . The quantity Kn,U will be referred as the complexity of S (n) on U . ˜ let us denote by Kn,x the number of M ˜ (n),i that contain x, which For x ∈ M will be referred as the complexity of S (n) at x. Finally, we define the complexity of S (n) as Kn := sup Kn,x . ˜ x∈M

˜ that for every n there exists an ε Remark 2.8. It follows from compactness of M such that if the diameter of U is less than ε, then Kn,U ≤ Kn . A very common assumption in the theory of singular hyperbolic dynamical systems is that the complexity Kn is a subexponential function of n, or at most of O(λn ) where λ is strictly less than the smallest expansion on the unstable cone. We also have to assume this property, see Assumption 1.3. 2.5. Further regularity properties In this section we discuss two further regularity properties, which are not new, and are in a sense known to Billiards experts, however, we could not locate a precise formulation and proof in the literature. 2.5.1. Smoothness of one-step singularities. Much of the difficulty in the discussion of multi-dimensional dispersing billiards is related to a phenomenon discovered in [5]: if we consider higher iterates of the dynamics, the singularity set will be non-smooth, i.e. its curvature blows up near certain “pathological” points. In the present work, however, it is important for us that such a pathology does not appear for the (non-iterated, or 1-step) dynamics itself. With the notations of Section 2.2.3: T −n S 0 may behave irregularly for n ≥ 2, but S = T −1 S 0 is smooth. This is also true for the secondary singularity set Γ, and even for every submanifold ˜ which is (similarly to a secondary singularity) the pre-image of an arbitrary in M {ϕ = const} set. Moreover, there is a universal upper bound for the curvature of all of these manifolds. This fact is stated in the following proposition. Proposition 2.9. There is a global constant KS such that for any ϕ0 ∈ [0, π2 ], the ˜ | ϕ(x) = ϕ0 }) at any of its points is at curvature of the submanifold T −1 ({x ∈ M most KS .

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The proof of this proposition is postponed until Section 3.4.1, since the precise notion of “curvature” used in the statement is discussed in Section 3.4 only. 2.5.2. Anisotropy near tangent collisions. A key feature of multi-dimensional dispersing billiard dynamics is the anisotropy of expansion in unstable directions. This means that near singularities the expansion is not only very strong, but also very direction-dependent. Indeed, certain unstable vectors are expanded enormously (of the order 1/ cos ϕ1 ), while others are only expanded moderately (of the order 1). There is in a sense only one strongly expanding direction, which is approximately the direction of perturbations within the plane of the trajectory (at the next collision). Here we need to make these statements precise, and draw the consequence that the strongly expanding direction is ‘just orthogonal’ to the secondary singularities – that is, the distance of nearby {ϕ1 = const} manifolds is increased by T by a factor of 1/ cos ϕ1 . At this point we are still using the Euclidean norm on the phase space, but the statements will also hold with the new Riemannian structure to be introduced in Section 3. Lemma 2.10. For any x ∈ M near a singularity, there is a tangent vector dx ∈ Tx M such that 1 |dϕ1 | ∼ kDT dxke ∼ kdxke . cos ϕ(T x) Proof. We work in the tangent space of M at T x, so for convenience we choose T x as time zero and denote quantities at T x without indices. We will say that a vector of T or J is in the “strongly scattering direction”, if it is in the plane spanned by n and v (orthogonal to T ∩ J ). Such vectors have the property that they are greatly expanded by V or V ∗ (exactly by the factor 1/ cos ϕ). We will use that for a tangent vector (dr, dv) of a front just after collision, which has the vector (dq, dw) as its trace on T M , we have dq = V dr (which is (2.16)) and dw = B − dr + V ∗ K cos ϕV dr , which comes e.g. from the combination of (2.8) and (2.17). Choose dx so that DT dx = (dq, dw) has dq pointing in the strongly scattering direction. Then kdqk = kV drk = cos1 ϕ kdrk, so the vector cos ϕV dr has length kdrk and is in the strongly scattering direction. Since K is positive definite and c < K < C, the vector K cos ϕV dr also has a component in the strongly scattering direction which is at least ckdrk long, and the other component is not longer than Ckdrk either. So V ∗ K cos ϕV dr is at least cosc ϕ kdrk long and points mainly in the strongly scattering direction. Since B − dr is of order kdrk only (by (2.12)), the whole of dw points mainly in the strongly scattering direction, meaning that kdwk ≈ |dϕ|. Now (2.15) gives kDT dxke ∼ |dϕ|. On the other hand, dx was chosen exactly so that (2.19) gives kDT dxke 1 ∼ , kdxke cos ϕ so dx is really the vector we are looking for.

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Corollary 2.11. Let γ1 and γ2 be two nearby {ϕ = const} manifolds: γ1 = {x ∈ M | ϕ(x) = ϕ1 }, γ2 = {x ∈ M | ϕ(x) = ϕ1 + dϕ}. Then dist(γ1 , γ2 ) ∼

1 dist(T −1 γ1 , T −1 γ2 ) . cos ϕ1

Proof. (2.18) implies the upper bound: no vector can be expanded more than ∼ cos1ϕ1 . The lower bound comes from Lemma 2.10: dist(γ1 , γ2 ) = |dϕ| ∼

1 1 kdxke ≥ dist(T −1 γ1 , T −1 γ2 ) . cos ϕ1 cos ϕ1

3. Riemannian structure and regularity Our aim now is to introduce a Riemannian structure on the billiard phase space, which will be different from the natural Riemannian structure of the manifold, and will exhibit, as a main feature, one-step expansion of unstable vectors. Similar metric-type quantities have been used before, see e.g. the quadratic form Q used in [17], or the pseudo-metric called “p-metric” in [5, 15, 19]. These are known to increase on the unstable cone, but are not well-behaved on general tangent vectors. The new feature of the metric we are about to introduce is that it comes from a true Riemannian structure, equivalent in a strong sense to the “original” Euclidean metric (see Appendix B). We will use this equivalence strongly. 3.1. Motivation The reason for this is the following. As mentioned in Section 2, the use of the Euclidean structure has the big advantage that the regularity properties A.4, A.5 and A.6 have already been proven in [5]. On the other hand, uniform hyperbolicity (A.2) is only true for a higher iterate of the dynamics, e.g. the length of an unstable vector may decrease with one application of the (derivative of the) dynamics, which leads to difficulties when trying to prove the growth properties A.7. The key feature of the Riemannian structure we are about to introduce is that the induced metric exhibits one step expansion on unstable vectors. It is important to see that we can not just use a higher iterate of the dynamics to achieve one-step expansion. The reason is the substantial difference between the singularity set of the 1-step dynamics and the higher iterates. When proving the growth condition A.7, we will need to estimate the measure of some δneighbourhood of the singularity set (within the unstable manifold). As discussed in Section 2.5.1, the 1-step singularities are uniformly smooth, so such an estimate can be based on a locally flat picture – both unstable manifolds and singularities can be pictured as affine subspaces in a Euclidean space. However, already for T 2 , the curvature of the singularity manifolds blows up, so such an estimate does not work – no matter how small a scale one chooses to work on. For higher iterates of the map, the singularity structure is extremely complicated, and we were unable to find any useful estimate on the measure of the δ-neighbourhood. The way

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out of this problem is the new metric, which makes the detailed understanding of higher-order singularities avoidable. 3.2. Chernov–Dolgopyat metric in two dimensions The main idea comes from [10], where the authors use a metric which measures infinitesimal distances on the front, rather than in the Poincaré section. Since expansion of an unstable front is monotonous, this kind of length of unstable vectors clearly grows from collision to collision. For the sake of easier understanding, we first discuss the 2-dimensional construction, and give the multi-dimensional generalization thereafter. In two dimensions, measuring distances “on the front” simply means using (dr, dv) instead of (dq, dw) = (dq, dϕ) (with the notations of Section 2). See footnote.2 It is important to note however, that (dr, dv) is the tangent vector of the front after collision, which results in an asymmetry in the behaviour of stable and unstable vectors. So the metric, which we will call the Chernov–Dolgopyat metric or C-D metric, is defined as k(dq, dϕ)kC−D := k(dr, dv)ke = k(dq cos ϕ, dϕ + Kdq)ke p = (dq cos ϕ)2 + (dϕ + Kdq)2 . Equivalently, the metric tensor has the form cos2 ϕ + K 2 K dq2 . K 1 dϕ2 2 ϕ+K 2 K To be absolutely precise: the matrix cos K is the matrix of the metric 1 tensor in the basis {e, f } where e ∈ T , f ∈ J , keke = kf ke = 1. (This basis is orthonormal in the Euclidean metric.) We will not rigorously prove hyperbolicity with respect to this metric here, since it will be done in Section 3.3 for the multi-dimensional case, and that is what we need. Instead, we discuss the relation of the C-D metric to the Euclidean. It is easy to see using (2.15) and (2.11) that for vectors of the unstable cone, the C-D metric is equivalent to the Euclidean: kdxkC−D ∼ kdxke for u-vectors. Also, kdxkC−D ≤ Ckdxke holds for every vector dx, since K is bounded. However, kdxkC−D can be much smaller than kdxke for some vectors (in the stable cone) ˜ . Indeed, the determinant of the matrix of the metric near the boundary of M ˜ . This non-equivalence has the inconvenient tensor is cos2 ϕ, which vanishes on ∂ M ˜ . More precisely, it is consequence that gC−D is not a Riemannian structure on M a Riemannian structure only on the inner part, and it cannot be extended to the boundary in a continuous non-vanishing manner. This is inconvenient for several reasons, e.g. no compactness arguments will work. gC−D (dq1 , dϕ1 ), (dq2 , dϕ2 ) = dq1

2 Notice

dϕ1

that we are not ignoring the velocity component, so this metric is not the so-called p-metric of billiards literature, which only measures configurational distances on the front.

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3.3. Generalization to high dimension The multi-dimensional generalization is quite straightforward. Consider a tangent vector (dq, dw) of the Poincaré section. The configurational part is dq ∈ T , while the velocity part, dw ∈ J is in another space. So we have to be more careful than in 2D where these were numbers that one can add. The counterpart of (dq, dw) on the front is (dr, dv) where dr ∈ J and dv ∈ J . The transition is via the operator V : J → T . The formulas of the transition are listed in Section 2.3.3 as (2.16) and (2.17), and they are dr = V −1 dq

(3.1) ∗

dv = dw + hv, niV Kdq . The C-D metric is defined almost like in two dimensions. However, for some reason – to be explained in Remark 3.3 below – we need to insert a small scaling factor εC−D > 0 which ensures that the velocity component is taken into account with a sufficiently small weight: Definition 3.1. k(dq, dw)kC−D := k(dr, εC−D dv)ke

= V −1 dq, εC−D dw + hv, niV ∗ Kdq . (3.2) e

Or, equivalently, the C-D metric tensor has the matrix (written in block form) −1∗ −1 V V + ε2C−D hv, ni2 KV V ∗ K ε2C−D hv, niKV gC−D = ε2C−D hv, niV ∗ K ε2C−D in the basis {e1 , . . . , ed−1 , f1 , . . . , fd−1 } where {e1 , . . . , ed−1 } is a basis of T , {f1 , . . . , fd−1 } is a basis of J , both are orthonormal with respect to the Euclidean metric, K is the matrix of the operator K : T → T with respect to the basis {e1 , . . . , ed−1 }, and V is the matrix of the operator V : J → T with respect to the bases {e1 , . . . , ed−1 } and {f1 , . . . , fd−1 }. The factor εC−D is needed due to another typical multi-dimensional phenomenon. In the following lemma we show that the billiard dynamics shows one-step uniform hyperbolicity with respect to the C-D metric, if εC−D is small enough. This is a very important advantage in comparison with the Euclidean metric, where we only have uniform hyperbolicity for some higher iterate of the dynamics. Lemma 3.2. If norms of tangent vectors are measured using the C-D metric and εC−D is small enough, then T exhibits uniform hyperbolicity in the sense of Condition A.2. Proof. The existence and properties of the invariant cone fields have already been established in [5], and are listed in Section 2.3.2. So we only need to show that there exists some global constant ΛC−D > 1 such that the k.kC−D -norm for any

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tangent vector (dr+ , dv + ) of any post-collision front that corresponds to some uvector is expanded at least by a factor ΛC−D during a free flight and a collision – that is, until the front becomes a post-collision front again. So let (dr1− , dv1− ) and (dr1+ , dv1+ ) denote the time-evolved tangent vector just before and after the next collision, respectively. The formulas of free propagation, (2.4) and (2.5) say that (dr1− , dv1− ) = (dr+ + τ dv + , dv + ), so k(dr1− , εC−D dv1− )k2e = |dr+ |2 + 2τ hdr+ , dv + i + τ 2 |dv + |2 + ε2C−D |dv + |2 .

(3.3)

Since dv + = B + dr+ and B + > c by (2.11), we have |dv + | ≥ c|dr+ | and hdr+ , dv + i ≥ 0, so (3.3) implies k(dr1− , εC−D dv1− )k2e |dv + |2 ≥ 1 + τ2 + 2 = 1 + τ2 + + 2 k(dr , εC−D dv )ke |dr | + ε2C−D |dv + |2

1 |dr + |2 |dv + |2

2 ≥ 1 + τmin

1 c2

+ ε2C−D

1 +1

(3.4)

whenever εC−D ≤ 1. In words: the expansion of the tangent vector during free flight is considerable, since the velocity component is non-negligible, and the flight is not very short. Similarly (2.6), the first of the collision formulas, together with B − < C from (2.12) imply that k(dr1+ , εC−D dv1+ )k2e kdr1− k2 1 ≥ = . 1 + ε2C−D C 2 k(dr1− , εC−D dv1− )k2e kdr1− k2 + ε2C−D C 2 kdr1− k2

(3.5)

This and (3.4) give the statement when εC−D is small enough. The argument for the uniform contraction of stable vectors would be completely analogous. Indeed, the minimum expansion along an unstable front from one pre-collision moment to the other can be obtained by multiplying the exact same expressions on the right hand sides of (3.4) and (3.5), now in opposite order. But that is exactly the inverse of stable contraction from one post-collision moment to the other. k(dr + ,dv + )k2

e 1 1 Remark 3.3. We note that in general k(dr− ≥ 1 is not true for an unstable − 2 1 ,dv1 )ke front: The tangent vector of the front may be contracted at collision. This does not happen if either the operator B − or K is close to a scalar. However, in general it may happen that hB − dr, V ∗ KV dri < 0 despite the fact that both B − and V ∗ KV

are positive definite. In such a case, (2.6) and (2.7) may give This is another typical multi-dimensional phenomenon.

k(dr1+ ,dv1+ )k2e k(dr1− ,dv1− )k2e

< 1.

The following two lemmas are about the relation of the C-D metric to the Euclidean. The statements are greatly different for unstable and stable vectors, which reflects the asymmetry in the definition of the metric.

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Lemma 3.4. The C-D and the Euclidean metric are equivalent for vectors of the unstable cone field. That is, there exists a global constant C < ∞ such that for any vector dx of any unstable cone Cxu , 1 kdxke ≤ kdxkC−D ≤ Ckdxke . C Proof. First let us note that, after having chosen εC−D according to Lemma 3.2, we may keep it fixed so that it becomes a global constant. Let dx = (dq, dw) be a vector of the unstable cone, and let (dr, dv) be its equivalent on the front. The transition is given by (2.16) and (2.17). Vectors of the unstable cone satisfy B + > c (from (2.11)), which means that k(dq, dw)kC−D = k(dr, εC−D dv)ke ∼ kdvk. But (2.15) says k(dq, dw)ke ∼ kdwk. Finally, a combination of (2.7), (2.16) and (2.17) give kdvk ∼ kdwk. Lemma 3.5. There exists a global constant C < ∞ such that for any vector dx of any stable cone Cxs , kDT dxke ≤ CkdxkC−D . Proof. Let us use the notation dx = (dq, dw) and DT dx = (dq1 , dw1 ). First, let the tangent vector of the front corresponding to DT dx be (dr1 , dv1 ). From (2.17) we have that |dw1 | ≤ |dv1 | + | cos ϕ1 V1∗ K1 dq1 |. Since kV1−1 k = 1, (2.16), (2.10) and (2.14) give |dv1 | ≤ C|dq1 |. Besides, since k cos ϕ1 V ∗ k = 1, (1.2) gives | cos ϕ1 V1∗ K1 dq1 | ≤ Kmax |dq1 |. These together imply that kDT dxke ≤ C|dq1 | .

(3.6)

Second, let the tangent vector of the front corresponding to dx be (dr, dv). The definition of the C-D metric implies that kdxkC−D ≥ |dr| .

(3.7)

Due to (3.6) and (3.7), it is enough to show that |dq1 | ≤ C|dr| ,

(3.8)

and this is what we will do. In order to prove (3.8) we invoke some notation from [5]. Given an invertible linear operator O (that may depend on the phase point we are considering), the relation c ≺ O ≺ C means that there exist global constants C1 , C2 > 0 such that kOk ≤ C1 and kO−1 k ≤ C2 , uniformly on the phase space. Furthermore, we invoke the key technical Lemma 4.3 from [5]: given two symmetric, positive definite operators K 0 : T → T and B 0 : J → J with c ≺ K 0 ≺ C and c ≺ B 0 ≺ C, we also have c ≺ B 0 V −1 + hn, viV ∗ K 0 ≺ C . (3.9) Now let us rewrite (2.4), (2.5), (2.9) and (3.1) as dr = dr1 − τ dv1− = dr1 − τ B1− dr1 = (I − τ B1− )V1−1 dq1

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where I is the identity operator. We use (2.8) to express B1− in terms of B1+ and obtain: dr = (I − τ B1+ )V1−1 + 2τ hn1 , v1 iV1∗ K1 dq1 . Now we may invoke (3.9) with B 0 = I −τ B1+ and K 0 = 2τ K1 to prove (3.8). To see that these operators are indeed bounded from above and below we refer to (2.14) on the one hand, and to our assumptions (1.1), (1.2), (1.3) and (1.4) on the other hand. Although the use of the C-D metric ensures uniform hyperbolicity, it has the disadvantage that it is not a true Riemannian structure. This can be seen exactly as in 2 dimensions: the determinant of the metric tensor (with respect to ˜ , where the metthe Euclidean) is cos2 ϕ, which vanishes on the boundary of M ric is degenerate. This has many unpleasant consequences. First of all, Chernov’s Condition A.1 about the dynamical system formally demands a true Riemannian ¯ compact. However, the problems with the non-Riemannian namanifold, with M ture of the C − D metric are deeper, and not just formal. Some details will be explained in Appendix B, Remark B.10. For these reasons, we will use a regularized version of the C-D metric structure, which will be truly Riemannian, and will not exhibit the unpleasant features of the original C-D metric. ˜ Definition 3.6. The Riemannian structure we use on the Poincaré phase space M is the “Regularized Chernov–Dolgopyat” metric tensor field defined by g := gC−D + εg ge . Here ge is the Euclidean metric tensor field (the natural Riemannian structure on ˜ ), and εg > 0 is an arbitrary constant. M The choice of εg will be based on the following proposition: Proposition 3.7. The “Regularized Chernov–Dolgopyat” metric tensor field g is ˜ . If εg is small enough, then T is uniformly a C 1 Riemannian structure on M hyperbolic with respect to g. That is, Condition A.2 holds. Proof. The fact that gC−D is a C 1 field of symmetric tensors of type (0, 2) is clear from the definition: it is built up of K, hn, viV , hn, viV ∗ , V −1 and V −1∗ , which are ˜ . gC−D is also all bounded and continuously differentiable up to the boundary of M 1 positive semi-definite, which is clear from (3.2). Since ge is a C Riemannian structure, g = gC−D + εg ge is also a C 1 field of symmetric tensors of type (0, 2), which is positive definite (everywhere) if εg > 0. One can check by direct calculation that the determinant of g (with respect to the Euclidean) is uniformly bounded away from 0. This altogether means that g is truely a Riemannian structure. To prove uniform hyperbolicity (i.e. that Condition A.2 holds), we still use the invariant cone field already introduced in [5] and described in Section 2.3.2, used in Lemma 3.2 as well. So we only need to see that vectors of the unstable cone are expanded, and vectors of the stable cone are contracted at least by a factor Λ > 1.

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For unstable vectors this is easy, since the C-D and the Euclidean metric are equivalent on unstable vectors (by Lemma 3.4), so the term εg ge in g is negligible if εg is small enough. So expansion is inherited from the C-D metric (Lemma 3.2). Indeed, kdxke ≤ CkdxkC−D for every vector dx of the unstable cone, thus q kdxk ≤ 1 + ε2g C 2 kdxkC−D , so 1 kDT dxk ≥ kDT dxkC−D ≥ ΛC−D kdxkC−D ≥ ΛC−D p kdxk , 1 + εg C which proves the statement for unstable vectors if εg is so small that ΛC−D q

> 1.

1 + ε2g C 2

The case of stable vectors dx is also easy, once we have the difficult Lemma 3.5 about the dynamical comparison of the metrics on stable vectors. Using that lemma (and Lemma 3.2 about the hyperbolicity of the C-D metric), we can simply write kDT dxk2 = kDT dxk2C−D + ε2g kDT dxk2e ≤ =

1 Λ2C−D

+

ε2g C 2

kdxk2C−D + ε2g C 2 kdxk2C−D Λ2C−D

kdxk2C−D

(3.10) (3.11)

which proves the statement for stable vectors if εg is so small that 1 + ε2g C 2 < 1 . Λ2C−D

3.4. Curvature bounds and Riemannian structure “Bounded curvature” is a commonly used regularity property in Dynamical Systems theory. In the literature there are many statements which claim that the curvature of certain submanifolds of the phase space or the configuration space is bounded. There is a variety of notions of curvature used in these statements. The essence of curvature bounds is always the fact that “if two points are near, then their tangent spaces are also near”, so one needs to compare vectors of different tangent spaces. This can be done without any special care if the containing manifold is Euclidean, but in general one would need to identify nearby tangent spaces through the parallel transport of the manifold. ˜ which is not EuSince we are going to use a Riemannian structure on M clidean, and even different from the natural Riemannian structure, we will now formulate precisely what we mean by curvature bounds. We use notation which is standard in differential geometry, see e.g. [14].

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The proper notion for the curvature of an unstable manifold is the curvature as of a submanifold, so it’s not a quantity of inner geometry. It should describe how fast the submanifold “bends away” from the geodesics tangent to it, e.g. a cylinder (surface) as a subset of R3 with a small radius should be considered heavily curved, although its inner geometry is Euclidean. Only this way can bounded curvature mean that the submanifold can be viewed (at the cost of an arbitrarily small error) as a plane, if the scale is small enough. The quantity which measures curvature in this sense is the second fundamental form. We will use this phrase many times, and abbreviate it as s.f.f. We have already used this quantity in describing fronts as subsets of Q. Since in Q fronts are one-codimensional, the notion of second fundamental form was easier there, but the generalization to higher codimensions is also known in differential geometry. Let M be a Riemannian manifold with Riemannian metric tensor field g, and let ∇ be the connection defined by g. Definition 3.8. The second fundamental form of a C 2 submanifold W at the point x is II : Tx × Tx → Nx , where Tx is the tangent space and Nx is the normal space of W at x, defined by II(v, w) = ∇⊥ v w. Here ⊥ means “component orthogonal to W ”. For this definition to make sense, at least w has to be a vector field, but the value of II(v,w) will only depend on the value of w at x, as long as w is a tangent vector field of W (at least in every point of W near x). II is bilinear. Remark 3.9. To clarify why this quantity is indeed the proper notion of curvature of W as a submanifold, i.e. the amount of non-flatness of W in M , here is a small picture about its meaning with coordinates. We will not use this picture later, all our proofs will be based on the definition. Let us choose {e1 , . . . , ek } to be an orthonormal basis of Tx and {n1 , . . . , nl } to be an orthonormal basis of Nx , and choose, as a coordinate chart, normal coordinates built from the basis {e1 , . . . , ek , n1 , . . . , nl }. Denote the coordinates as (x1 , . . . , xk , y 1 , . . . , y l ). Then in this coordinate chart the submanifold W (near the origin) will be the graph of a function f : Rk → Rl . In the Taylor polynomials of f the constant and the linear term are zero by the choice of the coordinate system, and the quadratic term is exactly the quadratic transformation from Rk to Rl defined by the components of II in the bases {e1 , . . . , ek }, {n1 , . . . , nl }. That is, the second degree Taylor polynomial of f at 0 is a b c y a = T2a (x1 , . . . , xk ) = IIbc x x

where II(eb , ec ) = II

∂ ∂ , ∂xb ∂xc

a = IIbc

∂ a = IIbc na . ∂y a

Now we can make Condition A.4 about curvature bounds precise:

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Condition 3.10. There should exist a constant KW < ∞ such that at any point x of any unstable manifold W , the second fundamental form of W at x (as a bilinear operator) is bounded by KW . That is, for any v, w ∈ Tx W , kII(v, w)k ≤ KW kvkkwk. The reader may check that – although differently formulated – this is exactly ˜ in Theorem 5.5 of [5], and – what was proven for the Euclidean metric of M although differently said – this is exactly what is used in [9] as the “bounded curvature” assumption. 3.4.1. Proof of Proposition 2.9 about bounded curvature of one-step singularities. The proof of Proposition 2.9 will be based on the following lemma. Let V and W be two smooth 1-codimensional submanifolds of a Riemannian manifold M, and let them be transversal. Then V ∩ W is also a 1-codimensional submanifold of V . Lemma 3.11. For every C < ∞ and α > 0 there exists a C 0 < ∞ such that if at a point x ∈ V ∩ W the s.f.f. of V and the s.f.f. of W are both bounded by C and the angle of V and W is at least α, then the s.f.f. of V ∩ W as a submanifold of V is bounded by C 0 (at x). Proof. Let us denote the covariant derivation within V by ∇V . Let v and w both be tangent vectors (vector fields) of V ∩ W . The quantity we wish to estimate is the component of ∇Vv w orthogonal to V ∩ W . But ∇Vv w is just the component of ∇v w parallel to V , that is, ∇Vv w = ∇v w − IIV (v, w) , where IIV is the s.f.f. of V . Due to our assumption, the length of IIV (v, w) is at most C|v||w|, and of course its component orthogonal to V ∩ W cannot be longer either. So it is enough to find an estimate for the other term, i.e. the component of ∇v w orthogonal to V ∩ W . Now denote the (unit) normal vectors of V and W by e and f . Our assumptions imply that the components of ∇v w in the direction of e and f are both bounded by C|v||w|. The statement is that the component of ∇v w within the plane of e and f is also bounded by some C 0 |v||w|. But this is clear since the angle of e and f is at least α, so any vector within their plane which is long, must have a long component is at least one of their directions. To prove Proposition 2.9 we apply the lemma with M the flow phase space of the billiard, and V = M , the Poincaré section phase space. W is chosen to be the 1-step singularity manifold of the flow dynamics. It is easy to see that the s.f.f. of V = M within M is bounded, since M is a compact smooth submanifold of M. We now only need to see that the s.f.f. of W is bounded in the points of M , and that M and W are uniformly transversal. Both of these can be seen easily, since the minimum free flight τmin was supposed to be nonzero - i.e. there are no corner points.

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3.5. Regularity properties with respect to the new Riemannian structure In Appendix B we consider the problem of having two different Riemannian structures on the same differentiable manifold. We show that if the regularity properties of Sections 2.4.1 and 2.5 are checked with the usual Riemannian structure, then they automatically follow for the new one. The result is the following proposition: Proposition 3.12. All the regularity properties stated in Sections 2.4.1 and 2.5 are satisfied also if the regularized Chernov–Dolgopyat metric tensor field g is used to define the Riemannian structure on M instead of the Euclidean structure ge . Specifically, • piecewise H¨ older continuity of the dynamics • bounded expansion away from the singularities • uniform transversality of stable and unstable cone fields • uniform alignment • uniform curvature bounds for stable and unstable manifolds • uniform distortion bounds • uniform absolute continuity • Proposition 2.9 about the smoothness of one-step singularities, and • Corollary 2.11 about the anisotropy near tangent collisions remain valid when the phase space is equipped with the regularized Chernov– Dolgopyat Riemannian structure instead of the Euclidean. The proof can be found in Appendix B.

4. Growth properties In this section we prove that the studied multidimensional dispersing billiard systems satisfy Chernov’s growth properties. More precisely, we show that there is some fixed integer n0 such that the n0 th iterate of the billiard map T satisfies Condition A.7. We recall that in a “d-dimensional” billiard the Poincaré phase space is 2(d − 1)-dimensional. Since we will be working with unstable manifolds, we introduce the notation m = d − 1 for their dimension. Throughout the section we will use the notation A[δ] to denote the (closed) δ-neighbourhood of a subset A of the phase space, or of an unstable manifold. In accordance with the exposition of Appendix A, it is worth introducing the following notations. Let Tˆ be a n0 th iterate of the original billiard map, i.e. Tˆ = T n0 , where ˆ = Γ(n0 ) , and n0 ∈ N is to be specified later. Thus the singularity set for Tˆ is Γ ˆ T expands unstable vectors (and contracts stable vectors) at least by a factor ˆ = Λn0 . Λ For δ0 > 0, we call W a δ0 -LUM if it is a local unstable manifold (LUM, see Appendix A) and diam W ≤ δ0 . For an open subset V ⊂ W and x ∈ V denote by V (x) the connected component of V containing the point x.

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ˆ (n) = ∅ (i.e., Let n ≥ 0. We call an open subset V ⊂ W a (δ0 , n)-subset if V ∩ Γ the map Tˆn is smoothly defined on V ) and diam Tˆn V (x) ≤ δ0 for every x ∈ V . Note that Tˆn V is then a union of δ0 -LUM-s. Proposition 4.1. There is a fixed δ0 > 0, furthermore, there exist constants α ∈ (0, 1) and β, D, κ, σ > 0 with the following property. For any 0 ≤ δ < 1 and any δ0 ˆ [δ] LUM W there is an open (δ0 , 0)-subset Vδ0 and an open (δ0 , 1)-subset Vδ1 ⊂ W \ Γ (one of these may be empty) such that the two sets are disjoint, mW (W \ (Vδ0 ∪ Vδ1 )) = 0 and ∀ε > 0 First Growth Property: n o mW x ∈ Vδ1 | ρ Tˆx, ∂ TˆVδ1 (x) < ε ˆ · mW x ∈ W | ρ(x, ∂W ) < ε/Λ ˆ ≤ αΛ + εβδ0−1 mW (W ) ;

(4.1)

Second Growth Property: n o mW x ∈ Vδ0 | ρ x, ∂Vδ0 (x) < ε ≤ Dδ −κ mW

x ∈ W | ρ(x, ∂W ) < ε

;

(4.2)

and Third Growth Property: mW (Vδ0 ) ≤ D mW

x ∈ W | ρ(x, ∂W ) < δ σ

.

(4.3)

Remark 4.2. Note that Proposition 4.1 is slightly stronger than Condition A.7. Most importantly, here we allow for arbitrary 0 ≤ δ < 1, while the condition requires only δ sufficiently small. Allowing for δ = 0 in the first growth property (note in such a case the second and the third growth properties are trivial) provides useful estimates, see also Remarks 4.4, 4.7 and Corollary 4.13. 4.1. Outline The first growth property is much more difficult than the other two. Reason for this is that in the second and the third growth properties we have a large amount of freedom, due to the fact that an arbitrary power of δ may appear (δ κ and δ σ , respectively). This allows for the use of quite crude measure estimates in their proof, see the exposition in Sections 4.2.2 and 4.3.2. The case of the first growth property is completely different. Here there is no δ, the inequality is sharp and thus there is very limited freedom in the measure estimates. The two terms appearing on the right hand side of (4.1) are responsible for two different effects. The first term estimates the measure of points that get close to the singularities. The second term corresponds to the fact that some components may grow large when applying Tˆ and may fail to have diameter less than δ0 . Thus one needs to partition these components. The second term estimates the measure of points that get close to these artificial boundaries. Handling the effect of this further chopping is rather standard, (see [8] and Section 4.4 below). Thus what is to be understood is how LUMs are expanded and, simultaneously,

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partitioned by singularities when iterates of T are applied. This will be the content of our Lemmas 4.3 and 4.5, which will be referred to as 1-step and n-step lemmas, respectively. Throughout the rest of the section we will consider the original billiard map T . Recall the concept of δ0 -LUM and (δ0 , n)-subset from above. We also introduce another notation: Given a (δ0 , n)-subset V , define a function rV,n on V by rV,n (x) = ρT n V (x) T n x, ∂T n V (x) . (4.4) Note that rV,n (x) is the radius of the largest open ball in T n V (x) centered at T n x. In particular, rW,0 (x) = ρW (x, ∂W ). Note that at the formulation of Condition A.7 an analogous quantity for Tˆ, the function rˆ has been used. However, throughout until the end of Section 4.3.2, we may forget Tˆ and rˆ and consider the quantities for the original billiard map T . First we describe how the above mentioned growth-fractioning process acts when the first iterate of T is applied. Given a δ1 -LUM W (the constant δ1 will be chosen later in Section 4.1.1) and some δ ≥ 0, we construct a subset Gδ (W ) ⊂ W , the (δ-)gap of W , that contains points that are δ-close to the first step singularity Γ in an appropriate sense. The complement of Gδ (W ) will be denoted by Fδ (W ) and will be referred to as the remaining part of W . The subscripts δ and/or the dependence on W will be sometimes omitted for brevity if no confusion arises. Then we show that this construction does not create too much new boundary: the sizes and shapes of the components of F and G can be controlled as expressed in our 1-step lemma. To formulate it, recall that KW,1 is the first step complexity from Definition 2.7, Λ is the factor of uniform expansion from Proposition 3.7, and introduce λ = Λ1/100 . Lemma 4.3 (1-step lemma). There are some global constants D1 , κ1 , σ1 > 0 with the following property. Consider a δ1 -LUM W with the corresponding gap G = Gδ (W ) and remaining part F = Fδ (W ) constructed for some 0 ≤ δ < 1. Then for every ε > 0: (G0). Fδ ⊂ W \ (Γ[δ] ), (G1). mW (rFδ ,1 < ε) ≤ λ2 K1,W · mW (rW,0 < ε/Λ), (G2). mW (rGδ ,0 < ε) ≤ D1 δ −κ1 mW (rW,0 < ε), (G3). mW (Gδ ) ≤ D1 mW (rW,0 < δ σ1 ). Remark 4.4. If δ = 0, we have Gδ = W ∩ Γ, which is of zero Lebesgue measure. Thus (G2) and (G3) are trivial in this case, however, (G1) is important. Now if we knew K1,W < Λ1/2 say, Lemma 4.3 would essentially imply the three Growth Properties of Proposition 4.1 for Tˆ = T . However, there is no reason for such a relation. We would like to emphasize that the necessity of using a higher iterate of T is a special feature of multidimensional dispersing billiards. In the two dimensional case recent important observations, see [10], made it possible to prove the growth

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properties for T itself, regardless of complexity. However, as the geometry is more complicated, it is not possible to adapt directly the exposition of [10] to higher dimensions. If d ≥ 3, complexity issues seem unavoidable, thus to gain enough expansion, it is essential to switch to some higher iterate of T . It seems, however, very difficult to consider higher iterates directly, as the higher order singularity manifolds do not possess uniform curvature bounds (see [5] and [2]). Thus we perform an inductive argument: given a sufficiently small LUM W and δ ≥ 0, we construct the n-gap Gnδ (W ) and its complement, the n-remaining set Fδn (W ) in an inductive manner. Then, with an inductive application of Lemma 4.3, we obtain our n-step lemma: n

Lemma 4.5 (n-step lemma). Let δn = δ13 . For any fixed integer n ∈ N, there exist global constants σn , κn > 0 and Dn > 0 with the following property. Consider an arbitrary δn -LUM W with the corresponding n-gap Gnδ = Gnδ (W ) and n-remaining part Fδn = Fδn (W ) constructed for some 0 ≤ δ < 1. Then for every ε > 0 we have: (Gn0). Fδn ⊂ W \ (Γ(n) )[δ] , (Gn1). mW (rFδn ,n < ε) ≤ λ4n Kn,W · mW (rW,0 < ε/Λn ), (Gn2). mW (rGnδ ,0 < ε) ≤ Dn δ −κn mW (rW,0 < ε), (Gn3). mW (Gnδ ) ≤ Dn mW (rW,0 < δ σn ). Remark 4.6. It is worth noting that Dn = D1 + Dn−1 K1 λ3 , σn = σ1 /3n−1 and κn = κn−1 /3 + d − 1 (thus, in particular κn ≤ κ1 + 3(d − 1)/2 for all n ∈ N). Remark 4.7. If δ = 0, we have Gnδ = W ∩ Γ(n) , which is of zero Lebesgue measure. Thus (Gn2) and (Gn3) are trivial in this case, however, (Gn1) is important. (See also Remark 4.4.) We will apply this n-step lemma for a fixed n = n0 , to be chosen below in Section 4.1.1. Considering Tˆ = T n0 , our Assumption 1.3 ensures that, for n0 chosen appropriately, statement (Gn1) implies the first Growth Property of Proposition 4.1. (Gn2) and (Gn3) will imply the second and third Growth Properties, respectively, with the choice of δ0 = δn0 , D = Dn0 , κ = κn0 and σ = σn0 . 4.1.1. Further remarks and how the constants are chosen. Before turning to the constructions and the proofs in detail, we close this subsection with some further remarks on the exposition in general. In the proof of Lemma 4.3 it is crucial that we can apply a locally flat picture. This is possible as LUM-s and the first step singularity manifolds (i.e., the components of S and Γ) possess uniform curvature bounds. Thus, on sufficiently short distance scales, we may regard the intersections of LUMs and first step singularity manifolds as the intersection of d − 1 dimensional flat disks with 2d − 3 dimensional hyperplanes in R2d−2 . Moreover, by the alignment property, this intersection is transversal. Recall that λ = Λ1/100 (where Λ > 1 is the factor of uniform expansion). We 1/3 choose δ1 in Lemma 4.3 in such a way that, for δ1 -LUMs, measure estimates based on the locally flat picture are accurate up to λ-precision. The reason for 1/3 is that,

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given a δ1 -LUM W , H¨ older continuity ensures that all the connected components 1/3 of T W are δ1 -LUMs, and thus can be regarded as locally flat pieces in the above sense. Two further requirements on the smallness of δ1 is that distortions on this scale should not exceed λ, cf. Proposition 3.12, and that KV,k ≤ Kk for k ≤ n0 1/3 and for any δ1 -LUM V , see Remark 2.8. n Now in Lemma 4.5 δn = δ13 . By virtue of Hölder continuity this implies that, for a δn -LUM W , the components of T k W for all k ≤ n are δ1 -LUMs, and thus satisfy the hypotheses of Lemma 4.3. Now when constructing Gδ we essentially cut out the δ-neighborhood of Γ from W (what exactly is happening is explained in Section 4.2 below). In the inductive construction of Gnδ we apply this to some component of T k W (k ≤ n), and pull back to W . More precisely, we apply the one step construction and, correspondingly, Lemma 4.3 to these components with δ 7→ δ 0 , where δ 0 is some suitable positive power of δ. Then H¨ older continuity ensures that we cut out some neighborhood of T −k Γ from W . The neighborhood of the singularity from which Fδn refrains will be smaller as we proceed with the induction. Thus what we can ensure in the end is that we are a certain positive distance away from the singularity – this is the δ that appears in the statement of Lemma 4.5. In particular, the measure estimates of Lemma 4.5 can be ensured in terms of this δ, despite of the fact that the distance to certain singularity components, cut out at earlier steps of the induction, will be much bigger than δ. Finally, as there are many global constants appearing in different arguments, some of which depend on some others, here we summarize how these constants are chosen to make the exposition more transparent. 1. We have seen that some εg can be chosen in Definition 3.6, which ensures that the good metric inherits uniform expansion/contraction of unstable/stable vectors from the C-D metric, with some factor Λ > 1. The new metric satisfies transversality, alignment and the curvatures of LUMs and first step singularity manifolds are uniformly bounded, see Proposition 3.12 – by the choice of εg the constants appearing in these statements are fixed for the rest of the argument. 2. The next constant to fix is n0 , the integer power of the one-step dynamics ˆ = Λn0 . By we use as the map Tˆ = T n0 . The expansion factor for Tˆ is Λ Assumption √ n01.3 we may ensure that “expansion prevails fractioning”, that is, ˆ for some α < 1. (cf. Kn0 ≤ Λ , which guarantees Kn0 λ4n0 = αΛn0 = αΛ (Gn1) from Lemma 4.5 and the First Growth Property from Proposition 4.1). 3. The next constant we choose is k0 , the integer we start the labelling of homogeneity strips (and, correspondingly, secondary singularities) with (cf. Formulas (2.2)). We want k0 to be so big that 2 • k 0 is big enough, P −2 • ≈ k10 is small enough k≥k0 k in comparison to some other global constants. Once k0 is fixed, the distortion bounds and the absolute continuity in Proposition 3.12 get a precise formulation.

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4. Finally, we fix the constant δ1 which specifies the δ1 -LUM-s for which Lemma 4.3 is to be proven. We choose δ1 small enough such that: k • For any fixed k = 1, . . . n0 , given any δ13 -LUM W , the set W \S (k) has at most Kk components. This is possible by complexity and continuation. 1/3 • δ1 -LUMs are locally flat up to λ precision and distortions of T are at most λ on them (see the discussion in Section 4.1.1 above). 4.2. The one-step construction and its properties 4.2.1. Construction of the gap Gδ . Since we will now need to introduce a sequence of global constants depending on each other, please recall Notation 2.1 concerning the convention on C-s. It is time to describe how the gap Gδ (W ) (and correspondingly, the remaining part Fδ (W )) are constructed. Pulling back the components of T W we get W \ S = KW,1 ∪i=1 Wi . KW,1 We will construct Gδ (W ) as ∪i=1 Gδ (Wi ). For fixed i the set Gδ (Wi ) will cover all points of Wi that are in the δ-neighborhood of (either primary, or secondary) singularities intersecting Wi . To treat the effect of secondary singularities consider T Wi , which lies, by definition, on a fixed scatterer. However, it may be partitioned by the homogeneity layers into countably many components to be denoted by T Wi,k = T Wi ∩ Ik (k ≥ k0 ). Here T Wi,k0 lies in the “middle of the phase space” while the further components lie in the vicinity of the boundary, cf. (2.2). Note, furthermore, that T Wi can be foliated by ϕ =const. hypersurfaces, corresponding to phase points where ϕ, the angle of incidence (cf. Notation 2.4) is fixed. These foliae will be denoted as γϕ . In particular, the hypersurfaces that separate the neighboring T Wi,k -s from each other are elements of this foliation. In other words, T Wi,k (k > k0 ) consists 1 of foliae γϕ with π2 − k12 < ϕ < π2 − (k+1) 2. Now we define: [ Gδ (Wi ) = Wi ∩ S [δ] ∪k Gδ (Wi,k ) , (4.5) where the construction of Gδ (Wi,k ) is described below. Let us, however, first note that the effect of the primary singularities is already taken care of in the first set appearing in the above expression. In Gδ (Wi,k ) we will consider the effect of secondary singularities as it appears in Wi,k . Fix a global constant C1 , the value of which will be chosen below (it is determined by the alignment property and the constants of Corollary 2.11). Consider T Wi,k which, by the above description, can be visualized as a narrow strip consisting of γϕ -s which satisfy the required bounds. Now T Gδ (Wi,k ) will consist of the two exterior substrips of T Wi,k , i.e. the sets made up of the foliae γϕ 1 2 with π2 − k12 < ϕ ≤ π2 − k12 +C1 δk 2 on the one hand, and with π2 − (k+1) ≤ 2 −C1 δk π 1 ϕ < 2 − (k+1)2 on the other hand. See Figure 2. Now when pulling back this set to W by T , Gδ (Wi,k ) will consist of two regions around two neighboring elements of the (secondary) singularity set Γ.

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Figure 2. Construction of the gap Gδ . Corollary 2.11 ensures that width of these regions is ∼ δ. Exploiting this fact along with the alignment property we may choose C1 above in such a way that Gδ covers W ∩ (Γ[δ] ). Remark 4.8. It is worth noting that, for k big enough (more precisely, for k > 1 Cδ − 5 ), the two exterior strips overlap. In such a case we cut out the full Wi,k : Gδ will consist of at most KW,1 thicker strips, corresponding to overlaps coming from big k, and narrower strips, coming from smaller k where there is no overlap. In particular, by straightforward calculations: 1

(i) the number of boundary components of Gδ does not exceed Cδ − 5 . (ii) the width, and consequently, the measure of the thicker strips does not exceed 3 Cδ 5 . This construction may seem too complicated at first sight. However, it has several advantages that will help us to prove Lemma 4.3 with relatively simple arguments. Most importantly, (most of) the boundary components of Gδ (and thus of Fδ ), defined this way, are pre-images of certain foliae γϕ as well. This ensures that Corollary 2.11 applies to them, which is very useful when proving (G1), i.e. when estimating the measure of points that will lie in the ε neighborhood of these boundary components. Note that for a simpler choice of Gδ – setting simply Gδ = Γ(δ) ∩ W , say – it would be much more difficult to check (G1) in lack of direct applicability of Corollary 2.11. Remark 4.9. It is useful to note that the construction of the gap does only depend ˜ ⊂ W we have on the singularity set. In particular, given δ1 -LUMs W and W ˜ ) = Gδ (W ) ∩ W ˜ for any δ. Gδ (W

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Figure 3. The statement of Sublemma 4.10.

Figure 4. The statement of Sublemma 4.11. 4.2.2. Proof of Lemma 4.3. Statement (G0) follows by the construction of Gδ and its properties described above. To prove statements (G1)–(G3), first we state two simple geometrical sublemmas. For both of them consider any nonempty bounded measurable set W ⊂ Rm , and a 1-codimensional plane E ⊂ Rm . E cuts Rm into two half-spaces, which we will call ‘left’ and ‘right’. Accordingly, W is cut into a ‘left’ and ‘right’ part, Wl and Wr (one of these may be empty). Our sublemmas will compare sets of points in W near different parts of the boundary. We will apply them with m = d − 1, i.e. on u-manifolds. For the proof see Appendix C. Sublemma 4.10. For any ε ≥ 0 Leb x ∈ Wl | ρ(x, ∂Wl ) < ε ≤ Leb x ∈ W | ρ(x, ∂W ) < ε

(4.6)

and the same holds for Wr . See Figure 3. Sublemma 4.11. For any ε ≥ 0 and 0 ≤ ξ ≤ 1, Leb x ∈ W | ρ(x, E) ≤ ξε \ x ∈ W | ρ(x, ∂W ) ≤ ε ≤ ξLeb x ∈ W | ρ(x, ∂W ) ≤ ε . (4.7) See Figure 4. Let us prove (G2) first. What appears in addition to {rW,0 < ε} on the left hand side is, by Remark 4.8, the ε neighborhood of finitely many hyperplanes 1 within W . Moreover, the number of these hyperplanes does not exceed Cδ − 5 . On the other hand, applying Sublemma 4.11 (with ξ = 1), we have an upper bound mW (rW,0 < ε) on the contribution of any such hyperplane. This proves (G2) with κ1 = 15 . To prove (G3) we first give an upper bound on its left hand side, i.e., on the measure of Gδ . We again invoke Remark 4.8. Gδ consists, on the one hand, of

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finitely many “thick” strips (of measure estimated by Remark 4.8/(ii)), the number of which is uniformly bounded above. Thus their contribution to mW (Gδ ) is not 3 1 more than Cδ 5 . On the other hand, we have at most Cδ − 5 many components of width (and thus measure) δ. Their overall contribution to mW (Gδ ) does not exceed 3 3 Cδ 5 . Altogether we have the upper bound Cδ 5 on the left hand side of (G3). 3 We complete the proof of (G3) with σ1 = 5(d−1) . The case when the whole of W is contained in the set Wc := {rW,0 < δ σ1 } is trivial. If, on the other hand, W \ Wc 6= ∅, then it is easy to see that Wc does necessarily contain two disjoint hemispheres of radius δ σ1 . Thus, for the measure that appears on the right hand side of (G3): 3

mW (Wc ) ≥ Γd−1 (δ σ1 )d−1 = Cδ 5 where Γd−1 is the volume of the (d−1)-dimensional unit ball. This means that (G3) holds in this case as well if D1 is chosen appropriately. To prove (G1) recall the notation for Wi from Section 4.2.1 for i = 1, . . . , KW,1 . Introduce furthermore Fi := Wi \ S [δ] , ε/Λ Fi := x ∈ Fi | ρW (x, ∂Fi ) < ε/Λ , ε Fi,+ := x ∈ Fi | ρT W (T x, T ∂Fi ) < ε .

(4.8)

Our first observation is that ε/Λ

ε Fi,+ ⊂ Fi

(4.9)

which follows from the fact that T expands distances on u-manifolds uniformly by a factor Λ. This is a trivial observation, nonetheless, it is important to emphasize how hard we worked for it (this is the reason why we had to introduce a new metric) and what an important role it plays. It is (4.9) that enables us to reduce the proof of the growth lemmas to estimates on the one step dynamics (i.e. to Lemma 4.3). Before proceeding we note that if we had only primary singularities, Fδ ∩ Wi would coincide with Fi , and the set of points in W for which rFδ ,1 < ε – that is, ε the set which appears on the left hand side of (G1) – would coincide with ∪i Fi,+ . The second key observation is that “the contribution of secondary singularities is negligible”, that is, for any i: ε/Λ mW x ∈ Fi | rFδ ,1 (x) < ε ≤ λmW (Fi ) . (4.10) To prove (4.10) consider the connected components of T (Fi ∩ Fδ ). These connected components have boundaries of two different types. On the one hand there is T ∂Fi , arising form primary singularities. On the other hand, recalling the details of the construction from Section 4.2.1, we see that the secondary singularities give rise to further boundary components, namely countably many foliae γϕk for

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ϕk ≈ π2 − k12 with k ≥ k0 3 . We may denote these foliae as T Ei,k , where Ei,k ⊂ Fi is a hyperplane by our convention of local flatness. It is worth noting that it is crucial that we may apply a locally flat approximation as only first step singularity manifolds appear, cf. Section 4.1.1. Corresponding to the above characterization of boundary components: ε x ∈ Fi | rFδ ,1 (x) < ε ⊂ Fi,+ ∪ ∪∞ x ∈ F | ρ (T x, T E ) < ε . i T W i,k k=k0 (4.11) Let us concentrate on the contribution of secondary singularities. Now we may reveal why it was so important to construct Gδ as explained in Section 4.2.1: we have that T Ei,k are themselves constant ϕ foliae. Thus, by consecutive applications of Corollary 2.11 and the alignment property: Cε x ∈ Fi | ρT W (T x, T Ei,k ) < ε ⊂ x ∈ Fi | ρW (x, Ei,k ) < 2 . k Applying Sublemma 4.11 (with ε → ε/Λ and ξ → with (4.9) implies

mW x ∈ Fi | rFδ ,1 (x) < ε ≤

CΛ k2 )

∞ X CΛ 1+ k2

plugged into (4.11), along ! ε/Λ

mW (Fi

).

k=k0

We may choose k0 so big that (4.10) holds. To complete the proof, note that by the continuation property and the convention on local flatness Fi can be considered as the result of the following process: start with W , cut it along a hyperplane, keep one of the two pieces, cut it again along a hyperplane and repeat the above for finitely many (at most K1,W ) times. (See Section 2.4.2, especially Figure 1.) Thus, by consecutive applications of Lemma 4.10 we see that ε/Λ

mW (Fi

) ≤ mW (rW,0 < ε/Λ) .

(4.12)

Plugging this into (4.10) and summing over i completes the proof of (G1). To terminate, we admit that we have cheated a little bit by using a locally flat picture, which is true only up to λ-precision. Thus, further λ factors appear on the right hand sides of the obtained estimates. As for statements (G2) and (G3), this can be swallowed in a suitably chosen D1 , while we have a prefactor λ2 K1,W in (G1), which corresponds exactly to the claim. Remark 4.12. In the Corollary to follow, and throughout the rest of the section, we will use the following notation. Whenever ∂T W appears, it is understood in terms of the modified phase space of Section 2.2.3, i.e., T W is cut by secondary singularities. 3 Having

a closer look at the exposition of Section 4.2.1, we see that (i) only finitely many such foliae contribute, nonetheless, their number is unbounded, more precisely, is bounded only in terms of δ, cf. Remark 4.8; (ii) for each k there are two such foliae, corresponding to the two exterior strips within the kth homogeneity layer. These details, however, do not modify the exposition, thus we disregard them, to avoid overcomplified notation.

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Corollary 4.13. The statement (G1) from Lemma 4.3 has a special significance. In particular, it can be formulated for δ = 0 (cf. Remark 4.4). As it contains no additional information on gap construction, this is the version that indeed estimates how much new boundary is created by the singularity manifolds when T is applied. We formulate it for future record. Given a δ1 -LUM W , for any ε > 0 we have: mW x ∈ W |ρ(T x, ∂T W ) < ε ≤ λ2 K1,W · mW (rW,0 < ε/Λ) . (4.13) 4.3. The n-step construction and its properties 4.3.1. Construction of the n-gap Gnδ . We construct Gnδ (W ) by induction. We will use the construction of Section 4.2 repeatedly. In particular, the first step of the induction is exactly the first step construction described there. Recall that at the nth step of the induction we need to treat δn -LUMs with n δn = δ13 . Now assume inductively that, given an arbitrary δn−1 -LUM, we already know how to construct the relevant n − 1-gap for 0 ≤ δ < 1. To proceed consider a δn -LUM W . Below we describe how for a given 0 ≤ δ < 1 the n-gap Gnδ (and correspondingly, the n-remaining part Fδn ) in W is to be constructed. As W is a δn -LUM, it is, in particular, a δ1 -LUM, thus the whole exposition of Section 4.2 applies to it. This means we may consider its 1-gap Gδ (W ), its primary components Wi and its secondary components Wi,k . Furthermore, the T Wi,k -s are δn−1 -LUMs by the Hölder continuity of the 1/3 dynamics (note δn−1 = δn ). By our inductive assumption the (n − 1)-gaps for any T Wi,k can be constructed for 0 ≤ δ < 1. In particular, let us construct the (n − 1)-gap for any such T Wi,k with δ → δ 1/3 . We will denote these (n − 1)-gaps lying in some fixed T Wi,k by Gi,k := Gn−1 (T Wi,k ) ⊂ T Wi,k . δ 1/3

(4.14)

Gnδ

To construct the n-gap for W , we need to identify all points that get close to some singularity manifold within the first n iterates. Points that are close to some singularity right now are contained in the 1-gap Gδ (W ). We need to add those points x for which T i x is close to some singularity manifold for some i = 1, . . . , n. These are exactly the preimages of points that start out from some T Wi,k and get close to some singularity within the first n − 1 iterates, in other words, the preimages of Gi,k . This is the reason for constructing the n-gap as: Gnδ (W ) = Gδ (W ) ∪ Gδ F (W ) , Gδ F (W ) = ∪i,k T −1 (Gi,k ) \ Gδ (W ) ,

(4.15)

where the superscript F stands for future. Remark 4.14. As a consequence of our observation in Remark 4.9, we have that the construction of the n-gap depends only on the (n-step) singularity set as well. ˜ ⊂ W we have Gn (W ˜ ) = Gn (W ) ∩ W ˜ for In particular, given δn -LUMs W and W δ δ any δ (and any n).

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4.3.2. Proof of Lemma 4.5. We will prove Lemma 4.5 for constants κn , σn and Dn satisfying the following recursive relations: σn = σn−1 /3 ,

κn = κn−1 /3 + (d − 1) ,

Dn = D1 + Dn−1 K1 λ3 .

(4.16)

This is the reason for the relations mentioned in Remark 4.6. All the statements (Gn0)–(Gn3) from Lemma 4.5 are proved by induction on n, nonetheless, these inductions are independent. Before performing the four inductions, let us mention that the general strategy for all of them is essentially the same. This strategy relies on the decomposition (4.15). In rough terms, to prove (Gn1)–(Gn3) we have to establish upper bounds on the measure of certain sets related to Gnδ (W ). The contribution of Gδ (W ) is simply estimated by Lemma 4.3 (i.e., Formulas (G0)–(G3)). To treat Gδ F (W ) we assume inductively that we already have the relevant bound on Gi,k , the (n − 1)-gap of the δn−1 -LUM T Wi,k , for any i and k. “Pulling back” the relevant estimates to W relies on two key observations. On the one hand, the T Wi,k are homogeneous u-manifolds, thus we have bounds on the distortions of T , which help us express estimates in terms of measures in W . On the other hand, by means of Corollary 4.13, we can reformulate our estimates in terms of distances on W . To prove (Gn0) we make the following observations. Any point in W that lies in the δ neighborhood of Γ belongs to Gδ (W ) by (G0). On the other hand, by the inductive assumption, if x0 ∈ T Wi,k (for some i, k fixed) lies in the δ 1/3 neighborhood of Γ(n−1) , then x0 ∈ Gi,k . Thus, by Hölder continuity of T , if x(= T −1 x0 ) ∈ W lies in the δ neighborhood of T −1 Γ(n−1) , it should belong to Gδ F . The fact that Γ(n) = Γ ∪ T −1 Γ(n−1) completes the proof of (Gn0). To prove (Gn3) we need to provide an upper bound on the measure of Gnδ (W ). On the one hand, by (G3): mW (Gδ ) ≤ D1 mW (rW,0 < δ σ1 ) .

(4.17)

To estimate the measure of Gδ F , we use our inductive assumption on Gi,k for any i, k fixed. Recall that Gi,k is the n − 1 gap for δ 1/3 in T Wi,k . Since by (4.16) we have (δ 1/3 )σn−1 = δ σn , this means that the inductive assumption – i.e., (Gn3) formulated for n → n − 1 and δ → δ 1/3 in T Wi,k – reads as mT Wi,k (Gi,k ) ≤ Dn−1 mT Wi,k (rT Wi,k ,0 < δ σn ) .

(4.18)

Now we are going to use that T Wi,k is, on the one hand, a δ1 -LUM, and, on the other hand, it is homogeneous. Thus we have that T distorts measures on it at most by a factor λ (recall how the constants are chosen from Section 4.1.1). Thus (4.17) implies: mWi,k (T −1 Gi,k ) ≤ Dn−1 λmWi,k (rWi,k ,1 < δ σn ) . We may sum first over k and then over i to obtain mW (Gδ F ) ≤ Dn−1 λ mW x ∈ W |ρ(T x, ∂T W ) < δ σn . where the right hand side is understood according to Remark 4.12.

(4.19)

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Figure 5. The statement of Sublemma 4.15. Now we apply Corollary 4.13 with ε → δ σn : mW x ∈ W |ρ(T x, ∂T W ) < δ σn ≤ K1,W λ2 mW (rW,0 < δ σn /Λ) ≤ K1 λ2 mW (rW,0 < δ σn ) .

(4.20)

Note that Dn = D1 + Dn−1 λ3 K1 (4.16). Furthermore, as D1 ≤ Dn and σn ≤ σ1 , the decomposition (4.15) along with the three inequalities (4.17), (4.19) and (4.20) altogether imply (Gn3). As a preparation for the inductive proof of (Gn2) we state the following geometric sublemma. For the proof see Appendix C. Sublemma 4.15. Let W ∈ Rm be any nonempty bounded measurable set, ε ≥ 0 and k > 1. Then Leb x ∈ W | ρ(x, ∂W ) ≤ kε ≤ k m Leb x ∈ W | ρ(x, ∂W ) ≤ ε . See Figure 5. Now we can prove (Gn2). By (4.15): x ∈ W |rGnδ ,0 (x) < ε ⊂ x ∈ Gδ |rGnδ ,0 (x) < ε ∪ x ∈ Gδ F |rGnδ ,0 (x) < ε . For the first component we apply (G2): mW (rGδ ,0 < ε) ≤ D1 δ −κ1 mW (rW,0 < ε) ,

(4.21)

F

while the second component, as a subset of Gδ , is at least δ away from the firststep singularity set Γ. This implies that T expands distances at most by a factor 1/δ on this set (due to 2. in Section 2.4.1 and Proposition 3.12). As a consequence, recalling also (4.15): x ∈ Gδ F |rGnδ ,0 (x) < ε ⊂ T −1 x ∈ T Wi,k |rGi,k ,0 (x) < ε/δ . (4.22) We estimate the contribution for fixed i and k. In particular, we assume inductively that Gi,k , as the (n − 1)-gap for δ → δ 1/3 within T Wi,k satisfies (Gn2) in the relevant form: mT Wi,k (rGi,k ,0 < ε/δ) ≤ Dn−1 δ −κn−1 /3 mT Wi,k (rT Wi,k ,0 < ε/δ) . We apply distortion bounds on T restricted to Wi,k the same way as in the proof of (Gn3): mWi,k (rT −1 Gi,k ,1 < ε/δ) ≤ Dn−1 λδ −κn−1 /3 mWi,k (rWi,k ,1 < ε/δ) ,

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and sum over k and i (recall Remark 4.12): mW x ∈ Gδ F |rGnδ ,0 (x) < ε ≤ Dn−1 λδ −κn−1 /3 mW x ∈ W | ρ(T x, ∂T W ) < ε/δ . (4.23) Applying Corollary 4.13 to the right hand side of (4.23) – the same way as in (4.20) – implies: mW x ∈ Gδ F |rGnδ ,0 (x) < ε ≤ Dn−1 K1 λ3 δ −κn−1 /3 mW (rW,0 < ε/δ) . Finally let us invoke Sublemma 4.15 with k = 1/δ to bound the right hand side from above: mW (rW,0 < ε/δ) ≤ δ −(d−1) mW (rW,0 < ε) , which completes the inductive proof of (Gn2) as δ −κn = δ −κn−1 /3 δ −(d−1) by (4.16). To prove (Gn1) we need to introduce some more notation. By definition, Fδ = W \ Gδ and Fδn = W \ Gnδ . Furthermore let ˜ i,k := Wi,k ∩ Fδ (⊂ Wi,k ⊂ W ) , W ˜ i,k \ Gi,k (⊂ T Wi,k ⊂ T W ) . Fi,k := T W

(4.24)

Note that ˜ i,k Fδ = ∪i,k W

and Fδn = ∪i,k T −1 Fi,k

where the unions are disjoint. This implies ˜ i,k |r ˜ x ∈ Fδ |rFδ ,1 (x) < ε = ∪i,k x ∈ W Wi,k ,1 (x) < ε ,

(4.25)

and

x ∈ Fδn |rFδn ,n (x) < ε = ∪i,k x ∈ T −1 Fi,k |rT −1 Fi,k ,n (x) < ε .

(4.26)

On the other hand – recalling also Remarks 4.9 and 4.14 – by (4.24) Fi,k is ˜ i,k . Thus we may assume inductively that (Gn1) the (n − 1) remaining part in T W holds for n − 1: 4(n−1) n−1 mT W mT W ). ˜ i,k (rFi,k ,n−1 < ε) ≤ Kn−1,T W ˜ i,k λ ˜ i,k (rT W ˜ i,k ,0 < ε/Λ

˜ i,k are homogeneous δ1 -LUM-s, the distortions of T are suitably bounded As the T W and we have n−1 4(n−1) mW λmW ). ˜ i,k (rT −1 Fi,k ,n < ε) ≤ Kn−1,T W ˜ i,k λ ˜ i,k (rW ˜ i,k ,1 < ε/Λ

(4.27)

Note that K.,. describes the effect of primary singularities, thus it can be defined for any (not necessarily homogeneous) unstable manifold, in particular, we may consider the quantity Kn−1,T Wi . We also have Kn−1,T W ˜ i,k ≤ Kn−1,T Wi

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by means of which, keeping i fixed, we sum (4.27) over k: mWi x ∈ Wi |rFδn ,n (x) < ε ≤ Kn−1,T Wi λ4(n−1) λmWi x ∈ Wi |rFδ ,1 (x) < ε/Λn−1 . (4.28) Here we have also used the characterizations (4.25) and (4.26). To bound the right hand side from the above, first we apply (G1) to Wi for i fixed (with ε → ε/Λn−1 ): mWi (rFδ ,1 < ε/Λn−1 ) ≤ K1,Wi λ2 mWi (rWi ,0 < ε/Λn ) . To proceed recall that, as long as only primary singularities are concerned, the Wi are the smooth components of W . Thus K1,Wi = 1 and mWi (rFδ ,1 < ε/Λn−1 ) ≤ λ2 mWi (rWi ,0 < ε/Λn ) .

(4.29)

By means of the continuation property, we apply, as we did in the proof of (4.12) in Section 4.2, Sublemma 4.10: mWi (rWi ,0 < ε/Λn ) ≤ λmW (rW,0 < ε/Λn ) . Here the additional λ factor appears as the error term of the locally flat estimate. This last formula gives, along with (4.29) and (4.28) and summation on i:   K1,W X mW (rF n ,n < ε) ≤  Kn−1,T Wi  λ4n mW (rW,0 < ε/Λn ) . δ

i=1

Finally, as a consequence of the continuation property we have   K1,W X  Kn−1,T Wi  = Kn,W i=1

which completes the inductive proof of (Gn1). 4.4. Proof of Proposition 4.1 To complete the proof of the Growth Properties for Tˆ = T n0 , choose n0 according ˆ to the exposition of Section 4.1.1. In particular, by Assumption 1.3, Kn0 λ4n0 = αΛ n0 ˆ ˆ for some α < 1 (here Λ = Λ is the factor of expansion for T ). The constants for which we prove Proposition 4.1 are the above mentioned α along with δ0 = δn0 , σ = σn0 , D = Dn0 and κ = κn0 chosen according to Lemma 4.5 (see also Remark 4.6). Let us consider an arbitrary δ0 (= δn0 )-LUM W , and an arbitrary 0 ≤ δ < 1. To prove the proposition, first we should tell what the sets Vδ0 and Vδ1 are. As W is a δn0 -LUM, we may apply the n(=n0 ) step construction of Section 4.3.1 to it and construct Gnδ 0 (W ) and Fδn0 (W ). Now define: Vδ0 = Gnδ 0 (W ) ,

Wδ1 = Fδn0 (W ) .

Note that as Gnδ 0 and Fδn0 make a partition of W , we have mW W \ (Vδ0 ∪ Wδ1 ) = 0 ,

(4.30)

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and by (Gn0) ˆ [δ] . Wδ1 ⊂ W \ Γ

(4.31) However, we cannot use as The reason is that, in terms of Tˆ, Wδ1 is not a (δ0 , 1)-subset. In particular, we should have that the components of TˆVδ1 have diameter less than δ0 . The construction of Section 4.3.1, on the other hand, 1/3 ensures only that the components of TˆWδ1 have diameter less than δ1 , which is much greater than δ0 . To obtain smaller components, we construct below Vδ1 by removing sets of zero mW -measure from Wδ1 . This, of course, does not spoil the validity of (4.30) and (4.31). To proceed we may reformulate statements (Gn1)–(Gn3) from Lemma 4.5 in terms of Tˆ = T n0 . Keeping also in mind how rW,n is defined, (Gn1) reads as n o mW x ∈ Wδ1 | ρ Tˆx, ∂ TˆWδ1 (x) < ε ˆ · mW x ∈ W | ρ(x, ∂W ) < ε/Λ ˆ ≤ αΛ (4.32) Wδ1

Vδ1 .

(Gn2) as n o mW x ∈ Vδ0 | ρ x, ∂Vδ0 (x) < ε ≤ Dδ −κ mW

x ∈ W | ρ(x, ∂W ) < ε

;

(4.33)

and (Gn3) as mW (Vδ0 ) ≤ D mW

x ∈ W | ρ(x, ∂W ) < δ σ

.

(4.34)

Note that (4.33) and (4.34) are exactly the Second and the Third Growth Properties, respectively. Furthermore, (4.32) is almost the First Growth Property, actually, it is an even better upper bound on the set of points that get close to the boundaries of TˆWδ1 . Recall that we need to partition the components of Wδ1 into smaller pieces to arrive at Vδ1 . We will see that the contribution of these additional boundary components can be estimated by the second term that appears on the right hand side of (4.1). By the exposition of Section 4.3.1, the set Wδ1 has finitely many components (the number of which, actually, depends on δ). Consider any such component and denote it by ∆1 : we would like to chop ∆1 into pieces which do not grow larger then δ0 in diameter when Tˆ is applied. Our argument will roughly follow [8]. 1/3 What we know is that Tˆ∆1 is a δ1 -LUM, thus by our convention on local flatness, in measure related calculations it can be considered as a piece of Rd−1 . Furthermore, it is a homogeneous LUM, thus the distortions of Tˆ restricted to ∆1 are suitably bounded. 0 We shall work in Rd−1 . Let us fix δ 0 = 2√δd−1 (which ensures that a hypercube 0 of side δ has diameter δ0 /2). For given numbers a1 , . . . , ad−1 such that 0 ≤ ai
0 depends on the dimension, the distortion bounds and the accuracy of the locally flat approximation. We chop all the ∆1 -s (the connected components of Wδ1 ) according to this machinery to get Vδ1 . After summation we get: n o n o mW x ∈ Vδ1 | ρ Tˆx, ∂ TˆVδ1 (x) < ε −mW x ∈ Wδ1 | ρ Tˆx, ∂ TˆWδ1 (x) < ε ≤ εδ0−1 βmW (W ) . (4.36) Now (4.36) along with (4.32) implies (4.1). This completes the proof of the First Growth Property, and thus of Proposition 4.1.

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4.5. The conditions of ergodicity Following the program outlined in the Introduction, the arguments presented in the preceding sections verify Chernov’s axioms from Appendix A with the only exception of Condition A.3, that is, the ergodicity of the billiard map T (and its higher iterates). As we already mentioned, to complete the proof of Theorem 1.7, we refer to [4] where Condition A.3 is verified. More precisely, we use the result in [4] to prove Proposition 4.16. The billiard dynamics satisfying the assumptions of Theorem 1.7 is ergodic, as well as all of its iterates. Now [4] proves (local) ergodicity based on a set of assumptions that are different and, literally, independent of the conditions in Appendix A. However, the assumptions (apart for some well-known regularity properties) in [4] follow from those of Appendix A with two exceptions. First, the setting of [4] is symmetric with respect to the roles of the stable and unstable direction, while Chernov’s axioms are formulated only in terms of the unstable direction. Second, [4] (specifically, Assumption A5 in that paper) requires some kind of proper alignment of unstable manifolds and negative time singularities. This does not coincide with the notion of Alignment mentioned in Section 2.4 and used in Section 4. 4.5.1. Regularity properties and growth lemma for stable manifolds. The standard argument for checking that certain statements, already proven for the unstable direction and the forward dynamics, can be reformulated for the stable direction and the backward dynamics is to refer to the time reflection symmetry of the billiard map (see e.g. [11]). This reasoning is correct if one measures distances with respect to the Euclidean metric, which is, indeed, symmetric with respect to time reflection – see Remark 2.5. Note, however, that the metric we use – the regularized Chernov–Dolgopyat metric of Definition 3.6 – no longer has this time reflection symmetry. Fortunately, the regularity properties of unstable manifolds – in particular conditions A.4 and A.5 from Appendix A – have all been verified for the Euclidean metric (see Section 2). Condition A.7 on the growth properties of unstable manifolds is the only property which is stated and proven in terms of the regularized Chernov–Dolgopyat metric and is not verified in terms of the Euclidean one. We do that now. The version of the growth lemma that appears among the assumptions of [4] (both in a stable and in an unstable form) is slightly weaker than Condition A.7; it is exactly the δ = 0 version of (A.2). Recalling the definitions for the function rV,k and that actually Tˆ = T n0 , this statement reads as (see also remarks 4.4, 4.7 and Corollary 4.13): mW x ∈ W |ρ(T n0 x, ∂T n0 W ) < ε ≤ α0 Λn0 · mW ρ(x, ∂W ) < ε/Λn0 + εβ0 δ0−1 mW (W ) , (4.37) where α0 ∈ (0, 1) and β0 > 0 are some global constants.

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To distinguish between the two metrics we will use ρE for the distances measured in the Euclidean metric and ρCD for the regularized Chernov–Dolgopyat metric. We claim that (4.37) with ρ = ρCD implies the same statement with ρ = ρE , with slightly worse constants. To see this, first recall from Section 3 that the two metrics are equivalent. This implies mW x ∈ W |ρE (T n x, ∂T n W ) < ε ≤ mW x ∈ W |ρCD (T n x, ∂T n W ) < Cε (4.38) and mW ρCD (x, ∂W ) < Cε/Λn ≤ mW ρE (x, ∂W ) < C 2 ε/Λn .

(4.39)

˜ n if Λ ˜ is slightly less than Λ, and n is large enough. But clearly C 2 ε/Λn ≤ ε/Λ Thus (4.38), (4.39) and (4.37) for the regularized C-D metric directly imply (4.37) ˜ and a (possibly) larger n0 . As mentioned for the Euclidean metric with Λ → Λ above, this statement is directly transferable to the inverse map and the stable direction. Remark 4.17. In an analogous way one could check Condition A.7 in full generality for the Euclidean metric. This means that the set of conditions from Appendix A can be verified for the Euclidean metric ρE . However, in our proof of the growth properties the use of ρCD is crucial (this is the only way we could reduce the statement with an inductive argument to the one step Lemma 4.3). Furthermore, our arguments throughout Section 4 use heavily the uniform curvature and distortion bounds for the metric ρCD , which is the reason for the necessity of the differential geometric analysis of Section 3. 4.5.2. Alignment for negative time singularities. Let us introduce the set Γ− = Γ0 ∪ T Γ0 . (Remember from Section 2.2.3 that Γ0 is the boundary of our phase space after introducing homogeneity layers.) We think of Γ− as the singularity set of the inverse dynamics T −1 , although it is important that it also contains the boundary of the phase space. The notion of alignment required in Assumption A.5 of [4] says roughly that unstable manifolds intersect Γ− transversally, and their angle at any intersection point is at least some global constant c. However, this rough form of the assumption is not satisfied by our systems: it is known that even in 2 dimensions unstable ˜ . Actually, for every component S of T ∂ M ˜, manifolds may be tangent to T ∂ M this happens exactly on one side of S (remember that S is a one-codimensional submanifold that cuts the phase-space into two pieces). Indeed, this is the side of S ˜ . The other side typically consists that contains images (under T ) of points near ∂ M ˜ ˜ , on the side which of images of points which were not near ∂ M , but near T −1 ∂ M eventually avoided the nearly-tangent collision. See Figure 6 for an explanation: ˜ . b is on the side of T −1 ∂ M ˜ which avoids the nearly T a is a phase point on ∂ M ˜ , while c tangent collision and travels directly to the neighbourhood of T 2 a ∈ T ∂ M 2 is first mapped near T a, and only thereafter near to T a. The violation of the ˜ containing T 2 c. rough form of Alignment happens on the side of T ∂ M

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Figure 6. Possible trajectories of points near a singularity. Now we can explain the refined form of the alignment assumption (Assumption A.5 from [4]): we only expect an unstable manifold (developing in time) to be transversal to Γ− at the time of their first encounter. The following proposition states this in a precise form. It implies the Alignment assumption of [4]. (The only difference is that the assumption in [4] is formulated in terms of distances instead of angles, since no smoothness of these subsets is formally assumed there.) Proposition 4.18. There exists a global constant c > 0 with the following property: Let W be an unstable manifold contained entirely in a connected component of ¯ ∩ Γ− (here W ¯ denotes the closure of W ). If the inverse M \ Γ− , and let x ∈ W image of x under T as a one-sided limit, x− :=

lim

y∈W, y→x

T −1 y

(4.40)

is not in Γ− , then the angle of W and any smooth component S of Γ− at x is at least c. ˜ , then this transversality is known from [5] and stated Proof. If S ⊂ ∂M = Γ0 ∪∂ M as (part of) the alignment property in Section 2.4. If S ⊂ T Γ0k for some k ∈ {k0 , k1 , . . . } (that is, S is a secondary singularity −1 of T ), then the extension of T −1 to S as a one-sided limit is the same for both sides, and the only possible inverse image x− of x is in Γ0k ⊂ Γ− , so there’s nothing to prove (the conditions of the proposition cannot hold). ˜ , and x− ∈ ˜ . With the So the only interesting case is when S ⊂ T ∂ M / ∂M 2 − notation of Figure 6 this corresponds to x = T a, x = a, and the limit in (4.40) is through points y = T b. Let us look at this case now. In high dimensions (when d ≥ 3) we have dim(W ) + dim(S) = (d − 1) + (2d − 3) > 2d − 2 = dim(M ), so transversality of W and S can only mean that there is a d − 1-dimensional subspace U ⊂ Tx S which is transversal to W in the sense that the angle between any (dq1 , dw1 ) ∈ U and any (dq2 , dw2 ) ∈ Tx W is at least c. Now d − 1-dimensional submanifolds can typically be considered as traces of fronts on the Poincaré phase space. (Remember from Section 2.3 that fronts are subsets of the flow phase space.) The (tangent space of the) unstable manifold W is known from Section 2.3.2 to correspond to a well-understood convex

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front described by (2.11) and (2.12). To prove transversality of S and W in the appropriate sense, it is enough to find another front, whose trace on the Poincaré phase space lies within S, and show that these two fronts are transversal. To see this recall from Section 2.3.4 that if two fronts with s.f.f.-s B1− and B2− (just before collision) satisfy B2− − B1− > c1 , −C1 < B1− < C1 , (4.41) then the angle of their traces in the Poincaré phase space is at least some c > 0 depending only on c1 and C1 (and the geometry of the billiard table). ˜ , which is T a in Figure 6. Let τ1 denote the Consider the point T −1 x ∈ ∂ M flight time from x− to T −1 x, and τ2 the flight time from T −1 x to x. The tangent space of T −1 W at x− (as a one-sided limit) is described by a convex front according to (2.11), so the pre-collision s.f.f. of its image at x satisfies 0 < B1−
1 with the following properties: • DT (Cxu ) ⊂ CTu x and DT (Cxs ) ⊃ CTs x whenever DT exists; • |DT (v)| ≥ Λ|v| ∀v ∈ Cxu ; • |DT −1 (v)| ≥ Λ|v| ∀v ∈ Cxs ; ¯ , their axes have the same di• these families of cones are continuous on M ¯ mensions across the entire M which we denote by du and ds , respectively; • du + ds = dim M ; • the angles between Cxu and Cxs are uniformly bounded away from zero: ∃ α > 0 such that ∀x ∈ M and for any dw1 ∈ Cxu and dw2 ∈ Cxs one has ^(dw1 , dw2 ) ≥ α . The

Cxu

are called the unstable cones whereas Cxs are called the stable ones.

