Annales Henri Poincaré - Volume 10

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Ann. Henri Poincaré 10 (2009), 1–17 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/010001-17, published online November 18, 2008 DOI 10.1007/s00023-008-0397-1

Annales Henri Poincar´ e

On the Ultraviolet Problem for the 2D Weakly Interacting Fermi Gas Giuseppe Benfatto

Abstract. We prove that the effective potential of the two-dimensional interacting continuous Fermi gas with infrared cutoff is an analytic function of the coupling strength near the origin. This is the starting point to study the infrared problem of the model without putting any ultraviolet cutoff, as usually done in the literature.

1. Introduction We consider a continuous system of two-dimensional fermions interacting with a 2 smooth integrable potential λ¯ v (x − y ), x, y ∈ TL , the two-dimensional torus of side L. v¯(x) is supposed to be bounded, smooth and with finite range. Since the spin will not play any role in this paper, we shall suppose that the fermions are spinless. In the mathematical physical literature the infrared problem for this system (sometimes called the jellium) is in general studied with an ultraviolet (UV) cutoff, whose introduction is motivated by the remark that the model can not be valid at high energies, see for example [3, 5]. This implies that the results could strongly depend on the cutoff scale, if the system were not stable in the UV region. Indeed, in these papers the interaction is not supposed to satisfy the usual stability condition, which would be sufficient at least to ensure the existence of the thermodynamical limit for the pressure, for any β, and of the correlation functions, for β large enough [8,12]. Moreover, even in the case of a stable potential, the techniques that are used to study the infrared problem do not allow to use this condition, but are based on the analytic dependence on λ of the effective potential. The aim of this paper is to prove that the UV cutoff can be safely removed. Note that the dimension plays an essential role; in fact, if d ≥ 3, it is well known that there are two-body interactions, which do not satisfy the stability condition,

2

G. Benfatto

Ann. Henri Poincaré

for which the pressure can not be defined, even for fermion systems [12]. In agreement with this difficulty, our proof can not be extended to d > 2, see remark 3) below. Let us now define more precisely the model. Given the inverse temperature β, 2 2 we define Λ = [0, β] × TL and, if x0 ∈ [0, β] and x ∈ TL , we put x = (x0 , x). 1 Moreover, we define Dβ,L = Dβ × DL with Dβ = {k0 = 2π β (n0 + 2 ) : n0 ∈ Z} and 2 DL = {k = 2π n : n ∈ Z }; the elements of Dβ,L will be denoted by k = (k0 , k). L Finally, given the particle mass m > 0 and the chemical potential μ, we shall define ε0 (k) = (k 2 /2m) − μ. We study the UV problem for the system of fermions, by analyzing the effective potential Vef f (ϕ) = log

P (dψ)e−V (ψ+ϕ) ,

(1.1)

where ϕ is the (Grassmannian) external field, P (dψ) is the (Grassmannian) Gaussian measure of covariance χ(k02 + ε0 (k)2 ) 1 e−ik · (x−y) guv (x − y) = P (dψ)ψx− ψy+ = , (1.2) 2 βL −ik0 + ε0 (k) k∈Dβ,L χ(t), defined for t > 0, being a smooth cutoff function, with a fixed infrared cutoff on scale 1. We can choose it, for example, by putting χ(t) equal to 1 for t ≥ 2 and equal to 0 for t ≤ 1. Finally the interaction term is given by + + − − dxdy v(x − y) ψx ψy ψy ψx + ν dxψx+ ψx− , (1.3) V (ψ) = λ Λ×Λ

Λ

v(x − y) = δ(x0 − y0 )¯ v (x − y ) ,

(1.4)

ν being a finite counterterm, which is in general introduced to fix the Fermi surface at the free theory value, when the infrared cutoff is lowered [3, 5]; hence it is a constant of order λ, whose precise value is of no importance in this paper. Vef f (ϕ) can be formally expanded in powers of ϕ; we shall write ∞ k k Vef f (ϕ) = −βL2 p(λ, ν) + ϕ+ ϕ− (1.5) dx dy W2k (x, y, λ, ν) xi yi , i=1

k=1

i=1

where we used the notation x ≡ (x1 , . . . , xk ). The pressure p(λ, ν) and the kernels W2k (x, y, λ, ν) have well defined formal representations as powers of λ and ν; we shall write: pn,m λn ν m , (1.6) p(λ, ν) = n+m≥1

W2k (x, y, λ, ν) =

λn ν m W2k,n,m (x, y) .

n+m≥1

The kernels W2k (x, y, λ, ν) are finite sums of products of delta functions of the difference between two space or time variables times suitable measurable functions of a subset of the x and y variables, determined by the delta functions; we shall

Vol. 10 (2009)

The UV Problem for 2D Fermi Gas

3

call, with an abuse of notation, L1 norm of W2k (x, y, λ, ν), and we shall denote by dxdy|W2k (x, y, λ, ν)|, the sum of the L1 norms of these measurable functions. The L1 norm is of course a uniform bound for the Fourier transform of the kernels. As it is well known, it is difficult to check directly the convergence of the power series in (1.6), because of the UV singularity of the propagator (1.2). Up to now, the convergence has been proved (in a suitable norm) only in the one-dimensional case [2]. However, it is possible to show that, if one expand in Feynman graphs the series coefficients, the L1 norm of each graph satisfies a C n+m /(n + m)! bound; since the number of graphs is of order (n + m)!2 , one gets a C n+m (n + m)! bound for the L1 norm of the series coefficients [6]. This result is obtained by using the fact that the singular part of the covariance (1.2) is different from zero only if x0 − y0 or x0 − y0 + β are positive, see Section 2. This argument works in all dimensions, but we shall prove that, in the two-dimensional case, we can use it in a more efficient way, so proving that the series coefficients indeed satisfy a C n+m bound. In order to get this result, we introduce in Section 2 a suitable UV regularization of the propagator, depending on a integer parameter N , which diverges as the cutoff is removed, and we add a superscript (N ) to all quantities in (1.6), to remind the dependence on N . Then we prove that the series coefficients satisfy bounds good enough to imply the power series convergence of p(N ) (λ) and bounds on the (N ) L1 norm of W2k (x, y, λ, ν), for λ and ν small enough, uniformly in N . These bounds are based on a multi-scale expansion, which allows us to prove also the (N ) (N ) convergence, as N → ∞, of p(N ) (λ), pn,m , as well as convergence of W2k,n,m (x, y) (N )

and W2k (x, y, λ, ν) in the L1 norm, to quantities satisfying (1.6). We could also prove the convergence at non coinciding points of the correlation functions and the fast decaying properties on scale 1 of the effective potential kernels. For simplicity, here we shall only prove the following theorem. Theorem 1.1. There are constants Ck , independent of L, β and N , such that (βL2 )−1

n+m ) , |p(N n,m | ≤ C0

(1.7)

dx dy |W2k,n,m (x, y)| ≤ Ckn+m .

(1.8)

(N )

Moreover, there exist constants pn,m and distributions W2k,n,m (x, y), with the (N )

same structure of W2k,n,m (x, y) as sum of products of delta functions times measurable functions, which satisfy the same bounds and are such that if λ and ν are small enough, ) λn ν m |p(N (1.9) lim n,m − pn,m )| = 0 , N →∞

n+m≥1

lim (βL2 )−1

N →∞

dx dy

(N )

λn ν m |W2k,n,m (x, y)

n+m≥1

− W2k,n,m (x, y)| = 0 .

(1.10)

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G. Benfatto


Remark. 1) The expansion used to prove this theorem could be easily used to prove that the results of [5] and [3] about the normal behavior of the weakly interacting Fermi system up to exponentially small temperatures (close to the expected phase transition) can be obtained without any fixed ultraviolet cutoff. 2) The proof of the theorem is indeed valid even for a system with a fixed UV cutoff on the k variables, for example a system of fermions on a fixed lattice. In this case, however, the UV problem is much milder and a much simpler procedure to get the same result through a different multi-scale expansion is described in App. A of [4]. Moreover, in [11] it has been shown that, if there is a UV cutoff on the k variables, no scale decomposition is needed, thanks to an improved technique to control fermion determinants. 3) The expansion which allows us to prove the theorem could be applied also to the case of spatial dimension grater than 2. However, if d > 2, the bounds (1.7) and (1.8) can not be proved, as we shall explain in Section 4. As we said above, this result should be expected, since the existence of the thermodynamical limit is strictly related with the stability of the potential for d ≥ 3 [12], so that we can at most make the hypothesis, in agreement with the C n+m (n + m)! bound of the series coefficients (see above), that the perturbative series are Borel summable, if the two-body potential v¯(x) is stable and λ > 0. However, our technique is not suitable for getting a result of this type. The plan of the paper is the following. In Section 2 we discuss the multiscale decomposition of the covariance, in Section 3 we describe the corresponding expansion of the effective potential and, finally, in Section 4 we prove the theorem.

2. The decomposition of the covariance Note that guv (x) = g(x) − gir (x) , where g(x) =

gir (x) =

1 βL2 1 βL2

k∈Dβ,L

k∈Dβ,L

e−ik · x

1

,

(2.2)

1 − χ(k02 + ε0 (k)2 ) . −ik0 + ε0 (k)

(2.3)

−ik0 + ε0 (k)

e−ik · (x−y)

(2.1)

As it is well known, the sum over k0 in the r.h.s. of (2.2) can be performed and one gets, if |x0 | ≤ β, θ(x0 ) 1 −ik · x−x0 ε0 (k) 1 − θ(x0 ) g(x) = 2 e − , (2.4) L 1 + e−βε0 (k) 1 + e+βε0 (k) k∈DL

Vol. 10 (2009)


5

where θ(t) is the step function, equal to 1 for t > 0 and equal to 0 for t ≤ 0 (this choice ensures that g(0, x) = limx0 →0− g(x0 , x), a required condition [10]). The function g(x), which represents the covariance of the free fermion gas without cutoffs, is antiperiodic in x0 of period β and periodic of period 2β; then 1 2 we shall consider it as defined on T2β × TL . This function has a singularity at the points x0 = 0, ±β, which can be described in the following way. Let h0 (t) be a smooth function on R, equal to 1 for |t| ≤ 1 and equal to 0 for |t| ≥ 2, and let us ˜ 0 (x0 ) the function on T1 , which is equal to h0 (x0 ) around x0 = 0 and equal call h 2β to h0 (x0 + β) around x0 = −β. Let us call also h1 (x) the function on TL , which is ˜ 0 (x0 )h1 (x). One can easily check that equal to h0 (x2 ) around x = 0, and f (x) = h 2

f (x)g(x) = G(x) + R1 (x) ,

(2.5)

where (L)

G(x) = h0 (x0 )h1 (x)θ(x0 )eμx0 δx0 /m (x)

(2.6) (L)

− h0 (x0 + β)h1 (x)θ(x0 + β)eμ(x0 +β) δ(x0 +β)/m (x) , R1 (x) = −h0 (x0 )h1 (x)

1 e−ik · x−x0 ε0 (k) L2 1 + e+βε0 (k)

(2.7)

k∈DL

+ h0 (x0 + β)h1 (x)

1 e−ik · x−(x0 +β)ε0 (k) , L2 1 + e+βε0 (k) k∈DL

and, if t > 0, we are defining (L) δt (x)

e−(x+nL)2 /(2t) 1 −ik · x−tk2 /2 . = 2 e = L 2πt 2 k∈DL

n∈

(2.8)

Z

By using (2.1) and (2.5), we can write: guv = f guv + (1 − f )guv = f g − f gir + (1 − f )guv ≡ G + R ,

(2.9)

where R = R1 − f gir + (1 − f )guv . It is easy to check that R(x) is a smooth 1 2 function on T2β × TL and that, given any integer M > 0, there exists a constant CM , independent of L and β, such that, if dβ (x0 , y0 ) and dL (x, y ) denote the 1 1 2 distances on Tβ (not T2β ) and TL , respectively, then |R(x − y)| ≤

CM . 1 + dβ (x0 − y0 )M + dL (x − y )M

(2.10)

As it is well known, the decomposition (2.9) of the covariance implies that, if we call PR (dψ) and PG (dψ) the Gaussian measures of covariance R(x − y) and

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G. Benfatto


G(x − y), respectively, then we can rewrite (1.1) in the form Vef f (ϕ) = log PR (dψ)eVG (ψ+ϕ) , VG (ϕ) = log PG (dψ)e−V (ψ+ϕ) .

(2.11) (2.12)

The decaying property (2.10) of R(x) implies, by using standard techniques based on the Gram–Hadamard inequality, that the integration in (2.11) gives no problem. Hence, in order to prove Theorem 1.1, it is sufficient to prove it for the functional VG (ϕ) of (2.12). We shall do that by a suitable scale decomposition of the covariance, that we are going to describe. If γ > 1 is the scaling parameter, we define, for any integer h ≥ 0,

Since

∞

θh (x0 ) = θ(x0 )u(γ h x0 ) ,

h=0 θh (x0 )

u(t) = h0 (t) − h0 (γt) .

(2.13)

Gh (x) ,

(2.14)

= θ(x0 )h0 (x0 ), we have G(x) =

∞ h=0

where, by (2.6), (L)

Gh (x) = θh (x0 )h1 (x)eμx0 δx0 /m (x)

(2.15) (L)

− θh (x0 + β)h1 (x)eμ(x0 +β) δ(x0 +β)/m (x) . It is easy to see that there are two constants A and κ, independent of L, β and h, such that

h/2 (2.16) |Gh (x − y)| ≤ Aγ h e−κγ dL (x,y) θh (x0 − y0 ) + θh (x0 − y0 + β) . This implies, in particular, that, given two points x, y ∈ Λ, Gh (x − y) = 0 ⇒ γ −h−1 < x0 − y0 < 2γ −h or

− β + γ −h−1 < x0 − y0 < −β + 2γ −h ,

(2.17)

that is Gh (x − y) can be different from 0 only if the time coordinates x0 and y0 are ordered on the interval [0, β] thought as a torus, so that x0 follows y0 in the positive direction. Let us now put N Gh (x) (2.18) G≤N (x) = h=0

and let us call P≤N (dψ) the gaussian measure with covariance G≤N (x − y). We shall regularize the functional (2.12), by putting (N ) (2.19) VG (ϕ) = log P≤N (dψ)e−V (ψ+ϕ) .

Vol. 10 (2009)


7

v r

v0

−1 0

hv Figure 1.

N

N +1

An example of tree.

3. The tree expansion An essential role in our analysis will be played by the tree expansion [1,7], in a form and with notations very similar to those used in [2]; we assume that the reader is enough familiar with this method to allow us to skip many technical details. We start with some definitions and notations. 1) Let us consider the family of all unlabeled trees which can be constructed by joining a point r, the root, with an ordered set of n ≥ 1 points, the endpoints of the tree (see Figure 1), so that only one line emerges from the root. The unlabeled trees are partially ordered from the root to the endpoints in the natural way (we shall use the symbol < to denote the order); n will be called the order of the unlabeled tree. We shall consider also the labeled trees with cutoff N (which in general will be simply called trees in the following); they are defined by associating some labels with the unlabeled trees, as explained in the following items. We (N ) shall denote Tn the set of labeled trees of order n and cutoff N . (N ) 2) Given τ ∈ Tn , we associate to each endpoint one of the two terms in the r.h.s. of (1.3) and we shall distinguish the two choices by saying that the endpoint is of type λ or ν, respectively. We shall call space-time points the corresponding integration variables; they will be ordered so that x2i−1 , x2i , i = 1, . . . , n4 , will denote the 2n4 space-time points associated to the n4 endpoints of type λ and x2n4 +j , j = 1, . . . , n − n4 , those associated to the endpoints of type ν. x will be the set of all space-time points. We shall use also the notation xj = (x0,j , xj ) to denote the time and space components of xj . 3) We associate to the endpoints 4n4 + 2n2 = nϕ fields, ordered in a fixed arbitrary way; we shall attach a label f = 1, . . . , nϕ to each field to distinguish

8

G. Benfatto

4)

5)

6)

7)

8)


them and we shall call Iτ the set of this labels. If f ∈ Iτ , xf and σf will denote σ the space-time and the σ label, respectively, of the corresponding field ψxff . We introduce a family of vertical lines, labeled by a frequency index h, which takes all the integer values between −1 and N + 1; the vertical lines are ordered from left to right as the frequency index increases. Furthermore the root must belong to the line with index −1, the endpoints must belong to a line with index ≥ 0 and, finally, any branch point must belong to a vertical line with index larger than −1 and smaller than N + 1. We call non trivial vertices of τ its branch points (this set is empty if n = 1 and, in this case, there is only one unlabeled tree); we call trivial vertices the points where the branches connecting two non trivial vertices intersect the family of vertical lines; finally, we call vertices the trivial or non trivial vertices and the endpoints (see the dots in Figure 1). Given a vertex v, we call hv the frequency index of the vertical line containing it. The first vertex of the tree (having frequency index 0) will be denoted v0 . Given a trivial or non trivial vertex v, sv will denote the number of lines branching from v (then sv = 1, if v is a trivial vertex). If v is an endpoint and v is the non trivial vertex immediately preceding it, there is the constraint hv = hv + 1; this follows from the fact that Gh (0, x − y ) = 0. Given a vertex v, we shall call the cluster of v the family of space-time points associated to all the endpoints following v, if v is not an endpoint, or v itself, otherwise. Finally, we denote Eh and EhT the expectation and the truncated expectation, respectively, with respect to the Gaussian measure with covariance Gh . We can expand the functional (2.19) as (N )

VG (ϕ) =

∞

(N )

V2k,n (ϕ) ,

(3.1)

k=0 n≥1 (N )

where V2k,n (ϕ) is the contribution of the terms of order 2k in the field and order n in the couplings λ, ν, which will further expanded in the following way. Given a tree (N ) τ ∈ Tn , we associate to it many different terms, each term being characterized by selecting, for any vertex v ∈ τ , a subset Pv ⊂ Iτ , so that the family {Pv , v ∈ τ } satisfies the following conditions. 1) If v is an endpoint, Pv coincides with the set of fields appearing in the corresponding interaction term. 2) If v is not an endpoint (v not e.p. in the following) and v 1 , . . . , v sv are the sv vertices immediately following v, then Pv ⊆ i=1 Pvi and, if v > v˜0 , the first non trivialvertex of τ , Pv = ∅. Moreover, if we define Qvi = Pv Pvi (so sv Qvi ), then Pvi \Qvi = ∅ for any i, if sv > 1. that Pv = i=1 Let us now define, for any set P ⊂ Iτ , σ σ ϕ(P ) = ϕxff , ψ(P ) = ψxff . (3.2) f ∈P

f ∈P

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9

One can show that (see [2] for details) (N ) V2k,n (ϕ) = dx ϕ(Pv0 ) W (τ, Pv0 , x) , (N )

τ ∈Tn

⎡ ⎢ W (τ, Pv0 , x) = ⎣

(3.3)

|Pv0 |=2k

⎤ ⎥ ⎦ W τ, {Pv }, x ,

(3.4)

v>v0 Pv v not e.p. n4 n2

W τ, {Pv }, x = (−ν)

− λv(x2i−1 − x2i )

(3.5)

i=1

1 T · E ψ(Pv1 \Qv1 ), . . . , ψ(Pvsv \Qvsv ) . s ! hv v not e.p. v

(N )

Note that there is no explicit dependence on N in W (τ, {Pv }, x), so that V2k,n (ϕ) (N )

depends on N only because the sum over the trees is restricted to Tn

.

4. Proof of Theorem 1.1 By the remark following (2.12), in order to prove the first part of Theorem 1.1, that is the bounds (1.7) and (1.8), it is sufficient to prove that, if we put λ0 = max{|λ|, |ν|}, (βL2 )−1 dx W (τ, Pv0 , x) ≤ (Ck λ0 )n . (4.1) (N ) |P |=2k v τ ∈Tn

0

To begin with, we note that, by proceeding as in App. 2 of [2] (where an essential role is played by the Gram–Hadamard inequality, used as in [9]) and by using (2.16), we get (h) h T e−κdT (P1 ,...,Ps ) , (4.2) Eh ψ(P1 ), . . . , ψ(Ps ) ≤ C j |Pj | γ 2 j |Pj | T

where T is an anchored tree graph between the clusters of space-time points from which the fields labeled by P1 , . . . , Ps emerge; this means that T is a set of lines connecting two points in different clusters, which becomes a tree graph if one identifies all the points in the same cluster. Moreover, if xi and yi , i = 1, . . . , s are the space-time coordinates of the two points connected by the lines belonging to T , we define: (h) dT (P1

. . . Ps ) =

s−1

γ h dβ (xi0 , y0i ) + γ h/2 dL (xi , y i ) .

(4.3)

i=1

Note that, if s = 1, the sum over T is void and must be understood as a trivial factor 1.

10

G. Benfatto


We can use the inequality (4.2) to bound the r.h.s. of (3.5). We get n 4 W τ, {Pv }, x ≤ (Cλ0 )n |v(x2i−1 − x2i )|

(4.4)

i=1

(hv ) 1 hv si=1 v [|P |−|P |] −κd (P ,...,P ) v1 v sv vi v Tv · γ 2 e , s ! v not e.p. v Tv sv [|Pvi | − |Qvi |] ≤ 2n. where we used the fact that v not e.p. i=1 Let us now observe that, if we select one tree graph Tv for each vertex v, except the endpoints, and we add, for each endpoint of type λ, the line connecting the two corresponding space-time points, we get a spanning tree for the all set of points x associate to τ . We can use this spanning tree (and the corresponding

(hv )

e−κdTv (Pv1 ,...,Pvsv ) or v(x2i−1 − x2i ) factors) to perform the integrations over the space-time points in the usual way, by ordering them in a way suggested by the spanning tree so that each integration involves only one difference of coordinates, except the last one, which gives a volume factor. As shown in App. 3 of [2], the sum over the possible choices of the spanning tree is controlled, up to a C n constant, by the 1/sv ! factors, hence we get: 2 −1 v n1 4 dxW τ, {Pv }, x ≤ (Cλ0 )n ¯ (βL ) hv sv · γ 2 i=1 [|Pvi |−|Pv |] γ −2hv (sv −1) , (4.5) v not e.p.

where ¯ v 1 = dx v¯(x). Note that v(x) is not a bounded function, see (1.4); hence, in the previous bound an essential role is played by the fact that all the spanning trees contain all the lines associated to the endpoints of type λ. Let us now call n4,v and n2,v the number of endpoints of type λ and ν, respectively, following the vertex v, if it is not an endpoint, nv their sum. and sv˜ [|Pv˜i | − |Pv˜ |] = Then, it is easy to check that, if v is not an endpoint, v˜≥v i=1 4n4,v + 2n2,v − |Pv | and that ˜ − 1) = n4,v + n2,v − 1. Then we can v ˜≥v (sv rewrite (4.5) as 2 −1 γ −d(n4,v ,n2,v ,|Pv |) , (4.6) dxW τ, {Pv }, x ≤ (Cλ0 )n (βL ) v not e.p.

where the vertex dimension d(n4 , n2 , p) is defined as 1 p d(n4 , n2 , p) = 2(n4 + n2 − 1) − (4n4 + 2n2 − p) = n2 − 2 + . 2 2 (N )

(4.7)

If, given the tree τ ∈ Tn , it turns out that d(n4,v , n2,v , |Pv |) > 0 for any v > v0 which is not an endpoint, then, by proceeding as in §3 of [2], one can perform the sum over the sets {Pv , v > v0 } and all the labels of τ of the l.h.s. of (4.1), obtaining a (Cλ0 )n bound. Hence, if this property were true for all trees of order n, one could get the bound (4.1), after exchanging the absolute value with

Vol. 10 (2009)


11

x

x y Figure 2.

The interaction terms.

the sums in the l.h.s., since the number of unlabeled trees of order n is bounded by 4n . The dependence on k of the constant Ck is related to the sum over Pv0 . However, one can see immediately that there are trees which contain vertices with zero or negative dimension. Luckily, this can happen only in a few cases: (a) |Pv | = 4, n2,v = 0 (b) |Pv | = 2, n2,v = 0, 1 (c) |Pv | = 0, n2,v = 0, 1, 2 (this can happen only in the vertices between v0 and the first non trivial vertex of τ ) Note that this result is strictly related with the space dimension. In fact our procedure works even for d > 2, the only difference being that, instead of (4.7), we get d(n4 , n2 , p) = −n4 (d/2 − 1) + n2 − d/2 − 1 + pd/4, so that, for any value of |Pv |, we have vertices with negative dimension, if n4,v is large enough. This makes completely useless the tree expansion. We want now to show that, if d = 2, we can improve our bounds, by summing the contributions of some trees in the l.h.s. of (4.1), before exchanging the absolute value with the sums. Note that, if we expand, given any vertex v, all the truncated expectations in the vertices v˜ ≥ v, we get a sum of connected Feynman graphs. These graphs can be drawn in the usual way by representing the two interaction terms of (1.3) as in Figure 2, where the solid lines correspond to fermions and the wiggling lines correspond to the two-body potential; moreover, the arrow on a fermion line coming in (out) the space-time point x means that the corresponding field has positive (negative) σ-label. It turns out that the graphs must have |Pv | external fermion lines, associated to the field variables Pv , one internal wiggling line for svin ˜ [|Pv˜i | − |Pv˜ |]}/2 internal fermion each endpoint of type λ following v and { i=1 lines, associated to propagators of scales hv˜ , for each v˜ ≥ v. Moreover, the set of all fermion lines can be partitioned uniquely in a family of paths, which are simple loops (fermionic loops in the following) or open paths connecting two external lines; in all paths the arrows define a precise orientation. However, thanks to the time ordering condition (2.17) (see the remark following it), the value of the graph is 0 if there is a fermionic loop containing a number of space-time points less than βγ hv /2, as it is easy to check. It follows that, if nv < n∗hv ≡ [βγ hv /2]

(4.8)

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G. Benfatto


z1

z2

zm−1

zm

y1

y2

ym−1

ym

Figure 3.

A ladder graph.

the value of the graph is exactly zero, if there is some fermionic loop. Moreover, because of the delta function in the time variables of the two-body potential and the fact that the connectivity of the graph can be ensured only through the wiggling lines, the orientation of all open fermion lines must be the same. The previous considerations imply in particular that, if v satisfies condition (a) above and (4.8), the only graphs giving a non vanishing contribution are those having the structure of a ladder graph, that is a graph with two open paths with the same orientation, connected by n4,v non intersecting wiggling lines, see Figure 3. Let us now suppose that v satisfies condition (b) or (c) and (4.8); in this case, it is possible to build graphs without loops only if n4,v = 0 and |Pv | = 2, so that nv = n2,v = 1, which is impossible if v > v0 . Hence a tree τ may give a non vanishing contribution only if there is no trivial or non trivial vertex satisfying (b) or (c), together with (4.8), except the trivial tree with nv = n2,v = 1. (N ) We are now ready to modify our expansion. Given a tree τ ∈ Tn and the sets {Pv , v ∈ τ }, let us consider the family Fτ of all vertices different from v0 , which satisfy condition (a), (4.8) and the further condition that, if v ∈ Fτ , there is no vertex v˜ < v, except possibly v0 , which also satisfies (a) and (4.8). We want to rebuild our expansion (3.3), by summing all terms associated to trees which are obtained from τ by substituting, for each v ∈ Fτ , the subtree starting from v with another arbitrary subtree with the same number nv of endpoints of type λ; we (N ) shall call Tv the family of all such subtrees starting from v. Moreover, we sum also over all the choices of the sets Pv˜ , v˜ > v, and we leave unchanged the set Pv , again for each v ∈ Fτ . In order to describe the result of such operation, we note that, if v ∈ Fτ and m = nv , ⎡ ⎤ ⎢ ⎥ 1 T Ehv˜ ψ(Pv˜1 \Qv˜1 ), . . . , ψ(Pv˜sv˜ \Qv˜sv˜ ) = (4.9) ⎣ ⎦ s ! v ˜ (N ) τ ∗ ∈Tv

v ˜>v Pv˜ v ˜ not e.p.

v ˜≥v v ˜ not e.p.

1 T E[hv ,N ] ψz+1 ψy+1 ; ψz+2 ψy+2 ψy−2 ψz−2 ; . . . , ψz+m−1 ψy+m−1 ψy−m−1 ψz−m−1 ; ψy−m ψz−m , m!

Vol. 10 (2009)


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T where E[h,N ] denotes the truncated expectation with respect to the Gaussian meaN sure with covariance G[h,N ] = h =h Gh and the fields involved in the truncated expectation are those associated to the endpoints following v, except the fields associate to Pv , that is ψz−1 , ψy−1 , ψz+m and ψy+m . This claim easily follows by comparing the multiscale expansion of log P[hv ,N ] (dψ) exp V (ψ+ϕ) with the ordinary cumulant expansion, P[h,N ] (dψ) being the measure with covariance G[h,N ] . (N ) the set of trees, whose definition differs from that of Let us now call Tñ (N )

Tn for the following reasons. 1) The number of endpoints is not fixed, but is at most n. 2) Besides the endpoints of type ν and λ (whose number is still denoted by n2 and n4 , respectively), there are also n∗4 endpoints, to be called of type λ∗ and order n∗v ≤ n; if Fτ is the set of endpoints of type λ∗ , the following constraint has to be satisfied: n4 + n2 + n∗v = n . (4.10) v∈Fτ

3) To each v ∈ Fτ we associate a frequency label hv ∈ [1, N ], satisfying the constraint that hv = hv + 1, if v is the higher vertex preceding v in the tree. 4) To each v ∈ Fτ we associate n∗v interaction terms of type λ; we shall call x∗v the set of corresponding 2n∗v space-time points. Moreover, if the set xv∗ is written as in the second line of (4.9) and Figure 3, with m = n∗v , and we put xm = x∗v , h = hv , we shall define the function 1 v(yi − zi ) m! i=1 m

(N )

v˜m,h (xm ) =

(4.11)

T + + + + − − + + − − − − ψ ψ ; ψ ψ ψ ψ ; . . . , ψ ψ ψ ψ ; ψ ψ · E[h z1 y1 z2 y2 y2 z2 zm−1 ym−1 ym−1 zm−1 ym zm . v ,N ] It follows that we can substitute (3.3) with a similar expansion: (N ) ˜ (N ) (τ, Pv , x) , V2k,n (ϕ) = dx ϕ(Pv0 ) W 0 (N )

τ ∈Tñ

⎡ ˜ (N ) (τ, Pv , x) = ⎢ W ⎣ 0

(4.12)

|Pv0 |=2k

⎤ ∗ ⎥ ˜ (N ) τ, {Pv }, x , ⎦W

(4.13)

v>v0 Pv v not e.p.

∗ where Pv means that the sum is constrained by the condition |Pv | ≥ 6, if nv < n∗hv , and n4

∗ (N ) (N ) n2 ˜ τ, {Pv }, x = (−ν) − λv(x2i−1 − x2i ) W (−λ)nv vñ∗ ,hv (x∗v ) v

i=1

v∈Fτ

1 T · E ψ(Pv1 \Qv1 ), . . . , ψ(Pvsv \Qvsv ) . (4.14) s ! hv v not e.p. v

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G. Benfatto


Note that, if v ∈ Fτ , the set Pv is not fixed, but has to be chosen in all possible ways among the sets of four fields, a couple with σ(f ) = + and a couple with σ(f ) = −, that one can select so that the two couples of fields belong to two different interaction terms of type λ, among those associated to v, see item 4) above. For the other vertices, Pv is defined as before. ˜ (N ) (τ, {Pv }, x) satisfies a bound similar to (4.5). The It is easy to see that W only difference is that now we do not have a spanning tree and a corresponding decaying factor to perform the integration inside the sets xv∗ associated to the endpoints of type λ∗ . However, if we shrink every set xv∗ to a point, we get a spanning tree by choosing, as before, a tree graph Tv or the interaction line for the other vertices; if we use this spanning tree to perform in the usual way the integrations over the differences of coordinates associated to its lines, we are left with the integrations over the sets xv∗ , v ∈ Fτ , with the condition that one of the space-time points from which the four external lines Pv emerge is fixed, in each set. It follows, by observing that, if v ∈ Fτ , the set Pv can be chosen in n∗v (n∗v − 1) different ways, and by using the constraint (4.10), that we have to multiply the (N ) r.h.s. of (4.5) by v∈Fτ ||˜ vn∗ ,hv ||, having defined v (N ) (N ) vm,h (xm )| , (4.15) ||˜ vm,h || = max sup d(xm \x∗i )|˜ i

x∗ i

where x∗i is one of the four points from which the four external lines of Pv emerge. On the other hand, the graph expansion of (4.9) gives rise to a sum of 2m−2 (m − 2)! ladder graphs with propagator G[h,N ] ; hence, by using (4.11) and (1.4), we get ⎤ ⎡ m m m (N ) m ||˜ vm,h || ≤ 2 sup ⎣ dzi,0 dzi ⎦ dyi |¯ v (yi − zi )| j

·

m−1

i=1,i =j

i=1

i=1

|G[h,N ] (zi+1 − zi )||G[h,N ] (zi+1,0 − zi,0 , yi+1 − yi )| .

(4.16)

i=1

By using (2.15) and (2.13), we can easily see that, uniformly in N ,

dx|G[h,N ] (x0 , x)| ≤ C h0 (γ h |x0 |) + h0 (γ h |x0 + β|) .

(4.17)

Therefore, if we integrate in (4.16) first the space coordinates, by using a spanning tree containing all fermion lines and one interaction line, then the time coordinates, by using the fact that h0 (γ h |x0 |) has support in an interval of size γ −h , we get ||˜ vm,h || ≤ C m γ −h(m−1) ¯ v m−1 v 1 , ∞ ¯ (N )

(4.18)

where ¯ v ∞ = supx |¯ v (x)|. ˜ (N ) (τ, {Pv }, x) satisfies a bound similar The bound (4.18) implies that W to (4.6). We could indeed even improve the bound, by using the γ −h(m−1) factor in the r.h.s. of (4.18), but we do not need this improvement. It will be sufficient

Vol. 10 (2009)


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to use the fact that now the set {Pv } has the important property that, if a vertex v has non positive dimension (that is it satisfies either condition (a), (b) or (c) above), then nv ≥ n∗hv , that is nv has to be very large, if hv is large. To exploit this property in an efficient way, let us select, given τ , the family Rτ of vertices such that nv ≥ n∗hv and there is no vertex v˜ > v with the same property. We note that, for any ε > 0, εβ hv ∗ ∗ 1= e−εnhv eεnhv ≤ eεn e− 2 γ , (4.19) v∈Rτ

so that, if we choose ε = 2β

v∈Rτ

−1

, we get −1 1 ≤ e2β n (4e−2 )|Rτ | γ −2hv ≤ C n γ −2hv . v∈Rτ

(4.20)

v∈Rτ

By using this bound, we get finally (N ) ˜ τ, {Pv }, x (βL2 )−1 dxW n −2hv γ ≤ (Cλ0 )

γ −d(n4,v ,n2,v ,|Pv |) .

(4.21)

v not e.p.

v∈Rτ

Let us call τ ∗ the minimal subtree of τ which contains v0 and the set Rτ of vertices. Our definitions imply that all the vertices of τ which have non positive dimension belong to τ ∗ and that the elements of Rτ are the endpoints of τ ∗ . Hence, we can distribute the γ −2hv factors in the r.h.s. of (4.21) along the paths connecting the vertices in Rτ to v0 , so to increase the dimension of all vertices of τ ∗ of at least two units. Since d(n4 , n2 , p) is ≥ −1 and grows linearly with p, this is sufficient to control the sum over the set {Pv }, see remark after (4.7). In order to complete the proof of Theorem 1.1, we define the constants pn,m (N ) (N ) and the distributions W2k,n,m (x, y) in a way analogous to pn,m and W2k,n,m (x, y), that is by using, instead of the expansion (4.12), the similar expansion ˜ (τ, Pv , x) , V2k,n (ϕ) = (4.22) dx ϕ(Pv0 ) W 0 τ ∈Tñ |Pv0 |=2k

(N ) where Tñ = N ≥0 Tñ and

⎡

˜ (τ, Pv , x) = ⎢ W ⎣ 0

⎤ ∗ ⎥ ˜ τ, {Pv }, x , ⎦W

(4.23)

v>v0 Pv v not e.p.

˜ (τ, {Pv }, x) being the expression obtained from (4.14) by substituting v˜(N∗ ) (x∗ ) W v n ,hv v

with vñ∗v ,hv (x∗v ) = vñ∗ ,hv (x∗v ). (∞) v

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In order to prove the bounds (1.9) and (1.10), it is sufficient to prove that there exists a constant δN , which goes to 0 as N → ∞, such that 2 −1 ˜ (N ) (τ, Pv , x) W a(βL ) dx 0 ˜ (N ) |Pv |=2k τ ∈Tn 0 ˜ W (τ, Pv0 , x) ≤ (Ck λ0 )n δN . (4.24) − τ ∈T˜ |Pv |=2k n

0

This is indeed an almost immediate consequence of the previous bounds. In fact, we can write the expression inside the modulus in the r.h.s. of (4.24) as the difference between

˜ (N ) (τ, Pv , x) − W ˜ (τ, Pv , x) W (4.25) Δ1,N (x) = 0 0 (N ) |P v0 |=2k τ ∈Tñ

and Δ2,N (x) =

(N ) τ ∈Tñ \Tñ

|Pv0 |=2k

˜ (τ, Pv , x) . W 0

(4.26)

On the other hand, by proceeding as in the proof of (4.18), it is easy to prove that v m−1 v 1 , ||˜ vm,h − v˜m,h || ≤ C m γ −N γ −h(m−2) ¯ ∞ ¯ (N )

which implies in a simple way that (βL2 )−1 dx|Δ1,N (x)| ≤ (Ck λ0 )n γ −N .

(4.27)

(4.28)

Let us now consider Δ2,N (x). The sum over the trees in the r.h.s. of (4.26) is restricted to trees which have at least one non trivial vertex with frequency index greater than N , say v¯. Hence we can extract from the bound of (βL2 )−1 dx ˜ (τ, Pv , x)|, which is of the form of the r.h.s. in (4.6), a factor γ −N/2 , without |W 0 affecting the bound of the sum over the trees; in fact this factor can be compensated by lowering of 1/2 the vertex dimension (which is ≥ 1) in all the vertices of the path joining v¯ with the root. It follows that (βL2 )−1 dx|Δ2,N (x)| ≤ (Ck λ0 )n γ −N/2 . (4.29) Therefore, the bound (4.24) is proved with δN = γ −N/2 .

Acknowledgements I want to thank A. Giuliani and V. Mastropietro for the fruitful discussions I had with them and for the critical reading of the first version of the manuscript.

Vol. 10 (2009)


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References [1] G. Benfatto, G. Gallavotti, Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems, J. Stat. Phys. 59 (1990), 541–664. [2] G. Benfatto, G. Gallavotti, A. Procacci, B. Scoppola, Beta functions and Schwinger functions for a many fermions system in one dimension, Comm. Math. Phys. 160 (1994), 93–171. [3] G. Benfatto, A. Giuliani, V. Mastropietro, Low temperature analysis of two dimensional Fermi systems with symmetric Fermi surface , Ann. Henri Poincaré 4 (2003), 137–193. [4] G. Benfatto, A. Giuliani, V. Mastropietro, Fermi liquid behavior in the 2D Hubbard model at low temperatures, Ann. Henri Poincaré 7 (2006), 809–898. [5] G. Disertori, V. Rivasseau, Interacting Fermi liquid in two dimensions at finite temperature I and II, Comm. Math. Phys. 215 (2000), 251–290 and 291–341. [6] J. Feldman, E. Trubowitz, Perturbation theory for many fermion systems, Helv. Phys. Acta 63 (1990), 156–260. [7] G. Gallavotti, Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods, Rev. Mod. Phys. 57 (1985), 471–562. [8] J. Ginibre, Some applications of functional integration in statistical mechanics, in Statistical Mechanics and Field Theory, C. De Witt and R. Stora (eds.), Gordon and Breach, New York (1970). [9] A. Lesniewski, Effective action for the Yukawa 2 quantum field Theory, Commun. Math. Phys. 108 (1987), 437–467. [10] J. W. Negele, H. Orland, Quantum Many-Particle Systems, Addison Wesley, New York (1998). [11] W. Pedra, M. Salmhofer, Determinat bounds and the Matsubara UV problem of many-fermion systems, Commun. Math. Phys. 282 (1987), 797–818. [12] D. Ruelle, Statistical Mechanics, W. A. Benjamin (1969). Giuseppe Benfatto Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133, Roma Italy e-mail: [email protected] Communicated by Joel Feldman. Submitted: June 30, 2008. Accepted: September 10, 2008.

Ann. Henri Poincaré 10 (2009), 19–34 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/010019-16, published online March 5, 2009 DOI 10.1007/s00023-009-0403-2


Borchers’ Commutation Relations for Sectors with Braid Group Statistics in Low Dimensions Jens Mund Abstract. Borchers has shown that in a translation covariant vacuum representation of a theory of local observables with positive energy the following holds: The (Tomita) modular objects associated with the observable algebra of a fixed wedge region give rise to a representation of the subgroup of the Poincaré group generated by the boosts and the reflection associated to the wedge, and the translations. We prove here that Borchers’ theorem also holds in charged sectors with (possibly non-Abelian) braid group statistics in low space-time dimensions. Our result is a crucial step towards the Bisognano– Wichmann theorem for Plektons in d = 3, namely that the mentioned modular objects generate a representation of the proper Poincaré group, including a CPT operator. Our main assumptions are Haag duality of the observable algebra, and translation covariance with positive energy as well as finite statistics of the sector under consideration.

Introduction Borchers has shown [4] that in a theory of local observables, which is translation covariant with positive energy, the modular objects associated with the observable algebra of a (Rindler) wedge region and the vacuum state have certain specific commutation relations with the representers of the translations. Namely, these commutation relations manifest that the corresponding unitary modular group implements the group of boosts which leave the wedge invariant, and that the corresponding modular conjugation implements the reflection about the edge of the wedge. Borchers’ theorem has profound consequences. For example in twodimensional theories it means that the modular objects generate a representation of the proper Poincaré group, under which the observables behave covariantly, and implies the CPT theorem. In higher dimensions, it is a crucial step towards the Bisognano–Wichmann theorem in the general context of local quantum physics [5, Supported by FAPEMIG.

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8, 10, 24, 26, 31, 34]. This theorem asserts that a certain class of Poincaré covariant theories enjoys the property of modular covariance, namely that the mentioned unitary modular group coincides with the representers of the boosts, and that the modular conjugation is a CPT operator (where ‘PT’ means the reflection about the edge of the wedge). The hypothesis under which Borchers’ theorem works is the double role played by the vacuum vector within a theory of local algebras: The vacuum is cyclic and separating for the local algebras, and it is invariant under the positive energy representation of the translation group under which these algebras are covariant. In a charged sector, i.e., a non-vacuum representation of the observables, this situation is not given. (This problem has been posed by Borchers in [6, Sect. VII.4].) In the case of permutation group statistics, one can use the field algebra instead of the observable algebra to recover the result. However, in lowdimensional space-time there may occur superselection sectors with braid group statistics [18, 21]. Then only in the Abelian case there is a field (C∗ ) algebra for which the vacuum is cyclic and separating. In the non-Abelian case, there is no such field algebra. Due to this complication, a general result corresponding to Borchers’ theorem has not been achieved yet. In the present article, we prove an analogue of Borchers’ theorem for a superselection sector corresponding to a localizable charge. The implementers of the boosts and the reflection which we find are the relative modular objects associated with the observable algebra of the wedge, the vacuum state and some specific state in the conjugate sector. We assume that the observable algebra satisfies Haag duality, see Eq. (7), and that the sector under consideration has finite statistics and positive energy, and is irreducible. We also need a slightly stronger irreducibility property (12), which may be ensured by requiring for example Lorentz covariance or the split property. We consider charges which are localizable in space-like cones1 , and admit the case of non-Abelian braid group statistics which can occur in low space-time dimensions, d = 2 and 3. It must be noted that in two dimensions, our result is already practically covered by the work of Guido and Longo [22]. Namely, they show how a certain condition of modular covariance in the vacuum sector allows, under the same hypothesis as in the present article, for the construction of a (ray) representation of the proper Poincaré group in charged sectors. But in two dimensions, their modular covariance condition is satisfied due to Borchers’ theorem (in the vacuum sector), so their analysis goes through, even in sectors with Braid group statistics. Also, it must be noted that in d = 2 the assumption of Haag duality excludes some massive models with braid group statistics as e.g. the anionic sectors of the CAR algebra [1], and together with the split property for wedges (expected to hold in massive models) excludes localizable charges altogether [29].

1 A space-like cone is, in d ≥ 3, a convex cone in Minkowski space generated by a double cone and a point in its causal complement, and in d = 2 the causal completion thereof, which is a wedge region.

Vol. 10 (2009)

Borchers’ Commutation Relations in Low Dimensions

21

Our result shall be used to derive the CPT and Bisognano–Wichmann theorems for particles with braid group statistics in three-dimensional space-time [30]. It would be gratifying to extend our analysis to soliton sectors in 2 dimensions, which would extend the range (and simplify the proof) of Rehrens’ CPT theorem for solitons [32].

1. General setting, assumptions and results We consider a theory of local observables, given by a family of von Neumann algebras A0 (O) of operators acting in the vacuum Hilbert space H0 , indexed by the double cones O in Minkowski space, and satisfying the conditions of isotony and locality: A0 (O1 ) ⊂ A0 (O2 ) if O1 ⊂ O2

and

A0 (O1 ) ⊂ A0 (O2 )

if

O1 ⊂ O2 ,

where the prime denotes the commutant or the causal complement, respectively. The vacuum Hilbert space H0 carries a strongly continuous unitary representation U0 of the group of space-time translations Rd with positive energy, i.e. its spectrum2 lies in the forward light cone. It has a unique, up to a phase, invariant vector Ω ∈ H0 , corresponding to the vacuum state. The representation U0 implements automorphisms under which the net O → A0 (O) is covariant: AdU0 (x) A0 (O) = A0 (x + O)

(1)

for all x ∈ R . (By AdU we denote the adjoint action of a unitary U .) Borchers’ theorem, which we wish to generalize to charged sectors, asserts that the representation U0 has specific commutation relations with certain algebraic objects, the so-called modular group and conjugation, which suggest a geometric interpretation of the latter. Let us recall Borchers’ commutation relations in this setting. Let W1 be the wedge defined as d

W1 := { x ∈ Rd : |x0 | < x1 } .

(2)

By the Reeh–Schlieder property, Ω is cyclic and separating for the von Neumann algebra A0 (W1 ) generated by all A0 (O), O ⊂ W1 . This allows for the definition of the Tomita operator [9], S0 , associated to A0 (W1 ): It is the closed anti-linear involution satisfying S0 AΩ = A∗ Ω ,

A ∈ A0 (W1 ) .

(3)

1/2

Its polar decomposition, S0 = J0 Δ0 , defines an anti-unitary involution J0 , the so-called modular conjugation, and a positive operator Δ0 giving rise to the socalled modular unitary group Δit 0 associated to the wedge W1 . By Tomita’s Theorem, see e.g. [9], the adjoint action of Δit 0 leaves A0 (W1 ) invariant, and the adjoint action of J0 maps A0 (W1 ) onto its commutant A0 (W1 ) . The mentioned theorem of Borchers now asserts that Δit 0 and J0 , together with the representation U0 of 2 By spectrum of a representation of the translation group we mean the energy-momentum spectrum, namely the joint spectrum of the generators.

22

J. Mund


the translations, induce a representation of the subgroup of P+ generated by the boosts λt and the reflection j associated to the wedge, and the translations. More precisely, let λt be the (rescaled) 1-boosts, leaving W1 invariant and acting on the coordinates x0 , x1 as cosh(2πt) sinh(−2πt) , (4) sinh(−2πt) cosh(2πt) and let j be the reflection about the edge of W1 , acting on the coordinates x0 , x1 as −1 and leaving the other coordinates unchanged (if d > 2). Then Borchers’ theorem asserts that −it = U0 (λt x) , Δit 0 U0 (x) Δ0

J0 U0 (x) J0 = U0 (jx)

(5) (6)

for all t ∈ R and x ∈ Rd . These relations constitute the group relations of the translations with the boosts and reflections, respectively. Modular theory further implies that J0 is an involution and commutes with the modular unitary group, implementing the group relations j 2 = 1 and jλt j −1 = λt . Altogether, U0 (x), Δit 0 and J0 constitute a representation of the subgroup of the Poincaré group generated by the translations, the boosts λt and the reflection j (which is the direct product of the proper Poincaré group in the time-like x0 -x1 plane and the translation group in the remaining d − 2 dimensions). Our aim is to find a similar result in a charged sector, that is in a representation of the abstract C ∗ -algebra generated by the local algebras A0 (O), which is inequivalent from the defining vacuum representation. We shall consider an irreducible representation π, which is localizable in space-like cones. That means that π and the vacuum representation are unitarily equivalent in restriction to the observable algebra associated with the causal complement of any space-like cone.3 We assume that the observable algebra satisfies Haag duality for space-like cones and wedges, i.e., regions which arise by a proper Poincaré transformation from W1 . Namely, denoting by K the class of space-like cones, their causal complements, and wedges, we require A0 (C ) = A0 (C) ,

C ∈ K.

(7)

A localizable representation can then be described by an endomorphism of the so-called universal algebra A generated by isomorphic images A(C) of the A0 (C), C ∈ K, see [19, 20, 22]. The family of isomorphisms A(C) ∼ = A0 (C) extends to a representation π0 of A, the vacuum representation. We then have A0 (C) = π0 A(C) ,

3 It

is known that every purely massive representation is localizable in space-like cones [11].

(8)

Vol. 10 (2009)


23

and the vacuum representation is faithful4 and normal on the local5 algebras A(C). The adjoint action (1) of the translations on the local algebras lifts to a representation by automorphisms αx : AdU0 (x) ◦ π0 = π0 ◦ αx , αx A(C) = A(x + C) .

(9) (10)

Our localizable representation π is then equivalent [13,19] with a representation of the form π0 ◦ ρ acting in H0 , where ρ is an endomorphism of A localized in some specific space-time region C0 ∈ K in the sense that ρ(A) = A if

A ∈ A(C0 ) .

(11)

We shall take the localization region of ρ to be properly contained in W1 , which implies by Haag duality (7) that ρ restricts to an endomorphism of A(W1 ). We shall require that this endomorphism of A(W1 ) be irreducible, namely that π0 A(W1 ) ∩ π0 ρA(W1 ) = C1 . (12) This is a slightly stronger requirement than irreducibility of the representation π0 ρ of A. It has been shown by Guido and Longo that irreducibility of π0 ρ, together with finite statistics, imply irreducibility in the sense of Eq. (12) if ρ is covariant under the (proper orthochronous) Poincaré group [23, Cor. 2.10] 6 or if ρ satisfies the split property [22, Prop. 6.3]. We further assume the representation π ∼ = π0 ρ to be translation covariant with positive energy. That means that there is a unitary representation Uρ of the translation group Rd with spectrum contained in the forward light cone such that AdUρ (x) ◦ π0 ρ = π0 ρ ◦ αx ,

x ∈ Rd .

(13)

We finally assume that ρ has finite statistics, i.e. that the so-called statistics parameter λρ [13] be non-zero. This holds automatically if ρ is massive [17], and implies [14] the existence of a conjugate morphism ρ¯ characterized, up to equivalence, by the fact that the composite sector π0 ρ¯ρ contains the vacuum representation π0 precisely once. Thus there is a unique, up to a factor, intertwiner Rρ ∈ A(C0 ) satisfying ρ¯ρ(A)Rρ = Rρ A for all A ∈ A. The conjugate ρ¯ shares with ρ the properties of covariance (13), finite statistics, and localization (11) in some space-like cone which we choose to be C0 . Using the normalization convention of [14, Eq. (3.14)], namely Rρ∗ Rρ = |λρ |−1 1 , the positive linear endomorphism φρ of A defined as φρ (A) = |λρ | Rρ∗ ρ¯(A)Rρ

(14)

is the unique left inverse [11, 14] of ρ. In the low-dimensional situation, d = 2, 3, the statistics parameter λρ may be a complex non-real number, corresponding 4 However, π is in general not faithful on the global algebra A due to the existence of global 0 intertwiners [20]. 5 We call the algebras A(C) “local” although the regions C extend to infinity in some direction, just in distinction from the “global” algebra A. 6 Although not explicitly mentioned in [23], the proof does not depend on covariance of ρ under the full Moebius group. See also [28, Thm. 2.2].

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to braid group statistics. We admit the case when its modulus is different from one (namely when ρ is not surjective), corresponding to non-Abelian braid group statistics. The modular objects for which we shall prove Borchers’ commutation relations are defined as follows. Let Sρ be the closed anti-linear operator satisfying Sρ π0 (A)Ω := π0 ρ¯(A∗ )Rρ Ω ,

A ∈ A(W1 ) ,

(15)

1/2

and denote the polar decomposition of Sρ by Sρ = Jρ Δρ . Sρ is just the relative Tomita operator [35] with respect to a certain pair of (non-normalized) states. Namely, consider the vacuum state ω0 := ( Ω, π0 ( · )Ω ), and the positive functional ϕρ := |λρ |−1 ω0 ◦ φρ = Rρ Ω, π0 ρ¯( · )Rρ Ω . The restriction of ϕρ to A(W1 ) is faithful and normal, and has the GNS-triple (H0 , π0 ρ¯, Rρ Ω). Thus, Sρ is the relative Tomita operator associated with the algebra A(W1 ) and the pair of states ω0 and ϕρ , see Appendix A. It is worth mentioning that the relative modular objects in charged sectors have been used, for different purposes, by Wiesbrock [36], by Longo [27] and by Bertozzini, Conti and Longo [2]. Our motivation to consider these objects (instead of the modular objects associated with A(W1 ) and one suitable state, e.g. ϕρ ) is that the so-defined relative modular unitary group Δit ρ implements the modular automorphism group associated with A(W1 ) and ω0 in the same way as the representation Uρ (x) implements the translations αx , see Eq. (26) below. This opens up the possibility to lift Borchers’ commutation relations (5) in the vacuum representation to the representation π0 ρ. In fact, pursuing this strategy, we shall find the following result. Let G be the subgroup of the proper Poincaré group generated by the translations, the boosts λt and the reflection j. Recalling that the representation Uρ may be shifted to a representation eik · x Uρ (x) whose spectrum has a Lorentz invariant lower boundary [7],7 we show under the above-mentioned assumptions: Theorem 1 (Commutation Relations.). Assume that the lower boundary of the spectrum of Uρ is Lorentz-invariant. Then Uρ (x), Δit ρ , Jρ and the counterparts for ρ¯ constitute a continuous (anti-) unitary representation8 of G. More specifically, there hold the commutation relations −it Δit = Uρ (λt x) , ρ Uρ (x) Δρ

Jρ Uρ (x) Jρ−1 −1 Jρ Δit ρ Jρ

= Uρ¯(jx) , =

Δit ρ¯

,

Jρ Jρ¯ = χρ 1 ,

(16) (17) (18) (19)

7 This is automatically the case if ρ is localizable in double cones and d > 2 by a result of Borchers [3], which is applicable since in this case ρ is implemented by local charged field operators [15]. It is also the case of course if Uρ extends to the Poincaré group. 8 Strictly speaking, a ray representation since J J is only a multiple of unity. ρ ρ ¯

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for all t ∈ R and x ∈ Rd . The complex number χρ in Eq. (19) has modulus one, conjugate χ ¯ρ = χρ¯ and is a square root of unity if ρ¯ = ρ. (Note that Eq. (18) corresponds to a standard property of modular objects, but needs to be proved for our relative modular objects.) We also show that this representation of G acts geometrically correctly on the wedge algebras, namely for W in the family W1 of translates of W1 and W1 , W1 := {x + W1 , x ∈ Rd } ∪ {x + W1 , x ∈ Rd } , there holds AdΔit ρ : π0 ρA(W ) → π0 ρA(λt W ) , AdJρ : π0 ρA(W ) → π0 ρ¯A(jW ) .

(20) (21) π0−1 (A0 (W1 ) )

To this end, observe that modular theory [9] and the relation = A(W1 ) imply that Δit 0 and J0 implement an automorphism σt of A(W1 ) and A(W1 ), and an anti-isomorphism from A(W1 ) onto A(W1 ) and vice versa, respectively, defined by AdΔit 0 ◦ π0 = π0 ◦ σt

(22)

AdJ0 ◦ π0 = π0 ◦ αj

(23)

A(W1 ).

on A(W1 ) ∪ By Borchers’ commutation relations, the same equations extend σt and αj to the family A(W ), W ∈ W1 , acting in a geometrically correct way: σt : A(W ) → A(λt W ) ,

(24)

αj : A(W ) → A(jW ) ,

(25)

W ∈ W1 , see [4, Lem. III.2]. But our representers Δit ρ and Jρ implement these isomorphisms σt and αj , respectively, in the direct product representation π0 ρ ⊕ π0 ρ¯, namely: Proposition 1 (Implementation properties). There holds AdΔit ρ ◦ π0 ρ = π0 ρ ◦ σt

(26)

AdJρ ◦ π0 ρ = π0 ρ¯ ◦ αj

(27)

on the family of algebras A(W ), W ∈ W1 . Since σt and αj act geometrically correctly, c.f. Eq.s (24) and (25), this implies that Δit ρ and Jρ act geometrically correctly, as claimed in Eq.s (20) and (21). In two space-time dimensions, our group G already coincides with the proper Poincaré group P+ , and our results therefore imply that the translations and the relative modular objects constitute an (anti-) unitary representation of the latter. By our assumption of Haag duality (7) for wedges, the so-called dual net Ad (O) := A(W ) W ⊃O

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J. Mund


is still local. (One needs to intersect in fact only the algebras of one “right wedge” of the form W1 +x and one “left wedge” of the form W1 +y.) The modular (anti-) automorphisms σt and αj act on it in a geometrically correct way, see [4, Prop. III.3]. If the original net satisfies Haag duality also for double cones, it coincides with the dual net. Then the implementation properties (26) and (27) hold, and therefore the representation of P+ constructed in Theorem 1 acts geometrically correctly, namely there holds for any double cone O: AdUρ (g) : π0 ρA(O) → π0 ρA(g O) ,

g ∈ P+↑ ,

AdJρ : π0 ρA(O) → π0 ρ¯A(j O) .

(28)

9 Here we have written Uρ (a, λt ) := Uρ (a)Δit ρ . In particular, Jρ is a CPT operator. Again, it must be noted that these results (in d = 2) are already implicit in the work of Guido and Longo [22], and also that the split property would exclude any charged sectors in our sense.

2. Proofs We now prove Theorem 1 and Proposition 1. Instead of proving Borchers’ commutation relations directly (e.g. paralleling Florig’s nice proof [16]), we show how they lift from the vacuum sector to our charged sector. We shall use some wellknown facts about relative modular objects, which we recall in the Appendix for the convenience of the reader, see also [35] for a review. Namely, the operator −it is in π0 A(W1 ) for t ∈ R, and we define Δit ρ Δ0 −it ∈ A(W1 ) . Zρ (t) := π0−1 Δit (29) ρ Δ0 This family of observables coincides with the Connes cocycle (Dϕρ : Dω0 )t with respect to the pair of weights ω0 and ϕρ , see Eq. (A.2). In the present context, it satisfies (30) AdZρ (t) ◦ σt ◦ ρ = ρ ◦ σt on A(W1 ) , see Proposition 1.1 in [27]. The definition (29) and Eq. (30) are analogous to well-known properties of the translation cocycles which we shall use in the sequel. Observe that for a ∈ W1– , the closure of W1 , we have W1 + a ⊂ W1 and W1 − a ⊂ W1 . Since ρ acts trivially on W1 , this implies that the operator Uρ (a)U0 (−a) is in π0 A(W1 ) which coincides with π0 A(W1 ) by Haag duality. This gives rise to the translation cocycle ∈ A(W ) , a ∈ W – . (31) Y (a) := π −1 U (a)U (−a) ρ

0

ρ

0

1

1

9 If the net does not satisfy Haag duality for double cones, it does not coincide with the dual net. Then our endomorphism ρ has two generally distinct extensions ρR/L to the dual net, according a choice of the right or left wedge [33]. (Each of them is localizable only in one type of wedges.) In this case, Jρ intertwines π0 ρR with π0 ρ¯L αj , and in Eq. (28) there appears ρR on one side and ρ¯L on the other side.

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By virtue of Eqs. (9) and (13), it satisfies the intertwiner relation AdYρ (x) ◦ αx ◦ ρ = ρ ◦ αx ,

x ∈ Rd .

(32)

The definitions of the cocycles Zρ (t) and Yρ (x), the intertwiner relations (30) and (32), and invariance of Ω under Δit 0 and U0 (x) imply the identities it A ∈ A(W1 ) , (33) Δρ π0 (A)Ω = π0 Zρ (t)σt (A) Ω , Uρ (x) π0 (A)Ω = π0 Yρ (x)αx (A) Ω , A ∈ A, (34) which we shall frequently use in the sequel. We shall also use the fact that Borchers’ theorem applied to the observable algebra implies that σt αλ−t x σ−t = αx

(35)

holds as an isomorphism from A(W ) onto A(W + x), x ∈ R , W ∈ W1 . Finally, we make the interesting observation that Sρ is the relative Tomita operator associated not only with the pair of states (ω0 , ϕρ ), but also with pair of states (ϕρ¯, ω0 ): d

Lemma 1. The span, D, of vectors of the form π0 [ρ(A)Rρ¯]Ω, A ∈ A(W1 ), is a core for the relative Tomita operator Sρ , and Sρ acts on D as Sρ π0 ρ(A)Rρ¯Ω = χρ π0 (A∗ )Ω ,

A ∈ A(W1 ) ,

(36)

¯ρ = χρ¯, and is a square root where χρ is a complex number of modulus one, with χ of unity if ρ¯ = ρ. Proof. Eq.s (30), (33) and (41) imply that for A ∈ A(W1 ) there holds Δit ρ π0 [ρ(A)Rρ¯]Ω = π0 ρσt AZρ¯(−t) Rρ¯ Ω . Thus, the domain D is invariant under the unitary group Δit ρ . It is therefore a core 1/2

for Δρ

and hence for Sρ . On this core, we have by definition

Sρ π0 ρ(A)Rρ¯ Ω = π0 ρ¯(Rρ∗¯)Rρ A∗ Ω .

But ρ¯(Rρ∗¯)Rρ is a self-intertwiner of ρ, hence a multiple of unity, χρ 1 . This proves Eq. (36). For the stated properties of χρ , see [20, Eq. (3.2)]. Since (H0 , π0 ρ, Rρ¯Ω) is the GNS triple for the (non-normalized) state ϕρ¯ and χρ Ω is a GNS vector for ω0 , the Lemma implies that Sρ is the relative Tomita operator associated with the pair of states (ϕρ¯, ω0 ). Proof of Theorem 1. To prove Eq. (16) of Theorem 1, let A ∈ A(W ) and a ∈ W – . 1

Using Eq.s (33), (34) and (35), we then have −it ˆ Δit ρ Uρ (λ−t a)Δρ π0 (A)Ω = π0 Yρ (a, t)αa (A) Ω , Yˆρ (a, t) := Zρ (t)σt Yρ (λ−t a)αλ−t a Zρ (−t) .

1

(37)

The intertwiner relations (30) and (32) imply that on A(W1 ) there holds AdYˆρ (a, t) ◦ αa ◦ ρ ≡ AdZρ (t) ◦ σt ◦ AdYρ (λ−t a) ◦ αλ−t a ◦ AdZρ (−t) ◦ σ−t ◦ ρ = ρ ◦ σt ◦ αλ−t a ◦ σ−t = ρ ◦ αa .

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J. Mund


That is, Yˆρ (a, t) satisfies the same intertwiner relation (32) on A(W1 ) as Yρ (a). On the other hand, Yˆρ (a, t) is also contained in A(W1 ). Therefore Yˆρ (a, t)Yρ (a)∗ is in (ρA(W1 )) ∩ A(W1 ) which is trivial by our assumption (12) of irreducibility. Thus Yˆρ (a, t) coincides with Yρ (a) up to a scalar function c(a, t). Hence Eq. (37) reads −it Δit ρ Uρ (λ−t a)Δρ π0 (A)Ω = c(a, t) π0 Yρ (a)αa (A) Ω ≡ c(a, t) Uρ (a) π0 (A)Ω . Since the vacuum is cyclic for π0 A(W1 ) by the Reeh–Schlieder property, this shows that −it = c(a, t) Uρ (a) Δit ρ Uρ (λ−t a)Δρ

(38)

for a ∈ W1– . By adjoining, we get an analogous equation for −a ∈ W1– . Since the closures of W1 and −W1 span the whole Minkowski space, this shows that there is a function c(a, t) such that Eq. (38) holds for all a ∈ Rd . It remains to show that c(a, t) ≡ 1. Eq. (38) gives us a ray representation of the group G generated by the boosts λt and the translations in the 0, 1-plane, defined by U (a, λt ) := Uρ (a)Δit ρ . (The group G is a subgroup of P+↑ in d = 3 and coincides with P+↑ in d = 2. The product in G is (a, λt ) · (a , λt ) = (a + λt a , λt+t ).) Now G is simply connected, and its second cohomology group is known to be trivial. Therefore there exists ˆ (g) := ν(g) U (g) is a true a function ν from G into the unit circle such that U representation of G. Eq. (38) then implies that c(a, t) = ν(a, 1 ) ν(λ−t a, 1 )−1 .

(39)

Since Uρ is a true representation of the translations, the restriction of ν to the translations is a one-dimensional representation, that is of the form ν(a, 1 ) = ˆ = ν ⊗ U and U differ by a eik · a . Therefore, the spectra of the representations U ˆ is invariant under the 1-boosts translation about a vector k. But the spectrum of U ˆ extends to a true representation of the (2-dimensional) Poincaré group G, since U and the lower boundary of the spectrum of U is also Lorentz invariant since it coincides with the spectrum of Uρ . This implies that k = 0 and hence, by Eq. (39), that c(a, t) ≡ 1. This completes the proof of Eq. (16) of the Theorem. We now prove Eq. (18) of the Theorem. For A ∈ A(W1 ), we have by Eq. (33) and the intertwiner relation (30) −it ∗ ∗ ρ ¯ Z Δit R S Δ π (A)Ω = π )Z (t)σ (−t) ρ ¯ (A Ω. (40) 0 0 ρ¯ t ρ ρ ρ¯ ρ ρ We shall now use a result of Longo [27]. Namely, we are in the situation where Propositions 1.3 and 1.4 in [27] apply, yielding Rρ∗ ρ¯ Zρ (−t) Zρ¯(−t) = σ−t (Rρ∗ ) .

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Applying σt , adjoining, and using the cocycle identity Zρ¯(t)σt (Zρ¯(−t)) = 1, see Eq. (A.1), yields (41) Zρ¯(t)σt ρ¯ Zρ (−t)∗ Rρ = Rρ . Hence Eq. (40) reads

−it ¯(A∗ )Rρ Ω ≡ Sρ π0 (A)Ω . Δit ρ¯ Sρ Δρ π0 (A)Ω = π0 ρ

Since Δit ρ maps the core π0 A(W1 )Ω of Sρ onto itself by Eq. (33), this shows that −it Δit = Sρ , ρ¯ Sρ Δρ

which implies Eq. (18) of the Theorem. For the proof of Eq. (17) we need the following Lemma. Lemma 2. For a in the closure of W1 , there holds Uρ¯(a)−1 Sρ Uρ (a) ⊂ Sρ . ˆρ¯ defined by Proof. First recall from [11, 17] that the representation U

ˆρ¯(x) π0 ρ¯(A)Rρ Ω := π0 ρ¯ αx (A)Yρ (x)∗ Rρ Ω U

(42)

(43)

ˆρ¯(x) ◦ π0 ρ¯ = π0 ρ¯ ◦ αx .10 The repimplements αx in the representation π0 ρ¯, i.e. AdU ˆ resentation Uρ¯ therefore coincides with Uρ¯ up to a one-dimensional representation c( · ). We now have, for a in the closure of W1 and A ∈ A(W1 − a),

ˆρ¯(a) π0 ρ¯(A∗ )Rρ Ω Sρ Uρ (a) π0 (A)Ω = π0 ρ¯ αa (A∗ )Yρ (a)∗ Rρ Ω = U ˆρ¯(a) Sρ π0 (A)Ω . =U ˆρ¯ and Uρ¯ coincide up to the character c as discussed above, we therefore Since U have Uρ¯(−a) Sρ Uρ (a) = c(a) Sρ on D := A0 (W1 − a)Ω . −it to this equation and using the by now established Eq.s (16) Applying Δit ρ¯ · Δρ and (18) of the Theorem, yields c(λt a) = c(a) or c((1 − λt )a) = 1. By the representation property of c, the same holds for −a ∈ W1– . Since W1– and −W1– span the whole Minkowski space and 1 − λt is invertible for t = 0, this shows that c is trivial. Since D is a core for the left hand side of relation (42), this completes the proof.

recall the argument in the present setting. The endomorphism α−x ◦ φρ ◦ βx , where βx := AdYρ (x) ◦ αx , is a left inverse of ρ and therefore coincides with φρ by uniqueness. This implies that the state ϕρ is invariant under the automorphism group βx and hence that

10 We

Uρρ ρ(A)Rρ ]Ω := π0 [¯ ρβx (A)Rρ ]Ω ¯ (x)π0 [¯ ˆρ¯(x) defined above coincides with defines a unitary representation of the translations. But U π0 ρ¯(Yρ (x)∗ )Uρρ (x), hence is a well-defined unitary operator. The implementation property is ¯ checked directly from the definition (43), and implies in turn the representation property.

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J. Mund


We are now ready to prove Eq. (17) of the Theorem. To this end, let a ∈ W1– and φ ∈ D := A0 (W1 − a)Ω. By Eq (16), we have for all t ∈ R it Δit ρ Uρ (a) φ = Uρ (λt a) Δρ φ .

(44) 1/2

Now by Lemma 2, the vector Uρ (a) φ is in the domain of the operator Δρ , hence the left hand side is bounded for t in the strip R − i[0, 1/2] and analytic in its interior. The same holds for the vector valued function t → Δit ρ φ on the right – hand side. Further, for a ∈ W1 the operator valued function t → Uρ (λt a) is normbounded on the strip R − i[0, 1/2] and analytic in its interior, and at t = −i/2 has the value Uρ (ja), see e.g. [25, Section V.4.1]. Therefore, Eq. (44) implies that Δρ1/2 Uρ (a) φ = Uρ (ja) Δρ1/2 φ . Multiplying with Jρ and using relation (42) of Lemma 2 yields Uρ¯(a) Sρ φ = Jρ Uρ (ja)Jρ−1 Sρ φ .

(45)

Since Sρ has dense range, this shows Eq. (17) for x = ja in the closure of jW1 and, by adjoining, also for arbitrary x. This completes the proof of Eq. (17) of the Theorem. To prove Eq. (19), note that Lemma 1 implies that Sρ = χρ Sρ−1 ¯ . −1/2 −1 −1 1/2 Using that Δρ¯ Jρ¯ = Jρ¯ Δρ by Eq. (18) and that Jρ¯ is anti-linear, one gets Eq. (19). This completes the proof of the Theorem. Proof of Proposition 1. We now turn to Eq. (26) of Proposition 1. On A(W1 ), this equation follows from Eq. (30) by applying π0 to the latter. Further, the fact −it that π0−1 (Δit ρ Δ0 ) is in A(W1 ) and hence commutes with A(W1 ) implies that it AdΔρ ◦ π0 = π0 ◦ σt on A(W1 ). Since ρ acts as the identity on A(W1 ), this implies Eq. (26) on A(W1 ). For translates of W1 or W1 , the equation follows from Borchers’ commutation relations, Eq.s (16) and (35). Before proving Eq. (27) of the proposition, we establish the following intertwiner properties of the relative modular conjugation. Lemma 3 (Intertwiner properties of Jρ ). The unitary operators Jρ J0 and J0 Jρ have the intertwiner properties π0 ρ¯(A) Jρ J0 = Jρ J0 π0 (A) ,

(46)

π0 (A) J0 Jρ = J0 Jρ π0 ρ(A)

(47)

for A ∈ A(W1 ). Proof. These are consequences of a standard result [35] which relates the conjugations of relative Tomita operators, see Eq. (A.3) in the Appendix. Here, in Eq. (46) Sρ is being considered as the relative Tomita operator associated with the pair of states (ω0 , ϕρ ), characterized by Eq. (15), and in Eq. (47) as the relative Tomita operator associated with the pair (ϕρ¯, ω0 ), characterized by Eq. (36) of Lemma 1.

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We are now ready to prove Eq. (27) of Proposition 1. By Eq.s (47) and Eq. (23) we have on A(W1 ) AdJρ ◦ π0 ρ ≡ AdJ0 ◦ Ad(J0 Jρ ) ◦ π0 ρ = π0 ◦ αj = π0 ρ¯ ◦ αj , since ρ¯ acts as the identity on αj A(W1 ) ≡ A(W1 ), while by Eq. (46) and Eq. (23) we have on A(W1 ) AdJρ ◦ π0 ρ = AdJρ ◦ π0 ≡ Ad(Jρ J0 ) ◦ AdJ0 ◦ π0 = π0 ρ¯ ◦ αj . This shows that Eq. (27) holds on A(W1 ) ∪ A(W1 ). Borchers’ commutation relations then imply that it holds on A(W ), W ∈ W1 , completing the proof of Proposition 1.

Appendix A. Relative Tomita operators We recall the relevant notions from relative Tomita theory, following [35]. (For the standard Tomita theory, see e.g. [9] and Eq. (3) above.) Let M be a von Neumann algebra and ϕ1 , ϕ2 two faithful normal positive functionals on M, and denote by σt1 and σt2 the respective modular automorphism groups. Then there exists a family of unitaries Z21 (t) ∈ M satisfying the intertwiner and cocycle properties σt2 (A) Z21 (t) = Z21 (t) σt1 (A) , Z21 (t + s) = Z21 (t)σt1 Z21 (s) ,

(A.1)

respectively, and characterized by a certain KMS property. These facts have been shown by Connes [12] and are reviewed in [35, Sect. I.3.1]. The family Z21 (t) is called the Connes-cocycle associated with the pair ϕ1 and ϕ2 and usually denoted by (Dϕ1 : Dϕ2 )t . This cocycle may be expressed in terms of the corresponding GNS representations as follows [35, Sect. I.3.11]. Let (Hi , πi , ξi ) be the GNS triples of ϕi , i = 1, 2. Then the operator S21 from H1 to H2 defined by S21 π1 (A)ξ1 := π2 (A∗ )ξ2 ,

A ∈ M,

is closable. We denote its closure by the same symbol, and its polar decomposition by 1/2 S21 = J21 Δ21 . These operators are called the relative Tomita modular objects associated with the pair ϕ1 and ϕ2 . Let now Δit 1 denote the unitary modular group of π1 (M) and −it ξ1 . Then Δit Δ is in π (M) and coincides with π1 (Z21 (t)), i.e. there holds [35, 1 21 1 Sect. I.3.11] −it . (A.2) Z21 (t) = π1−1 Δit 21 Δ1 Finally, as shown in [35, Sect. I.3.16], the unitary operator V21 := J21 J1 ≡ J2 J21 , where Ji is the modular conjugation of πi (M) and ξi , i = 1, 2, is an intertwiner from π1 to π2 , that means it satisfies π2 (A) V21 = V21 π1 (A) ,

A ∈ M.

(A.3)

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Acknowledgements I gratefully acknowledge financial support by FAPEMIG.

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[19] K. Fredenhagen, Generalizations of the theory of superselection sectors, The algebraic theory of superselection sectors. Introduction and recent results, D. Kastler (ed.), World Scientific, 1990. [20] K. Fredenhagen, K.-H. Rehren, and B. Schroer, Superselection sectors with braid group statistics and exchange algebras II: Geometric aspects and conformal covariance, Rev. Math. Phys. SI1 (1992), 113–157. [21] J. Fr¨ ohlich and P. A. Marchetti, Quantum field theories of vortices and anyons, Commun. Math. Phys. 121 (1989), 177–223. [22] D. Guido and R. Longo, Relativistic invariance and charge conjugation in quantum field theory, Commun. Math. Phys. 148 (1992), 521–551. [23] D. Guido and R. Longo, The conformal spin and statistics theorem, Commun. Math. Phys. 181 (1996), 11–35. [24] D. Guido and R. Longo, Natural energy bounds in quantum thermodynamics, Commun. Math. Phys. 218 (2001), 513–536. [25] R. Haag, Local quantum physics, second ed., Texts and Monographs in Physics, Springer, Berlin, Heidelberg, 1996. [26] B. Kuckert, Two uniqueness results on the Unruh effect and on PCT-symmetry, Commun. Math. Phys. 221 (2001), 77–100. [27] R. Longo, An analogue of the Kac-Wakimoto formula and black hole conditional entropy, Commun. Math. Phys. 186 (1997), 451–479. [28] R. Longo, On the spin-statistics relation for topological charges, Operator Algebras and Quantum Field Theory, S. Doplicher, R. Longo, J. Roberts, and L. Zsido (eds.), Int. Press, Cambridge, MA, 1997, pp. 661–669. [29] M. M¨ uger, Superselection structure of massive quantum field theories in 1 + 1 dimensions, Rev. Math. Phys. 10 (1998), 1147–1170. [30] J. Mund, The CPT and Bisognano–Wichmann theorems for Anyons and Plektons in d = 2 + 1, in preparation. [31] J. Mund, The Bisognano–Wichmann theorem for massive theories, Ann. H. Poinc. 2 (2001), 907–926. [32] K.-H. Rehren, Spin statistics and CPT for solitons, Lett. Math. Phys. 95 (1998), 110. [33] J. E. Roberts, Local cohomology and superselection structure, Commun. Math. Phys. 51 (1976), 107–119. [34] B. Schroer and H.-W. Wiesbrock, Modular theory and geometry, Rev. Math. Phys. 12 (2000), 139–158. [35] S. Strˇ atilˇ a, Modular theory in operator algebras, Abacus Press, Tunbridge Wells, England, 1981. [36] H.-W Wiesbrock, Superselection structure and localized Connes’ cocycles, Rev. Math. Phys. 7 (1995), 133–160.

34 Jens Mund Departamento de F´ısica Universidade Federal de Juiz de Fora 36036-900 Juiz de Fora, MG Brazil e-mail: [email protected] Communicated by Klaus Fredenhagen. Submitted: June 11, 2008. Accepted: August 4, 2008.

J. Mund




Dispersion Relations in the Noncommutative φ3 and Wess–Zumino Model in the Yang–Feldman Formalism Claus Döscher and Jochen Zahn Abstract. We study dispersion relations in the noncommutative φ3 and Wess– Zumino model in the Yang–Feldman formalism at one-loop order. Nonplanar graphs lead to a distortion of the dispersion relation. We find that the strength of this effect is moderate if the scale of noncommutativity is identified with the Planck scale and parameters typical for a Higgs field are employed. The contribution of the nonplanar graphs is calculated rigorously using the framework of oscillatory integrals.

1. Introduction We discuss dispersion relations for quantum field theories on the noncommutative Minkowski space, which is generated by coordinates q μ subject to the commutation relations [q μ , q ν ] = iσ μν . Here σ is an antisymmetric matrix. Such commutation relations are motivated from Gedanken experiments on limitations of the localization of experiments [11]. They are also obtained as a limit of open string theory in the presence of a constant background B-field [35]. We emphasize that for the space-time uncertainty relations derived in [11] it is crucial that σ is nondegenerate, in particular σ 0i = 0, i.e., one has space/time noncommutativity. Thus, we focus on this case. We remark that such a σ can not be obtained as a limit of string theory [36]. There are several inequivalent approaches to quantum field theory on the noncommutative Minkowski space (NCQFT). In the modified Feynman rules originally proposed in [15] for both the noncommutative Euclidean and the Minkowski space, one simply attaches a phase factor depending on the momenta, the so-called twisting, to each vertex. In cases where the twistings do not cancel, one speaks of a non-planar diagram. Then an oscillating phase remains in the loop integral.

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C. D¨ oscher and J. Zahn


It is part of the folklore of NCQFT that this makes the loop integral convergent. However, to the best of our knowledge, the precise meaning of these integrals has never been stated. They are not absolutely convergent and are, with the exception of the tadpole, no Fourier transformations. It is one of the goals of this paper to give a precise definition for such integrals. Furthermore, to the best of our knowledge, all calculations in this approach were done in the Euclidean setting. However, since there is no Osterwalder–Schrader theorem for field theories on the noncommutative Minkowski space, the relation between calculations in the Euclidean and the Lorentzian metric is obscure in the case of space/time noncommutativity. In fact there are hints that if such a relation exists at all, it must be quite complicated [4, p. 84 f.]. If one accepts the formal nature of the loop calculations and the transition to the Euclidean signature, the picture is as follows: If k is the outer momentum of a nonplanar loop, one can argue heuristically that an original f (Λ)-divergence, − 1 where Λ is the UV cutoff, becomes regularized to f ((kσ)2 2 ). Thus, a UVdivergence becomes an IR-divergence. This is the so-called UV-IR mixing first discussed in [28]. In the case of space/time noncommutativity this approach leads to a violation of unitarity [20]. In the case of space/space noncommutativity, some general results, in particular a PCT theorem, were proven, see [2] and references therein. The Hamiltonian approach [11, 24] leads to a unitary theory also in the case of space/time noncommutativity. In some cases these theories are UV-finite [5, 6]. However, in the case of space/time noncommutativity, the interacting field does, at tree level, not fulfill the classical equations of motion [4, 22]. In the case of electrodynamics, this leads to a violation of the Ward identity [31]1 . Another proposal is to consider Euclidean self-dual theories in the sense of [23] by adding a confining potential. In this approach the renormalizability of the φ4 model has been shown to all orders [21]. However, there is no indication that these models are related to NCQFT on Minkowski spacetime. Thus, the most promising approach to NCQFT in the case of space/time noncommutativity is the Yang–Feldman approach [39]. It can also be employed in situations where a Hamiltonian quantization is problematic. In particular, it was used in the context of nonlocal field theories, see, e.g., [26, 29]. In the context of NCQFT, it was first proposed in [7]. Here the UV-IR mixing manifests itself as a distortion of the dispersion relation in the infrared. In the case of the φ4 -model, this effect has been shown to be very strong [8]. This is to be expected, since the underlying UV-divergence is quadratic. Thus, it is natural to ask whether the effects are weaker in theories that are only logarithmically divergent2 . This is the 1 In

[22], a different time-ordering, with respect to light-cone coordinates was proposed. While Feynman rules can be formulated quite elegantly in this setting, actual computations seem to be rather involved. 2 One has to bear in mind that it is not clear if the usual power counting arguments can be applied in the Yang–Feldman approach, in particular in the presence of twisting factors. This will become clearer in Section 3.

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aim of the present paper where we consider the φ3 and the Wess–Zumino model at the one-loop level. It turns out that the effect is indeed quite weak if one uses the Planck scale as the scale of noncommutativity and uses parameters typical for a Higgs field. The contributions of the nonplanar graphs, which are made finite by an oscillating factor, are treated in a rigorous way by the use of the theory of oscillatory integrals [33]. To our knowledge this has not been done before. A remark on the issue of Lorentz invariance is in order here. We will see that the self-energy for an outer momentum k is of the form Σ(k 2 , (kσ)2 ). It is thus invariant under Lorentz transformations if σ transforms as a tensor, as has been proposed in [11]. Since the group velocity compares the energy of waves of different wavelengths, observed from a fixed reference frame, it should be computed for fixed σ. Thus, the dispersion relation can be distorted even though the theory is invariant under a boost of the reference frame3 . In the same context, one should remark that we do not use the concept of twisted Poincaré invariance [30,41] here. However, if the interaction term is given by the -product and the interacting field is defined via the the Yang–Feldman formalism, one would recover the results presented here, cf. [41]. The noncommutative φ3 -model has already been treated in [28, 32] in the context of the modified Feynman rules, in [6] in a Hamiltonian setting, and in [19] in the Euclidean self-dual setting. The noncommutative Wess–Zumino model was first discussed in [17] for space/space noncommutativity in the setting of the modified Feynman rules. It was shown that the UV-IR mixing is much weaker as in the φ4 -theory, so that the theory is renormalizable to all orders. The paper is organized as follows: In Section 2 we discuss how to compute momentum-dependent mass and field strength renormalization in the Yang– Feldman approach and to extract the corresponding group velocity. In Section 3 we apply this machinery to the noncommutative φ3 -model at second order, i.e., for one loop. In particular, we compute the distortion of the group velocity for parameters typical for a Higgs field. In Section 4 we treat the noncommutative Wess–Zumino model, also at one-loop order. We show and discuss the fact that the local SUSY current is not conserved in the interacting case. We also compute the momentum dependent mass and field strength normalization and show that the distortion of the group velocity is simply twice that of the φ3 -case. The oscillating integrals so far have only been calculated formally. A rigorous calculation in the sense of oscillatory integrals is presented in Section 5. It turns out that the formal results are indeed correct. We conclude with a summary and an outlook.

2. Dispersion relations in the Yang–Feldman formalism We want to discuss how to compute (possibly momentum dependent) mass and field strength renormalizations in the Yang–Feldman formalism. In this formalism, 3 See

also the discussion in [9], in particular the distinction between observer and particle Lorentz transformations.

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the interacting field is recursively defined as a formal power series in the coupling constant. As an example, we consider a commutative scalar theory and a localized mass term as interaction, i.e., we have the equation of motion ¯ 2 g(x)φ(x) , ( + m2 )φ(x) = −m where g is a test function. Making the ansatz φ=

∞

m ¯ 2n φn

n=0

for the interacting field, this leads to the equations ( + m2 )φ0 = 0 , ( + m2 )φn = −gφn−1 ,

n ≥ 1.

Obviously, φ0 is a free field. We identify it with the incoming field. Then the higher order terms are given recursively by φn = ΔR × (gφn−1 ) ,

n ≥ 1,

where × denotes the convolution and ΔR the retarded propagator at mass m. We define the observable ˆ , φ(f ) = d4 x f (x)φ(x) = d4 k fˆ(−k)φ(k) (1) where the hat denotes the Fourier transform. We are now interested in the Wightman two-point function φ(f )φ(h) (2) of the interacting field. The vacuum state here is the vacuum state for the free field φ0 , i.e., in order to compute the above, one has to express φ solely in terms ¯ 2, of φ0 and then determine the vacuum expectation value. At zeroth order in m we obtain the usual free two-point function 2 ˆ Δ ˆ + (k) φ0 (f )φ0 (h) = (2π) d4 k fˆ(−k)h(k) 2 2 ˆ = 2π d4 k fˆ(−k)h(k)θ(k (3) 0 )δ(k − m ) . At first order in m ¯ 2 , we get φ1 (f )φ0 (h) + φ0 (f )φ1 (h) ˆ g (k − l) Δ ˆ R (k)Δ ˆ + (l) + Δ ˆ + (k)Δ ˆ A (l) . = −(2π)2 d4 kd4 l fˆ(−k)h(l)ˆ Here ΔA is the advanced propagator. It has been shown in [14] that, under quite general assumptions, in the adiabatic limit g → 1, i.e., gˆ → (2π)2 δ, this becomes 0 2 ˆ )δ (k − m2 ) . (4) −2π d4 k fˆ(−k)h(k)θ(k

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Obviously, this can be interpreted as the first order term in an expansion of ¯ 2 , · ) around m2 , cf. (3). Δ+ (m2 + m When considering noncommutative field theories, the following changes have to be made: Fields and test functions are now functions of the noncommuting coordinates q μ , so that products are given by −ikq −ilq ˆ e f (q)h(q) = (2π)−4 d4 kd4 l fˆ(k)h(l)e i ˆ 2 kσl . = d4 k e−ikq d4 l fˆ(k − l)h(l)e (5) Here fˆ denotes the Fourier transform of the Weyl symbol of f (q). Alternatively, one could use functions of x and the Weyl–Moyal -product. The integral (trace) is defined as usual as d4 q f (q) = (2π)2 fˆ(0) . Then, analogously to (1), we have ˆ . φ(f ) = d4 q f (q)φ(q) = d4 k fˆ(−k)φ(k) The Yang–Feldman series can be set up exactly as before, i.e., φ0 is the free field and for n ≥ 1, we have4 φn (q) = d4 x ΔR (x)g(q − x)φn−1 (q − x) i −2 4 −ikq ˆ = (2π) d k ΔR (k)e d4 l gˆ(k − l)φˆn−1 (l)e 2 kσl . It was shown in [14] that also in this case one obtains (4) as the first order contribution to the two-point function in the adiabatic limit gˆ(k) → (2π)2 δ(k). 2.1. Interactions Now we consider truly interacting models. For simplicity we start with a scalar field theory on the ordinary Minkowski space. The coupling constant is denoted by λ. When computing the two-point function (2), one finds again (3) as the zeroth order contribution. In the models discussed in this paper, there is no O(λ) contribution5 . At second order, one finds the three terms (6) φ2 (f )φ0 (h) + φ0 (f )φ2 (h) + φ1 (f )φ1 (h) . 4 Here

the infrared cutoff was implemented by multiplying the “interaction term” m ¯ 2 φ(q) in the equation of motion with a “test function” g(q) from the left. One can also use more symmetric products, for details see [14]. 5 In the φ3 -model, the first order term φ is a product of two free fields φ , cf. section 3. The 1 0 two-point function at first order is thus a vacuum expectation value of a product of three free fields and vanishes. This is completely analogous to conventional quantization schemes. Of course the discussion can be done for models with a first order contribution, e.g. φ4 .

40



As we will see later, the third term is a contribution to the continuous spectrum and thus not interesting at the moment. In order to treat the first two terms, we notice that in the models discussed here, φ2 is formally of the form

ˇ × (gφ0 ) + n.o. , (7) φ2 = (2π)−2 ΔR × g Σ where n.o. stands for a term that is normal ordered and whose spectrum has no overlap with the positive or negative mass shell if the support of gˆ is chosen small enough. Thus, this term drops out in the first two terms in (6). As shown ˇ can be identified with the inverse Fourier transform of the self-energy. It below, Σ will in general be divergent and has to be renormalized, which we assume in the following. Then the first term in (7) is quite similar to φ1 in the case of a mass term as interaction. It is thus not very surprising that, using the same techniques as in [14], one can show (for details see [13, 40]) that in the adiabatic limit g → 1, one obtains ∂ ˆ 2 ˆ Δ+ (k) , d4 k fˆ(−k)h(k)Σ(k) (8) −(2π) ∂m2 for the first two terms in (6) under the condition that Σ(k) = Σ(−k) in a neighborˇ and can be identified hood of the mass shell. Here Σ is the Fourier transform of Σ with the self-energy. In the commutative case, Σ(k) is only a function of k 2 , and (8) corresponds to a mass and field strength renormalization δm2 = −λ2 Σ(m2 ) , δZ = −λ2

∂ Σ(m2 ) . ∂k 2

In the noncommutative case, a rigorous adiabatic limit meets serious difficulties because of UV-IR mixing effects. Intuitively, this can be understood as follows: Because of the UV-IR mixing, an infrared cutoff is also an ultraviolet cutoff. Hence, as long as the adiabatic limit is not carried out, there is no ultraviolet divergence and thus no need for renormalization6 . However, in the adiabatic limit, the ultraviolet divergences show up again, so one would have to deal with ultraviolet and infrared divergences at the same time. While this seems to be feasible for logarithmically divergent models, it will be quite difficult in the general case. For details we refer to [13, 40]. We thus take a pragmatic point of view and work formally, i.e., without infrared cutoff. In analogy to (7), we write φ2 in the form ˆ R (k)Σ(k)φˆ0 (k) + n.o. φˆ2 (k) = (2π)2 Δ and take this as an implicit definition of Σ (again, we assume Σ to be renormalized). If then Σ(k) = Σ(−k) in a neighborhood of the mass shell, we use (8) as the sum of the first two terms in (6). Now Σ(k) is in general not only a function of k 2 , 6 In

fact, this depends on the cutoff scheme. But for any reasonable cutoff scheme there are always terms that are finite under the infrared cutoff, but diverge in the adiabatic limit.

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but also of (kσ)2 . Thus, we obtain momentum-dependent mass and field strength renormalizations:

δm2 (kσ)2 = −λ2 Σ m2 , (kσ)2 , (9)

∂ δZ (kσ)2 = −λ2 2 Σ k 2 , (kσ)2 |k2 =m2 . (10) ∂k Remark 2.1. As suggested in [7], we do not subtract these terms (despite their naming), since they are neither local, nor, in general, divergent. We remark, however, that such a subtraction has been proposed in [25]. 2.2. The group velocity The sum of the zeroth order term (3) and the second order contribution (8) can be interpreted as the expansion (in λ) of

0 ˆ 2π d4 k fˆ(−k)h(k)θ(k )δ k 2 − m2 + λ2 Σ k 2 , (kσ)2 + O(λ4 ) . (11) This can be interpreted as a change of the dispersion relation. Remark 2.2. This modification of the dispersion relation is a manifestation of the breaking of particle Lorentz invariance, cf. the discussion in the introduction. However, particle Lorentz invariance of the asymptotic fields is a crucial ingredient of scattering theory and the LSZ relations, which are part of the foundations of quantum field theory. In this sense, the conceptual basis of the present approach is rather shaky. In the following, we will take a phenomenological standpoint and compute the distortion of the dispersion relation for different models in order to check if they are realistic. We now discuss how to extract the group velocity in the above setting. From (11), and allowing for a finite local mass and field strength renormalization, we get the dispersion relation

(12) F (k) = k 2 − m2 + λ2 Σ k 2 , (kσ)2 − α + βk 2 + O(λ4 ) = 0 . For a given spatial momentum k we want to compute the corresponding k 0 that solves (12) as a formal power series in λ. We find 1 2 Σ m , (k+ σ)2 − α + βm2 + O(λ4 ) . k 0 = ωk − λ 2 (13) 2ωk

2 Note that in ωk = |k| + m2 and k+ = (ωk , k) the bare mass m enters. The group velocity is then given by k

k + λ2 3 Σ m2 , (k+ σ)2 − α + βm2 ∇k 0 = ωk 2ωk

1 ∂ ∇(k+ σ)2 Σ m2 , (k+ σ)2 + O(λ4 ) . − λ2 2ωk ∂(kσ)2

42



By comparison with (13), we get ∇k 0 =

∇(k+ σ)2 ∂ k − λ2 Σ m2 , (k+ σ)2 + O(λ4 ) . 0 0 2 k 2k ∂(kσ)

In order to make things more concrete, we choose a particular σ, namely, 0 − 2 σ = σ0 = λnc . 0

(14)

Then we have 2

(kσ0 )2 = −λ4nc (k 2 + 2 |k⊥ | )

(15)

with k⊥ = (k1 , 0, k3 ). We also define k|| = (0, k2 , 0). Thus, in the case σ = σ0 , we find

2 k|| ∂ k⊥ 2 ∇k 0 = 0 + 0 1 + 2λ2 λ4nc Σ m , (k σ ) (16) + O(λ4 ) . + 0 k k ∂(kσ)2 Remark 2.3. This treatment differs slightly from the one given in [8]. There, Σ is not Taylor expanded in λ. Then the argument of Σ in (16) is not restricted to the mass m shell. It follows that by tuning α and β one can make the deviation arbitrarily small, which is not possible here. Remark 2.4. The modification of the dispersion relation can be interpreted as an effect of the momentum-dependent mass renormalization (9), since λ2 Σ in (16) can be replaced by −δm2 . The momentum-dependent field strength renormalization (10), on the other hand, multiplies, in momentum space, the free propagators, in particular the retarded propagator. In position space, this can be interpreted as a smearing of the source, and thus as a non-local effect. In [40], this is explained in more detail, and the effect is computed for the case of noncommutative supersymmetric electrodynamics. In particular, it is shown that, surprisingly, the range of this nonlocality is independent from the scale of noncommutativity.

3. The φ3 -model We now apply the above tools to the noncommutative φ3 -model and compute the momentum-dependent mass and field strength renormalization and the distortion of the group velocity at second order. We start from the equation of motion ( + m2 )φ = λφ2 . The Yang–Feldman ansatz φ = n λn φn , and the identification of φ0 with the incoming field then leads to φ1 = ΔR × (φ0 φ0 ) , φ2 = ΔR × (φ1 φ0 + φ0 φ1 ) . We substract the tadpole from the start, i.e., we use normal ordering and redefine φ1 = ΔR × ( :φ0 φ0:) .

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Now we want to compute the two-point function of the interacting field. At zeroth order, we find the usual result (3). At first order, there is no contribution. At second order, there are the three terms (6). We first focus on the sum of the first two terms. As discussed in the previous section, we treat it by computing the self-energy Σ(k). Performing the contractions in φ2 , we obtain ˆ R (k)φˆ0 (k) φˆ2 (k) = (2π)2 Δ

ˆ + (l)(1 + eikσl ) ˆ R (k − l) Δ ˆ + (−l) 1 + e−ikσl + Δ × d4 l Δ + n.o. Thus, Σ is given by ˆ R (k − l) Δ ˆ + (−l)(1 + e−ikσl ) + Δ ˆ + (l)(1 + eikσl ) . Σ(k) = d4 l Δ This can be split into a planar part not involving the phase factors and a nonplanar part. The planar part is precisely half of the self-energy of the commutative φ3 model. For the following consideration, it is important that we are only interested in Σ(k) in a small neighborhood of the mass shell. But also the loop momentum l is confined to the mass shell, so if (m − )2 < k 2 < (m + )2 , then either (k − l)2 < 2 ˆ R (k − l) is not met and the or (k − l)2 > (2m − )2 . Thus, the singularity of Δ i -prescription does not matter: One may simply write −1 −1 ˆ R (k − l) = (2π)−2 Δ . = (2π)−2 2 2 2 (k − l) − m k − 2k · l We begin by discussing the planar part ˆ R (k − l) Δ ˆ + (−l) + Δ ˆ + (l) . Σpl (k) = d4 l Δ

(17)

As usual, this expression is not well-defined. Because of the preceding remark, it is straightforward to show that at least formally Σpl (k) = Σpl (−k) in a neighborhood of the mass shell. It has been shown in [7] that ΔR · (Δ+ + Δ− ) = Δ2F − Δ2− holds. Here Δ2− is well-defined, while Δ2F has the usual logarithmic divergence. Alternatively, one may argue with the following formal calculation: Because of Lorentz invariance, we may choose k = (k0 , 0). Then 3 1 1 d l Σpl (k) = − (2π)−3 + 2ωl k02 − 2k0 ωl k02 + 2k0 ωl ∞ l2 , (18) dl = − 2(2π)−2 2 ωl (k0 − 4ωl2 ) 0 which diverges logarithmically. We note that it is necessary to consider the sum of the two terms in (17). The individual terms are linearly divergent. It is a priori not clear if the same cancellation takes place in the presence of the twisting factors,

44



i.e., in the nonplanar part. Hence, the validity of power counting arguments for noncommutative field theories in the Yang–Feldman formalism is doubtful. Finally, we remark that the field strength renormalization is finite. Using (10), one computes 2π 3− √ 3 . (19) δZ = (2π)−2 12m2 3.1. The nonplanar part We now want to discuss the nonplanar part of Σ(k), i.e.,

ˆ R (k − l) + Δ ˆ + (l)eikσl Δ ˆ R (k + l) , Σnp (k) = d4 l Δ

(20)

for k in a neighborhood of the mass shell. In particular, we want to show that it is finite and that Σnp (k) = Σnp (−k) there. Note that the above integral is neither absolutely convergent nor a Fourier transformation (since k does not only appear in the phase factor). In the following, we compute this integral in a formal way. In Section 5 we show that (20) can be defined as an oscillatory integral and that a calculation in this framework gives the same result as our formal calculation. First of all we note that if Σnp (k) is well defined, then it is invariant under the Lorentz transformation k → kΛ ,

σ → Λ−1 σΛT

−1

.

Thus, instead of computing the above at k, σ we may compute it at k = kΛ, σ = −1 Λ−1 σΛT . Since at the one-loop level we are only √ interested in Σnp (k) in a neigh borhood of the mass shell, we may choose k = ( k 2 , 0). Since σ is antisymmetric, k σ has vanishing time component. We denote its spatial component by k σ . Then we have 3 −ik σ · l e−ik σ · l e d l −3 √ √ + Σnp (k) = −(2π) 2ωl k 2 − 2 k 2 ωl k 2 + 2 k 2 ωl 3 1 d l = −2(2π)−3 cos(k σ · l) 2ωl k 2 − 4ωl2 ∞ sin l −(kσ)2 l2 −2 dl . (21) = −2(2π) ωl (k 2 − 4ωl2 ) l −(kσ)2 0 In the first step we used the the symmetry properties of the integrand. In the next 2 step we used (kσ)2 = (k σ )2 = − |(k σ )| . Obviously, the integral is finite and 2 2 only a function of k and (kσ) . Furthermore, Σnp (k) = Σnp (−k). In order to estimate the strength of the distortion of the dispersion relation, we calculate δm2 ((kσ)2 ) and δZ((kσ)2 ) numerically. We use the parameters σ = σ0 (cf. (14)), m = 10−17 λ−1 nc and λ = m. If λnc is identified with the Planck length, this corresponds to a mass of about 100 GeV, i.e., the estimated order of magnitude of the Higgs mass. The chosen value of λ is slightly above the expectation for the cubic term in the Higgs potential (∼ 0.6m). Figure 1 shows the relative mass

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m2 Δm2 kΣ2 0.96

0.97

0.98

0.99

2

4

6

8

10

ΛNC k 1017

Figure 1. The relative mass correction m−2 δm2 ((kσ)2 ) as a function of the perpendicular momentum k⊥ . correction m−2 δm2 ((kσ)2 ) as a function of the perpendicular momentum k⊥ , obtained with the numerical integration method of mathematica (for the definition of k⊥ , see Section 2.2). We see that the relative mass shift is of order 1 for small perpendicular momenta. This might look like a strong effect. However, we have the freedom to apply a finite mass renormalization in order to restore the rest mass. The important question is rather how strong the momentum dependence of the mass renormalization is. As can be estimated from Figure 1, it is at the %-level for perpendicular momenta of the order of the mass. As a consequence, also the distortion of the group velocity is of this order, as we will show below. The plot for δZ((kσ)2 ) for the same parameters is not very interesting, since δZ is constant, −1.32477 · 10−3 , within machine precision. This coincides with the planar contribution (19). The reason for this is easily understood: If one differentiates the integrand in (21) with respect to k 2 , one obtains a function that, even without the factor sin l −(kσ)2 , l −(kσ)2 is integrable. Without this factor, it would coincide with the corresponding planar expression obtained by differentiating (18). But the above factor deviates from 1 1 appreciably only for l ∼ (−(kσ)2 )− 2 , i.e., for very high energies, where the rest of the integrand is negligible. According to equation (16), the deviation of the group velocity from the phase velocity in the perpendicular direction is, to lowest order in λ, given by ∂ 2λ2 λ4nc ∂(kσ) 2 Σnp . Figure 2 shows this quantity for the same parameters as above. The deviation is biggest for small perpendicular momenta and at the %-level.

46



1 v k k0 k k0 102 1.2 1. 0.8 0.6 0.4 0.2

2

4

6

8

10

ΛNC k 1017

Figure 2. The distortion of the group velocity in perpendicular direction as a function of the perpendicular momentum k⊥ . We see that in the φ3 model the distortion of the dispersion relation is moderate for realistic masses and couplings. This is in sharp contrast to the situation in the φ4 model, where realistic dispersion relations could only be obtained for masses close to the noncommutativity scale [4]. 3.2. The 2-particle spectrum We now discuss the third term in (6). We obtain

4 ˆ Δ ˆ R (k)Δ ˆ A (k) (Δ+ · Δ+ )ˆ(k) + (Δ+ 2σ Δ+ )ˆ(k) . d4 k fˆ(−k)h(k) (2π) Here 2σ is the -product at 2σ, i.e., the product corresponding to the twisting factor eikσl . Like Δ+ · Δ+ , Δ+ 2σ Δ+ is a well-defined distribution, as can be seen in momentum space. It has its support above the 2m mass shell, thus this term corresponds to the multi-particle spectrum. Using Lorentz invariance as above, one can compute

sin −(kσ)2 14 k 2 − m2 √ . (Δ+ 2σ Δ+ )ˆ(k) = θ(k 2 − 4m2 )(2π)−3 2 k 2 −(kσ)2 In the limit (kσ)2 → 0, this gives back the commutative result. Note that devia√ tions from the commutative case become appreciable for −(kσ)2 ∼ k −2 , i.e. if k 2 λ2 or the transversal momentum k⊥ is of the order √nc . This is obviously no threat k2 to phenomenology.

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4. The Wess–Zumino model In this section we consider the Wess–Zumino model on the noncommutative Minkowski space. We use the standard supersymmetric noncommutative Minkowski space, in which the (anti-) commutators involving the fermionic variables θ, θ¯ are unchanged [10]. In order to arrive at the equations of motion for the component fields, we start from the Lagrangean in superfield form, taking particular care for the order of the fields in the different terms7 . In superfield form the Wess–Zumino model is given by the following Lagrangean8 : m λ ¯ θ2 θ¯2 + ΦΦ + ΦΦΦ |θ2 + h.c. . L = ΦΦ| 2 3 Here Φ is the chiral superfield √ ¯ μ φ + √i θ2 ∂μ χσ μ θ¯ − 1 θ2 θ¯2 φ , Φ = φ + 2θχ + θ2 F − iθσ μ θ∂ 4 2 where φ and F are complex scalar fields and χ is a Weyl spinor. In component fields the action is then, up to surface terms, 4 ¯σ μ χ − φ∗ φ + F ∗ F S = d q −i∂μ χ¯ 1 . + m φF − χχ + λ(φφF − χχφ) + h.c. 2 This leads to the equations of motion F + mφ∗ + λφ∗ φ∗ = 0 −φ + mF ∗ + λ(φ∗ F ∗ + F ∗ φ∗ ) − λχ ¯χ ¯=0 i¯ σ μ ∂ μ χ − mχ ¯ − λ(φ∗ χ ¯ + χφ ¯ ∗) = 0 . We eliminate F using its equation of motion. Furthermore, we introduce the Majorana spinor 1 1 χα ¯α˙ ) , ψ¯ = ψ † γ 0 = √ (χα , χ ψ=√ α ˙ χ ¯ 2 2 and the projectors 1 ∓ iγ5 P± = . 2 ¯ + ψ = χχ we get Using 2ψP ¯ − ψ − mλ(φφ + φ∗ φ + φφ∗ ) − λ2 (φ∗ φφ + φφφ∗ ) ( + m2 )φ = −2λψP (i∂/ − m)ψ = λP+ (φψ + ψφ) + λP− (φ∗ ψ + ψφ∗ ) . is important, since for example the tadpole corresponding to the interaction term φ∗ φφ∗ φ does not have a twisting factor, in contrast to the interaction term φ∗ φ∗ φφ, as has already been noted in [3]. 8 In the following, we use the conventions of [38], except for the metric, which we choose to have signature (+ − −−). Accordingly, we also changed the sign of σ 0 , and thus also of γ 0 and γ 5 . 7 This

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4.1. The SUSY current We first want to discuss the changes that noncommutativity brings in at the classical level. The equations of motion are the same, we only have to replace the usual product by the noncommutative one. But there are some changes for the currents. It is an interesting feature of noncommutative interacting theories that the local9 currents associated to symmetries are in general not conserved [16, 42]. Examples are the energy-momentum tensor in the φ4 -model [27] and in electrodynamics [18]. Here we show that the local current associated to the supersymmetry transformation is not conserved in the interacting case, i.e., for λ = 0. We discuss this in terms of the superfield Φ. The equation of motion is 1 ¯2 ¯ Φ + mΦ + λΦΦ = 0 . − D 4 The local supercurrent is given by Vαα˙ =

1 ¯ + i{∂/αα˙ Φ, Φ} ¯ . ¯ α˙ Φ] ¯ − i{Φ, ∂/αα˙ Φ} [Dα Φ, D 2

Here we used a symmetrized version of the usual current, since this is usually advantageous in the noncommutative case. By standard methods (see, e.g., [37]) one can show that ¯ α˙ Vαα˙ = 1 {Dα Φ, D ¯ − 1 {Φ, Dα D ¯ 2 Φ} ¯ ¯ 2 Φ} D 2 4 holds. Using the equation of motion, we get ¯ α˙ Vαα˙ = 2 Dα Φ, (mΦ + λΦΦ) − Φ, Dα (mΦ + λΦΦ) D = mDα Φ2 + λ [Dα Φ, Φ], Φ . The first term is already present in the commutative case. It does not affect the charge corresponding to the supersymmetry transformation, but simply expresses the fact that the theory is not conformal. The second term, however, is a genuinely noncommutative one. It also affects the SUSY charge. Since it is given by a commutator, the non-conservation of the charge is relevant only at the noncommutativity scale10 . Like the non-conservation of the local energy-momentum tensor, this effect does not show up in a perturbative treatment of the corresponding quantum theory, at least not at second order. 9 By

local we mean expressions that are polynomials of (derivatives) of fields, where the product is the appropriate algebra product, i.e., (5) in the present case. Using different products (nonlocal in our sense), it is possible to construct conserved currents, see, e.g., [1, 34]. 10 Such an effect is to be expected by heuristic considerations [12]: Charge conservation requires that the production of a particle with positive charge is always accompanied by the production of a particle with opposite charge at the same place. But because of the noncommutativity, it is not possible to localize two particles at the same place, see, e.g., the discussion in [5].

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4.2. The self energy Now we compute the self energy at the one-loop level. Using the equations of motion, the first terms in the Yang–Feldman series are

φ1 = −ΔR × 2ψ¯0 P− ψ0 + m(φ∗0 φ0 + φ0 φ∗0 + φ0 φ0 ) , (22)

∗ ∗ ψ1 = SR × P+ (φ0 ψ0 + ψ0 φ0 ) + P− (φ0 ψ0 + ψ0 φ0 ) , (23) and the analogous formulas for the conjugate fields. The second order component of φ is φ2 = −ΔR × 2ψ¯1 P− ψ0 + 2ψ¯0 P− ψ1 (24) + m(φ∗1 φ0 + φ∗0 φ1 + φ1 φ∗0 + φ0 φ∗1 + φ1 φ0 + φ0 φ1 ) + (φ0 φ0 φ∗0 + φ∗0 φ0 φ0 )

(25) (26)

Inserting (22) and (23) in (24) and (25) and contracting the free fields, one can write φ2 in the form

ˆ R (k) Σ(k)φˆ0 (k) + Σ (k)φˆ∗ (k) + n.o. , φˆ2 (k) = (2π)2 Δ 0

cf. (7). For the computation of the graphs involving fermions, we need the formulae11 ˆ R (k) , SˆR (k) = (−k/ − m)Δ ¯R (k) = (k/ − m)Δ ˆ R (k) , Sˆ ˆ 1 ˆ + (k) . ψ¯α (k)ψˆβ (p) = (2π)2 δ(k + p)(−k/ + m)βα Δ 2 The φ4 tadpole is obtained from the term (26) of φ2 . We find the quadratically divergent contribution −2 2 ˆ + (l)(1 + eikσl ) . 4 Σφ −tp (k) = −2(2π) λ d4 lΔ The φ3 tadpole is obtained from the term (25) by contracting the φ0 s in φ1 or φ∗1 among themselves. Due to the retarded propagator with zero momentum connecting the loop with the line, the mass appearing in the interaction term cancels and we get Σφ3 −tp (k) = 8(2π)−2 λ2

ˆ + (l) . d4 l Δ

Note that no twisting factor appears. The φ3 fish graph is obtained from the term (25) by contracting a φ0 in φ1 or φ∗1 with the outer φ∗0 (f ). We get

ˆ + (l)(1 + eikσl ) Δ ˆ R (k − l) + Δ ˆ R (k + l) . Σφ3 −fish (k) = 3m2 λ2 d4 lΔ 11 The

factor 1/2 in the last line is due to the Majorana nature of the fermions.

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The Yukawa tadpole is obtained from (25) by contracting the fermions in φ1 or φ∗1 . Since the trace of a single γ-matrix vanishes we only get a supplementary factor 4m and thus −2 2 ˆ + (l) . ΣYuk (k) = −8(2π) λ d4 l Δ The fermion fish graph is obtained from the term (24). The relevant part of φ2 , i.e., the part involving φ0 , is ˆ R (k) d4 ld4 l cos lσl φˆ2 (k) = −4Δ 2 i ˆ ˆ ¯ ˆ × ψ0 (k − l)P− SR (l)P+ ψ0 (l − l )φˆ0 (l )e− 2 kσl ¯0 (l − l )P+ Sˆ¯R (l)P− ψˆ0 (k − l)φˆ0 (l )e− 2i lσk . +ψˆ Contraction of the fermion fields now yields lσk 2ˆ ˆ −2(2π) ΔR (k)φ0 (k) d4 l cos 2

ˆ R (l)Δ ˆ + (k − l)e− 2i kσl × tr P− (−/l − m)P+ (k/ − /l − m) Δ

ˆ R (l)Δ ˆ + (−k + l)e− 2i lσk + tr P+ (/l − m)P− (−k/ + /l − m) Δ ˆ R (k)φˆ0 (k) d4 l cos lσk = −2(2π)2 Δ 2

ˆ R (k − l)Δ ˆ + (l)e− 2i lσk × tr P− (/l − k/ − m)P+ (/l − m) Δ

ˆ R (k + l)Δ ˆ + (l)e− 2i lσk . + tr P+ (k/ + /l − m)P− (/l − m) Δ With the usual γ matrix algebra, we get 2 ˆ + (l)(1 + eikσl ) Σψ−fish (k) = 2λ d4 l Δ

ˆ R (k − l) − (k + l) · lΔ ˆ R (k + l) . × (k − l) · lΔ Now we collect all our terms. The Yukawa tadpole and the φ3 tadpole cancel (this has to be so in order to have a vanishing VEV of φ1 ). Using ˆ + (l) = 0 , (l2 − m2 )Δ

ˆ A (l) = −(2π)−2 , (l2 − m2 )Δ

the combination of the other terms gives

2

2 2 ˆ + (l)(1 + eikσl ) Δ ˆ R (k − l) + Δ ˆ R (k + l) . Σ(k) = λ k + m d4 l Δ Apart from the prefactor (k 2 +m2 ), this is exactly the expression we already found for the φ3 -model. We remark that for the self-energy of the fermion, one obtains the same result.

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The prefactor is to be expected: Assuming that the non-renormalization the¯ θ2 θ¯2 -term gets renormalized. From the orem still holds, we know that only the ΦΦ| free equations of motion (1 + δZ)F − mφ∗ = 0 ,

(1 + δZ)φ + mF ∗ = 0

we get, at first order in δZ, ( + m2 )φ = −δZ( − m2 )φ . Note that in our terminology, this corresponds to both a field strength and a mass renormalization. Explicitly, we have, after subtracting the planar part, δm2 (s) = −2m2 Σnp (m2 , s) , δZ(s) = −Σnp (m2 , s) − 2m2

(27) ∂ Σnp (m2 , s) . ∂k 2

(28)

Here we used the Σnp from the previous section, cf. equation (21). From (27) we conclude that for σ = σ0 , m = 10−17 λ−1 nc , λ = 1 the distortion of the group velocity is twice as strong as in the φ3 -model. Identifying φ with the Higgs field, an effect of this magnitude might be measurable at the next generation of particle colliders. As was already discussed in the previous section, the second term in (28) is effectively constant for realistic momenta. The first term has already been plotted in Fig. 1, apart from the sign. As discussed in Remark 2.4, a momentum-dependent field strength renormalization leads to a nonlocal smearing. In order to estimate its strength, one has to compute the Fourier transform of Σnp . In [40], such a calculation is performed in the setting of noncommutative supersymmetric electrodynamics. Note that the mass and field strength renormalizations for the fermion component are exactly the same.

5. Calculation in the sense of oscillatory integrals The aim of this section is to show that (20) is well-defined in the sense of oscillatory integrals, and that a calculation is this sense yields the same result as the formal calculation done in Section 3.1. We use the theory of oscillatory integrals as given in [33]. We first state the main definitions and results. Let Ω be an open set in Rs . Definition 5.1. A phase function on Ω×Rt is a continuous function φ : Ω×Rt → R with 1. ∀λ ≥ 0, (k, l) ∈ Ω × Rt : φ(k, λl) = λφ(k, l), 2. φ is C ∞ on Ω × (Rt \{0}), 3. (∇k φ, ∇l φ) = (0, 0) on Ω × (Rt \{0}).

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Definition 5.2. A C ∞ function a : Ω × Rt → C is called symbol of order r ∈ R on Ω × Rt if ∀K ⊂ Ω compact and for all multiindices α, β the seminorms aK,α,β =

sup k∈K,l∈Rt

(1 + |l|)|β|−r |Dkα Dlβ a(k, l)|

are finite. The set of all such symbols with topology given by the seminorms will be denoted by Sym(Ω, t, r). A function a : Ω × Rt → C is called asymptotic symbol, if it can be written as a = a1 + a2 with a1 ∈ Sym(Ω, t, r) and a2 having compact support in l and the map k → a2 (k, · ) is C ∞ as a map from Ω to L∞ (Rt ). If r < r then Sym(Ω, t, r) ⊂ Sym(Ω, t, r ) and the C ∞ functions of compact support are dense in Sym(Ω, t, r) in the topology of Sym(Ω, t, r ). For a1 ∈ Sym(Ω, t, r1 ) and a2 ∈ Sym(Ω, t, r2 ) the product a1 · a2 is in Sym(Ω, t, r1 + r2 ) and similar for asymptotic symbols. Now we want to give a natural extension to expressions like dt l a(k, l)eiφ(k,l) if the integral is not absolutely convergent: Theorem 5.3. Let φ be a phase function. We can associate with φ a linear map from the asymptotic symbols to D (Ω) denoted by Tφ (a) and uniquely determined by: 1. If a has compact support in l then Tφ (a)(k) = dt l a(k, l)eiφ(k,l) and is a C ∞ function of k. 2. The restriction of Tφ to Sym(Ω, t, r) is a continuous function from Sym(Ω, t, r) to D (Ω). Furthermore, one can show that the singular support of Tφ (a) is contained in the set k|∃l ∈ Rt \{0} with ∇l φ(k, l) = 0 . (29) Remark 5.4. It is easy to see that the notion of asymptotic symbols can be generalized further. The function a could be split even further into a = a1 +a2 +a3 +. . .. For the additional terms, k → ai (k, · ) should again be a C ∞ map, having compact support in l, into some suitable space of functions or distributions. Example for such spaces would be L∞ (Rt ), which was already used for the asymptotic symbols, t C ∞ around l = 0.12 The important point is that or the elements ofs E (R ) which are iφ(k,l) the integrals d k f (k)ai (k, l)e should each be well defined for f ∈ D(Ω), one of these in the sense of oscillatory integrals, and their sum independent of the splitting. So one could even allow for some k → ai (k, · ) to be distributions instead of C ∞ maps. This could, of course, increase the singular support beyond (29). In our concrete case (20), we choose Ω to be an open neighbourhood of the mass shell m such that for k ∈ Ω we have (k ± l+ )2 = m2 . For example 12 As the phase function does not have to be smooth in l = 0, a (k. · ) should, e.g., not contain i derivatives of the δ function at that point.

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Ω = {k| m 2
0. Then only Δ and at the singularity we have k0 − ωl > 0. Thus, we may simply add ±i to the denominator of the first fraction in (32). Of course one then has to assume that f has compact support in {k ∈ R4 |k 2 > 4m2 , k0 > 0}. One can then proceed as above and obtains (21), but with (k 2 − 4ωl2 ± i ) in the denominator. Using 1 1 x±i = P x − iπδ(x), this can be split into real and imaginary part. The imaginary part resembles the usual imaginary parts for forward/backward scattering. ˆ R (k ± l+ ) is not compact. For spacelike k, the singular support of l → Δ ˆ Consider, e.g., k = (0, 0, 0, kz ). Then ΔR (k − l+ ) is singular on the hyperplane l3 = 2kz . Thus, it is not possible to use the framework indicated in Remark 5.4. One has to extend the framework further in order to accommodate for symbols whose singularities are not compactly supported. There are two natural Ans¨ atze for such an extension: 1. The distributions a could be approximated by a sequence of symbols (an )n∈N . For each an the oscillatory integral is well defined. The oscillatory integral for a can then be achieved if one calculates the limit n → ∞ after integrating, if this is well defined and to a large extent independent of the choice of the sequence. 2. One could see the relation ds kdt l f (k)gn (l)a(k, l)eiφ(k,l) (34) ds kdt l f (k)a(k, l)eiφ(k,l) = lim n→∞

for a sequence gn of symbols with compact support and approaching 1, as a definition. The right hand side of (34), with finite n, is even defined for a being some distribution. If the limit exists and is independent of the choice of the sequence gn out of some large class of sequences, this would be a reasonable extension. We would also like to mention the approach followed in [40]: There, the nonplanar loop integral is interpreted as a function F (k, y) of two independent variables k and y, where the twisting factor is written as e−iyl+ . One can show that the integral is a well-defined tempered distribution in R8 . The question is then if it is possible to restrict y to kσ. Whether the loop integral is well-defined is then a question that can be answered by computing F (k, y). The problem is that it is rather difficult to perform such a calculation analytically. Remark 5.6. The nonplanar loop integrals that appear in the setting of the modified Feynman rules can also be treated rigorously in the sense of oscillatory integrals. Since one is working in the Euclidean metric there, the symbols can not

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become singular, so that there are no problems for spacelike external momenta. However, as already mentioned in the introduction, it is not clear whether there is any relation between the results for Euclidean and Minkowski metric.

6. Summary and Outlook We discussed dispersion relations in the Yang–Feldman formalism at the one-loop level and computed them in the noncommutative φ3 and Wess–Zumino model. It turned out that the distortions of the dispersion relation were moderate for parameters typically expected for the Higgs field. We also showed that the local SUSY current is not conserved in the noncommutative Wess–Zumino model. A shortcoming of the present work is of course the lack of a systematic treatment of renormalizability. In the case of the noncommutative Euclidean space, it is usually argued that the IR-divergence induced by the UV-IR mixing can at most be of the same degree as the underlying UV-divergence, i.e., logarithmic in the two cases studied here. Then the integration over a non-planar subgraph poses no problem. However, in the present situation of the noncommutative Minkowski space we have the difficulties mentioned at the end of Section 5. To solve these, an extension of the mathematical framework of oscillatory integrals is needed.

Acknowledgements We would like to thank Klaus Fredenhagen for valuable comments and discussions. Financial support from the Graduiertenkolleg “Zuk¨ unftige Entwicklungen in der Teilchenphysik” is gratefully acknowledged. Part of this work was done while J. Zahn visited the Dipartimento di Matematica of the Universit` a di Roma “La Sapienza” with a grant of the research training network “Quantum Spaces – Noncommutative Geometry”. It is a pleasure to thank Sergio Doplicher for kind hospitality.

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Claus D¨ oscher and Jochen Zahn II. Institut f¨ ur Theoretische Physik Universit¨ at Hamburg Luruper Chaussee 149 D-22761 Hamburg Germany e-mail: [email protected] [email protected] Communicated by Raimar Wulkenhaar. Submitted: October 13, 2007. Accepted: August 11, 2008.




The Effect of Time-Dependent Coupling on Non-Equilibrium Steady States Horia D. Cornean, Hagen Neidhardt and Valentin A. Zagrebnov Abstract. Consider (for simplicity) two one-dimensional semi-infinite leads coupled to a quantum well via time dependent point interactions. In the remote past the system is decoupled, and each of its components is at thermal equilibrium. In the remote future the system is fully coupled. We define and compute the non equilibrium steady state (NESS) generated by this evolution. We show that when restricted to the subspace of absolute continuity of the fully coupled system, the state does not depend at all on the switching. Moreover, we show that the stationary charge current has the same invariant property, and derive the Landau–Lifschitz and Landauer–B¨ uttiker formulas.

1. Introduction The aim of this paper is to construct and study non equilibrium steady states for systems containing quantum wells, and to describe the quantum transport of electrons through them. Even though our results can be generalized to higher dimensions, we choose for the moment to work in a (quasi) one dimensional setting; let us describe it in some more detail. A quantum well consists of potential barriers which are supposed to confine particles. On both sides of the barriers are reservoirs of electrons. Carriers can pass through the barriers by tunneling. We are interested in the carrier transport through the barriers, as well as in the carrier distribution between these barriers. Models of such type are very often used to describe processes going on in nanoelectronic devices: quantum well lasers, resonant tunneling diodes, and nanotransistors, see [34]. The quasi one-dimensional geometry assumes that the carriers can freely move in the plane orthogonal to the transport axis, but these degrees of freedom are integrated out. Thus, separating variables we are dealing with an essentially one-dimensional physical system. To describe such a system we consider the transport model of a single band in a given spatially varying potential v, under the

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H. D. Cornean et al.


assumption that v and all other possible parameters of the model are constant outside a fixed interval (a, b), see [17, 18, 22]. odinger More precisely, in the Hilbert space H := L2 (R) we consider the Schr¨ operator 1 d 1 d f (x) + V (x)f (x) , x ∈ R , (1.1) (Hf )(x) := − 2 dx M (x) dx with domain 1 Dom (H) := f ∈ W 1,2 (R) : f ∈ W 1,2 (R) . (1.2) M It is assumed that the effective mass M (x) and the real potential V (x) admit decompositions of the form ⎧ ⎪ x ∈ (−∞, a] ⎨ma , (1.3) M (x) := m(x) x ∈ (a, b) ⎪ ⎩ x ∈ [b, ∞) mb where 0 < ma , mb < ∞, m(x) > 0, x ∈ (a, b), (m + 1/m) ∈ L∞ ((a, b)), and ⎧ ⎪ x ∈ (−∞, a] ⎨va (1.4) V (x) := v(x) x ∈ (a, b) , va ≥ vb , ⎪ ⎩ x ∈ [b, ∞) vb with va , vb ∈ R, v ∈ L∞ ((a, b)). The quantum well is identified with the interval (a, b), (or physically, with the three-dimensional region (a, b) × R2 ). The regions (−∞, a) and (b, ∞) (physically they correspond to (−∞, a) × R2 and (b, ∞) × R2 ), are the reservoirs. Schr¨ odinger operators with step-like potentials were firstly considered by Buslaev and Fomin in [9]. For that reason we call them Buslaev-Fomin operators. The inverse scattering problem for such Buslaev-Fomin operators was subsequently investigated in [1–3, 11, 12, 20]. For the first time these kind of operators were used by P¨ otz [30], in order to describe rigorously the quantum transport in mesoscopic systems. In [6], the Buslaev-Fomin operator was an important ingredient for a self-consistent quantum transmitting Schr¨ odinger-Poisson system, which was used to describe quantum transport in tunneling diodes. In a further step, this was extended to a socalled hybrid model which consists of a classical drift-diffusion part and a quantum transmitting Schr¨ odinger-Poisson part, see [7]. Hybrid models are effective tools of describing and calculating nanostructures like tunneling diodes, see [5]. Notice that to obtain a self-consistent description of carrier transport through quantum wells, one has to know the carrier distribution between the barriers in order to insert it into the Poisson equation, which serves to determine the corresponding electric field. One finds this manner of description of semiconductor devices in [19,24,31]. The important ingredient of this approach is a relation which assigns to each real electric potential v ∈ L∞ ((a, b)) a function of carrier density

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u ∈ L1 ((a, b)). The corresponding (nonlinear ) operator is called the carrier-density operator and we denoted it by N ( · ) : L∞ ((a, b)) −→ L1 ((a, b)),

N (v) = u .

In fact, the problem of definition of carrier-density operators in our case can be reduced to the problem of determining an appropriate density operator , which uniquely defines the corresponding states. To this end, notice that since we are going to mimic the transport in a manyelectron non-interacting system, the corresponding fermion states are quasi-free, see e.g [8]. In this case we have to associate with the one-particle system {H, H}, see (1.1), the gauge-invariant Fermi-algebra of the Canonical Anticommutation Relations CAR(H) generated by the creation-annihilation operators {c∗ (f ), c(f )}f ∈H in the fermionic Fock space Ff (H). Recall that if ω is a state on CAR(H), then there exist a bounded self-adjoint operator on H (called density operator) which yields (1.5) ω c∗ (f1 )c(f2 ) = (f2 , f1 )H , f1,2 ∈ H and 0 ≤ ≤ I . The converse is also true, therefore the state ω is called the gauge-invariant quasifree state associated with the density operator . It is known that dynamics τt , implemented by the one-particle Hamiltonian H: ωt := ω ◦ τt , −itH

(1.6)

itH

where t = e e , leaves these class of states invariant, i.e. ωt is gaugeinvariant and quasi-free. Remark 1.1. Hence, for the perfect Fermi-gas the quasi-free states are uniquely defined by the density operator (1.5). If ω =: ω(β,μ) is the grand-canonical equilibrium state for the inverse temperature β > 0 and the chemical potential μ ∈ R, then ω(β,μ) is a unique β-KMS state on CAR(H). This implies that the operator (β, μ) for the perfect fermions coincides with Fermi-density operator defined by the one-particle Hamiltonian H: −1 , (1.7) (β, μ) ≡ fβ (H − μ) := κ eβ(H−μ) + I where κ = 1 if we omit the spin degeneration and κ = 2 if not. These remarks motivate the following notations and definitions that we use below: Definition 1.2. For a quasi-free state we call a bounded non-negative operator in L2 (R) a density operator if it satisfies (1.5) and the product M (χ(a,b) ) is a traceclass operator on L2 (a, b), where M (χ(a,b) ) is the multiplication operator induced by the characteristic function χ(a,b) of the interval (a, b). Definition 1.3. A density operator (or quasi-free state) is called a steady state for H, if commutes with H, i.e., belongs to the commutant of the W ∗ -algebra generated by the spectral measure EH ( · ) of H. We call a steady state an equilibrium state if it belongs to the bicommutant of this algebra, cf. (1.7).

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To give a description of all possible steady states, one has to introduce the spectral representation of H. Taking into account results of [6], it turns out that the operator H is unitarily equivalent to the multiplication M induced by the independent variable λ in the direct integral L2 (R, h(λ), ν),

C, λ ∈ (−∞, va ] h(λ) := , (1.8) C2 , λ ∈ (va , ∞) and (with the usual abuse of notation) dν(λ) =

N

δ(λ − λj )dλ + χ[vb ,∞) (λ)dλ ,

λ ∈ R,

j=1

where it is assumed va ≥ vb , and {λj }N j=1 denote the finite number of simple eigenvalues of H which are all situated below the threshold vb . We note that 2 2 2 . L2 R, h(λ), ν ⊕N j=1 C ⊕ L [vb , va ], C ⊕ L (va , ∞), C The unitary operator Φ : L2 (R) −→ L2 (R, h(λ), ν) establishing the unitary equivalence of H and M is called the generalized Fourier transform. If is a steady state for H, then there exists a ν-measurable function R λ → ρ˜(λ) ∈ B h(λ) of non-negative bounded operators in h(λ) such that ν −supλ∈R ˜ ρ(λ) B(h(λ)) < ∞ and is unitarily equivalent to the multiplication operator M (˜ ρ) induced by ρ˜ via the generalized Fourier transform = Φ−1 M (˜ ρ)Φ .

(1.9)

The measurable family {˜ ρ(λ)}λ∈R is uniquely determined by the steady state up to a ν-zero set and is called the distribution function of the steady state. In other words, there is an one-to-one correspondence between the set of steady states and the set of distribution functions. When ρ is an equilibrium state, then ρ˜(λ) must be proportional to the identity operator in h(λ), hence ρ must be a function of H. Let us note that the same distribution function can produce quite different steady states in L2 (R). This is due to the fact that the generalized Fourier transform strongly dependents on H, in particular, on the potential v. Having a steady state for H one defines the carrier density in accordance with [6] as the Radon-Nikodym derivative of the Lebesgue continuous measure E(ω) E(ω) := Tr M (χω ) where ω is a Borel subset of (a, b). The quantity E(ω) can be regarded as the expectation value that the carriers are contained in ω. Therefore the carrier density u is defined by uρ (x) :=

Tr(M (χdx )) E(dx) = , dx dx

x ∈ (a, b) .

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The carrier density operator Nρ ( · ) : L∞ ((a, b)) −→ L1 ((a, b)) is now defined as Nρ (v) := uρ (x)

(1.10)

where v ∈ L∞ ((a, b)) is the potential of the operator H. The steady state is given by (1.9). Therefore the self-consistent description of the carrier transport through quantum wells is obtained if there is a way to determine physically relevant distribution functions ρ˜. One goal of this paper is to propose a time-dependent procedure allowing to determine those functions. 1.1. The strategy Let us describe the strategy. We start with a completely decoupled system which consists of three subsystems living in the Hilbert spaces Ha := L2 ((−∞, a]) ,

HI := L2 (I) ,

Hb := L2 ([b, ∞))

(1.11)

where I = (a, b). We note that H = Ha ⊕ HI ⊕ Hb .

(1.12)

With Ha we associate the Hamiltonian Ha 1 d2 (Ha f )(x) := − f (x) + va f (x) , 2ma dx2

f ∈ Dom (Ha ) := f ∈ W 2,2 ((−∞, a)) : f (a) = 0

(1.13) (1.14)

with HI the Hamiltonian HI , 1 d 1 d f (x) + v(x)f (x) , (HI f )(x) := − 2 dx m(x) dx 1 f ∈ W 1,2 (I) f ∈ Dom (HI ) := f ∈ W 1,2 (I) : m f (a) = f (b) = 0

(1.15) (1.16)

and with Hb the Hamiltonian Hb , 1 d2 (Hb f )(x) := − f (x) + vb f (x) , 2mb dx2

f ∈ Dom (Hb ) := f ∈ W 2,2 ((b, ∞)) : f (b) = 0 .

(1.17) (1.18)

In H we set (1.19) HD := Ha ⊕ HI ⊕ Hb where the sub-index “D indicates Dirichlet boundary conditions. The quantum subsystems {Ha , Ha } and {Hb , Hb } are called left- and right-hand reservoirs. The middle system {HI , HI } is identified with a closed quantum well. We assume that all three subsystems are at thermal equilibrium; according to Definition 1.3, the corresponding sub-states must be functions of their corresponding sub-Hamiltonians. The total state is the direct sum of the three sub-states. One example borrowed from the physical literature [17, 18], where the quasione-dimensional features of the problem along a given direction (for simplicity, the x-direction) is taken into account as follows: Assume the same temperature T

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in the three-dimensional sample. The quasi-free equilibrium sub-states a , I and b are spatially separated along the x-direction and are Fermi-densities a := fa (Ha − μa ) ,

I := fI (HI − μI ) ,

b := fb (Hb − μb )

(1.20)

which one obtains from (1.7) by integration over momenta in y and z directions [17]: fa (λ) := ca ln(1 + e−βλ ) ,

fI (λ) := cI ln(1 + e−βλ ) ,

fb (λ) := cb ln(1 + e−βλ ) ,

λ ∈ R, β := 1/(kT ), k is the Boltzmann constant, μa and μb are the chemical potentials of left- and right-hand reservoirs and μI the chemical potential of the quantum well. The constants ca , cI and cb are given by ca :=

m⊥ a , πβ

cI :=

m⊥ I , πβ

cb :=

m⊥ b πβ

(1.21)

⊥ ⊥ where m⊥ a , mI and mb are the electronic effective masses in the transversal direction appearing after two-dimensional integration mentioned above (see [13, 17, 18] for more details). We set (1.22) D := a ⊕ I ⊕ b . For the whole system {H, HD } the state D is a steady state because D commutes with HD (see Definition 1.3). In general, the state D cannot be represented as a function of HD which is characteristic for equilibrium states, but it is the direct sum of equilibrium sub-states. In any case, D is a special non-equilibrium steady state (NESS) for the system {H, HD }. Now here comes the main question: can we construct a NESS for {H, H} starting from D ? Let us assume that at t = −∞ the quantum system {H, HD } is described by the NESS D . Then we connect in a time dependent manner the left- and right-hand reservoirs to the closed quantum well {HI , HI }. We assume that the connection process is described by the time-dependent Hamiltonian

Hα (t) := H + e−αt δ(x − a) + e−αt δ(x − b) , The operator Hα (t) is defined by 1 d 1 d f (x) + V (x)f (x) , Hα (t)f (x) := − 2 dx M (x) dx

t ∈ R,

α > 0.

f ∈ Dom Hα (t) ,

where the domain Dom (Hα (t)) is given by Dom Hα (t) ⎧ ⎫ 1 1,2 (R) ⎨ ⎬ Mf ∈ W 1 1 f )(a + 0) − ( 2M f )(a − 0) = e−αt f (a) := f ∈ W 1,2 (R) : ( 2M . ⎩ ⎭ 1 1 ( 2M f )(b + 0) − ( 2M f )(b − 0) = e−αt f (b)

(1.23)

(1.24)

(1.25)

After a rather standard analysis, one can prove the following convergence in the norm resolvent sense: −1 = (HD − z)−1 (1.26) n − lim Hα (t) − z t→−∞

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and n − lim

t→+∞

Hα (t) − z

−1

= (H − z)−1 ,

67

(1.27)

z ∈ C \ R. Then we consider the quantum Liouville equation (details about the various topologies will follow later): ∂ α (t) = Hα (t), α (t) , ∂t for a fixed α > 0 satisfying the initial condition i

t ∈ R,

(1.28)

s- lim α (t) = D . t→−∞

Having found a solution α (t) we are interested in the ergodic limit 1 T α (t)dt . α = lim T →+∞ T 0 If we can verify that the limit α exists and commutes with H, then α is regarded as the desired NESS of the fully coupled system {H, H}. Inserting α into the definition of the carrier density operator Nρα we complete the definition of the carrier density operator. Finally, the steady state a allows to determine the corresponding distribution function {˜ ρα (λ)}λ∈R . 1.2. Outline of results The precise formulation of our main result can be found in Theorem 3.6, and here we only describe its main features in words. We need to introduce the incoming wave operator W− := s- lim eitH e−itHD P ac (HD ) t→−∞

(1.29)

where P ac (HD ) is the projection on the absolutely continuous subspace Hac (HD ) of HD . We note that Hac (HD ) = L2 ((−∞, a]) ⊕ L2 ([b, ∞)). The wave operator exists and is complete, that is, W− is an isometric operator acting from Hac (HD ) onto Hac (H) where Hac (H) is the absolutely continuous subspace of H (the range of P ac (H)). One not so surprising result is that α exists for all α > 0. In fact, if we restrict ourselves to the subspace Hac (H), then we do not need to take the ergodic limit, since the usual strong limit exist. The surprising fact is that s- lim α (t)P ac (H) = α P ac (H) = W− ρD W−∗ P ac (H) , t→∞

(1.30)

which is independent of α. The only α dependence can be found in α P d (H), where P d (H) is the projection on the subspace generated by the discrete eigenfunctions of H. But this part does not contribute to the stationary current as can be seen in Section 4. Here the ergodic limit is essential, because it kills off the oscillations produced by the interference between different eigenfunctions.

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Note that the case α = ∞ would describe the situation in which the coupling is suddenly made at t = 0 and then the system evolves freely with the dynamics generated by H (see [21], and the end of Section 3). The case α 0 would correspond to the adiabatic limit. Inspired by the physical literature which seems to claim that the adiabatic limit would take care of the above mentioned oscillations, we conjecture the following result for the transient current: Conjecture 1.4.

lim lim sup Tr α (t)P d (H)[H, χ] = 0 ,

α0

t→∞

where χ is any smoothed out characteristic function of one of the reservoirs. Before ending this introduction, let us comment on some other physical aspects related to quantum transport problems. Many physics papers are dealing with transient currents and not only with the steady ones. More precisely, they investigate non-stationary electronic transport in noninteracting nanostructures driven by a finite bias and time-dependent signals applied at their contacts to the leads, while they allow the carriers to self-interact inside the quantum well (see for example [25, 26] and references therein). A similar more abstract approach was used by Nier in [29]. An interesting open problem is to study the existence of NESS in the Cini (partition-free) approach [10,14,14–16]. Some nice results which are in the same spirit with ours have already been obtained in the physical literature [32, 33], even for systems which allow local self-interactions. Now let us describe the organization of our paper. Section 2 introduces all the necessary notation and presents an explicit description of a spectral representation of HD and H. Section 3 deals with the quantum Liouville equation, and contains the proof of our main result in Theorem 3.6. In Section 4 we define the stationary current and derive the Landau–Lifschitz and Landauer–B¨ uttiker formulas.

2. Technical preliminaries 2.1. The uncoupled system Let us start by describing the uncoupled system, and begin with the left reservoir. The spectrum of Ha is absolutely continuous and σ(Ha ) = σac (Ha ) = [va , ∞). The operator is simple. The generalized eigenfunctions ψa ( · , λ), λ ∈ [va , ∞), of Ha are given by ψa (x, λ) :=

sin(2ma qa (λ)(x − a)) , πqa (λ)

x ∈ (−∞, a] ,

λ ∈ [va , ∞)

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where

λ − va . 2ma The system of eigenfunctions {ψa ( · , λ)}λ∈[va ,∞) is orthonormal, that is, one has in distributional sense ∞ dx ψa (x, λ)ψa (x, μ) = δ(λ − μ) , λ, μ ∈ [va , ∞) . (2.1) qa (λ) :=

a

With the generalized eigenfunctions one associates the generalized Fourier transform Ψa : L2 ((−∞, a]) −→ L2 ([va , ∞)) given by a a (Ψa f )(λ) = f (x)ψa (x, λ)dx = f (x)ψa (x, λ)dx , −∞

−∞

f ∈ L2 ((−∞, a]). Using (2.1) a straightforward computation shows that the generalized Fourier transform is an isometry acting from L2 ((−∞, a]) onto L2 ([va , ∞)). 2 2 The inverse operator Ψ−1 a : L ([va , ∞) −→ L (−∞, a]) admits the representation ∞ ∞ sin(2ma qa (λ)(x − a)) (Ψ−1 f (λ)dλ , ψa (x, λ)f (λ)dλ = a f )(λ) = πqa (λ) va va f ∈ L2 ([va , ∞)). Since ψa ( · , λ) are generalized eigenfunctions of Ha one easily verifies that Ma = Ψa Ha Ψ−1 or Ha = Ψ−1 a a Ma Ψa where Ma is the multiplication operator induced by the independent variable λ in L2 ([va , ∞)) and defined by (Ma f )(λ) = λf (λ) ,

f ∈ Dom (Ma ) := f ∈ L2 ([va , ∞)) : λf (λ) ∈ L2 ([va , ∞)) . This shows that {L2 ([va , ∞)), Ma } is a spectral representation of Ha . For the equilibrium sub-state a = fa (Ha − μa ) one has the representation a = Ψ−1 a M fa ( · − μa ) Ψa where M (fa ( · − μa )) denotes the multiplication operator induced by the function fa ( · − μa ). Let us continue with the closed quantum well. The operator HI has purely discrete point spectrum {ξk }k∈N with an accumulation point at +∞. The eigenvalues are simple. The density matrix operator I = fI (HI − μI ) is trace class. One easily verifies that there is an isometric map ΨI : L2 ((a, b)) −→ L2 (R, C, νI ), ∞ dνI (λ) = k=1 δ(λ − ξk )dλ, such that {L2 (R, C, νI ), MI } becomes a spectral representation of HI where MI denotes the multiplication operator in L2 (R, C, νI ). Finally, the right-hand reservoir. The spectrum of Hb is absolutely continuous and σ(Hb ) = σac (Hb ) = [vb , ∞). The operator Hb is simple. The generalized eigenfunctions ψb ( · , λ), λ ∈ [vb , ∞) are given by ψb (x, λ) =

sin(2mb qb (λ)(x − b)) , πqb (λ)

x ∈ [b, ∞) ,

λ ∈ [vb , ∞)

70


where

qb (λ) =


λ − vb . 2mb

The generalized eigenfunctions {ψb ( · , λ)}λ∈[vb ,∞) perform an orthonormal system and define a generalized Fourier transform Ψb : L2 ([b, ∞)) −→ L2 ([vb , ∞)) by ∞ ∞ f (x)ψb (x, λ)dx = f (x)ψb (x, λ)dx , (Ψb f )(λ) := b

b

f ∈ L ([b, ∞)). The inverse Fourier transform Ψ−1 : L2 ([vb , ∞)) −→ L2 ([b, ∞)) b admits the representation ∞ ∞ sin(2mb qb (λ)(x − b)) −1 f (λ)dλ , (Ψb f )(x) = ψb (x, λ)f (λ)dλ = πqb (λ) vb vb 2

f ∈ L2 ([vb , ∞)). Denoting by Mb the multiplication operator induced by the independent variable λ in L2 ([vb , ∞)) we get Mb = Ψb Hb Ψ−1 b

or

Hb = Ψ−1 b Mb Ψb

which shows that {L2 ([vb , ∞)), Mb } is a spectral representation of Hb . The equilibrium sub-state b = fb (Hb − μb ) is unitarily equivalent to the multiplication operator M (fb ( · − μb )) induced by the function fb ( · − μb ) in L2 ([vb , ∞)), that is, b = Ψ−1 b M fb ( · − μb ) Ψb . 2.2. Spectral representation of the decoupled system A straightforward computation shows that the direct sum Ψ = Ψa ⊕ ΨI ⊕ Ψb defines an isometric map acting from L2 (R) onto L2 (R, h(λ), νD (λ)), dνD (λ) = ∞ k=1 δ(λ − ξk )dλ + χ[vb ,∞) (λ)dλ, such that HD becomes unitarily equivalent to the multiplication operator MD defined in L2 (R, h(λ), νD (λ)) (see (1.8)). Here we slightly change the definition of h(λ) such that it re-becomes C when λ hits an eigenvalue. This does not affect the absolutely continuous part. Hence {L2 (R, h(λ), νD (λ)), MD } is a spectral representation of HD . Under the ac = Ha ⊕ Hb of HD is unitarily equivamap Ψ the absolutely continuous part HD ac ac ), dνD (λ) = χ[vb ,∞) (λ)dλ. lent to the multiplication operator M in L2 (R, h(λ), νD 2 ac ac Therefore {L (R, h(λ), νD ), M } is a spectral representation of HD . 2 With respect to the spectral representation {L (R, h(λ), νD ), M } the distribution function {˜ ρD (λ)} of the steady state D is given by ⎧ ⎪ 0, λ ∈ R \ σ(HD ) ⎪ ⎪ ⎪ ⎪ (λ − μ ), λ ∈ σp (HD ) = σ(HI ) f ⎪ I I ⎨ ρ˜D (λ) := fb (λ − μb ), λ ∈ [vb , va ) \ σ(HI ) . ⎪ ⎪ ⎪ f (λ − μ ) 0 ⎪ b b ⎪ ⎪ , λ ∈ [va , ∞) \ σ(HI ) ⎩ 0 fa (λ − μa ) We note that M (˜ ρD ) = ΨD Ψ−1 .

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2.3. The fully coupled system The Hamiltonian H in (1.1) was investigated in detail in [6]. If va ≥ vb , then it turns out that the operator H has a finite simple point spectrum on (−∞, vb ), on [vb , va ) the spectrum is absolutely continuous and simple, and on [va , ∞) the spectrum is also absolutely continuous with multiplicity two. Denoting by {λp }N p=1 the eigenvalues on (−∞, vb ), we have a corresponding finite sequence of L2 -eigenfunctions {ψ(x, λj )}N j=1 . Moreover, one can construct a set of generalized eigenfunctions φa (x, λ), x ∈ R, λ ∈ [va , ∞), and φb (x, λ), x ∈ R, λ ∈ [vb , ∞) of H such that {φb ( · , λ)}λ∈[vb ,va ) and {φb ( · , λ), φa ( · , λ)}λ∈[va ,∞) generate a complete orthonormal systems of generalized eigenfunctions. More precisely: φa (x, λ)φa (x, μ)dx = δ(λ − μ) , λ, μ ∈ [va , ∞) R φb (x, λ)φb (x, μ)dx = δ(λ − μ) , λ, μ ∈ [vb , ∞) R φa (x, λ)φb (x, μ)dx = 0 , λ, μ ∈ [va , ∞) , R

see [6]. The existence of generalized eigenfunctions is shown by constructing solutions φa (x, λ) and φb (x, λ) of the ordinary differential equation −

1 d 1 d φp (x, λ) + v(x)φp (x, λ) = λφp (x, λ) , 2 dx m(x) dx

x ∈ R, λ ∈ [vb , ∞), p = a, b, obeying

ei2ma qa (λ)(x−a) + Saa (λ)e−i2ma qa (λ)(x−a) φa (x, λ) = Sba (λ)ei2mb qb (λ)(x−b) λ ∈ [va , ∞), and φb (x, λ) =

Sab (λ)e−i2ma qa (λ)(x−a) e−i2mb qb (λ)(x−b) + Sbb (λ)ei2mb qb (λ)(x−b)

x ∈ (−∞, a] x ∈ [b, ∞) , x ∈ (−∞, a] x ∈ [b, ∞) ,

λ ∈ [vb , ∞). The coefficients Saa (λ) and Sbb (λ) are called reflection coefficients while Sba (λ) and Sab (λ) are called transmission coefficients. The solutions φa (λ) and φb (λ) define the normalized generalized eigenfunctions of H by 1 φb (x, λ) , x ∈ R , λ ∈ [vb , ∞), φb (x, λ) := 4πqb (λ) 1 φa (x, λ) := φa (x, λ) , x ∈ R , λ ∈ [va , ∞) . 4πqa (λ) Having the existence of the generalized eigenfunctions one introduces the generalized Fourier transform Φ : L2 (R) −→ L2 (h(R, h(λ), ν) by λ)dx , f ∈ L2 (R) , λ ∈ σ(H) , (Φf )(λ) := f (x)φ(x, (2.2) R

72

where


⎧ φ(x, λj ) ⎪ ⎪ ⎪ ⎨φ (x, λ) λ) := b φ(x, ⎪ ⎪ φb (x, λ) ⎪ ⎩ φa (x, λ)


λ ∈ σp (H) , x ∈ R λ ∈ [vb , va ) , x ∈ R λ ∈ [vb , ∞),

(2.3)

x ∈ R,

see [6]. The inverse generalized Fourier transform Φ−1 : L2 (R, h(λ), ν) −→ L2 (R) is given by λ) g(λ), φ(x, dν(λ) , (2.4) (Φ−1 g) = h(λ) R

x ∈ R, g ∈ L2 (R, h(λ), ν) where · , · h(λ) is the scalar product in h(λ). Since Φ is an isometry from L2 (R) onto L2 (R, h(λ), ν) such that M = ΦHΦ−1 holds where M is the multiplication operator induced by the independent variable in L2 (R, h(λ), ν) one gets that {L2 (R, h(λ), ν), M } is a spectral representation of H. Under Φ the absolutely continuous part H ac becomes unitarily equivalent to the multiplication operator M in L2 (R, h(λ), ν ac ), dν ac = χ[vb ,∞) dλ. Hence ac {L2 (R, h(λ), ν ac ), M } is a spectral representation of H ac , as it was for HD . 2.4. The incoming wave operator We have already mentioned that W− as defined in (1.29) exists and is complete [35]. We will need in Section 4 the expression of the ”rotated” wave operator ΦW− Ψ−1 which acts from L2 (R, h(λ), ν ac ) onto itself. By direct (but tedious) computations − := ΦW− Ψ−1 acts as a multiplication operator, which means one can show that W − (λ)}λ∈R of isometries acting from h(λ) onto h(λ) such that there is a family {W that − (λ)f (λ) , f ∈ L2 R, h(λ), ν ac . − f )(λ) = W (W − (λ)}λ∈R is called the incoming wave matrix and can be explicitly The family {W calculated. One gets ⎧ ⎪ ⎨i λ ∈ [vb , va ] W− (λ) = (2.5) i 0 ⎪ λ ∈ (va , ∞) . ⎩ 0 −i Note that another possible approach to the spectral problem (and completely different) would be to construct generalized eigenfunctions for H out of those of HD by using the unitarity of W− between their subspaces of absolute continuity − (λ) and the formal intertwining identity φp ( · , λ) := W− ψp ( · , λ). In this case W would always equal the identity matrix.

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3. The quantum Liouville equation The time dependent operators Hα (t) from (1.23) are defined by the sesquilinear forms hα [t]( · , · ), hα [t](f, g)

f (x)g (x) + V (x)f (x)g(x) dx + e−αt f (a)g(a) + e−αt f (b)g(b) , (3.1) = R

f, g ∈ Dom (ha [t]) := W 1,2 (R), t ∈ R. Obviously, we have Hα (t) + τ ≥ I, τ :=

V L∞ (R) + 1. For each t ∈ R the operator Hα (t) can be regarded as a bounded operator acting from W 1,2 (R) into W −1,2 (R). Classical Sobolev embedding results ensure that (Hα (t) + τ )−1/2 maps L2 (R) into continuous functions, and it has an integral kernel G(x, x ; τ ) with the property that G( · , x ; τ ) ∈ L2 (R) for every fixed x . Let us introduce the operators Ba : L2 (R) −→ C and Bb : L2 (R) −→ C defined by: −1/2 (Ba f ) := Hα (t) + τ f (a) = G(a, x; τ )f (x)dx , f ∈ L2 (R) , (Ba∗ c)(x) = G(x, a; τ ) ,

R

c ∈ C,

(3.2) Ba∗ Ba

Bb∗ Bb

2

and are bounded in L (R) and and similarly for Bb . The operators correspond to the sesquilinear forms ! −1/2 " ba [t](f, g) := Hα (t) + τ f (a) (Hα (t) + τ )−1/2 g (a) , f, g ∈ Dom (ba [t]) = L2 (R), and ! −1/2 " bb [t](f, g) := Hα (t) + τ f (b) (Hα (t) + τ )−1/2 g (b) , f, g ∈ Dom (bb [t]) = L2 (R), respectively. We define the rank two operator B := Ba∗ Ba + Bb∗ Bb

(3.3)

= G( · , a; τ )G(a, · ; τ ) + G( · , b; τ )G(b, · ; τ ) : L2 (R) −→ L2 (R) . The resolvent (Hα (t) + τ )−1 admits the representation −1 = (H + τ )−1/2 (I + e−αt B)−1 (H + τ )−1/2 , Hα (t) + τ

t ∈ R,

α > 0 . (3.4)

3.1. The unitary evolution Let us consider a weakly differentiable map R t → u(t) ∈ W 1,2 (R). We are interested in the evolution equation ∂ (3.5) i u(t) = Hα (t)u(t) , t ∈ R , α > 0 . ∂t where Hα (t) is regarded as a bounded operator acting from W 1,2 (R) into W−1,2 (R). By Theorem 6.1 of [27] with evolution equation (3.5) one can associate a unique unitary solution operator or propagator {U (t, s)}(t,s)∈R×R leaving invariant

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the Hilbert space W 1,2 (R). By Theorem 8.1 of [23] we find that for g, h ∈ W 1,2 (R) the sesquilinear form (U (t, s)g, h) is continuously differentiable with respect t ∈ R and s ∈ R such that ∂ U (t, s)g, h = −i Hα (t)U (t, s)g, h , g, h ∈ W 1,2 (R) , (3.6) ∂t ∂ U (t, s)g, h = i Hα (s)g, U (s, t)h , g, h ∈ W 1,2 (R) . (3.7) ∂s 3.2. Quantum Liouville equation We note that α (t) := U (t, s)α (s)U (s, t) ,

t, s ∈ R ,

(3.8)

−1,2

1,2

(R) is differentiable and solves the quantum seen as a map from W (R) into W Liouville equation (1.28) satisfying the initial condition α (t)|t=s = α (s), provided α (s) leaves W 1,2 (R) invariant. Indeed, using (3.6) and (3.7) we find ∂ α (s)U (s, t)g, U (s, t)h = i U (s, t)Hα (t)g, α (s)U (s, t)h ∂t − i α (s)U (s, t)g, U (s, t)Hα (t)h = i Hα (t)g, α (t)h − i α (t)g, Hα (t)h , g, h ∈ W 1,2 (R), which yields i

∂ α (t)g, h = α (t)g, Hα (t)h − Hα (t)g, α (t)h , ∂t

g, h ∈ W 1,2 (R), t, s ∈ R. 3.3. Time dependent scattering We set U (t) := U (t, 0), t ∈ R and consider the wave operators Ω− := s- lim U (t)∗ e−itHD t→−∞

and Ω+ := s- lim U (t)∗ e−itH . t→+∞

Proposition 3.1. Let HD and Hα (t), t ∈ R, α > 0, be given by (1.11)–(1.19) and (1.24)–(1.25), respectively. Then the wave operator Ω− and the limit R− := s- lim U (t)∗ (HD + τ )−1 U (t)

(3.9)

Ran(Ω− )⊥ = Ker(R− ) .

(3.10)

t→−∞

exist. Moreover, Proof. We start with (3.9). Let us introduce the time-dependent identification operator −1 (HD + τ )−1 , t ∈ R . JD (t) := Hα (t) + τ

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We have d U (t)∗ JD (t)e−itHD f dt !

−1 " −itHD = iU (t)∗ (HD + τ )−1 − Hα (t) + τ e f + U (t)∗ J˙D (t)e−itH f

for f ∈ H where J˙D :=

d dt JD (t).

(3.11)

Hence we get

! U (s)∗ (HD + τ )−1 s −1 " −irHD e − Hα (r) + τ f dr (3.12) t + U (r)∗ J˙D (r)e−irHD f ds .

U (t)JD (t)e−itHD f − U (s)∗ JD (s)e−isHD f = i

t

0

Using (3.4) we find −1 = (H + τ )−1/2 QB (H + τ )−1/2 Hα (t) + τ αt −1/2 + eαt (H + τ )−1/2 Q⊥ + B)−1 Q⊥ B (e B (H + τ )

(3.13)

where QB is the orthogonal projection onto the subspace Ker(B). Note that Q⊥ B has rank 2. Taking into account (1.26) we get the representation (HD + τ )−1 = (H + τ )−1/2 QB (H + τ )−1/2 .

(3.14)

By (3.13) and (3.14) we obtain −1 (HD + τ )−1 − Hα (t) + τ αt −1/2 = −eαt (H + τ )−1/2 Q⊥ + B)−1 Q⊥ . B (e B (H + τ )

(3.15)

Since B is positive and invertible on Ker(B)⊥ we get the estimate # # −1 ⊥ #(HD + τ )−1 − Hα (t) + τ −1 # ≤ eαt Q⊥ QB , t ∈ R , α > 0 . BB

(3.16)

Using again (3.13) we have αt −1/2 J˙D (t) = αeαt (H + τ )−1/2 Q⊥ + B)−2 BQ⊥ (HD + τ )−1 . (3.17) B (e B (H + τ )

This gives the estimate −1 ⊥ QB .

J˙D (t) ≤ αeαt Q⊥ BB

(3.18)

Using (3.12), (3.16) and (3.18) we prove the existence of the limit $ − := s- lim U (t)∗ JD (t)e−itHD . Ω t→−∞

In fact, the convergence is in the operator norm: $ − − U (t)∗ JD (t)e−itHD = 0 . lim Ω

t→−∞

(3.19)

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Using the identity U (t)∗ JD (t)e−itHD − U (t)∗ e−itHD (HD + τ )−2 = U (t)∗ (Hα (t) + τ )−1 − (HD + τ )−1 e−itHD (HD + τ )−1 and (3.16) we get the estimate −1 ⊥ QB ,

U (t)∗ JD (t)e−itHD − U (t)∗ e−itHD (HD + τ )−2 ≤ eαt Q⊥ BB

which yields

$ − − U (t)∗ e−itHD (HD + τ )−2 = 0 , lim Ω

t→−∞

(3.20) (3.21)

$ − exists we get the existence of for all t ∈ R, α > 0. Since the wave operator Ω s- lim U (t)∗ e−itHD (HD + τ )−2 . t→−∞

Using that Ran ((HD + τ )−2 ) is dense in H, we prove the existence of Ω− . In particular, this proves that Ω− is isometric, i.e Ω∗− Ω− = I. Now let us prove that the operator in (3.9) exists. Note that the norm convergence in (3.19) yields the same property for adjoints: $ ∗− − eitHD JD (t)∗ U (t) = 0 . lim Ω

t→−∞

In particular

$ ∗− = s- lim eitHD JD (t)∗ U (t) . Ω t→−∞

In the quadratic form sense we get −1 −1 d d U (t)∗ Hα (t) + τ Hα (t) + τ U (t)f = U (t)∗ U (t)f dt dt f ∈ H, t ∈ R, α > 0. Hence −1 −1 U (t)∗ Hα (t) + τ U (t)f − U (s)∗ Hα (t) + τ U (s)f t d ∗ −1 (Hα (r) + τ ) = dr U (r) U (r)f dr s f ∈ H, t, s ∈ R, α > 0. By (3.4) we get −1 d Hα (t) + τ = αe−αt (H + τ )−1/2 (I + e−αt B)−2 B(H + τ )−1/2 dt which gives the estimate # # #d −1 # −1 ⊥ # # ≤ αeαt Q⊥ QB . (3.22) BB # dt Hα (t) + τ # Hence R− exists, and we even have convergence in operator norm: # # −1 lim #R− − U (t)∗ Hα (t) + τ U (t)∗ # = 0 . t→−∞

Taking into account the estimate (3.16) we find lim R− − U (t)∗ (HD + τ )−1 U (t)∗ = 0 .

t→−∞

(3.23)

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In particular we have # 2 # −2 − U (t)∗ Hα (t) + τ U (t)∗ # lim #R− t→−∞

2 = lim R− − U (t)∗ (HD + τ )−2 U (t)∗ = 0 . t→−∞

Using the identity ! −1 " −1 −2 Hα (t) + τ JD (t)∗ = (HD + τ )−1 − Hα (t) + τ + Hα (t) + τ and taking into account the estimate (3.16) we obtain 2 lim R− − U (t)∗ JD (t)∗ U (t) = 0 .

t→−∞

Hence we find $ ∗− = s- lim eitHD JD (t)∗ U (t) Ω

(3.24)

t→−∞

2 = s- lim eitHD U (t)U (t)∗ JD (t)∗ U (t) = s- lim eitHD U (t)R− t→−∞

t→−∞

itHD

U (t)f exist for elements

lim eitHD U (t)R− f = Ω∗− R− f

(3.25)

which shows in particular that the limit limt→−∞ e f ∈ Ran (R− ). More precisely: t→−∞

for all f . We are now ready to prove (3.10). Assume that f ⊥ Ran (Ω− ). Then using the definitions, the unitarity of U (t)∗ , and (3.23) we obtain: 0 = f, Ω− (HD + τ )−1 g = lim U (t)∗ (HD + τ )−1 U (t)f, U (t)∗ e−itHD g = (R− f, Ω− g) t→−∞

for g ∈ H. Hence f ⊥ Ran (Ω− ) implies R− f ⊥ Ran (Ω− ) = Ker(Ω∗− ). Thus Ω∗− R− f = 0. Using (3.25) we get 0 = lim eitHD U (t)R− f = R− f , t→−∞

thus f ∈ Ker(R− ). We have thus shown that Ran (Ω− )⊥ ⊂ Ker(R− ). Conversely, choose f ∈ Ker(R− ). We have (use (3.23)): f, Ω− (HD + τ )−1 g = lim f, U (t)∗ e−itHD (HD + τ )−1 g t→−∞ = lim U (t)∗ (HD + τ )−1 U (t)f, U (t)∗ e−itHD g t→−∞

= (R− f, Ω− g) = 0 , for all g. Thus Ω∗− f is orthogonal on a dense set (domain of HD ), thus equals zero. Therefore Ker(R− ) ⊂ Ran (Ω− )⊥ and (3.10) is proved. Remark 3.2. Note that Ω− is unitary if Ker(R− ) = {0}.

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Proposition 3.3. Let H and Hα (t), t ∈ R, α > 0, be given by (1.1)–(1.4) and (1.24)–(1.25), respectively. Then the wave operator Ω+ exists and is unitary. Proof. We introduce the identification operator −1 (H + τ )−1 , J(t) := Hα (t) + τ

t ∈ R.

In the quadratic form sense we get that d U (t)∗ J(t)e−itH f dt !

−1 " −itH −itH ˙ e f + U (t)∗ J(t)e f , (3.26) = iU (t)∗ (H + τ )−1 − Hα (t) + τ

t ∈ R, where ˙ := d J(t) . J(t) dt Taking into account (3.4) we find −1 = e−αt (H +τ )−1/2 (I +e−αt B)−1 B(H +τ )−1/2 , (3.27) (H +τ )−1 − Hα (t)+τ t ∈ R. Hence we have the estimate # # #(H + τ )−1 − Hα (t) + τ −1 # ≤ e−αt B ,

t ∈ R.

(3.28)

Moreover, we get ˙ = d Hα (t) + τ −1 (H + τ )−1 J(t) dr = αe−αt (H + τ )−1/2 (I + e−αt B)−2 B(H + τ )−3/2 which yields the estimate

# # ˙ # ≤ αe−αt B , #J(t)

(3.29)

t ∈ R.

Hence the strong limit $ + := s- lim U (t)∗ J(t)e−itH Ω t→+∞

exists. Moreover, the convergence is also true in operator norm: $ + − U (t)∗ J(t)e−itH = 0 . lim Ω

t→+∞

(3.30)

Using the identity U (t)∗ J(t)e−itH − U (t)∗ e−itH (H + τ )−2 ! " −1 = U (t)∗ Hα (t) + τ − (H + τ )−1 e−itH (H + τ )−1

(3.31)

and taking into account the estimate (3.28) we obtain $ + − U (t)∗ e−itH (H + τ )−2 = 0 . lim Ω

t→+∞

Hence Ω+ exists on a dense domain and is isometric, i.e. Ω∗+ Ω+ = I.

(3.32)

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Let us now prove that Ω+ is unitary. Since (3.30) holds also true for adjoints we find $ ∗ − eitH J(t)U (t) = 0 . lim Ω (3.33) + t→+∞

Hence, we have the representation $ ∗ = s- lim eitH J(t)U (t) . Ω +

(3.34)

t→+∞

Furthermore, in the quadratic form sense we have −1 −1 d d ∗ ∗ U (t) Hα (t) + τ Hα (t) + τ U (t) = U (t) U (t) dt dt = αe−αt U (t)∗ (H + τ )−1/2 (I + e−αt B)−2 B(H + τ )−1/2 U (t) . Hence we get

−1 −1 U (t)∗ Hα (t) + τ U (t) = Hα (0) + τ t +α e−αt U (t)∗ (H + τ )−1/2 (I 0 −αt

+e

(3.35)

B)−2 B(H + τ )−1/2 U (t)dt .

Using the estimate # # #U (t)∗ (H + τ )−1/2 (I + e−αt B)−2 B(H + τ )−1/2 U (t)# ≤ B ,

t ∈ R,

we find that the following weak integral exists and defines a bounded operator: ∞ α dt e−αt U (t)∗ (H + τ )−1/2 (I + e−αt B)−2 B(H + τ )−1/2 U (t) . (3.36) 0

Moreover, by the Cook argument it also implies the existence of the limit −1 U (t) . R+ := s- lim U (t)∗ Hα (t) + τ t→+∞

In fact, the convergence takes place in operator norm: # # −1 lim #R+ − U (t)∗ Hα (t) + τ U (t)# = 0 . t→+∞

Taking into account the estimate (3.28) we obtain lim R+ − U (t)∗ (H + τ )−1 U (t) = 0 .

t→+∞

which yields

2 − U (t)∗ J(t)U (t) = 0 . lim R+

t→+∞

By (3.33) we get

2 $ ∗+ − eitH U (t)R+

=0 lim Ω

t→+∞

which shows the existence of Ω∗+ f = s-limt→+∞ eitH U (t)f for f ∈ Ran (R+ ). Now in order to prove the unitarity of Ω+ it is enough to show that Ran (R+ ) is dense in H. Let us do that.

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From (3.35) we obtain −1 R+ = Hα (0) + τ ∞ +α dt e−αt U (t)∗ (H + τ )−1/2 (I + e−αt B)−2 B(H + τ )−1/2 U (t) , 0

which by the positivity of the integral it gives 0 ≤ (Hα (0) + τ )−1 ≤ R+ . Hence ! −1/2 " 1/2 Ker(R+ ) ⊆ Ker Hα (0) + τ , thus Ran (R+ )⊥ ⊆ Ran 1/2

!

Hα (0) + τ

−1/2 "⊥

= ∅.

1/2

2 Thus we get that Ran (R+ ) is dense in H which yields that Ran (R+ ) is dense in ∗ itH H. Therefore we have the representation Ω+ = s-limt→+∞ e U (t) which proves that Ω+ is unitary.

3.4. Time-dependent density matrix operator Now we are ready to write down a solution to our Liouville equation (1.28) which also obeys the initial condition at t = −∞. Let us introduce the notation: α (0) := Ω− D Ω∗− which defines a non-negative self-adjoint operator. Here D is given by (1.20)– (1.22). In accordance with (3.8), the time evolution of α (0) is given by α (t) = U (t)α (0)U (t)∗ = U (t)Ω− D Ω∗− U (t)∗ ,

t ∈ R,

(3.37)

where we have used the notation U (t) := U (t, 0) and the relation U (0, t) = U (t)∗ , t ∈ R. We now show that the initial condition is fulfilled. Proposition 3.4. Let HD and Hα (t), t ∈ R, α > 0, be given by (1.11)–(1.19) and (1.24)–(1.25), respectively. If D is a steady state for the system {H, HD } such that the operator $D := (HD + τ )4 D is bounded, then lim D − α (t) = 0 .

t→−∞

(3.38)

Proof. We write the identity: U (t)Ω− D Ω∗− U (t)∗ = U (t)Ω− (HD + τ )−2 $D (HD + τ )−2 Ω∗− U (t)∗ ,

(3.39)

t ∈ R. Taking into account (3.21) we find $ ∗ U (t)∗ . $ − $D Ω U (t)Ω− D Ω∗− U (t)∗ = U (t)Ω −

(3.40)

From (3.19) we get $ ∗ U (t)∗ − JD (t)e−itHD $D eitHD JD (t)∗ $ − $D Ω lim U (t)Ω −

t→−∞

$ ∗− U (t)∗ − JD (t)$ $ − $D Ω D JD (t)∗ = 0 . = lim U (t)Ω t→−∞

(3.41)

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Using (3.16) we get D JD (t)∗ − (HD + τ )−2 $D (HD + τ )−2 lim JD (t)$

t→−∞

= lim JD (t)$ D JD (t)∗ − D = 0 t→−∞

(3.42)

Taking into account (3.39)–(3.42) we prove (3.38). 3.5. Large time behavior on the space of absolute continuity We now are ready to prove the result announced in (1.30).

Proposition 3.5. Let H and Hα (t), t ∈ R, α > 0, be given by (1.1)–(1.4) and (1.24)–(1.25), respectively. Let W− be the incoming wave operator as defined in (1.29). If D is a steady state for the system {H, HD } such that the operator $D := (HD + τ )4 D is bounded, then s- lim α (t)P ac (H) = W− D W−∗ . t→+∞

Proof. Let us assume that the following three technical results hold true: s- lim U (t)∗ − eitH P ac (H) = 0 , t→+∞

(3.43)

(3.44)

(HD + τ )−2 (Ω∗− − I) is compact ,

(3.45)

s- lim (HD + τ )−2 (Ω∗− − I)eitH P ac (H) = 0 .

