Ann. Henri Poincar´e 5 (2004) 1 – 73 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010001-73 DOI 10.1007/s00023-004-0160-1
Annales Henri Poincar´ e
Non-Selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I Michael Hitrik and Johannes Sj¨ ostrand Abstract. This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength of the perturbation is h (or sometimes only h2 ) and bounded from above by hδ for some δ > 0. We get a complete asymptotic description of all eigenvalues in certain rectangles [−1/C, 1/C] + i[F0 − 1/C, F0 + 1/C].
1 Introduction In [20], A. Melin and the second author observed that for a wide and stable class of non-selfadjoint operators in dimension 2 and in the semi-classical limit (h → 0), it is possible to describe all eigenvalues individually in an h-independent domain in C, by means of a Bohr-Sommerfeld quantization condition. This result is quite remarkable since the corresponding conclusion in the selfadjoint case seems to be possible only in dimension 1 or under strong (and unstable) assumptions of complete integrability. The underlying reason for this result is the absence of small denominators which allows us to avoid the usual trouble with exceptional sets in the KAM theorem. As a next step, the second author noticed ([22]) that for non-selfadjoint operators of the form P (x, hDx ) + iQ(x, hDx ) it is possible to find a similar result, when P is selfadjoint, > 0 small and fixed and the classical bicharacteristic flow is periodic on each real energy surface. (Again, it is important that we are in dimension 2.) The method is similar to the one in [20] and uses non-linear CauchyRiemann equations, now in an “-degenerate” form. (See also [24] for a different extension.) It soon became quite clear that we run into a fairly vast program, and that logically one should start with even smaller perturbations, say = O(hδ ), for some δ > 0. The present work is planned to be the first in a series, devoted to small perturbations of selfadjoint operators in dimension 2. In addition to the challenge of doing plenty of things in dimension 2, that can usually only be done in dimension 1, we have been motivated by recent progress around the damped wave equation ([19], [2], [25], [14]), as well as the problem of barrier top resonances for the semi-classical Schr¨ odinger operator ([17]) where more complete results than the corresponding ones for eigenvalues of potential wells ([26], [3], [21]) seem possible. One long term goal of this series is to get improved results on the distribution of
2
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
resonances for strictly convex obstacles in R3 . See [30] (and references given there) for a first result on Weyl asymptotics for the real parts inside certain bands. In the case of analytic obstacles, much more can probably be said, especially in dimension 3 (and 2). Let M denote R2 or a compact real-analytic manifold of dimension 2. When M = R2 , let (1.1) P = P (x, hDx , ; h) be the Weyl quantization on R2 of a symbol P (x, ξ, ; h) depending smoothly on ∈ neigh (0, R) with values in the space of holomorphic functions of (x, ξ) in a tubular neighborhood of R4 in C4 , with |P (x, ξ, ; h)| ≤ Cm(Re (x, ξ))
(1.2)
there. Here m is assumed to be an order function on R4 , in the sense that m > 0 and (1.3) m(X) ≤ C0 X − Y N0 m(Y ), X, Y ∈ R4 . We also assume that m ≥ 1.
(1.4)
We further assume that P (x, ξ, ; h) ∼
∞
pj, (x, ξ)hj , h → 0,
(1.5)
j=0
in the space of such functions. We make the ellipticity assumption |p0, (x, ξ)| ≥
1 m(Re (x, ξ)), |(x, ξ)| ≥ C, C
for some C > 0. When M is a compact manifold, we let P = aα, (x; h)(hDx )α ,
(1.6)
(1.7)
|α|≤m
be a differential operator on M , such that for every choice of local coordinates, centered at some point of M , aα, (x; h) is a smooth function of with values in the space of bounded holomorphic functions in a complex neighborhood of x = 0. We further assume that aα, (x; h) ∼
∞
aα,,j (x)hj , h → 0,
(1.8)
j=0
in the space of such functions. The semi-classical principal symbol in this case is given by p0, (x, ξ) = aα,,0 (x)ξ α , (1.9)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
3
and we make the ellipticity assumption |p0, (x, ξ)| ≥
1 ξm , (x, ξ) ∈ T ∗ M, |ξ| ≥ C, C
(1.10)
for some large C > 0. (Here we assume that M has been equipped with some Riemannian metric, so that |ξ| and ξ = (1 + |ξ|2 )1/2 are well defined.) Sometimes, we write p for p0, and simply p for p0,0 . Assume P=0 is formally selfadjoint.
(1.11)
In the case when M is compact, we let the underlying Hilbert space be L2 (M, µ(dx)) for some positive real-analytic density µ(dx) on M . Under these assumptions, P will have discrete spectrum in some fixed neighborhood of 0 ∈ C, when h > 0, ≥ 0 are sufficiently small, and the spectrum in this region will be contained in a band |Im z| ≤ O(). The purpose of this work and later ones in this series, is to give detailed asymptotic results about the distribution of individual eigenvalues inside such a band. Assume for simplicity that (with p = p=0 ) p−1 (0) ∩ T ∗ M is connected.
(1.12)
∂ ∂ − px · ∂ξ be the Hamilton field of p. In this work, we will always Let Hp = pξ · ∂x assume that for E ∈ neigh (0, R):
The Hp -flow is periodic on p−1 (E) ∩ T ∗ M with period T (E) > 0 depending analytically on E.
(1.13)
∂ Let q = 1i ( ∂ )=0 p , so that
p = p + iq + O(2 m),
(1.14)
in the case M = R2 and p = p + iq + O(2 ξm ) in the manifold case. Let T (E)/2 1 q = q ◦ exp tHp dt on p−1 (E) ∩ T ∗ M. (1.15) T (E) −T (E)/2 Notice that p, q are in involution; 0 = Hp q =: {p, q}. In Section 3, we shall see how to reduce ourselves to the case when p = p + iq + O(2 ), −1
(1.16)
∗
near p (0) ∩ T M . An easy consequence of this is that the spectrum of P in {z ∈ C; |Re z| < δ} is confined to ] − δ, δ[+i]Re qmin,0 − o(1), Re qmax,0 + o(1)[, when δ, , h → 0, where Re qmin,0 = minp−1 (0)∩T ∗ M Re q and similarly for qmax,0 . We will mainly think about the case when q is real-valued but we will work under the more general assumption that Im q is an analytic function of p and Re q, in the region of T ∗ M , where |p| ≤ 1/|O(1)|.
(1.17)
4
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
Let F0 ∈ [Re qmin,0 , Re qmax,0 ]. The purpose of the present work is to determine all eigenvalues in a rectangle ]−
1 1 1 1 , [ + i ]F0 − , F0 + [, |O(1)| |O(1)| |O(1)| |O(1)|
(1.18)
for h ≤ O(hδ ),
(1.19)
where δ > 0 is any fixed number. (When the subprincipal symbol of P is zero, we can treat even smaller values of : h2 ≤ O(hδ ).) We will achieve this under the general assumption that T (0) is the minimal period of every Hp -trajectory in Λ0,F0 , where
Λ0,F0 := {ρ ∈ T ∗ M ; p(ρ) = 0, Re q(ρ) = F0 },
(1.20)
(1.21)
in the following three cases: I) The first case is when dp, dRe q are linearly independent at every point of Λ0,F0 .
(1.22)
This implies that every connected component of Λ0,F0 is a two-dimensional Lagrangian torus. For simplicity, we shall assume that there is only one such component. Notice that in view of (1.20), the space of closed orbits in p−1 (0) ∩ T ∗ M ; Σ := (p−1 (0) ∩ T ∗ M )/ ∼, where ρ ∼ µ if ρ = exp tHp µ for some t ∈ R, becomes a 2-dimensional symplectic manifold near the image of Λ0,F0 , and (1.22) simply means that Re q, viewed as a function on Σ, has non-vanishing differential along the image of Λ0,F0 . The image of Λ0,F0 is just a closed curve. The main results in this case are Theorems 6.2, 6.4 and they show that the eigenvalues form a distorted lattice. II) The second case is when F0 ∈ {Re qmin,0 , Re qmax,0 }. In this case, we again view Re q as a smooth function on Σ near the image of Λ0,F0 and assume that The Hessian of Re q is non-degenerate (positive or negative) at every point ρ ∈ Σ, with Re q(ρ) = F0 .
(1.23)
The main results in this case are given by Theorems 6.6, 6.7 which tell us that the eigenvalues form a distorted half-lattice. III) The third natural case would be when F0 is a critical value of Re q corresponding to a saddle point. We hope to study this case in the near future. The analyticity assumptions are introduced, because the optimal spaces are deformations of the usual L2 -space obtained by adding exponential weights with
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
5
exponents that are O(), and there are closely related Fourier integral operators with complex phase some of which have associated complex canonical transformations that are -perturbations of the identity. When ∼ hδ , 0 < δ < 1, appropriate Gevrey type assumptions would probably suffice, but in the case ∼ h we seem to need analyticity assumptions at one point, even though standard C ∞ -microlocal analysis would suffice for most of the steps. At the opposite extreme, small but independent of h, the analyticity assumptions seem necessary, and in order to avoid technicalities, we have chosen to assume analyticity independently of the size of . In the selfadjoint case there have been many works about operators whose associated classical flow is periodic ([31], [8], [5], [11], [9], [16]), and we follow one of the main ideas in those works, namely to use some sort of averaging procedure in order to reduce the dimension by one unit, so that in our case, we come down to a one-dimensional problem. The implementation of this is more complicated in our case because of the need to work in modified exponentially weighted spaces (after suitable FBI-transforms). It should also be pointed out that in the case when is small but independent of h ([22]), this does not seem to work and the problem remains two-dimensional. The same seems to be the case (for the whole scale of ) in other situations, when the Hp -flow is completely integrable without being periodic, or more generally when the energy surface p−1 (0) ∩ T ∗ M contains certain invariant Lagrangian tori. We intend to treat such situations later in this series. The plan of the paper is the following: In Section 2, we reexamine the Egorov theorem in a form suitable for us, and complete some observations of [13] about the two term version of this result. In Section 3 we perform dimension reduction by averaging. In Section 4 we make a complete reduction in the torus case (I) and determine the corresponding quasi-eigenvalues. In Section 5 we do the analogous work in the extreme case (II). In Section 6 we justify the earlier computations by treating an auxiliary global (Grushin) problem, and we obtain the two main results. In Section 7, we give a first application to barrier top resonances. In the appendix, we review some standard facts about FBI-transforms on manifolds. The next work(s) in this series (in addition to [22]) will remain in the case when the classical flow of the unperturbed part is periodic. We intend to study the saddle point case (III), and the case when q vanishes.
6
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
2 Quantization of canonical transformations between non-simply connected domains in phase space We first give an affirmative answer to a question asked in Appendix A of [13]. Let κ : neigh ((y0 , η0 ), T ∗ Rn ) → neigh ((x0 , ξ0 ), T ∗ Rn ) be an analytic canonical transformation and consider a corresponding Fourier integral operator n+N U u(x) = h− 2 (2.1) eiφ(x,y,θ)/ha(x, y, θ; h)u(y)dydθ, with a = a0 + O(h), a classical symbol in S 0,0 (see the appendix), and φ nondegenerate phase function in the sense of H¨ ormander [15] (without the homogeneity requirement in θ) which generates the graph of κ. (Since we work microlocally, φ, a are assumed to be defined near a fixed point (x0 , y0 , θ0 ) with φθ (x0 , y0 , θ0 ) = 0, (x0 , ξ0 ) = (x0 , φx (x0 , y0 , θ0 )), (y0 , η0 ) = (y0 , −φy (x0 , y0 , θ0 )).) We require U to be unitary: (2.2) U ∗ U = 1, microlocally near (y0 , η0 ), and we are interested in the improved Egorov property: If P U = U Q, where P = P w , Q = Qw are h-pseudodifferential operators of order 0, then P ◦ κ = Q + O(h2 ).
(2.3)
Here and in what follows we use the same letter to denote an operator and a corresponding Weyl symbol. In Appendix A of [13], it was shown that such U ’s exist and we shall answer the question raised there, by establishing the following proposition. (We learned from C. Fefferman that Jorge Silva has obtained essentially the same result in the framework of classical Fourier integral operators.) Proposition 2.1 Within the class of operators satisfying (2.1) and (2.2), the property (2.3) is equivalent to: a0| C has constant argument. φ
(2.4)
Here φ is defined in some open set D(φ) ⊂ R2n+N and Cφ = {(x, y, θ) ∈ D(φ); φθ (x, y, θ) = 0}. Proof. We first consider the special case of pseudodifferential operators, i.e., the case when κ is the identity. Then a0 is the principal symbol and (2.2) implies that |a0 | = 1 (after inserting an additional factor (2π)−n in front of the integral and taking the standard phase φ = (x − y) · θ). Write U −1 P U = P + U −1 [P, U ]. We see that (2.3) holds iff {p, a0 } = 0 for all p, i.e., iff a0 = Const. The proposition follows in the case of pseudodifferential operators since we also know in general
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
7
that the property (2.4) is invariant under changes of (φ, a) in the representation of the given operator. When φ is quadratic and a is constant, we have a metaplectic operator and κ is linear. In that case, we know that (2.3) holds, and using the special case of h-pseudodifferential operators, we see that we have equivalence between (2.3) and (2.4) in the case when κ is linear. Consider a smooth deformation of canonical transformations [0, 1] t → κt , with a deformation field Ha(t) , so that ∂t κt (ρ) = Ha(t) (κt (ρ)) where a(t) = a(t, x, ξ) is smooth and independent of h. Let A(t) = aw (x, hDx ) and consider a corresponding family of Fourier integral operators U (t) associated to κt : hDt U (t) + A(t) ◦ U (t) = 0.
(2.5)
Since A(t) are selfadjoint, unitarity of U (t) is conserved under the flow of (2.5). Let U (t) be such a unitary family. Proposition 2.2 We have (2.3) for one value of t iff we have it for all values of t. Proof. Suppose we have (2.3) for U (0). From (2.5) we get hDt (U (t)−1 ) = U (t)−1 A(t). Consider a family P (t) = U (t)P U (t)−1 . Then hDt P (t) + [A(t), P (t)] = 0, and on the level of Weyl symbols, we get ∂t P (t) + {a(t), P (t)} = O(h2 ), or in other words, (∂t + Ha(t) )P (t) = O(h2 ). This means that P (t) ◦ (κt (ρ)) = P (0) ◦ κ0 + O(h2 ) = P (ρ) + O(h2 ), where we used (2.3) for U (0) in the last step. Then P (t) fulfills (2.3) for all t. On the other hand, if U (t) fulfills (2.5), we know, using that the subprincipal symbol of A(t) is 0, that if we represent n+N i U (t) = h− 2 e h φt (x,y,θ)at (x, y, θ; h)u(y)dydθ, with φt , at depending smoothly on t, then the argument of at,0 |
C φt
is constant
along every curve in {(t, x, θ); (x, θ) ∈ Cφt } corresponding to a Ha(t) -trajectory:
8
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
t → (κt (ρ), κ0 (ρ)). This can be seen either by a direct computation leading to a real transport equation for the leading symbol, (using that e−iφ(x)/h ◦ aw (x, hDx ) ◦ eiφ(x)/h = (a(x, φ (x) + hDx ))w + O(h2 ), see Appendix A in [13]), or by using H¨ ormander’s definition ([15]) of the principal symbol of a Fourier integral operator, as well as a result of Duistermaat-H¨ ormander giving a real transport equation for the principal symbol for the evolution problem (2.5). In particular, if at | C has constant argument for one value of t, the same φt holds for all other values. For a given U associated to κ, choose κt and U (t) as in (2.5), so that κ0 is linear and U (1) = U . (We may assume for simplicity that (y0 , η0 ) = (x0 , ξ0 ) = (0, 0) and take κt (y, η) = 1t κ(t(y, η)).) Then using Proposition 2.2 and the above remark, we get the equivalences: [U satisfies (2.3).] ⇔ [U (0) satisfies (2.3).] ⇔ [The principal symbol of U (0) has constant argument.] ⇔ [The principal symbol of U has constant argument.] This gives Proposition 2.1. Let X, Y be analytic manifolds of dimension n equipped with analytic integration densities L(dx) = LX (dx), L(dy) = LY (dy). Let κ : ΩY → ΩX be a canonical transformation (and diffeomorphism), analytic for simplicity, where ΩY ⊂⊂ T ∗ Y, ΩX ⊂⊂ T ∗ X, are connected, open with smooth boundary. We do not assume ΩX , ΩY to be simply connected, so we may have finitely many closed cycles γ1 , . . . , γN ⊂ ΩY which generate the homotopy group of ΩY . T : L2 (Y ) → HΨ (Y ) be corresponding FBILet S : L2 (X) → HΦ (X), Y denote tubular complex neighborhoods transforms as in the appendix, where X, of X, Y and with associated canonical transformations: κS : T ∗ X ∩ {|ξ| < C} → ΛΦ , κT : T ∗ Y ∩ {|η| < C} → ΛΨ , where we equip HΦ , HΨ with the scalar products that make S, T unitary, and we can have C > 0 as large as we like. Choose C large enough, so that κS , κT are well defined on ΩX , ΩY respectively, and let X = πx κS ΩX ⊂ X, Ω Y = πy κT ΩY ⊂ Y . Ω Let κ : ΛΨ → ΛΦ be the lift of κ, so that κ = κS ◦ κ ◦ κ−1 T . Here ΛΦ,Ψ are restricted 2 2 X }. to ΩX,Y : ΛΨ = {(y, i ∂y Ψ); y ∈ ΩY }, ΛΦ = {(x, i ∂x Φ); x ∈ Ω
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
9
We shall define a multi-valued “Floquet periodic” Fourier integral operator U : L2 (Y ) → L2 (X) which is only microlocally defined from ΩY to ΩX and associated to κ. Requiring that U be microlocally unitary with the improved Egorov property, we will see that we can have the Floquet periodicity: γ∗ U = eiθ(γ) U,
(2.6)
where γ is a closed loop in ΩY joining some point ρ to itself, U denotes the operator U as it is defined near ρ and the left-hand side of (2.6) denotes the operator obtained from U by following the loop γ. We will then achieve (2.6) with θ(γ) = h−1 S(γ) + k(γ)π/2, where S(γ) = κ◦γ ξdx − γ ηdy is the difference of the actions of κ ◦ γ and γ, and k(γ) ∈ Z is a “Maslov index”, both quantities depending only on the homotopy class of γ. (Requiring only the unitarity of U , we could take θ(γ) = S(γ)/h.) When discussing the improved property (2.3), recall from [13] and [29], that on a manifold with a preferred positive density, we can define the Weyl symbol of a 0-th order h-pseudodifferential operator modulo O(h2 ) by taking the ordinary Weyl symbol for some system of local coordinates x1 , . . . , xn for which the preferred density reduces to the Lebesgue measure. Clearly Proposition 2.1 extends to this situation. We first notice that if − n+N 2 eiφ(x,y,θ)/h a(x, y, θ; h)u(y)dydθ V u(x) = h is an elliptic Fourier integral operator with leading symbol a0 (x, y, θ) = 0 on Cφ , then we can obtain V ∗ V = 1 + O(h) by multiplying a0 by a positive real-analytic function. The same remark applies to loc X ), V : HΨ (ΩY ) → HΦloc (Ω
where we put V = S ◦ V ◦ T −1 and represent it as in [20] by V u(x) = h−n eiψ(x,y)/h b(x, y; h)u(y)e−2Ψ(y)/h L(dy).
(2.7)
Here ψ(x, y) is the multi-valued grad-periodic function near πx,y Γ, with ∂x,y ψ = 0, ∂x,y b = 0 near πx,y (Γ), ∂x ψ(x, y) =
2 2 ∂x Φ(x), ∂y ψ(x, y) = ∂y Ψ(y) on πx,y (Γ), i i
Φ(x) + Ψ(y) + Im ψ(x, y) ∼ dist ((x, y), πx,y (Γ))2 , where Γ denotes the graph of κ . (In [20] the first equation holds only to infinite order on πx,y (Γ) and the present improvement follows from the analyticity of κ .)
10
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
Recall that Im ψ is single-valued, and that ξdx − ηdy, var(κ◦γ,γ) ψ = κ◦γ
(2.8)
γ
is the action difference, when γ is a closed curve in ΛΨ and ( κ ◦ γ, γ) denotes the Y , Ω X whenever so curve t → ( κ(γ(t)), γ(t)). Here we also identify ΛΨ , ΛΦ with Ω is convenient. Thus after multiplying b| π (Γ) by a positive real-analytic function, we may x,y assume that V ∗ V = 1 + O(h). (2.9) In order to have the improved Egorov property, we further need that locally on πx,y (Γ): arg b0 (x, y) = K(y) + Const., (notice that x = x(y) on Γ),
(2.10)
where K(y) is a grad-periodic function on πx,y (Γ), that we do not try to compute here, but whose existence we infer from Proposition 2.1 and the computation of V as S ◦ V ◦ T −1 , with V written microlocally with a real phase as in (2.1). We can find b0 satisfying (2.10) everywhere if we accept that b0 | π (Γ) is x,y multi-valued. More precisely, K is not globally well defined on πx,y (Γ) ΩY , but ω = dK is a well defined closed real 1-form on ΩY and we can find b0 | π (Γ) , x,y unique up to a constant factor of modulus 1, such that (2.9), (2.10) hold, though b0 will be multi-valued: γ∗ b0 = exp (i ω)b0 , (2.11) γ
where γ∗ b0 denotes the new locally defined symbol obtained by following b0 around the closed loop γ in πx,y (Γ) ΩY . Proposition 2.3 We have γ ω = k(γ) π2 for some integer k(γ) ∈ Z, for every closed loop γ ⊂ πx,y (Γ). Proof. Let γ be a closed loop and cover γ by small open topologically trivial sets 0, Ω 1, . . . , Ω N −1 with increasing index corresponding to the orientation of γ in Ω N = Ω 0 . Let Ωj be the corresponding regions in ΩY . In Ωj , the natural way. Let Ω we represent V by n+Nj Vj u(x) = h− 2 eiφj (x,y,θ)/haj (x, y, θ; h)u(y)dydθ. (2.12) θ∈RNj
For a given point in Ωj ∩ Ωj+1 , we have φj = φj+1 ,
aj+1 = rj+1,j eiαj+1,j π/2 aj + O(h),
rj+1,j > 0, αj+1,j ∈ Z,
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
11
at the corresponding points in Cφj , Cφj+1 , provided that we require all the fibervariable dimensions Nj to have the same parity. (Cf. [15].) This last property is easy to achieve since we can always add one fiber-variable. We conclude that π ω = (α1,0 + α2,1 + · · · + αN,N −1 ), 2 γ
and the proposition follows. Take V as above with b = b0 in (2.7), so that (2.9), (2.10) hold. Put 1
U = V (V ∗ V )− 2 .
(2.13)
= SU T −1 is of the form (2.7) with b = b0 + O(h). We have U ∗ U = 1 and Then U U satisfies (2.3). Since the unitarization is a local operation which commutes with multiplication by a constant factor of modulus 1, (2.11) becomes valid also for b: π
γ∗ b = eik(γ) 2 b.
(2.14)
Here we also used Proposition 2.3. Summing up, we get Theorem 2.4 Under the assumptions above on κ, we can find a microlocally defined multi-valued Fourier integral operator U associated to κ, and a corresponding lift = SU T −1 of the form (2.7), such that U is unitary: U ∗ U = 1 + O(e−1/(Ch) ), U satisfies the improved Egorov property (2.3), and γ∗ U = ei(S(γ)/h+k(γ)π/2) U, for every closed loop in ΩY , where k(γ) ∈ Z and S(γ) = ξdx − ηdy. κ◦γ
γ
3 Reduction by averaging along trajectories Let P , M be as in the introduction. We work in a neighborhood of p−1 (0) ∩ T ∗ M , and recall that P = P has the semi-classical principal symbol p = p + iq + O(2 ),
(3.1)
in a complex neighborhood of p−1 (0)∩T ∗ M . Let G0 be an analytic function defined near p−1 (0) ∩ T ∗ M such that Hp G0 = q − q,
(3.2)
12
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
where q is the trajectory average, defined in (1.15). We may take 1 G0 = T (E)
T (E)/2
−T (E)/2
(1R− (t)(t +
T (E) T (E) ) + 1R+ (t)(t − ))q ◦ exp tHp dt, (3.3) 2 2
on p−1 (E). We replace R4 by the new IR-manifold ΛG0 = exp (iHG0 )(R4 ),
(3.4)
which is defined in a complex neighborhood of p−1 (0) ∩ T ∗ M . Writing (x, ξ) = exp (iHG0 )(y, η), and using ρ = (y, η) as real symplectic coordinates on ΛG0 , we get p | Λ
G0
= p (exp (iHG0 )(ρ)) =
(3.5)
∞ (iHG0 )k (p ) = p + iq + O(2 ). k! k=0
Iterating this procedure, or looking more directly for G(x, ξ, ) as an asymptotic sum ∞ k Gk (x, ξ) (3.6) G∼ 0 −1
in some complex neighborhood of p (0) ∩ T ∗ M , we see that we can find G1 , G2 . . . such that if (3.7) ΛG = exp (iHG )(R4 ), and we again write ΛG (x, ξ) = exp (iHG )(y, η) and parametrize by the real variables (y, η), then p | Λ
G
= p + iq + 2 q2 + 3 q3 + · · · ,
(3.8)
where qj = qj , j ≥ 2. This means that we can transform p to p ◦ exp (iHG ) in such a way that we get a new leading symbol which Poisson commutes with the unperturbed leading symbol. As is well known in the selfadjoint case, this construction can be extended to the level of operators, and we may develop this globally in another paper. In the present work we will do it only after a reduction to a torus-like situation. After replacing p by p ◦ exp (iHG0 ) and correspondingly P , by U−1 ◦ P ◦ i U , where U is the Fourier integral operator U = e− h iG0 (x,hDx ) = e h G0 (x,hDx ) (defined microlocally near p−1 (0) ∩ T ∗M ), we may assume that our operator P is microlocally defined near p−1 (0) ∩ T ∗ M and has the h-principal symbol p = p + iq + O(2 ).
(3.9)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
13
This can be done in such a way that P=0 remains the original unperturbed operator. We refer to the beginning of Section 6 for the construction of U by means of an FBI-transform. Let γ0 ⊂ p−1 (0) ∩ T ∗ M be a closed Hp -trajectory and assume that T (0) is the minimal period of γ0 . Let g : neigh (0, R) → R be the analytic function defined by T (E) , g(0) = 0. (3.10) g (E) = 2π Then Hg◦p = g (p)Hp has a 2π-periodic flow and the same closed trajectories as Hp . Clearly 2π is the minimal period of γ0 when viewed as a Hg◦p -trajectory. Proposition 3.1 There exists an analytic canonical transformation κ : neigh ({τ = x = ξ = 0}, T ∗(St1 × Rx )) → neigh (γ0 , T ∗ M ), mapping {τ = x = ξ = 0} onto γ0 , such that g ◦ p ◦ κ = τ . Proof. Fix a point ρ0 ∈ γ0 and choose local symplectic coordinates (t, τ ; x, ξ) centered at ρ0 , with g ◦ p = τ . This means that {ξ, x} = 1, {t, x} = {t, ξ} = 0
(3.11)
Hτ t = 1, Hτ x = Hτ ξ = 0.
(3.12)
Now extend the definition of t, τ, x, ξ to a full neighborhood of γ0 , by putting τ = g ◦ p and requiring t, x, ξ to solve (3.12). Since the Hτ -flow is 2π-periodic (with 2π as the minimal period) near γ0 , we see that x, ξ are well defined singlevalued functions, while t becomes multi-valued in such a way that it increases by 2π each time we make a loop in the increasing time direction. (3.11) extends to a full neighborhood of γ0 . This is equivalent to the proposition. Notice that p ◦ κ = f (τ ),
(3.13)
where f := g −1 . From (3.9) we infer that p ◦ κ = f (τ ) + iq(τ, x, ξ) + O(2 ),
(3.14)
for a new function q which is independent of t (and obtained from the earlier one by composition with κ). If we let the Fourier integral operator U quantize κ as in Section 2, we get a new operator U −1 P U with leading semi-classical symbol p ◦ κ as in (3.14). (Here P is the new version of P ; P,new = U −1 P,old U .) Now write simply p, p , P for the transformed objects. Then P = P (t, x, hDt,x , ; h) is the formal Weyl quantization of a symbol P (t, x, τ, ξ, ; h) which has an asymptotic expansion (1.5) in the space of holomorphic functions in a fixed complex
14
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
neighborhood of {Im t = τ = x = ξ = 0} in T ∗ (S1 × C), with S1 = S 1 + iR, and we will use the same notation as in Section 1. (An exact value of the new symbol P (t, x, τ, ξ, ; h) cannot be easily defined, but we know how to define it mod O(e−1/(Ch) ). We shall however avoid using the full power of analytic pseudodifferential operators, and content ourselves with the knowledge of P mod O(h∞ ).) Now look for G(1) = G1 (t, τ, x, ξ) + 2 G2 (t, τ, x, ξ) + · · · such that p ◦ exp iHG(1) = f (τ ) + iq(τ, x, ξ) + O(2 ) is independent of t. Here the left-hand side can be written ∞ 1 (iHG(1) )k p , k!
k=0
and we get p + i2 HG1 (f (τ )) + O(3 ) = f (τ ) + iq(τ, x, ξ) − i2 f (τ )
∂ G1 + O(2 ) + O(3 ), ∂t
where the O(2 ) term is the same as in (3.14). It is clear that we can find G1 so that the 2 -term in this expression is independent of t. Looking at the O(3 )-term we then determine G2 and so on. (In this construction, we could have applied κ at the very beginning before replacing q by q by averaging, and then incorporated G0 into the expression G = G0 + G1 + · · · , and as already indicated, this could also have been done entirely (and in a full neighborhood of p−1 (0) ∩ T ∗ M ), before applying κ.) After replacing p by p ◦ exp iHG(1) , we are now reduced to the case when p = f (τ ) + iq(τ, x, ξ) + O(2 )
(3.15)
is independent of t, up to O(∞ ). Finally we remove the t-dependence from the lower order terms. After conjugating P by a Fourier integral operator V , which quantizes exp iHG(1) , we may assume that p in (3.15) is the principal symbol of P (and that it is independent of t). Look for an h-pseudodifferential operator A(t, x, hDt,x , ; h) with symbol A(t, x, τ, ξ, ; h) ∼
∞
ak (t, x, τ, ξ, )hk ,
(3.16)
k=1
such that the full (Weyl) symbol of e
i hA
P e
− hi A
=e
i h adA
∞ 1 i ( adA )k P P = k! h k=0
(3.17)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
15
is independent of t. Since A = O(h) we know that hi adA lowers the order in h by one (with the convention that a symbol = O(h−j ) is of order j), so (3.17) makes sense asymptotically. The subprincipal symbol of (3.17) is h(p1, (x, ξ) + {p , a1 }) = h(p1, (x, ξ) + f (τ )
∂ a1 (t, τ, x, ξ, ) + O()), ∂t
and we make this independent of t by successively determining the coefficients in the asymptotic series a1 (t, τ, x, ξ, ) =
∞
a1,j (t, τ, x, ξ)j .
j=0
After that we return to (3.17) and see that the construction of a2 , a3 , . . . is essentially the same. Actually, we do not have to do this construction in 2 steps, and we can view G(1) above as (a constant factor times) the leading symbol a0 = O(2 ) in A∼
∞
ak (t, x, τ, ξ, )hk ,
(3.18)
k=0
such that if P denotes the very first operator we get on S 1 × R, then the left-hand side of (3.17) has a symbol which is well defined as an asymptotic series in (, h) and is independent of t, up to O(h∞ ). This can be seen by first determining a0 from (3.17) (leading to a repetition of what we already did) and then the other terms. (When is small but fixed, the problem becomes more subtle and the break-up into two steps is more natural, with the first step being the one containing the new difficulties.) Summing up the discussion of this section, we have Proposition 3.2 Let P , M be as in Section 1. Let γ0 ⊂ p−1 (0) ∩ T ∗ M be a closed Hp -trajectory where T (0) is the minimal period and let κ be the canonical transformation of Proposition 3.1. Let U be a corresponding elliptic Fourier integral operator as in Section 2. Then there exist G(x, ξ, ) (independent of γ0 , κ, U ) with the asymptotic expansion (3.6) in the space of holomorphic functions in some fixed complex neighborhood of p−1 (0) ∩ T ∗ M and a symbol A(t, x, τ, ξ, ; h) as in (3.16), where ∞ ak,j (t, x, τ, ξ)j (3.19) ak ∼ j=0
in the space of holomorphic functions in a fixed complex neighborhood of Im t = τ = x = ξ = 0 in T ∗ (S1 × C)), such that if G, A also denote the corresponding Weyl quantizations, the operator i i P = e h A U −1 e− h G P e h G U e− h A = Ad
i
e h A U −1 e− h G
P
(3.20)
16
M. Hitrik and J. Sj¨ ostrand
has a symbol P (x, τ, ξ, ; h) ∼
∞
Ann. Henri Poincar´e
pk (x, τ, ξ, )hk
(3.21)
0
∞ independent of t (up to O(h∞ )). Here each pk = pk (x, τ, ξ, ) ∼ j=0 pk,j (x, τ, ξ)j in the space of holomorphic functions in a fixed complex neighborhood of τ, x, ξ = 0. Moreover (3.22) p0, = f (τ ) + iq(τ, x, ξ) + O(2 ). If q has a non-degenerate extreme value along γ0 , then the proposition is directly applicable (see Section 5), while in other situations (such as in Section 4), it is not global enough.
4 Normal forms and quasi-eigenvalues in the torus case Let P, M, p, q, q, Λ0,F0 be as in Section 1. After replacing q by q − F0 , we may assume that F0 = 0, so we consider Λ0,0 : p = 0, Re q = 0.
(4.1)
Notice that Λ0,0 is invariant under the Hp -flow. We assume that T (0) is the minimal period for all the closed trajectories in Λ0,0 and that dp, dRe q are independent at the points of Λ0,0 ,
(4.2)
so that Λ0,0 is a Lagrangian manifold and also a union of tori. Assume for simplicity that Λ0,0 is connected, so that it is equal to one single Lagrangian torus. In this section we work microlocally near Λ0,0 and proceed somewhat formally. In Section 6 we follow up with suitable function spaces and see how to justify the computation of the spectrum via a global Grushin problem. We have seen that we can reduce ourselves to the case when p = p + iq + O(2 ).
(4.3)
Assume from now on that q is real-valued or more generally that q is a function of p and Re q. We can make a real canonical transformation κ : neigh (ξ = 0, T ∗T2 ) → neigh (Λ0,0 , T ∗ M ), T2 = (R/2πZ)2 ,
(4.4)
such that p ◦ κ = p(ξ1 ), q ◦ κ = q(ξ) (with a slight abuse of notation). Recall that this can be done in the following way: Let ΛE,F be the Lagrangian torus given by p = E, Re q = F , for (E, F ) ∈ neigh (0, R2 ). Let γ1 (E, F ) be the cycle in ΛE,F corresponding to a closed Hp -trajectory with minimal period, and let γ2 (E, F ) be a second cycle so that γ1 , γ2 form a fundamental system of cycles on the torus ΛE,F . Necessarily γ2 maps to the simple loop given by Re q = F in the abstract quotient manifold p−1 (E)/RHp . Now it is classical (see [1]) that
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
17
we can find a real analytic canonical transformation κ : neigh (η = 0, T ∗ T2 ) (y, η) → (x, ξ) ∈ neigh (Λ0,0 , T ∗ M ), such that 1 ( ξdx − ξdx), ηj = 2π γj (E,F ) γj (0,0) where E, F depend on (x, ξ) and are determined by (x, ξ) ∈ ΛE,F , i.e., by E = p(x, ξ), F = Re q(x, ξ). We also know that here η1 = η1 (E) is a function of E only. Let us also recall that κ can be constructed as follows: We start by taking a first canonical transformation κ0 : neigh (ξ = 0, T ∗ T2 ) → neigh (Λ0,0 , T ∗ M ) such that the zero section is mapped to Λ0,0 and the lines {x2 = Const, ξ = 0} are mapped onto the closed Hp -trajectories in Λ0,0 . Then using κ0 , we can consider p, q as living on T ∗ T2 . ΛE,F is then given by ξ = φx , φ = φper (x, E, F ) + η1 x1 + η2 x2 , with det φx,(E,F ) = 0, with ηj = ηj (E, F ) as above (now being the actions/2π with respect to ξdx), and φper being (2πZ)2 -periodic. Moreover, φx (x, η) = 0, η = 0 for E = F = 0. It is easy to check, using that our functions are real-valued, that (E, F ) → (η1 (E, F ), η2 (E, F )) is a local diffeomorphism, so we can use η1 , η2 as new parameters replacing E, F , and write φ = φ(x, η). Consider κ1 : (
∂φ ∂φ , η) → (x, ) ∂η ∂x
which maps the zero section to itself. Then κ := κ0 ◦κ1 has the required properties. Let U be a corresponding Fourier integral operator, implementing κ, so that if we denote by P also the conjugated operator U −1 P U , we have a new operator with leading symbol (4.5) p = p(ξ1 ) + iq(ξ) + O(2 ). For the conjugated operator, we still have the property that P=0 is selfadjoint. From the assumption (4.2) about linear independence, we get ∂ξ1 p(0) = 0, ∂ξ2 Re q(0) = 0.
(4.6)
As in the section, we can find an h-pseudodifferential operator A with preceding ν symbol ∞ h a (x, ξ, ), a0 = O(2 ), such that formally ν ν=0 i
i
i
e h A P e− h A = e h adA (P ) =
∞ 1 i ( adA )k (P ) =: P , k! h
k=0
with P (x, ξ, ; h) independent of x1 , and leading symbol p = p(ξ1 ) + iq(ξ) + O(2 )
(4.7)
18
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
also independent of x1 . We recall that the symbol A(x, ξ, ; h) is a formal power series both in and h with coefficients all holomorphic in the same complex neighborhood of ξ = 0. This construction can be done in such a way that P=0 is selfadjoint. We next look for a further conjugation that eliminates the x2 -dependence in the symbol. a) We start by considering the general case, when the subprincipal symbol of P=0 is not necessarily 0, so that the complete symbol of P takes the form P (x2 , ξ; h) =
∞
hν pν (x2 , ξ, ),
(4.8)
ν=0
with p0 (x2 , ξ, ) = p = p(ξ1 ) + iq(ξ) + O(2 ),
(4.9)
and p1 (x2 , ξ, 0) not necessarily identically equal to 0. The easiest case is when h/ ≤ O(hδ1 ) for some δ1 > 0, so that we can consider h/ as an asymptotically small parameter. Look for ∞ h h hν bν (x2 , ξ, , ), B(x2 , ξ, , , h) = ν=0
(4.10)
with bν = O( + h/), such that on the operator level (with hDx instead of ξ), i i h e h B P e− h B =: P (hDx , , , h)
(4.11)
has a symbol independent of x. Notice that B(x2 , hDx , ; h) and p(hDx1 ) commute. On the symbol level we write h h P = p(ξ1 ) + (iq(ξ) + O() + p1 (x2 , ξ, ) + h p2 (x2 , ξ, ) + · · · ) (4.12) h h = p(ξ1 ) + (r0 (x2 , ξ, , ) + hr1 (x2 , ξ, , ) + · · · ), with
h h h r0 (x2 , ξ, , ) = iq(ξ) + O() + p1 = iq(ξ) + O() + O( ), r1 =
h p2 (x2 , ξ, ), . . .
Notice that rj = O(h/) for j ≥ 1. We shall treat h/ as an independent parameter. We use this and develop (4.11) to get, with adb c denoting the symbol of adb(x,hDx ) c(x, hDx ) = [b(x, hD), c(x, hD)],
Vol. 5, 2004
p(ξ1 ) +
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
∞ ∞
···
k=0 j1 =0
∞ ∞
h+j1 +···+jk
jk =0 =0
19
∞ i 1 i ( adbj1 )...( adbjk )r = p(ξ1 ) + hn rn , k! h h n=0
with rn being equal to the sum of all coefficients for hn resulting from all the expressions 1 i i h+j1 +···+jk ( adbj1 ) · · · ( adbjk )r , (4.13) k! h h with + j1 + · · · + jk ≤ n. The first term is r0 =
1 H k r0 = r0 ◦ exp (Hb0 ), k! b0
where we want r0 to be independent of x2 (in addition to x1 ). We get with b0 = O( + h/): h r0 = iq(ξ) + O( + h/) − i∂ξ2 q∂x2 b0 + O((, )2 ),
(4.14)
and using that ∂ξ2 q = 0, it is clear how to construct b0 = O( + h/) as a formal Taylor series in , h/, so that r0 = iq(ξ)+O(+h/) is independent of x (modulo a term O(h∞ )). i Assume for simplicity that the conjugation by e h b0 (x2 ,hDx ,,h/) has already been carried out, so that we are reduced to the case when r0 = iq(ξ)+O(+h/) is independent of x2 , and rj = O( + h/) for jν ≥ 1. Then hlook for a new conjugation exp hi adB , with B(x2 , ξ, , h/; h) = ∞ ν=1 h bν (x2 , ξ, , ). The new expression for the left-hand side of (4.11) becomes p(ξ1 ) +
∞ ∞ k=0 j1 =1
···
∞ ∞ jk =1 =0
h+j1 +···+jk
∞ 1 i i ( adbj1 )...( adbjk )r = p(ξ1 ) + hn rn , k! h h n=0
(4.15) with rn equal to the sum of all coefficients for hn resulting from the expressions (4.13) with + j1 + · · · + jk ≤ n and jν ≥ 1. Then r0 = r0 , r1 = r1 + Hb1 r0 = r1 − Hr0 b1 , . . . , rn = rn − Hr0 bn + sn , where sn only depends on b1 , . . . , bn−1 and is the sum of all coefficients of hn arising in the expressions (4.13) with +j1 +· · ·+jk ≤ n, j1 , . . . , jk , < n, jν ≥ 1. It is therefore clear how to find b1 , b2 , . . . successively with bj = O( + h/), such that all the rj are independent of x and = O( + h/). This completes the proof of (4.11). Summing up the discussion so far, if we do not make any assumption on the subprincipal symbol of P=0 and restrict the attention to h/ ≤ O(hδ1 ) for some δ1 > 0, then we can find B0 = b0 (x2 , hDx , , h/), b0 = O( + h/),
20
M. Hitrik and J. Sj¨ ostrand
and B1 =
∞
Ann. Henri Poincar´e
bν (x2 , hDx , , h/)hν , bν = O( + h/),
ν=1
such that
i i P := e h adB1 e h adB0 P
(4.16)
has a symbol independent of x: h h P = p(ξ1 ) + (r0 (ξ, , ) + hr1 (ξ, , ) + · · · ),
(4.17)
with r0 = iq(ξ) + O( + h/), and rν = O( + h/) for ν ≥ 1. Remaining in the general case, without any assumption on the lower order terms, we now assume merely that h/ ≤ δ0 for some sufficiently small δ0 > 0. This means that we can no longer construct b0 by a formal Taylor series in h/, i and we shall replace e h b0 (x2 ,hDx ,,h/) by a Fourier integral operator, constructed directly. Look for φ = φ(x2 , ξ, , h/) solving h h r0 (x2 , ξ1 , ξ2 + ∂x2 φ, , ) = r0 (·, ξ, , ),
(4.18)
where · denotes the average with respect to x2 . By the implicit function theorem, (4.18) has a solution with ∂x2 φ single-valued and O( + h/). If we Taylor expand (4.18), we get h h h h (∂ξ2 r0 )(x2 , ξ, , )∂x2 φ + (r0 (x2 , ξ, , ) − r0 (·, ξ, , )) = O(( , )2 ), and using also that h h ∂ξ2 r0 (x2 , ξ, , ) = i∂ξ2 q(ξ) + O( + ), we get, φ = φper + x2 ζ2 , with ζ2 = ζ2 (ξ, , h ) = O((, h/)2 ), and φper = O((, h/)) periodic in x2 . Put η = η(ξ, , h/) = (ξ1 , ξ2 + ζ2 ), and h ψ(x, η, , ) = x · η + φper , where φper is viewed as a function of η rather than ξ. Consider the canonical transformation κ : (ψη , η) → (x, ψx ),
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
21
which is ( + h/)-close to the identity and can be viewed as a family of transforms depending analytically on the parameter ξ1 . With ξ = ξ(η, , h ), we have by construction: h h h h (r0 ◦ κ)(y, η, , ) = r0 (·, ξ, , ) = r0 (·, η, , ) + O(2 + ( )2 ), (4.19) and this is a function of (y, η) which is independent of y. Notice that p(ξ1 ) is unchanged under composition with κ. We can quantize κ as a Fourier integral operator U and after conjugation by this operator, we may assume that we have a new operator P as in (4.12) with r0 = iq(ξ) + O( + h/) independent of x and with rj = O( + h/) . i As before, we can then make a further conjugation e h adB1 in order to remove the x-dependence completely and the conclusion is that if we make no assumption on the subprincipal symbol and restrict the attention to h/ ≤ δ0 , for δ0 > 0 small enough, then we can find a Fourier integral operator, i 1 −1 U u(x; h) = (4.20) e h (ψ(x,η)−y·η) a(x, η; h)u(y)dydη, (2πh)2 with ψ(x, η) = x · η + φper (x2 , η, , h/), φper = O( + h/), and B1 = such that
∞
h h bν (x2 , hDx , , )hν , bν = O( + ), ν=1 i P := e h adB1 AdU P
has a symbol independent of x as in (4.17), with the same estimates as there. b) We now assume that in the original problem, P=0 has subprincipal symbol 0. Then after a first time averaging, transportation to the torus, and the elimination of the x1 -dependence, we may assume that P (x2 , ξ, ; h) =
∞
hν pν (x2 , ξ, ),
(4.21)
ν=0
with p0 independent of x mod O(2 ): p0 (x2 , ξ, ) = p(ξ1 ) + iq(ξ) + O(2 ),
(4.22)
p1 (x2 , ξ, 0) = 0.
(4.23)
(Recall from Section 2 and the references given there, that the canonical transformations can be quantized in such a way that Egorov’s theorem holds modulo O(h2 ).) In analogy with (4.12), we have with p1 (x2 , ξ, ) = q1 (x2 , ξ, ), P
=
p(ξ1 ) + (iq(ξ) + O() + hq1 (x2 , ξ, ) +
=
p(ξ1 ) + (r0 (x2 , ξ, ,
h2 h2 p2 + h p3 + · · · )
h2 h2 ) + hr1 (x2 , ξ, , ) + h2 r2 + · · · ),
(4.24)
22
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
with h2 ) h2 r1 (x2 , ξ, , ) h2 r2 (x2 , ξ, , )
r0 (x2 , ξ, ,
= iq(ξ) + O() + = q1 (x2 , ξ, ) + =
h2 p2 ,
h2 p3 ,
h2 p4 , . . .
We first consider the case when h2 ≤ hδ1 ,
(4.25) i
h2
for some fixed δ1 > 0. A first conjugation by e h b0 (x2 ,hDx ,, ) , with b0 = O( + h2 /), allows us to make r0 independent of x2 , and we still have (4.24) with rj = O(1) for j ≥ 1. Then we look for a new conjugation exp hi adB1 with B1 (x2 , ξ, ,
∞ h2 h2 ; h) = hν bν (x2 , ξ, , ). ν=1
(4.26)
The conjugated operator (4.11) can be expanded as in (4.15) and as after that equation it is clear how to get bν = O(1) for ν ≥ 1, such that the resulting rn are independent of x2 , with r0 (ξ, , h2 /) = iq(ξ) + O( + h2 /). Summing up the discussion so far, if we assume that the subprincipal symbol of P=0 vanishes, and restrict the attention to the range (4.25) for some fixed 2 2 δ1 > 0, then we can find B0 = b0 (x2 , hDx , , h ) with b0 = O( + h ) and 2 B1 (x2 , hDx , , h ; h) with symbol (4.26), and bν = O(1), such that i i e h adB1 e h adB0 P = P
has the symbol p(ξ1 ) + (r0 (ξ, ,
h2 h2 ) + hr1 (ξ, , ) + · · · )
(4.27)
independent of x and with r0 = iq(ξ) + O( +
h2 ), rν = O(1), ν ≥ 1.
(4.28)
If we replace (4.25) by the weaker assumption, h2 ≤ δ0 , δ0 1,
(4.29)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
23
i
then again we have to replace the conjugation by e h B0 by that by a Fourier integral operator constructed as earlier: We solve (4.18) (with h/ replaced by h2 /) and get ∂x2 φ single-valued and O( + h2 /). Taylor expanding (4.18) and using that ∂ξ2 r0 (x2 , ξ, ,
h2 h2 ) = i∂ξ2 q(ξ) + O( + ),
we get φ = φper + x2 ζ2 , 2
2
with ζ2 = ζ2 (ξ, , h ) = O((, h )2 ) and φper = O( + h2 /) periodic in x2 . Again we put η = η(ξ, , h2 /) = (ξ1 , ξ2 + ζ2 ) and ψ(x, η, ,
h2 ) = x · η + φper .
The canonical transformation κ : (ψη , η) →
(x, ψx ) is ( + h2 /)-close to the 2 identity and with ξ = ξ(η, , h /), we have by construction (r0 ◦ κ)(y, η, ,
h2 h2 h2 h2 ) = r0 (·, ξ, , ) = r0 (·, η, , + O((, )2 ),
(4.30)
which is a function independent of y. Let U −1 be the corresponding Fourier integral operator as before. Then after replacing P by AdU P , we still have (4.24), where now r0 = iq(ξ) + O( + h2 /) is independent of x and rj = O(1) for j ≥ 1. i We can then make a further conjugation by e h B1 as before, and we get the following conclusion: Assume that the subprincipal symbol of P=0 vanishes and restrict the attention to the range (4.29). Then we can find an elliptic Fourier integral operator U −1 of the form (4.20) with ψ as above and B1 (x2 , hDx , , h2 /; h) with symbol (4.26), and bν = O(1), such that i e h adB1 AdU P = P (hDx , , h2 /; h)
(4.31)
has a symbol P (ξ, , h2 /; h) of the form (4.27), such that (4.28) holds. We finish this section by discussing what spectral results can be expected from the reductions above. The first reduction (as in Section 3) was to conjugate the original operator P by a Fourier integral operator eiG(x,hD,)/h , with G(x, ξ, ) ∼ (G0 (x, ξ) + G1 (x, ξ) + · · · ), defined in some complex neighborhood of p−1 (0) ∩ T ∗ M , to achieve that the leading symbol of the conjugated operator is of the form p + iq + O(2 ) and Poisson commutes with p. At least formally, the new operator also acts on L2 (M ) and we have no Floquet type conditions to worry about. Geometrically, this corresponds to the fact that a canonical transformation κ = exp HG with asingle-valued generator G = O() preserves actions along closed loops: κ◦γ ξdx = γ ηdy, for every closed loop γ. The second reduction was to take κ in (4.4) and to conjugate by the inverse of the corresponding Fourier integral operator U . Let α1 (=γ0 ) and α2 be the
24
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
fundamental cycles in Λ0,0 given by αj = κ ◦ βj , where β1 , β2 are the fundamental cycles in T2 {(x, 0) ∈ T ∗ T2 }, given by x2 = 0 and x1 = 0 respectively. Put Sj = ξdx, (4.32)
αj
so that Sj is the difference of actions, κ◦βj ξdx − βj ηdy, j = 1, 2. Since κ is a canonical transformation we know that if β is a closed loop homotopic to βj , then ξdx − ηdy = S . j κ◦β β As in [20] or as in Theorem 2.4, we see (at least formally) that if we want U u to be single-valued on M (possibly defined only microlocally near Λ0,0 ), then u should not necessarily be periodic on R2 (i.e., a function on T2 ) but a Floquet periodic function with iν·S
u(x − ν) = e 2πh +
iν·k0 4
u(x), ν ∈ (2πZ)2 , S = (S1 , S2 ), k0 ∈ Z2 .
(4.33)
The conjugated operator Ad −1 hi G P should therefore act on Floquet periodic U e functions as in (4.33). The further conjugations are by operators on the torus that conserve the property (4.33). This is clear from the definitions, and corresponds to the fact that a canonical transformation: (y, η) → (x, ξ), generated by ψ(x, η) = x·η + φper(x, η) and close to the identity, conserves actions. Indeed, on the graph of the transform, we have ξdx + ydη = dψ, so ξdx − ηdy = d(ψ − y · η) = d((x − y) · η + φper (x, η)), and (x − y) · η + φper (x, η) is single-valued on the graph. On the other hand the space of Floquet periodic functions as in (4.33), equipped with the L2 -norm over a fundamental domain of T2 , has the ON basis: i
ek (x) = e h x·(h(k−
k0 4
S )− 2π )
, k ∈ Z2 ,
(4.34)
and applying our reductions down to the operator P in the cases (a) and (b) above, we get formally (in the sense that we do not define the notion of quasi-eigenvalue): Proposition 4.1 Recall that we took F0 = 0 and that S, k0 are the actions and the Maslov indices in (4.32), (4.33). 1 1 a) In the general case, P has the quasi-eigenvalues in ] − |O(1)| , |O(1)| [+i] − 1 1 δ |O(1)| , |O(1)| [ for = O(h ), h/ 1: S h k0 , , ; h , k ∈ Z2 , P h(k − ) − 4 2π where P (ξ, , h ; h) is holomorphic in ξ ∈ neigh (0, C2 ), smooth in (0, R) and has the asymptotic expansion (4.17), when h → 0.
(4.35) h ,
∈ neigh
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
25
b) If we assume that P=0 has subprincipal symbol 0, then P has the quasi-eigen1 1 1 1 values in ] − |O(1)| , |O(1)| [+i] − |O(1)| , |O(1)| [ for = O(hδ ), h2 / 1: S h2 k0 , , ; h , k ∈ Z2 , P h(k − ) − 4 2π
(4.36)
where P(ξ, , h2 /; h) is holomorphic in ξ ∈ neigh (0, C2 ), smooth in and h2 / ∈ neigh (0, R) and has the asymptotic expansion (4.27), (4.28), when h → 0.
5 Quasi-eigenvalues in the extreme cases We make the assumptions of the case II in the introduction and assume, in order to fix the ideas, that 0 = F0 = Re qmin,0 . (5.1) Apply Proposition 3.2 and reduce P near γ0 to P = P(x, hDt,x , ; h) with symbol described in that proposition. Recall that P has the leading symbol p = f (τ ) + iq(τ, x, ξ) + O(2 ),
(5.2)
where q(τ, x, ξ) is equal to the original averaged function q, composed with the canonical transformation κ of Proposition 3.1. The assumptions (1.23) and (5.1) imply that Re q(0, x, ξ) ∼ |(x, ξ)|2 (5.3) on the real domain. Also recall that we have the assumption (1.17) which with (5.3) implies that q(τ, x, ξ) = g(τ, Re q(τ, x, ξ)) (5.4) on the real domain, for some analytic function g(τ,q) with g(0,0) = 0, Re g(τ,q) = q. We conclude that (x, ξ) → iq(τ, x, ξ) + O(), appearing in (5.2), has a nondegenerate critical point (x(τ, ), ξ(τ, )) = O(|τ |+) depending analytically on τ, and real when τ ∈ R, = 0. After composition with the (τ, )-dependent (symplectic) translation (x, ξ) → (x − x(τ, ), ξ − ξ(τ, )) and subtracting the corresponding critical value, we may assume that the critical point is (0, 0) and hence that p (τ, x, ξ) = f (τ ) + iq(τ, x, ξ, ),
(5.5)
Re q(τ, x, ξ, ) ∼ |(x, ξ)|2
(5.6)
q(τ, x, ξ, 0) = g(τ, Re q(τ, x, ξ, 0)),
(5.7)
with on the real domain, and
on the real domain, where g(τ, 0) = 0, Re g(τ, q) = q.
26
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
We shall next construct a (τ, )-dependent canonical transformation in the x, ξ-variables, which reduces p (τ, x, ξ) to a function of τ, , 12 (x2 + ξ 2 ). In doing so, we essentially follow Appendix B of [13], where the model was xξ rather than p0 := 12 (x2 + ξ 2 ). These two quadratic forms are equivalent up to a constant factor and composition by a linear complex canonical transformation, so the only difference is that the real domains are not the same. Let p(x, ξ) ∼ (x, ξ)2 be real and analytic in a neighborhood of (0,0). Lemma 5.1 There exists a real and analytic function f (E) defined near E = 0, with f (0) = 0, f (0) > 0, such that the Hamilton flow of f ◦ p is 2π-periodic, with 2π as its minimal period except at (0, 0). Proof. Consider, first for 0 < E 1, the action ξdx = E I(E) = p−1 (E)
−1 qE (1)
ηdy,
√ where qE (y, η) = E1 p( E(y, η)), so that q0 is a positive quadratic form (in the √ limit E → 0). Then qE is an analytic function of E in a neighborhood of 0 and consequently we have the same fact for I(E). If we let E describe a simple closed loop around 0 in neigh (0, C) \ {0}, then qE (y, η) transforms into qE (y, η) = qE (−y, −η) and it follows that I(E) transforms into itself. It follows that I(E) is analytic as a function of E. The period T (E) of the Hp -flow is given by T (E) = I (E) and the period of the Hf ◦p -flow is T (E)/f (E). It suffices to choose f with f (E) = T (E)/2π and f (0) = 0. In the following discussion, we replace p by f ◦ p, so that we get a reduction to the case when the Hp -flow is 2π-periodic. After composition with a real linear canonical transformation, we may assume that p(x, ξ) = p0 (x, ξ) + O((x, ξ)3 ), even though that is not really needed for the argument to follow. Consider the involution ι = exp (πHp ) with ι2 = id. Correspondingly, we have ι0 = exp (πHp0 ), so that ι0 (ρ) = −ρ. Let h(x, ξ) be a real-valued analytic function defined near (0, 0) with dh(0, 0) = 0, and put g = 12 (h − h ◦ ι). Then dg(0) = dh(0, 0) = 0, and g ◦ ι = −g.
(5.8)
Γ := g −1 (0) is a real curve passing through the origin, invariant under the action of ι. Let Γ also denote a corresponding complexification. If g0 , Γ0 are the corresponding objects for p0 , we may assume (though this is not essential), that dg(0, 0) = dg0 (0, 0) so that Γ, Γ0 are tangent at (0, 0). Since Γ is a curve, we have p| Γ = q 2 for some analytic function q, and similarly p0 | Γ = q02 . (We may assume that dq0 = dq = 0 at 0.) Let α : Γ0 → Γ be the 0 analytic diffeomorphism given by q ◦ α = q0 , so that p ◦ α = p0 on Γ0 . For neigh ((0, 0), C2 ) ρ = exp tHp0 (ν), ν ∈ Γ0 , t ∈ C,
(5.9)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
27
we put κ(ρ) = exp tHp (α(ν)).
(5.10)
With the precautions taken above, it is easy to see that the definition of κ(ρ) does not depend on how we choose ν ∈ Γ0 (unique up to the action of ι0 ) and t (unique mod (2π), once ν has been chosen.) As in [13], we see that some exceptional points ρ ∈ neigh ((0, 0), C2 ) cannot be represented as in (5.9), namely the ones = (0, 0) in the stable outgoing and incoming complex (Lagrangian) curves for the iHp0 flow, and if ρ converges to one of these lines, then in general |t| → ∞ for the t in (5.9), so a priori it is not clear then that the right-hand side of (5.10) is defined. These difficulties were analyzed and settled in [13], and at this point there is no difference with our situation, so we conclude that κ is a well-defined analytic map in a neighborhood of (0, 0): Lemma 5.2 With f, p as in Lemma 5.1, there exists an analytic canonical transformation κ : neigh ((0, 0), R2 ) → neigh ((0, 0), R2 ), with f ◦ p ◦ κ = p0 . If p depends smoothly (analytically) on some real parameters, and fulfills the assumptions above, then f, κ can be chosen to depend smoothly (analytically) on the same parameters. If p = p = O((x, ξ)2 ) is analytic in (x, ξ), depends smoothly on ∈ neigh (0, R) and satisfies the assumptions above for = 0, then we get f (E), κ (x, ξ), holomorphic in E and x, ξ, depending smoothly on with f ◦ p ◦ κ = p0 , but f , κ are no more necessarily real when = 0. Clearly Im f (E) = O(), Im κ (x, ξ) = O() when E, x, ξ are real. In our case the parameters are τ, and the above discussion gives: Proposition 5.3 For p (τ, x, ξ) in (5.5), we can find a canonical transformation (x, ξ) → κτ, (x, ξ) depending analytically on τ and smoothly on with values in the holomorphic canonical transformations: neigh ((0, 0), C2 ) → neigh ((0, 0), C2 ), and an analytic function g (τ, q) depending smoothly on such that κτ, (0, 0) = (x(τ, ), ξ(τ, )),
Moreover, κτ,0
1 p (τ, κτ, (x, ξ)) = f (τ ) + ig (τ, (x2 + ξ 2 )). 2 is real when τ is real and ∂ Re g (0, 0) > 0. ∂q
(5.11) (5.12)
(5.13)
As a matter of fact, as in Section 4, we will apply this result to a modification of p , containing also the leading lower order symbol. Before doing so, we recall how to treat lower order symbols in general for operators with leading symbol modelled on the 1-dimensional harmonic oscillator (similarly to what we did in Section 3 and as in [26]).
28
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
Consider a formal h-pseudodifferential operator Q(x, hDx ; h) with symbol Q(x, ξ; h) ∼ q0 (x, ξ) + hq1 (x, ξ) + · · · ,
(5.14)
defined in a neighborhood of (0, 0) ∈ R2 . As usual, q0 , q1 , . . . are supposed to be smooth and we assume q0 (x, ξ) = g0 (p0 (x, ξ)), (5.15) where g0 ∈ C ∞ (neigh (0, R)) satisfies g0 (0) = 0, g0 (0) = 0. (We do not assume g0 to be real-valued.) As in Section 3 we find a smooth function a0 (x, ξ), defined in a neighborhood of (0, 0), such that Hq0 a0 = q1 − q1 , (5.16) 2π 1 q1 ◦ exp (tHp0 )dt. Adding lower order where q1 is the trajectory average 2π 0 corrections, we see that there exists A(x, ξ; h) ∼ a0 (x, ξ) + ha1 (x, ξ) + · · ·
(5.17)
with all aj smooth in some common neighborhood of (0, 0), such that hDx ; h) eiA(x,hDx ;h) Q(x, hDx ; h)e−iA(x,hDx ;h) =: Q(x,
(5.18)
∼ q0 + h has a symbol Q q1 + · · · , with q0 = q0 and Hq0 qj = 0, ∀j.
(5.19)
This means that qj is a smooth function of p0 (x, ξ) and as is well known (and exploited for instance in [26]), the facts (5.18), (5.19) can be reformulated by saying that we have found A as in (5.17) such that eiA(x,hD;h) Q(x, hD; h)e−iA(x,hD;h) = g(p0 (x, hD); h), ∞ where g(E; h) ∼ 0 gj (E)hj in C ∞ (neigh (0, R)), with g0 as before. When g0 , qj are holomorphic in fixed neighborhoods of E = 0 and (x, ξ) = (0, 0), we get the corresponding holomorphy for gk , q . Now return to the operator P of the beginning of this section. Write the full symbol as P (τ, x, ξ, ; h) ∼ p (τ, x, ξ) + h p1 (τ, x, ξ, ) + h2 p2 (τ, x, ξ, ) + · · ·
(5.20)
a) Consider first the general case without any assumptions on the subprincipal symbol, and assume that (5.21) h < hδ , for some fixed δ > 0. Following the strategy of Section 4, we rewrite (5.20) as h h h P (τ, x, ξ; h) = f (τ ) + [(iq(τ, x, ξ) + O() + p1 (τ, x, ξ)) + h p2 + h2 p3 + · · · ]. (5.22)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
29
As before, we now treat h/ as an additional small parameter. Proposition 5.3 extends to the case when p is replaced by p + h p1 , so we have a canonical transformation (x, ξ) → κτ,,h/ (x, ξ) depending analytically on τ and smoothly on , h , equal to κτ, when h = 0, such that h 1 ( p + p1 )(τ, κτ,, h (x, ξ)) = f (τ ) + ig, h (τ, (x2 + ξ 2 )), 2 with g,0 = g appearing in Proposition 5.3. As in Section 4, we therefore obtain an elliptic Fourier integral operator U,h/ , which is a convolution in t, and such that the Fourier transform with ,h/ (τ ), is a 1-dimensional Fourier integral operator in x quantizing respect to t, U κτ,,h/. After conjugation of P by U,h/ , we get a new operator P of the same type, with symbol 1 P (τ, x, ξ, , h/; h) = f (τ ) + [ig, h (τ, (x2 + ξ 2 )) + h p2 + h2 p3 + · · · ], 2
(5.23)
where p2 , p3 , . . . also depend on h/. h After a further conjugation by eiA(hDt ,x,hDx ,, ;h) , where each term Aj in the h-asymptotic expansion: h h h A(τ, x, ξ, , ; h) ∼ A0 (τ, x, ξ, , ) + hA1 (τ, x, ξ, , ) + · · · is holomorphic in τ, x, ξ in a fixed neighborhood of (0, 0, 0) ∈ C3 and smooth in , h/, we get a new operator of the form 1 h P = f (hDt ) + iG(hDt , (x2 + (hDx )2 ), , ; h), 2 where
(5.24)
∞
h h G(τ, q, , ; h) ∼ Gj (τ, q, , )hj , 0
(5.25)
with Gj holomorphic in τ, q in a j-independent neighborhood of (0, 0) and smooth in , h/. Moreover G0 is equal to the term g,h/ (τ, q) in (5.23). Recalling that 1 1 2 2 2 (x + (hDx ) ) has the eigenvalues h( 2 + k2 ), k2 ∈ N, we get the conclusion: Proposition 5.4 Make the assumptions of case II in the introduction, and assume that F0 = Re qmin,0 (the case when F0 is a maximum being analogous). Then in a 1 1 1 1 rectangle ] − |O(1)| , |O(1)| [+i]F0 − |O(1)| , F0 + |O(1)| [, P has the quasi-eigenvalues: f
h(k1 −
S1 k0 )− 4 2π
S1 1 k0 h , h( + k2 ), , ; h , + iG h(k1 − ) − 4 2π 2 (k1 , k2 ) ∈ Z × N. (5.26)
30
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
Here f (τ ) is real-valued with f (0) = 0, f (0) > 0. The function G has the properties ∂ described in and after (5.25) and Re G0 (0, 0, 0, 0) = F0 , ∂q Re G0 (0, 0, 0, 0) > 0. Finally, k0 is a fixed integer. b) We next consider the case when the subprincipal symbol of P=0 vanishes, and assume that h2 < hδ , (5.27) for some fixed δ > 0. According to the improved Egorov theorem of Section 2, we know that p1 in (5.20) vanishes for = 0, so we can write h p1 (τ, x, ξ, ) = h p1 (τ, x, ξ, ) in (5.22) and treat this term as a lower order term, while we now 2 allow h p2 to be a correction to the leading terms. As in the corresponding case in Section 4, we get h2 / as an additional small parameter instead of h/, and the same procedure as in case a) now leads to (5.24), (5.25) with h/ replaced by h2 /. Proposition 5.5 Make the assumptions of Proposition 5.4 and assume in addition that the subprincipal symbol of P=0 vanishes. Then for in the range (5.27), P has the quasi-eigenvalues as described in the preceding proposition, with the only difference that “h/” in (5.26) should be replaced by “h2 /”.
6 Global Grushin problem Let P be as in Section 1. In Sections 4 and 5 we have constructed microlocal normal forms for P near a Lagrangian torus and near a closed Hp -trajectory, respectively. The purpose of this section is to justify the preceding microlocal constructions and computations, and to show that the quasi-eigenvalues of Proposition 4.1 and Propositions 5.4 and 5.5 give, modulo O(h∞ ), all of the true eigenvalues of P , in suitable regions of the complex plane. This will be achieved by studying an auxiliary global Grushin problem, well posed in a certain h-dependent Hilbert space, and the first and the main step for us will be to define this space globally. The actual setup of the Grushin problem and some of the details of the computations will be closely related to the corresponding analysis in [20]. When constructing the Hilbert space, we shall inspect all the steps of the microlocal reductions of Sections 3–5, and implement each step of the construction. In doing so, for simplicity, we shall concentrate on the case when M = R2 . In view of the results of the appendix, it will be clear how to extend the following discussion to the case of compact real-analytic manifolds. Also, in order to simplify the presentation, we shall assume throughout the section that the order function m, introduced in (1.2), is equal to 1. Again, it will be clear that the discussion below will extend to the case of a general order function. Throughout this section we shall assume that = O(hδ ), for some fixed δ > 0. Let G = G(x, ξ, ) be as in (3.6). We shall introduce an IR-manifold ΛG ⊂ C4 , which in a complex neighborhood of p−1 (0) ∩ R4 is equal to exp (iHG )(R4 ), and further away from p−1 (0) ∩ R4 agrees with the real phase space R4 . The
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
31
manifold ΛG will be -close to R4 , and when defining it, it will be convenient to work on the FBI transform side. We shall use the FBI-Bargmann transform −3/2 (6.1) T u(x) = Ch eiϕ(x,y)/h u(y) dy, x ∈ C2 , C > 0, where ϕ(x, y) = i/2(x − y)2 . Associated to T there is a complex linear canonical transformation κT , given by C4 (y, −ϕy (x, y)) −→ (x, ϕx (x, y)) ∈ C4 . It is well known, see [28], that κT maps R4 onto
2 ∂Φ0 (Im x)2 . x, ΛΦ0 := , x ∈ C2 , Φ0 (x) = i ∂x 2 The IR-manifold ΛG has already been defined near p−1 (0) ∩ R4 , and when constructing it globally, we require that the IR-manifold κT (ΛG ) should agree with ΛΦ0 outside a bounded set and that it is -close to that manifold everywhere. We define therefore ΛG so that the representation
2 ∂Φ 2 κT (ΛG ) = x, (6.2) , x ∈ C =: ΛΦ i ∂x holds true. Here the function Φ ∈ C ∞ (C2 ; R) is uniformly strictly plurisubharmonic, and is such that Φ(x) = Φ0 (x) + g(x, ), with g(x, ) ∈ C ∞ in both arguments and with a uniformly compact support with respect to x. Associated to ΛG we then introduce the corresponding Hilbert space H(ΛG ) which agrees with L2 (R2 ) as a space, and which we equip with the norm || u || := || T u ||L2Φ . Here L2Φ = L2 (C2 ; e−2Φ/h L(dx)), with L(dx) being the Lebesgue measure on C2 . Performing a contour deformation in the integral representation of P on the FBI-Bargmann transform side, as in [20], [28], we see that P = O(1) : H(ΛG ) → H(ΛG ),
(6.3)
and the leading symbol on the FBI transform side is then p ◦ κ−1 T
. Continuing
ΛΦ
to work on the FBI-Bargmann transform side, as in Section 2 of [20], we introduce a microlocally unitary semiclassical Fourier integral operator eG(x,hDx,)/h : L2 (R2 ) → H(ΛG ),
(6.4)
32
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
microlocally defined near p−1 (0) ∩ R4 , and associated to the complex canonical transformation exp (iHG ) : R4 → ΛG . The operator in (6.3) is then microlocally near p−1 (0), unitarily equivalent to the operator e−G(x,hDx,)/h P eG(x,hDx ,)/h : L2 → L2 , with the principal symbol p ◦ exp (iHG ) = p + iq + O(2 ).
(6.5)
This averaging procedure allows us therefore to reduce the further analysis to an operator P , microlocally defined near p−1 (0) ∩ R4 , which has the principal symbol (6.5), where q, as well as the O(2 )-term, are in involution with p. As explained in Section 4, at this stage the operator P acts on single-valued functions in L2 (R2 ). In the first part of this section we shall concentrate on the torus case of Section 4. We assume therefore that dp and dRe q are linearly independent on the set (6.6) Λ0,0 : p = 0, Re q = 0. We recall also the assumption that T (0) is the minimal period of every closed Hp trajectory in the Lagrangian torus Λ0,0 , and notice that in a neighborhood of Λ0,0 , p and Re q form a completely integrable system. Introduce a new Lagrangian 0,0 ⊂ ΛG defined by torus Λ 0,0 : p ◦ exp (−iHG ) = 0, Re q ◦ exp (−iHG ) = 0. Λ
(6.7)
0,0 by means of In what follows we shall often identify the tori Λ0,0 and Λ 0,0 when there is no risk of exp (iHG ), and we shall continue to write Λ0,0 for Λ confusion. Combining exp (iHG ) with the canonical transformation κ, introduced in (4.4), and given by the action-angle coordinates associated with p, Re q, we get a smooth canonical diffeomorphism
κ : neigh ξ = 0, T ∗ T2 → neigh (Λ0,0 , ΛG ) , (6.8) so that κ = exp (iHG ) ◦ κ. As in (4.32), we set ξ dx, j = 1, 2, Sj = αj
where α1 and α2 are the fundamental cycles in Λ0,0 , with α1 corresponding to a closed Hp -trajectory of the minimal period T (0). Introduce also the “Maslov indices” k0 (αj ) ∈ Z, j = 1, 2, of the cycles αj , defined as in Proposition 2.3. Let L2θ (T2 ) be the subspace of L2loc (R2 ) consisting of Floquet periodic functions u(x), satisfying u(x − ν) = eiθ·ν u(x),
2
ν ∈ (2πZ) ,
where θ =
k0 S + . 2πh 4
(6.9)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
33
Here S = (S1 , S2 ) and k0 = (k0 (α1 ), k0 (α2 )) ∈ Z2 . An application of Theorem 2.4 allows us to conclude that there exists a microlocally unitary multi-valued Fourier integral operator (6.10) U : L2θ (T2 ) → L2 (R2 ), microlocally defined from a neighborhood of ξ = 0 in T ∗ T2 to a neighborhood of Λ0,0 in R4 , and associated to κ in (4.4). Moreover, U satisfies the improved Egorov property (2.3). The composition eG(x,hDx,)/h ◦ U is then associated with κ in (6.8), and we have a Egorov’s theorem, still with the improved property (2.3). The operator P , acting in H(ΛG ) is therefore unitarily equivalent to an h-pseudodifferential operator microlocally defined near ξ = 0, acting in L2θ (T2 ), and which has the leading symbol p(ξ1 ) + iq(ξ) + O(2 ), independent of x1 . We shall continue to write P for the conjugated operator on T2 . From Section 4 we next recall that there exists an elliptic pseudodifferential operator of the form eiA/h , acting on L2θ (T2 ), such that after a conjugation by it, the full symbol of P becomes independent of x1 . Recall also that A is constructed as a formal power series in and h, with coefficients holomorphic in a fixed complex neighborhood of the zero section of T ∗ T2 . These formal power series are then realized as C ∞ -symbols, in view of our basic assumption = O(hδ ), δ > 0. Summing up the discussion so far, we have now achieved that, microlocally near Λ0,0 , the operator P : H(ΛG ) → H(ΛG ) is equivalent to an operator of the form P (x2 , ξ, ; h) ∼
∞
hν pν (x2 , ξ, )
(6.11)
ν=0
acting on L2θ (T2 ). Here pν (x2 , ξ, ) are holomorphic in a ν-independent complex neighborhood of ξ = 0, and p0 = p(ξ1 ) + iq(ξ) + O(2 ). Furthermore, P=0 is selfadjoint. Remark. It follows from the construction together with Theorem 2.4 that if the subprincipal symbol of P=0 vanishes, then p1 (x2 , ξ, 0) = 0. We must now implement the final conjugation of P , which removes the x2 dependence in the full symbol. In doing so, we shall first assume that we are in the general case, so that the subprincipal symbol of P=0 does not necessarily vanish. We shall work under the assumption h ≤ δ0 1.
(6.12)
34
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
As in Section 4, we write h h P = p(ξ1 ) + r0 x2 , ξ, , + hr1 x2 , ξ, , + ··· , h h r0 x2 , ξ, , = iq(ξ) + O() + p1 (x2 , ξ, ),
where
and rj = Oj (h/), j ≥ 1. Let us introduce a complexification of the standard 2 = T2 + iR2 . From the constructions of Section 4 we know that there 2-torus, T exists a holomorphic canonical transformation 2 × C2 (6.13) κ : neigh Im y = η = 0, T 2 × C2 (y, η) → (x, ξ) ∈ neigh Im x = ξ = 0, T with the generating function of the form h h ψ x, η, , = x · η + φper x2 , η, , ,
φper
h =O + ,
and such that
(r0 ◦ κ ) (y, η, , h/) = r0 (·, η, , h/) + O
h ,
(6.14)
2 (6.15)
is independent of y – see (4.19). It follows from (6.14) that κ is ( + h/)-close to the identity, and has the expression (y1 , η1 ; y2 , η2 ) −→ (x1 (y2 , η), η1 ; x2 (y2 , η), ξ2 (y2 , η)). In particular it is true that h Im x = O + ,
h Im ξ2 = O + ,
Im ξ1 = 0,
2 2 2 on the image of T ∗ T
. We introduce now an IR-manifold Λ ⊂ T × C , which ∗ 2 is equal to κ T T in a complex neighborhood of the zero section of T ∗ T2 , and outside another complex fixed neighborhood of ξ = 0, coincides with T ∗ T2 . in such a way that it remains In the intermediate region, we shall construct Λ ∗ 2 we have the an ( + h/)-perturbation of T T , and such that everywhere on Λ property =⇒ Im ξ1 = 0. (6.16) (x1 , ξ1 ; x2 , ξ2 ) ∈ Λ
and describing the conjugation of P by a Fourier integral When constructing Λ operator associated to κ , it is convenient to work on the FBI transform side. As
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
35
in Section 3 of [20], we notice that the FBI-Bargmann transformation introduced in (6.1) generates an operator from L2θ (T2 ) to the space of Floquet periodic holomorphic functions on C2 . We continue to denote this operator by T . Then after the application of the canonical transformation κT , associated to T , the cotangent 2 × C2 given by space T ∗ T2 becomes an IR-manifold ΛΦ1 ⊂ T 2 ∂Φ1 (Im x)2 = −Im x, Φ1 (x) = . i ∂x 2 Since T is a convolution operator acting separately in y1 and y2 , we see that ΛΦ1 : ξ =
= ΛΦ , κT (Λ)
ΛΦ : ξ =
2 ∂Φ , i ∂x
∂Φ where Φ is an ( + h/)-perturbation of Φ1 with the property that ξ1 = (2/i) ∂x 1 is real. It follows that Φ = Φ(Im x1 , x2 ) is independent of Re x1 . Using a standard cutoff function around Im x = 0, we modify Φ away from Im x = 0 to obtain a strictly plurisubharmonic function Φ which coincides with Φ1 further away from Im x = 0, in such a way that Φ remains an ( + h/)-perturbation of Φ1 and is still a function independent of Re x1 . We then define the global IR-manifold = κ−1 (ΛΦ ). Λ T Associated to κ , there is a Fourier integral operator U −1 introduced in (4.20),
U −1 = O(1) : L2 (T2 ) → H(Λ), is microlocally near ξ = 0 unitarily equivalent such that the action of P on H(Λ) to the operator U P U −1 : L2 (T2 ) → L2 (T2 ), whose Weyl symbol has the form h h p(ξ1 ) + r0 ξ, , + hr1 x2 , ξ, , + ··· .
(6.17)
h r0 = iq(ξ) + O +
Here
is independent of x, and
h rj = O + ,
j ≥ 1.
The corresponding statement is also true when considering the action on L2θ (T2 ), since U −1 preserves the Floquet property (6.9). on the torus side, there is an IR-manifold Associated to the IR-deformation Λ 4 Λ ⊂ C which is an (+ h/)-perturbation of ΛG near Λ0,0 , obtained by replacing exp (iHG ) ◦ κ(T ∗ T2 ) there by (T ∗ T2 ) = exp (iHG ) ◦ κ(Λ). exp (iHG ) ◦ κ ◦ κ
36
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
, which is ( + h/)-close to In such a way we get a globally defined IR-manifold Λ 4 ΛG and agrees with R outside a neighborhood of p−1 (0) ∩ R4 . Associated with ), defined similarly to H(ΛG ), and obtained we then have a Hilbert space H(Λ Λ by modifying the standard weight Φ0 (x) on the FBI-Bargmann transform side. 0,0 ⊂ Λ , with the property We also get a corresponding new Lagrangian torus Λ that microlocally near Λ0,0 , the original operator ) → H(Λ ) P : H(Λ is equivalent to an operator on L2θ (T2 ), whose complete symbol has the form (6.17). Taking into account the conjugation by an elliptic operator eiB1 /h on the torus side, which was constructed in Section 4 and which eliminates the x2 -dependence also in the terms rj with j ≥ 1, we get the following result. Proposition 6.1 We make all the assumptions of case I in the introduction, and recall that we also take F0 = 0. Assume that = O(hδ ), δ > 0 is such that h/ ≤ δ0 , ⊂ C4 , and a smooth Lagrangian torus 0 < δ0 1. There exists an IR-manifold Λ 0,0 ⊂ Λ , such that when ρ ∈ Λ is away from a small neighborhood of Λ 0,0 in Λ Λ , we have |Re P (ρ, h)| ≥
1 |O(1)|
or
|Im P (ρ, h)| ≥
. |O(1)|
(6.18)
is an + h -perturbation of R4 in the natural sense, and it is The manifold Λ equal to R4 outside a neighborhood of p−1 (0) ∩ R4 . We have ) → H(Λ ). P = O(1) : H(Λ There exists a smooth canonical transformation 0,0 , Λ ) → neigh (ξ = 0, T ∗ T2 ), κ : neigh (Λ 0,0 ) = T2 ×{0}. Associated to κ , there is a Fourier integral operator such that κ (Λ ) → L2 (T2 ), U = O(1) : H(Λ θ which has the following properties: ), χ2 ∈ 1) U is concentrated to the graph of κ in the sense that if χ1 ∈ C0∞ (Λ ∞ ∗ 2 C0 (T T ), are such that 0,0 , Λ )} = ∅, (suppχ2 × suppχ1 ) ∩ {(κ (y, η), y, η); (y, η) ∈ neigh(Λ then
) → L2θ (T2 ). χ2 (x, hDx ) ◦ U ◦ χ1 (x, hDx ) = O(h∞ ) : H(Λ
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
37
2) The operator U is microlocally invertible: there exists an operator V = O(1) : ) such that for every χ1 ∈ C0∞ (neigh(Λ 0,0 , Λ )), we have L2θ (T2 ) → H(Λ ) → H(Λ ). (V U − 1) χ1 (x, hDx ) = O(h∞ ) : H(Λ For every χ2 ∈ C0∞ (neigh(ξ = 0, T ∗ T2 )), we have (U V − 1) χ2 (x, hDx ) = O(h∞ ) : L2θ (T2 ) → L2θ (T2 ).
3) We have Egorov’s theorem: Acting on L2θ (T2 ), there exists P hDx , , h ; h with the symbol ∞ h h j P ξ, , ; h ∼ p(ξ1 ) + h rj ξ, , , j=0 with
|ξ| ≤
1 , |O(1)|
h r0 = iq(ξ) + O + ,
and rj = Oj
h +
,
j ≥ 1,
such that PU = U P microlocally, i.e., P U − U P χ1 (x, hDx ) = O(h∞ ), χ2 (x, hDx ) P U − U P = O(h∞ ), for every χ1 , χ2 as in 2). Remark. The estimate (6.18) holds true thanks to the property (6.16) of the final deformation, since then the term p(ξ1 ) does not contribute to the imaginary part of the symbol on the torus side. The bound (6.18) will allow us to reduce the 0,0 . spectral analysis of P to a small neighborhood of the Lagrangian torus Λ Using Proposition 6.1, we shall now proceed to describe the spectrum of P in a rectangle of the form
1 RC, = z ∈ C; |Re z| < , |Im z| < , (6.19) C C for a sufficiently large constant C > 0. We shall show that the eigenvalues in (6.19) are given by the quasi-eigenvalues of Proposition 4.1, modulo O(h∞ ). In doing so, let us consider the set of the quasi-eigenvalues, introduced in (4.35),
h k0 S + . Σ(, h) = P h(k − θ), , ; h ; k ∈ Z2 RC, , θ = 2πh 4
38
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
Then the distance between 2 elements of Σ(, h) corresponding to k, l ∈ Z2 , k = l, is ≥ h |k − l| /|O(1)|. Introduce h h δ := 1/4 inf dist (P (h(k − θ), , ; h), P(h(l − θ), , ; h)) > 0, k=l and consider the family of open discs
h Ωk (h) := z ∈ RC, ; z − P (h(k − θ), , ; h) < δ ,
k ∈ Z2 .
The sets Ωk (h) are then disjoint, and dist (Ωk (h), Ωl (h)) ≥ h |k − l| /|O(1)|. As a warm-up exercise, we shall first show that Spec (P ) in the set (6.19) is contained in the union of the Ωk (h). When z ∈ C is in the rectangle (6.19), let us consider the equation (P − z) u = v,
). u ∈ H(Λ
(6.20)
We notice here that the symbol of Im P =
P − P∗ , 2i
), is O(), and from Proposition 6.1 we know taken in the operator sense in H(Λ it is true that |Im P (ρ, h)| > 0,0 in Λ that away from any fixed neighborhood of Λ /C, provided that |Re P (ρ, h)| ≤ 1/C, where C > 0 is sufficiently large. Here we are using the same letters for the operators and the corresponding (Weyl) symbols, and P + P∗ ) → H(Λ ). : H(Λ Re P = 2 ). We shall also write p to denote the leading symbol of P=0 , acting on H(Λ Let us introduce a smooth partition of unity on the manifold Λ , 1 = χ + ψ1,+ + ψ1,− + ψ2,+ + ψ2,− . ) is such that χ = 1 near Λ 0,0 , and supp χ is contained in a small Here χ ∈ C0∞ (Λ ) are sup neighborhood of Λ0,0 where U P = P U . The functions ψ1,± ∈ C0∞ (Λ ported in regions, invariant under the Hp -flow, where ±Im P > /C, respectively. ) are such that Finally ψ2,± ∈ Cb∞ (Λ
suppψ2,± ⊂ ρ; ±Re P (ρ, h) > 1/C . . Moreover, we arrange so that the functions ψ1,± Poisson commute with p on Λ We shall prove that 1 (6.21) || v || + O(h∞ )|| u ||, || (1 − χ)u || ≤ O ). In doing so, we shall first derive where we let || · || stand for the norm in H(Λ a priori estimates for ψ1,+ u.
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
39
When N ∈ N, let ψ0 ≺ ψ1 ≺ · · · ≺ ψN ,
ψ0 := ψ1,+ ,
; [0, 1]), supported in an Hp -flow invariant region be cutoff functions in C0∞ (Λ where Im P ∼ , and which are in involution with p. Here standard notation f ≺ g means that supp f is contained in the interior of the set where g = 1. It is then true that in the operator norm, [P , ψj ] = [P=0 , ψj ] + O(h) = O(h2 ) + O(h) = O(h),
0 ≤ j ≤ N,
(6.22)
since ≥ h. For future reference we notice that in the case when the subprincipal symbol of P=0 vanishes, the Weyl calculus shows that [P=0 , ψj ] = O(h3 ), and since ≥ h2 , we still get (6.22). Here we have also used that the subprincipal symbol of ψj is 0, 0 ≤ j ≤ N . Near the support of ψj it is true that Im P ∼ , and an application of the semiclassical G˚ arding inequality allows us therefore to conclude that (Im (P − z)ψj u|ψj u) ≥
|| ψj u ||2 − O(h∞ )|| u ||2 . O(1)
). On the other hand, we have Here the inner product is taken in H(Λ (Im (P − z)ψj u|ψj u) = Im (ψj (P − z)u|ψj u) + ([P , ψj ]u|ψj u) , and since in the operator sense ψj (1 − ψj+1 ) = O(h∞ ), we see that the absolute value of this expression does not exceed O(1)|| (P − z)u || || ψj u || + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 . We get
≤
|| ψj u ||2 ≤ O(1)|| (P − z)u || || ψj u || + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 C O(1) || ψj u ||2 + || (P − z)u ||2 + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 , 2C
and hence, || ψj u ||2 ≤
O(1) || (P − z)u ||2 + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 . 2
Combining these estimates for j = 0, 1, . . . , N , we get || ψ0 u ||2 ≤
O(1) || (P − z)u ||2 + ON (1)hN || ψN u ||2 + O(h∞ )|| u ||2 , 2
40
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
and therefore
O(1) || v || + O(h∞ )|| u ||. The same estimate can be obtained for ψ1,− u, microlocally concentrated in a flow invariant region where Im P ∼ −, and a fortiori such estimates also hold in regions where Re P ∼ 1 and Re P ∼ −1. The bound (6.21) follows. || ψ1,+ u || ≤
Write next (P − z) χu = χv + w, w = [P , χ]u, where w satisfies
(6.23)
h || w || ≤ O || v || + O(h∞ )|| u ||.
0,0 . Applying the operator U of Here we have used (6.21) with a cutoff closer to Λ Proposition 6.1 to (6.23), we get P − z U χu = U χv + U w + T∞ u, where
) → L2 (T2 ). T∞ = O(h∞ ) : H(Λ θ
Using an expansion in Fourier series (6.25) below, we see that the operator P − z : L2θ (T2 ) → L2θ (T2 ) is invertible, microlocally in |ξ| ≤ 1/|O(1)|, with a microlocal inverse of the norm O(1/h), provided that z ∈ RC, avoids the discs Ωk (h). Using also the uniform boundedness of the microlocal inverse V of U , we get || χu || ≤
O(1) || v || + O(h∞ )|| u ||. h
(6.24)
Combining (6.21) and (6.24), we see that when z ∈ RC, is in the complement of the union of the Ωk (h), the operator ) → H(Λ ) P − z : H(Λ is injective. Since the ellipticity assumption (1.6) implies that it is a Fredholm ) → H(Λ ) is bijective. operator of index zero, we know that P − z : H(Λ We shall now let z vary in the disc Ωk (h) ⊂ RC, , for some k ∈ Z2 . We shall show that z ∈ Ωk (h) is an eigenvalue of P if and only if z = P (h(k −θ), , h ; h)+r, where r = O(h∞ ). In doing so, we shall study a globally well-posed Grushin ). problem for the operator P − z in the space H(Λ As a preparation for that, we shall introduce an auxiliary Grushin problem for the operator P − z, defined microlocally near ξ = 0 in T ∗ T2 . From (4.34), let us recall the functions el (x) =
S 1 i(l−θ)x 1 hi (h(l− k40 )− 2π )x , e e = 2π 2π
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
41
which form an ON basis for the space L2θ (T2 ), so that when u ∈ L2θ (T2 ), we have a Fourier series expansion, u(x) = u (l − θ)el (x). (6.25) l∈Z2
We also remark that el (x) are microlocally concentrated to the region of the phase space where ξ ∼ h l − k40 − S/2π. + : L2 (T2 ) → C and R − : C → L2 (T2 ), given Introduce rank one operators R θ θ − u− = u− ek . Here the inner product in the definition + u = (u|ek ) and R by R + is taken in the space L2 (T2 ). Using (6.25), it is then easy to see that the of R θ operator −z R − P := P (6.26) : L2θ (T2 ) × C → L2θ (T2 ) × C, + R 0 defined microlocally near ξ = 0, has a microlocal inverse there, which has the form + E(z) E E = (6.27) − E −+ (z) . E The following localization properties can be inferred from the construction of E: + = if ψ ∈ Cb∞ (T ∗ T2 ) has its support disjoint from ξ = 0, then it is true that ψ E − ψ = O(h∞ ) : L2 → C. We also find that O(h∞ ) : C → L2θ , and E θ −+ (z) = z − P h(k − θ), , h ; h . E (6.28) Using (6.25), we furthermore see that the following estimates hold true, = O(1) : L2 (T2 ) → L2 (T2 ), E θ θ h + = O(1) : C → L2θ (T2 ), E
− = O(1) : L2θ (T2 ) → C, E
−+ = O(h) : C → C, E so that h|| u || + || u− || ≤ O(1) (|| v || + h|| v+ ||) , when P
u u−
=
v v+
(6.29)
.
In (6.29), the norms of u and v are taken in L2θ (T2 ) and those of u− and v+ in C. ) → C and R− : C → H(Λ ) by Passing to the case of P , we define R+ : H(Λ + U χu = (U χu|ek ), R+ u = R
− u − = u − V ek . R− u− = V R
(6.30)
42
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
It is then true that ), χR− = R− + O(h∞ ) : C → H(Λ
(6.31)
decreasing the support of χ if necessary. We now claim that for z ∈ Ωk (h), the Grushin problem (P − z) u + R− u− = v, (6.32) R+ u = v+ ) × C for every (v, v+ ) ∈ H(Λ ) × C, with an has a unique solution (u, u− ) ∈ H(Λ a priori estimate, h|| u || + || u− || ≤ O(1) (|| v || + h|| v+ ||) .
(6.33)
), and those of u− and v+ in C. To Here the norms of u and v are taken in H(Λ verify the claim, we first see that as in (6.21), we have 1 (6.34) || (1 − χ)u || ≤ O || v || + O(h∞ ) (|| u || + || u− ||) . Here we have also used (6.31). Applying χ to the first equation in (6.32) we get (P − z) χu + R− u− = χv + w + R−∞ u− , R+ u = v+ , where w = [P , χ]u satisfies || w || ≤ O
(6.35)
h || v || + O(h∞ ) (|| u || + || u− ||) ,
and R−∞ = O(h∞ ) in the operator norm. Applying U to the first equation in (6.35) and using (6.30), we get − u− = U χv + U w + w− (P − z)U χu + R (6.36) + U χu = v+ . R where the L2θ (T2 )-norm of w− is O(h∞ ) (|| u || + || u− ||). We therefore get a mi in (6.26), and in view of (6.29) we crolocally well-posed Grushin problem for P obtain, h|| χu || + || u− || ≤ O(1) (|| v || + h|| v+ ||) + O(h∞ ) (|| u || + || u− ||) .
(6.37)
Combining (6.34) and (6.37), we get (6.33). We have thus also proved that the operator P − z R− ) × C ) × C → H(Λ (6.38) : H(Λ P= R+ 0
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
43
is injective, for z ∈ Ωk (h). Now P is a finite rank perturbation of P − z 0 , 0 0 which is a Fredholm operator of index zero. It follows that P is also Fredholm of index 0 and hence bijective, since we already know that it is injective. The inverse of P has the form E(z) E+ , (6.39) E= E− E−+ (z) and we recall that the spectrum of P in Ωk (h) will be the set of values z for which E−+ (z) = 0. We finally claim that the components E+ and E−+ (z) in (6.39) are given by
+ , and E−+ (z) = E −+ (z) = z − P h(k − θ), , h ; h , modulo terms that E+ = V E are O(h∞ ). Indeed, we need only to check that + ≡ 1, R+ V E
−+ ≡ 0, + + R− E (P − z) V E
(6.40)
∞
modulo O(h ), and at this stage the verification of (6.40) is identical to the corresponding computation from Section 6 of [20]. In particular, we get h E−+ (z) = z − P h(k − θ), , ; h + O(h∞ ), (6.41) and we have now proved the first of our two main results. Theorem 6.2 Let F0 be a regular value of Re q viewed as a function on p−1 (0) ∩ R4 . Assume that the Lagrangian manifold Λ0,F0 : p = 0, Re q = F0 is connected, and that T (0) is the minimal period of every closed Hp -trajectory in Λ0,F0 . When α1 and α2 are the fundamental cycles in Λ0,F0 with α1 corresponding to a closed Hp -trajectory of minimal period, we write S = (S1 , S2 ) and k0 = (k0 (α1 ), k0 (α2 )) for the actions and Maslov indices of the cycles, respectively. Assume furthermore that = O(hδ ), δ > 0, is such that h/ 1. Let C > 0 be sufficiently large. Then the eigenvalues of P in the rectangle |Re z| < are given by
1 , C
|Im z − F0 |
0.
(6.45)
Recalling the operators eG(x,hDx,)/h and U from (6.4) and (6.10), respectively, we see, as in the general case, that the symbol of Im P on H(ΛG ) is O(), and away from any fixed neighborhood of Λ0,0 in ΛG , we have |Im P (ρ, h)| ∼ , if |Re P (ρ, h)| < 1/|O(1)|. We write, as in Section 4, h2 h2 P (x2 , ξ, , h) = p(ξ1 ) + r0 x2 , ξ, , + hr1 x2 , ξ, , + ··· , where
h2 h2 = iq + O() + p2 (x2 , ξ, ), r0 x2 , ξ, , 2 h h2 r1 (x2 , ξ, ) = q1 (x2 , ξ, ) + p3 (x2 , ξ, ), rj (x2 , ξ, ) = O , j ≥ 2.
Using the canonical transformation κ, generated by the function h2 h2 ψ x, η, , = x · η + φper x2 , η, , ,
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
with φper = O( +
45
h2 ),
constructed in Section 4, we then argue similarly to the 2 × C2 which is an ⊂T general torus case. We thus introduce an IR-manifold Λ 2 ∗ 2 ∗ 2 ( + h /)-perturbation of T T , which agrees with κ(T T ) near ξ = 0, and we first further away from this set coincides with T ∗ T2 . When constructing Λ, ∗ 2 notice that κ(T T ) has the form Im x = Gξ (Re (x, ξ)),
Im ξ = −Gx (Re (x, ξ)),
2
where G = G(x2 , ξ, , h ) is such that h2 ∂ξ G, ∂x2 G = O + . As was observed in Section 4, the transformation κ conserves actions, and therefore the smooth function G is single-valued. We may assume that h2 G=O + . If we let χ(ξ) ∈ C0∞ (R2 ; [0, 1]) be a cutoff function with a small support and such by that χ = 1 in a small neighborhood of 0, we define Λ (Re (x, ξ)), G(Re (Re (x, ξ)), Im ξ = −G (x, ξ)) = χ(Re ξ)G(Re (x, ξ)). Im x = G ξ x such that Im ξ1 = 0 on We then obtain the desired globally defined IR-manifold Λ Λ. When acting on H(Λ), P is microlocally near ξ = 0 unitarily equivalent to an operator on L2 (T2 ), which has the form h2 h2 p(ξ1 ) + r0 ξ, , + hr1 x2 , ξ, , + ··· , where
h2 h2 r0 ξ, , = iq + O +
is independent of x. It follows, as in the general torus case, that on the Bargmann transform can be described by an FBI-weight Φ = Φ(Im x1 , x2 ) which does not side, Λ depend on Re x1 . Repeating the previous arguments, we obtain therefore a new associated to an IR-manifold Λ ⊂ C4 , and a globally defined Hilbert space H(Λ), → H(Λ) is Lagrangian torus Λ0,0 ⊂ Λ such that microlocally near Λ0,0 , P : H(Λ) 2 2 equivalent to an operator on Lθ (T ), described in (4.27), (4.28). Proposition 6.3 Assume that the subprincipal symbol of P=0 vanishes, and consider the range M h2 < = O(hδ ) for M 1, δ > 0. There exists an IR-manifold ⊂ C4 and a smooth Lagrangian torus Λ 0,0 ⊂ Λ such that when ρ ∈ Λ is away Λ
46
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
0,0 in Λ and |Re P (ρ, h)| < 1/C, for a sufficiently from a small neighborhood of Λ large C > 0, it is true that |Im P (ρ, h)| ∼ . is ( + h2 /)-close to R4 and it coincides with R4 outside a neighThe manifold Λ −1 borhood of p (0) ∩ R4 . There exists a canonical transformation 0,0 , Λ) → neigh(ξ = 0, T ∗ T2 ), κ : neigh(Λ 0,0 onto T2 , and an elliptic Fourier integral operator U : H(Λ) → mapping Λ 2 2 Lθ (T ) associated to κ , such that, microlocally near Λ0,0 , U P = P U . Here 2
h P = P(hDx , , ; h) has the Weyl symbol, depending smoothly on , h2 / ∈ neigh(0, R), ∞ h2 h2 P ξ, , ; h ∼ p(ξ1 ) + hj rj ξ, , . j=0 We have r0 = iq(ξ) + O(1)( + h2 /),
rj = O(1), j ≥ 1.
Repeating the arguments, leading to Theorem 6.2, and using Proposition 6.3 instead of Proposition 6.1, we then find first that the spectrum of P in a region of the form (6.19) is contained in the union of disjoint discs of radii h/|O(1)| around
the quasi-eigenvalues P h(k − θ), , h2 /; h . Furthermore, when z varies in such a disc corresponding to k ∈ Z2 , such that the corresponding quasi-eigenvalue falls into the region (6.19), an inspection of the previous arguments shows that the Grushin operator P − z R− × C → H(Λ) ×C : H(Λ) R+ 0 is bijective with the inverse of the norm O((h)−1 ) – see (6.33) for the precise a and R+ : H(Λ) → C are defined as in (6.30). priori estimate. Here R− : C → H(Λ) This leads to the following result. Theorem 6.4 Keep all the assumptions and notation of Theorem 6.2, and in addition assume that the subprincipal symbol of P=0 vanishes. Let = O(1)hδ for some fixed δ > 0 be such that h2 . Then the eigenvalues of P in the rectangle 1 1 1 1 − , + i F0 − , F0 + C C C C
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
47
are given by h2 k0 S zk = P h k − , , ; h + O(h∞ ), k ∈ Z2 . − 4 2π Here C > 0 is large enough, P (ξ, , h2 /; h) is holomorphic in ξ ∈ neigh(0, C2 ), smooth in and h2 / ∈ neigh(0, R), and as h → 0, there is an asymptotic expansion h2 h2 h2 P ξ, , ; h ∼ p(ξ1 ) + r0 ξ, , + hr1 ξ, , + ··· . We have h2 h2 r0 ξ, , = iq(ξ) + O + ,
h2 rj ξ, , = O(1), j ≥ 1.
We shall now turn to the case II from the introduction. Let us recall from Section 1, that if z ∈ Spec P is such that |Re z| ≤ δ → 0, then Im z ∈ inf Re q − o(1), sup Re q + o(1) , h → 0. (6.46) Σ
Σ
Here, as in Section 1, we write Σ = p−1 (0) ∩ R4 /exp (RHp ). Our purpose here is to show that the quasi-eigenvalues of Propositions 5.4 and 5.5 give, up to O(h∞ ), the actual eigenvalues in a set of the form |Re z| ≤
1 , |O(1)|
|Im z − F0 | ≤
, |O(1)|
when F0 ∈ {inf Σ Re q, supΣ Re q}. As we shall see, the analysis here will be parallel to the torus case just treated, so that in what follows we shall concentrate on the new features of the problem, and some of the computations that are essentially identical to the ones already performed, will not be repeated. In order to fix the ideas, we shall discuss the case when F0 = inf Re q, Σ
and we shall take F0 = 0. Recall from the beginning of this section that the original operator P acting on H(ΛG ), is microlocally unitarily equivalent to the operator P ∼
∞
hj pj (x, ξ, ),
j=0
acting on L2 and defined microlocally near p−1 (0) ∩ R4 , with p0 = p + iq + O(2 ),
(6.47)
48
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
and the functions q and O(2 )-term are in involution with p. Let γ1 , . . . , γN ⊂ p−1 (0) ∩ R4 be the finitely many trajectories such that Re q = 0 along γj , 1 ≤ j ≤ N . We know that T (0) is the minimal period of each γj , and if ρj ∈ Σ is the corresponding point, then the Hessian of Re q at ρj is positive definite, 1 ≤ j ≤ N . Associated to γj , we have the quantities S = S(γj ) and k0 = k0 (γj ), the action along γj and the Maslov index, respectively, defined as in Section 2, and we recall from [11] that these quantities do not depend on j. In what follows we shall work microlocally near a fixed critical trajectory, say γ1 . We let L2S (S 1 × R) be the space of locally square integrable functions u(t, x) on R × R such that 2π
|u(t, x)|2 dx dt < ∞.
0
and u(t − 2π, x) = eiS/h+ik0 π/2 u(t, x). Applying Theorem 2.4 to the canonical transformation κ of Proposition 3.1, we see that there exists an analytic microlocally unitary Fourier integral operator U0 : L2S (S 1 × R) → L2 (R2 ), associated to κ,
and defined microlocally from a neighborhood of {τ = x = ξ = 0} in T ∗ S 1 × R to a neighborhood of γ1 in R4 , so that we have the two-term Egorov property (2.3). Combining exp (iHG ) with κ, we get a smooth canonical transformation
(6.48) κ : neigh τ = x = ξ = 0, T ∗ S 1 × R → neigh(γ1 , ΛG ), where abusing the notation slightly, we write here γ1 ⊂ ΛG also for the image of γ1 under the complex canonical transformation exp (iHG ). The operator eG(x,hDx ,)/h ◦ U0 is then associated with κ in (6.48), and an application of Egorov’s theorem shows that, microlocally near γ1 , we get a unitary equivalence between the operator P acting on H(ΛG ) and operator
an h-pseudodifferential microlocally defined near τ = x = ξ = 0 in T ∗ S 1 × R , with the leading symbol p0 (τ, x, ξ, ) = f (τ ) + iq(τ, x, ξ) + O(2 ), independent of t. Taking into account an additional conjugation by the elliptic operator eiA/h , acting on L2S (S 1 × R), with A∼
∞
ak (t, τ, x, ξ, )hk ,
k=1
constructed as a formal power series in , h in Proposition 3.2, we see that microlocally near γ1 , the operator P : H(ΛG ) → H(ΛG ) is equivalent to an operator of the form ∞ P (τ, x, ξ, ) ∼ hk pk (τ, x, ξ, ), (6.49) k=0
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
49
acting on L2S (S 1 × R), whose full symbol is independent of t. We have p0 = f (τ ) + iq(τ, x, ξ) + O(2 ),
(6.50)
and Re q(0, x, ξ) ∼ x2 + ξ 2 on the real domain. We shall first consider the general case when the subprincipal symbol of the unperturbed operator P=0 does not necessarily vanish, and in doing so, it will be assumed that (6.51) h = O(1)hδ , δ > 0. As in Section 5, we write h h P = f (τ ) + iq(τ, x, ξ) + O() + p1 + h p2 + · · · . According to Proposition 5.3, there exists a holomorphic canonical transformation κσ,, h : neigh(0, C2 ) → neigh(0, C2 ),
depending analytically on σ ∈ neigh(0, C) and smoothly on , h ∈ neigh(0, R), such that h Im κσ,, h (y, η) = O + , when σ, y, η are real, and such that h y2 + η2 σ, κσ,, h (y, η) = f (σ) + ig, h σ, . p0 + p1 2 Here g, h (σ, q) is an analytic function, depending smoothly on , h/, for which
∂ Re g,0 (0, 0) > 0. ∂q We now lift the family of locally defined canonical transformations κσ,, h to a canonical transformation 1 × C (s, σ; y, η) Ξ, h : neigh Im s = 0, σ = y = η = 0, T ∗ S 1 × C
→ (t, τ ; x, ξ) ∈ neigh Im t = 0, τ = x = ξ = 0, T ∗ S given by Ξ, h (s, σ; y, η) = (t, τ ; x, ξ) = (s + h(y, σ, η), σ; κσ,, h (y, η)).
(6.52)
50
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
Here h(y, σ, η) is uniquely determined up to a function g = g(σ), and if ϕσ,, h (x, y, θ) is an analytic family of non-degenerate phase functions (in the sense of H¨ ormander) locally generating the family κσ,, h , then
Φ, h (t, x, s, y, θ, σ) := ϕσ,, h (x, y, θ) + (t − s)σ
is a non-degenerate phase function with θ, σ as fiber variables, such that Φ, h generates the graph of Ξ, h . 1 × C , which in ⊂ T∗ S Associated to Ξ, h , we introduce an IR-manifold Λ
a complex neighborhood of τ = x = ξ = 0, is equal to Ξ, h T ∗ S 1 × R , and
further away from this set agrees with T ∗ S 1 × R . In the intermediate region, we
in such a way that it remains an ( + h )-perturbation of T ∗ S 1 × R , construct Λ it is true that and so that everywhere on Λ, =⇒ τ ∈ R. (t, τ ; x, ξ) ∈ Λ
(6.53)
If we now use the standard FBI-Bargmann transformation, viewed as a mapping on L2S (S 1 × R), so that under the associated canonical transformation, T ∗ (S 1 × R) 1 × C); (τ, ξ) = −Im (t, x)}, then as before we see is mapped to {(t, τ ; x, ξ) ∈ T ∗ (S is described by that after an application of such a transformation, the manifold Λ a weight function Φ = Φ(Im t, x) which does not depend on Re t. At this stage, the situation is similar to the previously analyzed torus case, and, in particular, we see again that the form of the weight Φ(Im t, x) implies that the term f (τ ) in (6.50) gives no contribution to the imaginary part of the operator. Summing up the discussion so far, we arrive to the following result. Proposition 6.5 Make the assumptions of case II in the introduction, and assume that F0 = inf Re q = 0. Σ
Assume that = O(h ), for some δ > 0, is such that h . There exists a closed IR-manifold Λ ⊂ C4 and finitely many simple closed disjoint curves γ1 , . . . , γN ⊂ Λ, which are ( + h/)-close to the closed Hp -trajectories ⊂ p−1 (0) ∩ R4 , along which Re q = 0, such that when ρ is outside a small neighborhood of ∪N j=1 γj in Λ, then 1 or |Im P (ρ, h)| ≥ . (6.54) |Re P (ρ, h)| ≥ |O(1)| |O(1)| δ
This estimate is true away from an arbitrarily small neighborhood of ∪N j=1 γj , provided that the implicit constant in (6.54) is chosen sufficiently large. The manifold Λ coincides with R4 away from a neighborhood of p−1 (0) ∩ R4 and is ( + h/)close to R4 everywhere. For each j with 1 ≤ j ≤ N , there exists a canonical transformation
κ,j : neigh (γj , Λ) → neigh τ = x = ξ = 0, T ∗(S 1 × R) ,
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
51
whose domain of definition does not intersect the closure of the union of the domains of the κ,k for k = j, and an elliptic Fourier integral operator Uj = O(1) : H(Λ) → L2S (S 1 × R), associated to κ,j , such that, microlocally near γj , Uj P = Pj Uj . Here Pj = Pj (hDt , (1/2)(x2 + (hDx )2 ), , h ; h) has the Weyl symbol h h x2 + ξ 2 Pj τ, x, ξ, , ; h = f (τ ) + iGj τ, , , ; h , 2 with
∞ h h Gj τ, q, , ; h ∼ hl Gj,l τ, q, , ,
h → 0,
l=0
and Gj,l holomorphic in (τ, q) ∈ neigh(0, C2 ), smooth in , h/ ∈ neigh(0, R). Furthermore, Re Gj,0 (0, 0, 0, 0) = 0 and ∂ Re Gj,0 (0, 0, 0, 0) > 0. ∂q Take now small open sets Ωj ⊂ Λ, 1 ≤ j ≤ N , such that γj ⊂ Ωj and Ωj ∩ Ωk = ∅,
j = k.
Let χj ∈ C0∞ (Ωj ), 0 ≤ χj ≤ 1, be such that χj = 1 near γj , 1 ≤ j ≤ N . When z ∈ C satisfies 1 |Re z| ≤ , |Im z| ≤ , (6.55) C C and (P − z)u = v, it follows from (6.54) by repeating the arguments of the torus case, that N 1 χj u || ≤ O (6.56) || 1 − || v || + O(h∞ )|| u ||. j=1 We shall now discuss the setup of the global Grushin problem. Associated with each j , 1 ≤ j ≤ N , we have the quasi-eigenvalues given in Proposition normal form P 5.4, S S k0 k0 1 h , h k2 + z(j, k) := f h(k1 − ) − + iGj h(k1 − ) − , , ; h , 4 2π 4 2π 2 when 1 ≤ j ≤ N and k = (k1 , k2 ) ∈ Z × N. We also introduce an ON system of eigenfunctions of the (formally) commuting operators Pj , k0 S 1 i ek (t, x) = √ e h (h(k1 − 4 )− 2π )t ek2 (x), 2π
k = (k1 , k2 ) ∈ Z × N,
which forms an ON basis in L2S (S 1 × R). Here ek2 (x), k2 ∈ N, are the normalized eigenfunctions of 1/2(x2 + (hDx )2 ) with eigenvalues (k2 + 1/2)h.
52
M. Hitrik and J. Sj¨ ostrand
When 1 ≤ j ≤ N , let Mj = # z(j, k), |Re z(j, k)|
0, the Grushin operator P=
P − z R+
R− 0
: H(Λ) × CM1 × · · · × CMN → H(Λ) × CM1 × · · · × CMN
(6.57)
is bijective. Indeed, when v ∈ H(Λ) and v+ ∈ CM1 × · · · × CMN , let us consider (P − z)u + R− u− = v, (6.58) R+ u = v+ . As in (6.56), we get N 1 χj u || ≤ O || 1 − || v || + O(h∞ ) (|| u || + || u− ||) . j=1 Applying χj and then Uj , 1 ≤ j ≤ N , to the first equation in (6.58), we get Mj (Pj − z)Uj χj u + l=1 u− (j)(l)ek(j,l) = (6.59) Uj (χj v + [P , χj ]u) + R∞ u + R−,∞ (j)u− , (Uj χj u|ek(j,l) ) = v+ (j)(l), 1 ≤ l ≤ Mj ,
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
53
and here R∞ = R∞ (j) = O(h∞ ) and R−,∞ (j) = O(h∞ ) in the corresponding operator norms. For each j, 1 ≤ j ≤ N , we get a microlocally well-posed Grushin problem for Pj − z in L2S (S 1 × R), with inverse of the norm O(1/), and the global well-posedness of (6.58) follows. The inverse E of P in (6.57) has the form E(z) E+ E= , (6.60) E− E−+ (z) and a straightforward computation shows that E+ : CM1 × · · · × CMN → H(Λ) modulo O(h∞ ), is given by E+ v+ ≡
Mj N
v+ (j)(l)Vj ek(j,l) = R− v+ ,
j=1 l=1
and E−+ (z) ∈ L CM1 × · · · × CMN , CM1 × · · · × CMN is a block diagonal matrix with the blocks E−+ (z)(j) ∈ L(CMj , CMj ), 1 ≤ j ≤ N , of the form E−+ (z)(j)(m, n) ≡ (z − z(j, k(j, m))) δmn ,
1 ≤ m ≤ n ≤ Mj ,
modulo O(h∞ ). The computation of eigenvalues near the boundary of the band has therefore been justified, and we get the second of our two main results. Theorem 6.6 Assume that F0 = inf Re q Σ
is achieved along finitely many closed Hp -trajectories γ1 , . . . , γN ⊂ p−1 (0) ∩ R4 of minimal period T (0), and that the Hessian of Re q at the corresponding points ρj ∈ Σ, j = 1, . . . , N , is positive definite. Let us write S and k0 to denote the common values of the action and the Maslov index of γj , j = 1, . . . , N , respectively. Assume that = O(hδ ) for a fixed δ > 0, is such that h . Let C > 0 be sufficiently large. Then the eigenvalues of P in the set 1 1 1 1 − , + i F0 − , F0 + (6.61) C C C C are given by z(j, k) = f
k0 S h k1 − − 4 2π
1 k0 h S ,h + k2 , , ; h , + iGj h k1 − − 4 2π 2
54
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
modulo O(h∞ ), when 1 ≤ j ≤ N and (k1 , k2 ) ∈ Z×N. Here f (τ ) is real-valued with f (0) = 0 and f (0) > 0. The function Gj (τ, q, , h/; h), 1 ≤ j ≤ N , is analytic in τ and q in a neighborhood of (0, 0) ∈ C2 , and smooth in , h/ ∈ neigh(0, R), and has an asymptotic expansion in the space of such functions, as h → 0, ∞ h h Gj τ, q, , ; h ∼ Gj,l τ, q, , , hl . l=0
We have Re Gj,0 (0, 0, 0, 0) = F0 and ∂ Re Gj,0 (0, 0, 0, 0) > 0, ∂q
1 ≤ j ≤ N.
Remark. With obvious modifications, Theorem 6.6 describes the eigenvalues in the region (6.61), when F0 = supΣ Re q, if we assume that F0 is attained along finitely many trajectories of minimal period T (0), such that the transversal Hessian of Re q along the trajectories is negative definite. The treatment of the remaining case of the eigenvalues near the boundary of the band (6.61), when the subprincipal symbol of P=0 vanishes proceeds in full analogy with the previously analyzed torus case. Thus, restricting attention to the region M h2 < = O(hδ ), M 1, we find that the symbol of Im P , acting on H(ΛG ) is O(), and away from an arbitrarily small but fixed neighborhood of ∪N j=1 γj we have that |Im P (ρ)| ≥ /C when we restrict the attention to the region |Re P (ρ)| ≤ 1/C. When working microlocally near τ = x = ξ = 0 in T ∗ (S 1 ×R) and simplifying the operator (6.49) further, we use Proposition 5.3 to find a holomorphic canonical transformation κσ,, h2 : neigh(0, C2 ) → neigh(0, C2 )
depending analytically on σ ∈ neigh(0, C) and smoothly on , h2 / ∈ neigh(0, R), such that h2 y2 + η2 σ, κσ,, h2 (y, η) = f (σ) + ig, h2 σ, p0 + p2 . 2 1 × C) As before, associated to κσ,, h2 , we construct an IR-submanifold of T ∗ (S
which is ( + h2 /)-close to T ∗ (S 1 × R), and which has the property that τ is real along this submanifold. This leads to a new IR-manifold Λ ⊂ C4 such that on Λ, Im P has a symbol of modulus ∼ in the region |Re P | < 1/C, when away from the union of small neighborhoods Ωj of γj ⊂ Λ, 1 ≤ j ≤ N . In Ωj , P is equivalent to an operator constructed in Section 5, which has the form h2 x2 + (hDx )2 , , ; h , f (hDt ) + iGj hDt , 2
Vol. 5, 2004
with
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
55
∞ h2 h2 Gj τ, q, , ; h ∼ Gj,l τ, q, , hl . l=1
Again we see that we have a globally well-posed Grushin problem for P − z in the h-dependent Hilbert space H(Λ). The following result complements Theorem 6.6. Theorem 6.7 Make the assumptions of Theorem 6.6, and assume in addition that the subprincipal symbol of P=0 vanishes. Then for in the range h2 < hδ ,
δ > 0,
the eigenvalues of P in the set of the form 1 1 1 1 − , + i F0 − , F0 + , C C C C
C 1,
are given by 1 k0 k0 h2 S S ,h + k2 , , ; h , f h k1 − − + iGj h k1 − − 4 2π 4 2π 2 modulo O(h∞ ), when 1 ≤ j ≤ N and (k1 , k2 ) ∈ Z×N. Here f (τ ) is real-valued with f (0) = 0 and f (0) > 0. The function Gj (τ, q, , h2 /; h) for 1 ≤ j ≤ N , is analytic in τ and q in a neighborhood of (0, 0) ∈ C2 , and smooth in , h2 / ∈ neigh(0, R), and has an asymptotic expansion in the space of such functions, as h → 0, ∞ h2 h2 Gj,l τ, q, , Gj τ, q, , ; h ∼ hl , l=0
where Re Gj,0 (0, 0, 0, 0) = F0 and ∂ Re Gj,0 (0, 0, 0, 0) > 0. ∂q
7 Barrier top resonances in the resonant case Consider P = −h2 ∆ + V (x),
p(x, ξ) = ξ 2 + V (x), x, ξ ∈ R2 ,
(7.1)
and let us assume that V (x) is real-valued, and that it extends holomorphically to a set {x ∈ C2 ; |Im x| ≤ Re x/C}, for some C > 0, and tends to 0 when x → ∞ in that set. The resonances of P can be defined in an angle {z ∈ C; −2θ0 < arg z < 0} for some fixed small θ0 > 0, as the eigenvalues of P in the same region. iθ 2 e
0R
56
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
We shall assume that V (0) = E0 > 0, ∇V (0) = 0, and V (0) is a negative definite quadratic form. Assume also that the union of trapped Hp -trajectories in p−1 (E0 ) ∩ R4 is reduced to (0, 0) ∈ R4 . (We recall that a trapped trajectory is a maximal integral curve of the Hamilton vector field Hp , contained in a bounded set.) We are then interested in resonances of P near E0 , created by the critical point of V . After a linear symplectic change of coordinates, and a conjugation of P by means of the corresponding metaplectic operator, we may assume that as (x, ξ) → 0, p(x, ξ) − E0 =
2
λj 2 ξj − x2j + p3 (x) + p4 (x) + · · · , 2 j=1
λj > 0.
(7.2)
Here pj (x) is a homogeneous polynomial of degree j ≥ 3. For future reference we recall that according to the theory of resonances developed in [12], the resonances of P in a fixed h-independent neighborhood of E0 can also be viewed as the eigenvalues of P : H(ΛtG , 1) → H(ΛtG , 1), equipped with the domain H(ΛtG , ξ2 ). Here G ∈ C ∞ (R2 ; R) is an escape function in the sense of [12], t > 0 is sufficiently small and fixed, and ΛtG is a suitable IR-deformation of R4 , associated with the function G. The Hilbert space H(ΛtG , 1) consists of all tempered distributions u such that a suitable FBI transform T u belongs to a certain exponentially weighted L2 -space. We refer to [12] for the original presentation of the microlocal theory of resonances, and to [18] for a simplified version of the theory, which is applicable in the present setting of operators with globally analytic coefficients, converging to the Laplacian at infinity. Here we shall only remark that as in [17], the escape function G can be chosen such that G = x · ξ in a neighborhood of (0, 0), and such that Hp G > 0 on p−1 (E0 ) \ {(0, 0)}. Under the assumptions above, but without any restriction on the dimension and without any assumption on the signature of V (0), all resonances in a disc around E0 of radius Ch were determined in [23]. Here C > 0 is arbitrarily large and fixed. (See also [7].) Specializing the result of [23] to the present barrier top case, we may recall that in this disc, the resonances are of the form 1 1 λ1 h − i k2 + λ2 h + O(h3/2 ), h → 0, k = (k1 , k2 ) ∈ N2 . E0 − i k1 + 2 2 (7.3) Furthermore, in the non-resonant case, i.e., when λ · k = 0,
0 = k ∈ Z2 ,
(7.4)
a result of Kaidi and Kerdelhu´e [17] extended [23] to obtain all resonances in a disc around E0 of radius hδ , for each fixed δ > 0 and h > 0 small enough depending on δ. In this case, the resonances are given by asymptotic expansions in integer powers of h, with the leading term as in (7.3).
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
57
Throughout this section we shall work under the following resonant assumption, λ · k = 0, for some 0 = k ∈ Z2 .
(7.5)
In this case we shall show how to obtain a description of all the resonances in an energy shell of the form h4/5 |E − E0 | < O(1)hδ ,
δ > 0,
provided that we avoid an arbitrarily small half-cubic neighborhood of E0 −i[0, ∞). The starting point is a reduction to an eigenvalue problem for a scaled operator, as in [17], [20], [24]. In these works it was shown how to adapt the theory of [12] so that P can be realized as an operator acting on a suitable H(Λ)-space, where
Λ ⊂ C4 is an IR-manifold which coincides with T ∗ eiπ/4 R2 near (0, 0), and further away from a neighborhood of this point, it agrees with ΛtG . Furthermore, Λ has the property that on this manifold, p − E0 is elliptic away from a small neighborhood of (0, 0), and this neighborhood can be chosen arbitrarily small, provided that the constant in the elliptic estimate is taken sufficiently large. Using a Grushin reduction exactly as in [20], we may and will therefore reduce the study of resonances of P near E0 to an eigenvalue problem for P after the complex x scaling, which near (0, 0) is given by x = eiπ/4 x , ξ = e−iπ/4 ξ, , ξ ∈ R. Using (7.2) and dropping the tildes from the notation, we see that the principal symbol of the scaled operator has the form E0 − i p2 (x, ξ) + ie3πi/4 p3 (x) + ie4iπ/4 p4 (x) + · · · , (x, ξ) → 0, (7.6) where p2 (x, ξ) =
2
λj 2 ξj + x2j 2 j=1
(7.7)
is the harmonic oscillator. In what follows we shall therefore consider an h-pseudodifferential operator P on R2 , microlocally defined near (0, 0), with the leading symbol p(x, ξ) = p2 (x, ξ) + ie3πi/4 p3 (x) + · · · , (x, ξ) → 0, (7.8) and with the vanishing subprincipal symbol. We extend P to be globally defined as a symbol of class S 0 (R4 ) = Cb∞ (R4 ), with the asymptotic expansion P (x, ξ; h) ∼ p(x, ξ) + h2 p(2) (x, ξ) + · · · , in this space, and so that |p(x, ξ)| ≥ outside a small neighborhood of (0, 0).
1 , C
C > 0,
58
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
We shall be interested in eigenvalues E of P with |E| ∼ 2 , 0 < 1. It follows from [26] that the corresponding eigenfunctions are concentrated in a region where |(x, ξ)| ∼ , and so we introduce the change of variables x = y, hδ ≤ ≤ 1, 0 < δ < 1/2. Then 1 1 P (x, hDx ; h) = 2 P ((y, hDy ); h), 2
h h = 2 1.
The corresponding new symbol is 1 1 P ((y, η); h) ∼ 2 p((y, η)) + 2 h2 p(2) ((y, η)) + · · · , 2 to be considered in the region where |(y, η)| ∼ 1. The leading symbol becomes 1 p((y, η)) = p2 (y, η) + ie3πi/4 p3 (y) + O(2 ), 2 for (y, η) in a fixed neighborhood of (0, 0). Now the resonant assumption (7.5) implies that the Hp2 -flow is periodic on (E), for E ∈ neigh(1, R), with period T > 0 which does not depend on E. For p−1 2 z ∈ neigh(1, C), we shall then discuss the invertibility of 1/2 P (x, hDx ; h) − z in the range of , dictated by Theorem 6.4, and using h as the new semiclassical parameter. Indeed, all the assumptions of that theorem are satisfied in a fixed neighborhood of (0, 0), and outside such a neighborhood, we have ellipticity which guarantees the invertibility there. Proposition 7.1 Assume that (7.5) holds. When p3 is a homogeneous polynomial of degree 3 on R2 , we let p3 denote the average of p3 along the trajectories of the Hamilton vector field of p2 in (7.7), and assume that p3 is not identically zero. Let F0 ∈ R be a regular value of cos(3π/4)p3 restricted to p−1 2 (1), and assume that T is the minimal period of the Hp2 -trajectories in the manifold Λ1,F0 given by 3π Λ1,F0 : p2 = 1, cos p3 = F0 . 4 Assume that Λ1,F0 is connected. Let satisfy h2/5 = O(1)hδ ,
δ > 0.
(7.9)
Then for z in the set 1−
! ! 1 1 1 1 ,1 + , F0 + + i F0 − , |O(1)| |O(1)| |O(1)| |O(1)|
(7.10)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
59
the operator −2 P (x, hDx ; h) − z : L2 → L2 is non-invertible precisely when z = zk for some k ∈ Z2 , where the numbers zk satisfy 2 S h α h zk = P , , ; h + O(h∞ ), h(k − ) − h = 2. 4 2π 2 Here P ξ, , h ; h has an expansion as h → 0, ∞ h2 h2 j P ξ, , ; h ∼ p2 (ξ1 ) + , h rj ξ, , j=0
where r0 = ie
3πi/4
h2 p3 (ξ) + O +
.
The coordinates ξ1 = ξ1 (E) and ξ2 = ξ2 (E, F ) are the normalized actions of 3π ΛE,F : p2 = E, cos p3 = F, 4 for E ∈ neigh(1, R), F ∈ neigh(F0 , R), given by 1 η dy − η dy , ξj = 2π γj (E,F ) γj (1,F0 )
j = 1, 2,
(7.11)
with γj (E, F ) being fundamental cycles in ΛE,F , such that γ1 (E, F ) corresponds to a closed Hp2 -trajectory of minimal period T . Furthermore, Sj = η dy, j = 1, 2, S = (S1 , S2 ), (7.12) γj (1,F0 )
and α ∈ Z2 is fixed. Remark. In the case when the compact manifold Λ1,F0 has finitely many connected components Λj , 1 ≤ j ≤ M , with each Λj being diffeomorphic to a torus, the set of z in (7.10) for which the operator −2 P (x, hDx ; h) − z is non-invertible agrees with the union of the quasi-eigenvalues constructed for each component, up to an error which is O(h∞ ). In the following discussion, for simplicity it will be tacitly assumed that Λ1,F0 is connected. The reduction by complex scaling together with the scaling argument above and Proposition 7.1 allows us to describe the resonances E of the operator (7.1) in the set (7.13) h4/5 |E − E0 | = O(1)hδ , δ > 0,
60
by
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
2 α S h h k− − , , ; h + O(h∞ ), E = E0 − i P 4 2π 2
(7.14)
where we choose > 0 with |E − E0 | /2 ∼ 1. The description (7.14) is valid provided that we exclude sets of the form 3/2 E ∈ C, Re E − E0 − F0 |Im E|
0,
where the λj satisfy (7.5). In order to describe the Hp2 -flow, it is convenient to introduce the action-angle variables Ij ≥ 0, τj ∈ R/2πZ for p2 , given by xj =
" " 2Ij cos τj , ξj = − 2Ij sin τj .
(7.16)
λj Ij and the Hamilton flow is given by R t → (I(t), τ (t)), with Then p2 = I(t) = I(0), τ (t) = τ (0) + tλ, λ = (λ1 , λ2 ). In the original coordinates, this gives " xj (t) = 2Ij (0) cos(τj (0) + λj t) " (7.17) ξj (t) = − 2Ij (0) sin(τj (0) + λj t), and we get a combination of two rotations in (xj , ξj ), j = 1, 2, with minimal periods 2π/λj (except in the degenerate cases when one of the (xj , ξj ) vanishes). Avoiding the totally degenerate case when I = 0, we get trajectories with • minimal period 2π/λ2 when I1 (0) = 0, • minimal period 2π/λ1 when I2 (0) = 0, • minimal period T = −k20 2π/λ1 = k10 2π/λ2 , when both I1 (0) and I2 (0) are = 0.
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
61
Here we let k 0 = (k10 , k20 ) be the point satisfying (7.5), which has minimal norm and positive first component. The integers k in (7.5) are equally spaced on the straight line λ⊥ , and it will be convenient to represent them in the form nk 0 , n ∈ Z \ {0}. We shall now compute the averages xα along the Hp2 -trajectories of a 1 α2 monomial xα = xα 1 x2 . Using (7.17), we get α
xα = I(0) 2 2 α
I(0) 2 1 = |α| 2 2 T
T
|α| 2
1 T
T
(cos(τ1 (0) + λ1 t))α1 (cos(τ2 (0) + λ2 t))α2 dt
(7.18)
0
(ei(τ1 (0)+λ1 t) + e−i(τ1 (0)+λ1 t) )α1 (ei(τ2 (0)+λ2 t) + e−i(τ2 (0)+λ2 t) )α2 dt.
0
Here the integrand can be developed with the binomial theorem, α2 α1 α1 α2 i((2k1 −α1 )τ1 (0)+(2k2 −α2 )τ2 (0)) i((2k1 −α1 )λ1 +(2k2 −α2 )λ2 )t e , e k1 k2
k1 =0 k2 =0
and only the terms with (2k1 − α1 )λ1 + (2k2 − α2 )λ2 = 0 can give a non-vanishing contribution to the integral. This means that 2k − α = nk 0 for some n ∈ Z, i.e., α + nk 0 = 2k with 0 ≤ k ≤ α componentwise. We get I(0)α/2 x = |α|/2 2 α
α+nk0 =2k 0≤k≤α
α1 k1
α2 cos((2k1 − α1 )τ1 (0) + (2k2 − α2 )τ2 (0)), k2
(7.19) where it is understood that n ∈ Z, k ∈ N2 , and where we notice that if α + nk0 = 2k, 0 ≤ k ≤ α, then k := α − k also participates in the sum, since 0 ≤ k ≤ α and α − nk0 = 2k. Also notice that the cosine in (7.19) can be written in the form cos(nk0 · τ (0)). In order to find the non-vanishing terms in (7.19), we consider the “line” Z n → α + nk0 ∈ Z2 . The points on this line in the rectangle ([0, 2α1 ] × [0, 2α2 ]) ∩ N2 with even coordinates correspond to the terms in (7.19). Example 1. Let k 0 = (1, −1), corresponding for instance to λ = (1, 1). In this case the two components of α must have the same parity. For α = (2, 0) we have only one term with n = 0, k = (1, 0), and x21 = I1 (0). For α = (0, 2) we get similarly x22 = I2 (0). For α = (1, 1) we get two " terms with n = 1, k = (1, 0) and n = −1, k = (0, 1) respectively, and x1 x2 = I1 (0)I2 (0) cos(τ1 (0) − τ2 (0)). For |α| = 3 we get no non-vanishing terms. For α = (4, 0) we have one term with n = 0, k = (2, 0) and we get x41 = 32 I1 (0)2 . For α = (0, 4) we get similarly, x42 = 32 I2 (0)2 . For α = (2, 2) we get one term with n = 2, k = (2, 0) and one with n = −2, k = (0, 2), We also have a term with n = 0, k = (1, 1), and this leads to x21 x22 = I1 (0)I2 (0)(1 + 12 cos 2(τ1 (0) − τ2 (0))).
62
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
It follows from Example 1 that Proposition 7.1 does not apply when λ = Const. (1, 1), since in this case p3 ≡ 0. We shall therefore consider a different choice of the resonant frequencies. Example 2. Let us take k 0 = (2, −1), corresponding for instance to λ = (1, 2), and let |α| = 3. For α = (3, 0), (0, 3), (1, 2) it follows from (7.19) that xα = 0. For α = (2, 1) we get two terms, one with n = 1, k = (2, 0) and one with n = −1, k = (0, 1). It follows that x21 x2 = 2−1/2 I1 (0)I2 (0)1/2 cos(2τ1 (0) − τ2 (0)).
(7.20)
For future reference, we shall also describe how the averages xα can be computed after a suitable complex linear change of symplectic coordinates. Introduce x = √12 (y + iη) y = √12 (x − iξ) , . η = i√1 2 (x + iξ) ξ = √i2 (y − iη) In these coordinates p =
2 j=1
iλj yj ηj , and
exp (tHp )(y, η) = (eitλ1 y1 , eitλ2 y2 , e−itλ1 η1 , e−itλ2 η2 ), so that 1 y α η β = T
T
eiλ·(α−β)t dty α η β = 0
We apply this to xα =
1 2|α|/2
and get x = 2 α
y α η β if λ · (α − β) = 0, 0 otherwise.
α y k (iη)α−k , k
0≤k≤α
α (x − iξ)k (x + iξ)α−k . k =2k
−|α|
(7.21)
α+nk0 0≤k≤α
As before we check that for each term present there is also the complex conjugate. The computations of Examples 1 and 2 can be written like (7.21). We shall only do it for the last example with k 0 = (2, −1), α = (2, 1): x21 x2 =
1 1 Re ((x1 + iξ1 )2 (x2 − iξ2 )) = (x21 x2 + 2x1 ξ1 ξ2 − x2 ξ12 ). 4 4
(7.22)
We may assume that λ = (1, 2), so that p2 =
1 2 (x + ξ12 ) + (x22 + ξ22 ), 2 1
and we may then check directly that Hp2 x21 x2 = 0.
(7.23)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
63
From (7.22) and (7.23) it is clear that dp2 and dx21 x2 are linearly independent except on some set of measure 0. When computing the critical points of x21 x2 on p−1 2 (1), we shall first make use of the (I, τ )-coordinates. From (7.20) we recall that √ 1 (7.24) p2 = I1 + 2I2 , 2x21 x2 = I1 I22 cos(2τ1 − τ2 ). It follows from the Hamilton equations that θ := 2τ1 − τ2 is invariant under the Hp2 -flow, and we can therefore work in the coordinates I1 , I2 , θ. We have √ 1 1 1 −1 dp2 = dI1 + 2dI2 , 2dx21 x2 = (I22 cos θ)dI1 + I1 I2 2 (cos θ)dI2 − I1 I22 (sin θ)dθ. 2 (7.25) If θ ∈ πZ, I1 , I2 = 0, we have ∂θ x21 x2 = 0, and hence the differentials are linearly independent. Still with I1 , I2 = 0, let θ ∈ πZ, so that cos θ = ±1. Then the differentials are linearly dependent iff 1 2 , i.e., iff I1 = 4I2 . 0 = det 1/2 1 − 12 I2 2 I1 I2 This gives two closed trajectories inside the energy surface p2 = 1 and the corresponding values for x21 x2 : I1 =
2 1 1 , I2 = , 2τ1 − τ2 = 0; x21 x2 = √ , 3 6 3 3
(7.26)
and
2 1 −1 , I2 = , 2τ1 − τ2 = π; x21 x2 = √ . (7.27) 3 6 3 3 When I1 = 0 or I2 = 0, the question of linear independence of the differentials should be analyzed directly in the (x, ξ)-coordinates (or (y, η)-coordinates), and here we shall use (7.22). On the plane I1 = 0, corresponding to x1 = ξ1 = 0, we have dx21 x2 = 0, so here we have linear dependence, with the corresponding critical value x21 x2 = 0. On the plane I2 = 0, corresponding to x2 = ξ2 = 0, we have dx21 x2 = 14 (x21 − ξ12 )dx2 + 12 x1 ξ1 dξ2 , dp2 = x1 dx1 + ξ1 dξ1 , I1 =
and these differentials are independent, since we avoid the point x = ξ = 0. We shall now look at the nature of the critical points of x21 x2 , when viewed as a function on Σ := p−1 2 (1)/exp (RHp ). For the trajectories found in (7.26) and (7.27), we use θ and I2 as local coordinates on Σ, and using√(7.24) together with I1 = 1 − 2I2 , we get for θ = kπ, k = 0, 1, I2 = 1/6 and f = 2x21 x2 , 1 3 − 12 1 − 32 2 k 2 k 2 I ∂θ ∂I2 f = 0, ∂θ f = −(1 − 2I2 )I2 (−1) , ∂I2 f = −(−1) + I2 . 4 2 2 For k = 0 we therefore have a non-degenerate maximum and for k = 1 we get a non-degenerate minimum.
64
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
For the third trajectory, given by x1 = ξ1 = 0,
x22 + ξ22 = 1,
use that x21 x1 vanishes to the second order Hessian in p−1 2 (1) can be identified with the
we sal which is given by the matrix
1 2
x2 ξ2
(7.28)
there, and hence that the transverfree Hessian with respect to x1 , ξ1 ,
ξ2 . −x2
The eigenvalues are 12 and − 21 . Thus we have a non-degenerate saddle point. We summarize the discussion above in the following proposition. Proposition 7.2 Let p2 (x, ξ) =
1 2 x1 + ξ12 + (x22 + ξ22 ). 2
Then the Hp2 -flow is periodic in p−1 2 (E), for E ∈ neigh(1, R), with period T = 2π. If p3 (x) = a3,0 x31 + a1,2 x1 x22 + x21 x2 + a0,3 x32 , then we have
1 2 x1 x2 + 2x1 ξ1 ξ2 − x2 ξ12 . 4 The differential of p3 , restricted to p−1 2 (1), vanishes along three closed Hp2 trajectories, given by (7.26), (7.27), and (7.28). These critical trajectories are nondegenerate in the sense that the transversal √ Hessian of p3 is non-degenerate. The set of the critical values of p3 is {±(3 3)−1 , 0}, and the maximum and the minimum of p3 are attained along the trajectories (7.26) and (7.27), respectively. The transversal Hessian of p3 along (7.28) has the signature (1, −1). The minimal period of the trajectories in (7.26) and (7.27) is equal to T = 2π, and the minimal period in (7.28) is π. Let finally F0 be a regular value of p3 restricted to p−1 2 (1). Then the minimal period of every closed Hp2 -trajectory in the Lagrangian manifold p3 (x, ξ) =
Λ1,F0 : p2 = 1, p3 = F0 is equal to T = 2π. We now return to the operator P with principal symbol p in (7.1). Under the general assumptions from the beginning of this section, we shall assume that as (x, ξ) → 0, we have p(x, ξ) − E0 =
1 2 (ξ − x21 ) + (ξ22 − x22 ) + p3 (x) + O(x4 ), 2 1
where p3 (x) = a3,0 x31 + a1,2 x1 x22 + x21 x2 + a0,3 x32 . √ √ Let us write A1 = −(3 6)−1 , A2 = (3 6)−1 , and A3 = 0.