In our case, dim M = 2d − 2, and du = ds = d − 1. The property that the angle between stable and unstable cones is uniformly bounded away from zero is called transversality. Some notation and definitions. For any δ > 0 denote by Uδ the δ-neighbourhood of the closed set Γ ∪ ∂M . We denote by ρ the Riemannian metric in M and by m the Lebesgue measure (volume) in M . For any submanifold W ⊂ M we denote by ρW the metric on W induced by the Riemannian metric in M , by mW the Lebesgue measure on W generated by ρW and by diamW , the diameter of W in the ρW metric. LUM-s. To be able to formulate the further properties let us fix what we mean by the notion of local unstable manifolds. A submanifold W u ⊂ M homeomorphic to a du -dimensional ball is called a local unstable manifold (LUM) if (i) dim W u = du ,

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(ii) T −n is defined and smooth on W u for all n ≥ 0, (iii) ∀x, y ∈ W u we have ρ(T −n x, T −n y) → 0 exponentially fast as n → ∞. It is worth noting that this notion of LUM does not coincide with the standard concept of a local unstable manifold in Pesin theory, as formulated, for example, in [7]. In particular, unextendable unstable manifolds – maximal ones for which T −n is smooth for any n ≥ 0 – are LUMs as well. Here the term “local” simply refers to the presence of singularities, which put restriction on the size of any LUM. We denote by W u (x) (or just W (x)) a local unstable manifold containing x. Similarly, local stable manifolds (LSM) are defined. Condition A.3. SRB measure. The dynamics T has to have an invariant ergodic Sinai–Ruelle–Bowen (SRB) measure µ. That is, there should be an ergodic probability measure µ on M such that for µ-a.e. x ∈ M a LUM W (x) exists, and the conditional measure on W (x) induced by µ is absolutely continuous with respect to mW (x) . Furthermore, the SRB-measure should have nice mixing properties: the system (T n , µ) is ergodic for all finite n ≥ 1. In our case the SRB measure is simply the Liouville measure defined by (2.1). Absolute continuity and invariance of µ are straightforward, while ergodicity is proved in [4], based on the conditions mentioned in Section 4.5. Condition A.4. Uniformly bounded curvature. There should exist a global constant KW < ∞ such that the curvature of any unstable manifold at any of its points is at most KW . The meaning of the word “curvature” for the purpose of this condition is made precise in Section 3.4. Accordingly, the condition is formulated more precisely as Condition 3.10. Some notation. Denote by J u (x) = | det(DT |Exu )| the Jacobian of the map T restricted to W (x) at x, i.e. the factor of the volume expansion on the LUM W (x) at the point x. Let Γ(n) denote the singularity set of T n – that is, the smallest set ⊂ M for which T n is defined on M \ Γ(n) in the sense of Condition A.1. Condition A.5. Uniform distortion bounds. Let x, y be in one connected component of W \ Γ(n−1) , which we denote by V . Then log

n−1 Y i=0

J u (T i x) ≤ ϕ ρT n V (T n x, T n y) u i J (T y)

where ϕ( · ) is some function, independent of W , such that ϕ(s) → 0 as s → 0. Condition A.6. Uniform absolute continuity. Let W1 , W2 be two sufficiently small LUM-s, such that any LSM W s intersects each of W1 and W2 in at most one point. Let W10 = {x ∈ W1 : W s (x) ∩ W2 6= ∅}. Then we define a map h : W10 → W2 by sliding along stable manifolds. This map is often called a holonomy map. This has

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to be absolutely continuous with respect to the Lebesgue measures mW1 and mW2 , and its Jacobian (at any density point of W10 ) should be bounded, i.e. 1/C 0 ≤

mW2 (h(W10 )) ≤ C0 mW1 (W10 )

with some C 0 = C 0 (T ) > 0. Some further notation. Let δ0 > 0. We call W a δ0 -LUM if it is a LUM and diam W ≤ δ0 . For an open subset V ⊂ W and x ∈ V denote by V (x) the connected component of V containing the point x. Let n ≥ 0. We call an open subset V ⊂ W a (δ0 , n)-subset if V ∩Γ(n) = ∅ (i.e., the map T n is smoothly defined on V ) and diam T n V (x) ≤ δ0 for every x ∈ V . Note that T n V is then a union of δ0 -LUM-s. Define a function rV,n on V by rV,n (x) = ρT n V (x) T n x, ∂T n V (x) . (A.1) Note that rV,n (x) is the radius of the largest open ball in T n V (x) centered at T n x. In particular, rW,0 (x) = ρW (x, ∂W ). Now we formulate Chernov’s Growth Properties, in essentially (cf. Remark A.8) the same form as they appeared in [8] and [9]. In view of the curvature and distortion bounds (conditions A.4 and A.5) these conditions can roughly be seen as conditions about some piecewise linear expanding map on a union of flat hypersurfaces. Condition A.7. Growth of unstable manifolds. Let us assume there is a fixed δ0 > 0, furthermore, there exist constants α ∈ (0, 1) and β, D, κ, σ, ζ > 0 with the following property. For any sufficiently small δ > 0 and any δ0 -LUM W there is an open (δ0 , 0)-subset Vδ0 and an open (δ0 , 1)-subset Vδ1 ⊂ W \ Γ[δ] (one of these may be empty) such that the two sets are disjoint, mW (W \ (Vδ0 ∪ Vδ1 )) = 0 and ∀ε > 0 First Growth Property: mW (rVδ1 ,1 < ε) ≤ αΛ · mW (rW,0 < ε/Λ) + εβδ0−1 mW (W )

(A.2)

Second Growth Property: mW (rVδ0 ,0 < ε) ≤ Dδ −κ mW (rW,0 < ε)

(A.3)

and Third Growth Property: mW (Vδ0 ) ≤ D mW (rW,0 < ζδ σ ) .

(A.4)

Remark A.8. Note that these growth properties are slightly different from those assumed in [9], namely, there it was assumed that Vδ0 ⊂ Γ[δ] . However, it is easy to see, that the whole strategy of [9] works without this assumption. What is indeed important is that the set Vδ1 is disjoint both from Vδ0 and Γ[δ] , and that the measure of Vδ0 can be estimated by the third growth lemma. How Vδ0 and Γ[δ] are related is discussed in Section 4.2.1. Now we can formulate Theorem 2.1 from [9].

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Theorem A.9 (Chernov, 1999). Under the conditions A.1,. . . ,A.7, the dynamical system enjoys exponential decay of correlations and the central limit theorem for H¨ older-continuous functions. The properties stated in the theorem are defined in Definition 1.5 and Remark 1.8.

B. Equivalence of Riemannian structures and inherited regularity properties In this section we consider the problem of having two different Riemannian structures on the same differentiable manifold. The essence of the statements is that under the mildest possible regularity conditions (adequate differentiability of the metric tensor fields) the regularity properties of submanifolds and maps (uniform curvature and distortion bounds) are inherited from one Riemannian manifold (or rather: one Riemannian structure) to the other. That is, these notions are independent of the choice of the Riemannian structure. The goal is to prove Proposition 3.12. In the statements to come, M will always denote a C 2 differentiable manifold. g, g˜ will be C 1 Riemannian metric tensor fields on M , and W a C 2 smooth submanifold of M . Since the notions of ‘covariant differentiation’, ‘second fundamental form’, ‘orthogonality’ and ‘norm’ depend on the Riemannian structure, we ˜ k.k˜ to denote ˜ II, ˜ ⊥, will use ∇, II, ⊥, k.k to denote them when g is used, and ∇, them with respect to g˜. As before, the phrase ‘second fundamental form’ will be abbreviated as s.f.f. Definition B.1. Let M be a C 2 differentiable manifold, possibly with boundary. Two C 1 Riemannian metric tensor fields g and g˜ on M are said to be C 1 equivalent with constant K if both g˜ and ∇˜ g (as tensors) are bounded by K when g is used for the definition of norm and covariant derivation, and vice versa. In detail: for any x ∈ M , u, v, w ∈ Tx M |˜ g (v, w)| ≤ Kkvkkwk , |(∇u g˜)(v, w)| ≤ Kkukkvkkwk , |g(v, w)| ≤ Kkvk˜kwk˜, ˜ u g)(v, w)| ≤ Kkuk˜kvk˜kwk˜. |(∇ If translated √to norms of vectors, √ the first and third inequality of the definition say that kvk˜ ≤ Kkvk and kvk ≤ Kkvk˜, but for convenience we will omit the square root and use kvk˜ ≤ Kkvk, kvk ≤ Kkvk˜. This is fine since K ≥ 1. Lemma B.2. On a compact C 2 differentiable manifold M (possibly with boundary) any two C 1 Riemannian metric tensor fields g and g˜ are C 1 equivalent with some constant K (depending of course on the tensor fields).

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Proof. Since g˜ and g are continuous, the norm of g˜ with respect to g, k˜ gk = sup{|˜ g (v, w)| | kvk = kwk = 1} is a continuous function on M , so it has a finite maximum, since M is compact. Similarly, ∇˜ g is continuous, so its norm with respect to g, k∇˜ g k = sup{|(∇u g˜)(v, w)| | kuk = kvk = kwk = 1} is a continuous function on M , so it has a finite maximum. The same is true with g and g˜ interchanged. The greatest of these four maxima can be chosen as K. ˜ < ∞ (depending only on K1 , K2 Lemma B.3. For any K1 , K2 < ∞ there is a K and dim(M )), such that for any g, g˜ and W , if g and g˜ are C 1 equivalent with constant K1 and the s.f.f. of W with respect to g is bounded by K2 , then the s.f.f. ˜ of W with respect to g˜ is bounded by K. Before we can start the proof, we state and prove a sublemma: Sublemma B.4. The norm of the vector-valued tensor S defined by S(v, w) = ˜ v w − ∇v w is bounded by a constant K ˆ depending only on K1 and dim(M ). That ∇ ˆ is, kS(v, w)k ≤ Kkvkkwk. Proof. The fact that S is indeed a vector-valued tensor is known, see e.g. [14]. At any point x ∈ M we can take normal coordinates with respect to g. In this coordinate chart, the Christoffel-symbols of ∇ are zero (at the single point x). This has two consequences. First, the components of S are exactly the Christoffel˜ Second, in this coordinate chart, at x, the partial derivatives of the symbols of ∇. components of g˜ are exactly the components of ∇˜ g , and are thus bounded by K1 because of the C 1 equivalence we assumed. So are the components of g −1 , again by ˜ can be expressed in the equivalence of g and g˜. Since the Christoffel-symbols of ∇ terms of the above two, they can clearly be estimated using K1 and dim(M ). This implies a similar estimate for the norm of S. The estimate is clearly independent of the choice of the point x. Now we can turn to the proof of Lemma B.3. ˜ the s.f.f. of W with respect to g˜. Proof of Lemma B.3. We want to estimate II, We will use the definition of II, the definition of S and the fact that ⊥ W ∇u v = ∇ W u v + ∇u v = ∇u v + II(u, v) ,

where ∇W u v is parallel to W . We write ˜ ˜ ˜ u v)⊥ II(u, v) = (∇ = ∇u v + S(u, v)

⊥ ˜

⊥ ˜ = ∇W u v + II(u, v) + S(u, v) ⊥ ⊥ ˜ ˜ ˜ ⊥ = (∇W + S(u, v) . u v) + II(u, v) ˜ The first term is zero, and the other two can be overestimated if we omit the ⊥. So we get – using all the assumptions of the lemma and the statement of the

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sublemma ˜ kII(u, v)k˜ ≤ kII(u, v)k˜ + kS(u, v)k˜ ˆ ≤ K1 kII(u, v)k + K1 kS(u, v)k ≤ K1 K2 kukkvk + K1 Kkukkvk ˜ ˆ 1 kuk˜K1 kvk˜ = K 3 (K2 + K)kuk ˆ ≤ K1 K2 K1 kuk˜K1 kvk˜ + K1 KK kvk˜. 1

Lemma B.5. For any K < ∞ there is a K 0 < ∞ (depending only on K), such that for any g, g˜ and W if g and g˜ are C 1 equivalent with constant K and the s.f.f. of W with respect to g is bounded by K, then the restrictions of g and g˜ to W , g|W and g˜|W , are C 1 equivalent with constant K 0 . Proof. To have the equivalence, we first need to see that the norm with respect to g|W of g˜|W is bounded, and vice versa. This is obviously inherited from g and g˜, so K 0 = K would do. Second, we need that the derivatives are bounded. Let us denote, for a moment, the covariant derivative with respect to g|W by ∇g|W , and similarly for g˜. We need to see that the norm with respect to g|W of ∇g|W g˜|W is bounded (and vice versa), but this is of course the same as the norm with respect to g, so we introduce no new notation. To understand ∇g|W g˜|W , we first describe how ∇g|W acts on vectors. It is known (see e.g. [14] again) that when we split the covariant derivative ∇u v into tangential and orthogonal components using ∇u v = ∇W u v+II(u, v), the tangential component is nothing else than the covariant derivative with respect to the metric g|W tensor restricted to W , so ∇W u v = ∇u v. g|W We will express ∇ g˜|W by using the definition of covariant differentiation for a tensor, and the above fact. Let u, v, w be tangent vectors of W (at the same point). W W W (∇g| ˜|W )(v, w) = u g˜(v, w) − g˜(∇g| ˜(v, ∇g| u g u v, w) − g u w) = u g˜(v, w) − g˜(∇u v, w) + g˜ II(u, v), w − g˜(v, ∇u w) + g˜ v, II(u, w) = (∇u g˜)(v, w) + g˜ II(u, v), w + g˜ v, II(u, w) . These three terms can be readily estimated using the bounds on k∇˜ g k, k˜ g k and kIIk that we have assumed. We get W |(∇g| ˜|W )(v, w)| ≤ Kkukkvkkwk + KkII(u, v)kkwk + KkvkkII(u, w)k u g

≤ Kkukkvkkwk + KKkukkvkkwk + KkvkKkukkwk = (K + 2K 2 )kukkvkkwk . We got that k∇g|W g˜|W k ≤ K 0 = K + 2K 2 . The bound for k∇g˜|W g|W k˜ is exactly the same. Now we investigate how sensitive distortions are to the choice of the Riemannian structure. The following lemma states that if a map of one manifold

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to the other satisfies certain distortion bounds, then modifying the Riemannian structures on both manifolds up to C 1 equivalence, distortion bounds remain valid. Later we will apply the result to the restriction of the dynamics to an unstable manifold, which maps to another unstable manifold. In the lemma let M and M 0 be two C 2 differentiable manifolds. Let M carry two C 1 Riemannian metric tensor fields, g and g˜. Similarly, let M 0 carry two C 1 Riemannian metric tensor fields, g 0 and g˜0 . Let T : M → M 0 be a C 1 map and let ˜ the x, y ∈ M . The two metrics on M defined by g and g˜ are denoted by d and d, two metrics on M 0 defined by g 0 and g˜0 are denoted by d0 and d˜0 . The Jacobian of T with respect to (g, g 0 ) is denoted by J, and the Jacobian of T with respect to ˜ (˜ g , g˜0 ) is denoted by J. Lemma B.6. For any k > 0, K < ∞ and any h ∈ o(1) there exist k˜ > 0 and ˜ ∈ o(1) (depending only on h, K, k and dim(M )), such that h if • g and g˜ are C 1 equivalent (on M ) with constant K and • g 0 and g˜0 are C 1 equivalent (on M 0 ) with constant K and • d0 (T x, T y) ≥ k d(x, y) and • | log Jy − log Jx | ≤ h(d0 (T x, T y)) then ˜ y) and • d˜0 (T x, T y) ≥ k˜ d(x, ˜ d˜0 (T x, T y)). ˜ ˜ • | log Jy − log Jx | ≤ h( ˜ is similar in shape to h – it is obtained from h Remark B.7. The function h basically by linear rescaling and adding a linear term (see the end of the proof). We will not make use of this fact, but it could be useful in applications where the ˜ does not depend on T asymptotics is important. However, it is important that h (as long as the conditions are satisfied with the same k, K and h), so we will be able to apply the lemma to T n instead of T – actually, to all the T n simultaneously, ˜ getting the same h. Proof. The first statement follows from the third assumption and the equivalence of the metrics: 1 1 1 1 ˜ d˜0 (T x, T y) ≥ d0 (T x, T y) ≥ kd(x, y) ≥ k d(x, y) , K K K K so k˜ = Kk2 will do. For the second statement, let A be a parallelepiped (ordered dim(M )-tuple of tangent vectors) at x, and B a parallelepiped at y (let them be nondegenerate). With slight abuse of notation, we denote their images under the derivative of T by T A and T B. Denote the volume element (canonical dim(M )-form) associated to g, g˜ g 0 and g˜0 by V , V˜ , V 0 and V˜ 0 , respectively. Then we have V 0 (T A) V 0 (T B) V˜ 0 (T A) V˜ 0 (T B) Jx = ; Jy = ; J˜x = ; J˜y = , (B.5) V (A) V (B) V˜ (A) V˜ (B) which are of course independent of the choice of A and B.

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P. B´ alint and I. P. T´ oth ˜


˜0

V (.) Also, the ratios VV (.) (.) and V 0 (.) are independent of the argument – they are actually the square root of the appropriate determinant: q V˜ (.) p V˜0 (.) = detg g˜ ; = detg0 g˜0 V (.) V 0 (.)

which of course, depend on the base point. The (covariant) derivative of detg g˜ can be expressed (say, coordinate-vise) in terms of g˜ and ∇˜ g , so it is bounded by some constant K 0 < ∞ depending only on K and dim(M ). detg g˜ is also separated from zero, so the same is true for its logarithm with some K 00 < ∞. This implies that V˜ (B) V˜ (A) 1 1 ≤ K 00 d(x, y) . (B.6) log − log = log det g ˜ (y) − log det g ˜ (x) g g 2 V (B) V (A) 2 Similarly, V˜ 0 (T B) V˜ 0 (T A) − log 0 log 0 ≤ K 00 d0 (T x, T y) . V (T B) V (T A)

(B.7)

Knowing this, we can force these quantities to show up in the expression for log J˜y − log J˜x . First write J˜y V˜ 0 (T B) V˜ (A) V˜ 0 (T B) V 0 (T B) V (B) V˜ (A) V (A) V 0 (T A) = = 0 , V (T B) V (B) V˜ (B) V (A) V 0 (T A) V˜ 0 (T A) J˜x V˜ (B) V˜ 0 (T A) than take the logarithm to get ! ˜ 0 (T B) ˜ 0 (T A) V V log J˜x − log J˜y = log 0 − log 0 V (T B) V (T A) ! V˜ (A) V˜ (B) V 0 (T B) V 0 (T A) + log − log + log − log . V (A) V (B) V (B) V (A) The first and second term can be estimated using (B.7) and (B.6), while the third term is log Jy − log Jx by (B.5) and can be estimated using the last assumption of the lemma. We get | log J˜y − log J˜x | ≤ K 00 d0 (T x, T y) + K 00 d(x, y) + h d0 (T x, T y) . Now we use the equivalence of metrics and the third assumption to replace all distances by d˜0 (T x, T y), and get 1 | log J˜y − log J˜x | ≤ K 00 K d˜0 (T x, T y) + K 00 K d˜0 (T x, T y) k + sup h(s) | s ≤ K d˜0 (T x, T y) . ˜ = K 00 K(1 + 1 )t + sup{h(s) | s ≤ Kt} ∈ o(1) will do. So h(t) k

With these lemmas, we can prove two strong theorems about the invariance of the regularity properties with respect to the choice of a Riemannian structure. In both statements we think of the W as unstable manifolds, but this is not required. Actually, in the first statement, not even the presence of a dynamics is required.

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Theorem B.8. Let g and g˜ be two C 1 Riemannian structures on the compact man˜ < ∞ (depending only on g, g˜ ifold M . Then for any K < ∞ there exists a K and K) such that if {Wi }i∈I is any collection of submanifolds of M that have the second fundamental forms with respect to g bounded everywhere by K, then the ˜ second fundamental forms with respect to g˜ are bounded by K. Proof. This is an immediate consequence of Lemma B.2 and Lemma B.3.

Theorem B.9. Let M be a compact manifold with the C 1 Riemannian structure g. Let {Wi }i∈I be a collection of submanifolds of M with all s.f.f.-s bounded by some constant K < ∞. Let T be a map which is defined and C 2 smooth on ∪i∈I Wi , and suppose that each T Wi is also a submanifold of M with the s.f.f. bounded by K. Assume that T restricted to the Wi satisfies uniform distortion bounds in the sense that there is a function ϕ (independent of i) with lims→0 ϕ(s) = 0 such that for any x, y ∈ Wi J Wi (x) log Wi ≤ ϕ ρWi (T x, T y) , (B.8) J (y) where J Wi denotes the Jacobian of T restricted to Wi , and ρWi is the metric on the submanifold Wi induced by g. Assume also that the contraction by T along the Wi is uniformly limited: there is a k > 0 such that ||DT v|| ≥ k||v|| for any v in the tangent space of any Wi . Then, if g˜ is another C 1 Riemannian structure on M , then there exists a function ϕ˜ depending only on g, g˜, k, K and ϕ (so not depending on T and {Wi }i∈I ) with lims→0 ϕ(s) ˜ = 0, such that (B.8) is satisfied with ϕ˜ instead of ϕ, when J and ρ are defined with respect to g˜ instead of g. Proof. We apply Lemma B.6 to the restriction of T to each Wi . The conditions of this lemma are ensured by Lemmas B.2, B.3 and B.5. Now we summarize the results of this section by proving Proposition 3.12. Proof of Proposition 3.12. In the case of • piecewise H¨ older continuity of the dynamics • bounded expansion away from the singularities • transversality of stable and unstable cone fields • alignement • absolute continuity, and • Corollary 2.11 the statement is easy to see without detailed analysis: these depend only on the C 0 equivalence of metrics, which is the comparability of length. The statement about curvature bounds and smoothness of the one-step singularities follows from Proposition 3.7 (which claims that the C-D structure is indeed C 1 ), and Theorem B.8. The fact that distortion bounds are inherited follows from Proposition 3.7, the curvature bounds, and Theorem B.9. Here Theorem B.9 is applied directly (and

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simultaneously) to the iterates T n (restricted to the unstable manifold), about which the bounded distortion condition is formulated. Remark B.10. We note that the reason for using the regularized structure instead of the C-D metric tensor is deeper than the easiness and robustness of the proof we gave for the regularity properties. Actually, we have checked by explicit calculation, that the unstable manifolds do not satisfy the bounded curvature assumption, if the unregularized C-D metric is used: the curvature blows up near the boundary ˜ , even in the simplest 3-dimensional configurations. Surprisingly, this does not of M happen in 2D (despite the degeneracy of the metric), which is why Chernov and Dolgopyat could use this tool with such success in [10]. Blow-up of curvatures with respect to the C-D metric is – beside the anisotropy of the unstable expansion and the pathological behaviour of higher-order singularities – another typical multidimensional phenomenon.

C. Geometric lemmas Here we prove the geometric Sublemmas 4.10, 4.11 and 4.15. Please recall the relevant notation and the statement of the these sublemmas from Sections 4.2.2 and 4.3.2. Sublemmas 4.10 and 4.11 will be easy corollaries of Sublemma C.1 below. We denote the distance in Rm by ρ. Sublemma C.1. For any ε ≥ 0 and 0 ≤ ξ ≤ 1, Leb x ∈ Wl | ρ(x, E) ≤ ξε \ x ∈ W | ρ(x, ∂W ) ≤ ε ≤ ξLeb x ∈ Wr | ρ(x, ∂W ) ≤ ε . The same is true with Wl and Wr interchanged. Proof. Denote the two sets to compare by A and B, so A = x ∈ Wl | ρ(x, E) ≤ ξε \ x ∈ W | ρ(x, ∂W ) ≤ ε , B = x ∈ Wr | ρ(x, ∂W ) ≤ ε . At any point z ∈ E, denote the line orthogonal to E by ez . Denote the Lebesgue measure on ez by Lebez , and the Lebesgue measure on E by LebE . Finally, let Az = A ∩ ez , Bz = B ∩ ez . See Figure 7. We can calculate the measure of A and B as Z Z Leb(A) = Lebez (Az ) dLebE (z) , Leb(B) = Lebez (Bz ) dLebE (z) . E

E

To get the statement of the lemma, it is clearly enough to see that Lebez (Az ) ≤ ξLebez (Bz )

for any

z ∈E.

(C.1)

To see this, let Cz be the interval of length ξε in ez which is just left of E. Clearly, Az ⊂ Cz and thus Lebez (Az ) ≤ ξε.

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Figure 7. Notation in the proof of Sublemma C.1. If Cz is not entirely a subset of W , then either it is entirely outside W , or it contains a point of ∂W . In both cases, Az is empty and (C.1) is trivial. So, suppose Cz ⊂ W . Let v be the nearest point of ∂W ∩ ez on the right of E, and let Dz be the interval of length ε in ez just left of v. If d := ρ(v, z) ≥ ε, then Dz ⊂ Bz , so Lebez (Bz ) ≥ ε and (C.1) is trivial again. If not, then the intersection of Cz and Dz belongs to neither Az nor Bz , so the estimate still holds: Lebez (Az ) ≤ Lebez (Cz \ Dz ) ≤ max ξε − (ε − d), 0 = max ξd − (1 − ξ)(ε − d), 0 ≤ ξd and Lebez (Bz ) ≥ ρ(v, z) = d imply (C.1). The statement with Wl and Wr interchanged is the same (with other notation). Proof of Sublemma 4.10. The set on the left hand side can be decomposed as x ∈ Wl | ρ(x, ∂Wl ) ≤ ε = x ∈ Wl | ρ(x, ∂W ) ≤ ε ∪ x ∈ Wl | ρ(x, E) ≤ ε \ x ∈ W | ρ(x, ∂W ) ≤ ε .

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The measure of the second term can be estimated using Sublemma C.1 with ξ = 1. We get Leb x ∈ Wl | ρ(x, ∂Wl ) ≤ ε ≤ Leb x ∈ Wl | ρ(x, ∂W ) ≤ ε + Leb x ∈ Wr | ρ(x, ∂W ) ≤ ε , which is exactly what we need. The statement for Wr is the same.

Proof of Sublemma 4.11. The set on the left hand side can be decomposed into the parts to the left and right of E as x ∈ W | ρ(x, E) ≤ ξε \ x ∈ W | ρ(x, ∂W ) ≤ ε = x ∈ Wl | ρ(x, E) ≤ ξε \ x ∈ W | ρ(x, ∂W ) ≤ ε ∪ x ∈ Wr | ρ(x, E) ≤ ξε \ x ∈ W | ρ(x, ∂W ) ≤ ε . Both terms can be estimated using Sublemma C.1, and the result is Leb x ∈ W | ρ(x, E) ≤ ξε \ x ∈ W | ρ(x, ∂W ) ≤ ε ≤ ξLeb x ∈ Wr | ρ(x, ∂W ) < ε + ξLeb x ∈ Wl | ρ(x, ∂W ) < ε which is what we need.

Proof of Sublemma 4.15. We use the notation Hε = {x ∈ W | ρ(x, ∂W ) ≤ ε}, Hkε = {x ∈ W | ρ(x, ∂W ) ≤ kε}, Vε = Leb(Hε ), Vkε = Leb(Hkε ). We use the property of the Lebesgue measure that it can be obtained as the infimum of sums of volumes of spheres in a countable covering: (∞ ) ∞ X [ m m + Vε = inf Γm ri | Hε ⊂ Bri (yi ), yi ∈ R , ri ∈ R , i=1

i=1

where Br (y) denotes the sphere of radius r centered at y, and Γm is the volume of ∞ the m-dimensional δ > 0 there exist {yi }∞ i=1 and {ri }i=1 S∞unit sphere. So forPany ∞ m such that Hε ⊂ i=1 Bri (yi ) and Γm i=1 ri < Vε + δ. Now for every i, let xi be one of the points of ∂W which is the closest to yi . Such a point exists, since ∂W is compact, so the infimum defining ρ(yi , ∂W ) is obtained. Define zi = xi + k(yi − xi ) and Bi0 = Bkri (zi ). That is, Bi0 is obtained with the magnification of Bri (yi ) with a factor k, but with xi as the center of the magnification. See Figure 8 for S∞the notation. We claim that Hkε ⊂ i=1 Bi0 . This immediately implies the statement of the lemma, since it means that ∞ ∞ X X m m Vkε ≤ Γm (kri ) = k Γm (ri )m ≤ k m (Vε + δ) i=1

i=1

∞ by the choice of {yi }∞ i=1 and {ri }i=1 , and this holds for every δ > 0. To see the claim, choose any point c ∈ Hkε . Let a be one of the points of ∂W closest to c – again, such a point exists. Define b = a + c−a k . We can see that

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Figure 8. Notation for the proof of Sublemma 4.15. ρ(b, ∂W ) = ρ(b, a), because the existence of a point d ∈ ∂W with ρ(b, d) < ρ(b, a) would imply ρ(c, d) < ρ(c, a), which contradicts the choice of a. ) Notice that b ∈ Hε , because ρ(b, ∂W ) = ρ(b, a) = ρ(c,a) = ρ(c,∂W ≤ kε k k k = ε by the choice of c, and b ∈ W because c ∈ W and ρ(c, ∂W ) > ρ(c, b). This means that there is an i for which b ∈ Bri (yi ). We will show that for the same i, c ∈ Bi0 , and this completes the proof. To make the proof of c ∈ Bi0 transparent, we introduce the vectors e = xi − a, f = b − a and g = yi − xi . We will make use of the choice of xi and a through the inequalities ρ(yi , a) ≥ ρ(yi , xi ) ; ρ(c, xi ) ≥ ρ(c, a) . Our statement follows from ρ(zi , c) ≤ kρ(yi , b) . With the vectors introduced, the conditions can be written as |e + g| ≥ |g| ;

|kf − e| ≥ |kf |

and the statement becomes |e + kg − kf | ≤ k|e + g − f | . The conditions can be further rewritten as e2 + 2eg ≥ 0 ;

e2 − 2kef ≥ 0 ,

and the statement becomes (using k > 1) ke2 + e2 + 2keg − 2kef ≥ 0 . In this form, the statement is just the sum of k times the first condition and the second.

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References [1] D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems, Russian Mathematical Surveys 22 (1967), 103–167. [2] P. Bachurin, On the structure of the singularity manifolds of dispersing billiards, Arxiv preprint math.DS/0505620, 2005 - arxiv.org. [3] V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Arxiv preprint math.DS/0711.1960v1, 2007 - arxiv.org. [4] P. B´ alint, P. Bachurin and I. P. T´ oth, Local ergodicity of systems with growth properties including multi-dimensional dispersing billiards, to appear in Israel Journal of Mathematics; preprint available at http://www.renyi.hu/~bp/pub.html. [5] P. B´ alint, N. I. Chernov, D. Sz´ asz and I. P. T´ oth, Geometry of multi-dimensional dispersing billiards, Astérisque 286 (2003), 119–150. [6] P. B´ alint, N. I. Chernov, D. Sz´ asz and I. P. T´ oth, Multidimensional semi-dispersing billiards: Singularities and the fundamental theorem, Annales Henri Poincaré 3 (2002), 451 – 482. [7] L. Barreira and Ya. Pesin, Smooth ergodic theory and nonuniformly hyperbolic dynamics, with appendix by O. Sarig, Handbook of Dynamical Systems 1B, Elsevier (2006), 57–263. [8] N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete and Continuous Dynamical Systems 5 (1999), 425–448. [9] N. Chernov, Decay of correlations and dispersing billiards, Journal of Statistical Physics 94 (1999), 513–556. [10] N. Chernov and D. Dolgopyat, Brownian Brownian motion I., to appear in Memoirs AMS, available at http://www.math.umd.edu/~dmitry/brown11.pdf. [11] N. Chernov and R. Markarian, Chaotic billiards, Mathematical Surveys and Monographs 127 (2006) AMS, Providence, RI. [12] N. Chernov and L.-S. Young, Decay of correlations of Lorentz gases and hard balls, in Hard ball systems and the Lorentz gas, ed. Domokos Sz´ asz, Springer, Berlin (2000), 89–120. [13] A. Katok and J.-M.Strelcyn, Invariant manifolds, entropy and billiards; smooth maps with singularities, Lecture Notes in Mathematics 1222, Springer (1986). [14] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Wiley, 1963–1996. [15] A. Kr´ amli, N. Sim´ anyi and D. Sz´ asz, A “transversal” fundamental theorem for semidispersing billiards, Communications in Mathematical Physics 129 (1990), 535–560. [16] C. Liverani, Decay of correlations, Annals of Mathematics 142 (1995), 239–301. [17] C. Liverani and M. Wojtkowski, Ergodicity in Hamiltonian systems, Dynamics Reported (New Series) 4 (1995), 130–202. [18] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal of Mathematics 116 (2000), 223–248. [19] Ya. G. Sinai and N. Chernov, Ergodic properties of certain systems of 2–D discs and 3–D balls, Russian Mathematical Surveys (3) 42 (1987), 181–201. [20] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Mathematics 147 (1998), 585–650.

Vol. 9 (2008)


Péter B´ alint Mathematical Institute of the Technical University of Budapest Egry J´ ozsef u. 1 H-1111 Budapest Hungary e-mail: [email protected] Imre Péter T´ oth MTA-BME Stochastics Research Group Egry J´ ozsef u. 1 H-1111 Budapest Hungary e-mail: [email protected] Communicated by Viviane Baladi. Submitted: November 10, 2007. Accepted: May 23, 2008.

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A Multiplicity Result for the p-Laplacian Involving a Parameter Friedemann Brock∗ , Leonelo Iturriaga† and Pedro Ubilla‡ Abstract. We study existence and multiplicity of positive solutions for the following problem −Δp u = λf (x, u) in Ω , u=0 on ∂Ω where λ is a positive parameter, Ω is a bounded and smooth domain in RN , p ∈ (1, N ), f (x, t) behaves, for instance, like o(|t|p−1 ) near 0 and +∞, and satisfies some further properties. In particular, our assumptions allow us to consider both positive and sign changing nonlinearitites f , the latter describing logistic as well as reaction–diffusion processes. By using sub- and supersolutions and variational arguments, we prove that there exists a positive constant λ such that the above problem has at least two positive solutions for λ > λ, at least one positive solution for λ = λ and ole plays the fact that local minimizers no solution for λ < λ. An important rˆ of certain functionals in the C 1 -topology are also minimizers in W01,p (Ω). We give a short new proof of this known result.

1. Introduction and statement of the results During the last two decades the p-Laplacian operator, Δp , has received growing attention. This is due to the fact that it arises in various applications. For instance, in Fluid Mechanics, the shear stress τ and the velocity ∇u of certain fluids are related via an equation of the form τ (x) = a(x)∇p u(x), where ∇p u = |∇u|p−2 ∇u. Here p > 1 is an arbitrary real number. The case p = 2 corresponds to a Newtonian fluid, and models of Non–Newtonian fluids are given by p = 2. The equations of ∗

Supported by FONDECYT No 1050412 Partially supported by FONDECYT No 3060061, FONDAP Matem´ aticas aplicadas and Convenio de Desempe˜ no UTA-MECESUP 2 ‡ Supported by FONDECYT No 1040990 †

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F. Brock, L. Iturriaga and P. Ubilla


motion then involve div(a∇p u), which reduces to aΔp u = adiv∇p u, provided that a is a constant. Notice that the p-Laplacian also appears in the study of torsional creep (elastic case for p = 2, plastic case as p → ∞, see [23]), of flow through porous media (p = 32 , see [29]), or of glacial sliding (p ∈ (1, 43 ], see [26]). In this work, we will focus on the study the multiplicity of weak solutions of the problem −Δp u = λf (x, u) , u ≥ 0 in Ω (P )λ u=0 on ∂Ω , where Ω is a bounded domain in RN with C 2,β -boundary (β ∈ (0, 1)), and p ∈ (1, N ). Observe that if the nonlinearity f (x, t) behaves like o(|t|p−1 ) near zero and infinity, then via variational techniques it is not difficult to prove the existence of two positive solutions for λ large enough. Furthermore, under the same assumptions, a simple calculus involving the first eigenvalue of the p-Laplacian allows us to prove non–existence for λ small enough. Our goal is to specify the range of multiplicity for every λ ∈ (0, ∞). More precisely, we establish the existence of a positive constant λ such that the problem has at least two positive solutions for λ > λ, at least one positive solution for λ = λ and no positive solution for λ < λ. The proofs are based on variational arguments and the sub- and supersolutions technique. An important rˆ ole plays the fact that local minimizers of certain functionals in the C 1 -topology are also minimizers in W01,p (Ω). In the appendix we will give a short new proof of this known result. Our assumptions on the nonlinearity f will be the following: (H1 ) f : Ω × [0, +∞) −→ R is a measurable function and f (x, · ) is continuous, uniformly for a.e. x ∈ Ω, and |f (x, t)| ≤ C(1 + tr ) (H2 )

(H3 ) (H4 ) (H5 )

∀(x, t) ∈ Ω × [0, +∞) ,

(1.1)

for some numbers C > 0 and r ∈ [0, p∗ − 1), where p∗ = N p/(N − p). There exists a continuous nondecreasing function g : [0, +∞) → [0, +∞) satisfying g(0) = 0, and such that the mapping t −→ f (x, t) + g(t) is nondecreasing. lim t→0+ f (x, t)t1−p = 0, uniformly for every x ∈ Ω. lim supt→+∞ f (x, t)t1−p ≤ 0, uniformly for every x ∈ Ω. There holds either (i) f (x, t) > 0 for every (x, t) ∈ Ω × (0, +∞); or (ii) there exists δ1 > 0 and a ball Bε0 (x0 ) ⊂ Ω, (ε0 > 0, x0 ∈ Ω), such t that F (x, t) > 0 on Bε0 (x0 ) × (0, δ1 ], where F (x, t) := 0 f (x, s) ds, and there exists q > p − 1 such that the mapping t −→ t−q f (x, t) is strictly decreasing on (0, +∞) for a.e. x ∈ Ω.

Let us comment on the hypotheses above. A model for assumption (H1 ) is, for instance, f (x, u) = a(x)g(u), where a ∈ L∞ (Ω) and g ∈ C(R, R) is subcritical. (H1 ) and (H2 ) are standard in order to apply the sub- and super–solutions method (see [3], and Lemma 2.1 below).

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A Multiplicity Result for the p-Laplacian

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¯ > 0 such that The assumptions (H3 ) and (H4 ) ensure that there is a number λ ¯ and has no solution for 0 < λ < λ. ¯ Problem (P )λ has a positive solution for λ ≥ λ, Notice that (H3 ) is a natural condition to obtain two positive solutions, since there are well known uniqueness results in cases that the limit in (H3 ) is positive, see for instance [9, 19, 22]. For the existence of branches of positive solutions for asymptotically equi–diffusive problems, see Ambrosetti, Garc´ıa and Peral [3]. In particular, [3] deals with the equation −Δp u = λf (u) , u ∈ W01,p (Ω) where f satisfies, for instance, limt→+∞ f (x, t)t1−p = c > 0. We emphasize that this case is not considered here (compare with Hypothesis (H4 )). Finally, we use assumption (H5 ) in obtaining appropriate subsolutions of our problem (P )λ (see the proofs of the Lemmata 3.2 and 3.4 below). (H5 ) also ensures the existence of a solution if λ is large enough (see the proof of Lemma 3.3). Additionally, we suppose the following condition on the nonlinearity f which ¯ allows us to obtain a second solution of our Problem (P )λ for λ ≥ λ. (H6 ) There exist numbers c0 ≥ 0, δ0 > 0 such that the mapping t −→ f (x, t) + c0 tp−1 is nondecreasing for (x, t) ∈ Ω × (0, δ0 ]. Our main result is the following Theorem 1.1. Assume that f satisfies the conditions (H1 )–(H6 ). Then there exists a positive constant λ such that Problem (P )λ has at least two positive solutions for λ > λ, at least one positive solution for λ = λ and no positive solution for λ < λ. Let us first give a few applications of our main result. Let p − 1 < q, r > 0, and a1 , a2 ∈ L∞ (Ω), with a1 nonnegative and ess infΩ a2 > 0. Then the conclusions of Theorem 1.1 hold in any one of the following cases: 1. f1 (x, t) = a1 (x)tq (1 − a2 (x)tr ). Notice in this case, in order to apply Theorem 1.1, one first needs to show the equivalence with an appropriate truncated problem, (see Section 4). tq 2. f2 (x, t) = a2 (x) 1+t r . where q < p − 1 + r. 3. f3 (x, t) = a2 (x) ln(1 + tq ). Observe that particular cases of f1 have been considered by many authors, since this type of nonlinearity models, for instance, reaction–diffusion processes or logistic problems in population dynamics, (see [25, 30]). For example, when a1 (x) = a2 (x) = 1, a multiplicity result was obtained by Takeuchi in [31] under the restriction p > 2. Later Dong and Cheng [11] proved the same result for all p > 1. We notice that the Laplacian case was studied by Rabinowitz by combining critical point theory with the Leray–Schauder degree [28]. For more information about type f1 , see [10, 17, 20, 22, 30, 32, 33] and the cited references therein. We also observe that the functions f1 , f2 , f3 above may be written as f (x, t) = tq h(x, u) where the mapping t −→ h(x, t) is strictly decreasing on (0, +∞) for a.e. x ∈ Ω. Existence results for nonlinearities of this type have been analyzed by Ca˜ nada et

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al [6]. However, in contrast to the present paper, the authors did not study the multiplicity of solutions. On the other hand, it is interesting to compare our Problem (P )λ with a dual case, as it has been investigated in the classical paper of Ambrosetti, Brezis and Cerami [2]. The model equation with a concave-convex nonlinearity considered in [2] is −Δu = λuq + ur , u ∈ H01 (Ω), where 0 < q < 1 < r ≤ 2∗ . The authors ˜ > 0 such that the problem above has at least two proved that there exists λ ˜ at least one positive solution for λ = λ ˜ and no positive solutions for 0 < λ < λ, ˜ positive solution for λ > λ. Further results on this type of nonlinearities are given in [12,13]. The analogous situation for the p-Laplacian has been studied by Garc´ıa and Peral [16]. Finally note that in Ambrosetti, Garc´ıa and Peral also established in [3] the existence of sign-changing solutions. There are some recent papers that deal with the compactness of the branches of solutions of similar equations which are also relevant in the context of this paper: Cabré and Sanchon [5] considered nonnegative solutions of −Δp u = f (x, u). Assuming that f (x, u) grows like (1 + u)m , where 0 < m < m∗ and m∗ is some critical value, and introducing the notion of semi-stability, they proved that certain minimizers of the associated energy functional are semi-stable and bounded. Castorina et al [7] studied minimal solution branches (uλ , λ) of the equation −Δp u = λh(x)f (u), with 0 < u < 1, and 0 < λ < λ∗ , where h is positive and H¨ older continuous, f behaves like (1 − u)−m near u = 1, and λ∗ is some critical value. They showed that the mapping λ → uλ is non-decreasing, and composed by semistable solutions. Our work is organized as follows: In Section 2 we give some definitions and basics facts which we will be used throughout the article. The proof of Theorem 1.1 is presented in Section 3. Then we extend our result to even more general nonlinearities in Section 4. Finally we present a new proof of a well–known result, Lemma 2.2, in an Appendix, Section 5.

2. Preliminaries. Sub- and supersolutions Let λ1 (Ω) denote the first eigenvalue of the Dirichlet p-Laplacian. For convenience we extend the function f (x, t) for negative values of t, f (x, t) if (x, t) ∈ Ω × [0, +∞) , f (x, t) = 0 if (x, t) ∈ Ω × (−∞, 0) and we will work with the problem −Δp u = λf(x, u) (P)λ u=0 instead of (P )λ .

in Ω on ∂Ω ,

(2.1)

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Definition 2.1. A function u ∈ W 1,p (Ω) ∩ L∞ (Ω) is said to be a subsolution of (P)λ if ⎧ ⎨ Ω |∇u|p−2 ∇u · ∇φ dx ≤ λ Ω f(x, u)φ dx for every φ ∈ W01,p (Ω) with φ ≥ 0, and ⎩ u ≤ 0 on ∂Ω . (A function v ∈ W 1,p (Ω) is said to be less than or equal to w ∈ W 1,p (Ω) on ∂Ω when max{0; v − w} ∈ W01,p (Ω)). Furthermore, a function u ∈ W 1,p (Ω) ∩ L∞ (Ω) is said to be a supersolution of (P )λ if ⎧ ⎨ Ω |∇u|p−2 ∇u · ∇φ dx ≥ λ Ω f(x, u)φ dx for every φ ∈ W01,p (Ω) with φ ≥ 0, and ⎩ u ≥ 0 on ∂Ω . Finally, a function u ∈ W01,p (Ω) ∩ L∞ (Ω) which is both a sub- and a supersolution, is called a solution of problem (P)λ . Remark 2.1. First notice that every solution of (P)λ is nonnegative. To see this, use u− := max{0; −u} as a test function in (2.1). Integration by parts then leads to p |∇u− | dx = −λ f(x, u)u− dx ≤ 0 , 0≤ Ω

Ω

which implies that u− = 0 a.e. on Ω. Thus, u is a solution of problem (P)λ iff u is also a solution of problem (P )λ . By the Strong Maximum Principle we then obtain that every nontrivial solution of (P)λ is positive in Ω. Conditions (H1 ) and (H4 ) imply that there is a constant C > 0 which depends only on f and λ such that every solution u of problem (P)λ satisfies

u C 1,α (Ω) ≤ C .

(2.2)

Indeed, by condition (H4 ) we have that there is a constant cλ which depends only on λ such that λf(x, t) ≤ cλ + (1/2)λ1 (Ω)tp−1 ∀ (x, t) ∈ Ω × [0, +∞) . This inequality implies that all solutions of (P)λ are uniformly bounded in the W01,p (Ω)-norm. It is well-known that this implies that all solutions (P)λ are uniformly bounded in the L∞ -norm. By using the regularity results of Guedda and Veron [18] and Lieberman [24] we then obtain the estimate (2.2). Finally, condition (H3 ) implies that if u is nontrivial, then it is positive in Ω and satisfies ∂u on ∂Ω , (ν : exterior normal ) , (2.3) 0> ∂ν see [34]. The following auxiliary result is well-known (see e.g. [1]) and will be basic in our approach.

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Lemma 2.1. Consider Problem (P)λ under the hypotheses (H1 )–(H2 ). Let u, u ∈ W 1,p (Ω) ∩ L∞ (Ω) be, respectively, a subsolution and a supersolution of Problem (P)λ , with u(x) ≤ u(x) a.e. in Ω. Then there exists a minimal (and, respectively, a maximal) weak solution u∗ (resp. u∗ ) for Problem (P)λ in the “interval”

[u, u] = u ∈ L∞ (Ω) : u(x) ≤ u(x) ≤ u(x) a.e. in Ω . In particular, every weak solution u ∈ [u, u] of (P)λ also satisfies u∗ (x) ≤ u(x) ≤ u∗ (x) for a.e. x ∈ Ω. The following lemma is crucial in showing multiplicity of solutions. It has been shown in the case p = 2 by Brezis and Nirenberg in [4], in the case p > 2 by Guo et al. in [21], and in the general case by Garc´ıa Azorero et al. in [15]. The idea consists in analyzing a penalized minimization problem. The proofs given in [15] and [21] are quite technical since the constraint involves the gradient of the admissible functions. We will present a short new proof in the Appendix. Notice that our constraint merely involves a certain Lq -norm. Lemma 2.2. Let f : Ω × R → R be a Caratheodory function which satisfies |f (x, t)| ≤ C(1 + |t|r )

∀ (x, t) ∈ Ω × R ,

for some numbers C > 0 and r ∈ [0, p∗ − 1), and assume that u ∈ weak solution of −Δp u = f (x, u) in Ω . u=0 on ∂Ω

(2.4) W01,p (Ω)

Let I denote the energy functional associated with (2.5), that is

1 |∇v|p − F (x, v) dx , v ∈ W01,p (Ω) . I(v) := p Ω

is a

(2.5)

(2.6)

Assume finally that u is a local minimizer of I in C01 (Ω), that is, there exists a number ε > 0 such that I(v) ≥ I(u) for every v ∈ C01 (Ω) satisfying v − u C 1 (Ω) < ε. Then u is also a local minimizer in W01,p (Ω).

3. Proof of the main result From the hypotheses (H1 ), (H3 ) and (H4 ) it follows that there is a number λ0 > 0 such that λ1 (Ω)tp−1 ≥ λ0 f(x, t) ∀(x, t) ∈ Ω × [0, +∞) . (3.1) where λ1 (Ω) denotes the first eigenvalue of the Dirichlet p-Laplacian. Lemma 3.1. Problem (P)λ does not have any positive solution for λ < λ0 , where λ0 is given by (3.1).

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Proof. Suppose that (P)λ admits a positive solution uλ for some λ < λ0 . Using uλ as a test function in (P)λ we then obtain p |∇uλ | dx = λ upλ dx , f (x, uλ )uλ dx < λ1 (Ω) Ω

Ω

Ω

which contradicts to the variational characterization of λ1 (Ω).

Lemma 3.2. Suppose that for some λ > 0, Problem (P)λ admits a positive solution uλ . Then for every λ > λ , Problem (P)λ has at least one positive solution. Proof. We first construct a subsolution u. 1) Assume f satisfies (H5 ), (i). Since uλ is a solution of (P)λ , we have that −Δp uλ = λ f(x, uλ ) ≤ λf(x, uλ )

in

Ω,

that is uλ =: u is a subsolution of Problem (P)λ . 2) Assume that f satisfies (H5 ), (ii). Then there exists a number σ ∈ (0, 1) such that λ = λσ q−p+1 . Hence −Δp (σuλ ) = σ p−1 λ f(x, uλ ) < λf(x, σuλ )

in

Ω,

(3.2)

that is σuλ =: u is a subsolution. A supersolution is constructed as follows: Let e be the solution of the following problem, −Δp e = 1 in Ω (3.3) e=0 on ∂Ω , and set e0 := sup{e(x) : x ∈ Ω}. From (H4 ), and since e, u ∈ C 1 (Ω) and ∂e/∂ν < 0 on ∂Ω, we have for every k > 0 and large enough, ke(x) > u(x)

in

∂u ∂(ke) < on ∂Ω , ∂ν ∂ν and λ(e0 )p−1 f(x, t) < tp−1 ,

Ω,

(3.4)

λλ1 (Ω) < k (p−1)/2 λ0 (3.5) √ for all (x, t) ∈ Ω × ( k, +∞), where λ0 is the number in (3.1). Then the inequalities (3.1) and (3.5) imply λf(x, kt) < k p−1 Hence

∀(x, t) ∈ Ω × [0, e0 ] .

−Δp ke(x) = k p−1 > λf x, ke(x)

in

(3.6)

Ω,

(3.7)

that is, ke =: u is a supersolution of problem (P)λ . Finally, using Lemma 2.1 there exists a positive solution u of Problem (P)λ satisfying u ≤ u ≤ u.

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In the following we will work with the energy functional associated to problem (P)λ , that is

1 p |∇v| − λF (x, v) dx , v ∈ W01,p (Ω) , (3.8) Iλ (v) := p Ω t where F(x, t) := 0 f(x, s) ds, ((x, t) ∈ Ω × R). Lemma 3.3. Let λ := inf{λ : (P)λ has a positive solution}. Then 0 < λ < +∞, and (P)λ has a positive solution for every λ > λ and no positive solution for 0 < λ < λ. Proof. By our assumptions, Iλ is differentiable, bounded from below, and coercive. Hence there is a global minimizer of Iλ on W01,p (Ω) which is solution of (P)λ . In view of condition (H5 ), Iλ attains negative values if λ is large enough. Hence we have that Iλ (uλ ) < 0. Using the Lemmata 3.1 and 3.2 this implies that there is a number λ ∈ (0, +∞) such that problem (P)λ has a solution if λ > λ, and no solution if λ < λ. Lemma 3.4. Let λ > λ and suppose that Problem (P)λ has a unique positive solution uλ . Then uλ is a local minimizer of Iλ in C01 (Ω). Proof. The idea is to construct a sub- and a supersolution which are strictly separated from the solution uλ . Let λ < λ < λ, and define u and u as in the proof of Lemma 3.2. Since uλ ∈ C 1 (Ω), we may add the requirement that the function u = ke in (3.7) satisfies u > uλ ∂uλ ∂u > ∂ν ∂ν

in

Ω,

and

on ∂Ω .

For the construction of a subsolution u we split into two cases: 1) Assume (H5 ), (i). Then u = uλ ≤ uλ in Ω. Set

Ω−r := x ∈ Ω : dist (x, ∂Ω) > r ,

(r > 0) ,

and notice that Ω−r is a C 2 -domain for small enough r. Since uλ , uλ ∈ C01 (Ω), there is a number r > 0 such that |∇uλ |, |∇uλ | > 0 on Ω\Ω−2r and uλ (x), uλ (x) ∈ (0, δ0 ) on Ω \ Ω−2r . Together with assumption (H6 ) this in particular implies −Δp uλ + λc0 (uλ )p−1 = λ f(x, uλ ) + λc0 (uλ )p−1 < λ f(x, uλ ) + c0 (uλ )p−1 ≤ λ f(x, uλ ) + c0 (uλ )p−1 = −Δp uλ + λc0 (uλ )p−1

in

Ω \ Ω−2r ,

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in the distributional sense. The Strong Comparison Principle – see for instance Proposition 5.1 of [14], or Proposition 3 of [8] - then tells us that uλ < uλ

in

Ω \ Ω−2r ,

and

(3.9)

∂uλ ∂u > on ∂Ω . (3.10) ∂ν ∂ν Hence there is a number ε0 > 0 such that uλ ≥ ε0 + u on ∂Ω−r . This implies that if 0 < ε < ε0 then (uλ − uλ + ε)+ ∈ W01,p (Ω−r ). Hence 0≤ |∇uλ |p−2 ∇uλ − |∇uλ |p−2 ∇uλ · ∇(uλ − uλ + ε)+ dx Ω −r λ f(x, uλ ) − λf(x, uλ ) (u − uλ + ε)+ dx , (3.11) = 0>

λ

Ω−r

for these ε, where the inequality in (3.11) follows from p−2 |x| x − |y|p−2 y · (x − y) ≥ 0 ∀x, y ∈ RN .

(3.12)

Since λ < λ, uλ ≤ uλ and f(x, uλ (x)) > 0 in Ω−r , and since f is continuous in the second variable, there exists a number ε1 ∈ (0, ε0 ) such that

x ∈ Ω−r : uλ (x) + ε1 > uλ (x) . λ f x, uλ (x) − λf x, uλ (x) < 0 on In view of (3.11) this implies that u + ε1 ≤ uλ in Ω−r . 2) Assume (H5 ), (ii). Then u = σuλ is a subsolution of (P)λ , where λ = λσ q−p+1 , and σuλ ≤ uλ in Ω. Choosing r > 0 as in case 1) and taking into account that σ q f(x, uλ ) < f(x, σuλ ) in Ω, an analogous calculus shows that σuλ < uλ in Ω and that (3.10) holds with uλ replaced by σuλ . Now setting

A := v ∈ C01 (Ω) : u ≤ v ≤ u , in any of the above cases, we find that uλ is an interior point of A with respect to the C 1 -topology. It is well-known that this implies that uλ is a local minimizer of Iλ in C01 (Ω) (see [15], proof of Theorem 5.2). For the convenience of the reader we repeat the argument below. Let ⎧ ⎪ ⎨f (x, u(x)) if s < u(x) f (x, s) = f(x, s) if u(x) ≤ s ≤ u(x) , ⎪ ⎩ f (x, u(x)) if u(x) < s t F (x, t) := 0 f (x, s) ds, ((x, t) ∈ Ω × R), and define a functional I λ analogously as Iλ with F replaced by F . From our assumptions on the nonlinearity f and standard arguments it follows that I λ has a global minimizer u0 ∈ W01,p (Ω). Clearly u0 is a weak solution of −Δp u0 = λf (x, u0 ) in Ω , (3.13) on ∂Ω . u0 = 0

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Moreover, well-known regularity results (see e.g. [18]) show that u0 ∈ C01 (Ω). On the other hand, using (u − u0 )+ as a test function for (3.13) and (P)λ , we obtain, using (3.12), 0≤ (|∇u|p−2 ∇u − |∇u0 |p−2 ∇u0 ) · (∇u − ∇u0 ) dx {u>u0 } ≤λ f(x, u) − f (x, u0 ) (u − u0 ) = 0 , {u>u0 }

which implies that u0 ≥ u. Analogously one shows that u0 ≤ u. Hence we have that u0 ∈ A, which implies uλ = u0 , by our hypothesis. Since uλ is an interior point of A, there exists a number ε > 0 such that one has u ∈ A for every u ∈ C01 (Ω) satisfying uλ − u C 1 (Ω) < ε. Furthermore, we find that for every u ∈ A, F(x, u) − F (x, u) dx Iλ (u) − I λ (u) = −λ Ω u = −λ f(x, s) − f (x, s) ds dx Ω

0

which is a constant independent of u. Hence uλ is a local minimizer of Iλ in C01 (Ω). Proof of Theorem 1.1. We argue by contradiction, i.e., we suppose that there is a number λ > λ such that Problem (P)λ has a unique positive solution uλ . Then the Lemmata 3.4 and 2.2 imply that the solution uλ is a local minimizer of Iλ in W01,p (Ω). Furthermore, another local minimizer of Iλ on W01,p (Ω) is given by 0. Since the functional Iλ is coercive, it follows that Iλ satisfies the (PS)–condition. Then applying an extended version of the Mountain Pass Theorem due to Pucci and Serrin [27], we obtain the existence of a third critical point of Iλ , which contradicts our assumption. Finally we claim that problem (P)λ has a positive solution. By assumption (H3 ) there is a number t0 > 0 such that λ1 (Ω) p−1 t ∀(x, t) ∈ Ω × [0, t0 ] . (3.14) (λ + 1)f(x, t) ≤ 2 Let {λn } a strictly decreasing sequence with limn→∞ λn = λ, λn ∈ (λ, λ + 1), and let uλn be a positive solution of (P)λn , n = 1, 2, . . .. We will show that (3.14) implies that

uλn ∞ > t0 n = 1, 2, . . . . (3.15) In fact, assume that uλn ≤ t0 in Ω. Then condition (3.14) gives λ1 (Ω) |∇uλn |p dx = λn upλn dx , f(x, uλn )uλn dx ≤ 2 Ω Ω Ω which contradicts the variational characterization of λ1 (Ω). Hence we obtain (3.15). As in Remark 2.1 we then have that the functions uλn are equibounded in C 1,α (Ω). Thus by Arzela–Ascoli’s Theorem there is a subsequence of {uλn } which converges

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in C01 (Ω) to a function u ∈ C01 (Ω) which can be identified easily as a solution of problem (P)λ , and (3.15) shows that

u ∞ ≥ t0 .

(3.16)

This proves the claim.

4. Concluding remarks We conclude our work with some examples and comments of our main result. Assume first that f satisfies (H1 )–(H4 ), (H5 ), (ii), (H6 ), except condition (1.1), but instead suppose that there exists a number t0 > 0, such that f (x, t0 ) ≤ 0 for every x ∈ Ω. Then any solution of the modified problem, −Δu = λf(x, u) in Ω , (Pλ ) , (4.1) u=0 on ∂Ω where f is the truncated function f(x, t) =

f(x, t) f(x, t0 )

if t ≤ t0 if t > t0

satisfies u(x) ≤ t0 and is also a solution of (P )λ . Notice that the truncated function f satisfies (1.1). This means that we obtain at least two solutions for large enough λ even in situations when f has supercritical growth at infinity. However, it might be a difficult question, whether or not there are solutions of the original problem which are not solutions of the truncated problem in such a case. We illustrate this be means of a simple example: Let f (x, t) = a(x)tq |1 − t| , where p−1 < q < p∗ −2 and a ∈ L∞ (Ω). Then, it is easy to see that Problem (P )λ has a Mountain Pass solution for any λ > 0. By using the truncation argument at t0 = 1 we obtain the existence of a positive constant λ such that the Problem (P)λ has two solutions for λ > λ and no solution for 0 < λ < λ. This means in particular that the Mountain Pass solution uλ of the Problem (P )λ satisfies

uλ L∞ (Ω) > 1 for 0 < λ < λ. On the other hand, the nonexistence of solutions for the original problem for λ small enough can be established, if there exists a positive constant t0 such that f (x, t) ≤ 0 for all t ≥ t0 . Indeed, applying the Strong Maximum Principle, it is easy to see that any solution of the original problem is bounded from above by t0 , which means that the original and the truncated problem are equivalent in such a case. Finally we mention that we may also deal with nonlinearities which involve nonpositive perturbations:

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In fact, let f and k be functions satisfying conditions (H1 )–(H4 ), (H6 ), with f positive and k nonpositive on Ω × (0, ∞), and consider the problem −Δp u = λf (x, u) + k(x, u) in Ω , (K)λ . (4.2) u=0 on ∂Ω By slightly modifying the proofs of Lemma 3.2 and Lemma 3.4 in case 1), it easy to see that the conclusions of Theorem 1.1 hold for the family of problems (K)λ , (λ > 0). We emphasize that in the context of multiplicity, nonlinearities as in (4.2) have been studied only in the case that both f and k are homogeneous in the second variable, and more precisely, for f (x, t) = a(x)tq , k(x, t) = −b(x)tr , (a, b ∈ L∞ (Ω), a, b > 0, r > q > p − 1), (see [10]). However, a simple linear transformation shows that the families (P )λ and (K)λ are equivalent in such a case.

5. Appendix Proof of Lemma 2.2. First observe that due to the growth conditions on f and the smoothness of Ω we have that u ∈ C 1,α (Ω) for some α ∈ (0, 1), see [18]. Let q ∈ (r, p∗ − 1), where r is the number from (2.4), 1 |w(x) − u(x)|q+1 dx, w ∈ W01,p (Ω) , K(w) := q+1 Ω and

Sε := v ∈ W01,p (Ω) : K(v) ≤ ε ,

(ε > 0) .

Assume that the conclusion of Lemma 2.2 is not true. Then there exists for every ε ∈ (0, 1] a function vε ∈ Sε such that I(vε ) < I(u). Moreover, we may assume w.l.o.g., that vε is a global minimizer of I in Sε . In the following, let C be a generic positive constant which may vary from line to line, and is independent of ε, and let · , · denote the duality product between W01,p (Ω) and its dual space. We consider two cases. 1) Let K(vε ) < ε. Then vε is also a local minimizer of I in W01,p (Ω). Hence vε is a solution of (2.5), which implies that

vε C 1,α (Ω) ≤ C ,

(5.1)

where α ∈ (0, 1), and the constant C depends only on Ω, u L∞ and q. 2) Let K(vε ) = ε. Then there exists a number με ∈ R - a Lagrangean multiplier such that I (vε ) = με K (vε ), i.e., vε is a weak W01,p -solution of −Δp vε = γ(με , x, vε ) in Ω , (5.2) u=0 on ∂Ω , where γ is defined by

γ(s, x, t) := f (x, t) + s|t − u(x)|q−1 t − u(x) ,

(s, x, t) ∈ R × Ω × R .

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First assume that με > 0. Then there exists h ∈ W01,p (Ω) such that I (vε ), h < 0 and K (vε ), h < 0, which implies K (vε ), h = μ−1 ε I (vε ), h < 0. Using Taylor’s theorem, this means that there exists a number τ0 (= τ0 (ε)) > 0 such that K(vε + τ h) < K(vε ) and also I(vε + τ h) < I(vε ) for every τ ∈ (0, τ0 ). The first of these inequalities implies that (vε + τ h) ∈ Sε for these τ . Hence vε is not a global minimizer of I in Sε , a contradiction. It follows that με ≤ 0. First suppose με ∈ [−1, 0]. Since u ∈ L∞ (Ω), we have in view of (2.4), |γ(s, x, t)| ≤ C(1 + |t|q ) ∀(s, x, t) ∈ [−1, 0] × Ω × R where C is a positive constant depending only on u ∞ , q and Ω. Now, ε ≤ 1 implies that vε Lq+1 ≤ C( u ∞ , Ω, q). Hence, this inequality together with the equation (5.2) give us a W01,p (Ω)-estimate depending only on u ∞ , q and Ω. Thus vε is bounded in L∞ (Ω)-norm by a constant which does not depend on ε. Therefore vε is bounded in C 1,α (Ω)-norm by a constant which does not dependent on ε (see [24]). This implies (5.1) in this case. Suppose finally that με ≤ −1. Since there exists a number M > 0, which is independent on ε, such that γ(s, x, t) < 0

∀(s, x, t) ∈ (−∞, −1] × Ω × (M, +∞) ,

γ(s, x, t) > 0

∀(s, x, t) ∈ (−∞, −1] × Ω × (−∞, −M ) ,

and

the Maximum Principle tells us that |vε (x)| ≤ M in Ω. Using (vε − u)|vε − u|β−1 , with β ≥ 1, as a test function in (2.5) and (5.2) we obtain, 0 ≤ β (|∇vε |p−2 ∇vε − |∇u|p−2 ∇u) · ∇(vε − u)|vε − u|β−1 dx Ω f (x, vε ) − f (x, u) (vε − u) |vε − u|β−1 dx + με |vε − u|β+q dx . = Ω

Ω

In view of the bounds for u and vε and H¨ older’s inequality this implies −με vε − u qLβ+q (Ω) ≤ C|Ω|q/(q+β) , where C does not depend on β and ε. Passing to the limit β → +∞ this leads to −με vε − u qL∞ (Ω) ≤ C . Hence we have that

γ με , x, vε (x) ≤ C

in Ω ,

from which we again obtain (5.1). Thus we have shown that the functions vε , (0 < ε ≤ 1), are equibounded in C 1,α (Ω). Using Ascoli–Arzela’s Theorem we find a sequence εn 0 such that vεn → u

in C 1 (Ω) .

Since I(vεn ) < I(u), we then have that u is not a local minimizer of I in C01 (Ω), a contradiction.

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Acknowledgements We would like to thank the referees for carefully reading this paper and suggesting many useful comments. Part of this work was done while the second author was visiting the IMECC–UNICAMP/BRAZIL. The author thanks Professor Djairo de Figueiredo and all the faculty and staff of IMECC–UNICAMP for their kind hospitality.

References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), no. 4, 620–709. [2] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, Journal of Functional Anal. 122, (1994), no. 2, 519–543. [3] A. Ambrosetti, J. Garc´ıa Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations. J. Funct. Anal. 137 (1996), no. 1, 219–242. [4] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C.R.A.S. Paris 317 (1993), 465–472. [5] X. Cabré, M. Sanchon, Semi-stable and extremal solutions of reaction equations involving the p-Laplacian. Commun. Pure Appl. Anal. 6 (2007), no. 1, 43–67. [6] A. Ca˜ nada, P. Drabek, J. L. Gomez, Existence of positive solutions for some problems with nonlinear diffusion, Transactions of the AMS 349 (1997), no. 10, 4231–4249. [7] D. Castorina, P. Esposito, B. Sciunzi, Degenerate elliptic equations with singular nonlinearities. Calc. Var. PDE, to appear. [8] M. Cuesta, P. Tak´ aˇc, A strong comparison principle for the Dirichlet p-Laplacian. Reaction diffusion systems (Trieste, 1995), 79–87, Lecture Notes in Pure and Appl. Math., 194, Dekker, New York, 1998. [9] J. D´ıaz, J. Saa, Existence et unicité de solutions positives pour certaines ´ equations elliptiques quasilinéaires. (French) [Existence and uniqueness of positive solutions of some quasilinear elliptic equations]. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 12, 521–524. [10] W. Dong, A priori estimates and existence of positive solutions for a quasilinear elliptic equation. J. London Math. Soc. 72 (2005), no. 2, 645–662. [11] W. Dong, J. T. Chen, Existence and multiplicity results for a degenerate elliptic equation, Acta Mathematica Sinica 22 (2006), no. 3, 665–670. [12] D. G. de Figueiredo, J. P. Gossez, P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 199 (2003), no. 2, 452–467. [13] D. G. de Figueiredo, J. P. Gossez, P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity. J. Eur. Math. Soc. 8 (2006), no. 2, 269–286. [14] J. Fleckinger, P. Tak´ aˇc, Uniqueness of positive solutions for nonlinear cooperative systems with the p-Laplacian. Indiana Univ. Math. J. 43, no. 4, 1227–1253. [15] J. P. Garc´ıa Azorero, J. Manfredi, I. Peral, Sobolev versus H¨ older local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2 (2000), no. 3, 385–404.

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[16] J. Garc´ıa Azorero, I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian: Nonlinear Eigenvalues. Commun. in PDE 12 (1987), 1389–1430. [17] J. Garc´ıa Meli´ an, J. Sabina de Lis, Uniqueness to quasilinear problems for the pLaplacian in radially symmetric domains. Nonlinear Anal. 43 (2001), no. 7, Ser. A: Theory Methods, 803–835. [18] M. Guedda, L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13 (1989), no. 8, 879–902. [19] Z. Guo, J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 1, 189–198. [20] Z. Guo, H. Zhang, On the global structure of the set of positive solutions for some quasilinear elliptic boundary value problems. Nonlinear Anal. 46 (2001), no. 7, Ser. A: Theory Methods, 1021–1037. [21] Z. Guo, Z. Zhang, W 1,p versus C 1 local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286 (2003), 32–50. [22] S. Kamin, L. Véron, Flat core properties associated to the p-Laplace operator. Proc. Amer. Math. Soc. 118 (1993), no. 4, 1079–1085. [23] B. Kawohl, On a family of torsional creep problems. J. Reine Angew. Math. 410 (1990), 1–22. [24] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12 (1988), 1203–1219. [25] J. D. Logan, An introduction to Nonlinear Partial Differential Equations, J. Wiley & Sons, N.Y. (1994). [26] M.-C. Pélissier, M. L. Reynaud, Etude d’un mod` ele mathématique d’écoulement de glacier. C. R. Acad. Sci. Paris Sér. I Math. 279 (1974), 531–534. [27] P. Pucci, J. Serrin, Extensions of the mountain pass theorem. J. Funct. Anal. 59 (1984), no. 2 , 185–210. [28] P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 23 (1973/74), 173–186. [29] R. E. Showalter, N. J. Walkington, Diffusion of fluid in a fissured medium with microstructure. SIAM J. Math. Anal. 22 (1991), 1702–1722. [30] S. Takeuchi, Y. Yamada, Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian. Nonlinear Anal. 42 (2000), no. 1, Ser. A: Theory Methods, 41–61. [31] S. Takeuchi, Positive solutions of a degenerate elliptic equation with logistic reaction. Proc. Amer. Math. Soc. 129 (2001), no. 2, 433–441. [32] S. Takeuchi, Multiplicity result for a degenerate elliptic equation with logistic reaction. J. Differential Equations 173 (2001), no. 1, 138–144. [33] S. Takeuchi, Stationary profiles of degenerate problems with inhomogeneous saturation values. Nonlinear Anal. 63 (2005), e1009–e1016. [34] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), 191–202.

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Friedemann Brock American University of Beirut Department of Mathematics P.O. Box 11-0236 Beirut Lebanon e-mail: [email protected] Leonelo Iturriaga Instituto de Alta Investigaci´ on Universidad de Tarapac´ a Casilla 7 D Arica Chile e-mail: [email protected] Pedro Ubilla Departamento de Matem´ aticas y C. C. Universidad de Santiago de Chile Casilla 307 Correo 2 Santiago Chile e-mail: [email protected] Communicated by Rafael D. Benguria. Submitted: November 8, 2007. Accepted: May 15, 2008.




Third Derivative of the One-Electron Density at the Nucleus Søren Fournais, Maria Hoffmann-Ostenhof, and Thomas Østergaard Sørensen Abstract. We study electron densities of eigenfunctions of atomic Schr¨ odinger operators. We prove the existence of ρ (0), the third derivative of the spherically averaged atomic density ρ at the nucleus. For eigenfunctions with corresponding eigenvalue below the essential spectrum in any symmetry subspace we obtain the bound ρ (0) ≤ −(7/12)Z 3 ρ(0), where Z denotes the nuclear charge. This bound is optimal.

1. Introduction and results In a recent paper [5] the present authors (together with T. Hoffmann-Ostenhof (THO)) proved that electron densities of atomic and molecular eigenfunctions are real analytic away from the positions of the nuclei. Concerning questions of regularity of ρ it therefore remains to study the behaviour of ρ in the vicinity of the nuclei. A general (optimal) structure-result was obtained recently [2]. For more detailed information, two possible approaches are to study limits when approaching a nucleus under a fixed angle ω ∈ S2 , as was done in [2], and to study the spherical average of ρ (here denoted ρ ), which is mostly interesting for atoms. The existence of ρ (0), the first derivative of ρ at the nucleus, and the identity ρ (0) = −Z ρ(0) (see (1.12) below) follow immediately from Kato’s classical result [12] on the ‘Cusp Condition’ for the associated eigenfunction (see also [10, 15]). Two of the present authors proved (with THO) the existence of ρ (0), and, for densities corresponding to eigenvalues below the essential spectrum, a lower positive bound to ρ (0) in terms of ρ(0) in [9]. In the present paper we prove the existence of ρ (0) and derive a negative upper bound to it (see Theorem 1.2). We prove this bound for c 2008 by the authors. This article may be reproduced in its entirety for non-commercial purposes.

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eigenvalues below the essential spectrum in any symmetry subspace. In particular, it holds for the physical (fermionic) ground state (see Remark 1.4). A key role in the proof is played by the a priori estimate on 2nd order derivatives of eigenfunctions obtained in [6] (see also Remark 1.3 and Appendix B below). Furthermore our investigations lead to an improvement of the lower bound to ρ (0) (see Corollary 1.7). The bounds on ρ (0) and ρ (0) in terms of ρ(0) are optimal (see Remarks 1.5 and 1.8). We turn to the precise description of the problem. We consider a non-relativistic N -electron atom with a nucleus of charge Z fixed at the origin in R3 . The Hamiltonian describing the system is given by N 1 Z . (1.1) H = HN (Z) = −Δj − + |x | |x − xj | j i j=1 1≤i<j≤N

The positions of the N electrons are denoted by x = (x1 , x2 , . . . , xN ) ∈ R3N , where xj = (xj,1 , xj,2 , xj,3 ) denotes the position of the j’th electron in R3 , and Δj denotes the Laplacian with respect to xj . For shortness, we will sometimes write where Δ =

H = −Δ + V (x) ,

N j=1

(1.2)

Δj is the 3N -dimensional Laplacian, and

V (x) = VN,Z (x) =

N

−

j=1

Z + |xj |

1≤i<j≤N

1 |xi − xj |

(1.3)

is the complete (many-body) potential. With ∇j the gradient with respect to xj , ∇ = (∇1 , . . . , ∇N ) will denote the gradient with respect to x. It is a standard fact (see e.g. Kato [11]) that H is selfadjoint with operator domain D(H) = W 2,2 (R3N ) and quadratic form domain Q(H) = W 1,2 (R3N ). We consider eigenfunctions ψ of H, i.e., solutions ψ ∈ L2 (R3N ) to the equation Hψ = Eψ , (1.4) with E ∈ R. To simplify notation we assume from now on, without loss, that ψ is real. Apart from the wave function ψ itself, the most important quantity describing the state of the atom is the one-electron density ρ. It is defined by N N ˆ j )|2 dˆ ρ(x) = ρj (x) = |ψ(x, x xj , (1.5) j=1

j=1

R3N −3

where we use the notation ˆ j = (x1 , . . . , xj−1 , xj+1 , . . . , xN ) x

(1.6)

and dˆ xj = dx1 . . . dxj−1 dxj+1 . . . dxN

(1.7)

ˆ j ). and, by abuse of notation, identify (x1 , . . . , xj−1 , x, xj+1 , . . . , xN ) and (x, x

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We assume throughout when studying ρ that E and ψ in (1.4) are such that there exist constants C0 , γ > 0 such that |ψ(x)| ≤ C0 e−γ|x|

for all

x ∈ R3N .