(3.46)

and t→+∞

We will first use these estimates in order to prove the proposition, and then we will give their own proof. We write the identity: U (t)α (0)U (t)∗ P ac (H) = U (t)Ω− (HD + τ )−2 $D (HD + τ )−2 Ω∗− U (t)∗ P ac (H) = U (t)Ω− (HD + τ )−2 $D (HD + τ )−2 Ω∗− U (t)∗ − eitH P ac (H) + U (t)Ω− (HD + τ )−2 $D (HD + τ )−2 (Ω∗− − I)eitH P ac (H) + U (t)Ω− (HD + τ )−2 eitHD $D (HD + τ )−2 e−itHD eitH P ac (H) . Taking into account (3.44)–(3.46), and using the completeness of W− which yields W−∗ = s-limt→−∞ eitHD e−itH P ac (H), we get: s- lim U (t)α (0)U (t)∗ P ac (H) t→+∞

= s- lim U (t)Ω− (HD + τ )−2 eitHD $D (HD + τ )−2 W−∗ . t→+∞

Since (Ω− − I)(HD + τ )−2 is also compact (its adjoint is compact, see (3.45)), we have: s- lim (Ω− − I)(HD + τ )−2 eitHD P ac (HD ) = 0 . t→+∞

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Thus: s- lim U (t)α (0)U (t)∗ P ac (H) t→+∞

= s- lim U (t)eitHD (HD + τ )−2 $D (HD + τ )−2 W−∗ t→+∞

= s- lim U (t)eitHD D W−∗ = s- lim U (t)eitH P ac (H)W− D W−∗ . t→+∞

t→+∞

Finally, we apply (3.44) once again, and (3.43) is proved. Now let us prove the three technical results announced in (3.44)–(3.46). We start with (3.44). We have the identity: U (t)∗ − eitH (H + τ )−2 = U (t)∗ (H + τ )−2 e−itH − (H + τ )−2 eitH . (3.47) Then by adding and subtracting several terms we can write another identity: U (t)∗ − eitH (H + τ )−2

$ + eitH = U (t)∗ (H + τ )−2 e−itH − Ω (3.48)

itH $ + − J(0) e g + J(0) − (H + τ )−2 eitH . (3.49) + Ω By (3.32) we get $ + = 0 . lim U (t)∗ (H + τ )−2 e−itH − Ω

t→+∞

(3.50)

which shows that (3.48) tends to zero as t → +∞. Next, from (3.4), (3.26) and (3.29) we get U (t)∗ J(t)e−itH − J(0) t =i ds e−αs U (s)∗ (H + τ )−1/2 (I + e−αt B)−1 B(H + τ )−1/2 e−isH 0 t +α ds e−αs U (s)∗ (H + τ )−1/2 (I + e−αs B)−2 B(H + τ )−3/2 e−isH , 0

which yields $ + − J(0) Ω ∞ =i ds e−αs U (s)∗ (H + τ )−1/2 (I + e−αs B)−1 B(H + τ )−1/2 e−isH 0 ∞ +α ds e−αs U (s)∗ (H + τ )−1/2 (I + e−αs B)−2 B(H + τ )−3/2 e−isH . 0

$ + − J(0) is a compact Since B is a compact (rank 2) operator, we get that Ω operator. This fact immediately implies (via the RAGE theorem): $ + − J(0) eitH P ac (H) = 0 . (3.51) s- lim Ω t→+∞

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Furthermore, we have the identity: " ! −1 J(0) − (H + τ )−2 = Hα (0) + τ − (H + τ )−1 (H + τ )−1 = −(H + τ )−1/2 (I + B)−1 B(H + τ )−3/2 , which gives that J(0) − (H + τ )−2 is compact. Thus: s- lim J(0) − (H + τ )−2 eitH P ac (H) = 0 . t→+∞

(3.52)

Taking into account (3.50), (3.51) and (3.52) we find s- lim U (t)∗ − eitH (H + τ )−2 P ac (H) = 0 t→+∞

which proves (3.44). Next we prove (3.45) (the estimate (3.46) is just an easy consequence of (3.45) $ − . From (3.12), via the RAGE theorem). Using (3.21) we have Ω− (HD + I)−2 = Ω (3.15) and (3.17) we obtain: JD (0) − U (t)∗ JD (t)e−itHD 0 αs −1/2 −isHD = −i ds eαs U (s)∗ (H + τ )−1/2 Q⊥ + B)−1 Q⊥ e B (e B (H + τ ) t

+α

0

ds eαs U (s)∗ (H + τ )−1/2 (eαs + B)−2 B(H + τ )−1/2 (HD + τ )−1 e−isHD

t

$ and together with the fact that Q⊥ B is a rank 2 operator we find that Ω− − JD (0) ∗ ∗ $ is compact. Hence Ω− − JD (0) is compact, too. Moreover, using (3.15) we get " ! −1 JD (0) − (HD + τ )−2 = Hα (0) + τ − (HD + τ )−1 (HD + τ )−1 αt −1 ⊥ = −eαt (H +τ )−1/2 Q⊥ QB (H +τ )−1/2 (HD +τ )−1 B (e +B)

which shows that JD (0) − (HD + τ )−2 is compact. Hence JD (0)∗ − (HD + τ )−2 is compact. Now use the identity: ∗ $ − − JD (0) + JD (0) − (HD + τ )−2 , (HD + τ )−2 (Ω∗− − I) = Ω which proves (3.45). Finally, the relation (3.46) follows from (3.45) and the RAGE theorem. 3.6. The main result We are now ready to rigorously formulate and prove our main result, announced in the introduction: Theorem 3.6. Let H and Hα (t), t ∈ R, α > 0, be given by (1.1)–(1.4) and (1.24)– (1.25), respectively. Let W− be the incoming wave operator from (1.29). Further, let EH ( · ) and {λj }N j=1 be the spectral measure and the eigenvalues of H. If D is

84



a steady state for the system {H, HD } such that the operator $D := (HD + τ )4 D is bounded, then the limit 1 T α := s- lim dt α (t) (3.53) T →+∞ T 0 =

N

EH ({λj })Sα D Sα∗ EH ({λj }) + W− D W−∗

j=1

exists and defines a steady state for the system {H, H} where Sα := Ω∗+ Ω− . Remark 3.7. We stress once again that only the part corresponding to the pure N ∗ point spectrum pα := j=1 EH ({Λj })Sα D Sα EH ({λj }) of the steady state α depends on the parameter α > 0 while the absolutely continuous part ac α := W− D W−∗ does not. Note that with respect to the decomposition H = Hp (H) ⊕ Hac (H), one has α = pα ⊕ ac α . We note further that the assumption $D := (HD + τ )4 D is bounded is satisfied in applications, cf. (1.20)–(1.22). Proof. By Proposition 3.5 we have 1 T s- lim dt α (t)P ac (H) = W− D W−∗ . T →+∞ T 0 In particular, this yields: 1 T →+∞ T

(3.54)

T

dt P s (H)α (t)P ac (H) = 0 ,

s- lim

0

where P s (H) is the projection onto the singular subspace of H. Now we are going to prove 1 T ac P (H)α (t)P s (H)dt = 0 . (3.55) s- lim T →+∞ T 0 By (3.32) we find $ + = 0 , lim U (t)Ω− D Ω∗− U (t)∗ (I + H)−2 e−itH − U (t)Ω− D Ω∗− Ω

t→∞

which yields $ + eitH = 0 . lim U (t)Ω− D Ω∗− U (t)∗ (I + H)−2 − U (t)Ω− D Ω∗− Ω

t→∞

Let λj be an eigenvalue of H with corresponding to an eigenfunction φj . Then $ + φj = 0 . lim U (t)Ω− D Ω∗− U (t)∗ (I + H)−2 φj − eitλj U (t)Ω− D Ω∗− Ω

t→∞

This and the unitarity of Ω∗+ give: lim α (t)(I + H)−2 φj − eitλj e−itH Ω∗+ Ω− D Ω∗− Ω+ (H + τ )−2 φj = 0

t→∞

or

lim α (t)φj − eitλj e−itH Ω∗+ Ω− D Ω∗− Ω+ φj = 0 .

t→∞

(3.56)

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Hence we have 1 T 1 T dt P ac (H)α (t)φj = dt eitλj e−itH P ac (H)Ω∗+ Ω− D Ω∗− Ω+ φj . T 0 T 0 We use the decomposition 1 T ac P (H)α (t)φj T 0 1 T = dt eitλj e−itH EH (|λ − λj | < )P ac (H)Ω∗+ Ω− D Ω∗− Ω+ φj T 0 1 T + dt eitλj e−itH EH (|λ − λj | ≥ )P ac (H)Ω∗+ Ω− D Ω∗− Ω+ φj . T 0 If is small enough, then EH (|λ − λj | < )P ac (H) = 0. This yields the estimate: # # #1 T # # # ac P (H)α (t)φj # # #T 0 # # 2# ≤ #(H − λj )−1 EH (|λ − λj | ≥ )P ac (H)Ω∗+ Ω− D Ω∗− Ω+ φj # T which immediately shows that 1 T s- lim dt P ac (H)α (t)φj = 0 , (3.57) T →∞ T 0 and (3.55) is proved. Next, from (3.56) we easily obtain: 1 T s- lim dt P s (H)α (t)φj = EH ({λj }) Sα D Sα∗ EH ({λj }) . T →∞ T 0

(3.58)

Now put together (3.54), (3.55), (3.57) and (3.58), and the proof of (3.53) is over. Now the operator α is non-negative, bounded, and commutes with H. Hence α is a steady state for {H, H}. Corollary 3.8. Let H and Hα (t), t ∈ R, α > 0, be given by (1.1)–(1.4) and (1.24)– (1.25), respectively. Then with respect to the spectral representation {L2 (R, h(λ), ν), M } of H the distribution function {˜ ρα (λ)}λ∈R of the steady state α is given by ⎧ ⎪ 0, λ ∈ R \ σ(H) ⎪ ⎪ ⎪ ⎪ ρ , λ = λj , j = 1, . . . , N ⎪ ⎨ α,j fb (λ − μb ), (3.59) ρ˜α (λ) := λ ∈ [vb , va ) ⎪ ⎪ ⎪ fb (λ − μb ) 0 ⎪ ⎪ ⎪ , λ ∈ [va , ∞) ⎩ 0 fa (λ − μa ) where ρα,j := (Sα φj , φj ), j = 1, 2, . . . , N .

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Proof. Using the generalized Fourier transform (2.2)–(2.4) one has to consider the operator Φ−1 a Φ : L2 (R, h(λ), ν) −→ L2 (R, h(λ), ν). Using the representations −1 Φα Φ−1 = Φpα Φ−1 + Φac α Φ

and ac D

−1 ∗ −1 −1 = ΦW− ac = ΦW− Ψ−1 Ψac ΨW−∗ Φ−1 , Φac α Φ D W− Φ DΨ := a ⊕ b , we get −1 ∗ −1 M (˜ ρac . M (˜ ρac α ) = ΦW− Ψ D )ΨW− Φ

where

⎧ 0, ⎪ ⎪ ⎪ ⎨f (λ − μ ) b b ρãc D (λ) := ⎪ f (λ − μ ) 0 b b ⎪ ⎪ , ⎩ 0 fa (λ − μa

λ ∈ R \ σac (H) λ ∈ [vb , va )

(3.60)

λ ∈ [va , ∞)

Ψ is defined in Subsection 2.2. Taking into account (2.5) we prove (3.59).

Remark 3.9. Formula (3.59) shows that the steady state α becomes an equilibrium state if and only if the functions fb ( · ) and fa ( · ) as well as the chemical potentials μb and μa are equal. Indeed in this case there is a function fα such that α = fα (H). 3.7. The case of sudden coupling Let us compare our results with following model [4]. Assume that our system is not coupled for t < 0 and suddenly at t = 0 the system becomes fully coupled. In a more mathematical manner this can be modeled by the following family of self-adjoint operators:

HD t < 0 H(t) := H t ≥ 0. To the evolution equation i

∂ u(t) = H(t)u(t) , ∂t

t ∈ R,

(t, s)}(t,s)∈R×R it corresponds a unique unitary solution operator or propagator {U given by (t, s) := U (t)U (s)−1 , t, s ∈ R , U where

−itHD , t≤0 (t) = e U −itH , t > 0. e The time evolution of the density matrix operator is given by (t)∗ , (t)D U ∞ (t) := U

t > 0.

Clearly, limt→−∞ ∞ (t) − D = 0. Then using the identity: (t) = e−itH eitHD D e−itHD eitH ,

t > 0,

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we immediately get that s- lim ∞ (t)P ac (H) = W− D W−∗ . t→+∞

Hence we find 1 T →+∞ T

T

s- lim

0

dt ∞ (t)P ac (H) = W− D W−∗ .

As above, we can show that 1 s- lim T →+∞ T

T

dt P ac (H)∞ (t)P s (H) = 0 0

and 1 T →+∞ T

T

dt P s (H)∞ (t)P s (H) =

s- lim

0

N

EH ({λj })D EH ({λj }) .

j=1

Hence we find 1 T →+∞ T

s- lim

T

dt ∞ (t) = 0

N

EH ({λj })D EH ({λj }) + W− D W−∗ .

j=1

4. The stationary current, the Landau–Lifschitz and the Landauer–B¨ uttiker formula There are by now several proofs of the Landauer–B¨ uttiker formula in the NESS approach (see [4, 28]), and in the finite volume regularization approach (see [14– 16]). Here we give yet another proof in the NESS approach. In fact, we will only justify the so-called Landau–Lifschitz current density formula (see (4.4) in what follows), which was the starting point in [6] for the proof of the Landauer–B¨ uttiker formula (see Example 5.11 in that paper). Let us start by defining the stationary current, in the manner introduced in [4]. Let η > 0, and choose an integer N ≥ 2. Denote by χb the characteristic function of the interval (b, ∞) (the right reservoir). Without loss of generality, let us assume that H > 0. Definition 4.1. The trace class operator j(η) := i H(1 + ηH)−N , χb is called the regularized current operator. The stationary current coming out of the right reservoir is defined to be Iα := lim Tr α j(η) . η0

88



Now a few comments. The current operator is trace class because we can write it as i H(1 + ηH)−N − HD (1 + ηHD )−N , χb which clearly is trace class. Since pα does not contribute to the trace in the definition of the current, we will focus on the clearly α independent quantity:

I = lim Tr W− D W−∗ P ac (H)j(η) . η0

We start with a technical result: Lemma 4.2. Let χ a bounded, compactly supported function. Then the operator χ(1 + H)−2 is trace class. Proof. Choose a smooth and compactly supported function χ ˜ such that χχ ˜ = χ. Write −2 −1 −1 ˜ = χ(1+H)−1 χ(1+H) ˜ +χ(1+H)−2 [H, χ](1+H) ˜ . χ(1+H)−2 = χχ(1+H) −1 ˜ Since χ ˜ is smooth and compactly supported, the operator (1+H)−1 [H, χ](1+H) −1 −1 is Hilbert–Schmidt. The operators χ(1 + H) and χ(1 ˜ + H) are also Hilbert– Schmidt, thus χ(1 + H)−2 can be written as a sum of products of Hilbert–Schmidt operators, therefore it is trace class.

Next, let us now prove that we can replace the sharp characteristic function in the definition of the current by a smooth one. Let c > b+1. Choose any function φc ∈ C ∞ (R) such that 0 ≤ φc ≤ 1 ,

φc (x) = 1 if x ≥ c + 1 ,

supp(φc ) ⊂ (c − 1, ∞) .

(4.1)

Then let us prove the following identity: Lemma 4.3. Let c > b + 1 and let φc ∈ C ∞ (R) such that the conditions (4.1) are satisfied. If D is a steady state for the system {H, HD } such that the operator $D := (HD + τ )4 D is bounded, then

Tr W− D W−∗ P ac (H)j(η) % = iTr W− D W−∗ P ac (H) H(1 + ηH)−N , φc Big} . (4.2) Proof. First, the commutator [H(1+ηH)−N , φc ] defines a trace class operator; that d 1 is because now [H, φc ] = − 2m ( d φ + φc dx ), and (1 + ηH)−1 [H, φc ](1 + ηH)−1 is b dx c a trace class operator (we can write it as a sum of products of two Hilbert–Schmidt operators). We also use the identity W− D W−∗ P ac (H) = W− D (1 + HD )W−∗ P ac (H)(1 + H)−1 which is an easy consequence of the intertwining property of W− . Second, (4.2) would be implied by: % & Tr W− D W−∗ P ac (H) H(1 + ηH)−N , φc − χb = 0 .

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We see that φc − χb has compact support. If we write the commutator as the difference of two terms, both of them will be trace class. The first one is W− D W−∗ P ac (H)H(1 + ηH)−N (φc − χb )

= W− D (1 + HD )2 W−∗ P ac (H)H(1 + ηH)−N (1 + H)−2 (φc − χb )

and the second one is W− D W−∗ P ac (H)(φc − χb )H(1 + ηH)−N

= W− D (1 + HD )2 W−∗ P ac (H) (1 + H)−2 (φc − χb ) H(1 + ηH)−N . Now according to Lemma 4.2, (1 + H)−2 (φc − χb ) is a trace class operator. Thus the two traces will be equal due to the cyclicity property and the fact that H commutes with the steady state. We can now take the limit η 0: Lemma 4.4. The operator (1 + H)−2 [H, φc ] is trace class, and

I = iTr W− D W−∗ P ac (H)[H, φc ] ,

(4.3)

independent of φc . 1 d (2 dx φc −φc ), where both φc and φc are compactly Proof. Note that [H, φc ] = − 2m b d supported. Using the method of Lemma 4.2 one can prove that (1 + H)−2 dx φc is −2 ∗ ac trace class, hence (1 + H) [H, φc ] is trace class. Thus W− D W− P (H)[H, φc ] is trace class since we can write

W− D W−∗ P ac (H)[H, φc ] = W− D (1 + HD )2 W−∗ P ac (H)(1 + H)−2 [H, φc ] . In fact, using trace cyclicity one can prove that

Tr W− D W−∗ P ac (H)[H, φc ]

= Tr W− D (1 + HD )2 W−∗ P ac (H)(1 + H)−1 [H, φc ](1 + H)−1 % & = −Tr W− D (1 + HD )2 W−∗ P ac (H) (1 + H)−1 , φc . This last identity indicates the strategy of the proof. Write: % & Tr W− D W−∗ P ac (H) H(1 + ηH)−N , φc % & = Tr W− D (1 + HD )3 W−∗ P ac (H)(1 + H)−2 H(1 + ηH)−N , φc (1 + H)−1 . Now it is not so complicated to prove that (1 + H)−2 [H(1 + ηH)−N , φc ](1 + H)−1 converges in the trace norm to (1 + H)−2 [H, φc ](1 + H)−1 when η 0; we do not give details. Now use (4.2) and take the limit; we obtain:

I = iTr W− D (1 + HD )3 W−∗ P ac (H)(1 + H)−2 [H, φc ](1 + H)−1

= iTr W− D W−∗ P ac (H)[H, φc ] , where in the last line we used trace cyclicity.

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We are going to compute the trace in (4.3) using the spectral representation 1 d (− dx φc − of H. Let us compute the integral kernel of A := iW− D W−∗ P ac (H) 2m b d φc dx ) in this representation. We use (3.60), where we denote the diagonal elements ãc of ãc D (λ) by D (λ)pp (the other entries are zero). We obtain: ' ( d i ac d ˜ φc (x) + φc (x) ˜D (λ)pp φ˜p (x, λ) A(λ, p; λ , p ) = − φp (x, λ )dx 2mb dx dx R % & i ac =− ˜D (λ)pp φc (x) φ˜p (x, λ)φ˜p (x, λ )− φ˜p (x, λ)φ˜p (x, λ ) dx , 2mb R where in the second line we integrated by parts (remember that φc is compactly supported). In order to compute the trace, we put λ = λ , p = p , and integrate/sum over the variables. We obtain: φc (x)j(x)dx , I= R

where j(x) :=

1 mb

∞

vb

˜ ˜ ãc D (λ)pp Im φp (x, λ)φp (x, λ) dλ

(4.4)

p

is the current density, which is independent of x (the above imaginary part is a of two solutions of a Schr¨ odinger equation, see [6] for details). But ) Wronskian φ (x)dx = 1 for our class of cut-off functions, therefore the stationary current R c equals the (constant) value of its density. The Landauer–B¨ uttiker formula follows from the Landau–Lifschitz formula (4.4). Following [6] one gets jλ :=

1 T (λ) f (λ − μa ) − f (λ − μb ) , 2π

λ ∈ (va , ∞) ,

where T (λ) :=

qa (λ) qb (λ) |Sba (λ)|2 = |Sab (λ)|2 qa (λ) qb (λ)

is the so-called transmission coefficient and ∞ jλ dλ . j(x) = j = va

Acknowledgements H. Cornean and H. Neidhardt acknowledge support from the Danish F.N.U. grant Mathematical Physics and Partial Differential Equations. This work was initiated during a visit of H. C. and V. Z. at WIAS, and they are thankful for the hospitality and financial support extended to them during the work on this paper. We thank the referee for his useful remarks.

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References [1] T. Aktosun, On the Schr¨ odinger equation with steplike potentials, J. Math. Phys. 40 (11) (1999), 5289–5305. [2] T. Aktosun, Darboux transformation for the Schr¨ odinger equation with steplike potentials, J. Math. Phys. 41 (4) (2000), 1619–1631. [3] W. O. Amrein and P. Jacquet, Time delay for one-dimensional quantum systems with steplike potentials, Preprint, arXiv:quant-ph/0610198, (2006). [4] W. Aschbacher, V. Jakˇsić, Y. Pautrat and C.-A. Pillet, Transport properties of quasifree fermions, J. Math. Phys. 48 (3)(2007), 032101, 28. [5] M. Baro, N. Ben Abdallah, P. Degond and A. El Ayyadi, A 1D coupled Schr¨ odinger drift-diffusion model including collisions, J. Comput. Phys. 203 (1) (2005), 129–153. [6] M. Baro, H.-Chr. Kaiser, H. Neidhardt and J. Rehberg, A quantum transmitting Schr¨ odinger–Poisson system, Rev. Math. Phys. 16 (3) (2004), 281–330. [7] M. Baro, H. Neidhardt and J. Rehberg, Current coupling of drift-diffusion models and Schr¨ odinger–Poisson systems: dissipative hybrid models, SIAM J. Math. Anal. 37 (3) (2005), 941–981. [8] O. Bratteli and D. W. Robinson, Operator algebras and quantum-statistical mechanics. II, Springer Verlag, New York, 1981. Equilibrium states. Models in quantumstatistical mechanics, Texts and Monographs in Physics. [9] V. Buslaev and V. Fomin, An inverse scattering problem for the one-dimensional Schr¨ odinger equation on the entire axis, Vestnik Leningrad. Univ. 17(1) (1962), 56– 64. [10] M. Cini, Time-dependent approach to electron transport through junctions: General theory and simple applications, Phys. Rev. B. 22 (1980), 5887–5899. [11] A. Cohen, A counterexample in the inverse scattering theory for steplike potentials, Comm. Partial Differential Equations 7 (8) (1982), 883–904. [12] A. Cohen and T. Kappeler, Scattering and inverse scattering for steplike potentials in the Schr¨ odinger equation, Indiana Univ. Math. J. 34 (1) (1985), 127–180. [13] H. Cornean, K. Hoke, H. Neidhardt, P. N. Racec and J. Rehberg, A Kohn–Sham system at zero temperature, J. Phys. A: Math. Theor. 41 (38) (2008), 385304. [14] H. D. Cornean, P. Duclos, G. Nenciu and R. Purice, Adiabatically switched-on electrical bias and the Landauer–B¨ uttiker formula, J. Math. Phys. 49 (2008), 102106. [15] H. D. Cornean, A. Jensen and V. Moldoveanu, A rigorous proof of the Landauer– B¨ uttiker formula, J. Math. Phys. 46 (4) (2005), 042106, 28. [16] H. D. Cornean, A. Jensen and V. Moldoveanu, The Landauer–B¨ uttiker formula and resonant quantum transport,In Mathematical physics of quantum mechanics, volume 690 of Lecture Notes in Phys., pages 45–53, Springer, Berlin, 2006. [17] W. R. Frensley, Quantum transport, In Norman G. Einspruch and William R. Frensley, (eds.), Heterostructures and quantum devices. [18] W. R. Frensley, Boundary conditions for open quantum systems driven far from equilibrium, Rev. Modern Phys. 62 (1990), 745–791. [19] H. Gajewski, Analysis und Numerik von Ladungstransport in Halbleitern, Mitt. Ges. Angew. Math. Mech. 16(1) (1993), 35–57.

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[20] F. Gesztesy, R. Nowell and W. P¨ otz, One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics, Differential Integral Equations 10(3) (1997), 521–546. [21] V. Jakˇsić and C.-A. Pillet, Mathematical theory of non-equilibrium quantum statistical mechanics, J. Statist. Phys. 108(5–6) (2002), 787–829. [22] D. Kirkner and C. Lent, The quantum transmitting boundary method, J. Appl. Phys. 67 (1990), 6353–6359. [23] J. Kisy´ nski, Sur les opérateurs de Green des problèmes de Cauchy abstraits, Studia Math. 23 (1963/1964), 285–328. [24] P. A. Markowich, The stationary semiconductor device equations, Computational Microelectronics, Springer Verlag, Vienna, 1986. [25] V. Moldoveanu, V. Gudmundsson and A. Manolescu, Transient regime in nonlinear transport through many-level quantum dots, Phys. Rev. B. 76 (2007), 085330. [26] C. M. Na´ on, M. J. Salvay and M. L. Trobo, Effect of nonadiabatic switching of dynamic perturbations in one-dimensional Fermi systems, Phys. Rev. B. 70 (2004), 195109. [27] H. Neidhardt and V. A. Zagrebnov, Linear non-autonomous Cauchy problems and evolution semigroups, Preprint, arXiv:0711:0284, 2007. [28] G. Nenciu, Independent electron model for open quantum systems: Landauer– B¨ uttiker formula and strict positivity of the entropy production, J. Math. Phys. 48 (3) (2007), 033302, 8. [29] F. Nier, The dynamics of some quantum open systems with short-range nonlinearities, Nonlinearity 11 (4) (1998), 1127–1172. [30] W. P¨ otz, Scattering theory for mesoscopic quantum systems with non-trivial spatial asymptotics in one dimension, J. Math. Phys. 36 (4) (1995), 1707–1740. [31] S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer Verlag, Wien, 1984. [32] G. Stefanucci, Bound states in ab initio approaches to quantum transport: A timedependent formulation, Phys. Rev. B. 75, (2007), 195115. [33] G. Stefanucci and C.-O.Almbladh, Time-dependent partition-free approach in resonant tunneling systems, Phys. Rev. B. 69, (2004), 195318. [34] B. Vinter and C. Weisbuch, Quantum Semiconductor Structures: Fundamentals and Applications, Academic Press, Boston, 1991. [35] D. R. Yafaev, Mathematical scattering theory, Translations of Mathematical Monographs 105, American Mathematical Society, Providence, RI, 1992.

Horia D. Cornean Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G DK-9220 Aalborg Denmark e-mail: [email protected]

Vol. 10 (2009)


Hagen Neidhardt WIAS Berlin Mohrenstr. 39 D-10117 Berlin Germany e-mail: [email protected] Valentin A. Zagrebnov Université de la Méditerranée (Aix-Marseille II) and Centre de Physique Théorique - UMR 6207 Luminy - Case 907 F-13288 Marseille Cedex 9 France e-mail: [email protected] Communicated by Claude Alain Pillet. Submitted: July 11, 2008. Accepted: October 23, 2008.

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Sharp Asymptotics for the Neumann Laplacian with Variable Magnetic Field: Case of Dimension 2 Nicolas Raymond Abstract. The aim of this paper is to establish estimates of the lowest eigenvalue of the Neumann realization of (i∇ + BA)2 on an open bounded subset Ω ⊂ R2 with smooth boundary as B tends to infinity. We introduce a “magnetic” curvature mixing the curvature of ∂Ω and the normal derivative of the magnetic field and obtain an estimate analogous with the one of constant case. Actually, we give a precise estimate of the lowest eigenvalue in the case where the restriction of magnetic field to the boundary admits a unique minimum which is non degenerate. We also give an estimate of the third critical field in Ginzburg–Landau theory in the variable magnetic field case.

1. Introduction and statement of main results Let Ω be an open bounded subset of R2 with smooth boundary and A ∈ C ∞ (Ω, R2 ). We let: β =∇×A and for B > 0 and u ∈ H 1 (Ω):

|(i∇ + BA)u|2 dx

N qBA,Ω (u) = Ω

and we consider the associated selfadjoint operator, i.e. the Neumann realization of (i∇ + BA)2 on Ω. We denote by λ1 (BA) the lowest eigenvalue of this operator. By the minimax principle, we have: λ1 (BA) =

N qBA,Ω (u) . u2 (Ω)

inf1

u∈H

We first recall some properties of the harmonic oscillator on a half axis (see [8,13]).

96

N. Raymond


Harmonic oscillator on a half axis. For ξ ∈ R, we consider the Neumann realization hN,ξ in L2 (R+ ) associated with the operator d2 + (t + ξ)2 , D(hN,ξ ) = u ∈ B 2 (R+ ) : u (0) = 0 . (1.1) 2 dt One knows that it has compact resolvent and its lowest eigenvalue is denoted μ(ξ); the associated L2 -normalized and positive eigenstate is denoted by uξ = u( · , ξ) and is in the Schwartz class. The function ξ → μ(ξ) admits a unique minimum in ξ = ξ0 and we let: −

Θ0 = μ(ξ0 ) ,

(1.2)

u2ξ0 (0)

. (1.3) 3 Let us also recall identities established by [5, p. 1283–1284]. For k ∈ N∗ , we denote by Mk : (t + ξ0 )k |uξ0 (t)|2 dt . Mk = C1 =

t>0

Θ0 C1 , M3 = and 2 2 Let us state a result in the case where β is constant: M0 = 1 ,

M1 = 0 ,

M2 =

μ (ξ0 ) = 3C1 Θ0 . (1.4) 2

Theorem 1.1. Assuming that β = 1, we have the estimate: √ λ1 (BA) = Θ0 B − C1 κmax B + O(B 1/3 ) , where

κmax = max k(s), s ∈ ∂Ω

and k(s) denotes the curvature of the boundary at the point s. Moreover, the groundstate decays exponentially away from the points of maximal curvature. Remark 1.2. This result was first announced by a formal analysis in [5] and rigorously proved in the case of the disk (see [4]). Let us also mention that in [19], an estimate at the first order was rigorously proved (see also [21] for the problem in R2 and R2+ ). For higher order expansion in the case of constant magnetic field, one can finally mention [9, 10, 12, 13]. Our aim is to obtain a similar result when the the magnetic field is not constant. We will assume that β > 0 on Ω. We introduce: b = inf β Ω

and

b = inf β , ∂Ω

(1.5)

and we assume: Θ0 b < b .

(1.6)

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Estimate for the variable magnetic field. Let us state a first (rough) estimate concerning the first eigenvalue: Theorem 1.3. Assuming that β|∂Ω admits a unique and non degenerate minimum, we have: λ1 (BA) = Θ0 b B + O(B 1/2 ) . Remark 1.4. The first term was obtained by many authors (cf. [13, 19]) with a worse remainder estimate. Our assumption of non-degeneracy permits to find the optimal remainder O(B 1/2 ) (the improvement occurs for the lower bound) which is crucial to establish tangential Agmon estimates (see Section 4). Let us also state a tangential localization result of the first eigenfunctions: Proposition 1.5 (Tangential Agmon’s estimates for uB ). Let uB be an eigenfunction associated with the lowest eigenvalue of the Neumann realization of (i∇ + BA)2 . We have the control: exp α1 χ t(x) d s(x) B 1/2 |uB |2 + B −1 |(i∇ + BA)uB |2 dx ≤ CuB 2 , where χ is a smooth cutoff function in a neighborhood of the boundary, t(x) = d(x, ∂Ω), s(x) the curvilinear coordinate on the boundary and where d is the Agmon distance to the minimum of β defined in Section 4. Remark 1.6. This estimate improves the localization found in [13] by specifying the behaviour of uB near the minimum of β. In Section 4 we also get tangential Agmon estimates for Ds uB . All these localizations properties are essential to obtain the second correction term of Theorem 1.3. Theorem 1.7. Assuming that β|∂Ω admits a unique and non degenerate minimum in x0 , we have: λ1 (BA) = Θ0 b B + Θ1/2 b1/2 B 1/2 + O(B 2/5 ) , where Θ1/2 = Θ1/2 (x0 ) = −κ(x0 )C1 +

C1 − Θ0 ξ0 2

1 ∂β 3/4 (x0 ) + Θ0 b ∂t

1/2 3C1 ∂ 2 β (x ) . 0 2b ∂s2

Remark 1.8. 1. When β|∂Ω admits a finite set M of non degenerate minima, we have the same expansion by replacing Θ1/2 by minx∈M Θ1/2 (x). 2. Without assuming the non degeneracy of the minima, we believe that the conclusion of Theorem 1.7 is true by replacing Θ1/2 by minx∈M Θ1/2 (x). 3. The optimal remainder is certainly O(B 1/4 ) as suggested by the upper bound.

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4. The computations for the upper bound lead to conjecture the following expansion of the n-th eigenvalue: λn (BA) = Θ0 b B + Θn1/2 B 1/2 + O(B 1/4 ) . where:

Θn1/2

= −κ(x0 )C1 +

C1 − Θ0 ξ0 2

1 ∂β 3/4 (x0 ) + (2n − 1)Θ0 b ∂t

1/2 3C1 ∂ 2 β (x0 ) . 2b ∂s2

5. In the variable case and under the assumption of Theorem 1.7, the localization due to the curvature doesn’t play a role anymore; the effect of the curvature is small compared to the variation of the magnetic field. 6. This expansion with two terms of the first eigenvalue could be generalized at any order under the previous assumptions (unique and non degenerate minimum of β|∂Ω ) by using a Grushin approach (see [10]). 7. The case where the magnetic field (non degenerately) vanishes in Ω was treated in [18]. Moreover, the case where it non degenerately vanishes on the boundary remains open and should be an interesting problem. 8. Theorems 1.3 and 1.7 are also sensible under the hypothesis of regularity of the domain. When the domain has corners (see [6, Theorem 1.2]) and with a variable magnetic field, the ground state is not necessarily localized near the points of the boundary where the magnetic field is minimum. 9. The asymptotic behaviour in Theorems 1.3 and 1.7 is strongly dependent on the Neumann boundary condition we impose, as one can see in [15–17]. In particular, in certain cases, the localization is no more determined by the minimal points of β. 10. Finally, let us mention that there exists results in dimension 3. Indeed, the two terms asymptotics in the constant magnetic field case is given in [14, Theorem 1.2] and, until now, we only know the first term in the variable case (see [14, Theorem 4.4]). Constant magnetic field on the boundary. In [2,3], the case of the constant magnetic field on the boundary is treated. Nevertheless, this case is studied under a non degeneracy condition: it is assumed that the curvature of the boundary κ admits a unique maximum at x = x0 and that the normal derivative ∂β ∂t admits a unique minimum at x = x0 ; moreover, the min imum of ∂β − b κ has to be non degenerate. Here, we improve his result by using ∂t more generic assumptions; in particular, we will see that the quantity to maximize is the “magnetic curvature” defined by:

C1 1 ∂β (x) . κ ˜ (x) = C1 κ(x) + Θ0 ξ0 − 2 b ∂t More precisely, our result is the following:

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Theorem 1.9 (Upper bound: Constant magnetic field on ∂Ω). When the magnetic field is constant on the boundary, we have the upper bound:

C1 1 ∂β 1 − Θ0 ξ0 (x) b1/2 B 1/2 +O(B 1/3 ) , λ (BA) ≤ Θ0 b B − max C1 κ(x) − x∈∂Ω 2 b ∂t where κ(x) denotes the curvature of the boundary at x. Remark 1.10. 1. The corresponding lower bound could certainly be obtained by the techniques of [12]. 2. Assuming the existence of a unique and non degenerate maximum of the magnetic curvature κ ˜ , one could surely give an asymptotics at any order of λ1 (BA) and localization properties as for the constant magnetic field case (see [10]) which would improve the hypothesis of Aramaki. Organization of the paper. In Section 2 and 3, we will prove the Theorem 1.3 and give the upper bound of Theorem 1.7 and of Theorem 1.9. Then, we will see, in Section 4, that this first rough estimate gives information on the localization of the groundstates on the boundary near the mimimum of the magnetic field. In Section 5, we prove the lower bound of Theorem 1.7 thanks to a reduction to a degenerate case studied by S. Fournais and B. Helffer. Finally, we apply the previous results to give an estimate of the third critical field in Ginzburg–Landau theory.

2. A rough lower bound In order to get the lower bound in Theorem 1.3, we use a localization technique permitting the reduction to easier models. 2.1. Partition of unity For each 0 < ρ < 12 , B > 0, > 0 and C0 > 0, we consider a partition of unity (cf. [14]) for which there exists C = C(Ω, β, , C0 ) > 0 such that: 2 |χB (2.7) j | = 1 on Ω ; j

2 2ρ |∇χB j | ≤ CB

on

Ω.

(2.8)

j ∞ Each χB j is a C -cutoff function supported in Dj ∩ Ω. Moreover, we may assume that there exists a ball Dj = Djmin whose center is the minimum of β on the boundary and C0 B −ρ for radius. We may also assume that the balls which intersect the boundary have their centers on the boundary and that those one admit B −ρ for radius. The radius of all the other balls is assumed to be B −ρ . We will choose ρ, and C0 later for optimizing the error. We will use the following localization IMS formula (cf. [7]):

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Lemma 2.1. qBA (u) =

qBA (χB j u) −

j


2 |∇χB j |u ,

∀u ∈ H 1 (Ω) .

(2.9)

j

So, in order to minimize qBA (u), we are reduced to the minimization of qBA (v), with v supported in some Dj . 2.2. Estimates for the lower bound 2.2.1. Study inside Ω. Let j such that Dj does not intersect the boundary. It is well known that: B 2 2 u) ≥ B β(x)|χ u| dx ≥ bB |χB qBA (χB j j j u| dx . Ω

Ω

Having in mind (1.6), these terms will not play a role in the computation of the asymptotics. 2.2.2. Study at the boundary. In the next paragraph, we introduce boundary coordinates. Boundary coordinates. We choose a parametrization of the boundary: γ : R/(|∂Ω|Z) → ∂Ω . Let ν(s) be the unit vector normal to the boundary, pointing inward at the point γ(s). We choose the orientation of the parametrization γ to be counter-clockwise, so det γ (s), ν(s) = 1 . The curvature k(s) at the point γ(s) is given in this parametrization by: γ (s) = k(s)ν(s) . The map Φ defined by: Φ : R/(|∂Ω|Z)×]0, t0 [→ Ω (s, t) → γ(s) + tν(s) , is clearly a diffeomorphism, when t0 is sufficiently small, with image Φ R/(|∂Ω|Z)×]0, t0 [ = x ∈ Ω|d(x, ∂Ω) < t0 = Ωt0 . We let: A˜1 (s, t) = 1 − tk(s) A Φ(s, t) · γ (s) , ˜ t) = β Φ(s, t) , β(s, and we get:

A˜2 (s, t) = A Φ(s, t) · ν(s) ,

˜ t) . ∂s A˜2 − ∂t A˜1 = 1 − tk(s) β(s,

˜j = vj ◦Φ: Let j such that Bj intersect the boundary; we have, with vj = χB j u and v −1 qBA (vj ) = 1 − tk(s) |(i∂t + B A˜2 )˜ vj |2 + 1 − tk(s) |(i∂s + B A˜1 )˜ vj |2 dsdt .

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Approximation by a constant magnetic field on a domain with constant curvature. Locally, we can choose a gauge such that t ˜ t )dt , A˜2 = 0 . A˜1 (s, t) = 1 − t k(s) β(s, 0

We assume that the center of the ball Dj has the coordinates (sj , 0) and that the coordinates of the minimum are (0, 0). We let: ˜ j , 0) = β˜j and Δkj (s) = k(s) − kj . kj = k(sj ) , β(s We have:

˜ t) = (1 − tkj )β˜j − tΔkj (s)β(s, ˜ t) 1 − tk(s) β(s, ˜ t) − β˜j . + (1 − tkj ) β(s,

(2.10)

A˜1 (s, t) = A1,j (s, t) + Rj (s, t) ,

(2.11)

We write: with

A1,j (s, t) =

t − kj

t2 2

β˜j .

(2.12)

Control of the remainders. Therefore, we are reduced to compare qBA with the quadratic form associated with the Neumann problem on a domain with constant curvature (see [4, 10, 12, 13]). For all λ > 0, we get the inequality (with the Cauchy–Schwarz inequality): vj |2 dsdt qBA (vj ) ≥ (1 − λ) (1 − tkj )|∂t v˜j |2 + (1 − tkj )−1 |(i∂s + BA1,j )˜ − C Δkj (s)t |∂t v˜j |2 + |(i∂s + B A˜1 )˜ vj |2 dsdt B2 |Rj (s, t)˜ − vj |2 dsdt . λ We apply the result of the constant magnetic field on a domain with constant curvature to get the existence of C > 0 such that for all j such that Dj ∩ ∂Ω = ∅ (cf. [4, Theorem 6.1]): vj |2 dsdt (1 − tkj )|∂t v˜j |2 + (1 − tkj )−1 |(i∂s + BA1,j )˜ ≥ (Θ0 β˜j B − C1 kj B 1/2 − C)˜ vj 2 .

(2.13)

In order to control the remainders, we recall the Agmon estimates (cf. [1,10,12,13]): Proposition 2.2 (Normal Agmon’s estimates). Let uB be an eigenfunction associated with the lowest eigenvalue of the Neumann realization of (i∇ + BA)2 . We have the control the momenta of order n in the normal variable t: n t(x)n |uB |2 + B −1 |(i∇ + BA)uB |2 dx ≤ Cn B − 2 uB 2 .

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∂Ω C0 B −1/4 0 xjmin = (0, 0)

B −1/4 xj = (sj , 0) Ω

0

Figure 1. Partition of unity near the boundary. We choose ρ = 14 (see Figure 1) and notice that |Δj k(s)| = O(B −1/4 ) (uniformly in j). So, there exists C > 0 such that for all j: 1 2 Δkj (s)t |∂t v˜j |2 + |(i∂s + B A˜1 )˜ vj | dsdt ≤ CB 4 ˜ vj 2 . We let:

1 ∂2β (0, 0) . (2.14) 2 ∂s2 Using the assumption of non degeneracy of the minimum, we can choose 0 > 0 small enough such that α 2 ˜ 0) − β(0, ˜ 0) ≤ 3 αs2 s ≤ β(s, (2.15) 2 2 for all |s| ≤ 0 . To estimate the other remainder, we will distinguish between three cases: • j = jmin , • |sj | ≥ 0 , • C0 B −1/4 ≤ |sj | ≤ 0 . α=

Case 1: j = jmin . As

∂ β˜ (0, 0) = 0 , ∂s

we have, with (2.10) and (2.11): |Rjmin (s, t)| ≤ C(t2 + s2 t) .

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Consequently, using Proposition 2.2, we get: vjmin |2 dsdt ≤ CB −2 ˜ vjmin 2 . |Rjmin (s, t)˜ Taking λ = B −1/2 , we deduce: qBA (vjmin ) ≥ (Θ0 b B − CB 1/2 )˜ vjmin 2 . Case 2: |sj | ≥ 0 . We get: Thus, we find:

|Rj (s, t)| ≤ C (s − sj )t + t2 .

|Rj (s, t)˜ vj |2 dsdt ≤ C(B −3/2 2 + B −2 )˜ vj 2 .

Moreover, there exists b > b such that for all |sj | ≥ 0 , we have: β˜j ≥ b . We take λ = B −1/2 and deduce, using (2.13) and for B large enough, that for all j satisfying |sj | ≥ 0 : qBA (vj ) ≥ Θ0 b B˜ vj 2 . Case 3: C0 B −1/4 ≤ |sj | ≤ 0 . We use the inequality:

2 ∂ β˜ sup |β˜ − b | sup (0, s) ≤ C ∂s −1/4 −1/4 |s−sj |≤B |s−sj |≤B

to find with (2.11) and (2.10): 2 |Rj (s, t)˜ vj | dsdt ≤ C B −3/2 2

sup |s−sj

|≤B −1/4

As a consequence, we can write, with λ = B −1/2 : ˜ j ) − b − C 2 qBA (vj ) ≥ Θ0 b B + B Θ0 β(s

|β − b | + B

|s−sj

By non degeneracy, we have, for C0 ≥ 2 : |β˜ − b | ≤ 27 sup |s−sj

|≤B −1/4

inf

−2

˜ vj 2 .

sup |≤B −1/4

|s−sj |≤B −1/4

|β − b |

|β˜ − b | .

Indeed, we have, for all C0 ≥ 2 : inf

|s−sj |≤B −1/4

|β˜ − b | ≥

and sup |s−sj |≤B −1/4

|β˜ − b | ≤

α α inf s2 ≥ (sj − B −1/4 )2 2 |s−sj |≤B −1/4 2

3α 3α sup (sj + B −1/4 )2 . s2 ≤ 2 |s−sj |≤B −1/4 2

˜ vj 2 .

(2.16)

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Thus, we get, for C0 ≥ 2 :

2 2 sup|s−sj |≤B −1/4 |β˜ − b | sj + B −1/4 2 B −1/4 ≤3 = 3 1 + ≤ 27 . sj − B −1/4 sj − B −1/4 inf |s−sj |≤B −1/4 |β˜ − b | We deduce, for C0 ≥ 2 :

qBA (vj ) ≥ Θ0 b B + B(Θ0 − 27C 2 )

inf

|s−sj |≤B −1/4

|β˜ − b | v˜j 2 .

We will further use that there exists c > 0 such that for all C0 ≥ 2 : ˜ j ) − b v˜j 2 . qBA (vj ) ≥ Θ0 b B + cB β(s Indeed, we have, for all C0 ≥ 2 : |s−sj

inf

|≤B −1/4

|β˜ − b | ≥

1 ˜ β(sj ) − b . 27

We find, for > 0 small enough: qBA (vj ) ≥ (Θ0 b B + CB 1/2 )˜ vj 2 . We conclude that:

qBA (vj ) ≥ (Θ0 b B − CB 1/2 )

j bnd

vj 2 .

j bnd

Putting together this estimate and the estimate inside Ω, we have the lower bound in Theorem 1.3.

3. Models near a minimum of β and upper bounds 3.1. Model operator We fix k0 , k1 and α ≥ 0 and we wish to study the quadratic form on the Hilbert space L2 ((1 − k0 t)dtds) defined, for u ∈ C0∞ (Bk0 ) by: qk0 ,k1 ,α,B (u) = (1 − tk0 )|∂t u|2 s∈R 0 0 such that: |A˜1 (s, t)| ≤ Ct ,

1/2−2ρ

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Δk(s)t |∂t uB |2 + |(−i∂s + B A˜1 )uB |2 dsdt ≤ CB 1/4+ρ uB 2 .

Let us prove the upper bound for the first term (the second can be treated in the same way). We have: ∂t uB = χ (t)ψ(B 1/2 t)e−s

2

B 1/2−2ρ iξ0 B 1/2 s

e

+B 1/2 χ(t)ψ (B 1/2 t)e−s

2

B 1/2−2ρ iξ0 B 1/2 s

e

.

Thus we get: |∂t uB |2 ≤ 2|χ (t)ψ(B 1/2 t)|2 e−2s

2

B 1/2−2ρ

+ 2B|χ(t)ψ (B 1/2 t)|2 e−2s

2

B 1/2−2ρ

.

Then, we find: 2 1/2−2ρ dtds tΔk(s)|∂t uB |2 dtds ≤ C ts|χ (t)|2 |ψ(B 1/2 t)|2 e−2s B 2 1/2−2ρ + CB ts|χ(t)|2 |ψ (B 1/2 t)|2 e−2s B dtds . As ψ is in the Schwartz class, we get: 2 1/2−2ρ dtds = O(B −∞ )uB 2 . ts|χ (t)|2 |ψ(B 1/2 t)|2 e−2s B Then, we have after rescaling, for some C > 0 independent of B: 2 1/2−2ρ B ts|χ(t)|2 |ψ (B 1/2 t)|2 e−2s B dtds ≤ CBB −1/2 B −1/4+ρ uB 2 = CB 1/4+ρ uB 2 . Moreover, we have: (1 − tk0 )|∂t uB |2 + (1 − tk0 )−1 |(−i∂s + B A˜1 )uB |2 dsdt = (1 − tk0 )|∂t uB |2 + (1 − tk0 )−1 |(−i∂s + BA1 )uB |2 dsdt

+ (1 − tk0 )−1 (B 2 |RuB |2 + 2B(−i∂s + BA1 )uB RuB )dsdt . We get: 0 ,k1 qk0 ,k1 ,0,B (uB ) ≤ (Θ0 B + Θk1/2 B 1/2 + CB 1/2−2ρ )uB 2 ,

the crucial points being to estimate the term 2 1/2−2ρ 2 1/2−2ρ |e−s B |χ(t)|2 |ψ(B 1/2 t)|2 dtds |∂s2 e−s B 2 1/2−2ρ by O(B 1/2−2ρ )uB 2 and the term (Bt + B 1/2 ξ0 )∂s (χ(t)ψ(B 1/2 t)e−s B ) by O(B −∞ )uB 2 thanks to the fact that M1 = 0 (cf. (1.4)) and that ψ is in the Schwartz class. Using that: |R(s, t)| ≤ C(t3 + s4 t + st2 ) ,

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we find:

2 −1 2 (1 − tk B dsdt ) |Ru | + 2B(−i∂ + BA )u Ru 0 B s 1 B B ≤ CB 1/4+ρ uB 2 . and finally with ρ =

1 12 :

0 ,k1 B 1/2 + CB 1/3 )uB 2 . qBA (uB ) ≤ (Θ0 B + Θk1/2

Thus, after replacing k1 by its expression, the upper bound of Theorem 1.9 is proved. Remark 3.1. It follows from the identities (1.4) that: C1 − Θ0 ξ0 = M3 − ξ03 > 0 , 2 where M3 = t>0 (t + ξ0 )3 u20 dt. This remark permits to understand how the upper bound of Theorem 1.9 improves the one of Aramaki. 3.3. Non-degenerate case α > 0 3.3.1. Formal computation. We consider the operator H (cf.(3.19)):

−1

k0 t k0 t − 1 − 1/2 ∂t 1 − 1/2 ∂t B B

−2

2 k0 t k1 2 α 2 ∂s + 1 − 1/2 t + 1/2 s t − i 1/4 . t + ξ0 − B 2B 1/2 B B Formally, we write: H=

+∞

B −j/4 Hj .

j=1

Let us look for a quasimode expressed as: U=

+∞

B −j/4 Uj .

(3.21)

j=1

and a Taylor expansion of the lowest eigenvalue: λN 1 (B) =

+∞

Θj/4 B −j/4 .

j=1

Here, we have: H0 = −∂t2 + (t + ξ0 )2 , H1 = −2i∂s (t + ξ0 ) , H2 = k0 ∂t −

∂s2

k1 + 2(t + ξ0 ) αs t − t2 2 2

+ 2k0 t(t + ξ0 )2 .

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This leads us to solve: H0 U0 = λ0 U0 . We write U0 as U0 = u0 (t)ψ0 (s) and, as we look for λN 1 minimal, we take λ0 = Θ0 and u0 > 0 the associated normalized eigenvector. Then, we solve: H1 U0 + H0 U1 = Θ0 U1 + λ1 U0 . We can take Θ1/4 = 0 by writing U1 = u1 (t)ψ1 (s) with ψ1 = ∂s ψ0 and we find: (H0 − Θ0 )u1 = 2i(t + ξ0 )u0 . As M1 = 0 (see (1.4)), this last equation admits a unique solution u1 such that u u dt = 0. t>0 0 1 Finally, we consider: H0 U2 + H1 U1 + H2 U0 = Θ0 U2 + Θ1/2 U0 . Thus, we get: (H0 − Θ0 )U2 = −H1 U1 − H2 U0 + Θ1/2 U0 = 2i(t + ξ0 )u1 ∂s ψ1 − H2 U0 + Θ1/2 U0 . Multiplying by u0 and integrating with respect to t, one applies the formulas (1.4) and one solves:

k0 + k1 C1 − (k1 − k0 )Θ0 ξ0 ψ0 . −(1 − 4I2 )∂s2 ψ0 + αΘ0 s2 ψ0 = Θ1/2 + 2 where

(t + ξ0 )R0 (t + ξ0 )u0 u0 dt .

I2 = t>0

This last integral can be rewritten by letting v = R0 ((t + ξ0 )u0 ); we have: (H0 − Θ0 )v = (t + ξ0 )u0 . By computing, we get: 1 ∂u ( · , ξ0 ) = v . 2 ∂ξ Using the identities of [12], we find: −

1 − 4I2 =

μ (ξ0 ) = 3C1 Θ0 > 0 . 2

After rescaling, we let: −

ψ0 (s) = e

1/4 √ Θ0 αs2 2

√

3C1

and: 0 ,k1 ,α Θ1/2 = Θk1/2 =−

k0 + k1 3/4 √ C1 + (k1 − k0 )Θ0 ξ0 + 3C1 Θ0 α. 2

(3.22)

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3.3.2. Quasimode. For simplicity, we assume b = 1. We write: A˜1 = A1 + R , where

k1 A1 = t 1 − t + αs2 2 with α defined in (2.14) and k1 defined in (3.20). We let: uB (s, t) = χ(t)U (B 1/4 s, B 1/2 t)eiξ0 B

1/2

s

,

where U consists of the three first terms of (3.21). We have: qBA (uB ) ≤ (1 − tk0 )|∂t uB |2 + (1 − tk0 )−1 |(−i∂s + B A˜1 )uB |2 dsdt + C Δk(s)t |∂t uB |2 + |(−i∂s + B A˜1 )uB |2 dsdt . Moreover, we have: (1 − tk0 )|∂t uB |2 + (1 − tk0 )−1 |(−i∂s + B A˜1 )uB |2 dsdt = (1 − tk0 )|∂t uB |2 + (1 − tk0 )−1 |(−i∂s + BA1 )uB |2 dsdt

2 −1 2 + (1 − tk0 ) B |RuB | + 2B(−i∂s + BA1 )uB RuB dsdt . Using that U is in the Schwartz class, we get: 0 ,k1 ,α qk0 ,k1 ,α,B (uB ) ≤ (Θ0 B + Θk1/2 B 1/2 + C)uB 2 .

Moreover, we have: |A˜1 − A1 | ≤ C(s3 t + st2 + t3 ) . So, we get:

2 −1 2 (1 − tk0 ) B |RuB | + 2B(−i∂s + BA1 )uB RuB dsdt ≤ CB 1/4 uB 2 . Finally, we find: 0 ,k1 ,α qBA (uB ) ≤ (Θ0 B + Θk1/2 B 1/2 + CB 1/4 )uB 2 .

In particular, we have proved the upper bound in Theorem 1.3.

4. Tangential Agmon’s estimates We first observe that, for Φ a real Lipschitzian function and if u is in the domain of the Neumann realization of (i∇ + BA)2 , then we have, by integration by parts: (i∇ + BA)2 u, exp(2B 1/2 Φ)u = qBA exp(B 1/2 Φ)u − B|∇Φ| exp(B 1/2 Φ)u2 .

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Taking u = uB an eigenfunction attached to the lowest eigenvalue λ1 (BA), we get: λ1 (BA) exp(B 1/2 Φ)uB 2 = qBA exp(B 1/2 Φ)uB − B|∇Φ| exp(B 1/2 Φ)uB 2 .

(4.23)

4.1. Tangential Agmon’s estimates for uB We now use the lower bound found in Section 2; more precisely, for all > 0, there exists c > 0 and C > 0 such that, for all C0 > 0 sufficiently large, there exists C > 0 s.t for all u in the form domain of qBA : qBA (u) ≥ (bB − CB 1/2 ) χj u2 j int

+

Θ0 b B + c β(sj ) − b B χj u2

j bnd,j=jmin

+ (Θ0 b B − C B 1/2 )χjmin u2 . We choose u = exp(B 1/2 Φ)uB ; we recall that, by Theorem 1.3, we have the upper bound: λ1 (BA) ≤ Θ0 b B + CB 1/2 . Using these estimates in (4.23), we find the inequality by dividing by B: (C B −1/2 + |∇Φ|2 )|χjmin exp(B 1/2 Φ)uB |2 ˜ j ) − b − CB −1/2 − |∇Φ|2 |χj exp(B 1/2 Φ)uB |2 dsdt . c β(s ≥ j bnd

j=jmin

We choose Φ = α1 d(s) , where d is the Agmon distance associated with the metric (β(s, 0) − b )ds2 i.e.: |s| 1/2 β(σ, 0) − b d(s) = dσ . 0

On Djmin , we notice that

|∇Φ|2 ≤ CB −1/2 . Then, for j =

jmin , we consider the quantity: ˜ − b . ˜ j ) − b − CB −1/2 − α2 β(s) c β(s 1

For > 0 and α1 small enough, there exists c > 0 such that for j such that |sj | ≥ 0 and B large enough, we have: ˜ − b ≥ c . ˜ j ) − b − CB −1/2 − α2 β(s) c β(s 1 For C0 ≥ 2 , there exists c > 0 such that for j = jmin and |sj | ≤ 0 and B large enough, we have: ˜ − b ≥ c B −1/2 . ˜ j ) − b − CB −1/2 − α2 β(s) c β(s 1

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Indeed, due to the non degeneracy, we have (2.16). Thus, we get C > 0 and B0 > 0 such that for all B ≥ B0 : | exp(B 1/2 Φ)uB |2 . |χj exp(B 1/2 Φ)uB |2 ≤ C |s|≤C0 B −1/4

j bnd

We deduce Proposition 1.5 and have the following corollary: Corollary 4.1. For all n ∈ N, there exists C > 0 such that for all B large enough: s2n |uB |2 + B −1 |(i∇ + BA)uB |2 dx ≤ CB −n/2 |uB |2 dx . Ω

Ω

4.2. Agmon’s estimates for Ds uB We consider a partition of unity as in (2.7). We have the formula (2.9) and: 1/2 qBA (u) ≥ qBA (χB u2 . j u) − CB j

We use (4.23). We have λ1 (B) ≤ Θ0 b B + CB 1/2 . Thus, we get, using the inequalities of the previous section: 2 1/2 1/2 Θ0 b B + c β(sj ) − b B χj eB Φ uB qBA χjmin eB Φ uB + j=jmin

+ bB

2 1/2 χj eB Φ uB j int

2 1/2 B 1/2 Φ 2 1/2 ≤ (Θ0 b B + CB ) e uB + B ∇ΦeB Φ uB + CB 1/2 uB 2 , where Φ = α1 d(s). We have the control: 2 2 B 1/2 Φ 1/2 B 1/2 Φ 1/2 B uB ≤ CB uB ≤ CB |uB |2 dx , χjmin ∇Φe χjmin e Ω

Ω

Ω

and we deduce, for α1 small enough: B 1/2 Φ

qBA (χjmin e We introduce:

2 B 1/2 Φ 1/2 uB ) − Θ0 b B χjmin e uB ≤ CB |uB |2 .

(1 − k0 t)|∂t v|2

qapp (v) = t>0,s∈R

(4.24)

2 k1 + (1 − k0 t)−1 Bt + Bαs2 t + B 1/2 ξ0 − Ds − B t2 v dtds . 2

If we write: A˜1 (s, t) =

˜ t )dt , 1 − t k(s) β(s,

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˜ t) = (1 − tk1 ) + αs2 + O(t2 + st + s3 ) , 1 − tk(s) β(s,

and thus:

k1 A˜1 (s, t) = t − t2 + αs2 t + O(t3 + st2 + s3 t) . 2 Then, by the Cauchy–Schwarz inequality, we have for all λ > 0:

(4.25)

B2 Rv2 . qBA (v) ≥ (1 − λ)qapp (v) − λ For instance, we can estimate (st2 )2 |v|2 . Using the tangential (cf. Proposition 1.5) and normal Agmon estimates and letting: v = χjmin eB

we have: B

2

1/2

Φ

uB ,

s2 t4 |v|2 dsdt ≤ CB 2 B −1/2 B −2 v2 .

In the same way, we control the other remainders and by choosing λ correctly, we get: 1/4 qBA (v) ≥ qapp (v) − CB (4.26) |v|2 . Using the Cauchy–Schwarz inequality and again the Agmon estimates, we find: 2 qapp (v) ≥ (1 − B −1/2 )qapp (v) − CB 1/2 |v|2 , where

2 qapp (v) =

(1 − k0 t)|∂t v|2 + (1 − k0 t)−1 |(Bt + B 1/2 ξ0 − Ds )v|2 dx .

t>0,s∈R

Making a Fourier transform in the variable s and letting w = vˆ, we have: 2 qapp (v) = (1 − k0 t)|∂t w|2 + (1 − k0 t)−1 |(Bt + B 1/2 ξ0 − σ)w|2 dtdσ . t>0,σ∈R

Thus, we get (see [12, Chapter 6, Prop 6.2.1] or [13, Section 11]): 2 2 1/2 μ (ξ0 ) 2 1/2 |Ds v| − CB qapp (v) ≥ Θ0 b B |v| + B |v|2 . 2 Consequently, we get the upper bound: 2 1/2 Ds (χjmin eB Φ uB ) ≤ C |uB |2 . We deduce the following proposition: Proposition 4.2 (Tangential Agmon’s estimates for Ds uB ). With the previous notations, there exists C > 0 and α1 > 0 such that for all B large enough: 2 α1 B 1/2 χ(t(x))d(s(x)) 1/2 Ds uB dx ≤ CB |uB |2 dx , e Ω

Ω

where χ is a smooth cutoff function supported in [−t0 , t0 ].

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Corollary 4.3. For all n ∈ N, there exists C > 0 such that for all B large enough, we have: 2n 2 1/2−n/2 χ(t)s |Ds uB | dx ≤ CB |uB |2 dx . Ω

Ω

Remark 4.4. The tangential and normal Agmon estimates roughly say that |uB | 2 1/2 has the same behaviour as e−αs B u0 (B 1/2 t).

5. Refined lower bounds In this section, we prove the lower bound in Theorem 1.7. We consider a partition of unity as in (2.7) with ρ = 14 − η for η > 0. We have: 1/2−2η qBA (χB u2 . qBA (u) ≥ j u) − CB j

5.1. Control far from the minimum Let us first recall some the estimates we have proved. For j such that Dj does not intersect the boundary, we have: qBA (χj u) ≥ bB |χj u|2 dx . For j such that Dj intersect the boundary and j = jmin , we notice that, for B large enough: qBA (χj u) ≥ Θ0 b B |χj u|2 . 5.2. Reduction to a model near the minimum Using the inequalities of the previous section, we get: qBA (uB ) ≥ Θ0 b B χj uB 2 + qBA (χjmin uB ) − CB 1/2−2η uB 2 . j=jmin

By the normal and tangential Agmon estimates, we have proved in (4.26), with (4.24), (4.25) and the Cauchy–Schwarz inequality: qBA (χjmin uB ) ≥ qapp (χjmin uB ) − CB 1/4 uB 2 . In order to make the term in αs2 t disappear, we make the change of variables: t = λ(s)τ , where λ(s) = (1 + αs2 )−1/2 ; we have ∂s v =

∂τ ∂τ v˜ + ∂s v˜ , ∂s

∂t v =

∂τ ∂τ v˜ , ∂t

(5.27)

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where v˜ denotes the function v in the variables (τ, s) and we are reduced to the form: 1 − k0 τ λ(s) |∂τ v|2 q app (v) =

2 −1 Bτ + ξ0 λ(s)B 1/2 − λ(s)Ds − B k1 τ λ(s)3 + 1 − k0 τ λ(s) 2 2 + ατ sλ(s)3 Dτ v λ(s)−1 dτ ds , where we have omitted the tilde. Noticing that s2 = O(B 2ρ−1/2 ), on the support of v = χjmin uB , we make the approximations in L2 : −λ(s)Ds v = −Ds v + O s2 Ds v , τ 2 λ(s)3 v = τ 2 v + O s2 τ 2 v , sλ(s)3 τ Dτ v = sτ Dτ v + O(s3 τ )Dτ v . We first find: (1 − τ k0 )|∂τ v|2 q app (v) ≥

k1 τ 2 λ(s)3 + (1 − τ k0 )−1 Bτ + ξ0 λ(s)B 1/2 − λ(s)Ds − B 2 2 3 + αsλ(s) τ Dτ v λ(s)−1 dτ ds

k1 τ 2 2 λ(s)3 − C Δλ(s)τ |∂τ v| + Bτ + ξ0 λ(s)B 1/2 − λ(s)Ds − B 2 2 3 + αsλ(s) τ Dτ v λ(s)−1 dτ ds , where Δλ(s) = λ(s) − λ(0) . Let us consider the second term:

k1 τ 2 λ(s)3 Δλ(s)τ |∂τ v|2 + Bτ + ξ0 λ(s)B 1/2 − λ(s)Ds − B 2 2 3 + αsλ(s) τ Dτ v λ(s)−1 dτ ds .

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Coming back in the variables (t, s), this term becomes:

k1 t2 Δλ(s) 2 2 1/2 t |∂t v| + | B(1 + αs )t + ξ0 B − Ds − B v|2 dtds . λ(s) 2 Thus, the Agmon estimates give a control of the second term of order O(1). Then, by the Cauchy–Schwarz inequality, the Agmon estimates (for uB and Ds uB after having come back in the variables (s, t)) and using the same kind of analysis as in (4.26), we have:

k1 τ 2 2 −1 λ(s)3 (1 − τ k0 )|∂τ v| + (1 − τ k0 ) Bτ + ξ0 λ(s)B 1/2 − λ(s)Ds − B 2 2 3 + αsλ(s) τ Dτ v λ(s)−1 dτ ds

2 k1 τ 2 2 −1 1/2 ≥ (1 − τ k0 )|∂τ v| + (1 − τ k0 ) Bτ + ξ0 λ(s)B − Ds − B v 2 λ(s)−1 dτ ds − CB 1/4 v2 . We have finally, with v = χjmin uB : χj uB 2 qBA (uB ) ≥ Θ0 b B j=jmin

2 k1 τ 2 2 −1 1/2 + (1 − τ k0 )|∂τ v| + (1 − τ k0 ) Bτ + ξ0 λ(s)B − Ds − B v 2 λ(s)−1 dτ ds − CB 1/4 uB 2 − CB 1/2−2η uB 2 .

(5.28)

Moreover, thanks to the exponential decrease of uB away from the boundary (normal Agmon estimates), we can replace χjmin by a smooth cutoff function such that supp χjmin ⊂ {0 < t ≤ B −1/2+η and|s| ≤ B −1/4+η } , that is we assume χjmin is supported in rectangles rather than balls; the reason is technical and will appear in the next section. 5.3. Lower bound for the model τˆ sˆ , s = B 1/4 , to the study of: So, we are reduced, after the rescaling τ = B 1/2

k0 τˆ 1 − 1/2 |∂τˆ u|2 qmod (u) = B τˆ>0,ˆ s∈R

−1

k0 τˆ τˆ + ξ0 λ(B −1/4 sˆ) − Dsˆ + 1 − 1/2 B B 1/4 2

1/2 k1 αˆ s2 − 1/2 τˆ2 u dˆ τ dˆ s. 1 + 1/2 2B B

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Reduction to the euclidean measure. s2 1/2 , we make the change of In order to make disappear the measure (1 + Bαˆ 1/2 ) function defined by:

1/4 αˆ s2 v = 1 + 1/2 u = fB (ˆ s)u , B we have:

k0 τˆ qmod (u) = 1 − 1/2 |∂τˆ v|2 B τˆ>0,ˆ s∈R

−1

s) k0 τˆ τˆ + ξ0 λ(B −1/4 sˆ) − Dsˆ − fB (ˆ + 1 − 1/2 1/4 1/4 B B B fB (ˆ s) 2 k1 τ dˆ s. − 1/2 τˆ2 v dˆ 2B

Term in sˆ. We want to make a Fourier transform in the variable sˆ to be reduced to a problem on a half axis, but the term ξ0 λ(B −1/4 sˆ) is annoying; that is why we make it disappear with a change of gauge. We write: λ(B −1/4 sˆ) = 1 + rB (ˆ s) and we make −iφ(ˆ s) the change of gauge v → v˜ = ve , where

sˆ

ξ0 rB (σ) −

φ(ˆ s) = 0

fB (σ) dσ B 1/4 fB (σ) 1

to be reduced to: q v) = mod (˜

k0 τˆ |∂τˆ v˜|2 B 1/2 τˆ>0,ˆ s∈R

−1

2 k0 τˆ D k s ˆ 1 τˆ + ξ0 − + 1 − 1/2 − τˆ2 v˜ dˆ τ dˆ s, B B 1/4 2B 1/2

1−

where u = (χjmin uB )(B 1/2 τˆ, B 1/4 sˆ). We make a Fourier transform in the variable sˆ and we are reduced to a half axis problem in the normal variable:

k0 τˆ qn (w) = 1 − 1/2 |∂τˆ w|2 B τˆ>0

−1

2 k0 τˆ τˆ + ξ0 − σ − k1 τˆ2 w dˆ + 1 − 1/2 τ, 1/4 1/2 B B 2B with w = vˆ.

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Model on a half axis. We can apply the same kind of analysis as in [12, Chapter 6, Prop 6.2.1] or in [13, Section 11] to get the lower bound; there exists C > 0 such that for all B large enough:

μ (ξ0 ) 2 0 ,k1 σ B −1/2 + qn (w) ≥ Θ0 + Θk1/2 2

τˆk0 − CB −3/4+3η |w|2 1 − 1/2 dˆ τ. (5.29) B τˆ>0 Remark 5.1. In [12], the fact that the magnetic field is constant permits to be 0 ,k1 reduced to the case k0 = k1 = 1, thus Θk1/2 = −C1 . Let us just recall the main ideas of the proof. We consider first the (formal) ˆ 0τ )dˆ τ ): operator on L2 ((1 − Bk1/2

−1

d d k0 τˆ k0 τˆ h(σ, B) = − 1 − 1/2 1 − 1/2 dˆ τ dˆ τ B B

−2

2 τˆ2 k0 τˆ σ + 1 − 1/2 . τˆ + ξ0 − 1/4 − k1 1/2 B B 2B Then, we formally expand this operator in powers of B and, for |σ| ≤ M B η , with η > 0 small enough: h(σ, B) = h0 + B −1/4 h1 + B −1/2 h2 + O(B −3/4+3η ) , where d2 + (ˆ τ + ξ0 )2 , dˆ τ2 τ + ξ0 )σ , h1 = −2(ˆ h0 = −

h2 = k0 τˆ

d − k1 τˆ2 (ˆ τ + ξ0 ) + 2k0 τˆ(ˆ τ + ξ0 )2 + σ 2 . dˆ τ

Thus, as in Section 3.3.1, we compute a quasimode and obtain for some ψ: h(σ, B) − (λ0 + λ1 B −1/4 + λ2 B −1/2 ) ψ 2 = O(B −3/4+3η ) . k τ ˆ L (R ,(1− 0 )) +

B 1/2

Finally, we can prove that the previous operator admits only one eigenvalue strictly less than 1 thanks to a comparison with the harmonic oscillator on a half axis and, applying the spectral theorem, we get the bottom of the spectrum given in (5.29) (the values of σ such that |σ| ≥ M B η provide higher energies thanks to the nondegeneracy of ξ → μ(ξ) near ξ0 ).

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Return in the initial variables. Applying the Parseval formula, we get:

119

τˆk0 |v|2 1 − 1/2 dˆ τ dˆ s B sˆ∈R τ ˆ>0

μ (ξ ) τˆk0 0 −1/2 2 |Dsˆv˜| 1 − 1/2 dˆ +B τ dˆ s − CB −3/4+3η u2 . 2 B sˆ∈R

0 ,k1 v ) ≥ (Θ0 + Θk1/2 B −1/2 ) qmod (u) = q mod (˜

τ ˆ>0

We have:

2 s) v . |Dsˆv˜|2 = Dsˆ − φ (ˆ

s)| ≤ C sˆ2 B −1/2 ≤ CB −1/2+2η on the support of v, we get: As |φ (ˆ |Dsˆv˜|2 ≥ (1 − B −1/4+η )|Dsˆv|2 − B −1/4+η |v|2 . Moreover, we have: Dsˆv =

αˆ s fB (s)−3 u + fB (ˆ s)Dsû . 2B 1/2

We deduce: |Dsˆv˜|2 ≥ |Dsû|2 − CB −1/4+η (|u|2 + |Dsû|2 ) − B −1/4+η |Dsˆv|2 . Recalling that

dˆ τ dˆ s=

|v| = 2

αˆ s 1 + 1/2 B αˆ s 1 + 1/2 B

1/2 dtˆdˆ s, 1/2 |u|2 ,

where tˆ = B −1/2 t, and with the tangential Agmon estimates, we get: Dsˆv2 ≤ Cu2 ,

Dsû2 ≤ Cu2

and qmod (u) ≥ Θ0 u2 + Θk0 ,k1 B −1/2 u2

μ (ξ0 ) tˆk0 2 2 |Dsû + αΘ0 |ˆ su ˇ| + ˇ| dˆ s 1 − 1/2 dtˆ B −1/2 2 B − CB −3/4+3η ,

(5.30)

where u ˇ(tˆ, sˆ) = u(ˆ τ , sˆ) and, thanks to the Agmon estimates, we have replaced Dsû ˆ ˇ and τˆ by tˆ by noticing that Dsû = Dsû ˇ + ∂∂sˆt Dtû ˇ and λ(ˆ sB −1/4 )ˆ τ = tˆ. by Dsû We recognize the quadratic form of the harmonic oscillator and we have:

μ (ξ0 )αΘ0 μ (ξ0 ) 2 2 |Dsû |ˇ u|2 dˆ αΘ0 |ˆ su ˇ| + ˇ| dˆ s ≥ s. 2 2

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1 We take η = 20 and the lower bound of Theorem 1.7 follows from (5.30), (5.28) and (1.4) after having noticed that the estimates of Agmon give: 2 −cB η |χjmin uB | dx = 1 + O(e ) |uB |2 dx . Ω

Ω

6. Estimate for the third critical field of the Ginzburg–Landau functional In this section, we give an estimate of the third critical field of the Ginzburg– Landau functional in the case where the applied magnetic field denoted by β admits a unique and non degenerate minimum on the boundary of Ω. The constant magnetic field case has already been studied in details (see [11, 19–21]). Recall of properties of the functional. The Ginzburg–Landau functional is defined by:

κ2 |∇ × A − β|2 dx , |(i∇ + σκA)ψ|2 − κ2 |ψ|2 + |ψ|4 dx + (κσ)2 G(ψ, A) = 2 Ω Ω 1 for ψ ∈ H 1 (Ω, C) and A ∈ Hdiv (Ω, R3 ) where 1 (Ω, R3 ) = A ∈ H 1 (Ω, R3 ) : div(A) = 0 in Ω, A · ν = 0 on ∂Ω . Hdiv

We assume moreover that β = ∇ × F. Then, we recall the definitions of the critical fields (the first one was introduced in [20]): HC3 (κ) = inf σ > 0 : (0, F) is the unique minimizer of Gκ,σ , H C3 (κ) = inf σ > 0 : (0, F) is the unique minimizer of Gκ,σ for all σ > σ , H C3 (κ) = inf σ > 0 : (0, F) is a minimizer of Gκ,σ and

loc H C3 (κ) = sup σ > 0 | λ1 (κσF) < κ2 .

We have H C3 (κ) ≤ HC3 (κ) ≤ H C3 (κ) and

loc

H C3 (κ) ≤ H C3 (κ) . We can prove the following result (cf. [12]): Theorem 6.1. Let Ω be a bounded, simply connected domain with smooth boundary and suppose that the applied magnetic field β satisfies 0 < Θ0 b < b . Then, there exists κ0 > 0 such that for all κ ≥ κ0 : loc

H C3 (κ) = H C3 (κ) .

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Furthermore, if B → λ1 (BF) is strictly increasing for large B, then all the critical fields coincide for large κ and are given by the unique solution H of λ1 (κHF) = κ2 . Estimate of HC3 (κ) for large κ. Noticing that B → λ1 (BF) is strictly increasing for large B (it is due to the exponential decrease of the first eigenfunctions away from the boundary, still true in the case of variable magnetic field; see [12, Chapter 9, Section 6]), we deduce the following theorem: Theorem 6.2. Let Ω be a bounded, simply connected domain with smooth boundary and suppose that the applied magnetic field β has a unique and non degenerate minimum on ∂Ω and that: 0 < Θ0 b < b . Then, we have: Θ1/2 κ HC3 (κ) = − b1/2 3/2 + O(κ−7/20 ) . b Θ0 Θ 0

Acknowledgements I am deeply grateful to Professor B. Helffer for his help, advice and comments. I would also like to thank A. Kachmar for his attentive reading and suggestions which improved the presentation of the paper.

References [1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: Bounds on eigenfunctions of N-body Schr¨ odinger operators, Princeton University Press, 1982. [2] J. Aramaki, Upper critical field and location of surface nucleation for the Ginzburg– Landau system in non-constant applied field, Far East J. Math. Sci. 23(1) (2006), 89–125. [3] J. Aramaki, Asymptotics of the eigenvalues for the Neumann Laplacian with nonconstant magnetic field associated with supraconductivity, Far East J. Math. Sci. 25(3) (2007), 529–584. [4] P. Bauman and D. Phillips and D. Tang, Stable nucleation for the Ginzburg–Landau system with an applied magnetic field, Arch. Rational Mech. Anal. 142 (1998), 1–43. [5] A. Bernoff and P. Sternberg, Onset of superconductivity in decreasing fields for general domains, J. Math. Phys. 39 (1998), 1272–1284. [6] V. Bonnaillie, On the fundamental state energy for a Schr¨ odinger operator with magnetic field in domains with corners, Asympt. Anal. 41(3–4) (2005), 215–258. [7] H.-L. Cycon, R-G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators, SpringerVerlag, 1986.

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[8] M. Dauge and B. Helffer, Eigenvalues variation. I. Neumann problem for SturmLiouville operators, Journal of Differential Equations 104 (1993), 243–262. [9] M. del Pino, P. Felmer and P. Sternberg, Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity, Comm. in Math. Phys. 210 (2000), 413–446. [10] S. Fournais and B. Helffer, Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Annales de l’institut Fourier 56 (2006), 1–67. [11] S. Fournais and B. Helffer, On the third critical field in Ginzburg–Landau theory, Comm. in Math. Physics 266(1) (2006), 153–196. [12] S. Fournais and B. Helffer, Spectral methods in surface superconductivity, to appear, 2008. [13] B. Helffer and A. Morame, Magnetic bottles in connection with superconductivity, Proc. Indian. Sci. 185(21) (2001), 604–680. [14] B. Helffer and A. Morame, Magnetic bottles for the Neumann problem: Curvature effects in the case of dimension 3 (general case), Ann. Scient. E. Norm. Sup. 37(4) (2004), 105–170. [15] A. Kachmar, On the ground state energy for a magnetic Schr¨ odinger operator and the effect of the de Gennes boundary condition, J. Math. Phys. 47(7) (2006). [16] A. Kachmar, On the perfect superconducting solution for a generalized Ginzburg– Landau equation, Asympt. Anal. 54(3–4) (2007), 125–164. [17] A. Kachmar, On the stability of normal states for a generalized Ginzburg–Landau model, Asympt. Anal. 55(3–4)(2007), 145–201. [18] K. H. Kwek and X.-B. Pan, Schr¨ odinger operators with non-degenerately vanishing magnetic fields in bounded domains, Trans. AMS 10 (2002), 4201–4227. [19] K. Lu and X.-B. Pan, Eigenvalue problems of Ginzburg–Landau in bounded domains, J. Math. Phys. 40(6) (1999), 2647–2670. [20] K. Lu and X.-B. Pan, Estimates of the upper critical field for the Ginzburg–Landau equations of superconductivity, Physica D 127 (1999), 73–104. [21] K. Lu and X.-B. Pan, Gauge invariant eigenvalue problems on R2 and R2+ , Trans. AMS 352(2) (2000), 1247–1276. Nicolas Raymond Université Paris-Sud 11 Bˆ atiment 425 Laboratoire de Mathématiques F-91405 Orsay Cedex France e-mail: [email protected] Communicated by Christian Gérard. Submitted: June 26, 2008. Accepted: November 28, 2008.



Spectral and Scattering Theory of Friedrichs Models on the Positive Half Line with Hilbert–Schmidt Perturbations Hellmut Baumgärtel

Dedicated to Izrail Cudicoviˇ c Gohberg on his 80th birthday Abstract. The spectral theory of the Friedrichs model on the positive half line with Hilbert–Schmidt perturbations, equipped with distinguished analytic properties, is presented. In general, the (separable) multiplicity Hilbert space is assumed to be infinite-dimensional. The results include a spectral characterization of its resonances and the association of so-called Gamov vectors. Sufficient conditions are presented such that all resonances are simple poles of the scattering matrix. The connection between their residual terms and the associated Gamov vectors is pointed out.

1. Introduction The basic topic of this paper, the Friedrichs model, is a creation of K. O. Friedrichs, which can be traced back to [11] and [12]). By the results in these papers Friedrichs founded the perturbation theory of continuous spectra by the development of socalled stationary methods, which proved extremely important in scattering theory (also in quantum field theory). Later contributions (generalizations, applications, abstract developments) to this field are due to Friedrichs himself (see e.g. [13]) and many other authors. For general literature we refer to [3] and references therein. An important part of the perturbation theory of continuous spectra is the theory of resonances. It is especially interesting in quantum scattering theory because of the requirement to give rigorous interpretations of the so-called Breit– Wigner formulas, which describe bumps in quantum scattering cross sections (see e.g. [1, 9, 15] for contributions to the general theory of resonances). Since the Friedrichs model is algebraically soluble, i.e. the scattering matrix is explicitly

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given by the Livˇsic-matrix, it is especially suitable to study its resonances. There are many papers with results on decay properties of the model due to the resonances and on implications concerning the spectral properties of them (generalized eigenvalue problem). An early contribution in this direction is [16] (see also [2] and references therein). The paper presents the spectral theory of the Friedrichs model with separable multiplicity Hilbert space on the positive half line and with Hilbert–Schmidt perturbations, which have distinguished analytic properties. The results include a spectral characterization of the resonances w.r.t. the perturbed selfadjoint operator (by introducing a special boundary condition) and the association of so-called Gamov vectors (see e.g. [10] for this concept). The analytic structure of the scattering matrix is pointed out explicitly. Sufficient conditions are presented such that all resonances are simple poles. In this case their residual terms are closely related to the associated Gamov vectors. A foregoing paper [8] of the present one deals with the finite-dimensional Friedrichs model on the whole real line in the context of the Schwartz space approach. There the consideration is restricted to the eigenvalue problem (corresponding to that in Section 4).

2. Preliminaries 2.1. Assumptions We consider a Friedrichs model on the positive half line where the perturbation is of the Hilbert–Schmidt type. Put H0,+ := L2 (R+ , K, dλ), R+ := (0, ∞), where K is a separable multiplicity Hilbert space, and H := H0,+ ⊕E, where E is a separable Hilbert space, too. H0 denotes the multiplication operator on H0,+ , H0 f (λ) := λf (λ), f ∈ H0,+ . Let A be a selfadjoint compact operator on E. We consider the perturbation H := (H0 ⊕ A) + Γ + Γ∗ , where Γ ∈ L2 (H) is a Hilbert–Schmidt operator on H with ΓH0,+ = {0} and ΓE ⊆ H0,+ . Then Γ∗ E = {0} and Γ∗ H0,+ ⊆ E follow. That is, Γe(λ) = M (λ)e ,

e∈E,

and M (λ) is a Hilbert–Schmidt operator of L2 (E → K), well-defined a.e. on R+ , ∞ satisfying 0 M (λ)22 dλ < ∞ because of ∞ 2 Γ2 = M (λ)22 dλ . 0

H is selfadjoint, bounded below and dom H = dom H0 ⊕ E. We assume ΓE ⊂ dom H0 . In the finite-dimensional case where dim E < ∞, dim K < ∞ this includes the case of the usual (finite-dimensional) Friedrichs model, where Γ is a partial isometry with Γ∗ Γ = PE , the projection onto E, and ΓΓ∗ < 1l − PE , such that ∞ ∗ M (λ) M (λ)dλ = 1lE . 0

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For convenience in the following Γ E is identified with Γ without confusion. On this model we impose the following assumptions: (i) H0,+ = clo spa{E0 (Δ)f, f ∈ ΓE} and H = clo spa{E(Δ)e, e ∈ E}, where E0 ( · ), E( · ) denote the spectral measures of H0 , H, respectively. (The Δ denote Borel sets of the real line.) This means: ΓE is generating for H0,+ w.r.t. H0 and E is generating for H w.r.t. H. (In the case dim E < ∞ this implies dim E = dim K.) If dim E = ∞ then necessarily dim K = ∞. The set of eigenvalues of H is denoted by Sp . In the following we assume dim E ≤ ∞. The essential parameter of the model is the Hilbert–Schmidt-valued operator function M ( · ) on the positive half line. For this parameter we require the following condition of analytic continuability: (ii) The operator function M ( · ) is holomorphic on R+ and meromorphically continuable into a region R+ ⊂ G ⊂ C \ (−∞, 0], symmetric w.r.t. R+ , i.e. G = G. The operators M (z) are Hilbert–Schmidt, i.e. G z → M (z) ∈ L2 (E → K). The set of all poles of M ( · ) is denoted by P0 which is assumed to be finite. Further ker M (z) = {0} for z ∈ G \ P0 . Then z → M (z)∗ is also meromorphic on G, the set of poles is P0 . The set of poles of z → M (z)∗ M (z) is then given by P0 ∪ P0 =: P. We put P± := P ∩ C± and G± := G ∩ C± . 2.2. Livˇsic-matrix and partial resolvent We put ∞ M (λ)∗ M (λ) dλ , z ∈ C>0 , (1) Φ0 (z) := Γ∗ R0 (z)Γ = z−λ 0 where C>0 is the complex plane with cut [0, ∞), i.e. C>0 := {z : z ∈ C \ [0, ∞)} and R0 (z) := (z − H0 )−1 denotes the resolvent of H0 . Φ0 ( · ) is holomorphic there, Φ0 (z) ∈ L1 (E), M (λ)∗ M (λ) ∈ L1 (E). One has Γ∗ E0 (dλ)Γ = M (λ)∗ M (λ) , dλ

λ>0

and

Φ0 (z)∗ = Φ0 (z) , z ∈ C>0 . According to (ii) Φ0 is meromorphically continuable across R+ , where the path can be negatively or positively oriented. The negatively oriented continuation of Φ0 ( · ) into G− is given by Φ+ (z) := Φ0 (z) − 2πiM (z)∗ M (z) ,

z ∈ G− ,

and the positively oriented continuation into G+ by Φ− (z) := Φ0 (z) + 2πiM (z)∗ M (z) ,

z ∈ G+ .

The Φ+ ( · ), Φ− ( · ) are branches on the lower sheet G− and the upper sheet G+ , respectively, of an analytic function Φ( · ), where Φ0 ( · ) is the branch on the basic sheet C>0 . The whole domain of Φ( · ) (its Riemannian surface) is denoted by dom Φ. The union of the branches Φ0 and Φ± is denoted by Φ± , too. (Note that sometimes the basic sheet is called the first one and the lower sheet the second one.)

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The set of poles of Φ± ( · ) is P∓ . Note that Φ0 (λ − i0) − Φ0 (λ + i0) = 2πiM (λ)∗ M (λ) ,

λ > 0.

(2)

For brevity let Φ0 (λ ± i0) =: Φ± (λ) for λ > 0. The Livˇsic-matrix is defined by L0 (z) := z1lE − A − Φ0 (z) ,

z ∈ C>0 .

Then one has L0 (z)∗ = L0 (z) ,

z ∈ C>0 .

(3)

The analytic continuations, corresponding to those of Φ0 , are denoted by L± , and the whole analytic function by L, which is meromorphic on dom Φ. One has L+ (z)∗ = L− (z)

z ∈ G− ,

z ∈ G+ .

(4)

By straightforward calculation one obtains the identity R0 (z)Γ + PE = R(z)L0 (z) ,

z ∈ C>0 ,

Γ∗ R0 (z) + PE = L0 (z)R(z) ,

z ∈ C>0 ,

(5)

which implies where R(z) := (z − H)−1 denotes the resolvent of H. Hence one obtains PE R(z)PE E = L0 (z)−1 ,

z ∈ C>0 ,

(6)

that is, the so-called partial resolvent PE R(z)PE coincides with the inverse of the Livˇsic-matrix. (6) shows that L0 ( · )−1 is holomorphic on C± . Writing L(z) = z 1lE − z −1 A + Φ(z) = z 1lE − K(z) , where K(z) := z −1 (A + Φ(z)) is compact for all z ∈ dom Φ, one has L(z)−1 = z −1 (1lE − K(z))−1 . Since z → (1lE − K(z))−1 is holomorphic on C± , it follows that L(z)−1 is meromorphic on dom Φ, according to a well-known result for meromorphic operator functions (see e.g. Gochberg/Krein [11, p. 64]). This result implies (see [17], see also [4, p. 137]) that Sp , which is denumerable, can only accumulate at the point 0. Exactly the non-vanishing eigenvalues of H are the real-valued poles of L( · )−1 , for λ > 0 the poles of L± ( · )−1 , for λ < 0 the poles of L0 ( · )−1 . These poles are simple and in the case λ > 0 the residua of these poles for L+ ( · )−1 and L− ( · )−1 coincide (see [4, p. 139]). The set of nonreal poles of L+ ( · )−1 is denoted by R. Since L+ ( · )−1 is holomorphic on C>0 (basic sheet) one has R ⊂ G− , it is denumerable and accumulation points are at most from ∂G ∩ C− . The points of R are called resonances. To simplify the treatment it is assumed that the sets of poles of L+ ( · ) and L+ ( · )−1 are disjoint, i.e. we assume R ∩ P− = ∅.

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From (6) and (2) one obtains that the spectral measure of H, sandwiched by the projection PE , satisfies PE E(dλ)PE dλ1 −1 − L+ (λ)−1 ) = L± (λ)−1 M (λ)∗ M (λ)L∓ (λ)−1 , 2πi (L− (λ) = 0, λ < 0.

λ > 0,

(7)

Note that L− ( · )−1 − L+ ( · )−1 is holomorphic on R+ also at positive eigenvalues. Lemma 1. ζ0 ∈ G− \ P− is a pole of L+ ( · )−1 iff dim ker L+ (ζ0 ) > 0. Proof. If ζ0 is a pole of L+ ( · )−1 then one has 0 ∈ spec L+ (ζ0 ) = spec {ζ0 (1lE − K+ (ζ0 ))} and this means 0 is an eigenvalue of 1lE − K+ (ζ0 ), i.e. also an eigenvalue of L+ (ζ0 ), hence ker L+ (ζ0 ) ⊃ {0}. The converse is obvious. Lemma 2. The absolutely continuous spectrum of H coincides with [0, ∞). There is no singular continuous spectrum. Proof. Let Δ0 be a Borel set with Δ0 ∩ Sp = ∅. Then 1 E(Δ)e, E(Δ0 )E(Δ)e = e, L− (λ)−1 − L+ (λ)−1 e dλ = 0 , 2πi Δ∩Δ0 hence E(Δ0 )E(Δ)e = 0 and, according to (i), E(Δ0 )f = 0 for f ∈ H follows. Therefore the projection P onto the orthogonal complement of the eigenspace of H and the projection P ac onto the absolutely continuous subspace coincide. Further P ac E((−∞, 0)) = 0, i.e. the absolutely continuous spectrum is given by [0, ∞).

3. Spectral representations, Abelian wave operators, scattering matrix The spectral integral

H0,+ x :=

∞

E0 (dλ)Γf (λ) , 0

R+ λ → f (λ) ∈ E ,

∞ exists iff 0 M (λ)f (λ)2K dλ < ∞. According to assumption (i) the linear manifold M0,+ of all such spectral integrals is dense in H0,+ . It defines a spectral representation of H0,+ w.r.t. H0 which is given by the isometric isomorphism between H0,+ and L2 (R+ , Eˆλ , dλ), where the Hilbert space Eˆλ is the completion of E w.r.t. the scalar product e1 , e2 λ := (M (λ)e1 , M (λ)e2 )K . Note that ΓE is a spectral manifold w.r.t. E0 ( · ). If dim E < ∞ then Eˆλ = E for all λ. (See e.g. [3, p. 80 ff.] for details on spectral integrals and spectral representations.) The same procedure for H and H leads to a distinguished spectral representation of P ac H w.r.t. H. For convenience we denote in the following by E( · ) the

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spectral measure of H w.r.t. its absolutely continuous subspace. Then the spectral representation is defined by the spectral integral ∞ P ac H y := E(dλ)g(λ) , R+ λ → g(λ) ∈ E , (8) 0

where (8) exists iff

∞ 1 L− (λ)−1 − L+ (λ)−1 g(λ) dλ g(λ), 2πi 0 ∞ = M (λ)L± (λ)−1 g(λ)2K dλ < ∞ . 0

The set M of all spectral integrals (8) is dense in P ac H. That is, for these dense sets of spectral integrals we call f ( · ) the E0 -representer of x, x(λ) = M (λ)f (λ), and g( · ) the E-representer of y. Note that if y ∈ dom H then λ → λg(λ) is the E-representer of Hy, i.e. w.r.t. the spectral representation (8) H is represented by the multiplication operator M . These two spectral representations we call the natural spectral representations of P ac H, H0,+ w.r.t. H, H0 , respectively. Lemma 3. The operators Ω± , defined on M0,+ by the spectral integral

∞ ∞ Ω± E0 (dλ)Γf (λ) = E(dλ)L± (λ)f (λ) . 0

0

are isometric and ima Ω± = M. The inverse Ω−1 ± is given by

∞ ∞ E(dλ)g(λ) = E0 (dλ)ΓL± (λ)−1 g(λ) , Ω−1 ± 0

0

i.e. if λ → f (λ) is the E0 -representer of x ∈ H0,+ then the E-representer of Ω± x ∈ H is given by λ → L± (λ)f (λ). Conversely, if λ → g(λ) is the E-representer −1 g(λ). of y ∈ H then the E0 -representer of Ω−1 ± y ∈ H0,+ is given by λ → L± (λ) Ω± is an intertwining operator for the spectral measures E0 ( · ), E( · ), i.e. Ω± E0 (Δ) = E(Δ)Ω± ,

Δ Borel set .

(9)

Proof. Obvious by calculation. For example, the E0 -representer of the vector Ω−1 + e, e ∈ E, is given by λ → L+ (λ)−1 e , the E0 -representer of

ac Ω−1 + P Γe

(10)

by

λ → L+ (λ)−1 (λ − A)e ,

(11)

because (λ − A)e is the E-representer of P Γe: ∞ ∞ ∞ E(dλ)(λ − A)e = λE(dλ)e − E(dλ)Ae = P ac (He − Ae) = P ac Γe . ac

0

0

0

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By isometric extension and putting Ω± e = 0 for e ∈ E, one obtains a partial isometry on H, the initial space is H0,+ and the final space P ac H. Note that Lemma 3 implies that P ac H and H0 are unitarily equivalent. The so-called Abelian wave operators for H and H0 are defined by ∞ e−t e±itH e∓itH0 f dt W± f := s-lim→+0 0 ∞ E(dλ) 1lH − V R0 (λ ± i) f , (12) = s-lim→+0 0

where f ∈ H0,+ and V := H − H0 = Γ + Γ∗ . For the transformation of the time integrals in (12) into spectral integrals see e.g. [5, p. 361]]. Theorem 1. The Abelian wave operators W± exist and they coincide with the partial isometries Ω± , i.e. W± P0ac = Ω± P0ac , where P0ac denotes the projection onto the absolutely continuous subspace H0,+ of H0 ⊕ A. Proof. Let e ∈ E. Then by straightforward calculation one obtains ∞ ∞ E(dλ) Γe − Γ∗ R0 (λ ± i)Γe = E(dλ)L± (λ ± i)e . 0

0

Now

∞

lim

→+0

M (λ)L± (λ)−1 L± (λ ± i)e − L± (λ)e 2 dλ = 0 ,

0

hence W± Γe exists and one has W± Γe = Ω± Γe .

(13)

This implies that W± exists on clo{E0 (Δ)Γe, e ∈ E}, which, according to (i), coincides with H0,+ and one has W± E0 (Δ)f = E(Δ)W± f, f ∈ H0,+ (see e.g. [6]). Then (9) and (13) yield the assertion. Theorem 1 (together with Lemma 3) means that the Abelian wave operators act by application of the Livˇsic-matrix resp. its inverse on the corresponding representers. In general, operator functions with these properties are called wave matrices of Ω± , Ω∗± . Note that wave matrices are well-defined only if the spectral representations are fixed. Corollary 1. The wave matrices of Ω± , Ω∗± w.r.t. the natural spectral representations of H0,+ H are given by Ω± (λ) = L± (λ), Ω± (λ)∗ = L± (λ)−1 , λ ∈ R+ \ Sp . Remark 1. Since Ω± is isometric, in this case even the relation ∞ lim e−t e±itH e∓itH0 x − Ω± x2 dt = 0 , x ∈ H0,+ , →+0

(14)

0

is true. The condition(14) is strongly related to the existence of the strong wave operators (for details see [3, p. 101 f.]). If Γ is even trace class, Γ ∈ L1 (H), then it is well-known that the strong wave operators exist. In particular, in the finitedimensional case dim E < ∞ the strong wave operators exist.

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Note that in general for Hilbert–Schmidt perturbations of H0 ⊕ A there is no unitary equivalence between the absolutely continuous parts like in the trace class case. On the contrary, according to a Theorem of J.v.Neumann (see e.g. [18, p. 525]) there is a Hilbert–Schmidt perturbation V such that the absolutely continuous spectrum of (H0 ⊕ A) + V is empty. The scattering operator S is defined by S := Ω∗+ Ω− , it is unitary on H0,+ and can be represented by its scattering matrix w.r.t. a given spectral representation of H0,+ . Lemma 4. W.r.t. the natural spectral representation of H0,+ the corresponding scattering matrix SE ( · ) is given by SE (λ) = L+ (λ)−1 L− (λ) = 1lE − 2πiL+ (λ)−1 M (λ)∗ M (λ) ,

λ ∈ R+ \ Sp ,

i.e. if x ∈ H0,+ and f ( · ) is its E0 -representer then λ → SE (λ)f (λ) is the E0 representer of Sx. The scattering matrix S( · ) w.r.t. the original spectral representation of H0,+ given by the K-valued functions λ → x(λ) ∈ K for x ∈ H0,+ satisfies S(λ)M (λ)f (λ) = M (λ)SE (λ)f (λ) which implies, according to Lemma 4, S(λ) := 1lK − 2πiM (λ)L+ (λ)−1 M (λ)∗ ,

λ ∈ R+ \ Sp .

(15)

Since S(λ) is unitary for λ ∈ R+ \ Sp (which can be seen also by straightforward calculation), one has S(λ) = 1 , λ ∈ R+ \ Sp , Since S( · ) is meromorphic on R+ this means that the positive eigenvalues of H are holomorphic points for the scattering matrix, i.e. S( · ) is holomorphic on the whole positive half line. Corollary 2. The perturbed operator H has no positive eigenvalues. Proof. Obvious by the condition ker M (λ) = {0} for λ > 0 from (ii).

This means that positive eigenvalues of A are unstable under the perturbation Γ + Γ∗ , they disappear.

4. The (generalized) eigenvalue problem for H A Gelfand triplet is given by the Gelfand space and its topology. The Gelfand space D ⊂ P ac H, used ∞in the following for the extension of H, is defined to be the manifold of all g = 0 E(dλ)g(λ) ∈ M, such that g( · ) is holomorphic on G and λ → λg(λ) is also an E-representer. The Gelfand topology in D is defined by the collection of norms D g → |g|K := gH + sup g(z)E , z∈K

K ⊂ G,

K compact .

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Obviously D is dense in P ac H w.r.t. the Hilbert topology and D defines a Gelfand triplet D ⊂ P ac H ⊂ D× , (16) × where D denotes the set of all continuous antilinear forms w.r.t. the Gelfand topology. Note that E ⊕ ΓE ⊂ D , D ⊆ dom H . Defining Φ by Φ := PE⊥ D ⊂ D one obtains D =Φ⊕E,

D× = Φ× × E

(cartesian product) .

Further one has Φ ⊆ dom H0 . For D d = φ + e, φ ∈ Φ, e ∈ E and d× = {Φ× , e× } ∈ Φ× × E one obtains d | d× = φ | φ× + (e, e× )E . The Gelfand triplet (16) yields a unique extension H × ×

×

d | H d := Hd | d ,

d ∈ D,

(17) ×

on D

×

×

given by

×

d ∈D .

(18)

×

The eigenvalue equation for eigenvalues ζ0 ∈ G− \ P− of H reads then × × × H × d× e0 ∈ E , ζ0 ∈ G− \ P− , φ× d× 0 = ζ0 d0 , 0 := φ0 (ζ0 , e0 ), e0 , 0 ∈Φ . (19) This means × Hd | d× d ∈ D, 0 = ζ 0 d | d0 , which is equivalent with × ∗ (H0 e − ζ 0 e, e0 ) + Γe | φ× 0 = ζ 0 φ − H0 φ | φ0 − (Γ φ, e0 ) ,

d = φ + e.

Since e and φ vary independently we obtain two equations (i) ((ζ 0 − H0 )e, e0 ) − Γe | φ× 0 = 0, e ∈ E, ∗ (ii) (ζ 0 − H0 )φ | φ× = (Γ φ, e0 ), φ ∈ Φ. 0 × × The part φ0 of d0 depends on ζ0 , the candidate for an eigenvalue (and on e0 ). For this part of a solution of the eigenvalue equation (19) we impose the following boundary condition: • φ× 0 is required to be the analytic continuation into G− across R+ of a holomorphic vector antilinear form φ× 0 (z, e0 ) on C+ such that (ii) is an identity on C+ . This means: it is required that ∗ z ∈ C+ , φ ∈ Φ , (z − H0 )φ, φ× 0 (z, e0 ) H = (Γ φ, e0 )E , 0

or

φ, (z − H0 )φ× 0 (z, e0 ) H = (φ, Γe0 )H0 , 0

This implies or

z ∈ C+ ,

(z − H0 )φ× 0 (z, e0 ) = Γe0

−1 φ× Γe0 , z ∈ C+ . 0 (z, e0 ) = (z − H0 ) The solution of this eigenvalue problem is given by

φ ∈ Φ.

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Theorem 2. The antilinear form d× 0 is an eigensolution of (19) satisfying the boundary condition iff e0 satisfies L+ (ζ0 )e0 = 0, i.e. iff ζ0 is a resonance and e0 ∈ ker L+ (ζ0 ). The associated vector antilinearform on C+ is given by

φ | φ× , z ∈ C+ , φ ∈ Φ . (20) 0 (z, e) := φ, R0 (z)Γe H 0,+

The dimension of the corresponding eigenspace is dim ker L+ (ζ0 ), the geometric multiplicity of the eigenvalue 0 of L+ (ζ0 ). Proof. It is an improved version of the similar proof of [8, Theorem 1]. The first and crucial point in the proof is to show that (20) has an analytic continuation into G− across R+ . With φ := PE⊥ g one obtains

φ | φ× , z ∈ C+ . 0 (z, e0 ) = g, R0 (z)Γe0 H 0,+

Second, using the identity (5) one obtains g, R(z)L+ (z)e = Ψ− (z), L+ (z)e , where

∞

Ψ± (z) := 0

1 L− (μ)−1 M (μ)∗ M (μ)L+ (μ)−1 g(μ)dμ , z−μ

(21)

z ∈ C± .

Then the mentioned analytic continuability of (20) is obtained by inspection of (21) and this means that φ× 0 ( · , e0 ) satisfies (ii) for all ζ ∈ C+ ∪ (G− \ P− ). Now we consider the left hand side of (i) first on C+ . Then it reads (z − H0 )e, e0 − Γe, (z − H0 )−1 Γe0 = e, (z − H0 )e0 − e, Γ∗ (z − H0 )−1 Γe0 = e, L+ (z)e0 , where e ∈ E, z ∈ C+ . That is, the left hand side of (i) is analytically continuable into G− and holomorphic in G− \ P− . Therefore (i) is equivalent with L+ (ζ0 )e0 = 0 ,

ζ0 ∈ G− \ P ,

0 = e0 ∈ E

and this is the assertion.

Remark 2. This result reflects the effect of the boundary condition. If one omits this condition then the solutions d× 0 of the eigenvalue equation (19) w.r.t. the triplet (16) are given by

ζ0 ∈ G− \ P− , e0 ∈ E , (22) g | d× 0 (ζ0 , e0 ) := g(ζ0 ), e0 E , because of

Hg | d× 0 = (M g)(ζ0 ), e0 E = ζ0 g(ζ0 ), e0 E .

(Here M denotes the multiplication operator). That is, the boundary condition selects from (22) those solutions, where first ζ0 is a resonance and second the (multiplicity) vectors e0 are from ker L+ (ζ0 ).

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The Gelfand space D can be transformed to a Gelfand space G± by the wave operators Ω± . For example we use G+ := Ω∗+ D. Then × G+ ⊂ H0,+ ⊂ G+

(23)

is the corresponding Gelfand triplet. For the “back transformation” of the eigen× solution d× 0 (ζ0 , e0 ) of (19) to G+ , given by ∗ × × s× 0 (ζ0 , e0 ) := (Ω+ ) d0 (ζ0 , e0 ) ,

× s× 0 ∈ G+ ,

we obtain the following result. × Theorem 3. The eigenantilinear form s× 0 of H0 w.r.t. the Gelfand triplet G+ ⊂ × × H0,+ ⊂ G+ , associated to d0 is of pure Dirac type w.r.t. the point ζ 0 ∈ C+ :

G+ s → s | s× 0 (ζ0 , e0 ) := 2πi s(ζ 0 ), k0 K ,

where ζ0 ∈ R , Proof. Since

k0 := M (ζ0 )e0 ,

e0 ∈ ker L+ (ζ0 ) .

(24)

g := Ω+ s s | s× 0 (z, e0 ) = g, R(z)L+ (z)e0 it is an easy implication of (21). For z ∈ G− ) \ P− one has

−1 s | s× g(z), M (z)e0 . 0 (z, e0 ) = Ψ− (z), L+ (z)e0 + 2πi M (z)L+ (z)

If z = ζ0 is a resonance then the assertion follows.

Omitting the boundary condition the solutions s× 0 of the back transform for the eigenvalue equation w.r.t. the triplet (23) are given by

s | s× ζ0 ∈ G− \ P− , k0 ∈ K . (25) 0 (ζ0 , e0 ) := s(ζ0 ), k0 K , Note that

K s(λ) = M (λ)L+ (λ)−1 g(λ) , λ ∈ R+ , g ∈ D . (26) In the back transform the selection principle for the ζ and the multiplicity vectors k0 ∈ K again is given by (24).

5. Association of Gamov vectors to resonances According to (15) the scattering matrix S( · ) is meromorphically continuable into G− , where the poles are contained in R ∪ P− . If ζ0 is a resonance then there is a residual term S−1 , (27) z − ζ0 the first term in the main part of the Laurent expansion of S( · ) at ζ0 . Lemma 5. If ζ0 is a simple pole of S( · ) then ima S−1 ⊆ M (ζ0 ) ker L+ (ζ0 ) and the inclusion is dense w.r.t. · K

(28)

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Proof. First ζ0 is also a simple pole of L+ ( · )−1 , because it is a simple pole of z → M (z)L+ (z)−1 M (z)∗ , hence M (ζ0 )L−n M (ζ0 )∗ = 0 for n ≥ 2, where L−n is the n-th coefficient of the Laurent expansion of L+ ( · )−1 around ζ0 . This implies L−n M (ζ0 )∗ = 0 hence M (ζ0 )L∗−n = 0 and L−n = 0 follow. Now a simple calculation gives ima L−1 = ker L+ (ζ0 ). Therefore we obtain ima S−1 = ima M (ζ0 )L−1 M (ζ0 )∗ ⊆ ima M (ζ0 )L−1 = M (ζ0 ) ima L−1 = M (ζ0 ) ker L+ (ζ0 ) ,

and the inclusion is dense.

This means that ima S−1 is dense in the manifold of all selected multiplicity vectors, according to the boundary condition. It suggests that there is a connection between the residual term of the resonance in the scattering matrix and the corresponding eigenantilinear forms subjected to the boundary condition, at least in the case of simple poles. A connection of this type can be established explicitly by association of so-called Gamov vectors to resonances resp. to the corresponding eigenantilinear forms. To obtain this connection the inspection of the eigenantilinear form s× 0 (ζ0 , e0 ) given by G+ s → 2πi(s(ζ 0 ), k0 )K , corresponding to the resonance ζ0 and the Kvector k0 = M (ζ0 )e0 , suggests its extension to larger Gelfand spaces than G+ . First let G be the set of all x ∈ dom H0 such that x( · ) is meromorphic on G with poles in R ∪ P0 . G is equipped with the topology |x|K := xH0 ,+ + sup |x(z)| , z∈K

K ⊂ G \ (R ∪ P0 ) ,

(29)

where K is compact. G ⊂ H0,+ is a dense inclusion and G defines a Gelfand triplet. Then one has G+ ⊂ G . Note that the solutions of the (generalized) eigenvalue equation for H0 w.r.t. G+ and G coincide, i.e. the extension of the eigenantilinear form from G+ onto G is again continuous w.r.t. the topology of G. Note further that this is true for all parameters ζ0 , k0 , not only for the restricted values caused by the boundary condition. 2 By H± (R, K) ⊂ H0 := L2 (R, K, dλ) we denote the Hardy (sub)spaces. For 2 2 2 (R, K) =: H± . The projections onto H± are denoted by Q± . brevity we put H± Let P± be the projection given by the characteristic function χR± ( · ), such that P+ H0 can be identified with H0,+ . Note that P± and Q± are pairs of projections in generic position, this means P+ H0 ∩ Q+ H0 = P+ H0 ∩ Q− H0 = P− H0 ∩ Q+ H0 = P− H0 ∩ Q− H0 = {0} . This concept was introduced by Halmos [19]. Decisive results for this topic are due to Kato [18, p. 56 ff.] (see also [7, p. 4165 ff.]). These results imply that the 2 ⊂ H0,+ are dense. inclusions P+ H±

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2 ∩ G ⊂ H0,+ P+ H+

(30)

Lemma 6. The inclusion is dense. Proof. The inclusion Q+ P− H0 ⊂ Q+ H0 is dense because Q+ and P− are projections in generic position. Therefore the inclusions P+ Q+ P− H0 ⊂ P+ Q+ H0 ⊂ H0,+ are also dense. Further 2 P+ Q+ P− H0 ⊂ G ∩ P+ H+

because the functions of Q+ P− H0 are even holomorphic in C0 , upper sheet G+ and lower sheet G− are labeled by κ = 0, −, +, respectively. The poles of S+ ( · ) on G− are contained in R ∪ P− . Considering (33) on the basic sheet one obtains S0 (z) + S0 (z)∗ = 2 · 1lK ,

z ∈ C>0 ,

(34)

because of (3). The only poles of S0 ( · ) are the negative eigenvalues of H. Our special interest is directed to S+ ( · ) on the lower sheet and to its poles. As it is pointed out in Section 3 (p.8), R+ λ → S0 (λ + i0) =: S(λ) is the physical scattering matrix, which is unitary on R+ . The corresponding relation can be written in the form S0 (λ + i0)S+ (λ − i0)∗ = S+ (λ − i0)∗ S0 (λ + i0) = 1lK ,

λ > 0,

(35)

where we use S+ (λ − i0) = S0 (λ + i0). Since (35) is a relation for an analytic function on R+ , it is true for all z ∈ G− , i.e. S+ (z)S0 (z)∗ = S0 (z)∗ S+ (z) = 1lK ,

z ∈ G− ,

z ∈ C>0

holds. Using (34) one obtains −1 −1 S+ (z) = 2 · 1lK − S0 (z) = 1lK + 2πiM (z)L0 (z)−1 M (z)∗ ,

z ∈ G− . (36)

In the following we consider several conditions for the parameter M ( · ) together with some implications for the scattering matrix. First we introduce the condition (iii) The operator function M ( · ) is bounded at infinity together with its first derivative,, i.e. there is a radius R > 0 such that sup M (z) < ∞ ,

|z|>R

sup M (z) < ∞ .

|z|>R

(37)

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From (iii) one gets sup|z|>,z∈C>0 Φ(z) < ∞ for each > 0. This implies that there is R > 0 such that sup |z|>R,z∈G∓

and sup |z|>R,z∈G∓

Φ± (z) < ∞ .

L± (z)−1 < ∞ ,

sup |z|>R,z∈C>0

L0 (z)−1 < ∞ .

(38)

Since z → L(z)−1 is meromorphic this means that the set of its poles is bounded and accumulation points are at most from ∂G. Further from (38) one obtains sup |z|>R,z∈G−κ

Sκ (z)K < ∞ ,

κ = 0, ± ,

(39)

where G0 := C>0 and −κ := 0, ∓. That is, the set of poles of S( · ) is bounded and accumulation points are at most from ∂G. In particular, the set R of all resonances is bounded. A second condition is (iv) The operator function G z → z −γ M (z) is bounded at z = 0 for some γ ≥ 12 , i.e. there is 1 > δ > 0 and c > 0 such that M (z) ≤ c|z|γ ,

z ∈ Gδ := Kδ ∩ G ,

Kδ := {z : |z| < δ} .

Condition (iv) yields a result on the distribution of the poles of S+ ( · ). Lemma 8. Let H as before. Assume that G contains a sector G ⊃ Sβ := {z = |z|e−iφ ∈ C : |z| > 0, −β ≤ φ ≤ β} ,

0 < β ≤ π, α and condition (iv) is satisfied. Further let 0 < α < β. Then: if c < sin 2π then there is a region Sα,δ such that the estimation

−1 2πc2 S(z) ≤ 1 − , z ∈ Sα,δ , (40) sin α is valid. (Note that S( · ) means S+ ( · ) on G− and S0 ( · ) on G+ .) Proof. One has (z − H)−1 ≤

1 , |z| sin α

z ∈ S˜α := {z : α ≤ |arg z| ≤ π − α} ,

00 , i.e. Gκ = C>0 , κ = 0, ±. In particular, G− contains the negative half line. In this case, in general, the operator function L+ ( · )−1 is holomorphic on R− .

Lemma 9. Assume that M ( · ) satisfies (v). If 1 k + , k ∈ Z, 4 2 is holomorphic, i.e. there is no pole on R− . α =

then R− λ → L+ (λ)−1

(43)

Proof. In this case we have L+ (λ − i0) = L(λ) + 2πiM (λ + i0)∗ M (λ − i0) ,

λ < 0.

−1

Now, if λ := −λ0 , λ0 > 0 is a pole of L+ ( · − i0) then there is 0 = e0 ∈ E such that L(−λ0 ) + 2πiM (−λ0 + i0)∗ M (−λ0 − i0) e0 = 0 or

−λ0 − A + 0

∞

M (μ)∗ M (μ) ∗ dμ + 2πiM (−λ0 + i0) M (−λ0 − i0) e0 = 0 λ0 + μ

hence e0 , (λ0 + A)e0 = e0 ,

∞ 0

M (μ)∗ M (μ) dμ e0 λ0 + μ

−2iπα F (−λ0 )e0 2E . + 2πiλ2α 0 e

But cos 2πα = 0 for α = 1/4 + k/2, k ∈ Z. That is, for these values α it follows that F (−λ0 )e0 = 0 hence e0 = 0. Corollary 4. If the conditions (iii)–(v) are satisfied then R is bounded, the only accumulation point of R is z = 0 and the resonances around zero are contained in narrow sectors below the real line, i.e. R ∩ Kδ ⊂ {z ∈ C : z = |z|−iφ , |z| < δ, 0 < φ < α, π − α < φ < π} where sin α > 2πc2 . Additionally, if (43) is satisfied then S+ ( · ) is even holomorphic on R− . Proof. According to (v) S+ ( · ) is meromorphic on R− , i.e. the resonances cannot accumulate at points of R− , R is bounded because of (iii) and the sector assertion is implied by (iv).

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A class of operator functions M ( · ), satisfying the assumptions of Subsection 2.1 and the conditions (iii) and (v) is defined by the following properties: (vi) 1 α>− , 2 where F ( · ) is rational, F (0) = 0, P0 ∩ R = ∅, i.e. there are no poles on the real line, ∞ is not a pole of F ( · ) and F (∞) = 0. The first term of the main part of the Laurent expansion of F ( · ) at a pole ζ is of the type A(z − ζ)−n , where n > α + 32 . M (z) := z α F (z) ,

Note that conditions (iv) and (43) define subclasses of (vi) where α ≥ 12 and α = − 41 + n2 , n = 0, 1, 2, 3, . . ., respectively. For simplicity we consider these M ( · ) on the Riemannian surface of log z. In these cases Φ0 (z) can be calculated explicitly. If 2α =: m = 0, 1, 2, . . . then Φ0 (z) = z m log z · F (z)∗ F (z) − H(z) ,

z ∈ C>0 ,

(44)

where log(−1) = iπ and H( · ) is the sum of all main parts of the Laurent expansion at the poles of z m log z · F (z)∗ F (z) without the constant terms. H( · ) is holomorphic at z = 0. If 2α is not entire then e−2iπα 2α z F (z)∗ F (z) − H(z) , z ∈ C>0 , (45) sin 2πα where H( · ) and log(−1) are as before. Thus in all cases the basic sheet is characterized by log(−1) = iπ and the full analytic continuation Φ can be considered on the surface of log z. The first upper and lower sheets are of special interest. As before, the corresponding branches of Φ are denoted by Φ− and Φ+ , respectively. Now we extend the scattering operator S, originally a unitary operator on H0,+ , to an operator on H0 using the values S+ (λ) for λ < 0. We define Φ0 (z) = π

H0 f → Sf ∈ H0 by

(Sf )(λ) = Sι (λ)f (λ) :=

S(λ)f (λ) , λ > 0 , S+ (λ)f (λ) , λ < 0 .

If sup S+ (λ)K < ∞

(46)

λ∈R−

then S is a bounded operator on H0 and S ≤ max 1, sup S+ (λ)K

.

λ∈R−

If (43) is satisfied then the behaviour of S+ ( · ), which is holomorphic for λ < 0, at z = 0 is decisive for the boundedness of S. For example, if S( · ) is uniformly bounded in the neighborhood of z = 0 then S is bounded.

140

H. Baumg¨ artel


Lemma 10. Let M be from the class (vi), α = 0, ker M (0) = {0} and dim K < ∞. Then (47) sup S(z) < ∞ , > 0 , |z| 0 are satisfied then S is bounded. If additionally R is finite then it turns out that the resonance poles are necessarily simple. Theorem 4. Let M satisfy condition (iii) and assume S to be bounded and R to be finite.Then the poles of the scattering matrix S( · ) at R ∪ P− are necessarily simple and the Laurent representation of S( · ) in C 0, is the following derivation and DG operator with respect to the variable z = (x, v): ∂k k DG = CG (α1 . . . αk ) α1 , (6.8) ∂z . . . ∂z αk α1 ...αk : αj =1,...,6

where CG (α1 . . . αk ) =

1 k!

R6

dζ e−ζ

2

k

ζ αj .

(6.9)

j=1

Therefore, CG (α1 . . . αk ) is equal to zero for each sequence α1 . . . αk in which at least one index appears an odd number of times. Hypotheses H. In the present paper we assume that the probability density g = (0) f0 ∈ S(R3 × R3 ), thus (6.7) make sense for any n ≥ 1. As regard to the pair interaction potential φ, we assume that φ ∈ C ∞ (R3 ), that any derivative of φ is uniformly bounded (in order to be able to apply Proposition 5.1) and that φ is spherically symmetric. Remark 6.1. In this paper we consider a completely factorized N -particle initial state. Furthermore the one-particle state is a mixture and this automatically excludes the Bose statistics.

162

F. Pezzotti and M. Pulvirenti


Remark 6.2. We made the choice to expand fully the initial state f0 according to equation (6.5). Another possibility is to assume the (ε dependent) state f0 (which is a probability measure in the present case) as initial condition for the Vlasov (k) problem and, consequently, f0 = 0 for the problems (2.15). Now the coefficients f (k) (t) are ε dependent but this does not change deeply our analysis because f0 is smooth, uniformly in ε. Under hypotheses H, we can give a sense to the linear problem (2.15) for any k ≥ 1, by virtue of the following proposition, whose (straightforward) proof will be given in Appendix B. Proposition 6.1. Consider the following initial value problem: (∂t + v · ∇x ) γ = L(h)γ + Θ , γ(x, v; t)|t=0 = γ0 (x, v) ,

(6.10)

1 3 0 L (R × R3 ), R+ , with γ0 ∈ L1 (R3 × R3 ), h = h(x, v; t) is such that |∇ v h| ∈ C Θ = Θ(x, v; t) is such that Θ ∈ C 0 L1 (R3 × R3 ), R+ . Then, there exists a unique solution γ = γ(x, v; t) of (6.10), such that γ ∈ C 0 (L1 (R3 × R3 ), R+ ), given by an by L(h), explicit series expansion. Furthermore, denoting by Σh the flow generated we have that Σh (t, 0)γ0 ∈ C k R3 × R3 provided that ∇v h ∈ C k R3 × R3 and γ0 ∈ C k (R3 × R3 ). The main goal of the present paper is to compare the j-particle semiclassical (k) expansion associated with the N -particle flow, namely WN,j (t), k = 0, 1, 2, . . . , (k)

with the corresponding coefficients fj (t) of the expansion: (0)

(1)

(k)

fjε (t) = fj (t) + εfj (t) + · · · + εk fj (t) + · · · ,

(6.11)

(k)

where fj (t) is given by (4.9). The main result is the following. Theorem 6.1. Under the Hypotheses H, for all t > 0, for any integers k and j, the following limit holds in S (R3j × R3j ): (k)

(k)

WN,j (t) → fj (t) .

(6.12)

as N → ∞. Remark 6.3. As we shall see in the sequel, the convergence (6.12) is slightly (k) stronger than the convergence in S (R3j × R3j ). Indeed, the sequence WN,j (t) ∞ 3j 3j converges also when it is tested on functions in Cb (R × R ), namely, the space of functions which are uniformly bounded and infinitely differentiable. Such kind of convergence, which is natural in the present context, will be called Cb∞ -weak convergence. Proposition 6.2. Let γN (x, v; t) be a sequence in S (R3 ×R3 ) (for each t) satisfying: (∂t + v · ∇x ) γN = L(hN )γN + ΘN , (6.13) γN (x, v; t)|t=0 = γN,0 (x, v) ,

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where γN,0 , ΘN are sequences in S (R3 × R3 ). We assume that: i) hN (x, v; t) is a sequence of probability measures converging, 3 as N → ∞, ∞ 3 R and such that × R to a measure h(t)dxdv with a density h(t) ∈ C b |∇v h| ∈ C 0 L1 (R3 × R3 ), R+ . ii) for all u1 , u2 in Cb∞ (R3 × R3 ) , there exists a constant C = C(u1 , u2 ) > 0, not depending on N , such that:

u1 ∗ (u2 γN ) L∞ (R3 ×R3 ) < C < +∞

for any t .

(6.14)

iii) γN,0 → γ0 , as N → ∞, Cb∞ -weakly , γ0 = γ0 (x, v) is a function belonging to L1 (R3 × R3 ). ∞ iv) ΘN → Θ, as N → ∞, Cb -weakly , Θ = Θ(x, v; t) is a function belonging to 0 1 3 3 + C L (R × R ), R . Then: (6.15) γN → γ , as N → ∞ Cb∞ -weakly , where γ is the unique solution of the problem (6.10) in C 0 L1 R3 × R3 , R+ . For the proof, see Appendix B.

7. Convergence This section is devoted to the proof of Theorem 6.1. By (3.17) and (3.14), for k ≥ 0 we have: (k)

WN (ZN ; t) =

k

n≥0 r=0

r1 ...rn :

rj >0 rj =k−r

t

t1

dt1

tn−1

dt2 . . .

0

0

dtn 0

(r )

(r )

(r)

SN (t − t1 )TN 1 SN (t1 − t2 ) . . . TN n SN (tn )WN,0 (ZN ) .

(7.1)

It is useful to remind that the only non-vanishing terms in (7.1) are those for which all r1 , . . . , rn are even. According to (3.11) and (6.7), (r) WN,0 (ZN )

=

N

2s (0) DG,jj f0 (zj ) ,

(7.2)

s1 ...sN j=1 0≤sj ≤r

j sj =r k where DG,j is defined in (6.8) and the extra symbol j means that this operator acts on the variable zj . Defining the operator D2r as:

D0 = 1 , D

2r

=

N

s1 ...sN : j=1 0≤sj ≤r

j sj =r

2s

DG,jj ,

r ≥ 1,

(7.3)

164



we have: (r)

(0)

WN,0 (ZN ) = D2r WN,0 (ZN )

∀ r ≥ 0.

(7.4) (k)

In order to investigate the behavior of the j-particle functions WN,j (Zj ; t) when N → ∞, we consider the following object, for a given configuration Zj = (z1 . . . zj ): (k) (k) ωN,j (Zj ; t) = dZN WN (ZN ; t)μN (z1 |ZN ) . . . μN (zj |ZN ) . (7.5) R6N

In the end of the section, we will show that (7.5) is asymptotically equivalent to (k) WN,j (Zj ; t). From (7.1), (7.4) and (7.5), it follows that: ωN,j (Zj ; t) = (k)

k n≥0 r=0

r1 ...rn :

rj >0 rj =k−r

t

t1

dt1 0

0

dtn 0

(r )

tn−1

dt2 . . .

R6N

dZN μN,j (Zj |ZN )

(r )

(0)

SN (t − t1 )TN 1 SN (t1 − t2 ) . . . TN n SN (tn )D2r WN,0 (ZN ) ,

(7.6)

where μN,j (Zj |ZN ) = μN (z1 |ZN ) . . . μN (zj |ZN ) .

(7.7) (r )

Integrating by parts, reminding that each rj is even and that each TN j involves derivatives of order rj + 1, we have: ωN,j (Zj ; t) = (k)

(−1)n

n≥0

k

r=0 rn : rj >0 |rn |=k−r

t

dtn ord

(r ) (r ) (r ) EN D2r TN n (tn )TN n−1 (tn−1 ) . . . TN 1 (t1 )μN,j Zj |ZN (t) , (7.8) where rn is the sequence of positive integers r1 , . . . , rn , |rn | = the the the

n

j=1 rj and ZN (t) is t Hamiltonian flow defined in (5.2). Moreover tn = t1 . . . tn and ord dtn denotes integral over he simplex 0 < tn < tn−1 < · · · < t1 < t. Finally, EN stands for (0) expectation with respect to the N -particle density WN,0 and (r)

(r)

TN (t) = SN (−t)TN SN (t) .

(7.9)

Therefore, the objects we have to investigate in the limit N → ∞ are: νj (Zj ; t) = (k)

n≥0

(−1)n

k

r=0 rn : rj >0 |rn |=k−r

t

ord

dtn ηj (Zj ; t, r, rn , tn , ZN ) ,

(7.10)

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165

(0)

(for any configuration ZN , typical with respect to f0 ), where ηj is given by: ηj (Zj ; t, r, rn , tn , ZN )

(r ) (r ) (r ) = D2r TN n (tn )TN n−1 (tn−1 ) . . . TN 1 (t1 )μN,j Zj |ZN (t) .

Note that:

(0) νj (Zj ; t) = μN,j Zj |ZN (t) . (k)

We start by analyzing the behavior of νj lead to consider:

(7.11)

(7.12)

in the cases j = 1, 2, thus we are

η1 (z1 ; t, r, rn , tn , ZN )

(r ) (r ) (r ) = D2r TN n (tn )TN n−1 (tn−1 ) . . . TN 1 (t1 )μN z1 |ZN (t) ,

(7.13)

and η2 (z1 , z2 ; t, r, rn , tn , ZN )

(r ) (r ) (r ) = D2r TN n (tn )TN n−1 (tn−1 ) . . . TN 1 (t1 )μN,2 Z2 |ZN (t) .

(7.14)

(r )

It is useful to stress that the operators TN j (tj ) (j = 1, . . . , n) and D2r act as suitable distributional derivatives with respect to the variables ZN . To evaluate (r) η1 , let us first analyze the action of TN (τ ). By (7.9) and (3.18), for any function G = G(ZN ), we have: (r) TN (τ ) G (ZN ) (r) (r) = SN (−τ )(TˆN + RN ) SN (τ )G (ZN ) cr SN (τ )G (ZN ) SN (−τ ) Dxr+1 φ(xj − xl ) · Dvr+1 = (−1)r/2 j N j,l

+

·

1 N

N

l,j=1

k1 ,k2 ∈N3 |k1 |+|k2 |=r+1

Ck1 ,k2 SN (−τ )

∂ r+1 |k | |k |

∂xl1 ∂xj2

φ(xl − xj )

∂ r+1 |k | |k |

∂v l 1 ∂v j 2 × SN (τ ) G (ZN ) .

(7.15)

Note that the derivatives involved here are done with respect to the variables at time t = 0. Denoting by Dzrj any derivative of order r with respect to a variable zj at time t = 0, we observe that: SN (−t)Dzrj G(ZN ) = (Dzrj G) ZN (t) = Dzrj (t) SN (−t)G (ZN ) , (7.16)

166



where, by Dzrj (t), we denote the same derivative of order r with respect to the variable zj (t). Then, by (7.16) and (7.15): (r)

(r)

(r)

(TN (τ ) G) (ZN ) = SN (−τ ) (TˆN + RN )SN (τ ) G (ZN ) cr r+1 Dx φ xj (τ ) − xl (τ ) = (−1)r/2 N j,l

· Dvr+1 (τ )G (ZN ) j N 1 + N

l,j=1

·

∂ r+1 |k | |k |

∂v l 1 ∂v j 2

Ck1 ,k2

k1 ,k2 ∈N3 |k1 |+|k2 |=r+1

∂ r+1 |k | |k | ∂xl1 ∂xj2

φ

xl (τ ) − xj (τ )

(τ )G (ZN ) .

(7.17)

(r)

Therefore, in computing the action of TN (τ ), we have to consider derivatives with respect to the variables at time τ . As a consequence, we have to deal with a complicated function of the configuration ZN which, however, we do not need to make explicit, as we shall see in a moment. On the basis of the previous considerations, we compute the time derivative of (r ) (r ) (r ) η1 by applying the operators D2r TN n (tn )TN n−1 (tn−1 ) . . . TN 1 (t1 ) to the Vlasov equation: (7.18) (∂t + v1 · ∇x1 )μN (t) = ∇x1 φ ∗ μN (t) · ∇v1 μN (t) . In doing this we have to compute (r ) (r ) (r ) D2r TN n (tn )TN n−1 (tn−1 ) . . . TN 1 (t1 )μN z1 |ZN (t) μN z2 |ZN (t) .

(7.19)

(r )

Now we select the contribution in which each TN (t ) and D2r apply either on μN (z1 |ZN (t)) or to μN (z2 |ZN (t)). The other contribution involves terms in which are present products of derivatives with respect to the same variable. By Proposition 5.1 and Proposition 5.2 we expect those terms to be negligible (in the Cb∞ -weak sense) in the limit N → ∞. Therefore we obtain the following equation: (∂t + v1 · ∇x1 )η1 (z1 , t, r, rn , tn , ZN ) = L μN (t) η1 (z1 , t, r, rn , tn , ZN ) + ∇x1 φ ∗ η1 ( · , t, , rI , tI , ZN ) 0≤≤r 0≤m≤n

I⊂In : |I|=m, 00 |rn |=k−r

dtn η1 (z1 ; t, r, rn , tn , ZN )

t

ord

t1 =t

dtn ∂t + v1 · ∇x1 η1 (z1 ; t, r, rn , tn , ZN ) . (7.22)

In evaluating the first term on the right hand side of (7.22), we are lead to consider η1 evaluated in t = t1 . Thus, according to the expression of η1 (see (7.13)), we have to deal with: (r ) (r ) (7.23) TN 1 (t)μN z1 |ZN (t) = SN (−t)TN 1 μN (z1 |ZN ) . Therefore:

(r ) TN 1 (t)μN z1 |ZN (t) = (−1)r1 /2 cr1 Dxr1 +1 φ ∗ μN (t) (x1 ) · Dvr1 +1 μN z1 |ZN (t) 1 1 r1 +1 r1 /2 cr1 dx2 dv2 Dx φ(x1 − x2 ) · Dvr1 +1 = (−1) 1 1 × μN x1 , v1 |ZN (t) μN x2 , v2 |ZN (t) , (7.24) (r )

where the term involving off-diagonal derivatives, namely RN 1 (see (3.20)), disappears because both the derivatives and the empirical distribution are evaluated at time t. Hence we compute η1 in t = t1 and, inserting it in the first term of the right hand side of (7.22), we obtain: tn−1 t t2 k n (−1) dt2 dt3 . . . dtn η1 (z1 ; t, r, rn , tn , ZN ) n≥0

=

r=0 rn : rj >0 |rn |=k−r

(−1)r1 /2 cr1

0

0

(k−r1 )

1

t1 =t

dx2 dv2 Dxr1 +1 φ(x1 − x2 )

00 |rn |=k−r

ord

sup dydw u1 (x1 − y, v1 − w)u2 (y, w) x1 ,v1 (r ) (r ) D2r TN n (tn ) . . . TN 1 (t1 )μN y, w|ZN (t) k t = dtn (r ) (r ) sup dydw g(x1 , v1 , y, w)D2r TN n (tn ) . . . TN 1 (t1 )μN y, w|ZN (t) , (7.34)

x1 ,v1

where we used the notation g(x1 , v1 , y, w) := u1 (x1 −y, v1 −w)u2 (y, w) and, clearly, we have g(x1 , v1 , · , · ) ∈ Cb∞ (R3 ×R3 ) for any x1 and v1 and g( · , · , y, w) ∈ Cb∞ (R3 × R3 ) for any y and w. By some estimates which will be proven in Appendix C (r ) (see Lemma C.2), we are guaranteed that, applying the operator D2r TN n (tn ) . . . (r ) TN 1 (t1 ) on the empirical measure μN (t) and integrating versus a function in

170



Cb∞ (R3 × R3 ) we obtain a quantity uniformly bounded in N . This feature, by virtue of the good properties of the function g ensures that (7.34) is finite. (k) Let us now look at the initial datum for ν1 (t), in order to verify assumption iii). (k) From (7.28) we know that ν1 (0) = D2k μN ∈ S (R3 × R3 ). As regard to its limiting behavior, we find that: (k)

ν1 (t)|t=0 = D2k μN =

N

2s

DG,jj μN ,

(7.35)

n=1 I⊂IN sj :j∈I j∈I |I|=n 1≤sj ≤k

j sj =k

where IN = {1, . . . , N }. For our convenience, we have written the action of the operator D2k in a equivalent and slightly different way from that we used in (7.3). We realize that the only surviving term in the sum (7.35) is that with n = 1. Hence: ν1k (t)|t=0 =

N

2k DG,j μN =

j=1

N 1 2k 2k D δ(z − zj ) = DG μN . N j=1 G,j 1

(7.36)

Therefore we can conclude, by using the mean-field limit: (k)

2k (u, ν1 (t)|t=0 ) = (u, DG μN ) (0)

2k 2k = (DG u, μN ) → (DG u, f0 ) (0)

(k)

2k = (u, DG f0 ) = (u, f0 ) ,

∀u

in

Cb∞ (R3

as

N → ∞,

× R ). 3

(7.37)

(k) Thus, f0 plays the role of γ0 in Proposition 6.2 and it is in L1 R3 × R3 because (0) f0 ∈ S R3 × R3 . We conclude the convergence proof (for the one and two-particle functions) by induction. For k = 0 we know that, for any configuration ZN which is typical (0) with respect to f0 , we have: (0)

ν1 (t) = μN (t) → f (0) (t) ,

as

N → ∞,

(7.38)

in the weak sense of probability measures, and, as a consequence, the convergence holds Cb∞ − weakly. Moreover (0)

(0)

ν2 (t) = μN (t) ⊗ μN (t) → f2 (t) = f (0) (t) ⊗ f (0) (t) ,

as

N → ∞,

(7.39)

in the weak sense of probability measures, and, as a consequence, the convergence holds Cb∞ − weakly. We make the following inductive assumptions for all h < k: (h)

ν1 (t) → f (h) (t) ,

as

N → ∞,

Cb∞ − weakly ,

(7.40)

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(0)

for any configuration ZN which is typical with respect to f0 , and (h)

(h)

ν2 (t) → f2 (t) =

f (q) (t)f (h−q) (t) ,

0≤q≤h

Cb∞ − weakly ,

N → ∞,

as

(7.41)

(0)

for any configuration ZN which is typical with respect to f0 . Now we want to prove that (7.40) and (7.41) hold also for h = k. Thanks to (7.40), we can affirm that: () (k−) (∇x1 φ ∗ ν1 ) · ∇v1 ν1 → (∇x1 φ ∗ f () ) · ∇v1 f (k−) 0 0. • S(A, B; k) = 1 − 2P for some, and hence for all, k > 0. • AB ∗ = 0, i.e., L = 0. The last point shows that k-independent S-matrices arise exactly in the case of non-Robin boundary conditions. Below we are going to prove some properties of S-matrices that are relevant for the trace formula. In this context, for Robin boundary conditions an important role will be played by the spectrum σ(L) of the self adjoint matrix L (2.6). We shall make extensive use of the S-matrix extended to complex wave numbers k and therefore need the following. Lemma 3.1. Let A and B specify self adjoint boundary conditions for the Laplacian on the graph. Then the S-matrix (3.3) has the following properties: 1. S(A, B; k) can be continued into the complex k-plane as a meromorphic function, and has simple poles at the points of the set iσ(L) \ {0}. 2. S(A, B; k) is unitary for all k ∈ R. 3. S(A, B; k) is invertible for all k ∈ C \ [±iσ(L) \ {0}], and its inverse is S(A, B; −k).

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The Trace Formula for Quantum Graphs

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Proof. We henceforth extend the self adjoint endomorphism L of ran B ∗ , see (2.6), to an endomorphism of C2E by setting it to zero on (ran B ∗ )⊥ = ker B. We then diagonalise L utilising an appropriate unitary W , and denote the non-zero eigenvalues (counted with their multiplicities) by {λ1 , . . . , λd }. This leaves the eigenvalue zero with a multiplicity of 2E − d. There are r := dim ran B ∗ − d (orthonormal) eigenvectors of L in ran B ∗ and s := dim ker B = 2E − dim ran B ∗ eigenvectors in ker B, respectively, corresponding to the eigenvalue zero. Employing this diagonalisation in the representation (3.4) of the S-matrix then leads to the expression ⎛ λ1 −ik ⎞ − λ1 +ik ⎜ ⎟ .. ⎜ ⎟ . ⎟ ∗⎜ λd −ik (3.5) S(A, B; k) = W ⎜ ⎟W . − λd +ik ⎜ ⎟ ⎝ ⎠ 1r −1s Since the unitary W is independent of k the first statement of the lemma is obvious. The unitarity of S(A, B; k) for real k also follows immediately by observing that the diagonal entries in (3.5) are all of unit absolute value. The third statement follows in a completely analogous fashion from the representation (3.5). Knowing that the S-matrix is analytic in k, one would like to calculate its derivative. This in fact is required in the proof of the trace formula below. It is even possible to relate the derivative of S(k) to the S-matrix itself. Lemma 3.2. Under the same assumptions as in Lemma 3.1 one obtains for k ∈ C \ [±iσ(L) \ {0}], d 1 S(A, B; k) = − S(A, B; k) − S(A, B, k)−1 S(A, B; k) . dk 2k

(3.6)

We remark that for real k the unitarity of the S-matrix can be invoked to obtain from (3.6) that it is independent of k, iff it is self adjoint. Proof. Let us first assume that k ∈ R \ {0} and abbreviate S(A, B; k) as S(k). d We also denote derivatives w.r.t. k by a dash and use the relation dk [X(k)]−1 = −1 −1 −X(k) X (k)X(k) , which is true for any differentiable function X(k) taking values in GL(2E, C). Recall that the conditions imposed on A, B ensure that A ± ikB is invertible for k ∈ R \ {0}. Hence S (k) = (A + ikB)−1 (iB)(A + ikB)−1 (A − ikB) + (A + ikB)−1 (iB) = −i(A + ikB)−1 B S(k) − 1 .

(3.7)

The last line is obviously invariant under a replacement of A and B by CA and CB, respectively, where C ∈ GL(2E, C) is arbitrary.

198

J. Bolte and S. Endres


Now choose C(k) = (A + ikB)−1 ∈ GL(2E, C) and find that (see also [23]) C(k)A = −

1 S(k) − 1 2

and C(k)B = −

1 S(k) + 1 . 2ik

(3.8)

Inserting this into (3.7) finally leads to 1 S(k) + 1 S(k) − 1 2k 1 =− S(k) − S(k)∗ S(k) , 2k

S (k) =

(3.9)

which proves the statement for k ∈ R \ {0}. From Lemma 3.1 we infer that S(k) is analytic in a neighbourhood of k = 0 and that the right-hand side of (3.9) has a removable singularity at k = 0; hence (3.9) extends to all real k. Since, moreover, S(k) is unitary on R and analytic on C \ [±iσ(L)], and k1 as well as S(k)−1 are also analytic on this set, the full statement of the lemma follows by analytic continuation. From Lemma 3.1 we know that S(k) is meromorphic with finitely many poles on the imaginary axis. One can therefore perform power series expansions of the S-matrix in large parts of the complex k-plane. Below we want to specify two such expansions, and to this end we exclude an annulus containing iσ(L) \ {0}. This annulus is characterised by the two radii λmin / max := min / max |λ|; λ ∈ σ(L) \ {0} . Here we assume that σ(L) = {0}; otherwise L = 0 which is equivalent to the S-matrix being independent of k. We are now in a position to provide the announced expansions of the Smatrix. Lemma 3.3. Let the same conditions as in Lemma 3.1 be given and assume that L = 0. Then the following power series expansions converge absolutely and uniformly in any closed subsets of the specified regions: 1. For |k| > λmax , S(A, B; k) = 1 − 2P + 2

∞ 1 (iL)n . n k n=1

(3.10)

2. For |k| < λmin , S(A, B; k) = −1 + 2P˜ − 2

∞

n ˜ k n iL ,

(3.11)

n=1

˜ emerge from P and L, respectively, by replacing A, B with where P˜ and L −B, A.

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Proof. For the first expansion we refer to the representation (3.5) of the S-matrix and employ the expansion n ∞ 1 + ki λα iλα λα − ik − = = 1 + 2 , α = 1, . . . , d , λα + ik k 1 − ki λα n=1 ∞ valid for |k| > λα . Hence, for |k| > λmax the S-matrix is 1 + 2 n=1 k1n (iL)n on ran B ∗ and −1 on ker B. Since L = 0 on ker B, the relation (3.10) follows. For the second expansion we remark that from (3.3) one can readily deduce the relation S(A, B; k) = −S(−B, A; k1 ) for k ∈ R \ {0}, see [22]. Thus, (3.10) implies (3.11), and the domain |k| < λmin of convergence, in which S(k) is holomorphic, follows immediately. Lemma 3.3 also provides limiting expressions for the edge S-matrix as |k| → ∞ and |k| → 0, respectively, S∞ = 1 − 2P

and S0 = −1 + 2P˜ ,

which we shall use subsequently. Later we shall integrate expressions containing the S-matrix along contours in the upper complex half plane and, therefore, we need to estimate the norm of the S-matrix along the contours. To this end we introduce

min{λ ∈ σ(L); λ > 0} , if ∃λα > 0 + λmin := , (3.12) ∞, else and obtain the following. Lemma 3.4. Let k ∈ R and 0 < κ < λ+ min , then −1

S(k + iκ) , S(k − iκ)

λ+ min + κ ≤ max 1, + λmin − κ

(3.13)

in the operator norm. Furthermore, if κ > λmax , then S(k + iκ) ,

S(k − iκ)−1 ≤

κ + λmax . κ − λmax

(3.14)

Proof. From (3.5) one can read off the eigenvalues of S(k + iκ) as ±1 and −(λα + κ − ik)/(λα − κ + ik), α = 1, . . . , d. In absolute values the latter quantities, as functions of k ∈ R, are maximised at k = 0. Now suppose that λ+ min > 0 and let . Then the largest quantity among the |λ + κ|/|λ − κ| is the one on 0 < κ < λ+ α α min = ∞ the upper bound is one. The the right-hand side of (3.13). In the case λ+ min proof for S(k − iκ)−1 is completely analogous. If κ > λmax the same argument yields the bound (3.14).

200



4. The spectrum of the Laplacian The scattering approach to the quantisation of a finite, metric graph utilises a secular equation based on the edge S-matrix of the graph. Here we closely follow the original approach as developed by Kottos and Smilansky [21] for the case of (generalised) Kirchhoff boundary conditions, which was later generalised by Kostrykin and Schrader [23]. To keep the presentation sufficiently self-contained, we reproduce the relevant results below. We begin, however, with some general properties of Laplace spectra and finish this section with some remarks on the eigenvalue zero. 4.1. Preliminaries on the spectrum Given a Laplacian on a compact metric graph, one would naturally expect that its spectrum is discrete and has a finite lowest eigenvalue. Kuchment indeed proved [24] that any such self adjoint (negative) Laplacian is bounded from below, and that its resolvent is trace class. Thus, the spectrum is discrete and bounded from below. Subsequently, Kostrykin and Schrader [23] improved the lower bound. They showed that −Δ ≥ −s2 ,

(4.1)

where s ≥ 0 is the unique solution of slmin s tanh = λ+ max , 2 with lmin denoting the shortest edge length and

max{λ ∈ σ(L); λ > 0} , + λmax := 0,

(4.2)

if ∃λα > 0 . else

(4.3)

Remark 4.1. As an aside we should like to mention that the lower bound (4.1) is optimal in the following sense: Consider a trivial example of a metric graph given by an interval I of length l, and a Laplacian with domain specified by the choice A = λ12

and B = 12 ,

where λ > 0, such that L = λ12 and σ(L) = {λ}. (Equivalently, P = 0, Q = 12 and L = λ12 .) Hence, (2.7) leads to the Robin boundary conditions f (0) f (0) λ + = 0, f (l) −f (l) and this implies the quantisation condition 2 λ − ik e2ikl = 1 . λ + ik

(4.4)

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The solution k = iκ, with κ > λ > 0, representing the lowest Laplace eigenvalue −κ2 corresponds to a solution of the equation κl κ tanh = λ. 2 This condition is equivalent to (4.2), demonstrating that the bound (4.1) is sharp for this ‘quantum graph’. Kostrykin and Schrader also showed [23] that the number of negative Laplace eigenvalues is bounded by the number of positive eigenvalues of L (counted with their respective multiplicities). In the example above, the Robin Laplacian on an interval hence has at least one and at most two negative eigenvalues. For the trace formula one requires an a priori estimate on the number of eigenvalues. This is well known for the Dirichlet and the Neumann Laplacian, and in the case of Kirchhoff boundary conditions can be found in [33]. The same asymptotic law, however, holds also in the general case. Proposition 4.2. Given a self adjoint realisation of the Laplacian on a compact metric graph, the number of its eigenvalues kj2 ∈ R (counted with their multiplicities) fulfils the following asymptotic law, L (4.5) N (K) := # j; kj2 ≤ K 2 ∼ K , K → ∞ , π where L := le1 +· · ·+leE is the total length of the graph. In particular, the Laplacian has infinitely many eigenvalues that only accumulate at infinity. Proof. We prove the asymptotic law employing a variational characterisation of the eigenvalues based on the quadratic form (2.9), as well as an analogue of the Dirichlet-Neumann bracketing (see [31]). To this end we introduce two comparison operators. The first comparison operator is the direct sum of the Dirichlet operators on the edges, i.e., the Dirichlet-Laplacian of Section 2.2. The domain of the associated quadratic form is characterised by the condition Fbv = 0, and therefore is contained in the domain of (2.9). Moreover, on the Dirichlet-form domain both quadratic forms coincide. The comparison lemma devised in [31] hence implies that ND (K) ≤ N (K). Here ND (K) is the counting function for the eigenvalues of the DirichletLaplacian, which trivially fulfils the asymptotic law (4.5). As our second comparison operator we choose the direct sum of Robin Laplacians on the edges (see the example above). This operator is characterised by PR = 0, QR = 12E and L = λ12E with some λ ≥ 0. (λ = 0 in fact corresponds to = 0, Neumann Laplacians on the edges.) The boundary conditions are λFbv + Fbv and thus they decouple the edges. The respective eigenvalue counting function NR (K) can be determined from (4.4) and clearly obeys the asymptotic law (4.5). The associated form domain, characterised by the condition PR Fbv = 0, contains the domain of (2.9). Choosing λ = λ+ max , see (4.3), on the form domain of

202



(2.9) one finds QR [F ] ≤ QΔ [F ]. Therefore, the comparison lemma of [31] implies NR (K) ≥ N (K). Thus, ND (K) ≤ N (K) ≤ NR (K) and the upper and the lower bounds both fulfil the same asymptotic law, which proves (4.5). The further statement follows immediately. This result is independent of any details of the quantum graph, apart from its volume. This is analogous to the corresponding results for Laplacians on manifolds or domains. In general, the asymptotic growth of the number of eigenvalues is proportional to the volume of the manifold/domain and to K D , where D is the dimension of the manifold/domain. This type of results is often referred to as ‘Weyl’s law’ [35], and insofar Proposition 4.2 is the quantum graph version of Weyl’s law. 4.2. The secular equation Apart from the edge S-matrix, the scattering approach requires the metric information of the graph, which enters through ⎞ ⎛ ikl e 1 0 t(l; k) ⎟ ⎜ .. T (l; k) := with t(l; k) := ⎝ (4.6) ⎠, . t(l; k) 0 iklE e where k ∈ C. Both matrices are then used to introduce U (k) := S(A, B; k) T (l; k) .

(4.7)

The topological and the metric data entering U (k) are hence clearly separated. For real k the endomorphisms S(k), T (k) and U (k) of C2E are obviously unitary. We therefore denote the eigenvalues of U (k) by eiθ1 (k) , . . . , eiθ2E (k) . Following Lemma 3.1 we conclude that U (k) can be extended into the complex k-plane as a meromorphic function with poles at iσ(L) \ {0}. The determinant function F (k) := det 1 − U (k) , (4.8) on which the scattering approach is based (see [21]), hence is also meromorphic. Its poles are in iσ(L) \ {0}, but do not necessarily exhaust the entire set. Proposition 4.3 (Kostrykin, Schrader [23]). The determinant function (4.8) is meromorphic on the complex plane with poles in the set iσ(L) \ {0}. Furthermore, let kn ∈ C \ [iσ(L) ∪ {0}] with Im kn ≥ 0, then kn2 is an eigenvalue of −Δ, iff kn is a zero of the function (4.8), i.e., F (kn ) = 0. Moreover, the spectral multiplicity gn of the Laplace eigenvalue kn2 > 0 coincides with the multiplicity of the eigenvalue one of U (kn ). Proposition 4.3 establishes a close connection between zeros of the determinant function (4.8) and Laplace eigenvalues. Notice that although Laplace eigenvalues occur as squares, k 2 , the function (4.8) is not invariant under a change of sign in its argument. There exists, however, a functional equation under the substitution k → −k.

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Lemma 4.4. For all C \ [±iσ(L) \ {0}] the following identity holds: d λα − ik M 2ikL F (−k) , F (k) = (−1) e λ + ik α=1 α

203

(4.9)

where M = E + d + dim ker B and L = le1 + · · · + leE is the sum of all edge lengths. Proof. We decompose 1 − S(k)T (k) = −S(k) 1 − S(−k)T (−k) T (k) , so that after taking determinants, F (k) = (−1)2E det T (k) det S(k) F (−k) . Using the definition (4.6) of T (k) and the representation (3.5) of S(k) then yields (4.9). We remark that when k 2 ∈ R is a Laplace eigenvalue, then either k ∈ R when the eigenvalue is non-negative or, in the case of a negative eigenvalue, k = ±iκ with κ > 0. The relevant zeros of the determinant (4.8) are then ±k ∈ R or iκ, respectively. Unless κ ∈ σ(L) or k = 0, all Laplace eigenvalues are covered by Proposition 4.3. However, in order to count Laplace eigenvalues in terms of zeros of F (k) with their correct multiplicities one has to establish a connection between the order of the zero and the multiplicity of one as an eigenvalue of U (k). In the trace formula we shall need this connection for the non-negative eigenvalues and hence now consider the eigenphases θ(k) of U (k), defined through U (k)v(k) = eiθ(k) v(k)

with

v(k) = 1 ,

(4.10)

for k ∈ R. We then recall that U (k) = S(A, B; k)T (l; k) is analytic in C \ [iσ(L) \ {0}]. According to analytic perturbation theory (see, e.g., [16]) its eigenvalues, for which we keep the notation eiθ(k) , are continuous on this set and differentiable apart from possibly isolated points. Since, however, U (k) is real-analytic and normal for all k ∈ R, we can apply a sharpened version of analytic perturbation theory (see [26]) to conclude that the eigenvalues are real analytic for all k ∈ R and that there exists a choice of eigenvectors with the same property. The following statement is a generalisation of a result found in [5, 21] that is valid for k-independent S-matrices. Lemma 4.5. Let θ(k) be an eigenphase of U (k), k ∈ R, with associated normalised eigenvector v(k) = (v1 (k), . . . , v2E (k))T . Then 2E d L 2 θ(k) = li |vi (k)| − 2 v(k), 2 v(k) . dk L + k2 C2E i=1

(4.11)

204



Proof. Taking Lemma 3.2 into account we first observe that 1 S(k) − S ∗ (k) S(k)T (k) + iS(k)T (k)D(l) U (k) = − 2k L = −2i 2 U (k) + iU (k)D(l) , L + k2 where ⎞ ⎛ l1 D1 (l) 0 ⎟ ⎜ .. D(l) := , D1 (l) := ⎝ ⎠. . 0 D1 (l) lE We also employed the relation L 1 S(k) − S ∗ (k) = −2i 2 − 2k L + k2

(4.12)

(4.13)

(4.14)

that follows from (3.4). This then yields d L U (k)v(k) = −2i eiθ(k) v(k), 2 2 v(k) + i U ∗ (k)v(k), D(l)v(k) v(k), dk L +k ∗ + U (k)v(k), v (k) L = −2i eiθ(k) v(k), 2 v(k) + i eiθ(k) v(k), D(l)v(k) 2 L +k iθ(k) v(k), v (k) . (4.15) +e On the other hand, taking the derivative on the right-hand side of (4.10) and then multiplying with v(k) leads to d iθ(k) e v(k), v(k) = iθ (k)eiθ(k) + eiθ(k) v(k), v (k) . (4.16) dk

Comparing (4.15) and (4.16) proves the statement (4.11).

This lemma allows us to obtain an upper and a lower bound for the derivative of an eigenphase. Using lmax / min to denote the largest and the smallest edge length, respectively, and introducing

min{|λ|; λ ∈ σ(L) ∩ R− }, if ∃λα < 0 − λmin := , ∞, else in analogy to (3.12), we immediately get the following. Corollary 4.6. The derivative θ (k) of an eigenphase θ(k) is bounded from above and below according to lmin −

2 λ+ min

≤ θ (k) ≤ lmax +

2 λ− min

.

(4.17)

In particular, if lmin > 2/λ+ min the derivatives of all eigenphases are always positive.

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Proof. Obviously, lmin ≤

2E

li |vi (k)|2 ≤ lmax ,

i=1

since the eigenvector is supposed to be normalised. Moreover, after a diagonalisation of L, when W LW ∗ is diagonal with the eigenvalues λα on the diagonal, one obtains 2E λα L v(k), 2 v(k) = |wα (k)|2 2 , 2 L +k λα + k 2 α=1 where w(k) = W v(k). This first yields 1 L 1 − − ≤ v(k), 2 v(k) ≤ + , 2 L + k λmin λmin

and then finally (4.17).

As an important consequence of this corollary we are now able to extend the statement of Proposition 4.3 as required to count eigenvalues in terms of zeros of the determinant function. Proposition 4.7. Let the metric structure of the graph be such that lmin > 2/λ+ min . Then kn2 > 0 is an eigenvalue of the Laplacian with multiplicity gn , if ±kn ∈ R\{0} are zeros of the function F (k) of order gn . Proof. From Proposition 4.3 and Lemma 4.4 we know that the positive Laplace eigenvalues are in one-to-one correspondence with (pairs of) k values where U (k) has an eigenvalue one, and the respective multiplicities coincide. It hence remains to establish that the order of the corresponding zeros of F (k) are exactly these multiplicities: From the definition (4.8) of the function F (k) one obtains F (k) =

2E

1 − eiθj (k) ,

j=1

so Corollary 4.6 implies immediately.

d dk

1 − eiθj (k) = −iθj (k)eiθj (k) = 0, and the claim follows

We remark that since the scattering approach to the proof of the trace formula is based on counting zeros of the function F (k) on the real line with their multiplicities, the requirement lmin > 2/λ+ min is essential. Otherwise one might count Laplace eigenvalues with incorrect multiplicities. Whenever L has no positive part, however, the condition is empty. This is, e.g., the case for non-Robin boundary conditions.

206



4.3. The eigenvalue zero In general, zero is a Laplace eigenvalue as well as a zero of the determinant (4.8), and in so far Proposition 4.3 also applies to k0 = 0. The spectral multiplicity g0 , however, typically is different from the degree of k0 = 0 as a zero of F (k). For Kirchhoff boundary conditions it has been shown in [17] that the degree of the zero is E − V + 2, whereas the zero Laplace eigenvalue is non-degenerate, i.e., g0 = 1. Kurasov [25] subsequently linked this difference in the multiplicities to the topology of the graph by noticing that a suitable trace formula contains the quantity 1 1 1 − (E − V + 2) = (V − E) , 2 2 and hence the Euler characteristic of the graph. This observation was generalised to yield an index theorem for any quantum graph with non-Robin boundary conditions by Fulling, Kuchment and Wilson [9]. One can view the t-independent term 1 1 4 tr S = 2 (V −E) in the trace formula for the heat kernel (due to [29] for Kirchhoff boundary conditions and [18] for general non-Robin conditions) as a predecessor of this result. See also [2] for a more detailed discussion. We here wish to give a further characterisation of the spectral multiplicity of the zero eigenvalue in the case of a general self adjoint realisation of the Laplacian. To this end we first introduce, for k ∈ R \ {0}, the matrix ⎛ ⎞ 2i l1

k

⎜ 2 k +l1 ⎜ ⎜ ⎜ ⎜ ⎜ C(l; k) := ⎜ ⎜ 2 ki ⎜ 2 i +l1 ⎜ k ⎜ ⎜ ⎝ i

..

2 ki +l1

.

..

. 2 ki 2 ki +lE

lE 2 ki +lE

..

l1 2 ki +l1

.

..

.

2 ki 2 ki +lE

lE 2 ki +lE

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4.18)

in which all matrix entries not indicated are zero. This now enables us to formulate the following. Proposition 4.8. For any given self adjoint realisation of the Laplacian specified through A, B, zero is a Laplace eigenvalue, iff one is an eigenvalue of S(A, B; k) C(l; k) for one, and hence any, k ∈ R \ {0}. Moreover, the multiplicity of this eigenvalue one coincides with the spectral multiplicity g0 of the zero Laplace eigenvalue. Proof. Eigenfunctions of the Laplacian corresponding to the eigenvalue zero must be of the form F = (f1 , . . . , fE )T with fj (x) = αj + βj x ,

x ∈ [0, lj ] .

(4.19)

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Hence, the boundary values (2.2) take the form T Fbv = α1 , . . . , αE , α1 + β1 l1 , . . . , αE + β1 lE , T = β1 , . . . , βE , −β1 , . . . , −βE . Fbv We now employ the boundary conditions (2.3) by using the expressions (3.8) for a possible choice of A and B. The result can be rearranged to yield α α = C− (l; k) , (4.20) S(k)C+ (l; k) β β where, for any k ∈ R \ {0}, we have introduced 1E ± ki 1E , C± (l; k) := 1E ∓ ki 1E + D1 (l) with D1 (l) as defined in (4.13). We also use the abbreviations ⎛ ⎞ ⎛ ⎞ α1 β1 ⎜ .. ⎟ ⎜ .. ⎟ α := ⎝ . ⎠ and β := ⎝ . ⎠ . αE βE The matrices C± (l, k) are invertible for all k ∈ R \ {0}, with ⎛ ± i −l ±i k

1

i ⎜ ±2 k −l1

−1

C± (l; k)

⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ i1 ⎜ ±2 k −l1 ⎜ ⎜ ⎝

⎞

k

..

±2 ki −l1

. ± ki −lE ±2 ki −lE

..

−1 ±2 ki −l1

.

..

± ki ±2 ki −l1

..

v(k) := C− (l; k)

. −1 ±2 ki −l1

1 ±2 ki −lE

We now substitute

.

α β

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠

(4.21)

in (4.20) and obtain S(k)C+ (l; k)C− (l; k)−1 v(k) = v(k) . It is straight forward to check that C+ (l; k)C− (l; k)−1 = C(l; k), compare (4.18). The linearly independent eigenvectors of SC corresponding to the eigenvalue one then yield, via (4.21) and (4.19), coefficients αj and βj for as many linearly inde pendent Laplace eigenfunctions in L2 (Γ). We remark that the order N of k0 = 0 as a zero of the function (4.8) is the multiplicity of the eigenvalue one of 0 1E , U (0) = S0 1E 0

208



which is in no obvious way related to the multiplicity of the eigenvalue one of S(k)C(l; k) that appears in Proposition 4.8. In the case of non-Robin boundary conditions, where the edge S-matrix is independent of k, however, Fulling, Kuchment and Wilson [9] were able to relate the different multiplicities in the form of an index theorem. They showed, in particular, that then 1 1 g0 − N = tr S . 2 4 As mentioned above, this term will reappear in the trace formula.

5. The trace formula A trace formula expresses counting functions of Laplace eigenvalues in terms of sums over periodic orbits. Ideally, one would like to count Laplace eigenvalues kn2 , with their multiplicities gn , in intervals I in the form Tr χI (−Δ) = gn . (5.1) 2 ∈I kn

The sharp cut-off provided by the characteristic function χI of the interval I, however, cannot be dealt with. One therefore replaces (5.1) with a smooth cut-off and, moreover, performs this count in terms of the associated wave numbers kn , i.e., one seeks to find a representation for gn h(kn ) (5.2) n

in terms of sums over periodic orbits. One ambition then is to find a sufficiently large class of test functions h. It turns out that the following one parameter family of test functions is particularly suited for these purposes. Definition 5.1. For each r ≥ 0 the space Hr consists of all functions h : C → C satisfying the following conditions: • h is even, i.e., h(k) = h(−k). • For each h ∈ Hr there exists δ > 0 such that h is analytic in the strip Mr+δ := {k ∈ C; | Im k| < r + δ}. 1 • For each h ∈ Hr there exists η > 0 such that h(k) = O( (1+|k|) 1+η ) on Mr+δ . We stress that the trace formula will only take non-negative Laplace eigenvalues into account. Hence, the test functions in (5.2) will be evaluated at real arguments. We can therefore arrange the wave numbers kn > 0 corresponding to Laplace eigenvalues kn2 in ascending order. Proposition 4.2 then readily implies the existence of a constant C > 0 such that gn |h(kn )| ≤ CK sup |h(k)| . (5.3) 0 0, which must be sufficiently small so that in the set Cε,K = {k ∈ C; | Im k| ≤ ε, | Re k| ≤ K} the determinant function F (k) has only (finitely many) real zeros related to non-negative Laplace eigenvalues (compare Proposition 4.3). Then, by the argument principle, F 1 (k) h(k) dk = N h(0) + 2 gn h(kn ) . (5.4) 2πi ∂Cε,K F 0 0, respectively, as a zeros of F (k). Following Proposition 4.7, gn also is the multiplicity of the Laplace eigenvalue kn2 . The factor of two occurs since kn , −kn ∈ Cε,K and we have exploited the fact that h(k) is even. Based on this relation the trace formula emerges when one expresses F (k)/F (k) in terms of a suitable series (eventually leading to a sum over periodic orbits), and performs the limit K → ∞. The result of this procedure is summarised in the following statement. Proposition 5.2. Let lmin > 2/λ+ min and choose h ∈ Hr with any r ≥ 0. Then ∞ 1 +∞ gn h(kn ) = tr Λ(k)U l (k) h(k) dk , (5.5) N h(0) + 2 2πi −∞ n=1 l∈Z

where Λ(k) = −i

2L + iD(l) . L2 + k 2

Proof. Our strategy is to show that both sides of (5.5) equal ! +∞ F F 1 (k − iε) h(k − iε) − (k + iε) h(k + iε) dk . lim+ F F ε→0 2πi −∞

(5.6)

(5.7)

Beginning with the left-hand side, we have to show that in (5.4) the limit K → ∞ can be taken, followed by ε → 0. To this end we notice that the right-hand side of (5.4) is explicitly independent of ε, in the range described above that equation. Furthermore, from (5.3) we already know that the sum over kn converges absolutely in the limit K → ∞. In order to perform these limits on the left-hand side of (5.4), and thus producing (5.7), we have to estimate the contribution to the integral coming from the vertical parts of the contour.

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Lemma 3.3 implies that for k ∈ C with |k| > λmax the approximation F (k) = F∞ (k) + O(|k|−1 ) holds, where (5.8) F∞ (k) := det 1 − S∞ T (k) . Since F∞ depends on k only through the matrix entries eikle of T (k), see (4.6), it can be represented as b dn eiβn k , (5.9) F∞ (k) = 1 + n=1

with some b < ∞. Moreover, βn > 0 is a finite sum of edge lengths, and dn ∈ C is an appropriate coefficient. The expression (5.9) is first defined for k ∈ R, but can be readily extended to complex k. Since S (k) = O(|k|−2 ), see Lemma 3.2 and (k) + O(|k|−1 ). For sufficiently large |k| one eq. (4.14), we also find F (k) = F∞ (k)/F∞ (k). In order to estimate the can therefore approximate F (k)/F (k) by F∞ latter we employ (5.9) to obtain " " "F∞ (k)" ≤ b dmax βmax e−βmin Im k , (5.10) with dmax := max{|dn |} and βmax / min := max / min{βn }. Note that this bound is independent of Re k. Furthermore, we pick k(0) ∈ R such that F (k(0) ) = 0 as well as F∞ (k(0) ) = 0, and take advantage of the fact that F∞ (k), k ∈ R, is an almost periodic function (see, e.g., [4]). One can hence construct a (strictly increasing) sequence {k(j) ; j ∈ N0 } with |F∞ (k(j) ) − F∞ (k(0) )|
0.

(5.11)

Hence, in particular, F∞ (k(j) ) = 0. The estimate (5.10), moreover, implies that |F∞ (k(j) ) − F∞ (k(j) + iκ)|
0 such that " " " k(j) +iε F (k) " " " ∞ h(k) dk " ≤ Cε sup |h(k(j) + iκ)|; κ ∈ (−ε, +ε) . (5.13) " " k(j) −iε F∞ (k) " The third property of test functions required in Definition 5.1 then ensures that this integral vanishes as k(j) → ∞. A completely analogous reasoning applies when replacing k(j) with −k(j) . Hence, as K → ∞ the contributions coming from [−K + iε, −K − iε] and [K − iε, K + iε] to the integration along ∂Cε,K in (5.4) vanish, and the limits ε → 0 and K → ∞ can be interchanged. It remains to prove the equality of (5.7) with the right-hand side of (5.5). In order to achieve this we first recall from Proposition 4.3 that the function F (k) is holomorphic and zero-free in the strip {k ∈ C; 0 < Im k < ε1 }, where

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2 ε1 := min{λ+ min , κ1 } when −κ1 is the largest negative eigenvalue of −Δ; if −Δ ≥ 0 + we set ε1 := λmin . Hence, −1 d F (k + iε) = log det 1 − U (k + iε) = − tr 1 − U (k + iε) U (k + iε) , F dk when 0 < ε < ε1 . Furthermore, following Lemma 3.13 we obtain the bound λ+ + ε (5.14) U (k + iε) ≤ S(k + iε) T (k + iε) ≤ max 1, min e−εlmin λ+ − ε min

in the operator norm. Thus, when ε
2/λ+ min this condition + is solvable, providing some ε2 with 0 < ε2 < λmin such that (5.15) is true for all ε ∈ (0, ε2 ). For such ε the expansion ∞ −1 1 − U (k + iε) = U (k + iε)l l=0

holds. Moreover, a straight forward calculation based on the relation (4.12) produces tr[U (k)l U (k)] = tr[Λ(k)U (k)l+1 ], see (5.6). Therefore, after cyclic permutations under the trace, ∞ F (k + iε) = − tr Λ(k + iε)U l (k + iε) (5.16) F l=1

is true for all k ∈ R and ε ∈ (0, ε3 ), where ε3 := min{ε1 , ε2 }. Likewise, based on the relation −1 −1 −1 1 − U (k − iε) = − 1 − U −1 (k − iε) U (k − iε) that holds for 0 < ε < λ+ min , we obtain ∞ F (k − iε) = tr Λ(k − iε)U −l (k − iε) F

(5.17)

l=0

for k ∈ R and sufficiently small ε > 0. We now want to use the representations (5.16) and (5.17) in (5.7), interchange integration and summations, and finally perform the limit ε → 0. In order to achieve this we first notice that (for fixed ε) the estimate " " "tr Λ(k ± iε)U ±l (k ± iε) " ≤ 2E Λ(k ± iε)U ±l (k ± iε) (5.18) ≤ 2E Λ(k ± iε) U ±1 (k ± iε) l justifies to interchange integration and summation according to the dominated convergence theorem. Next, the contour of the integral tr Λ(k)U ±l (k) h(k) dk (5.19) Im k=±ε

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can be deformed into Im k = ±δ with a sufficiently small δ > 0; in particular, δ < r. We then fix δ and find " " " " ±l " " tr Λ(k)U (k) h(k) dk " " Im k=±ε +∞ " " "tr Λ(k ± iδ)U ±l (k ± iδ) " |h(k ± iδ)|dk ≤ (5.20) −∞

≤ 2E

λ+ min + δ −δlmin e λ+ min − δ

l

+∞

−∞

Λ(k ± iδ) |h(k ± iδ)|dk ,

when using (5.14) and (5.18) with δ instead of ε. Since the conditions in Definition 5.1 apply, the integral on the right-hand side is finite and the positive constant raised to the power l is smaller than one. The sum on l hence possesses an absolutely convergent majorant, uniform in ε in the range indicated. Thus the summation and the limit ε → 0 can be interchanged. Furthermore, another application of the dominated convergence theorem allows to perform the limit ε → 0 of (5.19) inside the integral, finally proving (5.5). In order to arrive at the trace formula itself, the sum on the right-hand side of (5.5) has to be reformulated as a sum over periodic orbits. The summation index l then denotes the topological length of the orbits and the trace of Λ(k)U l (k) is identified as a sum over the set Pl of periodic orbits with topological length l. The term with l = 0 plays a special role and will be treated separately, whereas contributions with negative l are related to those with −l in a simple way. Before we state the trace formula, however, we introduce some notation: The ‘volume’ of the graph is the sum L = le1 +· · ·+leE of all edge lengths. Furthermore, if h ∈ Hr is a test function, its Fourier transform is +∞ 1 ˆ h(x) = h(k) eikx dk . 2π −∞ We recall that the second and the third property required for h in Definition 5.1 ˆ guarantee that h(x) = O(e−rx ) as x → ∞. Moreover, the Fourier transform of a product A(k)h(k) is the convolution of the respective Fourier transforms, i.e., it reads +∞ ˆ ˆ # ˆ − y) h(y) A(x dy = Aˆ ∗ h(x) . (5.21) Ah(x) = −∞

Below this convolution will often have to be understood in a distributional sense, as the functions A(k), though being regular, not always decay sufficiently fast as k → ∞. We are now in a position to state the first variant of the trace formula. Theorem 5.3. Let Γ be a compact, metric graph with a self adjoint realisation −Δ(A, B; l) of the Laplacian, such that the condition lmin > 2/λ+ min is fulfilled.

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Furthermore, let h ∈ Hr be a test function with an arbitrary r ≥ 0. Then the following identity holds: +∞ ∞ 1 1 Im tr S(k) ˆ dk gn h(kn ) = L h(0) + g0 − N h(0) − h(k) 2 4π k −∞ n=0 (5.22) ∞ $ % ˆ ˆ ˆ ˆ h ∗ Ap (lp ) + h ∗ Ap (lp ) . + l=1 p∈Pl

Here Aˆp is the Fourier transform of the amplitude function Ap (k) associated with every periodic orbit p. This function is meromorphic with possible poles at the poles of the S-matrix, and has a Taylor expansion ∞ ap(j) k −j (5.23) Ap (k) = j=0

that converges for |k| > λmax . In general, only the sum over the topological lengths l on the right-hand side of (5.22) converges absolutely, but not the entire double sum over periodic orbits. Proof. We have to evaluate tr[Λ(k)U l (k)], and first notice that according to (4.6) and (4.7) the matrix entries of U are Uj1 j2 = Sj1 ωj2 eiklj1 , where

j + E, if 1 ≤ j ≤ E ωj := . j − E, if E + 1 ≤ j ≤ 2E

Hence, the indices j and ωj correspond to the two edge ends of the j-th edge. Local boundary conditions then imply that Sj1 ωj2 = 0 requires the edges with ends j1 and ωj2 to be adjacent. Therefore, when l > 0 the non vanishing terms in the multiple sum tr ΛU l =

2E

Λj0 j1 Sj1 ωj2 . . . Sjl ωj0 eik(lj1 +···+ljl )

(5.24)

j0 ,...,jl =1

correspond to the closed paths of topological length l on the graph. We then make use of the decomposition (5.6) of Λ and begin with the contribution of tr[D(l)U l (k)], which can be evaluated as in the case of Kirchhoff boundary conditions [21]: Due to the specific diagonal form of D the terms in (5.24) corresponding to closed paths related by cyclic permutations of their edges can be grouped together. This finally yields a sum over the periodic orbits of topological length l, lp# A1,p (k) eiklp . tr D(l)U l (k) = 2 p∈Pl

According to (5.24), the functions A1,p (k) result from multiplying the local Smatrices of the vertices visited along the periodic orbit p. Moreover, lp# is the

214



primitive length of p, i.e., the length of the primitive periodic orbit associated with p. Due to the relation (5.21) we therefore obtain +∞ 1 ˆ ∗ Aˆ1,p (lp ) . tr D(l)U l (k) h(k) dk = 2 lp# h 2π −∞ p∈Pl

The case of negative l follows by noticing that U −1 (k) = T −1 (k)S −1 (k) = T (−k)S(−k) , and thus

tr D(l)U −l (k) = 2 lp# A1,p (k) e−iklp , p∈Pl

leading to +∞ $ % 1 tr D(l) U l (k) + U −l (k) h(k) dk 2π −∞ % $ ˆ (l ) . ˆ ∗ Aˆ1,p (lp ) + h ˆ∗A =2 lp# h 1,p p p∈Pl l In order to calculate the contribution from tr[ L2L +k2 U (k)] we notice that ! L l −2i tr 2 U = tr[S T U l−1 ] L + k2

=

2E

(5.25) Sj 1 ωj2 Sj2 ωj3 . . . Sjl ωj1 eik(lj1 +···+ljl ) .

j1 ,...,jl =1

Again, the multiple sum can be viewed as a sum over the closed paths of topological length l, and the contributions of representatives of periodic orbits can be grouped together. Eventually, this leads to ! L l −2 tr 2 U (k) =2 A2,p (k) eiklp , (5.26) 2 L +k p∈Pl

where the functions A2,p (k) emerge from multiplying local S-matrix elements and their derivatives along the closed paths as specified in (5.25). Negative l are dealt with as above, so that the contribution from l ∈ Z \ {0} to the sum on the right-hand side of (5.5) yields the sum on the right-hand side of (5.22), with Ap (k) = lp# A1,p (k) + A2,p (k). The contribution coming from l = 0 finally is +∞ +∞ 2L 1 1 L +∞ tr Λ(k) h(k) dk = − tr h(k) dk h(k) dk + 2πi −∞ 2π −∞ L2 + k 2 π −∞ +∞ 1 Im tr S(k) ˆ dk . = 2L h(0) − h(k) 2π −∞ k Adding the contribution of the Laplace-eigenvalue zero to (5.5), after a rearrangement of the terms the result (5.22) follows.

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As mentioned in Theorem 5.3, the double sum over periodic orbits in (5.22) does not converge absolutely when h ∈ Hr with arbitrarily small r ≥ 0. In the following we are going to show that an absolutely convergent periodic orbit sum can be achieved under sharpened conditions on the test function. In order to formulate these conditions we introduce the function 2 1 l(κ) := log(2E) + artanh κ κ

κ λ+ min

,

0 < κ < λ+ min ,

(5.27)

which attains its unique minimum at some σ ∈ (0, λ+ min ). Moreover, the minimum can be bounded from below as l(σ) ≥ (2 + log(2E))/λ+ min so that, in particular, . l(σ) > 2/λ+ min Theorem 5.4. Let Γ be a compact, metric graph with a self adjoint realisation −Δ(A, B; l) of the Laplacian, such that the condition lmin > l(σ) is fulfilled. Furthermore, let h ∈ Hr be a test function with an arbitrary r ≥ σ. Then the following identity holds: +∞ 1 1 Im tr S(k) ˆ dk gn h(kn ) = L h(0) + g0 − N h(0) − h(k) 2 4π −∞ k n=0 % $ ˆ (l ) . ˆ ∗ Aˆp (lp ) + h ˆ∗A + h p p ∞

(5.28)

p∈P

The quantities appearing are the same as in Theorem 5.3. Here, however, the sum over periodic orbits converges absolutely. Proof. The task is to analyse the convergence of the sum over periodic orbits in (5.28) more closely. In Theorem 5.3 we only considered the convergence of the sum over l. This was based on the estimate (5.20), which shall now be refined. To this end we refer to (5.24) and (5.25), and notice that tr[Λ(k)U l (k)] is a sum consisting of (2E)l terms si1 i2 (k) . . . sil il+1 (k)h(k), each of which contains a product of l factors that are matrix elements of either S(k), iD(l), or i L22L +k2 . These factors are + all holomorphic in a strip 0 ≤ Im k < λmin and, following Lemma 3.4, in any strip 0 ≤ Im k ≤ κ with κ < λ+ min , they are bounded from above in absolute value by a + + + κ)/(λ constant times (λ+ min min − κ). Thus, in particular, for all κ < min{λmin , r} any product si1 i2 (k) . . . sil il+1 (k) h(k) is holomorphic for 0 ≤ Im k ≤ κ. Since here we are evaluating Fourier transforms at li1 + · · · + lil > 0, we can apply a suitable version of the Paley-Wiener theorem that is concerned with Fourier transforms of holomorphic functions in vertical strips of the upper half plane. This is an obvious, slight variation of Theorem IX.14 in [30], and implies that there exists Cκ > 0 such

216


that " " " "

+∞

−∞

" " l tr Λ(k)U (k) h(k) dk "" " 2E " " ≤ " i1 ,...,il =1

≤ Cκ


+∞

−∞

si1 i2 (k) . . . sil i1 (k) h(k) e

λ+ + κ −κlmin e 2E min λ+ min − κ

l = Cκ eκl

ik(li1 +···+lil )

l(κ)−lmin

" " dk ""

,

(5.29)

where we made use of the function defined in (5.27). We hence conclude that under the conditions stated the sum over periodic orbits converges absolutely. In particular, lmin > l(κ) must be fulfilled. The latter condition is optimised for κ = σ, at which the function l(κ) attains its minimum. For the above to hold, the condition h ∈ Hr with r ≥ σ must be satisfied. We remark that whenever an amplitude function Ap (k) associated with a periodic orbit is independent of k, as it is always true for non-Robin boundary ˆ p ). In any ˆ ∗ Aˆp (lp ) degenerates into a product Ap h(l conditions, the convolution h ˆ ∗ Aˆp (x) = case, the condition r ≥ σ imposed on the test function h ensures that h −σ|x| ), which enables the absolute convergence of the periodic orbit sum. O(e 5.2. The trace of the heat kernel The first trace formula of a quantum graph, due to Roth [29], expresses the trace of the heat kernel for the Laplacian with Kirchhoff boundary conditions in terms of a sum over periodic orbits. This has recently been extended to all non-Robin boundary conditions by Kostrykin, Potthoff and Schrader [18]. An application of Theorem 5.4 now allows us to produce an appropriate trace formula for all self adjoint realisations of the Laplacian. In such a case, however, negative Laplace eigenvalues −κ2n , with multiplicities gn− , may occur so that the trace of the heat kernel, 2 2 gn− eκn t + gn e−kn t , t > 0 , Tr eΔt := −κ2n 0 . Tr+ eΔt := 2 ≥0 kn

In the trace formula we hence have to choose the (entire holomorphic) function 2 ˆ = h(k) = e−k t , t > 0, which is in Hr for all r ≥ 0. Its Fourier transform is h(x) 2 1 −x /4t √ e . We also introduce the functions 4πt +∞ 2 1 Aˆp (lp − y) e−y /4t dy ap (t) := √ 4πt −∞ associated with the periodic orbits on the graph and obtain the following statement.