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
65
Proposition 7.3 The resonances of P in the domain 3 # $ % 4/5 δ z ∈ C; h
|z − E0 | = O(1)h \ {z; Re z − E0 − Aj |Im z|3/2 j=1
< η |Im z|3/2 }, (7.29) where δ, η > 0 are arbitrary but fixed, are given by ∞ 2 S h α h − , , 5 , (7.30) hj −2j rj k− ∼ E0 − i h(k1 − α1 /4) + 3 2 4 2π j=0 with
h2 h2 r0 ξ, , 5 = ie3πi/4 p3 (ξ) + O + 5 , 2 2 h h rj ξ, , 5 = O + 5 , j ≥ 1
analytic in ξ ∈ neigh(0, C2 ), and smooth in , h2 / ∈ neigh(0, R). We have k = (k1 , k2 ) ∈ Z2 , S = (S1 , S2 ) with S1 = 2π, and α = (α1 , α2 ) ∈ Z2 is fixed, and we choose > 0 with |E − E0 | ∼ 2 . The resonances in the set # 3/2 3/2 $ and h4/5 |z − E0 | = O(1)hδ , z ∈ C, Re z − E0 − A1 |Im z| < η |Im z| (7.31) are given by E0 plus ∞ 2 h h 1 α1 α 1 h 1 k1 − h k1 − + i3 − 1, 2 k2 + hj −2j Gj , , 5 , i 4 2 4 2 j=0 (7.32) with (k1 , k2 ) ∈ Z × N, α1 ∈ Z, and |E − E0 | ∼ 2 . The function G0 (τ, q, , h2 /5 ) is ∂ such that Re G(0, 0, 0, 0) = A1 and ∂q Re G0 (0, 0, 0, 0) > 0. An analogous description of resonances is valid in the domain (7.31) with A1 replaced by A2 . Here in (7.30) we have also used that when expressed in terms of the action coordinates from (7.11), it is true that p2 (ξ1 ) = ξ1 + 1. Remark. If we replace rj (ξ, , h2 /5 ) in (7.30) by rj (ξ + S/2π, , h2 /5 ), then we get ∞ 2 h α h ∼ E0 − i h(k1 − α1 /4) + 3 k− , , 5 . hj −2j rj 2 4 j=0 Now let us notice that the choice of is not unique, and replacing by λ, with λ ∼ 1, does not affect the resonances. It follows therefore that ξ τ , λ, rj (ξ, , τ ) = λ3−2j rj . (7.33) λ2 λ5
66
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
Using this, we define rj (ξ, 1, τ ) = 3−2j rj
ξ τ , , 5 2
,
when |ξ| ∼ 2 and |τ | ≤ O(5 ). Then (7.30) becomes
α ∼ E0 − i h(k1 − α1 /4) + , 1, h2 . hj rj h k − 4 j=0
A
∞
Function spaces and FBI-transforms on manifolds
Let X be a compact analytic manifold of dimension n. In this section we first review some parts of Section 1 in [27] about how to define global FBI-transforms on X, and function spaces associated to certain IR-deformations of the real cotangent space. After that we shall perform Bargmann type transforms which allow us to view the above-mentioned function spaces, microlocally in a bounded frequency region, as weighted spaces of holomorphic functions. The theory in [27] is an adaptation to the case of compact manifolds of the one in [12] and this as well as the Bargmann transform below are closely related to similar ideas and techniques, developed in [6], [4], [28], [32], [10]. We equip X with some analytic Riemannian metric so that we have a distance d and a volume density dy. Let φ(α, y) be an analytic function on {(α, y) ∈ T ∗ X × X; d(αx , y) < 1/C} (using the notation α = (αx , αξ ), αx ∈ X, αξ ∈ Tα∗x X) with the following two properties (A) and (B): (A) φ has a holomorphic extension to a domain of the form × X; |Im αx |, |Im y| < {(α, y) ∈ T ∗ X
1 1 1 , |Re αx − Re y| < , |Im αξ | < |αξ |} C C C (A.1)
and satisfies |φ| ≤ O(1)|αξ | there. denotes the cotangent space is some complexification of X and T ∗ X Here X in the sense of complex manifolds with pointwise fiber spanned by the pointwise & 2 2 (1,0)-forms. We write αξ = 1 + αξ with αξ defined by means of the dual metric, and as below, we shall often give statements in local coordinates whenever convenient and leave to the reader to check that the statements make sense globally. Notice that by the Cauchy inequalities, ∂αk x ∂α ξ ∂ym φ = Ok,,m (1)|αξ |1−|| , in a set of the form (A.1), with a slightly increased constant C.
(A.2)
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
67
The second assumption is (B) φ(α, αx ) = 0, (∂y φ)(α, αx ) = −αξ , Im (∂y2 φ)(α, αx ) ∼ |Re αξ |I. By Taylor’s formula, we have φ(α, y) = αξ · (αx − y) + O(1)αξ |αx − y|2 ,
(A.3)
and on the real domain, for d(αx , y) ≤ 1/C, with C sufficiently large, we have: Im φ(α, y) ∼ αξ (αx − y)2 .
(A.4)
The following example was found in a joint discussion with M. Zworski: Let exp x : Tx X → X be the geodesic exponential map. Then we can take i 2 φ(α, y) = −αξ · exp −1 αx (y) + αξ d(αx , y) . 2
(A.5)
be a closed I-Lagrangian manifold which is close to T ∗ X in Let Λ ⊂ T ∗ X ∞ the C -sense and which coincides with this set outside a compact set. Recall that “I-Lagrangian” means Lagrangian for the real symplectic form −Im σ, where σ= dαξj ∧dαxj is the standard complex symplectic form. This means that if we choose (analytic) coordinates y in X and let (y, η) be the corresponding canonical then Λ is of the form {(y, η) + iHG (y, η); (y, η) ∈ coordinates on T ∗ X and T ∗ X, T ∗ X} for some real-valued smooth function G(y, η) which is close to 0 in the C ∞ sense and has compact support in η. Here HG denotes the Hamilton field of G. Since Λ is close to T ∗ X, it is also R-symplectic in the sense that the restriction to Λ of Re σ is non-degenerate. (We say that Λ is an IR-manifold.) It follows that dα| Λ = dαx1 ∧ · · · ∧ dαxn ∧ dαξ1 ∧ · · · ∧ dαξn | = Λ
1 n σ |Λ n!
is a real non-vanishing 2n-form on Λ, that we view as a positive density. We also need some symbol classes. A smooth function a(x, ξ; h), defined on is said to be of class S m,k , if Λ or on a suitable neighborhood of T ∗ X in T ∗ X ∂xp ∂ξq a = O(1)h−m ξk−q .
(A.6)
m,k A formal classical symbol a ∈ Scl is of the form a ∼ h−m (a0 + ha1 + · · · ) 0,k−j where aj ∈ S is independent of h. Here and in the following, we let 0 < h ≤ h0 for some sufficiently small h0 > 0. When the domain of definition is real or equal to Λ, we can find a realization of a in S m,k (denoted by the same letter a) so that
a − h−m
N 0
hj aj ∈ S −(N +1)+m,k−(N +1) .
68
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
m,k When the domain of definition is a complex domain, we say that a ∈ Scl is a m,k formal classical analytic symbol (a ∈ Scla ) if aj are holomorphic and satisfy
|aj | ≤ C0 C j (j!)|ξ|k−j .
(A.7)
It is then standard, that we can find a realization a ∈ S m,k (denoted by the same letter a) such that ∂xk ∂ξ ∂ x,ξ a = Ok, (1)e−|ξ |/Ch, |a − h−m hj aj | ≤ O(1)e−|ξ| /C1 h ,
(A.8)
0≤j≤|ξ |/C0 h
where in the last estimate C0 > 0 is sufficiently large and C, C1 > 0 depend on m,k m,k and Scla also the classes of realizations of classical C0 . We will denote by Scl symbols. We say that a classical (analytic) symbol a ∼ h−m (a0 + ha1 + · · · ) is 3n n ,4
0,−k 4 elliptic, if a0 is elliptic, so that a−1 . Take such an elliptic a(α, y; h) ∈ Scla 0 ∈S and put i
e h φ(α,y) a(α, y; h)χ(αx , y)u(y)dy,
T u(α; h) =
(A.9)
where χ is smooth with support close to the diagonal and equal to 1 in a neighborhood of the same set. 3n n 4 ,4 , such that if According to [27] there exists b(α, x; h) ∈ Scla i ∗ Sv(x) = e− h φ (x,α) b(α, x; h)χ(αx , x)v(α)dα, (A.10) T ∗X
then ST u = u + Ru,
(A.11)
where R has a distribution kernel R(x, y; h) satisfying |∂xα ∂y R| ≤ Ck, e
− C1 h 0
.
(A.12)
Here we denote in general by f ∗ , the holomorphic extension of the complex conjugate of f . With Λ as above, we put TΛ u = T u| Λ ,
(A.13)
and define SΛ v by (A.10), but with T ∗ X replaced by Λ. Then, SΛ TΛ u = u + RΛ u,
(A.14)
where RΛ satisfies (A.12) (with a slightly larger C0 and under the assumption that Λ is sufficiently close to T ∗ X). In fact, using Stokes’ formula and the exponential decrease of ∂ of the symbols involved, we see that SΛ TΛ coincides up to an exponentially small error with ST .
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
69
Since Λ is I-Lagrangian, we can find locally a real-valued smooth function H(α) on Λ, such that dH = −Im (αξ · dαx )| Λ . (A.15) Indeed, −Im (αξ · dαx ) is a primitive of −Im σ and the latter vanishes on Λ, so the right-hand side of (A.15) is closed. We assume: The equation (A.15) has a global solution H ∈ C ∞ (Λ; R). Notice that this property is equivalent to Im (αξ · dαx ) = 0, for all closed curves γ ⊂ Λ.
(A.16)
(A.17)
γ
When (A.16) is fulfilled, H is well defined up to a constant, and we shall always choose H to be zero for large αξ . As in [27] we notice that (A.16) is fulfilled in the case of IR-manifolds gen R) in the following way: Let HG = H Im σ erated by a weight G ∈ C ∞ (T ∗ X; G be the Hamilton field of G with respect to Im σ, and assume that G = 0 in the region where |αξ | is large. Then for t real with |t| small enough, we can consider the IR-manifold Λt = exp (tHG )(Λ0 ), where Λ0 = T ∗ X. Then we get (A.16) with H = Ht given by t (exp (s − t)HG )∗ (G + HG , ω)ds, (A.18) Ht = 0
where ω = −Im (αξ · dαx ) The function H appears naturally in connection with TΛ . We have dα φ = αξ · dαx + O(|αx − y|), so (dα φ)(α, αx ) = αξ · dαx and −Im (dα φ)(α, αx )| Λ = dα H.
(A.19)
Definition. For m ∈ R, put H(Λ; αξ m ) = {u ∈ D (X); TΛ u ∈ L2 (Λ; e−2H/h |αξ |2m dα)}.
(A.20)
When Λ = T ∗ X we get the usual h-Sobolev spaces, and in particular the case m = 0 just gives L2 (X). For general Λ we get the same spaces, but the equivalence of the norm (A.21) uH(Λ,αξ m ) = TΛ uL2 (Λ;e−2H/h |αξ |2m dα) with the h-m-Sobolev norm uH(T ∗ X,αξ m ) is no longer uniform with respect to h, in general. Recall from [27] that if we choose another FBI-transform T of the same type as T but with different phase φ and amplitude a, then for Λ close enough to T ∗ X,
70
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
the definition (A.20) does not change if we replace T by T, and we get a new norm which is equivalent to the previous one, uniformly with respect to h. This follows from a fairly explicit description of TΛ TΛ−1 . We also know that T u = TT ∗ X u and TΛ u satisfy compatibility conditions similar to the Cauchy-Riemann equations for holomorphic functions. For the analysis in the most interesting region where ξ is bounded, it will be convenient to work with transforms which are holomorphic up to exponentially small errors, and for that we make a different choice of T , and take an FBI-transform as in [28], now with a global choice of phase (cf [4], [10], [32]). The function d(x, y)2 is analytic in a neighborhood of the diagonal in X × X, so we can consider it as a holomorphic function in a region × X; dist (x, y) < 1 , |Im x|, |Im y| < 1 }. {(x, y) ∈ X C C Put φ(x, y) = iλd(x, y)2 ,
(A.22)
where λ > 0 is a constant that we choose large enough, depending on the size of the neighborhood of the zero section in T ∗ X, that we wish to cover. |Im x| < 1/C, put For x ∈ X, i − 3n 4 (A.23) e h φ(x,y) χ(x, y)u(y)dy, u ∈ D (X), T u(x; h) = h × X; |Im x| < where χ is a smooth cut-off function with support in {(x, y) ∈ X 1/C, d(y, y(x)) < 1/C}. Here y(x) ∈ X is the point close to x, where X y → −Im φ(x, y) attains its non-degenerate maximum. We have the following facts ([28]): |Im x| < 1/C, is strictly The function Φ0 (x) = −Im φ(x, y(x)), x ∈ X, plurisubharmonic and is of the order of magnitude ∼ |Im x|2 . is an IR-manifold given by ΛΦ0 = κT (T ∗ X), ΛΦ0 := {(x, 2i ∂Φ0 ) ∈ T ∗ X} where κT is the complex canonical transform associated to T , given by with its (y, −φy (x, y)) → (x, φx (x, y)). Here and in the following, we identify X intersection with a tubular neighborhood of X which is independent of the choice of λ in (A.22). e−2Φ0 /h L(dx)), for L(dx) denoting a choice of Lebesgue meaIf L2Φ0 = L2 (X; sure (up to a non-vanishing continuous factor), then T = O(1) : L2 (X) → L2Φ0 , ∂ x T = O(e−1/Ch ) : L2 (X) → L2Φ0 . This means that up to an exponentially small error T u is holomorphic for u ∈ L2 (X) (and even for u ∈ D (X)). A natural is choice of Lebesgue measure might be (n!)−1 |π∗ (σ| ΛΦ )n |, where π : ΛΦ0 → X 0 the natural projection. ⊂ L2 (X) be the subspace of holomorphic functions. Assuming, Let HΦ0 (X) Φ0 is a Stein (“pseudoconvex”) domain, we can apply the wellas we may, that X 2 known L results of H¨ ormander for the ∂-operator and replace T by T = T + K,
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
71
so that T : L2 (X) → HΦ0 (X). In the where K = O(e−1/(Ch) ) : L2 (X) → L2Φ0 (X), main text we do not distinguish between T and T . Unitarity: Modulo exponentially small errors and microlocally, T is unitary a0 e−2Φ0 /h L(dx)), where L(dx) is chosen as indicated above, and L2 (X) → L2 (X; a0 (x; h) is a positive elliptic analytic symbol of order 0. be an IR-manifold as before, satisfying (A.16) (or the equivalent Let Λ ⊂ T ∗ X condition (A.17)). Then κT (Λ) = ΛΦ , where Φ = ΦΛ , can be normalized by the (Here is where we have to requirement that Φ = Φ0 near the boundary of X. choose λ large enough, depending on Λ. In the applications, for a given elliptic operator, Λ and T ∗ X will coincide outside a fixed compact neighborhood of the zero section, and the whole study will be carried out with a fixed λ.) Let Ω ⊂ T ∗ X be the open neighborhood of the 0-section, given by πx κT Ω = X and view also Ω as a subset of Λ in the natural sense, assuming that T ∗ X and Λ coincide in a neighborhood of the closure of the complement of Ω. If χ ∈ C0∞ (Ω), then the norm uH(Λ,αξ m ) is equivalent to the norm T uL2Φ + (1 − χ)TΛ uL2 (Λ;e−2H/h |αξ |2m dα) uniformly with respect to h.
Acknowledgments We would like to thank Anders Melin and Maciej Zworski for useful discussions. The first author gratefully acknowledges the support of the Swedish Foundation for International Cooperation in Research and Higher Education (STINT) as well as of the MSRI postdoctoral fellowship.
References [1] V. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, New York, 1989. [2] M. Asch and G. Lebeau, The spectrum of the damped wave equation, preprint, 1999. See http://www.math.u-psud.fr/∼biblio/rt/1999/. [3] D. Bambusi, S. Graffi and T. Paul, Normal forms and quantization formulae, Comm. Math. Phys. 207, 173–195 (1999). [4] L. Boutet de Monvel, Convergence dans le domaine complexe des s´eries de fonctions propres, C. R. Acad. Sci. Paris, S´erie A–B, 287, 855–856 (1978). [5] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Math. Studies 99, Princeton University Press, 1981. [6] L. Boutet de Monvel and J. Sj¨ ostrand, Sur la singularit´e des noyaux de Bergman et de Szeg¨o, Ast´erisque 34–35, 123–164 (1976).
72
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
[7] P. Briet, J.M. Combes, P. Duclos, On the location of resonances for Schr¨ odinger operators in the semiclassical limit. II. Barrier top resonances, Comm. P.D.E. 12, 201–222 (1987). [8] Y. Colin de Verdi`ere, Sur le spectre des op´erateurs elliptiques a bicaract´eristiques toutes p´eriodiques, Comment Math. Helv. 54, 508–522 (1979). [9] S. Dozias, Clustering for the spectrum of h-pseudodifferential operators with periodic flow on an energy surface, Journ. Funct. Anal. 145, 296–311 (1997). [10] F. Golse, E. Leichtnam, and M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. Ecole Norm. Sup. 29, 669–736 (1996). [11] B. Helffer and D. Robert, Puits de potentiel g´en´eralis´es et asymptotique semiclassique, Ann. Inst. H. Poincar´e 41, 291–331 (1984). [12] B. Helffer and J. Sj¨ ostrand, R´esonances en limite semiclassique, M´em. Soc. Math. France (N.S.) 24–25 (1986). [13] B. Helffer and J. Sj¨ ostrand, Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mem. Soc. Math. France (N.S.) 39, 1–124 (1989). [14] M. Hitrik, Eigenfrequencies for damped wave equations on Zoll manifolds, Asymptot. Analysis 31, 265–277 (2002). [15] L. H¨ormander, Fourier Integral Operators I, Acta Math. 127, 79–183 (1971). [16] V. Ivrii, Microlocal analysis and precise spectral asymptotics, Springer-Verlag, Berlin, 1998. [17] N. Kaidi and P. Kerdelhue, Forme normale de Birkhoff et r´esonances, Asymptot. Analysis 23, 1–21 (2000). [18] A. Lahmar-Benbernou and A. Martinez, On Helffer-Sj¨ ostrand’s theory of resonances, IMRN 13, 697–717 (2002). [19] G. Lebeau, Equation des ondes amorties, in Algebraic and Geometric Methods of Mathematical Physics (Kaciveli 1993), 73–109, Math. Phys. Stud., 19 Kluwer Acad. Publ., Dordrecht, 1996. [20] A. Melin, J. Sj¨ostrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Ast´erisque 284, 181–244 (2003) . [21] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. II. Quantum Birkhoff normal forms, Ann. Henri Poincar´e 1, 249–279 (2000). [22] J. Sj¨ ostrand, Perturbations of selfadjoint operators with periodic classical flow, RIMS Kokyuroku 1315 (April 2003), “Wave Phenomena and asymptotic analysis”, 1–23. Also: http://xxx.lanl.gov/abs/math.SP/0303023.
Vol. 5, 2004
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
73
[23] J. Sj¨ ostrand, Semiclassical resonances generated by a non-degenerate critical point, Springer LNM, 1256, 402–429. [24] J. Sj¨ ostrand, Resonances associated to a closed hyperbolic trajectory in dimension 2, preprint, September 2002, http://xxx.lanl.gov/abs/math.SP/0209147, Asymptotic Analysis, a` paraˆitre. [25] J. Sj¨ ostrand, Asymptotic distribution of of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36, 573–611 (2000). [26] J. Sj¨ ostrand, Semi-excited states in non-degenerate potential wells, Asymptot. Analysis 6, 29–43 (1992). [27] J. Sj¨ ostrand, Density of resonances for strictly convex analytic obstacles, Can. J. Math. 48, 397–447 (1996). [28] J. Sj¨ ostrand, Singularit´es analytiques microlocales, Ast´erisque 85 (1982). [29] J. Sj¨ ostrand and M. Zworski, Quantum monodromy and semiclassical trace formulae, J. Math. Pure Appl. 81, 1–33 (2002). [30] J. Sj¨ ostrand and M. Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183, 191–253 (1999). [31] A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44, 883–892 (1977). [32] M. Zworski, Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, Inv. Math. 136, 353–409 (1999). Michael Hitrik Department of Mathematics University of California Los Angeles, CA 90095–1555 USA email:
[email protected] Johannes Sj¨ostrand Centre de Math´ematiques UMR 7640 CNRS Ecole Polytechnique F-91128 Palaiseau France email:
[email protected] Communicated by Bernard Helffer submitted 13/03/03, accepted 06/10/03
Ann. Henri Poincar´e 5 (2004) 75 – 118 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010075-44 DOI 10.1007/s00023-004-0161-0
Annales Henri Poincar´ e
Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials Hideo Tamura Abstract. We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schr¨ odinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results.
1 Introduction We consider the relativistic massless particle moving in the two-dimensional space. We denote by x = (x1 , x2 ) a generic point in R2 and write D(A, V ) =
2
σj (−i∂j − Aj ) + V,
∂j = ∂/∂xj ,
j=1
for the Dirac operator, where A = (A1 , A2 ) : R2 → R2 and V : R2 → R are magnetic and scalar potentials respectively, and 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = 1 0 i 0 0 −1 are the Pauli spin matrices. The magnetic field b : R2 → R is defined by b = ∇ × A = ∂1 A2 − ∂2 A1 . The operator D(A, V ) acts on [L2 ]2 = [L2 (R2 )]2 . If A and V are bounded, then it is selfadjoint with domain [H 1 (R2 )]2 , where H s (R2 ) is the Sobolev space of order s. We also write L(A, V ) = (−i∇ − A)2 + V for the Schr¨ odinger operator. If A has further bounded derivatives, then L(A, V ) is selfadjoint with domain H 2 (R2 ) in L2 . If L(A, V )u = 0 has a bounded but not square integrable solution, then L(A, V ) is said to have a resonance at zero energy.
76
H. Tamura
Ann. Henri Poincar´e
Let b and V be given magnetic field and scalar potential. We assume that b, V ∈ C0∞ (R2 → R) are smooth functions with compact support. We define A(x) by (1.1) A(x) = (−∂2 ϕ(x), ∂1 ϕ(x)) , where ϕ(x) = (2π)−1
log |x − y| b(y) dy
(1.2)
and the integration without the domain attached is taken over the whole space. By definition, A satisfies ∇ × A = ∆ϕ = b, and hence it becomes the potential associated with field b. The function ϕ obeys ϕ(x) = α log |x| + O(|x|−1 ) as |x| → ∞, where α = (2π)−1
b(x) dx
is called the flux of b. The magnetic effect strongly appears when α ∈ Z is not an integer. We restrict ourselves to the case 0 < α < 1.
(1.3)
We make a brief comment on the other cases that α < 0 and α > 1 (Remark 8.1 at the end of Section 8). The potential A(x) is not necessarily expected to fall off rapidly and it has the long-range property at infinity even if b is of compact support. In fact, it behaves like A(x) = A0α (x) + O(|x|−2 ),
(1.4)
A0α (x) = α(−x2 /|x|2 , x1 /|x|2 ) = α(−∂2 log |x|, ∂1 log |x|)
(1.5)
where A0α is defined by
and it is often called the Aharonov-Bohm potential in physical articles. Let T = D(A, V ) = T0 + V , where T0 = D(A, 0) = σ1 ν1 + σ2 ν2 ,
(ν1 , ν2 ) = −i∇ − A,
is the Dirac operator without scalar potential V . We sometimes identify the coordinates ω = (ω1 , ω2 ) over the unit circle S with the azimuth angle from the positive x1 axis. According to this notation, we set τ (ω) = t (1, eiω ),
eiω = cos ω + i sin ω = ω1 + iω2 .
(1.6)
We denote by f (ω → ω ˜ ; E) the scattering amplitude of T for scattering from initial direction ω ∈ S to final one ω ˜ at energy E > 0. Roughly speaking, it is defined through the behavior at infinity of solution ψ = ψ(x; E, ω) to equation T ψ = Eψ, and the solution takes the asymptotic form ˜ ; E)τ (˜ ω )eiEr r−1/2 , ψ(rω ˜ ) ∼ ψin + f (ω → ω
r = |x| → ∞,
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
77
along direction ω ˜ = ω, where the first term ψin = τ (ω)eiEx·ω is the wave incident from ω and the second term denotes the scattering wave. The precise representation of it is given in Section 4. We study the scattering by electromagnetic fields with small support. We set Aε (x) = ε−1 A(x/ε),
bε (x) = ε−2 b(x/ε),
Vε (x) = ε−1 V (x/ε)
(1.7)
for 0 < ε 1 small enough. Then Aε satisfies ∇ × Aε = bε . Our aim here is to analyze the asymptotic behavior as ε → 0 of amplitude fε (ω → ω ˜ ; E) of Tε = D(Aε , Vε ). The problem is closely related to the resonance state at zero energy of magnetic Schr¨ odinger operators in a natural way. Let R(z; H) denote the resolvent (H − z)−1 of selfadjoint operator H. We write T0 = σ1 ν1 + σ2 ν2 as 0 ν1 − iν2 0 ν− T0 = = , ν1 + iν2 0 ν+ 0 where (ν1 , ν2 ) = −i∇ − A with A = (−∂2 ϕ(x), ∂1 ϕ(x)), ϕ being defined by (1.2). Since ν1 and ν2 satisfies the commutator relation [ν1 , ν2 ] = ν1 ν2 − ν2 ν1 = ib, a simple computation yields ν± ν∓ = ν12 + ν22 ± b = L(A, ±b), so that T02 is diagonalized as T02
=
L(A, −b) 0 0 L(A, b)
.
∗ The two Schr¨odinger operators L(A, ±b) = ν∓ ν∓ ≥ 0 are non-negative, but the spectral structure at zero energy is different. By (1.1), we have
ν+
=
ν1 + iν2 = −i∂1 + ∂2 ϕ + i(−i∂2 − ∂1 ϕ)
=
−i ((∂1 + ∂1 ϕ) + i(∂2 + ∂2 ϕ)) = −ie−ϕ ( ∂1 + i∂2 ) eϕ .
(1.8)
Hence L(A, −b)u = 0 has a bounded solution behaving like ρ(x) = e−ϕ(x) = |x|−α (1 + O(|x|−1 )),
|x| → ∞.
(1.9)
By assumption (1.3), ρ is not in L2 , and hence L(A, −b) has a resonance state at zero energy. On the other hand, L(A, b) does not have a resonance state. The amplitude fε is represented in terms of the boundary values R(E + i0; Tε ) = lim R(E + iδ; Tε ) δ↓0
78
H. Tamura
Ann. Henri Poincar´e
to the real axis of resolvent R(E + iδ; Tε ). We now define the unitary operator Jε : [L2 ]2 → [L2 ]2 by (Jε u)(x) = ε−1 u(x/ε), (1.10) then we have Tε = ε−1 Jε T Jε∗ for T = D(A, V ), and hence R(E + i0; Tε ) = εJε R(k + i0; T )Jε∗ ,
k = εE.
(1.11)
Thus the analysis relies on the behavior at low energy of resolvents R(k + i0; T0 ) = (T0 + k)R(k 2 + i0; T02) and R(k + i0; T ), and a basic role is played by the zero energy resonance of the magnetic Schr¨ odinger operator L(A, −b). We note that there is no fear of our confusing the operator Jε with the Bessel function Jν (x) in the argument below. We take the limit ε → 0 in a formal way. It follows from (1.4) that Aε is convergent to the Aharonov-Bohm potential A0α (x), and hence Tε = D(Aε , Vε ) → Dα = D(A0α , 0)
(1.12)
on [C0∞ (R2 \ {0})]2 . However A0α is strongly singular at the origin, and it has the δ-like field 2παδ(x) as a magnetic field. We know ([14, 19, 21]) that Dα is not essentially selfadjoint and it has the deficiency indices (1,1). According to the Krein theory, we can obtain a family of selfadjoint extensions {Hκ } with one real parameter κ, − ∞ < κ ≤ ∞. The element u = t (u1 , u2 ) in the domain D(Hκ ) is specified by the boundary condition u−1 + iκ u−2 = 0
(1.13)
at the origin under assumption (1.3), where u−1 = lim rα u1 (x), r→0
u−2 = lim r1−α e−iθ u2 (x) r→0
(1.14)
in the polar coordinate system (r, θ). If κ = ∞, then u−2 = 0 and the second component u2 (x) has a weak singularity near the origin for u ∈ D(H∞ ), while the first component u1 (x) has a weak singularity for κ = 0. The boundary condition in which both components remain bounded is not in general allowed ([14, 19]). In Section 2, we explicitly calculate the amplitude of Hκ after discussing the problem of selfadjoint extension in some detail. The amplitude fε in question is expected to converge to that of Hκ for some κ. We state the obtained results somewhat loosely. All the main theorems are ˜ ; E) the scattering amplitude of formulated in Section 5. We denote by gκ (ω → ω Hκ . As stated above, gκ can be calculated explicitly. If the scalar potential V (x) vanishes identically, then fε is shown to converge to g∞ (Theorem 5.1). However the situation changes as soon as V is added as a perturbation (Theorem 5.2). It is interesting that this occurs even for small perturbations. We here deal with only
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
79
the simple but generic case that T has neither bound state nor resonance state at zero energy. The definition of resonance state is given in Section 5. Roughly speaking, it means that the equation T u = 0 admits a bounded solution. We note that T does not have a resonance state for V small enough. The obtained result depends on the flux α of field b. The amplitude fε is proved to converge to g∞ for 0 < α < 1/2 and to g0 for 1/2 < α < 1. If α = 1/2, then fε is convergent to gκ for some κ determined from the resonance state ρ = e−ϕ of L(A, −b). A similar problem has been studied by the physical literature [2, Section 7.10] for the scattering outside the small disk {|x| < ε}, and it has shown that the limit takes a different form according as 0 < α < 1/2, α = 1/2 or 1/2 < α < 1. However the argument there is based on the explicit calculation using the Bessel functions, and the connection with zero energy resonance has not been recognized. As stated in the beginning, another motivation of this work comes from the study on the scattering of Dirac particles in the interaction of cosmic string with matter. This problem is mathematically formulated as follows (see [7] for the detail on the physical background). Let Aε , bε = ∇ × Aε and Vε be defined by (1.7). We consider two kinds of particles (for example, lepton and quark) moving in the magnetic field bε and interacting with each other through the scalar potential Vε . If we denote by w = t (u, v) = t (u1 , u2 , v1 , v2 ) the wave function of these two particles, then w obeys the equation Tε w = T0ε w + Vε w = Ew at energy E > 0, where T0ε 0 T0ε = , 0 T0ε
Vε =
0 Vε
Vε 0
(1.15)
,
T0ε = D(Aε , 0).
We assume that the wave function w has only u-wave as an incident wave. Then w behaves like w ∼ t (τ (ω), 0)eiEx·ω + wscat + o(r−1/2 ),
r → ∞,
where τ (ω) is defined by (1.6), and the scattering wave wscat takes the form ˜ ; E)t (τ (˜ ω ), 0) + f2ε (ω → ω ˜ ; E)t (0, τ (˜ ω )) eiEr r−1/2 (1.16) wscat = f1ε (ω → ω along direction ω ˜ . The amplitude f2ε (ω → ω ˜ ; E) describes the v-wave produced by incident u-wave, and it is an important physical quantity in the interaction of cosmic string with matter. We analyze the asymptotic behavior as ε → 0 of ˜ ; E). The asymptotic form is shown to take the form f2ε (ω → ω f2ε (ω → ω ˜ ; E) = Cα ε|2α−1| (1 + o(1)) ,
ε → 0,
for some constant Cα (Theorem 5.3). The constant is independent of incident and final directions ω and ω ˜ , but is different according as 0 < α < 1/2, α = 1/2 or
80
H. Tamura
Ann. Henri Poincar´e
1/2 < α < 1. A similar asymptotic form has been derived by the earlier work [7] in the special case that A(x) = A0α (x) is the Aharonov-Bohm potential and V (x) is the characteristic function of the unit disk. However the calculation there is again based on the explicit calculation using the Bessel functions, and the important role of zero energy resonance seems to have been completely hidden behind this explicit calculation. In this work we make clear from a mathematical point of view how the leading coefficient Cα is determined and how it is related to the resonance state ρ of L(A, −b) at zero energy. We confine ourselves to the positive energy case E > 0 for notational brevity, and we fix E > 0 throughout the whole exposition. The dependence on E does not matter. We end the section by noting that the obtained results easily extend to the operator σ1 ν1 + σ2 ν2 + mσ3 + V with mass m > 0.
2 Dirac operators with point-like fields In this section we calculate the scattering amplitude gκ (ω → ω ˜ ; E) of selfadjoint extension Hκ obtained from Dα defined by (1.12) after explaining briefly the Krein theory on the problem of selfadjoint extension. The problem of selfadjoint extension for two-dimensional Dirac operators with singular magnetic fields has already been studied by several authors. We refer to [14, 19, 21] for details, and, in particular, to [21] for the recent references. The argument here follows [23]. The operator Dα = D(A0α , 0) =
0 π+
π− 0
,
π± = π1 ± i π2 ,
(2.1)
2 defined over C0∞ (R2 \ {0}) is symmetric, where (π1 , π2 ) = −i∇ − A0α . The two operators π± are represented as π+ = eiθ −i∂r + r−1 (∂θ − iα) , π− = e−iθ −i∂r − r−1 (∂θ − iα) (2.2) in terms of polar coordinates (r, θ), and we have π+ π− = π12 + π22 = −∂r2 − r−1 ∂r + r−2 (−i∂θ − α)
2
for r = |x| > 0, and similarly for π− π+ . We denote by Dα and Dα∗ the closure and adjoint of Dα respectively, and we set Σ± = {u ∈ [L2 ]2 : (Dα∗ ∓ i) u = 0}. The pair (n+ , n− ), n± = dim Σ± , is called the deficiency indices of Dα . As is well known, Dα has selfadjoint extensions if and only if n+ = n− . (1)
We show that n+ = n− = 1. We denote by Hµ (z) = Hµ (z) the Hankel function of first kind, and all the Hankel functions are understood to be of first
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
81
kind throughout. If u = t (u1 , u2 ) ∈ [L2 ]2 solves (Dα − i) u = 0, then u2 satisfies (π+ π− + 1) u2 = 0 in R2 \ {0}, and u1 is given by u1 = −iπ− u2 . By formula, Hµ (z) satisfies (2.3) (d/dz) z ±µ Hµ (az) = ±az ±µ Hµ∓1 (az). The same formula is still true for Jµ (z). This formula yields π− H1−α (ir)eiθ = H−α (ir) = eiαπ Hα (ir). Hence we see that Σ+ is the one-dimensional space spanned by u+ = Nα t (−ieiαπ Hα (ir), H1−α (ir)eiθ ), where u+ is normalized as u+ L2 = 1. Similarly Σ− is also the one-dimensional space spanned by u− = Nα t (ieiαπ Hα (ir), H1−α (ir)eiθ ),
u− L2 = 1.
All the possible selfadjoint extensions are determined by the Krein theory ([8, 20]). Let U : Σ+ → Σ− be the unitary mapping defined by multiplication U u+ = eiζ u− with −π < ζ ≤ π. Then the selfadjoint extension HU associated with U is realized as the operator HU u = Dα v + icu+ − iceiζ u− acting on the domain D(HU ) = {u ∈ [L2 ]2 : u = v + cu+ + ceiζ u− , v ∈ D(D α ), c ∈ C}. We examine which boundary condition u ∈ D(HU ) satisfies at the origin. The Hankel function Hµ (z) with non-integer µ > 0 is represented as Hµ (z) = (i/ sin µπ) e−iµπ Jµ (z) − J−µ (z) (2.4) in terms of Bessel functions, and it behaves like Hµ (z) = (−i/ sin µπ) (2µ /Γ(1 − µ)) z −µ 1 + O(|z|2µ ) + O(|z|2 )
(2.5)
as |z| → 0. If v = t (v1 , v2 ) ∈ D(D α ), then v obeys v1 = o(|x|−α ) and v2 = o(|x|−(1−α) ) as |x| → 0, so that u = t (u1 , u2 ) ∈ D(HU ) has the limits u−1 and u−2 in (1.14). If we take account of the above asymptotic formula of Hankel functions, then the ratio κ = iu−1 /u−2 = 22α−1 Γ(α)/Γ(1 − α) tan(ζ/2) is calculated as a quantity independent of u. Thus we obtain the family of selfadjoint extensions {Hκ } parameterized by real number κ, − ∞ < κ ≤ ∞, and the operator has the domain D(Hκ ) = {u = (u1 , u2 ) ∈ [L2 ]2 : Dα u ∈ [L2 ]2 , u−1 + iκu−2 = 0},
(2.6)
82
H. Tamura
Ann. Henri Poincar´e
where Dα u is understood in the distribution sense, and u−1 and u−2 are defined by (1.14). We move to calculating the scattering amplitude of Hκ . It has already been calculated in the physical articles ([17]) for the special case κ = 0 or κ = ∞. We again note that ω ∈ S is often identified with the azimuth angle from the positive x1 axis. ˜ ; E), ω ˜ = ω, denote the scattering amplitude of Hκ Proposition 2.1 Let gκ (ω → ω for the scattering from initial direction ω into final one ω ˜ at energy E > 0. Then ei(˜ω−ω)/2 2κτα E 2α−1 −1/2 + gκ = − (2πiE) sin απ , (2.7) sin((˜ ω − ω)/2) i(κτα E 2α−1 − eiαπ ) where
τα = 21−2α Γ(1 − α)/Γ(α).
(2.8)
If, in particular, κ = 0 or κ = ∞, then g0 g∞
sin απ
ei(˜ω−ω)/2 , sin((˜ ω − ω)/2)
= − (2πiE)−1/2 sin απ
e−i(˜ω−ω)/2 , sin((˜ ω − ω)/2)
−1/2
= − (2πiE)
and if α = 1/2, then −1/2
gκ = − (2πiE)
ei(˜ω−ω)/2 2κ + sin((˜ ω − ω)/2) 1 + iκ
.
We need two lemmas to prove the proposition. Before stating the lemmas, we briefly discuss the problem of selfadjoint extensions for magnetic Schr¨odinger operator 2 (2.9) Lα = L(A0α , 0) = (−i ∇ − A0α ) with Aharonov-Bohm potential A0α . We know ([1, 13]) that Lα has the deficiency indices (2,2) as a symmetric operator on C0∞ (R2 \{0}), and the Krein theory again yields the family of all possible selfadjoint extensions {LU } parameterized by 2 × 2 unitary mapping U from one deficiency subspace to the other one. The selfadjoint operator LU is realized as a differential operator with some boundary conditions at the origin. If w is in the domain D(LU ), then w behaves like
w = w−0 r−α + w+0 rα + o(rα ) + w−1 r−(1−α) + w+1 r1−α + o(r1−α ) eiθ + o(r) for some coefficients w±k , k = 0, 1, and there exist 2 × 2 matrices B± for which the boundary condition is described as the relation w−0 w+0 B− + B+ =0 w−1 w+1
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
83
between these four coefficients. We distinguish the two operators by the following special notation : D(LAB ) = {w ∈ L2 : Lw ∈ L2 , w−0 = w−1 = 0} D(LZ ) = {w ∈ L2 : Lw ∈ L2 , w+0 = w−1 = 0}
(2.10)
among admissible selfadjoint extensions. The first operator LAB is known as the Aharonov-Bohm Hamiltonian ([3]). We denote by γ(x; ω) the azimuth angle from ω. The operator Lα defined by (2.9) admits the polar coordinate decomposition ⊕ hl , Lα l∈Z
where hl = −(d/dr)2 + (ν 2 − 1/4)r−2 with ν = |l − α|. If we define ϕ± (x; E, ω) = e∓iνπ/2 eilγ(x;∓ω) Jν (Er)
(2.11)
l∈Z
for ν = |l − α|, then ϕ± vanishes at the origin and solves Lα − E 2 ϕ± = 0. Thus ϕ± becomes the generalized eigenfunction of LAB with eigenvalue E 2 . The first lemma is due to [16] (see [3, 10] also). Lemma 2.1 Let ϕ+ (x; E, ω) be as above. Define ϕin (x; E, ω) = eiEx·ω eiα(γ(x;ω)−π)
(2.12)
for x = rθ, θ = ω. Then ϕ+ (x; E, ω) obeys ϕ+ (rθ; E, ω) = ϕin (rθ; E, ω) + g+ (ω → θ; E)eiEr r−1/2 (1 + o(1)) ,
r → ∞,
along direction θ, where −1/2
g+ (ω → θ; E) = − (2πiE)
sin απ
ei(θ−ω)/2 . sin((θ − ω)/2)
(2.13)
This lemma implies that ϕ+ (x; E, ω) is the outgoing eigenfunction of LAB , and g+ (ω → θ; E) defines the scattering amplitude. This is known as the AharonovBohm scattering amplitude ([3]). On the other hand, ϕ− (x; E, ω) is shown to be the incoming eigenfunction, but its asymptotic form is not required in the argument below. We move to the second lemma. The proof of this lemma uses the following formula for the Bessel functions : ±iEJν±1 (Er)ei(l±1)θ (l ≥ 1) ilθ = π± Jν (Er)e (2.14) ∓iEJν∓1 (Er)ei(l±1)θ (l ≤ 0) for ν = |l − α| with 0 < α < 1. This follows from (2.3) after a direct computation. The same formula remains true for the Hankel Hν (Er).
84
H. Tamura
Ann. Henri Poincar´e
Lemma 2.2 Let π+ be as in (2.2) and let g+ be as in Lemma 2.1. Then (π+ ϕ+ ) (rθ; E, ω) = Eeiω ϕin (rθ; E, ω) + Eeiθ g+ (ω → θ; E)eiEr r−1/2 (1 + o(1)) as r → ∞ along direction θ, θ = ω. Proof. We calculate I = (π+ ϕ+ )(x; E, ω)/E. Since eilγ(x;−ω) = eilθ eil(π−ω) for x = rθ, we obtain I= ie−iνπ/2 Jν+1 (Er)ei(l+1)θ eil(π−ω) − ie−iνπ/2 Jν−1 (Er)ei(l+1)θ eil(π−ω) l≥1
l≤0
by use of formula (2.14). We use the simple relation ei(l+1)θ eil(π−ω) = −ei(l+1)γ(x;−ω) eiω . If l ≥ 1, then ν + 1 = |l + 1 − α| and ie−iνπ/2 = −e−i|l+1−α|π/2 , and if l ≤ −1, then ν − 1 = |l + 1 − α| and ie−iνπ/2 = e−i|l+1−α|π/2 . If we take account of these relations, then we make a change of variables l + 1 → l to obtain that e−iνπ/2 eilγ(x;−ω) Jν (Er) − e−i(α−1)π/2 Jα−1 (Er)eiθ , I = eiω l =1
so that it equals
I = eiω ϕ+ (x; E, ω) + e−i(1−α)π/2 J1−α (Er) − e−i(α−1)π/2 Jα−1 (Er) eiθ . Hence it follows from (2.4) that I = eiω ϕ+ (x; E, ω) + e−iαπ/2 sin απH1−α (Er)eiθ .
(2.15)
The Hankel function Hµ (z), µ > 0, is known to behave like Hµ (z) = (2/iπ)1/2 e−iµπ/2 eiz z −1/2 1 + O(|z|−1 )
(2.16)
as |z| → ∞. This, together with Lemma 2.1, implies that I = eiω ϕin (x; E, ω) + g˜(ω → θ; E)eiEr r−1/2 (1 + o(1)) , where
−1/2
g˜ = eiω g+ (ω → θ; E) − 2i (2πiE)
sin απeiθ .
A simple computation yields
g˜ = (2πiE)−1/2 sin απ −e−i(θ−ω)/2 / sin((θ − ω)/2) + 2/i eiθ = g+ (ω → θ; E)eiθ . This proves the lemma.
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
85
Proof of Proposition 2.1. Let Dα = D(A0α , 0) be as in (2.1). We look for the solution ψ = (ψ1 , ψ2 ) to equation (Dα − E) ψ = 0 in the form ψ1 = ϕ+ (x; E, ω) + βκ Hα (Er),
ψ2 = (1/E) (π+ ψ1 ) (x; E, ω)
(2.17)
with some constant βκ . If ψ takes the above form, then it is easy to see that ψ solves the equation. The coefficient βκ is determined so as to satisfy the boundary condition (1.13) at the origin. Then ψ = ψ(x; E, ω) becomes the eigenfunction of selfadjoint operator Hκ and the amplitude gκ is determined through the asymptotic form of ψ(x; E, ω). We calculate the limits u−1 and u−2 defined by (1.14). The eigenfunction ϕ+ of LAB vanishes at the origin, so that u−1 = lim rα ψ1 = βκ (−i/ sin απ) (2α /Γ(1 − α)) E −α r→0
by (2.5). Since π+ Hα (Er) = −iEHα−1 (Er)eiθ = iEe−iαπ H1−α (Er)eiθ by (2.14), it follows from (2.15) that
ψ2 = eiω ϕ+ (x; E, ω) + e−iαπ/2 sin απ + ie−iαπ βκ H1−α (Er)eiθ
(2.18)
and hence
u−2 = (−i/ sin απ) e−iαπ/2 sin απ + ie−iαπ βκ 21−α /Γ(α) E −1+α .
Thus βκ is determined as
βκ = ieiαπ/2 sin απ κτα E 2α−1 /(κτα E 2α−1 − eiαπ ) ,
(2.19)
where τα is defined in (2.8). By Lemmas 2.1 and 2.2 and by (2.16), ψ(x; E, ω) behaves like ˜ ; E)τ (˜ ω )eiEr r−1/2 + o(r−1/2 ) ψ = τ (ω)ϕin (x; E, ω) + gκ (ω → ω
(2.20)
as r → ∞ along direction ω ˜ = ω, where τ (ω) is in (1.6), and gκ = g+ (ω → ω ˜ ; E) + 2(2πiE)−1/2 e−iαπ/2 βκ . This determines the desired amplitude and the proof is complete.
We end the section by making some additional comments on the outgoing eigenfunction ψ+ (x; E, ω) and the incoming one ψ− (x; E, ω) of H∞ . These eigenfunctions are used to represent the amplitude f (ω → ω ˜ ; E) of T = D(A, V ) in Section 4. The outgoing eigenfunction ψ+ = t (ψ+1 , ψ+2 ) is defined by (2.17) with β∞ = ieiαπ/2 sin απ, and we have ψ+1 = ϕ+ (x; E, ω) + β∞ Hα (Er),
ψ+2 = eiω ϕ+ (x; E, ω)
86
H. Tamura
Ann. Henri Poincar´e
by (2.18). This is expanded as ψ+1 (x; E, ω) =
l =0
ψ+2 (x; E, ω) =
eiω
e−iνπ/2 eilγ(x;−ω) Jν (Er) + eiαπ/2 J−α (Er),
e−iνπ/2 eilγ(x;−ω) Jν (Er).
(2.21)
l∈Z (2)
(2)
The Hankel function Hµ (z) of second kind is related to Hµ (z) through Hµ (z) = (2) (2) Hµ (z) for z ∈ R, and it satisfies H−µ (z) = e−iµπ Hµ (z). If we make use of these relations, a similar argument enables us to construct the incoming eigenfunction ψ− (x; E, ω) = t (ψ−1 , ψ−2 ) as ψ−1 = ϕ− (x; E, ω) + β ∞ Hα (Er),
ψ−2 = eiω ϕ− (x; E, ω)
with ϕ− defined by (2.11), and it admits the expansion ψ−1 (x; E, ω)
=
l =0
ψ−2 (x; E, ω)
= eiω
eiνπ/2 eilγ(x;ω) Jν (Er) + e−iαπ/2 J−α (Er),
eiνπ/2 eilγ(x;ω) Jν (Er).
(2.22)
l∈Z
3 Resolvent of selfadjoint extensions We here establish the relation between the two resolvents R(E +i0; Hκ ) and R(E + i0; H∞ ). We fix several new notation. We denote by ( , ) the scalar product in L2 2 or L2 , and write f ⊗ g = ( · , g)f for the integral operator with kernel f (x) g(y). This acts as (f ⊗ g) u = (u, g)f on u ∈ L2 . We also use a similar notation u ⊗ v = (uj ⊗ vk )1≤j,k≤2 ,
u = t (u1 , u2 ),
v = t (v1 , v2 ),
for a vector version over [L2 ]2 . We further define the two basic functions ξ+ (x; E) = t −ieiαπ Hα (Er), H1−α (Er)eiθ ,
ξ− (x; E) = t −ie−iαπ Hα (Er), H1−α (Er)eiθ
(3.1)
for E > 0. The second function may be written as
(2) ξ− (x; E) = t −ie−iαπ Hα(2) (Er), H1−α (Er)eiθ . If we repeat almost the same argument as in the previous section, then it is easy to see that these two functions solve (Dα − E) u = 0, and form a pair of linearly independent solutions. The aim here is to prove the following proposition.
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
87
Proposition 3.1 Let ξ± = ξ± (x; E) be as above. Then R(E + i0; Hκ ) = R(E + i0; H∞ ) − cκ E (ξ+ ⊗ ξ− ) , where
cκ = sin απ/(4(κτα E 2α−1 − eiαπ ))
with τα defined by (2.8). If, in particular, α = 1/2, then cκ = −1/(4(i − κ)). The proposition is proved at the end of this section. Let LAB and LZ be defined in (2.10), and let Aε and bε = ∇ × Aε be as in (1.9). We again set T0ε = D(Aε , 0), which is convergent to Dα = D(A0α , 0) as ε → 0 on [C0∞ (R2 \ {0})]2 by (1.12). We represent R(E + i0; H∞ ) in terms of resolvents of LAB and LZ . We repeat the same argument as used in Section 1 to obtain R(z 2 ; L−ε ) 0 R(z; T0ε ) = (T0ε + z) , L±ε = L(Aε , ±bε ), 0 R(z 2 ; L+ε ) for z, Im z = 0. According to the results in [23, Section 3], we have R(z; T0ε ) → R(z; H∞ ) and R(z; L+ε ) → R(z; LAB ),
R(z; L−ε ) → R(z; LZ ),
as ε → 0 in norm (in norm resolvent sense). We also have ER(E 2 + i0; LZ ) π− R(E 2 + i0; LAB ) R(E + i0; H∞ ) = . π+ R(E 2 + i0; LZ ) ER(E 2 + i0; LAB )
(3.2)
We now calculate the Green kernels of R(E 2 + i0; LAB ) and R(E 2 + i0; LZ ). To do this, we decompose L2 = L2 (0, ∞) ⊗ L2 (S), and we define the mapping Ul by (Ul f )(r) = (2π)−1/2 r1/2
0
2π
f (rθ)e−ilθ dθ : L2 → L2 (0, ∞)
for l ∈ Z. Then (Ul∗ g)(x) = (2π)−1/2 r−1/2 g(r)eilθ : L2 (0, ∞) → L2 , and R(E 2 + i0; LAB ) admits the decomposition R(E 2 + i0; LAB ) = ⊕ Rl , Rl = Ul∗ R(E 2 + i0; hl )Ul , l∈Z
where the domain of selfadjoint operator hl = −(d/dr)2 + (ν 2 − 1/4)r−2 ,
ν = |l − α|,
(3.3)
88
H. Tamura
Ann. Henri Poincar´e
is specified by the boundary condition lim r−(1/2−α) g(r) = 0 at the origin. Simir→0
larly we have ˜0 ⊕ R(E 2 + i0; LZ ) = R
⊕ Rl ,
˜ 0 = U ∗ R(E 2 + i0; ˜h0 )U0 , R 0
(3.4)
l =0
and the domain of selfadjoint operator ˜ 0 = −(d/dr)2 + (α2 − 1/4)r−2 h is specified by the condition lim r−(1/2+α) (g(r) − g0 r1/2−α ) = 0
r→0
with g0 = lim r−(1/2−α) g(r). The two functions r1/2 Jν (Er) and r1/2 Hν (Er) are r→0
linearly independent solutions to (hl − E 2 )g = 0 for E > 0. By formula, we know W (Jµ , J−µ )(z) = −2 sin µπ/(πz) for the Wronskian of Bessel functions, so that W (Hµ , Jµ )(z) = −2i/(πz),
W (Hµ , J−µ )(z) = −2ie−iµπ /(πz)
by (2.4). Thus we can construct the Green kernels Rl (x, y) ˜ 0 (x, y) R
= (i/4) Hν (E(r ∨ ρ))Jν (E(r ∧ ρ))eil(θ−ϕ) , = ieiαπ /4 Hα (E(r ∨ ρ))J−α (E(r ∧ ρ))
(3.5)
in the standard way, where r ∨ ρ = max (r, ρ) and r ∧ ρ = min (r, ρ) for (x, y) = (reiθ , ρeiϕ ). We are now in a position to prove Proposition 3.1. Proof of Proposition 3.1. According to the Krein theory ([8]), the two resolvents are related to each other through the relation in the proposition. We have only to calculate the constant cκ . We set t
(u1 , u2 ) = R(E + i0; Hκ )F
for F = t (f, 0) with f ∈ C0∞ (R2 \ {0}). Then u1 = v1 − ieiαπ cσHα (Er),
u2 = v2 + cσH1−α (Er)eiθ ,
c = −cκ E,
where t
(v1 , v2 ) = R(E + i0; H∞ )F = t (ER(E 2 + i0; LZ )f, π+ R(E 2 + i0; LZ )f )
by (3.2), and σ = (F, ξ− ) is the scalar product between F = t (f, 0) and ξ− . The constant cκ is determined by boundary condition u−1 + iκu−2 = 0, where
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
89
u−1 and u−2 are defined by (1.14). We calculate the limits u−1 and u−2 . Since t (v1 , v2 ) = R(E + i0; H∞ )F , v2 obeys v2 = o(r−(1−α) )eiθ , and hence it follows from (2.5) that u−2 = cσ (−i/ sin απ) 21−α /Γ(α) E α−1 . If we use (3.5) and (3.1), then v1 behaves like ˜ 0 f + o(1) = (σE/4)J−α (Er) + o(1), v1 = E R and hence
r → 0,
u−1 = σ E/4 − ceiαπ / sin απ (2α /Γ(1 − α)) E −α
by (2.5). Then cκ is determined as in the proposition.
4 Scattering amplitudes in the presence of scalar potentials The aim here is to derive the representation (4.6) below for the scattering amplitude f (ω → ω ˜ ; E) of T = D(A, V ) with scalar potential V ∈ C0∞ (R2 → R), where 2 A ∈ C ∞ (R → R2 ) is defined by (1.1). The derivation requires two lemmas. Lemma 4.1 Write ψ− (ω) for the incoming eigenfunction ψ− (x; E, ω), defined by (2.22), of H∞ . Let F (x) = t (f1 (r)eimθ , f2 (r)ei(m+1)θ ),
m ∈ Z,
for f1 , f2 ∈ C0∞ [0, ∞). Then 1/2
(R(E + i0; H∞ )F )(rω ˜ ) = (iE/8π)
(F, ψ− (˜ ω )) τ (˜ ω )eiEr r−1/2 + o(r−1/2 )
as r → ∞ uniformly in ω ˜ ∈ S, where (F, ψ− (ω)) is the scalar product in [L2 ]2 between F and ψ− (ω). Proof. We prove the lemma for the case m = 0 only. A similar argument applies to the other cases. Set t (u1 , u2 ) = R(E + i0; H∞ )F for F as in the lemma. Then u1 = Ev1 + π− v2 ,
u2 = π+ v1 + Ev2
by (3.2), where v1 = R(E 2 + i0; LZ )f1 ,
v2 = R(E 2 + i0; LAB )(f2 eiθ ).
˜ 0 f1 and v2 = R1 (f2 eiθ ). The two It follows from (3.3) and (3.4) that v1 = R ˜ operators R0 and R1 have the kernels (3.5). By assumption, f1 and f2 have compact support. Hence we have v1 = (ieiαπ /4)(f1 , J−α )Hα (Er),
v2 = (i/4)(f2 , J1−α )H1−α (Er)eiθ
90
H. Tamura
Ann. Henri Poincar´e
for |x| 1. Since π− H1−α (Er)eiθ = −iEH−α (Er) = −iEeiαπ Hα (Er) by (2.14), it follows from (2.16) that u1
=
(iE/4) eiαπ ((f1 , J−α ) − i(f2 , J1−α )) Hα (Er)
=
(iE/8π)1/2 eiαπ/2 ((f1 , J−α ) − i(f2 , J1−α ))eiEr r−1/2 + o(r−1/2 )
as r → ∞. The eigenfunction ψ− has the expansion (2.22), and we have (F, ψ− (˜ ω )) = (f1 , ψ−1 (˜ ω )) + (f2 eiθ , ψ−2 (˜ ω )) = eiαπ/2 ((f1 , J−α ) − i(f2 , J1−α )) . This yields the desired asymptotic form for u1 . We can show in a similar way that u2 also takes the asymptotic form in the theorem. Thus the proof is complete. We now introduce the Banach spaces B and B ∗ with norms uB =
∞ j=0
2
j
1/2
2
Ωj
|u(x)| dx
,
uB ∗ = sup
R>0
1 R
|x| 1/2, where L2s = L2 (R2 ; x2s dx) with x = (1 + |x|2 )1/2 . We use the notation o∗ (r−1/2 ) as r = |x| → ∞ to denote functions u obeying the bound 1 |u(x)|2 dx → 0, R → ∞. R |x| 2. (4.1) We define the auxiliary operator K as K = D(a, V ).
(4.2)
This is selfadjoint with domain D(K) = [H 1 (R2 )]2 , and we know ([11, 15]) that the boundary value R(E + i0; K) to the real axis exists as a bounded operator from [L2s ]2 into [L2−s ]2 for s > 1/2. We further introduce a basic cut-off function χ0 ∈ C0∞ (R2 → R) with the properties supp χ0 ⊂ {|x| < 2},
χ0 = 1 on {|x| < 1}.
(4.3)
We set χ+ (x) = χ0 (x/2) and χ− (x) = χ0 (x/4). We study the behavior at infinity of eigenfunction ψ(x; E, ω) of K. Since K = D(A0α , 0) = Dα over {|x| > 2} by (4.1), we have (1−χ+ ) (K − E) ψ+ = 0 for the outgoing eigenfunction ψ+ (ω) = ψ+ (x; E, ω) of H∞ . Hence the eigenfunction ψ = ψ(x; E, ω) with incident wave ϕin (x; E, ω) as in Lemma 2.2 is written as ψ = (1 − χ+ )ψ+ + R(E + i0; K)Π+ ψ+ ,
(4.4)
where Π+ = [Dα , χ+ ]. Similarly ψ+ (x; E, ω) is represented as ψ+ = (1 − χ− )ψ + R(E + i0; H∞ )Π− ψ with Π− = [Dα , χ− ]. Hence it follows from Lemma 4.2 that ψ = ψ+ − (iE/8π)1/2 (Π− ψ, ψ− (˜ ω )) τ (˜ ω )eiEr r−1/2 + o∗ (r−1/2 ).
(4.5)
We insert (4.4) into ψ on the right side of (4.5). Since Π− (1 − χ+ ) = 0 and Π∗− = −Π− , we obtain (Π− ψ, ψ− (˜ ω )) = −(R(E + i0; K)Π+ ψ+ (ω), Π− ψ− (˜ ω )). We recall that ψ+ obeys (2.20) with κ = ∞. Hence the amplitude f (ω → ω ˜ ; E) of K is given by f = g∞ (ω → ω ˜ ; E) + (iE/8π)1/2 (R(E + i0; K)Π+ ψ+ (ω), Π− ψ− (˜ ω )),
(4.6)
where g∞ is the amplitude of H∞ . The amplitude of T = D(A, V ) is shown to be represented in the same way. Since A and a have the same field b, we have the relation A = a + ∇h (4.7)
92
H. Tamura
Ann. Henri Poincar´e
for some function h ∈ C ∞ (R2 → R), and T = eih Ke−ih . The difference obeys A − a = O(|x|−2 ) at infinity, so that h falls off with h = O(|x|−1 ) and eih(x) = 1 + O(|x|−1 ). Thus T has the same scattering operator as K and hence the scattering amplitude of T is also represented as (4.6). To sum up, the amplitude f (ω → ω ˜ ; E) of T = D(A, V ) is defined through the asymptotic form ˜ ; E)τ (˜ ω )eiEr r−1/2 + o∗ (r−1/2 ) ψ = τ (ω)ϕin (x; E, ω) + f (ω → ω as r = |x| → ∞ of solution ψ to equation T ψ = (T0 + V ) ψ = Eψ, and it has the representation (4.6). In the mathematical scattering theory, it is standard to define the scattering amplitudes through integral kernels of scattering matrices after establishing the basic problems such as the existence and completeness of wave operators and the limiting absorption principle [9, 15, 18, 24, 25]. However, K has the special property that it admits the polar coordinate decomposition on {|x| > 2}. If we make use of this property, the Agmon-H¨ ormander theory ([5]) enables us to define directly the scattering amplitude through the asymptotic form of eigenfunction. We can show that these two representations defined in a different way coincide with each other, but we do not go into the details here.
5 Scattering by electromagnetic fields with small support In this section we formulate the results on the asymptotic behavior of amplitudes for the scattering by electromagnetic fields with small support. We obtain the three main theorems and the remaining four sections (Sections 6, 7, 8 and 9) are devoted to the proof of these theorems. ˜ ; E) the scattering Let Aε and Vε be defined by (1.7). We denote by fε (ω → ω amplitude of Tε = D(Aε , Vε ). If we set Kε = D(aε , Vε ),
aε = ε−1 a(x/ε),
(5.1)
then aε (x) = A0α (x) over |x| > 2ε, and the amplitude fε has the representation ˜ ; E) + (iE/8π)1/2 (R(E + i0; Kε )Π+ ψ+ (ω), Π− ψ− (˜ ω )), fε = g∞ (ω → ω
(5.2)
where Π± = [Dα , χ± ] with χ+ = χ0 (x/2) and χ− = χ0 (x/4) again. We have explicitly calculated the scattering amplitude gκ (ω → ω ˜ ; E) of Hκ in Proposition 2.1. It admits the representation ˜ ; E) + (iE/8π)1/2 (R(E + i0; Hκ )Π+ ψ+ (ω), Π− ψ− (˜ ω )) gκ = g∞ (ω → ω
(5.3)
in terms of resolvent R(E + i0; Hκ ). In fact, this is obtained by repeating almost the same argument as used to derive (4.6). We first deal with the case without electric fields.
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
93
Theorem 5.1 Assume that V = 0 identically. Then fε (ω → ω ˜ ; E) → g∞ (ω → ω ˜ ; E),
ε → 0,
for ω = ω ˜. Next we discuss the case when V ∈ C0∞ (R2 → R) does not vanish identically. We assume that V (x) ≥ 0, (5.4) so that the scalar product λ0 = (V ρ, ρ) > 0
(5.5) −ϕ(x)
is strictly positive for the resonance function ρ(x) = e defined by (1.9). The assumption (5.4) does not matter, but λ0 = 0 is important to the future argument. Before stating the second theorem, we define the resonance state of Dirac operator T = D(A, V ) at zero energy. The definition is different according as 0 < α ≤ 1/2 or 1/2 < α < 1. Definition 5.1. (1) Let 0 < α ≤ 1/2. Assume that the equation T v = 0 has a non-trivial solution such that v = t (v1 , v2 ) ∈ L2 × L∞ and v2 (x) = O(|x|−1+α ) at infinity. If v2 ∈ L2 , then T is said to admit a resonance state at zero energy, and if v2 ∈ L2 , then T has an eigenvalue at zero energy. (2) Let 1/2 < α < 1. Assume that T v = 0 has a non-trivial solution such that v = t (v1 , v2 ) ∈ L∞ × L2 and v1 (x) = O(|x|−α ) at infinity. If v1 ∈ L2 , then T is said to admit a resonance state at zero energy, and if v1 ∈ L2 , then T has an eigenvalue at zero energy. In the present work, we deal with only the case that T has neither eigenstates nor resonance states at zero energy. This case is simple but generic. Thus we always assume that T has neither eigenstates nor resonance states at zero energy.
(5.6)
If |V | 1 is small enough, then it can be shown that T fulfills (5.6). The lemma below plays an important role in proving the remaining two main theorems. This basic lemma is proved in Section 7. Lemma 5.1 Assume that (5.6) is fulfilled. Then : (1) Let 0 < α ≤ 1/2. Then there exists a unique solution e ∈ L∞ × L∞ to equation T e = 0 such that e = t (e1 , e2 ) obeys e1 = r−α + O(|x|−1−α ),
e2 = O(|x|−1+α )
(5.7)
at infinity, and e2 (x) behaves like e2 (x) = iλ2 r−1+α eiθ + O(|x|−2+α ), for some real constant λ2 .
|x| → ∞,
(5.8)
94
H. Tamura
Ann. Henri Poincar´e
(2) Let 1/2 < α < 1. Then there exists a unique solution e ∈ L∞ × L∞ to T e = 0 such that e = t (e1 , e2 ) obeys e1 = O(|x|−α ),
e2 = ir−1+α eiθ + O(|x|−2+α )
(5.9)
at infinity, and e1 (x) behaves like e1 (x) = λ1 r−α + O(|x|−1−α ),
|x| → ∞,
(5.10)
for some real constant λ1 . We are now in a position to state the second theorem. When the scalar potential V is added as a perturbation, the situation changes even for small perturbation. The limit heavily depends on the values α of fluxes and it changes discontinuously at half-integer flux α = 1/2. Theorem 5.2 Let V ∈ C0∞ (R2 ) satisfy (5.4), and assume that T fulfills (5.6). If ω = ω ˜ for incident and final directions w and ω ˜ , then one has the following asymptotic form as ε → 0 : (1) Let 0 < α < 1/2. Then fε (ω → ω ˜ ; E) → g∞ (ω → ω ˜ ; E). (2) Let α = 1/2 and let λ2 be as in (5.8) of Lemma 5.1. Then fε (ω → ω ˜ ; E) → gκ (ω → ω ˜ ; E) for κ = 1/λ2 (κ = ∞ provided that λ2 = 0). (3) Let 1/2 < α < 1. Then ˜ ; E) → g0 (ω → ω ˜ ; E). fε (ω → ω The third theorem is concerned with the scattering of Dirac particles appearing in the interaction of cosmic string with matter. We now consider the 2 × 2 ˜ ) in question is desystem (1.15) of Dirac equations. The amplitude f2ε (ω → ω fined through the asymptotic form of solution w to equation (1.15). The solution behaves like w
=
t
(τ (ω), 0)ϕin (x; E, ω) + f1ε (ω → ω ˜ ; E)t (τ (˜ ω ), 0)eiEr r−1/2 + f2ε (ω → ω ˜ ; E)t (0, τ (˜ ω ))eiEr r−1/2 + o∗ (r−1/2 ),
r → ∞,
for incident wave t (τ (ω), 0)ϕin (x; E, ω). The aim of the third theorem is to analyze ˜ ; E). the asymptotic behavior as ε → 0 of f2ε (ω → ω Theorem 5.3 Let V ∈ C0∞ (R2 → R) satisfy (5.4), and assume that T fulfills (5.6). Then the amplitude f2ε (ω → ω ˜ ; E) behaves like 1/2 iE Cα ε|2α−1| + o(ε|2α−1| ), ε → 0, f2ε = 8π
Vol. 5, 2004
where
Scattering of Dirac Particles by Electromagnetic Fields
95
α −α α 2 0 < α < 1/2, (2 E i /Γ(1 − α)) 2πλ2 , α = 1/2, 4E −1 iλ2 (1 + λ22 )−1 , Cα = 2 1−α α−1 1−α 2 E i /Γ(α) 2πλ1 , 1/2 < α < 1.
We end the section by making some comments on Theorems 5.2 and 5.3. (1) As stated in Section 1, a result similar to Theorem 5.2 has been obtained by Afanasiev [2, Section 7.10], where the behavior of amplitude has been analyzed for the scattering by the small obstacle {|x| < ε} under a certain impenetrable boundary condition in the background of the δ-like field 2παδ(x). As ε → 0, the amplitude fε is convergent to g∞ , gκ with κ = −1 or g0 according as 0 < α < 1/2, α = 1/2 or 1/2 < α < 1. (2) The assumption that A(x) and V (x) are smooth is not essential. The two theorems extend to the case of bounded electromagnetic fields, and the extension is possible even for singular magnetic potentials. For example, the theorems apply to the case that A(x) = A0α (x) is the Aharonov-Bohm potential and V (x) is the characteristic function of unit disk {|x| < 1}. If we consider (1.13) with κ = ∞ as the boundary condition at the origin, we can calculate λ1 and λ2 explicitly. In fact, if we set e(x) = t (e1 (r), e2 (r)eiθ ), then it follows from (2.2) that e solves e 1 + α r−1 e1 + iV e2 = 0,
e 2 + (1 − α)r−1 e2 + iV e1 = 0,
where e = (d/dr)e. We use the formula (2.14) to solve the equation above. If we take account of (5.7), then λ2 is determined as λ2 = −J1−α (1)/J−α (1) for 0 < α ≤ 1/2, while (5.9) yields λ1 = −J−α (1)/J1−α (1) for 1/2 < α < 1. (3) As a work related to Theorem 5.3, [7] has dealt with the case that the electric potential is λV (x) and A(x) is the Aharonov-Bohm potential A0α (x) with boundary condition (1.13) with κ = ∞ or κ = 0, where λ > 0 is a small coupling constant and V still denotes the characteristic function of the unit disk.
6 Behavior of resolvent at low energy The proof of all the theorems in the previous section is based on the behavior as ε → 0 of resolvent R(E + i0; Kε ). We first follow the idea from [6, Chapter I.1.2] to derive the basic representation for R(E + i0; Kε ). The derivation is done by repeated use of the resolvent identity. If we set K0ε = D(aε , 0), then Kε = K0ε +Vε , and we have R(E + i0; Kε ) = R(E + i0; K0ε ) − R(E + i0; Kε )Vε R(E + i0; K0ε ) by the resolvent identity. We have assumed that V (x) ≥ 0. If we further define Yε = Vε1/2 R(E + i0; K0ε )Vε1/2 : [L2 ]2 → [L2 ]2 ,
(6.1)
96
H. Tamura
Ann. Henri Poincar´e
then the resolvent identity yields the relation R(E + i0; Kε )Vε1/2 (1 + Yε ) = R(E + i0; K0ε )Vε1/2 . The operator 1 + Yε has the bounded inverse (1 + Yε )−1 : [L2 ]2 → [L2 ]2 , which follows from the fact that the outgoing solution to equation (Kε −E)u = 0 identically vanishes. Thus R(E + i0; Kε ) is represented as R(E + i0; K0ε ) − R(E + i0; K0ε )Vε1/2 (1 + Yε )−1 Vε1/2 R(E + i0; K0ε ) by the resolvent identity. Let Jε : [L2 ]2 → [L2 ]2 be again the unitary operator defined by (Jε u) (x) = ε−1 u(x/ε). We set Xε = Jε∗ Yε Jε . Since K0ε = ε−1 Jε K0 Jε∗ for K0 = D(a, 0), we have Xε = Jε∗ Yε Jε = V 1/2 R(k + i0; K0 )V 1/2 ,
k = εE > 0,
(6.2)
and hence R(E + i0; Kε ) = R(E + i0; K0ε ) − ε−1 Γε (E + i0)(1 + Xε )−1 Γε (E − i0)∗ , (6.3) where
Γε (E ± i0) = R(E ± i0; K0ε )Jε V 1/2 .
(6.4)
This is a basic representation. This section is devoted to the analysis on the behavior as ε → 0 of Xε as the first step towards proving the three theorems. By (4.7), the potential a : R2 → R2 takes the form a = (−∂2 ϕ(x), ∂1 ϕ(x)) + ∇h = A + ∇h for some h ∈ C ∞ (R2 → R) falling off like h = O(|x|−1 ) at infinity, and the field b = ∇ × a has support in {|x| < 1}. We set p = (p1 , p2 ) = −i∇ − a and write K0 as 0 p− K0 = σ1 p1 + σ2 p2 = p+ 0 in the matrix form, where p± = p1 ± ip2 . We define the Schr¨ odinger operators L± by (6.5) L± = L(a, ±b) = p21 + p22 ± b = (−i∇ − a)2 ± b. These are selfadjoint with domain D(L± ) = H 2 (R2 ) in L2 . Since i[p1 , p2 ] = i(p1 p2 − p2 p1 ) = −b, we have L± = p± p∓ = p∗∓ p∓ , and R(k + i0; K0 ) is represented as kR(k 2 + i0; L−) p− R(k 2 + i0; L+ ) R(k + i0; K0 ) = . p+ R(k 2 + i0; L− ) kR(k 2 + i0; L+ )
(6.6)
Thus the problem is reduced to the study on the behavior of R(k 2 + i0; L± ) as k → 0.
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
97
The two operators L± = p∗∓ p∓ ≥ 0 are non-negative, and since 0 < α < 1 by assumption (1.3), it follows by the Aharonov-Casher theorem ([4]) that L± have no bound states at zero energy. However, the spectral structure at zero energy is different in the sense that L− has a resonance state. The resonance state is defined as a bounded solution u to equation L− u = p− p+ u = 0. If u is such a solution, then a simple calculation using integral by parts shows that p1 u and p2 u are in L2 , so that p+ u = 0. By (4.7) (see also (1.8)), we have p+ = −ieih e−ϕ ( ∂1 + i∂2 ) eϕ e−ih .
(6.7)
Thus L− has the resonance state behaving like u(x) = e−ϕ eih = |x|−α 1 + O(|x|−1 ) at infinity. On the other hand, L+ = p+ p− does not have a resonance state. We note that if α > 1, L− has bound states at zero energy with multiplicity [α] by the Aharonov-Casher theorem again. We now introduce the following notation : η ∈ C0 (R2 ) is a continuous function with compact support and η0 ∈ C0 (R2 ) is a function compactly supported away from the origin. We further use the notation Op(εσ ) and op(εσ ) to denote the classes of bounded operators obeying the bound O(εσ ) and o(εσ ) in norm respectively. We make a brief review on the behavior at low energy of R(k 2 + i0; L±) obtained by ([23, Propositions 4.2 and 4.3]). We first consider L− . Let h(x) be as in (6.7). Then (6.8) ρ0 (x) = e−ϕ eih , solves L− ρ0 = 0 and behaves like
ρ0 (x) = |x|−α 1 + O(|x|−1 )
(6.9)
at infinity. We know ([23]) that L− has the one-dimensional resonance space spanned by ρ0 at zero energy. Proposition 6.1 Let ρ0 be as above and let γ0 be the constant defined by γ0 = −22(1−α) πΓ(1 − α)/Γ(α). Then
(6.10)
ηR(k 2 + i0; L− )η = γ− (k)i2α k −2α η(ρ0 ⊗ ρ0 )η + Op(ε0 )
for some coefficient γ− (k) obeying γ− (k) = −1/γ0 + o(1) as k → 0. Remark 6.1 (1) The proposition above corresponds to Proposition 4.3 in [23], where the resonance function ρ0 (x) is normalized as ρ0 (x) = (2πα)−1/2 e−ϕ eih , so that the constant γ− (k) undergoes a suitable change. (2) By elliptic estimate, ∇ηR(k 2 + i0; L−)η admits a similar asymptotic form under a natural modification.
98
H. Tamura
Ann. Henri Poincar´e
Next we move to L+ which has neither bound states nor resonance states at zero energy. We set L2com = {u ∈ L2 (R2 ) : supp u ⊂ BM },
BM = {|x| < M },
for M 1 fixed arbitrarily but sufficiently large. We have shown in [23] that there exists a limit G+ = lim R(k 2 + i0; L+ ) : L2com → L2−1 (6.11) k→0
as a bounded operator from L2com to L2−1 = L2 (R2 ; x−2 dx). We further know that the equation L+ = p+ p− u = 0 has a unique solution behaving like ω+l = rν eilθ + O(1),
|x| → ∞,
(6.12)
for l = 0, 1, where ν = |l − α| again. Proposition 6.2 Let the notation be as above. Then there exists γ+l (k) such that γ+l (k)i−2ν k 2ν η(ω+l ⊗ ω+l )η + Op(ε2 ), ηR(k 2 + i0; L+)η = ηG+ η + l=0,1
where the two constants γ+l (k), l = 0, 1, are bounded uniformly in k = εE > 0. This proposition has been obtained as Proposition 4.2 in [23]. We can make precise the behavior as k → 0 of the constant γ+l (k), but the argument below does not require such an asymptotic form. By (6.7), p+ = −2i eihe−ϕ ∂eϕ e−ih with ∂ = (1/2) ( ∂1 + i∂2 ). The CauchyRiemann operator ∂ has the fundamental solution (1/π) (x1 + ix2 )−1 . We denote −1 by ∂ the convolution operator ∂ and we define and p−1 −
−1
−1
= (1/π) (x1 + ix2 )
−1 ih −ϕ p−1 e e ∂ + = −(2i)
∗
−1 ϕ −ih
e e −1 ∗ = p+ . By definition, we have p± p−1 ± = 1.
Lemma 6.1 One has the relations p− G+ f = p−1 + f,
G+ p+ f = p−1 − f
for any bounded function f with compact support. Proof. We prove only the first relation. The second one follows by taking the 2 adjoint of both sides. Let f be as in the lemma, and set w1 = p−1 + f . Then w1 ∈ L and it solves p+ w1 = f . If, on the other hand, we set w2 = p− G+ f , then w2 satisfies p+ w2 = p+ p− G+ f = L+ G+ f = f.
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
99
Since w2 ∈ L2−1 by (6.11), it follows that w2 ∈ L2 . In fact, we have p− G+ f L2 < ∞ by a simple use of partial integration. Set w = eϕ e−ih (w1 − w2 ). Then ∂w = 0, so that w is an entire function. Note that eϕ = O(|x|α ) at infinity for 0 < α < 1. Since w1 − w2 ∈ L2 , we can easily show that w = 0, and hence w1 = w2 . Thus the lemma is obtained. Lemma 6.2 Let ω+0 be as in (6.12). Then one has p− ω+0 = 0. Proof. Set v0 = e−ih eϕ . Then p− v0 = 0 and the difference u = ω+0 − v0 is bounded. The function u solves p+ p− u = L+ u = L+ ω+0 − p+ p− v0 = 0. Hence it follows from Lemma 4.3 of [22] (or by the argument used in its proof) that p− u = 0. This implies that p− ω+0 = 0, and the proof is complete. Lemma 6.3 Let ω+1 be also as in (6.12). Then one has p− ω+1 = cρ0 for some c. Proof. Set u = p− ω+1 . Then u obeys the bound u = O(|x|−α ) at infinity, and it solves the equation L− u = p− L+ ω+1 = 0. This implies that u is in the resonance space of L− at zero energy. Since the resonance space is one-dimensional, the lemma follows at once. If we make use of the simple relation p+ R(k 2 ± i0; L− ) = R(k 2 ± i0; L+ )p+ , then we obtain from (6.6) that kR(k 2 + i0, L− ) R(k + i0; K0 ) = R(k 2 + i0; L+ )p+
p− R(k 2 + i0; L+ ) kR(k 2 + i0, L+ )
for k = εE > 0. Thus we combine Propositions 6.1, 6.2 and Lemmas 6.1, 6.2 and 6.3 to get the following proposition. Proposition 6.3 As ε → 0, ηR(k + i0; K0 )η takes the form ηR(k + i0; K0)η = η γ(ε) (˜ ρ0 ⊗ ρ˜0 ) ε1−2α + G0 + O(ε2(1−α) )G1 η + Op(ε), where ρ˜0 = t (ρ0 , 0) and 0 p−1 + G0 = , p−1 0 −
G1 =
0 ω+1 ⊗ cρ0
cρ0 ⊗ ω+1 0
,
c being as in Lemma 6.3, and γ(ε) = i2α E 1−2α γ− (εE) = −i2α E 1−2α (1/γ0 + o(1)) ,
ε → 0.
(6.13)
100
H. Tamura
Ann. Henri Poincar´e
In particular, Xε defined by (6.2) takes the form Xε = γ(ε) (q0 ⊗ q0 ) ε1−2α + Z0 + O(ε2(1−α) )Z1 + Op(ε), where and Z0 = V
q0 = V 1/2 ρ˜0 , 1/2
G0 V
1/2
and Z1 = V
1/2
G1 V
ρ˜0 = t (ρ0 , 0), 1/2
(6.14) (6.15)
.
7 Resonance at zero energy: proof of Lemma 5.1 The second step is to analyze the inversion of (1 + Xε )−1 which appears in representation (6.3) for the resolvent R(E + i0; Kε ) under consideration. We also prove Lemma 5.1 at the end of the section. As is easily seen from assumption (5.6), K = D(a, V ) = K0 + V has neither eigenstates nor resonance states at zero energy. Lemma 7.1 Assume that 0 < α ≤ 1/2. Let Z0 be as in Proposition 6.3. If (5.6) is fulfilled, then Z0 : [L2 ]2 → [L2 ]2 has the bounded inverse (1 + Z0 )−1 on [L2 ]2 . Proof. The operator Z0 is compact. Set Φ = ker (1 + Z0). It suffices to show that dim Φ = 0. The proof is done by contradiction. Assume that u = t (u1 , u2 ) ∈ Φ does not vanish identically. If we set v = t (v1 , v2 ) = G0 V 1/2 u for u as above, then V 1/2 v = Z0 u = −u, and v satisfies K0 v = V 1/2 u = −V v, so that v solves Kv = 0. We can easily see that v is notidentically zero. The first −1 ∗ 1/2 component v1 = p−1 u2 is in L2 . Since p−1 is the integral operator + V − = p+ with kernel −(2πi)−1 eϕ eih (x1 − ix2 )−1 ∗ e−ϕ e−ih , 1/2 u1 behaves like the second component v2 = p−1 − V
v2 (x) = −(2πi)−1 (u1 , V 1/2 ρ0 )eϕ eih (x1 −ix2 )−1 +O(|x|−2+α ) = O(|x|−1+α ) (7.1) as |x| → ∞. This implies that K has either eigenstates or resonance states at zero energy. This contradicts the assumption and the proof is complete. By assumption (5.5), λ0 = (V ρ0 , ρ0 ) = 0. This enables us to define P = λ−1 0 (q0 ⊗ q0 ),
q0 = V 1/2 ρ˜0 ,
(7.2)
as a projection on [L2 ]2 . Lemma 7.2 Assume that 1/2 < α < 1. Let Q = 1 − P and Σ = Ran Q. If (5.6) is −1 fulfilled, then QZ0 Q : Σ → Σ has the bounded inverse (1 + QZ0 Q) on Σ.
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
101
Proof. We again show by contradiction that dim Ψ = 0, where Ψ = {u ∈ Σ : QZ0 Qu = −u}. Assume that u not vanishing identically belongs to Ψ. We set v = t (v1 , v2 ) = G0 V 1/2 u − d˜ ρ0 , 1/2 ρ˜0 ) = λ−1 ˜0 = 0 and since where d = λ−1 0 (Z0 u, V 0 (Z0 u, q0 ). Since K0 ρ
V 1/2 v = Z0 u − P Z0 u = QZ0 u = −u, we see that v satisfies K0 v = V 1/2 u = −V v, and hence v solves Kv = 0. We also have that v = 0. The first component v1 behaves like v1 (x) = −dρ0 (x) + O(|x|−1−α ) = O(|x|−α ) at infinity. We claim that v2 ∈ L2 , which follows from (7.1). In fact, we have only to note that (u1 , V 1/2 ρ0 ) = (u, V 1/2 ρ˜0 ) = −(V v, ρ˜0 ) = −(V 1/2 Z0 u − dV ρ˜0 , ρ˜0 ) = 0 by the choice of constant d. Thus v ∈ L∞ × L2 becomes either eigenstate or resonance state. This proves the lemma. Remark 7.1 The converse statements of the two lemmas above are also true, although we do not prove it here. The proof is easy. Hence, if |V | 1 is small enough, then (5.6) is fulfilled. Lemma 7.3 (1)
Let 0 < α ≤ 1/2 and set q = (1 + Z0 )−1 q0 ∈ L2 × L2 .
Then q is represented as q = V 1/2 e with e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves Ke = 0 under the condition that e1 = r−α + O(|x|−1−α ), (2)
e2 = O(|x|−1+α ),
|x| → ∞.
(7.3)
Let 1/2 < α < 1 and set q = q0 − (1 + QZ0 Q)−1 QZ0 q0 .
Then q = V 1/2 e for some e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves Ke = 0 under the condition that e1 = O(|x|−α ), Proof.
e2 = −i (λ0 /2π) r−1+α eiθ + O(|x|−2+α ),
|x| → ∞.
(7.4)
(1) If we set e = ρ˜0 − G0 V 1/2 q, then it follows that q = q0 − Z0 q = V 1/2 e.
We assert that e has the desired properties. By definition, e satisfies
Ke = −V 1/2 q + V ρ˜0 − G0 V 1/2 q = V 1/2 (q0 − q − Z0 q) = 0 and obeys (7.3). Since K has neither eigenstates nor resonance states, it is easy to see that e uniquely solves Ke = 0. This proves (1).
102
H. Tamura
Ann. Henri Poincar´e
(2) This is verified in almost the same way as (1). We set r = −(1 + QZ0 Q)−1 QZ0 q0 . Then we have r = −QZ0 r − QZ0 q0 = −Z0 r − Z0 q0 + P Z0 r + P Z0 q0 and hence q is represented as q = q0 + r = V 1/2 e, where e = d1 ρ˜0 − G0 V 1/2 r − G0 V 1/2 q0
(7.5)
with constant d1 = 1 + (Z0 (r + q0 ), q0 )/λ0 . A simple calculation yields Ke = V 1/2 (d1 q0 − (r + q0 ) − Z0 (r + q0 )) = V 1/2 (d1 q0 − q0 − P Z0 (r + q0 )) = 0. It is easy to see that e1 = O(|x|−α ). We look at the second component e2 . If we note that (V 1/2 r, ρ˜0 ) = (Qr, V 1/2 ρ˜0 ) = (Qr, q0 ) = 0, then it follows from (7.1) that the second component of G0 V 1/2 r obeys O(|x|−2+α ). 1/2 The second component −p−1 q0 of the term −G0 V 1/2 q0 behaves like − V 1/2 −p−1 q0 = (2πi)−1 eϕ eih r−1 eiθ λ0 + O(|x|−2+α ). − V
This yields the coefficient −i(λ0 /2π) in (7.4). Thus we can show that e has the desired properties and the lemma is proved. We end the section by proving Lemma 5.1. Proof of Lemma 5.1. (1) Assume that 0 < α ≤ 1/2. Let q = t (q1 , q2 ) = (1 + Z0 )−1 q0 = V 1/2 e be as in Lemma 7.3, where e = ρ˜0 − G0 V 1/2 q. Then the second component e2 = 1/2 q1 behaves like −p−1 − V e2 = iλ2 r−1+α eiθ + O(|x|−2+α ),
|x| → ∞,
for some constant λ2 . We show that λ2 is real. To to this, we compute ((1 + Z0 )−1 q0 , q0 ) = (q, q0 ) = (V e, ρ˜0 ) = −(K0 e, ρ˜0 ) = −(p− e2 , ρ0 ). Recall the representation (2.2) for π− in terms of the polar coordinates. Since p− = π− on {|x| > 2} and since p+ ρ0 = 0, we have ((1 + Z0 )−1 q0 , q0 ) = i lim e−iθ e2 ρ0 ds = −2πλ2 R→∞
|x|=R
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
103
by partial integration. This yields λ2 = −((1 + Z0 )−1 q0 , q0 )/2π
(7.6)
and λ2 is real. This implies that e has all the desired properties. (2) We proceed to proving (2). Assume that 1/2 < α < 1. Let e be defined by (7.5) in the proof of Lemma 7.3. We calculate the constant d1 in (7.5). According to the argument in the proof of Lemma 7.3, we have d1
=
1 + ((r + q0 ), Z0 q0 )/λ0 = 1 + (V 1/2 e, Z0 q0 )/λ0
=
1 + (q, Z0 q0 )/λ0 = 1 + (q0 − (1 + QZ0 Q)−1 QZ0 q0 , Z0 q0 )/λ0 1 + (q0 , Z0 q0 ) − ((1 + QZ0 Q)−1 QZ0 q0 , QZ0 q0 ) /λ0 .
=
Thus d1 is real, and e1 behaves like e1 (x) = d1 r−α + O(|x|−1−α ). The desired solution is obtained as −(2π/λ0 )e, and then λ1 = −(2π/λ0 )d1 is also determined as a real number. This completes the proof.
(7.7)
8 Convergence of resolvent: proof of Theorems 5.1 and 5.2 In this section we prove Theorems 5.1 and 5.2 through a series lemmas. We recall that η0 ∈ C0 (R2 ) has support away from the origin. We also use the notation o2 (1) to denote remainder terms of which the L2 norm obeys the bound o(1) as ε → 0. We start by the following two lemmas. Lemma 8.1 Let ξ± = ξ± (x; E) be defined by (3.1). Then η0 R(E ± i0; H∞ )Jε η = β± η0 (ξ± ⊗ r˜0 ) ηε1−α + Op(ε), where r˜0 (x) = t (r0 (x), 0) with r0 (x) = |x|−α , and β± = ∓ 2α−2 /Γ(1 − α) E 1−α . Lemma 8.2 Let the notation be as in Lemma 8.1. Then η0 R(E ± i0; K0ε )Jε η = β± ((η0 ξ± + o2 (1)) ⊗ ρ˜0 ) ηε1−α + Op(ε) and, in particular, Γε (E ± i0) defined by (6.4) takes the form η0 Γε (E ± i0) = β± ((η0 ξ± + o2 (1)) ⊗ q0 ) ε1−α + Op(ε), where q0 = t (V 1/2 ρ0 , 0) ∈ [L2 ]2 is defined by (6.15).
(8.1)
104
H. Tamura
Ann. Henri Poincar´e
Proof of Lemma 8.1. We prove the lemma for the + case only. For brevity, we write ξ+ = t (ξ1 , ξ2 ), ξ1 = −ieiαπ Hα (Er), ξ2 = H1−α (Er)eiθ . The resolvent R(E+i0; H∞ ) is represented in terms of R(E 2 +i0; LAB ) and R(E 2 + i0; LZ ) by (3.2). We first consider R(E 2 + i0; LZ ). This admits the decomposition ˜0 ⊕ R(E 2 + i0; LZ ) = R ⊕ Rl l =0
˜ 0 and with respect to angular momentum (see (3.4)), and the Green kernels of R Rl are defined by (3.5). Since η0 has support away from the origin, we can take ε ˜ 0 Jε η has so small that |x| > ε|y| when x ∈ supp η0 and y ∈ supp η, and hence η0 R the kernel G(x, y) = ε(ieiαπ /4)η0 (x)Hα (E|x|)J−α (εE|y|)η(y) by a change of variables. This implies that ˜ 0 Jε η = β+ η0 (ξ1 ⊗ r0 )ηε1−α + Op(ε). Eη0 R A similar argument applies to Rl , l = 0, and we obtain η0 Rl Jε η = Op(ε) uniformly in l. Thus we have Eη0 R(E 2 + i0; LZ )Jε η = β+ η0 (ξ1 ⊗ r0 )ηε1−α + Op(ε). Since π+ ξ1 = Eξ2 by (2.14), we make use of this relation to obtain that η0 π+ R(E 2 + i0; LZ )Jε η = β+ η0 (ξ2 ⊗ r0 )ηε1−α + Op(ε). Similarly R(E 2 + i0; LAB ) is shown to obey η0 R(E 2 + i0; LAB )Jε η = Op(ε),
η0 π− R(E 2 + i0; LAB )Jε η = Op(ε).
This proves the lemma. Proof of Lemma 8.2. We again prove the lemma for the + case only. Set ζε (x) = ζ(x/ε),
ζ(x) = 1 − χ0 (x/2),
for the basic cut-off function χ0 (x) with property (4.3). Then we have supp ζε ⊂ {|x| > 2ε},
ζε = 1 on {|x| > 4ε}.
We may assume that ζε η0 = η0 for ε small enough, and we have η0 R(E + i0; K0ε )Jε η
=
η0 R(E + i0; H∞ )ζε Jε η
+
η0 R(E + i0; H∞ )Wε R(E + i0; K0ε )Jε η
(8.2)
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
105
by the resolvent identity, where Wε = H∞ ζε − ζε K0ε . By (4.1), H∞ = K0ε = Dα over |x| > 2ε. If we make use of relations ζε = Jε ζJε∗ and Dα = ε−1 Jε Dα Jε∗ , Wε equals the commutator Wε = [Dα , ζε ] = ε−1 Jε [Dα , ζ]Jε∗ . If we further use the relation Jε∗ R(E + i0; K0ε )Jε = εR(k + i0; K0 ) with k = εE, then we obtain η0 R(E + i0; K0ε )Jε η = η0 R(E + i0; H∞ )Jε ζη + Fε R(k + i0; K0)η,
(8.3)
where Fε = η0 R(E + i0; H∞ )Jε [Dα , ζ]. It follows from Lemma 8.1 that Fε is of the form 0 β+ η0 (ξ1 ⊗ r0 ) [π− , ζ]ε1−α Fε = + Op(ε) 0 β+ η0 (ξ2 ⊗ r0 ) [π− , ζ]ε1−α with ξ+ = t (ξ1 , ξ2 ) as in the proof of Lemma 8.1. Next we evaluate Fε R(k + i0; K0 )η. The operator ηR(k + i0; K0 )η admits the decomposition in Proposition 6.3 for η ∈ C0 (R2 ). We calculate : ρ0 ⊗ ρ˜0 ) ηε1−2α = (o2 (1) ⊗ ρ˜0 ) η, Fε (˜ 1−α Fε G0 η = β+ η0 (ξ+ ⊗ r˜0 )[ π− , ζ]p−1 + Op(ε), − ηε
O(ε2(1−α) )Fε G1 η = (o2 (1) ⊗ ρ˜0 ) η + Op(ε) for G0 and G1 as in Proposition 6.3. We combine these relations with Lemma 8.1. Then η0 (E + i0; K0ε )Jε η = β+ ((η0 ξ+ + o2 (1)) ⊗ r˜1 ) ηε1−α + Op(ε) with r˜1 = t (r1 , 0), where r1 = ζr0 + p−1 + [ζ, π+ ]r0 ,
r0 (x) = |x|−α .
Since ζπ+ r0 = 0, it is easy to see that p+ r1 = 0, and also r1 (x) behaves like r1 (x) = |x|−α + O(|x|−1−α ) at infinity. By uniqueness, this implies that r1 = ρ0 , and the proof is complete. Theorem 5.1 is obtained as an immediate consequence of the lemma below. Lemma 8.3 One has η0 R(E ± i0; K0ε )η0 → η0 R(E ± i0; H∞ )η0 ,
ε → 0,
in norm. Proof. We deal with the + case only. Let ζε be defined by (8.2). Since ζε η0 = η0 for ε small enough, we have η0 R(E + i0; K0ε )η0
=
η0 R(E + i0; H∞ )η0
+
η0 R(E + i0; K0ε )Wε∗ R(E + i0; H∞ )η0
(8.4)
106
H. Tamura
Ann. Henri Poincar´e
by the resolvent identity, where ∗
Wε∗ = (H∞ ζε − ζε K0ε ) = ζε H∞ − K0ε ζε = ε−1 Jε [ζ, Dα ]Jε∗ . We decompose the second term on the right side of (8.4) into the product F1ε F0ε F2ε of three operators, where F1ε = η0 R(E + i0; K0ε)Jε η, F2ε = ηJε∗ R(E + i0; H∞)η0 = (η0 R(E − i0; H∞ )Jε η)
∗
for some η ∈ C0 (R2 ), and F0ε = ε−1 [ζ, Dα ]. By Lemmas 8.1 and 8.2, F1ε and F2ε take the form Op(ε1−α ) Op(ε) Op(ε1−α ) Op(ε1−α ) F1ε = , F2ε = Op(ε1−α ) Op(ε) Op(ε) Op(ε) and F0ε equals
F0ε =
0 ε−1 [ζ, π− ] −1 0 ε [ζ, π+ ]
.
A simple computation yields F1ε F0ε F2ε = Op(ε1−α ). This proves the lemma.
Proof of Theorem 5.1. If we recall that fε and g∞ are represented by (5.2) and (5.3) respectively, then the theorem follows from Lemma 8.3 at once. We proceed to the proof of Theorem 5.2. We first accept the lemma below as proved to complete the proof of the theorem. Lemma 8.4 Assume that (5.6) is fulfilled. Recall that P : [L2 ]2 → [L2 ]2 is the projection defined by (7.2), and set Q = 1−P . Then (1+Xε )−1 obeys the following asymptotic form as ε → 0: (1) If 0 < α < 1/2, then (1 + Xε )−1 = (1 + Z0 )−1 + Op(ε1−2α ). (2) If α = 1/2, then (1 + Xε )−1 = (1 + Z0 )−1 + a (q ⊗ q) + op(ε0 ), where a = −i/(2π + iτ ),
τ = (q, q0 ),
q = (1 + Z0 )−1 q0 .
(8.5)
(3) If 1/2 < α < 1, then
(1 + Xε )−1 = δ+ (ε)P 1 + Op(ε2α−1 ) P
− δ+ (ε)Q (Q + QZ0 Q)−1 QZ0 + Op(ε2α−1 ) + Op(ε2(1−α) ) P
− δ+ (ε)P Z0 Q(Q + QZ0 Q)−1 + Op(ε2α−1 ) + Op(ε2(1−α) ) Q + Q (Q + QZ0 Q)−1 + Op(ε2α−1 ) Q,
where δ+ (ε) = 1/µ+ (ε),
µ+ (ε) = 1 + γ− (k)i2α k 1−2α λ0 ,
k = εE.
(8.6)
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
107
Proof of Theorem 5.2. The proof is based on the relation R(E + i0; Kε ) = R(E + i0; K0ε ) − ε−1 Γε (E + i0)(1 + Xε )−1 Γε (E − i0)∗ derived by (6.3). By Lemma 8.3, we have η0 R(E + i0; K0ε )η0 → η0 R(E + i0; H∞ )η0 ,
ε → 0,
in norm for the first operator on the right side. We analyze the second operator R(ε) = ε−1 η0 Γε (E + i0)(1 + Xε )−1 Γε (E − i0)∗ η0 . The behavior as ε → 0 of R(ε) takes a different form according as 0 < α < 1/2, α = 1/2 or 1/2 < α < 1. (1) Let 0 < α < 1/2. Then it follows from Lemmas 8.2 and 8.4 that R(ε) = O(ε−1 )O(ε2(1−α) ) = O(ε1−2α ), so that η0 R(E + i0; K0ε )η0 → η0 R(E + i0; H∞ )η0 ,
ε → 0,
and hence fε → g∞ . This proves (1). (2) If α = 1/2, then β± = ∓2−3/2 E 1/2 /π 1/2 by (8.1), so that β+ β− = −E/8π. By Lemmas 8.2 and 8.4 again, we have R(ε) → a0 η0 (ξ+ ⊗ ξ− )η0 , where
a0 = β+ β− τ + aτ 2 = −(E/4) (i + 2π/τ )−1 .
Since λ2 = −τ /2π by (7.6), it follows from Proposition 3.1 that η0 R(E + i0; K0ε )η0 → η0 R(E + i0; Hκ )η0 ,
κ = 1/λ2 .
This proves (2). (3) The final case is 1/2 < α < 1. Recall that q0 2 = V 1/2 ρ0 2 = (V ρ0 , ρ0 ) = λ0 by (5.5). Since P q0 = q0 and Qq0 = 0, we have by Lemmas 8.2 and 8.4 that R(ε) behaves like R(ε) = a1 (ε)η0 (ξ+ ⊗ ξ− )η0 + op(ε0 ),
a1 (ε) = ε−1 β+ β− ε2(1−α) δ+ (ε)λ0 .
108
H. Tamura
Ann. Henri Poincar´e
2 We calculate β+ β− = − 2α−2 E 1−α /Γ(1 − α) by (8.1). Since
γ− (k) → −1/γ0 = Γ(α)/ 22(1−α) πΓ(1 − α) in Proposition 6.1, it follows that ε1−2α δ+ (ε) → −γ0 i−2α E 2α−1 /λ0 and hence a1 (ε)
2 → γ0 i−2α E 2α−1 2α−2 E 1−α /Γ(1 − α) =
−(E/4) (π/Γ(α)Γ(1 − α)) i−2α = −(E/4) sin απ/eiαπ .
This, together with Proposition 3.1, implies that fε → g0 , and (3) is obtained. Thus the proof of the theorem is now complete. Proof of Lemma 8.4. By Proposition 6.3, we have 1 + Xε = 1 + Z0 + γ− (k)i2α k 1−2α (q0 ⊗ q0 ) + O(ε2(1−α) )Z1 + Op(ε) for k = εE > 0, where γ− (k) = −1/γ0 + o(1) as ε → 0. (1) Assume that 0 < α < 1/2. If K = K0 + V has neither bound nor resonance state at zero energy, then 1 + Z0 : [L2 ]2 → [L2 ]2 admits a bounded inverse by Lemma 7.1, and hence (1 + Xε )−1 takes the form as in the lemma. (2) If α = 1/2, we have 1 + Xε = 1 + Z0 + (i/2π) (q0 ⊗ q0 ) + op(ε0 ). Let q = (1 + Z0 )−1 q0 and τ = (q, q0 ) be as in (8.5). Then 1 + Xε = (1 + Z0 ) (1 + (i/2π) (q ⊗ q0 )) + op(ε0 ). A simple computation yields −1
(1 + (i/2π) (q ⊗ q0 )) with a as in the lemma. Hence (1 + Xε )
−1
= 1 + a (q ⊗ q0 )
takes the desired form.
(3) We deal with the case 1/2 < α < 1. We employ the method from [12], which has been applied to the analysis on the behavior at low energy of resolvents of Schr¨ odinger operators −∆ + V in two dimensions. We write µ(ε) and δ(ε) = 1/µ(ε) = O(ε2α−1 ),
ε → 0,
for µ+ (ε) and δ+ (ε) respectively. Then 1 + Xε = µ(ε)P + Q + Z0 + O(ε2(1−α) )Z1 + Op(ε)
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
109
by Proposition 6.3. If we use the two simple relations (µ(ε)P + Q)−1 = δ(ε)P + Q,
(1 + QZ0 P )−1 = 1 − QZ0 P,
then 1 + Xε takes the form 1 + Xε = (µ(ε)P + Q) (1 + QZ0 P ) Gε , and hence (1 + Xε )
−1
= G−1 ε (δ(ε)(P − QZ0 P ) + Q) ,
(8.7)
where Gε is represented in the form Gε = 1 + QZ0 Q + δ(ε)(1 − QZ0 )P Z0 + QOp(ε2(1−α) ) + Op(ε). We now set Σ0 = Ran P and Σ = Ran Q. The second factor on the right side of (8.7) has the matrix representation δ(ε)P 0 Σ0 Σ0 δ(ε)(P − QZ0 P ) + Q = : → , (8.8) −δ(ε)QZ0 P Q Σ Σ while Gε = (Gjk (ε))0≤j,k≤1 has the components G00 = P (1 + Op(ε2α−1 ))P,
G01 = P (δ(ε)Z0 + Op(ε))Q,
G10 = Q(−δ(ε)Z0 P Z0 + Op(ε2(1−α) ))P,
G11 = Q(1 + Z0 + Op(ε2α−1 ))Q.
By Lemma 7.2, Q + QZ0 Q : Σ → Σ has a bounded inverse, so that G−1 11 : Σ → Σ exists for ε small enough. If we take account of this fact, then G−1 ε = Eε = (Ejk (ε))0≤j,k≤1 can be calculated as −1 E00 = G00 − G01 G−1 , 11 G10
−1 E01 = − G00 − G01 G−1 G01 G−1 11 G10 11 ,
−1 E10 = − G11 − G10 G−1 G10 G−1 00 G01 00 , Hence (1 + Xε ) −1
(1 + Xε )
−1
−1 E11 = G11 − G10 G−1 . 00 G01
takes the form
= δ(ε)(E00 P − E01 QZ0 P ) + E01 Q + δ(ε)(E10 P − E11 QZ0 P ) + E11 Q
by use of (8.7) and (8.8). Each component Ejk (ε) behaves like : E00 E01
= P (1 + Op(ε2α−1 ))P, = P (−δ(ε)Z0 Q(Q + QZ0 Q)−1 + Op(ε2(2α−1) ) + Op(ε))Q,
E10 E11
= Q(δ(ε)(Q + QZ0 Q)−1 QZ0 P Z0 + Op(ε2(2α−1) ) + Op(ε2(1−α) ))P, = Q((Q + QZ0 Q)−1 + Op(ε2α−1 ))Q.
If we take account of these relations, (1 + Xε ) in the lemma, and the proof is complete.
−1
can be shown to take the form
110
H. Tamura
Ann. Henri Poincar´e
We end the section by making a brief comment on the case when α < 0 and α > 1. Remark 8.1 If we replace the magnetic potential A(x) by −A(x), the argument here extends to the case −1 < α < 0 without any essential change. If |α| > 1, then the magnetic Schr¨ odinger operator L(A, −b) has eigenstates at zero energy besides the resonance state by the Aharonov-Casher theorem [4], so that the norm convergence of resolvent η0 R(E + i0; Kε )η0 can not be expected ([23]). However the strong convergence can be expected, and hence Theorems 5.1 and 5.2 seem to remain true in the case |α| > 1 also.
9 Scattering in the interaction of cosmic string with matter The last section is devoted to proving Theorem 5.3. We begin by representing the ˜ ; E) in question in terms of the resolvent R(E + i0; Kε ) of amplitude f2ε (ω → ω K0ε 0 Vε 0 . Kε = K0ε + Vε = + Vε 0 0 K0ε If we decompose V into the product 1/2 0 V V 0 0 V= = V 0 0 V 1/2 V 1/2
V 1/2 0
= V1 V2 ,
then almost the same argument as used to derive (6.3) enables us to obtain −1
R(E + i0; Kε ) = R(E + i0; K0ε ) − ε−1 Γ1ε (E + i0) (1 + Xε )
Γ2ε (E − i0)∗ , (9.1)
where Xε = V2 R(k + i0; K0 )V1 with k = εE > 0, and Γ1ε (E + i0) = R(E + i0; K0ε )Jε V1 , A direct computation yields Γε (E + i0) 0 Γ1ε = , 0 Γε (E + i0)
Γ2ε (E − i0) = R(E − i0; K0ε )Jε V2 . Γ2ε =
0 Γε (E − i0) Γε (E − i0) 0
where Γε (E ± i0) is defined by (6.4). We further have 0 Xε (1 − Xε2 )−1 −1 Xε = , (1 + Xε ) = Xε 0 −Xε (1 − Xε2 )−1
−Xε (1 − Xε2 )−1 (1 − Xε2 )−1
We divide R(E + i0; Kε ) into the block form R(E + i0; Kε ) = (Rjk (E + i0; Kε ))1≤j,k≤2 , where Rjk (E + i0; Kε ) acts on [L2 ]2 . In particular, we have R21 (E + i0; Kε ) = −ε−1 Γε (E + i0)(1 − Xε2 )−1 Γε (E − i0)∗ .
,
.
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
111
We can represent f2ε (ω → ω ˜ ; E) as ˜ ; E) = (iE/8π)1/2 (R21 (E + i0; Kε )Π+ ψ+ (ω), Π− ψ− (˜ ω )) f2ε (ω → ω by repeating the same argument as in Section 4, and hence we have f2ε = −ε−1 (iE/8π)1/2 (Γε (E + i0)(1 − Xε2 )−1 Γε (E − i0)∗ Π+ ψ+ (ω), Π− ψ− (˜ ω )). (9.2) The argument here is based on this representation. Lemma 9.1 The operator K0 − V has a resonance at zero energy if and only if so does K = K0 + V , and the same statement is also true for an eigenstate. Proof. The lemma is easy to prove. For brevity, we consider the case 0 < α ≤ 1/2 only. A similar argument applies to the case 1/2 < α < 1. Let v+ = (v1 , v2 ) ∈ L2 × L∞ be a resonance state of K0 + V . If we set v− = (v1 , −v2 ), then v− solves (K0 − V ) v− = 0, and it becomes a resonance by Definition 5.1. The case of eigenstate is also shown in the same way. We keep the same notation as in the previous sections. The lemma above implies the existence of bounded inverses (1 − Z0 )−1 : [L2 ]2 → [L2 ]2 and (1 − QZ0 Q)−1 : Σ → Σ. The following lemma is verified in exactly the same way as in the proof of Lemmas 8.4. We skip the proof. Lemma 9.2 If (5.6) is fulfilled, then (1 − Xε )−1 has the following asymptotic form as ε → 0 : (1) If 0 < α < 1/2, then (1 − Xε )−1 = (1 − Z0 )−1 + Op(ε1−2α ). (2) If α = 1/2, then (1 − Xε )−1 = (1 − Z0 )−1 + a (q ⊗ q ) + op(ε0 ), where
a = i/(2π − iτ ),
τ = (q , q0 ),
q = (1 − Z0 )−1 q0 .
(9.3)
(3) If 1/2 < α < 1, then (1 − Xε )−1 = δ− (ε)P 1 + Op(ε2α−1 ) P
+ δ− (ε)Q (Q − QZ0 Q)−1 QZ0 + Op(ε2α−1 ) + Op(ε2(1−α) ) P
+ δ− (ε)P Z0 Q(Q − QZ0 Q)−1 + Op(ε2α−1 ) + Op(ε2(1−α) ) Q + Q (Q − QZ0 Q)−1 + Op(ε2α−1 ) Q, where δ− (ε) = 1/µ− (ε),
µ− (ε) = 1 − γ− (k)i2α k 1−2α λ0 ,
k = εE.
(9.4)
112
H. Tamura
Ann. Henri Poincar´e
Lemma 9.3 Let ξ± be defined by (3.1). Set I+ = (ξ+ , Π− ψ− (˜ ω )) , Then
I+ = −4eiαπ/2 /E,
I− = (ξ− , Π+ ψ+ (ω)) . I− = 4e−iαπ/2 /E.
Proof. We calculate I+ only. A similar computation applies to I− . For brevity, we write ξ+ = ξ = t (ξ1 , ξ2 ),
ψ− = ψ = t (ψ1 , ψ2 ),
χ− (x) = χ0 (x/4) = χ(x).
By (4.3), χ has support in {|x| < 8} and χ = 1 on {|x| < 4}. Since Π− ψ = [Dα , χ]ψ = [Dα − E, χ]ψ = (Dα − E) χψ for x = 0, I+ equals I+ = lim δ→0
|x|>δ
ξ1 (π− χψ2 − Eχψ1 ) + ξ2 (π+ χψ1 − Eχψ2 ) dx.
Note that (Dα − E) ξ = 0, and π+ and π− take the form π+ = eiθ (−i∂r + · · · ) ,
π− = e−iθ (−i∂r + · · · )
by (2.2). We integrate by parts to calculate I+ . Since χ = 1 on {|x| = δ}, we have iθ I+ = −i lim e ξ1 ψ 2 + e−iθ ξ2 ψ 1 ds, ds = δ dθ. δ→0
|x|=δ
By (2.5) and (2.22), the first term in the integrand obeys eiθ ξ1 (x)ψ 2 (x) = O(r1−2α ) + O(1),
and hence lim
δ→0
|x|=δ
r = |x| → 0,
eiθ ξ1 ψ 2 ds = 0,
because 0 < α < 1. On the other hand, the second term behaves like −1 iαπ/2
e−iθ ξ2 ψ 1 = (−i/ sin απ) (1/Γ(α)Γ(1 − α)) (Er/2)
e
(1 + o(1))
as |x| → 0. Since Γ(α)Γ(1 − α) = π/ sin απ by formula, we have −i lim e−iθ ξ2 (x)ψ 1 (x) ds = −4eiαπ/2 /E. δ→0
|x|=δ
This yields the desired value. We now define Iε by Iε = (1 − Xε2 )−1 q0 , q0 = (1 + Xε )−1 q0 , (1 − Xε∗ )−1 q0 .
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
113
Lemma 9.4 Let λ1 and λ2 be as in Lemma 5.1. Then one has the following statements: (1) If 0 < α < 1/2, then Iε = −2πλ2 + o(1),
ε → 0.
(2) If α = 1/2, then Iε = −2πλ2 (1 + λ22 )−1 + o(1),
ε → 0.
(3) If 1/2 < α < 1, then Iε = −γ02 (λ1 /2π)i−4α E 2(2α−1) ε2(2α−1) (1 + o(1)) ,
ε → 0.
We complete the proof Theorem 5.3, accepting this lemma as proved. Throughout the proof of the theorem, we use the notation O2 (ε) to denote remainder terms of which the L2 norm obeys O(ε). Proof of Theorem 5.3. We set η± = η0 ξ± + o2 (1) in Lemma 8.2. The amplitude f2ε (ω → ω ˜ ; E) is represented as (9.2). If we use Lemma 8.2, then a simple computation enables us to evaluate the amplitude as follows : f2ε
= + +
−(iE/8π)1/2 β− β+ (Π+ ψ+ (ω), η− )(η+ , Π− ψ− (˜ ω ))Iε ε1−2α −α 2 −1 O(ε )((1 − Xε ) O2 (ε), q0 ) O(ε−α )((1 − Xε2 )−1 q0 , O2 (ε)) + O(ε).
(9.5)
The leading term comes from the first term on the right side of (9.5). We first consider the case 1/2 < α < 1. If 1/2 < α < 1, then it follows from Lemmas 8.4 and 9.2 that (1 − Xε2 )−1 takes the form (1 − Xε2 )−1
= P Op(ε2(2α−1) )P + Q Op(ε0 )Q + P Op(ε2α−1 )Q + Q Op(ε2α−1 )P
and hence we have |((1 − Xε2 )−1 O2 (ε), q0 )| + |((1 − Xε2 )−1 q0 , O2 (ε))| = O(ε2α ), because Qq0 = 0. This implies that the three remainder terms on the right side of (9.5) obey O(εα ) = O(ε2α−1 )O(ε1−α ) = o(ε2α−1 ). Thus we have f2ε = −(iE/8π)1/2 β− β+ (Π+ ψ+ (ω), η− )(η+ , Π− ψ− (˜ ω ))Iε ε1−2α + o(ε2α−1 ). If we combine Lemmas 9.3 and 9.4, the desired asymptotic form is obtained after a little tedious computation of the leading constant Cα . Next we move to the case 0 < α ≤ 1/2. By Lemmas 8.4 and 9.2 again, (1 − Xε2 )−1 is bounded uniformly in ε, so that |((1 − Xε2 )−1 O2 (ε), q0 )| + |((1 − Xε2 )−1 q0 , O2 (ε))| = O(ε).
114
H. Tamura
Ann. Henri Poincar´e
Then the remainder terms on the right side of (9.5) obey O(ε1−α ) = o(ε1−2α ). Thus we have ω ))Iε ε1−2α + o(ε1−2α ). f2ε = −(iE/8π)1/2 β− β+ (Π+ ψ+ (ω), η− )(η+ , Π− ψ− (˜ We again combine Lemmas 9.3 and 9.4 to obtain the desired asymptotic form for the case 0 < α ≤ 1/2, and the proof is complete. It remains to prove Lemma 9.4. The proof requires two auxiliary lemmas. The first lemma below is proved in the same way as Lemma 7.3. We skip the proof. Lemma 9.5 (1)
If 0 < α ≤ 1/2, then q = (1 − Z0 )−1 q0 = V 1/2 e
for some e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves (K0 − V ) e = 0 under the condition that e1 = r−α + O(|x|−1−α ),
(2)
e2 = O(|x|−1+α ),
|x| → ∞.
If 1/2 < α < 1, then q = q0 + (Q − QZ0 Q)−1 QZ0 q0 = V 1/2 e
for some e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves (K0 − V ) e = 0 under the condition that e1 = O(|x|−α ),
e2 = i(λ0 /2π)r−1+α eiθ + O(|x|−2+α ),
|x| → ∞.
Lemma 9.6 Assume that 0 < α ≤ 1/2. Let τ and τ be the real numbers as in (8.5) and (9.3) respectively. Then one has τ = (q, q0 ) = (1 + Z0 )−1 q0 , q0 = −2πλ2 , τ = (q , q0 ) = (1 − Z0 )−1 q0 , q0 = −2πλ2 . Proof. We write e+ = t (e1 , e2 ) for e in Lemma 7.3 and e− for e in Lemma 9.5. Then it follows by uniqueness that e− is given as e− = t (e1 , −e2 ) for 0 < α ≤ 1/2. We prove the first relation only. The second relation is obtained in a similar way. By Lemma 7.3, τ = (V e, ρ˜0 ) and e solves Ke = (K0 + V ) e = 0. Hence τ = −(K0 e, ρ˜0 ) = −(p− e2 , ρ0 ). Note that p∗− ρ0 = p+ ρ0 = 0, and p− takes the form p− = e−iθ (−i∂r . . .). Hence we have τ = i lim e−iθ e2 ρ0 ds, ds = R dθ, R→∞
|x|=R
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
115
by partial integration. Since ρ0 (x) = r−α + O(r−1−α ) as |x| → ∞ and since e2 (x) = iλ2 eiθ r−1+α + O(r−2+α )
by Lemma 5.1, the desired relation follows from (7.6). t
Proof of Lemma 9.4. We again write e+ = (e1 , e2 ) for e in Lemma 7.3 and e− for e in Lemma 9.5. If 0 < α ≤ 1/2, then e− = t (e1 , −e2 ), and if 1/2 < α < 1, then e− = t (−e1 , e2 ). (1) Assume that 0 < α < 1/2. By Lemmas 8.4 and 9.2, it follows that Iε = ((1 + Z0 )−1 q0 , (1 − Z0 )−1 q0 ) + o(1),
ε → 0.
We further obtain Iε = (V e+ , e− ) + o(1) by Lemmas 7.3 and 9.5. The leading term on the right side equals (V e+ , e− ) = −((K0 − V )e+ , e− )/2,
(9.6)
because (K0 ± V )e± = 0. We assert that ((K0 − V )e+ , e− ) = 4πλ2 ,
(9.7)
which implies that Iε = −2πλ2 + o(1). We shall show (9.7). By definition, ((K0 − V )e+ , e− ) = ((p− e2 − V e1 ), e1 ) − ((p+ e1 − V e2 ), e2 ). We recall that p± = e±iθ (−i∂r . . .) for |x| 1. Hence we have −iθ e e2 e1 − eiθ e1 e2 ds, ((K0 − V )e+ , e− ) = −i lim R→∞
ds = R dθ,
|x|=R
by integration by parts. Thus Lemma 5.1 yields (9.7). (2) Assume that α = 1/2. According to Lemmas 8.4 and 9.2, we have −1
(1 + Xε ) Hence
q0 = (1 + aτ )q + o2 (1),
−1
(1 − Xε∗ )
q0 = (1 + a τ )q + o2 (1).
Iε = (1 + aτ )(1 + a τ )(q, q ) + o(1),
ε → 0.
We repeat the same argument as used in proving (1) to obtain that (q, q ) = ((1 + Z0 )−1 q0 , (1 − Z0 )−1 q0 ) = −2πλ2 . On the other hand, Lemma 9.6, together with (8.5), implies that 1 + aτ = 1 − iτ /(2π + iτ ) = 2π/(2π + iτ ) = (1 − iλ2 )−1 , and similarly 1 + a τ = (1 + iλ2 )−1 (see (9.3)). This proves (2).
116
H. Tamura
Ann. Henri Poincar´e
(3) Let 1/2 < α < 1. (3) is verified in almost the same way as (1). Since Qq0 = 0 and P q0 = q0 , it follows from Lemmas 8.4 and 9.2 that (1 + Xε )−1 q0 ∼ δ+ (ε) q0 − Q(Q + QZ0 Q)−1 QZ0 q0 , (1 − Xε∗ )−1 q0 ∼ δ− (ε) q0 + Q(Q − QZ0 Q)−1 QZ0 q0 , and hence we have Iε
= δ+ (ε)δ− (ε)(V e+ , e− ) + o(ε2(2α−1) ) = −δ+ (ε)δ− (ε)((K0 − V )e+ , e− )/2 + o(ε2(2α−1) )
by Lemmas 7.3 and 9.5. Note that e1 behaves like e1 (x) = −(λ1 λ0 /2π)r−α + O(|x|−1−α ),
|x| → ∞,
for the real number λ1 as in Lemma 5.1. Hence the scalar product ((K0 −V )e+ , e− ) is calculated as −iθ ((K0 − V )e+ , e− ) = −i lim −e e2 e1 + eiθ e1 e2 ds = −λ1 λ20 /π (9.8) R→∞
|x|=R
by use of partial integration. As is seen from (8.6) and (9.4), δ± (ε) = 1/µ± (ε) = ∓(γ0 /λ0 )i−2α E 2α−1 ε2α−1 (1 + o(1)), because γ− (k) → −1/γ0 as k = εE → 0. This, together with (9.8), yields the desired asymptotic form.
References [1] R. Adami and A. Teta, On the Aharonov-Bohm Hamiltonian, Lett. Math. Phys. 43, 43–53 (1998). [2] G.N. Afanasiev, Topological Effects in Quantum Mechanics, Kluwer Academic Publishers (1999). [3] Y. Aharonov and D. Bohm, Significance of electromagnetic potential in the quantum theory, Phys. Rev. 115, 485–491 (1959). [4] Y. Aharonov and A. Casher, Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field, Phys. Rev. A 19, 2461–2462 (1979). [5] S. Agmon and L. H¨ ormander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30, 1–38 (1976). [6] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Text and Monographs in Physics, Springer (1988).
Vol. 5, 2004
Scattering of Dirac Particles by Electromagnetic Fields
117
[7] M.G. Alford, J. March-Russell and F. Wilczek, Enhanced baryon number violation due to cosmic strings, Nucl. Phys. B 328, 140–158 (1989). [8] N. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert space, Vol. 2, Pitman (1981). [9] E. Balslev and B. Helffer, Limiting absorption and resonances for the Dirac operators, Adv. in Appl. Math. 13, 186–215 (1992). [10] M.V. Berry, R.G. Chambers, M.D. Large, C. Upstill and J.C. Walmsley, Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue, Eur. J. Phys. 1, 154–162 (1980). [11] A. Berthier and V. Georgescu, On the point spectrum of Dirac operators, J. Func. Anal. 71, 309–338 (1987). [12] D. Boll´e, F. Gesztesy and C. Danneels, Threshold scattering in two dimensions, Ann. Inst. H. Poincar´e 48, 175–204 (1988). [13] L. Dabrowski and P. Stovicek, Aharonov-Bohm effect with δ-type interaction, J. Math. Phys. 39, 47–62 (1998). [14] Ph. de Sousa Gerbert, Fermions in an Aharonov-Bohm field and cosmic strings, Phys. Rev. D 40, 1346–1349 (1989). [15] Y. Gˆ atel and D. Yafaev, Scattering theory for the Dirac operator with a long-range electromagnetic potential, J. Func. Anal. 184, 136–176 (2001). [16] C.R. Hagen, Aharonov-Bohm scattering amplitude, Phys. Rev. D 41, 2015– 2017 (1990). [17] C.R. Hagen, Aharonov-Bohm scattering amplitude with spin, Phys. Rev. Lett. 64, 503–506 (1990). [18] H.T. Ito, High-energy behavior of the scattering amplitude for a Dirac operator, Publ. RIMS. Kyoto Univ. 31, 1107–1133 (1995). [19] U. Percoco and V.M. Villalba, Aharonov-Bohm effect for a relativistic Dirac electron, Phys. Lett. A 140, 105–107 (1989). [20] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis, Self-Adjointness, Academic Press (1975). [21] Y.A. Sitenko, Self-adjointness of the two-dimensional massless Dirac Hamiltonian and vacuum polarization effects in the background of a singular magnetic vortex, Ann. Phys. 282, 167–217 (2000). [22] H. Tamura, Norm resolvent convergence to magnetic Schr¨ odinger operators with point interactions, Rev. Math. Phys. 13, 465–512 (2001).
118
H. Tamura
Ann. Henri Poincar´e
[23] H. Tamura, Resolvent convergence in norm for Dirac operator with Aharonov-Bohm field, J. Math. Phys. 44, 2967–2993 (2003). [24] B. Thaller, Dirac Equations, Texts and Monographs in Physics, Springer (1992). [25] O. Yamada, On the principle of limiting absorption for the Dirac operator, Publ. RIMS. Kyoto Univ. 8, 557–577 (1972/73). Hideo Tamura Department of Mathematics Okayama University Okayama 700–8530 Japan email:
[email protected] Communicated by Bernard Helffer submitted 05/05/03, accepted 31/07/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 119 – 133 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010119-15 DOI 10.1007/s00023-004-0162-z
Annales Henri Poincar´ e
On the Flux-Across-Surfaces Theorem for Short-Range Potentials Takeyuki Nagao Abstract. The Flux-Across-Surfaces theorem is established for three-dimensional Schr¨ odinger equation with short-range potentials V satisfying the decay condition |∂xα V (x)| ≤ Cα (1 + |x|)−σ , |α| ≤ 2 for some σ > 3. Exceptional cases are treated as well and the required decay index is σ > 5. Explicit conditions for the initial states are found. A stationary representation for the integrated flux is obtained and exploited in the proof. Spatial asymptotic expansion of the resolvent is employed to calculate the limit of the integrated flux at spatial infinity.
1 Introduction Particles moving in the three-dimensional Euclidean space under the influence of the potential are described by the time-dependent Schr¨ odinger equation i∂t u(t, x) = −u(t, x) + V (x)u(t, x). Choosing a scattering state as the initial condition, we consider the scattering for this equation. Namely, u(0, x) = f (x) with f belonging to the absolutely continuous subspace Hac (H) of the Hamiltonian H = − + V . The principal object of this paper is to prove the Flux-Across-Surfaces theorem, which provides us an alternative way of defining the scattering cross section in terms of the integrated flux for the particle. Given a cone K with vertex at the origin, we take a section Σρ = K ∩ {|x| = ρ} and define the integrated flux Iρ (f ) across this surface by the integral ∞ Iρ (f ) = J · n dSdt, (1.1) 0
Σρ
where J = 2Im(u∇u) is the quantum flux density and n is the outward unit normal on the surface Σρ . The integrated flux is interpreted as the expectation of the particle crossing the surface Σρ at sometime in the future and its limit at infinity limρ→∞ Iρ (f ) can be a reasonable substitute for the definition of the scattering cross section as long as the following limit relation is valid with F+ denoting the generalized Fourier transform associated with the Schr¨odinger operator H (cf. [7]). lim Iρ (f ) = F+ f 2L2 (K) .
ρ→∞
(1.2)
120
T. Nagao
Ann. Henri Poincar´e
The above identity is called the Flux-Across-Surfaces theorem, or the FAS theorem for short, and proved under various assumptions, including the free (V = 0) [5, 7], short-range [4, 6, 18], and long-range cases [3]. The usual proofs, however, require rather implicit assumptions on the initial state. As is pointed out by Dell’AntonioPanati [6], it is preferable to establish the FAS theorem under explicit decay and smoothness assumptions on the initial state. In this paper, we shall prove the FAS theorem for short-range potentials and formulate the condition on the initial state in terms of the weighted Sobolev spaces H m,s (R3 ) defined by the norm f H m,s (R3 ) = (1 − )m/2 (1 + |x|2 )s/2 f L2 (R3 ) . The potential V = V (x) is supposed to be a bounded real-valued function on R3 and satisfy the following condition with decay index at least σ > 3. |∂xα V (x)| ≤ C(1 + |x|)−σ ,
|α| ≤ 2.
(1.3)
Due to the possible existence of zero eigenvalues or zero resonance and the resulting singular temporal asymptotics of the propagator, we shall separate the problem into two cases. We factor the potential into the product V = vw with v(x) = |V (x)|1/2 and consider v and w as multiplication operators. The integral operator with kernel 1/(4π|x − y|) is denoted by G0 . By our assumption on the potential, the operator vG0 w is compact in L2 (R3 ). Definition 1.1 The case where 1 + vG0 w has a bounded inverse in L2 (R3 ) is called generic and the case where 1 + vG0 w has no bounded inverse in L2 (R3 ) is called exceptional. The main result of this paper is the following theorem. We state the theorem for the generic case and the statement for the exceptional case is enclosed by the parentheses. Theorem 1.1 Suppose that the potential V satisfies the decay condition (1.3) with σ > 3 (σ > 5). Then, for each f ∈ Hac (H) ∩ H 2,s (R3 ) with s > 3/2 (s > 5/2), we have lim Iρ (f ) = F+ f 2L2 (K) . ρ→∞
In our analysis, we will employ a stationary representation for the integrated flux Iρ (f ) and analyse the integrated flux by investigating the trace of the resolvent onto the sphere. We remark that the integrated flux can be written as ∞ 2Im∂r e−itH f |e−itH f L2 (Σρ ) dt, (1.4) Iρ (f ) = 0
where ∂r = |x|−1 x · ∂x is the radial derivative and f |gL2 (Σρ ) = Σρ f g¯dS with dS the surface measure on the surface Σρ . The trace operator τρ onto the sphere
Vol. 5, 2004
On the Flux-Across-Surfaces Theorem for Short-Range Potentials
121
Sρ = {|x| = ρ} will be omitted in various places. The first step in our approach is to establish the following representation, describing the integrated flux by means of the resolvents. ∞ Iρ (f ) = 2Im−i∂r R+ (λ)f |EH (λ)f L2 (Σρ ) dλ. (1.5) 0
As usual, R± (λ) are the boundary values of the resolvent R(z) = (H − z)−1 at λ > 0 from the upper (+) and the lower (−) half-planes. The spectral density EH is given by the formula EH (λ) = R+ (λ) − R− (λ) /(2πi). (1.6) In the generic case, the expression (1.5) is verified by showing the H-smoothness of the operator τρ Pac , where Pac is the projection onto the absolutely continuous spectrum of H. Because of the singularity of the resolvent at zero energy, τρ Pac may not be H-smooth in the exceptional case and we use a different method to justify the representation in this case. Making use of the spatial asymptotic expansion of the resolvent, we calculate the limit of the integrated flux limρ→∞ Iρ (f ). Specifically, we show that √ (∂r ∓ i λ)R± (λ)f L2 (Sρ ) → 0 as ρ → ∞ and by the asymptotic expansion of the resolvent we obtain the following formulae, relating the limiting behavior of the resolvent at spatial infinity with the generalized Fourier transform. ∞√ 2 lim λ R+ (λ)f L2 (Σρ ) dλ = πF+ f 2L2 (K) , ρ→∞
0
lim
ρ→∞
0
∞
√ λR± (λ)f |R∓ (λ)f L2 (Σρ ) dλ = 0.
The above three limit relations lead to the FAS theorem. Major benefit of our approach is that the above limit relations are valid also for the exceptional case, and the main difference between the generic and exceptional cases resides in establishing the stationary representation for the integrated flux, which permits us to reduce the analysis of the integrated flux into the trace estimates of the resolvent.
2 Generic case In this section, we shall prove the FAS theorem in the generic case. The FAS theorem for the unperturbed system is included in this case. Actually, most of the analysis are devoted to the investigation of the free system and the results for the perturbed system are deduced by simple perturbative arguments.
122
T. Nagao
Ann. Henri Poincar´e
Throughout this section, we suppose that the potential V = V (x) satisfies the following decay condition with a constant σ > 3. |∂xα V (x)| ≤ C(1 + |x|)−σ ,
|α| ≤ 2.
(2.1) 2
3
Further, we suppose that the operator 1+vG0 w is invertible in L (R ) (see Section 1 for the definition of this operator). Under these assumptions, H is selfadjoint with the same domain H 2 (R3 ) as the free Hamiltonian H0 = −, its absolutely continuous spectrum is σac (H) = σ(H0 ) = [0, ∞), H has no positive eigenvalues, and the singular continuous spectrum is absent. It is well known (cf. [1, 10, 15]) that as a function of z, the resolvent R(z) = (H − z)−1 is continuous in each closed quadrant Ω± = {z ∈ C : Rez ≥ 0, ±Imz ≥ 0} in the topology of B(L2s (R3 ); H 2,−s (R3 )) with any weight s > 1, where B(X; Y ) denotes the totality of bounded operators from a Banach space X into another Banach space Y with operator topology and L2s (R3 ) = H 0,s (R3 ). On several occasions we denote by R± (λ) = limε↓0 R(λ ± iε) the boundary values of the resolvent at λ > 0 and this notation is also used for the free resolvent R0 (z) = (H0 − z)−1 . Including these boundary values, the resolvent satisfies the estimate R(z)f H m,−s (R3 ) ≤ Cs z(m−1)/2 f L2s (R3 ) ,
z ∈ Ω± , 0 ≤ m ≤ 2
(2.2)
for s > 1 with z = (1 + |z|2 )1/2 and the resolvent identities R(z) − R0 (z) = −R0 (z)V R(z) = −R(z)V R0 (z)
(2.3)
hold for z ∈ Ω± as identities for operators in B(L2s (R3 ); H 2,−s (R3 )) with s > 1. We remark that the estimate (2.2) is, in general short-range scattering, only valid for z away from the origin (we only need s > 1/2 for this. See, e.g., [1]), but for the generic case we are dealing with, (2.2) is valid with all z ∈ Ω± if s > 1 (cf. [10, 17]). It is easy to see from the estimate (2.2) and the differentiability of the potential that V R(z)f H 2,s (R3 ) ≤ Cs f H 2,s (R3 ) ,
z ∈ Ω± , s > 1.
(2.4)
The weighted resolvent estimates (2.2) and (2.4) enable us to reduce the analysis to the unperturbed problem via the resolvent identity (2.3), and to handle the reduced problem we shall use the integral representation of the free resolvent R0 (z) = (H0 − z)−1 . For z from each Ω± , we denote by K ± (x, y) = K ± (z; x, y) the integral kernel of R0 (z), viz. √ e±i z|x−y| ± ± . (2.5) K (z; x, y)f (y)dy, K (z; x, y) = R0 (z)f (x) = 4π|x − y| R3 First of all, we state and prove the most important lemma in this paper, which handles integral operators similar to the free resolvent and gives us the decay ratio of the trace of such operators at spatial infinity.
Vol. 5, 2004
On the Flux-Across-Surfaces Theorem for Short-Range Potentials
123
kernel K(x, y) of the integral operator T f (x) = Lemma 2.1 Assume that the 3 K(x, y)f (y)dy, x ∈ R satisfies |K(x, y)| ≤ C|x − y|−α for some constant 3 R α with 0 ≤ α < 5/2. Then for any s > 3/2 there exists a constant Cs such that the following inequality holds for any ρ > 1. T f L2(Sρ ) ≤ Cs ρ1−α f L2s(R3 ) . Proof. We define u1 (x), u2 (x) by K(x, y)f (y)dy, u1 (x) = |y||x|/2
and show that they satisfy uj L2 (Sρ ) ≤ Cs ρ1−α f L2s(R3 ) , (j = 1, 2). First, we consider u1 (x). If |y| < |x|/2, then |x − y| > |x|/2, thus |u1 (x)| ≤ C|x|−α f L1 . Taking the square of the both sides and integrating over the spere |x| = ρ, we obtain u1 2L2 (Sρ ) ≤ Cρ2−2α f 2L1 ≤ Cs ρ2−2α f 2L2(R3 ) . Next, we estimate u2 (x). s Choose positive constants µ, ν in such a way that µ + ν = 1, 2αµ < 1, αν < 2. 2
2µ
Since, |x − y|2 = (|x| − |y|) + 2|x||y|(1 − x ˆ · yˆ) ≥ ||x| − |y|| with x ˆ = |x|−1 x, we see that −αν
|u2 (x)| ≤ C|x|
|y|>|x|/2
||x| −
|f (y)| dy. −x ˆ · yˆ)αν/2
|y||αµ (1
By the Schwarz inequality, we have |u2 (x)|2 ≤ C|x|−2αν g1 (x)
y2s |f (y)|2 dy, (1 − x ˆ · yˆ)αν/2
R3
with g1 defined by g1 (x) =
|y|>|x|/2
y−2s dy. ||x| − |y||2αµ (1 − x ˆ · yˆ)αν/2
Using the polar coordinate, g1 can be written as g1 (x) = βg2 (x), with g2 (x) =
∞
|x|/2
τ 2 (1 + τ 2 )−s dτ, ||x| − τ |2αµ
β= |y|=1
(1 − x ˆ · yˆ)−αν/2 dS(y).
By changes of variable, we see that g2 (x) ≤ |x|3−2s−2αµ
∞
1/2
τ 2−2s dτ, |1 − τ |2αµ
1
β = 2π −1
ν
[2|x||y|(1 − x ˆ · yˆ)]
(1 − t)−αν/2 dt
124
T. Nagao
Ann. Henri Poincar´e
and these integrals are finite if 2αµ < 1, αν < 2 and s > 3/2. Note that β is independent of x. Combining the above computations, we have y2s |f (y)|2 2 3−2s−2α |u2 (x)| ≤ C|x| dy ˆ · yˆ)αν/2 R3 (1 − x and integrating the both sides over the sphere |x| = ρ, we see that u2 2L2 (Sρ ) ≤ Cρ5−2s−2α f 2L2s (R3 ) ≤ Cρ2−2α f 2L2s (R3 ) as claimed. Applying Lemma 2.1 for the free resolvent, we obtain the following trace estimate which is uniform in the radius of the sphere. τρ R0 (z)f L2(Sρ ) ≤ Cs f L2s (R3 ) ,
s > 3/2, z ∈ Ω± , ρ > 1.
(2.6)
More detailed results will be obtained later in Proposition 2.2 for the generic case and in Proposition 3.1 for the exceptional case. Now, we are in a position to prove the smoothness of τρ . At this point, we remark that the classical trace estimate τρ f L2 (Sρ ) ≤ Cα f H α (R3 ) ,
α > 1/2,
(2.7)
is valid with a uniform constant Cα , where the uniformity refers to ρ > 1. One can deduce this uniform inequality from the estimate (2.2) applied to the free resolvent and the formula below, relating the trace with the spectral density of the free resolvent via the Fourier transform. |τρ f (x)|2 dS(x) = 2ρEH (ρ2 )fˆ|fˆL2 (R3 ) . 0 |x|=ρ
Lemma 2.2 For large γ > 0: (1) τρ is H0 -smooth. (2) τρ Pac is H-smooth. (3) τρ ∂r (H + γ)−1/2 is H0 -smooth. (4) τρ ∂r (H + γ)−1/2 Pac is H-smooth. Proof. (1) It suffices to show that τρ R0 (z)τρ∗ gL2 (Sρ ) ≤ CρgL2 (Sρ ) ,
z ∈ Ω± ,
(2.8)
where the adjoint τρ∗ is taken with respect to the inner product of L2 (Sρ ). This is equivalent to the following estimate. K ± (x, y)g(y)dS(y) ≤ CρgL2 (Sρ ) , z ∈ Ω± . |y|=ρ 2 L (Sρ )
ˆ · yˆ), so If |x| = |y| = ρ, |x − y|2 = 2ρ2 (1 − x (2.8) follows from Young’s inequality.
|y|=ρ
|K ± (x, y)|dS(y) ≤ ρ. Hence,
Vol. 5, 2004
On the Flux-Across-Surfaces Theorem for Short-Range Potentials
125
(2) In view of (2.8), we only have to prove τρ [R(z) − R0 (z)] τρ∗ gL2 (Sρ ) ≤ CgL2 (Sρ ) ,
z ∈ Ω± , ρ > 1.
(2.9)
This is an easy consequence of the identity R(z) − R0 (z) = −R0 (z)V R0 (z) + R0 (z)V R(z)V R0 (z),
(2.10)
combined with (2.2) and (2.6). (3) Set A = τρ ∂r (H + γ)−1/2 and A0 = τρ ∂r (H0 + γ)−1/2 . It is easy to see from (1) that A0 is H0 -smooth. Since x−s is H0 -smooth if s > 1, we only have to show that A − A0 has the following decomposition. A − A0 = Mρ x−σ ,
sup Mρ B(L2 (R3 );L2 (Sρ )) < ∞, ρ>1
(2.11)
where σ is the decay index for the potential. By our assumption on the potential, B = (H0 + γ) (H + γ)−1/2 − (H0 + γ)−1/2 xσ is a bounded operator in L2 (R3 ), so if we set Mρ = τρ ∂r (H0 + γ)−1 B, we see from the trace theorem (see (2.7) and the remark for it) that Mρ is bounded with norm uniformly bounded by a constant for ρ > 1. (4) We use the same A and A0 as in (3). Notice that (3) is equivalent to |ImAR0 (z)A∗ g|gL2 (Sρ ) | ≤ C(1 + ρ)g2L2 (Sρ ) ,
z ∈ Ω± , ρ > 1.
Hence, from (2.10) and (2.2), it is enough to show that AR0 (z)f L2 (Sρ ) ≤ Cs f L2s(R3 ) ,
s > 3/2, z ∈ Ω± , ρ > 1.