(1.8)

The a priori estimate [9, Theorem 1.2] (see also [9, Remark 1.7]) and (1.8) imply the existence of constants C1 , γ1 > 0 such that ∇ψ(x) ≤ C1 e−γ1 |x| for almost all x ∈ R3N . (1.9) Remark 1.1. Since ψ is continuous (see Kato [12]), (1.8) is only an assumption on the behaviour at infinity. The proofs of our results rely on some kind of decay-rate for ψ; exponential decay is not essential, but assumed for convenience. It is known to hold for eigenfunctions associated to non-threshold eigenvalues, in particular, for all eigenvalues below the essential spectrum in any symmetry subspace (see e.g. Froese and Herbst [7] and Simon [14]). Note that (1.8) and (1.9) imply that ρ is Lipschitz continuous in R3 by Lebesgue’s theorem on dominated convergence. In [5] we proved (with THO) that ρ is real analytic away from the position of the nucleus. More generally, for a molecule with K fixed nuclei at R1 , . . . , RK , Rj ∈ R3 , it was proved that ρ ∈ C ω (R3 \ {R1 , . . . , RK }); see also [3] and [4]. Note that the proof of analyticity does not require any decay of ρ (apart from ψ ∈ W 2,2 (R3N )). That ρ itself is not analytic at the positions of the nuclei is already clear for the groundstate of ‘Hydrogenic atoms’ (N = 1); in this case, ∇ρ is not even continuous at x = 0. However, as was proved in [2], eZ|x| ρ ∈ C 1,1 (R3 ). To obtain more information about the behaviour of the density at the positions of the nuclei one therefore has to study the regularity of other quantities, derived from ρ. One possibility is to study the function r → ρ(r, ω) := ρ(rω) for fixed ω = x ∈ S2 (r = |x|); results in this direction were derived by the authors (with THO) |x| in [2, Theorem 1.5]. In particular, for the case of atoms it was proved that for all ω ∈ S2 , ρ( · , ω) ∈ C 2,α ([0, ∞))

for all α ∈ (0, 1) ,

(1.10)

and the 1st and 2nd radial derivatives were investigated at r = 0. The main quantity studied in this paper is the spherical average of ρ, ρ(r) = ρ(rω) dω , r ∈ [0, ∞) . (1.11) S2

It follows from the analyticity of ρ mentioned above that also ρ ∈ C ω ((0, ∞)), and from the Lipschitz continuity of ρ in R3 that ρ is Lipschitz continuous in [0, ∞). The existence of ρ (0), the continuity of ρ at r = 0, and the Cusp Condition ρ (0) = −Z ρ(0) ,

(1.12)

follows from a similar result for ψ itself by Kato [12]; see [10, 15], and [9, Remark 1.13].

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To investigate properties of ρ and the derived quantities above it is essential that ρ satisfies a differential equation. Such an equation easily follows via (1.4) from N ˆ j )(H − E)ψ(x, x ˆ j ) dˆ ψ(x, x xj = 0 . (1.13) j=1

R3N −3

This implies that ρ satisfies, in the distributional sense, the (inhomogeneous oneparticle Schr¨ odinger) equation 1 Z − Δρ − ρ + h = 0 in R3 . 2 |x|

(1.14)

The function h in (1.14) is given by h(x) =

N

hj (x),

(1.15)

j=1

hj (x) =

N

ˆ j )| dˆ |∇ψ(x, x xj − 2

R3N −3

+

=1,=j

N =1,=j

R3N −3

+

R3N −3

Z ˆ j )|2 dˆ |ψ(x, x xj |x |

1 ˆ j )|2 dˆ |ψ(x, x xj (1.16) |x − x | 1 ˆ j )|2 dˆ |ψ(x, x xj − Eρj (x) . R3N −3 |xk − x |

1≤k 3 remains an open problem. The two main steps in the proof of Theorem 1.2 are Propositions 1.6 and 2.1 below. From the latter one sees that the existence of h (k−2) (0) is necessary to prove (k) h (0) is proved and this existence of ρ (0). In Proposition 1.6 the existence of result already heavily relies on the optimal regularity results for ψ (involving an a priori estimate for second order partial derivatives of ψ) obtained in [6] (see also Appendix B below). Remark 1.4. Note that the inequality ρ (0) ≤ −

7 3 Z ρ(0) 12

(1.26)

follows from (1.24) as soon as ψ is such that N ψ(0, · ), HN −1 (Z − 1) − E ψ(0, · ) j=1

−3 L2 (R3N ) ˆ x

≥ 0.

(1.27)

j

There are cases where (1.27) holds even if the assumption ε ≥ 0 does not. For instance when E is an embedded eigenvalue for the full operator H (ε < 0), but non-embedded for the operator restricted to a symmetry subspace. In particular, (1.26) holds for the fermionic ground state.

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Remark 1.5. Compare (1.12), (1.22), and Theorem 1.2 with the fact that for the ground state of ‘Hydrogenic atoms’ (N = 1), the corresponding density ρ1 (r) = c e−Zr satisfies ρ1(k) (0) = (−Z)k ρ1 (0) .

(1.28)

In fact, if (−Δ − Z/|x|)ψn = En ψn , En = −Z 2 /4n2 , n ∈ N, ψn (x) = e− 2 |x| φn (x), then (1.24) implies that the corresponding density ρn satisfies

7 3 5 ρn (0) = − Z + ZE ρn (0) − 4πZ|∇φn (0)|2 12 3

Z3 5 =− (1.29) 7 + 2 ρn (0) − 4πZ|∇φn (0)|2 . 12 n Z

For the ground state, i.e., for n = 1, E1 = −Z 2 /4, φ1 ≡ 1, this reduces to (1.28) with k = 3. Furthermore, for s - states (zero angular momentum), we get that ∇φn (0) = 0, since φn is radial and C 1,α (see (3.25) below). Taking n large in (1.29) illustrates the quality of the bound (1.25). The proof of Theorem 1.2 is based on the following result on h. Its proof is given in Section 3.1. Proposition 1.6. Let ψ ∈ L2 (R3N ) be an atomic eigenfunction, HN (Z)ψ = Eψ, satisfying (1.8), and let h be as defined in (1.15)–(1.16). Let ω ∈ S2 and h(r) = h(rω) dω. S2 Then both h and the function r → h(r, ω) := h(rω) belong to C 1 ([0, ∞)). ˆ j ) = e 2 |x| ψ(x, x ˆ j ), j = 1, . . . , N , Furthermore, with ϕj (x, x Z

N Z2 ˆ j )|2 dˆ ρ(0) + 4π h(0) = |∇j ϕj (0, x xj 4 3N −3 R j=1 + ψ(0, · ), HN −1 (Z − 1) − E ψ(0, · ) 2 3N −3 , L (Rx ˆ

N

j

)

4π Z3 ˆ j )|2 dˆ ρ(0) + Z h(0) + |∇j ϕj (0, x xj h (0) = −Z 12 3 j=1 R3N −3 − ψ(0, · ), HN −1 (Z − 1) − E ψ(0, · ) . 3N −3 L2 (Rx ˆ

j

(1.30)

)

(1.31)

As a byproduct of (1.30) we get the following improvement of (1.22). Corollary 1.7. Let ψ ∈ L2 (R3N ) be an atomic eigenfunction, HN (Z)ψ = Eψ, satisfying (1.8), with associated spherically averaged density ρ defined by (1.5) and (1.11). Let ε be given by (1.20), and assume ε ≥ 0.

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5 2 5Z 2 + ε ρ(0) ≥ Z 2 ρ(0) . ρ (0) ≥ 3 4 6

1393

(1.32)

Proof. Using the HVZ-theorem [13, Theorem XIII.17], (1.30) provides an improvement of the bound (1.21) for r = 0 to

2 Z + ε ρ(0) . (1.33) h(0) ≥ 4

This, using (1.19), gives (1.32).

Remark 1.8. For ‘Hydrogenic atoms’ (N = 1) (see Remark 1.5), (1.19) and (1.30) imply

Z2 8π 1 |∇φn (0)|2 , ρn (0) = (1.34) 5 + 2 ρn (0) + 6 n 3 which illustrates the quality of the bound (1.32) above (see also the discussion in Remark 1.5), and reduces to (1.28) with k = 2 for the ground state (n = 1, φ1 ≡ 1). We outline the structure of the rest of the paper. In Section 2, we use Proposition 1.6 and the equation for ρ (see (1.17)) to prove Theorem 1.2. In Section 3 we then prove Proposition 1.6. This is done applying the characterization of the regularity of the eigenfunction ψ up to order C 1,1 proved in [6] (see also Appendix B and Lemma 3.9) to the different terms in (1.15)–(1.16).

2. Proof of Theorem 1.2 That ρ ∈ C 3 ([0, ∞)) and the formula (1.23) follow from Proposition 2.1 below (with k = 1), using Proposition 1.6 and (1.19). The formula (1.24) then follows from (1.23) and Proposition 1.6. If ε ≥ 0, then the HVZ-theorem [13, Theorem XIII.17] implies that N

ψ(0, · ), HN −1 (Z − 1) − E ψ(0, · )

j=1

−3 L2 (R3N ) ˆ x j

≥ε

N j=1

ψ(0, · ), ψ(0, · ) L2 (R3N −3 ) = ερ(0) ,

(2.1)

ˆj x

which, together with (1.24), implies (1.25), since ρ(0) = 4πρ(0). It therefore remains to prove the following proposition. Proposition 2.1. Let ψ ∈ L2 (R3N ) be an atomic eigenfunction, HN (Z)ψ = Eψ, satisfying (1.8), with associated spherically averaged density ρ defined by (1.5) and (1.11), and let h be as defined in (1.15)–(1.16) and (1.18). Let k ∈ N ∪ {0}.

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If h ∈ C k ([0, ∞)) then ρ ∈ C k+2 ([0, ∞)), and ρ (k+2) (0) =

2 (k + 1) h(k) (0) − Z ρ (k+1) (0) . k+3

(2.2)

Proof. Let r > 0; multiplying (1.17) with r2 , integrating over [δ, r] for 0 < δ < r, and then taking the limit δ ↓ 0, using that h, ρ, and ρ are all continuous on [0, ∞) (see the introduction), it follows from Lebesgue’s theorem on dominated convergence that r 2 − Z ρ(s)s + h(s)s2 ds ρ (r) = 2 r 0 r 1 2 ρ(rσ)σ dσ + 2 (2.3) h(s)s2 ds . = −2Z r 0 0 Using again Lebesgue’s theorem on dominated convergence (in the form of Proposition 3.6 below) we get that for r > 0,

h(r) − ρ (r) = 2

1

Z ρ (rσ) + 2 h(rσ) σ 2 dσ .

(2.4)

0

Since ρ ∈ C 2 ([0, ∞)) (see the introduction), (2.4) extends to r = 0 by continuity and Lebesgue’s theorem. This finishes the proof in the case k = 0. For k ∈ N, applying Lebegue’s theorem to (2.4) it is easy to prove by induction that if h ∈ C k ([0, ∞)) then ρ ∈ C k+2 ([0, ∞)), and that for r ≥ 0,

ρ

(k+2)

(r) = 2 h(k) (r) −

1

(k+1) k+2 (k) Z ρ (rσ) + 2h (rσ) σ dσ .

(2.5)

0

In particular, (2.2) holds. This finishes the proof of the proposition. This finishes the proof of Theorem 1.2. It remains to prove Proposition 1.6.

3. Study of the function h 3.1. Proof of Proposition 1.6 It clearly suffices to prove the statements in Proposition 1.6 for each hj (j = 1, . . . , N ) in (1.16). Proposition 1.6 then follows for h by summation. We shall prove the statements in Proposition 1.6 for h1 ; the proof for the other hj is completely analogous.

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Recall (see (1.15)–(1.16)) that h1 is defined by h1 (x) = t1 (x) − v1 (x) + w1 (x) − Eρ1 (x) , ˆ 1 )|2 dˆ t1 (x) = |∇ψ(x, x x1 , v1 (x) =

w1 (x) =

R3N −3 N

k=2

R3N −3

k=2

R3N −3

N

+ ρ1 (x) =

2≤k 0, ω ∈ S2 , and x ˆ 1 such that V exists an > 0 and a neighbourhood U ⊂ S1 (ω) ⊂ R3N −3 of x in (1.3) is C ∞ on B3 (rω, ) × U ⊂ R3N . It follows from elliptic regularity [8] that ψ ∈ C ∞ (B3 (rω, ) × U ). In particular, if G : R3N → R is any of the integrands ˆ 1 ) of the in (3.2)–(3.5), then, for all ω ∈ S2 , the partial derivative ∂Gω /∂r(r, x ˆ 1 ) ≡ Gω (r, x ˆ 1 ) exists, and satisfies ˆ 1 ) → G(rω, x function (r, x ∂Gω ˆ 1 ) = ω · (∇1 G)(rω, x ˆ 1 ) for all r > 0 , x ˆ 1 ∈ S1 (ω) . (r, x (3.13) ∂r This assures that the hypothesis ii) in Proposition 3.6 will be satisfied in all cases below where we apply the proposition (the hypothesis i) is trivially satisfied). We illustrate how to apply this. Let ˆ 1 ) dˆ ˆ 1 ) dˆ G(rω, x x1 G(rω, x x1 . gω (r) = g(r, ω) = = R3N −3

S1 (ω)

Interchanging integration and differentiation as discussed above, we have, for ω ∈ S2 fixed and r > 0, that ˆ 1 ) dˆ ω · (∇1 G)(rω, x x1 . (3.14) gω (r) = S1 (ω)

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To justify this, and to prove the existence of limr↓0 gω (r) we will need to prove two things. First, we need to find a dominant to the integrand in (3.14), that is, a function Φω ∈ L1 (R3N −3 ) such that, for some R0 > 0,

ω · ∇1 G (rω, x ˆ 1 ) ≤ Φω (ˆ ˆ 1 ∈ S1 (ω) . x1 ) for all r ∈ (0, R0 ) , x (3.15) d and the This will, by Proposition 3.6, also justify the above interchanging of dr integral ((3.15) ensures that the hypothesis iii) is satisfied). We note that, with one exception, whenever we apply this, in fact Φ ≡ Φω will be independent of ω ∈ S2 , and therefore Φ ∈ L1 (R3N −3 × S2 ). ˆ 1 ∈ S1 (ω) fixed, the existence Secondly, we need to prove, for all ω ∈ S2 and x of ˆ1) . (3.16) lim ω · (∇1 G)(rω, x r↓0

This, by Lebesgue’s Theorem of Dominated Convergence, will prove the existence of limr↓0 gω (r). To study ˆ 1 ) dˆ g(r) = g(r, ω) dω = G(rω, x x1 dω (3.17) S2

S2

R3N −3

note that, by the above, the set 3N −3 2 S1 (ω) × {ω} R ×S \ ω∈S2

has measure zero, and that, also by the above, the partial derivative ∂ G/∂r(r, (xˆ1 , ω)) of the function

r, (ˆ ˆ1) r, (ˆ x1 , ω) → G x1 , ω) ≡ G(rω, x exists for all r > 0 and (ˆ x1 , ω) ∈ ∪ω∈S2 S1 (ω) × {ω} . As noted above, the dominants Φ we will exhibit below when studying g(r, ω) will (except in one case) be independent of ω ∈ S2 , and so can also be used to apply both Lebegue’s Theorem on Dominated Convergence, Proposition 3.6, and Fubini’s Theorem on the integral in (3.17). This implies that ˆ 1 ) dω dˆ lim ω · (∇1 G)(rω, x x1 , (3.18) lim g (r) = r↓0

R3N −3

S2

r↓0

as soon as we have proved the pointwise convergence of the integrand for all ω ∈ S2 ˆ 1 ∈ S1 (ω), and provided the mentioned dominant. (In the last, exceptional and x case, we will provide a dominant Φ ∈ L1 (R3N −3 × S2 )). In the sequel, we shall use all of this without further mentioning, apart from proving the existence of a dominant, and the existence of the pointwise limits of the integrands on S1 (ω). Also, for notational convenience, we shall allow ourselves to write all integrals over R3N −3 instead of over S1 (ω).

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3.3. Additional partial regularity For the existence of pointwise limits the following lemma will be essential; it gives detailed information about the structure of the eigenfunction ψ in the vicinity of the two-particle singularity x = 0. The lemma is reminiscent of more detailed results obtained earlier, see [3, Proposition 2], [4, Lemma 2.2], and [5, Lemma 3.1]. We need to recall the definition of H¨ older-continuity. Definition 3.8. Let Ω be a domain in Rn , k ∈ N, and α ∈ (0, 1]. We say that a function u belongs to C k,α (Ω) whenever u ∈ C k (Ω), and for all β ∈ Nn with |β| = k, and all open balls Bn (x0 , r) with Bn (x0 , r) ⊂ Ω, we have |Dβ u(x) − Dβ u(y)| ≤ C(x0 , r) . |x − y|α x,y∈Bn (x0 ,r), x=y sup

When k = 0 and α ∈ (0, 1) we also write C α (Ω) ≡ C 0,α (Ω). ˆ 01 ∈ S1 (ω). Then there exists an open neighbourhood Lemma 3.9. Let ω ∈ S2 and x 3N −3 0 ˆ 1 and > 0 such that of x U ⊂ S1 (ω) ⊂ R ˆ 1 ) with ˆ 1 ) = e− 2 |x| ϕ1 (x, x ψ(x, x

β 1,α ∂xˆ 1 ϕ1 ∈ C B3 (0, ) × U for all Z

(3.19) α ∈ (0, 1) ,

3N −3

β∈N

.

(3.20)

Proof. By the definition (3.12) of S1 (ω) there exists a neighbourhood U ⊂ S1 (ω) ⊂ ˆ 01 ∈ S1 (ω), and > 0 such that R3N −3 of x xj = xk

for j, k ∈ {2, . . . , N }

xj = 0 ,

xj = x for j ∈ {2, . . . , N }

for all

with

j = k ,

(3.21)

(x, x2 , . . . , xN ) ∈ B3 (0, ) × U ⊂ R3N .

Make the Ansatz ψ = e− 2 |x| ϕ1 . Using (1.4), (1.1), and that Δx (|x|) = 2|x|−1 , we get that ϕ1 satisfies the equation 2 Z x · ∇1 ϕ1 + + E − V 1 ϕ1 = 0 Δϕ1 − Z (3.22) |x| 4 Z

with ˆ1) = V1 (x, x

N j=2

where, due to (3.21),

−

1 Z + + |xj | |x − xk | N

k=2

2≤j 0, interchanging integration and differentiation as discussed above, we get, using (3.27), that

1 d ˆ 1 )|2 dˆ |ψ(rω, x x1 dr R3N −3 |rω − x2 |

2 = ψ ω · (∇1 ψ + ∇2 ψ) (rω, y + rω, x3 , . . . , xN ) R3N −3 |y| dy dx3 . . . dxN . (3.28) Note that, using (1.8), (1.9), and equivalence of norms in R3N , there exists positive constants C, c1 , c2 such that 2 ψ ω · (∇1 ψ + ∇2 ψ) (rω, y + rω, x3 , . . . , xN ) |y| 1 −c2 |(y,x3 ,...,xN )| ≤ C ec1 r e , (3.29) |y| which provides a dominant, independent of ω ∈ S2 , in the sense of (3.15), uniformly for r ∈ (0, R0 ) for any R0 > 0. Z Writing ψ = e− 2 |x| ϕ1 , the integrand in (3.28) equals

1 − Z|ψ|2 + 2ϕ1 e−Zr ω · {∇1 ϕ1 + ∇2 ϕ1 } (rω, y + rω, x3 , . . . , xN ) , |y| which has a limit as r ↓ 0 (for ω ∈ S2 and (y, x3 , . . . , xN ) fixed) since ϕ1 ∈ C 1,α by Lemma 3.9. This proves the existence of the limit of the integrand in (3.28),

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and therefore of the limit as r ↓ 0 of the derivative with respect to r of the first term in (3.4). Note that since S2 ω dω = 0, the terms proportional to ω vanish by integration over (y, x3 , . . . , xN , ω). The limit of the derivative of the spherical average of this term is therefore (after setting x2 = y) 1 ˆ 1 )|2 dˆ |ψ(0, x x1 dω . (3.30) −Z |x 2| S2 R3N −3 For the second term in (3.4), assume without loss that k = 2, = 3, and Z write ψ = e− 2 |x| ϕ1 as before. Then, for ω ∈ S2 fixed, and r > 0, interchanging integration and differentiation we get

1 d ˆ 1 )|2 dˆ |ψ(x, x x1 dr R3N −3 |x2 − x3 | 2 ˆ 1 ) ω · ∇1 ψ(rω, x ˆ 1 ) dxˆ1 . ψ(rω, x = (3.31) 3N −3 |x2 − x3 | R 1 ˆ 1 ) dˆ − Z|ψ|2 + 2ϕ1 e−Zr (ω · ∇1 ϕ1 ) (rω, x x1 . (3.32) = |x − x | 3N −3 2 3 R As above, (1.8) and (1.9) provides us with a dominant to the integrand in (3.31) in the sense of (3.15). Also as before, the integrand in (3.32) has a limit as r ↓ 0, since ϕ1 ∈ C 1,α by Lemma 3.9. This proves the existence of the limit as r ↓ 0 of the derivative of the second term in (3.4). Again, the term in (3.32) proportional to ω vanish when integrating over (ˆ x1 , ω), since S2 ω dω = 0. The limit of the derivative of the spherical average of this term is therefore 1 ˆ 1 )|2 dˆ |ψ(0, x x1 dω . (3.33) −Z |x − x3 | 2 3N −3 2 S R This proves the existence of limr↓0 w1 (r, ω) (for any ω ∈ S2 fixed), and of 1 (r). Furthermore, from (3.30) and (3.33), limr↓0 w N 1 ˆ 1 )|2 dˆ |ψ(0, x 1 (r) = −Z x1 dω lim w r↓0 S2 R3N −3 k=2 |xk | ⎤ ⎡ 1 ⎦ |ψ(0, x ⎣ ˆ 1 )|2 dˆ −Z x1 dω |x − x | 2 3N −3 k S R 2≤k 0 we get, by arguments as for w1 , that N 2Z ˆ 1 ) ω · ∇1 ψ(rω, x ˆ 1 ) dˆ ψ(rω, x v1 (r, ω) = x1 (3.34) 3N −3 |xk | k=2 R N Z ˆ 1 ) dˆ − Z|ψ|2 + 2ϕ1 e−Zr (ω · ∇1 ϕ1 ) (rω, x = x1 . (3.35) |x | 3N −3 k R k=2

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One provides a dominant to the integrand in (3.34) in a similar way as for w1 . We omit the details. Again, existence of the limit as r ↓ 0 of the integrand in (3.35) is ensured by Lemma 3.9. The last term in (3.35) again vanishes when taking the limit r ↓ 0 and then integrating over (ˆ x1 , ω), since S2 ω dω = 0, and so lim v1 (r, ω) r↓0

= −Z

N k=2

S2

R3N −3

Z ˆ 1 )|2 dˆ |ψ(0, x x1 dω |xk |

= −Z v1 (0) . This finishes the proof of Proposition 3.1.

3.5. Proof of Proposition 3.3 We proceed as in the proof of Proposition 3.1. Interchanging integration and differentiation as discussed in Section 3.2 we have, for ω ∈ S2 fixed and r > 0, that ˆ 1 ) dˆ t1 (r, ω) = ω · ∇1 (|∇ψ|2 )(rω, x x1 . (3.36) R3N −3

Again, to justify this and to prove the existence of limr↓0 t1 (r, ω) we need to prove two things: The existence of the pointwise limit as r ↓ 0 of the integrand ˆ 1 ∈ S1 (ω) fixed, and the existence of a dominant in the above, for ω ∈ S2 and x sense of (3.15). Pointwise limits We start with the pointwise limit. We allow ourselves to compute the integrals of the found limits, presuming the dominant found. We shall provide the necessary dominants afterwards. Z Recall from (the proof of) Lemma 3.9 that ϕ1 = e 2 |x| ψ satisfies the equation 2 Z x · ∇1 ϕ1 + + E − V 1 ϕ1 = 0 (3.37) Δϕ1 − Z |x| 4 where

V1 ∈ C ∞ B3 (0, ) × U

ˆ1. for some > 0 and U ⊂ S1 (ω) ⊂ R3N −3 some neighbourhood of x Using Lemma 3.9, we get that Δxˆ 1 ϕ1 ∈ C 1,α (B3 (0, ) × U ) for all α ∈ (0, 1). ˆ 1 ∈ S1 (ω) fixed, and some > 0, the From this and (3.37) follows that for any x ˆ 1 ) satisfies the equation function x → ϕ1 (x, x 2 Z x · ∇x ϕ1 = −Δxˆ 1 ϕ1 − + E − V 1 ϕ1 (3.38) Δx ϕ1 − Z |x| 4

(3.39) ≡ Gxˆ 1 , Gxˆ 1 ∈ C 1,α B3 (0, ) , α ∈ (0, 1) .

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ˆ 1 ) is wellFurthermore, from Lemma 3.9 (with β = 0) it follows that ∇1 ϕ1 (0, x defined. Note that for c ∈ R3 the function 1 1 (3.40) v(x) = |x|2 (c · ω) = |x|(x · c) 4 4 ˆ 1 ), the function u = uxˆ 1 = solves Δx v = c · ω. Therefore, with c = Z∇1 ϕ1 (0, x ˆ 1 ) − v satisfies the equation (in x, with x ˆ 1 ∈ S1 (ω) fixed) ϕ1 ( · , x

x ˆ 1 ) − ∇1 ϕ1 (0, x ˆ1) · ∇1 ϕ1 (x, x Δx u(x) = Z |x| 2 Z ˆ1) − ˆ 1 ) ≡ gxˆ 1 (x) . + E − V1 ϕ1 (x, x (3.41) − Δxˆ 1 ϕ1 (x, x 4 By the above, and Lemma A.1 in Appendix A (using that ∇1 ϕ1 is C α ), gxˆ 1 ∈ C α (R3 ) for all α ∈ (0, 1), and so, by standard elliptic regularity, u ∈ C 2,α (R3 ) for ˆ1) ˆ 1 ∈ S1 (ω), the function x → ϕ1 (x, x all α ∈ (0, 1). To recapitulate, for any x satisfies 1 2 x ˆ 1 ) = uxˆ 1 (x) + |x| c · ϕ1 (x, x , uxˆ 1 ∈ C 2,α (R3 ) , α ∈ (0, 1) . (3.42) 4 |x| Note that ˆ 1 ) = Z∇x uxˆ 1 (0) . c = Z∇1 ϕ1 (0, x

(3.43)

We now apply the above to prove the existence of the pointwise limit of the ˆ 1 ∈ S1 (ω). integrand in (3.36) for fixed ω ∈ S2 and x −Z |x| 2 ϕ1 , we have, for j = 2, . . . , N , First, since ψ = e 2 ˆ1) ω · ∇1 (|∇j ψ| )(rω, xˆ1 ) = ω · ∇1 (e−Z|x| |∇j ϕ1 |2 )(rω, x ˆ1) = −Ze−Zr |∇j ϕ1 |2 (rω, x ˆ1) . + e−Zr ω · ∇1 (|∇j ϕ1 |2 )(rω, x Because of (3.20) in Lemma 3.9, this has a limit as r ↓ 0 for fixed ω ∈ S2 (recall that j = 2, . . . , N ). The contribution to limr↓0 t1 (r) from this is N ˆ 1 )|2 dxˆ1 dω , |∇j ϕ1 (0, x (3.44) −Z j=2

S2

R3N −3

since terms proportional with ω vanish upon integration.

So we are left with considering ω · ∇1 |∇1 ψ|2 (see (3.36)). To this end, use Z again ψ = e− 2 |x| ϕ1 to get 2

Z Z ω · ∇x |∇1 ψ|2 = ω · ∇x − ωψ + e− 2 |x| ∇1 ϕ1 (3.45) 2 2 Z 2 −Z|x| 2 −Z|x| = ω · ∇x ψ +e |∇1 ϕ1 | − Z(ω · ∇1 ϕ1 )e ϕ1 . 4

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ˆ 1 )). We will study each of the three terms (We leave out the variables, (rω, x in (3.45) separately. Z For the first term in (3.45), again using ψ = e− 2 |x| ϕ1 , we have 2 Z 2 Z 2 − Z |x| Z3 ω · ∇x ψ = − ψ2 + ψe 2 (ω · ∇1 ϕ1 ) , (3.46) 4 4 2 ˆ 1 ∈ S1 (ω), since ϕ1 is C 1,α which has a limit as r ↓ 0 for fixed ω ∈ S2 and x (see (3.25)). The contribution to limr↓0 t1 (r) from this is Z3 Z3 ˆ 1 )|2 dˆ |ψ(0, x x1 dω = − ρ1 (0) . (3.47) − 4 S2 R3N −3 4 As for the second term in (3.45) we have

ω · ∇x e−Z|x| |∇1 ϕ1 |2 = −Ze−Z|x| |∇1 ϕ1 |2 + e−Z|x| (ω · ∇1 |∇1 ϕ1 |2 ) ,

(3.48)

ˆ 1 ∈ S1 (ω), since ϕ1 where the first term has a limit as r ↓ 0 for fixed ω ∈ S2 and x is C 1,α (see (3.25)). This contributes ˆ 1 )|2 dˆ |∇1 ϕ1 (0, x x1 dω (3.49) −Z S2

R3N −3

limr↓0 t1 (r).

to For the second term (in (3.48)), using that ϕ1 = u + 14 r2 (c · ω) (see (3.42)), we get that 1 |∇1 ϕ1 |2 = |∇1 u|2 + (ω · ∇1 u)(c · x) + r(c · ∇1 u) 2 3 1 + (c · x)2 + r2 (c · c) , 16 16 and so 3 1 ω · ∇1 |∇1 ϕ1 |2 = 2 ω, (D2 u)∇1 u + r(c · ω)2 + r(c · c) 8 8 1 2 + ω, (D u)ω (c · x) + (ω · ∇1 u)(c · ω) 2 + (c · ∇1 u) + r ω, (D2 u)c . Here, D2 u is the Hessian matrix of u = uxˆ 1 with respect to x, and · , · is the scalar product in R3 . Since ϕ1 ∈ C 1,α , and u ∈ C 2,α , all terms have a limit as ˆ 1 ∈ S1 (ω). We get r ↓ 0 for fixed ω ∈ S2 and x −Z|x| ˆ 1 ) = 2ω · (D2 uxˆ 1 )(0)∇x uxˆ 1 (0) ω · ∇1 |∇1 ϕ1 |2 )(rω, x lim (e r↓0

1 c · ∇x uxˆ 1 (0) + ω · ∇x uxˆ 1 (0) (c · ω) . + 2 (3.50) When integrating, this contributes 2Z ˆ 1 )|2 dˆ |∇1 ϕ1 (0, x x1 dω 3 S2 R3N −3

(3.51)

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to limr↓0 t1 (r); to see this, use (3.43) and Lemma A.2 in Appendix A, and that ω dω = 0. S2 For the third and last term in (3.45), using that ϕ1 = u+ 14 r2 (c · ω) (see (3.42)) and ω · ∇x = ∂r , we get that

ω · ∇x − Z(ω · ∇1 ϕ1 )e−Z|x| ϕ1 = Z 2 (ω · ∇1 ϕ1 )e−Z|x| ϕ1 1 − Ze−Z|x| ϕ1 ω, (D2 u)ω + (c · ω) 2 − Z(ω · ∇x ϕ1 )2 e−Z|x| . Again, since ϕ1 ∈ C 1,α , and u ∈ C 2,α , all terms have a limit as r ↓ 0 for fixed ˆ 1 ∈ S1 (ω). ω ∈ S2 and x The contribution from this to limr↓0 t1 (r) is, using Lemma A.2, and ω dω = 0, S2 Z − 3

S2

R3N −3

ˆ 1 )|2 dxˆ1 dω |∇1 ϕ1 (0, x Z ˆ 1 ) dxˆ1 dω . Δx uxˆ 1 (0)ϕ1 (0, x − 3 S2 R3N −3

(3.52)

Here we used that Tr(D2 f ) = Δf . This proves the existence of the pointwise limit of the integrand in (3.36) for ˆ 1 ∈ S1 (ω). Also, from (3.44), (3.47), (3.49) (3.51), and (3.52), fixed ω ∈ S2 and x

Z3 2 ˆ 1 )| dˆ ρ1 (0) |∇ϕ1 (0, x x1 dω + lim t1 (r) = − Z r↓0 4 2 3N −3 S R Z ˆ 1 )|2 dˆ + |∇1 ϕ1 (0, x x1 dω 3 S2 R3N −3 Z ˆ 1 ) dˆ Δx uxˆ 1 (0)ϕ1 (0, x x1 dω . (3.53) − 3 S2 R3N −3 Note that, due to (3.41) and ψ = e− 2 |x| ϕ1 , 2 Z ˆ1) − ˆ1) , + E − VN −1,Z−1 (ˆ x1 ) ψ(0, x Δx uxˆ 1 (0) = −Δxˆ 1 ψ(0, x 4 Z

ˆ 1 ) = VN −1,Z−1 (ˆ since V1 (0, x x1 ) (see (1.3)). This implies that

Z3 2 ˆ ρ1 (0) lim t1 (r) = − Z |∇ϕ1 (0, x1 )| dˆ x1 dω + r↓0 4 S2 R3N −3

Z Z2 ˆ 1 )|2 dˆ + ρ1 (0) |∇1 ϕ1 (0, x x1 dω + (3.54) 3 S2 R3N −3 4

2 2 ˆ 1 )| + VN −1,Z−1 (ˆ ˆ 1 )| dˆ x1 )−E |ψ(0, x − |∇xˆ 1 ψ(0, x x1 dω . S2

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Here we used that

ˆ 1 ) Δxˆ 1 ψ (0, x ˆ 1 ) dˆ − ψ(0, x x1 = R3N −3


R3N −3

ˆ 1 )|2 dˆ |∇xˆ 1 ψ(0, x x1 .

(3.55)

Before we provide a dominant in the sense of (3.15) to the integrand in (3.36) we now compute t1 (0). Note that for r > 0, ˆ 1 )|2 dˆ t1 (r) = |∇1 ψ(rω, x x1 dω (3.56) S2 R3N −3 N

+

j=2

S2

R3N −3

ˆ 1 )|2 dˆ |∇j ψ(rω, x x1 dω .

ˆ 1 ∈ S1 (ω) fixed, and j = 2, . . . , N , Use ψ = e− 2 |x| ϕ1 , then, for all ω ∈ S2 and x Lemma 3.9 gives that Z

ˆ 1 )|2 = e−Zr |∇j ϕ1 (rω, x ˆ 1 )|2 |∇j ψ(rω, x

r→0

−→

ˆ 1 )|2 . |∇j ϕ1 (0, x

(3.57)

ˆ 1 )|2 . |∇j ϕ1 (0, x

(3.58)

In particular, this proves the existence of ˆ 1 )|2 = |∇xˆ 1 ψ(0, x

N

ˆ 1 )|2 = |∇j ψ(0, x

j=2

N j=2

Similarly, ˆ 1 )|2 = |∇1 ψ(rω, x

Z2 ˆ 1 )|2 + e−Zr |∇1 ϕ1 (rω, x ˆ 1 )|2 |ψ(rω, x 4

Z ˆ 1 ) ω · ∇1 ϕ1 (rω, x ˆ1) − Ze− 2 r ψ(rω, x

(3.59)

r→0 Z ˆ 1 )|2 + |∇1 ϕ1 (0, x ˆ 1 )|2 − Zψ(0, x ˆ 1 ) ω · ∇1 ϕ1 (0, x ˆ1) . |ψ(0, x −→ 4 2

Using again S2 ω dω = 0, it follows from (3.56)–(3.59), and Lebesgue’s Theorem of Dominated Convergence that Z2 ˆ 1 )|2 dˆ ρ1 (0) + lim t1 (r) = |∇1 ϕ1 (0, x x1 dω r↓0 4 S2 R3N −3 ˆ 1 )|2 dˆ |∇xˆ 1 ψ(0, x x1 dω . (3.60) + S2

R3N −3

This proves (3.9). Combining this with (3.54) (using (3.58)) proves (3.10). Dominant We turn to finding a dominant to the integrand in (3.36). We shall apply results from [6], recalled in Appendix B.

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From the a priori estimate (B.8) in Theorem B.3 and (3.36) follows that, for ˆ 1 ) = x ∈ R3N (choose, e.g., R = 1, R = 2) almost all (x, x

ω · ∇1 |∇ψ|2 (x, x ˆ 1 ) ψL∞ (B3N ((x,ˆx1 ),2)) ˆ 1 ) ≤ C ∇ψ(x, x (3.61) N 3 3 x ∂ψ ∂ 2 Fcut 1,k ˆ 1 ) , 2 (x, x ψ + |x1 | ∂x,m ∂x1,k ∂x,m m=1 =1 k=1

with Fcut = F2,cut + F3,cut as in Definition B.2. Now, using the exponential decay of ψ (1.8) (assumed), and of ∇ψ (1.9), we get that there exist constants C, γ > 0 such that ∇ψ(x, x ˆ 1 ) ψL∞ (B3N ((x,ˆx1 ),2)) ≤ Ce−γ|ˆx1 |

for almost all

x ∈ R3 .

(3.62)

This provides a dominant (in the sense of (3.15)) for the first term in (3.61). We need to find a dominant for the second term in (3.61). First recall that Fcut = F2,cut + F3,cut . With F2 as in (B.1) we have F2,cut = F2 + (F2,cut − F2 ), with (F2,cut − F2 )(x) =

N i=1

+

Z χ(|xi |) − 1 |xi | 2 1

χ(|xi − xj |) − 1 |xi − xj | . 4

−

(3.63)

1≤i<j≤N

Note that ∂ 2 (χ(|x|) − 1)|x| is bounded in R3 for all second derivatives ∂ 2 , due to the definition (B.3) of χ. Using the exponential decay of ψ (1.8), and ∇ψ (1.9), we therefore get a dominant for the term N 3 3 x1,k ∂ψ ∂ 2 (F2,cut − F2 ) ˆ1) . (x, x 2 ψ |x1 | ∂x,m ∂x1,k ∂x,m m=1 =1 k=1

A tedious, but straightforward computation gives that N 3 3 ∂ψ x1,k ∂ 2 F2 2 ψ |x1 | ∂x,m ∂x1,k ∂x,m =1 k=1 m=1

N

1 x1 1 · ∇1 ψ − ∇i ψ = ψ 2 i=2 |x1 − xi | |x1 | x1 x1 − xi (∇1 ψ − ∇i ψ) · (x1 − xi ) . · − |x1 | |x1 − xi |3

(3.64)

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We first remark that, again using exponential decay of ψ and ∇ψ, we get the estimate N 1

x1 1 ˆ 1 ) ψ · ∇1 ψ − ∇i ψ (rω, x 2 |x − x | |x | 1 i 1 i=2 ⎛ ⎡ ⎞⎤ N N −c|xi | ' e ⎝ ⎣ ≤C e−c|xj | ⎠⎦ , (3.65) dist(L(ω), x ) i i=2 j=2,j=i

where dist(L(ω), y) is the distance from y ∈ R3 to the line L(ω) spanned by ω ∈ S2 . Note that for fixed ω ∈ S2 , the function e−c|y| /dist(L(ω), y) is integrable in R3 , and its integral is independent of ω ∈ S2 . Therefore the right side of (3.65) is integrable in R3N −3 for fixed ω ∈ S2 , and is integrable in R3N −3 ×S2 (all uniformly for r ∈ R). The argument is similar for the second term in (3.64). We are left with considering the terms in (3.61) with F3,cut . Let

f3 (x, y) = C0 Z(x · y) ln |x|2 + |y|2 (so that F3 (x) = i<j f3 (xi , xj ), see (B.2)). For all second derivatives ∂ 2 we easily get, due to the definition (B.3) of χ, ∂ 2 χ(|x|)χ(|y|)f3 (x, y) = χ(|x|)χ(|y|)∂ 2 f3 (x, y) + g3 (x, y)

= C0 Zχ(|x|)χ(|y|) ln |x|2 + |y|2 ∂ 2 (x · y) + g3 (x, y) , with g3 , g3 bounded on R6 . Therefore, defining, for all second derivatives ∂ 2 and with χ as above,

2F χ(|xi |)χ(|xj |) ln |x|2 + |y|2 ∂ 2 (x · y) , (3.66) ∂* 3,cut (x) := C0 Z 1≤i<j≤N

we get a dominant for the term N 3 3 2F x1,k ∂ψ ∂* ∂ 2 F3,cut 3,cut ˆ1) , 2 − (x, x ψ |x1 | ∂x,m ∂x1,k ∂x,m ∂x1,k ∂x,m m=1 =1 k=1

using the exponential decay of ψ and ∇ψ. We find that N 3 3 2F ∂ψ x1,k ∂* 3,cut 2 ψ |x | ∂x ∂x ∂x 1 ,m 1,k ,m =1 k=1 m=1

N x1 · ∇i ψ . = 2C0 Zψ χ(|x1 |)χ(|xi |) ln(|x1 |2 + |xi |2 ) |x1 | i=2 Note that, by the definition of χ, χ(|x|)χ(|y|) ln(|x|2 + |y|2 ) ≤ χ(|y|) | ln(|y|2 )| + 3 ,

(3.67)

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and so, again by the exponential decay of ψ and ∇ψ, N x1 2 2 ˆ 1 ) · ∇i ψ (rω, x χ(|x1 |)χ(|xi |) ln(|x1 | + |xi | ) 2C0 Zψ |x | 1 i=2 ⎞ ⎛ N N ' χ(|xi |) | ln(|xi |2 )| + 3 ⎝ e−c|xj | ⎠ ≤C i=2

1409

(3.68)

j=2

ˆ 1 ∈ S1 (ω). This provides a dominant for the for all ω ∈ S, r ∈ R and (almost) all x term N 3 3 2F x1,k ∂ψ ∂* 3,cut ˆ1) , 2 (x, x ψ |x1 | ∂x,m ∂x1,k ∂x,m m=1 =1 k=1

and we have therefore provided a dominant, in the sense of of (3.15), for the integrand in (3.36). This finishes the proof of Proposition 3.3.

Appendix A. Two useful lemmas The following lemma is Lemma 2.9 in [6]; we include it, without proof, for the convenience of the reader. (The proof is simple, and can be found in [6]). Lemma A.1. Let G : U → Rn for U ⊂ Rn+m a neighbourhood of a point (0, y0 ) ∈ Rn × Rm . Assume G(0, y) = 0 for all y such that (0, y) ∈ U . Let + x |x| · G(x, y) x = 0 , f (x, y) = 0 x = 0. Then, for α ∈ (0, 1], G ∈ C 0,α (U ; Rn ) ⇒ f ∈ C 0,α (U ) .

(A.1)

Furthermore, f C α (U ) ≤ 2GC α (U ) . The following lemma is used to evaluate certain integrals. Lemma A.2. Let a, b ∈ R3 and let A ∈ M3×3 (C). Then 1 4π (a · b) = (a · b) (ω · a)(ω · b) dω = dω , 3 3 2 S2 S 1 4π Tr(A) = Tr(A) ω · (Aω) dω = dω . 3 3 S2 S2 Proof. Both (A.2) and (A.3) follow from the identity 4π δi,j , ωi ωj dω = 3 2 S which we now prove.

(A.2) (A.3)

(A.4)

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For i = j, the integrand ωi ωj is odd as a function of ωj , which implies (A.4) in that case. For i = j we calculate, using rotational symmetry, 1 4π 2 . ωi dω = (ω12 + ω22 + ω32 ) dω = 3 3 2 2 S S This finishes the proof of (A.4) and by consequence of Lemma A.2.

Appendix B. Regularity of the eigenfunction ψ The following two theorems were proved in [6]. Theorem B.1 ([6, Theorem 1.1 for atoms]). Suppose ψ is a solution to Hψ = Eψ in Ω ⊆ R3N where H is given by (1.2). Let F = eF2 +F3 with N

1 Z |xi − xj | , − |xi | + 2 4 i=1 1≤i<j≤N Z (xi · xj ) ln(|xi |2 + |xj |2 ) , F3 (x) = C0

F2 (x) =

(B.1) (B.2)

1≤i<j≤N

where C0 =

2−π 12π .

Then ψ = Fφ3 with φ3 ∈ C 1,1 (Ω).

Definition B.2. Let χ ∈ C0∞ (R), 0 ≤ χ ≤ 1, with , 1 for |x| ≤ 1 χ(x) = 0 for |x| ≥ 2 .

(B.3)

We define Fcut = F2,cut + F3,cut ,

(B.4)

where F2,cut (x) =

N i=1

F3,cut (x) = C0

−

Z 1 χ(|xi |) |xi | + χ(|xi − xj |) |xi − xj | , 2 4 1≤i<j≤N Z χ(|xi |)χ(|xj |)(xi · xj ) ln(|xi |2 + |xj |2 ) ,

(B.5) (B.6)

1≤i<j≤N

and where C0 is the constant from (B.2). We also introduce φ3,cut by ψ = eFcut φ3,cut .

(B.7)

Theorem B.3 ([6, Theorem 1.5 for atoms]). Suppose ψ is a solution to Hψ = Eψ in R3N . Then for all 0 < R < R there exists a constant C(R, R ), not depending on ψ nor x0 ∈ R3N , such that for any second order derivative, ∂2 =

∂2 , ∂xi,k ∂xj,

i, j = 1, 2, . . . , N ,

k, = 1, 2, 3 ,

the following a priori estimate holds: ∂ 2 ψ − ψ ∂ 2Fcut L∞ (B3N (x0 ,R)) ≤ C(R, R )ψL∞ (B3N (x0 ,R )) .

(B.8)

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Acknowledgements Parts of this work have been carried out at various institutions, whose hospitality is gratefully acknowledged: Mathematisches Forschungsinstitut Oberwolfach (SF, TØS), Erwin Schr¨ odinger Institute (SF, TØS), Université Paris-Sud (TØS), and ´ (TØS). Financial support from the European Science Foundation Prothe IHES gramme Spectral Theory and Partial Differential Equations (SPECT), and EU IHP network Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems, contract no. HPRN-CT-2002-00277, is gratefully acknowledged. TØS was partially supported by the embedding grant from The Danish National Research Foundation: Network in Mathematical Physics and Stochastics, and by the European Commission through its 6th Framework Programme Structuring the European Research Area and the contract Nr. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action.

References [1] J.-M. Bony, Cours d’analyse. Th´ eorie des distributions et analyse de Fourier, Pub´ lications Ecole Polytechnique, Diffusion Ellipses, Paris, 2000. [2] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. Østergaard Sørensen, Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei, Ann. Henri Poincaré 8 (2007) no. 4, 731–748. [3] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. Østergaard Sørensen, The electron density is smooth away from the nuclei, Comm. Math. Phys. 228 (2002), no. 3, 401–415. [4] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. Østergaard Sørensen, On the regularity of the density of electronic wavefunctions, Mathematical Results in Quantum Mechanics (Taxco, 2001), Contemp. Math., vol. 307, Amer. Math. Soc., Providence, RI, 2002, pp. 143–148. [5] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. Østergaard Sørensen, Analyticity of the density of electronic wavefunctions, Ark. Mat. 42 (2004), no. 1, 87–106. [6] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. Østergaard Sørensen, Sharp regularity results for Coulombic many-electron wave functions, Comm. Math. Phys. 255 (2005), no. 1, 183–227. [7] R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N -body Schr¨ odinger operators, Comm. Math. Phys. 87 (1982/83), no. 3, 429–447. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer-Verlag, Berlin, 1983. [9] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. Østergaard Sørensen, Electron wavefunctions and densities for atoms, Ann. Henri Poincaré 2 (2001), no. 1, 77–100. [10] M. Hoffmann-Ostenhof and R. Seiler, Cusp conditions for eigenfunctions of nelectron systems, Phys. Rev. A 23 (1981), no. 1, 21–23.

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[11] T. Kato, Fundamental properties of Hamiltonian operators of Schr¨ odinger type, Trans. Amer. Math. Soc. 70 (1951), 195–211. [12] T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics, Comm. Pure Appl. Math. 10 (1957), 151–177. [13] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. [14] B. Simon, Schr¨ odinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. Erratum: “Schr¨ odinger Semigroups”, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 426. [15] E. Steiner, Charge densities in atoms, J. Chem. Phys. 39 (1963), no. 9, 2365–2366. Søren Fournais Department of Mathematical Sciences University of Aarhus Ny Munkegade, Building 1530 DK-8000 ˚ Arhus C Denmark On leave from: CNRS and Laboratoire de Mathématiques Université Paris-Sud Bˆ at 425 F-91405 Orsay Cedex France e-mail: [email protected] Maria Hoffmann-Ostenhof Fakult¨ at f¨ ur Mathematik Universit¨ at Wien Nordbergstraße 15 A-1090 Vienna Austria e-mail: [email protected] Thomas Østergaard Sørensen Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G DK-9220 Aalborg East Denmark e-mail: [email protected] Communicated by Claude Alain Pillet. Submitted: April 22, 2008. Accepted: July 7, 2008.



The Ponzano–Regge Asymptotic of the 6j Symbol: An Elementary Proof Razvan Gurau Abstract. In this paper we give a direct proof of the Ponzano–Regge asymptotic formula for the Wigner 6j symbol starting from Racah’s single sum formula. Our method treats halfinteger and integer spins on the same footing. The generalization to Minkowskian tetrahedra is direct. All orders subleading contributions can be computed in this setting. This result should be relevant for the introduction of renormalization scales in spin foam models.

1. Introduction The connection between the renormalization group, so successful in describing low energy physics, and the theory of loop quantum gravity is still an open question. A promising line of research is to explore in detail the relationship between the renormalization group and the spin foam quantization of gravity [13] (more precisely the group field theory (GFT) dual to spin foams). GFT [8] can be represented either in terms of group integrals or in terms of tensor models. The situation is highly reminiscent of the one encountered in noncommutative quantum field theory (NCQFT) [4, 6], where the group integral formulation is similar to the direct space representation [10] and the tensorial model is similar to the matrix base representation [5,15] (see also [7] for a beautiful derivation of NCQFT starting from GFT). Particularly, the dual GFT of 3D quantum gravity resembles a φ4 model. We can describe it in terms of a tensorial field theory, with a vertex weight given by Wigner’s 6j symbol, and a trivial propagator [8]. This setting, although very encouraging, is not yet adapted to renormalization. The triviality of the propagator makes the definition of scales unclear. The parallel problem in the NCQFT has been solved by the introduction of spectral scales, first in the matrix base [15], and then in the direct space [10]. Consequently, one may argue that the scales in GFT could be more readily accessible in the tensor model formulation. A deeper understanding of this model is then required.

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It is well known that the 6j symbol obeys the Ponzano–Regge asymptotic formula [14]1 . Several proofs of this formula exist [2, 9, 16], but they rely either on an algebraic definition or on a group integral definition of the 6j symbol. The goal of this paper is to present an alternative proof of this asymptotic formula, developed entirely in the discrete space of indexes of the tensor model, hence, in a formalism presumably better suited to the introduction of scales and renormalization. In the next section we give some notations and state our main theorem. The following section consists of its proof. The last section elaborates on the different generalizations of our method.

2. Notations and main theorem There are several ways of expressing the 6j symbol. The starting point of our derivation is Racah’s single sum formula j1 j2 j3 = Δ(j1 , j2 , j3 )Δ(J1 , j2 , J3 )Δ(J1 , J2 , j3 )Δ(j1 , J2 , J3 ) J1 J2 J3

min pj

(−1)t

max vi

(t + 1)! , (t − v i )! i j (pj − t)!

(1)

with v1 = j1 + j2 + j3 v2 = J1 + j2 + J3 v3 = J1 + J2 + j3 v4 = j1 + J2 + J3 p1 = j2 + J2 + j3 + J3 Δ(j1 , j2 , j3 ) =

p2 = j1 + J1 + j3 + J3

p3 = j2 + J2 + j1 + J1

(j1 + j2 − j3 )!(j1 − j2 + j3 )!(−j1 + j2 + j3 )! . (j1 + j2 + j3 + 1)!

(2)

The j’s and J’s are integers or halfintegers. The sum in eq. (1) is over all integers t such that all the arguments of the factorials are positive. The Δ(j1 , j2 , j3 ) factors are called triangle coefficients. This weight is associated with a tetrahedron with edges j1 , j2 , j3 , J1 , J2 , J3 labeled as in figure 1. The 6j symbol associated with a tetrahedron is non zero if and only if all its faces are triangles in euclidean space. That is, the numbers j1 , j2 , j3 (and their counterparts bordering any face of the tetrahedron) must obey j1 ≤ j2 + j3

j2 ≤ j1 + j3

j3 ≤ j1 + j2

j1 + j2 + j3

is even .

(3)

As a whole, the tetrahedron can be of three types: Euclidean, (its volume is a real number and it can be embedded in Euclidean 3 dimensional space), Minkowskian, (its volume is a complex number and it can be embedded in Minkowskian 3 dimensional space), or degenerate, (is its volume is zero). 1A

generalization can be found in [12].

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The Ponzano–Regge Asymptotic of the 6j Symbol

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J3 j2

j1

J1 J2

j3

Figure 1. Labeling of the tetrahedron. We can rewrite j2 , J3 , J1 in terms of j1 , J2 , j3 like j2 = j3 − j1

J3 = J2 − j1

J1 = J2 − j3 .

(4)

We will prove the following asymptotic formula for the 6j symbol for an Euclidean tetrahedron Theorem 1. Under a rescaling of all its arguments by a large k the 6j symbol associated to an Euclidean tetrahedron behaves like kj1 kj2 kj3 kJ1 kJ2 kJ3

3 1 π 1 1 =√ cos + kji + , (5) θji + kJi + θ Ji 4 i=1 2 2 12πk 3 V where V is the volume of the tetrahedron and θji and θJi are the exterior dihedral angles of the tetrahedron corresponding to the edges ji and Ji respectively. Asymptotic formulae exist also for the Minkowskian and degenerate tetrahedron. As their derivation follows the same steps we will only highlight the modifications one needs to take into account to treat these cases. In all sum and products in the sequel the 4from 1 to 4 while indexes of v run those of p run from 1 to 3. Thus for example (t−vi ) denotes i=1 (t−vi ) whereas 3 (pj − t) denotes j=1 (pj − t).

3. Proof of the main theorem Our proof builds on the techniques developed in [11]. We start by expressing all factorials in eq. (1) by Stirling’s formula. Step two consists in approximating the discrete sum over t in eq. (1) by an integral. In step three we give an asymptotic expression for this integral using a saddle point approximation2 , and obtain Theorem 1. 2 The

proofs of [9, 16] also use saddle point approximations for some integral representations of the 6j symbol.

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As we are interested only in the dominant behavior we will always consider only first order approximations, so that throughout this paper = will mean equal up to a multiplicative factor 1 + 1/k. 3.1. The prefactor We use Stirling’s formula n! =

√

1

2πe(n+ 2 ) ln(n)−n ,

(6)

to express all the factorials. Thus, a typical triangle coefficient will be Δ(kj1 , kj2 , kj3 ) =

2π [k(j1 + j2 + j3 ) + 1] e[k(j1 +j2 −j3 )+ 2 ] ln[k(j1 +j2 −j3 )]−k(j1 +j2 −j3 ) 1

e[k(j1 −j2 +j3 )+ 2 ] ln[k(j1 −j2 +j3 )]−k(j1 −j2 +j3 ) 1

1

e[k(−j1 +j2 +j3 )+ 2 ] ln[k(−j1 +j2 +j3 )]−k(−j1 +j2 +j3 ) e−[k(j1 +j2 +j3 )+ 2 ] ln[k(j1 +j2 +j3 )]+k(j1 +j2 +j3 ) , 1

(7)

were we separated the first term in the denominator. A straightforward computation gives Δ(kj1 , kj2 , kj3 ) 1

= 2πe 2

ln

(j1 +j2 −j3 )(j1 −j2 +j3 )(−j1 +j2 +j3 ) (j1 +j2 +j3 )3

ek[(j1 +j2 −j3 ) ln(j1 +j2 −j3 )+(j1 −j2 +j3 ) ln(j1 −j2 +j3 )+(−j1 +j2 +j3 ) ln(−j1 +j2 +j3 )] e−k[(j1 +j2 +j3 ) ln(j1 +j2 +j3 )] .

(8)

The prefactor of the sum in eq. (1) is a product of four such triangle coefficients. After some manipulations it can be put into the form (2π)2 eH(j,J)+kh(j,J) ,

(9)

with h(j, J) = j1 hj1 + j2 hj2 + j3 hj3 + J1 hJ1 + J2 hJ2 + J3 hJ3 (j1 + j2 − j3 )(j1 − j2 + j3 ) 1 hj1 = ln 2 (j1 + j2 + j3 )(−j1 + j2 + j3 ) (j1 + J2 − J3 )(j1 − J2 + J3 ) , (j1 + J2 + J3 )(−j1 + J2 + J3 )

(10)

and H(j, J) =

1 (hj + hj2 + hj3 + hJ1 + hJ2 + hJ3 ) . 2 1

(11)

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The Ponzano–Regge Asymptotic of the 6j Symbol

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3.2. The integral approximation We now turn our attention to the sum in eq. (1), which we denote by Σ. Denoting v1 = j1 + j2 + j3 etc. (hence without the scale factor k), and separating the t + 1 term in the numerator, the sum becomes Σ=

1 (2π)3

k min pj

eg(t) ,

(12)

k max vi

with

t 1 + t ln(t) − 1 g(t) = ıπt + ln(t + 1) + ln 2 (t − kvi ) (kpj − t)

− (t − kvi ) ln(t − kvi ) − 1 − (kpj − t) ln(kpj − t) − 1 t 1 = ıπt + ln(t + 1) + ln 2 (t − kvi ) (kpj − t) + t ln(t) − (t − kvi ) ln(t − kvi ) − (kpj − t) ln(kpj − t) , where in the last equality we have used i vi = j pj . We change variables to t = kx. The exponent rewrites as x3 1 ln 4 + k ıπx + x ln(x) (x − vi ) (pj − x) 2 k − (x − vi ) ln(x − vi ) − (pj − x) ln(pj − x) .

(13)

(14)

Taking out the k in the first logarithm we write the sum as Σ=

1 (2π)3

min pj

x= max vi

1 F (x)+kf (x) e , k2

(15)

where F (x) and f (x) can be read out of eq. (14). The sum (15) is identified as a Riemann sum. We approximate it by an integral and taking into account that one k −1 factor in eq. (15) plays the role of dx we have min pj 1 Σ= dx eF (x)+kf (x) . (16) (2π)3 k max vi Note that eq. (8) and (16) can be used to defined a {6j} symbol with not only integer and halfinteger entries, but also continuous entries. This continuous version is an analytic continuation of the symbol with (half-)integer entries3 . As k is a large parameter the integral (16) can be computes by a saddle point approximation. Taking into account eq. (9) we find the following contribution of 3 As

(16) is a first order approximation for the 6j symbol the Racah–Elliot the Biedenharn Elliot are expected to hold, replacing the sums by appropriate integrals.

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a saddle point xs to the value of the 6j symbol 1 1 √ eH(j,J)+F (xs )+k[h(j,J)+f (xs )] . 3 −f (x ) 2πk s 3.3. The saddle points The saddle points equation is f (x) = ıπ + ln(x) −

ln(x − vi ) +

ln(pj − x) = 0 ,

(17)

(18)

that is x(p1 − x)(p2 − x)(p3 − x) = −(x − v1 )(x − v2 )(x − v3 )(x − v4 ) .

(19)

The coefficients of x4 and x3 compute to zero. The saddle point equation becomes Ax2 − Bx + C = 0 , with A=−

pk pl +

k 0 under the extra assumption that the area of the group orbits vanishes on the past boundary, namely r0 = 0. In order to complete the proof of strong cosmic censorship for this class of spacetimes, it is therefore sufficient to restrict ourselves to the cases where r0 > 0. This is the content of Theorem 2 which states that T 2 -symmetric spacetimes with Λ > 0, non-vanishing collisionless matter and for which r0 > 0 are C 2 inextendible. Understanding extendibility or inextendibility of vacuum spacetimes is generally thought to be harder as is well illustrated by the difficult analysis of the vacuum Gowdy spacetimes. Theorem 1 states that vacuum T 2 -symmetric spacetimes with Λ > 0 and for which r0 > 0 on the past boundary are generically C 2 inextendible. Thus, Theorem 1 reduces the issue of strong cosmic censorship in this class of spacetimes to the cases where the area of the symmetry orbits takes all 1 More

is actually known since the existence of global areal foliations where r is considered as a time coordinate has been established. Note also that the existence of constant mean curvature foliations has been obtained for these spacetimes.

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T 2 -Symmetric Spacetimes with Λ > 0 and Matter

1427

positive values. This is an unexpected result as it was thought that proving r0 = 0 was a necessary step towards a proof of generic inextendibility. Assuming the spacetime to be extendible, we will first use the C 1 extension of the Killing fields to the Cauchy horizon [7] in order to infer initial data in a bounded null coordinate system for some energy functions defined for T 2 symmetric spacetimes. Analysing these energy functions along characteristics, we will obtain pointwise estimates in a fundamental domain of the universal cover of the spacetime. These pointwise estimates mean that the part of the Cauchy horizon which coincides with the past boundary of the fundamental domain is then everywhere regular. Using this regularity, we will evaluate the Raychaudhuri equation on the past boundary in order to obtain a certain rigidity of the horizon. In the vacuum case, this rigidity will be translated in a new set of initial data on the past boundary for the same energy functions. With these new initial data, we will carry another analysis of the energy functions along characteristics in order to extend our bounds to the tip of the universal cover. As before, the regularity of the horizon and the Raychaudhuri equation gives a rigidity of the horizon. This rigidity, now extended to the whole past boundary of the universal cover, will then imply that one of the metric function needs to be constant throughout the spacetime, which is a non-generic criterion. In the case of collisionless matter, the contradiction arises easily from the rigidity of the past boundary of the fundamental domain and the conservation of the flux of particles as in Theorem 13.2 of [10]. The proofs of Theorem 1 and Theorem 2 rely heavily on these rigidity properties of possible Cauchy horizons which follows once we have shown that Cauchy horizons are necessarily regular. This can be compared with the results of [11] where it was shown that spacetimes which admit a non-degenerate, regular, Cauchy horizon foliated by null closed geodesics have an extra Killing field which is null on the Cauchy horizon. One interesting feature of T 2 -symmetric spacetimes with Λ > 0 and r0 > 0 is that one directly obtains from the Einstein equations that regular Cauchy horizons are necessarily non-degenerate. Moreover, for these models, a Killing field null on the Cauchy horizon could not be one of the Killing fields that generates the T 2 -symmetry because the orbits would not be null. Therefore, the existence of such a Killing field will imply that these spacetimes are non generic. However, in the case of T 2 -symmetric spacetimes admitting a regular Cauchy horizon, it is not known whether the null geodesics are closed so the results of [11] do not apply directly after we have obtained the regularity of the Cauchy horizon.

2. Preliminaries 2.1. T 2 -symmetric spacetimes with spatial topology T 3 A spacetime (M, g) is said to be T 2 -symmetric if the metric is invariant under the action of the Lie group T 2 and the group orbits are spatial. The Lie algebra of T 2 is spanned by two commuting Killing fields X and Y everywhere non-vanishing

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and we may normalise them so that the area element r of the group orbits is given by: g(X, X)g(Y, Y ) − g(X, Y )2 = r2 . Moreover, we will require the geometric part of the initial data to be of the form (Σ, g, K) with Σ diffeomorphic to T 3 and g and K as well as the suitable initial data for the matter fields to be invariant under the action of T 2 . For convenience, we will suppose that the initial data is smooth. The class of initial data for the matter fields is described in Appendix B. It can be easily shown that the maximal Cauchy development (M, g) of such initial data has topology R × T 3 with metric: 2 ds2 = −Ω2 [dt2 − dθ2 ] + e2U dx + Ady + 2(G + AH)dθ 2 + e−2U r2 dy 2 + 2Hdθ , (1) where all functions depend only on t and θ and are periodic in the latter. The so-called Gowdy spacetimes correspond to the particular cases where the functions G and H vanish everywhere, resulting in the orbits of symmetry being orthogonal to the t and θ directions. This orthogonality implies that there is a natural way of prescribing a Lorentzian metric on the 2-manifold Q = M/T 2 . For general T 2 -symmetric spacetimes the orbits are not orthogonal to the t and θ directions but we can still prescribe a metric on the quotient manifold Q by the rule that the inner product of two vectors on the quotient is equal to the inner product of the unique vectors of the spacetime which project onto them orthogonally to the orbit. In the following, the lift of (Q, gQ ) to its universal cover will be denoted g) and more generally, we will write T the lift of T to Q, for any T . by (Q, By a change of coordinates of the form v = α(t + θ), u = β(t − θ), we may rewrite the metric (1) in null coordinates: 2 dv du − ds2 = −Ω2 dudv + e2U dx + Ady + (G + AH) α (v) β (u) 2 dv du − + e−2U r2 dy + H , (2) α (v) β (u) relabelling Ω suitably. By choosing α, β appropriately, we can ensure that Ω = 1 along a constant v ray or a constant u ray as needed later, but note that by doing so, we will also break the periodicity in θ. It is easy to see from the form of the metric that any constant u (or v) hypersurface is then a null hypersurface, however, in general, a constant (u, x, y) ray will not be a null ray because of the non-orthogonality of the orbits. Let us define the two null vectors which are orthogonal to the orbits by N(u) , N(v) : 1 ∂ ∂ ∂ + +H N(u) = G , (3) ∂u β ∂x ∂y 1 ∂ ∂ ∂ − +H G . (4) N(v) = ∂v α ∂x ∂y

Vol. 9 (2008)


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A computation shows that Ω−2 is an affine parameter for N(u) , N(v) i.e: ∇ N(u) Ω2

N(u) = 0, Ω2

∇ N(v) Ω2

N(v) = 0. Ω2

(5)

The affine lengths curves of N(u) and N(v) are therefore given of the integral respectively by u Ω2 du and v Ω2 dv. Finally by analogy with [2], we define the twist quantities: Hu (Gt + AHt ) Gu Gv Hv Γ=2 = 2 + 2 + 2A + , (6) α β α β α β Ht Hu Hv Π=2 =2 +2 . (7) αβ α β 2.2. The Einstein equations in null coordinates The Einstein equations give rise to the following system: Null constraint equations:

1 ∂v Ω−2 rv e−2U = − Ω−2 e2U A2v − 2re−2U Ω−2 Uv2 2r β − 2πre−2U (ρ + P1 − 2J1 ) , α

−2 −2U 1 −2 2U 2 = − Ω e Au − 2re−2U Ω−2 Uu2 ∂u Ω ru e 2r α − 2πre−2U (ρ + P1 + 2J1 ) . β

(8)

(9)

Evolution equations: r re2U Γ2 r3 e−2U Π2 ruv = 2πrΩ2 (ρ − P1 ) + Ω2 Λ + + , 2 8 Ω2 8 Ω2 1 1 1 Uuv = − (rv Uu + ru Uv ) + 2 e4U Au Av + Ω2 Λ 2r 2r 4 e2U Γ2 + + πΩ2 (ρ − P1 + P2 − P3 ) , 8 Ω2 1 Auv = −2(Av Uu + Au Uv ) + (Au rv + Av ru ) 2r r2 e−2U ΓΠ + + 4πrΩ2 e−2U S23 , 4 Ω2 1 1 (rv Uu + ru Uv ) ∂u ∂v log Ω = −Uu Uv − 2 e4U Au Av + 4r 2r 3e2U Γ2 3r2 e−2U Π2 − − − πΩ2 (ρ − P1 + P2 + P3 ) . 2 16 Ω 16 Ω2

(10)

(11)

(12)

(13)

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Auxiliary equations: ∂v

Γ 2U re Ω2

U

= 8πrΩe

β (S12 − J2 ) , α

α (S12 + J2 ) , β Π 3 −2U β Γ 2U 2 −U ∂v r e re A = 8πr Ωe (S13 − J3 ) , + v Ω2 Ω2 α Π 3 −2U β Γ ∂u r e (S13 + J3 ) . + 2 re2U Au = 8πr2 Ωe−U 2 Ω Ω α ∂u

Γ 2U re Ω2

= 8πrΩeU

(14) (15) (16) (17)

Γ and Π are given by (6) and (7) and ρ, Pk , Jk , Sjk are the components of the energy-momentum tensor in the orthonormal frame: −1/2 ∂ ∂ ∂ , , E0 = −g ∂t ∂t ∂t −1/2 ∂ ∂ ∂ ∂ ∂ E1 = −g , −G −H , ∂t ∂t ∂θ ∂x ∂y ∂ , E2 = e−U ∂x ∂ ∂ U −1 −A E3 = e r . (18) ∂y ∂x An introduction to collisionless matter can be found in Appendix A. Writing the energy-momemtum tensor in an invariant orthonormal frame will be useful later as it is difficult to interpret the energy conditions such as the dominant and the strong energy conditions for the matter in the basis associated with the (u, v, x, y) coordinates. In the following, we shall often use the notation rv = λ, ru = ν. 2.3. Global null coordinates on Q and global null To conclude the preliminaries, let us summarize the existence of Q coordinates as follows: Proposition 1. Let (M, g, f ) be the maximal Cauchy development of T 2 -symmetric initial data with non-negative cosmological constant Λ ≥ 0 and (possibly vanishing) Vlasov matter. Let Q denote the space of group-orbits: Q = M/T 2

(19)

and let π1 : M → Q denote the standard projection. Then, there exists smooth functions U Q , AQ , rQ , GQ , H Q and ΩQ defined on Q, with rQ and ΩQ strictly positive, and a smooth non-negative function f Q defined on Q × R3 such that the following holds:

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• There exists a globally defined coordinate system (t, θ, x, y) covering M such that g satisfies (1) for some smooth functions A, U , r and Ω, with r and Ω strictly positive, and such that all functions are independent of x and y and are periodic with period 1 in θ. • U , A, r, Ω, f are the pull-back of U Q , AQ , rQ , ΩQ and f Q by π1∗ . the universal cover of Q. Let U , A, r, G and H be the Let moreover denote by Q Q Q Q Q Q Q 3 lifts of U , A , r , G , H to Q and let f be the lift of f to Q × R . Let finally α and β be two smooth functions defined on R and such that α > 0 and β > 0. Then the following holds: and a smooth • There exists global null future-directed coordinates u and v on Q 2 strictly positive function Ω such that Ω α (v)β (u) is the lift of Ω2 to Q, • U , A, r, G, H, Ω, α, β and f satisfy the system (8)–(17), where in the case of non-vanishing Vlasov matter, ρ, Pk , Sjk are defined by (88)–(91) and all functions in the equations should be replaced by their tilde versions. In the above proposition, we have assumed smoothness of the initial data for convenience but one could easily obtain statements for initial data lying in a lower class of differentiability. Proof. The proof follows by standard arguments as found in [2, 3, 5] and a change of coordinates of the type v = α(t + θ), u = β(t − θ). The above proposition shows the existence of a global null coordinate system but note that we will allow ourselves to move to other global null coordinate on Q, systems, by rescaling u and v, as for instance in Proposition 2. In other words, by choosing α and β appropriately, we will simplify our analysis. The rules which dictate the change of coordinates are described in detail in Appendix D. In the rest of this article, we will, by an abuse of notation, drop the tilde on the functions defined on Q.

3. The theorems Theorem 1. Let (M, g) be the maximal Cauchy development of vacuum T 2 symmetric initial data with positive cosmological constant Λ > 0 as described above and suppose that: 1. There exists a maximal past-directed causal geodesic γ(t) such that r(t) → r0 > 0. 2. The Killing fields generating the T 2 -symmetry cannot be chosen so that they are mutually orthogonal on the initial Cauchy surface. Then (M, g) is past inextendible as a C 2 Lorentzian manifold. Assumption 2 is equivalent to the statement that the function A as defined in (1) is not constant in M.

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Theorem 2. Let (M, g) be the maximal Cauchy development of T 2 -symmetric initial data with positive cosmological constant Λ > 0 and collisionless matter as described above and suppose that: 1. There exists a maximal past directed causal geodesic γ(t) such that r(t) → r0 > 0. 2. The Vlasov field f does not vanish everywhere. Then (M, g) is past inextendible as a C 2 Lorentzian manifold. Note that for any maximal Cauchy development of T 2 -symmetric data with Λ ≥ 0 and collisionless matter, it is easy to see that r tends to the same limit r0 for all maximal past directed geodesics. Since the cases where Λ = 0 were treated in [9] and since the methods of [9] may also be applied in the cases where Λ > 0 and r(t) → 0 along past directed causal geodesics, we immediately obtain from Theorem 2 and Theorem 4.1 of [9]: Corollary 1. Let (M, g) be the maximal Cauchy development of T 2 -symmetric data with non-negative cosmological constant and collisionless matter and let f : P → R denote the Vlasov field and suppose that: 1. There exists a constant δ > 0 such that for any open U ⊂ P ∩ π −1 (Σ) we have that f does not vanish identically on U ∩ {p : g(p, X)2 + g(p, Y )2 < δ}. Then (M, g) is inextendible as a C 2 Lorentzian manifold. Finally, since future inextendibility holds for T 2 -symmetric spacetimes with non-negative cosmological constant and collisionless matter in view of the results of [7], we can replace past inextendible by past and future inextendible in Theorem 1, Theorem 2 and Corollary 1.

4. Initial data for the estimates In this section, we will show how we can construct appropriate initial data to perform estimates of the type found in [2] in bounded null coordinates. We therefore want to establish the existence of a bounded null coordinate system (u, v) such that along a certain null ray v = v1 , A, U and r as well as their first derivatives are bounded. First, we will need the following: 4.1. Extendibility of the Killing vector fields Lemma 1. Let (M, g) be the past maximal Cauchy development of T 2 -symmetric initial data with positive cosmological constant, in the vacuum or with Vlasov matter, as in Theorem 1 and 2. Suppose that r0 > 0 and that (M, g) is extendible as a Lorentzian manifold with C 2 metric. Let γ be a causal geodesic leaving (M, g) and p be the intersection of γ with the past boundary of M. Then r, U and A admit a C 1 extension along γ to p.

Vol. 9 (2008)


1433

Proof. We know from [7] that the Killing vector fields generating the surface of symmetry must have C 1 extensions to the Cauchy horizon in any C 2 extension of the spacetime. Suppose that γ is a causal geodesic leaving (M, g) and let p be the intersection of γ with the past boundary of M. We then have that g(X, X), g(X, Y ), g(Y, Y ) are C 1 functions along γ and in particular, they are bounded along γ until p. From (1), we have: g(X, X) = e2U , 2U

g(X, Y ) = e

A,

2 −2U

g(Y, Y ) = r e

(20) (21) 2 2U

+A e

.