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Theorem 5.5. Let Γ be a compact, metric graph with a self adjoint realisation −Δ(A, B; l) of the Laplacian, such that the condition lmin > l(σ) is fulfilled. Then the following identity holds: d √ L 1 1 λα λ2α t e Tr+ eΔt = √ + g0 − N − erfc |λα | t 2 2 |λ | 4πt α α=1 Re ap (t) . +2

(5.30)

p∈P

Here d is the number of non vanishing eigenvalues λα of L (counted with their multiplicities) and ∞ 2 2 e−y dy , x ≥ 0 , erfc(x) = √ π x is the error function complement. Moreover, as t →√0+ the trace of the heat kernel has a complete asymptotic expansion in powers of t, whose leading terms read √ L Tr eΔt = √ + γ + O( t) , t → 0+ . (5.31) 4πt Here N − 1 1 d+ − d− + , gn − γ0,j + γp,j − γ = g0 − 2 2 2 2 n j j where γ0,j and γp,j are the orders of the finitely many, non zero, purely imaginary zeros and poles, respectively, of the determinant function F and d± denotes the number of positive/negative eigenvalues of L. Proof. We first observe that the expression (4.14) implies Im tr S(k) = tr

d 2kL λα = 2k ; 2 2 2 L +k λ + k2 α=1 α

then we employ the representation erfc(|x|) =

2|x| −x2 e π

0

∞

e−y dy , 2 y + x2 2

x ∈ R.

Hence the relation (5.30) is an immediate consequence of (5.28), when the test function h is chosen as indicated above. In order to determine the small-t asymptotics we go back to the relation (5.4), 2 in which we use the test function h(k) = e−k t , t > 0. We then deform the contour into ∂Cβ,K , with β > max{λmax , s}, where s from (4.2) is such that −s2 yields a lower bound on the Laplace spectrum. Thus the contour now encloses all non real zeros and poles of the determinant function F and, therefore, in this process we pick up contributions from all poles of F /F on the imaginary axis. Having to perform the limit K → ∞ with β kept fixed, we need to estimate the contribution coming from the vertical parts of the contour, i.e., for | Re k| = K and ε < | Im k| < β. Firstly, F (k) is a polynomial in the matrix entries of

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S(k) and T (k). The latter are eikle , whereas the k-dependence of the former is given by (λα − ik)/(λα + ik), see (3.5). In the two strips ε < | Im k| < β, with neighbourhoods of the poles at iλα removed, all matrix entries are bounded, and hence F (k) is of polynomial growth in k. Secondly, for sufficiently large |k| we &2E again approximate F (k) by F∞ (k), see (5.8), and write F∞ (k) = j=1 (1 − uj (k)). Here u1 (k), . . . , u2E (k) are the eigenvalues of S∞ T (k), which satisfy |uj (k)| ≤ S∞ T (k) ≤ e−εlmin < 1 , for all k in the strips ε < | Im k| < β. Thus, in these strips, |F∞ (k)| > (1 − e−εlmin )2E > 0 . Thirdly, when |K| > β the factor e−tk is of the order e−tK in the strips. Hence, the integrand of (5.4) on the vertical parts of the contour with | Im k| > ε is 2 bounded by a polynomial times e−tK . As K → ∞, these parts of the contour therefore do not contribute to the integral. Hence, from (5.4) we obtain 2

N +2

∞

gn e−kn t = 2

n=1

1 + 2πi

2

γp,j eκp,j t −

j

2

2

γ0,j eκ0,j t

j

! F F −(k−iβ)2 t −(k+iβ)2 t (k − iβ) e (k + iβ) e − dk . F F

+∞

−∞

Here the non real zeros of F are denoted as iκ0,j and its poles as iκp,j ; the respective orders are γ0,j and γp,j . To proceed further we follow the argument leading from (5.14) to (5.16) and (5.17), as well as the subsequent discussion of interchanging integration and summation. As we are dealing with large β instead of small ε, due to (3.14) the estimate (5.14) is replaced by β + λmax e−εlmin , U (k + iβ) ≤ max 1, β − λmax Hence, for k ∈ R and β sufficiently large analogous relations to (5.16) and (5.17) are obtained, eventually leading to N +2

∞ n=1

gn e−kn t = 2

2

γp,j eκp,j t −

j

j

2

γ0,j eκ0,j t

+∞ ∞ 2 1 + tr Λ(k + iβ)U l (k + iβ) e−(k+iβ) t dk 2πi −∞ l=1 +∞ ∞ 2 1 + tr Λ(k − iβ)U −l (k − iβ) e−(k−iβ) t dk . 2πi −∞ l=0

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√ In the integrals with l =√0 we now replace β by β/ t, 0 < t ≤ 1, and change variables from k to q = k t, yielding ! +∞ 2 1 1 1 √ Il± (t, β) := tr Λ √ (q ± iβ) U ±l √ (q ± iβ) e−(q±iβ) dq . 2πi t −∞ t t These integrals can be bounded in analogy to (5.29), √ l Cβ β + tλmax −βlmin /√t ± √ |Il (t, β)| ≤ √ 2E e . t β − tλmax Summing over l = 0 then finally shows that these contributions can be estimated √ as being O( √1t e−βlmin / t ). The term with l = 0 can be calculated explicitly and yields the same contribution to the heat trace as the first term and the sum over σ(L) \ {0} on the right-hand side of (5.30). We add the contributions of negative Laplace eigenvalues and use that erfc(x) has a complete asymptotic expansion in x, with erfc(x) = 1 + O(x), as x → 0. The expansion (5.31) then follows immediately. At this point we recall that in the case of non-Robin boundary conditions γ = g0 − 12 N = 14 tr S, which has also been given an interpretation as (one half of) a suitable Fredholm index. In this case, therefore, the constant term in the small-t asymptotics of the heat kernel has a topological meaning. Finally, we should like to mention that a suitable Tauberian theorem (see, e.g., [15]) allows us to recover Weyl’s law L K , K → ∞, π see also Proposition 4.2, from the leading term in the expansion (5.31). N (K) ∼

6. Conclusions Our principal goal was to investigate spectra of general self adjoint realisations of Laplace operators on compact metric graphs, culminating in proofs of some trace formulae. In this context we achieved to allow for a large class of test functions, leading either to absolutely or conditionally convergent sums over periodic orbits, respectively, representing appropriate spectral functions of the Laplacian. Previous work on quantum graph trace formulae [17,18,21,29] was restricted to Laplacians with non-Robin boundary conditions. As compared to these cases there are some modifications we had to take care of. Firstly, non-Robin boundary conditions correspond to k-independent S-matrices and hence do not involve any derivatives of S(k). This is in line with the fact that L = 0, so that in the trace formula the contribution (5.26) to the periodic orbit sum is absent. Moreover, since therefore λ+ min = ∞, the restriction imposed on lmin in the general case is void so that any set of lengths can be attributed to the edges. What still remains

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to ensure an absolutely convergent periodic orbit sum in the case of non-Robin boundary conditions is the single requirement h ∈ Hr with r ≥ (log 2E)/lmin on ˆ the test functions. This implies h(x) = O(e−r|x| ), which in turn compensates for the growth in the number of periodic orbits entering the sum ˆ p) , Ap h(l p∈P,lp ≤

when → ∞. Another property of Laplacians with non-Robin boundary conditions is that they are non-negative. This fact is linked to the non-positivity of L which, as L = 0, is trivial. The determinant function (4.8) hence is entire holomorphic with only real zeros in the complex non-negative half plane, see Proposition 4.3. As we have shown in Theorem 5.3 the condition on the test functions can be relaxed to h ∈ Hr with any r ≥ 0, so that h(x) = O(e−δ|x| ) with some (arbitrarily small) δ > 0, when one is willing to accept a conditionally convergent sum. This has to be understood in the sense given in (5.22), i.e., where the terms are arranged as a double sum over the topological lengths of the orbits and over the periodic orbits of fixed topological length. For non-Robin boundary conditions a refined analysis of convergence had produced even more relaxed conditions to be demanded from the test functions, see [36]. An important application of the trace formula for quantum graphs with nonRobin boundary conditions was to prove an inverse theorem, very much in the sense of Kac’s famous question ‘Can one hear the shape of a drum?’ [14]. Gutkin and Smilansky [10] showed that under certain conditions, which include the requirement that the edge lengths be rationally independent, the Laplace spectrum determines the connectedness and the metric structure of a compact metric graph uniquely. In this sense isospectral quantum graphs are isomorphic. Gutkin and Smilansky made essential use of the trace of the wave group, √

Tr e−it

−Δ

+ c.c. = 2

∞

gn cos(kn t) ,

t = 0 ,

n=0

which can, in the case of non-Robin boundary conditions, be expressed as a sum of δ-singularities at the lengths lp of periodic orbits. Searching for these singularities then allows one to first identify the geometric length spectrum. In a second step a certain algorithm can be used to determine the connectedness and all individual edge lengths. After a slight modification this proof can now be taken over to the case of Robin boundary conditions almost verbatim. For this purpose one reads off from the trace formula (5.28) the distributional identity ∞

d 1 λα −|tλα | 1 e gn cos(kn t) = L δ(t) + g0 − N − 2 2 α=1 |λα | n=0 + Re Aˆp (lp − t) + Aˆp (lp + t) . p∈P

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Since, in general, the periodic orbit amplitudes are functions of k, there are no longer pure δ-singularities present at the lengths of periodic orbits. For large k the (0) amplitudes, however, possess the expansions (5.23) with leading terms ap . These do not vanish since they stem from the corresponding leading term S∞ = 1 − 2P of the S-matrix. Hence, the Fourier transforms Aˆp of the amplitudes have leading singularities of δ-type at the lengths lp of periodic orbits. Therefore, applying the algorithm of Gutkin and Smilanksy to this wave trace enables one to identify the graph connectivity and the edge lengths in the same way as previously. In summary, almost all spectral properties established so far for Laplacians on compact metric graphs with non-Robin boundary conditions carry over to arbitrary self adjoint realisations of the Laplacian. Therefore, there are many more quantum graphs model available that are suitable for further investigations in, e.g., the field of quantum chaos.

Acknowledgments J. Bolte would like to thank Stephen Fulling for very helpful discussions. We are also grateful to an anonymous referee for useful hints.

References [1] R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain. III. Eigenfrequency density oscillations, Ann. Physics (NY) 69 (1972), 76–160. [2] J. Bolte and S. Endres, Trace formulae for quantum graphs, Analysis on Graphs and its Applications, P. Exner, J. P. Keating, P. Kuchment, T. Sunada, and A. Teplyaev (eds.), Proc. Symp. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 247–259. [3] O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984), 1–4. [4] H. Bohr, Almost periodic functions, Chelsea Publishing Company, 1947. [5] G. Berkolaiko and B. Winn, Relationship between scattering matrix and spectrum of quantum graphs, preprint arxiv:math-ph/0801.4104 (2008). [6] Y. C. de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques. I, II, Compositio Math. 27 (1973), 83–106; ibid. 27 (1973), 159–184. [7] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39–79. [8] E. Doron and U. Smilansky, Semiclassical quantization of chaotic billiards: a scattering theory approach, Nonlinearity 5 (1992), 1055–1084. [9] S. A. Fulling, P. Kuchment, and J. H. Wilson, Index theorems for quantum graphs, J. Phys. A 40 (2007), 14165–14180. [10] B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A 34 (2001), 6061–6068.

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[11] S. Gnutzmann and U. Smilansky, Quantum graphs: Applications to quantum chaos and universal spectral statistics, Advances in Physics 55 (2006), 527–625. [12] M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. 12 (1971), 383–358. [13] F. Haake, Quantum Signatures of Chaos, Springer Verlag, Berlin, 2001. [14] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1–23. [15] J. Karamata, Neuer Beweis und Verallgemeinerung einiger Tauberian-S¨ atze, Math. Z. 33 (1931), 294–299. [16] T. Kato, Perturbation theory for linear operators, Springer Verlag, Berlin, 1995. [17] P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A 38 (2005), 4901–4915. [18] V. Kostrykin, J. Potthoff, and R. Schrader, Heat kernels on metric graphs and a trace formula, Adventure in Mathematical Physics, F. Germinet and P. D. Hislop (eds.), Contemp. Math., vol. 447, Amer. Math. Soc., Providence, RI, 2007, pp. 175–198. [19] T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett. 79 (1997), 4794–4797. [20] V. Kostrykin and R. Schrader, Kirchhoff ’s rule for quantum wires, J. Phys. A 32 (1999), 595–630. [21] T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Phys. (NY) 274 (1999), 76–124. [22] V. Kostrykin and R. Schrader, The inverse scattering problem for metric graphs and the traveling salesman problem, preprint, arxiv:math-ph/0603010 (2006). [23] V. Kostrykin and R. Schrader, Laplacians on metric graphs: eigenvalues, resolvents and semigroups, Quantum graphs and their applications (G. Berkolaiko, R. Carlson, S. A. Fulling, and P. Kuchment, eds.), Contemp. Math., vol. 415, Amer. Math. Soc., Providence, RI, 2006, pp. 201–225. [24] P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media 14 (2004), S107–S128. [25] P. Kurasov, Graph Laplacians and topology, Ark. Mat. 46 (2008), 95–111. [26] P. Lancaster and M. Tismenetsky, The Theory of Matrices, second ed., Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL, 1985. [27] E. Meinrenken, Semiclassical principal symbols and Gutzwiller’s trace formula, Rep. Math. Phys. 31 (1992), 279–295. [28] T. Paul and A. Uribe, The semi-classical trace formula and propagation of wave packets, J. Funct. Anal. 132 (1995), 192–249. [29] J.-P. Roth, Spectre du laplacien sur un graphe, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 793–795. [30] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [31] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978. [32] A. Selberg, Harmonic analysis and discontinuous and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.

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[33] M. Solomyak, On eigenvalue estimates for the weighted Laplacian on metric graphs, Nonlinear problems in mathematical physics and related topics, I, Int. Math. Ser. (N.Y.), vol. 1, Kluwer/Plenum, New York, 2002, pp. 327–347. [34] H.-J. St¨ ockmann, Quantum Chaos, Cambridge University Press, Cambridge, 1999. ¨ [35] H. Weyl, Uber die asymptotische Verteilung der Eigenwerte, Nachrichten der K¨ oniglichen Gesellschaft der Wissenschaften zu G¨ ottingen. Mathem.-physikal. Klasse (1911), 110–117. [36] B. Winn, A conditionally convergent trace formula for quantum graphs, Analysis on Graphs and its Applications, P. Exner, J. P. Keating, P. Kuchment, T. Sunada, and A. Teplyaev (eds.), Proc. Symp. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 491–501. Jens Bolte Department of Mathematics Royal Holloway, University of London Egham, TW20 0EX United Kingdom e-mail: [email protected] Sebastian Endres Institut f¨ ur Theoretische Physik Universit¨ at Ulm Albert-Einstein-Allee 11 D-89069 Ulm Germany e-mail: [email protected] Communicated by Jens Marklof. Submitted: May 20, 2008. Accepted: January 6, 2009.

Ann. Henri Poincaré 10 (2009), 225–274 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020225-50, published online May 22, 2009 DOI 10.1007/s00023-009-0407-y


Dynamics of Bianchi Type I Solutions of the Einstein Equations with Anisotropic Matter Simone Calogero and J. Mark Heinzle Abstract. We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an ‘anisotropy classification’ for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type A+ : Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types B+ and C+ : Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type D+ : Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions.

1. Introduction A pivotal feature in the study of spatially homogeneous cosmologies is the fact that the (asymptotic) dynamics of cosmological solutions of the Einstein equations is largely determined by the spatial geometry of the initial spacelike hypersurface. This is reflected in the systematics of the Bianchi classification of spatially homogeneous cosmologies. In contrast, a systematic analysis of the influence of the matter model on the (asymptotic) dynamics of solutions is not available at present. It is known that the dynamics of solutions (and the asymptotics towards the initial

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singularity, in particular) strongly depends on the matter source that is considered; the relatively simple behaviour of the vacuum and the perfect fluid case, see, e.g., [22], is replaced by a considerably more intricate behaviour in the case of anisotropic matter models [3, 6, 11, 15, 19]. The purpose of this paper is to analyze systematically and in detail in which way anisotropies of the matter source determine the dynamics of cosmological solutions. In our analysis we consider spatially homogeneous solutions of Bianchi type I. The reason for this choice is that Bianchi type I is the ‘foundation’ of the Bianchi classification. Generally, the dynamics of the ‘higher’ types in the Bianchi classification is based on the dynamics of the ‘lower’ types; for instance, the asymptotics of solutions of Bianchi types VIII and IX can only be understood in terms of solutions of Bianchi type I and II, see [7] for an up-to-date discussion. Likewise, our understanding of cosmological models in general is conjectured to be built on the dynamics of spatially homogeneous models and thus on the dynamics of Bianchi type I models in particular, see [8] and references therein. The dynamics of Bianchi type I cosmological models where the matter is a perfect fluid is well-known [22]. In this context, it is customary to assume that the the perfect fluid is represented by an energy density ρ and pressure p that obey a linear equation of state p = wρ with w = const ∈ (−1, 1). Each Bianchi type I perfect fluid solution isotropizes for late times; this means that each solution approaches a (flat) Friedmann–Robertson–Walker (FRW) solution for late times. Towards the initial singularity, each solution approaches a Kasner solution (Bianchi type I vacuum solution), i.e., to leading order in the limit t → 0 the metric is described by ds2 = −dt2 + a1 t2p1 dx1 ⊗ dx1 + a2 t2p2 dx2 ⊗ dx2 + a3 t2p3 dx3 ⊗ dx3 , where a1 , a2 , a3 are positive constants, and p1 , p2 , p3 are the Kasner exponents, which are constants that satisfy p1 + p2 + p3 = p21 + p22 + p23 = 1. Conversely (and importantly), for each Kasner solution (including the Taub solution, for which (p1 , p2 , p3 ) = (1, 0, 0) and permutations) there exists a Bianchi type I perfect fluid model that converges to this solution as t → 0; in other words, each Kasner solution is a possible past asymptotic state. In this paper we are concerned with Bianchi type I cosmological models with anisotropic matter. We do not specify the matter model explicitly; on the contrary, the assumptions that we impose on the matter source are so mild that we can treat several matter models at the same time, examples being matter models as different from each other as collisionless matter, elastic matter (for a wide variety of constitutive equations) and magnetic fields (aligned along one of the axes). The main results we derive are presented as Theorems 1–3 in Section 6 and summarized in Figure 7, but let us give a rough non-technical overview: First consider the asymptotic behaviour of models towards the future. For ‘conventional’ matter sources – where ‘conventional’ means that the isotropic state of these anisotropic matter models is energetically favorable – we find that each associated Bianchi type I solution isotropizes towards the future, i.e., each solution approaches a FRW solution for late times. However, there also exist matter models

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whose isotropic state is unstable. In this case, the future asymptotic behaviour is completely different; isotropization occurs but it is non-generic; generically, solutions approach different self-similar solutions, e.g., there exist solutions that approach Kasner solutions as t → ∞. It should be emphasized that these models are often compatible with the energy conditions. Second consider the asymptotic behaviour towards the singularity. We observe a rather diverse asymptotic behaviour, but the past dynamics of generic Bianchi type I solutions is again intimately connected with the Kasner states. Interestingly enough, the details of the past asymptotic dynamics of solutions are governed by one particular parameter, β, that describes certain ‘asymptotic properties’ of the matter (the properties of the matter under extreme conditions – extreme stress, etc. – that are found close to the spacetime singularity); this parameter can also be regarded as a measure for the degree of (dominant) energy condition violation under extreme conditions. (Clearly, we always assume that the dominant energy condition is satisfied under ‘normal conditions’ of the matter.) If the matter satisfies the dominant energy condition also under extreme conditions, then each Bianchi type I solution approaches a Kasner solution, and conversely, each Kasner solution is a possible past asymptotic state; the Kasner solutions are on an equal footing in this respect. (The case where the energy condition is satisfied only marginally is slightly different; the Taub solutions play a special role in that case.) If the dominant energy condition is violated under extreme conditions, this result breaks down; however, we must distinguish between ‘under-critical’ violations of the energy condition (corresponding to a small value of β) and ‘over-critical’ violations (corresponding to a large value of β). For an under-critical violation of the energy condition, each Bianchi type I solution approaches a Kasner state, but the converse is false, i.e., there exist Kasner solutions that are excluded as possible past asymptotic states. Only Kasner solutions that are ‘sufficiently different’ from the Taub solution, i.e., only Kasner solutions with (p1 , p2 , p3 ) sufficiently different from (1, 0, 0) and permutations, are attractors for Bianchi type I solutions. The set of potential asymptotic states is smaller if the degree of the energy condition violation is larger. Finally, for an over-critical violation of the energy condition, Bianchi type I solutions with anisotropic matter do not converge to a Kasner state as t → 0, but they follow a sequence of Kasner states (‘epochs’), in each of which their behaviour is approximately described by a Kasner solution. This is a type of behaviour that resembles the Mixmaster oscillatory behaviour that is expected for Bianchi type VIII and IX (vacuum) cosmologies (and generic cosmological models); note, however, that both the details and the origin of these oscillations are very different – in the present case, the oscillatory behaviour is not related to the Mixmaster map; it is a consequence of (over-critical) energy condition violation; in the Mixmaster case, the reason is the geometry of the problem. Independently of the energy conditions, there also exist a variety of Bianchi type I solutions with anisotropic matter that exhibit a past asymptotic behaviour that is not connected with Kasner asymptotics; however, these are non-generic solutions; we prove that there exist essentially three different types of non-generic asymptotics.

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The outline of the paper is as follows. In Section 2 we discuss the Einstein equations in Bianchi type I symmetry and introduce our main assumption on the anisotropic matter model: The stress-energy tensor is assumed to be represented by a function of the metric and the initial data of the matter source. We shall restrict ourselves to the study of diagonal models, i.e., solutions for which the metric is diagonal in the standard coordinates of Bianchi type I symmetric spacetimes. In Section 3 we rewrite the Einstein equations in terms of new variables and show that the essential dynamics is described by a system of autonomous differential equations on a four dimensional compact state space – the reduced dynamical system. Section 4 is by far the longest and most technical part of the paper. It contains a detailed analysis of the flow induced by the reduced dynamical system on the boundary of the state space. In Section 5 we study the local stability properties of the fixed points of the reduced dynamical system. Finally, in Section 6 we present the main results: Theorems 1–3 and Figure 7. The proofs are based on the results of Sections 4 and 5 and make use of techniques from the theory of dynamical systems. The purpose of Section 7 is to bring to life the general assumptions on the matter source that are made in Sections 2–5. We give three important examples of matter models to which our analysis applies straightforwardly: collisionless (Vlasov) matter, elastic matter and magnetic fields. Finally, Section 8 contains some concluding remarks together with an outlook of possible developments of the present work.

2. The Einstein equations in Bianchi type I We consider a spatially homogeneous spacetime of Bianchi type I. The spacetime metric can be written as ds2 = −dt2 + gij (t)dxi dxj

(i, j = 1, 2, 3) ,

(1)

where gij is the induced Riemannian metric on the spatially homogeneous surfaces t = const. By kij we denote the second fundamental form of the surfaces t = const. The energy-momentum tensor Tμν (μ, ν = 0, 1, 2, 3) contains the energy density ρ = T00 and the momentum density jk = T0k . Einstein’s equations, in units c = 1 = 8πG, decompose into evolution equations and constraints. The evolution equations are 1 (2a) ∂t gij = −2gil k l j , ∂t k ij = tr k k ij − T ij + δ ij (tr T − ρ) , 2 the Hamiltonian constraint reads (tr k)2 − k ij k ji − 2ρ = 0 ,

(2b)

and the momentum constraint is jk = 0 (which is due to the fact that Bianchi I spacetimes are spatially flat). The latter does not impose any restriction on the gravitational degrees of freedom, which are represented by gij and k ij , but only on the matter fields. In general, the system (2) must be complemented by equations describing the evolution of the matter fields.

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For certain matter models, the energy-momentum tensor takes the form of an explicit functional of the metric gij . Let us give some examples. For a (non-tilted) perfect fluid, the energy-momentum tensor reads Tμν dxμ dxν = ρ dt2 + p gij dxi dxj , where the energy density ρ and the pressure p are connected via a barotropic equation of state p(ρ). When we define n as n = (det g)−1/2 , the Einstein equations (2) imply p(ρ) + ρ = n(dρ/dn) (which is a special case of equation (3) below). Hence ρ is obtained as a function ρ = ρ(n). Clearly, prescribing ρ as a function of n is equivalent to prescribing an equation of state for the fluid (modulo a scaling constant). Likewise, for collisionless matter the components of the energy-momentum tensor are given as functions of the metric gij , which depend on the initial data for the matter (which is described by the distribution function of particles in phase space; see Section 7 for details). We thereby obtain the energy density ρ and the principal pressures pi (which are defined as the eigenvalues of T ij ) as ρ = ρ(gij ) and pi = pi (gjk ). Finally, for elastic matter the constitutive equation of state of the material determines ρ and pi as functions of gij ; for details we refer again to Section 7. Motivated by these examples we make our fundamental assumption on the matter model. Assumption 1. The components of the energy-momentum tensor are represented by smooth (at least C 1 ) functions of the metric gij . We assume that ρ = ρ(gij ) is positive (as long as gij is non-degenerate). Remark. We note that the particular form of the functions that represent the components of the energy-momentum tensor may depend on a number of external parameters or external functions that describe the properties of the matter, and/or the initial data of the matter field(s); see Section 7 and [21] for examples. By Assumption 1, the evolution equations of the matter fields in Bianchi type I are contained in the Einstein evolution equations (2a) via the contracted Bianchi identity. Remark. The regularity of T ij as a function of the spatial metric ensures that the Cauchy problem for the evolution equations with initial data at t = t0 is locally well-posed on a time interval (t− , t+ ), t− < t0 < t+ . The requirement ρ > 0 and the constraint equation (2b) imply that the mean curvature tr k never vanishes in the interval of existence. Without loss of generality we assume that tr k < 0 on (t− , t+ ), i.e., we consider an expanding spacetime; the case tr k > 0 is obtained from tr k < 0 by replacing t with −t. If the energy momentum tensor satisfies the inequality tr T + 3ρ > 0 (which corresponds to Assumption 3 in Section 3), then the mean curvature is non-decreasing, the solutions of (2) exist in an interval (t− , +∞) with t− > −∞, and the spacetime is future geodesically complete. By a time translation we can assume t− = 0. Under additional conditions on T ij the limit t → 0 corresponds to a curvature singularity. We refer to [16, 18] for a proof of these statements in the context of general spatially homogeneous cosmologies; for the special case we consider these results follow straightforwardly from our formulation of the equations, cf. (11).

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Lemma 1. Assumption 1 implies T ij = −2

∂ρ gjl − δ i j ρ . ∂gil

(3)

In particular, the functional dependence of the energy-momentum tensor on the metric gij is completely determined by the function ρ(gij ). Proof. Differentiating (2b) w.r.t. t and using the evolution equation for k ij we obtain (4a) ∂t ρ = k ji (T ij + δ ij ρ) . On the other hand, using ρ = ρ(gij ) and the evolution equation for the metric we obtain ∂ρ ∂t ρ = −2k ji gjl . (4b) ∂gil Equating the r.h. sides of (4a) and (4b) gives an identity between two polynomials in k ji with coefficients that depend only on the metric gij ; therefore the corresponding coefficients must be equal, which gives (3). Equation (3) will play a fundamental role in the following. We emphasize that the energy-momentum tensor must be independent of the second fundamental form for the proof of Lemma 1 to hold. We refer to Section 7 for a derivation of (3) within the Lagrangian formalism. Initial data for the Einstein-matter system are given by gij (t0 ), k ij (t0 ) and the initial values of the matter fields. Without loss of generality we can assume that gij (t0 ) and k ij (t0 ) are diagonal (by choosing coordinates adapted to an orthogonal basis of eigenvectors of k ij (t0 )). Assumption 2. We assume that there exists initial data for the matter such that T ij is diagonal in any orthogonal frame. Uniqueness of solutions of the evolution equations then implies that (gij , k ij , T ij ) remain diagonal for all times. We refer to solutions of this type as diagonal models; henceforth we restrict ourselves to these models.

3. Dynamical systems formulation We introduce a set of variables for which the Einstein evolution equations (2a) decouple. The reduced system we thereby obtain is a dynamical system on a bounded state space. Let tr k , 3 ki Σi = − i − 1 , H H=−

x = g 11 + g 22 + g 33 si =

g ii x

(dimensional variables) , (dimensionless variables) .

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There is no summation over the index i in these definitions. The division by H is not a restriction, since H > 0 for all solutions, cf. the remark in Section 2. The dimensionless variables satisfy the constraints Σ1 + Σ2 + Σ3 = 0 ,

s1 + s2 + s3 = 1 .

The transformation from the six variables (g ii , k ii ) to the variables (H, x, si , Σi ) that satisfy the constraints is one-to-one. The variable x can be replaced by n = (det g)−1/2 , since x = n2/3 (s1 s2 s3 )−1/3 . We define dimensionless matter quantities by Ω=

ρ , 3H 2

wi =

Ti pi = i, ρ ρ

w=

1 p (w1 + w2 + w3 ) = ; 3 ρ

there is no summation over i. The quantity p is the isotropic pressure; it is simply given as the average of the principal pressures pi (i = 1, 2, 3). The quantities wi (i = 1, 2, 3), which are the rescaled principal pressures, encode the degree of anisotropy of the matter. If wi = w ∀i, the matter is isotropic. The density is a function of the metric and can thus be expressed as ρ = ρ(n, s1 , s2 , s3 ). Equation (3) then entails ∂ log ρ − 1, ∂ log n ⎞ ⎛ ∂ log ρ ∂ log ρ ⎠. wi = w + 2 ⎝ − si ∂ log si ∂ log s j j w=

(5a) (5b)

We make the following simplifying assumption: Assumption 3. We suppose that the isotropic pressure and the density are proportional, i.e., we assume w = const, where w ∈ (−1, 1) .

(6)

According to Assumption 3, the density and the isotropic pressure behave like those of a perfect fluid with a linear equation of state satisfying the dominant energy condition. It seems natural to focus attention on anisotropic matter sources satisfying Assumption 3, since they generalize the behaviour of perfect fluid models widely used in cosmology. In the concluding remarks, see Section 8, we will indicate how to treat the more general case where w = const. Remark. In Assumption 3 the cases w = ±1 are excluded. The case w = 1 leads to different dynamics, which we refrain from discussing here. The case w = −1 is excluded since it comprises as a subcase the de Sitter spacetime which does not exhibit a singularity. Taking account of (5), Assumption 3 implies that ρ(n, s1 , s2 , s3 ) = n1+w ψ(s1 , s2 , s3 ) ,

(7)

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for some function ψ(s1 , s2 , s3 ). Moreover, for all i, the matter anisotropies wi are functions of (s1 , s2 , s3 ) alone and ⎞ ⎛ ∂ log ψ ∂ log ψ ⎠. (8) − si wi = wi (s1 , s2 , s3 ) = w + 2 ⎝ ∂ log si ∂ log s j j Finally we introduce a dimensionless time variable τ by d d = H −1 ; dτ dt

(9)

in the following a prime will denote differentiation w.r.t. τ . Expressed in the new variables, the Einstein evolution equations decouple into the dimensional equations

Ω H = −3H 1 − (1 − w) , x = −2 x 1 + sk Σk , (10) 2 k

and an autonomous system of equations 1 Σi = −3Ω (1 − w)Σi − (wi − w) 2 sk Σk si = −2si Σi −

(i = 1, 2, 3) ,

(11a)

(i = 1, 2, 3) .

(11b)

k

The Hamiltonian constraint results in Ω = 1 − Σ2 ,

where

Σ2 :=

1 2 Σ1 + Σ22 + Σ23 , 6

(12)

which implies Σ2 < 1. This constraint is used to substitute for Ω in (11). The reduced dimensionless dynamical system (11) encodes the essential dynamics of Bianchi type I anisotropic spacetimes where the matter is described by the quantities wi = wi (s1 , s2 , s3 ) and w = const; once this system is solved, the decoupled equations (10) can be integrated and the solution (g ii , k ii ) can be constructed. There exist useful auxiliary equations in connection with the system (11). In particular, the evolution equation for Ω is given by 2 wk Σk . (13) Ω = Ω 3(1 − w)Σ − k

Likewise, for ρ we get ρ = −ρ [3(1 + w) +

k

wk Σk ].

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233

The state space of the dimensionless dynamical system (11) is the fourdimensional bounded open connected set X given by X =

(Σ1 , Σ2 , Σ3 , s1 , s2 , s3 ) (Σ < 1) ∧ 2

Σk = 0

∧ (0 < si < 1 ∀i) ∧

k

sk = 1

.

k

This set can be written as the Cartesian product X = K×T of two two-dimensional bounded open connected sets,

2 K = (Σ1 , Σ2 , Σ3 ) Σ < 1 ∧ Σk = 0 . (14a) k

T=

(s1 , s2 , s3 ) (0 < si < 1 ∀i ) ∧

sk = 1

.

(14b)

k

The set K is the Kasner disc; it is typically depicted in a projection onto the plane with conormal (1, 1, 1), see Figure 1. The boundary of K is the Kasner circle ∂K = {(Σ2 = 1) ∧ (Σ1 + Σ2 + Σ3 = 0)}. The Kasner circle contains six special points, which are referred to as LRS points: The three Taub points T1 , T2 , T3 given by (Σ1 , Σ2 , Σ3 ) = (2, −1, −1) and permutations, and the three non-flat LRS points Q1 , Q2 , Q3 given by (Σ1 , Σ2 , Σ3 ) = (−2, 1, 1) and permutations. The six sectors of K are denoted by permutations of the triple 123 ; by definition, Σi < Σj < Σk holds in sector ijk . The set T is the interior of a triangle contained in the affine plane s1 +s2 +s3 = 1, see Figure 1. The boundaries form a triangle with the three sides s1 = 0, s2 = 0 and s3 = 0; the corners are given by (1, 0, 0), (0, 1, 0) and (0, 0, 1). The functions ψ(s1 , s2 , s3 ) and wi (s1 , s2 , s3 ) in (7) and (8) are understood as functions on the domain T. We conclude this section with a discussion of the energy conditions. Our basic assumption is ρ > 0. Then the dominant energy condition is expressed in the new matter variables as |wi (s1 , s2 , s3 )| ≤ 1, ∀(s1 , s2 , s3 ) ∈ T and ∀ i = 1, 2, 3; the weak energy condition is −1 ≤ wi ∀ i = 1, 2, 3; the strong energy condition is satisfied if the weak energy condition holds and w ≥ −1/3. We are mainly interested in matter models that satisfy the dominant energy condition; however we shall also discuss the more general case when the functions wi are merely bounded: Assumption 4. We assume that the rescaled principal pressures are bounded functions, sup |wi (s1 , s2 , s3 )| < ∞ , T

∀ i = 1, 2, 3 .

(15)

234


Q2 213

Σ1 = 1

321

Q3

Σ2

Σ3

=

123

)

−1

=

1

,←

T1

=

T3

1

, 0) ,← (→

Σ1

Σ2

s2

Q1

(0, →

231

=

Σ3

Σ1 = −1

Σ3


132

−1

T2 312

(a) The Kasner disc K

Σ2

s3

(→, 0, ←)

s1

(b) The space T

Figure 1. The four-dimensional state space X is the Cartesian product of the Kasner disc K and the set T. The latter is represented by (the interior of) a triangle; the center of the triangle is the point (s1 , s2 , s3 ) = (1/3, 1/3, 1/3); for the sides, the values of (s1 , s2 , s3 ) are given in the figure, where the arrows denote the directions of increasing values (from 0 to 1).

4. The flow on the boundaries of the state space The present section is devoted to a detailed analysis of the flow induced by (11) on the boundary of the state space. This analysis is rather technical but essential to understand the asymptotic behaviour (in particular toward the past) of solutions of (11). The state space X = K × T, see (14) and Figure 1, is relatively compact. Its boundary is the union of two three-dimensional compact sets, i.e., ∂X = (∂K × T) ∪ (K × ∂T) .

(16)

The intersection of these boundary components is the two-dimensional compact set ∂K × ∂T. 4.1. The vacuum boundary: ∂K × T Consider the boundary component ∂K × T, which is characterized by Σ2 = 1 (i.e., Ω = 0); we call this component the vacuum boundary – the reason for this terminology will become clear through the remarks in this subsection. The dynamical system (11) admits a regular extension from X to the vacuum boundary; we simply let Ω → 0, so that equation (11a) becomes Σi = 0 in this limit. This is independent of (s1 , s2 , s3 ) ∈ T, since wi is bounded on T by Assumption 4. The

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dynamical system (11) thus induces the system Σi = 0 , si = −2si Σi − sk Σk

(i = 1, 2, 3) ,

235

(17)

k

on the vacuum boundary. The vacuum boundary ∂K × T is a solid torus whose cross section is T; each cross section corresponds to (Σ1 , Σ2 , Σ3 ) = const and is an invariant subspace of (17). The fixed points of the system (17) are transversally hyperbolic and form a connected network of lines on ∂K × ∂T: • Kasner circles: There exist three circles of fixed points that can be interpreted as Kasner circles. The Kasner circles are located at the vertices of T, i.e., let (ijk) be a cyclic permutation of (123), then KCi is given by KCi :

Σ2 = 1 ,

(si , sj , sk ) = (1, 0, 0) .

• Taub lines: These are three lines of fixed points given by TLi :

(Σi , Σj , Σk ) = (2, −1, −1) ,

(si , sj , sk ) = (0, s, 1 − s), s ∈ [0, 1] .

• Non-flat LRS lines: These are three lines of fixed points given by QLi :

(Σi , Σj , Σk ) = (−2, 1, 1) ,

(si , sj , sk ) = (0, s, 1 − s), s ∈ [0, 1] .

Let (ijk) be a cyclic permutation of (123). On the Kasner circle KCi there exist three ‘Taub points’: Tii , Tij , and Tik , which are given by (Σi , Σj , Σk ) = (2, −1, −1), (−1, 2, −1), and (−1, −1, 2), respectively. The point Tij (Tik ) is the point of intersection of KCi with TLj (TLk ); Tii is an isolated Taub point since it does not lie on any of the Taub lines. Analogously, there exist three ‘non-flat LRS points’ on KCi : Qii , Qij , and Qik , which are given by (Σi , Σj , Σk ) = (−2, 1, 1), (1, −2, 1), and (1, 1, −2), respectively; the analogous comments apply. In Figure 2(a) we give a schematic depiction of the network of fixed points. Remark. Each of the fixed points on the Kasner circles is associated with a Kasner solution (Bianchi type I vacuum solution) of the Einstein equations, ds2 = −dt2 + a1 t2p1 dx1 ⊗ dx1 + a2 t2p2 dx2 ⊗ dx2 + a3 t2p3 dx3 ⊗ dx3 ,

(18)

where a1 , a2 , a3 are positive constants, and p1 , p2 , p3 are the so-called Kasner exponents, which are constants that satisfy p1 + p2 + p3 = p21 + p22 + p23 = 1. The relation between (Σ1 , Σ2 , Σ3 ) and (p1 , p2 , p3 ) for a Kasner fixed point is 3pi = Σi +1, i = 1, 2, 3. The fixed points on TLi and QLi represent the flat LRS Kasner solution (Taub solution) and the non-flat LRS Kasner solution, respectively; in these cases the Kasner exponents are (1, 0, 0) and permutations and (2/3, −1/3, −1/3) and permutations, respectively . Lemma 2. The solutions of the dynamical system (17) on the vacuum boundary are heteroclinic orbits, i.e., the α- and the ω-limit sets consist of one fixed point each; see Figure 2(b).

236


T23

Q22

KC2

Q23

T22

s

s

1

QL Q33

T32

s TL2

123

321

QL 3

Q31

T31

KC3

QL2

T33

231

3 TL

TL 1

Σ1

Q32

Σ3

213

Q21

T21


Q12

T13

132

312

Σ2

Q11 T11 Q13

T12

(a) Vacuum fixed point set

11 00 11 00 11 00 11 00

KC1

P

(b) Dynamics on the vacuum boundary

Figure 2. A schematic depiction of (a) the network of fixed points and (b) the flow of the dynamical system on the vacuum boundary ∂K × T. Orbits in the interior of ∂K × T connect a transversally hyperbolic source with a transversally hyperbolic sink; for instance, the α-limit (ω-limit) of the orbit through the point P ∈ ∂K × T in Figure 2(b) is a point in sector 312 of KC2 (KC3 ). Proof. Since Σi = const ∀i, the result follows by studying the sign of si for each sector ijk (and at the special points) of the Kasner circle separately. Remark. Not only the Kasner fixed points themselves, but each solution of the dynamical system (17) on the vacuum boundary can be interpreted as a Kasner solution. This is simply because (Σ1 , Σ2 , Σ3 ) ≡ const and Σ2 = 1. 4.2. The cylindrical boundary: K × ∂T The second component of ∂X is the set K×∂T; the set ∂T is a triangle and consists of the three sides {si = 0} (i = 1, 2, 3), hence K × ∂T = C1 ∪ C2 ∪ C3 ,

where

Ci := K × {si = 0} .

(19)

Each set Ci is compact; since it has the form of a cylinder, we call K × ∂T the cylindrical boundary. By Ci we denote the interior of Ci . Asymptotic properties of matter models. The dynamical system (11) admits a regular extension from X to the lateral surfaces of the cylinders Ci , cf. the discussion of the vacuum boundary; however, in general, the system is not extendible to the interior Ci , nor to the top/base surfaces. The simple reason for this is that, although wi (s1 , s2 , s3 ), i = 1, 2, 3, are bounded on T, cf. (15), in general these functions need not possess limits as (s1 , s2 , s3 ) converges to a point on ∂T. However,

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237

for reasonable matter models (such as collisionless matter and elastic matter), the matter anisotropies wi (s1 , s2 , s3 ) are functions on T that admit a unique extension to T. We make this a general assumption. Assumption 5. The matter anisotropies wi (s1 , s2 , s3 ) (i = 1, 2, 3) admit unique extensions from T to T. The extended functions are assumed to be sufficiently smooth on ∂T. (For the majority of our future purposes, continuity is a sufficient requirement; however, we will assume the functions to be C 1 to facilitate our analysis.) Assumption 5 ensures that the dynamical system (11) can be extended to Ci (i = 1, 2, 3) and thus to the entire boundary ∂X of the state space X . The next assumption is a simplifying assumption on the functions wi (i = 1, 2, 3) on ∂T. Assumption 6. We assume that wi (s1 , s2 , s3 ) ≡ v− = constant on Ci (i.e., when si = 0)x for all i = 1, 2, 3. Remark. Assumption 6 stems from basic physical considerations; in particular, it is satisfied for collisionless matter and for elastic matter, see Section 7. Loosely speaking, Assumption 6 means that the (rescaled) principal pressure in a direction i becomes independent of the values of gjj and gkk (i = j = k = i) in the limit gii → ∞ (⇔ g ii → 0), provided that gjj , gkk remain bounded. (The complementary statement is implicit in our previous assumptions: The fact that wi is well-defined for si = 1 means that the rescaled principal pressure in a direction i converges to a limit as gii → 0, when gjj and gkk remain bounded from below.) For some matter models like elastic matter, there exists a function v(z1 , z2 , z3 ), which is defined on T (z1 , z2 , z3 ), sufficiently smooth on ∂T, and symmetric in the arguments z2 and z3 , such that w1 (s1 , s2 , s3 ) = v(s1 , s2 , s3 ) ,

w2 (s1 , s2 , s3 ) = v(s2 , s3 , s1 ) , w3 (s1 , s2 , s3 ) = v(s3 , s1 , s2 )

(20a)

for all (s1 , s2 , s3 ) ∈ T. Equation (20a) follows from basic physical considerations; it is a consequence of the fact that there is no distinguished direction and reflects the freedom of permuting the axes. Assumption 6 is equivalent to the requirement that v(0, ζ1 , ζ2 ) = v− for all ζ1 , ζ2 . Definition 1. For matter models satisfying the symmetry property (20a), in slight abuse of notation we define a function v(s) on the interval [0, 1] by [0, 1] s → v(s) := v(s, 1 − s, 0) .

(20b)

The value of v at the endpoint s = 0 is v(0) = v− , cf. Assumption 6; the value at s = 1 we denote by v+ , i.e., v(1) = v+ . For more general matter models, the initial data for the matter fields might break the symmetry (20a), an example being collisionless matter. Let us denote by I the initial data of the matter fields and by I the space of possible initial data. Let σ be a permutation of the triple (123), which induces the permutation

238



(s1 , s2 , s3 ) → (sσ(1) , sσ(2) , sσ(3) ) on any ordered triple (s1 , s2 , s3 ) in R3 ; in slight abuse of notation we use the symbol σ for the induced permutation as well. Finally, let I(σ) denote the initial data arising from I by the (induced) permutation. (For instance, in the case of collisionless matter, I represents the distribution function f0 , and I(σ) = f0 ◦ σ, cf. Section 7). Then there exists u : I × T → R, I × (z1 , z2 , z3 ) → u[I](z1 , z2 , z3 ), sufficiently smooth, such that wi = u I(σ−1 ) ◦ σi , (20a ) i

where σi is a permutation with σi (1) = i. Equation (20a ) reduces to (20a) if there is no dependence on I ∈ I; then u[I] is simply replaced by v. (The symmetry of v in the second and third argument corresponds to the freedom of choosing even or odd permutations in (20a ).) Assumption 6 is equivalent to the requirement that u[I](0, ζ1 , ζ2 ) = v− for all I ∈ I, for all ζ1 , ζ2 . For matter models of this type Definition 1 is replaced by Definition 1 . For i = 1, 2, 3, let j, k be such that sgn(ijk) = +1. We define a function ui (s) on the interval [0, 1] by (20b ) [0, 1] s → ui (s) := u I(σ−1 ) (s, 1 − s, 0) . j

The value of ui at the endpoint s = 0 is ui (0) = v− , cf. Assumption 6; the value at s = 1 we denote by v+ , i.e., ui (1) = v+ . Equation (20b ) reduces to (20b) if there is no dependence on I and ui is replaced by v. Remark. The important fact that ui (1) is independent of i is a direct consequence of (20a ) and the identity w1 + w2 + w3 = 3w; see (21) and the discussion below. The dynamical system on Ci . In the following let (ijk) be a cyclic permutation of (123). We begin our analysis of the flow of the dynamical system on the cylindrical boundary (19) by fixing the notation. Consider the cylinder Ci . Since si = 0 on Ci , we may set sj = s, sk = 1 − s, s ∈ [0, 1]; the cylinder Ci is then given as the Cartesian product K × {0 ≤ s ≤ 1}. We call the set K × {s = 0} the base, and K × {s = 1} the top of the cylinder. The boundary of the top is the Kasner circle KCj , the boundary of the base is KCk , see Figure 3. The matter anisotropies on the cylinder Ci are represented by wi , wj , and wk . Assumption 6 implies that wi ≡ v− on Ci . When regarded as a function of s ∈ [0, 1], wj coincides with the function ui (s) of Definition 1 , i.e., wj (s1 , s2 , s3 ) = u[I(σ−1 ) ](sj , sk , si ) = u[I(σ−1 ) ](s, 1−s, 0) = ui (s) on Ci . Finally, wk = 3w−wi −wj , j j since wi + wj + wk = 3w. Evaluation of this identity at s = 1 (i.e., for (si , sj , sk ) = (0, 1, 0)) yields v− + ui (1) + v− = 3w, because wk = v− for (si , sj , sk ) = (0, 1, 0) by Assumption 6. Since, by definition, ui (1) = v+ , we obtain 2v− + v+ = 3w .

(21)

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239

Using this equation in the form 3w−v− = v− +v+ , the anisotropies on the cylinder Ci take the form wi = v− ,

wj = ui (s) ,

wk = v+ + v− − ui (s) ,

(22 )

where ui (s) satisfies ui (0) = v− and ui (1) = v+ (independently of i). In the simpler case (20a) we have wi = v− ,

wj = v(s) ,

wk = v+ + v− − v(s) ,

(22)

where v(s) satisfies v(0) = v− and v(1) = v+ , and again (21). However, by (20a), on Ci , wk can also be written as wk = v(1 − s, 0, s) = v(1 − s). Therefore, v(s) + v(1 − s) = v+ + v− ; this identity can be written in an alternative form, v− + v+ v− + v+ v(s) − = − v(1 − s) − , (23) 2 2 which states that v(s) − (v− + v+ )/2 is antisymmetric around s = 1/2. Using (22) we can write the dynamical system that is induced on Ci by (11) as s = −2s(1 − s)(Σj − Σk ) , 1 Σi = −3Ω (1 − w)Σi − [v− − w] , 2 1 (1 − w)Σj − v(s) − w , Σj = −3Ω 2 1 (1 − w)Σk − v+ + v− − v(s) − w , Σk = −3Ω 2

(24a)

(24b)

where we recall that Ω = 1 − Σ2 . The remainder of this section is devoted to a detailed analysis of the flow of the dynamical system (24). In the case (22 ) the function v(s) in (24) is simply replaced by ui (s). The qualitative dynamics of the resulting system remains unchanged because the key quantities in our analysis will turn out to be v− and v+ , which are the same for (22) and (22 ). In the following we analyze in detail the dynamical system (24) which is connected with the matter models satisfying (20a), (20b) and (22). The more general case, represented by (20a ), (20b ) and (22 ), leads to the same conclusions and will only be commented on sporadically. Flow on the lateral boundary of Ci . The flow on the lateral surface of Ci (which is given by Σ2 = 1) is independent of the matter quantities, since Ω = 0. Orbits on the lateral surface satisfy s ≶ 0 ⇔ Σj − Σk ≷ 0 , while s = 0 on the lines of fixed points TLi and QLi , see Figure 3.

240


KCj

⎞ si = 0 ⎝ sj = 1 ⎠ sk = 0 ⎛

TLi ⎛


sj

QLi

Σk

⎞

si = 0 ⎝ sj ↑ ⎠ sk ↓ ⎛ ⎞ si = 0 ⎝ sj = 0 ⎠ sk = 1

Σi

Σj − Σk < 0 Σj − Σk > 0

Σj

KCk Figure 3. The boundary component K×∂T consists of the three cylinders C1 ∪ C2 ∪ C3 . In this figure, Ci is depicted together with the flow of the dynamical system on the lateral boundary and the vacuum fixed points. Flow on the base/top of Ci and definition of the various cases. On the base of the cylinder Ci , where s = 0, we have wi = wj = v− and wk = v+ , see (22) or (22 ). The dynamical system induced on the base of Ci is thus 1 Σi Σi = −3Ω (1 − w) − (v− − w) , Σj Σj 2 1 Σk = −3Ω (1 − w)Σk − (v+ − w) ; (25) 2 since 2v− + v+ = 3w we have v+ − w = 2(w − v− ). Definition 2. We define the quantity β as v+ − w w − v− = . β := 2 1−w 1−w

(26)

This quantity will play an essential role in our analysis. It is not difficult to show that the solutions of the system (25) form a family of straight lines that are attracted by a common focal point, the fixed point Rk :

(Σi , Σj , Σk ) = β (−1, −1, 2) ,

(27)

see Figure 4. Analogously, on the top of the cylinder Ci (which is given by si = 0, sj = 1, sk = 0) there is a fixed point Rj , where (Σi , Σj , Σk ) = β (−1, 2, −1). The flow of the dynamical system that is induced on the top of Ci is represented by a family of straight lines focusing at Rj . Depending on the value of β, see (26), we distinguish several scenarios whose main characteristic is the position of the fixed point Rk in relation to the Kasner

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241

circle KCk (on the base of Ci ). In the following we discuss these scenarios in detail, in particular in view of their compatibility with the energy conditions. C+ β > 1 (and β < 2, see below). The point Rk lies beyond the Taub point on the Kasner circle (which is the point Tkk on KCk ), i.e., Σk |Rk > 2; see Figure 4(a). This is the case if and only if v+ > 1, which is not compatible with the dominant energy condition. B+ β = 1. The point Rk coincides with the Taub point Tkk , i.e., Σk |Rk = 2; see Figure 4(b). This is the case if and only if v+ = 1; accordingly, v− = (3w−1)/2 by (21), i.e., we have the chain (3w − 1)/2 = v− < w < v+ = 1. Imposing the dominant energy condition on v− leads to the restriction 1 w≥− . 3 At the same time the strong energy condition is satisfied. A+ β ∈ (0, 1). The point Rk lies between the center of the Kasner disk and the Taub point with a value Σk |Rk ∈ (0, 2); see Figure 4(c). This is the case if and only if (3w − 1)/2 < v− < w < v+ < 1. If w ≥ −1/3, the dominant (and strong) energy condition are automatically satisfied for v± and β can take any value in (0, 1). If w < −1/3, the condition v− ≥ −1 is stronger than the condition v− > (3w − 1)/2; the range of β is then restricted to β ∈ (0, 2(1 + w)/(1 − w)]. A 0 β = 0. The point Rk lies at the center of the Kasner disc. This is the case iff v− = v+ = w. A− β ∈ (−1, 0). The point Rk lies between the center of the Kasner disk and the non-flat LRS point (the point Qkk ) with a value Σk |Rk ∈ (−2, 0); see Figure 4(d). This is the case iff −1 + 2w < v+ < w < v− < (1 + w)/2. If w ≥ 0, the dominant (and strong) energy condition are automatically satisfied for v± and β can take any value in (−1, 0). If w < 0, the condition v+ ≥ −1 is stronger than the condition v+ > −1 + 2w; the range of β is then restricted to β ∈ [−(1 + w)/(1 − w), 0). B− β = −1. The point Rk coincides with the non-flat LRS point Qkk , i.e., Σk |Rk = −2; see Figure 4(e). This is the case iff −1 + 2w = v+ < w < v− = (1 + w)/2. The dominant energy condition requires w≥0 and at the same time the strong energy condition is satisfied. C− β < −1 (and β > −2, see below). The point Rk lies beyond the non-flat LRS point Qkk , i.e., Σk |Rk < −2; see Figure 4(f). This is the case if and only if v+ < −1 + 2w < w < (1 + w)/2 < v− . To ensure compatibility with the dominant energy condition we must presuppose w > 0. The quantity β cannot assume arbitrary values less than −1 without a violation of the energy conditions. Imposing the energy conditions on v± restricts the possible range; we obtain that β must be greater than or equal to the

242



maximum max{−(1 + w)/(1 − w), −2}. The extremal case would be β = −2, which corresponds to v− = 1 (⇔ v+ = 3w − 2). In this extremal case, the dominant energy condition requires w ≥ 1/3. Remark. In a nutshell: Iff β ∈ [−2, 1], then compatibility with the energy conditions is possible; in general it is required that w be sufficiently large (w ≥ 1/3 is always sufficient). Remark. For the function v(s) (where we recall v(0) = v− and v(1) = v+ ) we have the fundamental inequalities v− < w < v+

in the + cases

(β > 0)

v − = w = v+

in the case A 0

(β = 0)

v− > w > v+

in the − cases

(β < 0)

(28)

which will be used several times in the following. Remark. As we will see below, several characteristic properties of the flow of the dynamical system on the cylinder Ci change when |β| ≥ 2. Therefore, in the following, C+ will denote the case β ∈ (1, 2) only, and C− will denote the case β ∈ (−2, −1). The cases |β| ≥ 2 will be denoted by extra symbols: D+ β ≥ 2. The point Rk lies far beyond the Taub point on the Kasner circle, i.e., Σk |Rk ≥ 4. Like case C+ this case is not compatible with the dominant energy condition. D− β ≤ −2. The point Rk lies far beyond the non-flat LRS point Qkk , i.e., Σk |Rk ≤ −4. The only subcase that is compatible with the dominant energy condition is β = −2, which corresponds to v− = 1 (⇔ v+ = 3w − 2). In this case, the dominant energy condition requires w ≥ 1/3. Stability of the fixed points on the boundary of Ci . Combining the analysis of the flow on the lateral surface and on the base/top surface we obtain the stability properties of the fixed points on the Kasner circles KCj and KCk . (We simply put Figures 3 and 4 on top of each other.) Depending on the case there exist (transversally hyperbolic) sources and sinks on KCj /KCk ; the location of these sources and sinks on KCk is depicted in Figure 4 (the location of sources/sinks on KCj is obtained by a reflection w.r.t. the Σi -axis). In cases B+ and B− , Rk (which coincides with Tkk and Qkk , respectively) is not transversally hyperbolic; the center manifold reduction theorem implies that, in case B+ , the fixed point is a center saddle; in case B− , it acts as a sink, i.e., it is the ω-limit for a two-parameter set of orbits in Ci ; the analog holds for Rj . The point Rj [Rk ] lies in the interior of the top [base] of the cylinder, iff we are in one of the A scenarios. By (24) we obtain (29) (1 − s)−1 (1 − s) = 6β , s−1 s = 6β . Rj

Rk

In combination with the results on the flow on the base/top, see Figure 4, we obtain that Rj [Rk ] is a saddle in case A+ , while it is a sink in case A− .

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Σk

Σk

Σk

Tkk ≡ Rk

Rk Σk

=

Rk 2/β

Σi

Σi

Σi

Σj

Σj

Σj (a) C+

(b) B+

Σk

Σi

(c) A+

Σk

Σk

Σi

Σi

Σk

Rk

Σj (d) A−

243

Qkk ≡ Rk

Σj (e) B−

=

2/β

Rk

Σj (f) C−

Figure 4. Flow on the base of the cylinder Ci ; the orbits are straight lines focusing at Rk . The Kasner circle KCk consists of fixed points that can act as sources, saddles, or sinks for orbits in the interior of the cylinder Ci ; in the figures, bold continuous [dashed] lines denote transversally hyperbolic sources [sinks]. In the cases B+ and B− , Rk is not transversally hyperbolic. In the cases C+ and C− the point Rk is close to the Kasner circle since |β| < 2; increasing the value of |β| amounts to increasing the distance between Rk and the Kasner circle; in the case |β| > 2 the distance is greater than the Kasner radius. Accordingly, if β > 2 there appear transversally hyperbolic sinks in the lower half of the Kasner circle; if β < −2, some of the transversally hyperbolic sources in the upper half vanish. The flow on the top of the cylinder and the stability properties of the Kasner fixed points on KCj are obtained by a reflection w.r.t. the Σi -axis. Remark. For completeness we briefly discuss the case A 0 as well. In this case, β = 0 and the fixed points Rj /Rk are not hyperbolic. Center manifold analysis reveals that the character of these points depends on the derivative of v(s) at s = 0 and s = 1, i.e., on v (0) and v (1). As the antisymmetry property (23) of the function v(s) guarantees that v (0) = v (1), there exist two subcases: A 0+ v (0) = v (1) < 0. Rj /Rk are center saddles. This case resembles the case A+ . A 0− v (0) = v (1) > 0. Rj /Rk act as sinks. This case resembles the case A− .

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Apart from a special subcase (v(s) ≡ w), the degenerate case v (0) = v (1) = 0 is excluded by Assumption 7 below. (If we analyze the system (24) with the function ui (s) instead of v(s), cf. (22 ), there might exist more subcases of A 0 , because the signs of ui (0) and ui (1) might be different; however, since the focus of our analysis lies on the + and − cases and not on A 0 , we refrain from discussing these additional subcases further.) The remaining fixed points on the boundary of Ci are located on the lateral boundary: TLi and QLi . Using (13), (21), and (26) we find 3 (30a) Ω−1 Ω TL = 3(1 − v− ) = (1 − w)(2 + β) , i 2 3 3 Ω−1 Ω QL = 2(1 − v+ ) + (1 − v− ) = (1 − w) + (1 − v+ ) i 2 2 3 (30b) = (1 − w)(2 − β) . 2 Therefore, if β < 2 each fixed point on QLi acts as the source for exactly one orbit in the interior of Ci ; if β > 2, each point on QLi is the ω-limit set for one interior orbit. The analog is true for TLi : If β > −2, then each fixed point on TLi acts as the source for one interior orbit; if β < −2, each fixed point on TLi attracts one interior orbit as τ → ∞. (The proof of these statements is based on the center manifold reduction theorem, where we use that the lateral boundary is the center manifold of QLi .) The borderline cases β = ±2 cannot be dealt with by using local methods; these cases are discussed in Lemma 3 and Lemma 4 below. Dynamics in the interior of Ci . The dynamical system (24) on the cylinder Ci admits a non-negative function, M(i) := (Σi + β)2 ,

M(i) = −3Ω(1 − w)M(i) ,

(31)

which is strictly monotonically decreasing on Ci whenever Σi = −β. The plane Σi = −β itself, i.e., (32) Di = (s, Σi , Σj , Σk ) ∈ Ci Σi = −β is an invariant subset. It can be characterized as the unique plane, orthogonal to the base/top of the cylinder, whose boundary contains the points Rj and Rk . The plane Di intersects the cylinder Ci whenever −2 < β < 2. (When β = ±2, Di is tangent to Ci ; the intersection of the two is TLi or QLi . When |β| > 2, Di does not intersect Ci .) Lemma 3. Let γ be an orbit in the (interior of the) cylinder Ci such that γ ⊂ Di . The α-limit set of γ is • one of (where • a fixed • a fixed

the transversally hyperbolic sources on the Kasner circles KCj /KCk the points on ∂Di are excluded), or point on QLi (when β < 2), or point on TLi (when β > −2).

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Each transversally hyperbolic source on KCj /KCk is the α-limit set for a oneparameter family of orbits; each fixed point on QLi and TLi is the α-limit set for one orbit in Ci (under the given assumption on β). Proof. Assume first that |β| < 2. The orbit γ is either contained in the invari− ant subset S+ i = {Σi > −β} or in the invariant subset Si = {Σi < −β} of Ci . Since the function M(i) is monotonically decreasing on the invariant sets S± i , the monotonicity principle implies that the α-limit set of γ is contained on the boundary of S± i ; however, Di is excluded, since M(i) attains its minimum there. This leaves the lateral boundary of Ci as the only possible superset of the α-limit set of γ. From the structure of the flow on the boundary of Ci , which is depicted in Figure 3, we conclude that it is only the fixed points that come into question as possible α-limit sets. Combining these observations with the local analysis of the fixed points, which we performed in the previous subsection, leads to the state+ ment of the lemma. In the case β ≥ 2, the set S− i is empty and Si corresponds to the interior of the cylinder. The monotonicity principle, when applied to the function M(i) , implies that α(γ) is located on the lateral boundary of Ci , where QLi is excluded. The case β ≤ −2 is analogous and thus the claim of the lemma is proved. Lemma 4. Let γ ⊂ Ci , γ ⊂ Di . Then the ω-limit set ω(γ) of γ depends on the actual case under consideration (as characterized by the quantity β): • β > 2 (→ D+ ). ω(γ) is one of the transversally hyperbolic sinks on KCj /KCk or a point on QLi . • β = 2 (→ D+ ). ω(γ) ⊆ QLi . • β ∈ [−1, 2) (↔ A, B, C+ ). ω(γ) ⊆ Di . • β ∈ (−2, −1) (↔ C− ). ω(γ) is one of the transversally hyperbolic sinks on KCj /KCk or it lies on Di . • β ≤ −2 (↔ D− ). ω(γ) is one of the transversally hyperbolic sinks on KCj /KCk or it lies on TLi . Remark. In analogy to the statement of Lemma 3, each transversally hyperbolic sink on KCj /KCk is the ω-limit set for a one-parameter set of orbits; for β > 2, each fixed point on QLi attracts exactly one interior orbit, so that the set QLi , when regarded as a whole, attracts a one-parameter set of interior orbits; see (30); in contrast, for β = 2, QLi attracts every orbit in Ci , i.e., a two-parameter set. Proof. The proof is analogous to the proof of Lemma 3. Consider first the case + |β| < 2. On Si , which is the closure of the invariant subset defined in the proof of − Lemma 3, the function M(i) attains its maximum on TLi ; on Si , the maximum is attained on QLi ; therefore, these sets are excluded as possible ω-limits sets. Using the analysis of the flow on the boundary of the cylinder, the statement of the lemma ensues. The cases |β| ≥ 2 are analogous.

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Dynamics on the invariant plane Di . Lemma 4 states that the future dynamics of orbits in Ci is connected with the properties of the flow on Di . (This is for |β| < 2, i.e., for all cases A, B, C.) We thus proceed by investigating the flow of the dynamical system on this invariant plane. Let Di denote the set of fixed points on (the interior of) Di for the dynamical ¯ k ) of any fixed point P ∈ Di must satisfy ¯j, Σ system (24). The coordinates (¯ s, Σ ¯ ¯ Σj = Σk = β/2 and 1 v(¯ s) = (v− + v+ ) , (33) 2 where s¯ ∈ (0, 1). In the + and − cases (i.e., β ≶ 0) we have v− ≷ v+ ; therefore there exists at least one solution of equation (33). (Analogously, in the cases A 0+ and A 0− , existence of a solution is guaranteed by the fact that the derivatives of v(s), or ui (s), at s = 0 and s = 1 have the same sign.) Remark. Because v(s) − (v− + v+ )/2 is antisymmetric around s = 1/2, cf. (23), (0) one solution of equation (33) is s¯ = 1/2. Consequently, the point Pi with coor¯ ¯ dinates (¯ s, Σj , Σk ) = (1/2)(1, β, β) is always a fixed point on Di . (If we analyze the system (24) with the function ui (s) instead of v(s), cf. (22 ), s¯ = 1/2 will in general not be a solution.) We make the simplifying assumption that the zeros of the function v(s) − (v− + v+ )/2 are simple zeros: Assumption 7. We assume that the function v(s) either satisfies v (¯ s) = 0 ,

for all

s¯ ∈ [0, 1]

such that

v(¯ s) =

v− + v+ , 2

(34)

or v(s) ≡ w for all s. Assumption 7 is automatically satisfied for collisionless matter and elastic matter and helps to avoid unnecessary clutter in the analysis of the flow on Di ; requiring (34) is equivalent to assuming that the set of fixed points Di on Di is a discrete set of hyperbolic fixed points. In addition, it follows that the number of fixed points is always odd, i.e., #Di = 2d + 1, d ∈ N. Accordingly, we can write (−d) (−1) (0) (1) (d) Di = {Pi , . . . , Pi , Pi , Pi , . . . , Pi }, where the coordinates of these fixed points satisfy 0 < s¯(−d) < · · · < s¯(−1) < s¯(0) < s¯(1) < · · · < s¯(d) < 1. The only exception we admit is the case v(s) ≡ w, which is a subcase of A 0 ; in this special case, the set Di of fixed points is not a discrete set but coincides with the line Σj = Σk = 0 in Di ; this special case will be briefly commented on at the end of this section. Remark. The antisymmetry property (23) of v(s) guarantees that s¯n + s¯−n = 1, n = 0, 1, . . . , d. This is not true in general, if we analyze the system (24) with the function ui (s) instead of v(s), cf. (22 ).

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Let P ∈ Di be a fixed point in Di with coordinates (¯ s, β/2, β/2). The eigenvalues of the linearization of the dynamical system (24) at P are given by 64¯ s(1 − s¯) 9 v (¯ s) (35a) − (1 − w)(2 + β)(2 − β) 1 ± 1 − 16 3(1 − w)2 (2 − β) and

3 − (1 − w) ΩP . 2

(35b)

The former are associated with eigenvectors tangential to the plane Di , the latter is associated with an eigenvector transversal to Di . It follows that P is a sink if s) > 0, while P is a saddle if v (¯ s) < 0. Therefore, by Assumption 7, the set v (¯ (−d) (d) Di = {Pi , . . . , Pi } of fixed points can be regarded as an alternating sequence of saddles and sinks. In the + cases, inequality (28) implies the structure Di = {sink, saddle, sink, . . . , sink, saddle, sink}; in particular, d + 1 fixed points are sinks and d points are saddles. In the − cases, inequality (28) implies the structure Di = {saddle, sink, saddle, . . . , saddle, sink, saddle}; in particular, d + 1 fixed points are saddles and d points are sinks. Remark. In the cases A± the fixed points Rj and Rk can be added to the al(−d) (d) ternating sequence {Pi , . . . , Pi } of saddles and sinks; this follows from the local analysis of the fixed points Rj /Rk . Accordingly, in the case A+ , the se(−d) (d) quence {Rk , Pi , . . . , Pi , Rj } is of the type {saddle, sink, saddle, . . . , saddle, sink, saddle}, while in the case A− , it is of the type {sink, saddle, sink, . . . , sink, saddle, sink}. The cases A 0+ and A 0− can be subsumed under the cases A+ and A− , respectively, the only difference being that Rj /Rk are non-hyperbolic. The next step in our study of the flow of the dynamical system on the plane Di is to investigate the global dynamics. To this end we consider the function N = (1 − Σ2 )−1 κ(s) , where κ(s) is positive and satisfies the differential equation dκ(s) 1 v+ + v − s(1 − s) = v(s) − κ(s) . ds 2 2

(36)

(37)

By (28), in the + cases the function κ(s) goes to ∞ as s → 0 and s → 1; in the − cases, κ(s) → 0 in the limit s → 0 and s → 1. A straightforward computation shows that N is strictly monotonically decreasing on Di except on the fixed point set Di , i.e., 2 2 1−w β β Σj − N Di = − + Σk − N, (38) 2 2 2

248

and


N

Σj =Σk =β/2


9 (2 + β)2 (2 − β)2 (1 − w) 16 2 2 v− + v+ v− + v+ + v(1 − s) − × v(s) − N. 2 2

=−

The monotone function N on Di excludes the existence of periodic orbits, homoclinic orbits and heteroclinic cycles in (the interior of) Di . Using the monotone function in combination with the local analysis of the fixed points, it is possible to solve the global dynamics of the flow on Di . Since the relevant arguments are rather standard (because the theory of planar flows is well-developed) we omit the derivation here and merely summarize the results in Figure 5. Note that in the cases B+ , C+ , the flow on ∂Di is given by a heteroclinic cycle; in the case B+ it is represented as follows: Tjj ←−−−− Tjk ⏐ ⏐ ⏐ ⏐ Tkj −−−−→ Tkk Having solved the dynamics on Di we are now in a position to complete the analysis of the future dynamics of orbits in Ci . Dynamics in the interior of Ci (cont.) While Lemma 3 describes the past asymptotics of orbits in Ci \Di completely, the description of the future asymptotics by Lemma 4 is incomplete in the cases where |β| < 2. Exploiting the results of the previous subsection we are able to complete the discussion. Lemma 5. Consider either of the + cases (with β < 2), i.e., either of the cases A+ , B+ , C+ (or A 0+ ). Let γ be an orbit such that γ ⊂ Di . Then ω(γ) is one of the fixed points of Di . More specifically, of the #Di = 2d + 1 fixed points, there are d + 1 sinks that attract a two-parameter set of orbits each, while each of the d saddles attracts a one-parameter family. Proof. The flow on Di is known in detail from the analysis of the preceding subsection. We see that the only cases that require a careful analysis are B+ and C+ , since in these cases, the boundary ∂Di of the plane Di is a heteroclinic cycle, which has to be excluded as a possible ω-limit of γ. To this end we use the function N given by (36). Suppose that γ converges to the heteroclinic cycle ∂Di . A straightforward computation shows that 2 2 1−w Σi Σi 3 −1 Σj + + Σk + − (1 − w)(Σ2i − β 2 ) (39) N N =− 2 2 2 4 along γ. The term in square brackets on the r.h.s. of (39) is a positive function, which does not converge to zero along γ; on the contrary, for sufficiently large times τ this function is approximately equal to (3/2)(4 − β 2 ) for most of the time

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Rj

Rk

(a) B+ , C+

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(b) A+ (and A 0+ )

Rk

(c) A− (and A 0− )

(d) B− , C−

Figure 5. The figures display the flow of the dynamical system on the plane Di . We assume |β| < 2 (cases A, B, C) so that Di is an invariant subset lying in Ci . The number of fixed points in the interior of Di is necessarily odd, i.e., #Di = 2d + 1, d ∈ N; we depict the case d = 1, i.e., #Di = 3. [The case #Di = 1, which is not depicted, is commented on in square brackets.] Nongeneric orbits are represented by dashed lines. (a) The boundary ∂Di forms a heteroclinic cycle in cases B+ and C+ . Generic orbits converge to this heteroclinic cycle as τ → −∞. The central fixed (0) (±1) are sinks. [In the point Pi is a saddle; the two other points Pi case #Di = 1, ∂Di is the α-limit set, and the (unique) fixed point (0) Pi is the ω-limit set for all (non-trivial) solutions.] (b) In case A+ , the properties of the three fixed points Di are analogous. The points Rj /Rk are saddles; one interior orbit converges to Rj , one to Rk as τ → −∞. (Whether these two orbits converge to the sinks as τ → ∞, as depicted, or the to central fixed point, depends on the function v(s).) The past attractor for generic orbits consists of the Kasner points in the upper left/lower right corner. [In the case (0) #Di = 1, the fixed point Pi is the ω-limit set for all (non-trivial) solutions.] The flow in the case A 0+ resembles the flow in the case A+ ; however, Rj /Rk are center saddles. (c) In case A− , the central (0) (±1) fixed point Pi is a sink, the two other interior fixed points Pi are saddles; the points Rj /Rk are sinks. The Kasner points in the upper left/lower right corner are the α-limit for generic orbits. [In (0) the case #Di = 1, the fixed point Pi is a saddle, which leaves Rj /Rk as the exclusive future attractor for generic orbits.] The flow in the case A 0− resembles the flow in the case A− ; however, Rj /Rk are not hyperbolic. (d) The cases B− and C− are similar to A− ; however, the Kasner points in the lower left/upper right (0) corner are sinks. [In the case #Di = 1, the fixed point Pi is a saddle, which leaves these Kasner points as the generic future attractor.]

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(which is because the orbit γ spends most of its time in a neighborhood of the fixed points on ∂Di ). In contrast, the second term on the r.h.s. of (39) converges to zero as τ → ∞. It follows that N → 0 along γ. However, this is a contradiction, because N is infinite on ∂Di ; therefore, the heteroclinic cycle ∂Di is excluded as a ω-limit of γ. Lemma 6. Consider either of the cases A− , B− (or A 0− ). Let γ be an orbit such that γ ⊂ Di . Then ω(γ) is one of the fixed points on Di , i.e., one of the points Rj , Rk , or one of the points of the set Di . More specifically, of the #Di = 2d + 1 fixed points, there are d sinks that attract a two-parameter set of orbits each, while each of the d + 1 saddles attracts a one-parameter family. The fixed points Rj , Rk (which coincide with Qjj , Qkk in B− ) attract a two-parameter family of orbits each. Lemma 7. Consider the case C− . Let γ be an orbit such that γ ⊂ Di . Then ω(γ) is one of the fixed points of the set Di or one of the transversally hyperbolic sinks on the Kasner circles KCj /KCk . More specifically, of the #Di = 2d + 1 fixed points, there are d sinks that attract a two-parameter set of orbits each, while each of the d + 1 saddles attracts a one-parameter family. Each sink on the Kasner circles attracts a one-parameter family of orbits. Proof. The statements of the two previous lemmas follow from Lemma 4, the analysis of the flow on Di , and the local analysis of the fixed points. Lemma 3 describes the past dynamics, Lemma 4 (in the cases D) and Lemmas 5–7 (in the cases A, B, C) describe the future dynamics of orbits in Ci . The qualitative behaviour of orbits is depicted in Figure 6. Remark. In Assumption 7 we allow for a special case, defined by the requirement v(s) ≡ w for all s. In this special case, Di is not a discrete set of fixed points but a line of fixed points, the line Σj = Σk . It follows from (35) that the fixed points of Di are transversally hyperbolic sinks. Using the monotone function (36) and the simple structure of the flow on Di it is not difficult to show that ω(γ) ∈ Di for all orbits γ ⊂ Ci \Di .

5. Local dynamics In this section we perform a stability analysis of the fixed points in the (closure of the) state space X . This local dynamical systems analysis is an essential ingredient in order to understand the global dynamics of solutions in X . Fixed points on the boundary ∂X . In the previous section we have analyzed in detail the dynamical system (11) on ∂X . The fixed points of this system reside on the cylindrical boundary C1 ∪ C2 ∪ C3 ⊂ ∂X . Since Ci is given as si = 0, the direction orthogonal to Ci is given by the variable si . From (11) we obtain

(40) = −2 Σi − sΣj − (1 − s)Σk , s−1 si i

si =0

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Tjk

251

Rj

Tjj Di

Di

Di

Tkk Rk

Tkj

(a) C+

(b) B+

(c) A+ (and A 0+ )

Qjj

Rj

Di

Rk

(d) A− (and A 0− )

Di

Di

Qkk

(e) B−

(f) C−

Figure 6. A schematic depiction of the flow on the cylinder Ci . In contrast to Figure 5, to simplify the illustration we assume in this figure that the function v(s) is such that there is only one fixed point Di in the plane Di . where we again use the convention (sj , sk ) = (s, 1 − s). Evaluated at the fixed points we get TLi : s−1 QLi : s−1 (41a) i si si =0 = −6 , i si si =0 = 6 , Di : s−1 (41b) i si si =0 = 3β , −1 −1 Rj /Rk : si si si =0 = 6β , KCj /KCk : si si si =0 = −2(Σi − Σj/k ) , (41c) where β is given by (26); β encodes the dependence on the rescaled matter anisotropies, which are represented by v± , cf. (22). The results of (41) can be combined with the results of Section 4 to complete the stability analysis of the fixed points on the boundary of the state space. The results are summarized in Table 1, Table 2, and Table 3. Example. Take, for instance, the fixed point Rk in the case A+ : From Figure 4(c) we see that Rk is a sink on the base of the cylinder Ci ; see also Figure 6(c). Figure 5(b) shows that there exists exactly one orbit in Di (and thus in Ci ) that emerges from Rk . Together with (41c) it thus follows that Rk possesses a twodimensional unstable manifold (which lies in the interior of the state space X ), so

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Table 1. In the + cases, the fixed points on ∂X are α-limits for orbits of the interior of the state space X ; in this table we list how many interior orbits converge to a particular fixed point as τ → −∞. The fixed points on the Kasner circles that satisfy the condition in the table are transversally hyperbolic sources and thus act as the α-limit for a two-parameter set of orbits each. (The points that do not fulfill the condition do not attract any orbits.) The fixed points on QLi (where the end points Qij and Qik are excluded) act as the α-limit for a one-parameter set of interior orbits; hence, in total, a two-parameter set of orbits has an α-limit on QLi . In case B+ , the fixed points Rj /Rk (which coincide with Tjj /Tkk ) lie on ∂Di , which is a heteroclinic cycle. The collection of these cycles, ∂D1 ∪ ∂D2 ∪ ∂D3 , forms a so-called heteroclinic network. point ∈ Ci ⊂ ∂X

A+

point ∈ QLi d + 1 points ∈ Di d points ∈ Di Rj /Rk point P ∈ KCj /KCk [condition on P] point ∈ TLi

B+

1-parameter family 1-parameter family one orbit one orbit 1-parameter family 1-parameter family 1-parameter family part of ∂D i 2-parameter family 2-parameter family [1 < Σj/k |P ] [1 < Σj/k |P < 2] does not attract any interior orbit

C+ 1-parameter family one orbit 1-parameter family – 2-parameter family [1 < Σj/k |P < 2/β] as τ → −∞

that there exists a one-parameter family of orbits (in X ) that converge to Rk as τ → −∞; cf. Table 1. Fixed points in the interior of X . The dynamical system (11) possesses a set of fixed points in the interior of X (= K × T), which is given as the set of solutions of the equations Σ1 = Σ2 = Σ3 = 0 ,

w1 = w2 = w3 = w ;

(42)

the first condition defines a point in K; in general, the second condition defines a subset of T. Since the principal pressures coincide, each of these fixed points represents the flat isotropic Friedmann–Robertson–Walker (FRW) perfect fluid solution associated with the equation of state p = wρ. Assumption 8. For simplicity, we assume that the matter model is such that w1 = w2 = w3 = w defines a point in T; hence (42) defines an isolated fixed point in X , which we denote by F. Remark. This is an assumption out of convenience. (It is automatically satisfied for the matter models we consider in detail in Section 7 – collisionless matter and elastic matter.) The analysis of this section is equally valid when (42) has more than one solution. In that case, F denotes any of the multiple fixed points in X .

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Table 2. In the − cases, some fixed points on ∂X are α-limits, some are ω-limits for orbits of the interior of the state space X ; in this table we list how many interior orbits converge to a particular fixed point as τ → −∞ (denoted by α) or τ → ∞ (denoted by ω). In case B− , the fixed points Rj /Rk coincide with Qjj /Qkk . The proof that these points attract a three-parameter set of orbits requires center manifold analysis. point ∈ Ci ⊂ ∂X point ∈ QLi point P ∈ KCj /KCk [condition on P] point ∈ TLi d + 1 points ∈ Di d points ∈ Di Rj /Rk point P ∈ KCj /KCk [condition on P]

A− α α

ω ω ω ω

B−

C−

1-parameter family 1-parameter family 2-parameter family 2-parameter family [1 < Σj/k |P ] [1 < Σj/k |P ] does not attract any interior orbit as τ 2-parameter family 2-parameter family 3-parameter family 3-parameter family 3-parameter family 3-parameter family does not act as ω-limit [–]

1-parameter family 2-parameter family [1 < Σj/k |P ] → −∞ or τ → ∞ 2-parameter family 3-parameter family – 2-parameter family [Σj/k |P < 2/β]

Table 3. If |β| ≥ 2, the properties of the fixed points change considerably. If β ≥ 2 (D+ ), neither of the fixed points can be the α/ω-limit set of an interior orbit. (However, a fixed point can be contained in the α-limit set of some orbit γ ⊆ X .) If β ≤ −2 (D− ), then each fixed point P on KCj /KCk with 1 < Σj/k |P is a transversally hyperbolic source, while Σj/k |P < −1 yields a transversally hyperbolic sink. point ∈ Ci ⊂ ∂X point ∈ QLi point P ∈ KCj /KCk [condition on P] point ∈ TLi point P ∈ KCj /KCk [condition on P]

α α ω ω

D+

D−

not attractive not attractive [–] not attractive not attractive [–]

1-parameter family 2-parameter family [1 < Σj/k |P ] 1-parameter family 2-parameter family [Σj/k |P < −1]

To study the stability properties of F, it is useful to define a different set of coordinates on T. Let (ijk) be a cyclic permutation of the triple (123) and set si =

eti , 1 + eti + etj

sj =

etj , 1 + eti + etj

sk =

1 1+

eti

+ etj

.

(43)

This defines a coordinate system R2 (ti , tj ) → T, such that the point (ti , tj ) = (0, 0) corresponds to the center (s1 , s2 , s3 ) = (1/3, 1/3, 1/3) of T; the limit si → 0 corresponds to ti → −∞, sj → 0 to tj → −∞; the limit si → 1 corresponds to ti → ∞, sj → 1 to tj → ∞; finally, sk → 0 corresponds to a combined limit

254



ti → ∞, tj → ∞. In these coordinates the fixed point F is given by F : (Σ1 , Σ2 , Σ3 )|F = (0, 0, 0) ,

(ti , tj )|F := (t¯i , t¯j ) .

(44)

Using (ti , tj ) on T, the rescaled pressures (8) can be expressed as ∂ ∂ log ψ , wj = w + 2 log ψ , ∂ti ∂tj ∂ ∂ wk = w − 2 + log ψ , ∂ti ∂tj wi = w + 2

(45)

where ψ is regarded as a function of the two variables (ti , tj ). Since w1 = w2 = w3 at F, it follows that (t¯i , t¯j ) ∈ T is a critical point of ψ; we set ψ¯ = ψ(t¯i , t¯j ). The dynamical system (11) can be represented in the four variables (Σi , Σj , ti , tj ) as 1 Σn = −3Ω (1 − w)Σn − (wn − w) , tn = −2(2Σn + Σm ) , 2 (n, m) ∈ (i, j), (j, i) , where Ω = 1 − 13 (Σ2i + Σ2j + Σi Σj ). The matrix representing the linearization of (11) at the point F contains the Hessian h of ψ evaluated at (t¯i , t¯j ). The four eigenvalues of this matrix are given by ! 3 λ±± = (1 − w) −1 ± 1 − Λ± , 4 where Λ± are given by " 2 ¯ Λ± = 64 ψ tr h + hij ± (tr h + hij ) − 3 det h / 3(1 − w)2 . Lemma 8. If (i) h is positive definite, i.e., det h > 0 and tr h > 0, then all eigenvalues λ±± have negative real part and F is a hyperbolic sink. If (ii) h is negative definite, i.e., det h > 0 and tr h < 0, then the eigenvalues λ+± are positive and λ−± negative so that F is a hyperbolic saddle with a two-dimensional stable and a two-dimensional unstable manifold. If (iii) det h < 0, then λ±+ and λ−− have negative real part, whereas λ+− has positive real part, so that F is a hyperbolic saddle with a three-dimensional stable and one-dimensional unstable manifold. In the exceptional case (iv) det h = 0, there exists at least one zero eigenvalue. Proof. From the inequality det h ≤

1 tr h2 − h2ij , 4

(46)

we obtain (tr h + hij )2 − 3 det h ≥ 0, hence Λ± are real numbers. Studying the sign of Λ± in the different cases, the claim of the lemma follows immediately.

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6. Global dynamics On the state space X we define the positive function M = (1 − Σ2 )−1 ψ(s1 , s2 , s3 ) ,

(47)

where ψ(s1 , s2 , s3 ), (s1 , s2 , s3 ) ∈ T, is defined by (7). By a straightforward computation, where we use that M can be written as M = 3H 2 /n1+w (which √ makes it = −3n, which follows from d det g/dt = possible to apply (10) and the equation n √ 3H det g), we obtain M = −3(1 − w)Σ2 M . (48a) The computation of higher derivatives reveals that (wk − w)2 (48b) M Σ2 =0 = −9(1 − w)M k

on the subset Σ2 = 0 of the state space. Therefore, since w < 1 by Assumption 3, we obtain M < 0 when Σ2 = 0 and M |Σ2 =0 < 0 except at the point F, cf. Assumption 8. Hence we have proved that M is a strictly monotonically decreasing function along the flow of the dynamical system (11) on X \F. Lemma 9. Let γ be an orbit in X \F. Then the α-limit set α(γ) and the ω-limit set ω(γ) satisfy • either α(γ) = F or α(γ) ⊆ ∂X , and • either ω(γ) = F or ω(γ) ⊆ C1 ∪ C2 ∪ C3 ⊂ ∂X , cf. (19). Proof. The lemma is a direct consequence of the monotonicity principle. Since M is strictly monotone along the flow of the dynamical system (11) on X \F, the α/ω-limit set of γ must be contained in F ∪ ∂X . Because these limit sets are necessarily connected, we find that either α(γ) = F or α(γ) ⊆ ∂X , and likewise for ω(γ). Moreover, M |∂K×T = ∞ (i.e., M → ∞ along every sequence converging to a point on ∂K × T), hence this subset of ∂X is excluded for ω(γ), which leaves ω(γ) ⊆ K × ∂T = C1 ∪ C2 ∪ C3 . It is the quantity β, see (26), that determines the details of the asymptotics. Therefore we discuss the cases β > 0 and β < 0 separately; the former includes the + cases A+ , B+ , C+ , and the extreme case D+ , the latter includes the − cases A− , B− , C− , and the extreme case D− . The + cases Theorem 1 (Future asymptotics). In the + cases A+ , B+ , C+ , and D+ , the fixed point F is the future attractor of the dynamical system (11); every orbit in the state space X converges to the fixed point F as τ → +∞. Interpretation of Theorem 1. Since the fixed point F corresponds to a FRW perfect fluid solution associated with the equation of state p = wρ, the theorem states that each Bianchi type I model with anisotropic matter that satisfies v− < w < v+ , cf. (28), isotropizes toward the future and behaves, to first order, like an (infinitely diluted) isotropic perfect fluid solution.

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Remark. Theorem 1 can be proved under assumptions that are considerably weaker than the ones we made. (Assumptions 5–7 are rather irrelevant for the future asymptotics – they are tailored to the past asymptotics.) This fact will become obvious from the proof and is thus not further commented on. Proof. To prove Theorem 1 we must show that ω(γ) ⊆ C1 ∪ C2 ∪ C3 is impossible, see Lemma 9. To this end we use that ψ(s1 , s2 , s3 ) → ∞ as (s1 , s2 , s3 ) → ∂T (which we prove below). It then follows that M = supX M = ∞ on the cylindrical boundary and consequently on the entire boundary ∂X of the state space, cf. (47). To finish the proof we apply the monotonicity principle [22] for the function M on X \{F }: The monotonicity principle precludes the possibility that ω(γ) might be contained on ∂X for any orbit γ ⊆ X and the theorem is established. To show that ψ → ∞ as ∂T is approached, we use that v− < w < v+ , which holds by assumption, cf. (28). Employing the coordinates (43) on T and the representation (45) of the rescaled matter quantities, we find that there exists some > 0 such that for all tj there exists T > 0 and ∂ ∂ wi − w wi − w ≤ − , ≥ log ψ = log ψ = ∂ti 2 ∂ti 2 for all ti < −|T | and ti > T respectively. Consequently, log ψ → ∞ and thus ψ → ∞ as ti → ±∞. (Likewise, ψ → ∞ as tj → ±∞.) We thus conclude that ψ = ∞ on ∂T and the theorem is proved. Theorem 2 (Past asymptotics). Let γ ⊂ X \F be an orbit of the dynamical system (11). • In the + cases A+ , B+ , C+ , the α-limit set of γ is A+ one of the fixed points on ∂X ; B+ one of the fixed points on ∂X or, possibly, the heteroclinic network ∂D1 ∪ ∂D2 ∪ ∂D3 ; C+ one of the fixed points on ∂X or one of the heteroclinic cycles ∂D1 , ∂D2 , ∂D3 . Convergence to a fixed point is the generic scenario, i.e., the set of orbits that converge to the heteroclinic structures is a set of measure zero in X . Table 1 gives a complete list of the fixed points on ∂X together with the number of orbits that converge to the particular points. • In the case D+ , the α-limit set of γ is represented either by a heteroclinic cycle or by a heteroclinic sequence on ∂X and thus contains a sequence of Kasner points; in particular, α(γ) cannot be a fixed point. Interpretation of Theorem 2 and conclusions. Theorem 2 (in conjunction with the results of Table 1) entails that in each of the + cases, generic solutions approach the fixed points on the Kasner circle(s), which implies that the past attractor of the dynamical system (11) is located on the Kasner circle(s). Therefore, each generic Bianchi type I model with anisotropic matter, where the matter is supposed to satisfy the requirements of the + cases, approaches a Kasner solution toward the singularity. However, there are interesting differences among the

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cases. In case A+ each Kasner solution is a possible past asymptotic state, i.e., given an arbitrary Kasner solution (18), represented by the Kasner exponents (p1 , p2 , p3 ) = (1/3)(Σ1 + 1, Σ2 + 1, Σ3 + 1), p1 + p2 + p3 = p21 + p22 + p23 = 1, there exist anisotropic Bianchi type I solutions that converge to this Kasner solution as t → 0. In particular, each of the three Taub solutions, which are characterized by (pi , pj , pk ) = (1, 0, 0), represents a possible past asymptotic state. In case B+ this is no longer correct. Since the Taub points are (center) saddles there does not exist any solution that converges to a Taub solution as t → 0. However, the Taub points are part of a larger structure, the heteroclinic network ∂D1 ∪ ∂D2 ∪ ∂D3 , which represents a potential α-limit set for (a probably two-parameter set) of solutions. Accordingly, we conjecture that there exists a two-parameter set of solutions whose asymptotic behaviour is characterized by oscillations between different representations of the Taub solution. In case C+ (which is characterized by 1 < β < 2) there is an entire one-parameter set of Kasner solutions (including the Taub states) that are excluded as α-limit sets. The condition of Table 1 yields that a Kasner solution is excluded as a past asymptotic state for Bianchi type I models with anisotropic matter iff maxl Σl ≥ 2/β, or, equivalently, maxl pl ≥ (2 + β)/(3β). In other words, generic Bianchi type I models behave asymptotically like Kasner solutions as the singularity is approached; however, the set of possible Kasner limits is restricted to those that are “sufficiently different” from the Taub solutions. If β ≥ 2, i.e., in case D+ , this set is the empty set. Each point on the Kasner circles takes the role of a saddle point. This induces generic oscillatory asymptotic behaviour of Bianchi type I models. The α-limit of a generic solution contains a sequence of Kasner states, hence the solution undergoes a sequence of phases (“Kasner epochs”), in each of which the behaviour of the solution is approximately described by a Kasner solution, and “transitions” between these epochs. Remark. Theorem 2 (in conjunction with the results of Table 1) also implies that there exist (non-generic) Bianchi type I models whose asymptotic behaviour is not connected to the dynamics of any Kasner solution. There exist solutions whose asymptotic behaviour is characterized by (Σi , Σj , Σk ) → β(−1, 1/2, 1/2) as the singularity is approached, and solutions with (Σi , Σj , Σk ) → β(2, −1, −1), where the latter concerns the case A+ since β < 1 is required. Furthermore, in the cases B+ and C+ , there exist solutions whose asymptotic behaviour is oscillatory: An orbit approaching the heteroclinic cycles/network (∂D1 , ∂D2 , ∂D3 ) corresponds to a solution that is represented by a periodic sequence of “Kasner epochs” associated with the Kasner fixed points that lie on the cycles; in the Kasner epochs the solution is characterized by (Σi , Σj , Σk ) = (−β, β/2 ± 3/4 4 − β 2 , β/2 ∓ 3/4 4 − β 2 ). Remark. The distinction into the different + scenarios is intimately connected with the energy conditions; see the list of cases in Section 4. If the energy conditions hold (case A+ , β < 1), each Kasner solution represents a possible past asymptotic state for anisotropic Bianchi type I models. If the energy conditions are satisfied only marginally (case B+ , β = 1), i.e., at the onset of energy condition violation,

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the Taub solutions are excluded as past asymptotic states. Finally, in the case β > 1 (case C+ and case D+ ) the energy conditions are violated; in this context, the quantity β −1 can be regarded as a measure for the magnitude of the violation. Increasing the value of β − 1 from zero to one leads to an ever increasing exclusion of Kasner states from the past attractor; if β − 1 reaches one (β = 2), the set of Kasner states that represent past asymptotic states has shrunk to the empty set. This is the onset of oscillatory behaviour of generic solutions. Proof. In the proof we choose to restrict ourselves to the cases A+ , B+ , C+ , since these are the difficult cases; the proof in the case D+ is similar (but simpler). The first step to prove Theorem 2 is to show that α(γ) must be contained in ∂X . Using Lemma 9 this amounts to proving that α(γ) = F is impossible. Therefore, assume that there exists γ ⊂ X \F such that α(γ) = F. By Theorem 1 we know that ω(γ) = F, hence γ must be a homoclinic orbit. However, this a contradiction to the fact that the function M , which is well-defined in X and in particular at F, cf. (47), is strictly monotonically decreasing along γ. Consequently, α(γ) = F is impossible, and we conclude α(γ) ⊆ ∂X . In the next step of the proof we show that in a subset of

α(γ) is contained the boundary ∂X : α(γ) ⊆ (∂C1 ∪ ∂C2 ∪ ∂C3 ) ∪ D1 ∪ D2 ∪ D3 . By definition, ∂X is represented by the disjoint union

∂X = ∂C1 ∪ ∂C2 ∪ ∂C3 ∪ D1 ∪ D2 ∪ D3 # $% & # $% & [∂X ]1

[∂X ]2

' ( ∪ C1 ∪ C2 ∪ C3 \ D1 ∪ D2 ∪ D3 ∪ ∂K × T , # $% & # $% & [∂X ]3

[∂X ]4

see Section 4. Suppose that there exists an orbit γ such that α(γ) P ∈ [∂X ]3 ; w.l.o.g. P ∈ Ci (P ∈ Di ) for some i. Since the α-limit set α(γ) is invariant under the flow of the dynamical system, the entire orbit through P, γP , and its ω-limit must be contained in α(γ) as well. By Lemma 5, ω(γP ) is a point PDi of the set Di . However, PDi is a saddle in X , see Table 1; therefore, since PDi ∈ α(γ), either γ is contained in the unstable manifold of PDi , or the intersection of γ with the unstable manifold is empty. The first alternative contradicts the assumption P ∈ α(γ). In the second case we appeal to the Hartman-Grobman theorem to infer that α(γ) contains not only PDi but also points of the unstable manifold of PDi . (Consider a sequence τn such that τn → −∞ and γ(τn ) → PDi as n → ∞. By the Hartman-Grobman theorem there exists a sequence τ¯n = τn + δτn , with δτn > 0, τ¯n < τn−1 , such that the sequence γ(¯ τn ) possesses an accumulation point on the unstable manifold of PDi .) The local dynamical systems analysis, cf. Table 1, thus implies that α(γ) contains points that do not lie on Ci but in the interior of X . This, however, is a contradiction to Lemma 9. The assumption has led to a contradiction; therefore, the intersection of α(γ) and [∂X ]3 must be empty. The procedure to show that α(γ) ∩ [∂X ]4 = ∅ is identical, where we use the flow on

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∂K×T, see Figure 2(b), and the stability analysis of the fixed points as

summarized in Table 1. We have therefore shown that α(γ) ⊆ ∂C1 ∪∂C2 ∪∂C3 ∪ D1 ∪D2 ∪D3 . In the last step of the proof we investigate which structures on the set ∂C1 ∪ ∂C2 ∪ ∂C3 are potential α-limit sets. Since this set is two-dimensional, possible α-limit sets are fixed points, periodic orbits, homoclinic orbits, or heteroclinic cycles/networks. The analysis of Section 4, see Figures 3, 4, and Figure 6, shows that, in case A+ , there do not exist any other potential α-limit sets on ∂C1 ∪ ∂C2 ∪ ∂C3 than fixed points. (This concludes the proof of the theorem for case A+ .) In case B+ , there exists an entangled network of heteroclinic cycles, a ‘heteroclinic network’; in case C+ , there are three independent heteroclinic cycles: ∂D1 , ∂D2 , ∂D3 To complete the proof of the theorem we have to investigate whether α(γ) can coincide with ∂Di for some i in case C+ (where 1 < β < 2). Consider the heteroclinic cycle ∂Di for some i. At the point P on ∂Di given by (si , sj , sk ) = (0, 0, 1), Σi + β = 0, Σj = Σk , we consider the three-dimensional hyperplane HP : Σj = Σk orthogonal to ∂Di ; the natural axes on HP are si , sj , and Σi + β. We study the Poincaré map induced on HP by the dynamical system. The subspace si = 0 is an invariant subspace of HP (since Ci is an invariant subspace of X ). Equation (31) and Figure 5(a) reveal that P is a saddle point for the dynamical system on HP , the unstable manifold being the sj -axis, the stable manifold being the (Σi + β)-axis. A straightforward calculation shows that √

s−1 3 4 − β2 λ , λ ∈ [−1, 1] i si = 3β − along ∂Di . Under the assumptions of case C+ the r. h. side is strictly positive for −1 all λ, hence, for some > 0, s−1 i si ≥ 2 along ∂Di and si si ≥ in a sufficiently small neighborhood Ui of ∂Di in X . Therefore, as long as a solution is contained in this neighborhood Ui of ∂Di , we find that si (τ ) is bounded by some constant times eτ (for decreasing τ ). We infer that a solution either remains within Ui for all sufficiently small τ (τ → −∞) or |Σi + β| becomes large (and the solution leaves Ui in this way). Letting dς = −(3/2)(1 − w)Ωdτ (so that decreasing τ -time corresponds to increasing ς-time) we find that 2 d (Σi + β) = (Σi + β) − (wi − v− ) , dς 1−w

(49)

where wi = wi (s1 , s2 , s3 ) satisfies wi → v− as si → 0. Let us consider a simplified problem. A differential equation of the type f (ς) = f (ς) + g(ς) with a function g(ς) that converges to zero as ς → ∞ possesses a unique solution such that f (ς) → 0 as ς → ∞; if, initially, f is larger than this special solution, then f (ς) → ∞ as ς → ∞; if, initially, f is smaller, then f (ς) → −∞ as ς → ∞. Likewise, we obtain from (49) that the quantity Σi + β is increasing if it is positive and sufficiently large initially, while it is decreasing if it is negative and sufficiently small initially. In the former cases, the solution eventually leaves Ui , since Σi +β has become too large; in the latter case, the solution leaves Ui , because

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Σi + β has become too small. Consider a one-parameter set of initial data, e.g., si = const, sj = const in HP . Let (Σi + β)0 be an initial value of Σi + β such that the associated solution satisfies (Σi + β)(ς) > δ for sufficiently large ς (for some appropriately chosen δ). By continuous dependence on initial data, there exists an open interval of initial data containing (Σi + β)0 such that the analog holds for each initial datum of this interval. Let (Σi + β)pos denote the infimum of (Σi + β) of the maximally extended open interval. (This infimum exists because there exists an analogous maximally extended open interval comprising the small initial values of Σi + β which lead to (Σi + β)(ς) < −δ.) By construction, the solution with initial datum (Σi + β)pos must remain in Ui for all times, which implies that the α-limit of this solution is ∂Di . Accordingly, ∂Di is a possible α-limit set of orbits in X . To see that convergence to ∂Di is non-generic (in the sense stated in the theorem) we can invoke the Hartman-Grobman theorem for discrete flows (for the flow on HP ); we obtain that ∂Di has the character of a saddle: There exists solutions that converge to ∂Di , but generically, solutions are driven away from this cycle. Alternatively, we note that in order to converge to ∂Di a solution must satisfy ) ∞ ! 2 (Σi + β)(0) = (Σi + β)0 = e−ς wi si (ς), sj (ς), sk (ς) − v− dς , 1−w 0 which is a direct consequence of (49). Since (Σi + β)0 = o(1) as si → 0 (and uniformly in sj ), there cannot exist an open neighborhood of P in HP such that solutions of the Poincaré map with initial data of that neighborhood possess P (i.e., ∂Di ) as an α-limit set. This completes the proof of the theorem. The − cases To simplify the analysis of the − cases, we require the matter model to satisfy a typical ‘genericity’ assumption which concerns the neighborhood of the isotropic state of the matter: We assume definiteness of the Hessian h of ψ at the Friedmann point (i.e., we exclude the cases (iii)-(iv) of Lemma 8). The matter models we discuss in Section 7 satisfy the required assumption automatically. Lemma 10. In the − cases A− , B− , C− , and D− the fixed point F is a hyperbolic saddle with a two-dimensional stable and a two-dimensional unstable manifold. Proof. In terms of the coordinates (43) on T, the fixed point F is given by (Σ1 , Σ2 , Σ3 ) = (0, 0, 0) and (ti , tj ) = (t¯i , t¯j ), where (t¯i , t¯j ) is the (single) critical point of the (positive) function ψ on T, see (44). Using that ψ = 0 on ∂T (which we prove below) it follows that (t¯i , t¯j ) is a global maximum of ψ. Therefore, at least for a matter model with a generic characteristic function ψ, the Hessian h of ψ at (t¯i , t¯j ) is negative definite. Part (ii) of Lemma 8 then implies the statement we have wished to prove. It remains to show that ψ → 0 as ∂T is approached. To this end we use the representation (45) of the rescaled matter quantities and the fact that v− > w > v+ , which holds by assumption, cf. (28). Then there exists

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some > 0 such that for all tj there exists T > 0 and ∂ wi − w ≥ , log ψ = ∂ti 2

∂ wi − w ≤ − log ψ = ∂ti 2

for all ti < −|T | and ti > T respectively. Consequently, log ψ → −∞ and thus ψ → 0 as ti → ±∞. Hence ψ = 0 on ∂T. Theorem 3 (Future and past asymptotics). Let γ ⊂ X \F be an orbit of the dynamical system (11). In the − cases A− , B− , C− , and D− , one of the following possibilities occurs: • α(γ) is a fixed point on ∂X and ω(γ) is a fixed point on ∂X ; this is the generic case; • α(γ) is a fixed point on ∂X and ω(γ) = F; • α(γ) = F and ω(γ) is a fixed point on ∂X . Table 2 gives a complete list of the fixed points on ∂X together with the number of orbits that converge to the particular points. Proof. The theorem is essentially proved by repeating the first two steps of the proof of Theorem 2. The main difference is the character of the fixed point F, which is described in Lemma 10. Note that on ∂C1 ∪ ∂C2 ∪ ∂C3 (which is the relevant part of ∂X where α(γ)/ω(γ) must reside) there do not exist any structures (like periodic orbits or heteroclinic cycles) that qualify as possible α-/ω-limits except the fixed points of Table 2. Interpretation of Theorem 3 and conclusions. Theorem 3 (in conjunction with the results of Table 2) entails that in each of the − cases, generic solutions converge to fixed points on the Kasner circle(s) as τ → −∞, which implies that the past attractor of the dynamical system (11) is located on the Kasner circle(s). Therefore, each generic Bianchi type I model with anisotropic matter, where the matter is supposed to satisfy the requirements of the − cases, approaches a Kasner solution toward the singularity. In contrast, toward the future, the asymptotic behaviour of generic solutions is quite diverse. In particular, the asymptotic properties of the matter model, in the shape of the function v(s), play an important role, because these determine the number of fixed points in Di . Assuming the simplest case, #Di = 1, which is in accord with the examples of anisotropic matter models discussed in Section 7, we obtain the following results: In the cases A− and B− the behaviour of generic solutions towards the future is (Σi , Σj , Σk ) → β(2, −1, −1). In case C− , generic solutions converge to a Kasner solution towards the future. However, there is only a subset of Kasner solutions that qualify as possible future asymptotic states; it is only those Kasner solutions with maxl Σl > 2/β (which corresponds to maxl pl > (2 + β)/(3β) when written in terms of the Kasner exponents) that come into question. This set of Kasner solutions that qualify as future asymptotic states becomes larger with decreasing β. Finally, in the case D− (β ≤ −2), generic solutions converge to Kasner solutions toward the future, and conversely, each Kasner solution occurs as a future asymptotic state.

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Remark. Theorem 2 (in conjunction with the results of Table 2) also implies that there exist (non-generic) Bianchi type I models whose asymptotic behaviour is quite different. In particular, in cases A− , B− , C− , the orbits converging to Di as τ → ∞ give rise to solutions whose asymptotic behaviour toward the future is characterized by (Σi , Σj , Σk ) → β(−1, 1/2, 1/2). If #Di = 1, this behaviour is nongeneric; however, for matter models whose properties are such that #Di > 1, this asymptotic behaviour is shared by a generic set of solutions. The most interesting non-generic behaviour concerns isotropization: There exist non-generic solutions (a one-parameter set) that isotropize toward the past, and non-generic solutions (a one-parameter set) that isotropize toward the future. Remark. In Section 4 we have seen that the − cases A− , B− , C− (and the special case β = −2 of case D− ) are compatible with the energy conditions. However, despite this fact the assumptions of the − cases are representative of a matter model that is rather unconventional. This is due to the fact that the isotropic state of the matter is not energetically favorable and thus unstable (cf. the proof of Lemma 10, where we showed that the Hessian h of ψ is negative definite at the coordinates of F). The case A 0 The case A 0 represents a bifurcation between the + cases and the − cases. The analysis of this case is difficult for several reasons, the most obvious being the fact that center manifold analysis becomes ubiquitous, since several fixed points possess large center subspaces. Another problem is the fact that the asymptotic behaviour of the function ψ(s1 , s2 , s3 ) as (s1 , s2 , s3 ) → ∂T is undetermined by the assumptions of case A 0 ; there are several alternatives: ψ → 0, ψ → ∞, ψ might converge to some positive number, or ψ might not be convergent at all as (s1 , s2 , s3 ) → ∂T. Our expectation is that – under consistent assumptions on the behaviour of the function ψ on T – the subcase A 0+ resembles the case A+ , and the subcase A 0− resembles the case A− . However, a careful analysis is required to establish these vague expectations as facts; this goes beyond the scope of the present paper.

7. Anisotropic matter models In this section we present in detail three important examples of matter models to which one can apply the main results of this paper: Collisionless matter, described by the Vlasov equation, elastic matter, and magnetic fields. For a general introduction to collisionless matter and the Vlasov–Einstein system we refer to [1, 17]; the Bianchi type I case is discussed in detail in [6]. For a thorough discussion on the general relativistic theory of elasticity we refer to [2,4,9,10,20]; we will confine ourselves to deriving the energy-momentum tensor of elastic bodies.

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T3

=

R3

2/ β D2

∂D1

2

Σ1 = 2/β

∂D

3

D1

∂D1

Σ D2

T1

∂D

D2

2

D1

R1

∂D

∂D

Σ2

=

D3

β 2/

D1 D3

3

3

D3

263

R2 T2

(a) C+ (1 < β < 2)

(b) B+ (β = 1)

(c) A+ (0 < β < 1)

Q2

R2

R1

D1

D3

D3

D3 D1

D2

Q1

D2

D1

D2

R3 Q3

(d) A− (−1 < β < 0)

(e) B− (β = −1)

(f) C− (−2 < β < −1)

Figure 7. A schematic depiction of the projection onto the Kasner disc of interior orbits. By abuse of notation, we use the same letter to denote a fixed point and its projection onto the Kasner disc. The center of the disc is the projection of the fixed point F (which represents the isotropic solution). Generic orbits are in bold. In the cases B+ /C+ , the dashed lines represent the heteroclinic cycles/network. In the case B+ , the heteroclinic network could also attract orbits as τ → −∞ (not depicted). In the case A− , there are Kasner states which act both as α- and ω-limit of interior orbits.

7.1. Collisionless matter Consider an ensemble of particles with mass m that move along the geodesics of a spacetime (M, g¯). (The geodesic motion of the particles reflects the condition of absence of any interactions other than gravity; in particular, collisions are excluded.) The ensemble of particles is represented by a distribution function (‘phase space density’) f ≥ 0, which is defined on the mass shell, i.e., on the subset of the tangent bundle given by g¯(v, v) = −m2 , where v denotes the (future directed) four momentum. If (t, xi ) is a system of coordinates on M such that ∂t is timelike and ∂xi is spacelike, then the spatial coordinates v i of the four momentum are coordinates on the mass shell, and we can regard f as a function f = f (t, xi , v j ),

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i, j = 1, 2, 3. The distribution function f satisfies the Vlasov equation vj 1 ∂xj f − 0 Γjμν v μ v ν ∂vj f = 0 , (50) 0 v v where v 0 > 0 is determined in terms of the metric g¯μν and the spatial coordinates v i of the momentum via the mass shell relation g¯μν v μ v ν = −m2 . The energymomentum tensor is defined as ) T μν = f v μ v ν | det g¯| |v0 |−1 dv 1 dv 2 dv 3 . (51) ∂t f +

In the definition of a Bianchi type I solution of the Einstein–Vlasov system, where we use the notation and conventions of Section 2, f is assumed to be independent of xi and so (50) takes the form ∂t f + 2k jl v l ∂vj f = 0 .

(52)

It is well known, see, e.g., [12–14, 16], that the general solution of (52) is given by f = f (t, v j ) = f0 (vj ) ,

(53)

where f0 is an arbitrary function which corresponds to the initial data; usually, f0 is assumed to be compactly supported, so that the integral (51) is well defined. Using the spatial metric gij , cf. (1), we may replace det g¯ by det g in (51) and we can write |v0 | = m2 + gij v i v j . Accordingly, equation (51) means that the energy-momentum tensor is given as a function of the spatial metric gij (which depends on the initial data f0 ); we have thus verified Assumption 1 for collisionless matter (in Bianchi type I). If f0 satisfies the symmetry requirement f0 (v1 , v2 , v3 ) = f0 (−v1 , −v2 , v3 ) = f0 (−v1 , v2 , −v3 ) = f0 (v1 , −v2 , −v3 ) , and the metric is diagonal, then the energy-momentum tensor T μν is diagonal as well, which proves that Assumption 2 is satisfied. Therefore, the Bianchi type I Einstein–Vlasov system admits the class of diagonal models as solutions, cf. the remarks at the end of Section 2. The rescaled principal pressures are given by 2 !−1/2 * si f0 vi2 mx + k sk vk2 dv1 dv2 dv3 wi = ! 1/2 * 2 f0 mx + k sk vk2 dv1 dv2 dv3 ! −1/2 * 2 si f0 vi2 dv1 dv2 dv3 k sk vk = , (54) ! 1/2 * 2 f0 s v dv dv dv k 1 2 3 k k when we assume that the particles have zero mass (see, however, the remark at the end of this section). As a consequence, Assumption 3 holds with 1 (55) w= , 3

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and (7) takes the form 4/3

ρ = ρ(n, s1 , s2 , s3 ) = n

−1/6

(s1 s2 s3 )

) f0

1/2 sk vk2

dv1 dv2 dv3 .

(56)

k

Taking the limit si → 0 in (54) we find that * sj f0 vj2 [sj vj2 + sk vk2 ]−1/2 dv1 dv2 dv3 * wi = v− = 0 , wj = , f0 [sj vj2 + sk vk2 ]1/2 dv1 dv2 dv3

wk = 1 − wj (57)

on Ci , i.e., Assumptions 4–6 are satisfied (with v− = 0). In order to make contact with (20a ) we identify I with f0 and define !−1/2 * 2 z1 f0 (v1 , v2 , v3 ) v12 z v dv1 dv2 dv3 k k k (58) u[I](z1 , z2 , z3 ) := ! 1/2 * 2 f0 (v1 , v2 , v3 ) z v dv dv dv k 1 2 3 k k for all f0 and (z1 , z2 , z3 ) ∈ T. Using that I(σ) = f0 ◦ σ it is straightforward to show that (20a ) holds, i.e., wi = u[I(σ−1 ) ] ◦ σi leads to (54). The function ui (s) of i Definition 1 can be read off easily; equation (57) thus corresponds to (22 ) with v− = 0, v+ = 1; cf. also (21). We conclude that β = 1, i.e., an ensemble of massless collisionless particles constitutes an anisotropic matter model that is of class B+ . Finally we note that Assumption 7 is satisfied as well. One can show that ui (s) (which replaces v(s) in (34)) is strictly monotonically increasing, hence there exists only one solution of (33) and thus only one fixed point on Di , i.e., #Di = 1. Analogously, one can convince oneself that Assumption 8 holds. Remark. If we consider ensembles of collisionless particles with positive mass, m > 0, wi and w are functions of (s1 , s2 , s3 ) and an additional scale, which can be taken to be x or n = (det g)−1/2 (where we recall that x = n2/3 (s1 s2 s3 )−1/3 ). (In [6], x is replaced by z = m2 /(m2 + x).) In Section 8 we will show that the analysis of this paper carries over straightforwardly to this more general situation (the main reason being that the length scale is a monotone function). 7.2. Magnetic fields For an electromagnetic field represented by the antisymmetric electromagnetic field tensor Fμν the energy-momentum tensor is given by 1 1 μ β α μ μ α Tν =− F αF ν − δ ν F αF β . 4π 4 The equations for the field are the Maxwell equations ∇μ F μν = 0 ,

∇[σ Fμν] = 0 .

Consider specifically a purely magnetic field in a Bianchi type I spacetime with metric (1) that is aligned along, say, the third axis. In this case the electromagnetic

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field tensor takes the form

⎛

Fμν

0 ⎜0 =⎜ ⎝0 0

0 0 −K 0

0 K 0 0


⎞ 0 0⎟ ⎟, 0⎠ 0

where K determines the magnetic field: B 1 = 0, B 2 = 0, B 3 = K(g 11 g 22 g 33 )1/2 . For a diagonal metric, the Maxwell equations imply that K is a constant; hence the energy density ρ, 1 11 22 2 g g K , ρ= 8π is a function of the metric (which depends on the initial data for the magnetic field). Furthermore, T μν is diagonal and T 11 = T 22 = ρ , T 33 = −ρ .

(59)

Accordingly, Assumptions 1 and 2 are satisfied and diagonal models exist. It follows from (59) that w1 = 1, w2 = 1, w3 = −1, and w = 1/3. For these normalized anisotropic pressures all the remaining assumptions are satisfied straightforwardly and we have v− = 1, v+ = 1 and thus β = −2 . Therefore, in our classification, the asymptotic behaviour is of type D− . The conclusions are identical if we add an electric field parallel to the magnetic field. The Maxwell equations show that E 1 = E 2 = 0 and E 3 =L(g 11 g 22 g 33 )1/2 , where L = const. The energy density becomes 8πρ = g 11 g 22 (K 2 + L2 ), the energymomentum tensor remains a functional of the metric, and (59) and its consequences remain valid. Remark. The magnetic field considered in [11] is not aligned with one axis and thus not included in our analysis. In fact, when not aligned, the magnetic field has to rotate and its dynamics becomes non-trivial and has to be added to the system of equations; in addition, the energy-momentum tensor is no longer diagonal. 7.3. Elastic matter In elasticity theory, an elastic material in a completely relaxed state is represented by a three-dimensional Riemannian manifold (N, γ), the material space, where the points of N identify the particles of the material (in the continuum limit) and γ measures the distance between the particles in the completely relaxed state; let us denote by X A , A = 1, 2, 3, a system of local coordinates on N . The coordinates on the spacetime (M, g¯), on the other hand, are denoted by xμ , μ = 0, . . . , 3. The state of the elastic material is described by the configuration function ψ, which is a (smooth) map ψ : M → N , xμ → X A = ψ A (xμ ) , such that the kernel of the deformation gradient T ψ : T M → T N is generated by a (future-directed unit) timelike vector field u, i.e., ker T ψ = u or uμ ∂μ ψ A = 0.

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The vector field u is the matter four-velocity; by construction, ψ −1 (p) (i.e., the world-line of the particle p ∈ N ) is an integral curve. The pull-back of the material metric by the map ψ is the relativistic strain tensor hμν = ∂μ ψ A ∂ν ψ B γAB ; since hμν uμ = 0, h represents a tensor in the orthogonal complement u ⊥ of u in T M . hμν is a Riemannian metric on u ⊥ , hence hμν has three positive eigenvalues h1 , h2 , h3 . Note also that Lu hμν = 0, i.e., h is constant along the matter flow. The material is unstrained at the point x iff hμν = gμν holds x, where gμν = g¯μν + uμ uν is the Riemannian metric induced by the spacetime metric g¯ on u ⊥ . The scalar quantity (60) n = detg h = h1 h2 h3 is the particle density of the material. This interpretation is justified by virtue of the continuity equation ∇μ (nuμ ) = 0 . Definition 3. An elastic material in a Bianchi type I spacetime is said to be Bianchi type I symmetric iff Lξi hμν = 0, where ξi , i = 1, 2, 3, are the Killing vectors of the spacetime. Furthermore, if the matter four velocity u is orthogonal to the surfaces of homogeneity, the elastic material is said to be non-tilted. According to Definition 3 (and Lu hμν = 0), the strain tensor of a Bianchi type I symmetric and non-tilted elastic material satisfies h00 = h0k = 0 ,

hij = const .

(61)

in the coordinates of (1). Consequently, hi j = g ik hjk and thus h1 , h2 , h3 depend only on t. A specific choice of elastic material is made by postulating a constitutive equation, i.e., the functional dependence of the (rest frame) energy density ρ of the material on the configuration map, the deformation gradient and the spacetime metric. An important class of materials is the one for which this functional dependence enters only through the principal invariants of the strain tensor. In this case we have (62) ρ = ρ(q1 , q2 , q3 ) , where q1 = tr h ,

q2 = tr h2 ,

q3 = tr h3 ;

since n2 = (q31 − 3q1 q2 + 2q3 )/6, one of the principal invariants can be replaced by the particle density n. The materials described by (62) generalize the class of isotropic, homogeneous, hyperelastic materials from the classical theory of elasticity. In the following we shall refer to these material simply as elastic materials. The stress-energy tensor associated with these materials is obtained as the variation

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* with respect to the spacetime metric of the matter action SM = − |g| ρ. The general expression which results for the stress-energy tensor is ∂ρ (63) Tμν = 2 μν − ρ g¯μν , ∂¯ g which results in (3) for the metric (1). In Bianchi type I symmetry, and assuming that the elastic material is nontilted, it follows from (61) that the principal invariants of the strain are functions of the spatial metric g only, hence non-tilted elastic materials satisfy Assumption 1. In the following we introduce a natural class of constitutive equations that are compatible with Assumptions 3–8 as well. Remark. There exist elastic materials which do not satisfy these assumptions. For instance, for the constitutive equation considered in [3], which was taken from [9], the rescaled matter quantities (written in terms of the variables s1 , s2 , s3 ) do not have a continuous extension to the boundary ∂T of the space T. Therefore, strictly speaking, the analysis of this paper does not apply to this class of materials. However, the reason for this problem is that the variables (s1 , s2 , s3 ) are ill-adapted to this particular constitutive equation; replacing the ‘triangle variables’ (s1 , s2 , s3 ) by ‘hexagon variables’, cf. [3], remedies this deficiency and Assumptions 3–8 hold w.r.t. this alternative formulation of the problem. The dynamics of the elastic matter models investigated in [3] seems considerably more complicated than the dynamics of the matter models considered in this paper; however, modulo the necessary reformulation of the problem, it appears that the models of [3] are of type β = 2. We consider elastic materials whose constitutive equation depends only on the number density n and the dimensionless shear scalar s defined by A(h1 , h2 , h3 ) q1 −1 , (64) s = 2/3 − 3 = 3 G(h1 , h2 , h3 ) n where A and G denote the arithmetic and geometric mean functions, respectively. The fundamental inequality A(a1 , a2 , a3 ) ≥ G(a1 , a2 , a3 ), which holds for all real non-negative numbers a1 , a2 , a3 , with equality iff a1 = a2 = a3 , see [5], implies that s ≥ 0 and s = 0 iff h1 = h2 = h3 . Thus the shear scalar s measures deviations from isotropy. Using the requirement ρ = ρ(n, s) in (63) leads to ∂ρ 2 ∂ρ 1 − ρ gμν + 2/3 Tμν = ρ uμ uν + n hμν − q1 gμν . ∂n ∂s 3 n We restrict ourselves to constitutive equations of the quasi Hookean form ρ = ρˇ(n) + μ ˇ(n) f (s) ,

(65)

where ρˇ(n) is the unsheared energy density and μ ˇ(n) the modulus of rigidity. For such a constitutive equation we obtain 2ˇ μ(n) 1 (66) Tμν = ρ uμ uν + p gμν + 2/3 f (s) hμν − q1 gμν . 3 n

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Here, p denotes the isotropic pressure p, which is given by ρˇ μ ˇ 2 d 2 d p = pˇ(n) + νˇ(n) f (s) , pˇ = n , νˇ = n . dn n dn n

269

(67)

The quantity pˇ(n) is the unsheared pressure. Note that if the constitutive equation (65) depends only on n, i.e., if f (s) ≡ 0, then the elastic material reduces to a perfect fluid with energy density ρˇ and pressure pˇ. To specify the functions ρˇ and μ ˇ in the constitutive equation (65) we postulate a linear equation of state between the unsheared pressure and the unsheared energy density, i.e., pˇ = aˇ ρ , and a linear equation of state between the modulus of rigidity and the unsheared pressure, i.e., μ ˇ = b pˇ . By (67) this is equivalent to setting ρˇ = ρ0 na+1 ,

μ ˇ = ρ0 ab na+1

for some constant ρ0 > 0. Accordingly, the density ρ and the pressure p take the form

(68) ρ = ρ0 na+1 1 + ab f (s) , and p = aρ and the stress-energy tensor (66) becomes hμν ab f (s) 1 Tμν = ρ uμ uν + agμν + 2(3 + s) − gμν . (69) 1 + ab f (s) q1 3 We make a number of assumptions on the parameters a, b and the function f : (A1) a ∈ [−1, 1), a b ≥ 0. (A2) f (s) ≥ 0 and f (s) = 0 if and only if s = 0. (A3) f (s) > 0. Conditions (A1) and (A2) imply that the energy density is non-negative and has a global minimum at zero shear. Condition (A3) then corresponds to the (physically reasonable) assumption that the rest energy of the body is an increasing function of the shear. Remark. If b = 0, the modulus of rigidity μ ˇ vanishes and the elastic matter reduces to a perfect fluid with linear equation of state p = aρ. If a = 0 (so that p = 0), the choice of b is irrelevant, since ab = 0; this is clear because shear cannot occur for dust. Denoting by wi the rescaled principal pressures, equation (69) leads to hi ab f (s) pi 1 = a + 2 (3 + s) wi = − . (70) ρ 1 + ab f (s) q1 3 # $% & =: Q(s) (A4) We assume that supz>0 Q(z) < ∞ and that the limit Q∞ := limz→∞ Q(z) exists. For elastic matter that is Bianchi type I symmetric and non-tilted the relativistic strain is given by (61). We assume that hij (which plays the role of the initial data for the matter) is diagonal; by scaling the spatial coordinates we can

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achieve hij = δij . Substituting into (69) it follows immediately that T ij (t0 ) is diagonal (since gij (t0 ) is diagonal), hence T ij remains diagonal for all times by the evolution equations (2a). Therefore, the class of diagonal Bianchi type I models with elastic matter is well-defined, cf. the remarks at the end of Section 2. Using hij = δij we further obtain h1 = g 11 ,

h2 = g 22 ,

h3 = g 33 ,

q1 = x ,

n = (det g)−1/2 ,

(71a)

and therefore

x − 3 = (s1 s2 s3 )−1/3 − 3 . (71b) n2/3 Inserting (71) into (70) we find 1 Q(s) (72) wi (s1 , s2 , s3 ) = a + 2 si − 3 and w = a. This shows that the elastic materials under consideration satisfy Assumption 3. Furthermore, by (A4), the functions wi can be extended to ∂T; on Ci we obtain 2 1 1 wi = a − Q∞ , wj = a + 2 sj − Q∞ , wk = a + 2 sk − Q∞ ; 3 3 3 s=

hence Assumptions 4–6 hold as well, where v− = a−2Q∞ /3. Evidently, the rescaled pressures are of the form (20a) with 1 v(z1 , z2 , z3 ) = a + 2 z1 − Q∞ ; 3 the function v(s) is given by v(s) = a + 2(s − 1/3)Q∞ , see (20b); cf. also (22). We find that 2 4 4Q∞ . (73) v− = a − Q∞ , v+ = a + Q∞ , and β = 3 3 3(1 − a) (A1)–(A4) imply that β ≥ 0, i.e., elastic models are anisotropic models of the + type (A+ , B+ , C+ , or D+ ) or of the A 0 type. However, the latter case occurs only if Q∞ = 0, which implies that v(s) ≡ w (= a), which is the special subcase of A 0 introduced in Assumption 7. It is easy to see from the linearity of the function v(s) that Assumption 7 is satisfied for all values of Q∞ . In particular, there exists only one solution of (33) and thus only one fixed point on Di , i.e., #Di = 1. Remark. Assumption (A3) excludes the possibility Q∞ < 0, hence neither of the − cases can occur for the elastic materials. A formal way to obtain an elastic material that is of type − would be to allow f < 0; this is unphysical, however, since it implies that the rest energy of the elastic body has a maximum at zero shear (instead of a minimum). It remains to check the validity of Assumption 8. By (72), the equations wi = a ∀i are satisfied only if s1 = s2 = s3 = 1/3 (i.e., at the center of T) or at points (s1 , s2 , s3 ) where Q(s) = 0. By (A3), the latter equation has no solutions for finite values of s (i.e., in T), hence Assumption 8 holds.

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Remark. It is immediate from (72) that the dominant energy condition is satisfied if (A5) supz>0 Q(z) ≤ A(a) := min{ 34 (1 − a), 32 (1 + a)}. Clearly, 0 ≤ A(a) ≤ 1; A(a) = 0 if and only if a = ±1, while A(a) = 1 if and only if a = −1/3. If in addition a ≥ −1/3 holds, then the strong energy condition is also satisfied; note in particular that the validity of the strong energy condition does not impose any condition on the constant b. Example (John materials). f (s) = s. This class of elastic materials was introduced in [20]. In this case Q∞ = 1, hence β = 4/[3(1 − a)] by (73). Consequently, this class of materials is of type A+ if a ∈ [−1, −1/3), of type B+ if a = −1/3, of type C+ if a ∈ (−1/3, 1/3), and of type D+ if a ≥ 1/3. Condition (A5) is satisfied if and only if 1 (74) a = − , −1 ≤ b < 0 , 3 i.e., in case B+ . Example (Logarithmic materials). f (s) = log(ε + s) − log ε with ε > 0; for simplicity, ε = 3. Since Q∞ = 0, these materials are of type A 0 where v(s) ≡ w (= a). Condition (A5) reads ab ≤ A(a) , (75) hence these materials satisfy the dominant energy condition if b is sufficiently small (a = ±1). Example (Power-Law materials). f (s) = (ε + s)λ − ελ with ε > 0 and λ > 0; for simplicity, ε = 3. In this case Q∞ = λ, hence β = 4λ/[3(1 − a)] by (73). Consequently, this class of materials is of one of the types A+ , B+ , C+ , or D+ , depending on the values of a and λ. Since supz>0 Q(z) = λ min{1, ab 3λ }, condition (A5) is satisfied if

3−λ A(a) or if (ab ≤ 3−λ ) ∧ λ ≤ A(a) . (76) λ This condition reduces to (74) if λ = 1; while (76) is never satisfied if λ > 1, it holds for a certain range of a and √ b, if λ < 1. For instance, with a = 1/3 we have A(a) = 1/2 and (76) reads b ≤ 3, λ ≤ 1/2. 3−λ ≤ ab ≤

8. Concluding remarks In this paper we have discussed the dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The focus of our analysis has been the asymptotic behaviour of solutions in the limit of late times and toward the initial singularity. The matter model was not specified explicitly, but only through a set of mild assumptions that are motivated by basic physical considerations. In this way, our analysis applies to a wide variety of anisotropic matter models including matter models as different from each other as collisionless matter, elastic materials and magnetic fields.

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A basic assumption on the matter model was to require that the isotropic pressure p and the density ρ obey a linear equation of state p = wρ, where w = const ∈ (−1, 1), see Assumption 3. It is important to note, however, that this assumption is not necessary. On the contrary, based on the results derived in this paper it is relatively simple to treat nonlinear equations of state. Let us elaborate. Assume that ρ is given by a more general function instead of (7), i.e., ρ = ρ(n, s1 , s2 , s3 ). In this case we obtain from (5) that w = w(n, s1 , s2 , s3 ) and wi = wi (n, s1 , s2 , s3 ), i = 1, 2, 3; an interesting subcase is ρ(n, s1 , s2 , s3 ) = ϕ(n)ψ(s1 , s2 , s3 ); here, w = w(n) = (∂ log ϕ/∂n) − 1 and wi = wi (n, s1 , s2 , s3 ) is given by (8), where w is replaced by w(n). The Einstein evolution equations, written in terms of the dynamical systems variables of Section 3, decouple into an equation for H and a reduced system of equations for the remaining variables, where we can replace x by n, since x = n2/3 (s1 s2 s3 )−1/3 . Accordingly, the dynamical system that encodes the dynamics is given by (11), which is supplemented by the equation n = −3n. When we choose to compactify the variable n, i.e., when we replace n by N = n/(1 + n), the state space of this dynamical system is K × T × (0, 1). If we assume an equation of state such that w(N, s1 , s2 , s3 ) and wi (N, s1 , s2 , s3 ) possess well-defined limits as N → 0 and N → 1, we can extend the dynamical system to the boundaries X0 = K × T × {0} and X1 = K × T × {1} of the state space. The dynamical system on each of these boundary subsets coincides with the system (11) that we have discussed so extensively in this paper. Since the variable N is strictly monotone, the asymptotic dynamics of solutions of the dynamical system is associated with the limits N → 0 and N → 1. Accordingly, asymptotically, the flow of the boundary subsets X0 and X1 (and thus the results of the present paper) constitute the key to an understanding of the dynamics of the more general problem with nonlinear equations of state. It is conceivable that several of our assumptions can be relaxed (however, physics might disapprove), which could lead to interesting extensions of the results of this paper. For instance, if Assumption 8 is removed, then there exist several isotropic states of the matter and thus there might exist Bianchi type I solutions that isotropize both toward the past and toward the future. Assumption 1, on the other hand, could be replaced by the condition that the energy-momentum tensor depends not only on the spatial metric but also on the second fundamental form. In this way it is possible to extend the analysis to even larger classes of matter fields. This might lead to valuable generalizations of the results presented in this paper.

Acknowledgements We gratefully acknowledge the hospitality of the Institut Mittag–Leffler. Also, we would like to thank an anonymous referee for useful suggestions. S. C. is supported by Ministerio Ciencia e Innovaci´ on, Spain (Project MTM2008-05271).

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References [1] H. Andréasson, The Einstein–Vlasov system/Kinetic theory, Living Rev. Relativity 8 (2005), lrr-2005-2 . Online journal article, http://www.livingreviews.org/ llr-2005-2. [2] R. Beig, B. G. Schmidt, Relativistic elasticity, Class. Quantum Grav. 20 (2003), 889– 904. [3] S. Calogero, J. M. Heinzle, Dynamics of Bianchi I elastic spacetimes, Class. Quantum Grav. 24 (2007), 5173–5201. [4] B. Carter, H. Quintana, Foundations of general relativistic high-pressure elasticity theory, Proc. R. Soc. Lond. A. 331 (1972), 57–83. [5] G. Hardy, J. E. Littlewood, G. P´ olya, Inequalities, Cambridge University Press, Cambridge (1934). [6] J. M. Heinzle, C. Uggla, Dynamics of the spatially homogeneous Bianchi type I Einstein–Vlasov equations, Class. Quantum Grav. 23 No. 10 (2006), 3463–3489. [7] J. M. Heinzle, C. Uggla, Mixmaster: Fact and belief, Preprint (2009). [8] J. M. Heinzle, C. Uggla, N. R¨ ohr, The cosmological billiard attractor, Electronic archive: arXiv.org/gr-qc/0702141. To appear in Adv. Theor. Math. Phys. (2009). [9] M. Karlovini, L. Samuelsson, Elastic stars in general relativity: I. Foundations and equilibrium models, Class. Quantum Grav. 20 No. 16 (2003), 3613–3648. [10] J. Kijowski, G. Magli, Relativistic elastomechanics as a Lagrangian field theory, Journal Geom. Phys. 9 (1992), 207–223. [11] V. G. LeBlanc, Asymptotic states of magnetic Bianchi I cosmologies, Class. Quantum Grav. 14 No. 8 (1997), 2281–2301. [12] R. Maartens, S. D. Maharaj, Collision-free gases in Bianchi space-times Gen. Rel. Grav. 22 (1990), 595–607. [13] A. D. Rendall, Cosmic censorship for some spatially homogeneous cosmological models, Ann. Phys. 233 (1994), 82–96. [14] A. D. Rendall, The initial singularity in solutions of the Einstein–Vlasov system of Bianchi type I, J. Math Phys. 37 (1996), 438–451. [15] A. D. Rendall, C. Uggla, Dynamics of spatially homogeneous locally rotationally symmetric solutions of the Einstein–Vlasov equations, Class. Quantum Grav. 17 No. 22 (2000), 4697–4713. [16] A. D. Rendall, Global properties of locally spatially homogeneous cosmological models with matter, Math. Proc. Camb. Phil. Soc. 118 (1995), 511–526. [17] A. D. Rendall, An introduction to the Einstein–Vlasov system, Banach Center Publications 41 (1997), 35–68. [18] A. D. Rendall, Partial Differential Equations in General Relativity, Oxford University Press, Oxford (2008). [19] B. Saha, Bianchi type I universe with viscous fluid, Mod. Phys. Lett. A 20 (2005), 2127–2144. [20] A. S. Tahvildar-Zadeh, Relativistic and non-relativistic elastodynamics with small shear strains, Ann. Inst. H. Poincaré, Phys. theor. 69, No. 3 (1998), 275–307.

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[21] C. Uggla, R. T. Jantzen, K. Rosquist, Exact hypersurface-homogeneous solutions in cosmology and astrophysics, Phys. Rev. D 51 (1995), 5522. [22] J. Wainwright, G. F. R. Ellis, Dynamical Systems in Cosmology, Cambridge University Press, Cambridge (1997). Simone Calogero Departamento de Matem´ atica Aplicada Facultad de Ciencias Universidad de Granada E-18071 Granada Spain e-mail: [email protected] J. Mark Heinzle Gravitational Physics Faculty of Physics University of Vienna A-1090 Vienna Austria e-mail: [email protected] Communicated by Piotr T. Chrusciel. Submitted: October 28, 2008. Accepted: January 26, 2009.

Ann. Henri Poincaré 10 (2009), 275–337 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020275-63, published online May 22, 2009 DOI 10.1007/s00023-009-0406-z


Convergent Null Data Expansions at Space-Like Infinity of Stationary Vacuum Solutions Andrés E. Ace˜ na Abstract. We present a characterization of the asymptotics of all asymptotically flat, stationary solutions with non-vanishing ADM mass to Einstein’s vacuum field equations. This characterization is given in terms of two sequences of symmetric trace free tensors (we call them the ‘null data’), which determine a formal expansion of the solution, and which are in a one to one correspondence to Hansen’s multipoles. We obtain necessary and sufficient growth estimates on the null data to define an absolutely convergent series in a neighborhood of spatial infinity. This provides a complete characterization of all asymptotically flat, stationary vacuum solutions to the field equations with non-vanishing ADM mass.

1. Introduction ˜ , g˜μν , ξ μ ), where M ˜ is a A stationary vacuum spacetime is given by a triplet (M four-dimensional manifold, g˜μν is a Lorentzian metric with signature (+ − −−) that satisfy Einstein’s vacuum equations Ric[˜ g ] = 0, and ξ μ is a time-like Killing vector field with complete orbits. The metric can be written locally as ˜ ab d˜ xa )2 + V −1 h xa d˜ xb , g˜ = V (dt + γa d˜

a, b = 1, 2, 3 ,

(1)

˜ ab depend only on the spatial coordinates x where V , γa and h ã . As shown by Geroch [10] the description of this spacetime can be done in terms of fields defined ˜ which is obtained as the quotient on an abstract three-dimensional manifold N ˜ ab on M ˜ with respect to the trajectories of ξ μ . The fields V , γa , h ˜ can space of M ˜ . The latter be considered as pull-backs under the projection map of fields on N will be denoted by the same symbols. In the following we shall work in terms of ˜ ab on N ˜ ab is a negative definite metric on N ˜ , where h ˜. the fields V , γa , h ˜ imply that on N ˜ the quantity The vacuum Einstein’s field equations in M ˜ bγc ωa = −V 2 ãbc D

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is curl-free, i.e. ˜ [a ωb] = 0 , D ˜ ab and ãbc = ˜[abc] , ˜123 = ˜ is the covariant derivative with respect to h where D 1 ˜ ab | 2 . We are interested in the asymptotics of the spacetime at spatial infinity, | det h ˜ is diffeomorphic to the complement of a closed ball so it will be assumed that N ¯R (0) in R3 . Thus N ˜ is simply connected and there exists a scalar field ω such B that ˜ a ω = ωa . D Instead of working with V and ω it is convenient to use the combinations V 2 + ω2 − 1 , φ˜M = 4V ω φ˜S = , 2V introduced by Hansen [11]. In this setting Einstein’s vacuum field equations are then equivalent to ˜ φÃ , A = M, S , (2) Δh˜ φÃ = 2R[h] ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ (3) Rab [h] = 2 (Da φM )(Db φM ) + (Da φS )(Db φS ) − (Da φK )(Db φK ) , 1 1 where φ˜K = 4 + φ˜2M + φ˜2S 2 . Equations (2), (3) will be referred to as the stationary vacuum field equations. We are looking for solutions of (2) and (3). The ˜ ab , φ˜M , φ˜S ) (cf. [7] for a ˜, h ˜ , g˜μν , ξ μ ) can be reconstructed from (N spacetime (M detailed discussion). ˜ ab ) to ˜, h The asymptotic flatness condition is usually stated by assuming (N admit a smooth conformal extension in the following way: there exist a smooth ˜ ) such that Riemannian manifold (N, hab ) and a function Ω ∈ C 2 (N ) ∩ C ∞ (N ˜ N = N ∪ {i}, where i is a single point, Ω > 0 on ˜ ab hab = Ω2 h Ω|i = 0 ,

Da Ω|i = 0 ,

˜, N ˜, on N

Da Db Ω|i = −2hab |i ,

(4)

where D is the covariant derivative operator defined by h. This makes N diffeomorphic to an open ball in R3 , with center at the point i, which represents space-like ˜ to be asymptotically flat in the stated sense. infinity. From now on we assume N ˜ ¯R (0) Considering N to be diffeomorphic to the complement of a closed ball B 3 in R is natural in the present context. It corresponds to the idea of an isolated system, where the material sources are confined to a bounded region outside of ˜ is diffeomorphic to R3 which is vacuum. Lichnerowicz [14] has shown that if N ˜ then N is flat. Reula [15] has shown existence and uniqueness of asymptotically flat solutions to (2), (3), in terms of a boundary value problem, when data are prescribed on the ˜ . In order to be able to control the precise asymptotic behavior of the sphere ∂ N

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spacetime, however, it would be convenient to have a complete description of the asymptotically flat stationary vacuum solutions in terms of asymptotic quantities. Candidates for this task are Hansen’s multipoles [11]. With the previous assumptions Hansen proposes a definition of multipoles, which extends Geroch’s definition of multipoles for asymptotically flat static spacetimes [9] to the stationary case. He defines the conformal potentials 1 φA = Ω− 2 φÃ , A = M, S , (5) and two sequences of tensor fields near i through

1 P A = φA , PaA = Da P A , PaA2 a1 = C Da2 PaA1 − P A Ra2 a1 , 2 1 PaAs+1 ...a1 = C Das+1 PaAs ...a1 − s(2s − 1)PaAs+1 ...a3 Ra2 a1 , A = M, S , 2

(6) (7)

where Rab is the Ricci tensor of hab and C is the projector onto the symmetric trace free part of the respective tensor fields. The multipole moments are then defined as the tensors ν A = P A (i) ,

νaAp ...a1 = PaAp ...a1 (i) ,

A = M, S ,

p = 1, 2, 3, . . .

(8)

A

Keeping aside the monopoles, ν , we will denote the two sequences of remaining multipoles by A = {νaA1 , νaA2 a1 , νaA3 a2 a1 , . . .} , A = M, S . Dmp The multipole moments are proposed as a way to characterize solutions of (2), (3). So a natural question is to what extent do the multipoles determine the metric h and the potentials φM , φS . For this to be the case the metric and the potentials should be real analytic even at i in suitable coordinates and conformal rescaling. Beig and Simon [3] and Kundu [13] have shown that the metric and the potentials do extend in a suitable gauge as real analytic fields to i if it is assumed that (ν M )2 + (ν S )2 = 0 . As explained in [16] (cf. also [4]), in order for a solution of (2), (3) to lead to an ˜ it is necessary that ν S = 0. So, we assume from asymptotically flat spacetime M now on that (9) ν M = 0 , ν S = 0 . In [3] and [13] it is also shown that for given multipoles there is a unique formal expansion of a ‘formal solution’ to the stationary field equations, but it is not touched upon the convergence of the expansion. Bäckdahl and Herberthson [2] have found, assuming a given asymptotically flat solution of the stationary field equations, necessary bounds on the multipoles. The question that remains open is under which conditions a pair of sequences, taken as the multipoles, do indeed determine a convergent expansion of a stationary solution. This question has been studied for the axisymmetric case by B¨ ackdahl [1]. In the static case there is only one sequence of multipoles. Friedrich [8] has used as data a sequence of trace-free symmetric tensors, referred to as null data, which

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are different but related to the multipoles. He has shown that imposing certain types of estimates on the null data is necessary and sufficient for the existence of asymptotically flat static spacetimes. However, so far the existence question has never been answered for the general stationary case. Using a conformal factor that is specified later on, we define the following two sequences of trace-free symmetric tensors at infinity

Dnφ = C(Da1 φ)(i), C(Da2 Da1 φ)(i), C(Da3 Da2 Da1 φ)(i), . . . ,

DnS = Sa2 a1 (i), C(Da3 Sa2 a1 )(i), C(Da4 Da3 Sa2 a1 )(i), . . . , (10) where φ = φS and Sab is the trace free part of the Ricci tensor of h. These two sequences are referred to again as the null data. The purpose of this work is to derive, under the assumption (9), necessary and sufficient conditions for the null data Dnφ , DnS , to determine apart from gauge conditions (unique) real analytic solutions of (2) and (3) and thus to provide a complete characterization of all possible asymptotically flat solutions to the stationary vacuum field equations. This generalizes the work by Friedrich [8] from the static to the stationary case in a way discussed later on. ˜ to be considered diffeomorphic to the For the same reasons that justify N 3 complement of a closed ball in R , we shall treat the case in which N may comprise a small neighborhood of the point i, without worrying about the behavior of the ˜ a neighborhood of i covers an infinite solution in the large (note that in terms of h domain extending to space-like infinity). For our analysis it is convenient to remove the conformal gauge freedom and use, following Beig and Simon [3],

1 1 −2 2 2 2 ˜ ˜ 1 + 4φM + 4φS −1 . (11) Ω= m 2 With this conformal factor they derive fall-off conditions and then show that under some assumptions the rescaled metric can be extended in suitable coordinates on a suitable neighborhood of space-like infinity as a metric which is real analytic at i. The potentials φM and φS are then also real analytic at i, so that the multipoles are well defined. Using this gauge, and taking into account that the angular momentum monopole vanish, we get νM = m ,

νaM = 0 .

We express now the tensors in Dnφ , DnS in terms of an h-orthonormal frame ca , a = 1, 2, 3, at i. Denoting by Da the covariant derivative in the direction of ca ,

Dnφ∗ = C(Da1 φ)(i), C(Da2 Da1 φ)(i), C(Da3 Da2 Da1 φ)(i), . . . , (12)

(13) DnS∗ = Sa2 a1 (i), C(Da3 Sa2 a1 )(i), C(Da4 Da3 Sa2 a1 )(i), . . . . These tensors, which are defined uniquely up to rigid rotations in R3 , will be referred to as the null data of h in the frame ca .

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If the metric h and the potential φ are real analytic near i, then there exist constants M, r > 0 such that the components of these tensors satisfy the estimates M p! |C(Dap . . . Da1 φ)(i)| ≤ p , ap , . . . , a1 = 1, 2, 3 , p = 0, 1, 2, . . . , r M p! |C(Dap . . . Da1 Sbc )(i)| ≤ p , ap , . . . , a1 , b, c = 1, 2, 3 , p = 0, 1, 2, . . . . (14) r Although these estimates are similar to the Cauchy estimates, known to hold for the derivatives of analytic functions, they are not the same. The difference being that here the estimates are on the symmetric trace free part of the derivatives instead of being directly on the derivatives. These estimates are derived from Cauchy estimates in Section 3. Remarkably, the statement that these estimates are not only necessary but also sufficient to have an analytic solution of the stationary field equations is also true. This constitutes our main result, given in the following theorem. Theorem 1.1. Suppose m = 0 and ˆ φ = {ψa , ψa D n

ˆ nS D

1

2 a1

, ψa3 a2 a1 , . . .} ,

(15)

= {Ψa2 a1 , Ψa3 a2 a1 , Ψa4 a3 a2 a1 , . . .} ,

(16)

are two infinite sequences of symmetric, trace free tensors given in an orthonormal frame at the origin of a 3-dimensional Euclidean space. If there exist constants M, r > 0 such that the components of these tensors satisfy the estimates M p! |ψap ...a1 | ≤ p , ap , . . . , a1 = 1, 2, 3 , p = 1, 2, . . . , r M p! |Ψap ...a1 bc | ≤ p , ap , . . . , a1 , b, c = 1, 2, 3 , p = 0, 1, 2, . . . , r then there exists an analytic, asymptotically flat, stationary vacuum solution ˜ φ˜M , φ˜S ) with mass monopole m and zero angular momentum monopole, unique (h, 1 ˜ up to isometries, so that the null data implied by h = 14 m−4 [(1+4φ˜2M +4φ˜2S ) 2 −1]2 h 1 1 1 2 2 − and φS = 2 2 m[(1 + 4φ˜M + 4φ˜S ) 2 − 1] 2 φ˜S in a suitable frame ca as described above satisfy C(Daq . . . Da1 φS )(i) = ψaq ...a1 ,

aq , . . . , a1 = 1, 2, 3 ,

q = 1, 2, . . . ,

C(Daq . . . Da3 Sa2 a1 )(i) = Ψaq ...a1 ,

aq , . . . , a1 = 1, 2, 3 ,

q = 2, 3, . . . .

Two sequences of data of the form (15), (16), not necessarily satisfying any estimates, will be referred to as abstract null data. The type of estimates imposed here on the abstract null data does not depend on the orthonormal frame in which they are given. Since these estimates are necessary as well as sufficient, all possible asymptotically flat solutions of the stationary vacuum field equations are characterized by the null data. Corvino and Schoen [6] and Chru´sciel and Delay [5] have proven that it is possible to deform given general asymptotically flat vacuum data in an annulus in order to glue that data to stationary vacuum data in the asymptotic region. In

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relation with those works, as they need a family of asymptotically flat stationary solutions to perform the gluing procedure, our result gives a complete survey of the possible stationary asymptotics that can be attained beyond the known exact solutions. As both, the multipoles and the null data, determine the metric and the M S , Dmp potentials, there is a bijective map between them. The sets Dnφ , DnS and Dmp thus contain the same information. We prefer to work with the null data because the expressions are linear in φ and Sab . This work contains the static case as a special case. Starting from (1) the static case can be attained by making γa = 0, which gives ω = 0, φ˜S = 0 and φS = 0. This implies that all tensors in Dnφ are zero. Conversely, if all tensors in S are zero and by Xanthopoulos’ work [17] the Dnφ are zero then all tensors in Dmp spacetime is static. So we are left with DnS as the free data in the static case. Friedrich [8] has given the same result for the static case using a different conformal metric. Let us assume for now that we are in the static case, then ˘ which is conformally related to our metric h by Friedrich uses a metric h, ˘=Ω ˘ 2h , h where

(17)

1 1 4 (1 + m2 Ω) 2 + mΩ 2 ˘= Ω 2 . 1 1 (1 + m2 Ω) 2 + mΩ 2 + 1

˘ he defines a sequence of symmetric trace-free tensors D ˘ n in the same way Using h as we defined DnS in (10). He shows that imposing estimates of the type (14) on ˘ n is necessary and sufficient for the existence of an asymptotically the tensors in D flat static vacuum solution of the Einstein’s equations. To see that this result is equivalent to our result in the static case, we have to show that having estimates of the type (14) on the tensors in DnS imply estimates of the same type on the ˘ n and vice versa. This is done through Theorem 1.1 and relation (17). tensors in D If the tensors in DnS satisfy estimates of the type (14) then there exist h and Ω ˘ given by (17) is also analytic, thus the tensors in D ˘ n satisfy analytic, and then h estimates of the type (14), the converse is shown in the same way using Friedrich’s result. Hence this work generalizes the work by Friedrich [8] from the static to the stationary case. The procedure that we use in the present work follows similar steps and several of the technics in [8] will be used. For completeness we include them. Theorem 1.1 will be proven in terms of the conformal metric h. Thus we shall express in Section 2 the stationary vacuum field equations as ‘conformal stationary vacuum field equations’. In Section 3 we show, by going to the space-spinor formalism, that the abstract null data indeed determine the expansion coefficients of a certain type of formal expansions of solutions to the conformal stationary vacuum field equations uniquely. Showing convergence in this way appears difficult,

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however. For this reason the underlying geometry of the problem is used in Section 4 to cast the problem in a certain setting, where it becomes a characteristic initial value problem. The setting is necessarily singular, as the set where data is prescribed contains a vertex, but the convergence problem can now be handled. In Section 5 it is shown how to determine a formal solution to a subset of the conformal field equations from a given set of abstract null data. Then, in Section 6, the convergence of the series so obtained is shown. In Section 7 it is shown that the obtained solution satisfy the full set of conformal field equations. Finally, in Section 8, the convergence result is translated into a gauge which is regular near i, allowing us to prove Theorem 1.1.

2. The stationary field equations in the conformal setting The existence problem will be analyzed completely in terms of the conformally rescaled metric h, so we need to express the stationary field equations in terms of the conformal fields. By a constant conformal rescaling it can always be achieved that m = 1. For simplicity we use this scale from now on. If we directly transform the fields in (2) and (3) we arrive at a system of equations that is singular at i. To overcome this problem we follow the work of Beig and Simon [3]. Using (11) as the conformal factor, which together with (5) imply (18) Ω = φ2M + φ2S − 1 , and standard formulae for conformal transformations, they rewrite the stationary field equations, arriving at the following equivalent system of equations: 1 5 a a ΔφA = − R − Da ΩD Ω + 10(1 + Ω)πa φA , A=M,S , 2 2 1 2 Da Db Ω = −ΩRab − hab R + Ω + hab Dc ΩDc Ω 3 3 1 2 −4 Ω+ (Ω + 1)hab πc c − (Ω − 1)Da ΩDb Ω + 2Ω2 πab , 3 2 Da R = 7Db ΩDa Db Ω + 3Rab Db Ω + 4(3Ω − 2)πb b Da Ω 3 − Db ΩDb ΩDa Ω − 6Ωπab Db Ω − 2(7Ω + 4)Da πb b , 2 D[c Rb]a = 2(3Ω − 1)πd d ha[b Dc] Ω − ha[b Dc] ΩDd ΩDd Ω − 2(Ω − 1)ha[b πc]d Dd Ω − 2(2Ω + 1)ha[b Dc] πd d 1 + 2ha[b Dc] Dd ΩDd Ω + D[c ΩDb] Da Ω − (Ω − 4)πa[b Dc] Ω 2 1 + 2ΩD[c πb]a + Ra[b Dc] Ω + ha[b Rc]d Dd Ω , 2

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where πab ≡ Da φM Db φM + Da φS Db φS

(19)

has been introduced just as a useful notation. These equations are regular even at i. They form a quasi-linear, overdetermined system of PDE’s which implies, by applying formal derivatives to some of the equations, elliptic equations for all unknowns in a suitable gauge. Considering the fall-off conditions on the fields, Beig and Simon [3] deduced a certain smoothness of the conformal fields at i. Invoking a general theorem of Morrey on elliptic systems of this type they concluded that the solutions are in fact real analytic at i. Later Kennefick and O’Murchadha [12] showed that the fall-off conditions are reasonable, as they are implied by the spacetime being asymptotically flat. To avoid introducing additional constraints by taking derivatives, we shall deal with the system as it is. For our purposes it is convenient to make some changes to this system. We separate the Ricci tensor into its trace free part and the Ricci scalar, 1 Rab = Sab + hab R . 3

(20)

We also get rid of πab by using (19) in the other equations. From (18) we see that Ω, φM and φS are not independent, we use this equation to get rid of φM in the other equations. With these changes and the change of notation φS → φ the system of equations takes the form 1 1 2 a 5 R+ φ D ΩDa Ω Δφ = −φ (21) 2 1 + Ω − φ2 4 −(1 + Ω)φDa ΩDa φ + (1 + Ω)2 Da φDa φ , 1 (22) Da Db Ω = −ΩSab − (1 + Ω)hab R 3 1 1 1 + (−1 + Ω)φ2 Da ΩDb Ω + 2 1+Ω−φ 2 1 − (2 + 3Ω)φ2 hab Dc ΩDc Ω − 2Ω2 φD(a ΩDb) φ 3 4 + (1 + Ω)(2 + 3Ω)φhab Dc ΩDc φ + 2Ω2 (1 + Ω)Da φDb φ 3 4 − (1 + Ω)2 (2 + 3Ω)hab Dc φDc φ , 3 1 2(4+7Ω)φDb ΩDb Da φ−4(1+Ω)(4+7Ω)Db φDb Da φ (23) Da R = 1+Ω−φ2 + 3 + (−3 + 7Ω)φ2 Db ΩSba − 2Ω(4 + 7Ω)φDb φSba 1 2 2 + (4 + 7Ω)φ RDa Ω − (1 + Ω)(4 + 7Ω)φRDa φ 3 3

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1 2 1 φ −12 + (40 + 21Ω)φ2 Db ΩDb ΩDa Ω 2 2 3(1 + Ω − φ ) 2 − 2φ − 18(1 + Ω) + (46 + 61Ω + 21Ω2 )φ2 Db ΩDb φDa Ω + 2(1 + Ω) − 24(1 + Ω) + (52 + 61Ω + 21Ω2 )φ2 Db φDb φDa Ω − φ 12(1 + Ω) + (16 + 61Ω + 21Ω2 )φ2 Db ΩDb ΩDa φ + 4(1 + Ω) 6(1 + Ω) + (22 + 61Ω + 21Ω2 )φ2 Db ΩDb φDa φ −4(1 + Ω)2 (28 + 61Ω + 21Ω2 )φDb φDb φDa φ , 1 = ΩφDa D[b φDc] Ω − 2Ω(1 + Ω)Da D[b φDc] φ 1 + Ω − φ2 2 4 − (1 + Ω)φha[b Dc] Dd φDd Ω + (1 + Ω)2 ha[b Dc] Dd φDd φ 3 3 1 2 + 1 + (−1 + Ω)φ Sa[b Dc] Ω − Ω2 φSa[b Dc] φ 2 2 1 − Ωφ2 ha[b Sc]d Dd Ω + Ω(1 + Ω)φha[b Sc]d Dd φ 3 3 1 1 2 + (−2 + Ω)φ Rha[b Dc] Ω − (−2 + Ω)(1 + Ω)φRha[b Dc] φ 18 9

283

+

D[c Sb]a

(24)

+2φDa ΩD[b ΩDc] φ − 4(1 + Ω)Da φD[b ΩDc] φ 1 2 1 + φ 3 + 2(−5 + 3Ω)φ2 ha[b Dc] ΩDd ΩDd Ω 2 2 9(1 + Ω − φ ) 2 − φ 6(1 + Ω) + (−13 − 4Ω + 6Ω2 )φ2 ha[b Dc] φDd ΩDd Ω − 2(−7 − 4Ω + 6Ω2 )φ3 ha[b Dc] ΩDd ΩDd φ + 4(1 + Ω)2 3 + 2(−5 + 3Ω)φ2 ha[b Dc] φDd ΩDd φ + 2(1 + Ω) − 3(1 + Ω) + (−4 − 4Ω + 6Ω2 )φ2 ha[b Dc] ΩDd φDd φ −4(1 + Ω)2 (−7 − 4Ω + 6Ω2 )φha[b Dc] φDd φDd φ . Besides (21), (22), (23), (24) we need an equation for the metric or for the frame field and the connection coefficients. This equation is just (20), 1 Rab [h] = Sab + hab R , 3

(25)

where the expression on the left hand side is understood as the Ricci operator acting on the metric h. The system of equations (25), (21), (22), (23), (24), together with conditions (4) and the condition R|i = − (6 + 8Da φDa φ) |i ,

(26)

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implied by (4), will be referred to as the conformal stationary vacuum field equations for the unknown fields hab , φ, Ω, R, Sab .

(27)

3. The exact sets of equations argument To see that it is possible to construct solutions to the conformal stationary vacuum field equations from the null data we study expansions of the conformal fields (27) in normal coordinates. We assume from now on N to be small enough to coincide with a convex h-normal neighborhood of i. Let ca , a = 1, 2, 3, be an h-orthonormal frame field on N which is parallelly transported along the h-geodesics through i and let xa denote normal coordinates centered at i so that cb a ≡ dxb , ca = δ b a at i. We refer to such a frame as normal frame centered at i. Its dual frame will be denoted by χc = χc b dxb . In the following all tensor fields, except the frame field ca and the coframe field χc , will be expressed in terms of this frame field, so that the metric is given by hab ≡ h(ca , cb ) = −δab . With Da ≡ Dca denoting the covariant derivative in the ca direction, the connection coefficients with respect to ca are defined by Da cc = Γa b c cb . An analytic tensor field Ta1 ...ak on N has in the normal coordinates xa a normal expansion at i, which can be written 1 xcp . . . xc1 Dcp . . . Dc1 Ta1 ...ak (i) , (28) Ta1 ...ak (x) = p! p≥0

where we assume from now on that the summation convention does not distinguish between bold face and other indices. Since hab = −δab , it remains to be seen how to obtain normal expansions for φ, Ω, R, Sab ,

(29)

using the field equations and the null data. The algebra necessary for doing this simplifies considerably in the space-spinor formalism. To do the transition we introduce the constant van der Waerden symbols αAB a , αa AB , a = 1, 2, 3, A, B = 0, 1, which are symmetric in AB and whose components, if read as matrices, are 1 1 1 −1 0 −i 0 0 1 αAB 1 = √ , αAB 2 = √ , αAB 3 = √ , 0 1 0 −i 1 0 2 2 2 1 1 1 −1 0 i 0 0 1 , α2 AB = √ , α3 AB = √ . α1 AB = √ 0 1 0 i 1 0 2 2 2 The relation between tensors given in the frame ca and space-spinors is made by T a1 ...ap b1 ...bq → T A1 B1 ...Ap Bp C1 D1 ...Cq Dq ≡ T a1 ...ap b1 ...bq αA1 B1 a1 . . . αbq Cq Dq .

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With the summation rule also applying to capital indices we get δ b a = αb AB αAB a ,

−δab αa AB αb CD = −A(C D)B ≡ hABCD , a, b = 1, 2, 3 ,

A, B, C, D = 0, 1 ,

where the constant -spinor satisfies AB = −BA , 01 = 1. It is used to move indices according to the rules ιB = ιA AB , ιA = AB ιB , so that A B corresponds to the Kronecker delta. As the spinors are in general complex, we need a way to sort out those that arise from real tensors. For this we define

τ AA = 0 A 0 A + 1 A 1 A . Primed indices take values 0, 1 and the summation rule also applies to them. A bar denotes complex conjugation and indices acquire a prime under complex conjugation, an exception being A B , the complex conjugate of AB . We define + = τA A ..τH H ξ¯A ...H . ξA...H

Then a space spinor field TA1 B1 ...Ap Bp = T(A1 B1 )...(Ap Bp ) arises from a real tensor field Ta1 ...ap if and only if TA1 B1 ...Ap Bp = (−1)p TA+1 B1 ...Ap Bp .

(30)

Any spinor field TA...H admits a decomposition into products of totally symmetric spinor fields and epsilon spinors which can be written schematically in the form s × symmetrized contractions of T . (31) TA...H = T(A...H) + It will be important that if TA1 B1 ...Ap Bp arises from Ta1 ...ap then T(A1 B1 ...Ap Bp ) = C(Ta1 ...ap )αa1 A1 B1 . . . αap Ap Bp . To discuss vector analysis in terms of spinors, a complex frame field and its dual 1-form field are defined by cAB = αa AB ca ,

χAB = αAB a χa ,

so that h(cAB , cCD ) = hABCD . From this one sees that c00 and c11 are null vectors orthogonal to c01 . The derivative of a function f in the direction of cAB is denoted by cAB (f ) = f,a ca AB and the spinor connection coefficients are defined by ΓAB C D =

1 b a Γa c α AB αCH b αc DH , 2

so that

ΓABCD = Γ(AB)(CD) .

The covariant derivative of a spinor field ιA is then given by DAB ιC = cAB (ιC ) + ΓAB C D ιD . If it is required to satisfy the Leibniz rule with respect to tensor products, then covariant derivatives in the ca -frame formalism translate under contractions with

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the van der Waerden symbols into spinor covariant derivatives and vice versa. We also have (DCD DEF − DEF DCD )ιA = rA BCDEF ιB , 1 1 rABCDEF = SABCE − RhABCE DF 2 6 1 + SABDF − RhABDF CE , 6

(32) (33)

a b where R is the Ricci scalar of h and SABCD = Sab αAB αCD = S(ABCD) represents the trace free part of the Ricci tensor of h. Equations (21), (24) take in the space-spinor formalism the form 1 2 1 10 φ DP Q ΩDP Q Ω (34) DP B DAP φ = − AB φ R + 4 1 + Ω − φ2 4 −(1 + Ω)φDP Q ΩDP Q φ + (1 + Ω)2 DP Q φDP Q φ ,

DP A SBCDP =

1 1 + Ω − φ2

(35) ΩφDA P ΩD(BC DD)P φ − 2Ω(1 + Ω)DA P φD(BC DD)P φ

+ (1 + Ω)φDP Q ΩD(BC DP Q φD)A − 2(1 + Ω)2 DP Q φD(BC DP Q φD)A 1 + 1 + (−1 + Ω)φ2 DA P ΩSP BCD − Ω2 φDA P φSP BCD 2 1 + Ωφ2 DP Q ΩSP Q(BC D)A − Ω(1 + Ω)φDP Q φSP Q(BC D)A 2 1 1 + (1 + Ω)φ2 RD(BC ΩD)A − (1 + Ω)2 φRD(BC φD)A 6 3 + 2φ DA P φDP (B ΩDCD) Ω − DA P ΩDP (B ΩDCD) φ P P + 4(1 + Ω) DA ΩDP (B φDCD) φ − DA φDP (B φDCD) Ω 1 2 1 φ − 6 + (20 + 3Ω)φ2 DP Q ΩDP Q ΩD(BC ΩD)A + (1 + Ω − φ2 )2 24 1 − (14 + 23Ω + 3Ω2 )φ3 DP Q ΩDP Q φD(BC ΩD)A 6 1 + (1 + Ω) 6(1 + Ω) + (8 + 23Ω + 3Ω2 )φ2 DP Q φDP Q φD(BC ΩD)A 6 1 − φ −12(1 + Ω) + (26 + 23Ω + 3Ω2 )φ2 DP Q ΩDP Q ΩD(BC φD)A 12 1 + (1 + Ω)2 −6 + (20 + 3Ω)φ2 DP Q ΩDP Q φD(BC φD)A 3 1 2 2 PQ − (1 + Ω) 14 + 23Ω + 3Ω φD φDP Q φD(BC φD)A . 3

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Equations (22), (23) are translated into the space-spinor formalism by making the index replacements a → AB, b → CD, c → EF . We use equations (34), (35), the spinor version of equations (22), (23) and the theory of ‘exact sets of fields’ to prove the following result. Lemma 3.1. Let there be two given sequences ˆ φ = {ψA B , ψA B A B , ψA B A B A B , . . .} , D n 1 1 2 2 1 1 3 3 2 2 1 1 ˆ S = {ΨA B A B , ΨA B A B A B , ΨA B A B A B A B , . . .} , D n 2 2 1 1 3 3 2 2 1 1 4 4 3 3 2 2 1 1 of totally symmetric spinors satisfying the reality condition (30). Assume that there exists a solution h, φ, Ω, R, SABCD to the conformal stationary field equations ˆ nφ , D ˆ nS (25), (21), (22), (23), (24) satisfying (4), (26) so that the spinors given by D coincide with the null data Dnφ∗ , DnS∗ given by (12), (13) of the metric h in terms of an h-orthonormal normal frame centered at i, i.e., ψAp Bp ...A1 B1 = D(Ap Bp . . . DA1 B1 ) φ(i) ,

p ≥ 1,

ΨAp Bp ...A1 B1 = D(Ap Bp . . . DA3 B3 SA2 B2 A1 B1 ) (i) ,

(36) p ≥ 2.

(37)

Then the coefficients of the normal expansions (28) of the fields (29), i.e. DAp Bp . . . DA1 B1 φ(i) ,

DAp Bp . . . DA1 B1 Ω(i) ,

DAp Bp . . . DA1 B1 R(i) ,

DAp Bp . . . DA1 B1 SABCD (i) ,

p ≥ 0,

ˆ nS and satisfy the reality conditions. ˆ nφ , D are uniquely determined by the data D Proof. It holds φ(i) = 0, DAB φ(i) = ψAB and SABCD (i) = ΨABCD by assumption and the expansion coefficients for Ω and R of lowest order are given by (4) and (26). Assume the expansion coefficients of φ and Ω up to order p and the expansion coefficients of R and SABCD up to order p − 1 are known. To discuss the induction step we start with DAp+1 Bp+1 . . . DA1 B1 φ(i) and its decomposition in the form (31). By assumption, the totally symmetric part of it is given by ψAp+1 Bp+1 ...A1 B1 . The other terms in the decomposition contain contractions. Let us consider Ai contracted with Aj . We can commute the operators DAi Bi and DAj Bj with other covariant derivatives, generating by (32) and (33) only terms of lower order, until we have DAp+1 Bp+1 . . . DAi+1 Bi+1 DAi−1 Bi−1 . . . DAj+1 Bj+1 DAj−1 Bj−1 . . . DA1 B1 DP

Bi DP Bj φ(i) .

Equation (34) then shows how to express the resulting term by quantities of lower order that are already known. For DAp+1 Bp+1 . . . DA1 B1 Ω(i) and DAp Bp . . . DA1 B1 R(i) we just use the spinor versions of (22) and (23) to express them by quantities of lower order. Finally, dealing with DAp Bp . . . DA1 B1 SCDEF (i) is quite similar to DAp+1 Bp+1 . . . DA1 B1 φ(i). The symmetric term is known by the data. If a contraction is performed between a derivative index and one of C, D, E, F then (35) is

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used after interchanging derivatives. If the contraction is between two derivatives, the general identities 1 DH(A DH B) SCDEF = −2SH(CDE SF ) H AB + RSH(CDE hF ) H AB , 3 1 DAB DAB SCDEF = −2DF G DG H SCDEH + 3SGH(CD SE)F GH + RSCDEF , 2 implied by (32), (33), together with (35) show that the corresponding term can be expressed in terms of quantities of lower order. The induction step is completed. That the expansion coefficients satisfy the reality condition is a consequence of the formalism and the fact that they are satisfied by the data. In order to show the convergence of the formal series determined in the previous lemma we need to impose estimates on the free coefficients given by ˆ nφ , D ˆ nS . For this we have the following result. D Lemma 3.2. A necessary condition for the formal series determined in Lemma 3.1 ˆ φ, D ˆ S satisfy to be absolutely convergent near the origin is that the data given by D n n estimates of the type p!M , rp p!M |ΨAp Bp ...A1 B1 CDEF | ≤ p , r with some constants M, r > 0. |ψAp Bp ...A1 B1 | ≤

p = 1, 2, 3, . . . ,

(38)

p = 0, 1, 2, . . . ,

(39)

We skip the proof of this lemma because it uses the same argument as the proof of Lemma 3.2 in [8]. Lemma 3.1 shows that the null data determines a formal solution to the stationary field equations. As shown by Beig and Simon [3], the multipole moments do the same. Thus there is a bijective map Θ from the null data to the multipoles M S sequences, Θ : {Dnφ , DnS } → {Dmp , Dmp }. Instead of using this argument, we can try to gain more information on the relation starting from (6), (7). It is convenient to work in space-spinor form, that means that we are using the h-orthonormal frame and normal coordinates previously defined. We get the following result. Lemma 3.3. The spinor fields PAMp Bp ...A1 B1 , PASp Bp ...A1 B1 , near i, given by (6), (7), are of the form PAMp Bp ...A1 B1 (40) 1 1 − = − 1 + Ω − φ2 2 1 + 2Ω − φ2 D(Ap Bp . . . DA3 B3 SA2 B2 A1 B1 ) 2 − 1 − 1 + Ω − φ2 2 φD(Ap Bp . . . DA1 B1 ) φ − 3 1 1 + Ω − φ2 2 φ p − 2Ω2 D(Ap Bp ΩDAp−1 Bp−1 . . . DA1 B1 ) φ + 2

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− 3 − 1 + Ω−φ2 2 (1 + Ω) p−2Ω2 D(Ap Bp φDAp−1 Bp−1 . . . DA1 B1 ) φ + FAMp Bp ...A1 B1 , PASp Bp ...A1 B1

p ≥ 3, p ≥ 2,

= D(Ap Bp . . . DA1 B1 ) φ + FASp Bp ...A1 B1 ,

(41)

with symmetric spinor-valued functions FAMp Bp ...A1 B1 and FASp Bp ...A1 B1 . The function FAMp Bp ...A1 B1 , p ≥ 3, is at each point a real linear combination of symmetrized tensor products of D(Aq−1 Bq−1 . . . DA1 B1 ) φ ,

D(Aq Bq . . . DA3 B3 SA2 B2 A1 B1 ) ,

DAB Ω ,

2 ≤ q ≤ p −1 ,

with coefficients that depend on Ω and φ. The function FASp Bp ...A1 B1 , p ≥ 2, is a real linear combination of symmetrized tensor products of D(Aq−2 Bq−2 . . . DA1 B1 ) φ , Proof. From (18) we get

D(Aq Bq . . . DA3 B3 SA2 B2 A1 B1 ) ,

2 ≤ q ≤ p.

1 φM = 1 + Ω − φ 2 2 ,

and by direct calculations from (6), (7) we see that (40) is valid for p = 3 and that (41) is valid for p = 2, with the stated properties for FAM3 B3 A2 B2 A1 B1 and FAS2 B2 A1 B1 . Assuming that the lemma is true for p ≤ k, inserting (40) and (41) into the recursion relation (7), and using the symmetrized spinor version of (22), we see that the lemma is true for p = k + 1. Using (6), (7), (8) and the identification (36), (37) we get for the lower multipoles 1 M M νA = 0 , νA = − ΨA2 B2 A1 B1 − ψ(A2 B2 ψA1 B1 ) , 1 B1 2 B2 A 1 B1 2 1 M νA = − ΨA3 B3 A2 B2 A1 B1 − 3ψ(A3 B3 ψA2 B2 A1 B1 ) , 3 B3 A 2 B2 A 1 B1 2 S S = ψA 2 B2 A 1 B1 . νA1 B1 = ψA1 B1 , νA 2 B2 A 1 B1

order (42) (43) (44)

Also restricting (40) and (41) to i and with the identification (36), (37) we get 1 M νA = − ΨAp Bp ...A1 B1 − pψ(Ap Bp ψAp−1 Bp−1 ...A1 B1 ) p Bp ...A1 B1 2 + fAMp Bp ...A1 B1 , p ≥ 3 , S = ψAp Bp ...A1 B1 + fASp Bp ...A1 B1 , νA p Bp ...A1 B1

p ≥ 2,

(45)

(46)

where fAMp Bp ...A1 B1 , p ≥ 3, is a real linear combination of symmetrized tensor products of ψAq−1 Bq−1 ...A1 B1 ,

ΨAq Bq ...A1 B1 ,

2 ≤ q ≤ p − 1,

and fASp Bp ...A1 B1 , p ≥ 2, is a real linear combination of symmetrized tensor products of ψAq−2 Bq−2 ...A1 B1 , ΨAq Bq ...A1 B1 , 2 ≤ q ≤ p .