(2.12)
Observe that by (2.6), (2.12) is true if we replace A by A0 . To finish the proof, we use the decomposition (2.11) and the estimate (2.2), which is also valid for the free resolvent. As a consequence of the above lemma, the integrated flux Iρ (f ) is well defined as a function of f ∈ Hac (H) ∩ H 1 (R3 ) and it is continuous in the norm topology of H 1 (R3 ). Indeed, H-smoothness of τρ Pac is equivalent to ∞ τρ e−itH f 2L2 (Sρ ) dt ≤ Cρ f 2L2 (R3 ) , f ∈ Hac (H) −∞
and H-smoothness of τρ ∂r (H + γ)−1/2 Pac is equivalent to ∞ τρ ∂r e−itH f 2L2 (Sρ ) dt ≤ Cρ f 2H 1 (R3 ) , f ∈ Hac (H) ∩ H 1 (R3 ). −∞
126
T. Nagao
Ann. Henri Poincar´e
Hence the integral (1.4) is absolutely convergent and the integrated flux satisfies the estimate |Iρ (f )| ≤ Cρ f 2H 1 (R3 ) . The Plancherel theorem applied to Hilbert space-valued functions gives the identity (1.5). We note that the mappings t → τρ e−itH f and t → τρ ∂r e−itH f are L2 functions with values in L2 (Sρ ) if f ∈ Hac (H)∩H 1 (R3 ) and that the mappings λ → τρ R± (λ)f and λ → τρ ∂r R± (λ)f belong to L2 ((0, ∞); L2 (Sρ )) by the Fourier transform with respect to the variable t. To summarize the above consideration, we have obtained the following Proposition 2.1 The integrated flux Iρ (f ) is a well-defined function of f ∈ Hac (H) ∩H 1 (R3 ) and continuous in the norm topology of H 1 (R3 ). Moreover, the following identity holds if f belongs to Hac (H) ∩ H 1 (R3 ). ∞ Iρ (f ) = 2Im−i∂r R+ (λ)f |EH (λ)f L2 (Σρ ) dλ. (2.13) 0
Although the integrated flux satisfies the estimate |Iρ (f )| ≤ Cρ f 2H 1 (R3 ) , this is not satisfactory for our purpose since we are interested in the limiting behavior of the flux Iρ (f ) as ρ → ∞. It is obvious from the proof of Lemma 2.2 that the constant Cρ can be chosen as a linear function of ρ. Scaling argument shows that this is the correct order as ρ → ∞ for the unperturbed problem. In order to gain uniform control over the integrated flux Iρ (f ), we shall estimate the trace of the resolvent by means of Lemma 2.1. Proposition 2.2 For any s > 3/2 there exists a constant Cs > 0 such that for all z ∈ Ω± , ρ > 1, multi-indices α with |α| ≤ 1, and f ∈ H 2,s (R3 ) we have ∂xα R(z)f L2 (Sρ ) ≤ Cs z|α|/2−1 f H 2,s (R3 ) .
(2.14)
Proof. The proof is reduced to the unperturbed case, since by the resolvent identity (2.3) we have the decomposition R(z)f = R0 (z)gz ,
gz = f − V R(z)f,
(2.15)
and from (2.4), gz H 2,s (R3 ) ≤ Cf H 2,s (R3 ) with the constant C independent of z. We only prove the case for α = 0, since the other case |α| = 1 can be handled similarly. For the sake of simplicity, we suppose z ∈ Ω+ . As we have already seen in (2.6), the estimate (2.14) is true if we remove the decay factor z−1 , so we may assume that |z| > 1. In the polar coordinate, the formula (2.5) can be written as ∞ i√zr re f (x − rω)dS(ω)dr. (2.16) R0 (z)f (x) = 4π 0 |ω|=1 Multiplying the both sides by z and integrating by parts twice, we see that
|∂j ∂k f (y)| |∇f (y)| |zR0 (z)f (x)| ≤ C |f (x)| + dy . dy + 2 |x − y| R3 |x − y| R3 j,k=1,2,3
Vol. 5, 2004
On the Flux-Across-Surfaces Theorem for Short-Range Potentials
127
Taking the · L2 (Sρ ) norm of both sides and applying the trace estimate (2.7) and Lemma 2.1, we have R0 (z)f L2 (Sρ ) ≤ Cz−1 f H 2,s (R3 ) with C independent of ρ > 1 and z. Also as a consequence of Lemma 2.1, we obtain the first one of the three limit relations claimed in the introduction. Proposition 2.3 Assume that f ∈ H 2,s (R3 ) for some constant s > 3/2. Then, we have √ (2.17) lim (∂r ∓ i λ)R± (λ)f L2 (Sρ ) = 0, λ > 0. ρ→∞
Proof. By the same argument as in the proof of Proposition 2.2, we reduce the proof to the free system. We only deal with the case of R0+ . The integral kernel of the operator ∂r R0+ (λ) is given by
√ 1 (x − y) · x . Kr (λ; x, y) = G(x, y) i λ − K + (λ; x, y), G(x, y) = |x − y| |x − y||x| Notice that the function G satisfies the inequalities |y|2µ |y|2 0 ≤ 1 − G(x, y) ≤ min 2, ≤ 21−2µ µ 2|x||x − y| |x| |x − y|µ
(2.18)
for any µ with 0 ≤ µ ≤ 1. We can prove (2.17) by this simple observation. Now, split the kernel Kr into the sum Kr = K1 + K2 with √ K1 (λ; x, y) = i λG(x, y)K + (λ; x, y) and let T be the operator with integral kernel K1 . Since |K2 (λ; x, y)| ≤ C|x− y|−2 , we can neglect K2 in view of Lemma 2.1 and it is enough to show the following identity. √ lim T − i λR0+ (λ) f 2 = 0. (2.19) ρ→∞
L (Sρ )
Choose the constants µ, σ in such a way that σ + 2µ ≤ s (2.20) √ + and decompose the function in (2.19) as [T −i λR0 (λ)]f = Sg, where the integral kernel K3 of the operator S and the function g are defined by √ K3 (λ; x, y) = i λ[1 − G(x, y)]K + (λ; x, y)y−2µ , g(y) = y2µ f (y). 0 < µ < 1,
σ > 3/2,
From (2.18) and (2.20) we have |K3 (λ; x, y)| ≤ C|x|−µ |x − y|−1−µ ,
gL2σ (R3 ) ≤ Cf L2s (R3 ) ,
and thus, by Lemma 2.1, we see that as ρ → ∞, √ [T − i λR0+ (λ)]f L2 (Sρ ) = SgL2(Sρ ) ≤ Cρ−2µ gL2,σ (R3 ) → 0 .
128
T. Nagao
Ann. Henri Poincar´e
Computation of the limit limρ→∞ Iρ (f ) is based on the spatial asymptotic expansion of the resolvent. We shall state and prove a version of asymptotic expansion lemma suitable for our purpose. Some notations from scattering theory are required. Let F0 be the Fourier transform and F0 (λ) be its trace onto the unit sphere, namely, F0 f (ξ) = fˆ(ξ) = (2π)−3/2 e−ixξ f (x)dx, R3
√ F0 (λ)f (ω) = 2−1/2 λ1/4 (F0 f )( λω),
|ω| = 1, λ > 0.
We define the generalized Fourier transform F± by setting (F± f )(ξ) = 21/2 |ξ|−1/2 (F± (ξ 2 )f )(ξ/|ξ|), F± (λ)f = F0 (λ)[1 − V R± (λ)].
(2.21)
It is well known (see [1], for example) that for short-range potentials, F± is well defined as a partial isometry on L2 (R3 ) and its trace F± (λ) is a bounded operator from L2s (R3 ) to L2 (S1 ) for s > 1/2. Actually, the mapping f → F± (λ)f can be extended to a bounded operator from L2 (R3 ) to L2 (0, ∞; L2 (S1 )) satisfying the equality ∞
0
F± (λ)f 2L2 (Σ1 ) dλ = F± f 2L2 (K)
(2.22)
for any cone K and its section Σ1 by the unit sphere. Lemma 2.3 Suppose that f ∈ H 2,s (R3 ) for some constant s > 3/2. Then, the function R± (λ)f (x) can be decomposed as R± (λ)f (x) = u± (f ; x, λ) + E± (f ; x, λ), u± (f ; x, λ) =
√
πλ−1/4
e
(2.23)
√ ±i λr
(F± (λ)f ) (±ω), r where r = |x|, ω = |x|−1 x and the remainder term E± (f, x, λ) satisfies ∞√ lim λE± (f ; ·, λ)2L2 (Sρ ) dλ = 0. ρ→∞
(2.24)
(2.25)
0
Proof. By Proposition 2.2, we have √ λR± (λ)f 2L2 (Sρ ) ≤ Cλ−3/2 f 2H 2,s (R3 ) ,
ρ>1
and by the definition of u± we have the identity √ λu± (f ; ·, λ)2L2 (Sρ ) = πF± (λ)f 2L2 (S1 ) .
(2.26)
Hence the integrand in (2.25) is uniformly bounded by an L1 function and the integral is uniformly bounded by f 2H 2,s (R3 ) . Here, the uniformity refers to ρ > 1.
Vol. 5, 2004
On the Flux-Across-Surfaces Theorem for Short-Range Potentials
129
Therefore, by the dominated convergence, we only have to show the pointwise decay E± (f ; ·, λ)L2 (Sρ ) → 0 as ρ → ∞
(2.27)
for each λ > 0 and smooth rapidly decreasing functions f . In order to prove this, we apply the same reduction argument as in the proof of Proposition 2.2. For the free resolvent, it is well known that the error term E± has the decay E± (f ; x, λ) = O(|x|−2 ) as |x| → ∞, hence (2.27) follows by integration. The remaining two limit relations are direct consequence of Lemma 2.3. Proposition 2.4 Assume that f ∈ H 2,s (R3 ) for some constant s > 3/2. Then the following limit relations hold with K ± = {x ∈ R3 : ±x ∈ K}.
∞
lim
ρ→∞
0
√ ± 2 λ R (λ)f L2 (Σρ ) dλ = πF± f 2L2 (K ± ) ,
lim
ρ→∞
0
∞
√ λR± (λ)f |R∓ (λ)f L2 (Σρ ) dλ = 0.
(2.28)
(2.29)
Proof. By Lemma 2.3, we can replace R± (λ)f by u± (λ) = u± (f ; ·, λ). (2.28) follows from the identity √ λu± (λ)2L2 (Σρ ) = πF± (λ)f (±·)2L2 (Σ1 ) .
(2.30)
(2.29) follows from the identity √ √ λu± (λ)|u∓ (λ)L2 (Σρ ) = πe±2iρ λ F± (λ)f (±·)|F∓ (λ)f (∓·)L2 (Σ1 )
(2.31)
and the Riemann-Lebesgue lemma. Finally, we put all the above propositions together to prove the FAS theorem. Proof of Theorem 1.1. By Proposition 2.2, the integrand in (2.13) is uniformly bounded by Cλ−3/2 f 2H 2,s (R3 ) , which is integrable in λ on the half-line. Therefore, from Proposition 2.3 and the identity (1.6), followed by Proposition 2.4, we conclude that ∞√ λ + R (λ)f 2L2 (Σρ ) dλ = F+ f 2L2 (K) . lim Iρ (f ) = lim ρ→∞ ρ→∞ 0 π
130
T. Nagao
Ann. Henri Poincar´e
3 Exceptional case In this section, we shall prove the FAS theorem in the exceptional case. The decay index for the potential is taken to be σ > 5 and we suppose that 1 + vG0 w has no bounded inverse in L2 (R3 ). The operator τρ Pac may not be H-smooth in this case, since the resolvent R(z) has a singularity at z = 0. Hence, we take another approach to prove the representation (1.5). After establishing this identity, we can proceed exactly in the same way as in the generic case except for the increased weight s > 5/2. According to the work of Jensen-Kato [10], we have the following two propositions, describing the low-energy asymptotics of the resolvent and the long-time asymptotics of the propagator. The topology used here is slightly different from that of [10], but one can prove them in a similar manner as in [10]. Proposition 3.1 For any s > 5/2, we have for small z ∈ Ω+ \ {0}, ˜1 (z). R(z) = −z −1 B−2 − iz −1/2 B−1 + B0 + B
(3.1)
The operators B−2 to B0 belong to B(L2,s (R3 ); H 2,−s (R3 )) and the remainder term satisfies ˜ (3.2) B1 (z) 2,s 3 2,−s 3 = O(z 1/2 ), z → 0. B(L
(R );H
(R ))
Moreover, we have the following expressions for the first two coefficients. P0 denotes the projection onto the zero eigenspace of H, G3 is the operator with integral kernel |x − y|2 /(24π), and ψ is the resonant function for H. B−2 = P0 ,
B−1 = P0 V G3 V P0 − ·|ψψ.
(3.3)
A similar result holds for z ∈ Ω− . Proposition 3.2 For any s > 5/2, we have ˜ e−itH Pac = −(πit)−1/2 B−1 + D(t).
(3.4)
B−1 is the same operator as in Proposition 3.1 and the remainder term satisfies −3/2 ˜ D(t) ), B(L2,s (R3 );H 2,−s (R3 )) = O(t
t → ∞.
(3.5)
As we have seen in the proof of Proposition 2.2, the trace estimate (2.14) is a consequence of the inequality (2.4) and the estimate (2.14) for the free resolvent. From Proposition 3.1, we have for f ∈ Hac (H) ∩ H 2,s (R3 ) with s > 5/2, −1/2
V R(z)f H 2,s (R3 ) ≤ Cs z0
f H 2,s (R3 ) ,
z ∈ Ω± \ {0},
(3.6)
where we denote by z0 the function which equals z for |z| ≤ 1 and equals 1 for |z| > 1. This leads to the following trace estimate for the resolvent.
Vol. 5, 2004
On the Flux-Across-Surfaces Theorem for Short-Range Potentials
131
Proposition 3.3 For any s > 5/2 there exists a constant Cs > 0 such that for all z ∈ Ω± \ {0}, ρ > 1, multi-indices α with |α| ≤ 1, and f ∈ Hac (H) ∩ H 2,s (R3 ) we have −1/2 ∂xα R(z)f L2 (Sρ ) ≤ Cs z0 z|α|/2−1 f H 2,s (R3 ) . (3.7) The stationary representation (1.5) follows from Propositions 3.1 to 3.3 and the spectral decomposition. Proposition 3.4 Let s > 5/2. Then, the integrated flux Iρ (f ) is a well-defined function of f ∈ Hac (H) ∩ H 2,s (R3 ) and continuous in the norm topology of H 2,s (R3 ). Moreover, the following identity holds if f belongs to Hac (H) ∩ H 2,s (R3 ). ∞ 2Im−i∂r R+ (λ)f |EH (λ)f L2 (Σρ ) dλ. (3.8) Iρ (f ) = 0
Proof. First, we show that the integral (1.4) is absolutely convergent. The integrand is bounded by a constant for finite t if f ∈ H 2 (R3 ). Hence, in view of Proposition 3.2, it is enough to show that Im∂r B−1 f |B−1 f L2 (Σρ ) = 0.
(3.9)
This follows from the fact that the resonant function ψ is real-valued and that B−1 f (x) = cψ(x) for a constant c independent of x, if f ∈ Hac (H) ∩ H 2,s (R3 ). Similarly, one can show that the integral (3.8) is absolutely convergent. Now, we prove the identity (3.8). It is easy to see from the spectral decomposition that for f ∈ Hac (H) ∩ H 2,s (R3 ), ∞ −itH −itH 2Im∂r e f |e f L2 (Σρ ) = 2Im∂r e−itH f |EH (λ)f L2 (Σρ ) dλ. (3.10) 0
The integral on the right-hand side is absolutely convergent and its value is bounded by a constant independent of t. Multiplying the both sides of (3.10) by e−εt and integrating over t, we see that ∞ Iρ (f ) = lim 2Im−i∂r R(λ + iε)f |EH (λ)f L2 (Σρ ) dλ. ε↓0
0
From Proposition 3.3 and a similar argument as above about the resonant function, we see that the integrand is bounded by an L1 function of λ independent of ε. Therefore, (3.8) follows by the dominated convergence. One can use the resolvent estimate (3.7) to prove Propositions 2.3 and 2.4 for the exceptional case. All weights in the statements should be replaced by s > 5/2 and the condition f ∈ H 2,s (R3 ) in Proposition 2.4 by f ∈ Hac (H) ∩ H 2,s (R3 ), but the proofs remain unchanged. The proof of Theorem 1.1 is exactly the same as in the generic case.
132
T. Nagao
Ann. Henri Poincar´e
Acknowledgments The author is grateful to Prof. Kenji Yajima for helpful suggestions and constructive remarks, and also to Prof. Detlef D¨ urr for interesting discussion on Bohmian mechanics. Part of this work was done during the author’s stay at Mathematisches Institut Ludwig-Maximilians-Universit¨ at M¨ unchen as a program student and the author expresses his sincere gratitude to Prof. Heinz Siedentop for the hospitality.
References [1] S. Agmon, Spectral properties of Schr¨ odinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa, Ser. IV 2, 151–218 (1975). [2] S. Agmon and L. H¨ ormander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math. 30, 1–38 (1976). [3] W.O. Amrein and D.B. Pearson, Flux and scattering into cones for long range and singular potentials, Journal of Physics A 30, 5361–5379 (1997). [4] W.O. Amrein and J.L. Zuleta, Flux and scattering into cones in potential scattering, Helv. Phys. Acta 70, 1–15 (1997). [5] M. Daumer, D. D¨ urr, S. Goldstein, and N. Zangh`ı, On the flux-across-surfaces theorem, Lett. Math. Phys. 38, 103–116 (1996). [6] G. Dell’Antonio and G. Panati, Zero-energy resonances and the flux-acrosssurfaces theorem, submitted for publication, preprint mp arc 01-402 (2001). [7] D. D¨ urr and S. Teufel, On the role of flux in scattering theory, Stochastic processes, physics and geometry: new interplays, (Leipzig, 1999), CMS Conf. Proc. 28, 123–137 (2000). [8] Y. Gˆ atel and D. Yafaev, On solutions of the Schr¨ odinger equation with radiation conditions at infinity: the long-range case. Ann. Inst. Fourier 49, 1581– 1602 (1999). [9] H. Isozaki, Asymptotic properties of solutions to 3-particle Schr¨ odinger equations, Comm. Math. Phys. 222, 371–413 (2001). [10] A. Jensen and T. Kato, Spectral properties of Schr¨ odinger operators and time-decay of the wave functions, Duke Math. J. 46, 583–611 (1979). [11] A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys. 13, 717–754 (2001). [12] T. Kato, Wave operators and similarity for some non-selfadjoint operators, Mathematische Annalen 162, 258–279 (1965/1966).
Vol. 5, 2004
On the Flux-Across-Surfaces Theorem for Short-Range Potentials
133
[13] C.E. Kenig, A. Ruiz, and C.D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55, 329–347 (1987). [14] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1, 481–496 (1989). [15] S.T. Kuroda, An introduction to scattering theory, Aarhus Universitet Matematisk Institut, Aarhus (1978). [16] M. Murata, Asymptotic expansions in time for solutions of Schr¨ odinger-type equations, J. Funct. Anal. 49, 10–56 (1982). [17] I. Rodnianski and W. Schlag, Time decay for solutions of Schr¨ odinger equations with rough and time-dependent potentials, preprint, mp arc 01-369 (2001). [18] S. Teufel, D. D¨ urr, and K. M¨ unch-Berndl, The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs, J. Math. Phys. 40, 1901–1922 (1999). [19] D. Yafaev, On solutions of the Schr¨ odinger equation with radiation conditions at infinity, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–1990), Adv. Soviet Math. 7, 179– 204 (1991). odinger operators. [20] K. Yajima, The W k,p -continuity of wave operators for Schr¨ III. Even-dimensional cases m ≥ 4, J. Math. Sci. Univ. Tokyo 2, 311–346 (1995). Takeyuki Nagao Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 Japan email:
[email protected] Communicated by Gian Michele Graf Submitted 01/03/03, accepted 30/05/03
Ann. Henri Poincar´e 5 (2004) 135 – 168 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010135-34 DOI 10.1007/s00023-004-0163-y
Annales Henri Poincar´ e
On the Plancherel Formula for the (Discrete) Laplacian in a Weyl Chamber with Repulsive Boundary Conditions at the Walls∗ J.F. van Diejen
Abstract. It is known from early work of Gaudin that the quantum system of n Bosonic particles on the line with a pairwise delta-potential interaction admits a natural generalization in terms of the root systems of simple Lie algebras. The corresponding quantum eigenvalue problem amounts to that of a Laplacian in a convex cone, the Weyl chamber, with linear homogeneous boundary conditions at the walls. In this paper we study a discretization of this eigenvalue problem, which is characterized by a discrete Laplacian on the dominant cone of the weight lattice endowed with suitable linear homogeneous conditions at the boundary. The eigenfunctions of this discrete model are computed by the Bethe Ansatz method. The orthogonality and completeness of the resulting Bethe wave functions (i.e., the Plancherel formula) turn out to follow from an elementary computation performed by Macdonald in his study of the zonal spherical functions on p-adic simple Lie groups. Through a continuum limit, the Plancherel formula for the ordinary Laplacian in the Weyl chamber with linear homogeneous boundary conditions is recovered. Throughout this paper we restrict ourselves to the case of repulsive boundary conditions.
1 Introduction It is well known that the quantum eigenvalue problem for n Bosons on the line that interact pairwise through a delta-potential can be solved by the Bethe Ansatz method [LL, Mc, BZ, Y1, Y2, G1, G2, O]. From a physical point of view, this manybody system describes the n-particle sector of the quantized nonlinear Schr¨ odinger field theory (i.e., the quantum NLS). For an overview of the literature concerning both the mathematical and physical aspects of this model we refer to the collections [M, G4, KBI]. The Hamiltonian of the n-particle system in question is given by the Schr¨ odinger operator H = −∆ + g
δ(xj − xk ),
(1.1)
1≤j=k≤n ∗ Work supported in part by the Fondo Nacional de Desarrollo Cient´ ıfico y Tecnol´ ogico (FONDECYT) Grant # 1010217 and the Programa Formas Cuadr´ aticas of the Universidad de Talca.
136
J.F. van Diejen
Ann. Henri Poincar´e
where x1 , . . . , xn denote the position variables, ∆ = ∂x21 + · · · + ∂x2n , δ(·) refers to the delta distribution, and g represents a real coupling parameter determining the strength of the interaction. For g > 0 the pairwise interaction is repulsive and for g < 0 it is attractive. Mathematically, the eigenvalue problem for H (1.1) amounts to that of a free Laplacian −∆ with jump conditions on the normal derivative of the (continuous) wave function at the hyperplanes xj = xk , 1 ≤ j < k ≤ n. (Specifically, the jump of the normal derivative of the wave functions at the hyperplanes should be 2g times the value of the wave function.) By exploiting the permutation and translational symmetry, the eigenvalue problem at issue can be reduced to the form −∆ψ = ξ2 ψ
(where ξ2 := ξ12 + · · · + ξn2 ),
(1.2a)
for a domain of wave functions ψ = ψ(x; ξ) := ψ(x1 , . . . , xn ; ξ1 , . . . , ξn ) supported in the closure of the fundamental convex cone C = {x ∈ Rn | x1 > x2 > · · · > xn , x1 + · · · + xn = 0},
(1.2b)
and subject to linear homogeneous boundary conditions at the walls given by (1.2c) (∂xj − ∂xj+1 − g)ψ xj =xj+1 = 0, j = 1, . . . , n − 1. (Here the variable ξ ∈ RN plays the role of the spectral parameter.) The idea of the Bethe Ansatz method is now to construct the solution of this eigenvalue problem as a permutation-invariant linear combination of plane waves, with suitable coefficients such that the boundary conditions at the walls are satisfied. An important problem is the question of the orthogonality and completeness of the Bethe eigenfunctions in a Hilbert space setting. This problem is commonly referred to in the mathematically oriented literature as the Plancherel Problem. For the repulsive regime g > 0, the spectrum of the Hamiltonian is absolutely continuous; the corresponding Plancherel formula was demonstrated formally by Gaudin [G1, G2, G4]. For the attractive regime g < 0, one has both discrete and continuous spectrum; in this case the Plancherel problem was solved by Oxford [O] by building on work of Yang [Y1] and exploiting ideas from an analysis of a related Plancherel problem for the infinite volume XXX isotropic Heisenberg spin chain by Babbitt and Thomas [T, BT]. Thanks to a fundamental observation by Gaudin, it is known that the nBoson system with delta-potential interaction admits natural generalization in terms of the root systems of simple Lie algebras [G3, G4]. From this perspective, the original n-particle model with pairwise interaction corresponds to a root system of type An−1 (i.e., the Lie algebra sl(n; C)). Other classical root systems appear when restricting the particles to a half-line or by distributing them symmetrically around the origin. It turns out that the eigenfunctions of these generalized deltapotential models related to root systems can again be constructed with the Bethe Ansatz method [G3, GS, G, G4]. The corresponding Plancherel formula was proven
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
137
recently by Heckman and Opdam, who considered both the repulsive and the attractive regime [HO]. The aim of the present paper is to study a discrete version of the spectral problem for the Laplace operator with a delta-potential on root systems. Throughout the paper, we restrict ourselves to the repulsive case. More specifically, we consider a discrete Laplacian acting on lattice functions with support in the dominant cone of the weight lattice of the root system, subject to suitable repulsive boundary conditions. We construct the eigenfunctions of this discrete Laplacian through the Bethe Ansatz method. The resulting eigenfunctions turn out to correspond to (the parameter deformations of) the zonal spherical functions on p-adic Lie groups studied by Macdonald [M1, M3]. In particular, the Plancherel problem reduces in this discrete setting to an elementary calculation already carried out by Macdonald to prove the orthogonality of the spherical functions in question with respect to the Plancherel measure. Finally, we perform a continuum limit as the lattice spacing tends to zero and recover the repulsive case of the Plancherel formula for the Laplace operator with a delta-potential on root systems from [HO]. In this limit the discrete Laplacian converges in the strong resolvent topology to the Laplacian of the continuous model. To give rigorous meaning to our continuum limit in a Hilbert space sense, we employ techniques developed by Ruijsenaars in his study of the continuum limit of the infinite isotropic Heisenberg spin chain [R]. The material is organized as follows. Section 2 serves to prepare some basic definitions and notations from the theory of root systems that are needed to formulate the results. Section 3 recalls the eigenfunctions and exhibits the Plancherel formula for the Laplacian in the Weyl chamber with repulsive boundary conditions at the walls. Section 4 is devoted to the discretization of this Laplacian. Specifically, we introduce our discrete Laplacian on the dominant cone of the weight lattice endowed with linear homogeneous conditions at the boundary. The eigenfunctions of the discrete Laplacian are constructed with the Bethe Ansatz method and the Plancherel problem for the repulsive case is resolved by connecting to Macdonald’s theory of zonal spherical functions on p-adic Lie groups. In Section 5 it is shown how – by passing to the continuum limit – the eigenfunctions and the Plancherel formula for the (continuous) Laplacian in Section 3 can be recovered from the eigenfunctions and the Plancherel formula for the discrete Laplacian in Section 4. A few technical points concerning the proof of the Plancherel inversion formula in the continuous situation have been isolated in Appendix A. Furthermore, some crucial results due to Macdonald – which constitute the backbone of the proof for the Plancherel formula in the discrete situation – have been outlined in Appendix B at the end of the paper.
2 Preliminaries on root systems Throughout the paper we will make extensive use of the language of root systems. For a thorough treatment of the concepts and theory surrounding root systems the
138
J.F. van Diejen
Ann. Henri Poincar´e
reader is referred to the standard texts [B, H1, H2, K]. Here we restrict ourselves to recalling just the bare minimum of definitions, notations, and properties needed for our purposes. This section is probably best skipped at first reading and referred back to as needed.
2.1
Roots
Let E be a real finite-dimensional Euclidean vector space with the inner product denoted by ·, ·. For a nonzero vector α ∈ E, the action of the orthogonal reflection rα : E → E in the hyperplane through the origin perpendicular to it is given explicitly by (x ∈ E), (2.1) rα (x) = x − x, α∨ α where α∨ := 2α/α, α. By definition, a (crystallographic) root system is a nonempty subset R ⊂ E \ {0} satisfying the properties (i) rα (R) = R, ∀α ∈ R (reflection invariance), (ii) α, β ∨ ∈ Z, ∀α, β ∈ R (integrality).
(2.2)
A vector in R is referred to as a root. The roots generate an Abelian group Q := SpanZ (R) called the root lattice of R. The dimension of Q is called the rank of the root system. Here we will always assume that the ambient Euclidean space E is chosen minimal in the sense that dim(E) is equal to the rank of the root system (i.e., SpanR (R) = E). If one fixes a choice of normal vector generically, in the sense that the hyperplane through the origin perpendicular to it does not intersect R, then the hyperplane in question divides the root system in two subsets of equal size called the positive and negative roots: R = R+ ∪ R−
with
R− = −R+ .
(2.3)
The positive roots determine a nonnegative semigroup Q+ := SpanN (R+ ) of the root lattice. A positive root α is called simple if α − β ∈ R+ for any β ∈ R+ . Let us denote the simple roots by α1 , . . . , αN . These simple roots form a basis for Q and Q+ , i.e., Q = Z α1 ⊕ · · · ⊕ Z αN
and Q+ = N α1 ⊕ · · · ⊕ N αN .
(2.4)
(Hence, the number of simple roots N is equal to the rank of the root system.) It means that starting from the origin we can reach any vector in the root lattice Q by successive addition or subtraction of simple roots. One defines the height of a vector κ ∈ Q as (2.5) ht(κ) := κ, ρ∨ , with ρ∨ := α∈R+ α∨ /2. In the basis of simple roots the height reads ht(κ) = ht(k1 α1 + · · · + kN αN ) = k1 +· · ·+kN . In particular, for κ ∈ Q+ the height function ht(·) counts the number
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
139
of simple roots in κ. The (unique) positive root α0 such that ht(α) ≤ ht(α0 ) for all α ∈ R+ is called the maximal root of R. A root system is said to be irreducible if it cannot be decomposed as a direct orthogonal sum of two (smaller) root systems. Furthermore, a root system is called reduced if any half-line starting from the origin contains at most one single root α ∈ R. (This amounts to the condition that for any α ∈ R the multiple kα is a root if and only if k = 1 or k = −1.)
2.2
The Weyl group
The group W ⊂ O(E; R) generated by all reflections rα , α ∈ R is called the Weyl group of R. The first defining property (i) of a root system states that R is invariant with respect to the action of the Weyl group; the second defining property (ii) guarantees moreover that the root lattice Q is also invariant with respect to this action. In the case of an irreducible reduced root system, the action of the Weyl group splits up R in at most two orbits. More specifically, there are two possible situations: (i) either all roots have the same length, in which case the action of W on R is transitive, or (ii) the roots come in two different sizes, in which case R splits up in an orbit Rs consisting of the short roots and an orbit Rl consisting of the long roots. The reflections in the simple roots rj := rαj , j = 1, . . . , N are referred to as the simple reflections. They form a minimal set of generators for the Weyl group W . In other words, any Weyl group element w ∈ W can be decomposed (non-uniquely) in terms of simple reflections w = rj1 rj2 · · · rj
(2.6)
(with the indices j1 , . . . , j ∈ {1, . . . , N } not necessarily distinct). The number is referred to as the length of the decomposition. If, for given w ∈ W , the length is minimal then the corresponding decomposition is called reduced. An important property of Weyl groups (used frequently in our analysis below) is that a group element w ∈ W admits a reduced decomposition ending in the simple reflection rj (i.e., with rj in (2.6) equal to rj ) if and only if w(αj ) ∈ R− . Let us – for R both irreducible and reduced – define the following (length) functions on W (w)
:= |{α ∈ R+ | w(α) ∈ R− }|,
(2.7a)
s (w) l (w)
− := |{α ∈ R+ s | w(α) ∈ Rs }|, − := |{α ∈ R+ l | w(α) ∈ Rl }|,
(2.7b) (2.7c)
± ± ± where R± s := Rs ∩ R , Rl := Rl ∩ R , and | · | refers to the cardinality of the set in question. Clearly (w) = s (w) + l (w). (If all roots have the same length, then by convention Rs := R and Rl := ∅, so s (w) = (w) and l (w) = 0.) It turns out that the numbers (w), s (w) and l (w) count, respectively, the number of
140
J.F. van Diejen
Ann. Henri Poincar´e
simple reflections, the number of short simple reflections and the number of long simple reflections that appear in a reduced decomposition (2.6) of w into simple reflections. For later use, it will be convenient to split up the height function ht(·) (2.5) as a sum of partial height functions as well hts (κ) := κ, ρ∨ s , htl (κ) :=
κ, ρ∨ l ,
with
ρ∨ s :=
with
ρ∨ l
:=
α∨ /2,
α∈R+ s
α∈R+ l
∨
α /2.
(2.8a) (2.8b)
For κ = k1 α1 + · · · + kN αN , this gives hts (κ) =
1≤j≤N αj short
kj ,
htl (κ) =
kj ,
(2.9)
1≤j≤N, αj long
which for κ ∈ Q+ amounts to a count of, respectively, the number of short and long simple roots in κ.
2.3
Weights
The weight lattice P and its nonnegative dominant cone P + are the duals of the root lattice Q and its nonnegative semigroup Q+ , i.e., P P+
:= :=
{λ ∈ E | λ, α∨ ∈ Z, ∀α ∈ R}, {λ ∈ E | λ, α∨ ∈ N, ∀α ∈ R+ }.
(2.10a) (2.10b)
One has that Q ⊂ P but Q+ ⊂ P + (unless N = 1). A vector in P is called a weight. Furthermore, a weight in P + is called a dominant weight. The special dominant weights ω1 , . . . , ωN that are related to the simple roots via the duality ωj , α∨ k = δj,k are referred to as the fundamental weights. These fundamental weights form a basis for P and P + , i.e., P = Z ω1 ⊕ · · · ⊕ Z ωN
and P + = N ω1 ⊕ · · · ⊕ N ωN .
(2.11)
λ µ ⇐⇒ λ − µ ∈ Q+
(2.12)
The following definition ∀λ, µ ∈ P :
endows the weight lattice with a natural partial order. This partial order is usually referred to as the dominance order. The cone of dominant weights P + constitutes a fundamental domain for P with respect to the action of the Weyl group, in the sense that for any µ ∈ P the Weyl orbit W (µ) := {w(µ) | w ∈ W } (2.13)
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
141
intersects the dominant cone P + precisely once. For µ ∈ P, one defines wµ ∈ W as the unique shortest Weyl group element such that wµ (µ) ∈ P + .
(2.14)
The group element wµ admits a reduced decomposition ending in rj if and only if µ, α∨ j < 0 (i.e., if and only if the hyperplane perpendicular to αj separates µ and wµ (µ)). It is instructive to reformulate this criterion in terms of the partial order in Eq. (2.12): the group element wµ admits a reduced decomposition ending in rj if and only if rj (µ) µ. In particular, it means that any dominant weight λ is maximal in its Weyl orbit W (λ), i.e., ∀λ ∈ P + :
λ w(λ),
∀w ∈ W.
(2.15)
The stabilizer of a weight λ ∈ P is defined as Wλ := {w ∈ W | w(λ) = λ}.
(2.16)
The stabilizer Wλ is a subgroup of the Weyl group W that is generated by the simple reflections rj such that rj (λ) = λ.
3 Laplacian on the Weyl chamber In this section we review the solution of the spectral problem for the Laplacian in a Weyl chamber with repulsive boundary conditions at the walls and formulate the associated Plancherel theorem. Note. From now on it will always be assumed that our root system R is both irreducible and reduced. A helpful list of all irreducible root systems and their concrete properties can be found in Bourbaki’s tables [B].
3.1
Eigenvalue problem
The Weyl chamber is the open convex cone C = {x ∈ E | x, α > 0, ∀α ∈ R+ }.
(3.1)
It is bounded by the walls Cj = {x ∈ E | x, αj = 0 and x, α > 0, ∀α ∈ R+ \ {αj } }
(3.2)
perpendicular to the simple roots αj , j = 1, . . . , N . Let gs , gl be two generic (possibly complex) parameters and let us set gs if α ∈ Rs , gα := (3.3) gl if α ∈ Rl .
142
J.F. van Diejen
Ann. Henri Poincar´e
The generalization of the eigenvalue problem in Eqs. (1.2a)–(1.2c) to the case of an arbitrary root system R is given by −∇2x ψ(x; ξ) = ξ2 ψ(x; ξ),
x, ξ ∈ C,
with linear homogeneous boundary conditions at the walls of the form ∇x ψ, αj − gαj ψ = 0, j = 1, . . . , N. x∈Cj
(3.4a)
(3.4b)
Here ∇2x and ∇x denote the Laplacian and gradient on E, respectively, and ξ := ξ, ξ. Theorem 3.1 (Eigenfunction). The wave function Ψ0 (x; ξ) =
α, ξ − igα w eix,ξw , α, ξ w +
(3.5)
w∈W α∈R
with ξ w := w(ξ), solves the eigenvalue problem in Eqs. (3.4a), (3.4b). Theorem 3.1 is due to Gaudin, who constructed the wave function in question by means of the Bethe Ansatz Method [G3, G4]. It is clear that the linear combination of plane waves Ψ0 (x; ξ) (3.5) solves the eigenvalue equation in Eq. (3.4a), since −∇2x eix,ξw = ξ w , ξw eix,ξw = ξ, ξeix,ξw . To infer that the boundary conditions in Eq. (3.4b) are also satisfied it suffices to perform a small computation based on the action of the directional derivative on plane waves: ∇x eix,ξ , αj = iαj , ξeix,ξ . Specifically, the following sequence of elementary manipulations reveals that for x ∈ Cj ∇x Ψ0 , αj α, ξ − igα w iαj , ξ w eix,ξw = α, ξ w w∈W α∈R+ α, ξ − ig α w eix,ξw = (gαj + iαj , ξ w ) α, ξ w + w∈W
(i)
=
gαj
α∈R α=αj
α, ξ − igα w eix,ξw α, ξ w +
w∈W α∈R α=αj (ii)
=
gαj
1−
w∈W
=
gαj
α∈R α=αj
α, ξ − igα w eix,ξw α, ξ w +
w∈W α∈R
=
igαj α, ξ w − igα ix,ξ w e αj , ξw α, ξ w +
gαj Ψ0 .
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
143
In Steps (i) and (ii) one exploits the fact that the expressions under consideration are symmetrized with respect to the action of the Weyl group W . Notice in this connection that the relevant terms on the third and fifth line are built of factors that are (skew-)symmetric with respect to the simple reflection rj . Indeed, we have the skew-symmetry αj , rj (ξ w ) = −αj , ξ w (as rj (αj ) = −αj ) and the symmetries α, ξ − ig α, ξ − igα α w w rj = α, ξ α, ξw w + + α∈R α=αj
α∈R α=αj
(as the simple reflection rj permutes the positive roots other than αj ) and x, rj (ξ w ) = x, ξ w (as x ∈ Cj so rj (x) = x). When symmetrizing with respect to the action of the Weyl group the skew-symmetric parts involving αj , ξ w thus drop out.
3.2
Continuous Plancherel formula
Note. From now on we will restrict attention to the repulsive case of nonnegative parameters gs , gl (and hence gα ). Let H0 = L2 (C, dx) be the Hilbert space of square-integrable functions on the Weyl chamber equipped with the standard inner product (f, g)H0 = f (x)g(x)dx (∀f, g ∈ H0 ), (3.6) C
ˆ 0 (ξ) dξ) be the Hilbert space of square-integrable ˆ 0 = L2 (C, (2π)−N ∆ and let H functions on the Weyl chamber with respect to the positive weight function ˆ 0 (ξ) = ∆
igα −1 , 1+ α, ξ
(3.7)
α∈R
equipped with the normalized inner product 1 ˆ 0 (ξ)dξ (fˆ, gˆ)Hˆ 0 = g (ξ) ∆ fˆ(ξ)ˆ (2π)N C
ˆ 0 ). (∀fˆ, gˆ ∈ H
(3.8)
For f ∈ H0 , we now define the eigenfunction transform fˆ0 = F0 f by means of the pairing f (x)Ψ0 (x; ξ)dx, (3.9a) fˆ0 (ξ) = (F0 f )(ξ) := C
ˆ 0 we define the adjoint with Ψ0 (x; ξ) given by Eq. (3.5). Reversely, for fˆ ∈ H ˆ ˆ eigenfunction transform f0 = F0 f as 1 ˆ ˆ ˆ 0 (ξ)dξ. (3.9b) f0 (x) = (F0 f )(x) := fˆ(ξ)Ψ0 (x; ξ) ∆ (2π)N C
144
J.F. van Diejen
Ann. Henri Poincar´e
(So formally: fˆ0 (ξ) = (f, Ψ0 (ξ))H0 and f0 (x) = (fˆ, Ψ0 (x))Hˆ 0 .) For gs , gl = 0, the transformations F0 and Fˆ0 amount to the Fourier and inverse Fourier transformation on C, respectively. The following theorem generalizes this state of affairs to the case of general nonnegative parameter values gs , gl . Theorem 3.2 (Continuous Plancherel Formula). The eigenfunction transform F0 ˆ 0 , with (3.9a) constitutes a unitary Hilbert space isomorphism between H0 and H ˆ the inverse transformation given by F0 (3.9b), i.e., ˆ
F 0 ,F 0 ˆ 0, H0 ←→ H
Fˆ0 F0 = IH0 ,
F0 Fˆ0 = IHˆ 0 .
(3.10)
Below we will show that Theorem 3.2 arises as a degeneration of a more elementary “polynomial” Plancherel formula for a discretization of the eigenvalue problem in Eqs. (3.4a), (3.4b). The Plancherel formula of Theorem 3.2 is in agreement with the previous results due to Gaudin [G1, G2, G4] (for root systems of type A) and HeckmanOpdam [HO] (for arbitrary root systems), who showed that the transformation F0 ˆ 0 with left-inverse Fˆ0 (3.9b). The idea (3.9a) constitutes an isometry of H0 into H of the proof for this inversion formula outlined by Heckman and Opdam [HO] is far from elementary: it hinges on a deep result due to Peetre concerning the abstract characterization of differential operators as support preserving linear operators acting on spaces of smooth functions [P1, P2]. For the reader’s convenience, we have included a completely elementary proof of this inversion formula in Appendix A at the end of the paper. It follows from Theorems 3.1 and 3.2 that the operator −∇2x in the Weyl chamber, with repulsive boundary conditions at the walls of the form in Eq. (3.4b), determines a unique selfadjoint extension in H0 given by the pullback of the mulˆ 0 with respect to the eigenfunction tiplication operator fˆ(ξ) → ξ2 fˆ(ξ) in H transformation F0 . From this observation the following corollary is immediate. Corollary 3.3 (Spectrum and Self-adjointness). The operator −∇2x in the Weyl chamber C (3.1), with repulsive boundary conditions of the form in Eq. (3.4b) at the walls, is essentially selfadjoint in H0 and (its closure) has a purely absolutely continuous spectrum filling the nonnegative real axis.
4 Discrete Laplacian on the cone of dominant weights In this section we introduce a discrete Laplacian with repulsive boundary conditions on the cone of dominant weights and solve the associated spectral problem.
4.1
Action of the discrete Laplacian and boundary conditions
A nonzero dominant weight σ is called minuscule if σ, α∨ ≤ 1 for all α ∈ R+ and it is called quasi-minuscule if σ, α∨ ≤ 1 for all α ∈ R+ \ {σ} (without it being
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
145
minuscule). The number of minuscule weights is one less than the index |P/Q|, which means that there are no minuscule weights iff the root lattice Q fills the whole weight lattice P. A quasi-minuscule weight, on the other hand, always exists and it is moreover unique. Specifically, it is given by the dominant weight σ such that σ ∨ is the maximal root of the dual root system R∨ := {α∨ | α ∈ R}. We will now associate to a (quasi-)minuscule weight σ a discrete Laplace operator Lσ acting on the space C(P + ) of complex functions over the cone of dominant weights P + (2.10b). Definition (Discrete Laplacian). Let σ ∈ P + be (quasi-)minuscule and let ts , tl denote two generic complex parameters. The action of the discrete Laplace operator Lσ : C(P + ) −→ C(P + ) on an arbitrary lattice function ψ ∈ C(P + ) is defined as ψλ+ν (λ ∈ P + ), (4.1a) Lσ ψλ = ν∈W (σ) +
where for λ+ν ∈ P \P the value of ψλ+ν is determined by the boundary condition ψλ+ν
(wλ+ν )
= tss (wλ+ν ) tl l
ψwλ+ν (λ+ν)
s (ν) −htl (ν) + θλ+ν t−ht tl (1 s
−
(4.1b)
t−1 s )ψλ ,
with θµ := ht(wµ (µ) − µ) − (wµ ).
(4.1c)
To appreciate the structure of the above boundary conditions the following proposition is helpful. It exploits the decomposition of Weyl group elements in terms of simple reflections to disentangle the boundary conditions completely in terms of simple reflection relations. In this alternative characterization it turns out to be convenient to work with W invariant parameters tα , α ∈ R upon setting (cf. Eq. (3.3)) ts if α ∈ Rs , tα := (4.2) tl if α ∈ Rl . Proposition 4.1 (Boundary Reflection Relations). Let λ be a dominant weight and let σ ∈ P + be (quasi-)minuscule. Then the boundary conditions in Eqs. (4.1b), (4.1c) are equivalent to the requirement that ∀ν ∈ W (σ) such that λ + ν ∈ P \ P + , and for all simple roots αj such that λ + ν, α∨ j < 0, the following reflection relations are satisfied if ht (rj (λ + ν) − λ − ν) = 1, (I) tαj ψrj (λ+ν) ψλ+ν = tαj ψrj (λ+ν) + (tαj − 1)ψλ if ht (rj (λ + ν) − λ − ν) = 2, (II) or equivalently tαj ψλ+ν+αj ψλ+ν = tαj ψλ tαj ψλ+αj + (tαj − 1)ψλ
∨ if λ, α∨ j = 0 and ν, αj = −1, (Ia ) ∨ if λ, α∨ j = 1 and ν, αj = −2, (Ib ) ∨ if λ, α∨ j = 0 and ν, αj = −2. (II )
146
J.F. van Diejen
Ann. Henri Poincar´e
Proof. Let us first check that the reflection relations in (I), (II) and in (Ia ), (Ib ), ∨ (II ) are indeed equivalent. Since λ, α∨ j ≥ 0 (as λ is dominant) and ν, αj ≥ −2 with equality holding only when ν = −αj (as ν ∈ W (σ) with σ (quasi-)minuscule), the condition λ + ν, α∨ j < 0 breaks up in the three cases (Ia ), (Ib ) or (II ). It is readily verified that Cases (Ia ) and (Ib ) correspond to (I) and Case (II ) corresponds to (II). Indeed, we have: rj (λ + ν) = λ + ν + αj in Case (Ia ), ν = −αj and rj (λ + ν) = λ in Case (Ib ), and ν = −αj and rj (λ + ν) = λ + αj in Case (II ). Hence, the corresponding reflection relations match in each case. (Notice also that for σ minuscule we are always in Case (Ia ) (i.e., (I)); the Cases (Ib ) or (II ) (i.e., (II)) can only occur when σ is quasi-minuscule.) Next we verify that the conditions in the proposition amount to the boundary conditions in Eqs. (4.1b), (4.1c). To this end we exploit the decomposition in simple reflections to perform induction on the length of wλ+ν , starting from the trivial induction base (wλ+ν ) = 0. (Notice in this connection that (wλ+ν ) = 0 implies that λ + ν is dominant, which agrees with the fact that formally the r.h.s. of Eq. (4.1b) reduces in this situation to ψλ+ν .) For (wλ+ν ) > 0, there exists a simple reflection rj such that wλ+ν = wrj (λ+ν) rj with (wrj (λ+ν) ) = (wλ+ν ) − 1. One furthermore has that rj (λ + ν) λ + ν, i.e., λ + ν, α∨ j < 0. We thus fall in either one of the three cases (Ia ), (Ib ) or (II ), which are to be analyzed separately below. (Ia ) In this situation rj (λ+ν) = λ+rj (ν), which implies that wλ+ν = wλ+rj (ν) rj . By applying first the reflection relation in (Ia ) and then the induction hypothesis we get ψλ+ν
= tαj ψλ+rj (ν) s (wλ+rj (ν) ) l (wλ+rj (ν) ) tl ψwλ+rj (ν) (λ+rj (ν))
= tαj ts
−htl (rj (ν))
s (rj (ν)) + tαj t−ht tl s
(wλ+ν )
= tss (wλ+ν ) tl l +
θλ+rj (ν) (1 − t−1 s )ψλ
ψwλ+ν (λ+ν)
s (ν) −htl (ν) t−ht tl θλ+ν (1 s
− t−1 s )ψλ ,
which coincides with the expression on the r.h.s. of Eq. (4.1b). (Ib ) In this situation rj (λ + ν) = λ, which implies that wλ+ν = rj and ν = −αj ∈ Rs . We get from the reflection relation in (Ib ) ψλ+ν = tαj ψλ = ts ψλ , which corresponds to Eq. (4.1b) with s (wλ+ν ) = s (rj ) = 1, l (wλ+ν ) = l (rj ) = 0, hts (ν) = hts (−αj ) = −1, htl (ν) = htl (−αj ) = 0, and θλ+ν = θλ−αj = ht(αj ) − (rj ) = 0. (II ) In this situation rj (λ + ν) = λ + rj (ν) = λ + αj , which implies that wλ+ν = wλ+αj rj and ν = −αj ∈ Rs . By applying first the reflection relation in (II )
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
147
and then the induction hypothesis we get ψλ+ν
=
tαj ψλ+αj + (tαj − 1)ψλ
=
tαj ts
=
s (wλ+αj ) l (wλ+αj ) tl ψwλ+αj (λ+αj )
+ (tαj − 1)ψλ
(w ) tss (wλ+ν ) tl l λ+ν ψwλ+ν (λ+ν) s (ν) −htl (ν) + t−ht tl θλ+ν (1 − t−1 s s )ψλ ,
which coincides with the expression on the r.h.s. of Eq. (4.1b). Since all three cases lead to the boundary condition in Eqs. (4.1b), (4.1c), this completes the induction step (and therewith the proof of the proposition). Remark. (i) It is clear from the proof of the proposition that for σ minuscule θλ+ν = ht(wλ+ν (λ + ν) − λ − ν) − (wλ+ν ) = 0 (as we are always in Case (Ia )). Hence, in this situation the boundary condition in Eq. (4.1b) reduces to (wλ+ν )
ψλ+ν = tss (wλ+ν ) tl l
ψwλ+ν (λ+ν) .
(4.4)
Remark. (ii) The parameters ts and tl play the role of coupling parameters that determine the strength of the boundary conditions. There are two special extremal situations worth singling out. For ts , tl → 1 the action of Lσ reduces to that of a (n) free Laplacian Lσ : C(P + ) → C(P + ) with Neumann type boundary conditions:
L(n) σ ψλ =
ψwλ+ν (λ+ν) .
(4.5a)
ν∈W (σ) (d)
For ts , tl → 0 the action of Lσ reduces to that of a free Laplacian Lσ : C(P + ) → C(P + ) with Dirichlet type boundary conditions: L(d) σ ψλ = −Nσ (λ)ψλ +
ψλ+ν ,
(4.5b)
ν∈W (σ) λ+ν∈P +
where Nσ (λ) = 0 if σ is minuscule and Nσ (λ) is equal to the number of short (0) simple roots perpendicular to λ if σ is quasi-minuscule. Let Lσ : C(P) −→ C(P) denote the free Laplacian on the (full) weight lattice characterized by the action L(0) σ ψλ =
ψλ+ν
(λ ∈ P).
(4.6)
ν∈W (σ) (n)
(d)
(0)
The operators Lσ (4.5a) and Lσ (4.5b) can be seen as the reduction of Lσ (4.6) to the space of W invariant functions and W skew-invariant functions on P, respectively (upon restriction to the fundamental domain P + ).
148
4.2
J.F. van Diejen
Ann. Henri Poincar´e
Bethe Ansatz solution
Let Q∨ denote the dual root lattice SpanZ (R∨ ) and let us write TR for the torus E/(2πQ∨ ). It is evident that the plane waves ψλ (ξ) = exp(iλ, ξ), ξ ∈ (0) TR constitute a (Fourier) basis of eigenfunctions for the free Laplacian Lσ : C(P) → C(P) in Eq. (4.6). The corresponding eigenvalues are given by Eσ (ξ) = ν∈W (σ) exp(iν, ξ), ξ ∈ TR . Following the Bethe Ansatz method, we will now construct suitable linear combination of plane waves that satisfies the boundary conditions in Eqs. (4.1b), (4.1c). By construction, the resulting wave function will thus constitute an eigenfunction of our Laplacian Lσ (4.1a)–(4.1c). Specifically, as Bethe Ansatz wave function we take an arbitrary Weyl group invariant linear combination of plane waves of the form Ψλ (ξ) =
1 (−1)w C(ξ w )eiρ+λ,ξw , δ(ξ)
(4.7a)
w∈W
with (−1)w := det(w) = (−1)(w) , and δ(ξ) = (eiα,ξ/2 − e−iα,ξ/2 ),
(4.7b)
α∈R+
ρ
=
1 α. 2 +
(4.7c)
α∈R
(This wave function is W invariant in the sense that Ψλ (ξ w ) = Ψλ (ξ).) The following theorem matches the coefficients so as to meet the boundary conditions (4.1b), (4.1c). Theorem 4.2 (Bethe Wave Function). Let Lσ : C(P + ) → C(P + ) be the discrete Laplacian with boundary conditions defined in Eqs. (4.1a)–(4.1c). Then the Bethe Ansatz wave function Ψλ (ξ) (4.7a)–(4.7c) solves the eigenvalue equation Lσ ψ(ξ) = Eσ (ξ)ψ(ξ) with Eσ (ξ) = exp(iν, ξ), (4.8) ν∈W (σ)
provided that C(ξ) =
(1 − tα e−iα,ξ )
(4.9)
α∈R+
(or a scalar multiple thereof ). Proof. It suffices to check that the Bethe Ansatz wave function Ψλ (ξ) (4.7a)–(4.7c) satisfies the boundary conditions (4.1b), (4.1c), provided that C(ξ) is of the form stated by the theorem. To this end we compute C(ξ) from the boundary reflection relations of Proposition 4.1. Indeed, upon assuming the technical conditions detailed in the proposition, substitution of the Bethe Ansatz wave function in
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
149
the boundary reflection relations readily leads to the stated expression for the coefficients C(ξ). Specifically, we find in Case (I) that equating Ψλ+ν (ξ)
1 (−1)w C(ξ w )eiρ+λ+ν,ξw δ(ξ) w∈W wρ+λ+ν −1 = (−1) δ (ξ) (−1)wµ eiµ,ξ =
µ∈W (ρ+λ+ν)
(−1)w C(ξ w )
w∈Wρ+λ+ν
to tαj Ψrj (λ+ν) (ξ) = = =
tαj Ψλ+ν+αj (ξ) tαj (−1)w C(ξ w )eiαj ,ξw eiρ+λ+ν,ξw δ(ξ) w∈W tαj (−1)wρ+λ+ν δ −1 (ξ) (−1)wµ eiµ,ξ ×
µ∈W (ρ+λ+ν)
(−1) C(ξw )eiαj ,ξw w
w∈Wρ+λ+ν
leads to the relation
(−1)w C(ξw ) = tαj
w∈Wρ+λ+ν
(−1)w C(ξ w )eiαj ,ξw .
w∈Wρ+λ+ν
Because rj stabilizes ρ + λ + ν (i.e., rj ∈ Wρ+λ+ν ), the latter relation can be rewritten as (−1)w [C(ξw ) − C(rj (ξ w ))] w∈Wρ+λ+ν w −1 (αj )∈R+
= tαj
(−1)w [C(ξw )eiαj ,ξw − C(rj (ξ w ))e−iαj ,ξw ].
w∈Wρ+λ+ν w −1 (αj )∈R+
By induction on the cardinality of the stabilizer Wρ+λ+ν , starting from the smallest value |Wρ+λ+ν | = 2 (as it contains as subgroup the cyclic group of order 2 generated by rj ), one concludes that C(ξ) − C(rj (ξ)) = tαj [C(ξ)eiαj ,ξ − C(rj (ξ))e−iαj ,ξ ], or equivalently (assuming C(ξ) is nontrivial in the sense that it does not vanish identically) 1 − tαj e−iαj ,ξ C(ξ) = . (4.10) C(rj (ξ)) 1 − tαj eiαj ,ξ
150
J.F. van Diejen
Ann. Henri Poincar´e
From varying λ and ν, it is clear that the reflection relation in Eq. (4.10) should hold for all simple reflections rj , j = 1, . . . , N . We thus conclude that C(ξ) must in fact be of the form (1 − tα e−iα,ξ ), C(ξ) = c0 (ξ) α∈R+
where c0 (ξ) denotes an arbitrary W invariant overall factor (i.e., c0 (ξ w ) = c0 (ξ), ∀w ∈ W ). It remains to check that this choice for the coefficient C(ξ) is also compatible with the boundary conditions of Case (II). This follows from an analysis similar to that of Case (I). Indeed, we see by equating Ψλ+ν (ξ) = = =
Ψλ−αj (ξ) 1 (−1)w C(ξ w )eiρ+λ−αj ,ξw δ(ξ) w∈W 1 (−1)w [C(ξw )e−iαj ,ξw − C(rj (ξ w ))]eiρ+λ,ξw δ(ξ) w∈W w −1 (αj )∈R+
to the sum of tαj Ψrj (λ+ν) (ξ) = tαj Ψλ+αj tαj = (−1)w C(ξ w )eiρ+λ+αj ,ξw δ(ξ) w∈W tαj = (−1)w [C(ξ w )eiαj ,ξw − C(rj (ξ w ))e−2iαj ,ξw ]eiρ+λ,ξw δ(ξ) w∈W w −1 (αj )∈R+
and (tαj − 1)Ψλ (ξ) =
(tαj − 1) (−1)w C(ξ w )eiρ+λ,ξw δ(ξ) w∈W
=
(tαj − 1) δ(ξ)
(−1)w [C(ξw ) − C(rj (ξ w ))e−iαj ,ξw ]eiρ+λ,ξw ,
w∈W w −1 (αj )∈R+
that it is sufficient to require that C(ξ)e−iαj ,ξ − C(rj (ξ)) =
tαj [C(ξ)eiαj ,ξ − C(rj (ξ))e−2iαj ,ξ ] +(tαj − 1)[C(ξ) − C(rj (ξ))e−iαj ,ξ ].
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
151
Upon collecting the factors of C(ξ) and C(rj (ξ)) the latter relation can be rewritten as C(ξ)(1 − tαj eiαj ,ξ )(1 − e−iαj ,ξ ) = C(rj (ξ))(1 − tαj e−iαj ,ξ )(1 − e−iαj ,ξ ), which leads us back to the reflection relation in Eq. (4.10).
4.3
Discrete Plancherel formula
Next we will address the question of the orthogonality and completeness of the Bethe wave functions given by Theorem 4.2. Note. From now on it will always be assumed that the parameters lie in the (repulsive) domain 0 < ts , tl < 1 (unless explicitly stated otherwise). It is straightforward to rewrite the Bethe wave function of Theorem 4.2 as Ψλ (ξ) =
1 − tα e−iα,ξw eiλ,ξw . −iα,ξw 1 − e + w∈W
(4.11)
α∈R
From this expression it can be seen that the functions Ψλ (ξ) amount in essence to (a parameter deformation of) the zonal spherical functions on p-adic Lie groups computed by Macdonald [M1, M3]. (To make this connection with Macdonald’s work more explicit, the interested reader might want to compare Ψλ (ξ) (4.11) with [M1, Theorem (4.2.1)] and [M3, Eq. (10.1)].) The upshot is that the solution of the Plancherel problem is now a direct consequence of Macdonald’s orthogonality relations for the (deformed) spherical functions in question. To describe the result, some notation is needed. Let H = 2 (P + , ∆λ ) denote the Hilbert space of complex functions on the cone of dominant weights P + (2.10b) that are square-summable with respect to the positive weight function ∆λ =
hts (α) htl (α) tl hts (α) htl (α) tα ts tl
1 − ts
+
α∈R λ,α∨ =0
1−
(4.12)
(λ ∈ P + ). The standard inner product on H is given by (f, g)H =
fλ gλ ∆λ
(∀f, g ∈ H).
(4.13)
λ∈P +
ˆ = L2 (A, |W |−1 Vol(A)−1 ∆(ξ)dξ) ˆ Furthermore, let H denote the Hilbert space of complex functions on the Weyl alcove A = {ξ ∈ E | 0 < ξ, α < 2π, ∀α ∈ R+ }
(4.14)
152
J.F. van Diejen
Ann. Henri Poincar´e
that are square-integrable with respect to the positive weight function ˆ ∆(ξ) = =
|δ(ξ)|2 C(ξ)C(−ξ) 1 − eiα,ξ α∈R
(4.15a) (4.15b)
1 − tα eiα,ξ
ˆ reads (ξ ∈ A). The normalized inner product on H 1 ˆ ˆ fˆ(ξ)ˆ g (ξ)∆(ξ)dξ (f , gˆ)Hˆ = |W |Vol(A) A
ˆ (∀fˆ, gˆ ∈ H).
(4.16)
To the Bethe wave function Ψλ (ξ) in Theorem 4.2, we associate the integral ˆ given by the Fourier pairing transformation F : H → H fˆ(ξ) = =
(F f )(ξ) := (f, Ψ(ξ))H fλ Ψλ (ξ)∆λ
(4.17a) (4.17b)
λ∈P +
ˆ → H given by the Fourier (∀f ∈ H), and the adjoint integral transformation Fˆ : H pairing fλ
= =
(Fˆ fˆ)λ := (fˆ, Ψλ )Hˆ 1 ˆ fˆ(ξ)Ψλ (ξ)∆(ξ)dξ |W | Vol(A) A
(4.18a) (4.18b)
ˆ (∀fˆ ∈ H). Theorem 4.3 (Discrete Plancherel Formula). The eigenfunction transformation ˆ in Eqs. (4.17a), (4.17b) constitutes a unitary Hilbert space isomorF :H →H phism with the inverse transformation F −1 given by the adjoint eigenfunction ˆ → H in Eqs. (4.18a), (4.18b). transformation Fˆ : H Proof. The theorem is a direct consequence of the fact that the zonal spherical ˆ satisfying the orthogonality functions Ψλ (ξ), λ ∈ P + form an orthogonal basis of H relations [M1, M3] if λ = µ, ∆−1 λ (Ψλ , Ψµ )Hˆ = (4.19) 0 if λ = µ. To keep our treatment selfcontained, a brief outline of Macdonald’s proof of these orthogonality relations is isolated in Appendix B at the end of the paper. ˆσ : H ˆ→H ˆ be the multiplication operator Let E (Eˆσ fˆ)(ξ) := Eσ (ξ)fˆ(ξ),
(4.20)
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
153
where Eσ (ξ) stands for the eigenvalue in Eq. (4.8). It is a straightforward consequence of Theorem 4.2 and Theorem 4.3 that the discrete Laplace operator Lσ ˆσ with respect to the (4.1a)–(4.1c) is the pullback of the multiplication operator E transformation F . From this observation the following two corollaries are immediate. Corollary 4.4 (Spectrum). The discrete Laplace operator Lσ (4.1a)–(4.1c) has a purely absolutely continuous spectrum in the Hilbert space H given by the compact ¯ ⊂ C. set Spec(Lσ ) = {Eσ (ξ) | ξ ∈ A} Remark. (i) The complex conjugate of the function Eσ (ξ) is given by E−w0 (σ) (ξ), where w0 denotes the longest element in the Weyl group W (i.e., the unique element w0 ∈ W such that w0 (A) = −A). Corollary 4.5 (Adjoint). The adjoint of Lσ in H is given by L−w0 (σ) . In particular, this means that Lσ is selfadjoint if and only if w0 (σ) = −σ. This is for instance the case when σ is quasi-minuscule or when w0 = −Id. If w0 (σ) = −σ, then one can make the eigenvalue problem selfadjoint by passing to the operator (Lσ + L−w0 (σ) ). Remark. (ii) It is instructive to detail the contents of the Plancherel formula in Theorem 3.2 for the special parameter limit cases, corresponding to the free discrete Laplacians over P + with Neumann and Dirichlet type boundary conditions, exhibited in the second remark at the end of Section 4.1. For ts , tl → 1, the Bethe wave function Ψλ (ξ) (4.11) reduces to the monomial symmetric function (n) Ψλ (ξ) = |Wλ | mλ (ξ), with mλ (ξ) = eiµ,ξ , (4.21) µ∈W (λ)
ˆ and ∆(ξ) = 1, ∆λ = 1/|Wλ | (cf. Eq. (B.4) of Appendix B below). The eigenfunction transform F amounts in this situation to the W invariant part of the Fourier transformation on 2 (P): fˆ(ξ) = fλ mλ (ξ), (4.22a) λ∈P +
with the inversion formula fλ =
1 |W (λ)| Vol(A)
A
fˆ(ξ)mλ (ξ)dξ.
(4.22b)
For ts , tl → 0 the Bethe wave function Ψλ (ξ) (4.11) reduces to the Weyl character 1 (d) (−1)w eiρ+λ,ξw , (4.23) Ψλ (ξ) = χλ (ξ), with χλ (ξ) = δ(ξ) w∈W
2
ˆ and ∆(ξ) = |δ(ξ)| , ∆λ = 1. The eigenfunction transform F amounts in this situation to the W skew-invariant part of the Fourier transformation on 2 (P): fλ χλ (ξ), (4.24a) fˆ(ξ) = λ∈P +
154
J.F. van Diejen
with the inversion formula 1 fλ = |W | Vol(A)
A
fˆ(ξ)χλ (ξ)|δ(ξ)|2 dξ.
Ann. Henri Poincar´e
(4.24b)
5 The continuum limit In this section it is shown that the discrete Plancherel formula of Theorem 4.3 degenerates to continuous Plancherel formula of Theorem 3.2 in the continuum limit as the lattice distance tends to zero. The discrete Laplacian from Eqs. (4.1a)–(4.1c) degenerates in this limit – upon symmetrization and rescaling – in the strong resolvent sense to the continuous Laplacian from Eqs. (3.4a)–(3.4b). The approach in this section is inspired by Ruijsenaars’ proof of the fact that the ground-state representation of the infinite isotropic Heisenberg spin chain converges in the continuum limit to a free Boson gas [R]. Note. Throughout this section we will employ the parametrization tα = e−gα with > 0 and with gα positive (and W invariant, cf Eq. (3.3)).
5.1
Embedding
To perform the continuum limit, we embed the Hilbert space H = 2 (P + , ∆λ ) from 2 Section 4 isometrically in the Hilbert space H0 = L (C, dx) with standard inner product (f, g)H0 = C f (x)g(x)dx. This is done via the one-parameter family of embeddings J : H → H0 , > 0, which associate to a lattice function f ∈ H the staircase function f ∈ H0 defined by −N/2 1/2 ∆[−1 x] f[−1 x] . f (x) = (J f )(x) := det(P)
(5.1a)
Here det(P) := det(ω1 , . . . , ωN ) and for x ∈ C ∨ + [x] := [x, α∨ 1 ] ω1 + · · · + [x, αN ] ωN ∈ P
(where [x] denotes the integral part of a nonnegative real number x obtained via ˆ ˆ = L2 (A, |W |−1 Vol(A)−1 ∆(ξ)dξ) truncation). Similarly, the dual Hilbert space H 2 −N ∨ −1 ˆ = L (A, (2π) det(Q ) ∆(ξ)dξ) from Section 4 is embedded isometrically in ˆ 0 = L2 (C, (2π)−N dξ) with normalized inner product (fˆ, gˆ) ˆ = the Hilbert space H H0 1 ˆ g (ξ)dξ. This is done via the one-parameter family of embeddings (2π)N C f (ξ)ˆ ˆ →H ˆ 0 , > 0, which associate to a function fˆ ∈ H ˆ the rescaled function Jˆ : H ˆ 0 defined by fˆ ∈ H ˆ 1/2 (ξ)fˆ(ξ). fˆ (ξ) = (Jˆ fˆ)(ξ) := ∆ det(Q∨ ) N/2
∨ −N Here det(Q∨ ) := det(α∨ |W |Vol(A). 1 , . . . , αN ) = (2π)
(5.1b)
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
155
ˆ := Jˆ (H) ˆ ⊂H ˆ 0 . The eigenfunction transLet H := J (H) ⊂ H0 and let H ˆ ˆ ˆ (4.18a), (4.18b) form F : H → H (4.17a), (4.17b) and its inverse F : H → H ˆ lift under the embeddings J and J , respectively, to a corresponding transform ˆ and its inverse Fˆ : H ˆ → H of the form F : H → H fˆ (ξ) = (F f )(ξ) := f (x)Φ[−1 x] (ξ)dx (5.2a) C
and f (x) = (Fˆ fˆ)(x) :=
1 (2π)N
C
fˆ(ξ)Φ[−1 x] (ξ)dξ,
(5.2b)
with a kernel given by 1/2
()
ˆ 1/2 (ξ)χ (ξ)Ψ[−1 x] (ξ) Φ[−1 x] (ξ) = ∆[−1 x] ∆ A −1 1/2 () S1/2 (ξ w )ei[ x],ξw , = ∆[−1 x] χA (ξ)
(5.3a)
w∈W
where S (ξ) =
sin (α, ξ − igα ) 2 , sin 2 (α, ξ + igα ) +
(5.3b)
α∈R ()
and with χA (ξ) denoting the characteristic function of the rescaled alcove −1 A ⊂ C. It follows from Theorem 4.3 that the transform F and its inverse Fˆ define a unitary Hilbert space isomorphism between the closed subspaces H ⊂ H0 and ˆ 0 . In other words, we have the following commutative diagram of unitary ˆ ⊂ H H Hilbert space isomorphisms H J H
F ,Fˆ
←→ ˆ F ,F
←→
ˆ H Jˆ ˆ H
Fˆ F = IH
F Fˆ = IHˆ
Fˆ F = IH
F Fˆ = IHˆ
.
(5.4)
ˆ : H ˆ0 → H ˆ on the closed The orthogonal projections Π : H0 → H and Π ˆ ⊂ H ˆ 0 , respectively, are given explicitly by subspaces H ⊂ H0 and H −N (Π f )(x) = f (y) dy, (5.5a) det(P) T() ([−1 x]) ∨ ∨ with T() (λ) := {x ∈ E | λ, α∨ j ≤ x, αj < (λ, αj + 1), j = 1, . . . , N }, and by ˆ fˆ)(ξ) = χ() (ξ)fˆ(ξ). (Π (5.5b) A
If we extend the definitions of F and Fˆ in Eqs. (5.2a) and (5.2b) to arbitrary ˆ 0 , respectively, then clearly f ∈ H0 and fˆ ∈ H F Π = F
ˆ = Fˆ . and Fˆ Π
(5.6)
156
J.F. van Diejen
Ann. Henri Poincar´e
This gives rise to the following commutative diagrams of bounded transformations H0 ↑ J H
F
−→
F
−→
ˆ0 H ↑Jˆ ˆ H
H0 ↑ J H
Fˆ
←−
ˆ F
←−
ˆ0 H ↑Jˆ , ˆ H
(5.7)
ˆ 0 and Fˆ : H ˆ 0 → H0 being contractive operators in the sense that with F : H0 → H ˆ ˆ ∀f ∈ H0 and ∀f ∈ H0 and Fˆ fˆH0 ≤ fˆHˆ 0
F f Hˆ 0 ≤ f H0 1/2
(5.8)
1/2
(where · H0 := ·, ·H0 and · Hˆ 0 := ·, ·Hˆ ). 0
5.2
The continuum limit → 0: eigenfunction transform
For x and ξ in the interior of the Weyl chamber C, it is straightforward to check that in the limit → 0 the kernel function Φ[−1 x] (ξ) (5.3a) degenerates pointwise to Φ0 (x; ξ) = =
ˆ 1/2 (ξ) Ψ0 (x; ξ) ∆ 0 1/2 S0 (ξ w )eix,ξw ,
(5.9a)
w∈W
with S0 (ξ) =
α, ξ − igα . α, ξ + igα +
α∈R
So, formally the eigenfunction transform F (5.2a) and its adjoint Fˆ (5.2b) degenerate in this limit to fˆ0 (ξ) = (F0 f )(ξ) := f (x)Φ0 (x; ξ)dx (5.10a) C
and its adjoint f0 (x) = (Fˆ0 fˆ)(x) :=
1 (2π)N
C
fˆ(ξ)Φ0 (x; ξ)dξ,
(5.10b)
respectively. From the fact that |S0 (ξ)| = 1, combined with the Plancherel property of the Fourier transform on L2 (E), it follows that the integral transforms in Eqs. ˆ 0 and Fˆ0 : H ˆ 0 → H0 . (5.10a) and (5.10b) define bounded operators F0 : H0 → H The following two lemmas provide a precise meaning to the intuitive idea that for →0 ˆ → H ˆ0 and F → F0 , Fˆ → Fˆ0 . (5.11) H → H0 , H
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
157
Lemma 5.1 (Continuum Limit: the Hilbert Space). One has that s − lim Π = IH0 →0
ˆ = Iˆ s − lim Π H0
and
(5.12)
→0
(strongly). Proof. Since Π is a projection operator, it is obvious that Π H0 ≤ 1 uniformly ∀ > 0. Hence, for validating the first limit in Eq. (5.12), it suffices to show that lim→0 Π φ = φ for any φ in the dense subspace C0∞ (C) ⊂ H0 . It is obvious from the definition in Eq. (5.5a) that, for any test function φ ∈ C0∞ (C), the staircase approximation (Π φ)(x) converges pointwise to φ(x) when tends to 0. Moreover, since φ has compact support it is clear that the difference |Π φ − φ| admits an L2 upper bound that is uniform in (for ≤ 1 say). The desired convergence lim→0 Π φ − φH0 = 0 thus follows by the dominated convergence theorem of Lebesgue. To demonstrate the second limit in Eq. (5.12), we simply observe that ˆ0 for any fˆ ∈ H () ˆ fˆ − fˆ2ˆ = lim (2π)−N 1 − χA (ξ) |fˆ(ξ)|2 dξ, lim Π H 0
→0
→0
C
which converges to zero (again by Lebesgue’s dominated convergence theorem). Lemma 5.2 (Continuum Limit: the Eigenfunction Transform). One has that i) ∀f ∈ H0 :
lim (F f )(ξ) = (F0 f )(ξ),
ξ∈C
→0
(5.13a)
(pointwise) and that ii) s − lim Fˆ = Fˆ0
(5.13b)
→0
(strongly). Proof. i). The action of F on f ∈ H0 reads () −1 1/2 χA (ξ)S1/2 (−ξw ) f (x)∆[−1 x] e−i[ x],ξw dx. (F f )(ξ) = C
w∈W
For any x, ξ ∈ C, we have that for → 0 ()
χA (ξ) → 1,
S (ξ w ) → S0 (ξ w ), −1
∆[−1 x] → 1,
ei[
−1
x],ξw
→ eix,ξw (5.14)
pointwise. Since |e−i[ x],ξw | = 1 and ∆[−1 x] ≤ 1, the pointwise limit in Eq. (5.13a) follows by Lebesgue’s dominated convergence theorem. ˆ 0 is given by ii). The action of Fˆ on any fˆ ∈ H −1 1 () −1 (Fˆ fˆ)(x) = ∆ fˆ(ξ)S1/2 (ξ w )ei[ x],ξw χA (ξ)dξ. [ x] (2π)N C w∈W
158
J.F. van Diejen
Ann. Henri Poincar´e
The pointwise limit lim→0 (Fˆ fˆ)(x) = (Fˆ0 fˆ)(x) thus follows by dominated convergence from the pointwise convergence in Eq. (5.14) combined with the bounds −1 () |S (ξ w )| = 1, |e−i[ x],ξw | = 1 and |χA (ξ)| ≤ 1. It remains to show that the transition Fˆ → Fˆ0 converges in fact strongly. Since Fˆ is uniformly bounded in in view of Eq. (5.8), it suffices to show that lim→0 Fˆ φˆ = Fˆ0 φˆ for any φˆ in the ˆ 0 . The latter limit follows from the estimate dense subspace C0∞ (C) ⊂ H ˆ ≤ C(1 + x2N )−1 |(Fˆ φ)(x)|
(5.15)
uniformly in for sufficiently small. Indeed, the already established pointwise ˆ ˆ convergence lim→0 (Fˆ φ)(x) = (Fˆ0 φ)(x) combined with the L2 -bound in Eq. (5.15) guarantees the convergence of the limit in the Hilbert space H0 by the bounded convergence theorem. In order to verify the estimate in Eq. (5.15), we note that from the explicit formula for the action of Fˆ it is clear that 1/2 1 2N −1 2N ˆ ˆ ˆ S (ξ w )φ(ξ) |∇2N |dξ, (5.16) [ x] |(F φ)(x)| ≤ ξ N (2π) C w∈W
provided is sufficiently small so as to ensure that the support of φˆ is contained in −1 A. Now let ∂ξ1 , . . . , ∂ξN be the partial derivatives associated to an orthonormal basis e1 , . . . , eN of E. Then ∇2ξ = ∂ξ21 + · · · , ∂ξ2N . Hence, to show that the bound (5.15) follows from (5.16) it suffices to check that the partial derivatives m 1/2 ˆ The partial derivatives in ∂ξj j ∂ξmkk S (ξ) are bounded in on the support of φ. question are sums of products of expressions of the form sin (α, ξ − ig ) 1/2 α n 2 ∂ξjj ∂ξnkk . (5.17) sin 2 (α, ξ + igα ) The derivatives in Eq. (5.17) are in turn sums of products built of expressions of the form sin 2 (α,ξ−igα ) 1/2 sin (α,ξ+ig ) α,ej cos 2 (α,ξ−igα ) α,ej cos 2 (α,ξ+igα ) , sin 2 (α,ξ−igαα ) , , , sin (α,ξ+igα ) sin (α,ξ+igα ) sin (α,ξ+igα ) 2
2
2
2
and α, ej , which remain bounded as → 0. With the aid of Lemmas 5.1 and 5.2, we are now in the position to push through the continuum limit → 0 at the level of the Plancherel formula. ˆ 0 → H0 constitutes an Proposition 5.3 (Isometry). The transformation Fˆ0 : H ˆ isometry with left-inverse F0 : H0 → H0 . Proof. The transform Fˆ0 inherits from Fˆ the property that it is an isometry in ˆ0 view of Lemmas 5.1 and 5.2. Indeed, for any fˆ ∈ H Fˆ0 fˆH0
Eq. (5.13b)
=
Eq. (5.4)
=
lim Fˆ fˆH0
→0
ˆ fˆ ˆ lim Π H0
→0
Eq. (5.12)
=
Eq. (5.12)
=
ˆ fˆH0 lim Fˆ Π
→0
fˆHˆ 0 ,
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
159
ˆ 0 → H0 is an isometry. To see that F0 is a left-inverse of Fˆ0 , we whence Fˆ0 : H consider the identity (cf. the commutative diagram in Eq. (5.4)) ˆ φˆ F Fˆ φˆ = Π
(5.18)
for φˆ ∈ C0∞ (C). The l.h.s. of this identity is given by () −1 1/2 ˆ χ (ξ)S 1/2 (−ξ ) (Fˆ φ)(x)∆ e−i[ x],ξw dx. −1 A
w∈W
w
C
[
x]
(5.19)
ˆ We know from the second part of the proof of Lemma 5.2 that (Fˆ φ)(x) admits an L2 -bound that is uniform in for sufficiently small (cf. Eq. (5.15)) and that for ˆ By following the steps in the first part → 0 it converges pointwise to (Fˆ0 φ)(x). of the proof of Lemma 5.2, we readily infer from the expression in Eq. (5.19) that ˆ ˆ = (F0 Fˆ0 φ)(ξ) (pointwise). On the other hand, it follows from lim→0 (F Fˆ φ)(ξ) ˆ ˆ ˆ φ)(ξ) (the proof of) Lemma 5.1 that lim→0 (Π = φ(ξ). We thus conclude that for → 0 the identity in Eq. (5.18) degenerates to ˆ F0 Fˆ0 φˆ = φ, ˆ 0 and the operators whence F0 Fˆ0 = IHˆ 0 (since the subspace C0∞ (C) is dense in H involved are bounded). In other words, Proposition 5.3 states that Fˆ0 is a unitary Hilbert space ˆ 0 ) ⊂ H0 . The following ˆ 0 and the closed subspace Fˆ0 (H isomorphism between H ˆ ˆ proposition ensures that in fact F0 (H0 ) = H0 . ˆ 0 → H0 is surjective, Proposition 5.4 (Completeness). The transformation Fˆ0 : H ˆ 0 ) = H0 . i.e., Fˆ0 (H ˆ 0 → H0 it is enough to show that Proof. For proving the surjectivity of Fˆ0 : H ˆ F0 : H0 → H0 is injective (in view of Proposition 5.3). This injectivity is verified in Appendix A below. Combination of Propositions 5.3 and 5.4 entails that the transformation F0 : ˆ 0 constitutes a unitary Hilbert space isomorphism with inverse Fˆ0 : H ˆ0 → H0 → H H0 : F0 ,Fˆ0 ˆ 0, Fˆ0 F0 = IH0 , F0 Fˆ0 = I ˆ . (5.20) H0 ←→ H H0
The Plancherel formula in Theorem 3.2 is now immediate upon performing the ˆ 1/2 fˆ at the spectral side, so as to trade the uniform gauge transformation fˆ → ∆ 0 ˆ 0 (ξ)dξ. Lebesgue measure dξ for the Plancherel measure ∆
5.3
The continuum limit → 0: Laplacian
ˆσ, and E ˆ0 be multiplication operators in H ˆ 0 of the form Let E ˆ0 (ξ)fˆ(ξ), (Eˆσ, fˆ)(ξ) = Eˆσ, (ξ)fˆ(ξ) and (Eˆ0 fˆ)(ξ) = E
(5.21a)
160
J.F. van Diejen
with ˆσ, (ξ) = −2 E
Ann. Henri Poincar´e
ˆ0 (ξ) = ξ2 . 1 − cos(ν, ξ) and E
(5.21b)
ν∈W (σ)
ˆσ, and E ˆ0 We introduce the operators Lσ, and L0 in H0 as the pullbacks of E ˆ 0 and F0 : H0 → H ˆ 0: with respect to the eigenfunction transforms F : H0 → H Lσ, L0
:= Fˆ Eˆσ, F , ˆ0 F0 . := Fˆ0 E
(5.22a) (5.22b)
The operator L0 (5.22b) amounts to the Laplacian −∇2x in the Weyl chamber C with boundary conditions at the walls of the form in Eq. (3.4b), and the operator Lσ, (5.22a) corresponds to the lift of −2 |W (σ)| − Lσ /2 − Lw0 (σ) /2 from H to H : 1 (5.23) Lσ, J = 2 2|W (σ)| − Lσ − Lw0 (σ) , 2 where Lσ denotes the discrete Laplacian defined in Eqs. (4.1a)–(4.1c). The following proposition states that, in the continuum limit → 0, the discrete difference operator Lσ, (5.22a) tends (up to a positive factor) to the differential operator L0 (5.22b) in the strong resolvent sense. Proposition 5.5 (Continuum Limit: the Laplacian). Let z ∈ C \ [0, ∞). Then s − lim (Lσ, − zIH0 )−1 = (cσ L0 − zIH0 )−1 →0
for some positive constant cσ . Proof. From the limit 1 1 lim 2 1 − cos(ν, ξ) = →0 2 ν∈W (σ)
|ν, ξ|2 = cσ ξ2
ν∈W (σ)
ˆσ, (ξ) = cσ E ˆ0 (ξ) pointfor some positive constant cσ , one concludes that lim→0 E ˆ ˆ wise. Hence, for any f ∈ H0 and z ∈ C \ [0, ∞) lim (Eˆσ, − zIHˆ 0 )−1 fˆ = (cσ Eˆ0 − zIHˆ 0 )−1 fˆ
→0
strongly, by the dominated convergence theorem. The proposition now follows from the telescope (Lσ, − zIH0 )−1 f − (cσ L0 − zIH0 )−1 f H0 ≤ (Lσ, − zIH0 )−1 (Fˆ0 − Fˆ )F0 f H0 ˆ0 − zI ˆ )−1 ]F0 f H0 + Fˆ [(Eˆσ, − zIHˆ 0 )−1 − (cσ E H0 ˆ0 − zI ˆ )−1 F0 f H0 + (Fˆ − Fˆ0 )(cσ E H0 upon sending to zero (and invoking of Lemma 5.2).
Vol. 5, 2004
A
Plancherel Formula for the (Discrete) Laplacian
161
The inversion formula: Continuous case
In the proof of Proposition 5.4 we needed the fact that the transformation F0 : ˆ 0 in Eq. (5.10a) – or equivalently the transformation F0 : H0 → H ˆ 0 in H0 → H Eq. (3.9a) – is injective. In principle this injectivity follows from the analysis by Heckman and Opdam. Indeed, it was shown in Ref. [HO] that Fˆ0 F0 = IH0
(A.1)
upon restriction to the dense subspace C0∞ (C) ⊂ H0 (cf. also the comments just after Theorem 3.2). Since all operators involved are bounded, the inversion formula in Eq. (A.1) is readily extended from C0∞ (C) to the whole of H0 (by taking the ˆ 0 (and thus the transformation closure), whence the transformation F0 : H0 → H ˆ 0 ) is injective. F0 : H0 → H The proof of Eq. (A.1) indicated in [HO] is quite sophisticated and hinges on a deep result due to Peetre regarding the characterization of differential operators as support preserving operators on smooth test functions [P1, P2]. In this appendix we present an elementary proof for this inversion formula. Let φ ∈ C0∞ (C). Then (F0 φ)(ξ) = φ(x)Ψ0 (x; ξ)dx C = C0 (−ξ w ) φ(x)e−ix,ξw dx C
w∈W
=
˘ C0 (−ξ w ) φ(ξ w ),
(A.2a)
α, ξ − igα α, ξ +
(A.2b)
φ(x)e−ix,ξ dx.
(A.2c)
w∈W
where C0 (ξ) =
α∈R
and
˘ φ(ξ) = E
Substitution of φˆ = F0 φ into ˆ (Fˆ0 φ)(x)
= = =
1 ˆ φ(ξ)Ψ 0 (x; ξ)∆0 (ξ)dξ (2π)N C 1 ix,ξw ˆ ∆0 (ξ)dξ φ(ξ)C 0 (ξ w )e (2π)N C w∈W 1 1 ˆ eix,ξw dξ φ(ξ) (2π)N C (−ξ ) 0 C w w∈W
(A.3)
162
J.F. van Diejen
Ann. Henri Poincar´e
yields (Fˆ0 F0 φ)(x)
= =
1 (2π)N 1 (2π)N
w1 ,w2 ∈W
w∈W
E
C
C0 (−ξw1 ) ˘ φ(ξ w1 )eix,ξw2 dξ C0 (−ξw2 )
C0 (−ξ) ˘ φ(ξ)eix,ξw dξ, C0 (−ξw )
(A.4a)
where C0 (−ξ) C0 (−ξw )
=
α, ξ + igα α, ξ w α, ξ α, ξ w + igα + +
α∈R
=
α∈R+ ∩w −1 (R− )
α∈R
α, ξ + igα . α, ξ − igα
(A.4b)
The inversion formula Fˆ0 F0 φ = φ for φ ∈ C0∞ (C) is now immediate from Eqs. (A.4a), (A.4b) combined with the fact that for x ∈ C and w ∈ W φ(x) if w = Id, 1 C0 (−ξ) ˘ ix,ξw φ(ξ)e dξ = (A.5) N (2π) 0 if w = Id. E C0 (−ξ w ) To infer the equality in Eq. (A.5), let us first note that the case w = Id is clear as it amounts to the standard Fourier inversion formula on E. The case w = Id is verified with the aid of the following straightforward observations. ˘ (A.2c) is entire in ξ and rapidly (i) For φ ∈ C0∞ (C) the Fourier transform φ(ξ) decreasing on the tubular domain E − iC∨ , where C∨ denotes the open convex cone dual to C, generated by the positive roots (i.e., C∨ := SpanR+ (R+ )). (ii) The parameter restriction gα > 0 ensures that the quotient C0 (−ξ)/C0 (−ξ w ) (A.4b) is holomorphic and bounded on the tubular domain E − iCw , where Cw := {ξ ∈ E | ξ, α > 0, ∀α ∈ R+ ∩ w−1 (R− )}. ∨ −1 (iii) For all x ∈ C and ϑ ∈ C∨ (−C∨ ) = SpanR+ (R+ ∩ w−1 (R− )), w := C ∩ w one has that x, ϑw < 0.
Indeed, we conclude from (i) and (ii) and the Cauchy integral theorem that for an arbitrary but fixed ϑ ∈ Cw ∩ C∨ w C0 (−ξ) ˘ C0 (−ξ) ˘ ix,ξw φ(ξ)e φ(ξ)eix,ξw dξ dξ = (A.6) C (−ξ ) C E 0 E−isϑ 0 (−ξ w ) w for all s ≥ 0. Furthermore, it follows from (i), (ii) and (iii) that for s → ∞ the r.h.s. of Eq. (A.6) tends to zero, whence the case w = Id of the equality in Eq. (A.5) follows.
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
163
Note. To convince oneself that the cone Cw ∩C∨ w is nonempty for any w ∈ W \{Id}, we observe that it contains the nonzero vector ρ − ρw−1 (where ρ = α∈R+ α/2). Indeed, we have on the one hand that ρ ∈ C and ρw−1 ∈ w−1 (C), so ρ−ρw−1 ∈ Cw , while on the other hand ρ − ρw−1
= =
1 1 1 α− α+ α 2 2 2 + + + + − −1 −1 α∈R α∈R ∩w (R ) α∈R ∩w (R ) α, α∈R+ ∩w −1 (R− )
so ρ − ρw−1 ∈ C∨ w.
B Macdonald’s orthogonality relations In this appendix we outline the proof of Macdonald’s orthogonality relations [M1, M3] if λ = µ, ∆−1 λ (Ψλ , Ψµ )Hˆ = (B.1) 0 if λ = µ, for the Bethe wave functions Ψλ (ξ) of Theorem 4.2. (Here ∆λ is given by Eq. (4.12).) The proof, which follows Macdonald’s treatment in Ref. [M3, §10], hinges on two key lemmas. The first lemma states that the Bethe wave function expands triangularly with respect to the dominance order on the basis of monomial symmetric functions mµ (ξ) = ν∈W (µ) eiν,ξ , µ ∈ P + . Lemma B.1 (Triangularity). For any λ ∈ P + , the Bethe wave function Ψλ (ξ) (4.11) expands as Ψλ (ξ) = ∆−1 λ mµ (ξ) +
aλµ mµ (ξ),
µ∈P + , µ≺λ
for certain coefficients aλµ ∈ C. The second lemma describes a biorthogonality relation between the Bethe wave function Ψλ (ξ) and the monomial symmetric functions mµ (ξ) corresponding to dominant weights µ that are not bigger than λ in the dominance order. Lemma B.2 (Biorthogonality Relations). The Bethe wave functions Ψλ (ξ), λ ∈ P + and the monomial symmetric functions mµ (ξ), µ ∈ P + satisfy the biorthogonality relations 1 if µ = λ, (Ψλ , mµ )Hˆ = 0 if µ λ.
164
J.F. van Diejen
Ann. Henri Poincar´e
Before going into the proof of these lemmas, let us first observe that they imply the orthogonality relations in Eq. (B.1). Indeed, it is immediate from Lemmas −1 B.1 and B.2 that (Ψλ , Ψλ )Hˆ = (Ψλ , ∆−1 ˆ = ∆λ and that (Ψλ , Ψµ )H ˆ = 0 if λ mλ )H µ λ. But then (Ψλ , Ψµ )Hˆ must in fact vanish for all dominant weights µ = λ in view of the symmetry (Ψλ , Ψµ )Hˆ = (Ψµ , Ψλ )Hˆ , whence the orthogonality relations follow.
Proof of Lemma B.1 Starting point for the proof of the first lemma is the formula for Ψλ (ξ) in Eqs. (4.7a)–(4.7c) with C(ξ) of the form in Eq. (4.9): Ψλ (ξ) =
1 (−1)w eiρ+λ,ξw (1 − tα e−iα,ξw ). δ(ξ) + w∈W
α∈R
By expanding the product over the positive roots, one arrives at c 1 (−1)w (−1)|X| eiρ(X )−ρ(X)+λ,ξw tα , Ψλ (ξ) = δ(ξ) + w∈W
α∈X
X⊂R
with ρ(X) :=
1 α, 2
ρ(X c ) :=
α∈X
1 2
α.
α∈R+ \X
−1
Next, by exploiting the symmetry eiµ,ξw = eiw (µ),ξ and combining contributions of monomials with weights in the same Weyl orbit into Weyl characters χµ (ξ) = δ −1 (ξ) w∈W (−1)w eiρ+µ,ξw , one finds the expansion (−1)wX (−1)|X| tα χλ(X) (ξ), (B.2) Ψλ (ξ) = X⊂R+
α∈X
where we have introduced the notation wX := wρ(X c )−ρ(X)+λ ,
λ(X) := wX (ρ(X c ) − ρ(X) + λ) − ρ.
In the expansion (B.2) the factor χλ(X) (ξ) vanishes if the weight λ(X) is not dominant. Indeed, from the fact that ρ = ω1 + · · · + ωN (the sum of the fundamental weights) [B, H1], it follows that in such case the dominant weight ρ + λ(X) cannot be regular (i.e., it must have a nontrivial stabilizer), whence the skewsymmetrization with respect to the Weyl-group action in the definition of the Weyl character leads to zero. It follows from Eq. (B.2) that the Bethe function Ψλ (ξ) expands triangularly on the basis of Weyl characters: bλµ χµ (ξ), Ψλ (ξ) = µ∈P + , µ λ
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
165
for certain complex coefficients bλµ . Indeed, one has that λ(X)
= wX (λ) + wX (ρ(X c ) − ρ(X)) − ρ α+ = wX (λ) − α∈R+ ∩wX (X)
α
α∈R− ∩wX (X c )
wX (λ) λ (where in the last step we exploited the well-known fact that any dominant weight is maximal (with respect to the dominance order) in its Weyl orbit, i.e., for all λ ∈ P + and w ∈ W : λw λ [B, H1]). To compute the leading coefficient bλλ , it is needed to collect all terms in Eq. (B.2) for which λ(X) = λ. These terms correspond to those subsets X ⊂ R+ for −1 which wX (λ) = λ and wX (ρ) = ρ(X c ) − ρ(X), or equivalently, to those subsets X for which X = {α ∈ R+ | w(α) ∈ R− } with w ∈ Wλ . By summing the contributions of all such subsets X, one obtains that bλλ is given by the following Poincar´e type series of the stabilizer Wλ : (w) bλλ = tss (w) tl l . (B.3) w∈Wλ
The lemma now follows from the well-known fact that the Weyl characters expand unitriangularly on the monomials χλ = mλ + µ∈P + , µ≺λ cλµ χµ , combined with Macdonald’s celebrated product formula [M2, M3] for the Poincar´e type series in Eq. (B.3) 1 − tα tshts (α) thtl (α) (w) l tss (w) tl l = . (B.4) hts (α) htl (α) 1 − t t s w∈Wλ l α∈R+ λ,α∨ =0
Proof of Lemma B.2 Starting from the formula for the Bethe wave function Ψλ (ξ) in Eq. (4.11), we obtain upon taking the inner product with the monomial symmetric function mµ (ξ): 1 |W | Vol(A) |Wµ | 1 − tα e−iα,ξw1 ˆ ∆(ξ) eiλ,ξw1 × e−iµ,ξw2 dξ −iα,ξw1 1 − e A w1 ∈W w2 ∈W α∈R+ 1 − eiα,ξ 1 (i) dξ = eiλ,ξ−iµ,ξw Vol(T) |Wµ | 1 − tα eiα,ξ w∈W T α∈R+ ∞ 1 (ii) 1+ = eiλ−µw ,ξ (tnαα − tnαα −1 )einα α,ξ dξ, Vol(T) |Wµ | T + n =1
(Ψλ , mµ )Hˆ =
w∈W
α∈R
α
166
J.F. van Diejen
Ann. Henri Poincar´e
ˆ where T denotes the torus E/(2πQ∨ ). Here step (i) follows by plugging in ∆(ξ) (4.15a), (4.15b), exploiting the Weyl-group invariance of the integrand, and using the standard fact that the Weyl alcove A (4.14) constitutes a fundamental domain for the action of the Weyl group W on the torus T [B, H2, K]. Furthermore, step (ii) follows by expansion of the denominators with the aid of a geometric series. Clearly, the integral on the last line picks up the constant term of the integrand multiplied by the volume of the torus T. It is immediate that a nonzero constant term can occur only if λ − µw ∈ −Q+ (i.e., µw λ) for some w ∈ W . Now, if µ λ, then for all w ∈ W also µw λ (since µw µ, cf. the proof of Lemma B.1 above). Hence, in this situation the constant term vanishes. On the other hand, if µ = λ then the constant part of the term labelled by w is nonzero (namely equal to 1) if and only if w ∈ Wλ . By summing over all these contributions originating from the stabilizer Wλ the lemma follows.
Acknowledgments This paper was written in large part while the author was visiting the Graduate School of Science and Engineering of the Tokyo Institute of Technology, Tokyo, Japan (January–March, 2003). It is a pleasure to thank the Department of Mathematics, and in particular Professors K. Mimachi and N. Kurokawa, for the warm hospitality. Thanks are also due to S.N.M. Ruijsenaars for several helpful discussions.
References [BT] D. Babbitt and L. Thomas, Ground state representation of the infinite onedimensional Heisenberg ferromagnet, II. An explicit Plancherel formula, Commun. Math. Phys. 54, 255–278 (1977). [BZ]
E. Br´ezin and J. Zinn-Justin, Un probl`eme `a N corps soluble, C.R. Acad. Sci. Paris S´er. A-B 263, B670–B673 (1966).
[B]
N. Bourbaki, Groupes et alg`ebres de Lie, Chapitres 4–6, Hermann, Paris, 1968.
[G1]
M. Gaudin, Bose gas in one dimension, I. The closure property of the scattering wave functions, J. Math. Phys. 12, 1674–1676 (1971).
[G2]
M. Gaudin, Bose gas in one dimension, II. Orthogonality of the scattering states, J. Math. Phys. 12, 1677–1680 (1971).
[G3]
M. Gaudin, Boundary energy of a Bose gas in one dimension, Phys. Rev. A. 4, 386–394 (1971).
[G4]
M. Gaudin, La Fonction d’Onde de Bethe, Masson, Paris, 1983.
Vol. 5, 2004
Plancherel Formula for the (Discrete) Laplacian
167
[G]
E. Gutkin, Integrable systems with delta-potential, Duke Math. J. 49, 1–21 (1982).
[GS]
E. Gutkin and B. Sutherland, Completely integrable systems and groups generated by reflections, Proc. Nat. Acad. Sci. USA 76, 6057–6059 (1979).
[HO] G.J. Heckman and E.M. Opdam, Yang’s system of particles and Hecke algebras, Ann. Math. 145, 139–173 (1997); erratum ibid. 146, 749–750 (1997). [H1]
J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.
[H2]
J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.
[K]
R. Kane, Reflection Groups and Invariant Theory, CMS Books in Mathematics 5, Springer-Verlag, New York, 2001.
[KBI] V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993. [LL]
E.H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas, I. The general solution and the ground state, Phys. Rev. 130 (2), 1605–1616 (1963).
[M1] I.G. Macdonald, Spherical Functions of p-adic Type, Publ. of the Ramanujan Inst., No. 2, 1971. [M2] I.G. Macdonald, The Poincar´e series of a Coxeter group, Math. Ann. 199, 151–174 (1972). [M3] I.G. Macdonald, Orthogonal polynomials associated with root systems, S´em. Lothar. Combin. 45 (2000/01), Art. B45a, 40 pp. (electronic). [M]
D.C. Mattis (eds.), The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension, World Scientific, Singapore, 1994.
[Mc] J.B. McGuire, Study of exactly soluble one-dimensional N -body problems, J. Math. Phys. 5, 622–636 (1964). [O]
S. Oxford, The Hamiltonian of the Quantized Nonlinear Schr¨ odinger Equation, Ph. D. Thesis, UCLA, 1979.
[P1]
J. Peetre, Une caract´erisation abstraite des op´erateurs diff´erentiels, Math. Scand. 7, 211–218 (1959).
[P2]
J. Peetre, R´ectification a` l’article “Une caract´erisation abstraite des op´erateurs diff´erentiels”, Math. Scand. 8, 116–120 (1960).
168
J.F. van Diejen
Ann. Henri Poincar´e
[R]
S.N.M. Ruijsenaars, The continuum limit of the infinite isotropic Heisenberg chain in its ground state representation, J. Funct. Anal. 39, 75–84 (1980).
[T]
L. Thomas, Ground state representation of the infinite one-dimensional Heisenberg ferromagnet, J. Math. Anal. Appl. 59, 392–414 (1977).
[Y1]
C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19, 1312–1314 (1967).
[Y2]
C.N. Yang, S matrix for the one-dimensional N -body problem with repulsive or attractive delta-function interaction, Phys. Rev. 168 (2), 1920–1923 (1968).
J.F. van Diejen Instituto de Matem´ atica y F´ısica Universidad de Talca Casilla 747 Talca, Chile and Graduate School of Science and Engineering Tokyo Institute of Technology 2-12-1 Oh-okayama Meguro-ku Tokyo, 152-8551 Japan email:
[email protected] Communicated by Rafael D. Benguria Submitted 27/05/03, accepted 14/10/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 169 – 188 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010169-20 DOI 10.1007/s00023-004-0164-x
Annales Henri Poincar´ e
A Singular Expansion of Solution for a Regularized Compressible Stokes System Jae Ryong Kweon Abstract. A compressible Stokes system is considered in a sector of the plane. The continuity equation is regularized by adding the diffusion term −∆p. We give a high-order expansion of corner singularities for the regularized system when the corner singularities for the Laplacian are implemented. A solution formula is constructed in an abstract way, and new associate singular functions are introduced for extracting high-order corner singularities. In the expansion, the smoother parts of the associate singular functions are used.
1 Introduction Many issues for the compressible Stokes or Navier-Stokes systems in domains with singular boundaries are still open. The known results are very limited [10, 11, 14]. A reason is because the system is of mixed type which is neither elliptic nor hyperbolic. The boundary value problems of elliptic type have been exhaustingly investigated in such regions. For instance, see [6, 7, 9] for the Laplace or convection diffusion equations and [4, 6, 8, 9] for the incompressible Stokes equations. Note that there are numerous papers not listed in this paper. The investigation for the incompressible Stokes case is due to the known theory of elliptic problems while in the compressible case, such an adequate approach does not exist. In [10, 11] it is shown that the lowest-order corner singularity of the stationary compressible Stokes or Navier-Stokes system is the same as that of the Laplace equation near the concave vertices. In this case the continuity equation is solved for pressure along the characteristic lines directed by an ambient velocity vector. In [12, 14], when the domain is a convex polygon, the singular functions produced by the Stokes equations were considered in order to obtain a smoother part for the solution of the stationary Stokes or Navier-Stokes system. When it is not convex, the result is not true any more. To understand this difficulty in a different way, we regularize the system by adding − ∆p to the continuity equation. The goal of this paper is to consider the regularized compressible Stokes system in any sector of the plane and to give a high-order expansion of the corner singularities by the Laplacian. The system to be considered is − ∆p + κU · ∇p + λp + divu = −µ ∆u + ∇p
g in Ω,
=
f in Ω,
u, p =
0 on Γ.
(1.1)
170
J.R. Kweon
Ann. Henri Poincar´e
Here the unknowns are the velocity vector u = [u, v] and the pressure p; and λ are positive numbers, κ is the compressibility constant, µ is the viscosity constant; ¯ f and g are U is a given smooth vector function with UΓ = 0, say U ∈ C 1 (Ω); given functions. The region Ω is an infinite sector defined by Ω = {(r cos θ, r sin θ) : r = x2 + y 2 > 0, ω1 < θ < ω2 } where ωl , l = 1, 2, are two numbers satisfying ω1 < ω2 < ω1 + 2π. The angle of the sector is ω = ω2 − ω1 which is assumed 0 < ω ≤ 2π. The two rays that make up the boundary Γ of Ω are denoted Γl , l = 1, 2. Throughout this paper we shall denote by Z = κU · ∇ + λI. Note that there are no inflow and outflow due to UΓ = 0 and (1.1c). So a total mass of the gas is fixed. The equations in (1.1) are derived as follows. The second equation is derived by dropping the terms ν∇divu and ρU · ∇u in the momentum equation. The first equation is derived by the time-dependent continuity equation ρt + div(ρu) = 0. Using the relation ρ = ρ(p) we have ρ (p)pt + ρ(p)divu + ρ (p)u · ∇p = 0. When (U, P ) is a given ambient flow, a linearized continuity equation around the ambient flow is ρ (P )pt + ρ(P )divu + ρ (P )U · ∇p = 0. The derivative pt can be replaced by p(x, t)−p(x, t−∆t) /∆t when ∆t is small. When λ = κ/∆t with κ = ρ (P )/ρ(P ), we have κU · ∇p + λp + divu = g where g = λ P (x, t − ∆t). Adding −∆p, (1.1a) is obtained. One may extend the analysis of sector to a polygon but for simplicity, we will focus on the behavior of the solution near the origin of the sector Ω. Let χ ∈ C0∞ (R2 ) be a cutoff function which is identically 1 in a neighborhood of (0,0), and which satisfies χ(x, y) = 0 for r ≥ 1. Using this, (1.1) becomes − ∆(χp) + Z(χp) + div(χu) = χg − 2 ∇χ · ∇p + p(− ∆χ + U · ∇χ) + u · ∇χ,
(1.2)
−µ ∆(χu) + ∇(χp) = χ f − 2µ∇χ · ∇u − µu∆χ + p∇χ. It is seen that [χu, χp] is a weak solution of (1.2) with zero Dirichlet boundary data. From the property of χ, near r = 1 the solution [χu, χp] is sufficiently smooth and vanishes. In this sense, throughout this paper we assume that the solution [u, p] vanishes outside r = 1. Denote Ωa by a finite sector of Ω, truncated at radius a > 0. us means the norm of Hs (Ω) and us,Ωa the norm of Hs (Ωa ). Set γ0 = λ − |∇(κU)|∞ . We state a main result of this paper, which is shown in Section 4. For j = 1, 2, . . . let sj = jα + 1 with α = π/ω be given. Theorem 1.1 Suppose that the number γ0 is positive, in other words, either λ is sufficiently large or U is close to a constant. Assume that in (1.1), [u, p] vanishes for r ≥ 1. Then, if [f , g] ∈ L2 × L2 , then there is a unique solution [u, p] ∈ H10 × H10 of (1.1) satisfying √ √ √ µu1,Ω1 + ∇p1,Ω1 + γ0 p0,Ω1 ≤ C(f 0 + g0 ).
Vol. 5, 2004
An Expansion of Solution for a Regularized Compressible Stokes System
171
If 1 ≤ s < s1 and [f , g] ∈ Hs−2 × Hs−2 , then µ us,Ω1 + ps,Ω1 ≤ C(f s−2 + gs−2 ) for a constant C. For integer l ≥ 1, let sl < s < sl+1 . If [f , g] ∈ Hs−2 × Hs−2 and U ∈ Hs , then the solution [u, p] is split as follows: [u, p] = [us , ps ] + [uR , pR ],
[uR , pR ] = [u, p] − [us , ps ]
with l l−1 ˜ [Cj , Cj3 ]φj + [ηj,1 , τj,1 ]. [us , ps ] = j=1
j=1
Furthermore µuR s,Ω1 + pR s,Ω1 ≤ C(f s−2 +gs−2 ) where C is a constant. Here the coefficient [C˜j , Cj3 ] with C˜j = [Cj1 , Cj2 ] solves the system (4.14) and satisfies (4.16). The function φj is defined in Section 3, and for j ≥ 1, [ηj,1 , τj,1 ] is defined by ηj,1 = ηj − µ−1
j
Λi (χfs,j )φi ,
τj,1 = τj − −1
i=1
j
Λi (χgs,j )φi ,
i=1
where Λi is a bounded linear functional on Hs−2 for s > si (see Theorem A in Section 3), and where fs,1 = −C13 ∇φ1 , gs,1 = C˜1 · ∇φ1 − C13 Zφ1 , ··· fs,j = −(Cj3 ∇φj + ∇τj−1,1 ), gs,j = C˜j · ∇φj − Cj3 Zφj − (Zτj−1,1 + div ηj−1,1 ). The functions ηj and τj are the new associate singular functions which satisfy −µ∆ηj = χ fs,j in Ω, ηj = 0 on Γ and − ∆τj = χ gs,j in Ω, τj = 0 on Γ, respectively. In Theorem 1.1, we note that if s < s1 the solution [u, p] is as regular as permitted in the data [f , g] and that if s > s1 the solution is split with singular and regular parts. In the singular expansion when s > s2 , some new associate singular functions are needed in the expansion while they are not necessary in the Laplacian case [6, 7, 9]. Such singular information of solution might have an application to certain realistic modeling. Consider a high-speed flow over a body where a wall of the body is turned downward at the corner through a deflection angle [2]. If the flow is in high speed, say, supersonic or hypersonic, then at the corner, the flow properties may change drastically. It is thought that a complete understanding of the solution
172
J.R. Kweon
Ann. Henri Poincar´e
near the corner is an essential ingredient in studying such flows. Also, in the solid mechanics, when a solution domain is composed of different materials, the corner singularities occur at the intersections of their internal interfaces. In order to apply to problem (1.1) known results for the Laplacian, we define A : L2 ( or H−1 ) −→ H10 by u := AF where u is the solution of −∆u = F in Ω,
u = 0 on Γ.
(1.3)
In what follows, the following spaces and norms are used [1, 13]. Denote by L2 the space of all measurable functions u defined on Ω for which u0 := 1/2 2 . Let L20 be the set of the functions v in L2 which satisfies Ω |u(x)| dx v(x)dx = 0. Denote by |u|∞ = ess supx∈Ω |u(x)|. For integer l ≥ 0, define Ω Hl =
v ∈ L2 : vl :=
1/2 ∇α v20
0 we denote by Hs the space of all distributions u defined in Ω such that us < ∞ where s = n + σ is nonnegative and is not an integer. The norm is defined by
us =
un + |η|=n
Ω×Ω
|Dη u(x) − Dη u(y)|2 dx dy |x − y|2+2σ
1/2 .
We denote by Hs0 = Hs ∩ H10 and H−s the dual space of Hs0 normed by f −s =
sup
0=v∈Hs0
f, v vs
where , denote the duality pairing. s s s Denote by the bold face H = H ×H . Similar for other spaces. For simplicity, 2 2 we define [v, χ]s = vs + χs . We use C to denote a generic positive constant. Note that C may take different values in different places. The paper is organized as follows. In Section 2 a solution formula is constructed. Using this, existence and regularity are established in a fractional Sobolev space. In Section 3, based on the corner singularities of the Laplace equation, the solution is split into singular and regular parts up to the second-order of singularities. In Section 4 a high-order expansion of corner singularities is derived by introducing new associate singular functions.
Vol. 5, 2004
An Expansion of Solution for a Regularized Compressible Stokes System
173
2 Existence We consider some bilinear forms for (1.1) and associate operators with the forms. We define a solution operator and show an unique existence. When no corner singularity is split from the solution of (1.1), a best regularity result that the solution can have is established in a fractional Sobolev space (see Lemma 2.2). Let V = H10 , M = L2 and V = V × V . Define bilinear forms a(v, w) on V × V, b(χ, v) on M × V and c(p, χ) on V × V as follows:
a(u, v) = ∇u · ∇v dx, b(χ, v) = χ divv dx, Ω
Ω c(η, χ) = ∇p · ∇χ + −1 Zp χ dx. Ω
Using the forms, we associate the following problem: for [f , g] ∈ V × V , find [u, p] ∈ V × V such that µ a(u, v) − b(p, v) = f , v, ∀v ∈ V, c(p, χ) + b(χ, u) = g, χ, ∀χ ∈ V,
(2.1) (2.2)
where . , . is the duality pairing between V and V or V and V . We next associate with the forms a, b and c, operators A : V −→ V , defined by
div : V −→ M ,
∇ : M −→ V ,
C : V −→ V
(2.3)
Av, w = a(v, w), v, w ∈ V,
divv, χ = b(v, χ), v ∈ V, χ ∈ M,
∇χ, v = −b(v, χ), v ∈ V, χ ∈ M,
Cp, χ = c(p, χ), p ∈ V, χ ∈ V.
Using the operators, (2.1) and (2.2) can be expressed by µ Au + ∇p = f in V ,
Cp + divu = g in V .
(2.4) (2.5)
Note that C = A + Z. We next give some properties for the operators defined in (2.3) and define a mapping T which is used in constructing the solution formula. Lemma 2.1 (i) There is a constant α∗ > 0 such that α∗ vV ≤ AvV for all v ∈ V. (ii) Assume that U ∈ C 1 (Ω). Define T = C − µ−1 divA∇, where A (= A−1 ) is the solution operator defined in (1.3). If γ0 > 0, then T : V −→ V has a bounded inverse S := T −1 : V −→ V , with a bound S = sup
0=h∈V
ShV ≤ 1/ min{, γ0 }. hV
(2.6)
174
Proof.
J.R. Kweon
Ann. Henri Poincar´e
(i) is clear. We show (ii). From T η = h in V and using (2.3),
T η, χ = Cη, χ − µ−1 divA∇η, χ = c( η, χ) − µ−1 b(A∇η, χ),
∀χ ∈ V.
Let u = µ−1 A∇η. So µAu = ∇η in V and −b(u, η) = µa(u, u) ≥ µα∗ u2V = µ−1 α∗ A∇η2V . Since A : V −→ V is bounded and f V ≤ AAf V for all f ∈ V , we have
T η, η = c( η, η) − µ−1 b(A∇η, η) 1 ≥ ∇η2M + (λ − |∇(κU)|∞ )η2M + µ−1 α∗ A∇η2V 2 1 ≥ ∇η2M + (λ − |∇(κU)|∞ )η2M + µ−1 α∗ A−2 ∇η2V 2 ≥ ∇η2M + γ0 η2M , ∀η ∈ V. (2.7) Thus T is bounded below and has a bounded inverse S = T −1 . The estimation for S follows from (2.7). We are going to express u and p in terms of f and g. Indeed, let [f , g] ∈ H−1 × H−1 . From (2.4), u = µ−1 A(f − ∇p) ∈ H10 .
(2.8)
Using this, (2.5) and Lemma 2.1, we have T p + µ−1 divAf = g and p = −µ−1 SdivAf + Sg ∈ H10 .
(2.9)
Inserting the formula p of (2.9) into (2.8), u = µ−1 K1 Af − µ−1 A∇Sg ∈ H10 ,
(2.10)
where I is the identity operator and K1 = I + µ−1 A∇Sdiv. In a compact form the solution is given by u f =M p g
where M=
µ−1 K1 A
−µ−1 A∇S
−µ−1 SdivA
S
(2.11) .
For each fixed > 0, M is a solution operator for (1.1) which maps from H−1 ×H−1 into H10 × H10 . We next show an unique existence and establish a best regularity result that the solution can have when no corner singularity is split.
Vol. 5, 2004
An Expansion of Solution for a Regularized Compressible Stokes System
175
¯ Suppose that γ0 is positive. Suppose [u, p] vanishes Lemma 2.2 Let U ∈ C 1 (Ω). outside r = 1. If [f , g] ∈ L2 × L2 , then there is a unique solution [u, p] ∈ H10 × H10 of (1.1), satisfying √ √ √ µu1,Ω1 + ∇p1,Ω1 + γ0 p0,Ω1 ≤ C(f 0 + g0 ). (2.12) Furthermore there exists a constant C such that if 1 ≤ s < s1 and [f , g] ∈ Hs−2 × Hs−2 , then µ us,Ω1 + ps,Ω1 ≤ C(f s−2 + gs−2 ). Proof.
(2.13)
First, (2.12) follows from (2.1) and (2.2). Consider the diagram A
S
∇
A
H−1 −→ H10 −→ L2 −→ H10 ⊂ L2 −→ L2 ⊂ H−1 −→ H10 . div
(2.14)
The following mappings are well defined: A∇SdivA,
A∇S,
SdivA.
(2.15)
By the theory of corner singularity, if s < α + 1 and f ∈ Hs−2 , then Af ∈ Hs (see [6, 7, 9]). Hence the sequences of mappings can be considered: f ∈ Hs−2 ∩ H−1 −→Af ∈ Hs ∩ H10 −→divAf ∈ Hs−1 ∩ L2 −→SdivAf ∈ Hs ∩ H10 −→∇SdivAf ∈ H
s−1
(2.16) s
∩ H −→A∇SdivAf ∈ H ∩ 0
H10 ,
and g ∈ Hs−2 ∩ L2 −→Sg ∈ Hs ∩ H10 −→∇Sg ∈ Hs−1 ∩ H−1 −→A∇Sg ∈ Hs ∩ H10 .
(2.17)
Assuming that [f , g] ∈ Hs−2 × Hs−2 , it follows from (2.16) and (2.17) that K1 Af = [I + µ−1 A∇Sdiv]Af ∈ Hs , s
SdivAf ∈ H ,
A∇Sg ∈ Hs ,
s
Sg ∈ H .
By (2.6), S ∼ min{, γ0 }−1 ∼ −1 if γ0 > 0. So K1 A ∼ 1 + (µ)−1 and A∇S ∼ −1 . Using the operator M, µ us,Ω1 ≤ c1 −1 (f s−2 + gs−2 ),
(2.18)
where c1 is a constant, and
ps,Ω1 ≤ S µ−1 f s−2 + gs−2 ≤ c2 −1 (f s−2 + gs−2 ),
where c2 = C(divA, γ0 ). Thus (2.13) follows with c = max{c1 , c2 }.
176
J.R. Kweon
Ann. Henri Poincar´e
3 Decomposition We first cite a basic result on corner singularities for the Laplacian (see [6, 7, 9]). Let α = π/ω and define sj = jα + 1 for j = 1, 2, . . . Theorem A. There are linear functionals Λj and functions ψj , j = 1, 2, . . . , satisfying the following properties. (i) Λj is a bounded linear functional on Hs−2 for s > sj , but not for s ≤ sj . (ii) ψj ∈ Hs ∩ H10 for s < sj but not for s ≥ sj . Also ψj is smooth everywhere in Ω except at r = 0 and ∆ψj is smooth everywhere ¯ (iii) If sj < s < sj+1 , f ∈ Hs−2 , u is a solution of the Laplace problem: in Ω. −∆u = f in Ω, u = 0 on Γ, and u vanishes outside r = 1, then uj := u −
j
Λi (f )ψj ∈ Hs (Ω1 ), with uj s,Ω1 ≤ Cf s−2 .
i=1
There are many sets of singular functions that may be used in Theorem A. One set is given by the formula (see [7]) jα jα = integer, r sin[jα(θ − ω1 )], (ln r) rk sin[k(θ − ω1 )] + (θ − ω1 )rk cos[k(θ − ω1 )] ψj (x, y) = −(−1)j ω csck ω [−x sin ω1 + y cos ω1 ]k , jα = k = integer. Also specific formulas for the corresponding linear functionals Λj can be found in [7]. In this section, based on Theorem A, we split the first and second leading corner singularities of the Laplacian equation from the exact solution of (1.1). We show that the lowest-order corner singularity of (1.1) is the same as that of the Laplacian. We also show that the second leading singularity can be sorted out after some new associate singularity functions are subtracted from the regular solution of the first step. We shall denote by φj = χψj , j = 1, 2, . . . where χ is the smooth cutoff function. Step 1. s1 < s < s2 . We split the solution [u, p] of (1.1) as follows: u = C˜1 φ1 + u1 ,
p = C13 φ1 + p1
(3.1)
where C˜1 = [C11 , C12 ] and C13 are parameters to be constructed later. Inserting (3.1) into (1.1) and setting f1 = f + µ C˜1 ∆φ1 − C13 ∇φ1 , g1 = g + C13 ( ∆φ1 − Zφ1 ) + C˜1 · ∇φ1 , system (1.1) becomes −µ ∆u1 + ∇p1
=
f1 in Ω ,
− ∆p1 + Zp1 + divu1
=
g1 in Ω ,
u1 , p1
=
0
on Γ.
(3.2)
Vol. 5, 2004
An Expansion of Solution for a Regularized Compressible Stokes System
177
We next show a required regularity for f1 and g1 so that the linear functional Λ1 given in Theorem A can be applied. Lemma 3.1 If [f , g] ∈ Hs−2 × Hs−2 for s1 < s < s2 , then [f1 , g1 ] ∈ Hs−2 × Hs−2 . Proof. Since φ1 ∈ Hs for s < s1 , we have ∇φ1 ∈ Ht for t < α. Since 2α − 1 < α, we have ∇φ1 ∈ Hs−2 for s < s2 . Since ∆φ1 is sufficiently smooth, the required result follows. When Theorem A is applied to (3.2), it is required that Λ1 (f1 − ∇p1 ) = 0, Λ1 (g1 − Zp1 − divu1 ) = 0.
(3.3) (3.4)
Using (2.11) and letting K∗ = I + µ−1 ∇SdivA, f1 − ∇p1 = K∗ f1 − ∇Sg1 .
(3.5)
Letting ∇1 = ∇x , ∇2 = ∇y and Sij = −∇i S∇j , Mi = −∇i S∆ + −1 (∇i SZ − K∗ ∇i ),
(3.6)
Inserting f1 and g1 into (3.5), the vector equation (3.3) becomes λ1,11 C11 + λ1,12 C12 + λ1,13 C13 = h1,1 , λ1,21 C11 + λ1,22 C12 + λ1,23 C13 = h1,2 ,
(3.7)
where λ1,11 = µ Λ1 (K∗ ∆φ1 ) + Λ1 (S11 φ1 ), λ1,12 = Λ1 (S12 φ1 ), λ1,13 = Λ1 (M1 φ1 ), λ1,22 = µ Λ1 (K∗ ∆φ1 ) + Λ1 (S22 φ1 ), λ1,21 = Λ1 (S21 φ1 ), λ1,23 = Λ1 (M2 φ1 ), h1,1 = Λ1 (∇x Sg − K∗ f1 ), h1,2 = Λ1 (∇y Sg − K∗ f2 ), f = [f1 , f2 ]. We next derive the algebraic equation for (3.4). Set J1 = ZSdivA − divK1 A, J2 = I − ZS + µ−1 divA∇S, −1
J3 = −(J2 Z + µ
J1 ∇),
Ri = (ZS∇i − ∇i K1 )A∆ + J2 ∇i ,
(3.8)
178
J.R. Kweon
Ann. Henri Poincar´e
where K1 is defined in (2.10). Using (2.11) and (3.8), g1 − Zp1 − divu1 = µ−1 J1 f1 + J2 g1 = C11 (R1 φ1 ) + C12 (R2 φ1 ) + C13 (J2 ∆φ1 + J3 φ1 ) +µ−1 J1 f + J2 g.
(3.9)
Using (3.4), λ1,31 C11 + λ1,32 C12 + λ1,33 C13 = h1,3 ,
(3.10)
where λ1,31 = Λ1 (R1 φ1 ), λ1,32 = Λ1 (R2 φ1 ), λ1,33 = Λ1 (J2 ∆φ1 ) + Λ1 (J3 φ1 ), h1,3 = −Λ1 (µ−1 J1 f + J2 g). Before solving (3.6) and (3.10) for C1j , we need to show thatthe coefficients λ1,ij and h1,j are well defined and the determinant of the matrix λ1,ij 1≤i,j≤3 is not zero. Lemma 3.2 Let s1 < s < s2 . The following functions belong to Hs−2 : K∗ ∆φ1 , K∗ ∇φ1 , Sij φ1 , Mi φ1 , Ri φ1 . Proof.
Consider the following diagram A
S
∇
Hs−2 −→ Hs−1 −→ Hs−2 −→ Hs−1 −→ Hs−2 . div
(3.11)
Since ∆φ1 ∈ Hs−2 and K∗ = I + µ−1 ∇SdivA, we have K∗ ∆φ1 ∈ Hs−2 . Since ∇φ1 ∈ Ht for t < α and 2α − 1 < α, we have ∇φ1 ∈ Hs−2 . Using the diagram (3.11), we see that ∇S∇φ1 , ∇SZφ1 and K1 ∇φ1 belong to Hs−2 . Hence the functions Sij φ1 , Mi φ1 , Ri φ1 are in Hs−2 . Lemma 3.3 Suppose γ0 > 0. The mappings K∗ , J2 and J3 are nontrivial. Proof. Suppose K∗ f = 0 for a nonzero f . We have µf + ∇SdivAf = 0. Set p = −µ−1 SdivAf . So p|Γ = 0 and f − ∇p = 0. Using S = T −1 , Cp = µ−1 divA(∇p − f ) = 0. Since C = A + Z, we have ∇p20 + γ0 p20 ≤ 0, where γ0 = λ − |∇(kU)|∞ . So, if γ0 > 0, p ≡ 0 and f = 0. This is a contradiction. Suppose J2 g = 0 for a nonzero g. Without loss of generality, let g = 0 outside r = 1. Let p = Sg ∈ H10 (Ω1 ). Then J2 g = g − Zp + µ−1 divA∇p = 0. Let
Vol. 5, 2004
An Expansion of Solution for a Regularized Compressible Stokes System
179
u = −µ−1 A∇p. Then g − Zp − divu = 0. From p = Sg, we have Cp + divu = g. Thus ∆p = 0 in Ω1 . Since p|∂Ω1 = 0, p ≡ 0 on Ω1 , so Sg = 0. Since S is invertible, g = 0, which is also a contradiction. Suppose J3 p = 0 for a nonzero p. Without loss of generality we assume that p = 0 outside r = 1. Let u = −µ−1 A∇p. Since K1 = I + µ−1 A∇Sdiv, we have J3 p = −J2 Zp − (ZSdiv − divK1 )u = Zp + divu − ZS(Zp + divu) + µ−1 divA∇S(Zp + divu). Let q = S(Zp + divu) ∈ H10 (Ω1 ) and w = −µ−1 A∇q. Then J3 p = Z(p − q) + div(u − w) = 0. From q = S(Zp + divu), we have ∆q = 0 in Ω1 . Thus q ≡ 0 on Ω1 and Zp + divu = 0 in Ω1 . From u = −µ−1 A∇p, we have µ∆u = ∇p in Ω1 . So µ∇u20,Ω1 + γ0 p20,Ω1 ≤ 0. We have p ≡ 0, which is a contradiction. Using Lemmas 3.2 and 3.3 we obtain Lemma 3.4 (a) The coefficients λ1,ij are well defined. (b) Suppose that γ0 > 0. The determinant of the matrix (λ1,ij )1≤i,j≤3 is a quadratic equation in µ and if µ is not a root of the quadratic equation, the components of C1 solve (3.7) and (3.10). (c) If [f , g] ∈ Hs−2 × Hs−2 , then |C1 | =
3
|C1i | ≤ C(f s−2 + gs−2 ).
(3.12)
i=1
Proof. (a) From Lemma 3.2, the coefficients λ1,ij are well defined. (b) The determinant for the matrix (λ1,ij ) is expressed by d1 (, µ) = a21 (a2 + a3 )µ2 + (c1 + c2 )µ + c3
(3.13)
where ci are some constants and a1 = Λ1 (K∗ ∆φ1 ), a2 = Λ1 (J2 ∆φ1 ), a3 = Λ1 (J3 φ1 ). From Lemma 3.3, the numbers ai = 0. Let = −a3 /a2 . If µ is large enough or if µ is not a root of d1 (, µ) = 0, then d1 (, µ) = 0. Hence the equations (3.7) and (3.10) are solvable for C1j . Estimating h1,j , (3.12) follows. Using Lemmas 3.2 and 3.4, we obtain Theorem 3.1 Let s1 < s < s2 . Suppose γ0 > 0. Assume that [f , g] ∈ Hs−2 × Hs−2 . Let [u, p] be the solution of (1.1) with [u, p] ≡ [0, 0] for r ≥ 1. Then the solution is split as follows: [u, p] = C1 φ1 + [u1 , p1 ],
[u1 , p1 ] = [u, p] − C1 φ1 .
(3.14)
180
J.R. Kweon
Ann. Henri Poincar´e
The constant C1 = [C˜1 , C13 ] = [C11 , C12 , C13 ] is the solution of the equations (3.7), (3.10) and satisfies (3.12). The regular part satisfies µu1 s,Ω1 + p1 s,Ω1 ≤ C[f s−2 + gs−2 ]. Proof. Using Lemma 3.4, C1 solves (3.7) and (3.10), so (3.14) follows from (3.1). We next apply Theorem A to the equations −∆u1 = F and −∆p1 = G where F = µ−1 (f1 − ∇p1 ), G = −1 (g1 − Zp1 − divu1 ). Using (3.5) and (3.9), F = µ−1 (K∗ f1 − ∇Sg1 ),
G = −1 (µ−1 J1 f1 + J2 g1 ).
So µ u1 s,Ω1 ≤ CFs−2 ≤ C(K∗ f1 s−2 + ∇Sg1 s−2 ) p1 s,Ω1
≤ c1 −1 (f1 s−2 + g1 s−2 ), ≤ CGs−2 ≤ C(µ−1 J1 f1 s−2 + J2 g1 s−2 ) ≤ c2 (f1 s−2 + g1 s−2 ),
where ci are generic constants. The constants c1 and c2 are finite because K∗ and ∇S are bounded mappings from Hs−2 into Hs−2 , and J1 , J2 from Hs−2 into Hs−2 . Computing f1 s−2 and g1 s−2 and using (3.12), the inequality follows. Before stopping this step, using (3.6), (3.8), φj and Λj , we define the following numbers that will be used later: for integer j ≥ 1, λj,11 = µ Λj (K∗ ∆φj ) + Λj (S11 φj ), λj,12 = Λj (S12 φj ), λj,13 = Λj (M1 φj ), λj,22 = µ Λj (K∗ ∆φj ) + Λj (S22 φj ), λj,21 = Λj (S21 φj ), λj,23 = Λj (M2 φj ),
(3.15)
λj,31 = Λj (R1 φj ), λj,32 = Λj (R2 φj ), λj,33 = Λj (J2 ∆φj ) + Λj (J3 φj ). / Hs−2 . To extract the second leading corner Step 2. s2 < s < s3 . Recall that ∇φ1 ∈ singularity from [u1 , p1 ], the following terms −C13 ∇φ1 , C˜1 ·∇φ1 and −C13 Zφ1 must be removed in the functions f1 and g1 in (3.2), respectively. For this, let fs,1 = −C13 ∇φ1 ,
gs,1 = C˜1 · ∇φ1 − C13 Zφ1 .
(3.16)
Vol. 5, 2004
Define
An Expansion of Solution for a Regularized Compressible Stokes System
η1,1 = η1 − µ−1 Λ1 (χ fs,1 )φ1 , τ1,1 = τ1 − −1 Λ1 (χ gs,1 )φ1 ,
181
(3.17)
where η1 and τ1 are the solutions of the problems −µ∆η1 = χ fs,1 in Ω, η1 = 0 on Γ, − ∆τ1 = χ gs,1 in Ω, τ1 = 0 on Γ. Since ∇φ1 ∈ Ht−2 for s1 < t < s2 , and from Theorem A, we have [η1,1 , τ1,1 ] ∈ Ht × Ht for s1 < t < s2 . Define ¯ 1 = u1 − η1,1 , u p¯1 = p1 − τ1,1 . Then [¯ u1 , p¯1 ] ∈ Ht × Ht for s1 < t < s2 and solves −µ ∆¯ u1 + ∇¯ p1
= ¯f1 in Ω,
− ∆¯ p1 + Z p¯1 + div¯ u1 ¯ 1 , p¯1 u
= g¯1 in Ω, = 0 on Γ,
(3.18)
where ¯f1 = f + [µ C˜1 − Λ1 (χ fs,1 )]∆φ1 + (1 − χ)fs,1 − ∇τ1,1 ,
(3.19)
g¯1 = g + [ C3 − Λ1 (χ gs,1 )]∆φ1 + (1 − χ)gs,1 − (Zτ1,1 + div η1,1 ). We next show a regularity result for [η1,1 , τ1,1 ] and [¯f1 , g¯1 ]. Lemma 3.5 (i) If s1 < s < s2 , [η1,1 , τ1,1 ]s ≤ C[fs,1 , gs,1 ]s−2 ≤ C|C1 |. (ii) If s2 < s < s3 , [¯f1 , g¯1 ]s−2 ≤ C[f , g]s−2 and [η1,1 , τ1,1 ]s−1 ≤ C|C1 |, provided that [f , g] ∈ Hs−2 × Hs−2 . Proof. (i) Since ∇φ1 ∈ Ht for all t < 2α, and since 3α − 1 < 2α, we have ∇φ1 ∈ Ht for all t < 3α − 1. Using Theorem A, we have, for all t < s2 , [η1,1 , τ1,1 ]t ≤ C[χfs,1 , χgs,1 ]t−2 ≤ C|C1 |.
(3.20)
(ii) For s2 < s < s3 we have, using (3.20), ∇τ1,1 s−2 + Zτ1,1 + divη1,1 s−2 ≤ C[η1,1 , τ1,1 ]s−1 ≤ C|C1 |. Using (3.12), (3.19) and (3.21), [¯f1 , g¯1 ]s−2 ≤ [f , g]s−2 + C(µ + + 2)|C1 | +|C1 |(1 − χ)(|∇φ1 | + Zφ1 )s−2 +∇τ1,1 s−2 + Zτ1,1 + divη1,1 s−2 ≤ C[f , g]s−2 .
(3.21)
182
J.R. Kweon
Ann. Henri Poincar´e
We now subtract the second singular function φ2 from [¯ u1 , p¯1 ]. Define ¯ 1 − C˜2 φ2 , u2 = u
p2 = p¯1 − C23 φ2
(3.22)
where C2 = [C˜2 , C23 ] will be constructed soon. Using (3.22) and (3.18), −µ ∆u2 + ∇p2 − ∆p2 + Zp2 + divu2
= =
f2 in Ω , g2 in Ω ,
u2 , p2
=
0
(3.23)
on Γ,
where f2 = ¯f1 + µ C˜2 ∆φ2 − C23 ∇φ2 , g2 = g¯1 + C23 ( ∆φ2 − Zφ2 ) − C˜2 · ∇φ2 . We are going to construct C2 . In the view of Theorem A, we must require that Λ2 (f2 − ∇p2 ) = 0, Λ2 (g2 − Zp2 − divu2 ) = 0.
(3.24) (3.25)
From(3.24) and (3.25), and using the same procedures as used in Step 1, λ2,11 C21 + λ2,12 C22 + λ2,13 C23 = h2,1 , λ2,21 C21 + λ2,22 C22 + λ2,23 C23 = h2,2 , λ2,31 C31 + λ2,32 C32 + λ2,33 C33 = h2,3 ,
(3.26)
where the coefficients λ2,ij are given in (3.15) and h2,1 = Λ2 (∇x S¯ g1 − K∗ f¯11 ), g1 − K∗ f¯12 ), h2,2 = Λ2 (∇y S¯ h2,3 = −Λ2 (µ−1 J1¯f1 + J2 g¯1 ), where ¯f1 = [f¯11 , f¯12 ] and g¯1 are given in (3.19). We are going to solve the algebraic system (3.26) for C2j and show that λ2,ij and h2,j are well defined and its determinant is not zero. The following lemmas enable us to do them. Lemma 3.6 Let s2 < s < s3 . The following functions belong to Hs−2 : K∗ ∆φ2 , K∗ ∇φ2 , Sij φ2 , Mi φ2 , Ri φ2 . Proof. Since ∆φ2 ∈ Hs−2 and K∗ = I + (µ)−1 ∇SdivA, we have K∗ ∆φ2 ∈ Hs−2 . Since ∇φ2 ∈ Ht for t < 2α, and 3α − 1 < 2α, we have ∇φ2 ∈ Hs−2 . Furthermore ∇S∇φ2 , ∇SZφ2 and K1 ∇φ2 belong to Hs−2 . So Sij φ2 , Mi φ2 , Ri φ2 are in Hs−2 .
Vol. 5, 2004
An Expansion of Solution for a Regularized Compressible Stokes System
183
Lemma 3.7 (a) The coefficients λ2,ij are well defined. (b) Suppose γ0 > 0. The determinant of the matrix (λ2,ij )1≤i,j≤3 is not zero and the parameters C2j (j = 1, . . . , 3) is the solution of (3.26). (c) If [f , g] ∈ Hs−2 × Hs−2 , then |C2 | =
3
|C2i | ≤ C(f s−2 + gs−2 ).
(3.27)
i=1
Proof.
The proof is similar to the one of Lemma 3.4.
Using Lemmas 3.6 and 3.7, we obtain Theorem 3.2 Let s2 < s < s3 . Let [f , g] ∈ Hs−2 × Hs−2 . Suppose γ0 > 0. Let [u, p] be the solution of (1.1) with [u, p] ≡ [0, 0] for r ≥ 1. Then u2 = u −
2
C˜i φi − η1,1 ,
p2 = p −
i=1
2
Ci3 φi − τ1,1
(3.28)
i=1
where C1 is defined in Step 1, and C2 = [C˜2 , C23 ] is the solution of (3.26) and satisfies (3.27). Furthermore µu2 s,Ω1 + p2 s,Ω1 ≤ C(f s−2 + gs−2 )
(3.29)
where C is a constant. Proof. The proof is similar to the one of Theorem 3.1. Using (2.11) and (3.23), we have u2 = µ−1 (K∗ f2 − ∇Sg2 ) and p2 = −1 (µ−1 J1 f2 + J2 g2 ), so µu2 s,Ω1 ≤ c1 (f2 s−2 + −1 g2 s−2 ) and p2 s,Ω1 ≤ c2 (µ−1 f2 s−2 + g2 s−2 ). Estimating f2 s−2 , g2 s−2 , (3.29) follows.
4 High-order expansion Based on the corner singular functions φ2 , φ3 , φ4 , . . . , for the Laplace equation (see Theorem A) and also introducing new associate singular functions [ηj , τj ] to be defined below, we derive a high-order singular expansion for (1.1). Setting fs,2 = −(C23 ∇φ2 + ∇τ1,1 ), gs,2 = C˜2 · ∇φ2 − C23 Zφ2 − (Zτ1,1 + divη1,1 ), the functions f2 and g2 in (3.23) are rewritten by f2 = f + µ g2 = g +
2 i=1 2 i=1
C˜i ∆φi − Λ1 (χfs,1 )∆φ1 + (1 − χ)fs,1 + fs,2 ,
(4.1)
Ci3 ∆φi − Λ1 (χgs,1 )∆φ1 + (1 − χ)gs,1 + gs,2 .
(4.2)
184
J.R. Kweon
Ann. Henri Poincar´e
Let χ be a smooth function which vanishes for r > 1 and which is 1 in a neighborhood of the origin. Let j ≥ 2 be an integer. Assume that the parameters Ci = [Ci1 , Ci2 , Ci3 ] (1 ≤ i ≤ j) have been constructed in the previous steps. Define the associate singular functions [ηj , τj ] to satisfy −µ∆ηj = χfs,j in Ω,
ηj = 0 on Γ,
(4.3)
− ∆τj = χgs,j in Ω,
τj = 0 on Γ,
(4.4)
where fs,j = −(Cj3 ∇φj + ∇τj−1,1 ), gs,j = C˜j · ∇φj − Cj3 Zφj − (Zτj−1,1 + div ηj−1,1 ). Define the smoother part of the associate singular function as follows: −1
ηj,1 = ηj − µ
j
Λi (χfs,j )φi ,
(4.5)
Λi (χgs,j )φi .
(4.6)
i=1
τj,1 = τj − −1
j i=1
Next we subtract the smoother part [ηj,1 , τj,1 ] from [uj , pj ] so that the resulted right-hand sides at the (j + 1)th-step have an enough regularity. Define ¯ j = uj − ηj,1 , u
p¯j = pj − τj,1 .
(4.7)
Then [¯ uj , p¯j ] ∈ Ht × Ht for sj < t < sj+1 and solves −µ ∆¯ uj + ∇¯ pj
= ¯fj
in Ω,
− ∆¯ pj + Z p¯j + div¯ uj ¯ j , p¯j u
= g¯j
in Ω,
= 0
on Γ,
(4.8)
where ¯fj = f + µ
j
C˜i ∆φi + (1 − χ)
i=1
g¯j = g +
j
fs,i + Φj (χfs,j , φ) − ∇τj,1 ,
(4.9)
i=1
j
Ci3 ∆φi + (1 − χ)
i=1
j
gs,i + Φj (χgs,j , φ) − (Zτj,1 + divηj,1 ),
i=1
Φj (zj , φ) = − Λ1 (z1 )∆φ1 +
2 i=1
Λi (z2 )∆φi + · · · +
j
Λi (zj )∆φi .
i=1
Next we establish regularities for [fs,j , gs,j ], [ηj,1 , τj,1 ] and [¯fj , g¯j ].