(22)

From (22), we see that e−2U is bounded above since r0 > 0, which implies that U is bounded below. By (20) and (21), this implies that r, U and A are at least C 1 along γ. Using this lemma, we may then reduce the issue of inextendibility to this of orbits-orthogonal null geodesic inextendibility. 4.2. Reduction to orbits-orthogonal null geodesic inextendibility We will adapt the method of Proposition 13.1 of [10] to our geometry in order to prove the following lemma: Lemma 2. Let (M, g) be the past maximal Cauchy development of T 2 -symmetric initial data with positive cosmological constant, in the vacuum or with Vlasov matter, as in Theorem 1 and 2. Suppose that r0 > 0 and that (M, g) is extendible as a Lorentzian manifold with C 2 metric. Then there exists a null line, orthogonal to the orbits of symmetry, which leaves (M, g) and enters a C 2 extension of (M, g). Proof. Note that for any geodesic, the following quantities (conservation of angular momentum) are conserved: v˙ u˙ 2U − x˙ + Ay˙ + (G + AH) , (23) Jx = e α β v˙ u˙ − (24) + AJx . Jy = r2 e−2U y˙ + H α β Let H+ denote the past boundary of M in an extension M and let p ∈ H+ be such that there exists no null geodesic orthogonal to the orbits leaving the spacetime in a neighbourhood of p. Following [10], we can find a sequence of regular points pi ∈ H+ converging to p on H+ , and planes Oi , Ti such that Oi and Ti are respectively, the planes orthogonal and tangent to the orbits of symmetry. The planes Oi can also be regarded as the set of vectors with vanishing angular momentum Jx , Jy . Conservation of angular momentum implies that the planes Oi are null and their null generator Ki is necessarily tangential to Hp+i . We can then extract a subsequence Oi converging to a necessarily null plane O at p. We may then draw a convergent

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J. Smulevici


subsequence Ti converging to T . Since Ti and Oi are orthogonal, Ti is also null and there exists a null vector K ∈ O ∩ T . The achronality of H+ at p implies the existence of timelike geodesics entering M at p. If γ is such a geodesic, g(K, γ) ˙ = 0. Let Kj be a sequence of vectors tangential to the orbits of symmetry along γ and converging to K. The CauchySchwarz inequality implies:

(25) ˙ ≤ g(Kj , Kj )1/2 e−2U Jx2 + r−2 e2U (Jy − AJx )2 . g(Kj , γ) Since we have previously shown that U and A are bounded and Jx and Jy are constant along γ, as Kj goes to K, the right hand side goes to zero which is a contradiction. In order to provide the initial data necessary to perform the estimates of the next sections, we first need to understand some basic notions concerning the local and the global geometry of T 2 -symmetric spacetimes. First, we will need the following: 4.3. Monotonicity of r Lemma 3. Let (M, g) be the past maximal Cauchy development of T 2 -symmetric initial data with positive cosmological constant, in the vacuum or with Vlasov matter, as in Theorem 1 and 2. Then the gradient of r is timelike, which in double null coordinates means that: λν > 0 (26) and by a choice of orientation that: λ > 0,

ν > 0.

(27)

Proof. It is easy to see that Proposition 3.1 of [13] holds for T 2 -symmetric spacetimes with non-negative cosmological data and collisionless matter. It follows that either λν > 0 or the spacetime is flat. Since the term containing the cosmological constant on the right hand side of (10) ensures that the latter case does not occur, we have λν > 0. Choosing the time orientation such that the future corresponds to the expanding direction, we can assume λ > 0, ν > 0. 4.4. Global structure of Q In this section, we describe some global geometric features of Q. Lemma 4. Let (M, g) be the past maximal Cauchy development of T 2 -symmetric initial data with positive cosmological constant, in the vacuum or with Vlasov matter, as in Theorem 1 or 2. Suppose that r0 > 0. Then all integral curves of N(u) and N(v) are incomplete in the past direction. Proof. Since the dominant energy condition holds for Vlasov matter, we have ρ − P1 ≥ 0. Therefore all terms on the right-hand side of equation (10) are nonnegative and we have for all (u, v) and fixed u , u u 2 ΛΩ rdu ≤ 2ruv ≤ 2λ(u1 , v) (28) u

u

Vol. 9 (2008)


which implies, using the lower bound r ≥ r0 > 0, that: u 2λ(u , v) Ω2 du ≤ Λr0 u

1435

(29)

and similar inequalities hold for constant u null curves. Since Ω2 is an affine parameter for the integral curves of N(u) and N(v) , we have proved that their affine length is bounded.

v

=

r 0

Assuming r0 > 0, it is easy to see We may then draw a Penrose diagram for Q. that the range of the coordinates (t, θ, x, y) as in (1) is given by R × [0, 1]3 , using the classical energy estimates for T 2 -symmetric spacetimes as found in [2, 3, 5]. Taking a parametrisation u = α(t − θ), v = β(t + θ) such that Ω = 1 along a constant u and a constant v null curves, we can represent the Penrose diagram of by (see [10] for example): Q

r

= V =

u

r

=

U ,

, r0

4.5. The initial data We are now ready to build a null coordinate system and initial data tailored for the estimates of the next section. Proposition 2. Let (M, g) be the past maximal Cauchy development of T 2 symmetric initial data with positive cosmological constant, in the vacuum or with Vlasov matter, as in Theorem 1 or 2. Suppose that r0 > 0 and that there exists an integral curve of N(u) which leaves (M, g) and enters a C 2 extension of (M, g). with Then, there exists a double null bounded coordinate system (u, v) covering Q, null lines v = v1 , v = v0 , u = u1 , and a constant Ω0 > 0 such that the following holds: 1. Along v = v1 , r, U , A and their first derivatives may be continuously extended to u = U . 2. The null curve v = v0 is the image by the deck transformation of the null curve v = v1 .

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3. Moreover, the following holds:

sup

ν(u, v1 )

u∈(U,u1 ]

Ω(u, v1 ) = Ω0 , v1 − v 0 3 +√ = 1, r0 r0 λ(u1 , v) = 1 .

(30) (31) (32)

In order Proof. From Proposition 1, there exist global null coordinates (u, v) on Q. to build null coordinates such that the estimates of the proposition hold, we will rescale (u, v) and then exploit the regularity of ν along the null curve leaving ˜˜, v˜˜), satisfying the the spacetime to construct another null coordinate system, (u requirements of the proposition. Let thus γ be an integral curve of N(u) leaving the spacetime and suppose that in the original coordinate system (u, v), γ is given by v = v1 . Fix moreover a ∩ {(u, v)/ u ≤ u1 , v ≤ v1 }. Let v = v0 be the image null line u = u1 . Let T be Q by the deck transformation of the null line v = v1 . One may visualise T and the null curves v = v1 , v = v0 and u = u1 , in a Penrose diagram of Q:

v

u

=

v1

u

1

=

v = v0

T

We define a new coordinate system (˜ u, v˜) as follows. First define v˜ by: v1 v˜(v) = v1 − λ(u1 , v )dv = v1 − r(u1 , v1 ) + r(u1 , v) . (33) v

It follows immediately from the bound on r that v˜ is bounded and thus take values in (V, v1 ] for some finite V . Using v˜ as a replacement for v, one obtain a coordinate ˜ = rv˜ satisfies: system (u, v˜) for which λ ˜ 1 , v˜) = 1 . λ(u In the following, we drop the tilde on v˜. Let Ω0 be a strictly positive constant. Define u ˜ by: u1 2 Ω (u , v1 ) du . u ˜ = u1 − Ω20 u

(34)

(35)

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1437

From Lemma 4, the affine length of v = v1 is finite. Thus, u ˜ is bounded from u, v) defines a new below and takes value in (U, u1 ] for some finite U . Moreover, (˜ ˜ 2 = −2gu˜v satisfies: coordinate system for which Ω ˜ 2 (u, v1 ) = Ω20 . Ω

(36)

Note that since u ˜ is a function of u only, the change of coordinate does not affect equation (34). Note also that since r is a purely geometric quantity, it is invariant and therefore, that r0 is invariant. under a change of coordinates on Q, Since Ω is constant along v = v1 , it then follows that N(˜u) is parallely transported along v = v1 . From Lemma 1, r is a C 1 function along v = v1 up to u = U . In particular, ν˜ = ru˜ is bounded along v = v1 . Let A denote 0 ˜˜ by: √3 . We define u + sup(ν(u, v1 )) v1r−v r0 0 ˜˜ = A˜ u u.

(37)

˜˜ u ˜˜, v), Ω( ˜˜, v1 ) is also constant, the constant In the coordinate system given by (u Ω 0 being given by √A . Thus, N(u˜˜) is also parallely transported along v = v1 and it follows from Lemma 1 that r, A, U and their first derivatives admit continuous extension along v = v1 to u = U . Moreover, we have:

0 ˜ In particular, ν˜ ˜(u ˜, v1 ) v1r−v + 0

√3 r0

˜ sup ν˜ ˜(u ˜ , v1 )

ν˜ ν˜˜ = . A

(38)

≤ 1 and: v1 − v 0 3 +√ r0 r0

= 1.

˜ Thus, (u ˜, v) satisfies the requirements of the proposition.

(39)

5. Estimates in a fundamental domain of the universal cover In this section, we will assume that Proposition 2 holds in order to derive energy Let therefore (u, v) be the bounded null estimates in a fundamental domain of Q. coordinate system of Proposition 2, such that T = (U, u1 ] × (V, v1 ] and v = v1 is an integral curve of N(u) leaving the spacetime. Let v = v0 be the image by the deck transformation of v = v1 and define F by: F = (U, u1 ] × [v0 , v1 ] . We may represent F in a Penrose diagram:

(40)

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v v1

u 1

=

u

=

F

v = v0

v =

u

=

U

V

We will perform estimates in F, using the bounds we previously obtained along v = v1 in Section 4 and the bounds coming from the compactness of the intersection of u = u1 with F. 5.1. Uniform bounds on ν, λ in a fundamental domain From (10), it follows that ruv ≥ 0 and therefore ν and λ are bounded above by their values on the intersection of u = u1 with F and their values on v = v1 . 5.2. Uniform bounds on Uu , Uv , Au , Av The method follows [2, 3, 5] but the final Gronwall type argument needs to be modified to be adapted to the null bounded coordinate system and the facts that Λ > 0 and that matter fields may be present. We define the quantities G and H by: e4U 2 (Av + A2u ) , 4r e4U 2 H = r(Uv2 − Uu2 ) + (Av − A2u ) . 4r G = r(Uv2 + Uu2 ) +

(41) (42)

The positive quantities G + H, G − H satisfy: Γ2 e4U rAv e2U ΓΠ Au Av + rUv e2U 2 + 2 2r 2Ω 4 Ω2 2 2U 2 2 + rUv Ω Λ + e Av 4πΩ S23 + rUv 4πΩ (ρ − P1 + P2 − P3 ) , (43)

∂u (G + H) = −2Uv Uu λ + λ

Γ2 e4U rAu e2U ΓΠ Au Av + rUv e2U 2 + 2 2r 2Ω 4 Ω2 2 2U 2 2 + rUu Ω Λ + e Au 4πΩ S23 + rUu 4πΩ (ρ − P1 + P2 − P3 ) . (44)

∂v (G − H) = −2Uv Uu ν + ν

We can integrate (43) and (44) along constant v and u curves:

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v =

(u1 , v)

v1

F

u

=

u 1

(u, v1 )

(u, v)

v v

V

=

=

u

=

U

v0

We obtain:

u1

e4U 2Uu Uv λ − λ 2 Au Av 2r

[G + H](u, v) = [G + H](u1 , v) + u u1 2 Γ ΓΠ e2U 2U 2 rAv 2 + rUv Ω Λ − + rUv e 2Ω2 4 Ω uu1

2U − e Av 4πΩ2 S23 + rUv 4πΩ2 (ρ − P1 + P2 − P3 ) , u v1 e4U [G − H](u, v) = [G − H](u, v1 ) + 2Uu Uv ν − ν 2 Au Av 2r v v1 2 2U ΓΠ e 2U Γ 2 rAu 2 + rUu Ω Λ − + rUu e 2Ω2 4 Ω v v1

2U − e Au 4πΩ2 S23 + rUu 4πΩ2 (ρ − P1 + P2 − P3 ) .

(45)

(46)

v

Note that from the definition of the Vlasov matter2 we have: ρ ≥ P1 + P 2 + P 3 ,

P2 + P3 ≥ 2|S23 | ,

Pi ≥ 0 ,

∀i = 1, 2, 3 .

(47)

From the first inequality we obtain: (ρ − P1 + P2 − P3 ) ≤ 2(ρ − P1 ) . 4U 2|Uu Uv |+ e2r2

2U

e

(48)

Using these inequalities as well as |Au Av | ≤ |Ui | ≤ (G/r)1/2 , √ |Ai | ≤ 2 rG for i = u, v, we obtain: u1 1/2 u1 G Γ2 G λ + r e2U 2 [G + H](u, v) ≤ [G + H](u1 , v) + r r 2Ω u u u1 √ r|ΓΠ| √ + 2 rG + rGΩ2 Λ 4Ω2 u u1 √ √ 2 rG4πΩ2 S23 + rG8πΩ2 (ρ − P1 ) , (49) + u

2 See

Appendix A.

G r,

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u1 1/2 u1 G Γ2 G ≤ [G + H](u1 , v) + λ + r e2U 2 r r 2Ω u u u1 √ √ r|ΓΠ| √ + 2 rG + rGΩ2 Λ + rG12πΩ2 (ρ − P1 ) . 4Ω2 u

(50)

Using the inequality, 2ab ≤ a2 + b2 to estimate the product of the twist quantities, we obtain: u1 1/2 u1 G Γ2 G λ + r e2U 2 [G + H](u, v) ≤ [G + H](u1 , v) + r r 2Ω u u u1 1/2 2 2U G Γ e r Π2 e−2U r3 + 2 + r 8Ω2 8Ω2 u u1 √ √ + rGΩ2 Λ + rG12πΩ2 (ρ − P1 ) . (51) u

We then use equation (10) to estimate all the matter terms as well as the terms containing the twist quantities and the cosmological constant: u1 √ 6 G (52) λ + √ ruv G . [G + H](u, v) ≤ [G + H](u1 , v) + r r u Thus, using the compactness of {u1 } × [v0 , v1 ], there exists a positive stant C depending on the value of the metric functions on F ∩ {u = u1 } that: u1 √ 6 G [G + H](u, v) ≤ C + λ + √ ruv G . r r u Similarly, we have: v1 √ 6 G ν + √ ruv G , [G − H](u, v) ≤ C + r r v

consuch (53)

(54)

for some positive constant C . To close the estimates, one would like to apply a Gronwall type argument to the inequalities (53) and (54). One cannot do this directly, as the right-hand sides of (53) and (54) depends on G and not on, respectively, G + H and G − H. One is therefore tempted to add the√two inequalities, in order to obtain G on the left-hand side. Using the estimate G ≤ G+1 2 , we would obtain an inequality of the form: u1

(D + Eruv ) G (u , v)du G(u, v) ≤ C + v1u

(D + Eruv ) G (u, v )dv . + (55) v

Note that it is not possible a priori to apply directly Gronwall’s inequality to the above inequality, for instance by considering v to be fixed and applying a Gronwall argument as for a function of the variable u only, as by doing so, one would then need to estimate a volume integral of G. In the vacuum case with no

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cosmological constant, there are no terms containing ruv on the right-hand sides of (53) and (54). One can then take a supremum over one variable and apply a Gronwall argument to conclude, see [3]. However, when terms containing ruv appear, one cannot take a supremum anymore, as it is not known, a priori, that the integral of the supremum of ruv is bounded.√ This is why we will not apply the estimate G ≤ G+1 2 to the inequality (53). On the other hand, because of the identity (31), we will not need to be particularly careful with the terms arising from (54) in order to close the estimates. Thus, we √ in the inequality (54): may apply the estimate G ≤ G+1 2 v1 3 G [G − H](u, v) ≤ C + (56) ν + √ ruv (G + 1) , r r v v1 ν ruv + 3√ (57) [G − H](u, v) ≤ C + G (u, v )dv , r r v for some positive constant C depending on the bounds of the metric functions along v1 coming from Proposition 2. Adding (53) and (57), we obtain: √ 1 u1 6 G G(u, v) ≤ C + λ + √ ruv G (u , v)du 2 u r r v1 ν ruv 1 + 3√ + (58) G (u, v )dv 2 v r r and since v ≥ v0 for all (u, v) ∈ F, we have: √ 6 1 u1 G √ ruv G (u , v)du ≤C+ λ + 2 u r r ν ruv 1 v1 √ +3 + G (u, v )dv . 2 v0 r r

(59)

Since [v0 , v1 ] is compact and G continuous, there exists, for every u ∈ [u, u1 ], a vm (u ) ∈ [v0 , v1 ] such that G(u , vm (u )) = sup[v0 ,v1 ] G(u , .) and we can define F (u ) by F (u ) = sup[v0 ,v1 ] G(u , .) = G(u , vm (u )). We have: √ 6 1 u1 G G(u, v) ≤ C + λ + √ ruv G (u , v)du 2 u r r v1 ν ruv 1 + 3√ + F (u) (60) (u, v )dv , 2 r r v0 √ 6 1 u1 G ≤C+ λ + √ ruv G (u , v)du 2 u r r v1 − v 0 1 3 + F (u) sup ν +√ . (61) 2 r0 r0 T

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From equation (31) of Proposition fact that ruv ≤ 0 coming 2 and the

0 √3 + equation (10), we have that supT ν v1r−v r0 = 1. Thus, we obtain: 0 u1 √ 6 1 G G(u, v) ≤ C + λ + √ ruv G (u , v)du 2 u r r

1 + G u, vm (u) . 2 We can evaluate (62) at the point (u, vm (u)): √

6 1 u1 G F (u) = G u, vm (u) ≤ C + λ + √ ruv G u , vm (u) du 2 u r r 1 + F (u) . 2 It follows that: u1 √

6 G F (u) ≤ 2C + λ + √ ruv G u , vm (u) du r r u and therefore:

λ(u , vm (u)) F (u )du r 0 u u1 6ruv (u , vm (u)) + sup F du , √ r0 [u,u1 ] u u1 ≤ 2C + B F (u )du + D sup F ,

F (u) ≤ 2C +

from

(62)

(63)

(64)

u1

(65)

[u,u1 ]

u

for some constant B and D depending on the uniform bound on λ obtained in Section 5.1. Apply now an inequality of Gronwall type to obtain: u1

F (u) ≤ 2C + D sup F + 2C + D sup F B exp B(u − u) du . (66) [u,u1 ]

Note the trivial fact that: sup u ∈[u,u1 ]

We obtain: F (u) ≤ 2C + D

[u ,u1 ]

u

sup F

[u ,u1 ]

sup F + D

[u,u1 ]

=

sup F .

(67)

[u,u1 ]

sup F (u1 − U )B exp B(u1 − U ) [u,u1 ]

+ 2C(u1 − U )B exp B(u1 − U ) .

(68)

Therefore, there exist constants A , B , independent of u and v such that: F (u) ≤ A + B sup F . (69) [u,u1 ]

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This last inequality is true in particular at every um where F (u) reaches its maximum in [um , u1 ]: F (um ) ≤ A + B F (um ) , (70) √ B + B 2 +4A and by definition of um , we from which we obtain that F (um ) ≤ 2 obtain an upper bound on F , which implies an upper bound on G. This uniform bound on G gives us uniform bounds on the first derivatives of U and this implies that we can extend U continuously to the past boundary of F in M . We can then use the uniform bound on G to get uniform bounds on the first derivatives of A and extends A continuously. 5.3. Uniform bounds on Ω, Ωu , Ωv Observing that the integrals of the twist quantities and the matter terms that appear in (13) can be bounded using (10), we obtain bounds on Ωu /Ω, Ωv /Ω and then bounds above and below on log Ω. In particular we can extend Ω continuously to the past boundary of F to a strictly positive function.

6. The vacuum case In the following section, we will complete the proof of Theorem 1. In the vacuum case, the right-hand sides of all the auxiliary equations vanish. This implies that by choosing a linear combination of X and Y , we can ensure that Γ = 0 and ΩΠ2 r3 e−2U = K for some constant K. Note that in this case, the terms containing the twist quantities in equations (11) and (12) vanish. Moreover, in the fundamental domain F of Section 5, the terms containing the twist quantities in equation (10) are bounded. We therefore obtain bounds on Uuv , Auv and ruv . Finally, since there are no more matter terms in the right-hand sides of (9) and (8), we obtain bounds on ruu and rvv . We shall first show that Av = Uv = 0 on the past boundary before extending the estimates to the tip of the universal cover. The inextendibility criterion will then follow easily. 6.1. Uniform bounds on Avv , Uvv To get the bounds on Avv , Uvv , we first take the v derivative of (11) and (12): ru (71) Auvv = −2Avv Uu + Au Uvv + Avv + φ(u, v), 2r 1 1 Uuvv = − νUvv + 2 e4U Au Avv + ψ(u, v) , (72) 2r 2r where φ and ψ contain some previously bounded functions. Choose a null ray v = v belonging to F. The bounds previously found are valid along this ray.

vvWe can consider the system (71) as a differential equation in u for the vector cA Uvv : d Avv φ Avv =B + , (73) Uvv Uvv ψ du

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where B is 2 × 2 matrix whose coefficients are bounded. We can now integrate the last equation, take the norms and use Gronwall’s lemma to get the bounds on Avv , Uvv .

6.2. Value of λ, A, U , Av , Uv , Avv , Uvv on the past boundary Since r has been shown to have at least a C 1 extension to the boundary and is constant on u = U , we have λ = 0 on u = U . This implies from the Raychaudhuri equation (8) that Av =Uv =0. Repeating the argument of Section 6.1, we obtain bounds on the first derivatives of Avv and Uvv . Since Av and Uv are constant on u = U , we obtain that Avv and Uvv extend continuously to 0 on u = U .

6.3. Periodicity Applying the deck transformation to the original fundamental domain, we can bound the same quantities 3 in any fundamental domain and obtain in particular that A, U as well as their first and second v-derivatives have continuous extension on u = U , with Av = Uv = Avv = Uvv = 0.

6.4. Extension to the tip of the universal cover We can now apply a similar method while changing the initial data to obtain estimates for the tip of the universal cover. For p = (u, v) ∈ T , we integrate again equations (43) and (44) along constant v and constant u rays:

v

u 1

(u, v1 )

=

=

u

v1

=

u

v

=

U

(u, v)

V (U, v)

3 The

bounds on A and U do not depend on the fundamental domain by periodicity but all other bounds depend on the fundamental domain because of the broken periodicity.

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We obtain:

[G + H](u, v) = [G + H](U, v) − u + rUv Ω2 Λ , U

u

U

e4U 2Uu Uv λ + λ 2 Au Av 2r

(74)

[G − H](u, v) = [G − H](u, v1 ) + v1 − rUu Ω2 Λ .

1445

v

v1

2Uu Uv ν − ν

e4U Au Av 2r2

(75)

v

Similarly to the estimates of Section 5.2, we obtain: u ruv G [G + H](u, v) ≤ λ + 3 √ (G + 1) , r r U u λ ruv 3 + 3√ [G + H](u, v) ≤ Gdu + √ λ(u, v) r r0 r U and

[G − H](u, v) ≤ v

v1

ruv √ G ν + 6 √ G dv + C , r r

(76) (77)

(78)

for some positive constant C depending on the bounds of the metric functions along v1 as derived in Section 5. Therefore, we have: ruv ruv √ 1 v1 1 u λ G + 3√ (79) G≤D+ ν + 6 √ G dv . Gdu + 2 U r 2 v r r r Since we know that for every v ∈ [V, v0 ], G(., v) is a continuous function on the compact [U, u1 ], we can then apply the method of the end of Section 5.2, we only need to interchange the role of u and v in the argument. Therefore, we have derived a uniform bound on G in T , which gives us a bound on Ui , U , Ai and A. We can now continue as in Section 5 to get the same estimates. In particular, this implies a continuous extension of Au to 0 on v = V . From the Raychaudhuri equations, we obtain bounds on rvv , ruu valid in T . As in Section 6.1, we can obtain higher order estimates of A and U in every fundamental domain. The continuous extension of Auu and Uuu and the fact that Au is constant on v = V implies that Auu = 0 and Uuu = 0 everywhere on v = V . Once we have this information, we can apply again energy estimates of the type of Section 6.1 replacing Ai by Aii and Ui by Uii in the definition of H and G and taking the initial data on the past boundary. We obtain this way uniform bounds on the second derivatives of A and U which are independent of the fundamental domain. From equation (12), we obtain Auv = 0 on the past boundary and therefore Au = 0, Av = 0 everywhere on the past boundary using a continuity argument.

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6.5. The contradiction Equation (12) together with the C 1 initial data Au = 0 on u = U , Av = 0 on v = V form a well-posed characteristic initial-value problem with a homogeneous hyperbolic equation for A and trivial initial data. Standard arguments implies ˜ which is a contradiction in view of our initial that A is constant everywhere in Q assumption. Theorem 1 is thus proved.

7. The Vlasov case We will now proceed to the proof of Theorem 2. We will therefore suppose that the Vlasov field f does not vanish identically in (M, g), as in the statement of 1. Our argument is based on an adaptation to our geometry of the proof of Theorem 13.2 of [10] and on the estimates of Section 5. In the following the indices u ˆ, vˆ, x, y will ∂ ∂ , ∂y ) basis and be used to denote the components of tensors in the (N(u) , N(v) , ∂x indices (0,1,2,3) will be use to denote the components of tensors in the basis (18). bounded by two null curves, Let F denote a fundamental domain for Q v = v0 , v = v1 , v1 ≥ v0 and let F = F ∪ τ (F ) ∪ τ −1 (F ) where τ generates the deck transformations. F is bounded by two null curves, v = v2 , v = v−1 : F

v =

u

=

0

v2 u

v−

=

=

U

v 1

By a change of parametrisation, set Ω = 1 along v = v2 . Since T uˆvˆ , T vûˆ , T can be related to the components of curvature in a parallely transported null frame on a null geodesic entering a C 2 extension, they can be uniformly bounded along v = v2 . We either have pvˆ ≥ 1, in which case, pvˆ ≤ (pvˆ )2 and therefore N vˆ ≤ T vˆvˆ , or pvˆ ≤ 1, in which case pvˆ ≤ (puˆ )2 and N vˆ ≤ T uûˆ . It follows that N vˆ is bounded pointwise and thus the flux through v = v2 is bounded. By periodicity, the flux through v = v−1 is also bounded. From these bounds and conservation of particle current, it follows that the particle flux is uniformly bounded along any constant u null ray in F and approaches the initial flux through F ∩ Σ, as u → U . u û ˆ


Vol. 9 (2008)

Since f is not identically zero in (M, g), we have: v1 N uˆ rΩ2 dv > δ0 . lim u→U

Since

(80)

v0

v0

lim

u→U

1447

λ(u, v)dv = 0 , v−1 v2

λ(u, v)dv = 0 ,

lim

u→U

(81)

v1

there must exist for every u > U , v˜0 (u) and v˜1 (u) where v−1 ≤ v˜0 (u) ≤ v0 , v1 ≤ v˜1 (u) ≤ v2 , such that

lim λ u, v˜1 (u) = 0 . (82) u→U

Note that α (v) is bounded in F since it is a continuous function on the compact [v1 , v2 ]. Integrating equation (8) and using the bounds on Ω, U and r, α , we obtain that: v1 v˜1 (u) (ρ + P1 − 2J1 )dv ≤ lim β (u) (ρ + P1 − 2J1 )dv = 0 . (83) lim β (u) u→U

v0

u→U

v˜0 (u)

We will now use a result obtained in Section 3 of [13]. If we consider the Vlasov field f as a function of the space-time coordinates and (p1 , p2 , p3 ), where the pi are the components of the momentum of the geodesics in the orthonormal frame (18), the support of the Vlasov f in p2 , p3 is bounded as long as U and A are bounded: sup |p2 |, |p3 |/∃(u, v, p1 )/f (u, v, p1 , p2 , p3 ) = 0 ≤ M . (84) √ Using this and the fact that p0 −p1 ≥ √δβ implies β (p0 −p1 ) ≤ βδ (p0 −p1 )2 , we obtain: √ β 0 N − N1 , N uˆ = √ α Ω √ β p0 − p1 1 2 3 =√ dp dp dp p0 α Ω (p0 −p1 )≥ √δ β p0 − p1 1 2 3 + dp dp dp , p0 (p0 −p1 )≤ βδ (p0 − p1 )2 1 2 3 dp dp dp + Bδ , (85) N uˆ ≤ δ −1 A f p0 where A and B are constants which depend on M and the strictly positive lower bounds on α (v) and Ω.

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From (83), the first term on the right-hand side goes to zero as u → U and therefore choosing δ small enough contradicts (80). Theorem 2 has been therefore proved.

8. Comments While we still do not know whether, generically, r goes to 0 as one approaches the past boundary of T 2 -symmetric spacetimes with positive cosmological constant and with our without collisionless matter, our result shows that such knowledge is actually unnecessary as far as the C 2 formulation of strong cosmic censorship is concerned for the Einstein-Vlasov system. Indeed, Corollary 1 completes the proof of the C 2 formulation of strong cosmic censorship for this class of spacetimes. Moreover, Theorem 1 means that we only need to focus on the cases where r goes to 0 for the vacuum models. Note that the set of T 2 -symmetric spacetimes with positive cosmological constant for which r0 > 0 is non-empty. We describe briefly how one can build a family of solutions in Appendix E. It should be emphasized that the positivity of the cosmological constant plays an important role in our analysis. In particular, once we know that a possible Cauchy horizon needs to be regular everywhere and must include at least one side of the past boundary, as shown in Section 5, the Einstein equation (10) implies that the integral curve of the null generator of the horizon u = U is past incomplete, or equivalently, that the horizon is non-degenerate. This is an interesting fact since understanding the degeneracy of possible horizons is often thought to be difficult. In the vacuum case, the estimates of Section 6 shows that the Cauchy horizon must then coincide with the past boundary of the maximal Cauchy development. We therefore have a bifurcate Cauchy horizon which is similar to the bifurcate horizons of [11]. The techniques used to build the appropriate initial data for the estimates of Section 5 have relied heavily on the continuity of the curvature tensor. Therefore the methods used here are unlikely to be extended to the C 0 formulation of strong cosmic censorship [4, 6, 8].

Appendix A. The Vlasov equation Let P ⊂ T M denote the set of all future directed timelike vectors of length −1. P is classically called the mass shell. Let f denote a nonnegative function on the mass shell. The Vlasov equation equation for f is derived from the condition that f be preserved along geodesics. In coordinates, we therefore have: β γ pα ∂xα f − Γα βγ p p ∂pα f = 0 , α

(86)

where p denotes the momentum coordinates on the tangent bundle conjugate to xα . Moreover, in the case of T 2 -symmetry, f is assumed to be invariant under the action induced on P by the action of T 2 on M.

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The energy momentum tensor is defined by: Tαβ = pα pβ f ,

1449

(87)

π −1 (x)

where π : P → M is the natural projection from the mass shell to the spacetime −1 and the integral is with respect to the natural volume form on π (x). 0 i j In an orthonormal frame, with p = 1 + δij p p , the components of the energy-momentum tensor are given by: ρ(u, v) = p0 f (u, v, p)d3 p , (88) R3 (pk )2 Pk (u, v) = f (u, v, p)d3 p , (89) 0 R3 p pk f (u, v, p)d3 p , (90) Jk (u, v) = R3 pj pk Sjk (u, v) = f (u, v, p)d3 p . (91) 0 R3 p Both the dominant and the strong energy conditions are valid for Vlasov matter as a direct consequence of the definitions.

Appendix B. The class of initial data for the Vlasov field We will require that f has initially compact support in p2 and p3 . Since A, U and r are bounded on the initial Cauchy surface, it is equivalent to say that f has initially compact support in px and py . Note that this requirement is compatible with assumption 1 of Corollary 1 since we do not add any constraint on the support of f in p0 and p1 .

Appendix C. Conservation of particle current Define the particle current vector field N by: α N = pα f .

(92)

π −1 (x)

The Vlasov equation implies that N is divergence free and we obtain the conservation law: u2 v2 u2 v ˆ 2 2 u ˆ 2 2 N Ω r (u, v1 )du + N Ω r (u1 , v)dv = N vˆ Ω2 r2 (u, v2 )du u1 v1 u1 v2 N uˆ Ω2 r2 (u2 , v)dv , (93) + v1

where vˆ, u ˆ are indices for the components of tensors along the null vectors N(u) , ∂ ∂ , ∂y ). N(v) in the basis (N(u) , N(v) , ∂x

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Appendix D. Change of global null coordinates on Q In this appendix, we recall how rescaling of global null coordinates affect the metric functions and the energy-momentum components. Suppose therefore that where u0 and v0 (u, v) ∈ (U, u0 ] × (V, v0 ] is a global null coordinate system on Q, are real numbers and U and V are real numbers or −∞. Let f and g be smooth functions on (U, u0 ] × (V, v0 ] with f > 0 and g > 0. Let (u∗ (u), v ∗ (v)) = (f (u), g(v)). In the coordinate system defined by (u∗ , v ∗ ), the functions Ω∗ , U ∗ , A∗ , r∗ are given by: Ω2 , f g U∗ = U ,

Ω∗ 2 =

(94) (95)

∗

A = A,

(96)

∗

r = r.

(97) ∗

∗

With Γ and Π defined by (6) and (7), we have that ΩΓ∗ 2 = ΩΓ2 and ΩΠ∗ 2 = ΩΠ2 . The derivatives change in the obvious way, for instance ru∗ ∗ = rfu , and since f and g are functions of a single variable, ru∗ ∗ v∗ = fruv g . Finally, since ρ, Pk , Sik are defined in the fixed frame (18), they are left unchanged by the change of coordinates.

Appendix E. A family of solutions with r0 > 0 We consider vacuum T 2 -symmetric spacetimes with positive cosmological constant and assume that the metric functions independent of θ. Moreover, we consider the ansatz r = e2U , G = H = A = 0. Working in areal coordinates (t = r), the metric can be written as: ds2 = −

e2γ e2γ 2 δdt2 + dθ + t(dx2 + dy 2 ) . t t

(98)

The Einstein equations reduce to the system: 1 + δΛe2γ , 4t δt = −4δ 2 e2γ Λ .

γt =

(99)

From which we obtain:

δt 1 = −4 γt − , δ 4t δt 1 = −4γt + , δ t ln δ = −4γ + ln t + 2 ln A ,

(100) (101) (102)

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where A is a strictly positive constant. 4γ = ln t − ln δ + 2 ln A , 1 1 2γ = ln t − ln δ + ln A , 2 2 t . e2γ = A δ Inserting (105) in (99), we obtain: t 2 δt = −4δ A Λ, δ √ δt = −4δ 3/2 AΛ t , √ 1 δt − 3/2 = 2AΛ t 2δ and by integration: 1 4 √ = AΛt3/2 + C , 3 δ where the constant C satisfies: 4 AΛt3/2 + C > 0 , 3 4 C > − AΛt3/2 . 3 We obtain for the solution: 1 δ=

2 , 4 3/2 AΛt + C 3 √ 4 t 2γ 2 e =A =A AΛt + C t , δ 3 or for the metric components:

(104) (105)

(106) (107) (108)

(109)

(110) (111)

(112) (113)

4 C AΛt + √ , 3 t 1 e2γ √ . δ =A4 2 t 3 AΛt + C t e2γ =A t

(103)

(114) (115)

Suppose C < 0 and let D = −C. The previous coordinate system breaks down at t0 : 2/3 3 D . (116) t0 = 4 AΛ However, this system of coordinates is known to cover the whole past maximal Cauchy development of the initial data. Therefore, the area element of such solutions does not go to zero on the past boundary of the maximal Cauchy development. Nevertheless, one should note that these solutions have a Cauchy horizon

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at t = t0 , as one can easily get around the coordinate singularity with a coordi1 nate transformation of the type t = f (t) + θ, with f (t) = ± A(4/3AΛt+ C . These √ ) t

solutions are clearly excluded by the generic condition of Theorem 1.

Acknowledgements I would like to thank Mihalis Dafermos for suggesting this problem as well as for many useful advice and comments. I would also like to thank Alan Rendall for pointing out the existence of the example of Appendix E and John Stewart for his help along the way. Finally, I wish to gratefully acknowledge funding from EPSRC.

References [1] H. Andréasson, Global foliations of matter spacetimes with Gowdy symmetry, Commun. Math. Phys. 206 (1999), 337–365. [2] H. Andreasson, A. D. Rendall and M. Weaver, Existence of CMC and constant areal time foliations in T 2 symmetric spacetimes with Vlasov matter, Comm. Partial Differential Equations 29 (2004), no. 1–2, 237–262. [3] B. K. Berger, J. Isenberg, P. T. Chrusciel and V. Moncrief, Global foliations of vacuum spacetimes with T 2 isometry, Annals Phys. 260 (1997), 117–148. [4] D. Christodoulou, On the global initial value problem and the issue of singularities, Class. Quantum Grav. 16 (1999), A23–A35. [5] A. Clausen and J. Isenberg, Areal foliation and asymptotically velocity-term dominated behavior in T 2 symmetric space-times with positive cosmological constant, J. Math. Phys. 48 (2007), no. 8, 082501. [6] M. Dafermos, Stability and instability of the Cauchy horizon for the spherically symmetric Einstein–Maxwell-scalar field equations, Ann. Math. 158 (2003), 875–928. [7] M. Dafermos and A. D. Rendall, Inextendibility of expanding cosmological models with symmetry, Class. Quantum Grav. 22 (2005), L143–L147. [8] M. Dafermos, The interior of charged black holes and the problem of uniqueness in general relativity, Commun. Pure Appl. Math. 58 (2005), 445–505. [9] M. Dafermos and A. D. Rendall, Strong cosmic censorship for T 2 symmetric cosmological spacetimes with collisionless matter, gr-qc/0610075. [10] M. Dafermos and A. D. Rendall, Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter, gr-qc/0701034. [11] H. Friedrich, I. R´ acz and R. M. Wald, On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon, Commun. Math. Phys. 204 (1999), 691–707. [12] J. Isenberg and M. Weaver, On the area of the symmetry orbits in T 2 symmetric spacetimes, Class. Quantum Grav. 20 (2003), 3783–3796. [13] A. D. Rendall, Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry, Commun. Math. Phys. 189 (1997), 145–164.

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[14] H. Ringstr¨ om, Strong cosmic censorship for T 3 Gowdy spacetimes, to appear in Ann. Math. [15] M. Weaver, On the area of the symmetry orbits in T 2 symmetric spacetimes with Vlasov matter, Class. Quantum Grav. 21 (2004), 1079–1097. Jacques Smulevici University of Cambridge Department of Applied Mathematics and Theoretical Physics Wilberforce Road Cambridge CB3 0WA United Kingdom e-mail: [email protected] Communicated by Piotr T. Chrusciel. Submitted: February 2, 2008. Accepted: June 12, 2008.

Ann. Henri Poincaré 9 (2008), 1455–1477 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/081455-23, published online November 18, 2008 DOI 10.1007/s00023-008-0395-3


On the Rate of Quantum Ergodicity for Quantised Maps Roman Schubert Abstract. We study the distribution of expectation values and transition amplitudes for quantised maps on the torus. If the classical map is ergodic then the variance of the distribution of expectation values will tend to zero in the semiclassical limit by the quantum ergodicity theorem. Similarly the variance of transition amplitude goes to zero if the map is weak mixing. In this paper we derive estimates on the rate by which these variances tend to zero. For a class of hyperbolic maps we derive a rate which is logarithmic in the semiclassical parameter, and then show that this bound is sharp for cat maps. For a parabolic map we get an algebraic rate which again is sharp.

1. Introduction The behaviour of eigenfunctions and eigenvalues of Schr¨ odinger operators in the semiclassical limit depends strongly on the dynamical properties of the underlying classical Hamiltonian system. For a generic chaotic system it is conjectured that the eigenvalue and eigenfunction statistics behave universally. By the Bohigas Giannoni Schmidt conjecture, [5], the eigenvalue statistics can be modelled by random matrix theory, and Berry’s random wave model, [4], predicts that the eigenfunctions behave locally like random superposition of plane waves. One way of characterising the behaviour of eigenfunctions is by studying expectation values of observables and a fundamental result about them is the quantum ergodicity theorem. It states that if the underlying classical system is ergodic, then for sufficiently nice observables and almost all eigenfunction the expectation values tend to the classical mean value of the observable. This result goes back to Shnirelman, Zelditch and Colin de Verdière, [9, 31, 36], for the eigenfunctions of the Laplacian on compact manifolds, and has since then been generalised to many other situations. But the question on the rate by which the expectation values approach the classical mean has proved to be very difficult. For the eigenfunctions of the Laplacian on a compact manifold of negative curvature, Zelditch, [38], proved an upper

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bound on the mean behaviour of the expectation values which decays logarithmically in the semiclassical parameter. This bound has been recently generalised in [30], but the order could not be improved. It is believed that this bound is far from the optimal one, and this view is supported by conjectures, numerical studies and physical heuristics, see [1, 3, 16, 18, 29, 35]. Only for the case of the modular surface Luo and Sarnak were recently able to compute the variance of the matrix elements, [25], and so obtained the optimal bound. Since for Schr¨ odinger operators estimating the rate of quantum ergodicity seems to be a very hard problem it is interesting to look for simpler classes of system where this question might be attacked with more hope of success. This is the reason why we consider quantised maps on the two-dimensional torus in this article. For these maps we sometimes have better control on the semiclassical approximations, which will allow us to understand the rate of quantum ergodicity better. The first class of systems we study are perturbed cat maps. These are hyperbolic maps and we obtain for their quantisations the same logarithmic upper bound on the mean rate of quantum ergodicity as in [30, 38]. For the unperturbed cat maps Kurlberg and Rudnick, [23], showed that the variance of matrix elements decays much quicker for a special choice of basis of eigenfunctions, a so called Hecke eigenbasis. But since the eigenvalues of the cat map can have high multiplicities, there are many choices of a eigenbasis. We will show that there exists an eigenbasis for which the variance of expectation values is of logarithmic order in the semiclassical parameter. As will be explained below this is due to very particular properties of cat maps, and is probably not true for a generic perturbation of the cat-map. In our methods the hyperbolicity of the classical system is the source of large error terms in semiclassical approximations of time evolution, so one might wonder if we get better bounds if the system is ergodic, but not hyperbolic. The second class of quantised maps we study are exactly of this type. These are parabolic maps whose quantisations have been studied by Marklov and Rudnick, [27], who have derived sharp bounds on individual eigenfunctions. So they are interesting test cases for us, and we show that our general method produces sharp results in this case, too. Quantum ergodicity is a quite general and stable result, it relies only on a few properties, like the validity of some correspondence principle between quantum and classical mechanics, and it therefore has been proved in many different situations. The main general insight to be drawn from the results presented here is that in this general framework the logarithmic bound from Zelditch [30, 38] is sharp, as we show for the cat map, and in order to improve the bounds one needs additional assumptions. This paper is organised as follows. In the first two sections we describe the general setup. In the first section we recall some notions on ergodic maps on the torus. In the second section we recall the quantisation of maps on the torus and state the quantum ergodicity theorems relating variances of expectation values

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and transition amplitudes to ergodicity and weak mixing of the classical map. We then state a general result which relates the variance of expectation values and transition amplitudes to an average of the classical autocorrelation function, plus semiclassical error terms. This result shows that the quality of bounds we have on the error terms for the semiclassical time evolution determines the rate of quantum ergodicity, and it will be our main tool in the following sections. In the last two sections we then study explicit classes of maps and apply the method of Section 3 to them. In the first part of Section 4 we obtain a logarithmic upper bound on the rate for perturbed cat maps. In the second part we show that this bound is sharp for unperturbed cat maps. In the last section we get a bound on the rate of quantum ergodicity for parabolic maps, which is algebraic in the semiclassical parameter. By the results of Marklof and Rudnick, [27], this bound is sharp, too. Notation. We will use the notation e(x) := e2πix and eN (x) := e(x/N ). Furthermore we will write e(∗) for a phase factor, i.e., a number with |e(∗)| = 1, the value of e(∗) can change from line to line.

2. Maps on the torus Let T 2 := R2 /Z2 be the two-torus and dx the normalised Lebesgue measure on T 2 . We will study some classes of smooth symplectic maps on the torus Φ : T 2 → T 2 .

(1)

In this section we will recall ergodic notions we will need later on, see [34] for more details. For a ∈ L2 (T 2 ) we define the average a ¯ := a(x) dx (2) T2

and the autocorrelation function as a∗ (x)a Φt (x) dx − |¯ a|2 , C[a](t) :=

(3)

T2

for t ∈ Z. Properties like ergodicity, weak mixing and mixing can be expressed in terms of the autocorrelation function. The map is ergodic if for every a ∈ L2 (T 2 ) one has T −1 1 C[a](t) = 0 , (4) lim T →∞ T t=0 it is weak mixing if for every θ ∈ R and a ∈ L2 (T 2 ) one has T −1 1 e(θt)C[a](t) = 0 , T →∞ T t=0

lim

(5)

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and it is mixing if for every a ∈ L2 (T 2 ) lim C[a](t) = 0 .

t→∞

(6)

The map is called exponentially mixing if there is a γ > 0 such that for any a ∈ C ∞ (T 2 ) C[a](t) = Oa (e−γ|t| ) .

(7)

If a map Φ is mixing and the correlation functions for smooth observables decay at least like |C[a](t)| = O(t−1−δ ) for some δ > 0, then we can define for a ∈ C ∞ (T 2 ) the quantity ∞ C[a](t) . (8) V [a] := t=−∞

One can show that under the same condition on the decay of C[a](t) one has T −1 2 t 1 a Φ (x) − a V [a] = lim ¯ dx , (9) T →∞ T T 2 t=0 i.e., V [a] is actually the variance of the fluctuations of the time mean around the space mean. This representation shows as well that V [a] ≥ 0. Since we will only work with (8) and not with (9) we skip the easy proof of their equivalence.

3. Quantum maps and quantum ergodicity Let us first quickly review the setup for quantised maps on the 2-torus T 2 = R2 /Z2 , see [10, 13, 26] for some recent and more complete treatments, we follow here the presentation in [7]. The Hilbert space is constructed from generalised functions on the line by requiring periodicity in position and in momentum space. For x = (p, q) ∈ R2 let

i (pQ − qP ) , (10) T (p, q) := exp be the phase space translation operator acting on functions on R, where Qψ(q) := qψ(q) and P ψ(q) := i dψ(q) are the position and momentum operators, respecdq tively. The simultaneous periodicity condition reads T (1, 0)ψ = e(κ1 )ψ ,

T (0, 1)ψ = e(κ2 )ψ ,

(11)

where two constant phase factors κ = (κ1 , κ2 ) ∈ R2 /Z2 are allowed because the quantum mechanical state represented by a wave function does not depend on an overall phase. This set of equations has non-trivial solutions only when 1 =N ∈N 2π

(12)

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and then the vector space HN (κ) of solutions has dimension N and consists of distributions of the form ψ(q) = Ψ(k)δ q − (k + κ1 )/N , with Ψ(k + N ) = e(−κ2 )Ψ(k) . (13) k∈Z

The translation operators TN (n) := T (n1 /N, n2 /N ) ,

(14)

with 2π = 1/N and where n = (n1 , n1 ) ∈ Z2 , leave the space HN (κ) invariant and if one equips HN (κ) with the inner product ψ, φ =

N 1 ∗ Ψ (q)Φ(q) , N q=1

(15)

then the restrictions of TN (n) to HN (κ) become unitary. By slight abuse of notation we will denote the restrictions of TN (n) to HN (κ) as well by TN (n). So instead of one fixed Hilbert space, as in quantum mechanics on Rn , the quantisation of the 2-torus leads for each with 1/ = N ∈ N to a family of Hilbert spaces HN (κ) depending on an additional parameter κ. But for each class of maps we study we will fix for each N a particular choice of κ, κN , and therefore we will most of the time discard it from the notation. We will need some further properties of the translation operators TN (n), they satisfy (16) TN (m)TN (n) = eN ω(m, n)/2 TN (m + n) , where ω(m, n) = m1 n2 − m2 n1 , and TN∗ (n) = TN (−n). Furthermore their trace satisfies Tr TN (n) = 0 only if n = 0 mod N , and e(∗) if n = m mod N 1 ∗ Tr TN (n)TN (m) = , (17) N 0 otherwise and in the particular case that n = m the phase factor is e(∗) = 1. The quantisation of observables is now defined by a Weyl quantisation prescription using the translation operators. For a ∈ C ∞ (T 2 ) one defines the Weyl quantisation as a ˆ(n)TN (n) (18) OpN [a] := n∈Z2

with the Fourier coefficients

a ˆ(n) =

a(x)e − ω(n, x) dx .

(19)

T2

The function a is called the symbol of the operator OpN [a]. This quantisation prescription has natural properties, e.g., real valued symbols correspond to selfadjoint operators. Note that in particular TN (n) = OpN [e(ω(n, x))].

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Traces are related to the phase space average of the symbols, using (18) one a(x) dx + Oa (1/N ∞ ) , (20) Tr OpN [a] = T2

for a ∈ C ∞ (T 2 ), where the remainder depends on derivatives of a. One related more precise result we will need later on is Lemma 1. Let a, b ∈ C ∞ (T 2 ), then for all L ≥ 3 we have

|a|L |b|3+L 1 Tr OpN [a] OpN [b] = a(x)b(x) dx + O . N NL T2 where |a|L := |α|≤L supx∈T 2 |∂ α a(x)|. Proof. Using the definition (18) and (17) we obtain 1 Tr OpN [a] OpN [b] = a ˆ(n)ˆb(−n + mN )e(∗) N n∈Z2 m∈Z2 = a ˆ(n)ˆb(−n) n∈Z2

+

(21)

(22)

a ˆ(n)ˆb(−n + mN )e(∗) .

n∈Z2 m∈Z2 \{0}

Now we have for the first term a(x)b(x) dx = a ˆ(n)ˆb(−n) T2

(23)

n∈Z2

and by partial integration for m = 0 |ˆ a(n)| ≤ C|a|k (1 + |n|)−k , |ˆb(−n + mN )| ≤ C|b|L (1 + |n|)L (N |m|)−L ,

(24)

so with k = L + 3 and L ≥ 3 the remainder term converges and the result follows. Finally we look at the quantisation of maps. Let Φ : T 2 → T 2 be a symplectic diffeomorphism on the torus, we say that a sequence of unitary operators {UN }N ∈N , UN : HN → HN , is a quantisation of Φ if for all a ∈ C ∞ (T 2 ) we have ∗ OpN [a]UN − OpN [a ◦ Φ] = 0 . lim UN

N →∞

(25)

This property is a manifestation of the correspondence principle, it means that for large N the quantum system reproduces the classical system. In the mathematical literature this is usually referred to as the validity of an Egorov theorem. A more quantitative version is usually true, 1 −t t UN OpN [a]UN = OpN [a ◦ Φt ] + RN (t) . (26) N with RN (t) ≤ C for t in a finite interval. If we let t increase then the available bounds for RN (t) increase exponentially for generic hyperbolic maps.

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The concrete procedure by which the unitary operators UN are constructed depends on the map and we will discuss this for particular classes of examples below. In case that the classical map is ergodic the Egorov property (25) allows to prove a quantum ergodicity theorem. We will recall the statements now. In their formulation we will use the notation |θ|S 1 := min|θ + k| , k∈Z

(27)

for θ ∈ R. Theorem 1. Assume the map Φ : T 2 → T 2 is ergodic, and UN is a quantisation of Φ. Let for every N ∈ N, ψjN , j = 1, . . . , N , be an orthonormal basis of eigenfunctions with eigenvalues e(θjN ) of UN and let a ∈ C ∞ (T 2 ), then we have N 2 1 N ψj , OpN [a]ψjN − a = 0 . N →∞ N j=1

lim

Furthermore, Φ is ergodic if, and only if, for every a ∈ C ∞ (T 2 ) N 1 ψi , OpN [a]ψjN − δij a2 = 0 . lim lim sup δ→0 N →∞ N N N

(28)

(29)

|θi −θj |S 1 ≤δ

The result (28) was proved in [7], and (29) is a version for maps of a result in [37], see as well [32]. We will give below a simple proof in order to prepare for the estimates on the rate later on. The estimate (28) means that the variance of the distribution of the expectation values around its mean is going to 0. This implies that almost all of the expectation values tend to the mean value, which is the usual way in which quantum ergodicity is stated. The second estimate (29) implies of course the first one, but it is stronger and implies in addition that almost all of the near diagonal transition amplitudes have to tend to 0. Theorem 2. Assume the map Φ : T 2 → T 2 is weak mixing, and UN is a quantisation of Φ. Let for every N ∈ N, ψjN , j = 1, . . . , N , be an orthonormal basis of eigenfunctions of UN , with eigenvalues e(θiN ), and let a ∈ C ∞ (T 2 ), then we have for any θ ∈ R N 1 ψi , OpN [a]ψjN − δij a 2 = 0 . lim lim sup (30) δ→0 N →∞ N N N |θi −θj −θ|S 1 ≤δ

Conversely, if (30) holds for every a ∈ C ∞ (T 2 ) and θ ∈ R, then Φ is weak mixing. The analogue of this result for eigenfunctions of the Laplacian on compact Riemannian manifolds was given in [37, 39]. Our main objective in this paper is to obtain bounds on the rate by which the averages over expectation values and transition amplitudes in Theorems 1 and 2

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tend to zero. Our main tool to do so will be the next proposition. In its statement and in the following we will fix a function f ∈ S(R) which satisfies f, fˆ ≥ 0 ,

supp fˆ ∈ [−1, 1]

f (0) = 1 and

and then define for T > 0

t 1 fˆT (t) := fˆ T T

and

FT (θ) :=

f T (θ − n) .

(31)

(32)

n∈Z

Such a function can be easily constructed by choosing a function g > 0 with support in [−1/2, 1/2] and taking fˆ = g ∗ g, and then normalising g such that f (0) = 1. Note that we can always set a = 0 without loss of generality. This will shorten some formulas. Proposition 1. Let UN : HN → HN be a quantisation of the map Φ : T 2 → T 2 , and let ψjN , j = 1, . . . , N , be an orthonormal basis of eigenfunctions of UN with eigenvalues e(θjN ). Then for every a ∈ C ∞ (T 2 ) with a = 0 we have N 2 1 N ψj , OpN [a]ψjN ≤ fˆT (t)C[a](t) N j=1 t∈Z

1 ˆ fT (t)RN (t) N t∈Z Ca,L ˆ + L fT (t)|a ◦ Φt |L N

+

(33)

t∈Z

for T > 0, L ≥ 3, where RN (t) is the remainder term in (26) and fˆT is as in (32). Furthermore let θ ∈ R then for every δ > 0 there is a C > 0 such that N 1 ψi , OpN [a]ψjN 2 ≤ C e(θt)fˆT (t)C[a](t) N N N |θi −θj −θ|S 1 ≤δ/T

t∈Z

1 e(θt)fˆT (t)RN (t) (34) N t∈Z Ca,L +C L e(θt)fˆT (t)|a ◦ Φt |L . N

+C

t∈Z

This rather technical looking result will be the main tool in following investigations on the rate of quantum ergodicity. In (33) and (34) we have estimates on the variances of expectation values and transition amplitudes in terms of three quantities. The first term on the right hand side of (33) and (34) does not depend on N it is a purely classical quantity whose behaviour for large T is determined by the ergodic properties of the map Φ, as was discussed in Section 2. The remaining two terms on the right hand side of (33) and (34) depend on N and T , and they will determine how we can couple the two limits N → ∞ and T → ∞ in an optimal

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way. As can already be seen by the appearance of |a ◦ Φt |L the behaviour of these terms will differ sharply between hyperbolic and non-hyperbolic maps. The proof of Proposition 1 will rely on Lemma 2. Assume UN satisfy the assumptions of Proposition 1 and f be as in (31), ¯ = 0 and any θ ∈ R then for a ∈ C ∞ (T 2 ) with a N 2 1 FT (θiN − θjN − θ) ψiN , OpN [a]ψjN N i,j=1

=

t∈Z

1 −t t OpN [a]UN , fˆT (t) e(θt) Tr OpN [a]∗ UN N

(35)

with fˆT and FT given in (32), and in particular N 2 1 N 1 −t t ψj , OpN [a]ψjN ≤ OpN [a]UN . fˆT (t) Tr OpN [a]∗ UN N j=1 N

(36)

t∈Z

Proof. Let ψjN , e(θjN ), j = 1, . . . , N , be the eigenfunctions and eigenvalues of UN , then we have −t t OpN [a]UN = Tr OpN [a]∗ UN

N N ψj , Op[a]ψiN 2 e t(θjN − θiN ) .

(37)

i,j=1

Now we have by the Poisson summation formula fˆT (t)e t(θjN − θiN − θ) = f T (θjN − θiN − θ − n) = FT (θjN − θiN − θ) (38) t∈Z

n∈Z

for any T > 0. So if we multiply (37) with fˆT (t) e(tθ) and sum over t we obtain (35). If we use then furthermore that FT ≥ 0 and FT (0) ≥ 1 we obtain the estimate (36) by setting θ = 0, restricting the sum in (35) to the terms i = j and using that the remaining terms with i = j are positive. Proof of Proposition 1. By Lemma 2 we have to estimate t OpN [a]UN . Using (26) we obtain

1 N

−t Tr OpN [a]UN

1 1 1 −t t Tr OpN [a]UN Tr OpN [a] OpN [a ◦ Φt ] + 2 Tr OpN [a]RN (t) , OpN [a]UN = N N N (39) but OpN [a] is bounded and Tr IN = N , so 1 OpN [a] RN (t) . |Tr OpN [a]RN (t)| ≤ N2 N With Lemma 1 we then get

1 |a ◦ Φt |L t Tr OpN [a] OpN [a ◦ Φ ] = C[a](t) + O |a|L+3 N NL

(40)

(41)

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for L ≥ 3, and so we have 1 −t t Tr OpN [a]UN OpN [a]UN N

= C[a](t) + Oa

RN (t) N

|a ◦ Φt |L + O |a|L+3 NL

.

(42)

Together with (36) this gives (33). In order to prove (34) we use (35), since FT is positive and FT (0) ≥ 1 for any δ > 0 there is a C > 0 such that N 2 1 FT (θiN − θjN − θ) ψiN , OpN [a]ψjN N i,j=1

≥

1 1 CN

N ψi , OpN [a]ψjN 2

(43)

|θiN −θjN −θ|S 1 ≤δ/T

and now using (35) and (42) gives (34).

Proof of Theorem 1. Let us first prove (28). Taking the limit N → ∞ in (33) gives lim sup N →∞

N 2 1 N ψj , OpN [a]ψjN − a ≤ fˆT (t)C[a](t) . N j=1

(44)

t∈Z

Now we take the limit T → ∞ and by ergodicity, see (4), the right hand side tends to zero. The proof of (29) follows the same line, but using (34) with θ = 0 instead. Taking the limit N → ∞ gives N 1 ψi , OpN [a]ψjN 2 ≤ C lim sup (45) fˆT (t)C[a](t) N →∞ N |θi −θj |S 1 ≤δ/T

t∈Z

and taking T → ∞ implies that the right hand side tends to 0. But on the left hand side T → ∞ is equivalent to δ → 0. To prove the converse part, we use that from the proof of Proposition 1 we have 1 −t t Tr OpN [a]UN OpN [a]UN = C[a](t) (46) lim N →∞ N and so by (35) N 2 1 FT (θiN − θjN − θ) ψiN , OpN [a]ψjN = fˆT (t) e(θt)C[a](t) . (47) N →∞ N i,j=1

lim

t∈Z

For θ = 0 this reads N 2 1 FT (θiN − θjN ) ψiN , OpN [a]ψjN = fˆT (t)C[a](t) . N →∞ N i,j=1

lim

(48)

t∈Z

Now if (29) holds, then the left hand side of (48) has to tend to 0 for T → ∞, so the right hand side has to go to 0, too. This is valid for all a ∈ C ∞ (T 2 ) with a = 0

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and since C ∞ (T 2 ) is dense in L2 (T 2 ) and fˆ ≥ 0 this means the map Φ is ergodic by (4). We will skip the proof of Theorem 2, as it follows the same line as the proof of Theorem 1, one uses (47) for all θ ∈ R and weak mixing, (5), instead of ergodicity.

4. Hyperbolic maps 4.1. Cat maps A cat map, or a hyperbolic toral automorphism, on the 2-torus T 2 = R2 /Z2 is given by a matrix A ∈ SL(2, Z) with |Tr A| > 2 acting on T 2 as

p p → A mod Z2 . q q The condition |Tr A| > 2 ensures that the map is hyperbolic. These maps are ergodic and rapidly mixing, i.e., for a ∈ C ∞ (T 2 ) C[a](t) = Ok (e−kγ|t| ) for all k > 0, where γ > 0 is the Liapunov exponent of A. These maps have been quantised on HN (κ) with κ = (0, 0) in [19] for even N and for odd N if the off-diagonal terms of A are even. The problems for odd N can be avoided by choosing HN (κ) with κ = (1/2, 1/2) for these N , [7,14]. So therefore we will fix our choice of κ in this section from now on, HN = HN ((0, 0)) for even N and HN = HN ((1/2, 1/2)) for odd N . The map A can then be quantised using the metaplectic representation and the resulting sequence of unitary operators UN satisfies ∗ OpN [a]UN = OpN [a ◦ A] , (49) UN see [7, 10, 13, 21]. So here the Egorov theorem holds exactly. 4.2. Perturbed cat maps We will now consider a class of Anosov maps, the perturbed cat maps introduced in [2]. We will rely mainly on the recent study by Bouclet and De Bièvre in [8] of these maps. Let A ∈ SL(2, Z) be a cat map and g ∈ C ∞ (T 2 ) a real valued function, and consider the Hamiltonian flow φt : T 2 → T 2 generated by g. One can define then (50) Φε := φε ◦ A : T 2 → T 2 which for small ε is a small perturbation of the Anosov map A, and hence by structural stability Φε will be an Anosov map, too. The quantisation of Φε is now defined as (51) Uε,N := e−2πiN ε OpN [g] UN , where UN is the quantisation of A. In [8] it is then shown that there is a constant Γ > 0 such that for t ∈ Z 1 −t t Uε,N OpN [a]Uε,N − OpN [a ◦ Φtε ] ≤ Ca eΓ|t| . (52) N In fact the estimates in [8] are more precise, and Γ is estimated quite explicitly, but the estimate (52) is sufficient for our purpose.

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Theorem 3. Let Uε,N be the sequence of quantum maps (51) and ψjN , j = 1, . . . , N an orthonormal basis of eigenfunctions of Uε,N for every N ∈ N. Then for every a ∈ C ∞ (T 2 ) there is a constant Ca such that N 2 1 N 1 ψj , OpN [a]ψjN − a ≤ Ca . N j=1 ln N

(53)

The same result has been recently proved for the baker’s map too, see [15]. For cat maps much stronger results are known for a special choice of a basis of eigenfunctions, due to their arithmetic nature, see [21, 23] and Section 4.3 below. Proof. We will use (33). The Egorov estimate (52) gives RN (t) ≤ CeΓ|t| and since the map is hyperbolic there is a constant Γ > 0 such that |a◦Φtε |3 ≤ Ce3Γ |t| . −γ|t| Furthermore Φε is exponentially mixing, i.e., C[a](t) ≤ Ce , and so (54) fˆT (t)C[a](t) = O(1/T ) , fˆT (t)RN (t) = O(eΓT ) , t∈Z

t∈Z

and

fˆT (t)|a ◦ Φtε |3 = O(e3Γ T ) .

(55)

t∈Z

Therefore we obtain N 2 1 N ψj , Op[a]ψjN − a ≤ O(1/T ) + O eΓT /N + O e3Γ T /N 3 , N j=1

(56)

and the right hand side becomes minimal for the choice of T = δ ln N , with δ < min{Γ, Γ }, which gives then (53). By the same methods one can derive an upper bound on the transition amplitudes, strengthening the estimates (29) and Theorem 2. Theorem 4. Let Uε,N be the sequence of quantum maps (51) and ψjN , j = 1, . . . , N , an orthonormal basis of eigenfunctions of Uε,N for every N ∈ N. Then for any δ > 0 and every a ∈ C ∞ (T 2 ) there is a constant Ca such that 1 N

|θiN −θjN −θ|S 1 ≤δ/ ln N

N ψi , OpN [a]ψjN − δij a2 ≤ Ca 1 , ln N

(57)

for every θ ∈ R. We skip the proof since it is identical to the one of Theorem 3, only using (34) instead of (33).

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4.3. Cat maps: The logarithmic bound is sharp In this section we want to show that the upper bound in Theorem 3 is sharp for unperturbed cat maps. The unperturbed cat map has very special properties, in particular for certain values of N the quantum map has very large multiplicities of the eigenvalues. This means that there are many bases of eigenfunctions and quantum ergodic properties can, and do, depend on the choice of a basis of eigenfunctions. In this section we will show that for a given observable OpN [a] one can choose the basis of eigenfunctions such that the upper bound in Theorem 3 becomes sharp. The special properties of cat maps are due to the fact that they are periodic, [19]. Let A ∈ SL(2, Z) be a cat map and UN : HN → HN its quantisation, the quantum period P (N ) ∈ N is defined to be the smallest positive integer such that there is a φN ∈ R/Z with P (N )

UN

= e(φN )IN

(58)

with IN the identity operator. The origin of these quantum periods are periods of the action of A on Z2 modulo N , for a given A ∈ SL(2, Z) one defines T (N ) to be the smallest integer such that AT (N ) ≡ I mod N , i.e., such that there exists a matrix AN with integer entries such that AT (N ) = I + N AN .

(59)

Then one has either P (N ) = 2T (N ) or P (N ) = T (N ), depending on N and A, as was shown in [19]. P (N ) is on average of order N , [20], but there exist N where it is of order ln N . More precisely, there exists a sequence Nk , k ∈ N, such that 2 P (Nk ) = ln Nk + O(1) (60) γ where γ is the Liapunov exponent of A, [6, 22]. Now the relation (58) implies that the eigenvalues of UN are P (N )’th roots of unity shifted by φN and since the dimension of HN is N , a quantum period much shorter than N implies large multiplicities of the eigenvalues and therefore there are many different choices for an orthonormal basis of eigenfunctions of UN . For an observable a ∈ C ∞ (T 2 ) we define the quantum average over one period as P (N )−1 P 1 −t t OpN [a] := UN OpN [a]UN . (61) P (N ) t=0 P

P

−1 Then we have UN OpN [a] UN = OpN [a] and so the averaged operator commutes with the quantised cat map, P

P

OpN [a] UN = UN OpN [a] . If a is real valued, then OpN [a] and OpN [a]

P

(62)

are selfadjoint, and so by (62) we

can choose an orthonormal basis of joint eigenfunctions of OpN [a]

P

and UN .

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We will say that a function a ∈ C ∞ (T 2 ) is a trigonometric polynomial of degree R if a ˆ(n) e ω(n, x) . (63) a(x) = n∈Z2 , |n|≤R

We can now give the main result of this section. Theorem 5. Let A ∈ SL(2, Z) be hyperbolic and denote its quantisation by UN : HN → HN . Let Nk , k ∈ N, be a sequence with P (Nk ) = γ2 ln Nk + O(1), then for every R > 0 there exists a KR > 0 such that if a(x) is a real valued trigonometric polynomial of degree R and ψjNk , j = 1, . . . Nk , an orthonormal basis of joint eigenfunctions of OpNk [a]

P

and UNk , then we have

Nk 2 N 1 1 ψ k , OpN [a]ψ Nk − a V [a] , ¯ = j j k Nk j=1 P (Nk )

(64)

for k ≥ KR . Recall that V [a] = 0 if, and only if, there is a b ∈ C ∞ (T 2 ) such that a = b−b◦A, see, e.g., [33, Proposition 4.14]. So there are many function with V [a] > 0, and therefore the bound from Theorem 3 is sharp. P. Kurlberg has as well obtained logarithmic lower bounds on the variance of expectation values for cat maps, [24]. The variance of expectation values depends strongly on the choice of the basis of eigenfunctions, in [23] it is shown that for a different choice than ours, a so called Hecke basis, the variance of expectation values is of order 1/N . That is the behaviour which is expected for a generic Anosov system. But there is a point of view from which the logarithmic decay looks natural. In [12, 16] it is argued that the decay of the variance is of order 1/TH (N ), where TH (N ) is the so called Heisenberg-time. Now the Heisenberg-time is inversely proportional to the mean level spacing, and so TH (N ) ∼ P (N ), if we measure the level spacing without multiplicities, which means that (64) coincides with the expressions in [12, 16]. The basic idea we use here, namely the averaging over one quantum period, is the same that was used in [17] to prove the existence of scarred eigenstates of cat maps. The proof of Theorem 5 proceeds in several steps. We first reduce it to the computation of a trace. Without loss of generality we will assume in the following that a ¯ = 0. Lemma 3. Let ψjN , j = 1, . . . N , be an orthonormal basis of joint eigenfunctions of OpN [a]

P

and UN , then N 2 P 1 N 1 ψj , OpN [a]ψjN = Tr OpN [a] OpN [a]∗ . N j=1 N

(65)

In general for an arbitrary basis the right hand side of (65) is an upper bound of the left hand side, so our choice of basis maximises the variance.

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Proof. Since ψjN are eigenfunctions of OpN [a]

P

1469

and orthonormal we have ψjN ,

P

OpN [a] ψkN = 0 if j = k. For j = k we use that ψjN are eigenfunctions of UN to P

get ψjN , OpN [a] ψjN = ψjN , OpN [a]ψjN . Using these two relations gives P

P

Tr OpN [a] OpN [a]∗ =

N N ψj , OpN [a]P ψkN 2 j,k=1

=

N N ψj , OpN [a]P ψjN 2

(66)

j=1 N N ψj , OpN [a]ψjN 2 . = j=1

Furthermore, by using that OpN [a] P

P

Tr OpN [a] OpN [a]∗ =

P

1 P (N )

commutes with UN , we obtain

P (N )−1

P

−t t Tr OpN [a] UN OpN [a]∗ UN

t=0 P

= Tr OpN [a] OpN [a]∗ .

(67)

Now we turn our attention to short periods (60). In the definition of the P

average OpN [a] , equation (61), we averaged from t = 0 to t = P (N ) − 1, but P

P

P

−T T since UN and OpN [a] commute we have OpN [a] = UN OpN [a] UN for any T ∈ Z, which means we can shift the average to t ∈ [−T, P (N ) − 1 − T ] ∩ Z for any T , and we will do this for the choice

T = [P (N )/2] ,

(68)

which is the integer part of P (N )/2. Lemma 4. Let Nk , k ∈ N, be a sequence satisfying (60). Then for every R > 0 there exist a KR > 0 such that for k ≥ KR whenever there are two m, n ∈ Z2 with |m|, |n| ≤ R and (69) At n = m mod Nk , for some t ∈ [−[P (Nk )/2], P (Nk ) − 1 − [P (Nk )/2]] ∩ Z, then At n = m .

(70)

Proof. The relation At n = m mod Nk means that there is a αk ∈ Z2 such that At n − m = Nk αk

(71)

and we start by deriving a bound on the size of αk . Since A is hyperbolic and |t| ≤ P (Nk )/2 = γ1 ln Nk + O(1) we find |At n − m| ≤ R(1 + Ceγ|t| ) ≤ R(1 + C Nk ) ,

(72)

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˜ > 0, independent of k, such and since by (71) |αk | = |At n − m|/Nk there is a R that ˜. |αk | ≤ R

(73)

So the αk can only have finite size, and now we want to show that actually only αk = 0 can occur for k large enough, that will prove the Lemma. Let Wu/s be the stable and unstable manifolds of the fixed-point at 0, these are straight lines through the origin with irrational slopes, see [17], and by definition they satisfy for t ≥ 0 d(Wu , At n) ≤ Ce−γt ,

d(Ws , A−t n) ≤ Ce−γt ,

(74)

where d(Wu/s , A±t n) denotes the Euclidean distance between the manifolds Ws/u and the points A±t n. On the other hand, since the slopes of Wu/s are irrational and the slopes of the lines R τ → m+τ α, for α = 0, α, m ∈ Z2 , are rational, the distance between Wu/s and the points m + N α grows linearly in N . So for N large enough there is a ˜ such that for all m, α ∈ Z2 with |m| ≤ R, |α| ≤ R ˜ constant C depending on R, R and α = 0 we have d(Wu/s , m + N α) ≥ CN .

(75)

But since At n = m + αk Nk the only way the two inequalities (74) and (75) can hold simultaneously is if Nk e−γ|t|

(76)

and so Nk has to be bounded. So the only case remaining allowing an infinite series of Nk is the case with αk = 0 and hence At n = m. Lemma 5. Let us write n ∼A m if n, m are in the same orbit of A on Z2 . Assume Nk , k ∈ N, is a sequence for which the quantum periods satisfy (60), then for every R > 0 there exist a KR > 0 such that for m, n ∈ Z2 \{0} with |m|, |n| ≤ R and k ≥ KR , we have P 1 Tr TNk (n)∗ TNk (m) = Nk

1 P (Nk )

0

if

n ∼A m ,

otherwise .

(77)

∗ Proof. Since TN (n) = OpN [e(ω(n, x))] we have by Egorov’s Theorem UN TN (n)UN −1 −1 = OpN [e(ω(n, A x))], and because A is symplectic ω(n, A x) = ω(An, x), so ∗ = TN (An). Iterating this result gives UN TN (n)UN −t t UN TN (n)UN = TN (At n) ,

(78)

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1471

for all t ∈ Z. Using this and the discussion before Lemma 4 about the average gives P

TN (m) =

1 P (N )

1 = P (N )

P (N )−1−[P (N )/2] −t t UN TN (m)UN

t=−[P (N )/2]

(79)

P (N )−1−[P (N )/2]

TN (At m)

t=−[P (N )/2]

and so P 1 Tr TNk (n)∗ TNk (m) Nk

=

1 P (Nk )

P (Nk )−1−[P (Nk )/2]

t=−[P (Nk )/2]

1 Tr TNk (n)∗ TNk (At m) . Nk

(80)

But Tr TNk (n)∗ TNk (At m) = 0 only if n ≡ At m mod Nk , see (17), and by Lemma 4 this is for the t range we sum over and k ≥ KR only the case if n = At m, i.e., n ∼A m. In that case we have Tr TNk (n)∗ TNk (At m) = Nk and so the result follows. Proof of Theorem 5. Let a be a real valued trigonometric polynomial of order R, P then by Lemma 3 we have to compute N1k Tr OpNk [a]∗ OpNk [a] . By the definition of OpN [a] we have P 1 Tr OpNk [a]∗ OpNk [a] Nk

=

a ˆ(n)∗ a ˆ(m)

n,m∈Z2 \{0}

P 1 Tr TNk (n)∗ TNk (m) , Nk

where we used that a ˆ(0) = a ¯ = 0 by assumption, and then Lemma 5 gives P 1 1 Tr OpNk [a]∗ OpNk [a] = a ˆ(n)∗ a ˆ(m) Nk P (Nk ) 2 m: m∼ n n∈Z

for k ≥ KR . But

a ˆ(n)∗ a ˆ(m) =

m: m∼A n

(81)

(82)

A

a ˆ(n)∗ a ˆ(At n) ,

(83)

t∈Z

where the sum over t is finite since a ˆ(m) = 0 for |m| > R, and so P 1 1 Tr OpNk [a]∗ OpNk [a] = a ˆ(n)∗ a ˆ(At n) . Nk P (Nk ) 2 t∈Z n∈Z ˆ(n) e(ω(n, x)) one finds On the other hand side, using a(x) = n a ∗ a ˆ(n) a ˆ(At n) C[a](t) = n∈Z2

(84)

(85)

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and therefore P 1 1 1 V [a] . Tr OpNk [a]∗ OpNk [a] = C[a](t) = Nk P (Nk ) P (Nk )

(86)

t∈Z

5. Parabolic maps In this section we will study quantisations of parabolic maps. The interest in parabolic maps from our point of view is that they can be ergodic, but are not hyperbolic. Since hyperbolicity is the main obstacle in the control of the quantum time evolution for large times, we expect that for parabolic maps we get much stronger results. This is indeed the case, and as we will show below one again gets optimal results in some cases. Our example will be the parabolic map studied by Marklof and Rudnick in [27], see as well [7, 11]. Let α ∈ R, then the map Ψα : T 2 → T 2 is defined by

p p+α → mod 1 . (87) Ψα : q q + 2p If α is irrational this map is uniquely ergodic but not weak mixing and not hyperbolic. This map is quantised in [27] and it is shown that their quantisation UN satisfies the Egorov estimate −t t OpN [a]UN − OpN [a ◦ Ψtα ] ≤ Ca UN

|t| N

(88)

for t ∈ Z. In order to study the rate of quantum ergodicity, we need an estimate on the rate of classical ergodicity. Lemma 6. Let a ∈ C ∞ (T 2 ) and C[a](t) be the autocorrelation function of the map (87) and assume that α satisfies a Diophantine condition, i.e., there are C, γ > 0 such that |kα − l| ≥ C/|k|γ for all k, l ∈ Z\{0}. Then we have for f ∈ S(R)

1 t

1 ˆ f C[a](t) = O , (89) T T T t∈Z

where fˆ denotes the Fourier-transform of f . Furthermore, if a(p, q) depends only on p then

1 t

1 ˆ , for all M ∈ N . (90) f C[a](t) = OM T T TM t∈Z

Proof. We have Ψtα

p p + tα : → , q q + 2tp + αt(t − 1)

(91)

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1473

and with a(x) = n∈Z2 a ˆ(n)e(nx) (notice that we use here the standard Fourier series, not the one twisted with ω used in the quantisation procedure) we get C[a](t) = a ˆ(n)ˆ a(m) e(nx)e mΨtα (x) dx . (92) T2

n,m∈Z2 \{0}

Then we find e(nx)e mΨtα (x) dx T2

= δ(−n1 , m1 + 2tm2 )δ(−n2 , m2 )e m1 αt + m2 αt(t − 1) ,

(93)

where δ(m, n) denotes the Kronecker delta, and therefore C[a](t) =

a ˆ(−m1 − 2tm2 , −m2 )ˆ a(m1 , m2 )e m1 αt + m2 αt(t − 1) .