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Equations (42), (43), (44), (45) and (46) give a nonlinear map Θ, that can be read as a map ˆ S } → {D ˆM , D ˆS } ˆ φ, D Θ : {D n n mp mp of the set of abstract null data into the set of abstract multipoles (i.e., sequences of symmetric spinors not necessarily derived from a metric). It is now possible to show that the map can be inverted. ˆ nφ , D ˆ nS } Corollary 3.4. The map Θ that maps sequences of abstract null data {D M S ˆ ˆ onto sequences of abstract multipoles {Dmp , Dmp } is bijective. Proof. From (43), (44) we see that fAM3 B3 A2 B2 A1 B1 = 0, fAS2 B2 A1 B1 = 0, with this and the stated properties for fAMp Bp ...A1 B1 and fASp Bp ...A1 B1 an inverse for Θ can be constructed inverting the relations (45) and (46) recursively. Hence, for a given metric h, the sequences of multipoles and the sequences of null data in a given standard frame carry the same information on h. As said, we prefer to work with the null data because they are linear in φ and SABCD .

4. The characteristic initial value problem After showing that the null data determine the solution, one would have to show that the estimates (38), (39) imply Cauchy estimates for the expansion coefficients p!M , Ap , Bp , . . . , A1 , B1 = 0, 1 , p = 0, 1, 2, . . . , rp where T is any of φ, Ω, R, SABCD . This would ensure the convergence of the normal expansion at i. The induction procedure used so far for calculating the expansion coefficients from the null data generates additional non-linear terms each time one interchanges a derivative or uses the conformal field equations. Thus, it does not seem suited for deriving estimates. Instead, we use the intrinsic geometric nature of the problem and the data to formulate the problem as a boundary value problem to which Cauchy-Kowalevskaya type arguments apply. As the fields h, φ, Ω, R, SABCD are real analytic in the normal coordinates xa and a standard frame cAB centered at i, they can be extended near i by analyticity into the complex domain and considered as holomorphic fields on a complex analytic manifold Nc . Choosing Nc to be a sufficiently small neighborhood of i, we can assume the extended coordinates, again denoted by xa , to define a holomorphic coordinate system on Nc which identifies Nc with an open neighborhood of the origin in C3 . The original manifold N is then a real, 3-dimensional, real analytic submanifold of the real, 6-dimensional, real analytic manifold underlying Nc . Under the analytic extension the main differential geometric concepts and formulas remain valid. The coordinates xa and the extended frame, again denoted by cAB , satisfy the same defining equations and the extended fields, denoted again by h, φ, Ω, R, SABCD , satisfy the conformal stationary vacuum field equations as before. |DAp Bp . . . DA1 B1 T (i)| ≤

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The analytic function Γ = δab xa xb on N extends to a holomorphic function on Nc . On N it vanishes only at i, but the set

Ni = p ∈ Nc |Γ(p) = 0 , is an irreducible analytical set such that Ni \{i} is a 2-dimensional complex submanifold of Nc . It is the cone swept out by the complex null geodesics through i and we will refer to it as the null cone at i. Now let u → xa (u) be a null geodesic through i such that xa (0) = 0. Its tangent vector is then of the form x˙ AB = ιA ιB with a spinor field ιA = ιA (u) satisfying Dx˙ ιA = 0 along the geodesic. Then φ(u) = φ x(u) , (47) a b A B C D (48) S0 (u) = x˙ x˙ Sab x(u) = ι ι ι ι SABCD x(u) , are analytic functions of u with Taylor expansion φ(u) =

∞ 1 p dp φ u (0) , p! dup p=0

S0 (u) =

∞ 1 p dp S0 u (0) , p! dup p=0

where dp φ (0) = x˙ ap . . . x˙ a1 Dap . . . Da1 φ(0) = ιAp ιBp . . . ιA1 ιB1 D(Ap Bp . . . DA1 B1 ) φ(i) , dup dp S 0 (0) = ιAp ιBp . . . ιA1 ιB1 ιC ιD ιE ιF D(Ap Bp . . . DA1 B1 SCDEF ) (i) . dup This shows that knowing these expansion coefficients for initial null vectors ιA ιB covering an open subset of the null directions at i is equivalent to knowing the null ˆ nφ , D ˆ nS of the metric h. data D Our problem can thus be formulated as the boundary value problem for the conformal stationary vacuum equations with data given by the functions (47), (48) on Ni , where the ιA ιB are parallelly propagated null vectors tangent to Ni . Ni is not a smooth hypersurface but an analytic set with a vertex at the point i, and we need a setting in which the mechanism of calculating the expansion coefficients allows us to derive estimates on the coefficients from the conditions imposed on the data. That is done in the following subsections. 4.1. The geometric gauge We need to choose a gauge suitably adapted to the singular set Ni . The coordinates and the frame field will then necessarily be singular and the frame will no longer define a smooth lift to the bundle of frames but a subset which becomes tangent to the fibres over some points. π → N with We will use the principal bundle of normalized spin frames SU (N ) − A structure group SU (2), which is the group of complex 2 × 2 matrices (s B )A,B=0,1 satisfying (49) AB sA C sB D = CD , τAB sA C s¯B D = τCD .

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The 2 : 1 covering homomorphism of SU (2) onto SO(3, R) is performed via SU (2) sA B → sa b = αa AB sA C sB D αCD b ∈ SO(3, R) . Under holomorphic extension the map above extends to a 2 : 1 covering homomorphism of the group SL(2, C) onto the group SO(3, C), where SL(2, C) denotes the group of complex 2 × 2 matrices satisfying only the first of conditions (49). A point δ ∈ SU (N ) is given by a pair of spinors δ = (δ0A , δ1A ) at a given point of N which satisfies (δA , δB ) = AB ,

+ (δA , δB ) = τAB ,

(50)

and the action of the structure group is given for s ∈ SU (2) by δ → δ · s where

(δ · s)A = sB A δB .

The projection π maps a frame δ into its base point in N . The bundle of spin p frames is mapped by a 2 : 1 bundle morphism SU (N ) − → SO(N ) onto the bundle π

SO(N ) −→ N of oriented, orthonormal frames on N so that π ◦ p = π. For any A )A,B=0,1 with an element of spin frame δ we can identify by (50) the matrix (δB the group SU (2). With this reading the map p will be assumed to be realized by E F δB cEF ∈ SO(N ) , SU (N ) δ → p(δ)AB = δA

where cAB denotes the normal frame field on N introduced before. We refer to p(δ) as the frame associated with the spin frame δ. π Under holomorphic extension the bundle SU (N ) − → N is extended to the π A A principal bundle SL(Nc ) − → Nc of spin frames δ = (δ0 , δ1 ) at given points of Nc which satisfy only the first of conditions (50). Its structure group is SL(2, C). The π π bundle SU (N ) − → N is embedded into SL(Nc ) − → Nc as a real analytic subbundle. The bundle morphism p extends to a 2 : 1 bundle morphism, again denoted by p, π

π

→ Nc onto the bundle SO(Nc ) −→ Nc of oriented, normalized frames of SL(Nc ) − of Nc with structure group SO(3, C). We shall make use of several structures on SM (Nc ). With each α ∈ sl(2, C), i.e., α = (αA B ) with αAB = αBA , is associated a vertical vector field Zα tangent to the fibres, which is given at δ ∈ SL(Nc ) by d Zα (δ) = dv (δ · exp(vα))|v=0 , where v ∈ C and exp denotes the exponential map sl(2, C) → SL(2, C). The C3 -valued soldering form σ AB = σ (AB) maps a tangent vector X ∈ Tδ SL(Nc ) onto the components of its projection Tδ (π)X ∈ Tπ(δ) Nc in the frame p(δ) associated with δ so that Tδ (π)X = σ AB , X p(δ)AB . It follows that σ AB , Zα = 0 for any vertical vector field Zα . The sl(2, C)-valued connection form ω A B on SL(Nc ) transforms with the adjoint transformation under the action of SL(2, C) and maps any vertical vector field Zα onto its generator so that ω A B , Zα = αA B . With xAB = x(AB) ∈ C3 is associated the horizontal vector field Hx on SL(Nc ) which is horizontal in the sense that ω A B , Hx = 0 and which satisfies

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293

σ AB , Hx = xAB . Denoting by HAB , A, B = 0, 1, the horizontal vector fields satisfying σ AB , HCD = hAB CD , it follows that Hx = xAB HAB . An integral curve of a horizontal vector field projects onto an h-geodesic and represents a spin frame field which is parallelly transported along this geodesic. A holomorphic spinor field ψ on Nc is represented on SL(Nc ) by a holomorphic spinor-valued function ψA1 ...Aj (δ) on SL(Nc ), given by the components of ψ in the frame δ. We shall use the notation ψk = ψ(A1 ...Aj )k , k = 0, .., j, where (. . . . . .)k denotes the operation ‘symmetrize and set k indices equal to 1 the rest equal to 0’. These functions completely specify ψ if ψ is symmetric. They are then referred to as the essential components of ψ. ˆ of SL(Nc ) 4.2. The submanifold N Using the available geometrical structure we construct a three-dimensional subˆ of SL(Nc ) in such a way that it induces coordinates in Nc . By the manifold N construction procedure the induced coordinates are suitable adapted to the set Ni . We start by choosing a spin frame δ ∗ such that π(δ ∗ ) = i and p(δ ∗ )AB = cAB . The curve C v → δ(v) = δ ∗ · s(v) ∈ SL(Nc ) , 1 0 s(v) = exp(vα) = , v 1

α=

0 1

0 0

∈ sl(2, C) ,

(51)

defines a vertical, 1-dimensional, holomorphic submanifold I of SL(Nc ) on which v defines a coordinate. The associated family of frames eAB (v) at i is given by eAB (v) = sC A (v)sD B (v)cCD , and explicitly by e00 = c00 + 2vc01 + v 2 c11 ,

e01 (v) = c01 + vc11 ,

e11 (v) = c11 .

We perform the following construction in a neighborhood of I. If it is chosen small enough all the following statements will be correct. The set I is moved with the flow of H11 to obtain a holomorphic 2-manifold U0 of SL(Nc ). We denote by w the parameter on the integral curves of H11 that vanishes on I, and we extend v to U0 by assuming it to be constant on the integral curves of H11 . All these integral curves are mapped by π onto the null geodesic γ(w) with affine parameter w and tangent vector γ (0) = c11 at γ(0) = i. The parameter v specifies which frame fields are parallelly propagated along γ. U0 is now moved with the flow of H00 to obtain a holomorphic 3-submanifold ˆ of SL(Nc ). We denote by u the parameter on the integral curves of H00 that N ˆ by assuming them to be constant along vanishes on U0 and we extend v and w to N the integral curves of H00 . The functions z 1 = u, z 2 = v, z 3 = w define holomorphic ˆ . We denote again π the restriction of the projection to N ˆ. coordinates on N The projections of the integral curves of H00 with a fixed value of w sweep out, together with γ, the null cone Nγ(w) near γ(w), which is generated by the null geodesics through the point γ(w). On the null geodesics u is an affine parameter which vanishes at γ(w) while v parametrizes the different generators. The set W0 = {w = 0} projects onto Ni \γ and will define the initial data set for our

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ˆ ≡ N ˆ \U0 problem. The map π induces a biholomorphic diffeomorphism of N ˆ onto π(N ). The singularity of the gauge at points of U0 consists in π dropping rank on U0 , where ∂v = Zα . The null curve γ(w) will be referred to as the the singular generator of Ni in the gauge determined by the spin frame δ ∗ resp. the corresponding frame cAB at i. ˆ, The soldering an the connection form pull back to holomorphic 1-forms on N AB A which will be denoted again by σ and ω B . If the pull back of the curvature ˆ is denoted again by ΩA B , then the form ΩA B = 12 rA BCDEF σ CD ∧ σ EF to N soldering and the connection form satisfy the structural equations dσ AB = −ω A C ∧ σ CB − ω B C ∧ σ AC , dω A B = −ω A C ∧ ω C B + ΩA B . ˆ is constructed, and in terms of the coordinates z a , we Using the way in which N AB AB a ˆ , where get σ = σ a dz on N ⎞ ⎛ ⎞ ⎛ 1 O(u3 ) O(u2 ) 1 σ 00 2 σ 00 3 (σ AB a ) = ⎝ 0 σ 01 2 σ 01 3 ⎠ = ⎝ 0 u + O(u3 ) O(u2 ) ⎠ as u → 0 . 0 0 1 0 0 1 ˆ there exist unique, holomorphic vector fields eAB which satisfy On N σ AB , eCD = hAB CD . If one writes eAB = ea AB ∂za , then ⎞ ⎛ ⎛ 1 1 e1 01 e1 11 (ea AB ) = ⎝ 0 e2 01 e2 11 ⎠ = ⎝ 0 0 0 1 0 We shall write

O(u2 ) 1 2u + O(u) 0

⎞ O(u2 ) O(u) ⎠ 1

as

u → 0.

ea AB = e∗a AB + eâ AB ,

with singular part 1 e∗a AB = δ1a A 0 B 0 + δ2a (A 0 B) 1 + δ3a A 1 B 1 , u ˆ which satisfy and holomorphic functions eâ AB on N (52) eâAB = O(u) as u → 0 . ˆ by ω A B = ΓCD A B σ CD with ΓCDAB ≡ We define the connection coefficients on N ωAB , eCD , so that ΓABCD = Γ(AB)(CD) , and from the definition of the frame ˆ, Γ00AB = 0 on N and it follows that

Γ11AB = 0 on

U0 ,

ˆ ABCD , ΓABCD = Γ∗ABCD + Γ

with singular part 1 Γ∗ABCD = − (A 0 B) 1 C 0 D 0 , u ˆ which satisfy ˆ ABCD on N and holomorphic functions Γ ˆ ABCD = O(u) as u → 0 . Γ

(53)

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4.3. Tensoriality and expansion type ˆ into Nc is singular on U0 , not every holomorphic As the induced map π of N a ˆ of a holomorphic function on function of the z can arise as a pull-back to N Nc . The latter must have a special type of expansion in terms of the z a which reflects the particular relation between the ‘angular’ coordinate v and the ‘radial’ coordinate u. We take from [8] the following definition and lemma. ˆ is said to be of v-finite expansion Definition 4.1. A holomorphic function f on N type kf , with kf an integer, if it has in terms of the coordinates u, v, and w a Taylor expansion at the origin of the form f=

∞ 2m+k ∞ f p=0 m=0

fm,n,p um v n wp ,

n=0

where it is assumed that fm,n,p = 0 if 2m + kf < 0. Lemma 4.1. Let φA1 ...Aj be a holomorphic, symmetric, spinor-valued function on SL(Nc ). Then the restrictions of its essential components φk = φ(A1 ...Aj )k , 0 ≤ ˆ satisfy k ≤ j, to N ∂v φk = (j − k)φk+1 ,

k = 0, . . . , j ,

on

U0 ,

(where we set φj+1 = 0) and φk is of expansion type j − k. 4.4. The null data on W0 As we have seen, prescribing the null data is equivalent to knowing φ and S0 in the null cone. Now we need to know how this fit into our particular gauge. For this we derive an expansion of the restriction of φ and S0 to the hypersurface W0 . Consider the normal frame cAB on Nc near i which agrees at i with the frame associated with δ ∗ and denote the null data of h in this frame by

∗ ∗ . . . DA φ(i), p = 1, 2, 3, . . . , Dnφ∗ = D(A p Bp 1 B1 )

∗ ∗ . . . DA S∗ (i), p = 0, 1, 2, 3, . . . . DnS∗ = D(A 1 B1 ABCD) p Bp Choose now a fixed value of v and consider s(v) as in (51), then the vector H00 (δ ∗ · s) projects onto the null vector e00 = sA 0 sB 0 cAB at i and is tangent to a null geodesic η = η(u, v) on Ni with affine parameter u, u = 0 at i. The integral curve of H00 through δ ∗ · s projects onto this null geodesic. Using the explicit expression for s = s(v) follows that ∞ 1 m m φ(u, v) = φ|η(u,v) = u D00 φ|η(0,v) m! m=0 =

=

∞ 1 m A m Bm ∗ ∗ u s 0 s 0 . . . sA1 0 sB1 0 D(A . . . DA φ(i) m Bm 1 B1 ) m! m=0 2m ∞ m=0 n=0

ψm,n um v n ,

(54)

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A. E. Ace˜ na

with ψm,n


1 2m ∗ ∗ = . . . DA φ(i) , D(A m Bm 1 B1 ) n m! n

0 ≤ n ≤ 2m .

In the same way S0 (u, v) = S0000 |η(u,v)

(55)

∗ |η(u,v) = = sA 0 sB 0 sC 0 sD 0 SABCD

∞ 2m+4

Ψm,n um v n ,

m=0 n=0

with Ψm,n =

1 2m + 4 ∗ ∗ . . . DA S∗ (i) , D(A 1 B1 ABCD)n m Bm m! n

0 ≤ n ≤ 2m .

This shows how to determine φ(u, v), S0 (u, v) from the null data Dnφ∗ , DnS∗ and vice versa.

ˆ 5. The conformal stationary vacuum field equations on N Now we can use the frame calculus in its standard form. Given the fields Ω, φ, R and SABCD , and using the frame eAB and the connection coefficients ΓABCD ˆ , we set on N rABCDEF ≡ eCD (ΓEF AB ) − eEF (ΓCDAB ) + ΓEF

a CD e EF

RABCDEF

AAB ΣAB ΦAB

C ΓDKAB

+ ΓEF

K

D ΓCKAB

− ΓCD

+ ΓEF

K

B ΓCDAK

− ΓCD K B ΓEF AK − tCD GH EF ΓGHAB ,

and we define there the quantities tAB EF ΣABCD and HABCD by tAB EF

K

K

E ΓKF AB

CD ,

− ΓCD K F ΓEKAB

RABCDEF , AAB , ΣAB , ΦAB , ΠAB ,

≡ 2ΓAB E (C ea D)E − 2ΓCD E (A ea B)E − ea CD,b eb AB + ea AB,b eb CD , 1 1 ≡ rABCDEF − SABCE − RhABCE DF 2 6 1 + SABDF − RhABDF CE , 6 ≡ DAB φ − φAB , ≡ DAB Ω − ΩAB , 1 2 1 10 P φ ΩP Q ΩP Q ≡ D B φAP + AB φ R + 4 1 + Ω − φ2 4 PQ 2 PQ − (1 + Ω)φΩ φP Q + (1 + Ω) φP Q φ ,

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ΠAB ≡ DAB R −

297

1 2(4 + 7Ω)φΩP Q DAB φP Q 1 + Ω − φ2

− 4(1 + Ω)(4 + 7Ω)φP Q DAB φP Q + 3 + (−3 + 7Ω)φ2 ΩP Q SP QAB − 2Ω(4 + 7Ω)φφP Q SP QAB 1 2 2 + (4 + 7Ω)φ RΩAB − (1 + Ω)(4 + 7Ω)φRφAB 3 3 1 2 1 − φ − 12 + (40 + 21Ω)φ2 ΩP Q ΩP Q ΩAB 2 2 3(1 + Ω − φ ) 2 − 2φ − 18(1 + Ω) + (46 + 61Ω + 21Ω2 )φ2 ΩP Q φP Q ΩAB + 2(1 + Ω) − 24(1 + Ω) + (52 + 61Ω + 21Ω2 )φ2 φP Q φP Q ΩAB − φ 12(1 + Ω) + (16 + 61Ω + 21Ω2 )φ2 ΩP Q ΩP Q φAB + 4(1 + Ω) 6(1 + Ω) + (22 + 61Ω + 21Ω2 )φ2 ΩP Q φP Q φAB 2 PQ − 4(1 + Ω) (7 + 3Ω)(4 + 7Ω)φφ φP Q φAB , 1 ΣABCD ≡ DAB ΩCD + ΩSABCD + (1 + Ω)RhABCD 3 1 1 1 + (−1 + Ω)φ2 ΩAB ΩCD − 1 + Ω − φ2 2 − Ω2 φ(ΩAB φCD + ΩCD φAB ) + 2Ω2 (1 + Ω)φAB φCD 1 4 − (2 + 3Ω) φ2 ΩP Q ΩP Q − (1 + Ω)φΩP Q φP Q 3 4 2 PQ + (1 + Ω) φP Q φ hABCD , HABCD ≡ DP A SBCDP −

1 ΩφΩA P D(BC φD)P − 2Ω(1 + Ω)φA P D(BC φD)P 1 + Ω − φ2

+ (1 + Ω)φΩP Q D(BC φP Q D)A − 2(1 + Ω)2 φP Q D(BC φP Q D)A 1 + 1 + (−1 + Ω)φ2 ΩA P SP BCD − Ω2 φφA P SP BCD 2 1 + Ωφ2 ΩP Q SP Q(BC D)A − Ω(1 + Ω)φφP Q SP Q(BC D)A 2 1 1 + (1 + Ω)φ2 RΩ(BC D)A − (1 + Ω)2 φRφ(BC D)A 6 3 + 2φ φA P ΩP (B ΩCD) − ΩA P ΩP (B φCD) P P + 4(1 + Ω) ΩA φP (B φCD) − φA φP (B ΩCD)

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1 2 1 φ − 6 + (20 + 3Ω)φ2 ΩP Q ΩP Q Ω(BC D)A 2 2 3(1 + Ω−φ ) 8 1 − (14 + 23Ω + 3Ω2 )φ3 ΩP Q φP Q Ω(BC D)A 2 1 + (1 + Ω) 6(1 + Ω) + (8 + 23Ω + 3Ω2 )φ2 φP Q φP Q Ω(BC D)A 2 1 − φ − 12(1 + Ω) + (26 + 23Ω + 3Ω2 )φ2 ΩP Q ΩP Q φ(BC D)A 4 + (1 + Ω)2 − 6 + (20 + 3Ω)φ2 ΩP Q φP Q φ(BC D)A 2 PQ − (1 + Ω) (7 + Ω) (2 + 3Ω) φφ φP Q φ(BC D)A . −

The tensor fields on the left hand side have been introduced as labels for the equations and for discussing in an ordered manner the interdependencies of the equations. In terms of these tensor fields, the conformal stationary vacuum equations read tAB EF

a CD e EF

ΦAB

= 0, = 0,

RABCDEF = 0 , ΠAB = 0 ,

AAB = 0 , ΣABCD = 0 ,

ΣAB = 0 , HABCD = 0 .

The first equation is Cartan’s first structural equation with the requirement that the metric connection be torsion free. The second equation is Cartan’s second structural equation, requiring the Ricci tensor to coincide with the appropriate combination of the trace free tensor Sab and the scalar R. The third and fourth equations define the symmetric spinors φAB and ΩAB respectively. The rest of the equations have already been considered. We want to calculate, using our particular gauge, a formal expansion of the conformal fields using the initial data in the form φ(u, v), S0 (u, v). As the system of conformal stationary vacuum field equations is an overdetermined system, we have to choose a subsystem of it. In the rest of this section we choose a particular subsystem, writing the chosen equations in our gauge, and at the end we see how a formal expansion is determined by these equations and the initial data. 5.1. The A00 = 0 equation The first equation that needs particular attention is the equation A00 = 0. In our gauge it reads ∂u φ = φ00 . This equation is used in the following to calculate φ00 each time we know φ as a function of u. In particular, as φ will be prescribed on W0 as part of the initial data, this equation allows us to calculate φ00 there immediately. 5.2. The ‘∂u -equations’ We now present what we will refer to as the ‘∂u -equations’. These equations are chosen because they have the following features. They are a system of PDE’s for the ˆ A1CD , Ω, ΩAB , φA1 , R, S1 , S2 , S3 and S4 , which comprise set of functions eâ A1 , Γ

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all the unknowns with the exceptions of the free data φ, S0 and the derived function φ00 . They are all interior equations on the hypersurfaces {w = w0 } in the sense that only derivatives in the directions of u and v are involved, in particular, if we consider the hypersurface {w = 0}, they are all inner equations in Ni . Also they split into a hierarchy that will be presented in the following section. The ∂u -equations: Equations tAB EF 00 ea EF = 0 : 1 1 ˆ 0101 + 2Γ ˆ 0100 eˆ1 01 , eˆ 01 = −2Γ u 1 1ˆ ˆ ∂u eˆ2 01 + eˆ2 01 = Γ ˆ2 01 , 0100 + 2Γ0100 e u u ˆ 1101 + 2Γ ˆ 1100 eˆ1 01 , ∂u eˆ1 11 = −2Γ 1ˆ ˆ ˆ2 01 . ∂u eˆ2 11 = Γ 1100 + 2Γ1100 e u

∂u eˆ1 01 +

Equations RAB00EF = 0 : ˆ 0100 + 2 Γ ˆ 0100 − 2Γ ˆ2 ∂u Γ 0100 u ˆ 0101 + 1 Γ ˆ 0101 ˆ 0101 − 2Γ ˆ 0100 Γ ∂u Γ u ˆ 0111 + 1 Γ ˆ 0111 ˆ 0111 − 2Γ ˆ 0100 Γ ∂u Γ u ˆ 1100 + 1 Γ ˆ 1100 ˆ 1100 − 2Γ ˆ 0100 Γ ∂u Γ u ˆ 1101 − 2Γ ˆ 0101 ˆ 1100 Γ ∂u Γ

1 S0 , 2 1 = S1 , 2 1 1 = S2 − R , 2 12 =

= S1 , = S2 +

1 R, 12

ˆ 1111 − 2Γ ˆ 0111 = S3 . ˆ 1100 Γ ∂u Γ Equation Σ00 = 0 : ∂u Ω = Ω00 . Equations ΦA0 = 0 : ∂u φ01 =

1 ˆ 0101 φ00 + 2Γ ˆ 0100 φ01 , (∂v φ00 − 2φ01 ) + eˆ1 01 ∂u φ00 + eˆ2 01 ∂u φ00 − 2Γ 2u

1 ˆ 0111 φ00 + Γ ˆ 0100 φ11 (∂v φ01 − φ11 ) − eˆ1 01 ∂u φ01 − eˆ2 01 ∂v φ01 = −Γ 2u 2 1 2 10 1 1 φ Ω00 Ω11 − φΩ01 − 2(1 + Ω)φ01 − φ R+ 4 1 + Ω − φ2 2 2 2 − (1 + Ω)φ (Ω00 φ11 + Ω11 φ00 ) + 2(1 + Ω) φ00 φ11 .

∂u φ11 −

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Equations Σ00CD = 0 :

∂u Ω01

1 1 + (−1 + Ω)φ2 Ω200 2 − 2Ω2 φΩ00 φ00 + 2Ω2 (1 + Ω)φ200 , 1 1 1 + (−1 + Ω)φ2 Ω00 Ω01 = −ΩS1 + 2 1+Ω−φ 2 2 2 − Ω φ (Ω00 φ01 + Ω01 φ00 ) + 2Ω (1 + Ω)φ00 φ01 ,

∂u Ω00 = −ΩS0 +

1 1 + Ω − φ2

1 ∂u Ω11 = −ΩS2 − (1 + Ω)R 3 1 1 3 − (11 + 9Ω)φ2 Ω00 Ω11 + 2 3(1 + Ω − φ ) 2 2 + 2(2 + 3Ω) φΩ01 − 2(1 + Ω)φ01 2 + (8 + 20Ω + 9Ω ) φ(Ω00 φ11 + Ω11 φ00 ) − 2(1 + Ω)φ00 φ11 . Equation Π00 = 0 : 1 2(4 + 7Ω)φ(Ω11 ∂u φ00 − 2Ω01 ∂u φ01 + Ω00 ∂u φ11 ) 1 + Ω − φ2 − 4(1 + Ω)(4 + 7Ω)(φ11 ∂u φ00 − 2φ01 ∂u φ01 + φ00 ∂u φ11 ) 1 3 + (−3 + 7Ω)φ2 (Ω11 S0 − 2Ω01 S1 + Ω00 S2 ) = 2 1+Ω−φ − 2Ω(4 + 7Ω)φ(φ11 S0 − 2φ01 S1 + φ00 S2 ) 1 + (4 + 7Ω)φ φΩ00 − 2(1 + Ω)φ00 R 3 1 + φ2 − 12 + (40 + 21Ω)φ2 (Ω00 Ω11 − Ω201 )Ω00 2 2 3(1 + Ω − φ ) − 2φ − 18(1 + Ω) + (46 + 61Ω + 21Ω2 )φ2 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )Ω00 + 4(1 + Ω) − 24(1 + Ω) + (52 + 61Ω + 21Ω2 )φ2 (φ00 φ11 − φ201 )Ω00 − 2φ 12(1 + Ω) + (16 + 61Ω + 21Ω2 )φ2 (Ω00 Ω11 − Ω201 )φ00 + 4(1 + Ω) 6(1 + Ω)+(22 + 61Ω + 21Ω2 )φ2 (Ω00 φ11 −2Ω01 φ01 + Ω11 φ00 )φ00 − 8(1 + Ω)2 (7 + 3Ω)(4 + 7Ω)φ(φ00 φ11 − φ201 )φ00 .

∂u R −

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Equations H0(ABC)k = 0, k=0,1,2,3 : 1 (∂v S0 − 4S1 ) − eˆ1 01 ∂u S0 − eˆ2 01 ∂v S0 2u 1 Ω φΩ ∂ ∂ + − 2(1 + Ω)φ φ − Ω φΩ − 2(1 + Ω)φ φ 01 01 u 00 00 00 u 01 1 + Ω−φ2 ˆ 0100 S1 ˆ 0101 S0 + 4Γ = −4Γ 1 1 1 + (−1 + Ω)φ2 (Ω01 S0 − Ω00 S1 ) − Ω2 φ(φ01 S0 − φ00 S1 ) − 2 1+Ω−φ 2 + 2 φΩ00 − 2(1 + Ω)φ00 (Ω00 φ01 − Ω01 φ00 ) ,

∂u S 1 −

1 (∂v S1 − 3S2 ) − eˆ1 01 ∂u S1 − eˆ2 01 ∂v S1 2u 1 + − (1 + Ω) φΩ11 − 2(1 + Ω)φ11 ∂u φ00 2 3(1 + Ω − φ ) + (2 + 5Ω) φΩ01 − 2(1 + Ω)φ01 ∂u φ01 −(1 + 2Ω) φΩ00 − 2(1 + Ω)φ00 ∂u φ11 1 1 2 (∂v φ01 − φ11 ) + eˆ 01 ∂u φ01 + eˆ 01 ∂v φ01 − 2Ω φΩ00 − 2(1 + Ω)φ00 2u ˆ 0101 S1 + 3Γ ˆ 0100 S2 ˆ 0111 S0 − 2Γ = −Γ 2 1 ˆ 0111 φ00 − Γ ˆ 0100 φ11 Ω φΩ00 − 2(1 + Ω)φ00 Γ − 2 1+Ω−φ 3 1 + 1 + (−1 + Ω)φ2 (Ω01 S1 − Ω00 S2 ) − Ω2 φ(φ01 S1 − φ00 S2 ) 2 1 1 − Ωφ2 (Ω11 S0 − 2Ω01 S1 + Ω00 S2 ) + Ω(1 + Ω)φ(φ11 S0 − 2φ01 S1 + φ00 S2 ) 6 3 1 − (1 + Ω)φ φΩ00 − 2(1 + Ω)φ00 R 18 + 2 φΩ01 − 2(1 + Ω)φ01 (Ω00 φ01 − Ω01 φ00 ) 1 2 1 + φ − 6 + (20 + 3Ω)φ2 (Ω00 Ω11 − Ω201 )Ω00 2 2 9(1 + Ω − φ ) 4 1 − (14 + 23Ω + 3Ω2 )φ3 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )Ω00 2 + (1 + Ω) 6(1 + Ω) + (8 + 23Ω + 3Ω2 )φ2 (φ00 φ11 − φ201 )Ω00 1 − φ − 12(1 + Ω) + (26 + 23Ω + 3Ω2 )φ2 (Ω00 Ω11 − Ω201 )φ00 2 + (1 + Ω)2 − 6 + (20 + 3Ω)φ2 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )φ00 2 2 − 2(1 + Ω) (7 + Ω) (2 + 3Ω) φ(φ00 φ11 − φ01 )φ00 ,

∂u S 2 −

302

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1 1 (∂v S2 − 2S3 ) − eˆ1 01 ∂u S2 − eˆ2 01 ∂v S2 + ∂u S 3 − 2u 3(1 + Ω − φ2 ) − 2(1 + Ω) φΩ11 − 2(1 + Ω)φ11 ∂u φ01 + Ω φΩ01 − 2(1 + Ω)φ01 ∂u φ11 1 (∂v φ01 −φ11 ) + eˆ1 01 ∂u φ01 + eˆ2 01 ∂v φ01 + 2(2 + 3Ω) φΩ01 −2(1 + Ω)φ01 2u 1 1 2 − (2 + 5Ω) φΩ00 − 2(1 + Ω)φ00 ∂v φ11 + eˆ 01 ∂u φ11 + eˆ 01 ∂v φ11 2u ˆ 0100 S3 ˆ 0111 S1 + 2Γ = −2Γ 1 2 ˆ 0111 φ00 − Γ ˆ 0100 φ11 − − (2 + 3Ω) φΩ01 − 2(1 + Ω)φ01 Γ 2 1+Ω−φ 3 2 ˆ 0111 φ01 − Γ ˆ 0101 φ11 + (2 + 5Ω) φΩ00 − 2(1 + Ω)φ00 Γ 3 1 + 1 + (−1 + Ω)φ2 (Ω01 S2 − Ω00 S3 ) − Ω2 φ(φ01 S2 − φ00 S3 ) 2 1 2 − Ωφ2 (Ω11 S1 − 2Ω01 S2 + Ω00 S3 ) + Ω(1 + Ω)φ(φ11 S1 − 2φ01 S2 + φ00 S3 ) 3 3 1 − (1 + Ω)φ φΩ01 − 2(1 + Ω)φ01 R 9 + 2 φΩ11 − 2(1 + Ω)φ11 (Ω00 φ01 − Ω01 φ00 ) 1 2 2 + φ − 6 + (20 + 3Ω)φ2 (Ω00 Ω11 − Ω201 )Ω01 2 2 9(1 + Ω − φ ) 4 1 − (14 + 23Ω + 3Ω2 )φ3 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )Ω01 2 + (1 + Ω) 6(1 + Ω) + (8 + 23Ω + 3Ω2 )φ2 (φ00 φ11 − φ201 )Ω01 1 − φ − 12(1 + Ω) + (26 + 23Ω + 3Ω2 )φ2 (Ω00 Ω11 − Ω201 )φ01 2 + (1 + Ω)2 − 6 + (20 + 3Ω)φ2 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )φ01 2 2 − 2(1 + Ω) (7 + Ω) (2 + 3Ω) φ(φ00 φ11 − φ01 )φ01 , 1 (∂v S3 − S4 ) − eˆ1 01 ∂u S3 − eˆ2 01 ∂v S3 2u 1 + − (1 + Ω) φΩ11 − 2(1 + Ω)φ11 ∂u φ11 1 + Ω − φ2 1 1 2 ∂v φ11 + eˆ 01 ∂u φ11 + eˆ 01 ∂v φ11 + (2 + 3Ω) φΩ01 − 2(1 + Ω)φ01 2u − (1 + 2Ω) φΩ00 − 2(1 + Ω)φ00 eˆ1 11 ∂u φ11 + eˆ2 11 ∂v φ11 + ∂w φ11

∂u S 4 −

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ˆ 0111 S2 + 2Γ ˆ 0101 S3 + Γ ˆ 0100 S4 = −3Γ 1 ˆ 0111 φ01 − Γ ˆ 0101 φ11 + 2(2 + 3Ω) φΩ01 − 2(1 + Ω)φ01 Γ 2 1+Ω−φ ˆ 1111 φ01 − Γ ˆ 1101 φ11 − 2(1 + 2Ω) φΩ00 − 2(1 + Ω)φ00 Γ 1 − 1 + (−1 + Ω)φ2 (Ω01 S3 − Ω00 S4 ) + Ω2 φ(φ01 S3 − φ00 S4 ) 2 1 + Ωφ2 (Ω11 S2 − 2Ω01 S3 + Ω00 S4 ) − Ω(1 + Ω)φ(φ11 S2 − 2φ01 S3 + φ00 S4 ) 2 1 + (1 + Ω)φ φΩ11 − 2(1 + Ω)φ11 R 6 − 2φ Ω11 (Ω01 φ01 − Ω11 φ00 ) + φ11 (Ω00 Ω11 − Ω201 ) + 4(1 + Ω) φ11 (Ω01 φ01 − Ω00 φ11 ) + Ω11 (φ00 φ11 − φ201 ) 1 2 1 φ − 6 + (20 + 3Ω)φ2 (Ω00 Ω11 − Ω201 )Ω11 + 2 2 3(1 + Ω − φ ) 4 1 − (14 + 23Ω + 3Ω2 )φ3 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )Ω11 2 + (1 + Ω) 6(1 + Ω) + (8 + 23Ω + 3Ω2 )φ2 (φ00 φ11 − φ201 )Ω11 1 − φ − 12(1 + Ω) + (26 + 23Ω + 3Ω2 )φ2 (Ω00 Ω11 − Ω201 )φ11 2 + (1 + Ω)2 − 6 + (20 + 3Ω)φ2 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )φ11 2 2 − 2(1 + Ω) (7 + Ω) (2 + 3Ω) φ(φ00 φ11 − φ01 )φ11

5.3. The ∂u -equations hierarchy The system of ∂u -equations splits into two groups, referred to as G1 and G2. Each of these groups splits into a hierarchy, which is defined as follows: G1.1: R000001 = 0, G1.2: t01 EF 00 e2 EF = 0, G1.3: t01 EF 00 e1 EF = 0, R010001 = 0, Σ00 = 0, Σ0000 = 0, Σ0001 = 0, Φ00 = 0, H0000 = 0, G1.4: R110001 = 0, Σ0011 = 0, Φ10 = 0, Π00 = 0, H0001 = 0, G1.5: R000011 = 0, G1.6: R010011 = 0, G1.7: t11 EF 00 e1 EF = 0, G1.8: t11 EF 00 e2 EF = 0, G2.1: H0011 = 0, G2.2: R110011 = 0, G2.3: H0111 = 0.

304

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For dealing with the unknowns we separate them into three groups, x1 , x2 and x3 . The unknowns involved in G1 are collected in x1 , that is x1 = (ˆ e1 01 , eˆ2 01 , ˆ 0100 , Γ ˆ 0101 , Γ ˆ 0111 , Γ ˆ 1100 , Γ ˆ 1101 , Ω, Ω00 , Ω01 , Ω11 , φ01 , φ11 , R, S1 , S2 ). eˆ1 11 , eˆ2 11 , Γ The set x2 consist of the unknowns of x1 plus φ, S0 and φ00 . The unknowns in G2 ˆ 1111 , S3 , S4 ). So all the unknowns are included are collected in x3 , that is x3 = (Γ in the union of x2 and x3 . The defining property of the hierarchy is the following feature. If φ and S0 are prescribed on {w = w0 } then G1.1 reduces to an ODE. Once we have its solution, G1.2 reduces to an ODE. Given its solution, G1.3 reduces to a system of ODE’s, with coefficients that are calculated by operations interior to {w = w0 } from the previously known or calculated functions. This procedure continues till G1.8. So, given φ and S0 on {w = w0 } and the appropriate initial data on U0 ∩ {w = w0 }, the set x1 can be determined on {w = w0 } by solving a sequence of ODE’s in the independent variable u. The process to be followed with G2 is very similar, with the exception that to solve G2.3 it is necessary to know also ∂w φ11 on {w = w0 }, this problem can be overcome solving G1 recursively and then analyzing G2. 5.4. The ‘∂w -equations’ Our initial data, φ and S0 , is prescribed on W0 , and to determine their evolution off W0 we need the equation A11 = 0, which reads ∂w φ + eˆ1 11 ∂u φ + eˆ2 11 ∂v φ = φ11 , and the equation H1(ABC)0 + H0(ABC)1 = 0, which is given by ∂w S0 − ∂u S2 + eˆ1 11 ∂u S0 + eˆ2 11 ∂v S0 1 − (2 + 5Ω) φΩ11 − 2(1 + Ω)φ11 ∂u φ00 2 3(1 + Ω − φ ) − 4(1 + Ω) φΩ01 − 2(1 + Ω)φ01 ∂u φ01 + (2 + Ω) φΩ00 − 2(1 + Ω)φ00 ∂u φ11 1 (∂v φ01 − φ11 ) + eˆ1 01 ∂u φ01 + eˆ2 01 ∂v φ01 − 2Ω φΩ00 − 2(1 + Ω)φ00 2u ˆ ˆ = 4Γ1101 S0 − 4Γ1100 S1 2 1 ˆ 0111 φ00 − Γ ˆ 0100 φ11 Ω φΩ00 − 2(1 + Ω)φ00 Γ + 1 + Ω − φ2 3 1 + 1 + (−1 + Ω)φ2 (Ω11 S0 − Ω00 S2 ) − Ω2 φ(φ11 S0 − φ00 S2 ) 2 2 1 + Ωφ2 (Ω11 S0 − 2Ω01 S1 + Ω00 S2 ) − Ω(1 + Ω)φ(φ11 S0 − 2φ01 S1 + φ00 S2 ) 3 3 1 + (1 + Ω)φ φΩ00 − 2(1 + Ω)φ00 R 9 + 2 φΩ00 − 2(1 + Ω)φ00 (Ω00 φ11 − Ω11 φ00 )

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1 2 2 φ − 6 + (20 + 3Ω)φ2 (Ω00 Ω11 − Ω201 )Ω00 2 2 9(1 + Ω − φ ) 4 1 − (14 + 23Ω + 3Ω2 )φ3 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )Ω00 2 + (1 + Ω) 6(1 + Ω) + (8 + 23Ω + 3Ω2 )φ2 (φ00 φ11 − φ201 )Ω00 1 − φ − 12(1 + Ω) + (26 + 23Ω + 3Ω2 )φ2 (Ω00 Ω11 − Ω201 )φ00 2 + (1 + Ω)2 −6 + (20 + 3Ω)φ2 (Ω00 φ11 − 2Ω01 φ01 + Ω11 φ00 )φ00 2 2 − 2(1 + Ω) (7 + Ω) (2 + 3Ω) φ(φ00 φ11 − φ01 )φ00 . +

These two equations will be referred to as the ‘∂w -equations’. 5.5. The initial conditions for the ∂u -equations The initial conditions for the ∂u -equations follow from our gauge conditions (52), (53) which imply eâ A1 |I = 0 , ˆ A1CD |I = 0 , Γ

a = 1, 2 ,

A = 0, 1 ,

A, C, D = 0, 1 .

From (4) we get Ω|I = 0 , ΩAB |I = 0 ,

A, B = 0, 1 , 2

R|I = −6 − 8∂u φ|I ∂u ∂v2 φ|I + 4 (∂u ∂v φ|I ) , and from the required spinorial behavior in order to have analytic solutions, as discussed in Section 4.3, 1 (56) φA1 |I = ∂u ∂v1+A φ|I , A = 0, 1 , 2 (4 − k)! k ∂v S0 |I , k = 1, 2, 3, 4 , Sk |I = 4! where A00 = 0 has been used. 5.6. Calculating the formal expansion As the system of equations is overdetermined, we have chosen a subsystem in order to calculate a formal expansion of the solution. It will be shown later on that the expansion obtained using this subsystem lead to a formal solution of the full system of equations. We prescribe φ and S0 on W0 as our datum and the initial conditions on I for the ∂u -equations are given in Section 5.5. Following what has been said in Section 5.3 we successively integrate the subsystems on G1 to determine all components of x1 on W0 . We give now an inductive argument involving G1 and the ∂w -equations to k x2 |W0 can be determined for all k. show that ∂w

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k From our initial data and what has been said we know already ∂w x2 |W0 for k = 0. As inductive hypothesis we assume as known p ∂w x2 |W0 ,

0 ≤ p ≤ k − 1,

k ≥ 1.

k−1 ∂w

Applying formally to the ∂w -equations, and restricting them to W0 , we find k k k φ|W0 and ∂w S0 |W0 in terms of known functions. We apply formally ∂w to G1. ∂w k This is a system of PDE’s where the unknowns are ∂w x1 . Keeping the hierarchy and considering the functions that we already know on W0 , it again becomes a sequence of ODE’s, which can be integrated on W0 given the appropriate initial conditions on I. The initial conditions for the frame coefficients and the connection coefficients are obtained from the gauge requirements (52), (53) which imply k a ∂w eˆ A1 |I = 0 , kˆ ΓA1CD |I ∂w

= 0,

a = 1, 2 ,

A = 0, 1 ,

A, C, D = 0, 1 .

From the spinorial behavior as discussed in Section 4.3 we obtain the following set of initial conditions. 1 k k φ01 |I = ∂u ∂v ∂w φ|I , ∂w 2 1 k k ∂w φ11 |I = ∂u ∂v2 ∂w φ|I , 2 1 k k ∂w S1 |I = ∂v ∂w S0 |I , 4 1 2 k k ∂ ∂ S0 |I . ∂w S2 |I = 12 v w By restricting the equations Σ11 = 0, Σ11CD = 0 and Π11 = 0 to U0 and using that Ω|I = 0, ΩAB |I = 0 we get k ∂w Ω|I = 0 , k ΩA1 |I = 0 , A = 0, 1 , ∂w 8 1 2 , ∂w Ω00 |U0 = − R + (φΩ φ − 2φ φ + 2φ ) (57) 00 11 00 11 01 2 3 3(1 − φ ) U0 8 ∂w R|U0 = 3Ω00 S4 + (58) (φΩ00 − 2φ00 )∂w φ11 1 − φ2 1 + 4φ01 ∂w φ01 − 2φ11 ∂w φ00 − φRφ11 3

8 2 2 . + (3 + 11φ φ )Ω φ − 28φ(φ φ − φ ) 11 00 11 00 11 01 3(1 − φ2 )2 U0 k−1 to (57), (58) and evaluating them at I by using the known Applying ∂w functions from the inductive hypothesis and the previously stated initial conditions k k Ω00 |I and ∂w R|I . we get ∂w

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Now we have all the needed initial conditions, thus we know k x2 |W0 ∂w

and the induction step is completed. k The procedure with G2 is quite similar. Once we know ∂w x2 |W0 for all k, G2.1 reduces to an ODE, which can be integrated on W0 given the corresponding initial condition. Once we know the solution of G2.1, G2.2 also reduces to an ODE, and finally also G2.3 reduces to an ODE. The initial conditions for G2 are given in Section 5.5. The inductive step is very similar to the inductive step for x2 . We assume p x3 |W0 , ∂w

0 ≤ p ≤ k − 1,

k ≥ 1,

k ∂w

to the equations in G2. If we stick to the to be known. We apply formally hierarchy this system again reduces in the prescribed order to a system of ODE’s k x3 , which can be integrated given the corresponding initial conditions. Those for ∂w are ˆ 1111 |I = 0 , ∂k Γ w

1 3 k ∂ ∂ S0 |I , 24 v w 1 4 k k ∂ ∂ S0 |I , ∂w S4 |I = 24 v w obtained from (53) and Section 4.3. Now we know k ∂w x3 |W0 and the induction step is complete. If we now call X any of the quantities included in x2 and x3 , that is, X comprises all the unknown quantities that we are solving for, the procedure just k X|W0 for all k. Expanding these functions around stated shows that we know ∂w i = {u = 0, v = 0, w = 0} gives k S3 |I = ∂w

p X|i ∂um ∂vn ∂w

∀ m, n, p ,

and the procedure gives a unique sequence of expansion coefficients for all the functions in X. Lemma 5.1. The procedure described above determines at the point O = (u = 0, v = 0, w = 0) from the data φ, S0 , given on W0 according to (54), (55), a unique sequence of expansion coefficients p f (O) , ∂um ∂vn ∂w

m, n, p = 0, 1, 2, . . . ,

ˆ ABCD ,φ,φAB ,Ω,ΩAB ,R,Sk . where f stands for any of the functions eâ AB ,Γ If the corresponding Taylor series are absolutely convergent in some neighborhood P of O, they define a solution to the equation A00 = 0, to the ∂u -equations and to the ∂w -equations on P which satisfies on P ∩U0 equations (56) and Σ11 = 0, Σ11CD = 0, Π11 = 0.

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By Lemma 4.1 we know that all spinor-valued functions should have a specific v-finite expansion type. The following lemma, whose proof is quite similar to the proof in [8], will be important for the convergence proof. Lemma 5.2. If the data φ, S0 are given on W0 as in (54), (55), the formal expansions of the fields obtained in Lemma 5.1 correspond to ones of functions of v-finite expansion types given by keˆ1 AB = −A − B ,

keˆ2 AB = 3 − A − B ,

kΓˆ 01AB = 2 − A − B ,

AB = 01, 11 ,

kΓˆ 11AB = 1 − A − B ,

A, B = 0, 1 ,

kφ = 0 ,

kφAB = 2 − A − B ,

A, B = 0, 1 ,

kΩ = 0 ,

kΩAB = 2 − A − B ,

A, B = 0, 1 ,

kR = 0 , kSj = 4 − j ,

j = 0, 1, 2, 3, 4 .

6. Convergence of the formal expansion In the previous section we have seen how to calculate a formal expansion for eâ AB , ˆ ABCD , φ, φAB , Ω, ΩAB , R, Sk given φ|W and S0 |W , or, what is the same, given Γ 0 0 the null data. From Lemma 3.2 we know which are the necessary conditions on the null data in order to have analytic solutions of the conformal field equations. In this section we show that those conditions, (38) and (39), are also sufficient for the formal expansion determined in the previous section to be absolutely convergent. So we start considering the abstract null data as given by two sequences ˆ nφ = {ψA B , ψA B A B , ψA B A B A B , . . .} , D 1

1

2

2

1

1

3

3

2

2

1

1

ˆ nS = {ΨA B A B , ΨA B A B A B , ΨA B A A B A B A B , . . .} , D 2 2 1 1 3 3 2 2 1 1 4 4 2 3 3 2 2 1 1 of totally symmetric spinors satisfying the reality condition (30) and we construct φ|W0 and S0 |W0 , by setting in the expansions (54), (55) ∗ ∗ . . . DA φ(i) = ψAm Bm ...A1 B1 , D(A m Bm 1 B1 ) ∗ D(A m Bm

∗ . . . DA S∗ (i) 1 B1 ABCD)

m ≥ 1,

= ΨAm Bm ...A1 B1 ABCD ,

m ≥ 0.

Observing Lemma 3.2, one finds as a necessary condition for the functions φ, S0 on W0 to determine an analytic solution to the conformal static vacuum field equations that its non-vanishing Taylor coefficients at the point O satisfy estimates of the form 2m M m!n! m , m ≥ 0 , 0 ≤ n ≤ 2m , (59) |∂um ∂vn φ(0)| ≤ n r 2m + 4 M m!n! m , m ≥ 0 , 0 ≤ n ≤ 2m + 4 . (60) |∂um ∂vn S0 (0)| ≤ n r This conditions are also sufficient for φ(u, v) and S0 (u, v) to be holomorphic functions on W0 . So the null data gives rise to two analytic functions, φ and S0 , on W0 .

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309

From A00 = 0 we have φ00 = ∂u φ, so having φ|W0 we have φ00 |W0 , which is also an analytic function on W0 . Following Lemma 6.1 in [8], we can derive from (59), (60), slightly different type of estimates for φ(u, v), S0 (u, v), which are more convenient in our case. Lemma 6.1. Let e be the Euler number. For given ρφ , ρS0 , both in R, such that 0 < ρφ < e2 , 0 < ρS0 < e2 , there exist positive constants c˜φ , rφ , c˜S0 , rS0 , so that (59), (60), imply estimates of the form |∂um ∂vn φ| ≤ c˜φ

rφm−1 m!ρnφ n!

, m ≥ 0 , 0 ≤ n ≤ 2m , (m + 1)2 (n + 1)2 rSm0 m!ρnS0 n! , m ≥ 0 , 0 ≤ n ≤ 2m + 4 . |∂um ∂vn S0 | ≤ c˜S0 (m + 1)2 (n + 1)2

(61) (62)

We can present our estimates. Lemma 6.2. Assume φ = φ(u, v), S0 = S0 (u, v) are holomorphic functions defined on some open neighborhood U of O = {u = 0, v = 0, w = 0} in W0 = {w = 0} which have expansions of the form φ(u, v) =

2m ∞

ψm,n um v n ,

m=0 n=0

S0 (u, v) =

∞ 2m+4

Ψm,n um v n ,

m=0 n=0

so that its Taylor coefficients at the point O satisfy estimates of the type (61), (62) with some positive constants c˜φ , rφ , c˜S0 , rS0 , and ρφ < 13 , ρS0 < 13 . Then there exist positive constants r, ρ, ceâ AB , cΓˆ ABCD , cφ , cφAB , cΩ , cΩAB , cR , cSk so that the expansion coefficients determined from φ and S0 in Lemma 5.1 satisfy for m, n, p = 0, 1, 2, . . . p |∂um ∂vn ∂w f (O)| ≤ cf

rm+p+qf (m + p)!ρn n! , (m + 1)2 (n + 1)2 (p + 1)2

(63)

where f stands for any of the functions ˆ ABCD , φ, φAB , Ω, ΩAB , R, Sk , eâ AB , Γ and qeâ AB = qΓˆ ABCD = qφ = qΩ = qΩAB = −1 ,

qφAB = qR = qSk = 0 .

Remark. Taking into account the v-finite expansion types of the functions f (Lemma 5.2), we can replace the right hand sides in the estimates above by zero if n is large enough relative to m. This will not be pointed out at each step and for convenience the estimates will be written as above.

310

A. E. Ace˜ na


We take the following four lemmas from [8]. The first states the necessary part of the estimates, and the other three are needed in order to manipulate the estimates in the proof of Lemma 6.2. Lemma 6.3. If f is holomorphic near O, there exist positive constants c, r0 , ρ0 such that p |∂um ∂vn ∂w f (O)| ≤ c

rm+p (m + p)!ρn n! , (m + 1)2 (n + 1)2 (p + 1)2

m, n, p = 0, 1, 2, . . .

for any r ≥ r0 , ρ ≥ ρ0 . If in addition f (0, v, 0) = 0, the constants can be chosen such that p f (O)| ≤ c |∂um ∂vn ∂w

rm+p−1 (m + p)!ρn n! , (m + 1)2 (n + 1)2 (p + 1)2

m, n, p = 0, 1, 2, . . .

for any r ≥ r0 , ρ ≥ ρ0 . Lemma 6.4. For any non-negative integer n there is a positive constant C, C > 1, independent of n so that n k=0

1 1 ≤C . (k + 1)2 (n − k + 1)2 (n + 1)2

In the following C will always denote the constant above. Lemma 6.5. For any integers m, n, k, j, with 0 ≤ k ≤ m, and 0 ≤ j ≤ n resp. 0 ≤ j ≤ n − 1 holds m+n m n−1 m+n m n ≤ resp. ≤ . k j k+j k j k+j Lemma 6.6. Let m, n, p be non-negative integers and fi , i = 1, . . . , N , be smooth complex valued functions of u, v, w on some neighborhood U of O whose derivatives satisfy on U (resp. at a given point p ∈ U ) estimates of the form l fi | ≤ ci |∂uj ∂vk ∂w

rj+l+qi (j + l)!ρk k! (j + 1)2 (k + 1)2 (l + 1)2

for 0 ≤ j ≤ m, 0 ≤ k ≤ n, 0 ≤ l ≤ p, with some positive constants ci , r, ρ and some fixed integers qi (independent of j, k, l). Then one has on U (resp. at p) the estimates p |∂um ∂vn ∂w (f1 · · · · · fN )| ≤ C 3(N −1) c1 · · · · · cN

rm+p+q1 +...qN (m + p)!ρn n! . (64) (m + 1)2 (n + 1)2 (p + 1)2

Remark. This lemma remains true if m, n, p are replaced in (64) by integers m , n , p with 0 ≤ m ≤ m, 0 ≤ n ≤ n, 0 ≤ p ≤ p. The factor C 3(N −1) in (64) can be replaced by C (3−s)(N −1) if s of the integers m, n, p vanish.

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Proof of Lemma 6.2. The proof is by induction, following the procedure which led to Lemma 5.1. A general outline is as follows. We start leaving the choice of the constants r, ρ, cf , open. We use the induction hypothesis and the equations that lead to Lemma 5.1 to derive estimates for the derivatives of the next order. These estimates are of the form rm+p+qf (m + p)!ρn n! p |∂um ∂vn ∂w f (O)| ≤ cf Af , (65) (m + 1)2 (n + 1)2 (p + 1)2 with certain constants Af which depend on m, n, p and the constants cf , r and ρ. Sometimes superscripts will indicate to which order of differentiability particular constants Af refer. In the way we will have to make assumptions on r to proceed with the induction step. We shall collect these conditions and the constants Af , or estimates for them, and at the end it will be shown that the constants cf , r and ρ can be adjusted so that all conditions are satisfied and Af ≤ 1. This will complete the induction proof. In order not to write long formulas that do not add to the understanding of the procedure, we state here some properties that are used to simplify the estimates: • As a corollary of Lemma 6.6 we have: If rj+l−1 (j + l)!ρk k! l |∂uj ∂vk ∂w g| ≤ cg (j + 1)2 (k + 1)2 (l + 1)2 for 0 ≤ j ≤ m, 0 ≤ k ≤ n, 0 ≤ l ≤ p, where g is φ or Ω, and if r > C3 2 2 12 2 [cΩ + (cΩ + 4cφ ) ], then m n p 1 rm+p (m + p)!ρn n! 1 ∂u ∂v ∂w ≤ 1

. 1 + Ω − φ2 C 3 1 − C 3 c + C 3 c2 (m + 1)2 (n + 1)2 (p + 1)2 Ω r r φ 1

• If r ≥ C 3 [cΩ + (c2Ω + 2c2φ ) 2 ] then 1−

C3 r

1 cΩ +

C3 2 r cφ

≤ 2.

(66)

• After calculating the estimates and using (66) we find that all the A’s satisfy inequalities of the form A≤α+

9 αi i=1

ri

,

where α, αi are constants that do not depend on r. If αi = 0 then we have to show that we can make α ≤ 1. If the αi ’s are not zero we can take a constant a, 0 < a8 < 1, and require that α ≤ a and then choose r large enough such that i=1 αrii ≤ 1 − a. In the estimates that follows, we shall not write the explicit expressions for the αi ’s, as they do not play any role if we are able to make r big enough at the end of the procedure.

312

A. E. Ace˜ na


From now on we consider that a function in a modulus sign is evaluated at the origin O. From the analyticity of φ00 (u, v) we also get that, for given ρφ00 ∈ R, 0 < ρφ00 < 13 , there exist positive constants cφ00 , rφ00 , such that |∂um ∂vn φ00 | ≤ cφ00

rφm00 m!ρnφ00 n! , (m + 1)2 (n + 1)2

m ≥ 0,

0 ≤ n ≤ 2m + 2 .

As φ(0, v) = 0 the inequalities (61), (62) are maintained if we change the constants for bigger constants. We choose cφ = max{˜ cφ , cφ00 } , 64 cS0 = max c˜S0 , C 3 c2φ00 . 3

(67) (68)

Also we require the constants r, ρ to satisfy r ≥ max{rφ , rS0 , rφ00 } , ρ ≥ max{ρφ , ρS0 , ρφ00 } ,

(69)

but we leave the choice of the precise value open. So we have rm−1 m!ρn n! , m ≥ 0 , 0 ≤ n ≤ 2m , (m + 1)2 (n + 1)2 rm m!ρn n! 0 |∂um ∂vn ∂w S0 | ≤ cS0 , m ≥ 0 , 0 ≤ n ≤ 2m + 4 , (m + 1)2 (n + 1)2 rm m!ρn n! 0 |∂um ∂vn ∂w φ00 | ≤ cφ00 , m ≥ 0 , 0 ≤ n ≤ 2m + 2 . (m + 1)2 (n + 1)2 0 |∂um ∂vn ∂w φ| ≤ cφ

ˆ ABCD |U = 0 follows From the frame properties eâ AB |U0 = 0, Γ 0 pˆ ΓABCD | = 0 , |∂u0 ∂vn ∂w

p a |∂u0 ∂vn ∂w eˆ AB | = 0 .

The conditions on the conformal factor, Ω|I = 0, ΩAB |I = 0, give 0 |∂u0 ∂vn ∂w Ω| = 0 ,

0 |∂u0 ∂vn ∂w ΩAB | = 0 .

Using Lemma 4.1 we get the relations: 1 1+A ∂ φ00 |U0 , A = 0, 1 , 2 v (4 − k)! k ∂v S0 |U0 , k = 1, 2, 3, 4 , = 4!

φA1 |U0 = Sk |U0

(70) (71)

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which imply

0 φA1 | |∂u0 ∂vn ∂w

≤

ρn+1+A (n+1+A)! 1 , 2 cφ00 (n+2+A)2

0,

n≤1−A n>1−A

313

n

ρ n! Am=0,p=0 , (n + 1)2 φA1 1 cφ00 1+A 1 cφ00 1+A = ρ hA,n ≤ ρ , 2 cφA1 2 cφA1 (n+1+A)! (n+1)2 , 0 ≤ n ≤ 1 − A 2 n! (n+2+A) = ≤ 1, 0, n>1−A

= cφA1 Am=0,p=0 φA1 hA,n

(72)

and similarly 0 |∂u0 ∂vn ∂w Sk | ≤ cSk

Am=0,p=0 ≤ Sk

ρn n! Am=0,p=0 , (n + 1)2 Sk

cS0 k ρ . cSk

(73)

Taking into account that R is a scalar and the initial condition R|i = −6 − 16 φ00 φ11 − φ201 |i , we get 6 + 73 ρ2 c2φ00 , n = 0 0 n 0 36 |∂u ∂v ∂w R| ≤ 0, n>0 n ρ n! Am=0,p=0 , = cR (n + 1)2 R 1 73 2 2 ρ Am=0,p=0 ≤ c 6 + φ00 . R cR 36 We have obtained so far the estimates for m = 0, p = 0 and general n. Now we should consider the equations in G1 to get in an inductive form estimates 0 x1 |, considering as for the quantities in x1 , that means, estimates for |∂um ∂vn ∂w l n 0 known estimates of this type for |∂u ∂v ∂w x1 | with 0 ≤ l < m. And once we have this estimates we should do the same procedure with G2 to get estimates for 0 x3 |. These estimates, i.e. estimates for p = 0, can be obtained from the |∂um ∂vn ∂w estimates for general p that appears later replacing C 3 by C 2 and p by 0. The estimates for general p are also more restrictive, so we do not enumerate the estimates for p = 0 here. We continue with the induction procedure by considering that the estimates l X| for 0 ≤ l < p, and try to determine conditions for are satisfied for |∂um ∂vn ∂w performing the induction step. p−1 to the equation A11 = 0 and taking We start by formally applying ∂um ∂vn ∂w the modulus at the origin. We get p p−1 p−1 1 p−1 2 φ| ≤ |∂um ∂vn ∂w φ11 | + |∂um ∂vn ∂w (ˆ e 11 ∂u φ)| + |∂um ∂vn ∂w (ˆ e 11 ∂v φ)| . |∂um ∂vn ∂w

314

A. E. Ace˜ na


To estimate the terms in the r.h.s. of this inequality we have, using the induction hypothesis, p−1 |∂um ∂vn ∂w φ11 | ≤ cφ11

rm+p−1 (m + p − 1)!ρn n! , (m + 1)2 (n + 1)2 p2

p−1 1 |∂um ∂vn ∂w (ˆ e 11 ∂u φ)|

p−1 m n m n p−1 l 1 p−l−1 ≤ eˆ 11 ||∂um−j+1 ∂vn−k ∂w φ| |∂uj ∂vk ∂w j k l j=0 k=0 l=0 p−1 m p−1 m n j m+pl ≤ j=0 k=0 l=0

j+l

ceˆ1 11 cφ rm+p−2 (m + p)!ρn n! (j + + 1)2 (l + 1)2 (m − j + 2)2 (n − k + 1)2 (p − l)2 rm+p−2 (m + p)!ρn n! ≤ C 3 ceˆ1 11 cφ , (m + 2)2 (n + 1)2 p2 ×

1)2 (k

and similarly p−1 2 |∂um ∂vn ∂w (ˆ e 11 ∂v φ)| ≤ C 3 ceˆ2 11 cφ

rm+p−3 (m + p − 1)!ρn (n + 1)! . (m + 1)2 (n + 2)2 p2

p Using these inequalities and writing |∂um ∂vn ∂w φ| in the form (65), we obtain 2 1 (p + 1) (p + 1)2 (m + 1)2 C 3 c ceˆ1 11 cφ Ap≥1 = + φ 11 φ cφ p2 (m + p) p2 (m + 2)2 r (p + 1)2 (n + 1)3 C 3 ρceˆ2 11 cφ . p2 (m + p)(n + 2)2 r2

Taking into account the v-finite expansion types of the terms involved, we see that = 0 if n > 2m, and thus Ap≥1 φ p≥1 9 4 C3 C3 cφ11 (αφ )i 1 2 c Ap≥1 ≤ + c + 2 ρc c + . c = 4 φ11 eˆ 11 φ eˆ 11 φ φ cφ r r2 cφ ri i=1 The procedure with the rest of the equations is similar to the one presented for the equation A11 = 0, the only difference being that if an equation is singular with p−1 , and then u−1 terms we have first to multiply it by u, formally apply ∂um+1 ∂vn ∂w estimate the modulus. Therefore we shall not repeat the details that led from the equations to the estimates, as we shall not state the v-finite expansion type at each step. What we will state is which equation is used for deriving that particular estimate. p−1 to the equation D11 φ00 = D00 φ11 , which folApplying formally ∂um ∂vn ∂w lows from A00 = 0 and A11 = 0, we obtain Ap≥1 φ00

p≥1 9 cφ11 (αφ00 )i ≤4 + . cφ00 ri i=1

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315

p−1 Multiplying H1(ABC)0 + H0(ABC)1 = 0 by u and formally applying ∂um+1 ∂vn ∂w we get

Ap≥1 S0 ≤

9 (αSp≥1 )i 4 16 0 . cS2 + C 3 (cφ00 cφ11 + c2φ01 ) + i cS0 3 r i=1

In the same way as we used (70), (71) to obtain (72), (73) we get 1 cφ00 1+A ρ , 2 cφA1 cS ≤ 0 ρk . cSk

Am=0,p≥1 ≤ φA1 Am=0,p≥1 Sk

Restricting Σ11 = 0 and Σ11CD = 0 to U0 we find that on U0 Ω = 0 , Ω01 = 0 , Ω11 = 0 , 8 1 (φΩ00 φ11 − 2φ00 φ11 + 2φ201 ) . ∂w Ω00 = − R + 3 3(1 − φ2 ) Taking formal derivatives of these equations we get Am=0 = 0, Ω

Am=0 Ω01 = 0 ,

Am=0 Ω11 = 0 ,

and ≤ Am=0,p≥1 Ω00

m=0,p≥1 9 (αΩ )i 4 1 00 cR + 32C 2 (cφ00 cφ11 + c2φ01 ) + . i 3 cΩ00 r i=1

Restricting Π11 = 0 to U0 gives 8 ∂w R = 3Ω00 S4 + − 2φ11 ∂w φ00 + 4φ01 ∂w φ01 + (φΩ00 − 2φ00 )∂w φ11 1 − φ2 1 8 − φRφ11 + φ11 (3 + 11φ2 )Ω00 φ11 − 28φ(φ00 φ11 − φ201 ) , 2 2 3 3(1 − φ ) so that Am=0,p≥1 R

9 m=0,p≥1 (αR )i 64C 2 2 ≤ (cφ00 cφ11 + cφ01 ) + . i cR r i=1

We complete the calculation of the A’s by using the ∂u -equations. We have to calculate the estimates in the order given by the hierarchy presented in Section 5.3 but for simplicity we present the estimates in the order the ∂u -equations were stated in Section 5.2.

316

tAB EF

A. E. Ace˜ na a 00 e EF


= 0: Am≥1 eˆ1 01 ≤

9 (αem≥1 )i ˆ1 01

,

ri

i=1

1 cΓˆ 0100 (αeˆ2 01 )i ≤ + , 2 ceˆ2 01 ri i=1 m≥1

9

Am≥1 eˆ2 01

Am≥1 eˆ1 11 ≤

9 (αem≥1 )i ˆ1 11

,

ri

i=1

9 (αem≥1 cΓˆ 1100 ˆ2 11 )i + . ceˆ2 11 ri i=1

Am≥1 eˆ2 11 ≤ RAB00EF = 0:

m≥1

)i (αΓˆ 2 cS0 0100 + , i 3 cΓˆ 0100 r i=1 9

Am≥1 ≤ ˆ Γ 0100

≤ Am≥1 ˆ Γ 0101

≤ Am≥1 ˆ Γ 0111

cS1 cΓˆ 0101

1100

≤ Am≥1 ˆ Γ 1101

9 (αm≥1 ) i ˆ Γ 0101

ri

i=1

cΓˆ 0111

+

1111

m≥1

)i (αΓˆ cR 0111 + , i 6cΓˆ 0111 r i=1

cS1 cΓˆ 1100

+

4 cΓˆ 1101

9 (αm≥1 ) i ˆ Γ 1100

ri

i=1

cS2 +

1 cR 12

,

+

9 (αm≥1 ) i ˆ Γ 1101

i=1

m≥1

)i (αΓˆ 4cS3 1111 + . i cΓˆ 1111 r i=1 9

≤ Am≥1 ˆ Γ

,

9

cS2

≤2 Am≥1 ˆ Γ

+

Σ00 = 0: Am≥1 ≤ Ω

9 (αm≥1 )i Ω

i=1

ri

.

Φ00 = 0: Am≥1 φ01 ≤

9 (αφm≥1 )i cφ00 01 ρ+ . i cφ01 r i=1

Φ10 = 0: Am≥1 φ11

9 (αφm≥1 )i cφ01 11 ≤ ρ+ . i cφ11 r i=1

ri

,

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317

Σ00CD = 0: Am≥1 Ω00 ≤

m≥1 9 (αΩ )i 00

i=1

Am≥1 Ω01

≤

m≥1 9 (αΩ )i 01

i=1

Am≥1 Ω11 ≤

ri ri

,

,

m≥1 9 (αΩ )i 4 11 cR + 32C 3 (cφ00 cφ11 + c2φ01 ) + . i 3cΩ11 r i=1

Π00 = 0: Am≥1 R

9 m≥1 (αR )i 64C 3 2 ≤ (cφ00 cφ11 + cφ01 ) + . i cR r i=1

H0(ABC)k = 0: 9 (αSm≥1 )i cS0 1 ρ+ , i cS1 r i=1 9 (αSm≥1 )i 1 8 3 2 2 ≤ , ρcS1 + C (cφ00 cφ11 + cφ01 ) + i cS2 3 r i=1 9 (αSm≥1 )i 1 8 3 3 ≤ , ρcS2 + C 3 ρ(cφ00 cφ11 + 2c2φ01 ) + cφ01 cφ11 + i cS3 3 2 r i=1 9 (αSm≥1 )i 1 8 3 4 ≤ . ρcS3 + 4C cφ11 2ρcφ01 + cφ00 + cφ11 + i cS4 3 r i=1

Am≥1 ≤ S1 Am≥1 S2 Am≥1 S3 Am≥1 S4

We now have to show that all the constants can be chosen in a way that makes all the A’s less or equal than 1. So, introducing a constant a, 0 < a < 1, the following inequalities need to be satisfied:

4

cφ11 ≤ a, cφ

1 cφ00 ρ ≤ 1, 2 cφ01 1 cφ00 2 ρ ≤ 1, 2 cφ11 cS0 cS0 2 cS0 3 cS0 4 ρ ≤ 1, ρ ≤ 1, ρ ≤ 1, ρ ≤ 1, cS1 cS2 cS3 cS4 1 73 2 2 6 + ρ cφ00 ≤ 1 , cR 36 cφ11 4 16 3 2 4 ≤ a, cS2 + C (cφ00 cφ11 + cφ01 ) ≤ a , cφ00 cS0 3 4 1 cR + 32C 2 (cφ00 cφ11 + c2φ01 ) ≤ a , 3 cΩ00

(74) (75) (76) (77) (78) (79)

318

A. E. Ace˜ na


1 64C 2 (cφ00 cφ11 + c2φ01 ) ≤ a , cR cΓˆ 1100 1 cΓˆ 0100 2 cS0 cS1 ≤ a, ≤ a, ≤ a, ≤ a, 2 ceˆ2 01 ceˆ2 11 3 cΓˆ 0100 cΓˆ 0101 1 1 cS1 ≤ a, cS2 + cR ≤ a , 2 cΓˆ 0111 6 cΓˆ 1100 4 1 cS3 ≤ a, cS2 + cR ≤ a , 4 cΓˆ 1101 12 cΓˆ 1111 cφ00 ρ ≤ a, cφ01 cφ01 ρ ≤ a, cφ11 4 1 cR + 32C 3 (cφ00 cφ11 + c2φ01 ) ≤ a , 3 cΩ11 1 64C 3 (cφ00 cφ11 + c2φ01 ) ≤ a , cR cS0 1 8 3 2 ρ ≤ a, cS1 ρ + C (cφ00 cφ11 + cφ01 ) ≤ a , cS1 cS2 3 1 4 3 2 cS2 ρ + C (2ρcφ00 cφ11 + 4ρcφ01 + 3cφ01 cφ11 ) ≤ a , cS3 3 1 4 3 cS3 ρ + C cφ11 (8cφ00 + 6ρcφ01 + 3cφ11 ) ≤ a . cS4 3

(80) (81) (82) (83) (84) (85) (86) (87) (88) (89) (90)

Now we have to show that we can choose the constants such that these inequalities will be satisfied. We start by setting ρ cφ01 ≡ cφ00 , a with which we satisfy (74) and (84). Next we set cφ11 ≡

ρ2 cφ , a2 00

so that (75) and (85) are satisfied. We continue by setting cS1 cS3 cS4

ρ ρ2 ≡ cS0 , cS2 ≡ 2 a a ρ3 8 ≡ 3 cS0 + 3+ a 3 ρ4 8 ≡ 4 cS0 + 6+ a 3

16 C 3 2 c cS0 + , 3 a φ00 7 C 3 c2φ00 , 2a a 5 + 4 2 C 3 c2φ00 . a ρ

With this we satisfy (76), (88), (89) and (90).

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319

Inequalities (77), (80) and (87) are satisfied with ρ2 73 cR ≡ max 128 3 C 3 c2φ00 , 6 + ρ2 c2φ00 . a 36 With this definition for cR we set 2 4 2ρ 2 cΩ00 ≡ cR + 64C 2 cφ00 , 3a a

cΩ11

4 ≡ 3a

cR +

ρ2 64C 3 2 c2φ00 a

,

so (79) and (86) are respectively satisfied. Using the previous definitions we set also 2 1 cΓˆ 0100 ≡ cS0 , cΓˆ 0101 ≡ cS1 , 3a a 1 1 2 cΓˆ 0111 ≡ cS2 + cR , cΓˆ 1100 ≡ cS1 , a 6 a 4 1 4 cΓˆ 1101 ≡ cS2 + cR , cΓˆ 1111 ≡ cS3 , a 12 a 1 2 ceˆ2 01 ≡ 2 cS0 , ceˆ2 11 ≡ 2 cS1 , 3a a and (81), (82) and (83) are satisfied. There are three inequalities that we have not yet considered, (78). These are now reduced to cφ 4ρ2 00 ≤ a3 , cφ 4ρ2 ≤ a3 , 1 16 3 2 1 2 4ρ 1 + C cφ00 2 + ≤ a3 . cS0 3 a Taking into consideration now (67), (68), (69) we see that these inequalities can be satisfied if we define 1 ρ ≡ max{ρφ , ρS0 , ρφ00 } < , 3 1 1 , (8ρ2 ) 3 < 1 . a ≡ max 2 Now we choose some positive constants cΩ , cΩ01 , ceˆ1 01 , ceˆ1 11 , that are not restricted by the procedure. Finally we choose r so large that 1 r > max rφ , rS0 , rφ00 , C 3 cΩ + (c2Ω + 2c2φ ) 2 and that all the A’s are less or equal than 1. The induction proof is completed. The following lemma states the convergence result. The proof follows as the one given in [8].

320

A. E. Ace˜ na


Lemma 6.7. The estimates (63) for the derivatives of the functions f and the expansion types given in Lemma 5.2 imply that the associated Taylor series are 2 1 , |u| + |w| < αr , for any real number absolutely convergent in the domain |v| < αρ α, 0 < α ≤ 1. It follows that the formal expansion determined in Lemma 5.1 defines indeed a (unique) holomorphic solution to the conformal static vacuum field equations which induces the data φ, S0 on W0 .

ˆ 7. The complete set of equations on N We have seen in Section 5 how to calculate a formal expansion for our fields using a subset of the conformal stationary vacuum field equations. In the previous section we have shown that these formal expansions are convergent in a neighborhood of infinity. In this section we shall show that these fields satisfy the complete system of conformal stationary vacuum field equations. First, we prove that the conformal stationary vacuum field equations are satisfied in the limit as u → 0. Second, we derive a subsidiary system of equations, for which the first result provides the initial conditions, and which allows us to prove that the complete system is satisfied. ˆ ABCD , φ, φAB , Ω, ΩAB , R, Sk , whose exLemma 7.1. The functions eâ AB , Γ pansion coefficients are determined by Lemma 5.1, with expansions that converge on an open neighborhood of the point 0, neighborhood that we assume to coinˆ , satisfy the complete set of conformal field equations on the set U0 in cide with N the sense that the fields tAB CD EF , RABCDEF , AAB , ΣAB , ΦAB , ΠAB , ΣABCD , ˆ \U0 have vanishing limit as u → 0. HABCD calculated from these functions on N Proof. Taking into account which equations have already been used to determine the formal expansions, and the symmetries of the equations, it is left to show that t01 EF 11 , RAB0111 , A01 , Σ01 , Π01 , Σ01CD , H1(BCD)k=1,2,3 , have vanishing limit ˆ \U0 as u → 0, and that in the same limit ΦAB = −ΦBA . on N Because σ AB , eEF = hAB EF then t01 EF 11 = 2Γ01 (E 1 1 F ) − 2Γ11 (E (0 1) F ) − σ EF a ea 11,b eb 01 − ea 01,b eb 11 , and using the way in which the coordinates and the frame field were constructed, we see that t01 EF 11 = O(u) , as u → 0 . We now consider RAB0111

1 1ˆ 1 ˆ 0 ∂v Γ11AB − Γ SAB11 − RA1 B1 + 111(A B) 6 2u u 1ˆ 1 1 1 0 0 01 + A B − 2 eˆ 11 + Γ0111 − t01 11 + O(u) . 2u u u

1 =− 2

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Using that ˆ 0111 − 1 ∂v eˆ2 11 − 1 eˆ1 11 + O(u2 ) t01 01 11 = Γ 2 2u we get RAB0111 =

1 ˆ 11AB − 2Γ ˆ 111(A B) 0 + A 0 B 0 − 2 eˆ1 11 − ∂v eˆ2 11 + 4Γ ˆ 0111 ∂v Γ 2u u 1 1 − SAB11 − RA1 B1 + O(u) , 2 6

so that lim RAB0111 =

u→0

1 ˆ 11AB − 2∂u Γ ˆ 111(A B) 0 ∂u ∂v Γ 2 ˆ 0111 ) + A 0 B 0 (−∂u2 eˆ1 11 − ∂u ∂v eˆ2 11 + 4∂u Γ 1 − SAB11 + RA1 B1 . 6 u=0

For the case A = B = 0 we get from the ∂u -equations that ˆ 1101 , ∂u2 eˆ1 11 = −2∂u Γ

ˆ 1100 , ∂u ∂v eˆ2 11 = ∂u ∂v Γ

ˆ 0111 = ∂u Γ

1 4

1 S2 − R , 6

on U0 , and so limu→0 R000111 = 0. Using the ∂u -equations and that ∂v S2 = 2S3 on U0 , ˆ 1111 = S3 , ∂u Γ

ˆ 1101 = 2S3 , ∂u ∂v Γ

on U0 , and so limu→0 R010111 = 0. As ∂v S3 = S4 on U0 , limu→0 R110111 = 0. We take now the limit of A01 as u goes to 0, lim A01 =

u→0

1 ∂u ∂v φ − φ01 . 2 u=0

Using that φ01 = 12 ∂v φ00 on U0 and that we have A00 = 0 as part of the ∂u equations we get limu→0 A01 = 0. With the same procedure we get limu→0 Σ01 = 0. Now we consider ΦAB . As AAB = 0 on U0 then DP

B φAP |U0

= −DP A φBP |U0 ,

so ΦAB |U0 = −ΦBA |U0 and as we already have Φ10 = 0 then ΦAB = 0 on U0 .

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We now take the limit as u goes to 0 of the combination Π01 − 12 ∂v Π00 . For the limits of the derivatives involved we have at {u = 0} D00 φAB = ∂u φAB , 1 D01 φ01 = (∂u ∂v φ01 − ∂u φ11 ) , 2 1 D01 φ11 = ∂u ∂v φ11 , 2 D11 φ11 = ∂w φ11 , D00 R = ∂u R , 1 D01 R = ∂u ∂v R , 2 D11 R = ∂w R . We also use that on U0 ∂v φk = (2 − k)φk+1 , ∂v Sk = (4 − k)Sk+1 , We have already used the equations Σ11 = 0, Σ11CD = 0 restricted to U0 , finding that Ω, ΩA1 are zero on U0 . Furthermore we use Φ00 = 0, that says that on U0 ∂u ∂v φ00 = 4∂u φ01 . So we get for the limit

1 lim Π01 − ∂v Π00 = 0 . u→0 2 Considering that from the ∂u -equations we already have Π00 = 0 we get lim Π01 = 0 .

u→0

We apply a similar procedure to the Σ01AB equations. We take the limit as u goes to zero of the combinations Σ0100 − Σ0001 , 2Σ0101 − ∂v Σ0001 + Σ0011 , 2Σ0111 − ∂v Σ0011 . Using what has already been said together with the following limits at {u = 0} D00 ΩAB = ∂u ΩAB , 1 D01 Ω01 = (∂u ∂v Ω01 − ∂u Ω11 ) , 2 1 D01 Ω11 = ∂u ∂v Ω11 , 2 D11 Ω11 = ∂w Ω11 ,

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we see that the limits vanishes, which imply lim Σ01AB = 0 .

u→0

Finally we consider the limit as u goes to zero of the combinations 4H1(ABC)1 − ∂v H1(ABC)0 − ∂v H0(ABC)1 + 2H0(ABC)2 , 12H1(ABC)2 − ∂v2 H1(ABC)0 − ∂v2 H0(ABC)1 − 2∂v H0(ABC)2 + 4H0(ABC)3 , 24H1(ABC)3 − ∂v3 H1(ABC)0 − ∂v3 H0(ABC)1 − 2∂v2 H0(ABC)2 − 8∂v H0(ABC)3 , and using what has been said together with: the limits D00 Sk = ∂u Sk , 1 D01 Sk = ∂u ∂v Sk − (4 − k)∂u Sk+1 , 2 D11 Sk = ∂w Sk , the equality on U0 ∂v Sk = (4 − k)Sk+1 , and the equations D00 φA1 = DA1 φ00 , we find that those limits are all zero, and considering the equations that we have used to calculate the unknowns we get lim H1(ABC)k = 0 ,

u→0

k = 1, 2, 3 .

This completes the proof that the complete system of conformal field equations are satisfied in the limit as u → 0. ˆ ABCD , φ, φAB , Ω, ΩAB , R, Sk , corresponding Lemma 7.2. The functions eâ AB , Γ to the expansions determined in Lemma 5.1 satisfy the complete set of conformal ˆ. vacuum field equations on the set N ˆ the quantities t01 EF 11 , RAB0111 , A01 , ΣA1 , Proof. We have to show that on N ΠA1 , ΣA1CD , H1(BCD)k=1,2,3 vanish, and that ΦAB = −ΦBA . For this we derive a system of subsidiary equations for these fields. The values of the fields at U0 , given by Lemma 7.1, are the initial conditions for the subsidiary system of equations, and they are used throughout the proof. Using the definitions of AAB and ΦAB : DAB ACD − DCD AAB = −tAB EF and in particular

CD DEF φ

+ AD ΦBC + BC ΦDA ,

1 ˆ 0100 A01 , A01 = 2Γ ∂u + u which implies A01 = 0, and from that AAB = 0. This also shows that ΦAB = −ΦAB , and as we already know that Φ10 = 0 then ΦAB = 0.

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Following the proof of Lemma 5.5 in [8] we find that 1 ˆ 0100 t01 AB 11 + 2R(A 00111 0 B) , ∂u + t01 AB 11 = 2Γ u

(91)

which directly shows that t01 11 11 = 0. Also following the proof of Lemma 5.5 in [8] and taking into account that SABCD and R satisfy the contracted Bianchi identity then 1 ˆ 0100 RAB0111 − 1 H1AB0 − 1 ΠAB , ∂u + (92) RAB0111 = 2Γ u 2 6 from which R000111 = 0, which also gives t01 01 11 = 0. It is still left to show that t01 00 11 , RA10111 , ΣA1 , ΠA1 , ΣA1CD , H1(BCD)k=1,2,3

(93)

ˆ. vanish on N Using the definitions of ΣAB and ΣABCD , DAB ΣCD − DCD ΣAB = −tAB EF

CD DEF Ω

− ΣABCD + ΣCDAB ,

and from that 1 ˆ A100 Σ01 + ΣA100 . Σ01 A 0 = 2Γ (94) u At this point the expressions became to long to be treated by hand, so we resort to a computer program for tensor manipulations. For ΣABCD we obtain ∂u ΣA1 +

DEF ΣCDAB − DCD ΣEF AB = tCD P Q EF DP Q ΩAB − 2ΩP

(A RB)P CDEF

+ Ω (DE HF ABC + CF HDABE ) + SABCD ΣEF − SABEF ΣCD R 1 + (hABCD ΣEF − hABEF ΣCD ) + (1 + Ω) (hABCD ΠEF − hABEF ΠCD ) 3 3 1 3 1 − (1 − Ω)φ2 ΩEF ΣCDAB − ΩCD ΣEF AB + 6 (1 + Ω − φ2 ) + ΩAB (ΣCDEF − ΣEF CD ) − 6Ω2 φ φEF ΣCDAB − φCD ΣEF AB + φAB (ΣCDEF − ΣEF CD ) + 4 (2 + 3Ω) φ2 hABCD ΣEF P Q ΩP Q − hABEF ΣCDP Q ΩP Q − 8 (1 + Ω) (2 + 3Ω) φ hABCD ΣEF P Q φP Q − hABEF ΣCDP Q φP Q 1 1 2 2 ΩAB (ΩCD ΣEF − ΩEF ΣCD ) + 2 2 1−φ 2 (1 + Ω − φ ) + Ωφ 2 + Ω − 2φ2 φAB (ΩCD ΣEF − ΩEF ΣCD ) + ΩAB (φCD ΣEF − φEF ΣCD ) − 2Ω 2(1 + Ω)2 − (2 + 3Ω)φ2 φAB (φCD ΣEF − φEF ΣCD )

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1 2 φ (−1 + 3φ2 )ΩP Q ΩP Q + 4φ 3(1 + Ω)2 − (5 + 6Ω)φ2 ΩP Q φP Q 3 − 4(1 + Ω) (1 + Ω)(5 + 6Ω) − (7 + 9Ω)φ2 φP Q φP Q (hABCD ΣEF − hABEF ΣCD ) , −

which implies 1 0 ˆ C100 Σ01AB − Ω0C H10AB − S00AB Σ1C C Σ01AB = 2Γ u R 1 − h00AB Σ1C − (1 + Ω) h00AB Π1C 3 3 1 3 1 − (1 − Ω)φ2 (Ω00 Σ1CAB + ΩAB Σ1C00 ) + 2 6 (1 + Ω − φ )

∂u ΣC1AB +

(95)

− 6Ω2 φ (φ00 Σ1CAB + φAB Σ1C00 ) − 4 (2 + 3Ω) φ2 h00AB ΣC1P Q ΩP Q + 8 (1 + Ω) (2 + 3Ω) φh00AB ΣC1P Q φP Q 2 1 1 1 − φ2 ΩAB Ω00 Σ1C + − 2 2 (1 + Ω − φ2 ) 2 − Ωφ 2 + Ω − 2φ (φAB Ω00 + ΩAB φ00 ) Σ1C 2 + 2Ω 2 (1 + Ω) − (2 + 3Ω) φ2 φAB φ00 Σ1C 1 2 + φ2 −1 + 3φ2 ΩP Q ΩP Q + 4φ 3 (1 + Ω) − (5 + 6Ω) φ2 ΩP Q φP Q 3 − 4 (1 + Ω) (1 + Ω) (5 + 6Ω) − (7 + 9Ω) φ2 φP Q φP Q h00AB Σ1C .

Now with ΠAB DCD ΠAB − DAB ΠCD = tAB EF CD DEF R 1 + − 2 (4 + 7Ω) φΩGH − 2(1 + Ω)φGH DEF φGH tAB EF CD 2 1+Ω−φ − 4 (4 + 7Ω) φΩGH − 2(1 + Ω)φGH φGE RE HABCD − (3 − 3φ2 + 7Ωφ2 )ΩEF − 2Ωφ(4 + 7Ω)φEF (BC HDAEF + AD HBCEF ) 1 + φ (4 + 7Ω) φ (ΩCD ΠAB − ΩAB ΠCD ) − 2(1 + Ω) (φCD ΠAB − φAB ΠCD ) 3 1 + φ2 (4 + 7Ω) R (ΣABCD − ΣCDAB ) + 2 (4 + 7Ω) φ 3 DCD φEF ΣABEF − DAB φEF ΣCDEF 2 2 EF EF SCD ΣABEF − SAB ΣCDEF + 3 − 3φ + 7Ωφ

326

A. E. Ace˜ na


1 2 φ −12 + 40φ2 + 21Ωφ2 ΩEF ΩEF (ΣABCD − ΣCDAB ) 6 (1 + Ω − 2 − φ − 18(1 + Ω) + (46 + 61Ω + 21Ω2 )φ2 ΩEF φEF (ΣABCD − ΣCDAB ) 3 2 + (1 + Ω) − 24(1 + Ω) + (52 + 61Ω + 21Ω2 )φ2 φEF φEF 3 (ΣABCD − ΣCDAB ) 1 + φ2 −12 + 40φ2 + 21Ωφ2 ΩEF (ΩCD ΣABEF − ΩAB ΣCDEF ) 3 2 − φ − 18(1 + Ω) + (46 + 61Ω + 21Ω2 )φ2 φEF (ΩCD ΣABEF − ΩAB ΣCDEF ) 3 2 − φ 12(1 + Ω) + (16 + 61Ω + 21Ω2 )φ2 ΩEF (φCD ΣABEF − φAB ΣCDEF ) 3 4 + (1 + Ω) 6(1 + Ω) + (22 + 61Ω + 21Ω2 )φ2 φEF 3 (φCD ΣABEF − φAB ΣCDEF ) 1 + φ2 −3 + 7φ2 R (ΩAB ΣCD − ΩCD ΣAB ) 3 2 − φ − 7(1 + Ω)2 + (11 + 14Ω)φ2 R (φAB ΣCD − φCD ΣAB ) 3 + 2φ −3 + 7φ2 ΩEF (DAB φEF ΣCD − DCD φEF ΣAB ) − 4 − 7(1 + Ω)2 + (11 + 14Ω)φ2 φEF (DAB φEF ΣCD − DCD φEF ΣAB ) + −1 + φ2 −3 + 7φ2 ΩEF (SEF AB ΣCD − SEF CD ΣAB ) − 2φ − 4 − 14Ω − 7Ω2 + 2(2 + 7Ω)φ2 φEF (SEF AB ΣCD − SEF CD ΣAB ) 1 2 1 + φ − 24 + (59 + 21Ω)φ2 + 21φ4 ΩEF ΩEF 3 3 (1 + Ω − φ2 ) 2

+

1

2 φ2 )

(ΩAB ΣCD − ΩCD ΣAB ) − 2φ − 18(1 + Ω) + (13 + 19Ω)φ2 + (61 + 42Ω)φ4 ΩEF φEF (ΩAB ΣCD − ΩCD ΣAB ) + 2φ2 − 3(1 + Ω)(19 + 14Ω + 7Ω2 ) + (113 + 164Ω + 63Ω2 )φ2 φEF φEF (ΩAB ΣCD − ΩCD ΣAB ) − φ 12(1 + Ω) + (−17 + 19Ω)φ2 + (61 + 42Ω)φ4 ΩEF ΩEF (φAB ΣCD − φCD ΣAB ) + 4φ2 − 3(1 + Ω)(9 + 14Ω + 7Ω2 ) + (83 + 164Ω + 63Ω2 )φ2 ΩEF φEF (φAB ΣCD − φCD ΣAB )

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+ 4φ(1 + Ω) (1 + Ω)2 (61 + 42Ω) − 3(39 + 75Ω + 28Ω2 )φ2 φEF φEF (φAB ΣCD − φCD ΣAB ) , and we get 1 0 ˆ A101 A Π01 = 2Γ (96) u 1 (3 − 3φ2 + 7Ωφ2 )ΩEF − 2Ωφ (4 + 7Ω) φEF 0A H10EF + 2 1+Ω−φ 1 1 + φ (4 + 7Ω) φΩ00 − 2(1 + Ω)φ00 ΠA1 + φ2 (4 + 7Ω) RΣA100 3 3

∂u ΠA1 +

+ 2 (4 + 7Ω) φ∂u φEF ΣA1EF + (3 − 3φ2 + 7Ωφ2 )S00 EF ΣA1EF 1 2 1 2 2 EF + ΩEF ΣA100 2 6 φ (−12 + 40φ + 21Ωφ )Ω 2 (1 + Ω − φ ) 2 − φ − 18(1 + Ω) + (46 + 61Ω + 21Ω2 )φ2 ΩEF φEF ΣA100 3 2 + (1 + Ω) − 24(1 + Ω) + (52 + 61Ω + 21Ω2 )φ2 φEF φEF ΣA100 3 1 + φ2 (−12 + 40φ2 + 21Ωφ2 )ΩEF Ω00 ΣA1EF 3 2 − φ − 18(1 + Ω) + (46 + 61Ω + 21Ω2 )φ2 φEF Ω00 ΣA1EF 3 2 − φ 12(1 + Ω) + (16 + 61Ω + 21Ω2 )φ2 ΩEF φ00 ΣA1EF 3 4 + (1 + Ω) 6(1 + Ω) + (22 + 61Ω + 21Ω2 )φ2 φEF φ00 ΣA1EF 3 1 2 − φ2 (−3 + 7φ2 )RΩ00 ΣA1 + φ − 7(1 + Ω)2 + (11 + 14Ω)φ2 Rφ00 ΣA1 3 3 − 2φ(−3 + 7φ2 )ΩEF ∂u φEF ΣA1 + 4 − 7(1 + Ω)2 + (11 + 14Ω)φ2 φEF ∂u φEF ΣA1 − −1 + φ2 (−3 + 7φ2 )ΩEF SEF 00 ΣA1 EF 2 2 + 2φ − 4 − 14Ω − 7Ω + 2(2 + 7Ω)φ φ SEF 00 ΣA1 1 2 1 − φ − 24 + (59 + 21Ω)φ2 + 21φ4 ΩEF ΩEF Ω00 ΣA1 2 3 3(1 + Ω − φ ) 2 − 2φ − 18(1 + Ω) + (13 + 19Ω)φ2 + (61 + 42Ω)φ4 ΩEF φEF Ω00 ΣA1 + 2φ2 − 3(1 + Ω)(19 + 14Ω + 7Ω2 ) + (113 + 164Ω + 63Ω2 )φ2 φEF φEF Ω00 ΣA1

328

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− φ 12(1 + Ω) + (−17 + 19Ω)φ2 + (61 + 42Ω)φ4 ΩEF ΩEF φ00 ΣA1 + 4φ2 − 3(1 + Ω)(9 + 14Ω + 7Ω2 ) + (83 + 164Ω + 63Ω2 )φ2 ΩEF φEF φ00 ΣA1 + 4φ(1 + Ω) (1 + Ω)2 (61 + 42Ω) − 3(39 + 75Ω + 28Ω2 )φ2 EF φ φEF φ00 ΣA1 . Finally, with HABCD , 1 (97) DEF HEF CD = − tEF HI E G DHI SCDF G − 2SE(F GC RE D) HF H G 2 EF 1 1 φΩ − 2(1 + Ω)φEF − (1 + Ω)DGH φEF tI C GH DI + 2 1+Ω−φ 3 2 − ΩDGH φF I tE IGH CD − (1 + Ω)φF G RG E H CDH + ΩφF G RGH EHCD 3 1 EF GH − Ωφ RGF EHCD + φΩ φΩ − 2(1 + Ω)φEF H(CD)EF 3 1 + (1 − φ2 + Ωφ2 )ΩEF − 2Ω2 φφEF HEF CD 2 1 + (−2 + Ω)φ φΩE (C ΠD)E − 2(1 + Ω)φE (C ΠD)E 18 1 + (−2 + Ω)φ2 RΣE(CD) E + ΩφDCD φEF ΣGEF G 18 2 1 − (1 + Ω)φDE (C φF G ΣD)EF G + (1 − φ2 + Ωφ2 )SCD EF ΣGEF G 3 2 EF 1 2 EF G − Ωφ S ΣGEF G (C ΣD)EF G − 2 φΩCD − 2(1 + Ω)φCD φ 3 − 4φΩEF φCD ΣGEF G + 2φΩEF φF G ΣEGCD 1 2 1 + φ (3 − 10φ2 + 6Ωφ2 )ΩEF (1 + Ω − φ2 )2 18 ΩEF ΣG(CD) G + 2ΩG (C ΣD)GEF 2 + φ3 (7 + 4Ω − 6Ω2 )φEF ΩEF ΣG(CD) G + ΩG (C ΣD)GEF 9 2 − φ 6(1 + Ω) − (13 + 4Ω − 6Ω2 )φ2 ΩEF φG (C ΣD)GEF 9 2 − (1 + Ω) 3(1 + Ω) + 2(2 + 2Ω − 3Ω2 )φ2 φEF φEF ΣG(CD) G 9 4 + (1 + Ω)2 (3 − 10φ2 + 6Ωφ2 )φEF φG (C ΣD)GEF 9 1 1 − φ2 (−3 + φ2 )RΩE (C ΣD)E − φ (1 + Ω)2 + (1 − 2Ω)φ2 RφE (C ΣD)E 18 9

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2 + φ3 ΩEF DG (C φEF ΣD)G − φ(1 − φ2 )ΩEF DCD φF G ΣEG 3 4 + (1 + Ω)(1 + Ω − 2φ2 )φEF DG (C φEF ΣD)G + 2 (1 + Ω)2 − (1 + 2Ω)φ2 3 1 φEF DCD φF G ΣEG − (1 − φ2 )φ2 ΩEF S G EF (C ΣD)G 3 1 2 2 EF G + (1 − φ ) Ω S F CD ΣEG 2 2 − φ − (1 + Ω)2 + (1 + 2Ω)φ2 φEF S G EF (C ΣD)G 3 + Ωφ(2 + Ω − 2φ2 )φEF S G F CD ΣEG + 2φ (ΩCD − 2φφCD ) ΩE F φEG ΣF G 1 + − φ2 3 − 13φ3 + 3Ωφ2 + 3φ4 ΩEF ΩEF ΩG (C ΣD)G 2 3 9(1 + Ω − φ ) − 4φ3 (5 + 8Ω + 2φ2 − 6Ωφ2 )ΩEF φEF ΩG (C ΣD)G + 4φ2 3(1 + Ω)3 + (4 − 2Ω − 9Ω2 )φ2 φEF φEF ΩG (C ΣD)G + 2φ (1 + Ω)(3 − 8φ2 ) − 2(1 − 3Ω)φ4 ΩEF ΩEF φG (C ΣD)G + 8(1 + Ω)φ2 3Ω(2 + Ω) + (7 − 9Ω)φ2 ΩEF φEF φG (C ΣD)G − 8(1 + Ω)φ 2(1 + Ω)2 (−1 + 3Ω) + 3(3 − 4Ω2 )φ2 φEF φEF φG (C ΣD)G , where the l.h.s. is

1 H11CD + H110(C D) 0 DEF HEF CD = ∂u H11CD + u 1 a ˆ 0100 H11CD − Γ ˆ 010C H110D ∂v + eˆ 01 ∂a H10CD − 2Γ − 2u ˆ 010D H110C + Γ ˆ 011C H100D + Γ ˆ 011D H100C + Γ ˆ 1100 H10CD . −Γ

Equations (91), (92), (94), (95), (96), (97) are the system of subsidiary equations for the quantities (93). The expressions on the right hand sides of these equations are homogeneous functions of the quantities (93). Together with Lemma 7.1 this implies that all the expansion coefficients of the quantities (93) vanish on U0 . As the functions (93) are necessarily holomorphic, this implies that they vanish ˆ. on N

8. Analyticity at space-like infinity Our gauge is singular and thus the holomorphic solution of Lemma 6.7 does not cover a full neighborhood of the point i. To show that we can indeed get a holomorphic solution in a hole neighborhood of i we go to a normal frame field based on the frame cAB at i and the corresponding normal coordinates xa . The argument follows with some modifications the line of the corresponding argument in [8].

330

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The geodesic equation for z a (s) = (u(s), v(s), w(s)), Dz˙ z˙ = 0, can be written in the form z˙ a = mAB ea AB , m ˙ AB = −2mCD ΓCD (A B mB)E . The initial conditions for the geodesics to start at i are u|s=0 = 0 ,

w|s=0 = 0 ,

and we have to prescribe v0 = v|s=0 , in order to determine the ∂u -∂w -plane where the tangent vector is. The components of the tangent vector to the geodesic at i are given by mAB |s=0 = mAB 0 , and by regularity and the geodesic equations we have ˙ s=0 ≡ u˙ 0 , m00 0 = u|

m01 0 = 0,

m11 ˙ s=0 ≡ w˙ 0 . 0 = w|

We can identify the frame eAB with its projection into Ti Nc , then mAB 0 eAB = m∗AB cAB = xa ca , where as defined cAB = αa AB ca , and we get √ 1 i x1 = √ w˙ 0 + (v02 − 1)u˙ 0 , x2 = √ w˙ 0 + (v02 + 1)u˙ 0 , x3 = 2v0 u˙ 0 , 2 2 or, inverting the relations u˙ 0 (xa ) = −

x1 + ix2 √ , 2

v0 (xa ) = −

x1

x3 , + ix2

δab xa xb . w˙ 0 = √ 2(x1 + ix2 )

Here we see that in order to have a well defined vector we need x1 + ix2 = 0, or, what is the same, u˙ 0 = 0. This correspond to the singular generator of Ni in the cAB -gauge. The vectors xa ca cover all directions at i except those tangent to the complex null hyperplane (c1 + ic2 )⊥ = {a(c1 + ic2 ) + bc3 |a, b ∈ C}. As we have used a frame formalism, we need also to determine the normal frame centered at i and based on the frame cAB at i. As we already have the frame fields eAB , we write the equation for the normal frame cAB , Dx˙ cAB = 0 as an equation for the transformation tA B ∈ SL(2, C) that relates the frames eAB and cAB , cAB = tC A tD B eCD . The equation can be written as t˙A B = −mDE ΓDE A C tC B ,

(98)

and the initial condition comes from having to take eAB |i (v) to cAB |i , tA B |s=0 = sA B (−v0 ) .

(99)

Following the proofs of Lemma 7.1, Lemma 7.2 and Lemma 7.3 in [8] we arrive at the following two lemmas. Lemma 8.1. For any given initial data u˙ 0 , v0 , w˙ 0 , with u˙ 0 = 0, there exist a number t = t(u˙ 0 , v0 , w˙ 0 ) and unique holomorphic solutions z a (s) = z a (s, u˙ 0 , v0 , w˙ 0 ) of the initial value problem for the geodesic equations with initial conditions as described above which is defined for |s| < 1/t. The functions z a (s, u˙ 0 , v0 , w˙ 0 ) are in fact

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holomorphic functions of all four variables (s, u˙ 0 , v0 , w˙ 0 ) in a certain P1/t (0) × U , where U is a compactly embedded subset of (C\{0}) × C × C. Lemma 8.2. Along the geodesic corresponding to s → z a (s, u˙ 0 , v0 , w˙ 0 ) equations (98) have a unique holomorphic solution tA B (s) = tA B (s, u˙ 0 , v0 , w˙ 0 ) satisfying the initial conditions (99). The functions tA B (s) = tA B (s, u˙ 0 , v0 , w˙ 0 ) are holomorphic in all four variables in the domain where the z a (s, u˙ 0 , v0 , w˙ 0 ) are holomorphic. Following the discussion in [8] it can be seen that, as |x| ≡ δab x ¯a xb → 0, 1 2 x + ix = 0, x1 + ix2 √ + O(|x|3 ) , 2 x3 v(xc ) = − 1 + O(|x|2 ) , x + ix2 δab xa xb + O(|x|3 ) . w(xc ) = √ 2(x1 + ix2 ) u(xc ) = −

This gives for the forms χAB = χAB c dxc dual to the normal frame cAB χAB (xc ) = αAB a + χ ÂB a dxa , ÂB a = O(|x|2 ) as |x| → with holomorphic functions χ ÂB a (xc ) which satisfy χ 0. Also the coefficients ca AB = dxa , cAB of the normal frame in the normal coordinates satisfy ca AB (xc ) = αa AB + câ AB , with holomorphic functions câ AB (xc ) which satisfy câ AB = O(|x|2 ) as |x| → 0. The three 1-forms αa AB dxa are linearly independent and thus for small |xc | the coordinate transformation xa → z a (xc ), where defined, is nondegenerate. This means that all the tensor fields entering the conformal stationary vacuum field equations can be expressed in terms of the normal coordinates xc and the normal frame field cAB . Now we can derive our main result. Proof of Theorem 1.1. The coordinates xa cover a domain U in C3 on which the frame vector fields cAB = ca AB ∂/∂xa exist, are linearly independent and holomorphic. Also in U the other tensor fields expressed in terms of the xa and cAB are holomorphic. However U does not contain the hypersurface x1 + ix2 = 0 but the boundary of U becomes tangent to this hypersurface at xa = 0. We want to see that the solution indeed cover a domain containing an open neighborhood of the origin. We still have the gauge freedom to perform with some tA B ∈ SU (2) a rotation δ ∗ → δ ∗ · t of the spin frame. Whit this rotation is associated the rotation cAB → ctAB = tC A tD B cCD ˆ was done based on the of the frame cAB at i. The construction of the submanifold N t frame cAB , starting now with cAB all the previous constructions and derivations

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can be repeated as far as the estimates for the null data in the cAB -gauge can be translated to the same type of estimates for the null data in the ctAB -gauge. We will denote u , v , w and etAB the analogues in the new gauge of the coordinates u, v, w and the frame eAB . The set Ni is invariant under this rotation. The sets {w = 0} and {w = 0} are both lifts of the set Ni to the bundle of spin frames. The coordinates u and u are both affine parameters on the null generators of Ni , which vanish at i. The coordinates v, v both label the null generators of Ni . The frame vectors e00 and et00 are auto-parallel vector fields tangent to the null generators. If v and v label the same generator η of Ni , then et00 (v ) = f 2 e00 (v) at i, with some f = 0. Furthermore, as e00 and et00 are auto-parallel, then et00 = f 2 e00 must hold along η, with f constant along the geodesic. This means that at i sC 0 (v )sD 0 (v )tE C tF

D cEF

= f 2 sC 0 (v)sD 0 (v)cCD ,

and absorbing the undetermined sign in f , tE C sC 0 (v ) = f sE 0 (v) . We can write tA B ∈ SU (2) as a −¯ c (tA B ) = , c a ¯

a, c ∈ C ,

(100)

|a|2 + |c|2 = 1 .

(101)

This gives with (100) v =

−c + av , a ¯ + c¯v

f=

1 , a ¯ + c¯v

resp. v =

c+a ¯v , a − c¯v

f = a − c¯v .

(102)

As du, e00 = 1 = du , et00 we have for the affine parameter along η u = f 2 u .

(103)

With (102), (103) holds η(u , v ) = η(u, v). If c = 0 then v → ∞ as v → a/¯ c. So the null generator in the cAB -gauge, where we need information, is contained, excepting the origin, in the regular domain of the ctAB -gauge. ˆ φ, D ˆS Let us consider now the abstract null data given in the cAB -gauge D n n t φt St ˆn , D ˆn , satisfying estimates of the form (38), (39). In the cAB -gauge we have D with terms given by t = tGmAm tHmBm . . . tG1A1 tH1B1 ψGm Hm ...G1 H1 , ψA m Bm ...A1 B1

ΨtAm Bm ...A1 B1 CDEF = tGmAm tHmBm . . . tG1A1 tH1B1 tIC tJD tKE tKL ΨGm Hm ...G1 H1 IJKL .

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Using the essential components of ψ and ψ t 2m 2m (Gm t ψ(Am Bm ...A1 B1 )n = t (Am tHmBm . . . tG1A1 tH1 )jB1 )n ψ(Gm Hm ...G1 H1 )j j j=0

= The numbers T2m

2m n

j

n (t)

=

− 12 1 2m 2m 2 j=0

2m n

12

2m j

j 12

T2m j n (t)ψ(Gm Hm ...G1 H1 )j .

t(Gm(Am tHmBm . . . tG1A1 tH1 )jB1 )n

satisfy |T2m j n (t)| ≤ 1 ,

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

0 ≤ j ≤ 2m ,

0 ≤ n ≤ 2m ,

as they represent the matrix elements of a unitary representation of SU (2). So we get m!M t |ψA | ≤ m , m = 1, 2, 3, . . . , m Bm ...A1 B1 r where r = r/4. In the same way we get |ΨtAm Bm ...A1 B1 CDEF | ≤

m!M , rm

m = 0, 1, 2, 3, . . . ,

where M = 16M . So the estimates for the null data on the cAB -gauge translate into the same type of estimates for the null data on the ctAB -gauge. Assuming now c = 0 in (101), we have two possibilities for getting the solution in the ctAB -gauge: i. Using the solution in the cAB -gauge we can determine, where possible, the coordinate and frame transformation to the ctAB -gauge. In particular, the singular generator of Ni in the ctAB -gauge will coincide with the regular gena/¯ c. We are thus able to erator of Ni in the cAB gauge on which v = −¯ determine near the singular generator in the ctAB -gauge the expansion of the solution in terms of the coordinates u , v , w and the frame field etAB . ˆ nφt , D ˆ nSt in the ct -gauge, one can repeat all the steps ii. Using the null data D AB of the previous sections to show the existence of a solution to the conformal stationary vacuum field equations in the coordinates u , v , w of the ctAB gauge. All the statements made about the solution in the cAB -gauge apply also to this solution, in particular statements about domains of convergence. The formal expansions of the fields in terms of u , v , w are uniquely determined ˆ nφt , D ˆ nSt , thus the solutions obtained by the two methods are holoby the data D morphically related to each other on certain domains by the gauge transformation obtained in (i). As done with the solution in the cAB -gauge, the solution in the

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ctAB -gauge can be expressed in terms of the normal coordinates xat and the normal frame field ctAB . The xat cover a certain domain Ut ⊂ C3 and the frame field ctAB is non-degenerate. All the tensor fields expressed in terms of xat and ctAB are holomorphic on Ut . Then the solution in the cAB -gauge and the solution in the ctAB -gauge are related on certain domains by the transformation xat = t−1a b xb ,

ctAB = tC A tD B cCD ,

which gives the transformation corresponding to the rotation of normal coordinates. We can extend this as a coordinate and frame transformation to the solution obtained in (ii) to express all fields in terms of xa and cAB . With this extension all fields are defined and holomorphic on t−1 Ut . Then the solution obtained in the cAB -gauge and the solution in the ctAB -gauge are genuine holomorphic extensions of each other, as one covers the singular generator of the other one away from the origin in a regular way. Let now xa∗ = 0 be an arbitrary point in C3 . We want to show that the solution extends in the coordinates xa to a domain which covers the set sxa∗ for 0 < s < for some > 0. That is the case in the cAB -gauge as far as xa∗ = (α, iα, β), α, β ∈ C. We need to see what happens if xa∗ = (α, iα, β), with α = 0 or β = 0. If xa∗ = (α, iα, β) and α = 0, we consider the ctAB -gauge, where tAB is given by (101) with a = 0, c = 1. The normal coordinates in the two gauges are related by x1t = −x1 , x2t = x2 , x3t = −x3 . The holomorphic transformation (x1t , x2t , x3t ) → (−x1 , x2 , −x3 ) maps Ut onto a subset of C3 , denoted by t−1 Ut , which has nonempty intersection with U . After the transformation the two solutions coincide on t−1 Ut ∩ U . Under this transformation, the singular set {x1 + ix2 = 0} in the cAB -gauge correspond to the set {x1t − ix2t = 0}, which is covered in a regular way in a neighborhood of i in the ctAB -gauge. So the set t−1 Ut ∪ U admits a holomorphic extension of our solution in the coordinates xa and the frame cAB . In this extension there exist such that sxa∗ , xa∗ = (α, iα, β) with α = 0, is covered by the solution for 0 < s < . We need also to consider the case α = 0, that is, xa∗ = (0, 0, β), β = 0. In this case we use the ctAB -gauge, where tAB is given by (101) with a = √12 , c = √i2 . The normal coordinates are related by x1t = x1 ,

x2t = −x3 ,

x3t = x2 .

The argument follows the same lines as for the α = 0 case. Thus the set U can be extended so that the points sxa∗ with 0 < s < are covered by U and all fields are holomorphic on U in the coordinates xa . Then it can be assumed U to contain a punctured neighborhood of the origin in which the solution is holomorphic in the normal coordinates xa and the normal frame cAB . Then the solution is in fact holomorphic on a full neighborhood of the origin xa = 0, which represents the point i, as holomorphic functions in more than one dimension cannot have isolated singularities.

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By Lemma 3.1 we have from null data satisfying the reality conditions a formal expansion of the solution with expansion coefficients satisfying the reality conditions. By the various uniqueness statements obtained in the lemmas, this expansion must coincide with the expansion in normal coordinates of the solution obtained above. This implies the existence of a 3-dimensional real slice on which the tensor fields satisfy the reality conditions. It is obtained by requiring the coordinates xa to assume values in R3 .

9. Conclusions We have seen how to determine a formal expansion of an asymptotically flat stationary vacuum solution to Einstein’s field equations using a minimal set of freely specifiable data, the null data. These data are given by two sequences of symmetric trace free tensors at space-like infinity. We have obtained necessary and sufficient conditions on the null data for the formal expansion to be absolutely convergent, hence showing that the null data characterize all asymptotically flat, stationary vacuum solutions to the field equations with non-vanishing ADM mass. This work contains the static case as a particular case, and is a generalization of Friedrich’s work [8] from the static to the stationary case. Corvino and Schoen [6] and Chru´sciel and Delay [5] have shown that it is possible to produce vacuum initial data that is fairly general in the interior region and exactly static or stationary in the exterior region. The present work shows which are the possible exteriors. It is a long standing conjecture that Hansen‘s multipoles [11], which are relevant because they have nice geometrical transformation properties under change of conformal factor, do characterize an asymptotically flat stationary vacuum solutions to the field equations in the way we have shown the null data do. This have been shown in the axisymmetric case [1] and some steps have been achieved in the general case, like showing that the multipoles determine a formal expansion of a solution [3, 13], or necessary bounds on the multipoles if the solution exist [2], but general conditions on the multipoles for the expansion to be convergent has not been found yet. As there is a bijective correspondence between the null data and Hansen’s multipoles, although the relation is highly non linear, it would be nice if this correspondence could be exploited to get necessary and sufficient conditions on the multipoles to determine a convergent expansion.

Acknowledgements I would like to thank Helmut Friedrich for presenting me this problem and for guidance during this work. The author is supported by a PhD scholarship from the International Max Planck Research School.

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References [1] T. B¨ ackdahl, Axisymmetric stationary solutions with arbitrary multipole moments, Class. Quantum Grav. 24 (9) (2007), 2205–2215. [2] T. B¨ ackdahl and M. Herberthson, Calculation of, and bounds for, the multipole moments of stationary spacetimes, Class. Quantum Grav. 23 (20) (2006), 5997–6006. [3] R. Beig and W. Simon, On the multipole expansion for stationay space-times, Proc. R. Soc. Lond. A 376 (1765) (1981), 333–341. [4] R. Beig and W. Simon, The multipole structure of stationary space-times, J. Math. Phys. 24 (5) (1983), 1163–1171. [5] P. Chru´sciel and E. Delay, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mémoires de la SMF 94 (2003), vi–103. [6] J. Corvino and R. Schoen, On the asymptotics for the vacuum Einstein constraint equations, J. Differ. Geom. 73 (2) (2006), 185–217. [7] S. Dain, Initial data for stationary spacetimes near spacelike infinity, Class. Quantum Grav. 18 (20) (2001), 4329–4338. [8] H. Friedrich, Static vacuum solutions from convergent null data expansions at spacelike infinity, Ann. Henri Poincaré 8 (5) (2007), 817–884. [9] R. Geroch, Multipole moments. II. Curved space, J. Math. Phys. 11 (8) (1970), 2580– 2588. [10] R. Geroch, A method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (6) (1971), 918–924. [11] R. Hansen, Multipole moments of stationary space-times, J. Math. Phys. 15 (1) (1974), 46–52. ´ Murchadha, Weakly decaying asymptotically flat static and [12] D. Kennefick and N. O stationary solutions to the Einstein equations, Class. Quantum Grav. 12 (1) (1995), 149–158. [13] P. Kundu, On the analyticity of stationary gravitational fields at spatial infinity, J. Math. Phys. 22 (9) (1981), 2006–2011. [14] A. Lichnerowicz, Sur les équations relativistes de la gravitation, Bulletin de la S.M.F. 80 (1952), 237–251. [15] O. Reula, On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations, Commun. Math. Phys. 122 (4) (1989), 615–624. ats[16] W. Simon, Die Multipolstruktur station¨ arer R¨ aume in der allgemeinen Relativit¨ theorie, PhD thesis, Formal- und Naturwissenschaftlichen Fakult¨ at der Universit¨ at Wien, 1980. [17] B. Xanthopoulos, Multipole moments in general relativity, J. Phys. A: Math. Gen. 12 (7) (1979), 1025–1028.

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Andrés E. Ace˜ na Max Planck Institute for Gravitational Physics Am M¨ uhlenberg 1 D-14476 Golm Germany e-mail: [email protected] Communicated by Piotr T. Chrusciel. Submitted: December 9, 2008. Accepted: January 21, 2009.

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Ann. Henri Poincaré 10 (2009), 339–356 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/020339-18, published online May 22, 2009 DOI 10.1007/s00023-009-0411-2


Octonionic Twists for Supermembrane Matrix Models Jens Hoppe, Douglas Lundholm and Maciej Trzetrzelewski Abstract. A certain G2 × U (1) invariant Hamiltonian arising from the standard membrane matrix model via conjugating any of the supercharges by a cubic, octonionic, exponential is proven to have a spectrum covering the whole half-axis R+ . The model could be useful in determining a normalizable zero-energy state in the original SO(9) invariant SU (N ) matrix model.

1. Introduction Despite considerable effort [3, 7, 8, 12, 13, 15–19, 21–25, 29–31, 35, 38, 39, 43, 48, 49] during the last decade, and crucial relevance to M-theory [4, 46] membrane theory [5, 14, 20, 45] reduced Yang–Mills theory [2, 6, 9, 11], existence, uniqueness and structure of zero-energy states in Spin(9) × SU (N ) invariant supersymmetric matrix models are not really understood to a degree that one could call satisfactory. In this paper we consider models with G2 × U (1) × SU (N ) symmetry that we obtain by deforming (cp. [10, 36]) the Spin(9) × SU (N ) models, and which we believe to be relevant both from the point of view of deformation theory and possible relations between ground states, as well as because (for the fixed value of the deformation parameter that we take) the Hamiltonian is slightly simpler, and therefore a good testing ground for new approaches. The model is introduced in Section 2 by deforming the Spin(9) × SU (N ) invariant one via a particular cubic exponential. In Section 3, with the help of various propositions that are proved in Section 4, this model is shown to share a central feature of the original theory, namely that the Hamiltonian, in contrast with the discreteness of the spectrum (cp. [34, 40]) for the purely bosonic theory, has an essential spectrum covering the whole positive axis (cp. [41,44]). A summary of the results is presented in Section 5. In the appendices some background material is provided, and the deformation we introduce put into a slightly more general context.

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2. The deformed model To find a normalizable state annihilated by the supercharges 1 i ∂ 1 j jk Qβ := iδαβ fABC zB z¯C + iΓαβ − fABC xjB xkC Γαβ λαA 2 ∂xjA 2 ∂ j + 2δαβ − ifABC xjB z¯C Γαβ λ†αA ∂zA (and by their hermitian conjugates) is a difficult task; the (xjA )A=1,...,N j=1,...,7

2

−1

(1) and

−1 (λαA )A=1,...,N α=1,...,8 2

Grassmann varizA = x8A + ix9A are bosonic coordinates, ables, fABC totally antisymmetric structure constants of SU (N ), Γjk := 12 [Γj , Γk ], and (Γj )j=1,...,7 (purely imaginary, antisymmetric) matrices satisfying {Γj , Γk } = 2δ jk 18×8 , in a particular representation given by iΓjα8 = δαj , iΓjkl = −cjkl , totally anti-symmetric octonionic structure constants. In [26] conjugation by the exponent of 1 (2) g(x) := fABC xjA xkB xlC (iΓjkl )ββ 6 was shown to remove the third term in (1) (extending an observation made in [45]; kl note that j Γjαβ Γjkl ββ = Γαβ for fixed β, cp. Appendix B). Correspondingly, defin†

ing Hk := {Q(k), Q(k) } ≥ 0, where (choosing β = 8) k Q(k) := ekg(x) Q8 e−kg(x) = Q8 + fABC xjB xlC Γjl α8 λαA , 2 gives

(3)

Hk = −ΔR9(N 2 −1) + (k − 1)2 V1...7 + V89 + z¯A fABC xjB fA B C xjB zA + 2fAA E xjE (δα8 δα j − δαj δα 8 )λαA λ†α A + 2(k − 1)fEAA xjE iΓj ll λlA λ†l A + fEAA zE λαA λαA + fEAA z¯E λ†αA λ†αA .

(4)

The potential terms for the x- resp. z-coordinates are given by 1 1 V1...7 = fABC xjB xlC fAB C xjB xlC resp. V89 = fABC z¯B zC fAB C zB z¯C . 2 4 While for large k, ˆ := −Δx + V1...7 − 2fEAA xjE cjll λlA λ† , H (5) lA

appears to be the relevant operator (having rescaled x → (k − 1)−1/3 x) 2 we will, ˜ which is of the form (cp. (4)) in this note, exclusively study Hk=1 =: H, ˜ = −Δx + HD + V89 + f zλλ + f z¯λ† λ† , H (6) HD := −Δz + z¯A fABC xjB fA B C xjB zA +2fAA E xjE (δα8 δα j −δαj δα 8 )λαA λ†α A . 1 cp.

Appendix A point (and [10] in general) was discussed with B. Durhuus and J. P. Solovej, -which we gratefully acknowledge. 2 This

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The operator HD (= HD (x) ≥ 0) arises from the second line of (1) alone, and we will heavily use that its spectrum and eigenfunctions are known [27]. The operator V89 +zf λλ+ z¯f λ† λ† appearing in (6) and involving only the z-coordinates will be denoted by K. We also note that, regardless of the choice of k, the bosonic part of Hk (first line in (4)) has a strictly positive and purely discrete spectrum (this is easily proved along the lines of [34, 40]). The bosonic part of HD describes two sets of n := N 2 −1 harmonic oscillators whose frequencies ωA are the square root of the eigenvalues of the parametrically x-dependent, positive semidefinite frequency matrix SAA (x) := fABC xjB fA B C xjB ,

(7)

while its fermionic part, 2WαA βB (x)λαA λ†βB , that is linear in xjA , has eigenvalues arising from those of 2WαA βB , which are {±2ωA (x)}A=1,...,n as well as 6n times the eigenvalue zero – altogether leading to the exact zero-energy state(s) of HD [27] ψx =

n A=1

ωA (x) − 1 ωA (x)¯zA zA (−ω1 ) (−ω ) e 2 eα1 A1 (x) . . . eαl All (x)λα1 A1 . . . λαl Al , π

(8)

(where, n ≤ l ≤ 7n, ωl>n := 0, and we have diagonalized S via zA = RAB (x)zB , T 2 (ω) S = R [ωA ]R), that involves the eigenvectors e (x) of the matrix W (x) corresponding to eigenvalue ω. Excited states of HD are obtained by acting with the bosonic creation operators (i.e. multiplying ψx by the corresponding Hermite polynomials) and/or adding fermions corresponding to positive eigenvalues 2ω (i.e. (+ω) multiplying ψx by eαA λαA ). Alternatively, thinking of the coordinate point x = (xjA ) as a tuple (Xj ) of traceless hermitian N × N matrices, with Xj = xjA TA in a basis {TA } s.t. [TA , TB ] = ifABC TC , the matrix (operator) S can also be written as

S(x) =

7

adXj ◦ adXj = Xj , [Xj , · ] ,

j=1 2 acting on i · su(N ) ∼ is = Rn , E ↔ (eA ). In particular, its lowest eigenvalue ωmin given by 2 = min eA SAB eB = min [Xj , E]2 , ωmin e∈S n−1

E=1

j

where · here denotes the corresponding norm on i · su(N ) ∼ = Rn . Hence, for N > 2, S(x) will have zero-modes not only when all matrices Xj commute, but (of qualitative significance) for the larger space of configurations where all the Xj are simultaneously block-diagonalizable.

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˜ 3. Continuity of the spectrum of H In this section we formulate and prove the main theorem of the paper. We will make use of three propositions and one lemma (which are proved in Section 4 in order not to break the flow of the text). Main theorem. For any λ ≥ 0 there exists a sequence (Ψt ) of rapidly decaying smooth SU (N )-invariant functions such that Ψt = 1 and ˜ (H − λ)Ψt → 0 as t → ∞ . ˜ (when restricted to the physical Hilbert space) It follows that the spectrum of H consists of the whole non-negative real line. This is clearly similar to the the case for the original Hk=0 . However, because of the terms that vanish for Hk=1 , together with the convenient structure of the remaining terms noted in the previous section, we are able to construct such a sequence (Ψt ) explicitly without resorting to the gauge fixing procedure used in [41, 44]. ˜ as In the following, we write H ˜ = −Δx + HD (x) + K , H (9) where

HD (x) = −4∂z¯ · ∂z + z¯ · S(x)z + 2W (x)λλ†

(10)

and

1 f z¯zf z z¯ + f zλλ + f z¯λ† λ† . (11) 4 ˜ is an unbounded operator, it is considered to be We also point out that, since H defined as a differential operator on the Schwartz class S of smooth functions of rapid decay, and then extends by closure or Friedrichs extension to a self-adjoint operator in H = L2 (R9n ) ⊗ F. Our candidate for the sequence Ψt will be wavefunctions given by the minimal fermion number ground state ψx of HD (x) multiplied by some gauge invariant cutoff function χt . Formally, it is convenient to write the Hilbert space H as a constant fiber direct integral (see [37]) over the x-coordinates,

⊕ H= h dx , K(z) =

7n

where (writing dx = d x, dz = d

n

R7n n x8 d x9 for

the integration measures)

⊕ h := L2 (R2n ) ⊗ F = F dz R2n

is the z-coordinate Hilbert space on which the operator HD (x) + K acts in each point x. Hence, for any Ψ(x, z) = χ(x)ψx (z) ∈ H, we have

2 2 2 2 ΨH = |χ(x)| ψx h dx = |χ(x)| ψx (z)2F dz dx . (12) R7n

R7n

R2n

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We also write the ground state (8) of HD (x) in a more compact notation, ψx (z) = π − 2 s(x) 4 e− 2 z¯ · S(x) n

1

1

1/2

z

ξx ,

(13)

where s := det S, and ξx ∈ Fn (i.e. n fermions) is the normalized fermionic eigenvector satisfying 1 ωA ξx = − tr S(x) 2 ξx . (14) W (x)λλ† ξx = − A

We note the following: Proposition 1. ψx is smooth (also in x), rapidly decaying, and SU (N )-invariant. Proposition 2. ψx h = 1, and |z|k ψx h ≤ positive constants Ck .

Ck , k/2 ωmin (x)

for k = 1, 2, 4 and some

Hence, by choosing an appropriate cut-off function χt for the x-coordinates such that ωmin (x) → ∞ as t → ∞, we can make the terms in K(z) arbitrarily small. The following proposition shows that such a choice is indeed possible. Proposition 3. For any λ ≥ 0 and t sufficiently large there exist SU (N )-invariant cut-off functions χt ∈ C0∞ (R7n ) such that ∀x ∈ supp χt ωmin (x) ≥ c1 t ,

c2 t ≤ |x| ≤ c3 t ,

(15)

and, as t → ∞, χt L2 (R7n ) = 1 ,

∂jA χt L2 (R7n ) ≤ c4 ,

(−ΔR7n − λ)χt L2 (R7n ) → 0 ,

(16)

where (here, and in the following) ck=1,2,3,... are some positive constants. As a final preparation before proving the main theorem, we state the following lemma which ensures that also certain derivatives tend to zero. Lemma 4. ∂jA ψx h ≤

c5 ωmin (x)

2 and ∂jA ψx h ≤

c6 2 ωmin (x)

on supp χt .

3.1. Proof of the main theorem Motivated by the expression (9) and the above preparations, we define Ψt (x, z) := χt (x)ψx (z) , where χt ∈ C0∞ (R7n ) is chosen according to Proposition 3. We note that Ψt is in ˜ and by (12) has Ψt = 1. Acting with H ˜ on Ψt (x, z), we obtain the domain of H 2 ˜ t = −ψx Δx χt − 2 HΨ ∂jA χt ∂jA ψx − χt ∂jA ψx + χt K(z)ψx , (17) j,A

j,A

(where we used the fact that HD (x)ψx = 0). Subtracting λΨt from this equation and using Propositions 2, 3 and Lemma 4 (and that any operator on F is bounded)

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to estimate the norms of the terms on the r.h.s. as t → ∞, we find ψx (−Δx − λ)χt 2 = (−Δx − λ)χt 2L2 (R7n ) → 0 , 2

c5 c7 c8 2 2 ∂jA χt ∂jA ψx ≤ |∂jA χt | dx ≤ 2 ∂jA χt 2L2 (R7n ) ≤ 2 → 0 , ωmin t t 2

c6 c9 2 χt ∂jA ψx 2 ≤ |χt |2 dx ≤ 4 χt 2L2 (R7n ) → 0 , 2 ωmin t and

χt K(z)ψx 2 ≤

|χt |2 (c10 |z|4 ψx h + c11 |z|ψx h )2 dx

≤

c10 C4 c11 C1 + 1/2 2 ωmin ωmin

2 |χt |2 dx ≤

c12 χt 2L2 (R7n ) → 0 . t

˜ − λ)Ψt → 0 as t → ∞. Hence, (H ˜ − λ does not have a bounded It follows that, for any λ ≥ 0, the operator H ˜ inverse. Together with H ≥ 0 from supersymmetry, this proves the theorem.

4. Proofs Here we present detailed proofs of the propositions and lemma that were stated in the previous section. 4.1. Proof of Proposition 1 It is obvious from (13) that ψx ∈ S(R2n ) ⊗ Fn =: Sn . Smoothness in x for the scalar (bosonic) part of ψx follows from our requirement that ωmin (x) > 0, i.e. s > 0 for every x we consider. As for the fermionic part ξx , smoothness follows by considering Fn as a real space of dimension 8n n and, for each point x, viewing ξx as the (up to sign) unique normalized eigenvector of the linear map ξ → W (x)λλ† ξ with eigenvalue − A ωA (x). (A consistent choice of sign can be made because we will only be working on orientable subsets of R7n .) Smoothness of ξx now follows from smoothness of W (x) and the implicit function theorem. Also note that any x-derivatives ∂jA ψx , ∂jA ∂kB ψx , etc. still lie in Sn . ˜ Rx (Rz) = ψx (z), ψx is SU (N )-invariant (covariant) in the sense that Rψ ˜ where R (resp. R) ∈ SU (N ) → SO(n) (resp. Spin(Fn )). This follows from the uniqueness of ψx at each point x and covariance of the operator HD (x), i.e. UR HD (x)UR† = HD (RT x), where U denotes the corresponding unitary representation of SU (N ) on h.

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4.2. Proof of Proposition 2 Since ψx (z) is Gaussian in the z-coordinates, the evaluation of the moments |z|k ψx 2h = |z|2k ψx is straightforward. We find that ψx 2h = ξx 2F = 1, 2

|z|ψx h =

1 , ωA A

2 1 , and ωA A A 2 1 1 1 1 4 2 |z| ψx = k1 + k3 4 + k2 3 2 h ωA ωA ωA ωA A A A A 2 4 1 1 1 + k4 + k5 2 ωA ωA ωA

1 2 2 1 |z| ψx = 3 2 + 2 h 2 ωA

A

A

A

for some combinatorial factors k1 , . . . , k5 . For example, the evaluation of |z|2 Ψx 2h goes as follows 2 2 |z| ψx = |z|4 ψx h

2 2 1/2 1/2 1 = π −n s 2 |u| + |v|2 e−u · S u e−v · S v dn udn v

|u|4 e−u · S u dn u + 2π −n s 2 2 n 1 = 2π − 2 s 4 ˜2B e− C ωC u˜C dn u ˜ u ˜2A u = 2π − 2 s 4 n

1

1/2

A,B

+ 2π −n s

1 2

u ˜2A e−

1

|u|2 e−u · S

1/2

2 u n

d u

2 ˜2C C ωC u

dn u ˜

A

2 1 1·3 1 1 =2 +2 2 +4 22 ωA 2ωA 2ωB 2ωA A A 0. A basis for the Lie algebra i · su(N ) of traceless hermitian N × N ωmin (ˆ matrices is given by N − 1 diagonal ones, hk , together with the off-diagonal eij := Eij + Eji ,

fij := i(Eij − Eji ) ,

(1 ≤ i < j ≤ N ), where Eij denotes the standard basis of matrices. For any diagonal matrix Λ = diag(λ1 , . . . , λN ) we have [Λ, eij ] = −i(λi − λj )fij

and

[Λ, fij ] = i(λi − λj )eij .

(18)

ˆ 1 := diag(m, m − 1, . . . , −m + 1, −m) (or any other traceless diagonal Let e.g. X ˆ 2 := eij . Now, take any fixed matrix with all entries different), and X i<j E= αk hk + βij eij + γij fij ∈ isu(N ) i<j

k

i<j

ˆ 1 and X ˆ 2 . Then, by (18), [X ˆ 1 , E] = 0 implies and require E to commute with both X βij = γij = 0, i.e. E must be diagonal, E = diag(λ1 , . . . , λN ). Again, by (18), ˆ 2 ] = 0 implies (λi − λj )fij = 0, i.e. all λi are equal. Tracelessness then [E, X i<j implies that E = 0. Hence, ˆ 1 , E]2 + [X ˆ 2 , E]2 > 0 [X for all E = 0, and since S n−1 is compact it also follows that 2 ˆ 1 , E]2 + [X ˆ 2 , E]2 =: c > 0 , ωmin (ˆ x) ≥ min [X E=1

ˆ1, x ˆ 2 as usual. where x ˆ = (ˆ x1 , x ˆ2 , x3 , . . . , x7 ) ∈ R7n , and x ˆ1 ↔ X ˆ2 ↔ X n n n−1 → R+ , Now, consider the map F : R × R × S F (x1 , x2 , e) := [X1 , E]2 + [X2 , E]2 . ˆ2 , · ) ≥ c. Furthermore, note that for any R ∈ We know from the above that F (ˆ x1 , x x1 , Rˆ x2 , · ) SU (N ) → SO(n), since F (Rx1 , Rx2 , e) = F (x1 , x2 , RT e), we have F (Rˆ ≥ c > 0 as well. Then, because F is continuous and S n−1 compact, there exists x1 ) and an R > 0 such that F (x1 , x2 , · ) ≥ c/2 for all x1 , x2 in the balls BR (Rˆ x2 ), respectively. Also, by compactness of SU (N ), there is an > 0 such BR (Rˆ that c ∀(x1 , x2 ) ∈ B (Rˆ F (x1 , x2 , · ) ≥ x1 ) × B (Rˆ x2 ) ∀R ∈ SU (N ) . 2 Therefore, defining D1 := B (Rˆ x1 ) × B (Rˆ x2 ) × B1 (0)5 ⊆ R7n , R∈SU (N )

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we find that ωmin (x) ≥ c/2 for all x ∈ D1 . Furthermore, we have (|ˆ x1 | − )2 + (|ˆ x2 | − )2 ≤ |x|2 ≤ (|ˆ x1 | + )2 + (|ˆ x2 | + )2 + 5 on D1 . By rescaling this set (note that F is homogeneous of degree 2), Dt := tD1 , 1 we reach the conditions (15). It is also useful to note that ωmin ≤ ωmax = S 2 op ≤ c13 |x| (where · op denotes the operator norm). 4.3.2. Construction of the function χt . We set χt (x1 , . . . , x7 ) := μt (x1 , x2 )ηt (x3 ) . . . ηt (x6 )ζt (x7 ) , where μt , ηt and ζt are to be defined below. Given some spherically symmetric bump function η ∈ C0∞ (Rn ) with support on the unit ball B1 (0) and unit L2 -norm, ηL2 (Rn ) = 1, we define ηt (x) := t−n/2 η(x/t) so that supp ηt ⊆ Bt (0), ηt L2 (Rn ) = 1, and ∂ α ηt L2 (Rn ) = t−|α| ∂ α ηL2 (Rn ) for any partial derivative multi-index α. The function ζt is chosen to be asymptotically a gauge invariant solution to the Helmholtz equation, namely we take (for the case λ = 0 we instead take ζt := ηt ) ζt (x) := At ρt (x)h(x) , −1 where At := ρt hL2 (Rn ) , ρt is a cut-off function ρt (x) := ρ(x/t) such that ρ ∈ C0∞ (Rn ) is spherically symmetric, 0 ≤ ρ ≤ 1, ρ = 1 on B1/2 (0) and ρ = 0 outside B1 (0), and 1 1 h(x) := 1 sinn−3 (λ 2 |x|) λ 2 |x| satisfies (Δ + λ)h = 0 (see [42]), with sinp (x) :=

∞ k=0

(−1)k x2k+1 = cp x(1−p)/2 J(1+p)/2 (x) . 2(3 + p)4(5 + p) · · · 2k(2k + 1 + p)

Since the Bessel functions Jk behave asymptotically as [1] 1 2 π Jk (x) = cos x − (2k + 1) + O , πx 4 x one finds

R

h2 dx = c14 BR (0)

and

0

1

2 J(n−2)/2 (λ 2 r)rdr = c15 R + o(R)

|∂A h|2 dx ≤ c16

BR (0)

R

1

2 Jn/2 (λ 2 r)rdr ≤ c17 + c18 R . 0

Hence, At ≤ c19 /t1/2 → 0, t → ∞, and ∂A ζt L2 (Rn ) ≤ At (∂A ρt )hL2 (Rn ) + ρt ∂A hL2 (Rn ) c19 c20 (c15 t)1/2 + (c17 + c18 t)1/2 → c21 . ≤ 1/2 t t

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2 Furthermore, ( A ∂A + λ)ζt = At (Δρt )h + 2At A (∂A ρt )(∂A h), so that c19 c22 c20 1/2 1/2 (c (c t) + 2n + c t) (−Δ − λ)ζt L2 (Rn ) ≤ 1/2 → 0. 15 17 18 t2 t t Lastly, we set μt (x1 , x2 ) := t−n μ−1 L2 (R2n ) μ(x1 /t, x2 /t), where

η (x1 − Rˆ x1 )η (x2 − Rˆ x2 ) dμH (R) μ(x1 , x2 ) := R∈SU (N )

and μH denotes some Haar measure on SU (N ). Then μt ∈ C0∞ (R2n ), supp μt × Bt (0)5 ⊆ Dt , μt L2 (R2n ) = 1, ∂ α μt L2 (R2n ) ≤ cα /t|α| , and μt (Rx1 , Rx2 ) = μt (x1 , x2 ) for all R ∈ SU (N ). Hence, χt ∈ C0∞ (R7n ) is SU (N )-invariant, supp χt ⊆ Dt , and χt 2L2 (R7n ) = μt 2L2 (R2n ) ηt 8L2 (Rn ) ζt 2L2 (Rn ) = 1 . Furthermore, as t → ∞, ∂jA χt L2 (R7n )

⎧ ⎨ ∂jA μt L2 (R2n ) ≤ c23 /t → 0 , ∂A ηt L2 (Rn ) ≤ c24 /t → 0 , = ⎩ ∂A ζt L2 (Rn ) ≤ c21 ,

j = 1, 2 j = 3, 4, 5, 6 j=7

and (−Δx − λ)χt L2 (R7n ) ≤ Δ(x1 ,x2 ) μt L2 (R2n ) +

6

Δxj ηt L2 (Rn )

j=3

+ (Δx7 − λ)ζt L2 (Rn ) → 0 .

4.4. Proof of Lemma 4 2 Here, we will denote the partial derivatives ∂jA ψx , ∂jA ψx by ψx and ψx , respectively. Since ψx is a zero-energy state of HD , we have in particular that ∂jA (HD ψx ) = 0, which can be equivalently written as −HD ψx = z¯ · S (x)z + 2W (x)λλ† ψx =: Φx . In the following we will need an estimate on the norm of Φx ∈ Sn . We have, using Propositions 2 and 3, z · S (x)zψx h + 2 W (x)λλ† ψx h Φx h ≤ ¯ ≤ S (x)op(Rn ) |z|2 ψx h + 2 W (x)λλ† ψx op(F )

≤ c25 |x|

h

t C2 + c26 ≤ c27 + c26 =: c5 . ωmin (x) t

¯ D )⊥ =: P+ where H ¯ D denotes the self-adjoint Now, we note that Φx ∈ (ker H ¯ extension of HD . To see this, consider any Φ0 ∈ ker HD . We have ¯ D ψ , Φ0 h = −ψ , H ¯ Φ0 h = 0 . Φx , Φ0 h = −H x x D =0

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¯ D on P+ is 2ωmin > 0, we have Therefore, since the lowest eigenvalue of H ¯ D |−1 op(h) ≤ 1 and H P+ 2ωmin ¯ −1 Φx h ≤ H ¯ D |−1 op(h) Φx h ≤ c5 . ψx h = H D P+ ωmin 2 (HD ψx ) = 0, i.e. Similarly, taking the second derivative we have ∂jA ˜x . −HD ψx = z¯ · S (x)zψx + 2 z¯ · S (x)z + 2W (x)λλ† ψx =: Φ

˜ x ∈ P+ ∩ Sn , but we will also need an estimate on Just as above, we see that Φ |z|2 ψx h . For this, we recall from the proof of Proposition 2 that 2 1 3 1 tr S − 2 =: T (x) , |z|2 ψx , |z|2 ψx h = tr S −1 + 2 2 Note that T is smooth for s > 0 and homogeneous of degree −2, because S is homogeneous of degree 2. It follows that, if x = re with e ∈ S 7n−1 , then T (x) = r−4 T (e) and 2 1 2 Re |z|2 ψx , |z|2 ψx h + 2 |z|2 ψx h = T (x) ≤ 4 sup |T | , r K0 where K0 := S 7n−1 ∩ {x ∈ R7n : s(x) ≥ (c1 /c3 )2n } is compact and we have used that ωmin /|x| ≥ c1 /c3 by Proposition 3. Hence, 2 2 2 |z| ψx ≤ c28 + ψx |z|4 ψx ≤ c29 1 + ωmin ψx h , h 4 h h |x|4 ωmin so that −1 ˜ Φx h ψx h = HD 1 c30 |z|2 ψx h + c31 S op |z|2 ψx h + c32 ψx h ≤ ωmin 1 c30 C2 c33 |x| c32 c5 1 2 1 + ωmin + 2 ψx h 2 + ≤ ωmin ωmin ωmin ωmin 2 and thus, ωmin ψx h ≤ c6 for some constant c6 .

5. Summary We have introduced G2 ×U (1)×SU (N ) invariant matrix models as deformations of the standard Spin(9)×SU (N ) invariant models by conjugating a supercharge with a cubic, octonionic, exponential. Furthermore, similarly to what has been shown for the original models, we have proved that the spectrum of the corresponding ˜ covers the whole positive half-axis by finding sequences of states Hamiltonian H ˜ − λ for any λ ≥ contradicting existence of a bounded inverse to the operator H 0. However, contrary to the case for the original models, we have constructed such sequences explicitly, without fixing the gauge. Making use of the convenient ˜ we could configure the states to annihilate some structure of terms appearing in H, terms, while, related to having the possibility of making the lowest eigenvalue of a

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certain frequency matrix S arbitrarily large, other terms could be made arbitrarily small – using a gauge invariant asymptotic solution to the Helmholtz equation, with support on a set of matrices that are not simultaneously block-diagonalizable.

Acknowledgements This work was supported by the Swedish Research Council and the Marie Curie Training Network ENIGMA (contract MRNT-CT-2004-5652). J. Hoppe and D. Lundholm would like to thank V. Bach for collaboration on related subjects. D. Lundholm would also like to thank H. Kalf, L. Svensson and M. Bj¨ orklund for discussions.

Appendix A In this appendix we give notation and conventions used in the paper (cp. e.g. [28]). The supermembrane matrix theory is a quantum mechanical model with N = 16 supersymmetries, SU (N ) gauge invariance and Spin(9) symmetry. The theory involves real bosonic variables xsA (coordinates) and real fermionic ones θαA (Majorana spinors) with s = 1, . . . , 9, α = 1, . . . 16 and A = 1, . . . , N 2 − 1 spatial, spinor and color indices respectively. The corresponding supercharges and the Hamiltonian of the model are 1 1 s st + fABC xsB xtC γαβ θβA , γ st = [γ s , γ t ] , Qα = psA γαβ 2 2 1 s θαA θβB xsC , H = psA psA + (fABC xsB xtC )2 + ifABC γαβ 2 i s xsA JA , JA = fABC xsB psC − θαB θαC . (19) {Qα , Qβ } = δαβ H + 2γαβ 2 Here psA are momenta conjugate to xsA , [xsA , ptB ] = iδst δAB , γ s are 16 × 16 dimensional, real matrices s.t. {γ s , γ t } = 2δ st 116×16 , θαA are Grassmann numbers s.t. {θαA , θβB } = δαβ δAB , and fABC are SU (N ) structure constants (real, anti2 symmetric). The operators are defined on the Hilbert space H = L2 (R9(N −1) )⊗F, where F is the irreducible representation of θ’s, while the physical (gauge invariant) Hilbert space consists of states |ψ satisfying JA |ψ = 0 which corresponds to the Gauss law in unreduced N = 1 super Yang–Mills theory. Such singlet constraint is an essential requirement for the model to be supersymmetric which is apparent in Eq. (19). However, the necessity of the constraint follows also from simply counting the fermionic and bosonic degrees of freedom. Let us consider the Fock space formulation of the model. For the case at hand 2 there are 9(N 2 − 1) bosonic degrees of freedom, however there are 16 2 (N − 1) 2 fermionic ones. The mismatch is equal to N − 1, which is exactly the number of constraints coming from the Gauss law.

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There are many ways in which one can single out 8 out of 16 fermions (which is required in order to obtain an irreducible Fock representation F). We will follow the convention in [45] and introduce complex spinor variables λαA := √12 (θαA + iθ8+α A ) i.e.3 1 θαA = √ (λαA + λ†αA ) , 2

1 θα+8 A = √ (λαA − λ†αA ) . i 2

We then also split the coordinates xsA into (xjA , zA , z¯A ) where zA = x8A + ix9A and j = 1, . . . , 7. After this is done the Spin(9) symmetry of (19) is not explicit, however now an arbitrary wavefunction Ψ(x, z, z¯) can be written as Ψ(x, z, z¯) = ψ + ψαA λαA +

1 ψαA βB λαA λβB + . . . , 2!

with ψα1 A1 ...αl Al complex-valued and square integrable. The above sum is finite and truncates when the number of fermions is more than 8(N 2 − 1).4 It now follows that the Hamiltonian (19) can be written in terms of nonhermitian (“cohomology”) charges Qα := √12 (Qα + iQ8+α ), i ∂ 1 − fABC xjB xkC Γjk Qβ = iδαβ fABC zB z¯C + iΓjαβ αβ λαA 2 ∂xjA 2 ∂ + 2δαβ − ifABC xjB z¯C Γjαβ λ†αA , (20) ∂zA so that, on the physical Hilbert space, {Qα , Q†β } = δαβ H ,

{Qα , Qβ } = 0 ,

{Q†α , Q†β } = 0 .

Here, Γj are 8 × 8, purely imaginary, antisymmetric matrices satisfying {Γj , Γk } = 2δ jk 18×8 . We have chosen the following representation of γ s matrices 0 iΓj 0 18×8 18×8 0 8 9 γj = = = , γ , γ , −iΓj 0 18×8 0 0 −18×8 implying

γ

jk

=

γ

j9

=

Γjk 0 0 −iΓj

0 Γjk

,

−iΓj 0

γ

j8

=

,

γ

89

=

iΓj 0 0 18×8

0 −iΓj

−18×8 0

, ,

3 Other choices of 8 fermions are possible, e.g. Majorana–Weyl spinors (see [47]). From now on the spinor indices α, β, . . . run from 1 to 8. † 4 Note that in this notation λ αA is a fermionic creation operator while λαA fermionic annihilation operator.

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and

γ

jkl

=

γ jk9 =

iΓjkl , 0 0 , −Γjk


0 −iΓjkl

γ

Γjk 0

γ j89 =

jk8

=

0 Γjk

Γjk 0

iΓj 0

0 iΓj

, ,

where γ st := 12 [γ s , γ t ], γ stu := 16 (γ s [γ t , γ u ] + cycl.) and Γjk , Γjkl respectively. It is here where the octonions enter, in choosing the representation iΓjα8 = δαj , j iΓkl = −cjkl with totally antisymmetric octonionic structure constants.5 This is also natural from the view of representation theory of Clifford algebras since the representations of Γj are uniquely given by left or right multiplication on the octonion algebra (see e.g. [32]). Furthermore, because the automorphism group of the octonions is given by the exceptional group G2 (which is also the subgroup of ˆ and H ˜ Spin(7) fixing a chosen spinor index), the deformed Hamiltonians Hk , H, will be G2 invariant.

Appendix B Starting from the 9-dimensional Fierz identity (see e.g. [2]) s st s st s st t t γαβ γαst β + γαs β γαβ + γαβ γα β + γα β γαβ = 2(δαα γββ − δββ γαα ) ,

which holds for all t = 1, . . . , 9, α, α , β, β = 1, . . . , 16, and using the representation in Appendix A with α, α , β = 1, . . . , 8, β = 9, . . . , 16 (then redefining β := β − 8), we obtain the corresponding 7-dimensional Fierz identity j jk k k k k k Γjαβ Γjk α β + Γα β Γαβ = δαβ Γα β + δα β Γαβ − δαβ Γα β − δα β Γαβ − 2δαα Γββ

for all k = 1, . . . , 7, α, α , β, β = 1, . . . , 8. From this identity it follows that j jk k k k k Γjαβ Γjk α β − Γα β Γαβ = −2(δαα Γββ + δββ Γαα − δα β Γαβ + δαβ Γα β ) .

Multiplying this equation with Γlβ β˙ , summing over β , and taking α = β = β˙ to be fixed, we obtain = Γkl . Γjαβ˙ Γjkl αβ˙ β˙ β˙

Appendix C In this appendix we consider deformed Hamiltonians from a more general viewpoint and show how one could be led to the particular deformation considered in this paper. 5 Explicitly,

cijk = +1 for (ijk) = (123), (165), (246), (435), (147), (367), (257).

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Let us consider the algebra of N > 1 supersymmetric quantum mechanics, {Qα , Qβ } = δαβ H, and the corresponding cohomology supercharges 6 1 Qαβ := √ (Qα + iQβ ) , 2 We have {Qαβ , Qμν } = 0 ,

1 Q†αβ = √ (Qα − iQβ ) . 2

{Qαβ , Q†μν } = δ(αβ)(μν) H .

The deformed Hamiltonian Hαβ (k) := {Qαβ (k), Q†αβ (k)} (no sum over α,β) given by deformed cohomology supercharges Qαβ (k) := ekg(x) Qαβ e−kg(x) , where k ∈ R, and g(x) is some operator s.t. [Qβ , g(x)] commutes with g(x), becomes

2 Hαβ (k) = H − 2ik Qα , Qβ , g(x) − 2k 2 Qβ , g(x) . Substituting the supercharges (19) for the particular model considered here, we obtain 2 st Hαβ (k) = H + k 2 ∂sA g(x) + kγαβ ∂sA g(x)ptA − ∂tA g(x)psA − k(γ st γ u )αβ fABC xsA xtB ∂uC g(x) s t + 2ik∂sA ∂tB g(x)γαα γββ θα A θβ B .

Now, say we are interested in a particular deformation where g(x) is cubic in x (so that (∂g)2 is quartic). Because γ stu is totally antisymmetric, a natural choice is 1 stu , g(x) = fABC xsA xtB xuC γαβ 6 with α < β. Taking e.g. (α, β) = (8, 16) and choosing the representation of γ s matrices as in Appendix A, we find that 1 1 g(x) = fABC xjA xkB xlC iΓjkl 8,8 = cjkl fABC xjA xkB xlC , 6 6 and that H8,16 (k) becomes precisely Hk in (4).

References [1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1968. [2] M. Baake, P. Reinicke, V. Rittenberg, Fierz identities for real clifford algebras and the number of supercharges, J. Math. Phys. 26 (1985), 1070. [3] V. Bach, J. Hoppe, D. Lundholm, Dynamical symmetries in supersymmetric matrix models, Documenta Math. 13 (2008), 103–116, arXiv:0706.0355 [4] T. Banks, W. Fischler, S. Shenker and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D55 (1997), 6189, arXiv:hep-th/9610043. 6 We consider the (αβ) as distinct pairs of indices with α and β in disjoint subsets of the index set. This is a common construction of non-hermitian charges involving the complex structure i, but other variations are possible; see e.g. [33] and references therein.

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[26] J. Hoppe, On the construction of zero energy states in supersymmetric matrix models II, arXiv:hep-th/9709217. [27] J. Hoppe, On the construction of zero energy states in supersymmetric matrix models III, arXiv:hep-th/9711033. [28] J. Hoppe, Membranes and Matrix Models, arXiv:hep-th/0206192. [29] V. G. Kac, A. V. Smilga, Normalized vacuum states in N = 4 supersymmetric Yang– Mills quantum mechanics with any gauge group, Nucl. Phys. B571 (2000), 515–554, arXiv:hep-th/9908096. [30] A. Konechny, On asymptotic Hamiltonian for SU(N) matrix theory, JHEP 9810 (1998), 018, arXiv:hep-th/9805046. [31] W. Krauth, H. Nicolai, M. Staudacher, Monte Carlo approach to m-theory, Phys. Lett. B431 (1998), 31-41, arXiv:hep-th/9803117. [32] H. B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, Princeton, New Jersey, 1989. [33] D. Lundholm, On the geometry of supersymmetric quantum mechanical systems, J. Math. Phys. 49 (2008), 062101, arXiv:0710.2881. [34] M. L¨ uscher, Some analytic results concerning the mass spectrum of Yang–Mills gauge theories on a torus, Nucl. Phys. B219 (1983), 233–261. [35] G. Moore, N. Nekrasov, S. Shatashvili, D-particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000), 77–95, arXiv:hep-th/9803265. [36] M. Porrati, A. Rozenberg, Bound states at threshold in supersymmetric quantum mechanics, Nucl. Phys. B515 (1998), 184–202, arXiv:hep-th/9708119. [37] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators, Academic Press, New York, 1978. [38] S. Sethi, M. Stern, D-brane bound states redux, Commun. Math. Phys. 194 (1998), 675–705, arXiv:hep-th/9705046. [39] S. Sethi, M. Stern, Invariance theorems for supersymmetric Yang–Mills theories, Adv. Theor. Math. Phys. 4 (2000), 487–501, arXiv:hep-th/0001189. [40] B. Simon, Some quantum mechanical operators with discrete spectrum but classically continuous spectrum, Ann. Phys. 146 (1983), 209–220. [41] A. Smilga, Super Yang Mills quantum mechanics and supermembrane spectrum, Proc. 1989 Trieste Conf. M. Duff, C. Pope and E. Sezgin (eds.) (Singapore: World Scientific). [42] M. Trzetrzelewski, Large N behavior of two dimensional supersymmetric Yang–Mills quantum mechanics, J. Math. Phys. 48 (2007), 012302, arXiv:hep-th/0608147. [43] M. Trzetrzelewski, The number of gauge singlets in supersymmetric Yang–Mills quantum mechanics, Phys. Rev. D76 (2007), 085012, arXiv:0708.2946. [44] B. de Wit, W. L¨ uscher, H. Nicolai, The supermembrane is unstable, Nucl. Phys. B320 (1989), 135–159. [45] D. de Wit, J. Hoppe, H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B305 [FS23] (1988), 545–581. [46] E. Witten, Bound states of strings and p-branes, Nucl. Phys. B460 (1996), 335–350, arXiv:hep-th/9510135.

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[47] J. Wosiek, On the SO(9) structure of supersymmetric Yang–Mills quantum mechanics Phys. Lett. B619 (2005), 171–176, arXiv:hep-th/0503236. [48] J. Wosiek, Supersymmetric Yang–Mills quantum mechanics in various dimensions, Int. J. Mod. Phys. A20 (2005), 4484–4491, arXiv:hep-th/0410066. [49] P. Yi, Witten index and threshold bound states of d-branes, Nucl. Phys. B505 (1997), 307–318, arXiv:hep-th/9704098. Jens Hoppe and Douglas Lundholm Department of Mathematics Royal Institute of Technology, KTH S-100 44 Stockholm Sweden e-mail: [email protected] [email protected] Maciej Trzetrzelewski Department of Mathematics Royal Institute of Technology KTH, 100 44 Stockholm Sweden and Institute of Physics Jagiellonian University Reymonta 4 P-30-059 Krak´ ow Poland e-mail: [email protected] Communicated by Yosi Avron. Submitted: November 5, 2008. Accepted: January 19, 2009.



Unbounded Orbits for Semicircular Outer Billiard Dmitry Dolgopyat and Bassam Fayad Abstract. We show that the outer billiard around a semicircle has an open ball escaping to infinity.

1. Introduction An outer billard map F is defined outside a closed convex curve Γ in the following way. Let z be a point on the plane. Consider the supporting line L(z) from z to Γ such that Γ lies on the right of L. F (z) lies on L(z) so that the point of contact divides the segment [z, F (z)] in half. If Γ contains segments then F (z) is not defined if L(z) contains a segment. In this case F (z) is defined almost everywhere but it is discontinuous. In [14] Moser outlined the proof of the fact that if Γ is smooth and strictly convex then all trajectories are bounded (the complete proof (for C 6 -curves) was later given in [3]). Moser also asked [14, 15] what happens if Γ is only piecewise smooth, mentioning in particular that even in the case where Γ is quadrangle, the question of boundedness of the orbits was open. The majority of the subsequent papers on this subject dealt with the most degenerate case when Γ is a polygon. In this case F (z) is obtained from z by reflection around a vertex. In [11,12,23] boundedness of the trajectories was proved for the so called quasi-rational polygons, a class including rational polygons as well as regular n-gones. Since affine equivalent curves have conjugated outer billiards all triangular outer billiards have bounded (in fact, periodic) orbits. It was proved in [5] that if Γ is a trapezoid then all trajectories are bounded. Schwartz [18, 19] considers kites – quadrangles with vertices (−1, 0), (0, 1), (A, 0) and (0, −1), and proves that for all irrational A, there exists an unbounded orbit (if A is rational then the orbits are bounded since the kite is rational). It is believed that unbounded orbits exist for almost all N -gonal outer billiards for N ≥ 4 but this question is far from settled. In particular it is unknown if there exists a polygon with an infinite

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z0 z2

z3

z1 z4 Figure 1. Four iterations of semicircular outer billiard. measure of unbounded orbits (it seems that for kites almost all orbits are periodic but that case is quite special because a foliation by lines parallel to the x axis is preserved by F 2 ). An intermediate case that consists of curves containing both segments and strictly convex pieces received much less attention. Numerical simulations reported in [21] show that for the outer billiard around a semicircle there is a large set of unbounded trajectories. In this paper we show that this numerical conclusion is indeed correct. Theorem 1. The outer billiard around a semicircle has an open ball escaping to infinity.

2. Main ingredients In all what follows, the billiard curve Γ will be fixed for definiteness to be the semicircle given by the upper part of the unit circle of R2 , and F will denote the outer billiard map around Γ. Let 1 denote the halfline {x ≥ 1, y = 0}. Let D be the infinite region bounded by 1 , F 2 1 and {x = x0 } where x0 is a large constant. We shall show in the appendix that F 2 is a graph of a function y = h(x) where h(x) = 2 + O(1/x2 ). Thus D = (x, y) : x ≥ x0 , 0 ≤ y ≤ h(x) . Denote by F the first return map to D under F 2 . Theorem 1 will of course follow if we show the existence of balls that escape to infinity under the iteration by F. The proof of this fact consists of two parts. We can rescale the coordinates in D and think of it as a cylinder (where the boundaries are identified by F 2 ). A further change of coordinates allows to derive a normal form expression for the return map F that consists of a periodic ¯ : D → D, and an asymptotic expansion in powers of (piecewise) linear part L

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1/R, where R denotes the radial coordinate in the fundamental domain (we call a map of D periodic if it commutes with integer translations of R). The normal form will have singularity lines corresponding to the discontinuities of F 2 and of its derivative that result from the flat piece and the corners in Γ. This normal form is presented at the end of this section while its proof, based on lengthy but straightforward computations, is deferred to Appendix A. ¯ = Lπ ˆ where π is ¯ is periodic there is a map L ˆ : T2 → T2 such that π L Since L the projection of the cylinder D to the torus. Hence, we can reduce the dynamics ¯ to a dynamics on T2 and look for an escaping orbit on D which projects to of L the simplest orbit on T2 , namely a periodic orbit. In our case, we will exhibit a ˆ that moves up in D by two units in the R direction under each fixed point of L ¯ iteration of L. The question is hence that of the stability of this orbit when the additional terms of the normal form are considered, which constitutes the second ˆ happens part of the proof. The first crucial observation is that the linear part L ¯ to be elliptic so that the escaping orbit of L is actually accompanied by a small ball around it. ¯ the radial coordinate R of the points in the escaping ball goes Under L, to infinity linearly with time, and viewed on T2 , our problem becomes similar to that of establishing the stability of a periodic point under a time dependent perturbation. If the perturbation was independent of time, it would be possible to derive the result from Moser’s theorem on the stability of elliptic fixed points. Time dependent perturbations were studied in [4, 16] but there it was assumed that the perturbation vanishes at the fixed point. This is not true in our case, however there are two features which considerably simplify the problem. (1) The unperturbed map is globally linear so that the approximation by the linear map does not become worse as we move away from the fixed point. (2) Rather than a general smooth function, the perturbation has a special form. Namely, if we use a coordinate system centered around the escaping orbit ¯ the main term of our return map will be the linear elliptic map L, ¯ of L, 1 ¯ the term of order O( R¯ n ), where Rn ∼ 2n is the radial coordinate at time n, is quadratic, the term of order O( R¯12 ) = O( n12 ) is cubic and so on (see n

Lemma 2). The O( n12 )-terms clearly do not alter the stability displayed in the linear picture, and the O( n1 )-perturbation contains only one resonant term which may cause divergence (rather than infinitely many resonant terms which might have appeared for a general smooth perturbation). In our case, the only resonance is related to the fact that the small perturbation of an area preserving map can be area contracting or area expanding (and in the latter case stability is clearly impossible). Stability is thus insured by the nullity of the resonant term, which we obtain a posteriori due to the area preservation property of the outer billiard map. Finally, a particular attention has also to be given to the verification that the candidate escaping ball stays away from the singularity lines of the map.

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Figure 2. Dynamics of L. R decreases on the shaded parallelograms and increases on white parallelograms. The thick solid lines are discontinuities of L. The thick dashed lines are singularities of higher order terms which are invisible in the linear approximation. Also shown is an escaping orbit which projects to a constant orbit of T2 . We now specify with two statements the two principal moments of the proof discussed above. Define on the cylinder C = [R0 , ∞) × T the map 4 8 L(R, φ) = R − + {φ − R}, {φ − R} . 3 3 We will use the notation [x] for the integer part of x and {x} for its fractional part x − [x]. Lemma 2. There exists a smooth change of coordinates G : (x, y) ∈ D → (R, φ) ∈ C = [R0 , ∞) × T of the form 1 1 1 2 y R= x− +O , φ= +O (1) 3 6 x 2 x so that the following holds.1 (a) The singularities of F are O(1/R) close to one of the following curves • the singularities of L2 • {φ − R2 } = 14 and {φ − R2 } = 34 ˜ ˜ ˜ = L(R, θ). ˜ φ) • {φ˜ − R2 } = 14 and {φ˜ − R2 } = 34 where (R, (b) If (R, φ) is O(1/R) far from the singularities then F(R, φ) = L2 (R, φ) + P {R}, φ /[R], Q {R}, φ /[R] + O(1/R2 ) where P, Q : [0, 1] × T → R are piecewise polynomials of degree 2 (c) The map F preserves a measure with density 1 + W (φ)/R + O(R−2 ) where W is an affine function. Lemma 3. Any map satisfying the conclusions of Lemma 2 has an open ball of points escaping to infinity. 1 We

use the same notation for F and G ◦ F ◦ G−1 .

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Remark. A more explicit expression for G in Lemma 2 is given by formulas (14), (15), (18), (19), (21) and (23) of Appendix A. The reader will notice that those formulas are rather cumbersome however for our proof (that is for Lemma 3) we only need the properties listed in Lemma 2. Lemma 3 is proven in Section 3. We follow [16] but our case is simpler since we deal with a perturbation of the linear system and the first order perturbative terms are polynomials while [16] considers an arbitrary perturbation. The proof of Lemma 2 is given in Appendix A.

3. Construction of unbounded orbits Following [16] we first observe that the limit map L2 has an escaping orbit given by Rn = 2n and φn = φ0 = 7/8, namely L2 (Rn , φ0 ) = (Rn+1 , φ0 ). Notice that the latter escaping orbit remains away from the singularity lines. Define 5 8 −3 3 L = dL = . −1 1 Notice that the trace of L is equal to −2/3 which implies that it is elliptic, and so is L2 . Hence a full ball will accompany the escaping point to infinity. To deal with the higher order perturbative terms, a first observation is that F has no singularities in balls of sufficiently small but fixed radius around the escaping points (2n, φ0 ). We will therefore consider a point {RN , φN } in a small neighborhood of {2N, φ0 } and study its dynamics. For n ≥ N , we will denote {Rn , φn } the n − N iterate of {RN , φN }, and introduce Un = Rn − 2n, υn = φn − φ0 . Let s be such that cos(2πs) = −1/3. We can introduce a suitable complex coordinate zn = Un + i(aUn + bυn ) such that DF becomes a rotation by angle 2πs near the origin. In these coordinates F takes the following form in a small neighborhood of (0, 0) zn+1 = ei2πs zn +

A(zn ) + O(N −2 ) N

(2)

where A(z) = w1 + w2 z + w3 z¯ + w4 z 2 + w5 z z¯ + w6 z¯2 . Lemma 4. (a) We have that Re(e−i2πs w2 ) = 0. (b) There exists √ > 0 and a constant C such that if |zN | ≤ , then for every n ∈ [N, N + N ] |zn | ≤ |zN | + CN −1 . Part (b) is the main result of the lemma. Part (a) is an auxiliary statement needed in the proof of (b). Namely, part (a) says that the resonant coefficient mentioned in Section 2 vanishes. Before we prove this lemma, let us observe that it implies that √ for sufficiently large N , all the points |zN | ≤ /2 are escaping orbits. Indeed by [ N ] applications of lemma 4 there is a constant C such that 1 |zl | ≤ + CN − 2 2

362



for every l ∈ [N, 2N ]. It now follows by induction on k that if l ∈ [2k N, 2k+1 N ] then |zl | ≤ k where

j k C 1 √ √ k = + 2 2 N j=0

(N has to be chosen large so that k ≤ for all k). This proves Lemma 3. Proof of Lemma 4. Let n ¯ = n − N. For n ¯≤ zn = ei2πn¯ s zN +

√ N equation (2) gives

n ¯ −1 3 1 i2πms e A(ei2π(¯n−m−1)s zN +¯n−m−1 ) + O(N − 2 ) . N m=0

In particular for these values of n we have zn = ei2πs(n−N ) zN + O

1 √ N

(3)

.

Substituting this into (3) gives zn = ei2πn¯ s zN +

n ¯ −1 1 1 i2πms e A(ei2π(¯n−m−1)s zN ) + O . N m=0 N

To compute the sum above expand A as a sum of monomials and observe that n ¯ −1

α β e−i2π(¯n−m−1)s z¯N ei2πms ei2π(¯n−m−1)s zN

m=0

is bounded for α + β ≤ 2 unless α = β + 1 (that is α = 1, β = 0). Therefore n ¯ ˜2 + O N −1 zn = ei2πn¯ s zN 1 + w N

(4)

where w ˜2 = e−i2πs w2 . Consider now the disc DN around 0 of radius N −0.4 . Then by (4) Area(F n¯ DN ) n ¯ = 1 + 2Re(w ˜2 ) + O N −0.6 . Area(DN ) N On the other hand there exists z ∈ DN such that denoting z = F n¯ z we have 1 + W (z)/N Area(F n¯ DN ) = + O N −2 = 1 + O N −1.4 Area(DN ) 1 + W (z )/(¯ n + N) since W (z) − W (z ) = O N −0.4 . Comparing those two expressions for the ratio of areas we obtain that Re(w ˜2 ) = 0. This proves part (a) of Lemma 4. Part (b) now follows from (4).

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4. Open questions In this paper we consider a very simple piecewise smooth curve – the semicircle and prove that there is a positive measure set of escaping orbits. This is just the first step in the study of outer billiards around piecewise smooth curves. In the current section we discuss some of the immediate questions raised by this work. Question 1. Prove that unbounded orbits exist generically for the following classes of curves (a) circular caps; (b) curves consisting of finitely many strictly convex pieces and finitely many segments; (c) unions of two circular arcs. We observe that since our proof depends on the existence of an elliptic fixed point for a certain auxiliary map which is an open condition we also obtain the existence of unbounded orbits for caps close to the semicircle. However the limiting cases of caps close to the full circle appear to be much more difficult. It is not difficult to extend Lemma 2 to small piecewise smooth perturbations of the identity in the plane corresponding to curves (a)–(c) above (see the Appendix), however proving that the limiting map has unbounded orbits is more complicated. Question 2. Let P be the set of maps of the cylinder R × T of the form F(R, φ) = id + L {R}, φ where L is piecewise linear which are invertible and area preserving. Is it true that a generic element of P has unbounded orbits? We observe that the affirmative answer to Question 2 would be a significant step in answering cases (a) and (b) of Question 1, however it would give little for case (c) (because in cases (a) and (b) the outer billiard map is discontinuous while in case (c) it is continuous but not smooth). Another interesting direction of research is to describe different possible types of behavior for outer billiards. In particular we say that the orbit {zn } is oscillatory if lim sup |zn | = +∞ , lim inf |zn | < ∞ . We say that an oscillatory orbit is erratic if in addition lim inf d(zn , Γ) = 0 . We observe that unbounded orbits constructed in [18, 19] are erratic and it is conjectured there that for outer billiard around kites every orbit is either periodic or erratic. Question 3. Does generic outer billiard for classes (a)–(c) of Question 1 have (a) oscillatory (in particular erratic) orbits; (b) infinite measure of bounded (in particular quasiperiodic) orbits; (c) bounded non-quasiperiodic orbits?

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Concerning parts (b) and (c) of Question 3 we observe that the semicircular outer billiard has infinitely many elliptic periodic orbits close to R ∈ N, φ = 1/2 (that are periodic points for the linear part) however verifying their KAM stability requires checking a nonzero twist condition, and thus computing an asymptotic of the orbits with a higher precision than what it is done in the Appendix. As it is the case for Question 1, the natural first step in investigating Question 3 is to study the limiting maps of Question 2. In case the limiting map is elliptic the dynamics is piecewise isometric for a suitable metric. We refer the reader to [7] for a survey of general properties of piecewise isometries and to [8] and references wherein for an interesting case study with an emphasis on existence of unbounded orbits. Question 4. Does there exist a curve Γ such that the limiting map of Lemma 2 has hyperbolic linear part? Thus Question 4 raises the problem of existence of a curve such that the outer billiard dynamics is chaotic near infinity. We note that [6] gives an example of a curve such that the outer billiard is chaotic near the curve itself. See also [2, 10] for the discussion of unstable orbits near the boundary of piecewise smooth outer billiards. In conclusion we mention that there are several mechanical systems (with collisions) of which the dynamics for large energies is given by a small piecewise smooth perturbation of an integrable map (see [2, 9, 20, 24], for example) and all the questions discussed here are interesting for these systems as well.

Appendix A. Normal form Here we prove Lemma 2. The proof consists of three parts. In Subsection A.1 we derive the formulas for F 2 in polar coordinates. In Subsection A.2 we make a coordinate change to simplify the expression of F 2 . Our approach follows closely the computations of the normal form for small perturbations of the id (cf [1, 13]), however the resulting normal form is different due to the presence of singularities. In Subsection A.3 we use the coordinates of Subsection A.2 to compute the first return map of F 2 inside D. A.1. Semicircular outer billiard. Here we obtain the asymptotic expansion of F near infinity. Consider coordinates in which the semidisc is given by {x2 + y 2 ≤ 1, y ≥ 0} . F is piecewise smooth with discontinuities at the following halflines 1 = {x ≥ 1, y = 0} ,

2 = {x = 1, y ≥ 0} ,

3 = {x = −1, y ≤ 0} .

Inside its continuity domains F can be describes as follows:

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V II 2

I

3

1 1

III 2

3

IV

VI Figure 3. Continuity regions for F 2 . between 1 and 2 -reflection about O1 = (1, 0); between 2 and 3 -reflection about a tangency point to the circular part; between 3 and 1 -reflection about O2 = (−1, 0). Let j = F −1 j . Denote by Rj the reflection about Oj . Observe that far from the origin F looks like the reflection about the origin. Therefore we are interested in F 2 which is close to id. F 2 has six continuity domains. Region Region Region Region Region Region

I : between 1 and 3 we have F 2 = T R1 ; V : between 3 and 2 we have F 2 = R2 R1 ; II : between 2 and 1 we have F 2 = R2 T ; III : between 1 and 2 we have F 2 = R1 T ; V I : between 2 and 3 we have F 2 = T 2 ; IV : between 3 and 1 we have F 2 = T R2 .