Vol. 5, 2004
An Expansion of Solution for a Regularized Compressible Stokes System
185
Lemma 4.1 Let j ≥ 2 be an integer. (a) For sj < s < sj+1 , we have [fs,j , gs,j ]s−2 j ≤ C i=1 |Ci | and [ηj,1 , τj,1 ]s ≤ C[fs,j , gs,j ]s−2 . (b) For sj+1 < s < sj+2 , we j have [ηj,1 , τj,1 ]s−1 ≤ C i=1 |Ci | and [¯fj , g¯j ]s−2 ≤ C[f , g]s−2 , provided that [f , g] ∈ Hs−2 × Hs . Proof. The proof follows by an induction argument. The case j = 2 easily follows. Assume that they are true for j − 1. If s < sj , then φj ∈ Hs , and [ηj−1,1 , τj−1,1 ] ∈ Hs × Hs for all s < sj by the assumption, so the functions: ∇φj , Zφj , ∇τj−1,1 , Zτj−1,1 , divηj−1,1 belong to Ht or Ht for t < jα. Hence, since jα − (sj+1 − 1) = 1 − α > 0, we have [fs,j , gs,j ] ∈ Hs−2 × Hs−2 for all s < sj+1 . Thus, when we apply Theorem 3.1 to (4.3)–(4.6), we have [ηj,1 , τj,1 ] ∈ Hs × Hs for s < sj+1 . Since j 2 Φj (χfs,j , φ)s−2 ≤ C |Λ1 (χfs,1 )| + |Λi (χfs,2 )| + · · · + |Λi (χfs,j )| i=1
i=1
≤ C[f , g]s−2 and similarly Φj (χgs,j , φ)s−2 ≤ C[f , g]s−2 , we have [¯fj , g¯j ]s−2 ≤ [f , g]s−2 + C(µ + + 1)
j
|Ci | + C[ηj,1 , τj,1 ]s−1
i=1
+[Φj (χfs,j , φ), Φj (χgs,j , φ)]s−2 ≤ C[f , g]s−2 . Thus the inequality follows. We split the (j + 1)-th singular function φj+1 from the solution [¯ uj , p¯j ] of (4.8): ¯ j = uj+1 + C˜j+1 φj+1 , u
p¯j = pj+1 + Cj+1,3 φj+1
(4.10)
where C˜j+1 = [Cj+1,1 , Cj+1,2 ] and Cj+1,3 are parameters to be determined later. Inserting (4.10) into (4.8), system (4.8) becomes −µ ∆uj+1 + ∇pj+1 − ∆pj+1 + Zpj+1 + divuj+1
= =
fj+1 in Ω, gj+1 in Ω ,
uj+1 , pj+1
=
0
(4.11)
on Γ,
where fj+1 = ¯fj + µ C˜j+1 ∆φj+1 − Cj+1,3 ∇φj+1 , gj+1 = g¯j + C˜j+1 · ∇φj+1 + Cj+1,3 ( ∆φj+1 − Zφj+1 ). As in the previous steps, we construct the parameter Cj+1 and establish the regularity of the solution [uj+1 , pj+1 ] of (4.11).
186
J.R. Kweon
Ann. Henri Poincar´e
Let l = j + 1, for simplicity. Requiring that Λl (fl − ∇pl ) = 0,
(4.12)
Λl (gl − Zpl − divul ) = 0,
(4.13)
one can have, like the previous steps, λl,11 Cl1 + λl,12 Cl2 + λl,13 Cl3 = hl,1 , λl,21 Cl1 + λl,22 Cl2 + λl,23 Cl3 = hl,2 , λl,31 Cl1 + λl,32 Cl2 + λl,33 Cl3 = hl,3 ,
(4.14)
where the coefficients λl,ij are given in (3.15) and gj − K∗ f¯j1 ), hl,1 = Λl (∇x S¯ hl,2 = Λl (∇y S¯ gj − K∗ f¯j2 ), hl,3 = −Λl (µ−1 J1¯fj + J2 g¯j ),
(4.15) ¯fj = [f¯j1 , f¯j2 ].
Using (2.11) and like Lemmas 3.4 and 3.7, one can show that the coefficients λl,ij are well defined. The determinant is of the form dl (µ, ) = a21 (a2 + a3 )µ2 + (c1 + c2 )µ+ c3 , where a1 = Λl (K∗ ∆φl ), a2 = Λl (J2 ∆φl ), a3 = −Λl (J2 Zφl ), which are not zero by Lemma 3.3. So the determinant is not zero by the proof of Lemma 3.4. Thus Cl = [Cl1 , Cl2 , Cl3 ] solves (4.14) and satisfies |Cl | ≤ C(¯fj s−2 + ¯ gj s−2 ) ≤ C[f , g]s−2 .
(4.16)
Since [fl , gl ] ∈ Hs−2 × Hs−2 (s < sl+1 ), and using (2.12) and (4.11), µul = K1 Afl − A∇Sgl , pl = −µ−1 SdivAfl + Sgl , which are in Hs (Ω) and Hs (Ω), respectively. Summarizing all results obtained and the ones of Section 3, we obtain: Theorem 4.1 Suppose that γ0 > 0, in other words, either λ is large enough or U is close to a constant function. Assume that [u, p] vanish for r > 1. For an integer l ≥ 1, if sl < s < sl+1 and [f , g] ∈ Hs−2 × Hs−2 , then [u, p] = [us , ps ] + [uR , pR ] with [uR , pR ] = [u, p] − [us , ps ] and [us , ps ] =
l i=1
[C˜i , Ci3 ]φi +
l−1
[ηi,1 , τi,1 ].
i=1
Here [ηi,1 , τi,1 ] is the smoother part of the new associate singular function [ηi , τi ] defined by (4.3) and (4.4), respectively, the constant [C˜i , Ci3 ] with C˜i = [Ci1 , Ci2 ] is the solution of (4.14) and satisfies (4.16). Moreover, the regular part [uR , pR ] satisfies µuR s,Ω1 + pR s,Ω1 ≤ C(f s−2 + gs−2 ) where C is a constant.
Vol. 5, 2004
Proof.
An Expansion of Solution for a Regularized Compressible Stokes System
The proof is similar to the ones of Theorems 3.2 and 3.3.
187
Thus Theorem 1.1 is obtained, which follows from a combination of Lemma 2.2 and Theorem 4.1.
References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] J.D. Anderson, Jr., Fundamentals of Aerodynamics, 2nd ed., McGraw-Hill, New York, 1991. [3] H. Beir˜ ao da Veiga, An Lp -theory for the n-dimensional, Stationary, Compressible Navier-Stokes Equations, and the Incompressible Limit for Compressible Fluids. The Equilibrium Solutions, Commun. Math. Phys. 109, 229– 248 (1987). [4] M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or threedimensional domains with corners. Part I: Linearized Equations, SIAM J. Math. Anal. 20, 74–97 (1989). [5] V. Girault, P.-A. Raviart, Finite element methods for Navier-Stokes equations: Theory Algorithms, Springer-Verlag, 1986. [6] P. Grisvard, Elliptic problems in non-smooth domains, Pitman Advanced Publishing Program, Boston. London. Melbourne, 1985. [7] R.B. Kellogg, Corner singularities and singular perturbations, Ann. Univ. Ferrara – Sez. VII – Sc. Mat., Vol. XLVII, 177–206 (2001). [8] R.B. Kellogg, J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal. 21, 397–431 (1976). [9] V.A. Kozlov, V.G. Maz’ya, J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, AMS, 2001. [10] J.R. Kweon, R.B. Kellogg, Compressible Stokes problem on non-convex polygon, J. Differential Equations 176, 290–314 (2001). [11] J.R. Kweon, R.B. Kellogg, Regularity of solutions to the Navier-Stokes equations for compressible barotropic flows on a polygon, Arch. Rational Mech. Anal. 163 1, 35–64 (2002). [12] J.R. Kweon, A regularity result of solution to the compressible Stokes equations on a convex polygon, to appear in ZAMP. [13] J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag Berlin Heidelberg New York, 1972.
188
J.R. Kweon
Ann. Henri Poincar´e
[14] S.A. Nazarov, A. Novotny, K. Pileckas, On steady compressible Navier-Stokes equations in plane domains with corners, Math. Ann. 304 1, 121–150 (1996). Jae Ryong Kweon1 Department of Mathematics Pohang University of Science and Technology Pohang 790–784 Korea email:
[email protected] Communicated by Rafael D. Benguria submitted 14/11/02, revised 12/08/03, accepted 04/10/03
To access this journal online: http://www.birkhauser.ch
1 This
work was supported by Korea Research Foundation Grant(KRF–2001–015–DS0002)
Ann. Henri Poincar´e 5 (2004) 189 – 201 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010189-13 DOI 10.1007/s00023-004-0165-9
Annales Henri Poincar´ e
Outgoing Radiation from an Isolated Collisionless Plasma Simone Calogero Abstract. The asymptotic properties at future null infinity of the solutions of the relativistic Vlasov-Maxwell system whose global existence for small data has been established by the author in a previous work are investigated. These solutions describe a collisionless plasma isolated from incoming radiation. It is shown that a non-negative quantity associated to the plasma decreases as a consequence of the dissipation of energy in form of outgoing radiation. This quantity represents the analogue of the Bondi mass in general relativity.
1 Introduction and main results The dynamics of a collisionless plasma in interaction with the mean electromagnetic field generated by the charges is described by the relativistic Vlasov-Maxwell system. In this model the unknowns are the electromagnetic field (E, B) and a set of N non-negative functions fα which give the distributions in phase space of N different species of particles. The system consists of the Vlasov equation ∂t fα + pα · ∂x fα + qα (E + pα ∧ B) · ∂p fα = 0,
∀ α = 1, . . . , N,
coupled to the Maxwell equations with charge density ρ and current density j given by ρ(t, x) = R3
N
qα fα dp,
j(t, x) =
α=1
N
R3
qα fα pα dp.
(1.1)
α=1
In the previous equations, t ∈ R is the time, x ∈ R3 , p ∈ R3 are the position and the momentum of the particles, qα the charge of a particle of species α, p pα = , m2α + p2
p2 ≡ |p|2
denotes the relativistic velocity and mα is the mass of a particle of species α. Units are chosen so that the speed of light is equal to unity. The relativistic Vlasov-Maxwell system has several applications in plasma physics and in astrophysics, where it is used for instance to model the dynamics of the solar wind. Many mathematical problems remain unsolved. For example, existence of global classical solutions is known only under certain restrictions on the size of the initial data (see [2, 5, 6, 10]); global existence for large data has
190
S. Calogero
Ann. Henri Poincar´e
been proved for a modified version of the system in which the particle density is forced to have compact support in the momentum (see [4]). An important feature of the dynamics, which is due to its relativistic character, is the presence of radiation fields. The radiation is defined as the part of the electromagnetic field which carries energy to null infinity. It is distinguished in outgoing radiation, which propagates energy to the future null infinity I + , and incoming radiation, which propagates energy to the past null infinity I − . The latter can be interpreted as a flux of energy flowing in onto the system from I − . For an isolated system the incoming radiation should be ruled out by appropriate boundary conditions, which will be now briefly discussed. The amount of energy Ein (v1 , v2 ) carried to I − by the incoming radiation in the interval [v1 , v2 ] of the advanced time, v = t + |x|, can be formally calculated by the limit v2 Ein (v1 , v2 ) = − lim [S · k](s − r, x)dx ds, (1.2) r→+∞
v1
|x|=r
where k = x/|x| and S is the Poynting vector, S = (4π)−1 (E ∧ B). It should be emphasized that (1.2) is only a formal definition, since it is not known in general whether the above limit exists for a solution of the relativistic Vlasov-Maxwell system. Analogously, the limit u2 [S · k](s + r, x)dx ds (1.3) Eout (u1 , u2 ) = lim r→+∞
u1
|x|=r
gives the energy which is propagated to I + by the outgoing radiation in the interval [u1 , u2 ] of the retarded time, u = t − |x|. A solution of the relativistic Vlasov-Maxwell system is isolated from incoming radiation if Ein (v1 , v2 ) = 0, for all v1 , v2 ∈ R. In [2] it was proved that for small data this system admits solutions which satisfy this property. These solutions are defined by replacing the Maxwell equations with the retarded part of the field. The resulting system has been called retarded relativistic Vlasov-Maxwell system and reads ∂t fα + pα · ∂x fα + qα (Eret + pα ∧ Bret ) · ∂p fα = 0, ∀ α = 1, . . . , N, dy , (∂x ρ + ∂t j)(t − |x − y|, y) Eret (t, x) = − |x − y| 3 R dy , Bret (t, x) = (∂x ∧ j)(t − |x − y|, y) |x − y| 3 R
(1.4) (1.5) (1.6)
where ρ and j are given by (1.1). The purpose of this paper is to derive information about the asymptotic behaviour at future null infinity of such isolated solutions, i.e., to study the properties of the outgoing radiation generated by the plasma. Let us first recall, for sake of reference, the global existence result of [2]. Define P = sup{|p| : (x, p) ∈ suppfα (t), t ∈ R, 1 ≤ α ≤ N }
Vol. 5, 2004
Outgoing Radiation from an Isolated Collisionless Plasma
191
and denote by BR (0) the sphere in R6 with center in the origin and radius R > 0 and by λ the set of constants {qα , mα }. Theorem 1 Let fαin (x, p) ≥ 0 be given in C02 (R3 × R3 ) such that fαin = 0 for 2 (x, p, α) ∈ BR (0)c × {1, . . . , N }. Define ∆ = α |µ|=0 ∇µ fαin ∞ , where µ ∈ N6 is a multi-index. Then there exists a positive constant ε = ε(R, λ) such that for ∆ ≤ ε the retarded relativistic Vlasov-Maxwell system has a unique global solution {fα } ∈ (C 2 )N which satisfies fα (0, x, p) = fαin (x, p). Moreover Fret = (Eret , Bret ) ∈ C 1 (R × R3 ) and there exists a positive constant C = C(R, λ) such that P ≤ C and the following estimates hold for all (t, x) ∈ R × R3 : |Fret (t, x)| ≤ C∆(1 + |t| + |x|)−1 (1 + |t − |x||)−1 ,
(1.7)
ρ(t, x) ≤ C∆(1 + |t| + |x|)−3 .
(1.8)
The estimate (1.7) shows that the solution of Theorem 1 is isolated from incoming radiation in the sense specified above. We note that the statement of Theorem 1 differs from the main result of [2] in two aspects. Firstly in [2] only the case of a single species of particles is considered. However the restriction to this case has been made only to simplify the notation and the generalization of the result of [2] to the case of a mixture is straightforward. Secondly we claim here that the distribution functions are twice continuously differentiable, whereas in [2] we only proved that they are C 1 . We shall return to this point at the end of the introduction. We state now the main results of this paper. Let us define 1 2 2 |Eret | + |Bret | + e(t, x) = m2α + p2 fα dp, 8π 3 R α 1 (Eret ∧ Bret ) + p(t, x) = 4π α
R3
pfα dp,
the local energy and momentum of a solution of (1.4)–(1.6), respectively. We also set [e − p · k](u + |x|, x) dx (1.9) M∨ (u) = R3
∨
and note that M (u) is non-negative. Theorem 2 Let {fα } ∈ (C 2 )N be a solution of the retarded relativistic VlasovMaxwell system with data as stated in Theorem 1, such that Fret ∈ C 1 (R × R3 ) and fα (t, x, p) = 0, ∀x ∈ R3 : |x| ≥ R + a|t|, α = 1, . . . , N, (1.10) for some a ∈ [0, 1). Then the limit F rad (u, k) =
lim
|x|→+∞
|x|Fret (u + |x|, x)
(1.11)
192
S. Calogero
Ann. Henri Poincar´e
exists and is attained uniformly in k = x/|x| ∈ S 2 and u ∈ K, for all K ⊂ R compact. Moreover the radiation field F rad = (E rad , B rad ) satisfies the following algebraic properties: E rad · B rad = 0,
|E rad | = |B rad |,
E rad · k = B rad · k = 0, (E rad ∧ B rad ) · k = |E rad |2 . Theorem 3 Let {fα } ∈ (C 2 )N be a solution of the retarded relativistic VlasovMaxwell system with data as stated in Theorem 1 and such that Fret ∈ C 1 (R× R3 ). Assume the following estimates hold (i) P ≤ C (ii) |Fret (t, x)| ≤ C(1 + |t| + |x|)−1 (1 + |t − |x||)−1 (iii) ρ(t, x) ≤ C(1 + |t| + |x|)−3 for some positive constant C and for all (t, x) ∈ R × R3 . Then M∨ ∈ C 1 and the following equation is satisfied: d 1 M∨ (u) = − |E rad (u, k)|2 dk. (1.12) du 4π S 2 Note that the conclusions of Theorems 2 and 3 apply to the solution of Theorem 1. However the proofs do not require the solution to be small. In fact the proofs of Theorems 2 and 3 do not make use explicitly of the Vlasov equation either. The only tools which enter into play are the continuity equation, ∂t ρ + ∇x · j = 0, and the local energy conservation law, ∂t e+∇x ·p = 0, which are of course satisfied by any “good” matter model. However since the existence of global solutions of the retarded relativistic Vlasov-Maxwell system which satisfy the assumptions of Theorems 2 and 3 is known by Theorem 1, we restrict ourselves to consider this case. Let us now comment the meaning of the results stated above. In Theorem 2 it is claimed that the retarded field generated by the charge distribution is asymptotically null and outwardly directed along the future pointing null geodesics. (We recall that an electromagnetic field (E, B) is said to be null if E · B = 0 and |E| = |B|, cf. [11], page 322.) This result follows essentially from [7]. However, for sake of completeness and to help the reader who is not familiar with the formalism used in [7], we give in Section 2 a complete proof of Theorem 2 adapted to our case. The property (1.10), which for the solution of Theorem 1 follows from the estimate P ≤ C, is in turn a special case of the assumption made in [7] that the matter has to be contained in a timelike world-tube. With regard to Theorem 3, it shows that the function M∨ plays in the context of the retarded relativistic Vlasov-Maxwell system the same role as the Bondi mass in general relativity (cf. [1, 12]). In fact, M∨ is non-increasing and its variation
Vol. 5, 2004
Outgoing Radiation from an Isolated Collisionless Plasma
193
on the interval [u1 , u2 ] equals the energy dissipated in form of outgoing radiation in such interval of the retarded time, as it follows from (1.3) and Theorem 2. Although the Bondi mass loss formula in general relativity is extensively studied, it seems the first time that its generalization to plasma physics is considered and that (1.12) appears in the literature. It should be mentioned, however, that a rigorous mathematical derivation of the Bondi formula in general relativity is a much more difficult task (cf. [3, 8]). Let us now deal with the technical point concerning the regularity of the solution. We consider for simplicity the system for a single species of particle and denote by fret the unique global solution for small C 2 data. In [2] we stated that fret ∈ C 1 ; here we claim that fret ∈ C 2 . This gain of regularity can be easily understood by appealing to the smoothing effect which was pointed out in [9]. We recall that the solution of the Vlasov equation can be represented as fret = f in (X(0), P (0)) where X(s), P (s) are the characteristics of (1.4) and are given by s P(τ ) dτ, X(s) = x + P (s) = p + t
t s
Eret (τ, X) + P ∧ Bret (τ, X) dτ.
It was observed in [9] that the time integral of the field evaluated on the characteristics is one derivative smoother than the field itself provided that X(s) is a timelike curve. The latter condition is satisfied by the solution of [2] in virtue of the estimate P ≤ C. Hence the characteristics are C 2 and since f in is also given as a C 2 function, then the solution of the Vlasov equation itself is twice continuously differentiable.
2 Algebraic properties of the radiation field In this section we prove Theorem 2. Let us denote by φ any of the components of the electromagnetic field, i.e., we set φ = Ei or Bi and define
−(∂t ji + ∂xi ρ) for the electric field E, F = for the magnetic field B. (∂x ∧ j)i Then F ∈ C 1 (R × R3 ) and by means of (1.10), F (t, x) = 0 for |x| ≥ R + a|t|. The retarded field defined by (1.5), (1.6) has the form dy , F (t − |x − y|, y) φ(t, x) = |x − y| Ξa (t,x) where Ξa (t, x) = {y ∈ R3 : |y| ≤ R + a|t − |x − y||}, which is a compact set for any fixed t ∈ R, x ∈ R3 . We have the following
194
S. Calogero
Lemma 1
Ann. Henri Poincar´e
lim
|x|→+∞
|x| φ(u + |x|, x) =
F (u + k · y, y) dy, Ωa (u)
uniformly in (u, k) ∈ K × S 2 , for all K ⊂ R compact, where Ωa (u) = {y ∈ R3 : |y| ≤ (1 − a)−1 (R + a|u|)}. Proof. The crucial point to prove this lemma is that for t = u + |x| and for |x| large, i.e., where we need to evaluate the function φ, the domain of integration Ξa (t, x) is contained in Ωa (u), a compact set whose measure depends only on the fixed u. Let ρ = |x|−1 and define g(ρ) = [1 − 2ρk · y + ρ2 |y|2 ]−1/2 =
h(ρ) = =
|x| , |x − y|
ρ−1 [1 − ρk · y − (1 − 2ρk · y + ρ2 |y|2 )1/2 ] |x| − k · y − |x − y|, for ρ = 0
and put h(0) = 0, h (0) = 32 (k · y)2 − 12 |y|2 , so that h ∈ C 1 (R) (here the prime denotes the derivative with respect to ρ). Then we have F (u + k · y + h(ρ), y) g(ρ) dy. lim |x| φ(u + |x|, x) = lim ρ→0
|x|→+∞
Ωa (u)
Setting Gu,k (ρ, y) = F (u + k · y + h(ρ), y)g(ρ), we have to prove that lim Gu,k (ρ, y) dy − Gu,k (0, y) dy = 0, ρ→0
Ωa (u)
Ωa (u)
uniformly in (u, k) ∈ K × S 2 . By the mean value theorem we have Gu,k (ρ, y) = Gu,k (0, y) + ρ Ru,k (ρ, y), where the remainder is bounded as |Ru,k (ρ, y)| ≤ sup{|Gu,k (τ, y)|, 0 ≤ τ ≤ ρ}. Hence
Gu,k (ρ, y) dy − Ωa (u)
Gu,k (0, y) dy
Ωa (u)
≤ ρ sup |Gu,k (τ, y)|, 0 ≤ τ ≤ ρ, y ∈ Ωa (u) Vol[Ωa (u)]. Since G is C 1 and Ωa (u) is compact, the lemma is proved.
Vol. 5, 2004
Outgoing Radiation from an Isolated Collisionless Plasma
195
By means of Lemma 1, the radiation field is continuous and is given by Eirad (u, k) = − (∂i ρ + ∂t ji )(u + k · y, y) dy, (2.1) Ωa (u)
(∂x ∧ j)i (u + k · y, y) dy.
Birad (u, k) =
(2.2)
Ωa (u)
It remains to prove the algebraic properties of (E rad , B rad ). In (2.1) we replace the identities ∂t ji (u + k · y, y) = ∂u ji (u + k · y, y), ∂i ρ(u + k · y, y) =
∂yi [ρ(u + k · y, y)] + ∂y · [j(u + k · y, y)]ki −∂u (j · k)(u + k · y, y)ki ,
the second one being a consequence of the continuity equation. After integrating by parts we get rad Ei (u, k) = ∂u [(j · k)ki − ji ](u + k · y, y) dy. Ωa (u)
Let M denote the vector [(j · k)k − j](u + k · y, y) dy.
M (u, k) =
(2.3)
Ωa (u)
Since the integrand function in (2.3) is C 1 and vanishes on the boundary of Ωa (u), then we have E rad = ∂u M . Analogously, from (2.2) we get B rad = ∂u N , where (j ∧ k)(u + k · y, y) dy.
N (u, k) = Ωa (u)
The next lemma describes the algebraic properties of the vectors M , N . Lemma 2 ∀(u, k) ∈ R × S 2 the vector fields defined by (2.3), (2.4) satisfy (1) M (u, k) · k = N (u, k) · k = 0 (2) M (u, k) · N (u, k) = 0 (3) |M (u, k)| = |N (u, k)| (4) (M (u, k) ∧ N (u, k)) · k = |M (u, k)|2 .
(2.4)
196
S. Calogero
Ann. Henri Poincar´e
Proof. The proof of (1) is straightforward. For (2) we put j = j(u + k · y, y) and j = j(u + k · y , y ) for short and write M ·N = [j − (j · k)k] · (j ∧ k) dy dy = j · (j ∧ k) dy dy = − j · (j ∧ k) dy dy, where it is understood that the integrals are over the set Ωa (u). Then interchanging y and y we get M · N = −M · N , i.e., M · N = 0. To prove (3) we write |M |2 = (j − (j · k)k) · (j − (j · k)k) dy dy = [j · j − (j · k)(j · k)] dy dy
and |N |2 =
(j ∧ k)(j ∧ k) dy dy.
In the previous equation we use the following rule of vector calculus (a ∧ b) · (c ∧ d) = (a · c)(b · d) − (a · d)(b · c), which is valid for any vectors a, b, c, d and the identity (3) follows at once. To prove (4) we write (M ∧ N ) · k = [((j · k)k − j) ∧ (j ∧ k)] · k dy dy = − [((j · k)k − j) ∧ k] · (j ∧ k) dy dy = (j ∧ k)(j ∧ k) dy dy = |N |2 = |M |2 . The following lemma permits to relate the algebraic properties of the vectors M , N to the ones of the radiation field and concludes the proof of Theorem 2. To simplify the notation we suppress the dependence on k and denote by an upper dot the differentiation with respect to u. Lemma 3 Let M (u), N (u) be C 1 vector fields satisfying the properties (1)–(4) of Lemma 2. Then ∀u ∈ R: (a) N˙ (u) · k = M˙ (u) · k = 0
Vol. 5, 2004
Outgoing Radiation from an Isolated Collisionless Plasma
197
(b) M˙ (u) · N˙ (u) = 0 (c) |M˙ (u)| = |N˙ (u)| (d) (M˙ (u) ∧ N˙ (u)) · k = |M˙ (u)|2 . Proof. The identity (a) is proved at once by differentiating (1) with respect to u. To prove the other identities we consider a coordinate system in which the z-axis is parallel to k. In this frame the vectors M and N have the form M (u) = (m1 (u), m2 (u), 0), N (u) = (n1 (u), n2 (u), 0). Now, because of (2), (3) and (4) of Lemma 2: m1 (u)n1 (u) + m2 (u)n2 (u) = 0,
(2.5)
m1 (u)2 + m2 (u)2 = n1 (u)2 + n2 (u)2 , m1 (u)n2 (u) − m2 (u)n1 (u) = m1 (u)2 + m2 (u)2 .
(2.6) (2.7)
After some elementary algebra, (2.5)–(2.7) give m1 = n2 , m2 = −n1 . Hence the vectors M, N can be represented in the following form: M (u) = (q(u), r(u), 0) ⇒ M˙ = (q, ˙ r, ˙ 0), N (u) = (−r(u), q(u), 0) ⇒ N˙ = (−r, ˙ q, ˙ 0), by which the properties (b), (c) and (d) follow at once.
3 Bondi mass of the plasma In this section we prove Theorem 3. Let us introduce m∨ (r, u) = [e − p · k](u + |x|, x) dx.
(3.1)
|x|≤r
Note that m∨ (·, u) is non-decreasing and so its limit as r → +∞ exists. We first prove (1.12) assuming that ( ) m∨ (r, u) converges as r → +∞ for all u ∈ R. Assume ( ) holds. Then M∨ (u) is well defined as improper integral and we have M∨ (u) = lim m∨ (r, u). r→+∞
Let K be a generic compact subset of R. Evaluating the energy conservation law ∂t e = −∂x · p on the future light cone corresponding to the value u of the retarded time we have ∂u e(u + |x|, x) = −∂x · p(u + |x|, x).
198
S. Calogero
Ann. Henri Poincar´e
In the previous equation we use the identity ∂x · p(u + |x|, x) = ∂x · [(p(u + |x|, x)] − ∂u (p · k)(u + |x|, x) and so doing we get ∂u (e − p · k)(u + |x|, x) = −∂x · [p(u + |x|, x)].
(3.2)
We now integrate (3.2) on the region |x| ≤ r, and use the Gauss theorem to transform the right-hand side into a surface integral over the sphere of radius r. Since f (t, x, p) is supported on the region |x| ≤ R + a|t|, with a ∈ [0, 1) then for r large enough, f (u + |x|, x, p) vanishes on Sr = {x : |x| = r} and so we get ∂u (e − p · k)(u + |x|, x) dx lim r→+∞ |x|≤r = −(4π)−1 lim (Eret ∧ Bret ) · k (u + r, x) dSr . (3.3) r→+∞
Sr
The integral in the left-hand side of (3.3) is equal to ∂u m∨ (r, u), ∀r > 0. By Theorem 2 and (3.3) , ∂u m converges to the right-hand side of (1.12), uniformly in u ∈ K, as r → +∞. Hence, using ( ) we infer that m∨ (r, u) converges uniformly in u ∈ K and also d d lim m∨ (r, u) = M∨ (u). lim ∂u m∨ (r, u) = r→+∞ du r→+∞ du Thus the proof of (1.12) is complete if we show that the property ( ) above is satisfied. We note that to this purpose, a direct use of the estimate (ii) in Theorem 3 is not enough, since it entails |E(u + |x|, x)| = O(|x|−1 ), as |x| → ∞, which is too weak to bound M∨ . To solve this problem we rewrite M∨ in a way that the decay at past null infinity, which is faster by means of the absence of incoming radiation, enters into the estimates (in other words, we make the advanced time v appear instead of the retarded time u). For this purpose we introduce, besides m∨ (r, u), the following function (e + p · k)(v − |x|, x) dx. (3.4) m∧ (r, v) = |x|≤r
We also set M∧ (v) = lim m∧ (r, v). r→+∞
(3.5)
Lemma 4 Under the assumptions of Theorem 3, M∧ (v) is continuously differentiable and satisfies d M∧ (v) = 0, ∀v ∈ R. (3.6) dv
Vol. 5, 2004
Outgoing Radiation from an Isolated Collisionless Plasma
199
Proof. We split M∧ (v) into four parts as follows: M∧ (v) = I1 (v) + I2 (v) + I3 (v) + I4 (v), where
1 (|Eret |2 + |Bret |2 )(v − |x|, x) dx, 8π R3 I2 (v) = m2α + p2 fα (v − |x|, x, p) dp dx, I1 (v) =
R3
α
1 4π
I3 (v) = I4 (v) =
|p|≤C
α
R3
(Eret ∧ Bret ) · k (v − |x|, x) dx,
R3
|p|≤C
p · k fα (v − |x|, x, p) dp dx.
Since I3 (resp. I4 ) is dominated by I1 (resp. I2 ), it suffices to estimate I1 (v) and I2 (v). By means of (ii), I1 (v) is uniformly bounded in v ∈ K. For I2 (v) we use that fα (v − |x|, x, p) = 0 for |x| ≥ R + a|v − |x||, which implies fα (v − |x|, x, p) = 0 for |x| ≥ (1 − a)−1 (R + a|v|). Thus |I4 (v)| ≤ CVol[{x : |x| ≤ (1 − a)−1 (R + a|v|)}] ≤ C,
∀v ∈ K.
Since m∧ (·, v) is non-decreasing, the limit (3.5) exists for all v ∈ R and by the previous estimates, M∧ (v) converges, as improper integral, uniformly in v ∈ K. To prove (3.6) we repeat the argument which led to (3.2), now evaluating on the past light cone. So doing we get ∂v (e + p · k)(v − |x|, x) = −∂x · [p(v − |x|, x)]. Integrating and using the Gauss theorem we obtain, lim ∂v (e + p · k)(v − |x|, x) dx r→+∞ |x|≤r (Eret ∧ Bret ) · k (v − r, x) dSr , = −(4π)−1 lim r→+∞
(3.7)
(3.8)
Sr
where again we used that f (v − r, x, p) vanishes on Sr for large r. The estimate (ii) implies that the right-hand side of (3.8) tends to zero uniformly in v ∈ K and so we get 0 = lim ∂v (e + p · k)(v − |x|, x) dx r→+∞ |x|≤r d d lim M∧ (v), (e + p · k)(v − |x|, x) = = dv r→+∞ |x|≤r dv where the uniform convergence has been used to shift the derivative.
200
S. Calogero
Ann. Henri Poincar´e
Note that in the proof of Lemma 4 the estimate (iii) in Theorem 3 has not been used. This will become important for the completion of the proof of ( ). The identity (3.6) represents the counterpart of (1.12) on the backward cones of light. In fact, from one hand the conservation of M∧ (v) is due to the absence of incoming radiation and, on the other hand, the Bondi mass M∨ (u) decreases as a consequence of the emission of outgoing radiation. We are able now to complete the proof of ( ). Integrating (3.7) between u and u + 2|x| we get u+2|x| (e + p · k)(u + |x|, x) − (e + p · k)(u − |x|, x) = − ∂x · [p(v − |x|, x)] dv. u
Integrating in the region |x| ≤ r we get (e + p · k)(u + |x|, x) dx = |x|≤r
|x|≤r
(e + p · k)(u − |x|, x) dx
− Now we use the identity u+2|x| p(v − |x|, x) dv] ∂x · [ u
=
u+2|x|
u
|x|≤r
(3.9)
∂x · [p(v − |x|, x)] dv dx.
2p · k (u + |x|, x)
u+2|x|
+ u
∂x · [p(v − |x|, x)] dv.
Substituting into (3.9) and using the Gauss theorem we obtain m∨ (r, u) = m∧ (r, u) + q(r, u). where
q(r, u) = −
u+2r
Sr
u
α
Sr
(3.10)
p · k (v − r, x) dv dSr
u+2r 1 (Eret ∧ Bret ) · k (v − r, x) dv dSr 4π Sr u u+2r − p · k fα (v − r, x) dv dSr .
= −
u
|p|≤C
Using (ii) in the first term and (iii) in the second term, we conclude that limr→+∞ |q(r, u)| is bounded for all u ∈ R and so the property ( ) follows from (3.10) and Lemma 4. This concludes the proof of Theorem 3. Acknowledgments. The results presented in this paper have been obtained while the author was preparing his PhD thesis at the Albert Einstein Institute in Potsdam, which is thereby acknowledged for the hospitality. Support from the European Network HYKE (contract HPRN-CT-2002-00282) is also acknowledged.
Vol. 5, 2004
Outgoing Radiation from an Isolated Collisionless Plasma
201
References [1] H. Bondi, M.G.J. van der Burg, A.W.K. Metzner, Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems, Proc. R. Soc. London, Ser. A 269, 21–52 (1962). [2] S. Calogero, Global Small Solutions of the Vlasov-Maxwell System in the Absence of Incoming Radiation, Indiana Univ. Math. Journal, (to appear) Preprint: math-ph/0211013. [3] D. Christodoulou, S. Klainerman, The global non-linear stability of the Minkowski space, Princeton Mathematical series 41 (1993). [4] R. Glassey, W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rat. Mech. Anal. 92, 59–90 (1986). [5] R. Glassey, W. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys. 113, 191–208 (1987). [6] R. Glassey, J. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys. 119, 353–384 (1988). [7] J.N. Goldberg, R.P. Kerr, Asymptotic Properties of the Electromagnetic Field, J. Math. Phys. 5, 172–176 (1964). [8] S. Klainerman, F. Nicol´ o, The Evolution Problem in General Relativity, Birkh¨auser (Basel) (2003). [9] S. Klainerman, G. Staffilani, A new approach to study the Vlasov-Maxwell system, Comm. Pure Appl. Anal. 1, 1, 103–125 (2002). [10] G. Rein, Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Comm. Math. Phys. 135, 41–78 (1990). [11] J.L. Synge, Relativity: The Special Theory, North-Holland, Amsterdam (1965). [12] R.M. Wald, General relativity, (Chicago, IL: The University of Chicago Press) (1984). Simone Calogero Department of Mathematics Chalmers University S-412 96 G¨ oteborg Sweden email:
[email protected] Communicated by Vincent Rivasseau submitted 13/07/03, accepted 11/09/03
Ann. Henri Poincar´e 5 (2004) 203 – 233 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020203-31 DOI 10.1007/s00023-004-0166-8
Annales Henri Poincar´ e
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems N´andor Sim´ anyi∗
Abstract. We consider the system of N (≥ 2) hard balls with masses m1 , . . . , mN and radius r in the flat torus TνL = Rν /L · Zν of size L, ν ≥ 3. We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection (m1 , . . . , mN ; L) of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case ν = 2. The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.
1 Introduction Hard ball systems or, a bit more generally, mathematical billiards constitute an important and quite interesting family of dynamical systems being intensively studied by dynamicists and researchers of mathematical physics, as well. These dynamical systems pose many challenging mathematical questions, most of them concerning the ergodic (mixing) properties of such systems. The introduction of hard ball systems and the first major steps in their investigations date back to the 40’s and 60’s, see Krylov’s paper [K(1942)] and Sinai’s ground-breaking works [Sin(1963)] and [Sin(1970)], in which the author – among other things – formulated the modern version of Boltzmann’s ergodic hypothesis (what we call today the Boltzmann-Sinai ergodic hypothesis) by claiming that every hard ball system in a flat torus is ergodic, of course after fixing the values of the trivial flow-invariant quantities. In the articles [Sin(1970)] and [B-S(1973)] Bunimovich and Sinai proved this hypothesis for two hard disks on the two-dimensional unit torus T2 . The generalization of this result to higher dimensions ν > 2 took fourteen years, and was done by Chernov and Sinai in [S-Ch(1987)]. Although the model of two hard balls in Tν is already rather involved technically, it is still a so-called strictly dispersive billiard system, i.e., such that the smooth components of the boundary ∂Q of the configuration space are strictly concave from outside Q. (They are bending away from Q.) The billiard systems of more than two hard balls in Tν are no longer strictly dispersive, but just semi-dispersive (strict concavity of the smooth components of ∂Q is lost, merely concavity persists), and this circumstance causes a lot of additional technical troubles in their study. In the series of my ∗ Research
supported by the National Science Foundation, grant DMS-0098773.
204
N. Sim´ anyi
Ann. Henri Poincar´e
joint papers with A. Kr´ amli and D. Sz´ asz [K-S-Sz(1989)], [K-S-Sz(1990)], [K-SSz(1991)], and [K-S-Sz(1992)] we developed several new methods, and proved the ergodicity of more and more complicated semi-dispersive billiards culminating in the proof of ergodicity of four billiard balls in the torus Tν (ν ≥ 3), [K-S-Sz(1992)]. Then, in 1992, Bunimovich, Liverani, Pellegrinotti and Sukhov [B-L-P-S(1992)] were able to prove the ergodicity for some systems with an arbitrarily large number of hard balls. The shortcoming of their model, however, is that, on one hand, they restrict the types of all feasible ball-to-ball collisions, on the other hand they introduce some additional scattering effect with the collisions at the strictly concave wall of the container. The only result with an arbitrarily large number of balls in a flat unit torus Tν was achieved in the twin papers of mine [Sim(1992I-II)], where I managed to prove the ergodicity (actually, the K-mixing property) of N hard balls in Tν , provided that N ≤ ν. The annoying shortcoming of that result is that the larger the number of balls N is, larger and larger dimension ν of the ambient container is required by the method of the proof. On the other hand, if someone considers a hard ball system in an elongated torus which is long in one direction but narrow in the others, so that the balls must keep their cyclic order in the “long direction” (Sinai’s “pen-case” model), then the technical difficulties can be handled, thanks to the fact that the collisions of balls are now restricted to neighboring pairs. The hyperbolicity of such models in three dimensions and the ergodicity in dimension four have been proved in [S-Sz(1995)]. The positivity of the metric entropy for several systems of hard balls can be proven relatively easily, as was shown in the paper [W(1988)]. The articles [L-W(1995)] and [W(1990)] are nice surveys describing a general setup leading to the technical problems treated in a series of research papers. For a comprehensive survey of the results and open problems in this field, see [Sz(1996)]. Pesin’s theory [P(1977)] on the ergodic properties of non-uniformly hyperbolic, smooth dynamical systems has been generalized substantially to dynamical systems with singularities (and with a relatively mild behavior near the singularities) by A. Katok and J-M. Strelcyn [K-S(1986)]. Since then, the so-called Pesin’s and Katok-Strelcyn’s theories have become part of the folklore in the theory of dynamical systems. They claim that – under some mild regularity conditions, particularly near the singularities – every non-uniformly hyperbolic and ergodic flow enjoys the Kolmogorov-mixing property, shortly the K-mixing property. Later on it was discovered and proven in [C-H(1996)] and [O-W(1998)] that the above-mentioned fully hyperbolic and ergodic flows with singularities turn out to be automatically having the Bernoulli mixing (B-mixing) property. It is worth noting here that almost every semi-dispersive billiard system, especially every hard ball system, enjoys those mild regularity conditions imposed on the systems (as axioms) by [K-S(1986)], [C-H(1996)], and [O-W(1998)]. In other words, for a hard ball flow (M, {S t }, µ) the (global) ergodicity of the system actually implies its full hyperbolicity and the B-mixing property, as well. Finally, in our joint venture with D. Sz´ asz [S-Sz(1999)], we prevailed over the difficulty caused by the low value of the dimension ν by developing a brand
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
205
new algebraic approach for the study of hard ball systems. That result, however, only establishes complete hyperbolicity (nonzero Lyapunov exponents almost everywhere) for N balls in Tν . The ergodicity appeared to be a harder task. We note, however, that the algebraic method developed in [S-Sz(1999)] is being further developed in this paper in order to obtain ergodicity, not only full hyperbolicity. Consider the ν-dimensional (ν ≥ 2), standard, flat torus TνL = Rν /L · Zν as the vessel containing N (≥ 2) hard balls (spheres) B1 , . . . , BN with positive masses m1 , . . . , mN and (just for simplicity) common radius r > 0. We always assume that the radius r > 0 is not too big, so that even the interior of the arising configuration space Q is connected. Denote the center of the ball Bi by qi ∈ Tν , and let vi = q˙i be the velocity of the i-th particle. We investigate the uniform motion of the balls B1 , . . . , BN inside the container Tν with half a unit 1 N 1 2 of total kinetic energy: E = . We assume that the collisions i=1 mi ||vi || = 2 2 between balls are perfectly elastic. Since – beside the kinetic energy E – the total ν momentum I = N i=1 mi vi ∈ R is also a trivial first integral of the motion, we make the standard reduction I = 0. Due to the apparent translation invariance of the arising dynamical system, we factorize the configuration space with respect to uniform spatial translations as follows: (q1 , . . . , qN ) ∼ (q1 + a, . . . , qN + a) for all ν translation vectors The configuration space Q of the arising flow is then a∈T . ν N the factor torus (T ) / ∼ ∼ = Tν(N −1) minus the cylinders Ci,j = (q1 , . . . , qN ) ∈ Tν(N −1) : dist(qi , qj ) < 2r (1 ≤ i < j ≤ N ) corresponding to the forbidden overlap between the i-th and j-th spheres. Then it is easy to see that the compound configuration point q = (q1 , . . . , qN ) ∈ Q = Tν(N −1) \
Ci,j
1≤i<j≤N
moves in Q uniformly with unit speed and bounces back from the boundaries ∂Ci,j of the cylinders Ci,j according to the classical law of geometric optics: the angle of reflection equals the angle of incidence. More precisely: the post-collision velocity v + can be obtained from the pre-collision velocity v − by the orthogonal reflection across the tangent hyperplane of the boundary ∂Q at the point of collision. Here we must emphasize that the phrase “orthogonal” should be understood with respect to the natural Riemannian metric (the so-called mass metric) N ||dq||2 = i=1 mi ||dqi ||2 in the configuration space Q. For the normalized Liouville measure µ of the arising flow {S t } we obviously have dµ = const · dq · dv, where dq is the Riemannian volume in Q induced by the above metric and dv is the surface measure (determined by the restriction of the Riemannian metric above) on the
206
N. Sim´ anyi
sphere of compound velocities ν(N −1)−1
S
=
ν N
(v1 , . . . , vN ) ∈ (R )
:
N i=1
mi vi = 0 and
N
Ann. Henri Poincar´e
2
mi ||vi || = 1 .
i=1
The phase space M of the flow {S t } is the unit tangent bundle Q × Sd−1 of the configuration space Q. (We will always use the shorthand notation d = ν(N − 1) for the dimension of the billiard table Q.) We must, however, note here that at the boundary ∂Q of Q one has to glue together the pre-collision and post-collision velocities in order to form the phase space M, so M is equal to the unit tangent bundle Q × Sd−1 modulo this identification. A bit more detailed definition of hard ball systems with arbitrary masses, as well as their role in the family of cylindric billiards, can be found in §4 of [SSz(2000)] and in §1 of [S-Sz(1999)]. We denote the arising flow by (M, {S t }t∈R , µ). In the series of articles [K-S-Sz(1989)], [K-S-Sz(1991)], [K-S-Sz(1992)], [Sim(1992-I)], and [Sim(1992-II)] the authors developed a powerful, three-step strategy for proving the (hyperbolic) ergodicity of hard ball systems. First of all, all these proofs are inductions on the number N of balls involved in the problem. Secondly, the induction step itself consists of the following three major steps: Step I. To prove that every non-singular (i.e., smooth) trajectory segment S [a,b] x0 with a “combinatorially rich” (in a well-defined sense) symbolic collision sequence is automatically sufficient (or, in other words, “geometrically hyperbolic”, see below in §2), provided that the phase point x0 does not belong to a countable union J of smooth sub-manifolds with codimension at least two. (Containing the exceptional phase points.) The exceptional set J featuring this result is negligible in our dynamical considerations – it is a so-called slim set. For the basic properties of slim sets, see §2 below. Step II. Assume the induction hypothesis, i.e., that all hard ball systems with N balls (2 ≤ N < N ) are (hyperbolic and) ergodic. Prove that then there exists a slim set S ⊂ M (see §2) with the following property: For every phase point x0 ∈ M \ S the entire trajectory S R x0 contains at most one singularity and its symbolic collision sequence is combinatorially rich, just as required by the result of Step I. Step III. By using again the induction hypothesis, prove that almost every singular trajectory is sufficient in the time interval (t0 , +∞), where t0 is the time moment of the singular reflection. (Here the phrase “almost every” refers to the volume defined by the induced Riemannian metric on the singularity manifolds.) We note here that the almost sure sufficiency of the singular trajectories (featuring Step III) is an essential condition for the proof of the celebrated theorem on local ergodicity for algebraic semi-dispersive billiards proved by B´ alint-ChernovSz´asz-T´oth in [B-Ch-Sz-T (2002)]. Under this assumption the theorem of [B-ChSz-T (2002)] states that in any algebraic semi-dispersive billiard system (i.e., in
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
207
a system such that the smooth components of the boundary ∂Q are algebraic hypersurfaces) a suitable, open neighborhood U0 of any sufficient phase point x0 ∈ M (with at most one singularity on its trajectory) belongs to a single ergodic component of the billiard flow (M, {S t }t∈R , µ). In an inductive proof of ergodicity, steps I and II together ensure that there exists an arc-wise connected set C ⊂ M with full measure, such that every phase point x0 ∈ C is sufficient with at most one singularity on its trajectory. Then the cited theorem on local ergodicity (now taking advantage of the result of Step III) states that for every phase point x0 ∈ C an open neighborhood U0 of x0 belongs to one ergodic component of the flow. Finally, the connectedness of the set C and µ(M \ C) = 0 easily imply that the flow (M, {S t }t∈R , µ) (now with N balls) is indeed ergodic, and actually fully hyperbolic, as well. The main result of this paper is the Theorem. In the case ν ≥ 3 for almost every selection (m1 , . . . , mN ; L) of the outer geometric parameters from the region mi > 0, L > L0 (r, ν), where the interior of t the phase space is connected, it is true that the billiard flow (Mm,L , {S }, µm,L ) of the N -ball system is ergodic and completely hyperbolic. Then, following from the results of Chernov-Haskell [C-H(1996)] and Ornstein-Weiss [O-W(1998)], such a semi-dispersive billiard system actually enjoys the B-mixing property, as well. Remark 1. We note that the main result of this paper and that of [Sim(2003)] nicely complement each other. They precisely assert the same, almost sure ergodicity of hard ball systems in the cases ν ≥ 3 and ν = 2, respectively. It should be noted, however, that the proof of [Sim(2003)] is primarily dynamical-geometric (except the verification of the Chernov-Sinai Ansatz), whereas the novel parts of the present proof are fundamentally algebraic. Remark 2. The above inequality L > L0 (r, ν) corresponds to physically relevant situations. Indeed, in the case L < L0 (r, ν) the particles would not have enough room even to freely exchange positions. The paper is organized as follows: §2 provides all necessary prerequisites and technical tools that will be required by the proof of the theorem. Based on the results obtained in [S-Sz(1999)], the subsequent Section §3 carries out Step I of the inductive strategy outlined above, but for the case when the outer geometric parameters (m1 , . . . , mN ; L) are incorporated in the algebraic process as variables. (Just as the positions and velocities of the particles!) Finally, the closing Section §4 utilizes a “Fubini type argument” by proving Step I for almost every (with respect to the Lebesgue measure of the (m1 , . . . , mN ; L)-space) hard ball system (N ≥ 2, ν ≥ 3). This will finish the inductive proof of the theorem, for Steps II and III of the induction strategy are easy consequences of some earlier results.
208
N. Sim´ anyi
Ann. Henri Poincar´e
2 Prerequisites 2.1
Cylindric billiards
Consider the d-dimensional (d ≥ 2) flat torus Td = Rd /L supplied with the usual Riemannian inner product . , . inherited from the standard inner product of the universal covering space Rd . Here L ⊂ Rd is assumed to be a lattice, i.e., a discrete subgroup of the additive group Rd with rank(L) = d. The reason why we want to allow general lattices, other than just the integer lattice Zd , is that otherwise the hard ball systems would not be covered. The geometry of the structure lattice L in the case of a hard ball system is significantly different from the geometry of the standard lattice Zd in the standard Euklidean space Rd , see later in this section. The configuration space of a cylindric billiard is Q = Td \ (C1 ∪ · · · ∪ Ck ), where the cylindric scatterers Ci (i = 1, . . . , k) are defined as follows. Let Ai ⊂ Rd be a so-called lattice subspace of Rd , which means that rank(Ai ∩ L) = dimAi . In this case the factor Ai /(Ai ∩ L) is a sub-torus in Td = Rd /L which will be taken as the generator of the cylinder Ci ⊂ Td , i = 1, . . . , k. Denote by d Li = A⊥ i the ortho-complement of Ai in R . Throughout this paper we will always assume that dimLi ≥ 2. Let, furthermore, the numbers ri > 0 (the radii of the spherical cylinders Ci ) and some translation vectors ti ∈ Td = Rd /L be given. The translation vectors ti play a role in positioning the cylinders Ci in the ambient torus Td . Set
Ci = x ∈ Td : dist (x − ti , Ai /(Ai ∩ L)) < ri . In order to avoid further unnecessary complications, we always assume that the interior of the configuration space Q = Td \ (C1 ∪ · · · ∪ Ck ) is connected. The phase space M of our cylindric billiard flow will be the unit tangent bundle of Q (modulo the natural gluing at its boundary), i.e., M = Q × Sd−1 . (Here Sd−1 denotes the unit sphere of Rd .) The dynamical system (M, {S t }t∈R , µ), where S t (t ∈ R) is the dynamics defined by the uniform motion inside the domain Q and specular reflections at its boundary (at the scatterers), and µ is the Liouville measure, is called a cylindric billiard flow. We note that the cylindric billiards – defined above – belong to the wider class of so-called semi-dispersive billiards, which means that the smooth components ∂Qi of the boundary ∂Q of the configuration space Q are (not necessarily strictly) concave from outside of Q, i.e., they are bending away from the interior of Q. As to the notions and notations in connection with semi-dispersive billiards, the reader is kindly referred to the article [K-S-Sz(1990)]. Throughout this paper we will always assume – without explicitly stating – that the considered semi-dispersive billiard system fulfills the following conditions: intQ is connected, and the d-dim spatial angle α(q) subtended by Q at any of its boundary points q ∈ ∂Q is uniformly positive.
(2.1.1) (2.1.2)
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
209
We note, however, that in the case of hard ball systems with a fixed radius r of the balls (see below) the non-degeneracy condition (2.1.2) only excludes countably many values of the size L of the container torus TνL = Rν /L · Zν from the region L > L0 (r, ν) where (2.1.1) is true. Therefore, in the sense of our theorem of “almost sure ergodicity”, the non-degeneracy condition (2.1.2) does not mean a restriction of generality.
2.2
Hard ball systems
Hard ball systems in the flat torus TνL = Rν /L · Zν (ν ≥ 2) with positive masses m1 , . . . , mN are described (for example) in §1 of [S-Sz(1999)]. These are the dynamical systems describing the motion of N (≥ 2) hard balls with a common radius r > 0 and positive masses m1 , . . . , mN in the flat torus of size L, TνL = Rν /L · Zν . (Just for simplicity, we will assume that the radii have the common value r.) The center of the i-th ball is denoted by qi (∈ TνL ), its time derivative is vi= q˙i , i = 1, . . . , N . One uses the standard reduction of kinetic N energy E = 12 i=1 mi ||vi ||2 = 12 . The arising configuration space (still without the removal of the scattering cylinders Ci,j ) is the torus ν TνN L = (TL )
N
= {(q1 , . . . , qN ) : qi ∈ TνL , i = 1, . . . , N }
supplied with the Riemannian inner product (the so-called mass metric) v, v =
N
mi vi , vi
(2.2.1)
i=1 N
in its common tangent space RνN = (Rν ) . Now the Euklidean space RνN with the inner product (2.2.1) plays the role of Rd in the original definition of cylindric billiards, see §2.1 above. The generator subspace Ai,j ⊂ RνN (1 ≤ i < j ≤ N ) of the cylinder Ci,j (describing the collisions between the i-th and j-th balls) is given by the equation (2.2.2) Ai,j = (q1 , . . . , qN ) ∈ (Rν )N : qi = qj , see (4.3) in [S-Sz(2000)]. Its ortho-complement Li,j ⊂ RνN is then defined by the equation N Li,j = (q1 , . . . , qN ) ∈ (Rν ) : qk = 0 for k = i, j, and mi qi + mj qj = 0 , (2.2.3) see (4.4) in [S-Sz(2000)]. Easy calculation shows that the cylinder Ci,j (describing the overlap of the i-th and j-th balls) is indeed spherical and the radius of its base m m sphere is equal to ri,j = 2r mi i+mjj , see §4, especially formula (4.6) in [S-Sz(2000)]. N
The structure lattice L ⊂ RνN is clearly the lattice L = (L · Zν )
= L · ZN ν .
210
N. Sim´ anyi
Ann. Henri Poincar´e
Due to the presence of an extra invariant quantity I = N i=1 mi vi , one usually N makes the reduction i=1 mi vi = 0 and, correspondingly, factorizes the configuration space with respect to uniform spatial translations: (q1 , . . . , qN ) ∼ (q1 + a, . . . , qN + a),
a ∈ TνL .
(2.2.4)
The natural, common tangent space of this reduced configuration space is then Z=
(v1 , . . . , vN ) ∈ (Rν )N :
N i=1
mi vi = 0
=
⊥ Ai,j = (A)⊥
(2.2.5)
i<j
supplied again with the inner product (2.2.1), see also (4.1) and (4.2) in [SSz(2000)]. The base spaces Li,j of (2.2.3) are obviously subspaces of Z, and we take A˜i,j = Ai,j ∩ Z = PZ (Ai,j ) as the ortho-complement of Li,j in Z. (Here PZ denotes the orthogonal projection onto the space Z.) Note that the configuration space of the reduced system (with the identification (2.2.4)) is naturally the torus RνN /(A + L · ZνN ) = Z/PZ (L · ZνN ).
2.3
Collision graphs
Let S [a,b] x be a nonsingular, finite trajectory segment with the collisions σ1 , . . . , σn listed in time order. (Each σk is an unordered pair (i, j) of different labels i, j ∈ {1, 2, . . . , N }.) The graph G = (V, E) with vertex set V = {1, 2, . . . , N } and set of edges E = {σ1 , . . . , σn } is called the collision graph of the orbit segment S [a,b] x. For a given positive number C, the collision graph G = (V, E) of the orbit segment S [a,b] x will be called C-rich if G contains at least C connected, consecutive (i.e., following one after the other in time, according to the time-ordering given by the trajectory segment S [a,b] x) subgraphs.
2.4
Trajectory branches
We are going to briefly describe the discontinuity of the flow {S t } caused by a multiple collisions at time t0 . Assume first that the pre-collision velocities of the particles are given. What can we say about the possible post-collision velocities? Let us perturb the pre-collision phase point (at time t0 − 0) infinitesimally, so that the collisions at ∼ t0 occur at infinitesimally different moments. By applying the collision laws to the arising finite sequence of collisions, we see that the postcollision velocities are fully determined by the time-ordering of the considered collisions. Therefore, the collection of all possible time-orderings of these collisions gives rise to a finite family of continuations of the trajectory beyond t0 . They are called the trajectory branches. It is quite clear that similar statements can be said regarding the evolution of a trajectory through a multiple collision in reverse time. Furthermore, it is also obvious that for any given phase point x0 ∈ M there are two, ω-high trees T+ and T− such that T+ (T− ) describes all the possible
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
211
continuations of the positive (negative) trajectory S [0,∞) x0 (S (−∞,0] x0 ). (For the definitions of trees and for some of their applications to billiards, cf. the beginning of §5 in [K-S-Sz(1992)].) It is also clear that all possible continuations (branches) of the whole trajectory S (−∞,∞) x0 can be uniquely described by all pairs (B− , B+ ) of ω-high branches of the trees T− and T+ (B− ⊂ T− , B+ ⊂ T+ ). Finally, we note that the trajectory of the phase point x0 has exactly two branches, provided that S t x0 hits a singularity for a single value t = t0 , and the phase point S t0 x0 does not lie on the intersection of more than one singularity manifolds. In this case we say that the trajectory of x0 has a “simple singularity”.
2.5
Neutral subspaces, advance, and sufficiency
Consider a nonsingular trajectory segment S [a,b] x. Suppose that a and b are not moments of collision. Definition 2.5.1 The neutral space N0 (S [a,b] x) of the trajectory segment S [a,b] x at time zero (a < 0 < b) is defined by the following formula:
N0 (S [a,b] x) = W ∈ Z : ∃(δ > 0) such that ∀α ∈ (−δ, δ) V (S a (Q(x) + αW, V (x))) = V (S a x) and V S b (Q(x) + αW, V (x)) = V (S b x) . (Z is the common tangent space Tq Q of the parallelizable manifold Q at any of its points q, while V (x) is the velocity component of the phase point x = (Q(x), V (x)).) It is known (see (3) in §3 of [S-Ch (1987)]) that N0 (S [a,b] x) is a linear subspace of Z indeed, and V (x) ∈ N0 (S [a,b] x). The neutral space Nt (S [a,b] x) of the segment S [a,b] x at time t ∈ [a, b] is defined as follows: Nt (S [a,b] x) = N0 S [a−t,b−t] (S t x) . It is clear that the neutral space Nt (S [a,b] x) can be canonically identified with N0 (S [a,b] x) by the usual identification of the tangent spaces of Q along the trajectory S (−∞,∞) x (see, for instance, §2 of [K-S-Sz(1990)]). Our next definition is that of the advance. Consider a non-singular orbit segment S [a,b] x with the symbolic collision sequence Σ = (σ1 , . . . , σn ) (n ≥ 1), meaning that S [a,b] x has exactly n collisions with ∂Q, and the i-th collision (1 ≤ i ≤ n) takes place at the boundary of the cylinder Cσi . For x = (Q, V ) ∈ M and W ∈ Z, W sufficiently small, denote TW (Q, V ) := (Q + W, V ). Definition 2.5.2 For any 1 ≤ k ≤ n and t ∈ [a, b], the advance α(σk ) : Nt (S [a,b] x) → R of the collision σk is the unique linear extension of the linear functional α(σk ) defined in a sufficiently small neighborhood of the origin of Nt (S [a,b] x) in the following way: α(σk )(W ) := tk (x) − tk (S −t TW S t x).
212
N. Sim´ anyi
Ann. Henri Poincar´e
Here tk = tk (x) is the time moment of the k-th collision σk on the trajectory of x after time t = a. The above formula and the notion of the advance functional αk = α(σk ) : Nt S [a,b] x → R has two important features: (i) If the spatial translation (Q, V ) → (Q + W, V ) is carried out at time t, then tk changes linearly in W , and it takes place just αk (W ) units of time earlier. (This is why it is called “advance”.) (ii) If the considered reference time t is somewhere between tk−1 and tk , then the neutrality of W with respect to σk precisely means that W − αk (W ) · V (x) ∈ Aσk , i.e., a neutral (with respect to the collision σk ) spatial translation W with the advance αk (W ) = 0 means that the vector W belongs to the generator space Aσk of the cylinder Cσk . It is now time to bring up the basic notion of sufficiency (or, sometimes it is also called geometric hyperbolicity) of a trajectory (segment). This is the utmost important necessary condition for the proof of the fundamental theorem for algebraic semi-dispersive billiards, see Theorem 4.4 in [B-Ch-Sz-T(2002)]. Definition 2.5.3 (i) The nonsingular trajectory segment S [a,b] x (a and b are supposed not to be moments of collision) is said to be sufficient if and only if the dimension of Nt (S [a,b] x) (t ∈ [a, b]) is minimal, i.e., dim Nt (S [a,b] x) = 1. (ii) The trajectory segment S [a,b] x containing exactly one singularity (a so-called “simple singularity”, see 2.4 above) is said to be sufficient if and only if both branches of this trajectory segment are sufficient. Definition 2.5.4 The phase point x ∈ M with at most one (simple) singularity is said to be sufficient if and only if its whole trajectory S (−∞,∞) x is sufficient, which means, by definition, that some of its bounded segments S [a,b] x are sufficient. In the case of an orbit S (−∞,∞) x with a simple singularity, sufficiency means that both branches of S (−∞,∞) x are sufficient.
2.6
No accumulation (of collisions) in finite time
By the results of Vaserstein [V(1979)], Galperin [G(1981)] and Burago-FerlegerKononenko [B-F-K(1998)], in a semi-dispersive billiard flow with the property (2.1.2) there can only be finitely many collisions in finite time intervals, see Theorem 1 in [B-F-K(1998)]. Thus, the dynamics is well defined as long as the trajectory does not hit more than one boundary components at the same time.
Vol. 5, 2004
2.7
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
213
Slim sets
We are going to summarize the basic properties of codimension-two subsets A of a connected, smooth manifold M with a possible boundary. Since these subsets A are just those negligible in our dynamical discussions, we shall call them slim. As to a broader exposition of the issues, see [E(1978)] or §2 of [K-S-Sz(1991)]. Note that the dimension dim A of a separable metric space A is one of the ˇ three classical notions of topological dimension: the covering (Cech-Lebesgue), the ˇ small inductive (Menger-Urysohn), or the large inductive (Brouwer-Cech) dimension. As it is known from general general topology, all of them are the same for separable metric spaces. Definition 2.7.1 A subset A of M is called slim if and only if A can be covered by a countable family of codimension-two (i.e., at least two) closed sets of µ-measure zero, where µ is a smooth measure on M . (Cf. Definition 2.12 of [K-S-Sz(1991)].) Property 2.7.2 The collection of all slim subsets of M is a σ-ideal, that is, countable unions of slim sets and arbitrary subsets of slim sets are also slim. Proposition 2.7.3. (Locality) A subset A ⊂ M is slim if and only if for every x ∈ A there exists an open neighborhood U of x in M such that U ∩ A is slim. (Cf. Lemma 2.14 of [K-S-Sz(1991)].) Property 2.7.4 A closed subset A ⊂ M is slim if and only if µ(A) = 0 and dim A ≤ dim M − 2. Property 2.7.5. (Integrability) If A ⊂ M1 × M2 is a closed subset of the product of two smooth manifolds with possible boundaries, and for every x ∈ M1 the set Ax = {y ∈ M2 : (x, y) ∈ A} is slim in M2 , then A is slim in M1 × M2 . The following propositions characterize the codimension-one and codimension-two sets. Proposition 2.7.6 For any closed subset S ⊂ M the following three conditions are equivalent: (i) dim S ≤ dim M − 2; (ii) intS = ∅ and for every open connected set G ⊂ M the difference set G \ S is also connected; (iii) intS = ∅ and for every point x ∈ M and for any open neighborhood V of x in M there exists a smaller open neighborhood W ⊂ V of the point x such that for every pair of points y, z ∈ W \ S there is a continuous curve γ in the set V \ S connecting the points y and z. (See Theorem 1.8.13 and Problem 1.8.E of [E(1978)].)
214
N. Sim´ anyi
Ann. Henri Poincar´e
Proposition 2.7.7 For any subset S ⊂ M the condition dim S ≤ dim M − 1 is equivalent to intS = ∅. (See Theorem 1.8.10 of [E(1978)].) We recall an elementary, but important lemma (Lemma 4.15 of [K-S-Sz(1991)]). Let R2 be the set of phase points x ∈ M \ ∂M such that the trajectory S (−∞,∞) x has more than one singularities. Proposition 2.7.8 The set R2 is a countable union of codimension-two smooth sub-manifolds of M and, being such, it is slim. The next lemma establishes the most important property of slim sets which gives us the fundamental geometric tool to connect the open ergodic components of billiard flows. Proposition 2.7.9 If M is connected, then the complement M \ A of a slim Fσ set A ⊂ M is an arc-wise connected (Gδ ) set of full measure. (See Property 3 of §4.1 in [K-S-Sz(1989)]. The Fσ sets are, by definition, the countable unions of closed sets, while the Gδ sets are the countable intersections of open sets.)
2.8
The subsets M0 and M#
Denote by M# the set of all phase points x ∈ M for which the trajectory of x encounters infinitely many non-tangential collisions in both time directions. The trajectories of the points x ∈ M \ M# are lines: the motion is linear and uniform, see the appendix of [Sz(1994)]. It is proven in lemmas A.2.1 and A.2.2 of [Sz(1994)] that the closed set M \ M# is a finite union of hyperplanes. It is also proven in [Sz(1994)] that, locally, the two sides of a hyper-planar component of M \ M# can be connected by a positively measured beam of trajectories, hence, from the point of view of ergodicity, in this paper it is enough to show that the connected components of M# entirely belong to one ergodic component. This is what we are going to do in this paper. Denote by M0 the set of all phase points x ∈ M# the trajectory of which does not hit any singularity, and use the notation M1 for the set of all phase points x ∈ M# whose orbit contains exactly one, simple singularity. According to Proposition 2.7.8, the set M# \ (M0 ∪ M1 ) is a countable union of smooth, codimension-two (≥ 2) submanifolds of M, and, therefore, this set may be discarded in our study of ergodicity, please see also the properties of slim sets above. Thus, we will restrict our attention to the phase points x ∈ M0 ∪ M1 .
2.9
The “Chernov-Sinai Ansatz”
An essential precondition for the theorem on local ergodicity by B´ alint-ChernovSz´asz-T´oth (Theorem 4.4 of [B-Ch-Sz-T(2002)]) is the so-called “Chernov-Sinai Ansatz” which we are going to formulate below. Denote by SR+ ⊂ ∂M the set of all phase points x0 = (q0 , v0 ) ∈ ∂M corresponding to singular reflections (a
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
215
tangential or a double collision at time zero) supplied with the post-collision (outgoing) velocity v0 . It is well known that SR+ is a compact cell complex with dimension 2d − 3 = dimM − 2. It is also known (see Lemma 4.1 in [K-S-Sz(1990)]) that for ν-almost every phase point x0 ∈ SR+ the forward orbit S (0,∞) x0 does not hit any further singularity. (Here ν is the Riemannian volume of SR+ induced by the restriction of the natural Riemannian metric of M.) The Chernov-Sinai Ansatz postulates that for ν-almost every x0 ∈ SR+ the forward orbit S (0,∞) x0 is sufficient (geometrically hyperbolic).
2.10
The theorem on local ergodicity
The theorem on local ergodicity by B´ alint-Chernov-Sz´ asz-T´oth (Theorem 4.4 of [B-Ch-Sz-T(2002)]) claims the following: Let (M, {S t }t∈R , µ) be a semi-dispersive billiard flow with (2.1.1)–(2.1.2) and with the property that the smooth components of the boundary ∂Q of the configuration space are algebraic hyper-surfaces. (The cylindric billiards automatically fulfill this algebraicity condition.) Assume – further – that the Chernov-Sinai Ansatz holds true, and a phase point x0 ∈ (M \ ∂M) ∩ M# is given with the properties (i) S (−∞,∞) x has at most one singularity, and (ii) S (−∞,∞) x is sufficient. Then some open neighborhood U0 ⊂ M of x0 belongs to a single ergodic component of the flow (M, {S t }t∈R , µ). (Modulo the zero sets, of course.)
3 Non-sufficiency occurs on a codimension-two set. The case ν ≥ 3 The opening part of this section contains a slightly modified version of Lemma 4.43 from [S-Sz(1999)]. The reason why we had to modify the recursion for the sequence C(N ) (from C(N ) = (N/2) · max {C(N − 1), 3} to C(N ) = (N/2) · (2C(N − 1) + 1)) is that our Corollary 3.5 (below) requires (2C(N ) + 1)-richness instead of the usual C(N )-richness. In the present paper the sequence C(N ) always denotes the one defined by the recursion in Lemma 3.1 instead of the one defined in Lemma 4.43 of [S-Sz(1999)]. This should not cause any confusion. We note that the upcoming lemma is purely combinatorial. Lemma 3.1 Define the sequence of positive numbers C(N ) recursively by taking C(2) = 1 and C(N ) = (N/2) · (2C(N − 1) + 1) for N ≥ 3. Let N ≥ 3, and suppose that the symbolic collision sequence Σ = (σ1 , . . . , σn ) for N particles is C(N )-rich. Then we can find a particle, say the one with label N , and two indices 1 ≤ p < q ≤ n such that
216
(i) (ii) (iii) (iv)
N. Sim´ anyi
Ann. Henri Poincar´e
N ∈ σp ∩ σq , N∈ / q−1 j=p+1 σj , σp = σq =⇒ (∃j) (p < j < q & σp ∩ σj = ∅), and Σ is (2C(N − 1) + 1)-rich on the vertex set {1, . . . , N − 1}.
(Here, just as in the case of derived schemes, we denote by Σ the symbolic sequence that can be obtained from Σ by discarding all edges containing N .) Proof. The hypothesis on Σ implies that there exist subsequences Σ1 , . . . , Σr of Σ with the following properties: (1) For 1 ≤ i < j ≤ r every collision of Σi precedes every collision of Σj , (2) the graph of Σi (1 ≤ i ≤ r) is a tree (a connected graph without loop) on the vertex set {1, . . . , N }, and (3) r ≥ C(N ). Since every tree contains at least two vertices with degree one and C(N ) = (N/2) · {2C(N − 1) + 1}, there is a vertex, say the one labeled by N , such that N is a degree-one vertex of Σi(1) , . . . , Σi(t) , where 1 ≤ i(1) < · · · < i(t) ≤ r and t ≥ 2C(N − 1) + 1. Thus (iv) obviously holds. Let σp the edge of Σi(1) that contains N and, similarly, let σq be the edge of Σi(t) containing the vertex N . Then the fact t ≥ 3 ensures that the following properties hold: (i) N ∈ σp ∩ σq , (iii) σp = σq =⇒ ∃j p < j < q & σp ∩ σj = ∅, σj = σp . Let σp , σq (1 ≤ p < q ≤ n) be a pair of edges σp , σq (1 ≤ p < q ≤ n) fulfilling (i) and (iii) and having the minimum possible value of q −p . Elementary inspection shows that then (ii) must also hold for σp , σq . Lemma 3.5.1 is now proved. Let us fix a triplet (Σ, A, τ ) of the discrete (combinatorial) orbit structure with Property (A) (just as in [S-Sz(1999)], see Definition 3.31 there), and assume that Σ = (σ1 , . . . , σn ) is C(N )-rich, i.e., it contains at least C(N ) consecutive, connected collision graphs. We also consider the complex analytic manifold Ω (Σ, A, τ ) of all complex (Σ, A, τ )-orbits ω (Definition 3.20 in [S-Sz(1999)]) and the open, dense, connected domain D (Σ, A, τ ) ⊂ C(2ν+1)N +1 of all allowable initial data x = x(ω), see Definition 3.18 in [S-Sz(1999)]. Let, finally, Q(x) be a common irreducible divisor of the polynomials P1 (x), . . . , Ps (x) from (4.3) in [S-Sz(1999)]. (If such a common divisor exists.) In this section we will need several results about such common irreducible divisors Q(x) of the polynomials P1 (x), . . . , Ps (x). The first of them, as it is classically known from algebraic geometry (see, for example, [M(1976)]), is that the solution set V = {Q(x) = 0} of the equation Q(x) = 0 is a so-called irreducible (or, indecomposable) complex algebraic variety of codimension 1 in C(2ν+1)N +1 , which means that V is not the union of two, proper algebraic sub-varieties. Secondly, the smooth part S of V turns out to be a connected complex analytic manifold, while the non-smooth part V \ S of V
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
217
is a complex algebraic variety of dimension strictly less that dimV = (2ν + 1)N , see [M(1976)]. Finally, if a polynomial R(x) vanishes on V , then Q(x) must be a divisor of R(x). (The last statement is a direct consequence of Hilbert’s theorem on Zeroes, see again [M(1976)].) The first result, specific to our current dynamical situation, is Proposition 3.2 The polynomials P1 (x), . . . , Ps (x) of (4.3) in [S-Sz(1999)] are homogeneous in the masses m1 , . . . , mN and, consequently, any common divisor Q(x) of these polynomials is also homogeneous in the masses. Proof. It is clear that the complex dynamics encoded in the orbits ω ∈ Ω (Σ, A, τ ) only depends on the ratios of masses m2 /m1 , m3 /m1 , . . . , mN /m1 . Consequently, all algebraic functions fi (x) (i = 1, . . . , s) featuring the proof of Lemma 4.2 of [SSz(1999)] are homogeneous of degree 0 in the masses. Since the rational function Pi (x) ∈ K0 = C(x) Qi (x) is the product α of all conjugates of fi (x) (see the proof of the lemma just cited), Pi (x) is also homogeneous of degree 0 in the variables m1 , . . . , mN . we get that Qi (x) Then elementary algebra yields that both Pi (x) and Qi (x) are homogeneous (of the same degree) in the masses. Since any factor of a homogeneous polynomial is easily seen to be also homogeneous, we get that the common divisor Q(x) of P1 (x), . . . , Ps (x) is also homogeneous in the variables m1 , . . . , mN . Our next result, specific to our dynamics, that will be needed later is Proposition 3.3 Let ν ≥ 3, (Σ, A, τ ) be a discrete orbit structure with Property (A) and a C(N )-rich symbolic collision sequence Σ = (σ1 , . . . , σn ). Denote by P1 (x), . . . , Ps (x) the polynomials of (4.3) in [S-Sz(1999)] just as before, and let Q(x) be a common irreducible divisor of P1 (x), . . . , Ps (x). (If such a common be an extended discrete orbit structure with divisor exists.) Let, finally, Σ, A, ρ Property (A) and an extended collision sequence Σ = (σ0 , σ1 , . . . , σn ). We claim that the irreducible (indecomposable) solution set V of the equation Q(x) = 0 cannot even locally coincide with any of the following singularity manifolds C defined by one of the following equations: (1) vi00 − vj00 2 = 0, (2) vi00 − vj00 ; q˜i00 − q˜j00 − L · a0 = 0, (3) mi0 + mj0 = 0, i.e., the irreducible polynomial Q(x) is not equal to any of the (irreducible) polynomials on the left-hand sides of (1)–(3). (In [S-Sz(1999)] these equations fea .) Consequently, the open subset ture Definition 3.18 of the domain D Σ, A, ρ V ∩ D Σ, A, ρ of V is connected and dense in V .
218
N. Sim´ anyi
Ann. Henri Poincar´e
Remark. The first point where we (implicitly) use the condition ν ≥ 3 is the irreducibility of the polynomial vi00 − vj00 2 on the left-hand side of (1). Indeed, in the case ν = 2 this polynomial splits as √ vi00 − vj00 2 ≡ vi00 1 − vj00 1 + −1 vi00 2 − vj00 2 √ · vi00 1 − vj00 1 − −1 vi00 2 − vj00 2 . However, it is easy to see that, in the case ν ≥ 3, all polynomials on the left of (1)–(3) are indeed irreducible. Proof. First of all, we slightly reformulate the negation of the statement of the proposition. Fix one of the three equations of (1)–(3) above, and denote the irreducible polynomial on its left-hand side by R(x). By using the quadratic (or linear) equation R(x) = 0, we eliminate one variable xj out of x by expressing it as an algebraic function xj = g(y ) of the remaining variables y of x, so that the algebraic function g only contains finitely many field operations and (at most one) square root. After this elimination xj = g(y ), the meaning of R(x) ≡ Q(x) (i.e., the negation of the assertion of the proposition) is that all algebraic functions fi (x) ≡ f˜i (y ) in the proof 4.2 of [S-Sz(1999)] (i = 1, . . . , s, constructed for (Σ, A, τ ), not of Lemma for Σ, A, ρ ) are identically zero in terms of y , meaning that every complex orbit segment ω ∈ Ω (Σ, A, τ ), with the initial data x(ω) in the solution set of R(x) = 0, is non-sufficient, see also the “Dichotomy Corollary” 4.7 in [S-Sz(1999)]. Thus, the negation of the proposition means that no orbit segment ω ∈ Ω (Σ, A, τ ) in the considered singularity is sufficient. Now we carry out an induction on the number of balls N quite in the spirit of the proof of Key Lemma 4.1 of [S-Sz(1999)]. Indeed, the statement of the proposition is obviously true in the case N = 2, for in that case there are no non-sufficient (complex) trajectories ω ∈ Ω (Σ, A, τ ), i.e., the greatest common divisor of the polynomials P1 (x), . . . , Ps (x) is 1. Assume now that N ≥ 3, ν ≥ 3, and the statement of Proposition 3.3 has been proven for all values N < N . Suppose, however, that the statement with N balls and Property is false for some (Σ, A, τ ) and extension Σ, A, ρ (A), i.e., that there exists a common irreducible divisor Q(x) of all the polynomials P1 (x), . . . , Ps (x), and Q(x) happens to be one of the irreducible polynomials on the left-hand side of (1), (2), or (3). By using the C(N )-richness of Σ = (σ1 , . . . , σn ), we select a suitable label k0 ∈ {1, 2, . . . , N }, say k0 = N , for the above, substitution mN = 0 along the lines of Lemma by also ensuring the exis 3.1 tence of the derived schemes (Σ , A , τ ) and Σ , A , ρ for the (N −1)-ball-system {1, 2, . . . , N − 1} and properties (i)–(iv) (of Lemma 3.1) for Σ , see Corollary 4.35 ˜ x) the polynomial obtained of [S-Sz(1999)] and Lemma 3.1 above. Denote by Q( from Q(x) after the substitution mN = 0. ˜ x) is not constant. Lemma 3.4 The polynomial Q(
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
219
˜ x) ≡ c ∈ C. The case c = 0 means that mN is a divisor of Proof. Assume that Q( Q(x), thus mN ≡ Q(x) (for Q(x) is irreducible), which is impossible, since Q(x) has to be one of the polynomials on the left-hand side of (1), (2), or (3). If, however, c = 0, then we have that Q(x) ≡ c + mN S(x) with some nonzero polynomial S(x). (S(x) has to be non-zero, otherwise Q(x) would be a constant, not an irreducible polynomial.) However, this contradicts to the proved homogeneity of the polynomial Q(x) in the masses, see Proposition 3.2 above. This finishes the proof of the lemma. Remark. If one takes a look at the equations (1), (2), (3), he/she immediately ˜ x) ≡ Q(x) (when Q(x) is the polynomial on the left-hand realizes that either Q( ˜ x) ≡ mi0 when side of (1) or (2), or Q(x) ≡ mi0 + mj0 and N ∈ {i0 , j0 }), or Q( Q(x) ≡ mi0 + mj0 and N = j0 . In this way one can directly and easily verify Lemma 3.4 without the above “involved” algebraic proof. The reason why we still included the above proof is that later on in this section (in the proof of Sub-lemma 3.7) we will need the idea of the presented proof. The next lemma will use Definition 3.5 Suppose that two indices 1 ≤ p < q ≤ n and two labels of balls i, j ∈ {1, . . . , N } are given with the additional requirement that if i = j, then i ∈ q−1 l=p+1 σl . Following the proof of Lemma 4.2 of [S-Sz(1999)], denote by Q1 (x), Q2 (x), . . . , Qν (x) (x ∈ C(2ν+1)N +1 ) the polynomials with the property that for every vector of initial data x ∈ D (Σ, A, τ ) and for every k, k = 1, . . . , ν, the following equivalence holds true: ∃ ω ∈ Ω such that x(ω) = x & (vip (ω))k = vjq−1 (ω) k
⇐⇒ Qk (x) = 0. Our next lemma is a strengthened version of Lemma 4.39 of [S-Sz(1999)]: Lemma 3.6 Assume that the combinatorial-algebraic scheme (Σ, A, τ ) has Property (A), and use the assumptions and notations of the above definition. We claim that the polynomials Q1 (x), Q2 (x), . . . , Qν (x) do not have any non-constant common divisor. In other words, the equality vip (ω) = vjq−1 (ω) only takes place on an algebraic variety with at least two codimensions. Remark. Lemma 4.39 of [S-Sz(1999)] asserted that at least one of the above polynomials Qk (x) is nonzero. Then, by the permutation symmetry of the components k ∈ {1, . . . , ν}, all of these polynomials are actually nonzero. Proof. Induction on the number N ≥ 2. 1. Base of the induction, N = 2: First of all, by performing the substitution L = 0, we can annihilate all adjustment vectors, see (I), (IV), (VII) of Lemma 4.21 in [S-Sz(1999)], and Remark 4.22 there. Then, an elementary inspection shows that for any selection of positive real masses (m1 , m2 ), indeed, the
220
N. Sim´ anyi
Ann. Henri Poincar´e
equality vip (ω) = vjq−1 (ω) only occurs on a manifold with ν − 1 (≥ 2) codimensions in the section Ω (Σ, A, τ , m) of Ω (Σ, A, τ ) corresponding to the selected masses, since any trajectory segment of a two-particle system with positive masses and A = 0 has a very nice, totally real (and essentially ν-dimensional) representation in the relative coordinates of the particles: The consecutive, elastic bounces of a point particle moving uniformly inside a ball of radius 2r of Rν . Therefore, the statement of the lemma is true for N = 2. Assume now that N ≥ 3, and the lemma has been successfully proven for all smaller numbers of balls. By re-labeling the particles, if necessary, we can achieve that (i) N = i, N = j and (ii) if i = j, then the ball i has at least one collision between σp and σq with a particle different from N . For the fixed combinatorial scheme (Σ, A, τ ), select a derived scheme (Σ , A , τ ) corresponding to the substitution mN = 0, see Definition 4.11 and Corollary 4.35 in [S-Sz(1999)]. Our induction step is going to be a proof by contradiction. Assume, therefore, that the nonzero polynomials Q1 (x), Q2 (x), . . . , Qν (x) do have a common irreducible divisor R(x). According to Proposition 3.2 above, the (irreducible) ˜ x) polynomial R(x) is homogeneous in the variables m1 , . . . , mN . Denote by R( the polynomial that we obtain from R(x) after the substitution mN = 0. Similarly to Lemma 3.4 above, we claim ˜ x) is not constant. Sub-lemma 3.7. The polynomial R( Remark. The reason why we cannot simply apply Lemma 3.4 is that in the proof of that lemma we used the assumption that the irreducible polynomial Q(x) was one of the polynomials on the left-hand side of (1), (2), or (3) of Proposition 3.3. Right here we do not have such an assumption. ˜ x) ≡ c, where c ∈ C is a constant, i.e., R(x) ≡ c+mN ·S(x). Proof. Suppose that R( In the case c = 0 the polynomial mN ≡ R(x) would be a common divisor of all the polynomials Q1 (x), Q2 (x), . . . , Qν (x), meaning that in the considered N -ball system Ω (Σ, A, τ ) the equation mN (ω) = 0 implies the equality vip (ω) = vjq−1 (ω). This, in turn, means that in the (N − 1)-ball system {1, . . . , N − 1} (with the discrete algebraic scheme (Σ , A , τ )) the equality vip (ω) = vjq−1 (ω) is an identity, thus contradicting to the induction hypothesis. Therefore c = 0, and in the expansion R(x) ≡ c + mN · S(x) of the irreducible polynomial R(x) we certainly have S(x) ≡ 0, and this means that R(x) is not homogeneous in the mass variables, thus contradicting to Proposition 3.2. This finishes the proof of the sub-lemma. ˜ Finishing the proof of Lemma 3.6. Denote by Qk (x) the polynomial that we obtain from Qk (x) after the substitution mN = 0 (k = 1, . . . , ν), and by Tk (x) the polynomial constructed for the (N − 1)-ball system {1, . . . , N − 1} (with the discrete algebraic scheme (Σ , A , τ )) along the lines of Lemma 4.2 of [S-Sz(1999)],
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
221
describing the event (vip (ω))k = (vjq−1 (ω))k in this subsystem (k = 1, . . . , ν). It follows from the induction hypothesis that the zero set Wk of the polyno˜ k (x) (in the phase space of the (N − 1)-ball system (Σ , A , τ )) has a mial Q codimension-two intersection with the singularities of the (Σ , A , τ ) system. In deed, otherwise we would have (vip )k ≡ vjq−1
on some (irreducible) singularity k
manifold of the (Σ , A , τ ) subsystem. Then, by the symmetry with respect to the coordinates k = 1, 2, . . . , ν, we would have vip ≡ vjq−1 on a codimensionone singularity of the subsystem (Σ , A , τ ), contradicting to the induction hypothesis. This means that the polynomial Tk (x) vanishes on the zero set Wk of ˜ k (x), so the non-constant common divisor R( ˜ x) of Q ˜ k (x) is a common divisor of Q T1 (x), . . . , Tν (x), contradicting to the induction hypothesis. This finishes the proof of Lemma 3.6. Continuing the proof of Proposition 3.3. Denote by P˜1 (x), . . . , P˜t (x) the “Pi polynomials” of the N -ball system (Σ, A, τ ) with the constraint mN = 0 constructed the same way as the polynomials P1 (x), . . . , Ps (x) in (4.3) of [S-Sz(1999)] for the general case mN ∈ C, see also the proof of Lemma 4.2 in the cited paper. It follows from the algebraic construction of these polynomials that the irreducible polyno˜ x) is a common divisor of P˜1 (x), . . . , P˜t (x). Recall that, according to our mial Q( indirect assumption made right before Lemma 3.4, Q(x) is a common, irreducible divisor of the polynomials P1 (x), . . . , Ps (x) and, at the same time, Q(x) is one of the polynomials on the left-hand side of (1), (2), or (3) in Proposition 3.3. The ˜ x) was obtained from Q(x) by the substitution mN = 0. polynomial Q( Let us focus now on Lemma 4.9 of [S-Sz(1999)]. The non-sufficiency of the N -ball system (Σ, A, τ ) with the side condition mN = 0 comes from two sources: Either from the parallelity of the relative velocities in (2) of Lemma 4.9, or from the non-sufficiency of the (N − 1)-ball part of the orbit segment ω with the combinatorial scheme (Σ , A , τ ). The first case takes place on a complex algebraic set of (at least) 2 codimensions, thanks to our original assumption ν ≥ 3 and Lemma 3.6 above. Concerning the application of the “non-equality” Lemma 3.6 (ω) are not equal, the above, we note here that once the velocities vipp (ω) and viq−1 q p q−1 p (ω) − vipp (ω) and vN (ω) − viq−1 (ω) = vN (ω) − viq−1 (ω) are relative velocities vN q q
p q−1 not parallel, unless the common velocity vN (ω) = vN (ω) belongs to the complex p q−1 line connecting the different velocities vip (ω) and viq (ω), which is a codimensionp ˜ x) = 0 with (ω). Therefore, the equation Q( (ν − 1) condition on the velocity vN ˜ ˜ the irreducible common divisor Q(x) of the polynomials P1 (x), . . . , P˜t (x) can only ˜ x) should describe the non-sufficiency of the (Σ , A , τ )-part of the system, thus Q( lack the kinetic and mass variables corresponding to the ball with label N , as the following sub-lemma states: ˜ x) of the polynomials Sub-lemma 3.8. The irreducible common divisor Q(
P˜1 (x), . . . , P˜t (x) does not contain the variables with label N .
222
N. Sim´ anyi
Ann. Henri Poincar´e
Proof. Let D = D Σ, A, τ mN = 0 ⊂ C(2ν+1)N be the open, connected and dense domain in C(2ν+1)N defined analogously to Definition 3.18 (of [S-Sz(1999)]) but incorporating the constraint mN = 0, see also Lemma 3.19 in [S-Sz(1999)]. Let, further, N ⊂ D be a small, complex analytic submanifold of D with complex dimension (2ν + 1)N − 1, holomorphic to the unit open ball of C(2ν+1)N −1 , and ˜ x) (x ∈ D) vanishes on N . (Such a manifold N ⊂ D such that the polynomial Q( exists by the induction hypothesis of Proposition 3.3.) We split the vectors x ∈ D as x = (y , z ), so that the z-part precisely contains the variables bearing the ball label N . We may assume that N ⊂ B1 × B2 , where B1 and B2 are small, open balls in the spaces of the components y and z, respectively. Assume, to the contrary of the statement of the sub-lemma, that the polyno˜ x) ≡ Q( ˜ y , z ) does depend on the component z. Then, for typical but fixed mial Q( y , the “slice” { y0 } × B2 intersects the manifold N in a set of comvalues y0 of plex codimension one. However, this fact clearly contradicts our earlier observation that the non-sufficiency of the orbit segments ω ∈ D = D Σ, A, τ mN = 0 imposes a codimension-2 condition on the coordinates z bearing the label N . This contradiction finishes the proof of the sub-lemma. Finishing the proof of Proposition 3.3. If Q(x) is the left-hand side of (1) or (2) ˜ x) ≡ in 3.3, then we arrive at the conclusion that the irreducible polynomial Q( ˜ ˜ Q(x) divides P1 (x), . . . , Pt (x), and N = i0 , N = j0 by Sub-lemma 3.8. This means, however, that the statement of the proposition is false for the (N − 1)ball system with the discrete algebraic scheme (Σ , A , τ ), contradicting to our induction hypothesis. If, however, the polynomial Q(x) is mi0 + mj0 , then in the case if N ∈ {i0 , j0 } we arrive at a contradiction just the same way as above. If N ∈ {i0 , j0 }, ˜ x) ≡ mi0 , and mi0 is a common divisor of all polynomials say N = j0 , then Q( ˜ ˜ P1 (x), . . . , Pt (x) describing the non-sufficiency of the (Σ , A , τ ) subsystem with the N − 1 balls {1, 2, . . . , N − 1}. This means that the above (Σ , A , τ ) subsystem is always non-sufficient, provided that mi0 = 0. In the case N ≥ 4 it follows from Lemma 4.1 of [S-Sz(1999)] (applied to the (N − 2)-ball system {1, 2, . . . , N } \ {i0 , N }) and from the “non-equality” Lemma 3.6 that almost every (Σ , A , τ )orbit with mi0 = 0 is in fact sufficient. One easily checks by inspection that, in the case N = 3, actually every orbit of the 2-ball system {1, 2} with the side condition mi0 = 0 is hyperbolic (sufficient). The obtained contradiction finishes the inductive proof of Proposition 3.3. Corollary 3.9. Keep all the notations and assumptions of Proposition 3.3, except that we assume now that Σ = (σ1 , . . . , σn ) is (2C(N ) + 1)-rich and the singularity manifold C is defined by one of the following equations: (1) vikk − vjkk 2 = 0, (2) vikk − vjkk ; q˜ikk − q˜jkk − L · ak = 0, (3) mik + mjk = 0
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
223
with some k, 1 ≤ k ≤ n. Let Q(x) be a common irreducible divisor of the polynomials P1 (x), . . . , Ps (x) in (4.3) of [S-Sz(1999)] constructed for the entire discrete structure (Σ, A, τ ) as above. We again claim the same thing: The manifold C and the solution set of Q(x) = 0 cannot locally coincide. Consequently, an open, dense, and connected part of the irreducible variety V = {Q(x) = 0} belongs to the domain D (Σ, A, τ ) of the allowable initial data. Proof. We write Σ = (σ1 , . . . , σn ) in the form Σ = (Σ1 , σk , Σ2 ), where Σ1 = (σ1 , . . . , σk−1 ), Σ2 = (σk+1 , . . . , σn ). Then either Σ1 or Σ2 is C(N )-rich. If Σ2 turns out to be C(N )-rich, then we can directly apply the proposition after a simple time shift 0 −→ k. In the other case, when we only know that Σ1 is C(N )rich, beside the time shift 0 −→ k an additional time-reversal is also necessary to facilitate the applicability of the proposition. Another consequence of Proposition 3.3 is Corollary 3.10. Let ν ≥ 3, (Σ, A, τ ) be a discrete orbit structure with Property (A) and a C(N )-rich symbolic collision sequence Σ = (σ1 , . . . , σn ). Denote by P1 (x), . . . , Ps (x) the polynomials of (4.3) of [S-Sz(1999)] just as before, and let Q(x) be a common irreducible divisor of P1 (x), . . . , Ps (x). (If such a common be an extended discrete orbit structure with divisor exists.) Let, finally, Σ, A, ρ Property (A) and an extended collision sequence Σ = (σ0 , σ1 , . . . , σn ). According to Lemma 3.1, we can find a particle, say the one with label N , and two indices 1 ≤ p < q ≤ n such that (i) (ii) (iii) (iv)
N ∈ σp ∩ σq , q−1 N∈ / j=p+1 σj , σp = σq =⇒ (∃j) (p < j < q & σp ∩ σj = ∅), and Σ is (2C(N − 1) + 1)-rich on the vertex set {1, . . . , N − 1}.
(Here, just as in the case of derived schemes, we denote by Σ the symbolic sequence that can be obtained from Σ by discarding all edges containing N .) Denote by ˜ x) the polynomial that we obtain from Q(x) after the substitution mN = 0. Q( ˜ x) ≡ 0, otherwise there would not be any sufficient orbit segment (Obviously, Q( ω ∈ Ω (Σ, A, τ ) with mN = 0.) We claim that none of the irreducible polynomials on the left-hand side of (1) vi00 − vj00 2 = 0, (2) vi00 − vj00 ; q˜i00 − q˜j00 − L · a0 = 0, (3) mi0 + mj0 = 0 ˜ x). is a divisor of Q( ˜ x). According to SubProof. Consider and fix an irreducible factor R(x) of Q( lemma 3.8, the polynomial R(x) ≡ R(y , z ) does not contain any variable bearing the label N , i.e., R(x) ≡ R(y ).
224
N. Sim´ anyi
Ann. Henri Poincar´e
Assume, to the contrary of the statement that we want to prove, that the irreducible polynomial R(x) ≡ R(y ) is identical to one of the irreducible polynomials on the left-hand side of (1), (2), or (3). In particular, we have that N = i0 , of Sub-lemma 3.8, the algebraic N = j0 . As we saw in the course of the proof variety V = y ∈ C(2ν+1)(N −1)+1 R(y ) = 0 , defined by one of the equations (1), (2), or (3), describes the non-sufficiency of the derived system Ω (Σ , A , τ ) that one obtains from the original Ω (Σ, A, τ ) by taking mN = 0, i.e., for any point y ∈ D (Σ , A , τ ) it is true that y ∈ V if and only if there is some non-sufficient complex orbit segment ω ∈ Ω (Σ , A , τ ) with y (ω) = y. However, this statement contradicts to the assertion of Proposition 3.3. ˜ x) cannot be a constant c = 0, otherwise Remark. Note that the polynomial Q( the original polynomial Q(x) = c + mN · S(x) would not be homogeneous in the mass variables, see also the proof of Lemma 3.4. The main result of this section is Key Lemma 3.11. Keep all the notations and notions of this section. Assume that ν ≥ 3 and the symbolic collision sequence Σ = (σ1 , . . . , σn ) of the discrete algebraic frame (Σ, A, τ ) (with Property (A)) is C(N )-rich. We claim that all orbit segments ω ∈ Ω (Σ, A, τ ) are sufficient apart from an algebraic variety of codimension-two (at least two, that is), i.e., the polynomials P1 (x), . . . , Ps (x) of (4.3) of [S-Sz(1999)] do not have a non-constant common divisor. Proof. The inductive proof employs many of the ideas of the proof of Proposition 3.3 and it will use the statement of the proposition itself. (More precisely, the statement of Corollary 3.9 is to be used.) Indeed, the assertion of this key lemma is trivially true in the case N = 2, for in that case there are no non-sufficient, complex orbit segments ω ∈ Ω (Σ, A, τ ) at all. Assume that N ≥ 3, and the statement of the key lemma has been successfully proven for all smaller values (2 ≤) N < N . Our induction step is going to be a proof by contradiction. Suppose, therefore, that the polynomials P1 (x), . . . , Ps (x) do have a common irreducible divisor Q(x). Following the assertion of Lemma 3.1, select a suitable label k0 ∈ {1, . . . , N } for the substitution mk0 = 0 so that a derived scheme (Σ , A , τ ) (with Property (A)) exists for the arising (N − 1)-ball system {1, . . . , N } \ {k0 } with a symbolic sequence Σ , possessing the properties (1)–(4) of Lemma 3.1, the same way as we did in the proof of Proposition 3.3. Without loss of generality, we may assume that k0 = N . Consider now the original system Ω (Σ, A, τ ) with the constraint mN = 0. After the substitution mN = 0 the polynomial Q(x) becomes a new, non-constant ˜ x), see the proof of Sub-lemma 3.7 above. Let S(x) be an irreducible polynomial Q( ˜ divisor of Q(x). The (indecomposable) algebraic variety V = {S(x) = 0} has one ˜ = ΩmN =0 of Ω (Σ, A, τ ), and for every x ∈ V codimension, in the submanifold Ω there exists a non-hyperbolic complex orbit segment ω ∈ Ω (Σ, A, τ ) with x(ω) = x
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
225
and mN (ω) = 0. As far as the non-sufficiency of the orbits ω ∈ Ω (Σ, A, τ ) with mN (ω) = 0 is concerned, we again take a close look at Lemma 4.9 of [S-Sz(1999)]. We saw earlier (see the proof of Lemma 3.6, which clearly carries over to the models subjected to the side condition mN = 0) that the parallelity of the relative p q−1 velocities vN (ω) − vipp (ω) and vN (ω) − viq−1 (ω) takes place on a manifold with q codimension at least two in our case of ν ≥ 3. Therefore, according to Lemma 4.9 of [S-Sz(1999)], the “codimension-one event” x(ω) ∈ V (⇐⇒ S(x(ω)) = 0) for orbits with mN (ω) = 0 can only be equivalent to the non-sufficiency of the {1, . . . , N − 1}-part trunc(ω) ∈ Ω (Σ , A , τ ) of the system. In this way it follows from Lemma 4.9 that the irreducible polynomial S(x) lacks all variables bearing the label N , see also the statement and the proof of Sub-lemma 3.8. We conclude that for every x ∈ V (i.e., with S(x) = 0) there exists an orbit segment ω ∈ Ω (Σ, A, τ ) with mN (ω) = 0, x(ω) = x, and a non-sufficient truncated segment ω = trunc(ω) ∈ Ω (Σ , A , τ ). According to Proposition 3.3 above (applied to the (N − 1)-ball system {1, . . . , N − 1} with the algebraic scheme (Σ , A , τ )), the variety {S(x) = 0} does not even locally coincide with the singularities of the complex dynamics Ω (Σ , A , τ ). This means that a codimension-one family of complex orbit segments ω = trunc(ω) ∈ Ω (Σ , A , τ ), x(ω) ∈ V , is not sufficient. This, in turn, contradicts the induction hypothesis of the proof of Key Lemma 3.11 by actually finishing it.
4 Finishing the proof of ergodicity From C back to R First of all, we transfer the main result of the previous section (Key Lemma 3.11) from the complex set-up back to the real case. This result will be an almost immediate consequence of Key Lemma 3.11. Fix a discrete algebraic scheme (Σ, A, τ ) for N balls with Property (A) (see Definition 3.31 in [S-Sz(1999)]) and a C(N )-rich symbolic collision sequence Σ = (σ1 , . . . , σn ). (The definition of the threshold C(N ) can be found in Lemma 3.1.) Denote by ΩR = ΩR (Σ, A, τ ) the set of all elements ω ∈ Ω (Σ, A, τ ) for which (1) all kinetic functions q˜ik (ω) j , vik (ω) j , mi (ω), and L(ω) take real values, i = 1, . . . , N ; k = 0, 1, . . . , n; j = 1, . . . , ν; (2) τk (ω) = tk (ω) − tk−1 (ω) > 0 for k = 1, . . . , n; (3) out of the two real roots of (3.8) of [S-Sz(1999)] the root τk is always selected as the smaller one, k = 1, . . . , n. It is clear that either ΩR = ΩR (Σ, A, τ ) is a ((2ν + 1)N + 1)-dimensional, real analytic submanifold of Ω = Ω (Σ, A, τ ), or ΩR = ∅. Of course, we will never investigate the case ΩR = ∅. Consider the corresponding polynomials P1 (x), . . . , Ps (x) of (4.3) of [S-Sz(1999)] describing the non-sufficiency of the complex orbit segments ω ∈ Ω (Σ, A, τ ), along the lines of Lemma 4.2 of [S-Sz(1999)], in terms of the kinetic
226
N. Sim´ anyi
Ann. Henri Poincar´e
data x = x(ω). According to the statement in the third paragraph on p. 61 of [S-Sz(1999)], these polynomials Pi (x) admit real coefficients. By Key Lemma 3.11, the greatest common divisor of P1 (x), . . . , Ps (x) is 1, hence the common zero set x ∈ R(2ν+1)N +1 P1 (x) = P2 (x) = · · · = Ps (x) = 0 of these polynomials does not contain any smooth real submanifold of (real) dimension (2ν + 1)N . In this way we obtained Proposition 4.1. Use all the notions, notations and assumptions from above. There exists no smooth, real submanifold M of ΩR with dimR M = dimR ΩR − 1 (= (2ν + 1)N ) and with the property that all orbit segments ω ∈ M are nonsufficient. (For the concept of non-sufficiency, please see §2.) The “Fubini-type” argument Our dynamics Ω (Σ, A, τ ) has the obvious feature that the variables mi = mi (ω) (i = 1, . . . , N ) and L(ω) (the so-called outer geometric parameters) remain unchanged during the time-evolution. Quite naturally, we do not need Proposition 4.1 directly but, rather, we need to use its analog for (almost) every fixed (N +1)-tuple (m1 , . . . , mN ; L) ∈ RN +1 . This will be easily achieved by a classical “Fubini-type” product argument. The result is Proposition 4.2. Use all the notions, notations and assumptions from above. Denote by
N S = N S (Σ, A, τ ) = ω ∈ ΩR (Σ, A, τ ) dimC N (ω) > ν + 1 the set of all non-sufficient, real orbit segments ω ∈ ΩR = ΩR (Σ, A, τ ). (For the definition of the complex neutral space N (ω), please see (3.21) in [S-Sz(1999)].) Finally, we use the notation
= m, and L(ω) = L L) = ω ∈ ΩR m(ω) ΩR (m, for any given (N + 1)-tuple (m, L) = (m1 , . . . , mN , L) ∈ RN +1 . We claim that N +1 for almost every (m, L) ∈ R (for which ΩR (m, L) = ∅) the intersection N S ∩ L) has at least 2 codimensions in ΩR (m, L). ΩR (m, Remark 4.3. As it is always the case with such algebraic systems, the exceptional zero-measure set of the parameters (m, L) turns out to be a countable union of smooth, proper submanifolds of RN +1 . Proof of Proposition 4.2. It is clear that the statement of the proposition is a local one, therefore it is enough to prove that for any small, open subset U0 ⊂ ΩR of ΩR = ΩR (Σ, A, τ ) the set
(m, L) ∈ RN +1 dimR (N S ∩ ΩR (m, L) ∩ U0 ) ≥ 2νN − 1 of the “bad points” (m, L) has zero Lebesgue measure. The points ω ∈ U0 can be identified locally (in U0 ) with the vector x = x(ω) ∈ DR = D (Σ, A, τ )∩ΩR of their
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
227
initial coordinates. After this identification the small open set U0 ⊂ ΩR naturally becomes an open subset U0 ⊂ DR . Furthermore, we split the points x ∈ U0 as x = ((m, L), y ), where y contains all variables other than m1 , . . . , mN , L. In this way we may assume that U0 has a product structure U0 = B0 × B1 of two small open balls, so that B0 ⊂ RN +1 , while the open ball B1 ⊂ R2νN contains the y -parts of the points x = ((m, L), y ) ∈ U0 . Assume that the statement of the proposition is false. Then there exists a small open set U0 = B0 × B1 ⊂ RN +1 × R2νN (with the above splitting) and there is a positive number 0 such that the set
L) × B1 ) ∩ N S contains a A0 = (m, L) ∈ B0 ((m, (2νN − 1)-dimensional, smooth, real submanifold with inner radius > 0 has a positive Lebesgue measure in B0 . Then one can find an orthogonal projection P : R2νN → H onto a hyperplane H of R2νN such that, by taking Π(x) = Π ((m, L), y ) = P (y ), (Π : R(2ν+1)N +1 → H), the set
L) × B1 ) ∩ N S] contains A1 = (m, L) ∈ B0 Π [((m, an open ball of radius > 0 /2 in H has positive Lebesgue measure in B0 . By the Fubini theorem the set ˜ [N S ∩ (B0 × B1 )] Π has positive Lebesgue measure in B0 × H, where ˜ x) = Π ˜ ((m, Π( L), y) = ((m, L), P (y )) ∈ B0 × H for x ∈ B0 × B1 . However, dimR (B0 × H) = (2ν + 1)N , and dimR (N S ∩ ΩR ) ≤ (2ν + 1)N − 1 (according to Proposition 4.1). Thus, we obtained that the real algebraic set ˜ (N S ∩ (B0 × B1 )) ⊂ B0 × H Π has dimension strictly less than dimR (B0 × H) = (2ν + 1)N , yet it has positive Lebesgue measure in the space B0 × H. The obtained contradiction finishes the proof of Proposition 4.2. Finishing the proof of the theorem We will carry out an induction with respect to the number of balls N (≥ 2). For N = 2 the system is well known to be a strictly dispersive billiard flow (after the obvious reductions m1 v1 + m2 v2 = 0, m1 ||v1 ||2 + m2 ||v2 ||2 = 1 (m1 , m2 > 0), and after the factorization with respect to the uniform spatial translations, as usual) and, as such, it is proved to be ergodic by Sinai in [Sin(1970)], see also the paper [S-W(1989)] about the case of different masses.
228
N. Sim´ anyi
Ann. Henri Poincar´e
Assume now that N ≥ 3, ν ≥ 3, and the theorem has been successfully proven for all smaller numbers of balls N < N . Suppose that a billiard flow
t M, {S t }t∈R , µ = Mm,L , µm,L , Sm,L is given for N balls and outer geometric parameters (m, L) = (m1 , . . . , mN , L) (mi > 0, L > 0) in such a way that, besides the always assumed properties (2.1.1)– (2.1.2), (∗) the vector (m, L) of geometric parameters is such that for any subsystem 1 ≤ i1 < i2 < · · · < iN ≤ N (2 ≤ N ≤ N ) and for any C(N )-rich discrete algebraic scheme (Σ, A, τ ) (with Property (A)),for this subsystem mi1 , . . . , miN , L it is true that the parameter vector mi1 , . . . , miN , L does not belong to the zero-measured exceptional set of parameters featuring Proposition 4.2. According to Lemma 4.1 of [K-S-Sz(1990)], the set R2 ⊂ M of the phase points with at least two singularities on their trajectories is a countable union of smooth submanifolds of M with codimension two, so this set R2 can be safely discarded in the proof, for it is slim, see also §2 about the slim sets. Secondly, by the induction hypothesis and by Theorem 5.1 of [Sim(1992-I)] (adapted to the case of different masses) there is a slim subset S1 ⊂ M such that for every phase point x ∈ M \ S1 (i) S (−∞,∞) x contains at most one singularity, and (ii) S (−∞,∞) x contains an arbitrarily large number of consecutive, connected collision graphs. (In the case of a singular trajectory S (−∞,∞) x we require that both branches contain an arbitrarily large number of consecutive, connected collision graphs.) Then, by Proposition 4.2 just proved, there is another slim subset S2 ⊃ S1 of M such that (H) for every x ∈ M\S2 the trajectory S (−∞,∞) x contains at most one singularity and it is sufficient (or, geometrically hyperbolic). According to Theorem 6.1 of [Sim(1992-I)] (easily adapted to the case of different masses) and Proposition 4.2, the so-called Chernov-Sinai Ansatz (see §2) holds true, i.e., for almost every singular phase point x ∈ SR+ the positive semitrajectory S (0,∞) x is non-singular and sufficient. This is the point where the fundamental theorem for algebraic semidispersive billiards (Theorem 4.4 in [B-Ch-Sz-T(2002)]) comes to play! According to that theorem, by also using the crucial conditions (H) and the Ansatz above, it is true that for every phase point x ∈ (intM) \ S2 some open neighborhood Ux of x in M belongs to a single ergodic component of the considered billiard t flow Sm,L . Since the set (intM) \ S2 contains an arc-wise connected set C with full µ-measure (see Proposition 2.7.9 above), we get that the entire set C belongs to a single ergodic component of the flow ergodicity theorem.
t Sm,L . This finishes the proof of the
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
229
5 Concluding remark: The irrational mass ratio Due to the natural reduction N i=1 mi vi = 0 (which we always assume), in §§1–2 we had to factorize the configuration space with respect to spatial translations: (q1 , . . . , qN ) ∼ (q1 + a, . . . , qN + a) for all a ∈ Tν . It is a remarkable fact, howN ever, that (despite the reduction i=1 mi vi = 0) even without this translation factorization the system still retains the Bernoulli mixing property, provided that the masses m1 , . . . , mN are rationally independent. (We note that dropping the above-mentioned configuration factorization obviously introduces ν zero Lyapunov exponents.) For the case N = 2 (i.e., two disks) this was proven in [S-W(1989)] by successfully applying D. Rudolph’s following theorem on the B-property of isometric group extensions of Bernoulli shifts [R(1978)]: Suppose that we are given a dynamical system (M, T, µ) with a probability measure µ and an automorphism T . Assume that a compact metric group G is also given with the normalized Haar measure λ and left invariant metric ρ. Finally, let ϕ : M → G be a measurable map. Consider the skew product dynamical system (M × G, S, µ × λ) with S(x, g) = (T x, ϕ(x) · g), x ∈ M , g ∈ G. We call the system (M × G, S, µ × λ) an isometric group extension of the base (or factor) (M, T, µ). (The phrase “isometric” comes from the fact that the left translations ϕ(x) · g are isometries of the group G.) Rudolph’s mentioned theorem claims that the isometric group extension (M × G, S, µ × λ) enjoys the B-mixing property as long as it is at least weakly mixing and the factor system (M, T, µ) is a B-mixing system. But how do we apply this theorem to show that the typical system of N hard N balls in Tν with i=1 mi vi = 0 is a Bernoulli flow, even if we do not make the factorization (of the configuration space) with respect to uniform spatial translations? It is simple. The base system (M, T, µ) of the isometric group extension (M ×G, S, µ×λ) will be the time-one map of the factorized (with respect to spatial translations) hard ball system. The group G will be just the container torus Tν with its standard Euklidean metric ρ and normalized Haar measure λ. The second component g of a phase point y = (x, g) ∈ M × G will be just the position of the center of the (say) first ball in Tν . Finally, the governing translation ϕ(x) ∈ Tν is quite naturally the total displacement 0
1
v1 (xt )dt
(mod Zν )
of the first particle while unity of time elapses. In the previous sections the Bmixing property of the factor map (M, T, µ) has been proven successfully for typical geometric parameters (m1 , . . . , mN ; L). Then the key step in proving the Bproperty of the isometric group extension (M × G, S, µ × λ) is to show that the latter system is weakly mixing. This is just the essential contents of the article [S-W(1989)], and it takes advantage of the assumption of rational independence of the masses. Here we are only presenting to the reader the outline of that proof. As a matter of fact, we not only proved the weak mixing property of the extension
230
N. Sim´ anyi
Ann. Henri Poincar´e
(M × G, S, µ × λ), but we showed that this system has in fact the K-mixing property by proving that the Pinsker partition π of (M × G, S, µ × λ) is trivial. (The Pinsker partition is, by definition, the finest invariant, measurable partition of the dynamical system with respect to which the factor system has zero metric entropy. A dynamical system is K-mixing if and only if its Pinsker partition is trivial, i.e., it consists of only the sets with measure zero and one, see [K-S-F(1980)].) In order to show that the Pinsker partition is trivial, in [S-W(1989)] we constructed a pair of measurable partitions (ξ s , ξ u ) for (M × G, S, µ × λ) made up by open and connected sub-manifolds of the local stable and unstable manifolds, respectively. It followed by standard methods (see [Sin(1968)]) that the partition π is coarser than each of ξ s and ξ u . Due to the S-invariance of π, we have that π is coarser than S nξs ∧ S nξu . (∗) n∈Z
n∈Z
In the final step, by using now the rational independence of the masses, we showed that the partition in (∗) is, indeed, trivial.
References [B-Ch-Sz-T(2002)] P. B´alint, N. Chernov, D. Sz´ asz, I.P. T´oth, Multidimensional semidispersing billiards: singularities and the fundamental theorem, Ann. Henri Poincar´e 3, No. 3, 451–482 (2002). [B-F-K(1998)]
D. Burago, S. Ferleger, A. Kononenko, Uniform estimates on the number of collisions in semi-dispersing billiards, Annals of Mathematics 147, 695–708 (1998).
[B-L-P-S(1992)]
L. Bunimovich, C. Liverani, A. Pellegrinotti, Yu. Sukhov, Special Systems of Hard Balls that Are Ergodic, Commun. Math. Phys. 146, 357–396 (1992).
[B-S(1973)]
L.A. Bunimovich, Ya.G. Sinai, The fundamental theorem of the theory of scattering billiards, Math. USSR-Sb. 19, 407– 423 (1973).
[C-H(1996)]
N.I. Chernov, C. Haskell, Nonuniformly hyperbolic K-systems are Bernoulli, Ergod. Th. & Dynam. Sys. 16, 19–44 (1996).
[E(1978)]
R. Engelking, Dimension Theory, North Holland (1978).
[G(1981)]
G. Galperin, On systems of locally interacting and repelling particles moving in space, Trudy MMO 43, 142–196 (1981).
[K(1942)]
N.S. Krylov, The Processes of Relaxation of Statistical Systems and the Criterion of Mechanical Instability, Thesis, Moscow, (1942);
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
231
Republished in English by Princeton University Press, Princeton N.J. (1979). [K-S(1986)]
A. Katok, J.-M. Strelcyn, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, Lecture Notes in Mathematics 1222, Springer (1986).
[K-S-F(1980)]
I.P. Kornfeld, Ya.G. Sinai, S.V. Fomin, Ergodic Theory, Nauka, Moscow (1980).
[K-S-Sz(1989)]
A. Kr´amli, N. Sim´anyi, D. Sz´ asz, Ergodic Properties of SemiDispersing Billiards I. Two Cylindric Scatterers in the 3-D Torus, Nonlinearity 2, 311–326 (1989).
[K-S-Sz(1990)]
A. Kr´amli, N. Sim´anyi, D. Sz´ asz, A “Transversal” Fundamental Theorem for Semi-Dispersing Billiards, Commun. Math. Phys. 129, 535–560 (1990).
[K-S-Sz(1991)]
A. Kr´amli, N. Sim´anyi, D. Sz´ asz, The K-Property of Three Billiard Balls, Annals of Mathematics 133, 37–72 (1991).
[K-S-Sz(1992)]
A. Kr´amli, N. Sim´anyi, D. Sz´ asz, The K-Property of Four Billiard Balls, Commun. Math. Phys. 144, 107–148 (1992).
[L-W(1995)]
C. Liverani, M. Wojtkowski, Ergodicity in Hamiltonian systems, Dynamics Reported 4, 130–202 (1995), arXiv:math.DS/9210229.
[M(1976)]
D. Mumford, Algebraic Geometry I. Complex Projective Varieties, Springer Verlag, Berlin Heidelberg (1976).
[O-W(1998)]
D. Ornstein, B. Weiss, On the Bernoulli Nature of Systems with Some Hyperbolic Structure, Ergod. Th. & Dynam. Sys. 18, 441–456 (1998).
[P(1977)]
Ya. Pesin, Characteristic Exponents and Smooth Ergodic Theory, Russian Math. surveys 32, 55–114 (1977).
[R(1978)]
D.J. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. d’Anal. Math. 34, 36–50 (1978).
[Sim(1992)-I]
N. Sim´ anyi, The K-property of N billiard balls I, Invent. Math. 108, 521–548 (1992).
[Sim(1992)-II]
N. Sim´ anyi, The K-property of N billiard balls II, Invent. Math. 110, 151–172 (1992).
232
N. Sim´ anyi
Ann. Henri Poincar´e
[Sim(2003)]
N. Sim´ anyi, Proof of the Boltzmann-Sinai Ergodic Hypothesis for Typical Hard Disk Systems, Inventiones Mathematicae 154 No. 1, 123–178 (2003).
[Sin(1963)]
Ya.G. Sinai, On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics, Soviet Math. Dokl. 4, 1818–1822 (1963).
[Sin(1968)]
Ya.G. Sinai, Dynamical systems with countably multiple Lebesgue spectrum II, Amer. Math. Soc. Transl. 68 No. 2, 34–38 (1968).
[Sin(1970)]
Ya.G. Sinai, Dynamical Systems with Elastic Reflections, Russian Math. Surveys 25, 2, 137–189 (1970).
[St(1973)]
I. Stewart, Galois Theory, Chapman and Hall, London (1973).
[S-Ch(1987)]
Ya.G. Sinai, N.I. Chernov, Ergodic properties of certain systems of 2-D discs and 3-D balls, Russian Math. Surveys 42 No. 3, 181–207 (1987).
[S-Sz(1995)]
N. Sim´anyi, D. Sz´ asz, The K-property of Hamiltonian systems with restricted hard ball interactions, Mathematical Research Letters 2 No. 6, 751–770 (1995).
[S-Sz(1999)]
N. Sim´anyi, D. Sz´ asz, Hard ball systems are completely hyperbolic, Annals of Mathematics 149, 35–96 (1999).
[S-Sz(2000)]
N. Sim´anyi, D. Sz´ asz, Non-integrability of Cylindric Billiards and Transitive Lie Group Actions, Ergod. Th. & Dynam. Sys. 20, 593–610 (2000).
[S-W(1989)]
N. Sim´ anyi, M. Wojtkowski, Two-particle billiard system with arbitrary mass ratio, Ergod. Th. & Dynam. Sys. 9, 165–171 (1989).
[Sz(1994)]
D. Sz´asz, The K-property of ‘Orthogonal’ Cylindric Billiards, Commun. Math. Phys. 160, 581–597 (1994).
[Sz(1996)]
D. Sz´asz, Boltzmann’s Ergodic Hypothesis, a Conjecture for Centuries, Studia Sci. Math. Hung 31, 299–322 (1996).
[V(1979)]
L.N. Vaserstein, On Systems of Particles with Finite Range and/or Repulsive Interactions, Commun. Math. Phys. 69, 31– 56 (1979).
[VDW(1970)]
B.L. van der Waerden, Algebra I, Frederick Ungar Publ. Co. (1970).
Vol. 5, 2004
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems
233
[W(1988)]
M. Wojtkowski, Measure theoretic entropy of the system of hard spheres, Ergod. Th. & Dynam. Sys. 8, 133–153 (1988).
[W(1990)]
M. Wojtkowski, Linearly stable orbits in 3-dimensional billiards, Commun. Math. Phys. 129 No. 2, 319–327 (1990).
N´ andor Sim´ anyi University of Alabama at Birmingham Department of Mathematics Campbell Hall Birmingham, AL 35294 USA email: simanyimath.uab.edu Communicated by Eduard Zehnder submitted 17/10/02, accepted 01/12/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 235 – 244 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020235-10 DOI 10.1007/s00023-004-0167-7
Annales Henri Poincar´ e
Bel-Robinson Energy and Constant Mean Curvature Foliations Lars Andersson∗
Abstract. An energy estimate is proved for the Bel-Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an L∞ bound on the second fundamental form, and a bound on a spacetime version of Bel-Robinson energy.
1 Introduction ¯ , g¯) be a 3 + 1-dimensional C ∞ maximal globally hyperbolic vacuum Let (M ¯ has compact Cauchy sur(MGHV) space-time, which is spatially compact, i.e., M faces. One of the main conjectures (the CMC conjecture, see [5] for background) concerning spatially compact MGHV spacetimes states that if there is a constant ¯ , then there is mean curvature (CMC) Cauchy surface M0 , in such a spacetime M ¯ a foliation in M of CMC Cauchy surfaces with mean curvatures taking on all geometrically allowed values. Specifically, in case the Cauchy surface M0 is of Yamabe type −1 or 0, then the mean curvatures take all values in (−∞, 0), or (0, ∞), depending on the sign of the mean curvature of M0 , while in case M0 is of Yamabe type +1, the mean curvatures take on all values in (−∞, ∞). The only progress towards proving the CMC conjecture so far has been made under conditions of symmetry, cf. [11], or curvature bounds [3], [6]. One approach to the CMC conjecture is to view it as a statement about the global existence problem for the Einstein vacuum field equations ¯ αβ = 0, R
(EFE)
in the CMC time gauge. It is known that in the CMC gauge with zero shift, the (EFE) are non-strictly hyperbolic [8] while in other gauges such as wave coordinates, the (EFE) form a system of quasi-linear wave equations for the metric g¯. In this context, it has been conjectured that the Cauchy problem for the (EFE) is well posed for data in H 2 × H 1 (the H 2 conjecture, see [10]). From this point of view it is interesting to consider continuation principles for the (EFE), in CMC gauge. In this note we will use a scaling argument to prove an energy estimate for CMC foliations. The energy we consider is a version of the ∗ Supported in part by the Swedish Research Council, contract no. R-RA 4873-307, the NSF, contract no. DMS 0104402, and the Erwin Schr¨ odinger Institute, Vienna.
236
L. Andersson
Ann. Henri Poincar´e
¯ , the energy expression Bel-Robinson energy. For a spatial hypersurface M in M we consider is defined by (|E|2 + |B|2 )µg , Q(M ) = M
where E, B are the electric and magnetic parts of the Weyl tensor (defined w.r.t. the timelike normal T of M ). Roughly speaking, Q bounds Cauchy data (g, K) on M in H 2 × H 1 . Here, g is the induced metric on M and K is the second fundamental form of K. Therefore, if the H 2 conjecture is true, apriori bounds for the Bel-Robinson energy can be expected to be relevant to the global existence problem for the (EFE). Let H = trK denote the mean curvature, and assume the CMC gauge condition H = t. Define the spacetime Bel-Robinson energy of a CMC foliation FI = {Mt , t ∈ I} by Q(FI ) = dt N (|E|2 + |B|2 )µg , (1.1) I
Mt
where N is the lapse function. We are now ready to state our main result ¯ , g¯) be a MGHV space-time, and let I = (t− , t+ ) with −∞ < Theorem 1.1. Let (M ¯ , g¯). Let t0 ∈ I. t− < t+ < 0, be such that there is a CMC foliation FI in (M Suppose that lim supt→t+ Q(t) = ∞. Then at least one of the following holds: 1. lim suptt+ Q(F[t0 ,t) ) = ∞, 2. lim suptt+
||K(t)||L∞ |H(t)|
= ∞.
The time reversed statement with t+ replaced by t− also holds. Remark 1.1. Let (M, γ) be a compact hyperbolic 3-manifold with sectional cur¯ = (0, ∞) × M is flat. It vature −1. Then the metric γ¯ = −dρ2 + ρ2 γ on M follows from the work of Andersson and Moncrief [7] that for small perturbations ¯ , γ¯), there is a global CMC foliation F[t ,t) in the expanding direction, and of (M 0 for this foliation, the Bel-Robinson energy decays as Q(t) = O(H 2 (t)), which implies that the space-time Bel-Robinson energy Q(F ) is bounded in this case. It is interesting to consider the behavior of Q(F[t0 ,t) ) when t0 decreases. The proof of Theorem 1.1 is based on a scaling argument, which we now sketch. The statement of the theorem is symmetric in time, but here we consider only the future time direction, the argument in the reverse direction is similar. Suppose for a contradiction there is a constant Λ < ∞ so that Q(F[t0 ,t∗ ) ) ≤ Λ, ||K||2L∞ /H 2 ≤ Λ for all t ∈ [t0 , t∗ ), and that lim sup Q(t) = ∞ . tt∗
Vol. 5, 2004
Bel-Robinson Energy
237
An energy estimate shows that rh (t)Q(t) ≤ C, where rh is an L1,p harmonic radius, for some fixed p, 3 < p < 6, and hence if rh is bounded from below there is nothing to prove. Suppose for a contradiction that rh → 0 as t t∗ . The combination rh Q is scale invariant, and hence by rescaling g¯ to g¯ = rh−2 g¯, we get a sequence of metrics g with Q bounded. Q bounds g in L2,2 and hence we may pick out a subsequence of (g , N ), which converges weakly to a solution (g∞ , N∞ ) of the static vacuum Einstein equations, ∆N = 0, 2
∇ N = N Ric.
(1.2a) (1.2b)
It follows from our assumptions that the limit g∞ is complete, and the limiting N∞ is bounded from above and below. Then by [2], g∞ must be flat, with infinite harmonic radius, which contradicts rh = 1, by the weak continuity of rh on L2,2 . We conclude that in fact rh is bounded away from zero. and hence that Q does not blow up, which proves the theorem.
2 Preliminaries ¯ we denote its timelike normal T and induced For a space-like hypersurface M in M metric and second fundamental form (g, K). We assume all fields are C ∞ unless otherwise stated. Let lower case greek indices run over 0, . . . , 3 while lower case latin indices run over 1, . . . , 3. We work in an adapted frame eα , with e0 = ∂t . Our convention for K is Kab = − 21 LT g¯ab , so that if the mean curvature H = trK is negative, T points in the expanding direction. We will sometimes use an index T to denote contraction with T , for example uT = uα T α . In a nonflat spatially compact, globally hyperbolic, vacuum spacetime, the maximum principle implies uniqueness of constant mean curvature (CMC) Cauchy ¯ is contained in at most one CMC Cauchy surfaces. In particular, each x ∈ M surface, and for each t ∈ R, there is at most one Mt with mean curvature t. ¯ is a foliation FI = Let I ⊂ R be an interval. A CMC foliation FI in M {Mt , t ∈ I} such that for each t ∈ I, Mt is a C ∞ CMC Cauchy surface with mean curvature t. When convenient we will write g(t), K(t) for the data induced on Mt . Introducing coordinates xα with x0 = t, the lapse and shift N, X of the foliation are defined by ∂t = N T + X. We may without loss of generality assume X = 0. ¯ if there is no interval I containing We call FI a maximal CMC foliation in M ¯ , we write I as a strict subset with a CMC foliation FI . Given a foliation in M ¯ F for the support of F . M ¯ contains a compact, constant mean curvature (CMC) Cauchy Assume that M surface M0 with mean curvature H 0 < 0. By standard results there is then an interval I = (t− , t+ ) ⊂ R, H 0 ∈ I, such that there is a CMC foliation FI , and by uniqueness, MH 0 = M0 . Hence if FI is a maximal CMC foliation, then I has nonempty interior.
238
2.1
L. Andersson
Ann. Henri Poincar´e
The Bel-Robinson energy
¯ , g¯) and let ∗ W denote its (left) dual (in vacuum Let W be the Weyl tensor of (M ∗ ∗ ¯ , g¯) is defined by W = W ). The Bel-Robinson tensor Q of (M Qαβγδ = Wαµγν Wβµδ ν + ∗ W αµγν ∗ W β
µ ν δ .
(2.3)
Then Q is totally symmetric and trace-less, and in vacuum, Q has vanishing divergence. Let Eαβ = WαT βT , Bαβ = ∗ W αT βT be the electric and magnetic parts of the Weyl tensor. Then E, B are symmetric, t-tangent (i.e., EαT = BαT = 0) and trace invariant, g ab Eab = g ab Bab = 0. In vacuum, we have Eab = Ricab + HKab − Kac K cb ,
Bab = −curlKab ,
where for a symmetric tensor in dimension 3, 1 st ( ∇t Asb + bst ∇t Asa ). 2 a Recall that for symmetric traceless tensors in dimension 3, the Hodge system A → (divA, curlA) is elliptic. The following identities, see [7], relate Q to E and B, curlAab =
QT T T T = Eab E ab + Bab B ab = |E|2 + |B|2 ,
(2.4a)
QaT T T = 2(E ∧ B)a ,
(2.4b)
1 QabT T = −(E × E)ab − (B × B)ab + (|E|2 + |B|2 )gij , 3 where by definition, for symmetric tensors A, B in dimension 3,
(2.4c)
(A ∧ B)a = abc Abd Bdc , 1 1 (A × B)ab = acd bef Ace Bdf + (A · B)gab − (trA)(trB)gab . 3 3 From equation (2.4a) it follows that QT T T T ≥ 0 with equality if and only if W = 0. ¯ . The Bel-Robinson energy Q(t) of Mt ∈ F w.r.t. the Let F be a foliation in M time-like normal T , is defined by QT T T T µg = (|E|2 + |B|2 )µg . Q(t) = Q(Mt ) = Mt
Mt
An application of the Gauss law gives in vacuum, N QαβT T π αβ µg , ∂t Q(t) = −3 Mt
¯ α Tβ . A computation shows that the only nonzero components of where παβ = ∇ παβ are πab = −Kab , πT a = N −1 ∇a N . Thus N QαβT T π αβ = −N QabT T K ab − QaT T T ∇a N.
(2.5)
Vol. 5, 2004
Bel-Robinson Energy
239
3 Proof of Theorem 1.1 We will assume that the complement of points 1., 2. of Theorem 1.1 holds, and prove from this that Q(t) does not blow up. Assume for a contradiction there is a constant Λ > 1 so that for t ∈ [t0 , t∗ ), Q(F[t0 ,t) ) ≤ Λ,
||K(t)||2L∞ ≤ Λ, H 2 (t)
(3.6)
and that lim suptt∗ Q(t) = ∞. We let Ls,p denote the Lp Sobolev spaces and write H s for Ls,2 . We will sometimes use subindices x or t, x to distinguish function spaces defined w.r.t. space or space-time. For a foliation FI , we may without loss of generality assume that T = N −1 ∂t , where N > 0 is the lapse function of the foliation. Then g¯ is of the form g¯ = −N 2 dt2 + gab dxa dxb . The lapse function satisfies −∆N + |K|2 N = 1,
(3.7)
which using the maximum principle implies the estimate 1/||K||2L∞ ≤ N ≤ 3/H 2 .
(3.8)
¯ . From (2.5) we get Let F be a foliation in M |∂t Q| ≤ C1 (||∇N ||L∞ + ||N ||L∞ ||K||L∞ )Q, ˆ = K − (H/3)g with C1 a universal constant. By assumption, |K|2 /H 2 ≤ Λ. Let K ˆ 2 ≥ H 2 /3, we get be the traceless part of K. Using |K|2 = H 2 /3 + |K| |∂t Q| ≤ C(||∇N ||L∞ +
Λ )Q. |H|
(3.9)
We may assume without loss of generality that ||∇N ||L∞ ≥ Λ/|H|, since otherwise there would be nothing to prove. Therefore we may absorb Λ/|H| in the constant in (3.9) to get (3.10) |∂t Q| ≤ C||∇N ||L∞ Q.
3.1
The blowup
Choose once and for all a fixed p satisfying 3 < p < 6.
(3.11)
240
L. Andersson
Ann. Henri Poincar´e
On the Riemannian manifold (Mt , gt ), let rh (x) denote the L1,p harmonic radius of (Mt , gt ) at x ∈ Mt ; thus rh (x) is the radius of the largest geodesic ball about x on which there is a harmonic coordinate chart in which the metric coefficients gab satisfy rh (x)−3/p ||gab − δab ||Lp (Bx (rh (x))) + rh (x)(p−3)/p ||∂gab ||Lp (Bx (rh (x))) ≤ C, (3.12) where C is a fixed constant (say C = 1), cf. [1, 4]. By Sobolev embedding, in Bx (rh (x)), the C β norm of gab is controlled, for β = 1 − 3p . The presence of the factors of rh (x) in (3.12) means the estimate (3.12) is scale invariant. It follows from this that rh (x) scales as a distance. It is well known that the Laplacian in such a local harmonic coordinate chart on Bx (rh (x)) has the form ∆u = g ab ∂a ∂b u. Thus, within Bx (rh (x)), ∆ is given in these local coordinates as a non-divergence form elliptic operator, with uniform C β control on the coefficients g ab , and uniform bounds on the ellipticity constants. We have the following standard (interior) Lp elliptic estimate for this Laplace operator, cf. [9, Thm. 9.11]. Let B = Bx (rh (x)) and B = Bx ( 12 rh (x)). Then ||N ||L2,p (B ) ≤ C(rh (x), p)[||∆N ||Lp (B) + ||N ||Lp (B) ].
(3.13)
We drop the dependence on p, since p is fixed. We need to make explicit the dependence of the constant C on rh (x). This is done by a standard scaling argument. Thus, assume (by rescaling if necessary), that rh (x) = 1. Then (3.13) becomes ||N ||L2,p (B ) ≤ C[||∆N ||Lp (B) + ||N ||Lp (B) ]. By Sobolev embedding, since p > 3 is fixed, and B = B( 12 ), we have ||∇N ||L∞ (B ) ≤ c · ||N ||L2,p (B ) , so that ||∇N ||L∞ (B ) ≤ Co [||∆N ||Lp (B) + ||N ||Lp (B) ]. and in particular, ||∇N ||L∞ (B ) ≤ Co [||∆N ||L∞ (B) + ||N ||L∞ (B) ],
(3.14)
where Co is an absolute constant, (i.e., independent of N , given control on ∆ from definition of rh = 1). Now we put in scale factors to make (3.14) scale invariant and write (3.14) as rh (x)||∇N ||L∞ (B ) ≤ Co [rh (x)2 ||∆N ||L∞ (B) + ||N ||L∞ (B) ].
(3.15)
Note that the function N is itself scale invariant. Each term in (3.15) is invariant under scaling, and thus (3.15) holds in any scale. Therefore, it holds in the metric g(t).
Vol. 5, 2004
Bel-Robinson Energy
241
Let rh = rh (t) = inf inf rh (x). s≤t x∈Ms
From the lapse equation,
∆N = N |K|2 − 1.
ˆ 2 we find Using (3.8) and |K|2 = H 2 /3 + |K| 0 ≤ ∆N ≤ 3 Thus, we have
ˆ 2 |K| |K|2 ≤ 3 ≤ 3Λ. H2 H2
1 rh ||∇N ||L∞ (B ) ≤ 3Co rh2 Λ + 2 . H
(3.16)
In particular, this gives the estimate ||∇N (t)||L∞ ≤ C(Λ, t∗ )/rh (t).
(3.17)
Integrating (3.10), (recall we absorbed the term Λ/|H| in (3.9) into the constant), gives t1 Q(t1 ) ≤ Q(t0 ) + C ds||∇N (s)||L∞ Q(s). t0
We may without loss of generality assume the last term is bigger than 1, so we may absorb Q(t0 ) into C. Multiplying both sides by rh , and using (3.16) we have t1 1 rh Q(t1 ) ≤ C ds rh2 Λ + 2 Q(s). H (s) t0 We may without loss of generality assume rh ≤ 1/|H|, since otherwise there would be nothing to prove, and therefore we can absorb the term rh2 Λ into the constant. Then we have t1 1 rh Q(t1 ) ≤ C ds 2 Q(s). H (s) t0 The inequality (3.8) implies
N ≥ 3Λ−1 H −2 ,
which by the definition of the spacetime Bel-Robinson energy Q(F[t0 ,t1 ) ), see (1.1), gives rh (t1 )Q(t1 ) ≤ CQ(F[t0 ,t1 ) ). (3.18) By assumption, lim suptt∗ Q(t) = ∞, which by the assumed bound on Q(F[t0 ,t1 ) ) implies lim rh (t) = 0. tt∗
We will show that this contradicts (3.6).
242
L. Andersson
Ann. Henri Poincar´e
Suppose then that there is an increasing sequence of times ti , ti → t∗ as i → ∞, so that ri = rh (ti ) satisfy limi→∞ ri = 0 (recall that by construction rh (t) is decreasing). Now we have from (3.18) and our assumptions, ri Q(ti ) ≤ C.
(3.19)
Now we introduce the blowup scale. Let g¯i = ri−2 g¯. We will denote the scaled versions of g, K by gi , Ki . We scale the coordinates as ti = ri−1 t, xi = ri−1 x , so that the coordinate components of gi are scale invariant. Then |Ki | ≤ Λri , while the lapse N does not scale, Ni (xi ) = N (x). After translating the time coordinate as ti = ri−1 (t − ti ), we focus our attention on the time interval ti ∈ [−1, 0]. We further translate the space coordinate so that the center of the coordinate system (0,0) is the point where the harmonic radius achieves its minimum value. Since rh Q is scale invariant, we have rh Q = Q , and hence the inequality Q (ti ) ≤ C
(3.20)
holds. This means in view of the definition of the Bel-Robinson energy that at the blowup scale, Rici is bounded in L2 . By construction rh ≥ 1 and by [4], it follows from the Ricci bound that gi is bounded in L2,2 loc . Similarly the Hodge system . relating K to B leads to Ki bounded in L1,2 loc The Einstein vacuum equation is scale invariant, and therefore holds at the blowup scale. We will argue in the next section, that the above bounds on gi , Ki allow us to pick out a weakly convergent subsequence of (gi , Ni ) with limit g∞ , N∞ solving the static vacuum Einstein equation, cf. equation (1.2) below, with g∞ complete.
3.2
Weak convergence
Let g¯i be the sequence of rescaled spacetime metrics. We consider rescaled time ti in the interval [−1, 0]. By construction, the L1,p harmonic radius satisfies ri (0) = 1, and ri (t) ≥ 1 for t ∈ [−1, 0]. Equation (3.8) implies that the rescaled lapse is bounded from above and below, 1 t2− Λ
≤ Ni ≤
3 . t2+
(3.21)
By (3.20), we have Q (t) ≤ C, for t ∈ [−1, 0], and hence we have (gi , Ki ) bounded 1,2 in L∞ ([−1, 0]; L2,2 loc × Lloc ). It follows that there is a subsequence which converges 1,2 weak- to a limit (g∞ , K∞ ) ∈ L∞ ([−1, 0]; L2,2 loc × Lloc ), with corresponding spacetime metric g¯∞ . By passing to a further subsequence if necessary, which we still denote using the index i, we may assume that gi (0) g∞ (0) weakly in L2,2 loc . Let us consider the properties of this limit. First note that |Ki | ≤ Λri → 0 as i → ∞, and hence K∞ ≡ 0. The relation ∂ti gi = −2Ni Ki holds in the limit
Vol. 5, 2004
Bel-Robinson Energy
243
and since K∞ ≡ 0, we conclude that g∞ is time-independent, so that the limiting spacetime metric g¯∞ is static. The lapse equation now implies that the limiting lapse function satisfies (3.22a) ∆∞ N∞ = 0, where ∆∞ is the Laplace operator defined w.r.t. g∞ . The rescaled spacetime metrics g¯i are solutions of the Einstein vacuum equation and the evolution equation for K, ∂t K = −∇2 N + N (Ric + HK − 2K:K), where (K : K)ab = Kac K cb , holds weakly in the limit. In view of the fact that g → Ric is weakly continuous on L2,2 loc and K∞ ≡ 0, we get the equation 0 = −∇2∞ N∞ + N∞ Ric∞ .
(3.22b)
By construction, g∞ is complete, and hence in view of (3.22) we have a complete solution of the static Einstein equations with N∞ > 0. It follows by [2, Theorem 3.2] that g∞ is flat and N∞ is constant. In particular, rh [g∞ ](0) = ∞. Now, since rh is by definition the L1,p harmonic radius, 3 < p < 6, the map g → rh is weakly continuous on L2,2 loc and hence by construction rh [g∞ ](0) = 1. This is a contradiction, and it follows that in fact lim inf i→∞ ri > 0, which by the BR energy estimate (3.19) implies that Q(t) does not blow up. This completes the proof of Theorem 1.1. Acknowledgments. The problem studied in this paper was suggested by Mike Anderson. I am grateful to him for many helpful suggestions, and to Vince Moncrief and Jim Isenberg for useful discussions on the topic of this paper.