(94)

(m1 ,m2 )∈Z2 \{0}

Now we split C[a](t) into two parts, C[a](t) = C 0 [a](t)+C 1 [a](t), such that C 0 [a](t) contains only the terms with m2 = 0 a ˆ(−m, 0)ˆ a(m, 0)e(mαt) . (95) C 0 [a](t) = m∈Z\{0}

The second term satisfies |C 1 [a](t)| ≤ CK (1 + |t|)−K

(96)

for all K ∈ N since the Fourier-coefficients a ˆ(n) are quickly decreasing and therefore

1 1 t fˆ C 1 [a](t) = O . (97) T T T t∈Z

For the first term we find 1 t

fˆ C 0 [a](t) = T T t∈Z

|ˆ a(m, 0)|2

m∈Z\{0}

1 t

fˆ e(mαt) T T

and by the Poisson summation formula we obtain 1 t

fˆ f T (mα − n) = OM (|m|γM T −M ) e(mαt) = T T t∈Z

(98)

t∈Z

(99)

n∈Z

since f ∈ S(R) and by the Diophantine condition on α. And since the Fouriercoefficients a ˆ(n) are quickly decreasing we find 1 t

(100) fˆ C 0 [a](t) = OM (T −M ) T T t∈Z

Combining the two estimates for C 0 [a](t) and C 1 [a](t) gives the lemma.

Combining the Egorov estimate and this lemma we then obtain from Proposition 1.

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Theorem 6. Let UN be the quantisation of the map (87) due to [27] with a Diophantine α, and ψjN , j = 1, . . . N , a orthonormal basis of eigenfunctions. Then we have N 2 1 1 N ψj , Op[a]ψjN − a ¯ ≤ Ca 1/2 , (101) N j=1 N and if a(p, q) depends on p only then we have the stronger estimate N 2 1 N 1 ψj , Op[a]ψjN − a ¯ ≤ Ca,ε 1−ε , N j=1 N

(102)

for every ε > 0. Proof. By the Egorov estimate (88) and since |a ◦ Ψtα |k ≤ Ck |t|k , fˆT (t)RN = O(T ) , fˆT (t)|a ◦ Ψtα |k = O(T k ) . t∈Z

(103)

t∈Z

If we use then (89) we obtain by Proposition 1

3

N 2 1 T T 1 N ψj , Op[a]ψjN − a ¯ = O + O +O , N j=1 T N3 N

(104)

and so the choice T = N 1/2 gives (101). If we have instead the faster decay (90) we get

3

N 2 1 N 1 T T N ψj , Op[a]ψj − a ¯ = OM +O +O , (105) M 3 N j=1 T N N

for every M ∈ N and so by choosing T = N ε , with ε small enough, and M large enough we obtain (102). The results in [27] show that the estimate (101) is optimal, so we again obtain a sharp estimate. The analysis in [27] is much more detailed and provides sharp estimates for the rate of quantum ergodicity for individual eigenfunctions. But Theorem 6 might still be of some interest because the proof is of a more dynamical nature, and therefore may be easier to extend to other cases. One further class of systems where one could apply the same methods is given by perturbed Kronecker maps, which were recently studied by Rosenzweig, [28]. Here the proof would be very similar to the one of (102), and we would get the same rate Oε (1/N 1−ε ). But in [28] a stronger bound on individual eigenfunctions is given, so our method does not give an optimal result. The parabolic maps Ψα are not weak mixing, so by Theorem 2 the off-diagonal matrix elements ψiN , OpN [a]ψjN are not going to 0. In fact, using the same methods as in the proof of Lemma 6, one finds for Ψtα that for θ = kα, k ∈ Z\{0},

1 t

1 2 ˆ f C[a](t)e(θt) = |ˆ a(k, 0)| + O . (106) T T T t∈Z

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From this result together with the techniques used in the proof of Theorem 6 one can derive N 1 ψi , OpN [a]ψjN 2 = |ˆ lim a(k, 0)|2 , (107) N →∞ N N N 1/2 |θi −θj −θ/N |S 1 ≤δ/N

for any δ > 0, where ψiN , e(θiN ), i = 1, . . . , N are the eigenvectors and eigenvalues of UN , and θ = kα.

Acknowledgements This paper grew out of a section in a preliminary version of [30], I would like to thank the referee who suggested to split the paper and as well asked if a logarithmic rate is sharp for the cat map. I am indebted to Stéphane Nonnemacher for many constructive remarks. Further thanks go to Zeev Rudnick, Jon Keating and Brian Winn for helpful remarks. This work was fully supported by the EPSRC Grant GR/T28058/01.

References [1] R. Aurich and M. Taglieber, On the rate of quantum ergodicity on hyperbolic surfaces and for billiards, Physica D 118 (1998), 84–102. [2] M. Basilio de Matos and A. M. Ozorio de Almeida, Quantization of Anosov maps, Ann. Physics 237 (1995), no. 1, 46–65. [3] A. B¨ acker, R. Schubert, and P. Stifter, Rate of quantum ergodicity in euclidean billiards, Phys. Rev E 57 (1998), 5425–5447, erratum: ibid. 58 (1998), 5192. [4] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), no. 12, 2083–2091. [5] O. Bohigas, M.-J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984), no. 1, 1–4. [6] F. Bonechi and S. De Bièvre, Exponential mixing and | ln | time scales in quantized hyperbolic maps on the torus, Comm. Math. Phys. 211 (2000), no. 3, 659–686. [7] A. Bouzouina and S. De Bièvre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. Math. Phys. 178 (1996), no. 1, 83–105. [8] J. Bouclet and S. De Bièvre, Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms, Ann. Henri Poincaré 6 (2005), no. 5, 885–913. [9] Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), no. 3, 497–502. [10] S. De Bièvre, Quantum chaos: a brief first visit, Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), Contemp. Math., vol. 289, Amer. Math. Soc., Providence, RI, 2001, 161–218.

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R. Schubert


[11] S. De Bièvre and M. Degli Esposti, Egorov theorems and equidistribution of eigenfunctions for the quantized sawtooth and baker maps, Ann. Inst. H. Poincaré Phys. Théor. 69 (1998), no. 1, 1–30. [12] T. O. de Carvalho, J.P. Keating, and J.M. Robbins, Fluctuations in quantum expectation values for chaotic systems with broken time reversal symmetry, J. Phys. A (1998), 5631–5640. [13] M. Degli Esposti and S. Graffi, Quantum maps, The Mathematical Aspects of Quantum Maps, Lecture Notes in Physics, vol. 618, Springer, 2003, 49–90. [14] M. Degli Esposti, S. Graffi, and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys. 167 (1995), no. 3, 471–507. [15] M. Degli Esposti, S. Nonnenmacher, and B. Winn, Quantum variance and ergodicity for the baker’s map, Comm. Math. Phys. 263 (2006), no. 2, 325–352. [16] B. Eckhardt, S. Fishman, J. Keating, O. Agam, J. Main, and K. M¨ uller, Approach to ergodicity in quantum wave functions, Phys. Rev. E 52 (1995), no. 6, 5893–5903. [17] F. Faure, S. Nonnenmacher, and S. De Bièvre, Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys. 239 (2003), no. 3, 449–492. [18] M. Feingold and A. Peres, Distribution of matrix elements of chaotic systems, Phys. Rev. A (3) 34 (1986), no. 1, 591–595. [19] J. H. Hannay and M. V. Berry, Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating, Phys. D 1 (1980), no. 3, 267–290. [20] J. P. Keating, Asymptotic properties of the periodic orbits of the cat maps, Nonlinearity 4 (1991), no. 2, 277–307. [21] P. Kurlberg and Z. Rudnick, Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J. 103 (2000), no. 1, 47–77. [22] P. Kurlberg and Z. Rudnick, On quantum ergodicity for linear maps of the torus, Comm. Math. Phys. 222 (2001), no. 1, 201–227. [23] P. Kurlberg and Z. Rudnick, On the distribution of matrix elements for the quantum cat map, Ann. of Math. 161 (2005), no. 1, 489–507. [24] P Kurlberg, private communication, 2005. ´ [25] W. Luo and P. Sarnak, Quantum variance for Hecke eigenforms, Ann. Sci. Ecole Norm. Sup. (4) 37 (2004), no. 5, 769–799. [26] J. Marklof and S. O’Keefe, Weyl’s law and quantum ergodicity for maps with divided phase space, Nonlinearity 18 (2005), no. 1, 277–304, With an appendix “Converse quantum ergodicity” by Steve Zelditch. [27] J. Marklof and Z. Rudnick, Quantum unique ergodicity for parabolic maps, Geom. Funct. Anal. 10 (2000), no. 6, 1554–1578. [28] L. Rosenzweig, Quantum unique ergodicity for maps on the torus, Ann. Henri Poincaré 7 (2006), no. 3, 447–469. [29] Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195–213. [30] R. Schubert, Upper bounds on the rate of quantum ergodicity, Ann. Henri Poincaré 7 (2006), no. 6, 1085–1098. ˇ [31] A. I. Snirel’man, Ergodic properties of eigenfunctions, Uspehi Mat. Nauk 29 (1974), no. 6 (180), 181–182.

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[32] T. Sunada, Quantum ergodicity, Progress in inverse spectral geometry, Trends Math., Birkh¨ auser, Basel, 1997, 175–196. [33] M. Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloqium, IMPA, 1997. [34] P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer Verlag, New York, 1982. [35] M. Wilkinson, A semiclassical sum rule for matrix elements of classically chaotic systems, J. Phys. A 20 (1987), no. 9, 2415–2423. [36] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), no. 4, 919–941. [37] S. Zelditch, Quantum transition amplitudes for ergodic and for completely integrable systems, J. Funct. Anal. 94 (1990), no. 2, 415–436. [38] S. Zelditch, On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys. 160 (1994), no. 1, 81–92. [39] S. Zelditch, Quantum mixing, J. Funct. Anal. 140 (1996), no. 1, 68–86. Roman Schubert School of Mathematics University of Bristol Bristol BS8 1TW United Kingdom e-mail: [email protected] Communicated by Jens Marklof. Submitted: May 31, 2008. Accepted: September 19, 2008.



On Matrix Elements for the Quantized Cat Map Modulo Prime Powers Dubi Kelmer Abstract. The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously expected. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.

1. Introduction The quantum cat map is a model for a quantum system with underlying chaotic dynamics that was originally introduced by the physicists Hannay and Berry [10]. This model can be used to study the semiclassical properties of such systems [2, 6, 15, 16]. The classical dynamics underlying this model is the discrete time iteration of a hyperbolic map, A ∈ SL(2, Z), on the torus, T2 = R2 /Z2 . In order to quantize the cat map, for every integer N (playing the role of the inverse of Planck’s constant) the Hilbert space of states is HN = L2 (Z/N Z). For every smooth real valued function f there is a quantum observable, i.e., a Hermitian operator OpN (f ) : HN → HN . The quantum evolution is given by a unitary operator UN (A) on HN . For generic quantum systems with underlying chaotic dynamics, it is believed that matrix elements of smooth observables tend to the phase space average of the observable in the semiclassical limit. In order to test this phenomenon in the quantum cat map model, Kurlberg and Rudnick introduced hidden symmetries of this This material is based upon work supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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model, a group of commuting operators that commute with UN (A), they called Hecke operators [15]. They showed that for any sequence of Hecke eigenfunctions (i.e., joint eigenfunctions of all Hecke operators), the corresponding matrix elements converge to the phase space average as N → ∞. To be more precise they showed [15, Theorem 1] that for any f ∈ C ∞ (T2 ) and ψ ∈ HN a Hecke eigenfunction the matrix elements satisfy 1 OpN (f )ψ, ψ − f f, N − 4 + . 2 T

They remarked [15, Remark 1.2] that the exponent of 14 is not optimal and that 1 the correct bound should be O(N − 2 + ), in accordance to the second and fourth moments. For N prime (and consequently also for N square free) this is indeed the correct bound [5, 9]. Remark 1.1. We note that without the arithmetic symmetries these bounds hold only if the spectral degeneracies are sufficiently small. In fact, there are sequences of eigenfunctions (where the degeneracies are exceptionally large) that don’t converge to the phase space average at all. For these eigenfunctions the matrix elements localize around short periodic orbits in the sense that the corresponding limiting measure contains a component that is supported on the periodic orbit [6]. In [16] Kurlberg and Rudnick went on to investigate the fluctuation of the normalized matrix elements, √ (N ) OpN (f )ψj , ψj − f dx , (1.1) Fj = N T2

where ψj are Hecke eigenfunctions and N → ∞ through primes. For this purpose they introduced the quadratic form Q(n) = ω(nA, n) (with ω(n, m) = n1 m2 − n2 m1 the standard symplectic form) and used it to define twisted Fourier coefficients. For a smooth function f ∈ C ∞ (T2 ) with Fourier coefficients fˆ(n) for n ∈ Z2 , the twisted coefficients are given by (−1)n1 n2 fˆ(n) . f # (ν) = (1.2) Q(n)=ν

Conjecture (Kurlberg–Rudnick [16]). As N → ∞ through primes, the limiting (N ) distribution of the normalized matrix elements Fj is that of the random variable Xf = f # (ν)Tr(Uν ) ν=0

where Uν are independently chosen random matrices in SU(2) endowed with Haar probability measure. As evidence, the second and fourth moment were computed to show agreement with this conjecture. In particular, the moment calculation implies that the limiting distribution is not Gaussian, in contrast to generic chaotic systems where the fluctuations are believed to be Gaussian [4, 7].

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In this paper we further study the matrix elements for the cat map for composite N . In fact, it is sufficient to understand the case of prime powers (see [15, Section 4.1]), and so we restrict ourselves to this case. For N a prime power, we give an explicit formula for the matrix elements as a weighted sum of certain exponential sums. We then use this formula to show that there are sequences of eigenfunctions such that the matrix elements decay like N −1/3 rather then the expected rate of N −1/2+ . We further show that when N = pk with k > 1, the matrix elements have a limiting distribution as p → ∞. This distribution is not Gaussian and it is also different from the (conjectured) distribution for k = 1. Instead of behaving like traces of random elements from SU(2), here the normalized matrix elements vanish for half of the eigenfunctions and for the rest they behave like 2 cos(θ) where the angle is chosen at random. 1.1. Results For every N = pk denote by

C(pk ) = B ∈ SL(2, Z/pk Z)|AB = BA (mod pk ) , the group of Hecke operators. For ν ∈ Z and χ a character of C(pk ) define the exponential sum Epk (ν, χ) = epk (νx)χ β(x) , x∈X(pk )

where

X(pk ) = x ∈ Z/pk Z| Tr(A)2 − 4 x2 = 1 (mod p)

and β : X(pk ) → C(pk ) is an injection of X(pk ) into C(pk ) given by a rational function (defined by (3.1)). ˆ k ) of Theorem 1.2. For each prime power pk , there is a subset Cˆ0 (pk ) ⊂ C(p ˆ0 (pk )| |C characters, with limp→∞ pk = 1 such that 1. For any χ ∈ Cˆ0 (pk ) there is a unique Hecke eigenfunction ψ, s.t., χ is a joint eigenvalue. 2. For this eigenfunction, and any elementary observable fn (x) = exp(2πin · x) with Q(n) ≡ 0 (mod p) Q(n) (−1)n1 n2 E k , χχ0 , Oppk (fn )ψ, ψ = ± #C(pk ) p 2 where χ0 is a fixed character of C(pk ) and the sign (±) depends on p, k but not on ψ. If we consider nontrivial prime powers (i.e., k > 1) we can use elementary methods to evaluate these sums. In particular we find that there are matrix ele1 ments that decay much slower then the expected rate of N − 2 + . Theorem 1.3. There are smooth observables f ∈ C ∞ (T2 ), and sequences of Hecke 1 eigenfunctions satisfying | OpN (f )ψj , ψj − T2 f | N − 3 .

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We note, however, that these exceptional matrix elements are quite rare, in the sense that for a fixed observable the number of matrix elements decaying slower 1 then N − 2 + is bounded by O(pk−1 ) (see Corollary 4.2). Remark 1.4. In [17] Olofsson studied the supremum norm of Hecke eigenfunctions for the quantized cat map. He showed that for composite N the supremum norm 1 can be of order N 4 , which is much larger then the case of N prime (or square free) where all Hecke eigenfunctions satisfy ψ ∞ N [8, 14]. Although the two phenomena look similar, there does not seem to be any apparent connection between them. At least in the sense that the eigenfunctions with large matrix elements are usually not the eigenfunctions with large supremum norm. For nontrivial prime powers, we can also show that the exponential sums Epk (ν, χ) (and hence also the matrix elements) have a limiting distribution as p → ∞. (See [13] for similar results on twisted Kloosterman sums). To simplify the discussion we will assume from here on that the observable f is a trigonometric (N ) polynomial and let Fj be the normalized matrix element as in (1.1). Let μ denote the measure on [0, π) defined by π 1 π 1 μ(f ) = f + f (θ)dθ . 2 2 2π 0 Theorem 1.5. Let f be a trigonometric polynomial. For any k > 1, as p → ∞ (pk ) through primes, the limiting distribution of the normalized matrix elements Fj is that of the random variable Yf = 2 f # (ν) cos(θν ) ν=0

where θν are independently chosen from [0, π) with respect to the measure μ. Remark 1.6. As mentioned above, there can be exceptionally large matrix elements (N ) N 1/6 are not bounded. Such matrix elements would cause the for which Fj moments (above the 6’th moment) to blow up as N → ∞. Nevertheless, since the number of exceptional matrix elements is of limiting density zero, they do not influence the limiting distribution (see Section 2.5 for more details). 1.2. Outline The outline of the paper is as follows: In Section 2 we provide some background on the cat map and its quantization and on the notion of a limit distributions. In Section 3 we compute the formulas for the matrix elements proving Theorem 1.2. In Section 4 we compute the exponential sums appearing in these formulas for non trivial prime powers, and establish the limiting distribution as p → ∞. Then in Section 5 we deduce both of the results on the matrix elements (Theorems 1.3 and 1.5) from the analysis of the exponential sums.

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2. Background The full details for the cat map and it’s quantization can be found in [15]. We briefly review the setup and go over our notation. 2.1. Classical dynamics The classical dynamics are given by the iteration of a hyperbolic linear map A ∈ SL(2, Z). p x= ∈ T2 → Ax (mod 1) . q Given an observable f ∈ C ∞ (T2 ), the classical evolution defined by A is f → f ◦A. 2.2. Quantum kinematics For doing quantum mechanics on the torus, one takes Planck’s constant to be 1/N , as the Hilbert space of states one takes HN = L2 (Z/N Z), where the inner product is given by: 1 φ(y)ψ(y) .

φ, ψ = N y∈Z/N Z

For n = (n1 , n2 ) ∈ Z define elementary operators TN (n) acting on ψ ∈ HN via: 2

TN (n)ψ(y) = e2N (n1 n2 )eN (n2 y)ψ(y + n1 ) ,

(2.1)

where eN (x) = e . For any smooth classical observable f ∈ C ∞ (T2 ) with Fourier expansion f (x) = n∈Z2 fˆ(n)e2πin · x , its quantization is given by OpN (f ) = fˆ(n)TN (n) . 2πix N

n∈Z2

2.3. Quantum dynamics For any A ∈ SL(2, Z), we assign unitary operators UN (A), acting on L2 (Z/N Z) having the following important properties: • “Exact Egorov”: For A ≡ I (mod 2), and any f ∈ C ∞ (T2 ) UN (A)−1 OpN (f )UN (A) = OpN (f ◦ A) . • The map A → UN (A) is a representation of SL(2, Z/N Z): If C ≡ AB (mod N ) then UN (A)UN (B) = UN (C). We will make use of the following formula for UN (A), (valid for odd N and any A ∈ SL(2, Z)) [12, Proposition 1.4]. σN (A) TÑ (m)TÑ (−mA) (2.2) UN (A) = | kerN (A − I)|N 2 m∈(Z/N Z)

where σN (A) = Tr(UN (A)) is the character of the representation,

| kerN (A − I)| = # n ∈ (Z/N Z)2 |n(A − I) ≡ 0 (mod N ) , and TÑ (n) = (−1)n1 n2 TN (n) are twisted elementary operators.

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Remark 2.1. The twisted operators TÑ (n) have the convenient feature that UN (A)∗ TÑ (n)UN (A) = TÑ (nA) for any A ∈ SL(2, Z/N Z) (without the parity condition). 2.4. Hecke eigenfunctions Let α > α−1 be the eigenvalues of A in a (real) quadratic extension K/Q. Then the eigenvectors v± A = α±1 v± . Denote vectors v± = (c, α±1 − a) are corresponding √ 2 + by D = Tr(A) − 4 ∈ Z so that D = α − α−1 . Consider the ring O = Z[α] and denote by ι : O → Mat(2, Z) the map sending β = n + mα → B = n + mA (this map is a ring homomorphism as α and A have the same minimal polynomial). For any integer N the norm map, NK/Q : K ∗ → Q∗ , induce a well defined map NN : (O/N O)∗ → (Z/N Z)∗ . Let C(N ) = ker NN be its kernel, then its image ι(C(N )) ⊂ SL(2, Z/N Z) is a commutative subgroup of SL(2, Z/N Z) that commutes with A (mod N ). The Hecke operators are then {UN (B)|B ∈ ι(C(N ))}, and Hecke eigenfunctions are joint eigenfunctions of UN (A) and all the Hecke operators. The eigenvalues corresponding to each Hecke eigenfunction define a character χ of C(N ) i.e., UN (ι(β))ψ = χ(β)ψ. We canthus decompose our Hilbert space into a direct sum of joint eigenspaces HN = χ Hχ , parameterized by the characters of C(N ). We say that a character χ appears with multiplicity one in the decomposition when the corresponding eigenspace is one dimensional. 2.5. Limit distribution We recall the notion of a limiting distribution for a sequence of points on the line. (N ) For each N let {Fj }N j=1 be a set of points on the line. We say that these points have a limiting distribution Y (where Y : Ω → R is some random variable on a probability space Ω) if for any segment [a, b] ⊂ R the limit

(N ) # j|a ≤ Fj ≤ b = Prob Y ∈ [a, b] . lim N →∞ N From this definition it is strait forward that making an arbitrary change in a density zero set of points (i.e., changing SN points for each N with SNN → 0), does not affect the limiting distribution. An equivalent condition for having a limiting distribution Y , is that for any (N ) continues bounded function g the average N1 j g(Fj ) converges as N → ∞ to Ω g(Y (ω))dω. Note that the condition that the test function g is bounded is (N )

necessary unless both the variable Y and the points Fj (N ) Fj

are uniformly bounded.

In particular, if the points are not uniformly bounded then their moments don’t necessarily converge to the moments of Y .

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2.6. Notation We use the notation e(x) = e2πix . For any N ∈ N we denote by eN ( · ) the character x of Z/N Z given by eN (x) = e( N ). When there is no risk of confusion we will slightly abuse notation and write eN ( ab ) for eN (ab−1 ) (where b−1 denotes the inverse of b modulo N ). For example, for N odd and a ∈ Z we may write e2N (a) = (−1)a eN ( a2 ).

3. Formulas for matrix elements For N a prime power we give formulas for the matrix elements of elementary observables explicitly as exponential sums. When N is prime these formulas appeared in [16] (for primes that split in O) and in [12] (for inert primes). We will make use of the following parametrization of the Hecke operators. For any integer 1 ≤ l ≤ k we define subgroups Cp (k, l) ⊂ C(pk ) by

Cp (k, l) = β ∈ C(pk )|β ≡ 1 (mod pl ) . For notational convenience we will also define Cp (k, k + 1) = {1}. Let

X(pk ) = x ∈ Z/pk Z|Dx2 = 1 (mod p) then the map

√ Dx + 1 β(x) = √ Dx − 1

(3.1)

is a bijection between X(pk ) and C(pk ) \ Cp (k, 1) with inverse map given by 1+β(x) x = √D(1−β(x)) (mod pk ) (note that for β = 1 (mod p) the inverse map is indeed well defined). For every character χ of C(pk ) and any ν ∈ (Z/pk Z)∗ we have the exponential sum Epk (ν, χ) = epk (νx)χ β(x) . x∈X(pk )

To prove Theorem 1.2 we will show that for any n ∈ Z2 with Q(n) = ν = 0 (mod p), and for every character χ of C(pk ) that appears with multiplicity one, the corresponding matrix element is given by νx ±1 χχ0 β(x) , T˜pk (n)ψ, ψ = e k p #C(pk ) 2 k x∈X(p )

(where χ0 is a fixed character of C(pk ) and the sign is −1 when p is inert and k is odd and +1 otherwise). We can then take our set Cˆ0 (pk ) to be the set of characters appearing with multiplicity one. This set is of order pk if p is inert (Lemma 3.3) k ˆ and of order pk − pk−1 if p splits (Lemma 3.1). Hence, indeed C0p(pk ) = 1 + O( p1 ). We will compute the matrix elements separately for the inert and split cases.

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3.1. Split case When p is split, we can give explicit formulas for the Hecke eigenfunctions and use them to compute the matrix elements. Since we assume that p splits in O, y 0 there is a matrix M ∈ SL(2, Z/pk Z) satisfying that M −1 AM = ( 0 y−1 ) (mod pk ). Consequently, the Hecke group is given by x 0 −1 k ∗ C(pk ) = M |x ∈ (Z/p Z) M , 0 x−1 which is naturally isomorphic to (Z/pk Z)∗ . We recall that x 0 Upk ψ(y) = χ0 (x)ψ(xy) , 0 x−1

(3.2)

where χ0 is a fixed character of (Z/pk Z)∗ [15, Section 4.3]. ∗ k Lemma 3.1. For any character χ of (Z/pk Z) (extended to a function on Z/p Z p by setting χ(px) = 0), the function ψ = p−1 Upk (M )χ is a normalized joint

eigenfunction of all Hecke eigenfunctions with eigenvalue χχ0 . Furthermore, if χ is not trivial on the subgroup Cp (k, k − 1) then this is the only eigenfunction. Proof. The first assertion is an immediate consequence of (3.2). For the second part, assume that ψ is an eigenfunction with eigenvalue χχ0 , and that χ is not trivial on Cp (k, k − 1). Then there is x0 ∈ Cp (k, k − 1) with χ(x0 ) = 1. Now, let φ = Upk (M )−1 ψ, then for any x ∈ (Z/pk Z)∗ , x 0 χχ0 (x)φ(y) = Upk φ(y) = χ0 (x)φ(xy) , 0 x−1 hence φ(xy) = χ(x)φ(y). For any y ≡ 0 (mod p) we have that x0 y ≡ y (mod pk ) (as x0 ≡ 1 (mod pk−1 )). Consequently, φ(y) = φ(x0 y) = χ(x0 )φ(y) implying that φ(y) = 0. On the other hand, for y ≡ 0 (mod p) we have φ(y) = χ(y)φ(1) so φ is uniquely determined (up to normalization). Remark 3.2. In the case that the character χ is trivial on the group Cp (k, l) (but not on Cp (k, l − 1)) then the above argument implies that the corresponding eigenspace is of dimension k − l + 1. Proof of Theorem 1.2 (split case). Let χ be a character not trivial on Cp (k, k − 1). p Then, ψ = p−1 Upk (M )χ is an eigenfunction with character χχ0 , where A = y

0

M ( 0 y−1 )M −1 (mod pk ). Consequently, for any (twisted) elementary observable p T˜pk (n)ψ, ψ = Upk (M )∗ T˜pk (n)Upk (M )χ, χ p−1 p ˜ Tpk (m)χ, χ = p−1 with m = nM (mod pk ).

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Now, let d = y − y −1 so that d2 ≡ D (mod pk ) (recall Tr(A) ≡ y + y −1 (mod pk )). Then m m x + m1 1 1 2 epk epk (m2 x)χ T˜pk (m)χ, χ = k p 2 x x∈(Z/pk Z)∗ dt + 1 dm1 m2 1 t χ epk = k p 2 dt −1 k t∈X(p )

where we made the change of variables 2x = m1 (dt − 1). Finally, notice that for m = nM (mod pk ) we have that Q(n) = ω(nA, n) ≡ m1 m2 (y − y −1 ) ≡ dm1 m2 Hence indeed T˜pk (n)ψ, ψ =

1 #C(pk )

epk

x∈X(pk )

Q(n)x 2

(mod pk ) .

χ β(x) .

3.2. Inert case First we show that for p inert, any joint eigenspace is one dimensional. Lemma 3.3. For N = pk and p inert, the dimension of any joint eigenspace satisfies dim Hχ ≤ 1. Proof. The trace of the quantum propagators satisfy [12, Corollary 1.6]

Tr Upk (B) 2 = # n ∈ (Z/pk Z)2 : n(B − I) ≡ 0 (mod pk ) . For p inert, the group C(pk ) is of order #C(pk ) = pk−1 (p + 1), and the groups

Cp (k, l) = β ∈ C(pk )|β ≡ 1 (mod pl ) , k

) are of order #Cp (k, l) = #C(p = pk−l . Moreover, for any β ∈ Cp (k, l)\Cp (k, l+1) #C(pl ) we have |Tr(Upk (ι(β)))|2 = p2l . Consequently

k−1 2 Tr Upk ι(β) = pk +

2 Tr Upk ι(β) + p2k

l=1 β∈Cp (k,l)\Cp (k,l+1)

β∈C(pk )

= pk + k

k−1

(pk−l − pk−l−1 )p2l + p2k

l=1 2k−1

=p +p

− pk + p2k = pk #C(pk ) .

On the other hand, if we denote by nχ = dim Hχ then 2 2 1 U ι(β) nχ . k Tr = p #C(pk ) k χ β∈C(p )

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Comparing the two expressions 1 n2χ = k #C(p ) k χ


we get 2 nχ . Tr Upk ι(β) = pk = dim H = χ

β∈C(p )

Since nχ are non negative integers this implies nχ ≤ 1.

After establishing this fact, following the idea of Gurevich and Hadani [9], we can write the matrix elements of elementary observables as Tpk (n)ψj , ψj = Tr Tpk (n)Pχj , with Pχj =

1 #C(pk )

¯j Upk ι(β) χ

β∈C(pk )

the projection operator to the (one dimensional) eigenspace spanned by ψj . We then use formula (2.2) for Upk (ι(β)) in order compute Tr(Tpk (n)Upk (ι(β))). However, in order to do this we first need to give a formula for the character of the representation σ(B) = Tr(Upk (B)) (appearing in (2.2)), for any B ∈ ι(C(pk )). ˆ k ) such that for any β ∈ C(pk ), we Proposition 3.4. There is a character χ0 ∈ C(p have Tr Upk ι(β) = (−1)k (−p)l χ0 (β) , with 1 ≤ l ≤ k the maximal integer such that β ≡ 1 (mod pl ). Proof. For any 1 ≤ l ≤ k + 1 consider the subgroup of characters

ˆ k )|χ(β) = 1, ∀β ∈ Cp (k, l) . Cˆ (l) (pk ) = χ ∈ C(p ˆ k) For 1 ≤ l ≤ k, the group Cˆ (l) (pk ) is the kernel of the restriction map from C(p k #C(p ) l−1 to Cˆp (k, l) and hence of order = p (p + 1) (and for l = k + 1 we have #Cp (k,l)

Cˆ (k+1) (pk ) = Cˆ (k) (pk ) = C(pk )). We will first prove the following: For each 1 ≤ l ≤ k + 1 there is a character χl ∈ Cˆ (l) (pk ) and a subset Sl ⊂ Cˆ (l−1) (pk ) of order #Sl = pl−1 such that for any β ∈ C(pk ) \ Cp (k, l), Tr Upk ι(β) = (−1)k+l+1 χl χl+1 . . . χk (β) χ(β) . χ∈Sl

First for l = k +1 we take the character to be the trivial character and the set ˆ k ) to be the set of characters that appear in the decompoSk+1 ⊂ Cˆ (k) (pk ) = C(p sition of Hpk (there are pk such characters each appearing with multiplicity one). Then indeed Tr(Upk (ι(β))) = χ∈Sk+1 χ(β). If k = 1 the sum is over all but one ˆ of the characters, say χ0 ∈ C(p), and hence Tr(Up (ι(β))) = −χ0 (β) as claimed. For k > 1 we proceed by induction as follows.

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We assume the assertion is true for 1 < l ≤ k + 1 and show that it is true for l − 1. For any β ∈ C(pk ) \ Cp (k, l), by our assumption Tr Upk ι(β) = (−1)k+l+1 χl χl+1 . . . χk (β) χ(β) χ∈Sl

with Sl ⊂ Cˆ (l−1) (pk ) of order #Sl = pl−1 . The order #Cˆ (l−1) (pk ) = pl−1 + pl−2 hence the complement Slc in Cˆ (l−1) (pk ) is of order pl−2 . Now, if β ∈ Cp (k, l − 1) then the sum over all characters in Cˆ (l−1) (pk ) vanish, and hence χ∈Sl χ(β) = − χ∈S c χ(β). We thus have that l χ(β) . Tr Upk ι(β) = (−1)k+l χl χl+1 . . . χk (β) χ∈Slc

On the other hand, for β ∈ Cp (k, l −2)\Cp (k, l −1) we have that |Tr(Upk (ι(β)))| = pl−2 , which could happen only if χ(β) takes the same value for all χ ∈ Slc . Now take c ˆ (l−2) (pk ) χl−1 to be any character from Slc and let Sl−1 = χ−1 l−1 Sl . Then Sl−1 ⊆ C l−2 and is of order p χ(β) . Tr Upk ι(β) = (−1)k+l−1 χl−1 χl χl+1 . . . χk (β) χ∈Sl−1

Now, let χ0 = χ1 · χ2 · · · χk and let β ∈ Cp (k, l) \ Cp (k, l + 1). Since β ∈ Cp (k, l + 1) we have, Tr Upk ι(β) = (−1)k+l χl+1 χl+2 . . . χk (β) χ(β) . χ∈Sl+1

ˆ (l) k On the other hand we also assume β ∈ Cp (k, l), hence, for alll χ ∈ Sl+1 ⊂ C (p ) we have χ(β) = 1 implying that χ∈Sl+1 χ(β) = #Sl+1 = p . Also for any m ≤ l, χm ∈ C (m) (pk ) ⊂ C (l) (pk ), so χm (β) = 1. We thus get that indeed Tr Upk ι(β) = (−1)k (−p)l χ0 (β) . Proposition 3.5. Let n ∈ Z2 and B ∈ ι(C(pk )). For B ≡ I (mod p) the trace Tr(T˜pk (n)Upk (B)) = 0. Otherwise, there is x ∈ X(pk ) such that B = ι(β(x)) and Q(n)x Tr T˜pk (n)Upk (B) = (−1)k χ0 β(x) epk − . 2 Proof. Use formula (2.2) for Upk (B) to get that Tr T˜pk (n)Upk (B) =

σpk (B) | kerpk (B − I)|pk

Tr T˜pk (n)T˜pk (m)T˜pk (−mB) .

m∈(Z/pk Z)2

Note that up to a phase T˜pk (n)T˜pk (m)T˜pk (−mB) = eiα T˜pk (n − m(B − I)) and recall that Tr(T˜pk (n)) = 0 unless n ≡ 0 (mod pk ) (see e.g., [15, Lemma 4]). Hence,

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the only summand that does not vanish is the one satisfying n = m(B − I) (mod pk ). We can assume n = 0 (mod p), so that the trace vanishes whenever B ≡ I (mod p). Otherwise, B = ι(β) for some β ∈ C(pk ) \ Cp (k, 1) and σpk (B) = (−1)k χ0 (β) so that ω(m, mB) k ˜ Tr Tpk (n)Upk (B) = (−1) χ0 (β)epk − , 2 with m = n(B − I)−1 (mod pk ).

Now recall the parametrization C(pk ) \ Cp (k, 1) = β(x)|x ∈ X(pk ) , with β(x) =

√ √Dx+1 . Dx−1

We claim that for B = ι(β(x)) and m = nB we have that

ω(m, mB) = Q(n)x. To show this substitute (β(x) − 1)−1 = 1)−1 β(x) =

√

√

Dx−1 2

and (β(x) −

Dx+1 . 2

Consequently we get √ √ x √ Dx − 1 Dx + 1 , nι = ω nι( D), n . ω(m, mB) = ω nι 2 2 2 √ Recall that D = (α − α−1 ) so that indeed √ ω nι( D), n = ω n(A − A−1 ), n = 2ω(nA, n) = 2Q(n) .

Proof of Theorem 1.2 (inert case). For every character χ let 1 Pχ = Upk (B)χ(B) ¯ , k #C(p ) k B∈C(p )

be the projection operator to the (one dimensional) eigenspace corresponding to χ. Let ψ be the corresponding Hecke eigenfunction. Then 1 T˜pk (n)ψ, ψ = Tr T˜pk (n)Pχ = Tr T˜pk (n)Upk (B) χ(B) ¯ . k #C(p ) k B∈C(p )

Now from the Proposition 3.5 Tr T˜pk (n)Upk (B) = (−1)k χ0 (B)epk implying that (−1)k T˜pk (n)ψ, ψ = #C(pk )

β∈C(pk )

Q(n)x − , 2

Q(n)x ¯ β(x) . epk − χ0 β(x) χ 2

After a change of variables x → −x we get

(−1)k T˜pk (n)ψ, ψ = E k Q(n)/2, χχ ¯0 . #C(pk ) p

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4. Analysis of the exponential sums In this section we compute the exponential sums Epk (ν, χ) for any prime power k > 1. This can be done using elementary methods (see, e.g., [11, Section 12.3] or [1, Chapter 1.6]), however, since the setup here is slightly different we will perform this computation in full. We then evaluate all mixed moments of these exponential sums to deduce their limiting distribution. 4.1. Computation of exponential sums For ν ∈ Z/pk Z its “square root” (modulo pk ) is the set

Sq(ν, pk ) = x ∈ Z/pk Z|x2 = ν (mod pk ) . Note that for ν = 0 (mod p) this set contains two or zero elements, for ν ≡ 0 (mod pk ) it contains p[k/2] elements (and for ν = pl ν˜ with ν˜ coprime to p it contains zero or 2pl/2 elements). Proposition 4.1. For k = 2l even

Epk (ν, χ) = pl

epk (νx)χ β(x) ,

2t +ν

χ x∈Sq( νD ,pl ) 2 Dx =1(p)

√ where tχ ∈ Z/pl Z satisfies that χ(1 + pl Dx) = epl (tχ x) For k = 2l + 1 odd Epk (ν, χ) = pl epk (νx)χ β(x) G(x) 2t +ν

χ x∈Sq( νD ,pl ) Dx2 =1(p)

√ 2 where tχ ∈ Z/pl+1 Z satisfies χ(1 + pl Dx + p2l D 2 x ) = epl+1 (tχ x), and G(x) is the Gauss sum given by G(x) = ep f (x)y 2 + g(x)y , y∈Z/pZ 2t x 2tχ +ν with f (x) = Dx2χ−1 and g(x) = p−l (ν−tχ Dx22 −1 ). (Notice that for x ∈ Sq( νD , pl ) we have (ν − tχ Dx22 −1 ) ≡ 0 (mod pl ), hence p−l (ν − tχ Dx22 −1 ) gives a well defined

residue modulo p).

Proof. First for k = 2l, write the sum as Epk (ν, χ) = epk ν(x + pl y) χ β(x + pl y) . x∈X(pl ) y∈Z/pl Z

Replace β(x + pl y) ≡ β(x)(1 + Epk (ν, χ) =

β l β (x)p y)

(mod p2l ) to get β epk (νx)χ β(x) epl (νy)χ 1 + pl (x)y . β l l

x∈X(p )

y∈Z/p Z

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D. Kelmer √ √Dx+1 Dx−1

Differentiating β(x) = takes the form

we get

β β (x)

Ann. Henri Poincaré √

2 D = − Dx 2 −1 , so that the inner sum

√ epl (νy)χ 1 − pl D

y∈Z/pl Z

2y Dx2 − 1

.

√ (2l, l). Hence, for any The map x → 1 + pl Dx is an isomorphism of Z/pl Z and Cp√ character χ of C(p2l ) there is tχ ∈ Z/pl Z such that χ(1 + pl Dx) = epl (tχ x). We can thus write the inner sum as 2ytχ 2tχ epl (νy)epl − e ν − = y . l p Dx2 − 1 Dx2 − 1 l l y∈Z/p Z

y∈Z/p Z

2t +ν

χ , pl ) in which case it is equal pl . This sum vanishes unless x ∈ Sq( νD Now for k = 2l + 1, we start again by writing

Epk (ν, χ) =

epk ν(x + pl y) χ β(x + pl y) ,

x∈X(pl ) y∈Z/pl Z

and replace

β 1 β (x)p2l y 2 β(x + p y) ≡ β(x) 1 + (x)pl y + β 2 β

l

(mod p2l+1 ) .

√ 2 It easy to verify that the map x → 1 + pl Dx + p2l D 2 x is an isomorphism of l+1 Z/p Z with Cp (2l + 1, l). Consequently, for every character χ of C(pk ), there is tχ ∈ Z/pl+1 Z such that √ D χ 1 + pl Dx + p2l x2 = epl+1 (tχ x) . 2 By differentiating β(x) =

√ √Dx+1 Dx−1

(twice), we get that

√ β β 2(y − xpl y 2 ) (x)y + pl (x)y 2 = D − β 2β Dx2 − 1 2 2(y − xpl y 2 ) lD +p 2 Dx2 − 1

(mod pl+1 ) ,

implying that the inner sum is of the form 2tχ l 2tχ x 2 y + p y epl+1 νy − . Dx2 − 1 Dx2 − 1 l+1 y∈Z/p

Z

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2t +ν

χ This sum vanishes unless x ∈ Sq( νD , pl ). To see this make a change of summation variable y → y + py0 to get that 2 l 2tχ x 2 y+p y epl+1 νy − tχ Dx2 − 1 Dx2 − 1 y∈Z/pl+1 Z 2tχ 2tχ l 2tχ x 2 = epl y + p y e ν− νy − . y l+1 0 p Dx2 − 1 Dx2 − 1 Dx2 − 1 l+1

Z

y∈Z/p

2t

2t

Now unless ν − Dx2χ−1 ≡ 0 (mod pl ) we can find y0 so that epl ((ν − Dx2χ−1 )y0 ) = 1, 2tχ +ν , pl ) the inner sum given by pl implying that the sum must vanish. For x ∈ Sq( νD times the Gauss sum ep f (x)y 2 + g(x)y . G(x) = y∈Z/pZ

In particular this computation implies that for most characters the exponential sum has square root cancellation. Corollary 4.2. For any character χ with 2tχ ≡ −ν (mod p) there is θ = θ(χ, ν) ∈ [0, π) such that Epk (ν, χ) = pk/2 cos(θ(ν, χ)). 2t +ν

χ Proof. The condition 2tχ ≡ −ν (mod p) implies that νD = 0 (mod p). Hence for any 1 ≤ l ≤ k 2tχ +ν 2tχ + ν l 2 νD = (mod p) #Sq ,p = 0 otherwise νD

Now for k = 2l even, recall that Epk (ν, χ) = pl

x∈Sq(

2tχ +ν νD

epk (νx)χ β(x) . ,pl )

2t +ν

χ If νD = (mod p) this sum vanishes. Otherwise it is a sum over two elements of absolute value pl = pk/2 hence indeed Epk (ν, χ) = 2pk/2 cos(θ(ν, χ)). For k = 2l + 1 odd we have Epk (ν, χ) = pl epk (νx)χ β(x) G(x) .

x∈Sq(

2tχ +ν νD

,pl )

The condition 2tχ ≡ −ν (mod p) implies that the Gauss sum G(x) is not a trivial √ 2tχ +ν sum and hence of order p. As before, the sum Epk (ν, χ) either vanishes (if νD = 1 (mod p)) or it is a sum of two elements of absolute value pl+ 2 = pk/2 .

On the other hand, if 2tχ ≡ −ν (mod p2l ) for some l ≤ k2 then the sum contains pl elements and could be much larger. Moreover, in the odd case, if 2tχ ≡ 2tχ +ν −ν (mod pl+1 ) then g(x) ≡ 0 (mod p). Also, in this case any x ∈ Sq( νD , pl ) 2t x satisfies x ≡ 0 (mod p) and hence also f (x) = Dx2χ−1 ≡ 0 (mod p). So that in

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this case the Gauss sum |G(x)| = p rather then particular we get


√

p and the sum is even bigger. In

ˆ 3 ) with tχ ≡ −ν (mod p2 ), Corollary 4.3. For ν ∈ Z and χ ∈ C(p |Ep3 (ν, χ)| = p2 . 4.2. Equidistribution of exponential sums We now show that as p → ∞ the normalized exponential sums p−k/2 Epk (ν, χ) become equidistributed with respect to the measure π 1 1 π + f (θ)dθ . (4.1) μ(f ) = f 2 2 2π 0 2t +ν

χ For fixed ν and a character χ, if νD is not a square modulo p then the sum Epk (ν, χ) = 0 (or equivalently θ(ν, χ) = π2 ). The following lemma shows that this happens for roughly half the characters, and that this behavior is independent for different values of ν.

Lemma 4.4. Fix a finite set of nonzero distinct integers ν¯ = {ν1 , . . . , νr }. Then, 1 1 t − νj 1 # t ∈ Z/pZ|∀j, ≡ (mod p) = r + O √ . p Dνj 2 p Proof. We can write r t − νj t − νj ≡ (mod p) = χ2 +1 , 2r # t ∈ Z/pZ|∀j, Dνj Dνj t j=1 with χ2 the quadratic character modulo p. Now expand the right hand side ⎛ ⎞ r t − νj t − νj ⎠. χ2 ⎝ χ2 +1 = Dν Dν j j t j=1 t J⊆{1,...,r}

j∈J

Where the sumis over all subsets J ⊆ {1, . . . , r}. The contribution of the empty set is exactly t 1 = p, while for nonempty J we get an exponential sum of t−ν the form t χ2 ( j∈J Dνjj ). Since we assumed all νj are distinct, the polynomial t−ν g(t) = j∈J Dνjj is not a square and we can apply the Weil bounds t χ2 (g(t)) = √ O( p) [18]. Consequently, we have that indeed t − νj √ ≡ (mod p) = p + O( p) . 2r # t ∈ Z/pZ|∀j, Dνj Next we need to show that for the rest of the characters (when the exponential sum does not vanish) the angles θ(χ, ν) become equidistributed (independently) in [0, π]. We will do that by computing all mixed moments. However, we recall that there are exceptional characters for which the normalized exponential sums are not bounded causing the moments to blow up. For that reason we first restrict ourself to a set of “good” characters (of limiting density one) for which the sums are bounded and only then we calculate the moments.

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Fix a finite set of r nonzero distinct integers ν¯ = {ν1 , . . . , νr }, and define the set of “good” characters to be

ˆ k )|∀j, 2tχ ≡ −νj (mod p) , ν ) = χ ∈ C(p Spk (¯ where tχ is determined by χ as above. Then for any character χ ∈ Spk (¯ ν ), we can write Epk (νj , χ) = pk/2 cos(θ(νj , χ)) with θ(νj , χ) ∈ [0, π). Furthermore, for any νj there are precisely pk−2 (p ± 1) characters with 2tχ ≡ νj (mod p) (this ˆ k ) to Cˆp (k, 1)). Hence, is the size of the kernel of the restriction map from C(p |Spk (¯ ν )| ˆ k ). = 1 + O( 1 ) and the set Spk (¯ ν ) is of (limiting) density one inside C(p k p

p

Before we proceed to calculate the moments we will need to set some notations. For any k define the set

r Y (pk , ν¯) = x ¯ ∈ X(pk ) |ν1 (Dx21 − 1) = νj (Dx2j − 1), ∀2 ≤ j ≤ r For every fixed set of integers n ¯ = {n1 , . . . , nr } let ⎧ ⎨ ¯) = x ¯ ∈ Y (pk )| β(xj )nj ≡ 1 Y0 (pk , ν¯, n ⎩ j

⎫ ⎬ (mod pk ) . ⎭

For notational convenience we will sometimes use the notation Spk , Y (pk ), Y0 (pk ) where the dependence on ν¯ and n ¯ is implicit. We will also denote by Y (pk ) (re k spectively Y0 (p )) the elements of Y (pk ) (respectively Y0 (pk )) with all xj = 0 (mod p). Lemma 4.5. As p → ∞, the number of points in Y (pk ) satisfy #Y (pk ) = pk + O(pk− 2 ) . 1

Proof. For any t ∈ Z/pk Z satisfying ∀j, t = νj (mod p) we have that % t−ν

2r ∀j, Dνjj ≡ (mod p) 2 # x ¯|∀j, νj (Dxj − 1) = t = 0 otherwise On the other hand if t ≡ νj (mod p) for some j, then

# x ¯|∀j, νj (Dx2j − 1) = t ≤ 2r pk−1 (as there are at most two possibilities for xi with i = j and at most 2pk−1 possibilities for xj ). We thus have

# x ¯ ∈ (Z/pk Z)r |νj (Dx2j − 1) = t #Y (pk ) = t∈ (Z/pk Z)∗

t − νj ≡ (mod p) + O(pk−1 ) . = 2r pk−1 # t ∈ (Z/pZ)∗ |∀j, Dνj

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Also note that #Y (pk ) = #Y (pk ) + O(pk−1 ). To conclude the proof we use the estimate t − νj √ ≡ (mod p) = p + O( p) , 2r # t ∈ Z/pZ|∀j, Dνj

from Lemma 4.4. Lemma 4.6. As p → ∞, the number of points in Y0 (pk ) satisfy #Y0 (pk ) = O(pk−1 ) .

Proof. To prove this bound we will show that there is a nonzero polynomial F (t) ν (Dx2 −1) with integer coefficients such that for any x ¯ ∈ Y0 (pk ), with b = 1 2 1 we have F (b−1 ) ≡ 0 (mod pk ) (recall that for x1 ∈ X(pk ) we have Dx21 = 1 (mod p) and hence b = 0 (mod p) is invertible). This would imply that b can take at most deg F values modulo p, implying that #Y0 (pk ) ≤ 2r deg(F )pk−1 . Now to define F , consider the formal polynomial in the variables β1±1 , . . . βr±1 given by ⎛ ⎞ r σ n ⎝ βj j j − 1⎠ . G(β1 , . . . βr ) = σ∈{±1}r

j=1

Recall that if a polynomial in two variables x, y is symmetric under permutation then it can be written as a polynomial in the symmetric polynomials σ1 = x + y, σ2 = xy (see e.g., [3, Chapter 6]). The polynomial G is symmetric under any substitution βj → βj−1 and hence there is another polynomial F˜ in r variables with integer coefficients, satisfying G(β1 , . . . , βr ) = F˜ (β1 + β1−1 , . . . , βr + βr−1 ) . Define the polynomial F (t) = F˜ (2 + ν1 t, . . . , 2 + νr t). For any x1 , . . . , xr 2b−ν with x2j = νj Dj (mod pk ) we have β(xj ) + β(xj )−1 = 2 + νj b−1 (mod pk ) (recall β(x) =

√ √Dx+1 ). Dx−1

Hence,

G β(x1 ), . . . , β(xr ) = F˜ (2 + ν1 b−1 , . . . , 2 + νr b−1 ) = F (b−1 ) . Now, if in addition β(x1 )n1 · · · β(xr )nr ≡ 1 (mod pk O) then indeed F (b−1 ) = G(β(x1 ), . . . , β(xr )) ≡ 0 (mod pk ). It remains to show that F (t) is not the zero polynomial. To do this, we think of it as a complex valued polynomial, and note that for it to be identically zero there has to be some choice of signs σ ∈ {±1}r so that the function σj nj & r 2t + νj β Gσ (t) = νj D j=1

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satisfies Gσ (t) ≡ 1. Assume that there is such a choice σ, so the derivative Gσ (t) must also vanish. But we have ' r νj , G (t) = −Gσ (t) σj nj 2 (2t + ν ) t j j=1 so as t → − ν21 the term

ν1 t2 (2t+ν1 )

blows up while the rest of the terms remain

bounded (recall that all νj are different). In particular Gσ (t) is not identically zero. Remark 4.7. The bound #Y0 (pk ) = O(pk−1 ) is probably not optimal. Notice that if the polynomial F (t) defined above is separable (i.e., if it has no multiple roots) then there are at most deg F solutions to F (t) ≡ 0 (mod pk ) and the corresponding bound would be #Y0 (pk ) = O(1). We now preform the moment calculation establishing the limiting distribution of the exponential sums (when running over characters in Spk ). Proposition 4.8. Let μ be as in (4.1) and let g ∈ C([−1, 1]r ) be any continuous function then 1 g cos θ(ν , χ) , . . . , cos θ(ν , χ) 1 r p→∞ pk χ∈Spk g cos(θ1 ), . . . , cos(θr ) dμ(θ1 ) · · · dμ(θr ) . = lim

[0,π]d

Proof. We will give the proof for k = 2l even, the odd case is analogous. Since we can always approximate the function g by polynomials, it is sufficient to show this holds for all monomials of the form g(x) = (2x1 )m1 . . . (2xr )mr . We thus need to show that mj mj 1 lim k 2 cos θ(νj , χ) 2 cos(θ) = dμ(θ) . p→∞ p [0,π] j j χ∈Spk

With out loss of generality we can also assume that all the mj are nonzero (since μ is a probability measure, if mj = 0 then the corresponding factor is 1 and we can consider the same problem for r − 1 instead of r). In this case the right hand side is given by π π mj mj dθ 1 2 cos(θ) 2 cos(θ) dμ(θ) = . 2 0 π 0 j j m

The integral in each factor is 12 ( njj ) for mj = 2nj even and it is zero otherwise.

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Now fix a character χ ∈ Spk and let tχ ∈ Z/pl Z as above. If Sq( then 2 cos θ(νj , χ) = p

−k/2

2tχ +νj l νj D , p )

=∅

Epk (νj , χ) = 0. Otherwise, ν x j j χ β(xj ) , 2 cos θ(νj , χ) = p−k/2 Epk (νj , χ) = 2 epk 2 2t +ν

with xj ∈ Sq( νχj D j , pl ) (recall that for χ ∈ Spk we know 2tχ + ν = 0 (mod p)). Hence, the only contributions to the sum mj 2 cos θ(νj , χ) , χ∈Spk

j 2t +ν

comes from characters χ such that for all j there is xj ∈ Sq( χνD j , pl ) (equivalently, there is xj ∈ (Z/pl Z)∗ satisfying νj (Dx2j − 1) ≡ 2tχ (mod pl )). Also note that if we multiply χ by any character that is trivial on Cp (k, l) this does not change tχ . Let Cˆ (l) (pk ) be the group of characters that are trivial on Cp (k, l), ˆ k ) be a representative of C(p ˆ k )/Cˆ (l) (pk ) with and for any b ∈ Z/pl Z let χb ∈ C(p tχb = b. We thus have that mj mj 1 1 2 cos θ(νj , χ) 2 cos θ(νj , χχb ) = r k k p 2 p l j (l) k j ˆ x ¯∈Y (p ) χ∈C

χ∈Spk

(p )

ν1 (Dx21 −1)

where b = b(¯ x) = . 2 Now use the formula, m m m 2 cos(θ) = cos (m − 2n)θ . n n=0

The main contribution comes from the terms where in each factor mj − 2nj = 0. This vanishes unless all mj are even in which case it is given by m j 1 mj 1 1 = +O √ , nj 2r pk 2 nj p l ˆ (l) (pk ) j x ¯∈Y (p ) χ∈C

j

where we used Lemma 4.5 to get that #Y (pl ) · #Cˆ (l) (pk ) = pk + O(pk− 2 ). It thus remains to bound the rest of the terms, which is reduced to the vanishing (in the limit p → ∞) of the sums 1 cos nj θ(νj , χχb ) , k p l (l) k j 1

ˆ x ¯∈Y (p ) χ∈C

(p )

for any nonzero integers {n1 , . . . , nr }. For any x ¯ ∈ Y (pl ) we have that n νx j j cos nj θ(νj , χχb ) = 2 epk χχb β(xj )nj 2

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ν (Dx2 −1) with b = 1 2 1 . When expanding the product j cos(nj θ(νj , χχb )) we get a sum over 2r terms, each of the form ⎞ ⎛ ⎛ ⎞ r r ±n ν x j j j⎠ epk ⎝ χχb ⎝ β(xj )±nj ⎠ . 2 j=1 j=1 We thus need to bound the exponential sum coming from each term. We will now bound the corresponding sum ⎞ ⎛ ⎛ ⎞ r r 1 nj νj xj ⎠ χχb ⎝ epk ⎝ β(xj )nj ⎠ , pk 2 l (l) k j=1 j=1 ˆ x ¯∈Y (p ) χ∈C

(p )

(the same bound obviously holds when changing any nj to −nj ). Rewrite this sum as ⎞ ⎛ ⎛ ⎞ ⎞ ⎛ r r r n ν x 1 j j j ⎠ χb ⎝ epk ⎝ β(xj )nj ⎠ χ⎝ β(xj )nj ⎠ , pk 2 l j=1 j=1 (l) k j=1 x ¯∈Y (p )

r

ˆ χ∈C

(p )

and note that the inner sum vanishes unless j=1 β(xj )nj ≡ 1 (mod pl ) in which case it is equal #Cˆ (l) (pk ) = pk−l . We can thus rewrite this sum as ⎞ ⎛ ⎛ ⎞ r r 1 n ν x j j j ⎠ χb ⎝ epk ⎝ β(xj )nj ⎠ , pl 2 l j=1 j=1 x ¯∈Y0 (p )

which is trivially bounded by p−l #Y0 (pl ) = O( p1 ) (Lemma 4.6).

Remark 4.9. The above proof also gives the rate at which the fluctuations of the normalized exponential sums approach their limiting distribution. If one takes the test function g in Proposition 4.8 to be smooth then the rate of convergence is O( √1p ). This rate comes from the bound on the error term in Lemma 4.5 which seems to be a sharp bound.

5. Back to matrix elements We can now deduce Theorems 1.3 and 1.5 from Theorem 1.2 and the analysis of the exponential sums. Proof of Theorem 1.3. Let f (x) = e2πin · x be any elementary observable. Take N = p3 to be a prime cubed. Then by Corollary 4.3 there is a character satisfying 2 |Ep3 ( Q(n) 2 , χ)| = p . Let ψ be a Hecke eigenfunction corresponding to χ, then by Theorem 1.2 we get 1 Q(n) 1 OpN (f )ψ, ψ = 3 Ep ,χ = N −1/3 . #C(p3 ) 2 p±1

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Proof of Theorem 1.5. Let f be a trigonometric polynomial and write f= fˆ(n)e(n · x) , |n|≤R

for some fixed R > 0. Let {ν1 , . . . , νr } = {Q(n)|0 < |n| ≤ R}, and consider the random variable r Yf = 2 f # (νj ) cos(θj ) , j=1

with θj chosen independently from [0, π) with respect to μ. We need to show that (pk )

as p → ∞ the limiting distribution of Fj is that of Yf . For any character χ of C(pk ) consider the weighted sum of the corresponding exponential sums r ν k j , χχ0 . Fχ(p ) = f # (νj )p−k/2 Epk 2 j=1 (pk )

By Proposition 4.8, as p → ∞ the limiting distribution of Fχ as χ runs through Spk (hence, also as χ runs through the whole group of characters) is that of Yf . Now, for p sufficiently large (i.e., p > max{νj }) and χj ∈ Cˆ0 (pk ), we have r ν pk/2 j (pk ) . E , χ = f # (νj ) χ Fj k j 0 #C(pk ) p 2 j=1 ν

If we further assume that χj χ0 ∈ Spk (¯ ν ) then |Epk ( 2j , χχ0 )| ≤ 2pk/2 , and hence k 1 (pk ) Fj = Fχ(pj ) + O . p The set of characters {χj ∈ Cˆ0 (pk )|χj χ0 ∈ Spk } is again of density one, hence, the (pk )

limiting distribution of Fj

(pk )

is the same as of Fχj

concluding the proof.

References [1] B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1998, A Wiley-Interscience Publication. [2] F. Bonechi and S. De Bièvre, Controlling strong scarring for quantized ergodic toral automorphisms, Duke Math. J. 117 (2003), no. 3, 571–587. [3] N. Bourbaki, Algebra. II. Chapters 4–7, Elements of Mathematics (Berlin), Translated from the French by P. M. Cohn and J. Howie, Springer Verlag, Berlin, 1990. [4] B. Eckhardt, S. Fishman, J. Keating, O. Agam, J. Main, and K. M¨ uller, Approach to ergodicity in quantum wave functions, Phys. Rev. E 52 (1995), no. 6, 5893–5903. [5] D. Esposti, Mirko and S. Graffi, and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys. 167 (1995), no. 3, 471–507.

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[6] F. Faure, S. Nonnenmacher, and S. De Bièvre, Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys. 239 (2003), no. 3, 449–492. [7] M. Feingold and A. Peres, Distribution of matrix elements of chaotic systems, Phys. Rev. A (3) 34 (1986), no. 1, 591–595. [8] S. Gurevich and R. Hadani, Heisenberg realizations, eigenfunctions and proof of the Kurlberg–Rudnick supremum conjecture, preprint 2005. [9] S. Gurevich and R. Hadani, Proof of the Kurlberg-Rudnick rate conjecture, C. R. Math. Acad. Sci. Paris 342 (2006), no. 1, 69–72. [10] J. H. Hannay and M. V. Berry, Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating, Phys. D 1 (1980), no. 3, 267–290. [11] H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. [12] D. Kelmer, Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, to appear in Ann. of Math. [13] D. Kelmer, Distribution of twisted kloosterman sums modulo prime powers, preprint arXiv:0801.4162, 2008 [14] P. Kurlberg, Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps, Ann. Henri Poincaré 8 (2007), no. 1, 75–89. [15] P. Kurlberg and Z. Rudnick, Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J. 103 (2000), no. 1, 47–77. [16] P. Kurlberg and Z. Rudnick, On the distribution of matrix elements for the quantum cat map, Ann. of Math. (2) 161 (2005), no. 1, 489–507. [17] R. Olofsson, Large supremum norms and small shannon entropy for hecke eigenfunctions of quantized cat maps, preprint 2008. [18] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. USA 34 (1948), 204–207. Dubi Kelmer School of Mathematics Institute for Advanced Study 1 Einstein Drive Princeton New Jersey 08540 USA e-mail: [email protected] Communicated by Jens Marklof. Submitted: March 3, 2008. Accepted: August 11, 2008.



Mean Field Limit for Bosons and Infinite Dimensional Phase-Space Analysis Zied Ammari and Francis Nier Abstract. This article proposes the construction of Wigner measures in the infinite dimensional bosonic quantum field theory, with applications to the derivation of the mean field dynamics. Once these asymptotic objects are well defined, it is shown how they can be used to make connections between different kinds of results or to prove new ones.

1. Introduction The bosonic quantum field theory relies on two different bases: On one side the quantization of a symplectic space, the approach followed for example by Berezin in [5], Kree-Raczka in [34]; on the other side the gaussian stochastic processes presentation also known as the integral functional point of view followed for example by Glimm-Jaffe in [25] and Simon in [43]. Both approaches have to be handled in order to tackle on the most basic problems in constructive quantum field theory (see [3, 15]). The interaction of constructive quantum field theory with other fields of mathematics like pseudodifferential calculus (see [6] or [35]) or stochastic processes (see [2, 38]) is often instructive. In the recent years the mean field limit of N -body quantum dynamics has been reconsidered by various authors via a BBGKY-hierarchy approach (see [4,16, 17, 19, 20, 45] and [21] for a short presentation) mainly motivated by the study of Bose–Einstein condensates (see [12]). Although this was present in earlier works around the so-called Hepp method (see [32] and [24]), the relationship with the microlocal or semiclassical analysis in infinite dimension has been neglected. Difficulties are known in this direction: 1) The gap between the inductive and projective construction of quantized observable in infinite dimension; 2) the difficulties to built algebras of pseudodifferential operators which contain the usual hamiltonians

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Z. Ammari and F. Nier


and preserve some properties of the finite dimensional calculus like a Calderon– Vaillancourt theorem, a good notion of ellipticity or the asymptotic positivity with a G˚ arding inequality; 3) even when step 2) is possible, no satisfactory Egorov theorem is available. Recall the example of an N -body Schr¨ odinger hamiltonian 1 V (xi − xj ) , on RdN , HN = −Δ + N 1≤i<j≤N

and consider the time-evolved wave function ΨN (t) = e−itHN ψ ⊗N ,

ψ ∈ L2 (Rd ) .

The 1-particle marginal state, the quantum analogous of the one particle empirical distribution in the classical N -body problem, is given by N 1 1 I ⊗ · · · I ⊗ I ⊗

A ⊗I ⊗ · · · ⊗ I ΨN (t) . Tr A (t) = ΨN (t) , N i=1 i

The mean field limit says that in the limit N → ∞, the marginal state evolves according to a non-linear Hartree equation

with

1 (t) = |z(t)z(t)| + o(1) , as N → ∞ ,

2 i∂t z = −Δz + (V ∗ |z| )z on Rt × Rd z(t = 0) = ψ .

By setting N = 1ε and in the Fock space framework with ε-dependent CCR (i.e.: [a(g), a∗ (f )] = ε g, f ), the problem becomes 1 HN = ∇a∗ (x)∇a(x) dx + V (x − y)a∗ (x)a∗ (y)a(x)a(y) dxdy ε Rd R2d 1 ε = H ε ε t e−itHN = e−i ε H , Tr A1 (t) = ΨN (t) , dΓ(A)ΨN (t) = ΨN (t) , pA (z)W ick ΨN (t) , where pA is the polynomial pA (z) = z , Az . Higher order marginals, taking into accounts correlations, can be defined after using the polynomials pA (z) = z ⊗k , Az ⊗k with A ∈ L (L2 (Rkd )) . On this example, the scaling of the hamiltonian, of the time scale and of the observables as Wick operators enters formally in the ε-dependent semiclassical analysis. The Hepp method concerns the evolution of squeezed coherent states [12, 24, 32], which amounts in the finite dimensional case to the phase-space evolution of a gaussian state according to the time dependent quadratic approximation of the non linear hamiltonian, centered on the solution to the classical hamiltonian equation. We refer the reader to [13] for accurate developments of such an approach in the finite dimensional case.

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In the nineties and as a byproduct of the development of microlocal analysis, alternative and more flexible methods were introduced in order to study the semiclassical limit with the help of Wigner (or semiclassical) measures (see [10, 21, 29, 36, 46]). Such objects are defined by duality and rely on the asymptotic positivity of the ε-dependent quantizations. It gives a weak but more flexible form of the principal term of the semiclassical (here mean-field) approximation. Via the introduction of probability measures on the symplectic phase-space, it provides an interesting way to analyze the relationship between the two basic approaches to quantum field theory. Further in finite dimension, the Wick, anti-Wick and Weyl quantizations are asymptotically equivalent in the limit ε → 0. This is not so obvious in infinite dimension. Several attempts have been tried to develop an infinite dimensional Weyl pseudodifferential calculus with an inductive approach. Lascar in [35] introduced an algebra and a notion of ellipticity in this direction, making more effective the general presentation of [34]. The works of Helffer–Sj¨ ostrand in [28,31] and Amour– Kerdelhué–Nourrigat in [1] about the pseudodifferential calculus in large dimension motivated by the analysis of the thermodynamical limit enter in this category. With such an approach, it is not clear that the infinite dimensional phase-space is well explored and that no information is lost in the limit ε → 0. Meanwhile this inductive approach is limited by Hilbert–Schmidt type restriction like in Shale’s theorem about the quasi-equivalence of gaussian measures. It is known after [26] that the nonlinear transformations which preserve the quasi-equivalence with a given gaussian measure within the Schr¨ odinger representation are very restricted and do not cover realistic models. Hence no Egorov theorem can be expected with Weyl observables. Simple remarks suggests alternative point of views. The Wick calculus with polynomial symbols present encouraging specificities: It contains the standard hamiltonians, it makes an algebra under more general assumptions (the Hilbert– Schmidt condition can be relaxed) and allows some propagation results when tested on appropriate states (see [19, 20]). Meanwhile the Wigner measures in the limit ε → 0 can be defined very easily via the separation of variables as weak distribution, in a projective way which fits with the stochastic processes point of view. After reviewing and sometimes simplifying or improving known results and techniques about the mean field limit, our aim is to show the interests of the extension to the infinite dimensional case of Wigner measures: • After the introduction of the small parameter ε → 0 and the definition of Weyl operator W (z), z ∈ Z the phase-space, choosing between the quantization of symplectic space and the stochastic processes point of view is no more a question of general principles nor of mathematical taste. It is a matter of scaling. The symplectic geometry arises when considering macroscopic phasespace translation W ( zε ), while the operator W (z) is used with this scaling in the introduction of Wigner measures via their characteristic function. Corrections to the mean field limit considered for example in [11] with a stochastic

1506



processes point of view can be interpreted within this picture: They attempt to give a better information on the shape of the state in a small phase-space scale. • Once the Wigner measures are well defined as Radon measures, it is possible to make explicit the relationship between different kinds of results and to extend them in a flexible way. It accounts for the propagation of chaos (result obtained via the BBGKY approach) according to the classical hamiltonian dynamics in the phase-space. Actually we shall prove in a very general framework that the propagation of squeezed coherent states as derived via the Hepp method implies a weak version of the mean field limit for product states. Further propagation results can be obtained for some non standard mixed states without reconsidering a rather heavy analysis process. • The comparison between the Wick, Weyl and anti-Wick quantization can be analyzed accurately in the infinite dimensional case. With the Wick calculus, complete asymptotic expansions can be proved after testing with some specific states. The relationship of such results with the propagation of Wigner measures works in a rather general setting but has to be handled with care. • The gap between the projective and inductive approaches can be formulated accurately in the limit ε → 0. We shall explain in the examples the possibility of a dimensional defect of compactness. This work is presented and illustrated with examples simpler than more realistic models considered in other works like [4, 16, 17, 24, 32] with more singular interaction potentials. That was our choice in order to make the correspondence between various approaches more straightforward and to pave the way for further improvements. We hope that this information will be valuable for other colleagues and useful for further developments. The outline of this article is the following. In Section 2, standard notions about the symmetric Fock space are recalled and Wick calculus is specified. In Section 3 the Weyl and Anti-Wick calculus are introduced in a projective way after recalling accurately (most of all the scaling) of finite dimensional semiclassical calculus. The Section 4 recalls the distinction between coherent states and product or Hermite states, and their properties when measured with different kinds of observables. The two methods used to derive the mean field dynamics, the Hepp method and the analysis through truncated Dyson expansions, are reviewed within our formalism and with some variations in Section 5. The Wigner measures are introduced in Section 6 with the extension of some finite dimensional properties and specific infinite dimensional phenomena. Finally examples and applications are detailed in Section 7, in particular: 1) reconsidering a simple presentation of the Bose–Einstein condensation shows an interesting example of what we call the dimensional defect of compactness; 2) a general result says that the propagation of squeezed coherent states, which can be attacked via the Hepp method, implies a slightly weaker form of the propagation of chaos (formulated with product states and Wick observables); 3) the mean field dynamics can be easily derived for some states which present some asymptotically vanishing correlations.

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2. Fock space and Wick quantization After introducing the symmetric Fock space with ε-dependent CCR’s, an algebra of observables resulting from the Wick quantization process is presented. 2.1. Fock space Consider a separable Hilbert space Z endowed with a scalar product ., . which is anti-linear in the left argument and linear in the right one and with the associated norm |z| = z, z. Let σ = Im., . and S = Re., . respectively denote the canonical symplectic and the real scalar product over Z . The symmetric Fock space on Z is the Hilbert space n ∞ Z = Γs (Z ) , H = n

n=0

where Z is the n-fold symmetric tensor product. Almost all the direct sums and tensor products are completed within the Hilbert framework. This is omitted in the notation. On the contrary, a specific alg superscript will be used for the algebraic direct sums or tensor products. n Z onto the closed subspace n For any n ∈ N, the orthogonal projection of Z willbe denoted by Sn . For any (ξ1 , ξ2 , . . . , ξn ) ∈ Z n , the vector ξ1 ∨ ξ2 ∨ n Z will be · · · ∨ ξn ∈ 1 ξ1 ∨ ξ2 ∨ · · · ∨ ξn = Sn (ξ1 ⊗ ξ2 · · · ⊗ ξn ) = ξσ(1) ⊗ ξσ(2) · · · ⊗ ξσ(n) . n! σ∈Σn

n,alg Z and a The family of vectors (ξ1 ∨ · · · ∨ ξn )ξi ∈Z is a generating family of n Z . Thanks to the polarization identity total family of ⎛ ⎞⊗n n 1 ε1 · · · εn ⎝ εj ξj ⎠ , (1) ξ1 ∨ ξ2 ∨ · · · ∨ ξn = n 2 n! ε =±1 j=1 i

the same property holds for the family (z ⊗n )n∈N,z∈Z . ik jk Z → Z , k = 1, 2, the notation A1 A2 For two operators Ak : stands for i +i j1 +j2 1 2 A1 A2 = Sj1 +j2 ◦ (A1 ⊗ A2 ) ◦ Si1 +i2 ∈ L Z, Z . 0 Z = C λ → λz ∈ Z = Any z ∈ Z is identified with the operator |z : 1 Z while z| denotes the linear form Z ξ → z , ξ ∈ C. The creation and annihilation operators a∗ (z) and a(z), parameterized by ε > 0, are then defined by: √ a(z)| n Z = εn z| ⊗ I n−1 Z a∗ (z)| n Z = ε(n + 1) Sn+1 ◦ ( |z ⊗ I n Z ) = ε(n + 1) z I n Z .

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Each of (a(z))z∈Z and (a∗ (z))z∈Z are commuting families of operators and they satisfy the canonical commutation relations (CCR): a(z1 ), a∗ (z2 ) = εz1 , z2 I . (2) We also consider the canonical quantization of the real variables Φ(z) = √12 (a∗ (z)+ a(z)) and Π(z) = Φ(iz) = i√1 2 (a(z) − a∗ (z)). They are self-adjoint operators on H and satisfy the identities: Φ(z1 ), Φ(z2 ) = iεσ(z1 , z2 )I , Φ(z1 ), Π(z2 ) = iεS(z1 , z2 )I . The representation of the Weyl commutation relations in the Fock space W (z1 )W (z2 ) = e− 2 σ(z1 ,z2 ) W (z1 + z2 ) iε

(3)

= e−iεσ(z1 ,z2 ) W (z2 )W (z1 ) , is obtained by setting W (z) = eiΦ(z) . The generating functional associated with this representation is given by 2 ε Ω, W (z)Ω = e− 4 |z| , where Ω is the vacuum vector (1, 0, . . .) ∈ H . The total family of vectors E(z) = √ ∗ 1 W iε2z Ω = e ε [a (z)−a(z)] Ω, z ∈ Z , have the explicit form −

|z|2 2ε

= e−

|z|2 2ε

E(z) = e

∞ 1 a∗ (z)n Ω εn n! n=0 ∞

z ⊗n ε−n/2 √ . n! n=0

(4)

The number operator is also parametrized by ε > 0, N| n Z = εnI| n Z . It is convenient to introduce the subspace Hf in =

alg n

Z

n∈N

of H , which is a set of analytic vectors for N . For any contraction S ∈ L (Z ), |S|L (H ) ≤ 1, Γ(S) is the contraction in H defined by Γ(S)| n Z = S ⊗ S · · · ⊗ S . More generally Γ(B) can be defined by the same formula as an operator on Hf in for any B ∈ L (Z ). Meanwhile, for any self-adjoint operator A : Z ⊃ D(A) → Z ,

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1509

the operator dΓ(A) is the self-adjoint operator given by it

e ε dΓ(A) = Γ(eitA ) n I ⊗ · · · ⊗

A ⊗··· ⊗ I . dΓ(A)| n,alg D(A) = ε k=1

k

For example N = dΓ(I) . 2.2. Wick operators In this subsection we consider the Wick symbolic calculus on (homogeneous) polynomials. We will show some product and commutation formulas useful later for the application. For example time evolved Wick observables can be expressed as ε-asymptotic expansion of quantized Wick symbols. For a detailed exposition on more general Wick polynomials we refer the reader to [15]. A (p, q)-homogeneous polynomial function of z ∈ Z is defined as P (z) = q,alg p,alg ⊗q ⊗p (z , z ), where is a sesquilinear form on ( Z)× ( Z ), with ¯ q λp P (z). Owing to the polarization formula (1) and the identity P (λz) = λ 1 1 (η ⊗q , ξ ⊗p ) = [e2iπθ η + e2iπϕ ξ]⊗q , [e2iπθ η + e2iπϕ ξ]⊗p e2iπ(qθ−pϕ) dθ dϕ 0

0

the correspondence → P is a bijection when the set of forms is restricted to the q,alg p,alg sesquilinear forms on ( Z )×( Z ). Any of the continuity properties of P are thus encoded by the continuity properties of the sesquilinear form with the following hierarchy (from the weakest to the strongest) | (η1 ∨ · · · ∨ ηq , ξ1 ∨ · · · ∨ ξp )| ≤ C |η1 |Z . . . |ηq |Z |ξ1 |Z . . . |ξp |Z , ηi ∈ Z , | (φ, ψ)| ≤ C |φ| q Z |ψ| p Z ,

ψ∈

p

ξj ∈ Z

Z ,

φ∈

q

(5) Z (6)

1≤i,j≤K

ci,j (φi , ψj ) ≤ C

ci,j φi | ⊗ ψj

1≤i,j≤K

K ∈ N,

cij ∈ C ,

, (

φi ∈

q

q

Z

Z ,

)∗ ⊗(

p

Z) p

ψj ∈

Z . (7)

For example, when p = q = 1 the two first ones define L (Z ), while the third one defines the space of Hilbert–Schmidt operators. By Taylor expansion any (p, q)homogeneous polynomial P admits Gˆ ateaux differentials and we set ∂zk ∂zk P (z)[u1 , . . . , uk , v1 , . . . , vk ] = ∂¯u1 · · · ∂¯uk ∂v1 · · · ∂vk P (z)

where ∂¯u , ∂v are the complex directional derivatives relative to u, v ∈ Z .