Thus regions I–IV look like the four coordinate quadrants while regions V and V I are small buffers between them (it is easy to see that when the orbit of F 2 visits the last two regions it leaves them immediately). We call the union of regions I, V and II the upper region and the union of regions III, IV and V I the lower region. We consider coordinates (r, θ) which are polar coordinates in the upper region and polar coordinate shifted by π in the lower region. Thus both in upper and the lower region 0 ≤ θ ≤ π. Our choice is motivated by the wish to make F 2 in the upper and the lower region look similar. We shall need the formulas for j and j in polar coordinates.

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Proposition 5. The discontinuity lines of F 2 are given by the following equations 1 1 π 3 π 1 1 ⊂ {θ = 0} , 3 ⊂ θ = − + O − + O ⊂ θ = , , 2 2 r r2 2 r r2 1 1 2 π 3 − + O ⊂ θ = , , 1 ⊂ θ = π − + O 2 r r2 2 r r2 1 π 1 . 3 ⊂ θ = − + O 2 r r2 Proof. The result for j follows by direct computation. To obtain the result for j observe that the preimage of a halfline (t) = u + vt satisfies F −1 (t) = w − 2u − vt + O(1/t) where w is the vector where the supporting line of the semidisc has slope v (because the midpoint of the segment [(t), F −1 (t)] is O(1/t) close to w). Alternatively one can compute F −1 explicitly and obtain the following parametric equations for j :

2 1 − t, 2 1 − , t≥1 , 1 = t t 1 − 3t2 4t 2 = , − t , t > 0 , 3 = {x = 3, y ≥ 0} . 1 + t 2 1 + t2 Proposition 6. In our coordinates R1 and R2 have the same form given by 1 1 2 sin2 θ 2 sin θ 2 sin 2θ +O + + O R(r, θ) = r − 2 cos θ + , θ + r r2 r r2 r3 and

T (r, θ) =

r, θ +

2 +O r

1 r3

.

Proof. The proof is based on elementary computations that we describe for R1 in the region I, the other cases being similar. Let A be a point in region I and let (r, θ) and (x, y) be its polar and cartesian coordinates respectively. Denote 1/2 A˜ = F (A). Then x ˜ = 2 − x and y˜ = −y. Hence r˜ = (4 + x2 − 4x + y 2 ) = 2 1/2 θ 2 = r − 2 cos θ + 2−2 cos + O(1/r ). (r2 + 4 − 4r cos θ) r As a direct consequence of Proposition 6 we get Proposition 7. In the regions I–IV , F 2 takes the following form 1 1 a1 (θ) b(θ) b1 (θ) 2 F (r, θ) = r + a(θ) + +O + 2 +O ,θ + r r2 r r r3

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where a(θ) = −2 cos θ ,

b(θ) = 2(1 + sin θ) ,

(5)

b1 (θ) = 4 cos θ(1 + sin θ) and

a1 (θ) = 2 sin2 θ in regions I and IV, a1 (θ) = 2 sin2 θ + 4 sin θ in regions II and III .

In the region V I F 2 is given by

2

F (r, θ) =

4 r, θ + + O r

1 r3

(6)

while in the region V it is given by2 π 1 1 8 4 −θ +O + O F 2 (r, θ) = r + − 4 , θ + . r 2 r2 r r3

(7)

Proof. This follows from the previous proposition by a direct computation, since F 2 = T R in the regions I and IV and F 2 = RT in the regions II and III. The fact that F 2 = R2 and F 2 = T 2 in the regions V and V I respectively directly yields (6) and (7). Remark. (5) can also be obtained by a simpler computation as follows. Let A0 be a point, say, in region I, with polar coordinates (r, θ). Denote A1 = F A0 , A2 = F 2 A0 and let B1 and B2 be the midpoints of A0 A1 and A1 A2 respectively. −→

−→

Then the triangles A0 A1 A2 and B1 A1 B2 are similar so that A0 A2 = 2B1 B2 . −→

If r is large then B1 and B2 are close to the points where the lines with slope OA0 touch the semicircle, that is B1 = O1 , B2 ≈ (− sin θ, cos θ). Therefore −→ A0 A2 ≈ 2 − (1 + sin θ), cos θ . −→

a and b are the radial and the angular components of A0 A2 giving (5). −→

Computing how far is B1 B2 from the limiting vector allows to express the higher order terms in terms of the curvature of Γ and its derivatives, however for the semicircle it seems simpler to use the explicit formulas of Proposition 6. A.2. Normal form coordinates In this section we make a coordinate change to simplify the outer billiard map near infinity. In particular we refine the result of [22] about the asymptotics of the outer billiard orbits. Proposition 8. There exists a piecewise smooth change of coordinates G : (r, θ) → (ρ, ψ), with discontinuity lines 1 and 1 and the lines θ = π/2 in the upper and lower regions, such that F 2 takes the following normal form in the variables (ρ, ψ). 2 Notice

that in region V we have θ = the same order.

π 2

+O

1 r

so the second and the third terms in (7) are of

368



In the regions I–IV we have ρn+1 = ρn + O(1/ρ3n ) , ψn+1 = ψn +

1 c u + 2 + 3 +O ρn ρn ρn

1 ρ4n

(8)

,

(9)

where c = cupper = 14 inside the upper region and c = clower = − 14 in the lower region. In the regions V and V I, we have 1 1 u2 + 2 +O , (10) ψn+1 = ψn + ρn ρn ρ3n u1 1 (11) + 64(ψn − ) + O 1/ρ2n , in region V ρn+1 = ρn + ρn 3 v1 ρn+1 = ρn + (12) + O 1/ρ2n , in region V I ρn where u1 , v1 , u2 are constants that may be different in the upper and lower regions.3 Moreover, for (r, θ) ∈ D, we have the following expression for the Jacobian of G 1 cupper 1 +O Jac(G) = (1 − θ) + . (13) 2 2r r2 Proof. We will look for a coordinate change of the form Φ3 (θ) (14) ρ = rΦ1 (θ) + Φ2 (θ) + r Ψ1 (θ) Ψ2 (θ) + ψ = Ψ(θ) + . (15) r r2 This is a usual expression of a change of coordinates in view of a normal form for a twist map and the order of the 1/r powers is determined by the order of the perturbative terms required in (8) and (9) that in term correspond to what is needed to obtain in the sequel an expression of the return map to D as in Lemma 2. Notice that (8) and (9) should be viewed as functions of the lower and upper coordinates (r, θ) rather than the original polar coordinates. Remark. We will observe that ρ has a discontinuity when the orbit crosses from the upper to the lower region, a fact that translates into the discontinuity of the derivative of our coordinate change Φ1 at the same crossing. This fact is crucial since we shall see in the next subsection that the main change to ρ will come from crossing between the regions. Given the expression of F 2 in Proposition 7, observe that iterating by F 2 in the regions I–IV yields 1 1 Φ1 b2 + aΦ1 b + a1 Φ1 + Φ2 b ρn+1 − ρn = Φ1 b + aΦ1 + Φ1 b1 + +O 2 rn rn2 3 The

explicit values of these constants is not necessary for the sequel.

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so that if we require that Φ1 a =− , Φ1 b 1 Φ b2 Φ2 = − a1 Φ1 + Φ1 (ab + b1 ) + 1 b 2

(16) (17)

then ρn+1 − ρn = O( r12 ), and since 1 ≤ Φ1 ≤ 2 and −1 ≤ Φ2 ≤ 1 it follows that n ρn+1 − ρn = O( ρ12 ). Given the expressions of a, b, a1 , b1 in the different regions, we n can choose Φ1 (θ) = 1 + sin θ , Φupper 2

(18)

= 1 − | cos θ| ,

Φlower 2

= | cos θ| − 1 .

(19)

Likewise, expanding ψn+1 − ψn we get 2 1 Ψ b Ψ1 b aΨ1 b b1 ψn+1 − ψn = Ψ +Ψ 2 + + 2 − 2 +O . rn rn 2 rn rn rn rn3 So if we require that Ψ = then, using the computational fact

Φ 1 b1 b

1 bΦ1

=

(Φ1 b) 2 ,

(20) we obtain

1 Φ2 + bΦ21 Ψ1 + bΦ1 Φ1 Ψ1 ψn+1 − ψn = + +O ρn ρ2n

1 ρ3n

.

Observe that (20) is satisfied by 1 t2 t 1 2 1 , + Ψ(θ) = − 3 + 2 + (1 + t) 3 3 (1 + t)3 (1 + t) (1 + t)

(21)

where t = tan θ2 . (To obtain (21) observe that the change of variables t = tan 2s transforms θ 2 tan−1 θ ds 1 + t2 Ψ(θ) = = dt (22) 2 (1 + t)4 0 2(1 + sin s) 0 2t ds 2 since sin s = 1+t 2 , dt = 1+t2 . See also [22].) Next, Ψ1 is defined by Ψ1 (0) = 0 and

(Ψ1 Φ1 ) =

c Φ2 − bΦ1 bΦ1

(23)

and cupper and clower are chosen so that Ψ1 (π) = Ψ1 (0) = 0 both in the upper and lower regions. The values cupper = −clower = 14 are then obtained from the π Φ2 π using the same change of variables as in (22). computation of 0 bΦ1 1 and 0 bΦ 1

370



Finally, Φ3 and Ψ2 are chosen in a similar fashion as Φ1 , Φ2 , Ψ, Ψ1 to guarantee that 1 1 1 c u ρn+1 − ρn = O + 2 + 3 +O and ψn+1 − ψn = . 3 ρn ρn ρn ρn ρ3n We do not have to explicit the functions Φ3 and Ψ2 since we do not need to know the constant u for the sequel. Observe that our coordinate change is designed to simplify the map in the regions I–IV so they do not bring much simplification in the buffer regions V and V I. However in those regions the angular coordinate equals to π/2 + O(1/ρ), so we can use the Taylor expansion around π/2. Namely equations (6)–(7) and the facts that Φ1 (π/2) = 2 and Ψ (π/2) = 1/8 imply (10)–(11) for some constants u1 and u2 that may be different in the lower and upper regions and that we will not need to know explicitly for the rest of the proof. To obtain the Jacobian estimate (13), recall that in D we have θ = O(1/r). Thus Jac(G) = |Φ1 Ψ (θ) + (Ψ1 Φ1 ) (0)/r| + O(1/r2 ). Next, due to (20) Φ1 Ψ =

1 b

= 12 (1 − θ) + O(θ2 ) while (Ψ1 Φ1 ) (0) =

cupper . 2

Remark. The explicit expression for Ψ is quite complicated, however in the computations of Section A.3 we shall only use (20) and the fact that π 1 2 = . (24) Ψ(π) = , Ψ 3 2 3 Remark. A similar argument shows that a change of variables of the type ρ=

k−1

r1−j Φj (θ) ,

ψ=

j=0

k−1

r−j Ψj (θ)

j=0

brings F 2 to the forms ρn+1 = ρn + O ρ−k ,

ψn+1 = ψn +

k

cj ρ−j + O ρ−(k+1) .

j=0

A.3. Proof of Lemma 2 According to our division of the plane into upper and lower regions, we will rep˜ be the region bounded by resent F as a composition of two maps. Namely let D the line y = 0, x ≤ −˜ x0 (where x ˜0 x0 used in the definition of D) and its image by F 2 . ˜ We represent F = F2 F1 where F1 corresponds to the passage from D to D ˜ and F2 corresponds to the passage from D to D. ˜ that make them, up to We introduce changes of coordinates in D and D identification of their boundary lines, diffeomorphic to half cylinders of the form

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φ, φ˜ ∈ T, ρ, ρ˜ ≥ ρ0 + O(1). Equation (9) shows that these changes of coordinates ˜ → (˜ ˜ where are given by (ρ, ψ) → (ρ, φ) and (˜ ρ, ψ) ρ, φ) 1 1 upper lower ˜ ˜ ˜ ψ+O ψ+O φ = ρψ − c , φ = ρ˜ψ − c 2 ρ ρ˜2 ˜ Conversely, where we have used the bounds ψ = O( ρ1 ) in D and ψ˜ = O( ρ1˜ ) in D. observe that lower ˜ ˜ 1 1 φ φ cupper φ ˜= φ + c + O + O ψ= + , ψ . ρ ρ2 ρ3 ρ˜ ρ˜2 ρ˜3 ˜ and give an expression We will now study the iteration by F 2 from D to D ˜ = F −2 (D) ˜ and observe that until entering D ˜ the for F1 . We first introduce D 2 normal form of F in the (ρ, ψ) coordinates can be used, albeit a special care must be given to an eventual passage in region V . The discontinuity lines of the differential of F 2 that limit the region V are l2 and l3 . Their equations in the (ρ, ψ) coordinates become 1 1 1 +O 2 ⊂ ψ = − , 3 4ρ ρ2 (25) 1 3 1 +O . 3 ⊂ ψ = − 3 4ρ ρ2 This is due to the fact that 1/(2r) = 1/ρ + O(1/ρ2 ) in a O(1/r)-neighborhood of the vertical axis (that contains 2 and 3 ), and the equalities Ψ(π/2) = 1/3 and Ψ (π/2) = 1/8. Until entering region V we have 1 ρk = ρ + O , ρ2 1 k kcupper ku + + O . ψk = ψ + + 2 3 ρ ρ ρ ρ3 The value cupper = 1/4 allows to compute the entrance times n and n to the ˜ respectively. Namely region V and to D π ρ π 1 −φ− n= Ψ ρ−Ψ cupper − φ = , (26) 2 2 3 12 2ρ 1 −φ− (27) n = Ψ(π)ρ − Ψ(π)cupper − φ = 3 6 unless the orbit comes close to a discontinuity. (This precaution is discussed at the end of the section). ρ 1 1 if 3 − φ − 12 ∈ [1/4, 3/4] and υ = 0 We let υ = u1 − 64 ρ3 − φ − 12 otherwise, which corresponds to points that visit (respectively do not visit) the

372


ρ


/ρ + O(1/ρ2 ). It then follows from (11) 1 υ ρn+1 = ρ + + O . ρ ρ2 1 Define now υ = 2ρ 3 − φ − 6 . We get 1 υ ρn = ρ + + O , ρ ρ2 1 n (n − 1)cupper u2 nu n ˜υ + 2 + 3 − 3 +O ψn = ψ + + 2 ρ ρ ρ ρ ρ ρ3

region V , since ψn = 1/3 − that

3

−φ−

1 12

where n ˜ = n − n is the time remaining after going through the region V . Hence 1 2 υ Z(ρ, φ) ψn = − + +O 2 3 ρ ρ ρ3 where Z(ρ, φ) = (−φ − 16 − υ)cupper + u2 + 23 u − υ3 . Going back to the upper region polar coordinates and using that Ψ(π) = 2/3, Ψ (π) = 1/2, Ψ (π) = 1, Φ1 (π) = 1, Φ1 (π) = −1, Φ1 (π) = 0, Φ2 (π) = 0, Φ2 (π) = 0, we get 1 P1 rn = ρ − 2υ + +O , (28) ρ ρ2 1 2υ Q1 + 2 +O . (29) θn = π − ρ ρ ρ3 Here and in the sequel Pi and Qi , i = 1, 2, . . . will denote piecewise polynomials of degree 2 in the variables υ, φ that are not necessary to explicit for the rest of the proof. Switching to the lower region coordinates and iterating once more by F 2 we get 1 P2 rñ+1 = ρ − 2 − 2υ + +O , (30) ρ ρ2 1 2 2υ Q2 + 2 +O . (31) θñ+1 = − ρ ρ ρ ρ3 Consequently (14) and (15) yield

1 P3 +O , ρ ρ2 1 1 υ Q3 = − + 2 +O . ρ ρ ρ ρ3

ρñ+1 = ρ − 4υ + ψñ+1

˜ coordinates of the iterate of (ρ, φ) inside D ˜ as This finally gives the (˜ ρ, φ) 1 P4 ρ˜ = ρ − 4 + 4φ˜ + +O , ρ ρ2 1 2ρ 1 Q4 + +O φ˜ = φ − + 3 6 ρ ρ2 1 where we used that 1 − υ = φ − 2ρ 3 + 6 .

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Repeating similar computations in the lower region, with this difference that in the last iteration before entering D, the −2 discontinuity term of equation (30) is replaced by the +2 term (see (5)) and cupper is replaced by clower in (27) we ˆ the image in D of (˜ ˜ ∈D ˜ get, denoting by (ˆ ρ, φ) ρ, φ) 1 P5 ρˆ = ρ˜ + 4φˆ + +O , ρ˜ ρ˜2 1 2˜ ρ 1 Q5 − +O . φˆ = φ˜ − + 3 6 ρ˜ ρ˜2 Introducing 2ρ 1 − , R= 3 6 1 we get up to O ρ -terms

ρ 7 ˜ = 2˜ R + , 3 6

ρ 1 ˆ = 2ˆ R − 3 6

˜ ˜ ∼ R − 4 + 8φ , R 3 3 ˆ ˜ , R ˆ∼R ˜ − 4 + 8φ . φˆ ∼ {φ˜ − R} 3 3 This shows that the linear part of F has the required form. To prove the statement about the singularities in the upper region, we must find the equation of the preimage of 2 , so we use the normal form of F 2 and (25) to find that a point (ρ, φ) would hit 2 after n = ρ/3 + O(1) iterations if and only if 1 1 1 1 1 − = . φ+n+ +O 3 4ρ ρ 12 ρ2 Thus ρ 1 1 1 n= − − φ+ +O . 3 12 4 ρ φ˜ ∼ {φ − R} ,

1 = R/2 we have that the preimage of 2 is O( R1 )-close to {φ − R/2} = Since ρ3 − 12 3/4. Likewise the preimage of 3 is O( R1 )-close to {φ − R/2} = 1/4. The computations in the lower regions are similar. As for the density of the invariant measure of the return map, start by denoting F the original map in the coordinate system (r, θ). Let h1 : (ρ, ψ) → (r, θ) denote the inverse map of the conjugacy obtained in subsection A.2, and let h2 : (ρ, φ) → (ρ, ψ) denote the inverse of the rescaling used in Subsection A.3. −1 N +1 ◦ h1 ◦ h2 . Since F preserves the area We are interested in F = h−1 2 ◦ h1 ◦ F element, we get that F preserves the density Jac(h1 ◦ h2 )(ρ − 2φ + O(1/ρ))dρdφ (this is because due to (14) and (15) we have that the r coordinate of h1 ◦ h2 (ρ, φ) is ρ − 2φ + O(1/ρ)). Next, (13) implies that Jac(h1 ) in D equals 1 1 4φ − c 2 + 2θ − c/ρ + O +O =2+ ρ2 ρ ρ2

374



with c = cupper . Finally, since ψ = φ/ρ + cφ/ρ2 + O(1/ρ3 ) we have that Jac(h2 ) = 1 2c 1 ρ (1 − ρ ) + O( ρ3 ). As a consequence the density preserved by F is of the form 2 − 5c/ρ + O(1/ρ2 ). This ends the proof of Lemma 2. Remark. A similar argument shows that in Lemma 2 O(1/ρ2 )-terms are piecewise polynomials of degree 3, O(1/ρ3 )-terms are piecewise polynomials of degree 4 etc. We shall not use this fact here, but it could be useful in other problems, for example, for verification of KAM stability of bounded periodic orbits.

Acknowledgements The authors are grateful to Raphaël Krikorian and Richard Schwartz for useful conversations and comments on the manuscript. The first author is grateful to the University of Paris 13 for its hospitality.

References [1] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Encyclopaedia Math. Sci. 3 (1993), Springer, Berlin. [2] P. Boyland, Dual billiards, twist maps and impact oscillators, Nonlinearity 9 (1996), 1411–1438. [3] R. Douady, Thèse de 3-ème cycle, Université de Paris 7, 1982. [4] R. Douady, Systèmes dynamiques non autonomes: d´ emonstration d’un th´ eorème de Pustylnikov, J. Math. Pures Appl. 68 (1989), 297–317. [5] D. Genin, Regular and Chaotic Dynamics of Outer Billiards, Penn State Ph.D. thesis (2005). [6] D. Genin, Hyperbolic outer billiards: a first example, Nonlinearity 19 (2006), 1403– 1413. [7] A. Goetz, Piecewise isometries – an emerging area of dynamical systems, in Fractals in Graz 2001, P. Grabner and W. Woess (editors) Trends Math., Birkh¨ auser, Basel, 2003, 135–144. [8] A. Goetz, A. Quas, Global properties of piecewise isometries, to appear in Erg. Th. Dyn. Syst. [9] I. V. Gorelyshev, A. I. Neishtadt, On the adiabatic perturbation theory for systems with impacts, J. Appl. Math., Mech. 70 (2006), 4–17. [10] E. Gutkin, A. Katok, Caustics in inner and outer billiards, Comm. Math. Phys. 173 (1995), 101–133. [11] E. Gutkin, N. Simanyi, Dual polygonal billiards and necklace dynamics, Comm. Math. Phys. 143 (1992), 431–449. [12] R. Kolodziej, The antibilliard outside a polygon, Bull. Polish Acad. Sci. Math. 37 (1989),163–168. [13] P. Lochak, C. Meunier, Multiphase Averaging for Classical Systems, Springer Appl. Math. Sci. 72 (1988).

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[14] J. Moser, Stable and random motions in dynamical systems, Annals of Math. Studies 77 (1973), Princeton University Press, Princeton, NJ. [15] J. Moser, Is the solar system stable?, Math. Intelligencer 1 (1978/79), no. 2, 65–71. [16] L. D. Pustylnikov, Stable and oscillating motions in nonautonomous dynamical systems-II, (Russian) Proc. Moscow Math. Soc. 34 (1977), 3–103. [17] L. D. Pustylnikov, Poincaré models, rigorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism, Russian Math. Surveys 50 (1995), 145–189. [18] R. Schwartz, Unbounded orbits for outer billiards-1, J. of Modern Dyn. 1 (2007), 371–424. [19] R. Schwartz, Outer billiards on kites, to appear in Annals of Math. Studies. [20] A. J. Scott, C. A. Holmes, G. J. Milburn, Hamiltonian mappings and circle packing phase spaces, Phys. D 155 (2001), 34–50. [21] S. Tabachnikov, Outer billiards, Russian Math. Surv. 48 (1993), no. 6, 81–109. [22] S. Tabachnikov, Asymptotic dynamics of the dual billiard transformation, J. Statist. Phys. 83 (1996), 27–37. [23] F. Vivaldi, A. V. Shaidenko, Global stability of a class of discontinuous dual billiards, Comm. Math. Phys. 110 (1987), 625–640. [24] P. Wright, A simple piston problem in one dimension, Nonlinearity 19 (2006), 2365– 2389. Dmitry Dolgopyat Department of Mathematics University of Maryland College Park MD 20742–4015 USA e-mail: [email protected] Bassam Fayad Laboratoire Analyse, Géométrie et Applications UMR 7539 Institut Galilée Université Paris 13 99, Av. J-B. Clément F-93430 Villetaneuse France e-mail: [email protected] Communicated by Domokos Szasz. Submitted: April 19, 2008. Accepted: January 11, 2009.



On Confining Potentials and Essential Self-Adjointness for Schr¨ odinger Operators on Bounded Domains in Rn Gheorghe Nenciu and Irina Nenciu Abstract. Let Ω be a bounded domain in Rn with C 2 -smooth boundary, ∂Ω, of co-dimension 1, and let H = −Δ + V (x) be a Schr¨ odinger operator on (Ω). We seek the weakest conditions we can find Ω with potential V ∈ L∞ loc on the rate of growth of the potential V close to the boundary ∂Ω which guarantee essential self-adjointness of H on C0∞ (Ω). As a special case of an abstract condition, we add optimal logarithmic type corrections to the known 3 condition V (x) ≥ 4d(x) 2 where d(x) = dist(x, ∂Ω). More precisely, we show that if, as x approaches ∂Ω, 3 1 1 1 V (x) ≥ − − − · · · d(x)2 4 ln(d(x)−1 ) ln(d(x)−1 ) · ln ln(d(x)−1 ) where the brackets contain an arbitrary finite number of logarithmic terms, then H is essentially self-adjoint on C0∞ (Ω). The constant 1 in front of each logarithmic term is optimal. The proof is based on a refined Agmon exponential estimate combined with a well-known multidimensional Hardy inequality.

1. Introduction Consider a particle in a bounded domain Ω in Rn , n ≥ 1, in the presence of a potential V . At the heuristic level, if V (x) → ∞ as x approaches the boundary ∂Ω, then the particle is confined in Ω and never visits the boundary. At the classical level, this indeed happens when V (x) → ∞ as x → ∂Ω (see, e.g. [17, Theorem X.5]). At the quantum level, the problem is much more complicated due to the possibility that the particle tunnels through the infinite potential barrier and “sees” the boundary. The fact that the particle never feels the boundary amounts We wish to thank F. Gesztesy, A. Laptev, M. Loss and B. Simon for useful comments and suggestions. I.N.’s research was partly supported by the NSF grant DMS 0701026.

378

G. Nenciu and I. Nenciu


to saying that V determines completely the dynamics: there is no need for boundary conditions. At the mathematical level, by Stone’s Theorem, the problem is then finding conditions on the rate of growth of V (x) as x → ∂Ω which ensure that the Schr¨ odinger operator H = −Δ + V

(1.1)

is essentially self-adjoint on C0∞ (Ω). Let us note here that oscillations of the potential could also play a role in the essential self-adjointness problem due to the possibility of coherent reflections by an appropriately chosen sequence of potential barriers (see [17], the Appendix to Chapter X.1). In this paper we will not consider oscillatory potentials, but rather focus on potentials which grow to infinity at the boundary of the domain. The problem has a long and distinguished history; for details and further references, we send the reader to [6] and [17] and the review papers [4,5,15]. In the 1-dimensional case (say, Ω = (0, 1)) there exists a well-developed theory of essential self-adjointness of Sturm-Liouville operators, which is based on limit point/limit circle Weyl type criteria (see e.g. [6, 17] and the references therein). In particular if, under appropriate regularity conditions, 1 3 , (1.2) V (x) ≥ · 4 d(x)2 where d(x) = dist x, {0, 1} , then H is essentially self-adjoint on C0∞ (0, 1). The constant 34 is optimal, in the sense that if for some ε > 0, 3 1 0 ≤ V (x) ≤ −ε · , 4 d(x)2 near 0 and/or 1, then H is not essentially self-adjoint on C0∞ (0, 1) (see Theorem X.10 in [17]). Many results have been generalized from one to higher dimensions – see, for example, a comprehensive review of these results in [5]. In particular, if Ω is a bounded domain with C 2 boundary ∂Ω of codimension 1, and if V satisfies (1.2) as x approaches ∂Ω, with d(x) = dist(x, ∂Ω), then H defined as in (1.1) is essentially self-adjoint on C0∞ (Ω). Moreover, Theorem 6.2 in [5] implies that for the case at hand the essential self-adjointness of H is assured by a weaker condition, namely 1 3 c (1.3) V (x) ≥ · − 2 4 d(x) d(x) with some c ∈ R+ . This raises the following optimality question: While among 1 power-type growth conditions, 34 · d(x) 2 is optimal both in the exponent and in the constant, does a growth condition of the type 1 3 V (x) ≥ · 1 − m d(x) , lim m(t) = 0 , m(t) ≥ 0 t→0+ 4 d(x)2 still imply essential self-adjointness of H? It turns out that this is false – see the counterexample in the proof of Theorem 3. So the question of optimality should 1 be refined to asking whether 34 · d(x) 2 is the leading term of a (possibly formal)

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379

asymptotic expansion near ∂Ω of a critical potential Vc such that V ≥ Vc near ∂Ω implies essential self-adjointness of H on C0∞ (Ω). This would amount to finding the form and size of sub-leading terms in the asymptotic expansion of Vc . The main result of this note is the affirmative answer to this optimality question. Namely, we show that for bounded domains Ω in Rn , n = 1, 2, 3, . . . having C 2 boundary of codimension 1, and for potentials V satisfying 3 1 1 1 V (x) ≥ − − − · · · (1.4) d(x)2 4 ln(d(x)−1 ) ln(d(x)−1 ) · ln ln(d(x)−1 ) as x approaches ∂Ω, the Schr¨ odinger operator H is essentially self-adjoint on C0∞ (Ω), and that the constants 1 in front of each logarithmic term on the righthand side of (1.4) are optimal (for a precise statement, see Theorem 3). Two remarks are in order here. The first one is that we are interested in optimality rather than generality. Accordingly, and also in order not to obscure the main ideas of our proofs by technicalities, we consider the simplest case, which is still the most interesting from a physical point of view: a bounded domain Ω ⊂ Rn with C 2 boundary of co-dimension 1. In addition, we only consider scalar Schr¨ odinger operators with regular (L∞ loc ) potentials, and we consider only what one can think of as the “isotropic” case, i.e. we seek conditions on V (x) which depend only on d(x), and not on the specific point x0 of the boundary that x approaches, or the direction along which x → x0 . In this setting the proofs are short and elementary. Concerning more general situations, we note here that many results about the essential self-adjointness problem of second order elliptic operators of general form on arbitrary domains in Rn can be found in [5] (see especially Corollary 3.3 and its applications) – and then one can consider again the above optimality question. At the price of technicalities, one may be able to extend the results of the present note to more general situations, e.g. boundaries with components of higher co-dimension, local singularities of the potential or second order elliptic operators of general form. Reducing the regularity of the boundary ∂Ω below C 2 seems to require a finer analysis – in particular, of multidimensional Hardy inequalities on domains with less smooth boundaries (see e.g. [7, 8, 16] and references therein for results in this area). The second remark concerns the method of proof. While the proofs in [5] are based on his theory of semimaximal operators, our method of proof is based on the observation that essential self-adjointness follows (via the fundamental criterion for self-adjointness, see, e.g., [2,17]) from Agmon type results on exponential decay of eigenfunctions (see [1, Theorem 1.5a]). As stated, the result in [1] does not lead to optimal growth conditions on the potential. One has both to strengthen the exponential decay estimates, and to combine them with multidimensional Hardy inequalities [7]. So our basic technical result is an exponential estimate of Agmontype – see Theorem 4. Here the point is that our condition (Σ.2) below is strictly weaker than the corresponding condition (3.12) from Brusentsev [5]. The paper is organized as follows. In Section 2 we state the problem and the main results. Section 3 contains the proof of the Agmon-type Theorem 4. While

380



some of the results in this section go back to Agmon [1] and are well-known (e.g. the identity in Lemma 3.2), we give complete proofs for the reader’s convenience. Finally Section 4 contains the proofs of Theorems 1 and 2.

2. Main results Let Ω be a bounded domain in Rn , n ≥ 1, with C 2 -smooth boundary, ∂Ω, of co-dimension 1. We consider the function d(x) = dist(x, ∂Ω),

for x ∈ Ω ,

(2.1)

where “dist” denotes the usual, Euclidean distance in R . As is well-known (see, for example, the Appendix to Chapter 14 in [13]), d is Lipshitz and differentiable a.e. in Ω. More importantly for us here, there exists a constant n

dΩ > 0

(depending only on the domain Ω)

such that for x ∈ Ω with d(x) < dΩ , d is twice-differentiable and |∇d(x)| ≤ 1 .

(2.2)

Remark 2.1. Actually |∇d(x)| = 1 for x ∈ Ω with d(x) < dΩ , see for example [13], or Lemma 6.2 in [5], but in the proofs below we use only (2.2). In Ω we consider the Schr¨ odinger operator H = −Δ + V with V ∈ L∞ loc (Ω), ∞ defined on D(H) = C0 (Ω). As explained in the Introduction, we are seeking growth conditions on V close to ∂Ω ensuring essential self-adjointness of H. These will be given in terms of functions G described below: Condition (Σ). A function G : (0, ∞) → R is said to satisfy condition (Σ) if it is C 1 (0, ∞) and such that: (Σ.1) There exists d0 > 0, d0 ≤ dΩ , such that 1 0 ≤ G (t) ≤ , for t ∈ (0, d0 ) and t G (t) = 0 for t ≥ d0 . (Σ.2) For any ρ0 ≤

d0 2 ,

∞

4−n e−2G(2

−n

ρ0 )

=∞

(2.3)

n=1

We can now formulate our main result: Theorem 1. Consider an open, bounded domain Ω ⊂ Rn with C 2 -smooth boundary, and the Schr¨ odinger operator H = −Δ + V , L∞ loc

and domain D(H) = with V ∈ G satisfying condition (Σ) such that

C0∞ (Ω).

(2.4)

Assume that there exists a function

V = V1 + V 2 ,

(2.5)

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where

2 1 1 · ≥ G d(x) and 2 4 d(x) Then H is essentially self-adjoint in L2 (Ω). V1 (x) +

V2 ∈ L∞ (Ω) .

(2.6)

Theorem 1 follows from the fundamental criterion for self-adjointness (see, for example, [2,17]), a multidimensional Hardy inequality [7], and a (refined) Agmontype exponential estimate (see Theorem 4 in Section 3). We now turn to various examples of functions G satisfying condition (Σ), and the associated criteria for essential self-adjointness of H in terms of the growth of the potential at the boundary of the domain. The first, simplest example of a function G satisfying condition (Σ) is the one for which at sufficiently small t: G(t) = ln t , which leads to the classical bound 1 3 , V1 (x) ≥ · 4 d(x)2

as

x → ∂Ω .

(2.7)

The second example is (again for t sufficiently small) G(t) = ln t − c · t , This choice of G leads, through (2.6), to 1 3 c˜ , − V1 (x) ≥ · 4 d(x)2 d(x)

c ∈ R+ .

as

x → ∂Ω ,

(2.8)

for all c˜ < 2c. This is the lower bound obtained by Brusentsev in [5, Theorem 6.2] for the case at hand. The next example is (again for sufficiently small t) of the form G(t) = ln t + f (u) du t

with f (u) ≥ 0 ,

lim uf (u) = 0 ,

u→0

and

lim

t→0

This leads to a bound on V of the form 1 c˜ 3 · f d(x) , − V (x) ≥ · 2 4 d(x) d(x)

f (u) du < ∞ .

(2.9)

x → ∂Ω ,

(2.10)

t

as

with f as above and all c˜ < 2. Although this result does not appear in an explicit form in [5] it can still be obtained from Corollary 3.3 in [5]. Note that, since we c˜ required that uf (u) → 0 as u → 0, the second term d(x) · f (d(x)) in (2.10) is of 1 1 lower order than d(x)2 , and thus does not contradict the optimality of 34 · d(x) 2. The last example is our main hierarchy of essential self-adjointness conditions. Let p ∈ Z, p ≥ 2, and iteratively define L1 (t) = ln(1/t) ,

Lp (t) = ln Lp−1 (t) ,

(2.11)

382



ep−1 where each Lp is defined for t ∈ (0, e−1 . Then we have p ) with e1 = e and ep = e the following result:

Theorem 2. Consider an open, bounded domain Ω ⊂ Rn with C 2 -smooth boundary, and the Schr¨ odinger operator H = −Δ + V, (2.12) ∞ with V ∈ L∞ loc and domain D(H) = C0 (Ω). Let p ∈ Z, p ≥ 2 and assume that

V = V1 + V 2 , where p 1 3 1 − V1 (x) ≥ · 4 d(x)2 d(x)2 j=2

j−1

(2.13)

Lk d(x)

−1

k=1

−

1 · f d(x) , d(x)

(2.14)

for all x with d(x) < min(e−1 p , dΩ ), with f satisfying (2.9), V1 bounded from below on Ω, and V2 ∈ L∞ (Ω). Then H is essentially self-adjoint in L2 (Ω). Remark 2.2. Let K be a positive constant. Rewriting V (x) as V (x) = V1 (x) + K + V2 (x) − K one sees that, in order to obtain the result above, it is sufficient to prove Theorem 2 with V1 bounded from below by some (appropriately chosen) positive constant; this is exactly how we prove the theorem – see Section 4.

j−1 Note that, for any given j ≥ 2, each term 1t · ( k=1 Lk (t))−1 is non-integrable, and hence a higher order correction than the integrable term f (t). Further note

j−1 −1 that the domain on which is well defined shrinks to the j≤p ( k=1 Lk (t)) empty set as p → ∞. 1 The term 14 · d(x) 2 in (2.6) comes from the additional “barrier” given by the uncertainty principle of quantum mechanics via the Hardy inequality (see (4.2) below). The fact that Hardy inequalities appear here is not surprising since, as expressions of the uncertainty principle, they play a key role in various aspects of the spectral analysis of Schr¨ odinger and Dirac operators like stability, selfadjointness, etc (see e.g. [10–12, 14] and the references therein). During the last decade a large body of literature about improvements to Hardy inequalities has appeared (see e.g. the references in [3,9,11,18]). In particular, in [3] (under suitable conditions) the following optimal improvement of (4.2) was proved:

∞ i 2 d(x) |ϕ(x)| 1 |∇ϕ(x)|2 dx ≥ Xk2 1+ dx (2.15) 4 Ω d(x)2 D Ω i=1 k=1

where D is a sufficiently large constant, and Xk (t), t > 0 are defined recursively by X1 (t) = (1 − ln t)−1 , Xk (t) = X1 Xk−1 (t) .

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However, this improvement of Hardy’s inequality does not lead to an improvement of the result in Theorem 2 (which according to Theorem 3 is already optimal at the level of logarithmic subleading terms). Indeed, at the level of the leading term, as t → 0, Xk2 = L−2 k and so the contribution of the logarithmic terms in (2.15) can be absorbed in the last (integrable) term on the right-hand side of (2.14). As we will show in Section 4, the theorem follows from Theorem 1 with the following choice, for sufficiently small t, of G function: p 1 Lj (t) + f˜(u) du , (2.16) Gp (t) = ln t + · 2 j=2 t where f˜ also satisfies (2.9). Our last result is about the optimality of (2.14). With the hypotheses of Theorem 2, it is well-know that the constant 34 in front of the first term on the right-hand side of (2.14) is optimal. We claim that the constant 1 in front of each logarithmic term in the sum above is also optimal, in the following precise sense: Theorem 3. Given p ≥ 2 and a constant c > 1, there exist potentials V for which H = −Δ + V is not essentially self-adjoint, and which grow close to the boundary ∂Ω as −1

p−1 j−1 1 1 3 − Lk d(x) V (x) ≥ · 4 d(x)2 d(x)2 j=2 k=1 (2.17) −1

p−1 1 − c· · Lk d(x) . d(x)2 k=1

We end this section with a discussion of condition (Σ) and its relation with condition (3.12) from Corollary 3.3 in [5]. We comment first on condition (Σ.1). Note that (Σ.2) implies that G(t) → −∞ as t → 0. So G (t) ≥ 0 in (Σ.1) only adds that G(t) → −∞ monotonically which is not a real restriction as far as we are not considering (as already stated in the Introduction) the effect of oscillations of the potential. In fact, if one considers potentials which grow monotonically as x → ∂Ω one may impose even a stronger condition that G (t) is monotonically increasing to ∞ as t → 0. Consider now G (t) ≤ 1t in (Σ.1). This is again harmless (as far as it does not contradict (Σ.2)!) since if G1 (t) ≥ G2 (t) then Theorem 1 with G(t) = G2 (t) gives a stronger result than with G(t) = G1 (t). The crucial condition is (Σ.2) and this is to be compared with Brusentsev’s condition (3.12) from Corollary 3.3. We show now that Brusentsev’s condition (3.12) is (at least for G(t) satisfying (Σ.1)) strictly stronger than (Σ.2). Notice that we have restricted our attention to the situation when his matrix A ≡ I. Comparing functions, we see that in Brusentsev’s notation the function which determines the growth of the potential at the boundary is η(x), and that we are therefore interested in showing that, if η(x) = −G d(x) , (2.18)

384



satisfies condition (3.12) in [5], then G must satisfy our condition (Σ.2). Condition (3.12) in Brusentsev guarantees that there exists a constant C > 0 such that |∇η(x)| · e−η(x) ≤ C .

(2.19)

If we recall that for x with d(x) small enough, |∇d(x)| = 1, then we get from (2.18) and (2.19) that d G(t) e = G (t)eG(t) ≤ C , dt for all 0 < t < dΩ . But since G(t) → −∞ as t → 0+, we can integrate, for all n greater than some fixed integer NΩ , 2−n ρ0 −n eG(2 ρ0 ) = G (t)eG(t) dt ≤ 2−n · Cρ0 . 0

Plugging this into the series from (2.3) we get ∞

4−n e−2G(2

−n

∞

≥

ρ0 )

n=1

4−n · 4n (Cρ0 )−2 = +∞ ,

n≥NΩ

thus showing that G satisfies (Σ.2). Conversely, recall the Gp defined in (2.16). As we will show in Section 4, the function Gp satisfies (Σ). Take now the simplest case G(t) = G2 (t) with f˜ ≡ 0 i.e. G(t) = ln t + 12 ln ln 1t for sufficiently small t and set η(x) = −G d(x) . Then as t = d(x) → 0+ −η(x)

|∇η(x)|e

= G (t) · e

G(t)

1 1 2 1 1 = ln 1− → +∞ , t 2 ln 1t

and hence η does not satisfy condition (3.12) from [5].

3. Agmon-type estimates Proposition 3.1. Let ψ be a weak solution of Hψ = Eψ , i.e. ψ ∈

1 Hloc(Ω)

and satisfies ψ, (H − E)ϕ = 0 ,

for every

ϕ ∈ C0∞ (Ω)

(3.1)

Let g ∈ C 1 (Ω) be a real-valued function for which there exists a constant c > 0 such that (3.2) ϕ, (H − E)ϕ − |ϕ(x)|2 |∇g(x)|2 dx ≥ c ϕ 2 Ω

for every ϕ ∈ C0∞ (Ω).

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For ρ > 0, small enough, set Ωρ = {x ∈ Ω | d(x) > ρ}. Then there exists a constant K = K(c) < ∞, independent of ρ, such that 1 K(c) + |∇g(x)| |eg(x) ψ(x)|2 dx . |eg(x) ψ(x)|2 dx ≤ (3.3) ρ Ω2ρ Ωρ \Ω2ρ ρ Since this might be of independent interest and the proof is the same, we will actually prove this proposition in a slightly more general context. Indeed, consider the Schr¨ odinger operator with magnetic potential on Ω H = ( p − a)2 + V ,

V ∈ L∞ loc (Ω) ,

1 a ∈ Cloc (Ω) ,

p = −i∇ ,

(3.4)

defined on D(H) = C0∞ (Ω) and, for ϕ, ψ ∈ W 1,2 , the associated quadratic form p − a)ϕ · ( p − a)ψ dx + ϕ¯ · V ψ dx . (3.5) h[ϕ, ψ] = ( Ω

Ω

Note that if ϕ and ψ are both in C02 (Ω), then h[ϕ, ψ] = ϕ(x) (Hψ)(x) dx . Ω

One of the main technical ingredients is the following simple identity [1]: Lemma 3.2. Let ψ be a weak solution of Hψ = Eψ, and let f = f¯ ∈ C01 (Ω). Then |2 ψ . (h − E)[f ψ, f ψ] = ψ, |∇f (3.6) Proof. Consider first f ∈ C0∞ and let ϕ ∈ C0∞ . Then (h − E)[ϕ, f ψ] = (H − E)ϕ, f ψ = f (H − E)ϕ, ψ . Since [f, p − a] = i∇f on C0∞ , we get that [f, H] = f, ( p − a)2 = i ( p − a) · ∇f + ∇f · ( p − a) , and so, if we remember that ψ is a weak solution, (h − E)[ϕ, f ψ] = [f, H]ϕ, ψ = ϕ, [H, f ]ψ . Since f ψ ∈ W01,2 (Ω) and C0∞ is dense in the W 1,2 topology, the identity above implies that (h − E)[f ψ, f ψ] = ψ, f [H, f ]ψ = Re ψ, f [H, f ]ψ 1 ψ, f [H, f ] − [H, f ]f ψ = (3.7) 2 1 ψ, f, [H, f ] ψ . = 2 Finally, a straightforward computation shows that f, [H, f ] = −i f, ( p − a) · ∇f + ∇f · ( p − a) = −i (2i∇f · ∇f ) = 2|∇f |2 , which completes the proof.

386



Proof of Proposition 3.1. As in [1], we will now choose a function f to plug into the formula (3.6). More precisely, let f = eg φ , where g ∈ C 1 (Ω), real-valued, is the function from the statement of the proposition, and φ ∈ C0∞ (Ω), 0 ≤ φ ≤ 1, is a cut-off function, 0, x ∈ / Ωρ φ(x) = 1 , x ∈ Ω2ρ . Taking φ of the form φ(x) = k(d(x)) where 0, 0 ≤ t ≤ ρ k(t) = 1 , t ≥ 2ρ one sees that for ρ small enough (say ρ < |∇φ| ≤

dΩ 2 )

K1 , ρ

(3.8)

with K1 an absolute constant. Then |∇f |2 = f 2 |∇g|2 + m , where m = 2f eg ∇g · ∇φ + e2g |∇φ|2 . Estimating directly leads to:

|ψ, mψ | ≤ ψ, |m|ψ = |ψ|2 (2e2g φ|∇g| |∇φ| + e2g |∇φ|2 ) dx Ω g 2 K1 K1 ψe ≤ 2|∇g| + dx ρ Ωρ \Ω2ρ ρ where inthe last inequality we used (3.8), as well as the fact that ∇φ ≡ 0 on (Ω \ Ωρ ) Ω2ρ . But now recall that the Agmon condition (3.2) was that (h − E)[ϕ, ϕ] − |ϕ(x)|2 |∇g(x)|2 dx ≥ c ϕ 2 , Ω

with c independent of ϕ and ρ. Using the density of C0∞ in W01,2 , we obtain that (h − E)[f ψ, f ψ] − f ψ, |∇g|2 f ψ ≥ c f ψ 2 .

(3.9)

Since (h − E)[f ψ, f ψ] − f ψ, |∇g|2 f ψ = ψ, mψ , we obtain g 2 K1 ψe 2|∇g| + K1 dx ≥ |ψ, mψ | ≥ c f ψ 2 dx , ρ Ωρ \Ω2ρ ρ Ω which, if we recall the choice of f made at the beginning of the proof, leads directly to the claim of the proposition.

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Theorem 4. Consider an open, bounded domain Ω ⊂ Rn with C 2 -smooth boundary, and the Schr¨ odinger operator H = −Δ + V , L∞ loc

(3.10)

C0∞ (Ω).

with V ∈ and domain D(H) = Assume that there exist E ∈ R and c > 0 such that ϕ, (H − E)ϕ − |∇g(x)|2 |ϕ(x)|2 ≥ c ϕ 2 , (3.11) Ω

for all ϕ ∈ C0∞ (Ω), where g(x) = G(d(x)) for some G satisfying condition (Σ). If ψ is a weak solution of Hψ = Eψ, then ψ ≡ 0. Proof. Let d0 > 0 be the constant that appears in condition (Σ) for the function G from the hypothesis. Fix, for the time being, 0 < ρ0 ≤ d0 /2, and let ρ > 0 be such that 2ρ ≤ ρ0 . Then define a “normalized” G function: Gρ (t) = G(t) − G(ρ) , and set

gρ (x) = Gρ d(x) .

Note that for all x ∈ Ω we have

∇gρ (x) = G d(x) ∇d(x) .

(3.12)

This, together with condition (Σ.1) for G, and the fact that |∇d(x)| ≤ 1 for d(x) < dΩ , implies in particular that |∇gρ (x)| ≤

1 d(x)

for x ∈ Ω \ Ωd0 /2 .

(3.13)

On the other hand, look at x ∈ Ωρ0 . Since ρ0 ≤ d(x) , condition (Σ.1) implies that gρ (x) ≥ Gρ (ρ0 ) = G(ρ0 ) − G(ρ) ,

(3.14)

and so e2gρ (x) ≥ e2G(ρ0 ) · e−2G(ρ) , Therefore e2G(ρ0 ) · e−2G(ρ)

|ψ(x)|2 dx ≤

Ω ρ0

for all x ∈ Ωρ0 .

|egρ (x) ψ(x)|2 dx ≤ Ω ρ0

(3.15)

|egρ (x) ψ(x)|2 dx , Ω2ρ

where we used the fact that 2ρ ≤ ρ0 and so Ωρ0 ⊂ Ω2ρ . Now note that ∇gρ = ∇g, and so gρ satisfies (3.11) with the same E and c as g. In particular, one can apply Proposition 3.1 and obtain 1 K(c) + ∇gρ (x) |egρ (x) ψ(x)|2 dx . |egρ (x) ψ(x)|2 dx ≤ ρ ρ Ω2ρ Ωρ \Ω2ρ

388



Since 0 < ρ < 2ρ < ρ0 ≤ d0 /2, it follows that Ωρ \ Ω2ρ ⊂ Ω \ Ωd0 /2 and so (3.13) implies that ˜ K(c) 1 K(c) + |∇gρ (x)| |egρ (x) ψ(x)|2 dx ≤ 2 |ψ(x)|2 dx , ρ ρ ρ Ωρ \Ω2ρ Ωρ \Ω2ρ where we also used the fact that, for x ∈ Ωρ \ Ω2ρ , 2ρ G (t) dt ≤ gρ (x) ≤ Gρ (2ρ) = G(2ρ) − G(ρ) = ρ

ρ

Putting it all together, we get that 2 −2G(ρ) 2 K2 (c, ρ0 ) · ρ e |ψ(x)| dx ≤

Ωρ \Ω2ρ

Ω ρ0

2ρ

1 dt = log 2 . t

|ψ(x)|2 dx .

(3.16)

for some constant K2 (c, ρ0 ). Now, let n ≥ 1 be an integer, and set ρn =

1 ρ0 . 2n

So 2ρn = ρn−1 , and we get M

(Ωρn \ Ω2ρn ) =

n=1

M

(Ωρn \ Ωρn−1 ) = ΩρM \ Ωρ0 ⊂ Ω .

n=1

So using (3.16) successively with ρ = ρn , 1 ≤ n ≤ M , and summing leads to

M −n 2 −n −2G(2 ρ0 ) 2 ρ0 K2 (c, ρ0 ) 4 e |ψ(x)| dx ≤ |ψ(x)|2 dx < ∞ . (3.17) Ω ρ0

n=1

Ω

But from condition (Σ.2) we know that the series and so we find that |ψ(x)|2 dx = 0 .

n≥1

4−n e−2G(2

−n

ρ0 )

diverges, (3.18)

Ω ρ0

But ρ0 > 0 was arbitrary, and so by taking ρ0 → 0 it follows that |ψ(x)|2 dx = 0 ,

(3.19)

Ω

as claimed.

4. Proofs of the main theorems Our strategy in approaching Theorem 1 consists of combining Agmon-type decay estimates for (weak) eigenfunctions (see Theorem 4) with multidimensional Hardy inequalities. More precisely, for H as above, the fundamental criterion for selfadjointness tells us that Theorem 1 can be proved as follows:

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Lemma 4.1. With the hypotheses of Theorem 4, there exists an E < 0 such that, for any ψ ∈ L2 (Ω), the condition ψ, (H − E)ϕ = 0 , for every ϕ ∈ C0∞ (Ω) (4.1) implies that ψ ≡ 0. Proof. In view of Theorem 4 the only thing to be proved is that for ψ ∈ L2 (Ω), (3.1) 1 implies that ψ is a weak solution of (H − E)ψ = 0 i.e. ψ ∈ Hloc (Ω). This is an interior regularity result for elliptic equations and follows from general theory. In 2 (Ω). Indeed let our simple setting one can see by elementary means that ψ ∈ Hloc ∞ ˜ ˜ ˜ Ω ⊂ Ω, dist(Ω, ∂Ω) > 0. Then V ∈ L (Ω) and from ψ, (H − E)ϕ = 0 it follows ψ, (−Δ + 1)ϕ = ψ, (V − E + 1)ϕ ≤ K ˜ ϕ

Ω,E ˜ which via the Riesz lemma implies for all ϕ ∈ C0∞ (Ω), ψ, (−Δ + 1)ϕ = Φ, ϕ ˜ This means that the distribution (−Δ + 1)Ψ on C ∞ (Ω) ˜ is for some Φ ∈ L2 (Ω). 0 2 ˜ represented by a L (Ω) function and the proof is finished. The following multidimensional Hardy inequality will allow us to complete the proof of our main theorem: Theorem 5 (Multidimensional Hardy Inequality). Let Ω ⊂ Rn be a bounded open set with C 2 -smooth boundary. Then there exists a constant A = A(Ω) ∈ R such that 1 |ϕ(x)|2 dx ≤ |∇ϕ(x)|2 dx + A ϕ 2 (4.2) 4 Ω d(x)2 Ω for every ϕ ∈ C0∞ (Ω). This particular form of the Hardy inequality in domains in Rn can be found, for example, in [7]. Now the proof of Theorem 1 follows very quickly. Proof of Theorem 1. From the fundamental criterion for self-adjointness (via Lemma 4.1) and the Agmon-type Theorem 4, we conclude that what we must show in order to complete the proof is that there exist E ∈ R, as well as c > 0 and a function g(x) = G(d(x)) with G satisfying (Σ) such that |∇g(x)|2 |ϕ(x)|2 dx ≥ c ϕ 2 , (4.3) ϕ, (H − E)ϕ − Ω

C0∞ (Ω).

for all ϕ ∈ Recall that under the hypotheses of Theorem 1, the potential V = V1 + V2 with V2 ∈ L∞ (Ω) and 2 1 1 , V1 (x) ≥ G d(x) − · 4 d(x)2

390



for some G satisfying (Σ). Using exactly this G to define the g we need, and applying the result of the multidimensional Hardy inequality above, we get that for E ∈ R ϕ, (H − E)ϕ − |∇g(x)|2 |ϕ(x)|2 dx Ω 2 1 ≥ V1 (x) − G d(x) + · |ϕ(x)|2 dx 2 4d(x) Ω + − V2 L∞ − A − E ϕ 2 ≥ − V2 L∞ − A − E ϕ 2 . On the way we have used the fact that |∇g(x)|2 ≤ G (d(x))2 . So choosing, for example, E = − V2 L∞ − A − 1 (4.4) leads to (4.3) being satisfied with c = 1. This is exactly what we needed, and concludes our proof. Proof of Theorem 2. As already explained in Section 2, Theorem 2 follows directly from Theorem 1 and a choice of function G which for small t coincides with (see (2.16)): p 1 ln t + · Lj (t) + f˜(u) du , 2 j=2 t where we recall that the functions Lj were defined in (2.11), and f˜, which is to be found, must satisfy (2.9). More precisely, let ⎛ ⎞−1 p k−1 ⎝ Lp (t) = Lj (t)⎠ (4.5) k=2

j=1

defined for 0 < t < e−1 p . Notice that limt→0 Lp (t) = 0 and moreover integrable at zero. So if we define 1 Lp (t)2 , for 0 ≤ t ≤ e−1 f (t) + 4t p f˜(t) = 0, for t ≥ e−1 p ,

Lp (t)2 t

is

(4.6)

then it satisfies (2.9). Let now h(t) be a smooth function with the properties: t, for 0 ≤ t ≤ d20 (4.7) h(t) = 3 4 d0 , for t ≥ d0 , and 0 < h (t) ≤ 1 for all 0 < t < d0 . Here d0 ≤ min{e−1 p , dΩ } and in addition is sufficiently small such that for t ∈ (0, d0 ) 2 1 1 − Lp (t) − tf˜(t) ≥ . 2 3

(4.8)

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We claim that Gp (t) = ln h(t) +

391

p 1 · Lj h(t) + f˜(u) du , 2 j=2 h(t)

(4.9)

satisfies all the needed conditions. To check that Gp satisfies (Σ), first note that, for any k ≥ 1, ⎛ ⎞−1 k−1 1 Lj (t)⎠ , Lk (t) = − ⎝ t j=1 and so, for t ∈ (0, d0 ) Gp (t) =

1 1 · 1 − Lp h(t) − h(t)f˜ h(t) h (t) , h(t) 2

(4.10)

while for t ≥ d0 , Gp (t) = 0. Then (Σ.1) follows from (4.10) , (4.8) and the properties of h(t). To check (Σ.2), note that from (4.9) for t < d20 (take into account that d0 < 1 and for t < d20 , h(t) = t) ⎛ ⎞−1 p−1 −n ˜ −2 f (u) du −2 0+ ·⎝ Lj (2−n ρ0 )⎠ . e−2Gp (2 ρ0 ) ≥ 4n ρ0 e j=1

If we define, for x ∈ R large enough, the log-log functions ln0 (x) = x, lnk (x) = −1 1 ln(lnk−1 (x)), then note that for all 1 ≤ j ≤ p − 1 and n ≥ N (ρ0 ) = 1−ln 2 ln ρ0 −1 (remember that 2ρ0 < ep ) −1 Lj (2−n ρ0 ) = lnj (2n ρ−1 0 ) = lnj−1 (n ln 2 + ln ρ0 ) ≤ lnj−1 n .

But then ∞ −n 4−n e−2Gp (2 ρ0 ) ≥ const. n=0

∞ n=N (ρ0 )

1 = +∞ , n ln(n) ln2 (n) · · · lnp−2 (n)

where the divergence of the latter series is an elementary consequence of the integral test. Since supt≥ d0 Gp (t)2 < ∞, in view of the remark following Theorem 2, all 2 that remains to be done in order to apply Theorem 1 is to show that for t ∈ (0, d20 ) (2.14) implies (2.6) with G(t) = Gp (t). Taking into account (4.10) it is sufficient to check that for t ∈ (0, d20 ): 2 1 1 1 1 1 ˜ − 2 Lp (t) − f (t) ≥ 2 1 − Lp (t) − tf (t) (4.11) t2 t t t 2 Doing the algebra one gets the condition f˜(t) f (t) f˜(t) Lp (t)2 ˜ − ≥ −2 + + tf (t) + Lp (t) 2 t t 4t t

392



Now taking into account (4.6) and that from (4.8), for t ∈ (0, d20 ), Lp ≤ 23 , tf˜(t) ≤ 1 3 one has f˜(t) f˜(t) Lp (t)2 ˜ f˜(t) Lp (t)2 f (t) + ≤ − + , + t f (t) + L (t) =− p 2 2 t 4t t t 4t t and the proof is finished. −2

Finally we turn to the proof of our optimality theorem: Proof of Theorem 3. In order to achieve this, we will work in 1 dimension, on the interval (0, 1), and construct such a potential close to 0. In this case, let α ∈ R and consider the wave function ⎛ ⎞− 12 p−1 1 ψp,α (x) = x− 2 · ⎝ Lj (x)⎠ · Lp (x)α . (4.12) j=1

First note that ψp,α grows as x → 0+ for all α ∈ R, but that 1 2 ψp,α (x) dx = ∞ ⇐⇒ α ≥ − . 2 0+ A direct calculation shows that (x) = Vp,α (x)ψp,α (x) , ψp,α

with p−1 3 1 1 Vp,α (x) = · 2 − 2 · 4 x x j=1

j

(4.13) −1

Lk (x)

k=1

⎛ ⎞−1 p 1 + 2α + o(1) · 2 · ⎝ Lj (x)⎠ x j=1

(4.14)

where the o(1) comes from a sum of terms which are of lower order (in the same spirit as in the previous proof). In this case, they are ⎛ ⎛ ⎞−2 −1 ⎞2

j p−1 p 2 1 1 ⎝ α ⎠ + · · Lk (x) ·⎝ Lj (x)⎠ 2 4 x2 x j=1 j=1 k=1

⎛ ⎞−1 −1 ⎞ ⎛ p

j p−1 α ⎝ ⎠·⎝ − 2· Lk (x) Lj (x)⎠ x j=1 j=1 k=1

j −1 k −1 p−1 j 1 1 + · 2 Ll (x) Ll (x) 2 x j=1 k=1 l=1 l=1 −1 k −1

p p α − 2 Ll (x) Lm (x) . x m=1 k=1

l=1

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Further note that the other (decreasing at 0+) solution of φp,α (x) = Vp,α (x)φp,α (x) is given by the usual relation

φp,α (x) = ψp,α (x) · 0 −2 ψp,α (y)

x

1 dy . 2 (y) ψp,α

Since ∼y as y → 0+ for any given > 0, we see that φp,α (x) → 0 as x → 0+, and so in particular φp,α and ψp,α are indeed two independent solutions. But for α < − 21 , they are both in L2 (0+) and so we are in the limit-circle case and Hp,α = −Δ + Vp,α is not essentially self-adjoint on (0, 1). But this is exactly the type of potential we were looking for: given a constant c > 1, pick an α < − 2c < − 12 . Thus Hp,α is not essentially self-adjoint, but for x close enough to the boundary ∂Ω, equation (4.14) together with our choice of α implies that Vp,α satisfies (2.17), as claimed in the theorem. Finally, the potentials Vp,α can also be used in several space dimensions to construct counterexamples 1−

References [1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schr¨ odinger operators. Mathematical Notes, 29, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. [2] N. I. Akhiezer, I. M. Glazman, Theory of linear operators in Hilbert space. Vol II. Translated from the third Russian edition by E. R. Dawson. Translation edited by W. N. Everitt. Monographs and Studies in Mathematics, 10. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. [3] G. Barbatis, S. Filippas, A. Tertikas, Series expansions for Lp Hardy inequalities, Indiana Univ. Math. J. 52 (2003), 171–190. [4] M. Braverman, O. Milatovic, M. Shubin, Essential self-adjointness of Schr¨ odinger type operators on manifolds, Russ. Math. Surveys 57 (2002), 641–692. [5] A. G. Brusentsev, Selfadjointness of elliptic differential operators in L2 (G), and correction potentials, Trans. Moscow Math. Soc. 65 (2004), 31–61. [6] E. Coddington, N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1955. [7] B. Davies, A review of Hardy inequalities, The Mazćya anniversary collection, Vol. 2 (Rostock, 1998), 55–67, Oper. Theory Adv. Appl., 110, Birkh¨ auser, Basel, 1999. [8] B. Davies, The Hardy constant, Quart. J. Math. Oxford (2) 46 (1996), 417–431. [9] J. Dolbeault, M. Esteban, M. Loss, L. Vega, An analytical proof of Hardy-like inequalities related to the Dirac operator, J. Funct. Anal. 216 (2004), 1–21. [10] M. Esteban, M. Loss, Self-adjointness via partial Hardy-like inequalities, in Proceedings of the QMath10 Conference: Mathematical Results in Quantum Mechanics. World Scientific 2008, 41–47.

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[11] R. Frank, E. H. Lieb, R. Seiringer, Hardy–Lieb–Thirring inequalities for fractional Schr¨ odinger operators, J. Amer. Math. Soc. 21 (2008), 925–950. [12] F. Gesztesy, On non-degenerate ground states for Schr¨ odinger operators, Rep. Math. Phys. 20 (1984), 93–109. [13] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer, 1983. [14] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, J. Tidblom, Manyparticle Hardy inequalities, J. Lond. Math. Soc. (2) 77 (2008), 99–114. [15] H. Kalf, U.-V. Schminke, J. Walter, R. W¨ ust, On the spectral theory of Schr¨ odinger and Dirac operators with strongly singular potentials, in Lecture Notes in Mathematics 448 (1975), 182–226. [16] A. Laptev, A. Sobolev, Hardy inequalities for simply connected planar domains, arXiv:math/0603362 [17] M. Reed, B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press , New York-London, 1975. xv+361 pp. [18] A. Tertikas, N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math. 209 (2007), 407–459. Gheorghe Nenciu Department of Theoretical Physics and Mathematics University of Bucharest P.O. Box MG 11 RO-077125 Bucharest Romania and Institute of Mathematics “Simion Stoilow” of the Romanian Academy 21, Calea Grivit¸ei RO-010702 Bucharest, Sector 1 Romania e-mail: [email protected] Irina Nenciu Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 S. Morgan Street Chicago, IL USA and Institute of Mathematics “Simion Stoilow” of the Romanian Academy 21, Calea Grivit¸ei RO-010702 Bucharest, Sector 1 Romania e-mail: [email protected] Communicated by Claude Alain Pillet. Submitted: November 18, 2008. Accepted: January 19, 2009.



A Lower Bound on the First Spectral Gap of Schr¨ odinger Operators with Kato Class Measures Hendrik Vogt Abstract. We study Schr¨ odinger operators on Rn formally given by Hμ = −Δ − μ, where μ is a positive, compactly supported measure from the Kato class. Under the assumption that a certain condition on the μ-volume of balls is satisfied and that Hμ has at least two eigenvalues below the essential spectrum σess(Hμ ) = [0, ∞), we derive a lower bound on the first spectral gap of Hμ . The assumption on the μ-volume of balls is in particular satisfied if μ is of the form μ = aσM , where M is a compact (n−1)-dimensional Lipschitz submanifold of Rn , σM the surface measure on M , and 0 ≤ a ∈ L∞ (M ).

1. Introduction and main results There is extensive literature on estimates for the first spectral gap of Schr¨ odinger operators. Many papers concentrate on one of the following two extreme situations: On the one hand, for Schr¨ odinger operators with convex potentials on convex domains the gap turns out to be relatively large (see, e.g., [13,14] and the references therein); on the other hand, in tunneling situations, where the potential has two or more separated wells (minima) of equal depth, the gap becomes exponentially small with respect to a separation parameter (see, e.g., [6, 7, 12]). For Schr¨ odinger operators on Rn with bounded potentials, Kirsch and Simon proved in [8; Sec. 4] that the first spectral gap always admits a lower bound that is comparable to the gap size in tunnelling situations, without particular assumptions on the shape of V . In [9], Kondej and Veselic proved a similar result in dimension n = 2 for potentials given by certain measures supported on compact curves. This is the only gap estimate known to us for Schr¨ odinger operators with a singular interaction given by a measure. In the present paper we generalise the gap estimate of [9] to higher dimensions, for operators Hμ = −Δ − μ on Rn (defined as a form sum, cf. Section 2). We

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assume that μ is a positive, compactly supported measure on the Borel σ-algebra of Rn satisfying the volume bound μ B(x, r) ≤ cμ rn−α (x ∈ Rn , r > 0) (1.1) for some cμ > 0, α ∈ [0, 2). (Up to a change in the constant cμ , we could equivalently assume this bound for r ≤ 1 only since μ has compact support.) Assumption (1.1) implies that μ is in the Kato class; conversely, μ being in the Kato class implies (1.1) for α = 2 and some cμ > 0 (cf. Proposition 2.2). In Example 2.3(b) we will show that the surface measure on a compact (n−1)-dimensional Lipschitz submanifold of Rn satisfies (1.1) with α = 1. The proof of the gap estimate in [8] uses the fact that the eigenfunctions of the Schr¨ odinger operator are Lipschitz continuous if the potential is bounded. For singular potentials, Lipschitz continuity is no longer true in general. This difficulty is overcome in [9] by means of a method that relies on detailed knowledge of the geometry of spt μ, such as curvature bounds. Theorem 1.1 below provides a better (though not sharp) gap estimate, based only on the volume estimate (1.1). Since μ has compact support, we have σess(Hμ ) = [0, ∞) by [3; Thm. 3.1], so σ(Hμ ) \ [0, ∞) consists of isolated eigenvalues of finite multiplicity. In the main results of this paper, Theorems 1.1 and 1.3 below, we will assume that (A) Hμ has at least two negative eigenvalues; the lowest two are denoted by λ0 = −κ20 and λ1 = −κ21 ( > λ0 ). The proofs of these theorems are given in Section 6. Theorem 1.1. Let n ≥ 2 and suppose that μ ≥ 0 is a compactly supported measure on Rn satisfying (1.1) for some cμ > 0, 0 ≤ α < 2. Let d denote the diameter of the smallest closed ball containing spt μ. If assumption (A) is satisfied then there exist C, p, q, β > 0 depending only on n and α such that λ1 − λ0 ≥

C|λ0 | e−βκ0 (d+1) . (cμ + 1)p (d + 1)q

(1.2)

More precisely, for α < 1 one can choose β = n + 1, for α = 1 any β > n + 1, and n−1 + 2 for α > 1. For n ≥ 5, the factor |λ0 | on the right hand side of (1.2) β = 2−α can be omitted. Remarks 1.2. (a) In [9; Thm. 4.3], a gap estimate similar to the above is proved for the case n = 2, μ = cσim γ , where c > 0 and γ is a C 2 -curve without self-intersections, parameterised by arc length (cf. Example 2.3(c)). An application of Theorem 1.1 in this situation yields a slightly better estimate: Firstly, in [9] the constant C depends on the curvature of γ, and β is not given explicitly. Secondly, there is a factor |λ1 | in the gap estimate of [9]; replacing this factor by |λ0 | is in fact achieved by a simple trick (see the argument leading to equation (6.3)). Thirdly, in [9] the estimate contains an additional factor that behaves like |λ0 |8 for small |λ0 |.

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(b) One can hardly expect that the value of β given in Theorem 1.1 is sharp. Computable examples of tunneling situations might lead to the conjecture that (1.2) always holds with β = 1, but it does not seem possible to prove this with the method presented in this paper. (c) The attentive reader will note that the estimate (1.2) is not scaling invariant: It is easy to see that for s > 0 the operators Hμ and Hμs are unitarily equivalent, where μs is defined by μs (A) := s2−n μ(sA). Moreover, under the assumptions of Theorem 1.1 one computes that μs satisfies (1.1) with cμs = s2−α cμ , that the smallest closed ball containing spt μs has diameter d/s, and that the lowest two eigenvalues of Hμs are s2 λ0 and s2 λ1 . Applying Theorem 1.1 to μs , with s = εd for some ε > 0, we thus obtain the scaling invariant estimate Cε |λ0 | p e−(1+ε)βκ0 d , λ1 − λ0 ≥ 2−α (εd) cμ + 1 where Cε = C/( 1ε + 1)q . In dimension n = 1 we obtain a much simpler result. Theorem 1.3. Let μ ≥ 0 be a compactly supported measure on R, d the diameter of spt μ. If assumption (A) is satisfied then λ1 − λ0 ≥

|λ0 | e−2κ0 d d||μ|| + 1

and

λ1 − λ0 ≥

κ1 e−2κ0 d . d(κ1 d + 1)

(1.3)

The paper is organised as follows. In Section 2 we recall some basic results on form small measures and the (extended) Kato class of measures. In Section 3 we use a ground state transformation to show the representation 2 ϕ1 λ1 − λ0 = ϕ0 ∇ ||ϕ1 ||−2 2 ϕ0 2 of the lowest spectral gap, where ϕj is an eigenfunction corresponding to the eigenvalue λj , for j = 1, 2. We demonstrate how this representation can be used to prove the main results, given the following two ingredients: (i) an estimate of the modulus of continuity of the eigenfunctions, (ii) a pointwise estimate from below for the ground state ϕ0 of Hμ . These ingredients are provided in Sections 4 and 5, respectively. Since they are of independent interest, we will prove estimates that are sharper than necessary for the proof of Theorem 1.1. The proofs of Theorems 1.1 and 1.3 are given in Section 6. In the appendix we provide an estimate on the convolution kernel of (κ2 − Δ)−1 (where κ > 0), needed in Section 5, that we did not find in the literature.

2. Form small measures and Kato class measures Throughout this section let μ ≥ 0 be a measure on the Borel σ-algebra of Rn . We recall the definition of the operator Hμ for form small μ and some results on the (extended) Kato class of measures. In the following we write for brevity Lp for Lp (Rn ), and similarly W21 , Cc∞ , etc.

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The measure μ is called form small with respect to the Laplacian on Rn if μ does not charge sets of zero capacity and there exist γ ∈ [0, 1), c ∈ R such that 2 2 |u| dμ ≤ γ |∇u| dx + c |u|2 dx (u ∈ W21 ) . (2.1) Here and in the following, we tacitly assume that a quasi-continuous representative of u is chosen if we write u ∈ W21 ; then the integral |u|2 dμ is unambiguously defined. It is well-known that, under condition (2.1), 1 2 D(τμ ) := W2 , τμ (u) := |∇u| dx − |u|2 dμ (2.2) defines a closed quadratic form τμ in L2 . The domain D(τμ ) is dense in L2 , so we can define the Schr¨ odinger operator Hμ as the selfadjoint operator in L2 associated with τμ . In accordance with [15; p. 114] we say that μ is in the extended Kato class if there exists κ > 0 such that Gκ ∗μ ∈ L∞ , where Gκ is the convolution kernel of the free resolvent (κ2 − Δ)−1 . (It is automatic that then μ does not charge sets of zero capacity.) We say that μ is Kato small if limκ→∞ ||Gκ ∗μ||∞ < 1. By [15; Thm. 3.1], a Kato small measure is also form small. The measure μ is in the (proper) Kato class if ||Gκ ∗ μ||∞ → 0 as κ → ∞. In the following let kt denote the convolution kernel of etΔ , i.e., kt (x) = 2 n n (4πt)− 2 exp(− |x| 4t ) for t > 0, x ∈ R . Moreover, for an operator B on L2 and p, q ∈ [1, ∞] we denote by ||B||p→q the norm of B|Lp ∩L2 regarded as an operator from Lp to Lq . Proposition 2.1. Let α > 0, γ ∈ [0, 1) and assume that α kt ∗ μ dt ≤ γ . 0

−tHμ

Then μ is Kato small, and ||e

||1→1 ≤

∞ 1 1−γ

for all t ∈ [0, α].

Proof. Arguing as in [17; Prop. 4.7(b)] one finds κ > 0 such that ∞ −κ2 t e kt ∗ μ dt < 1 , ||Gκ ∗ μ||∞ = 0

∞

so μ is Kato small. We recall the approximation of Hμ given in [15]: For j, m ∈ N, m ≥ κ let μj := 1 B(0,j) μ, Vj,m := (m2 − κ2 )Gm ∗ μj . Then α α 2 tΔ 2 (m e V dt = − κ )G ∗ k ∗ μ dt j,m m t j 0 0 ∞ ∞ α 2 2 −1 ≤ ||m (m − Δ) ||∞→∞ kt ∗ μ dt ≤ γ , 0

∞

and by [16; Thm. 1(c) and the last formula line of part (i) of its proof ] we obtain 1 that ||e−tHVj,m ||1→1 ≤ 1−γ for all t ∈ [0, α], j, m ∈ N, m ≥ κ. Moreover, HVj,m → Hμj in the strong resolvent sense as m → ∞, by [15; Cor. 2.4(a)], and Hμj → Hμ

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in the strong resolvent sense as j → ∞, by [15; Thm. 3.3(a)]. We thus conclude 1 for all t ∈ [0, α]; cf. [15; proof of Cor. 2.4(b)]. that ||e−tHμ ||1→1 ≤ 1−γ In the next result we show the relation between the Kato class condition and the volume bound (1.1). We will need the following fact: Assume that m : [0, ∞) → [0, ∞) is increasing, m(0) = 0, μ(B(x, r)) ≤ m(r) for all x ∈ Rn , r > 0. Then for any decreasing function f : [0, ∞) → [0, ∞) and all x ∈ Rn , one can estimate ∞ f (|x − y|) dμ(y) ≤ f (r) dm(r) ; (2.3) Rn

0

cf. [5; p. 179]. Proposition 2.2. Let 0 ≤ α ≤ 2. Assume that there exists c ≥ 0 such that t α ks ∗ μ ds ≤ ct1− 2 (t > 0) .

(2.4)

∞

0

1 Then the volume estimate (1.1) holds with cμ = c/g(e1 ), where g(x) := 0 ks (x) ds. Conversely, if α < 2 (α ≤ 1 in the case n = 1) then (1.1) implies (2.4) with n 2 2−α π − 2 Γ( n−α c = 2−α 2 + 1)cμ ; in particular, μ is in the Kato class. Proof. Let r > 0. For y ∈ Rn \ {0} we compute, substituting s = r2 t, that 1 r2 y y dt = r2−n g =: gr (y) . ks (y) ds = r2−n kt r r 0 0 Since gr ∗ μ is lower semicontinuous, we thus obtain from assumption (2.4) that sup(gr ∗ μ) ≤ cr2−α . Moreover, gr (y) ≥ gr (re1 ) = r2−n g(e1 ) for 0 < |y| ≤ r, so we infer for all x ∈ Rn that 2−n r g(e1 )μ B(x, r) ≤ gr (x − y) dμ(y) ≤ gr ∗ μ(x) ≤ cr2−α , B(x,r)

and the first assertion follows. Conversely, assume that (1.1) holds and that α < 2. Then for s > 0 we obtain 1 by (2.3), substituting r = (4sρ) 2 , that ∞ ∞ n−α n−α n r2 n ||ks ∗ μ||∞ ≤ (4πs)− 2 e− 4s cμ drn−α = cμ (4πs)− 2 e−ρ (4s) 2 dρ 2 0 0

n n−α −α − +1 . = cμ (4s) 2 π 2 Γ 2 (This computation is also valid in the case n = α = 1 if one replaces drn−α and n−α dρ 2 with the Dirac measure at 0.) Integration from 0 to t now yields the second assertion. If n = 1 and μ is finite, then assumption (1.1) is trivially satisfied with α = 1 and cμ = ||μ||. We now give examples in dimension n ≥ 2 where assumption (1.1) is satisfied with α = 1.

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Example 2.3. (a) (cf. [3; Thm. 4.1(iv)]) Let U ⊆ Rn−1 be open and bounded, f : U → Rn Lipschitz continuous, and |f (x) − f (y)| ≥ |x − y| (x, y ∈ U ) . Let μ = σf (U ) , i.e., ϕ f (y) g(y) dy ϕ dμ =

ϕ ∈ Cc (Rn ) ,

U

1 2

with g := (det(f f )) the square root of the Gram determinant. We show that then (1.1) holds with α = 1, cμ = 2n−1 ωn−1 ||g||∞ , where ωn−1 denotes the volume of the (n−1)-dimensional unit ball. Indeed, let x ∈ Rn , r > 0. There exists y0 ∈ U such that r0 := dist(x, f (U )) = |x − f (y0 )|. If r < r0 then μ(B(x, r)) = 0 ≤ cμ rn−1 . If r ≥ r0 then for y ∈ U with f (y) ∈ B(x, r) we have |y − y0 | ≤ |f (y) − f (y0 )| ≤ |f (y) − x| + |x − f (y0 )| ≤ r + r0 ≤ 2r , and it follows that 1 B(x,r) f (y) g(y) dy ≤ μ B(x, r) = U

||g||∞ dy = ωn−1 (2r)n−1 ||g||∞ .

B(y0 ,2r)

(b) Let M be a compact (n−1)-dimensional Lipschitz submanifold of Rn , σM the surface measure on M and 0 ≤ a ∈ L∞ (M ). Then M can be covered by finitely many relatively open subsets that can be parameterised as in (a), so μ = aσM satisfies (1.1) with α = 1. (c) Let n = 2, N ∈ N, γ1 , . . . , γN Lipschitz curves in R2 , γj : Ij → R2 parameterised by arc length on a compact interval Ij ⊆ R (j = 1, . . . , N ), and assume that |γj (s) − γj (t)| ≥ 12 |s − t| for all j ∈ {1, . . . , N }, s, t ∈ Ij . In particular, each curve γj is intersection free, but the different curves may intersect. If, e.g., γ : I → R2 is a curve (with a compact interval I ⊆ R) that is piecewise C 1 and parameterised by arc length, then γ can be split into finitely many parts γ1 , . . . , γN satisfying the above. Let now N Γ := γj , a ∈ L∞ (im Γ) , μ := aσim Γ . j=1

By part (a), applied with fj (t) := γj (2t), we obtain that then (1.1) holds with α = 1, cμ = N ||a||∞ 21 ω1 · 2 = 8N ||a||∞ . It follows that the spectral gap estimate of [9] can be obtained as a special case of Theorem 1.1. The following representation of the eigenfunctions of Hμ is extracted from [3; proof of Cor. 2.3]. We include the proof for the reader’s convenience. Lemma 2.4. Assume that μ is form small, ϕ an eigenfunction of Hμ , Hμ ϕ = −κ2 ϕ for some κ > 0. Then ϕ = Gκ ∗ (ϕμ).

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Proof. The form smallness of μ implies that D(Hμ ) ⊆ W21 ⊆ L2 (μ), so ϕ ∈ L2 (μ). By [3; Lemma 2.2] we obtain that u := Gκ ∗ (ϕμ) ∈ W21 and 2

u, vκ := ∇u, ∇v + κ u, v = ϕv dμ (v ∈ W21 ) . (2.5) Moreover, for v ∈ W21 we have

0 = (Hμ + κ2 )ϕ, v = (τμ + κ2 )(ϕ, v) = ϕ, vκ −

ϕv dμ .

(2.6)

Combining (2.5) and (2.6) we conclude that ϕ, vκ = u, vκ for all v ∈ W21 and therefore ϕ = u. We use the above representation for proving the following estimates of the L2 -norm against the L∞ -norm of the eigenfunctions of Hμ (cf. [9; Lemma 5.5]); these estimates will be needed in the proofs of the main results. Proposition 2.5. Assume that μ is form small, ϕ a bounded eigenfunction of Hμ , Hμ ϕ = −κ2 ϕ for some κ > 0. (a) If μ is finite then ||ϕ||22 ≤ κ−2 ||μ||||ϕ||2∞ . (b) If n = 1 and spt μ ⊆ B[0, R] for some R > 0 then ||ϕ||22 ≤ (2R + κ1 )||ϕ||2∞ . n 2 (c) If n ≥ 5 and spt μ ⊆ B[0, R] for some R > 0 then ||ϕ||22 ≤ 2n−4 n−4 ωn R ||ϕ||∞ . Proof. Without loss of generality assume that ||ϕ||∞ = 1. By Lemma 2.4 we have ϕ = Gκ ∗ (ϕμ). (a) Since ||Gκ ||1 = ||(κ2 − Δ)−1 ||1→1 = κ−2 , we obtain that ||ϕ||22 ≤ ||ϕ||1 ||ϕ||∞ = ||Gκ ∗ (ϕμ)||1 ≤ κ−2 ||μ|| . (b) Outside spt μ we have (κ2 − Δ)ϕ = 0 and hence (κ2 − Δ)|ϕ| ≤ 0 in the distributional sense, i.e., |ϕ| is a subsolution of the equation (κ2 − Δ)u = 0. Moreover, u(x) := e−κ(|x|−R) defines a solution of this equation on R \ {0}, and |ϕ| ≤ u on [−R, R]. Since ϕ = Gκ ∗ (ϕμ) vanishes at ∞, we conclude that |ϕ| ≤ u on R and therefore R ∞ 1 1 dx + 2 e−2κ(x−R) dx = 2R + . ||ϕ||22 ≤ 2 κ 0 R (c) Arguing as in (b) we obtain that |ϕ| is subharmonic on Rn \ B[0, R] and R n−2 ) for |x| ≥ R and hence vanishes at ∞. Therefore, |ϕ(x)| ≤ ( |x|

2(n−2) ∞ R n ∧1 ≤ dx = ωn R + nωn R2(n−2) r−2(n−2) rn−1 dr |x| R

1 n n 2(n−2) 4−n R = 1+ = ωn R + nωn R ωn R n . n−4 n−4

||ϕ||22

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3. The ground state transformation Throughout this section, μ ≥ 0 is a Kato class measure on Rn . It follows from [2; Thm. 3.2.(ii)] that then the eigenfunctions of Hμ are in C0 (Rn ), the space of continuous functions vanishing at ∞. Assume that λ0 := inf σ(Hμ ) < 0 is an eigenvalue of Hμ . It is well-known that λ0 is non-degenerate and that a corresponding eigenfunction ϕ0 can be chosen such that ϕ0 ≥ 0. Lemma 2.4 implies that then inf B[0,R] ϕ0 > 0 for all R > 0. We define the unitarily transformed (and shifted) form τμ in L2 (ϕ20 ) := L2 (Rn , ϕ20 dx) by D( τμ ) := u ∈ L2 (ϕ20 ); ϕ0 u ∈ D(τμ ) , τμ (u) := (τμ − λ0 )(ϕ0 u) . In the case μ = V dx, with V from the Kato class, the following result is already proved in [4; Prop. 4.4S]; see [9; Thm. 3.1] for the case μ = cσM , with M ⊆ Rn a compact C 2 -manifold of codimension 1 (cf. Example 2.3(b)). Proposition 3.1. For all u ∈ D( τμ ) one has ∇u ∈ L2 (ϕ20 )n and τμ (u) = |∇u|2 ϕ20 dx .

(3.1)

Proof. We first assume that u ∈ W21 ∩ L∞ . Then ϕ0 u ∈ W21 and ϕ0 uu ∈ W21 since ϕ0 ∈ W21 ∩ L∞ . Therefore, by the product rule, ∇(ϕ0 u) · ∇(ϕ0 u) dx = |∇u|2 ϕ20 dx + ∇ϕ0 · ∇(ϕ0 uu) dx , and hence τμ (u) −

|∇u|2 ϕ20 dx = τμ (ϕ0 u) − λ0 |ϕ0 u|2 dx − |∇u|2 ϕ20 dx = ∇ϕ0 · ∇(ϕ0 uu) dx − ϕ0 · ϕ0 uu dμ − λ0 ϕ0 · ϕ0 uu dx = (τμ − λ0 )(ϕ0 , ϕ0 uu) = (Hμ − λ0 )ϕ0 · ϕ0 uu dx = 0 .

Thus we have shown the assertion for u ∈ D := W21 ∩ L∞ . Now observe that D ⊇ ϕ−1 0 Cc∞ (ψ ∈ Cc∞ implies ϕ0−1 ψ ∈ D since ϕ0−1 ∈ W2,loc1 ∩ L∞,loc ). Since Cc∞ is a core for τμ , it follows that D is a core for τμ . Let u ∈ D( τμ ), (uk ) ⊆ D, uk → u in D( τμ ). Then uk → u in L2 (ϕ20 ), and by (3.1) applied to uk − uk ∈ D we obtain that (∇uk )k is a Cauchy sequence in L2 (ϕ20 )n . This implies ∇u ∈ L2 (ϕ20 )n in the distributional sense, ∇uk → ∇u in L2 (ϕ20 )n , and we conclude that (3.1) holds for all u ∈ D( τμ ). In the in addition that μ has compact support and that following assume λ1 := inf σ(Hμ ) \ {λ0 } < 0; then λ1 is an eigenvalue of Hμ since σess(Hμ ) =

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[0, ∞) by [3; Thm. 3.1]. Let ϕ1 be an associated eigenfunction, ϕ1 real-valued. By Proposition 3.1 we then obtain (λ1 − λ0 )||ϕ1 ||22 = (τμ − λ0 )(ϕ1 ) = τμ

ϕ1 ϕ0

2 ϕ1 = ϕ0 ∇ . ϕ0 2

(3.2)

We now describe how this formula can be used for the estimation of λ1 − λ0 . The ansatz is largely the same as in [8] and [9]; we will indicate the differences below. As in [8; p. 405] we normalise ϕ0 , ϕ1 such that ||ϕ0 ||∞ = ||ϕ1 ||∞ = 1, sup ϕ1 = 1. Note that then inf ϕ1 < 0 since ϕ0 , ϕ1 are orthogonal. Since ϕ1 ∈ C0 (Rn ), there exist x0 , x1 ∈ Rn such that ϕ1 (x0 ) = min ϕ1 , ϕ1 (x1 ) = max ϕ1 . By Lemma 2.4 we have ϕ1 = G√|λ | ∗ (ϕ1 μ), and hence (|λ1 | − Δ)ϕ1 = 0 on Rn \ spt μ. This implies 1 that ϕ1 has no positive maxima and no negative minima outside spt μ, and thus x0 , x1 ∈ spt μ. In the following we first assume that n ≥ 2. Let R be the radius of the smallest closed ball containing spt μ; for simplicity suppose that spt μ ⊆ B[0, R]. Let ε := 14 inf B(0,R+1) ϕ0 . Then 0 < ε ≤ 14 . Since ϕ1 ∈ C0 (Rn ), there exists δ ∈ (0, 1] such that |ϕ1 (x) − ϕ1 (y)| ≤ ε for |x − y| ≤ δ. (Explicit estimates from below for ε and δ will be given in the next two sections.) It follows that ϕ1 ε 1 ≤ ≤ ϕ0 ϕ0 4

on B(x0 , δ) ,

ϕ1 3 ≥ ϕ1 ≥ 1 − ε ≥ ϕ0 4

on B(x1 , δ) .

(3.3)

In [8] and [9], similar estimates were proved by means of gradient estimates on ϕ0 and ϕ1 . Let now T be the convexhull of B(x0 , δ) ∪ B(x1 , δ), and T := x ∈ T ; 0 ≤

x − x0 , x1 − x0 ≤ |x1 − x0 |2 . Then T is a tube connecting the two points x0 and x1 . By the Cauchy–Schwarz inequality we obtain from (3.2) that (λ1 − λ0 )||ϕ1 ||22 ≥ T

ϕ1 2 1 inf ϕ2 ϕ20 ∇ ≥ ϕ0 |T | T 0

ϕ1 2 ∇ , ϕ0 T

where |T | denotes the volume of T . (This estimate leads to better results than the corresponding estimate in [8; eq. (4.4)] and [9; eq. the fundamental (16)].) Using 1 1 ≥ theorem of calculus, we infer from (3.3) that T ∇ ϕ ω δ n−1 . Moreover, n−1 ϕ0 2 |T | = |x1 − x0 |ωn−1 δ n−1 , so we conclude that (λ1 −

λ0 )||ϕ1 ||22

≥

1 2Rωn−1 δ n−1

2

(4ε)

1 ωn−1 δ n−1 2

2 =

2ωn−1 2 n−1 ε δ . R

(3.4)

For n = 1, the above estimation simplifies considerably: If spt μ ⊆ [−R, R] then from (3.2) we obtain, again using the Cauchy–Schwarz inequality and the

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H. Vogt


fundamental theorem of calculus, that 2 R 2 2 ϕ1 ϕ0 ∇ (λ1 − λ0 )||ϕ1 ||2 ≥ ϕ0 −R 2 R ϕ1 1 1 2 ∇ inf ϕ0 inf ϕ2 . ≥ ≥ 2R [−R,R] ϕ0 2R [−R,R] 0 −R

(3.5)

4. Hölder continuity of the eigenfunctions Throughout this section we assume that n ≥ 2 and that μ ≥ 0 is a Kato class measure on Rn . Then the eigenfunctions of Hμ are uniformly continuous since they are in C0 (Rn ) by [2]. In the main result of this section, Theorem 4.3 below, we will show that the eigenfunctions are H¨ older continuous if the volume bound (1.1) older exponent depending on α. is satisfied for some cμ > 0, α ∈ [0, 2), with a H¨ We start with an estimate of the modulus of continuity of the eigenfunctions of Hμ that is rather simple but holds in great generality. Let G0 be the fundamental solution of −Δ given by 1 ln 1 if n = 2 , G0 (x) = 2π 1 |x| 2−n if n ≥ 3 . (n−2)σn−1 |x| For r0 > 0 let

+ ρ1,r0 := G0 − G0 (r0 e1 ) ,

Then ρ2,r0 (x) =

1 1−n σn−1 |x|

ρ2,r0 := |∇(G0 − ρ1,r0 )| .

for |x| > r0 , ρ2,r0 (x) = 0 otherwise.

Proposition 4.1. Let ϕ be an eigenfunction of Hμ , Hμ ϕ = −κ2 ϕ for some κ > 0. Let ε > 0. If r0 > 0 satisfies ||ρ1,r0 ∗ μ||∞ ≤ 4ε then ε |ϕ(x) − ϕ(y)| ≤ ε||ϕ||∞ for all x, y ∈ Rn such that |x − y| ≤ ||ρ2,r0 ∗ μ||−1 ∞ . 2 Proof. Let g1 := (Gκ − Gκ (r0 e1 ))+ , g2 := Gκ − g1 . By Lemma 2.4 we have ϕ = Gκ ∗ (ϕμ) = (g1 + g2 ) ∗ (ϕμ) . Since g1 ∗ (ϕμ) = ϕ − g2 ∗ (ϕμ) is continuous, we can estimate |g1 ∗ (ϕμ)(x) − g1 ∗ (ϕμ)(y)| ≤ 2||g1 ∗ (ϕμ)||∞ and hence |ϕ(x) − ϕ(y)| ≤ |g2 ∗ (ϕμ)(x) − g2 ∗ (ϕμ)(y)| + 2||g1 ∗ (ϕμ)||∞ ≤ |x − y| · |||∇g2 | ∗ (ϕμ)||∞ + 2||g1 ∗ (ϕμ)||∞ ≤ |x − y| · |||∇g2 | ∗ μ||∞ + 2||g1 ∗ μ||∞ ||ϕ||∞ for all x, y ∈ Rn . Observe that ∇Gκ (z) = 0

∞

−n 2

(4πt)

2z − 4t

e−

|z|2 4t

−κ2 t

dt

(4.1)

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for all z ∈ Rn \{0}, and that this formula also holds for κ = 0. (For n ≥ 3 the latter is clear; for n = 2 it can easily be seen by computing the integral.) We infer that |∇g2 | ≤ ρ2,r0 and g1 ≤ ρ1,r0 . By the assumption on r0 we thus obtain from (4.1) that ε ||ϕ||∞ , |ϕ(x) − ϕ(y)| ≤ |x − y| · ||ρ2,r0 ∗ μ||∞ + 2 · 4 and the assertion follows. Remarks 4.2. (a) The above estimate of the modulus of continuity does not depend on κ. It is clear that for large κ one obtains a better estimate using (4.1) without further estimating g1 and |∇g2 |. (b) Let r0 > 0 be as in Proposition 4.1 and assume that μ is finite. Obviously, 1 r01−n ||μ||, so it follows that ||ρ2,r0 ∗ μ||∞ ≤ ||ρ2,r0 ||∞ ||μ|| = σn−1 σn−1 n−1 εr . |ϕ(x) − ϕ(y)| ≤ ε||ϕ||∞ if |x − y| ≤ 2||μ|| 0 For the proof of Theorem 1.1 we will need the following application of Proposition 4.1. In the case of Example 2.3(c) (where n = 2 and α = 1), a related result is given in [9; Prop. 6.8]. Theorem 4.3. Assume that the measure μ is finite and that (1.1) holds for some cμ > 0, 0 ≤ α < 2. Let ϕ be an eigenfunction of Hμ and ε > 0. Then |ϕ(x) − ϕ(y)| ≤ ε||ϕ||∞ where

for all

|x − y| ≤ δ ,

⎧σ 1−α n−1 n−1 1−α ⎪ ||μ||− n−α c− μ n−α ε ⎪ 2 n−α ⎪

−1 ⎪ n−2 ⎨σ 1 n−1 4 1 n−1 ε n−1 c μ δ= 2n cμ ln 1 + σn−1 ||μ|| ε ⎪ ⎪ 1 2−α ⎪ α−1 1 ⎪ α−1 2−α 2−α ε ⎩ 1 (σ n−1 ) 2−α n−1 2 4 cμ

if

α < 1,

if

α = 1,

if

α > 1.

Proof. Let r0 > 0. In order to apply Proposition 4.1, we have to estimate ||ρ1,r0 ∗ μ||∞ and ||ρ2,r0 ∗ μ||∞ . Let m(r) := cμ rn−α (r ≥ 0). Using (2.3) we estimate r0 r0 r0 ||ρ1,r0 ∗ μ||∞ ≤ ρ1,r0 (re1 ) dm(r) = ρ1,r0 (re1 )m(r)|0+ − m(r) dρ1,r0 (r) 0 0 r0 r0 1 1−n cμ cμ r2−α . m(r) r dr = r1−α dr = = σ σ σ (2 − α) 0 n−1 n−1 0 n−1 0 Setting

1 2−α ε 2−α r0 := σn−1 (4.2) 4 cμ we thus obtain ||ρ1,r0 ∗ μ||∞ ≤ 4ε , so ε (4.3) |ϕ(x) − ϕ(y)| ≤ ε||ϕ||∞ for all |x − y| ≤ ||ρ2,r0 ∗ μ||−1 ∞ 2 by Proposition 4.1.

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Recall that σn−1 ρ2,r0 (x) ≤ (|x| ∨ r0 )1−n for all x ∈ Rn . In the case α > 1 we obtain by (2.3) that ∞ ∞ (r ∨ r0 )1−n dm(r) = r01−n m(r0 ) + r−α (n − α)cμ dr σn−1 ||ρ2,r0 ∗ μ||∞ ≤ 0

r0

n − α 1−α n − 1 1−α r0 r = cμ r01−α + cμ = cμ , α−1 α−1 0 hence ε ε 1α−1 1α−1 ||ρ2,r0 ∗ μ||−1 σn−1 r0α−1 = ∞ ≥ 2 2n−1 cμ 2n−1

2−α 4

α−1 2−α

ε σn−1 cμ

1+ α−1 2−α

by (4.2), and the assertion follows from (4.3). 1 n−α . Then m(r ) = c r n−α = ||μ|| and hence, again Let now r1 := ( ||μ|| 1 μ 1 cμ ) by (2.3), r1 σn−1 ||ρ2,r0 ∗ μ||∞ ≤ (r ∨ r0 )1−n dm(r) . 0

In the case α < 1 we estimate

σn−1 ||ρ2,r0 ∗ μ||∞ ≤

r1

r1

r1−n dm(r) = 0

r−α (n − α)cμ dr

0

n−1 1−α n − α 1−α n−α r1 ||μ|| n−α cμn−α , = 1−α 1−α so as above the assertion follows from (4.3). Finally, let α = 1. Then r1 r−1 (n − 1)cμ dr σn−1 ||ρ2,r0 ∗ μ||∞ ≤ r01−n m(r0 ∧ r1 ) +

= cμ

r0 ∧r1

r1 r0 ∧ r1

n−1

r1 r1 r1 = cμ ∧1 + (n − 1) ln ∨1 = cμ f , r0 r0 r0 =

r01−n cμ (r0

∧ r1 )

n−1

+ (n − 1)cμ ln

with f (x) := (x ∧ 1)n−1 + (n − 1) ln(x ∨ 1). We show that f (x) ≤ n ln(1 + x) by distinguishing the cases x < 1 and x ≥ 1: If x < 1 then 1 ln(1 + x) ≤ n ln(1 + x) f (x) = xn−1 + 0 ≤ x ≤ ln 2 since n ≥ 2. If x ≥ 1 then f (x) = 1 + (n − 1) ln x, so we have to show that hn (x) := n ln(1 + x) − (n − 1) ln x ≥ 1. Observe that hn (x) ≥ h2 (x) and that h2 is increasing on [1, ∞), hence hn (x) ≥ h2 (1) = 2 ln 2 − ln 1 ≥ 1. We conclude that

r1 cμ r1 ncμ ||ρ2,r0 ∗ μ||∞ ≤ f ln 1 + ≤ . σn−1 r0 σn−1 r0 n−1 and r = By (4.3) this implies the assertion since r1 = ( ||μ|| 0 cμ ) 1

σn−1 ε 4 cμ .

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Remark 4.4. Theorem 4.3 implies that the eigenfunctions of Hμ are Lipschitz continuous if α < 1. In general, the eigenfunctions are not Lipschitz continuous if α = 1: One can show that for n ≥ 2 and μ = cσM , where M = BRn−1 (0, 1) × {0} and c > 0 is large enough for the ground state ϕ0 to exist, the logarithmic factor given in Theorem 4.3 reflects the correct behaviour of the modulus of continuity of ϕ0 .

5. Estimate of ϕ0 from below In this section we prove a pointwise estimate of the ground state ϕ0 of Hμ from below. Throughout the section we assume that t ≤ ctθ (t > 0) k ∗ μ ds (5.1) s ∞

0

for some c > 0, θ ∈ (0, 1] (cf. Proposition 2.2); in particular, μ is in the Kato class. As mentioned in Section 3, μ being in the Kato class implies that the eigenfunctions of Hμ are in C0 (Rn ). The following result provides an estimate of the L∞ -norm of the eigenfunctions. Proposition 5.1. Assume that (5.1) holds for some c > 0, θ ∈ (0, 1]. Let ϕ be an eigenfunction of Hμ , Hμ ϕ = −κ2 ϕ for some κ > 0. If |ϕ| dμ < ∞ then ||ϕ||∞ ≤ cn (c, θ, κ) |ϕ| dμ , where 1 , 2κ 1 e c2 (c, θ, κ) = ln 1 + (4c) θ κ−2 , 2π n−2 n−2 1 ωn−2/n (4π)− 2 (4c) 2θ cn (c, θ, κ) = (n ≥ 3) . n−2 Before proving this proposition, we state and prove the main result of this section, the announced pointwise estimate of ϕ0 . c1 (c, θ, κ) =

Theorem 5.2. Assume that spt μ ⊆ B[0, R] for some R > 0 and that (5.1) holds for some c > 0, θ ∈ (0, 1]. Let 0 ≤ ϕ0 ∈ D(Hμ ) be the ground state of Hμ , ||ϕ0 ||∞ = 1, κ > 0 such that Hμ ϕ0 = −κ2 ϕ0 . (a) If n = 1 then ϕ0 (x) ≥ e−κ(|x|+R) for all x ∈ R. (b) If n = 2 then 1 1 (4c)− 2θ (|x| + R)−1 ∧ 1 e−κ(|x|+R) (x ∈ R2 ) . ϕ0 (x) ≥ 2e (c) If n ≥ 3 then ϕ0 (x) ≥ cn (4c)− where cn =

n−2 2θ

(|x| + R)2−n e−κ(|x|+R)

n−2 2/n 1 2 σn−1 ωn (4π)

2

= 2n−3 ( n2 ) n Γ( n2 )

n−2 n

(x ∈ Rn ) , .

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H. Vogt


Proof. Lemma 2.4 implies that ϕ0 dμ ϕ0 (x) = Gκ (x − y)ϕ0 (y) dμ(y) ≥ Gκ (|x| + R)e1

(5.2)

B(0,R)

for all x ∈ Rn . 1 −κ|x| (a) For n = 1 we have Gκ (x) = 2κ e (x ∈ R), and ϕ0 dμ ≥ 2κ by Proposition 5.1, so the assertion follows from (5.2). (b) Let x ∈ R2 \ {0}. Proposition A.1(b) from the appendix and Proposition 5.1 yield −κ|x| 1 1 1 ln 1 + κa −κ|x| 2π ln 1 + κ|x| e e Gκ (x) ϕ0 dμ ≥ , 2 = 1 e 2e ln 1 + κb ln 1 + (4c) 2θ 1 2π

κ

1 2θ

= (4c) . By (5.2) the assertion follows if ln(1+x)/ ln(1+y) ≥ xy ∧1 with a = for all x, y > 0. In the case x ≥ y the latter inequality is clear; for x < y it is equivalent to x1 ln(1 + x) ≥ y1 ln(1 + y), which is a consequence of the concavity of ln, so the proof of (b) is complete. (c) Let x ∈ Rn \ {0}. Proposition A.1(a) from the appendix and Proposition 5.1 yield n−2 n−2 1 |x|2−n e−κ|x| · (n − 2)ωn2/n (4π) 2 (4c)− 2θ , Gκ (x) ϕ0 dμ ≥ (n − 2)σn−1 1 |x| , b

so the assertion follows from (5.2).

For the proof of Proposition 5.1 we need the following estimate of ||Tμ (t)||p→∞ . Lemma 5.3. Assume that (5.1) holds for some c > 0, θ ∈ (0, 1]. Let t0 := 2(4c)− θ . Then 1

||Tμ (t0 )||p→∞ ≤ 2(2πt0 )− 2p = 2(4π)− 2p (4c) 2θp n

n

n

(1 ≤ p ≤ ∞) .

− θ1

Proof. Let t1 := 12 t0 = (4c) . By assumption (5.1) we have t1 1 kt ∗ μ dt ≤ ctθ1 = . 4 ∞

0

From Proposition 2.1 it follows that ||Taμ (t1 )||1→1 ≤ 1/(1 − a4 ) for all 0 ≤ a < 4. n Moreover, ||T0μ (t1 )||1→∞ = ||et1 Δ ||1→∞ = (4πt1 )− 2 , so by the Stein interpolation theorem we infer as in [15; Thm. 5.1] that 1

1

2 2 ||T0μ (t1 )||1→∞ ≤ 2 2 (4πt1 )− 4 . ||Tμ (t1 )||1→2 ≤ ||T2μ (t1 )||1→1 1

n

Hence, by duality, ||Tμ (t0 )||1→∞ ≤ ||Tμ (t1 )||21→2 ≤ 2(4πt1 )− 2 = 2(2πt0 )− 2 . n

n

The assertion now follows from Riesz-Thorin interpolation with the estimate

2 1 2 ||Tμ (t0 )||∞→∞ ≤ ||Tμ (t1 )||1→1 ≤ < 2. 1 − 1/4

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Proof of Proposition 5.1. By Lemma 2.4 we have ϕ = Gκ ∗(ϕμ) = (κ2 −Δ)−1 (ϕμ). 1 . For n ≥ 2 we estimate, In the case n = 1, the assertion follows since ||Gκ ||∞ = 2κ 1 −θ with t0 = 2(4c) as in Lemma 5.3, 2 ||ϕ||∞ = ||e−κ t0 Tμ (t0 )ϕ||∞ ≤ Tμ (t0 )(κ2 − Δ)−1 : M (Rn ) → L∞ (Rn ) |ϕ| dμ , where M (Rn ) is the Banach space of finite Borel measures on Rn . We now have to show that the norm of Tμ (t0 )(κ2 − Δ)−1 is less or equal cn (c, θ, κ). First suppose that n = 2. Then for all p ∈ [1, ∞) we have ∞ 2 ||(κ2 − Δ)−1 : M (Rn ) → Lp (Rn )|| ≤ e−κ t ||etΔ : M (Rn ) → Lp (Rn )|| dt 0 ∞ 2 2 1 ≤ (4πt)− 2 (1− p ) e−κ t dt 0 ∞ 2 1 1 −p −1 p s p −1 e−s ds . = κ (4π) Since

∞ 0

0

s

1 p −1

−s

e

ds =

Γ( p1 )

= pΓ(1 +

1 p)

≤ p, we conclude by Lemma 5.3 that

||Tμ (t0 )(κ2 − Δ)−1 : M (Rn ) → L∞ (Rn )|| ≤ 2(4π)− 2p (4c) 2θp · κ− p (4π) p −1 p 1 1 1 p (4c) θ κ−2 p =: f (p) . = 2π For the proof of the case n = 2 it remains to show that there exists p ∈ [1, ∞) such 1 1 that f (p) ≤ c2 (c, θ, κ). With a := (4c) θ κ−2 this inequality reads pa p ≤ e ln(1 + a). If a ≤ e then a ≤ e ln(1 + a), so the inequality holds with p = 1. For a > e we take p = ln a; then 1 1 pa p = pe p ln a = ln a · e1 ≤ e ln(1 + a) . Let now n ≥ 3. For m ∈ M (Rn ) we have (κ2 − Δ)−1 m = Gκ ( · − y) dm(y) (Bochner integral of y → Gκ ( · − y) ∈ L1 (Rn )) and hence 2 −1 ||Tμ (t0 )(κ − Δ) m||∞ ≤ ||Tμ (t0 )Gκ ( · − y)||∞ dm(y) 2

2

2

1

≤ sup ||Tμ (t0 )G0 ( · − y)||∞ · ||m|| . y∈Rn

Using Lemma 5.4 below and Lemma 5.3, we deduce that n−2 2 1 −2 n n ωn n ||Tμ (t0 )||1→∞ ||Tμ (t0 )||∞→∞ 2(n−2) n−2 n−2 1 −2 ωn n · 2(4π)− 2 (4c) 2θ ≤ 2(n − 2)

||Tμ (t0 )(κ2 −Δ)−1 : M (Rn ) → L∞ (Rn )|| ≤

and conclude the proof.

In the following interpolation lemma we implicitly use the fact that G0 is in n the weak Lebesgue space L n−2 ,w .

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H. Vogt


Lemma 5.4. Let n ≥ 3. Let T : L1 + L∞ (Rn ) → L∞ (Rn ) be a bounded linear 1 |x − y|2−n (x ∈ Rn \ {y}). Then operator, y ∈ Rn , f (x) := (n − 2)G0 (x − y) = σn−1 ||T f ||∞ ≤

n−2 2 1 − n2 n n ωn ||T ||1→∞ ||T ||∞→∞ . 2

Proof. Without loss of generality assume that y = 0. Let r0 > 0, f1 := (f − 1 r02−n , f (r0 e1 ))+ , f2 := f − f1 . Then ||f2 ||∞ = f (r0 e1 ) = σn−1

r0 r0 1 1 − (r2−n − r02−n ) rn−1 dr = (r − r02−n rn−1 ) dr = ||f1 ||1 = r2 , 2 n 0 0 0 and hence ||T f ||∞ ≤ ||T f1 ||∞ + ||T f2 ||∞ ≤ ||T ||1→∞

n−2 2 1 2−n r + ||T ||∞→∞ r . 2n 0 σn−1 0 1/n

1 Let now a := n1 ||T ||1→∞ , b := σn−1 ||T ||∞→∞ and r0 := ( ab ) (without loss of generality T = 0, so a = 0). Then we obtain that

n−2 n−2 −n 2 2 a + br0 a+a ||T f ||∞ ≤ r0 = r0 2 2

2 n ||T ||∞→∞ n n 1 = ||T ||1→∞ , σn−1 ||T ||1→∞ 2n

and the assertion follows.

6. Proofs of the main results We are now in a position to prove our main theorems stated in Section 1. Proof of Theorem 1.1. Let R := d2 and assume without loss of generality that spt μ ⊆ B[0, R]. As in Section 3 we normalise ϕ0 , ϕ1 such that ||ϕ0 ||∞ = ||ϕ1 ||∞ = 1 and sup ϕ0 = sup ϕ1 = 1. Let ε := 14 inf B[0,R+1] ϕ0 . Then ε ∈ (0, 14 ]. Let δ˜ > 0 be ˜ and let δ := δ˜ ∧ 1. as in Theorem 4.3, so that |ϕ1 (x) − ϕ1 (y)| ≤ ε for |x − y| ≤ δ, According to (3.4) we can estimate λ1 − λ 0 ≥

2ωn−1 2 n−1 ε δ ||ϕ1 ||−2 2 . R

−1 By Proposition 2.5(a) we have ||ϕ1 ||−2 . Moreover, 2 ≥ |λ1 |||μ|| ||μ|| = μ B[0, R] ≤ cμ Rn−α

and hence α−n−1 2 n−1 ε δ ≥ γκ21 , κ20 − κ21 = λ1 − λ0 ≥ 2ωn−1 |λ1 |c−1 μ R

(6.1)

(6.2)

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411

where γ := 2ωn−1 (cμ + 1)−1 (d + 1)α−n−1 ε2 δ n−1 . This implies κ20 ≥ (1 + γ)κ21 , and γ 1 therefore κ20 − κ21 ≥ κ20 − 1+γ κ20 = 1+γ κ20 . Since γ ≤ 2ωn−1 ( 14 )2 ≤ 1, we conclude that γ γ κ2 ≥ κ20 = ωn−1 |λ0 |(cμ + 1)−1 (d + 1)α−n−1 ε2 δ n−1 . (6.3) λ1 − λ0 ≥ 1+γ 0 2 In the case n ≥ 5 we have ||ϕ1 ||22 ≤ 6ωn Rn by Proposition 2.5(c); then (6.1) yields 2ωn−1 λ1 − λ0 ≥ (d + 1)−n−1 ε2 δ n−1 . (6.4) 6ωn We are going to show that ε2 δ n−1 ≥ C0 (cμ + 1)−p0 (d + 1)−q0 e−βκ0 (d+1) ,

(6.5)

with β as in the assertion of the theorem and constants C0 , p0 , q0 > 0 depending only on n and α. Then (6.3) implies (1.2) with C = ωn−1 C0 , p = p0 + 1, q = q0 + n + 1 − α, and the additional assertion for n ≥ 5 follows from (6.4). Recall from Proposition 2.2 that (5.1) holds with θ = 1 − α2 and c = n 2 −α · π − 2 Γ( n−α 2−α 2 2 + 1)cμ . Hence, by Theorem 5.2 and the definition of ε we have 1 − 2−α 1 (2R + 1)−1 e−κ0 (2R+1) if n = 2 , 8e (4c + 1) ε ≥ cn n−2 − 2−α (2R + 1)2−n e−κ0 (2R+1) if n ≥ 3 . 4 (4c) We infer that there exists a constant cn,α > 0 (depending only on n and α) such that n−1 ε ≥ cn,α (cμ + 1)− 2−α (d + 1)−(n−1) e−κ0 (d+1) . (6.6) −(1−α) −1 cμ by (6.2). AcAssume now that α < 1. Then ||μ||− n−α c− μ n−α ≥ R −1 ˜ cording to Theorem 4.3 we thus have δ ≥ cñ,α (cμ + 1) (d + 1)−(1−α) ε =: δ1 , with cñ,α > 0 depending only on n and α. Without loss of generality cñ,α ≤ 4; then δ1 ≤ 1 since ε ≤ 14 . We obtain that 1−α

n−1

n−1 (cμ + 1)−(n−1) (d + 1)−(n−1)(1−α) εn+1 , ε2 δ n−1 ≥ cñ,α −1 so from (6.6) we deduce that (6.5) holds with β = n + 1, p0 = n2−α + n − 1, q0 = 2 n−1 n+1 n − 1 + (n − 1)(1 − α) and C0 = cñ,α cn,α . 1 In the case α > 1 we have δ˜ ≥ cñ,α cμε+1 2−α =: δ2 . Again, a suitable choice of cñ,α ensures that δ2 ≤ 1, and in the same way as above we deduce (6.5), now n−1 n−1 β with β = 2−α + 2, p0 = β 2 − β − 2, q0 = (n − 1)β and C0 = cñ,α cn,α . Finally assume that α = 1. Given β > n + 1, there exists α ˜ ∈ (1, 2) such that n−1 n−1 ˜ ≤ cμ Rα−1 rn−α˜ for all x ∈ Rn , r > 0, β = 2− α ˜ + 2. Then μ(B(x, r)) ≤ cμ (r ∧ R) and from the preceding paragraph we obtain that 2

n−1 β α−1 ˜ cμ + 1)−(β ε2 δ n−1 ≥ cñ, α ˜ cn,α ˜ (R

2

−β−2)

(d + 1)−(n−1)β e−βκ0 (d+1) .

˜ ˜ cμ +1 ≤ (R+1)α−1 (cμ +1), we deduce by straightforward Using the inequality Rα−1 computation that (6.5) holds with p0 = β 2 − β − 2, q0 = β 2 − β − n − 1 and n−1 β C0 = cñ, α ˜ cn,α ˜.

412

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Proof of Theorem 1.3. Normalising μ, ϕ0 and ϕ1 as in the proof of Theorem 1.1, we obtain from (3.5) and Theorem 5.2(a) that λ1 − λ0 ≥

1 1 −2κ0 d inf ϕ2 ||ϕ1 ||−2 ||ϕ1 ||−2 2 ≥ e 2 . d [−d/2,d/2] 0 d

The second assertions thus follows from Proposition 2.5(b). Proposition 2.5(a) |λ1 | −2κ0 d e , so the first assertion follows as in the argument implies that λ1 − λ0 ≥ d||μ|| leading to equation (6.3).

A. Appendix Here we prove the estimate of Gκ from below that is needed in the proof of The2π n/2 orem 5.2. We assume that n ≥ 2 and recall that σn−1 = Γ(n/2) is the (n−1)dimensional volume of the unit sphere in Rn . Proposition A.1. Let κ > 0, and let Gκ be the convolution kernel of (κ2 − Δ)−1 on L2 (Rn ). (a) If n ≥ 3 then Gκ (x) ≥ (b) If n = 2 then Gκ (x) ≥

1 2−n −κ|x| e for all x ∈ Rn (n−2)σn−1 |x| 1 1 −κ|x| for all x ∈ R2 \ 2π ln(1 + κ|x| )e

Proof. By [10; eq. (VII,10;15)] we have Gκ (x) = (2π)− 2 κ n

n−2 2

|x|−

n−2 2

K n−2 (κ|x|) 2

\ {0}. {0}.

x ∈ Rn \ {0} ,

(A.1)

where Kν is the modified Bessel function of the third kind of order ν ≥ 0. In particular we see that Gκ (x) = κn−2 G1 (κx), and that it therefore suffices to show the assertion for κ = 1. (a) By [1; 9.6.9] we obtain from (A.1) that G1 (x) =

1 |x|2−n + o(|x|2−n ) as (n − 2)σn−1

x → 0.

Let ε > 0, ϕε (x) :=

1−ε |x|2−n e−|x| − G1 (x) (n − 2)σn−1

x ∈ Rn \ {0} .

Then there exists δ > 0 such that ϕε < 0 on B(0, δ) \ {0}, and ϕε (x) → 0 as |x| → ∞. Moreover, a straightforward computation shows that (1 − Δ)ϕε (x) = 1−ε − (n−2)σ (n − 3)|x|1−n e−|x| ≤ 0 for x = 0. Thus ϕε has no positive maxima. We n−1 conclude that ϕε < 0 on Rn \ {0} for all ε > 0, and for ε → 0 we obtain (a). 1 K0 (|x|), so we must show that K0 (r) ≥ (b) By (A.1) we have G1 (x) = 2π 1 −r ln(1 + r )e for all r > 0. Let a := 2e−γ , where γ = 0.577 . . . is the Euler– Mascheroni constant. (Then 1 < a < 2.) From [1; 9.6.13] it follows that K0 (r) = ln ar + o(r) as r → 0. Let f (r) := ln(1 + ar )e−r for all r > 0. One easily sees that f (r) = (1 − a r) ln r + O(r) as r → 0, so g := f − K0 < 0 on (0, ε) for some ε > 0. We will

Vol. 10 (2009)


413

prove that g < 0 on (0, ∞); then the claim follows. A straightforward computation shows that r 1 1 a h e−r (A.2) g (r) + g (r) − g(r) = f (r) + f (r) − f (r) = r r r(a + r) a + 2 − (1 + s) ln(1 + 1s ), and that

2 1 2 1 − 1+ h (s) = (s > 0) . (1 + s)3 a s

for all r > 0, with h(s) :=

1 1 a 1+s

We obtain that h is concave on (0, s0 ) and convex on (s0 , ∞), where s0 = ( a2 − 1)−1 . Since h(s) → −∞ as s → 0 and h(s) → 1 as s → ∞, we infer that there exists s1 ∈ (0, s0 ) such that h < 0 on (0, s1 ) and h > 0 on (s1 , ∞). Now, if g has a positive maximum at r > 0 then the left hand side of (A.2) is negative, and hence ar ∈ (0, s1 ). Similarly, if g has a negative minimum at r > 0 then ar ∈ (s1 , ∞). Thus, to the left of a positive maximum there cannot be any negative minimum. Since g < 0 on (0, ε) and g(r) → 0 as r → 0, we conclude that g has no positive maximum. This implies that g < 0 on (0, ∞) since g(r) → 0 as r → ∞.

Acknowledgements The author thanks Ivan Veselić for introducing him to the subject and for some valuable discussions.

References [1] M. Abramovitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1972. [2] Ph. Blanchard and Z. M. Ma, Semigroup of Schr¨ odinger operators with potentials given by Radon measures, in Stochastic processes, physics and geometry (Ascona and Locarno, 1988), pp. 160–195, World Sci. Publ., Teaneck, NJ, 1990. ˇ [3] J. F. Brasche, P. Exner, Yu. A. Kuperin and P. Seba, Schr¨ odinger operators with singular interactions, J. Math. Anal. Appl. 184 (1994), no. 1, 112–139. [4] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schr¨ odinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), no. 2, 335–395. [5] E. B. Davies, Lp spectral independence and L1 analyticity, J. London Math. Soc. (2) 52 (1995), no. 1, 177–184. [6] E. M. Harrell, On the rate of asymptotic eigenvalue degeneracy, Comm. Math. Phys. 60 (1978), no. 1, 73–95. [7] E. M. Harrell, Double wells, Comm. Math. Phys. 75 (1980), no. 3, 239–261. [8] W. Kirsch and B. Simon, Comparison theorems for the gap of Schr¨ odinger operators, J. Funct. Anal. 75 (1987), no. 2, 396–410. [9] S. Kondej and I. Veselić, Lower bounds on the lowest spectral gap of singular potential Hamiltonians, Ann. Henri Poincaré 8 (2007), no. 1, 109–134.

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[10] L. Schwartz, Théorie des distributions, Nouvelle éd. Hermann, Paris, 1966. [11] B. Simon, Schr¨ odinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. [12] B. Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. (2) 120 (1984), no. 1, 89–118. [13] I. M. Singer, B. Wong, S.-T. Yau and S. S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schr¨ odinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 319–333. [14] R. G. Smits, Spectral gaps and rates to equilibrium for diffusions in convex domains, Michigan Math. J. 43 (1996), no. 1, 141–157. [15] P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109–138. [16] J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann. 229 (1977), 163–171. [17] J. Voigt, Absorption semigroups, their generators and Schr¨ odinger semigroups, J. Funct. Anal. 67 (1986), no. 2, 167–205. Hendrik Vogt Fachrichtung Mathematik Technische Universit¨ at Dresden D-01062 Dresden Germany e-mail: [email protected] Communicated by Claude Alain Pillet. Submitted: May 8, 2008. Accepted: February 20, 2009.



High Frequency Dispersive Estimates in Dimension Two Simon Moulin

Abstract. We prove dispersive estimates at high frequency in dimension two for both the wave and the Schr¨ odinger groups for a very large class of realvalued potentials.

1. Introduction and statement of results The purpose of√ this note is to prove dispersive estimates at high frequency for the wave group eit G and the Schr¨ odinger group eitG , where G denotes the self-adjoint realization of the operator −Δ + V on L2 (R2 ) and V is a real-valued potential which decays at infinity in a way that G has no real resonances nor eigenvalues in an interval [a0 , +∞), a0 > 0. In fact, we are looking for as large as possible class of potentials for which we have dispersive estimates similar to those we do for the free operator G0 . Hereafter G0 denotes the self-adjoint realization of the operator −Δ on L2 (R2 ). It turns out that in dimension two one can get such dispersive estimates at high frequency for potentials satisfying |V (x)|dx ≤ C < +∞ . (1.1) sup 1/2 y∈R2 R2 |x − y| Clearly, (1.1) is fulfilled for potentials V ∈ L∞ (R2 ) satisfying |V (x)| ≤ Cx−δ ,

∀x ∈ R2 ,

(1.2)

with constants C > 0, δ > 3/2. Given any a > 0, set χa (σ) = χ1 (σ/a), where χ1 ∈ C ∞ (R), χ1 (σ) = 0 for σ ≤ 1, χ1 (σ) = 1 for σ ≥ 2. Our first result is the following

416

S. Moulin


Theorem 1.1. Let V satisfy (1.1). Then, there exists a constant a0 > 0 so that for every a ≥ a0 , 0 < 1, 2 ≤ p < +∞, we have the estimates √

eit

G

G−3/4− χa (G) L1 →L∞ ≤ C |t|−1/2 ,

√ it G

e

−3α/4

G

−α/2

χa (G) Lp →Lp ≤ C|t|

,

t = 0 ,

(1.3)

t = 0 ,

(1.4)

where 1/p + 1/p = 1, α = 1 − 2/p. The estimate (1.3) is proved in [2] under the assumption (1.2). Moreover, if in addition one supposes that G has no strictly positive resonances, it is shown in [2] that (1.3) holds for any a > 0 still under (1.2). In dimension three an analogue of (1.3) is proved in [2, 4] for potentials satisfying (1.2) with δ > 2, and extended in [3] to a large subset of potentials satisfying |V (x)|dx ≤ C < +∞ . (1.5) sup 3 |x − y| 3 y∈R R In dimensions n ≥ 4 there are very few results. In [1], an analogue of (1.3) is proved for potentials belonging to the Schwartz class, while in [12] dispersive estimates with a loss of (n − 3)/2 derivatives are obtained for potentials satisfying (1.2) with δ > (n + 1)/2. Recently, in [8] dispersive estimates at low frequency have been proved in dimensions n ≥ 4 for a very large class of potentials, provided zero is neither an eigenvalue nor a resonance. Our second result is the following Theorem 1.2. Let V satisfy (1.1). Then, there exists a constant a0 > 0 so that for every a ≥ a0 , we have the estimate

eitG χa (G) L1 →L∞ ≤ C|t|−1 ,

t = 0 .

(1.6)

Note that the estimate (1.6) (for any a > 0) is proved in [10] for potentials satisfying (1.2) with δ > 2. An one dimensional analogue of (1.6) is proved in [6] for potentials V ∈ L1 . In dimension three an analogue of (1.6) (for any a > 0) is proved in [11] for potentials satisfying (1.5) with C > 0 small enough, and in [5] for potentials V ∈ L3/2− ∩ L3/2+ , 0 < 1, not necessarily small. In dimensions n ≥ 4, an analogue of (1.6) (for any a > 0) is proved in [7] for potentials satisfying (1.2) with δ > n as well as the condition V ∈ L1 . This result has been recently extended in [9] to potentials satisfying (1.2) with δ > n − 1 and V ∈ L1 . Note also the work [13], where an analogue of (1.6) (for any a > 0) with a loss of (n − 3)/2 derivatives is obtained for potentials satisfying (1.2) with δ > (n + 2)/2. In [9] dispersive estimates at low frequency have been also proved in dimensions n ≥ 4 for a very large class of potentials, provided zero is neither an eigenvalue nor a resonance. To prove Theorem 1.1 we use the same idea we have already used in [8] to prove low frequency dispersive estimates in dimensions n ≥ 4. The key point is

Vol. 10 (2009) High Frequency Dispersive Estimates in Dimension Two

the following estimate which holds in all dimensions n ≥ 2: ∞ √ h

V eit G0 ψ(h2 G0 )f L1 dt ≤ γn Cn (V )h−(n−3)/2 f L1 ,

h > 0,

417

(1.7)

−∞

where ψ ∈ C0∞ ((0, +∞)), γn > 0 is a constant independent of V , h and f , and |V (x)|dx Cn (V ) := sup < +∞ . (1.8) (n−1)/2 n y∈R Rn |x − y| Our approach is based on the observation that if Cn (V )h−(n−3)/2 1 ,

(1.9)

then (1.7) implies (under reasonable assumptions on the potential) a similar estimate for the perturbed wave group, namely ∞ √ n (V )h−(n−3)/2 f 1 .

V eit G ψ(h2 G)f L1 dt ≤ C (1.10) h L −∞

When n = 3, (1.9) is fulfilled for small potentials and all h, when n ≥ 4, (1.9) is fulfilled for large h (i.e. at low frequency) without extra restrictions on the potential, while for n = 2, (1.9) is fulfilled for small h (i.e. at high frequency) again without restrictions on the potential others than (1.1). Note that (1.10) may hold without (1.9). Indeed, when n = 3, (1.10) is proved in [5] for potentials V ∈ L3/2− ∩ L3/2+ and all h > 0, and then used to prove the three dimensional analogue of (1.6). In the present paper we adapt this approach to the case of dimension two, and show that (1.6) follows from (1.10) for potentials satisfying (1.1) only, provided the parameter a is taken large enough (see Section 3). Note finally that it would be quite irrealistic to expect that the dispersive estimates above could hold for potentials satisfying (1.1) at frequencies smaller than the critical level a0 . In fact, in this range of frequencies one could hardly do better than the already existing results. Recall that for frequencies belonging to an odinger interval [ε, a0 ], 0 < ε 1, dispersive estimates for the wave and the Schr¨ groups have been proved respectively in [2] for potentials satisfying (1.2) with δ > 3/2 and in [10] for potentials satisfying (1.2) with δ > 2. For small frequencies in [0, ε] dispersive estimates for the Schr¨ odinger group have been proved in [10] for potentials satisfying (1.2) with δ > 3 supposing additionally that zero is neither an eigenvalue nor a resonance.

2. Proof of Theorem 1.1 Let ψ ∈ C0∞ ((0, +∞)) and set √

Φ(t; h) = eit

G

√

ψ(h2 G) − eit

G0

ψ(h2 G0 ) .

We will first show that (1.3) and (1.4) follow from the following

418

S. Moulin


Proposition 2.1. Let V satisfy (1.1). Then, there exist positive constants C and h0 so that for 0 < h ≤ h0 we have

Φ(t; h) L1 →L∞ ≤ Ch−1 |t|−1/2 , Writing σ −3/4− χa (σ) =

a−1

t = 0 .

(2.1)

ψ(σθ)θ−1/4+ dθ ,

0

where ψ(σ) = σ 1/4− χ1 (σ) ∈ C0∞ ((0, +∞)), and using (2.1) we get √

eit

G

√

−3/4−

G−3/4− χa (G) − eit G0 G0 χa (G0 ) L1 →L∞ a−1 √ ≤

Φ(t; θ) L1 →L∞ θ−1/4+ dθ 0

≤ C|t|−1/2

a−1

θ−3/4+ dθ ≤ C|t|−1/2 ,

(2.2)

0

provided a is taken large enough. Clearly, (1.3) follows from (2.2) and the fact that it holds for G0 . To prove (1.4), observe that an interpolation between (2.1) and the trivial bound

Φ(t; h) L2 →L2 ≤ C yields

Φ(t; h) Lp →Lp ≤ Ch−α |t|−α/2 ,

t = 0 ,

(2.3)

for every 2 ≤ p ≤ +∞, p and α being as in Theorem 1.1. Now we write a−1 −3α/4 σ χa (σ) = ψ(σθ)θ−1+3α/4 dθ , 0

and use (2.3) to obtain (for 0 < α ≤ 1) √

eit

G

√

−3α/4

G−3α/4 χa (G) − eit G0 G0 χa (G0 ) Lp →Lp −1 a √ ≤

Φ(t; θ) Lp →Lp θ−1+3α/4 dθ 0

−α/2

≤ C|t|

a−1

θ−1+α/4 dθ ≤ C|t|−α/2 ,

(2.4)

0

provided a is taken large enough. Now, (1.4) follows from (2.4) and the fact that it holds for G0 . Proof of Proposition 2.1. We will first prove the following Lemma 2.2. Let V satisfy (1.1). Then, there exist positive constants C and h0 so that for 0 < h ≤ h0 we have

ψ(h2 G) − ψ(h2 G0 ) L1 →L1 ≤ Ch1/2 .

(2.5)


Proof. We will make use of the formula ∂ϕ 2 2 (z)(h2 G − z 2 )−1 zL(dz) , ψ(h G) = π C ∂ z¯

419

(2.6)

where L(dz) denotes the Lebesgue measure on C, ϕ ∈ C0∞ (C) is an almost analytic 2 continuation of ϕ(λ) = ψ(λ ) supported in a small complex neighbourhood of supp ϕ and satisfying ∂ϕ (z) ≤ CN |Im z|N , ∀N ≥ 1 . ∂ z¯ For ±Im λ ≥ 0, Re λ > 0, set R0± (λ) = (G0 − λ2 )−1 , We have the identity

R± (λ) = (G − λ2 )−1 .

R± (λ) 1 + V R0± (λ) = R0± (λ) .

(2.7)

R0± (λ)

are given in terms of the It is well known that the kernels of the operators zero order Hankel functions by the formula ± R0 (λ) (x, y) = ±i4−1 H0± (λ|x − y|) . Moreover, the functions H0± satisfy the bound |H0± (λ)| ≤ C|λ|−1/2 e−|Im λ| ,

|λ| ≥ 1 ,

±Im λ ≥ 0 ,

(2.8)

while near λ = 0 they are of the form ± H0± (λ) = a± 0,1 (λ) + a0,2 (λ) log λ ,

(2.9)

where a± 0,j are analytic functions. In particular, we have |H0± (λ)| ≤ C|λ|−1/2 ,

Re λ > 0 ,

±Im λ ≥ 0 .

(2.10)

Using these bounds we will prove the following Lemma 2.3. Let V satisfy (1.1). Then, there exist constants C > 0 and 0 < h0 ≤ 1 ±Im z ≥ 0}, we have the estimates so that for z ∈ C± ϕ := {z ∈ supp ϕ,

V R0± (z/h) L1 →L1 ≤ Ch1/2 , ±

V R (z/h) L1 →L1 ≤ Ch

1/2

0 < h ≤ 1 , (2.11) 0 < h ≤ h0 ,

,

(2.12)

R0± (z/h) L1 →L1

R± (z/h) L1 →L1

−2

≤ Ch |Im z| 2

,

≤ Ch2 |Im z|−2 ,

0 < h ≤ 1, Im z = 0 ,

(2.13)

0 < h ≤ h0 , Im z = 0 .

(2.14)

Proof. Applying Schur’s lemma and using (1.1) and (2.10), we get that the norm in the left-hand side of (2.11) is upper bounded by |V (x)dx |V (x)||H0± (z|x − y|/h)|dx ≤ Ch1/2 sup ≤ C h1/2 . sup 1/2 y∈R2 R2 y∈R2 R2 |x − y|

420

S. Moulin


Similarly, the norm in the left-hand side of (2.13) is upper bounded by ∞ sup |H0± (z|x − y|/h)|dx = c2 h2 σ|H0± (zσ)|dσ . y∈R2

R2

0

−1 ≤ |z| ≤ C, ±Im z > 0, we have |H ± (zσ)| ≤ C|σ|−1/2 e−σ|Im z| for σ ≥ 1. Since C 0 We obtain ∞ ∞ σ|H0± (zσ)|dσ ≤ C σ 1/2 e−σ|Im z| dσ ≤ C|Im z|−2 . 1

1

For 0 < σ ≤ 1, we use

|H0± (zσ)|

≤ C| log |zσ|| ≤ C|σ|−1 to obtain

1

σ|H0± (zσ)|dσ ≤ C ,

0

which clearly implies (2.13). To prove (2.12) and (2.14) we will make use of the identity (2.7). It follows from (2.11) that there exists a constant 0 < h0 ≤ 1 so that for 0 < h ≤ h0 the operator 1 + V R0± (z/h) is invertible on L1 with an inverse satisfying

1 + V R± (z/h) −1 1 1 ≤ C , z ∈ C± (2.15) ϕ , 0 L →L with a constant C > 0 independent of z and h. Clearly, (2.12) follows from (2.11) and (2.15), while (2.14) follows from (2.13) and (2.15). To prove (2.5) we rewrite the identity (2.7) in the form −1 , R± (z/h) − R0± (z/h) = −R0± (z/h)V R0± (z/h) 1 + V R0± (z/h) and hence, using Lemma 2.3 and (2.15), we get

h−2 R± (z/h) − h−2 R0± (z/h) L1 →L1 ≤ Ch1/2 |Im z|−2 , 0 < h ≤ h0 ,

z ∈ C± ϕ ,

Im z = 0 .

It is easy now to see that (2.5) follows from (2.6) and (2.16).

(2.16)

We will now derive (2.1) from the following Proposition 2.4. Let V satisfy (1.1). Then, there exist positive constants C and h0 so that we have, for 0 ≤ s ≤ 1/2, f, g ∈ L1 , √

eit G0 ψ(h2 G0 )f L∞ ≤ Ch−3/2 |t|−1/2 f L1 , h > 0 , t = 0 , ∞ √ |t|s |x − y|−s V eit G0 ψ(h2 G0 )f (x) |g(y)| dtdxdy

(2.17)

≤ Ch

(2.18)

R2

R2

R2 −∞ −1/2

f L1 g L1 , ∞

R2 −∞ s−1/2

≤ Ch

h > 0, √ it G

|t|s |x − y|/h−s |V e

f L1 g L1 ,

ψ(h2 G)f (x)||g(y)|dtdxdy

0 < h ≤ h0 .

(2.19)


421

As in [13], we introduce the functions ψ1 ∈ C0∞ ((0, +∞)), ψ1 = 1 on supp ψ, ψ(σ) = σ 1/2 ψ(σ), ψ1 (σ) = σ −1/2 ψ1 (σ). Set t √ ψ1 (h2 G0 ) sin (t − τ ) G0 V eiτ G ψ(h2 G)dτ . Φ2 (t; h) := −h 0

Then using Duhamel’s formula for the wave equation √ √ √ √ √ t sin (t − τ ) G0 sin(t G0 ) √ it G it G0 √ e =e +i √ G − G0 − V eiτ G dτ , G0 G0 0 multiplying by ψ1 (h2 G0 ) on the left and by ψ(h2 G) on the right, we get the identity Φ(t; h) − Φ2 (t; h) = Φ(t; h) + ψ1 (h2 G0 ) √ √ √ sin(t G0 ) √ × eit G0 + i √ ( G − G0 ) − eit G ψ(h2 G) G0 √

= eit

G

√

ψ(h2 G) − eit √

− ψ1 (h2 G0 )eit

G0

√

ψ(h2 G0 ) + ψ(h2 G0 )eit

G0

ψ(h2 G)

G

ψ(h2 G) ˜ 2 G) + iψ˜1 (h2 G0 ) sin(t G0 )ψ(h − iψ1 (h2 G0 ) sin(t G0 )ψ(h2 G) . √ √ 2 G0 ) = 0 and By adding iψ1 (h2 G0 ) sin(t G0 )ψ(h2 G0 ) − iψ1 (h2 G0 ) sin(t G0 )ψ(h using the commuting properties and ψ = ψ1 ψ, we obtain √ Φ1 (t; h) := Φ(t; h) − Φ2 (t; h) = ψ1 (h2 G) − ψ1 (h2 G0 ) eit G ψ(h2 G) + ψ1 (h2 G0 ) cos t G0 ψ(h2 G) − ψ(h2 G0 ) 2 G) − ψ(h 2 G0 ) . (2.20) + iψ1 (h2 G0 ) sin t G0 ψ(h By Proposition 2.4 and (2.5), we have

Φ1 (t; h)f L∞ −1

≤ Ch ≤h

−1/2

|t|

t/2

0

t

t/2 −1/2

+

Φ(t; h)f L∞ ,

t

1/2

Φ2 (t; h)f, g

(2.21)

√

(t − τ )1/2 sin (t − τ ) G0 ψ1 (h2 G0 )g L∞ V eiτ G ψ(h2 G)f L1 dτ

+h ≤ Ch

f L1 + Ch

1/2

√

V sin (t − τ ) G0 ψ1 (h2 G0 )g 1 τ 1/2 eiτ G ψ(h2 G)f L∞ dτ L

g L1

∞

V eiτ

√

ψ(h2 G)f L1 dτ −∞ ∞ √

1/2 iτ G

V sin (t − τ ) G0 ψ1 (h2 G0 )g 1 dτ ψ(h2 G)f L∞ h sup τ e L G

t/2≤τ ≤t

−1

≤ Ch

g L1 f L1 + Ch

−∞

1/2

g L1

sup τ 1/2 eiτ

t/2≤τ ≤t

√

G

ψ(h2 G)f L∞ ,

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S. Moulin


for t > 0, which clearly implies t1/2 Φ2 (t; h)f L∞ ≤ Ch−1 f L1 + Ch1/2

sup τ 1/2 eiτ

√

t/2≤τ ≤t

G

ψ(h2 G)f L∞ .

(2.22)

By (2.20)–(2.22), we conclude t1/2 Φ(t; h)f L∞ ≤ Ch−1 f L1 + Ch1/2 t1/2 Φ(t; h)f L∞ + Ch1/2

sup τ 1/2 Φ(τ ; h)f L∞ .

(2.23)

t/2≤τ ≤t

Taking h small enough we can absorb the second and the third terms in the righthand side of (2.23), thus obtaining (2.1). Clearly, the case of t < 0 can be treated in the same way. √

Proof of Proposition 2.4. The kernel of the operator eit G0 ψ(h2 G0 ) is of the form Kh (|x − y|, t), where ∞ eitλ J0 (σλ)ψ(h2 λ2 )λdλ = h−2 K1 (σh−1 , th−1 ) , (2.24) Kh (σ, t) = (2π)−1 0

where J0 (z) = (H0+ (z) + H0− (z))/2 is the Bessel function of order zero. It is shown in [13] (Section 2) that Kh satisfies the estimates (for all σ, h > 0, t = 0) |K1 (σ, t)| ≤ C|t|−s σs−1/2 , |Kh (σ, t)| ≤ Ch

−3/2

−s s−1/2

|t|

σ

,

∀s ≥ 0 ,

(2.25)

0 ≤ s ≤ 1/2 .

(2.26)

Clearly, (2.17) follows from (2.26) with s = 1/2. It is easy also to see that (2.18) follows from (1.1) and the following Lemma 2.5. For all 0 ≤ s ≤ 1/2, σ, h > 0, we have ∞ t/hs |Kh (σ, t)| dt ≤ Ch−1 σ/hs−1/2 , −∞ ∞ |t|s |Kh (σ, t)| dt ≤ Ch−1/2 σ s−1/2 .

(2.27) (2.28)

−∞

Proof. Clearly, (2.28) follows from (2.27). It is also clear from (2.24) that it suffices to prove (2.27) with h = 1. When 0 < σ ≤ 1, this follows from (2.25). To see that, we can split the integral into two parts: |t| ≤ 1 and |t| ≥ 1 and then use (2.25) with s for the first part and s + 1 + for the second part. We conclude by using that σ1+ ≤ 1 for the bound of the second part. Let now σ ≥ 1. We decompose K1 as K1+ + K1− , where K1± is defined by replacing in (2.24) the function J0 by H0± /2. Recall that ± H0± (z) = e±iz b± 0 (z), where b0 (z) is a symbol of order −1/2 for z ≥ 1. Using this fact and integrating by parts m times, we get |K1± (σ, t)| ≤ Cm σ −1/2 t ± σ−m .

(2.29)


By (2.29), we obtain ∞ ts |K1± (σ, t)|dt ≤ 2σ s

423

∞ |K1± (σ, t)|dt + |t ± σ|s |K1± (σ, t)|dt −∞ −∞ ∞ ∞ ≤ Cm σ s−1/2 t ± σ−m dt + Cm σ −1/2 t ± σs−m dt

−∞

∞

−∞

≤ Cσ

s−1/2

−∞

,

which clearly implies (2.27) in this case. To prove (2.19) we will use the formula ∞ √ eitλ ϕh (λ) R+ (λ) − R− (λ) dλ , eit G ψ(h2 G) = (iπh)−1

(2.30)

0

where ϕh (λ) = ϕ1 (hλ), ϕ1 (λ) = λψ(λ2 ). Combining (2.30) together with (2.7), we get √ ∞ ± V Ph± (t − τ )Uh± (τ )dτ , (2.31) V eit G ψ(h2 G) = (iπh)−1 ±

where Ph± (t) = Uh± (t)

∞

0 ∞

=

−∞

eitλ ϕ h (λ)R0± (λ)dλ , −1 eitλ ϕh (λ) 1 + V R0± (λ) dλ ,

0

1 (hλ), ϕ 1 ∈ C0∞ ((0, +∞)) is such that ϕ 1 = 1 on supp ϕ1 . The where ϕ h (λ) = ϕ ± kernel of the operator Ph (t) is of the form A± (|x − y|, t), where h ∞ −1 eitλ ϕ h (λ)H0± (σλ)dλ = h−1 A± (2.32) A± 1 (σ/h, t/h) . h (σ, t) = ±i4 0

In the same way as in the proof of Lemma 2.5 one can see that the function A± h satisfies the estimate ∞ 1/2 s−1/2 |t|s A± σ (1 + hs σ −s ) , h (σ, t) dt ≤ Ch −∞

0 ≤ s ≤ 1/2 ,

0 < h ≤ 1,

(2.33)

where s = 0 if 0 ≤ s < 1/2, s = if s = 1/2. Clearly, it suffices to prove (2.19) with s = 0 and s = 1/2. For these values of s, using (1.1), (2.31) and (2.33), we obtain ∞ √ −s |t|s |x − y|/h V eit G ψ(h2 G)f (x) |g(y)| dtdxdy 2 2 −∞ R R ∞ ∞ −s ≤ Ch−1 |x − y|/h (|t − τ |s + |τ |s ) ±

R2

R2

−∞

−∞

× V P ± (t − τ )U ± (τ )f (x) |g(y)| dτ dtdxdy h

h

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S. Moulin

≤ Ch−1

± ± |Ah (|x

R2

± ∞

R2

∞

R2

R2

R2

R2

∞

−∞

∞


|V (x)| |x − y|/h

−∞

−s

(|t − τ |s + |τ |s )

× − x |, t − τ )| Uh± (τ )f (x ) |g(y)| dτ dtdx dxdy −s ≤ Ch−1 |V (x)| |x − y|/h |g(y)|

∞ ± U (τ )f (x ) dτ dx dxdy |τ |s |A± (|x − x |, τ )|dτ h h −∞ −∞ −s + Ch−1 |V (x)| |x − y|/h |g(y)| ×

×

R2

±

−∞ −1/2

≤ Ch

|A± h (|x

R2

∞

|τ | Uh± (τ )f (x ) dτ

dx dxdy − x |, τ )|dτ −∞ −s |V (x)| |x − y|/h |x − x |s−1/2

R2

±

R2

R2

s

R2

∞ ± s −s Uh (τ )f (x ) dτ dx dxdy |g(y)| × 1 + h |x − x | −∞ −s −1/2 + Ch |V (x)| |x − y|/h |x − x |−1/2 |g(y)| ×

R2

±

∞

−∞

R2

R2

|τ | Uh± (τ )f (x ) dτ

s

dx dxdy := I1 + I2 .

(2.34)

To estimate I1 when s = 1/2, set q = (2)−1 , 1/p + 1/q = 1, and observe that in view of (1.1) we have the bound −1/2 |V (x)| |x − y|/h |x − x |− dx R2

|V (x)| |x − y|/h

≤

−p/2

1/p

|V (x)||x − x |

dx

R2

≤ C1

|V (x)| |x − y|/h R2

−1/2

1/p

1/q dx

R2

dx

|V (x)||x − y|−1/2 dx

≤ C1 h1/(2p)

−1/2

1/p ≤ C2 h1/2− .

R2

Thus, we obtain I1 ≤ C hs−1/2

±

R2

R2

∞

−∞

|Uh± (τ )f (x )||g(y)|dτ dx dy .

(2.35)

To estimate I2 when s = 1/2, we use the inequality −1/2

|x − y|/h

|x − x |−1/2 ≤ |x − y|/h

−1/2

(|x − y|−1/2 + |x − x |−1/2 ) .


We get −1/2

I2 ≤ C h

±

R2

∞

−∞

R2

425

|τ |s |x − y|/h−s |Uh± (τ )f (x )||g(y)|dτ dx dy . (2.36)

On the other hand, by the identity −1 −1 = 1 − V R0± (λ) 1 + V R0± (λ) , 1 + V R0± (λ) we obtain h (t) − Uh± (t) = ϕ Since

∞

−∞

V Ph± (t − τ )Uh± (τ )dτ .

(2.37)

ϕ h (t) = h−1 ϕ 1 (t/h) ,

we have

∞

−∞

|t|s |ϕ h (t)|dt ≤ Chs .

(2.38)

Using (2.37) and (2.38), in the same way as in the proof of (2.34)–(2.36), we obtain with s = 0 or s = 1/2, ∞ |t|s |x − y|/h−s |Uh± (t)f (x)||g(y)|dtdxdy ≤ Chs f L1 g L1 R2 R2 −∞ ∞ + Chs+1/2 |Uh± (τ )f (x )||g(y)|dτ dx dy R2 R2 −∞ ∞ + Ch1/2 |τ |s |x − y|/h−s |Uh± (τ )f (x )||g(y)|dτ dx dy . (2.39) R2

R2

−∞

Taking h small enough we can absorb the second and the third terms in the righthand side of (2.39) and get the estimate ∞ |t|s |x − y|/h−s |Uh± (t)f (x)||g(y)|dtdxdy R2

R2

−∞

≤ C hs f L1 g L1 .

(2.40)

Now (2.19) follows from (2.34)–(2.36) and (2.40).

3. Proof of Theorem 1.2 Set Ψ(t; h) = eitG ψ(h2 G) − eitG0 ψ(h2 G0 ) . As in the previous section, one can derive (1.6) from the following Proposition 3.1. Let V satisfy (1.1). Then, there exist positive constants C and h0 so that for 0 < h ≤ h0 , we have

Ψ(t; h) L1 →L∞ ≤ Ch1/2 |t|−1 ,

t = 0 .

(3.1)

426

S. Moulin


Proof. We will derive (3.1) from (2.19). To this end, we will use the identity itλ2

e

2 2

∞

ϕ(h λ ) = −∞

eiτ λ ζh (t, τ )dτ ,

(3.2)

where ϕ ∈ C0∞ ((0, +∞)), ϕ = 1 on supp ψ1 , the functions ψ and ψ1 being as in the previous section, and ∞ 2 eitλ −iτ λ ϕ(h2 λ2 )dλ = h−1 ζ1 (th−2 , τ h−1 ) . (3.3) ζh (t, τ ) = (2π)−1 0

We deduce from (3.2) the formula itG

e

2

∞

ψ(h G) = −∞

ζh (t, τ )eiτ

√

G

ψ(h2 G)dτ .

(3.4)

Given any integer m ≥ 0, integrating by parts m times and using the well known bound ∞ itλ2 −iτ λ ≤ C|t|−1/2 , ∀t = 0 , τ ∈ R , e φ(λ)dλ −∞

where φ ∈

C0∞ (R),

one easily obtains the bound

|ζ1 (t, τ )| ≤ Cm |t|−m−1/2 τ m ,

∀t = 0 ,

τ ∈ R.

(3.5)

By (3.3) and (3.5), |ζh (t, τ )| ≤ Cm h2m |t|−m−1/2 τ /hm ,

∀t = 0 ,

τ ∈ R,

h > 0,

(3.6)

for every integer m ≥ 0, and hence for all real m ≥ 0. By (2.5), (2.20) and (3.4), we get Ψ(t; h)f, g ≤ Ch1/2 Ψ(t; h)f L∞ g L1 ∞ + |ζh (t, τ )| cos(τ G0 )ψ1 (h2 G0 ) ψ(h2 G) − ψ(h2 G0 ) f, g dτ −∞ ∞ 2 G) − ψ(h 2 G0 ) f, g dτ + |ζh (t, τ )| sin(τ G0 )ψ1 (h2 G0 ) ψ(h −∞ ∞ τ √ +h |ζh (t, τ )| V eiτ G ψ(h2 G)f, sin (τ −τ ) G0 ψ1 (h2 G0 )g dτ dτ . −∞

0

(3.7) Using (3.6) with m = 1/2 and (2.27) with s = 1/2 together with (2.5), we obtain that the first integral (and similarly for the second one) in the right-hand side of


427

(3.7) is upper bounded by ∞ Ch|t|−1 τ /h1/2 |Khc (|x − y|, τ )| |g(x)| R2 R2 −∞ × ψ(h2 G) − ψ(h2 G0 ) f (y) dτ dxdy ≤ C|t|−1 |g(x)| ψ(h2 G) − ψ(h2 G0 ) f (y) dxdy ≤ Ch

1/2

R2 −1

|t|

R2

f L1 g L1 ,

√ where Khc (|x − y|, t) denotes the kernel of the operator cos t G0 ψ1 (h2 G0 ). The last term is upper bounded by ∞ τ 1/2 s Kh |x − y|, (τ − τ ) |τ /h|1/2 + (τ − τ )/h Ch2 |t|−1 R2

R2

−∞

0

√

ψ(h2 G)f (x)||g(y)|dτ dτ dxdy ∞ √ 3/2 −1 ≤ Ch |t| |τ |1/2 |V eiτ G ψ(h2 G)f (x)||g(y)|dτ R2 R2 −∞ ∞ × |Khs (|x − y|, τ )|dτ dxdy −∞ ∞ √ 2 −1 + Ch |t| |V eiτ G ψ(h2 G)f (x)||g(y)|dτ R2 R2 −∞ ∞ × τ /h1/2 |Khs (|x − y|, τ )|dτ dxdy −∞ ∞ 1/2 −1 ≤ Ch |t| |τ |1/2 |x − y|/h−1/2 × |V eiτ

× |V e

iτ

√

G

R2 G

R2

−∞

2

ψ(h G)f (x)||g(y)|dτ dxdy ∞ √ + Ch|t|−1 |V eiτ G ψ(h2 G)f (x)||g(y)|dτ dxdy R2

≤ Ch

1/2

−1

|t|

R2

−∞

f L1 g L1 ,

√ where Khs (|x−y|, t) denotes the kernel of the operator sin(t G0 )ψ1 (h2 G0 ), and we have used (2.19) together with the fact that the function Khs (σ, t) satisfies (2.27). Thus, we obtain Ψ(t; h)f, g ≤ Ch1/2 Ψ(t; h)f L∞ g L1 + Ch1/2 |t|−1 f L1 g L1 , which clearly implies (3.1), provided h is taken small enough.

References [1] M. Beals, Optimal L∞ decay estimates for solutions to the wave equation with a potential, Commun. Partial Diff. Equations 19 (1994), 1319–1369.

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[2] F. Cardoso, C. Cuevas and G. Vodev, Dispersive estimates of solutions to the wave equation with a potential in dimensions two and three, Serdica Math. J. 31 (2005), 263–278. [3] P. D’ancona and V. Pierfelice, On the wave equation with a large rough potential, J. Funct. Analysis 227 (2005), 30–77. [4] V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential, Commun. Partial Diff. Equations 28 (2003), 1325–1369. [5] M. Goldberg, Dispersive bounds for the three dimensional Schr¨ odinger equation with almost critical potentials, GAFA 16 (2006), 517–536. [6] M. Goldberg and W. Schlag, Dispersive estimates for Schr¨ odinger operators in dimensions one and three, Commun. Math. Phys. 251 (2004), 157–178. [7] J.-L. Journé, A. Soffer and C. Sogge, Decay estimates for Schr¨ odinger operators, Commun. Pure Appl. Math. 44 (1991), 573–604. [8] S. Moulin, Low-frequency dispersive estimates for the wave equation in higher dimensions, Asymptot. Anal. 60 (2008), 15–27. [9] S. Moulin and G. Vodev, Low-frequency dispersive estimates for the Schr¨ odinger group in higher dimensions, Asymptot. Anal. 55 (2007), 49–71. [10] W. Schlag, Dispersive estimates for Schr¨ odinger operators in two dimensions, Commun. Math. Phys. 257 (2005), 87–117. [11] I. Rodnianski and W. Schlag, Time decay for solutions of Schr¨ odinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), 451–513. [12] G. Vodev, Dispersive estimates of solutions to the Schr¨ odinger equation in dimensions n ≥ 4, Asymptot. Anal. 49 (2006), 61–86. [13] G. Vodev, Dispersive estimates of solutions to the wave equation with a potential in dimensions n ≥ 4, Commun. Partial Diff. Equations 31 (2006), 1709–1733. Simon Moulin Université de Nantes Département de Mathématiques UMR 6629 du CNRS 2, rue de la Houssinière BP 92208 44332 Nantes Cedex 03 France e-mail: [email protected] Communicated by Christian Gérard. Submitted: April 2, 2008. Accepted: December 19, 2008.

Ann. Henri Poincaré 10 (2009), 429–451 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/030429-23, published online May 27, 2009 DOI 10.1007/s00023-009-0415-y


Entropy and Decay of Correlations for Real Analytic Semi-Flows Frédéric Naud Abstract. We consider real analytic suspension semi-flows Φtτ : Xτ → Xτ over uniformly expanding real-analytic map of the interval. We show that for any Φtτ -invariant equilibrium measure μτg related to an analytic potential g, there exists a Banach space B 0 (g) of test functions such that for generic observables in B 0 (g), the corresponding correlation functions Corg (F, G)(t) cannot decay faster than O(e−2hg t ), where hg is the measure theoretic entropy of μτg . This statement implies the existence of essential spectrum for the Perron–Frobenius operator associated to the semi-flow, when acting on any reasonable Banach space.

1. Introduction Since the early work of Ruelle and Pollicott [18,21], on the statistical properties of hyperbolic flows, we have learned that, unlike uniformly hyperbolic maps, flows do not necessarily mix at an exponential rate i.e. correlation functions do not necessarily decay exponentially. Actually, as demonstrated by Ruelle and Pollicott, one can construct suspension semi-flows with arbitrarily slow decay rate. However, the conjecture stating that correlations should decay exponentially for generic hyperbolic flows still remains unsolved. The work of Dolgopyat [8, 9] showed that this is the case in a symbolic setting and for low dimensional Anosov flows, the prototype being geodesic flows on negatively curved surfaces. Liverani [13] later extended this result to higher dimensional contact Anosov flows, overcoming the difficult issue of the low regularity of hyperbolic foliations. Using a pseudo-differential approach, Tsujii [24] showed that generic suspensions over uniformly expanding maps of the circle have exponential decay of correlations and went a step further by giving an asymptotic expansion with an explicit upper bound on the error term, involving the expansion rate of the flow. He recently obtained the same result [25], unconditionally, for contact Anosov flows. The key argument of Tsujii’s work is the

430

F. Naud


obtention of quasi-compactness for the Perron–Frobenius operator (acting on a suitable Hilbert space) i.e. its essential spectral radius is strictly smaller than 1. In this paper we detail a large class of suspension semi-flows for which we obtain an explicit exponential lower bound for generic correlation functions. This lower bound is related to measure theoretic entropy and shows that a small entropy of semi-flows implies a slow decay of correlations. Another related consequence is that the spectrum of the Perron–Frobenius operator of the flow, when acting on any Banach space that contains C k -compactly supported observables (k can be taken arbitrarily large), always has an essential part. More precisely, the essential spectral radius of the Perron–Frobenius operator is bounded from below by a quantity which do not depends on the regularity of observables, in sharp contrast with the case of uniformly hyperbolic maps. In particular, real analytic semi-flows cannot exhibit super-exponential decay of correlations, even for analytic observables. In the following we assume that the reader has some familiarity with basic ergodic theory. We refer the reader for example to [15,26] for more information on equilibrium measures and various definitions of topological and measure theoretic entropy of maps and flows. Let us be more precise. Let T : [0, 1] → [0, 1] be a topologically mixing, uniformly expanding1 Markov map of the interval. Let τ be a positive continuous function on [0, 1], this will be our return time or roof function. The suspension (branched) manifold Xτ is the set Xτ = (x, u) ∈ [0, 1] × R+ : 0 ≤ u ≤ τ (x) / x, τ (x) ∼ (T x, 0) , that is the set of points (x, u) under the graph of τ where we identify (x, τ (x)) with (T x, 0) for all x. The suspension flow Φtτ : Xτ → Xτ is then defined for all points (x, u) ∈ Xτ by Φtτ (x, u) = (x, u + t) , with the above identifications in mind. Let g be a Hölder continuous function on [0, 1], then there exists a unique equilibrium measure μg on [0, 1], i.e. a T -invariant probability measure maximizing the topological pressure 1 P (g) = sup gdμ , hμ (T ) + μ∈Minv

0

where Minv is the set of T -invariant probability measures and hμ (T ) is the measure theoretic entropy. From μg we can construct a natural Φtτ -invariant probability measure μτg on Xτ by setting for all F continuous on Xτ , 1 τ (x) 1 τ F dμg = 1 F (x, u)dudμg (x) . τ dμg 0 0 Xτ 0 1 Say

at least of class C 2 for the moment, we will provide some specific conditions in Section 2.

Vol. 10 (2009)

Entropy and Semi-Flows

431

This invariant measure for the flow is in turn an equilibrium measure and conversely, all equilibrium measure (with respect to a H¨ older potential2 on Xτ ) for t older). Of Φτ can be obtained this way, provided that τ has enough regularity (H¨ special interest are the examples of the potential g = − log |T | which gives the Sinai–Ruelle–Bowen measure on Xτ , and the potential g = −htop τ , where htop is the topological entropy of the suspension flow, which gives the measure of maximal entropy. Given such a measure μτg and smooth enough observables F, G on Xτ , we want to study the precise asymptotic behaviour of the correlation function (F ◦ Φtτ )Gdμτg − F dμτg Gdμτg , Corg (F, G)(t) = Xτ

Xτ

Xτ

as t → +∞. One way to approach this problem is through the Laplace transform +∞

e−st Cor(F, G)(t)dt , L Corg (F, G) (s) = 0

which is a holomorphic function in the half plane {Re(s) > 0}. In our setting, with suitable regularity conditions (see next section for details) this Laplace transform enjoys a meromorphic continuation to a wider halfplane or even to C. Poles are often called Ruelle–Pollicott resonances, and they indicate the asymptotic behaviour of the correlation function. For example, exponential decay of correlations implies analytic continuation of L[Corg (F, G)](s) in a half-plane {Re(s) > −} for some > 0. We refer the reader to [3, 19] for exponential decay of correlations results for semi-flows. It has been shown in various settings [16, 17] that these poles do not depend on observables (only the residues do) and can be viewed as poles of the meromorphic continuation of a generalized Fredholm determinant, intrinsically defined by an infinite Euler product on periodic orbits of the flow. To avoid “boundary problems” we will focus on spaces of C k observables which are supported “below” the ceiling function τ . More precisely, fix 0 > 0 and 0 < τ0 < min τ (x) . x∈[0,1]

Set

X0 ,τ0 = (x, u) ∈ [0, 1] × R+ : 0 ≤ u ≤ τ0 .

Denote by C0k (X0 ,τ0 ) the space of continuous functions f on X0 ,τ0 , of class C k on the interior, whose partial derivatives up to the order k have a continuous extension to X0 ,τ0 . Partial derivatives being denoted by Dα f , |α| ≤ k we also require that for all x ∈ [0, 1] we have Dα f (x, 0 ) = Dα f (x, τ0 ) = 0 . Clearly any function in C0k (X0 ,τ0 ) has a well defined extension to Xτ . In the next section, we will define the relevant class of real-analytic suspension flows and we assume from now on that we work in this class. Fix an arbitrarily large k. 2 On

the branched manifold Xτ , we say that a function is H¨ older on Xτ if its lifting to the covering V = {(x, u) ∈ [0, 1] × R+ : 0 ≤ u ≤ τ (x)} is H¨ older.

432

F. Naud


Given a real analytic potential g on [0, 1] , we will construct a Banach space Bk (g) ⊂ C0k (X0 ,τ0 ) of observables on Xτ for which the following theorem holds. Theorem 1.1. Fix μτg an equilibrium measure for a given analytic potential g on [0, 1]. There exists a generic Gδ -subset G0 of Bk (g) × Bk (g) such that for all (F, G) ∈ G0 , we have: 1. The Laplace transform L[Corg (F, G)](s) has a meromorphic extension to C. 2. For all ε > 0, L[Corg (F, G)](s) has infinitely many Ruelle–Pollicott resonances in the strip − 2hg − ε ≤ Re(s) ≤ 0 , where hg = hμτg (φ1τ ) is the measure theoretic entropy of the flow Φtτ with respect to μτg . The first statement is of course nothing new. The second result shows the existence of a natural “barrier” due to a critical strip with infinitely many resonances. The width of this critical strip is controlled by measure theoretic entropy. To our knowledge, this is the first result of this kind in the literature. It has some immediate consequences on correlations as the next corollary shows. We recall the usual Ω notation of number theory: a real function f (t) is said to be Ω(g(t)) as t → +∞ if there do not exist a constant C ≥ 0 and a real number T ≥ 0 such that for all t ≥ T , |f (t)| ≤ Cg(t). We denote this property by the formula f (t) = Ω g(t) . Corollary 1.2. Fix μτg an equilibrium measure for a given analytic potential g on [0, 1]. There exists a generic Gδ -subset G0 of Bk (g) × Bk (g) such that for all (F, G) ∈ G0 , we have the following. For all integer q ≥ 0, and complex numbers ωi , integers mi ≥ 0 and complex exponents3 λi with i = 1, . . . , q, we have for all ε > 0, q

mi λi t ωi t e = Ω e(−2hg −ε)t . Corg (F, G)(t) − i=1

Let us derive this corollary from Theorem 1.1. Fix a small ε = ε0 > 0 and assume that this Ω-lower bound does not hold for a given (F, G) ∈ G0 as above. That means that the Laplace transform of the correlation function is simply for Re(s) > 0, q

(−1)mi +1 mi ! ωi + L[R](s) , L Corg (F, G) (s) = (λi − s)mi +1 i=1 q where R(t) = Corg (F, G)(t) − i=1 ωi tmi eλi t is bounded for all t ≥ T by |R(t)| ≤ Ce(−2hg −ε0 )t . 3 Here

the exponents λi can be repeated according to multiplicity.

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Clearly L[R](s) has an analytic extension to the half-plane {Re(s) > −2hg − ε0 }, and therefore L[Corg (F, G)](s) has a meromorphic extension with finitely many poles to {Re(s) > −2hg − ε0 }, which contradicts Theorem 1.1. The above statement essentially shows that generic correlation functions with observables in Bk (g) cannot enjoy finite asymptotic expansions (in the polynomials × exponentials scale) with an arbitrarily exponentially small error term: one cannot be more precise than O(e−2hg t ). In particular, for most observables, the rate of mixing is never super-exponential. Without major changes to the proof, a similar statement can be obtained for Frechet spaces of C ∞ smooth observables, as well as real analytic (see Section 4). At this point, we would like to put some emphasis on the difference with uniformly hyperbolic maps: there exist Anosov diffeomorphisms which mix at a super-exponential rate for analytic observables. Indeed, on the two-torus T2 = R2 /Z2 , hyperbolic matrices in SL2 (Z) act as Anosov diffeomorphisms. Using Fourier series and the algebraic action of SL2 (Z) on Fourier components, one can show that real analytic4 observables mix at a super-exponential speed. A similar behaviour can be observed for expanding maps z → z p on the unit circle. One can also derive from Theorem 1.1 an interesting spectral consequence. Under the above assumptions, define the Perron–Frobenius operator Ptg for t ≥ 0 as the unique bounded linear map on L2 (Xτ , dμτg ) such that for all F ∈ L2 (Xτ , dμτg ), G ∈ L2 (Xτ , dμτg ), we have F ◦ Φtτ Gdμτg = F Ptg G dμτg . Xτ

Xτ

Then we have the following. Corollary 1.3. Let H(g) be a Ptg -invariant Banach space with5 C0k (X0 ,τ0 ) ⊂ H(g) ⊂ L2 (Xτ , dμτg ), and assume that (for all t large) Ptg : H(g) → H(g) acts as a strongly continuous semi-group of bounded operators. Then for all t large enough, we have ρess (Ptg ) ≥ e−2hg t , where ρess (Ptg ) is the essential spectral radius. Proof. Remark first that under the above hypothesis, Bk (g) ⊂ H(g). Assume the contrary i.e. there exists t0 large such that ρess (Ptg0 ) < e−2hg t0 . Choose μ > 0 such that ρess (Ptg0 ) < μt0 < e−2hg t0 . t0 By holomorphic functional calculus, one can show that there exist two P -invariant closed subspaces V, E such that H(g) = V E, where the spectral radius of Ptg0 |E is smaller than μt0 and V is the (finite) sum of the finite dimensional Ptg0 eigenspaces related to the eigenvalues of modulus larger than μt0 . For all s large, 4 Real 5 Here

analytic functions on the torus have exponentially decaying Fourier coefficients. we suppose that the embedding H(g) ⊂ L2 (Xτ , dμτg ) is continuous.

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V is Psg -invariant since Ptg0 and Psg commute. The strong continuity and the semigroup property then imply that Psg |V is exp(sA) where A is a linear map from the finite dimensional space V to itself. Now writing Ps = Pn(s)t0 ◦ Ps−n(s)t0 where n(s) = [s/t0 ], we get that for all > 0, there exist constants C, C such that Ps GL2 ≤ CPs−n(s)t0 L2 Pn(s)t0 GH(g) ≤ C μ(1+)s GH(g) , for all G ∈ E. Using a Jordan block decomposition for A and the above estimate we get an asymptotic expansion of correlation functions Corg (F, G)(s) with an error term of magnitude O(μ(1+)s ), which contradicts Corollary 1.2 for small , provided that we have chosen (F, G) in the appropriate set G0 . In a very similar setting, when μτg is the Lebesgue measure, Tsujii essentially shows in [24] that for generic ceiling functions, one can construct a Hilbert space H with the above properties such that ρess (Pt ) ≤ e−

tλmax 2

,

where λmax is related to the positive Liapounov exponent of the flow. It is easy to check that our lower bound is indeed smaller than his upper bound. Both of our lower bound and Tsujii’s upper bound are likely to be not optimal but they reveal the influence of quantities such as entropy and Liapounov exponents on the generic rate of mixing in a rather general setting. In the case of contact Anosov flows [25], Tsujii has proved similar estimates. Theorem 1.1 is actually a consequence of the next central result. We recall that if f is a function defined on [0, 1], f (n) (x) denotes the sum f (n) (x) = f (x) + f (T x) + · · · + f (T n−1 x) . We define a dynamical zeta function Zg (s) associated to the weight g and the semi-flow as defined above by +∞

1 e−sτ (n) (x)+g(n) (x)−nP (g) , Zg (s) = exp − n n 1 − ((T n ) (x))−1 n=1 T x=x

which is known to be convergent for Re(s) large. Theorem 1.4. Following the notations above, Zg (s) extends as an entire function to C and for all > 0, the entire function Zg (s) has infinitely many zeros inside the strip − 2hg − ≤ Re(s) ≤ 0 . We believe that the above existence and localization result is of interest in itself. It may be useful in other situation e.g. to prove existence of spectra for certain families of complex weighted transfer operators. The paper is organized as follows. In Section 2, we give the promised definition of the real analytic class of flows we will focus on. We then build various function spaces and recall how the Laplace transform is related to the resolvent of an analytic family of trace class transfer operators. Analytic Fredholm theory

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then relates the Ruelle–Pollicott resonances to zeros of Zg (s). A priori bounds for this function are then derived using the classical Weyl inequalities. In Section 3 we prove a summation formula involving closed orbits of the flow and its Ruelle– Pollicot resonances. The main theorem is then proved using this technology and some ergodic theoretic arguments. The final proof of Theorem 1.1 is postponed to Section 4 and follows easily from the main result by a Baire category argument. Although the central objects of the proof (transfer operators, dynamical determinants) have been used for a long time, our technique is fairly new in the field. A careful reading shows that the real analytic regularity is used only to get the a priori bounds, and the main ideas should work in great generality and without much regularity. We expect these results to be extended to general Anosov flows, provided that we can build a good function space on the manifold to carry out this program. The critical part is the obtention of the appropriate a priori estimates. In that direction, the ideas of Gouëzel–Liverani [11], Butterley–Liverani [7], Baladi–Tsujii [2], and also the short paper of Liverani–Tsujii [14], on dynamical zeta functions, should be very helpfull. It would also be interesting to study similar questions for partially hyperbolic maps (for examples skew products over an expanding map).

2. Function spaces, transfer operators and a priori bounds Let us define more precisely the class of semi-flows we will work with. We need first to be more specific with the expanding map T . We assume that [0, 1] = ∪pi=1 Ii , where each Ii = [ai , bi ] is a closed interval and Int(Ii ) ∩ Int(Ij ) = ∅ if i = j. For i = 1, . . . , p, the map Ti := T |[ai ,bi ) is continuous, and extend continuously to Ii as a one-to one map Ti : Ii → [0, 1]. We assume also that there exists a bounded connected open set (a domain) C ⊃ Ω0 ⊃ [0, 1] such that the following holds. 1. The contracting inverse branches of T , denoted by γi := Ti−1 : [0, 1] → [0, 1], have all an holomorphic extension to Ω0 , continuous on Ω0 . 2. There exists a compact set K0 such that for all 1 ≤ i ≤ p, γi (Ω0 ) ⊂ K0 ⊂ Ω0 . 3. The ceiling function τ has an holomorphic extension to Ω0 continuous on Ω0 . All the above assumptions imply that T is Markov, eventually expanding, topologically mixing and semi-conjugated to the full shift on p symbols σ : Σ → Σ with Σ = {1, . . . , p}N . In addition, the intersection of two different intervals Ii and Ij , if not empty, cannot contain periodic points of T (only pre-periodic points). Therefore there is a one-to-one correspondence between periodic points of the shift map σ and periodic points of T . From now on we assume that we work under the above framework. The Banach space of holomorphic functions on Ω0 , continuous on Ω0 , endowed with

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the supremum norm is denoted by H0 . A larger function space on which part of the analysis will be done is the classical Bergmann space H 2 (Ω0 ) defined as H 2 (Ω0 ) = f holomorphic on Ω0 : |f (x + iy)|2 dxdy < +∞ . Ω0 2

It is easy to check that H0 is a subset of H (Ω0 ) and that the embedding H0 → H 2 (Ω0 ) is continuous. The Bergmann space H 2 (Ω0 ) is in addition a Hilbert space. In the next sections we will allow us to shrink the domain Ω0 : under the above hypothesis, the system of inverses branches (γi )1≤i≤p is automatically6 complex contracting i.e. there exists a complex domain U with [0, 1] ⊂ U ⊂ Ω0 and an integer n0 such that sup sup |γi (z)| < 1 , i∈{1,...,p}n0 z∈U

where i = (i1 , . . . , in0 ) and γi = γi1 ◦ . . . γin0 . This property implies (see the proof in [4]) that all sufficiently small ε-neighbourhoods of [0, 1] have property 2) of the above definition. An ε-neighbourhood of [0, 1] has a C 1+1 Jordan boundary ∂Ω0 (clearly, one can find a parametrization ϕ : [0, 2π] → ∂Ω0 of class C 1 with a non vanishing Lipschitz derivative). This has the nice consequence that Ω0 is a Dinismooth simply connected domain, see [20] Chapter 3 Theorem 3.5, and hence is conformally equivalent to the unit disc via a conformal map ψ : Ω0 → D where ψ, ψ have a continuous extension to the boundary ∂Ω0 . Therefore, if Ω0 is an ε-neighborhood of [0, 1], we have a natural isometry between the two Bergmann spaces 2 H (D) → H 2 (Ω0 ) I: f

→ ψ (f ◦ ψ) with the additional feature that I maps holomorphic functions on D with a continuous extension to D into H0 . Polynomials being dense in H 2 (D), the space H0 is therefore dense in H 2 (Ω0 ). The class of transfer operators we will focus on is the following. Let g ∈ H0 , with g real valued on [0, 1]. For all f ∈ H 2 (Ω0 ), s ∈ C we define the transfer operator Ls,g by p

e−sτ (γi z)+g(γi z) f (γi z) . Ls,g (f )(z) = i=1

This is clearly a bounded linear operator on H 2 (Ω0 ). Let us state what we need about the spectral theory of such operators. Proposition 2.1. Using the above notations, we have the following. 1. Acting on the Hilbert space H 2 (Ω0 ), the operators Ls,g are trace class operators. 6 It follows from the fact that T : [0, 1] → [0, 1] is expanding and that we have only a finite number of inverse branches, see the paper [4].

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2. Let (λn (s))n≥1 (repeated with multiplicities and ordered with decreasing modulus) be the sequence of eigenvalues of Ls,g , then there exists ρ0 < 1 such that |λn (s)| ≤ C0 eC0 |s| (ρ0 )

n−1

,

where C0 > 0 depends only on the semi-flow and g. 3. Let P (g) denote the topological pressure of g. There exist a unique T -invariant 1 probability measure νg and a unique function Wg ∈ H0 with 0 Wg dνg = 1, such that for all continuous f on [0, 1], 1 1 P (g) L0,g (f )dνg = e f dνg and L0,g (Wg ) = eP (g) Wg . 0

0

In addition, Wg (x) = 0 for all x ∈ [0, 1]. 4. For all > 0 one can find C > 0 such that Lns,g H 2 ≤ C eC|s| en(P (−Re(s)τ +g)+) .