References [1] Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 no. 2, 429–445 (1990). [2]
, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds. I, Geom. Funct. Anal. 9 no. 5, 855–967 (1999).
[3]
, On long-time evolution in general relativity and geometrization of 3-manifolds, Comm. Math. Phys. 222 no. 3, 533–567 (2001).
[4] Michael T. Anderson and Jeff Cheeger, C α -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 no. 2, 265–281 (1992). [5] Lars Andersson, The global existence problem in general relativity, grqc/9911032, to appear in “50 Years of the Cauchy problem in General Relativity”, eds. Piotr T. Chru´sciel and Helmut Friedrich.
244
[6]
L. Andersson
Ann. Henri Poincar´e
, Constant mean curvature foliations of flat space-times, Comm. Anal. Geom. 10 no. 5, 1125–1150 (2002).
[7] Lars Andersson and Vincent Moncrief, Future complete vacuum spacetimes, gr-qc/0303045, to appear in “50 Years of the Cauchy problem in General Relativity”, eds. Piotr T. Chru´sciel and Helmut Friedrich. [8] Yvonne Choquet-Bruhat and Tommaso Ruggeri, Hyperbolicity of the 3 + 1 system of Einstein equations, Comm. Math. Phys. 89 no. 2, 269–275 (1983). [9] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, second ed., Springer-Verlag, Berlin, 1983. [10] Sergiu Klainerman, Geometric and Fourier methods in nonlinear wave equations, preprint, Princeton, 2001. [11] Alan D. Rendall, Constant mean curvature foliations in cosmological spacetimes, Helv. Phys. Acta 69 no. 4, 490–500, (1996), Journ´ees Relativistes 96, Part II (Ascona, 1996), gr-qc/9606049. Lars Andersson Department of Mathematics University of Miami Coral Gables, FL 33124 USA email:
[email protected] Communicated by Sergiu Klainerman Submitted 25/07/03, accepted 27/01/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 245 – 260 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020245-16 DOI 10.1007/s00023-004-0168-6
Annales Henri Poincar´ e
On the Uniqueness of AdS Space-Time in Higher Dimensions Jie Qing
Abstract. In this paper, based on an intrinsic definition of asymptotically AdS spacetimes, we show that the standard anti-de Sitter space-time is the unique strictly stationary asymptotically AdS solution to the vacuum Einstein equations with negative cosmological constant in dimension ≤ 8. Instead of using the positive energy theorem for asymptotically hyperbolic spaces with spin our approach appeals to the classic positive mass theorem for asymptotically flat spaces.
1 Introduction Recently, there has been some interest in the study of space-times that satisfy Einstein equations with negative cosmological constant in association with the so-called AdS/CFT correspondence. With the presence of a negative cosmological constant, the anti-de Sitter space-time replaces the Minkowski space-time as the ground state of the theory. Boucher, Gibbons and Horowitz demonstrated that in 3 + 1 dimensions, the only strictly stationary asymptotically AdS space-time that satisfies the vacuum Einstein equations with negative cosmological constant is the anti-de Sitter space-time in [BGH] (see also [CS]). Another class of globally static asymptotically locally AdS space-times, the AdS solitons, are also important in the theory. In [GSW], Galloway, Surya and Woolgar proved a uniqueness theorem of the AdS solitons. Later, in [ACD], Anderson, Chru´sciel and Delay improved the uniqueness theorem of AdS solitons. In [Wa1], the uniqueness result of [BGH] was generalized to higher dimensions when the space-time is static and the static slice is of spin structure. Proofs in [BGH] and [Wa1] all appeal to the positive energy theorem for asymptotically hyperbolic spaces (see [CH], [Wa3] and some early references therein). Our proof of the uniqueness of the AdS space-time instead appeals to the classic positive mass theorem for asymptotically flat spaces. By this approach we may use the classic positive mass theorem of Schoen and Yau [SY] to drop the spin structure assumption on the static slice in dimension less than 8. The anti-de Sitter space-time in n + 1 dimensions is given by (Rn+1 , gAdS ) where 1 gAdS = −(1 + r2 )dt2 + dr2 + r2 dσ0 (1.1) 1 + r2
246
J. Qing
Ann. Henri Poincar´e
in coordinates (t, r, θ) ∈ R × [0, ∞) × S n−1 and dσ0 is the standard round metric on a unit (n − 1)-sphere. It is a static solution to the vacuum Einstein equation 1 Ric − R g + Λg = 0 2
(1.2)
with negative cosmological constant Λ = − 12 n(n − 1). The staticity implies that √ (Rn+1 , gAdS ) can be constructed by a triple (Rn , gH , 1 + r2 ) where gH is the hyperbolic metric on Rn and (1.3) ∇2 1 + r2 = 1 + r2 gH on the hyperbolic space (Rn , gH ). The simplest examples of space-times that are asymptotically the same as the anti-de Sitter space-time at the infinity are the so-called Schwarzschild-AdS space-times whose metrics are given by + = −(1 + r2 − gM
M 1 )dt2 + 2 rn−2 1+r −
M r n−2
dr2 + r2 dσ0 .
(1.4)
They also satisfy the vacuum Einstein equation (1.2), but the difference is that on ∂ while this is the AdS space-time there is an everywhere time-like Killing field ∂t not so on the Schwarzschild-AdS space-times. In other words, the AdS space-time is strictly stationary, but the Schwarzschild-AdS space-times are not. We always assume in this paper that every orbit of the time-like Killing field is complete for stationary space-times. We will follow the idea in [AM] to give a definition of asymptotically AdS space-times (see Definition 2.1). One can find a good discussion of the comparisons of different definitions of asymptotically AdS space-times in [CS]. Then we show Theorem 1.1 Suppose that (Y n+1 , g) is a strictly stationary asymptotically AdS space-time satisfying the causality axiom. And suppose that g satisfies the vacuum Einstein equation with negative cosmological constant. Then (Y n+1 , g) is static, i.e., n+1 =R×Σ Y (1.5) g = −V dt2 + h where V > 0 on Σ and
√ √ ∆ V =n V √ √ Ric[h] + nh = ( V )−1 ∇2 V
(1.6)
on the Riemannian manifold (Σ, h). We adopt the definition of causality axiom from [Ca], which simply requires no closed time-like curves in the space-time. By the Frobenius Theorem, staticity is locally equivalent to θ = ω ∧ dω = 0 where ω is the dual of the given Killing
Vol. 5, 2004
Asymptotically AdS
247
vector field X. Instead of using topological assumptions to write ∗θ = dψ to prove the vanishing of θ in classic Lichnerowicz argument (see [BGH], [Ca]) we observe that 1 1 d( ω) = − 2 iX θ, (1.7) V V which allows us to calculate the boundary integral to show the vanishing of θ from the behavior of X near the infinity. We adopt the method of Fefferman and Graham [FG], [G] to construct a preferable coordinate system near the infinity which allows us to know the asymptotic behavior of both the metric g and the Killing field X near the infinity rather precisely. √ Our next goal is to prove the static solution (Σn , h, V ) must be the same √ as (Rn , gH , 1 + r2 ) for some choice of coordinates. Namely, Theorem 1.2 Suppose that a space-time is asymptotically AdS and space-like geodesically complete. And suppose that the asymptotically AdS space-time √ √ is a static solution satisfying (1.5) and (1.6). Then (Σn , h, V ) = (Rn , gH , 1 + r2 ) for some choice of coordinates in dimension 3 ≤ n ≤ 7. First we would like to remark here that the large class of static solutions in dimension 4 constructed by Anderson, Chru´sciel and Delay in [ACD] have different asymptotic behaviors than the one we imposed here. Our approach is similar to the one used to prove the uniqueness of conformally compact Einstein manifolds in √ [Q]. We use the global defining function ( V + 1)−1 to turn (Σ, h) into a compact ¯ which has the round sphere as its totally umbilical boundary and manifold (Σ, h) whose scalar curvature is non-negative. The nonnegativity of the scalar curvature √ n(n − 1)(V − |∇ V |2 − 1) follows from the application of the strong maximum principle and the following Bochner formula: Lemma 1.3 Suppose that an asymptotically AdS space-time is a static solution satisfying (1.5) and (1.6). Then √ √ √ √ √ ∇ V −∆(V − |∇ V |2 − 1) = 2|∇2 V − V h|2 − √ · ∇(V − |∇ V |2 − 1). (1.8) V One should compare (1.8) with an identity of Lindblom [L] for n = 3 (see also [BGH] [BS] for its applications). Then we appeal to the recent work in [Mi](see also [ST]) to conclude that it has to be scalar flat, which implies √ √ ∇2 V = V h. (1.9) Note that [Mi] relies on the classic positive mass theorem of Schoen and Yau [SY]. Theorem 1.2 then follows from the following lemma similar to a theorem of Obata in [Ob]. Lemma 1.4 Suppose that (M n , g) is a complete Riemannian manifold. And suppose that there is a positive function φ such that ∇2 φ = φg. Then (M, g) is isometric to (Rn , gH ).
248
J. Qing
Ann. Henri Poincar´e
In summary we prove Main Theorem Suppose that an asymptotically AdS space-time (Y n+1 , g) satisfies the causality axiom and is space-like geodesically complete. And suppose that (Y n+1 , g) is a strictly stationary solution to the vacuum Einstein equations with negative cosmological constant. Then (Y n+1 , g) is the standard AdS space-time for 3 ≤ n ≤ 7.
2 Asymptotically AdS space-times We assume through out this paper that a space-time is always orientable and connected as a manifold. In this section we will start with an intrinsic definition of asymptotically AdS space-times and derive some properties of a strictly stationary asymptotically AdS space-time. We will then prove a lemma of Lichnerowicz type similar to the one in [BGH]. We note that, in fact, it was asked whether the uniqueness theorem in their paper [BGH] still holds if one uses the definition of asymptotically AdS space-times proposed by Ashtekar and Magnon in [AM] (see also [Ha]). Let us first introduce the AdS space-time in general dimensions. The anti-de Sitter space-time in (n + 1) dimensions can be given by (Rn+1 , gAdS ) where gAdS = −(1 + r2 )dt2 +
1 dr2 + r2 dσ0 1 + r2
(2.1)
dσ0 is the unit round metric on S n−1 , t ∈ (−∞, +∞), and r ∈ [0, +∞). In the following we will adopt the definition of an asymptotically AdS space-time given by Ashtekar and Magnon in [AM]. To do so, let us first discuss what a conformal completion for a space-time is following the idea of Penrose in [Pe]. Suppose that Y n+1 is a manifold with boundary ∂Y n+1 = X n . Then Ω is said to be a defining function of X n in Y n+1 if a) Ω > 0, in Y n+1 ; b) Ω = 0, on X n ; and c) dΩ = 0 on X n . A space-time (Y n+1 , g) has a C k conformal completion if Y n+1 is a manifold with boundary X n and the metric Ω2 g for a defining function Ω of X n in Y n+1 extends in C k to the closure of Y n+1 . Definition 2.1 A space-time (Y n+1 , g) of dimension (n + 1) is said to be asymptotically AdS if 1) (Y n+1 , g) has a C k conformal completion and its boundary ∂Y n+1 = X n is topologically R × S n−1 ; 2) the space-time (Y n+1 , g) satisfies the Einstein equation with a negative cosmological constant Λ 1 Rab − Rgab + Λgab = 8πGTab 2 where Ω−n Tab admits a C k extension to the closure of Y n+1 ; 3) (X n , Ω2 g|T X n ) is conformal to (R × S n−1 , g0 ) where g0 = −dt2 + dσ0 .
(2.2)
Vol. 5, 2004
Asymptotically AdS
249
For the convenience, from now on, we will always assume Λ=−
n(n − 1) . 2
(2.3)
First, by definition, defining functions for X n in Y n+1 are not unique and two different defining functions differ by a positive function on the closure of Y n+1 . Therefore only the class of quadratic forms Ω2 g|T X n up to a conformal factor is determined by g. Second, by requiring the fall-off of the energy-stress tensor, one can compute that the sectional curvature of g would asymptotically go to −|dΩ|2Ω2 g , and conclude |dΩ|2Ω2 g |X n = 1 > 0. Therefore X n is a time-like hypersurface in (Y n+1 , Ω2 g). Finally the conformal flatness of the Lorentz metric Ω2 g|T X n should depend only on the Lorentz metric g. Hawking in [Ha] suggested that local conformal flatness is a necessary boundary condition, see more detailed discussions on this in dimension 4 by Chru´sciel and Simon in [CS]. But we choose to impose the global conformal flatness here to accommodate the existence of a time-like Killing vector field. We next want to choose a coordinate system near the boundary for an asymptotically AdS space-time. What we will do is mostly an analogue to the Euklidean cases which have been established in [FG], [G]. First, we construct a special defining function, at least in a tubular neighborhood of the boundary for each given metric in the class [−dt2 + dσ0 ] on the boundary by solving a first-order PDE. Namely, Lemma 2.1 Suppose (Y n+1 , g) is an asymptotically AdS space-time, and Ω is a defining function. Then, for each metric gˆ = e2φ g0 where g0 = −dt2 + dσ0 , there is a unique defining function s in a tubular neighborhood of the boundary X n in Y n+1 such that a) s2 g|T X n = gˆ; b) |ds|s2 g = 1 in the tubular neighborhood. Proof. Set s = ew Ω. Then ds = ew (dΩ + Ωdw) and
|ds|2s2 g = |ds|2e2w Ω2 g = e−2w |ds|2Ω2 g = |dΩ + Ωdw|2Ω2 g = |dΩ|2Ω2 g + 2Ω(dΩ, dw)Ω2 g + Ω2 |dw|2Ω2 g .
Thus, the requirement |ds|s2 g = 1 is equivalent to solving 2(dΩ, dw)Ω2 g + Ω|dw|2Ω2 g =
1 − |dΩ|2Ω2 g Ω
.
(2.4)
250
J. Qing
Ann. Henri Poincar´e
The boundary condition is determined as follows: if we denote Ω2 g|T X n = e2ψ g0 , then w|X n = φ − ψ. (2.5) It is easily seen that (2.4) and (2.5) is non-characteristic. Notice that both dΩ and dw are space-like. Lemma 2.2 Suppose (Y n+1 , g) is an asymptotically AdS space-time. Suppose that s is the special defining function obtained in Lemma 2.1 for which s2 g|T X n = g0 . Then g = s−2 (ds2 + gs ) (2.6) where gs = −(1 +
s2 2 2 s2 ) dt + (1 − )2 dσ0 + O(sn ). 4 4
(2.7)
Proof. The proof again is adopted from the argument given in [FG], [G]. By the fall-off condition of the energy-momentum tensor Tab one can rewrite the equation (2.2) in coordinates R × S n−1 × [0, ) near the boundary as 1 hab +(1−n)hab −hcdhcd hcd −shcd hac hbd + shcd hcd hab −2sRab [h] = O(sn ), (2.8) 2 where h stands for gs for convenience. The signature of gs here does not make any difference in terms of solving the expansion of gs . Therefore, similar to what is known for Euklidean case, all odd-order terms of order ≤ n−1 vanish and all evenorder terms of order ≤ n − 1 is determined by the metric g0 on X n . Moreover, when n is odd, the nth order term is traceless; when n is even, in general one would need to add one more term in the order of sn log s which is traceless and determined by g0 while the trace part of the nth order is also determined by g0 . By comparing to the AdS space gAdS = s−2 (ds2 − (1 +
s2 2 2 s2 ) dt + (1 − )2 dσ0 ) 4 4
which is of the same boundary metric −dt2 + dσ0 , we may complete the proof. Remark 2.3 In the above argument, it is clear that a weaker fall-off condition of the energy-momentum would imply a weaker control of the asymptotic of the metric g. Next we will follow [BGH] to restrict ourselves to the so-called strictly stationary space-time. That is to assume, for an asymptotically AdS space-time, there ∂ is a global everywhere time-like Killing field which approaches ∂t asymptotically towards the boundary. In [BGH] it was demonstrated that a strictly stationary asymptotically AdS space-time (by their definition in dimension 3) which solves the vacuum Einstein equations with negative cosmological constant must be a
Vol. 5, 2004
Asymptotically AdS
251
static one. Before proceeding to prove the staticity we want to study the asymp∂ totic behavior of a Killing field that approaches ∂t at the infinity. We will use the favorable coordinates constructed in Lemma 2.2. Denote the Killing field by X = a(s, t, σ)
∂ ∂ ∂ , + b(s, t, σ) + ci (s, t, σ) ∂s ∂t ∂θi
(2.9)
where (s, t, θ1 , . . . , θn−1 ) ∈ [0, ) × R × S n−1 . For similar computations, please see [Wa2]. First of all, by the boundary condition, we know that b(0, t, θ) = 1,
ci (0, t, θ) = 0, ∀i = 1, . . . , n − 1.
(2.10)
∂ ∂ Computing Xg( ∂s , ∂s ) one gets
a ∂a = . ∂s s
(2.11)
Therefore a(s, t, θ) = sa(t, θ). For convenience we denote by t = θ0 and b = c0 , ∂ , ∂θ∂α ) one gets and use Greek letters to include zero. Computing Xg( ∂s gs (
∂a ∂ ∂ ∂cβ +s , ) = 0. ∂θα ∂θβ ∂s ∂θα
s ∂a b(s, t, θ) = 1 − ugu0β du ∂θβ 0 s ∂a cα (s, t, θ) = − uguαβ du . ∂θβ 0
Therefore
(2.12)
In the other directions one gets gs (
∂ ∂ γ ∂ ∂ γ , )c + gs ( , )c = 0. ∂θα ∂θγ ,β ∂θβ ∂θγ ,α
(2.13)
Notice that, if we denote the Christoffel symbols of metric gs on the slices by Γα βγ ¯ a for the ones of metric g, then and Γ bc
¯α Γα βγ = Γβγ ,
¯ α = 1 hαγ ∂ hγβ − 1 δαβ , Γ sβ 2 ∂s s
(2.14)
where again, for convenience, we use h = gs . Thus hαγ (
∂cγ ∂cγ ∂ − cδ Γγδβ ) + hβγ ( − cδ Γγδα ) = 2ahαβ − sa hαβ . ∂θβ ∂θα ∂s
(2.15)
Taking s = 0 in (2.15) we immediately see that a(t, θ) = 0 and surprisingly get ∂ X = ∂t in this neighborhood. Moreover (2.15) implies gs is independent of t in this neighborhood. Let us summarize what we obtained in a lemma.
252
J. Qing
Ann. Henri Poincar´e
Lemma 2.4 Suppose that (Y n+1 , g) is an asymptotically AdS space-time and that s is the special defining function such that s2 g|T X = −dt2 + dσ0 . If X is a Killing ∂ ∂ field that approaches to ∂t at the infinity, then X = ∂t and gs is independent of t in a tubular neighborhood of the boundary. A strictly stationary asymptotically AdS space-time (Y n+1 , g) comes with a free R1 action. In fact the action is proper if the asymptotically AdS space-time satisfies the causality axiom (cf. [Ca]) since the behavior at the spatial infinity is described in the above Lemma 2.4. The causality axiom excludes any timelike closed curves in the space-time. Therefore, by Theorem 1.11.4 in [DK], we know that Y is a R1 -principle bundle over the smooth orbit space Y /R. Thus topologically, Y = R × M , since Y is orientable and connected, where M is a smooth n-manifold with boundary S n−1 in the light of Lemma 2.4. Now let us discuss the staticity of a space-time. A good reference for this discussion is [Ca]. A space-time is said to be static if there is an everywhere time-like Killing field whose trajectories are everywhere orthogonal to a family of space-like hypersurfaces. We emphasize that our definition of staticity adopted from [Ca] requires not only (2.17) below but also the existence of a global integral hypersurface. Let us introduce some notations. Let {ea } be an orthonormal frame and {wa } be its co-frame. Suppose X = k a ea is an everywhere time-like Killing field and let ω = ka wa . In this notation X is a Killing field if ka,b + kb,a = 0.
(2.16)
To apply Frobenius’ Theorem, we need to ask first that the differential ideal generated by the differential ω be closed under exterior differentiation, i.e., θ = ω ∧ dω = 0.
(2.17)
After (2.17) is established, we would like to say such connected space-time becomes R × Σ where Σ is a static slice (topologically the same as M ). This is easily seen because the following. First, take any global space-like hypersurface, say M of Y , such M exists since Y topologically is R×M where R is the action generated by the given time-like Killing field. Then, any piece of embedded space-like hypersurface is a graph over M and the function of the graph is “reference time” relative to M . For a maximal integral hypersurface Σ which is everywhere orthogonal to the Killing field through a given point in the space-time, we claim that, the “reference time” never reaches infinite in any bounded region. Simply because Σ may be considered at the same time in a “real time”. Hence, relative to Σ, we may read the “real time” at each point on M and it never gets to infinite in any bounded region. Therefore the maximal integral hypersurface Σ has to reach to the space infinity where we have better idea what happens. Thus the maximal integral hypersurface is global static slice. The metric may written as g = −V dt2 + gΣ where V = −k a ka and gΣ is the metric of Euklidean signature on the slice Σ . Now we are ready to prove the following lemma of Lichnerowicz type (cf. [BGH] [Ca]). Our proof is adapted for
Vol. 5, 2004
Asymptotically AdS
253
general dimensions and uses no additional topological assumptions. Notice that the definition of an asymptotically AdS space-time in [BGH] is different from ours in this note. Lemma 2.5 Any strictly stationary asymptotically AdS space-time (Y n+1 , g) which satisfies the causality axiom and the vacuum Einstein equations with negative cosmological constant Λ is static. To prove this lemma we observe Lemma 2.6
iX θ ω = −d( ) V2 V
(2.18)
where V = −k a ka . Proof. We simply compute d(
dω 1 1 iX θ ω )= − 2 dV ∧ ω = − 2 (dω(−V ) + dV ∧ ω) = − 2 . V V V V V
Because iX ω = ω(X) = ka k a = −V and
iX dw = iX (ka,b wb ∧ wa ) = ka,b iX wb wa − ka,b wb iX wa = ka,b k b wa − ka,b k a wb = −2k b kb,a wa = dV.
Proof of Lemma 2.5. Let us consider the Hodge dual ∗θ of θ . Since d∗ θ = (k[a,b kc] )c wa ∧ wb =
2 c k Rc[a kb] wa ∧ wb = 0 3
(2.19)
due to the fact that Rab = nηab where ηab is the standard Minkowski metric (please see Chapter 6 in Part II of [Ca]) and iX (∗θ) = ∗(θ ∧ ω) = 0, it follows
(2.20)
iX θ ∧ ∗θ ω iX (θ ∧ ∗θ) ∧ ∗θ) = − =− . (2.21) V V2 V2 The next step is to integrate over a space-like hypersurface Σ whose boundary is a large (n − 1)-sphere S n−1 = {s = , t = c} in the preferable coordinates. We therefore have |θ|2 ω(N ) ω ∧ ∗θ (2.22) dσ = − 2 n−1 V V Σ S d(
254
J. Qing
Ann. Henri Poincar´e
where N is the unit normal of Σ , and dσ is the volume element of Σ in the space-time. Notice that ω(N ) > 0 and θ is space-like since θ ∧ ω = 0. Now let us ∂ recall that the Killing field X is just ∂t in the preferable coordinates. Thus ω|Sn−1 = −V dt + g0k dθk = −V dt + sn−2 τ0k dθk = n−2 τ0k dθk where g = s−2 (ds2 + (1 +
s2 2 2 s2 ) dt + (1 − )2 dσ0 + sn τ ) 4 4
and V = s−2 (1 +
s2 2 ) − sn−2 τ00 . 4
(2.23)
(2.24)
(2.25)
Then dω = −dV ∧ dt + dg0k ∧ dθk , θ = −V dt ∧ dg0k ∧ dθk − sn−2 τ0k dθk ∧ dV ∧ dt + sn−2 τ0k dθk ∧ dg0l ∧ θl , (2.26) and
∂V ∂g0k + g0k ) ∗ (dt ∧ ds ∧ dθk ) ∂s ∂s = Csn−5 ∗ (dt ∧ ds ∧ dθk )
∗θ|Sn−1 = (−V
(2.27)
k ∧ · · · ∧ dθn−1 = Cdθ1 ∧ · · · dθ where C stands for some function on S n−1 . Therefore ω ∧ ∗θ|Sn−1 = O(n )dθ1 ∧ dθ2 ∧ · · · ∧ dθn−1 . (2.28) V This implies that θ = 0 on the hypersurface Σ. But Σ is arbitrary, so θ = 0 on Y n+1 , which finishes the proof in the light of the discussions before the statement of this lemma. Let us conclude this section by making it clear what a static asymptotically AdS space-time which satisfies the vacuum Einstein equations with negative cosmological constant Λ is. We first state an observation in the following lemma. Lemma 2.7 Under the assumption of Lemma 2.5, in the preferable coordinate sys∂ is orthogonal tem at the infinity, indeed, a slice of constant t is a static slice, i.e., ∂t to the slice of constant t. Proof. Consider the conformal completion (Y n+1 , g¯) where g¯ = ds2 +gs . By the construction of the preferable coordinate system, each curve γ(s) = (s, t, θ1 , . . . , θn−1 ) is a geodesic from the point (0, t, θ1 , . . . , θn−1 ) in the space-time (Y n+1 , g¯). On the other hand, a static slice Σ of (Y n+1 , g) is still a maximum integral hypersurface ∂ everywhere with respect to g¯. Because g¯ = ds2 + gs which is orthogonal to ∂t is independent of t, such Σ is totally geodesic in (Y n+1 , g¯). Therefore a geodesic emanating from a boundary point (0, t0 , θ1 , . . . , θn−1 ) with respect to the metric g¯ stays in a static slice. Thus a slice of constant t coincides with a static slice. So the proof is complete. We summarize our result in the following theorem:
Vol. 5, 2004
Asymptotically AdS
255
Theorem 2.8 Suppose that (Y n+1 , g) is a space-like geodesically complete spacetime. And suppose (Y n+1 , g) is a strictly stationary asymptotically AdS space-time that satisfies the causality axiom and the vacuum Einstein equations with negative . Then Y n+1 = R × Σn , cosmological constant Λ = − n(n−1) 2 g = −V dt2 + h and, on Σ,
√ √ ∆ V =n V √ √ Ric[h] + nh = ( V )−1 ∇2 V ,
(2.29)
(2.30)
where h is the metric of Euklidean signature induced from g on a static slice Σn . Moreover (Σn , h) is complete and conformally compact of the same regularity as of the conformal completion of (Y n+1 , g) with the conformal infinity (S n−1 , [dσ0 ]) where V −1 h|T S n−1 = dσ0 .
3 Static asymptotically AdS space-times In this section we study static asymptotically AdS space-times. We will prove the uniqueness of static asymptotically AdS space-times. In dimension 3 + 1, with a bit restrictive definition of asymptotically AdS space-times, the uniqueness was first proved in [BGH] (see also [CS]). Then, assuming spin structure for n > 3, the uniqueness of static solutions (M n , g, V ) to the vacuum Einstein equations with negative cosmological constant was established in [Wa1] (also see the definition of a static solution (M, g, V ) in [CS], [Wa1]). Our proof will not use the spin structure in dimensions higher than three, but instead will rely on a recent work of Miao [Mi] (see also [ST]) which in turn depends on the classic positive theorem of Schoen and Yau [SY] for asymptotically flat manifolds. By Theorem 2.8 in the previous section, a static asymptotically AdS spacetime which satisfies the vacuum Einstein equations with negative cosmological √ constant is given by a static solution (Σ, h, V ) in our notation. Therefore, by Lemma 2.2 in the previous section, we know that h = s−2 (ds2 + (1 − V = s−2 ((1 + and
s2 2 ) dσ0 + τ sn + o(sn )), 4
s2 2 ) − αsn + o(sn )), 4
√ V − |∇ V |2 − 1 = nαsn−2 + o(sn−2 )
where α = −Trdσ0 τ (these were known in [Wa1]).
(3.1)
(3.2)
(3.3)
256
J. Qing
Ann. Henri Poincar´e
To motivate our argument in this section we recall the following fact about 2 2 2 2 1+|x| the static solution (B n , ( 1−|x| 2 ) |dx| , 1−|x|2 ) associated with the AdS space-time (Rn+1 , gAdS ). Namely, if one uses the global defining function 1 − |x|2 1 , = u= √ 2 V +1 then
u2 h = |dx|2 . √ Therefore, for a static solution (Σ, h, V ), if we denote u = global defining function for S n−1 in Σ and u2 h = where
√ 1 , V +1
then u is a
1 s2 √ (ds2 + (1 − )2 dσ0 + τ sn + o(sn )) 4 s2 ( V + 1)2
√ √ √ s2 ( V + 1)2 = ( s2 V + s)2 = s2 V + 2s s2 V + s2 = 1 + 2s + O(s2 ).
So
u2 h = (1 + 2s + O(s2 ))ds2 + (1 + 2s + O(s2 ))dσ0 + O(s2 ).
(3.4)
2
Thus (Σ, u h) is a compact manifold with the standard (n−1)-sphere as its boundary and the second fundamental form for ∂Σ in Σ is dσ0 (i.e., the boundary is totally umbilical). In the light of (2.30) one may compute the scalar curvature for u2 h as follows: n−2 n−2 4(n − 1) ∆u 2 + R[h]u 2 ) n−2 √ = n(n − 1)(V − |∇ V |2 − 1),
R[u2 h] = u−
n+2 2
(−
(3.5)
which goes to zero as s → 0 by (3.3). We observe the following lemma which will allow us to apply the Strong maximum principle to conclude that R[u2 h] ≥ 0. Namely, Lemma 3.1
√ √ 2 √ √ √ ∇ V 2 2 −∆(V − |∇ V | − 1) = 2|∇ V − V h| − √ · ∇(V − |∇ V |2 − 1). (3.6) V
Proof. We simply compute √ −∆V = 2(−nV − |∇ V |2 ) and
√ √ √ √ ∆|∇ V |2 = 2(|∇2 V |2 + ( V )i ( V )ijj ) √ √ √ √ √ = 2(|∇2 V |2 + ( V )−1 ( V )i ( V )ij ( V )j )).
Vol. 5, 2004
Asymptotically AdS
257
Therefore √ −∆(V − |∇ V |2 − 1) √ √ √ √ √ √ ( V )i √ = 2|∇2 V − V h|2 − √ (2( V )( V )i − 2( V )ij ( V )j ) (3.7) V √ √ √ √ ∇ V = 2|∇2 V − V h|2 − √ · ∇(V − |∇ V |2 − 1). V √ Theorem 3.2 Suppose that (Σn , h, V ) is a geodesically complete and conformally compact static solution to √ the vacuum Einstein equations with negative cosmological constant, i.e., (Σn , h, V ) comes from Theorem 2.8 in the previous section. √ 2 2 2 2 1+|x| Then (Σn , h, V ) = (B n , ( 1−|x| 2 ) |dx| , 1−|x|2 ) for some choice of coordinate in dimension 3 ≤ n ≤ 7. Proof. We consider the defining function u = √V1+1 and the compact manifold (Σ, u2 h). From (3.4) we know that ∂Σ = S n−1 and u2 h|∂Σ = dσ0 . Moreover, from (3.4), we know that the standard round S n−1 is the boundary of (Σ, u2 h) and has the second fundamental form dσ0 (i.e., it is totally umbilical). On the other hand, by (3.5) and (3.3), the scalar curvature R[u2 h] goes to zero as s → 0 when n > 2. Using the above Lemma 3.1 and the strong maximum principle, we therefore conclude that R[u2 h] ≥ 0. Now we appeal to the recent work of Miao [Mi] (see also works of Shi and Tam [ST]). We apply the work in [Mi] to the manifold (M, G) where Ω = Σ and g− = u2 h, and M \ Ω = Rn \ B n and g+ is the Euklidean metric. By Corollary 5.1 in [Mi], for example, √ we then conclude that R[u2 h] ≡ 0 for 3 ≤ n ≤ 7. √ n Back to (Σ , h, V ), in the light of (3.6) in Lemma 3.1 and V −|∇ V |2 −1 = 0, we observe that √ √ ∇2 V = V h. (3.8) Similar to what was proved in [Ob], we prove that (3.8)√implies that (Σn , h) is isometric to the standard hyperbolic space form and V = 1 + r2 for some choice of coordinates in the following lemma. Then the proof of theorem is complete. Lemma 3.3 Suppose that (M n , g) is a complete Riemannian manifold. And suppose that there is a positive function φ such that ∇2 φ = φg. (3.9) √ Then (M n , g) = (Rn , gH ) and φ = c 1 + r2 for some choice of coordinates. Proof. First one observes that, φ has one and only one global minimum point p0 on M , since it is strictly convex. Due to the homogeneity of (3.9), one may assume that φ(p0 ) = 1. Let us consider a geodesic γ(s) emanating from p0 and
258
J. Qing
Ann. Henri Poincar´e
parameterized with its length s. Then, along this geodesic, for φ(γ(s)), we have φ − φ = 0 φ(0) = 1 (3.10) φ (0) = 0. ∂ Thus φ(s) = cosh s. Now take an othonormal base X 0 = ∂s , X 1 , X 2 , . . . , X n−1 at p0 and parallel translate them along γ(s). We want to calculate d(expp0 )(sX 0 ) (X k ). That is, we compute the Jacobi field Y k (s) along γ(s) such that k k 0 0 0 0 ∇X ∇X Y + R(Y , X )X = 0 (3.11) Y k (0) = 0 k k ∇X 0 Y (0) = X (0). k i Let Y (s) = fi (s)X (s). Then (3.11) becomes i j fi X + fi R0i0j X = 0 (3.12) fi = 0 fi = δik .
Notice that, by (3.9) and Ricci identity, φa,bc − φa,cb = φd Rdabc = sinh sR0abc = φc δab − φb δac ,
(3.13)
which gives us R0i0j = Plugging into (3.12), we have
1 (φj δi0 − φ0 δij ) = −δij . sinh s fi − fi = 0 fi (0) = 0 fi (0) = δik .
(3.14)
(3.15)
Thus Y k (s) = sinh sX k (s). To show that (M, g) is a hyperbolic space form, we use the exponential map expp0 which takes the tangent space Tp0 M onto M in the light of completeness, thus gives a nice global coordinate chart. Next we want to calculate the metric g under these coordinates. Let us use spherical coordinates for Tp0 M , that is, (s, v) ∈ [0, ∞)×S n−1 and expp0 (sv) ∈ M . By the above calculations of Jacobi fields, we immediately have g = ds2 + (sinh s)2 dσ0 ,
(3.16)
which is the hyperbolic metric. So (M, g) is a hyperbolic space √ form. Finally let us point out that, if we denote r = sinh s, then φ = cosh s = 1 + r2 . Acknowledgment. The author is deeply indebted to Professor P.T. Chru´sciel and the referee for their careful reading of the paper and many corrections of statements.
Vol. 5, 2004
Asymptotically AdS
259
References [ACD] M. Anderson, P.T. Chru´sciel and E. Delay, Non-trivial, static, geodesically complete, vacuum space-times with a negative cosmological constant, arXiv: hep-th/0211006. [AM]
A. Ashtekar and A. Magnon, Asymptotically anti-de Sitter space-times, Class. Quantum Grav. 1, L39–L40 (1984).
[BS]
R. Beig and W. Simon, On the spherical symmetry of a static perfect fluid in general relativity, Lett. Math. Phys. 21 no. 3, 245–250 (1991).
[BGH] W. Boucher, G.W. Gibbons and G.T. Horowitz, Uniqueness theorem for anti-de Sitter spacetime, Phys. Review D. (3) 30, no. 12, 2447–2451 (1984). [Ca]
B. Carter, Black hole equilibrium states, Part II, in “Black Holes”, edited by C. DeWitt and B. DeWitt (New York, 1973).
[CH]
P.T. Chru´sciel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Preprint math.DG/0110035.
[CS]
P.T. Chru´sciel and W. Simon, Towards the classification of static vacuum spacetimes with negative cosmological constant, J. Math. Phys. 42, no. 4, 1779–1817 (2001).
[DK]
J.Duistermaat and J. Kolk, “Lie groups”, Springer-Verlag, Berlin, New York, 2000.
[FG]
C. Fefferman, and C.R. Graham, Conformal invariants, in The mathematical heritage of Elie Cartan, Ast´erisque, 1985, 95–116.
[GSW] G.J. Galloway, S. Surya and E. Woolgar, On the geometry and mass of static, asymptotically AdS space-times,and uniqueness of the AdS soliton, arXiv: hep-th/0204081. [G]
C.R. Graham, Volume and Area renormalizations for conformally compact Einstein metrics. The Proceedings of the 19th Winter School “Geometry and Physics” (Srn`i, 1999). Rend. Circ. Mat. Palermo (2) Suppl. No. 63 (2000), 31–42.
[Ha]
S. Hawking, The boundary conditions for gauged supergravity, Phys. Lett. B126, no. 3-4, 175–177 (1983).
[L]
L. Lindblom, Static uniform-density stars must be spherical in general relativity, J. Math. Phys. 29, no. 2, 436–439 (1988).
[Mi]
P. Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, ArXiv: math-ph/0212025.
260
J. Qing
Ann. Henri Poincar´e
[Ob]
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan. 14, no. 3, 333–340 (1962).
[Pe]
R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett. 10, 66–68 (1963).
[Q]
J. Qing, On the rigidity for conformally compact Einstein manifolds, International Mathematics Research Notices 21, 1141–1153 (2003), ArXiv: math.DG/0305084.
[SY]
R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. Topics in calculus of variations (Montecatini Terme, 1987), 120–154, Lecture Notes in Math., 1365, Springer, Berlin, 1989.
[ST]
Y. Shi and L. Tam, Positive mass theorem and the boundary behaviors of a compact manifolds with nonnegative scalar curvature, arXiv: math.DG/0301047.
[Wa1]
X. Wang, Uniqueness of AdS space-time in any dimension, arXiv: math.DG/0210165.
[Wa2]
X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8, no. 5–6, 671–688 (2001).
[Wa3]
X. Wang, The mass of Asymptotically hyperbolic manifolds, J. Diff. Geo. 57, no. 2, 273–299 (2001).
Jie Qing Department of Mathematics University of California, Santa Cruz Santa Cruz, CA 95064 USA email:
[email protected] Communicated by Piotr T. Chrusciel Submitted 17/10/03, accepted 07/11/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 261 – 287 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020261-27 DOI 10.1007/s00023-004-0169-5
Annales Henri Poincar´ e
D´eformation Elliptique de la M´etrique de Randall et Sundrum Michel Gaudin
0 Introduction Dans sa version originelle, le mod`ele de Randall et Sundrum1) pour la hi´erarchie de masse consid`ere un univers global V5 , de dimension cinq, a` la topologie d’un cylindre aplati formant comme un ruban avec ses deux faces identifi´ees. La base en est le cercle [−π ϕ +π] modulo la sym´etrie de parit´e. Les deux bords du ruban sont les images de deux parois de dimension quatre, fronti`eres de V5 , l’une V (0) dite “cach´ee”, l’autre V (π) dite “visible” comme support des champs de particules. Les ´equations d’Einstein postul´ees dans V5 , dont les sources ont pour support les parois, font donc intervenir des singularit´es de type δ (ϕ) , δ (ϕ − π) (modulo 2π) dans les composantes du tenseur d’impulsion, ce qui ne pose pas trop de difficult´e ´etant donn´ee la lin´earit´e du premier membre dans les d´eriv´ees secondes de la m´etrique. Cependant il vaut mieux s’assurer dans ces probl`emes non-lin´eaires des relations de conservation sur la zone singuli`ere par une r´egularisation, ce qui revient `a introduire une ´epaisseur de la paroi. Mais alors, le bord de V5 n’est plus mat´erialis´e par un pic de la pression exactement localis´e, et ne subsiste que la notion g´eom´etrique due `a la sym´etrie impos´ee et `a la topologie. Sur le cercle de base la paroi diffuse est seulement une zˆone de variation rapide du tenseur des contraintes au voisinage de 0 et π. Celui-ci est d´ecrit soit par les scalaires de pression et densit´e dans l’hypoth`ese de fluide parfait dans V5 (isotropie locale dans V4 ), soit par un champ scalaire contrˆol´e par un puits de potentiel donn´e pour cr´eer les localisations ad hoc. Le probl`eme de la paroi diffuse V (0) a ´et´e trait´e par Ichinose2) en postulant le potentiel standard biquadratique V (φ) poss´edant un maximum central en φ = 0, et deux minima sym´etriques φ = ±φ0 . Une m´ethode perturbative en fonction d’un petit param`etre d’´epaisseur permet de d´eterminer num´eriquement la m´etrique et de reproduire les traits du mod`ele R.S. a` une paroi. Le r´esultat ne d´epend que de la forme qualitative et des deux param`etres du potentiel entre les deux minima. L’analyse perturbative devient inutile si l’on a la chance d’obtenir une solution exacte pour un potentiel convenable. Or ceci est possible dans le mod`ele d’Ichinose avec le potentiel sinuso¨ıdal le plus simple V (φ) = V0 cos
πφ φ0
V0 > 0
(1)
262
M. Gaudin
Ann. Henri Poincar´e
o` u l’intervalle −φ0 φ φ0 est suffisant pour couvrir V5 . Il s’agit donc l` a d’une solution exacte, a` une variable, du champ dit de sine-Gordon coupl´e `a la gravit´e. La solution de ce mod`ele dit “ap´eriodique” sugg`ere une extension naturelle au mod`ele p´eriodique `a deux parois d´ependant en outre d’un param`etre continu. C’est au prix de l’introduction d’un second champ dans un second potentiel, mais coupl´e `a la gravit´e de V5 avec le signe oppos´e (M → −M ), ce qui g´en´eralise naturellement le choix des constantes cosmologiques oppos´ees de R.S. Cette hypoth`ese un peu hardie ´etant admise, les r´esultats de R.S. en d´ecoulent, n’´etant qu’un cas limite du mod`ele d´eform´e propos´e. La notion stricte de paroi s’´evanouit par l` a-mˆeme, n’´etant qu’une forme limite de localisation approch´ee dans V5 .
1 Rappels et notations La m´etrique de R.S. est formellement analogue a` une m´etrique cosmologique R.W. o` u le rˆ ole du temps cosmique est tenu par la cinqui`eme variable d’espace: V5 :
ds2
avec x0 ≡
= ≡
dy 2 + e−2σ(y) d s˜2 gµν dxµ dxν
y,
µ = 0, 1, . . . , 4
V4 : d˜ s2 g00
= g˜µν dxµ dxν
(3)
g0µ = 0 (µ = 0)
= 1
= e−2σ g˜µν ,
(µ, ν = 0) √ Le facteur de r´eduction est b = e et le d´eterminant g = b4 . Si V4 est une vari´et´e de pure gravit´e quelconque gµν
(2)
∂0 g˜µν=0
(4)
−σ
V4 :
˜ µν = 0 R
les ´equations de V5 en pr´esence d’une source s’´ecrivent 2 6σ − 3σ gµν = κ Tµν 6σ 2 = κ T00
(5)
(6)
avec la notation du couplage κ−1 = M 3 . La source de V5 peut ˆetre le champ scalaire φ (y) 1 Tµν = ∂µ φ∂ν φ − gµν (∂φ.∂φ + 2V (φ)) 2
(7)
dont la conservation est assur´ee par l’´equation K.G. ∇2 φ −
∂V = 0. ∂φ
(8)
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
263
On peut la consid´erer aussi comme un pseudo-fluide isotrope, de pression p, densit´e ρ Tµν = pgµν + (ρ − p) uµ uν (9) o` u uµ est un courant unitaire de V5 du genre espace, u.u = 1. Si n−1 est le volume sp´ecifique, le courant nuµ est conserv´e, soit √ (10) n g = nb4 = Cste . La conservation ∇µ Tµν = 0 ´equivaut alors a` d ρb4 dρ p=ρ−n = . dn d (b4 )
(11)
La structure (6) impose ´evidement que φ, p, ρ ne d´ependent que de y, avec u0 = 1, et l’on a la correspondance. −p =
1 2 φ + V, 2
2 φ = ρ − p,
1 2 φ − V. 2 −2V = ρ + p ρ=
(12)
et l’´equation K.G. ´equivalente a` (11) φ − 4σ φ −
∂V =0. ∂φ
A l’aide de (6) et (12), les ´equations du probl`eme `a une paroi pos´e par Ichinose s’´ecrivent 2 3σ 2= κφ . (13) 3 σ − 4σ = 2κV (φ) On peut g´en´eraliser au cas d’un espace V4 qui soit un espace d’Einstein ˜ g´en´eral de constante cosmologique Λ ˜ µν = Λ ˜ g˜µν . R
(14)
Les ´equations de Friedman-Lemaˆıtre3) pour V5 s’´ecrivent 2 ˜ Λ 1 ˜ 2σ 3 b κρ = 3σ 2 − Λe − 2 = b b 2
b 6 b
=
κ (2p − ρ)
Pour un VD+1 on aurait
˜ D (D − 1) b2 − Λ b D (D − 1) b
.
(15)
= 6 σ 2 − σ
= 2 κρb2 = κ (Dp − (D − 2) ρ)
(16)
264
M. Gaudin
Ann. Henri Poincar´e
dont une solution particuli`ere bien connue montre que les espaces d’Einstein sont des “sph`eres g´en´eralis´ees” b = sh y ⇐⇒ ds2 = dy 2 + sh2 y d s˜2 D−1 ˜ Λ = (D − 1) ⇒ Λ = −D κρ = κp = − Λ . 2
(17)
2 Le mod`ele ap´eriodique La solution a` deux parois donn´ee par R.S. pour la fonction continue σ (y), paire et p´eriodique de p´eriode 2yc = 2πrc σ (y) = k |y|
,
−yc y yc
(18)
correspond a` une densit´e ρ (y) constante et positive κρ = 6k 2
(19)
tandis que la pression pr´esente deux pics delta d’intensit´es oppos´ees sur les parois κp (y) = κρ + 6k (−δ (y) + δ (y − yc ))
(20)
(y mod 2yc ) . 3 u la constante cosmologique Λ= Entre les parois, on a κp = κρ = − Λ o` 2 2 −4k < 0 caract´erise V5 (un adS si V4 est un M4 , mais les ´equations restent valides si V4 est un trou noir). La loi de conservation (11), ´ecrite sous la forme 4σ (p − ρ) = −ρ ≡ 0
(21)
implique que la fonction σ soit consid´er´ee comme la limite d’une fonction continue impaire anti-p´eriodique de sorte que l’on ait ,,
lim σ (y)δ(y) = k “ (y)δ(y) = 0
(22)
(idem en yc ). Ceci rappel´e, nous partons maintenant du mod`ele d’Ichinose avec champ u la paroi “physique” scalaire et potentiel, dans la limite ap´eriodique (rc → ∞) o` est envoy´ee `a l’infini de la coordonn´ee y, ce qui ne veut pas dire qu’elle n’existe plus mais qu’elle se situe `a l’horizon. Introduisons une unit´e de longueur y0 (de l’ordre de la longueur de Planck −1 MPl dans la th´eorie de Kaluza4 )) et la coordonn´ee sans dimension u = y/y0 , ainsi que le param`etre sans dimension = ky0 , de sorte que l’on a u ≡ ky .
(23)
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
265
En terme de la coordonn´ee u, les ´eqs. (13) restent formellement inchang´ees dσ en rempla¸cant V par V y02 avec d´esormais σ = . du Inversant la d´emarche d’Ichinose qui cherche `a r´esoudre pour le potentiel standard donn´e, nous proposons la fonction d’essai suivante, qui constitue une r´egularisation tr`es simple, approchant R.S. `a la limite → 0, u → ∞, u fini σ (u) = log ch u
ou b (u) =
⇒ σ = th u → σ =
1
(ch u)
= e−σ
(24)
κ = φ2 2 3 ch u
on introduit le champ sans dimension ϕ
π ϕ κ 1 φ, d’o` u ϕ = , qui s’int`egre en eu = tg + , soit th u = ϕ≡ 3l ch u 4 2 sin ϕ et 1 π π = cos ϕ = ϕ , − < ϕ < + . (25) ch u 2 2 Le champ ϕ ob´eit ` a l’´equation du pendule ap´eriodique et repr´esente le demi angle de rotation a` partir de l’´equilibre stable ϕ = 0, la variable u jouant le rˆ ole du temps m´ecanique. On d´eduit de (13) le potentiel V (φ) v (ϕ)
2κ 2 y V (φ) = σ − 4σ 2 3 0 = cos2 ϕ − 42 sin2 ϕ = − (1 + 4) sin2 ϕ
≡
ou encore
(26)
4κy02 V = (1 − 4) + (1 + 4) cos 2ϕ . (27) 3 Nous avons donc le r´esultat annonc´e en (1) pour le potentiel avec 2 φ0 π 3 φ0 = et V = (1 + 4) . 0 2 κ πy0 On d´eduit des relations (12) les expressions de la densit´e et de la pression 2κy02 ρ = ϕ2 − v (ϕ) = 42 th2 u = 42 sin2 ϕ 0 32 . 2κy0 p = −ϕ2 − v (ϕ) = 42 2 (2 + 1) = − + (2 + 1) sin2 ϕ 3 ch2 u (28) On voit comment le mod`ele R.S. a` une paroi est atteint comme limite du pr´ec´edent lorsque et y0 tendent vers z´ero de sorte que lim /y0 = k, puisque l’on a 1 1 lim = δ (y) 2 2y0 ch (y/y0 ) 2κp 2κρ = 4k 2 ; lim = 4k 2 − 4kδ (y) . (29) lim 3 3 2v ≡
266
M. Gaudin
Ann. Henri Poincar´e
Il est int´eressant d’exprimer la m´etrique dans la coordonn´ee angulaire ϕ qui π π ram`ene la base sur l’intervalle d’une p´eriode − < ϕ < + 2 2 ds2
=
y02 du2 +
=
y02
d˜ s2 2
(ch u)
dϕ2 + (cos ϕ)2 d˜ s2 . cos2 ϕ
(30)
π ne sont atteintes par les g´eod´esiques 2 de V5 qu’` a la limite d’un temps propre (de V4 ) infini: si dt2 ≡ −d˜ s2 Les extr´emit´es de l’intervalle ϕ = ±
|u| = |ky| ∝ log |t| .
(31)
Il s’agit donc d’un horizon, infranchissable dans la coordonn´ee ϕ. La m´etrique y est toujours singuli`ere puisque le d´eterminant y est soit nul 1 1 > soit infini < . Cependant si 2 est entier un prolongement π4 4 p´eriodique est possible, qui ne respecte la signature que si est entier. Une seconde forme limite de la m´etrique est a` remarquer pour → +∞ lim
lim → ∞ 2Kr02 3
√ lim ϕ = ϕ0
y02 = r02 ds2 lim V
= =
r02 dϕ20 1
+ −
2
e−ϕ0 4ϕ20
d˜ s2
.
(32)
Ne subsiste du potentiel que le voisinage du maximum central (potentiel parabolique invers´e) tandis que les deux puits sont envoy´es `a l’infini. Ceci montre que sur tout l’intervalle de variation du param`etre = ky0 (0 < < ∞) la d´eformation continue du mod`ele n’a pas d’autre effet que de modifier les largeur et profondeur relatives du pic de pression et du facteur de r´eduction. Ce mod`ele `a une paroi ne permet pas de d´efinir le facteur de r´eduction des masses, sans l’introduction d’une coupure ad hoc, fixant la distance de la paroi visible au voisinage de l’horizon. On aura not´e que c’est le terme “cin´etique” φ2 du champ scalaire qui simule le pic delta de la pression en y = 0. Il est donc impossible de simuler le second pic de signe oppos´e selon R.S. en introduisant un second champ scalaire, sans changer le signe du couplage gravitationnel. Ce fait semble moins choquant lorsqu’il concerne seulement les constantes cosmologiques de signe oppos´e des deux parois V (0) et V (π), mais il est incontournable. Cette hypoth`ese essentielle admise, nous montrons dans la section suivante comment l’introduction de deux champs et de deux potentiels analogues a` (27) permet de construire un mod`ele p´eriodique r´egularis´e qui tend vers celui de R.S. dans la limite consid´er´ee plus haut.
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
267
3 Le mod`ele des deux pendules Le mod`ele exact `a une paroi trait´e dans la section pr´ec´edente est celui du champ scalaire “sine-Gordon” coupl´e `a la gravit´e dans V5 . La variation du champ avec la cinqui`eme coordonn´ee spatiale est r´egie par l’´equation du pendule ap´eriodique dans la variable de temps de l’analogie m´ecanique. Cette ´equation (25), ϕ = cos ϕ, est un cas fronti`ere de celle du pendule p´eriodique ϕ2 + k 2 sin2 ϕ = 1
(33)
d´ependant du module k dans la notation traditionnelle que nous garderons. Pour ´eviter toute confusion avec le param`etre de masse de R.S., on notera d´esormais ce dernier kRS (par ex. l = y0 kRS selon (23)) . • Si 0 < k 2 < 1, le pendule fait un tour complet autour de son axe (2ϕ → 2ϕ + 2π) dans une p´eriode not´ee 2K, o` u K (k) est l’int´egrale elliptique compl`ete (notation de Whittaker et Watson, chap. XXII)
π 2
K (k) = 0
−1/2 dϕ 1 − k 2 sin2 ϕ
(34)
avec le comportement au voisinage de k = 1 K (k) ∝ log
4 , k
k 2 + k 2 = 1 .
(35)
1 • Si k > 1, le pendule oscille avec une amplitude angulaire |ϕ| sin . k π pour les petits mouLa demi-p´eriode est ReK (k) = k −1 K k −1 , a` la limite 2k vements k 1. La solution de (33) est la fonction “amplitude elliptique” de module k (W.W., p. 494) −1
2
ϕ = ϕ+π
=
am (u, k) ⇒ ϕ = dn (u, k)
(36)
am (u + 2K) .
On restaure ainsi dans la m´ethode du champ scalaire d’Ichinose une p´eriode 2K en u, ou 2Ky0 = 2yc en y, qui correspond certes `a la p´eriode π du potentiel, mais a` condition que le prolongement p´eriodique soit possible, c’est-`a-dire pour k 2 < 1. La d´eformation elliptique permet de retomber sur le mod`ele R.S. p´eriodique r´egularis´e. On postule un lagrangien source de V5 d´ependant d’un couple de champs scalaires {φ1 , φ2 } avec la structure suivante 1 −1 5 √ ∂φ1 .∂φ1 + V1 (φ1 ) κ L12 = d x g 2 1 − ∂φ2 .∂φ2 − V2 (φ2 ) + V (φ1 , φ2 ) (37) 2
268
M. Gaudin
Ann. Henri Poincar´e
qui donne le tenseur d’impulsion Tµν
=
∂µ φ1 ∂ν φ1 − ∂µ φ2 ∂ν φ2 1 − gµν (∂φ1 ∂φ1 − ∂φ2 .∂φ2 + 2 (V1 − V2 + V )) 2
(38)
dont la conservation est assur´ee par les deux relations
2 φ1 −
∂ ∂ (V1 + V ) = 0, 2 φ2 − (V2 − V ) = 0. ∂φ1 ∂φ2
(39)
et qui dans l’hypoth`ese de la seule d´ependance en u s’´ecrivent φ1 − 4σ φ1 − ∂ (V1 + V ) = 0, etc. . . ∂φ1 Les ´equations du probl`eme g´en´eralisant (13) sont 2 κ 2 = σ 3 φ1 − φ2 (40) 2 σ − 4σ 2 = 2κ 3 y0 (V1 (φ1 ) − V2 (φ2 ) + V (φ1 φ2 )) o` u les potentiels vont ˆetre d´etermin´es par la m´ethode inductive pr´ec´edente. La fonction d’essai σ (u) ou b (u) qui ´etend naturellement (24) au cas p´eriodique est l
b (u) = (dn (u, k)) ou σ = −l log dn u
(41)
o` u la fonction dn u = dn (−u) = dn (u + 2K) est positive, oscillant entre le maximum dn (0) = 1, et le minimum dnK = k , avec la propri´et´e dn u dn (u + K) = k qui entraˆıne donc b (u + K) =
(42)
k 2l . b (u) ¯
A la limite ap´eriodique k → 1, k → 0, K → ∞ avec k ∝ 4e−K = e−K , ¯ = K − log 4 et dn (u, 1) = 1 etc. . . K ch u On d´eduit de (41) σ
=
σ
=
dn sn cn = k 2 dn dn sn2 cn2 k 2 2 k 2 − cn2 − sn2 + k 2 ≡ dn dn2 dn2 −
On peut encore ´ecrire = dn2 u − dn2 (u + K) σ σ 2 = 2 1 + k 2 − dn2 u − dn2 (u + K)
.
.
(43)
(44)
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
269
D’o` u les identifications fond´ees sur (40) et (44), dans les notations sans diκ 2κy02 V, φ et v = mension d´ej`a utilis´ees en (25) et (26), ϕ = 3l 3 ϕ1 = dn u ou ϕ1 = am u (45) ϕ2 = dn (u + K) ou ϕ2 = am (u + K) v1 − v2 + v ≡ σ − 4σ 2
≡ −42 1 + k 2 + (4 + 1) dn2 u + (4 − 1) dn2 (u + K) . On d´eduit de (45) sin ϕ1 = tg
ϕ1 =
sn u
tg ϕ1 =
sc u
tg ϕ2 =
sc u 1 − cs u k
.
(46)
(47)
D’o` u la relation n´ecessaire entre ϕ1 (u) et ϕ2 (u) k tg ϕ1 tg ϕ2 = −1
(48)
qui entraˆıne encore selon (42) 1 − k 2 sin2 ϕ1 1 − k 2 sin2 ϕ2 = k 2 ou ϕ1 ϕ2 = k . Enfin pour les potentiels 2v1 (ϕ1 ) ≡
(49)
k 2 (1 + 4) cos 2ϕ1 = (1 + 4) 2dn2 u − 1 − k 2
2v2 (ϕ2 ) ≡
k 2 (1 − 4) cos 2ϕ2 = (1 − 4) 2dn2 (u + K) − 1 − k 2
(50)
a condition d’avoir la contrainte ` v (ϕ1 , ϕ2 ) = 0
(51)
au cours du mouvement, ce qui justement est r´ealis´e en (48). • Analogie des deux pendules Les angles ou amplitudes 2ϕ1 et 2ϕ2 , fonction du temps u, sont les angles avec la verticale du champ de pesanteur de deux pendules de mˆeme p´eriode 2K, ob´eissant a la mˆeme ´equation horaire (33) ou ` 2ϕ + k 2 sin 2ϕ = 0,
(52)
mais d´ecal´es d’une demi-p´eriode K, ce qui est une question de condition initiale. La relation (48) est alors conserv´ee au cours du temps.
270
M. Gaudin
Ann. Henri Poincar´e
La liaison entre les deux pendules n’est pas dynamique, mais g´eom´etrique. On peut se repr´esenter deux masses ponctuelles identiques oscillant (ou en rotation) sur deux grands-cercles verticaux de la mˆeme sph`ere, les deux plans formant en di`edre d’angle ψ donn´e par k = cos ψ. La relation (48) peut s’´ecrire cos γ ≡ cos ϕ1 cos ϕ2 + cos ψ sin ϕ1 sin ϕ2 = 0 .
(53)
Or ϕ1 , ϕ2 sont les angles avec la verticale des deux masses vues du z´enith de la sph`ere (2ϕ1 , 2ϕ2 angles vus du centre de rotation). π En vertu de (53), γ = ± , l’angle entre les deux masses vu du z´enith est 2 droit; les masses restent en positions conjugu´ees, ou si l’on veut, en quatrature vues du z´enith. Stabilisons cette liaison par un potentiel entre les deux pendules, minimal en quadrature 1 v (ϕ1 , ϕ2 ) = v0 cos2 γ 0 . (54) 2 Nous avons ∂v = v0 cos γ (− sin ϕ1 cos ϕ2 + k cos ϕ1 cos ϕ2 ) . ∂ϕ1 Utilisant (41) on trouve ∂v k = v0 cos γ = v0 cos γ ∂ϕ1 dn u
ϕ2
(55)
v´erifions les relations de conservation (39) ϕ1 − 4σ ϕ1 = soit 1 2
∂v1 ∂v + ∂ϕ1 ∂ϕ1
= = =
on en d´eduit
∂v =0 ∂ϕ1
1 ∂ (v1 + v) 2 ∂ϕ1
sn cn dn dn 1 − (4 + 1) k 2 sn cn = − k 2 (4 + 1) sin 2ϕ1 2 1 ∂v1 + 2 ρ∂ϕ1
(56)
∂v = 0 sur la trajectoire ∂ϕ2
(57)
−k 2 sn cn − 4k 2
,
ce qui r´esulte en effet de cos γ = 0, ∀v0 > 0. Le puits stabilisateur v (ϕ1 , ϕ2 ) poss`ede une ´equipotentielle minimale, ligne de points paraboliques qui constitue une trajectoire pour la condition initiale opposition d’une demi-p´eriode, ou quadrature vue du z´enith.
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
271
Prenant la limite v0 → +0, il est permis d’oublier ce potentiel mutuel et de consid`erer les champs scalaires φ1 et φ2 comme ind´ependants, avec des tenseurs s´epar´ement conserv´es, pourvu que les potentiels V1 et V2 soient donn´es par (50). On notera qu’il sont de mˆeme forme sinuso¨ıdale, mais d’intensit´e diff´erente sauf dans la r´egion 1, ou d’intensit´e oppos´ee pour 1.
4 La m´etrique d´eform´ee de V5 3/2 Deux champs scalaires (ϕ1 , ϕ2 ), de dimension M , r´egis par deux potentiels πφ semblables en cos , coupl´es `a la gravit´e d’un V5 avec deux signes oppos´es, φ0 donnent lieu a` la m´etrique suivante dans l’hypoth`ese o` u ils ne d´ependent que de la cinqui`eme coordonn´ee: 2
ds2 = y02 du2 + (dn u) d˜ s2 o` u le param`etre ne d´epend que de φ0, 2 κ 2φ0 κ = M −3 = 3 π
(58)
(59)
tandis que le module elliptique k d´epend de l’intensit´e des potentiels une fois fix´ee πφ1 l’unit´e de longueur y0 . Si l’on ´ecrit V1 (φ1 ) = W1 cos , etc. . . , on a selon (50) φ0 2 2 2 kφ0 k 3 κ 4φ0 W1 = (1 + 4) = 1+ 2y0 κ πy0 3 π . (60) 1 + 4 W1 /W2 = 1 − 4 Du module k, on calcule K demi-p´eriode de la coordonn´ee u et yc = Ky0 , demi-p´eriode en y = y0 u. Revenant `a la m´etrique (58) on note qu’une translation u → u + K ´equivaut a changer le signe de et l’´echelle des longueurs dans V4 ` ds2 = y02 du2 +
1 2
(dn u)
k 2 d˜ s2
ce qui d´efinit formellement le facteur de r´eduction ¯ ¯ b(K) −K K ≡ K − log 4 b(0) = k ∼ e si K 1.
(61)
(62)
Il est int´eressant d’exprimer la m´etrique de V5 dans la coordonn´ee angulaire ϕ1 = ϕ 2 dϕ2 ds2 = y02 s . (63) + 1 − k 2 sin2 ϕ d˜ 2 2 1 − k sin ϕ On passe de ϕ1 ` a ϕ2 , comme en (61).
272
M. Gaudin
Ann. Henri Poincar´e
La m´etrique est p´eriodique et reguli`ere, ∀l, si k 2 < 1. Enfin pour la pression et la densit´e nous avons les formules 2 1 22 dn 22 κρ = 2 = 2 1 + k 2 − dn2 u − dn2 (u + K) 0 3 y0 dn y0 1 κ (ρ − p) = 2 dn2 u − dn2 (u + K) 3 y0 κp 2 y0 = 22 1 + k 2 − (1 + 2) dn2 u + (1 − 2) dn2 (u + K) . 3 On encore, si l’on veut, sachant que ϕ1 et ϕ2 sont li´es 1 2 k 4 sin2 2ϕ 22 k 2 2 sin ϕ1 + sin2 ϕ2 − 1 ≡ κρ = 2 2 2 2 3 2y0 1 − k sin ϕ y0 1 k 2 2 κ (p − ρ) = 2 = sin ϕ1 − sin2 ϕ2 . 3 y0
(64)
(65)
(66)
On v´erifie ais´ement que la pression (65) donne la limite de R.S., formule (20), pour → 0, k → 0 avec lim = kRS , lim Ky0 = yc , lim y0 = y. y0 • Remarque sur l’extension a` un V4 qui soit un espace d’Einstein de constante ˜ cosmologique Λ. Selon les ´eqs. de F.L. (15), il suffit d’effectuer le remplacement ˜ 2σ κρ → κρ + 2Λe κp → ˜ 2σ κp + Λe (67) 1 2 e2σ = dn (u + K) . k 2 ˜ = 0, ne va fonctionner ici que pour = 1, La m´ethode utilis´ee dans le cas Λ 2 la modification ne portant que sur Φ2 et V2 ϕ2 2 v2 Posant
→ ϕ2 2 → v2
− +
˜ 2 2σ Λy 0 3 e 2 2σ ˜ Λy0 e .
(68)
˜ 2 ˜ = Λ . y0 λ 3 k 2
le nouveau champ ϕ2 devient
˜ dn2 (u + K) soit ϕ2 = 1 − λ ˜ am (u + K) . = 1 − λ ϕ2 2
2 D’autre part le coefficient
de dn (u+ K) dans v2 (formule (50)) s’obtient en ˜ =3 λ ˜−1 . rempla¸cant (1 − 4) par 1 − 4 + 3λ
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
273
Il est remarquable que le second champ, terme cin´etique et potentiel, dis˜ = 1, c’est-`a-dire pour les valeurs paraisse pour λ 2
˜ = 3k > 0 Λ y02
(69)
˜ assez petit. ce qui d´etermine k pour Λ Pour cette valeur du module, la m´etrique dS 2 = y02 du2 + dn2 ud˜ s2
(70)
est donc celle d’un V5 coupl´e `a un seul champ scalaire de sine-Gordon avec le signe ˜ usuel, pourvu que V4 soit une espace d’Einstein de constante Λ. Si cette constante cosmologique devait ˆetre de l’ordre observ´e R0−2 o` u R0 est le rayon de l’univers V4 , on aurait k ∼
y0 10−33 = = 10−60 R0 1027
ce qui entraˆınerait K ∼ 140, alors que la hi´erarchie de masse exigeant qyc ∼ 35 dans la limite R.S. ce qui nous donnerait plutˆ ot selon la formule (80) pour l = 1, K = 2qyc ∼ 70. Le facteur de r´eduction donn´e par la contrainte cosmologique (69) serait donc de 10−30 au lieu de 10−15 . L’int´erˆet de cette propri´et´e, probablement ind´ependante de la dimension, vient de ce que (70) g´en´eralise celle des espaces d’Einstein (cf. (17)) usuels au cas d’une source de type sine-Gordon. Il serait alors possible d’avoir un scalaire de courbure tr`es grand pour un adS (V5 rempla¸cant le V4 pr´ec´edent) et une constante cosmologique tr`es petite pour une section V4 de rapidit´e convenable. • La constante de gravitation de V4 −2 Si κ = M −3 est la constante de couplage pour V5 , celle de V4 est κ ˜ = MPl et l’on d´efinira la masse q par κ ˜ = κq . (71) Dans le mod`ele de R.S. on a la relation suivante entre q et kRS kRS = 1 − e−2yc kRS ∼ 1 . q Dans le mod`ele d´eform´e nous avons encore yc e−2σ(y) dy q −1 = −yc
(72)
(73)
274
M. Gaudin
Ann. Henri Poincar´e
c’est-`a-dire 1 2qy0
= =
π 2
− 12 (dn u) du = dϕ 1 − k 2 sin2 ϕ 0 0 2 1+k π − 12 (k ) P− 12 2 2k K
2
(74)
o` u Pν (z) d´esigne la fonction de Legendre d’ordre ν r´eel, holomorphe pour z > −1. Pour k = 0 (mod`ele ap´eriodique), on a ∀ 1 Γ + 12 (75) qy0 = √ π Γ () et notamment pour voisin de z´ero qy0 ∝ , ce qui entraˆıne dans ce mod`ele limite ` une paroi q = kRS . a Pour k petit, la formule (75) donne le terme dominant, la correction ´etant de l’ordre k ∝ 4e−K , a` condition d’exclure le voisinage de = 0. La non-uniformit´e est claire: 1 . (76) ∀k > 0 , lim qy0 = →0 2K La condition de validit´e de la formule (75) est K 1. Enfin, selon Szeg¨o8) , uniform´ement en k pour 1 1 − 12 qy0 ∝ . (76 ) k π Si l’´etude des excitations confirme que le facteur de r´eduction dans ce mod`ele est (k ) , celui-ci sera exponentiellement petit, soit pour assez grand ∀k (0 < k < 1), soit pour k assez petit pour que l’on ait K assez grand. Dans le premier cas 1, la formule (76 ) nous donne K (∀k ou K) . (77) qyc ∝ k π Dans le second cas K 1, l’expression (77) ci-dessus reste encore une bonne estimation de l’ordre de grandeur (approximation de Stirling) pour 1; 1 1 une ´evaluation correcte de l’int´eEnfin pour K fini, c’est-` a-dire = O K grale (74) se fait en coupant l’intervalle en et en notant que l’approximation deux, dn u ∝ ch1u est valide uniferm´ement sur 0, K 2 , pour K assez grand. √ K −K En effet on a dn 2 , k = k ∝ 2e 2 ∝ ch1K on en d´eduit pour K fini, 2 K 1 K2 K2 1 1 −2u −2K 1 − e−2K (78) ∝ e du + e e2u du ∝ 2qy0 2 0 0
Vol. 5, 2004
soit
D´eformation Elliptique de la M´etrique de Randall et Sundrum
1 1 = 1 − e−2K qyc K
275
(79)
on a donc l’identification K = kRS yc , qui montre l’´equivalence du mod`ele d´eform´e et du mod`ele limite R.S. lorsque est de l’ordre de K −1 . q c Pour 12 , voici quelques valeurs du rapport qy ecroissant en gros K ≈ kRS , d´ 1 comme √ qyc = 12 = π2 K qyc 1 = 1 = 2E(k) K (80) qyc 4 = 32 = 3π(1+k 2 ) K qyc 1 = √kπ . K
5 Excitations et ´echelle de masse Sans entrer dans les justifications physiques, nous consid´erons ici l’´equation de Klein-Gordon pour une onde r´eelle ψ scalaire sur V5 ∇2 ψ = 0 .
(81)
gµν en coordonn´ees harPar exemple, si V4 est plat, la fluctuation de spin 2, δ˜ moniques est r´egie, `a l’approximation lin´eaire, par l’´equation (81). On peut aussi consid´erer ψ comme la fluctuation (scalaire) d’un coefficient m´etrique diagonal relatif a` une sixi`eme dimension. Nous consid´erons plus g´en´eralement l’´equation K.G. massive sur V5
soit
1 √ √ ∂µ ( gg µν ∂ν ψ) = M52 ψ g
(82)
˜ 2 ψ + e4σ ∂0 e−4σ ∂0 ψ = M52 ψ e2σ ∇
(83)
dont nous cherchons une solution factoris´ee de masse m2 dans V4 ˜ 2 ψ = m2 ψ. ∇
(84)
Pour la d´ependance dans la cinqui`eme coordonn´ee x0 = y = y0 u, en notant σ =
dσ , du
nous obtenons l’´equation en ψ (u) ψ − 4σ ψ + y02 m2 e2σ − M52 ψ = 0.
(85)
276
M. Gaudin
Ann. Henri Poincar´e
Prenant comme fonction inconnue χ = b2 ψ ≡ e−2σ ψ .
(86)
Nous obtenons pour χ (u) −χ + y02 M52 − m2 e2σ + 4σ − 2σ χ = 0,
(87)
et utilisant la formule (12)–(13) pour la pression σ − 2σ 2 −χ + y02 M52 +
2κ 3 p
= − κ3 y02 p − m2 e2σ χ
(88)
= 0.
Puisque l’´equation de KG r´esulte du principe variationnel √ δ ∂ψ · ∂ψ gd5 x = 0
(89)
qui d´erive forc´ement, si elle est valide, du lagrangien d’origine a` l’approximation quadratique, il convient d’exiger la convergence des int´egrales • si m2 = 0
+∞
2
ψ (u) e
−2σ(u)
−∞
du
≡
+∞
χ2 e2σ du
(90)
−∞
• ou si m2 = 0,
+∞
χ2 (u) du.
−∞
Ces conditions ach`event de d´eterminer la fluctuation m´etrique comme une fonction d’onde r´eelle d’´etat li´e en m´ecanique quantique, ce qui illustre en passant l’id´ee primitive de Klein qui voit la relativit´e g´en´erale, notamment en dimension cinq, englobant la m´ecanique ondulatoire de la premi`ere quantification. Dans le cas particulier des masses nulles M52 = 0, m2 = 0, on a la solution triviale ψ = Cste ou “mode z´ero”, ce qui donne l’onde li´ee χ (u) = (ch 1u)2 dans ce mod`ele ap´eriodique. Dans le mod`ele p´eriodique, l’onde apparaˆıt plutˆ ot comme l’´etat fondamental d’une bande (impulsion nulle). Mod`ele ap´eriodique (une paroi) Selon (24) et (28), l’´equation (88) en χ s’´ecrit 2 (2 + 1) 2 2 2 −χ + y02 M52 + 42 − − y m (ch u) χ=0. 0 ch2 u
(91)
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
Dans la coordonn´ee x = th u = sin ϕ, notre ´equation devient 2 4µ2 dχ y02 m2 2 d χ + 2 (2 + 1) − 1−x − 2x + χ=0 +1 dx2 dx 1 − x2 (1 − x2 ) o` u l’on a pos´e
4µ2 = 42 + y02 M52
(µ > 0) .
277
(92)
(93)
Le cas de masse nulle dans V4 , m2 = 0, nous donne l’´equation des fonctions de Legendre associ´ees dont les param`etres 2 et 2µ sont a priori des r´eels positifs quelconques. Avec la d´efinition de Hobson (cf. W.W., p. 326) de la fonction µ 1+x 1 1−x 2µ (x) = F −2, 2 + 1, 1 − 2µ; P2 Γ (1 − 2µ) 1 − x 2 on a g´en´eriquement un comportement singulier aux extr´emit´es de l’intervalle [−1, +1] −µ 1 − x2 ∼ (ch u)2µ a l’exception bien connue des polynˆ ` omes et fonctions de Legendre associ´ees pour 2 = entier positif, 2µ = entier positif, 0 ≤ µ ≤ . Cette classe tr`es particuli`ere est celle des fonctions p´eriodiques r´eguli`eres dans la 2µ variable angulaire ϕ P2 , int´egrables selon la norme (90) relative `a m2 = 0; Mais, 2 selon la norme m = 0 qui est +∞ +∞ dx χ2 (u) du = P 2 (x) , (94) 1 − x2 −∞ −∞ la fonction relative a` µ = 0 est exclue. Le param`etre 2 n’ayant pas de raison d’ˆetre quantifi´e, la classe des fonctions propres convenable est, pour donn´e, la suite finie index´ee par l’entier r, avec max r = [2] − 1 soit
−2+r χr = ar P2 (x)
([ ] = partie enti`ere) ;
r = 0, 1, 2, . . . , [2] − 1
et explicitement χr =
ar 22 Γ (2
r − d 2 1 − x2 1 − x2 2 r + 1) dx
(95)
on a aussi χr
−− r2 1 − x2 × (Polynˆ ome de Gegenbauer de degr´e r) 1 ≡ × (polynˆ ome en th u) , (ch u)2−r =
(96)
278
M. Gaudin
Ann. Henri Poincar´e
ce qui montre nettement le comportement `a l’infini. Pour ≤ 12 , il n’existe qu’une seule solution qui est le mode z´ero s’´ecrivant ∀ χ0 (u) =
a0 (ch u)2
= a0 (cos ϕ)2
(97)
avec la normalisation (94) `a l’unit´e a−2 0 =
√ Γ (2) . π Γ 2 + 12
Avec la croissance du param`etre 2 apparaissent des ´etats nouveaux formant une suite finie orthogonale de dimension [2], ce qui r´esulte ´evidemment de l’´equation diff´erentielle (91) ou d’un calcul direct partant de (95). Cette propri´et´e n’a pas de rapport avec l’orthogonalit´e de la suite infinie des fonctions de Gegenbauer, ou des fonctions de Legendre r´eguli`eres Pnm appartenant au mˆeme indice m. Il s’agit en fait d’une forme limite des fonctions ellipso¨ıdales qui constituent une d´eformation des harmoniques sph´eriques. Nos fonctions jouent plutˆ ot le rˆ ole de la suite eimϕ pour donn´ee. Selon (99), nous avons 2µ = − + r > 0, d’o` u le spectre en vertu de (93) −y02 M52 = r (4 − r) ≥ 0 .
(98)
En dehors du mode z´ero, on obtient un specre tachyonique dans V5 , ou si l’on veut, de masse nulle dans un V6 de signature adS. On n’a pas calcul´e le coefficient de normalisation ar () de ces fonctions peu usit´ees. La norme usuelle des fonctions de Legendre correspond au choix ar ≡ 1 +∞ −∞
mais notre norme
+∞ −∞
χ2r (u) chdu 2u
=
2 4+1
·
Γ(4−r+1) , r!
(99)
χ2r (u) du semble plus difficile `a obtenir.
Cette ´equation d’onde de masse nulle a ´et´e ´etudi´ee comme KG sur une brane DS par Bertola and all [10]. • Cas m2 = 0 Nous ne nous attarderons pas sur le cas des excitations massives dans ce mod`ele `a une paroi. En effet la force r´epulsive due au potentiel −m2 (ch u)2 les ´eloigne de la paroi, et le probl`eme physique est d´eplac´e vers le voisinage de la seconde paroi, ici rejet´ee `a l’infini. Outre le mode z´ero et les r´esonances tachyoniques peu perturb´ees, les ondes instables sont calculables par la m´ethode W.K.B. Elles sont de carr´e sommable bien que dans le continu, mais les deux courants oppos´es sous-jacents ne le sont pas, puisque constants.
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
279
Pour terminer ce paragraphe, il reste a` montrer que la forme limite de l’´equation (91) des excitations ap´eriodiques est l’´equation donn´ee par Randall et Sundrum. Avec lim y0 = q = kRS , apr`es division de (91) par y02 , nous avons 2 d2 χ 2q (1 + 2) y 2 2 2 χ=0 (100) − 2 + M5 + 4q − − m ch dy y0 y0 ch2 yy0 qui, a` la limite y0 → 0, devient l’´equation R.S. −
d2 χ 2 2 2 2q|y| χ=0 + M + 4q − 4qδ (y) − m e 5 dy 2
(101)
en prenant M5 = 0. Mod`ele elliptique (` a deux parois). Selon l’expression (65) pour la pression, l’´equation (88) s’´ecrit y02 m2 k 2 2 2 −χ + 4µ − 2 (1 + 2) dn u + 2 (1 − 2) 2 − χ=0 dn u (dn u)2 o` u l’on a pos´e
4µ2 = y02 M52 + 42 1 + k 2 ,
(102)
(103)
dont la forme limite pour k → 0 est (91). La fonction potentielle v (u) =
−2 (1 + 2) dn2 u + 2 (1 − 2) dn2 (u + K) −
y02 m2
2
(dn u)
(104)
est 2K-p´eriodique et r´eguli`ere sur une p´eriode. Mode z´ero Pour m2 = 0, M52 = 0, on a ´evidemment le mode z´ero χ = e−2σ = (dn u)2 , dont la norme est donn´ee par la formule (74) en rempla¸cant par 2, avec la bonne estimation (75) lorsque k est petit. Selon le principe de continuit´e avec le module, il existe sˆ urement une classe de solutions de spectre discret tachyonique correspondant a` (95); nous y reviendrons. Pour m2 = 0, le probl`eme est de d´ecrire la classe de solutions χ qui soient p´eriodiques (r´eelles) et, en cons´equence, r´eguli`eres sur l’axe r´eel puisque v (u) est r´egulier. Une propri´et´e importante de ce potentiel est la suivante (105) v u + K, ; m2 ≡ v u; −, m2q avec la d´efinition m2q =
m2 k 2
(106)
280
M. Gaudin
Ann. Henri Poincar´e
ou si k est petit, ∀ non nul, ¯
m2 ∝ m2q e−2K .
(107)
Cette propri´et´e traduit la sym´etrie de ce mod`ele dans l’´echange des deux parois (ou des deux potentiels φ1 et φ2 ) u
←→ u + K
κ
←→ −κ
,
←→ −
(108)
(ou M ←→ −M )
a condition de changer l’´echelle des masses par le facteur de r´eduction, selon (106). ` Il faudrait faire une ´etude d´etaill´ee du potentiel en fonction des deux param`etres et k, dans le domaine d’int´erˆet k 10, mais nous reviendrons sur le comportement qualitatif qui est assez simple: deux puits de potentiel s´epar´es par une barri`ere, dont les profondeurs sont dans le rapport du facteur r´eduction. Les ´etats physiques sont li´es dans le plus profond qui est celui de la seconde paroi. • Echelle de masse dans la limite R.S. Nous examinons la limite R.S. dans le voisinage du second puits: il suffit d’effectuer la translation d’une demi-p´eriode K sur l’´equation (102). Posant χ (u + K) = ξ (u), nous avons k 2 2 2 2 2 2 −ξ + 4µ − 2 (1 + 2) 2 + 2 (1 − 2) dn u − y0 mq (dn u) ξ = 0. (109) dn u nous avons l’approximation uniforme si k 1 1 2 (1 − 2) 2 2 2 4µ + ξ=0 (110) − y 0 mq 2 ch2 u (ch u)
Dans la r´egion |u| < −ξ +
K 2,
c’est l’´equation analogue a` (100) avec → −, la limite R.S., lim /y0 = q, nous obtenons −
m → mq . Apr`es division par y02
d2 ξ 2 + M5 + 4q 2 + 4qδ (y) − m2q e−2q|y| ξ = 0 . 2 dy
(111)
Distinguons les deux cas de parit´e, le potentiel δ ´etant inop´erant sur les ´etats impairs. L’onde li´ee d´ecroissant `a l’infini en exp (−2µ |y|) est exactement
m ξ + (y) Jν qq e−q|y|
(112) ξ − (y) (y) Jν mq e−q|y| q o` u l’on a pos´e 2µ = νq, soit ν=
1/2 M2 ≥ 2. 4 + 25 q
(113)
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
281
on a les conditions spectrales mq mq mq Jν + 2Jν =0 q q q mq Jν =0 q
• Parit´e +: • Parit´e −:
(114) (115)
Pour la parit´e +, ce sont les extrema de x2 Jν (x), donc entre les z´eros de Jν . On ´ecrira dans les deux cas m(n) = qxn q
n = 1, 2, 3, . . .
(116)
o` u la parit´e de (n − 1) sera celle de l’´etat. Dans le cas M52 = 0, ν = 2, on aura J1 (xn ) = 0 n impair (parite´ +) n = 1, 3, . . . J2 (xn ) = 0 n pair (parite´ −) n = 2, 4, . . . Asymptotiquement avec n xn ∼
3 n+ 2
x = 3, 83 . . . π 1 x2 = 5, 13 . . . 2 x = 7, 01 . . . 3
(117)
En tout cas, dans la limite R.S., nous avons le r´esultat essentiel de la hi´erarchie de masse avec la d´efinition (106) de mq m2 = m2q k 2 = x2 q 2 k 2
(118)
m2 (n) ∼ x2n q 2 e−2K = x2n m2c
(119)
mc = qe−K ,
(120)
soit encore avec qui serait la masse de r´ef´erence du mod`ele standard. L’´equation (110) et sa forme limite (111) montrent clairement comment l’excitation massive d´ecrite par l’onde ψ dans V5 est li´ee `a la seconde paroi (visible), par un potentiel attractif induit de largeur q −1 qui en mesure l’´epaisseur. Ce m´ecanisme qui d´etermine la paroi physique r´esulte de la dynamique des champs dans les potentiels choisis. Les excitations de masse nulle sont li´ees `a la premi`ere paroi en appelant mc = qe−qyc la masse de r´ef´erence du mod`ele standard. Pour obtenir mc de l’ordre de 100 GeV, il suffit donc d’avoir qyc = K ∼ 40 au voisinage de la limite R-S, mais le truc de la r´eduction exponentielle fonctionne ∀k en vertu de l’estimation (77), pourvu que atteigne 10 ou 100.
282
M. Gaudin
Ann. Henri Poincar´e
Contribution au propagateur des ´etats massifs La contribution des ´etats massifs – `a l’exclusion du mode z´ero – au propagateur de la fluctuation m´etrique (champ scalaire ψ) s’´ecrit en repr´esentation impulsion p et dans le cas statique p4 = 0 G (p) =
∞
1 ≡ G+ + G− . 2 + m2 x2 p c n n=1
(121)
Dans le cas M5 = 0 (ν = 2), nous avons dx J1 (x) 1 2G+ (p) = 2π (C) J1 (x) p2 + m2c x2
(122)
o` u l’int´egrale de Cauchy porte sur une fonction impaire m´eromorphe dans tout le plan et born´ee uniform´ement si Ju x > 0. Le contour (C) est l’union de (C+ ) et (C− ) , (C+ ) entourant dans Rex > 0 la suite {+xn }, (C− ) dans Rex < 0 la suite {−xn }, toujours positivement. Le contour (C) est ´equivalent a` un circuit n´egatif autour des trois pˆ oles x = 0, x = ±i mpc , et l’on obtient 2G+ (p) =
1 I1 mc p I1
p mc
−
1 I2 1 ≡ 2 p mc p I1
p mc
>0.
De mˆeme pour la parit´e (−) 1 1 I3 p dx J2 2G− (p) = ≡ >0. 2π J2 p2 + m2c x2 mc p I2 mc
(123)
(124)
D’o` u les comportements `a petite et `a grande impulsion • p mc
5 1 p2 G (p) = − +O 24m2c 144 m4c
• p mc 1 +O G (p) = mc p
mc p3
p2 m6c
(125)
(126)
on en d´eduit la correction a` la loi de Newton en 1r due au mode z´ero. Ce type de correction a ´et´e consid´er´e par Tanaka et Montes [3] dans le mod`ele de Randall Sundrum. 1 G (p) e−i p · r d3 p G(r) = 2 2π ∞ 1 −xn mc r e ≡ . (127) r n=1
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
• A tr`es courte distance r m−1 c 2 1 G (r) ∼ +O πmc r2
1 r
283
(128)
terme qui n’est plus une correction, mais domine la loi de Newton. • A distance sup´erieure a` quelques longueurs de Compton m−1 c , la loi n’admet qu’une correction n´egligeable en exp − (x1 mc r) due a` la masse la plus basse. • Dans la zˆone de transition π2 mc r de l’ordre de 1, la loi asymptotique (117) donne l’estimation e−mx1 r . (129) G (r) ∼ π 1 − e − 2 mc r Pour des transferts d’impulsion petits devant mc , l’interaction r´esiduelle ´equivaut a` une interaction de contact selon (125) 5π δ r (130) G (r) ∼ 6m2c ce qui ´evoque naturellement l’interaction de Fermi, v´ehicul´ee par un boson de Higgs de masse O (mc ) dans le mod`ele standard, mais ici due `a tout le spectre d’excitation. • Etude qualitative du cas 1 On pourrait ´etudier les choses en d´etail en fonction des deux param`etres. Pour fini, k petit les r´esultats sont essentiellement inchang´es. Examinons seulement le cas de d´eformation extrˆeme 1. Le potentiel est en gros constitu´e de deux puits de profondeurs comparables centr´es en u = 0 et u = K, s´epar´es par une barri`ere k a k1 , avec X = 1 pour u = K en n = K 2 . Posons X = dn2 u , croissant avec u de k ` 2 . 2 + 1 y 2 m2 + (2 − 1) X + 0 X v (X) = − 2k (131) X k ∂v 2 + 1 y02 m2 −1 = − 2k 2 − 1 − X + . (132) ∂X X2 k = 42 1 + k 2 + 2k 2 + y02 m2 −v (u = 0) − 2k 2 −v (u = K) = 42 1 + k 2 + y02 m2q (133) = 82 k −v u = K + y02 m2q k 2 On voit que
admet un unique z´ero pour 1 √ 2 + 1 e Xmax = k + ··· 1+ 2 − 1 4 1 = 1+O 2
∂v ∂X
(134)
284
M. Gaudin
Ann. Henri Poincar´e
d’o` u le maximum de la barri`ere centr´ee en n = K eriode). Pour que les 2 (quart de p´ profondeurs des puits soient comparables, on doit avoir y02 m2q de l’ordre de 42 . Posons y02 m2q = 42 ξ 2 on a les termes dominants a` l’approximatiojn exponentielle O k 2 , en supposant k nettement diff´erent de 1 : −v (u = 0) ∼ 42 1 + k 2 + 2k 2 − 2k 2 −v (u = K) ∼ 42 1 + k 2 + ξ 2 (135) ∼ 82 k . −v u = K 2 La barri`ere disparaˆıt dans la limite k → 1 (k → 0) o` u le facteur de r´eduction exponentiel est d’autant moins efficace. Supposons encore k assez petit pour que les deux puits d´ecouplent pratiquement les deux ´etats d’´energie commune −4µ2 ≡ −42 , si M5 = 0. Dans ce cas extrˆeme, ∀ assez grand, on a les deux probl`emes d´ecoupl´es `a l’ordre dominant (n´egligeant devant 2 ) • Paroi u + 0
−
K 2
u
−X −
K 2
2 (2 + 1) X + 42 X = 0 ch2 u
(136)
qui nous donne le mode z´ero X0 . • Paroi u = K. Posant v = K − u, |v| K 2 ξ2 2 X=0. −X + 4 1 − 2 ch v L’approximation WKB donne pour le spectre ξn 1/2 2 ξ 1 2 dv = n + − 1 π. 2 ch2 v Par une approximation assez grossi`ere π 1 = y 0 mq . 2ξn = n + 2 2
(137)
(138)
(139)
Utilisant la valeur de q estim´ee en (76), pour grand et k petit qy0 ∼ π , l’estimation (139) nous donne √ 1 π (n 1) (140) mq = 2q n + 2 2 ´eliminant √ la d´ependance explicite en , formule tr`es analogue au spectre (116) au facteur 2 pr`es.
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
285
6 Conclusion L’introduction de deux champs scalaires φ r´egis par deux potentiels p´eriodiques de mˆeme forme cos π φφ0 , coupl´es `a la gravit´e d’un V5 avec des signes oppos´es, a permis de construire une sorte de d´eformation du mod`ele de Randall et Sundrum d´ependant de deux param`etres. Ce sont la p´eriode 2φ0 des champs (li´ee au param`etre ), et la p´eriode 2Ky0 = 2yc de la cinqui`eme dimension y (K est li´ee au module k), qui sont reli´ees entre elles par les constantes de gravitation κ de V5 et κ ˜ de V4 . (A la limite R.S. on a seulement y0 = 3π4 2 κ ˜ φ20 .) Selon l’id´ee d´evelopp´ee par Ichinose en introduisant un potentiel la paroi est une notion d´eriv´ee ; elle est n´ecessairement diffuse, la paroi-section de R.S. ´etant un concept limite. La p´eriodicit´e, trait essentiel de la th´eorie de Klein, r´esulte habituellement de la des petits mouvements au voisinage d’un minimum du potentiel. 2 dynamique k 1 . Avec l’analogie du pendule circulaire dont l’angle de rotation 2ψ est toujours croissant k 2 < 1 , l’hypoth`ese topologique du cercle pour la cinqui`eme dimension porte de fa¸con ´equivalente sur le champ qui est un angle. L’image des deux pendules en opposition illustre la liaison cin´ematique des deux champs, n´ecessairement li´es puisqu’ils ne d´ependent que d’une seule variable dans cette solution sp´eciale. Cette liaison ´equivaut `a une contrainte g´eom´etrique sans transfert d’´energie. Dans cette analogie m´ecanique o` u la cinqui`eme variable d’espace joue le rˆole de “temps”, le ph´enom`ene des deux parois diffuses, a` la limite R.S. d’´epaisseur nulle, apparaˆıt comme dˆ u a` l’approche du mouvement pendulaire ap´eriodique (limite k → 1, K → +∞). En alternance, les pendules passent quasiment une demi-p´eriode au voisinage de l’´equilibre instable (le z´enith 2ϕ1 = ±π, ϕ1 petit, ϕ2 grand, et vice-versa). D’o` u les pics cin´etiques, ou de pression, caract´erisant les parois. Ce syst`eme pendulaire sert d’horloge de r´ef´erence pour la coordonn´ee u = y/y0 et les champs scalaires φ, tandis que la courbure de V5 ou le facteur de dy r´eduction e−σ = dτ mesure le rapport du temps m´ecanique des pendules dy au temps propre dτ de V4 , sur une g´eod´esique nulle de V5 . La source de cette courbure est le diff´erentiel de densit´e d’´energie et de pression entre les deux champs, qui en fait ne constituent qu’une seule entit´e. Toute particule test massive est de masse rapidement variable sur une p´eriode, de l’ordre de la masse de Planck au voisinage de la paroi cach´ee. Cette masse n’est stabilis´ee `a une valeur physique observable que dans le voisinage de la paroi visible `a laquelle l’excitation massive est comme li´ee. Le couplage avec signes oppos´es apparaˆıt certes comme un artifice. Mais il pr´eserve cependant une sym´etrie d’´echange des deux champs ; le syst`eme global est invariant par la triple op´eration: a) M ←→ −M (ou κ ←→ −κ du fait de la dimension impaire), b) φ1 ←→ φ2 , c) ←→ −, ou inversion du facteur d’´echelle.
286
M. Gaudin
Ann. Henri Poincar´e
Cette variation math´ematique sur le th`eme de la m´etrique R-S ne change rien sur le fond de l’interpr´etation et semble indiquer une certaine stabilit´e structurelle du mod`ele, tout en accentuant le rˆ ole d’une hypoth`ese artificielle. Concernant la m´etrique, il n’est pas sans int´erˆet de voir s’introduire simplement et exactement une p´eriodisation de la variable de rapidit´e (ou angle hyperbolique) g´en´eralisant les espaces d’Einstein (`a constante cosmologique) qui sont d´ej` a des sph`eres g´en´eralis´ees (ou hyperbolo¨ıdes), au prix d’une source de type sine-Gordon. Si la cinqui`eme coordonn´ee devait ˆetre identifi´ee `a la variable compactifi´ee de K.K., l’unification g´eom´etrique gravit´e-´electromagn´etique serait `a transposer dans ce contexte. La p´eriode 2yc est alors d´etermin´ee. 2π yc = √ y˜ ; α
√ = G = longueur de Planck. ˜ M
y˜ =
√ 2 G yc e2 = Cste de structure fine. ≡ . (141) e π Puisque le syst`eme global est formellement invariant par changement de coordonn´ee, on peut appliquer a` la solution particuli`ere obtenue la transformation sp´eciale qui pr´eserve l’hypoth`ese cylindrique de la th´eorie de Kaluza. Ce qui sugg`ere, en pr´esence d’un champ e.m additionnel A (x), la m´etrique invariante de jauge, d´ependant d’un champ scalaire a (x) α=
2 √ √ ds2 = dy − 2 GAdx + e−2σ(y−2 Ga) d˜ s2
(142)
inchang´ee par √ 2 G S (x) y→y+ e
,
ea → ea + S
,
eA → eA + ∂S
(143)
sur une p´eriode 2yc , l’action S augmente de 2π. Choisir a (x) ≡ 0, fixe la jauge. Mais la projection V5 → V4 , avec le d´ecouplage exact de Kaluza, Thiry, etc. . . qui ´etait dˆ u a` l’hypoth`ese cylindrique, n’est plus qu’approximative et n´ecessite la moyenne pr´ealable sur une p´eriode. Le potentiel Aµ n’est plus un vecteur de Killing de V5 , sinon en moyenne, puisque l’on a dans V5 , 1 ∇µ Aν + ∇ν Aµ = + √ σ gµν G
(144)
ce qui revient en gros a` remplacer dans les formules de Thiry le champ de V4 Fµν par Fµν + √1G σ e−2σ g˜µν . D’o` u une formule analogue a` celle de Thiry avec le terme de pression suppl´ementaire dˆ u a` la source scalaire de sine-Gordon. Le champ F admet comme source un courant qui est proportionnel a` A dans la jauge particuli`ere a (x) ≡ 0, ce qui d´etermine implicitement une action S.
Vol. 5, 2004
D´eformation Elliptique de la M´etrique de Randall et Sundrum
287
L’´equation de Klein-Gordon associ´ee en pr´esence du champ A dans la jauge fix´ee sort ainsi sous la forme r´eelle qu’avait not´ee Schr¨odinger ˜ 2Ψ ¯ + e 2 A · A − m2 Ψ ¯ =0 2 ∇ (145) dans la jauge telle que le courant conserv´e soit justement proportionnel a` A 2 ¯ A =0. (146) ∂ Ψ
References [1] L. Randall et R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), Phys. Rev. Lett. 83, 4690 (1999), preprints hep-ph/9905221–9906064. [2] S. Ichinose, Class. Quantum Grav. 18, 421–432 (2001). Voir aussi S. Kobayashi, K. Koyama et J. Soda, preprint hep-th/0107025 [3] T. Tanaka et X. Montes, Nucl. Phys. B 582, 259 (2000). [4] Pour les ´equations de Friedmann-Lemaˆıtre, r´ef´erences cit´ees dans Essais Cosmologiques J.P. Luminet. L’invention du Big Bang, Le Seuil (1997). [5] Th. Kaluza Sitz. Preuss. Akad., 966 (1921), Modern Kaluza-Klein Theories Appelquist, Chodos and Freund, A.W.P. (1987). [6] O. Klein, Zeitschrift f¨ ur Physik 37, 895 (1926). [7] J.M. Souriau, Five-dimensional Relativity. Nuovo Cimento XXX, 2 (1963). [8] Whittaker and Watson, Modern Analysis, IV`eme ´ed. Cambridge U.P. (1958). [9] G. Szeg¨ o, Orthogonal Polynomials R.I. AMS (1939). [10] M. Bertola, J. Bros, V. Gorini, U. Moschella, R. Schaefer, Decomposing quantum fields on branes, Nuclear Physics B581, 575–603 (2000). Michel Gaudin Service de Physique Th´eorique Orme des Merisiers CEA/Saclay F-91191 Gif sur Yvette France email:
[email protected] Communicated by Vincent Rivasseau Submitted 07/09/01, accepted 25/11/03
Ann. Henri Poincar´e 5 (2004) 289 – 326 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020289-38 DOI 10.1007/s00023-004-0170-z
Annales Henri Poincar´ e
Conformal Transformations and the SLE Partition Function Martingale Michel Bauer and Denis Bernard∗
Abstract. We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick’s theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half-plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half-plane is the SLE hull, we show that the partition function reduces to a useful local martingale known to probabilists, thereby unraveling its CFT origin.
1 Introduction Since its very origins, the statistical mechanics of two-dimensional critical systems has seen a deep interplay between physics and mathematics. This was already true for Onsager’s solution of the 2d Ising model and the computation of the magnetization by Yang [19]. In the 80’s, the link between physics and mathematics was mainly through representation theory, affine Lie algebras and the Virasoro algebra playing the most central roles. Two-dimensional conformal field theories [4] have led to an enormous amount of exact results, including the computation of multi-point correlators and partial classifications. The study of multi-fractal properties of conformally invariant critical clusters has been less systematic, but has nevertheless produced a number of remarkable successes (see, e.g., refs.[18, 6, 10] and references therein), the famous Cardy formula giving the probability for the existence of a connected cluster percolating between two opposite sides of a rectangle in two-dimensional critical percolation [5] being one of the highlights. Cardy’s formula is now a theorem [22]. More recently, probability theory, stochastic processes to be precise, have started to play an important role, due to a beautiful connection between Brownian motion and critical clusters discovered by Schramm [21]. This connection is ∗ Member
of the CNRS
290
M. Bauer and D. Bernard
Ann. Henri Poincar´e
via the Loewner evolution equation, which describes locally growing domains Kt (called hulls) in the upper half-plane implicitly by prescribing the variation of the normalized uniformizing map for the complement. In this way, the growth of the hull is encoded by a real continuous function. Taking this function to be a Brownian sample path leads to stochastic (chordal) Loewner evolutions (SLE) of growing hulls whose properties are those expected for conformally invariant critical clusters. There is a single parameter, denoted κ, which is the time scale for the Brownian motion. This has led to important probabilistic theorems, among which Brownian intersection exponents [15]. Moreover, this framework has opened a way to prove that critical lattice statistical models possess a conformally invariant scaling limit. The link between SLE and standard conformal field theory (CFT) was obscure for several years, but recently we proposed a direct connection [1]. The idea is to couple CFT to SLE via boundary conditions, namely to look at a boundary CFT in the random geometry of the complement of the hull in the upper halfplane. The crucial observation is that if one inserts at the origin (where the hull starts to grow) a primary boundary operator (leading to a boundary state |ω) of appropriate weight in a CFT of appropriate central charge, and then lets the hull grow, the corresponding conformal state is a local martingale in the sense of probability theory, i.e., a quantity whose probabilistic average is time-independent1 . In this way, many quantities computed by probabilistic methods can be shown to be directly related to correlation functions of CFT [2]. Another relation between CFT and chordal SLE which uses the restriction property has been presented in ref. [12]. The purpose of this paper is twofold. The first is SLE independent. We give a rigorous construction of the CFT operator implementing finite local conformal transformations that fix a point. This amounts to show how to go from certain sub-algebras of the Virasoro algebra to a corresponding Lie group via exponentiation. As a first application, we use coordinates on these groups to build coherent state highest weight representations of the Virasoro algebra. We observe a striking similarity with the representations of the Virasoro algebra that appear in matrix models [8]. This is a pedestrian implementation of the geometric ideas ` a la BorelWeil presented in [3]. Under some global conditions, one can multiply operators corresponding to local conformal transformations fixing different points, leading to an embryonic version of the Virasoro group (which is ill defined in the CFT context: the central extension of the group of diffeomorphisms of the circle is not what is needed). As a byproduct, we give a theorem which does for the Virasoro algebra what Wick’s theorem does for oscillator algebras. This kind of computation could have been made right at the early stages of CFT, in the 80’s. It seems that certain analogous 1 Under certain boundedness conditions: technically, nice linear forms applied to this state are time-independent in mean.
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
291
formulæ were derived at that time [25], but we have not been able to trace those back in the published literature. These purely algebraic considerations have applications to SLE. The uniformization of the growing hull Kt is given, close to the point at infinity, by a suitably normalized local conformal transformation kt . This leads immediately to a clean definition of the conformal state Gkt |ω describing the growing hull Kt . The invertible operator Gkt is then shown to satisfy a stochastic differential equation2 which implies that Gkt |ω is a local martingale. We give a brief account of the proof, using the above-mentioned coherent state representations of the Virasoro algebra, that Gkt |ω is the generating function of all SLE martingales in a precise algebraic sense and that these martingales build a certain highest weight representation of the Virasoro algebra with a non trivial character. This is an elaboration of [3]. Finally, we turn to the partition function martingale. If a CFT is coupled via boundary conditions not only to the growing hull Kt but also to a fixed (deterministic) hull A disjoint from Kt , the CFT partition function contains a universal contribution corresponding to some kind of interaction between A and Kt . This is by definition a local martingale. We use Wick’s theorem for the Virasoro algebra to give yet another illustration that the SLE quantities computed by Lawler, Schramm and Werner [16] are in fact deeply rooted in CFT. For κ = 8/3, this martingale computes the probability that Kt never touches A. The previous paragraph is definitely not a claim that mathematicians have rediscovered things that were known to theoretical physicists. Quite the opposite is true: the discoveries of probabilists have motivated us to go back to the foundations of CFT to realize that maybe certain basic construction had not been given enough attention and that some CFT jewels had been left dormant.
2 (Chordal) SLE and CFT The aim of this section is to recall basic definitions of stochastic Loewner evolutions (SLE) and its generalizations that we shall need in the following. Most results that we recall can be found in [20, 14, 15, 16]. See [7] for a nice introduction to SLE for physicists and [23] for pedagogical summer school notes. A hull in the upper half-plane H = {z ∈ C, z > O} is a bounded simply connected subset K ⊂ H (for the usual topology of C) such that H \ K is open, connected and simply connected. The local growth of a family of hulls Kt parameterized by t ∈ [0, T [ with K0 = ∅ is related to complex analysis in the following way. The complement of Kt in H is a domain Ht which is simply connected by hypothesis, so that by the Riemann mapping theorem Ht is conformally equivalent to H via a map ft . This map can be normalized to behave as ft (z) = z + 2t/z + O(1/z 2): 2 In our previous papers, this equation was used as a heuristic definition of G . We had to kt leave aside analytical questions of existence of solutions, relying on physical intuition.
292
M. Bauer and D. Bernard
Ann. Henri Poincar´e
the P SL2 (R) automorphism group of H allows to impose ft (z) = z + O(1/z) for large z, and then the coefficient of 1/z is fixed to be 2t by a time reparametrization. The crucial condition of local growth leads to the Loewner differential equation ∂t ft (z) =
2 , ft (z) − ξt
ft=0 (z) = z
with ξt a real function. For fixed z, ft (z) is well defined up to the time τz ≤ +∞ for which fτz (z) = ξτz . Then Kt = {z ∈ H : τz ≤ t}. √ (Chordal) stochastic Loewner evolutions is obtained [21] by choosing ξt = κ Bt with Bt a normalized Brownian motion and κ a real positive parameter so that E[ξt ξs ] = κ min(t, s). Here and in the following, E[· · · ] denotes expectation value. The next section, which also contains basic definitions that the reader can refer to, is devoted to a careful discussion of the implementation of finite local conformal transformations in conformal field theory. In the rest of this section, we simply assume that such an implementation is possible, and we derive a direct connection between SLE and CFT. SLE is defined via an ordinary differential equation, but for our reinterpretation in terms of conformal field theories, it is useful to define kt (z) ≡ ft (z) − ξt which satisfies the stochastic differential equation dkt =
2dt − dξt . kt
We observe that the conditions at spatial infinity satisfied by kt imply that its germ there, which determines it uniquely, belongs to the group N− of germs of holomorphic functions at ∞ of the form z + m≤−1 fm z m+1 , the group law being composition. In this way, the Loewner equations describe trajectories on N− in a time-dependent left-invariant vector field, whose value at the identity element is (2/z − ξ˙t )∂z . Due to the fact that ξt is almost surely nowhere differentiable, this observation has to be taken with a grain of salt. We let f ∈ N− act on O∞ , the space of germs of holomorphic functions at infinity, by composition, γf · F ≡ F ◦ f . Observe that γg◦f = γf · γg so this is an anti-representation. Ito’s formula gives 2dt κ − dξt + (γkt · F ) dγkt · F = (γkt · F ) kt 2 from which we derive γk−1 · dγkt = dt t
2 κ ∂z + ∂z2 z 2
− dξt ∂z .
The operators ln = −z n+1 ∂z are represented in conformal field theories by operators Ln which satisfy the Virasoro algebra vir c [Ln , Lm ] = (n − m)Ln+m + (n3 − n)δn+m,0 [c, Ln ] = 0. 12
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
293
The representations of vir are not automatically representations of N− , one of the reasons being that the Lie algebra of N− contains infinite linear combinations of the ln ’s. However, as we shall see in the next section, highest weight representations of vir can be extended in such a way as to become representations of N− . We take this for granted for the moment and associate to γf an operator Gf acting on appropriate representations and satisfying Gg◦f = Gf Gg and κ 2 + dξt L−1 . L dG = dt −2L + G−1 kt −2 kt 2 −1 The basic observation is the following [1]: Let |ω be the highest weight vector in the irreducible highest weight repreand conformal weight hκ = 6−κ sentation of vir of central charge cκ = (6−κ)(8κ−3) 2κ 2κ . Then Gkt |ω is a local martingale. Assuming appropriate boundedness conditions on v|, the scalar v|Gkt |ω is a martingale. In particular E[v|Gkt |ω] is time-independent. This is a direct consequence of the fact that for this special choice of central charge and weight, the irreducible highest weight representation is degenerate at level 2 and (−2L−2 + κ2 L2−1 )|ω = 0. Then κ dGkt |ω = Gkt dt −2L−2 + L2−1 + dξt L−1 |ω = dξt Gkt |ω . 2 In probabilistic terms, a random variable whose Ito derivative contains only a dξt contribution (no dt) is called a local martingale. We refer the interested reader to the mathematical literature [13]. From the definition of Ito integrals, dξt and Gkt are independent, so that naively dE[v|Gkt |ω] = 0 for any v|. A word of caution is needed here. Before talking about E[v|Gkt |ω], we should in principle show that v|Gkt |ω is an integrable random variable. Then v|Gkt |ω is a martingale. This is true for instance if v| is a finite excitation of ω|, but this condition is far too restrictive for probability theory and for conformal field theory as well3 . This result can be interpreted as follows. Take a conformal field theory in Ht . The correlation functions in this geometry can be computed by looking at the same theory in H modulo the insertion of an operator representing the deformation from H to Ht . This operator is Gkt . Suppose that the central charge is cκ and the boundary conditions are such that there is a boundary changing primary operator of weight hκ inserted at the tip of kt (the existence of this tip is more or less a consequence of the local growth condition). Then in average the correlation functions of the conformal field theory in the fluctuating geometry Ht are timeindependent and equal to their value at t = 0. 3 We shall often drop the term local, even if the notion of martingale, though closely related to the notion of local martingale, is more restrictive. In particular, the time-independence of expectations is always true for martingales.
294
M. Bauer and D. Bernard
Ann. Henri Poincar´e
We call Gkt |ω a generating function for conserved quantities because for any time-independent bra v| satisfying the integrability condition, the scalar E[v|Gkt |ω] is time-independent. We shall see later that in an algebraic sense, all conserved quantities for chordal SLE are of this form.
3 Conformal transformations in conformal field theory A (rather provocative) definition of (boundary) conformal field theory is that it is the representation theory of the Virasoro algebra vir. The Virasoro algebra has a sub-algebra n− , with generators the Ln ’s n < 0, which is closely related to N− , the group of germs of conformal transformations that fix ∞. This is crucial for the construction of Gkt . Our goal in this section is to show that indeed, N− acts on sufficiently many physically relevant representations of vir to be able to make sense of conformal field theories in the fluctuating geometry Ht . In the same spirit, the group N+ of germs of conformal transformations that fix 0 is closely related to the sub-algebra n+ of vir with generators the Ln ’s n > 0. This group will also play an important role in the forthcoming discussion.
3.1
Background
The theories we shall study will mostly be boundary conformal field theories, and we shall talk of field or operator without making always explicit whether the argument is in the bulk or on the boundary. The basic principles of conformal field theory state that the fields can be classified according to their behavior under (local) conformal transformations. Then the correlation functions in a region U are known once they are known in a region U0 and an explicit conformal map f from U to U0 that preserves boundary conditions is given. Primary fields have a very simple behavior under conformal transformations: for a bulk primary field ϕ of weight (h, h), ϕ(z, z)dz h dz h is invariant, and for a boundary conformal field ψ of weight δ, ψ(x)|dx|δ is invariant. So the statistical averages in U and U0 are related by · · · ϕ(z, z) · · · ψ(x) · · ·U h
= · · · ϕ(f (z), f (z))f (z)h f (z) · · · ψ(f (x))|f (x)|δ · · ·U0 . Such a behavior is described as local conformal covariance. In a local theory, small deformations are generated by the insertion of a local operator, the stress tensor. Local conformal covariance can then be rephrased: the stress tensor of a conformal field theory is not only conserved and symmetric, but also traceless, so that it has only two independent components, one of which, T , is holomorphic (except for singularities when the argument of T approaches the argument of other insertions), and the other one, T , is anti-holomorphic (again
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
295
except for short distance singularities). The field T itself is not a primary field in general, but a projective connection: c · · · T (z) · · ·U = · · · T (f (z))f (z)2 + Sf (z) · · ·U0 . 12 2 (z) (z) In this formula, c is the central charge and Sf (z) = ff (z) − 12 ff (z) is the Schwarzian derivative of f at z. If U is a non empty simply connected region strictly contained in C, the Riemann mapping theorem states that U0 can be chosen to be unit disk D or equivalently the upper half-plane H – then the point at infinity is a boundary point. This second choice will prove most convenient for us in the sequel. In boundary conformal field theory, T and T are not independent: they are related by analytic continuation. The relationship is expressed most simply in the upper half-plane. The vectors fields z n+1 ∂z and z n+1 ∂z are generators of infinitesimal conformal transformations in C but only the combination z n+1 ∂z + z n+1 ∂z ≡ − n preserves the boundary of H, that is, the real axis. Write z = x + iy and for a while write T (x, y) for what we usually write T (z). Choosing boundary conditions such that there is no flow of energy momentum across the boundary x = 0, T (x, y) is real along the real axis, and by the Schwarz reflection principle has an analytic extension to the lower half-plane as T (x, −y) ≡ T (x, y) = T (x, y). Due to this property, most contour integrals involving T and T in the upper half-plane can be seen as contour integrals involving only T but in the full complex plane. Using conformal field theory in H to express correlators in any simply connected region strictly contained in C has another advantage: one can use the formalism of radial quantization in a straightforward way. The statistical averages are replaced by quantum expectation values: ˆ · · · |Ω. ˆ z) · · · ψ(x) · · · T (z) · · · ϕ(z, z) · · · ψ(x) · · ·H = Ω| · · · Tˆ (z) · · · ϕ(z, r
In this formula, |Ω is the vacuum and r denotes radial ordering: the fields are ordered from left to right from the farthest to the closest to the origin. The integral dzz n+1 Tˆ (z) along any contour of index 1 with respect to 0, defines an operator Ln (note again that from the point of view of contour integrals in the upper halfplane, Ln involves T and T ). The fact that the stress tensor is the generator of infinitesimal conformal maps implies that ˆ ˆ [Ln , ψ(x)] = xn+1 ∂x + δ(n + 1)xn ψ(x) n+1 [Ln , ϕ(z, ˆ z) ˆ z)] = z ∂z + h(n + 1)z n + z n+1 ∂z + h(n + 1)z n ϕ(z, n+1 c [Ln , T (z)] = z ∂z + 2(n + 1)z n T (z) + (n3 − n)z n−2 12 c 3 [Ln , Lm ] = (n − m)Ln+m + (n − n)δn+m,0 . 12 Except for the anomalous central c-term, the commutation relations of the Ln ’s are those of the n ’s. Let us take this opportunity to recall that the crucial point
296
M. Bauer and D. Bernard
Ann. Henri Poincar´e
to implement symmetries in quantum mechanics is to have the symmetries act well on operators, i.e., that the adjoint action represents the symmetries. Hence symmetries in quantum mechanics act projectively, and this leaves room for central terms such as c in vir. The advantage of the operator formulation of conformal field theory is that one can use the powerful methods of representation theory, applied to the Virasoro algebra.
3.2
Some representation theory
Basic references for this standard material are for instance [9]. In the sequel we denote by h the (maximal) Abelian sub-algebra of vir generated by L0 and c, by n− (resp. n+ ) the nilpotent4 Lie sub-algebra of vir generated by the Ln ’s, n < 0 (resp. n > 0) and by b− (resp. b+ ) the Borel Lie sub-algebra of vir generated by the Ln ’s, n ≤ 0 (resp. n ≥ 0) and c. If g is any Lie algebra, we denote by U(g) its universal enveloping algebra. Then a representation of g is the same as a left U(g)-module. Let us describe representations of vir by starting with the simplest ones, which we call positive energy representations. These are representations whose underlying space M splits as a direct sum M = m≥0 Mm of finite-dimensional subspaces such that Ln maps Mm to Mm−n for any m, n ∈ Z (with the convention that Mm ≡ {0} for m < 0) and L0 is diagonalizable on each Mm . If M has positive energy, we can define the contravariant representation of vir whose underlying space is the little graded dual of M , which we define as M ∗ ≡ ∗ ∗ m≥0 Mm , where Mm is the standard algebraic dual of the finite-dimensional space Mm . Observe that one can view Ln acting on M as a collection of linear maps Ln : Mm → Mm−n indexed by m. For each of these maps, one can take the ∗ ∗ algebraic transpose t Ln : Mm−n → Mm , defined (as usual for finite-dimensional t ∗ . We define Ln acting on spaces) by Ln y, x ≡ y, Ln x for (x, y) ∈ Mm × Mm−n ∗ t ∗ ∗ M by the collection L−n : Mm → Mm−n . We decide that c is the multiplication by the same scalar on M ∗ as on M . The representation property is checked by a simple computation. Note that M ∗∗ is canonically isomorphic to M as a virmodule. The most important examples of positive energy representations are highest weight modules and their contravariants. A vir highest weight module M is a representation of the Virasoro algebra which contains a vector v such that (i) Cv is a one-dimensional representation of h and is annihilated by n+ and (ii) the smallest sub-representation of M containing v is M itself, i.e., all states in M can be obtained by linear combinations of strings of generators of vir acting on v. Because Cv is a one-dimensional representation of b+ , all states in M can be obtained by linear combinations of strings of generators of n− acting on v. On such a representation, the generator c acts on M as multiplication 4 Triangular would be more accurate, but we keep this definition by analogy with finitedimensional Lie algebras.
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
297
by a scalar, which we denote by c again and call the central charge. The number h such that L0 v = hv is called the conformal weight of the representation. One can write M = m≥0 Mm where L0 acts on Mm by multiplication by h + m, M0 = Cv and Mm is finite-dimensional with dimension at most p(m), the number of partitions of m. For convenience, we define Mm ≡ {0} for m < 0. Then Ln maps Mm to Mm−n for any m, n ∈ Z. By construction, highest weight cyclic modules have positive energy. The existence of highest weight modules for given c and h is ensured by a universal construction using induced representation. Let R(c, h) denote the onedimensional representation of h, of central charge c and conformal weight h. View R(c, h) as a representation of b+ where n+ act trivially. This turns R(c, h) into a left U(b+ )-module. For any g, U(g) acts on itself on the left and on the right, so by restriction, we can view U(vir)
as a left U(vir)-module and as a right U(b+ )module. Then V (c, h) ≡ U(vir) U (b+ ) R(c, h) is a left U(vir)-module, called the Verma module with parameters (c, h). As a U(n− )-module, V (c, h) is isomorphic to U(n− ) itself, so the number of states in V (c, h)n is exactly p(n). Any highest weight cyclic module M with parameters (c, h) is a quotient of V (c, h). The contravariant M ∗ of a highest weight module is not always highest weight: U(vir)M0∗ is always irreducible, hence is a proper submodule of M ∗ if M is not irreducible.
3.3
Completions
In the following, we shall often need to deal with infinite linear combinations of Virasoro generators. For instance, formally T (z) = n Ln z −n−2 . So we make some new definitions. We denote by n+ the formal completion of n+ which is made of arbitrary (not necessarily finite) linear combinations of Ln ’s, n > 0. The Lie algebra structure on n+ extends to a Lie algebra structure on n+ if we define
m>0
am L m ,
n>0
bn Ln ≡
k>0
(m − n)am bn Lk .
m>0,n>0, m+n=k
As usual with formal power series, this works because for fixed k, the sum m>0,n>0, is a finite sum. m+n=k
We can go one step further and define vir+ as the direct sum n+ ⊕ b− , which is still a Lie algebra with the obvious definition. One can make analogous definitions for n− , b+ , b− , n− ⊕ b+ . All these Lie algebras are contained in n− ⊕ h ⊕ n+ , but we shall not (!) try to put a Lie algebra structure on that space. Note that vir, n− , n+ , b− and b+ are graded Lie algebras, so their universal enveloping algebras are graded too (the grading should not be confused with the
298
M. Bauer and D. Bernard
Ann. Henri Poincar´e
filtration which exists for any Lie algebra). We denote by U(vir)n , U(n− )n , U(n+ )n , U(b− )n and U(b+ )n the subspace of degree n in eachof the corresponding algebras. Using the grading, one checks that U(n+ ) ≡ n>0 U(n+ )n , the formal completion5 of U(n+ ) has a natural associative algebra structure which extends that of degrees U(n+ ). Actually, U(n+ ) is made of formal series of monomials of arbitrary in the Ln ’s, n > 0. One can make analogous remarks for U(n− ) ≡ n r but |fB (z)| < R 8 . For such z’s, first the composition fA∪B (z) = f ˜ ◦ fB (z) can be computed by inserting the A −1 series expansions, and second G−1 f ˜ GfB T (z)GfB GfA˜ is well defined, given by A
c absolutely convergent series, and is equal to T (fA∪B (z))fA∪B (z)2 + 12 SfA∪B (z). Of course, the roles of A and B could be interchanged, and we could first ˜ ≡ fA (B) by f ˜ remove A by fA which is regular around 0 and fixes 0 and then B B which is regular around ∞ and such that fB˜ (z) = z + O(1). As they uniformize the same domain, we know that fA˜ ◦ fB and fB˜ ◦ fA differ by a (real) linear fractional transformation: there is an h ∈ P SL2 (R) such that fB˜ ◦ fA = h ◦ fA˜ ◦ fB . Suppose that fA and fB are given. There is some freedom in the choice of fA˜ and fB˜ : namely we can replace fA˜ by h0 ◦ fA˜ where h0 is a linear fractional transformation fixing 0, and fB˜ by h∞ ◦ fB˜ where h∞ is a linear
8 Such z’s exist in the above geometry, for instance in a small neighborhood of the segment of the real axis that separates A and B. In such a region, radial ordering is also preserved by the maps.
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
311
fractional transformation such that h∞ (z) = z +O(1) at infinity, i.e., a translation. A simple computation shows that unless there is a z such that fA (z) = ∞ and fB (z) = 0, there is a unique choice of fA˜ and fB˜ such that fB˜ ◦ fA = fA˜ ◦ fB . This commutative diagram was introduced in ref. [16]. In the sequel, we shall concentrate on this generic situation. So we deduce that for z’s in some open
Figure 3: The generic commutative diagram. −1 −1 −1 set, G−1 ˜ . As the modes Ln of T fA˜ GfB T (z)GfB GfA˜ = GfB˜ GfA T (z)GfA GfB generate all states in a highest weight representation, the operators GfB GfA˜ and GfA GfB˜ have to be proportional: they differ at most by a factor involving the central charge c. We write GfB GfA˜ = Z(A, B) GfA GfB˜ , or
−1 G−1 fA GfB = Z(A, B) GfB˜ Gf ˜ . A
(11)
As implicit in the notation, Z(A, B) depends only on A and B: a simple computation shows that it is invariant if fA is replaced by h0 ◦ fA and fB by h∞ ◦ fB . Formula (11) plays for the Virasoro algebra the role that Wick’s theorem plays for collections of harmonic oscillators. We call Z(A, B) a partition function for the following reason: we can write −1 ˆ Ω G−1 GfB GfA˜ Ω fA˜ GfB · · · T (z) · · · r 1 ˆ Ω G−1 G−1 GfB GfA˜ Ω . = fB fA · · · T (z) · · · ˜ Z(A, B) r But |Ω is annihilated by b+ and Ω| is annihilated by n− so 1 ˆ (z) · · · GfB Ω , · · · T (z) · · ·HA∪B = · · · T Ω G−1 fA Z(A, B) r and
Z(A, B) = Ω G−1 G . f BΩ fA
312
6.2
M. Bauer and D. Bernard
Ann. Henri Poincar´e
Computation of the partition function
The computation of Z(A, B) goes along the following lines. If one changes A by a small amount, the variation of fA can be written as δfA = vA (fA ). In order to keep the initial properties of A and B, we impose that vA is a vector field holomorphic in the full plane but for cuts along the real axis with positive discontinuities, satisfies the Schwarz reflexion principle (vA (z) = vA (z)), and is such that the ˜ We may open disk of convergence of its power series expansion at 0 contains B. µA (x)dx with µA (x) positive. write vA in terms of its discontinuities as vA (z) = z−x Similar considerations hold if B is distorted slightly, we write δfB = vB (fB ) and vB satisfies corresponding conditions. Then we know that −1 −1 δ(G−1 G ) = v (u)T (u)duG G − G G vB (v)T (v)dv A fA fB fA fB fA fB 0 ∞ −1 = Z(A, B) vA (u)T (u)duGfB˜ G−1 − G G v (v)T (v)dv . f B ˜ f˜ f˜ B A
0
A
∞
By hypothesis, we can deform the small contour around 0 to a contour in a region where vA and fB˜ have a convergent expansion, and the small contour around ∞ to a contour in a region where vB and fA˜ have a convergent expansion. Then we may conjugate, with the result δ(G−1 fA GfB ) Z(A, B)
= −
c GfB˜ ( vA (u)(T (fB˜ (u))fB˜ (u)2 + SfB˜ (u))du 12 c vB (v)(T (fA˜ (v))fA˜ (v)2 + SfA˜ (v))dv)G−1 fA˜ . 12
Taking the vacuum expectation value yields c vA (u)SfB˜ (u)du − vB (v)SfA˜ (v)dv . δ log Z(A, B) = 12 The explicit value of log Z(A, B) can be computed by means of several formulæ. The most symmetrical ones are obtained if A and B are both described by integrating infinitesimal deformations of H. Consider two families of hulls, As and Bt that interpolate between the trivial hull and A or B respectively. We arrange that fAs and fBt are generic, so that unique fAs,t and fBt,s exist, which satisfy fBt,s ◦ fAs = fAs,t ◦ fBt . ∂f ∂f Define vector fields by vAs and vBt by ∂sAs = vAs (fAs ) and ∂tBt = vBt (fBt ). Now set As,t = fBt (As ) and Bt,s = fAs (Bt ), and define vector fields vAs,t and vBt,s by
∂fAs,t ∂s
= vAs,t (fAs,t ) and σ L(Aσ , Bτ ) ≡ − ds 0
∂fBt,s ∂t
0
τ
= vBt,s (fBt,s ). Set dt dw dz vAs,t (w) Γw
Γz
6 vB (z) (z − w)4 t,s
(12)
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
313
where the contours Γw and Γz are simple contours in C of index 1 with respect to 0, such that the bounded component of C\Γz contains the cuts of fB−1 , the t,s bounded component of C\Γw contains Γz and the unbounded component contains the cuts of fA−1 as described on Fig. (4). We observe that the kernel is a four order s,t pole, i.e., is proportional to the two-point correlation function for the stress energy tensor in the plane geometry. Moreover, by writing vAs,t and vBt,s as integrals of their (positive) discontinuities one sees that L(A, B) is positive. We claim that c Z(Aσ , Bτ ) = exp L(Aσ , Bτ ). 12
Γw
cut for A s,t
cut for B t,s
Γz O
Figure 4: Integration contours intrication. This formula is very symmetrical, but it does not make clear that log Z(Aσ , Bτ ) really depends only on Aσ and Bτ , not on the full trajectories As , s ≤ σ and Bt , t ≤ τ . The following steps are also useful to show that eq. (12) has the correct variational derivative. We start by the change of variable z = fAs,t (ζ), which is valid for z in a simply connected neighborhood of Γw containing the origin, hence on Γz . Taking the t-derivative of fBt,s ◦ fAs = fAs,t ◦ fBt , we obtain vBt,s (fAs,t (ζ)) = ∂fA
(ζ)
∂fAs,t (ζ) + fA s,t (ζ)vBt (ζ). ∂t
s,t is a holomorphic function of ζ in a neighborhood of the origin conBut ∂t taining the ζ integration contour, so in eq. (12) we may replace vBt,s (z)dz by fA s,t (ζ)vBt (ζ)fA s,t (ζ)dζ.
314
M. Bauer and D. Bernard
Hence
L(Aσ , Bτ ) =
−
0
−
dt 0
σ
dw
−1 fA (Γz )
Γw
dt 0
dζ vAs,t (w)
s,t
τ
ds 0
τ
ds
=
σ
−1 fA (Γz )
Ann. Henri Poincar´e
6fA s,t (ζ)2 (fAs,t (ζ) − w)4
dζ vA (fAs,t (ζ))fA s,t (ζ)2 vBt (ζ) . s,t
vBt (ζ) (13)
s,t
In the second line, the w integral has been computed by the residue formula. This is legitimate because, by hypothesis, vAs,t (w) is holomorphic in the bounded component of C\Γw . We can now make use of a useful identity for the variations of the Schwarzian derivative. From its definition one checks that S(f + εv(f ))(f ) = εv (f ) + O(ε2 ). Combined with the cocycle property S(f + εv(f ))(z)dz 2 = S(f + εv(f ))(f )df 2 + S(f )(z)dz 2 this yields d S(f + εv(f ))(z)|ε=0 = v (f (z))f (z)2 . dε Finally
L(Aσ , Bτ )
= −
0
dt 0
dt
τ
τ
ds
= −
σ
0
−1 fA (Γz ) s,t
−1 fA (Γz )
dζ
d SfAs,t (ζ)vBt (ζ) ds
dζ SfAσ,t (ζ)vBt (ζ).
(14)
s,t
The roles of Aσ and Bτ could be interchanged to remove the Γw and t integrations, leading to σ ds dw vAs (w)SfBτ,s (w) L(Aσ , Bτ ) = 0 τ dt dz vBt (z)SfAσ,t (z). = − 0
Using these formulæ, it is apparent that L(Aσ , Bτ ) does not depend on the detailed way the hulls are built: only the final hulls count. It is also clear that setting c L(Aσ , Bτ ) A ≡ Aσ , A ∪ δA = Aσ+dσ , B ≡ Bτ , B ∪ δB = Bτ +dτ , the variation of 12 is exactly the one of log Z(A, B). So we have proved σ c log Z(Aσ , Bτ ) = ds dw vAs (w)SfBτ,s (w) 12 0 τ c dt dz vBt (z)SfAσ,t (z). = − 12 0 We present two examples in Appendix E. The quantity Z(A, B) has a remarkable interpretation [24] in terms of the Brownian loop-soup [17]: it is the probability that no loop of the soup intersects both A and B, with λ = −c the parameter of the soup.
Vol. 5, 2004
6.3
Conformal Transformations and the SLE Partition Function Martingale
315
Factorization of unity and Virasoro vertex operators
Consider a hull A whose closure contains neither 0 nor ∞. There is a one parameter family of maps uniformizing the complement of A in H and which are regular both at the origin and at infinity. Let us pick one of them, which we call fA (z). Since fA (z) is regular at the origin, we may implement it in conformal field theory by GA+ fA (0)−L0 with GA+ in N+ . Alternatively, since it is also regular at infinity, we may implement it by GA− fA (∞)−L0 with GA− ∈ N− . The product VA ≡ GA− fA (∞)−L0 fA (0)L0 G−1 A+ is the Virasoro analogue of what vertex operators of dual or string models are for the Heisenberg or the affine Kac-Moody algebras. It does not depend on the representative one chooses in the one parameter family. This product is well defined and non trivial in positive energy representation. It may be thought of as the factorization of the identity since the conformal transformation it implements is the composition of two inverse conformal maps.
7 The partition function martingale We now come to the application that has motivated most of our investment in the explicit implementation of conformal transformations. For the convenience of the reader, we start with a quick reminder of [1] phrased in a more rigorous setting. Remember that cκ = (6−κ)(8κ−3) and hκ = 6−κ 2κ 2κ . The Verma module V (cκ , hκ ) is κ 2 not irreducible, and (−2L−2 + 2 L−1 ) acting on the highest weight state is another highest weight generating a sub-representation. We quotient V (cκ , hκ ) by this subrepresentation and denote by |ω the highest weight state in the quotient. Then (−2L−2 + κ2 L2−1 )|ω = 0.
7.1
Ito’s formula for Gkt
The maps ft and kt = ft − ξt that uniformize the growing hull Kt fix the point at infinity, so that there are well-defined elements Gft , Gkt ∈ N− ⊂ U(n− ) implementing them in CFT. The maps are related by a change of the constant coefficient in the expansion around ∞, so the operators are related by Gkt = Gft eξt L−1 . 2 , the The map ft satisfies the ordinary differential equation ∂t ft (z) = ft (z)−ξ t 2 corresponding vector field being v(f ) = f −ξt whose expansion at infinity reads v(f ) = 2 m≤−2 f m+1 ξt−m−2 , so that G−1 ft dGft = −2dt
Lm ξt−m−2
m≤−2
= −2e
ξt L−1
L−2 e−ξt L−1 dt.
316
M. Bauer and D. Bernard
Ann. Henri Poincar´e
ξt L−1 To get G−1 which kt dGkt it remains only to compute the Ito derivative of e κ 2 −ξt L−1 ξt L−1 de = L−1 dξt + 2 L−1 dt. Finally, reads e κ 2 G−1 kt dGkt = (−2L−2 + L−1 )dt + L−1 dξt 2 as announced in Section 2. In particular, dGkt |ω = L−1 dξt Gkt |ω, so that Gkt |ω is a (generating function of) local martingale(s).
7.2
The partition function martingale
We have given an explicit formula for Z(A, B), but motivated by the martingale generating function, we shall sandwich G−1 fA GfB not with the vacuum Ω but with another highest weight state, namely |ω. Using the Virasoro-Wick theorem, one computes that (remember that ω| is annihilated by n− , but |ω is not annihilated by b+ , the L0 part contributes) −1 hκ ω|G−1 ˜ (0) . fA GfB |ω = Z(A, B)ω|Gf ˜ |ω = Z(A, B)fA A
Observe that while the vacuum expectation value depends only on the hulls, the expectation value in a non conformally invariant state depends on the choices of fA and fB . We apply the results of Section 6 to the case when B is the growing hull Kt and A is another disjoint hull. From the previous computation we know that ω|G−1 fA Gkt |ω is a local martingale. We start from fA and ft to build a commutative diagram as before, with maps denoted by fA˜t and f˜t uniformizing respectively ft (A) and fA (Kt ), and satisfying f˜t ◦ fA = fA˜t ◦ ft . Now −1 ξt L−1 ω|G−1 |ω fA Gkt |ω = ω|GfA Gft e ξt L−1 = Z(A, Kt )ω|G−1 |ω fA˜t e −1 = Z(A, Kt )ω| e−ξt L−1 GfA˜t efA˜t (ξt )L−1 |ω.
From eq. (5) we know that the operator e−ξt L−1 GfA˜t efA˜t (ξt )L−1 corresponds to the map z → fA˜t (ξt + z) − fA˜t (ξt ), so that −1 ω| e−ξt L−1 GfA˜t efA˜t (ξt )L−1 |ω = fA˜t (ξt )hκ . 2 From the Loewner equation vKt (z) = z−ξ and t t 2 L(A, Kt ) = − dτ dz SfAτ (z) z − ξτ 0 t dτ SfAτ (ξτ ) . = −2 0
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
Finally hκ ω|G−1 exp − ˜t (ξt ) fA Gkt |ω = fA
c 6
317
t
0
dτ SfAτ (ξτ ),
were fAτ ◦ fτ uniformizes the two hull geometry corresponding to A ∪ Kτ and fAτ is normalized to ensure the commutativity of the uniformization diagram as explained before. It should be noted that the randomness in the above formula is explicit through the appearance of ξt but also implicit through fA˜τ which is a random function. This local martingale was discovered without any recourse to representation theory by Lawler, Schramm and Werner [16], but we hope to have convinced the reader that it is nevertheless deeply rooted in CFT. For the sake of completeness, we shall give two illustration of how this machinery is used to compute explicit probabilities. The following discussion does not claim originality, as the derivations merely sketch the ones given in [16].
7.3
Restriction
We already know that ω|G−1 fA Gkt |ω is a local martingale. One can show that it is a true martingale for κ ≤ 4, let us just note that the region κ ≤ 4 is also the one for which, almost surely, the SLE hull Kt is a simple curve that avoids the real axis at all positive times. For the rest of this section assume κ ≤ 4. Suppose that A is bounded and choose a very large semi circle CR of radius R in H centered at the origin. Let τR be the first time when Kt touches either A or CR . Then τR is a stopping time. It is crucial to normalize fA correctly, and one does so by imposing that it fixes 0 (as already done) and that moreover fA (z) = z + O(1) close to ∞, which by use of the commutative diagram ensures that ensures that fA˜t (z) = z + O(1) close to ∞ as well. These three conditions fix fA completely. Then we claim that fA˜ (ξτR ) is 0 if the SLE hull hits A at τR and goes to 1 for τR
large R if the SLE hull hits CR at τR . Indeed, when the hull approaches A, one or more points on A˜t approach ξt , and at the hitting time, a bounded connected component is swallowed ξt (this uses the normalization of fA ) indicating that the derivative has to vanish there. On the other hand, if CR is hit first, then A˜τR is dwarfed so that (this uses again the normalization of fA ) fA˜τ is close to the R identity map away from A˜τR and in particular at the point ξτR . The behavior of the other factor in the martingale, Z(A, Kt ), is much harder to control, so we now restrict to κ = 8/3, which is the same as cκ = 0 because κ ≤ 4. So the partition function martingale fA˜ (ξt )h8/3 , at t = τR , is 0 if A is hit before CR and close to t 1 if the opposite is true. But the expectation of a martingale is time-independent, h8/3 = fA (0)5/8 . so that the probability that Kt does not hit A is fA˜ (ξt )|t=0 t
318
M. Bauer and D. Bernard
7.4
Ann. Henri Poincar´e
Locality
Let us consider again the case when B is the SLE hull Kt and A another disjoint hull. We may apply the Virasoro-Wick theorem to G−1 fA Gkt to get −1 G−1 ˜t Gf . fA Gkt = Z(A, Kt ) Gk At
Here k˜t is a uniformizing map of the image of the SLE hull by fA and it defines the SLE growth in H \ A. Its lift Gk˜t in N− depending locally on kt is random. A simple computation shows that its Ito derivative is G−1 ˜t = (−2L−2 + ˜ dGk k t
κ 2 κ−6 L )f (0)2 dt + L−1 fA (0)dt + L−1 fA (0)dξt . t t 2 −1 A t 2
Hence, for κ = 6, Gk˜t is statistically equivalent to Gkt up to a time reparametrisation, dt → ds = fA (0)2 dt. This expresses the locality property of critical percot lation.
A
Proof of identity (1)
We start with the proof of eq. (1): the operators Am ≡
f (w) f (w)n+2
n≥m
Ln
0
dwwm+1
satisfy the zero curvature equation ∂Ak ∂Al − = [Ak , Al ]. ∂fk ∂fl
Integration by parts gives ∂ ∂fl
dww 0
m+1
f (w) m+1 ∂ = f (w)n+2 n + 1 ∂fl
so
1 dww =−(m+1) n+1 f (w) 0
m
∂Al ∂Ak − = (k − l) Lj ∂fk ∂fl j
dw 0
dw 0
wl+m+1 f (w)n+2
wk+l+1 . f (w)j+2
On the other hand, [Ak , Al ] =
(m − n)Lm+n
m,n
duuk+1 0
f (u) f (u)m+2
dvv l+1 0
f (v) . f (v)n+2
Split this sum in two pieces by splitting m − n = (m + 1) − (n + 1). In the sum involving m + 1 use uk k+1 f (u) (m + 1) duu = (k + 1) du . m+2 f (u) f (u)m+1 0 0
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
319
In the sum involving n + 1 use dvv l+1
(n + 1) 0
f (v) = (l + 1) f (v)n+2
dv 0
vl , f (v)n+1
interchange the dummy variables m and n, and also u and v. This leads to [Ak , Al ] =
Lm+n
m,n
du
dv
0
0
f (v)((k + 1)uk v l+1 − (l + 1)ul v k+1 ) . f (u)m+1 f (v)n+2
Up to now, the contours in the u and v planes where independent. But if they are adjusted in such a way that |f (v)| < |f (u)|, we can fix j = m + n and sum over m to obtain f (v)((k + 1)uk v l+1 − (l + 1)ul v k+1 ) Lj du dv . [Ak , Al ] = (f (u) − f (v))f (v)j+2 0 0 j Inside the u-plane contour, the singularities of the u-integrand consist now in a simple pole at u = v, and taking the residue leads to [Ak , Al ] = (k − l)
Lj
j
dw 0
wk+l+1 ∂Al ∂Ak = − . f (w)j+2 ∂fk ∂fl
This concludes the proof.
B Proof of identity (3) We continue with the proof of eq. (3): G−1 f Lm Gf
c = 12
dww
m+1
0
Sf (w) +
n≥m
Ln
dwwm+1 0
f (w)2 f (w)n+2
m ∈ Z.
Observe that if we extend the summation over all n’s, the integrals with n < m vanish anyway. Defining Lm (f ) to be the right-hand side, one way to prove this identity could be the tedious check that both sides have the same variation when f is changed into f + εz k+1 , i.e., ∂Lm (f ) f (w) = Ll dwwk+1 , Lm (f ) k ≥ 1. ∂fk f (w)l+2 0 l≥k
This can be done, but it is simpler to consider the variation of Gf and Lm (f ) when f is changed to f + εv(f ). If v(f ) = l≥1 vl f l+1 , we know that the variation of
320
M. Bauer and D. Bernard
Gf is −Gf
l≥1 vl Ll .