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Definition 2.1. For p, q ∈ N, the set of (p, q)-homogeneous polynomial functions on Z which satisfy the continuity condition (6) is denoted by Pp,q (Z ): ! ˜b = 1 1 ∂ p ∂ q b(z) ∈ L (p Z , q Z ) , p! q! z z b(z) ∈ Pp,q (Z ) ⇔ b(z) = z ⊗q , ˜bz ⊗p . The subspace of Pp,q (Z ) made of polynomials b such that ˜b is a compact operator ˜b ∈ L ∞ (p Z , q Z ) (resp. b ∈ L r (p Z , q Z )) is denoted by P ∞ (Z ) (resp. p,q r Pp,q (Z )). Remark 2.2. In the case of Z = Cd the symbol is often written b(z, z). Of course our polynomials have an holomorphic and antiholomorphic part but we prefer to keep the notation b(z). The symbol b is simply considered as a function of the point z ∈ Z . The writing b(z, z) would suggest that Z is endowed with a complex conjugation operator, which is not necessary at this level. p Z into It will be sometimes convenient to consider ˜b as an operator from q Z with the obvious convention for symmetric operators ˜b = Sq ˜bSp . Owing p q to the condition ˜b ∈ L ( Z , Z ) for b ∈ Pp,q (Z ), this definition implies that j k any differential ∂z ∂z b(z) at the point z ∈ Z equals # " # " q! p! z ⊗q−j | I j Z ˜b z ⊗p−k ∂zj ∂zk b(z) = I k Z (p − k)! (q − j)! k j ∈L Z , Z . (8) We will mainly work with fixed homogeneity degrees p, q but the key statement of this section (Proposition 2.7) says that ⊕alg p,q∈N Pp,q (Z ) is an algebra of symbols with the same explicit product formula as in the finite dimensional case. With any “symbol” b ∈ Pp,q (Z ), a Wick monomial bW ick can be associated according to: (9) bW ick : Hf in → Hf in , n n+q−p " # p+q n!(n + q − p)! W ick Z, Z , ∈L ε 2 ˜b I n−p Z b| n Z = 1[p,+∞) (n) (n − p)! with ˜b = (p!)−1 (q!)−1 ∂zp ∂zq b(z) . Here are the basic symbol-operator correspondence: √ z, ξ ←→ a∗ (ξ) 2S(ξ, z) ←→ Φ(ξ) z, Az ←→ √ ξ, z ←→ a(ξ) 2σ(ξ, z) ←→ Π(ξ) |z|2 ←→

dΓ(A) N.

Other examples can be derived from the next propositions. The first one is a direct consequence of the definition (9). Proposition 2.3. The following identities hold true on Hf in for every b ∈ Pp,q (Z ): ∗ (i) bW ick = ¯bW ick .

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W ick (ii) C(z)b(z)A(z) = C W ick bW ick AW ick , if A ∈ Pα,0 (Z ), C ∈ P0,β (Z ). W ick i εt dΓ(A) W ick −i εt dΓ(A) b e = b(e−itA z) , if A is a self-adjoint operator (iii) e on Z . Proposition 2.4. (i) The Wick operator associated with b(z) = Z , equals

$p i=1

z, ηi ×

$q j=1

ξj , z, ηi , ξj ∈

bW ick = a∗ (η1 ) · · · a∗ (ηp )a(ξ1 ) · · · a(ξq ) . (ii) For b ∈ Pp,q (Z ) and z ∈ Z the equality % p+q k!j! k−p+j−q + ε 2 |z| b(z) z ⊗j , bW ick z ⊗k = δk−p,j−q (k − p)!(j − q)!

(10)

+ denotes δα,β 1[0,+∞) (α) where δα,β is holds for any k, j ∈ N. The symbol δα,β the standard Kronecker symbol.

Proof. (i) is a direct consequence of Proposition 2.3 with (z , ξ)W ick = a∗ (ξ) and (ξ, z)W ick = a(ξ) . (ii) This comes directly from the definition (9) of bW ick . The next result specifies the boundedness properties of bW ick . Lemma 2.5. For b ∈ Pp,q (Z ), the estimate bW ick

L(

k

Z,

j

q

Z)

p + ≤ δk−p,j−q (jε) 2 (kε) 2 ˜b

L(

p

Z,

q

Z)

,

1 p q ∂ ∂ b, with ˜b = p!q! z z¯

(11)

.

(12)

holds for any k, j ∈ N. This implies − q2

N

bW ick N

−p 2 L (H )

≤ ˜b

L(

p

Z,

q

Z)

k j Proof. A consequence of (10) is bW ick ( Z ) ⊂ Z with j = k − p + q. For k Z and j = k − p + q, write ψ∈ bW ick ψ j Z √ j!k! p+q ε 2 Sj (b ⊗ I k−p Z )ψ j = (k − p)! Z % % q p j! k! b ⊗ I k−p Z ≤ (jε) 2 (kε) 2 q (j − q)!j (k − p)!k p

L(

k

Z,

j

Z)

|ψ| k Z .

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An important property of our class of Wick polynomials is that a composition ick ick ◦ bW with b1 , b2 ∈ ⊕alg of bW p,q Pp,q (Z ) is a Wick polynomial with symbol in 1 2 alg ⊕p,q Pp,q (Z ). In the following we prove this result and specifies the Wick symbol of the product. For b ∈ Pp,q (Z ), specific cases with j = 0 or k = 0 of (8) imply k ∗ j k ∂z b(z) ∈ and ∂zj b(z) ∈ Z Z , for any fixed z ∈ Z . For two symbols bi ∈ Ppi ,qi (Z ), i = 1, 2, and any k ∈ N, the new symbol ∂zk b1 .∂z¯k b2 is now defined by ∂zk b1 . ∂z¯k b2 (z) = ∂zk b1 (z), ∂z¯k b2 (z) ( k Z )∗ , k Z . (13) We also use the following notation for multiple Poisson brackets: {b1 , b2 }(k) = ∂zk b1 .∂z¯k b2 − ∂zk b2 .∂z¯k b1 , {b1 , b2 } = {b1 , b2 }(1) . These operations with polynomialsare easier q to handle than there corresponding p versions for the operators ˜bi ∈ L ( i Z , i Z ). Nevertheless their explicit operator expressions as contracted products allow to check that ⊕alg p,q Pp,q (Z ) is stable w.r.t these operations . Lemma 2.6. Fix p1 , p2 , q1 and q2 in N. For two polynomials bi ∈ Ppi ,qi (Z ), i = 1, 2, set ˜bi = (pi !qi !)−1 ∂zpi ∂z¯qi bi and for any k ∈ {0, . . . , min{p1 , q2 }} k 1 ˜b1 ˜b2 = ∂zp1 +p2 −k ∂z¯q1 +q2 −k ∂zk b1 .∂z¯k b2 . (p1 + p2 − k)!(q1 + q2 − k)! Then k

˜b1 ˜b2 =

q2 ! p1 ! Sq1 +q2 −k (˜b1 ⊗ I q2 −k Z )(I p1 −k ⊗ ˜b2 ) (p1 − k)! (q2 − k)! p +p −k q1 +q 1 2 2 −k Z, Z , ∈L

(14)

with the estimate k

˜b1 ˜b2

L(

p

1 +p2 −k

Z,

q

1 +q2 −k

Z)

q2 ! p1 ! ˜b1 ˜b2 ≤ . (p1 − k)! (q2 − k)! L ( p1 Z , q1 Z ) L ( p2 Z , q2 Z ) q p1 Z and φ ∈ 2 Z , introduce the vector Proof. For ψ ∈ k " # (q2 − k)! k Z ∂z bφ (z) ∈ z ⊗q2 −k , φ = z ⊗q2 −k | ⊗ I k Z φ = q2 !

(15)

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with bφ (z) = z q2 , φ and the form ψ, z

⊗p1 −k

(p1 − k)! k ∂z bψ (z) ∈ := p1 !

k

∗ Z

,

with

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bψ (z) = ψ , z ⊗p1 .

The identity ψ, z ⊗p1 −k , z ⊗q2 −k , φ ( k Z )∗ , k Z = ψ ⊗ z ⊗q2 −k , z ⊗p1 −k ⊗ φ p1 +q2 −k Z ⊗p1

⊗q2

(16)

⊗n

and φ = η with ξ, η ∈ Z . Since (ξ )ξ∈Z is a is obviously true nwhen ψ = ξ Z with the polarization identity (1), the identity (16) holds for total space of p1 q Z . After noticing the relations all φ ∈ 2 Z and all ψ ∈ p1 ! q2 ! ψ, z ⊗p1 −k , ∂zk b2 (z) = z ⊗q2 −k , φ , ∂zk b1 (z) = (p1 − k)! (q2 − k)! with ψ = ˜b∗ z ⊗q1 and φ = ˜b2 z ⊗p2 , the identity (16) leads to 1

∂zk b1 .∂z¯k b2 (z) =

' & p1 ! q2 ! z ⊗q1 +q2 −k , (˜b1 ⊗ I q2 −k Z ) (I p1 −k Z ⊗ ˜b2 )z ⊗p2 +p1 −k . (p1 − k)! (q2 − k)!

Therefore ∂zk b1 .∂z¯k b2 is a continuous homogeneous polynomial in Pp1 +p2 −k,q1 +q2 −k (Z ) with the associated operator given by (14). The estimate (15) follows immediately by (14). Proposition 2.7. The formulas (i) ⎞W ick ⎛ min{p1 ,q2 } k #W ick " ε ick W ick ∂zk b1 .∂z¯k b2 ⎠ bW b2 =⎝ = eε∂z ,∂ω¯ b1 (z)b2 (ω) |z=ω , 1 k! k=0

(17) (ii)

⎛ ick W ick , b2 ]=⎝ [bW 1

max{min{p1 ,q2 } , min{p2 ,q1 }}

k=1

⎞W ick k

ε {b1 , b2 }(k) ⎠ k!

,

(18)

hold for any bi ∈ Ppi ,qi (Z ), i = 1, 2 as identities on Hf in . Remark 2.8. This result has exactly the form of the finite dimensional formula. Lemma 2.6 gives the relation with the writing which can be found in [19]. Proof. The second statement (ii) is a straightforward consequence of the first one (i). Let us focus on (i) which will be proved in several steps. Step 0: Before proving the identity, first notice that both sides are well defined. Actually, for any b ∈ Pp,q (Z ), the operator bW ick sends Hf in into itself. Hence,

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ick ick the product bW ◦ bW is well defined as an operator Hf in → Hf in . Finally we 1 2 know from Lemma 2.6 that eε∂z ,∂ω b1 (z)b2 (ω) z=ω belongs to ⊕alg p,q Pp,q (Z ). q

Step 1: Consider b1 (z) = η , z and b2 (z) = z , ξ , q ∈ N. The formula a(η)a∗ (ξ)q = a∗ (ξ)q a(η) + εqη , ξa∗ (ξ)q−1 is exactly ick W ick b2 = (b1 b2 )W ick + ε(∂z b1 .∂z¯b2 )W ick . bW 1 p

q

Step 2: Consider b1 (z) = βp (z) = η , z and b2 (z) = z , ξ , p, q ∈ N. The induction is already initialized for p = 1 according to Step 1. Assume that the formula is true for p − 1 and all q ∈ N and compute ⎡ ⎤ min{p−1,q} k ε W ick ick W ick W ick ⎦ ∂ k βp−1 , ∂z¯k b2 = β1W ick ⎣ = β1W ick βp−1 b2 βpW ick bW 2 k! z k=0 ⎡ ⎤ min{p−1,q} k (p − 1)! ε q! k = a(η) ⎣ η , ξ a∗ (ξ)q−k a(η)p−1−k ⎦ k! (q − k)! (p − 1 − k)! k=0

min{p−1,q}

=

∗ q−k εk q!(p − 1)! k η , ξ a (ξ) a(η)p−k k! (q − k)!(p − 1 − k)! k=0 , +ε(q − k)η , ξa∗ (ξ)q−k a(η)p−(k+1)

min{p,q}

=

k=0

k εk η, ξ q!(p − 1)! k 1[1,p] (k) 1[0,p−1] (k) + k!(q − k)!(p − 1 − k)! (p − k)

× a∗ (ξ)q−k a(η)p−k

min{p,q}

=

k=0

W ick εk k ∂z βp , ∂z¯k b2 . k!

We used several times the relation ∂zj βn (z) =

n! n−j η , z η |⊗j (n − j)!

and its dual version for ∂z¯j b2 . Step 3: From Step 2, the statement (ii) of Proposition 2.3 leads to a∗ (ξ 1 )q1 a(η 1 )p1 a∗ (ξ 2 )q2 a(η 2 )p2 min{p1 ,q2 }

=

k=0

#W ick εk " k q p q p ∂z z, ξ 1 1 η 1 , z 1 .∂z¯k z, ξ 2 2 η 2 , z 2 k!

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for any ξ 1 , ξ 2 , η 1 , η 2 ∈ Z and any p1 , q1 , p2 , q2 ∈ N . Again the polarization formula (1) in the form ⎡ ⎛ ⎞⎤n n n 1 a (ξi ) = n ε1 · · · εn ⎣a ⎝ εj ξj ⎠⎦ , 2 n! ε =±1 i=1 j=1 i

yields the result for any b (z) =

p -

z , ξi

i=1

q -

ηj , z ,

= 1, 2 ,

j=1

that is for any ˜b in the form ˜b = |ξ ∨ · · · ∨ ξ η ∨ · · · ∨ η | , 1 p 1 q

= 1, 2 .

(19)

Step 4: We want to check the identity ick ick ψn , bW ◦ bW ψn = 1 2

min{p1 ,q2 }

p=0

εp ψ n , (∂zp b1 ∂zp b2 )W ick ψn p!

n n for any ψn ∈ Z and any ψn ∈ Z , n, n ∈ N. W ick From the definition of b , the left-hand side equals ick ick ψn , bW ◦ bW ψn 1 2 & #" # ' " ˜b2 I n−p = Cn,n ,p1,2 ,q1,2 ,ε ψn , ˜b1 I n+q2 −p2 −p1 Z ψn 1 Z &" # # ' " I n −q1 Z ψn , ˜b2 I n−p1 Z ψn . = Cn,n ,p1,2 ,q1,2 ,ε ˜b∗1 Similarly and owing to Lemma 2.6, every term of the right-hand side satisfies ψ n , (∂zp b1 ∂zp b2 )W ick ψn & #" # , ' ."

˜b2 ˜b1 ⊗I q2−p p −p n−p −p +p , = Cn,n ψ I ψn ⊗ I ,p,p 1 1 2 n ,q ,ε Z Z Z 1,2 1,2 &" # # ' "

˜b∗ ⊗I n−p1 ψ n , I p1−p Z ⊗ ˜b2 ⊗ I n−p1−p2 +p Z ψn . = Cn,n ,p,p 1 1,2 ,q1,2 ,ε Z Hence for fixed ψn , ψn ∈ Hf in , both side are sesquilinear continuous expression of (˜b1 , b˜2 ) when the first factor is considered with the ∗-strong topology of operators and the second one with the strong topology. The operators (19) for which the equality is true, form a total family for these topologies. This can be proved in two steps: approximate first any finite rank operators by linear combinations of the specific rank one operators (19) and then any bounded operators by finite rank operators. Thus the equality holds for any b ∈ Pp ,q (Z ), = 1, 2 .

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Remark 2.9. The formulas (17) and (18) make sense with ε-dependent symbols. One can work with polynomials in ε b(z, ε) =

n

εα bα (z) ,

bα ∈ Pp,q (Z )

α=0

or with asymptotic sums b(z, ε) ∼

∞

εα bα (z)

bα ∈ Pp,q (Z ) .

α=0

The expression (17) and (18) take then the form ick W ick bW b2 1

∼

∞

⎛ εj ⎝

j=0

⎛

∞ W ick W ick b1 ∼ , b2 εj ⎝ j=1

α+β+k=j

α+β+k=j

⎞W ick 1 k ∂ b1,α .∂z¯k b2,β ⎠ k! z ⎞W ick 1 k ∂ b1,α .∂z¯k b2,β − ∂zk b2,β .∂z¯k b1,α ⎠ , k! z

/

/ for b1 ∼ α εα b1,α ∈ Pp1 ,q1 (Z ) and b2 ∼ β εβ b2,β ∈ Pp2 ,q2 (Z ) . Here (p1 , q1 ) (resp. (p2 , q2 )) does not depend on α (resp. β). We have the following useful result. Proposition 2.10. For any b ∈ ⊕alg p,q∈N Pp,q (Z ) we have: (i) bW ick is closable and the domain of its closure contains 0 1 H0 = vect W (z)φ, φ ∈ Hf in , z ∈ Z . √

(ii) By setting E(z) = W (

2z iε )Ω

according to (4), the identity b(z) = E(z) , bW ick E(z)

(20)

holds for every z ∈ Z . (iii) For any z0 ∈ Z the identity √ √ ∗ W ick 2 2 W ick z0 b z0 = b(z + z0 ) W W iε iε holds on H0 where b( · + z0 ) ∈ ⊕alg p,q∈N Pp,q (Z ) . Proof. (i) bW ick is closable by Proposition 2.3 (i). It is enough to consider b ∈ Pp,q (Z ) when we prove that H0 is a core for the closure of bW ick . The last

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statement is deduced from the estimate ∞ 1 bW ick Φ(z)n ϕ(k) ≤ |˜b|L ( p Z , q Z ) |ϕ(k) | k Z n! H n=0 2 ∞ √ p+q ( 2ε)n (n + k)! ε(n + k + q) 2 |z|n × n! k! n=0 0 such that the estimate δ W (z1 ) − W (z2 ) (N + 1)−δ/2 ≤ Cδ |z1 − z2 | min(ε|z1 |, ε|z2 |)δ + max(1, ε)δ , holds for all ε > 0, and all z1 , z2 ∈ Z . Proof. We have by Weyl’s relation W (z1 ) − W (z2 ) (N + 1)−δ/2 ≤ W (z1 − z2 ) − I (N + 1)−δ/2 + eiεσ(z1 ,z2 ) − 1 .

(24)

The estimate |eis − 1| ≤ Cδ |s|δ , leads to eiεσ(z1 ,z2 ) − 1 = eiεσ(z1 −z2 ,z2 ) − 1 = eiεσ(z1 ,z2 −z1 ) − 1 ≤ Cδ εδ |z1 − z2 | min(|z1 |, |z2 |)δ . δ

The first part of the r.h.s. in (24) is estimated via a complex interpolation argument. Indeed, for δ = 0 notice that |W (z1 − z2 ) − I| ≤ 2 and for δ = 1 the

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estimate eis − 1 ≤ C1 |s| combined with the spectral theorem yields W (z1 − z2 ) − I (N + 1)−1/2 ψ ≤ C1 |Φ(z1 − z2 )|(N + 1)−1/2 ψ ≤ C1 Φ(z1 − z2 )(N + 1)−1/2 ψ . Now by the number estimate (12) we obtain W (z1 − z2 ) − I (N + 1)−1/2 ≤ C max(1, ε) |z1 − z2 | .

3.2. Finite dimensional Weyl quantization The finite dimensional Weyl calculus provides us a collection of results on the Weyl quantization. We specify here the relation between the Weyl quantization defined on Z via (23) and the usual semiclassical Weyl quantization within the Schr¨ odinger representation on Rd . For p ∈ P the orthogonal projector I −p is denoted by p⊥ . Let Γs (pZ ) denotes the symmetric Fock space over pZ . The separation of variables in finite dimensions extends to general symmetric Fock spaces owing to the canonical isomorphism of Fock spaces Tp : H = Γs (Z ) → Γs (pZ ) ⊗ Γs (p⊥ Z ) ,

(25) ⊥

for any finite dimensional projector p ∈ P, with Tp Ω = ΩpZ ⊗ Ωp Z when ΩpZ ⊥ and Ωp Z are the vacuum vectors of the corresponding Fock spaces. We will omit the notation Tp and identify directly the tensor products. Fix p ∈ P. The tensor decomposition of the Weyl quantization comes from the Weyl relation which implies W (ξ + ξ ) = W (ξ)W (ξ ) = Wp (ξ) ⊗ Wp⊥ (ξ ) for any (ξ, ξ ) ∈ pZ × p⊥ Z . The symbols Wp stands for the Weyl operator defined on the Fock space Γs (pZ ) and the Weyl quantization of b ∈ S (F ), for any finite eyl dimensional complex subspace F of Z , is denoted by bW . Hence the Weyl F quantization of b ∈ Scyl (Z ) based on pZ equals √ eyl bW eyl = F [b](z)W ( 2πz) Lp (dz) = bW ⊗ IΓs (p⊥ Z ) . pZ pZ

In order to apply directly the finite dimensional results on Weyl quantization, we need to specify the correspondence of representations. On Rd the Weyl quantization is often introduced as 6 5 (x−y).ξ x+y dξdy , ξ u(y) bW eyl (x, hDx )u(x) = ei h b . d 2 (2πh) d R √ √ By a simple conjugation with a dilatation, it becomes aW eyl ( hx, hDx ) where the position (x) and frequency (ξ) variables play the same role. An equivalent definition can be given with the help of the phase translations : W eyl , [τx0 ,ξ0 u](x) = eiξ0 (2x−x0 )/2 u(x−x0 ) . τ(x0 ,ξ0 ) = ei(ξ0 x−x0 Dx ) = ei(ξ0 x−x0 ξ)

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It reads


√ √ √ √ F [b](y, η)e2iπ(y.( hx)+η. hDx ) dydη bW eyl ( hx, hDx ) = ∗ d T R F [b](y, η)τ(−2π√hη,2π√hy) dydη . = T ∗ Rd

The symplectic form [[ , ]] and the scalar product ( , ) on T ∗ Rd are defined according to (x, ξ), (y, η) = ξ.y − x.η = −Im x + iξ , y + iη = −σ(x + iξ, y + iη) (x, ξ), (y, η) = x.y + ξ.η = Re x + iξ , y + iη = S(x + iξ, y + iη) . After noting

.√ , √ √ √ hx + h∂x , hx − h∂x = 2h ,

the correspondence with the definition (23) is summarized in the next table T ∗ Rd L2 (Rd )

pZ ∼ Cd Γs (pZ ) ∼ Γs (Cd ) ,

z1 , z2 = S(z1 , z2 ) + iσ(z1 , z2 ) z = eiθ (x+iξ) (x1 , ξ1 ) , (x2 , ξ2 ) = ξ1 .ξ2 + x1 .x2 = S(z1 , z2 ) (x1 , ξ1 ), (x2 , ξ2 ) = ξ1 .x2 − x1 .ξ2 = −σ(z1 , z2 ) ⎞ ⎛ d d √ √ a(z) = αj ej ⎠ αj ( h∂xj+ hxj ) a(z) = a ⎝ j=1

j=1

⎛ ⎞ d a∗ (z) = a∗ ⎝ αj ej ⎠

a∗ (z) =

j=1

a(z1 ), a∗ (z2 ) = ε z1 , z2 1 Φ(z0 ) = √ a(z0 ) + a∗ (z0 ) 2 W (z0 ) = eiΦ(z0 ) √ 2 z0 Ω E(z0 ) = W iε

d

√ √ αj (− h∂xj+ hxj )

j=1

a(z1 ), a∗ (z2 ) = 2h z1 , z2 √ 2h(x0 .x + ξ0 .Dx )

ε = 2h z0 = x0 +iξ0

z0 i

θ=0

τ(−√2hξ0 ,√2hx0 )

= ξ0 −ix0

τ( √x0 , √ξ0 ) (π −d/4 e− h

z0⊗n , |z0 | = 1

x2 2

)

h

Hermite function n x2 z0 .(−∂x + x) π −d/4 e− 2

−1/2

∩ D NpZ k ,

k∈N

∗ ∪ D NpZ k

k∈N

(n!)

S (Rd ) ,

S (Rd )

Once this is fixed, the general results on the semiclassical Weyl–Hörmander pseudodifferential calculus (see for example [8, 33] for the general introduction and [37,39,41] for the small parameter version) can be applied for any fixed p ∈ P.

Vol. 9 (2008)


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The notion of slow and temperate metric and weight depend only on the symplectic structure which is given by σ(z1 , z2 ) = Imz1 , z2 . With such a metric the gain function λ is given on pZ by λ(z)2 =

gzσ (T ) T ∈pZ \{0} gz (T )

gzσ (T ) =

|[[T, S]]| |σ(T, S)| = sup . g(S) g(S) S∈pZ \{0} S∈pZ \{0}

with

inf

2

2

sup

With a slow and temperate metric g and a slow and temperate weight m, is associated a symbol class usually denoted S(m, g). After writing X = (x, ξ) ∈ T ∗ Rd for the complete phase-space variable, the differential operator DX is (Dx , Dξ ) = (i−1 ∂x , i−1 ∂ξ ). In the composition formula of symbols, the differential operator ih 2 [[DX1 , DX2 ]] appears. After recalling ∂z =

1 (∇x + i∇ξ ) 2

and ∂z =

1 (∇x − i∇ξ ) 2

it equals ih ε [[DX1 , DX2 ]] = (∂z1 .∂z2 − ∂z1 .∂z2 ) . 2 2 We refer to [39] for an explicit semiclassical writing of the Weyl–H¨ ormander calculus within the Bony–Lerner presentation [8] and with a general version of the Beals criterion following Bony–Chemin [7]. Proposition 3.2. Let g be a slow and temperate metric on pZ , dimC (pZ ) = d and let m1 and m2 be two slow and temperate weights for g. For b ∈ eyl k SpZ (m , g), = 1, 2, the operator bW ,pZ acts continuously on ∩k∈N D(NpZ ) k ∗ and on ∪k∈N D(NpZ ) . eyl W eyl The symbol b1 #ε/2 b2 of bW 1,pZ ◦ b2,pZ satisfies b1 #ε/2 b2 (z) = e 2 (∂z1 .∂z2 −∂z1 .∂z2 ) b1 (z1 )b2 (z2 ) z1 =z2 =z #j 1 "ε (∂z1 .∂z2 − ∂z1 .∂z2 ) b1 (z1 )b2 (z2 ) = j! 2 ε

0≤j 0 and the seminorm pkd depend essentially on the dimension d = dimpZ . A variation of it holds when b ∈ F −1 (Mb (pZ )), when Mb (pZ ) denotes the set of bounded (Radon) measures on pZ and comes directly from (30). Proposition 3.7. For any p ∈ P and any b ∈ F −1 (Mb (pZ )), the Anti-Wick and Weyl observables are asymptotically the same: lim bA−W ick − bW eyl

ε→0

L (H )

= 0.

Proof. Recall that bW eyl can be defined for any b ∈ S (pZ ) as a continuous k k ) ∼ S (Rd ) to ∪k∈N D(NpZ )∗ ∼ S (Rd ), with d = operator from ∩k∈N D(NpZ dimpZ and (30) is still valid for such a symbol. Assume F b = ν ∈ Mb (pZ ). The identity & # '" √ 2 επ 2 W eyl A−W ick ψ , W ( 2πξ)ϕ 1 − e− 2 |ξ| dν(ξ) ψ , (b −b )ϕ = pZ

holds for any ϕ, ψ ∈

k ∩k∈N D(NpZ ).

bW eyl − bA−W ick

L (H )

This implies " # 2 επ 2 ε→0 1 − e− 2 |ξ| d |ν| (ξ) → 0 . ≤ pZ

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3.5. Weyl quantization and specific Wick observables In finite dimension, that is for any fixed p ∈ P, polynomially bounded symbols can be introduced after considering the class of symbols ∪s∈R SpZ (zs , gp ) where gp is 2 either the metric |dz|2 or |dz| z 2 on pZ . According to Proposition 3.2 it is an algebra with the Moyal product, #ε/2 , associated with the composition of Weyl quantized observable with a complete asymptotic expansion of b1 #ε/2 b2 . For any m, q ∈ N, the finite dimensional space Pm,q (pZ ) of (m, q)-homogeneous polynomials on Z is contained in SpZ (zm+q , gp ). The comparison between the Weyl and Wick quantizations is symmetric to (29) (see [6]): ⎞W ick ⎛ 2 ∀b ∈

⊕alg m,q Pm,q (pZ

),

eyl bW pZ

⎜ = ⎝b ∗

pZ

|z|pZ

e− ε/2 ⎟ ⎠ (πε/2)dimpZ

.

For polynomials the deconvolution is possible and we get for any m, q ∈ N and any b ∈ Pm,q (pZ ) eyl ick W eyl − bW ε−1 (bW pZ pZ ) = cpZ (ε)

where the symbol c(ε) equals

⎡⎛

⎢⎜ c(ε) = ε−1 ⎣⎝b ∗

pZ

|z|2 pZ

⎞

⎤

e ⎟ ⎥ ⎠ − b⎦ (πε/2)dimpZ ε/2

and is uniformly bounded in SpZ (zm+q−2 , gp ) w.r.t ε ∈ (0, ε). The space Pm,q (pZ ) is identified with a subspace of Pm,q (Z ) and even of r (Z ) for any r ∈ [1, +∞] with any Pm,q ∀b ∈ Pm,q (pZ ) ,

∀z ∈ Z , b(z) = b(pz + p⊥ z) = b(pz) ˜b = p⊗q ◦ ˜b ◦ p⊗m = Γs (p)˜bΓs (p) .

After tensoring the finite dimensional comparison with IΓs (p⊥ Z ) , we have proved Proposition 3.8. For any p ∈ P, any m, q ∈ N, the class of symbols Pm,q (pZ ) 1 is contained in Pm,q (Z ) ∩SpZ (zm+q , gp ). Moreover the operator ε−1 (bW ick − eyl with cε uniformly bounded in SpZ (zm+q−2 , gp ) w.r.t bW eyl ) can be written cW ε 2 ε ∈ (0, ε). (The metric gp can be either |dz|2 or |dz| z 2 on pZ .)

4. Coherent and product states √

We distinguish the coherent states E(z) = W ( iε2 z)Ω (resp. the projector |E(z)E(z)|) from the product or Hermite state z ⊗k (resp. the projector |z ⊗k z ⊗k |). Although they are intimately related, the asymptotics of coherent state E(z) tested on Wick, Weyl or Anti-Wick observables encoded exactly the

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geometry of the phase-space Z , while the asymptotics of the product states z ⊗k , kε → 0 keeps track of the gauge invariance ∀θ ∈ [0, 2π] ,

|(eiθ z)⊗k (eiθ z)⊗k | = |z ⊗k z ⊗k |

with variations according to the quantization. Proposition 4.1. Fix z, ξ ∈ Z with |z| = 1. (i) The convergence lim V [z ⊗k , z ⊗k−m ](ξ) =

ε→0 kε→1

1 2π

2π

e2πiS(z

θ

,ξ) −imθ

e

dθ ,

0

holds for any fixed m ∈ N by setting z θ = eiθ z . √ (ii) The coherent state E(z) = W ( iε2 z)Ω satisfies ε|ξ|2 ε→0 V E(z), E(z) (ξ) = e2πiS(ξ,z) e− 2 → e2πiS(ξ,z) .

Proof. i) Set j = k − m and compute V [z ⊗k , z ⊗j ](ξ) with ξ = √ξ2π according to Proposition 3.5: 2 5 6 # " ε #m/2 2 ξ j! (m) " ε

m Lj |ξ , z|2 ξ , z e−ε|ξ | /4 V [z ⊗k , z ⊗j ] √ = (i)m k! 2 2 2π % % ∞ (−1)s j! k! m 1[0,j] (s) = (i) s m+s s!(s + m)! (j − s)!k (j − s)!k s=0 5 6 2s+m 2 2 εk m |ξ , z|2s ξ , z e−ε|ξ | /4 . 2 /∞ Cs The bounds (εk) ≤ C and s=0 s!(s+m)! < ∞ imply 5 6 ∞ ξ (−1)s m |ξ , z|2s ξ , z , lim V [z ⊗k , z ⊗j ] √ = (i)m 2s+m ε→0 2 2π s!(s + m)! s=0 2 kε→1 /∞ tk owing to Lebesgue’s theorem. A simple series expansion et = k=0 k! for t = √ θ

i 2S(z , ξ ) gives 2π √ ∞ θ 1 (−1)s m |ξ , z|2s ξ , z . ei 2S(z ,ξ ) e−imθ dθ = (i)m 2s+m 2π 0 2 2 s!(s + m)! s=0 ii) is a straightforward consequence of (22).

The next result specifies the similarity and the differences between the product states and the coherent states in the mean-field or semiclassical limit. Theorem 4.2. Let z ∈ Z and m ∈ N be fixed with |z| = 1 and set z θ = eiθ z for θ ∈ [0, 2π]. The next limits exist as ε → 0, kε → 1.

Vol. 9 (2008)


(i) For b ∈ Scyl (Z ), lim z ⊗k−m , bW eyl z ⊗k = lim z ⊗k−m , bA−W ick z ⊗k =

ε→0 kε→1

ε→0 kε→1

1 2π

2π

1531

b(z θ )e−imθ dθ .

0

Meanwhile the coherent state E(z) satisfies lim E(z) , bW eyl E(z) = lim E(z) , bA−W ick E(z) = b(z) . ε→0

ε→0

(ii) For b ∈ Pp,q (Z ), with p, q ∈ N fixed, lim z ⊗k−m , bW ick z ⊗k = δp−q,m b(z) =

ε→0 kε→1

1 2π

2π

b(z θ )e−imθ dθ .

0

Meanwhile the coherent state E(z) satisfies ∀ε > 0 , E(z) , bW ick E(z) = b(z) . Proof. Set j = k − m, with m ∈ N fixed. For (i), fix b ∈ Scyl (Z ). The definition of bW eyl in (23), says √ z ⊗j , bW eyl z ⊗k = F [b](ξ) z ⊗j , W ( 2πξ) z ⊗k Lp (dξ) pZ = F [b](ξ) V [z ⊗k , z ⊗j ](ξ) Lp (dξ) . pZ ⊗k

Since F [b] ∈ S (pZ ) and V [z , z ⊗j ](ξ) converges pointwise according to Proposition 4.1, Lebesgue’s theorem yields 5 6 2π 1 ⊗j W eyl ⊗k i2πS(z θ ,ξ) −imθ z = F [b](ξ) e e dθ Lp (dξ) lim z , b ε→0 2π 0 pZ kε→1 2π 1 = b(z θ )e−imθ dθ . 2π 0 The limit with Anti-Wick observables is a consequence of the formula (30): √ 2 επ 2 ⊗j A−W ick ⊗k z = F [b](ξ) z ⊗j , W ( 2πξ)z ⊗k e− 2 |ξ|pZ Lp (dξ) . z , b pZ

The statement about the coherent state E(z) is even simpler by referring to Proposition 4.1 (ii). Let us prove (ii). The statement (ii) of Proposition 2.4 gives % p+q k!j! + ⊗j W ick ⊗k z , b ε 2 z ⊗q , bz ⊗p z = δk−p,j−q (k − p)!(j − q)! % % k! j! (εk)p+q z ⊗q , bz ⊗p . = δm,p−q p (k − p)!k (j − q)!k q 3 3 j! k! We conclude again with (k−p)!k p (j−q)!kq → 1 as k → ∞.

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5. An example of a dynamical mean-field limit In order to illustrate the general presentation, the standard example of the mean field derivation of the Hartree equation from the non relativistic Hamiltonian of bosons with a quartic interaction is considered. Two standard methods are considered: The coherent state method (see [24, 32] or [12] for a rapid presentation) also known as Hepp method and the propagation of chaos approach with a truncated Dyson expansion according to [16, 17, 19, 20, 45]. d Consider Z = L2C (Rd , dx) and take V ∈ L∞ R (R , dx) such that V (−x) = ⊗2 ⊗2 ˜ ˜ ∈ is associated with the operator Q V (x). The polynomial Q(z) = z , Qz 2 Z ) defined by L( ˜ : ⊗2 Z Q

→

⊗2 Z , 1 V (x1 − x2 ) u(x1 )w(x2 ) . 2

u(x1 )w(x2 ) → The Hamiltonian is defined as

Hε = dΓ(−Δ) + QW ick , where −Δ is the Laplacian of Rd , while Hε0 denotes the free Hamiltonian dΓ(−Δ). It is well known that Hε is a self-adjoint operator on H (see [24]) and the quantum t t time-evolution group is denoted by Uε (t) = e−i ε Hε while Uε0 (t) = e−i ε H0 = Γ(eitΔ ) stands for the free dynamics. Although the Wick quantization of non continuous polynomials has not been introduced here, this Hamiltonian appears as the Wick quantization of the energy functional 2 |∇z| dx + Q(z) . (31) h(z) = Rd

It is also well known that the mean field limit is the nonlinear dynamics issued from the Hartree equation i∂t zt = −Δzt + V ∗ |zt |2 zt = ∂z h(zt )

(32)

with initial condition z0 = z ∈ Z . An important property of the dynamical groups Uε (t) and Uε0 (t) is that they preserve the number Uε (t)∗ N Uε (t) = N ,

[Hε , N ] = [Hε0 , N ] = [QW ick , N ] = 0 .

Remark 5.1. All the results of this section can be easily adapted with a selfadjoint operator A on Z and a polynomial Q(z) ∈ ⊕alg n∈N Pn,n (Z ). Nevertheless it is better to stick to this concrete presentation which fits better with a widely studied problem. 5.1. Propagation of squeezed coherent states (Hepp method) In finite dimension it is nothing but checking the propagation of gaussian wave packets. Although it works only for some specific states it is a direct and very

Vol. 9 (2008)


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flexible method. Moreover it agrees very well with the phase-space geometric intuition. Extensions with more singular potentials or about the long time behaviour have been carried out in [24, 32]. Proposition 5.2. For any z0 ∈ Z , the estimate √ ω(t) 2 −i εt Hε i ε e zt U2 (t, 0)Ω E(z0 ) − e W iε

≤ C eC|V |L∞ z0

2

(|t|+1)

ε1/2

H

holds with 2

i∂t zt = −Δzt + (V ∗ |zt | )zt , zt=0 = z0 t ω(t) = Q(zs ) ds 0 iε∂t U2 (t, 0) = dΓ(−Δ) + Q2 (t)W ick U2 (t, 0) , U2 (0, 0) = I , 1 . 2 ∂z Q(zt ) , z ⊗2 Q2 (t, z) = 2 , + z ⊗2 , ∂z2 Q(zt ) + 2 z , ∂z ∂z Q(zt )z , ' & 2 ˜ zt⊗2 , z ⊗2 ∈ P2,0 (Z ) , ∂z Q(zt ) , z ⊗2 = 2 Q ' & ˜ z ∨ zt ∈ P1,1 (Z ) . z , ∂z ∂z Q(zt )z = 4 z ∨ zt , Q

(33) (34) (35)

(36)

Proof. This proposition says that the evolution of a coherent state is well described after applying a time dependent (real) affine Bogoliubov transformation like the ones considered in Proposition 2.12. It is sufficient that √ ω(t) 2 i εt Hε i ε zt U2 (t, 0)Ω e e W iε √ ω(t) 2 −itΔ i εt Hε itΔ i ε e =e Γ(e )e W zt Γ(e−itΔ )U2 (t, 0)Ω iε ˆε (0, t) = ei εt Hε Γ(eitΔ ), U ˆ2 (t, 0) = remains close enough to E(z0 ) . The quantities U −itΔ −itΔ )U2 (t, 0) and zˆt = e zt solve the differential equations Γ(e ˆε (0, t) = −U ˆε (0, t)Γ(e−itΔ )QW ick Γ(eitΔ ) = −U ˆε (t, 0)Q(t) ˆ W ick , iε∂t U

(37)

ˆ2 (t, 0) = Γ(e−itΔ )Q2 (t)W ick Γ(eitΔ )U ˆ2 (t, 0) , ˆ2 (t, 0) = Q ˆ 2 (t)W ick U iε∂t U

(38)

−itΔ

i∂t zˆt = e

(V ∗ |e

itΔ

zˆt | )e 2

itΔ

ˆ zˆt ) , zˆt = ∂z Q(t,

zˆ0 = z0 ,

(39)

after setting ˆ z) = Q(eitΔ z) and Q ˆ 2 (t, z) = Q2 (t, eitΔ z) . Q(t,

(40)

1534



ˆ2 (t, 0) are derived in [24, Proposition 4.1] and in particular The main properties of U ˆ we know that U2 (t, 0)Ω belongs to the domain of the closure of any bW ick with b ∈ ⊕alg p,q∈N Pp,q (Z ). The differentiation of the Weyl relation (3) on Hf in says √ √ √ 2 2 zˆt = −Reˆ zˆt zt , i∂t zˆt + 2Φ(i∂t zˆt ) W iε∂t W iε iε √ . , 2 ˆ zˆt ) + a∗ ∂z Q ˆ t (ˆ ˆ t (ˆ zˆt = −Re zˆt , ∂z Q(t, zt ) + a ∂z Q zt ) W iε √ . W ick , 2 ˆ ˆ W zˆt . = −Re zˆt , ∂z Q(t, zˆt ) + 2Re z , ∂z Qt (ˆ zt ) iε The translation property (iii) of Proposition 2.10 then leads to i εt Hε i

e

e

ω(t) ε

√ 2 zt U2 (t, 0)Ω − E(z0 ) W iε √ ω(s) 2 1 tˆ ˆ2 (s, 0)Ω ds Uε (0, s)ei ε W zˆs A (s)W ick U = iε 0 iε

after testing both sides on Hf in and setting ˆ zˆs ) + 2Re z , ∂z Q ˆ s (ˆ ˆ z + zˆs ) − ω (s) + Re zˆs , ∂z Q(s, zs ) A (s, z) = −Q(s, ˆ 2 (s, z) +Q

ˆ s (ˆ ˆ s (ˆ ˆ z s ) + z , ∂z Q ˆ 2 (s, z) . ˆ z + zˆs ) + Q(ˆ z s ) + ∂z Q zs ) , z + Q = −Q(s, The last equality is given by our choice of ω(t) in (34). It suffices to find a uniform estimate w.r.t s ∈ [0, t] of the squared norm ˆ2 (s, 0)Ω ε−1 A (s)W ick U

2 H

& ' ˆ2 (0, s)A (s)W ick,∗ A (s)W ick U ˆ2 (s, 0)Ω . = ε−2 Ω , U

(41)

The important point is that the symbol A (s) vanishes at the second order at z = 0. More precisely it can be written A (s) = A1,2 (s) + A2,1 (s) + A2,2 (s) with Ap,q (s) ∈ Pp,q (Z ) A˜p,q (s)

L(

p

Z,

q

and 4−p−q

Z)

≤ Cp,q |V |L∞ |z0 |

.

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1535

Owing to Proposition 2.7 and Lemma 2.6 the operator A (s)W ick,∗ A W ick (s) takes the form 2 A (s)W ick,∗ A (s)W ick = εk Bk,p,q (s)W ick with k=0

B˜k,p,q (s)

L(

p

Z,

q

6−2k≤p+q≤8 2

Z)

2

≤ Ck,p,q |V |L∞ z0 .

The estimate of every term & ' ˆ2 (0, s)Bk,p,q (s)W ick U ˆ2 (s, 0)Ω , εk−2 Ω , U

p + q ≥ 6 − 2k

is given by the Lemma 5.3 below and yields the result.

ˆ 2 defined by (36) (40) Lemma 5.3. Consider the time dependent Wick operator Q ˆ2 (s, 0) and parametrized by z0 ∈ Z . Consider the associated unitary operator U defined by (38). For any p, q ∈ N, there exists a constant Cp,q such that the estimate & ' p+q ˆ2 (0, s)bW ick U ˆ2 (s, 0)Ω ≤ Cp,q eCp,q |V |L∞ z0 2 (|s|+1) ˜b Ω, U ε 2 q p L(

Z,

Z)

holds for any b ∈ Pp,q (Z ) and any s ∈ R . Proof. By introducing an anti-unitary operator Jz = z, the R-linear operator ˆ 2 (t) can be written ∂z Q ˆ 2 (t)z = R(t)z + R2 (t)z . ∂z Q The definitions (36) (40) ensure that R(t) is a bounded operator strongly continuous with respect to t ∈ R and that R2 (t) is a Hilbert–Schmidt operator which depends continuously on t ∈ R in the Hilbert–Schmidt norm. Moreover the following uniform estimates hold 2

2

|R(t)|L (Z ) ≤ 2 |V |L∞ |z0 | ,

|R2 (t)|L 2 (Z ) ≤ 2 |V |L∞ |z0 | .

Hence the equation ˆ 2 (t)Φ2 = R(t)Φ2 + R2 (t)JΦ2 i∂t Φ2 = ∂z Q defines a dynamical system of bounded R-linear operators with the estimate |Φ2 (t2 , t1 )|LR (Z ) ≤ e4|t2 −t1 ||V |L∞ |z0 | . 2

More precisely the Duhamel formula −i

Φ2 (t2 , t1 ) = T e

@ t2 t1

R(s) ds

−i

t2

T e−i

@ t2 t

R(s) ds

R2 (t)JΦ2 (t, t1 ) dt

t1

implies that the R-linear operator Φ2 (t2 , t1 ) can be written Φ2 (t2 , t1 ) = B(t2 , t1 ) + B2 (t2 , t1 )J

with

|B(t2 , t1 )|L (Z ) +|B2 (t2 , t1 )|L 2 (Z ) ≤ C |V |L∞ |z0 | (|t2 −t1 |+1)eC|t2 −t1 ||V |L∞ |z0 | . 2

2

1536



According to Proposition 2.12, for any c ∈ ⊕p+q=m Pp,q (Z ) and any t ∈ R, the polynomial c(t, z) = c(Φ2 (0, t)z) belongs to ⊕p+q=m Pp,q (Z ) with

|∂zq ∂zp c(t, z)|L ( p Z , q Z )

p+q=m 1 Cm |V |L∞ z0 ≤ Cm e 1

2

(|t|+1)

|∂zq ∂zp c(z)|L ( p Z , q Z ) .

p+q=m

Applying the characteristic method, that is differentiating c(z) = c(t, Φ2 (t, 0)z), shows that c(z, t) solves the equation ˆ 2 (t, z) − ∂z Q ˆ 2 (t, z)∂z c(t, z) = 0 . i∂t c(t, z) + ∂z c(t, z).∂z Q ˆ2 (t, 0)Ω ∈ Thanks to the Wick calculus in Proposition 2.7 and the fact that U k ∩k∈N D(N ) (see [24, Proposition 4.1]), this leads to ˆ2 (0, t)c(t)W ick U ˆ2 (t, 0)Ω i∂t U # " ˆ2 (t, 0)Ω ˆ2 (0, t) ε−1 cW ick (t), Q ˆ 2 (t)W ick + i∂t c(t)W ick U =U "0 1 #W ick ˆ2 (t, 0)Ω . ˆ 2 (t) (2) ˆ2 (0, t) ε c(t), Q U =U 2 Take b ∈ ⊕p+q=m0 Pp,q (Z ) and apply this result with c defined by c(s, z) = b(z), which means

c Φ2 (0, s)z = c(s, z) = b(z) or c(z) = b Φ2 (s, 0)z ∈ ⊕p+q=m0 Pp,q (Z ) |∂zq ∂zp c(z)|L ( p Z , q Z )

≤

with

1 2 1 Cm eCm0 |V |L∞ z0 (|s|+1) 0

p+q=m0

×

|∂zq ∂zp b(z)|L ( p Z , q Z ) .

p+q=m0

This leads to ' & ˆ2 (s, 0)Ω ˆ2 (0, s)bW ick U Ω, U s7 " # 8 W ick W ick ˆ ˆ = Ω,c U2 (t, 0) Ω dt Ω + Ω , ∂t U2 (0, t)c(t) 0 8 s7 "0 1(2) #W ick iε ˆ ˆ ˆ U2 (t, 0)Ω dt . Ω , U2 (0, t) c(t), Q2 (t) =− 2 0

Vol. 9 (2008)


1537

ˆ 2 (t)} vanishes when m0 < 2 or belongs to By noticing that the symbol {c(t), Q ⊕p+q=m0 −2 Pp,q (Z ) with 0 1 ˆ 2 (t) (2) ∂zq ∂zp c(t), Q p q L(

p+q=m0 −2

2

≤ C |V |L∞ |z0 |

Z,

Z)

|∂zq ∂zp c(t)|L ( p Z , q Z )

p+q=m0 1 ≤ C |V |L∞ |z0 | Cm eCm0 |V |L∞ z0 0 1

2

2

(2|s|+1)

|∂zq ∂zp b|L ( p Z , q Z )

p+q=m0

the result is proved by induction on m0 and by using xn ≤ n!ex for x > 0.

5.2. Truncated Dyson expansion We focus now on the propagation of chaos point of view which has been considered by several authors in [4, 16, 17, 20]. In the bosonic setting Hermite states tested on some Wick observable is exactly the BBGKY hierarchy. For example the reduced one particle density matrix can be defined as Tr[1 A] = Tr[dΓ(A)] = Tr[A W ick ] with A (z) = z , Az . While reproducing the Dyson expansion analysis of [20], we check here that a full asymptotic expansion can be written, when Wick observables are tested after the suitable number truncation. The strategy of the proof in [20] relies on an analysis of the Schwinger–Dyson expansion of a time evolved observable Uε (t)∗ O Uε (t) given by Uε (t)∗ O Uε (t) = Ot +

∞ 5 6n i n=1

ε

0

t

dt1 · · · 0

tn−1

ick ick dtn QW , · · · [QW , Ot ] · · · tn t1

(42)

ick = Uε0 (s)∗ QW ick Uε0 (s). The commutation where Ot = Uε0 (t)∗ O Uε0 (t), QW s relation in Proposition 2.3 (iii) yields W ick ick QW = (eisΔ z)⊗2 , Q(eisΔ z)⊗2 , s

or shortly Qs (z) = Q(eisΔ z) and we shall set more generally for b ∈ Pp,q (Z ) and s∈R bs ∈ Pp,q (Z ) : ∀z ∈ Z , bs (z) = b(eisΔ z) . Although the convergence of the series can be proved as an operator acting on k Z , with k ∈ N fixed, the ε-asymptotic analysis is done with its truncated version tn−1 −1 5 6n t i ∗ ick ick Uε (t) OUε (t) = Ot + dt1 · · · dtn QW , · · · [QW , Ot ] · · · tn t1 ε 0 0 n=1 5 6 t t−1 i ick dt1 · · · dt Uε (t )∗ Uε0 (t ) QW ,··· + t ε 0 0 ick · · · [QW , Ot ] · · · Uε0 (t )∗ Uε (t ) . (43) t1 The Poisson brackets analogue of the multicommutators will be necessary.

1538



Definition 5.4. For n, r ∈ N, r ≤ n and any fixed b ∈ Pp,q (Z ), the polynomial (n) Cr (t1 , . . . , tn ) is defined by Cr(n) (tn , . . . , t1 , t) 1 = r 2

{i: εi =2}=r

0

Qtn , . . . , {Qt1 , bt }(ε1 ) · · ·

1(εn )

εi ∈{1,2}

∈ Pp−r+n,q−r+n (Z ) ,

(44)

A (n) (n) and Cr (t1 , . . . , tn , t, z) denotes its values at z ∈ Z while Cr (t1 , . . . , tn , t) or A (n) simply Cr denotes the associated operator according to Definition 2.1 . We shall prove. Theorem 5.5. Fix p, q ∈ N and assume b ∈ Pp,q (Z ). Then the asymptotic expansion tn−1 t −1 ∞ . ,W ick ∗ W ick r n Uε (t) = ε i dt1 · · · dtn Cr(n) (tn , . . . , t1 , t) Uε (t) b r=0

0

n=0

0

+ ε R (ε, t) k−p+q k Z ) with the uniform holds for any ∈ N and any δ > 0 in L ( Z , estimate |R (ε, t)|L ( k Z , k−p+q Z ) ≤ C,δ kε ≤ 1 + δ/2

when

and

4(1 + 2δ)|t| |V |L∞ ≤ 1 .

A particular case takes a more explicit form. Theorem 5.6. Take b ∈ Pp,q (Z ). Let z ∈ Z be such that |z| = 1 and call zt the solution to (32) with z0 = z. (i) Then the expansion

−1 ⊗k−m z , Uε (t)∗ bW ick Uε (t) z ⊗k = δp−q,m εr β (r) (t, z, k, ε) + Ot (ε ) , (45) r=0

holds as ε → 0, kε → 1 by setting β (0) (t, z, k, ε) = b(zt ), β

(r)

(t, z, k, ε) =

k−p+r

n

i

n=r

···

k!(k − m)! εp+q+2(n−r) (k − (p + n − r))!

t

dt1 · · · 0

tn−1

dtn Cr(n) (tn , . . . , t1 , t; z) , 0

and as soon as 4|t| |V |L∞ < 1 .

(46)

Vol. 9 (2008)


1539

(ii) More generally the limit lim z ⊗k−m , Uε (t)∗ bW ick Uε (t) z ⊗k = δp−q,m b(zt ) ε→0, kε→1

holds for all times t ∈ R. Corollary 5.7. In the specific case m = 0, q = p, the expansion (45) takes the form

z

⊗k

∗ W ick

, Uε (t) b

Uε (t) z

⊗k

=

−1

ε

s=0

s

∞ n=0

t

dt1 · · ·

n

i

0

tn−1

dtn 0

⎡ ⎤ s (n) ⎣ αjs−j,n (kε)Cs−j (tn , . . . , t1 , t; z)⎦ + O(ε ) , j=0

where the coefficients p+n−r−1

αjr,n (κ)

are polynomials in κ given by

αjr,n (κ)εj = κ(κ − ε)(κ − 2ε) · · · κ − (p + n − r − 1)ε ,

j=0

and the convention that αjr,n = 0 when j ≥ (p + n − r) or r > n. Proof. We are considering the particular case p = q, m = 0. Setting κ = kε = (k − m)ε gives: k!εp+(n−r) = κ(κ − ε)(κ − 2ε) · · · κ − (p + n − r − 1)ε . (k − (p + n − r))! Putting together the terms of order εs , s less than − 1 in Theorem 5.5 (ii), yields the result. Before proving Theorem 5.5 and Theorem 5.6, let us collect some technical preliminaries. Lemma 5.8. For b ∈ Pp,q (Z ) the identity n #W ick " 1 W ick W ick W ick r (n) Q , · · · , [Q , b ] = ε (t , . . . , t , t) , C n 1 tn t1 t r εn r=0 (n)

holds with the symbols Cr (t1 , . . . , tn , t) defined according to (44) in Definition 5.4. Proof. Proposition 2.7 provides the induction formula 1 (n−1) Cr(n) = {Qtn , Cr(n−1) } + {Qtn , Cr−1 }(2) , 2

(47)

(l)

with Cr = 0 if l < r or r < 0. In particular, we get 0 1 (n) C0 = Qtn , . . . , {Qt1 , bt } . A simple iteration of (47) yields the result.

1540



Lemma 5.9. Let b belong to Pp,q (Z ). (i) The estimate B1 Ξ

L(

p+1

Z,

q+1

Z)

≤ (p + q) |V |L∞ |b|L ( p Z , q Z ) ,

p+1 q+1 1 p+1 q+1 B1 = 1 holds by setting Ξ ∂z¯ {Qs , bt }(1) ∈ L ( Z, Z ). (p+1)! (q+1)! ∂z (ii) Similarly, the inequality B2 Ξ

L(

p

Z,

q

B2 = holds with Ξ

Z)

≤ p(p − 1) + q(q − 1) |V |L∞ |b|L ( p Z , q Z )

1 1 p q (2) p! q! ∂z ∂z¯ {Qs , bt }

.

A (n) (iii) For any n ∈ N and r ∈ {0, 1, . . . , n}, the operator Cr associated with the (n) symbol Cr (tn , . . . , t1 , t) ∈ Pp+n−r,q+n−r (Z ) according to Definition 5.4 satisfies A (n) Cr L(

p+n−r

Z,

q+n−r

Z)

≤ 2n−r Cnr (p + n − r)2r

(p + n − r − 1)! |V |nL∞ |b|L ( p Z , q Z ) , (p − 1)!

when p ≥ q with a similar expression when q ≥ p (replace (p + n − r, p − 1) with (q + n − r, q − 1)) . Proof. The statements (i) and (ii) are particular cases of Lemma 2.6. The estimate in (iii) is a consequence of (i)(ii) and the definition (44). Proof of Theorem 5.5. Set j = k−p+q. Since Uε (t) and Uε0 (t) preserve the number ick the equality like QW t Uε (t)∗ bW ick Uε (t) tn−1 −1 5 6n t i ick ick W ick dt1 · · · dtn QW , · · · [QW , bt ]··· = tn t1 ε 0 0 n=0 5 6 t t−1 i ick dt1 · · · dt Uε (t )∗ Uε0 (t ) QW ,··· + t ε 0 0 ick W ick · · · [QW , bt ] · · · Uε0 (t )∗ Uε (t ) , t1

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1541

k j derived from (43) holds in L ( Z , Z ). Then Lemma 5.8 implies Uε (t)∗ bW ick Uε (t) t −1 in dt1 · · · = n=0

0

t−1

dt1 · · · 0

0

t

0

n

. ,W ick εr Cr(n) (tn , . . . , t1 , t)

(48)

r=0

. ,W ick () dt Uε (t )∗ Uε0 (t )ε C (t , . . . , t1 , t) Uε0 (t )∗ Uε (t ) (49) −1

t−1

dt Uε (t )∗ Uε0 (t )

dt1 · · ·

+ i

dtn

0

t

+ i

tn−1

0

,W ick εr Cr() (t , . . . , t1 , t) Uε0 (t )∗Uε (t ) . .

r=0

(50) Keep untouched the part (48)–(49) and iterate the Dyson series on the third term (50). While doing so, use the formula

−1 ick . ,W ick QW tn+1 , εr Cr(n) (tn , . . . , t1 , t) ε r=0

=

−1

. ,W ick εr Cr(n+1) (tn+1 , . . . , t1 , t)

r=0

+

1(2) ,W ick ε . 0 (n) Qtn+1 , C (tn+1 , . . . , t1 , t) , 2

(51)

inductively for n = , + 1, . . . , M − 1. After M − steps, collecting the factors of ε yields Uε (t)∗ bW ick Uε (t) t M −1 in dt1 · · · = 0

n=0

+

M

t

dt1 · · ·

n

i

0

n=

.0

dtn

0

0

tn−1

min(−1,n)

tn−1

,W ick . εr Cr(n) (tn , . . . , t1 , t)

dtn Uε (tn )∗ Uε0 (tn )

ε 2

1(2) ,W ick 0 (n−1) Qtn , C−1 (tn−1 , . . . , t1 , t) Uε (tn )∗ Uε (tn ) tM −1 t + iM dt1 · · · dtM Uε (tM )∗ Uε0 (tM ) ×

0

×

−1 r=0

(52)

r=0

0

,W ick . εr Cr(M ) (tM , . . . , t1 , t) Uε0 (tM )∗ Uε (tM ) .

(53)

(54)

1542



Assume that for δ > 0 there exists a constant Cδ such that ∞

(1 + δ)n

dt1 · · ·

0

r=0

n=

t

tn−1

···

A (n) dtn Cr (tn , . . . , t1 , t) L(

0

p+n−r

Z,

< Cδ .

q+n−r

(55)

Z)

According to Lemma 2.5, the first term (52) of (52) (53) (54) provides in k Uε (t)∗ bW ick Uε (t) k Z the partial sum of a convergent series in L ( Z , k−p+q Z ) when kε ≤ 1 + 2δ . With the same argument the remainder term (54) vanishes as M → ∞ and kε ≤ 1 + 2δ . By referring to Lemma 5.9 (ii) and again to Lemma 2.5 the factor of ε in (53) is associated with a series which converges k k−p+q in L ( Z , Z ) as M → ∞ uniformly w.r.t. (k, ε) when kε ≤ 1 + 2δ . The sum of the series is simply denoted by R (t, ε). Let us prove (55) to finish the proof of (ii). Lemma 2.5 and Lemma 5.9 say ∞

n

(1 + δ)

r=0

n=

≤ ≤

∞

dt1 · · ·

0

tn−1

0

A (n) dtn Cr (tn , . . . , t1 , t) L(

n=

|tn | A (n) Cr (tn , . . . , t1 , t) max tn ≤···≤t1 ≤t n! r=0

∞

(1 + δ)n (1 + δ)n

r=0

n=

× |V ≤

t

∞

|nL∞

≤ 2p

∞

L(

Z,

p+n−r

q+n−r

Z,

Z)

q+n−r

Z)

r (p + n − r − 1)! 2n−r |tn | r Cn (p + n − r)(p + n − r − 1) n! (p − 1)!

|˜b|L ( p Z , q Z )

(1 + δ)n |t|n

n=

p+n−r

2n−r p−1 (p + n)2r Cn−r+p−1 |V |nL∞ |˜b|L ( p Z , q Z ) r! r=0

(1 + δ)n 4n |t|n (n + p)2 |V |nL∞ |˜b|L ( p Z , q Z ) .

n=

The last r.h.s. is finite whenever 4|t||V |L∞ < (1 + δ)−1 . The condition (1 + 2δ)4|t||V |L∞ ≤ 1 is sufficient and provides the uniform bound Cδ in (55) .

Proof of Theorem 5.6. Set j = k − m. By Theorem 5.5, the right-hand side of (45) k k−p+q vanishes when m = p−q and the convergence of the series in L ( Z , Z)

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combined with Proposition 2.4-ii) implies

z ⊗j ,Uε (t)∗ bW ick Uε (t) z ⊗k % −1 ∞ k!j! εp+q+2(n−r) + r n δk−(p+n−r), = ε i j−(q+n−r) (k − (p + n − r))!(j − (q + n − r))! r=0 n=0 t tn−1 × dt1 · · · dtn Cr(n) (tn , . . . , t1 , t; z) + Oδ (ε ) , 0

0

when kε ≤ 1 + 2δ , for any δ > 0. By considering the limit ε → 0, kε → 1 every factor % k!j! εp+q+2(n−r) (k − (p + n − r))!(j − (q + n − r))! converges to 1. Therefore this proves (ii) for small times t such that 4|t||v|L∞ < 1 up to the identification of the first term as b(zt ). From our definitions we know ' & b(zt ) = zt⊗q , ˜bzt⊗p = bt (e−isΔ zs )

s=t

.

By setting ws = e−isΔ zs , the quantity b(zt ) equals b(zt ) = bt (w0 )+

t

∂s bt (ws ) ds = bt (w0 )+

0

t

∂s ws .∂z bt (ws )+∂z bt (ws ).∂s ws ds . 0

Moreover the equation (32) has the equivalent form with the vector ws = e−isΔ zs and ws i∂s ws = e−isΔ ∂z Q(zs ) = ∂z Qs (ws )

− i∂s ws = ∂z Qs (ws ) .

Hence we get

t

{Qt1 , bt } (wt1 ) dt1 .

b(zt ) = bt (w0 ) + i 0

An induction with w0 = z and the convergence of the series already checked yields b(zt ) =

∞ n=0

t

dt1 · · ·

n

i

0

tn−1

(n)

dtn C0 (tn , . . . , t1 , t; z) . 0

Now let us prove the limit (i) for all times by following the argument in [20,45]. Assume that the result is true for |t| ≤ 4|VK|L∞ . Let s be such that |s| < 1/4|V |L∞ . The convergence of the series given in Theorem 5.5 and the fact that Uε (t) preserves

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the number gives ⊗j z , Uε (t + s)∗ bW ick Uε (t + s) z ⊗k s sn−1 & ∞ n ' W ick n r = i ε ds1 · · · dsn z ⊗j , Uε (t)∗ Cr(n) (sn , . . . , s1 , s) Uε (t) z ⊗k =

n=0 ∞ n=0

r=0 s

0

ds1 · · ·

in 0

0 sn−1

& ' (n) W ick dsn z ⊗j , Uε (t)∗ C0 (sn , . . . , s1 , s) Uε (t) z ⊗k

0

+ Os (ε)

(56)

with an absolutely and uniformly convergent series /∞ in the (56) when kε is close to 1. Hence the limit ε → 0, εk → 1 and the sum n=0 in (56) can be interchanged when 4|s||V |L∞ < 1. An induction on K = 0, 1, 2 . . . finishes the proof. 5.3. Coherent states and Wick observables We show here that information on the propagation of coherent states can be directly deduced from the results about Hermite states. Proposition 5.10. For any z0 ∈ Z and any b ∈ Pp,q (Z ), the limit lim Uε (t)E(z0 ) , bW ick Uε (t)E(z0 ) = b(zt ) ε→0

holds for any t ∈ R when zt denotes the solution to the Hartree equation (32). Proof. By symmetry, one can assume m = p − q ≥ 0. Recall that E(z0 ) = |z0 |2 /∞ −n/2 ε√ e− 2ε z ⊗n and start first with |z0 | = 1. Since Uε (t) preserves the n=0 n! 0 number, one gets ∞ −n −1 ε with an ε−1 Uε (t)E(z0 ) , bW ick Uε (t)E(z0 ) = e−ε n! n=m an ε−1 = εm/2 n(n − 1) . . . (n − m + 1) z0⊗n−m , Uε (t)∗ bW ick Uε (t)z0⊗n . By Lemma 2.5 the quantity an ε−1 satisfies p+q+m p ˜b ≤ nε ˜b p q . |an ε−1 | ≤ (nε) 2 q p L(

Z,

Z)

−1

L(

Z,

Z)

Hence Lemma A.1 applied here with λ = ε and μ = p reduces the problem to the proof of s2 e− 2 √ √ ds . lim a (λ) λ→∞ R [ λs+λ] 2π The uniform estimate 7 8p |s| p √ a[ λs+λ] (λ) ≤ Cp 1 + √ ≤ Cp s λ √ and the pointwise convergence induced by Theorem 5.6 with z = z0 , k = [ λs + λ] and ε = λ−1 yields the result.

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For a general |z0 | > 0, write E(z0 ) = e− 2ε 1

with z0 = tion, with

z0 |z0 |

and ε =

ε |z0 |2

∞ (ε )−n/2 ⊗n √ (z0 ) = E (z0 ) n! n=0

. By replacing the ε-quantization by the ε -quantiza-

bW ick,ε = |z0 |

−p−q W ick

for b ∈ Pp,q (Z )

b

Hε = |z0 | dΓε (−Δ) + |z0 |4 QW ick,ε and " # 2 (iε∂t u = Hε u) ⇔ iε ∂t u = dΓε (−Δ)u + |z0 | QW ick,ε u . 2

Hence the previous result applied with E (z0 ), |z0 | = 1 and the ε -quantization implies p+q b(zt ) lim Uε (t)E(z0 ) , bW ick Uε (t)E(z0 ) = |z0 | ε→0

where zt solves i∂t zt = −Δzt + |z0 | (V ∗ |zt | )zt , 2

2

zt=0 = z0 =

z0 . |z0 |

Since this mean field equation preserves the norm |zt | like (32) does for |zt |, this implies −1 −1 p+q b(zt ) = b(zt ) . zt = |z0 | zt = |zt | zt and |z0 | Remark 5.11. Another proof can be obtained directly from Proposition 5.2 after checking uniform number estimates for U2 (t, 0)Ω. But working in this direction is more efficient with the help of Wigner measures.

6. Wigner measures: Definition and first properties The notion of Wigner (or semiclassical) measures is well established in the finite dimensional case. We refer the reader to [10, 22, 23, 29, 36, 46] for details. The extension that we propose here to the infinite dimensional case follows a projective approach. 6.1. Wigner measures of normal states Consider the algebra of cylindrical sets Bcyl (Z ) = {X(p, E) = p−1 (E), p ∈ P, E ∈ B(pZ )} where B(pZ ) denotes for any p ∈ P the set of Borel subsets of pZ . A cylindrical measure μ is a mapping defined on Bcyl (Z ) such that: • μ(Z ) = 1, • For any p ∈ P, μp (A) = μ(p−1 (A)) for A ∈ B(pZ ) defines a probability measure μp on B(pZ ). The family of measures {μp }p∈P is often called a weak distribution. This notion is often introduced within the framework of real Hilbert spaces (or more generally real topological vector spaces). This makes no difference at this level. The real structure on Z , namely the real scalar product S, is useful for the

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application of Bochner’s theorem. For any ξ ∈ Z the function z → e−2πi S(z,ξ) is a cylindrical measurable function and the Fourier transform of μ is well defined by e−2πi S(z,ξ) dμ . F [μ](ξ) = Z

Bochner’s theorem characterizes the Fourier transform of a weak distribution. It says (see for example [3]) that a function G is the Fourier transform of a weak distribution if and only if • G is normalized: G(0) = 1, /N • G is of positive type: i,j=1 λi λj G(ξi − ξj ) ≥ 0, • For any p ∈ P, the restricted function G|pZ is continuous. An important point is that Z is a separable Hilbert space. Hence the σ-algebra generated by the cylindrical sets, that is containing Bcyl (Z ), is nothing but the Borel σ-algebra, B(Z ), associated with the norm topology on Z . A probability measure well defined on B(Z ) will be shortly called a probability measure on Z . The tightness Prokhorov’s criterion (see [42]) has within this setting the next simple form. Lemma 6.1 (See [44]). A cylindrical measure μ on Z extends to a probability measure on Z if and only if for any η > 0 there exists Rη > 0 such that ∀p ∈ P,

μ ({z ∈ Z , |pz| ≤ Rη }) ≥ 1 − η .

By recalling that for any R > 0 the ball {z ∈ Z : |z| ≤ R} is weakly compact, this can be reinterpreted by saying that a weak distribution μ extends as a Borel probability measure if and only if its outer extension is a Radon measure on Z endowed with the weak topology (see [42]). Consider a family (ρε )ε∈(0,¯ε) of non negative trace class operators on H such that Tr[ρε ] = 1, or equivalently normal states O → Tr[ρε O] on the space of all bounded operators L (H ) . An additional number estimate assumption allows to associate with such a family, Wigner probability measures on Z . Theorem 6.2. Let (ε )ε∈(0,¯ε) be a family of normal states on L (H ) parametrized by ε. Assume Tr[N δ ρε ] ≤ Cδ uniformly w.r.t. ε ∈ (0, ε) for some fixed δ > 0 and Cδ ∈ (0, +∞). Then for every sequence (εn )n∈N with limn→∞ εn = 0 the exists a subsequence (εnk )k∈N and a Borel probability measure μ on Z such that εnk W eyl εnk A−W ick lim Tr[ρ b ] = lim Tr[ρ b ]= b(z) dμ(z) , k→∞

k→∞

Z

−1

for all b ∈ ∪p∈P F (Mb (pZ )). @ Moreover this probability measure μ satisfies Z |z|2δ dμ(z) < ∞. Remark 6.3. a) By introducing the reduced density matrix εp ∈ L 1 (Γs (pZ )) defined for p ∈ P as a partially traced operator Tr[εp A] = Tr[ε (A ⊗ IΓs (p⊥ Z ) )], one could consider the Husimi function μεp of εp which is its finite dimensional

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Wick symbol. It is known that this makes a weak probability distribution which admits weak limits after extracting subsequences εnk → ∞. The number estimate implies in finite dimension that such a limit is a probability measure. Our results say essentially two things: First after a proper extraction of subsequences, the family (μp )p∈P makes a weak distribution, i.e. the convergence can hold simultaneously for all the non countable family p ∈ P. Secondly the weak distribution is a Borel probability measure. @ 2δ b) The estimate Z |z| dμ(z) < +∞ will be proved in the more precise form 2 δ 1 + |z| dμ(z) ≤ lim inf Tr εnk (1 + N )δ ≤ Cδ < +∞ . Z

εnk →∞

Contrary to the finite dimensional case, the first inequality is not an equality even when the right-hand side converges. Examples are given in Section 7.4. c) For a non negative trace-class operator , the assumption , . C ≥ Tr[N δ ] = sup Tr[A] = sup Tr N δ/2 1[0,k] (N )1[0,k] (N )N δ/2 k∈N A ∈ L (H ) δ 0≤A≤N , . = sup Tr 1/2 1[0,k] (N )N δ 1/2 k∈N

implies (1 + N )δ/2 (1 + N )δ/2 ∈ L 1 (H ) with a norm estimate. Reciprocally, assuming (1+N )δ/2 (1+N )δ/2 L 1 ≤ C implies that the quantity Tr[N δ ] defined as the above supremum is bounded by C. Such an equivalence is no more true when ≥ 0 is not assumed and the second version has to be considered (see Proposition 6.4). Proof. i) The Proposition 3.7 implies ε→0 Tr ε bW eyl − Tr ε bA−W ick ≤ bW eyl − bA−W ick → 0 , for fixed b ∈ ∪p∈P F −1 (Mb (pZ )). Hence the result is true when it is proved after considering simply the Anti-Wick observables. ii) Consider for ε > 0 the function , . , επ2 2 . √ Gε (ξ) = Tr ε W ( 2πξ) e− 2 |ξ| = Tr ε (e2iπS(ξ,.) )A−W ick . The positive type property and the normalization come from Gε (0) = Tr [ε ] = 1 ⎤ ⎡ ⎛ ⎞ 2 A−W ick N N ⎥ ⎢ λi λj Gε (ξi − ξj ) = Tr ⎣ε ⎝ λk e2iπS(ξk ,.) ⎠ ⎦ ≥ 0. i,j=1

k=1

The continuity when ξ is restricted to any fixed finite dimensional pZ can be written with uniform estimates w.r.t ε ∈ (0, ε¯). Consider the estimate Tr ε (1 + N )δ1

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≤ Cδ1 with δ1 ∈ (0, min(1, 2δ)). Write for any ξ, η ∈ Z √ (N + 1)δ1 /2 (N + 1)δ1 /2 √ W ( 2πη) − W ( 2πξ) |Gε (η) − Gε (ξ)| = Tr ρε (N + 1)δ1 /2 (N + 1)δ1 /2 επ 2

επ 2

+ e− 2 |η| − e− 2 |ξ| √ √ ≤ W ( 2πη) − W ( 2πξ) (N + 1)−δ1 /2 2

2

− επ2 |η|2

+ e

2

2

− επ2 |ξ|2

−e

L (H )

Tr (N + 1)δ1 ρε

.