(1)

Proof. The first statement has been proved by various authors for various purposes. See for example [4, 5, 10, 12, 23]. The proof is based on explicit estimates of the singular values through an explicit basis. The second statement is derived directly from [4, 5] (after a conformal change of variables) though we can also find it implicitly in [12, 23]. One can also obtain a similar statement without using conformal representation from [6], Theorem 5.13. The last statement is a corollary of the classical Ruelle–Perron–Frobenius theorem, see for example [1, 15]. Notice that the equilibrium measure of the potential g is here μg = Wg νg . The only non-standard point may be the fact that Wg ∈ H0 , which follows easily from the fact that eP (g) is in the spectrum of the adjoint operator of L0,g acting on H0 . Indeed, eP (g) is therefore in the spectrum of L0,g acting on H0 , is an eigenvalue by compactness, and uniqueness is enough to conclude. Let us now say a few words about the bound 4) which is also a folklore estimate. Given f ∈ H 2 (Ω0 ), one can write

(n) (n) Lns,g (f )(z) = e−sτ (γi z)+g (γi z) f (γi z) , i∈{1,...,p}n

where we have used the notation h(n) (x) = h(x) + h(T x) + · · · + h(T n−1 x). By uniform contraction of inverse branches, we can show in a fairly standard way that there exist constants C1 , C2 > 0 such that for all n ≥ 1 and all i ∈ {1, . . . , p}n , d (n) d (n) sup (τ ◦ γi z) ≤ C1 and sup (g ◦ γi z) ≤ C2 . z∈Ω0 dz z∈Ω0 dz Fix any η ∈ [0, 1], we have therefore |Lns,g (f )(z)| ≤ e(|s|C1 +C2 )diam(Ω0 )

i∈{1,...,p}n

e−Re(s)τ

(n)

(γi η)+g (n) (γi η)

|f (γi z)| .

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Since we have the limit ⎛

lim ⎝ n→∞


⎞1/n e−Re(s)τ

(n)

(γi η)+g

(n)

(γi η) ⎠

= eP (−Re(s)τ +g) ,

i∈{1,...,p}n

we obtain the estimate simply by using the reproducing kernel of the Bergmann space to show that supz∈K0 |f (z)| ≤ Cf H 2 , which concludes the proof. We can now give the promised definition of the Banach space Bk (g). Let g be a real-valued, real-analytic function on [0, 1] (the potential). Using ε-neighborhoods of the unit interval, we can assume that Ω0 is small enough such that both g and Wg−1 belong to H0 .7 We now define Bk as the topological tensor product8 0k ([0 , τ0 ]), Bk := H0 ⊗C where C0k ([0 , τ0 ]) is the Banach space of C k functions on [0 , τ0 ], having all derivatives up to the order k vanishing on the boundary, and endowed with its obvious norm. The topological tensor product Bk is the completion of the algebraic tensor product H0 ⊗ C0k ([0 , τ0 ]) for the norm

F Bk = inf fi H0 gi C k : F = fi ⊗ gi , i

i

where the simple tensors f ⊗ g are simply identified with the pointwise product (f ⊗ g)(x, u) = f (x)g(u) . In other words, Bk is the space of functions F on Ω0 × [0 , τ0 ] that can be written as the sum of an infinite series +∞

λi fi (x)gi (u) , F (x, u) = i=1

with i |λi | < +∞, and (fi , gi ) ∈ H0 × C k ([0 , τ0 ]), the sequences fi H0 , gi C k being bounded. This is by definition a Banach space, and one can check that Bk ⊂ C0k (X0 ,τ0 ). One of the motivations for the definition of Bk is the fact that the averaging operator defined for F ∈ C0k (X0 ,τ0 ) and x ∈ [0, 1] by τ (x) τ0 M(F )(x) = F (x, u)du = F (x, u)du 0

0

extends as a bounded linear map from Bk to H0 . The proof follows readily from the definition of the norm on Bk . Remark in addition that our space Bk , unlike 7 We will use that fact in the last section where we show that generically residues are nonvanishing. 8 Notice that the C k regularity assumption (in the flow direction) that we have here is not necessary for our analysis. It is just assumed to show that observables in Theorem 1.1 can be chosen as smooth as we want.

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spaces built by Tsujii, may not be invariant by the Perron–Frobenius operator Ptg of the flow, but this is not a trouble for our purpose. Transfer operators are related to correlation functions by the following remarkable formula (see the proof in Pollicott [16, 19]). We assume for simplicity that the topological pressure P (g) = 0. The general case follows just by replacing g by g −P (g). Now let F, G ∈ Bk . Then the Laplace transform of Corg (F, G)(t) is given for Re(s) large enough by +∞ 1

1 M(esu F )Lns,g Wg M(e−su G) Wg−1 dμg . L Corg (F, G) (s) = 1 τ dμg n=0 0 0 If Re(s) > 0, using the estimate (1) and properties of the pressure functional, we can rewrite the above formula as 1

1 L Corg (F, G) (s) = 1 M(esu F )R(s) Wg M(e−su G) Wg−1 dμg , (2) τ dμg 0 0 where R(s) = (I − Ls,g )−1 . Since s → Ls,g is an analytic family of compact operators, the celebrated analytic Fredholm theory implies automatically that the resolvent R(s) has a finitely meromorphic9 continuation to C, and therefore L[Corg (F, G)](s), for all F, G ∈ Bk . In our case, since Ls,g are trace class operators, we know (Fredholm theory, see [22]) that poles of the meromorphic continuation of the resolvent R(s) are (with the same multiplicities) zeros of the entire function Zg (s) := det(I − Ls,g ) . It is now clear that to prove our theorem on Ruelle–Pollicott resonances, it will be more convenient to work with Zg (s) instead of the resolvent. We will recall in the next section why the determinant Zg (s) is indeed the zeta function defined in the introduction. Before we move to the next section, a key a priori bound is needed. Proposition 2.2. Using the above notations, there exists a constant C > 0 such that for all s ∈ C, we have 2

|Zg (s)| ≤ CeC|s| . Proof. By definition of the determinant Zg (s), we have for all s ∈ C, Zg (s) = 1 − λn (s) , n∈N

so that we can bound log |Zg (s)| ≤

N

n=1

+∞

log 1 + |λn (s)| + log 1 + |λn (s)| , n=N +1

9 Finitely meromorphic means that it admits a finite Laurent expansion at each singularity with residues given by finite rank operators.

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where N will be adjusted later. Using the bound from Proposition 2.1, we have for |s| large, C0 eC0 |s| N log |Zg (s)| ≤ 2N C0 |s| + (ρ0 ) . (3) 1 − ρ0 Setting N = O(|s|) such that we have N | log(ρ0 )| ≥ C0 |s| , the right hand term of (3) becomes O(|s|2 ) and the proof is done.

3. A summation formula and the main result In that section we use the same notations as above and we assume for simplicity that the weight g ∈ H0 satisfies P (g) = 0. This is the case of the popular weights g = − log |T | and g = −htop τ . The proof of Theorem 1.4 is based on the following summation formula. Let ϕ be a C ∞ -function, compactly supported in (0, +∞). Define ψ(s) for all s ∈ C by +∞ ψ(s) = esx ϕ(x)dx . 0

Let Rg be the set of zeros of Zg (s) (the Ruelle–Pollicott resonances), repeated with multiplicities. Theorem 3.1. Using the above notations, we have the identity

z∈Rg

(n) +∞

1 τ (n) (x)eg (x) (n) ψ(z) = ϕ τ (x) , n n 1 − ((T n ) (x))−1 n=1

T x=x

where the right sum has to be understood as a sum over all T -periodic points of period n. This formula may be viewed as a Selberg-like trace formula for semi-flows. Notice that the sum on the right hand side is actually finite, whereas the left hand side is convergent by integration by parts and the growth bound on Rg proved in the next proposition. Before we can give a proof of Theorem 3.1, it is necessary to recall a few basic facts about the Fredholm determinant Zg (s). Proposition 3.2. The following holds. • For all Re(s) > 0 we have the absolutely convergent expression. +∞

1 e−sτ (n) (x)+g(n) (x) Zg (s) = exp − . n n 1 − ((T n ) (x))−1 n=1 T x=x

• There exists a constant C > 0 such that for all |s| ≥ 1, N (R) := #{z ∈ Rg : |z| ≤ R} ≤ CR2 .

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Proof. By Lidskii’s theorem, the trace of a trace class operator is indeed the sum of its eigenvalues, therefore whenever the spectral radius of Ls,g is smaller than 1, we have +∞

1 n det(I − Ls,g ) = exp − Tr(Ls,g ) . n n=1 A crude estimate on the spectral radius using (1) shows that the above formula is obviously valid for all Re(s) > 0, since for Re(s) > 0, P (−Re(s)τ + g) < P (g) = 0. By a folklore result (see for example [4] for a proof that fits our setting), the trace of Lns,g can be computed as Tr(Lns,g )

e−sτ (n) (x)+g(n) (x) = . 1 − ((T n ) (x))−1 n T x=x

Notice that we use here the fact that periodic points of T are in one-to-one correspondence with fixed points of compositions of inverse branches γi . This concludes the proof of the first claim. The second part follows readily by Jensen’s formula applied to Zg (s) on the disc D(0, 2R). We have indeed 2π (l) 1 (2R)l |Zg (0)| ≤ log |Zg (2Reiθ )|dθ , (log 2)N (R) + log l! 2π 0 where l is the multiplicity of 0 as a zero of Zg (s). Using the a priori bound of Proposition 2.2, we obtain as R → ∞, N (R) = O(R2 ). The proof is done. Let us proceed to the proof of the summation formula. Fix A > 0. We claim that according to the above notations, we have A+i∞ (n) +∞

Zg (s) 1 1 τ (n) (x)eg (x) (n) ds = ψ(s) ϕ τ (x) . (4) 2iπ A−i∞ Zg (s) n n 1 − ((T n ) (x))−1 n=1 T x=x

Notice that by definition, ψ(A + ir) = 2πF−1 (eAu ϕ)(r), where F−1 is the inverse Fourier transform defined for any f in the Schwartz space by +∞ 1 f (u)eiur dr . F−1 (f )(r) = 2π −∞ As a consequence ψ(s) is in the Schwartz space on the line {Re(s) = A}. Remark that we have also for all Re(s) ≥ A > 0 the absolutely convergent expression (n) (n) +∞ Zg (s) 1 τ (n) (x)eg (x)−sτ (x) = . Zg (s) n=1 n n 1 − ((T n ) (x))−1

T x=x

This yields the bound for all Re(s) ≥ A > 0, +∞ (n) (n) Zg (s) 1 τ (n) (x)eg (x)−Aτ (x) ≤ = CA . Zg (s) n 1 − ((T n ) (x))−1 n=1

T n x=x

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Therefore the above contour integral is clearly convergent. Intertwining summations using the Lebesgue theorem, we then write A+i∞ Zg (s) ds ψ(s) Zg (s) A−i∞ +∞ Zg (A + ir) =i dr ψ(A + ir) Zg (A + ir) −∞ (n) (n) +∞

(n) 1 τ (n) (x)eg (x)−Aτ (x) +∞ −1 Au = 2iπ F (e ϕ)(r)e−irτ (x) dr . n ) (x))−1 n 1 − ((T −∞ n n=1 T x=x

Fourier’s inversion formula concludes the proof of (4). The rest of the proof will follow from a contour deformation argument and the residue theorem. To achieve this, another (crude) a priori bound is required. Lemma 3.3. There exist a constant C > 0 and arbitrarily large radii r > 0 such that Zg (s) |ds| ≤ Cr5 . |s|=r Zg (s) Proof. Since Zg (s) is an entire function of order at most 2, we can use the Hadamard Factorization theorem to write for all s ∈ C, s 1 s 2 s Zg (s) = sl eQ(s) 1− eρ+2(ρ) , ρ ρ∈Rg \{0}

where Q is a polynomial of degree less than 2 and l is the multiplicity of 0 as a zero of Zg (s). This allows us to express the log-derivative for all s ∈ Rg as

Zg (s) l s2 = + Q (s) + . 2 Zg (s) s ρ (s − ρ) ρ∈Rg \{0}

Assume that r ∈ {|ρ|, ρ ∈ Rg }. We have the bound

Zg (s) 1 d|s| |ds| ≤ O(r2 ) + r2 . Zg (s) 2 |ρ| |s − ρ| |s|=r |s|=r ρ∈Rg \{0}

Since we have10

|s|=r

d|s| = |s − ρ|

|s|=r

2πr d|s| ≤ , |s − |ρ|| |r − |ρ||

we are left to estimate carefully the sum

1 1 . 2 |ρ| |r − |ρ|| ρ∈Rg \{0}

10 This

is of course a very rough estimate, but enough for our purpose.

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Fix R > 0 and divide the segment [R, 2R] into 1 + [4CR2 ] intervals of same length, where C is precisely the constant given by Proposition 3.2. By the box principle, there exist at least one these intervals, denoted by I0 , such that I0 ∩ {|ρ|, ρ ∈ Rg } = ∅. Choose r to be the middle point of I0 , set a = inf I0 and b = sup I0 . There exist constants C1 , C2 > 0 independent of r such that a ≤ r − C1 /r and b ≥ r + C2 /r. We can now view the above sum as a Stieltjes integral and split it as a +∞

1 1 dN (x) dN (x) = + . 2 (r − x) 2 (x − r) |ρ|2 |r − |ρ|| x x + 0 b ρ∈Rg \{0}

Using an integration by parts, the first term is bounded by a 1 dN (x) r O(1) + O(log r) = O(r log r) . ≤ 2 r − a 0+ x C1 By remarking that (x − r)−1 ≤ 2x−1 whenever x ≥ 2r, we estimate the second term by +∞ +∞ 2r r dN (x) dN (x) dN (x) ≤ + 2 = O(r) + O(1) . 2 2 x (x − r) C2 r x x3 b 2r As conclusion, the total contribution of all terms is of size O(r4 log r) and the proof is done. To end the proof of the summation formula, we simply use the residue theorem to write A+iY (r)

Zg (s) Zg (s) 1 1 ds = ds , ψ(s) ψ(ρ) − ψ(s) 2iπ A−iY (r) Zg (s) 2iπ Γ(r) Zg (s) |ρ|≤r

where r is chosen as in the above lemma, and the integration contour is defined by Γ(r) = z ∈ C : |z| = r and Re(z) ≤ A , |A ± iY (r)| = r . Repeated integration by parts show that for all N ≥ 0, one can find a constant CN such that for all Re(s) ≤ A, |ψ(s)| ≤

CN . (1 + |s|)N

Set N = 6, The result of Theorem 3.1 now follows by taking the limit as r → +∞. We can now prove Theorem 1.4. Let ϕ0 be C ∞ positive test function whose support is [−1, +1] and such that ϕ0 (0) = 1 and 0 ≤ ϕ0 (x) ≤ 1 for all −1 ≤ x ≤ 1. For all d, γ > 0 set x−d ϕγ,d (x) = ϕ0 . γ

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The idea is to use this test function for the summation formula and work by contradiction. In the following of the proof, d will be a large number (to be precisely adjusted later on) while γ will be a small positive number (depending on d) of size O(e−μd ), where μ > 0 will be chosen later. For all s ∈ C we set +∞ ϕγ,d (x)esx dx . ψγ,d (s) = 0

For Re(s) ≤ 0 repeated integration by parts and a change of variable show that for all integer n ≥ 0, one can find a constant Cn such that |ψγ,d (s)| ≤ γCn

eRe(s)(d−γ) . (1 + γ|s|)n

We now assume that there are only finitely many Ruelle–Pollicott resonances inside the strip − ρ ≤ Re(s) ≤ 0 for some ρ > 0. The first step is to estimate the size of the sum over resonances as d → +∞. We can split the total sum into two parts:

ψγ,d (z) = ψγ,d (z) + ψγ,d (z) . z∈Rg

Re(z) 0. As a consequence, as k → +∞, we get for some constant C 1 (k) 1 (k) lim τ (xk ) = lim τ (η) = τ dμg . k→+∞ k k→+∞ k [0,1]

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In a similar way we obtain that 1 (k) g (xk ) = k→+∞ k


lim

gdμg . [0,1]

The lemma is proved.

We now return to the proof of the main theorem. If we set dk = τ (k) (xk ) where xk is as in the above lemma, we get the formula 1 (k) gdμg (xk ) [0,1] g (k) (xk ) Ak dk kg e =e , where Ak = 1 (k) . and lim Ak = k→+∞ τ dμg (xk ) [0,1] kτ Recall that by Abramov’s formula the metric entropy of the suspension flow with respect to the measure μτg is given by hg =

hμg (T ) , τ dμg [0,1]

while the topological pressure

P (g) = hμg (T ) +

gdμg = 0 . [0,1]

Therefore we have lim Ak = −hg .

k→+∞

This yields the lower bound dk eg

(k)

(xk )

≥ dk e−dk (hg +) ,

for all > 0 and k large. Gathering all previous estimates, we have obtained for all k large, 1 dk e−dk (hg +) ≤ O(γk−1 e−ρdk ) + O(γk ) . 2 Set γk = e−μdk , we have a contradiction as k → +∞ provided ρ − μ > hg + and μ > hg + . Setting μ = hg + 2 for all ρ > 2hg + 3, we get a contradiction and the main result follows.

4. Residues and generic observables In this section we show the following result. Proposition 4.1. There exists a Gδ -dense subset G of Bk × Bk such that for all (F, G) ∈ G, the Laplace transform L[Corg (F, G)](s) has the same set of poles11 as the set of zeros of Zg (s). 11 The

set of poles is the same but the multiplicity may differ.

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Combined with Theorem 1.4, the proposition above gives the main result of s = M(esu F ) Theorem 1.1. Let us prove the above result. Given F, G ∈ Bk , set F −su and Gs = M(e G)Wg . Let ρ ∈ Rg be a zero of Zg (s). Since 1 is an eigenvalue of the transfer operator Lρ,g acting on H 2 (Ω0 ), the resolvent s → (I − Lρ,g )−1 must be singular at s = ρ. The analytic Fredholm theorem tells us that this singularity is a pole of finite order i.e. we have the Laurent expansion at ρ (I − Lρ,g )−1 =

P2 P1 Pk + + ··· + + analytic terms , s − ρ (s − ρ)2 (s − ρ)k

where k ≥ 1, P1 , . . . , Pk are finite rank operators on H 2 (Ω0 ), Pk = 0. There is a priori no connection between12 the algebraic multiplicity of 1 as an eigenvalue of Lρ,g (which is the same as the multiplicity of ρ as a zero of Zg (s)) and the order s s and G k of the pole at s = ρ of the resolvent. Writing Taylor expansions for F at s = ρ, and using Formula 2 one can observe that

Ck (F, G) C1 (F, G) + ··· + L Cor(F, G) (s) = + analytic terms , s−ρ (s − ρ)k where Ck (F, G) = 1 0

1 τ dμg

1

ρ Pk G ρ dνg , F

0

the other terms Ci (F, G) being (computable) polynomial functions of Pj , dj dj dsj (Fs )|s=ρ and dsj (Gs )|s=ρ . For all ρ ∈ Rg , we set Φρ (F, G) = Ck (F, G). Each Φρ is clearly a continuous bilinear form on Bk × Bk . We claim that they are all non-trivial (non identically vanishing) on Bk × Bk , which will not surprise the reader but needs to be checked at this point. Let F (x, u) = f (x)ϕ(u) and G(x, u) = h(x)ϕ(u)Wg−1 , with f, h ∈ H0 and where ϕ ∈ C0k [0 , τ0 ] is chosen such that τ0

C±ρ =

e±ρu ϕ(u)du = 0 .

0

We have Cρ C−ρ Φρ (F, G) = 1 τ dμg 0

1

f (x)Pk h(x)dνg (x) . 0

Since Pk ≡ 0 and H0 is dense in H 2 (Ω0 ), there exists a choice of h ∈ H0 such that h = Pk (h) = 0. It remains to check that we can adjust f ∈ H0 so that 1 f hdνg = 0 . 0

The measure νg has a Gibbs property and thus is non-vanishing on every open subset of [0, 1], which allows us to conclude by a standard approximation argument. 12 Simple

finite dimensional examples are enough to show that without an additional hypothesis, anything can occur.

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Indeed, since h is non zero and analytic on Ω0 , there exists x0 ∈ [0, 1] such that h = 0. Choose δ > 0 such that for all |x − x0 | ≤ δ, | h(x0 )| . | h(x) − h(x0 )| ≤ 2 Let ψ be a positive continuous function on [0, 1] with ψ(x0 ) > 0 and supp(ψ) ⊂ [x0 − δ, x0 + δ]. Clearly we have 1 ψdνg > 0 , 1

and we can set ψ = ψ 1

0

. We now write 1 ψ hdνg = h(x0 ) + ψ h− h(x0 ) dνg , 0

ψdνg

−1

0

0

and a simple estimate shows that 1 | h(x0 )| > 0. ψ hdνg ≥ 2 0 We can now use the classical Weierstrass theorem to approximate uniformly close enough ψ by a polynomial Q on [0, 1] so that 1 Q hdνg = 0 . 0

We have shown that each functional Φρ is non trivial. Its null set Nρ = (F, G) ∈ Bk × Bk : Φρ (F, G) = 0 is a closed subset whose complementary set Nρc is open and dense. Applying Baire’s theorem, G :=

Nρc ρ∈Rg

is a generic subset of Bk × Bk where the desired property holds.

Further remarks on regularity. As mentioned in the introduction, it is possible to obtain a similar result with additional regularity. Instead of the space of C k observables Bk , one can work without major modifications with the Frechet space of C ∞ observables 0∞ ([0 , τ0 ]) . B∞ := H0 ⊗C If one wishes to obtain results with real analytic observables, here is an outline of what should be done. Let U0 be a bounded open set in C such that U0 ⊃ |z| ≤ sup |τ (w)| . w∈Ω0

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We then define our space of observables Bω by ! Bω := F : Ω0 × U0 → C, holomorphic, continuous on Ω0 × U0 , " such that F (x, 0) = F x, τ (x) = 0, ∀x ∈ [0, 1] , endowed with the supremum norm. This is clearly a Banach space, as a closed subset of the classical space of bounded holomorphic functions on Ω0 × U0 . The averaging operator M : Bω → H0 is well defined through the contour integral 1 F z, tτ (z) τ (z)dt . M(F )(z) = 0

To reproduce the preceding arguments, we need to check that the bilinear functionals Φρ are non trivial on Bω × Bω . We therefore choose observables of the type u F (z, u) = f (z)ψβ , τ (z) where ψβ (x) = x(1 − x)e−βt , β ∈ R, f ∈ H0 . Clearly F ∈ Bω if we shrink Ω0 enough such that τ −1 ∈ H0 , and one can compute 1 ρu M(e F )(z) = f (z)τ (z) etρτ (z) ψβ (t)dt . 0

By a direct calculation, the map Hβ (s) =

1

ets ψβ (t)dt = H0 (s − β)

0

is an entire function of s and has its possibles zeros included in a half-plane Re(s) ≤ β + M , for some M > 0 that can be estimated explicitly. We then choose β 0, this is compatible with the fact that we did not specify, for the local version of the theorem, the requirement for asymptotic flatness, and hence are in a case where the mass is not necessarily positive. Next we consider the general case where ta is not hypersurface orthogonal. In view of Proposition 12, we can assume that A > 0 and z not locally constant on any open set. Then it is clear that the set MA is in fact dense in M: for if there exists an open set on which z = A, then Proposition 12 implies that A = 0 identically on M. Therefore, the set (Ml ∪ Ml ) ∩ MA is non-empty as long as Ml ∪ Ml is non-empty; this latter fact can be assured since by assumption (A4) that ta is timelike at some point p ∈ M, whereas la and la are non-co¨ıncidental null vectors, so in a neighborhood of p, we must have la ta = 0 = la ta . It is on this set that we consider the next proposition. Proposition 14. Assuming A > 0. Let p ∈ U ⊂ Ml ∩ MA such that ta , na , ba and la are well-defined on U, with normalization la ta = 1. Then the four vector fields form a holonomic basis, and U can be isometrically embedded into a Kerr–Newman space-time.

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Before giving the proof, we first record the metric for the Kerr–Newman solution in Kerr coordinates 2M r − q 2 2 (26) ds = − 1 − 2 dV 2 + 2drdV + (r2 + a2 cos2 θ)dθ2 r + a2 cos2 θ 2 (r + a2 )2 − (r2 − 2M r + a2 + q 2 )a2 sin2 θ sin2 θ 2 dφ + r2 + a2 cos2 θ 2a(2M r − q 2 ) sin2 θdV dφ . − 2a sin2 θdφdr − 2 r + a2 cos2 θ Notice that the metric is regular at r = M ± M 2 − a2 − q 2 the event and Cauchy horizons. Proof. We first note that in Ml , we have the normalization na = (y 2 + z 2 )(la + U la ) + (A + y 2 )ta . For the proof, it suffices to establish that the commutators between na , ba , la , ta vanish and that the vectors are linearly independent (for holonomy), and to calculate their relative inner-products to verify that they define coordinates equivalent to the Kerr coordinates above. First we show that the commutators vanish. The cases [t, · ] are trivial. Since we fixed la ta = 1, we have that 0 = tb ∇b (la ta ) = Kt tb lb = Kt so that Kt = 0 and thus [t, l] = [t, l] = 0. Since y and z are geometric quantities defined from Hab , and U is a function only of y and z, they are symmetric under the action of ta , therefore [t, n] = 0. Similarly, to evaluate [t, b], it suffices to consider [t, ∇z]. Using (13) we see that ∇z is defined by the volume form, the metric, and the vectors ta , la , la , all of which symmetric under t-action, and thus [t, b] = 0. The remaining cases require consideration of the connection coefficients. In view of the normalization condition imposed, ∇a y = −la + U la , so (18a) implies θC¯1 P¯ = 1, θC1 P = −U . Recall the null structure equation ¯ . −δθ = −ζθ + η(θ − θ) Using 0 = δ(θC¯1 P¯ ) = (δθ)C¯1 P¯ + θη C¯1 P¯ we have ¯ = 0. C¯1 P¯ (θη + ζθ − ηθ + η θ) Applications of (19) allows us to replace +η θ¯ by −ηθ in the brackets, and so, since θC¯1 P¯ = DC1 P = 0, we must have ζ = η, which considerably simplifies calculations. Next we write y2 + z2 1

ηC1 C¯12 P P¯ 2 m (ηC1 P m ¯ a − η¯C¯1 P¯ ma ) = i ¯ a − c.c ba = −i 2 2 A−z A−z

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by expanding ∇a z in tetrad coefficients, and where c.c. denotes complex conjugate. Then, since Dz = 0, ¯ − c.c . ¯ a − c.c + ηC1 C¯ 2 P P¯ 2 [D, δ] −i(A − z 2 )[l, b] = D(ηC1 C¯1 P P¯ 2 )m 1

We consider the commutator relation, simplified appropriately in view of computations above and in the proof of Lemma 10, 1 ¯ ¯ [D, δ] = −(Γ213 + θ)δ = Γ123 − ¯ ¯ δ¯ C1 P together with the structure equation (D + Γ123 )η = θ(η − η) and the relations in (19) and (17), we get ¯ D(ηC1 C¯ 2 P P¯ 2 )m ¯ a + ηC1 C¯ 2 P P¯ 2 [D, δ] 1

1

¯ a − η|C1 P |2 m ¯ a + ηD(C1 C¯12 P P¯ 2 )m ¯a = (D + Γ123 )ηC1 C¯12 P P¯ 2 m = θ(η − η)C1 C¯ 2 P P¯ 2 m ¯ a − η|C1 P |2 m ¯ a + η(θC¯ 3 P¯ 3 + 2θC1 C¯ 2 P P¯ 2 )m ¯a 1

1

1

= 0. Hence [l, b] = 0. In a similar fashion, we write 1 na = |C1 P |2 la + (A + y 2 + |C1 |2 − 2C1 C¯2 y)la + (A + y 2 )ta . 2 From the fact that ba ∇a y = 0 and from the known commutator relations, we have 1 [n, b] = [C1 C¯1 P P¯ l, b] + (A + y 2 + |C1 |2 − 2C1 C¯2 y)[l, b] + (A + y 2 )[t, b] , 2 of which the second and third terms are already known to vanish. We evaluate [C1 C¯1 P P¯ l, b] in the same way we evaluated [l, b], and a calculation shows that it also vanishes. To evaluate [l, n], we need to calculate [l, l]. To do so we write ¯a. ta = −U la − la − η¯C1 P ma − η C¯1 P¯ m Since [l, t] = 0, we infer ¯ [l, l] = −[l, U l + η¯C1 P m + η C¯1 P¯ m]

1 2¯ 2¯ = −DU l − l, η¯C C1 P P m − c.c . |C1 P |2 1 Notice that in the proof above for [l, b] = 0 we have demonstrated that [l, η¯C12 C¯1 P 2 P¯ m] = 0, so [l, l] = −DU l +

D(|C1 P |2 ) ¯ . (¯ η C1 P m + η C¯1 P¯ m) |C1 P |2

Direct computation yields DU =

y − C1 C¯2 2yU − 2 y2 + z2 y + z2

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and D(C1 C¯1 P P¯ ) = 2y (recall that we set Dy = 1) so we conclude that y − C1 C¯2 2y [l, l] = − 2 l− 2 (l + t) . y + z2 y + z2 So, using the decomposition for na given above [l, n] = l, (y 2 + z 2 )l + (y 2 + z 2 )U l + (A + y 2 )t = 2yl + (y − C1 C¯2 )l + 2yt + (y 2 + z 2 )[l, l] = 0. Having checked the commutators, we now calculate the scalar products between various components. A direct computation from the definition yields y2 + z2 A − z2 l · n = A − z2

b·n = 0

b2 =

t·n =

b·l = 0 l·t = 1

l2 = 0

(|C1 | − 2C1 C¯2 y)(z − A) y2 + z2 2

2

t2 = −1 −

|C1 | − 2C1 C¯2 y y2 + z2 2

t·b = 0

and

A − z2

2 ¯ C |C n2 = (A − z 2 ) A + y 2 − 2 | − 2C y . 1 1 2 y + z2 A simple computation shows that the determinant of the matrix of inner products yields | det | = (y 2 + z 2 )2 = 0

and therefore the vector fields are linearly independent. Thus we have shown that they form a holonomic basis. To construct the local isometry to Kerr–Newman space-time, we define coordinates attached to the holonomic vector fields t, l, b, n with the following rescalings. First, since A > 0, we can define a > 0 such that A = a2 . Then we can define the coordinates r, θ, V, φ by t = ∂V l = ∂r 1 ∂θ b= a sin θ n = −a∂φ .

y=r z = a cos θ

Notice that we can define θ from z in a way that makes sense since z 2 ≤ A. Applying the change of coordinates to the inner-products above we see that in r, θ, V, φ the metric is identical to the one for the Kerr coordinate of Kerr–Newman space-time. Furthermore, we see that n, or ∂φ , defines the corresponding axial Killing vector field.

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To finish this section, we need to show that the results we obtained in Propositions 13 and 14 can be extended to the manifold M, rather than restricted to (Ml ∪ Ml ) in the former and (Ml ∪ Ml ) ∩ MA in the latter. We shall need the following lemma (Lemma 6 in [12]; the lemma and its proof can be carried over to our case essentially without change, we reproduce them here for completeness): Lemma 15. The vector field na is a Killing vector field on the entirety of M. The set M \ MA = {na = 0}. Furthermore, • If A = 0, then M \ (Ml ∪ Ml ) = {ta = 0}. • If 0 < A ≤ (C1 C¯2 )2 − |C1 |2 , then M \ (Ml ∪ Ml ) = {either na − y+ ta = 0 or na − y− ta = 0} where y± = 2(C1 C¯2 )2 − |C1 |2 ± 2C1 C¯2 (C1 C¯2 )2 − |C1 |2 − A . • If A > (C1 C¯2 )2 − |C1 |2 , then M \ (Ml ∪ Ml ) = ∅. Proof. First consider the case A = 0. By Proposition 12, we have z = 0. So the definition (22) and (15) show that na vanishes identically. Furthermore, since MA = ∅ in this case, we have that na is a (trivial) Killing vector field on M vanishing on M \ MA . It is also clear from (15) that ta = 0 ⇐⇒ ta la = ta la = 0 in this case, proving the first bullet point. Now let A > 0. Then Proposition 14 shows that na is Killing on (Ml ∪ Ml ) ∩ MA , and does not co¨ıncide with ta . Since MA is dense in M (see paragraph immediately before Proposition 14), we have that na is Killing on Ml ∪ Ml (the overline denotes set closure). We wish to show that Ml ∪ Ml = M. Suppose not, then the open set U = M \ Ml ∪ Ml is non-empty. In U, ta la = ta la = 0, so by (13), ∇a y = 0 in U. Taking the real part of the third identity in Proposition 7, we must have y = C1 C¯2 in U, which by Lemma 10 implies A = (C1 C¯2 )2 − |C1 |2 . Consider the vector field defined on all of M given by na − [2(C1 C¯2 )2 − |C1 |2 ]ta . As it is a constant coefficient linear combination of non-vanishing independent Killing vector fields on Ml ∪ Ml , it is also a non-vanishing Killing vector field. However, on U, the vector field vanishes by construction. So we have Killing vector field on M that is not identically 0, yet vanishes on an non-empty open set, which is impossible (see Appendix C.3 in [20]). Therefore na is a Killing vector field everywhere on M. Now, outside of MA , we have that z 2 = A reaches a local maximum, so ∇a z must vanish. Therefore from (22) and (15) we conclude that na vanishes outside MA also, proving the second statement in the lemma. For the second a third bullet points, consider the function U = 12 (∇y)2 . By definition it vanishes outside Ml ∪ Ml . Using Lemma 10 we see that A + y 2 + |C1 |2 − 2C1 C¯2 y = 0 outside Ml ∪ Ml . The two bullet points are clear in view of the quadratic formula and (22). Now we can complete the main theorem in the same way as [12].

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Proof of the Main Theorem. In view of Propositions 13 and 14, we only need to show that the isometry thus defined extends to M \ (Ml ∪ Ml ) in the case of Reissner–Nordström and M \ [(Ml ∪ Ml ) ∩ MA ] in the case of Kerr–Newman. Lemma 15 shows that those points we are interested in are fixed points of Killing vector fields, and hence are either isolated points or smooth, two-dimensional, totally geodesic surfaces. Their complement, therefore, are connected and dense, with local isometry into the Kerr–Newman family. Therefore a sufficiently small neighborhood of one of these fixed-points will have a dense and connected subset isometric to a patch of Kerr–Newman, whence we can extend to those fixed-points by continuity.

4. Proof of the main global result To show Corollary 3, it suffices to demonstrate that the global assumption (G) leads to the local assumption (L). By asymptotic flatness and the imposed decay rate (the assumption that the mass and charge at infinity are non-zero), we can assume that there is a simply connected region MH near spatial infinity such that H2 = 0. It thus suffices to show that MH = M. Suppose not, then the former is a proper subset of the latter. Let p0 ∈ M be a point on ∂MH . We see that Theorem 2 applies to MH , with C1 taken to be qE + iqB and C3 = M/(qE − iqB ). In particular, the first equation in Proposition 7 shows that, by continuity, t2 = −1 at p0 . Let δ be a small neighborhood of p0 such that ta is everywhere time-like in δ with t2 < − 41 , then the metric g induces a uniform Riemannian metric on the bundle of orthogonal subspaces to ta , i.e. ∪p∈δ {v ∈ Tp M|g(v, t) = 0}. Now, consider a curve d γ : (s0 , 1] → δ such that γ(s) ∈ MH for s < 1, γ(1) = p0 , and ds γ(s) has norm 1 and is orthogonal to t. Consider the function (qE + iqB )P ◦ γ. By assumption, |(qE + iqB )P ◦ γ| ∞ as s 1. Since Lemma 10 guarantees that z is bounded in MH , and hence by continuity, at p0 , we must have that y blows up as we approach p0 along γ. However, d (y ◦ γ) = |∇ d y| ≤ C |∇a y∇a y| < C < ∞ ds ds γ where the constant C comes from the uniform control on g acting as a Riemannian metric on the orthogonal subspace to ta (note that ta ∇a y = 0 since y is a quantity derivable from quantities that are invariant under the t-action), and C arises because by Lemma 10, ∇a y∇a y is bounded for all |y| > 2M , which we can guarantee for s sufficiently close to 1. So we have a contradiction: y ◦ γ blows up in finite time while its derivative stays bounded. Therefore MH = M.

Appendix A. Tetrad formalisms The null tetrad formalism of Newman and Penrose is used extensively in the calculations above, albeit with slightly different notational conventions. In the following,

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a dictionary is given between the standard Newman–Penrose variables (see, e.g. Chapter 7 in [19]) and the null-structure variables of Ionescu and Klainerman [6] which is used in this paper. Following Ionescu and Klainerman [6], we consider a space-time with a natural choice of a null pair {l, l}. Recall that the complex valued vector field m is said to be compatible with the null pair if g(l, m) = g(l, m) = g(m, m) = 0 ,

g(m, m) ¯ =1

where m ¯ is the complex conjugate of m. Given a null pair, for any point p ∈ M, such a compatible vector field always exist on a sufficiently small neighborhood of p. We say that the vector fields {m, m, ¯ l, l} form a null tetrad if, in addition, they ¯ b lc ld = i (we can always swap m and m ¯ by the have positive orientation abcd ma m obvious transformation to satisfy this condition). The scalar functions corresponding to the connection coefficients of of the null tetrad are defined, with translation to the Newman–Penrose formalism, in Table 1. The Γ-notation is defined by Γαβγ = g(∇eγ eβ , eα ) where for e1 = m, e2 = m, ¯ e3 = l, and e4 = l. It is clear that Γ(αβ)γ = 0, i.e. it is antisymmetric in the first two indices. Two natural3 operations are then defined: the under-bar (e.g. θ ↔ θ) corresponds to swapping the indices 3 ↔ 4 ¯ corresponds to swapping (e.g. Γ142 ↔ Γ132 ), and complex conjugation (e.g. θ ↔ θ) the numeric indices 1 ↔ 2 (e.g. Γ142 ↔ Γ241 ). We note that θ, θ, ϑ, ϑ, ξ, ξ, η, η, ζ are complex-valued, while ω and ω are real-valued; thus these scalar functions, along with their complex conjugates, define 20 out of the 24 rotation coefficients: the only ones not given a “name” are Γ121 , Γ122 , Γ123 , Γ124 , among which the first two are related by complex-conjugation, and the latter-two by under-bar. The directional derivative operators are given by: D = la ∇a ,

D = la ∇a ,

δ = ma ∇a ,

δ¯ = m ¯ a ∇a

¯ (their respective symbols in Newman–Penrose notation are D, Δ, δ, δ). The spinor components of the Riemann curvature tensor can be given in terms of the following: let Wabcd be the Weyl curvature tensor, Sab be the traceless Ricci 3 Buyers beware: the operations are only natural in so much as those geometric statements that are agnostic to orientation of the frame vectors. Indeed, both the under-bar and complex conjugation changes the sign of the Levi-Civita symbol; while for the complex conjugation it is of less consequence (since the complex conjugate of −i is i, the sign difference is most naturally absorbed), for the under-bar operation one needs to take care in application to ascertain that sign-changes due to, say, the Hodge star operator is not present in the equation under consideration. In particular, generally co¨ ordinate independent geometric statements (such as the relations to be developed in this section) will be compatible with consistent application of the under-bar operations, while statements dependent on a particular choice of foliation or frame will usually need to be evaluated on a case-by-case basis.

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Table 1. Dictionary of Ricci rotation coefficients vs. Newman– Penrose spin coefficients vs. Ionescu–Klainerman connection coefficients.

g(∇m ¯ l, m) g(∇m ¯ l, m) g(∇m l, m) g(∇m l, m) g(∇l l, m) g(∇l l, m) g(∇l l, m) g(∇l l, m) g(∇l l, l) g(∇l l, l) g(∇m l, l)

Γ-notation Γ142 Γ132 Γ141 Γ131 Γ144 Γ133 Γ143 Γ134 Γ344 Γ433 Γ341

Newman–Penrose −ρ μ ¯ −σ ¯ λ −κ ν¯ −τ π ¯ −2 + Γ214 2γ + Γ123 −2β + Γ211

Ionescu–Klainerman θ θ ϑ ϑ ξ ξ η η ω ω ζ = −ζ

tensor, and R be the scalar curvature, we can write Ψ2 = W (l, m, l, m) ¯ Ψ−2 = Ψ2 = W (l, m, l, m) Ψ1 = W (m, l, l, l) ¯ Ψ−1 = Ψ1 = W (m, l, l, l) Ψ0 = W (m, ¯ l, m, l)

(27a) (27b) (27c) (27d) (27e)

Φ11 = S(l, l)

(27f)

Φ11 = S(l, l)

(27g)

Φ01 = S(m, l)

(27h)

Φ01 = S(m, l)

(27i)

Φ00 = S(m, m) 1 ¯ . Φ0 = S(l, l) + S(m, m) 2

(27j) (27k)

Notice that the quantities ΨA , A ∈ {−2, −1, 0, 1, 2} are automatically anti-selfdual: replacing Wabcd ↔ ∗ Wabcd we have ΨA (∗ W ) = (−i)ΨA (W ), this follows from the orthogonality properties of the null tetrad, as well as the orientation requirement (m, m, ¯ l, l) = i. Using this notation, we can write the null structure equations, which are equivalent to the Newman–Penrose equations. We derive them from the definition of the Riemann curvature tensor: Rαβμν = eμ (Γαβν ) − eν (Γαβμ ) + Γρ βν Γαρμ − Γρ βμ Γαρν + (Γρ μν − Γρ νμ )Γαβρ

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and that 1 1 Rαβμν = Wαβμν + (Sαμ gβν +Sβν gαμ −Sαν gβμ −Sβμ gαν )+ R(gαμ gβν −gβμ gαν ) . 2 12 So from R1441 = W1441 = −Ψ2 we get ¯ − Ψ2 (D + 2Γ124 )ϑ − (δ + Γ121 )ξ = ξ(2ζ + η + η) − ϑ(ω + θ + θ)

(28a)

by taking under-bar of the whole expression, we get for a similar expression for R1331 = −Ψ2 (in the interest of space, we omit the obvious changes of variables here). For R1442 = − 12 S44 (and analogously R1332 = − 12 S33 ) we have ¯ + ξ(2ζ¯ + η¯) − 1 Φ11 . Dθ − (δ¯ + Γ122 )ξ = −θ2 − ωθ − ϑϑ¯ + ξη 2

(28b)

From R1443 = −Ψ1 − 12 S14 1 (D + Γ124 )η − (D + Γ123 )ξ = −2ωξ + θ(η − η) + ϑ(¯ η − η¯) − Ψ1 − Φ01 . (28c) 2 From R1431 = 12 S11 we get 1 (D + 2Γ123 )ϑ − (δ + Γ121 )η = η 2 + ξξ − θϑ + ϑ(ω − ¯θ) + Φ00 . 2 From R1432 = −Ψ0 +

1 12 R

(28d)

we have

¯ + θ(ω − θ) − Ψ0 + R . Dθ − (δ¯ + Γ122 )η = ξ ¯ξ + η η¯ − ϑϑ 12

(28e)

From R1421 = −Ψ1 + 12 S41 we have ¯ + η(θ − θ) ¯ + ξ(θ − ¯θ) − Ψ1 + 1 Φ01 . (δ¯ + 2Γ122 )ϑ − δθ = ζθ − ζϑ 2

(28f)

Using R3441 = −Ψ1 − 12 S41 we get 1 ¯ −ζ)+ϑ(¯ ¯ ¯θ +ω)− ξϑ−Ψ ¯ (D +Γ124 )ζ −δω = ω(ζ +η)+ θ(η η − ζ)−ξ( 1 − Φ01 . (28g) 2 R ¯ From R3443 = Ψ0 + Ψ0 − S34 + we get 12

¯ + ξ ¯ξ − η¯η − η¯ ¯ − η) − (Ψ0 + Ψ ¯ 0 ) + Φ0 − R (28h) η + ζ(¯ η − η¯) + ζ(η Dω + Dω = ξξ 12 ¯ 0 we have and lastly from R3421 = Ψ0 − Ψ ¯ δ+Γ ¯ 122 )ζ = (ϑϑ−ϑ ¯ ¯ ¯θ−θθ)+ω(θ− ¯ ¯ ¯θ)−(Ψ0 −Ψ ¯ 0 ) . (28i) (δ−Γ121 )ζ−( ϑ)+(θ θ)−ω(θ− In this formalism, we can also write the Maxwell equations: let 1

¯ m) = Hab la lb Υ0 = H(l, l) + H(m, 2 Υ1 = H(l, m) = Hab la mb ¯ −1 = Υ = H(m, l) = H ¯ ab ma lb Υ 1

(29a) (29b) (29c)

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be the spinor components of the Maxwell two-form Hab . Maxwell’s equations becomes DΥ0 − (δ − Γ121 )Υ−1 = ¯ξΥ1 − 2¯θΥ0 − (ζ − η)Υ−1 (D + Γ123 )Υ1 − δΥ0 = (ω − ¯θ)Υ1 + 2ηΥ0 − ϑΥ−1

(30a) (30b)

and their under-bar counterparts. We also need the Bianchi identities ∇[e Rab]cd = 0 . Note that this implies ∇e Webcd = ∇[c Sd]b −

1 gb[c ∇d] R =: Jbcd 12

which gives 1 seab J srt rtcd 6 using the orientation condition (m, m, ¯ l, l) = i we calculate ∇[e Wab]cd =

1 1 (δ¯ + 2Γ122 )Ψ2 − (D + Γ124 )Ψ1 + δΦ11 − (D + Γ124 )Φ01 (31a) 2 2 = −(2ζ¯ + η¯)Ψ2 + (4θ + ω)Ψ1 + 3ξΨ0 1 ¯ 00 ¯ 01 + ζ + 1 η Φ11 + ξΦ0 + 1 ξΦ − θ¯ + ω Φ01 − ϑΦ 2 2 2 1 1 (D + 2Γ123 )Ψ2 − (δ + Γ121 )Ψ1 + (D + 2Γ124 )Φ00 − (δ + Γ121 )Φ01 (31b) 2 2 = (2ω − ¯θ)Ψ2 + (ζ + 4η)Ψ1 + 3ϑΨ0 1 1¯ 1 ϑΦ ζ + η − θΦ − ϑΦ − + ξΦ + Φ01 00 0 11 01 2 2 2 1 1 ¯ 01 − 1 DR (31c) −(δ¯ + Γ122 )Ψ1 − DΨ0 − DΦ0 + (δ − Γ121 )Φ 2 2 24 ¯ 2 + (2¯ ¯ 1 + 3θΨ0 + 2ξ Ψ ¯ = −ϑΨ η + ζ)Ψ 1 1 1 1 ¯ 0 + ¯θΦ11 + ϑΦ ¯ 00 ¯ 01 + θΦ − (ζ + η)Φ 2 2 2 1¯ 1 1 ¯ ¯Φ01 − ξ Φ − ξΦ 01 − η 2 2 2 01 1 ¯ 0 + (D + Γ123 )Φ01 − 1 δΦ0 + 1 δR (31d) (D + Γ124 )Ψ1 + δ Ψ 2 2 24 ¯ ¯ ¯ 1 − 3η Ψ ¯ 0 + (ω − 2θ)Ψ = −2ϑΨ 1 + ξΨ2 1 1¯ 1 ¯ 1 ¯ + (ω − ¯θ)Φ01 − θΦ − ϑΦ − ϑΦ 01 2 2 01 2 01 2 1 + η¯Φ00 + ηΦ0 . 2

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In addition, we can also take the trace of the Bianchi identities, which gives 0 = ∇e Webc b = Jbc b and evaluates to 1 −δΦ0 − (δ¯ + 2Γ122 )Φ00 + (D + Γ123 )Φ01 + (D + Γ124 )Φ01 + δR (31e) 4 ¯ ¯ = (¯ η + η¯)Φ00 + 2(η + η)Φ0 + (ω − 2θ − θ)Φ 01 + (ω − 2θ − θ)Φ01 ¯ 01 + ξΦ + ξΦ11 ¯ − ϑΦ − ϑΦ 01

11

¯ 01 − (δ¯ + Γ122 )Φ01 + 1 DR DΦ0 + DΦ11 − (δ − Γ121 )Φ 4 ¯ ¯ ¯ ¯ ¯ 00 = −ϑΦ00 − 2(θ + θ)Φ0 + ξΦ01 + (ζ + 2¯ η + η¯)Φ01 − ϑΦ ¯ + (ζ + 2η + η)Φ ¯ 01 + (2ω − θ − ¯θ)Φ11 . + ξΦ

(31f)

01

A simple identification using Table 1 and the definitions for various spinor components of the Riemann and traceless Ricci tensors shows that one can recover all of the Bianchi identities in Newman–Penrose formalism from the above six equations through the action of complex-conjugation and under-barring. Lastly, to complete the formalism, we record the commutator relations η − η¯)δ − ωD + ωD (32a) [D, D] = (η − η)δ¯ + (¯ ¯ + (η + ζ)D + ξD [D, δ] = −ϑδ¯ − (Γ124 + θ)δ ¯ = Γ121 δ¯ + Γ122 δ + (θ¯ − θ)D + (θ¯ − θ)D . [δ, δ]

(32b) (32c)

References [1] D. Bini, C. Cherubini, R. T. Jantzen, and G. Miniutti. The Simon and SimonMars tensors for stationary Einstein–Maxwell fields. Classical and Quantum Gravity, 21:1987–1998, 2004. [2] G. Bunting. Proof of the uniqueness conjecture for black holes. PhD thesis, University of New England, Australia, 1983. [3] B. Carter. Black hole equilibrium states. Gordon and Breach, 1973. [4] D. Christodoulou and S. Klainerman. The global nonlinear stability of the Minkowski space. Princeton University Press, 1993. [5] R. Debever, N. Kamran, and R. G. McLenaghan. Exhaustive integration and a single expression for the general solution of the type D vacuum and electrovac field equations with cosmological constant for a nonsingular aligned Maxwell field. Journal of Mathematical Physics, 25(6):1955–1972, 1984. [6] A. D. Ionescu and S. Klainerman. On the uniqueness of smooth, stationary black holes in vacuum. preprint arXiv:0711.0040v1 [gr-qc], 2007. [7] A. D. Ionescu and S. Klainerman. Uniqueness results for ill posed characteristic problems in curved space-times. preprint arXiv:0711.0042v1 [gr-qc], 2007. [8] W. Israel. Event horizons in static vacuum space-times. Physical review, 164(5):1776– 1779, 1967.

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[9] W. Israel. Event horizons in static electrovac space-times. Communications in mathematical physics, 8:245–260, 1968. [10] N. Voje Johansen and F. Ravndal. On the discovery of Birkhoff’s theorem. preprint arXiv:physics/0508163v2 [physics.hist-ph], 2005. [11] R. P. Kerr. Gravitational field of a spinning mass as an example of algebraically special metrics. Physical Review Letters, 11(5):237–238, 1963. [12] M. Mars. A spacetime characterization of the Kerr metric. Classical and Quantum Gravity, 16:2507–2523, 1999. [13] M. Mars. Uniqueness properties of the Kerr metric. Classical and Quantum Gravity, 17:3353–3373, 2000. [14] P. O. Mazur. Proof of uniqueness of the Kerr–Newman black hole solution. Journal of Physics A, 15:3173–3180, 1982. [15] E. T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash, and R. Torrence. Metric of a rotating, charged mass. Journal of Mathematical Physics, 6(6):918–919, 1965. [16] D. C. Robinson. Uniqueness of the Kerr black hole. Physical Review Letters, 34(14):905–906, 1975. [17] W. Simon. Characterizations of the Kerr metric. General Relativity and Gravitation, 16(5):465–476, 1984. [18] W. Simon. The multiple expansion of stationary Einstein–Maxwell fields. Journal of Mathematical Physics, 25(4):1035–1038, 1984. [19] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herit. Exact Solutions of Einstein’s Field Equations. Cambridge University Press, second edition, 2002. [20] R. M. Wald. General Relativity. University of Chicago Press, 1984. Willie W. Wong 408 Fine Hall Princeton University Princeton, NJ USA e-mail: [email protected] Communicated by Piotr T. Chrusciel. Submitted: November 16, 2008. Accepted: February 9, 2009.



Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction Roberto Conti and Gerardo Morsella Abstract. Given an inclusion B ⊂ F of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets B 0 ⊂ F0 , giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of F implies that of the scaling limit of B. As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion A ⊂ B of local nets with the same canonical field net F, we find sufficient conditions which entail the equality of the canonical field nets of A 0 and B 0 .

1. Introduction Local quantum physics is an approach to Quantum Field Theory (QFT) based only on observable quantities [19]. It has been very successful in the mathematical description of superselection sectors and of the global gauge group of a given QFT [17]. Also, the mathematical tools that are available in this setting are well suited for providing a detailed analysis of subsystems, an issue that is central in order to obtain an intrinsic description of the observable system one starts with [10–12]. In another direction, the recently proposed algebraic approach to the renormalization group [8] (see also Section 2) has opened the possibility of studying the short distance limit in the local quantum physics framework, and has started to convey new insight into our understanding of physically relevant issues such as confinement of colour charges and renormalization of pointlike fields [1, 3, 14]. Dealing with the general problem of understanding the scaling limit A0 of a given local net A, it is natural to ask whether there exists an efficient way to compute it in practical situations. Work supported by MIUR, GNAMPA-INDAM, the EU and SNS.

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Loosely speaking, starting with a given local net, one would like to mod out the degrees of freedom that play no role at short scale, and obtain a smaller and hopefully simpler net which has the same scaling limit of the original net. In turn, it is unlikely that a local net always contains a “large” subnet containing the whole information about scaling, however the notion of convergent scaling limit that we use at some crucial point of the main text is clearly an evolution of this naive idea. In fact, the very concept of the scaling algebra involves some redundancy in the choice of the scaling functions (as shown by the presence of a big kernel of the scaling limit representation), so that one could expect that, at least in particular cases, the consideration of some appropriate subalgebra of the scaling algebra would suffice. It might also be the case that one knows that the given local net can be realized as fixpoints of a larger net, and then wonder if the scaling limit of the fixpoints can be computed as the fixpoints of the scaling limit. In both cases, we are thus led to the problem of comparing the scaling limit of a system with that of a subsystem, and this paper came out as an attempt to understand this relationship. It has been shown in [8] that, for a given theory A, there are only three possibilities: either A has a trivial scaling limit, or a unique non trivial one, or several non-isomorphic ones. In the case of a subsystem A ⊂ B the situation is slightly more complicated, but one of our most basic observations is that there always exists a bijective correspondence between the sets of scaling limit states of A and B, and that for the corresponding scaling limit nets A0 and B0 one has a subsystem A0 ⊂ B0 . In this situation it seems natural to expect that if B has a unique scaling limit, the same should be true for A, at least under suitable assumptions. In Section 2 we provide a criterion for this to happen, and use it to show that fixed point nets in free field theory have a unique scaling limit. Another aspect of the problem is to study the inclusions A0 ⊂ B0 , and for instance one can ask whether it is possible to find necessary/sufficient conditions on A and B ensuring that A0 = B0

(1)

for every scaling limit state. At first sight, one could expect that the situation becomes somehow easier to handle if one knows that (A(O) and B(O) are factors and) [B : A] < ∞, but it has not really become important to employ this condition yet. In turn, the index of an inclusion is not necessarily preserved in the scaling limit, but there are cases in which the inequality [B0 : A0 ] ≤ [B : A] holds true. For instance, consider the free massive scalar field and its Z2 -fixpoints. In this case, after scaling the index remains the same, as shown in Section 2. On the other hand, tensorizing the Lutz model [21] with a massive free field one gets that the index of the scaling limit is smaller than the original one [15]. Also, the relation [B0 : A0 ] ≤ [B : A] is compatible with equation (3) below.

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Tensor products provide simple examples of subsystems, for which some questions can be answered. For instance, let us assume that A2 has trivial scaling limit.1 It is then natural to ask under which conditions the scaling limits of B := A1 ⊗ A2 and A := A1 ⊗ C A1 satisfy equation (1). A set of sufficient conditions for this to happen, expressed in terms of nuclearity properties, has been found in [15]. The fact that nuclearity plays a role in this context is not surprising, as it provides invariants which depend on the localization region, and therefore should be able to encode the fact that A(O) and B(O) become “closer” at small distances. Notice also that a certain (graded) tensor product decomposition plays a critical role in the classification of subsystems in [10, 11]. This work heavily relies on the DR-reconstruction [17]: given an observable net A, there exist a canonical field net F(A) and a compact group G(A) of automorphisms of F(A) such that A = F(A)G(A) (therefore A is a subsystem of F(A)). Some functoriality aspects of the reconstruction have been investigated in [12], and a classification result for subsystems of F(A) has been obtained in [10, 11]. The study of the scaling limits of A and F(A) is discussed in [14]. In typical cases, it holds F(A)0 = F(A0 )H , A0 =

G(A)/N F(A)0

(2) ,

(3)

with G(A)/N = G(A0 )/H. Here, N is the counterpart of the charges that disappear in the scaling limit, while H corresponds to the confined charges (i.e., those which appear only in the scaling limit). In Section 3 we show F(A)N 0 = F(A)0 , (4) which is again a case in which (1) holds. Also, notice that here both nets involved satisfy (twisted) Haag duality. In the remaining part of this work, we investigate the scaling limit of subsystems A ⊂ B of the form FK ⊂ FG , where F = F(A) = F(B) is a graded-local field net acted upon by the compact groups G ⊂ K. Of course, this situation includes the case of a field net and its gauge-invariant observable subnet recalled above, but, for example, also the subnet generated by the local energy-momentum tensor fits in. In this framework, we discuss the general relations between the groups appearing in equations (2), (3) associated to A and B. We then apply the results on classification of subsystems in [10,11] to gain some insights on the structure of the inclusion A0 ⊂ B0 , and in particular on the relation between the canonical field nets in the scaling limit F(A0 ) and F(B0 ). The content of this paper is as follows. In Section 2 we show that for a subsystem A ⊂ B with a conditional expectation E : B → A there is a one-to-one correspondence between the sets of scaling limit states of A and those of B. This 1 We remark that the role played by Haag duality in relation to the triviality of the scaling limit is not completely understood, as no examples of nets satisfying it and having trivial scaling limit are known to date.

488



entails the somewhat curious fact that the sets of scaling limit states of any two theories are in bijective correspondence [15]. As another consequence, we show that the scaling limit of the Z2 -fixed point net of the free massive scalar field coincides with the Z2 -fixed point net of the free massless scalar field. Then we readily adapt the argument in order to deal with more general free fields. In Section 3 we prove equation (4). Finally in Section 4 we present a detailed discussion of the scaling limit of subsystems of the form FK ⊂ FG , illustrating the main results with several examples.

2. Scaling limit for subsystems We start by recalling some known facts to be used in the following, also to fix our terminology and notation. Definition 2.1. By a graded-local net with gauge symmetry we mean a quadruple (F, α, β, ω), where: (i) O → F(O) is a net of unital C∗ -algebras over double cones in Minkowski spacetime; (ii) α is an automorphic action on F of a geometrical symmetry group Γ (the Poincaré group or its normal subgroup of translations) such that, for each double cone O, αγ (F(O)) = F(γ · O), γ ∈ Γ; (iii) β is an automorphic action on F of a compact group G commuting with α and such that, for each double cone O, βg (F(O)) = F(O), g ∈ G; (iv) ω is a pure state on F which is α- and β-invariant; (v) there exists an element k in the centre of G with k 2 = e such that, by defining 1 F± := F ± βk (F ) , F ∈ F , 2 for Fi ∈ F(Oi ), i = 1, 2, with O1 spacelike from O2 , there holds F1,+ F2,± = F2,± F1,+ ,

F1,− F2,− = −F2,− F1,− .

We need also a spatial version of the above concepts. Definition 2.2. A graded-local net with gauge symmetry in the vacuum sector will be a graded local net with gauge symmetry such that: (i) for each O, F(O) is a von Neumann algebra acting on the Hilbert space H; (ii) there is a strongly continuous unitary representation U of Γ on H such that αγ = Ad U (γ), and such that the joint spectrum of the generators of the representation of the translations subgroup R4 x → U (x) is contained in the closed forward light cone; (iii) there is a strongly continuous unitary representation V of G on H commuting with U and such that βg = Ad V (g); (iv) ω is the vector state induced by a U - and V -invariant unit vector Ω ∈ H which is cyclic for the quasi-local algebra F = O F(O) (closure in the norm topology).

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∼ Z2 , we will simply speak of a graded-local net. In the parIf G = {e, k} = ticular case in which G is trivial (and therefore k = e), we will use the traditional notation A instead of F, and we will refer to the triple (A, α, ω A ) as a local net (in the vacuum sector if it applies). If (F, α, β, ω F ) is a graded-local net with gauge symmetry, then one obtains a local net by defining A(O) := F(O)G := F ∈ F(O) : βg (F ) = F, g ∈ G . Moreover, an Haag dual net will be a local net in the vacuum sector such that A(O) = A(O ) , where as usual A(O ) is defined as the C∗ -algebra generated by the A(O1 ) for all double cones O1 ⊂ O . We recall the construction of the scaling algebra of a graded-local net with gauge symmetry in the vacuum sector F [8, 14]: we consider the C∗ -algebra of all bounded functions F : R+ → F, with norm F := supλ>0 F λ , endowed with the automorphic actions of Γ and G defined by αγ (F )λ := αγλ (F λ ) ,

β g (F )λ := βg (F λ ) ,

γ ∈ Γ, g ∈ G, λ > 0 ,

where γλ = (Λ, λx) if γ = (Λ, x). Then F(O) is the C∗ -subalgebra of the functions F such that 1. F λ ∈ F(λO) for all λ > 0; 2. lim αγ (F ) − F = 0; γ→(1,0)

3. lim β g (F ) − F = 0. g→e

In the particular case in which F = A is a local net, the third condition above is of course void because of the triviality of G. We denote by F the quasi-local C∗ -algebra defined by the net O → F(O). Remark 2.3. According to property 3. above, the scaling algebra F associated to (F, α, β, ω) depends on the action β of G. Since we do not require β to be faithful, it factors through an action β˜ of G/N , where N := {g ∈ G : βg (F ) = F, ∀F ∈ F}, ˜ ω). However, ˜ associated to (F, α, β, and one could consider the scaling algebra F thanks to the fact that the canonical projection G → G/N is open, it turns out ˜ = F. straightforwardly that actually F Next we introduce scaling limits. We define states ω μ , μ > 0, on F by ω μ (F ) := ω(F μ ), and we denote by SL(ω F ) the set of weak* limit points of (ω μ )μ>0 . We shall write SL(ω F ) = (ω 0,ι )ι∈IF , where IF is an appropriate index set. Each ω 0,ι will be called a scaling limit state of ω,2 and we denote by (π0,ι , H0,ι , Ω0,ι ) the GNS triple induced by ω 0,ι . According to the results in [14, Sec. 3], (F, α, β, ω 0,ι ) is a graded-local net with gauge symmetry, and by defining (5) F0,ι (O) := π0,ι F(O) 2A

more general definition of scaling limit state has been given in [1].

490



one gets a graded-local net with gauge symmetry in the vacuum sector, called a scaling limit net of F. The notation ω0,ι = Ω0,ι , ( · )Ω0,ι will be systematically employed in the following. For a general analysis of the notion of subsystem see [12, 20, 24]. Definition 2.4. Given two graded-local nets (F, αF , β F , ω F ), (B, αB , β B , ω B ), we say that they form an inclusion of graded-local nets, and write for brevity B ⊂ F, if: (i) (ii) (iii) (iv)

B(O) ⊂ F(O) for each double cone O; αγF (B) = αγB (B), for all B ∈ B, γ ∈ Γ; βkFF (B) = βkBB (B), for all B ∈ B; ω F (B) = ω B (B) for all B ∈ B.

Accordingly, when there is no danger of confusion, we will omit indices F, B and write simply α, k and ω. In the above situation, if (F, αF , β F , ω F ) is a gradedlocal net in the vacuum sector, it follows easily, by a Reeh–Schlieder type argument, that Ω is separating for F(O) for each O, and therefore it is clear that by restricting B(O), UF (γ) and VF (k) to HB := BΩF ⊂ HF , one gets a graded-local net in the vacuum sector, which is isomorphic to (B, αB , β B , ω B ) (see e.g. [10], top of page 93), and therefore it will be identified with (B, αB , β B , ω B ) when no ambiguities arise. In the sequel, we also assume the existence of a conditional expectation of nets E : F → B, meaning that E is a conditional expectation on the quasi-local algebra F onto the quasi-local algebra B, which in restriction to every F(O) is a conditional expectation onto B(O), and such that αγ E = Eαγ , βk E = Eβk and ω ◦ E = ω. It follows from the last property that if B, F are in the vacuum sector, E restricts to a normal conditional expectation of F(O) onto B(O). Such a conditional expectation exists if, e. g., F and B satisfy twisted Haag duality on their respective vacuum spaces [12, sec. 3] (see also [20]). Our setup includes in particular the case where F is a Doplicher–Roberts field net over a local net of observables B, so that E is obtained by taking the average over the compact global gauge group [17]. Now we wish to examine the possible relations between the scaling algebras B F(O) and B(O) and the scaling limit states SL(ω F ) = (ω F 0,ι )ι∈IF and SL(ω ) = B (ω 0,ι )ι∈IB associated to F and B respectively. It is clear that since F and B satisfy conditions (i)-(iii) of Definition 2.4, the same is true for F and B. It is then easy to see that the map E defined on F by E(F )λ := E(F λ ) ,

F ∈ F,

λ>0

(6)

is a conditional expectation of nets from F onto B, commuting with α and β k . Moreover, if E is faithful, then also E is: if, for each λ > 0, E(F ∗ F )λ = E(F ∗λ F λ ) = 0, then F λ = 0, i.e. F = 0.

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Proposition 2.5. Let B ⊂ F be an inclusion of graded-local nets and E : F → B a conditional expectation as before. Then SL(ω F ) = SL(ω B ) ◦ E, and there is a B bijective correspondence between IB and IF defined by mapping ω B 0,ι ∈ SL(ω ), F ι ∈ IB , to ω B 0,ι ◦ E ∈ SL(ω ). B B B Proof. Let ω B 0,ι ∈ SL(ω ), ι ∈ IB . Then, since ω μ ◦ E(F ) = ω (E(F μ )) = F F B F B ω (F μ ) = ω μ (F ), we have that ω 0,ι ◦ E ∈ SL(ω ). Also, if ω 0,ι ◦ E = ω B 0,κ ◦ E, B B B then ω B (B) = ω ◦ E(B) = ω ◦ E(B) = ω (B) for all B ∈ B, and the map 0,ι 0,ι 0,κ 0,κ defined in the statement is injective. F F Conversely, let ω F 0,ι ∈ SL(ω ), ι ∈ IF . Then ω 0,ι is a weak* limit point of B F F (ω F μ )μ>0 , and therefore ω 0,ι := ω 0,ι B is a weak* limit point of (ω μ B)μ>0 . F F B B B But, for B ∈ B, ω μ (B) = ω (B μ ) = ω (B μ ) = ω μ (B), and then ω 0,ι ∈ SL(ω B ), F B so that ω F 0,ι = ω 0,ι ◦ E = ω 0,ι ◦ E. This also shows that the above defined map is surjective, concluding the proof.

As a consequence of the above proposition, E is a conditional expectation of B B B B F the nets (F, αF , β F , ω B 0,ι ◦ E) and (B, α , β , ω 0,ι ). Also, denoting by π0,ι and π0,ι B F B the scaling limit representations defined by ω 0,ι and ω 0,ι = ω 0,ι ◦ E respectively, F B we see that π0,ι is the representation induced from π0,ι via E. Remark 2.6. It follows form the previous result that if B ⊂ F, even without assumF ing the existence of a conditional expectation of F onto B the map ω F 0,ι → ω 0,ι B induces a canonical bijection between IF and IB . In order to see this, assume, for simplicity, that B and F are local nets, and consider, as in [15, Prop. 3.5], the tensor product theory G := B ⊗ F, and the conditional expectations E B : G → B B ⊗ C1, E F : G → F C1 ⊗ F given respectively by E B (B ⊗ F ) = ω F (F )B, E F (B ⊗ F ) = ω B (B)F . According to the previous proposition, we have a bijection F F between IF and IG induced by ω F 0,ι → ω 0,ι ◦ E , and a bijection between IG and G IB induced by ω G 0,ι → ω 0,ι B, where, with a slight abuse, we identify B with the (isomorphic) subalgebra of G consisting of the functions λ → B λ ⊗ 1, B ∈ B. It F F is then sufficient to show that ω F 0,ι ◦ E B = ω 0,ι B, but this follows at once from F F F E (B λκ ⊗ 1) = lim ω B (B λκ ) = ω F ωF 0,ι E (B) = lim ω 0,ι (B) . κ

κ

The case in which B and F are genuinely graded-local nets can be handled in a similar way up to the replacement of the tensor product and of the slice maps with their Z2 -graded versions.3 At first sight this result may appear somewhat counter-intuitive, but it should be observed that this does not imply that states which corresponds to one another 3 We also notice that the above argument shows the existence of a canonical bijection between the sets of scaling limit states of any pair of nets B, F, without assuming B ⊂ F, such bijection being F F given by ω F 0,ι → ω 0,ι ◦ E B. On the other hand, one can show that the scaling limit states of a net are in one-to-one correspondence with a suitable subset of the pure states of the center of the scaling algebra, which is universal, i.e. independent of the underlying net [2, Sec. 2.1].

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in this bijection also share analogous physical interpretations: in particular, it may happen that all the scaling limit states of, say, B give rise to isomorphic scaling limit nets, while the same is not true for the corresponding states of F, see also the remark following Proposition 3.5 in [15]. In view of the above proposition, we indentify IB and IF and denote both simply by I, and for each ι ∈ I we denote by B0,ι and F0,ι the scaling limit nets obtained by the corresponding states in SL(ω B ) and SL(ω F ). Now one can show the existence of a conditional expectation on every scaling limit theory. Proposition 2.7. Given an inclusion B ⊂ F of graded-local nets in the vacuum sector, there is, for each ι ∈ I, an inclusion B0,ι ⊂ F0,ι of scaling limit nets. Furthermore, if a conditional expectation of nets E : F → B is given, there exists a conditional expectation of nets E0,ι : F0,ι → B0,ι uniquely defined by F F E0,ι (π0,ι (F )) := π0,ι (E(F )), F ∈ F. Moreover, if e0,ι := [B0,ι Ω0,ι ], one has E0,ι (F )e0,ι = e0,ι F e0,ι ,

F ∈ F0,ι .

(7)

Proof. Thanks to the Reeh–Schlieder property for F0,ι , the net B0,ι is isomorphic F to the net O → π0,ι (B(O)) ⊂ F0,ι (O), which gives the inclusion B0,ι ⊂ F0,ι . In order to show the existence of the conditional expectation E0,ι , we start by observing that, given B 1 , B 2 ∈ B, F ∈ F one has F F F π0,ι (B 1 )Ω0,ι , π0,ι (F )π0,ι (B 2 )Ω0,ι = ω 0,ι (B ∗1 F B 2 ) = ω 0,ι E(B ∗1 F B 2 ) = ω 0,ι B ∗1 E(F )B 2

B B B = π0,ι E(F ) π0,ι (B 1 )Ω0,ι , π0,ι (B 2 )Ω0,ι , which, taking into account the above mentioned isomorphism, shows that the F F map π0,ι (F ) → π0,ι (E(F )) is well-defined and σ-weakly continuous, and therefore extends uniquely to a σ-weakly continuous map F0,ι → B0,ι which is easily seen to be a conditional expectation of nets. (0,ι) (0,ι) (0,ι) (0,ι) and ω0,ι ◦ E0,ι = The properties αγ E0,ι = E0,ι αγ , βk E0,ι = E0,ι βk ω0,ι follow at once from the analogous properties for E. In order to prove (7), it is clear, by normality, that it is sufficient to prove F F E0,ι (π0,ι (F ))e0,ι = e0,ι π0,ι (F )e0,ι , F ∈ F. This is shown by choosing, for Φ ∈ F F H0,ι , a sequence Bn ∈ π0,ι (B) such that Bn Ω0,ι converges to e0,ι Φ, and then by evaluating

F F Φ, E0,ι π0,ι (F ) e0,ι Φ = e0,ι Φ, E0,ι π0,ι (F ) e0,ι Φ

F (F )Bn Ω0,ι = lim Ω0,ι , E0,ι Bn∗ π0,ι n→+∞ F = lim Ω0,ι , Bn∗ π0,ι (F )Bn Ω0,ι n→+∞ F F = e0,ι Φ, π0,ι (F )e0,ι Φ = Φ, e0,ι π0,ι (F )e0,ι Φ , which gives the statement.

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Remark 2.8. The above discussion carries over to the case where B ⊂ F is an inclusion of graded-local nets with gauge symmetry, meaning that condition (iii) in Definition 2.4 is replaced by the following F (iii’) βgB (B) = βφ(g) (B), for all B ∈ B, g ∈ GB , where φ : GB → GF is a continuous homomorphism such that φ(kB ) = kF , and there exists a conditional expectation of nets E : F → B such that βgB (E(F )) = F E(βφ(g) (F )) for all F ∈ F, g ∈ GB . Notice in fact that, if F ∈ F(O), then E(F ) defined by (6) is still an element of B(O), since F lim sup βgB E(F λ ) − E(F λ ) = lim sup E βφ(g) (F λ ) − E(F λ ) GB g→e λ>0

GB g→e λ>0

≤

lim

F sup βφ(g) (F λ ) − F λ = 0 .

GB g→e λ>0

For instance, this situation applies if N ⊂ GF is a closed normal subgroup, B := FN and E is the average on N , in which case one can assume GB = GF (and therefore φ = id), whose action factors through GF /N .4 When N = GF , A := B is the local net of observables associated to F, and we recover the existence of a conditional expectation from F0,ι to the observable scaling limit net A0,ι used in the proof of Lemma 5.1 in [14]. It is worth pointing out that our treatment is by no means limited to nets of von Neumann algebras. This should be already clear from the above discussion, and it is further exemplified by the next theorem which, following closely the arguments expounded in [9], is presented in the setting of nets of C ∗ -algebras, and where, as an application of the above results, we compute the scaling limit of the even part of the free scalar field. A von Neumann algebraic version will follow from the more general Theorem 2.11 afterwards. We will use the description of the scalar field net in terms of time-zero fields and locally Fock representations as in [9], as well as the main results of that paper. In particular, we consider the case where Γ = Rd , d = 3, 4. Also, we associate to the free scalar field of mass m ≥ 0 the net of C∗ -algebras O → F(m) (O) obtained by considering the elements of the canonical net of von Neumann algebras of the free scalar field of mass m which are norm-continuous under translations. This net is conveniently isomorphically represented on the Fock space of the massless scalar field by local normality, see [9] for details. Since we deal with nets of C∗ -algebras, in the following result it is understood that the scaling limit of a net F of C∗ -algebras is defined as F0,ι (O) = π0,ι (F(O)), without weak closure. Furthermore, we denote by Fr (O) := ∩O1 ⊃O¯ F(O1 ) the outer regularized net of F. Theorem 2.9. Let (F(m) , α(m) , ω (m) ) be the net associated to the free neutral scalar field of mass m ≥ 0 in d = 3, 4 spacetime dimensions, as described above, and let 4 We stress that, due to Remark 2.3, the scaling algebra of B when it is thought with an action of GF coincides with the scaling algebra obtained when B is thought with an action of GF /N .

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(A(m) , α(m) , ω (m) ) be the subnet of fixed points under the Z2 action defined by the involutive automorphism β W (f ) = W (−f ) , f ∈ D(Rd−1 ) . (m)

(m)

Then each outer regularized scaling limit net (A0,ι;r , α(m; 0,ι) , ω0,ι ) is isomorphic to the net (A(0) , α(0) , ω (0) ) of Z2 -fixed points of the net associated to the massless scalar field. (m)

(m)

Proof. In view of the above results, each net (A0,ι;r , α(m; 0,ι) , ω0,ι ) is a subnet (m)

(m)

of some outer regularized scaling limit net (F0,ι;r , α(m; 0,ι) , ω0,ι ). Let then φ : (m) F0,ι;r

→ F(0) be the isomorphism onto the net of the massless scalar field, whose (m)

existence is proven in [9]. We will show that φ A0,ι;r is an isomorphism onto A(0) . (m)

We begin by showing that φ(A0,ι;r ) ⊂ A(0) . To this end, let A ∈ A(m) (O); then, since in the chosen representation β is weakly continuous, being implemented by the unitary operator eiπN (with N the number operator of the massless scalar field), we have, for a suitable net (λκ )κ ⊆ R+ , β φ π0,ι (A) = w- lim βσλ−1 (Aλκ ) κ κ

= w- lim σλ−1 β(Aλκ ) κ κ

(Aλκ ) = φ π0,ι (A) , = w- lim σλ−1 κ κ

where, in the second equality, we have used the fact that β commutes with the dilations σλ as defined in [9, Eq. (2.6)], and in the third equality the fact that Aλ is β-invariant for each λ. Therefore φ(π0,ι (A)) ∈ A(0) (O) = F(0) (O)Z2 , and then ⎛ ⎞ (m) (m) φ A0,ι;r (O) = φ ⎝ A0,ι (O1 )⎠ ⊂ A(0) (O) ¯ O1 ⊃O

thanks to the outer regularity of A(0) (O). Conversely, φ being an isomorphism, any element A ∈ A(0) (O) is of the form ¯ and for some F ∈ F(m) (O1 ). For such an element, A = φ(π0,ι (F 1 )) for any O1 ⊃ O 1 define A1 := E(F 1 ) ∈ A(m) (O1 ), where E = 12 (id + β) is a conditional expectation of F(m) onto A(m) , which is obviously weakly continuous and commuting with the dilations. Then, arguing as above, we have E(F 1,λκ ) φ π0,ι (A1 ) = w- lim σλ−1 κ κ = E φ π0,ι (F 1 ) = φ π0,ι (F 1 ) = A , (m)

so that φ(A0,ι;r ) = A(0) .

Remark 2.10. The net A(m1 ) ⊗ A(m2 ) with m1 = m2 has no nontrivial subsystems thanks to the results in [10], but, according to the above theorem and the results

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in [15], any of its scaling limits is isomorphic to A(0) ⊗A(0) , which has, for instance, the subsystem obtained by taking the fixpoint net with respect to the natural action of SO(2). Therefore, this simple example shows that subsystems can appear in the scaling limit which are not related to subsystems already existing at finite scales. As anticipated in [15, Thm. 4.6] the argument in Theorem 2.9 carries over to the case of multiplets of free fields acted upon by a compact Lie group G. More precisely, consider a finite symmetric and generating set Δ of irreducible representations of G and a mass function μ : Δ → [0, +∞) such that μ(v) = μ(¯ v ). Let then F(μ) denote the graded-local net with gauge symmetry in the vacuum sector generated by a v-multiplet of free scalar fields of mass μ(v) for each v ∈ Δ, and A(μ) ⊂ F(μ) the fixed point net of F(μ) under the natural action of G. Again, this net is isomorphically represented on the Fock space H(0) corresponding to (μ) (μ) μ(v) = 0 for each v ∈ Δ (see [15] for details). Furthermore, denote by A0,ι ⊂ F0,ι the corresponding inclusion of scaling limit nets. As shown in [15, thm 4.3] there (μ) is a spatial net isomorphism θ : F0,ι → F(0) such that for each F ∈ F(μ) , (μ) θ π0,ι (F ) = w- lim δλ−1 (F λκ ) , κ κ

for a suitable net (λκ )κ ⊂ R+ , and where δλ is the adjoint action of the dilation group on H(0) . (μ)

Theorem 2.11. There is a net isomorphism between A0,ι and A(0) , obtained from θ by restriction. Proof. Since the action of G on F(μ) is μ-independent [15], the same is true for the conditional expectation E : F(μ) → A(μ) obtained by averaging with respect (μ) (μ) (μ) to G. Therefore, if E0,ι : F0,ι → A0,ι is the conditional expectation given by Proposition 2.7, in order to generalize the above argument it is sufficient to show (μ) (μ) that θ ◦ E0,ι = E ◦ θ. By normality of θ and E0,ι , it is sufficient to check this (μ)

equation on elements π0,ι (F ) with F ∈ F(μ) (O) for some O. This follows at once from the computation (μ) (μ) E(F λκ ) θ ◦ E0,ι π0,ι (F ) = w- lim δλ−1 κ κ (μ) = w- lim E δλ−1 (F λκ ) = E ◦ θ π0,ι (F ) , κ κ

where in the last equality the norm boundedness of δλ−1 (F λκ ) and the normality κ of E were used. The essential point in the above proofs is the existence of conditional expec(μ) (μ) (μ) tations E0,ι : F0,ι → A0,ι and E : F(0) → A(0) intertwining the action of the (μ)

isomorphism θ : F0,ι → F(0) . In fact if we consider the general situation of an inclusion B ⊂ F with a conditional expectation of nets E : F → B as discussed

496



above, and we assume that F has a unique (quantum) scaling limit, with isomorphisms φι,ι : F0,ι → F0,ι , and that the conditional expectations E0,ι : F0,ι → B0,ι introduced in Proposition 2.7 satisfy φι,ι ◦ E0,ι = E0,ι ◦ φι,ι ,

(8)

a similar argument shows that φι,ι (B0,ι ) = B0,ι , so that B has a unique scaling limit too. This happens in particular if F has a convergent scaling limit as introduced in [2]: we say that a net F has a convergent scaling limit if there exists an inclusion ˆ there exists limλ→0 ω(F ) ˆ ⊂ F such that for each F ∈ F of graded-local nets F λ and such that, for each scaling limit state ω 0,ι , in the corresponding scaling limit ˆ = F0,ι (O). It is easily seen that if a theory has representation one has π0,ι (F(O)) a convergent scaling limit then the scaling limit is unique. Proposition 2.12. Let B ⊂ F be an inclusion of graded-local nets in the vacuum sector, such that F has convergent scaling limit, and let E : F → B be a conditional expectation of nets. Then equation (8) holds. Furthermore B has convergent scaling ˆ ˆ ⊂ F(O). limit, if E(F(O)) Proof. It is straightforward to show that the unitary Vι,ι : H0,ι → H0,ι defined by ˆ, ˆ )Ω0,ι = π0,ι (F ˆ )Ω0,ι , F ˆ ∈F Vι,ι π0,ι (F ˆ ∈ implements a net isomorphism φι,ι : F0,ι → F0,ι , and there holds, for each F ˆ F(O), ˆ ) = π0,ι E(F ˆ ) = E0,ι ◦ φι,ι π0,ι (F ˆ) , φι,ι ◦ E0,ι π0,ι (F ˆ so that equation (8) follows thanks to π0,ι (F(O)) = F0,ι (O). Furthermore, using the normality of E0,ι , it is direct to verify that B has a ˆ ˆ convergent scaling limit by setting B(O) := E(F(O)).

3. Inclusions with coinciding scaling limits In the previous section we have discussed situations in which an inclusion of nets gives rise to a proper inclusion of nets in the scaling limit. For completeness, in the present section we provide a general construction of an inclusion of nets such that the corresponding inclusion of scaling limit nets is trivial. Let (F, α, β, ω) be a graded-local net with gauge symmetry in the vacuum sector. Then the quintuple (F, U, V, Ω, k) is a QFTGA according to [14], where of the translations and of G for the scaling limit the representations U0,ι and V0,ι F0,ι are introduced. As V0,ι is not necessarily faithful, we define the closed normal subgroup N := g ∈ G : V0,ι (g) = I . (9) The scaling limit net F0,ι is then obviously covariant with respect to the natural representation V0,ι of the factor group G0,ι := G/N .

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Proposition 3.1. With the above notation, let B be the subsystem of fixed points of F under N , with its natural action of a second countable G. Then for the associated scaling limit net there holds B0,ι = F0,ι . Proof. From B(O) ⊆ F(O), B0,ι (O) ⊆ F0,ι (O) readily follows. In order to prove the reverse inclusion, take F ∈ F(O) and define dn β n (F ) , B := N

where the integral is performed with respect to the normalized Haar measure on N and is understood in Bochner sense, cfr. [25]. This is well defined, since, by definition of F(O), n ∈ N → β n (F ) ∈ F(O) is a continuous function on a compact space, and then its range, being metrizable, is separable. We obtain then that B ∈ F(O). Furthermore, as the function F ∈ F(O) → F λ ∈ F(λO) is norm continuous, and a Bochner integral is a norm limit of Lebesgue sums, we get Bλ = dn β n (F ) = dn β n (F )λ = dn βn (F λ ) N

N

λ

so that, for m ∈ N ,

N

βm (B λ ) =

dn βmn (F λ ) = B λ , N

having used the invariance of the measure dn. This shows then that B ∈ B(O). Then, using again the norm continuity of n → β n (F ), that of π0,ι , and the defini , we get tion of V0,ι π0,ι (B) = dn π0,ι β n (F ) = dn V0,ι (n)π0,ι (F )V0,ι (n)∗ = π0,ι (F ) , N

N

where the last equality follows from the definition of N . Thus we get π0,ι (F ) ∈ B0,ι (O) and the statement of the proposition. Remarks 3.2. (i) At first sight, one might think that the above result is a trivial consequence of Lemma 5.1.(i) of [14], but some subtleties in the definition of the relevant scaling algebras prevent the application of the cited result. This is because the definition of the scaling limit net F0,ι really depends on both the underlying net F and the group G acting on it, through the requirement of norm continuity of functions g ∈ G → β g (F ), F ∈ F. So, if one was willing to apply Lemma 5.1.(i) of [14] with A = B, he should define a new scaling net ˜ associated to the datas (F, N ), i.e. in the same way as F but requiring F now only continuity of n ∈ N → β n (F). In general, this would result in a much bigger net than F. Then, application of the cited result would lead to ˜ 0,ι (O)N/N˜ , where now N ˜ is a normal subgroup of N , defined in B0,ι (O) = F the obvious way. Also, the scaling limit net would now be acting on a new Hilbert space, in general much bigger than our H0,ι . There are however cases in which the scaling net F does not really depend on the group G, and then

498



the result of [14] can be applied straightforwardly. For instance, this is the case if G is a finite group, so that the continuity requirement is empty, which ˜ = F, N ˜ = N and finally B0,ι (O) = F0,ι (O)N/N = F0,ι (O). entails F (ii) The group N is really non-trivial, in general: if φi , i = 1, . . . , n, are charged generalized free scalar fields with mass measure dρ(m2 ) = c dm2 , on which a compact gauge group G ⊆ U (n) acts, and if F(O) is generated by the fields 2n(O) φi (f ) with suppf ⊂ O, where n(O) → +∞ as the radius of O shrinks (g) = I to 0, then the scaling limit net F0 is trivial [15], and therefore V0,ι for all g ∈ G, and N = G. More generally, in [15] examples are constructed where N is any closed subgroup of an arbitrary compact Lie group G. (iii) For any net O → C(O) such that B(O) ⊆ C(O) ⊆ F(O), we define the associated “interpolated” scaling algebras as C(O) := F ∈ F(O) : F λ ∈ C(λO) , and the corresponding scaling limit net as C0,ι (O) := π0,ι C(O) . Then it follows at once from the above proposition that C0,ι (O) = F0,ι (O).

4. Scaling of subsystems and Doplicher–Roberts reconstruction Inside a net of local observables, there are operators with a specific physical interpretation like the energy momentum tensor, or Noether currents associated to (central) gauge symmetries, and the relations between the given net and the subsystem generated by such operators have been thoroughly investigated in [10–12] from the point of view of Doplicher–Roberts (DR) theory. In the present context, it is therefore natural to analyse the scaling limit of such subsystems and characterize them as subsystems of the scaling limit. 4.1. General properties As a first step in this direction, in this section we deal with the following abstract situation: we consider an inclusion A ⊂ B of Haag dual and Poincaré covariant nets in the vacuum sector as defined in Section 2. We also require that the vacuum Hilbert space HB is separable. Thanks to the results in the appendix of [23] (see also the remark in Section 4 of [14]), the main results in [17] can be applied to A and B, and we further assume that for the corresponding DR canonical covariant field nets one has F(A) = F(B). Therefore for the canonical DR gauge groups one has that G(B) is a closed subgroup of G(A). Let us fix a scaling limit state ω B 0,ι of B. According to the results in [14], there exists a scaling limit F(B)0,ι of F(B) and a quotient G(B)0,ι of G(B) by a normal closed subgroup N (B)0,ι defined in analogy to (9), such that B0,ι = G(B) F(B)0,ι 0,ι . Furthermore if B0,ι satisfies Haag duality and if its vacuum Hilbert space is separable, denoting by F(B0,ι ) the canonical DR field net of B0,ι , one has that F(B)0,ι is a fixed point net of F(B0,ι ) with respect to a certain normal closed

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subgroup H(B0,ι ) of the canonical DR group G(B0,ι ). Thanks to what was shown B in Section 2, using the corresponding scaling limit state ω A 0,ι := ω 0,ι A of A we get similar relations for the nets A0,ι , F(A)0,ι and F(A0,ι ). Summarizing, we get the following result. Proposition 4.1. With the above notations, the following diagram of inclusions of nets holds: B0,ι ⊂ F(B)0,ι ⊂ F(B0,ι ) ∪ ∪ ∪ (10) A0,ι ⊂ F(A)0,ι ⊂ F(A0,ι ) Proof. As noted above, the horizontal lines follow from the results in [14], while the first column is a consequence of the discussion in Section 2. The second column is immediate from the definition of the scaling limit net and the fact that G(B) ⊂ G(A), and finally the third column follows from the first and [12]. Notice that, even if F(A) = F(B), the results of [14] do not allow to conclude that F(A)0,ι = F(B)0,ι because of the fact that the construction of F(B)0,ι depends on G(B) (and similarly for F(A)0,ι ), see [14, Def. 2.2]. For completeness we also analyse the relations between the different gauge groups that arise in diagram (10). According to [14, Sec. 2, 5] and to the previous discussion, we have groups G(A), N (A)0,ι , G(A)0,ι , G(A0,ι ) and H(A0,ι ) such that G(A)/N (A)0,ι = G(A)0,ι = G(A0,ι )/H(A0,ι ), and similarly for B. Theorem 4.2. Under the standing assumptions, we have that N (B)0,ι is a subgroup of N (A)0,ι and that there exists a morphism φ : G(B0,ι ) → G(A0,ι ) such that φ(H(B0,ι )) ⊂ H(A0,ι ), and such that the quotient morphism φ˜ on G(B0,ι )/H(B0,ι ) ˜ = G(B)/N (B)0,ι is given by φ(gN (B)0,ι ) = gN (A)0,ι . Moreover, if F(B0,ι ) = F(A0,ι ), then φ is injective, and if in addition F(B)0,ι = F(A)0,ι , then N (B)0,ι = N (A)0,ι ∩ G(B), H(B0,ι ) = H(A0,ι ), and φ˜ is injective too. Proof. As already remarked, we have F(A) ⊂ F(B) and that G(B) is a subgroup of G(A). If ω 0,ι = limκ ω λκ , it is easy to see that N (A)0,ι = g ∈ G(A) : lim βg (F λκ ) − F λκ Ω = 0, ∀F ∈ F(A)(O) , κ

O

which immediately implies the inclusion N (B)0,ι ⊂ N (A)0,ι . The existence of the morphism φ, as well as its injectivity in the case F(B0,ι ) = F(A0,ι ), are direct consequences of the application of [12, Thm. 2.3] to the commuting square of inclusions provided by A0,ι , B0,ι , F(A0,ι ) and F(B0,ι ). Since φ is given by the restriction to F(A0,ι ) of automorphisms of F(B0,ι ), and H(B0,ι ) is the subgroup of G(B0,ι ) which leaves F(B)0,ι pointwise invariant, and similarly for H(A0,ι ), it is clear that φ(H(B0,ι )) ⊂ H(A0,ι ). This also implies that φ˜ is the restriction to F(A)0,ι of automorphisms of F(B)0,ι , and therefore it coincides with the map gN (B)0,ι → gN (A)0,ι .

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We assume now that F(B)0,ι = F(A)0,ι and F(B0,ι ) = F(A0,ι ). It follows immediately that H(B0,ι ) = H(A0,ι ). We show that N (A)0,ι ∩ G(B) ⊂ N (B)0,ι , the reverse inclusion being trivial. Let g ∈ N (A)0,ι ∩ G(B), i.e. g ∈ G(B) and lim βg (F λκ ) − F λκ Ω = 0 , ∀F ∈ F(A)(O) . κ

O

Then, if F ∈ F(B)(O) we can find a norm-bounded sequence F n ∈ F(A)(O) such that π0,ι (F n ) converges strongly to π0,ι (F ) as n → +∞. We have then lim βg (F λκ ) − F λκ Ω ≤ π0,ι β g (F − F n ) Ω0 + π0,ι (F − F n )Ω0 , κ

which, together with the fact that β g is unitarily implemented in π0,ι , readily gives g ∈ N (B)0,ι .The injectivity of φ˜ then clearly follows from N (A)0,ι ∩ G(B) = N (B)0,ι . 4.2. Field nets with trivial superselection structure in the scaling limit Until now we have employed the minimal set of assumptions on the scaling limit nets which allow us to make sense of the elements in diagram (10). In order to proceed further in the discussion of its properties, it is useful at this point to apply the general machinery that has recently become available in the theory of subsystems, which requires some rather natural additional assumptions on the scaling limit nets, see Definition 4.6. Partial results on the problem of deriving such assumptions from suitable hypotheses on the underlying nets at scale λ = 1 are discussed below in this section. We hope to give a more thorough analysis of these issues somewhere else. Below, we present some examples which corroborate the natural conjecture that, at least in typical cases, we have an equality in the last column of diagram (10). In Subsection 4.3, we outline a strategy for proving that F(B0,ι ) = F(A0,ι ). Thanks to Theorem 4.7, the main point will be to show that A0,ι = G(A) F(B)0,ι . Example 4.3. Let B be the net generated by a G-multiplet of massive free scalar fields. Then F(B) = B and G(B) is trivial [18]. Let also A = BG , so that F(A) = B = F(B) and G(A) = G. From the arguments in [15] it is possible to prove that, for each scaling limit state of B, F(A)0,ι = F(B)0,ι , and therefore diagram (10) trivially reduces to B0,ι = F(B)0,ι = F(B0,ι ) ∪ A0,ι ⊂ F(A)0,ι = F(A0,ι ) The equality F(A)0,ι = F(B)0,ι is obtained in the following way: one first observes that F(B)0,ι B(0) , the net generated by a corresponding G-multiplet of massless free scalar fields transforming under the same representation [9,15]. We recall that in the non-standard free field representation used in [15] (see also Section 2), for each double cone O based on the time zero plane one has B(0) (O) = B(O). Then,

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for each such double cone O and each DR G-multiplet ψj ∈ B(0) (O), we define, for a continuous compactly supported function h on R4 , dx h(x)αλx δλ (ψj ) , (αh ψj )λ := R4

so that αh ψj ∈ F(A)(O1 ) for suitable O1 ⊃ O. One then shows, using the same arguments as in the proof of [9, Thm. 3.1], that π0,ι (αh ψj ) ∈ F(A)0,ι (O1 ) converges strongly to ψj as h → δ, and therefore, by outer regularity, ψj ∈ F(A)0,ι (O), which entails F(B)0,ι ⊂ F(A)0,ι . The converse inclusion being trivial, the conclusion follows. Example 4.4. The equality F(B0,ι ) = F(A0,ι ), which holds in the above example, can be deduced under suitable assumptions from the fact that F(B)0,ι = F(A)0,ι , as shown e.g. by Theorem 4.7 and Remark 4.8. The latter condition is trivially satisfied if for instance G(B) is open in G(A), or if [G(A) : G(B)] is finite. A discussion of more general conditions under which this is true seems to be of independent interest but for the time being it will be postponed. Example 4.5. Suppose that F is a dilatation covariant graded-local net satisfying the Haag–Swieca compactness condition. Since F is considered to have a trivial gauge group, it is net-isomorphic to any of its scaling limit F0,ι through φ π0,ι (F ) = s- lim δλ−1 (F λκ ) , (11) κ κ

where (δλ )λ>0 are dilatations on F, see [8, Prop. 5.1] (in this reference only observable nets are considered, but the generalization to nets having normal commutations relations is not difficult). Assume now that F = F(A) is obtained as the DR field net of an observable net A, with gauge group G = G(A). Of course the scaling limit F(A)0,ι of F(A) satisfies F(A)0,ι ⊂ F0,ι . We show that the converse inclusion also holds. It suffices to show that π0,ι (F(O)) ⊂ π0,ι (F(A)(O)) , where π0,ι is, as before, the scaling limit representation of F. Consider then F ∈ F(O) and F = φ(π0,ι (F )) ∈ F(O). Defining F˜ ≡ βψ (F ) := G dg ψ(g)βg (F ), where ˜ = δλ (F˜ ), we have F ˜ ∈ F(A)(O) and φ(π0,ι (F ˜ )) = F˜ converges ψ ∈ C(G), and F λ strongly to F as ψ → δ. Therefore, φ being spatial, we obtain the desired inclusion and F(A)0,ι = F0,ι . Finally, if we also assume that F = F(A) = F(B), with A ⊂ B, it is possible to show that F(A)0,ι = F(B)0,ι = F(B0,ι ) = F(A0,ι ) F. To this end, we notice that the isomorphism φA between F(A)0,ι and F, defined as in (11), is just the restriction to F(A)0,ι of the analogous isomorphism φB between F(B)0,ι and F. Thus F(A)0,ι = F(B)0,ι F and the conclusion follows from the isomorphisms A A0,ι , B B0,ι . Therefore diagram (10) becomes in this case B0,ι ∪ A0,ι

⊂

F(B)0,ι ⊂ F(A)0,ι

F(B0,ι ) = F(A0,ι ) =

502



In the remaining part of this section we give a closer look at the situation in which for the nets of von Neumann algebras in the scaling limit there holds F(A)0,ι = F(B)0,ι . Actually, we discuss the seemingly more general case in which A0,ι is the fixpoint net of F(B)0,ι under a compact group action. Definition 4.6. We say that a graded-local net with gauge symmetry in the vacuum sector (F, α, β, Ω) has trivial superselection structure if (i) Ω is cyclic and separating for F(O) for each O (Reeh–Schlieder property); (ii) with Z := (I +ik)/(1+i), there holds F(O ) = ZF(O) Z ∗ for each O (twisted Haag duality); (iii) if (Δ, J) are the modular objects associated to (F(WR ) , Ω), WR being the right wedge, there holds Δit = U (ΛWR (t)), JU (Λ, a)J = U (jΛj, ja) and, for each O, JF(O)J = F(jO), where ΛWR is the one parameter group of boost leaving WR invariant and j is the reflection with respect to the edge of WR (geometric modular action); ¯ 1 ⊂ O2 there exists a type I factor (iv) for each pair of double cones O1 , O2 with O NO1 ,O2 such that F(O1 ) ⊂ NO1 ,O2 ⊂ F(O2 ) (split property); (v) there exists at most one fermionic irreducible DHR representation with finite statistics π of FZ2 inequivalent to the vacuum, and such that every DHR representation of FZ2 is the direct sum of copies of the vacuum representation and of π (triviality of the superselection structure). For a discussion of the above properties, in particular the last one, we refer the reader to [10, 11]. Theorem 4.7. Let A ⊂ B be an inclusion of local nets satisfying the standing assumptions, and suppose furthermore that F(B0,ι ) satisfies the properties (i)–(v) in Definition 4.6, and that A0,ι = F(B)Q 0,ι for some compact group Q of internal symmetries of the net F(B)0,ι . Then F(A0,ι ) = F(B0,ι ), and therefore A0,ι is the fixpoint net of F(B0,ι ) under a compact group of internal symmetries. Proof. As a first step, we show that F(B)0,ι ⊂ F(A0,ι ). To this end, we observe that, since A0,ι = F(B)Q 0,ι , F(B)0,ι inside F(B0,ι ) is generated (by A0,ι and) by the Hilbert spaces of isometries in F(B0,ι ) implementing the cohomological extensions to B0,ι of the covariant DHR sectors of A0,ι corresponding to the irreducible representations of the compact group Q (see [12, 17]). On the other hand, the copy of F(A0,ι ) in F(B0,ι ) is generated by A0,ι and the Hilbert spaces of isometries implementing the cohomological extensions to B0,ι of the covariant DHR endomorphisms with finite statistics of A0,ι [12, Thm. 3.5]. Therefore we obtain that B0,ι ⊂ F(A0,ι ) and then the conclusion follows immediately from [17, Thm. 3.6.a)] and [11, Thm. 3.4]. Remark 4.8. According to the general discussion at the beginning of this section, the condition A0,ι = F(B)Q 0,ι is automatically satisfied for Q = G(A)0,ι if F(A)0,ι = F(B)0,ι .

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It would be interesting to know conditions on B (and on ω 0,ι ) which guarantee that F(B0,ι ) satisfies the assumptions in Definition 4.6. It follows from the discussion in [14] that, since we already assumed that B0,ι satisfies Haag duality, assumptions (ii) and (iii) for F(B0,ι ) can be deduced from analogous assumptions on B. It is also reasonable to expect that suitable nuclearity requirements on B imply assumption (iv) for F(B0,ι ), see also the paragraph following Proposition 4.13. For what concerns assumption (i), the Reeh–Schlieder property in the scaling limit can be deduced for the algebras B0,ι (W ) associated to wedges. Finally, Theorem 4.7 of [12] allows to deduce property (v) for F(B0,ι ) from the absence of sectors with infinite statistics for B0,ι , however it is not clear how to obtain the latter property from the properties of B. 4.3. Convergent scaling limits We now turn to the discussion of the validity of the equality A0,ι = F(B)Q 0,ι , where actually Q will be a closure of G(A) in a suitable topology, and we provide a sufficient condition for it which has some conceptual flavour. In order to do this we appeal to the notion of convergent scaling limit introduced in Section 2, which is suggested by the experience with models in the perturbative approach to QFT, where there is usually no need of generalized subsequences in calculating the scaling limit of vacuum expectation values. We start by showing that, under the additional assumption that G(B) is a normal subgroup of G(A), the action of G(A) lifts to the scaling algebra F(B) and to each scaling limit theory F(B)0,ι . For simplicity, we also assume that the geometrical symmetry group Γ coincides with the translations group, but the arguments below carry over to more general choices. Lemma 4.9. Let A ⊂ B be an inclusion of Haag dual local nets with HB separable and with G(B) normal in G(A). Then the equation β γ (F )λ := βγ (F λ ) ,

γ ∈ G(A) ,

F ∈ F(B) ,

λ > 0,

defines an automorphic action of G(A) on F(B), which is unitarily implemented in the representation π0,ι corresponding to any scaling limit state ω 0,ι . Proof. Let F ∈ F(B)(O) and γ ∈ G(A), g ∈ G(B). We get sup βg βγ (F λ ) − βγ (F λ ) = sup βγ −1 gγ (F λ ) − F λ = β γ −1 gγ (F ) − F , λ>0

λ>0

−1

and, since γ gγ ∈ G(B) by assumption, the right hand side converges to zero as g → e; analogously sup αλx βγ (F λ ) − βγ (F λ ) = αx (F ) − F , λ>0

converges to zero as x → 0. Therefore we get β γ (F ) ∈ F(B)(O), and obviously γ ∈ G(A) → β γ ∈ Aut(F(B)) is a group homomorphism, albeit not pointwise norm continuous. Let then ω 0,ι be a scaling limit state of F(B). Since ω ◦ βγ = ω, it

504



follows ω 0,ι ◦β γ = ω 0,ι , and thus there exists a (not strongly continuous in general) unitary representation γ → V0,ι (γ) on HF(B)0,ι such that AdV0,ι (γ)(π0,ι (F )) = π0,ι (β γ (F )), where π0,ι is the scaling limit representation associated to ω 0,ι . Theorem 4.10. Let A ⊂ B be an inclusion of local nets satisfying the standing assumptions with G(B) normal in G(A). Moreover, suppose that B has a convergent ˆ ⊂ F(B) is globally invariant with respect to scaling limit and that the algebra B G(A) the action of G(A) defined in Lemma 4.9. Then A0,ι = F(B)0,ι . Proof. The inclusion A0,ι (O) ⊆ F(B)0,ι (O)G(A) is trivial: given A = π0,ι (A), A ∈ A(O), we obviously have Aλ ∈ F(B)(λO)G(A) and therefore, by definition of β, β γ (A) = A, which entails A ∈ F(B)0,ι (O)G(A) . In order to prove the converse inclusion, let F ∈ F(B)0,ι (O)G(A) , and note ˆ such that that, since F(B)0,ι (O)G(A) ⊂ B0,ι (O), we can choose elements F n ∈ B π0,ι (F n ) converges strongly to F as n → +∞. We define An,λ := dγ γF n,λ γ −1 , G(A)

where the integral is defined in the weak topology. It is plain that An,λ ∈ A(λO) and An,λ ≤ F n , and furthermore −1 αλx (An,λ ) − An,λ = dγ γ αλx (F n,λ ) − F n,λ γ ≤ αx (F n ) − F n , G(A) which gives An ∈ A(O). Moreover, for fixed n ∈ N, it is possible to find a sequence (λm )m∈N such that π0,ι (F ) − π0,ι (A ) Ω0,ι = lim dγ F n,λm − γF n,λm Ω n n m→+∞ G(A) (12) ≤

lim

m→+∞

G(A)

dγ (F n,λm − γF n,λm )Ω .

ˆ so that, for each γ ∈ G(A), ˆ β γ (F ) ∈ B Thanks to the G(A)-invariance of B, n there holds ∗ lim (F n,λ − γF n,λ )Ω2 = lim ω F n,λ − β γ (F n )λ F n,λ − β γ (F n )λ λ→0 λ→0 2 (13) = π0,ι (F n ) − AdV0,ι (γ) π0,ι (F n ) Ω0,ι 2 ≤ 4 π0,ι (F n ) − F Ω0,ι . Therefore, by applying Lebesgue’s dominated convergence theorem to (12), we conclude that π0,ι (F n ) − π0,ι (An ) Ω0,ι ≤ 2 π0,ι (F n ) − F Ω0,ι , which, together with the fact that Ω0,ι is separating for the local algebras, gives us that π0,ι (An ) converges strongly to F as n → +∞.

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Corollary 4.11. Under the same assumptions as in Theorem 4.10, and assuming that F(B0,ι ) satisfies the properties (i)–(v) in Definition 4.6, there holds F(A0,ι ) = F(B0,ι ), and A0,ι is the fixpoint net of F(B0,ι ) under a compact group of internal symmetries. Proof. If Q denotes the closure, in the strong operator topology on HF(B)0,ι , of the group of unitaries {V0,ι (γ) : γ ∈ G(A)}, it is an easy consequence of Theorem 4.10 that A0,ι = F(B)Q 0,ι . Furthermore F(B)0,ι ⊂ F(B0,ι ) satisfies the split property [11, Sec. 2], and therefore Q is compact [16], so that Theorem 4.7 gives the statement. The fact that the scaling limit is convergent has been checked in [2] for the theory of a single massive free scalar field. The G(A)-invariance condition used in the above theorem can possibly be shown for the theory of a multiplet of free scalar fields in the following way. If F is the field net generated by such a multiplet, it should be possible, using the same techniques as in [2], to construct a ˆ ⊂ F which is globally invariant under G(A) = Gmax , the maximal C∗ -subalgebra F group of internal symmetries of F,5 and on which a given normal subgroup G(B) ⊂ ˆ G(B) ⊂ B, one has that B ˆ is ˆ := F G(A) acts strongly continuously. Then if B ˆ G(A)-invariant thanks to the normality of G(B) in G(A). Moreover, B has the two properties required in the definition of a convergent scaling limit. It is plain ˆ as this holds for F, ˆ while the that there exists limλ→0 ω(B λ ) for each B ∈ B, ˆ ˆ by = B0,ι (O) follows from the analogous property for F property π0,ι (B(O)) averaging in the usual way with respect to the strongly continuous action of G(B). Finally, we notice that the fact that the scaling limit is convergent for the free scalar field depends on the nuclearity properties of the theory. Without the assumption of a convergent scaling limit, the above proof breaks down because of the necessity of interchanging the integral on G(A) in equation (12) with the limit along a generalized sequence (λκ )κ , which is not guaranteed under the present conditions. One can only speculate that additional assumptions (e.g. nuclearity) may provide further insight on this issue. Anyway, if one cannot take the limit under the integral sign in the above discussion, we have to leave open G(A) the possibility that A0,ι F(B)0,ι , in which case we are left with two mutually exclusive possibilities: either there exists some compact group Q “larger” than (the strong operator closure of) G(A) acting on F(B)0,ι such that A0,ι = F(B)Q 0,ι , or there is no such group. In the former case the principle of gauge invariance is restored at the price of “enlarging” the group G(A). As an illustration of the physical meaning of such situation, consider the particular case in which A = FG (i.e. G(B) is trivial and G = G(A), F = F(A) = F(B) = B): then the existence of Q would mean that A0,ι is the fixed point net of the “wrong” scaling limit field net F0,ι , defined without any reference to the action of G, and would imply that it is possible to create “new” sectors of A0,ι by looking at the scaling limit of states 5G ∗ max is the group of unitaries U on HF such that U F(O)U = F(O), U U (γ) = U (γ)U for each γ ∈ Γ, and U Ω = Ω.

506



where a region of radius λ contains an amount of charge which grows unboundedly as λ → 0. Such sectors should however not be regarded as confined, as they could be created by performing suitable operations at finite, albeit small, scales. A thorough analysis of the structure of such sectors is of considerable interest in itself, and would require going beyond the framework of [14]. 4.4. On the scaling of Noether currents As an application of the above results, we discuss the scaling limit of nets generated by local implementations of symmetries [7]. Let B be a local net and let A ⊂ B be the dual of the net generated by the local implementations of translations of F(B). The validity of the equality A = F(B)Gmax , where Gmax is the maximal group of internal symmetries of F(B), has been thoroughly discussed in [10, 11]. Theorem 4.12. Let A ⊂ B satisfy the standing assumptions, where A is the dual of the net generated by the canonical local implementers of the translations of F(B), and suppose furthermore that F(B0,ι ) satisfies the properties (i)–(v) in Definition 4.6, that G(B) is normal in Gmax , and that B has a convergent scaling limit ˆ is Gmax -invariant. Let A˜0,ι be the dual of the net generated by the such that B local implementations of translations of F(B0,ι ). Then A˜0,ι ⊂ A0,ι . Proof. Thanks to Corollary 4.11, one has F(A0,ι ) = F(B0,ι ), and A0,ι = ˜ max the maximal group of internal symmetries F(B0,ι )G(A0,ι ) , and therefore, with G ˜ G of F(B0,ι ), A˜0,ι = F(B0,ι ) max ⊂ A0,ι . In short, the above result states that the scaling limit of the net generated by the local energy-momentum tensor contains the net generated by the local energymomentum tensor of the scaling limit.6 It is likely that in favourable circumstances A˜0,ι = A0,ι . This is trivially illustrated by the example of the fixpoints of the free field net discussed in the previous section. In the case in which the scaling limit of B is not convergent, one has to look back at Theorem 4.7. In turn, one should be able to show by similar methods an analogous result for more general Noether currents, cf. [11]. An issue that should be taken into account is the fact that the split property is not necessarily preserved in the scaling limit. This somehow unpleasant feature, although strictly speaking ruled out by our assumptions, can partly justify at a heuristic level the possibly strict inclusion of nets that we obtained in Theorem 4.12. In fact, in that case one cannot even define the local implementers of the scaling limit although it still makes sense to consider the scaling limit of the net generated by the Noether charges of the original theory. Examples of local nets satisfying the split property but whose scaling limit does not satisfy it can be easily found. 6 This conclusion is supported by some preliminary calculations performed directly on the universal localizing maps that are used to construct the canonical local implementers.

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Proposition 4.13. Let B be the dual of the local net generated by a generalized free field with a mass measure dρ(m) = i δ(m − mi ), such that e−γmi < ∞ (14) i

for all γ > 0. Then the split property holds for B but for none of its scaling limit nets B0,ι . Proof. Since condition (14) implies i m4i e−δmi < ∞ for each δ > 0, the split property for B follows from [16, p. 529] and [13, Cor. 4.2]. By [22, Thm. 4.1] each scaling limit net B0,ι (contains a subnet that) does not satisfy the Haag–Swieca compactness condition, and thus, by [6, Prop. 4.2], it does not satisfy the split property either. A related problem is to provide conditions on B ensuring that F(B0,ι ) enjoys the split property. By the results in [5, Cor. 4.6], some conditions on B0,ι are known to imply suitable nuclearity properties of F(B0,ι ), which in turn imply the split property by [6, Sec. 4]. On the other hand, the methods employed in [4, Thm. 4.5] to prove nuclearity properties of the scaling limit theory B0,ι starting from certain phase space behaviour of the underlying theory B can possibly be adapted to show that (some of) the conditions on B0,ι considered in [5] follow from appropriate nuclearity requirements on B. 4.5. Preserved sectors The notion of preserved DHR sector has been introduced in [14, Def. 5.4]. In the spirit of the present paper a natural question concerns the relation between the preservation of DHR morphisms with finite statistics of A and B. Clearly, the cohomological extension property of morphisms plays again a crucial role. In particular, it is reasonable to expect that if all the morphisms of B are preserved, then the same will be true for the morphisms of A, since the Hilbert spaces of isometries in F(B) implementing the cohomological extension to B of a given morphism of A would satisfy the preservation condition. Possible applications of such result include a generalization of the theorem in the previous section, where we replace F(A)0,ι and F(B)0,ι with the subnets generated by the isometries associated to the (scaling limits of the) preserved sectors, which should be automatically independent of the gauge groups, and therefore coincide. However, what is missing in the above argument is the fact that the analysis in [14] has been carried out only for irreducible morphisms (while the extension maps in general irreducible morphisms to reducible ones) and, although there is no apparent obstruction for extending it to the reducible case, in the reminder of this short section we will limit ourselves to some simple remarks, postponing a thorough analysis of this point to future work. We consider the situation outlined at the beginning of Section 4.1, i.e. an inclusion A ⊂ B of Haag dual and Poincaré covariant nets in the vacuum sector

508



A B with F = F(A) = F(B), and scaling limit states ω B 0,ι of B and ω 0,ι = ω 0,ι A of A. In the following result we make use of the notion of asymptotic containment, introduced in [14, Def. 5.2].

Proposition 4.14. Let ξ be a ω A 0,ι -preserved class of DHR morphisms of A, and let ψj (λ) ∈ F(λO) be an associated scaled multiplet which is asymptotically contained in F(A). Then the cohomological extension ξˆ of ξ to B is ω B 0,ι -preserved, with ψj (λ) an associated scaled multiplet asymptotically contained in F(B). Proof. If ρλ is the DHR morphism of A in the class ξ localized in λO implemented ˆ still by the multiplet ψj (λ), then its cohomological extension ρˆλ is in the class ξ, localized in λO and also implemented by ψj (λ). Now, since F(A) ⊂ F(B), it is immediate to conclude that ψj (λ) is asymptotically contained in F(B), and therefore ξˆ is ω B 0,ι -preserved. We define F(A)pres 0,ι as the net generated by A0,ι and the scaling limits of scaled multiplets asymptotically contained in F(A) associated to ω A 0,ι -preserved sectors of A, see Prop. 5.5 in [14]. Likewise we define F(B)pres 0,ι with respect to the scaling limit state ω B 0,ι . Corollary 4.15. With the above notations, there holds A0,ι ∩ B0,ι

⊂ F(A)pres 0,ι ∩ ⊂ F(B)pres 0,ι

⊂ F(A)0,ι ∩ ⊂ F(B)0,ι

pres In some cases one has F(A)pres 0,ι = F(B)0,ι . For free fields this follows from Example 4.3 and the fact that all sectors of the fixpoint net of the free field are preserved. Another example is given by a dilation invariant theory satisfying the Haag–Swieca compactness condition, where it follows easily from the results pres of [8, 14], that F(A)pres 0,ι = F(B)0,ι = F(A)0,ι = F(B)0,ι = F.

5. Final comments We end this paper with few comments on further possible extension of the results presented above, in addition to those already mentioned in the main body. Given a subsystem A ⊂ B as in Section 4, we assumed that F(A) = F(B). However in general it holds F(A) ⊂ F(B) [12] and, in the situation considered in [10, 11], it is shown that F(A) = F(B) ⊗ C (graded tensor product) for a suitable net C . Therefore, in order to treat this more general framework, one should generalize the results about the scaling limit of tensor product theories in [15]. Another natural example of subsystem, to which most of our results don’t apply, is provided by the inclusion A ⊂ Ad of a net into its dual, a situation that arises typically when there are spontaneously broken symmetries. The analysis of the structure of such subsystems in the scaling limit has some interest as it could

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possibly simplify the study of the relations between the superselection structures of A and of A0,ι . For instance, sufficient conditions on A which imply essential duality, but not duality, of A0,ι are known, so it would be interesting to know when the scaling limit of Ad coincides with the dual of A0,ι . We conclude by mentioning few very intriguing but rather speculative ideas. In [14] it has been shown that it is possible to formulate conditions on the scaling limit of a theory which imply the equality of local and global intertwiners. There are other long-standing structural problems in superselection theory that could hopefully be related one way or another to the short distance properties of the theory. Just to give some example, we cite here the problem of recovering pointlike Wightman fields with specific physical interpretation out of local algebras (i.e., a full quantum Noether theorem), and that of ruling out the existence of sectors with infinite statistics.

Acknowledgements G. Morsella would like to thank the Scuola Normale Superiore for a grant that made the final stages of this collaboration possible, and the University of Newcastle for the kind hospitality extended to him. R. Conti thanks the Scuola Normale Superiore for supporting a short stay in Pisa.

References [1] H. Bostelmann, C. D’Antoni, G. Morsella, Scaling algebras and pointlike fields. A nonperturbative approach to renormalization, Comm. Math. Phys. 285 (2009), no. 2, 763–798. [2] H. Bostelmann, C. D’Antoni, G. Morsella, arXiv:0812.4762, to appear on Comm. Math. Phys. [3] D. Buchholz, Quarks, gluons, colour: Facts or fiction?, Nucl. Phys. B 469 (1996), no. 1–2, 333–353. [4] D. Buchholz, Phase space properties of local observables and structure of scaling limits, Ann. Inst. H. Poincaré Phys. Theor. 64 (1996), no. 4, 433–459. [5] D. Buchholz, C. D’Antoni, Phase space properties of charged fields in theories of local observables, Rev. Math. Phys. 7 (1995), no. 4, 527–557. [6] D. Buchholz, C. D’Antoni, R. Longo, Nuclear maps and modular structures. II. Applications to quantum field theory, Comm. Math. Phys. 129 (1990), no. 1, 115– 138. [7] D. Buchholz, S. Doplicher, R. Longo, On Noether’s theorem in quantum field theory, Ann. Phys. 170 (1986), no. 1, 1–17. [8] D. Buchholz, R. Verch, Scaling algebras and renormalization group in algebraic quantum field theory, Rev. Math. Phys. 7 (1995), no. 8, 1195–1239. [9] D. Buchholz, R. Verch, Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples, Rev. Math. Phys. 10 (1998), no. 6, 775–800.

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[10] S. Carpi, R. Conti, Classification of subsystems for local nets with trivial superselection structure, Comm. Math. Phys. 217 (2001), no. 1, 89–106. [11] S. Carpi, R. Conti, Classification of subsystems for graded-local nets with trivial superselection structure, Comm. Math. Phys. 253 (2005), no. 2, 423–449. [12] R. Conti, S. Doplicher, J. E. Roberts, Superselection theory for subsystems, Comm. Math. Phys. 218 (2001), no. 2, 263–281. [13] C. D’Antoni, S. Doplicher, K. Fredenhagen, R. Longo, Convergence of local charges and continuity properties of W ∗ -inclusions, Comm. Math. Phys. 110 (1987), no. 2, 325–348. [14] C. D’Antoni, G. Morsella, R. Verch, Scaling algebras for charged fields and shortdistance analysis for localizable and topological charges, Ann. Henri Poincaré 5 (2004), no. 5, 809–870. [15] C. D’Antoni, G. Morsella, Scaling algebras and superselection sectors: Study of a class of models, Rev. Math. Phys. 18 (2006), no. 5, 565–594. [16] S. Doplicher, R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Math. 75 (1984), no. 3, 493–536. [17] S. Doplicher, J. E. Roberts, Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Comm. Math. Phys. 131 (1990), no. 1, 51–107. [18] W. Driessler, Duality and absence of locally generated superselection sectors for CCRtype algebras, Comm. Math. Phys. 70 (1979), no. 3, 213-220. [19] R. Haag, Local quantum physics, IInd edition, Springer-Verlag, Berlin, 1996. [20] R. Longo, K.-H. Rehren, Nets of subfactors, Rev. Math. Phys. 7 (1995), no. 4, 567– 597. [21] M. Lutz, Ein lokales Netz ohne Ultraviolettfixpunkte der Renormierungsgruppe, diploma thesis, Hamburg University (1997). [22] S. Mohrdieck, Phase space structure and short distance behaviour of local quantum field theories, J. Math. Phys. 43 (2002), no. 7, 3565–3574. [23] J. E. Roberts, Localization in algebraic field theory, Comm. Math. Phys. 85 (1982), no. 1, 87–98. [24] E. H. Wichmann, On systems of local operators and the duality condition, J. Math. Phys. 24 (1983), no. 6, 1633–1644. [25] K. Yosida, Functional analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer Verlag, Berlin, 1995.

Roberto Conti School of Mathematical and Physical Sciences The University of Newcastle NSW 2308 Australia e-mail: [email protected]

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Gerardo Morsella Scuola Normale Superiore Piazza dei Cavalieri, 7 I-56126 Pisa Italy e-mail: [email protected] Present address of both authors: Dipartimento di Matematica Universit` a di Roma 2 Tor Vergata via della Ricerca Scientifica I-00133 Roma Italy e-mail: [email protected] [email protected] Communicated by Klaus Fredenhagen. Submitted: August 29, 2008. Accepted: March 23, 2009.

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Ann. Henri Poincaré 10 (2009), 513–548 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/030513-36, published online May 22, 2009 DOI 10.1007/s00023-009-0416-x


Bethe Ansatz for the Universal Weight Function Luc Frappat, Sergey Khoroshkin, Stanislav Pakuliak and ´ Eric Ragoucy Abstract. We consider universal off-shell Bethe vectors given in terms of Drin ) [10, 12]. We investigate ordering propfeld realization of the algebra U q (gl N erties of the product of the transfer matrix and these vectors. We derive that these vectors are eigenvectors of the transfer matrix if their Bethe parameters satisfy the universal Bethe equations [1].

1. Introduction Algebraic Bethe ansatz for quantum integrable models with gl2 symmetry as well as hierarchical Bethe ansatz for models with higher rank symmetries solves the eigenvalue problem for the set of the commuting quantum integrals of motion. The eigenvectors in these methods are built from the matrix elements of the monodromy matrix which satisfies Yang–Baxter relation defined by the quantum Rmatrix. Quantum integrable models solvable by these methods correspond to the different monodromy matrices and quantum R-matrices. Monodromy matrices can be obtained by considering the different representations of the quadratic algebras which have the same type of defining relations as monodromy matrices have. In this case one can use the generating series of the elements of this quadratic algebra as kind of the universal monodromy matrix and try to reformulate the Bethe ansatz on the universal level without specification to any concrete representations or concrete integrable model. Such an approach to find the eigenvalues for the quantum integrable models with different boundary conditions and symmetries was elaborated in [1] using certain analytical assumptions on the structure of these eigenvalues. This method is called analytical Bethe ansatz and by construction was unable to yield an information on the structure of the corresponding eigenvectors. In case of the models with glN symmetry the method to build the eigenvectors from the matrix elements

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of the monodromy matrices was designed in [15, 18–20] generalizing an approach of the hierarchical Bethe ansatz formulated in [14]. Authors of these papers used the universal monodromy which satisfies the commutation relations of the glN ) to construct Yangian or Borel subalgebra in the quantum affine algebra Uq (gl N the universal Bethe vectors of the hierarchical Bethe ansatz in terms of the matrix elements of the corresponding fundamental L-operators. Quantum affine algebra and their rational analogs – the Yangian doubles – have an alternative to the L-operator description [17] in term of currents [2, 5]. For the quantum affine algebras, it was proved in [11] that the modes of Drinfeld currents coincide with Cartan–Weyl generators of these algebras constructed from the finite set of Chevalley generators. One may address the question whether it is possible to construct the universal Bethe vectors from the current generators of the quantum affine algebra which serves as symmetry for some quantum integrable models. This problem was investigated in [7] on a rather general level and it was shown that in order to build the universal Bethe vectors from Cartan–Weyl or current generators of the algebra one has to consider different types of Borel subalgebras in the quantum affine algebras. In this paper it was suggested to construct universal off-shell Bethe vectors for arbitrary quantum affine algebra as certain projections of products of the currents onto intersections of Borel subalgebras of different types. The generating parameters of the currents become after this identification the Bethe parameters. The papers [12, 16] contain detailed analysis of ). In particular, these projecthese projections for quantum affine algebra Uq (gl N tions are explicitly expressed via entries of the fundamental monodromy matrix and are identified with off-shell Bethe vectors of the nested Bethe ansatz [14]. An algebraic Bethe ansatz always uses a special vector which is annihilated by some ideal in the symmetry algebra (bare vacuum) and Bethe vectors are obtained by the application of the universal Bethe vectors to this vector. From the representation theory point of view we will call such bare vacuum a weight singular vector. The Cartan–Weyl generators have a good property: their products can be ordered in a natural way. If we are able to express the commuting integrals as well as the universal Bethe vectors in terms of these generators, we may rise the question: what is special in the universal Bethe vectors if their Bethe parameters satisfy the universal Bethe equations. In this paper we address this question for the N ). We found that the Cartan–Weyl ordering of the quantum affine algebra U q (gl product of the universal transfer matrix and the universal Bethe vector produces the same Bethe vectors modulo elements of the ideal which annihilates the bare vacuum if the Bethe parameters satisfy the universal Bethe equations from [1]. All our calculations are performed on the level of generating series and the main technical trick which helps us to perform the ordering calculations is the identity (3.30) which is a particular case of more general relations between offshell Bethe vectors obtained in the paper [13] using the technique of the generating series. The paper is composed as follows. In Section 2, all necessary statements for N ) are given and the the different realizations of the quantum affine algebra U q (gl

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main assertion of the paper is formulated as the Theorem 1. Section 3 collects propositions which describe the ordering of the generating series of the Cartan– Weyl or current generators. There, an identity (3.30) is formulated: it is a new type of hierarchical relation between universal off-shell Bethe vectors expressed in terms of the current generators. Section 4 is devoted to the inductive proof of the main Theorem 1.

2. Basic algebraic structures ) as a quantum double 2.1. U q (gl N Let q be a complex parameter not equal to zero or to a root of unity. Let Eij ∈ End(CN ) be a matrix with the only nonzero entry equal to 1 at the intersection of the i-th row and j-th column. Let R(u, v) ∈ End(CN ⊗ CN ) ⊗ C[[v/u]], u−v Eii ⊗ Eii + (Eii ⊗ Ejj + Ejj ⊗ Eii ) R(u, v) = −1 qu − q v 1≤i≤N 1≤i<j≤N (2.1) q − q −1 + (uE ⊗ E + vE ⊗ E ) ij ji ji ij qu − q −1 v 1≤i<j≤N

be a trigonometric R-matrix associated with the vector representation of glN . ) with unit as a quantum We will consider an associative algebra U q (gl N double [4] of its Borel subalgebra generated by the modes L+ i,j [k], k ≥ 0, 1 ≤ ∞ N + + i, j ≤ N of the L-operators L (z) = k=0 i,j=1 Eij ⊗ Li,j [k]z −k , L+ j,i [0] = 0, 1 ≤ i < j ≤ N subject to the relations R(u, v) · L+ (u) ⊗ 1 · 1 ⊗ L+ (v) = 1 ⊗ L+ (v) · L+ (u) ⊗ 1 · R(u, v) (2.2) with a standard coproduct N + Δ L+ L+ i,j (u) = k,j (u) ⊗ Li,k (u) .

(2.3)

k=1

) and call it a standard Borel subalWe denote this subalgebra Uq (b+ ) ⊂ U q (gl N N gebra of U q (glN ). In (2.2) 1 = i=1 Eii . N ) is generated According to the general theory [4] the whole algebra U q (gl by the modes of the L-operator L+ (z) and by the modes of the dual L-operator −∞ N − −k , L− L− (z) = i,j [0] = 0, 1 ≤ i < j ≤ N . The dual k=0 i,j=1 Eij ⊗ Li,j [k]z Borel subalgebra Uq (b− ) has the same algebraic and coalgebraic properties (2.2) and (2.3) with L+ (z) replaced by L− (z) everywhere. The commutation relation between opposite Borel subalgebras R(u, v) · L+ (u) ⊗ 1 · 1 ⊗ L− (v) = 1 ⊗ L− (v) · L+ (u) ⊗ 1 · R(u, v) , (2.4) can be calculated using the non-degenerated pairing between these subalgebras.

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N ) with dropped The standard description of the quantum affine algebra Uq (gl gradation element and at vanishing central element can be obtained from the ) by imposing one more relation algebra U q (gl N − L+ i,i [0]Li,i [0] = 1 i = 1, . . . , N .

(2.5)

Here we shall not assume this restriction. We shall require only invertibility of the zero modes of the diagonal matrix elements of L-operators.

N) 2.2. Current realization of U q (gl ) [5] one has to introduce, To obtain the current realization of the algebra U q (gl N according to [2], the Gauss coordinates of the L-operators. There are two different ways of introducing Gauss decompositions of L-operators. Each of these two possibilities leads to hierarchical relations for the universal Bethe vectors given by the hierarchical Bethe ansatz. One type of hierarchy occurs when the smaller algebra L-operator is embedded in the upper-left corner of the bigger algebra L-operator. The second type corresponds to the embedding in the down-right corner [16, 20]. In this paper we will use the latter embedding. Then, Gauss coordinates F± b,a (t), ± ± Ea,b (t), b > a and kc (t) are given by the decomposition: ±

L (z) =

1+

N

F± b,a (z)Eab

·

N

ka± (z)Eaa

·

1+

a=1

a 0 (equivalently, by all modes of Uq (b+ ) · E+ i,j (t), 1 ≤ i < j ≤ N ). Equalities in Uq (b+ ) modulo element from the ideal J we denote by the symbol ‘ ∼J ’. It is clear that W · v = 0 for any element W ∈ J and arbitrary weight singular vector v. We call a (universal) transfer matrix the trace of L-operator ⎛ ⎞ N N N + + ⎝ki+ (t) + ⎠ TN (t) = L+ F+ (2.30) i,i (t) = j,i (t) kj (t) Ei,j (t) . i=1

i=1

F+ j,i (t),

j=i+1

E+ i,j (t)

The Gauss coordinates coincide with the projections of the corresponding composed currents and can be expressed through modes of the currents from subalgebras Uf+ and Ue+ (see section 3.1). Note that the presentation (2.30) of the transfer matrix TN (t) is normal ordered according to the circular ordering N + (2.22) and TN (t) ∼J i=1 ki (t). Definition 2.3. Let B be the left ideal of Uq (b+ ), generated by all elements of the form UF+ · b, where b are the modes of the series bij (t¯[¯n] ) =

ki+ (tij ) + ki+1 (tij )

−

ni−1 ni+1 −1 ni i i q − qti+1 q − q −1 tim /tij 1 − ti−1 m /tj m /tj . i i q −1 − qtim /tij m=1 q − q −1 ti−1 1 − ti+1 m /tj m=1 m /tj m=1 m=j

(2.31) Here i = 1, . . . , N − 1, j = 1, . . . , ni . Equalities in Uq (b+ ) modulo elements from the ideal B we denote by the symbol ‘ ∼B ’. We call this ideal the Bethe ideal and equations for the set of the Bethe parameters {tij } bij (t¯[¯n] ) = 0 , the universal Bethe equations.

i = 1, . . . , N − 1 ,

j = 1, . . . , ni ,

(2.32)

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The main statement of this paper is Theorem 1. A formal series identity with coefficients in Uq (b+ ) TN (t) · WN (t¯[¯n] ) − WN (t¯[¯n] ) · τN (t; t¯[¯n] ) ∼J,B 0

(2.33)

is valid modulo elements from the ideals J and B. Here τN (t; t¯[¯n] ) =

N

ni q − q −1 ti−1 q −1 − qtij /t j /t 1 − tij /t 1 − ti−1 j /t j=1 j=1

ni−1

ki+ (t)

i=1

(2.34)

is an eigenvalue of the universal transfer matrix. One may understand the universal Bethe equations (2.32) as a relation between Bethe parameters t¯[¯n] with coefficients in the commutative subalgebra Uk+ ⊂ Uq (b+ ) generated by the modes of the commuting Cartan currents ki+ (t). In a representation with a generating singular vector v these modes are being replaced by the corresponding eigenvalues of v. Proving the statement of Theorem 1 we will try to present the product TN (t) · WN (t¯[¯n] ) in the normal ordered form according to the ordering given in Definition 2.1. After performing this ordering we will observe that subtraction of the ordered product WN (t¯[¯n] ) · τN (t; t¯[¯n] ) results only in the terms which belong either to the ideal J or to the ideal B. For any weight singular vector v, let wVN (t¯[¯n] ) = WN (t¯[¯n] ) v be the weight N )-module V generated by v and function taking value in the U q (gl wVN (t¯[¯n] )

= β(t¯[¯n] )

N n a−1

λa (ta−1 ) wVN (t¯[¯n] )

(2.35)

a=2 =1

be the corresponding modified weight function [10]. Here β(t¯[¯n] ) =

N −1

a=1 1≤i >···>i1 >i

=1

Proof. First equality(3.5) was proved in [10] using induction with respect to j − i from the formula Pf+ Fi+1,i (t) = F+ i+1,i (t) [2]. Here we shall prove (3.6). We apply − the projection Pf to both sides of the relation (2.18) to obtain Pf− Fj,i (t) = Si Pf− Fj,i+1 (t) +(q−q −1 ) Pf− Fi+1,i (t) · Pf− Fj,i+1 (t) . (3.7) Using this relation recursively and the Lemma 3.2, we get Pf− Fj,i (t) + (q − q −1 )j−i−1 F− j,i (t) +

j−1

− (q − q −1 )j− F− ,i (t) · Pf Fj, (t) = 0 .

(3.8)

=i+1

From the identity

Pf− Fi+1,i (t) = −F− i+1,i (t)

(3.9)

one proves that equality (3.6) is a solution of this recurrence relation, and coincide with it for i = j − 1. Proceeding in analogous way, we may relate the projections of the dual composed currents Pe± (Ei,j (t)) with Gauss coordinates E± i,j (t), but here we shall need only the relations between different dual Gauss coordinates or analog of the ˆ ˆ ˆ Lemma 3.2 for E± i,j (t). Set SA (B) = AB − qBA and denote SEi [0] ≡ Si . 2 See

also the proof of the analogous Lemma 3.4 below.

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Lemma 3.4. We have ˆ ± (q − q −1 )E± i,j (t) = Si Ei+1,j (t) ,

i < j − 1.

(3.10)

Proof. Let us consider the commutation relations between the following matrix elements of L-operators

− (t − s) L± j,i+1 (t), Li+1,i (s)

± ± − = (q − q −1 ) tL− i+1,i+1 (s)Lj,i (t) − sLi+1,i+1 (t)Lj,i (s) ,

± (qt − q −1 s)L− i+1,i+1 (s)Lj,i+1 (t) − − −1 )sL± = (t − s)L± j,i+1 (t)Li+1,i+1 (s) + (q − q i+1,i+1 (t)Lj,i+1 (s) .

Choosing the coefficients at the zero power of the spectral parameter s in these − − relations and taking into account that L− i+1,i [0] = −ki+1 [0]Ei [0] and Li+1,i+1 [0] = − ki+1 [0] we obtain ± ± ˆ ± (q − q −1 )L± (3.11) j,i (t) = Ei [0]Lj,i+1 (t) − qLj,i+1 (t)Ei [0] = Si Lj,i+1 (t) . In order to obtain (3.10) we shall use an explicit expression for the matrix elements of the L-operator (2.9) in terms of the Gauss coordinates. The relation (3.11) ± (t) and Ei [0] for i = implies (3.10) for j = N due to the commutativity of kN 1, . . . , N − 2. Next, the relations (3.11) for j = N − 1 and (3.10) for j = N imply ± (3.10) for j = N − 1 due to the commutativity of the Gauss coordinates kN −1 (t), ± ± kN (t) and FN,N −1 (t) with Ei [0] for i = 1, . . . , N − 3. The statement of the lemma follows by induction over j. 3.2. Basic notations Let ¯l and r¯ be two collections of nonnegative integers satisfying a set of inequalities la ≤ ra ,

a = 1, . . . , N − 1 .

(3.12)

Denote by [¯l, r¯] a set of segments which contain positive integers {la + 1, la + 2, . . . , ra − 1, ra } including ra and excluding la . The length of each segment is equal to ra − la . For a given set [¯l, r¯] of segments we denote by t¯[¯l,¯r] the sets of variables −1 N −1 t¯[¯l,¯r] = {t1l1 +1 , . . . , t1r1 ; t2l2 +1 , . . . , t2r2 ; . . . ; tN lN −1 +1 , . . . , trN −1 } .

(3.13)

For any a = 1, . . . , N − 1 we denote the sets of variables corresponding to the segments [la , ra ] = {la + 1, la + 2, . . . , ra } as t¯a[la ,ra ] = {tala +1 , . . . , tara }. All the variables in t¯a[la ,ra ] have type a. For the segments [la , ra ] = [0, na ] we use the shorten notations t¯[¯0,¯n] ≡ t¯[¯n] and t¯a ≡ t¯a . [0,na ]

[na ]

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For a collection of variables t¯[¯l,¯r] we consider the ordered products of the currents ⎛ ⎞ ←− ←− ⎝ F(t¯[¯l,¯r] ) = Fa (ta )⎠ (3.14) N −1≥a≥1 ra ≥>la −1 1 1 = FN −1 (tN rN −1 ) · · · F1 (tr1 ) · · · F1 (tl1 +1 ) ,

where the series Fa (t) ≡ Fa+1,a (t) are defined by (2.10). As particular cases, we have F(t¯a[la ,ra ] ) = Fa (tara ) · · · Fa (tala +2 )Fa (tala +1 ). The product (3.14) is a formal series over the ratios tbk /tcl with b < c and a a ti /tj with i < j taking values in the algebra UF . −→ ←− Symbol a Aa (resp. a Aa ) means the ordered products of noncommutative entries Aa , such that Aa is on the right (resp. on the left) from Ab for b > a: ←− −→ Aa = Aj Aj−1 · · · Ai+1 Ai , Aa = Ai Ai+1 · · · Aj−1 Aj . j≥a≥i

i≤a≤j

Consider the permutation group Sn and its action on the formal series of n variables defined for the elementary transpositions σi,i+1 as follows π(σi,i+1 )G(t1 , . . . , ti , ti+1 , . . . , tn ) =

q −1 − q ti /ti+1 G(t1 , . . . , ti+1 , ti , . . . , tn ) . q − q −1 ti /ti+1

The q-depending factor in this formula is chosen in such a way that each product Fa (tn ) · · · Fa (t1 ) is invariant under this action. Summing the action over all the 1 group of permutations we obtain the operator Symu = n! σ∈Sn π(σ) acting as follows3 q −1 − q tσ( ) /tσ() 1 Sym t¯ G(t¯) = G(σ t) . (3.15) −1 t n! q − q /t σ( ) σ() σ∈Sn

σ( )

The product is taken over all pairs (, ), such that conditions < and σ() > σ( ) are satisfied simultaneously. We call the operator Symu a q-symmetrization. The operator Symu is the group average with respect to the action π, so that Sym

t¯

Sym

t¯

( · ) = Sym

t¯

(·).

An important property of q-symmetrization is the relation s!(n − s)! Sym(t1 ,...,tn ) = π(σ) Sym(t1 ,...,ts ) Sym(ts+1 ,...,tn ) , n! (s)

(3.16)

(3.17)

σ∈Sn

3 Normalization

of the q-symmetrization used here differs from the one used in the papers [12,16] 1 . by the combinatorial factor n!

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where s ∈ [0, n] is fixed and the sum is taken over the subset

! Sn(s) = σ ∈ Sn | σ(1) < · · · < σ(s) ; σ(s + 1) < · · · < σ(n) .

Denote by S[¯l,¯r] = S[l1 ,r1 ] × · · · × S[lN −1 ,rN −1 ] the direct product of the groups S[la ,ra ] permuting integer numbers la + 1, . . . , ra . The q-symmetrization over the whole set of variables t¯[¯l,¯r] is defined by the formula Sym

=

t¯[l,¯ ¯ r]

G(t¯[¯l,¯r] )

⎛

σ∈S[l,¯ ¯ r ] 1≤a≤N −1

⎞

⎜ 1 ⎜ ⎝ (ra − la )!

−1

σ a ( )

q −q taσa ( ) /taσa () ⎟ σ ⎟ G( t¯[¯l,¯r] ) , q −q −1 taσa ( ) /taσa () ⎠ (3.18)

where the set σ t¯[¯l,¯r] is defined as σ¯ t[¯l,¯r]

= {t1σ1 (l1 +1) , . . . , t1σ1 (r1 ) ; −1 −1 , . . . , tN }. t2σ2 (l2 +1) , . . . , t2σ2 (r2 ) ; . . . ; tN σ N −1 (lN −1 +1) σ N −1 (rN −1 )

(3.19)

We say that the series G(t¯[¯l,¯r] ) is q-symmetric, if it is invariant under the action π of each group S[la ,ra ] with respect to the permutations of the variables tala +1 , . . . , tra for a = 1, . . . , N − 1: Sym

G(t¯[¯l,¯r] ) t¯[l,¯ ¯ r]

= G(t¯[¯l,¯r] ) .

(3.20)

The q-symmetrization G(t¯[¯l,¯r] ) = Symt¯[l,¯ Q(t¯[¯l,¯r] ) of any series Q(t¯[¯l,¯r] ) is a q¯ r] symmetric series, which follows from (3.16). Let s¯ = {s1 , . . . , sN −1 } be a set of nonnegative integers satisfying la ≤ sa ≤ ra for a = 1, . . . , N − 1. The set of integers s¯ divides the set of the variables t¯[¯l,¯r] into two subsets t¯[¯s,¯r] ∪ t¯[¯l,¯s] . Using the property of the projections (2.25) we can present any product of the currents in a normal ordered form (in the sense of Definition 2.1): (ra − la )! ··· F(t¯[¯l,¯r] ) = (sa − la )!(ra − sa )! lN −1 ≤sN −1 ≤rN −1 l1 ≤s1 ≤r1 1≤a≤N −1 (3.21) + − ¯ ¯ ¯ Zs¯(t[¯l,¯r] ) Pf F(t[¯s,¯r] ) · Pf F(t[¯l,¯s] ) , × Sym t¯ ¯ [l,¯ r]

where Zs¯(t¯[¯l,¯r] ) =

N −2

a=1

sa m1 −ma−1

This ordered product was called the dual string in the paper [13]. Denote the negative projections of the q-symmetrized dual strings (3.26) as follows − ˜m ¯[m] S Sym t¯[m] ( t ) . (3.27) Em1 ,m2 ,...,mN −1 (t¯[m] ¯ ¯ ) = Pf ¯ ¯ Denote by Dn1 ,...,nN −1 (t¯[¯n] ) the elements of Uf− defined by the recursive relations ⎛ Sym

t¯[¯ n]

⎜ ⎜ ⎜ ⎝

···

n1 ≥m1 ≥0 nN −1 ≥mN −1 ≥0 m1 ≥···≥mN −1

⎞

⎟ ⎟ ¯ ¯ n] ) · Em1 ,...,mN −1 (t¯[m] Zm ¯ (t¯[¯ n] )Dn1 −m1 ,...,nN −1 −mN −1 (t[m,¯ ¯ )⎟ = 0 . ⎠

(3.28)

It was proved in [13] that the coefficients Dn1 ,...,nN −1 (t¯[¯n] ) defined by (3.28) are non-zero only iff n1 ≥ n2 ≥ · · · ≥ nN −1 ≥ 0 and can be defined uniquely by means

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of (3.28) from the initial condition D0,...,0 (t¯[¯n] ) = 1. In particular, D 1,...,1,0,...,0 (t1 , . . . , tN −1 ) = −E 1,...,1,0,...,0 (t1 , . . . , tN −1 ) m times

m times

=−

m−1 a=1

1 1−

ta /ta+1

Pf− Fm+1,1 (t1 ) .

(3.29)

Denote by t¯[¯s ] the following collection of the formal variables −1 −1 , . . . , tN t¯[¯s ] = {t21 , . . . , t2s2 ; . . . ; tN sN −1 } , 1

excluding the variables of type 1. We formulate without proof the following Proposition 3.5 ([13]). There is a formal series identity −1 N na ! ¯ WN (t[n] ) = Sym t¯[n] Zs¯(t¯[¯n] ) s ! s¯ a=1 a

× Dn1 −s1 ,...,nN −1 −sN −1 (t¯[¯s,¯n] ) · WN −1 (t¯[s ] ) · F1 (t1s1 ) · · · F1 (t11 ) .

(3.30)

In this paper we will need the following corollary of this proposition. Let t¯[¯n ]m be the following collection of formal variables t¯[¯n ]m = {t21 , . . . , t2n2 −1 ; . . . ; m+1 N −1 m −1 tm , . . . , tm+1 , . . . , tN 1 , . . . , tnm −1 ; t1 nm+1 ; . . . ; t1 nN −1 } .

(3.31)

Note that t¯[¯n ]1 ≡ t¯[¯n ] . Corollary 3.6. m N −1 (na ) Sym Pf+ F(t¯[¯n] ) = Pf+ F(t¯[¯n ]1 ) · F(t¯1[n1 ] ) −

t¯[¯ n]

Pf− Fm+1,1 (t1n1 )

m=1 a=1

× where Zm (t¯[¯n] ) =

m−1 a=1

Pf+

F(t¯[¯n ]m ) · F(t¯1[n1 −1] ) · Zm (t¯[¯n] ) + W ,

⎛ 1 ⎝ a 1 − tna /ta+1 na+1

na+1 −1

q − q −1 tana /ta+1 j

j=1

1 − tana /ta+1 j

⎞ ⎠

(3.32)

m+1 q − q −1 tm nm /tj

nm+1

j=1

m+1 1 − tm nm /tj

(3.33) and the terms W in (3.32) are such that Pf+ (: TN (t) · W :) = 0. Recall that F(t¯1[n1 ] ) = F1 (t1n1 ) · · · F1 (t11 ) and F(t¯1[n1 −1] ) = F1 (t1n1 −1 ) · · · F1 (t11 ). The first term in the right hand side of (3.32) corresponds to the term with all sa = na in (3.30). Each of the terms in the summation over m in (3.32) corresponds to the following values of sm in the general formula (3.30): s1 = n1 − 1, . . . , sm = nm − 1 and sm+1 = nm+1 , . . . , sN −1 = nN −1 . The corresponding elements D1,...,1,0,...,0 are given by (3.29), which brings in (3.32) the product of the −1 rational factors (1 − tana /ta+1 . Other rational factors are given by the series na+1 ) ¯ Zs¯(t[¯n] ) for these particular values of s¯.

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The general structure of the terms W which are not presented explicitly in the right hand side of (3.32) can be described as follows. The structure of the coefficients Em1 ,m2 ,...,mN −1 (t¯[m] ¯ ) (3.27) implies that these terms will have on the left the negative projections of the string S˜m ¯ (t¯[m] ¯ ) which contains, at least, the product of two currents Fc1 ,1 and Fc2 ,1 or the product of the several negative projections of the strings of type (3.27). Since the projection of the string can be always factorized to the product of the of the currents [12], the general projection structure of the terms W will be W = Pf− (Fc1 ,1 ) · Pf− (Fc2 ,1 ) · W . The elements W are some elements of UF and their exact structure is unimportant. The reason why Pf+ (: TN (t) · W :) = 0 will be explained in the next subsection. Note that the identity (3.30) can be proved directly using only the ordering relations (3.21) and the rules of calculations of the negative projections from the product of currents. Indeed, for an arbitrary product of currents F(t¯[¯n] ), these ordering relations can be written in the form Pf− (F ) · Pf+ (F ) , (3.34) Pf+ F(t¯[¯n] ) = F(t¯[¯n] ) − where the number of currents in the product F is less than in the original product F(t¯[¯n] ). Thus, one can replace recursively the positive projection Pf+ (F ) by the right hand side of the relation (3.34) up to the obvious identity Pf+ (Fi (t)) = Fi (t)−Pf− (Fi (t)) valid for arbitrary simple current Fi (t). Calculating the negative projections Pf− (F ) to obtain the projections of the strings of type (3.27), we can prove the identity (3.30) by brute force calculations. The technique of the generating series developed in [13] yields more elegant way of proving this and many other similar identities. Example 3.7. Let us present an example of the general formula (3.30) in the case N = 3, n1 = 2 and n2 = 2. To reduce the formula we will use shorthand notations Pf± ( · ) = [ · ]± . We also denote t2i = si and t1i = ti and Sym below will be the q-symmetrization over variables ti and si .

+

+ F2 (s2 )F2 (s1 )F1 (t2 )F1 (t1 ) = F2 (s2 )F2 (s1 ) F1 (t2 )F1 (t1 ) ⎞ ⎛ 2 −1

−

+ q t − qs 2 j⎠ F2 (s2 )F2 (s1 ) F1 (t1 ) − 2 Sym ⎝ F1 (t2 ) t − s 2 j j=1 $ %

−

+ s2 q −1 t2 − qs1 − 4 Sym F3,1 (t2 ) F2 (s1 ) F1 (t1 ) s2 − t2 t2 − s1 ⎧ ⎛ $ % ⎨

−

− 1

− F1 (t1 ) − F1 (t2 )F1 (t1 ) + 4 Sym ⎝ F1 (t2 ) ⎩ 2 ⎞ 2

+ q −1 ti − qsj ⎠ × F2 (s2 )F2 (s1 ) ti − sj i,j=1

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$

−

−

− s2 s2 q −1 t2 − qs2 F3,1 (t2 ) F1 (t1 ) + F1 (t2 ) s2 − t2 s2 − t1 t2 − s2 % 2

−

−

+ q −1 ti − qs1 s2 F2 (s1 ) F3,1 (t2 )F1 (t1 ) × F3,1 (t1 ) − s2 − t2 ti − s1 i=1 $$ %

−

− 1

− F3,1 (t2 ) F3,1 (t1 ) − F3,1 (t2 )F3,1 (t1 ) + 4 Sym 2 ⎫ %⎬ −1 s2 q t2 − qs1 s1 × . ⎭ s1 − t1 s2 − t2 t2 − s1

+ 4 Sym

The terms in curly brackets correspond to the term W in (3.32). − 3.4. The action of L+ a,b (t) onto Pf (Fc,d (t))

N ), generated by all elements of Definition 3.8. Let I be the right ideal of U q (gl the form Fi [n] · Uq (b+ ) such that i = 1, . . . , N − 1 and n < 0. We denote equalities modulo elements from the ideal I by the symbol ‘ ∼I ’. Proposition 3.9. We have an equivalence − L+ a,b (t) · Fc,d (s) ∼I δa,c

(q − q −1 )s + Ld,b (t) . s−t

(3.35)

One of our technical tools will be the rule of commuting the negative projections of the composed currents Pf− (Fc,d (t )) with matrix elements of the fundamental L-operator L+ a,b (t). We need a result of this calculation only modulo elements from the ideal I and call this as action of Pf− (Fc,d (t )) onto L+ a,b (t). Due to the relation (3.6), in order to calculate the action of the matrix elements L+ a,b (t) onto Pf− (Fc,d (t )) one has to calculate first the action of the matrix elements L+ a,b (t) − onto Gauss coordinates Fc,d (t ). Proof of Proposition 3.9 will be done considering each fixed a. Fix a and consider c < a. This case is simple. Formulas (3.6) can be inverted to express the Gauss coordinates F− c,d (s) in terms of the modes of the currents Fd [nd ], . . . , Fc−1 [nc−1 ]. But the L-operator modes L+ a,b [n] simply commute with − − these current modes and so L+ (t) · F (s) = F (s) · L+ a,b c,d c,d a,b (t) ∈ I. The case when a = N is also simple. For this choice, we have also b = N and + − L+ N,N (t) ≡ kN (t) commutes with the Gauss coordinates Fc,d (s) for c = 2, . . . , N −1. − It means that L+ N,N (t) · Fc,d (s) ∼ 0 for c < N . Let c = N . Taking into account − − − that FN,d (s) = Ld,N (s)LN,N (s)−1 and the commutation relation qt − q −1 s − (q − q −1 )s + Ld,N (s)L+ Ld,N (t)L− N,N (t) − N,N (s) (3.36) t−s t−s we prove the statement of the proposition for a = N . − L+ N,N (t)Ld,N (s) =

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Let a < N and c > a. First consider the case when b > c > a. We have − + − + − + + L+ F+ j,a (t)Fc,d (s)kj (t)Eb,j (t) a,b (t) · Fc,d (s) = Fb,a (t)Fc,d (s)kb (t) + j=b+1 + (t) with due to the commutativity of the Gauss coordinates kb+ (t) and kj+ (t)Eb,j modes of the currents Fd [nd ], . . . , Fc−1 [nc−1 ] or with Gauss coordinates F− c,d (s). The statement of the proposition follows from the lemma.

Lemma 3.10. For b > c > d > a and b > c > a ≥ d − − F+ b,a (t)Fc,d (s) ∼I Fb,a (t)Fc,d (s) ∈ I .

Proof is based on the commutation relations of the composed currents Fb,a (t) and Fc,d (s). They are t−s q −1 t − qs Fb,a (t) Fc,d (s) = Fc,d (s) Fb,a (t) , −1 qt − q s t−s q −1 t − qs Fb,a (t) Fc,d (s) = Fc,d (s) Fb,a (t) , t−s Fb,a (t) Fc,d (s) = Fc,d (s) Fb,a (t) , d > a ,

d < a, d = a,

and they take into account the Serre relations (2.12) (see details in Appendix A of the paper [12]). The product Fb,a (t) Fc,d (s) has no pole for d ≥ a and has first order pole at the point t = s in the case d < a, but the residue at this point of this product is zero. It means that commuting negative projections of the current Pf− (Fc,d (s)) through the total currents Fb,a (t) no higher currents will be created and the result of commutation will belong to the right ideal I. Because of the relation between negative Gauss coordinates and the negative projections of the composed currents given by (3.6) the same statement will be true for the Gauss coordinates. Next we consider the cases when c > a and c ≥ b. The statement of Proposition 3.9 will be proved by a double induction over c starting from c = N and over b starting from c. Let c = N and b = N . Then using again the fact − − −1 and the commutation relations F− N,d (s) = Ld,N (s)kN (s) − L+ a,N (t)Ld,N (s) =

t−s (q − q −1 )s − + L− L (t)L+ d,N (s)La,N (t) + −1 d,N (t) − qs q t − qs a,N

q −1 t

(3.37)

− we obtain the inclusion L+ a,N (t)FN,d (s) ∈ I. Before considering other cases we prove the following lemma.

Lemma 3.11. For b < c and arbitrary a < b and d < c we have − L+ a,b (t) · Ld,c (s) +

(q − q −1 )2 ts + Ld,c (t) · L− a,b (s) ∼I 0 , (t − s)2

a = d ,

− L+ a,b (t) · La,c (s) ∼I 0 ,

a = d.

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Proof. The case of a = d follows from the relation t−s (q − q −1 )s − + − L L (s)L+ (s)L (t) + a,c (t) . a,b qt − q −1 s a,c qt − q −1 s a,b The case a < d follows from two relations +

(q − q −1 ) + + − La,b (t), L− s La,c (t)L− d,c (s) = d,b (s) − t La,c (s)Ld,b (t) , t−s +

(q − q −1 )t − + − La,c (t), L− Ld,c (s)L+ d,b (s) = a,b (s) − Ld,c (t)La,b (s) . t−s The case a > d can be proved analogously. − L+ a,b (t)La,c (s) =

Return to the proof of Proposition 3.9. Keep c = N and consider b = N − 1. Then − + − − −1 L+ a,N −1 (t)FN,d (s) = La,N −1 Ld,N (s)kN (s) (q − q −1 )ts + − −1 Ld,N (t)L− a,N −1 (s)kN (s) (t − s)2 (q − q −1 )ts + − =− L (t) F− N −1,a (s)kN −1 (s) d,N (t − s)2 − − − (s)k (s)E (s) kN (s)−1 , + F− N,a N N −1,N ∼I −

(3.38)

where we used Lemma 3.11. Now the first term in the right hand side of (3.38) belongs to the ideal I because the second index of L+ d,N (t) is bigger than the first − index of FN −1,a (s). The second term corresponds to the case c = b = N considered above, and thus also belongs to I. Reducing b and using the Lemma 3.11 we proved the statement for all b < N = c. Let now c = N − 1. For the negative Gauss coordinate F− N −1,d (s) we can use the formula − − − − −1 F− (s) = L (s) − L (s)E (s) kN . (3.39) N −1,N −1 (s) N −1,d d,N −1 d,N − To prove that the product L+ a,N −1 (t)Ld,N −1 (s) ∈ I we can use the same arguments − + − as for the product L+ a,N (t)Ld,N (s). The fact that La,N −1 (t)Ld,N (s) ∈ I was already + − proved above. Continuing we check that La,b (t)Ld,N −1 (s) ∈ I for all b < N − 1. For general c we have to use instead of the formulae (3.39) the relation − − − −1 F− + L− c,d (s) = Ld,c (s)kc (s) d,c+1 (s)Xc+1 + · · · + Ld,N (s)XN , −

(3.40)

where the explicit form of the elements Xj ∈ Uq (b ) is not important. At last we have to check the case a = c < N . The case a = c = N was − considered above. According to (3.40) the consideration of L+ a,b (t)Fa,d (s) reduces − − −1 to the analysis of the product L+ . We have a,b (t)Ld,a (s)ka (s) −1 +

(q − q ) − + + + La,b (t), L− (s) = (s)L (t) − s L (t)L (t) . t L a,a a,a d,a d,b d,b t−s

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N − − − − −1 − −1 Since L− = 1 + j=a+1 F− and L+ a,a (s)ka (s) j,a (s)kj (s)Ea,j (s)ka (s) d,b (t)Fj,a (s) ∈ I the statement of the Proposition 3.9 is proved. Corollary 3.12. We have an equivalence (q − q −1 )c−d−1 s + − L+ Ld,b (t) . F (t) · P (s) ∼ δ c,d I a,c a,b f t−s

(3.41)

Proof. Let us apply the matrix element L+ a,b (t) to both side of (3.6). The first term gives the right hand side of (3.41). Other terms produce products of the Kronecker’s symbols δa,i1 δd,i2 δi1 ,i3 · · · δi−1 ,c which are zero due to the restriction d < i1 < · · · < i < c. If = 1 then δa,i1 δd,c = 0 since d < c. Let us explain why Pf+ (: TN (t) · W :) = 0, where W are the terms not shown explicitly in the right hand side of (3.32). Due to Corollary 3.12 the action of L+ a,b (t) onto the product of two negative projections of the currents Pf− (Fc1 ,1 ) · Pf− (Fc2 ,1 ) is proportional to the product of delta-symbols: δa,c1 δ1,c2 . But since c2 > 1 this is zero modulo elements of the right ideal I, which obviously satisfies Pf+ (I) = 0.

4. Ordering of the universal objects The proof of main Theorem 1 consists of a detailed analysis of the circular ordering of the product of the transfer matrix and of the universal Bethe vectors expressed ). In the in terms of the current generators of the quantum affine algebra U q (gl N next two subsections we will perform such an analysis to the case N = 2 and prove 2 ). Then we go on by induction over N . main Theorem 1 for the algebra U q (gl In what follows, besides the right ideal I and the left ideal J introduced in Definition 3.8 and Definition 2.2, we will also use the following ideal K. ) ideal generated by the Definition 4.1. We denote by K the two-sided U q (gl N − elements which have at least one arbitrary mode ki [n], i = 1, . . . , N , n ≤ 0, of the negative Cartan current ki− (t). ) modulo element of the ideal K are denoted by the Equalities in U q (gl N symbol ‘ ∼K ’. N ) modulo the right ideal I, the left ideal J and the twoEqualities in U q (gl sided ideal K will be denoted by the symbol ‘ ≈’. 2) 4.1. Ordering for U q (gl 2 ) is generated by the modes of the Gauss coordinates k ± (t), The algebra U q (gl 1 ± ± ± k2 (t), E12 (t), F21 (t) in the L-operator realization or by the modes of the currents k1± (t), k2± (t), E1 (t), F1 (t) in the current realization. The standard quantum 2 ) can be obtained from U q (gl 2 ) by imposing the restriction affine algebra Uq (gl + − ki [0] ki [0] = 1, i = 1, 2. To simplify further formulas we shall not use index of the single simple root in the notation of the Gauss coordinates and the currents, that is

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± ± ± we use the following identification: E± 12 (t) ≡ E (t), F21 (t) ≡ F (t), E1 (t) ≡ E(t), + + ± −1 F1 (t) ≡ F (t). Let ψ (t) = k1 (t)k2 (t) . ) is given by the relation The universal transfer matrix for U q (gl 2 + T2 (t) = L+ 11 (t) + L22 (t) with

+ + + + L+ 11 = k1 (t) + F (t)k2 (t)E (t) ,

+ L+ 22 (t) = k2 (t) ,

while a universal weight function is a projection Pf+ (F(t¯)) = Pf+ (F (tn ) · · · F (t1 )). We avoid to use upper index in the notation of the formal variables tj . ) Proposition 4.2. There is a formal series equality in the algebra U q (gl 2 n n −1 q −1 − qti /t q − q t /t i T2 (t) · Pf+ F(t¯) ≈ Pf+ F(t¯) k1+ (t) + k2+ (t) 1 − t /t 1 − t /t i i i=1 i=1 % $ (q−q −1 )t1 /t + + + + n Sym t¯ F (t)k2 (t)F (tn ) · · · F (t2 ) ψ (t1 ) 1−t1 /t $ % (q−q −1 )tn /t + + − n Sym t¯ F (t)k2 (t)F (tn−1 ) · · · F (t1 ) . (4.1) 1−tn /t Proof of Proposition 4.2 is based on the special presentation of the universal weight function given by the Corollary 3.6. In this case we have Pf+ F(t¯) = F (tn ) · · · F (t1 )−n Sym t¯ Pf− F (tn ) · F (tn−1 ) · · · F (t1 ) +W , (4.2) where W are the terms which have on the left the product of at least two negative projections of the currents F (t). Since Pf− (F (t)) = −F− (t) these terms can be equally described as having on the left the product of at least of two negative Gauss coordinates F− (t). As was explained above and as we will see explicitly below, the terms W are characterized by the property that T2 (t) · F ∈ I. We will order the product of T2 (t) and each summand in the right hand side of (4.2) separately. For the ordering of the first term we use the relation

(q − q −1 )t1 + ψ (t1 ) − ψ − (t1 ) , E+ (t), F (t1 ) = t − t1

so that +

k1 (t) + k2+ (t) + F+ (t)k2+ (t)E+ (t) · F (tn ) · · · F (t1 ) n n q −1 t − qti qt − q −1 ti + + k1 (t) + k2 (t) = F (tn ) · · · F (t1 ) t − ti t − ti i=1 i=1 % $ (q − q −1 )t1 + + n Sym t¯ F+ (t)k2+ (t) F (tn ) · · · F (t2 ) ψ (t1 ) t − t1 % $ (q − q −1 )t1 − − n Sym t¯ F+ (t)k2+ (t) F (tn ) · · · F (t2 ) ψ (t1 ) t − t1 + F+ (t)k2+ (t) F (tn ) · · · F (t1 ) E+ (t) .

(4.3)

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Here we used the exchange relations of the Cartan currents ki+ (t) and the total currents F (tj ) and q-symmetrization in second and third terms of the right hand side of (4.3) appears when commuting Cartan currents ψ ± (ti ) with total currents F (tj ). Observe that the last term in (4.3) belongs to the ideal J and the next to last term belongs to the ideal K. At this point we benefit from the absence of the 2 ). Otherwise we would have to take into restriction (2.14) in the algebra U q (gl account the zero modes of the currents ψ − (t1 ) if we were considering the standard 2 ). quantum affine algebra Uq (gl According to (3.21) F (tn ) · · · F (t1 ) ∼I Pf+ F (tn ) · · · F (t1 ) and equality (4.3) implies the equivalence

k1+ (t) + k2+ (t) + F+ (t)k2+ (t)E+ (t) · F (tn ) · · · F (t1 ) n n −1 q −1 − qti /t q − q t /t i k1+ (t) + k2+ (t) ≈ Pf+ F (tn ) · · · F (t1 ) 1 − t /t 1 − t /t i i i=1 i=1 % $ (q − q −1 )t1 /t + + + + n Sym t¯ F (t)k2 (t) F (tn ) · · · F (t2 ) ψ (t1 ) . (4.4) 1 − t1 /t

Consider now the ordering of the product T2 (t)(Pf− (F (tn )) · F (tn−1 ) · · · F (t1 )). To perform this we will use the following specialization of the Proposition 3.9. For arbitrary element X ∈ UF and m ≥ 1 we have − L+ 11 (t) · Pf F (tm ) · · · F (t1 ) · X ∼I 0 , (q − q −1 )t1 /t + − F (t) k2+ (t) · X . L+ 22 (t) · Pf F (tm ) · · · F (t1 ) · X ∼I δm1 1 − t1 /t

(4.5)

Applying (4.5) to the second term in the right hand side of (4.2) we obtain T2 (t) · Pf− F (tn ) · F (tn−1 ) · · · F (t1 ) ∼I

(q − q −1 )tn /t + F (t) k2+ (t) · F (tn−1 ) · · · F (t1 ) . 1 − tn /t

(4.6)

Since any equality modulo ideal I implies the related equality modulo all the ideals I, J and K, the relations (4.4) and (4.6) imply the statement of Proposition 4.2.

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Corollary 4.3. Relation (4.1) can be considered as the equality T2 (t) · Pf+ F(t¯) n n q −1 − qti /t + q − q −1 ti /t + + ¯ k1 (t) + k2 (t) ∼J Pf F(t) 1 − ti /t 1 − ti /t i=1 i=1 % $ + (q − q −1 )t1 /t + + + + n Sym t¯ Pf F (t)k2 (t) F (tn ) · · · F (t2 ) ψ (t1 ) 1 − t1 /t $ % + (q − q −1 )tn /t + + − n Sym t¯ Pf F (t)k2 (t) F (tn−1 ) · · · F (t1 ) , 1 − tn /t in Uq (b+ ) modulo elements of the ideal J. The sum of the last two terms in the above equality belongs to the left ideal B (see Definition 2.3). Proof. Left hand side of the equality (4.1) belongs to Uq (b+ ) while the right hand side does not. We can imposing projection Pf+ onto this right hand side to cancel all the terms which belongs to the ideal I. 2) 4.2. Proof of Theorem 1 for U q (gl Let us compare the last two lines in (4.1). They contain so called ‘unwanted’ terms. In order to cancel them we will use the following properties of q-symmetrization. For any formal series G(t1 , . . . , tn ) on n formal variables ti we have n Sym =

t¯ G(t1 , . . . , tn ) n n −1

q − q tm /tj Sym q −1 − qtm /tj m=1 j=m+1

t¯\tm

G(t1 , . . . , tm−1 , tm+1 , . . . , tn , tm ) (4.7)

and n Sym =

t¯

G(t1 , . . . , tn )

n m−1 m=1 j=1

q − q −1 tj /tm Sym q −1 − qtj /tm

t¯\tm

G(tm , t1 , . . . , tm−1 , tm+1 , . . . , tn ) ,

(4.8)

where q-symmetrization in the right hand sides of this formal series identities runs over (n − 1) variables t¯ \ tm = {t1 , . . . , tm−1 , tm+1 , . . . , tm }. Note that formulas (4.7) and (4.8) are particular cases of the property (3.17) for s = n and s = 1, respectively. Using formulas (4.7) and (4.8) we may write the difference of unwanted terms in (4.1) as a sum ⎛ n (q − q −1 )tm /t n Sym t¯\tm ⎝F+ (t)k2+ (t) F (tn ) · · · F (tm+1 )F (tm−1 ) · · · F (t1 ) 1 − tm /t m=1 ⎞⎞ ⎛ n m−1 q − q −1 tj /tm q − q −1 tm /tj + ⎠⎠ . ⎝ ψ (t ) − m q −1 − qtm /tj q −1 − qtj /tm j=m+1 j=1

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Applying to this equality the projection Pf+ , as it was explained in the proof of Corrolary 4.3, we obtain the elements from the left ideal B. Latter elements are vanishing if the Bethe parameters satisfy the universal Bethe equations [1] ψ + (tm ) =

n q − q −1 tj /tm k1+ (tm ) , = + k2 (tm ) j=m q −1 − qtj /tm

m = 1, . . . , n .

Thus, Theorem 1 in the case N = 2 is proved.

(4.9)

4.3. General case 4.3.1. Preliminary exchange relations. To consider the general case of Theorem 1, N ) → U q (gl N +1 ), defined by the rule4 we need the embedding Ψ : U q (gl [N ] Ψ Li,j (t) = Li+1,j+1 (t) , i, j = 1, . . . , N . (4.10) N +1 Let TN (t) = i=2 Lii (t) be the universal transfer matrix for the algebra ¯ U q (glN ) and WN (t[¯n ] ) be the universal weight function, where n ¯ is a set of the ¯ positive integers {n2 , . . . , nN } and t[¯n ] is an associated set of the formal variables: N t¯[¯n ] = {t21 , . . . , t2n2 ; . . . ; tN 1 , . . . , tnN } .

Using the result of Corollary 3.6 we present the U q (gl N +1 )-universal weight function WN +1 (t¯[¯n] ) in the form ¯ (t[¯n ] ) · F(t¯[n1 ] ) − S , WN +1 (t¯[¯n] ) = WN

(4.11)

where S contains the sum of terms as in the right hand side of (3.32) and the redundant terms W such that : TN +1 (t) · W : ∼I 0. We consider the product ¯ TN +1 (t) · WN +1 (t¯[¯n] ) = TN +1 (t) · WN (t[¯n ] ) · F(t¯[n1 ] ) − TN +1 (t) · S .

(4.12)

Since TN +1 (t) = L11 (t) + TN (t) we rewrite the first term in the right hand side of (4.12) modulo terms from the ideal I: ¯ (t[¯n ] ) · F(t¯[n1 ] ) + L11 (t) · F(t¯[¯n] ) . TN (t) · WN

(4.13)

To obtain (4.13) we use the result of the following Lemma 4.4. ¯ L11 (t) · WN (t[¯n ] ) · F(t¯[n1 ] ) ∼I L11 (t) · F(t¯[¯n] ) .

4 We

(4.14)

omit writing explicitly superscript ‘+’ of the L-operator and their Gauss coordinate, assuming that they are always from the standard positive Borel subalgebra Uq (b+ ).

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¯ N ) embedded Proof. The universal Bethe vector WN (t[¯n ] ) for the algebra U q (gl of the currents F2 (t), . . . , FN (t) into U q (glN +1 ) by (4.10) depends on the modes and is given by the projection Pf+ F(t¯[¯n ] ) of the product of these currents. According to the ordering rules (3.21) we may replace this projection by the product of the total currents F(t¯[¯n ] ) subtracting the terms which have on the left the negative projection of the currents Pf− (Fc,d (t)) with 2 ≤ d < c ≤ N + 1. According to (3.41) the product L11 (t)Pf− (Fc,d (t )) ∼I 0. This proves the statement of the lemma.

Now we apply the result of Proposition 3.9 to the second term in the right hand side of (4.12). We have N TN +1 (t) · S ∼I Sym t¯[¯n] L1,m+1 (t) · Pf+ F(t¯[¯n ]m ) · F(t¯1[n1 −1] ) m=1

(q − q −1 )m t1n1 /t ¯ Zm (t[¯n] ) (na ) × 1 − t1n1 /t a=1 m

(4.15)

where the sets of the formal variables t¯[¯n ]m are defined by (3.31) and a rational series Zm (t¯[¯n] ) is defined by (3.33). We can simplify the right hand side of (4.15) replacing the projection Pf+ F(t¯[¯n ]m ) by the product of the currents F(t¯[¯n ]m ) due to the following Lemma 4.5.

L1,m+1 (t) · Pf+ F(t¯[¯nm ] ) ∼I L1,m+1 (t) · F(t¯[¯nm ] ) .

Proof of this lemma is analogous to the proof of Lemma 4.4.

(4.16)

Using the explicit form of the matrix elements L1,1 (t) and L1,m+1 (t) in terms of the Gauss coordinates L1,1 (t) = k1+ (t) +

N +1

+ + F+ j,1 (t)kj (t)E1,j (t) ,

j=2 + L1,m+1 (t) = F+ m+1,1 (t)km+1 (t) +

N +1

+ + F+ j,1 (t)kj (t)Em+1,j (t) ,

j=m+2

we can present the product in the left hand side of (4.12) as the sum of the terms modulo ideal I ¯ TN +1 (t) · WN +1 (t¯[¯n] ) ∼I TN (t) · WN (t[¯n ] ) · F(t¯[n1 ] ) + k1+ (t) · F(t¯[¯n] ) j−2 N +1 (4.17) + + + + Fj,1 (t)kj (t) E1,j (t) · F(t¯[¯n] ) − Aj−1 − Em+1,j (t) · Am , + j=2

m=1

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$

% t1n1 /t . 1 − t1n1 /t a=1 (4.18) Calculation of (4.17) modulo elements of the ideal I is useful since now to continue ordering we have to exchange the Gauss coordinates E+ m+1,j with total currents. These exchange relations are based on the formulae given by Lemma 3.4: −1 m+2−j ˆ Sm+1 Sˆm+2 · · · Sˆj−2 E+ E+ ) (4.19) m+1,j (t) = (q − q j−1,j (t) , Am = (q − q −1 )m

m

(na ) Sym

t¯[¯ n]

F(t¯[¯n ]m ) · F(t¯1[n1 −1] ) · Zm (t¯[¯n] )

where the screening operators is defined as Sî (B) = Ei [0] B−qB Ei [0], the formulae

and ψi+ (t)

+

(q − q −1 )t /t + Ei,i+1 (t), Fj (t ) ∼K δi,j ψi (t ) , 1 − t /t

(4.20)

Ei [0], Fj (t ) ∼K δi,j (q − q −1 ) ψi+ (t ) .

(4.21)

+ ki+ (t) ki+1 (t)−1 .

= Recall that the symbol ‘∼K ’ means an equality Here ) with modulo terms of the ideal K which composed from the elements of U q (gl N − − − −1 any mode of the negative Cartan current ψi (t) = ki (t) ki+1 (t) . The ordering of the first term for j = 2 in the right hand side of (4.17) can be 2 ) since the Gauss coordinate E+ (t) performed as in the case of the algebra Uq (sl 1,2 does not commute only with the currents F1 (t1 ). This term is equal to (modulo elements from the ideals K and J) ⎛ 1 + ⎝F(t¯[¯n ]1 ) · F(t¯1[1,n ] ) t1 /t ψ1+ (t11 ) n1 (q − q −1 )F+ 2,1 (t)k2 (t) Sym t¯[¯ 1 n] 1 − t11 /t ⎞ n2 −1 1 2 1 q − q t /t tn1 /t n1 j ⎠ − F(t¯[¯n ]1 ) · F(t¯1[n1 −1] ) . (4.22) 1 − t1n1 /t j=1 1 − t1n1 /t2j The ordering of the Gauss coordinates E+ m+1,j (t) with total currents is more involved. To perform this ordering we have to use besides (4.19), (4.20), (4.21) also the relation Sî ψ + (t) = Ei [0] ψ + (t) − q ψ + (t) Ei [0] = (q − q −1 )ψ + (t)E+ (t) . (4.23) i+1

i+1

i+1

i+1

i

Fix the jth term of the sum in the right hand side of (4.17) and denote ¯ n] ) − (q − q −1 )j−1 Aj−1 − Rj = E+ 1,j (t) · F(t[¯

j−2

(q − q −1 )m E+ m+1,j (t) · Am . (4.24)

m=1

E+ m+1,j (t)

We will exchange the Gauss coordinates with total current. As above we will calculate modulo elements of the ideals K and J. We will describe this calculation as the sequence of the steps.