Ann. Henri Poincar´e
Now
2 f (w) = vl Ll , Ln dwwm+1 n+2 f (w) 0 n l≥1
+
vl
l≥1
((l − n)Ll+n
n
c δl+n,0 (l3 − l)) 12
dwwm+1 0
f (w)2 . f (w)n+2
term involving the central charges we sum over n, then l and get For the m+1 dww f (w)2 v (f (w)). For the remaining terms, for fixed l we replace the 0 dummy variable n by n − l, leading to f (w)2 vl (2l − n)Ln dwwm+1 , f (w)n−l+2 0 n c 12
l≥1
which is the same as f (w)2 (f (w)v (f (w)) − (n + 2)v(f (w))) Ln dwwm+1 . f (w)n+3 0 n Finally f (w)2 (f (w)v (f (w)) − (n + 2)v(f (w))) vl Ll , Lm (f ) = Ln dwwm+1 f (w)n+3 0 n l≥1 c + dwwm+1 f (w)2 v (f (w)). 12 0 It is easily seen that this is nothing but dLm (f + εv(f )) , dε |ε=0 which shows that G−1 f Lm Gf
and
c 12
dwwm+1 Sf (w) + 0
n≥m
Ln
dwwm+1 0
f (w)2 , f (w)n+2
which coincide at f (z) = z, have the same tangent map. Convexity ensures that they coincide everywhere.
C
A few properties of Gf
The expansion of Gf in powers of the fm ’s has an important property that is already apparent in the expansion above. Let I = (i1 , i2 , . . . ) be a sequence of
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
321
non-negative integers with finitely many nonzero terms. Let Em be the sequence made of zeroes except for a single 1 in the m-th position, so that I = m im Em . We define |I| ≡ i (which we call degree), d(I) ≡ mi (which we call m m m m im and LI ≡ m Limm (with the convention that grading) I! ≡ im !, fI ≡ m fm L1 factors are on the utmost right, then L2 , and so on). Then we claim that Gf =
(−)|I| I!
I
fI (LI + lower order terms)
where “lower order terms” mean f -independent linear combinations of LJ ’s with |J| < |I| but d(J) = d(I). The same statement would be true if we had chosen the opposite convention to order the Lm ’s in LI . The statement that d(J) = d(I) is simply that a dilation on z multiplies Ll by λl but divides fl by the same factor. Alternatively, one can check that the factor m+1 f (w) f (w)n+2 that appears in eq. (2) is a polynomial in the fl ’s of grading 0 dww n − m. The proof that |J| < |I| is obtained by taking a commuting limit: we set fm ≡ εϕm and Λm ≡ εLm (think of ε as ). Then in the limit ε → 0 keeping the ϕm ’s fixed, on the one hand the Λm ’s commute, and on the other hand f (w) dwwm+1 f (w) n+2 = δn,m so that the differential system defining Gf reduces to 0 ∂Gf ∂ϕm
= −GΛm , with solution Gf = e−
m
ϕm Λm
. This implies that in the ε expan-
(−)|I| I I! ϕI (ΛI
+ O(ε)). But expressed in sion in terms of ϕm ’s and Λm ’s, Gf = terms of fm ’s and Lm ’s the result is ε-independent. This means that the coefficient of ϕI εk involves only ΛJ ’s with |J| = |I| − k. This concludes the proof. An analogous computation would show that G−1 f =
1 fI (LI + lower order terms). I! I
We can rephrase these results as follows: Gf
=
(−)|I| I
G−1 f
=
I!
LI (fI + higher order terms),
1 LI (fI + higher order terms), I!
(15) (16)
I
where “higher order terms” mean L-independent linear combinations of fJ ’s with |J| > |I| but d(J) = d(I). In particular the polynomials in the fm ’s that appear as coefficients of the LI ’s in the above expansions form a basis of the space of all polynomials in the fm ’s. These observations will be useful for the application to representation theory in Section 5.
322
M. Bauer and D. Bernard
Ann. Henri Poincar´e
We can also write down a general recursive formula. We define PI by Gf ≡ (−)|I| m+1 f (w) I I! fI PI and combinatorial coefficients CJ (m, n) by 0 dww f (w)n+2 ≡ (−)|J| m−n dw J J! fJ CJ (m, n). The integrand can we written as w w times a function l in which each fl is multiplied by z : CJ (m, n) = 0 unless d(J) = n− m. The partial
differential equations for Gf lead to difference equations for the PI ’s. One gets PK+Em =
I+J=K
K! CJ (m, m + d(J))PI Lm+d(J) . I!J!
One finds PEm = Lm , PEm +En = Lm Ln + (n + 1)Lm+n , . . . .
D Final steps for the proof of (8) We start from eq. (7), repeated here for convenience: PLn y +
c 12
0
f (w)n+1 − h dw 2 Py w f (w) 0 0 f (z)2 f (w)n+1 =− dw m+2 dzz m+1 Gf Ll y, x . w f (w) f (z)l+2 0 m=n 0
Sf (w) dwf (w)n+1 f (w)
l≥1
(z)2 Now, fix m and concentrate on l≥1 0 dzz m+1 ff(z) l+2 Gf Ll y, x. From the m+1 Lagrange formula, one can expand z f (z) in powers of f (z) as
z
m+1
f (z) =
k+1
f (z)
0
k≥m
Define hm (z) ≡ z
f (z) −
um+1 f (u)2 . f (u)k+2
m+1
du
k+1
f (z)
du 0
k, m≤k≤0
um+1 f (u)2 . f (u)k+2
(17)
By definition, hm (z) is a O(z 2 ) and its z expansion reads hm (z) =
z
j+1
fj−m (j − m + 1) −
j≥1
0 k=m
um+1 f (u)2 du f (u)k+2 0
f (v)k+1 dv j+2 v 0
.
On the other hand, by construction, hm (z) is such that dzz 0
m+1
f (z)2 = f (z)l+2
dz 0
hm (z)f (z) f (z)l+2
for
l = 1, 2, . . . .
(18)
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
323
so, using again eq. (2), l≥1
dzz m+1
0
=−
f (z)2 Gf Ll f (z)l+2
fj−m (j − m + 1) −
j≥1
E
0 k=m
um+1 f (u)2 du f (u)k+2 0
f (v)k+1 dv j+2 v 0
∂Gf . ∂fj
Two explicit computations
Let a and b be real positive numbers
E.1 Example 1: Two slits We define the hull Bb to be the segment ]i0, ib] and Aa the segment [ia, i∞[ in H. Assuming that 0 ≤ b < a ≤ ∞ we compute L(Aa , Bb ). We interpolate between the empty hull and Bb (resp. Aa ) by Bβ , β ∈]0, b] (resp. Aα , α ∈ [a, ∞[). To uniformize H\Bβ we take the map fBβ (z) = (z 2 +β 2 )1/2 and for H\Aα the map fAα (z) = (z −2 + α−2 )−1/2 . Observe that fBβ maps Aα to αβ Aγ where γ = (α2 − β 2 )1/2 while fAα maps Bβ to Bδ , where δ = (α2 −β 2 )1/2 . One checks that fBδ ◦ fAα = 1−b12 /a2 fAγ ◦ fBβ , so we get a commutative diagram by taking fAα,β = 1−b12 /a2 fAγ and fBβ,α = fBδ . 2
2
2
2
2
3(z +2γ ) 3(z +2(a −β )) Now SfAα,β (z) = SfAγ (z) = − 2(z 2 +γ 2 )2 so SfAa,β (z) = − 2(z 2 +a2 −β 2 )2 . On
the other hand L(Aa , Bb ) = −
d dβ fBβ
=
b
dβ 0
β fBβ
so vBβ (z) =
β z.
To resume,
dzvBβ (z)SfAa,β (z) =
b
dβ
dz
0
β 3(z 2 + 2(a2 − β 2 )) . z 2(z 2 + a2 − β 2 )2
The relevant z-integral encircles the singularity at 0 and no other, so b β L(Aa , Bb ) = 3 0 dβ a2 −β 2 . Finally 3 L(Aa , Bb ) = − log(1 − b2 /a2 ). 2
E.2 Example 2: A slit and a half-disc We keep the definitions above for Aa , Aα , α ∈ [a, ∞[) and fAα . But now Bb is the intersection of the disc of center 0 and radius b with H, and to interpolate between the empty hull and Bb we use the half-discs Bβ , β ∈]0, b]. To uniformize H\Bβ we choose the map fBβ (z) = z+β 2 /z. Observe that fBβ maps Aα to Aγ where now γ = (α2 − β 2 )/α. The Schwarzian derivative is insensitive to the precise normalization 3(z 2 +2(a2 −β 2 )2 /a2 ) of fAα,β , so we can compute it by using fAγ : SfAa,β (z) = − 2(z 2 +(a2 −β 2 )2 /a2 )2 .
324
M. Bauer and D. Bernard
d dβ fBβ
fBβ −
"
fB 2 −4β 2 β
z−
Ann. Henri Poincar´e
√
z 2 −4β 2
On the other hand = so vBβ (z) = , where the β β square root is defined to ensure the appropriate properties of vBβ : this vector field is holomorphic in H with negative imaginary part, real on the real axis away from the cut and satisfies the Schwarz reflexion principle. Hence # b z − z 2 − 4β 2 3(z 2 + 2(a2 − β 2 )2 /a2 ) L(Aa , Bb ) = dβ dz . β 2(z 2 + (a2 − β 2 )2 /a2 )2 0 The relevant z-integral encircles the cut [−2β, 2β] and no other singularity. We may compute it with the help of the residue formula, because the integrand is meromorphic in the unbounded component of the complement of the integration contour, regular at infinity but with double poles at z = ±i(a2 − β 2 )/a. The index is −1 for both, and the residue is the same as well. This leads to b dβ β 2 (β 2 + 2a2 ) L(Aa , Bb ) = 3 . a4 − β 4 0 β Finally L(Aa , Bb ) =
1 + b2 /a2 3 log . 4 (1 − b2 /a2 )3
We observe in these two examples that L(A, B) becomes singular when A and B have a contact. As expected L(A, B) is positive. Acknowledgments. We take this opportunity to warmly thank Wendelin Werner for many illuminating explanations on the probabilistic and geometric intuition motivating SLE constructions and Misha Gromov for his questions on finite conformal transformations in conformal field theory. Work supported in part by EC contract number HPRN-CT-2002-00325 of the EUCLID research training network.
References [1] M. Bauer, D. Bernard, SLE growth processes and conformal field theories, Phys. Lett. B543, 135–138 (2002). [2] M. Bauer, D. Bernard, Conformal field theories of stochastic Loewner evolutions, arXiv:hep-th/0210015, Commun. Math. Phys. 239, 493–521 (2003). [3] M. Bauer, D. Bernard, SLE martingales and the Virasoro algebra, arXiv:hepth/0301064, Phys. Lett. B557, 309–316 (2003). [4] A. Belavin, A. Polyakov, A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241, 333–380 (1984). [5] J. Cardy, Critical percolation in finite geometry, J. Phys. A25, L201–206 (1992).
Vol. 5, 2004
Conformal Transformations and the SLE Partition Function Martingale
325
[6] J. Cardy, Conformal invariance and percolation, arXiv:math-ph/0103018. [7] J. Cardy, Conformal invariance in percolation, self-avoiding walks and related problems, arXiv:cond-mat/0209638. [8] F. David, Mod. Phys. Lett. A5, 1019 (1990), R. Dijkgraaf, H. Verlinde, E. Verlinde, Loop equations and Virasoro constraints in nonperturbative 2-d quantum gravity, Nucl. Phys. B348 435 (1991), V. Kazakov, Mod. Phys. Lett. A4 2125 (1989). [9] J. Dixmier, Alg`ebres enveloppantes. Gauthier-Villars, Paris 1974; V.G. Kac and A.K. Reina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Adv. Series in Math. Phys., vol. 2, World Scientific, Singapore, 1987. [10] B. Duplantier, Conformally invariant fractals and potential theory, Phys. Rev. Lett. 84, 1363–1367 (2000). [11] B. Duplantier, Higher conformal multifractality, J. Stat. Phys. 110, 691–738 (2003). [12] R. Friedrich and W. Werner, Conformal fields, restriction properties, degenerate representations and SLE, C.R. Acad. Sci. Paris, Ser I Math, arXiv:math.PR/0209382; Conformal restriction, highest weight representations and SLE, arXiv:math-ph/0305061, to appear in Commun. Math. Phys. [13] I. Karatzas, S.E. Shreve, Brownian motion and stochastic calculus, GTM 113, Springer, (1991). [14] G. Lawler, Introduction to the Stochastic Loewner Evolution, URL http://www.math.duke.edu/∼jose/papers.html, and references therein. [15] G. Lawler, O. Schramm, W. Werner, Values of Brownian intersections exponents I: half-plane exponents, Acta Mathematica 187, 237–273 (2001), arXiv:math.PR/9911084; G. Lawler, O. Schramm, W. Werner, Values of Brownian intersections exponents II: plane exponents, Acta Mathematica 187, 275–308 (2001), arXiv:math.PR/0003156; G. Lawler, O. Schramm, W. Werner, Values of Brownian intersections exponents III: two-sided exponents, Ann. Inst. Henri Poincar´e 38, 109–123 (2002), arXiv: math.PR/0005294. [16] G. Lawler, O. Schramm, W. Werner, Conformal restriction: the chordal case, arXiv:math.PR/0209343. [17] G. Lawler, W. Werner, The Brownian loop soup, arXiv: math.PR/0304419.
326
M. Bauer and D. Bernard
Ann. Henri Poincar´e
[18] B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Stat. Phys. 34, 731–761 (1983). [19] L. Onsager, Phys. Rev. 65, 117 (1944), L. Onsager, Nuovo Cimento 6, supplement, 261 (1949), C.N. Yang, Phys. Rev. 85, 808 (1952). [20] S. Rhode, O. Schramm, Basic properties of SLE, and references therein, arXiv:math.PR/0106036. [21] O. Schramm, Israel J. Math., 118, 221–288 (2000). [22] S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C.R. Acad. Sci. Paris 333 239–244 (2001). [23] W. Werner, Lectures notes of the 2002 Saint Flour summer school. [24] W. Werner, private communication. [25] P.A. Wiegmann, private communication. Michel Bauer and Denis Bernard Service de Physique Th´eorique de Saclay CEA/DSM/SPhT Unit´e de recherche associ´ee au CNRS CEA-Saclay F-91191 Gif-sur-Yvette France email:
[email protected] email:
[email protected] Communicated by Vincent Rivasseau Submitted 13/06/03, accepted 21/10/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 327 – 346 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020327-20 DOI 10.1007/s00023-004-0171-y
Annales Henri Poincar´ e
Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization Ernst Binz, Reinhard Honegger and Alfred Rieckers Abstract. For an arbitrary (possibly infinite-dimensional) pre-symplectic test function space (E, σ) the family of Weyl algebras {W(E, σ)}∈R , introduced in a previous work [1], is shown to constitute a continuous field of C*-algebras in the sense of Dixmier. Various Poisson algebras, given as abstract (Fr´echet-) *-algebras which are C*-norm-dense in W(E, 0), are constructed as domains for a Weyl quantization, which maps the classical onto the quantum mechanical Weyl elements. This kind of a quantization map is demonstrated to realize a continuous strict deformation quantization in the sense of Rieffel and Landsman. The quantization is proved to be equivariant under the automorphic actions of the full affine symplectic group. The relationship to formal field quantization in theoretical physics is discussed by suggesting a representation dependent direct field quantization in mathematically concise terms.
1 Introduction The asymptotic correspondence between classical and quantum physics is intimately connected with the name of N. Bohr, but it was P.A.M. Dirac, who first shaped this general principle into a limiting equality of the scaled quantum commutator with the classical Poisson bracket. The various concise mathematical formulations for such a limit → 0 developed much later. Initiated especially by the seminal paper [2] there has been in the last years an extensive study of various forms of the so-called deformation quantization (let us here only mention [3] with its references), which associates a quantum mechanical algebraic structure with a rather arbitrary, finite-dimensional Poisson manifold P. More precisely, one considers in this context often a Poisson algebra P of functions on P, which is not – as in traditional Hilbert space quantization – mapped into a set of selfadjoint operators, but which acquires for itself a deformed non-commutative product, replacing the usual pointwise commutative product of functions. The C*-algebraic version of deformation quantization, the so-called strict deformation quantization, (cf., e.g., [4], [5], [6] [7], [8]), is a combination of the theoretical framework of algebraic quantum theory and the deformation concept. Based on [8] we introduce in Section 2 the concepts of strict deformation quantization and its strengthened version of a continuous quantization in a slightly generalized form. There, the classical Poisson algebra P is mapped, for each value of , into a C*-algebra A . That means, that the basic principles of algebraic quantum theory remain unchanged, whereas the transition between classical and quantum
328
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
theory is formulated in more concise terms. In the continuous (strict) quantization the family {A | ∈ I} is used to construct a continuous field of C*-algebras and especially strong continuity properties are imposed on the corresponding quantization map for the classical limit → 0. Under certain assumptions the inverse of the strict quantization map pulls back the C*-algebraic product and defines a deformed, non-commutative product in the original set of phase space functions. In this manner the connection to the deformation idea is made evident. Most examples and methods of strict deformation quantization treated in the literature are executed on the level of finite degrees of freedom, resp. of locally compact groups, avoiding general C*-algebraic concepts by the use of special Hilbert space representations. The point, in which the various ansatzes differ from each other, is mostly the choice of the classical function algebra. For certain finitedimensional Poisson manifolds, given by locally compact abelian groups, Rieffel has developed a strict deformation quantization, which starts with a rather large function algebra, comprising the almost periodic functions. He develops a mathematically concise version of the Moyal products by means of oscillatory integrals [9], [4], [5]. It is shown, that this coincides with Weyl’s famous quantization of phase space functions, cf., e.g., [8]. The work of Weaver [10] generalizes in some sense Rieffel’s construction from Rn -actions to infinite-dimensional Hilbert space actions. The image of the quantization map is there part of a special, Hilbert space dependent von Neumann algebra. The phase space, and thus the test function space given by the pre-dual, carries a non-degenerate symplectic form, expressed by the imaginary part of the scalar product. Our subsequent developments employ the very flexible and universal construction of the C*-Weyl algebra [11], [1]: To a general pre-symplectic test function space (E, σ) (a real vector space E equipped with an anti-symmetric bilinear form σ), which expresses the degrees of freedom of a physical system with possible superselection rules, we have associated for each ∈ R a C*-algebra W(E, σ). It is generated by abstract Weyl elements W (f ), f ∈ E, satisfying the Weyl relations W (f )W (g) = exp{− 2i σ(f, g)}W (f + g) , W (f )∗ = W (−f ) ,
∀f, g ∈ E .
(1.1)
The requirement that the representations of W(E, σ) reproduce all projective, unitary realizations of the vector group E, determines W(E, σ) uniquely. The appropriate C*-norm, to complete the linear hull of the W (f ), f ∈ E, is obtained by maximizing the values of the projectively positive-definite functions on E, as is shortly recapitulated in the present Section 3. The thus obtained observable algebra is such an extreme generalization of the simple Weyl algebra for non-degenerate σ (and = 0), commonly used in algebraic quantum field theory, that it covers also the case of classical fields (at value = 0 resp. σ = 0). It corresponds to the formal field quantization in so far as it constitutes the smallest mathematically sufficient structure which contains a set of basic elements. In regular Hilbert space representations the Weyl elements
Vol. 5, 2004
Field-theoretic Weyl Quantization
329
may, in fact, be replaced by the unbounded field operators in their role as basic elements, provided that additional conventions, concerning the ordering of field operator products, are introduced. For discussing quantization proper, we have to specify the classical Poisson algebra P, considered as a sub-*-algebra of W(E, 0) in our scheme. The elements of the purely algebraically introduced P are infinite series of Weyl elements, which converge in a certain Fr´echet-topology. In the more explicit representation of P in terms of phase space functions a locally convex vector space topology τ is introduced on E in Section 4, and its topological dual Eτ becomes a Poisson manifold. In this manner the Poisson algebra P can be realized as a *-algebra of differentiable functions, densely contained in the continuous almost periodic functions on Eτ . This is, in some sense, a much smaller function algebra than Rieffel’s, but it is used also for infinite-dimensional E and is directly connected with the classical field observables. Only at this stage we have completed the scenario for introducing a (strict) quantization of the Poisson manifold P ≡ Eτ . Since we have now at our disposal a family of C*-Weyl algebras W(E, σ), ∈ R, the most suggestive quantization map relates the classical Weyl elements W 0 (f ) ( = 0) with the quantum mechanical ones W (f ) ( = 0), under preservation of the test function. We investigate at first this so-called symmetrical or Weyl quantization in Section 5 in terms of a global continuous quantization. The proof of the continuity properties with respect to the Planck parameter requires more mathematical technicalities than the verification of the Dirac condition. The fact, that the latter essentially may be reduced to the Dirac condition for Weyl elements, belongs to the merits of the present approach. We find that the concise form of the field-theoretic Weyl quantization enjoys all of the desirable properties: It is not only a strict deformation quantization, but also a continuous quantization, distinguished by the norm-continuity of all products of quantized observables, and is related with a continuous field of C*-Weyl algebras; and it commutes with the affine symplectic actions in the pertinent observable algebras. In the final Section we indicate shortly the relationship to a direct Hilbert space quantization of the (smeared) fields, in order to clarify further the connection with usual quantum field theory.
2 Strict and continuous deformation quantization We introduce here a C*-algebraic version of deformation quantization as it has been developed by Rieffel and Landsman. We take for the values of the Planck parameter a subset I ⊂ R, which contains = 0, and for which I0 := I\{0} accumulates at 0. Definition 2.1 (Strict Deformation Quantization) A strict quantization (A , Q )∈I of the (complex) Poisson algebra (P, {., .}) consists for each value ∈ I of a linear, *-preserving quantization map Q : P → A , where A is a
330
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
C*-algebra with norm . , such that Q0 is the identical embedding of P into A0 , and such that for all A, B ∈ P the following conditions are satisfied: (a) [Dirac’s Condition] The -scaled commutator [X, Y ] := proaches the Poisson bracket as → 0:
i (XY
− Y X) ap-
lim [Q (A), Q (B)] − Q ({A, B}) = 0 .
→0
(b) [von Neumann’s Condition] In the limit → 0 one has the asymptotic product homomorphy: lim Q (A)Q (B) − Q (AB) = 0 .
→0
(c) [Rieffel’s Condition] I → Q (A) is continuous. The strict quantization (A , Q )∈I is called a strict deformation quantization, if Q is injective and Q (P) is closed with respect to the product of A for each ∈ I0 (the latter is equivalent for Q (P) being a sub-*-algebra of A ). The basic condition is Dirac’s condition, which is part of every physically relevant quantization prescription in one form or the other. The C*-algebraic framework has been proved valuable especially for systems with infinitely many degrees of freedom. Von Neumann’s condition is independent from Dirac’s condition in virtue of a different scaling. Rieffel’s condition reinforces the smoothness demand. Note that we did not include the richness condition that the *-algebraic span of Q (P) be dense in A , since the latter is always obtainable by restricting A to the smallest sub-C*-algebra containing Q (P). A strict (deformation) quantization of a Poisson algebra is, of course, nonunique. In the case of a strict deformation quantization one may stay, if one wishes to, in the space of classical observables P equipping it with the deformed, in general non-commutative product A· B := Q−1 (Q (A)Q (B)) for A, B ∈ P. This renders P into a *-algebra with product · , which is *-algebraically isomorphic to the image Q (P), a strategy, which has acquired much attention in the literature. Even stronger continuity conditions, which may be of interest in quantum field theory, are expressible by means of continuous fields of C*-algebras in the sense of J. Dixmier [12, Chapter 10]. We present the pertinent notions and results in a way, adapted to our context of -dependent quantization. For our subset I ⊆ R we denote by ∈I A the cartesian product of the family of C*-algebras A , ∈ I, which may be considered as a bundle over the base manifold I. The elements K of ∈I A are then sections I → K() ∈ A , which we write explicitly as [ → K()] ∈ ∈I A . If the *-algebraic operations (scalar multiplication, addition, product, *-operation) are taken pointwise, then the cartesian product ∈I A becomes a *-algebra. Clearly, for each ∈ I the point evaluation α defined by α (K) := K() is a *-algebraic homomorphism from ∈I A onto A .
Vol. 5, 2004
Field-theoretic Weyl Quantization
331
Definition 2.2 (Continuous Field of C*-Algebras) A continuous field of C*algebras ({A }∈I , K) consists of a sub-*-algebra K of ∈I A satisfying: (a) I → K() is continuous for all K ∈ K. This ensure the notion of continuous sections for the elements of K. (b) For each ∈ I the set {K() | K ∈ K} is dense in A . (c) Let K ∈ ∈I A . If for each 0 ∈ I and each ε > 0 there exists a H ∈ K and a neighborhood U0 of 0 so that K() − H() < ε ∀ ∈ U0 , then K ∈ K. If K = [ → K()] ∈ K and u : I → C is continuous, then it follows that [ → u()K()] ∈ K. Moreover, one has always {K() | K ∈ K} = A which strengthens (b). The next result from [12] is essential for the construction of continuous fields of C*-algebras (cf. Subsection 5.1). Lemma 2.3 Let D be a sub-*-algebra of ∈I A such that (a) and (b) of the above Definition are fulfilled (with K replaced by D). Then there exists a unique continuous field of C*-algebras ({A }∈I , K) with K such that D ⊆ K. Moreover, K consists of those K ∈ ∈I A which satisfy: For each 0 ∈ I and each ε > 0 there is an H ∈ D and a neighborhood U0 of 0 so that K() − H() < ε ∀ ∈ U0 . In order to associate a C*-algebra with a given continuous field of C*-algebras K, Dixmier restricts himself to the continuous sections K ∈ K for which I → K() vanishes at infinity. This sub-*-algebra K∞ becomes a C*-algebra if one introduces the C*-norm Ksup := sup∈I K() . There is, however, a larger C*algebra naturally associated with K and more fitting to our Weyl quantization, namely the C*-algebra Kb of the bounded continuous sections K ∈ K, i.e., with Ksup < ∞. In our notion of a continuous quantization we do not, however, specify such a global C*-algebra and work instead directly with the continuous field of C*algebras. (This generalizes slightly [8, Definition II.1.2.5], where K∞ is preselected). Definition 2.4 (Continuous Quantization) Let be given a Poisson algebra (P,{.,.}), a continuous field of C*-algebras ({A }∈I , K), and a linear, *-preserving map Q : P → K. Then ({A }∈I , K; Q) is called a continuous quantization of (P, {., .}), if the following conditions are valid: P ⊆ A0 , and α0 (Q(A)) = A for all A ∈ P, and furthermore, Dirac’s condition is fulfilled for Q := α ◦ Q. Q is denoted a global quantization map. If ({A }∈I , K; Q) is a continuous quantization, then (A , Q )∈I is a strict quantization (but in general not a strict deformation quantization). To prove this it remains only to check the validity of von Neumann’s condition: For A, B ∈ P
332
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
put K := Q(A)Q(B) − Q(AB) ∈ K. Then, when applying the point evaluation *-homomorphisms α , one concludes that K() = Q (A)Q (B) − Q (AB). Now von Neumann’s condition follows immediately from the continuity of → K() in Definition 2.2(a). For the converse reasoning, let us mention without proof: Suppose (A , Q )∈I to be a strict quantization of the Poisson algebra P with fulfilled richness condition, then there exists a continuous quantization ({A }∈I , K; Q) of P with Q = α ◦ Q ∀ ∈ I, if and only if I → P is continuous for each polynomial P of the Q (A), A ∈ P.
3 General C*-Weyl algebras We recapitulate here some results from [1] for later use. Let be given an arbitrary pre-symplectic space (E, σ) and an ∈ R. We start from the linear hull ∆(E, σ) := LH{W (f ) | f ∈ E}
(3.1)
of linearly independent W (f ), f ∈ E, called Weyl elements. Equipped with the twisted product and the *-operation according to the Weyl relations (1.1), the linear hull ∆(E, σ) becomes a *-algebra. Its identity is given by 1 := W (0), and every W (f ) is unitary. Let C(E, σ) be the convex set of the normalized, projectively positivedefinite functions C : E → C. Normalization means C(0) = 1, and projecn tive positive-definiteness is zi zj exp{ 2i σ(fi , fj )}C(fj − fi ) ≥ 0 for arbitrary i,j=1
n ∈ N, fj ∈ E, and zj ∈ C. The latter is a generalization of the notion of positivedefinite functions, familiar from harmonic analysis. We extend each C ∈ C(E, σ) to the unique linear functional ωC on ∆(E, σ) satisfying ωC ; W (f ) = C(f ) for all f ∈ E. By the definition of C(E, σ), every state on the *-algebra ∆(E, σ) is of type ωC with a unique C ∈ C(E, σ). On the *-algebra ∆(E, σ) there exists a unique C*-norm . , given by A = sup{
ωC ; A∗ A | C ∈ C(E, σ)} ,
(3.2)
such that every representation of ∆(E, σ) is . -continuous. Consequently, if . is a further C*-norm on ∆(E, σ) with A ≤ A for all A ∈ ∆(E, σ), then . = . . The Weyl algebra W(E, σ) is the completion of the *-algebra ∆(E, σ). with respect to the C*-norm .. It is simple, if and only if σ is non-degenerate and = 0. Every representation and each state of the *-algebra ∆(E, σ) extends . continuously to the completion W(E, σ), the extension of which are denoted by the same symbol. Thus the mapping C → ωC is an affine bijection from C(E, σ) onto the state space of W(E, σ). C is called the characteristic function of the state ωC .
Vol. 5, 2004
Field-theoretic Weyl Quantization
333
Because of the linear independence of the Weyl elements W (f ), f ∈ E, we may define the vector space norm .1 on ∆(E, σ) by n n zk W (fk )1 := |zk | k=1
(3.3)
k=1
for arbitrary n ∈ N, and zk ∈ C, but different fk ’s from E. The inequality A ≤ A1 ,
∀A ∈ ∆(E, σ) ,
(3.4)
1
is immediate. The .1 -completion ∆(E, σ) of ∆(E, σ) together with the .1 continuous extension of the *-algebraic operations from ∆(E, σ) turns out to be a Banach-*-algebra. The estimation (3.4) yields that the identical map from ∆(E, σ) onto itself extends .1 -. -continuously to a mapping from ∆(E, σ)
1
1
into W(E, σ). This mapping is injective, and thus ∆(E, σ) is a sub-*-algebra of W(E, σ) consisting of those A ∈ W(E, σ) which possess the unique decom∞ zk W (fk ) with different fk ’s from E and summable coefficients position A = k=1
zk ∈ C, i.e., A1 =
∞ k=1
|zk | < ∞. 1
The Banach-*-algebra ∆(E, σ) is the twisted group algebra of the discrete vector group E with respect to the multiplier exp{− 2i σ(f, g)} occurring in the Weyl relations (1.1). The Weyl algebra W(E, σ) is its enveloping C*-algebra and thus the twisted group C*-algebra of E (cf. also [13], [11], [14], and references therein). Finally we present two additional facts concerning characteristic functions and states. The first one is proved, e.g., in [15, Theorem 3.4]. The second one is shown in [16], it generalizes the well-known fact from harmonic analysis that the product of two positive-definite functions is positive-definite, too. Proposition 3.1 The following assertions are valid: (a) A Gaussian function E f → exp{− 41 || s(f, f )}, where s is a positive symmetric R-bilinear form on E, is an element of C(E, σ), if and only if σ(f, g)2 ≤ s(f, f ) s(g, g) for all f, g ∈ E. (b) Suppose = 1 + 2 , j ∈ R. If C1 ∈ C(E, 1 σ) and C2 ∈ C(E, 2 σ), then C1 C2 ∈ C(E, σ) (pointwise product), and hence there exists a unique state ω on W(E, σ) with characteristic function ω; W (f ) = C1 (f )C2 (f ) for all f ∈ E.
4 Classical field-theoretic setup For the classical case = 0 the Weyl relations (1.1) imply the product formula W 0 (f + g) = W 0 (f )W 0 (g) = W 0 (g)W 0 (f ) for the Weyl elements W 0 (f ), f ∈ E.
334
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e 1
Thus the *-algebra ∆(E, 0) from Eq. (3.1), the Banach-*-algebra ∆(E, 0) , as well as the C*-Weyl algebra W(E, 0) are commutative. We are going to elaborate, how a given non-trivial pre-symplectic form σ on E gives rise to the construction of a Poisson bracket {., .} on suitable sub-*-algebras P satisfying ∆(E, 0) ⊆ P ⊆ 1
∆(E, 0) . Since the Weyl elements W 0 (f ), f ∈ E are by assumption linearly independent, the ansatz {W 0 (f ), W 0 (g)} := σ(f, g)W 0 (f + g) ,
∀f, g ∈ E ,
(4.1)
leads to a well-defined Poisson bracket {., .} on the commutative *-algebra ∆(E, 0) by means of complex bilinear extension. Let us henceforth write A = f ∈E zf W 0 (f ) instead of A = k zk W 0 (fk ) with different fk ’s. If A ∈ ∆(E, 0), then we have zf = 0 up to finitely many f ∈ E. 1 By Section 3, A = f zf W 0 (f ) ∈ ∆(E, 0) with A1 = f |zf | < ∞ and with at most countably many non-vanishing coefficients zf ∈ C. Let be given a semi-norm κ on E. Then for each n ∈ N0 the definition 1
n
Pκn := {A ∈ ∆(E, 0) | Aκ < ∞} leads to a Banach space with respect to the norm n n zf W 0 (f )κ := κ(f )m |zf | . f ∈E
Consequently, Pκ∞ :=
m=0 f ∈E
Pκn turns out to be a Fr´echet space with respect to the
n∈N
metrizable locally convex Hausdorff topology τκ arising from the increasing system 1 n 0 of norms .κ , n ∈ N. For n = 0 we re-obtain Pκ0 = ∆(E, 0) and .κ = .1 . n n ∞ Obviously, ∆(E, 0) is .κ -dense in Pκ and τκ -dense in Pκ . Conversely, for fixed n ∈ N {∞} the spaces Pκn are in inverse-order-preserving correspondence with the semi-norms κ on E. Lemma 4.1 Pκn constitutes a sub-*-algebra of the commutative Banach-*-algebra 1 ∆(E, 0) for each n ∈ N {∞}. Furthermore: (a) For each n ∈ N it holds Aκ = A∗ κ and ABκ ≤ cn Aκ Bκ for all A, B ∈ Pκn with some constant cn ≥ 1 defined in Eq. (4.2) below. (Since cn > 1 for n ≥ 2, Pκn is only a Banach-*-algebra for a norm equivalent to n .κ .) n
n
n
n
n
(b) Pκ∞ is a Fr´echet-*-algebra (product and *-operation are [jointly] τκ -continuous).
Vol. 5, 2004
Field-theoretic Weyl Quantization
335
n Proof. Anκ = A∗ nκ is immediate κ(f ) = κ(−f since ), thus0 Pκ is invariant 0 under the *-operation. Let A = f uf W (f ) and B = g vg W (g) be arbitrary 1 elements of Pκn (where uf , vg ∈ C). Then AB = f,g uf vg W 0 (f + g) ∈ ∆(E, 0) . We show AB ∈ Pκn . From the semi-norm property κ(f + g) ≤ κ(f ) + κ(g) we obtain that n n m n ABκ ≤ κ(f + g)m |uf | |vg | ≤ κ(f ) + κ(g) |uf | |vg | m=0 f,g∈E
m=0 f,g∈E
n m
m m−k k = κ(f ) |u | κ(g) |v | f g f g k m=0 k=0
n
n
(4.2)
which yields AB ∈ Pκn . Now the rest is immediate.
≤ sup{( nk ) | k = 0, 1, . . . , n} Aκ Bκ < ∞ , =: cn
In order to extend our above Poisson bracket {., .} from ∆(E, 0) to suitable *-algebras Pκn , we suppose the existence of a semi-norm ς on E such that |σ(f, g)| ≤ c ς(f ) ς(g) ,
∀f, g ∈ E ,
(4.3)
for some constant c > 0. Especially, ς has to be a norm for non-degenerate σ. Theorem 4.2 With the notations introduced above it holds that n−1
{A, B}ς
n
n
≤ c cn−1 Aς Bς ,
∀A, B ∈ ∆(E, 0) ,
∀n ∈ N .
So the Poisson bracket {., .} extends continuously to the jointly continuous mapping Pςn × Pςn −→ Pςn−1 , (A, B) −→ {A, B} . (4.4) Thus (Pς∞ , {., .}) is a Poisson algebra with jointly τς -continuous Poisson bracket {., .}. Proof. We have {A, B} = f,g σ(f, g)uf vg W 0 (f + g) ∈ ∆(E, 0) for A = f uf 0 W 0 (f ) and B = g vg W (g) from ∆(E, 0) by Eq. (4.1). Estimation (4.3) and proceeding similarly to the proof of Lemma 4.1 ensures that n−1
{A, B}ς
≤
n−1
ς(f + g)m |σ(f, g)| |uf | |vg |
m=0 f,g∈E
n−1 m
m m−k+1 k+1 ≤c ς(f ) |u | ς(g) |v | f g f g k m=0 k=0 n−1 n n ≤ c sup{ k | k = 0, 1, . . . , n − 1} Aς Bς < ∞ . = cn−1 (see Eq. (4.2)) This yields the stated results.
336
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
The Poisson bracket {., .} may be realized indeed in terms of a bivector field. For this let be given a locally convex Hausdorff vector space topology τ on E. The topological dual Eτ is interpreted as the (flat) phase space manifold of our classical field theory. The commutative C*-Weyl algebra W(E, 0) is *-isomorphic to the C*-algebra of the almost periodic, σ(Eτ , E)-continuous functions on Eτ [1]. The Weyl elements W 0 (f ) are realized by the periodic functions W 0 (f ) : Eτ → C ,
F → exp{iF (f )} = W 0 (f )[F ] .
(4.5)
For A : Eτ → R the total differential dF A ∈ TF∗ Eτ = E, where the cotangent space TF∗ Eτ is equipped with the σ(Eτ , E)-topology, is given by dF A[G] := d dt A[F + tG]|t=0 for all G ∈ TF Eτ ≡ Eτ , F ∈ Eτ , provided existence. For a Cvalued function A on Eτ we put dF A := dF A1 + idF A2 with its real and imaginary parts A1 resp. A2 . Our above Poisson bracket {., .} now is given in terms of the constant bivector field F → ΣF arising from σ (more details are found in [17]), {A, B}[F ] = ΣF (dF A, dF B) := −σ(dF A1 , dF B1 ) − iσ(dF A1 , dF B2 ) − iσ(dF A2 , dF B1 ) + σ(dF A2 , dF B2 ) . Especially, Eq. (4.1) is reproduced for the periodic functions W 0 (f ) from Eq. (4.5). The presented Poissonian structure is independent of the chosen locally convex topology τ on E, or equivalently, from the phase space Eτ . Consequently, the only essential ingredients of the algebraized classical field theory are the presymplectic test function space (E, σ) and a semi-norm ς satisfying (4.3), the remaining C*- and Poisson algebraic structure is a functor.
5 Field-theoretic Weyl quantization We select, for given pre-symplectic space (E, σ) and semi-norm ς satisfying (4.3), the two cases P = ∆(E, 0) resp. P = Pς∞ for a Poisson algebra (P, {., .}), both being dense in the commutative, classical Weyl algebra W(E, 0). After having specified the observable algebras of the quantized systems (with possibly intrinsic classical observables, i.e., superselection rules) as the C*-Weyl algebras W(E, σ), = 0, we may now proceed to a quantization proper. Dirac’s original notion as well as its mathematical explication of a quantization in Section 2 indicate for our special case a linear, *-preserving correspondence Q : P → W(E, σ), which should display certain asymptotic properties for → 0. The most suggestive, but certainly not only, choice for Q is the prescription (5.1) Q ( k zk W 0 (fk )) := k zk W (fk ) , zk ∈ C , fk ∈ E , which is well defined because of the Weyl elements being linearly independent. We do not, however, study this quantization map directly but prefer the global point of view of a continuous quantization, which deals with all values of the Planck parameter simultaneously. A specific quantization is then gotten by fixing the value of . The demonstration of the correct → 0 asymptotics comes afterwards.
Vol. 5, 2004
5.1
Field-theoretic Weyl Quantization
337
Continuous field of C*-Weyl algebras
By means of the Weyl relations it is immediately checked that (5.2) ∆WF (E, σ) := LH{ [ → exp{−is}W (f )] | (s, f ) ∈ R × E} (R × E the cartesian product) constitutes a sub-*-algebra of ∈R W(E, σ). Lemma 5.1 The generating elements [ → exp{−is}W (f )], where (s, f ) ∈ R × E, of the *-algebra ∆WF (E, σ) are linearly independent. Proof. An arbitrary element K ∈ ∆WF (E, σ) possesses the form K=
p m
zj,l [ → exp{−isj,l }W (gj )] ,
(5.3)
j=1 l=1
where m, p ∈ N and zj,l ∈ C, and where the g1 , . . . , gm ∈ E are different, and for each j ∈ {1, . . . , m} the sj,1 , . . . , sj,p ∈ R are different. Since the Weyl elements are linearly independent, we have K = 0, if and only if 0 = l zj,l exp{−isj,l } for all ∈ R and all j = 1, . . . , m. But the maps → exp{−is}, s ∈ R, constitute an orthonormal base of the Hilbert space of almost periodic functions on R [18]. The Weyl relations imply that W (f )∗ W (g)∗ W (f )W (g) = exp{−iσ(f, g)}1 ,
∀f, g ∈ E .
Thus, the *-algebra ∆WF (E, σ) is *-algebraically generated by the sections [ → W (f )], f ∈ E, if and only if σ = 0. Theorem 5.2 (Continuous Weyl C*-Field) There exists a unique continuous field of C*-algebras ({W(E, σ)}∈R , K) such that [ → W (f )] ∈ K for each f ∈ E. Proof. Let σ = 0. We show that D := ∆WF (E, σ), which is algebraically generated by the [ → W (f )], fulfills the assumptions of Lemma 2.3. Since {K() | K ∈ ∆WF (E, σ)} = ∆(E, σ) is dense in W(E, σ) by Section 3, part (b) of Definition 2.2 is already fulfilled. We turn to part (a). We put lim sup G() := lim sup{G() | ∈ [−λ + 0 , λ + 0 ]\{0}}, for a →0
λ0
map R → G() ≥ 0, and analogously for lim inf G(). →0
Suppose to be given an arbitrary element K =
n k=1
zk [ → exp{−isk }
W (fk )] of ∆WF (E, σ) with different tuples (sk , fk ) ∈ R × E. Let EK := LHR {f1 , . . . , fn }. Then W(EK , σ) is a sub-C*-algebra of W(E, σ) [1], and it suffices to evaluate the C*-norm K() with the states on ∆(EK , σ). One easily constructs a positive symmetric R-bilinear form s on EK so that σ(f, g)2 ≤
338
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
s(f, f ) s(g, g) for all f, g ∈ EK (the construction of such a form s may fail for infinite-dimensional E, that is the reason why we go over to EK ). By Proposition 3.1(a), Cs ∈ C(EK , ( − 0 )σ) for all ∈ R, where Cs (f ) := exp{− 41 | − 0 | s(f, f )}. By part (b) of the same Proposition there exists for every C ∈ C(EK , 0 σ) and each ∈ R a unique state ωC on W(EK , σ) with the characteristic function s ωC ; W (f ) = C(f )C (f ). Eq. (3.2) yields ; K()∗ K() = ωC
n zj zk exp{i(sj − sk + 12 σ(fj , fk ))}
j,k=1 2
× C(fk − fj )Cs (fk − fj ) ≤ K() for all ∈ R. Taking the limit → 0 we obtain for every C ∈ C(EK , 0 σ) that 0 ; K(0 )∗ K(0 ) = lim ωC ; K()∗ K() ωC →0
=
lim inf ωC ; K()∗ K() →0
2
≤ lim inf K() , →0
from which with Eq. (3.2) we get the estimation K(0 )0 ≤ lim inf K() . →0
Because of the linear independence of the Weyl elements we conclude that m wj W 0 (gj ) ∈ ∆(E, 0 σ) with wj ∈ C but different gj ’s the for arbitrary A = j=1
inverse image of α0 is given by α−1 0 (A) = {K ∈ ∆WF (E, σ) | K(0 ) = A} p zj,l exp{−i0 sj,l } = wj for j = 1, . . . , m . = K from Eq. (5.3) | l=1
Defining KA :=
m j=1
wj [ → W (gj )] ∈ α−1 0 (A), we obtain an injective linear
mapping A → KA from ∆(E, 0 σ) into ∆WF (E, σ). Thus for K ∈ α−1 0 (A) we get the estimation K() ≤ KA () + K() − KA () ≤ KA () +
p m
|zj,l | |exp{−isj,l } − exp{−i0 sj,l }| .
j=1 l=1
Since lim sup |exp{−is} − exp{−i0 s}| = 0 for any s ∈ R it follows that →0
lim sup K() ≤ lim sup KA () . →0
→0
Interchanging the roles of K and KA finally yields that the expression lim sup K() = lim sup KA () =: A0 , →0
→0
∀K ∈ α−1 0 (A) ,
Vol. 5, 2004
Field-theoretic Weyl Quantization
339
depends only on A ∈ ∆(E, 0 σ). One immediately checks that A → A0 defines a further C*-norm on ∆(E, 0 σ). Thus for each K ∈ ∆WF (E, σ) we have the estimation K(0 )0 ≤ lim inf K() ≤ lim sup K() = K(0 )0 . →0
→0
By Section 3 .0 = .0 , and consequently, → K() is continuous at 0 . For σ = 0 the Weyl algebra field is constant, and for D one may take the constant sections [ → A] with A ∈ ∆(E, 0), or A ∈ W(E, 0). It is immediate that [ → exp{−is}W (f )] ∈ Kb for each tuple (s, f ) ∈ R × E for the C*-algebra Kb of bounded continuous sections for our continuous field of C*-Weyl algebras ({W(E, σ)}∈R , K). Consequently, ∆WF (E, σ) is a sub∗ *-algebra of Kb , the .sup -closure of which is denoted by CWF (E, σ). Consider arbitrary tuples (sk , fk ) ∈ R × E and coefficients zk ∈ C satis∞ fying k |zk | < ∞. By Section 3 it holds K() := zk exp{−isk }W (fk ) ∈ k=1
1
∆(E, σ) , even if some of the tuples (sk , fk ) coincide. Hence we obtain the section K = [ → K()] =
∞
zk [ → exp{−isk }W (fk )] ∈
k=1
Put Kn () :=
n
∈R W(E, σ) .
(5.4)
zk exp{−isk }W (fk ) ∈ ∆WF (E, σ) for all n ∈ N. Then for
k=1
each ε > 0 there exists an m ∈ N such that K() − Km () ≤
∞
|zk | < ε
k=m+1
∗ uniformly for ∈ R. Definition 2.2(c) implies K ∈ K. CWF (E, σ) being complete leads to:
Lemma 5.3 K from Eq. (5.4) (with
k
∗ |zk | < ∞) is an element of CWF (E, σ).
Because of the linear independence stated in Lemma 5.1, we may introduce in our *-algebra ∆WF (E, σ) the vector space norm .1 by n n zk [ → exp{−isk }W (fk )]1 := |zk | k=1
(5.5)
k=1
for different tuples (sk , fk ) ∈ R × E and arbitrary zk ∈ C and n ∈ N. The estimation Ksup ≤ K1 , ∀K ∈ ∆WF (E, σ) , already has been established in the above proof. So, similarly as for the Weyl algebra in Section 3 we perform the .1 -completion of ∆WF (E, σ), which is de1
noted by ∆WF (E, σ) . The .1 -continuous extension of the *-algebraic operations
340
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
1
in ∆WF (E, σ) equips ∆WF (E, σ) with the structure of a Banach-*-algebra. We 1
see that ∆WF (E, σ) consists of those sections K from Eq. (5.4), for which the tuples (sk , fk ) ∈ R × E are different, and for which K1 = k |zk | < ∞. From Lemma 5.3 we conclude that we have the inclusions 1
∗ ∆WF (E, σ) ⊆ ∆WF (E, σ) ⊆ CWF (E, σ) ⊆ Kb ,
(5.6)
in the sense of being sub-*-algebras. The identity is given by [ → 1 ], and for every (s, f ) ∈ R × E the continuous section [ → exp{−is}W (f )] is a unitary element. ∗ (E, σ) are deferred to [19]. Further investigations concerning the C*-algebra CWF
5.2
Strict deformation quantization
The continuous field of C*-Weyl algebras ({W(E, σ)}∈R , K) from the previous Subsection 5.1 leads to a definite global quantization mapping Q : ∆(E, 0) → ∆WF (E, σ), which is given by the complex linear extension of Q(W 0 (f )) := [ → W (f )] ,
∀f ∈ E .
Q is well defined and injective, since the Weyl elements W 0 (f ), f ∈ E, as well as the sections [ → exp{−is}W (f )], (s, f ) ∈ R × E, are linearly independent. It is immediately checked that Q is an isometry with respect to the two norms .1 . Thus Q extends .1 -.1 -continously to a linear, *-preserving, surjective isometry 1
1
Q : ∆(E, 0) −→ ∆WF (E, σ) .
(5.7)
By means of the point evaluation α from Section 2 for our continuous field of C*-Weyl algebras let us define for each ∈ R the quantization map 1
1
Q := α ◦ Q : ∆(E, 0) −→ ∆(E, σ) ⊆ W(E, σ) . Then, Q ( k zk W 0 (fk )) = k zk W (fk ) for k |zk | < ∞, which coincides with (5.1). The quantization map Q is a linear, *-preserving .1 -.1 -isometry from 1
1
∆(E, 0) onto ∆(E, σ) . Obviously, Q0 is just the identity for the classical case = 0. 1
Lemma 5.4 The jointly continuous bracket Pς1 × Pς1 → ∆(E, 0) , (A, B) −→ {A, B} from Theorem 4.2 (choose n = 1 in Eq. (4.4)) fulfills the Dirac condition from Definition 2.1(a) with respect to the Banach-*-algebra norms .1 on 1
∆(E, σ) respectively, i.e., lim [Q (A), Q (B)] − Q ({A, B})1 = 0 ,
→0
∀A, B ∈ Pς1 . 1
(5.8)
Furthermore, Q (P) is a .1 -dense sub-*-algebra of ∆(E, σ) and a . dense sub-*-algebra of W(E, σ) for every ∈ R.
Vol. 5, 2004
Field-theoretic Weyl Quantization
341
Proof. The Poisson bracket relations (4.1) and the Weyl relations (1.1) yield that [Q (W 0 (f )), Q (W 0 (g))] − Q ({W 0 (f ), W 0 (g)}) 1 = i exp{− 2i σ(f, g)} − exp{ 2i σ(f, g)} W (f + g) − σ(f, g)W (f + g)1 i exp{− i σ(f, g)} − 1 →0 exp{ 2 σ(f, g)} − 1 2 −→ 0 , = i − i − σ(f, g) where we used the differential limits
exp{± 2i σ(f, g)} − 1 d exp{± 2i σ(f, g)} lim = →0 d
=0
i = ± σ(f, g) . 2
The mean value theorem of differential calculus ensures exp{± i σ(f, g)} − 1 1 c 2 ≤ |σ(f, g)| ≤ ς(f ) ς(g) , 2 2 which leads for every f, g ∈ E and all 0 = ∈ R to the domination [Q (W 0 (f )), Q (W 0 (g))] − Q ({W 0 (f ), W 0 (g)}) ≤ 2c ς(f ) ς(g) . 1 0 0 1 Since for A = f uf W (f ) and B = g vg W (g) from Pς the dominant is summable, we may exchange the limit → 0 with f,g . . . by Lebesgue’s dominated convergence theorem in order to obtain Eq. (5.8). 1
∞ It remainsto prove that Q (A)Q (B) ∈ Q (Pς ) ⊆ ∆(E, 0) for A, B ∈ ∞ 0 0 Pς . For A = f uf W (f ) and B = g vg W (g), it is Q (A)Q (B) = Q (C) 1 i 0 with C ∈ ∆(E, 0) given by C := f,g uf vg exp{− 2 σ(f, g)}W (f + g). With i exp{− σ(f, g)} = 1 we conclude as in the proof of Lemma 4.1 that Cn ≤ ς n2 n cn Aς Bς for each n ∈ N, thus C ∈ Pς∞ . 1
The estimation A ≤ A1 for all A ∈ ∆(E, σ) implies the validity of Dirac’s condition for the C*-norms . . As in the above proof one may show that von Neumann’s condition holds with respect to the norms .1 for all A, B ∈ 1
∆(E, 0) , and thus for the C*-norms . . (The latter follows also automatically from our continuous field of C*-Weyl algebras by Section 2.) Summarizing we have shown the following results: Theorem 5.5 (Continuous Quantization) ({W(E, σ)}∈R , K; Q) constitutes a continuous quantization of the Poisson algebra (Pς∞ , {., .}). Theorem 5.6 (Strict Deformation Quantization) (W(E, σ), Q )∈R constitutes a strict deformation quantization of (P, {., .}), where P = ∆(E, 0) or P = Pς∞ . If the quantization maps Q resp. Q are restricted to an arbitrary sub-Poisson algebra P˜ with ∆(E, 0) ⊂ P˜ ⊂ Pς∞ (proper inclusions), then we obtain again a ˜ {., .}) continuous quantization Q, but only a strict quantization (Q )∈R of (P, ˜ (possibly Q (P) is not invariant under products, cf. Definition 2.1).
342
5.3
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
Affine symplectic actions
Let symp(E, σ) be the symplectic group on (E, σ) (its elements T are R-linear the character group bijections on E with σ(f, g) = σ(T f, T g) ∀f, g ∈ E), and E (semidirect of the vector group E. The affine symplectic group symp(E, σ) E product) consists of pairs (T, χ) with group multiplication (T1 , χ1 ) · (T2 , χ2 ) := (T1 T2 , χ2 (χ1 ◦ T2 )). For each ∈ R there exists an automorphic action (T, χ) → αT,χ on W(E, σ) with the *-automorphisms αT,χ satisfying αT,χ (W (f )) = χ(f )W (T f ) ∀f ∈ E, [1], a combination of gauge with Bogoliubov transformations [20]. The action is .1 -isometric on the invariant sub-*-algebras ∆(E, σ) 1
and ∆(E, σ) . For the classical case = 0, α0 constitutes even a group of Poisson automorphisms on (∆(E, 0), {., .}), which in general cannot be extended to the larger Poisson algebra Pς∞ . As an immediate consequence of the construction of the quantization maps Q in the previous Subsection we get the equivariance of the strict deformation quantization from Theorem 5.6 with respect to the family of actions (α )∈R , that is, αT,χ (Q (A)) = Q (α0T,χ (A)) ,
1
∀A ∈ ∆(E, 0) .
are τ -continuous for some locally convex If T ∈ symp(E, σ) and χ ∈ E Hausdorff topology τ on E, then the dual operator T to T acts bijectively on Eτ , and there exists a G ∈ Eτ with χ(.) = exp{iG(.)} [21]. One easily deduces that the *-automorphism α0T,χ is the pullback of the affine symplectic diffeomorphism F → T F + G on the phase space manifold Eτ , i.e., α0T,χ (A)[F ] = A[T F + G] ∀F ∈ Eτ , where A ∈ W(E, 0) is considered as an almost periodic function on Eτ as in Section 4.
6 Direct field quantizations For = 0 one defines in a regular representation Π of W(E, σ) the field operators by differentiation with respect to the parameter t ∈ R [20]: d , f ∈E. (6.1) ΦΠ (f ) ≡ ΦΠ (f ) := −i Π (W (tf )) dt t=0 The unbounded, selfadjoint field operators may differ essentially from each other in the various representations of W(E, σ). In the GNS representation over an ordered state (e.g., a condensed boson state [20] or macroscopic coherent state [22]) they may even exhibit a classical part in addition to the quantum mechanical part. This clearly transcends the purely algebraic regime of the theory. Let us indicate, how our algebraic quantization theory provides, nevertheless, also a strategy for a representation dependent Weyl quantization, leading eventually to the quantization of field expressions.
Vol. 5, 2004
Field-theoretic Weyl Quantization
343
For a family Π ≡ (Π )∈R of regular, faithful representations Π of the W(E, σ) the Π-dependent quantization mappings may be defined by the linear and .1 -continuous extension of 0 QΠ (W (f )) := Π (W (f )) ,
= 0, ∀f ∈ E .
(6.2)
In virtue of the norm preservation of any faithful representation, we may transcribe the previous abstract results into the language of operator quantizations, as may be shortly indicated. In a completely analogous manner as before we introduce a continuous family ({Π (W(E, σ))}∈R , KΠ ) of represented C*-Weyl algebras and a global quantization map Qw Π : P → KΠ . Corollary 6.1 Let Π ≡ (Π ) be an arbitrary family of regular, non-degenerate, and faithful representations Π of W(E, σ), 0 = ∈ R. Then the following assertions are valid: (a) ({Π (W(E, σ))}∈R , KΠ ; Qw Π ) constitutes a continuous quantization of the Poisson algebra (P, {., .}). (b) (Π (W(E, σ)), Qw Π, )∈R constitutes a strict deformation quantization of (P, {., .}). In order to deal with the unbounded fields it is certainly desirable to make the representations from the family Π compatible with each other, e.g., by setting Π := Π≡1 ◦ β , where β is a *-isomorphism from W(E, σ) onto W(E, σ), = 0. The latter may be realized, e.g., by β (W (f )) = W 1 (T f ) for an R-linear bijection T on E with σ(T f, T g) = σ(f, g), [17]. In the almost periodic function realization one has for the classical field prodn ucts Φ0 (f1 ) . . . Φ0 (fn ) = (−i)n ∂t1∂...∂tn W 0 ( k tk fk )|tk =0 . Our presented version of the Weyl quantization suggests the extension of QΠ , = 0, from (6.2) to unbounded field polynomials by the linear extension of n 0 ∂n Π 0 0 n Π Q W ( tk fk ) Q (Φ (f1 ) . . . Φ (fn )) := (−i) ∂t1 . . . ∂tn k=1 t =...=tn =0 1 (6.3) n n ∂ n = (−i) Π W ( tk fk ) , ∂t1 . . . ∂tn k=1
t1 =...=tn =0
n ∈ N and fk ∈ E arbitrary, with the tk being real parameters. Since there are used the derivatives of the quantized (represented) Weyl operators, the quantization map for field polynomials given in Eq. (6.3) is (at least in quantum optics) still called ‘Weyl quantization’ (or ‘symmetric quantization’). There are in use, however, further quantization prescriptions for fields, here sym˜ Π (Φ0 (f )) = Φ (f ) bolized by a tilde, which satisfy the first quantization step Q Π for every f ∈ E in an unchanged manner. The quantization of a field monomial is, however, of the general form 0 0 ˜Π Q (Φ (f1 ) . . . Φ (fn )) = Pn (f1 , . . . , fn ) ,
344
E. Binz, R. Honegger and A. Rieckers
Ann. Henri Poincar´e
where the choice of the real-multilinear operator-valued Pn , n ≥ 2, determines ˜ Π . A normally ordered monomial in the creation the special kind of quantization Q and annihilation operators, introduced by means of a complex structure, is just a special real-multilinear operator expression of the fields. It is elaborated in [23], how the various operator orderings define direct field quantizations. They all lead back to a decorated Weyl quantization of the form 0 Qw Π, (W (f )) := w(, f )Π (W (f )) ,
∀f ∈ E ,
(6.4)
where the w(, f ) are certain numerical factors. With some modifications of the foregoing arguments it is shown in [23] that these are again strict and continuous deformation quantizations, which all of them refer to the described continuous field of C*-Weyl algebras and which are equivalent, in the sense of [8], to the Weyl quantization. Acknowledgment. This work has been supported by the Deutsche Forschungsgemeinschaft.
References [1] E. Binz, R. Honegger, and A. Rieckers, Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic form, Preprint Mannheim, T¨ ubingen, 2003. [2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerovicz, and D. Sternheimer, Deformation theory and quantization, J. Oper. Th. 3, 237–269 (1980). [3] M. DeWilde and P.B.A. Lecompte, Existence of star-products and of formal deformations of a Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7, 487–496 (1983). [4] M.A. Rieffel, Deformation quantization for actions of Rd , Mem. Amer. Math. Soc. 106 (1993). [5] M.A. Rieffel, Quantization and C*-algebras, In R.S. Doran, editor, C*Algebras: 1943–1993, pages 67–97. Contemp. Math. 167, Providence, RI, Amer. Math. Soc., 1994. [6] M.A. Rieffel, Questions on quantization, Berkeley, quant-ph/9712009, 1998. [7] N.P. Landsman, Strict quantization of coadjoint orbits, Amsterdam, mathph/9807027, 1998. [8] N.P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer, New York, 1998.
Vol. 5, 2004
Field-theoretic Weyl Quantization
345
[9] J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45, 99–124 (1949). [10] N. Weaver, Deformation quantization for Hilbert space actions, Commun. Math. Phys. 188, 217–232 (1997). [11] J. Manuceau, M. Sirugue, D. Testard, and A. Verbeure, The smallest C*algebra for canonical commutation relations, Commun. Math. Phys. 32, 231– 243 (1973). [12] J. Dixmier, C*-Algebras, North-Holland, Amsterdam, 1977. [13] C.M. Edwards and J.T. Lewis, Twisted group algebras I, II, Commun. Math. Phys. 13, 119–141 (1969). [14] H. Grundling, A group algebra for inductive limit groups. Continuity problems of the canonical commutation relations, Acta Appl. Math. 46, 107–145 (1997). [15] D. Petz, An Invitation to the Algebra of Canonical Commutation Relations, volume 2 of Leuven Notes in Mathematical and Theoretical Physics, Leuven University Press, Leuven Belgium, 1990. [16] R. Honegger and A. Rieckers, Partially classical states of a Boson field, Lett. Math. Phys. 64, 31–44 (2003). [17] E. Binz, R. Honegger, and A. Rieckers, Field-theoretic Weyl quantization of large Poisson algebras, Preprint Mannheim, T¨ ubingen, 2003. [18] F. Riesz and B. Sz.-Nagy, Vorlesungen u ¨ber Funktionalanalysis, Deutscher Verlag der Wissenschaften, Berlin, 1982.
VEB
[19] E. Binz, R. Honegger, and A. Rieckers, Infinite dimensional Heisenberg group algebra and field-theoretic deformation quantization, Preprint Mannheim, T¨ ubingen, 2003. [20] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, volume II, Springer-Verlag, New York, 1981. [21] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I, II, Springer-Verlag, New York, 1963, 1970. [22] R. Honegger and A. Rieckers, The general form of non-Fock coherent boson states, Publ. RIMS Kyoto Univ. 26, 397–417 (1990). [23] R. Honegger and A. Rieckers, Some continuous field quantizations, equivalent to the C*-Weyl quantization. to appear in Publ. RIMS Kyoto Univ., 2004.
346
E. Binz, R. Honegger and A. Rieckers
Ernst Binz Institut f¨ ur Mathematik und Informatik Universit¨ at Mannheim D-68131 Mannheim Germany email:
[email protected] Reinhard Honegger Institut f¨ ur Mathematik und Informatik Universit¨ at Mannheim D-68131 Mannheim Germany and Institut f¨ ur Theoretische Physik Universit¨ at T¨ ubingen D-72076 T¨ ubingen Germany email:
[email protected] Alfred Rieckers Institut f¨ ur Theoretische Physik Universit¨ at T¨ ubingen D-72076 T¨ ubingen Germany email:
[email protected] Communicated by Joel Feldman Submitted 07/10/03, accepted 07/11/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e
Ann. Henri Poincar´e 5 (2004) 347 – 379 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020347-33 DOI 10.1007/s00023-004-0172-x
Annales Henri Poincar´ e
Density of States and Thouless Formula for Random Unitary Band Matrices Alain Joye Abstract.We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can be considered as a unitary version of the Anderson model. It is also related with orthogonal polynomials on the unit circle. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density.
1 Introduction The stability of quantum dynamical systems generated by time periodic Hamiltonians is sometimes characterized by means of the spectral properties of the corresponding unitary evolution operator over a period, also called monodromy operator, see [Be, Ho1, Co3]. Unfortunately, even for this relatively simple timedependence, except for certain specific models, e.g., [Co2, DF, Bo], it is rarely the case that one has enough information about the actual monodromy operator so that a complete spectral analysis can be performed. Therefore, one resorts to different approximation techniques in some specific regimes to say something about the spectrum. For example, KAM inspired techniques, see, e.g., [Be, Co1, DS, ADE, DLSV, GY], or adiabatic related approaches, see, e.g., [Ho2, Ho3, Ho4, N1, J, N2], have been used to tackle this problem. In case the complexity of the monodromy operator is important enough to forbid of a complete description of it, one may resort to a statistical modelization. It is the case in particular in the study of the quantum dynamics of electrons confined to a ring threaded by a time-dependent magnetic flux, see, e.g., the paper [BB] and references therein. A modelization of this dynamics by means of an effective random monodromy operator taking into account the details of the metallic structure of the ring is considered and tested numerically in [BB]. We refer the reader to this paper and [BHJ] for a more detailed account of the construction of the monodromy operator. Motivated by this approach, the spectral analysis of a class of random and deterministic unitary operators, which contains the above monodromy operator, is performed in [BHJ]. The main characteristics of these unitaries is that, when expressed as matrices in some basis, they display a band structure: more precisely they are five-diagonal. The coefficients of the matrix are determined by an infinite
348
A. Joye
Ann. Henri Poincar´e
set of triples {rk , αk , θk }k∈Z , where rk ’s are reflection coefficients in ]0, 1[ and αk ’s and θk ’s are phases. For example, in the statistical modelization of the physical situation mentioned above, the phases are considered as random, whereas the reflection coefficients are deterministic. While the construction of the set of unitaries studied in [BHJ] is patterned after the above-mentioned physical model, we believe it contains sufficiently many parameters to be useful for a wider class of problems. Another motivation in favor of the spectral analysis of such unitary operators stems from the recent paper [CMV] where it is shown that certain infinite matrices associated with the construction of orthonormal polynomials on the unit circle display the same five-diagonal structure as our set of monodromy operators. Under certain conditions, these matrices define unitary operators which actually form a subset of those considered in [BHJ]. The authors of [CMV] show that these matrices are to orthogonal polynomials with respect to a measure on the circle what Jacobi matrices are to orthogonal polynomials with respect to a measure on the real line. Orthogonal polynomials on the circle are determined by an infinite set of complex numbers {ak }k∈N such o recurrence that |ak | < 1, called reflection coefficients, through the so-called Szeg¨ relations, see, e.g., [G] or [BGHN]. And indeed, we will see that |ak | = rk , for all k ∈ N. Therefore, once given the expression of the five-diagonal matrix in terms of these reflection coefficients, the orthogonality measure on the circle coincides with the spectral measure of the corresponding unitary operator. These operators are further shown in [CMV] to be unitarily equivalent to unitary operators introduced almost ten years ago in [GT] for the study of the same orthonormal polynomials on the unit circle. The matrix form of the latter operators displays a different structure, namely that of a Hessenberg matrix: it has zero coefficients for indices i, j when i ≥ j − 1 only. Although more complicated, this structure can allow for operator theoretical approaches of orthogonal polynomial on the circle as, e.g., in [GT] or [GNV]. Note in particular that in [GT], properties of random polynomials defined by means of random reflection coefficients ak are investigated through the corresponding random unitary operator, whereas some of the perturbative analyses performed in [GNV] and [BHJ] bear strong resemblance. Nevertheless, we emphasize that the operators under consideration in [BHJ] and the present paper are more general than those constructed in [GT] and [CMV] and therefore their spectral analysis is richer. In particular in the random case, the way randomness appears in the coefficients of the matrix elements may lead to different characteristics of the spectral measure due to the availability of one more random variable. The goal of the present paper is to pursue the analysis of such random unitaries in the random setting considered in the paper [BHJ]: the phases (αk , θk ) are random variables and the reflection coefficients rk are all set to r ∈]0, 1[. This means that the phases of the matrix elements of the five-diagonal operators are random whereas the deterministic moduli depend on the parameter r only. Hence, if the phases are all set to zero, what we will call the “free case”, the unitary
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
349
operator depends on the reflection coefficient r ∈]0, 1[. Note that, specializing to the (random) orthogonal polynomials setting, this means we consider cases with |ak | = r for all k’s whereas the argument of the ak ’s are random. Also the free case is linked to the so-called Geronimus polynomials, constructed by means of constant (complex) reflection coefficients ak = a ∈ C, for all k. However, while the analysis of [BHJ] focused on spectral issues, i.e., proving singularity of the almost sure spectrum by means of a unitary version of the Ishii-Pastur theorem and the positivity of the Lyapunov exponent obtained via Furstenberg’s Theorem, the main object of the present study is the density of states measure and its links with the corresponding Lyapunov exponent. The Lyapunov exponent here is of course characterizing the asymptotic behavior of generalized eigenvectors of the unitary operator. More precisely, expressing the density of states as the density of eigenvalues of a series of unitary operators restricted to “boxes”, we are able to state this relation as what is known as a Thouless formula. This formula allows to compute the Lyapunov exponent by means of the density of states and to recover the a.c. component of the density of states measure by means of a derivative of the Lyapunov exponent. A consequence of our version of Thouless formula is the extension of some results of [BHJ] providing, in particular, an explicit value of the Lyapunov exponent in these cases. We also prove the validity of the Thouless formula for the deterministic free case, by explicit computations of the relevant quantities. When applied to the orthogonal polynomials setting, the existence of the density of states measure can be expressed as the determination of a sequence of random polynomials with a distribution of zeros converging to a measure whose support is the support of the orthogonality measure, almost surely. These polynomials are associated with the random orthogonal polynomials, but they do not coincide with them as the zeros of the former are, by construction, on the unit circle whereas those of the latter lie strictly in the unit disk. Such polynomials are also constructed in [GT] by suitable truncations of the Hessenberg matrix considered. Our Thouless formula relates the potential of the density of states measure, see, e.g., [SaT], [StT] for these notions, with the Lyapunov exponent. Actually, the Lyapunov exponent is essentially the limit of the potentials of the distributions of zero of the random polynomials mentioned above and the density of states is the equilibrium measure in the external field given by the Lyapunov exponent, see below. The existence of the limit almost surely is a consequence of the ergodic properties of the phase distributions. Let us also note here that a Thouless formula is proven for the unitary random operator studied in [GT]. The Lyapunov exponent there characterizes the asymptotics of the difference equation corresponding to the Szeg¨o relations associated with random complex ak ’s. In the second part of the paper, we further assume that some natural linear combination of the original phases {ηk } are i.i.d. random variables, in order to take advantage of the analogy of our unitary matrices with the one-dimensional discrete Schr¨odinger operator. In that case, we characterize the support of the density of states in terms of that of the distribution of the ηk ’s. Finally, we provide
350
A. Joye
Ann. Henri Poincar´e
an effective criterion ensuring analyticity of the integrated density of states in terms of the exponential decay rate of the Fourier coefficients of the distribution of these phases. This result relies on some kind of propagation estimates for the free evolution. The above-mentioned assumption on the phases makes (αk , θk ) correlated random variables. In particular, in the orthogonal polynomials language, this means that when the phase of each reflection coefficient ak (of constant modulus) is given by a sum of k i.i.d. random phases, the almost sure support of the random orthogonality measure can be determined. The plan of the paper is as follows. Section 2 is devoted to the definition of the model and its basic properties. In particular, the link with the constructions of [CMV] to describe orthogonal polynomials on the unit circle is recalled there. The density of states is introduced in the next section and Thouless formula is proven in Section 4. The statements about the support of the density of states and its analyticity properties are made in Section 5, whereas an appendix contains some technical items. The main results will be expressed in the general framework described above. We shall content ourselves with commenting on their translation in the orthogonal polynomial language, where appropriate, except in Section 4 where a little bit more material about potential theory is provided.
2 The model We present here the unitary matrices we will be concerned with and recall some of their basic properties to be used later. The unitary operator we consider has the following explicit form in the canonical basis {ϕk }k∈Z of l2 (Z) Uω ϕ2k
Uω ϕ2k+1
ω
ω
= irte−iη2k ϕ2k−1 + r2 e−iη2k ϕ2k ω ω + irte−iη2k+1 ϕ2k+1 − t2 e−iη2k+1 ϕ2k+2 ω
ω
= −t2 e−iη2k ϕ2k−1 + itre−iη2k ϕ2k ω ω + r2 e−iη2k+1 ϕ2k+1 + irte−iη2k+1 ϕ2k+2 ,
(2.1)
for any k ∈ Z. According to [BHJ], the random phases {ηkω }k∈Z are functions of some physically relevant i.i.d. random variables {(θkω , αω k )}k∈Z on the torus given by ω ω + αω (2.2) ηkω = θkω + θk−1 k − αk−1 , for all k ∈ Z and the coefficients r, t ∈]0, 1[ are interpreted as reflection and transition coefficients linked by r2 + t2 = 1. We will identify the operator and its matrix representation (2.1). Let us recall that these parameters are assumed to be different from their extreme values 0 and 1, because in case r = 1 ⇐⇒ t = 0 the operator Uω is diagonal and if r = 0 ⇐⇒ t = 1, it is unitarily equivalent to the direct sum of two shifts. Let us finally mention that our Uω is a particular case
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
351
of the construction in Section 2 of [BHJ] that we briefly recall below, in order to make contact with the matrices considered in [CMV].
2.1
Link with orthogonal polynomials
Consider the set of 2 × 2 unitary matrices defined for any k ∈ Z by rk e−iαk itk −iθk Sk = e , itk rk eiαk
(2.3)
parameterized by αk , θk in the torus T and the real parameters tk , rk , the reflection and transition coefficients, linked by rk2 + t2k = 1. Then, let Pj be the orthogonal projector on the span of ϕj , ϕj+1 in l2 (Z), and let us introduce Ue , Uo two 2 × 2 block diagonal unitary operators on l2 (Z) defined by Ue = P2k S2k P2k and Uo = P2k+1 S2k+1 P2k+1 . (2.4) k∈Z
k∈Z
In matrix representation in the canonical basis, . .. S−2 Ue = S0 S2
..
(2.5)
.
and similarly for Uo , with S2k+1 in place of S2k . Note that the 2 × 2 blocks in Ue are shifted by one with respect to those of Uo along the diagonal. The unitary operator (2.6) U = Uo Ue coincides with (2.1) in case tk = t ⇐⇒ rk = r, for any k ∈ Z. Actually, a supplementary phase factor appears in the off-diagonal elements of all Sk ’s in the original definition of [BHJ]. We omit it here, as this phase is shown to be irrelevant in the spectral analysis of U , see Lemma 3.2 in [BHJ]. Without entering into the details, orthogonal polynomials on the unit circle with respect to a measure µ are determined
∞by a set of ak ’s such that |ak | < 1 for all k ∈ N, and we shall assume that k=0 |ak | = ∞, which is equivalent to saying that the corresponding Hessenberg matrix is the matrix representation of a unitary operator, [GT], Lemma 2.2. The equivalent five-diagonal matrix F of [CMV] described below is unitary as well. This matrix is constructed in the same way as (2.6) is, by means of blocks of the type (2.3) for k ≥ 0 of the form iγk −i(π/2−γk ) 2 −|a |a |e 1 − |a | |e i 1 − |ak |2 k k k = −i Θk = 1 − |ak |2 |ak |e−iγk i 1 − |ak |2 |ak |ei(π/2−γk ) (2.7)
352
A. Joye
Ann. Henri Poincar´e
where ak = |ak |eiγk , see Section 3 of [CMV]. This corresponds to the particular choices (2.8) θk = π/2, αk = π/2 − γk , rk = |ak |. The definition of F is supplemented by particular “boundary conditions” at zero of the type (3.5) described below, as it is infinite in one direction only. One of the main properties of the matrix F shown in [CMV] is that the determinant of its principal n×n submatrices coincides with the n-th (monic) orthogonal polynomial, as is also true for the corresponding Hessenberg matrix. This property makes the analogy between Jacobi matrices and such F matrices all the more striking. Note that despite the fact that the above matrix is infinite in one direction only whereas ours is infinite in both directions, a “duplication procedure” described in Section 3 of [BHJ] allows to go from the former to the latter case modulo a finite rank perturbation. Hence claims about the spectrum of the doubly infinite matrix also hold for the previous matrix, modulo Birman-Krein’s theorem on finite rank perturbations and multiplicity considerations. From now on, we shall stick to doubly infinite matrices and we further make the choice rk = r ∈]0, 1[, for all k ∈ Z.
2.2
Ergodic properties
More precisely, let us introduce a probabilistic space (Ω, F , P), where Ω is identified with {TZ }, T being the torus, and P = ⊗k∈Z Pk , where P2k = P0 and P2k+1 = P1 for any k ∈ Z are probability distributions on T and F the σ-algebra generated by the cylinders. We introduce a set of random vectors on (Ω, F , P) given by βk = (θk , αk ) : Ω → T2 , k ∈ Z, θkω = ω2k , αω k = ω2k+1 .
(2.9)
The random vectors {βk }k∈Z are thus i.i.d on T2 . We denote by Uω the random unitary operator corresponding to the random infinite matrix (2.1). In analogy with Jacobi matrices describing the discrete Schr¨ odinger equation, we will also denote the vector ϕk by the site k, k ∈ Z. Introducing the shift operator S on Ω by S(ω)k = ωk+2 , k ∈ Z,
(2.10)
we get an ergodic set {S j }j∈Z of translations. With the unitary operator Vj defined on the canonical basis of l2 (Z) by Vj ϕk = ϕk−2j , ∀k ∈ Z,
(2.11)
we observe that for any j ∈ Z US j ω = Vj Uω Vj∗ .
(2.12)
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
353
Therefore, our random operator Uω is a an ergodic unitary operator. Now, general arguments on the properties of the spectral resolution of ergodic operators Eω (∆), where ∆ is a Borel set of the torus T, ensure that this projector is weakly measurable, as well as Eωx (∆) = Pωx Eω (∆), where x = p.p., a.c. and s.c., denote the pure point, absolutely continuous and singular continuous components, see [CL], chapter V. The analysis performed in [BHJ] for the case where {(θkω , αω k )}k∈Z are uniformly distributed on the torus shows that the a.c. component of the spectrum of Uω is almost surely empty.
2.3
Lyapunov exponent
Let us proceed by recalling some facts concerning the Lyapunov exponent. It is shown in [BB] and [BHJ] that generalized eigenvectors defined by Uω ψ = eiλ ψ, ψ= ck ϕk , ck ∈ C, λ ∈ C
(2.13)
k∈Z
in our unitary setting can be computed by means of 2 × 2 transfer matrices due to the structure of the matrix Uω . They are such that for all k ∈ Z, ([BHJ]) c2k−2 c2k = T (k) (2.14) c2k+1 c2k−1 where the randomness lies in the phases ηk (λ) ≡ ηkω (λ) defined by ηk (λ) = ηk + λ,
(2.15)
and T (k)11 T (k)12 T (k)21 T (k)22
= −e−iη2k−1 (λ) (2.16)
r −iη2k−1 (λ) = i e −1 t
r = i ei(η2k (λ)−η2k−1 (λ)) − e−iη2k−1 (λ) t
1 r2 = − 2 eiη2k (λ) + 2 ei(η2k (λ)−η2k−1 (λ)) + 1 − e−iη2k−1 (λ) . t t
Note the properties T (k) ≡ T (η2k (λ), η2k−1 (λ)) i(η2k −η2k−1 )
(2.17)
is independent of λ. whereas det T (k) = e Therefore, knowing, e.g., the coefficients (c0 , c1 ), we compute for any k ∈ N, c0 c0 c2k = T (k) . . . T (2)T (1) ≡ Φ(k) c2k+1 c1 c1 c−2k c0 c = T (−k + 1)−1 . . . T (−1)−1 T (0)−1 ≡ Φ(−k) 0 . (2.18) c−2k+1 c1 c1
354
A. Joye
Ann. Henri Poincar´e
The dynamical system at hand being ergodic and the determinant of the transfer matrices being of modulus one, we get the existence of a deterministic Lyapunov exponent γ(eiλ ), for any λ ∈ C, such that 1 ln Φ(k) = γ(eiλ ) a.s. k→±∞ |k| lim
(2.19)
Writing eiλ = z ∈ C\{0}, we also know from classical arguments, see, e.g., [CFKS], that γ is a subharmonic function of z.
3 Density of states Following the standard approach in the self-adjoint case, we start by a definition of the density of states by averaging over the phases and invoking the Riesz-Markov theorem. Then we relate the density of states with alternative definitions in terms of the density of eigenvalues of truncations of the original operator to l2 ([M, N ]), as N − M → ∞. Definition. The density of states is the (non-random) measure dk on T defined by f (eiλ )dk(λ) := E[ϕ0 |f (Uω )ϕ0 + ϕ1 |f (Uω )ϕ1 ]/2, (3.1) T
for any continuous function f : S 1 → C. The average over the ϕ0 and ϕ1 matrix elements is motivated by the forms of the matrix (2.1) and shift (2.10). Note also that this definition makes dk a probability measure. Now we turn to the definition of appropriate finite size unitary matrices constructed from (2.1). There are several possible constructions suited to our purpose. Those we use below result from considering Uω provided with boundary conditions at certain sites forbidding transitions through these sites, in the more general definition (2.6) with variable reflection and transition coefficients. More precisely, such a boundary condition at site N corresponds to imposing tN = 0 whereas all other tk ’s are kept equal to their common value t. Therefore, one immediately gets that the matrix takes a block structure which decouples the sites with indices smaller than N from those with indices larger than N . Let us drop temporarily the sub- and super-scripts ω in the notation. Fix N ∈ Z and consider the unitary operator U 2N on l2 (Z) obtained from the original operator U by imposing the following boundary conditions at the sites 2N . Let / {2N, 2N + 1} where U 2N be defined by (2.1) for k ∈ η2N −1 = η2N = η2N +1 = η2N +2 = 0
(3.2)
and, for k ∈ {2N, 2N + 1} U 2N ϕ2N = itϕ2N −1 + rϕ2N U 2N ϕ2N +1 = rϕ2N +1 + itϕ2N +2 .
(3.3)
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
355
Similarly, a boundary condition imposed at site 2N + 1 defines U 2N +1 by (2.1) for k∈ / {2N, 2N + 1, 2N + 2, 2N + 3} where η2N +1 = η2N +2 = 0
(3.4)
and, for k ∈ {2N, 2N + 1, 2N + 2, 2N + 3} U 2N +1 ϕ2N = irte−iη2N ϕ2N −1 + r2 e−iη2N ϕ2N + itϕ2N +1 U 2N +1 ϕ2N +1 = −t2 e−iη2N ϕ2N −1 + irte−iη2N ϕ2N + rϕ2N +1 U 2N +1 ϕ2N +2 = rϕ2N +2 + irte−iη2N +3 ϕ2N +3 − t2 e−iη2N +3 ϕ2N +4 U 2N +1 ϕ2N +3 = +itϕ2N +2 + r2 e−iη2N +3 ϕ2N +3 + irte−iη2N +3 ϕ2N +4 .
(3.5)
For any M ∈ Z, the corresponding operator U M has a the block structure mentioned above and it is unitary. Then, given (M, N ) ∈ Z2 such that M + 4 < N , one defines a unitary matrix U M,N on l2 (Z) by imposing boundary conditions at sites M and N . By construction, U M,N contains an isolated (N − M ) × (N − M ) unitary block on l2 ([M + 1, N ]) we denote by V M,N . Remark. In the definition of the boundary conditions, we put some phases equal to zero around the sites 2N and 2N + 1, in order to avoid having to deal with random boundary conditions later. We could have set them equal to any other value, without changing the main properties of the construction. Introducing the characteristic function χM,N of the set [M + 1, N ] ∈ Z, we denote by the same symbol the projector on the sites [M + 1, N ], corresponding to the multiplication operator by χM,N . Therefore V M,N = χM,N U M,N = U M,N χM,N = χM,N U M,N χM,N . We now consider two measures related to finite matrices as follows. ˜ M,N on T are defined by Definitions. The measures dkM,N and dk f (eiλ )dkM,N (λ) := tr (f (V M,N ))/(N − M ) T ˜ M,N (λ) := tr (χM,N f (U )χM,N )/(N − M ), f (eiλ )dk
(3.6)
(3.7) (3.8)
T
for any continuous function f : S 1 → C. Notice that dkM,N is nothing but the counting measure on T associated with the ˜ M,N is associated with the projection spectrum of the finite block V M,N , and dk of U on [M + 1, N ]. This former operator is unitary whereas the latter is not. We denote the trace norm by · 1 and first show a slight generalization of [GT] allowing to get Lemma 3.1 With the above notations, assume
(U M,N − U )χM,N 1 = o(N − N ), as N − M → ∞,
(3.9)
356
A. Joye
Ann. Henri Poincar´e
then 1 tr (f (V M,N )) − tr (χM,N f (U )χM,N ) = 0. N −M→∞ N − M lim
(3.10)
Remark. The hypothesis is satisfied in particular if Rank(U M,N − U ) < ∞ and uniformly bounded in (N, M ), as is the case with the definitions of U M,N above by means of (3.3, 3.5). Proof. We first note that it is enough to consider functions which are polynomials in z and z¯, z ∈ S 1 . Any f ∈ C(S 1 ) can be approximated by trigonometric polynomials
R PR = j=−R gj eij· in such a way that if > 0 is given, there exists R( ) < ∞ so that (3.11) sup f (θ) − PR() (θ) ≤ . θ∈T
Hence we get using (3.6), tr (f (V M,N ) − χM,N f (U )χM,N ) = tr (χM,N (f (U M,N ) − f (U ))χM,N ) = tr (χM,N (PR() (U M,N ) − PR() (U ))χM,N ) +tr (χM,N ((f − PR() )(U M,N ) − (f − PR() )(U ))χM,N ),
(3.12)
where the trace norm of the last term is bounded by 2 (N −M ), so that it becomes negligeable when divided by (N − M ). We are thus to consider z s and z¯s , with s ∈ N. We can write for any s ≥ 1 U s − (U N,M )s =
s−1
U j (U − U N,M )(U N,M )s−j−1 ,
(3.13)
j=0
so that χM,N (U s − (U N,M )s )χM,N =
s−1
χM,N U j (U − U N,M )χM,N (U N,M )s−j−1 . (3.14)
j=0
Therefore, tr (χM,N (U s − (U N,M )s )χM,N ) s (U − U N,M )χM,N 1 ≤ . N −M N −M
(3.15)
The same result is true if s < 0, with all unitaries replaced by their adjoints. Thus, −R( ) ≤ s ≤ R( ) and the hypothesis on the trace norm of (U − U N,M )χM,N yield the result. Then, restoring the dependence on ω in the notation, we get by the same arguments as in the self adjoint case, that the density of states is almost surely ˜ M,N as N − M → ∞. the limit in the vague sense of the measures dkM,N and dk A proof is provided in the appendix for completeness.
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
Proposition 3.1 For any continuous function f : S 1 → C, iλ ˜ ω lim f (e )dk M,N (λ) = f (eiλ )dk(λ) N −M→∞
T
a.s. ,
357
(3.16)
T
and the support of the density of states dk coincides with Σ, the a.s. spectrum of Uω . Remark. The two previous results show that there exists a series of polynomials whose asymptotic distribution of zeros converges to the measure dk, as announced in the introduction. These polynomials are the characteristic polynomial of the unitary matrix V M,N . As we noted earlier, there is some freedom in the definition of the boundary conditions giving rise to these matrices, therefore this series of polynomials is not unique. Observe also that the difference between these polynomials and the orthogonal polynomials only lies in the boundary conditions used to define V M,N , as recalled at the end of Section 2.1.
4 Thouless formula The link between the density of states and the Lyapunov exponent is provided by an analysis of the spectrum of the finite unitary matrices V M,N . It reads Theorem 4.1 [Thouless Formula ] For any z ∈ C \ {0} γ(z) = 2 ln |z − eiλ |dk(λ ) + ln(1/t2 ) − ln |z|.
(4.1)
T
Remarks. 0) The identity γ(1/¯ z) = γ(z) holds. i) It follows from the above formula, as in Theorem 4.6 in [GT], that the integrated density of states is continuous and satisfies λ ln(2/t2 ) , where N (λ) = dk(λ ), (4.2) |N (λ1 ) − N (λ2 )| ≤ | ln |eiλ1 − eiλ2 || −π by an argument of Craig and Simon [CS]. ii) In case z = eiλ ∈ S 1 , the formula can be cast into the form iλ γ(e ) = ln(sin2 ((λ − λ )/2))dk(λ ) + ln(4/t2 ),
(4.3)
T
from which we recover the estimate 0 ≤ γ(eiλ ) ≤ ln(4/t2 ) that follows from the form of the transfer matrices (2.16). The proof of this version of Thouless formula is given at the end of the section and its translation in terms of potentials of measures is given after the proof. We proceed with a Corollary and an application of this formula. The Corollary essentially expresses the radial derivative of the Lyapunov exponent as the Poisson integral of the density of states measure dk, which allows to recover the a.c. component of dk by a limiting procedure.
358
A. Joye
Ann. Henri Poincar´e
Corollary 4.1 For any > 0 and any λ ∈ T,
lim γ(eiλ e− ) = γ(eiλ ), ∂ 1 − |eiλ e± |2 γ(eiλ e± ) = ∓ dk(λ) ≡ ∓P [dk](eiλ e± ). iλ − eiλ e± |2 ∂ |e T
→0+
(4.4) (4.5)
Therefore, if n(λ)dλ/2π denotes the a.c. component of dk(λ), lim
→0+
∂ ∂ γ(eiλ e− ) = n(λ ) = γ(eiλ ), ∂ ∂
(4.6)
where the limit and the derivative exist for Lebesgue almost all λ ∈ T. Remark. As in [CS], it follows also from the subharmonicity of γ(z), that if γ(eiλ0 ) = 0, then γ : S 1 → R+ is continuous at eiλ0 . Proof. Let us first consider the second statement with lower indices only. We compute iλ − 2 (4.7) γ(e e ) = + ln(1/t ) + ln(1 + e−2 − e− 2 cos(λ − λ ))dk(λ), T
which we can differentiate under the integral sign as long as > 0 to get −2e−2 + e− 2 cos(λ − λ ) ∂ iλ − γ(e e ) = 1 + dk(λ) −2 − e− 2 cos(λ − λ ) ∂ T 1+e 1 − e−2 dk(λ) = P [dk](eiλ e− ). (4.8) = −2 − e− 2 cos(λ − λ ) 1 + e T The existence for almost all λ ∈ T of the limit and the first equality in (4.6) is a direct consequence of the above equality. The existence and equality with the derivative at zero for such λ follows from the mean value Theorem. To get the first statement, notice that 1 + e−2 − e− 2 cos(x) > 2e− (1 − cos(x)) in formula (4.7) above yields 0 ≤ − ln((1 + e−2 − e− 2 cos(λ − λ ))/4) < − ln(2e− (1 − cos(λ − λ ))/4) = (4.9)
− ln((1 − cos(λ − λ ))/2), where the last function is in L1 (T, dk) by Thouless formula. Therefore, an application of the dominated convergence Theorem shows we can take the limit → 0 inside the integral to get the result. We consider now the properties of Uω characterized by i.i.d. phases θkω and in the definition (2.2), assuming one set of phases is uniformly distributed on T. In that situation, not only can we prove the transfer matrices have a (positive) Lyapunov behavior, but we can also exactly compute the Lyapunov exponent αω k
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
359
γ(eiλ ). This shows that in this situation, the spectrum of Uω is almost surely singular, in view of the unitary version of the Ishii-Pastur Theorem proven in [BHJ]. This strengthens the corresponding results of [BHJ], Theorem 4.1 and Propositions 5.4. There Furstenberg’s Theorem is applied to prove positivity of the Lyapunov exponent, so that no value for γ(eiλ ) is provided. Theorem 4.2 Let (θkω )k∈Z and (αω k )k∈Z be i.i.d. on T and assume the distribution ’s is uniform on T. Then, for any λ ∈ T, of either the θkω ’s or the αω k dk(λ) = dλ/2π,
and γ(eiλ ) = ln(1/t2 ) > 0,
(4.10)
therefore, σ(Uω )a.c = ∅ and σ(Uω )sing. = S 1 almost surely.
(4.11)
Remark. The assumption on the distribution of the phases actually implies that the ηk ’s are i.i.d. and uniform on T , see Lemma 4.1 below. This explains why the a.s. spectrum coincides with S 1 and why the density of states is flat. Proof of Theorem 4.2. We first use the following lemma of purely probabilistic nature proven in the appendix. Lemma 4.1 Under the hypotheses of Theorem 4.2, the ηkω ’s are i.i.d. and uniform on T . Then we show the density of states is uniform for uniformly distributed phases. Expanding (2.2) in the ηk (ω)’s we can write for any n = 0, ϕj |Uωn ϕj = (Uω )j,k1 (Uω )k1 ,k2 . . . (Uω )kn−1 ,j k=k1 ,k2 ,...,kn−1
=
k
exp −i
pl ηl (ω) (U0 )j,k1 (U0 )k1 ,k2 . . . (U0 )kn−1 ,j , (4.12)
l∈L
where U0 corresponds to Uω when all phases ηk = 0 and where L is a finite set of indices depending on j, k, n and pl are integers. Observing that the variables ηk (ω)’s all appear with the same sign in (2.1), no compensation can take place between contributions of different matrix elements above and one at least among the integers pl , for l ∈ L is strictly positive when n = 0. Using independence and the characterization E(e−imηk (ω) ) = δm,0 of the uniform distribution, we get E(ϕj |Uωn ϕj ) = δn,0 =⇒ einλ dk(λ) = δn,0 (4.13) T
and the first statement follows. The second equality is a consequence of Thouless formula together with the identity 2π ln |1 − eiλ |dλ = 0. (4.14) 0
360
A. Joye
Ann. Henri Poincar´e
The singular nature of the almost sure spectrum of Uω comes from the unitary version of Ishii-Pastur Theorem proven as Theorem 5.3 in [BHJ], which is independent of the properties of the common distributions of the αk ’s and θk ’s and only requires ergodicity. Finally, Proposition 3.1 yields the result about the support of the a.s. singular spectrum. We compute here, for the sake of completeness, the density of states and Lyapunov exponent for the deterministic free operator U0 corresponding to Uω in case ηk = 0, ∀k ∈ Z. In this case, equation (3.16) of Proposition 3.1 becomes a definition of the free density of states dk0 , provided the limit exists. That the limit exists, is the content of the next Lemma 4.2 The free density of states dk0 exists when defined for any f ∈ C(S 1 ) by iλ f (e )dk0 (λ) = lim f (eiλ )dk˜M,N (λ). (4.15) T
N −M→∞
T
As we know essentially everything about the purely a.c. operator U0 , we can also use a direct approach to perform these computations. In particular, the integrated density of states of U0 can be defined as the distribution function on T of the band functions yielding the spectrum Σ0 of U0 . This direct approach of the density of states coincides with the above definition, see the proofs of Proposition 4.1 and Lemma 4.2 in the appendix. We note here that the spectrum of U0 consists in the set 2 2 (4.16) Σ0 = {e±i(arccos(r −t cos(y))) , y ∈ T}. We get in particular that Σ0 is the support of the density of states whereas Σc0 is that of the Lyapunov exponent: Proposition 4.1 If N0 , dk0 and γ0 denote the integrated density of states, the density of states and Lyapunov exponents of U0 , respectively. We have for λ ∈ T ] − π, π], √ | sin(λ)| dλ if |λ| < arccos(r2 − t2 ) 2π t4 −(r 2 −cos(λ))2 (4.17) dk0 (λ) = 0 otherwise
2 1 arccos r −cos(λ) if λ ∈ [− arccos(r2 − t2 ), 0] 2 2π
2t (4.18) N0 (λ) = 1 − 1 arccos r −cos(λ) if λ ∈ [0, arccos(r2 − t2 )] 2π t2 2 2 02
if |λ| ≤ arccos(r − t ) iλ γ0 (e ) = (4.19) cosh−1 r −cos(λ) otherwise. t2 Finally, Thouless formula (4.1) holds true for these quantities with z = eiλ , λ ∈ T. Remarks. Note that the density of dk0 (λ) diverges as 1/ |λ − arccos(r2 − t2 )| at the band edges and behaves as 1/2πt as λ → 0.
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
361
The integrated density of states N0 (λ) tends to its values 0 and 1 as |λ − arccos(r2 − t2 )| at the band edges. Also, in keeping with the fact that U0 becomes a shift if t = 1 and the identity as r = 1, N0 (λ) becomes linear in λ as t → 1 and a step function as r → 1. The Lyapunov exponent, where non zero, is equivalently given by 2 2 − cos(λ) 2 r − cos(λ) r γ0 (eiλ ) = ln + − 1 . (4.20) t2 t2 It is an even C ∞ function of λ on {|λ| > arccos(r2 − t2 )}, strictly increasing on 2 2 iλ [arccos(r − t ), π]. And dγ0 (e )/dλ behaves as 1/ λ − arccos(r2 − t2 ) as λ → arccos(r2 − t2 )+ . Given Lemma 4.2 above, it is clear that Thouless formula holds for the above quantities. A direct proof of this fact is nevertheless given in the appendix. Finally, in terms of orthogonal polynomials, the free case is related to the choice of constant reflection coefficients ak = a ∈ C, for all k, which yields the Geronimus orthogonal polynomials on the circle. For any such choice, the corresponding five diagonal operator equals −U0 , see (2.8), (2.2), and depends on |a| only. The spectral picture corresponds to the one above, rotated by π. This is in agreement with the accounts of this special case given in [G] and [GNV] for example, modulo a point mass or eigenvalue stemming from the boundary condition at the origin which we don’t consider here, see Section 2.1.
4.1
Proof of Thouless formula
We now turn to the proof of Theorem 4.1. Writing down explicitly the effect of the boundary conditions at N > M on the coefficients of the eigenvector (2.13) we obtain the following relations, which depend on the parity of N and M . Let ψ M,N = χM,N ψ and consider V M,N ψ M,N = eiλ ψ M,N
in l2 [M + 1, N ].
(4.21)
We get by inspection, Lemma 4.3 Assume (4.21) is satisfied. Then, if M is even 1 −it(r − e−iλ ) cM+2 iλ = cM+1 b1 (e ) ≡ cM+1 2 . cM+3 (r − eiλ ) + r(r − e−iλ ) t If M is odd,
cM+1 cM+2
1 = cM+1 b2 (e ) ≡ cM+1 it iλ
it eiλ − r
(4.22)
.
Similarly, if N is even, 1 (r − eiλ ) + r(r − e−iλ ) cN −2 = cN b3 (eiλ ) ≡ cN 2 . −it(r − e−iλ ) cN −1 t
(4.23)
(4.24)
362
A. Joye
If N is odd,
cN −1 cN
= cN −1 b4 (eiλ ) ≡ cN −1
1 it
eiλ − r it
Ann. Henri Poincar´e
.
(4.25)
These relations together with the formulas (2.18) allow to describe the spectrum of V M,N in a convenient manner. Corollary 4.2 Let M < N be fixed and consider non zero vectors a1 , a2 ∈ C2 such that aj (eiλ ) ∈ (bj+2 (eiλ )C)⊥ , j = 1, 2. Then, eiλ ∈ σ(V M,N ) iff a1 (eiλ )|T (N/2 − 1) . . . T (M/2 + 2)b1 (eiλ ) = 0, a2 (e )|T ((N + 1)/2 − 1) . . . T (M/2 + 2)b1 (e ) = 0, iλ
iλ
M, N even M even , N odd
a1 (e )|T (N/2 − 1) . . . T ((M + 1)/2 + 1)b2 (e ) = 0, M odd , N even a2 (eiλ )|T ((N + 1)/2 − 1) . . . T ((M + 1)/2 + 1)b2 (eiλ ) = 0, M, N odd (4.26) iλ
iλ
Remark. In particular, a possible choice for the aj ’s is a1 (eiλ ) = b1 (e−iλ ), a2 (eiλ ) = b2 (e−iλ ).
(4.27)
Each of the above quantities denotes a matrix element of a product of transfer matrices of the type (2.18), which depend on eiλ , and will be linked in the limit N − M → ∞ to the Lyapunov exponent. Let eiλ = z ∈ C \ {0} and n0 , m0 ∈ Z. Defining Φm0 ,n0 (z) = T (n0 − 1) . . . T (m0 + 2),
(4.28)
one sees that the matrix elements aj (z)|Φm0 ,n0 (z)bk (z) correspond to those in the above corollary for values N = 2n0 , N = 2n0 − 1, M = 2m0 , M = 2m0 + 1, depending on the choice of indices j, k. Lemma 4.4 For any z ∈ C \ S 1 and any indices j, k = 1, 2 lim
n0 −m0 →∞
1 ln |aj (z)|Φm0 ,n0 (z)bk (z) | = 2(n0 − m0 ) ln |z − eiλ |dk(λ ) + ln(1/t) − ln(|z|1/2 ),
(4.29)
T
Proof. We note that for any k ∈ Z, there exist 2 × 2 matrices A(k), B(k), C(k) such that (with z = eiλ ) 0 0 iη2k T (k) = zA(k) + B(k) + C(k)/z, where A(k) = (4.30) 0 − −et2 (k)
(k)
Also, for any j = 1, 2, there exist vectors bj , aj , k = −1, 0, 1 such that ak (z) =
zak + ak + ak
(1)
(0)
bk (z) =
(1) zbk
(0) bk
+
(−1)
+
/z,
(−1) bk /z,
(4.31)
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
(−1)
363
(1)
where b2 = a2 = 0 are the only zero vectors with the choice (4.27). Thus, taking into account the above property, Pj,k (z) = z n0 −m0 +(1−k) aj (z)|Φm0 ,n0 (z)bk (z)
(4.32)
is a polynomial in z of degree 2(n0 − m0 ) + 2 − (k + j). Let pj,k be the coefficient of the highest power of z of Pj,k . Then, because of corollary 4.2, we can write
degPj,k
Pj,k (z) = pj,k
(z − eiλl ),
(4.33)
l=0
where {eiλl } is the set of eigenvalues of V M,N and we compute (2−j)
|pj,k | = |aj
|
n 0 −1
(1)
A(l)bk | =
l=m0 +2
K0 t2(n0 −m0 )
(n0 −m0 )−2 K1 −it 0 0 0 = 2(n0 −m0 ) (4.34) 0 1 r 1 t
where K0 , K1 are some constants that depend on j, k and t. Therefore, for any z ∈ C \ S1, degPj,k ln |z − eiλl | ln |Pj,k (z)| = ln(1/t2 ) + lim n0 −m0 →∞ (n0 − m0 ) n0 −m0 →∞ (n0 − m0 )
(4.35)
lim
l=0
Introducing the continuous function fz : S 1 → R given by fz (x) = ln |z − x|, the last term can be written deg Pj,k
lim
n0 −m0 →∞
l=0
fz (eiλj ) tr (fz (V M,N )) =2 = 2 lim M−N →∞ n 0 − m0 N −M
T
fz (eiλ )dk(λ )
(4.36) by application of Lemma 3.1 and Proposition 3.1. This ends the proof of the lemma. Then we make use the following easy lemma Lemma 4.5 If Φ : C2 → C2 is linear and aj , bj ∈ C2 , j = 1, 2 are such that span (a1 , a2 ) = span (b1 , b2 ) = C2 , then Φ := maxj,k |aj |Φbk | is a norm for Φ, noting that its hypothesis is satisfied by ak (z), bj (z), for all z = −1, and of the fact that the Lyapunov exponent is defined independently of the norm used in (2.19) to deduce that (4.29) actually equals half the Lyapunov exponent. Finally, the fact that both the Lyapunov exponent and the right-hand side of (4.29) are subharmonic and coincide on C \ S 1 implies the relation (4.1) on C as well, by classical arguments, see [CS]. This ends the proof of the Thouless formula.
364
4.2
A. Joye
Ann. Henri Poincar´e
Link with potentials of measures
We now express the Thouless formula as a property of the potential of the density of states measure. Following [SaT], we briefly and informally recall the main definitions. The (logarithmic) potential of a probability measure µ on the circle is defined by (4.37) p(dµ; z) = − ln |z − eiλ |dµ(λ), T
the (logarithmic) energy of such a measure is defined by I(dµ) = − ln |eiθ − eiλ |dµ(λ)dµ(θ),
(4.38)
T2
whereas the energy E of a set Σ ⊆ S 1 is E = inf{I(dµ) | supp dµ ⊆ Σ}.
(4.39)
In case an external field Q coming from a weight w(z) = e−Q(z) , z ∈ S 1
(4.40)
is added, the weighted energy of the measure is defined by ln |eiθ − eiλ |dµ(λ)dµ(θ) + 2 Q(eiλ )dµ(λ) Iw (dµ) = − T2
(4.41)
T
and the weighted energy Ew of a set Σ is defined as above, with Iw in place of I. Now, the equilibrium measure of a set Σ is the unique measure dµΣ realizing the infimum of the energy Ew , when finite. These quantities are defined according to
n the electrostatic analogy. For example, if dµA = j=1 n1 δzj , where zj ∈ S are the zeros (with multiplicity) of some monic polynomial A, µA is the distribution of the zeros of A and its potential equals 1 ln |z − zj | = − ln |A(z)|1/n , n j=1 n
p(dµA ; z) = −
(4.42)
and if Σ = S 1 , the equilibrium measure dµS 1 is the normalized Lebesgue measure so that 0 if |z| ≤ 1 . (4.43) p(dµS 1 ; z) = − ln |z| if |z| > 1 Hence we can cast our Thouless formula for dk under the form p(dk; z) + γ(z)/2 = ln(1/t)
∀z ∈ S 1 ,
(4.44)
which, in view of Theorem I.3.3 of [SaT] and the subharmonicity of γ says that the density of states measure dk is the equilibrium measure on S 1 for the weight given
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
365
by w(z) = e−γ(z)/2 . More generally, we can observe a similarity between the proof or our Thouless formula and Theorem III.4.1 in [SaT]. This theorem essentially says, in a deterministic framework, that if {An }n≥0 is a sequence of asymptotically extremal monic polynomials for a weight w (i.e., such that the asymptotic behavior as n → ∞ of (supz∈S 1 |w(z)n An (z)|)1/n is essentially given by a constant), then we have equivalence between lim
n→∞,n∈N
|An (z0 )|1/n = e−p(dµw ;z0 )
(4.45)
and lim dµAn = dµw ,
(4.46)
n→∞
in the vague sense, where dµw denotes the weighted equilibrium measure corresponding to w and N denotes an infinite subsequence of N. Regarding the definition of dk and the proof of Thouless formula, on the one hand we have that p(dk; z) =
lim
M−N →∞
p(dµ∆V M,N ; z)
(4.47)
where ∆V M,N (z) = det(z − V M,N ) is such that dµ∆V M,N → dk vaguely, and, on the other hand, that this potential is related to the Lyapunov exponent in such a way that dk is the equilibrium measure corresponding to the weight w(z) = e−γ(z)/2 . Hence, in our random setting, we can say our construction selects the asymptotically extremal monic polynomials allowing a discrete approximation of the equilibrium measure associated to the external field given by the Lyapunov exponent.
5 Properties of the density of states We mentioned several times the analogy between our unitary operator Uω and Jacobi matrices corresponding to the self-adjoint case. In this section we slightly drift away from the physical motivations underlying the study of (2.1) and consider more closely the links between these cases. The analogy is made clearer by the following Lemma which will be useful later. Lemma 5.1 Denoting unitary equivalence by , we have U ω D ω S0 , and
S0 =
..
.
rt r2 rt −t2
ω
with Dω = diag {e−iηk } −t2 −rt r2 −tr
(5.1)
rt r2 rt
−t2 −rt r2
−t2
−tr
..
U0 ,
(5.2)
.
where the translation along the diagonal is fixed by ϕ2k−2 |S0 ϕ2k = −t2 , k ∈ Z.
366
A. Joye
Ann. Henri Poincar´e
Remarks. In some sense, the Lemma says that, up to unitary equivalence, Uω is a unitary analog of the one-dimensional discrete random Schr¨ odinger operator where the a.c. unitary S0 plays the role of the discrete Laplacian, the pure point diagonal operator Dω plays the role of the potential on the sites, and the operator sum is replaced by a product. We also recall that tridiagonal unitary matrices are spectrally uninteresting as they either correspond to a shift of to infinite direct sums of blocks of size one or two, see Lemma 3.1 in [BHJ]. The Lemma also shows that our operator Uω is essentially a product of an absolutely continuous unitary and a pure point unitary, whereas it was constructed in Section 2 of [BHJ] as a product of two pure point unitaries. Proof. Let us define a collection of rank two operators by Pj = |ϕj ϕj | + |ϕj+1 ϕj+1 |, j ∈ Z, and the unitary V by the direct sum V =
⊕
P2j−1
j∈Z
ir t −it r
(5.3)
P2j−1 .
(5.4)
It is just a matter of computation to check that we can write Uω = (Uω U0−1 )U0 ≡ V −1 Dω V U0 = V −1 Dω (V U0 V −1 )V ≡ V −1 (Dω S0 )V, (5.5) with the required properties for S0 and Dω .
ηkω
Now, forgetting that the phases are in general correlated random variables, see (2.2), if we consider them as i.i.d., but not necessarily uniformly distributed on T, we get some unitary Anderson-like model. This is where we depart from the physical motivation, as it is recalled in Lemma 4.2 in [BHJ] that independence of the ηk ’s is associated with a uniform distribution.
5.1
Support of the density of states
Nevertheless, assuming the random phases {ηkω }k∈Z are i.i.d. according to the measure dµ on T, we can characterize the almost sure spectrum of Uω in term of the support of µ and of the spectrum Σ0 of U0 . Theorem 5.1 Under the above hypotheses, the almost sure spectrum of Uω consists in the set (5.6) Σ := exp(i suppµ)Σ0 = {eiα Σ0 | α ∈ suppµ}. Remarks. In the case where the ηk (ω) are i.i.d. and uniform on T, we recover the fact that the almost sure spectrum of Uω is S 1 . We recall that in the orthogonal polynomial setting, the hypothesis implies each phase γk of the reflection coefficients is given by a sum of i.i.d. phases, see (2.2), (2.8).
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
367
Proof. To show that Σ belongs to the almost sure spectrum, we simply construct Weyl sequences corresponding to the corresponding quasi-energies, with probability one. We know from Section 6 of [BHJ] that for any eiλ ∈ Σ0 , there exists a generalized eigenvector ψλ such ψλ = cj (λ)ϕj , U0 ψλ = eiλ ψλ , and 0 < K < |cj (λ)| < 1/K, ∀j ∈ Z, (5.7) j∈Z
for some K > 0. The last property can be checked also by means of the transfer matrices (2.16) Let α ∈ suppµ. Then, for all > 0, there exists a set I α such that |I | ≤ , and µ(I ) > 0. With the notation ω(k) = ηk (ω), k ∈ Z, we define for all n ∈ N and k ∈ Z, An (k) = {ω(kn) ∈ I , ω(kn + 1) ∈ I , . . . , ω(kn + n − 1) ∈ I }.
(5.8)
Due to the assumed independence, we have for any k, P(An (k)) = µ(I )n > 0 so that for any n > 0, by Borel-Cantelli, P(∪k∈Z An (k)) = 1. Let ∆n (k) = {kn, kn + 1, . . . , kn + n − 1} denote the set of indices appearing in An (k) and consider now ψn,k (λ) = cj (λ)ϕj = χ(∆n (k))ψ(λ), (5.9) j∈∆n (k)
where χ(∆n (k)) is the projector on the span of {ϕj }j∈∆n (k) Because of (5.7), − + (λ) + Rk(n+1) , U0 ψn,k (λ) = eiλ ψn,k (λ) + Rkn
(5.10)
where the vectors Rj± have at most four components close to the index j and
Rj± ≤ R, where R is uniform in j.
(5.11)
Also, by construction of An (k), U0 and Uω , we have
Uω ψn,k (λ) − eiα U0 ψn,k (λ)
≤
(Uω − eiα U0 )χ(∆n (k))| ψn,k (λ)
=
O( ) ψn,k (λ) ,
(5.12)
where the estimate O( ) is uniform in n and k. Therefore, for all > 0 and all n > 0, there exists, with probability one, a k such that An (k) and the corresponding ψn,k (λ) have the above properties so that
Uω ψn,k (λ) − ei(α+λ) ψn,k (λ) / ψn,k (λ)
= ( (Uω − eiα U0 )ψn,k (λ) + eiα (U0 − eiλ )ψn,k (λ) )/ ψn,k (λ)
≤ O( ) + 2R/ ψn,k (λ) = O( + 1/n).
(5.13)
It remains to chose n = [1/ ] to conclude that ei(α+λ) ∈ σ(Uω ) almost surely.
368
A. Joye
Ann. Henri Poincar´e
Let us now show that S 1 \ Σ belongs to the resolvent set of Uω . In order to do so we use Lemma 5.1 Therefore, we can consider as well the spectrum of the product Dω S0 to which the perturbation theory recalled in Chap.1, §11 of [Yaf] for example, applies. In particular, dropping the ω in the notation as randomness plays no role here, if we know that for all j ∈ Z, ηj ∈ [α, β] ⊂ T, then σ(D) ⊆ (δ1 , δ2 ) where (δ1 , δ2 ) denotes the corresponding arc on the unit circle swept in the positive direction from δ1 ∈ S 1 to δ2 ∈ S 1 . We denote by |(δ1 , δ2 )| the length on the torus of this arc. Since σ(S0 ) = Σ0 corresponds to the 2 2 2 2 symmetric arc (e−i arccos(r −t ) , ei arccos(r −t ) ), perturbation theory tells us that after (multiplicative) perturbation by S0 , the spectrum of U DS0 is a subset of an arc of wider aperture than (δ1 , δ2 ). Quantitatively, Theorem 8, p.65 in [Yaf] tells 2 2 2 2 us that the arc (ei arccos(r −t ) δ2 , e−i arccos(r −t ) δ1 ) belongs to the resolvent set of 2 2 2 2 U , provided |(δ1 , δ2 )| < |(ei arccos(r −t ) , e−i arccos(r −t ) )|. This condition simply insures that the subset of the resolvent set we are talking about is not reduced to the empty set. This is enough to get the result in case the support of µ is such that Σ is connected. In case this set is not connected, as |Σ0 | > 0, it consists of a finite set of connected components, each of which can be associated with the convex hull of sufficiently far apart subsets of the support of µ. Denoting these subsets by mj , j = 1, . . . , N and the associated arcs on S 1 by (M1 (j), M2 (j)), we have that the spectrum of D is the disjoint union of subsets σj satisfying σj ⊆ (M1 (j), M2 (j)). The same argument as above says that the spectrum of DS0 is confined to the 2 2 2 2 finite union of arcs ((ei arccos(r −t ) M1 (j), (e−i arccos(r −t ) M2 (j)), which ends the proof of the Theorem.
5.2
Analyticity of the density of states
Without really entering the delicate analysis of the smoothness of the density of states, we can further exploit the relation (4.12) in order to obtain, at the price of some combinatorics, a condition on the common distribution of the ηk ’s ensuring the analyticity of the density of states. Recall that a function f on T is analytic, if and only if its Fourier coefficients fˆ satisfy an estimate of the form |fˆ(n)| ≤ Ae−B|n| , ∀n ∈ Z,
(5.14)
for some positive constants A, B. We have Theorem 5.2 Assume the ηk ’s are distributed according to a law that has an analytic density f characterized by the estimate (5.14) with A, B > 0. Then, if B > ln(1 + 2rt) + ln A,
(5.15)
the density of states dk admits an analytic density, so that the integrated density of states N is analytic as well. Remarks. As fˆ(0) = T f (η)dη = 1, A ≥ 1. When the Theorem applies, it prevents the Lyapunov exponent from being zero on a set of positive measure.
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
369
This result has to be compared with Proposition VI.3.1. of [CL] stating a similar result for the d-dimensional Anderson model. As an immediate consequence, using r2 + t2 = 1, we get the following Corollary 5.1 If the ηk ’s have an analytic density f , characterized by (5.14) with B > ln A, then there exist r+ (f ) and r− (f ) in ]0, 1[ such that the density of states is analytic provided the reflection coefficient r satisfies 1 > r > r+ (f ) or 0 < r < r− (f ). If B > ln(2A), The density of states is analytic ∀r ∈ [0, 1]. Remark. It is easy to check that in both the extreme cases r = 1 and r = 0, the density of states is analytic. Indeed, if r = 1, dk(λ) = f (λ)dλ, where f is the density of the ηk ’s, whereas if r = 0, dk(λ) = dλ/(2π). Proof of Theorem 5.2. By hypothesis, for any n ∈ Z, |Φη (n)| = eiηn f (η)dη ≤ Ae−B|n| . Then, in (4.12) above,
(5.16)
T
l∈L
|Eϕj |Uωn ϕj | ≤ An e−Bn
pl = n, so that using independence
|(U0 )j,k1 ||(U0 )k1 ,k2 | . . . |(U0 )kn−1 ,j |
(5.17)
k1 ,k2 ,...,kn−1
Here the sum carries over a set of indices that form paths of length n + 1 from index j to index j. The allowed paths are those giving rise to non zero matrix elements (U0 )l,m in the sum above. In order to compute this last sum, we proceed as follows. Let us introduce more general j-dependent subsets Cn−1 (j) of indices of Zn−1 that appear in the computation of the matrix element ϕ0 |Uωn ϕj . This set consists of paths of the form {k0 = 0, k1 , k2 , . . . , kn−1 , kn = j} of length n + 1 in Z from 0 to j with the condition that km+1 − km ∈ {0, +1, −1, +2}
if
km is odd
km+1 − km ∈ {0, +1, −1, −2}
if
km is even,
(5.18)
for all m = 0, 1, . . . , n − 1. Let us define Sn−1 (j) := |(U0 )0,k1 ||(U0 )k1 ,k2 | . . . |(U0 )kn−1 ,j |,
(5.19)
Cn−1 (j)
where the matrix elements |(U0 )l,m | are given by r2 , rt and t2 respectively, when |l − m| equals 0, 1 and 2 respectively. This quantity actually gives a crude upper bound on the probability to go from site 0 to j in n time steps, under the free evolution. It is crude in the sense that it does not take the phases into account during that free evolution. We are actually interested in the computation of Sn−1 (0) and of the similar quantity appearing in the computation of ϕ1 |Uωn ϕ1 , which correspond the sum
370
A. Joye
Ann. Henri Poincar´e
in the right-hand side of (5.17), in the asymptotic regime n → ∞. The case of the matrix element ϕ1 |Uωn ϕ1 being similar, we only consider Sn−1 (0). The plan is to use a transfer matrix formalism to evaluate the generating function associated with Sn−1 (j) and then to compute the asymptotics of Sn−1 (0). In view of (5.17), the following proposition implies the theorem. Proposition 5.1 For some constant c > 0, Sn−1 (0) =
c(r + t)2n √ (1 + o(1)) as n → ∞. n
(5.20)
Proof of Proposition 5.1. Let
Pn (x) =
Sn−1 (j)xj
(5.21)
−2n≤j≤2n
be this generating function which we split into two parts Pn (x) = Pn+ (x) + Pn− (x) where Sn−1 (j)xj . (5.22) Pn± (x) = −2n≤j≤2n j
even odd
Clearly we have for n = 0, 1, P0+ (x) = r2 , P0− (x) = 0, P1+ (x) = r2 + t2 x−2 , P1− (x) = rt(x + x−1 ).
(5.23)
It is readily shown by induction that a transfer matrix allows to compute Pn (x) for any n: Lemma 5.2 For any n ≥ 0, 2 + Pn+1 (x) r + t2 x−2 = − rt(x + x−1 ) Pn+1 (x)
rt(x + x−1 ) r2 + t2 x2
Pn+ (x) Pn− (x)
,
with P0+ (x) = r2 , P0− (x) = 0. Denoting by T (x) the transfer matrix defined in this Lemma, and introducing the parameter τ = t/r ∈]0, ∞[, (5.24)
we rewrite it as T (x) = r2
1 + τ 2 x−2 τ (x + x−1 ) τ (x + x−1 ) 1 + τ 2 x2
.
(5.25)
We will consider first the case t = r ⇐⇒ τ = 1. The case τ = 1, for which more can be said about Sn−1 (j), see Proposition 5.2, is dealt with below.
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
371
5.2.1 Case τ = 1 The eigenvalues of T (x) are given by r2 times λ± (x), where λ± (x) = 1 + τ (x2 + x−2 )/2 ± (1 + τ (x2 + x−2 )/2)2 − (1 − τ 2 )2 ,
so that n
2n
T (x) = r A(x)
with A(x) =
λn+ (x) 0
0 λ− (x)n
A(x)−1
λ+ (x) − (1 + τ 2 x2 ) λ− (x) − (1 + τ 2 x2 ) τ (x + x−1 ) τ (x + x−1 )
(5.26)
(5.27) .
(5.28)
For the moment, x is just book keeping parameter, so that we ignore the potential problems of the definition of A(x) in case the eigenvalues are degenerate and we further compute 2 + r Pn (x) n = T (x) (5.29) Pn− (x) 0 r2n τ (x + x−1 ) = 2 (1 + τ (x2 + x−2 )/2)2 − (1 − τ 2 )2 λ+ (x)n+1 − λ− (x)n+1 − (λ+ (x)n − λ− (x)n )(1 + τ 2 x2 ) × . τ (x + x−1 )(λ+ (x)n − λ− (x)n ) We note at this point that one checks, using the binomial Theorem, that despite the presence of square roots in the expressions for Pn± (x), these quantities actually are given by finite Laurent expansions in x, as they should. Focusing on Pn+ (x) we √ can rewrite with the shorthand · for the square root of the denominator above Pn+ (x) 2n
=
−1
r τ (x + x √ 2 ·
(5.30) √ 2 ) · n n τ −2 2 n n (λ+ (x) + λ− (x) ) . (λ+ (x) − λ− (x) ) (x + x ) + 2 2
The quantity of interest to us is Sn−1 (0), the coefficient of x0 in the expansion of Pn+ (x). Substituting eiθ for x in Pn+ , we get a trigonometric polynomial whose zero’th Fourier coefficient is obtained by integration Sn−1 (0) = Pn+ (eiθ )dθ/(2π). (5.31) T
It remains to perform the asymptotic analysis of the above integral as n → ∞. It is a matter of routine to verify the following properties: The eigenvalues, as functions of θ ∈ T ] − π, π], are continuous. If τ < 1, they are real valued, with discontinuity of the derivative at θ = ±π/2, where they cross and are given by 1 − τ 2 . At all other values of θ, they are C ∞ and they satisfy λ+ (eiθ ) > λ− (eiθ ), with λ+ (eiθ ) > 1 − τ 2 .
(5.32)
372
A. Joye
Ann. Henri Poincar´e 2
If τ > 1, the eigenvalues become complex conjugate. Let θc = arccos( τ τ−2 2 )/2 be the critical value where the square root becomes zero. If θ ∈ [θc , π − θc ] ∪ [−π + θc , −θc ], the eigenvalues are complex conjugate, of modulus |1−τ 2 |. Otherwise they are real valued, and satisfy (5.32) as well. Therefore, the asymptotics as n → ∞ of (5.31) is determined by λ+ only. Moreover, in both cases, ln(λ+ (eiθ )) admits non degenerate maxima at θ = 0 and π, where λ+ reaches its maximum value (1 + τ 2 ). Therefore, Laplace’s method yields the asymptotics of the proposition. 5.2.2 Case τ = 1 The course of the proof being the same, it is presented in the appendix. However, instead of computing Sn−1 (0) as n → ∞, we can get exact forms for all Sn−1 (j)’s. The proposition we actually show is Proposition 5.2 Sn−1 (j)
=
Sn−1 (j)
=
1 2n − 1 , −2n ≤ j ≤ 2(n − 1), j even 2n j/2 + n 1 2n − 1 , −2n + 1 ≤ j ≤ 2n − 1, j odd (5.33) 2n (j − 1)/2 + n
Remark. √ Of course, Stirling’s formula for n large yields proposition 5.1 with r = t = 1/ 2: 1 2n − 1 2n Sn−1 (0) = n (5.34) √ . n 2 πn
6 Appendix Proof of Proposition 3.1. We have by definition, T
ω
˜ f (eiλ )dk M,N (λ) =
1 N −M
N
ϕj |f (Uω )ϕj ,
(6.1)
j=M+1
where, depending on the parity of M and N and due to the fact that f is uniformly bounded, the right-hand side can be rewritten as N/2 1 1 ) ϕ2k |f (Uω )ϕ2k + ϕ2k+1 |f (Uω )ϕ2k+1 + Of ( N −M N −M k=(M+1)/2 N/2 1 1 = ϕ0 |f (US k (ω) )ϕ0 + ϕ1 |f (US k (ω) )ϕ1 + Of ( ). N −M N −M k=(M+1)/2
(6.2)
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
373
Now, by the Birkhoff theorem, there exists Ωf of measure one such that for all ω ∈ Ωf , 1 N −M→∞ N − M
N/2
lim
ϕj |f (US k (ω) )ϕj =
k=(M+1)/2
1 E(ϕj |f (Uω )ϕj ), ∀j ∈ Z, (6.3) 2
therefore, 1 1 tr (χM,N f (Uω )) → (E(ϕ0 |f (Uω )ϕ0 + ϕ0 |f (Uω )ϕ0 )) . N −M 2
(6.4)
Then, C(S 1 ) being separable, we have the existence of a countable set of {fj }j∈N , dense in C(S 1 ), for which the above is true, on a set of probability one, which proves the almost sure convergence stated in the proposition. f such that Now assume eiλ0 ∈ Σ and take a continuous non-negative f (eiλ0 ) = 1 and f |Σ = 0. Then f (Uω ) = 0 a.s. so that f (eiλ )dk(λ) = 0 and 0 eiλ0 ∈ supp k. Conversely, if eiλ ∈ supp k, there exists a non-negative continuous f with f (eiλ0 ) = 1 and f (eiλ )dk(λ) = 0. Hence, a.s., ϕ0 |f (Uω )ϕ0 + ϕ1 |f (Uω )ϕ1 = 0, therefore, by ergodicity, ϕj |f (Uω )ϕj = 0 a.s. for any j and f (Uω ) = 0. As f is continuous and equals one at eiλ0 , we get that eiλ0 ∈ Σ. Proof of Lemma 4.1. We only deal with the case where the θkω ’s are i.i.d. and ω uniform, the other case being similar. Let Φη (n) = E(einηk ) be the characteristic function of the random variable ηkω , and similarly for αω k , and Φθ (n) = δn,0 . Then, using independence, Φη (n) = Φθ (n)2 Φα (n)Φα (−n) = δn,0 |Φα (n)|2 = δn,0 ,
(6.5)
so that the ηk ’s are uniformly distributed. Consider now Φηk0 ,ηk1 ,...,ηkj (n0 , n1 , . . . , nj ) = E(ei
j
l=0
kl ηl
).
(6.6)
We can assume the kj ’s are ordered and we observe that ηk and ηk+j are independent as soon as j ≥ 2, see (2.2). Therefore, we can consider consecutive indices kl and deal with Φηk ,ηk+1 ,...,ηk+j (n1 , n2 , . . . , nj ) = E(e
(6.7)
in0 θk−1 +i(n0 +n1 )θk +···+i(nj−1 +nj )θk+j−1 +nj θj )
E(f (α, n)),
where the second expectation contains αk ’s only. Then Φηk ,ηk+1 ,...,ηk+j (n1 , n2 , . . . , nj ) = Φθ (n0 )Φθ (n0 + n1 ) . . . Φθ (nj−1 + nj )Φθ (nj )E(f (α)) = δn0 ,0 δn1 ,0 . . . δnj ,0 E(f (α, n)) = δ E(f (α, 0)) = δ , n,0
with the obvious notation, which yields the result.
n,0
(6.8)
374
A. Joye
Ann. Henri Poincar´e
Proof of Proposition 4.1. We first prove this Proposition with the definition of the density of states as the distribution function of the “band functions” of U0 , to be defined below. Then we will see in the course of the proof of Lemma 4.2 below the equivalence with the definition as an average counting measure. The proof of Proposition 6.2 in [BHJ] shows that U0 on l2 (Z) is unitarily equivalent to the operator multiplication by the matrix 2 r − t2 e2ix 2itr cos(x) V (x) = (6.9) on L2 (T) L2+ (T) ⊕ L2− (T), 2itr cos(x) r2 − t2 e−2ix √ by the unitary mapping that sends ϕk → eikx / 2π, and where L2± (T) is the subspace generated by even/odd harmonics {eikx }k∈Z . The eigenvalues of V (x) are (6.10) λ± (x) = e±iα(x) , where α(x) = arccos(r2 − t2 cos(2x)). We note that λ± (x) = λ± (−x) and V (x) = JV (−x)J where J =
0 1
1 0
.
(6.11)
Hence, the corresponding eigenvectors χ± (x) satisfy V (x)χ± (x) = λ± (x)χ± (x) and V (x)Jχ± (−x) = λ± (x)Jχ± (−x),
(6.12)
so that χ± (x) and Jχ± (−x) are linearly dependent. This is in keeping with the fact that the subspace of generalized eigenvectors is of dimension 2, see (2.14). Also, one checks that for any phase β ∈] − arccos(r2 − t2 ), 0[∪]0, arccos(r2 − t2 )[, α−1 (β) = {x1 , x2 , −x2 − x1 } ⊂] − π, π[.
(6.13)
Therefore, due to (6.12), only half these points contribute for the computation of the density of states. We can now compute the integrated density of states N0 (β) as follows: Taking into account the normalization by a factor 1/2π in the definition (3.1), the fact that supp k ⊂ [− arccos(r2 − t2 ), arccos(r2 − t2 )] and the symmetries, we have for any β ∈ [− arccos(r2 − t2 ), 0] N0 (β) =
1 4π
=
1 2π
T
dλχ{−α(λ)(r2 −cos(β))/t2 }
1 arccos 2π
r2 − cos(β) t2
(6.14)
.
(6.15)
A similar computation for β ∈ [0, arccos(r2 − t2 )) yields (4.18). Therefore, dk0 is absolutely continuous w.r.t. Lebesgue and, for any |λ| < arccos(r2 − t2 ), dk0 (λ) = N (λ)dλ, from which the result on the density of states follows. In order to obtain the Lyapunov exponent, it is enough to observe that the transfer matrices (2.14) T ,
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
375
now independent of k, are of determinant one and trace equal to 2(r2 − cos(λ))/t2 . Then, an explicit computation of the eigenvalues of T together with definition (2.19) yield γ0 (eiλ ). In order to prove the last statement, we first rewrite the right-hand side of Thouless formula with dk0 (λ ) above as 1 1 ln((x − y)2 ) √ dx + ln 2 (6.16) 2π −1 1 − x2 by means elementary manipulations, changing variables to x = (r2 − cos(λ ))/t2 and introducing y = (r2 − cos(λ))/t2 ∈ [−1, (r2 + 1)/t2 ]. Hence we are to show that (6.16) above equals 0 if y ≤ 1 and ln(y + y 2 − 1) if y > 1. That this is true follows from standard manipulations: differentiation w.r.t. y, deformation of contours of integration in the complex plane and computation of residues. Proof of Lemma 4.2. We use freely the notations above. Let us introduce the eigenprojectors P± (x) associated with λ± (x) such that V (x) = P+ (x)λ+ (x) + P− (x)λ− (x).
(6.17)
These quantities are analytic in x, in a strip including the real axis. Let f ∈ C(S 1 ) and let us compute by means of (6.9) and the definition of L2± (T) tr χM,N |f (U0 )χM,N = ϕj |f (U0 )ϕj
=
1 2π even
j M <j≤N
+
odd
j M <j≤N
M<j≤N
1 2π
T
T
! 1 1 dx (f (λ (x))P (x) + f (λ (x))P (x)) + + − − 0 0 ! 0 0 dx. (f (λ+ (x))P+ (x) + f (λ− (x))P− (x)) 1 1
(6.18)
The summand being independent of j and uniformly bounded, we can rewrite the above trace as N − M gets large as N −M f (λ+ (x)) tr P+ (x) + f (λ− (x)) tr P− (x)dx + O(1) 4π T N −M = f (λ+ (x)) + f (λ− (x))dx + O(1). (6.19) 4π T Hence, with λ± (x) = e±iα(x) as in (6.10), and taking into account the properties of α, we get 1 iλ f (e )dk0 (λ) = f (eiα(x) ) + f (e−iα(x) )dx 4π T T π/2 1 = f (eiα(x) ) + f (e−iα(x) )dx, (6.20) 2π −π/2
376
A. Joye
Ann. Henri Poincar´e
which is easily seen to coincide with the “direct” definition of dk0 in the above proof. Proof of Proposition 5.2. As in that case a common term 21n can be factorized, see (5.17), we compute the generating function of |Cn−1 (j)|, the cardinal of the set of relevant indices. Using the same symbols as above, we consider this time |Cn−1 (j)|xj , (6.21) Pn (x) = −2n≤j≤2n
which we split into two parts Pn (x) = Pn+ (x) + Pn− (x) that satisfy for n = 0, 1, P0+ (x) = 1, P0− (x) = 0, P1+ (x) = 1 + x−2 , P1− (x) = x + x−1 .
(6.22)
As above, Lemma 6.1 For any n ≥ 0, + Pn+1 (x) 1 + x−2 = − x + x−1 Pn+1 (x)
x + x−1 1 + x2
Pn+ (x) Pn− (x)
,
with P0+ (x) = 1, P0− (x) = 0. By diagonalization of the corresponding transfer matrix, we get 0 0 n T (x) = A(x) A(x)−1 0 (x−1 + x)2n
where A(x) =
1 + x2 x + x−1 −1 −(x + x ) 1 + x2
(6.23)
(6.24)
and we compute
Pn+ (x) Pn− (x)
(x2 + 1)2n−1 1 1 = . = T (x) 0 x x2n n
(6.25)
Using the binomial theorem we obtain for Pn± (x) Pn+ (x)
=
n−1
2l
x
l=−n
Pn− (x)
=
n−1 l=−n
hence the end result.
2n − 1 l+n
2l+1
x
2n − 1 l+n
,
(6.26)
Acknowledgments. It is a pleasure to thank O. Bourget and R. Bacher for useful discussions, D. Damanik for pointing out reference [CMV] to me and B. Simon for comments on a previous version of the manuscript.
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
377
References [ADE]
J. Asch, P. Duclos, P. Exner, Stability of driven systems with growing gaps, quantum rings, and Wannier ladders , J. Stat. Phys. 92, 1053–1070 (1998).
[BB]
G. Blatter, D. Browne, Zener tunneling and localization in small conducting rings, Phys. Rev. B 37, 3856 (1988).
[BHJ]
O. Bourget, J.S. Howland and A. Joye, Spectral Analysis of Unitary Band Matrices, Commun. Math. Phys. 234, 191–227 (2003).
[Be]
J. Bellissard, Stability and Instability in quantum mechanics, in Trends and Developments in the Eighties, 1–106, Albeverio and Blanchard eds, World Scientific (1985).
[Bo]
O. Bourget, Floquet Operators with Singular Continuous Spectrum, J. Math. Anal. Appl. 276, 28–39 (2002).
[BGHN] A. Bultheel, P. Gonz´ alez-Vera, E. Hendriksen, O. Njastad, “Orthogonal Rational Functions”, Cambridge Monographs on Applied and Computational Mathematics, (1999). [CFKS] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schr¨ odinger Operators, Springer Verlag, 1987. [CL]
R. Carmona, J. Lacroix, Spectral theory of random Schrodinger Operators, Birkh¨ auser, 1990.
[CMV]
M.J. Cantero, L. Moral and L. Vel´ azquez, Five-Diagonal Matrices and Zeros of Orthogonal Polynomials on the Unit Circle, Linear Algebra and Its Applications 326 C, 29–56 (2003).
[Co1]
M. Combescure, The Quantum Stability Problem for Time-periodic Perturbations of the Harmonic Oscillator, Ann. Inst. H. Poincar´e 47, 451– 454 (1987).
[Co2]
M. Combescure, Spectral properties of a periodically kicked quantum Hamiltonian, J. Stat. Phys. vol. 59, 679–690 (1990).
[Co3]
M. Combescure, Recurrent versus diffusive quantum behaviour for time dependent Hamiltonians, Operator theory: advances and applications, vol. 57, Birkh¨ auser Verlag (1992).
[CS]
W. Craig, B. Simon, Subharmonicity of the Lyaponov Index, Duke Math. Jour. 50, 551–560 (1983).
[DF]
S. DeBi`evre, G. Forni, Transport properties of kicked and quasiperiodic Hamiltonians, J. Statist. Phys. 90, 1201–1223 (1998).
378
A. Joye
Ann. Henri Poincar´e
[DLSV] P. Duclos, O. Lev, P. Stovicek, M. Vittot, Weakly regular Hamiltonians with pure point spectrum, Rev. Math. Phys. 14, 531–568 (2002). [DS]
P. Duclos, P. Stovicek, Floquet Hamiltonians with pure point spectrum, Commun. Math. Phys. vol. 177, 327–347 (1996).
[G]
Ya. Geronimus, “Polynomials Orthogonal on the Circle and Interval”, Int. Ser. of Monographs in Pure and Applied Mathematics, Pergamon Press, Oxford, (1960).
[GNV]
L. Golinskii, P. Nevai, W. Van Assche, Perturbation of orthogonal polynomials on an arc of the unit circle, J. Approx. Theory 83, 392–422 (1995).
[GR]
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 4th Edition, Academic Press, 1965.
[GT]
J.S. Geronimo, A. Teplyaev, A Difference Equation Arising from the Trigonometric Moment Problem Having Random Reflection CoefficientsAn Operator Theoretic Approach, J. Func. Anal. 123, 12–45 (1994).
[GY]
S. Graffi, K. Yajima, Absolute Continuity of the Floquet Spectrum for a nonlinearly Forced Harmonic Oscillator, Comm. Math. Phys., vol. 215, no 2, 245–250 (2000).
[Ho1]
J. Howland, Quantum Stability, in “Schr¨ odinger Operators” Lecture Notes in Physics, E. Baslev Edt. Springer 403, 101–122 (1992).
[Ho2]
J. Howland, Floquet operators with singular continuous spectrum, I, Ann. Inst. H. Poincar´e Phys. Th´eor., vol. 49, 309–323 (1989).
[Ho3]
J. Howland, Floquet operators with singular continuous spectrum, II, Ann. Inst. H. Poincar´e Phys. Th´eor., vol. 49, 325–334 (1989).
[Ho4]
J. Howland, Floquet operators with singular continuous spectrum, III, Ann. Inst. H. Poincar´e Phys. Th´eor., vol. 69, 265–273 (1998).
[J]
A. Joye, Absence of absolutely continuous spectrum of Floquet operators, J. Stat. Phys., vol. 75, 929–952 (1994).
[N1]
G. Nenciu, Floquet operators without absolutely continuous spectrum, Ann. Inst. H. Poincar´e Phys. Th´eor., vol. 59, 91–97 (1993).
[N2]
G. Nenciu, Adiabatic theory: stability of systems with increasing gaps, Ann. Inst. H. Poincar´e Phys. Th´eor., vol. 67, 411–424 (1997).
[SaT]
E.B. Saff, V. Totik, “Logarithmic Potentials with External Fields”, Springer, 1997.
Vol. 5, 2004
Density of States and Thouless Formula for Random Band Matrices
379
[StT]
H. Stahl, V. Totik, “General Orthogonal Polynomials”, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, 1992.
[Yaf]
D. Yafaev, Mathematical Scattering Theory, General Theory, Transl. of Math. Mono. 105 (1992).
Alain Joye Institut Fourier Universit´e de Grenoble 1, BP 74 F-38402 Saint-Martin d’H`eres Cedex France email:
[email protected] Communicated by Eugene Bogomolny Submitted 07/11/03, accepted 15/01/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 381 – 403 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020381-23 DOI 10.1007/s00023-004-0173-9
Annales Henri Poincar´ e
On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels Claudio Fern´ andez and Georgi Raikov Abstract. We consider the three-dimensional Schr¨ odinger operators H0 and H± where H0 = (i∇+A)2 −b, A is a magnetic potential generating a constant magnetic field of strength b > 0, and H± = H0 ± V where V ≥ 0 decays fast enough at infinity. Then, A. Pushnitski’s representation of the spectral shift function (SSF) for the pair of operators H± , H0 is well defined for energies E = 2qb, q ∈ Z+ . We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels 2qb, q ∈ Z+ . Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. R´esum´e. On consid` ere les op´erateurs de Schr¨ odinger tridimensionnels H0 et H± o` u H0 = (i∇ + A)2 − b, A est un potentiel magn´etique engendrant un champ magn´etique constant d’intensit´e b > 0, et H± = H0 ±V o` u V ≥ 0 d´ ecroˆıt assez vite ` a l’infini. Alors, la repr´esentation obtenue par A. Pushnitski de la fonction du d´ecalage spectral pour les op´erateurs H± , H0 est bien d´efinie pour des ´energies E = 2qb, q ∈ Z+ . On ´ etudie le comportement du repr´esentant associ´e de la classe d’´equivalence d´ etermin´ee par la fonction du d´ecalage spectral, au voisinage des niveaux de Landau 2bq, q ∈ Z+ . En r´ eduisant l’analyse ` a l’investigation de l’asymptotique des valeurs propres d’une famille d’op´erateurs de Toeplitz compacts, on ´etablit une relation entre le type des singularit´es de la fonction du d´ecalage spectral aux niveaux de Landau et la vitesse de la d´ecroissance de V ` a l’infini.