We have found by Lemma 3.1 two constants δ1 ∈ (0, 1) and Cδ 1 > 0 such that ∀ξ, η ∈ Z , |Gε (η) − Gε (ξ)| ≤ Cδ 1 |η − ξ|δ1 (|η|2 + |ξ|2 )δ1 /2 + 1 , (57) holds uniformly w.r.t. ε ∈ (0, ε) and we recall the uniform estimate |Gε (ξ)| ≤ 1. Hence for any ε ∈ (0, ε), Gε is the Fourier transform of a weak distribution με such that ε A−W ick = Tr b b(z) dμε (z) Z

holds for all b ∈ ∪p∈P F −1 (Mb (pZ )). iii) Actually the uniform estimate (57) allows to apply an Ascoli type argument after considering sequence (εn )n∈N such that limn→∞ εn = 0: • Since Z is separable, it admits a countable dense set N = {ξ , ∈ N}. For any ∈ N the sequence Gεn (ξ ) remains in {σ ∈ C, |σ| ≤ 1}. Hence by a diagonal extraction process there exists a subsequence (εnk )k∈N such that for all ∈ N, Gεnk (ξ ) converges in {σ ∈ C, |σ| ≤ 1} as k → ∞. Set G(ξ ) = lim Gεnk (ξ ) k→∞

for all ∈ N. • The uniform estimate (57) implies that the limit G is uniformly continuous on any set N ∩ {z ∈ Z : |z| ≤ R}. Hence it admits a continuous extension still denoted G in (Z , | |Z ). An “epsilon/3”-argument shows that for any ξ ∈ Z limk→∞ Gεnk (ξ) exists and equals G(ξ). • Finally G is a normalized function of positive type as a limit of such functions. Finally the uniform estimates |Gε (ξ)| ≤ 1 and |G(ξ)| ≤ 1 allow to test the convergence against any ν ∈ Mb (pZ ) and to apply the Parseval identity with b = F −1 (ν). From any sequence (εn )n∈N such that limn→∞ εn = 0, one can extract a subsequence (εnk )k→∞ and find a weak distribution such that the limit lim Tr εnk bW eyl = lim Tr εnk bA−W ick = b(z) dμ(z) nk →∞

nk →∞

Z

holds for any b ∈ F L1 (pZ , Lp (dz)) and therefore for any b ∈ Scyl (Z ). iv) The Prokhorov’s criterion for μ in the form in Lemma 6.1 is again stated a consequence of the uniform number estimate Tr N δ ε ≤ Cδ . Fix any p ∈ P and set d = dimp. The operators Np = NpZ ⊗ IΓs (p⊥ Z ) = dΓ(IpZ ) ⊗ IΓs (p⊥ Z ) =

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dΓ(p), Np⊥ = IpZ ⊗ dΓ(Ip⊥ Z ) = dΓ(p⊥ ) and N = dΓ(I) make a commuting family of non negative operators such that N = Np + Np⊥ . Thus the inequality 6s 5 5 6s dε dε +N + Np ≥ 1+ 1+ 2 2 ε δ holds for any s ≥ 0. Hence the estimate Tr N ≤ Cδ implies 5 5 6δ 6δ dε dε ε ε Tr 1 + + Np +N ≤ Tr 1 + ≤ Tr ε (2 + N )δ ≤ Cδ , 2 2 with Cδ > 0 independent of ε and p as soon as ε ≤ d1 . Let χ ∈ C ∞ (pZ ) be a non negative function on pZ , such that χ ≡ 0 in a neighborhood of {|z| ≤ 1}. For any R ≥ 1 the estimates (1 + R2 )δ χ(R−1 z) ≤ 1 (1 + |z|2 )δ holds with uniform estimates of the left-hand side in SpZ (1, |dz| ). The pseudodifz 2 2

ferential calculus in pZ with the metric |dz| z 2 , provides the inequality of bounded operators on Γs (pZ ) W eyl (1 + R2 )δ −1 (1 + R2 )δ A ◦ BR ◦ A − Cε ≤ χ(R z) ≤ 1 + Cε (1 + |z|2 )δ ,W eyl . W eyl with A = (1+|z|2 )−δ/2 , BR = χ(R−1 z) and |BR |L (Γs (pZ )) ≤ C , 2

with a constant C > 0 independent of ε ∈ (0, d1 ) and R ≥ 1. By Proposition 3.3, there exists a constant C > 0 independent of ε ∈ (0, d1 ) (and R ≥ 1) such that 5 6δ dε A2 ◦ 1 + + NpZ − IΓs (pZ ) ≤ C ε . 2 L (Γs (pZ ))

Hence the inequality (1 + R2 )δ χ(R−1 pz)W eyl ≤ (1 + 2Cε)A−δ after tensorization with IΓs (p⊥ Z ) and testing on the normal state ε yields (1 + R2 )δ Tr ε χ(R−1 pz)W eyl ≤ Cδ

with a uniform constant Cδ

with respect to ε ∈ (0, d1 ) and R ≥ 1. After taking the limit nk → ∞, εnk → 0, we get 1{|pz|≥R} (z) dμ(z) ≤ χ(R−1 pz) dμ(z) = lim Tr εnk χ(R−1 pz)W eyl Z

≤

Z Cδ

(1

nk →∞

2 −δ

+R )

.

This inequality is valid for any p ∈ P and the Prokhorov’s criterion of Lemma 6.1 is satisfied. The weak distribution μ is a probability measure on Z .

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2δ

v) First the function z is Borel measurable in Z . Take p ∈ P and R ≥ 1 and take now χ0 ∈ C0∞ (pZ ), such that 0 ≤ χ0 ≤ 1 and χ0 ≡ 1 in a neighborhood of 0. Consider the estimates (1 + N )δ ≥ (1 + Np )δ ≥ (1 + Np )δ/2 χ0 (R−1 pz)W eyl (1 + Np )δ/2 − Cp ε(1 + Np )δ " W eyl #δ 2 −1 ≥ (1 + |pz| ) χ0 (R pz) − Cp ε(1 + N )δ where the two last inequalities are again derived from the finite dimensional Weyl calculus (with a uniform control w.r.t. R ≥ 1). After taking the limit nk → ∞, εnk → 0, this implies W eyl " #δ 2 δ 2 −1 εnk −1 (1 + |pz| ) χ0 (R pz) 1 + |pz| χ0 (R pz) dμ(z) = lim Tr nk →∞

Z

≤ lim inf Tr εnk (1 + N )δ ≤ Cδ . nk →∞

Taking the supremum w.r.t R ≥ 1 and then w.r.t a countable increasing sequence (pn )n∈N , pn ∈ P, such that supn∈N pn = IZ , yields (1 + |z|2 )δ dμ(z) ≤ Cδ < +∞ . Z

6.2. Complex Wigner measures, pure sequences More general families of trace class operators can be considered by linear decomposition ε = λεR+ εR+ − λεR− εR− + iλεI+ εI+ − iλεI− εI− , (58) with 1 1 λεR− εR− = |ε +(ε )∗ |−ε −(ε )∗ λεR+ εR+ = |ε +(ε )∗ |+ε +(ε )∗ , 4 4 1 ε 1 ε ε ε ∗ ε ε ∗ ε ε λI+ I+ = | −( ) |−i +i( ) , λI− I− = |ρε −(ε )∗ |+iε −i(ε )∗ , 4 4 such that λε• ≥ 0, ε• ≥ 0, Tr [ε• ] = 1 and λεR+ + λεR− + λεI+ + λεI− ≤ 4 |ε |L 1 (H ) . Proposition 6.4. Let (ε )ε∈(0,ε) be a family of trace class operators such that (1 + N )δ/2 ε (1 + N )δ/2

L 1 (H )

≤ Cδ

(59)

uniformly for some δ > 0 and some Cδ < +∞. Then for any sequence (εn )n∈N such that limn→∞ εn = 0, one can extract a subsequence (εnk )k∈N and find a (complex) Borel measure μ on Z such that εnk W eyl εnk A−W ick ] = lim Tr[ρ b ]= b(z) dμ(z) , (60) lim Tr[ρ b k→∞

k→∞

for all b ∈ ∪p∈P F −1 (Mb (pZ )). @ δ This measure satisfies Z z d |μ| (z) < +∞ .

Z

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After assuming additionally the stronger uniform estimate Tr (1 + N )δ |ε + (ε )∗ | + Tr (1 + N )δ |ε − (ε )∗ | ≤ Cδ , @ 2δ this measure satisfies Z z d |μ| (z) < +∞ .

(61)

Proof. Owing to the estimate λεR+ + λεR− + λεI+ + λεI− ≤ 4 |ε |L 1 (H ) ≤ 4Cδ , with all the λε• non negative, the extraction of a subsequence allows to reduce the analysis to the case when all the λε•n converge: limn→∞ λε•n = λ0• ∈ [0, +∞) . n→∞ → 0 and If one λ0• equals 0 then |λε•n ε•n |L 1 (H ) = λε•n W eyl ε ε limεn →0 Tr b (λ•n •n ) = 0 , for all b ∈ Scyl (Z ), and the corresponding term does not contribute to the asymptotic measure μ. Hence the problem is now reduced to the case when all the λ0• are positive, and therefore for N0 > 0 large enough, all the (λε•n )n>N0 are uniformly positive. Set in this case 1 0 n εn εn εn c = min λεR+ , λR− , λI+ , λI− ; n > N0 > 0 for N0 > 0 large enough. The decomposition (58) implies ε ε ε − λεR− rR−,δ + iλεI+ rI+,δ − iλI− rI−,δ (1 + N )δ/4 ε (1 + N )δ/4 = λεR+ rR+,ε

with

ε r•,δ = (1 + N )δ/4 ε• (1 + N )δ/4 ≥ 0 .

εn All the terms r•,δ are estimated in the same way as follows. For k ∈ N, consider the quantity: εn Tr 1[0,k] (N ) rR+,δ 1[0,k] (N ) . , 1 = εn Tr |εn + (εn )∗ | + εn + (εn )∗ (1 + N )δ/2 1[0,k] (N ) 4λR+ 1 |εn + (εn )∗ |(1 + N )δ/2 1[0,k] (N ) 1 ≤ 4c L (H ) 1 (1 + N )δ/2 εn (1 + N )δ/2 + . 2c L 1 (H )

The polar decomposition εn + (εn )∗ = Uεn |εn + (εn )∗ | provides the inequality |εn + (εn )∗ |(1 + N )δ/2 1[0,k] (N ) ≤2

Uε∗n (1

L 1 (H )

−δ/2

+ N)

L (H )

(1 + N )δ/2 εn (1 + N )δ/2

L 1 (H )

Therefore, this yields . , 1 . , εn εn (1+N )δ/2 εn (1+N )δ/2 1[0,k] (N ) ≤ Tr r•,δ = sup Tr 1[0,k] (N ) r•,δ c k∈N

.

L 1 (H )

.

Hence the four families of normal state (ε•n )n>N0 fulfill the assumptions of Theorem 6.2, with δ replaced with δ/2 and in the symmetric writing of Remark 6.3 c).

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Hence four Borel probability measures, μR+ , μR− , μI+ and μI− exist and a subsequence (εnk )k∈N can be extracted so that , . W eyl εnk = • b dμ• , lim Tr b with the estimates

k→∞

@ Z

μ=

z

Z

2(δ/2)

λ0R+ μR+

dμ• < +∞. We conclude by taking

+ λ0R− μR− + iλ0I+ μI+ − iλ0I− μI− .

Finally the last statement with the exponent 2δ comes from the operator inequalities 1 n (1 + N )δ/2 εR± (1 + N )δ/2 ≤ (1 + N )δ/2 |εn + (εn )∗ | (1 + N )δ/2 , and 2c 1 n (1 + N )δ/2 εI± (1 + N )δ/2 ≤ (1 + N )δ/2 |εn − (εn )∗ | (1 + N )δ/2 , 2c while considering the case when all the λ0• are positive. Definition 6.5. For a family (ε )ε∈(0,ε) , satisfying (59), the set of Borel measures μ which satisfy (60) is denoted M (ε , ε ∈ (0, ε)) or simply M (ε ). Such a family (ε )ε∈(0,ε) (resp. a sequence (εn )n∈N ) is said pure if M (ε , ε ∈ (0, ε)) (resp. M (εn , n ∈ N)) has a single element μ. When the family (ε )ε∈(0,ε) is pure the limit in (60) can be written with limε→0 instead of limnk →∞ . This provides a characterization of M (ε ) = {μ}. For simplicity, we shall often assume that the family (ε )ε∈(0,ε) is pure, when the reduction to such a case can be done after extracting a suitable sequence. 6.3. Countably separating sets of observables In order to identify a Wigner measure of μ ∈ M (ε ) it is sufficient to test on a “dense set” of observables. The good notion is given by the Stone–Weierstrass theorem for L1 spaces. It can be recovered from the standard Stone–Weierstrass theorem for continuous functions in our case. Lemma 6.6 (cf [14]). Let ν be a Borel probability measure on a separable Banach space X and let {fn , n ∈ N} be a countable set of bounded ν-measurable functions which separates the points ∀x, y ∈ X ,

∃n ∈ N ,

fn (x) = fn (y) .

Then for any p ∈ [0, ∞), the algebra generated by {fn , n ∈ N} is dense in Lp (X, dν). Since “the” Wigner measure is not known a priori, the good notion of “dense set” that we shall use is the following. Definition 6.7. A subset D ⊂ ∪p∈P F −1 (Mb (pZ )) is said countably separating whenever it contains a countable subset, D ⊃ D0 ∼ N, which separates the point of Z : ∀x, y ∈ Z , ∃f ∈ D0 , f (x) = f (y) .

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Proposition 6.8. Let μ1 be a bounded Borel measure on Z and let (ε )ε∈(0,ε) be a family of operators which fulfills the assumptions of Definition 6.5. The two next statements are equivalent: 1. M (ε ) = {μ1 }. 2. There exists a countably separating subset D ⊂ ∪p∈P F −1 (Mb (pZ )) such that ∀b ∈ D , lim Tr ε bW eyl = lim Tr ε bA−W ick = b(z) dμ1 (z) . ε→0

ε→0

Z

Remark 6.9. A similar equivalence is obtained for μ1 ∈ M ( ) after a subsequence extraction. ε

Proof. Assume μ ∈ M (ε ). There exists a sequence (εnk )k∈N and a Borel measure μ such that (60) holds for any b ∈ ∪p∈P F −1 Mb (pZ ). In particular this holds for any b ∈ D: b(z) dμ(z) = lim Tr εnk bW eyl = b(z) dμ1 (z) . k→∞

Z

Z

The set D is dense in L (Z , d|μ1 |) and in L1 (Z , d|μ|) so that the above equality of the extreme sides extend to any bounded Borel function. This implies μ = μ1 . 1

The next examples will be useful in the application and allow to reconsider an inductive point of view. Proposition 6.10. Let (p )∈N be an increasing sequence of projectors in P such that sup p = IZ and let the family of operators (ε )ε∈(0,ε) satisfy the assumptions of Definition 6.5. Then the identity M (ε ) = {μ} is equivalent to any of the next statement @ 1. For all b ∈ ∪∈N S (p Z ), the quantity Tr[ε bW eyl ] converges to Z b(z) dμ(z) as ε → 0. @ 2. For all b ∈ Scyl (Z ), the quantity Tr[ε bW eyl ] converges to Z b(z) dμ(z) as ε → 0. Proof. It suffices to notice that ∪∈N S (p Z ), and therefore Scyl (Z ), is countably separating because the weak topology separates the points. 6.4. Orthogonality argument Complex Wigner measures are especially interesting while considering the joint measure associated with two families of vectors (uε )ε∈(0,ε) and (v ε )ε∈(0,ε) . Introduce the notation εuv = |uε v ε | . Proposition 6.11. Assume that the family of vectors (uε )ε∈(0,ε) and (v ε )ε∈(0,ε) satisfy the uniform estimates (1 + N )δ/2 uε

H

+ (1 + N )δ/2 v ε

H

≤C,

|uε |H = |v ε |H = 1

1554



for some fixed δ > 0 and C > 0. Assume further that any μ ∈ M (εuu ) and any ν ∈ M (εvv ) are mutually orthogonal. Then the family (εuv )ε∈(0,ε) is pure with M εuv , ε ∈ (0, ε) = {0} i.e. lim uε , bW eyl v ε = lim uε , bA−W ick v ε = 0 ε→0

for any b ∈ F

−1

ε→0

(Mb (pZ )) and any p ∈ P.

Proof. Assume M (uu ) = {μ} and M (εvv ) = {ν} with μ ⊥ ν. Take η > 0. There exist two bounded closed subsets K1 and K2 such that μ(K1 ) ≥ 1 − η ,

ν(K2 ) ≥ 1 − η ,

K1 ∩ K2 = ∅ .

Since K1 and K2 are compact in the weak topology, K1 ⊂ K2 , K2 open in the weak topology, there exists a finite covering of K1 of the form 1 1 K 0 K 0 ∪ |pk (z − zk )| ≤ 2rk ∩ K2 = ∅ K1 ⊂ ∪ |pk (z − zk )| ≤ rk , k=1

k=1

with pk ∈ P, zk ∈ Z and rk > 0 for all k ∈ {1, . . . , K}. By choosing for any k a function χk ∈ C0∞ (pk Z ) such that χk (pk (z)) ≡ 1 when |pk (z − zk )| ≤ rk and /N χk (pk z) = 0 when |pk (z − zk )| ≥ 2rk the sum χ(z) = k=1 / χ kχ(pk(pz) z) defines a k k k cylindrical function χ ∈ Scyl (Z ) such that χ ≡ 1 on K1 and χ ≡ 0 on K2 . Take now any b ∈ Scyl (Z ) and write & ε W eyl ε W eyl ε ' u ,b = uε , (bχ)W eyl v ε + uε , b(1 − χ) v v W eyl ε ≤ b(1 − χ) u + (bχ)W eyl v ε H . H

From the Weyl pseudodifferential calculus we get . W eyl , W eyl ε 2 + Cbχ b(1 − χ) u ≤ Tr εuu (1 − χ)2 |b|2 H @ where the right-hand side converges to Z |b|2 (1 − χ)2 (z) dμ(z) as ε → 0. The property χ ≡ 1 on K1 with μ(K1 ) ≥ 1 − η implies W eyl ε 2 2 lim sup b(1 − χ) u ≤ η |b|L∞ H

ε→0

2

2

and with the symmetric argument lim supε→0 (bχ)W eyl v ε H ≤ η |b|L∞ . Hence we get √ ∀η > 0 , lim sup uε , bW eyl v ε ≤ 2 |b|L∞ η ε→0

for any b ∈ Scyl (Z ). This implies M (εuv , ε ∈ (0, ε)) = {0} .

A straightforward consequence is the next proposition. Proposition 6.12. Make the same assumptions as in Proposition 6.11 with the additional condition M (εuu ) = {μu } and M (εvv ) = {μv }. Then the family of trace class operators (εu+v,u+v )ε∈(0,ε) satisfies M (εu+v,u+v ) = {μu + μv } .

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Proof. Write simply ε u + v ε , bW eyl (uε + v ε ) = uε , bW eyl uε + v ε , bW eyl v ε + uε , bW eyl v ε + v ε , bW eyl uε , and take the limit of every term as ε → 0.

6.5. Wigner measure and Wick observables Up to some additional assumption on the state and by restricting the class of Wick observables, we check in this subsection that testing with Weyl, (or Anti-Wick) and Wick observables provides the same asymptotic information as ε → 0. 2 Fix once and for all p ∈ P, the choice of the metric gp = |dz|2 or gp = |dz| z 2 . From Proposition 3.8 we know that the class of symbols ∪p∈P,s∈R SpZ (zs , gp ) and ⊕alg m,q∈N Pm,q (Z ) both contain all the classes Pm,q (pZ ), with a good comparison of Weyl and Wick quantizations on these smaller sets. In the limit ε → 0, this ∞ comparison can be carried out to any b ∈ ⊕alg m,q∈N Pm,q (Z ). Theorem 6.13. Assume that the family of operators (ε )ε∈(0,ε) satisfies (1 + N )δ/2 ε (1 + N )δ/2

L 1 (H )

≤ Cδ

uniformly w.r.t ε ∈ (0, ε) for any δ > 0. s

1. For any fixed β ∈ ∪p∈P,s∈R SpZ (z , gp ), the families (β W eyl ε )ε∈(0,ε) and (β A−W ick ε )ε∈(0,ε) satisfy the assumptions of Definition 6.5 and 0 1 M (β W eyl ε ) = M (β A−W ick ε ) = βμ , μ ∈ M (ε ) (62) ∞ W ick ε )ε∈(0,ε) satisfies the 2. For any fixed β ∈ ⊕alg m,q∈N Pm,q (Z ) the family (β assumptions of Definition 6.5 and 0 1 (63) M (β W ick ε ) = βμ , μ ∈ M (ε ) .

A particular case holds when the measure is tested with b = 1. Corollary 6.14. Assume the uniform estimate (1 + N )δ/2 ε (1 + N )δ/2 Cδ for all δ > 0 and further M (ε ) = {μ}. 1. The equality lim Tr β W eyl ε = lim Tr β A−W ick ε =

ε→0

ε→0

∞ holds for any β ∈ ⊕alg m,q∈N Pm,q (Z ).

β(z) dμ(z) Z

holds when β ∈ ∪p∈P,s∈R SpZ (zs , gp ) 2. The limit lim Tr β W ick ε = ε→0

Z

β(z) dμ(z)

L 1 (H )

≤

1556



Proof of Theorem 6.13. 1) The relation (29) extends to any b ∈ SpZ (zs , gp ) and implies ε−1 (bW eyl − bA−W ick ) = c(ε)W eyl with c(ε) uniformly bounded in SpZ (zs−2 , gp ). The result for β A−W ick can be deduced from the one for β W eyl . Take p ∈ P, s ≥ 0 (this contains the case s < 0) and β ∈ SpZ (zs , gp ). Let Np = NpZ ⊗ IΓs (p⊥ Z ) and Np⊥ = IΓs (pZ ) ⊗ Np⊥ Z . Our assumption on (ε )ε∈(0,ε) and the commutations [Np⊥ , Np ] = [Np⊥ , β W eyl ] = 0 imply for any δ > 0 (1 + N )δ/2 β W eyl ε (1 + N )δ/2 = ABA RC δ/2

A = (1 + N )

with (1 + Np )−δ/2 (1 + Np⊥ )−δ/2

B = (1 + Np )δ/2 β W eyl (1 + Np )−δ/2−s A = (1 + Np )δ/2+s (1 + Np⊥ )δ/2 (1 + N )−δ−s R = (1 + N )δ+s ε (1 + N )δ+s −δ/2−s

C = (1 + N )

and

.

The factors A, A and C are uniformly bounded operators when δ > 0 (and s) is fixed. The trace class norm of the factor R is uniformly bounded by Cδ+s . Finally the Weyl pseudodifferential calculus on pZ implies that B = γ W eyl with γ(ε)

uniformly w.r.t uniformly bounded in SpZ (1, gp ) and therefore |B|L (H ) ≤ Cδ,s ε ∈ (0, ε). Hence the family (β W eyl ε )ε∈(0,ε) satisfies the assumptions of Definition 6.5. Let μ1 belong to M (β W eyl ε ). After extracting the proper sequence (εn )n∈N such that limn→∞ εn = 0, one can assume W eyl W eyl εn = lim Tr b β b(z) dμ1 (z) and n→∞ Z b(z) dμ(z) lim Tr bW eyl εn = n→∞

Z

for any b ∈ Scyl (Z ). But the finite dimensional pseudodifferential calculus implies bW eyl β W eyl = (bβ)W eyl + OL (H ) (εn ) with bβ ∈ Scyl (Z ). This implies b(z) dμ1 (z) = b(z)β(z) dμ(z) Z

Z

for all b ∈ Scyl (Z ). According to Proposition 6.10 this implies μ1 = βμ. 2) Since the ∪p∈P,s∈R SpZ (zs , gp ) contains ∪p∈P ⊕alg m,q∈N Pm,q (pZ ) , the ∞ (Z ) such that ˜b = Γ(p)˜bΓ(p) result is proved for any polynomial symbol b ∈ Pm,q ∞ for some finite dimensional projector p ∈ P. Consider now a general b ∈ Pm,q (Z ) with m, q ∈ N. By Lemma 2.5, the operator (1 + N )δ/2 bW ick (1 + N )−δ/2−m/2−q/2 δ+m+q

is uniformly bounded for any δ > 0. Since the trace class norm of (1+N ) 2 ε (1+ δ+m+q is uniformly bounded w.r.t ε ∈ (0, ε), the family (β W ick ε ) satisfies the N) 2

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assumptions of Definition 6.5. Introduce now an increasing sequence (p )∈N of P such that sup∈N p = I and consider for ∈ N β (z) = β(p z) ,

˜ ⊗m . β˜ = p⊗q ◦ β ◦ p

˜ ˜ Since β˜ is a compact operator, q the finite rank operator β converges to β in the m Z, Z ). The uniform estimates norm topology in L ( (β − β )W ick (1 + N )−m/2−q/2

L (H )

≤ C β˜ − ˜

2 m/2+q/2 1 + |z| |β(z)| + |β (z)| ≤ C

with

and the convergence ∀b ∈ Scyl (Z ) ,

lim Tr bW eyl βW ick εn =

n→∞

L(

m

Z,

q

Z)

,

lim β (z) = β(z) ,

→∞

Z

b(z)β (z) dμ(z)

@ after extracting a sequence (εn )n∈N , limn→∞ εn = 0, with Z (1+|z|2 )m/2+q/2 dμ(z) < +∞, lead to ∀b ∈ Scyl (Z ) , lim Tr bW eyl β W ick εn = b(z)β(z) dμ(z) . n→∞

Z

The previous results provide the behaviour of limε→0 Tr β W ick ε for β ∈ ∞ ε ⊕alg m,q∈N Pm,q (Z ) when M ( ) = {μ}. The next result checks the other way. Proposition 6.15. Assume that (ρε )ε∈(0,¯ε) is a family of normal states satisfying for any C > 0 there exist KC > 0 such that ∞ Ck Tr[N k ρε ] ≤ KC < ∞ [k/2]!

k=0

holds uniformly w.r.t ε ∈ (0, ε). Assume that there exists a Borel measure μ such that lim Tr bW ick ε = b(z) dμ(z) ε→0

holds for any b ∈

∞ ⊕alg m,q Pm,q (Z

Z

). This implies M (ε ) = {μ} .

Remark 6.16. A similar result for non self-adjoint trace-class operators with complex valued measure can be obtained by replacing the quantities Tr[N k ε ] with Tr[N k |ε + (ε )∗ |] + Tr[N k |ε − (ε )∗ |] like in (61). Proof. It is enough to prove the following statement: √ ε lim Tr W (ξ)ρ = e 2iS(ξ,z) dμ . ε→0

Z

1558



It is done when the right-hand side of ⎡ W ick ⎤ √ ∞ √ n | εξ| 2S(ξ, z) i √ Tr ⎣ hn ρε ⎦ Tr W (ξ)ρε = n n! 2 | εξ| n=0

(64)

is proved to be an absolutely convergent series, uniformly w.r.t. ε ∈ (0, ε¯). With Tr W (ξ)ρε = lim Tr W (ξ)1[0,M ] (N ) ρε M →∞ ⎡ ⎤ W ick √ ∞ √ | εξ|n 2S(ξ, z) i √ Tr ⎣ hn 1[0,M ] (N ) ρε ⎦ (65) = lim n n! M →∞ 2 | εξ| n=0 and ⎡

⎤ W ick √ 2S(ξ, z) i √ Tr ⎣hn 1[0,M ] (N ) ρε ⎦ | εξ| ≤ Mn

W ick √ 2S(ξ, z) i √ (N + 1)−n/2 hn (N + 1)−n/2 | εξ|

, L (H )

with Mn = Tr [(1 + N )n ε ], Lemma 3.4 implies −n/2

(N + 1)

W ick √ i 2S(ξ, z) √ hn (N + 1)−n/2 | εξ|

(1 + 2 2(k + j)ε)n n! ≤ sup n/2 (jε + 1)n/2 [n/2]! (kε + 1) k,j∈N n! . ≤ 8n [n/2]!

L (H )

This leads to ⎡ ⎤ W ick √ √ ∞ √ ∞ | εξ|n 2S(ξ, z) (4 ε|ξ|)n i ε √ Tr ⎣hn Mn 1[0,M ] (N ) ρ ⎦ ≤ n 2 n! [n/2]! | εξ| n=0 n=0 0. Hence we can take the limit M → ∞ inside in all the terms of (65). This leads to (64) with a uniformly absolutely convergent series in the right-hand side according to (66) and our initial assumption.

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Thus the sum and the limit as ε → 0 can be interchanged in (64): ⎡ W ick ⎤ √ ∞ √ |ξ|n 2S(ξ, z) i ε √ lim Tr ⎣ εn hn lim Tr W (ξ)ρ = ρε ⎦ n n! ε→0 ε→0 2 | εξ| n=0 ∞ √ n 1 i 2S(ξ, z) dμ n! Z n=0 √ = e 2iS(ξ,z) dμ . =

Z

The last equality follows owing to the dominated convergence theorem and ∞ δk ε δ|1pZ z|2 Tr ρ dΓ(1pZ )k < ∞ , e dμ = lim ε→0 k! Z k=0

for any δ > 0 and any p ∈ P. This completes the proof.

7. Examples and applications of Wigner measures 7.1. Finite dimensional cases The first examples are given by Theorem 4.2 1. For any z ∈ Z the family of operators ε = |E(z)E(z)| has a unique Wigner measure M |E(z)E(z)| , ε ∈ (0, ε) = {δz } . 2. For any z ∈ Z and any m ∈ Z the family of operators ε = |z ⊗kε −m z ⊗kε | with |z| = 1 and limε→0 εkε = 1 has a unique Wigner measure 2π ⊗kε −m ⊗kε 1 M |z z | , ε ∈ (0, ε) = e−imθ δeiθ z dθ . 2π 0 3. In case 1) and 2) the convergence can be tested with Weyl, Anti-Wick or Wick observables according to Proposition 6.4 and Theorem 6.13. Beside the explicit calculation of Theorem 4.2 these results can be considered through an inductive approach since E(z) or z ⊗n lie in Γs (Cz). The natural extension comes from Proposition 6.10-1) with a proper choice of the first term in the increasing sequence (p )∈N . Proposition 7.1. Assume that the family (ε )ε∈(0,ε) satisfies the assumptions of Definition 6.5. Assume further that there exists a finite dimensional space p0 ∈ P such that ε = Γ(p0 )Γ(p0 ) = εp0 ⊗ |ΩΩ| for all ε ∈ (0, ε) with εp0 ∈ L 1 (Γs (p0 Z )). Then the Wigner measures of (ε )ε∈(0,ε) are given by C D , M (ε ) = μ1 ⊗ δ0,p⊥ 0 Z

μ1 ∈ M (εp0 ) .

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7.2. Superpositions Two kinds of superpositions can be considered: 1) convex or linear combination of trace class operators; 2) convex or linear combination of wave functions. The first one is the simplest. Proposition 7.2. 1. Let (M, π) be a probability space. Let (ε (m))ε∈(0,ε),m∈M be a family of operators such that (1 + N )δ/2 ε (m)(1 + N )δ/2

L 1 (H )

≤ Cδ (m)

for π-almost every m ∈ M with Cδ ∈ L1 (M, dπ) for some δ > 0. Assume ε further M @ ( ε(m), ε ∈ (0, ε)) = {μ(m)} for π-almost every m ∈ M , then the family ( M (m) dπ(m))ε∈∈(0,ε) satisfies the assumptions of Definition 6.5 and 5 6 E M ε (m) dπ(m) , ε ∈ (0, ε) = μ(m) dπ(m) . M

M

2. Any bounded Borel measure on Z can be achieved as a Wigner measure. @ Proof. 1) Set ε = M ε (m) dπ(m) and write (1 + N )δ/2 ε (1 + N )δ/2 ≤ Cδ (m) dπ(m) . L 1 (H )

M

Then apply Lebesgue’s convergence theorem to Tr bW eyl ε = Tr bW eyl ε (m) dπ(m) . M

2) After reducing the problem to the case when μ is a Borel probability measure on Z , apply 1) with M = Z , π = μ, m = z and ε (z) = |E(z)E(z)|. The second type of superposition requires an orthogonality property. It is given by Proposition 6.12. Here are a few examples /L 1. Take uε = E(z ) for = 1, . . . , L, with L ∈ N fixed, and set uε = L−1/2 =1 uε . When the z are distinct, the family (|uε uε |)ε∈(0,ε) has a unique Wigner measure F ! L ε ε −1 δz . M (|u u |) = L =1

z⊗kε

2. Take for any ∈ {1, . . . , L}, = with |z | = 1 and limε→0 εkε = 1. The family (|uε uε |)ε∈(0,ε) has a unique Wigner measure: ! F L 2π M (|uε uε |) = (2πL)−1 δeiθ z dθ . uε

=1

0

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⊗kε

√ 3. For z ∈ Z and uε = E(z)+|z with |z| = 1 and limε→0 εkε = 1, the family 2 (|uε uε |)ε∈(0,ε) has a unique Wigner measure:

E 2π 1 1 ε ε δz + M (|u u |) = δeiθ z dθ . 2 4π 0

4. All this examples can be tested with Weyl, Anti-Wick or Wick observables according to Proposition 6.4 and Theorem 6.13. 7.3. Propagation of chaos and propagation of (squeezed) coherent states Let us go back to the example of Section 5 where Uε (t) = e−i ε Hε with Hε = ˜ = 1 V (x1 − x2 ) and zt solution to i∂t zt = −Δz + (V ∗ |zt |2 )zt . dΓ(−Δ) + QW ick , Q 2 Theorem 5.6, Proposition 5.10 and Proposition 6.15 imply: 1. For any z0 ∈ Z with |z0 | = 1, the family (|Uε (t)z0⊗kε Uε (t)z0⊗kε |)ε∈(0,ε) with limε→0 εkε = 1 is pure with E # 1 2π " # " δeiθ zt dθ = M |zt⊗kε zt⊗kε | M |Uε (t)z0⊗kε Uε (t)z0⊗kε | = 2π 0 t

2. For any z0 ∈ Z , the family (|Uε (t)E(z0 )Uε (t)E(z0 )|)ε∈(0,ε) is pure with M |Uε (t)E(z0 )Uε (t)E(z0 )| = {δzt } = M |E(zt )E(zt )| . These results are derived from the results for product states after testing with Wick observable (any b ∈ ⊕alg m,q Pm,q (Z )) . Actually it is possible to recover the second one directly from the Hepp method. For any b ∈ Scyl (Z ), Proposition 5.2 implies lim Tr bW eyl |Uε (t)E(z0 )Uε (t)E(z0 )|

ε→0

√ − W

2 zt iε

U2 (t, 0)Ω

W

√

2 zt iε

U2 (t, 0)Ω

= 0.

By the finite dimensional Weyl quantization, the second term equals U2 (t, 0)Ω , b(. − zt )W eyl U2 (t, 0)Ω and it suffices to check that the family (|U2 (t, 0)ΩU2 (t, 0)Ω|)ε∈(0,ε) admits the unique Wigner measure δ0 . This is a consequence of Lemma 5.3 which first says |N k U2 (t, 0)Ω|H ≤ Ck for any k ≥ 0 and then limε→0 U2 (t, 0)Ω , bW ick U2 (t, 0)Ω = 0 when b(0) = 0 . 7.4. Dimensional defect of compactness In the last example the mean field propagation of Wigner measure attached with Uε (t)E(z0 ) can be proved directly without using the result on Wick observables. As a corollary, this provides the result for Wick observables bW ick ∞ when b ∈ ⊕alg m,q Pm,q (Z ) according to Theorem 6.13. The result for a general alg b ∈ ⊕m,q Pm,q (Z ) is still true but comes from a direct proof or from Proposition 5.10.

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A natural question is whether the result of Theorem 6.13 can be extended to any observable bW ick with b ∈ ⊕alg m,q Pm,q (Z ). The answer is no, because in the infinite dimensional case there can be some defect of compactness w.r.t to the dimension variable. Here is a typical example. Consider a family (zε )ε∈(0,ε) such that zε converges weakly to 0. There exists a constant C > 0 such that |zε | ≤ C for all ε ∈ (0, ε) and the family (E(zε ))ε∈(0,ε) satisfies the assumptions of Proposition 6.15. The Wigner ∞ measures μ ∈ M (|E(zε E(zε )|)) are determined by testing on any b ∈ Pm,q (Z ). But Theorem 4.2 says E(zε ) , bW ick E(zε ) = b(zε ) = z ⊗q , ˜bz ⊗m . ε

ε

When m + q ≥ 1 the operator ˜b is compact, the right-hand side converges to 0 as ε → 0. According to Proposition 6.15 this implies M |E(zε )E(zε )| = {δ0 } . W ick Meanwhile testing with N = dΓ(I) = |z|2 implies 2 E(zε ) , N E(zε ) = |zε | where the right-hand side can reach any possible limit in [0, C]. 7.5. Bose–Einstein condensates The thermodynamic limit of the ideal Bose Gas presented within a local algebra presentation in [9] can be reconsidered by introducing a small parameter ε → 0. Namely, the large domain limit where bosonic particles are moving freely in a domain Λ, with volume |Λ| → ∞, can be formulated with |Λ| = 1ε and ε → 0. For a fixed particle density the total number of particle is O( 1ε ) coherent with a mean field approach. Before considering any dynamical problem, Wigner measures of εdependent Gibbs states bring some interesting presentation of the Bose–Einstein condensation. Consider the Laplace operator 0H0 = −Δx on 1the ε-dependent torus Rd /(ε−1/d Z)d with spectrum σ(H0 ) = ε2/d |2πn|, n ∈ Zd . The one particle space is Z ε = L2 (Rd /(ε−1/d Z)d ) and the bosonic Fock space is H ε = Γs (Z ε ). For the inverse temperature β = kB1T > 0 and a chemical potential μ, the Gibbs grand canonical equilibrium state is associated with the operator e−βdΓ(H0 −μI) = Γ(e−β(H0 −μI) ), which is trace class if and only if μ < 0 (see [9, Proposition 5.2.27]). This Gibbs state on L (H ε ) is given by 1 ωε (A) = Tr [ε A] with ε = Γ(e−β(H0 −μ) ) , μ < 0 . −β(H 0 −μ) )] Tr[Γ(e It is convenient to introduce the parameter z = eβμ and this Gibbs state restricted to the CCR-algebra (the C ∗ -algebra generated by the Weyl operators W1 (f ), f ∈ Z ε ) is the gauge-invariant quasi-free state given by the two-point function: ωε (a∗1 (f )a1 (g)) = g , ze−βH0 (1 − ze−βH0 )−1 f . The index 1 means that

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the CCR are written at this level in their initial form: [a1 (g), a∗1 (f )] = g , f . This is proved in [9, Proposition 5.2.28] with the straightforward rewriting . , ωε W1 (f ) = exp − f , (1 + ze−βH0 )(1 − ze−βH0 )−1 f /4 . The mean field analysis consists here in introducing a(f ) = ε1/2 a1 (f ) and W (f ) = W1 (ε1/2 f ): ωε a∗ (f )a(g) = ε g , ze−βH0 (1 − ze−βH0 )−1 f . , ωε W (f ) = exp − ε f , (1 + ze−βH0 )(1 − ze−βH0 )−1 f /4 . Further a rescaling motivated by the observation of the phenomena on a large scale, is implemented with f (x) = ε1/2 ϕ(ε1/d x) = Dε ϕ. After conjugating with the unitary transform Γ(Dε ) : H = Γs (Z ) → H ε = Γs (Z ε ), with Z = L2 (Rd /Zd ) we are led to consider the asymptotic behaviour as ε → 0 of the normal state ε = Γ(Dε )∗ ε Γ(Dε ) =

2/d 1 Γ(e−β(−ε Δ−μ) ) Tr[Γ(e−β(−ε2/d Δ−μ) )]

which satisfies . ε , 2/d 2/d Tr ε W (f ) = exp − f , (1 + zeβε Δ )(1 − zeβε Δ )−1 f Z 4 . ε , 2 2/d 2/d ε = e− 4 |f |Z exp − f , zeβε Δ (1 − zeβε Δ )−1 f Z 2 ε ∗ 2/d βε2/d Δ (1 − zeβε Δ )−1 f Z . Tr a (f )a(g) = ε g , ze The above expressions are explicit after the decomposition in the Fourier basis / f = n∈Zd fn e2iπn.z of any element f ∈ Z . For a given z < 1 and β > 0 the rescaled particle density is given by εz +ε 1−z

n∈Zd \{0}

ze−βε |2πn| εz + νε (β, z) . = 2/d |2πn|2 −βε 1−z (1 − ze ) 2/d

2

(67)

One checks easily for ε ≥ ε and z ≤ z < 1

ze−β|2πu| ν (β, z) ≤ νε (β, z) → ν0 (β, z) = du −β|2πu|2 Rd 1 − ze and ∀ε ∈ [0, 1) , νε (β, z) ≥ νε (β, z ) . ε

ε→0

2

Here comes the discussion about the Bose–Einstein condensation. In dimension d ≥ 3 (this restriction may change with an alternative Hamiltonian H0 = λ(Dx )), the quantity 2 e−β|2πu| du < +∞ . ν0 (β, 1) = −β|2πu|2 Rd 1 − e is well defined. We focus on the case d ≥ 3.

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The previous discussion imply ∀ε > 0 ,

∀z ∈ (0, 1) ,

νε (β, z) ≤ ν0 (β, 1)

while any total density can be achieved by (67). The Bose–Einstein condensation zε ε occurs while considering the limit ε → 0 with the constraint 1−z + νε (zε , β) = ν ε with β > 0 and ν > 0 fixed. There are two possible cases: εzε • ν ≤ ν0 (β, 1): Then limε→0 zε = z < 1 and limε→0 1−z = 0. ε εzε • ν > ν0 (β, 1): The inequality ν − ν0 (β, 1) ≤ 1−ε ≤ ν leads to zε = 1 − ε ν−ν0 (β,1) + o(ε) . The proportion 1 − ν0 (β, 1)/ν of the gas lies in the ground state n = 0 of the one-body Hamiltonian. This is the Bose–Einstein condensation phenomenon. It is interesting to reconsider this limit ε → 0 with β > 0 and ν > 0 fixed (d ≥ 3) within the Wigner measure point of view. This is possible owing to the explicit formula ⎡ ⎤ 2/d 2 , . −βε |2πn| √ 2 2 zε e 2 ⎦ , (68) |fn | Tr ε W ( 2πf ) = e−επ |f |Z exp ⎣−επ 2 (1 − zε e−βε2/d |2πn|2 ) n∈Zd / where f = n∈Zd fn e2iπn.x . Remember that the characteristic function of Wigner measures are determined after considering the limit ε → 0 of the above expression for any fixed f ∈ Z . Hence the problem is reduced to the application of Lebesgue’s theorem in the argument of the exponential. For any n = 0 the quantity

2/d

2

|2πn| zε e−βε 2/d |2πn|2 (1−zε e−βε )

converges to 0 as ε → 0 because

d/2 < 1 and zε ≤ 1. Hence we get . , √ επ 2 zε 2 lim Tr ε W ( 2πf ) = lim exp − |f0 | . ε→0 ε→0 1 − zε With the constraint

εzε 1−zε

≤ ν < +∞, there are two possibilities

• First = 0 implies ν ≤ ν0 (β, 1) and M (ε ) = {δ0 }. εzε • The second case limε→0 1−z = ν − ν0 (β, 1) > 0 implies ε . , √ 2 2 2 2 lim Tr ε W ( 2πf ) = e−π (ν−ν0 (β,1))|f0 | = e−π (ν−ν0 (β,1))|f,1 | . εzε limε→0 1−z ε

ε→0

Hence the Wigner measure of the family (ε )ε>0 equals γν ⊗ δ0 on Z = ⊥ C1 × {1} where γν is the gaussian measure −

|z1 |2

e ν−ν0 (β,1) γν (z1 ) = , (π(ν − ν0 (β, 1))d/2

z1 ∈ C .

Our scaled observables can measure asymptotically only the Bose–Einstein phase in a non trivial way. The rest of the state provides the factor δ0 . While testing with the observable (|z|2 )W ick = N , the dimensional defect of compactness phenomenon already illustrated in Subsection 7.4 occurs again: only the density of the condensate remains.

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Remark 7.3. i) It is possible to consider various dispersion relations H0 = λ(Dx ) and the discussion about the dimension may change. Other boundary conditions (here periodic boundary conditions are considered) and the discussion about the convergence of limε→0 zε = 1 may change a little bit. We refer the reader to [9] for the case of Dirichlet boundary conditions. ii) From (68) it is possible to consider the limit for any fixed f ∈ Z as ε → 0 with various behaviours of zε . This provides asymptotically a weak distribution. But the uniform tightness assumption Tr ε (1 + N )δ ≤ C is not satisfied. The scaling has to be adapted differently to the dimension d = 2 or d = 1 by taking care of the singularity at the momentum 0, in order to allow a non trivial Wigner measure in the thermodynamic and mean field limit. 7.6. Application 1: From the propagation of coherent states to the propagation of chaos via Wigner measures In the previous sections we showed how the propagation of (squeezed) coherent states can be derived from the propagation of Hermite states or directly via the Hepp method. The Hepp method is very flexible (see [24] for example) and therefore it is interesting to know whether a result for coherent states provides an information for product states or more general states. Here is a simple and abstract result which relies on some gauge invariance argument. Theorem 7.4. Let Uε be a unitary operator on H possibly depending on ε ∈ (0, ε) which commutes with the number operator [N, Uε ] = 0. Assume that for a given z ∈ Z such that |z| = 1, there exists zU ∈ Z such that M |Uε E(z)Uε E(z)| = {δzU } . @ 1 δ Then for any non negative @ function ϕ ∈ L (R, ds) such that R ϕ(s)(1 + |s|) ds < ∞ for some δ > 0 and R ϕ(s) ds = 1, the state εϕ =

∞

ε1/2 ϕ ε1/2 (n − ε−1 ) |Uε z ⊗n Uε z ⊗n |

n=0

satisfies the conditions of Definition 6.5 and 2π 1 δeiθ zU dθ . M εϕ = 2π 0 Proof. Owing to the relation Γ(e−iθ )bW eyl Γ(eiθ ) = e−iθN bW eyl eiθN = b(e−iθ .)W eyl . Our assumptions imply M Γ(eiθ )|Uε E(z)Uε E(z)|Γ(e−iθ ) = δeiθ zU

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for any θ ∈ R. The assumptions of Definition 6.5 are satisfied because Uε preserves the number. After taking the average w.r.t θ ∈ [0, 2π]: 2π 1 σε = Γ(eiθ )|Uε E(z)Uε E(z)|Γ(e−iθ ) dθ 2π 0 this implies

2π 1 M (σ ) = δeiθ zU dθ 2π 0 where the right-side is an extremal point of the convex set of Borel probability measure which are invariant after the natural action of S 1 on Z : S 1 × Z (γ, z) → γz ∈ Z . Again the commutation [Uε , N ] = 0 and the expression (4) for E(z) imply 2π σ ε = (2π)−1 Uε |Γ(eiθ )E(z)Γ(eiθ )E(z)|Uε∗ dθ ε

0

= (2π)−1

2π

Uε |E(eiθ z)E(eiθ z)|Uε∗ dθ

0

1 ∞ e− ε |Uε z ⊗n Uε z ⊗n | . = εn n! n=0

For any b ∈ Scyl (Z ), the quantity 1 ∞ e− ε Uε z ⊗n , bW eyl Uε z ⊗ = Tr bW eyl σ ε n n! ε n=0 @ 2π converges as ε → 0 to (2π)−1 0 b(eiθ zU ) dθ . By Lemma A.1 this implies 2 2π − s2 −1 e −1 ∀b ∈ Scyl (Z ) , lim a[ε−1/2 s+ε−1 ] (ε ) √ ds = (2π) b(eiθ zU ) dθ , ε→0 R 2π 0

where [t] is the integer part of t ∈ R and an (ε−1 ) = Uε z ⊗n , bW eyl Uε z ⊗n . s2

Call γ the Gaussian measure e− 2 √ds on R. For any finite subdivision I = 2π {I1 . . . , IL } of R = I1 " . . . " IL with intervals, the states −1 −1/2 −1/2 s+ε−1 ] s+ε−1 ] |Uε z ⊗[ε Uε z ⊗[ε | dγ(s) σIε = γ(I ) I

satisfy the assumptions of Definition 6.5 with the gauge invariance Γ(eiθ )σIε Γ(e−iθ ) = σIε . Moreover the state L −1/2 −1/2 ε s+ε−1 ] s+ε−1 ] |Uε z ⊗[ε Uε z ⊗[ε | dγ(s) = γ(I )σIε σ = R

=1

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1567

@ 2π is a finite barycenter of the σIε with a unique Wigner measure (2π) 0 δeiθ zU dθ. Since I is finite (or countable), from any sequence (σIεn ) with limn→∞ εn = 0, one can extract a subsequence (εnk )k∈N such that εn

M (σI k , k ∈ N) = {ν } . Since the measure μU is an extremal point in the convex set of gauge invariant probability measures, all the ν have to be identical to μU . Since this holds for any sequence (εn )n∈N , we have proved for any interval I = (α, β) with α < β, M (σIε , ε ∈ (0, ε)) = {μU }. Now take ψ ∈ L1 (R, γ) and consider the state σ εψ

−1/2

= R

|Uε z ⊗[ε

s+ε−1 ]

−1/2

Uε z ⊗[ε

s+ε−1 ]

| dγ(s) =

L

γ(I )σIε .

=1

@

If there exists δ > 0 such that R (1 + |s|)δ ψ(s) dγ(s) < +∞, the family (σ εψ )ε∈(0,ε) satisfy the assumption of Definition 6.5. Let (εn )n∈N be a sequence such that M (σ εψn , n ∈ N) = {ν}. Fix b ∈ Scyl (Z ). The function ψ can be approximated in L1 (R, dγ) by ψc ∈ Cc0 (R). After choosing a finite subdivision I such that the diameter of any I intersecting the support of ψc is bounded by Δ one gets @ L . , . , ψ (t) dt ε I c W eyl εn W eyl Tr b σI ≤ Cb ω(ψc )Δ + |ψ − ψc |L1 (R,γ) σ ψ − Tr b γ(I ) =0

where ω(ψc ) is the continuity modulus of ψc . Hence the right-hand side can be made arbitrarily small, uniformly with respect to εn , while we know that the second term of the left-hand side converges when ψc and I are fixed. We have proved W eyl εn = b(z) dν(z) = lim Tr b b(z) dμU (z) n→∞

Z

Z

for any b ∈ Scyl (Z ) and this proves ν = μU . Since this holds for any ν ∈ M (σ εψ ), we obtain M (σ εψ ) = {μU } . The result for εϕ comes from εϕ

−

σ εψ L 1 (H )

≤ ϕ−

k∈Z

ε

−1/2

Ikε

ε→0

→ 0

ϕ(t) dt 1Ikε L1 (R,ds)

√ s2 with Ikε = [ε1/2 k−ε−1/2 , ε1/2 (k+1)−ε−1/2 ] and ψ(s) = ϕ(s) 2πe 2 . The condition @ (1 + |s|)δ ϕ(s)ds < +∞ ensures that M (εϕ ) is well defined. R

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7.7. Application 2: Propagation of correlated states This a simple application of the orthogonality of Wigner measures combined with the results of Subsection 7.3. Let Hε = dΓ(−Δ) + QW ick be the Hamiltonian studied in Section 5 and 2 let zt denote the solution to i∂t zt = −Δzt + (V ∗ |zt | )zt . The family of integers (kε )ε∈(0,ε) is assumed to satisfy limε→0 εkε = 1. /L ⊗kε 1. Let z0, ∈ Z , = 1, . . . , L, satisfy |z0, | = 1 and set uε = L−1/2 =1 z0, , t ε −i ε Hε u (t) = e uε . At any time t ∈ R the identity ! F L 2π M |uε (t)uε (t)| = (2πL)−1 δeiθ zt, dθ =1

0

as soon as z1,t , . . . , z,t are linearly independent. In particular this holds for any t ∈ R when L = 2 and z0,1 and z0,2 are linearly independent. 2. Let z0 ∈ Z satisfy |z0 | = 1 and set uε = 2−1/2 z0⊗kε + 2−1/2 E(z0 ) and ε t uε (t) = e−i ε H uε . Then

E 2π ε 1 1 ε δz + δeiθ zt dθ . M |u (t)u (t)| = 2 t 4π 0 3. Moreover the convergence can be tested with Weyl, Anti-Wick and Wick operators according to Theorem 6.2 and Theorem 6.13 .

Appendix A. Normal approximation We prove a technical lemma which is a slight adaptation of the normal approximation to the Poisson distribution. Recall that for all −∞ ≤ α < β ≤ ∞ we have the well known fact: β − s2 λn −λ e 2 √ ds . e = (69) lim λ→∞ n! 2π α β α n 1+ √ ≤ λ ≤1+ √ λ

λ

Lemma A.1. Let {an (λ)}n∈Z,λ>0 be a family of complex numbers with an (λ) = 0 if n < 0. Assume that there exist μ ∈ N and Cμ > 0 such that: & n '−μ sup |an (λ)| ≤ Cμ . λ n∈N,λ>0 Then the equality s ∞ λn −λ e− 2 √ lim e an (λ) = lim a (λ) √ ds λ→∞ λ→∞ R [ λs+λ] n! 2π n=0 2

holds whenever one of the two limits exists.

(70)

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1569

Proof. Notice that both the series and the integral in (70) are absolutely convergent −μ for finite values of λ. By hypothesis a ñ (λ) = an (λ) nλ are bounded and moreover they satisfy ∞ " & n 'μ # λn −λ e a ñ (λ) 1 − = 0, λ→∞ n! λ n=0 μ √ 2 − s2 λs + λ] [ e √ ds = 0 a ˜√ (λ) 1 − lim λ→∞ R [ λs+λ] λ 2π

lim

(71)

(72)

since we may bound uniformly for λ large each of the terms inside the sum and the integral respectively by Cμ1

∞ λn −λ μ e n < Cμ0 , n! n=0

and

Cμ2

s2

e− 2 |s| √ ds < Cμ0 , 2π

∀λ > 1 .

μ

R

Therefore there is no restriction if we assume all an (λ) bounded by 1 since if we prove (70) for a ñ (λ) then it holds for an (λ) by the limits (71)–(72). For all h > 0 there exists α < β such that

∞

β

s2

e− 2 √ ds < h/7 , 2π

α

−∞

s2

e− 2 √ ds < h/7 . 2π

Now by (69) we have lim

λ→∞

1+ √β ≤ n λ

λn −λ e = n!

∞

β

s2

e− 2 √ ds , 2π

lim

λ→∞

n α √ λ ≤1+ λ

λ

λn −λ e = n!

α

−∞

s2

e− 2 √ ds 2π

Therefore there exists λ1 such that for all λ > λ1 we have 1+ √β ≤ n λ

λn −λ e ≤ h/6 , n!

n α √ λ ≤1+ λ

λ

Let denote Iα,β (λ) =

@β α

−s

λn −λ e ≤ h/6 . n!

2

a[√λs+λ] (λ) e√2π2 ds. We obtain for all λ > λ1 :

∞ s ∞ λn −λ e− 2 e an (λ) − a[√λs+λ] (λ) √ ds n! 2π −∞ n=0 2

≤

√ α< n−λ λ2 we have √ α< n−λ 2 and e−tV ∈ L1 (Q, dμ) for all t > 0. Let Vn = 1l{|V |≤n} V and Hn = H0 + Vn . Then the semigroups e−tHn converge strongly on H when n → ∞ to a strongly continuous semigroup on H denoted by e−tH . Its infinitesimal generator H has the following properties: i) H is the closure of H0 + V defined on D(H0 ) ∩ D(V ), ii) H is bounded below: H ≥ −c − ln e−δV L1 (Q,dμ) , where c and δ depend only on the constants C and T in Definition 2.2. We will also need the following result [14, Theorem 2.21]. Proposition 2.4. Let e−tH0 be a hypercontractive semigroup. Let V, Vn be real measurable functions on Q such that Vn → V in Lp (Q, dμ) for some p > 2, e−tV , e−tVn ∈ L1 (Q, dμ) for each t > 0 and e−tVn L1 is uniformly bounded in n for each t > 0. Then for b large enough (H0 + Vn + b)−1 → (H0 + V + b)−1

in norm .

The following lemma (see [13, Lemma V.5] for a proof) will be used later to show that a given function V on Q verifies e−tV ∈ L1 (Q, dμ). Lemma 2.5. Let for κ ≥ 1, Vκ , V be functions on Q such that for some n ∈ N: V − Vκ Lp (Q,dμ) ≤ C1 (p − 1)n κ− ,

∀ p ≥ 2,

Vκ ≥ −C2 − C3 (ln κ)n .

(2.3)

Then there exists constants κ0 , C4 and α > 0 such that

α μ q ∈ Q|V (q) ≤ −C4 (ln κ)n ≤ e−κ , ∀κ ≥ κ0 . Consequently e−tV ∈ L1 (Q, dμ), ∀t > 0 with a norm depending only on t and the constants Ci in (2.3). The following theorem of Nelson (see [13, Theorem 1.17]) establishes a connection between contractions on h and hypercontractive semigroups on the Q-space representation L2 (Q, dμ) associated to a conjugation c. Theorem 2.6. Let r ∈ B(h) be a self-adjoint contraction commuting with c. Then i) U Γ(r)U ∗ is a positivity preserving contraction on Lp (Q, dμ), 1 ≤ p ≤ ∞. 1 1 ii) if r ≤ (p − 1) 2 (q − 1)− 2 for 1 < p ≤ q < ∞ then U Γ(r)U ∗ is a contraction p q from L (Q, dμ) to L (Q, dμ). Combining Theorem 2.6 with Theorem 2.3, we obtain the following result.

1584

C. Gérard and A. Panati


Theorem 2.7. Let h be a Hilbert space with a conjugation c. Let a be a self-adjoint operator on h with [a, c] = 0 , a ≥ m > 0 . (2.4) 2 Let L (Q, dμ) be the Q-space representation of Γ(h) and let V be a real function on Q with V ∈ Lp (Q, dμ) for some p > 2 and e−tV ∈ L1 (Q, dμ) for all t > 0. Then: i) the operator sum H = dΓ(a) + V is essentially self-adjoint on D(dΓ(a)) ∩ D(V ). ii) H ≥ −C, where C depends only on m and e−V Lp (Q,dμ) , for some p depending only on m. Note that by applying Theorem 2.6 to a = (q − 1)− 2 1lh for q > 2, we obtain the following lemma about the Lp properties of finite vectors in Γ(h) (see [13, Theorem 1.22]). 1

Lemma 2.8. Let ψ ∈ ⊗ns h and q ≥ 2. Then U ψLq (Q,dμ) ≤ (q − 1)n/2 ψ . 2.7. Perturbations of Lp -contractive semigroups The following theorem is shown in [9, Section II.2]. Theorem 2.9. Let e−tH0 be an Lp -contractive semigroup and V a real measurable function on Q such that V ∈ Lp0 (Q, dμ) for some p0 > 2 and e−δV ∈ L1 (Q, dμ) for some δ > 0. Then H0 + V is essentially self-adjoint on A(H0 ) ∩ Lq (Q, dμ) for any ( 12 − p10 )−1 ≤ q < ∞ where A(H0 ) is the space of analytic vectors for H0 . 2.8. Pseudodifferential calculus on L2 (Rd ) We denote by S(Rd ) the Schwartz class of functions on Rd and by S (Rd ) the Schwartz class of tempered distributions on Rd . We denote by H s (Rd ) for s ∈ R the Sobolev spaces on Rd . 1 We set as usual D = i−1 ∂x and s = (s2 + 1) 2 . 1 For p, m ∈ R and 0 ≤ < 2 ,we denote by Sp,m the class of symbols a ∈ C ∞ (R2d ) such that |∂xα ∂kβ a(x, k)| ≤ Cα,β k p−|β| x m−(1−)|α|+|β| ,

α, β ∈ Nd .

The symbol class S0p,m will be simply denoted by S p,m . The symbol classes above are equipped with the topology given by the seminorms equal to the best constants in the estimates above. For a ∈ Sp,m , we denote by Op1,0 (a) (resp. Op0,1 (a)) the Kohn–Nirenberg (resp. anti Kohn–Nirenberg) quantization of a defined by: ei(x−y)k a(x, k)u(y)dydk , Op1,0 (a)(x, D)u(x) := (2π)−d 0,1 −d Op (a)(x, D)u(x) := (2π) ei(x−y)k a(y, k)u(y)dydk ,

Vol. 9 (2008)

Space-Cutoff P (ϕ)2 Models with Variable Metric

1585

which are well defined as continuous maps from S(Rd ) to S (Rd ). We denote by Opw (a) the Weyl quantization of a defined by: x+y , k u(y)dydk . Opw (a)(x, D)u(x) := (2π)−1 ei(x−y)k a 2 We recall that as operators from S(Rd ) to S (Rd ): Op0,1 (m)∗ = Op1,0 (m) ,

Opw (m)∗ = Opw (m) .

We will also need the following facts (see [7, Theorem 18.5.4]): w Op (b1 ), iOpw (b2 ) = Opw ({b1 , b2 }) + Opw (Sp1 +p2 −3,m1 +m2 −3(1−2) ) ,

(2.5)

Opw (b1 )Opw (b2 ) + Opw (b2 )Opw (b1 ) = 2Opw (b1 b2 ) + Opw (Sp1 +p2 −2,m1 +m2 −2(1−2) ) ,

(2.6)

if bi ∈ Spi ,mi and { , } denotes the Poisson bracket. The following two propositions will be proved in Appendix C. Proposition 2.10. Let b ∈ S 2,0 a real symbol such that for some C1 , C2 > 0 b(x, k) ≥ C1 k 2 − C2 . Then: i) Opw (b)(x, D) is self-adjoint and bounded below on H 2 (Rd ). ii) Let C such that Opw (b)(x, D)+C > 0 and s ∈ R. Then there exist mi ∈ S 2s,0 for i = 1, 2, 3 such that s w Op (b)(x, D) + C = Opw (m1 )(x, D) = Op1,0 (m2 )(x, D) = Op0,1 (m3 )(x, D) . Proposition 2.11. Let aij , c are real such that: [aij ](x) ≥ c0 1l , [aij ] − 1l ,

c(x) −

m2∞

Set: b(x, k) :=

c(x) ≥ c0

∈S

0,−μ

for some

for some

c0 > 0 ,

m∞ , μ > 0 .

(2.7)

ki aij (x)kj + c(x) ,

1≤i,j≤d

and h :=

Di aij (x)Dj + c(x) = Opw (b) .

1≤i,j≤d

Then: i) 1

1

ω := h 2 = Opw (b 2 ) + Opw (S 0,−1−μ ) . ii) there exists 0 < < 12 such that: ω, i ω, i x = Opw (γ)2 + Opw (r−1− ) ,

for

0,− 12

γ ∈ S

,

r−1− ∈ S0,−1− .

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3. The space-cutoff P (ϕ)2 model with variable metric In this section we define the space-cutoff P (ϕ)2 Hamiltonians with variable metric and we prove some of their basic properties. 3.1. The P (ϕ)2 model with variable metric For μ ∈ R we denote by S μ the class of symbols a ∈ C ∞ (R) such that |∂xα a(x)| ≤ Cα x μ−α ,

α ∈ N.

Let a, c two real symbols such that for some μ > 0: a − 1 ∈ S −μ ,

a(x) > 0 ,

c − m2∞ ∈ S −μ ,

c(x) > 0 ,

(3.1)

where the constant m∞ ≥ 0 has the meaning of the mass at infinity. For most of the paper we will assume that the model is massive i.e. m∞ > 0. The existence of the Hamiltonian in the massless case m∞ = 0 will be proved in Theorem 3.2. We consider the second order differential operator h = Da(x)D + c(x) , which is self-adjoint on H (R). Clearly h ≥ m for some m > 0 if m∞ > 0 and for m = 0 if m∞ = 0. Note that h is a real operator i.e. [h, c] = 0, if c is the standard conjugation. The one-particle space is 2

h = L2 (R, dx) , and the one-particle energy is 1 ω := Da(x)D + c(x) 2 ,

acting on h .

The kinetic energy of the field is H0 := dΓ(ω) ,

acting on

Γ(h) .

To define the interaction we fix a real polynomial with x-dependent coefficients: P (x, λ) =

2n

ap (x)λp ,

a2n (x) ≡ a2n > 0 ,

(3.2)

p=0

and a measurable function g with: g(x) ≥ 0 ,

∀ x ∈ R,

and set for 1 ≤ κ < ∞ an UV-cutoff parameter: Vκ := g(x) : P x, ϕκ (x) : dx , where : : denotes the Wick ordering and ϕκ (x) are the UV-cutoff fields. In the massive case, they are defined as: ϕκ (x) := φ(fκ,x ) ,

(3.3)

Vol. 9 (2008)


for

√

2ω − 2 χ 1

1587

ω ∞

δx , x ∈ R . (3.4) κ 1 Here χ ∈ C0∞ (R) is a cutoff function equal to 1 near 0, ω∞ = (D2 + m2∞ ) 2 , and δx is the δ distribution centered at x. In the massless case we take: ω √ 1 δx , x ∈ R . fκ,x = 2ω − 2 χ κ Note that one can also use the above definition in the massive case (see Lemma 6.4). Note also that since ω is a real operator, fκ,x are real vectors, which implies that Vκ is affiliated to Mc . Therefore in the Q-space representation associated to c, Vκ becomes a measurable function on (Q, μ). We will see later that under appropriate conditions on the functions gap (see Theorems 3.1 and 3.2) the functions Vκ converge in L2 (Q, dμ) when κ → ∞ to a function V which will be denoted by g(x) : P x, ϕ(x) : dx . V := fκ,x =

R

3.2. Existence and basic properties We consider first the massive case m∞ > 0. Theorem 3.1. Let ω = (Da(x)D + c(x)) 2 where a, c > 0 and a − 1, c − m2∞ ∈ S −μ for some μ > 0. Assume that m∞ > 0 . Let: 2n P (x, λ) = ap (x)λp , a2n (x) ≡ a2n > 0 . 1

p=0

Assume: gap ∈ L2 (R) ,

for

0 ≤ p ≤ 2n ,

g(ap )2n/(2n−p) ∈ L1 (R)

for

0 ≤ p ≤ 2n − 1 .

Then H = dΓ(ω) +

R

g ∈ L1 (R) ,

g ≥ 0,

(3.5)

g(x) : P x, ϕ(x) : dx = H0 + V

is essentially self-adjoint and bounded below on D(H0 ) ∩ D(V ). Proof. We apply Theorem 2.7 to a = ω. We need to show that V ∈ Lp (Q) for some p > 2 and e−tV ∈ L1 (Q) for all t > 0. The first fact follows from Lemma 6.2 and Lemma 2.8. To prove that e−tV ∈ L1 (Q) we use Lemma 2.5: we know from Lemma 6.2 i) that V − Vκ L2 (Q) ∈ O(κ− ) for some > 0. Since V Ω and Vκ Ω are finite particle vectors, we deduce from Lemma 2.8 that for all p ≥ 2 one has V − Vκ Lp (Q) ≤ C(p − 1)n κ− . Hence the first estimate of (2.3) is satisfied. The second follows from Lemma 7.1.

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We now consider the massless case m∞ = 0. For simplicity we assume that a(x) ≡ 1. Theorem 3.2. Let ω = (D2 + c(x)) 2 where c > 0 and c ∈ S −μ for some μ > 0. Let: 2n ap (x)λp , a2n (x) ≡ a2n > 0 . P (x, λ) = 1

p=0

Assume: g

is compactly supported , for

0 ≤ p ≤ 2n ,

∈ L (R)

for

0 ≤ p ≤ 2n − 1 .

2n/(2n−p)

g(ap )

(3.6)

gap ∈ L (R) , 2

Then

1

H = dΓ(ω) + R

g ≥ 0,

(3.7)

g(x) : P x, ϕ(x) : dx = H0 + V

is essentially self-adjoint on A(H0 ) ∩ Lq (Q, dμ) for q large enough, where A(H0 ) is the space of analytic vectors for H0 . Remark 3.3. It is not necessary to assume that g is compactly supported. In fact if we replace the cutoff function χ in the proof of Lemma 6.5 by the function x −μ/2 we see that Lemma 6.5 still holds if: c(x) ≥ C x −μ ,

for some

C > 0.

(3.8)

Similarly Lemma 6.6 ii) still holds if we replace the conditions gap ∈ L2 (R) ,

g

compactly supported ,

by gap x pμ/2 ∈ L2 (R) . The estimate iii) in Lemma 6.6 is replaced by: 1 1 h x −μ/2 ω − 2 F δx ∈ O (lnκ) 2 , 2 k

uniformly in x ∈ R .

Following the proof of Lemma 7.1, we see that Theorem 3.2 still holds if we assume (3.8), g ∈ L1 (R) and if conditions (3.7) hold with ap replaced by ap x pμ/2 . Remark 3.4. We believe that H is still bounded below in the massless case. For example using arguments similar to those in Lemma 6.5, one can check that the second order term in formal perturbation theory of the ground state energy E(λ) of H0 + λV is finite. Proof. Since ω ≥ 0 is a real operator, we see from Theorem 2.6 that e−tH0 is an Lp contractive semigroup. Applying Theorem 2.9, it suffices to show that V ∈ Lp (Q) for some p > 2 and e−δV ∈ L1 (Q) for some δ > 0. The first fact follows from

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1589

Lemma 6.6 and Lemma 2.8. To prove that e−tV ∈ L1 (Q) we use again Lemma 2.5: the fact that for all p ≥ 2 V − Vκ Lp (Q) ≤ C(p − 1)n κ− , follows as before from Lemma 6.6. The second condition in (2.3) follows from Lemma 6.6 iii), arguing as in the proof of Lemma 7.1.

4. Spectral and scattering theory of P (ϕ)2 Hamiltonians In this section, we state the main results of this paper. We consider a P (ϕ)2 Hamiltonian as in Theorem 3.1. We need first to state some conditions on the eigenfunctions and generalized eigenfunctions of h = ω 2 . These conditions will be needed to obtain higher order estimates in Section 8, an important ingredient in the proof of Theorems 4.3, 4.4 and 4.5. We will say that the families {ψl (x)}l∈I and {ψ(x, k)}k∈R form a basis of (generalized) eigenfunctions of h if: ψl ( · ) ∈ L2 (R) ,

l ≤ m2∞ ,

hψl = l ψl ,

l∈I

l∈I

k ∈ R, ψ( · , k) = (k + 1 ψ( · , k) ψ( · , k)dk = 1l . |ψl ) ψl + 2π R 2

ψ( · , k) ∈ S (R) ,

m2∞ )ψ( · , k) ,

Here I is equal either to N or to a finite subset of N. The existence of such bases follows easily from the spectral theory and scattering theory of the second order differential operator h, using hypotheses (3.1). Let M : R → [1 + ∞[ a locally bounded Borel function. We introduce the following assumption on such a basis: (BM 1) M −1 ( · )ψl ( · )2∞ < ∞ , l∈I

(BM 2) M −1 ( · )ψ( · , k)∞ ≤ C ,

∀ k ∈ R.

For a given weight function M , we introduce the following hypotheses on the coefficients of P (x, λ): (BM 3) gap M s ∈ L2 (R) ,

2n

g(ap M s ) 2n−p+s ∈ L1 (R) ,

∀ 0 ≤ s ≤ p ≤ 2n − 1 .

Remark 4.1. Hypotheses (BMi) for 1 ≤ i ≤ 3 have still a meaning if M takes values in [1, +∞], if we use the convention that (+∞)−1 = 0. Of course in order for (BM3) to hold M must take finite values on supp g. Remark 4.2. The results below still hold if we replace (BM2) by (BM 2 ) |M −1 ( · )ψ( · , k)∞ ≤ C sup(1, |k|−α ) , for some 0 ≤ α
1.

+ h.c.) and A = dΓ(a). Let τ = σpp (H) + m∞ N∗

be the set of thresholds of H. Then: i) the quadratic form [H, iA] defined on D(H) ∩ D(A) uniquely extend to a bounded quadratic form [H, iA]0 on D(H m ) for some m large enough. ii) if λ ∈ R\τ there exists > 0, c0 > 0 and a compact operator K such that 1l[λ−,λ+] (H)[H, iA]0 1l[λ−,λ+] (H) ≥ c0 1l[λ−,λ+] (H) + K . iii) for all λ1 ≤ λ2 such that [λ1 , λ2 ] ∩ τ = ∅ one has: dim1l[λ1 ,λ2 ] (H) < ∞ . Consequently σpp (H) can accumulate only at τ , which is a closed countable set. iv) if λ ∈ R\(τ ∪ σpp (H)) there exists > 0 and c0 > 0 such that 1l[λ−,λ+] (H)[H, iA]0 1l[λ−,λ+] (H) ≥ c0 1l[λ−,λ+] (H) . Theorem 4.5 (Scattering theory). Let H be as in Theorem 3.1 and assume that the hypotheses of Theorem 4.4 hold. Let us denote by hc (ω) the continuous spectral subspace of h for ω. Then: 1. The asymptotic Weyl operators: W ± (h) := s- lim eitH W (e−itω h)e−itH t±∞

exist for all

h ∈ hc (ω) ,

and define a regular CCR representation over hc (ω). 2. There exist unitary operators Ω± , called the wave operators: Ω± : Hpp (H) ⊗ Γ hc (ω) → Γ(h) such that W ± (h) = Ω± 1l ⊗ W (h)Ω±∗ , h ∈ hc (ω) , H = Ω± H|Hpp (H) ⊗ 1l + 1l ⊗ dΓ(ω) Ω±∗ .

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Remark 4.6. Appendices A and B are devoted to conditions (BM1), (BM2). For example condition (BM1) is always satisfied for M (x) = x α if α > 12 and is satisfied for M (x) = 1 if h has a finite number of eigenvalues (see Proposition A.1). Concerning condition (BM2), we show in Lemma A.3 that it suffices to consider the case where a(x) ≡ 1. For example if c(x) − m2∞ ∈ O( x −μ ) for μ > 2 and h has no zero energy resonances, then (BM2) is satisfied for M (x) = 1 (see Proposition B.3). If c(x) − m2∞ ∈ O( x −μ ) for 0 < μ < 2, is negative near infinity and has no zero energy resonances, then (BM2) is satisfied for M (x) = x μ/4 (see Proposition B.10). If c(x) − m2∞ is positive near infinity, holomorphic in a conic neighborhood of R and has no zero energy resonances, then (BM2) is satisfied for M (x) = 1 in {|x| ≤ R} and M (x) = +∞ in {|x| > R} (see Proposition B.14). Remark 4.7. A typical situation in which all the assumptions are satisfied is when a(x) − 1, c(x) − m2∞ and g, ap are all in the Schwartz class S(R). Proofs of Theorems 4.3, 4.4 and 4.5. It suffices to check that H belongs to the class of abstract QFT Hamiltonians considered in [4]. We check that H satisfies all the conditions in [4, Theorem 4.1], introduced in [4, Section 3]. Since ω ≥ m > 0, condition (H 1) in [4, Subsection 3.1] is satisfied. The interaction term V is clearly a Wick polynomial. By Theorem 3.1, H is essentially selfadjoint and bounded below on D(H0 ) ∩ D(V ), i.e. condition (H2) in [4, Subsection 3.1] holds. Next by Theorem 8.1 the higher order estimates hold for H, i.e. condition (H 3) in [4, Subsection 3.1] is satisfied. The second set of conditions concern the one-particle energy ω. Conditions 1 (G1) in [4, Subsection 3.2] are satisfied for S = S(R) and x = (x2 + 1) 2 . This follows immediately from the fact that ω ∈ Op(S 1,0 ) shown in Proposition 2.10 and pseudodifferential calculus. Condition (G2) in [4, Subsection 3.2] has been checked in Proposition 2.11. Let us now consider the conjugate operator a. To define a without ambiguity, we set e−ita := F −1 ut F, where ut is the unitary group on L2 (R, dk) generk ated by the vector field − k

· ∂k . We see that ut preserves the spaces S(R) and FD(ω) = D( k ). This implies first that a is essentially self-adjoint on S(R), by Nelson’s invariant subspace theorem. Moreover eita preserves D(ω) and [ω, a] is bounded on L2 (R). By [1, Proposition 5.1.2], ω ∈ C 1 (a) and condition (M1 i) in [4, Subsection 3.2] holds. We see also that a ∈ Op(S 0,1 ), so conditions (G3) and (G4) in [4, Subsec1 tion 3.2] hold. For ω∞ = (D2 + m2∞ ) 2 , we deduce as above from pseudodifferential calculus that −1 [ω, ia]0 = ω∞ D −1 D2 + Op(S 0,−μ ) . Since χ(ω) − χ(ω∞ ) is compact, we obtain that −1 χ(ω)[ω, ia]0 χ(ω) = χ2 (ω∞ )ω∞ D −1 D2 + K ,

where

K

is compact .