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1. According to (4.19) the Gauss coordinate E+ m+1,j (t) is composed from the zero mode of the currents Ei (t) with i = m+1, . . . , j −2 and the Gauss coordinate E+ j−1,j (t). It means that this Gauss coordinate will have nontrivial commutation relations only with the currents Fa (tas ) for a = m+1, . . . , j −1. The move a of the Gauss coordinates E+ m+1,j (t) through the total current Fa (ts ) with a = m + 1, . . . , j − 2 produces the terms which belong to the ideal J. These terms appear after commuting zero modes Ea [0] with the current Fa (tas ) and can be neglected since we perform the calculation modulo elements of the ideal J. The nontrivial terms, which we will consider, are created after exj−1 ). At changing the Gauss coordinate E+ m+1,j (t) with the currents Fj−1 (ts + j−1 the first step the Cartan currents ψj−1 (ts ) together with a rational factor (q−q −1 )tj−1 /t 1 1−tj−1 /t 1

appear according to (4.20) after an exchange of E+ j−1,j (t) and

) in the place of the latter current. This Cartan the total current Fj−1 (tj−1 s current should be moved to the right of the product of type j −1 currents. All j−1 ) the terms resulting from the exchange relation of E+ j−1,j (t) and Fj−1 (ts can be presented as the q-symmetrization over the type j − 1 formal variables of the single term which appears after commutation of E+ j−1,j (t) with j−1 Fj−1 (t1 ). Due to properties (3.16) this q-symmetrization is absorbed into ‘global’ q-symmetrization entering into definitions of the elements Am (4.18) producing factor nj−1 due to definition of the q-symmetrization. The product of the current F(t¯[¯n] ) in (4.24) can be also presented as the ‘global’ q-symmetrization over the set of the variables t¯[¯n] . 2. The sequence of the screening operators Sˆm+1 · · · Sˆj−2 which enter the formula for the Gauss coordinate E+ m+1,j (t) are applied according to equation + (4.23) to replace the Cartan current ψj−1 (tj−1 1 ) on the right of the product j−1 j−1 + of j − 1-type currents by the factor ψj−1 (t1 )E+ m+1,j−1 (t1 ). j−1 + 3. The Gauss coordinate Em+1,j−1 (t1 ) should be moved through the total currents of type j − 2 to produce q-symmetrization over variables of the same j−2 + + type and the factor ψj−2 (tj−2 1 )Em+1,j−2 (t1 ) on the right of the product of (q−q −1 )tj−2 /tj−1

this type current accompanied with a rational factor 1−tj−21/tj−11 . Note 1 1 that the factor nj−2 appears also due to definition of the q-symmetrization. a 4. After moving all Gauss coordinates E+ m+1,a (t1 ) to the right we are left with a product of total currents where all the currents Fa (ta1 ) have been replaced by the Cartan currents ψa+ (ta1 ) with a = m + 1, . . . , j − 1 together with the (q−q −1 )ta /ta+1

rational factors na 1−ta /t1a+11 . Then, we have to move all these Cartan 1 1 currents to the right of all total currents producing the rational factors of the Bethe parameters.

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These calculations result to the following exchange relations of the Gauss ¯ n ]m ) · F(t¯1 coordinate E+ m+1,j (t) and the product of the currents F(t[¯ [n1 −1] ) ¯ n ]m ) · F(t¯1[n −1] ) E+ m+1,j (t)F(t[¯ 1 j−1

∼J,K (q − q −1 )j−m−1

(na ) Sym

t¯[¯ n]

¯j F(t¯N [nN ] ) · · · F(t[nj ] )

a=m+1

¯m+1 ¯m ¯1 × F(t¯j−1 [1,nj−1 ] ) · · · F(t[1,nm+1 ] )F(t[nm −1] ) · · · F(t[n1 −1] ) j−1 tj−1 + a 1 /t ψa (t1 ) , Ym (t¯[¯n] ) × 1 − tj−1 1 /t a=m+1 where a rational series Ym (t¯[¯n] ) is defined by the relation n −1 j−2 na m m+1 a a+1 −1 a a+1 /t q − q t /t q − q −1 tm t 1 1 k 1 k /t1 Ym (t¯[¯n] ) = . a+1 a+1 m+1 1 − ta1 /t1 k=2 1 − tak /t1 1 − tm k /t1 a=m+1 k=1 (4.25) Recall that the notation t¯a[1,na ] means the collection of the variables {ta2 , . . . , tana }. We get finally the following presentation of the product TN +1 (t) · WN +1 (t¯[¯n] ): ¯ (t[¯n ] ) · F(t¯[n1 ] ) TN +1 (t) · WN +1 (t¯[¯n] ) ≈ TN (t) · WN

+ WN +1 (t¯[¯n] ) · k1+ (t)

n1 −1 n2 q − qt1 /t q − q −1 t2 /t i

i=1

+

N +1 j=2

(q − q −1 )j−1

j−1

1 − t1i /t

i

i=1

1 − t2i /t

+ (na ) F+ j,1 (t)kj (t) · Rj ,

(4.26)

a=1

where the symbol ‘≈’ means an equality modulo the ideals I, J and K (see Definition 4.1). The elements Rj have the structure tj−1 j j−1 1 1 /t ¯ ¯ ¯ ¯ ) · · · F( t )F( t ) · · · F( t ) Y ( t ) Rj = Sym t¯[¯n] F(t¯N 0 [¯ n ] [nN ] [1,n1 ] [nj ] [1,nj−1 ] 1 − tj−1 1 /t j−1

¯j ¯j−1 ¯1 ¯ n] ) ψa+ (ta1 )−F(t¯N [nN ] ) · · · F(t[nj ] )F(t[nj−1 −1] ) · · · F(t[n1 −1] ) · Zj−1 (t[¯

a=1

t1n1 /t 1−t1n1 /t

j−2 ¯j ¯j−1 ¯m+1 ¯m ¯1 − F(t¯N [nN ] ) · · · F(t[nj ] )F(t[1,nj−1 ] ) · · · F(t[1,nm+1 ] )F(t[nm −1] ) · · · F(t[n1 −1] ) m=1

j−1 t1n1 /t tj−1 + a 1 /t ψa (t1 ) . × Zm (t¯[¯n] ) Ym (t¯[¯n] ) 1 − t1n1 /t 1 − tj−1 1 /t a=m+1 The series Zm (t¯[¯n] ) is defined by the relation (3.33).

(4.27)

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4.3.2. Proof of Theorem 1. The following proposition generalizes Proposition 4.2 and serves as a main statement which is proved by induction over N . N) Proposition 4.6. There is a formal series equality in the algebra U q (gl TN (t) · WN (t¯[¯n] ) ≈ WN (t¯[¯n] ) ·

N

na q − q −1 ta−1 /t q −1 − qtak /t k ka+ (t) 1 − ta−1 /t k=1 1 − tak /t k a=1 k=1 na−1

⎛ N j−1 + ¯N −1 ¯j (na ) (q − q −1 )j−1 F+ + Sym t¯[¯n] ⎝ j,1 (t)kj (t) · F(t[nN −1 ] ) · · · F(t[nj ] ) j=2 a=1

×

¯1 ¯ n] ) F(t¯j−1 [nj−1 −1] ) · · · F(t[n1 −1] ) · Zj−1 (t[¯

×

ψ1+ (t1n1 )

n 1 −1 k=1

∼B WN (t¯[¯n] ) ·

N

tj−1 nj−1 /t

1 − tj−1 nj−1 /t

⎞ n2 q −1 − qt1k /t1n1 1 − t2k /t1n1 −1 ⎠ q − q −1 t1k /t1n1 q −1 − qt2k /t1n1 k=1

na q − q −1 ta−1 /t q −1 − qtak /t k 1 − ta−1 /t k=1 1 − tak /t k k=1

na−1

ka+ (t)

a=1

(4.28)

if the set {tij } of the Bethe parameters satisfies the set of the universal Bethe equations, i = 2, . . . , N − 1, j = 1, . . . , ni : ki+ (tij ) + ki+1 (tij )

=

ni−1 ni+1 −1 ni i i q − qti+1 q − q −1 tim /tij 1 − ti−1 m /tj m /tj . (4.29) i i−1 i+1 q −1 − qtim /tj m=1 q − q −1 tm /tij m=1 1 − tm /tij m=j

Proof. We will proof this proposition by induction over N taking as the base of the induction the statement of Proposition 4.2. We assume that the equality (4.28) is N ) embedded into U q (gl N +1 ) by the relation (4.10) and valid for the algebra U q (gl N +1 ). prove from the relation (4.26) that (4.28) is valid also for the algebra U q (gl N) First we rewrite the induction assumption (4.28) for the algebra U q (gl embedded into U q (glN +1 ) by (4.10). It takes the form ¯ TN (t) · WN (t[¯n ] ) ¯ (t[¯n ] ) · ≈ WN

N +1 a=2

na q − q −1 ta−1 /t q −1 − qtak /t k ka+ (t) 1 − ta−1 /t k=1 1 − tak /t k k=1 na−1

+ Q (4.30)

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where in ‘unwanted’ terms ⎛ j−1 N +1 + ¯N ¯j (na ) F+ (q − q −1 )j−1 Q = Sym t¯[¯n] ⎝ j,2 (t)kj (t) · F(t[nN ] ) · · · F(t[nj ] ) a=2

j=3

×

¯2 ¯ n ] ) F(t¯j−1 [nj−1 −1] ) · · · F(t[n2 −1] ) · Zj−1 (t[¯

×

ψ2+ (t2n2 )

n 2 −1 k=1

tj−1 nj−1 /t

1 − tj−1 nj−1 /t

⎞ n3 q −1 − qt2k /t2n2 1 − t3k /t2n2 −1 ⎠ q − q −1 t2k /t2n2 q −1 − qt3k /t2n2

(4.31)

k=1

parameters t¯an¯ a with a = 3, . . . , N satisfy the universal Bethe equations (4.29) for i = 3, . . . , N while parameters t¯2n¯ 2 are free. We substitute (4.30) into (4.26). First term in the right hand side of (4.30) together with the second term in (4.26) produces ‘wanted’ terms N +1 na−1 na q − q −1 ta−1 /t q −1 − qtak /t + k ¯ . WN +1 (t[¯n] ) · ka (t) 1 − tak /t 1 − ta−1 /t k a=1 k=1

k=1

The terms in the right hand side of (4.30) which belongs to the ideal J will be ¯ again in the same ideal after multiplication to the right of TN (t) · WN (t[¯n ] ) by the ¯ product of the first type currents F(t[n1 ] ) (see (4.26)). This is because the currents Ea (t) commute with the current F1 (t ) for a = 2, . . . , N . Fix parameters t¯2n¯ 2 from the condition that the ordered product : Q · F(t¯1[n1 ] ) : of the unwanted terms and the currents of the first type F1 (t1k ) belong to the ideal B. This results into Bethe equations n2 n1 n3 k2+ (t2k ) q − q −1 t2m /t2k 1 − t1m /t2k q −1 − qt3m /t2k = + 2 2 2 −1 2 −1 1 k3 (tk ) m=k q − qtm /tk m=1 q − q tm /tk m=1 1 − t3m /t2k

(4.32)

for the set of parameters t¯2n¯ 2 . Now we will examine the structure of the terms Rj given by (4.27) by the conditions that parameters t¯an¯ a with a = 2, . . . , N are bounded by the universal Bethe equations (4.29) and (4.32). We replace in Rj the Cartan currents ψa+ (ta1 ), a = 2, . . . , j−1 by the right hand sides of the universal Bethe equations. Each Bethe na qta1 −q−1 ta equation introduces the factor =2 a under q-symmetrization. This allows q −1 ta 1 −qt to use the following property of the q-symmetrization, which is a consequence of (4.7) and (4.8): n q −1 − qt1 /t Sym t¯ G(t1 , t2 , . . . , tn ) = Sym t¯ G(tn , t1 , . . . , tn−1 ) (4.33) −1 q − q t1 /t =2

for arbitrary formal series G of the formal variables (t1 , . . . , tn ). The variables {ta1 , ta2 , . . . , tana } are replaced by the permuted sets {tana , ta1 , . . . , tana −1 } for a =

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m + 1, . . . , j − 1. Using an identity j−2

j−2 1 1 1 − a+1 a+1 1 a a t − t t − t t n1 a=1 na+1 − tna na a=1 na+1

1 t − tj−1 nj−1 −

1 t − tj−1 nj−1

j−2 j−2 m−1 1 1 1 = 0, a+1 1 a t − tn1 m=1 a=m+1 tna+1 − tana a=1 ta+1 na+1 − tna

and the explicit forms of the series Zm and Ym we observe that the element Rj has the structure ¯j ¯j−1 ¯1 ¯ n] ) Rj = F(t¯N [nN ] ) · · · F(t[nj ] ) · F(t[nj−1 −1] ) · · · F(t[n1 −1] ) · Zj−1 (t[¯ ×

ψ1+ (t1n1 )

n 1 −1 k=1

tj−1 nj−1 /t 1 − tj−1 nj−1 /t

n2 q −1 − qt1k /t1n1 1 − t2k /t1n1 −1 q − q −1 t1k /t1n1 q −1 − qt2k /t1n1 k=1

belongs to the Bethe ideal B (see Definition 2.3). This proves the statement of the Proposition. The statement of Theorem 1 follows from Proposition 4.6. The element Rj vanishes if we impose one more universal Bethe equation for the set of the parameters {t¯1 } [n1 ]

n1 n2 q − q −1 t1m /t1j q −1 − qt2m /t1j . + 1 = 1 k2 (tj ) m=j q −1 − qt1m /tj m=1 1 − t2m /t1j

k1+ (t1j )

(4.34)

Since the left hand side and first term in the right hand side of (4.28) belong to the N ) the equality between them is valid standard Borel subalgebra Uq (b+ ) in U q (gl modulo elements of the ideal J and the Bethe ideal B. This finishes the proof of Theorem 1.

Acknowledgements This work was partially done when the third author (S.P.) visited Laboratoire d’Annecy-Le-Vieux de Physique Théorique in 2006 and 2007. These visits were possible due to the financial support of the CNRS-Russia exchange program on mathematical physics. He thanks LAPTH for the hospitality and stimulating scientific atmosphere. We are thankful to referee for valuable comments. Work by second (S.K) and third (S.P.) authors was supported in part by RFBR grant 0801-00667 and grant for support of scientific schools NSh-3036.2008.2. Third author (S.P.) was also supported in part by RFBR-CNRS grant 07-02-92166-CNRS.

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References [1] D. Arnaudon, N. Crampe, A. Doikou, L. Frappat, E. Ragoucy, Spectrum and Bethe ansatz equations for the Uq (gl(N )) closed and open spin chains in any representation, Annales Henri Poincaré 7 (2006), 1217–1268. [2] J. Ding, I. B. Frenkel, Isomorphism of two realizations of quantum affine algebra ), Commun. Math. Phys. 156 (1993), 277–300. Uq (gl N [3] J. Ding, S. Khoroshkin, Weyl group extension of quantized current algebras, Transformation Groups 5 (2000), 35–59. [4] V. Drinfeld, Quantum groups, In “International Congress of Mathematicians (Berkley, 1986)”, Amer. Math. Soc., Providence RI, 1987, 798–820. [5] V. Drinfeld, New realization of Yangians and quantum affine algebras, Sov. Math. Dokl. 36 (1988), 212–216. [6] B. Enriquez, On correlation functions of Drinfeld currents and shuffle algebras, Transformation Groups 5 (2000), no. 2, 111–120. [7] B. Enriquez, S. Khoroshkin, S. Pakuliak, Weight functions and Drinfeld currents, Commun. Math. Phys. 276 (2007), 691–725. [8] B. Enriquez, V. Rubtsov, Quasi-Hopf algebras associated with sl2 and complex curves, Israel J. Math 112 (1999), 61–108. 3 ), Theor. and Math. Phys. [9] S. Khoroshkin, S. Pakuliak, Weight function for Uq (sl 145 (2005), no. 1, 1373–1399. [10] S. Khoroshkin, S. Pakuliak, V. Tarasov, Off-shell Bethe vectors and Drinfeld currents, Journal of Geometry and Physics 57 (2007), 1713–1732. [11] S. Khoroshkin, V. Tolstoy, On Drinfeld realization of quantum affine algebras, Journal of Geometry and Physics 11 (1993), 101–108. [12] S. Khoroshkin, S. Pakuliak, A computation of an universal weight function for the ), J. Math. Kyoto Univ. 48, no. 2, 277–322. Preprint quantum affine algebra Uq (gl N arXiv:0711.2819 [math.QA]. [13] S. Khoroshkin, S. Pakuliak, Generating series for nested Bethe vectors, SIGMA 4, (2008), 081, 23 pages, arXiv:0810.3131. [14] P. Kulish, N. Reshetikhin, Diagonalization of GL(N ) invariant transfer matrices and quantum N -wave system (Lee model), J. Phys. A: Math. Gen. 16 (1983), L591–L596. [15] E. Mukhin, V. Tarasov, A. Varchenko, Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006, no. 8, P08002, 1–44. [16] A. Oskin, S. Pakuliak, A. Silantyev, On the universal weight function for the quantum ), Preprint arXiv:0711.2821 [math.QA]. affine algebra Uq (gl N [17] N. Reshetikhin, M. Semenov-Tian-Shansky, Central extentions of quantum current groups, Lett. Math. Phys. 19 (1990), 133–142. [18] V. Tarasov, A. Varchenko, Jackson integrals for the solutions to Knizhnik-Zamolodchikov equation, Algebra and Analysis 2 (1995) no.2, 275–313. [19] V. Tarasov, A. Varchenko, Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), 1–135. [20] V. Tarasov, A. Varchenko, Combinatorial formulae for nested Bethe vectors, arXiv:math/0702277 [math.QA].

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´ Luc Frappat and Eric Ragoucy Laboratoire d’Annecy-le-Vieux de Physique Théorique LAPTH UMR 5108 du CNRS associée ` a l’Université de Savoie BP 110 F-74941 Annecy-le-Vieux Cedex France e-mail: [email protected] [email protected] Sergey Khoroshkin Institute of Theoretical & Experimental Physics 117259 Moscow Russia e-mail: [email protected] Stanislav Pakuliak Laboratory of Theoretical Physics JINR 141980 Dubna, Moscow reg. Russia e-mail: [email protected] Communicated by Petr Kulish. Submitted: October 30, 2008. Accepted: February 16, 2009.


Ann. Henri Poincaré 10 (2009), 549–571 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/030549-23, published online June 15, 2009 DOI 10.1007/s00023-009-0408-x


Scattering at Zero Energy for Attractive Homogeneous Potentials Jan Derezi´ nski and Erik Skibsted Abstract. We compute up to a compact term the zero-energy scattering matrix for a class of potentials asymptotically behaving as −γ|x|−μ with 0 < μ < 2 and γ > 0. It turns out to be the propagator for the wave equation on μπ . the sphere at time 2−μ

1. Introduction and results The paper is devoted to a study of the zero-energy scattering matrix S(0) for a class of radial potentials on Rd with d ≥ 2. This class consists of the potentials of the form V (x) := −γ|x|−μ + W (|x|), where 0 < μ < 2 and γ > 0 and W (r) is a fast decaying perturbation. We will show that the leading term of S(0) can be computed and is an interesting Fourier integral operator. This paper can be considered as a companion to a series of papers [4–6], where the low-energy scattering theory has been developed for a somewhat more general class of potentials. Note however, that this paper can be read independently of [4–6]. Before stating our main result, which deals with quantum scattering, let us say a few words about its classical analog. Consider the equations of motion in a strictly homogeneous potential V (r) = −γr−μ . It turns out that this problem is exactly solvable at zero energy. The (non-collision) zero-energy orbits are given by the implicit equation (in polar coordinates) −1+ μ2 r(t) μ θ(t) = sin 1 − , (1.1) 2 rtp The research of J. D. is supported in part by the grant N N201 270135. Part of the research was done during a visit of both authors to the Erwin Schr¨ odinger Institute.

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J. Derezi´ nski and E. Skibsted


μπ see [4, Example 4.3]. Whence the deflection angle of such trajectories equals − 2−μ . In particular, for attractive Coulomb potentials it equals −π, which corresponds to the well-known fact that in this case zero-energy orbits are parabolas (see [15, p. 126] for example). One can ask whether a similar behavior can be seen at the quantum level. Our analysis shows that indeed this is the case. Our main result is stated in terms of the unitary group eiθΛ generated by a certain self-adjoint operator Λ on L2 (S d−1 ). The operator Λ is defined by setting ΛY = (l + d/2 − 1)Y if Y is a spherical harmonic of order l. eiθΛ can be called the propagator for the wave equation on the sphere. Note that for any θ, the distributional kernel of eiθΛ can be computed explicitly and its singularities appear at ω · ω = cos θ. This is expressed in the following fact [16]:

Proposition 1.1. eiθΛ equals 1. cθ I, where I is the identity, if θ ∈ π2Z; 2. cθ P , where P is the parity operator (given by τ (ω) → τ (−ω)), if θ ∈ π(2Z+1); d 3. the operator whose Schwartz kernel is of the form cθ (ω · ω − cos θ + i0)− 2 if θ ∈]π2k, π(2k + 1)[ for some k ∈ Z; d 4. the operator whose Schwartz kernel is of the form cθ (ω · ω − cos θ − i0)− 2 if θ ∈]π(2k − 1), π2k[ for some k ∈ Z. We also remark that for all θ, the operator eiθΛ belongs to the class of Fourier integral operators of order 0 in the sense of H¨ ormander [11, 12]. Let us now briefly recall some points of the time-dependent scattering theory for Schr¨ odinger operators. Set H0 := − 12 Δ and H = H0 + V (x). If the potential V (x) is short-range, following the standard formalism, we can define the usual scattering operator. In the long-range case the usual formalism does not apply. Nevertheless, one can use one of the modified formalisms, which leads to a modified scattering operator S. Clearly, H0 commutes with S, and hence S can be written in terms of the direct integral S ⊕S(λ)dλ . (1.2) ]0,∞[

The operators S(λ), called scattering matrices, are defined up to a set of measure zero. In the short-range case, one can chose S(λ) to be continuous for λ > 0, which fixes the value of positive energy scattering matrices uniquely. In the long-range case we need to use modified scattering operators, in whose definition there is a freedom of choosing an arbitrary phase factor depending on λ. One can however also choose S so that S(λ) is continuous for λ > 0, which fixes S(λ) up to a phase factor. Actually, we will use the modified formalism also in the short-range case. To us a “scattering matrix” will always mean a “modified scattering matrix”, defined up to a phase factor, also in the short-range case.

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The above description of scattering theory applies to a rather general class of potentials. Under more restrictive assumptions described in [5] one can show that the modified scattering operator can be chosen so that S(λ) is continuous down to λ = 0, which allows us to define the zero-energy scattering matrix S(0). This is one of the main results of [5]. One can compute the scattering matrix in terms of asymptotics of generalized eigenfunctions. If the potential is radial this is particularly convenient. Explicitly the asymptotics of the regular solution of the stationary Schr¨ odinger equation for the energy λ and the angular momentum sector l determines the scattering phase shift, denoted σl (λ). Again, the usual definition of the scattering phase shift applies only to the short-range case. In the long-range case one needs to introduce a modified scattering phase shift. Our construction differs from these by a trivial term (i.e. an l-independent term), cf. [5, Theorems 7.3 and 7.4]. Suppose that ]0, ∞[ r → V (r) is a continuous real function such that for some positive constants , κ and C V (r) + γr−μ ≤ Cr−1− μ2 − , r > 1 ; (1.3) |V (r)| ≤ Cr−2+κ ,

r ≤ 1.

(1.4)

With these assumptions, which will be the main assumptions used in this paper, one can show that our (appropriately modified) phase shift σl (λ) is continuous in λ down to λ = 0. We have the following relationship between the phase shift and the scattering matrix: (1.5) S(λ)Y = ei2σl (λ) Y , where Y is any spherical harmonic Y of order l. For positive energies (1.5) is a wellknown identity valid under rather general assumptions. For a (partial) justification under the above conditions we refer to [5]. However let us stress that in this paper we can (and will) avoid time-dependent formalisms completely, and in fact we take (1.5) as the definition of S(λ), in particular for the limiting case λ = 0. Whence we define the (modified) scattering matrix through the (modified) phase shift. Here is the main result of our paper: Theorem 1.2. Assume the conditions (1.3) and (1.4) on the potential V (r). Then, for some c ∈ R and a compact operator K on L2 (S d−1 ), we have μπ

S(0) = eic e−i 2−μ Λ + K . We prove Theorem 1.2 by a careful one-dimensional WKB-analysis, simultaneously in each angular momentum sector. Therefore our results do not follow easily from the literature on 1-dimensional Schr¨ odinger operators that we know. Consider the potential V (r) equal exactly to −γ|x|−μ , and the corresponding Hamiltonian Hμ := H0 − γr−μ . It is not difficult to show that Hμ is an analytic family of operators for Re μ ∈]0, 2[. In the preprint version of our paper we

552



formulated a conjecture that in the case of Hμ μπ

S(0) = eic e−i 2−μ Λ , without the compact error term, or alternatively, that the terms o(l0 ) in Proposition 3.2 vanish identically. A special case of this conjecture is the formula S(0) = eic P in the attractive Coulomb case, which has been known for a long time, see [20]. (Here, (P τ )(ω) = τ (−ω)). Recently, the above conjecture has been proven by R. Frank, see [7].

2. Propagator of the wave equation on the sphere 2.1. Distributional kernel of the propagator For any 1 ≤ i < j ≤ d, define the corresponding angular momentum operator Lij := −i(xi ∂xj − xj ∂xi ) . Set L2 :=

L2ij ,

Λ :=

L2 + (d/2 − 1)2 .

1≤i<j≤d

Note that Λ is a self-adjoint operator on L2 (S d−1 ) and its eigenfunctions with eigenvalue l + d/2 − 1 are the lth order spherical harmonics for l = 0, 1, . . . . For any θ one can compute exactly the integral kernel of eiθΛ . Although the result already appears in the literature, see [16, Chapter 4, (2.13)], we shall for the readers convenience give its complete derivation (this proof is different from Taylor’s). Note that the operator appears naturally when we solve the wave equation on the sphere, therefore we call it the propagator of the wave equation on the sphere. First we need to introduce some notation about distributions. For any > 0 and s ∈ R, the expression s R ∈ y → (y ± i )− 2 defines uniquely a function on a real line, which can be viewed as a distribution in S (R). It is well-known that for any φ ∈ S(R) there exists a limit s s lim (y ± i )− 2 φ(y)dy =: (y ± i0)− 2 φ(y)dy , 0

which defines a distribution in S (R). In the sequel we will treat this distribution s as if it were a function, denoting it by (y ± i0)− 2 . Note that for s, > 0 we have the identity s ∞ s−2 s e∓iπ 4 (y ± i )− 2 = eit(±y+i) t 2 dt . (2.1) Γ(s/2) 0 We shall in this section show the following result: Proposition 2.1. 1. If θ = π2k, k ∈ Z, then eiθΛ = (−1)kd times the identity. 2. If θ = π(2k + 1), k ∈ Z, then eiθΛ = eiπ(2k+1)(d/2−1) P , where P is the parity operator.

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3. If θ ∈]π2k, π(2k + 1)[, k ∈ Z, then eiθΛ has the distributional kernel eiθΛ (ω, ω ) = (2π)−d/2 sin θ Γ(d/2)e−iπ/2 (−ω · ω + cos θ − i0)−d/2 . 4. If θ ∈]π(2k − 1), π2k[, k ∈ Z, then eiθΛ has the distributional kernel eiθΛ (ω, ω ) = (2π)−d/2 sin θ Γ(d/2)e−iπ/2 (−ω · ω + cos θ + i0)−d/2 . 2.2. Tchebyshev and Gegenbauer polynomials Recall that the Tchebyshev polynomials (of the first kind) are defined by the identity Tn (cos φ) := cos nφ , n = 0, 1, . . . . Let |t| < 1. An elementary calculation yields the following generating function of Tchebyshev polynomials: ∞ 2tl 2 Tl (w) . (2.2) − ln(1 − 2wt + t ) = l l=1

Gegenbauer polynomials are defined by the generating function [1, 14] ∞

1 (d−2)/2 = tl Cl (w) . (1 − 2wt + t2 )(d−2)/2 l=0

(2.3)

The left hand sides of (2.2) and (2.3) look different. But after simple manipulations (involving differentiation of both sides) they become quite similar: ⎧

l ⎨T0 (w) + ∞ d = 2; −1 l=1 t 2Tl (w) , −t + t = (2.4) d ⎩ ∞ l+ d2 −1 2l+d−2 (d−2)/2 (t − 2w + t−1 ) 2 t C (w) , d ≥ 3 . l=0 l d−2 By substituting t = eiθ for Im θ > 0, we rewrite this as ⎧

ilθ ⎨T0 (w) + ∞ l=1 e 2Tl (w) , −i2 sin θ d = ⎩ ∞ i(l+ d2 −1)θ 2l+d−2 (d−2)/2 2d/2 (cos θ − w) 2 (w) , l=0 e d−2 Cl

d = 2; (2.5) d ≥ 3.

2.3. Projection onto lth sector of spherical harmonics It is well-known that the integral kernel of the projection onto lth sector of spherical harmonics in L2 (S d−1 ) can be computed explicitly. This fact is usually presented in the literature as the addition theorem for spherical harmonics, see e.g. Theorem 2, Sect. 2 of [14]. In the case d = 3 it can also be found in [17]. In what follows dˆ y will denote the natural measure on the unit sphere S d−1 . 2π d/2 . Note that for this measure the area of S d−1 equals sd−1 = Γ(d/2) Proposition 2.2. Let Y be an lth order spherical harmonic in L2 (S d−1 ).

554


1. In the case d = 2, 1 T0 (ˆ x · yˆ)Y (ˆ y )dˆ y = δl0 Y (ˆ x) ; 2π 1 S 1 Tn (ˆ x · yˆ)Y (ˆ y )dˆ y = δln Y (ˆ x) , S1 π


(2.6) n = 1, 2, . . . .

2. In the case d ≥ 3, (d − 2 + 2l)Γ(d/2 − 1) (d−2)/2 Cn (ˆ x · yˆ)Y (ˆ y )dˆ y = δln Y (ˆ x) . 4π d/2 S d−1

(2.7)

Proof. The case (2.6) is elementary. In the proof below we restrict ourselves to d ≥ 3. Let us first recall the formula for the Green’s function in Rd for d ≥ 3: Gd (x) = −

Γ(d/2 − 1) 1 =− , sd−1 (d − 2)|x|d−2 4π d/2 |x|d−2

(2.8)

It satisfies ΔGd = δ0 , where δ0 is Dirac’s delta at zero. Recall also the 3rd Green’s identity: if Δg = 0 and Ω is a sufficiently regular domain containing x, then g(y)∇y Gd (x − y)ds(y) − (∇g)(y)Gd (x − y)ds(y) . (2.9) g(x) = ∂Ω

∂Ω

We extend Y to Rd by setting g(x) = |x|l Y (ˆ x). Note that Δg(x) = 0 ,

x ˆ∇x g(x) = lg(x) .

By (2.3), for |x| < |y|, Gd (x − y) = −

∞ Γ(d/2 − 1) (d−2)/2 C (ˆ xyˆ)|x|n |y|−d+2−n , 4π d/2 n=0 n

∞ Γ(d/2 − 1) yˆ · ∇y Gd (x − y) = (d − 2 + n)Cn(d−2)/2 (ˆ xyˆ)|x|n |y|−d+1−n . 4π d/2 n=0

We apply (2.9) to the unit ball, so that |y| = 1 and |x| < 1: l |x| Y (ˆ x) = g(ˆ y )ˆ y · ∇Gd (x − yˆ)dˆ y− (ˆ y · ∇g)(ˆ y )Gd (x − yˆ)dˆ y S d−1

S d−1

∞ Γ(d/2 − 1) (d − 2 + n + l) Y (ˆ y )Cn(d−2)/2 (ˆ xyˆ)|x|n dˆ y . (2.10) = 4π d/2 n=0 S d−1 Comparing the powers of |x| on both sides of (2.10), we obtain (2.7).

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2.4. Proof of Proposition 2.1 be the orthogonal projection onto lth order spherical harmonics on S d−1 . Let Qd−1 l We multiply (2.5) by Γ(d/2)2−1 π −d/2 , set w = ω · ω and use Proposition 2.2. We obtain ∞

−i sin θ Γ(d/2) = Qd−1 (ω, ω )ei(l+d/2−1)θ l (2π)d/2 (cos θ − ω · ω )d/2 l=0 = eiθΛ (ω, ω ) . Replace θ with θ + i , where θ is real and positive. For small we have cos(θ + i ) ≈ cos θ − i sin θ . Now sin θ > 0 for θ ∈]π2k, π(2k + 1)[ and sin θ < 0 for θ ∈]π(2k − 1), π2k[, which ends the proof for the case θ ∈ R \ πZ. The case θ ∈ πZ is obvious. 2.5. Propagator as a FIO The operator eiθΛ is an interesting explicit example of a Fourier integral operator (whenceforth abbreviated FIO) in the sense of H¨ ormander [11, 12]. As a side remark, let us check this directly. (The material of this subsection will not be used in what follows.) Let X be a smooth compact manifold of dimension n. Let us recall some basic definitions related to Fourier integral operators on X, cf. [12]. We say that X × X × Rk (x, x , θ) → φ(x, x , θ) ∈ R is a non-degenerate phase function if it is a function homogeneous of degree 1 in θ, smooth and satisfying ∇φ = 0 away from θ = 0, and such that (x, x , θ) ∈ X × X × Rk | ∇θ φ(x, x , θ) = 0 is a smooth manifold on which ∇∇θ1 φ, . . . , ∇∇θk φ are linearly independent. Let χ be a smooth and homogeneous transformation on T∗ X \ X×{0}. We say that it is associated to a non-degenerate phase function φ iff two pairs (x, ξ), (x , ξ ) ∈ T∗ X \ X×{0} satisfy χ(x , ξ ) := (x, ξ) exactly when ξ = ∇x φ(x, x , θ) , ξ = −∇x φ(x, x , θ) , 0 = ∇θ φ(x, x , θ) .

(2.11)

The transformation χ is automatically canonical, that is, it preserves the symplectic form of T∗ X. We say that a smooth function X × X × Rk (x, x , θ) → u(x, x , θ) is an amplitude of order m iff ∂xα ∂xα ∂θβ u = O θm−|β| .

556



Recall from [12] that an operator U from C ∞ (X) to D (X) is called a Fourier integral operator of order n k m− + 2 2 iff in local coordinate patches its distributional kernel can be written as U (x, x ) = eiφ(x,x θ) u(x, x , θ) dθ , (2.12) where θ ∈ Rk are auxiliary variables, the function φ is a non-degenerate phase function, and u is an amplitude of order m. If the phase of U is associated to a canonical transformation χ, we say that U itself is associated to χ. Let W F (v) denote the wave front set of a distribution v, as defined in [12, Section 2.5]). Let us remark that under appropriate additional assumptions on a FIO U , for all v ∈ D (X) we have W F (U v) ⊆ χ W F (v) ; see [12, Proposition 2.5.7 and Theorem 2.5.14]. (Note that these additional assumptions are fulfilled for the example U = Uθ given below.) Theorem 2.3. The operator Uθ := eiθΛ is a FIO of order 0. Proof. If θ ∈ πZ, then eiθΛ is a so-called point transformation. But point transformations given by diffeomorphisms of the underlying manifold are always FIO of order zero. Assume that θ ∈ / πZ. Consider e.g. the case θ ∈]π2k, π(2k + 1)[. By (2.1) and Proposition 2.1 the kernel of Uθ can then be written as ∞ d−2 Uθ (ω, ω ) = C eit(ω · ω −cos θ) t 2 dt . (2.13) 0

If we compare (2.13) with the definition of a FIO given above, we see that t(ω · ω − cos θ) is a non-degenerate phase function. We also have n = d − 1, m = d−2 2 and k = 1. Thus Uθ is a FIO of order d−2 d−1 1 − + = 0. 2 2 2

Let us describe the canonical transformation associated to the FIO Uθ . Let (ω, ξ) ∈ T∗ (S d−1 ). It is enough to assume that |ξ| = 1. Then the canonical transformation χθ associated to Uθ is given by χθ (ω , ξ ) = (ω, ξ), where ω = ω cos θ − ξ sin θ , ξ = ω sin θ + ξ cos θ .

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3. Main result 3.1. Scattering matrix at positive energies Throughout the paper we fix μ ∈]0, 2[ and γ > 0, and impose the conditions (1.3) and (1.4) on the potential V (r). It will be convenient to fix R0 > 0 such that V (r) < 0 for r > R0 . Since our potential is radial, to define the scattering operator we can use the scattering phase shift formalism. This formalism is, at least under some mild additional conditions on the potential, equivalent to the usual time-dependent formalism of scattering theory, see [5] for an elaboration. In the paper we will need just the scattering matrix at zero energy. Let us however start with defining the scattering matrix at a positive energy. Let l ∈ N ∪ {0} have the meaning of a total angular momentum and λ > 0 be the energy. Introduce the notation − 1)2 − 4−1 . r2 Consider the reduced Schr¨ odinger equation on the half-line ]0, ∞[ for energy λ: Vl (r) = 2V (r) +

(l +

d 2

−u + Vl u = 2λu . One can show that all real solutions of (3.1) satisfy r −μ 14 −μ 12 lim (λ + γr ) u(r) − C sin (2λ + 2γ˜ r ) d˜ r+D =0 r→∞

(3.1)

(3.2)

R0

for some C > 0 and D ∈ R. The regular solution is the solution satisfying lim r−l−

r→0

d−1 2

u(r) = 1 .

(3.3)

(The existence and uniqueness of the regular solution is usually proven by studying an integral equation of Volterra type, cf. [15].) Now the phase shift at energy λ is defined in terms of the constant D for the regular solution by σl (λ) =D+

√ 2

∞

λ + γr−μ −

R0

d − 3 + 2l π . (3.4) λ − V (r) dr − 2λR0 + 4

We define the (modified) scattering matrix at energy λ as the unitary operator on L2 (S d−1 ) that on lth order spherical harmonics Y acts as S(λ)Y = ei2σl (λ) Y . Note that the above definition is adapted to the long-range case. However, we use it also in the short-range case, because it makes possible to take the limit as λ 0. σl (λ) defined above is also consistent with the convention adopted in [5]. For comparison, let us mention the standard definition of the phase shift in the short-range case. (3.2) needs to be replaced with 1 lim u(r) − C sin (2λ) 2 r + Dsr = 0 , (3.5) r→∞

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and (3.4) with

d − 3 + 2l π. 4 In particular under (1.3) and (1.4) with μ ∈]1, 2[ √ ∞ √ σlsr (λ) = σl (λ) + 2 λ − λ − V (r) dr . σlsr (λ) = Dsr +

(3.6)

(3.7)

R0

The integral to the right in (3.7) does not have a (finite) limit as λ 0 in this case. 3.2. Scattering matrix at zero energy It turns out that under the conditions (1.3) and (1.4) the definition of the scattering matrix can be extended to zero energy. The definitions of σl (0) and S(0) are special cases of the definitions of σ(λ) and S(λ) for λ > 0 described in Subsection 3.1. Explicitly, consider the zero-energy case of (3.1) −u + Vl u = 0 .

(3.8)

It follows from the WKB-analysis given in the bulk of Subsection 3.3 that all real solutions of (3.8) satisfy r −μ 14 −μ 12 (2γ˜ r ) d˜ r+D =0 (3.9) lim (γr ) u(r) − C sin r→∞

R0

for some C > 0 and D ∈ R. Consider D corresponding to the regular solution which is fixed by the requirement (3.3). Now we define the (modified) zero-energy phase shift as ∞ d − 3 + 2l π. (3.10) 2γr−μ − −2V (r) dr + σl (0) = D + 4 R0 We define the (modified) scattering matrix at energy 0 by S(0)Y = ei2σl (0) Y , where Y is any lth order spherical harmonic. Note that it follows from the proof given in the bulk of this section (as well as from [5]) that σl (0) = lim σl (λ) , λ0

S(0) = s− lim S(λ) . λ0

The following theorem is the main result of the paper: Theorem 3.1. For a certain compact operator K on L2 (S d−1 ), we have μπ

S(0) = eic0 e−i 2−μ Λ + K , where c0 =

√ ∞ 4 2γ 1− μ2 R0 2γr−μ − −2V (r) dr . +2 2−μ R0

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3.3. One-dimensional WKB-analysis We shall show the following asymptotics: Proposition 3.2. The phase shift obeys c μπ σl (0) = − l + + o(l0 ) , (3.11) 2(2 − μ) 2 √ ∞ c πμ(d − 2) 2 2γ 1− μ2 =− + R0 2γr−μ − −2V (r) dr . + 2 4(2 − μ) 2−μ R0 Clearly Theorem 3.1 is a consequence of Proposition 3.2. This subsection is devoted to the main part of the proof of Proposition 3.2. It is based on detailed 1-dimensional analysis. For convenience, let us note that the effective potential Vl of (3.8) for the case V (r) = −γr−μ is given by k(k + 1) d−3 . , k := l + r2 2 Abusing slightly notation, we shall henceforth denote this expression by Vk , and similarly σk (0) := σl (0). Note that now we have V0 (r) = −2γr−μ . In the case V = −γr−μ , there is for k > 0 a unique zero, say, denoted r0 , of the effective potential Vk . Explicitly, 1 k(k + 1) 2−μ Vk (r0 ) = 0 for r0 = . (3.12) 2γ Vl (r) = −2γr−μ +

For later applications, let us notice that Vk (r0 ) = −(2 − μ)

k(k + 1) . r03

(3.13)

Clearly Vk is positive to the left of r0 and negative to the right of r0 . Proposition 3.3. The regular solution satisfies (up to multiplication by a positive constant) r 1 π u(r) = (−Vk )− 4 (r) sin −Vk (˜ r) d˜ r + + o(k 0 ) + O(r−k ) , (3.14) 4 r0 where o(k 0 ) signifies a vanishing term that is independent of r and k > 0. 3.3.1. Scheme of proof of Proposition 3.3. We shall first concentrate on the case where V = −γr−μ ; the general case will be treated by the same scheme (to be discussed later). We introduce a partition of ]0, ∞[ into four subintervals given as follows in terms of 1 , 2 , 3 ∈ (0, 1] to be fixed later: 1. 2. 3. 4.

I1 I2 I3 I4

1

=]0, r1 ], r1 = r0 k − 2−μ . =]r1 , r2 ], r2 = r0 (1 − k −2 ). =]r2 , r3 ], r3 = r0 (1 + k −3 ). =]r3 , ∞[.

560



In each of the intervals Ij where j = 2, 3 or 4, we shall specify a certain model Schr¨ odinger equation together with its two linearly independent solutions φ± j . In terms of these, we can construct exact solutions to the reduced equation −u + Vk u = 0

(3.15)

by the method of variation of parameters, cf. for example [9]. Our subject of study is formulas for the regular solution u = uk . Specifically, in the interval I1 we shall use a comparison argument to get estimates of the regular solution at r = r1 . Then − we shall use a connection formula to get estimates of the “coefficients” a+ 2 and a2 of the ansatz + − − (3.16) u = a+ j φj + aj φj with j = 2 at the same point r = r1 . Next, using the differential equation for a+ 2 and a− 2 we shall derive estimates of these quantities at r = r2 . Proceeding similarly we shall consecutively represent u by (3.16) on I3 and I4 using connection formulas at r2 and r3 , and eventually get estimates in the interval I4 , and whence derive the relevant asymptotics of u. Suppose φ− and φ+ solve the same one-dimensional Schrödinger equation, say, −φ + Aφ = 0 . The variation of parameter method for the equations (3.15) and (3.16) yields + + d φ− φ+ 0 0 a a = (Vk − A) . (3.17) d + d − − + − − φ a φ a φ φ dτ dτ dτ (We have omitted the subscript j). We introduce the notation W (φ− , φ+ ) for the d + d − Wronskian W (φ− , φ+ ) = φ− dr φ − φ+ dr φ . Then we write B = Vk − A and transform (3.17) into + d a+ a = N , a− dr a− where N=

B W (φ− , φ+ )

φ− φ+ −(φ+ )2

(φ− )2 −φ− φ+

.

For a positive increasing continuous function f on I (to be specified), we introduce the matrix T = diag(1, f −1 ). We compute B f (φ− )2 φ− φ+ −1 TNT = . −f −1 (φ+ )2 −φ− φ+ W (φ− , φ+ ) r Introducing the operator (Mj z)(r) = rj−1 Nj (r )z(r ) dr , j ≥ 2, acting on continuous functions z( · ) : Ij → R2 , the above differential equation is solved by + + ∞ aj aj m Mj zj ; zj = (r) − zj = − (rj−1 ) . a− a j j m=1

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Whence we have the bound + ∞ m aj Tj (r) ≤ (r) − z Tj Mj Tj−1 Tj zj (r) ; j − aj

(3.18)

m=1

to the right Tj is considered as an operator acting as (Tj z)(r ) = (Tj )(r )z(r ). Using that fj is increasing, we can estimate r −1 (Tj Mj Tj z)(r) ≤ (Tj Nj Tj−1 )(r ) z(r ) dr , rj−1

which applied repeatedly in (3.18) yields the following bound for r ∈ Ij : + a Tj (r) (r) − zj a− r

≤

exp

=

rj−1

r

exp rj−1

(Tj Nj Tj−1 )(r ) dr

r)zj sup Tj (˜

−1

(Tj Nj Tj−1 )(r ) dr

r˜∈Ij

Tj (rj−1 )zj .

−1

(3.19)

We specify in the following φ± j , Aj , Bj and fj for j = 2, 3 and 4; in all cases = 1:

+ W (φ− j , φj )

Re interval I2 . We define − 14 ±

−2 φ± Vk 2 (r) = 2 1

e

r √ r1

Vk dr

,

(3.20a)

compute 2 − 14 14 5 Vk 1 Vk B2 = − Vk Vk = − + , 16 Vk 4 Vk 2 5 Vk 1 Vk A2 = Vk + − , 16 Vk 4 Vk and let f2 (r) =

√ φ+ 2 rr Vk dr 2 (r) 1 . = e − φ2 (r)

(3.20b)

(3.20c)

Re interval I3 . We define (in terms of the Airy function, cf. [9] and [10, Definition 7.6.8]) √ −1 1 πζ Ai − ζ 2 (r − r0 ) ; ζ := |Vk (r0 )| 6 , (3.21a) φ+ 3 (r) = √ πi −1 2πi − 2 φ3 (r) = πe 6 ζ Ai − ζ e 3 (r − r0 ) √ πi 2πi (3.21b) + πe− 6 ζ −1 Ai − ζ 2 e− 3 (r − r0 ) ,

562


compute B3 (r) = Vk (r) − Vk (r0 ) + Vk (r0 )(r − r0 ) =

r


(r − r˜)Vk (˜ r) d˜ r,

(3.21c)

if r < r0 ; if r ≥ r0 .

(3.21d)

r0

A3 (r) = Vk (r0 ) + Vk (r0 )(r − r0 ) , and let

3 exp − 43 ζ 3 (r0 − r) 2 , f3 (r) = 1,

Re interval I4 . We define 1

−4 φ− 4 (r) = (−Vk ) 1

π , 4 r0 r π cos −Vk dr + , 4 r0

−4 φ+ sin 4 (r) = (−Vk )

r

−Vk dr +

(3.22a) (3.22b)

compute 1 1 5 B4 = − (−Vk )− 4 (−Vk ) 4 = − 16 2 5 Vk 1 Vk A4 = Vk + − , 16 Vk 4 Vk

Vk Vk

2 +

1 Vk , 4 Vk

(3.22c)

and let f4 = 1 .

(3.22d)

3.3.2. Details of proof of Proposition 3.3. We start implementing the scheme outlined in Subsubsection 3.3.1. In the interval I1 we shall use a standard comparison argument. With Vk ˜ ˜ ˜ k+1 replaced by V = k(k+1) r 2 , the regular solution is given by the expression u = r and the corresponding Riccati equation ψ = V − ψ2

(3.23)

˜

is solved by ψ = φφ = k+1 r . We fix 1 ∈]0, 1] (actually 1 > 0 can be chosen arbitrarily) and notice the following uniform bound in r ∈ I1 : Vk (r) =

k(k + 1) 1 + O(k −1 ) . 2 r

(3.24)

Using (3.24), we can find C > 0 such that with k ± := k(1 ± Ck −1 ) and ± ± := k (kr2 +1) there are estimates ≤ Vk+ (r) Vk (r) , r ∈ I1 . ≥ Vk− (r)

Vk± (r)

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563

Now, by using [2, Theorem 1.8] and the Riccati equation, it follows that the regular solution u of (3.15) is positive in I1 , and that v := uu obeys the bounds + ≤ k r+1 v(r) , r ∈ I1 . − ≥ k r+1 We conclude the uniform bound k + 1 1 + O(k −1 ) , r ∈ I1 . v(r) = r The connection formula at r = r1 reads + − − a+ 1 j φj + aj φj = , cj + − − v r=r a+ j (φj ) + aj (φj ) r=r j−1

(3.25)

j = 2.

(3.26)

j = 2.

(3.27)

j−1

Obviously, (3.26) is solved for the coefficients by + − cj (−φ− aj j ) + φj v = + + a− (φ+ W (φ− j j ) − φj v j , φj ) r=r r=r j−1

, j−1

Next, from (3.20a) we compute 1 Vk ) = ± V − φ± (φ± k 2 2 . 4 Vk

(3.28)

We substitute these expressions and (3.25) in the right hand side of (3.27) and obtain + 2k 1 + O k −1 a2 (r1 ) = c2 . (3.29) O k −1 a− r1 2 (r1 ) To apply (3.19), we notice that −1 1 1 −1 − + T2 N2 T2 = B2 φ2 φ2 = B2 O Vk 2 . −1 −1 Whence (for the first inequality below we assume that the integral is bounded in k so that the inequality exp x − 1 ≤ Cx applies – this will be justified by (3.31)), + + a2 a2 T2 (r2 ) (r2 ) − − (r1 ) a− a 2 2 2 r2 − 12 5 Vk k 1 Vk = exp O Vk + − dr − 1 O 16 V 4 V r k k 1 r1 r2 /r0 2 Vk k |V | ≤ C1 r0 + k3 ds (changing variables r = r0 s) 5 r1 2 2 r1 /r0 Vk Vk r2 /r0 −6 s s−4 −1 ≤ C2 r1 ds 5 + 3 (s−2 − s−μ ) 2 (s−2 − s−μ ) 2 r1 /r0 = C2 r1−1

r2 /r0

1/2

1/2

· · · ds +

· · · ds r1 /r0

564


≤

C3 r1−1

r2 /r0

max

2−μ − 52

(1 − s

)

ds,

3

−1

s

ds

r1 /r0

1/2

≤ C4 k 2 2 −1

1/2


k ; r1

(3.30)

we need here 3 2 − 1 < 0 . 2

(3.31)

We conclude by combining (3.29) and (3.30): + 3 + O(k 2 2 −1 ) 1 + O(k − 1 ) √ 2k a2 (r2 ) r2 . = c2 3 a− r1 O(k −1 ) + e2 r1 Vk dr O(k 2 2 −1 ) 2 (r2 )

(3.32)

Next we repeat the above procedure passing from the interval I2 to I3 . The first issue is the connection formula (3.26) with j = 2 replaced by j = 3. The left hand side can be estimated using (3.28), (3.32) and the following estimates (where (3.31) is used): 1 k(k + 1) 1 − (1 − k −2 )2−μ 2 Vk (r2 ) = r2 2 1 k = (2 − μ) 2 k − 2 1 + O(k −2 ) , (3.33) r0 2 O kr3 Vk (r2 ) 2 = = r2−1 O k 2 . (3.34) Vk (r2 ) Vk (r2 ) Notice that (3.33) dominates (3.34) (by (3.31) again), so that 2 1 k 1 Vk Vk − (r2 ) = (2 − μ) 2 k − 2 1 + O(k −2 ) . 4 Vk r0 We conclude that 3 (φ+ 2 ) (r2 ) 1 + O(k 2 2 −1 ) + φ2 (r2 ) 2 1 k 3 = (2 − μ) 2 k − 2 1 + O(k −2 ) + O(k 2 2 −1 ) . r0

v(r2 ) =

(3.35)

By (3.26) and (3.27) with j = 2 replaced by j = 3, up to multiplication by a positive constant, + − (−φ− a3 3 ) + φ3 v = . (3.36) + a− (φ+ 3 r=r 3 ) − φ3 v r=r 2

2

Vol. 10 (2009)


565

It remains to examine the asymptotics of φ± 3 and their derivatives at r2 . For that we notice the asymptotics as r − r0 → −∞, cf. [9, Appendix B] and [10, (7.6.20)], 3 exp − 23 ζ 3 (r0 − r) 2 3 + φ3 = 1 + O ζ −3 (r0 − r)− 2 , (3.37a) 3 1 2ζ 2 (r0 − r) 4 2 3 3 − 3 ζ (r0 − r) 2 1 exp 3 3 2 1 + O ζ −3 (r0 − r)− 2 , (3.37b) ) = ζ (r − r) (φ+ 0 3 1 3 2ζ 2 (r0 − r) 4 2 3 3 exp 3 ζ (r0 − r) 2 3 − 1 + O ζ −3 (r0 − r)− 2 , (3.37c) φ3 = 3 1 ζ 2 (r0 − r) 4 2 3 3 −3 ζ (r0 − r) 2 1 exp − 3 − 32 3 (φ3 ) = −ζ (r0 − r) 2 (r − r) 1 + O ζ . (3.37d) 0 3 1 ζ 2 (r0 − r) 4 √ 3 2 3 Since ζ 3 (r0 − r2 ) 2 2 − μk 1− 2 2−μ , cf. (3.13), these asymptotics are applicable. By the same computation, (3.35) can be rewritten as 1 3 (3.38) v(r2 ) = ζ 3 (r0 − r2 ) 2 1 + O(k −2 ) + O(k 2 2 −1 ) . Whence, in conjunction (3.36), we obtain (up to multiplication by a positive constant) + 3 3 exp 23 ζ 3 (r0 − r2 ) 2 1 + O(k −2 ) + O(k 2 2 −1 ) a3 (r2 ) = . (3.39) 3 3 a− exp − 23 ζ 3 (r0 − r2 ) 2 O(k −2 ) + O(k 2 2 −1 ) 3 (r2 ) Next, to apply (3.19) with j = 3 we need the following asymptotics of φ± 3 and their derivatives as r − r0 → +∞, cf. [9, Appendix B] and [10, (7.6.20) and (7.6.21)]: −3 3 2 3 π − 32 − 14 − 32 2 + φ+ ζ = ζ (r − r ) (r − r ) (r − r ) sin +O ζ , (3.40a) 0 0 0 3 3 4 −3 3 1 3 2 3 π − 32 2 (r − r ) 4 2 + ζ ) = ζ (r − r ) (r − r ) cos , (3.40b) +O ζ (φ+ 0 0 0 3 3 4 −3 3 2 3 π − 32 − 14 − 32 2 + ζ = ζ (r − r ) (r − r ) (r − r ) cos +O ζ , (3.40c) φ− 0 0 0 3 3 4 −3 3 1 3 2 3 π − 32 2 (r − r ) 4 2 + ζ ) = −ζ (r − r ) (r − r ) sin +O ζ . (3.40d) (φ− 0 0 0 3 3 4 In particular, T3 N3 T3−1 = B3 ζ −2 O(k 0 )

uniformly in r ∈ I3 .

In conjunction with (3.19), (3.13) and the fact that 4+2μ Vk (r) = O k − 2−μ

uniformly in r ∈ I3 ,

(3.41)

566

we obtain



+ + a3 a3 T3 (r3 ) ) − ) (r (r 3 2 − − a3 a3 4+2μ 2 1+μ ≤ C1 (r3 − r0 )3 + (r0 − r2 )3 k − 2−μ k 3 2−μ a+ 3 (r2 ) ≤ C2 k 3 −3 min(2 ,3 ) a+ 3 (r2 ) ; 4

(3.42)

here we need

4 − 3 min( 2 , 3 ) < 0 , (3.43) 3 cf. (3.31). At this point let us for convenience take 3 = 2 , so that (3.43) simplifies and in conjunction with (3.31) leads to the single requirement 2 4 > 2 = 3 > . (3.44) 3 9 We conclude that (up to multiplication by the positive constant a+ 3 (r2 )) + 4 a3 (r3 ) 1 + O(k 3 −32 ) = . (3.45) 4 − a3 (r3 ) O(k 3 −32 ) Next we need to study the connection formula passing from I3 to I4 ; a little linear algebra takes it to the form + + a3 a4 W (φ− , φ+ ) W (φ− , φ− ) 4 3 4 3 = r = r3 . + + − + − , a− , φ ) W (φ , φ ) a W (φ 4 3 4 3 4 3

So we need to compute the appearing Wronskians. To this end we note the following uniform asymptotics for r ∈ [r0 , r3 ], which are readily obtained from (3.13) and (3.41) (recall that by now 3 = 2 ): (3.46a) Vk (r) = Vk (r0 )(r − r0 ) 1 + O(k −2 ) , Vk (r) = Vk (r0 ) 1 + O(k −2 ) , (3.46b) 1 3 − 2 −Vk (r) = ζ (r − r0 ) 2 1 + O(k ) , (3.46c) r 3 2 −Vk (r ) dr = ζ 3 (r − r0 ) 2 1 + O(k −2 ) 3 r0 3 5 2 = ζ 3 (r − r0 ) 2 + O(k 1− 2 2 ) . (3.46d) 3 Due to (3.46c) and (3.46d), the asymptotics (3.40a)–(3.40d) at the point r = r3 can be written in terms of r3 π −Vk (r ) dr + θ := 4 r0 as 3 φ+ 1− 52 2 3 (r3 ) ) + O(k −2 ) + O(k 2 2 −1 ) , (3.47a) 1 = sin θ + O(k − 4 (−Vk (r3 )) 3 (φ+ 1− 52 2 3 ) (r3 ) ) + O(k −2 ) + O(k 2 2 −1 ) , (3.47b) 1 = cos θ + O(k (−Vk (r3 )) 4

Vol. 10 (2009)


φ− 3 (r3 ) (−Vk (r3

1 ))− 4

(φ− 3 ) (r3 )

(−Vk (r3 ))

1 4

567

5 3 = cos θ + O(k 1− 2 2 ) + O(k −2 ) + O(k 2 2 −1 ) ,

(3.47c)

5 3 = − sin θ + O(k 1− 2 2 ) + O(k −2 ) + O(k 2 2 −1 ) .

(3.47d)

Next, using that −Vk

(r3 ) = O(k 2 2 −1 ) , 3

3

(−Vk ) 2

cf. (3.46a) and (3.46b), we obtain for the functions φ± 4 − 14 φ+ sin(θ) , 4 (r3 ) = − Vk (r3 ) 1 3 4 cos(θ) + O(k 2 2 −1 ) , (φ+ 4 ) (r3 ) = − Vk (r3 ) − 14 cos(θ) , φ− 4 (r3 ) = − Vk (r3 ) 14 3 − sin(θ) + O(k 2 2 −1 ) . (φ4 ) (r3 ) = − − Vk (r3 )

(3.48a) (3.48b) (3.48c) (3.48d)

The matrix of Wronskians is readily computed using (3.40a)–(3.40d) and (3.47a)–(3.47d). In combination with (3.45), we obtain (using in the second step (3.44)) + 3 4 5 a4 (r3 ) − 1 = O(k −2 ) + O(k 2 2 −1 ) + O(k 3 −32 ) + O(k 1− 2 2 ) (r ) a− 3 4 = O(k −2 ) + O(k 2 2 −1 ) + O(k 3 −32 ) . 3

4

(3.49)

Now we estimate in I4 using (3.49) (and mimicking partially (3.30)) + + a a4 4 − (r) − − (r3 ) a a4 4 2 r 1 5 Vk 1 Vk −2 ≤ C1 + exp O (−Vk ) − dr − 1 16 Vk 4 Vk r3 2 r/r0 − V − Vk k ds (changing variables r = r0 s) ≤ C2 r0 5 + 3 2 2 (−V ) (−V ) r3 /r0 k k r/r0 5 3 μ/2−1 ≤ C3 r0 sμ/2−2 (1 − sμ−2 )− 2 + (1 − sμ−2 )− 2 ds r3 /r0

≤

μ/2−1 C4 r0

2 μ/2−1

≤ C5 r0

2

− 52

(1 − sμ−2 )

sμ/2−2 ds +

ds

r3 /r0

3

k 2 2

= O(k 2 2 −1 ) . 3

∞

(3.50)

568



By the same type of estimation we also deduce that for fixed k there exist k > 0 and a± 4 (∞) ∈ R such that ± −k a± ). 4 (r) = a4 (∞) + O(r

By applying (3.50) with r = ∞ in combination with (3.49) (and using an elementary trigonometric formula), we conclude that (3.14) is true. The general case. It remains to prove (3.14) under Conditions (1.3) and (1.4). All previous constructions and estimates carry over, so below we consider only some additional estimates that are needed. Denoting U (r) = 2V (r) − 2γr−μ , the functions φ± j and fj and the potentials Aj are exactly the same, while the potentials Bj are given as the old Bj plus U , j = 2, 3, 4. Re interval I1 . We notice that (3.24) is valid (here with Vk defined upon replacing 2γr−μ → 2V ). Whence we can proceed exactly as before. Re interval I2 . In addition to (3.30) we need the following estimation (assuming in the last step that μ2 + < 1):

r2

r1

−1 |U O(Vk 2 )| dr O

≤ C1

k r0 r1

k r1

μ

r−1− 2 −

r2 /r0

1

r1 /r0

1− μ −

k r0 2 ≤ C2 r1 k 2 k . ≤ C3 k − 2−μ r1

ds (changing variables r = r0 s)

Vk2 μ

s− 2 −

r2 /r0

1

r1 /r0

(1 − s2−μ ) 2

ds (3.51)

Re interval I3 . In addition to (3.42) we need the following estimation

r3

|U ζ −2 | dr r3 μ 2 1+μ ≤ C1 k 3 2−μ r−1− 2 − dr r2

r2

≤ C2 k

2 1+μ 3 2−μ

−μ 2 −

r0

(k −2 + k −3 )

≤ C3 k 3 −2 − 2−μ . 1

2

Due to (3.44) the right hand side of (3.52) vanishes.

(3.52)

Vol. 10 (2009)


569

Re interval I4 . In addition to (3.50) we need the following estimation: r U O (−Vk )− 12 dr r3

≤

C1 r0−

r/r0

s−1− 1

r3 /r0

(1 − sμ−2 ) 2

ds (changing variables r = r0 s)

≤ C2 k − 2−μ . 2

(3.53)

This ends the proof of (3.14). 3.4. End of proof of Proposition 3.2 We need the following elementary identity: Lemma 3.4. Let μ < 2. Then ∞ √ 2−π . r−μ − r−2 − r−μ dr = 2−μ 1

(3.54)

1

Proof. We first substitute r = s μ−2 and then s = sin2 φ. Thus the left hand side of (3.54) equals 1 π2 3 √ 1 − cos φ 1 2 s− 2 1 − s − 1 ds = − 1 dφ 2−μ 0 2−μ 0 sin2 φ π/2 1 − cos φ 2−π 2 − φ . = = 2−μ sin φ 2−μ 0 Proof of Proposition 3.2. Using Proposition 3.3 we calculate from (3.10) r r π kπ σk (0) = lim −Vk (˜ r)d˜ r+ − −2V (˜ r)d˜ r+ + o(k 0 ) r→∞ 4 2 r R0 ∞ 0 −Vk (r) − −V0 (r) dr = r0 ∞ −V0 (r) − −2V (r) dr + R0 r0

−

R0

−V0 (r)dr +

(k + 12 )π + o(k 0 ) . 2

Now (using Lemma 3.4) ∞ −Vk (r) − −V0 (r) dr = k(k + 1) r0

1

∞

r−μ − r−2 −

√ r−μ dr

2−π = k(k + 1) ; 2−μ √ R0 2 2 2γ 1− μ2 R −V0 (r)dr = − k(k + 1) + . 2−μ 2−μ 0 r0

570

Thus, σk (0) −


∞

R0

−V0 (r) −


−2V (r) dr

√ (k + 12 )π 2 2γ 1− μ2 π + + R + o(k 0 ) = − k(k + 1) 2−μ 2 2−μ 0 √ (k + 12 )πμ 2 2γ 1− μ2 + R =− + o(k 0 ) . 2(2 − μ) 2−μ 0

References [1] G. Andrews, R. Askey and R. Roy, Special functions, Cambridge University, Cambridge 1999. [2] G. Birkhoff, G. C. Rota, Ordinary differential equations, (fourth edition) New York, Wiley 1989. [3] J. Derezi´ nski, C. Gérard, Scattering theory of classical and quantum N -particle systems, Texts and Monographs in Physics, Berlin, Springer 1997. [4] J. Derezi´ nski, E. Skibsted, Classical scattering at low energies, in Perspectives in Operator Algebras and Mathematical Physics, Theta Series in Advanced Mathematics 8 (2008), 51–83. [5] J. Derezi´ nski, E. Skibsted, Quantum scattering at low energies, to appear in Journal of Functional Analysis. [6] S. Fournais, E. Skibsted, Zero energy asymptotics of the resolvent for a class of slowly decaying potentials, Math. Z. 248 (2004), 593–633. [7] R. Frank, A note on low energy scattering for homogeneous long range potentials, arXiv 0812.2916, to appear in Annales Henri Poincaré. [8] Y. Gatel, D. Yafaev, On the solutions of the Schr¨ odinger equation with radiation conditions at infinity: the long-range case, Ann. Inst. Fourier, Grenoble 49, no. 5 (1999), 1581–1602. [9] E. Harrell, B. Simon, The mathematical theory of resonances whose widths are exponentially small, Duke Math. J. 47, no. 4 (1980), 845–902. [10] L. H¨ ormander, The analysis of linear partial differential operators. I, Berlin, Springer 1990. [11] L. H¨ ormander, The analysis of linear partial differential operators. II–IV, Berlin, Springer 1983–85. [12] L. H¨ ormander, Fourier integral operators. I, Acta Math. 127 (1971), 79–183. [13] H. Isozaki, J. Kitada, Scattering matrices for two-body Schr¨ odinger operators, Scientific papers of the College of Arts and Sciences, Tokyo Univ. 35 (1985), 81–107. [14] C. M¨ uller, Analysis of spherical symmetries in Euclidean spaces, Springer 1998. [15] R. G. Newton, Scattering theory of waves and particles, New York, Springer 1982. [16] M. E. Taylor, Noncommutative harmonic analysis, Mathematical Surveys and Monographs no. 22, AMS 1986. [17] S. V. Vladimirov, Equations of mathematical physics, Moscow, Nauka 1967.

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[18] D. Yafaev, The low energy scattering for slowly decreasing potentials, Comm. Math. Phys. 85, no. 2 (1982), 177–196. [19] D. Yafaev, The scattering amplitude for the Schr¨ odinger equation with a long-range potential, Comm. Math. Phys. 191, no. 1 (1998), 183–218. [20] D. Yafaev, On the classical and quantum Coulomb scattering, J. Phys. A 30, no. 19 (1997), 6981–6992. Jan Derezi´ nski Deptartment of Math. Methods in Physics Warsaw University Ho˙za 74 PL-00-682, Warszawa Poland e-mail: [email protected] Erik Skibsted Institut for Matematiske Fag Aarhus Universitet Ny Munkegade DK-8000 Aarhus C Denmark e-mail: [email protected] Communicated by Claude Alain Pillet. Submitted: November 28, 2008. Accepted: March 2, 2009.

Ann. Henri Poincaré 10 (2009), 573–575 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/030573-3, published online June 11, 2009 DOI 10.1007/s00023-009-0410-3


A Note on Low Energy Scattering for Homogeneous Long-Range Potentials Rupert L. Frank

Abstract. We explicitly calculate the scattering matrix at energy zero for attractive, radial and homogeneous long-range potentials. This proves a conjecture by Derezinski and Skibsted.

The very interesting, recent paper [3] is concerned with scattering at zero energy for long-range potentials. It is shown that if V = −α|x|−μ + W for some α > 0, 0 < μ < 2 and a sufficiently fast decaying, radial function W , then the scattering matrix for the pair (−Δ + V, −Δ) at zero energy is given by ⎛ exp ⎝−i

πμ 2−μ

−ΔSd−1 +

d−2 2

2

⎞ ⎠+K

for some compact operator K (depending on V ). Moreover, it is conjectured that if V = −α|x|−μ (that is, if W ≡ 0), then K ≡ 0. The purpose of this note is to prove this conjecture. Theorem 1. Let d ≥ 2, 0 < μ < 2 and α > 0. For λ > 0 let S(λ) be the scattering matrix for the pair (−Δ − α|x|−μ , −Δ) at energy λ in the sense of [2]. Then ⎛ πμ s − lim S(λ) = exp ⎝−i λ→0+ 2−μ

−ΔSd−1 +

d−2 2

2

⎞ ⎠.

It was shown in [2] that S(0) := s − limλ→0+ S(λ) exists and is given through the phase shifts of the solutions of the energy zero equations corresponding to fixed angular momentum. To be more precise, for any l ∈ N0 there exists a function f

574

R. L. Frank

satisfying

−f +

(l − 1 + d/2)2 − 1/4 α − μ 2 r r lim r−l−

lim

r→∞


f = 0 in

R+ ,

d−1 2

f (r) = 1 , √ 2 α 2−μ r 2 + Dl r−μ/4 f (r) − Cl sin =0 2−μ r→0

for some Cl > 0, and in terms of this function the action of S(0) on a spherical harmonic Y of order l is given by

π S(0)Y = exp i2 Dl + (d − 3 + 2l) Y . 4 (Note that the 2π-ambiguity in the definition of Dl does not affect the formula for S(0).) In view of these facts and recalling that the Laplacian acts on spherical harmonics of order l as multiplication by l(l + d − 2), Theorem 1 will follow if we π can prove that Dl = − π(d−2+2l) 2(2−μ) + 4 modulo 2π. This equality is the assertion of the following lemma applied to ν = l + d−2 2 . Lemma 2. Let ν ≥ 0, 0 < μ < 2 and α > 0. The function √ 2ν 2 − μ 2−μ 1/2 2 α 2−μ 2ν 2 √ 2ν +1 r r J 2−μ f (r) := Γ 2−μ 2−μ α satisfies

−f +

ν 2 − 1/4 α − μ 2 r r

√ 2 α 2−μ r 2 lim r−μ/4 f (r) − C sin r→∞ 2−μ √ 2ν + 1 2ν + 1) ((2 − μ)/ α) 2−μ 2 > 0. with C := π −1/2 Γ( 2−μ

f =0

lim r−ν−1/2 f (r) = 1 , π πν + − =0 2−μ 4 r→0

(1)

in

R+ ,

(2) (3) (4)

Proof. We will show that the function f given in (1) is the unique solution of the initial value problem (2)–(3). Then the asymptotics (4) follow from the asymptotics 2 π π˜ ν + Jν˜ (s) = sin s − + o(1) , s → ∞ , πs 2 4 of Bessel functions [1, (9.2.1)]. Let f denote any solution of the initial value problem (2)–(3) and define g by √ 2−μ

2 α 1/2 . f (r) =: r g br 2 , b := 2−μ This definition is motivated by the asymptotics (4): If we want to arrive at a function which behaves asymptotically like an inverse square root times an oscillating

Vol. 10 (2009)

Low Energy Scattering

575

function – which is the asymptotic behavior of any Bessel function –, then (4) 2−μ suggests to look at r−1/2 f (r) as a function of r 2 . This is essentially what we call g, and we are about to show that g is indeed a Bessel function. A short computation shows that equation (2) in terms of g becomes 2 2ν −1 s−2 g + g = 0 , g +s g − 2−μ which is Bessel’s equation with parameter ν˜ := 2ν/(2 − μ). Hence g is a linear combination of Jν˜ and Yν˜ . Boundary condition (3) in terms of g becomes lim s−˜ν g(s) = b−˜ν .

s→0

Since Jν˜ (s) ∼ (s/2)ν˜ /Γ(˜ ν + 1) for all ν˜ ≥ 0 and Yν˜ (s) ∼ −(1/π)Γ(˜ ν )(s/2)−˜ν for ν˜ > 0, resp. Y0 (s) ∼ (2/π) ln s as s → 0 [1, (9.1.7–9)], we conclude that g(s) = Γ(˜ ν + 1)(2/b)ν˜ Jν˜ (s) . This proves the lemma.

Acknowledgements The author wishes to thank R. Seiringer for useful discussions. Support through DFG grant FR 2664/1-1 and U.S. NSF grant PHY 06 52854 is gratefully acknowledged.

References [1] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Reprint of the 1972 edition. Dover Publications, New York, 1992. [2] J. Derezi´ nski, E. Skibsted, Quantum scattering at low energies, Preprint (2008). [3] J. Derezi´ nski, E. Skibsted, Scattering at zero energy for attractive homogeneous potentials, to appear in Ann. Henri Poincaré (2009). Rupert L. Frank Department of Mathematics Princeton University Washington Road Princeton, NJ 08544 USA e-mail: [email protected] Communicated by Claude Alain Pillet. Submitted: December 7, 2008. Accepted: March 2, 2009.



Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation Marcel Griesemer and David G. Hasler

Abstract. For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of α3/2 , α being the fine structure constant. A suitably chosen ground state vector depends analytically on α3/2 and it is twice continuously differentiable with respect to the nuclear coordinates.

1. Introduction When a neutral atom or molecule made from static nuclei and non-relativistic electrons is coupled to the (UV-cutoff) quantized radiation field, the least point of the energy spectrum becomes embedded in the continuous spectrum due to the absence of a photon mass, but it remains an eigenvalue [13, 18]. This ground state energy E depends on the parameters of the system, such as the fine-structure constant, the positions of static nuclei, or, in the center of mass frame of a translation invariant model, the total momentum. The regularity of E as a function of these parameters is of fundamental importance. For example, the accuracy of the Born-Oppenheimer approximation, a pillar of quantum chemistry, depends on the regularity of E and on the regularity of the ground state projection as functions of the nuclear coordinates. If E were an isolated eigenvalue, like it is in quantum mechanical description of molecules without radiation, then analyticity of E with

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respect to any of the aforementioned parameters would follow from regular perturbation theory. But in QED the energy E is not isolated and the analysis of its regularity is a difficult mathematical problem. In the present paper we study the problem of regularity, described above, in a large class of models of matter and radiation where the Hamiltonian H(s) depends analytically on complex parameters s = (s1 , . . . , sν ) ∈ Cν and is defined for values of s from a complex neighborhood of a compact set K ⊂ Rν . Important properties of H(s) are, that H(¯ s) = H(s)∗ and that, for s ∈ K, the lowest point, E(s), of the spectrum of H(s) is a non-degenerate eigenvalue. Under further assumptions, described below, we show that E(s) and the projection operator associated with the eigenspace of E(s) are real-analytic functions of s in a neighborhood of K. In particular, they are of class C ∞ in this neighborhood. We apply this result to the Hamiltonian of a molecule with static nuclei and non-relativistic electrons that are coupled to the quantized radiation field in dipole approximation. By our choice of atomic units, this Hamiltonian depends on the fine-structure constant α only though a factor of α3/2 in front of the dipole interaction operator. Hence the role of the parameter s may be played by α3/2 or, after a well-known unitary deformation argument [15], by the nuclear coordinates. The general theorem described above implies that the ground state energy, if it is a non-degenerate eigenvalue, depends analytically on α3/2 and the nuclear coordinates. The ground state projection is analytic in α3/2 , and twice continuously differentiable with respect to the nuclear coordinates. This paper thus also gives an answer to the question about the presence of α-dependent and logarithmically divergent coefficients in an expansion of the ground state energy in powers of α1/2 [4]: they do not occur when the dipole approximation is used. A further consequence of our main result concerns the accuracy of the adiaodinger equabatic approximation to the time evolution Uτ generated by the Schr¨ tion d i ϕt = H(t/τ )ϕt , t ∈ [0, τ ] , dt in the limit τ → ∞. If H(s) satisfies the assumptions of our result mentioned above with K = [0, 1], then the ground state projection P (s) is of class C ∞ ([0, 1]) and hence the adiabatic theorem without gap assumption implies that supt∈[0,τ ] (1 − P (t))Uτ (t)P (0) = o(1) as τ → ∞ [2,22]. Previously, in all applications of the adiabatic theorem without gap assumption the differentiability of P (s) was enforced or provided by the special form H(s) = U (s)HU (s)−1 of H(s) where U (s) is a unitary and (strongly) differentiable operator [1, 2, 21]. We now describe our main result in detail. We consider a class of Hamiltonians Hg (s) : D ⊂ H → H depending on a parameter s ∈ V , where V = V is a complex neighborhood of some point s0 ∈ Rν . For each s ∈ V , Hg (s) = Hat (s) ⊗ 1 + 1 ⊗ Hf + gW (s) , with respect to H = Hat ⊗ F, where Hat is an arbitrary complex Hilbert space and F denotes the symmetric Fock space over L2 (R3 × {1, 2}). We assume that

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Hat (¯ s) = Hat (s)∗ for all s ∈ V and that (Hat (s))s∈V is an analytic family of type (A). This means that the domain D of Hat (s) is independent of s ∈ V and that s → Hat (s)ϕ is analytic for all ϕ ∈ D. We assume, moreover, that Eat (s0 ) = inf σ(Hat (s0 )) is a simple and isolated eigenvalue of Hat (s0 ). The operator Hf describes the energy of the bosons, and gW (s) the interaction of the particle system described by Hat (s) and the bosons. In terms of creation and annihilation operators Hf = |k|a∗λ (k)aλ (k)d3 k λ=1,2

and W (s) =

Gs¯(k, λ)∗ ⊗ a(k, λ) + Gs (k, λ) ⊗ a∗ (k, λ)d3 k ,

λ=1,2

where, (k, λ) → Gs (k, λ), for each s ∈ V is an element of L2 (R3 × {1, 2}, L(Hat )). We assume that s → Gs is a bounded analytic function on V and that 1 sup (1) Gs (k, λ)2 2+2μ d3 k < ∞ |k| s∈V λ=1,2

for some μ > 0. Based on these assumptions we show that a neighborhood V0 ⊂ V of s0 and a positive constant g0 exist such that for s ∈ V0 and all g ∈ [0, g0 ) the operator Hg (s) has a non-degenerate eigenvalue Eg (s) and a corresponding eigenvector ψg (s) that are both analytic functions of s ∈ V0 . Moreover Eg (s) = inf σ(Hg (s)) for s ∈ R∩V . Before commenting on the proof of this result we briefly review the literature. In [4] the dependence on α of the ground state and the ground state energy, E, is studied for non-relativistic atoms that are minimally coupled to the quantized radiation field. It is shown that E and a suitably chosen ground state vector have expansions in asymptotic series of powers of α with α-dependent coefficients that may diverge logarithmically as α → 0. Smoothness is not expected and hence the dipole approximation seems necessary for our analyticity result. Much earlier, in [9], Fr¨ ohlich obtained results on the regularity of the ground state energy with respect to the total momentum P for the system of a single quantum particle coupled linearly to a quantized field of massless scalar bosons. Let H(P ) denote the Hamiltonian describing this system at fixed total momentum P ∈ R3 . The spectrum of H(P ) is of the form [E(P ), ∞) but E(P ) is not an eigenvalue for P = 0 [9] (see [14] for similar results on positive ions). For a non-relativistic particle of mass M , Fr¨ ohlich shows that P → E(P ) is differentiable a.e. in {|P | < √ ( 3 − 1)M }, and that ∇E(P ) is locally Lipschitz [9, Lemma 3.1]. This work was recently and independently continued by Alessandro Pizzo and Thomas Chen for systems with a fixed ultraviolet cutoff [7, 19]. After a unitary, P -dependent transformation of the Hamiltonian H(P ), Pizzo obtains a ground state vector older continuous with respect to P uniformly in an infrared cutoff φσ (P ) that is H¨ σ > 0. Chen studied the regularity of E(P, σ) for a non-relativistic particle coupled β minimally to the quantized radiation field [7]. He estimates |∂|p| (E(|p|, σ) − p2 /2)|

580



uniformly in σ > 0 for β ≤ 2. He also asserts that E(p, σ) is of class C 2 even for σ = 0. In [15] Hunziker proves analyticity with respect to the nuclear coordinates for non-relativistic molecules without radiation. The ground state energy is isolated but the Hamiltonian is not analytic with respect the nuclear coordinates. It only becomes analytic after a suitable unitary deformation (see Section 3) introduced by Hunziker. The results of the present paper are derived using the renormalization technique of Bach et al. [3,6], in a new version that we take from [10]. Like the authors of [10] we use a simplified renormalization map that consists of a Feshbach-Schur map and a scaling transformation only. In the corresponding spectral analysis the Hamiltonian is diagonalized, with respect to Hf , in a infinite sequence of renormalization steps. In each step the off-diagonal part becomes smaller, thanks to (1), and the spectral parameter is adjusted to enforce convergence of the diagonal part. This method provides a fairly explicit construction of an eigenvector of H(s), even for complex s, where H(s) is not self-adjoint. We show first, that the parameters of the renormalization analysis can be chosen independent of s and g in neighborhoods of s = s0 and g = 0, second, that all steps of the renormalization analysis preserve analyticity, and third, that all limits taken are uniform in s, which implies analyticity of the limit functions. On a technical level, these three points are the main achievements of this paper. The result that analyticity with respect to the spectral parameter is preserved under the renormalization map has been used in all previous papers working with the renormalization techniques of Bach et al. If asked about a proof, its authors would argue, that the renormalization map, as a map of operator kernels, is given by explicit formulas. A proof spelling out this argument, to our best knowledge, has never been published. Our proof of Proposition 17, provides a simple alternative argument. The δ-calculus introduced in [3] does not solve this problem. It seems unlikely that another approach, not based on a renormalization analysis would yield a result similar to ours. The proof of analyticity requires the construction of an eigenvector for complex values of s where H(s) is not self-adjoint and hence, variational methods, for example, are not applicable. There is, of course, the tempting alternative approach to first introduce a positive photon mass σ to separate the least energy from the rest of the spectrum. But the neighborhood of analyticity obtained in this way depends on the size of σ and vanishes in the limit σ → 0. We conclude this introduction with a description of the organization of this paper along with the strategy of our proof. In Section 2 we introduce the class of Hamiltonians (H(s))s∈V , we formulate all hypotheses, and state the main results. In Section 3 they are applied to nonrelativistic QED in dipole approximation to prove our results mentioned above on the regularity with respect to α3/2 and the nuclear coordinates. Section 4 describes the smoothed Feshbach transform Fχ (H) of an operator H and the isomorphism Q(H) between the kernels of Fχ (H) and H (isospectrality

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of the Feshbach transform). The transform H → Fχ (H) was discovered in [3], and generalized to the form needed here in [12]. In Section 5 we perform a first Feshbach transformation on H(s) − z to obtain an effective Hamiltonian H (0) [s, z] on Hred = P[0,1] (Hf )F, F being the Fock-space. We show that H (0) (s, z) is analytic in s and z. By the isospectrality of the Feshbach transform, the eigenvalue problem for H(s) is now reduced to finding a value z(s) ∈ C such that H (0) (s, z(s)) has a nontrivial kernel. In Section 6 we introduce a Banach space Wξ and a linear mapping H : Wξ → L(Hred ). The renormalization transformation Rρ is defined on a polydisc B(ρ/2, ρ/8, ρ/8) ⊂ H(Wξ ) as the composition of a Feshbach transform and a rescaling k → ρk of the photon momenta k. ρ ∈ (0, 1) is the factor by which the energy scale is reduced in each renormalization step. Rρ takes values in L(Hred ) and, like the Feshbach transform, it is isospectral. In Section 7 it is shown that the analyticity of a family of Hamiltonians is preserved under the renormalization transformation. This is one of the key properties on which our strategy is based. Sections 8 and 9 are devoted to the solution of our spectral problem for H (0) (s, z) by iterating the Renormalization map. Since this procedure is pointwise in s with estimates that hold uniformly on V , we drop the parameter s for notational simplicity. In Sections 8 we define H (n) [s, z] = Rn H (0) [s, z] for values of the spectral parameter z from non-empty sets Un (s). These sets are nested, Un (s) ⊃ Un+1 (s), and they shrink to a point, ∩n Un (s) = {z∞ (s)}. Since H (n) (z∞ ) → const Hf as n → ∞ in the norm of L(Hred ) and since the vacuum Ω is an eigenvector Hf with eigenvalue zero, it follows, by the isospectrality of R, that zero is an eigenvalue of H (n) (z∞ ) for all n. In Section 9 a vector ϕn in the kernel of H (n) (z∞ ) is computed by compositions of scaling transformations and mappings Q(H (k) (z∞ )), k ≥ n, applied to Ω. ϕgs = Q(H (0) (z∞ ))ϕ0 is an eigenvector of H with eigenvalue z∞ . In Section 10 we show that s → z∞ (s) is analytic and that Q(H (n) (z∞ (s))) maps analytic vectors to analytic vectors. Since the vacuum Ω is trivially analytic in s, it follows that ϕgs (s) is analytic in s. In the Appendices A and B we collect technical auxiliaries and for completeness we give a proof of H (n) (z∞ ) → const Hf as n → ∞, although this property is not used explicitly.

2. Assumptions and main results We consider families of (unbounded) closed operators Hg (s) : D(Hg (s)) ⊂ H → H, s ∈ V , where V ⊂ Cν is open, symmetric with respect to complex conjugation and V ∩ Rν = ∅. The Hilbert space H is a tensor product H = Hat ⊗ F ,

F=

∞ n=0

Sn (⊗n h) ,

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of an arbitrary, separable, complex Hilbert space Hat and the symmetric Fock space F over the Hilbert space h := L2 (R3 × {1, 2}; C) with norm given by h2 := |h(k, λ)|2 d3 k , h ∈ h . λ=1,2

Here S0 (⊗ h) := C and for n ≥ 1, Sn ∈ L(⊗n h) denotes the orthogonal projection onto the subspace left invariant by all permutation of the n factors of h. To simplify our notation we set d3 k , |k| := |k| , k := (k, λ) , dk := 0

λ=1,2

throughout the rest of this paper. For each s ∈ V , the operator Hg (s) is a sum Hg (s) = Hat (s) ⊗ 1 + 1 ⊗ Hf + gW (s) ,

(2)

of a closed operator Hat (s) in Hat , the second quantization, Hf , of the operator ω on L2 (R3 × {1, 2}) of multiplication with ω(k) = |k| , and an interaction operator gW (s), g ≥ 0 being a coupling constant. The operator W (s) is the sum W (s) = a(Gs¯) + a∗ (Gs ) of an annihilation operator, a(Gs¯), and a creation operator, a∗ (Gs ), associated with an operator Gs ∈ L(Hat , Hat ⊗ h). The creation operator, a∗ (Gs ), as usual, is defined as the closure of the linear operator in H given by √ a∗ (Gs )(ϕ ⊗ ψ) := nSn (Gs ϕ ⊗ ψ) , if ϕ ∈ Hat and ψ ∈ Sn−1 (⊗n−1 h). The annihilation operator a(Gs ) is the adjoint of a∗ (Gs ). Hypotheses I below will imply that Hg (s) is well defined on D(Hat (s)) ⊗ D(Hf ) and closable. To formulate it, some preliminary remarks are necessary. Let L2 (R3 , L(Hat )) be the Banach space of (weakly) measurable and square integrable functions from R3 to L(Hat ). Every element T of this space defines a linear operator T : Hat → L2 (R3 , Hat ) by (T ϕ)(k) := T (k)ϕ . This operator is bounded and T ≤ T 2 . Since L2 (R3 , Hat ) Hat ⊗ h, we may consider T as an element of L(Hat , Hat ⊗ h) and hence L2 (R3 , L(Hat )) as a subspace embedded in L(Hat , Hat ⊗ h). Hypothesis I. The mapping s → Gs is an bounded analytic function on V with values in L2 (R3 , L(Hat )), and there exists a μ > 0, such that 1 sup Gs (k)2 dk < ∞. 2+2μ |k| s∈V

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By Lemma 25, a# (Gs )(Hf + 1)−1/2 ≤ Gs ω , where 2 Gs 2ω := Gs (k) (|k|−1 + 1) dk . R3

Hence Hypothesis I implies that W (s) and W (s)∗ are well defined on Hat ⊗D(Hf ). It follows that Hg (s) is defined on D(Hat (s)) ⊗ D(Hf ) and that the adjoint of this operator has a domain which contains D(Hat (s)) ⊗ D(Hf ). This subspace is dense because Hat (s) is closed. That is, Hg (s) : D(Hat (s)) ⊗ D(Hf ) ⊂ H → H has a densely defined adjoint, and hence it is closable. Hypothesis II. (i) Hat (s) is an analytic family of operators in the sense of Kato and Hat (s)∗ = s) for all s ∈ V . In particular, Hat (s) is self-adjoint for s ∈ Rν ∩ V . Hat (¯ (ii) There exists a point s0 ∈ V ∩ Rν such that Eat (s0 ) := inf σ(Hat (s0 )) is an isolated, non-degenerate eigenvalue of Hat (s0 ). For the notion of an analytic family of operators in the sense of Kato we refer to [20]. The definition given there readily generalizes to several complex variables. We recall that a function of several complex variables is called analytic if it is analytic in each variable separately. By Hypothesis II, (ii), and the Kato–Rellich theorem of analytic perturbation theory [20], there is exactly one point Eat (s) of σ(Hat (s)) near Eat (s0 ), for s near s0 , and this point is a non-degenerate eigenvalue of Hat (s). Moreover, for s near s0 , there is an analytic projection onto the eigenvector of Eat (s), which is given by −1 1 z − Hat (s) Pat (s) := dz , 2πi |Eat (s)−z|= for > 0 sufficiently small. We set P at (s) = 1 − Pat (s). Hypothesis III. Hypothesis II holds and there exists a neighborhood U ⊂ V × C of (s0 , Eat (s0 )) such that for all (s, z) ∈ U, |Eat (s) − z| < 1/2 and q+1 < ∞. sup sup P (s) at q≥0 Hat (s) − z + q (s,z)∈U

Remarks. 1. Hypothesis III is satisfied, e.g., if Hypothesis II holds and Hat (s) is an analytic family of type (A), see Corollary 3. 2. The condition |Eat (s) − z| < 1/2 is needed in the proof of Theorem 13 and related to the constant 3/4 in the construction of χ. Since s → Eat (s) is continuous, it can always be met by choosing U sufficiently small. However, the smaller we choose U the smaller we will have to choose the coupling constant g. Optimal bounds on g could possibly be obtained by scaling the operator such that the gap in Hat is comparable to one. We are now ready to state the main results.

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Theorem 1. Suppose Hypotheses I, II and III hold. Then there exists a neighborhood V0 ⊂ V of s0 and a positive constant g0 such that for all s ∈ V0 and all g < g0 the operator Hg (s) has an eigenvalue Eg (s) and a corresponding eigenvector ψg (s) that are both analytic functions of s ∈ V0 such that Eg (s) = inf σ Hg (s) for s ∈ V0 ∩ Rν . Remark. For s ∈ V0 ∩ Rν and g sufficiently small, the eigenvalue Eg (s) is nondegenerate by Hypothesis II (ii) and a simple overlap estimate [5]. Corollary 2. Assume Hypotheses I and II are satisfied and that there exists a C such that for all s ∈ V Re ϕ, Hat (s)ϕ ≥ −Cϕ, ϕ , for ϕ ∈ D Hat (s) . (3) Then the conclusions of Theorem 1 hold. Proof. It suffices to verify Hypothesis III, then the corollary will follow from Theorem 1. For all s ∈ V , z ∈ C with |z − Eat (s0 )| ≤ 1, q ≥ q ∗ := C + |Eat (s0 )| + 2, and ϕ ∈ D(Hat (s)) with ϕ = 1,

Hat (s) − z + q ϕ ≥ Re ϕ, Hat (s) − z + q ϕ ≥ q − C − |Eat (s0 )| − 1 ≥ 1 . Since Hat (s)∗ = Hat (¯ s) an analog estimate holds for Hat (s)∗ . This proves that B1 (Eat (s0 )) ⊂ ρ(Hat (s) + q) for s ∈ V , q ≥ q ∗ , and that q∗ q+1 ≤ sup . (4) ∗ q − C + |Eat (s0 )| + 1 s∈V,|z−Eat (s0 )|≤1,q≥q ∗ Hat (s) − z + q We now turn to the case where 0 ≤ q ≤ q ∗ . The set Γ := (s, z) ∈ Cν × C|z ∈ ρ H(s) P at (s)H is open and (Hat (s) − z)−1 P at (s) is analytic on Γ [20]. On the other hand

γ := s0 , Eat (s0 ) − q |0 ≤ q ≤ q ∗ is a compact subset of Γ. It follows that the distance between γ and the complement of Γ is positive. Thus if g and δ > 0 are small enough, then (s, z − q) : |s − s0 | ≤ δ, |z − Eat (s0 )| ≤ δ, 0 ≤ q ≤ q ∗ is a compact subset of Γ on which (H(s) − z)−1 P at (s) is uniformly bounded. Comparing with (4) we conclude that for δ < 1 so small that Bδ (s0 ) ⊂ V the Hypothesis III holds with U = Bδ (s0 ) × Bδ (E(s0 )) The following corollaries prove the assertions in the introduction.

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Corollary 3. Suppose Hypothesis I holds and let Hat (s) be an analytic family of s) for all s ∈ V . If E(s0 ) = inf σ(H(s0 )) is a nontype (A) with Hat (s)∗ = Hat (¯ degenerate isolated eigenvalue of Hat (s0 ), then the conclusions of Theorem 1 hold. Proof. By Corollary 2 it suffices to show that (3) holds. To this end we set T (s) := Hat (s) − Eat (s0 ) −1

and R := (T (s0 ) + 1) . Since T (s) is an analytic family of type (A), the operators T (s)R and RT (s) are bounded and weakly analytic, hence strongly analytic [16]. It follows that (T (s) − T (s0 ))R → 0 and R(T (s) − T (s0 )) → 0 as s → 0. By abstract interpolation theory R1/2 T (s) − T (s0 ) R1/2 → 0 s → s0 . We choose ε > 0 so that Bε (s0 ) ⊂ V and sup R1/2 T (s) − T (s0 ) R1/2 ≤ 1/2 . |s| 0 we set dK 2 m+n , ; W0,0 Wm,n := Ls B |K|2+2μ 1/2 dK 2 wm,n (K) wm,n μ := |K|2+2μ B m+n where B := {k ∈ R3 × {1, 2} : |k| ≤ 1} and |K| :=

m+n

|kj | ,

j=1

dK :=

m+n

dkj .

j=1

That is, Wm,n is the space of measurable functions wm,n : B m+n → W0,0 that are symmetric with respect to all permutations of the m arguments from B m and the n arguments from B n , respectively, such that wm,n μ is finite. For given ξ ∈ (0, 1) and μ > 0 we define a Banach space Wξ := Wm,n m,n∈N

wμ,ξ :=

m,n≥0

ξ −(m+n) wm,n μ ,

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w0,0 μ := w0,0 , as the completion of the linear space of finite sequences w = (wm,n )m,n∈N ∈ m,n∈N Wm,n with respect to the norm wμ,ξ . The spaces Wm,n will often be identified with the corresponding subspaces of Wξ . Next we define a linear mapping H : Wξ → L(Hred ). For finite sequences w = (wm,n ) ∈ Wξ the operator H(w) is the sum H(w) := Hm,n (w) m,n

of operators Hm,n (w) on Hred , defined by H0,0 (w) := w0,0 (Hf ), and, for m+n ≥ 1, Hm,n (w) := Pred a∗ (k (m) )wm,n (Hf , K)a(k˜(n) )dK Pred , B m+n

where Pred := P[0,1] (Hf ), K = (k (m) , k˜(n) ), and m k (m) = (k1 , . . . , km ) ∈ R3 × {1, 2} , k˜(n)

a∗ (k (m) ) =

n = (k˜1 , . . . , kñ ) ∈ R × {1, 2} ,

a(k˜(n) ) =

m i=1 n

a∗ (ki ) , a(k˜i ) .

i=1

By the continuity established in the following proposition, the mapping w → H(w) has a unique extension to a bounded linear transformation on Wξ . Proposition 14 ([3]). (i) For all μ > 0, m, n ∈ N, with m + n ≥ 1, and w ∈ Wm,n , 1 Hm,n (w) ≤ (PΩ⊥ Hf )−m/2 Hm,n (w)(PΩ⊥ Hf )−n/2 ≤ √ m n wm,n μ . m n (ii) For all μ > 0 and all w ∈ Wξ H(w) ≤ wμ,ξ H(w) ≤ ξwμ,ξ ,

if

w0,0 = 0 .

(34)

In particular, the mapping w → H(w) is continuous. Proof. Statement (ii) follows immediately from (i) and ξ ≤ 1. For (i) we refer to [3], Theorem 3.1. Given α, β, γ ∈ R+ we define neighborhoods, B(α, β, γ) ⊂ H(Wξ ) of the operator Pred Hf Pred ∈ L(Hred ) by − 1∞ ≤ β, w − w0,0 μ,ξ ≤ γ . B(α, β, γ) := H(w)|w0,0 (0)| ≤ α, w0,0 Note that w0,0 (0) = Ω, w0,0 (Hf )Ω = Ω, H(w)Ω. The definition of B(α, β, γ) is motivated by the following Lemma and by Theorem 16. Lemma 15. Suppose ρ, ξ ∈ (0, 1) and μ > 0. If H(w) ∈ B(ρ/2, ρ/8, ρ/8), then (H(w), H0,0 (w)) is a Feshbach pair for χρ .

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599

Proof. The assumption H(w) ∈ B(ρ/2, ρ/8, ρ/8) implies, by Proposition 14, that ρ H(w) − H0,0 (w) ≤ ξ . 8 3 For r ∈ [ 4 ρ, 1], |w0,0 (r)| ≥ r − w0,0 (r) − w0,0 (0) − r − |w0,0 (0)| ρ ≥ r 1 − sup |w 0,0 (r) − 1| − 2 r 3ρ ρ ρ ρ ≥ 1− − ≥ . 4 8 2 8 By the spectral theorem, 8 1 H0,0 (w)−1 |`Ran χρ = w0,0 (Hf )−1 |`Ran χρ ≤ sup ≤ . ρ r∈[ 3 ρ,1] |w0,0 (r)| 4

Since χρ ≤ 1, it follows from the estimates above that H0,0 (w)−1 χρ H(w) − H0,0 (w) χρ |`Ran χρ ≤ ξ < 1 . This implies the bounded invertibility of H0,0 (w) + χρ H(w) − H0,0 (w) χρ |`Ran χρ = H0,0 (w) 1 + H0,0 (w)−1 χρ H(w) − H0,0 (w) χρ |`Ran χρ . The other conditions on a Feshbach pair are now also satisfied, since H(w) − H0,0 (w) is bounded on Hred . The renormalization transformation we use is a composition of a Feshbach transformation and a unitary scaling that puts the operator back on the original Hilbert space Hred . Unlike the renormalization transformation of Bach et al [3], our renormalization transformation involves no analytic transformation of the spectral parameter. Given ρ ∈ (0, 1), let Hρ = Ran χ(Hf ≤ ρ). Let w ∈ Wξ and suppose (H(w), H0,0 (w)) is a Feshbach pair for χρ . Then Fχρ H(w), H0,0 (w) : Hρ → Hρ is iso-spectral with H(w) in the sense of Theorem 7. In order to get a isospectral operator on Hred , rather than Hρ , we use the linear isomorphism Γρ : Hρ → H1 = Hred ,

Γρ := Γ(Uρ ) Hρ ,

where Uρ ∈ L(L (R × {1, 2})) is defined by 2

3

(Uρ f )(k) := ρ3/2 f (ρk) . Note that Γρ Hf Γ∗ρ = ρHf , and hence Γρ χρ Γ∗ρ = χ1 . The renormalization transformation Rρ maps bounded operators on Hred to bounded linear operators on Hred

600



and is defined on those operators H(w) for which (H(w), H0,0 (w)) is a Feshbach pair with respect to χρ . Explicitly, Rρ H(w) := ρ−1 Γρ Fχρ H(w), H0,0 (w) Γ∗ρ , which is a bounded linear operator on Hred . In [3], Theorem 3.3, it is shown that w → H(w) is one-to-one. Hence w ∈ Wξ is uniquely determined by the operator H(w) and the domain of Rρ , as described above, is a well-defined subset of L(Hred ). By Lemma 15 it contains the ball B(ρ/2, ρ/8, ρ/8). The following theorem describes conditions under which the Renormalization transform may be iterated. Theorem 16 (BCFS [3]). There exists a constant Cχ ≥ 1 depending only on χ, such √ that the following holds. If μ > 0, ρ ∈ (0, 1), ξ = ρ/(4Cχ ), and β, γ ≤ ρ/(8Cχ ), then Rρ − ρ−1 · Ω : B(ρ/2, β, γ) → B(α , β , γ ) , where γ2 γ2 , β = β + Cβ , γ = Cγ ρμ γ , α = Cβ ρ ρ with Cβ := 32 Cχ , Cγ := 128Cχ2 . This theorem is a variant of Theorem 3.8 of [3], with additional information from the proof of that theorem, in particular from Equations (3.104), (3.107) and (3.109). Another difference is due to our different definition of the renormalization transformation, i.e., without analytic deformation of the spectral parameter.

7. Renormalization preserves analyticity This section provides one of the key tools for our method to work, Proposition 17 below, which implies that analyticity is preserved under renormalization. It is part (a) of the following proposition that is nontrivial and not proved in the papers of Bach et al. (see Theorem 2.5 of [3] and the remark thereafter). Proposition 17. Let S be an open subset of Cν+1 , ν ≥ 0. Suppose σ → H(wσ ) ∈ L(Hred ) is analytic on S, and that H(wσ ) belongs to some ball B(α, β, γ) for all σ ∈ S. Then: (a) H0,0 (wσ ) is analytic on S. (b) If for all σ ∈ S, (H(wσ ), H0,0 (wσ )) is a Feshbach pair for χρ , then Fχρ (H(wσ ), H0,0 (wσ )) is analytic on S. Proof. Suppose (a) holds true. Then H0,0 (wσ ) and W = H(wσ ) − H0,0 (wσ ) are analytic function of σ ∈ S and hence so is the Feshbach map Fχρ H(wσ ), H0,0 (wσ ) −1 χρ W χρ . = H0,0 (wσ ) + χρ W χρ − χρ W χρ H0,0 (wσ ) + χρ W χρ This proves (b) and it remains to prove (a).

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Recall from Section 6 that B = {k ∈ R3 × {1, 2} : |k| ≤ 1} and let P1 denote the projection onto the one boson subspace of Hred , which is isomorphic to L2 (B). Then P1 H(wσ )P1 , like H(wσ ), is analytic and P1 H(wσ )P1 = P1 H0,0 (wσ )P1 + P1 H1,1 (wσ )P1 = Dσ + Kσ ,

(35)

σ where Dσ denotes multiplication with w0,0 and Kσ is the Hilbert Schmidt operator with kernel ˜ = wσ (0, k, k) ˜ . Mσ (k, k) 1,1

Our strategy is to show first that Kσ and hence P1 H0,0 (wσ )P1 = P1 H(wσ )P1 −Kσ is analytic. Then we show that H0,0 (wσ ) is an analytic operator on Hred . Step 1: Kσ is analytic. (n)

For each n ∈ N let {Qi }i be a collection of n measurable subsets of B such that B=

n

(n)

Qi

,

(n)

Qi

(n)

∩ Qj

= ∅,

i = j ,

(36)

i=1

and

const . (37) n (n) Let χi denote the operator on L2 (B) of multiplication with χQ(n) . Then for i = j, (n)

|Qi | ≤

i

(n)

(n)

(n)

(n)

χi Dσ χj = 0 because χi and χj have disjoint support and commute with Dσ . Together with (35) this implies that (n)

(n)

χi Kσ χj

(n)

(n)

= χi P1 H(wσ )P1 χj

,

for i = j .

Since the right hand side is analytic, so is the left hand side and hence (n) (n) Kσ(n) = χi Kσ χj i =j (n)

is analytic. It follows that σ → ϕ, Kσ ψ is analytic for all ϕ, ψ in L2 (B). Now let ϕ, ψ ∈ C(B). Then (n) ϕ, Kσ ψ − ϕ, Kσ ψ n (n) (n) = ϕ(x)ψ(y)Mσ (x, y) χi (x)χi (y)dxdy B×B i=1 n 1/2 (n) ≤ ϕ∞ ψ∞ Kσ HS |Qi |2 −→ 0 , (n → ∞) , i=1

uniformly in σ, because the Hilbert Schmidt norm Kσ HS is bounded uniformly in σ (in fact, it is bounded by γ). This proves that ϕ, Kσ ψ is analytic for all ϕ, ψ ∈ C(B). Since C(B) is dense in L2 (B), an other approximate argument using

602



supσ Kσ < ∞ shows that ϕ, Kσ ψ is analytic for all ϕ, ψ ∈ L2 (B). Therefore σ → Kσ is analytic [16]. σ (|k|) is an analytic function of σ. Step 2: For each k ∈ B, w0,0

For each n ∈ N let fk,n ∈ L2 (B) denote a multiple of the characteristic σ (|k|) as a function of B1/n (k) ∩ B with fn,k = 1. By the continuity of w0,0 function of k σ σ w0,0 (|k|) = lim |fk,n (x)|2 w0,0 (|x|)dx (38) n→∞ B ∗ = lim a (fk,n )Ω, H0,0 (wσ )a∗ (fk,n )Ω . n→∞

∗

Since a (fk,n )Ω ∈ P1 Hred the expression · · · , before taking the limit, is an anσ , this function is Lipschitz continuous alytic function of σ. By assumption on w0,0 with respect to |k| uniformly in σ. Therefore the convergence in (38) is uniform σ in σ and hence w0,0 (|k|) is analytic by the Weierstrass approximation theorem from complex analysis. σ (Hf ) is analytic. Step 3: H0,0 (wσ ) = w0,0

By the spectral theorem σ ϕ, w0,0 (Hf Pred )ϕ =

σ w0,0 (λ)dμϕ (λ) .

[0,1] σ By an application of Lebesgue’s dominated convergence theorem, using supσ w0,0 < ∞, we see that the right hand side, we call it ϕ(σ), it is a continuous function of σ. Therefore σ ϕ(σ)dσ = w0,0 (λ)dσ dμϕ (λ) Γ

[0,1]

Γ

for all closed loops Γ : t → σ(t) in S, with σj constant for all but one j ∈ {1, . . . , ν + 1}. The analyticity of σ → ϕ(σ) now follows from the analyticity of σ σ (λ) and the theorems of Cauchy and Morera. By polarization, w0,0 (Hf Pred ) is w0,0 weakly analytic and hence analytic.

8. Iterating the renormalization transform In Section 5 we have reduced, for small |g|, the problem of finding an eigenvalue of Hg (s) in the neighborhood U0 (s) := {z ∈ C|(s, z) ∈ U} of Eat (s) to finding z ∈ C such that H (0) [s, z] has a non-trivial kernel. We now use the renormalization map to define a sequence H (n) [s, z] := Rn H (0) [s, z] of operators on Hred , which, by Theorem 7, are isospectral in the sense that KerH (n+1) [s, z] is isomorphic to KerH (n) [s, z]. The main purpose of the present section is to show that the operators H (n) [s, z] are well-defined for all z from non-empty, but shrinking sets Un (s) {z∞ (s)}, (n → ∞). In the next section it will turn out that H (n) [s, z∞ (s)] has a non-trivial kernel and hence that z∞ (s) is an eigenvalue of Hg (s). The construction

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603

of the sets Un (s) is based on Theorems 13 and 23, but not on the explicit form of H (0) [s, z] as given by (18). Moreover, this construction is pointwise in s and g, all estimates being uniform in s ∈ V and |g| < g0 for some g0 > 0. We therefore drop these parameters from our notations and we now explain the construction of H (n) [z] making only the following assumption: (A) U0 is an open subset of C and for every z ∈ U0 , H (0) [z] ∈ B(∞, ρ/8, ρ/8) . The polydisc B(∞, ρ/8, ρ/8) ⊂ H(Wξ ) is defined in terms of ξ := and μ > 0, where ρ ∈ (0, 1) and Cχ is given by Theorem 16. By Lemma 15, we may define H (1) [z], . . . , H (N ) [z], recursively by H (n) [z] := Rρ H (n−1) [z]

√

ρ/(4Cχ )

(39)

provided that H (0) [z], . . . , H (N −1) [z] belong to B(ρ/2, ρ/8, ρ/8). Theorem 16 gives us sufficient conditions for this to occur: by iterating the map (β, γ) → (β , γ ) starting with (β0 , γ0 ), we find the conditions n

γn := (Cγ ρμ ) γ0 ≤ ρ/(8Cχ ) n−1 Cβ μ 2k (Cγ ρ ) γ02 ≤ ρ/(8Cχ ) , βn := β0 + ρ

(40) (41)

k=0

for n = 0, . . . , N − 1. They are obviously satisfied for all n ∈ N if Cγ ρμ < 1 and if β0 , γ0 are sufficiently small. Let this be the case and let E (n) (z) := Ω, H (n) [z]Ω . Then it remains to make sure that |E (n) (z)| ≤ ρ/2 for n = 0, . . . , N − 1. This is achieved by adjusting the admissible values of z step by step. We define recursively, for all n ≥ 1, Un := z ∈ Un−1 : |E (n−1) (z)| ≤ ρ/2 . (42) If z ∈ UN , H (0) (z) ∈ B(∞, β0 , γ0 ), and ρ, β0 , γ0 are small enough, as explained above, then the operators H (n) (z) for n = 1, . . . , N are well defined by (39). In addition we know from Theorem 16 that H (n) (z) ∈ B(∞, βn , γn ), and that (n−1) (n) (z) Cβ 2 E (z) − E (43) ≤ ρ γn−1 =: αn . ρ This latter information will be used in the proof of Lemma 19 to show that the sets Un are not empty. We summarize: Lemma 18. Suppose that (A) holds with ρ ∈ (0, 1) so small, that Cγ ρμ < 1. Suppose β0 , γ0 ≤ ρ/(8Cχ ) and, in addition, β0 +

Cβ /ρ ρ γ2 ≤ . 1 − (Cγ ρμ )2 0 8Cχ

(44)

604



If H (0) [z] ∈ B(∞, β0 , γ0 ) for all z ∈ U0 , then H (n) [z] is well defined for z ∈ Un , and 1 H (n) [z] − E (n−1) (z) ∈ B(αn , βn , γn ) , for n ≥ 1 (45) ρ with αn , βn , and γn as in (40), (41), and (43). The next lemma establishes conditions under which the set U0 and Un are non-empty. We introduce the discs Dr := {z ∈ C||z| ≤ r} and note that Un = E

(n−1) −1

(Dρ/2 ).

Lemma 19. Suppose that (A) holds with U0 Eat and ρ ∈ (0, 4/5) so small that Cγ ρμ < 1 and B(Eat , ρ) ⊂ U0 . Suppose that α0 < ρ/2, β0 , γ0 ≤ ρ/(8Cχ ) and that (44) hold. If z → H (0) [z] ∈ L(Hat ) is analytic in U0 and H (0) [z] − (Eat − z) ∈ B(α0 , β0 , γ0 ) for all z ∈ U0 , then the following is true. (a) For n ≥ 0, E (n) : Un → C is analytic in Un◦ and a conformal map from Un+1 onto Dρ/2 . In particular, E (n) has a unique zero, zn , in Un . Moreover, B(Eat , ρ) ⊃ U1 ⊃ U2 ⊃ U3 ⊃ · · · . (b) The limit z∞ := limn→∞ zn exists and for := 1/2 − ρ/2 − α1 > 0, ∞ 1 n |zn − z∞ | ≤ ρ exp αk . 2ρ2 k=0

Remark. We call a function f : A → B conformal if it is the restriction of an analytic bijection f : U → V between open sets U ⊃ A and V ⊃ B, and f (A) = B. Proof. Since H (0) is analytic on U0 , it follows, by Theorem 17, that H (n) is analytic on Un◦ for all n ∈ N. In particular E (n) is analytic on Un◦ . To begin with we prove: (I1 ) U1 ⊂ B(Eat , ρ) and E (0) : U1 → Dρ/2 conformally . By assumption on H (0) (z), |E (0) (z) − (Eat − z)| ≤ α0 , Hence, if z ∈ E

(0) −1

∀z ∈ U0 .

(46)

◦ (Dρ/2+ ) then

|Eat − z| ≤ α0 + ρ/2 + < ρ , provided > 0 is chosen sufficiently small. This proves that U1 ⊂ E (0) B(Eat , ρ). Since E (0) is continuous, it follows that E E (0) : E (0)

−1

◦ ◦ (Dρ/2+ ) → Dρ/2+

(0) −1

−1

◦ (Dρ/2+ )⊂

◦ (Dρ/2+ ) is open in C. If

is a bijection ,

(47)

then it is conformal on U1 . So it suffices to prove (47). To this end we use Rouche’s ◦ . Then Eat − z − w has exactly one zero z ∈ B(Eat , ρ) theorem. Let w ∈ Dρ/2+ and for all z ∈ ∂B(Eat , ρ), |Eat − z − w| ≥ ρ − |w| ≥ ρ/2 > α0 .

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Since, by (46),

605

(0) E (z) − w − (Eat − z − w) ≤ α0 ,

for all z ∈ B(Eat , ρ), it follows that E (0) (z) − w, like (Eat − z − w) has exactly one −1 ◦ ) ⊂ B(Eat , ρ). zero z ∈ B(Eat , ρ). This proves (47) because E (0) (Dρ/2+ Next we prove, by induction in n, that (In )

E (n−1) : Un → Dρ/2 conformally .

For n = 1, this follows from I1 . Suppose In , holds. First note that αn ≤ α1 = (Cβ /ρ)γ02 , Ineq. (44), Cχ ≥ 1, and ρ < 4/5 imply αn + ρ/2 < 1/2 .

(48)

Thus we can choose a positive such that αn + ρ/2 + 2 < 1/2 .

(49)

◦ ◦ ◦ ◦ ◦ ◦ := Dρ/2+ and D− := Dρ/2−ρ , so that D− ⊂ Dρ/2 ⊂ D+ . We claim We define D+ that

E (n)

−1

◦ (D+ ) ⊂ E (n−1)

−1

◦ (D− )

(50)

and that E (n) : E (n)

−1

◦ ◦ (D+ ) → (D+ )

is a bijection .

(51)

Suppose (50) and (51) hold. Then by (50) and the induction Hypothesis In , −1 −1 ◦ ◦ ) ⊂ Un◦ . Since E (n) is continuous on Un◦ , it follows that E (n) (D+ ) is E (n) (D+ (n) open. Since E is analytic, (51) implies In+1 . It remains to prove (50) and (51). (50) follows from (45) and (49): if |E (n) (z)| < ρ/2 + and |E (n) (z) − ρ−1 E (n−1) (z)| ≤ αn , then |E (n−1) (z)| < ρ/2 − ρ. ◦ To prove (51) we use Rouche’s Theorem. Let w ∈ D+ . Then, by (49), ρw ∈ ◦ (n−1) D− and the induction Hypothesis In implies that E (z) − ρw has exactly one zero z ∈ E (n−1)

−1

◦ (D− ). On the other hand, by (49),

−1 (n−1) ρ E (z) − ρw ≥ ρ−1 |E (n−1) (z)| > αn ,

−1 ◦ ∀z ∈ ∂ E (n−1) (D− ) .

Since, by (45), (n) E (z) − w − ρ−1 E (n−1) (z) − ρw ≤ αn ,

∀z ∈ Un , −1

it follows that E (n) (z) − w, like E (n−1) (z) − ρw, has exactly one zero z ∈ E (n−1) ◦ (D− ). Therefore, (51) follows from (50). (b) By (a), Uk+1 contains zk and all subsequent terms of the sequence (zn )∞ n=1 . Thus, to prove that (zn )∞ n=1 converges, it suffices to show that the diameter of Un tends to zero as n tends to infinity. To this end, let F (k) denote the inverse of the

606



function E (k) : Uk+1 → Dρ/2 . Then diam(Un+1 ) = diam F (n) (Dρ/2 ) = diam E (at) ◦ F (0) ◦ E (0) · · · ◦ F (n−1) ◦ E (n−1) ◦ F (n) (Dρ/2 ) , (52) where we used that z → E (at) (z) := Eat − z is an isometry. We want to estimate (52) from above. Let k ≥ 1. For all z ∈ Dρ/2 , by (45), ρz − E (k−1) F (k) (z) ≤ ραk , (53) and hence |E (k−1) ◦ F (k) (z)| ≤ ραk + ρ2 /2 ≤ ρ/2 − ρ, where := 1/2 − ρ/2 − α1 is positive by (48). This shows that E (k−1) ◦ F (k) maps Dρ/2 into Dρ/2−ρ . By Cauchy’s integral formula and by (53), (k−1) 1 E (k−1) ◦ F (k) (w) − ρw (k) ∂z E (54) ◦ F (z) − ρz = dw 2πi ∂Dρ/2 (z − w)2 ≤ αk /(22 ) ,

for z ∈ Dρ/2−ρ .

It follows that |(E (k−1) ◦ F (k) ) (z)| = ρ + αk /(22 ) for z ∈ Dρ/2−ρ . A similar estimate yields |(E (at) ◦ F (0) ) (z)| ≤ 1 + α0 /(2ρ2 ) for z ∈ Dρ/2−ρ . Using these estimates and (52) we obtain diam(Un+1 ) ≤ 1 + α0 /(2ρ2 ) diam E (0) ◦ F (1) ◦ · · · ◦ F (n−1) (Dρ/2−ρ ) ≤ ρn−1

n−1

1 + αk /(2ρ2 ) diamDρ/2−ρ

k=0

≤ ρ exp n

∞

2

αk /(2ρ ) ,

k=0

where we used that 1 + x ≤ exp(x) in the last inequality. This proves (b).

The following results will allow us to show that z∞ (s) = inf σ(H(s)), if s ∈ R. Corollary 20. Suppose the assumptions of Lemma 19 hold, Eat ∈ R, and H (0) (z)∗ = H (0) (z) for all z ∈ B(Eat , ρ). Then for all n ≥ 0, Un+1 ∩ R is an interval and ∂x E (n) (x) < 0 on Un+1 ∩ R. Proof. Using an induction argument and the definition of the renormalization transformation one sees that H (n) (z)∗ = H (n) (z) for z ∈ Un . In particular, E (n) (z) = E (n) (z) for all z ∈ Un . This together with E (n) : Un+1 → Dρ/2 being a homeomorphism, c.f. Lemma 19, implies that [an+1 , bn+1 ] := (E (n) )−1 [−ρ/2, ρ/2] = Un+1 ∩ R is indeed an interval. Moreover, by Lemma 19, Eat − ρ < a1 < a2 < · · · ≤ z∞ .

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607

We prove by induction that for all n ∈ N, ∂x E (n) (x) < 0 on [an+1 , bn+1 ] .

(55)

We begin with n = 0. By assumption on H [z], |E (z) − (Eat − z)| ≤ α0 for z ∈ U0 . For z = Eat − ρ, which belongs to U0 by choice of ρ, we obtain 1 |E (0) (Eat − ρ) − ρ| ≤ α0 < ρ , 2 by assumption on α0 . This proves that E (0) (Eat − ρ) > ρ/2. Since |E (0) (x)| ≥ ρ/2 for x ∈ [Eat − ρ, a1 ] the function E (0) must be positive on this interval. On the other hand it is a diffeomorphism from [a1 , b1 ] onto [−ρ/2, ρ/2] by Lemma 19. It follows that ∂x E (0) (x) < 0 for x ∈ [a1 , b1 ]. To prove (55) for n ≥ 1 suppose that (0)

∂x E (n−1) (x) < 0

on

[an , bn ] .

(0)

(56)

be the inverse of E : Un+1 → Dρ/2 . Setting z = 0 in (54) we obtain (n−1) ρ ραn ∂x E ◦ F (n) (x) − ρx x=0 ≤ ≤ 2α1 < ρ . 2 (ρ/2)2 This shows that 0 < E (n−1) ◦ F (n) (0) 1 = ∂x E (n−1) F (n) (0) . (∂x E (n) )(F (n) (0)) Let F

(n)

(n)

Hence (∂x E (n) )(F (n) (0)) has the same sign as (∂x E (n−1) )(F (n) (0)), which is negative by induction hypothesis (56). Since E (n) : [an+1 , bn+1 ] → [−ρ/2, ρ/2] is a diffeomorphism, ∂x E (n) (x) < 0 for all x ∈ [an+1 , bn+1 ]. Proposition 21. Suppose the assumptions of Lemma 19 are satisfied, Eat is real and H (0) [z]∗ = H (0) [z] for z ∈ B(Eat , ρ). Then, there exists an a < z∞ such that H (0) [x] has a bounded inverse for x ∈ (a, z∞ ). Proof. Let [an , bn ] = Un ∩ R, c.f. Corollary 20. Then, by Lemma 19, a1 < a2 < a3 < · · · < z∞ and limn→∞ an = z∞ . We show that H (n) [x] is bounded invertible for x ∈ [an , an+1 ). By a repeated application of the Feshbach property, Theorem 7 (i), it will follow that H (n−1) [x], . . . , H (0) [x] are also bounded invertible for x ∈ [an , an+1 ). (n) Let x ∈ [an , an+1 ). Then both H (n) [x] and H0,0 [x] are self-adjoint and, by (34) and (45), (n) (n) H (n) [x] = H0,0 [x] + H (n) [x] − H0,0 [x] ≥ E (n) (x) − ξγn , (57) (n)

where we have used that H0,0 [x] ≥ E (n) (x), which follows from βn < 1. Since the function E (n) is decreasing on [an+1 , bn+1 ] with a zero in this interval, we know that E (n) (an+1 ) > 0. On the other hand, by construction of Un , |E (n) | ≥ ρ/2 on [an , an+1 ). Therefore (57) implies that H (n) [x] ≥ (ρ/2 − ξγn ) > (ρ/2 − ξρ/8) > 0, which proves that H (n) [x] is bounded invertible.

608



9. Construction of the eigenvector Next we show that zero is an eigenvalue of H (0) [z∞ ]. In fact, we will show that zero is an eigenvalue of H (n) [z∞ ] for every n ∈ N. To this end we define (n) −1 Qn [z] := χρ − χρ Hχρ [z] χρ W (n) [z]χρ , for z ∈ Un , (n)

where W (n) = H (n) − H0,0 . By the definition of H (n) [z] and by Lemma 8 (c), H (n−1) [z]Qn−1 [z]Γ∗ρ = ρΓ∗ρ χ1 H (n) [z] (58) and moreover, if H (n) [z]ϕ = 0 and ϕ = 0 then Qn−1 [z]Γ∗ρ ϕ = 0 by Theorem 7. Thus if 0 is an eigenvalue of H (n) [z], then it is an eigenvalue of H (n−1) [z] as well, and the operator Qn−1 [z]Γ∗ρ maps the corresponding eigenvectors of H (n) [z] to eigenvectors of H (n−1) [z]. Theorem 22. Suppose the assumptions of Lemma 19 hold. Then the limit ϕ(0) = lim Q0 [z∞ ]Γ∗ρ Q1 [z∞ ] . . . Γ∗ρ Qn [z∞ ]Ω n→∞

(0)

exists, ϕ

= 0 and H [z∞ ]ϕ(0) = 0. Moreover, ∞ (0) γl , ϕ − Q0 [z∞ ]Γ∗ρ Q1 [z∞ ] . . . Γ∗ρ Qn [z∞ ]Ω ≤ C (0)

l=n+1

where C = C(ρ, ξ, γ0 ). Remark. By Theorem 22 and by Proposition 10 (ii), Qχ (ϕat ⊗ ϕ(0) ) is an eigenvector of Hg with eigenvalue z∞ . Proof. For k, l ∈ N with k ≤ l we define ϕk,l ∈ Hred by ϕk,l := Qk [z∞ ]Γ∗ρ Qk+1 [z∞ ]Γ∗ρ · . . . · Ql−1 [z∞ ]Γ∗ρ Ql [z∞ ]Ω , and we set ϕk,k−1 := Ω. Step 1: There is a constant C < ∞ depending on ξ, ρ and k, l ∈ N with k ≤ l ϕk,l − ϕk,l−1 ≤ Cγl .

n

γn such that, for all

By definition of ϕk,l and since Ω = Γ∗ρ χρ Ω ϕk,l − ϕk,l−1 =

l−1

Qn [z∞ ]Γ∗ρ Ql [z∞ ] − χρ Ω ,

n=k

where the empty product in the case k = l is to interpret as the identity operator. Since on Un , Qn Γ∗ρ = Qn ≤ Qn − χρ + 1 ≤ exp Qn − χρ it follows that l−1 ϕk,l − ϕk,l−1 ≤ exp Qn [z∞ ] − χρ Ql [z∞ ] − χρ , (59) n=k

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and hence it remains to estimate Qn [z∞ ] − χρ . By definition of Qn , on Un , (n) −1 (n) (n) χρ H − H0,0 χρ (60) Qn − χρ = −χρ Hχρ and by estimates in the proof of Lemma 15, (n) −1 8 1 (n) H , H (n) − H0,0 ≤ ξγn . χρ ≤ χρ ρ1−ξ

(61)

Equation (60), combined with the estimates (59), and (61) prove Step 1 with ⎛ ⎞ 8 ξ 8 ξ exp ⎝ γn ⎠ . C := ρ1−ξ ρ1−ξ n≥0

Step 2: For all k ∈ N, the limit ϕk,∞ := lim ϕk,n n→∞

exists, the convergence being uniform in s, and ϕk,∞ = 0 for k sufficiently large. Summing up the estimates from Step 1 for all l with l ≥ n + 1 we arrive at ϕk,∞ − ϕk,n ≤ C

∞

γl → 0 ,

n → ∞,

l=n+1

uniformly in s. Specializing this inequality to n = k − 1 so that ϕk,n = ϕk,k−1 = Ω, we see that ϕk,∞ − Ω < 1 = Ω and hence ϕk,∞ = 0 for sufficiently large k. Step 3: For all k ∈ N, H (k) [z∞ ]ϕk,∞ [z∞ ] = 0 ,

and ϕk,∞ [z∞ ] = 0 .

Since H (k) [z∞ ] is a bounded operator and by (58), H (k) [z∞ ]ϕk,∞ = lim H (k) [z∞ ]ϕk,n n→∞ n−k+1 (n+1) = lim ρΓ∗ρ χ1 H [z∞ ]Ω . n→∞

(62)

(n+1)

Using H (n+1) [z∞ ]Ω = E (n+1) (z∞ )Ω + (H (n+1) [z∞ ] − H0,0 [z∞ ])Ω and ρ (n+1) |E (n+1) (z∞ )| ≤ , H (n+1) [z∞ ] − H0,0 [z∞ ] ≤ γn ≤ γ0 , 2 we see that the limit (62) vanishes because limn→∞ ρn = 0. From ϕk−1,n = (Qk−1 [z∞ ]Γ∗ρ )ϕk,n , the boundedness of the operator Qk , and from Step 2 it follows that, ϕk−1,∞ = Qk−1 [z∞ ]Γ∗ρ ϕk,∞ . Since ϕk,∞ belongs to the kernel of H (k) [z∞ ], as we have just seen, it follows from Theorem 7 that ϕk−1,∞ = 0 whenever ϕk,∞ = 0. Iterating this argument starting with k so large that, by Step 2, ϕk,∞ = 0, we conclude that ϕk,∞ = 0 for all k ∈ N.

610



10. Analyticity of eigenvalues and eigenvectors This section is devoted to the proof of Theorem 1. It is essential for this proof, that a neighborhoods V0 ⊂ V of s0 and a bound g1 on g can be determined in such a way that the renormalization analysis of Sections 8 and 9, and in particular the choices of ρ and ξ are independent of s ∈ V0 and g < g1 . Once V0 and g are found, the assertions of Theorem 1 are derived from Proposition 17 and the uniform bounds of Sections 8 and 9. Proof of Theorem 1. Let μ > 0 and U ⊂ Cν+1 be given by Hypothesis I and Hypothesis III, respectively. For the renormalization procedure to work, we first choose ρ ∈ (0, 4/5) and a open neighborhood V0 ⊂ V of s0 , both small enough, so that Cγ ρμ < 1 and B Eat (s), ρ ⊂ z|(s, z) ∈ U , if s ∈ V0 . (63) √ This is possible since s → Eat (s) is continuous. Let ξ = ρ/(4Cχ ). Next we pick small positive constants α0 , β0 , and γ0 such that ρ ρ ρ α0 < , β0 ≤ , γ0 ≤ , (64) 2 8Cχ 8Cχ and in addition

Cβ /ρ ρ γ2 ≤ . 1 − (Cχ ρμ )2 0 8Cχ By Theorems 13 and 23, there exists a g1 > 0 such that for |g| ≤ g1 Hg(0) [s, z] − Eat (s) − z ∈ B(α0 , β0 , γ0 ) , for (s, z) ∈ U , β0 +

(65)

(0)

where Hg [s, z] is analytic on U. We define U0 := U Un := (s, z) ∈ Un−1 : |E n−1 (s, z)| ≤ ρ/8 , and

Un (s) := z|(s, z) ∈ Un , n ∈ N . Then, by (64), (65), and (63) the assumptions of Lemma 19 are satisfied for s ∈ V0 and U0 = U0 (s). It follows that, for all n ∈ N, H (n) [s, z] = Rn H (0) [s, z] is welldefined for (s, z) ∈ Un , and that Un (s) = ∅. By Proposition 17, H (n) [s, z] is analytic in Un◦ . Step 1: z∞ (s) = limn→∞ zn (s) exists and is analytic on V0 . Since H (n) [s, z] is analytic on Un◦ , so is E (n) (s, z). Let zn (s) denote the unique zero of the function z → En (s, z) on Un (s) as determined by Lemma 19. That is, E (n) s, zn (s) = 0 . By the implicit function theorem zn (s) is analytic in s. The application of the implicit function theorem is justified since z → E (n) (s, z) is bijective in a neighborhood of zn (s), and thus in this neighborhood ∂z E (n) (s, z) = 0. By Lemma 19 (b),

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zn (s) converges to z∞ (s) uniformly in s ∈ V0 . This implies the analyticity of z∞ (s) on V0 , by the Weierstrass approximation theorem of complex analysis. Step 2: For s ∈ V0 , there exists an eigenvector ψ(s) of H(s) with eigenvalue z∞ (s), such that ψ(s) depends analytically on s. Since H (n) [s, z] is analytic on Un◦ , it follows, by Proposition 17, that (n)

Qn [s, z] = χρ (s) − χρ (s)Hχρ [s, z]

−1

χρ (s)W (n) [s, z]χρ (s)

is analytic on Un◦ , where W (n) := H (n) − H0,0 . Hence, by Step 1, s → Qn [s, z∞ (s)] is analytic on V0 . It follows that ϕ0,n (s) := Q0 s, z∞ (s) Γ∗ρ Q1 s, z∞ (s) . . . Γ∗ρ Qn s, z∞ (s) Ω (n)

is analytic on V0 . From Theorem 22 we know that these vectors converge uniformly on V0 to a vector ϕ(0) (s) = 0 and that H (0) [s, z∞ (s)]ϕ(0) (s) = 0. Hence ϕ(0) (s) is analytic on V0 and, by the Feshbach property (Proposition 10 (ii)), the vector ψ(s) = Qχ s, z∞ (s) ϕat (s) ⊗ ϕ(0) (s) is an eigenvector of H(s) with eigenvalue with z∞ (s). Since ϕat is analytic on V0 we conclude that ψ is analytic on V0 as well. Step 3: For s ∈ V0 ∩ Rν , z∞ (s) = infσ(H(s)). Let s ∈ V0 ∩ Rν . Then H(s) is self-adjoint and its spectrum is a half line [E(s), ∞). By Step 2, z∞ (s) ≥ E(s). We use Proposition 21 to show that z∞ (s) > E(s) is impossible. Clearly Eat (s) ∈ R, and H (0) [s, z]∗ = H (0) [s, z] for z ∈ B(Eat (s), ρ) is a direct consequence of the definition of H (0) and the self-adjointness of H(s). Hence there exists a number a(s) < z∞ (s) such that H (0) [s, x] has a bounded inverse for all x ∈ (a(s), z∞ (s)). It follows, by Theorem 7, that (a(s), z∞ (s)) ∩ σ(H(s)) = ∅. Therefore z∞ (s) = E(s).

Appendix A. Neighborhood of effective Hamiltonians The purpose of this section is to prove the following theorem. Theorem 23. Let Hypotheses I, II, and III hold for some μ > 0 and U ⊂ C × C. For every ξ ∈ (0, 1) and every triple of positive constants α0 , β0 , γ0 , there exists a positive constant g1 such that for all g ∈ [0, g1 ) and all (s, z) ∈ U, (Hg (s) − z, H0 (s) − z) is a Feshbach pair for χ(s), and Hg(0) [s, z] − Eat (s) − z ∈ B(α0 , β0 , γ0 ) . (66) By Theorem 13 we know that we can choose g sufficiently small such that the Feshbach property is satisfied. To prove (66) we explicitly compute the sequence

612



(0)

of kernels w = (wm,n ) ∈ Wξ such that Hg [s, z] = H(w). To this end we recall that, by (18) and (19),

χgW χ1 , (67) Hg(0) [s, z] = (Eat − z) + Hf + χ1 gW − gW χ(Hg − z)−1 χ at

and we expand the resolvent (Hg −

z)−1 χ

χ1 gW − gW χ(Hg − z)−1 χgW χ1 χ

in a Neumann series. We find that

at

=

∞

(−1)L−1 g L χ1 (W F )L−1 W χ1 at ,

L=1

where F = χ(H0 − z)−1 χ is a function of Hf , that is, F = F (Hf ) with F (r) :=

χ2 (s, r) , Hat (s) − z + r

(68)

and χ(s, r) = P at (s) ⊗ 1 + Pat (s) ⊗ χ1 (r). Since W = a(G) + a∗ (G), the Lth term in this series is a sum of 2L terms. We label them by L-tuples σ = (σ1 , σ2 , . . . , σL ), with σi ∈ {−, +}, and we set a+ (G) := a∗ (G), a− (G) := a(G). With these notations $ L−1 % σL L−1 σj a (G)F (Hf ) a (G)χ1 W χ1 at = . (69) χ1 χ1 (W F ) σ∈{−,+}L

j=1

at

Next we use a variant of Wick’s theorem (see [6]) to expand each term of the sum (69) in a sum of normal ordered terms. Explicitly, this means that in each term of (69) the pull through formulas f (Hf )a∗ (k) = a∗ (k)f (Hf + |k|) ,

a(k)f (Hf ) = f (Hf + |k|)a(k) ,

and the canonical commutation relations are used to move all creation operators to the very left, and all annihilation operators to the right of all other operators. To write down the result we introduce the multi-indices m, p, n, q := (m1 , p1 , n1 , q1 , . . . , mL , pL , nL , qL ) ∈ {0, 1}4L , which run over the sets IL := {m, p, n, q ∈ {0, 1}4L |ml + pl + nl + ql = 1}. The numbers ml , pl , nl , ql may be thought of as flags that indicate the position of the operator aσl (k) in a given normal-ordered term: ml = 1 (nl = 1) if it is a noncontracted creation (annihilation) operator, pl = 1 (ql = 1) if it is a contracted creation (annihilation) operator. We obtain χ1 (W F )L−1 W χ1 at (70) & ' ' & L L ∗ ml nl ˜ ˜ ˜ Vm,p,n,q (Hf , km , kn ) , a (kml ) a(knl ) dkm dkn = m,p,n,q∈IL

l=1

l=1

Vol. 10 (2009)

with


613

Vm,p,n,q (r, km , kñ ) = χ1 r + r0 (m, n) ' $ &L−1 G(kml )ml G∗ (kñl )nl a∗ (G)pl a(G)ql F Hf + r + rl (m, n) × l=1

%

× G(kmL )

mL

∗

nL ∗

G (kñL )

pL

χ1 r + rL (m, n) ,

ql

a (G) a(G)

(71)

at,Ω

where Aat,Ω := (Ω, Aat Ω), Ω ∈ F being the vacuum vector. Moreover dk˜m :=

km := (m1 k1 , . . . , mL kL ) ,

L

dkl

l=1,ml =1

dkñ :=

kñ := (n1 k˜1 , . . . , nL k˜L ) ,

L

dk˜l

l=1,nl =1

and rl (m, n) =

|ki | +

i≤l

|k˜i | .

i≥l+1 nl =1

ml =1

Upon summing (70) for L = 1 through ∞ we collect all terms with equal numL L bers M = |m| := l=1 ml and N = |n| := l=1 nl of creation and annihilation operators, respectively. To this end we need to relabel the integration variables. That is, we distribute the M + N variables k1 , . . . , kM ∈ R3 × {1, 2} and k˜1 , . . . , kÑ ∈ R3 × {1, 2} into the M + N arguments of Vm,p,n,q (r, · , · ) designated by ml = 1 and nl = 1. Explicitly this is done by σm (k1 , . . . , kM ) = (m1 km(1) , . . . , mL km(L) ) ,

m(l) =

l

mj .

j=1

We obtain (−1)L−1 g L χ1 (W F )L−1 W χ1 at L≥1

=

M +N ≥1

a∗ (k (M ) )w ˆM,N (Hf , K)a(k˜(N ) ) dK

B M +N

where w ˆM,N (r, K) =

L≥M +N

(−1)L−1 g L

m,p,n,q∈IL

|m|=M,|n|=N

Vm,p,n,q r, σm (k (M ) ), σn (k˜(N ) )

(72)

614



(0)

and K = (k (M ) , k˜(N ) ). Hence Hg

= H(w) with w0,0 (r) = Eat − z + r + (−1)L−1 g L L≥1

V0,p,0,q (r) ,

(73)

p,q∈{0,1}2L

pl +ql =1

and wM,N (r, K) given by the symmetrisation of w ˆM,N (r, K) with respect to 3 3 ˜ ˜ k1 , . . . , kM ∈ R and k1 , . . . , kN ∈ R , respectively. It remains to show that H(w) − (Eat − z) belongs to the ball B(α0 , β0 , γ0 ) for g sufficiently small. To this end we need the following estimates on the operatorvalued function (68) and on its derivative, F (r) = −

χ2 (s, r) Pat (s) ⊗ 2χ1 (s, r)∂r χ1 (s, r) . + (Hat (s) − z + r)2 Hat (s) − z + r

(74)

Lemma 24. Let Hypothesis I and III hold for some μ > 0 and U. Then C0 := sup sup (Hf + 1)F (Hf + r) < ∞ (s,z)∈U r≥0

C1 := sup sup (Hf + 1)F (Hf + r) < ∞ , (s,z)∈U r≥0

for F given by (68). Proof. To show that C0 is finite we estimate χ2 (s, Hf + r) sup (Hf + 1) H (s) − z + H + r at f r≥0 P at (s) ⊗ 1 + Pat (s) ⊗ χ21 (r + q) = sup (q + 1) Hat (s) − z + q + r r,q≥0 P at (s) ≤ sup (q + 1) Hat (s) − z + q + r r,q≥0 χ21 (r + q) Pat (s) . (q + 1) + sup Eat (s) − z + q + r r,q≥0 By Hypothesis III, both terms are bounded on U. Similarly C1 is estimated using (74). Proof of Theorem 23. Let Hypothesis I and III hold for some μ > 0 and U. Let 0 < ξ < 1. By Theorem 13 we know that there exists a g0 > 0 such that for all |g| < g0 , (Hg − z, H0 − z) on U is a Feshbach pair for χ. Let (s, z) ∈ U. First we derive upper bounds for Vm,p,n,q and ∂r Vm,p,n,q . Inserting (Hf + 1)−1 (Hf + 1) in front of F (Hf + r + rl (m, n)) we obtain, from Lemma 24, that |Vm,p,n,q (r, km , kñ )| &L ' ≤ G(kml )ml G(knl )nl Gpωl +ql C0L−1 l=1

sup s:(s,z)∈U

Pat (s) .

(75)

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615

Let Cat := sups:(s,z)∈U Pat (s). Similarly, using (71), (74) and (75) we estimate |∂r Vm,p,n,q (r, km , kñ )| &L ' ≤ 2χ1 ∞ · G(kml )ml G(knl )nl Gpωl +ql C0L−1 Cat

(76)

l=1

$ &j−1 ' ml ∗ ˜ nl ∗ pl ql + G(kml ) G (knl ) a (Gg ) a(G) F Hf + r + rl (m, n) j=1 l=1 × G(kmj )mj G∗ (kñj )nj a∗ (G)pj a(G)qj F Hf + r + rj (m, n) ⎧ ⎫ ⎨ L−1 ⎬ G(kml )ml G∗ (kñl )nl a∗ (G)pl a(G)ql F Hf + r + rl (m, n) × ⎩ ⎭ l=j+1 % × G(kmL )mL G∗ (kñL )nL a∗ (G)pL a(Gg )ql at,Ω &L ' ml nl pl +ql ≤ G(kml ) G(knl ) Gω Cat C0L−2 2χ1 ∞ C0 +(L−1)C1 . (77) L−1

l=1

With the help of (75) and (77) we can now prove the theorem. From (73) and (75) it follows that |w0,0 (0) − (Eat − z)| ≤

∞

gL

L=2

V0,p,0,q (0)

p,q∈{0,1}2L

pl +ql =1

≤

∞ L=2

gL

L−1 GL Cat ω C0

p,q∈{0,1}2L

pl +ql =1

≤ Cat

∞

L−1 2L g L GL , ω C0

L=2

which can be made smaller than any positive α0 for small g. Estimate (77) implies that − 1∞ = sup |w 0,0 [r] − 1| w0,0 r

≤ sup r

∞ L=2

gL

p,q∈{0,1}2L

pl +ql =1

|∂r V0,p,0,q (r)|

616

M. Griesemer and D. G. Hasler ∞

≤

gL

L=2


L−2 2C0 χ1 ∞ + (L − 1)C1 GL ω Cat C0

p,q∈{0,1}2L

pl +ql =1 ∞

≤

L−2 L 2C0 χ1 ∞ + (L − 1)C1 , g L GL ω 2 Cat C0

L=2

which can be made smaller than any positive β0 for small g. It remains to show that (wM,N )M +N ≥1 μ,ξ ≤ γ0 for g sufficiently small. By (72) gL Vm,p,n,q μ (78) wM,N μ ≤ L≥M +N

m,p,n,q∈IL

|m|=M,|n|=N

where, by a triangle inequality and by (75) and (77) 1/2 dK Vm,p,n,q μ ≤ Vm,p,n,q (K)2∞ |K|2+2μ B M +N 1/2 dK 2 + ∂r Vm,p,n,q (K)∞ |K|2+2μ B M +N +N +N ) ≤ GM GL−(M SL , μ ω

with SL :=

Cat C0L−2

(C0 +

(79)

2χ1 ∞ C0

+ (L − 1)C1 ), and 1/2 dk G(k)2 2+2μ . |k| R3

Gμ := Combining (78) and (79) and find ∞ g L SL wM,N μ ≤ L=1

+N +N ) GM GL−(M , μ ω

m,p,n,q∈IL

|m|=M,|n|=N

where the condition L ≥ M + N has been relaxed to L ≥ 1. Therefore (wM,N )M +N ≥1 μ,ξ = ξ −(M +N ) wN,M μ ≤

M +N ≥1 ∞ L

g SL GL ω

L=1

≤

∞ L=1

≤

∞ L=1

ξ −(M +N )

M +N ≥1

⎛ ⎝ g L SL GL ω

G−1 ω Gμ

m,p,n,q∈IL

|m|=M,|n|=N

ξ −1 G−1 ω Gμ

m,p,n,q∈I1

L −1 g L SL GL G−1 . ω 2 + 2ξ ω Gμ

m1 +n1

⎞L ⎠

M +N

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617

This can be made smaller than any positive γ0 for small coupling g. It follows that we can find a g1 > 0 such that on U, (66) holds for all g ∈ [0, g1 ). This concludes the proof.

Appendix B. Technical auxiliaries Let L2 (R3 × {1, 2}, L(Hat )) be the . Banach space of (weakly) measurable functions T : R3 × {1, 2} → L(Hat ) with T (k)2 dk < ∞, and let 1/2 2 −1 T ω := T (k) (|k| + 1)dk . Lemma 25. If T ∈ L2 (R3 × {1, 2}, L(Hat )), then 1/2 T (k)2 |k|−1 dk , a(T )(Hf + 1)−1/2 ≤ a∗ (T )(Hf + 1)−1/2 ≤ T ω . For a proof of this lemma see, e.g., [5]. Lemma 26. Suppose the function F : U → L(Hat , L2 (R3 ; Hat )), s → Fs is uniformly bounded and suppose for a.e. k ∈ R3 and all s ∈ U , there exists an operator Fs (k) ∈ L(Hat ) such that Fs (k)ϕ = (Fs ϕ)(k) for all ϕ ∈ Hat . If for a.e. k ∈ R3 , the function s → Fs (k) ∈ L(Hat ) is analytic, then F is analytic. Proof. Let h ∈ L2 (R3 ) and ϕ1 , ϕ2 ∈ Hat , and suppose γ is a nullhomotopic closed curve in U . Then (h ⊗ ϕ1 , Fs ϕ2 )ds = h(k) ϕ1 , Fs (k)ϕ2 dkds γ γ = h(k) ϕ1 , Fs (k)ϕ2 dsdk = 0 , γ

where we interchanged the order of integration, which is justified since F is uniformly bounded. It follows that s → (h ⊗ ϕ1 , Fs ϕ2 ) is analytic. By linearity we conclude that s → (ψ, Fs ϕ2 ) is analytic for all ψ in a dense linear subset of Hat ⊗h. This and the uniform boundedness imply strong analyticity, see for example the remark following Theorem 3.12 of Chapter III in [16]. Proposition 27. Let R s → T (s) be an analytic family. Suppose there exists an isolated non-degenerate eigenvector E(s) with analytic projection operator P (s). Let P (s) := 1 − P (s) and let Γ := (s, z) ∈ R × C | T (s) − z is a bijection from D T (s) ∩ Ran P (s)

to Ran P (s)with bounded inverse . Then Γ is open and (s, z) → (T (s) − z)−1 P (s) is analytic on Γ.

618



Proof. Let (s0 , z0 ) ∈ Γ. There exists in a neighborhood of s0 a bijective operator U (s) : H → H, analytic in s, such that U (s)P (s)U (s)−1 = P (s0 ) and hence U (s)P (s)U (s)−1 = P (s0 ) ( [20] Thm. XII.12). The operator T(s) = U (s)T (s)U (s)−1 is an analytic family. It leaves Ran P (s0 ) invariant and thus T(s)|`Ran P (s0 ) : Ran P (s0 ) ∩ D(T(s)) → Ran P (s0 ) is an analytic family as well. By this and the fact that (T(s0 ) − z0 )|`Ran P (s0 ) is bijective with bounded inverse since (s0 , z0 ) ∈ Γ, it follows by [20] Thm. XII.7 that in a neighborhood of (s0 , z0 ), (T(s)−z)|`Ran P (s0 ) is bijective with bounded inverse and (T(s)−z)−1 P (s0 ) is analytic in both variables. Thus in this neighborhood also the function (T (s)−z)|`P (s) is bijective with bounded inverse and (T (s) − z)−1 P (s) is an analytic function of two variables. Theorem 28. Suppose the assumptions of Lemma 19 hold. Then in the norm of L(Hred ), lim H (n) (z∞ ) = λHf . n→∞

for some λ ∈ C. Proof. We recall the notations H (n) (z∞ ) = H(w(n) (z∞ )) and E (n) (z∞ ) = (n) w0,0 (z∞ , 0). Using the decomposition (n) H (n) (z∞ ) = H (n) (z∞ ) − w0,0 (z∞ , Hf ) (n) + w0,0 (z∞ , Hf ) − E (n) (z∞ ) + E (n) (z∞ ) , the theorem will follow from Steps 1 and 2 below. (n)

Step 1: limn→∞ H (n) (z∞ ) − w0,0 (z∞ , Hf ) = 0 and limn→∞ E (n) (z∞ ) = 0. From Lemma 18 we know that H (n) (z) − ρ−1 E (n−1) (z) ∈ B(αn , βn , γn ) ,

(80)

for z ∈ Un . By (34) this implies that (n)

(n)

H (n) (z∞ ) − w0,0 (z∞ , Hf ) ≤ w(n) (z∞ ) − w0,0 (z∞ )μ,ξ ≤ γn → 0 (n → ∞) . By (80), |E (n) (z)| ≤ ραn+1 + ρ|E (n+1) (z)| ,

z ∈ Un .

Iterating (81), we find |E (n) (z)| ≤

m

ρk αn+k + ρm |E (n+m) (z)| ,

k=1

which yields, |E (n) (zn+m )| ≤

∞ k=1

ρk αn+k .

(81)

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619

Since E (n) is continuous and limn→∞ zn = z∞ , we arrive at ∞ |E (n) (z∞ )| ≤ ρk αn+k → 0 , (n → ∞) . k=1

Step 2: There exists a λ ∈ C such that (n) lim w0,0 (z∞ , r) − w(n) (z∞ , 0) = λr , n→∞

uniformly in 0 ≤ r ≤ 1. (n)

To abbreviate the notation, we set T (n) (z∞ , r) := w0,0 (z∞ , r) − w(n) (z∞ , 0). From [3] (3.105–3.107), we have T (n) (z∞ , r) = ρ−1 T (n−1) (z∞ , ρr) + e(n−1) (z∞ , r) , with e(n−1) (z∞ , 0) = 0 and sup |∂r e(n) (z∞ , r)| + |e(n) (z∞ , r)| ≤ Cγn2 .

(82)

(83)

r∈[0,1]

Iterating (82), we arrive at T (n) (z∞ , r) = ρ−n T (0) (z∞ , ρn r) +

n−1

ρ−(n−1−k) e(k) (z∞ , ρn−1−k r) .

(84)

k=0

To prove Step 2 we now show that, uniformly in r ∈ [0, 1], ∞ (n) (0) (k) lim T (z∞ , r) = r ∂r T (z∞ , 0) + ∂r e (z∞ , 0) . n→∞

k=0

Note that the series on the right hand side converges by (83). Given > 0 we choose K so large that ∞ Cγk2 ≤ . (85) k=K

By (84) and the triangle inequality, we find for n ≥ K, (suppressing z∞ ) ∞ (n) (0) (k) ∂r e (0) T (r) − r ∂r T (0) − k=0

K −(n−1−k) (k) n−1−k ≤ ρ−n T (0) (ρn r) − r∂r T (0) (0) + e (ρ r) − r∂r e(k) (0) ρ k=0

+

∞ k=K+1

|ρ−(n−1−k) e(k) (ρn−1−k r)| +

∞

|r∂r e(k) (0)| .

k=K+1

The first two terms on the right hand side converge to zero as n tends to infinity because T (n) (0) = 0 and e(k) (0) = 0. The last term on the right hand side is bounded by , which follows from Eqns. (83) and (85). Using again (83) and (85)

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we see that the first term on the last line is bounded by as well, since, by the mean value theorem, α−1 |e(n) (αr)| ≤ supξ∈[0,1] |e(n) (αξ)|r for α, r ∈ [0, 1].

Acknowledgements We thank J¨ urg Fr¨ ohlich and Israel Michael Sigal for numerous discussions on the renormalization technique. M. Griesemer also thanks Volker Bach for explaining the results of [4], and Ira Herbst for the hospitality at the University of Virginia, where large parts of this work were done.

References [1] J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition: Two-level system coupled to quantized radiation field, Phys. Rev. A 58 (6) (1998), 4300–4306. [2] J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition, Comm. Math. Phys. 203 (2) (1999), 445–463. [3] V. Bach, T. Chen, J. Fr¨ ohlich and I. M. Sigal, Smooth Feshbach map and operatortheoretic renormalization group methods, J. Funct. Anal. 203 (1) (2003), 44–92. [4] V. Bach, J. Fr¨ ohlich and A. Pizzo, Infrared-finite algorithms in QED: The groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys. 264 (1) (2006), 145–165. [5] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (2) (1998), 299–395. [6] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Renormalization group analysis of spectral problems in quantum field theory, Adv. Math. 137 (2) (1998), 205–298. [7] T. Chen, Infrared renormalization in non-relativistic QED and scaling criticality, J. Funct. Anal. 254 (10) (2008), 2555–2647. [8] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and atoms: Introduction to quantum electrodynamics, John Wiley & Sons, 1989. [9] J. Fr¨ ohlich, On the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. H. Poincaré Sect. A (N.S.) 19 (1973), 1–103. [10] J. Fr¨ ohlich, M. Griesemer and I. M. Sigal, Spectral renormalization group, arXiv:0811.2616. [11] M. Griesemer, Non-relativistic matter and quantized radiation, In J. Derezinski and H. Siedentop, editors, Large Coulomb Systems, volume 695 of Lect. Notes Phys., pages 217–248. Springer, 2006. [12] M. Griesemer and D. Hasler, On the smooth Feshbach–Schur map, J. Funct. Anal. 254 (9) (2008), 2329–2335. [13] M. Griesemer, E. H. Lieb and M. Loss, Ground states in non-relativistic quantum electrodynamics, Invent. Math. 145 (3) (2001), 557–595. [14] D. Hasler and I. Herbst, Absence of ground states for a class of translation invariant models of non-relativistic QED, Comm. Math. Phys. 279 (3) (2008), 769–787. [15] W. Hunziker, Distortion analyticity and molecular resonance curves, Ann. Inst. H. Poincaré Phys. Théor. 45 (4) (1986), 339–358.

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[16] T. Kato, Perturbation theory for linear operators, Classics in Mathematics. SpringerVerlag, Berlin, 1995. Reprint of the 1980 edition. [17] S. G. Krantz, Function theory of several complex variables, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, second edition, 1992. [18] E. H. Lieb and M. Loss, Existence of atoms and molecules in non-relativistic quantum electrodynamics, Adv. Theor. Math. Phys. 7 (4) (2003), 667–710. [19] A. Pizzo, One-particle (improper) states in Nelson’s massless model, Ann. Henri Poincaré 4 (3) (2003), 439–486. [20] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. [21] S. Teufel, Effective N -body dynamics for the massless Nelson model and adiabatic decoupling without spectral gap, Ann. Henri Poincaré 3 (5) (2002), 939–965. [22] S. Teufel, A note on the adiabatic theorem without gap condition, Lett. Math. Phys. 58 (3) (2002), 261–266 (2002). Marcel Griesemer Fachbereich Mathematik Universit¨ at Stuttgart D-70569 Stuttgart Germany e-mail: [email protected] David G. Hasler Department of Mathematics College of William & Mary Williamsburg, VA 23187-8795 USA e-mail: [email protected] Communicated by Claude Alain Pillet. Submitted: November 24, 2008. Accepted: March 4, 2009.

Ann. Henri Poincaré 10 (2009), 623–671 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/040623-49, published online June 30, 2009 DOI 10.1007/s00023-009-0424-x


Regularity Conditions for Einstein’s Equations at Spatial Infinity Juan Antonio Valiente Kroon Abstract. The regular finite initial value problem at spatial infinity is used to obtain regularity conditions on the freely specifiable parts of initial data sets for the vacuum Einstein equations with non-vanishing second fundamental forms. These conditions ensure that the solutions of the conformal Einstein equations extend smoothly through the sets where spatial infinity touches null infinity. For simplicity, the conformal metric of the initial data set is assumed to be analytic, although the results presented here could be extended to a setting where the conformal metric is only smooth. The analysis given here is a generalisation of the analysis of the regular finite initial value problem first carried out by Friedrich, in the case of time symmetric initial data sets.

1. Introduction The regular finite initial value problem near spatial infinity introduced in [11] provides a valuable tool for analysing the properties of the gravitational field in the regions of spacetime “close” to both spatial and null infinity. This initial value problem makes use of the so-called extended conformal Einstein field equations and general properties of conformal structures. It is such that both the equations and the data are regular at the conformal boundary – the regular finite initial value problem at spatial infinity. Whereas the standard compactification of asymptotically flat spacetimes introduced by Penrose [15] considers spatial infinity as a point, the approach used in [11] represents spatial infinity as an extended set with the topology [−1, 1] × S2 . This so-called cylinder at spatial infinity is obtained as follows: starting from an asymptotically Euclidean initial data set for the Einstein vacuum equations, ˜ ij , χ ˜h ˜ij ), one performs a conformal compactification procedure to obtain a com(S, pact manifold, S, with a singled out point, i, representing the infinity of the initial hypersurface – for simplicity it is assumed there is only one asymptotic end. In a

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second stage, the point i is blown up to a 2-sphere. This blowing up of i is achieved by lifting a neighbourhood of i to the bundle of space-spinor dyads. In the final step, one uses a congruence of timelike conformal geodesics to obtain Gaussian coordinates in a neighbourhood of the initial hypersurface. The conformal geodesics are conformal invariants. Their use to construct a gauge renders a canonical class of conformal factors for the development of the initial data which can be written entirely in terms of initial data quantities. Hence, the location of the conformal boundary is known a priori. The conformal boundary described by the canonical conformal factor contains a null infinity with the usual structure and a spatial infinity which extends in the spatial dimension – so that one can speak of the cylinder at spatial infinity. The sets {±1} × S2 will be called critical sets as they can be regarded as the collection of points where null infinity “touches” spatial infinity. Null infinity and spatial infinity do not meet tangentially at the critical points. As a consequence, part of the propagation equations implied by the conformal field equations degenerates at these sets. The analysis in [11] has shown that the solutions to the conformal field equations develop certain types of logarithmic singularities at the critical sets. These singularities form an intrinsic part of the conformal structure. It could be the case that these logarithmic singularities do not affect the regularity of the rest of the spacetime – and in particular the smoothness of null infinity – but the hyperbolic nature of the propagation equations suggests the contrary. In [20] it has been shown that there is another class of logarithmic singularities forming at the critical sets. Generalisations of these calculations were carried out in [19, 21, 22], and seem to point out that stationary solutions play a prominent role in the discussion of the structure of spatial infinity. One of the crucial features of the formalism of the regular finite initial value problem at spatial infinity is that it allows to relate the behaviour of solutions to the conformal field equations at the critical sets to properties of the initial data. In particular, under the assumption of a time symmetric initial data set with an analytic conformal metric, it was possible to find conditions on the free initial data near spatial infinity which ensure that for solutions developing from these data, singularities cannot occur. Given the need to gain further insight into the results for initial data sets with a non-vanishing second fundamental form given in [21, 22], in this article a generalisation of the discussion of [11] is carried out: it is studied how to construct conditions on free data of a class of non-time symmetric initial data so that their development does not have the type of logarithmic singularities discussed in [11]. The understanding of the logarithmic singularities in [19–22] is an outstanding open problem. The need to extend the results of [11] to initial data sets with non-vanishing extrinsic curvature is fundamental. Spacetimes with non-vanishing ADM angular momentum do not admit time symmetric slices. Thus, the analysis of [11] excludes cases of physical and practical relevance like Bowen–York data or Kerr data.

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The main result ˜ ij , χ ˜h Assume that one has some asymptotically Euclidean initial data set (S, ˜ij ) for the vacuum Einstein field equations. Furthermore, assume that the data set is maximal so that χ ˜ii = 0. As in the previous paragraphs, let S = S˜ ∪ {i} be ˜ Let (S, hij , χij ) be the conformally rescaled initial the point compactification of S. data set induced by the conformal compactification. Under the so-called conformal Ansatz, the freely specifiable information in χij is given in terms of a h-tracefree symmetric tensor ˚ Φij . Following the discussion in [6], the higher order multipoles of χij can be prescribed by means of a complex function λ. The real part, λ(R) , of λ gives rise to a contribution to the second fundamental form, χij [λ(R) ], which can be regarded as containing the higher order mass multipoles. On the other hand, the imaginary part, λ(I) , produces a contribution, χij [λ(I ], which can be interpreted as the higher order angular momentum multipoles of the second fundamental form. In order to be carry out our analysis, it will necessary to assume that hij and χij possess a particular behaviour around the point at infinity. The precise nature of these assumptions will be discussed in the main text. In any case, it is fundamental that both hij and χij can be expanded around i in terms of powers of a suitable radial coordinate r. These requirements can be phrased in terms of conditions on the freely specifiable data, and in particular on the complex function λ. The class of data considered in the present analysis includes the Bowen–York data [2]. In order to state the main result of the present analysis, one requires the following tensors: 1 kl (3) bij ≡ j Dk ril − hil Dk r , 4 kl

cij [λ(R) ] ≡ (j D|k χl|i) [λ(R) ] , with χij [λ(R) ] the part of the second fundamental form constructed out of λ(R) , and rij , (3) r, Dk , respectively the Ricci tensor, the Ricci scalar and the Levi-Civita connection of hij , while ijk denotes the volume form on S. The tensor bij is the Hodge dual of the Cotton(–York–Bach) tensor of hij . Theorem 1. For the class of data under consideration, the solution to the regular finite initial value problem at spatial infinity is smooth through the critical sets only if C (Dip · · · Di1 bjk )(i) = 0 C Diq · · · Di1 cjk [λ(R) ]i1 j l (i) = 0

p = 0, 1, 2, . . . q = 0, 1, 2, . . .

is satisfied by the initial data. If the above conditions are violated at some order p or q, then the solution will develop logarithmic singularities at the critical sets. In the above expression C denotes the operation of taking the symmetric trace-free part of the relevant tensor. An alternative statement of the theorem written in the more natural language of space spinors will be given in the main

626



text. It will be shown in the main text that the condition on cij is actually a condition on the freely specifiable data ˚ Φij . The regularity conditions given in Theorem 1 provide a correction to a conjecture concerning the form that the generalisation of the time symmetric conditions obtained in [11] should have. It turns out that the generalisation given in that reference is not satisfied by Kerr initial data. As it will be discussed in the main text, the assumptions on the initial data which are being made here exclude Kerr data from our considerations. However, it should be pointed out that that “standard” Kerr data can be shown to satisfy the regularity conditions of Theorem 1. This seems to suggest that the result holds for more general classes of data that the ones being considered here. Overview of the article The article is structured as follows. In Section 2, in order to fix notation and conventions some relevant preliminaries concerning the construction of spacetimes from an initial value problem are reviewed. These include the construction of asymptotically Euclidean solutions to the vacuum Einstein constraint equations from free data on a compact manifold. There is also a brief review of the solutions to the momentum constraint on flat space and also in the more general case where the conformal metric is not flat. Of particular relevance is Subsection 2.4 where the assumptions made on the class of initial data are spelled out. Section 3 contains a brief discussion of the construction of the bundle manifold Ca which provides a convenient alternative representation of the asymptotic region of the initial manifold: one in which the point at infinity i is blown up to a set which is topologically a 2-sphere. In particular Subsections 3.3 and 3.5 discuss aspects of normal expansions at infinity. The ideas of these subsections are essential for the sequel. The techniques in Section 3 are slight adaptations of the original constructions given in reference [11] – there is a more recent detailed discussion in [12]. The purpose of this section is to introduce necessary notation and to serve as a quick reference for the reader. Section 4 discusses the solutions of the momentum constraint on the bundle manifold Ca . This section builds on an analysis given in references [21] and [22]. However, the particular level of detail required for our analysis, in particular with regards to the non-flat case, is not available elsewhere in the literature. Section 5 builds on the results of the previous section to provide a detailed analysis of the normal expansions of the Weyl spinor near infinity. The main results of this section are presented in Theorems 3 and 4. These theorems can be regarded as the generalisation of Theorem 4.1 in [11] to initial data sets with a non-vanishing second fundamental form. Section 6 gives a brief overview of the ideas behind the so-called regular finite initial value problem at spatial infinity. Again, the discussion is kept to a minimum and has the purpose of introducing notation and ideas which will be of relevance in the sequel. In particular, Subsection 6.3 contains the crucial result of [11] stating that the solution to the conformal propagation equations will generically develop

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logarithmic singularities at the sets where null infinity touches spatial infinity. How to eliminate such singularities by means of conditions on the class of initial data under consideration is the concern of the rest of the article. Section 7 provides the main result of the article, Theorem 6, which recasts the technical regularity conditions in terms of conditions on the Cotton tensor of the conformal metric and on the curl of the second fundamental form. This theorem brings together the discussions of Section 5 and 6. Section 8 discusses the connections of the conditions obtained in the main result to similar – but crucially not identical – conditions which have been obtained from an analysis of purely radiative spacetimes – see [9] and also the concluding remarks in [11]. Finally, a discussion of possible generalisations of the main result to other classes of data is given. A particularly desirable generalisation would be to a class of data including stationary solutions. The article contains two appendices. The first one contains a number of spinorial identities which are used throughout the main text. The second appendix contains a discussion of the asymptotic expansions of the solutions of the vacuum Einstein constraint equations for the class of data under consideration. Appendix B briefly reviews some relevant results obtained in [6] and extends some of the analysis therein as the data required to render our analysis non-trivial turns out to be slightly more general than the one considered in the aforementioned reference. One of the main challenges of the analysis presented in this article is to bring together results and ideas obtained in different frameworks and cast them in a common language which allows, in turn, to obtain new results and hopefully also valuable new insights into the structure of the gravitational field near spatial infinity.

2. Preliminaries ˜ g˜) This article is concerned with properties of asymptotically flat spacetimes (M, solving the Einstein vacuum field equations ˜ μν = 0 . R

(1)

The metric g˜μν will be assumed to have signature (+, −, −, −) and μ, ν, . . . are ˜ g˜) will be thought spacetime indices taking the values 0, . . . , 3 . The spacetime (M, of as the development of some initial data prescribed on an asymptotically Eu˜ The data on S˜ are given in terms of a 3-metric clidean Cauchy hypersurface S. ˜ hij of signature (−, −, −) and a symmetric tensor field χ ˜ij representing the second ˜ As mentioned in the introduction i, j, k, . . . fundamental form induced by g˜μν on S. will be spatial tensorial indices taking the values 1, 2, 3. The Einstein vacuum field equations imply the constraint equations (3)

r˜ − (χ ˜ii )2 + χ ˜ij χ ˜ij = 0 , i ˜ ˜ Dχ ˜ij − Dj χ ˜ii = 0 ,

(2) (3)

628



˜ denotes the Levi-Civita connection and (3) r˜ the Ricci scalar of the metric where D ˜ ij , χ ˜ ij . The initial data (S, ˜h ˜ij ) will be assumed to be asymptotically Euclidean – h for simplicity the situation with only one asymptotically flat end will be considered. On the asymptotically flat end it will be assumed that coordinates {y i } can be introduced such that 1 1 2m ˜ hij = − 1 + , χ ˜ij = O as |y| → ∞ , δij + O |y| |y|2 |y|2 ˜ In with |y|2 = (y 1 )2 + (y 2 )2 + (y 3 )2 and m a constant – the ADM mass of S. addition to these asymptotic flatness requirements, it will be assumed that there is a 3-dimensional, orientable, smooth compact manifold (S, h), a point i ∈ S, a ˜ with the diffeomorphism Φ : S \ {i} −→ S˜ and a function Ω ∈ C 2 (S) ∩ C ∞ (S) properties Ω(i) = 0 ,

Dj Ω(i) = 0 ,

Dj Dk Ω(i) = −2hjk (i) ,

Ω > 0 on S \ {i} , ˜ ij . hij = Ω2 Φ∗ h

(4a) (4b) (4c)

˜ ij – that is, S \{i} will be The last condition shall be, sloppily, written as hij = Ω2 h ˜ ˜ ˜ identified with S. Under these assumptions (S, h) will be said to be asymptotically Euclidean and regular. Suitable punctured neighbourhoods of the point i will be ˜ It should be clear from the context whether i mapped into the asymptotic end of S. denotes a point or a tensorial index. 2.1. The constraint equations In order to discuss the asymptotic properties of the solutions of the vacuum Einstein equations (1), the latter will be rewritten in terms of a suitable conformal factor Ω and a conformal metric gμν such that gμν = Ω2 g˜μν . The first and second fundamental forms determined by the metrics gμν and g˜μν on S˜ are related via ˜ ij , hij = Ω2 h

˜ ij ) , χij = Ω(χ ˜ij + Σh

where Σ denotes the derivative of Ω in the direction of the future directed g-unit ˜ ij χ normal of S. If χ = hij χij , χ ˜=h ˜ij , one has that Ωχ = χ ˜ + 3Σ . In terms of the fields Ω, hij and χij , the constraint equations take the form 1 1 2ΩDi Di Ω − 3Di ΩDi Ω + Ω2(3) r − 3Σ2 − Ω2 χ2 − χij χij + 2ΩΣχ = 0 , (5) 2 2 Ω3 Di (Ω−2 χij ) − Ω Dj χ − 2Ω−1 Dj Σ = 0 , (6)

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where D denotes the Levi-Civita connection and ric h. In the sequel it shall be assumed that Σ = 0,

χ=0

(3)

629

r the Ricci scalar of the met-

on S˜ .

(7)

That is, the hypersurface S˜ will be assumed to be maximal with respect to both ˜ ij and hij . h 2.2. Solving the constraint equations Under the assumptions (7) the conformal constraint equations take the form 1 (3) 1 Δ− r ϑ = χij χij ϑ , with ϑ = Ω−1/2 , (8) 8 8 (9) Di Ω−2 χij = 0 , with Δ ≡ Dk Dk . In what follows it shall be assumed that the above equations are solved using an adaptation of the so-called conformal method. Namely: 1. choose a smooth, negative definite metric hij on a 3-dimensional, orientable, smooth compact manifold S and pick a point i ∈ S. Set S˜ = S \ {i}. 2. Find a smooth, symmetric tensor field ψij on S˜ which is trace-free with respect to hij and satisfies Di ψij = 0 .

(10)

The tensor ψij can be obtained by means of a York-splitting. Given a smooth, ˜ set symmetric, trace-free tensor ˚ Φij on S, 2 ψij = Di vj + Dj vi − hij Dk v k + ˚ Φij , 3 = (Lv)ij + ˚ Φij where vi is some 1-form on S˜ and (Lv)ij is the conformal Killing operator of hij . Given the above Ansatz, then equation (10) implies the following elliptic system for vi : 1 Φkj . Δvj + Dj Dk v k + rjk v k = −Dk ˚ 3

(11)

3. Setting χij = ϑ−4 ψij in equation (8) one obtains the Lichnerowicz equation 1 (3) 1 Δ− r ϑ = ψij ψ ij ϑ−7 , (12) 8 8 that is, an elliptic equation for ϑ. The fields hij , Ω = ϑ−2 and χij = Ω2 ψij so constructed render a solution to the conformal constraints (5) and (6). It is important to recall that if φ is a positive scalar field on S then the transitions hij → φ4 hij ,

ψij → φ−2 ψij ,

Ω → φ2 Ω ,

χij → φ2 χij ,

(13)

630



provide another solution to the conformal constraint with the same physical data ˜ ij , χ ˜ij . This freedom in the conformal gauge will be used in the sequel to obtain h simplifications in the analysis of asymptotic expansions. Given a 3-metric hij consider a suitable a > 0 such that Ba ≡ Ba (i) is a strictly convex normal neighbourhood of i and let xi be normal coordinates with origin at i based on an h-orthonormal frame ea . The asymptotic condition (4a) implies |x|ϑ → 1 as x → 0. Accordingly, one has that 1 χij = O(1) , ψij = as x → 0 , |x|4 where the following transformation rules have been taken into account: χ ˜ij = Ω−1 χij = Ωψij . 2.3. The momentum constraint in flat space The analysis of the solutions of the momentum constraint in flat space given in [6] is now briefly recalled. In Section 4 it will be recast in a form – in terms of space spinors – which is convenient for the application discussed in this article. In this section assume that S is flat in at least a neighbourhood Ba of i. Let xk be a Cartesian coordinate system with origin at i. In these coordinates the metric is given by hij = −δij . We shall often write r = |x|. Let ni ≡ xi /|x| and complement it with complex vectors mi , mi to form a basis of the tangent bundle of R3 , by requiring that mi mi = mi mi = ni mi = ni mi = 0 ,

mi mi = −1 .

Recall that in this construction one has freedom of performing rotations mi → eiθ mi with θ independent of r. The metric hij can be written in the form hij = −ni nj − mi mj − mi mj , ˚ij can be written as while an arbitrary trace-free tensor ψ √ √ ˚ij = 1 ξ(3ni nj − δij ) + 2ηn(i mj) + 2ηn(i mj) + μmi mj + μmi mj , (14) ψ 3 r with ξ=

1 3˚ i j r ψij n n , 2

η=

√

˚ij ni mj , 2r3 ψ

˚ij mi mj . μ = r3 ψ

In the previous expression ξ is real, while η and μ are complex functions of spin ˜ be an arbitrary complex C ∞ function on Ba . weight 1 and 2 respectively. Let λ 2˜ Let λ ≡ ð λ, where ð denotes the eth-operator – see e.g. [16,18]. Let λ(R) ≡ Re(λ)

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and λ(I) ≡ i Im(λ). Let 3 − P i nj − P j ni − (δ ij − 5ni nj )P k nk , 4 2r 3 ≡ 3 ni jkl Jk nl + nj ikl Jk nl , r A ≡ 3 3ni nj − δ ij , r 3 ≡ 2 Qi nj + Qj ni − (δ ij − ni nj )Qk nk , 2r

ψPij ≡

(15a)

ψJij

(15b)

ij ψA ij ψQ

(15c) (15d)

where P i (the linear momentum), J i (the angular momentum), Qi (conformal momentum) and A are real constant vectors, respectively, scalars. Furthermore, let 2

ξ = ð λ(R) ,

(16a)

η = −2r∂r ðλ(R) + ðλ(I) , μ = 2r∂r (r∂r λ

(R)

) − 2λ

(R)

(16b) + ððλ

(R)

− r∂r λ

(I)

,

(16c)

˚ij [λ] ˜ the tensor obtained by substituting (16a)–(16c) into (14). and denote by ψ Then ˚ij [P ] + ψ ˚ij [J] + ψ ˚ij [A] + ψ ˚ij [Q] + ψ ˚ij [λ] ˜ , ˚ij = ψ (17) ψ satisfies the flat space momentum constraint ˚ij = 0 . ∂iψ

(18)

Conversely, any smooth solution to equation (18) is of the form (17) – cfr. The˜ which is smooth on orem 14 in [6]. In the present article the complex function λ Ba \ {i} will be taken to be of the form ˜ (2) , ˜=λ ˜ (1) + 1 λ λ r

(19)

with λ(1) , λ(2) ∈ C ∞ (Ba ). In analogy to equation (19) let also 1 λ = λ(1) + λ(2) . r ˜ non-smooth at i. The functions λ(1) Note that the factor 1/r makes the function λ (2) and λ are calculated using the ð operator. Accordingly, they are non-smooth. 2.4. Assumptions on the freely specifiable data In order to obtain a non-trivial outcome from the analysis to be discussed in the sequel, one has to choose a suitable class of initial data. If the initial data is to be constructed following the procedure discussed in Section 2.2, then the freely specifiable data is given in terms of a negative-definite 3-metric hij and an h-trace free symmetric tensor ˚ Φij .