1 Introduction In this paper we analyze the singularities of the spectral shift function (SSF) for the three-dimensional Schr¨ odinger operator with constant magnetic field, perturbed by an electric potential which decays fast enough at infinity. Let us recall the definition of the abstract SSF for a pair of self-adjoint operators. First, let us consider two self-adjoint operators T0 and T acting in the same Hilbert space, such that T − T0 ∈ S1 where S1 denotes the space of trace class operators. Then, there exists a unique function ξ(.; T , T0 ) ∈ L1 (R) such that the Lifshits-Kre˘ın trace formula Tr(φ(T ) − φ(T0 )) = ξ(E; T , T0 )φ (E)dE, φ ∈ C0∞ (R), (1.1) R
holds (see, e.g., [17, Theorem 8.3.3]). Let now H0 and H be two lower-bounded selfadjoint operators acting in the same Hilbert space. Assume that for some γ > 0,
382
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
and λ0 ∈ R lying strictly below the infima of the spectra of H0 and H, we have that (1.2) (H − λ0 )−γ − (H0 − λ0 )−γ ∈ S1 . Set ξ(E; H, H0 ) :=
−ξ((E − λ0 )−γ ; (H − λ0 )−γ , (H0 − λ0 )−γ ) 0 if E ≤ λ0 .
if
E > λ0 ,
Then, similarly to (1.1), Tr(φ(H) − φ(H0 )) =
R
ξ(E; H, H0 )φ (E)dE,
φ ∈ C0∞ (R),
(see [17, Theorem 8.9.1]). The function ξ(.; H, H0 ) is called the SSF for the pair of the operators H and H0 . Evidently, it does not depend on the particular choice of γ and λ0 in (1.2). If E lies below the infimum of the spectrum of H0 , then the spectrum of H below E could be at most discrete, and we have ξ(E; H, H0 ) = −N (E; H)
(1.3)
where N (E; H) denotes the number of eigenvalues of H in the interval (−∞, E), counted with the multiplicities. On the other hand, for almost every E in the absolutely continuous spectrum of H0 , the SSF ξ(E; H, H0 ) is related to the scattering determinant det S(E; H, H0 ) for the pair (H, H0 ) by the Birman-Kre˘ın formula det S(E; H, H0 ) = e−2πiξ(E;H,H0 ) (see [2] or [17, Section 8.4]). A survey of various asymptotic results concerning the SSF for numerous quantum Hamiltonians is contained in [15]. In the present paper the role of H0 is played by the operator H0 := (i∇ + A)2 − b, is essentially self-adjoint on C0∞ (R3 ). Here the magnetic potential A = which bx2 bx1 − 2 , 2 , 0 generates the constant magnetic field B = curl A = (0, 0, b), b > 0. It is well known that σ(H0 ) = σac (H0 ) = [0, ∞) (see [1]), where σ(H0 ) denotes the spectrum of H0 , and σac (H0 ) its absolutely continuous spectrum. Moreover, the so-called Landau levels 2bq, q ∈ Z+ := {0, 1, . . .}, play the role of thresholds in σ(H0 ). For x = (x1 , x2 , x3 ) ∈ R3 we denote by X⊥ = (x1 , x2 ) the variables on the plane perpendicular to the magnetic field. We assume that V satisfies x = (X⊥ , x3 ) ∈ R3 , (1.4) with C0 > 0, m⊥ > 2, m3 > 1, and x := (1 + |x|2 )1/2 , x ∈ Rd , d ≥ 1. Most of our results will hold under a more restrictive assumption than (1.4), namely V ≡ 0,
V ∈ C(R3 ),
V ≡ 0,
0 ≤ V (x) ≤ C0 X⊥ −m⊥ x3 −m3 ,
V ∈ C(R3 ),
0 ≤ V (x) ≤ C0 x−m ,
m > 3,
x ∈ R3 .
(1.5)
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
383
Note that (1.5) implies (1.4) with any m3 ∈ (0, m) and m⊥ = m − m3 . In particular, we can choose m3 ∈ (1, m − 2) so that m⊥ > 2. Set H± := H0 ± V so that the electric potential ±V has a definite sign. Obviously, inf σ(H+ ) = 0, inf σ(H− ) ≥ −C0 . The role of the perturbed operator H is played in this paper by H± . By (1.4) and the diamagnetic inequality (see, e.g., [1]), V 1/2 (H0 − λ0 )−1 with λ0 < 0 is a Hilbert-Schmidt operator. Therefore, the resolvent identity implies (H± − λ0 )−1 − (H0 − λ0 )−1 ∈ S1 for λ0 < inf σ(H± ) ≤ inf σ(H0 ), i.e., (1.2) holds with H = H± , H0 = H0 , and γ = 1, and, hence, the SSF ξ(.; H± , H0 ) exists. A priori the SSF ξ(E; H± , H0 ) is defined only for almost every E ∈ R. In Sec˜ H± , H0 ) of the equivalence class tion 2 below we introduce a representative ξ(.; determined by ξ(.; H± , H0 ), which is well defined and uniformly bounded on each compact subset of the complement of the Landau levels. Moreover, ξ˜ is continuous on R \ 2bZ+ everywhere except at the eigenvalues, isolated, or embedded in the continuous spectrum, of the operator H± . The main goal of the paper is the study of the asymptotic behaviour as λ → 0 ˜ of ξ(2bq + λ; H± , H0 ) with fixed q ∈ Z+ . Our results establish the asymptotic ˜ coincidence of ξ(2bq + λ; H± , H0 ) with the traces of certain functions of compact Toeplitz operators. Many of the spectral properties of those Toeplitz operators are well known, which allows us to describe explicitly the asymptotics as λ → 0 ˜ of ξ(2bq + λ; H± , H0 ) in several generic cases. These asymptotic results admit an interpretation directly in the terms of the SSF, which is independent of the choice of the representative of the equivalence class. In particular, these results reveal the link between the type of the singularities of the SSF at the Landau levels, and the decay rate of V at infinity. The paper is organized as follows. In Section 2 we introduce the representative ξ˜ of the SSF. In Section 3 we formulate our main results, summarize some known spectral properties of compact Toeplitz operators, and obtain as corollaries explicit asymptotic formulas describing the singularities of the SSF at the Landau levels. Section 4 contains preliminary estimates. The proofs of our main results can be found in Section 5. Finally, in Section 6 we prove some of the corollaries of the main results.
2 A. Pushnitski’s representation of the SSF 2.1. In this subsection we introduce some basic notations used throughout the paper. We denote by S∞ the class of linear compact operators acting in a fixed Hilbert space. Let T = T ∗ ∈ S∞ . Denote by PI (T ) the spectral projection of T associated with the interval I ⊂ R. For s > 0 set n± (s; T ) := rank P(s,∞) (±T ).
384
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
For an arbitrary (not necessarily self-adjoint) operator T ∈ S∞ put n∗ (s; T ) := n+ (s2 ; T ∗ T ),
s > 0.
(2.1)
If T = T ∗ , then evidently n∗ (s; T ) = n+ (s, T ) + n− (s; T ),
s > 0.
(2.2)
Moreover, if Tj = Tj∗ ∈ S∞ , j = 1, 2, then the Weyl inequalities n± (s1 + s2 , T1 + T2 ) ≤ n± (s1 , T1 ) + n± (s2 , T2 )
(2.3)
hold for each s1 , s2 > 0. Further, we denote by Sp , p ∈ [1, ∞), the Schatten-von Neumann class of compact ∞ 1/p operators for which the norm T p : = p 0 sp−1 n∗ (s; T ) ds is finite. If T ∈ Sp , p ∈ [1, ∞), then the following elementary inequality n∗ (s; T ) ≤ s−p T pp
(2.4)
holds for every s > 0. If T = T ∗ ∈ Sp , p ∈ [1, ∞), then (2.2) and (2.4) imply n± (s; T ) ≤ s−p T pp,
s > 0.
Finally, we define the self-adjoint operators Re T := 1 ∗ 2i (T − T ). Evidently, n± (s; Re T ) ≤ 2n∗ (s; T ),
(2.5) 1 2 (T
+ T ∗ ) and Im T :=
n± (s; Im T ) ≤ 2n∗ (s; T ).
(2.6)
2.2. In this subsection we summarize several results due to A. Pushnitski on the representation of the SSF for a pair of lower-bounded self-adjoint operators (see [8]–[10]). dt Let I ∈ R be a Lebesgue measurable set. Set µ(I) := π1 I 1+t 2 . Note that µ(R) = 1. Lemma 2.1. [8, Lemma 2.1] Let T1 = T1∗ ∈ S∞ and T2 = T2∗ ∈ S1 . Then 1 n± (s1 + s2 ; T1 + t T2 ) dµ(t) ≤ n± (s1 ; T1 ) + T2 1 , s1 , s2 > 0. πs 2 R
(2.7)
Let H± and H0 be two lower-bounded self-adjoint operators acting in the same Hilbert space. Let λ0 < inf σ(H± ) ∪ σ(H0 ). First of all, assume that (1.2) holds with H = H± for some γ > 0. Further, let
Finally, suppose that
V := ±(H± − H0 ) ≥ 0,
(2.8)
V 1/2 (H0 − λ0 )−1/2 ∈ S∞ .
(2.9)
V 1/2 (H0 − λ0 )−γ ∈ S2
(2.10)
holds for some γ > 0. For z ∈ C with Im z > 0 set T (z): = V 1/2 (H0 − z)−1 V 1/2 .
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
385
Lemma 2.2. (see, e.g., [8, Lemma 4.1]) Let (2.8)–(2.10) hold. Then for almost every E ∈ R the operator-norm limit T (E + i0) := n − limδ↓0 T (E + iδ) exists, and by (2.9) we have T (E + i0) ∈ S∞ . Moreover, Im T (E + i0) ∈ S1 . Theorem 2.1. [8, Theorem 1.2] Let (1.2) with H = H± , and (2.8)–(2.10) hold. Then for almost every E ∈ R we have ξ(E; H± , H0 ) = ± n∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t). R
Remark. The representation of the SSF described in the above theorem has been generalized to non-sign-definite perturbations in [6] in the case of trace-class perturbations, and in [10] in the case of relatively trace-class perturbations. These generalizations are based on the concept of the index of orthogonal projections. We will not use them in the present paper. Suppose now that V satisfies (1.4). Then relations (1.2) and (2.8)–(2.10) hold with V = V , H0 = H0 , and γ = γ = 1. For z ∈ C, Im z > 0, set T (z) := V 1/2 (H0 − z)−1 V 1/2 . By Lemma 2.2, for almost every E ∈ R the operator-norm limit T (E + i0) := n − lim T (E + iδ) (2.11) δ↓0
exists, and Im T (E + i0) ∈ S1 .
(2.12)
For trivial reasons the limit in (2.11) exists and (2.12) holds for each E < 0. In Corollary 4.3 below we show that this is also true for each E ∈ [0, ∞) \ 2bZ+ . Hence, by Lemma 2.1, the quantity R n∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t) is well defined for every E ∈ R \ 2bZ+ . Set ˜ ξ(E; H± , H0 ) = ± n∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t), E ∈ R \ 2bZ+ . R
(2.13)
By Theorem 2.1 we have ˜ ξ(E; H± , H0 ) = ξ(E; H± , H0 )
(2.14)
for almost every E ∈ R. Remark. In [4] it is shown that the function ξ˜ defined on R \ 2bZ+ is continuous away from the eigenvalues of the operator H± . Note that, in contrast to the case b = 0, we cannot rule out the possibility of existence of embedded eigenvalues, by imposing short-range assumptions of the type of (1.4) or (1.5): Theorem 5.1 of [1] shows that there are axisymmetric potentials V of compact support such that below each Landau level 2bq, q ∈ Z+ , there exists an infinite sequence of eigenvalues of H− which converges to 2bq. On the other hand, generically, the only possible accumulation points of the eigenvalues of the operators H± are the Landau levels (see [1, Theorem 4.7], [5, Theorem 3.5.3 (iii)]). Further information of the location of these eigenvalues can be found in [4].
386
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
3 Main results 3.1. In this subsection we formulate our main results. To this end we need some more notations. Introduce the Landau Hamiltonian 2 2 ∂ bx2 ∂ bx1 h(b) := i − + i + − b, (3.1) ∂x1 2 ∂x2 2 i.e., the two-dimensional Schr¨odinger operator with constant scalar magnetic field b > 0, essentially self-adjoint on C0∞ (R2 ). It is well known that σ(h(b)) = ∪∞ q=0 {2bq}, and each eigenvalue 2bq, q ∈ Z+ , has infinite multiplicity (see, e.g., [1]). For x, x ∈ R2 denote by Pq,b (x, x ) the integral kernel of the orthogonal projection pq (b) onto the subspace Ker (h(b) − 2bq), q ∈ Z+ . It is well known that Pq,b (x, x ) =
b Lq 2π
b|x − x |2 2
b (3.2) exp − (|x − x |2 + 2i(x1 x2 − x1 x2 )) 4
(see [7] or [12, Subsection 2.3.2]) where Lq (t) :=
q 1 t dq (tq e−t ) q (−t)k e , = k q! dtq k!
t ∈ R,
q ∈ Z+ ,
k=0
b are the Laguerre polynomials. Note that Pq,b (x, x) = 2π for each q ∈ Z+ and x ∈ R2 . Introduce the orthogonal projections Pq : L2 (R3 ) → L2 (R3 ), q ∈ Z+ , by (Pq u)(X⊥ , x3 ) = Pq,b (X⊥ , X⊥ ) u(X⊥ , x3 ) dX⊥ , u ∈ L2 (R3 ). (3.3) R2
Assume that (1.4) holds. Set W (X⊥ ) :=
R
V (X⊥ , x3 )dx3 ,
X ⊥ ∈ R2 .
(3.4)
X ⊥ ∈ R2 ,
(3.5)
If, moreover, V satisfies (1.5), then 0 ≤ W (X⊥ ) ≤ C0 X⊥ −m+1 , where C0 = C0
R
x−m dx. For q ∈ Z+ and λ > 0 introduce the operator 1 ωq (λ) := √ pq W pq . 2 λ
Evidently, ωq (λ) is self-adjoint and non-negative in L2 (R2 ). Lemma 3.1. Let U ∈ Lr (R2 ), r ≥ 1, and q ∈ Z+ . Then pq U pq ∈ Sr .
(3.6)
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
387
Proof. If U ∈ L∞ (R2 ), then evidently pq U pq ≤ U L∞ . If U ∈ L1 (R2 ), we write pq U pq = pq |U |1/2 ei arg U |U |1/2 pq , check that pq |U |1/2 22 =
b U L1 , 2π
ei arg U |U |1/2 pq 22 =
b U L1 , 2π
b U L1 . Interpolating, we get pq U pq rr ≤ and conclude that pq U pq 1 ≤ 2π b r 2π U Lr which implies the desired result.
Remark. The proof of Lemma 3.1 follows the idea of the proof of [11, Lemma 5.1]. We include it here in order to make the exposition self-contained. If λ > 0, and V satisfies (1.4) with m⊥ > 2 and m3 > 1, then Lemma 3.1 with U = W implies ωq (λ) ∈ S1 . Theorem 3.1. Assume that (1.5) is valid. Let q ∈ Z+ , b > 0. Then the asymptotic estimates ˜ (3.7) ξ(2bq − λ; H+ , H0 ) = O(1), and ˜ − λ; H− , H0 ) ≤ −n+ ((1 + ε); ωq (λ)) + O(1), −n+ ((1 − ε); ωq (λ)) + O(1) ≤ ξ(2bq (3.8) hold as λ ↓ 0 for each ε ∈ (0, 1). Suppose that the potential V satisfies (1.4). For λ > 0 define the matrixvalued function w11 w12 (3.9) , X ⊥ ∈ R2 , Wλ = Wλ (X⊥ ) := w21 w22 where w11 :=
R
√ V (X⊥ , x3 ) cos ( λx3 )dx3 , 2
w12 = w21 :=
R
w22 :=
R
√ V (X⊥ , x3 ) sin2 ( λx3 )dx3 ,
√ √ V (X⊥ , x3 ) cos ( λx3 ) sin ( λx3 )dx3 .
Introduce the operator 1 Ωq := √ pq Wλ pq . (3.10) 2 λ Evidently, Ωq (λ) is self-adjoint and non-negative in L2 (R2 )2 . Moreover, using the fact that ωq (λ) ∈ S1 , it is easy to check that Ωq (λ) ∈ S1 as well. Theorem 3.2. Assume that (1.5) is valid. Let q ∈ Z+ , b > 0. Then the asymptotic estimates 1 ˜ Tr arctan ((1 ± ε)−1 Ωq (λ)) + O(1) ≤ ξ(2bq + λ; H± , H0 ) π 1 ≤ ± Tr arctan ((1 ∓ ε)−1 Ωq (λ)) + O(1) π hold as λ ↓ 0 for each ε ∈ (0, 1). ±
(3.11)
388
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
The proofs of Theorems 3.1 and 3.2 can be found in Section 5. In the following ˜ subsection we will describe explicitly the asymptotics of ξ(2bq − λ; H− , H0 ) and ˜ ξ(2bq + λ; H± , H0 ) as λ ↓ 0 under generic assumptions about the behaviour of W (X⊥ ) as |X⊥ | → ∞. 3.2. Relations (3.8) and (3.11) allow us to reduce the analysis of the behaviour as ˜ λ → 0 of ξ(2bq + λ; H± , H0 ), to the study of the asymptotic distribution of the eigenvalues of Toeplitz-type operators pq U pq . The following three lemmas concern the spectral asymptotics of such operators. Lemma 3.2. [11, Theorem 2.6] Let the function U ∈ C 1 (R2 ) satisfy the estimates 0 ≤ U (X⊥ ) ≤ C1 X⊥ −α ,
|∇U (X⊥ )| ≤ C1 X⊥ −α−1 ,
X ⊥ ∈ R2 ,
for some α > 0 and C1 > 0. Assume, moreover, that U (X⊥ ) = u0 (X⊥ /|X⊥ |)|X⊥ |−α (1 + o(1)),
|X⊥ | → ∞,
where u0 is a continuous function on S1 which is non-negative and does not vanish identically. Then for each q ∈ Z+ we have n+ (s; pq U pq ) =
b
X⊥ ∈ R2 |U (X⊥ ) > s (1 + o(1)) = 2π ψα (s; u0 , b) (1 + o(1)),
where |.| denotes the Lebesgue measure, and ψα (s) = ψα (s; u0 , b) := s
−2/α
b 4π
S1
u0 (t)2/α dt,
s > 0.
s ↓ 0,
(3.12)
Remark. Theorem 2.6 of [11] contains a considerably more general result than Lemma 3.2. For the sake of the simplicity of exposition, here we reproduce only the special case of asymptotically homogeneous U . Lemma 3.3. [13, Theorem 2.1, Proposition 4.1] Let 0 ≤ U ∈ L∞ (R2 ). Assume that ln U (X⊥ ) = −µ|X⊥ |2β (1 + o(1)),
|X⊥ | → ∞,
for some β ∈ (0, ∞), µ ∈ (0, ∞). Then for each q ∈ Z+ we have n+ (s; pq U pq ) = ϕβ (s)(1 + o(1)), where ϕβ (s) = ϕβ (s; µ, b) :=
b | ln s|1/β if 0 < β 2µ1/β 1 β= ln (1+2µ/b) | ln s| if β −1 | ln s| if β−1 (ln | ln s|)
s ↓ 0,
< 1, 1, 1 < β < ∞,
s ∈ (e, ∞). (3.13)
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
389
Lemma 3.4. [13, Theorem 2.2, Proposition 4.1] Let 0 ≤ U ∈ L∞ (R2 ). Assume that the support of U is compact, and that there exists a constant C > 0 such that U ≥ C on an open non-empty subset of R2 . Then for each q ∈ Z+ we have n+ (s; pq U pq ) = ϕ∞ (s) (1 + o(1)), where
ϕ∞ (s) := (ln | ln s|)−1 | ln s|,
s ↓ 0,
s ∈ (e, ∞).
(3.14)
Remark. For each β ∈ (0, ∞] and c > 0 we have ϕβ (cs) = ϕβ (s)(1 + o(1)) as s ↓ 0. Employing Lemmas 3.2, 3.3, 3.4, and the above remark, we find that (3.8) immediately entails the following corollary. Corollary 3.1. Let (1.5) hold with m > 3. i) Assume that the hypotheses of Lemma 3.2 hold with U = W and α = m − 1. Then we have √ b ˜ ξ(2bq − λ; H− , H0 ) = − X⊥ ∈ R2 |W (X⊥ ) > 2 λ (1 + o(1)) 2π √ = −ψm−1 (2 λ; u0 , b) (1 + o(1)), λ ↓ 0, (3.15) the function ψα being defined in (3.12). ii) Assume that the hypotheses of Lemma 3.3 hold with U = W . Then we have √ ˜ ξ(2bq − λ; H− , H0 ) = −ϕβ ( λ; µ, b) (1 + o(1)), λ ↓ 0, β ∈ (0, ∞), the functions ϕβ being defined in (3.13). iii) Assume that the hypotheses of Lemma 3.4 hold with U = W . Then we have √ ˜ ξ(2bq − λ; H− , H0 ) = −ϕ∞ ( λ) (1 + o(1)), λ ↓ 0, the function ϕ∞ being defined in (3.14). ˜ Remark. In the special case q = 0 when −ξ(−λ; H− , H0 ) coincides for almost every λ > 0 with the eigenvalue counting function for the operator H− (see (1.3)), relation (3.15) was established for the first time in [16]. Here we use a different approach related to the one developed in [11]. Similarly, the combination of Theorem 3.2 with Lemmas 3.2–3.4 yields the following corollary. Corollary 3.2. i) Let (1.5) hold with m > 3. Assume that the hypotheses of Lemma 3.2 are fulfilled for U = W and α = m − 1. Then we have √ b ˜ arctan ((2 λ)−1 W (X⊥ ))dX⊥ (1 + o(1)) ξ(2bq + λ; H± , H0 ) = ± 2 2π R2 √ 1 =± ψm−1 (2 λ; u0 , b) (1 + o(1)), λ ↓ 0. 2 cos (π/(m − 1))
390
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
ii) Let (1.5) hold with m > 3. Suppose in addition that V satisfies (1.4) for some m⊥ > 2 and m3 > 2. Finally, assume that the hypotheses of Lemma 3.3 are fulfilled for U = W . Then we have √ 1 ˜ ξ(2bq + λ; H± , H0 ) = ± ϕβ ( λ; µ, b) (1 + o(1)), 2
λ ↓ 0,
β ∈ (0, ∞).
iii) Let the assumptions of the previous part be fulfilled, except that the hypotheses of Lemma 3.3 are replaced by those of Lemma 3.4. Then we have √ 1 ˜ ξ(2bq + λ; H± , H0 ) = ± ϕ∞ ( λ) (1 + o(1)), 2
λ ↓ 0.
The proof of Corollary 3.2 can be found in Section 6. 3.3. In this subsection we present a possible interpretation of our results directly in the terms of the SSF ξ(.; H± , H0 ) which is invariant of the choice of the representative of the equivalence class determined by the SSF. For λ > 0, and q ∈ Z+ , introduce the averaged values of the SSF Ξ± q,< (λ) :=
1 λ
Ξ± q,> (λ) :=
1 λ
2bq
2bq−λ
ξ(s; H± , H0 )ds =
1 λ
ξ(s; H± , H0 )ds =
1 λ
2bq+λ
2bq
λ
0
ξ(2bq − t; H± , H0 )dt,
λ
0
ξ(2bq + t; H± , H0 )dt.
± Since ξ(.; H± , H0 ) ∈ L1loc (R), the quantities Ξ± q,< (λ) and Ξq,> (λ) are well defined for every λ > 0. Applying (2.14), we find that the asymptotic bound Ξ+ q,< (λ) = O(1) as λ ↓ 0 follows from (3.7). Further, Corollary 3.1 i) implies
Ξ− q,< (λ) = −
√ m−1 ψm−1 (2 λ; u0 , b) (1 + o(1)), m−2
λ ↓ 0,
m > 3,
while Corollary 3.1 ii)–iii) entails √ Ξ− q,< (λ) = − ϕβ ( λ; µ, b) (1 + o(1)),
λ ↓ 0,
β ∈ (0, ∞].
Finally, it follows from Corollary 3.2 i) that Ξ± q,> (λ) = ±
√ m−1 1 ψm−1 (2 λ; u0 , b) (1 + o(1)), λ ↓ 0, m > 3, 2 cos (π/(m − 1)) m − 2
while Corollary 3.2 ii)–iii) implies Ξ± q,> (λ) = ±
√ 1 ϕβ ( λ; µ, b) (1 + o(1)), λ ↓ 0, 2
β ∈ (0, ∞].
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
391
4 Preliminary estimates −1 d2 For z ∈ C with Im z > 0, define the operator R(z) := − dx bounded in 2 − z 3
L2 (R), as well as the operators
Tq (z) := V 1/2 Pq (H0 − z)−1 V 1/2 ,
q ∈ Z+ ,
bounded in L2 (R3 ) (see (3.3) for the definition of the orthogonal projection Pq ). Rz (x) = The √ operator √ R(z) admits the integral √ kernel Rz (x3 − x3 ) where √ i z|x| ie /(2 z), x ∈R, the branch of z being chosen so that Im z > 0. Moreover, Tq (z) = V 1/2 pq (b) ⊗ R(z − 2bq) V 1/2 . For λ ∈ R, λ = 0, define R(λ) as the operator with integral kernel Rλ (x3 − x3 ) where √ e− √−λ|x| if λ < 0, 2√ −λ x ∈ R. (4.1) Rλ (x) := lim Rλ+iδ (x) = iei √λ|x| if λ > 0, δ↓0 2 λ
Evidently, if w ∈ L2 (R) and λ = 0, then wR(λ)w¯ ∈ S2 . For E ∈ R, E = 2bq, q ∈ Z+ , set Tq (E) := V 1/2 pq (b) ⊗ R(E − 2bq) V 1/2 . Proposition 4.1. Let E ∈ R, q ∈ Z+ , E = 2bq. Let (1.4) hold. Then Tq (E) ∈ S2 , and Tq (E) 22 ≤ C1 b/|E − 2bq| (4.2) with C1 independent of E, b, and q. Moreover, lim Tq (E + iδ) − Tq (E) 2 = 0.
(4.3)
δ↓0
Proof. The operator Tq (E) admits the representation Tq (E) = M (Gq ⊗ JE−2bq ) M
(4.4)
where M : L2 (R3 ) → L2 (R3 ) is the multiplier by V (X⊥ , x3 )1/2 X⊥ m⊥ /2 x3 m3 /2 , Gq : L2 (R2X ) → L2 (R2X⊥ ) is the operator with integral kernel ⊥
−m⊥ /2 )X⊥ , X⊥ −m⊥ /2 Pq,b (X⊥ , X⊥
X⊥ , X⊥ ∈ R2 ,
(see (3.2) for the definition of Pq,b ), while Jλ : L2 (Rx3 ) → L2 (Rx3 ) is the operator with integral kernel x3 −m3 /2 Rλ (x3 − x3 )x3 −m3 /2 ,
x3 , x3 ∈ R,
λ ∈ R \ {0}.
(4.5)
392
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
Evidently, Tq (E) 22 ≤ M 4 Gq 22 JE−2bq 22 . By (1.4) we have M 4 ≤ C02 . Further, JE−2bq 22
= R
|RE−2bq (x3 − x3 )|2 x3 −m3 x3 −m3 dx3 dx3 1 ≤ 4|E − 2bq|
R
−m3
x3
2 dx3
.
b Finally, since Gq ≤ 1, we have Gq 22 ≤ Gq 1 = 2π X⊥ −m⊥ dX⊥ . Hence, R2 2 2 C (4.2) holds with C1 = 8π0 R x3 −m3 dx3 X⊥ −m⊥ dX⊥ . To prove (4.3), we R2 write 2 2 Tq (E + iδ) − Tq (E) 2 = V (X⊥ , x3 )V (X⊥ , x3 )|Pq,b (X⊥ , X⊥ )| R2
R2
R
R
|RE−2bq+iδ (x3 − x3 ) − RE−2bq (x3 − x3 )|2 dx3 dx3 dX⊥ dX⊥ ,
note that limδ↓0 RE−2bq+iδ (x) = RE−2bq (x) for each x ∈ R, and that the integrand in the above integral is bounded from above for each δ > 0 by the L1 (R6 )-function 1 2 , x3 )|Pq,b (X⊥ , X⊥ )| , V (X⊥ , x3 )V (X⊥ |E − 2bq|
(X⊥ , x3 , X⊥ , x3 ) ∈ R6 .
Therefore, the dominated convergence theorem implies (4.3). Remark. Using more sophisticated tools than those of the proof of Proposition 4.1, it is shown in [4] that for E = 2bq we have not only Tq (E) ∈ S2 , but also Tq (E) ∈ S1 . We will not use this fact here. Corollary 4.1. Assume that (1.4) holds. Let E ∈ R, q ∈ Z+ , E = 2bq. Then Im Tq (E) ≥ 0. Moreover, if E < 2bq, then Im Tq (E) = 0. Proof. The non-negativity of Im Tq (E) follows from the representation Im Tq (E + iδ) = δV 1/2 Pq ((H0 − E)2 + δ 2 )−1 Pq V 1/2 ,
δ > 0,
and the limiting relation Im Tq (E) = n − limδ↓0 Im Tq (E + iδ), E = 2bq, which on its turn is implied by (4.3). Moreover, if E < 2bq, then (4.1) entails Tq (E) = Tq (E)∗ so that Im Tq (E) = 0. Corollary 4.2. Under the assumptions of Corollary 4.1 we have Im Tq (E) ∈ S1 . Furthermore, if E > 2bq, then b (E − 2bq)−1/2 V (x)dx. (4.6) Im Tq (E) 1 = Tr Im Tq (E) = 4π R3
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
393
Proof. Bearing in mind the representation (4.4), we find that the inclusion Im Tq (E) ∈ S1 would be implied by the inclusion Gq ∈ S1 and Im JE−2bq ∈ S1 . The first inclusion follows from Lemma 3.1, and the second one from the obvious fact that rank Im JE−2bq ≤ 2. Further, the first equality in (4.6) follows from the non-negativity of the operator Im Tq (E) which is guaranteed by Corollary 4.1. Since the operator Im Tq (E) with E > 2bq admits the kernel 1 √ V (X⊥ , x3 ) cos E − 2bq(x3 − x3 ) Pq,b (X⊥ , X⊥ ) 2 E − 2bq , x ), (X , x ), (X , x ) ∈ R3 , V (X⊥ ⊥ 3 ⊥ 3 3 the Mercer theorem (see, e.g., the lemma on pp. 65–66 of [14]) implies the second equality in (4.6). Proposition 4.2. Let q ∈ Z+ , λ ∈ R, |λ| ∈ (0, b], and δ > 0. Assume that V satisfies (1.4). Then the operator series Tq+ (2bq + λ + iδ) :=
∞
Tl (2bq + λ + iδ),
(4.7)
Tl (2bq + λ)
(4.8)
l=q+1
Tq+ (2bq + λ) :=
∞ l=q+1
are convergent in S2 . Moreover, Tq+ (2bq + λ) 22 ≤
∞ C0 b (2b(l − q) − λ)−3/2 V (x)dx. 8π R3
(4.9)
l=q+1
Finally,
lim Tq+(2bq + λ + iδ) − Tq+ (2bq + λ) 2 = 0. δ↓0
(4.10)
Proof. For each q , q ∈ Z+ such that q + 1 ≤ q < q < ∞ we have −1 q q √ √ 2 d2 + 2 Tl (2bq + λ) 2 = V pl ⊗ − 2 + 2b(l − q) − λ V 2 dx 3 l=q l=q −2 q d2 ≤ C0 Tr pl ⊗ − 2 + 2b(l − q) − λ V dx3 l=q
q dη C0 b V (x)dx = 2 (2π)2 (η 2 + 2b(l − q) − λ) R3 l=q R q C0 b (2b(l − q) − λ)−3/2 V (x)dx. = 8π R3
l=q
(4.11)
394
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
−3/2 Since the numerical series ∞ is convergent, and S2 is a l=q+1 (2b(l − q) − λ) Hilbert (hence, complete) space, we conclude that (4.11) entails the convergence in S2 of the operator series in (4.8), as well as the validity of (4.9). The convergence of the series in (4.7) is proved in exactly the same manner. Finally, (4.10) follows from the estimate Tq+ (2bq + λ + iδ) − Tq+ (2bq + λ) 22 ∞ dη 2 C0 b ≤δ V (x)dx (2π)2 (η 2 + 2b(l − q) − λ)2 ((η 2 + 2b(l − q) − λ)2 + δ 2 ) R3 l=q+1 R ∞ ∞ C0 b dη ≤ δ2 2 (2b(l − q) − λ)−7/2 V (x)dx. 2π (η 2 + 1)4 R3 0 l=q+1
For E = 2bq + λ with q ∈ Z+ , and λ ∈ R, |λ| ∈ (0, b], set Tq− (E) := T (E) − Tq (E) − Tq+ (E) (see (4.8)). Note that if q = 0, then Tq− (E) = 0, and if q−1 q ≥ 1, then Tq− (E) = l=0 Tl (E). Corollary 4.3. For E = 2bq + λ with q ∈ Z+ and λ ∈ R, |λ| ∈ (0, b] the operatornorm limit (2.11) exists, and T (E + i0) = Tq− (E) + Tq (E) + Tq+ (E).
(4.12)
Re T (E + i0) = Re Tq− (E) + Re Tq (E) + Tq+ (E),
(4.13)
Moreover,
Im T (E + i0) = Im
Tq− (E)
+ Im Tq (E) ∈ S1 .
(4.14)
q Proof. Let δ > 0. Evidently, T (E + iδ) = l=0 Tl (E + iδ) + Tq+ (E + iδ) (see (4.7)). Proposition 4.1 implies that n − limδ↓0 ql=0 Tl (E + iδ) = Tq− (E) + Tq (E), while Proposition 4.2 implies that n − limδ↓0 Tq+ (E) = Tq+ (E). Combining the above two relations, we get (4.12). Relation (4.13) follows immediately from (4.12) and Tq+ (E) = Tq+ (E)∗ , while (4.14) is implied by (4.12) and Corollaries 4.1 and 4.2.
5 Proof of the main results 5.1. This subsection contains a general estimate which will be used in the proofs of all our main results. Informally speaking, we show that we can replace the operator T (E + i0) by Tq (E) in the r.h.s of (2.13) when we deal with the first ˜ asymptotic term of ξ(E; H± , H0 ) as the energy E approaches a given Landau level 2bq, q ∈ Z+ .
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
395
Proposition 5.1. Assume that (1.4) holds. Let E = 2bq+λ with q ∈ Z+ , and λ ∈ R, |λ| ∈ (0, b]. Then the asymptotic estimates n± (1 + ε; Re Tq (E) + t Im Tq (E)) dµ(t) + O(1) R ≤ n± (1; Re T (E + i0) + t Im T (E + i0)) dµ(t) R ≤ n± (1 − ε; Re Tq (E) + t Im Tq (E)) dµ(t) + O(1) (5.1) R
hold as λ → 0 for each ε ∈ (0, 1). Proof. Using (4.13) and (4.14), and applying the Weyl inequalities (2.3), we get n± (1 + ε; Re Tq (E) + t Im Tq (E)) dµ(t) R − n∓ (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t) R ≤ n± (1; Re T (E + i0) + t Im T (E + i0)) dµ(t) R ≤ n± (1 − ε; Re Tq (E) + t Im Tq (E)) dµ(t) R + n± (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t). (5.2) R
In order to conclude that (5.2) implies (5.1), it remains to show that n± (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t) = O(1), λ → 0,
(5.3)
R
for each ε > 0. Employing (2.7) and (2.3), we find that n± (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t) R
2 Im Tq− (E) 1 επ 2 Im Tq− (E) 1 , ≤ n± (ε/4; Re Tq− (E)) + n± (ε/4; Tq+(E)) + επ ≤ n± (ε/2; Re Tq− (E) + Tq+ (E)) +
ε > 0.
(5.4)
If q = 0, and, hence, Tq− (E) = 0, we need only to apply (2.5) with p = 2, and (4.9), in order to get n± (ε/4; Tq+(E)) ≤ 16ε−2 Tq+ (E) 22 ≤
∞ 2C0 b −3/2 (2b(l − q) − λ) V (x)dx, ε2 π R3 l=q+1
(5.5)
396
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
which combined with (5.4) yields (5.3). If q ≥ 1 we should also utilize the estimate n± (ε/4; Re Tq− (E)) ≤ 32ε−2 Tq−(E) 22 ≤ 32ε−2 qC1 b
q−1
(2b(q − l) + λ)−1
(5.6)
l=0
which follows from (2.6), (2.4) with p = 2, and (4.2), as well as the estimate Im Tq− (E) 1 ≤
q−1 b (2b(q − l) + λ)−1/2 V (x)dx, 2π R3
(5.7)
l=0
which follows from (4.6). Thus, in the case q ≥ 1, estimate (5.3) is implied by from the combination of (5.4)–(5.7). 5.2. In this subsection we complete the proof of the first part of Theorem 3.1. Since Im Tq (2bq−λ) = 0 and Re Tq (2bq−λ) = Tq (2bq−λ) ≥ 0 if λ > 0, Proposition 5.1 implies immediately the following corollary. Corollary 5.1. Under the hypotheses of Proposition 5.1 the asymptotic estimates n− (1; Re T (2bq − λ + i0) + t Im T (2bq − λ + i0)) dµ(t) = O(1), (5.8) R
and ≤
R
n+ (1 + ε; Tq (2bq − λ)) + O(1) n+ (1; Re T (2bq − λ + i0) + t Im T (2bq − λ + i0)) dµ(t) ≤ n+ (1 − ε; Tq (2bq − λ)) + O(1)
(5.9)
hold as λ ↓ 0 for each ε ∈ (0, 1). Now the combination of (2.13) and (5.8) yields (3.7). 5.3. In this section we complete the proof of the second part of Theorem 3.1. For q ∈ Z+ and λ > 0 define Oq (λ) : L2 (R3 ) → L2 (R3 ) as the operator with integral kernel 1 , x ), √ V (X⊥ , x3 ) Pq,b (X⊥ , X⊥ ) V (X⊥ (X⊥ , x3 ), (X⊥ , x3 ) ∈ R3 . 3 2 λ Proposition 5.2. Under the hypotheses of Theorem 3.1 the asymptotic estimates n+ ((1 + ε)s; Oq (λ)) + O(1) ≤ n+ (s; Tq (2bq − λ)) ≤ n+ ((1 − ε)s; Oq (λ)) + O(1) (5.10) hold as λ ↓ 0 for each ε ∈ (0, 1) and s > 0.
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
397
Proof. Fix s > 0 and ε ∈ (0, 1). Then the Weyl inequalities entail n+ ((1 + ε)s; Oq (λ)) − n− (εs; Tq (2bq − λ) − Oq (λ)) ≤ n+ (s; Tq (2bq − λ)) ≤ n+ ((1 − ε)s; Oq (λ)) + n+ (εs; Tq (2bq − λ) − Oq (λ)). In order to get (5.10), it suffices to show that n± (t; Tq (2bq − λ) − Oq (λ)) = O(1),
λ ↓ 0,
(5.11)
for every fixed t > 0. Denote by T˜q the operator with integral kernel −
1 , x ), V (X⊥ , x3 ) |x3 − x3 | Pq,b (X⊥ , X⊥ ) V (X⊥ 3 2
(X⊥ , x3 ), (X⊥ , x3 ) ∈ R3 .
(5.12)
Pick m ∈ (3, m), and write ˜ q,m−m ⊗ J˜(0) M ˜ m,m ˜ m,m G T˜q = M m
˜ m,m is the multiplier by the bounded function V (X⊥ , x3 )X⊥ (m−m )/2 where M ˜ q,m−m : L2 (R2 ) → L2 (R2 ) is the operator with integral x3 m /2 , (X⊥ , x3 ) ∈ R3 , G kernel
−(m−m )/2 X⊥ −(m−m )/2 Pq,b (X⊥ , X⊥ )X⊥ ,
X⊥ , X⊥ ∈ R2 ,
(0) and J˜m : L2 (R) → L2 (R) is the operator with integral kernel 1 − x3 −m /2 |x3 − x3 |x3 −m /2 , 2
x3 , x3 ∈ R.
˜ q,m−m is compact, and Since m − m > 0, Lemma 3.1 implies that the operator G (0) ˜ ˜ since m > 3 we have Jm ∈ S2 . Finally, since Mm,m is bounded, we find that the operator T˜q is compact. Further, ˜ q,m−m ⊗ J˜(λ) M ˜ m,m ˜ m,m G Tq (2bq − λ) − Oq (λ) = M m (λ) where J˜m , λ > 0, is the operator with integral kernel √ √ λ|x3 − x3 | 1 −m /2 − 12 λ|x3 −x3 | x3 −m /2 , e sinh − √ x3 2 λ
x3 , x3 ∈ R.
(λ) Applying the dominated convergence theorem, we easily find that limλ↓0 J˜m − (0) J˜m 2 = 0. Therefore, T˜q = n − limλ↓0 (Tq (2bq − λ) − Oq (λ)). Fix t > 0. Choosing
398
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
λ > 0 so small that Tq (2bq − λ) − Oq (λ) − T˜q < t/2, and applying the Weyl inequalities, we get n± (t; Tq (2bq − λ) − Oq (λ)) ≤ n± (t/2; Tq (2bq − λ) − Oq (λ) − T˜q ) + n± (t/2; T˜q ) = n± (t/2; T˜q ). (5.13) Since the r.h.s. of (5.13) is finite and independent of λ, we conclude that (5.13) entails (5.11). Proposition 5.3. Assume that (1.4) holds. Then for each q ∈ Z+ , λ > 0, and s > 0 we have (5.14) n+ (s; Oq (λ)) = n+ (s; ωq (λ)) (see (3.6) for the definition of the operator ωq (λ)). Proof. Define the operator K : L2 (R3 ) → L2 (R2 ) by , x )u(X , x ) dx dX , Pq,b (X⊥ , X⊥ ) V (X⊥ (Ku)(X⊥ ) := ⊥ 3 3 ⊥ 3 R2
R
X ⊥ ∈ R2 ,
where u ∈ L2 (R3 ). The adjoint operator K ∗ : L2 (R2 ) → L2 (R3 ) is given by Pq,b (X⊥ , X⊥ )v(X⊥ ) dX⊥ , (X⊥ , x3 ) ∈ R3 , (K ∗ v)(X⊥ , x3 ) := V (X⊥ , x3 ) R2
2
2
where v ∈ L (R ). Obviously, 1 Oq (λ) = √ K ∗ K, 2 λ
1 ωq (λ) = √ K K ∗ . 2 λ
Since n+ (s; K ∗ K) = n+ (s; K K ∗ ) for each s > 0, we get (5.14). Putting together (2.13), (5.9), (5.10), and (5.14), we get (3.8). Thus, we are done with the proof of Theorem 3.1. 5.4. In this subsection we complete the proof of Theorem 3.2. Proposition 5.4. Let q ∈ Z+ and b > 0. Assume that (1.5) holds. Then the asymptotic estimates (5.15) n± (s; Re Tq (2bq + λ)) = O(1) are valid as λ ↓ 0 for each s > 0. Proof. The operator Re Tq (2bq + λ + i0) admits the integral kernel √ 1 , x ), − √ V (X⊥ , x3 ) sin λ|x3 − x3 | Pq,b (X⊥ , X⊥ ) V (X⊥ 3 2 λ (X⊥ , x3 ), (X⊥ , x3 ) ∈ R3 . Arguing as in the proof of Proposition 5.2, we find that n− limλ↓0 Re Tq (2bq + λ) = T˜q (see (5.12)). Fix s > 0. Choosing λ > 0 so small that Re Tq (2bq+λ)−T˜q < s/2, and applying the Weyl inequalities, we get n± (s; Re Tq (2bq + λ)) ≤ n± (s/2; T˜q ) which implies (5.15).
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
399
Taking into account Propositions 5.1 and 5.4, and applying the Weyl inequalities and the evident identities 1 n± (s; tT )dµ(t) = Tr arctan (s−1 T ), s > 0, π R where T = T ∗ ≥ 0, T ∈ S1 , we obtain the following Corollary 5.2. Let q ∈ Z+ , b > 0. Assume that V satisfies (1.5). Then the asymptotic estimates 1 Tr arctan ((1 + ε)−1 Im Tq (2bq + λ)) + O(1) π ≤ n± (1; Re Tq (2bq + λ) + t Im Tq (2bq + λ))dµ(t) R
1 Tr arctan ((1 − ε)−1 Im Tq (2bq + λ)) + O(1) π are valid as λ ↓ 0 for each ε ∈ (0, 1). ≤
(5.16)
Proposition 5.5. Assume that (1.4) holds. Then for each q ∈ Z+ , λ > 0, and s > 0, we have n+ (s; Im Tq (2bq + λ)) = n+ (s; Ωq (λ)) (5.17) (see (3.10) for the definition of the operator Ωq (λ)). Consequently, Tr arctan (s−1 Im Tq (2bq + λ)) = Tr arctan (s−1 Ωq (λ))
(5.18)
for each q ∈ Z+ , λ > 0. Proof. The proof is quite similar to that of Proposition 5.3. Define the operator K : L2 (R3 ) → L2 (R2 )2 by Ku := v = (v1 , v2 ) ∈ L2 (R2 )2 , where
v1 (X⊥ ) :=
R2
v2 (X⊥ ) :=
R2
R
u ∈ L2 (R3 ),
√ , x )u(X , x ) dx dX , Pq,b (X⊥ , X⊥ ) cos( λx3 ) V (X⊥ ⊥ 3 3 ⊥ 3
√ Pq,b (X⊥ , X⊥ ) sin( λx3 ) R , x )u(X , x ) dx dX , V (X⊥ ⊥ 3 3 ⊥ 3
X ⊥ ∈ R2 .
Evidently, the adjoint operator K∗ : L2 (R2 )2 → L2 (R3 ) is given by √ (K∗ v)(X⊥ , x3 ) := cos( λx3 ) V (X⊥ , x3 ) Pq,b (X⊥ , X⊥ )v1 (X⊥ ) dX⊥ √ + sin( λx3 ) V (X⊥ , x3 )
R2
R2
Pq,b (X⊥ , X⊥ )v2 (X⊥ ) dX⊥ ,
(X⊥ , x3 ) ∈ R3 ,
400
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
where v = (v1 , v2 ) ∈ L2 (R2 )2 . Obviously, 1 Im Tq (2bq + λ) = √ K∗ K, 2 λ
1 Ωq (λ) = √ K K∗ . 2 λ
Since n+ (s; K∗ K) = n+ (s; K K∗ ) for each s > 0, we get (5.17). Now the combination of (2.13), (5.1), (5.16), and (5.18) yields (3.11).
6 Proof of Corollary 2.2 Introduce the matrix-valued functions W 0 (1) W (X⊥ ) := , 0 0 W
(2)
(X⊥ ) = W
(2)
(X⊥ ; λ) := Wλ (X⊥ ) − W
(1)
(X⊥ ) =
−w22 w21
w12 w22
(see (3.4) and (3.9) for the definitions of W and Wλ , respectively), as well as the operators 1 (j) Ω(j) pq , q (λ) : √ pq W 2 λ
λ > 0,
q ∈ Z+ ,
j = 1, 2,
(j)
compact in L2 (R2 )2 . Evidently, Ωq (λ) ∈ S1 , j = 1, 2. Proposition 6.1. (i) Let (1.5) hold with m ∈ (3, 4]. Then for each q ∈ Z+ , s > 0, and δ > 4−m 2 , we have −δ Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) ), λ ↓ 0, (6.1) q (λ)) = O(λ (see (3.10) for the definition of the operator Ωq (λ)). (ii) Let (1.4) hold with m⊥ > 2, m3 > 2. Then for each q ∈ Z+ and s > 0 we have Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) λ ↓ 0. (6.2) q (λ)) = O(1), Proof. Applying the Lifshits-Kre˘ın trace formula (1.1), we easily get Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) q (λ)) 2 −1 = ξ(E; s−1 Ωq (λ); s−1 Ω(1) dE, q (λ))(1 + E ) R
s > 0. (6.3)
Vol. 5, 2004
Therefore, ≤
R
Singularities of the Magnetic Spectral Shift Function
401
Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) q (λ))
|ξ(E; s−1 Ωq (λ); s−1 Ω(1) q (λ))|dE ≤
1 1 Ωq (λ) − Ω(1) Ω(2) q (λ) 1 = q (λ) 1 s s (6.4)
(see [17, Theorem 8.2.1]). Further, b (2) Ωq (λ) 1 ≤ √ w22 (X⊥ )2 + w12 (X⊥ )2 dX⊥ 2π λ R2 √ b b √ ≤ |wj2 (X⊥ )|dX⊥ ≤ √ V (X⊥ , x3 )| sin( λx3 )|dx3 dX⊥ . 2π λ j=1,2 R2 π λ R2 R (6.5) , and m ∈ Assume now that V satisfies (1.5) with m ∈ (3, 4]. Pick δ > 4−m 2 (−2δ + 2, m − 2). Then we have √ V (X⊥ , x3 )| sin ( λx3 )|dx3 dX⊥ λ−1/2 R2 R −δ −(m−m ) ≤ λ C0 X⊥ dX⊥ x3 −m |x3 |−2δ+1 dx3 . (6.6) R2
R
Since m − m > 2 the integral with respect to X⊥ ∈ R2 is convergent, and since m + 2δ − 1 > 1 the integral with respect to x3 is convergent as well. Now the combination of (6.3)–(6.6) entails (6.1). Further, suppose that V satisfies (1.4) with m⊥ > 2 and m3 > 2. Then √ λ−1/2 V (X⊥ , x3 )| sin ( λx3 )|dx3 dX⊥ R2 R ≤ C0 X⊥ −m⊥ dX⊥ x3 −m3 |x3 |dx3 . (6.7) R2
R
Putting together (6.3)–(6.5), and (6.7), we get (6.2). Now note that if V satisfies (1.5) with m ∈ (3, 4], we can choose 4−m 4, then it satisfies (1.4) with m⊥ > 2 and m3 > 2, and, hence, (6.2) is valid. Finally, −1 ωq (λ)) Tr arctan (s−1 Ω(1) q (λ)) = Tr arctan (s ∞ n+ (st; ωq (λ)) dt, = 1 + t2 0
s > 0,
λ > 0, (6.9)
402
C. Fern´ andez and G. Raikov
Ann. Henri Poincar´e
(see (3.6) for the definition of the operator ωq (λ)). Putting together (6.9), (6.8), and (6.2), we conclude that Corollary 3.2 follows easily from Theorem 3.2 and Lemmas 3.2–3.4. Acknowledgments. The authors are grateful to the anonymous referee whose valuable remarks contributed to the improvement of the text. Georgi Raikov was partially supported by the Chilean Science Foundation Fondecyt under Grant 1020737.
References [1] J. Avron, I. Herbst, B. Simon, Schr¨ odinger operators with magnetic fields. I. General interactions, Duke Math. J. 45, 847–883 (1978). ˇ Birman, M.G. Kre˘ın, On the theory of wave operators and scattering [2] M.S. operators, Dokl. Akad. Nauk SSSR 144 (1962), 475–478 [in Russian]; English translation in Soviet Math. Doklady 3 (1962). ˇ Birman, D.R. Yafaev, The spectral shift function. The papers of M.G. [3] M.S. Kre˘ın and their further development, (Russian) Algebra i Analiz 4, 1–44 (1992); English translation in St. Petersburg Math. J. 4, 833–870 (1993). [4] V. Bruneau, A. Pushnitski, G.D. Raikov, Spectral shift function in strong magnetic fields, Algebra i Analiz 16, 207–238 (2004). [5] C. G´erard, I. L aba, Multiparticle Quantum Scattering in Constant Magnetic Fields, Mathematical Surveys and Monographs, 90, AMS, Providence, RI, 2002. [6] F. Gesztesy, K. Makarov, The Ξ operator and its relation to Kre˘ın’s spectral shift function, J. Anal. Math. 81, 139–183 (2000). [7] L. Landau, Diamagnetismus der Metalle, Z. Physik 64, 629–637 (1930). [8] A. Pushnitski˘ı, A representation for the spectral shift function in the case of perturbations of fixed sign, Algebra i Analiz 9, 197–213 (1997) [in Russian]; English translation in St. Petersburg Math. J. 9, 1181–1194 (1998). [9] A. Pushnitski, Estimates for the spectral shift function of the polyharmonic operator, J. Math. Phys. 40, 5578–5592 (1999). [10] A. Pushnitski, The spectral shift function and the invariance principle, J. Funct. Anal. 183, 269–320 (2001). [11] G.D. Raikov, Eigenvalue asymptotics for the Schr¨ odinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips, Commun. P.D.E. 15, 407–434 (1990); Errata: Commun. P.D.E. 18, 1977–1979 (1993).
Vol. 5, 2004
Singularities of the Magnetic Spectral Shift Function
403
[12] G.D. Raikov, M. Dimassi, Spectral asymptotics for quantum Hamiltonians in strong magnetic fields, Cubo Mat. Educ. 3, 317–391 (2001). [13] G.D. Raikov, S. Warzel, Quasi-classical versus non-classical spectral asymptotics for magnetic Schr¨ odinger operators with decreasing electric potentials, Rev. Math. Phys. 14, 1051–1072 (2002). [14] M. Reed, B. Simon, Methods of Modern Mathematical Physics. III. Scattering Theory, Academic Press, New York, 1979. [15] D. Robert, Semiclassical asymptotics for the spectral shift function, In: Differential Operators and Spectral theory, AMS Translations Ser. 2 189, 187–203, AMS, Providence, RI, 1999. [16] A.V. Sobolev, Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field, perturbed by a decreasing electric field. I, Probl. Mat. Anal. 9, 67–84 (1984) [in Russian]; English translation in: J. Sov. Math. 35, 2201–2212 (1986). [17] D.R. Yafaev, Mathematical scattering theory. General theory, Translations of Mathematical Monographs, 105 AMS, Providence, RI, 1992. Claudio Fern´ andez Departamento de Matem´aticas Facultad de Matem´ aticas Pontificia Universidad Cat´ olica de Chile Av. Vicu˜ na Mackenna 4860 Santiago Chile email:
[email protected] Georgi D. Raikov Departamento de Matem´aticas Facultad de Ciencias Universidad de Chile Las Palmeras 3425 Santiago Chile email:
[email protected] Communicated by Bernard Helffer submitted 23/09/03, accepted 15/01/04
Ann. Henri Poincar´e 5 (2004) 405 – 434 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030405-30 DOI 10.1007/s00023-004-0174-8
Annales Henri Poincar´ e
Exact Solution of the AVZ-Hamiltonian in the Grand-Canonical Ensemble Stephan Adams and Jean-Bernard Bru Abstract. The thermodynamic behavior of the Angelescu-Verbeure-Zagrebnov (AVZ) Hamiltonian [1], also called the superstable Bogoliubov model, is solved for any temperature and any chemical potential. It is found that its thermodynamics coincides with one for the Mean-Field Gas for small chemical potential or high temperature. However, for large chemical potential or low temperature, a non-conventional Bose condensation appears with, even at zero-temperature, a (non-zero) particle density outside the condensate. Following [2], the analysis in the present paper corresponds to the main technical step to deduce, in the canonical ensemble, a new microscopic theory of superfluidity at all temperatures explained in [3].
1 Introduction Let an interacting homogeneous gas of n spinless bosons with mass m be enclosed 3
in a cubic box Λ = × L ⊂ R3 . We denote by ϕ (x) = ϕ (x) a (real) two-body α=1
interaction potential and we assume that: (A) ϕ (x) ∈ L1 R3 . (B) Its (real) Fourier transformation λk = d3 xϕ (x) e−ikx , k ∈ R3 , R3
satisfies: λ0 > 0 and 0 ≤ λk = λ−k ≤
lim λk for k ∈ R3 .
k→0+
(C) The interaction potential ϕ (x) satisfies: λ0 λ0 + g00 ≥ 0, or (C2) : + g00 < 0, 2 2 where the (effective coupling) constant g00 equals 2 1 3 λk d k < 0, g00 ≡ − 3 εk 4 (2π) R3 (C1) :
(1.1)
with the one-particle energy spectrum defined by εk ≡ 2 k 2 /2m. The last conditions (C1)–(C2) will be important at the end of this paper and first appeared in the study of the Weakly Imperfect Bose Gas [1, 4–6].
406
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
The (non-diagonal) AVZ-Hamiltonian [1], also called the superstable Bogoliubov Hamiltonian, is defined for λ0 > 0 by SB B HΛ,λ ≡ HΛ,0 + UΛMF , 0 >0
(1.2)
where UΛMF B HΛ,λ 0
≡
λ0 2V
a∗k1 a∗k2 ak2 ak1 =
k1 ,k2 ∈Λ∗
λ0 2 NΛ − NΛ , 2V
≡ TΛ + UΛD + UΛN D + UΛBMF ,
and NΛ
≡
(1.3) (1.4)
a∗k ak ,
k∈Λ∗
TΛ
≡
εk a∗k ak ,
k∈Λ∗
UΛD UΛN D UΛBMF
≡ ≡ ≡
1 2V 1 2V
k∈Λ∗ \{0}
λk a∗0 a0 a∗k ak + a∗−k a−k ,
(1.5)
2 λk a∗k a∗−k a20 + a∗0 ak a−k ,
(1.6)
k∈Λ∗ \{0}
λ0 ∗2 2 λ0 ∗ a a + a0 a0 2V 0 0 V
a∗k ak .
(1.7)
k∈Λ∗ \{0}
SB acts on the boson Fock space The Hamiltonian HΛ,λ 0 +∞
(n)
FΛB ≡ ⊕ HB , n=0
(n)
with HB defined as the symmetrized n-particle Hilbert spaces (n) (0) HB ≡ L2 (Λn ) symm , HB ≡ C, see [7, 8]. Using periodic boundary conditions, let 2πnα ∗ 3 , nα = 0, ±1, ±2, . . . , α = 1, 2, 3 Λ ≡ k ∈ R : kα = L ∗ be the set of wave vectors. Also, note that a# k = {ak or ak } are the usual boson 1 creation / annihilation operators in the one-particle state ψk (x) = V − 2 eikx , k ∈ Λ∗ , x ∈ Λ, acting on the boson Fock space FΛB . Under assumptions (A) and (B) SB on the interaction potential ϕ (x) the Hamiltonian HΛ,λ is superstable [8]. 0 To fix the notations, β > 0 is the inverse temperature, µ the chemical potential, ρ > 0 the fixed full particle density. Before we embark on the rigorous results of this model, it may be useful to give briefly its origin and history.
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
407
B First note that, for λ0 > 0, the Hamiltonian HΛ,λ (1.4) is the Bogoliubov 0 >0 Hamiltonian [9–13], so-called the Weakly Imperfect Bose Gas. It is the starting point of the first microscopic theory of superfluidity proposed in 1947 by Bogoliubov [9–11]. Resuming the observations of [1, 2, 6], in various respects the BoB is not appropriate as the model of superfluidity. The first goliubov model HΛ,λ 0 problem of this theory was highlighted by Angelescu, Verbeure and Zagrebnov in B . 1992 [1]. It concerns the instability1 for positive chemical potential µ > 0 of HΛ,λ 0 In a sense the Bogoliubov theory is a series of recipes, which, after the first ansatz B , give a formula (the second ansatz) saving the theory from instaleading to HΛ,λ 0 bility for µ > 0 and from the gap in the spectrum. For a more detailed discussion of this problem, see the review [6]. Therefore, a “minimal” stabilization of the Bogoliubov Hamiltonian is to add the “forward scattering” interactions between particles above zero-mode. This apSB . Their proach was first developed in papers [1,14,15] and leads to the model HΛ,λ 0 main object was of course to correct the instability for positive chemical potenB but also to find a gapless Bogoliubov tials of the Bogoliubov Hamiltonian HΛ,λ 0 SB , spectrum. In [1], they use a Bogoliubov approximation partially applied on HΛ,λ 0 MF i.e., they save the Mean-Field interaction UΛ (1.3), whereas in [15], the authors use a “generalized” Bogoliubov approximation. This “generalized” ap √ Bogoliubov √
proximation corresponds to partially change the operators a0 / V , a∗0 / V by a
suitable function b (c) , b (c) in (1.2) except in the Mean-Field interaction UΛMF . Then, they prove a Bose condensation in zero-mode via second-order phase transition and a linear asymptotic of the elementary excitation spectrum in condensed phase for k → 0, see also discussions in Section 3.4 of [6]. Here we show that the two procedures [1,15] are inexact, in the sense that they SB . For are equivalent to some drastic modifications of the original Hamiltonian HΛ,λ 0 example, as Bogoliubov did, they were forced in [1] to add some additional assumptions to find a gapless spectrum. As it is explained with more details in [2], it was SB , in the grand-canonical ensemble, had unlikely that the exact solution of HΛ,λ 0 a gapless spectrum even in the presence of Bose-condensation. In fact, we prove that, on the thermodynamic level, the spectrum always has a gap in the grandcanonical ensemble (see below Remark 2.4). The main problem of their methods (Bogoliubov et al.) is to assume, a priori, the Bose condensation by directly doing the Bogoliubov approximation with an arbitrary choice of c or b (c) , without exactly solving it in terms of the thermodynamic behavior. As the review [6] explained in the “outline” section, we should be discouraged “from performing sloppy manipulations with Bose condensations, quantum fluctuations and different kinds of ans¨ atze”. The analysis in the present paper provides another strong warning in doing it. SB correActually, in the grand-canonical ensemble, the Hamiltonian HΛ,λ 0 sponds to a weaker truncation than the Bogoliubov one. This non-diagonal Bose 1 The
corresponding grand-canonicale pressure is infinite
408
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
gas is here rigorously solved at the thermodynamic level in the grand-canonical ensemble. We do not search for a gapless spectrum in the grand-canonical ensemble. However, this result represents a crucial technical step to find a new microscopic theory of superfluidity with a gapless spectrum. The philosophy of this new approach is explained in [2] and comes from a constructive criticism of the Bogoliubov theories. In particular, the paper [2] gives the (new) physical arguments leading to SB as a tool in the grand-canonical ensemble. use the superstable Hamiltonian HΛ,λ 0 Then, using the present paper by choosing λ0 > 0 as an arbitrary parameter, we provide in [3] a new theory of superfluidity with a gapless spectrum at any particle densities and temperatures. It leads us to a deeper understanding of the Bose condensation phenomenon in liquid helium: coexistence in the superfluid liquid of particles inside and outside the Bose condensate (even at zero temperature), Bose/Bogoliubov distribution, “Cooper-type pairs” in the Bose condensate. In the next section, we present the exact thermodynamic behavior of the SB in the grand-canonical ensemble. In particular, the infinite Hamiltonian HΛ,λ 0 volume pressure is explicitly found via variational problems in theorem 2.2. These variational problems are then solved in theorem 2.3 leading to the exact phase diagram. The corresponding phase transition is finally explained by the last theorem of Section 2 (theorem 2.5). It concerns the existence of a non-conventional Bose condensation for large chemical potential µ or high inverse temperatures β. Meantime, even for β → +∞, i.e., for a zero-temperature, only a fraction of the full density is in the condensate: there is a coexistence of particles inside and outside the condensate (see (2.20)). Note that this last phenomenon is already known as the depletion of the condensate. Some discussions corresponding to this problem can be found in [6, 12, 13, 16–21]. To simplify our purpose, the proofs are given in Section 3. They are technically based on two papers [22, 23]. First we use the proof of the exactness of the Bogoliubov approximation in the grand-canonical ensemble for a superstable gas [8], as done by Ginibre [22]. Then, we use the “superstabilization” method [23]. Note that we recall the Bogoliubov u-v transformation in the appendix.
2 Thermodynamics in the grand-canonical ensemble SB In this section we give the thermodynamic behavior of the AVZ-model HΛ,λ in 0 the grand-canonical ensemble. Before entering this study recall the definitions of SB the grand-canonical pressure pSB Λ (β, µ) and particle density ρΛ (β, µ) associated SB with HΛ,λ0 :
pSB Λ (β, µ) ≡ ρSB Λ (β, µ) ≡
SB 1 ln T rFΛB e−β (HΛ,λ0 −µNΛ ) , βV NΛ (β, µ) = ∂µ pSB Λ (β, µ) . V H SB Λ,λ0
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
409
Here − H SB (β, µ) represents the (finite volume) grand-canonical Gibbs state for Λ SB HΛSB . From the superstability of the Hamiltonian HΛ,λ it follows that pSB Λ (β, µ) 0 is defined for every pair (β, µ) ∈ QS ≡ {β > 0} × {µ ∈ R} , even in the thermodynamic limit [8].
2.1
The grand-canonical pressure
The first step is to use the Bogoliubov approximation, i.e., √ √ a0 / V → c ∈ C, a∗0 / V → c ∈ C, SB SB SB (µ) ≡ HΛ,λ −µNΛ . Since the model HΛ,λ is superstable for the Hamiltonian HΛ,λ 0 0 0 [8], Ginibre [22] proves the exactness of the Bogoliubov approximation in the sense that SB SB (β, µ) = sup p (β, µ, c) ≡ lim p (β, µ, c) , pSB (β, µ) = lim pSB sup Λ Λ Λ
Λ
c∈C
c∈C
(2.1) with pSB Λ (β, µ, c) ≡
SB 1 ln T rFB e−βHΛ,λ0 (µ,c) , pSB (β, µ, c) ≡lim pSB Λ (β, µ, c) . (2.2) Λ βV
Here
+∞
(n)
FB ≡ ⊕ HB,k=0 n=0
(n)
is the boson Fock space of the symmetrized n-particle Hilbert spaces HB,k=0 for non-zero momentum bosons. Note that SB B (µ, c) = HΛ,λ (µ, c) + HΛ,λ 0 0
λ0 2 NΛ,k=0 − NΛ,k=0 with λ0 > 0, 2V
(2.3)
B where HΛ,λ (µ, c) is defined by (A.1) in the appendix, and NΛ,k=0 is the operator 0 of the number of particles outside the zero-mode.
Remark 2.1 The applicability of the Ginibre’s proof [22] on the exactness of the Bogoliubov approximation concerns any superstable Bose systems of linear form of order 4 in operators a0 , a∗0 . The main difficulty is to control the upper bound of the pressure in the thermodynamic limit. This is mostly performed by some algebra: Taylor expansion around a0 , a∗0 , and then explicit calculations or estimations in relation with the superstability property. Briefly, this approach leads to an quadratic form in operator δa0 ≡ a0 − a0 and δa∗0 , allowing to use the standard reasoning of the Approximation Hamiltonian Method (see [24, 25]).
410
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
Then, the second step to find the thermodynamic limit pSB (β, µ) of the pressure uses [23] and gives the following result: Theorem 2.2 Let pB 0 (A.9) be the thermodynamic limit of the pressure of the nonB (α, c), then superstable Hamiltonian HΛ,0
p
SB
(β, µ) =sup p c∈C
SB
(β, µ, c) = sup x=|c|2 ≥0
inf
α≤0
pB 0
(µ − α) (β, α, x) + 2λ0
2
, (2.4)
for any (β, µ) ∈ QS . To discuss this theorem, let us consider the Mean-Field Hamiltonian HΛMF = TΛ + UΛMF ≡ TΛ +
λ0 2 NΛ − NΛ , 2V
(2.5)
see [26–32]. Then, by theorem 2.2 and (A.9), we get the following lower bound for the pressure:
2 (µ − α) pSB (β, µ) ≥ inf pB 0 (β, α, 0) + α≤0 2λ0
2 (µ − α) P BG = inf p (β, α) + (2.6) = pMF (β, µ) , α≤0 2λ0 where pP BG (β, α) and pMF (β, µ) are the (infinite volume) pressures respectively for the Perfect Bose Gas and the Mean-Field Bose Gas, see [23, 26–32]. Let α (x) ≡ α (β, µ, x) be the solution of
2 2 (µ − α) (µ − α) (2.7) = pB inf pB 0 (β, α, x) + 0 (β, α, x) + α≤0 2λ0 2λ0 α=α(x) for any fixed x ≥ 0. Thus we have
2 (µ − α) (µ − α) B B ∂α p0 (β, α, x) + = ρ0 (β, α, x) − =0 2λ0 λ0 α=α(x)≤0 α=α(x)≤0 (2.8) for chemical potentials µ ≤ µc (β, x) ≡ λ0 ρB 0 (β, 0, x) ,
(2.9)
whereas for µ ≥ µc (β, x) and α ≤ 0 the corresponding derivative is negative: ρB 0 (β, α, x) −
(µ − α) ≤ 0 which implies α (x) = 0. λ0
(2.10)
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
411
Here, ρB 0
(β, α, x)
≡
∂α pB 0 +
(β, α, x) = x +
1 3
(2π)
R3
1 3
(2π)
x2 λ2k
R3
B B fk,0 + Ek,0 2Ek,0
f k,0B d3 k B eβEk,0 − 1 Ek,0
d3 k.
(2.11)
P BG (β, α ≤ 0) is the critical density of the Perfect Bose Note that ρB 0 (β, α, 0) = ρ Gas. = Since all functions depend only on x = |c|2 , in the following we denote by x x (β, µ) the solution of the first variational problem of Theorem 2.2:
2 (µ − α) pSB (β, µ) = inf pB ) + , (2.12) 0 (β, α, x α≤0 2λ0
and we solve it via the following theorem. Theorem 2.3 For any β > 0, there exists a unique µc (β) such that 2 (µ − α (0)) B = pMF (β, µ) , for µ ≤ µc (β) . p0 (β, α (0) , 0) + 2λ0
SB p (β, µ) = 2 (µ − α (x)) B , for µ > µc (β) . p0 (β, α (x) , x) + 2λ0 x= x>0 The function µc (β) is bijective from [a, +∞) to R+ and we denote by βc (µ) ≥ 0 the inverse function of µc (β), see Figure 2.1. Here a = 0 if (C1) holds whereas if (C2) is satisfied a = µ0 ≡ µc (β = ∞) < 0. The pressure pSB (β, µ) is continuous for µ = µc (β) . Remark 2.4 Since one x) µc (β) or β > βc (µ) ,
412
S. Adams and J.-B. Bru
λ 0=0
Ann. Henri Poincar´e
λ0 θ c =1/βc (µ)
λ0
(C2) holds (C2) holds (C1) holds
λ0
(C1) holds
+ −0.5φ(0)
µ0
0
µ
Figure 2.1: Illustration of the critical temperature θc = 1/βc as a function of µ. Each curve corresponds to a different value of λ0 (from λ0 = +∞ to 0+ ): when (C1) holds, i.e., λ0 is sufficiently large, the curve starts at µ = 0 in contrast to the cases where (C2) holds. The term λ0 = 0 in this figure corresponds to the model B HΛ,0 where the pressure diverges for µ = −ϕ (0) /2 and not to the Perfect Bose Gas. = x (β, µ) and α (x) = where ρB 0 (β, α (x) , x) is defined by (2.11) and with x α (β, µ, x) the solutions of the variational problems. Now our main results concern the particle densities inside and outside the zero-mode, and are given in the following theorem. Theorem 2.5 Under the assumptions of the previous two theorems it follows for µ = µc (β) or β = βc (µ): (i) A non-conventional Bose condensation induced by the non-diagonal interaction UΛN D for large chemical potentials (high particle densities), or low temperatures: ∗ a0 a0 = 0 for µ < µc (β) or β < βc (µ) . (β, µ) = x (β, µ) = lim > 0 for µ > µc (β) or β > βc (µ) . Λ V SB H Λ,λ0
(ii) No Bose condensation (of any type I, II or III [33–35]) outside the zero-mode: ∗ ak ak ∗ ∀k ∈ Λ \ {0} , lim (β, µ) = 0. Λ V SB HΛ,λ 0 1 ∗ lim lim a a H SB (β, µ) = 0. k k δ→0+ Λ V Λ,λ0 {k∈Λ∗ ,0 0, lim a∗k ak H SB (β, µ) = Λ
Λ,λ0
1 . eβ(εk −α(0)) − 1
But for µ > µc (β) or β > βc (µ) , i.e., in the presence of a Bose condensation, we get another one, which we call the Bogoliubov distribution, for a corresponding chemical potential α ( x) < 0: lim a∗k ak H SB (β, µ) Λ,λ0 Λ 2 2 f x λk k,0B + = x=x,α=α(x) E B eβEk,0 − 1 B B fk,0 + Ek,0 2Ek,0 k,0 for any k ∈ Λ∗ such that k ≥ δ for all δ > 0. Remark 2.6 (a) Assuming condition (C2), a discontinuity of the densities appears because the direct term of repulsion λ0 2 λ0 ∗2 2 a0 a0 = N0 − N0 , with N0 ≡ a∗0 a0 , (2.14) 2V 2V in (1.2) becomes too weak to beat the attraction induced by UΛN D (1.6). The nondiagonal interaction UΛN D express itself via the effective coupling constant g00 (1.1) [5, 6]. (b) The depletion (Bogoliubov distribution) coincides with the one found for λ0 > 0 [6, 21] at a chemical potential α = 0 (high density regimes) in the thermodynamic B (1.4), so-called the Weakly Imbehavior of the Bogoliubov Hamiltonian HΛ,λ 0 >0 perfect Bose Gas.
414
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
We conclude this section by additional comments about the nontriviality of the Bose condensate at low temperatures, when (C1) holds. Indeed, in this case note that the corresponding kinetic part only turns on the Bose condensation phenomenon via the Bose distribution. Indeed, the solution of the variational problem α ( x) (2.7) for a Bose condensate density x (β, µ) , becomes zero when we reach the critical temperature as for the Mean-Field Bose Gas, but switches again to strictly negative values for x > 0 (Section 3). As soon as the Bose condensate appears, the non-diagonal interaction UΛN D becomes sufficiently important to drastically change all thermodynamic properties of the system by instantly switching the usual Mean-Field Bose Gas to a system of quasi-particles: the Bose-Einstein condensation becomes non-conventional. Whereas the non-diagonal interaction UΛN D is not strong enough to imply alone the Bose-condensation at the critical temperature of the Mean Field Gas when (C1) is satisfied, it strongly dominates all thermodynamics. In the case (C1), the origin of the nontriviality of the Bose condensate is then delicate compare to other Bose systems as for example the Weakly Imperfect Bose Gas [4, 6, 21]. In particular, when (C1) holds, this phenomenon is rather different from the thermodynamic behavior of the Weakly Imperfect Bose Gas [4, 6, 21] ole anymore. where UΛN D plays no rˆ All these previous arguments in the grand-canonical ensemble are valid only for λ0 > 0 and λk > 0 for k < A. Indeed in the limit λ0 0 (and λk 0), notice that (C1) holds but UΛN D does not exist anymore, and we obtain the Perfect Bose Gas, where the Bose condensation is conventional. For more information about this different but expected thermodynamic behavior, see [2].
2.3
The particle density as parameter in the grand-canonical ensemble
Let us consider the fixed particle density ρ in the grand-canonical ensemble which defines a unique chemical potential µβ,ρ satisfying ρSB (β, µβ,ρ ) = ρ.
(2.15)
Actually, at a fixed inverse temperature β the function µβ,ρ is the inverse function of the mean particle density ρSB (β, µ) of the superstable Bogoliubov Hamiltonian. (i) Let
ρc,inf (β) ≡
lim
ρSB (β, µ) .
lim
ρSB (β, µ) .
µ→µ− c (β)
ρc,sup (β) ≡
µ→µ+ c (β)
(2.16)
Recall that µc (β) and βc (µ) are defined in Theorem 2.3 (Figure 2.1). Through (iv) of Theorem 2.5 combined with (2.13) and (3.34), we deduce that ρc (β) ≡ ρc,inf (β) = ρc,sup (β) ≤ ρP BG (β, 0) if condition (C1) is satisfied. At fixed particle density ρ note that we can also define by βc (ρ) the unique critical inverse temperature.
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
415
If condition (C1) is satisfied, an illustration of the behavior of βc (ρ) for a fixed density is performed in Figure 2.2. Because of the interaction note that the critical inverse temperature βc (ρ) is always smaller or equal than the one for the MeanField Bose Gas (which is equal to the one for the Perfect Bose Gas). Unfortunately, at very high densities we are not able to prove an exact equality or the opposite, see Figure 2.2.
βc (ρ)
β 0
ρc (β)
ρ
Figure 2.2: Illustration of the critical inverse temperature βc (ρ) when condition (C1) is satisfied. The dotted line corresponds to the phase digram of the MeanField Bose Gas. The difference with the Mean-Field Bose Gas is always greater or equal to zero. It may be zero for all β > 0 (only at high densities, we are not able to prove an exact equality or the opposite). (ii) By Remark 3.1 and (2.13) we have x) < 0 for ρ ∈ / [ρc,inf (β) , ρc,sup (β)] or β = βc (ρ) . µβ,ρ − λ0 ρ = α ( (iii) Combining Theorem 2.3 with (2.13) and (2.17) we get B λ0 SB + ρ2 p (β, µβ,ρ ) = p0 (β, α (x) , x) 2 x= x
(2.17)
(2.18)
for any ρ > 0, where α ( x) < 0 is the unique solution of the Bogoliubov density equation: ρ = ρB ) for ρ ∈ / [ρc,inf (β) , ρc,sup (β)] or β = βc (ρ) . 0 (β, α, x
(2.19)
(iv) For ρ < ρc,inf (β), one has µβ,ρ < µc (β) , whereas µβ,ρ > µc (β) for ρ > ρc,sup (β), and Theorem 2.5 is still valid for any ρ ∈ / [ρc,inf (β) , ρc,sup (β)], i.e., for
416
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
µβ,ρ = µc (β) or β = βc (ρ). In fact, Theorem 2.5 is verified even for µ = µc (β) or β = βc (µ) but only if (C1) is satisfied (all densities are continuous). Then, for any ρ > ρc,sup (β) there is only one Bose condensation in the zero mode, whereas for ρ < ρc,inf (β) ≤ ρP BG (β, 0) the system behaves as the MeanField Bose Gas (2.5) with no Bose condensations. In particular, note that the result (ii) of Theorem 2.5 excludes any coexistence of non-conventional and conventional Bose condensation, as it appears for high densities in the Weakly Imperfect Bose Gas [6, 21]. Actually, for ρ ∈ [ρc,inf (β) , ρc,sup (β)] , when (C2) holds, the question of Bose condensation is still open in the grand-canonical ensemble. This unsolved problem appears also in the study of the Weakly Imperfect Bose Gas [4, 6, 21] and is quite similar. In this regime, when (C2) is satisfied, it should simply be a coexistence of two phases, see for example Section 4 in [31]. In fact we explain in [3] that this question is not relevant in the canonical ensemble, in the sense that the canonical thermodynamics of the AVZ-Hamiltonian should correspond to the grand-canonical one with the case (C1). Indeed, the superstable interaction UΛMF (1.3) is just a constant in the canonical ensemble. This question is related to the continuity of the particle density which is only satisfied when (C1) holds, see also Remark 2.7. (v) To conclude, note that we have a non-zero particle density outside the zeromode for any ρ > 0 even for zero-temperature: 1 lim lim a∗k ak H SB (β, µβ,ρ ) Λ,λ0 β→+∞ Λ V ∗ k∈Λ \{0} 1 2 2 x λ k 3 d k > 0, = 3 x=x B B (2π) f 2E + E k,0 α=α(x) k,0 k,0 3 R
∀k ∈ Λ∗ , k ≥ δ > 0, lim lim a∗k ak H SB (β, µβ,ρ ) Λ,λ0 β→+∞ Λ 2 2 x λk = > 0. 2E B f + E B x=x k,0 α=α( x) k,0 k,0
(2.20)
In the regime ρ > ρc,sup (β) , the system follows the Bogoliubov distribution (v) of Theorem 2.5, whereas in the absence of the Bose condensation, i.e., for ρ < ρc,inf (β), the (standard) Bose distribution holds. (vi) If we analyze the system as a function of the parameter λ = λ0 in UΛMF (1.3) for a fixed density ρ ∈ / [ρc,inf (β) , ρc,sup (β)] we can find ∂λ0 x = 0 by direct computations (see also [3]). An illustration of the behavior of x for a fixed density is performed in Figure 2.3. This is also true for α ( x). Actually, in [3] we prove that the solutions α ( x) = (λ0 ) of the variational problems (2.7) and (2.12) are also solutions in α ( x, λ0 ) and x the canonical ensemble of other variational problems and they do not depend on the
Vol. 5, 2004
λ 0=0
Exact Solution of the AVZ-Hamiltonian
λ0
x(β,µ ) 2
417
λ0
(C2) holds (C2) holds (C1) holds
x(β,ρ )
2
λ0
(C1) holds
+ −0.5φ(0)
µ β,ρ
0
µ β,ρ
µ
Figure 2.3: Illustration of x (β, µ) and x (β, µβ,ρ ) = x (β, ρ) for a fixed particle density ρ > 0 in the grand-canonical ensemble. Each curve corresponds to a different value of λ0 in UΛMF (from λ0 = +∞ to 0+ ): when (C1) holds, i.e., λ0 is sufficiently large, there is no jump in contrast with the cases where (C2) holds. B The term λ0 = 0 in this figure corresponds to the model HΛ,0 where the pressure diverges for µ = −ϕ (0) /2 and not to the Perfect Bose Gas. / [ρc,inf (β) , ρc,sup (β)]. Indeed, parameter λ0 in UΛMF for a fixed particle density ρ ∈ the particle density ρSB (β, µ) depends on λ0 for fixed µ, and then µβ,ρ = µβ,ρ (λ0 ) at a fixed density ρ ∈ / [ρc,inf (β) , ρc,sup (β)]. However, as an example, the solution x = x (β, µ, λ0 ) of the variational problem (2.12) is such that x = x (β, ρ) = x (β, µβ,ρ (λ0 ) , λ0 ) does not depend on λ0 ≥ 0 (Figure 2.3). At fixed particle densities ρ > 0, all other densities in the grand-canonical ensemble do not depend on the value λ = λ0 in UΛMF outside the phase transition. Note that the first term in conditions (C1) and (C2) only comes from UΛMF (see Remark 3.4). Remark 2.7 This last phenomenon is expected since for any λ0 > 0 the HamilSB B tonians HΛ,λ (1.2) and HΛ,0 differ only by a constant on the symmetrized n0 (n=[ρV ])
, i.e., in the canonical ensemble. Consequently, in particle Hilbert spaces HB the canonical ensemble all densities do not depend on the value λ = λ0 in UΛMF for a fixed particle density. This phenomenon is true in the grand-canonical ensemble, briefly because of the strong equivalence of ensembles (see [32,36–39] for the notion of strong equivalence). Here n = [ρV ] is defined as the integer of ρV .
418
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
3 Proofs The aim of this section is to give the promised details of the proofs of Theorems 2.2, 2.3 and 2.5.
3.1
Proof of Theorem 2.2
By (2.1) the proof consists of getting the thermodynamic limit pSB (β, µ, c) of the SB pressure pSB Λ (β, µ, c) associated with the Hamiltonian HΛ,λ0 (µ, c). We calculate the pressure pSB Λ (β, µ, c) via another related Hamiltonian defined in part 1. In 2. we consider the specific free energy densities in the canonical ensemble, whereas in 3. we show convexity, such that weak-equivalence of ensembles implies the theorem in part 4. 1. Let
B (γ, c) ≡ H B (γ, c) + γ |c|2 V − λ0 |c|4 V − |c|2 , H Λ,λ0 Λ,λ0 2
(3.1)
B (γ, c) and H SB (µ, c) are well defined on the see (A.1). The Hamiltonians H Λ,λ0 Λ,λ0 boson Fock space FB for any fixed c ∈ C. Here we use two chemical potentials γ B (γ, c) and H SB (µ, c). From the appendix, and µ respectively for the models H Λ,λ0 Λ,λ0 B (γ, c) is diagonalizable by the Bogoliubov canonical u-v the Hamiltonian H Λ,λ0 B transformation, see (A.7), and one gets a perfect Bose gas with a spectrum Ek,λ 0 (A.5). We then have pB Λ,λ0
(β, γ, c) =
pB Λ,λ0
for
λ0 (β, γ, c) − γ |c| + 2 2
2
γ ≤ |c| λ0 +
min
k∈Λ∗ \{0}
2
|c| |c| − V 4
! (3.2)
εk ,
see (A.6) and (A.8) in the appendix. The thermodynamic limit follows as λ0 2 2 B pB x , β, γ, x = |c| ≡lim pB λ0 Λ,λ0 (β, γ, c) = pλ0 (β, γ, x) − γx + Λ 2 cf. (A.9) for γ ≤ xλ0 = lim Λ
2. Note that
B H
" |c|2 λ0 +
# min ∗
k∈Λ \{0}
εk
and λ0 > 0.
$ SB %
0, HΛ,λ (γ, c) , N (µ, c) , NΛ,k=0 = 0. Λ,k=0 = Λ,λ0 0
(3.3)
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
419
B However, for a fixed particle density ρ1 > 0, let fΛ,λ (β, ρ1 , c) and fΛSB (β, ρ1 , c) 0 be the free-energy densities: " #
1 B (0,c) (n,k=0) −β H B Λ,λ0 e ln T r (β, ρ , c) ≡ − fΛ,λ , (n) 1 0 HB,k=0 βV " # (3.4)
(n,k=0) SB 1 e−βHΛ,λ0 (0,c) ln T rH(n) , fΛSB (β, ρ1 , c) ≡ − B,k=0 βV
where
(n)
A(n,k=0) ≡ A HB,k=0 is the restriction of any operator A acting on the boson Fock space FB (n = [ρ1 V ] is defined as the integer of ρV ). Note that
pB Λ,λ0 (β, γ, c) =
+∞ n n 1 B eβV {γ V −fΛ,λ0 (β, V ,c)} . ln βV n=0
(3.5)
B The free-energy density fΛ,λ (β, ρ1 , c) is in fact well defined for any ρ1 > 0 and 0 β > 0 in the thermodynamic limit, i.e., 2 B (β, ρ1 , c) < +∞. fλB0 β, ρ1 , x = |c| ≡ lim fΛ,λ 0 Λ
From (2.3) and (3.1) we then have λ0 2 ρ1 λ0 B fΛSB (β, ρ1 , c) = fΛ,λ ρ1 − − µ |c|2 + (β, ρ , c) + 1 0 2 V 2
|c|2 |c| − V 4
! ,
which gives λ0 λ0 2 f SB β, ρ1 , x = |c| ≡ lim fΛSB (β, ρ1 , c) = fλB0 (β, ρ1 , x) + ρ21 − µx + x2 . Λ 2 2 3. Notice that we do not know if the specific free energy fλB0 (β, ρ1 , x) is convex as a function of ρ1 , which is crucial in order to use [23] for our proof. It is the next step of the proof. By (3.2), there is a unique solution γΛ (ρ1 ) of ∂γ pB Λ,λ0 (β, γΛ (ρ1 ) , c) = ρ1
(3.6)
at all densities ρ1 > 0. By direct computations of (3.6) done via (3.3), the corresponding thermodynamic limit γ (ρ1 ) ≡lim γΛ (ρ1 ) = Λ
< xλ0 for ρ1 < ∂γ pB λ0 (β, λ0 x, x) , xλ0 for ρ1 ≥ ∂γ pB λ0 (β, λ0 x, x) ,
420
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
is an increasing continuous function of ρ1 > 0. By (3.5) we also have pB λ0 (β, γ (ρ1 ) , x)
B ≡ lim pB Λ,λ0 (β, γΛ (ρ1 ) , c) = γ (ρ1 ) ρ1 − fλ0 (β, ρ1 , x) Λ
= sup γ (ρ1 ) t − fλB0 (β, t, x) . (3.7) t>0
Therefore for any ρ1 > 0, ∂ρ1 fλB0 (β, ρ1 , x) = γ (ρ1 ) is an increasing function of ρ1 > 0, i.e., fλB0 (β, ρ1 , x) is a convex function of ρ1 > 0. B (γ, c) for 4. The weak equivalence of ensembles is then verified by the model H Λ,λ0 2 each fixed x = |c| ≥ 0, and using [23] combined with (2.3) and (3.1) we directly find
2 (µ − γ) λ0 2 SB B p (β, µ, c) = inf . (3.8) pλ0 (β, γ, x) + + µx − x γ≤xλ0 2λ0 2 x=|c|2 Therefore the theorem follows by (2.1), (3.3) and the last equality, if we take α = γ − xλ0 ≤ 0 in the expression for the infimum and finally switch from pB λ0 (β, γ, x) B to p0 (β, α, x).
3.2
Proof of Theorem 2.3
From Theorem 2.2 we get pSB (β, µ) =sup {Fβ (α (x) , x)} = {Fβ (α (x) , x)} x≥0
,
x= x
where the function Fβ (α, x) is given by (µ − α)2 (µ − α)2 Fβ (α, x) ≡ pB = ξ0 (β, α, x) + η0 (α, β) + . 0 (β, α, x) + 2λ0 2λ0 2 (µ − α) F∞ (α, x) ≡ lim Fβ (α, x) = η0 (α, x) + . β→∞ 2λ0 (3.9) (β, α, x) = ξ (β, α, x) + η (α, x) is defined by (A.9) in the We recall that pB 0 0 0 appendix. So, we have to evaluate the sign of ∂x {Fβ (α (x) , x)} = {∂x Fβ (α, x)} + {∂x α (x) ∂α Fβ (α, x)} (3.10) α=α(x)
α=α(x)
to obtain x = x maximizing the function Fβ (α (x) , x). The proof is then divided in four parts. First we get in 1. an easier expression of the derivative of the functional Fβ (α(x), x): the second term of (3.10) is in fact zero.
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
421
In the second step 2. we study the solution α(x) and the corresponding critical chemical potential µc (β, x) (2.9). In 3. and 4. we get the first results for β → ∞ and then for arbitrary finite β. 1. Through (3.9) one has ∂α Fβ (α, x) = ρB 0 (β, α, x) −
(µ − α) . λ0
(3.11)
Then, by (2.8)–(2.10), one has {∂α Fβ (α, x)} = 0 for µ < µc (β, x) . α=α(x) µc (β, x) . {∂α Fβ (α, x)}
(3.12)
α=α(x)=0
{∂x α (x) ∂α Fβ (α, x)}
Therefore
=0
α=α(x)
for any fixed µ and so (3.10) can be written as
∂x {Fβ (α (x) , x)} = {∂x Fβ (α, x)}
.
(3.13)
α=α(x)
Notice that α (x) = α (β, µ, x) is also a function of the inverse temperature and chemical potential and, in the same way, we get {∂ α (x) ∂ F (α, x)} = 0. β α β α=α(x) (3.14) = 0. {∂µ α (x) ∂α Fβ (α, x)} α=α(x)
2. By (2.9) and (2.11) note that lim µc (β, x) = +∞.
x→+∞
Moreover we have ∂x ρB 0 (β, α, x) =1+ −
1
1 3
2 (2π) 3
4 (2π)
R3
√
xλ2k
" 1+
2 B
εk − α (εk − α + 2xλk )3/2 eβEk,0 − 1 R3 " # εk − α + xλk β d3 k. λk εk − α + 2xλk sinh2 βE B /2 k,0
#
d3 k
(3.15)
422
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
Since the last in term in (3.15) vanishes when β → ∞ for all x ≥ 0 and all α ≤ 0, ∂x ρB 0 (β, α, x) > 0 for sufficiently high β. Thus P BG (β, 0) > 0 inf µc (β, x) = λ0 inf ρB 0 (β, 0, x) = µc (β, 0) = λ0 ρ
x≥0
x≥0
(3.16)
for sufficiently high β > 0 and the critical chemical potential µc (β, x) is an increasing function of x ≥ 0. Consequently there is for µ > µc (β, 0) a solution xµ > 0 of (3.17) µc (β, xµ ) = µ,
such that α (x) =
0 , for 0 ≤ x ≤ xµ , < 0 , for x > xµ > 0,
(3.18)
and for all x2 > x1 > xµ , α (x2 ) < α (x1 ) and
lim α (x) = −∞.
x→+∞
(3.19)
To summarize the behavior of α (x) = α (β, µ, x) : lim α (β, µ, x)
=
−∞ for β, µ fixed,
lim α (β, µ, x)
=
−∞ for β, x fixed,
lim α (β, µ, x)
=
−∞ for µ, x fixed.
x→+∞
µ→−∞ β→0+
(3.20)
3. We consider now the limit β → ∞. To analyze the derivative of the functional F∞ (α(x), x) we only have to consider the partial derivative with respect to x, because we get the same results for F∞ (α(x), x) as in (3.12) and (3.13) for the functional Fβ (α(x), x). Thus by (3.9) we have for any α ≥ 0 ∂x lim Fβ (α, x) = ∂x F∞ (α, x) = α + Ω (α, x) , β→+∞
where Ω (α, x) ≡
1 3
2 (2π)
R3
√ εk − α λk 1 − √ d3 k ≥ 0. εk − α + 2xλk
(3.21)
(3.22)
By direct computations of the partial derivatives with respect to α and x, we find that Ω (α, x) is a strictly increasing concave function of x ≥ 0 for any fixed α ≤ 0 with ϕ (0) , (3.23) Ω (α, 0) = 0 and lim Ω (α, x) = x→+∞ 2 whereas for any fixed x > 0, Ω (α, x) is a strictly increasing convex function of α ≤ 0 with √ εk 1 lim Ω (α, x) = 0 ≤ Ω (0, x) = λk 1 − √ d3 k. (3.24) 3 α→−∞ εk + 2xλk 2 (2π) R3
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
Via (3.16) one has
lim
β→+∞
423
inf µc (β, x)
x≥0
= 0,
i.e., we have to consider the cases µ > 0 and µ ≤ 0. 3.1. Let us first discuss the case µ > 0. By (3.17)-(3.19), there is xµ > 0 such that 0 , for 0 ≤ x ≤ xµ . α (x) = (3.25) < 0 , for x > xµ > 0. Combining (3.23)–(3.24) with the previous relation, we get Ω (0, xµ ) ≥ Ω (0, x) > 0 for 0 < x ≤ xµ and µ > 0 and the lower bound sup {F∞ (α (x) , x)} = {F∞ (α (x) , x)} > sup {F∞ (α (x) , x)} = F∞ (0, xµ ) x≥0 0≤x≤x x= x
µ
(3.26) x) < 0 for µ > 0. This first result proves the which implies x > xµ > 0 and α ( theorem for µ > 0 and β → ∞. 3.2. If µ ≤ 0 the condition (2.8) is always satisfied and gives an expression for α = α (x), i.e., α (x) = µ − λ0 ρB 0 (β, α (x) , x). Hence, since the second term in (2.11) vanishes in the limit β → ∞ we can rewrite (3.21): ∂x {F∞ (α (x) , x)} = {∂x F∞ (α, x)} = µ − λ0 ρB 0 (β, α(x), x) + Ω(α(x), x) α=α(x)
2 2 λ0 x λk 3 = µ + Ω (α, x) − λ0 x − k . (3.27) d 3 α=α(x) B B 2 (2π) f E + E k,0 k,0 k,0 3 R
Moreover, notice that ∂x2 F∞
(α, x) = −λ0 +
1 3
2 (2π)
R3
λ2k (εk − α − xλ0 ) d3 k √ 3/2 εk − α (εk − α + 2xλk )
< ∂x2 F∞ (α, x)
, (3.28) x=0
for any α ≤ 0 and x > 0, see (3.15) with β → +∞ for the derivative of the density ρB 0 (β, α, x). We also have λ2k 1 2 3 2 d k ≤ ∂x F∞ (α, x) = −λ0 + ∂x F∞ (α, x) 3 (εk − α) 2 (2π) x=0 x=0,α=0 R3 " # λ0 + g00 , = −2 2
424
S. Adams and J.-B. Bru
2 see (1.1). Therefore, by fixing the sign of ∂x F∞ (α, x)
Ann. Henri Poincar´e
the assumptions x=0,α=0
(C1)–(C2) imply two different behaviors for the solution x of the variational problem. (C1) If condition (C1) is satisfied, we find via the two previous expressions that ∂x2 F∞ (α, x) < 0 for all α ≤ 0 and x > 0, which via (3.27) implies µ + Ω (α (x) , x) − λ0 x −
λ0 x2 3
2 (2π)
R3
λ2 k d3 k B B Ek,0 fk,0 + Ek,0
< µ + Ω (α (x) , 0) = µ + Ω (α (0) , 0) = µ ≤ 0, (3.29) for x > 0, i.e., ∂x {F∞ (α (x) , x)} < 0 for all µ ≤ 0. Therefore we get x = 0. x>0
Actually, by combining the results for µ > 0 and µ ≤ 0, notice that > 0 for µ > 0, x = lim x = 0, lim x µ→0− µ→0+ x = 0 for µ ≤ 0,
(3.30)
at infinite inverse temperature (β → ∞). (C2) Assuming now condition (C2), there is a critical value α0 < 0 such that the upper bound of (3.28) becomes positive: ∂x2 F∞ (α, x) > 0, (3.31) x=0
for any α0 < α ≤ 0. Since for µ = 0 one has α (0) = 0 from (2.8), by (3.28) and (3.31) we have ∂ {F (α (x) , x)} = 0, x ∞ µ=0,x=0 (3.32) > 0, ∂x2 {F∞ (α (x) , x)} µ=0,x 0, because of continuity. Therefore from the definition of > 0 for µ = 0. Actually, there is a µ0 < 0 such that pSB (β,µ) and (3.9) we have x x > 0 for µ ≥ 0. lim x = lim− x . µ→0 µ→0+ x > 0 for µ0 < µ ≤ 0. (3.33) lim x > lim x = 0. µ→µ+ µ→µ− 0 0 x = 0 for µ < µ0 .
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
425
4. Now we consider the case of finite inverse temperatures β < +∞. By (3.9), (3.13) and (3.14) one has ∂ {F (α (x) , x)} = {∂ ξ (β, α, x) + ∂ F (α, x)} x β x 0 x ∞ α=α(x) (i) < {∂x F∞ (α, x)} , α=α(x) lim ξ0 (β, α (x) , x) = 0, x→+∞ ∂β {Fβ (α (x) , x)} = {∂β ξ0 (β, α, x)} < 0, (ii) α=α(x) lim ξ (β, α, x) = 0, β→+∞
0
for fixed µ ∈ R. By (i) for fixed µ, if x = 0 for β → ∞, then x = 0 for any β ≥ 0. Let µ > 0. By definition of Fβ (α, x) one has Fβ (α, x) > F∞ (α, x) , for µ > 0 and any fixed α ≤ 0. Since by (ii) the function Fβ (α (x) , x) is monotonically decreasing for β ∞, we find that x) , x > 0) < sup {Fβ (α (x) , x)} , Fβ (α (0) , 0) < F∞ (α ( x≥0
for sufficiently high β and large µ > 0, i.e., x > 0. Since one has (3.20), ∂x ξ0 (β, α, x) < 0 (i) and ∂β ∂x ξ0 (β, α, x) =
1 3
(2π)
R3
B
B βEk,0 e Ek,0 B d k 2 ∂x Ek,0 > 0, B βEk,0 1−e 3
for µ > 0, there is an inverse temperature βc (µ) > 0 such that x > 0 for β > βc (µ > 0) and x = 0 for β < βc (µ > 0) . The function βc (µ > 0) is bijective so we define by µc (β) > 0 the inverse function of βc (µ). Note that if µ > µc (β, 0) (2.9) then the arguments done in 2. (cf. (3.15)–(3.19)) and 3.1. still work. So, x > 0 for µ > µc (β, 0) . Consequently µc (β) ≤ µc (β, 0) = λ0 ρB 0 (β, 0, 0) ,
(3.34)
and βc (µ) is a strictly increasing function from [0, +∞) to [0, +∞) . If the condition (C2) is verified, the arguments done here in 4. for µ > 0 work also for µ > µ0 and the function βc (µ > µ0 ) is bijective. In particular the inverse function µc (β) of βc (µ) verifies: lim µc (β) = µ0 < 0,
β→+∞
and (3.33) holds for β > 0. An illustration of the critical temperature θc (µ) = 1/βc (µ) as a function of µ is given by Figure 2.1.
426
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
Remark 3.1 The solution x =x (β, µ) of (2.12) always satisfies ∂
|c|2 p
SB
(β, µ, c)
|c|2 = x
=
∂x pB 0
(β, α, x)
x =x (β, µ) = 0 for µ < µc (β) ,
= 0, for µ > µc (β) ,
x= x,α=α( x)
(3.35) see (3.9) and (3.13). Moreover, from the previous proof we can see that the solution α ( x) = α (β, µ, x ) of (2.7) is always strictly negative for any µ = µc (β) or β = βc (µ). In particular, one always has (2.8) for α = α ( x), x = x (2.12), and µ = µc (β) or β = βc (µ). Remark 3.2 Actually, as an extension for finite β of (3.30) and (3.33) we get two behaviors for x depending on conditions (C1) and (C2): = 0 for µ ≤ µc (β) . x = lim x = 0. lim x (C1) : µ→µ− µ→µ+ c (β) c (β) x > 0 for µ > µc (β) . or
= 0 for µ < µc (β) . x < lim x . 0 = lim x (C2) : µ→µ− µ→µ+ c (β) c (β) x > 0 for µ > µc (β) .
.
Remark 3.3 If condition (C1) holds, using arguments from the proof of Theorem 2.3 (3.2. and 4.) we have µc (β) ≥
inf µc (β, x) .
x≥0
Therefore, for sufficiently high β, i.e., for low temperatures (compare (3.16) and (3.34)), we have µc (β) = µc (β, 0) = λ0 ρP BG (β, 0) . P BG We recall that µc (β, x) = λ0 ρB (β, 0) is the 0 (β, 0, x) is defined by (2.9) and ρ critical density of the Perfect Bose Gas.
Remark 3.4 Notice that the proof of Theorem 2.3 does not depend on the fact that λ0 is the Fourier transformation of the interaction potential for k = 0. Actually, one could have taken as an arbitrary (strictly positive) parameter satisfying either (C1) or (C2). In [3], we explain that λ0 has no physical relevance for a fixed particle density and is then taken arbitrary large enough such that only (C1) holds with a strict inequality.
Vol. 5, 2004
3.3
Exact Solution of the AVZ-Hamiltonian
427
Proof of Theorem 2.5
1. Before we prove Theorem 2.5 for µ = µc (β) or β = βc (µ), one useful lemma is shown. Lemma 3.5 For any sequence {EΛ }Λ⊂R3 of subsets EΛ ⊆ Λ∗ such that if k ∈ EΛ then −k ∈ EΛ we have 1 ∗ B lim ak ak H SB (β, µ) = ∂γ p0 (β, α, γ, x) , Λ,λ0 Λ V α( x),γ=0,x= x ∗ k∈EΛ ⊆Λ
with pB 0 (β, α, γ, x) −1 3 B 1 1 −βEk,0 3 B ≡ ln 1 − e d k + fk,0 − Ek,0 d k 3 3 (2π) β 2 (2π) 3 3 R \E R \E 1 −1 B 1 −βEk,0 3 B 3 fk,0 − Ek,0 d k ln 1 − e d k+ + 3 α→α+γ (2π)3 β 2 (2π) E E + α + γ lim χEΛ (0) x, Λ
for any α ≤ 0 and γ ≤ 0. Here χEΛ denotes the characteristic function of EΛ and the set E is given by E ≡lim EΛ ⊆ R3 . Λ
Proof. Let pSB Λ (β, µ, γ) ≡
SB 1 ln T rFΛB e−βHΛ,λ0 ,γ (µ) βV
be the pressure associated with the perturbed (superstable) Hamiltonian SB HΛ,λ (µ) defined by: 0 ,γ SB SB HΛ,λ (µ) ≡ HΛ,λ − µNΛ − γ 0 ,γ 0
a∗k ak .
k∈EΛ ⊆Λ∗ SB Since HΛ,λ (µ) is superstable, its pressure is well defined and convex for any real 0 ,γ µ and γ. Consequently Theorems 2.2–2.3 are still valid for γ ∈ R:
pSB (β, µ, γ) ≡ = =
lim pSB Λ (β, µ, γ) Λ
(µ − α) sup inf (β, α, γ, x) + α≤0 2λ0 x≥0
2 (µ − α) pB 0 (β, α, γ, x) + 2λ0 pB 0
2
α=αγ ( xγ ),x= xγ
,
(3.36)
428
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
with the corresponding pressure pB xγ ) and x γ are the cor0 (β, α, γ, x) . Here αγ ( responding solutions of the variational problems. We also have ∂γ pSB Λ (β, µ, γ) =
1 V
k∈EΛ ⊆Λ∗
a∗k ak H SB
Λ,λ0 ,γ
(β, µ) ,
and, via the Griffiths lemma [40, 41], we get: lim Λ
1 V
k∈EΛ ⊆Λ∗
a∗k ak H SB (β, µ) = ∂γ lim pSB (β, µ, γ) Λ Λ,λ0
Λ
γ=0
= ∂γ pSB (β, µ, γ)
. (3.37) γ=0
From Remark 3.1 for µ = µc (β) or β = βc (µ) combined with (3.36) we get
2 ( x )) (µ − α γ γ ∂γ pSB (β, µ, γ) = ∂γ pB xγ ) , γ, x γ ) + 0 (β, αγ ( 2λ0 = ∂γ pB 0 (β, α, γ, x) α=αγ ( xγ ),x= xγ
for |γ| sufficiently small and µ = µc (β) or β = βc (µ) . Consequently the limit (3.37) combined with the last equation for γ = 0 gives the lemma. 2. Now we are in position to prove the five statements (i)–(v) of Theorem 2.5: (i) Using Lemma 3.5 for EΛ = {0} combined with Remark 3.2 for µ = µc (β), one gets (i). (ii) Let a, b be two arbitrary positive real numbers, with b > a ≥ 0. Lemma 3.5 with EΛ = {k ∈ Λ∗ : k ∈ (a, b)} implies 1 1 lim a∗k ak H SB (β, µ) = ξβ,µ (k) χ(a,b) (k) d3 k 3 Λ,λ0 Λ V (2π) {k∈Λ∗ ,k∈(a,b)} R3
(3.38) where χ(a,b) (k) is the characteristic function of (a, b) , and where ξβ,µ (k) is a continuous function on k ∈ R3 defined by ξβ,µ (k) ≡
1 eβ(εk −α(0))
−1
,
(3.39)
for µ < µc (β) or β < βc (µ) whereas for µ > µc (β) or β > βc (µ) 2 2 f λ x k k,0B + ξβ,µ (k) ≡ βEk,0 B B B Ek,0 e −1 2Ek,0 fk,0 + Ek,0
x= x,α( x)
.
(3.40)
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
429
This last result then implies the first limit in (ii). Moreover for a = 0 and b = δ it also gives the second limit of (ii) by taking the limit δ → 0+ in (3.38). (iii) Since
NΛ = a∗0 a0 +
a∗k ak ,
k∈Λ∗ \{0}
the limit is deduced from (2.13) and (i). (iv) is a direct consequence of Remark 3.2 combined with (i) and (iii). (v) Notice that the mean particle values a∗k ak H SB are defined on the discrete Λ,λ0
set Λ∗ ⊂ R3 . Below we denote by
gβ,µ,Λ (k) ≡ a∗k ak H SB (β, µ)
(3.41)
Λ,λ0
a continuous interpolation of these values from the set Λ∗ to R3 and we define by gβ,µ (k) the corresponding thermodynamic limit: gβ,µ (k) ≡ lim gβ,µ,Λ (k) for k ∈ R3 \ {0} . Λ
(3.42)
This limit exists at least almost surely. In fact, using correlation inequalities [7,42, 43] as it is done for the Weakly Imperfect Bose Gas [21], we can prove the existence of the thermodynamic limit (3.42) for any (β, µ) ∈ QS with an uniformly bound for all k ∈ R3 \ {0}. For any interval (a, b) with 0 < a < b, we have the convergence of the Riemann sums to the integral: lim Λ
1 V
a∗k ak H SB (β, µ) =
{k∈Λ∗ ,k∈(a>0,b)}
Λ,λ0
1 3
(2π)
gβ,µ (k) χ(a,b) (k) d3 k,
R3
which combined with (3.38) implies
1
gβ,µ (k) χ(a,b) (k) d k =
3
(2π)
3
R3
1 3
(2π)
ξβ,µ (k) χ(a,b) (k) d3 k
(3.43)
R3
with the continuous function ξβ,µ (k) defined by (3.39) and (3.40). Since the relation (3.43) is valid for any interval (a, b) ⊂ R with 0 < a < b one gets gβ,µ (k) = ξβ,µ (k) , k ∈ R3 , k ≥ δ > 0. By this and (3.41)–(3.43) combined with (3.39)–(3.40) we finally get the statements in (v) for k ≥ δ > 0.
430
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
Appendix: The Bogoliubov u-v transformation In this subsection we recall the Bogoliubov canonical u-v transformation by applying it on the Bogoliubov approximation [22] B HΛ,λ (α, c) 0 1 2 εk − α + λ0 |c| a∗k ak + = 2 ∗ k∈Λ \{0}
+
1 2
k∈Λ∗ \{0}
2
λk |c|
$
a∗k ak + a∗−k a−k
%
k∈Λ∗ \{0}
$ % λ0 4 2 2 (A.1) |c| V − |c| λk c2 a∗k a∗−k + c2 ak a−k − α |c| V + 2
B B of HΛ,λ (α) ≡ HΛ,λ − αNΛ (1.4) for any λ0 ≥ 0. Then, we compute the corre0 0 sponding pressure
pB Λ,λ0 (β, α, c) =
B 1 ln T rFB e−βHΛ,λ0 (α,c) . βV
(A.2)
After the canonical gauge transformation to boson operators ak e−i arg c , k ∈ Λ∗ \ {0} ,
(A.3)
2
B B (α, c) depends only on x ≡ |c| . Since HΛ,λ (α, c) is a bilinear form note that HΛ,λ 0 0 ∗ in boson operators {ak , ak }k∈Λ∗ \{0} , the Bogoliubov canonical u-v transformation diagonalizes it by using a new set of boson operators {bk , b∗k }k∈Λ∗ \{0} defined by
ak = uk bk − vk b∗−k , a∗k = uk b∗k − vk b−k ,
(A.4)
with real coefficients {uk = u−k }k∈Λ∗ \{0} and {vk = v−k }k∈Λ∗ \{0} satisfying: u2k − vk2 = 1, 2uk vk =
xλk εk , u2k + vk2 = B . B Ek,λ0 Ek,λ0
Here fk,λ0 = ε* k − α + x (λ0 + λk ) , + 2 fk,λ − x2 λ2k = (εk − α + xλ0 ) (εk − α + x (λ0 + 2λk )), 0
B Ek,λ = 0
where we recall that x ≡ |c|2 . Thus u2k
1 = 2
! fk,λ0 1 + 1 , vk2 = B 2 Ek,λ0
(A.5)
! fk,λ0 −1 . B Ek,λ 0
2
Notice that fk,λ0 ≥ xλk and, |c| and α satisfy the inequality: 2
α ≤ |c| λ0 +
min
k∈Λ∗ \{0}
εk .
(A.6)
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
431
The Hamiltonian (A.1) becomes: HB Λ,λ0 (α, c) =
B Ek,λ b∗ b + 0 k k
k∈Λ∗ \{0}
1 2
B 2 Ek,λ0 − fk,λ0 − α |c|
k∈Λ∗ \{0}
λ0 + 2
|c|2 |c| − V 4
! . (A.7)
Therefore, the pressure pB Λ,λ0 (β, α, c) (A.2) equals 2 2 β, α, x ≡ |c| + η α, x ≡ |c| , pB (β, α, c) = ξ Λ,λ Λ,λ 0 0 Λ,λ0 −1 B 1 ξΛ,λ0 (β, α, x) = ln 1 − e−βEk,λ0 , βV k∈Λ∗ \{0} λ0 2 1 x B fk,λ0 − Ek,λ + αx − x − , ηΛ,λ0 (α, x) = 0 2V 2 V ∗
(A.8)
k∈Λ \{0}
and has the following thermodynamic limit: 2 pB ≡ lim pB λ0 β, α, x ≡ |c| Λ,λ0 (β, α, c) = ξλ0 (β, α, x) + ηλ0 (α, x) , Λ −1 B 1 −βEk,λ 0 ln 1 − e d3 k, ξλ0 (β, α, x) ≡ lim ξΛ,λ0 (β, α, x) = 3 Λ (2π) β R3 1 λ0 B fk,λ0 − Ek,λ d3 k + αx − x2 , ηλ0 (α, x) ≡ lim ηΛ,λ0 (α, x) = 3 0 Λ 2 2 (2π) R3
B with Ek,λ , fk,λ0 ≥0 defined by (A.5) and α ≤ xλ0 (cf. (A.6). 0 ≥0
(A.9)
Acknowledgments. The work was supported by DFG grant DE 663/1-3 in the priority research program for interacting stochastic systems of high complexity. Special thanks first go to T. Dorlas and the DIAS for the very nice stay there where this work was finished. J.-B. Bru thanks Institut f¨ ur Mathematik, Technische Universit¨ at Berlin, and its members for their warm hospitality during the academic year 2001-2002 and more precisely S. Adams. J.-B. Bru also wants to express his gratitude to N. Angelescu, A. Verbeure and V.A. Zagrebnov for their useful discussions. And the second author thanks the P. master Dukes and Dido for their help in writing/correcting this article. The authors especially thank the referee for helpful remarks and suggestions.
432
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
References [1] N. Angelescu, A. Verbeure and V.A. Zagrebnov, On Bogoliubov’s model of superfluidity J. Phys. A: Math.Gen. 25, 3473 (1992). [2] S. Adams and J.-B. Bru, Critical Analysis of the Bogoliubov Theory of Superfluidity, Physica A 332, 60–78 (2004). [3] S. Adams and J.-B. Bru, A New Microscopic Theory of Superfluidity at all Temperatures, Annales Henri Poincar´e 5, 437–479 (2004). [4] J.-B. Bru and V.A. Zagrebnov, Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas, J. Phys. A: Math. Gen. A 31, 9377 (1998). [5] J.-B. Bru and V.A. Zagrebnov, Quantum interpretation of thermodynamic behaviour of the Bogoliubov weakly imperfect Bose gas, Phys. Lett. A 247, 37 (1998). [6] V.A. Zagrebnov and J.-B. Bru, The Bogoliubov Model of Weakly Imperfect Bose Gas, Phys. Rep. 350, 291 (2001). [7] O. Brattelli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. II, 2nd ed. Springer-Verlag, New York (1996). [8] D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin-Reading, NewYork (1969). [9] N.N. Bogoliubov, On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947). [10] N.N. Bogoliubov, About the theory of superfluidity, Izv. Akad. Nauk USSR 11, 77 (1947). [11] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, Bull. Moscow State Univ. 7, 43 (1947). [12] N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 1: Quantum Statistics, Gordon and Breach Science Publishers, New York-London-Paris (1970). [13] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, p. 242-257 in: Collection of papers, Vol. 2, Naukova Dumka, Kiev, (1970). [14] N. Angelescu and A. Verbeure, Variational solution of a superfluidity model, Physica A 216, 386 (1995). [15] N. Angelescu, A. Verbeure and V.A. Zagrebnov, Superfluidity III, J. Phys. A: Math.Gen. 30, 4895 (1997). [16] N.N. Bogoliubov and D.N. Zubarev, Wave function of the ground-state of interacting Bose-particles, JETP 28, 129 (1955).
Vol. 5, 2004
Exact Solution of the AVZ-Hamiltonian
433
[17] D.N. Zubarev, Distribution function of non-ideal Bose-gas for zero temperature, JETP 29, 881 (1955). [18] Yu.A. Tserkovnikov, Theory of the imperfect Bose-Gas for non-zero temperature, Doklady Acad. Nauk USSR 143, 832 (1962). [19] V.N. Popov, Functional Integrals and Collective Excitations, Univ. Press, Cambridge (1987). [20] H. Shi and A. Griffin, Finite-temperature excitations in a dilute Bosecondensated gas, Phys. Rep. 304, 1 (1998). [21] J.-B. Bru and V.A. Zagrebnov, On condensations in the Bogoliubov Weakly Imperfect Bose-Gas, J. Stat. Phys. 99, 1297 (2000). [22] J. Ginibre, On the Asymptotic Exactness of the Bogoliubov Approximation for many Bosons Systems, Commun. Math. Phys. 8, 26 (1968). [23] J.-B. Bru, Superstabilization of Bose Systems I: Thermodynamic Study, J. Phys. A: Math.Gen. 35, 8969 (2002). [24] N.N. Bogoliubov (Jr), J.G. Brankov, V.A. Zagrebnov, A.M. Kurbatov and N.S. Tonchev, The Approximating Hamiltonian Method in Statistical Physics (Publ. Bulgarian Akad. Sciences, Sofia, 1981). [25] N.N. Bogoliubov (Jr), J.G. Brankov, V.A. Zagrebnov, A.M. Kurbatov and N.S. Tonchev, Some classes of exactly soluble models of problems in Quantum Statistical Mechanics : the method of the approximating Hamiltonian, Russian Math. Surveys 39, 1 (1984). [26] K. Huang, Statistical Mechanics, Wiley, New York (1963). [27] E.B. Davies, The thermodynamic limit for an imperfect boson gas, Commun. Math. Phys. 28, 69 (1972). [28] M. Fannes and A. Verbeure, The condensed phase of the imperfect Bose gas, J. Math. Phys. 21, 1809 (1980). [29] M. van den Berg, J.T. Lewis and Ph. de Smedt, Condensation in the Imperfect Boson Gas, J. Stat. Phys. 37, 697 (1984). [30] E. Buffet and J.V. Pul`e, Fluctuations Properties of the Imperfect Boson Gas, J. Math. Phys. 24, 1608 (1983). [31] J.T. Lewis, J.V. Pul`e and V.A. Zagrebnov, The Large Deviation Principle for the Kac Distribution, Helv. Phys. Acta 61, 1063 (1988). [32] Vl.V. Papoyan and V.A. Zagrebnov, The ensemble equivalence problem for Bose systems (non-ideal Bose gas), Theor. Math. Phys. 69, 1240 (1986). [33] M. van den Berg and J.T. Lewis, On generalized condensation in the free boson gas, Physica A 110, 550 (1982).
434
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
[34] M. van den Berg, On boson condensation into an infinite number of low-lying levels, J. Math. Phys. 23, 1159 (1982). [35] M. van den Berg, J.T. Lewis and J.V. Pul`e, A general theory of Bose-Einstein condensation, Helv. Phys. Acta 59, 1271 (1986). [36] J.-B. Bru, Superstabilization of Bose systems II: Bose condensations and equivalence of ensembles, J. Phys. A: Math.Gen. 35, 8995 (2002). [37] H.O. Georgii, Large Deviations and the Equivalence of Ensembles for Gibbsian Particle Systems with Superstable Interaction, Probab. Th. Rel. Fields 99, 171 (1994). [38] H.O. Georgii, The equivalence of ensembles for classical systems of particles, Journal of Stat. Phys. Vol. 80, 1341–1378 (1994). [39] S. Adams, Complete Equivalence of the Gibbs ensembles for one-dimensional Markov-systems, Journal of Stat. Phys., Vol. 105, Nos. 5/6, 879–908 (2001). [40] R. Griffiths, A Proof that the Free Energy of a Spin System is extensive, J. Math. Phys. 5, 1215 (1964). [41] K. Hepp E. and H. Lieb, Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field, Phys. Rev. A 8, 2517 (1973). [42] M. Fannes and A. Verbeure, Correlation Inequalities and Equilibrium States I, Commun. Math. Phys. 55, 125 (1977). [43] M. Fannes and A. Verbeure, Correlation Inequalities and Equilibrium States II, Commun. Math. Phys. 57, 165 (1977). S. Adams Institut f¨ ur Mathematik Fakult¨ at II, SEK. MA 7-4 Technische Universit¨ at Berlin Strasse des 17. Juni 136 D-10623 Berlin, Germany email:
[email protected] J.-B. Bru School of Theoretical Physics Dublin Institute for Advanced Studies 10 Burlington Rd. Dublin 4, Ireland email:
[email protected] Communicated by Vincent Pasquier Submitted 31/03/03, accepted 01/12/03
Ann. Henri Poincar´e 5 (2004) 435 – 476 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030435-42 DOI 10.1007/s00023-004-0175-7
Annales Henri Poincar´ e
A New Microscopic Theory of Superfluidity at All Temperatures Stephan Adams and Jean-Bernard Bru
Abstract. Following the program suggested in [1], we propose a new microscopic theory of superfluidity for all temperatures and densities. In particular, the corresponding phase diagram of this theory exhibits: (i) a thermodynamic behavior corresponding to the Perfect Bose Gas for small densities or high temperatures, (ii) the “Landau-type” excitation spectrum in the presence of non-conventional Bose condensation for high densities or small temperatures, (iii) a depletion of the Bose condensate with the formation of “Cooper-type pairs”, even at zero-temperature (experimentally, an estimate of the fraction of condensate in liquid 4 He at T=0 K is 9%, see [2, 3]). In contrast to Bogoliubov’s last approach and while warning that the full interacting Hamiltonian is truncated, the analysis performed here is rigorous by involving a complete thermodynamic analysis of a non-trivial continuous gas in the canonical ensemble.
1 Introduction The first microscopic theory of superfluidity was originally proposed in 1947 by Bogoliubov in [4–8]. A recent analysis of the Bogoliubov theory has already been performed in the review [9], itself containing a summary of [10–15]. The critical analysis performed in [1] leads us to use a truncation of the full Hamiltonian within the framework of the canonical ensemble. The resulting model, different from the Bogoliubov one and defined in Section 2, is here rigorously solved at the thermodynamic level in the canonical ensemble (Section 2.2). In the case of homogeneous systems, this analysis provides a new (canonical) theory of superfluidity with a gapless spectrum at all particle densities and temperatures, leading us to a deeper understanding of the Bose condensation phenomenon in liquid helium explained in Section 3. Actually, at any temperatures T ≥ 0 below a critical temperature Tc , the corresponding Bose gas is a mixture of particles inside and outside the Bose condensate, i.e., there is a depletion of the Bose condensate. Even at zero-temperature, our interpretation is that two Bose subsystems coexist: the Bose condensate and a second system, denoted here as the Bogoliubov system. This comes from a nondiagonal interaction, which, in particular, implies an effective attraction between
436
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
bosons in the zero kinetic energy state, i.e., in the Bose condensate [12]. In contrast with the (conventional) Bose-Einstein condensation, these bosons pair up via the Bogoliubov system to form “Cooper-type pairs” or interacting (virtual) pairs of particles. This Bose condensation constituted by Cooper-type pairs is non-conventional [9, 11, 12, 14, 16, 17], i.e., turned on by the Bose distribution but completely transformed by interaction phenomena. The coherency due to the presence of the Bose condensation is not enough to make the Perfect Bose Gas superfluid, see discussions in [4–6]. The spectrum of elementary excitations has to be collective. In this theory, the particles outside the Bose condensate (the Bogoliubov system, Remark 3.2) follow a new distribution, different from the Bose distribution, which we call the Bogoliubov distribution. The Bogoliubov system coming from the depletion of the Bose condensate is a model of “quasi-particles” or linked pairs of particles with the Landau-type excitation spectrum. Therefore, following Landau’s criterion of superfluidity [18, 19] it is a superfluid gas. The corresponding “quasi-particles” are created from two particles respectively of momenta p and −p (p = 0) through the Bose condensate (p = 0) combined with phenomena of interaction. The theoretical critical temperature where the Landau-type excitation spectrum holds equals Tc ≈ 3.14 K. For the liquid 4 He, the superfluid liquid already disappears at Tλ = 2.17 K, but the Henshaw-Woods spectrum1 [20] does not λ change drastically when the temperature crosses Tλ : there is a temperature T where the Landau-type excitation spectrum persists for Tλ < T < Tλ . For a complete description of this theory in relation with liquid 4 He, see Sections 3.3 to 3.4. The phenomenon of Cooper pairs between two fermions corresponds to the phenomenological explanation given for the existence of superfluidity and Bose condensation in 3 He [21–23]. Therefore, at the end (Section 3.5), we explain how this theory may also be a starting point for a microscopic theory of superfluidity for 3 He within the framework of Fermi systems. Before finishing this short introduction, we recall again that this analysis is based on a truncation of the full interacting Bose gas in the canonical ensemble. This unique truncation hypothesis is still not proven in this paper, but we show that the theory is, at least, self-consistent. In fact, the aim of the present paper is to give the exact solution of a non-diagonal continuous model far from the Perfect Bose Gas in the canonical ensemble at all temperatures and densities. Note that this analysis is technically based on three papers [24–26]. We use the “superstabilization” method [24, 25] to analyze the corresponding model in the canonical ensemble from the grand-canonical one. This study is possible since the exact solution of the (non-diagonal) AVZ-Hamiltonian [27], also called the superstable Bogoliubov Hamiltonian, is found in the grand-canonical ensemble by the paper [26].
1 measure of the excitation spectrum in liquid (1961).
4 He
by a weak-inelastic neutron scattering
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
437
2 Our model for superfluidity We give here our proposal for a model for superfluidity. In particular, we first explain the philosophy of this model and then, we solve it in the canonical ensemble.
2.1
Setup of the appropriate model
Let an interacting homogeneous gas of n spinless bosons with mass m be enclosed 3
in a cubic box Λ = × L ⊂ R3 . We denote by ϕ (x) = ϕ (x) a (real) two-body α=1
interaction potential and we assume that: (A) ϕ (x) ∈ L1 R3 . (B) Its (real) Fourier transformation λk =
d3 xϕ (x) e−ikx , k ∈ R3 ,
R3
satisfies: λ0 > 0 and 0 ≤ λk = λ−k ≤ lim + λk for k ∈ R3 . k→0
The one-particle energy spectrum is εk ≡ 2 k 2 /2m and, using periodic boundary conditions, Λ∗ ≡
2πnα , nα = 0, ±1, ±2, . . . , α = 1, 2, 3 k ∈ R3 : kα = L
∗ is the set of wave vectors. Let a# k = {ak or ak } be the usual boson creation/ 1 annihilation operators in the one-particle state ψk (x) = V − 2 eikx , k ∈ Λ∗ , x ∈ Λ, acting on the boson Fock space +∞
(n)
FΛB ≡ ⊕ HB , n=0
(n)
with HB defined as the symmetrized n-particle Hilbert spaces (n) (0) HB ≡ L2 (Λn ) symm , HB = C, see [28, 29]. Therefore, the corresponding Hamiltonian of the system acting on the boson Fock space FΛB is equal to Λ + UΛMF , HΛ,λ0 >0 = TΛ + U
(2.1)
438
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
with TΛ
≡
εk a∗k ak
k∈Λ∗
Λ U UΛMF
1 2V
≡
λ0 2V
≡
λq a∗k1 +q a∗k2 −q ak1 ak2 ,
k1 ,k2 ,q=0∈Λ∗
a∗k1 a∗k2 ak2 ak1 =
k1 ,k2 ∈Λ∗
Here
NΛ ≡
λ0 2 NΛ − NΛ . 2V
(2.2)
a∗k ak
k∈Λ∗
is the particle number operator. Under assumptions (A) and (B) on the interaction potential ϕ (x), the full Hamiltonian HΛ,λ0 >0 is superstable [29]. Without any Bose condensation, the model should be equal to the MeanField model, i.e., the Perfect Bose gas in the canonical ensemble. Whereas, in Λ should play a crucial role on presence of Bose condensation, the interaction U the thermodynamics. Formally, the Mean-Field interaction UΛMF does not change the “physical properties” of a Bose system (cf. [1, 24, 25]). The “physical” effect of the interaction potential should express itself by the other terms of interaction, Λ . i.e., by U Within the framework of the canonical ensemble, considering the existence of a Bose condensation on the zero-kinetic energy state in liquid 4 He, originally suggested by Fritz London in 1938 [30], one should partially truncate the full interaction without taking into account the Mean-Field interaction since it is a constant in the canonical ensemble. This procedure implies the non-diagonal Hamiltonian: B HΛ,0 ≡ TΛ + UΛD + UΛN D
(2.3)
with UΛD UΛN D
≡ ≡
1 2V 1 2V
k∈Λ∗ \{0}
λk a∗0 a0 a∗k ak + a∗−k a−k ,
(2.4)
2 λk a∗k a∗−k a20 + a∗0 ak a−k .
(2.5)
k∈Λ∗ \{0}
Note that, with the usual Bogoliubov approximation √ √ a0 / V → c, a∗0 / V → c,
(2.6)
B on HΛ,0 , the new Hamiltonian does not commute with the particle number operator NΛ . To solve this problem in the canonical ensemble, Bogoliubov [6] suggests
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
439
a different but similar way corresponding to a “canonical Bogoliubov approximation”. We first use the new set of operators ζk = a∗0 (N0 + I)
−1/2
−1/2
ak , ζk∗ = a∗k (N0 + I)
a0 , k ∈ Λ ∗ .
(2.7)
The set {ζk }k∈Λ∗ \{0} satisfies the Canonical Commutation Relations. Then, for B HΛ,0 the new Bogoliubov approximation corresponds to do the following transformations: 1/2 1/2 (N0 − I) N0 N 2 2 → |c| , 0 → |c| , N0 ≡ a∗0 a0 . (2.8) V V It implies a bilinear form in Bose-operators {ζk }k∈Λ∗ \{0} :
B HΛ,0 (c) =
εk ζk∗ ζk +
k∈Λ∗ \{0}
+
1 2
1 2
2
λk |c|
∗ ζk∗ ζk + ζ−k ζ−k
k∈Λ∗ \{0}
∗ λk c2 ζk∗ ζ−k + c2 ζk ζ−k .
(2.9)
k∈Λ∗ \{0}
This Hamiltonian commutes with the particle number operator NΛ . After the canonical gauge transformation to boson operators ζk e−i arg c , k ∈ Λ∗ \ {0} , 2
B (c) only depends on x ≡ |c| . Then, the Bogoliubov canonithe model HΛ,0 cal u-v transformation diagonalizes it by using a new set of boson operators {bk , b∗k }k∈Λ∗ \{0} defined by
ζk = uk bk − vk b∗−k , ζk∗ = uk b∗k − vk b−k .
(2.10)
The real coefficients {uk = u−k }k∈Λ∗ \{0} and {vk = v−k }k∈Λ∗ \{0} satisfy: xλk εk , u2k + vk2 = . u2k − vk2 = 1, 2uk vk = εk (εk + 2xλk ) εk (εk + 2xλk ) B (c = 0) corresponds to the perfect Bose gas It follows that the Hamiltonian HΛ,0 of quasi-particles defined by 2 BG x ≡ |c| = HP εk (εk + 2xλk )b∗k bk Λ,0 k∈Λ∗ \{0}
+
1 2
εk (εk + 2xλk ) − (εk + xλk ) .
(2.11)
k∈Λ∗ \{0}
In other words, if we consider that this “canonical Bogoliubov approximation” is true, we directly get the well-known Bogoliubov gapless spectrum for a Bose con2 densate density x = |c| > 0.
440
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
The Bogoliubov procedures [4–8] always involved the truncation of the MeanB Field interaction UΛMF . It implied the Bogoliubov Hamiltonian HΛ,λ (A.1), see 0 >0 Appendix A. From the observations of [1,9,27], in various respects the Bogoliubov B is not appropriate as the model of superfluidity. For example, in order model HΛ,λ 0 B , but to deduce its gapless spectrum, Bogoliubov applied (2.6) or (2.8) on HΛ,λ 0 >0 he also needed additional approximations which were shown to be not legitimate. The main problem of the previous attempts (Bogoliubov et al., see for example [4–8, 27, 31, 32]) is to assume, a priori, the Bose condensation by directly doing the Bogoliubov approximation with an arbitrary choice of |c|2 , without exactly solving it in terms of the thermodynamic behavior. In particular, the “canonical B Bogoliubov approximation” (2.8) applied on HΛ,0 has to be proven. For example, Bogoliubov first needed the inexact assumption of 100% of Bose condensate at zero-temperature in the canonical ensemble (Appendix A) and he realized (also with Zubarev) the difficulty with this ansatz: his u-v transformation (2.10) implies a depletion of the Bose condensate due to repulsion between particles. Therefore, 2 what is our value of x = |c| after the approximation (2.8)? Actually, these questions are solved in the next subsection since the canonB is rigorously ical thermodynamic behavior of the non-diagonal Hamiltonian HΛ,0 performed. In particular, it is shown that the “canonical Bogoliubov approximaB corresponds, tion” (2.8) is true in the following sense: the thermodynamics of HΛ,0 at the thermodynamic level, to the perfect Bose gas (2.11) of quasi-particles for k ∈ Λ∗ \ {0} with a Bose condensate density x
(β, ρ) on k = 0 (cf. Theorems 2.2 and 2.3). Note that this new approach is originally explained in [1] and comes from a constructive criticism of the Bogoliubov theories. Moreover, before going further, we want to stress that our approach is different from the Bogoliubov one. In the canonical ensemble, the theory of the present paper is new and distinct from the canonical Bogoliubov theory (Appendix A) in many aspects: B B (2.3) and HΛ,λ = • First, they are based on two separate Bose gases HΛ,0 0 >0 B BMF (A.1). Both of this theories have behind them, in the correHΛ,0 + UΛ sponding truncation, the fundamental hypothesis originally given by Fritz London [30] about existence in the system of a Bose condensation in the zero-mode. • Secondly, the Bogoliubov theories must include other tricks in order to find the Landau-type excitation spectrum (Appendix A). However, his additional ansatz in the canonical ensemble are unnecessary for our approach (see also Section 3). • Finally, the canonical Bogoliubov theory could have been exact only at zerotemperature, whereas our results below conclusion) are valid at any temperatures T ≥ 0 for a fixed particle density ρ > 0. B solves, in the canonical ensemble, In fact we prove in this paper that the model HΛ,0 the problems of the previous Bogoliubov theories and implies a new microscopic theory of superfluidity at all temperatures explained in Section 3.
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
441
To fix the notations, β > 0 is here the inverse temperature, µ the chemical potential, ρ > 0 the fixed full particle density, whereas n = [ρV ], defined as the integer of ρV , is the number of particles in the canonical ensemble. Then, T = (kB β)−1 ≥ 0 is the temperature where kB is the Boltzmann constant. Here − H X (β, ρ) and − H X (β, µ) represent the (finite volume) canonical Λ Λ and grand-canonical Gibbs state respectively for some Hamiltonian HΛX : (n) X −βHΛ (−) e T rH(n) B − H X (β, ρ) ≡ (n) , Λ X e−βHΛ T rH(n) B
− H X Λ
X T rFΛB (−) e−β (HΛ −µNΛ ) (β, µ) ≡ X T rFΛB e−β (HΛ −µNΛ )
(n)
where A(n) ≡ A HB is the restriction of any operator A acting on the boson Fock (n) space FΛB to HB .
2.2
Thermodynamics in the canonical ensemble
B The aim of this section is to examine the Hamiltonian HΛ,0 (2.3) in the canonical B ensemble specified by (β, ρ). The model HΛ,0 turns out to be not sufficient for a microscopic theory of superfluidity in the grand-canonical ensemble because of its instability in presence of Bose condensation (Appendix B). The terms of repulsion B are not strong enough to prevent the system from collapse in the Hamiltonian HΛ,0 B in the grand-canonical ensemble. However, we explain here that the Bose gas HΛ,0 can be solved in the canonical ensemble by superstabilizing it [24,25] in the grandcanonical ensemble. The principle is the following: SB B (i) We denote by HΛ,λ the superstabilization of the model HΛ,0 which is defined by SB B ≡ HΛ,0 + HΛ,λ
λ 2V
k1 ,k2 ∈Λ∗
B a∗k1 a∗k2 ak2 ak1 = HΛ,0 +
λ 2 NΛ − NΛ . 2V
(2.12)
B the Mean-Field interaction UΛMF This procedure adds to the Hamiltonian HΛ,0 SB (2.2) with a sufficiently large parameter λ > 0. The model HΛ,λ is just a technical tool. For λ = λ0 it corresponds to the Angelescu-Verbeure-Zagrebnov Hamiltonian [27], also called the superstable Bogoliubov Hamiltonian. It is rigorously solved in the grand-canonical ensemble (see Appendix C.1) and the corresponding proof does not depend on the fact that λ0 is the Fourier transformation of the interaction potential for k = 0 [26]. The only constraint is to have the superstability of the SB model HΛ,λ , which is verified for large enough λ > 0.
442
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
(ii) Then, we use the notion of strong equivalence of ensembles (see [25, 33–35]) SB for the model HΛ,λ with the arbitrary parameter λ taken such that λ + g00 > 0. 2
(2.13)
Indeed, in the canonical ensemble, note that for a given density ρ, i.e., on the (n=[ρV ]) SB B , the Hamiltonians HΛ,λ and HΛ,0 differ only by a conHilbert space HB stant, i.e., their canonical thermodynamics are equal to each other. In Appendix C, we explain that the strong equivalence between canonical and grand-canonical SB if (2.13) is satisfied. Therefore the canonensembles is verified for the model HΛ,λ B ical thermodynamic properties of HΛ,0 correspond for a fixed particle density ρ SB to the one of HΛ,λ for a fixed density ρ in the grand-canonical ensemble (in [26]: Section 2.3). Remark 2.1 Here the (effective coupling) constant g00 equals 2 1 3 λk g00 ≡ − d k < 0, 3 εk 4 (2π)
(2.14)
R3
with the one-particle energy spectrum εk ≡ 2 k 2 /2m. We explain later in Section 3.2 the quantum interpretation given by [12] of the constant g00 . B Now we give all promised properties of the Hamiltonian HΛ,0 in the canonical ensemble. To simplify our purpose, the proofs are given in Appendix C. B 1. Let fΛ,0 (β, ρ) be the corresponding free-energy density defined for a fixed particle density ρ > 0 by (n=[ρV ]) B 1 B −βHΛ,0 e ln T rH(n) fΛ,0 (β, ρ) ≡ − . (2.15) B βV
Recall that the “canonical Bogoliubov approximation” (2.8) implies the model B B (c) corHΛ,0 (c) (2.9). Here, for technical considerations, we use the operator H Λ,0 B responding in HΛ,0 (c) to replace again the operators {ζk }k∈Λ∗ \{0} by {ak }k∈Λ∗ \{0} . B (c) is well defined on the boson Fock space The Hamiltonian H Λ,0
+∞
(n )
≡ ⊕ HB,k1 =0 FB n1 =0
(n )
of the symmetrized n1 -particle Hilbert spaces HB,k1 =0 for non-zero momentum bosons. The Bogoliubov canonical u-v transformation gives also for k ∈ Λ∗ \ {0} the perfect Bose gas (2.11) of quasi-particles for a Bose condensate (k = 0) density x. We then consider its (infinite volume) free-energy density defined by (n1 =[ρ1 V ],k=0) 1 B e−β HΛ,0 (c) ln T rH(n1 ) f0B (β, ρ1 , x) ≡lim − , (2.16) Λ B,k=0 βV
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
443
for any β > 0, ρ1 > 0 and x = |c|2 ≥ 0, where (n )
A(n1 ,k=0) ≡ A HB,k1 =0 is the restriction of any operator A acting on FB . The (infinite volume) pressure of this gas of quasi-particles is
pB 0 (β, α, x)
≡ = =
B (c)−α −β H Λ,0
a∗ k ak +x
1 k∈Λ∗ \{0} ln T rFB e βV sup α [ρ1 + x] − f0B (β, ρ1 , x) ρ1 >0 −1 1 3 B 1 1 −βEk,0 B ln 1 − e f d k, + − E αx + k,0 k,0 3 β 2 (2π) lim Λ
R3
(2.17) for α ≤ 0 with B fk,0 = εk − α + xλk , Ek,0 =
(εk − α) (εk − α + 2xλk ).
(2.18)
Then we get our first main result: Theorem 2.2 The thermodynamic limit f0B (β, ρ) exists for any β > 0 and ρ > 0. B (i) Moreover, the Hamiltonians HΛ,0 (2.3) is equivalent, at the thermodynamic level, to the perfect Bose gas (2.11) of quasi-particles for k ∈ Λ∗ \ {0} with a density x = x
(β, ρ): B B B . f0 (β, ρ) = inf f0 (β, ρ − x, x) = f0 (β, ρ − x, x) x∈[0,ρ]
x= x 0, i.e., for a chemical potential µβ,ρ (C.5).