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This implies that ρaω ≥ 0 and τa (ω) = {m∞ }, hence (M1 ii) in [4, Subsection 3.2] holds. Property (C) in [4, Subsection 3.2] follows from the fact that ω − ω∞ ∈ Op(S 1,−μ ) and pseudodifferential calculus. Finally property (S) in [4, Subsection 3.2] can be proved as explained in [4, Subsection 3.2]. The last set of conditions concern the decay properties of the Wick kernel of V . We see that condition (D) in [4, Subsection 3.2] is satisfied, using Lemma 6.2 and the fact that x s gap ∈ L2 (R) for all 0 ≤ p ≤ 2n. Applying then [4, Theorem 4.1] we obtain Theorems 4.3, 4.4 and 4.5.

5. Higher dimensional models In this section we briefly discuss similar models in higher space dimension, when the interaction term has an ultraviolet cutoff. We work now on L2 (Rd , dx) for d ≥ 2 and consider 1 Di aij (x)Dj + c(x) , ω = h 2 . h= 1≤i,j≤d

where aij , c satisfy (2.7). The free Hamiltonian is as above H0 = dΓ(ω) , acting on the Fock space Γ(L2 (Rd )). Since d ≥ 2 it is necessary to add an ultraviolet cutoff to make sense out of the formal expression g(x)P x, ϕ(x) dx . Rd

We set

1 ω δx , ϕκ (x) := φ ω − 2 χ κ where χ ∈ C0∞ ([−1, 1]) is a cutoff function equal to 1 on [− 12 , 12 ] and κ 1 is an 1 ultraviolet cutoff parameter. Since ω − 2 χ( ωκ )δx ∈ L2 (Rd ), ϕκ (x) is a well defined selfadjoint operator on Γ(L2 (Rd )). If P (x, λ) is as in (3.2) and g ∈ L1 (Rd ), then V := g(x)P x, ϕκ (x) dx , Rd

is a well defined selfadjoint operator on Γ(L2 (Rd )). 2n(2n−p)

Lemma 5.1. Assume that g ≥ 0, g ∈ L1 (Rd )∩L2 (Rd ) and gap ∈ L2 (Rd ), gap L1 (Rd ) for 0 ≤ p ≤ 2n − 1. Then Lp (Q, dμ) , V is bounded below . V ∈ 1≤p 0. Lemma 6.1. Let g ∈ S(R). Then for κ < ∞: p p p g(x) : ϕκ (x) : dx = r r=0 wp,κ (k1 , . . . , kr , kr+1 , . . . , kp )a∗ (k1 ) . . . a∗ (kr )a(−kr+1 ) . . . a(−kp )dk1 . . . dkp , where: −p/2

wp,κ (k1 , . . . , kp ) = (2π)

g(x)

p

e−ikj x mκ (x, kj )dx

(6.1)

j=1

and mκ (x, k) is the anti Kohn–Nirenberg symbol of ω − 2 χ( ωκ∞ ). 1

Proof. If mκ (x, k) is the anti Kohn–Nirenberg symbol of ω − 2 χ( ωκ∞ ) we have: 1 ω 1 ∞ δx (k) = (2π)− 2 e−ixk mκ (x, k) . F ω− 2 χ κ Note that it follows from Proposition 2.10 that mκ ∈ S −r,0 for each r ∈ N. We 1 observe moreover that ω − 2 χ( ωκ∞ ) is a real operator which implies that mκ (x, k) = mκ (x, −k) and hence 1 e−ikx mκ (x, k) a∗ (k) + a(−k) dk , ϕκ (x) = (2π)− 2 1

from which the lemma follows.

We extend the above notation to κ = ∞ by denoting by m∞ (x, k) the anti 1 Kohn–Nirenberg symbol of ω − 2 and by wp,∞ the function in (6.1) with mκ re1 placed by m∞ . Note that by Proposition 2.10 m∞ ∈ S − 2 ,0 so wp,∞ is a well defined function on Rd if g ∈ S(R). To study the properties of wκ,p it is convenient to introduce the following maps: Tκ : S(R) → S (Rp ) , g → wκ,p .

1 ≤ κ ≤ ∞,

Lemma 6.2. i) Tκ is bounded from L2 (R) to L2 (Rp ) for each 1 ≤ κ ≤ ∞ and there exists > 0 such that Tκ − T∞ B(L2 (R),L2 (Rp )) ∈ O(κ− ) .

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1595

ii) the map Dxi s T∞ x −s is bounded from L2 (R) to L2 (Rp ) for each s ≥ 0 and 1 ≤ i ≤ p. iii) one has 1 fκ,x ∈ 0 (ln κ) 2 , uniformly for x ∈ R . Proof. The operator Tκ has the distribution kernel p e−ikj x mκ (x, kj ) , (2π)−p/2 j=1

hence for f ∈ S(R ) we have: p eikj x mκ (x, kj )f (k1 , . . . , kp )dk1 · · · dkp . Tκ∗ f (x) = (2π)−p/2 p

j=1 ∞

∞

If R : C (R ) → C (R) is the operator of restriction to the diagonal p

Rf (x) = f (x, . . . , x) , we see that

Tκ∗ f = RMκ Fp−1 f ,

where Mκ =

p

Op1,0 (mκ )(xj , Dxj ) ,

j=1

and we have denoted by Fp the unitary Fourier transform on L2 (Rp ). Since F is the unitary Fourier transform on L2 (R), we have with obvious identification Γ(F) = Fp . Since Op1,0 (m) = Op0,1 (m)∗ , we see that ω 1 ∞ ω− 2 , Mκ = Γ χ 2 κ |⊗p s L (R) where we have used the Fock space notation. This yields ω 1 1 1 1 −1 ∞ 2 2 Tκ∗ = RΓ χ ω∞2 F −1 Γ(Fω∞ ω − 2 F −1 ) =: Tκ0∗ Γ(Fω∞ ω − 2 F −1 ) , κ where Tκ0 is the analog of Tκ with ω replaced by ω∞ . This yields: 1

2 Tκ = Γ(Fω − 2 ω∞ F −1 )Tκ0 . 1

(6.2)

1 2

− 12

By pseudodifferential calculus, we know that ω ω∞ ∈ Op(S 0,0 ) and hence is 0 . bounded on D( x s ) for all s. Therefore it suffices to prove i) and ii) for Tκ0 , T∞ i) for Tκ0 is shown in [2, Lemma 6.1]. To check ii) for Tκ0 for integer s we use that 0 (g)(k1 , . . . , kp ) T∞

= gˆ(k1 · · · + kp )

p

−1

ω∞2 (ki ) .

i=1 0 Then ∂ks1 T∞ (k1 , . . . , kp ) is a sum of terms

∂ks11 gˆ(k1

··· +

−1 kp )∂ks12 ω∞2 (k1 )

p i=2

−1

ω∞2 (ki )

1596

C. Gérard and A. Panati −1


for s1 + s2 = s. We note that ∂ks ω∞2 ∈ O( k − 2 −s ) for all s ∈ N. This implies that 0 (g) ∈ L2 (Rp ). This proves ii) for integer s. We extend if ∂ks gˆ ∈ L2 (R) then ∂ks1 T∞ it to all s ≥ 0 by interpolation. 1

−1

1

Finally a direct computation shows that ω∞2 χ( ωκ∞ )δx = O((ln κ) 2 ), which 1

2 is bounded on L2 (R). implies iii) since ω − 2 ω∞ 1

The following proposition follows easily from Lemmas 6.1 and 6.2. Proposition 6.3.

i) Assume that gap ∈ L2 (R) for 0 ≤ p ≤ 2n. Then lim Vκ =: V exists in Lp (Q, dμ) .

κ→∞

1≤p 0 , c ∈ S −μ . We set as above h = D2 + c(x) ,

1

ω1 = (h + 1) 2 .

Lemma 6.5. Let χ ∈ C0∞ (R). Then: 1

i)

ω12 χ(x)ω − 2 , 1

ω1 χ(x)ω −1

are bounded .

If F ∈ C0∞ (R) then 1 h ii) ω1δ χ(x), F ω − 2 ∈ O(κδ−3/2 ) κ2

∀ 0 ≤ δ < 3/2 .

Proof. Set χ = χ(x). Then χD2 χ = Dχ2 D − χ χ and hence χD2 χ ≤ CD2 + Cχ1 , for χ1 ∈ C0∞ (R). This implies that χ(h + 1)χ ≤ C(D2 + χ1 ) ≤ Ch , since c(x) > 0. Therefore ω1 χω −1 is bounded, which proves the second statement of i). Since ω1 χ2 ω1 ≤ C(h + 1), we also have χω1 χ2 ω1 χ ≤ Cω 2 , which by Heinz theorem implies that χω1 χ ≤ Cω and proves the first statement of i). To prove ii) we write using (2.1): 1 h ω1δ χ, F ω− 2 2 κ −1 −1 h i h δ − 12 ˜ = ∂ z F (z) z − 2 ω1 [χ, h]ω dz ∧ d z . z− 2 2πκ2 C κ κ

1598


Ann. Henri Poincaré δ+ 1

Since [χ, h] = 2Dχ − χ we see using i) that ω1δ [χ, h]ω − 2 = ω1 2 B, where B is bounded. Using the bound h α (z − κh2 )−1 ∈ O(κ−2α )|Imz|−1 for z ∈ supp F˜ , we obtain ii). 1

To define the interaction in the massless case, we set: √ 1 h ϕκ (x) := 2φ ω − 2 F δx x ∈ R, κ2 where F ∈ C0∞ (R) equals 1 near 0, κ 1 is again an UV cutoff parameter, and: g(x) : P x, ϕκ (x) : dx . Vκ := R

Lemma 6.6. Assume that g is compactly supported and gap ∈ L2 (R) for 0 ≤ p ≤ 2n. Then: 1 i) ω − 2 F ( κh2 )δx ∈ L2 (R) for x ∈ supp g so the UV cutoff fields ϕκ (x) are well defined. ii) Vκ converges in 1≤p 0 such that: V − Vκ Lp (Q,dμ) ≤ C(p − 1)n κ− , iii) one has −1 ω 2 F h δx ∈ O (lnκ) 12 , κ2

∀ p ≥ 2.

uniformly for

x ∈ supp g .

The function V in Lemma 6.6 will be denoted by: g(x) : P x, ϕ(x) : dx . V =: R

Proof. To simplify notation we set Fκ = F ( κh2 ). We take χ ∈ C0∞ (R) equal to 1 on supp g. Then for x ∈ supp g, we have ω − 2 Fκ δx = ω − 2 Fκ χδx = ω 2 Fκ ω −1 χω1 ω1−1 δx ∈ L2 (R) , 1

1

1

since ω1−1 δx ∈ L2 and ω −1 χω1 is bounded by Lemma 6.5 i). To prove ii) we may assume that P (x, λ) = λp . We express the kernel wp,κ (k1 , . . . , kp ) as in Lemma 6.1 and set wp,κ =: Tκ g. Since g = χp g, we have wp,κ = Tκ χp g, and hence wp,κ = T˜κ g, where: 1 −1 T˜κ∗ = RΓ(χω − 2 Fκ F −1 ) = RΓ(ω1 2 )Γ a(κ)F −1 , 1

for a(κ) = ω12 χFκ ω − 2 . We set also 1

1 T˜∞ = RΓ(χω − 2 F −1 ) ,

and we claim that ∗ ∈ O(κ− ) for some T˜κ∗ − T˜∞

which clearly implies ii).

> 0,

(6.5)

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If we set

1

1599

a0 (κ) = Fκ ω12 χω − 2 , then using Lemma 6.5 ii) , we obtain: a(κ) = a0 (κ) + a1 (κ) ,

and a1 (κ) ∈ O(κ−δ ) ,

a0 (κ) ∈ O(1) , Clearly on ⊗p h, one has: Γ(a0 + a1 ) =

1

for some

δ > 0.

aI(1) ⊗ · · · ⊗ aI(p) =: Γ(a0 ) + S(κ) ,

(6.6)

(6.7)

I⊂{1,...,p}

for I(j) = 1lI (j). By (6.6) the terms in (6.7) for I = ∅ are O(κ−δ ) hence S(κ) is −1

−1

O(κ−δ ). Since by Lemma 6.2 RΓ(ω1 2 ) is bounded, it follows that RΓ(ω1 2 )S(κ) is O(κ−δ ). Therefore we only have to estimate 1 1 1 −1 −1 −1 RΓ(χω − 2 ) − RΓ(ω1 2 )Γ a0 (κ) = RΓ(ω1 2 ) − RΓ(ω1 2 Fκ ) Γ(ω12 χω − 2 ) . 1

By Lemma 6.5 i) , Γ(ω12 χω − 2 ) is bounded, and by Lemma 6.2 1

−1

−1

RΓ(ω1 2 ) − RΓ(ω1 2 Fκ ) ∈ O(κ− ) . This completes the proof of ii). It remains to prove iii). We write for x ∈ supp g: ω − 2 Fκ δx = ω − 2 Fk χδx 1

1

= ω − 2 χFκ δx + ω − 2 [Fκ , χ]ω1 ω1−1 δx 1

1

1 2

− 12

(6.8)

= ω − 2 χω1 ω1 Fκ δx + ω − 2 [Fκ , χ]ω1 ω1−1 δx . 1

−1

1

1

By Lemma 6.4 ii) , ω1 2 Fκ δx ∈ O((lnκ) 2 ), uniformly for x ∈ supp g. Moreover 1 2

by Lemma 6.5, ω − 2 χω1 is bounded, hence the first term in the r.h.s. of (6.8) is 1 O(lnκ) 2 . Next ω1−1 δx is in L2 (R) uniformly in x, so the second term is O(κ−δ ) for some δ > 0 by Lemma 6.5 ii). This completes the proof of iii). 1

7. Lower bounds In this section we prove some lower bounds on the UV cutoff interaction Vκ . As explained in Section 3, Vκ is now considered as a function on Q. In all this section we assume that m∞ > 0. As consequence we prove Proposition 7.2, which will be needed in Section 8. We recall from (3.2) that: P (x, λ) =

2n p=0

for a2n (x) ≡ a2n > 0.

ap (x)λp ,

1600



mod Lemma 7.1. Let fκ,x and fκ,x be defined in (3.4), (6.3). Assume that

g ≥ 0,

2n

g ∈ L1 (R) ,

gap2n−p ∈ L1 (R) ,

0 ≤ p ≤ 2n − 1 .

Then there exists C > 0 such that if 2n 2n−p dx , g(x)|ap (x)| D2 := C 1 + sup

D3 = C 1 +

g(x)dx ,

0≤p≤2n−1

one has

g(x) : P x, φ(fκ,x ) : dx ≥ −D2 − D3 (ln κ)n ,

∀κ ≥ 2 ,

mod and the analogous result for fκ,x replaced by fκ,x . mod being the same, using Proof. We prove the lemma for fκ,x , the proof for fκ,x Lemma 6.4 ii) instead of Lemma 6.2 iii). Note first from by Lemma 6.2 iii) fκ,x ∈ 1 O((ln κ) 2 ) uniformly in x. We will use the inequality

ap bn−p ≤ bn + C an

∀ > 0 ,

a, b ≥ 0 ,

(7.1)

valid for n, p ∈ N with p ≤ n. In fact (7.1) follows from λp ≤ λn + C ,

∀ > 0 ,

λ ≥ 0,

by setting λ = ba−1 . We recall the well-known Wick identities:

[n/2]

: φ(f )n :=

m=0

m n! 1 φ(f )n−2m − f 2 . m!(n − 2m!) 2

(7.2)

We apply (7.2) to f = fκ,x . Picking first small enough in (7.1) we get: 1 : φ(fκ,x )2n :≥ φ(fκ,x )2n − C(ln κ)n . 2 Using again (7.1) for = 1, we get also: | : φ(fκ,x )p : | ≤ C2 |φ(fκ,x )|p + (ln κ)p/2 0 ≤ p < 2n , which yields: 1 : P x, φ(fκ,x ) : ≥ 2

φ(fκ,x )2n − C

−C

(ln κ)n +

2n−1

ap (x)|φ(fκ,x )|p

p=0 2n−1

ap (x)(ln κ)p/2

.

p=0

Using again (7.1), we get: 2n−p

2n

ap (x)|φ(fκ,x )|p = ap (x) 2n−p |φ(fκ,x )|p ≤ φ(fκ,x )2n + C ap (x) 2n−p , 2n−p 2n ap (x)(ln κ)p/2 = ap (x) 2n−p (ln κ)p/2 ≤ C (ln κ)n + ap (x) 2n−p ,

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1601

which yields for small enough: 2n−1 2n : P x, φ(fκ,x ) :≥ −C ap (x) 2n−p − C(ln κ)n .

p=0

Integrating this estimate we obtain the lemma.

As a consequence of Lemma 7.1, we have the following proposition, which allows to control a lower order polynomial by the P (ϕ)2 Hamiltonian H. Proposition 7.2. Let P (x, λ) be as in (3.2). Let H = dΓ(ω) + g(x) P x, ϕ(x) : dx R

and Q(x, λ) =

2n−1

br (x)λr

r=0 2n 2n−r

where gbr ∈ L2 (R), gbr sup gap 2 +

0≤p≤2n

∈ L1 (R). Let D > 0 such that

sup 0≤r≤2n−1

gbr 2 + g1 2n

+

sup 0≤p≤2n−1

Then ±

R

gap2n−p 1 +

2n

sup 0≤r≤2n−1

gbr2n−r 1 ≤ D .

g(x) : Q x, ϕ(x) : dx ≤ H + C(D) .

Proof. Set R(x, λ) = P (x, λ) ± Q(x, λ) and W = g(x) : R x, ϕ(x) : dx , Wκ = g(x) : R x, ϕκ (x) : dx . R

R

It follows from Lemma 6.2, Lemma 2.8, and Lemma 7.1 that W, Wκ satisfy the conditions in Lemma 2.5 with constants Ci depending only on D. It follows then from Theorem 2.3 that H ± g(x) : Q x, ϕ(x) : dx = H0 + W ≥ −C(D) , R

for some constant C(D) depending only on D.

8. Higher order estimates This section is devoted to the proof of higher order estimates for variable coefficients P (ϕ)2 Hamiltonians. Higher order estimates are important for the spectral and scattering theory of H, because they substitute for the lack of knowledge of the domain of H. The higher order estimates were originally proved by Rosen [11] in the con1 stant coefficients case ω = (D2 + m2 ) 2 for g ∈ C0∞ (R) and P (x, λ) independent

1602



on x. The proof was later extended in [2] to the natural class g ∈ L1 (R) ∩ L2 (R). The extension of these results to x-dependent polynomials is straightforward. Analysing closely the proof of Rosen, one notes that a crucial role is played by 1 the fact that the generalized eigenfunctions of the one-particle energy (D2 + m2 ) 2 , namely the exponentials eik · x are uniformly bounded both in x and k. To extend Rosen’s proof to the variable coefficients case, it is convenient to diagonalize the one-particle energy ω in terms of eigenfunctions and generalized eigenfunctions of ω 2 = Da(x)D + c(x). However some bounds on eigenfunctions and generalized eigenfunctions are needed to replace the uniform boundedness of the exponentials in the constant coefficients case. These bounds are given by conditions (BM1), (BM2). In this section, we will prove the following theorem. Theorem 8.1. Let H be a variable coefficients P (ϕ)2 Hamiltonian as in Theorem 3.1. Assume that hypotheses (BM1), (BM2), (BM3) hold. Then there exists b > 0 such that for all α ∈ N, the following higher order estimates hold: N α (H + b)−α < ∞ , H0 N α (H + b)−n−α < ∞ , −1

N (H + b) α

1−α

(N + 1)

(8.1)

< ∞.

The rest of the section is devoted to the proof of Theorem 8.1. 8.1. Diagonalization of ω Let h, ω as in Theorem 3.1. By Subsection A.3, h is unitarily equivalent (modulo a constant term) to a Schr¨ odinger operator D2 + V (x) for V ∈ S −μ . Applying then standard results on the spectral theory of one dimensional Schr¨ odinger operators, we know that there exists {ψl }l∈I and {ψ( · , k)}k∈R such that ψl ( · ) ∈ L2 (R) , hψl = (λl + m2∞ )ψl ,

ψ( · , k) ∈ S (R) ,

λl < 0 ,

ψl ∈ L2 (R) ,

hψ( · , k) = (k 2 + m2∞ )ψ( · , k) , k ∈ R∗ , 1 ψ( · , k) ψ( · , k)dk = 1l . |ψl )(ψl | + 2π R l∈I

Moreover using the results of Subsection A.2 and the fact that h is a real operator we can assume that ψ l = ψl , ψ(x, k) = ψ(x, −k) . (8.2) The index set I equals either N or a finite subset of N depending on the number of negative eigenvalues of D2 + V . Let ˜ := l2 (I) ⊕ L2 (R, dk) , h

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and ˜, W : L2 (R, dx) → h

1 W u := ((ψl |u))l∈I ⊕ √ ψ(y, k)u(y)dy . 2π R Clearly W is unitary and W ωW ∗ =: (˜ ωd ⊕ ω ˜c) , for 1 1 ˜ c = (k 2 + m2∞ ) 2 . ω ˜ d = ⊕l∈I (λl + m2∞ ) 2 , ω If we set c˜ = W cW ∗ , then it follows from (8.2) that c˜ (ul )l∈I ⊕ u(k) = (ul )l∈I ⊕ u(−k) ,

(8.3)

i.e. c˜ is the direct sum of the canonical conjugation on l2 (I) and the standard conjugation on L2 (R, dk) used for the constant coefficients P (ϕ)2 model. 8.2. Reduction of H We will consider in the rest of this section the transformed Hamiltonian: ˜ := Γ(W )HΓ(W )∗ . H ˜ In this subsection we determine the explicit form of H. ˜ c˜). We can extend ˜ Let (Q, μ ˜) be the Q-space associated to the couple (h, 2 2 ˜ to a unitary map T : L (Q, dμ) → L (Q, ˜ d˜ Γ(W ) : Γ(h) → Γ(h) μ). ˜ d˜ μ) for all 1 ≤ p ≤ ∞ and Lemma 8.2. T is an isometry from Lp (Q, μ) to Lp (Q, T 1 = 1. Proof. If F is a real measurable function on Q, and m(F ) the operator of multiplication by F on Γ(h), then m(T F ) = Γ(W )m(F )Γ(W )∗ , which shows that T is positivity preserving. Since T 1 = T ∗ 1 = 1, T is doubly Markovian, hence a contraction on all Lp spaces (see [13]). We use the same argument for T −1 . ˜ we have: Coming back to H ˜ =H ˜ 0 + V˜ , H for

˜ 0 := Γ(W )H0 Γ(W )∗ = dΓ(˜ H ωd ⊕ ω ˜ c ) , V˜ := Γ(W )V Γ(W )∗ . We know from Lemma 6.4 that V is the limit in 1≤p 0 1 uniformly in n, we know that N 2 (Nnτ +1)−1 is bounded uniformly in n. This implies that (Nnτ + 1)−1 converges in norm to (N τ + 1)−1 when n → ∞. To prove ii) we follow the proof of [14, Proposition 4.8]: we have seen above 1 1 that N − 2 (H0,n − H0 )N − 2 tends to 0 in norm. Moreover ωd ⊕ ω([k]νn ) ≥ C > 0 uniformly w.r.t. n. This implies that e−tH0,n is hypercontractive with hypercontractivity bounds uniform in n. This implies that if W ∈ Lp (Q, dμ) and e−T W ∈ L1 (Q, dμ) there exists C such that N ≤ C(H0,n + W + C), uniformly in n. Writing (H0,n +W +C)−1 −(H0 +W +C)−1 = (H0,n +W +C)−1 (H0 −H0,n )(H0 +W +C)−1 and using the above bound, we obtain that (H0,n + W + C)−1 converges in norm to (H + W + C)−1 . Moreover it follows from Theorem 2.3 ii) that the constant C above depend only on e−tW L1 for some t > 0. Since by (8.9) e−tVm is uniformly bounded in L1 (Q), we see that (H0,n + Vm + C)−1 converges in norm to (H0 + Vm + C)−1 when n → ∞, uniformly w.r.t. m. Again by (8.9) Vm → V in Lp for some p > 2 and e−tVm is uniformly bounded in L1 , so by Proposition 2.4 we obtain that (H0 + Vm + C)−1 converges to (H0 + V + C)−1 when m → ∞, which completes the proof of the lemma. Let us denote simply by ωn the operator ωd ⊕ωc (([k]νn ). Since [ωn , πn∗ πn ] = 0, we have H0,n = Un dΓ(ωn |hn ) ⊗ 1l + 1l ⊗ dΓ(ωn |h⊥ ) Un∗ , n where Un : Γ(hn ) ⊗ Γ(h⊥ n ) → Γ(h) is the exponential map. This implies that ˆ n ⊗ 1l + 1l ⊗ dΓ(ωn |h⊥ ) , Un∗ Hn Un = H n

ˆnτ ⊗ 1l + 1l ⊗ dΓ(ωnτ |h⊥ ) , Un∗ Nnτ Un = N n

ˆ n = dΓ(ωn |h ) + Vn , N ˆnτ = dΓ(ωnτ |h ). for H n n Proposition 8.5. Assume hypotheses (BMi) for i = 1, 2, 3. Set for J = {1, . . . , , s} ⊂ N and Ki ∈ I R: VnJ := ada(K1 ) . . . ada(Ks ) Vn . Then there exists b, c > 0 such that for all λ1 , λ2 < −b (Hn − λ2 )− 2 VnJ (Hn − λ1 )− 2 ≤ c 1

1

s

F (Ki ) ,

1

where F : I R → R+ satisfies for each δ > 0: |F (x, K)|2 ω(K)−δ dK ≤ C . I R

Proof. We have using (8.7): Vn =

g(x) : P x, ϕn (x) : dx ,

(8.10)

1608


where p

: ϕn (x) :=

p r p r=0

r

a∗ (πn∗ πn W ω − 2 δx ) 1

1

p


a(πn∗ πn W ω − 2 δx ) . 1

(8.11)

r+1

We note that

πn∗ πn =

|eδ eδ | ,

|δ|≤κn

which yields : ϕn (x)p : =

p p

r

r=0

r

a∗ (δi )

δ1 ,...,δp ∈Tn 1

p r+1

a(δi )

r

p

mn (x, δi )

1

mn (x, δi ) ,

(8.12)

r+1

where mn (x, δ) = (eδ |W δx )h . Let for k ∈ R:

1 1 Cn (k) := [k]νn − νn−1 , [k]νn + νn−1 , 2 2 be the cell of Γn centered at [k]νn . Using (8.3) we get: 1 1 νn2 Cn (γ) (k 2 + m2∞ )− 4 ψ(x, k)dk if δ = γ ∈ Γn , mn (x, δ) := 1 (λl + m2∞ )− 4 ψl (x) , if δ = l ∈ I .

(8.13)

Then as in [2, 11], we obtain that s J g(x) rn (x, Ki ) : P (s) x, ϕn (x) : dx , Vn = R

where P

(s)

(x, λ) =

1

d s ( dλ ) P (x, λ)

rn (x, K) =

and

νn Cn (k) ω(k )− 2 ψ(x, k )dk ψl (x) if K = l ∈ I .

We note that

VnJ = Πn

R

1

if K = k ∈ R ,

g(x) : Rn x, K1 , . . . , Ks , ϕ(x) : dx ,

for Rn (x, K1 , . . . , Ks , λ) = P (s) (x, λ)

s

rn (x, Ki ) .

1

Since assumptions (BM1), (BM2) are satisfied, we know that: |ψ(x, k)| ≤ CM (x) , where

|ψl (x)| ≤ Cl M (x) , 2 l∈I l

< ∞.

uniformly for x, k ∈ R , uniformly for x ∈ R ,

l∈I,

(8.14)

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Let us now prove corresponding bounds on the functions rn (x, K). We consider first the case K = l ∈ I: we have: |rn (x, l)| ≤ Cl M (x) ,

uniformly in x, l .

(8.15)

If K = k ∈ R we get: ω(k )− 2 |ψ(x, k )| ≤ Cω(k)− 2 M (x) , 1

1

uniformly for n ∈ N ,

k ∈ Cn (k) ,

x∈R ,

which yields: |rn (x, k)| ≤ Cω(k)− 2 M (x) , 1

If we set:

F (K) =


k, x ∈ R .

ω(k)− 2 if K = k ∈ R , l if K = l ∈ I ,

(8.16)

1

(8.17)

and collect (8.15), (8.16) we get: |rn (x, K)| ≤ CF (K)M (x) ,


K ∈I R ,

x ∈ R . (8.18)

We note that by condition (BM3), we have: gap M s ∈ L2 ,

2n

g(ap M s ) 2n−p+s ∈ L1 ,

0 ≤ s ≤ p ≤ 2n − 1 .

If we apply the arguments in Proposition 7.2 to the polynomial Qn (x, K1 , . . . , Ks , λ) = P

(s)

(x, λ)

s

F (Ki )−1 rn (x, Ki ) ,

1

using the bound (8.18), we obtain that e−t(V ±Wn )

is uniformly bounded in L1 (Q) ,

for

Wn (K1 , . . . Ks ) =

(8.19)

g(x) : Qn x, K1 , . . . , Ks , ϕ(x) : dx .

By (8.9) ii) , this implies that e−t(Vn ±Πn Wn )

is uniformly bounded in L1 (Q) .

Applying then Theorem 2.7 to a = ωn and using (8.19) we get that there exists C > 0 such that s ±VnJ ≤ F (Ki )(Hn + C) , uniformly in n . 1

To complete the proof of the proposition it remains to check that for each δ > 0 F (K)2 ω(K)−δ dK < ∞ , I R

which follows from (8.17) since

2 l∈I l

< ∞.

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Proof of Theorem 8.1. We follow the proof in [11]. This proof consists in first proving higher order estimates for the cutoff Hamiltonians Hn and Nnτ , with constants uniform in n. The corresponding estimates for the Hamiltonians without cutoffs are then obtained by the principle of cutoff independence [11, Proposition 4.1]. The convergence results needed to apply [11, Proposition 4.1] are proved in Lemma 8.4. The estimates for the cutoff Hamiltonians rely on three kinds of intermediate results: The first ([11, Lemma 4.2], [11, Corollary 4.3]) consists of identities expressing expectation values of (powers of) N τ in terms of Wick monomials. These identities carry over directly to our case, replacing R by I R, a (k) by a (K) and the mesure dk by dK. The second [11, Proposition 4.5] is the generalized pull through formula which also carries over to our case. The last is the bound in [11, Lemma 4.4] which is replaced in our case by Proposition 8.5. Carefully looking at the proof of the higher order estimates for the cutoff Hamiltonians in [11, Theorem 4.7] and [11, Corollary 4.8] we see that it relies on the fact that the F (K)2 ω(K)−δ dK < ∞ , I R

(in [11] F (K) equals simply ω(k)− 2 ), which is checked in Proposition 8.5. This completes the proof of Theorem 8.1. 1

Acknowledgements We thank Fritz Gestesy, Erik Skibsted, Martin Klaus and especially Dimitri Yafaev for very helpful correspondence on generalized eigenfunctions for one dimensional Schr¨ odinger operators.

Appendix A. In this section we will give sufficient conditions on the functions a, c in the definition of ω for conditions (BM1), (BM2) to hold. A.1. Sufficient conditions for (BM1) Proposition A.1. Let h = Da(x)D + c(x) be as in Theorem 3.1. Then: i) condition (BM1) is satisfied for M (x) = x α for α > 12 . ii) if h has a finite number of eigenvalues, condition (BM1) is satisfied for M (x) = 1. Proof. ii) is obvious. To prove i) we take an orthonormal basis {ψl }l∈I of the point spectrum subspace of h, and set ul = D s x −α ψl for some s > 12 . By Sobolev’s theorem we have x −α ψl 2∞ ≤ Cul 22 ,

uniformly in l ∈ I .

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Next


l∈I

ul 22 = Tr

1611

|ul )(ul |

l∈I

= Tr D s x −α

|ψl )(ψl | x −α D s

l∈I

= Tr D s x −α 1l]−∞,m2∞ ] (h) x −α D s . We use then that 1l]−∞,m2∞ ] (h) D m is bounded for all m ∈ N by elliptic regularity, which implies that 1l]−∞,m2∞ ] (h) x −α D s is Hilbert–Schmidt if α > 12 . This completes the proof of the proposition. A.2. Generalized eigenfunctions In this subsection we show that if h = Da(x)D + c(x) is a second order differential operator as in Theorem 3.1 satisfying (BM2), then the generalized eigenfunctions ψ(x, k) can be chosen to satisfy additionally the following reality condition: ψ(x, k) = ψ(x, −k) ,

k a.e.

Lemma A.2. Assume that the family {φ( · , k)}k∈R satisfies assumption (BM2). Then there exists a family {ψ( · , k)}k∈R of generalized eigenfunctions of h satisfying (BM2) and additionally: ψ(x, k) = ψ(x, −k) ,

k a.e.

(A.1)

Let {φ(x, k)}k∈R be a basis of generalized eigenfunctions for h. To such a family one can associate a unitary map: Wφ : 1l[m2∞ ,+∞[ (h)L2 (R, dx) → L2 (R, dx) , defined by Wφ u(x) = (2π)−1

eikx φ(y, k)u(y)dy dk ,

(A.2)

which satisfies Wφ h = (D2 + m2∞ )Wφ . Note that if ψ satisfy (A.1) and Wψ is defined as in (A.2), then Wψ is a real operator i.e. Wψ u = Wψ u , u ∈ L2 (R) . Proof. Let us define the unitary operator Ω : L2 (R, dx) → L2 (R+ , dk) ⊗ C2 obtained from Wφ by Fourier transform: Ωu(k) = (2π)−1 φ( · , k)|u , φ( · , −k)|u , satisfying Ωh = (k 2 + m2∞ ) ⊗ 1lC2 Ω. Set ˜ k) := φ(x, −k) x , φ(x,

k ∈ R.

(A.3)

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˜ · , k)}k∈R is a family of generalized eigenfunctions of h. Therefore we Clearly {φ( can introduce the unitary map ˜ · , k)|u , φ( ˜ · , −k)|u . ˜ Ωu(k) = (2π)−1 φ( ˜ −1 , then S commutes with (k 2 + m2∞ ) ⊗ 1lC2 and is unitary, so: If S = ΩΩ ⊕ S= S(k)dk , R+

for S(k) ∈ U (C2 ). Using that (φ(x, k), φ(x, −k)) = Ωδx for x ∈ R, the similar identity for φ˜ and (A.3), we obtain that φ(x, k), φ(x, −k) = S(k)T φ(x, k), φ(x, −k) , x ∈ R , k > 0 , (A.4) where T (z1 , z2 ) = (z2 , z1 ). Iterating this formula we obtain the identity: T ST = S −1 .

(A.5)

Let us find the generalized eigenfunctions ψ(x, k) under the form ψ(x, k), ψ(x, −k) = A(k) φ(x, k), φ(x, −k) x ∈ R , k > 0 . Clearly {ψ( · , k)}k∈R will be a basis of generalized eigenfunctions of h as soon as A(k) ∈ U (C2 ). Using (A.4) we see that it will satisfy (A.1) if A(k) = T A(k)S(k)T .

(A.6)

To solve (A.6), we deduce first from (A.5) that n

T S T = S −n ,

n ∈ N.

−1

Therefore T g(S)T = g(S ) if g is a polynomial. By the standard approximation argument and the spectral theorem for unitary operators this extends to all measurable functions on the unit circle. 1 For z = eiθ , −π < θ ≤ π, we set z α = eiαθ . Setting A(k) = (S)− 2 (k), we get 1

T AST = T S 2 T = S − 2 . 1

Since z − 2 = z − 2 , we obtain that S − 2 (k) = A(k). Therefore A(k) satisfies (A.6). 1 Moreover since z − 2 preserves the unit circle, A(k) is unitary. Therefore the family {ψ( · , k)}k∈R is a basis of generalized eingenfunctions of h. Moreover since the matrix A(k) is unitary, all entries have modulus less than 1, which implies that if {φ( · , k)}k∈R satisfies (BM2), so does {ψ( · , k)}k∈R . 1

1

1

A.3. Reduction to the case of the constant metric We show in this subsection that in order to verify condition (BM2) we can reduce ourselves to the case a(x) ≡ 1. We have then h = D2 + V + m2∞ ,

for V (x) = c(x) − m2∞ ∈ S −μ ,

which will allow to use standard results on generalized eigenfunctions for Schr¨ odinger operators in one dimension.

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1613

Let ψ : R → R be a diffeomorphism with ψ > 0. We denote by ψ −1 the inverse of ψ. To ψ we associate the unitary map Tψ : L2 (R) → L2 (R) Tψ u(x) := ψ (x) 2 u ◦ ψ(x) . 1

Lemma A.3. Let a, c satisfying (3.1) and set g = a 2 and ψ = φ−1 for x 1 ds . φ(x) = g(s) 0 1

Then i)

Da(x)D + c(x) = Tψ∗ D2 + c˜(x) Tψ , where

c˜(x) = c ◦ ψ(x) +

1 2 1 (g ) + gg 4 2

◦ ψ(x) .

ii) c˜ − m2∞ ∈ S −μ . ˜ , then Da(x)D + iii) if D2 + c˜(x) satisfies (BM1), (BM2) for a weight function M − 12 ˜ ˜ (x) = c(x) satisfies (BM1), (BM2) for M (x) = ((ψ ) M ) ◦ ψ −1 (x). If M x α , then M (x) x α . Proof. Let ψ : R → R be a diffeomorphism as above. We have 1 ψ −1 −1 ∂x T ψ u = T ψ ◦ ψ u + ψ ◦ ψ ∂ x u. 2 ψ Choosing ψ as in the lemma we get g(x) = ψ ◦ ψ −1 (x) and 1 ∂x T = T g(x)∂x + g (x) =: T A . 2 This yields

1 1 A∗ − g A − g 2 2 1 1 = A∗ A − (A∗ g + g A) + (g )2 2 4 1 2 1 ∗ = A A + (g ) + gg . 4 2

−∂x a(x)∂x =

This easily implies the first statement of the lemma. Next from (3.1) we get that g − 1 ∈ S −μ hence φ(x) − x ∈ S 1−μ from which ψ(x) − x ∈ S 1−μ follows. This implies that ( 14 (g )2 + 12 gg ) ◦ ψ ∈ S −2−μ and c ◦ ψ − m2∞ ∈ S −μ . Statement iii) is obvious.

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Appendix B. In this section we recall some results about generalized eigenfunctions for onedimensional Schr¨ odinger operators, taken from [16, 17]. For the reader’s convenience, we will sketch some of the proofs. These results are used to obtain some sufficient conditions for (BM2). We saw in Subsection A.3 that we can reduce ourselves to considering a Schr¨ odinger operator: h = D2 + V (x) + m2∞ ,

V ∈ S −μ

for μ > 0 .

It turns out that condition (BM2) is really a condition on the behavior of generalized eigenfunctions ψ(x, k) for k near 0. In this respect the potentials fall naturally into two classes, depending on whether μ > 2 or μ ≤ 2. This distinction is also relevant to condition (BM1). In fact by the Kato– Agmon–Simon theorem (see [10, Theorem XIII. 58]) if V ∈ S −μ for μ > 0 h has no strictly positive eigenvalues. As is well known h has a finite number of negative eigenvalues if μ > 2. Therefore condition (BM1) is always satisfied for M (x) ≡ 1 if μ > 2. Results of Subsections B.1, B.2, B.3 are standard results. We used the reference [16]. Results of Subsections B.4, B.5, B.6 are easy adaptations from those in [17]. For −π ≤ a < b ≤ π, we denote by Arg]a, b[ the open sector {z ∈ C| a < argz < b}. The corresponding closed sector (with 0 excluded) will be denoted by Arg[a, b]. For α ∈ R the function z α is defined by (reiθ )α = rα eiαθ , for −π < θ ≤ π. B.1. Jost solutions for quickly decreasing potentials For two solutions f, g of the equation −u + V u = ζ 2 u , the Wronskian W (f, g) = f (x)g(x) − f (x)g (x) is independent on x. We start by recalling a well-known fact about existence of Jost solutions. Proposition B.1. Assume V ∈ S −μ (R) for μ > 2. Then for any ζ ∈ Arg[0, π] there exist unique solutions θ± (x, ζ) of −u + V u = ζ 2 u , with asymptotics

θ± (x, ζ) = e±iζx 1 + o(1) ,

θ± (x, ζ) = ±iζe±iζx 1 + o(1)

when x → ±∞. They satisfy the estimates |θ± (x, ζ) − e±iζx | ≤ e∓Imζx C x −μ+1 , uniformly for ±x ≥ 0 and ζ ∈ Arg[0, π]. Moreover one has: θ± (x, ζ) = θ± (x, −ζ) .

Vol. 9 (2008)


1615

Proof. Uniqueness of θ± is obvious since the Wronskian of two solutions vanishes at ±∞. We look for θ± (x, ζ) as solutions of the Volterra equations: θ± (x, ζ) = e±iζx + K± θ± (x, ζ) , where:

+∞

K+ (ζ)u(x) = xx K− (ζ)u(x) =

−∞

(B.1)

ζ −1 sin ζ(y − x) V (y)u(y)dy ,

ζ −1 sin ζ(x − y) V (y)u(y)dy .

Using the bound −1 ζ sin ζ(y − x) e−Imζy ≤ Cye−Imζx , we obtain that K+ (ζ) n u(x) ≤ e−Imζx (n!)−1 C

0 ≤ x ≤ y, n

+∞

y|V (y)|dy

,

for x ≥ 0 ,

x

which gives the estimate

+∞ |θ+ (x, ζ) − eiζ · x | ≤ e−Imζx eC x |y||V (y)|dy − 1 ,

proving the desired bound for θ+ (x, ζ). The case of θ− (x, ζ) is treated similarly. The last identity follows from uniqueness. We recall additional identities between Jost solutions θ± ( · , ζ) if ζ = k > 0. We first set: w(k) := W θ+ ( · , k), θ− ( · , k) . Next by computing the Wronskian below at ±∞, we get that: W θ± ( · , k), θ± ( · , −k) = ±2ik . Clearly θ− (x, k) = m++ (k)θ+ (x, k) + m+− (k)θ+ (x, −k) , θ+ (x, k) = m−− (k)θ− (x, k) + m−+ (k)θ− (x, −k) .

(B.2)

We set

m(k) := (2ik)−1 w(k) . We can express the coefficients in (B.2) using Wronskians and get m+− (k) = m−+ (k) = (2ik)−1 W θ+ ( · , k)θ− ( · , k) = m(k) , m++ (k) = −(2ik)−1 W θ+ ( · , −k), θ− ( · , k) , m−− (k) = −(2ik)−1 W θ+ ( · , k), θ− ( · , −k) .

(B.3)

(B.4)

Using the identity θ± (x, −k) = θ± (x, k) and iterating the identities (B.2), we obtain m(−k) = m(−k) , m++ (k) = −m−− (k) , (B.5) |m(k)|2 = 1 + |m++ (k)|2 = 1 + |m−− (k)|2 .

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B.2. Resolvent and spectral family Proposition B.2. The family {ψ( · , k)} defined by ! m(k)−1 θ+ (x, k) k > 0 , ψ(x, k) := m(−k)−1 θ− (x, −k) k < 0

(B.6)

is a family of generalized eigenfunctions of h. Proof. Since θ± ( · , ζ) ∈ L2 (R± ) for Imζ > 0, we obtain by the standard argument that the resolvent (h − z)−1 has kernel ! −w(ζ)−1 θ+ (x, ζ)θ− (y, ζ) , y ≤ x , R(x, y, z) = −w(ζ)−1 θ− (x, ζ)θ+ (y, ζ) , x ≤ y , for ζ 2 = z, Imζ > 0 and w(ζ) = W θ+ ( · , ζ), θ− ( · , ζ) . The zeroes of w lie on iR+ and correspond to negative eigenvalues of h. If E(λ) = 1l]−∞,λ] (h), then from (2iπ) We obtain that for λ > 0 4πk

dE (λ) = R(λ + i0) − R(λ − i0) . dλ

dE dλ (λ)

has a kernel satisfying:

dE (x, y, λ) = m(k)−1 θ+ (x, k)θ− (y, −k) dλ + m(−k)−1 θ+ (x, −k)θ− (y, k) ,

for

y ≤ x,

where k 2 = λ. Note that dE dλ (λ) is both real and self-adjoint hence dE (y, x, λ). dλ Using the identities (B.2) and (B.5), we obtain that 4πk

dE dλ (x, y, λ)

dE (x, y, λ) = |m(k)|−2 θ+ (x, k)θ+ (y, −k) + θ− (x, k)θ− (y, −k) dλ

=

(B.7)

for k 2 = λ. Setting ψ± (x, k) := m(k)−1 θ± (x, k) k > 0 ,

(B.8)

we obtain 4πk

dE (x, y, λ) = ψ+ (x, k)ψ + (y, k) + ψ− (x, k)ψ − (y, k) , dλ

(B.9)

for k 2 = λ, which shows that {ψ( · , k)}k∈R defined in (B.6) is a family of generalized eigenfunctions of h.

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B.3. Condition (BM2) for quickly decreasing potentials Let us now consider in more details the Volterra integral equations (B.1) for ζ = k > 0. Let F± be the Banach space of C 1 functions on R± bounded with bounded derivatives equipped with the obvious norm. The operators (1l − K± (k))−1 are bounded on F± and ]0, +∞[ k → (1l − K± (k))−1 ∈ B(F± ) is norm continuous. It follows that k → θ± ( · , k) ∈ F± is continuous on ]0, +∞[, and hence w(k) is continuous on ]0, +∞[. Moreover when k → 0, (1l − K± (k))−1 converges in B(F± ) to (1l − K± (0))−1 , where +∞ K+ (0)u(x) = (y − x)V (y)u(y)dy , 0 x (x − y)V (y)u(y)dy . K− (0)u(x) = −∞

Therefore lim w(k) =: w(0) exists

k→0

and w(0) = 0 iff there exists a solution u of −u + V u = 0 , with asymptotics: u (x) → 0 for x → ±∞ ,

u(x) → u± ,

u± = 0 .

Such a solution is called a zero energy resonance for h. Recall that condition (BM2’) is introduced in Remark 4.2. Proposition B.3. Assume that V ∈ S −μ for μ > 2. Then: 1) if h has no zero energy resonance, then h satisfies (BM2) for M (x) ≡ 1. 2) if h has a zero energy resonance and |w(k)| ≥ C|k|3/2− in |k| ≤ 1 for some > 0 then h satisfies (BM2’) for M (x) ≡ 1. Remark B.4. Assume V ∈ S −μ for μ > 3. Then if h has a resonance, |w(k)| ≥ C|k| (see [16, Prop.7.13]). Proof. For k > 0 we deduce from (B.2) that: 1 x ≥ 0, m(k) θ+ (x, k) , ψ(x, k) = m−− (k) m(k) θ− (x, k) + θ− (x, −k) x ≤ 0

and ψ(x, −k) =

1 m(k) θ− (x, k) , m++ (k) m(k) θ+ (x, k)

x ≤ 0, + θ+ (x, −k) x ≥ 0 .

By Proposition A.2 the functions θ± (x, k) are uniformly bounded in k > 0 and x ∈ R± . Moreover by (B.5) m++ (k) ≤ 1 , m−− (k) ≤ 1 . m(k) m(k)

1618



Therefore it suffices to bound m(k)−1 . Using the integral equations (B.1), we obtain that: θ± (0, k) = 1 + O(k −1 ) , θ± (0, k) = ±ik + O(1) when k → +∞. Therefore w(k) = 2ik(1 + O(k −1 )) and m(k)−1 is bounded near +∞. If w(0) = limk→0 w(k) = 0, then limk→0 m(k)−1 = 0 and hence m(k)−1 is uniformly bounded on ]0, +∞[. If |w(k)| ≥ C|k|−3/2+ for |k| ≤ 1, we get instead 1 that |m(k)|−1 ≤ C|k|− 2 + for |k| ≤ 1. This completes the proof of the theorem. B.4. Quasiclassical solutions for slowly decreasing potentials Let V ∈ S −μ for 0 < μ ≤ 2. For Imζ ≥ 0, we set 1 F (x, ζ) := V (x) − ζ 2 2 , 1

where z 2 is defined as in the beginning of Section B. We see that F (x, ζ) is holomorphic in ζ in the two sectors Arg]0, π/2[ and Arg]π/2, π[ and C ∞ in x if ζ belongs to the above sectors. It is continuous in ζ in the closed sectors Arg[0, π/2] and Arg[π/2, π] but may not be continuous across Argζ = π/2 depending on the value of V (x). 1

1

Note that (z 2 ) = z 2 if Argz = π, which implies that F (x, ζ) = F (x, −ζ) ,

ζ ∈ Arg]0, π/2[∪Arg]π/2, π[ ,

We set also

x

F (y, ζ)dy ,

S(a, x, ζ) :=

F (x, k) = F (x, −k) k > 0 . (B.10) a ∈ R.

(B.11)

a

Proposition B.5. For ζ ∈ Arg[0, π], ζ = 0 there exist unique solutions η± (x, ζ) of −u + V u = ζ 2 u ,

(E)

with asymptotics

1 η+ (x, ζ) = F (x, ζ)− 2 e−S(0,x,ζ) 1 + o(1) , 1 η+ (x, ζ) = −F (x, ζ) 2 e−S(0,x,ζ) 1 + o(1) , x → +∞ , 1 η− (x, ζ) = F (x, ζ)− 2 eS(0,x,ζ) 1 + o(1) , 1 η− (x, ζ) = F (x, ζ) 2 eS(0,x,ζ) 1 + o(1) , x → −∞ .

We have η ± (x, ζ) = η± (x, −ζ) . For > 0 let R() be such that |V (y)| ≤ /2 for |y| ≥ R(). Then the following estimates are valid: |η± (x, ζ)| ≤ C()|ζ|− 2 , 1

|η± (x, ζ)| ≤ C()|ζ| 2 , 1

uniformly in

± x ≥ ±R() ,

|k| ≥ .

(B.12)

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Proof. We follow [17] and treat only the case of η+ . For |ζ| ≥ and x ≥ R(), we (x, ζ)) of the form look for (η+ (x, ζ), η+ 1 η+ (x, ζ) = F (x, ζ)− 2 e−S(0,x,ζ) u1 (x, ζ) + u2 (x, ζ) , 1 η+ (x, ζ) = F (x, ζ)− 2 e−S(0,x,ζ) F (x, ζ) − V (x)(4F 2 )−1 u1 (x, ζ) (B.13) − F (x, ζ) + V (x)(4F 2 )−1 u2 (x, ζ) . We find that (u1 ( · , ζ), u2 (x, ζ)) has to satisfy the following Volterra equation: +∞ e−2S(x,y,ζ) M (y, ζ) u1 (y, ζ) + u2 (y, ζ) dy , u1 (x, ζ) = − x (B.14) +∞ M (y, ζ) u1 (y, ζ) + u2 (y, ζ) dy , u2 (x, ζ) = 1 + x

for

M (x, ζ) = (32)−1 4V (x)F (x, ζ)−3 − 5(V )2 (x)F (x, ζ)−5 . Uniformly for |ζ| ≥ and y ≥ x ≥ R(), we have: |M (x, ζ)| ≤ C() x −2−μ ,

|e−S(x,y,ζ) | ≤ 1 .

The equation (B.14) can be solved by iteration and we obtain as in the proof of Proposition B.1 that: |u1 (x, ζ)| ≤ (eC() x

|u2 (x, ζ)| ≤ e

−1−μ

C() x −1−μ

− 1) ,

,

uniformly for |ζ| ≥ and y ≥ x ≥ R(). Since 1

1

C1 ()|ζ| 2 ≤ |F (x, ζ)| ≤ C2 ()|ζ| 2 , ( · , ζ). we obtain the desired bounds on η+ ( · , ζ), η+ To prove uniqueness of η+ ( · , ζ), we verify that the Wronskian of two solutions computed at x = +∞ vanishes. The fact that η ± ( · , ζ) = η± ( · , −ζ) follows from (B.10).

As in Subsection B.1, we compute some Wronskians. Lemma B.6. For |k| ≥ we have: R() 2 1 W η+ ( · , k), η+ ( · , −k) = 2isgn(k)e−2Re 0 (V (y)−k ) 2 dy , 0 2 1 2 W η− ( · , k), η− ( · , −k) = −2isgn(k)e−2Re −R() (V (y)−k ) dy . Proof. From Proposition B.5 we obtain that: η+ (x, k) ∼ (−ik)− 2 e−S(0,x,k) , 1

η+ (x, k) ∼ −(−ik) 2 e−S(0,x,k) , 1

Using that (ik) 2 (−ik)− 2 = i , 1

1

(−ik) 2 ( ik)− 2 = −i for k > 0 , 1

1

x → +∞ .

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we obtain η+ (x, k)η+ (x, −k) − η+ (x, k)η+ (x, −k) ∼ 2ie−2ReS(0,x,k) ,

x → +∞ .

(B.15)

If k ≥ and y ≥ R() we have V (y) − k < 0 so Re(V (y) − k ) = 0. Letting x → +∞ in (B.14) we obtain the first identity for k > 0 and replacing then k by −k for all k = 0. The proof of the second identity is similar, using instead 2

η− (x, k) ∼ (−ik)− 2 eS(0,x,k) , 1

2

1

η− (x, k) ∼ (−ik) 2 eS(0,x,k) ,

x → −∞ .

Proposition B.7. Set for |k| ≥ : 1

θ+ (x, k) := η+ (x, k)|k| 2 eRe 1 2

θ− (x, k) := η− (x, k)|k| e

Re

R() 0

(V (y)−k2 )dy

,

(B.16)

0

(V (y)−k2 )dy −R()

.

Then we have:

θ± (x, k) = θ± (x, −k) W θ± ( · , k), θ± ( · , −k) = ±2ik ,

and |θ± (x, k)| ≤ C ,

|θ± (x, k)| ≤ C |k| ,

uniformly for

|k| ≥ ,

±x ≥ 0 .

Proof. The first statement follows from Lemma B.6. To prove the second statement we use Proposition B.5. In fact by (B.12), the bounds in the second statement are valid uniformly for |k| ≥ and ±x ≥ ±R(). Let us first fix C 1 such that for ≥ C 1, we have R() = 0. Hence the bounds in the second statement are valid uniformly for |k| ≥ C and ±x ≥ 0. It remains to check the bounds uniformly for ≤ |k| ≤ C and ±x ∈ [0, R()]. We have θ± ± R(), k + θ± ± R(), k ≤ C() . Writing the differential equation satisfied by θ± as a first order system, we see that this bound extends to ±x ∈ [0, R()] uniformly for ≤ |k| ≤ C. B.5. Resolvent and spectral family Proposition B.8. We set as in Subsection B.1: w(k) := W θ+ ( · , k), θ− ( · , k) , m(k) := (2ik)−1 w(k) . The family {ψ( · , k)} defined by ! m(k)−1 θ+ (x, k) k > 0 , ψ(x, k) := m(−k)−1 θ− (x, −k) k < 0

(B.17)

is a family of generalized eigenfunctions of h in |k| ≥ . Proof. As in Subsection B.2, we can since η± ( · , ζ) ∈ L2 (R± ) for Imζ > 0 write the kernel of (h − z)−1 as: ! −r(ζ)−1 η+ (x, ζ)η− (y, ζ) , y ≤ x , R(x, y, z) = −r(ζ)−1 η− (x, ζ)η+ (y, ζ) , x ≤ y .

Vol. 9 (2008)


for ζ 2 = z, Imζ > 0 and

1621

r(ζ) = W η+ ( · , ζ), η− ( · , ζ) .

The zeroes of w lie on iR+ and correspond to negative eigenvalues of h. We write the kernel of the spectral family dE dλ (x, y, λ) using the functions θ± (x, ±k). Using (B.16) we obtain: dE (x, y, λ) = m(k)−1 θ+ (x, k)θ− (y, −k) 4πk dλ + m(−k)−1 θ+ (x, −k)θ− (y, k) , for y ≤ x, where k 2 = λ and By Proposition B.7 the algebraic identities used in the proof of Proposition B.2 are satisfied by θ± ( · , k). Repeating the above proof we obtain the proposition. B.6. Bounds on generalized eigenfunctions away from k = 0 The following result shows that generalized eigenfunctions are always uniformly bounded in |k| ≥ for > 0. Proposition B.9. Assume V ∈ S −μ for μ > 0. Then for {ψ(x, k)}k∈R defined in (B.17) one has for all > 0: ψ( · , k)∞ ≤ C

uniformly for

|k| ≥ .

Proof. Arguing as in the proof of Theorem B.3 it suffices by Proposition B.7 to verify that 2|k| |m(k)|−1 = , w(k) is uniformly bounded for |k| ≥ . We first claim that w(k) is a continuous function of k in |k| ≥ . In fact writing the Volterra integral equation (B.14) as a fixed point equation in an appropriate Banach space of continuous functions, we see that for a fixed x ≥ R(), u1 (x, k) and u2 (x, k) are continuous functions of k in |k| ≥ . The same holds for η+ (x, k), (x, k). Using the differential equation satisfied by η+ ( · , k), we see that k → η+ (0, k)) is continuous in k. Using the same argument for η− ( · , k), we (η+ (0, k), η+ obtain the continuity of w(k) in |k| ≥ . We note that w(k) does not vanish in |k| ≥ since w(k) = 0 would imply that k 2 is an eigenvalue of h which is impossible if V ∈ S −μ . Therefore |m(k)|−1 is locally bounded in |k| ≥ . It remains to bound |m(k)|−1 near infinity. We use the notation in the proof of Proposition B.5. Let us pick C 1 such that R(C) = 0. Then for k ≥ C we have: F (x, k) = −ik 1 + 0 x −μ |k|−2 , M (x, k) = O x −2−μ |k|−3 . Using the fact that u1 , u2 are uniformly bounded in x ≥ 0 and k ≥ C, we obtain from (B.14) that u1 (0, k) = O(|k|−3 ) ,

u2 (0, k) = 1 + O(|k|−3 ) ,

1622


which yields

1 η+ (0, k) = (−ik)− 2 1 + O(|k|−2 ) ,

The same argument gives 1 η− (0, k) = (−ik)− 2 1 + O(|k|−2 ) ,


1 η+ (0, k) = −(−ik) 2 1 + O(|k|−2 ) . 1 η− (0, k) = (−ik) 2 1 + O(|k|−2 ) ,

and hence W (η+ ( · , k), η− ( · , k)) = −2+0(|k|−2 ). Using that θ± (x, k) = η± (x, k)|k| 2 for k ≥ C, we obtain that |w|(k) ∼ 2|k| when k → ∞, which shows that |m|−1 (k) is uniformly bounded near infinity. 1

B.7. Condition (BM2) for slowly decreasing potentials In this subsection we give some classes of slowly decreasing potentials for which condition (BM2) holds. As in Subsection B.3 the possible existence of zero energy resonances has to be taken into account. For quickly decreasing potentials, the definition of zero energy resonances is connected with the integral equation (B.1) for ζ = 0. For slowly decreasing potentials, we have to consider instead the integral equations (B.14). This leads to the following definition: Assume that v ∈ S −μ for 0 < μ < 2 is such that |V (x)| ≥ c x −μ for |x| large enough. We will say that h has a zero energy resonance if there exists a solution of −u + V u = 0 , with asymptotics: x 1 u(x) = u± V (x)−1/4 e∓ 0 (V (s)) 2 ds 1 + o(1) , x → ±∞ , x 1 u (x) = ∓u± V (x)1/4 e± R (V (s)) 2 ds 1 + o(1) , x → ±∞ , for constants u± = 0. Potentials negative near infinity We consider first the case of potentials which are negative near infinity. We assume that V ∈ S −μ for 0 < μ < 2 and: V (x) ≤ −c x −μ

in |x| ≥ R ,

for some

c, R > 0 .

(B.18)

Proposition B.10. Assume that V ∈ S −μ for 0 < μ < 2 satisfies (B.18) and has no zero energy resonance. Then condition (BM2) holds for M (x) = x μ/4 . Proof. By Proposition B.9 it suffices to consider the region |k| ≤ 1. We fix R as (B.18) and define the functions η± (x, k) using the phase S(±R, x, ζ). We will consider only the + case. We first claim that (x, k) ∈ O(1) , θ+ (x, k) ∈ O x μ/4 , θ+ uniformly in x ≥ −R ,

|k| ≤ 1 .

(B.19)

Clearly it suffices to prove the statement in x ≥ R, since we can extend the bound to [−R, R] using the differential equation satisfied by θ+ . Let us prove (B.19). We

Vol. 9 (2008)


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will simply write “f (x, k) ∈ O( x )” for “f (x, k) ∈ O( x ) uniformly in x ≥ R, |k| ≤ 1”. The function F (x, k) is smooth in |x| ≥ R and one has |F (x, k)| ≥ c x −μ/2 . This implies that M (x, k) ∈ O( x −2+μ/2 ), from which we get u1 (x, k) ∈ O x −1+μ/2 , u2 (x, k) ∈ O(1), and

η+ (x, k) ∈ O x μ/4 ,

η+ (x, k) ∈ O(1) .

This proves (B.19). Next as in Subsection B.3, we can set U = (u1 , u2 − 1) and consider the equations (B.14) as a fixed point equation: 1l + T (k) U = F , in the Banach space B=

"

U = (v1 , v2 ) |vi continuous,

sup | x

1−μ/2

vi (x)| < ∞ .

[R,+∞[

For R large enough, T (k) < 12 uniformly in |k| ≤ 1 and k → T (k) is norm continuous. It follows that k → U (k) ∈ B is continuous up to k = 0. Therefore (u1 ( · , k), u2 ( · , k) − 1) has a limit (u1 ( · , 0), u2 ( · , 0) − 1) in B when k → 0. This implies also that (η+ ( · , k), η+ ( · , k)) converges locally uniformly when k → 0 to the pair (η+ ( · , 0), η+ ( · , 0)) obtained from (u1 ( · , 0), u2 ( · , 0)) by formula (B.13) for k = 0. We see that η+ (x, 0) is a solution of −u + V (x)u = 0 , with asymptotics: 1 + o(1) , x 1 η+ (x, 0) = −V (x)1/4 ei R (−V (s)) 2 ds 1 + o(1) , η+ (x, 0) = V (x)−1/4 ei

x R

1

(−V (s)) 2 ds

x → +∞ .

By the convergence result above (and its analog for η− ( · , k)), we also see that lim m(k) =: m(0) = cW η+ ( · , 0), η− ( · , 0) , k→0

for some c = 0. Clearly m(0) = 0 iff h admits a zero energy resonance. Using (B.17), (B.19) and Proposition B.9, we obtain then that |ψ(x, k)| ≤ C x μ/4 ,

uniformly for x ∈ R ,

which completes the proof of the proposition.

k ∈ R,

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Potentials positive near infinity Let us now consider the case of potentials which are positive near infinity. The following lemma is shown in [17, Theorem 4]. Lemma B.11. Assume that V ∈ S −μ for 0 < μ < 2 is positive near infinity, more precisely: (B.20) V (x) ∼ q0 |x|−μ , x → ∞ , q0 > 0 . Then there exists unique solutions η± (x, 0) of −u + V u = 0 , with asymptotics: η+ (x, 0) ∼ V (x)− 4 e− 1

η− (x, 0) ∼ V (x)− 4 e− 1

x a

x a

1

(V (y)) 2 dy (V

1 (y)) 2

dy

, ,

η+ (x, 0) ∼ −V (x) 4 e− 1

η− (x, 0) ∼ V (x) 4 e− 1

x a

x −a

1

(V (y)) 2 dy

(V

1 (y)) 2

dy

,

,

x → +∞ , x → −∞ ,

where a 1 is such that V (x) > 0 in |x| ≥ a. Lemma B.12. Assume in addition to (B.20) that there exists θ, R > 0 such that V extends holomorphically to D(R, θ) = {z ∈ C||z| > R , |Argz| < θ} and satisfies |V (z)| ≤ C(1 + |z|)−μ , Then for any ±x ≥ R, s → 0.

(η± (x, s), η± (x, s))

z ∈ D(R, θ) .

converges to (η± (x, 0), η± (x, 0)) when

Proof. We check that the assumptions of [17, Theorem 7] are satisfied. We consider the two parts D± (R, θ) = D(R, θ)∩{±Rez > 0} of D(R, θ) and set z = log(±z) for z ∈ D± (R, θ). Applying Hadamard three lines theorem to F (z ) = V (ez )(±eμz )− q0 , we obtain V (z) ∼ q0 (±z)−μ when |z| → +∞ , uniformly in D(R, θ0 ) ∩ {±Rez > 0} for all 0 < θ0 < θ. Similarly it follows from Cauchy’s inequalities that |∂zk V (z)| ≤ C(1 + |z|)−k−μ ,

z ∈ D(R, θ0 ) .

Applying then [17, Theorem 7], we obtain the lemma.

Lemma B.13. The functions η± (x, k) are uniformly bounded for |x| ≤ R, |k| ≤ 1. Proof. We consider only the case of η+ (x, k). Let φ0 (x, k), φ1 (x, k) the two regular solutions of (E) with boundary conditions: φ0 (0, k) = 1 ,

φ0 (0, k) = 0 ,

φ1 (0, k) = 0 ,

φ1 (0, k) = 1 .

Clearly φi (x, k), φi (x, k) are uniformly bounded and continuous in {(x, k)| |x| ≤ R, |k| ≤ C}. We have η+ ( · , k) = a1 (k)φ0 ( · , k) + a0 (k)φ1 ( · , k) , for ai (k) = W η+ ( · , k), φi ( · , k) . By Lemma B.12, ai (k) converges to ai (0) when k → 0.

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Proposition B.14. Assume that V ∈ S −μ for 0 < μ < 2 satisfies the hypotheses of Lemma B.12 and has no zero energy resonance. Then for each R > 0 condition (BM2) is satisfied for 1 for |x| ≤ R , M (x) = +∞ for |x| > R . We refer the reader to Remark 4.1 for the meaning of (BM2) if M takes its values in [0, +∞]. Proof. The uniform boundedness of η± (x, k) and hence of θ± (x, k) for |x| ≤ R and |k| ≤ 1 is shown in Lemma B.13. We have to show that limk→0 m(k) =: m(0) = 0. The limit exists and equals cW (η+ ( · , 0), η− ( · , 0)) for some c = 0 by Lemma B.12. By Lemma B.11 m(0) = 0 iff h has a zero energy resonance.

Appendix C. C.1. Proof of Proposition 2.10 We forget the superscript w to simplify notation. We recall the following characterization of Op(S p,m ) known as the Beals criterion: M ∈ Op(S p,m ) iff M : S(Rd ) → S(Rd ) and β D −p+|α| x −m+|β| adα x adD M

for all α, β ∈ Nd . (C.1) The topology given by the norms of the multicommutators with Op(m) in (C.1) is the same as the original topology on S p,m . We will need also similar objects for symbols and operators depending on a real parameter s ≥ 0. We say that m(s, x, ξ) belongs to S p,m,k if k |∂xα ∂kβ m(s, x, k)| ≤ Cα,β k 2 + s k −p+|α| x −m+|β| , α, β ∈ Nd , is bounded on L2 (Rd ) ,

uniformly for s ≥ 0. By the result recalled above, we see that M (s) ∈ Op(S p,m,k ) iff M (s) : S(Rd ) → S(Rd ) and −k β 2 d D 2 + s

D −p+|α| x −m+|β| adα x adD M (s) is bounded on L (R ) , (C.2) uniformly for s ≥ 0. Let us now prove Proposition 2.10. By elliptic regularity, we know that h is self-adjoint and bounded below on H 2 (Rd ) and (h + s)−1 preserves S(Rd ). β −1 Computing multicommutators adα on S(Rd ), we first see by induction x adD (h+s) 2 α β −1 −α on α, β that ( D + s ) x D (h+s) x D −β ∈ O(1), uniformly in s ≥ 0. β The same computations show then that ( D 2 + s ) D |α| x |β| adα x adD (h + −1 2 d s) is uniformly bounded on L (R ), which by the Beals criterion show that (h + s)−1 ∈ Op(S 0,0,−2 ) .

(C.3)

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Using the formula λ−α = cα

+∞

s−α (λ + s)−1 ds ,


for λ ≥ 0 ,

α ∈]0, 1[

(C.4)

0

we obtain that h−α ∈ Op(S −2α,0 ) for α ∈]0, 1[. Using also that hn ∈ Op(S 2n,0 ) for integer n, we obtain ii). C.2. Proof of Proposition 2.11 Let us first prove i). We use the notation in Subsection C.1. Set T (s) = Op((b + s)−1 ). By pdo calculus and (C.3), we get that 1

(h + s)T (s) − 1l ∈ Op(S 0,−1−μ,− 2 ) hence

(h + s)−1 − T (s) ∈ Op(S 0,−1−μ,−3/2 ) . Using (C.4) for = 12 this implies that h− 2 − Op(b− 2 ) ∈ Op(S −2,−1−μ ) . 1

1

Next we write using again pdo calculus: h 2 = hh− 2 = hOp(b− 2 ) + Op(S 0,−1−μ ) = Op(b 2 ) + Op(S 0,−1−μ ) , 1

1

1

1

which proves i). Let us now prove ii). By Proposition 2.10, we know that ω = Op(c) ,

1

for c − b 2 ∈ S 0,−1−μ ,

where b is defined in Proposition 2.10. By pseudodifferential calculus, we obtain that:

ω, i ω, i x = Op c, c, x

+ Op(S 0,−2 ) . Since c − k ∈ S 1,−μ , we get: 2

ξ (ξ|x)2 − +S 0,−1−μ . c, c, x = k , k , x +S 0,−1−μ = x −1 k 2 k 2 x 2 We pick 0 < 1 and write: 2 ξ (ξ|x)2 − = d2 (x, ξ) − x −2 , for d(x, ξ) k 2 k 2 x 2 2 12 ξ (ξ|x)2 −2 = − + x

. k 2 k 2 x 2 Using that d2 ∈ S 0,0 and d2 ≥ x −2 , we see easily that d ∈ S0,0 , hence x − 2 d ∈ 1

0,− 12

S

. Using again (2.6), we get: 1 2 Op x −1 d2 = Op x − 2 d + Op(S0,−3+4 ) .

Choosing > 0 small enough and setting γ = x − 2 d, we obtain the proposition. 1

Vol. 9 (2008)


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C.3. A technical lemma Lemma C.1. Let h = Da(x)D + c(x) for a, c as in (3.1), h∞ = D2 + m2∞ . Set 1

2 ω = h 2 , ω ∞ = h∞ . Let χ ∈ C0∞ (R) and F ∈ C ∞ (R) with F ≡ 0 near 0 and F ≡ 1 near ∞. Then for C large enough ω ω ∞ F ω∞ ∈ O(1) . χ κ Cκ 1

Proof. We know from Proposition 2.10 ii) that ω and ω∞ and hence [ω, ω∞ ] belong to Opw (S 1,0 ). Using formula (2.1), we deduce from this fact that for χ ∈ C0∞ (R): ω χ (C.5) , ω∞ ∈ O(1) . κ We take χ1 ∈ C0∞ (R) with χ1 χ = χ and set ω ω ω ˜ ∞ = χ1 ω ∞ χ1 . κ κ We first see that ω ω ω (ω∞ − ω , ω∞ (1 − χ1 ) ∈ O(1) , ˜∞) = χ χ κ κ κ by (C.5). We claim also that for F ∈ C0∞ (R): ω ω ω ˜∞ ∞ −F χ F ∈ O(κ−1 ) . κ κ κ

(C.6)

(C.7)

In fact we write using (2.1): ω ω ω ˜∞ ∞ χ −F F κ κ κ −1 ω ω∞ −1 −1 ω ˜∞ i z− ∂ z F˜ (z)χ κ (ω∞ − ω ˜∞) z − dz ∧ d z = 2π C κ κ κ ω i ω∞ −1 −1 ω = z− (ω∞ − ω ∂ z F˜ (z)χ κ χ ˜∞) 2π C κ κ κ −1 ω ˜∞ dz ∧ d z × z− κ ω∞ −1 −1 ω ω∞ −1 i χ + , ω∞ z − ∂ z F˜ (z) z − κ 2π C κ κ κ −1 ω ˜∞ × κ−1 (ω∞ − ω ˜∞) z − dz ∧ d z . κ This is easily seen to be O(κ−1 ) using the fact that (z − a)−1 , a(z − a)−1 are O(|Imz|−1 ) for z ∈ supp F˜ . We note then that ω ˜ ∞ ≤ c1 κ ,

1628



for some c1 > 0 since ω∞ ≤ c0 ω. Hence if G(s) = F (C −1 s) for F as in the lemma and C is large enough, we have G( ω˜κ∞ ) = 0. Applying then (C.7) to F = 1 − G, we obtain that ω ω ∞ (C.8) χ G ∈ O(κ−1 ) . κ κ We write: ω ω ω ω ω ω ∞ ∞ ∞ χ G ω∞ = ω∞ χ G + χ , ω∞ G κ κ κ κ κ κ ω ω∞ ω ω ∞ G + χ , ω∞ G . = ω∞ ω −1 ωχ κ κ κ κ The first term in the last line is O(1) using (C.8), the second also using (C.5). This completes the proof of the Lemma.

References [1] W. Amrein, A. Boutet de Monvel, W. Georgescu, C0 -groups, commutator methods and spectral theory of N -body hamiltonians, Birkh¨ auser, Basel, Boston, Berlin, 1996. [2] J. Derezinski, C. Gérard, Spectral and scattering theory of spatially cut-off P (ϕ)2 Hamiltonians, Comm. Math. Phys. 213 (2000), no. 1, 39–125. [3] J. Dimock, The P (ϕ)2 model with variable coefficients, Ann. Phys. 154 (1984), no. 2, 120–149. [4] C. Gérard, A. Panati, Spectral and scattering theory for some abstract QFT Hamiltonians, preprint (2008). [5] V. Georgescu, On the spectral analysis of Quantum Field Hamiltonians, J. Funct. Anal. 245 (2007), 89–143. [6] J. Glimm, A. Jaffe, Boson quantum field theory models, in Mathematics of Contemporary Physics R. Streater (ed.) (1972) Academic Press. [7] L. H¨ ormander, The analysis of linear partial differential operators, vol. III, Springer Verlag Berlin, Heidelberg, New York, (1985). [8] A. Klein, L. Landau, Singular perturbations of positivity preserving semigroups via path space techniques, J. Funct. Anal. 20 (1975), 44–82. [9] A. Klein, L. Landau, Construction of a unique selfadjoint generator for a symmetric local semigroup, J. Funct. Anal. 44 (1981), 121–137. [10] M. Reed, B. Simon, Methods of modern mathematical physics, vol. 4, Academic Press (1978). [11] L. Rosen, The (φ2n )2 Quantum field theory: Higher order estimates, Comm. Pure Appl. Math. 24 (1971), 417–457. [12] I. Segal, Construction of non linear local quantum processes I, Ann. Math. 92 (1970), 462–481. [13] B. Simon, The P (φ)2 Euclidean (Quantum) field theory, Princeton University Press, 1974. [14] B. Simon, R. Høgh-Krohn, Hypercontractive semigroups and two dimensional selfcoupled Bose fields, J. Funct. Anal. 9 (1972), 121–180.

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[15] S. Teufel, D. D¨ urr, K. M¨ unch-Bernandl, The flux across surfaces theorem for short range potentials and wave functions without energy cutoffs, J. Funct. Anal. 40 (1999), no. 4, 1901–1922. [16] D. Yafaev, Mathematical scattering theory (Analytic Theory), book in preparation. [17] D. Yafaev, The low energy scattering for slowly decreasing potentials, Comm. Math. Phys. 85 (1982), 177–196. Christian Gérard and Annalisa Panati Laboratoire de Mathématiques Université de Paris XI F-91405 Orsay Cedex France e-mail: [email protected] [email protected] Communicated by Vincent Rivasseau. Submitted: June 26, 2008. Accepted: September 3, 2008.

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Annales Henri Poincaré - Volume 9

Annales Henri Poincaré - Volume 2

Annales Henri Poincaré - Volume 11

Annales Henri Poincaré - Volume 4

Annales Henri Poincaré - Volume 3

Annales Henri Poincaré - Volume 6

Annales Henri Poincaré - Volume 10

Annales Henri Poincaré - Volume 1

Annales Henri Poincaré - Volume 5

Annales Henri Poincaré - Volume 4 - Supplements

Henri Poincaré

Henri IV

Henri Charriere

Encyclopaedia Judaica, Volume 9

Annales - Livre VI 31

Annales du DALF

Henri Michaux: Icebergs

Les Annales Akashiques

Les Annales des Heechees

Business Plans Handbook Volume 9

1001 Arabian Nights - Volume 9

Advances in Computers, Volume 9

Drama for Students Volume 9

Philosophy in Process, Volume 9

Progress in Optics, Volume 9

Advances in Immunology Volume 9

Topics in Stereochemistry, Volume 9

Archiv für Religionsgeschichte: Volume 9

Tafseer-e-Siddiqi (Volume 9)

Advances in Genetics Volume 9

Henri Poincaré : impatient genius

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