444
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
Also the solution α ( x) of (ii) is originally defined as the solution of the variational problem (C.1) for µ = µβ,ρ (C.5). It is the unique solution of the Bogoliubov density equation:
) for ρ > 0. ρ = ρB 0 (β, α, x Here ρB 0 (β, α, x)
≡ ∂α pB 0 (β, α, x) = x +
+
1 3
(2π)
R3
B 2Ek,0
1 3
(2π)
x2 λ2k fk,0 +
R3
B Ek,0
(2.19) f d3 k k,0B βEk,0 B Ek,0 e −1
d3 k.
(2.20)
Moreover, there is a particle density ρc (β) (C.3)-(C.4) such that the solution x
(β, ρ) = 0 for ρ ≤ ρc (β) , whereas for ρ > ρc (β) , 0 < x
(β, ρ) < ρ (even for β → +∞). For a fixed particle density ρ, there is also a critical inverse temperature βc (ρ), see Appendix C.1. An illustration of βc (ρ) is performed in Figure 2.1. Note that ∂ρ f0B (β, ρ) = α ( x) and ∂ρ f0B (β, ρ) < 0 for ρ = ρc (β) or β = βc (ρ) (Remark C.3). B corresponds For ρ ≤ ρc (β), note that the thermodynamic behavior of HΛ,0 to the Perfect Bose Gas (excitation spectrum εk ). B 2. Now we give our main result for the thermodynamic behavior of HΛ,0 in the canonical ensemble (β, ρ).
Theorem 2.3 (i) A non-conventional Bose condensation induced by the non-diagonal interaction UΛN D for high particle densities, or low temperatures: ∗ a0 a0 = 0 for ρ ≤ ρc (β) or β ≤ βc (ρ) . lim (β, ρ) = x
(β, ρ) = > 0 for ρ > ρc (β) or β > βc (ρ) . Λ V B H Λ,0
(ii) No Bose condensation (of any type I, II or III [36–38]) outside the zero-mode for any particle densities or temperatures: ∗ ak ak ∗ ∀k ∈ Λ \ {0} , lim (β, ρ) = 0 Λ V B HΛ,0 lim lim 1 a∗k ak H B (β, ρ) = 0 δ→0+ Λ V Λ,0 ∗ {k∈Λ ,0 0. B are defined by (2.18) for a chemical potential given by the Here fk,0 and Ek,0 solution α = α ( x) at a fixed particle density ρ > 0. (iv) There is no discontinuity of the particle densities (density in the zero-mode (i) or outside the zero-mode (iii)). (v) For ρ ≤ ρc (β) or β ≤ βc (ρ) one has the Bose distribution for a corresponding chemical potential α (0) < 0:
∀k ∈ Λ∗ : k ≥ δ > 0, lim a∗k ak H B (β, ρ) = Λ
Λ,0
1 eβ(εk −α(0))
−1
.
But for ρ > ρc (β) or β > βc (ρ) , i.e., in the presence of a Bose condensation, we get another one, which we call the Bogoliubov distribution, for a corresponding chemical potential α ( x) < 0: 2 2 λ f x k,0 ∗ k + lim ak ak H B (β, ρ) = B Λ,0 x= x,α=α( x) Λ E B eβEk,0 B B f +E −1 2E k,0
k,0
k,0
k,0
for any k ∈ Λ∗ such that k ≥ δ for any δ > 0. The illustrations of the particle densities inside and outside the zero-mode are given in Figures 2.2 and 2.3 respectively.
446
S. Adams and J.-B. Bru
x(β,ρ )
Ann. Henri Poincar´e
2
Bose Condensation
0
ρc (β)
ρ
Figure 2.2: Illustration of the non-conventional Bose condensate density x
(β, ρ) as a function of ρ. The dashed dotted line corresponds to a zero-temperature, i.e., for β → +∞. The straight line is x = ρ. Remark 2.4 For ρ > ρc (β), there is a non-conventional Bose condensation whereas no Bose condensation (of any type I, II, or III [36–38]) appears outside the zero-mode at all densities ρ > 0 (Theorem 2.3). In contrast to the Bogoliubov theories (see for example [1]), the theory is self-consistent with the corresponding truncation of the full Hamiltonian in the canonical ensemble.
3 A new microscopic theory of superfluidity B The aim of this section is to explain why the model HΛ,0 can imply a new microscopic theory of superfluidity for Bose systems. It is essential here to note that in the canonical ensemble the conditions relating to the interaction potential ϕ (x) may be relaxed as follows. The model is independent of the Fourier transformation of ϕ (x) for k = 0, which may be infinite for some specific interaction potentials. However, the (effective coupling) constant g00 (2.14) and ϕ (0) have to exist.
3.1
Landau-type excitation spectrum in the presence of Bose condensation
In order to obtain a microscopic theory of superfluidity we have to get a Landautype excitation spectrum [18, 19] as Bogoliubov did [4–8] for a suitable choice of c-numbers.
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
447
ρ − x(β,ρ ) 2
0
ρ
ρc (β)
Figure 2.3: Illustration of the particle density outside the zero-mode 2 ρ − | x (β, ρ)| as a function of ρ. Note that for ρ < ρc (β), x
(β, ρ) = 0. The dashed dotted line is the Bogoliubov system density at β → +∞, i.e., at zerotemperature. 1. As Landau’s predictions [18,19], at high densities ρ > ρc (β) (C.3) (or sufficiently B low temperatures) the Bose gas HΛ,0 is equivalent to a “gas of collective elementary excitations” or “quasi-particles” (2.11) for k ∈ Λ∗ \ {0} with a density x
(β, ρ) of Bose condensate on k = 0, cf. Theorems 2.2 and 2.3. Consequently, as stated in Section 2.1, the spectrum of excitations, which is macroscopically relevant, equals the Bogoliubov spectrum at inverse temperatures β > 0 and particle densities ρ > 0: 2 2 ε k = k /2m for β ≤ βc (ρ) or ρ ≤ ρc (β) , (3.1) EkB (β, ρ) = εk (εk + 2 xλk ) for β > βc (ρ) or ρ > ρc (β) , see (2.11). The collective excitation spectrum EkB (β, ρ) has no gap for any densities or temperatures as expected in Section 2.1. The main difficulties are to find B in the canonx
(β, ρ), i.e., the thermodynamic properties of the Hamiltonian HΛ,0 ical ensemble. B , since our Note that we do not rigorously know the exact spectrum of HΛ,0 analysis is only based on its thermodynamic properties. In infinite volume, this B question also implies the problem of definition of lim HΛ,0 ! Λ
2. Now, to find the exact Landau-type excitation spectrum from (3.1), i.e., to get the “phonons” part and the “rotons” one, we can reason along the standard lines of Bogoliubov microscopic theory of superfluidity, see [4–9]. For this approach, we have to assume some specific conditions relating to the two-body interaction potential ϕ (x). In particular, the two-body potential
448
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
ϕ (x) should verify (A)-(B). Here λk is spherically-symmetric, i.e., λk = λk , and additionally, as Bogoliubov did, we assume the absolute integrability of x2 ϕ (x) ∈ 3 1 L R . Actually, we need here the last assumption and the Fourier transformation of ϕ (x) for k = 0 in order to have a Taylor expansion λk = λ0 +
1 2 2 k λ0 + o k , 2
(3.2)
of λk allowing us to analyze EkB (β, ρ) for small k (phonon part). Here λ0 ≤ 0 −1 is the second derivative for k = 0 and |λk | ≤ const. k . Let ρ > ρc (β) or β > βc (ρ), i.e., x
(β, ρ) > 0 (cf. (i) of Theorem 2.3). Then the collective spectrum of excitations EkB (β, ρ) in this domain of (β, ρ) verifies: 1/2 2 λ0 x
k = w k , for k → 0+ . EkB (β, ρ) = (3.3) m εk = 2 k 2 /2m , for k → +∞. The gapless spectrum EkB (β, ρ) is phonon-like for small k (ρ > ρc (β)), whereas for large wave-vectors it behaves like the single-particle excitations εk . Since λk attains its maximum at k = 0, one can choose the potential ϕ (x) in such a way that εk εk + 2 xλk = 0 at k = krot = 0, (3.4) i.e., the spectrum EkB (β, ρ) has a local (“roton”) minimum at krot . On the other hand, one gets: EkB (β, ρ) ≥ k
2 2m
1/2
1/2
≡ k v0 (β, ρ) . min εk + 2λ2k x k
(3.5)
The Bogoliubov spectrum EkB (β, ρ) is a Landau-type excitation spectrum for ρ > ρc (β) or β > βc (ρ) and an illustration is given by Figure 3.1. Remark 3.1 The famous Landau’s criterion of superfluidity of 1941 [18, 19] gives the following critical velocity: B Ek (β, ρ) 0 , for β ≤ βc (ρ) or ρ ≤ ρc (β) . inf = v0 (β, ρ) = > 0, for β > βc (ρ) or ρ > ρc (β) . k k
3.2
Two complementary Bose systems: Cooper-type pairs and gas of quasi-particles
We give here the quantum interpretation of the canonical thermodynamic propB erties of the model HΛ,0 . First note that, in terms of particle densities, we obtain (see Theorem 2.3 and Figures 2.2 and 2.3):
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
449
E Bk (β,ρ)
hw k h2k2 2m h k v 0 (β,ρ) 0
k rot
k
Figure 3.1: The Bogoliubov spectrum EkB (β, ρ) for β > βc (ρ) or ρ > ρc (β) . • A non-conventional Bose condensation appears with the density 0 < x
(β, ρ) < ρ for ρ > ρc (β) (even with β → +∞), whereas at all densities ρ > 0 there is no Bose condensation (of any type I, II, or III [36–38]) outside the zero-mode. • Even for zero-temperature, we have a non-zero particle density outside the zero-mode for any ρ > 0: 1 lim lim a∗k ak H B (β, ρ) = Λ,0 β→+∞ Λ V ∗ k∈Λ \{0} 2 2 1 x λk 3 k > 0, d (2π)3 x=x B B f 2E + E k,0 α=α( x)
k,0 k,0 (3.6) R3 ∗ ∗ ∀k ∈ Λ : k ≥ δ > 0, lim lim a a
(β, ρ) = B k k HΛ,0 β→+∞ Λ 2 2 x λk > 0. 2E B f + E B x=x k,0 α=α(x)
k,0 k,0 In the regime ρ > ρc (β) , the system follows the Bogoliubov distribution (v) of Theorem 2.3, whereas in the absence of the Bose condensation, i.e., for ρ ≤ ρc (β), the (standard) Bose distribution holds. 1. The origin of the Bogoliubov distribution and also of (3.6) is a phenomenon of interaction. Actually, it is known since [12] that the collection of particles outside
450
S. Adams and J.-B. Bru
k
k’=0
k’=0
λk −k
Ann. Henri Poincar´e
k
λk k’=0
k’=0
−k
Figure 3.2: Non-diagonal-interaction vertices corresponding to UΛN D . the zero-mode imposes, through the non-diagonal interaction UΛN D , a glue-like attraction between particles in the zero-mode. A natural way to see this phenomenon is to remark that the non-diagonal interaction UΛN D (see Figure 3.2) implies an effective interaction term gΛ,00 for bosons with k = 0, see Figure 3.3. Evaluated via a Fr¨ ohlich transformation in the second order [12] (see also the review [9]), gΛ,00 is strictly negative. The corresponding thermodynamic limit lim gΛ,00 = g00 < 0 Λ
remarkably gives (2.14). In particular, this effective attraction term g00 amazingly B plays a crucial rˆ ole in the rigorous thermodynamic analysis of HΛ,0 (see Section 2.2 or in [26]: proof of Theorem 2.3). It is also essential in the rigorous study B , of the Weakly Imperfect Bose Gas, i.e., the Bogoliubov Hamiltonian HΛ,λ 0 >0 see [9–11, 13]. The Bose condensate with the density x
(β, ρ) and the remaining system with the density {ρ − x
> 0}, called here the Bogoliubov system, only exist via this gluelike attraction g00 (Figure 3.3). In fact, the particles inside the condensate pairs up via the Bogoliubov system to form “Cooper-type pairs” or interacting (virtual) pair of particles. This Bose condensation constituted by Cooper-type pairs is then nonconventional [9, 11, 12, 14, 16, 17], i.e., completely transformed by the non-diagonal interaction UΛN D . Remark 3.2 The existence of particles outside the Bose condensate even at zerotemperature is well known as the depletion of the Bose condensate, see for example [7, 8, 39–43]. But we go a step further in our interpretation. We consider this behavior as the coexistence of two subsystems: the Bose condensate and the previously defined Bogoliubov system. We agree that the term “Bogoliubov system” is only a personnel terminology, which stands for the “out-of-condensate Bosons”. 2. As it was claimed by Bogoliubov [4–6], the coherency due to the presence of the Bose condensation is not enough to make the Perfect Bose Gas superfluid. The spectrum of elementary excitations is not collective in this case: it corresponds B , following Landau’s to individual movements of particles. In the Bose gas HΛ,0
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures k
k=0
451
k=0
λk
λk −k
k=0
k=0
k=0
k=0
g k=0
Λ,00
k=0
Figure 3.3: Effective interaction for the zero-mode induced by the non-diagonal interaction UΛN D . criterion of superfluidity [18, 19] (Remark 3.1), the Bogoliubov system is here superfluid due to phenomena of interactions, which change, in the presence of the Bose condensate, the behavior of individual particles into an ideal Bose gas of “quasi-particles” with the given spectrum EkB (β, ρ). Indeed, through the Bose condensate, the non-diagonal interaction UΛN D combined with the diagonal interaction UΛD creates quasi-particles from two particles respectively of modes k and −k (k = 0). this can be seen via the Bogoliubov u-v transformation Formally, B (c) |c|2 = x>0 , cf. (2.10). This gas of quasi-particles or linked pairs applied to HΛ,0 of particles, i.e., the Bogoliubov system, exists if and only if the non-conventional Bose condensate exists too. 3. Also for high densities ρ > 0 we have
(β, ρ)} = 0, lim {ρ − x
ρ→+∞
(3.7)
cf. Theorem 2.3, Figure 2.2. Actually, the non-diagonal interaction UΛN D implies an effective repulsion term 1 λp λq 1 x
(β, ρ) gpq ≡ lim gΛ,pq = + ≥ 0, (3.8) Λ 4 εp εq inside each quasi-particle [9, 12], i.e., inside each couple of particles respectively with modes q and −q (q = 0) (Figure 3.4). The larger the Bose condensate den-
452
S. Adams and J.-B. Bru q
k=0
Ann. Henri Poincar´e
p
λq
λp
−q
k=0
−p
q
p
g −q
Λ ,pq
−p
Figure 3.4: Effective interaction outside the zero-mode induced by the non-diagonal interaction UΛN D . sity x
(β, ρ), the stronger the effective repulsion term gpq . The raise of the nonconventional Bose condensate progressively destroys the Bogoliubov one, see (3.7). The Bose condensate and the Bogoliubov system still remain in competition with each other. Remark 3.3 We could have denoted the Bogoliubov system as the Bogoliubov condensate, for example at zero-temperature. This notion can be confusing since it has nothing to do with a macroscopic occupation of individual particles on some modes k < δ (δ → 0+ ) [36–38]. However, considering this gas as a system of linked pairs of particles, the momentum of all these quasi-particles is always zero. This fact is similar to a usual Bose condensation (seen here as a macroscopic occupation of the zero-momentum) but in a gas of linked pairs of particles and the notion of Bogoliubov condensate may have a sense within this framework.
3.3
Microscopic theory of superfluidity of 4 He?
1. A microscopic interpretation at all temperatures T = (kB β)−1 ≥ 0 of Landau’s theory of superfluidity follows from the Landau-type excitation spectrum EkB (β, ρ) (3.1)–(3.5) (cf. Figure 3.1). Note that Landau’s theory of superfluidity of quantum liquids [4, 6, 7, 44–47] is based on the following principles: • quantum liquid is still fluid even for zero-temperature; • at low temperatures, apart translations (flow), the state of this liquid is entirely described by the spectrum of collective (elementary) excitations;
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
453
• through thermodynamic data [47, 48] (e.g., specific heat capacity) this spectrum for 4 He should be a phonon-like for the long-wave length collective excitations and should be above a straight line with positive slope with (“roo −1
ton”) minimum in the vicinity of krot 2 A
(Figure 3.1).
B HΛ,0
2. The thermodynamic behavior of the Bose gas is also close to the liquid He. This helium liquid is a Bose system with strong interactions. The interaction potential Uth (r) is of Lennard-Jones type [29] and was found by Slater et Kirkwood [49] using the electronic structure of 4 He (see Figure 3.5 with Uth (r) in Kelvin and also [50]). 4
Uth(K)
2.6 3
4.4
o
r(A)
0
−9 K Figure 3.5: The theoretical interaction potential of 4 He. The exact formula for the interaction potential Uth (r) given in [50] is valid only for strictly positive r, whereas close to zero it is given by a polynomial interaction like in Figure 3.5. A caricature of this interaction is the hard sphere interaction potential [51, 52]. This approximation gives surprisingly good estimates of the exB we have perimental condensate fraction: 9% at T = 0 K [2, 3]. In our model HΛ,0 to mimic an interaction potential ϕ (x) close to Uth (r). In particular, in contrast with the hard sphere potential the value of ϕ (x) for x = 0 has to be given and has not to be infinite. A standard way to do it is to cut Uth (r) when r → 0+ as follows: Uth (r) for r > rmin . ϕHe (x) = ϕHe (r = x) = Uth (rmin ) for 0 ≤ r ≤ rmin . This implies a Fourier transformation λ0 (rmin ) of ϕHe (x) for the mode k = 0 which drastically depends on rmin (specially when rmin → 0+ ), i.e., lim λ0 (rmin ) = rmin →0+
454
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
B +∞, but it has no influence on the canonical thermodynamic behavior of HΛ,0 . Moreover, for k = 0 the influence of rmin corresponds only to a small (specially when rmin → 0+ ) perturbation of the Fourier transformation of Uth (r) . In fact one should choose rmin 0. B Then, the thermodynamics of the theoretical Bose gas HΛ,0 is qualitatively 4 quite similar to the one of the liquid He: −1 • for small densities ρ ≤ ρc (β) or high temperatures T ≥ Tc ≡ (kB βc (ρ)) the thermodynamic behavior corresponds to the Perfect Bose Gas, • a non-conventional Bose condensation constituted of Cooper-type pairs appears via a second order transition (no discontinuity of the Bose condensate density) and the spectrum of excitations becomes a Landau-type excitation spectrum in this regime, i.e., for high densities ρ > ρc (β) or small temperatures T < Tc , • a coexistence of particles inside and outside the Bose condensate, even at zero-temperature as it is experimentally found in [2, 3].
As explained above, note that the Bose condensation becomes non-conventional with the formation of Cooper-type pairs via the term of attraction g00 , i.e., because of quantum fluctuations, see Figures 3.3 and 3.4. The importance of quantum fluctuations in helium systems corresponds also to the qualitative explanation for a liquid state at such extreme temperatures [50]. Quantitatively, the critical density ρc (β) is approximately given by ρc (β) ≈ ρP BG (β, 0), cf. (C.3)–(C.4) and Figure 2.1. The theoretical temperature of the BG phase transition Tc verifies Tc ≥ TP = 3.14 K (critical temperature evaluated c for a Perfect Gas of helium particles). The physical reason is that the non-diagonal interaction UΛN D implies an effective attraction in the zero-mode (see Figure 3.3), which helps the formation of the Bose condensation. However, Tc is quite close to BG : TP c Tc ≈ 3.14 K. (In fact we are able to prove an exact equality at small densities but we have no rigorous proof of a such result at very high densities). The experimental transition of the normal liquid 4 He (called 4 He I) to superfluid phase 4 He II (called the “λtransition”, discovered by Kapitza [53] and Allen, Misener [54] in 1938) takes place at a lower temperature Tλ = 2.17 K (along the vapor pressure curve), which is not B . However, note that the Henshaw-Woods so far from the one of the model HΛ,0 spectrum (experimental Landau-type excitation spectrum [20]) does not change drastically when the temperature crosses Tλ , whereas there is no superfluidity for these temperatures. λ > Tλ such that the exRemark 3.4 This means that there is a temperature T λ even if Landau’s criperimental “quasi-particle” system still exists for T < T terion of superfluidity (Remark 3.1) experimentally fails at these temperatures λ. Tλ < T < T
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
455
3. To resume, this analysis is not a complete theory of “real superfluidity”. In particular, the Bogoliubov phonon-maxon-roton dispersion branch is only a part of the spectrum of the full quantum-mechanical Hamiltonian of the helium system. Therefore, this theory fails in being a complete description of all thermodynamics of liquid helium. For example, at temperatures Tλ < T < Tc , a Bose condensation B still exists in HΛ,0 but not for liquid helium even if the system of “quasi-particles” λ (Remark 3.4). However, this theory resists in liquid helium for Tλ < T < T is an interesting mathematical approach to a microscopic theory of many-body interacting boson systems leading to a better understanding of such superfluid systems.
3.4
Additional interpretations of this microscopic theory of superfluidity
B Let us examine other interpretations of the Bose system HΛ,0 in relation with the 4 B liquid He. In fact, we give here two interpretations of the Bose gas HΛ,0 obtained by following or not Landau’s criterion of superfluidity [18, 19] (Remark 3.1). As B is a caricature and may contain only explained above, note that the model HΛ,0 a small part of the physical properties of real liquid helium. The sole purpose of B these discussions is to give some new directions in light of the Bose gas HΛ,0 . 1. It is known [53, 54] that below the critical temperature Tλ of the λ-transition, two fluids (4 He II phase) coexist: the normal fluid and the superfluid liquid. Later justified within the framework of phenomenological Landau’s theory [18,19,47], the picture suggested by Tisza and Landau was to interpret the condensate of frozen in momentum space bosons with p = 0 as a “superfluid component”, and the rest of particles as a “normal component” which is the carrier of the total entropy of the system. Experimentally, a Bose condensate was discovered in 4 He II. The apparition of this Bose condensate transition and the one of the superfluid liquid are strongly correlated to each other. Indeed, from [55–57] if γs is the fraction of superfluid liquid and γ0 the one of the condensate, one has
γs (T) ∼ (Tλ − T) ∼ γ0 (T) , for T → T− λ, η
(3.9)
see Figure 3.6. However, even for zero-temperature the superfluid liquid is not in a full Bose condensate phase which is in contradiction with the assumption of Tisza and Landau. B 2. Following Landau’s criterion of superfluidity [18, 19], the theory based on HΛ,0 might be understood as a microscopic theory of the superfluid liquid. Within this framework, it allows us to understand the close connection between the Bose condensate with density x
(β, ρ) and the Bogoliubov system with density {ρ − x
> 0}. These two systems may compose together the superfluid liquid, which coexists with the normal liquid for non-zero temperature at any positive velocity. Note that Landau’s criterion of superfluidity [18, 19] confronts an initial problem expressed by Remark 3.4 and also a second one: the application of this criterion to the Henshaw-Woods spectrum [20] gives for the critical velocity v0 ≈ 60 m/s
456
S. Adams and J.-B. Bru γ (%)
Ann. Henri Poincar´e
γ (%)
s
0
100
9
0
Tλ
0
Tλ
Figure 3.6: The fractions, γs of superfluid liquid and γ0 of the Bose condensate, as a function of the temperature T for 4 He
(Remark 3.1), whereas superfluidity in capillaries disappears when velocity is of the order of few cm/s. Moreover, it depends sensitively on the diameter of the channel. The attempts to explain these “misfittings” are concentrated around the idea that the Landau-type spectrum experimentally discovered by Henshaw and Woods [20] is only a part of a plethora of other types of “elementary” excitations not covered by the Bogoliubov theory, see [3, 57]. B , we have seen in Section 3.2 that Within the framework of the model HΛ,0 the Bose condensate has to exist in order to have the superfluidity property via the Bogoliubov system. Indeed, as soon as the non-conventional Bose condensate disappears, the collective phenomenon involved in the formation of the superfluid gas (Bogoliubov system) also vanishes. The introduction of a velocity in an inhomogeneous gas (in capillaries) may change the individual spectrum εk by increasing it. Then, the effective attraction g00 ((2.14), Figure 3.3) becomes smaller, i.e., the non-conventional Bose condensate and the (superfluid) Bogoliubov one may be destroyed for velocities sufficiently large but smaller than v0 (Remark 3.1). Note that the non-conventional Bose condensate constituted of Cooper-type pairs may be changed into a conventional Bose-Einstein condensation as it exists for the Perfect Bose Gas. An experimental study of the spectrum of excitations and also of the Bose condensation phenomenon should be interesting at different velocities. Actually, the collective behavior of this system should be quite delicate to save. A velocity may destroy the Cooper-type pairs and the quasi-particles. The important point is the following: the bigger the density of non-conventional Bose condensate, the stronger the robustness of Cooper-type pairs and quasi-particles to any perturbations. At temperatures T < Tλ even if the Bose condensate exists, its density may be not sufficiently important to keep the collective behavior for any positive velocities: some quasi-particles and Cooper-type pairs may be destroyed and a fraction λ (Remark 3.4) the therof normal fluid appears. At temperatures Tλ < T < T
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
457
mic fluctuations become sufficiently strong to destroy the non-conventional Bose condensate. Consequently, even if the quasi-particle gas resists in liquid helium for λ (Remark 3.4), it is quite unstable and any perturbation of the quasiTλ < T < T particles (like any positive velocity) may quickly destroy the collective system and switch it to a standard liquid where no superfluidity exists. 3. Note that this last conjecture may seem a little naive since the previous discussions are just phenomenological interpretations. Therefore, to conclude we examine B without taking into account Landau’s another interpretation of the Bose gas HΛ,0 criterion of superfluidity [18, 19], which is a phenomenological explanation of superfluidity. ηB −1 If γ0B (T) ∼ (Tc − T) at temperatures T = (kB β) → T− c is the fraction of Bose condensate for a fixed density ρ > 0, then via Theorem 2.3, the fraction γsB (T) = 1 − ρn /ρ satisfies: ηB
where
γsB (T) ∼ (Tc − T) ∼ γ0B (T) , for T → T− c , 1 fk,0 3 ρn (T) = k . d B /T Ek,0 (2π)3 x= x,α=α( x) B e E − 1 k,0 3
(3.10) (3.11)
R
The relation (3.10) is strangely similar to (3.9), see Figure 3.6. The fraction γsB (T) B . Therefore, at a may be considered as the superfluid fraction of the Bose gas HΛ,0 fixed density ρ > 0, the superfluid density ρs equals 2 2 x λk 1 3 ρs (T) = x + k , d 3 x= x,α=α( x) (2π) 2E B f + E B R3
k,0
k,0
k,0
whereas ρn (3.11) is the density of normal fluid which is the carrier of the total entropy of the system. Note that lim ρn = 0 and within this framework there is T→0+
=x
(T). See 100% of superfluid liquid at zero-temperature with a density ρs > x (i) of Theorem 2.3 to see the Bose condensate density at a fixed density ρ > 0. In fact, this conjecture has to be analyzed via the corresponding Hamiltonian with an external velocity field as it has been recently performed with dilute trapped Bose gases at zero-temperature [58].
3.5
Concluding remarks: superfluidity of Fermi systems
The superfluidity of a Fermi system, i.e., the 3 He liquid, was discovered in 1972 for sufficiently low temperatures [59, 60]. All the previous theories concern Bose systems. However, it is remarkable to see that, via the effective coupling constant B implies g00 < 0 (Figure 3.3), the non-diagonal interaction UΛN D of the model HΛ,0 an attraction between particles in the zero-mode. By analogy, it is well known that the phenomenon of superconductivity comes from the effective electron-electron interaction in the BCS theory which results
458
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
from the electron-phonon (non-diagonal) interaction in the second order of perturbation theory, see e.g., [61, 62]. Thus, in a superconductor, electrons can pair up in the metal crystal via phonons to form Cooper pairs which can then condense into a superconducting state. This phenomenon corresponds also to the explanation given for the existence of superfluidity in 3 He [21, 22]. Indeed, by cooling the liquid to a low enough temperature, 3 He atoms can pair up, making it a boson, and therefore superfluidity can be achieved. B , we found exactly the same kind of behavior on bosons: In the Bose gas HΛ,0 a system of linked pair of particles and Cooper-type pairs. Therefore, it should be interesting to study a similar Hamiltonian within the framework of Fermi systems. Of course, the main difference comes from the Fermi distribution. In particular, the critical density ρP BG (β, 0) for the Perfect Bose Gas does not exist for the B Perfect Fermi Gas. For the Bose system HΛ,0 , the corresponding kinetic part only turns on the Bose condensation phenomenon via the Bose distribution. Indeed, the corresponding “chemical potential” α ( x) , as solution of the variational problem (C.1) for a Bose condensate density x
(β, ρ) , becomes zero when we reach the critical density as for the Perfect Bose Gas, but switches again to strictly negative values for x
> 0 (in [26]: proof of Theorem 2.3). As soon as the Bose condensate appears, the non-diagonal interaction UΛN D becomes sufficiently important to drastically change all thermodynamic properties of the system by instantly switching the usual Perfect Bose gas to a gas of quasi-particles: the Bose-Einstein condensation becomes non-conventional in correlation with the creation of the Bogoliubov system and the formation of Cooper-type pairs (Section 3.2). Whereas the non-diagonal interaction UΛN D is not strong enough to imply alone the Bose-condensation at the critical temperature or density of the Perfect Bose Gas, for very small temperatures it strongly dominates all thermodynamics of the system. The non-diagonal interaction UΛN D obviously has a strong impact on the system (see for example the divergence of the grand-canonical pressure of B , Appendix B). It would have implied the non-conventional Bose condensation HΛ,0 without the Bose distribution at sufficiently low temperatures or high densities. In particular, if the Fermi distribution now holds, a similar non-diagonal interaction characterizing by an effective attraction g00 (like in (2.14), Figure 3.3) would drastically oppose the repulsion from the Pauli exclusion principle and would finally become strong enough at sufficiently low temperatures to imply alone the superfluid gas of quasi-particles explained above. This means of course that the critical temperature for the corresponding Fermi system should be quite lower B . Experimentally, the critical temperature of 3 He than that of the Bose model HΛ,0 is very low in comparison with that of 4 He (2.17 K): it is only two milli Kelvin for 3 He [59, 60]. We reserve this analysis on Fermi systems for another paper. To conclude, notice also that the 3 He liquid forms, at sufficiently low temperatures, several superfluid phases (A&B), which are much richer properties than those of the superfluid 4 He. For a complete review of properties of 3 He at low temperatures, see [63, 64].
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
Appendix A
459
The canonical Bogoliubov theory of superfluidity [6, 7, 46, 65–67]
Since the atoms of 4 He are bosons, a plausible conjecture links the superfluidity property with the Bose-Einstein condensation phenomenon predicted for the Perfect Bose Gas by Einstein in 1925 [68]. It was originally suggested by Fritz London in 1938 [30], since the transition of the normal liquid 4 He (called He I) to superfluid phase He II takes place at a temperature Tλ (2.17 K) very close to the one of the corresponding Perfect Bose Gas (3.14 K). Experimentally, a fraction of Bose condensate in liquid 4 He was only found in the sixties, almost 30 years after the London’s idea of genius, via deep-inelastic neutron scattering, see [2,3]. In 1941, Landau [18,19] understood for the first time that the properties of quantum liquids like 4 He (or 3 He) can be entirely described by the spectrum of collective excitation, which for liquid 4 He, has two branches : “phonons” for longwavelength excitations and “rotons” for a relatively short-wavelength collective excitations. Guided by the Landau’s idea that (at least) the low energy part of the spectrum in liquid 4 He is defined by coherent collective movements of the system instead of individual ones, Bogoliubov tried to find a physical (or mathematical) mechanism which as in crystals with phonons, favors the collective motions of the “helium jelly”, via some kind of ordering or coherence: the Bose condensation phenomenon suggested by London. Note that the spectrum of the Perfect Bose Gas does not satisfy the Landau criterion of superfluidity and from more recent experiments, recall that the Bose condensate represents at T = 0 K only 9% of the system, whereas there is 100% of Bose-Einstein condensation in the Perfect Bose Gas! In fact, Bogoliubov accepted the crucial role played by the Bose condensation mechanism in superfluid liquid 4 He, but he insisted that “an energy level scheme based on the solution of the quantum mechanical many-body problem with interactions, must be found” (in [7]: Part 3.4). More precisely, his aim was to deduce the Landau-type excitation spectrum from the full interacting gas HΛ,λ0 >0 (2.1). It was a very challenging program, and inspired by all the previous observations, Bogoliubov proposed his microscopic theory of superfluidity in 1947 [4–8]. His Weakly Imperfect Bose Gas was the starting point for this theory in the canonical or grand-canonical ensembles. This model arises from the truncation of the full interacting gas HΛ,λ0 >0 (2.1) by assuming the existence of a Bose condensation on the zero-mode for a weak enough interaction ϕ (x) . The most important terms in (2.1) should be those in which at least two operators a∗0 , a0 appear. This procedure implies his Weakly Imperfect Bose Gas, i.e., the Bogoliubov Hamiltonian (see [7], Part 3.5, eq. (3.81)) defined by B B ≡ HΛ,0 + UΛBMF , HΛ,λ 0 >0
(A.1)
460
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
cf. (2.3). Here the interaction UΛBMF ≡
λ0 ∗2 2 λ0 ∗ a a + a0 a0 2V 0 0 V
a∗k ak , λ0 > 0,
(A.2)
k∈Λ∗ \{0}
comes directly from the Bogoliubov truncation of the Mean-Field interaction UΛMF . Remark A.1 From the beginning, the Bogoliubov theories are not self-consistent. This fact was first highlighted by Angelescu-Verbeure-Zagrebnov [27] in the grandcanonical ensemble. Actually, in the grand-canonical ensemble the Bogoliubov B manifests, for high densities, a conventional Bose-Einstein conmodel HΛ,λ 0 >0 densation [9–11, 13, 14] on modes k = 2π/L = 0, which corresponding terms in (2.1) have been neglected in the truncation of the full interaction. B B seems to be “close” to the Bogoliubov model HΛ,λ , but The model HΛ,0 0 >0 their thermodynamics are in fact very different in the thermodynamic limit. For B is drastically example, in the grand-canonical ensemble, the Bose system HΛ,0 instable at high densities, i.e., the terms of repulsion are not strong enough to prevent the system from collapse for a chemical potential µ > −ϕ (0) /2 (Appendix B exists B), whereas the thermodynamics of the Bogoliubov Hamiltonian HΛ,λ 0 >0 in infinite volume for any µ ≤ 0 [27]. Actually, the interaction UΛBMF has a crucial and, unfortunately, a nasty B , see discussion in impact on the thermodynamics of the Bogoliubov model HΛ,λ 0 >0 [1]. In the canonical ensemble, he then proposed to use the operators {ζk }k∈Λ∗ (2.7) in order to apply his ingenious approximation (2.8). However, a direct application of (2.8) does not give the Landau-type excitation spectrum, again because of the interaction term UΛBMF . In fact, Bogoliubov suggested to eliminate the operator N0 from (A.2) at the cost of further approximations, see [6, 7] and discussion in [65]. He drastiB by using the following cally changed the original Bogoliubov Hamiltonian HΛ,λ 0 >0 approximation
N0 N02 + 2V V
k∈Λ∗ \{0}
ζk∗ ζk =
N0 (NΛ − N0 ) N2 N02 + Λ, 2V V 2V
(A.3)
i.e., the assumption, experimentally inexact for the liquid 4 He [2, 3], that 100% of Bose condensation occurs for T = 0 K. Note that this last ansatz gives a SB B superstabilized version HΛ,λ (2.12) of HΛ,0 with λ = λ0 .2 This model was not explicitly proposed by Bogoliubov since, meantime, he applied (2.8) combined with N0 NΛ 2 ≈ |c| = ρ ≈ V V 2 without
the last term of (2.12)
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
461
B in the canonical ensemble, in order to find HΛ,0 (ρ) (2.9) and the Landau-type SB with λ = λ0 have excitation spectrum. Actually, we recall that the model HΛ,λ ingeniously been proposed in the grand-canonical ensemble in [27]. It has recently been solved in [26]. Therefore, Bogoliubov had to take a completely condensed particle density to get the Landau-type excitation spectrum. This last assumption is not true for B (at least not in the grand-canonical the original Bogoliubov Hamiltonian HΛ,λ 0 >0 ensemble, see [10–14]). The Bogoliubov theory and the approach proposed here are applicable to the weakly interacting gas, but it does not mean that we reach the dilute limit. Following a perturbation theory, the depletion of the condensate is zero in the lowest-order: see for example (2.108)-(2.109) in Section 2.4 of [9]. We could replace N0 by NΛ everywhere it appears apart from the lowest-order Hartree term. This analysis is done after considering the creation/annihilation operators on the ground state as arbitrary chosen complex numbers. However, the crucial point is that, by keeping the creation/annihilation operators on the ground state, the nondiagonal interaction (2.5) of the Bogoliubov Hamiltonian implies a contribution in the second order, see [9, 12] and Section 3.2. This contribution is absolutely not negligible: this term is able to drastically change the thermodynamics of the system [9, 10, 12, 26]. It is the origin of the depletion of the condensate fraction and also of the quasi-particle system with the Landau-type excitation spectrum (Section 3.2). Bogoliubov (and Zubarev) early noticed the difficulty with his ansatz of 100% of Bose condensate. Some discussions corresponding to this problem can be found in [7, 8, 39–43].
Remark A.2 In the grand-canonical ensemble (β, µ) , the operators {ζk }k∈Λ∗ (2.7) are useless and we only need the Bogoliubov approximation (2.6). Then, from the 2 hypothesis µ = λ0 |c| > 0, which is shown to be not legitimate [27], Bogoliubov obtained the Landau-type excitation spectrum. For more details concerning the Bogoliubov theories of superfluidity, see [1, 9].
Appendix B
Thermodynamics in the grand-canonical ensemble
B We explore here the thermodynamic behavior of the Hamiltonian HΛ,0 (2.3) in the grand-canonical ensemble. Even if this study turns out to be useless to explain the superfluidity phenomenon, from the mathematical point of view it highlights the B , see discussion in the last subsection of this appendix. unusual behavior of HΛ,0 The pressure in the grand-canonical ensemble (β, µ) and the grand-canonical particle density are respectively given by B 1 ln T rFΛB e−β (HΛ,0 −µNΛ ) , pB Λ,0 (β, µ) ≡ βV NΛ ρB (β, µ) ≡ (β, µ) = ∂µ pB Λ,0 Λ,0 (β, µ) . V HB Λ,0
462
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
B.1 An upper bound for the grand-canonical pressure Regrouping terms in (2.3) one has B HΛ,0 = HΛI +
1 2V
k∈Λ∗ \{0}
where
HΛI =
∗ ∗ a0 ak + a∗−k a0 ≥ HΛI , λk a∗0 ak + a∗−k a0
k∈Λ∗ \{0}
λk 1 εk − Nk − 2V 2V
λk N0 .
k∈Λ∗ \{0}
Hence we obtain pB Λ,0
(β, µ) ≤
−1 λ 1 −β εk − 2Vk −µ (β, µ) ≡ ln 1 − e βV k∈Λ∗ \{0} −1 1 ln 1 − eβ(µ−µsup,Λ ) + , βV pIΛ
for µ < µsup,Λ ≡ −
1 2V
λk < 0.
(B.1)
k∈Λ∗ \{0}
B.2 A lower bound for the grand-canonical pressure using the Bogoliubov approximation B The corresponding lower bound for the Bogoliubov Hamiltonian HΛ,λ (A.1) 0 >0 found in [27] remains valid even for λ0 = 0 and one gets B pB Λ,0 (β, µ) ≥ sup pΛ,0 (β, µ, c) ,
(B.2)
c∈C
where pB Λ,0 (β, µ, c) is defined by 2 2 β, µ, x ≡ |c| + η µ, x ≡ |c| , (β, µ, c) = ξ pB Λ,0 Λ,0 Λ,0 −1 B 1 ξΛ,0 (β, µ, x) = ln 1 − e−βEk,0 , βV k∈Λ∗ \{0} 1 B fk,0 − Ek,0 + µx, ηΛ,0 (µ, x) = 2V ∗
(B.3)
k∈Λ \{0}
B with Ek,0 , fk,0 defined by (2.18) for α = µ ≤ 0. Therefore one has to analyze the lower bound sup pB Λ,0 (β, µ, c). c∈C
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
Lemma B.1 We have sup c∈C
pB Λ,0
(β, µ, c) =
463
P BG pB (β, µ) ; for µ ≤ µsup,Λ < 0 Λ,0 (β, µ, 0) = pΛ +∞ ; for µ > µsup,Λ ,
BG where pP (β, µ) = pB Λ Λ,0 (β, µ, 0) is the grand-canonical pressure for the Perfect Bose Gas.
Proof. Through (2.18) and (B.3), one gets that for µ ≤ 0: −1/2 1 1 2λk (i) ∂x ηΛ,0 (µ, x) = µ + λk − λk 1 + x , 2V 2V εk − µ ∗ ∗ k∈Λ \{0}
k∈Λ \{0}
∂x ηΛ,0 (µ, 0) = µ < 0; (ii)
∂x2 ηΛ,0
1 (µ, x) = 2V
Since lim
x→+∞
1 2V
k∈Λ∗ \{0}
k∈Λ∗ \{0}
√ λ2k εk − µ
3/2
(εk − µ + 2xλk )
> 0.
−1/2 2λk λk 1 + x = 0, εk − µ
even in the thermodynamic limit, (i) implies µ ≤ ∂x ηΛ,0 (µ, x) ≤ µ − µsup,Λ for all x ≥ 0 and lim {∂x ηΛ,0 (µ, x) − µ + µsup,Λ } = 0 .
x→+∞
So, we get with (ii)
sup {ηΛ,0 (µ, x)} = x≥0
ηΛ,0 (µ, x = 0) ; for µ ≤ µsup,Λ +∞ ; for µ > µsup,Λ .
(B.4)
Therefore, for β → ∞ (zero-temperature) the corresponding pressure pB (β, µ, c) (B.3) attains its supremum at c = 0 if µ ≤ µsup,Λ whereas sup Λ,0 pB Λ,0 (β, µ, c) does not exist for any µ > µsup,Λ . By (2.18) and (B.3) note that (i)
∂x ξΛ,0 (β, µ, x) < 0 and
(ii)
∂β ξΛ,0 (β, µ, x) < 0 and
c∈C
lim ξΛ,0 (β, µ, x) = 0,
x→+∞
lim ξΛ,0 (β, µ, x) = 0.
(B.5)
β→+∞
Hence via (B.4) and (B.5) the lemma holds.
Consequently, combining (B.2) with Lemma B.1, we find B pB Λ,0 (β, µ) ≥ pΛ,0 (β, µ, 0) ,
(B.6)
for any µ ≤ µsup,Λ , whereas for µ > µsup,Λ the pressure pB Λ,0 (β, µ) does not exist.
464
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
B.3 Thermodynamic behavior of the model Via the previous upper bound and (B.6) we get B I pB Λ,0 (β, µ, 0) ≤ pΛ,0 (β, µ) ≤ pΛ (β, µ) ,
for µ < µsup,Λ , which gives B P BG pB (β, µ) 0 (β, µ) =lim pΛ,0 (β, µ) = p Λ
(B.7)
in the thermodynamic limit for 1 µ < µsup ≡ lim µsup,Λ = − ϕ (0) < 0, Λ 2
(B.8)
and which can be extended by continuity of the pressure to µ ≤ µsup . Here pP BG (β, µ) is the infinite volume pressure for the Perfect Bose Gas. From (B.7) and Griffiths lemma [69, 70] the infinite volume particle density ρB 0 (β, µ) equals the one for the Perfect Bose Gas ρP BG (β, µ) for µ < µsup and therefore lim
µ→µ− sup
P BG ρB (β, µsup ) < +∞, 0 (β, µ) = ρ
(B.9)
i.e., it is not possible to reach high densities regimes in the grand-canonical ensemble (β, µ). B are, in a way, trivial Hence the thermodynamic properties of the model HΛ,0 for rather negative chemical potential µ ≤ µsup,Λ : they are equivalent to the Perfect Bose Gas. The non-diagonal interaction UΛN D (2.5) is not able to change the system for sufficiently negative chemical potential µ ≤ µsup . This fact is not surprising B since it is exactly the same for the Bogoliubov Hamiltonian HΛ,λ (A.1) for 0 >0 µ ≤ µsup , see the corresponding lower and upper bounds in [27] and discussions in [9, 11]. However, as soon as the non-diagonal interaction UΛN D beats the kinetic part for µ > µsup by attracting particles in the zero-mode (cf. [9, 12], Figures 3.2, 3.3 and 3.4), the system becomes unstable, i.e., all particles collapse in the zero-mode because of the absence of strong enough repulsion terms such as λ ∗2 2 λ 2 a0 a0 = N0 − N0 , with N0 ≡ a∗0 a0 . 2V 2V In fact this term of repulsion is crucial to induce the non-conventional Bose condensation mechanism without any instability in the grand-canonical ensemble (β, µ).
Appendix C
Proofs
The aim of this appendix is to give the promised details of the proofs of Theorems 2.2 and 2.3. But first, we quickly sum up the thermodynamic behavior of the SB (2.12), i.e., of the AVZ-Hamiltonian if superstable Bogoliubov Hamiltonian HΛ,λ λ = λ0 in the grand-canonical ensemble (β, µ).
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
465
C.1 Grand-canonical thermodynamics of the superstable Bogoliubov Hamiltonian [26] Note that the condition (2.13) corresponds to the assumption (C1) in [26] with a strict inequality. In 1. we consider the chemical potential µ as a fixed parameter, whereas in 2. the full particle density ρ is fixed in the grand-canonical ensemble. SB are denoted in 1. The grand-canonical pressure and density associated with HΛ,λ SB the thermodynamic limit as p (β, µ) and NΛ ρSB (β, µ) ≡lim (β, µ) = ∂µ pSB (β, µ) Λ V H SB Λ,λ
respectively. They are defined and explicitly found in [26] for any (β, µ) ∈ {β > 0}× {µ ∈ R}. We have to solve two variational problems. The first one is characterized by α (x) ≡ α (β, µ, x) ≤ 0, i.e., the unique solution of ( ' ( ' (µ − α)2 (µ − α)2 B B inf p0 (β, α, x) + = p0 (β, α, x) + (C.1) α≤0 2λ 2λ α=α(x) for any fixed x ≥ 0, where pB 0 (β, α, x) is defined by (2.17). Whereas the second variational problem directly related to pSB (β, µ) is (( ' ' (µ − α)2 SB B p (β, µ) = sup inf p0 (β, α, x) + α≤0 2λ x≥0 ( ' 2 (µ − α) B = inf p0 (β, α, x , (C.2)
) + α≤0 2λ which solution x
=x
(β, µ) is also unique. Then, via direct calculations, for any β > 0, there is a unique µc (β) such that the pressure pSB (β, µ) equals 2 (µ − α (0)) , for µ ≤ µc (β) . pB 0 (β, α (0) , 0) + 2λ ( ' 2 pSB (β, µ) = (µ − α (x)) B (β, α (x) , x) + , for µ > µc (β) . p 0 2λ x= x>0 The function µc (β) is bijective from R+ to R+ ( lim µc (β) = 0) and we denote β→+∞
by βc (µ) ≥ 0 the inverse function of µc (β). The pressure pSB (β, µ) is continuous for µ = µc (β) .
466
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
Remark C.1 The solution x
=x
(β, µ) of (C.2) always satisfies B = 0, for µ ≥ µc (β) , ∂x p0 (β, α, x) x= x,α=α( x) < 0, x
= 0, for µ ≤ µc (β) , ∂x pB 0 (β, α, x) x>0,α=α(x)
and the solution α ( x) = α (β, µ, x
) of (C.1) is always strictly negative for any µ = µc (β) or β = βc (µ). 2. Let ρc (β) =
lim
µ→µ− c (β)
ρSB (β, µ) =
lim
µ→µ+ c (β)
ρSB (β, µ) ≤ ρP BG (β, 0) ,
(C.3)
SB be the critical density of HΛ,λ , where ρP BG (β, 0) is the critical density of the Perfect Bose Gas (see (2.20) with x = 0). Then, ρc (β) ≈ ρP BG (β, 0) and for sufficiently large β, i.e., for small temperatures, we have
ρc (β) = ρP BG (β, 0) .
(C.4)
By fixing the particle density ρ in the grand-canonical ensemble, we define a unique chemical potential µβ,ρ satisfying ρSB (β, µβ,ρ ) = ρ.
(C.5)
Actually, at a fixed inverse temperature β the function µβ,ρ is the inverse function of the mean particle density ρSB (β, µ) of the AVZ-Hamiltonian. By βc (ρ) we denote the critical inverse temperature for a fixed particle density ρ (Figure 2.1). For ρ ≤ ρc (β), note that µβ,ρ ≤ µc (β) or β ≤ βc (ρ) whereas µβ,ρ > µc (β) , or β > βc (ρ), for ρ > ρc (β). Therefore we get the following properties: (i)
x) < 0, µβ,ρ − λρ = α ( SB (β, α (x) , x) p (β, µβ,ρ ) = pB 0
+ x= x
λ 2 ρ , 2
(C.6)
for any ρ > 0, where α ( x) < 0 is the unique solution of the Bogoliubov density equation ρ = ρB (β, α, x
) (2.19)–(2.20). 0 (ii) All the thermodynamic properties found for a fixed chemical potential µ are also valid for a fixed particle density ρ by using the corresponding chemical potential µβ,ρ . See for example Theorem 2.6 of [26]: for any ρ > ρc (β) there is only one Bose condensation in the zero mode characterized by x
=x
(β, µβ,ρ ) = x
(β, ρ) < ρ, whereas for ρ < ρc (β) the system behaves as the Mean-Field Bose Gas with no Bose condensations. No more Bose condensations outside the zero-mode appears for any ρ > 0.
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
467
C.2 Proof of Theorem 2.2 SB 1. Remark that HΛ,λ commutes with the particle number operator
SB B HΛ,λ , NΛ = HΛ,0 , NΛ = 0.
Let fΛSB (β, ρ) be the free-energy density defined for a fixed particle density ρ > 0 by (n=[ρV ]) SB 1 e−βHΛ,λ ln T rH(n) fΛSB (β, ρ) ≡ − . B βV B Notice that fΛ,0 (β, ρ) (2.15) and fΛSB (β, ρ) are related to each other by B fΛ,0 (β, ρ) = fΛSB (β, ρ) −
λ 2 ρ ρ − . 2 V
(C.7)
B SB The two models HΛ,0 and HΛ,λ are equivalent in the canonical ensemble, in the sense that their (infinite volume) free-energy densities at fixed densities differ only by a constant. Actually their Gibbs states are equal to each other for all (β, ρ). Therefore we only need to compute fΛSB (β, ρ) in the thermodynamic limit to deduce f0B (β, ρ). This is our next step.
2. Since the particle density ρSB (β, µ) as the derivative of the pressure pSB (β, µ) is continuous (as a function of µ) [26], using a Tauberian theorem proven in [71], the existence of pSB (β, µ) already implies the convexity of the thermodynamic limit f SB (β, ρ) for ρ > 0 of fΛSB (β, ρ) and the weak equivalence of the canonical and grand-canonical ensemble: pSB (β,µ) = sup µρ − f SB (β,ρ) = µρSB (β,µ) − f SB β,ρSB (β,µ) ,µ ∈ R, ρ>0 f SB (β,ρ) = sup µρ − pSB (β,µ) = µβ,ρ ρ − pSB (β,µβ,ρ ), ρ > 0 . µ∈R
(C.8) With (C.6) the Legendre transformation (C.8) implies an explicit expression of the corresponding free-energy density: λ SB B f (β, ρ) = α (x) ρ − p0 (β, α (x) , x) + ρ2 , 2 x= x i.e., via (C.7) we also have f0B (β, ρ) = α (x) ρ − pB (β, α (x) , x) 0
(C.9) x= x
with pB 0 (β, α, x) defined by (2.17). 3. Now we give an interpretation of this last equality to show that x
and α ( x) are also solutions of variational problems in the canonical ensemble. Meantime, we explicitly find another, more natural, expression for f0B (β, ρ).
468
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
2 (2.16) is well defined for any 3.1. The free-energy density f0B β, ρ1 , x = |c| ρ1 > 0 and β > 0. Moreover, since the particle density ρB 0 (β, α, x) (2.20) as the derivative of the pressure pB (β, α, x) (2.17) is continuous as a (strictly increasing) 0 function of α ≤ 0, for each fixed x ≥ 0, the function f0B (β, ρ1 , x) is convex for ρ1 > 0. Therefore, via (2.17) we find f0B (β, ρ1 , x) = sup α [ρ1 + x] − pB 0 (β, α, x) α≤0
= α (ρ1 , x) (ρ1 + x) − pB 0 (β, α (ρ1 , x) , x) , (C.10) for ρ1 ≡ ρ − x > 0, with α (ρ1 , x) defined as a solution of the Bogoliubov density equation ρ = ρB 0 (β, α (ρ1 , x) , x) (2.19)-(2.20) for ρ1 ≡ ρ − x > 0. 3.2. Since x
(β, ρ) (C.2) always satisfies x
< ρ even for β → +∞ [26], for x = x
the solution α ( x) < 0 of (C.1) is also the unique solution of the Bogoliubov density equation (2.19), see Appendix C.1. Therefore , (C.11) {α (ρ − x, x) = α (x)} x= x
which by (C.9) and (C.10) implies f0B
(β, ρ) =
f0B
(β, ρ − x, x)
.
(C.12)
x= x
, x
) in (C.12) can be understood as the result The free-energy density f0B (β, ρ − x B of the canonical Bogoliubov approximation applied on HΛ,0 . The equality (C.12) B finally means that the non-diagonal model HΛ,0 is thermodynamically equivalent BG to HP x) (2.11) for k ∈ Λ∗ \ {0} as stated in Section 2. Λ,0 ( 3.3. Moreover, by (C.10)-(C.11), one directly gets B
∂ρ1 f0 (β, ρ1 , x) = α ( x) , ρ1 =ρ− x,x= x
and
B
∂x f0 (β, ρ1 , x)
ρ1 =ρ− x,x= x
= α ( x) −
∂x pB 0
(β, α, x)
. α=α( x),x= x
Therefore, by using Remark C.1 the previous statements imply B B B f0 (β, ρ − x, x) , = inf f0 (β, ρ) = f0 (β, ρ − x, x) x= x 0. The solution x
=x
(β, µβ,ρ ) = x
(β, ρ) of the variational problem (C.2) is also solution of (C.13) for a fixed density ρ > 0. From (C.11), note that the solution α ( x) of the variational problem (C.1) is the solution in the canonical
> 0. Now we add some remarks to highlight ensemble of (C.10) with ρ1 ≡ ρ − x the important points in order to prepare the discussions of the next subsection.
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
469
Remark C.2 By (C.7), (C.8) and (C.12) we obtain µβ,ρ =
∂ρ f0B
B (β, ρ) + λρ = ∂ρ f0 (β, ρ − x, x)
+ λρ. x= x
Remark C.3 Via (C.6) combined with Remark C.2 it immediately follows that α ( x) = ∂ρ f0B (β, ρ) and ∂λ {α ( x)} = 0. Note that ∂ρ f0B (β, ρ) < 0 for any ρ = ρc (β) (Remark C.1). Remark C.4 For a fixed density ρ we have via (C.13) ∂λ x
= 0. We can see this results using Remark C.1 combined with Remark C.3. An illustration of the behavior of x
for a fixed density is performed in Figure C.1.
x(β,µ ) 2
λ x(β,ρ )
2
0
µ
β,ρ
λ
µ
β,ρ
µ
Figure C.1: Illustration of the density x
(β, µ) as a function of µ for two different parameters λ. The lower curve is for the larger value of λ. At a fixed particle
(β, ρ) is density ρ > 0 in the grand-canonical ensemble, the density x
(β, µβ,ρ ) = x constant as a function of λ.
C.3 Proof of Theorem 2.3 The aim of this subsection is now to deduce the canonical thermodynamic behavior B SB of the model HΛ,0 from the grand-canonical thermodynamic properties of HΛ,λ . This is done by using the notion of strong equivalence [25, 33–35].
470
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
B SB In fact, the two models HΛ,0 and HΛ,λ are equivalent in the canonical ensemB does not ble, see (C.7), i.e., their Gibbs states are equal for all (β, ρ). Since HΛ,0 SB depend on λ, one has to check if the grand-canonical densities for HΛ,λ depends on λ for any fixed particle density. Actually, the solutions α ( x) = α ( x, λ) and x
(λ) of the variational problems (C.1) and (C.2) are the key points of this first study. This is done via Remarks C.3 and C.4 (Figure C.1): the solutions α ( x, λ) and x
(λ) are also solutions in the canonical ensemble of the variational problems (C.10) with
> 0, and (C.13) respectively and they do not depend on λ for a fixed ρ1 ≡ ρ − x particle density ρ > 0. Consequently, all densities in the grand-canonical ensemble do not depend on λ for a fixed full particle density ρ. The parameter λ has no influence on the “physical” thermodynamic behavior of the system for a fixed particle density. Thus in the canonical ensemble the value of λ can be chosen freely as an arbitrary parameter as explained in the beginning SB is the “superstabilization” [24] of the model of Section 2.2. The Hamiltonian HΛ,λ SB such that H for a λ Λ,λ
SB SB = HΛ, HΛ,λ + λ
δ 2 >0 NΛ − NΛ with δ = λ − λ 2V
and
λ λ + g00 > + g00 > 0, 2 2 SB cf. (2.13)–(2.14). Because of the last inequality the model HΛ, satisfies the weak λ equivalence of ensembles for any density ρ > 0, and therefore the Hamiltonian SB satisfies the strong equivalence of ensembles [25] for any ρ > 0 and λ > 0 HΛ,λ sufficiently large. The strong equivalence is understood as follows: Let us consider by AΛ a (positive) quasi-local operator acting on +∞ B FΛB ⊂ F∞ ≡ ⊕ L2 Rnd symm n=0
such that lim AΛ H SB (β, ρ) < +∞ and lim AΛ H SB (β, µ) < +∞, Λ
Λ
Λ,λ
Λ,λ
(C.14)
for any β > 0 and ρ > 0. For β > 0, ρ > 0 and µΛ,β,ρ defined by NΛ (β, µΛ,β,ρ ) = ρ, V H SB Λ,λ
it follows from [25] that lim AΛ H B (β, ρ) =lim AΛ H SB (β, ρ) =lim AΛ H SB (β, µΛ,β,ρ ) , Λ
Λ
Λ,0
Λ,λ
Λ
Λ,λ
(C.15)
SB . Therefore the i.e., the strong equivalence of ensemble is verified by the model HΛ,λ SB B correspond thermodynamic properties in the canonical ensemble of HΛ,λ and HΛ,0 for a fixed particle density ρ to the one described in [26] with a chemical potential given by µ = µβ,ρ =lim µΛ,β,ρ (all densities are continuous). Λ
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
471
C.4 Additional remarks 1. Looking more closely at Theorem 2.6 of [26], the reader may be confused by the problem of the non-continuity of the grand-canonical particle density in the phase transition regime, if λ + g00 < 0, (C.16) 2 holds instead of (2.13) (in [26]: see condition (C2) and (iv) of Theorem 2.6). This in fact appears, because a (direct) coupling constant λ/2 satisfying (C.16) is too small to restore on f SB (β, ρ) the problem of strict convexity of f0B (β, ρ), see (C.7). This comes from the effective attraction g00 on the zero-mode arising from the non-diagonal interaction UΛN D (2.5) (cf. [9, 12], Figures 3.2, 3.3 and 3.4). On the other hand, for λ large enough, i.e., (2.13) is satisfied, the free-energy density f SB (β, ρ) becomes strictly convex. Thus, in this case the grand-canonical density is continuous and the two ensembles are in fact strong equivalent, cf. (C.15). 2. For low dimensions d = 1, 2, the effective coupling constant equals g00 = −∞ and (C.16) is satisfied for any λ > 0. Therefore the effective attraction g00 on the zero-mode should imply the existence of a non-conventional Bose condensation for d = 1, 2 (see [9, 12], Figures 3.2, 3.3 and 3.4). However the method used here to find the canonical thermodynamic properties fails since λ is never large enough to satisfy the condition (2.13). Acknowledgments. The work was supported by DFG grant DE 663/1-3 in the priority research program for interacting stochastic systems of high complexity. Special thanks first go to T. Dorlas and the DIAS for the very nice stay there where this work was finished. J.-B. Bru thanks Institut f¨ ur Mathematik, Technische Universit¨ at Berlin, and its members for their warm hospitality during the academic year 2001–2002 and more precisely S. Adams. J.-B. Bru also wants to express his gratitude to N. Angelescu, A. Verbeure and V.A. Zagrebnov for their useful discussions. And the second author thanks the P. master Dukes and Dido for their help in writing/correcting this article. The authors especially thank the referee for helpful remarks and suggestions.
References [1] S. Adams and J.-B. Bru, Critical Analysis of the Bogoliubov Theory of Superfluidity, Physica A 332, 60–78 (2004). [2] Bose-Einstein condensation, ed. A. Griffin, D.W. Snoke and S. Stringari, Cambridge Univ. Press, Cambridge (1996). [3] A. Griffin, Excitations in a Bose-Condensated Liquid, Cambridge Univ. Press, Cambridge (1993).
472
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
[4] N.N. Bogoliubov, On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947). [5] N.N. Bogoliubov, About the theory of superfluidity, Izv. Akad. Nauk USSR 11, 77 (1947). [6] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, Bull. Moscow State Univ. 7, 43 (1947). [7] N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 1: Quantum Statistics, Gordon and Breach Science Publishers, New York-London-Paris (1970). [8] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, p. 242–257 in: Collection of papers, Vol. 2, Naukova Dumka, Kiev, (1970). [9] V.A. Zagrebnov and J.-B. Bru, The Bogoliubov Model of Weakly Imperfect Bose Gas, Phys. Rep. 350, 291 (2001). [10] J.-B. Bru and V.A. Zagrebnov, Exact phase diagram of the Bogoliubov Weakly Imperfect Bose gas, Phys. Lett. A 244, 371 (1998). [11] J.-B. Bru and V.A. Zagrebnov, Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas, J. Phys. A: Math. Gen. A 31, 9377 (1998). [12] J.-B. Bru and V.A. Zagrebnov, Quantum interpretation of thermodynamic behaviour of the Bogoliubov weakly imperfect Bose gas, Phys. Lett. A 247, 37 (1998). [13] J.-B. Bru and V.A. Zagrebnov, Thermodynamic Behavior of the Bogoliubov Weakly Imperfect Bose Gas, p. 313 in: Mathematical Results in Statistical Mechanics, eds S. Miracle-Sole and al., World Scientific, Singapore (1999). [14] J.-B. Bru and V.A. Zagrebnov, On condensations in the Bogoliubov Weakly Imperfect Bose-Gas, J. Stat. Phys. 99, 1297 (2000). [15] V.A. Zagrebnov, Generalized condensation and the Bogoliubov theory of superfluidity, Cond. Matter Phys. 3, 265 (2000). [16] J.-B. Bru and V.A. Zagrebnov, Exactly soluble model with two kinds of BoseEinstein condensations, Physica A 268, 309 (1999). [17] J.-B. Bru and V.A. Zagrebnov, A model with coexistence of two kinds of Bose condensations, J. Phys. A: Math.Gen. 33, 449 (2000). [18] L.D. Landau, The theory of superfluidity of Helium II, J. Phys. (USSR) 5, 71 (1941). [19] L.D. Landau, On the theory of superfluidity of Helium II, J. Phys. (USSR) 11, 91 (1947).
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
473
[20] D.G. Henshaw and A.D.B. Woods, Modes of Atomic Motions in Liquid Helium by Inelastic Scattering of Neutrons, Phys. Rev. 121, 1266 (1961). [21] A.J. Leggett, p. 13 in: Modern Trends in the Theory of Condensed Matter, eds A. Pekalski and J. Przyslawa, Springer-Verlag, Berlin (1980). [22] A.J. Leggett, J. Phys. (Paris) Colloq. 41, C7-19 (1980). [23] A.J. Leggett, Superfluidity, Rev. Mod. Phys. 71, S318 (1999). [24] J.-B. Bru, Superstabilization of Bose Systems I: Thermodynamic Study, J. Phys. A: Math.Gen. 35, 8969 (2002). [25] J.-B. Bru, Superstabilization of Bose systems II: Bose condensations and equivalence of ensembles, J. Phys. A: Math. Gen. 35, 8995 (2002). [26] S. Adams and J.-B. Bru, Exact solution of the AVZ-Hamiltonian in the grandcanonical ensemble, Annales Henri Poincar´e 5, 405–435 (2004). [27] N. Angelescu, A. Verbeure and V.A. Zagrebnov, On Bogoliubov’s model of superfluidity J. Phys. A: Math. Gen. 25, 3473 (1992). [28] O. Brattelli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol II, 2nd ed. Springer-Verlag, New York (1996). [29] D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin-Reading, NewYork (1969). [30] F. London, The λ-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy Nature 141, 643 (1938). [31] N. Angelescu and A. Verbeure, Variational solution of a superfluidity model, Physica A 21, 386 (1995). [32] N. Angelescu, A. Verbeure and V.A. Zagrebnov, Superfluidity III, J. Phys. A: Math.Gen. 30, 4895 (1997). [33] H.O. Georgii, Large Deviations and the Equivalence of Ensembles for Gibbsian Particle Systems with Superstable Interaction, Probab. Th. Rel. Fields 99, 171 (1994). [34] H.O. Georgii, The equivalence of ensembles for classical systems of particles, Journal of Stat. Phys. Vol. 80, 1341–1378 (1994). [35] S. Adams, Complete Equivalence of the Gibbs ensembles for one-dimensional Markov-systems, Journal of Stat. Phys., Vol. 105, Nos. 5/6, 879–908 (2001). [36] M. van den Berg and J.T. Lewis, On generalized condensation in the free boson gas, Physica A 110, 550 (1982).
474
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
[37] M. van den Berg, On boson condensation into an infinite number of low-lying levels, J. Math. Phys. 23, 1159 (1982). [38] M. van den Berg, J.T. Lewis and J.V. Pul`e, A general theory of Bose-Einstein condensation, Helv. Phys. Acta 59, 1271 (1986). [39] N.N. Bogoliubov and D.N. Zubarev, Wave function of the ground-state of interacting Bose-particles, JETP 28, 129 (1955). [40] D.N. Zubarev, Distribution function of non-ideal Bose-gas for zero temperature, JETP 29, 881 (1955). [41] Yu.A. Tserkovnikov, Theory of the imperfect Bose-Gas for non-zero temperature, Doklady Acad. Nauk USSR 143, 832 (1962). [42] V.N. Popov, Functional Integrals and Collective Excitations, Cambridge Univ. Press, Cambridge (1987). [43] H. Shi and A. Griffin, Finite-temperature excitations in a dilute Bosecondensated gas, Phys. Rep. 304, 1 (1998). [44] R.P. Feynman, Application of Quantum Mechanics to Liquid Helium, p. 199 in: Progress in Low Temperature Physics, Vol. 1, Ch. 2. ed. J. Gorter, NorthHolland, Amsterdam (1955). [45] R.P. Feynman and M. Cohen, Energy spectrum of the Excitations in Liquid Helium, Phys. Rev. 102, 1189 (1956). [46] E.H. Lieb, The Bose fluid, in: Lectures in Theoretical Physics, Vol. VII C, ed. W.E. Briffin, University of Colorado, Boulder (1965). [47] I.M. Khalatnikov, An Introduction to the Theory of Superfluidity, BenjaminReading, New York (1965). [48] J. Wilks, An introduction to liquid helium, Claredon, Oxford (1970). [49] J.C. Slater and J.G. Kirkwood, The Van der Waals Forces in Gases, Phys. Rev. 37, 682 (1931). [50] K. Huang, Statistical Mechanics, Wiley, New York (1963). [51] O. Penrose, On the Quantum Mechanics of Helium II, Phil. Mag. 42, 1373 (1951). [52] O. Penrose and L. Onsager, Bose-Einstein condensation and Liquid Helium, Phys. Rev. 104, 576 (1956). [53] P. Kapitza, Nature 141, 74 (1938). [54] J.F. Allen and A.D. Misener, Nature 141, 75 (1938).
Vol. 5, 2004
A New Microscopic Theory of Superfluidity at All Temperatures
475
[55] L. Aleksandrov, V.A. Zagrebnov, Zh.A. Kozlov, V.A. Parfenov and V.B. Priezzhev, High energy neutron scattering and the Bose condensate in He II, Sov. Phys.-JETP 41, 915 (1975). [56] E.V. Dokukin, Zh.K. Kozlov, V.A. Parfenov and A.V. Puchkev, Investigation of the temperature dependence of the density of the Bose condensate in helium-4 in connection with the superfluidity phenomenon, Sov. Phys.-JETP 48, 1146 (1978). [57] N.M. Blagoveshchenskii, I.V. Bogoyavlenskii, L.V. Karnatsevich, V.G. Kolobrodov, Zh.A. Kozlov, V.B. Priezzhev, A.V. Puchkov, A.N. Skomorokhov and V.S. Yarunin, Structure of the excitation spectrum of liquid 4 He, Phys. Rev. B 50, 16550 (1994). [58] E.H. Lieb, R. Seiringer and J. Yngvason, Superfluidity in dilute trapped Bose gases, Phys. Rev. B 66, 134529 (2002). [59] D.D. Osheroff, R.C. Richardson, and D.M. Lee, Evidence for a New Phase of Solid He-3, Phys. Rev. Lett. 28, 885 (1972). [60] D.D. Osheroff, W.J. Gully, R.C. Richardson, and D.M. Lee, New Magnetic Phenomena in Liquid He-3 below 3 mK, Phys. Rev. Lett. 29, 920 (1972). [61] C. Kittel, Quantum Theory of Solids, John Wiley and Sons Inc., New York (1963). [62] N.N. Bogoliubov, V.V. Tolmachev and D.V. Shirkov, A New Method in the Theory of Superconductivity, Consultants Bureau Inc., New York (1959). [63] A.J. Leggett, A theoretical description of the new phases of liquid 3He, Rev. Mod. Phys. 47, 331 (1975). [64] D. Vollhardt and P. W¨ olfle, The Superfluid Phases of Helium 3, Taylor and Francis, London (1990). [65] N.N. Bogoliubov, Kinetic equations in the theory of superfluidity, JETP 18, 622 (1948). [66] M. Girardeau, Variational Method for the Quantum Statistics of Interacting Particles, J. Math. Phys. 3, 131 (1962). [67] A.J. Kromminga and M. Bolsterli, Perturbation Theory of Many-Boson Systems, Phys. Rev. 128, 2887 (1962). [68] A. Einstein, Sitzungsberichte der Preussischen Akademie der Wissenschaften I, 3 (1925). [69] R. Griffiths, A Proof that the Free Energy of a Spin System is extensive, J. Math. Phys. 5, 1215 (1964).
476
S. Adams and J.-B. Bru
Ann. Henri Poincar´e
[70] K. Hepp and E.H. Lieb, Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field, Phys. Rev. A 8, 2517 (1973). [71] R.A. Minlos and A.Ja. Povzner, Thermodynamic limit for entropy, Trans. Moscow Math. Soc. 17, 269 (1967). S. Adams Institut f¨ ur Mathematik Fakult¨ at II, SEK. MA 7-4 Technische Universit¨ at Berlin Strasse des 17. Juni 136 D-10623 Berlin, Germany email:
[email protected] J.-B. Bru School of Theoretical Physics Dublin Institute for Advanced Studies 10 Burlington Rd. Dublin 4, Ireland email:
[email protected] Communicated by Vincent Pasquier Submitted 31/03/03, accepted 30/01/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 477 – 521 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030477-45 DOI 10.1007/s00023-004-0176-6
Annales Henri Poincar´ e
A Mountain Pass for Reacting Molecules Mathieu Lewin Abstract. In this paper, we consider a neutral molecule that possesses two distinct stable positions for its nuclei, and look for a mountain pass point between the two minima in the non-relativistic Schr¨ odinger framework. We first prove some properties concerning the spectrum and the eigenstates of a molecule that splits into pieces, a behavior which is observed when the Palais-Smale sequences obtained by the mountain pass method are not compact. This enables us to identify precisely the possible values of the mountain pass energy and the associated “critical points at infinity” (a concept introduced by Bahri [2]) in this non-compact case. We then restrict our study to a simplified (but still relevant) model: a molecule made of two interacting parts, the geometry of each part being frozen. We show that this lack of compactness is impossible under some natural assumptions about the configurations “at infinity”, proving the existence of the mountain pass in these cases. More precisely, we suppose either that the molecules at infinity are charged, or that they are neutral but with dipoles at their ground state.
Introduction In this paper, we study in the non-relativistic quantum Schr¨ odinger framework the case of a molecule that possesses two distinct stable positions for its nuclei, as this is for instance the case for HCN and CNH. Our purpose is somewhat simple: can we obtain a critical point of the energy by using the classical mountain pass method between the two minima? Experiment suggests that this is the case (at least for the HCN↔CNH reaction). Indeed, such mountain pass points are frequently computed by chemists who need to understand the possible behavior of the molecule: it corresponds to a “transition state” during an infinitely slow reaction leading from one minimum to the other. But as far as we know, this problem has never been tackled from the mathematical point of view for the N -body quantum problem, or even in the context of the classical Hartree or Thomas-Fermi type models which are approximations of the exact theory. For a neutral molecule, the proof that there is a minimum with regards to the position of the nuclei can be found in the fundamental work of E.H. Lieb and W.E. Thirring [25] for the Schr¨ odinger model, and in a series of papers by I. Catto and P.-L. Lions [4, 5, 6, 7] for approximate models (Hartree or Thomas-Fermi type), the latter being really more complicated due to the non linearity of these models. In these two works, the authors had to prove that minimizing sequences are compact, the non-compactness behavior being related to the fact that the molecule can split into parts, each moving away from the others. Remark that
478
M. Lewin
Ann. Henri Poincar´e
binding does not occur for the Thomas-Fermi model (see the works of E. Teller [31], E.H. Lieb and B. Simon [24], and the references in [4]) and that the result is not known for the Hartree-Fock model, except in very special cases [7]. Let us make some comment on a tool used in these proofs that cannot be simply adapted to our setting. A common idea in these two works is to average over all the possible orientations of each piece in order to simplify the computation of the interaction energy between them, by suppressing the multipoles. To show that the energy can be lower than the energy at infinity, a new term using the correlation between the electrons is then created in [25] to obtain a Van Der Waals term of the form −C/R6 (R is the distance between the molecules), while a very detailed computation of the (exponentially small) combined energy is done in [4, 5, 6, 7] to conclude that the system can bind. Because of the preliminary averaging, the conclusion is that there exists some orientation of the molecules for which this is true, but this position is unknown a priori. In the case of the mountain pass method that we propose here, the noncompactness is obviously also due to a possible splitting of the molecule. However, we want to insist on the fact that we cannot use in this setting the same idea of averaging over all the rotations of the molecules, because we have to pull down the energy along a path. In other words, a comparison between the energies is not sufficient to conclude, and a precise information on the directions on which the energy decreases is needed. This is why we failed to treat the problem in its full generality and we had to add some hypothesis about the configurations “at infinity”. Nevertheless, we wish to ameliorate this first work in the future, and hope that it will stimulate further results. The results proved in this paper are the following. First, we study the spectrum and the eigenfunctions of the Hamiltonian when a molecule splits into parts. We obtain some bounds on the eigenvalues and the bottom of the essential spectrum which allow to show that the “electrons remain in the vicinity of the nuclei” when a fixed excited state is studied. In other words, no electron is lost during the process. This is obtained by a non-isotropic exponential decay of the electronic density, which is shown to be uniform when the distance between the molecules grows. We also specify the behavior of the associated wavefunctions and define the “critical points at infinity”, a concept introduced by A. Bahri [2]. Some parts of this first result are necessary for our min-max problem. Then, we prove a result that enables to identify the possible behavior of the non-compact min-maxing paths. As it is suggested by the intuition, it is shown that the optimal energy of the mountain pass corresponds in this case to a system where the molecule is split into independent parts (the electrons are shared among them), each being at its ground state. This Morse information on the critical points at infinity is rather intuitive. As announced, we were unable to treat the general case and we end this article by showing that this non-compactness behavior is impossible in the special case of
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
479
two interacting molecules with fixed nuclei. This is done under the hypothesis that we are in the easy case of two charged molecules at infinity, or in the more difficult case of two neutral molecules at infinity, but with dipoles at their ground state. This enables to obtain the required result for many practical situations. As explained before, the crucial step is to evaluate the interaction energy between the molecules and we use here a multipole expansion, even for wavefunctions that are not a simple tensor product of two ground states as in [25]. Finally, the expected result is deduced from the fact that the critical points of the dipole/dipole interaction energy which have a nonnegative energy have a Morse index which is at least 2. From a practical point of view, the study of this mountain pass method is really important. As mentioned above, the main idea is that a path leading from one minimum to the other represents an infinitely slow chemical reaction. The mountain pass energy is then interpreted as the lowest energy threshold for the reaction to happen. The numerical computation of this energy and of the optimum (even the whole path) is then a prime necessity for chemists, who have to understand the possible behaviors of the molecule (see for instance [28, 9] for chemical and numerical aspects). However, chemists only consider paths on which the molecule is at its ground state all along it, which leads to obvious problems of smoothness in the case of degeneracy of the first eigenvalue of the Hamiltonian, and can obstruct convergence. For mathematical reasons, we were thus forced to abandon this hypothesis and relax the problem by considering that the wavefunction can vary independently of the nuclear geometry, in order to obtain a critical point with respect to nuclei’s variations. Since we shall show that our min-max energy is in fact the same that the one used in practice, this approach could also be interesting for numerical computations. We conclude with a few words on the mathematical tools used in this paper. As in [4, 5, 6, 7], the proof is guided by P.-L. Lions’ Concentration-Compactness ideas [26], although the localization of the electrons is given by the uniform exponential decay of Theorem 2, and not by this theory. Let us remark that the physical intuition is somewhat often related to the behavior of the electronic density. For instance, when the molecule splits into parts, the latter becomes a sum of functions localized near the nuclei. But this point of view is not sufficient to understand the problem since the main object is not the density, but the wavefunction. The latter will not split into sums, but into sums of tensor products of wavefunctions in lower dimensions (see the work of G. Friesecke [10] for a very clear explanation of this phenomenon). Therefore, we use a variant of N -body geometric methods for Schr¨ odinger operators [30, 29, 18] that enables to relate the behavior of the wavefunction to those of the associated electronic density. This method is used in [10] and enabled G. Friesecke to notice an interesting link between the celebrated HVZ Theorem [18, 32, 33] and the Concentration-Compactness method [26]. Moreover, the HVZ Theorem (which enables to identify the bottom of the spectrum as the ground state energy of the same system but with an electron
480
M. Lewin
Ann. Henri Poincar´e
removed), and Zhislin’s Theorem (which states the existence of excited states for positive or neutral molecules) are abundantly used in this article. Finally, we use the results and methods developed by G. Fang and N. Ghoussoub [8, 14] which enable to obtain Morse information on the Palais-Smale sequences, related to the fact that the deformed object are paths (i.e. deformations of [0; 1]). We also use the duality theory developed in [14] which permits to locate critical points. The paper is organized as follows. In the next section, we describe the model in detail and recall known results on the Hamiltonian and its eigenfunctions. Then, in Section 2, we present our results without proof: for the sake of clarity, we have brought all the proofs together in the last section.
1 The model 1.1
Framework
We consider here a positive or neutral molecule with N non relativistic electrons, and M nuclei of charges Z1 +· · ·+ZM ≥ N . The nuclei are supposed to be correctly described by a classical model (Born-Oppenheimer approximation) and are thus represented as pointwise charges at R1 , . . . , RM ∈ R3 . In what follows, we let R = (R1 , . . . , RM ) ∈ Ω := (R3 )M \ (∪i=j {Ri = Rj }) and
Z = (Z1 , . . . , ZM ) ∈ (N∗ )M ,
|Z| = Z1 + · · · + ZM ≥ N.
The system is described by the purely coulombic N -body Hamiltonian H N (R, Z) =
N 1 − ∆xi + VR (xi ) + 2 i=1 VR (u) = −
1≤i<j≤N
M j=1
1 + |xi − xj |
1≤i<j≤M
Zi Zj , |Ri − Rj |
Zj . |u − Rj |
Its operator domain is the Sobolev space Ha2 (R3N , C), and its quadratic form domain is Ha1 (R3N , C). Throughout this paper, the subscript a indicates that we consider wavefunctions Ψ which are antisymmetric under interchanges of variables (expression of the Pauli exclusion principle): ∀σ ∈ SN , Ψ(x1 , . . . , xN ) = (σ)Ψ(xσ(1) , . . . , xσ(N ) ). The quantum energy of the system in a state Ψ ∈ Ha1 (R3N , C) is the associated quadratic form E N (R, Ψ) = Ψ, H N (R, Z)Ψ .
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
481
We refer the reader to [22, 3] for a description of this model and a detailed explanation of the Born-Oppenheimer approximation. The properties of H N (R, Z) and its eigenfunctions are recalled below. For the sake of simplicity, we have neglected the spin and the dynamic of the nuclei, as in [25]. We would like also to mention that all the results in this paper can be adapted to the caseof smeared nuclei, 1 1 is replaced by R3 |x−y−R dµi (y) that is to say when the Coulomb potential |x−R i| i| 1 3 where µi is a probability measure on R . Of course, |Ri −Rj | has to be replaced by 1 R6 |Ri +y−(Rj +z)| dµi (y)dµj (z). Remark that, in contrast to many other papers dealing with minimization, we work here with complex-valued wavefunctions, a hypothesis that plays a role in our results (see for instance Theorem 4 and the associated remarks). Z and N being fixed such that N ≤ |Z|, for each R ∈ Ω, the problem E N (R, Z) = min{E N (R, Ψ), ||Ψ||L2 = 1} has a solution Ψ, which is the ground state of the N electrons interacting with the M nuclei localized at the Ri . For neutral molecules (N = |Z|), it is also known that the problem E N = min E N (R, Z) R∈Ω
admits a solution [25], proving the stability of neutral molecules. We shall assume that (R, Ψ) and (R , Ψ ) are two local minima of E N . We then consider the classical mountain pass method c = inf max E N (γ(t)) γ∈Γ t∈[0;1]
(1)
where Γ = {γ ∈ C 0 ([0; 1], Ω × SHa1 (R3N )), γ(0) = (R, Ψ), γ(1) = (R , Ψ )} SHa1 (R3N ) = {Ψ ∈ Ha1 (R3N ), ||Ψ||L2 = 1} and want to show that c is a critical value of E N . As mentioned in the introduction, the physical interpretation of this minmax method is that paths γ ∈ Γ represent an infinitely slow reaction leading from one minimum to the other. c is thus interpreted as the lowest energy threshold for passing from (R, Ψ) to (R , Ψ ). In practice, the following definition is used c = inf max E N (r(t), Z) r∈R t∈[0;1]
where
R = {r ∈ C 0 ([0; 1], Ω), r(0) = R, r(1) = R }.
As explained in the introduction, the function R → E N (R, Z) is continuous but not necessary differentiable and this is why we shall study the min-max method (1). However, it will be shown that in fact c = c .
482
1.2
M. Lewin
Ann. Henri Poincar´e
Properties of H N (R, Z)
Let us now recall some well-known facts about the spectrum of H N (R, Z). We introduce inf sup H(R, Z)Ψ, Ψ . (2) λN d (R, Z) = dim(V )=d
Ψ∈V, ||Ψ||L2 =1
In the sequel, we shall denote for all d ≥ 1
E 0 (R, Z) = λ0d (R, Z) :=
1≤i<j≤M
Zi Zj , |Ri − Rj |
Σ0 (R, Z) = +∞.
For a wavefunction Ψ ∈ Ha1 (R3 , C), the electronic density and the electronic kinetic energy density are respectively defined by |Ψ(x, x2 , . . . , xN )|2 dx2 . . . dxN ρΨ (x) = N tΨ (x) = N
R3(N −1)
R3(N −1)
|∇Ψ(x, x2 , . . . , xN )|2 dx2 . . . dxN .
We have brought together the main known results in the following Theorem 1. We assume N ≥ 1. The following results are known: 1. (Self-adjointness [20]) H N (R, Z) is self-adjoint on L2a (R3N ) with operator domain Ha2 (R3N ) and quadratic form domain Ha1 (R3N ). 2. We have [19] σess H N (R, Z) = [ΣN (R, Z); +∞) Zi Zj N ∀d ≥ 1, λN d (R, Z) ≤ Σ (R, Z) ≤ |Ri − Rj | 1≤i<j≤M
3. (HVZ Theorem [18, 32, 33, 19]) We have ΣN (R, Z) = E N −1 (R, Z). N 4. (Compactness below the essential spectrum) If λN d (R, Z) < Σ (R, Z), then N λd (R, Z) is an eigenvalue of finite multiplicity and in particular, there exists a Ψd ∈ Ha2 (R3N ) such that
H N (R, Z)Ψd = λN d (R, Z)Ψd . It is locally lipschitz [21], i.e., Ψd ∈ C 0 (R3N )
and
3N |∇Ψd | ∈ L∞ ), loc (R
and real analytic [13] on U N \ {xi = xj }, where U = R3 \ {Ri }M i=1 . If ρd is the associated electronic density, then [13] ρd ∈ C ω (U ) ∩ C 0,1 (R3 ).
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
483
5. (Zhislin Theorem [33]) For positive and neutral molecules N ≤ |Z|, then N λN d (R, Z) < Σ (R, Z)
for all d ≥ 1, so that N N σ H N (R, Z) = {λN 1 (R, Z) ≤ · · · ≤ λd (R, Z) ≤ · · · } ∪ [Σ (R, Z); +∞). 6. (Negative molecules [19, 23]) For negative molecules N > |Z|, there exists a N δ such that λN δ (R, Z) = Σ (R, Z), and δ = 1 when N ≥ 2|Z| + M . Note that the functions ΣN (R, Z) and λN d (R, Z) (d ≥ 1) are continuous with respect to R.
2 The results In this section, we present the results that we have obtained concerning the mountain pass method defined above. As mentioned, all the proofs are postponed to the next section.
2.1
The spectrum of a molecule that splits into pieces
We begin the study by some general results about the spectrum and the behavior of the eigenstates when the molecule splits into pieces, that is to say when |Ri −Rj | → +∞ for some i and j. As mentioned above, this splitting of the molecule will be shown to be the main reason for the possible lack of compactness of Palais-Smale sequences. Although only the case of ground states will be necessary for the sequel, we tackle here arbitrary excited states. In this section, we consider a positive or neutral molecule (N ≤ |Z|). We fix a 2 ≤ p ≤ M (number of pieces). Let X1 , . . . , Xp : R+ → R3 be p functions that satisfy |Xi (t) − Xj (t)| ≥ t ∗ p for
p all i = j and t large enough. Let be m = (m1 , . . . , mp ) ∈3 (N ) such that some rj,k ∈ R and zj,k ∈ N for j=1 mj = M . We fix a positive constant R0 and
mj j = 1, . . . , p and k = 1, . . . , mj such that |Z| := pj=1 k=1 zj,k ≥ N , |rj,k | ≤ R0 and rj,k = rj,l when k = l. We then let zj = (zj,1 , . . . , zj,mj ),
Z = (z1 , . . . , zp ),
r˜j (t) = (Xj (t) + rj,1 , . . . , Xj (t) + rj,mj ),
rj = (rj,1 , . . . , rj,mj ), R(t) = (˜ r1 (t), . . . , r˜p (t)).
We also introduce ωj = {(rk ) ∈ B(0, R0 )mj , rk1 = rk2 if k1 = k2 }. and
U(R) = R3 \
p
j=1
B(Xj , R) .
484
M. Lewin
Ann. Henri Poincar´e
Z3
Z2 ×
X1 Z4
Z1
O ×
Z6
Z5 × X3
t
Y X2 r2,1 ×
*
+
Figure 1: An example with M = 6, p = 3, m1 = 3, m2 = 1 and m3 = 2. 2.1.1 Spectrum and uniform exponential decay We have the following result: Theorem 2 (Spectrum and uniform exponential decay). For all 1 ≤ N ≤ |Z| and all d ≥ 1, we have p 1. lim E N (R(t), Z) = min E Nj (rj , zj ), N1 + · · · + Np = N . t→+∞ j=1 p p Nj 2. limsup λN (R(t),Z) ≤ min λ (r ,z ),N + ···+ N = N, δ = d . j j 1 p j d δj t→+∞ j=1 j=1 3. inf lim inf ΣN (R(t), Z) − λN d (R(t), Z) > 0. rj ∈ωj t→+∞
4. Let ΨR be an eigenfunction associated to the eigenvalue λN d (R, Z), with associated densities ρR and tR . Then there exist positive constants R1 , C and α, depending only on N , d, R0 , p such that ρR (x) ≤ C exp(−αδ(x)) and tR (x) ≤ C exp(−αδ(x))
on U(R1 )
where δ(x) = min{|x − Xj |, j = 1, . . . , d}. The first part 1) identifies the limit of the ground state energy. This type of result is rather intuitive and classical. However, since we do not know a reference in this precise setting, a proof will be given in the next section. The interpretation of the last part 4) is that if a neutral or positively charged molecule splits into parts, then for a fixed excited state, the electrons remain in the vicinity of the nuclei.
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
485
For the sake of simplicity, let us denote, for r = (r1 ,...,rp ) and z = (z1 ,...,zp ) p p N λδjj (rj , zj ), N1 + · · · + Np = N, δj = d . ΛN d (r, z) := min j=1
j=1
2.1.2 Behavior of the wavefunctions, critical points at infinity Now that we have some bounds on the eigenvalues and the bottom of the essential spectrum, we want to prove a result describing the behavior of the eigenfunctions. This will enable us to define the “critical points at infinity” of the model, a concept that was introduced by A. Bahri [2]. The right-hand side of Theorem 2-1), or more generally an equality like c=
p
N
λδjj (rj , zj ),
(3)
j=1
is rather standard from P.-L. Lions’ Concentration-Compactness point of view: when the molecule splits into pieces, then the energy of an electronic excited states becomes a sum of excited states energies of the pieces. In other words, a “critical point at infinity” would be a system constituted by p molecules in some excited state, each being infinitely far from the others, so that the interactions between them vanish. p Let us consider a sequence tn → +∞, some r = (r1 , . . . , rp ) ∈ j=1 ωj , n n and denote by R = R(tn ) := (X1 (tn ) + r1 , . . . , Xp (tn ) + rp ), Xj = Xj (tn ). For the electronic density, such a configuration is then clearly obtained in the case
of dichotomy, that is to say ρn pj=1 ρnj where ρnj is essentially supported in the vicinity of Xjn (see the exponential decay of Theorem 2). But the behavior of the wavefunction Ψn is less simple since these functions will not split into sum of functions, but into sums of antisymmetric tensor products of wavefunctions in lower dimensions. In other words, a simple way to represent these non interacting molecules in terms of the wavefunction is to take Ψn = τX1n · ψ1 ∧ · · · ∧ τXpn · ψp
(4)
where each ψj is an eigenfunction of H Nj (rj , zj ) associated to the eigenvalue N λδjj (rj , zj ). We have used here the notation τv · Ψ(x1 , . . . , xN ) := Ψ(x1 − v, . . . , xN − v) and we recall that the tensor product is defined for ψ ∈ L2a (R3N1 ) and ψ ∈ L2a (R3N2 ) by 1 ε(σ)ψ(x1σ )ψ (x2σ ). ψ ∧ ψ (x1 , . . . , xN1 +N2 ) = √ N !N1 !N2 ! σ∈S N
where x1σ := (xσ(1) , . . . , xσ(N1 ) ) and x2σ = (xσ(N1 +1) , . . . , xσ(N ) ).
486
M. Lewin
Ann. Henri Poincar´e
With (4), one easily sees that ρn = pj=1 τXjn · ρj with obvious notations, and that H N (Rn , Z)Ψn − cΨn → 0 in L2 (R3N ) as n → +∞. N When a λδjj (rj , zj ) is degenerated, we can obtain the same behavior by taking a wavefunction which is a sum of such antisymmetric tensor products Ψn ∈
p
N τXjn · ker H Nj (rj , zj ) − λδjj (rj , zj ) .
j=1
To simplify notations, we shall denote τn · (ψ1 ∧ · · · ∧ ψp ) := τX1n · ψ1 ∧ · · · ∧ τXpn · ψp , N so that Ψn = τn · Ψ where Ψ ∈ pj=1 ker H Nj (rj , zj ) − λδjj (rj , zj ) . Suppose now that a molecule splits into two identical pieces: r1 = r2 and z1 = z2 , and that N1 = N2 . At infinity, we shall obtain two molecules with the same configurations of the nuclei, but not the same number of electrons. Since there is no reason to distinguish the two states obtained by inverting the electrons between the two molecules, a wavefunction can be a sum of these two states with the same energies. We are thus led to introduce the following definition. p n Definition 1. Let rn = (r1n , . . . , rpn ) ∈ j=1 ωj be such that rj → rj ∈ ωj , and n n R = R(tn ) := (X1 (tn ) + r1 , . . . , Xp (tn ) + rp ), Xj = Xj (tn ), for some tn → +∞. Let c be such that the set p N AN λδjj (rj , zj ) = c (Nj , δj ) ∈ (Np )2 , N1 + · · · + Np = N, (5) c (r, z) = j=1
is not empty. The sequence (Rn , Ψn ) in Ω × SHa1 (R3N ) converges to a critical point at infinity of energy c if there exists some p N ker H Nj (rj , zj ) − λδjj (rj , zj ) Ψ∈ (6) (Nj ,δj )∈AN c (r,z)
j=1
such that ||Ψn − τn · Ψ||Ha1 (R3N ) → 0. To justify the term critical point, we remark that one can prove Lemma 1. Let be c and Ψ that satisfy (5) and (6). Then (H N (Rn , Z) − c)(τn · Ψ) → 0 in L2 (R3N ). Details will be given later on.
(7)
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
487
Saying differently, a critical point at infinity is a class τK (r, Ψ) = τX1 ,...,Xp (r, Ψ), Xj ∈ R3 , |Xi − Xj | ≥ K
Nj p Nj ker H (r , z ) − λ (r , z ) , rj ∈ ωj and where Ψ ∈ (Nj ,δj )∈AN j j j j j=1 (r,z) δ j c p 1 3Nj τX1 ,...,Xp is defined on each Ω × j=1 Ha (R ) by τX1 ,...,Xp · r = (X1 + r1 , · · · , Xp + rp ) τX1 ,...,Xp · (ψ1 ∧ · · · ∧ ψp ) = (τX1 ψ1 ) ∧ · · · ∧ (τXp ψp ). A sequence (Rn , Ψn ) converges to this critical point at infinity when there exists a Kn → +∞ such that lim d[(Rn , Ψn ); τKn (r, Ψ)] = 0.
n→+∞
p Let us now fix a sequence tn → +∞ and some rn = (r1n , . . . , rpn ) ∈ j=1 ωj such that rjn → rj ∈ ωj , and denote Rn = R(tn ), Xjn = Xj (tn ). We then have the following result concerning the behavior of the eigenfunctions Theorem 3. We assume 1 ≤ N ≤ |Z| and d ≥ 1. Let (Ψn ) be a sequence of wavefunctions such that n n H N (Rn , Z) · Ψn = λN d (R , Z) · Ψ . n Then, up to a subsequence, we have limn→+∞ λN d (R , Z) := c with
ΛN 1 (r, z) ≤ c =
p j=1
N
λδ j (rj , zj ) ≤ ΛN d (r, z) j
n n for some (Nj , δj ) ∈ AN c (r, z), and (R , Ψ ) converges to a critical point at infinity of energy c.
2.2
The mountain pass method: a general result
Let us now come back to our mountain pass method, and consider again a neutral molecule (N = |Z|). Recall that (R, Ψ) and (R , Ψ ) are two local minima of (R, Ψ) → E N (R, Ψ), and that c and c are defined by c = inf max E N (γ(t)) γ∈Γ t∈[0;1]
Γ = {γ ∈ C 0 ([0; 1], Ω × SHa1 (R3N )), γ(0) = (R, Ψ), γ(1) = (R , Ψ )}. c = inf max E N (r(t), Z) r∈R t∈[0;1]
R = {r ∈ C 0 ([0; 1], Ω), r(0) = R, r(1) = R }.
(8)
488
M. Lewin
Ann. Henri Poincar´e
The following result enables to identify c in the case of lack of compactness: Theorem 4. We assume N = |Z|. We have c = c . There exists a min-maxing sequence (Rn , Ψn ) ∈ Ω × Ha1 (R3N ) such that:
If i=j |Rin − Rjn | is bounded then, up to a translation, (Rn , Ψn ) converges strongly in Ω × Ha1 (R3N ) to some critical point (R, Ψ) of E N such that H N (R, Z) · Ψ = c · Ψ,
c = λN 1 (R, Z).
If i=j |Rin − Rjn | is not bounded, there exists a 2 ≤ p ≤ M , some Xjn ∈ R3 with j = 1, . . . , p and a R0 > 0 such that, changing the indices if necessary, Rn = (X1n + r1n , . . . , Xpn + rpn ), ||rjn || ≤ R0 , Z = (z1 , . . . , zp ), and limn→+∞ |Xin − Xjn | = +∞, rjn → rj . Then c = ΛN 1 (r, z) = min
p
E Nj (rj , zj ), N1 + · · · + Np = N
j=1
and (Rn , Ψn ) converges up to a subsequence to a critical point at infinity of energy c = ΛN 1 (r, z). As a consequence, in the non-compact case, the molecule splits into pieces, the electrons being shared among them and at their ground state. We also believe that the rj correspond to positions of the nuclei with a Morse index equal to 0, but this is not necessary for the sequel. This result should be seen as the first step towards concluding the existence of a critical point of energy c, by proving that the second case in Theorem 4 does not happen. Unfortunately, we met with serious difficulties when trying to solve this general problem. This is why the compactness will be shown in the next section for the special case of two interacting molecules with fixed nuclei. Remark. Throughout this paper, we work with complex-valued wavefunctions Ψ. Although in other situations (minimization for instance) one often works with real-valued functions without any change, this is not the case here. In particular, the equality c = c is very easily obtained in this setting, while one can prove that this is also true for real-valued wavefunctions, but for well-chosen ground states Ψ and Ψ only. See the proof for more details.
2.3
Compactness in the case of two interacting molecules
Now that we have identified the critical points at infinity for the mountain pass method, the next step is to show that min-maxing paths cannot approach these critical points. We study here the case of two interacting molecules with fixed nuclei. The parameters are then
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
489
• the distance between the two molecules (denoted by α in the sequel), • the orientation of each molecule (represented by two rotations u and u ), • the electronic wavefunction. u u ... r ... ... . α ..
... r ... ... ... ... ... O
-
Figure 2: Two molecules with fixed nuclei.
m So we consider r = (r1 , . . . , rm ) ∈ B(0, R0 )m and r = (r1 , . . . , rm ) ∈ B(0, R0 ) such that r1 = r1 = 0, ri = rj and ri = rj for i = j, and some z = (z1 , . . . , zm ), z = (z1 , . . . , zm ). We denote by Z = (z, z ), and introduce
R(α, u, u ) = (u · r, α v + u · r ). where v is a fixed vector of norm 1, α ∈ R, and u, u are rotations in R3 . We have used the notation u · r = (u · r1 , . . . , u · rm ). We suppose now that N = |Z| and define E N (α, u, u , Ψ) := E N (R(α, u, u ), Ψ). In [25], it is proved that E N admits a minimum on R × (SO3 (R))2 × SHa1 (R3N ). As in the previous sections, we shall assume that E N possesses two local minima M and M . Up to a rotation of each molecule, we may suppose that α(M ) > 0 and α(M ) > 0. We then consider c = inf max E N (γ(t)) γ∈Γ t∈[0;1]
where Γ is the set of all the continuous functions γ : [0; 1] → X := (0; +∞) × (SO3 (R))2 × SHa1 (R3N ) such that γ(0) = M and γ(1) = M . 2.3.1 The mountain pass method We begin this section by stating a result which is the analogue of Theorem 4 in this special setting. Theorem 5. We have • either there exists a critical point (α, u, u , Ψ) of E N on X, such that H N (R(α, u, u ), Z) · Ψ = c · Ψ, c = λN 1 (R(α, u, u ), Z),
• or
c = min E N1 (r, z) + E N2 (r , z ), N1 + N2 = N .
Roughly speaking, the non compactness of min-maxing sequences is related to the existence of two gradient lines going from a local minimum to some critical point at infinity of index 0. The idea is that an “optimal path” has to follow these
490
M. Lewin
Ann. Henri Poincar´e
lines, and then to connect the two critical points at infinity. Since the molecule is split here into two independent parts, the problem is now to find two mountain pass paths connecting each configuration of the two molecules – two similar problems of lower dimension. When the position of the nuclei in each molecule is fixed, these paths can be obtained by only applying some rotations. In other words, the minima always belong to the same connected component and this is why the situation will be simpler in this setting. When the position is not fixed, even if we may assume the existence of such paths (by induction), the situation is much more complicated and we hope to come back to this more general issue in the future. The proof of Theorem 5 is very similar to the one of Theorem 4, and will be omitted. In order to prove that the second case in Theorem 5 does not happen, we need some information on the directions on which the energy decreases near the critical points at infinity. We shall thus need an expansion of the interaction energy between the two molecules when α grows. The terms involving in these expansion are classical. Let us first recall the definitions of the first multipoles. Definition 2. Let be R = (R1 , . . . , RM ) ∈ Ω, Z =(Z1 , . . . , ZM ), and ρ ∈ L1 (R3 ) ∩ S(R3 \ {Rj }) a non-negative function such that R3 ρ = N > 0. Then
M 1. the total density of charge is the measure ρ˜ := ρ − j=1 Zj δRj . The total charge is q := R3 ρ˜ = N − |Z|,
M 2. the dipole moment is the vector P := R3 x˜ ρ(x) dx = R3 xρ(x) dx − j=1 Zj Rj , 3. the quadrupole moment is the matrix Q := R3 xxT − 13 |x|2 I ρ˜(x) dx. When ρ is the electronic density associated to some eigenstate Ψ, we shall use the notations ρΨ , ρ˜Ψ , PΨ and QΨ . This multipoles will be used in the expansion of the interaction energy. To illustrate this point, we give here the following Lemma 2. We assume that N1 and N2 are such that N1 + N2 = N , E N1 (r, z) < ΣN1 (r, z) and E N2 (r , z ) < ΣN2 (r , z ). Let ψ1 and ψ2 be two ground states of respectively H N1 (r, z) and H N2 (r , z ). Denoting Ψ(α, u, u ) = (u · ψ1 ) ∧ (ταv · u · ψ2 ), we have E N (R(α, u, u ), Ψ(α, u, u )) = E N1 (r, z) + E N2 (r , z ) +
q1 q2 uP1 · v u P2 · v (uP1 ) · (u P2 ) − 3(uP1 · v )(u P2 · v ) + q2 − q1 + 2 α α α2 α3 T 1 3(q2 uQ1 uT + q1 u Q2 u )v · v + + O 2α3 α4
for all u, u ∈ SO3 (R) and when α goes to +∞.
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
491
In this result, qk , Pk and Qk are respectively the total charge, the dipole and the quadrupole moment associated to the electronic densities ρk of the states ψk . The terms of this expansion can be interpreted respectively as the energies of the molecules, and the interaction energy between them, which decomposes into the charge/charge (1/α), dipole/charge (1/α2 ), dipole/dipole and charge/ quadrupole (1/α3 ) terms. We are now able to state our main compactness results. As mentioned above, we had to add some hypothesis about the molecules “at infinity”, concerning their multipoles in their ground state. 2.3.2 The case of charged molecules at infinity Our first result will concern the case of monopoles at infinity, that is to say when the molecules are charged. Theorem 6 (Charged molecules at infinity). Let us assume that E N1 (r, z) + E N2 (r , z ) = min {E n1 (r, z) + E n2 (r , z ), n1 + n2 = N }
(9)
for some N1 and N2 with (N1 − |z|)(N2 − |z |) = 0. Then the case 2) in Theorem 5 does not happen. Therefore c is a critical value of E N on X. Remark. By (9), we have for instance
µ := E N1 (r, z) − E |z| (r, z) < E |z | (r , z ) − E N2 (r , z ) := µ for some N1 , N2 such that N1 + N2 = N and N1 < |z|. This can be viewed as a comparison between oxydo-reduction potentials. So (9) will be probably true if one molecule is a oxidant and the other is a reductor. 2.3.3 The case of neutral molecules with dipole moments at infinity If the two molecules at infinity are neutral, the first term involving in the expansion of the interaction energy is the dipole/dipole term. This is why we shall now consider the case of molecules that possess some dipole moment in their ground state (experiment suggests that this is the case for every non symmetric molecule). Let us introduce the following definition Definition 3. Let be R = (R1 , . . . , RM ) ∈ Ω, Z = (Z1 , . . . , ZM ) and N > 0 such N that λN 1 (R, Z) < Σ (R, Z). We shall say that the molecule (R, Z, N ) possesses a dipole moment at its ground state if PΨ = 0 for all ground state Ψ. Since V := ker H N (R, Z) − E N (R, Z) is finite-dimensional, let us notice that this implies min{|PΨ |, Ψ ∈ V, ||Ψ||L2 = 1} > 0.
492
M. Lewin
Ann. Henri Poincar´e
We then have the following result: Theorem 7 (Neutral molecules with dipole moments at infinity). Let us assume that
(H1) E |z| (r, z) + E |z | (r , z ) < E N1 (r, z) + E N2 (r , z ) for all N1 , N2 such that N1 + N2 = N and (N1 − |z|)(N2 − |z |) = 0, (H2) the two molecules (r, z, |z|) and (r , z , |z |) possess a dipole moment at their ground state,
(H3) E |z| (r, z) or E |z | (r , z ) is non-degenerated. Then the case 2) in Theorem 5 does not happen. Therefore c is a critical value of E N on X. Remark. (H3) is a purely mathematical restriction that simplifies the proof. Let us explain the general idea of the proof. Recall that the dipole/dipole interaction energy can be written F (P, P )/α3 (see Lemma 2). It is shown in Appendix 2 that the critical points of F which have a non-negative energy have a Morse index which is at least one. If a path approaches a critical point at infinity then, to pull down the energy along the path, one may use either the rotations of the molecules if the dipole/dipole interaction energy is positive (thanks to this Morse index information on F ), or the distance between them if it is negative (because α → F (P, P )/α3 is then increasing). This is why min-maxing paths do not approach the critical points at infinity, and give thus a compact Palais-Smale sequence. Obviously, this general idea does not suffice to lead the proof and there are some other difficulties (essentially due to the complexity of the model) that are explicited in the next section. Remark. This general information on the Morse index is probably true for the others multipoles interaction energies, a fact that could be used to treat the general case.
3 Proofs 3.1
Proof of Theorems 2 and 3
3.1.1 Preliminaries We shall use the following lemma, which is an adaptation of results in [15, 16, 11, 12, 13], and which is proved in Appendix 1. Lemma 3. Let ΨR be an eigenfunction associated to the eigenvalue λN d (R, Z) and ρR be the electronic density. We introduce R = ΣN (R, Z) − λN d (R, Z). Then 1. ρR satisfies the inequation 1 − ∆ρR + VR ρR + R ρR ≤ 0. 2
(10)
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
493
2. With R1 () := max R0 + 1, R0 + 2N p and C() := ∪pj=1 {x, |x − Xj | = R1 ()}, and if r > 2R1 (R ), then we have ≤
ρR (x)
||ρR ||L∞ (C(R ))
p
e−
√
R /p(|Xj −x|−R1 (R ))
j=1
≤ ≤
√ p||ρR ||L∞ (C(R )) e− R /p(δ(x)−R1 (R )) √ M e− R /p(δ(x)−R1 (R )) ,
(11)
on U(R1 (R )), where δ(x) = min{|x − Xj |, j = 1, . . . , d}, and M = M (p, N, R1 (R )). The explicit bound (11) has been written in order to show the dependence of all the constants with regard to R . It is clearly not optimal. It shows a nonisotropic exponential decay of the electronic density, which will be uniform if R 0. This type of bounds is studied in the work of Agmon [1] and we do not know if one can use his formalism to obtain the same result. Isotropic exponential bounds for N -body eigenfunctions are frequently seen in the literature, but surprising is the fact that such non-isotropic bounds has not yet been noticed. The next two lemmas will be useful to prove the exponential decay of Theorem 2. Lemma 4. For all α > 0, there exists a constant M = M (α, N, R0 ) such that ρR (y) dy tR (x) ≤ M B(x,α)
on U(R0 + 1/2 + α).
Proof of Lemma 4 – see [16]. Lemma 5. For all j = 1, . . . , p, d ≥ 1 and n ≤ |zj |, we have inf (Σn (rj , zj ) − λnd (rj , zj )) > 0.
rj ∈ωj
Proof. We have ˜ n (rj , zj ) ˜ n (rj , zj ) − λ Σn (rj , zj ) − λnd (rj , zj ) = Σ d ˜n (rj , zj ) and Σ ˜ n (rj , zj ) are the dth eigenvalue and the bottom of the eswhere λ d sential spectrum of the Hamiltonian with the nuclei interaction removed ˜ n (rj , zj ) = H
n 1 − ∆xi + Vrj (xi ) + 2 i=1
1≤i<j≤n
1 . |xi − xj |
494
M. Lewin
Ann. Henri Poincar´e
By Zhislin’s Theorem, it is known that ˜ n (rj , zj ) > 0 ˜ n (rj , zj ) − λ Σ d for all rj ∈ ωj and, since this function is continuous with regard to rj , inf (Σn (rj , zj ) − λnd (rj , zj )) > 0.
rj ∈ωj
In the next result, we use both HVZ and Zhislin’s Theorems. This lemma will be useful in the proof of Theorem 2 to construct test functions. Lemma 6. If the minimum p p Nj min λδj (rj , zj ), N1 + · · · + Np = N, δj = d . j=1
j=1
is attained for N1 , . . . , Nj and δ1 , . . . , δp , then necessarily N
N
λδjj (rj , zj ) < Σδjj (rj , zj ) for all j = 1, . . . , p. Proof of Lemma 6. Remark that by definition λ0δj (rj , zj ) < Σ0 (rj , zj ) = +∞ for all δj . We argue by contradiction and suppose that there exists a k such that Nk > 0 k and λN (rk , zk ) = ΣNk (rk , zk ) = E Nk −1 (rk , zk ). Theorem 1 implies Nk ≥ |zk | + 1. δ
k p Since j=1 (Nk − |zk |) = N − |Z| ≤ 0, there exists a l = k such that Nl < |zl |. / {k, l}, δk = 1, Nk = Nk − 1, δl = δk δl , and We then let δj = δj , Nj = Nj for j ∈ Nl = Nl + 1. We obtain p j=1
N λδjj (rj , zj )
−
p j=1
N
λδ j (rj , zj ) = j
Nl +1 l λN (rl , zl ) δl (rl , zl ) − λδ l
≥
E Nl (rl , zl ) − λδN l +1 (rl , zl )
=
ΣNl +1 (rl , zl ) − λδN l +1 (rl , zl )
>
0
l
l
since Nl + 1 ≤ |zl | (Zhislin Theorem), which is a contradiction.
3.1.2 Proof of Theorem 2 We are now able to prove Theorem 2. We first prove 2). Suppose that the right-hand side is pattained for some N1 , . . . , Nj and δ1 , . . . , δp such that N1 + · · · + Np = N and j=1 δj = p. For the
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
495
sake of simplicity, we may assume that Nj > 0 for all j = 1, . . . , p. By Lemma 6 and Theorem 1, there exist eigenfunctions Ψkj ∈ L2a (R3Nj ) satisfying
N
H Nj (rj , zj )Ψkj = λk j (rj , zj )Ψkj ,
R
3Nj
Ψkj Ψlj = δkl
for all j = 1, . . . , p and k = 1, . . . , δj . If Vj = span(Ψkj , k = 1, . . . , δj ) ⊂ L2a (R3Nj ) then we have N
max
Ψ∈Vj , ||Ψ||2L =1
H Nj (rj , zj )Ψ, Ψ = λδjj (rj , zj ).
We now consider a sequence tn → +∞ such that limn→+∞ λN d (R(tn ), Z) = 2 3N (R(t), Z). If Ψ ∈ L (R ), we introduce lim supt→+∞ λN d = τXj (tn ) · Ψkj Ψk,n j 2 3Nj V˜jn = span(Ψk,n ). j , k = 1, . . . , δj ) ⊂ La (R
(we recall that τv is the translation by v). Now, let be Wn = V˜1n ∧ · · · ∧ V˜pn = span(Ψ1k1 ,n ∧ · · · ∧ Ψkpp ,n , 1 ≤ kj ≤ δj ) which is a space of dimension
Ψ=
d
j=1 δj
= d. If
ck1 ,...,kp Ψ1k1 ,n ∧ · · · ∧ Ψkpp ,n ∈ Wn ,
1≤kj ≤δj
and
|ck1 ,...,kp |2 = 1, we have
H N (R, Z)Ψ, Ψ =
p
k ,n
k ,n
|ck1 ,...,kp |2 H Nj (rj , zj )Ψj j , Ψj j + en
j=1 1≤kj ≤δj
where en is the interaction energy between the p molecules. It is the sum of three terms en = e1n + e2n + e3n . e1n is the interaction between electrons in different molecules, and contains terms like kj kj kj kj (Ψj1 1 Ψj1 1 )(x, . . . )Ψj2 2 Ψj2 2 )(y, . . . ) dx dy. |x − y + Xj2 (tn ) − Xj1 (tn )|
496
M. Lewin
Ann. Henri Poincar´e
with j1 = j2 . e2n is the interaction between electrons and nuclei of different molecules, and contains terms like
k
k
zj2 ,i (Ψj1j1 Ψj1j1 )(x, . . . ) dx dy |x − rj2 ,i + Xj2 (tn ) − Xj1 (tn )|
with j1 = j2 . Finally, e3n is the interaction between nuclei of different molecules zj1 ,kj1 zj2 ,kj1 . e3n = |r − r j1 ,kj1 j2 ,kj2 + Xj2 (tn ) − Xj1 (tn )| j <j 1
2
1≤kj ≤mj
It is now easy to see that each of this term tends to 0 as n → +∞. By definition, we have λN d (R, Z) ≤ ≤
p
j=1
≤
max
|ck1 ,...,kp |2 =1
p
max
|ck1 ,...,kp
H N (R, Z)Ψ, Ψ
|2 =1
N
1≤kj ≤δj
|ck1 ,...,kp |2 λkjj (rj , zj ) + max en
N
λδjj (rj , zj ) + max en .
j=1
We may now pass to the limit as n → +∞ in this inequality and obtain the bound p p N lim sup λN λδjj (rj , zj ), N1 + · · · + Np = N, δj = d . d (R(t), Z) ≤ min t→+∞ j=1
j=1
We then prove simultaneously 1) 3) 4) by induction on N = 1, . . . , |Z|. For N = 1, it is known that Zi Zj , Σ1 (R, Z) = E 0 (R, Z) = |Ri − Rj | 1≤i<j≤M
and so lim Σ1 (R(t), Z) =
t→+∞
p
j=1 1≤k 0
ri ∈ωi t→+∞
r1 ∈ω1
by Lemma 5. The uniform exponential decay is then a consequence of Lemmas 3 and 4. Let tn → +∞ be such that limn→+∞ E 1 (R(tn ), Z) = lim inf t→+∞ E 1 (R(t), Z), and φn ∈ L2 (R3 ) be such that H 1 (R(tn ), Z)φn = E 1 (R(tn ), Z)φn . By the uniform exponential decay, we may write φn = supp(φnj ) ⊂ B(Xj , rn /3), and ||αn ||H 1 → 0. Then H 1 (R(tn ), Z)φn , φn =
p
˜ 1 (rj , zj )φn , φn + H j j
j=1
p
p j=1
j=1 1≤k 0
inf
j=1,...,p rj ∈ωj n≤|zj |
by Lemma 5. The uniform exponential decay 4) is then a consequence of Lemmas 3 and 4. We now prove the inequality p lim inf E N (R(t), Z) ≥ min E Nj (rj , zj ), N1 + · · · + Np = N t→+∞ j=1
by using a variant of classical N -body geometric methods for Schr¨ odinger operators [30, 29, 19], which is used in [10]. Let tn → +∞ be such that limn E N (R(tn ), Z) = lim inf t E N (R(t), Z), and Ψn an associated sequence of ground states, with densities ρΨn and tΨn . We denote Rn = R(tn ) and Xjn = Xj (tn ). Due to the uniform exponential decay, one has ρΨn = lim tΨn = 0. (12) lim n→+∞
U (tn /3)
n→+∞
U (tn /3)
Let ξn ∈ C ∞ (R3 , [0; 1]) be a cut-off function such that ξn ≡ 0 on U(tn /3), ξn ≡ 1 on R3 \U(tn /3−1), and ||∇ξn ||∞ ≤ 1, ||∆ξn ||∞ ≤ 2. We then introduce χn (x1 ,...,xN ) = N ˜ ˜ i=1 ξn (xi ) and Ψn = χn Ψn . Using (12), it is then easy to see that ||Ψn − Ψn ||H 1 → 0, and n ˜ n − λN ˜ H N (Rn , Z) · Ψ 1 (R , Z) · Ψn
=
−
N
(2∇xi χn · ∇xi Ψn + Ψn ∆xi χn )
i=1
→ 0 in L2 (R3N ), n ˜ n ) − λN ˜ E (R , Ψ 1 (R , Ψn )||Ψn ||L2 (R3N ) = N
n
N i=1
|Ψn |2 |∇xi χn |2 → 0.
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
499
Now, we may write ξn = pj=1 ξnj where Supp(ξnj ) ⊂ B(Xj , tn /3), and ˜n = Ψ ξnk1 (x1 ) · · · ξnkN (xN )Ψn := Ψkn1 ,...,kN . 1≤kj ≤p
1≤kj ≤p
Since the Ψkn1 ,...,kN have disjoint supports, ˜ n |2 = ˜ n) = |Ψ |Ψkn1 ,...,kN |2 , E N (Rn , Ψ E N (Rn , Ψkn1 ,...,kN ), 1≤kj ≤p
1≤kj ≤p
n k1 ,...,kN →0 H N (Rn , Z) − λN 1 (R , Z) · Ψn
in L2 (R3N ) for all k1 , . . . , kN . To end the proof of Theorem 2, it suffices to bound ˜ n ) from below by the appropriate constant.. We now fix k1 , . . . , kN and E N (Rn , Ψ introduce Cj = {i, ki = j}, Nj = |Cj |. Remark that Ψkn1 ,...,kN is antisymmetric in (xi )i∈Cj for all j = 1, . . . , p. Then p 1 |∇xi Ψkn1 ,...,kN |2 − E N (Rn , Ψkn1 ,...,kN ) = Vr˜j (xi )|Ψkn1 ,...,kN |2 2 j=1 i∈Cj i∈Cj |Ψk1 ,...,kN |2 n + E 0 (rj , zj ) + en + |xk − xl | k,l∈Cj
where en =
1≤j=j ≤p
i∈Cj
|Ψk1 ,...,kN |2 n + en , Vr˜j (xi )|Ψkn1 ,...,kN |2 + |xi − xi | i∈Cj i ∈Cj
en
being the interaction energy between nuclei in different molecules, which easily tends to 0 as n → +∞. Now |Ψk1 ,...,kN |2 3|zj | n + en → 0 |en | ≤ |Ψkn1 ,...,kN |2 + 3 t t n n 1≤j=j ≤p
i∈Cj i ∈Cj
i∈Cj
as n → +∞. Finally, since Ψkn1 ,...,kN is antisymmetric in (xi )i∈Cj for all j = 1, . . . , p and thanks to the translation invariance of the Hamiltonian, p E Nj (rj , zj ) ||Ψkn1 ,...,kN ||2L2 + en . E N (Rn , Ψkn1 ,...,kN ) ≥ j=1
Passing to the limit, we obtain ˜ n ) ≥ min lim E N (Rn , Ψ
n→+∞
p
which ends the proof of Theorem 2.
j=1
E Nj (rj , zj ), N1 + · · · Np = N
500
M. Lewin
Ann. Henri Poincar´e
3.1.3 Proof of Theorem 3 The proof uses exactly the same N -body geometric method as the end of the n N n proof of Theorem 2, but with λN 1 (R , Z) replaced by λd (R , Z). If we suppose N n that limn→+∞ λd (R , Z) = c, then passing to the limit and using Theorem 2 n N c ≤ lim sup λN d (R , Z) ≤ Λd (r, z). n→+∞
We have
N n H (R , Z) − c · Ψkn1 ,...,kN → 0
in L2 (R3N ) for all k1 , . . . , kN . Since all the interaction terms tend to 0 (see the proof of Theorem 2), we obtain p H Nj (Xjn + rjn , zj )Cj − c · Ψkn1 ,...,kN → 0 j=1
where the Hamiltonian H Nj (Xjn + rjn , zj )Cj acts on the variables (xi )i∈Cj . Due to the translation invariance, we obtain p ˜ k1 ,...,kN → 0 H Nj (rjn , zj )Cj − c · Ψ n j=1
˜ kn1 ,...,kN (x1 , . . . , xN ) = Ψkn1 ,...,kN (X n + xi ). But due to the exponential dewhere Ψ ki ˜ k1 ,...,kN is precompact in H 1 (R3N ) and converges up to a subsequence cay of Ψn , Ψ n ˜ k1 ,...,kN such that to some Ψ p ˜ k1 ,...,kN = c · Ψ ˜ k1 ,...,kN . H Nj (rj , zj )Cj Ψ j=1
p Nj on the tensor We thus have either c is an eigenvalue of H (r , z ) j j Cj j=1 p ˜ k1 ,...,kN = 0. product j=1 L2a (R3Nj )Cj (with an obvious notation), or Ψ
N
p p Nj j (rj , zj )Cj = (rj , zj )Cj so that Lemma 7. We have σ j=1 H j=1 σ H
p Nj σess H (r , z ) = [Σ; +∞) with j j C j j=1 Σ
= >
for all d ≥ 1.
min
j=j0
ΛN d (r, z)
E Nj (rj , zj ) + ΣNj0 (rj0 , zj0 ), 1 ≤ j0 ≤ p
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
501
Proof of Lemma 7. The fact that the spectrum of pj=1 H Nj (rj , zj )Cj is the sum
p Nj (rj , zj )Cj is standard (see for instance [27], Theorem VIII-33). Supj=1 σ H
Nj (rj , zj ) + ΣNj0 (rj0 , zj0 ) for some 1 ≤ j0 ≤ p. If pose now that Σ = j=j0 E N
N
j0 λd j0 (rj0 , zj0 ) < ΣNj0 (rj0 , zj0 ) then obviously Σ > ΛN d (r, z). If λd (rj0 , zj0 ) = Nj0 N Σ (rj0 , zj0 ) then Σ > Λd (r, z) by Lemma 6.
p Nj As a consequence, if c is an eigenvalue of (rj , zj )Cj , it is necesj=1 H sary below its essential spectrum. It is then easy to see that this implies
˜ k1 ,...,kN ∈ Ψ
p
N ker H Nj (rj , zj ) − λδjj (rj , zj )
j=1
Cj
N
for some λδjj (rj , zj ) < ΣNj (rj , zj ).
Now, we have ||Ψn − τn · Ψ||Ha1 (R3N ) → 0 where Ψ =
3.2
k1 ,...,kN
˜ k1 ,...,kN . Ψ
Proof of Theorem 4
We may suppose c > max(E N (R, Ψ), E N (R , Ψ )). Let us first prove the equality c = c . Indeed, c ≤ c is obvious. Let be rn ∈ R a sequence such that mn := maxt∈[0;1] E N (rn (t), Z) → c as n → +∞. For each n ∈ N, we define cn = inf max E N (rn (t), ψ(t)) ψ∈ΓΨ t∈[0;1]
ΓΨ = ψ ∈ C ([0; 1], SHa1 (R3N )), ψ(0) = Ψ, ψ(1) = Ψ .
0
We may now apply the methods of [8] to obtain some sequences tk ∈ [0; 1] and (Ψk )k≥1 such that 1.
lim E N (rn (tk ), Ψk ) = cn ,
k→+∞
2. H N (rn (tk ), Z) · Ψk − E N (rn (tk ), Ψk ) · Ψk → 0 in L2 (R3N ), 3. E N (rn (tk ), Ψk ) ≤ λN 1 (rn (tk ), Z) + k ,
lim k = 0
k→+∞
1, 2) correspond to the classical fact that one can obtain min-maxing sequences that are almost critical. On the other hand, 3) is a consequence of the less-known fact that one can obtain Palais-Smale sequences with Morse-type information related to the dimension of the homotopy-stable class used in the min-max method, which is 1 here (paths are deformations of [0; 1]). Since we are in C = R2 , eigenvectors N always have an even Morse index and this is why λN 1 appears in 3) and not λ2 . 1 3N The fact that such a sequence (Ψk ) is precompact in Ha (R ) is now a simple consequence of Theorem 1-4). Indeed, the compactness below the essential spectrum is nothing else but the Palais-Smale condition of E N with Morse-type information introduced in [14]. We have the following general lemma, whose proof is postponed until the end of the proof of Theorem 4.
502
M. Lewin
Ann. Henri Poincar´e
Lemma 8. We assume that Z = (Z1 , . . . , ZM ) is such that N ≤ |Z|. Let (Rn , Ψn ) be a sequence in Ω × SHa1 (R3N ) such that 1. Rn → R ∈ Ω 2. lim E N (Rn , Ψn ) = c, n→+∞
3. H N (Rn , Z) · Ψn − E N (Rn , Ψn ) · Ψn → 0 in L2 (R3N ), n 4. there exist d0 ≥ 1 and n → 0 such that E N (Rn , Ψn ) ≤ λN d0 (R , Z) + n . Then (Ψn ) is precompact in Ha1 (R3N ) and converges, up to a subsequence, to an eigenfunction Ψ of H N (R, Z) associated to λN d (R, Z) with d ≤ d0 . Applying this result, we obtain, by passing to the limit as k → +∞, c ≤ cn = λN 1 (rn (tn ), Z) ≤ mn for some tn ∈ [0; 1] and so c = c . We now prove the alternative of the Theorem. We introduce Fc (R0 ) = (R, Ψ) ∈ Ω × SHa1 (R3N ), |Ri − Rj | ≤ R0 , E N (R, Ψ) ≥ c i=j
Γ(α) =
! γ ∈ Γ, max E (γ(t)) ≤ c + α . N
t∈[0;1]
We have the following alternative: either there exist R0 > 0 and α > 0 such that, Fc (R0 ) ∩ γ([0; 1]) = ∅ for all γ ∈ Γ(α), or for all R0 > 0 there exists a min-maxing sequence γn ∈ Γ such that Fc (R0 ) ∩ γn ([0; 1]) = ∅. First Case: there exist R0 > 0 and α > 0 such that, Fc (R0 ) ∩ γ([0; 1]) = ∅ for all γ ∈ Γ(α). Since (R, Ψ) and (R , Ψ ) do not belong to Fc (R0 ), we may apply the methods of [14] (Fc (R0 ) is a set which is dual to the homotopy-stable class Γ(α) with boundary B = {(R, Ψ), (R , Ψ )}) to obtain a sequence (Rn , Ψn ) ∈ Ω × SHa1 (R3N ) such that 1. 2. 3.
lim d((Rn , Ψn ), Fc (R0 )) = 0,
n→+∞
lim E N (Rn , Ψn ) = c,
n→+∞
lim ∇R E N (Rn , Ψn ) = 0,
n→+∞
4. H N (Rn , Z) · Ψn − E N (Rn , Ψn ) · Ψn → 0 in L2 (R3N ), n 5. E N (Rn , Ψn ) ≤ λN 1 (R , Z) + n ,
lim n = 0
n→+∞
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
503
Remark that 2, 3, 4, 5) correspond to the fact that one can obtain min-maxing sequences that are almost critical, and with Morse-type information. On the other hand, 1) is the consequence of the duality theory developed in [14] that enables to locate the critical
points. Due to 1), i=j |Rin − Rjn | is bounded. Up to a translation, we may suppose n n Rn → R ∈ Ω (since λN 1 (R , Z) → +∞ when d(R , ∂Ω) → 0 due to the nun clei/nuclei repulsion). Now Ψ converges up to a subsequence to a Ψ in H 1 (R3N ) by Lemma 8. Second Case: for all R0 > 0 there exists a min-maxing sequence γn ∈ Γ such that Fc (R0 ) ∩ γn ([0; 1]) = ∅. Let (rn ) be a sequence in R such that rn → +∞. For each rn , there exists a γn such that, for instance, c ≤ max E N (γn (t)) ≤ c + t∈[0;1]
1 n
˜ n (t)) and fix n. The and Fc (rn ) ∩ γn ([0; 1]) = ∅. We now write γn (t) = (Rn (t), Ψ n n set Kn = {t ∈ [0; 1], |Ri (t) − Rj (t)| ≤ rn } is a compact subset of [0; 1] such that max E N (γ(Kn )) < c. We now introduce # " ˜n ΓΨ = ψ ∈ C 0 ([0; 1], SHa1 (R3N )), ψ|Kn ≡ Ψ |Kn ˜ n (Kn ), and which is an homotopy-stable class of dimension 1 with boundary Ψ cn = inf max E N (Rn (t), ψ(t)) ψ∈ΓΨ t∈[0;1]
(13)
so that
1 . n Applying the methods of [14], we may find a sequence tk ∈ [0; 1] \ Kn and Ψkn ∈ SHa1 (R3N ) such that tk → t¯ and
1. i=j |Rn,i (tk ) − Rn,j (tk )| ≥ rn 2. lim E N (Rn (tk ), Ψkn ) = cn c ≤ cn ≤ c +
k→+∞
3. H N (Rn (tk ), Z) · Ψkn − E N (Rn (tk ), Ψkn ) · Ψkn → 0 in L2 (R3N ) 4. E N (Rn (tk ), Ψkn ) ≤ λN with lim αk = 0 1 (Rn (tk ), Z) + αk k→+∞
By Lemma 8, (Ψkn )k∈N is precompact in Ha1 (R3N ) and converges, up to a subsequence, to some Ψn such that
n n 1. i=j |Ri − Rj | ≥ rn 2. lim E N (Rn , Ψn ) = c n→+∞
n n 3. H N (Rn , Z) · Ψn = λN 1 (R , Z) · Ψ where Rn := Rn (t¯).
504
M. Lewin
Ann. Henri Poincar´e
Since i=j |Rin − Rjn | → +∞, there exists a 2 ≤ p ≤ M , some Xjn ∈ R3 with j = 1, . . . , p and a R0 > 0 such that (changing the indices if necessary and up to a subsequence) Rn = (X1n + r1n , . . . , Xpn + rpn ), ||rjn || ≤ R0 , Z = (z1 , . . . , zp ), and limn→+∞ |Xin − Xjn | = +∞, rjn → rj . Passing to the limit, we obtain, by Theorem 2 n N c = lim λN 1 (R , Z) = Λ1 (r, z). n→+∞
We now simply apply Theorem 3 to obtain the convergence to a critical point at infinity of energy c, as defined in the corresponding section. Let us now prove Lemma 8. N Proof of Lemma 8. Let m be an integer such that λN m (R, Z) > λm−1 (R, Z) = N N n N λd0 (R, Z). Due to the fact that λd (R , Z) → λd (R, Z) as n → +∞, we have n N λN m (R , Z) > λd0 (R, Z) ≥ c for n large enough. $m−1 n N n n n Let be Vn = i=1 ker H(R , Z) − λi (R , Z) , and (ψ1 , . . . , ψm−1 ) an n n N n orthonormal basis of V , ψi being eigenfunctions of H (R , Z) (Theorem 1-4)). Due for instance to the uniform exponential decay of Theorem 2, one easily sees that each ψin is precompact and converges up to a subsequence in Ha1 (R3N ) to a $m−1 ψi , with span(ψi ) = i=1 ker H(R, Z) − λN i (R, Z) . Now we can write Ψn = ΨV n + Ψ(V n )⊥ with an obvious definition. Since E N (Rn , Ψn ) is bounded, it is a classical fact that (Ψn ) is bounded in Ha1 (R3N ) and so, up to a subsequence, Ψn Ψ weakly in Ha1 (R3N ). Since dim(Vn ) = m − 1, (ΨVn ) is precompact in H 1 (R3N ) and converges to a ΨV ∈ V . By difference, Ψ(V n )⊥ Ψ(V )⊥ weakly in H 1 (R3N ). Since limn→+∞ E N (Rn , Ψn ) = c, H N (Rn , Z) · Ψn − c · Ψn → 0 in L2 (R3N ), so we obtain (H N (Rn , Z) − c) · ΨVn + (H N (Rn , Z) − c) · Ψ(Vn )⊥ → 0 which implies (H N (Rn , Z) − c) · ΨVn → 0 and (H N (Rn , Z) − c) · Ψ(Vn )⊥ → 0 in L2 (R3N ). Finally, E N (Rn , Ψ(Vn )⊥ ) − c||Ψ(Vn )⊥ ||2L2 (R3N ) → 0. n N n Because min E N (Rn , S(Vn )⊥ ) = λN m (R , Z) > λd0 (R , Z) ≥ c, this implies n ||Ψ(Vn )⊥ ||L2 → 0 and then ||Ψ(Vn )⊥ ||H 1 → 0. Thus Ψ converges in H 1 (R3N ) to a Ψ = ΨV which is an eigenfunction of H N (R, Z) that belongs to V .
3.3
The case of two interacting molecules
3.3.1 Proof of Lemma 2 We shall use the following lemma:
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
505
Lemma 9 (Multipole expansion). There exists a constant C such that, for all R and h ∈ R3 with R + h = 0, % % % 1 1 C |h|3 eR · h 3(eR · h)2 − |h|2 %% % ≤ − − + % |R + h| % |R|3 |R + h| |R| |R|2 2|R|3 with eR = R/|R|. Proof of Lemma 9. It suffices to show % % 3 % % 1 t2 2 % ≤ √ C|t| %√ (3x − 1 + xt + − 1) % % 1 − 2xt + t2 2 1 − 2xt + t2 for all t ∈ R and x ∈ [−1; 1] (take x= −(eR · h)/|h| andt = |h|/|R|). We thus √ 2 introduce f (x, t) = 1 − 1 − 2xt + t2 1 + xt + t2 (3x2 − 1) . One easily computes ∂f ∂x (x, t)
=
3t3 (5x2 −2xt−1) √ , 2 1−2xt+t2
so that
max |f (x, t)| ≤ max{f1 (t), f2 (t), f3 (t), f4 (t)}
x∈[−1;1]
where f1 (t) = |f (x1 (t), t)|11−1≤x1 ≤1 (t),√ f2 (t) = |f (x2 (t), t)|1√ 1−1≤x2 ≤1 (t), f3 (t) = t− t2 +5 t+ t2 +5 |f (−1, t)|, f4 (t) = |f (1, t)|, x1 (t) = and x2 (t) = . It is now easy 5 5 to conclude that |f (x, t)| ≤ C|t|3 for some constant C > 0. We are now able to prove Lemma 2. Proof of Lemma 2. Let ξα ∈ C ∞ (R3 , [0; 1]) be a cut-off function such that ξα ≡ 0 on R3 \ B(O, α/3), ξα ≡ 1 on B(O, α/3 − 1), ||∇ξα ||∞ ≤ 1, ||∆ξα ||∞ ≤ 2. We Nj introduce ψ˜jα (x1 , . . . , xNj ) = k=1 ξα (xk )ψj (x1 , . . . , xNj ), ψjα = ψ˜jα /||ψ˜jα ||L2 and ˜ Ψ(α, u, u ) = (u · ψ1α ) ∧ (ταv · u · ψ2α ). Due to the exponential decay of Theorem 2, one has % % % % N ˜ R(α, u, u ), Ψ(α, u, u ) − E N (R(α, u, u ), Ψ(α, u, u ))% ≤ Ce−aα %E % N % % % %E 1 (r, z) − E N1 (r, ψ1α )% , %E N2 (r , z ) − E N2 (r , ψ2α )% ≤ Ce−aα for some C, a > 0. Let us recall that, by definition, & N1 !N2 ! ψ ∧ ψ (x1 , . . . , xN ) = σ(C)ψ(xC )ψ (xC ) N! |C|=N1
where xC = (xi1 , . . . , xiN1 ) when C = {i1 < · · · < iN1 }, σ(C) = ±1. Applying this ˜ we obtain on the right functions with disjoint supports. We shall equality to Ψ, therefore only study the expansion of E N (R(α, u, u ), (u · ψ1α ) ⊗ (ταv · u · ψ2α )) .
506
M. Lewin
Ann. Henri Poincar´e
We have (using the notation x1 = (x1 , . . . , xN1 ) and x2 = (xN1 +1 , . . . , xN )) E N (R(α, u, u ), (uψ1α ) ⊗ (ταv u ψ2α )) = E N1 (ur, uψ1α ) + E N2 (α v + u r , ταv u ψ2α ) N1 N2 m m zi zj |ψ1α (x1 )|2 |ψ2α (x2 )|2 1 2 + dx dx + v + u · x2j − u · x1i | |α v + u · rj − u · ri | 3N |α i=1 j=1 R i=1 j=1 N1 N2 m m α 1 2 α 2 2 α 1 2 α 2 2 z |ψ (x )| |ψ (x )| zi |ψ1 (x )| |ψ2 (x )| 2 j 1 + − dx1 dx2 · x1 − u · r | · r − u · x1 | |α v + u |α v + u 3N i R j j i i=1 j=1
i=1 j=1
so we obtain E N (R(α, u, u ), (uψ1α ) ⊗ (ταv u ψ2α )) = E N1 (r, ψ1α ) + E N2 (r , ψ2α ) m m α zi zj ρα 1 (x)ρ2 (y) + dx dy + v + u · y − u · x| |α v + u · rj − u · ri | R6 |α i=1 j=1 − N1
m i=1
R3
m zj ρα z i ρα 1 (x) 2 (y) dy − N dx 2 |α v + u · y − u · ri | v + u · rj − u · x| 3 |α j=1 R
α where the ρα k are the electronic densities associated to ψk , and finally
E N (R(α, u, u ), (uψ1α ) ⊗ (ταv u ψ2α )) = E N1 (r, ψ1α ) + E N2 (r , ψ2α ) ρ˜α ρα 1 (x)˜ 2 (y) dx dy + |α v + u · y − u · x| 6 R
m
m α α where ρ˜α ˜α 1 (x) = ρ1 (x) − 2 (y) = ρ2 (y) − j=1 zi δri (y) are the i=1 zi δri (x) and ρ α and ρ . Now, by Lemma total densities of charge associated to the distributions ρα 1 2 9, we have R6
−
q1 q2 ρ˜α ρα (uP1α ) · v (u P2α ) · v 1 (x)˜ 2 (y) dx dy = + q2 − q1 2 |α v + u · y − u · x| α α α2
3(uP1α · v )(u P2α · v ) − (uP1α ) · (u P2α ) 3(q2 uQ1 uT + q1 u Q2 u )v · v + α3 2α3 1 |u y − ux|3 |˜ ρα ρα 1 |(x)|˜ 2 |(y) dx dy +O α3 |α v + u · y − u · x| R6 T
But we have 3C |u y − ux|3 |˜ ρα ρα 1 |(x)|˜ 2 |(y) dx dy ≤ |x|3 (|˜ ρ1 | + |˜ ρ2 |)(x)dx. |α v + u · y − u · x| α R3 R6 −aα for some It suffices to notice that |Pkα − Pk | ≤ Ce−aα and |Qα k − Qk | ≤ Ce C, a > 0 to end the proof.
Vol. 5, 2004
A Mountain Pass for Reacting Molecules
507
3.3.2 Proof of Theorem 6 Let us suppose that we are in the second case of Theorem 5, and that c > max{E N (M ), E N (M )}. By the proof of Theorem 4, we obtain a sequence αn → +∞ and paths γn such that N ΛN 1 ≤ c ≤ max E (γn (t)) ≤ c + t∈[0;1]
1 n
and E N (α(t), u(t), u (t), Ψ(t)) < c when α(t) ≤ αn . Let tn1 and tn2 be respectively the minimum and the maximum of {t, α(t) ≥ αn }. By the definition of c, we have 0 < tn1 < tn2 < 1. For the sake of simplicity, we introduce unj = u(tnj ), u nj = u (tnj ), and Ψnj = Ψ(tnj ). n The idea of the proof is now to connect M1n = (αn , un1 , u 1 , Ψn1 ) and M2n = n n n (αn , u2 , u 2 , Ψ2 ) by a path on which α is constant, with a maximum energy that is below c. α *