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2.4.1. The conformal metric. For ease of the presentation, the results presented here will be restricted to the case of conformal metrics hij which are analytic in a neighbourhood, Ba , of spatial infinity. This assumption is non-essential and with recursive arguments it should be possible to extend all the results presented here to the smooth – i.e. C ∞ – setting. The analyticity of hij is explicitly used in the proof of part (ii) of Theorem 3 and of part (i) of Theorem 4, where the expansions of a certain tensor associated to the massless part of the Weyl tensor are calculated. The analyticity of hij is restrictive in the following sense: generic stationary spacetimes do not have a smooth conformal metric. Instead, as shown in [4], the conformal metric of stationary data is of the form (1)

(2)

hij = hij + r3 hij , (1)

(2)

with hij and hij analytic tensors. Due to the presence of the term r3 , the metric in this case will be C 2,α (Ba ), 0 < α < 1 – that is, their second order derivatives are Hölder continuous. The regularity of this conformal metric is conjectured to be optimal for strictly stationary data – i.e. stationary data which is not static. Throughout, it will be assumed that in normal coordinates centred at i one has that ˆ ij , hij = −δ ij + h ˆ ij , hij = −δij + h ˆ ij ∈ C ω (Ba ), h ˆ ij ∈ C ω (Ba ). It is recalled that normal coordinates satisfy with h ˆ ij = 0 , xi h

ˆ ij = 0 . xi h

The latter relations imply that xl xi Γlij = 0 . The analysis of the solutions to the momentum constraint performed in the appendix B requires the technical assumption ˆ ij = δij h ˆ ij = 0 . (20) δ ij h The conformal gauge freedom (13) can be used to obtain a representative in the conformal class of hij which is in the so-called cn (conformal normal)gauge – see [11]. Working with a metric in the cn-gauge renders a number of useful simplifications in the present discussion. In order to define the cn-gauge, consider on Ba solutions (xi (t), bi (t)) of the conformal geodesic equations x˙ k Dk x˙ i = −2(bk x˙ k )x˙ i + (hkl x˙ k x˙ l )hmi bm , 1 x˙ k Dk bi = (bk x˙ k )bi − (hkl bk bl )hmi x˙ m + lik x˙ k , 2 with initial conditions x(0) = i ,

hij x˙ i x˙ j = −1 ,

(21a) (21b)

bi (0) = 0 ,

where xi (t) denotes a curve in Ba through i, bi (t) is a 1-form along that curve, and 1 lik ≡ rik − rhik , 4

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is the Schouten tensor of the metric hij . If a neighbourhood Ba of i is taken to be small enough, then there is a unique conformal rescaling of the form given by (13) such that the conformal metric is analytic, keeps the metric and connection unchanged at i, and such that if the solution to the conformal geodesic equations (21a) and (21b) is written in terms of the rescaled metric one has bi x˙ i = 0 ,

on

Ba .

(22)

A metric in the conformal gauge of hij satisfying (22) will be said to be in the cn-gauge. From here on, it will always be assumed that the metric hij is in the cn-gauge. An important consequence of working in the cn-gauge is that both the Ricci scalar and Ricci tensor of hij vanish at i. Accordingly, ˆ ij = O(r3 ) , h

∂k hij = O(r2 ) .

In addition, it will be proved – see Lemma 3 – that if hij is in the cn-gauge then ˆ ij = r2 h ˇ ij so that h ˇ ij , hij = −δij + r2 h

ˇ ij , hij = −δ ij + r2 h

ˇ ij = O(r) is analytic. Note that because of the condition (20) one has that where h ij ˇ δ hij = 0. 2.4.2. Free data for the second fundamental form. The h-trace free symmetric tensor ˚ Φij , containing the free data for the second fundamental form will be con˚ij , solution of the flat space momentum constraint structed out of the tensor ψ given in equation (17). Thus, let ˚ij − 1 hij hkl ψ ˚kl . ˚ Φij ≡ ψ 3 Thus, ˚ Φij is specified by prescribing the constant vectors P i , J i , Qi , the constant scalar A, and the function λ. In order to obtain a vector v i and a scalar ϑ solving the equation (12) admitting an expansion in powers of r near i one has to set P i = 0 and consider a λ of the form given in equation (19). A more extended discussion of these issues is given in [6]. However, it turns out that if one admits a contribution ˚ij [λ] with λ given by equation (19), to the second fundamental form of the type ψ the solutions to the elliptic equation (11) will not have an asymptotic expansion in Ba consisting purely of powers of r. In order for this to be the case, Im(λ(1) ) and Re(λ(2) ) have to be related to the conformal metric hij in a particular way. This issue is discussed in more detail in the appendix. Throughout the main body of the article, it will be assumed that such conditions are satisfied, and accordingly, the solutions v i of equation (11) have expansions in Ba purely in powers of r – see below for the precise details. Examples of classes of data where these are valid are axially symmetric data and conformally flat data – in which case v i = 0.

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2.4.3. Consequences of the assumptions on the freely specifiable data. In order to make precise the idea that a given function or tensor field over S admits an expansion around i in terms of powers of r, introduce the following function spaces: E ∞ (Ba ) = f = f1 + rf2 | f1 , f2 ∈ C ∞ (Ba ) , Q∞ (Ba ) = v i ∈ C ∞ (Ba , R3 ) | xi v i = r2 v, v ∈ C ∞ (Ba ) . Theorem 15 in [6] states that if the function λ determining the higher mul˚ij is of the form given by equation (19) – i.e. λr ∈ E ∞ (Ba ) – and tipoles of ψ i ˚ij ψ ˚ij ∈ E ∞ (Ba ). If hij and λ are such that the solutions, v i , of P = 0, then r8 ψ equation (11) are of the form v i = rs v1i + v2i , for some integer s with v1i ∈ Q∞ (Ba ), v2i ∈ C ∞ (Ba ), and such that r8 ψij ψ ij ∈ E ∞ (Ba ), with ψij = ˚ Φij [A, J, Q, λ(1) , λ(2) /r] + (Lh v)ij [A, J, Q] + (Lh v)ij [λ(1) ] + (Lh v)ij [λ(2) ] , then Theorem 1 in [6] renders m U + W , U ∈ C ω (Ba ) , W ∈ E ∞ (Ba ) , W (i) = . (24) r 2 Furthermore, because of the use of the cn-gauge, U = 1 + O(r4 ). The discussion in appendix B renders a detailed description of the structure of the vectors v i [A, J, Q, λ(1) , λ(2) /r]. The function U satisfies the equation U 1 Δ − (3) r = 4πδ(i) , 8 |x| ϑ=

with δ(i) the Dirac’s delta distribution with support on i, whereas the function W satisfies −7 U 1 1 +W . Δ − (3) r W = ψij ψ ij 8 8 |x|

3. The manifold Ca In [11] a representation of the region of spacetime close to null infinity and spatial infinity has been introduced. The standard representation of this region of spacetime depicts i0 as a point. In contrast, the representation introduced in [11] depicts spatial infinity as a cylinder – the cylinder at spatial infinity. The technical and practical grounds for introducing this description have been discussed at length in that reference. The original construction in [11] was carried out for the class of time symmetric initial data sets with analytic conformal metric hij . However, as discussed in [21, 22], the construction can be adapted to settings with a non-vanishing second fundamental form. The purpose of this section is, primarily, to introduce notation that will be used in the sequel and to provide enough

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background material to follow the discussion in the sequel. In any case, the reader is referred to [11, 12] for a thorough discussion of the details. 3.1. The blowing up of i The construction given in [11] makes use of a blow up of the point i ∈ S to a 2-sphere. This blow up requires the introduction of a particular bundle of spinframes over Ba . In what follows, a space spinor formalism analogous to a tensorial 3+1 decomposition will be used. Consider the (unphysical, conformally rescaled) spacetime (M, gμν ) obtained as the development of the initial data set (S, hij , χij ). Let SL(S) be the set of spin dyads δ = {δA }A=0,1 = {oA , ιA } on S which are normalised with respect to the alternating spinor AB in such a way that 01 = 1. The set SL(S) has a natural bundle structure where S is the base space, and its structure group is given by SL(2, C) = tAB ∈ GL(2, C) | AC tAB tC D = BD , √ acting on SL(S) by δ → δ · t = {δA tAB }B=0,1 . Now, let τ = 2e0 , where e0 is the future g-unit normal of S and

τAA = g(τ, δA δ A ) = A0 A 0 + A1 A 1

a AA a is its spinorial counterpart – that is, τ = τ a ea = σAA ea where σAA τ denote the Infeld-van der Waerden symbols and {ea }, a = 0, . . . , 3 is an orthonormal frame. The spinor τAA enables the introduction of space-spinors – sometimes also called SU (2) spinors, see [1, 7, 17]. It defines a sub-bundle SU (S) of SL(S) with structure group A SU (2, C) = tAB ∈ SL(2, C) | τAA tAB t B = τBB ,

and projection π onto S. The spinor τ AA allows to introduce spatial van der Waerden symbols via (A

σaAB = σa

A τ

B)A

,

a σAB = τ(B

A

σ aA)A ,

i = 1, 2, 3 .

The latter satisfy hab = σaAB σbAB ,

a b −δab σAB σCD = −A(C D)B ≡ hABCD ,

with hab = h(ea , eb ) = −δab . The bundle SU (S) can be endowed with a su(2, C)valued connection form ω ˇ AB compatible with the metric hij and 1-form σ AB , the solder form of SU (S). The solder form satisfies by construction h ≡ hij dxi ⊗ dxj = hABCD σ AB ⊗ σ CD ,

(25)

where σ AB = σiAB dxi – note that the σiAB are not the spatial Infeld-van der Waerden symbols, σaAB . Now, given a spinorial dyad δ ∈ SU (S) one can define an associated vec tor frame via ea = ea (δ) = σaAB δA τB B δ B , a = 1, 2, 3. We shall restrict our attention to dyads related to frames {ej }j=0,...,3 on Ba such that e3 is tangent to ˇ denote the horizontal vector field on SU (S) the h-geodesics starting at i. Let H projecting to the radial vector e3 .

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The fibre π −1 (i) ⊂ SU (S) (the fibre “over” i) can be parametrised by choosing a fixed dyad δ ∗ and then letting the group SU (2, C) act on it. Let ˇ satis(−a, a) ρ → δ(ρ, tAB ) ∈ SU (S) be the integral curve to the vector H fying δ(0, tAB ) = δ(tAB ) ∈ π −1 (i). With this notation one defines the set

Ca = δ(ρ, tAB ) ∈ SU (Ba ) |ρ| < a, tAB ∈ SU (2, C) , which is a smooth submanifold of SU (S) diffeomorphic to (−a, a) × SU (2, C). The ˇ is such that its integral curves through the fibre π −1 (i) project onto vector field H the geodesics through i. From here it follows that the projection map π of the bundle SU (S) maps Ca into Ba . Let I 0 ≡ π −1 (i) = {ρ = 0} denote the fibre over i. It can be seen that I 0 ≈ SU (2, C). On the other hand, for p ∈ Ba \{i} it turns out that π −1 (p) consists of an orbit of U (1) for which ρ = |x(p)|, and another for which ρ = −|x(p)|, where xi (p) denote normal coordinates of the point p. In order to understand better the structure of the manifold Ca it is useful to quotient out the effect of U (1). It turns out that I 0 /U (1) ≈ S2 . Hence, one has an extension of the physical manifold S˜ by blowing up the point i to S2 . The manifold Ca inherits a number of structures from SU (S). In particular, the solder and connection forms can be pulled back to smooth 1-forms on Ca . These ˇ AB . They satisfy the structure equations shall be again denoted by σ AB and ω relating them to the so-called curvature form determined by the curvature spinor 1 1 1 1 sABCE − (3) rhABCE DF + sABDF − (3) rhABDF CE , rABCDEF = 2 12 2 12 where sABCD = s(ABCD) is the spinorial counterpart of the tracefree part of the Ricci tensor of hij and (3) r its Ricci scalar. These satisfy the 3-dimensional Bianchi identity 1 DAB sABCD = DCD (3) r . 6 3.2. Calculus on Ca In the sequel tAB and ρ will be used as coordinates on Ca . Consequently, one ˇ = ∂ρ . Vector fields relative to the SU (2, C)-dependent part of the has that H coordinates are obtained by looking at the basis of the (3-dimensional) Lie algebra su(2, C) given by 1 1 1 0 i 0 −1 i 0 u1 = , u2 = , u3 = . i 0 1 0 0 −i 2 2 2 In particular, the vector u3 is the generator of U (1). Denote by Zi , i = 1, 2, 3 the Killing vectors generated on SU (S) by ui and the action of SU (2, C). The vectors Zi are tangent to I 0 . On I 0 one sets X+ = −(Z2 + iZ1 ) ,

X− = −(Z2 − iZ1 ) ,

X = −2iZ3 ,

(26)

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and extend these vector fields to the rest of Ca by demanding them to commute ˇ = ∂ρ . For latter use it is noted that with H [X, X+ ] = 2X+ ,

[X, X− ] = −2X− ,

[X+ , X− ] = −X .

The vector fields are complex conjugates of each other in the following sense: given a real valued function W , X− W = X+ W . More importantly, it can be seen that ˇ X± span the tangent space at p. for p ∈ Ba \ {i} the projections of the fields H, A frame cAB = c(AB) dual to the solder forms σ CD is defined so that it does not pick components along the fibres – i.e. along the direction of X. These requirements imply σ AB , cCD = hAB CD ,

− cCD = c1CD ∂ρ + c+ CD X+ + cCD X− ,

(27)

where · , · denotes the action of a 1-from on a vector. Let α± and α be 1-forms on Ca annihilating the vector fields ∂τ , ∂ρ and having with X± the non-vanishing pairings α+ , X+ = α− , X− = α, X = 1 . Furthermore, from the properties of the solder form σ AB one finds that 1 1 c1AB = xAB , c+ ˇ+ c− ˇ− AB = zAB + c AB , AB = yAB + c AB , ρ ρ with constant spinors xAB , yAB and zAB given by √ 1 xAB ≡ 2o(A ιB) , yAB ≡ − √ ιA ιB , 2 and analytic spinor fields satisfying cˇα AB = O(ρ) ,

cˇα 01 = 0 ,

(28)

1 zAB = √ oA oB , 2

α = 1, ± .

Accordingly, one can write ˇ± ˇ± cˇ± y yAB + c z zAB . AB = c Furthermore, using the structure equations it can be shown that in fact cˇ1AB = 0. By virtue of the relations (27) and (28), the solder forms σ AB descend to forms ni dxi , mi dxi , mi dxi spanning the tangent space of the points of Ba with nonvanishing pairings ni ni = −1, mi mi = −1, and such that ni = xi /r. Note that ni , mi and mi are not smooth functions with respect to the coordinates xi . The connection coefficients are defined by contracting the connection form ω ˇ AB with the frame cAB . In general, one writes 1 ∗ A 1 ∗ γCD AB ≡ ˇ ω AB , cCD = γCD ˇCD AB , γABCD = (AC xBD + BD xAC ) . B +γ ρ 2 ∗ The spinor γABCD denotes the singular part of the connection coefficients. The regular part of the connection can be related to the frame coefficients cAB via commutator equations. The smooth part γˇABCD of the connection coefficients can be seen to satisfy γˇ11CD = 0 , γˇABCD = O(ρ) .

638



Furthermore, from an analysis of the structure equations one obtains that γˇ1100 = −ˇ γ0011 . Let f be a smooth function on Ca DAB f = cAB (f ) . Similarly, let μAB represent both a smooth spinor field on Ca . Then the covariant derivative of μAB is given by DAB μCD = cAB (μCD ) − γAB EC μED − γAB ED μCE . Analogous formulae hold for higher valence spinors. 3.3. Normal expansions at i In [11] a certain type of expansions of analytic fields near i has been discussed. These techniques can be extended to the smooth setting. If f ∈ C ∞ (Ba ), then one has that fi1 ···ik xi1 · · · xik , f∼ k≥0

which is to be interpreted as f=

m

fi1 ···ik xi1 · · · xik + fR ,

k=0

m with fR ∈ C ∞ (Ba ), fR = o(rm ), for all m ≥ 0. The sum k=0 fi1 ···ik xi1 · · · xik , with fi1 ···ik , k = 0, . . . , m real constants, is the Taylor polynomial of degree m of f . Now, suppose that T i1 ···irj1 ···js is a smooth tensorial field of rank(r, s) on Ba with components T ∗a1 ···arb1 ···bs with respect to the frame e∗a on which the normal coordinates xi are based. Let V = xj ∂j be the radial vector which is tangent to geodesics through i and satisfying Vi V i = −1. Let also ni be defined by V i = |x|ni . By construction one has that V k Dk e∗a = 0, thus following the procedure described in Section 3.3 of [11] one obtains T ∗a1 ···arb1 ···bs (q) =

m 1 p lp |x| n (q) · · · nl1 (q)Dlp · · · Dl1 T ∗a1 ···arb1 ···bs (i) + T ∗a1 ···arb1 ···bs R , p! p=0

(29)

with q ∈ Ba , and (T ∗a1 ···arb1 ···bs )R ∈ C ∞ (Ba ) and (T ∗a1 ···arb1 ···bs )R = o(rm ). The first term in the right-hand side of expression (29) will the called the analytic part of T ∗a1 ···arb1 ···bs . Analogous expansions can be obtained for smooth analytic spinor fields. Suppose ξA1 B1 ···Al Bl denotes the components of a smooth even rank ∗ associated to e∗a . In analogy to spinorial field with respect to the spin frame δA

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expression (29) one can introduce the expansion ∗ ξA (q) 1 B1 ···Al Bl

=

m ∗ 1 p C p Dp ∗ |x| n · · · nC1 D1 DCp Dp · · · DC1 D1 ξA + ξA , 1 B1 ···Al Bl 1 B1 ···Al Bl R p! p=0

(30) with nAB = v AB (q), q ∈ Ba , with the derivatives of the spinor field evaluated at ∗ ∗ ) ∈ C ∞ (Ba ), (ξA ) = o(rm ). Again, the the point i, and (ξA 1 B1 ···Al Bl R 1 B1 ···Al Bl R first term of the right-hand side of equation (30) will be referred to as the analytic part. The detailed structure of the summands in expression (30) can be untangled using a decomposition in terms of irreducible spinors. 3.4. An orthonormal basis for functions on SU (2, C) The lift of the expansion (30) from Ba to Ca introduces in a natural way a class of functions associated with unitary representations of SU (2, C). Namely, given tAB ∈ SU (2, C), define 1/2 1/2 m m B ) (B j A t 1 (A1 · · · t m j Am )k , Tm k (t B ) = j k T0 00 (tAB ) = 1 , with j, k = 0, . . . , m and m = 1, 2, 3, . . .. The expression (A1 · · · Am )k means that the indices are symmetrised and then k of them are set equal to 1, while the remaining ones are set to 0. Details √ about the properties of these functions can be found in [8, 11]. The functions m + 1Tm jk form a complete orthonormal set in the Hilbert space L2 (μ, SU (2, C)), where μ denotes the normalised Haar measure on SU (2, C). In particular, any analytic complex-valued function f on SU (2, C) admits an expansion f (tAB ) =

m ∞ m

fm,k,j Tm kj (tAB ) ,

m=0 j=0 k=0

with complex coefficients fm,k,j . Under complex conjugation the functions transform as Tm jk = (−1)j+k Tm m−jm−k . The action of the differential operators (26) on the functions Tm kj is given by XTm kj = (m − 2j)Tm kj ,

X+ Tm kj = j(m − j + 1)Tm kj−1 ,

X− Tm kj = − (j + 1)(m − j)Tm kj+1 .

A function f is said to have spin weight s if Xf = 2sf .

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Such a function has a simplified expansion of the form f=

∞ m

fm,k Tm km/2−s .

m≥|2s| k=0

3.5. Normal expansions at I 0 In the sequel it will be necessary to be able to relate fields in Ca with fields in Ba . Crucially, one will require to be able to lift smooth fields defined on Ba to Ca . The procedure for analytic fields has been described in [11]. Here we just state the analogous results for smooth spinorial fields. Denote by ξj = ξ(A1 B1 ···Al Bl )j , 0 ≤ j ≤ l the essential components of a spinorial field ξA1 B1 ···Al Bl . The function ξj has spin weight s = l − j and a unique expansion on Ca of the form ξj =

m

ξj,p ρp + (ξj )R ,

(31)

p=0

for all m, with p+l

ξj,p =

2q

ξj,p;2q,k T2q k2q−l+j ,

(32)

q=max{|l−j|,l−p} k=0

and complex coefficients ξj,p;2q,k . In particular one has that 1/2 −1/2 √ 2p + 2l 2p + 2l ∗ ξj,p;2q+2l,k = ( 2)p D(Cp Dp · · · DC1 D1 ξA (i) . 1 B1 ···AP Bp ) k p+j Hence, one has the symmetry ξ0,p;2p+2l,k = ξ2l,p;2p+2l,k which will play an important role in the sequel. Another lengthy, but straightforward calculation shows that E

ξj,p;2p+2,k = Kp,j,k D(Ap Bp · · · DA1 |E| ξABC)j (i) ,

(33)

with Kp,j,k a constant depending on p, j, k. If l is even, then ξA1 ···Al is associated to a real spatial tensor if and only if the following reality conditions hold: ξj = (−1)j ξ 2l−j ,

ξj,p;2q,k = (−1)r+q+k ξ 2l−j,p;2q,2q−k .

The discussion in the following sections will require to consider smooth spinorial fields ξA1 B1 ···Ar Br with essential components ξj = ξ(A1 B1 ···Ar Br )j , 0 ≤ j ≤ 2r of spin weight s = r − j with expansions which are more general than those given in equations (31) and (32). Accordingly, they do not descend to smooth spinors on Ba . In this case, instead of (32), one considers the more general expression ξj,p =

q(p) 2q q=|r−j| k=0

ξj,p;2q,k T2q kq−r+j ,

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where 0 ≤ |r − j| ≤ q(p) ≤ ∞. In this case one speaks of an expansion of type q(p). If f , g have expansion types q(p), q (p), respectively, then the product f q will have expansion type max0≤j≤p {q (j) + q(p − j)}, while the sum f + g will have expansion type max{q(p), q (p)}. In particular, ρf will have expansion type q(p) − 1. 3.6. Consequences of the cn-gauge for normal expansions In the sequel we shall make use of a number of technical results implied by the cn-gauge. The following lemma has been proved in [11]. Lemma 1. In the cn-gauge one has that r = O(ρ2 ) ,

type(r) = p , type(sABCD ) = p + 1 ,

sABCD = O(ρ) ,

type(ˇ γABCD ) = p ,

γˇABCD = O(ρ2 ) ,

type(ˇ c± AB ) = p ,

2 cˇ± AB = O(ρ ) ,

type(U − 1) = p − 1 ,

U = 1 + O(ρ4 ) .

In addition, if the function W in the equation ϑ = U/|x| + W satisfies W ∈ E ∞ (Ba ), then the procedure of Subsection 3.5 gives Lemma 2.

m + O(ρ) 2 Remark. By means of an adequate choice of the centre of mass it is possible to set W = m/2 + O(ρ2 ) – see e.g. [21]. This fact will not be used here. type(W ) = p ,

W =

Finally, we note the following lemma. Its proof follows by inspection. Lemma 3. In normal coordinates based around i, an analytic metric in the cn-gauge is of the form ˇ ij , hij = −δij + r2 h ˇ ij = O(r). with h

4. The solutions to the momentum constraint on Ca In this section the lifts of the solutions to the momentum constraint introduced in Section 2.3 will be analysed. The procedure described here is taken from [21, 23]. 4.1. The conformally flat case In Section 2.3 the solutions of the flat space momentum constraint have been discussed by introducing a certain frame {ni , mi , mi }. This frame is related to an orthonormal frame by means of the relations 1 i ei1 = √ (mi + mi ) , ei2 = √ (mi − mi ) , ei3 = ni . 2 2

642



Inverting these relations one can readily rewrite {ni , mi , mi } in terms of the orthonormal frame {eia }a=1,2,3 . This leads to a direct transcription into spinorial objects. Let {na , ma , ma } denote the components of the frame {ni , mi , mi } with respect to {eia }, a = 1, 2, 3. Using the spatial Infeld symbols one obtains to the following transcription rules: √ √ na → xAB , ma → 2zAB , ma → 2yAB . Thus, formula (14) lifts to

√ ˚ABCD = ξ(3xAB xCD + hABCD ) + 2η (xAB yCD + xCD yAB ) ρ3 ψ 1 √ + 2η1 (xAB zCD + xCD zAB ) + 2μ2 (yAB yCD ) + 2μ2 (zAB zCD ) . (34)

Alternatively, one could rewrite the previous formula in terms of the totally symmetric spinors iABCD , i = 0, 1, . . . , 4 as: ˚0 0 ˚ 1 ˚ 2 ˚ 3 ˚ 4 ˚ABCD = ψ (35) ρ3 ψ ABCD + ψ1 ABCD + ψ2 ABCD + ψ3 ABCD + ψ4 ABCD √ √ ˚1 ≡ 2 2η1 , ψ ˚2 ≡ 6ξ, ψ ˚3 ≡ −2 2η, ψ ˚4 ≡ μ. In order to show ˚0 ≡ μ, ψ with ψ the equivalence of the expressions (34) and (35) one makes use of the spinorial identities given in appendix A. Following the notation of Section 2.3 one writes ˚A ˚P ˚Q ˚J ˚ABCD = ψ +ψ +ψ +ψ . ψ ABCD

ABCD

ABCD

ABCD

˚A ˚P ˚Q ˚J The explicit form of the terms ψ ABCD , ψABCD , ψABCD , and ψABCD has been given in [21]. Their detailed structure will not be required here. With regards ˜ one has that ˚ABCD derived from the complex function λ, to the part of ψ 2 (R) ξ = X− λ ,

(36a)

η1 = −2ρ∂ρ X− λ(R) + X− λ(I) , μ2 = 2ρ∂ρ (ρ∂ρ λ

(R)

) + X+ X− λ

(R)

(36b) − 2λ

(R)

− ρ∂ρ λ

(I)

,

(36c)

with 2˜ 2 ˜ + X 2 Im(λ) ˜ = λ(R) + λ(I) . λ = X+ Re(λ) λ = X+ + By direct inspection of the previous expressions one has the following lemma.

Lemma 4. The lifts to Ca of the solutions to the flat momentum constraint satisfy ˚ABCD [A] = ρ−3 ΞABCD [A] , type ΞABCD [A] = p , ψ ˚ABCD [J] = ρ−3 ΞABCD [J] , ψ type ΞABCD [J] = p + 1 , ˚ABCD [Q] = ρ−3 ΞABCD [Q] , ψ type ΞABCD [Q] = p , ˚ABCD [λ(1) ] = ρ−3 ΞABCD [λ(1) ] , ψ type ΞABCD [λ(1) ] = p , ˚ABCD [λ(2) /r] = ρ−3 ΞABCD [λ(2) /r] , ψ type ΞABCD [λ(2) /r] = p + 1 . ˚ABCD is totally symmetric, then its projection to Ba Note that because ψ is associated to an h-trace free symmetric tensor. In other words, borrowing the ˚ABCD . terminology of Section 2.4, ˚ Φab = σaAB σbCD ψ

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4.2. The non-conformally flat case A detailed discussion of the solutions, v i , of equation (11) has been given in appendix B. From this analysis one can readily deduce the expansion types of the lift to Ca of the various parts of v i . ˚ij [Re(λ(2) )/r]. In Consider the vector v i [Re(λ(2) )/r] produced by the seed ψ appendix B it is shown that under the assumptions of Section 2.4 v i Re(λ(2) )/r = v1i Re(λ(2) )/r + v2i Re(λ(2) )/r , with v1i [Re(λ(2) )/r] ∈ Q∞ (Ba ), v1i [Re(λ(2) )/r] = O(r4 ), and v2i [Re(λ(2) )/r] ∈ C ∞ (Ba ). In what follows, the affixed [Re(λ(2) )/r] will be suppressed for ease of reading. As xi v i = r2 u, u ∈ C ∞ (Ba ), one can write v1i = runi + wmi + w mi , with w a C ∞ complex function. Thus, one has that v i = rni + (wmi + w mi ) . Accordingly, the lift of v i to Ca is found to be 2 , vAB = ρuxAB + (wzAB + wyAB ) + vAB 2 of expansion type p + 1. In what follows write uAB = uxAB , with v, u, vAB ÃB = wyAB . A calculation renders wAB = wzAB and w 2 ˜CD) + D(AB vCD) D(AB vCD) = ux(AB xCD) + ρD(AB uCD) + D(AB wCD) + D(AB w = ρ−3 ρ3 ux(AB xCD) + ρ4 D(AB uCD) + ρ3 D(AB wCD) 2 . + ρ3 D(AB w ˜CD) + ρ3 D(AB vCD)

It follows that type(ρ3 u) = p−2, type(ρ4 D(AB uCD) ) = p−2, type(ρ3 D(AB wCD) ) = 2 p − 1 and type(ρ3 D(AB vCD) ) = p − 1. Accordingly, one can write D(AB vCD) Re(λ(2) )/r = ρ−3 ΦvABCD Re(λ(2) )/r , type ΦvABCD Re(λ(2) )/r = p − 1 . Using similar arguments one obtains the following Theorem 2. Under the assumptions of Section 2.4, the lifts vAB of the vectors v i solving equation (11) are of the form: D(AB vCD) [A] = ρ−3 ΦvABCD [A] , type ΦvABCD [A] = p−1 , type ΦvABCD [J] = p , D(AB vCD) [A] = ρ−3 ΦvABCD [J] , type ΦvABCD [Q] = p , D(AB vCD) [Q] = ρ−3 ΦvABCD [Q] , D(AB vCD) Re(λ(1) ) = ρ−3 ΦvABCD Re(λ(1) ) , type ΦvABCD Re(λ(1) ) = p−1 , D(AB vCD) Im(λ(1) ) = ρ−3 ΦvABCD Im(λ(1) ) , type ΦvABCD Im(λ(1) ) = p−1 ,

644



D(AB vCD) Re(λ(2) )/r = ρ−3 ΦvABCD Re(λ(2) )/r , type ΦvABCD Re(λ(2) )/r = p−1 , D(AB vCD) Im(λ(2) )/r = ρ−3 ΦvABCD Im(λ(2) )/r , type ΦvABCD Im(λ(2) )/r = p .

5. Structure of the Weyl tensor on Ca The rescaled Weyl tensor plays a fundamental role in the discussion of the asymptotic properties of the gravitational field. In this section we study its structure using the manifold Ca . The discussion of this section is a generalisation of the analysis of Section 4 in reference [11]. Let Cμνλρ denote the Weyl tensor of the metric gμν . The tensor Wμνλρ = Ω−1 Cμνλρ will be will be called the rescaled Weyl tensor. Consider now a g-orthonormal frame field {ea }, a = 0, 1, 2, 3 such that e0 corresponds to the g-normal to S. In what follows, the (frame) indices a, b, c, . . . will be assumed to take the values 1, ∗ 2, 3. Let Wμνλρ = 12 Wμναβ αβ λρ , and define wab ≡ Wa0b0 ,

∗ ∗ wab ≡ Wa0b0 .

The spinorial counterpart of the rescaled tensor Wμνλρ is given in terms of WAA BB CC DD = φABCD A B C D + φA B C D AB CD , where φABCD is the spinorial counterpart of a self-dual tensor. At the level of initial ∗ – the data, φABCD , is fully described in terms of the spinors wABCD and wABCD ∗ spinorial counterparts of the spatial tensors wab and wab . One has that ∗ φABCD = wABCD + iwABCD .

Furthermore, using the conformal constraint equations – see e.g. [10] – one finds that wABCD = Ω−2 D(AB DCD) Ω + Ω−1 sABCD + Ω−1 χEF EF χ(ABCD) − χEF (AB χCD)EF , √ ∗ wABCD = −iΩ−1 2DF (A χBCD)F ,

(37a)

(37b)

where χABCD is the spinorial counterpart of the second fundamental form χij . For maximal data one has χEF EF = 0. If ψABCD is the spinorial counterpart of the rescaled second fundamental form ψij then χABCD = Ω2 ψABCD .

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∗ As the magnetic part, wABCD , is linear in ψABCD , it will prove useful to consider ∗ : the following splitting of wABCD ∗ ∗ ∗ ∗ wABCD = wABCD [A, J, Q] + wABCD [λ(R) ] + wABCD [λ(I) ] . ∗ ∗ In this last formula wABCD [λ(R) ] denotes the part of wABCD obtained from (R) ∗ (I) ψij [λ ] while wABCD [λ ] is the part calculated from ψij [λ(I) ].

5.1. The massive and massless parts of φABCD Using formula (24), the conformal factor Ω can be written as Ω=

|x|2 . (U + |x|W )2

Let now Ω ≡

|x|2 . U2

One can use Ω to define the massless part of φABCD . Namely, φABCD = φ ABCD + φABCD ,

(38)

with 1 2 U D(AB DCD) |x|2 − 4U D(AB |x|2 DCD) U − 2|x|2 D(AB DCD) U |x|4 + 6|x|2 D(AB U DCD) U + |x|2 U 2 sABCD , √ |x|6 EF 2 2 F = − 6 ψ (AB ψCD)EF + 2 D (A |x|2 ψBCD)F U√ U √ 2 2|x| F 2|x|2 F −4 D (A U ψBCD)F + D (A ψBCD)F . 3 U U2

φ ABCD =

φ ABCD

∗ and wABCD . One finds From here it is direct to obtain expressions for wABCD that

|x|6 EF ψ (AB ψCD)EF , U6 6 |x| EF = φ ψ (AB ψCD)EF . ABCD + U6

wABCD = φ ABCD −

(39a)

∗ wABCD

(39b)

The massive part is given by • φ•ABCD = φ • ABCD + φABCD ,

(40)

646


with φ • ABCD =

φ• ABCD =

1 |x|4


3 U W D(AB |x|2 DCD) |x|2 + U W |x|D(AB DCD) |x|2 − 2|x| + 2|x| W D(AB |x|2 DCD) U − 3U D(AB |x|2 DCD) W + 2|x|3 − U D(AB DCD) W − W D(AB DCD) U + 6D(AB U DCD) W + U W sABCD + |x|4 − 2W D(AB DCD) W + 6D(AB W DCD) W + W 2 sABCD , 5 |x|6 6U |x|W + 15U 4 |x|2 W 2 + 20U 3 |x|3 W 3 U 6 (U + |x|W )6 + 15U 2 |x|2 W 4 + 6U |x|3 W 5 + |x|3 W 6 ψ EF (AB ψCD)EF √ 2 2 2U |x|W + |x|2 W 2 DF (A |x|2 ψBCD)F − 2 2 U (U + |x|W ) √ 4 2|x|2 3U |x|2 W 2 + U 2 |x|W + |x|3 W 3 DF (A U ψBCD)F + 3 3 U (U + |x|W ) √ √ 4 2|x|2 W 4 2|x|3 F + D (A |x|ψBCD)F + DF (A W ψBCD)F (U + |x|W )3 (U + |x|W )3 √ 2|x|2 2U |x|W + |x|4 W 2 DF (A ψBCD)F , − 2 2 U (U + |x|W )

respectively, the time symmetric and non-time symmetric parts of the massive part of the Weyl spinor. It will turn out necessary to refine further the decomposition of the massless ∗ . Due to the linearity magnetic part of φABCD – which will be denoted by wABCD of the momentum constraint, it is possible to consider individually the various ∗ [A], parameters in the second fundamental form. These will be denoted by wABCD ∗ wABCD [J], etc. From expression (40) one can see that φ•ABCD = O(|x|−3 ) unless the ADM mass vanishes. Thus, in order to discuss the behaviour of φ•ABCD near i one needs to introduce a suitable rescaling1 . To this end let κ = |x|κ with κ (i) = 1 smooth. Consider the lifts to Ca of the spinorial fields φ˘ABCD = κ3 φABCD ,

φ˘ABCD = κ3 φABCD ,

φ˘•ABCD = κ3 φ•ABCD ,

and so on. Let φ˘ABCD denote any of the aforementioned spinorial fields. From general principles one would expect φ˘ABCD to be of expansion type p + 2. That is, its essential components φ˘j , j = 0, . . . , 4, have normal expansions near I 0 of the is worth noticing that on the other hand, under suitable assumptions φABCD is an analytic spinor on Ba . These considerations will be retaken in Section 8.

1 It

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form φ˘j ∼

647

2q ∞ 2p+4 1˘ φj,p;2q,k T2q kq+j−2 ρp , p! p=0 q=|2−j| k=0

with φ˘j,p;2q,k ∈ C. The symbol ∼ is to be understood in the sense described in Section 3.3. It turns out that the actual normal expansions have a more restricted form. The following generalisation of parts (i) and (ii) of Theorem 4.1 in [11] will be proved. Theorem 3. The analytic lifts φ˘ABCD , φ˘ABCD and φ˘•ABCD to Ca have expansion ∗ [A, J, Q] is of expansion type p − 1. In addition: type p, whereas w ˘ABCD of φ˘• = φ˘• satisfy (i) The expansion coefficients φ˘• j

j,p;2q,k

φ˘•0,p;2p,k

=

φ˘•4,p;2p,k

(ii) The expansion coefficients

,

(ABCD)j

p = 0, 1, 2, . . . ,

w ˘j,p;2p,k

k = 0, . . . , 2p .

satisfy the antisymmetry condition

w ˘0,p;2p,k

= −w ˘4,p;2p,k , p = 0, 1, 2, . . . , k = 0, . . . , 2p . ∗ coefficients w ˘j,p;2p,k [A, Q, Re(λ)] satisfy the antisymmetry

(iii) The expansion dition ∗ ∗ ˘4,p;2p,k A, Q, Re(λ) , w ˘0,p;2p,k A, Q, Re(λ) = −w

∗ w ˘j,p;2p,k [J, Im(λ)]

(iv) The expansion coefficients ∗ ∗ J, Im(λ) = w ˘4,p;2p,k J, Im(λ) , w ˘0,p;2p,k

p = 0, 1, 2, . . . ,

con-

k = 0, . . . , 2p .

satisfy the symmetry condition

p = 0, 1, 2, . . . ,

k = 0, . . . , 2p .

The proof of the various parts of the theorem will be given in the following subsections. It consists, essentially, of an analysis of the various terms in expressions (38) and (40). 5.1.1. Proof of the part (i) of Theorem 3. Start by considering the term φ˘ • ABCD = ˘• . This term coincides, formally, with the time symmetric φ κ3 φ • ABCD ABCD discussed in reference [11]. Note however, that in that reference, W is the lift of an analytic function on Ba , while in the case treated here it is the lift of a function belonging to E ∞ (Ba ). However, by virtue of Lemma 2 one has that type(W ) = p, and hence the argument in [11], which only requires W having this expansion type can be reproduced – this particular argument is independent of the assumption of analyticity. This will not be repeated here. One obtains ) = p, type(φ˘ • ABCD

and the symmetry φ˘ •

0,p;2p,k

= φ˘ • 4,p;2p,k ,

p = 0, 1, 2, . . . ,

φ• ABCD .

k = 0, . . . , 2p .

Now, consider the term From Lemma 4 and Theorem 2 it follows ˚ that the lift, ψABCD to Ca of ψij = ψij [A, Q, J, λ] + (Lv)ij satisfies ψABCD = ρ−3 ΦABCD , with ΦABCD a spinorial field of expansion type p+1. Hence, it follows that ψ EF (AB ψCD)EF = ρ−6 ΦEF (AB ΦCD)EF , where the spinor ΦEF (AB ΦCD)EF

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has expansion type p + 2. Recall that multiplication by a scalar function of the form 1 + O(ρ) does not change the expansion type or symmetries of the various terms. Using the observations in the previous paragraph one has that the expression 5 ρ9 6U ρW + 15U 4 ρ2 W 2 + 20U 3 ρ3 W 3 + 15U 2 ρ2 W 4 + 6U ρ3 W 5 6 6 U (U + ρW ) + ρ3 W 6 ψ EF (AB ψCD)EF , – which is essentially the lift of |x|3 times the terms in the fifth and sixth lines of formula (40) – has at most expansion type p − 2. Similarly, |x|3 2U |x|W + |x|2 W 2 DF (A |x|2 ψBCD)F 2 2 U (U + |x|W ) lifts to Ca as

2ρ2 2U ρW + ρ2 W 2 xF (A ΦBCD)F , 2 + ρW ) which can be checked to have expansion type of at most p − 1. Now, U 2 (U

|x|3 (3U |x|2 W 2 + U 2 |x|W + |x|3 W 3 )DF (A ψBCD)F U 2 (U + |x|W )3 lifts to ρ2 3U ρ2 W 2 + 3U 2 ρW + ρ3 W 3 DF (A ΦBCD)F , U 2 (U + ρW )3 hence, using that in the cn-gauge one has that type(U − 1) = p − 1 one concludes that the whole term has at most expansion type p − 2. The term |x|5 DF (A |x|ψBCD)F (U + |x|W )3

lifts to

ρ2 xF ΦBCD)F , (U + |x|W )3 (A

which can be seen to have at most expansion type p − 1. The lift of |x|6 DF (A W ψBCD)F (U + |x|W )3

is given by

ρ3 DF (A W ΦBCD)F . (U + ρW )3

Now, DAB W has expansion type p + 1, and hence DF (A W ΦBCD)F has expansion type p + 2. Accordingly, the overall expansion type of the term is p − 1. Finally, the lift of |x|5 (2U |x|W + |x|4 W 2 )DF (A ψBCD)F U 2 (U + |x|W )2 is given by U 2 (U

ρ2 (2U ρW + ρ4 W 2 )DF (A ΦBCD)F + ρW )2 ρ −3 2 (2U ρW + ρ4 W 2 )xF (A ΦBCD)F . U (U + ρW )2

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Hence, noticing that DF (A ΦBCD)F and xF (A ΦBCD)F have both expansion type p + 1 and p + 1 one concludes that the overall expansion type of the term is at most p − 2. Thus, one has that type(φ˘• ABCD ) = p − 1 , and hence the symmetry ˘• φ˘• 0,p;2p,k = φ4,p;2p,k ,

p = 0, 1, 2, . . . ,

k = 0, . . . , 2p ,

holds trivially. This proves part (i) of Theorem 3. 5.1.2. Proof of the part (ii) of Theorem 3. Recall now the expression (39a). The ˘ lift to Ca of φ˘ ABCD is identical to that of the time symmetric φABCD discussed in [11]. The argument used in that reference to analyse the expansion type and symmetries of the spinor φABCD uses in an essential manner the analyticity of the conformal metric hij to introduce a complex null cone formalism. To this end a 3-dimensional complex analytic metric manifold (B, h) was introduced. In this setting h defines a complex valued non-degenerate scalar product. The complex manifold (B, h) contains (Ba , h) as a real Riemannian subspace. The spinor-dyad bundle SU (Ba ) has a complex analytic extension to a bundle SL(B) of spin frames on B with structure group SL(2, C). One can analytically extend the solder and connection forms to B in the same way it was done for the metric hij . Crucial is now to consider the complex null cone, N , generated by the geodesics through i. As one is restricted to work with analytic functions, one considers the analytic conformal factor Ω obtained from setting W = 0 in Ω = ρ2 /(U + ρW )2 . Because Ω = ρ2 /U , one then has that the null cone N corresponds to the locus of points on B such that Ω = 0. A suitable frame (i.e. coordinates and a frame) adapted to the geometry of N can be introduced on B. This results in a formalism which allows to calculate at the point i. A similar formalism has been used with great effect to discuss the convergence of multipole expansions of static spacetimes – see [13]. The complex null cone formalism is a powerful machinery to calculate the properties of the expansions of φ˘ABCD . Still, the required calculations extend over 10 pages; it will be omitted. The key results are that ) = p, type(φ˘ ABCD

and ˘ φ˘ 0,p;2p,k = −φ4,p;2p,k ,

p = 0, 1, 2, . . .

k = 0, . . . , 2p .

Remark. This is the only part of the proof of Theorem 3 where the analyticity of hij is used in an essential way. Still, it is expected that a similar result would follow in the smooth setting. This however, would involve lengthy induction arguments which will not be considered in this article. To conclude the analysis it is noticed that the lift to Ca of κ3 |x|6 EF ψ (AB ψCD)EF U6

is given by

κ ρ3 EF Φ (AB ΦCD)EF . U6

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As already discussed, ΦABCD has expansion type p+1, and hence ΦEF (AB ΦCD)EF has expansion type p + 2. Accordingly, the expansion type of the whole term is p − 1. Hence, the part of w ˘j,p;2p,k coming from ψABCD satisfies trivially the required antisymmetry condition. This concludes the proof of part (ii) of Theorem 3. 5.1.3. Proof of part (iii) and (iv) of Theorem 3. Notice that using equation (39b) one has √ √ √ 2|x|2 F 2|x|2 F 2 2 F ∗ 2 D U ψ + D (A ψBCD)F . wABCD = 2 D (A |x| ψBCD)F −4 BCD)F (A U U3 U2 As in the proofs of parts (i) and (ii) consider, one by one, the lifts of the various ∗ ∗ terms on w ˘ABCD = κ3 wABCD . One notes that the lift to Ca of |x|3 F 2ρ F D (A |x|2 ψBCD)F is x ΦBCD)F , (41) 2 U U 2 (A which has expansion type p as ΦABCD has expansion type p + 1. Multiplication by the lift of κ3 will not alter the expansion type. The lift to Ca of |x|5 F ρ2 F D U ψ is D (A U ΦBCD)F . BCD)F (A U3 U3 Now, using Lemma 1 one can conclude that in the cn-gauge type(DAB U ) = p. Hence type(DF (A U ΦBCD)F ) = p + 1. Accordingly, the overall expansion type of the term is p − 1. Finally, the lift of |x|5 F ρ F ρ2 F D ψ is − 3 x Φ + D (A ΦBCD)F . (42) BCD)F BCD)F (A U2 U 2 (A U2 The expansion type of the first term of the latter expression has already been discussed – it is, modulo a constant, the same as the term (41). To analyse the second term, note that although DAB ΦCDEF has expansion type p + 2 if ΦCDEF has expansion type p + 1, one finds that DF (A ΦBCD)F has expansion type p + 1 – for this, expand DAB ΦCDEF in terms of symmetric irreducible terms and then contract indices. Multiplication by ρ2 renders a term with expansion type p − 1. ∗ Summarising, it has been found that the only terms in w ˘ABCD contributing −2 F towards an expansion of type p are proportional to ρU x (A ΦBCD)F . The spinor ΦABCD is totally symmetric. Hence one can write it as ΦABCD = Φ0 0ABCD + Φ1 1ABCD + Φ2 2ABCD + Φ3 3ABCD + Φ4 4ABCD . Furthermore, noting that the following relations hold 1 1 xF (A 0BCD)F = √ 0ABCD , xF (A 1BCD)F = √ 1ABCD , 2 2 2 xF (A 2BCD)F = 0 , 1 1 xF (A 3BCD)F = − √ 3ABCD , xF (A 4BCD)F = − √ 4ABCD , 2 2 2

(43a) (43b) (43c)

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one finds that 1 1 1 1 xF (A ΦBCD)F = √ Φ0 0ABCD + √ Φ1 1ABCD − √ Φ3 3ABCD − √ Φ4 4ABCD . 2 2 2 2 2 2 Now, ΦABCD has expansion type p + 1. Accordingly Φj ∼

∞

p+1

2q 1 Φj,p;2q,k T2q kq−2+j ρp . p!

p=0 q=max{|1−j|,2−p} k=0

A quick comparison with formulae (36a)–(36b) shows that the expansion coefficients of Φ0 and Φ4 are essentially those of μ2 and μ2 . Now, from Lemma 4 and Theorem 2 it follows that all the contributions of type p in ΦABCD come from ˜ (2) is a complex function with ˜ (2) where λ ˚ABCD [J, λ(2) /r]. Recall that λ(2) = ð2 λ ψ ∞ real and imaginary parts in C (Ba ). Accordingly, write λ(2) /ρ ∼

p+1 2q ∞

(fp;2q,k + gp;2q,k ) T2q kq−2 ρp ,

(44)

p=1 q=2 k=0

where fp;2q,k , gp;2q,k ∈ C. The coefficients fp;2q,k satisfy the conditions fp;2q,k = (−1)k+q f p;2q,2q−k ,

p = 1, . . . ,

q = 2, . . . , p + 1 ,

k = 0, . . . p ,

˜ (2) while the coefficients gp;2q,k so that they are associated with the real part of λ satisfy the conditions gp;2q,k = (−1)k+q+1 g p;2q,2q−k ,

p = 1, . . . ,

q = 2, . . . , p + 1 ,

k = 0, . . . p

˜ (2) . A direct calculation so that they are associated with the imaginary part of λ reveals that p+1 2q ∞ 1 2p(p − 1)βq − (q − 1)(q − 2)− 2 fp;2q,k T2q kq−2 ρp , μ2 Re(λ(2) )/ρ ∼ p! p=1 q=2 k=0

μ2 Im(λ(2) )/ρ ∼ −

p+1 2q ∞ p=1 q=2 k=0

with βq =

1 gp;2q,k T2q kq−2 ρp , (p − 1)!

(q − 1)q(q + 1)(q + 2) .

Another short computation using the rules (43a)–(43c) shows that Φ0,p;2p+2,k Re(λ(2) )/ρ = −Φ4,p;2p+2,k Re(λ(2) )/ρ , Φ0,p;2p+2,k Im(λ(2) )/ρ = Φ4,p;2p+2,k Im(λ(2) )/ρ . The latter relations imply the antisymmetry and symmetry conditions, ˘4,p;2p,k Re(λ(2) )/ρ , w ˘0,p;2p,k Re(λ(2) )/ρ = −w w ˘0,p;2p,k Im(λ(2) )/ρ = −w ˘4,p;2p,k Im(λ(2) )/ρ .

652



˚BCD)F one obtains that Finally, applying a similar argument to ρU −2 xF (A ψ Φ0 [J] = Φ4 [J] so that the symmetry conditions are trivially satisfied. This proves points (iii) and (iv) of Theorem 2. 5.2. Some further results Let

√ cABCD ≡ i 2DF (A χBCD)F .

(45)

The following lemma is a direct consequence of the proof of Theorem 3. Lemma 5. For the class of data under consideration type(cABCD ) = p + 1 . The essential components cj of cABCD are of the form cj ∼

∞

2p+2

2q 1 cj,p;2q,k T2q kq−2+j ρp . p!

p=0 q=max{|1−j|,2−p} k=0

Now, recall the expansion (44). The coefficients fp;2q,k satisfy the following lemma. Lemma 6. For the class of initial data under consideration: (i) ∗ Re(λ(2) )/ρ = 0 ⇔ fp;2p+2,k = 0 . w ˘j,p;2p,k (ii)

cj,p;2p+2,k Re(λ(2) )/ρ = 0 ⇔ fp;2p+2,k = 0 .

Proof. The proof of (i) follows directly from the analysis carried in the proof of parts (iii) and (iv) of Theorem 3. The proof of part (ii) follows directly from (i) ∗ by using that wABCD = −Ω−1 cABCD . Finally, to state the main result of this article one needs the following generalisation of part (iii) of Theorem 4.1 in [11]. Theorem 4. For the class of initial data under consideration one has that: (i) = 0 , p = 2, 3, . . . , k = 0, . . . , 2p , w ˘0,p;2p,k if and only if D(Ap Bp · · · DA1 B1 bABCD) (i) = 0 , (ii)

∗ w ˘0,p;2p,k Re(λ(2) )/ρ = 0 , if and only if

E

D(Ap Bp · · · DA1 |E| cABC)

p = 0, 1, 2, . . . .

p = 2, 3, . . . ,

Re(λ(2) )/ρ (i) = 0 ,

k = 0, . . . , 2p p = 1, 2, 3, . . . .

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Remark 1. The class of data under consideration automatically satisfies cABCD (i) = 0 . Proof of Theorem 4. From the proof of part (ii) of Theorem 3 one has that type(φ˘ type(φ˘ ABCD ) = p , ABCD ) = p − 1 . Hence, in order to prove part (i) of Theorem 4 one only has to consider the expan˘ sion of the term φ˘ ABCD . This is formally identical to the spinor φABCD considered in [11] under the assumption of an analytic conformal metric. Thus, (i) follows from the analysis based on the complex null cone formalism given in the aforementioned reference and will be omitted here. In order to prove part (ii) observe that consistent with Lemma 5 ∞ 1 (p) p cj ρ , cj ∼ p! p=0 with (p)

cj

=

2p+2 k=0

p

cj,p;2p+2,k T2p+2 kp−1+j +

2q

cj,p;2q,k T2q kq+j ,

q=max{|1−j|,2−p} k=0

p = 1, 2, . . . . Making use of expression (33), one has that E

cj,p;2p+2,k = Kp,j,k D(Ap Bp · · · DA1 |E| cABC)j (i) ,

j = 0, . . . , 4 ,

k = 0, . . . , 2p ,

and Kp,j,k are some constants. Part (ii) now follows from Lemma 6.

Remark 2. The spinor cABCD has expansion type p + 1. Hence, cj,p;2p+4,k = 0 and D(Ap Bp · · · DA1 B1 cABCD) (i) = 0 .

6. The spacetime Friedrich gauge The ultimate goal of the framework developed in [11] is to gain control over the evolution of the gravitational field in a neighbourhood of spacelike infinity which extends to null infinity. This problem is generically known as the initial value problem near spatial infinity. In slight contrast to the analysis of the non-linear stability of the Minkowski spacetime – see e.g. [3, 14] – the aim of the initial value problem near spatial infinity as discussed in [11] is not only showing that the outgoing null geodesics starting close to spatial infinity are complete, but also to analyse under which conditions on the initial data, the resulting spacetime will admit a smooth conformal extension through null infinity – and hence giving rise to an asymptotically simple spacetime. In the standard representation of spatial infinity as a point, the direct formulation of an initial value problem for the conformal field equations with data prescribed in a neighbourhood, say Ba of infinity, renders a problem which although

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local is also singular – this can be seen in a very poignant way by considering the expressions for the massless and massive parts of the Weyl spinor discussed in Section 5. The formulation of the initial value problem near spatial infinity presented in [11] employs gauge conditions based on timelike conformal geodesics. The conformal geodesics are autoparallel with respect to a Weyl connection – i.e. a torsionfree connection which is not necessarily the Levi-Civita connection of a metric. An analysis of Weyl connections in the context of the conformal field equations has been given in [10]. In terms of this gauge based on conformal geodesics – which shall be called the Friedrich gauge or F-gauge for short – the conformal factor of the spacetime can be determined explicitly in terms of the initial data for the Einstein vacuum equations. Hence, provided that the congruence of conformal geodesics and the fields describing the gravitational field extend in a regular manner to null infinity, one has complete control on the location of null infinity. In addition, the F-gauge renders a particularly simple representation of the propagation equations. Using this framework, the singular initial value problem at spatial infinity can be reformulated into another problem where null infinity is represented by an explicitly known hypersurface and where the data are regular at spacelike infinity. The construction of the bundle manifold Ca and the blowing up of the point i ∈ Ba to the set I 0 ⊂ Ca briefly described in Section 3 is the first step in the construction of the regular setting. The next step in the construction is to introduce a rescaling in the frame bundle so that fields that are singular at I 0 become regular. 6.1. The manifold Ma,κ Following the discussion of [11] assume that in the development of data prescribed on Ba the timelike spinor τ AA introduced in Section 3 is tangent to a congruence of timelike conformal geodesics which are orthogonal to Ba . The canonical factor rendered by the consideration of this congruence of conformal geodesics is given in terms of an affine parameter τ of the conformal geodesics by κ2 τ 2 2Ω −1 , (46) Θ=κ Ω 1− 2 , with ω = ω |Dα ΩDα Ω| where Ω = ϑ−2 and ϑ solves the Lichnerowicz equation (12). The function κ > 0 – which will be taken to be of the form κ = κ ρ, with κ smooth, κ (i) = 1 – expresses the remaining conformal freedom in the construction. Consistent with the scalings δA → κ1/2 δA induced by the function κ one considers the set Ca,κ = κ1/2 Ca of scaled spinor dyads. Furthermore, define the bundle manifold

ω(q) ω(q)

≤τ ≤ Ma,κ = (τ, q) q ∈ Ca,κ , − , κ(q) κ(q) which, assuming that the congruence of null geodesics and the relevant fields extend adequately, can be identified with the development of Ba up to null infinity – that is, the region of spacetime near null and spatial infinity. In addition to Ca,κ one

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defines the sets:

I = (τ, q) ∈ Ma,κ ρ(q) = 0, |τ | < 1 ,

I ± = (τ, q) ∈ Ma,κ ρ(q) = 0, τ = ±1 ,

ω(q) I ± = (τ, q) ∈ Ma,ω q ∈ Ca,ω , ρ > 0, τ = ± , κ(q) which will be referred to as, respectively, the cylinder at spatial infinity, the critical sets (where null infinity touches spatial infinity) and future and past null infinity. In order to coordinate the hypersurfaces of constant parameter τ , one extends the coordinates (ρ, tAB ) off Ca,κ by requiring them to be constant along the conformal geodesics – i.e. one has a system of conformal Gaussian coordinates. 6.2. The conformal propagation equations On the manifold Ma,κ it is possible to introduce a calculus based on the derivatives ∂τ and ∂ρ and on the operators X+ , X− and X. The operators ∂ρ , X+ , X− and X originally defined on Ca can be suitably extended to the rest of the manifold in a standard way. A frame cAA and the associated spin connection coefficients of the Weyl connection, ΓAA BC , will be used. The gravitational field is, in addition, described by the spinorial counterparts of the Ricci tensor of the Weyl connection, ΘAA BB , and of the rescaled Weyl tensor, φABCD . In order to describe the conformal propagation equations consider the vectors υ = (cAB , ΓABCD , ΘABCD ) ,

φ = (φABCD ) ,

where cAB , ΓABCD , ΘABCD are the space-spinor versions of the spacetime spinors cAA , ΓAA BC , ΘAA BB . The relation between space-spinors and spacetime spinors is implemented by suitable contractions with the spinor τ AA . This will not be elaborated further – the interested reader is referred to [7, 11, 17]. The explicit form of the propagation equations will not be required. Only general properties will be used. Schematically one has ∂τ υ = Kυ + Q(υ, υ) + Lφ , √ AB μ 2E∂τ φ + A cAB ∂μ φ = B(ΓABCD )φ ,

(48a) (48b)

where E denotes the (5 × 5) unit matrix, AAB cμAB are (5 × 5) matrices depending on the coordinates, and B(ΓABCD ) is a linear (5 × 5) matrix valued function with constant entries of the connection coefficients ΓABCD . In addition to the above propagation equations it is essential to consider the following constraint equations derived from the Bianchi identities: F AB cμAB ∂μ φ = H(ΓABCD ) , where now F AB cμAB denote (3 × 5) matrices, and H(ΓABCD ) is a (3 × 5) matrix valued function of the connection with constant entries. Equations (48a) and (48b) can be cast as a symmetric hyperbolic system on a neighbourhood N ⊂ Ma,κ of the initial hypersurface. Hence, given data that extends smoothly to I 0 , one obtains a unique smooth solution on N . The

656



metric can be seen to degenerate as ρ → 0. A consequence of this is AAB c1AB = 0 on I ∩ N – i.e. the coefficient associated with the ρ derivatives in the Bianchi propagation equations (48b). The restriction of the propagation equations to I implies an interior system on I which determines υ and φ on I uniquely from the restriction of the initial data to I 0 . Differentiating the propagation equations repeatedly with respect to ρ and restricting the result to I one obtains a hierarchy of interior equations for υ (p) = ∂ρp υ|I from where it is possible to determine formal expansions 1 1 υ (p) ρp , φ = φ(p) ρp , υ= p! p! p≥0

p≥0

of the solution at I. The calculation of the lowest order terms υ (0) shows that the matrix AAB c1AB is positive definite on I and extends smoothly to I ± , but it looses rank there. The entries of the vectors υ (p) and φ(p) on I can be seen to have definite spin-weights and hence admit very particular expansions in terms of the functions Tj kl – as described in Section 3.5. The use of these time dependent normal expansions reformulates the problem of calculating the vector υ (p) into a problem of linear ordinary differential equations. Once the entries of the vectors υ (p) and φ(p) have been expanded using the functions Tj kl , the interior equations are reduced to systems of ordinary differential equations for the τ -dependent expansion coefficients. The task of solving these equations reduces, in the end, to solving a hierarchy of ordinary differential equations of the form yα = Cα yα + bα ,

(49)

where Cα is a 2 × 2 matrix and yα and bα are column vectors. The components of yα consist of pth-order ρ-derivatives of certain components of the Weyl spinor φABCD . In equation (49) note the presence of a multi-index α = (p, q, k) indicating the order in the expansions, ρp , and the harmonic T2q kl to which the term in question is associated to. In what follows, in order to ease the discussion, the multi-index α will often be suppressed. For a given multi-index α = (p, q, k), the components of the vector b are calculated from the lower order solutions υ (q) and φ(q) , 0 ≤ q ≤ p − 1. The solutions to these equations can be written in the form τ X −1 (s)b(s)ds , (50) y(τ ) = X(τ )X −1 (0)y0 + X(τ ) 0

where y0 = y(0) and X(τ ) – again suppressing the relevant multi-index – denotes the fundamental matrix of the system of ordinary differential equations. The matrices X have been explicitly calculated in [11]. As p increases, the explicit expressions of the entries of the vector b become more complicated. One can, nevertheless, implement the aforediscussed procedure in a computer algebra system – see e.g. [20, 22]. Now, due to the degeneracy of AAB c1AB on I, the ordinary differential equations for the vector y are singular at τ = ±1. As a consequence, for a given p, there are certain choices of the multi-index α∗ = (p, p, k), k = 0, . . . , 2p, of for which the fundamental system develops logarithmic singularities at τ = ±1.

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For all other allowed values of the multi-index, the fundamental matrices can be written explicitly in terms of Jacobi polynomials. It can be seen that the vector b for the corresponding multi-index α∗ vanishes. Hence the logarithmic divergences cannot be cancelled out by the integral in (50). Thus, in general the functions υ (p) and φ(p) develop logarithmic singularities for all p ≥ 2 on I. 6.3. Regularity conditions Given the aforementioned state of affairs, can one find conditions on the initial data so that the functions yα extend smoothly to I ± ? An inspection of the term X(τ )X −1 (0)y0 for the singular multi-index values α∗ shows that there are conditions on the initial data for which the observed singularities at I ± do not arise. This analysis is completely general as long as the initial data is expandable in powers of ρ near I 0 . With the aim of formulating the main result of this article, it will be necessary to provide a more precise description of the conditions on the data that have been mentioned in the previous paragraph. Following the conventions of Section 5 consider the spinor φ˘ABCD = κ3 φABCD , and let (p) φ˘j = ∂ρp φ˘j |ρ=0 .

Observing Theorem 3 one expands (p) φ˘j =

p 2q

aj,p;2q,k T2q kq−2+j ,

q=|2−j| k=0

with τ -dependent complex coefficients aj,p;2q,k . The coefficients aj,p;2q,k can be determined along I from their value at I 0 by solving a set of transport equations implied by the conformal field equations as described in Section 6. The analysis of [11] implies the following result. Theorem 5. For the class of data under consideration one has that τ ds p+2 p−2 a0,p;2p,k (τ ) = (1 − τ ) (1 + τ ) C0,k + C1,k , p−1 (1 − s)p+3 0 (1 + s) τ ds , a4,p;2p,k (τ ) = (1 + τ )p+2 (1 − τ )p−2 C0,k + C1,k p−1 (1 + s)p+3 (1 − s) 0 with C0,k and C1,k constants. In particular, a0,p;2p,k and a4,p;2p,k extend analytically through τ = ±1 if and only if a0,p;2p,k (0) = a4,p;2p,k (0) , with k = 0, . . . , 2p.

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7. The main result In [11] it was shown that for the class of time symmetric data sets with an analytic conformal metric, the conditions of Theorem 5 conditions can be reformulated in terms of the vanishing of the Cotton tensor and its symmetrised higher order derivatives at infinity. This is an asymptotic condition on the freely specifiable data. The main result in this article is a generalisation of the result described in the previous paragraph to the class of data with a non-vanishing second fundamental form. This main result brings together the discussion in Section 5 (on the structure of initial data for the Weyl tensor near infinity), in particular of Theorems 3 and 4, with the discussion of Section 6 (on the regular finite initial value problem at spatial infinity). From Theorem 3 it follows readily that ∗ ∗ Re(λ(2) /ρ) + w Im(λ(2) /ρ) , ˘0,p;2p,k +w ˘0,p;2p,k ˘0,p;2p,k a0,p;2p,k (0) = φ˘•0,p;2p,k + w ∗ ∗ Re(λ(2) /ρ) + w Im(λ(2) /ρ) . a4,p;2p,k (0) = φ˘•0,p;2p,k − w ˘0,p;2p,k −w ˘0,p;2p,k ˘0,p;2p,k Hence the condition a0,p;2p,k (0) = a4,p;2p,k (0) ,

k = 0, . . . , 2p ,

of Theorem 5 implies w ˘0,p;2p,k = 0,

∗ Re(λ(2) /ρ) = 0 , w ˘0,p;2p,k

on Ca independently of the choice of κ. These last two conditions can be readily reformulated in terms of the tensors bABCD and cABCD using Theorem 4. In this way one obtains the main result of the article. Theorem 6. For the class of data under consideration, the solution to the regular finite initial value problem at spatial infinity is smooth through I ± only if the conditions D(Ap Bp · · · DA1 B1 bABCD) (i) = 0 , E D(Aq Bq · · · DA1 |E| cABC) [λ(R) ](i)

= 0,

p = 0, 1, 2, . . .

(51a)

q = 0, 1, 2, . . .

(51b)

are satisfied by the free initial data. If the above conditions are violated at some order p or q, then the solution will develop logarithmic singularities at I ± . Remark 3. This result is a generalisation of Theorem 8.2 in [11]. Remark 4. The result in Theorem 6 which is expressed in the language of spacespinors can be readily reformulated in terms of spatial tensors. Using standard transcription rules one finds the tensorial version of the main theorem given in the introductory section.

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8. Conclusions In order to bring further insight into Theorem 6, some connections with the construction of data for purely radiative spacetimes are pointed out. In reference [9] it has been shown how time symmetric Cauchy initial data with vanishing mass can be used to obtain analytic data at past null infinity. The development of this data renders a purely radiative spacetime. This construction is possible because if the mass vanishes, the point i satisfying the boundary conditions (4a)–(4c) can be interpreted not only as the spatial infinity of a vacuum spacetime but also as the past timelike infinity of the causal future of i. The crucial point in the analysis is to obtain conditions on the Cauchy data so that the spinor wABCD is analytic at i. The Cauchy–Kowalevskaya theorem can, in turn, be used to obtain analytic data on past null infinity close to i− (purely radiative data). The condition for the analyticity of wABCD is given by D(Aq Bq · · · DA1 B1 bABCD) (i) = 0 ,

q = 0, 1, 2, . . . ,

that is, the same as condition (51a) in Theorem 6. The way the analysis in reference [9] could be generalised to the case of non-time symmetric Cauchy data has been briefly discussed in the concluding remarks of [11]. That analysis will not be repeated here, but under the extra assumption that the Cauchy data has no linear ∗ momentum, necessary and sufficient conditions for φABCD = wABCD + iwABCD to be analytic at i are, in addition to (51a), that D(Ap Bp · · · DA1 B1 ) cABCD (i) = 0 ,

p = 0, 1, 2, . . . .

(52)

Note that condition (52) is much stronger than condition (51b). Indeed, from the discussion in Subsection 3.3 one has that E

D(Ap Bp · · · DA1 B1 ) cABCD = 0 ⇒ D(Ap Bp · · · DA1 |E| cABC) = 0 , but not conversely. Furthermore, condition (52) involves the whole of the free data ˚ij and not only the part depending on Re(λ(2) /ρ). ψ Condition (52) was suggested as the supplementary condition to be imposed on initial data sets with non-vanishing second fundamental form in order to obtain regularity at the critical sets of the cylinder at spatial infinity. However, it has been shown that initial data for the Kerr spacetimes does not fulfil it2 . On the other hand, it can be shown that condition (51b) is satisfied by Kerr data obtained from the line element in Boyer-Lindquist coordinates. In [5] it has been shown that for this “standard” Kerr initial data set one has λ(R) = 0, so that (51b) is satisfied trivially. In view of this, it is natural to ask whether the assertions of Theorem 6 are true for more general classes of initial data. In particular, it is important to see how stationary initial data fits into this picture. In [9] it has been shown that the condition (51a) for the spinor bABCD is satisfied by time symmetric static data. It is conjectured that “standard” stationary data satisfies the conditions (51a)–(51b). 2 S.

Dain: private communication.

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In [4] it has been shown that there is a slice and a gauge for which in a neighbourhood Ba of infinity, the conformal metric of stationary data is of the form (1) (2) hij = hij + r3 hij , (53) (1)

(2)

with hij and hij analytic. A conformal metric of this form is not smooth at i – i.e. at r = 0. In fact, it follows directly that it is C 2,α . Similarly, it can be seen that the corresponding tensor ψij is such that ψij = r−5 ψij + r−4 ψij , (1)

(1)

(2)

(54)

(2)

with ψij = O(r2 ) and ψij = O(r2 ) analytic. A choice of hij and ψij consistent with (53) and (54) provide a natural ground for generalising the results in this article. Unfortunately, this analysis is complicated by the fact that up to date there is no generalisation of the analysis in [6] for conformal metrics which are non-smooth.

Appendix A. Spinorial identities Let xAB ≡

√

0

1

2(A B) ,

1 yAB ≡ − √ A1 B 1 , 2

(E

kABCD = (A B F C G D)

H)k

1 zAB ≡ √ A0 B 0 , 2

.

The following identities are used the main text: 1 4 , 2 ABCD 1 = − 2ABCD , 2 1 0 = ABCD . 2

x(AB xCD) = 22ABCD ,

y(AB yCD) =

x(AB yCD) = −3ABCD ,

y(AB zCD)

x(AB zCD) = 1ABCD ,

z(AB zCD)

Also, xAF xBF =

1 AB 2

1 xAF yBF = − √ yAB , 2

1 xAF zBF = √ zAB , 2

and 1 1 xF (A 0BCD)F = √ 0ABCD , xF (A 1BCD)F = √ 1ABCD , 2 2 xF (A 2BCD)F = 0 , 1 1 xF (A 3BCD)F = − √ 3ABCD , xF (A 4BCD)F = − √ 4ABCD . 2 2

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Appendix B. Technical results concerning solutions of the constraints A detailed analysis of solutions to the vacuum Einstein constraint equations such that near spacelike infinity they admit asymptotic expansions of the form ˜k h χ ˜kij 2m ij ˜ , χ ˜ij ∼ hij ∼ 1 + δij + r˜ r˜ r˜k k≥2

k≥2

˜ k and χ where h ˜kij are smooth functions on S2 has been given in reference [6]. The ij results given in the latter reference are fundamental building blocks of the analysis carried out in the present article. This appendix offers a summary of the results of reference [6] together with some extensions of their work as the class of initial data considered in the present article turns out to be more general than the one in reference [6]. B.1. Solutions to the Hamiltonian constraint ˜ S˜ = S \ Ba , is said to be in E ∞ (Ba ) if on Ba one can A function f ∈ C ∞ (S), write f = f1 + rf2 with f1 , f2 ∈ C ∞ (Ba ), with r2 = δij xi xj . One could define in an analogous fashion the spaces E k (Ba ) and E ω (Ba ). Theorem 1 in [6] then states that: Theorem 7. Let hij be a smooth metric on S with positive Ricci scalar. Assume that ψij is smooth in S˜ and satisfies on Ba r8 ψij ψ ij ∈ E ∞ (Ba ) . Then there exists on S˜ = S \ {i} a unique solution ϑ to the Lichnerowicz equation (12), which is positive, satisfies limr→0 rϑ = 1 and has in Ba the form ϑ=

1 (u1 + ru2 ) , r

u1 ∈ Ba ,

u2 ∈ E ∞ (Ba ) ,

u1 = 1 + O(r2 ) .

A class of symmetric trace-free tensors ψij solving the momentum constraint and satisfying the condition r8 ψij ψ ij ∈ E ∞ (Ba ) has been discussed in Section 4.3 of [6] and their asymptotic properties in Corollary 5. Unfortunately this class of tensors ψij is not general enough for our purposes. Accordingly, further analysis is required. B.2. Solutions to the momentum constraint B.2.1. On the solutions to the flat space momentum constraint. As in the main ˚ij a solution to the Euclidean momentum constraint and write text, denote by ψ ˚ij = ψ ˚ij [P ] + ψ ˚ij [A] + ψ ˚ij [J] + ψ ˚ij [Q] + ψ ˚ij [λ] ˜ , ψ ˚ij depending on P i , etc. It can be seen where the [P ] denotes the part of ψ ˜ ∈ E ∞ (Ba ) and P i = 0 then that – see Theorem 15 in reference [6] – that if rλ

662



˜ made in the main text – ˚ij ψ ˚ij ∈ E ∞ (Ba ). This result justifies the choice of λ r8 ψ cfr. Ansatz (19). The h-trace-free part ˚ij − 1 hij hkl ψ ˚kl ˚ Φij = ψ 3 ˚ij will be used to prescribe the freely-specifiable data in the solutions to the of ψ ˜ ∈ E ∞ (Ba ) then momentum constraint. As mentioned in the main text, if rλ (1) (2) (1) ˜ (2) ∞ ˜ ˜ ˜ ˜ λ = λ + λ /r, with λ , λ ∈ C (Ba ). Accordingly, one can write without loss of generality ˜ (1) ∼ λ

αi1 ···ik xi1 · · · xik ,

˜ (2) ∼ λ

k≥2

βi1 ···ik xi1 · · · xik

k≥2

with αi1 ···ik , βi1 ···ik ∈ C, symmetric. It is recalled that the above expressions are to be interpreted as ˜ (1) = λ

m

(1)

˜ , αi1 ···ik xi1 · · · xik + λ R

k≥2

˜ (2) = λ

m

(2)

˜ , βi1 ···ik xi1 · · · xik + λ R

k≥2

˜ (1) = o(rm ) and λ ˜ (2) = o(rm ) for all m. Let λ ≡ ð2 λ, ˜ λ(1) ≡ ð2 λ ˜ (1) , with λ R R ˜ (2) . λ(2) ≡ ð2 λ The expressions given in Section 2.3 for the various possible solutions of the momentum constraint are given in terms of its components with respect to a frame {ni , mi , mi }. This choice is not convenient for the subsequent discussion of the solutions of equation (11). Therefore, the expressions involving the parts of ˜ are reformulated in terms of a the solution arising from the complex function λ Cartesian basis related to the normal coordinates {xi }. The lengthy calculations, ˚ij [Re(λ(1) )] is a sum of terms of which are omitted here, reveal that the tensor ψ the form (1)

(1)

B Ak Re(αi1 ···ik )xi1 · · · xik xi xj + k3 Re(αi1 ···ik )xi1 · · · xik δij r5 r (1) C + k3 x(i Re(αj)i1 ···ip−1 )xi1 · · · xp−1 , r (1)

(1)

(1)

where Ak , Bk and Ck are constants depending on k ≥ 2 such that the whole term is δ-trace free and δ-divergence free. The contributions due to the imaginary

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˚ij [Im(λ(1) )] which is a sum of terms of the form part of λ(1) render a tensor ψ

(1,0)

Dm kl x(i j) xk Im(αli1 ···im−1 )xi1 · · · xim−1 r4 (1,2)

+

Dm j j kl x(i j) xk Im(αli1 ···im−3 j1 j2 1 2 )xi1 · · · xim−3 + · · · r2 (1,0) Em + xk kl(i Im(αj)li1 ···im−2 )xi1 · · · xim−2 r2 (1,2) + Em xk kl(i Im(αj)li1 ···im−4 j1 j2 (1,0)

(1,2)

(1,0)

j1 j2

)xi1 · · · xim−4 + · · ·

,

(1,2)

where again Dk , Dk , . . . and Ek , Ek , . . . are constants making the whole term δ-divergence free – note that the terms are δ-trace free by construction. ˚ij [Re(λ(2) )/r] is a sum of terms of the form Similarly, the tensor ψ (2)

(2)

Ak B Re(βi1 ···ik )xi1 · · · xik xi xj + k4 Re(βi1 ···ik )xi1 · · · xik δij r6 r (2) C + k4 x(i Re(βj)i1 ···ip−1 )xi1 · · · xp−1 , r (2)

(2)

(2)

where Ak , Bk and Ck are constants depending on k ≥ 2 such that the term ˚ij [Im(λ(2) )/r] is a is δ-trace free and δ-divergence free. In addition, one has that ψ sum of terms of the form

(2,0)

Dm kl x(i j) xk Im(βli1 ···im−1 )xi1 · · · xim−1 r5 (2,2)

+

Dm j j kl x(i j) xk Im(βli1 ···im−3 j1 j2 1 2 )xi1 · · · xim−3 + · · · r3 (2,0) Em + xk kl(i Im(βj)li1 ···im−2 )xi1 · · · xim−2 r3

(2,2)

+ (2,0)

(2,2)

Em r

xk kl(i Im(βj)li1 ···im−4 j1 j2 (2,0)

(2,2)

j1 j2

)xi1 · · · xim−4 + · · ·

where Dk , Dk , . . . and Ek , Ek , . . . are constants making the term δdivergence free – note that the individual terms are δ-trace free by construction. For future reference it will be convenient to write ˚ij Im(λ(1) ) = r−4˚ ˚ij Re(λ(1) ) = r−5˚ Ξij Re(λ(1) ) , Ξij Im(λ(1) ) , ψ ψ ˚ij Re(λ(2) )/r = r−6˚ ˚ij Im(λ(2) )/r = r−5˚ Ξij Re(λ(2) )/r , ψ Ξij Im(λ(2) )/r , ψ

664



with ˚ Ξij [Re(λ(1) )], ˚ Ξij [Im(λ(1) )], ˚ Ξij [Re(λ(2) )/r], ˚ Ξij [Im(λ(2) )/r] ∈ C ∞ (Ba ). It is noticed that xi xj ˚ xi xj ˚ Ξij Re(λ(1) ) = r2˚ Ξ Re(λ(1) ) , Ξij Im(λ(1) ) = 0 , Ξij Re(λ(2) )/r = r2˚ Ξ Re(λ(2) )/r , Ξij Im(λ(2) )/r = 0 , xi xj ˚ xi xj ˚ Ξ[Re(λ(2) )/r] some scalar functions. for ˚ Ξ[Re(λ(1) )], ˚ B.2.2. General issues concerning the solutions to the non-flat momentum constraint. An analysis of the solutions to the elliptic equation for the vector v i – see equation (11) – for free data of the form ˚ Φij [A, J, Q, Re(λ(1) ), Im(λ(2) )/r], has been given in Theorem 16 and of reference [6]. If the conformal metric hij admits no conformal Killing vectors on S, then there exists a unique vector v i ∈ W 2,q , q > 1 solving equation (11). If the conformal metric admits conformal Killing vectors one can guarantee the existence of a solution only if the constants A, J i and Qi satisfy a particular relation. In order to discuss the asymptotic expansions of the vector v i recall the definition of Q∞ (Ba ) given in Section 2.4.3. Furthermore, introduce for m ∈ N, m ≥ 1 the following real spaces of functions: Qm = v i ∈ C ∞ (R3 , R3 ) | v i ∈ Pm , v i xi = r2 v with v ∈ Pm−1 , where Pm denotes the space of homogeneous polynomials of degree m – that is, v ∈ Pm if and only if v = vi1 ···im xi1 · · · xim , with vi1 ···im ∈ R totally symmetric. It turns out that a convenient way of grouping the different terms in the free specifiable data is the following: ˚ij = ψ ˚ij [A, J, Q] + ψ ˚ij Re(λ(1) ), Im(λ(2) )/r + ψ ˚ij Re(λ(2) )/r, Im(λ(1) ) . ψ We shall proceed to analyse the asymptotic expansions of the solutions to (11) implied by each of these terms. However, first an analysis of the source term Di ˚ Φij will be required. Φij . As seen in Sections 2.4 and 3.6 , if the conformal metric B.2.3. Analysis of Di ˚ hij satisfies the cn-gauge, then one can write ˇ ij , hij = −δij + r2 h

ˇ ij . hij = −δij + r2 h

ˇ ij = 0, δij h ˇ ij = 0. A In addition, the technical assumption (20) is recalled: δ ij h calculation renders ˇ jk + xj h ˇ ik − xk h ˇ ij + 1 r2 ∂i h ˇ kj + ∂j h ˇ ik − ∂k h ˇ ij Γkij = xi h 2 ˇ kij . ˜ kij + r2 Γ =Γ In addition, let ˜ l = hlk Γ ˜ kij , Γ ij One readily verifies that ˇ l , xj Γ ˇl , ˜ lij = r2 h ˜ lij = r2 h xi Γ j i

ˇ kij . ˇ l = hlk Γ Γ ij

ˇ ij , ˜ lij = r2 h xl Γ

ˇ ij . ˜ lij = −xl hij h hij Γ

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Now, recall that ˚ij − 1 hi hkl Di ψ ˚kl , Φij = Di ψ Di ˚ 3 j and that l ˚ ˚ij = ∂k ψ ˚ij − Γl ψ ˚ Dk ψ ki lj − Γ kj ψil ,

so that ki l ˚ ˚ij = r2 h ˇ ik ∂k ψ ˚ij − hki Γl ψ ˚ Di ψ ki lj − h Γ kj ψil .

(55)

In the last equation it has been used that by construction ˚ij = 0 . δ ik ∂k ψ Now, consider ˚ Φij [Re(λ(2) )/r], which is the most singular contribution of λ to the ˇ ik ∂k ψ ˚ij is given by a sum of seed tensor ˚ Φij . A direct calculation shows that r2 h terms of the form (2)

(2)

An ˇ ik nBn ˇ ik h δik Re(βi1 ···in )xi1 · · · xin xj + h δij Re(βi1 ···in−1 k )xi1 · · · xin−1 4 r r2 (2) (2) Cn ˇ ki Cn ˇ ki i1 in−1 + δ Re(β )x · · · x + h h δkj Re(βi1 ···in−1 i )xi1 · · · xin−1 ki i ···i j 1 n−1 2r2 2r2 (2) Cn (n − 1) ˇ ki + xj h Re(βi1 ···in−2 ki )xi1 · · · xin−2 . r2 Hence, if the technical condition (20) holds, one can conclude that 1 Di ˚ Φij Re(λ(2) )/r = 2 Sj Re(λ(2) )/r , r with Sj Re(λ(2) )/r ∈ Q∞ (Ba ) , Sj Re(λ(2) )/r = O(r4 ) . A similar analysis can be carried out with the contributions due to Re(λ(1) ), Im(λ(1) ) and Im(λ(2) )/r. One obtains the following lemma. Lemma 7. For the class of initial data under consideration expressed in the cngauge, and assuming that condition (20) holds, one has that: 1 Sj [A] , Sj [A] ∈ Q∞ (Ba ) , Sj [A] = O(r0 ) , r 1 Φij [J] = 3 Sj [J] , Di ˚ Sj [J] ∈ Q∞ (Ba ) , Sj [J] = O(r2 ) , r 1 Φij [Q] = Sj [Q] , Di ˚ Sj [Q] ∈ Q∞ (Ba ) , Sj [Q] = O(r) , r 1 1 Φij Re(λ(1) ) = Sj Re(λ(1) ) , Φij Im(λ(1) ) = 2 Sj Im(λ(1) ) , Di ˚ Di ˚ r r 1 1 Φij Re(λ(2) )/r = 2 Sj Re(λ(2) )/r , Di ˚ Φij Im(λ(2) )/r = 3 Sj Im(λ(2) )/r , Di ˚ r r Di ˚ Φij [A] =

666


Sj Re(λ(1) ) ∈ Q∞ (Ba ) , Sj Im(λ(1) ) ∈ Q∞ (Ba ) , Sj Re(λ(2) )/r ∈ Q∞ (Ba ) , Sj Im(λ(2) )/r ∈ Q∞ (Ba ) ,


Sj Re(λ(1) ) = O(r4 ) , Sj Im(λ(1) ) = O(r3 ) , Sj Re(λ(2) )/r = O(r4 ) , Sj Im(λ(2) )/r = O(r3 ) .

˚ij [A, J, Q]. The case has been discussed in Corollary 5 of [6]. B.2.4. Free data ψ Using the results of Lemma 7 which hold for the cn-gauge and assuming condition (20), a direct application of the techniques of Section 4.2 of reference [6] render v i [A] = rv1i [A] + v2i [A] , 1 v i [J] = v1i [J] + v2i [J] , r 1 i i v [Q] = v1 [Q] + v2i [Q] , r with v1i [A] ∈ Q∞ (Ba ) ,

v1i [A] = O(1) ,

v2i [A] ∈ C ∞ (Ba ) ,

v1i [J] ∈ Q∞ (Ba ) ,

v1i [J] = O(r2 ) ,

v2i [J] ∈ C ∞ (Ba ) ,

v1i [Q] ∈ Q∞ (Ba ) ,

v1i [Q] = O(r) ,

v2i [Q] ∈ C ∞ (Ba ) .

˚ij [Re(λ(1) ), Im(λ(2) )/r]. This case has not been discussed in B.2.5. Free data ψ complete generality in [6], but in view of the results of Lemma 7 it is essentially a direct application of Theorem 17 there. One obtains that v i Re(λ(1) ) = rv1i Re(λ(1) ) + v2i Re(λ(1) ) , 1 v i Im(λ(2) )/r = v1i Im(λ(2) )/r + v2i Im(λ(2) )/r , r with v1i Re(λ(1) ) ∈ Q∞ (Ba ) , v1i Re(λ(1) ) = O(r4 ) , v2i Re(λ(1) ) ∈ C ∞ (Ba ) , v1i Im(λ(2) )/r ∈ Q∞ (Ba ) , v1i Im(λ(2) )/r = O(r3 ) , v2i Im(λ(2) )/r ∈ C ∞ (Ba ) . ˚ij [Re(λ(2) )/r, Im(λ(1) )]. The discussion of the asymptotic strucB.2.6. Free data ψ ˚ij [Re(λ(2) )/r, ture of solutions of equation (11) when the free data is given by ψ Im(λ(1) )] is not directly covered by the techniques of [6]. The reason for this is the appearance of terms with even powers of 1/r in the source terms Di ˚ Φij . As it will be shown, these cases – which are the ones of more relevance for the present work – can be analysed under further assumptions on the free data. The analysis in Section 5 shows that free data of the form ˚ij Re(λ(1) ), Im(λ(2) )/r ˚ij [A, J, Q] + ψ ψ does not contribute to the regularity condition discussed in the present article. In order to have a non-trivial contribution from the second fundamental form in the

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regularity conditions at the critical points I ± one has to consider free data of the form ˚ij Re(λ(2) )/r, Im(λ(1) ) . ψ The complications concerning this case will be discussed in the sequel. In a neighbourhood Ba of infinity, the form that the conformal metric acquires in normal coordinates can be used to decompose the differential operator appearing in equation (11), namely 1 Lh v i = Dk Dk v i + Di Dk v k + rik v k 3 ˆ h , where in the form Lh = L0 + L 1 L0 vi = ∂ k ∂k vi + ∂i ∂ j vj , 3 1 ˆ jk jk ˆ ˆ ∂i ∂k vj + B jki ∂j vk + Aj i vj , Lh vi = h ∂j ∂k vi + h 3 with Aj i and B kji functions of the metric coefficients and their first and second derivatives. They are smooth functions and satisfy Aj i = O(r), B kji = O(r2 ). Properties of the operators Δ0 and Lh . As before, let Pm denote the real linear space of homogeneous polynomials of degree m. Likewise, let Hm denote the space harmonic polynomials of degree m. If αi1 ···im xi1 · · · xim ∈ Pm , αi1 ···im = α(i1 ···im ) , then it is an harmonic polynomial if and only if αi1 ···im is trace free. It is well known that the space Pm can be written as a direct sum Pm = Hm ⊕ r2 Hm−2 ⊕ r4 Hm−4 ⊕ · · · . Let s ∈ Z, Δ0 = ∂ k ∂k ,

Δ0 : rs Pm → rs−2 Pm ,

defines a bijective linear map if either s > 0 or s < 0, |s| is odd and m + s ≥ 0 – see [6]. If one wants to discuss the bijectivity of the Laplacian for functions of the form rs pm with s < 0, |s| even and pm ∈ Pm , one has to restrict the domain and the range sets. To this end define Pm,s ≡ Hm ⊕ r2 Hm−2 · · · ⊕ rm−|s|−1 H|s|+1 . Elaborating from the proof of Lemma 3 in [6] one finds the following lemma. Lemma 8. Let s ∈ Z, with s < 0, |s| even, then Δ0 : rs Pm,s → rs−2 Pm,s is a bijective mapping if m + s ≥ 0. The operator L0 has nice properties with regard to the spaces Qm . Indeed, if s ∈ Z, then L0 : rs Qm → rs−2 Qm

668



is a bijective linear mapping again if s > 0 or s < 0, |s| is odd and m + s ≥ 0. As in the case of the Laplacian, in order to obtain bijectivity for s < 0, |s| even, one has to restrict both the domain and the range. Let Qm,s ≡ v i ∈ C ∞ (R3 , R3 ) | v i ∈ Pm,s , v i xi = r2 v, v ∈ Pm,s . Following the arguments of [6] one can prove the following lemma – cfr. Lemma 11 in [6]. Lemma 9. Let s ∈ Z, with s < 0, |s| even, then L0 : rs Qm,s → rs−2 Qm,s is a bijective mapping if m + s ≥ 0. ˚ij [Re(λ(2) )/r, Concluding the analysis. For solutions to equation (11) with seed ψ (1) Im(λ )], Lemma 7 renders that Di ˚ Φij Re(λ(2) )/r, Im(λ(1) ) = r−2 Sj Re(λ(2) )/r, Im(λ(1) ) , Sj Re(λ(2) )/r, Im(λ(1) ) = O(r3 ) , with Sj [Re(λ(2) )/r, Im(λ(1) )] ∈ C ∞ (S). For the ease of notation, in what follows the affix [Re(λ(2) )/r, Im(λ(1) )] will be dropped. One can write for some arbitrary m∈N m si(k) + siR , si(k) ∈ Pk , Si = k=3

with

siR

m

= o(r ). The latter suggests considering vi =

m

i i v(k) + vR ,

k=s

so that

Lh vi = L0

m

i v(k)

ˆh +L

k=3

m

i v(k)

i . + Lh v R

k=3

Using Lemma 12 in reference [6] one readily finds that m m m i ˆh ˆ h (v i ) = r−2 L v(k) = ui(k) , L (k) k=3

k=3

k=3

with ui(k) = O(rk+3 ), ui(k) ∈ Qk (Ba ). The functions ui(k) can also be written as ui(k) =

m j=k+3

ui(k,j) + ui(k)R ,

ui(k,j) ∈ Pj ,

ui(k,j) = o(rm ) ,

Vol. 10 (2009)


so that

ˆh L

m

i v(k)

= r−2

k=3

m m

ui(k,j) + r−2

k=3 j=k+3

= r−2

j−3 m

m

669

ui(k)R ,

k=3

ui(k,j) + r−2

j=6 k=3

m

ui(k)R .

k=3

i The above analysis suggests calculating the polynomials v(k) recursively if one can find solutions to the equations i = r−2 si(k) , 6 ≤ k ≤ 8 L0 v(k) (56) ⎛ ⎞ k−3 i = r−2 ⎝si(k) − (57) ui(j,k) ⎠ , 9 ≤ k ≤ m . L0 v(k) j=3 i From Lemma 9 one sees that one can only find polynomials v(k) solving (56) and (57) if the right-hand sides are in r−6 Qk,3 . Thus, in general, solutions to ˚ij [Re(λ(2) )/r, Im(λ(1) )] cannot be expected to be smooth equation (11) with seed ψ at i, and its asymptotic expansions may have terms involving the function ln r. The conditions

⎛ ⎝si(k) −

k−3

si(k) ∈ Qk,3 , ⎞ ui(j,k) ⎠ ∈ Qk,3 ,

6 ≤ k ≤ 8, 9 ≤ k ≤ m,

j=6

can, in principle, be reformulated as conditions on the free data hij and Re(λ(2) )/r, Im(λ(1) ), so that they cannot be independent of each other. We shall neither be concerned with a detailed analysis of these conditions nor with an explicit formulation of them as this goes beyond the scope of the present work. Here, it will be assumed that they are satisfied. Nevertheless, it is important to remark that there ˚ij ), with hij smooth for which this is the case: exists a class of free-data, (hij , ψ axial symmetric data – see [5]. The proof that this is the case is done by methods different to the ones discussed here and exploits that in axial symmetry, one has an explicit solution, ψij , to the momentum constraint given in terms of a potential. i Once the polynomials v(k) have been determined, one is left to analyse the i behaviour of the remainder vR . The remainder satisfies the equation m i Lh v R = r−2 ΞiR − ui(k)R . k=8

Using the methods of Theorem 17 in [6] one concludes that the right-hand side of this equation is in C m+s−2,α (Ba ). Standard results on elliptic regularity then i ∈ C m+s,α (Ba ). Since m is arbitrary, one can conclude, by using a further yield vR i ∈ C ∞ (Ba ). argument that vR

670



Summarising, if v i [Re(λ(2) )/r, Im(λ(1) )] is the solution to equation (11) with ˚ij [Re(λ(2) )/r, Im(λ(1) )], then seed given by ψ v i Re(λ(2) )/r, Im(λ(1) ) = v1i Re(λ(2) )/r, Im(λ(1) ) + v2i Re(λ(2) )/r, Im(λ(1) ) , with

v1i Re(λ(2) )/r, Im(λ(1) ) ∈ Q∞ (Ba ) , v1i Re(λ(2) )/r, Im(λ(1) ) ∈ C ∞ (Ba ) .

v1i Re(λ(2) )/r, Im(λ(1) ) = O(r3 ) ,

B.2.7. The condition ψij ψ ij ∈ E ∞ (Ba ). In order to be able to use Theorem 7 with the ψij which has been constructed in the previous paragraphs, one has to verify that ˚ij [A, J, Q] + ψ ˚ij [λ(1) ] + ψ ˚ij [λ(2) ] ψij = ψ + (Lv)ij [A, J, Q] + (Lh v)ij [λ(1) ] + (Lh v)ij [λ(2) ] , satisfies r8 ψij ψ ij ∈ E ∞ (Ba ). A lengthy calculation using the ideas of Section 4.3 in [6] together with Lemma 7 shows that this is the case.

Acknowledgements This research is funded by an EPSRC Advanced Research Fellowship. I thank H. Friedrich and S. Dain for valuable discussions. I thank CM Losert-VK for a careful reading of the manuscript.

References [1] A. Ashtekar, Lectures on non-perturbative canonical gravity, World Scientific, 1991. [2] J. M. Bowen & J. W. York Jr., Time-asymmetric initial data for black holes and black-hole collisions, Phys. Rev. D 21, 2047 (1980). [3] D. Christodoulou & S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, 1993. [4] S. Dain, Initial data for stationary spacetimes near spacelike infinity, Class. Quantum Grav. 18, 4329 (2001). [5] S. Dain, Asymptotically flat and regular Cauchy data, in The conformal structure of space-time. Geometry, analysis, numerics, edited by J. Frauendiener & H. Friedrich, volume 604 of Lect. Notes. Phys., page 161, Springer, 2003. [6] S. Dain & H. Friedrich, Asymptotically flat initial data with prescribed regularity at infinity, Comm. Math. Phys. 222, 569 (2001). [7] J. Frauendiener, Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations. I. The conformal field equations, Phys. Rev. D 58, 064002 (1998). [8] H. Friedrich, On purely radiative space-times, Comm. Math. Phys. 103, 35 (1986). [9] H. Friedrich, On static and radiative space-times, Comm. Math. Phys. 119, 51 (1988).

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[10] H. Friedrich, Einstein equations and conformal structure: Existence of anti-de Sittertype space-times, J. Geom. Phys. 17, 125 (1995). [11] H. Friedrich, Gravitational fields near space-like and null infinity, J. Geom. Phys. 24, 83 (1998). [12] H. Friedrich, Smoothness at null infinity and the structure of initial data, in 50 years of the Cauchy problem in general relativity, edited by P. T. Chru´sciel & H. Friedrich, Birkh¨ auser, 2004. [13] H. Friedrich, Static vacuum solutions from convergent null data expansions at spacelike infinity, Ann. Henri Poincaré 8, 817 (2007). [14] S. Klainerman & F. Nicol` o, Peeling properties of asymptotically flat solutions to the Einstein vacuum equations, Class. Quantum Grav. 20, 3215 (2003). [15] R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett. 10, 66 (1963). [16] R. Penrose & W. Rindler, Spinors and space-time. Volume 2. Spinor and twistor methods in space-time geometry, Cambridge University Press, 1986. [17] P. Sommers, Space spinors, J. Math. Phys. 21, 2567 (1980). [18] J. Stewart, Advanced general relativity, Cambridge University Press, 1991. [19] J. A. Valiente Kroon, Does asymptotic simplicity allow for radiation near spatial infinity?, Comm. Math. Phys. 251 (2004). [20] J. A. Valiente Kroon, A new class of obstructions to the smoothness of null infinity, Comm. Math. Phys. 244, 133 (2004). [21] J. A. Valiente Kroon, Time asymmetric spacetimes near null and spatial infinity. I. Expansions of developments of conformally flat data, Class. Quantum Grav. 23, 5457 (2004). [22] J. A. Valiente Kroon, Time asymmetric spacetimes near null and spatial infinity. II. Expansions of developments of initial data sets with non-smooth conformal metrics, Class. Quantum Grav. 22, 1683 (2005). [23] J. A. Valiente Kroon, Asymptotic properties of the development of conformally flat data near spatial infinity, Class. Quantum Grav. 24, 3037 (2007). Juan Antonio Valiente Kroon School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS United Kingdom e-mail: [email protected] Communicated by Piotr T. Chrusciel. Submitted: May 12, 2009. Accepted: May 13, 2009.

Ann. Henri Poincaré 10 (2009), 673–709 c 2009 Birkh¨ auser Verlag Basel/Switzerland 1424-0637/040673-37, published online June 12, 2009 DOI 10.1007/s00023-009-0420-1


Multilinear Eigenfunction Estimates for the Harmonic Oscillator and the Nonlinear Schr¨ odinger Equation with the Harmonic Potential Takafumi Akahori and Kenichi Ito Abstract. We consider the Cauchy problem for the cubic nonlinear Schr¨ odinger equation with the harmonic potential. We prove global wellposedness below the energy class in energy subcritical cases. The main ingredients for the proof are a multilinear eigenfunction estimate for the harmonic oscillator and the I-method.

1. Introduction We study the defocusing cubic nonlinear Schr¨ odinger equation with the harmonic potential (1.1) i∂t u = (−Δ + |x|2 )u + |u|2 u , n where u = u(x, t) is a complex-valued function on R × R, n ≥ 1. This equation describes a Bose–Einstein condensate, see [4] and references therein. We are interested in global well-posedness of the Cauchy problem for the equation (1.1). The natural space for the initial data is s Σs (Rn ) := f : Rn → C : (1 − Δ + |x|2 ) 2 f ∈ L2 (Rn ) which is equipped with the norm s

f Σs (Rn ) := (1 − Δ + |x|2 ) 2 f L2 (Rn ) . Cauchy problems for Schr¨ odinger equations with the harmonic potential and power-nonlinearities have been studied by many authors [3, 5, 10]. In particular, when n = 1, the Cauchy problem for the equation (1.1) is globally well-posed for the data in L2 (Rn ) thanks to the mass conservation law u(t)L2 = u(0)L2 .

(1.2)

674

T. Akahori and K. Ito


Moreover, when n = 2, 3, by the mass conservation law (1.2) and the energy conservation law 1 1 E u(t) := u(t)2Σ1 + u(t)4L4 = E u(0) , 2 4

(1.3)

the global well-posedness for (1.1) holds for the data in Σ1 (Rn ). Recently, when n = 4, Killip, Visan and Zhang [10] proved the global well-posedness for the radial data in Σ1 (Rn ). However, there has been no result on the global well-posedness below the energy class Σ1 (Rn ) when n ≥ 2. We remark that the case where n = 2 corresponds to the L2 -critical case. Indeed, when n = 2, the equation (1.1) is odinger equation transformed into the L2 -critical nonlinear Schr¨ i∂t v = −Δv + |v|2 v via the transformation i

|x|2

e 1+t2 v(x, t) = u 1 + t2

x √ , arctan t . 1 + t2

The case where n ≤ 3 corresponds to the energy subcritical case, and the case where n = 4 corresponds to the energy critical case (cf. [10]). Our purpose here is to prove global well-posedness below the energy class in energy subcritical cases n = 2, 3. The difficulty is that the energy conservation law (1.3) does not make sense. To overcome this difficulty, we employ the I-method developed by Colliander, Keel, Staffilani, Takaoka and Tao [6, 7]. Although the I-method has wide application [1, 6, 7], we will encounter some difficulties in its application. These come from our little knowledge of multilinear estimates for the eigenfunctions of −Δ + |x|2 (in spite that the eigenfunctions are explicitly given by the Hermite functions). Thus, we need to study the multilinear eigenfunction estimate for the harmonic oscillator. Throughout this paper, we set H = −Δ + |x|2 and denote the distinct eigenvalues λ1 < λ2 < · · · . We know that λk = 2k + n, the multiplicity of λk is k+n of H by 2 n and L (R ) is the direct sum of the eigenspaces. We also use Pk to denote n−1 the projection onto the eigenspace of λk . We use S(Rm ) to denote the Schwartz class on Rm . In particular, S denotes the Schwartz class on R. For A, B ≥ 0 and a ∈ R, we use the notations A B, A ∼ B, A B, a+ and a− (cf. [6]). The notation A B means that there exists some universal constant C > 0 such that A ≤ CB. Then, A ∼ B means that A B A. Moreover, A B means that CA ≤ B for some universal constant C > 1. On the other hand, a+ and a− respectively mean a + ε and a − ε for some universal constant 0 < ε 1. Now, our main result is the following.

Vol. 10 (2009) Nonlinear Schr¨ odinger Equation with Harmonic Potential

675

2(2−s) Theorem 1.1. Let n = 2, 3, θ(s) := 1 + s − n2 and γ(s) := 1 − 2+s−[n−2−s] , where ≥0 n [ · ]≥0 denotes the non-negative part. If s > n+1 satisfies the condition 1 γ(s) , (1.4) (1 − s) 1 + < θ(s) 2

then the Cauchy problem for (1.1) is globally well-posed in Σs (Rn ). Moreover, for 1 ), the solution satisfies a time any T > 0 and any 0 < α < γ(s) − 2(1 − s)(1 + θ(s) growth estimate sup u(t)Σs (Rn ) (1 + T )

0≤t≤T

1−s α

u(0)Σs (Rn ) .

(1.5)

Remark 1. When n = 2, (1.4) is reduced to s3 + 4s2 − 2s − 2 > 0, and when n = 3, 8s3 + 10s2 − 17s + 1 > 0. Thus, we can take s < 1. This paper is organized as follows. In Section 2, we study the multilinear eigenfunction estimate for the harmonic oscillator. In Section 3, we introduce the Bourgain space X s,b and related estimates. In Section 4, we recall the I-method and give an almost conservation law. In Section 5, we discuss the local theory which is necessary for the proof of Theorem 1.1. In Section 6, we give the proof of Theorem 1.1. In Section 7, we prove the almost conservation law. Sections A, B, C are supplements of Section 2.

2. Multilinear eigenfunction estimates In this section, we prove the following theorem which plays an important role to prove Theorem 1.1. Theorem 2.1. Let m ∈ N. If C0 > 0 is sufficiently large, then for any ν > 0 there exists Cν > 0 such that for λkj , j = 1, 2, . . . m, with C0 λkj ≤ λk0

m

−ν

≤ C P u (x)P u (x) · · · P u (x) dx λ uk L2 (2.1) λk0 0 λk1 1 λkm m ν k0

Rn

k=0

for all u0 , u1 , . . . , um ∈ L . 2

This theorem is a harmonic oscillator version of Lemma 2.6 in [2]. For the proof, we need microlocal techniques. The first step for the proof of Theorem 2.1 is a construction of an FIO (Fourier Integral Operator) parametrix to the linear Schr¨ odinger equation i∂t u = (−Δ + |x|2 )u ,

u( · , 0) = u0 .

We have the explicit solution to (2.2) by the Mehler formula: n/2 2 2 cos 2t ei(x/ cos 2t−y)ξ+i(x +ξ )(tan 2t)/2 u0 (y) dydξ , u(x, t) = 2π

(2.2)

(2.3)

676



nonetheless, if we follow the strategy of [2], we will soon find that a non-negligible error term appears and the strategy is not successful. This is because the classical trajectories for the harmonic oscillator, which are reflected in the phase (x/ cos 2t− y)ξ + (x2 + ξ 2 )(tan 2t)/2, are essentially different from those without potentials. Our main idea to control this error term is to construct a parametrix U (t) to (2.2) in the following symmetric form: U (t) = P (t) + Q(t) = pΦt (t, x, x , Dx ) + F −1 ◦ qΨt (t, ξ, ξ , Dξ ) ◦ F ,

(2.4)

where pΦt (t, x, x , Dx ), qΨt (t, ξ, ξ , Dξ ) are FIOs with amplitudes p, q and phase functions Φt (x, y, ξ) = ϕ(x, y, ξ) − t(y 2 + ξ 2 ), Ψt (ξ, η, y) = ψ(ξ, η, y) − t(y 2 + η 2 ), respectively. We assume that, for small |t|, p(t, x, y, ξ) and q(t, ξ, η, y) are supported in conic regions in R3 away from {ξ = 0} and {y = 0}, respectively. Then, the method of [2] applies for P (t), since the classical trajectories for harmonic oscillator resemble those for the free particles in a conic region in the (x, ξ)-phase space away from {ξ = 0}. The non-negligible error mentioned above comes from Q(t), but, if we flip x and ξ in the phase space, then Q(t) is of the same form as P (t), and hence we can deal with Q(t) similarly to P (t). Note that, instead of constructing a parametrix (2.4), we might as well just decompose the Mehler formula (2.3) into two parts as in (2.4), but in this paper we do not use (2.3) since the argument in Appendix B is more general so that (2.2) may include lower order perturbations. Note that, since the multiplicity of λk is k+n n−1 , in order to prove Theorem 2.1, it suffices to consider uj = ekj , where ekj is an eigenfunction associated to λkj . Also note that, if χ1 ∈ S and χ1 (0) = 1, then it is clear that ekj = χ1 (H − λjk )ekj , where χ1 (H − λjk ) is an approximate projection. Using the parametrix U (t), we obtain the following proposition. Proposition 2.2. There exists χ1 ∈ S such that χ1 (0) = 1 and that χ1 (H − λ) = Kλ + F −1 ◦ Lλ ◦ F + Rλ , where, if we let Kλ , Lλ , and Rλ be the integral kernels of Kλ , Lλ , and Rλ , respectively, then √ √ 1 Kλ ( λx, λy) = λ− 2 eλΦ(x,y) aλ (x, y) , √ √ 1 Lλ ( λξ, λη) = λ− 2 eλΨ(ξ,η) bλ (ξ, η) , |(1 + |x|2 + |y|2 ) 2 ∂xα ∂yβ Rλ (x, y)| ≤ CαβkN λ−N k

with |∂xα ∂yβ aλ (x, y)| ≤ Cαβ , |∂ξα ∂ηβ bλ (ξ, η)|

≤ Cαβ ,

supp aλ ⊂ (x, y); |x| < 1 − ε1 , |y| < 1 − ε1 , supp bλ ⊂ (ξ, η); |ξ| < 1 − ε1 , |η| < 1 − ε1

(2.5) (2.6)


677

and |∇x Φ(x, y)| ≥ c > 0 ,

|∂xα Φ(x, y)| ≤ Cα ,

(2.7)

|∇ξ Ψ(ξ, η)| ≥ c > 0 ,

|∂ξα Ψ(ξ, η)|

(2.8)

≤ Cα

for λ ≥ 1, (x, y) and (ξ, η). Proof of Proposition 2.2. The strategy of the proof is similar to that in [12]. Our parametrix consists of two parts, the FIO part and the Fourier-transformed FIO part, and hence the resulting integral operator, which describes the approximate projection, consists of corresponding two parts. For the details, see Section C. Now, we are in a position to prove Theorem 2.1. Proof of Theorem 2.1. For simplicity, we only consider the trilinear case. The other cases are similar to this case. Fix ε ∈ (0, 1). Then, there are essentially three cases:

λk1 , λk2 ≤ λεk0 ,

λk2 ≤ λεk0 ≤ λk1 ≤

1 λk , M 0

λεk0 ≤ λk1 , λk2 ≤

1 λk M 0

for sufficiently large constant M > 0. We only deal with the second intermediate case, since the others are treated as extreme ones. As stated above, we may assume uj = ekj . Then, we have Pλk0 u0 (x)Pλk1 u1 (x)Pλk2 u2 (x) dx Rn = (Kλk0 ek0 )(Kλk1 ek1 )ek2 dx + (Kλk0 ek0 )(F −1 Lλk1 Fek1 )ek2 dx + (Lλk0 e k0 ) (FKλk1 F −1 e k1 ) ∗ e k2 d¯ξ + (Lλk0 e k0 ) (Lλk1 e k1 ) ∗ e k2 d¯ξ + (Kλk0 ek0 )(Rλk1 ek1 )ek2 dx + (F −1 Lλk0 Fek0 )(Rλk1 ek1 )ek2 dx (2.9) + (Rλk0 ek0 )ek1 ek2 dx .

Since λk1 ≥ λεk0 , we clearly have

(Kλ ek )(Rλ ek )ek dx + (F −1 Lλ Fek )(Rλ ek )ek dx

0 1 2 0 1 2 k0 k1 k0 k1

+

(Rλk0 ek0 )ek1 ek2 dx

≤ CN λ−N k0 .

678



On the other hand, the first term in the right-hand side of (2.9) is 1 −1 −1 2 2 n− 1 n−1 λk0 2 λk12 eiλk0 (Φ(x,y0 )+λk0 λk1 Φ(λk0 λk1 x,y1 )) 1

−1

1

1

1

× aλk0 (x, y0 )aλk1 (λk20 λk12 x, y1 )ek0 (λk20 y0 )ek1 (λk21 y1 )ek2 (λk20 x) dy0 dy1 dx . If M is large enough, then (2.7) implies that 1

c

− 12 2

≤ ∂x Φ(x, y0 ) + λ−1 k0 λk1 Φ(λk0 λk1 x, y1 ) ≤ 2C . 2 Thus, each time we integrate by parts in x, the order of the integral with respect

− 1−ε

to λk0 is lowered at least by λk0 2 . Hence the first term in (2.9) is O(λ−∞ k0 ). Next, the second term in (2.9) is 1 −1 −1 2 2 n− 1 n− 1 λk 0 2 λk 1 2 eiλk0 (Φ(x,y0 )+λk0 λk1 xξ1 +λk0 λk1 Ψ(ξ1 ,η1 )) 1

1

1

× aλk0 (x, y0 )bλk1 (ξ1 , η1 )ek0 (λk20 y0 ) ek1 (λk21 η1 )ek2 (λk20 x) dy0 d¯η1 d¯ξ1 dx . By the support property in (2.6) and an argument similar to the above, we conclude that the second term in (2.9) is O(λ−∞ k0 ). The remaining terms in (2.9) are similarly estimated. Thus, we have completed the proof.

3. Bourgain spaces We first introduce the space Σs,p (Rn ) which is defined by s Σs,p (Rn ) := f : Rn → C : (1 − Δ + |x|2 ) 2 f ∈ Lp (Rn ) . When p = 2, we simply write Σs = Σs,2 . The space Σs,p (Rn ) is a Banach space equipped with the norm s

f Σs,p (Rn ) := (1 − Δ + |x|2 ) 2 f Lp (Rn ) . We have an analogue of the Sobolev embedding for the harmonic oscillator. Lemma 3.1 (Lemma 2.8 in [10]). For given 0 ≤ s ≤ 1 and 1 < p ≤ q < ∞, we have f Lq (Rn ) f Σs,p (Rn ) provided s −

n p

= − nq , where the implicit constant depends only on n, s and p.

Now, we define the quantity s

uX s,b (Rn ×R) := ∂t b (1 + H) 2 eitH uL2t L2x (Rn ×R) H s 2 = (1 + λk ) 2 τ + λk b Ft→τ Pk (u) k∈N

L2τ L2x (Rn ×R)

12 .


679

s,b Then, the space XH (Rn × R) is defined as the completion of the Schwartz space s,b , so that we simply by · X s,b . We will often drop the H from the notation XH H

write X s,b . We also need a time localized version of X s,b (Rn × R): Let L ⊂ R be an interval. Then, we define X s,b (Rn × L) := u : Rn × L → C : ∃ u ∈ X s,b (Rn × R) s.t. u |L = u . The norm of this space is given by uX s,b (Rn ×L) := inf uX s,b (Rn ×R) : u ∈ X s,b (Rn × R), u |L = u . Lemma 3.2. For any 2 ≤ r < ∞, we have uLr (R;L2 (Rn )) u

0, r−2 2r

XH

(Rn ×R)

,

where the implicit constant depends only on n and r. Furthermore, for any b > 12 , we have uL∞ (R;L2 (Rn )) uX 0,b (Rn ×R) , H

where the implicit constant depends only on n and b. Proof of Lemma 3.2. We can derive the first inequality by the Gagliardo– Nirenberg inequality and Lemma 3.1. On the other hand, we can derive the second inequality by the embedding H b (R) → L∞ (R). Although we do not have Strichartz estimates globally in time, we have the following estimate. Lemma 3.3. Let b0 > 12 , 0 ≤ b ≤ b0 and 0 < η ≤ 1 (η ≥ have

1 2

if n = 1). Then, we

uLr (R;Lq (Rn )) uX 0,b (Rn ×R) with 2r = 1 − η bb0 and n( 12 − 1q ) = (1 − η) bb0 , where the implicit constant depends only on n, b0 , b and η. Proof of Lemma 3.3. We first consider the case b = b0 . Let ρ be a suitable bumpfunction supported on [0, 2] with j∈Z

ρ(t − j) ≡ 1 .

680



We set ρj (t) := ρ(t − j). Then, the support of ρj is in [j, j + 2]. By the Strichartz estimate (cf. Lemma 2.1 in [10]), we have itτ i(t−j)H −i(t−j)H e e e F ρ u (τ ) dτ ρj uLr (R;Lq ) = t→τ j

≤

R

R

R

Lr (R;Lq )

itH e Ft→τ e−itH ρ(t)u(t + j) (τ ) r dτ L ([0,2];Lq ) Ft→τ e−itH ρ(t)u(t + j) (τ ) 2 dτ L

τ

2b0

R

Ft→τ e−itH ρ(t)u(t + j) (τ )2 2 dτ L

12

= ρj uX 0,b0 (Rn ×R) . Hence, since r ≥ 2, by the above estimate, we have ⎞ r1 ⎞ r1 ⎛ ⎛ r r ρj uLr (R;Lq ) ⎠ ⎝ ρj uX 0,b0 (Rn ×R) ⎠ uLr (R;Lq ) ⎝ j∈Z

⎛ ⎝

j∈Z

⎞ 12 ρj uX 0,b0 (Rn ×R) ⎠ uX 0,b0 (Rn ×R) . 2

j∈Z

Thus, we have proved the lemma in the case b = b0 . Interpolating this and the trivial estimate uL2 (R;L2 ) uX 0,0 (Rn ×R) , we obtain the desired estimate for all 0 ≤ b ≤ b0 . Lemma 3.4. Let s ∈ R. For any 0 ≤ b ≤ b < |L| < 1, we have

and any interval L ⊂ R with

1 2

uX s,b (Rn ×L) |L|b−b uX s,b (Rn ×L) , where the implicit constant depends only on n, b and b .

Proof of Lemma 3.4. See Lemma 2.11 in [13]. We refer the eigenfunction expansion f = sion. We also use the coarse scale expansion PN (f ) := f= N ;dyadic

N ;dyadic

Besides the above decomposition, we use u= N ;dyadic L;dyadic

k

Pk (f ) to the fine scale expan-

Pk (f ) .

(3.1)

k∈N; √ N ≤ λk 12 . Set L = [t0 , t1 ] for t0 , t1 ∈ R with t0 < t1 . Let u be a solution of (1.1) belonging to 2(2−s) and γ2 < s+1+2[n−2−s]≥0 , X s,b (Rn ×L). Then, for any γ1 < 1− 2+s−[n−2−s] ≥0 we have

E(IN u)(t1 ) − E(IN u)(t0 ) N −γ1 + IN u4X 1,b (Rn ×L) + N −γ2 + IN u6X 1,b (Rn ×L) , where the implicit constant depends only on n, s, b, γ1 and γ2 . We will give the proof in Section 7 below. In what follows, for given dyadic numbers N1 , N2 , N3 , N4 , we let 1st, 2nd, 3rd, 4th be a permutation of the indices 1, 2, 3, 4 such that N1st ≥ N2nd ≥ N3rd ≥ N4th . As the Lemma 12.1 in [8], we have an interpolation lemma.

682


Lemma 4.2 (Interpolation lemma). Let b > following estimates:

1 2

> b > 0. Suppose that we have the

3

u1 u2 u3 X 1,−b (Rn ×R)


uj X 1,b (Rn ×R)

j=1

and u1 u2 u3 X s,−b (Rn ×R)

3

uj X s,b (Rn ×R) ,

j=1

where the implicit constants depend only on n, b, b and s. Then, we have s IN (u1 u2 u3 )X 1,−b (Rn ×R)

3

s IN uj X 1,b (Rn ×R) ,

j=1

where the implicit constant depends only on n, b, b and s (independent of N ). Proof of Lemma 4.2. By the triangle inequality, we have s (u1 u2 u3 )X 1,−b (Rn ×R) IN s ≤ IN PN1 (u1 )PN2 (u2 )PN3 (u3 ) N1 ,N2 ,N3 ;dyadic N ,N ,N N 1 2 3 X 1,−b (Rn ×R) s PN1 (u1 )PN2 (u2 )PN3 (u3 ) + IN N1 ,N2 ,N3 ;dyadic N1st N 1,−b n X

(4.2)

.

(R ×R)

We consider the first term on the right hand side of (4.2) whose square is equal to 2 s PN1 (u1 )PN2 (u2 )PN3 (u3 ) IN N1 ,N2 ,N3 ;dyadic N ,N ,N N 1 2 3 X 1,−b (Rn ×R) 1 2 τ + λ −b m( = (1 + λ ) λk )Ft→τ Pk (4.3) k k k∈N N ,N ,N ;dyadic 1 2 3 N ,N ,N N 1 2 3 2 × PN1 (u1 )PN2 (u2 )PN3 (u3 ) . L2τ L2x (Rn ×R)


683

In (4.3), applying Theorem 2.1 to the factor Pk PN1 (u1 )PN2 (u2 )PN3 (u3 ) , we √ see that the most contributory case is λk N . Thus, we may assume that m( λk ) ∼ 1 and hence, by the first assumption of the lemma, (4.3) is estimated by 2 3 PNj (uj ) 1,b n j=1 Nj N ;dyadic X

∼

3

(R ×R)

2 1 (1 + λk ) 2 τ + λk b Ft→τ Pk (uj ) 2

Lτ L2x (Rn ×R)

j=1 k∈N λk N

3 s 2 IN uj

X 1,b (Rn ×R)

j=1

.

On the other hand, we consider the second term on the right hand side of (4.2). We suppose that N1 = N1st . Then, we easily see that 2 P (u ) N1 1 s,b n N1 N ;dyadic X (R ×R) 2 s (4.4) = (1 + λk1 ) 2 τ1 + λk1 b Ft→τ Pk1 (u1 ) L2τ1 L2x (Rn ×R)

k1 ∈N λk1 N

s 2 N −2(1−s) IN u1

X 1,b (Rn ×R)

.

Since s IN (u1 u2 u3 )X 1,−b (Rn ×R) N 1−s u1 u2 u3 X s,−b (Rn ×R) ,

by the second assumption of the lemma and (4.4), we obtain the desired result. Thus, we have completed the proof.

5. Local theory We develop a local theory for the equation (1.1). The argument here is standard. We first prove the following estimate. n Lemma 5.1. Let n−2 2 < s < 2 , 0 < ε < 1 − s and ε1 > 0. Then, we have 2 |u| u s,− 1−ε u s, 1+ε u2 s, 1 − 2+2s−n + ε1 +ε(1+ε1 ) n n 2 2 X

(R ×R)

X

(R ×R)

X

2

4

where the implicit constant depends only on n, s, ε and ε1 .

2(1+ε)

(Rn ×R)

,

684


Proof of Lemma 5.1. Set b = (3.1), we have

1+ε 2 .

|u|2 uX s,b−1 (Rn ×R) =

≤

By the duality and the dyadic decomposition

sup w∈X −s,1−b ; w X −s,1−b =1

sup

w∈X −s,1−b ; N1 ,...,N4 ;dyadic w X −s,1−b =1


R

R

Rn

uuuw

PN1 (u)PN2 (u)PN3 (u)PN4 (w)

.

Rn

(5.1)

In the view of Theorem 2.1, we may assume that N1st ∼ N2nd . We further estimate the right hand side of (5.1) by |PN1 (u)||PN2 (u)||PN3 (u)||PN4 (w)| . (5.2) sup w∈X −s,1−b ; N1 ,...,N4 ;dyadic w X −s,1−b =1 N1st ∼N2nd

R

Rn

Here, by the symmetry, we may assume that N3 ≤ N2 ≤ N1 . Then, for sufficiently older inequality, the summand of (5.2) is estimated by small ε1 > 0, by the H¨ PN1 (u)

4 n−2(s−ε1 )

Lt

× PN3 (u)

PN2 (u)

n s−ε1

Lx

4(1+ε) 2+2(s−ε1 )−n−ε{n−2(s−ε1 )}

Lt

4(1+ε) 2+2(s−ε1 )−n−ε{n−2(s−ε1 )}

Lx

4(1+ε) n−2(s−ε1 )+ε{n−2(s−ε1 )+4}

Lt

(5.3)

2n n−2(s−ε1 )

Lt

× PN4 (w)

2n n−2(s−ε1 )

Lx

.

n s−ε1

Lx

2n

By the embedding Σs−ε1 (Rn ) → L n−2(s−ε1 ) (Rn ), Lemma 3.2 and Lemma 3.3, we estimate (5.3) by PN1 (u)

X 0,

× PN3 (u) Hence, since

N1−s

PN2 (u)

1+ε 2

X

s−ε1 , 1 − 2

X

s−ε1 , 1 − 2

2+2(s−ε1 )−n−ε{n−2(s−ε1 )} 4(1+ε)

2+2(s−ε1 )−n−ε{n−2(s−ε1 )} 4(1+ε)

PN4 (w)

1−ε X 0, 2

.

(5.4)

N4−s , (5.2) is estimated by ⎛ ⎞

sup w∈X −s,1−b ; w X −s,1−b =1

⎜ ⎜ ⎝

+

⎟ ⎟ ⎠

N1 ,...,N4 ;dyadic N1 ,...,N4 ;dyadic N1 ∼N2 N1 ∼N4

× PN1 (u)X s,b × PN3 (u)

X

N2−ε1

s, 1 − 2

PN2 (u)

X

s, 1 − 2

(5.5)

2+2(s−ε1 )−n−ε{n−2(s−ε1 )} 4(1+ε)

2+2(s−ε1 )−n−ε{n−2(s−ε1 )} 4(1+ε)

PN4 (w)X −s,1−b .

We estimate the part of the case N1 ∼ N2 by 2

uX s,b u

X

2

s, 1 − 2

2+2(s−ε1 )−n−ε{n−2(s−ε1 )} 4(1+ε)

= uX s,b u

X

ε +ε(1+ε1 ) s, 1 − 2+2s−n + 1 2 4 2(1+ε)

−ε1 where we have used N2−ε1 ∼ N1st and the boundedness of PNj in X s,b .

,


685

On the other hand, we estimate the part of the case N1 ∼ N4 by sup PN1 (u)X s,b PN4 (w)X −s,1−b w∈X −s,1−b ; N1 ,N4 ;dyadic w X −s,1−b =1 N1 ∼N4

× u

(5.6)

2 X

.

ε +ε(1+ε1 ) s, 1 − 2+2s−n + 1 2 4 2(1+ε)

Since for each N1 the number of dyadic numbers N4 with N1 ∼ N4 is comparable to 1, by the Schwartz inequality, we estimate (5.6) by ⎛ ⎞ 12 ⎛ ⎞ 12 2 2 ⎝ sup PN (u) s,b ⎠ ⎝ PN (w) −s,1−b ⎠ w∈X −s,1−b ; w X −s,1−b =1

X

X

N ;dyadic

N ;dyadic

2

× u

X

ε +ε(1+ε1 ) s, 1 − 2+2s−n + 1 2 4 2(1+ε)

2

∼ uX s,b u

X

ε +ε(1+ε1 ) s, 1 − 2+2s−n + 1 2 4 2(1+ε)

.

Thus, we have completed the proof.

Combining the standard contraction argument in the Bourgain space (cf. [9]) with Lemmas 5.1 and 3.4, we obtain the following local well-posedness. Proposition 5.2 (Local well-posedness). For any n2 > s > n−2 2 , the Cauchy problem for the equation (1.1) is locally well-posed for the data in Σs (Rn ). In particular, there exists b > 12 and interval L such that the solution belongs to X s,b (Rn × L). For the proof of Theorem 1.1, we need an estimate for the regularized solution. Proposition 5.3. Let L = [t0 , t1 ] be an interval with |L| = t1 − t0 < 1 and u be a solution on L with u ∈ X s,b (Rn ×L) for some b > 12 . Then, for any 0 < θ < s− n−2 2 , we have IN uX 1,b (Rn ×L) ≤ C1 IN u(t0 )Σ1 (Rn ) + C2 |L|θ IN u3X 1,b (Rn ×L)

(5.7)

for some constants C1 , C2 > 0 independent of L. Moreover, there exists ε0 > 0 depending only on C1 , C2 and θ such that setting −2

δ = ε0 IN u(t0 )Σ1θ(Rn ) , we have IN uX 1,b (Rn ×[t0 ,t0 +δ]) ≤ 10C1 IN u(t0 )Σ1 (Rn ) .

(5.8)

Proof of Proposition 5.3. Applying Lemma 5.1 and Lemma 4.2 to the integral equation of IN u, we obtain the first claim. Next, we prove the second claim. If (5.8) fails, then since IN uX 1,b (Rn ×[t0 ,t0 +T ]) is continuous in T , there exists 0 < δ0 < δ such that IN uX 1,b (Rn ×[t0 ,t0 +δ0 ]) = 10C1 IN u(t0 )Σ1 (Rn ) .

(5.9)

686



Combining this with (5.7), we have IN uX 1,b (Rn ×[t0 ,t0 +δ0 ]) ≤ C1 IN u(t0 )Σ1 (Rn ) + C2 δ ε IN u3X 1,b (Rn ×[t0 ,t0 +δ0 ]) 3 ≤ C1 IN u(t0 )Σ1 (Rn ) + 1000C13 C2 εε0 IN u(t0 )−2 Σ1 (Rn ) IN u(t0 )Σ1 (Rn )

(5.10)

= C1 IN u(t0 )Σ1 (Rn ) + 1000C13 C2 εε0 IN u(t0 )Σ1 (Rn ) . Then, taking ε0 > 0 so small that 1000C13 C2 εε0 ≤ C1 ,

we see that (5.10) contradicts (5.9) and hence the second claim follows.

6. Proof of main theorem In this section, we prove Theorem 1.1. The main ingredients of the proof are the almost conservation law (Proposition 4.1) and the bootstrap argument below. Proposition 6.1 (Bootstrap principle, cf. Proposition 1.21 in [13]). Let I ⊂ R be an interval, H(t) be a hypothesis at t ∈ I, and C(t) be a conclusion at t. Suppose that we have the followings: . (a) H(t) implies C(t) for each t ∈ I. . (b) If C(t0 ) holds for some t0 ∈ I, then H(t) holds for all t in some neighborhood of t0 . . (c) Let {tn }n∈N ⊂ I with C(tn ) holds and limn→∞ tn = t. C(t) holds. . (d) H(t) holds at least one t ∈ I. Then C(t) holds for all t ∈ I. Now, for any given T > 0, we set a hypothesis of the bootstrap argument s s sup IN u(t)Σ1 (Rn ) ≤ C3 IN u(0)Σ1 (Rn )

(6.1)

t∈[0,T ]

for some C3 > 0 and N > 0 to be chosen, and the conclusion s sup IN u(t)Σ1 (Rn ) ≤ t∈[0,T ]

C3 s I u(0)Σ1 (Rn ) . 2 N

(6.2)

We will prove this conclusion. Since IN u(t)Σ1 is continuous in t, conditions (b) and (c) easily follow. Moreover, by Proposition 5.7, (d) holds for sufficiently large −2

C3 and any T < ε0 IN u(0)Σ1θ with ε0 in Proposition 5.7. Thus, (a) remains. First, we have that there exists C4 > 0 such that s s s u(t)2Σ1 (Rn ) ≤ 2E(IN u)(t) + IN u(t)2L2 (Rn ) IN s s ≤ 2E(IN u)(t) + C4 IN u(0)2Σ1 (Rn ) ,


687

where we have used the relation s u(t)L2 (Rn ) ≤ u(t)L2 (Rn ) = u(0)L2 (Rn ) ≤ u(0)Σs (Rn ) IN s ≤ C4 IN u(0)Σ1 (Rn ) .

Thus, taking C32 ≥ 100C4 , it suffices to show that C32 s I u(0)2Σ1 (Rn ) 100 N for any t ∈ [0, T ]. Now, for ε0 in Proposition 5.7, we set δ0 := ε0 (C3 IN u(0)Σ1 )−2/θ , tj := jδ0 and Lj := [tj−1 , tj ]. Then, t ∈ [0, T ] belongs to one of the intervals L1 , . . . , Lκ , κ := T /δ0 . Suppose that t ∈ Lj0 for some j0 . Then, we have s u)(t) ≤ E(IN

s s s E(IN u)(t) = E(IN u)(t) − E(IN u)(tj0 −1 ) s s + E(IN u)(tj0 −1 ) − E(IN u)(tj0 −2 )

......

(6.3)

s s s + E(IN u)(t1 ) − E(IN u)(0) + E(IN u)(0) .

By Proposition 4.1 (we denote the implicit constant in Proposition 4.1 by C), the s hypothesis (6.1) and EN (u)(0) ≤ C5 IN u(0)2Σ1 (Rn ) , the right hand side of (6.3) is bounded by 4 6 ! −γ1 + −γ2 + j0 C N 2C3 IN u(0)Σ1 (Rn ) + N 2C3 IN u(0)Σ1 (Rn ) s + C5 IN u(0)2Σ1 (Rn ) # T " ≤ C N −γ1 + (2C3 )4 IN u(0)4Σ1 (Rn ) + N −γ2 + (2C3 )6 IN u(0)6Σ1 (Rn ) δ0 s + C5 IN u(0)2Σ1 (Rn ) .

(6.4)

Thus, for C32 ≥ 200C5 , it suffices to show that 2 T s C3 IN u(0)Σ1 (Rn ) θ ε0 # " × C N −γ1 + (2C3 )4 IN u(0)2Σ1 (Rn ) + N −γ2 + (2C3 )6 IN u(0)4Σ1 (Rn ) ≤

(6.5)

C32 . 200

s Since IN u(0)Σ1 (Rn ) ≤ C6 N 1−s u(0)H s (Rn ) , the left hand side of (6.5) is bounded by 2 " 2 T C C3 C6 u(0)Σs (Rn ) θ N θ (1−s)−γ1 +2(1−s)+ (2C3 )4 C62 u(0)2Σs (Rn ) ε0 # 2 +N θ (1−s)−γ2 +4(1−s)+ (2C3 )6 C64 u(0)4Σs (Rn ) . (6.6)

688



If the powers of N are negative in the parentheses, then, taking N so large that 2 2 T N θ (1−s)−γ1 +2(1−s)+ 1 and T N θ (1−s)−γ2 +4(1−s)+ 1, we see that the conclusion (6.2) holds. In order the powers to be negative, it suffices to take s such that 2 (1 − s) − γ1 + 2(1 − s) < 0 , θ which is the condition in Theorem 1.1. Then, by (4.1) and (6.2), we have s s sup u(t)Σs sup IN u(t)Σ1 IN u(0)Σ1

t∈[0,T ]

t∈[0,T ]

1−s

N 1−s u(0)Σs (1 + T ) γ1 −2(1−s)− θ (1−s) u(0)Σs , 2

which gives (1.5). Moreover, by (1.5), we can prove the global well-posedness.

7. Almost conservation law We prove Proposition 4.1. By a standard density argument, we may assume that the functions below are smooth (cf. Section 3 in [6]). First note that we have i∂t v(t) − Hv(t) − |v(t)|2 v(t) ∂t v(t) dx . ∂t E v(t) = − M

In particular, for a solution u to the equation (1.1), we have IN (|u|2 u) − |IN u|2 IN u HIN u dx ∂t E(IN u) = M IN (|u|2 u) − |IN u|2 IN u IN (|u|2 u) dx . + M

Thus, for a solution u to the equation (1.1), we have E(IN u)(t1 ) − E(IN u)(t0 ) = χ[t0 ,t1 ] (t)∂t E(IN u)(t) dt R χ[t0 ,t1 ] IN (|u|2 u) − |IN u|2 IN u HIN u = R Rn + χ[t0 ,t1 ] IN (|u|2 u) − |IN u|2 IN u IN (|u|2 u) R

(7.1)

Rn

= [T erm1 ] + [T erm2 ] , where χ[t0 ,t1 ] is the characteristic function on [t0 , t1 ]. We separately estimate the terms T erm1 and T erm2 in the next subsections.


689

7.1. Estimate of T erm1 We set uN = IN u. Then, applying the decomposition (3.2), we have T erm1 =

R

N1 ,...,N4 ; L1 ,...,L4 ; dyadic dyadic

HPN4 ,L4 (uN ) −

R

Rn

Rn

χ[t0 ,t1 ] IN PN1 ,L1 (u)PN2 ,L2 (u)PN3 ,L3 (u)

! χ[t0 ,t1 ] PN1 ,L1 (uN )PN2 ,L2 (uN )PN3 ,L3 (uN )HPN4 ,L4 (uN ) . (7.2)

We will also use a finer scale expression T erm1 =

M

N1 ,...,N4 ; L1 ,...,L4 ; k√ 1 ,...,k4 ; dyadic dyadic Nj ≤ λk n+1 = 23 and hence the decay factor is N −s+ 2 + . n=2 5 On the other hand, when n = 3, the decay factor is N −3s+ 2 + . Using the estimates (7.5)–(7.8) and summing over the Nj and Lj indices, we obtain the estimate T erm1,Case3 N −s+ 2 +2[n−2−s]≥0 + IN u 1

4 1 X 1, 2 +

.

β Finally, we consider Case 4. We divide the case into subcases N3rd N1st and β 1 N3rd N1st for some β < 1 to be chosen (we will choose β < 2 ). β . As in the Case 3, we use We first consider the case where N3rd N1st the coarse expression (7.2) and consider the first term only. By the symmetry, we may assume that N1 ≤ N2 ≤ N3 . Moreover, in the view of the regularity, we only consider the hardest case N4 ∼ N1st . Under these assumptions, by the 7n H¨ older inequality and the embedding (Lemma 3.1) Σn−2, 8n−14 (Rn ) → L7 (Rn ), we estimate the summand of the first term in (7.2) by 28 28 7 PN1 ,L1 (χ[t0 ,t1 ] u) 28−9n PN2 ,L2 (u) 3n n−2, 7n 8n−14 Lt Lx2 Lt Σx (7.9) 2 28 7 HPN ,L (IN u) 28 7 . × PN3 ,L3 (u) 3n 4 4 2 3n 2 Lt

Lx

Lt

Lx


691

7n 28 28 By Lemma 3.3 (( 8n−14 , 28−9n ) and ( 72 , 3n ) are admissible), (7.9) is estimated by PN1 ,L1 (χ[t0 ,t1 ] u) n−2, 1 + PN2 ,L2 (u) 0, 1 + 2 X X 2 2 × PN3 ,L3 (u) 0, 1 + HPN4 ,L4 (IN u) 0, 1 + X 2 X 2 (7.10) n−2−s −s −s 0+ N1 N2 N3 N4 L1 χ[t0 ,t1 ] uN 1, 21 − uN 3 1, 1 −

X −s+[n−2−s]≥0 −s+1 0+ 4 N3rd N1st L1st uN 1, 1 + . X 2

X

2

n with n ≥ 2, the power of N3rd is negative on the right hand side of Since s > n+1 β , we estimate the above by (7.10). Hence, using the relation N3rd N1st −s+1−β(s−[n−2−s]≥0 )+

N1st

4 L0− 1st uN

1

X 1, 2 +

.

In order that −s + 1 − β(s − [n − 2 − s]≥0 ) < 0, it is required that 1>s>

1 + (n − 2)β . 1 + (n − 1)β

(7.11)

Thus, assuming (7.11), we obtain a decay factor N −s+1−β(s−[n−2−s]≥0 )+ . β Next, we consider the Case 4 with N3rd N1st . In this case, we use the fine scale expression (7.3). We further divide the case into two subcases where N4 ≤ N3rd and N4 ≥ N2nd . Because of the operator H, the former case is clearly better than the latter one. Therefore it suffices to consider the case where N4 ∼ N1st . For the labor-saving, we make some observation. If N1st ∼ N2 , then L1st ∼ |τ1 + λk1 | + · · · + |τ4 + λk4 | ≥ |λk1 − λk2 + λk3 − λk4 | 2 , ≥ |λk4 + λk2 | − |λk3 + λk1 | ∼ N1st

which contradicts to N1st L1st . Thus, by the symmetry of the odd j subindices of Nj , we may assume that N4 ∼ N3 N2 , N1 . Note that M is independent of k1 and k2 in this case. Moreover, we have good localization of the maximal kj index 2 . Indeed, if the latter condition fails, such that |k3 − k4 | L1st or |k3 − k4 | N3rd 2 i.e., |k3 − k4 | N3rd , then we have L1st |τ1 + λk1 | + · · · + |τ4 + λk4 | ≥ |λk1 − λk2 + λk3 − λk4 | = 2|k1 − k2 + k3 − k4 | ≥ |k3 − k4 | − |k1 − k2 | ∼ |k3 − k4 | , √ √ which is the former condition. Moreover, dividing by | λ3 + λ4 | ∼ N1st , we see that |k3 − k4 | L1st implies

L1st

(7.12)

λk3 − λk4 N1st 2 implies and that |k3 − k4 | N3rd

N 2

λk3 − λk4 3rd . N1st

(7.13)

692



Then, we have that

m( λ )

1 k4

− 1 =

m( λk4 ) − m( λk3 )

m( λk3 )

m( λk3 )

N31−s 1

1−s λk3 − θ( λk3 − λk4 ) dθ λk3 − λk4 .

m N 0 L1st Hence, when |k3 − k4 | L1st , from | λk3 − λk4 | N N ∼ λk3 we 1st 1st have

m( λ )

L1st k4

sup − 1 2 (7.14)

m( λk3 )

N1st k3 ,k4 |k3 −k4 |L1st

2 and when |k3 − k4 | N3rd , from | λk3 − λk4 | N3rd N1st ∼ λk3 ,

sup k3 ,k4 2 |k3 −k4 |N3rd

m( λ )

N k4

3rd − 1 .

m( λk3 )

N1st

(7.15)

2 We first consider the case where |k3 − k4 | N3rd :

R

Rn

PN1 ,L1 (χ[t0 ,t1 ]uN )PN2 ,L2 (uN ) m( λk4 ) × − 1 Pk3 ,L3 (uN )HPk4 ,L4 (uN ) . m( λk3 ) k ,k ; 3 4 √ 2 Nj ≤

(7.16)

λkj 0

⎛

⎜ ×⎜ ⎝

2 Pk,L3 (uN )

1 X 0, 2 +

√ k∈N; λk ∼N1st

3

⎞ 12 ⎛

−2 −1 2 L1st N1st N3 N4 uN 4

⎟ ⎜ ⎟ ⎜ ⎠ ⎝

2 HPk,L4 (uN )

⎞ 12 ⎟ ⎟

(7.21)

X 0, 2 + ⎠

√ k∈N; λk ∼N1st

1

+

1

X 1, 2 +

3

−2 2 ∼ L1st N1st uN 4 +

1

X 1, 2 +

.

Since L1st N1st , we obtain a decay factor N − 2 + . Thus, we have estimated T erm1 by 1

max{N −s+ 2 +2[n−2−s]≥0 + , N −s+1−β(s−[n−2−s]≥0 )+ , 1

N −1+2β+ , N − 2 + }IN u4 1

1

X 1, 2 +

.

(7.22)

We choose β such that the second and third factors in the braces in (7.22) coincide, i.e., β=

2−s , 2 + s − [n − 2 − s]≥0

n which is less than 12 if s > n+1 . Also, if s > is arbitrary, we have proved that

T erm1 N

2−s −1+ 2+s−[n−2−s]

≥0

n n+1 ,

+

(7.23)

(7.11) is fulfilled. Thus, since u

IN u4X s,b (Rn ×L) .


695

7.2. Estimate of T erm2 We consider the T erm2 . T erm2 := χ[t0 ,t1 ] IN (|u|2 u) − |IN u|2 IN u IN (|u|2 u) dxdt R Rn = χ[t0 ,t1 ] IN PN1 ,L1 (u)PN2 ,L2 (u)PN3 ,L3 (u) PN4 ,L4 (UN ) R

N1 ,...,N4 ; L1 ,...,L4 ; dyadic dyadic

Rn

!

−

R

Rn

χ[t0 ,t1 ] PN1 ,L1 (uN )PN2 ,L2 (uN )PN3 ,L3 (uN )PN4 ,L4 (UN )

(7.24)

where uN := IN u and UN := IN (|u|2 u). We may ignore the sum with N1st N and N2nd N1st (cf. Case 1 and Case 2 above). Since, by Lemma 5.1 and Lemma 4.2, we have UN

1

X 1,− 2 +

= IN (|u|2 u)

1

X 1,− 2 +

IN u3

1

X 1, 2 +

,

(7.25)

we can apply the same estimate as the Case 3 above and obtain the decay factor N −s−1+2[n−2−s]≥0 . Indeed, because of the absence of the operator H which corresponds to two times derivative, T erm2 is better than T erm1 in the regularity. 1 − However, in order to employ the estimate (7.25), we make a loss of L42 , which corresponds to the loss of one half derivative in the Case 3 above. Thus, we have obtained the estimate T erm2 N −s−1+2[n−2−s]≥0 IN u6X s,b (Rn ×L) .

Appendix A. FIOs and ΨDOs We set x; y; ξ = (1+x2 +y 2 +ξ 2 ) 2 and use S(x; y; ξm , x; y; ξ−2 (dx2 +dy 2 +dξ 2 )) to denote the set of functions p ∈ C ∞ (Rn × Rn × Rn ) satisfying, for multi-indices α, β, γ, 1

|∂xα ∂yβ ∂ξγ p(x, y, ξ)| ≤ Cαβγ x; y; ξm−|α|−|β|−|γ| . Then we introduce the set of amplitudes " # / supp p , Γm (R3n ) = p ∈ S x; y; ξm , x; y; ξ−2 (dx2 + dy 2 + dξ 2 ) : 0 ∈ % & Γ∞ (R3n ) = Γm (R3n ) , Γ−∞ (R3n ) = Γm (R3n ) ⊂ S(R3n ) . m∈R

m∈R ∞

Phase function is a function ϕ ∈ C (R

3n

\ {0}, R) satisfying

2

ϕ(λx, λy, λξ) = λ ϕ(x, y, ξ) for (x, y, ξ) ∈ R3n \ {0}, λ > 0, and ∇y ϕ(x, y, ξ), ∇ξ ϕ(x, y, ξ) = 0

696



for (x, y, ξ) ∈ R2n × (Rn \ {0}). For any p ∈ Γm (R3n ) and any phase function ϕ we define the FIO pϕ (x, x , Dx ) by pϕ (x, x , Dx )u(x) = eiϕ(x,y,ξ) p(x, y, ξ)u(y) dyd¯ξ , u ∈ S(Rn ) , where d¯ξ = (2π)−n dξ and x is a dummy variable. We denote the set of such FIOs by Opϕ Γm (R3n ). In particular, when ϕ(x, y, ξ) = (x−y)ξ, we call it ΨDO (PseudoDifferential Operator) and simply write p(x, x , Dx ) = pϕ (x, x , Dx ), Op Γm (R3n ) = Opϕ Γm (R3n ). The operators Op Γ−∞ (R3n ) and Opϕ Γ−∞ (R3n ) coincide with the operators with kernels in the Schwartz class. We first prove the invariance of Op Γm (R3n ) under the Fourier transform. We set % S x; y; ξm x − ym , x; y; ξ−2 x − y2 (dx2 + dy 2 + dξ 2 ) Πm (R3n ) = m

and

Γm (R2n ) = S x; ξm , x; ξ−2 (dx2 + dξ 2 ) , 1

where x; ξ = (1 + |x|2 + |ξ|2 ) 2 . For amplitudes in Πm (R3n ) we can define the FIOs Opϕ Πm (R3n ) in a usual manner. For any p ∈ Γm (R2n ) we have x + y , ξ ∈ Πm (R3n ) , p˜(x, y, ξ) = p 2 and thus, in particular, we can define the Weyl quantization x + y , ξ u(y) dyd¯ξ . pw (x, Dx )u(x) = p˜(x, x , Dx )u(x) = ei(x−y)ξ p 2 It is clear that Γm (R3n ) ⊂ Πm (R3n ) .

(A.1)

We see, by direct computation, that F ◦ pw (x, Dx ) ◦ F −1 = pw (−Dξ , ξ) .

(A.2)

Proposition A.1. For any p ∈ Πm (R3n ) there exists the unique q ∈ Γm (R2n ) such that p(x, x , Dx ) = q w (x, Dx ) , that is, Op Πm (R3n ) = Opw Γm (R2n ) . Proof of Proposition A.1. See, e.g., [11]. Proposition A.2. For any p ∈ Γm (R2n ) there exists q ∈ Γm (R3n ) such that pw (x, Dx ) = q(x, x , Dx ) ,

(A.3)


697

that is, Opw Γm (R2n ) ⊂ Op Γm (R3n ). Combining it with (A.1) and (A.3), one obtains Op Γm (R3n ) ⊂ Op Πm (R3n ) = Opw Γm (R2n ) ⊂ Op Γm (R3n ) . In particular, from (A.2), it follows that F ◦ Op Γm (R3n ) ◦ F −1 = Op Γm (R3n ) . Proof of Proposition A.2. Let p ∈ Γm (R2n ). Take cutoff functions χ ∈ C0∞ (Rn ) and χ1 ∈ C0∞ (R3n ) with χ = 1 near 0 and χ1 = 1 near 0, and decompose x + y x−y x+y ,ξ = χ 1 − χ1 (x, y, ξ) p ,ξ p 2 ε|(x, y, ξ)| 2 x−y x+y 1 − χ1 (x, y, ξ) p ,ξ + 1−χ ε|(x, y, ξ)| 2 x+y + χ1 (x, y, ξ)p ,ξ . 2 Take ε > 0 small, then the first and the third terms belong to Γm (R3n ) and Γ−∞ (R3n ), respectively. For the second term, we can easily see that, by partial integrations, the operator kernel x+y x−y 1 − χ1 (x, y, ξ) p , ξ d¯ξ ei(x−y)ξ 1 − χ ε|(x, y, ξ)| 2 belongs to S(R2n ).

Next we prove that, if the phase function of an FIO has the leading term (x − y)ξ near x = y, then the FIO is actually a ΨDO: Proposition A.3. Let ϕ be a phase function and suppose that ∇x ϕ(x, x, ξ) = ξ ,

ϕ(x, x, ξ) = 0 ,

|∇ξ ϕ(x, y, ξ)| ≥ c|x − y| .

Then we have the identity Opϕ Γm (R3n ) = Op Γm (R3n ) . Here the classical amplitudes and the classical symbols correspond. Moreover, if we put q(x, ξ) = p(x, x, ξ) ∈ Γm (R2n ) for any p ∈ Γm (R3n ), then pϕ (x, x , Dx ) − q(x, Dx ) ∈ Op Γm−2 (R3n ) .

(A.4)

Proof of Proposition A.3. Let p ∈ Γm (R3n ). We may assume p = 0 near (x, y, ξ) = 0. Put ϕθ (x, y, ξ) = θ(x − y)ξ + (1 − θ)ϕ(x, y, ξ) ,

θ ∈ [0, 1]

698



and take χ ∈ C0∞ (Rn ) vanishing near 0. If θ = 0 or 1, for which ϕθ is certainly a phase function, then we have x−y pϕθ (x, x , Dx )u(x) = eiϕθ (x,y,ξ) χ p(x, y, ξ)u(y) dyd¯ξ ε|(x, y, ξ)| x−y p(x, y, ξ)u(y) dyd¯ξ . + eiϕθ (x,y,ξ) 1 − χ ε|(x, y, ξ)| By the partial integrations, the second term in the right-hand side is rewritten by ' (N ∇ξ ϕθ iϕθ (1 − χ)p u dyd ¯ξ = K(x, y)u(y)dy e i∇ξ · |∇ξ ϕθ |2 x−y for some K ∈ S. Since χ( ε|(x,y,ξ)| )p(x, y, ξ) ∈ Γm (R3n ), we may assume supp p ⊂ {|x − y| ≤ ε|(x, y, ξ)|} for arbitrary small ε > 0. Then, in particular, since

(∇y ϕθ )(x, x, ξ) = −ξ = 0 for ξ = 0 , and ∇y ϕθ is homogeneous of degree 1, ϕθ may be supposed to be a phase function at least on supp p uniformly in θ ∈ [0, 1]. Thus pϕθ (x, x , Dx ) is well-defined for any θ ∈ [0, 1], and is C ∞ with respect to θ as an operator-valued function. We apply the Taylor expansion around θ = 1 to pϕθ (x, x , Dx )u(x). By direct differentiation, ∂θl pϕθ (x, x , Dx )u(x) ⎤l ⎡ ϕjk (x, y, ξ)(xj − y j )(xk − y k )⎦ eiϕθ (x,y,ξ) p(x, y, ξ)u(y) dyd¯ξ , (A.5) = ⎣−i j,k

where ϕ(x, y, ξ) = (x − y)ξ + ϕjk (x, y, ξ) =

ϕjk (x, y, ξ)(xj − y j )(xk − y k ) ,

j,k 1

(1 − t)(∂xj ∂xk ϕ) y + t(x − y), y, ξ dt .

0

Note that ∂ξm ϕθ (x, y, ξ) =

⎡ ⎣δkm + (1 − θ)

⎤ ∂ξm ϕjk (x, y, ξ)(xj − y j )⎦ (xk − y k ) ,

j

k

and that, if ε is small, then (1 − θ)∂ξm ϕjk (x, y, ξ)(xj − y j ) is small on supp p by the homogeneity. Hence the above relation is invertible on supp p: there is ajk θ (x, y, ξ) homogeneous of degree 0 such that jk aθ (x, y, ξ)∂ξk ϕθ (x, y, ξ) . xj − y j = k


699

Insert it to one of x − y in (A.5) and integrate it by parts, then we have the order of ∂θl pϕθ (x, x , Dx ) lowered by 1. Repeating this argument, we consequently obtain ∂θl pϕθ (x, x , Dx ) ∈ Opϕθ Γm−2l (R3n ) uniformly in θ ∈ [0, 1]. Put

ql (x, x , Dx ) = ∂θl pϕθ (x, x , Dx ) θ=1 ∈ Op Γm−2l (R3n ) , then by the Taylor expansion we have N (−1)l

pϕ (x, x , Dx ) =

l!

l=0

ql (x, x , Dx ) +

(−1)N +1 N!

0

1

θN ∂θN +1 pϕθ (x, x , Dx ) dθ .

Thus, if q(x, y, ξ) ∼

∞ (−1)l l=0

l!

ql (x, y, ξ) ,

then q ∈ Γm (R3n ) and pϕ (x, x , Dx ) ≡ q(x, x , Dx ) mod Op Γ−∞ (R3n ) . We have verified Opϕ Γm (R3n ) ⊂ Op Γm (R3n ) . The converse inclusion follows similarly. The other assertions of the proposition are clear from the above argument. The following proposition is not necessarily needed in this paper, since the operator q(x, Dx ) in the following proposition is of simple form in this paper and the composition with an FIO is computed directly. However, for perturbed Hermite operators this formula would be needed. We just put it here without a proof. Proposition A.4. Let p ∈ Γm (R3n ), q ∈ Γl (R2n ), and ϕ be a phase function satisfying ∂x ϕ(x, y, ξ) = 0

for all

(x, y, ξ) = 0 .

Then

e−iϕ(x,y,ξ) q(x, Dx ) eiϕ(x,y,ξ) p(x, y, ξ) 1

q (α) x, ∂x ϕ(x, y, ξ) Dzα p(z, y, ξ)eiρ(z,x,y,ξ)

+ RN (x, y, ξ) , = α! z=x |α| 0. Proposition B.1. The Hamilton–Jacobi equation (B.2) with initial conditions (B.3) and (B.4) has the solution ϕ(x, y, ξ) defined at least on a conic region D=

(x, y, ξ) ∈ R

3n

|x − y| R

Our goal is now to show that if R > 0 is sufficiently large, then the maximum in (5.27) is always given by the sup|y|≤R -term for all t ≥ T0 . The claimed uniform exponential decay for Q(t) then follows easily. We now show that there is indeed such an R > 0. To this end, we first claim that we can take R > 0 sufficiently large such that h(t) (x)
R

and

t ≥ T0 .

(5.28)

Indeed, since |φ(x)| 1, we deduce |V (t) (x)| |x|−2 , by using Newton’s theorem and the fact that Q(t) are radial functions with Q(t) L2x 1. Furthermore, we have |V (t) (x)| 1 by Hardy’s inequality and Q(t) H˙ 1 1. Therefore, we obtain x the uniform pointwise bound |V (t) (x)|

1 , 1 + |x|2

for t ≥ T0 .

(5.29)

Vol. 10 (2009)

On Stability of Pseudo-Conformal Blowup for Hartree NLS

Plugging this estimate into (5.24), we deduce the uniform bound 1 dy, for t ≥ T0 , h(t) (x) eδ|x−y| |G(x − y)| 1 + |y|2

1201

(5.30)

R4

whence (5.28) follows by taking R > 0 sufficiently large. Furthermore, we note that Eq. (5.1) combined with a bootstrap arguments shows that Q(t) Hx1 1 yields Q(t) Hxs C(s) for any s ≥ 1 and some constants C(s). In particular, by Sobolev’s embedding, we conclude the uniform bound 1,

Q(t) L∞ x

for t ≥ T0 .

(5.31)

With the help of the uniform bound (5.28) and (5.31), we now deduce the claimed uniform exponential along the lines of the Slaggie–Wichmann argument. First, we observe that e−δ|x−y| = sup e−δ|x−y| e−δ|y−z| ,

(5.32)

z∈R4

which directly follows from the triangle inequality and the definition of the supremum. Thus, we have m(t) (x) = sup m(t) (z)e−δ|x−z| .

(5.33)

z∈R4

Now assume that R > 0 such that (5.28) holds. Then, by (5.26), we have |Q(t) (x)| < m(t) (x) whenever |x| > R. This fact and (5.32) imply sup |Q(t) (y)|e−δ|x−y| < sup m(t) (y)e−δ|x−y|

|y|>R

|y|>R

≤ sup m(t) (y)e−δ|x−y| = m(t) (x). y∈R4

Hence the sup|y|>R -term in (5.27) is strictly less than m(t) (x); and therefore m(t) (x) = sup |Q(t) (y)|e−δ|x−y| .

(5.34)

|y|≤R

Using the uniform bound (5.31) and sup|y|≤R eδ|y| 1, we deduce that m(t) (x) e−δ|x| ,

for t ≥ T0 .

(5.35)

Going back to (5.26) and noting that (5.30) also shows that h L∞ 1 for x t ≥ T0 , we complete the proof of Lemma 5.1. (t)

Finally, we are in the position to derive the following improved decay estimate for ∂t Q(t) for t 1. Lemma 5.2. For {Q(t) }t∈[T0 ,∞) as in Theorem 5 and T0 > 0 sufficiently large, we have

∂t Q(t) Hx1 t−k−1 ,

for t ≥ T0 .

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Proof. Note that Q(t) satisfies for all t ∈ [T0 , ∞),

G(Q(t) , t) = 0, where G :

Hr1 (R4 )

× [T0 , ∞) →

Hr1 (R4 )

G(u, t) = u + (−∆ + 1)−1 g(u, t),

is given by

with

g(u, t) = −

(5.36)

φ(t−k | · |k ) 2 ∗ |u| u. | · |2 (5.37)

Differentiating equation (5.36) with respect to t yields

−1 ∂t Q(t) = − ∂u G(Q(t) , t) ∂t G(Q(t) , t)

(5.38)

for all t sufficiently large, while using the fact that ∂u G(Q(t) , t) is invertible for (Q(t) , t) close (Q(∞) , ∞). Furthermore, by continuity, we have that

∂u G(Q(t) , t)−1 H 1 →H 1 1, for t sufficiently large. Using that ∂t G(Q(t) , t) = (−∆ + 1)−1 ∂t g(Q(t) , t), we therefore get

∂t Q(t) H 1 (−∆ + 1)−1 ∂t g(Q(t) , t) H 1 ∂t g(Q(t) , t) L2 Next, we note

⎛

(∂t g(Q(t) , t))(x) = −kt−k−1 ⎝

Rd

⎞ φ (t−k |x − y|k ) |x − y|k |Q(t) (y)|2 dy ⎠ Q(t) (x). |x − y|2 (5.39)

Let now m = k − 2. If −2 ≤ m ≤ 0 then, by Young’s and Hardy’s inequality and using that |φ (r)| 1, we obtain

−k k

φ (t | · | ) (t) 2

∗ |Q | 1. (5.40)

| · |−m L∞ Hence we conclude that

∂t gt (Q(t) , t) L2 t−k−1 Q(t) L2 t−k−1 .

(5.41)

whenever m = k − 2 ∈ (−2, 0], i.e., for k ∈ (0, 2]. It remains to prove such a bound when k > 2. To this end, we use the uniform exponential decay stated in Lemma 1. For m = k − 2 > 0, we recall the elementary inequality |x − y|m ≤ max{1, 2m }(|x|m + |y|m ). Next, by using Lemma 1, we deduce the pointwise bound ⎛ ⎞ |∂t g(Q(t) , t)|(x) t−k−1 ⎝ (|x|m + |y|m )e−2δ|y| dy ⎠ e−δ|x| t

−k−1

Rd m

(|x| + 1)e−δ|x| ,

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for some δ > 0. This shows that ∂t g(Q(t) , t) L2 t−k−1 for t large and if k > 2. This completes the proof of Lemma 5.2.

Acknowledgements J. K. is partially supported by National Science Foundation Grant DMS-0757278 and a Sloan Foundation Fellowship. E. L. gratefully acknowledges partial support by the National Science Foundation Grant DMS-0702492. P.R. was supported by the Agence Nationale de la Recherche, ANR Projet Blanc OndeNonLin and ANR jeune chercheur SWAP.

References [1] Burq, N., Gérard, P., Tzvetkov, N.: Two singular dynamics of the nonlinear Schr¨ odinger equation on a plane domain. Geom. Funct. Anal. 13(1), 1–19 (2003). MR MR1978490 (2004h:35206) [2] Bourgain, J., Wang, W.: Construction of blowup solutions for the nonlinear Schr¨ odinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 197–215 (1997/1998) [3] Cazenave, T.: Semilinear Schr¨ odinger equations. Courant Lecture Notes in Mathematics, vol. 10. New York University Courant Institute of Mathematical Sciences, New York (2003). MR MR2002047 (2004j:35266) [4] Cˆ ote, R.: Construction of solutions to the L2 -critical KdV equation with a given asymptotic behaviour. Duke Math. J. 138(3), 487–531 (2007). MR MR2322685 (2008d:35192) [5] Carmona, R., Simon, B.: Pointwise bounds on eigenfunctions and wave packets in N -body quantum systems. V. Lower bounds and path integrals. Commun. Math. Phys. 80(1), 59–98 (1981) [6] Erdo˘ gan, M.B., Schlag, W.: Dispersive estimates for Schr¨ odinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. II. J. Anal. Math. 99, 199–248 (2006) [7] Elgart, A., Schlein, B.: Mean field dynamics of boson stars. Comm. Pure Appl. Math. 60(4), 500–545 (2007). MR MR2290709 [8] Fr¨ ohlich, J., Jonsson, B.L.G., Lenzmann, E.: Effective dynamics for boson stars. Nonlinearity 20(5), 1031–1075 (2007) [9] Fr¨ ohlich, J., Lenzmann, E.: Mean-field limit of quantum Bose gases and nonlinear ´ Hartree equation, Séminaire: Equations aux Dérivées Partielles. 2003–2004, Sémin. ´ ´ Equ. Dériv. Partielles, Ecole Polytech., Palaiseau, 2004, pp. Exp. No. XIX, 26. MR MR2117050 (2005m:81079) [10] Fr¨ ohlich, J., Lenzmann, E.: Blowup for nonlinear wave equations describing boson stars. Comm. Pure Appl. Math. 60(11), 1691–1705 (2007). MR MR2349352 [11] Fr¨ ohlich, J., Tsai, T.-P., Yau, H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Commun. Math. Phys. 225(2), 223–274 (2002). MR MR1889225 (2003e:81047)

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[12] Ginibre, J., Velo, G.: On a class of nonlinear Schr¨ odinger equations with nonlocal interaction. Math. Z. 170, (2), 109–136 (1980). MR MR562582 (82c:35018) [13] Hislop, P.D.: Exponential decay of two-body eigenfunctions: a review. In: Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999) (San Marcos, TX), Electron. J. Differ. Equ. Conf., vol. 4. Southwest Texas State University, pp. 265–288 (2000) (electronic) [14] Hundertmark, D., Lee, Y.-R.: Exponential decay of eigenfunctions and generalized eigenfunctions of a non-self-adjoint matrix Schr¨ odinger operator related to NLS. Bull. Lond. Math. Soc. 39(5), 709–720 (2007). MR MR2365218 [15] Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical focusing nonlinear Schr¨ odinger equations in one dimension. J. Am. Math. Soc. 19(4), 815–920 (2006) (electronic). MR MR2219305 (2007b:35301) [16] Lenzmann, E.: Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE 2(1), 1–27 (2009) [17] Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/77) [18] Lemou, M., Méhats, F., Raphaël, P.: Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov–Poisson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(5), 825–833 (2007). MR MR2348054 (2008f:35040) [19] Martel, Y.: Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Am. J. Math. 127(5), 1103–1140 (2005). MR MR2170139 (2007a:35128) [20] Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schr¨ odinger equations with critical power. Duke Math. J. 69(2), 427–454 (1993) [21] Merle, F.: Nonexistence of minimal blow-up solutions of equations iut = −∆u − k(x)|u|4/N u in RN . Ann. Inst. H. Poincaré Phys. Théor. 64(1), 33–85 (1996). MR MR1378233 (97g:35073) [22] Martel, Y., Merle, F.: Nonexistence of blow-up solution with minimal L2 -mass for the critical gKdV equation. Duke Math. J. 115(2), 385–408 (2002). MR MR1944576 (2003j:35281) [23] Weinstein, M.I.: Modulational stability of ground states of nonlinear Schr¨ odinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985) [24] Weinstein, M.I.: Nonlinear Schr¨ odinger equations and sharp interpolation estimates. Commun. Math. Phys. 87(4), 567–576 (1982/83). MR MR691044 (84d:35140)

Joachim Krieger 4N67 Rittenhouse Lab Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104, USA e-mail: [email protected]

Vol. 10 (2009)


Enno Lenzmann Department of Mathematics Massachusetts Institute of Technology Building 2, Office 378 77 Massachusetts Avenue Cambridge, MA 02139, USA e-mail: [email protected] Pierre Raphaël Institut de Mathématiques Université Paul Sabatier Toulouse 31062 Toulouse Cedex 9, France e-mail: [email protected]; [email protected] Communicated by Rafael D. Benguria. Received: July 16, 2008. Accepted: February 2, 2009.

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Ann. Henri Poincaré 10 (2009), 1207–1222 c 2009 Birkh¨ auser Verlag AG, Basel/Switzerland 1424-0637/09/061207-16, published online August 15, 2009 DOI 10.1007/s00023-009-0009-8


The Ground State of Relativistic Ions in the Limit of High Magnetic Fields Doris H. Jakubassa-Amundsen Abstract. We consider the pseudorelativistic no-pair Brown–Ravenhall operator for the description of relativistic one-electron ions in a homogeneous magnetic field B. It is shown for central charge Z ≤ 87 that their ground √ state energy decreases according to B as B → ∞, in contrast to the nonrelativistic behaviour.

1. Introduction A relativistic atomic electron of mass m in a magnetic field B = ∇ × A resulting from a vector potential A is described by the Dirac operator H [18], H = DA + V,

DA = αpA + βm,

(1.1)

− γx

is the Coulomb field where pA = p − eA, α, β are Dirac matrices and V = generated by a point nucleus of charge Z fixed at the origin. The coordinate and momentum of the electron are denoted, respectively, by x and p (with x = |x| = 2 2 x1 + x2 + x23 ), and the field strength is γ = Ze2 . Relativistic units ( = c = 1) are used, with e2 ≈ 1/137.04 being the fine structure constant. A widely used pseudorelativistic operator which accounts for the spin degrees of freedom is the Brown–Ravenhall operator hBR . It can be obtained from a projection of H onto the positive spectral subspace of the electron at V = 0 [3] (see also [7] for its mathematical analysis). The Brown–Ravenhall operator in a magnetic field [4,10,14] is given by hBR = EA + V1 + V2 V1 = −γ AE

1 AE , x

V2 = −γ AE

σpA 1 σpA AE , EA + m x EA + m

(1.2)

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D. H. Jakubassa-Amundsen

where EA = |DA | is the kinetic energy operator, EA = (σpA )2 + m2 = p2A − eσB + m2 ≥ m,


AE =

EA + m . 2EA

(1.3)

Here, σ = (σ1 , σ2 , σ3 ) is the vector of Pauli spin matrices. hBR acts in the Hilbert space L2 (R3 ) ⊗ C2 and its form domain is H1/2 (R3 ) ⊗ C2 , where H1/2 denotes a Sobolev space. For B bounded or B ∈ L2 (R3 ), it has been shown [10,12] that the potential V1 + V2 is relatively form bounded with respect to EA , γ |(ψ, (V1 + V2 )ψ)| ≤ (ψ, EA ψ) + γ cB ψ2 (1.4) γc for ψ ∈ H1/2 (R3 ) ⊗ C2 , where γc = π2 , cB = π2 eB∞ , respectively, γc = 2 2 π − , cB = c() B2 with > 0 arbitrarily small and some constant c() for B bounded, respectively, B ∈ L2 (R3 ). For subcritical potential strength, γ < γc , the form bound is smaller than one such that hBR is bounded from below, allowing for its extension to a self-adjoint operator. Most recently [15], it was shown that for locally bounded A the critical potential strength for the semiboundedness of hBR could be increased to γ˜c = 2/( π2 + π2 ). However, no explicit B-dependence of the lower bound of hBR was provided. In the following, we take B = Be3 to be a constant magnetic field along the e3 -axis, generated by A(x) =

B (−x2 , x1 , 0) 2

(1.5)

which obeys ∇ · A = 0. A matter of interest is the behaviour of the ground state energy when B → ∞, which until now was only rigorously studied for systems with small nuclear charge. In this nonrelativistic limit the Dirac operator H turns into the Pauli operator HP = (σpA )2 + V. For this operator, using a scaling property by which the pair (B, γx ) changes into (B0 , λx ), where B0 is a fixed field and λ → 0 as B → ∞, the ground-state asymptotics could be investigated by means of a perturbative treatment of the electric potential [2]. It was derived that, as conjectured earlier [17], the ground-state energy decreases according to (ln B)2 as B → ∞, the error being of the order of ln B · ln(ln B). Recently, the Dirac operator itself was studied for finite magnetic fields and small nuclear charge (including the simultaneous limits B → ∞ and Z → 0) in the context of the lowest bound state’s diving into the negative continuum for sufficiently large B [5]. A similar scaling as in [2] was introduced, and an upper and lower bound for the ground-state energy was provided 1 which behave like B 2 . As concerns the Brown–Ravenhall operator hBR , its ground state was estimated with the help of a variational wavefunction, similar to the one used in the Schr¨ odinger case [16], with the result that the variationally determined 1 ground-state energy decreases asymptotically like B 2 [12]. Recalling that the variational energy is an upper bound while the form boundedness (1.4) provides a

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lower bound, the ground-state energy of hBR is thus, like that of the Dirac oper1 ator, sandwiched between bounds which decrease like B 2 for B → ∞. The aim of the present work is to prove a stronger statement. Theorem 1. Let hBR = EA + V1 + V2 be the Brown–Ravenhall operator for an electron in a high central Coulomb field of strength 0.1 γ < π2 (Z ≤ 87) and in a homogeneous magnetic field B. Then the ground-state energy behaves like √ (1.6) Eg ∼ −c B (B → ∞) where c > 0 is some constant.

2. Scaling Property Let us apply the scaling introduced by [2] to the Brown–Ravenhall operator, and set B =: µ0 B0 with µ0 > 0 and B0 some fixed constant field. Then µ0 → ∞ as 1 B → ∞. Define x ˜k := µ02 xk (k = 1,2,3) such that 1

1

pk = −i∂xk = −iµ02 ∂x˜k = µ02 p˜k

(2.1)

1 µ0 B0 −1 −1 (−µ0 2 x ˜2 , µ0 2 x ˜1 , 0) =: µ02 A0 (˜ x). A(x) = 2 √ Then with m/ µ0 =: m, ˜ 1

1

˜ BR Ã + V˜1 + V˜2 ) =: µ 2 h hBR = µ02 (E 0 where Ã := E

p˜2A − eσB0 + m ˜ 2,

1 V˜1 := −γ AE˜ AE˜ , x ˜

(2.2)

Ã = p ˜ − eA0 (˜ p x)

Ã ˜ A 1 σp σp A˜ V˜2 := −γAE˜ EA˜ + m ˜ x ˜ EA˜ + m ˜ E

(2.3)

Ã , m. and AE˜ follows from (1.3), where EA , m is replaced by E ˜ If we introduce 1 − ˜g := µ 2 Eg then the corresponding eigenvalue the scaled ground-state energy E 0 equation turns into ˜ BR ψg = E ˜ g ψg . (2.4) h In order to prove Theorem 1 we thus have to establish two items, ˜ BR for small m (i) the existence of a ground state of h ˜ ≥ 0 and fixed B0 > 0, ˜ ˜g (0) of (ii) the convergence of Eg (m) ˜ as m ˜ → 0 to the ground-state energy E BR ˜ h at m ˜ = 0. As a consequence of (ii), 1

˜g (m) ˜g (0), ˜ =E lim Eg /µ02 = lim E

µ0 →∞ 1

(2.5)

m→0 ˜

1

˜g (0) = −c B 12 as B → ∞, where c := −E ˜g (0)/B 2 . implying Eg ∼ µ02 E 0

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3. Existence of the Ground State Lemma 1. Let hBR be the Brown–Ravenhall operator for an electron of mass m in a constant magnetic field B0 . Then for 0.1 γ < π2 there exists mc > 0 such that hBR has a discrete ground state below m for 0 ≤ m < mc . BR Equivalently, hBR − m has a discrete ground state below 0 for m ∈ − := h [0, mc ).

Proof. First we show that the spectrum of hBR is discrete below m, if nonempty. From EA ≥ m we have σ(EA ) ⊂ [m, ∞) and hence σess (EA ) ⊂ [m, ∞). Actually m ∈ σess (EA ). This follows from m ∈ σess (DA ) (when m > 0) for a constant magnetic field [18, p. 202]. The spectrum is invariant if the unitary Foldy– Wouthuysen transformation U0 is applied to DA , which gives U0 DA U0−1 = βEA [4] and proves the statement. Since the magnetic field does not restrict the electronic motion parallel to the field, σ(EA ) is continuous and hence σess (EA ) = [m, ∞). From (3.7) below it follows that this holds also for m = 0. For m > 0 it was proven in [10] that σess (hBR ) = σess (EA ). The respective proof, using the compactness of the difference of the resolvents of hBR and EA , has to be extended to the case m = 0. This is straightforward because the compactness property relies basically on the invertibility of the operators EA + µ and hBR + µ for a suitable µ > 0 as well as on the relative form boundedness (1.4) of the potential V1 + V2 with respect to EA with form bound smaller than 1. For m = 0, (1.4) remains valid and thus the semiboundedness of hBR . An additional ingredient of the proof is the compactness of the operator K := χ0 (EA + µ)−1 , where χ0 is a bounded nonnegative function in coordinate space with χ0 → 0 as x → ∞. Since an error occurred in [10], we give that proof anew for m ≥ 0. 2 := (p − eA)2 + m2 be the free Schr¨ odinger operator (increased by Let SA 2 m ) in a magnetic field and decompose K = [χ0 (SA + µ)−1 ] (SA + µ) (EA + µ)−1 .

(3.1)

The compactness of the first factor can be shown with the help of the diamagnetic inequality. One has the pointwise inequality [1] for any A ∈ L2,loc (R3 ) and ψ ∈ L2 (R3 ) ⊗ C2 , 1 1 (3.2) SA + µ ψ (x) ≤ Ep + µ |ψ| (x) where Ep = p2 + m2 . Upon multiplication with χ0 it follows that |(χ0 (SA + µ)−1 ψ)(x)| ≤ (χ0 (Ep + µ)−1 |ψ|)(x). The operator χ0 (Ep + µ)−1 is compact as a product of two bounded functions of x respective p tending to zero at infinity. Therefore χ0 (SA + µ)−1 is compact, too [1, Thm 2.2]. Concerning the boundedness of the remaining factor in (3.1) we have for |B| ≤ B0 and ψ ∈ H1/2 (R3 ) ⊗ C2 , 2 SA ψ2 = (ψ, (EA + eσB) ψ) ≤ EA ψ2 + eB0 ψ2 ,

(3.3)

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√ whence SA ψ ≤ EA ψ + eB0 ψ. With ψ := (EA + µ)−1 ϕ it follows that 1 1 1 1 ϕ ≤ (EA ) ϕ + eB0 ϕ + µ ϕ (SA +µ) EA +µ EA +µ EA +µ EA +µ ≤ cϕ (3.4) with some constant c. Having thus established that σess (hBR ) = σess (EA ) for m ≥ 0 and γ < π2 , we have σess (hBR ) = [m, ∞) under the same conditions. Next we show that σ(hBR ) ∩ (−∞, m) = σd (hBR ) = ∅. (a) Case m = 0 In [12], it was shown numerically for relativistic atoms (Z ≥ 20; using the scaling and eB2 0 = 1) that there exists a trial function ψt ∈ L2 (R3 ) ⊗ C2 , √ 2 2 1 ψt (x) = Nt e−eB0 /4 e−Zef f x3 +1/(eB0 ) (3.5) 0 with = x21 + x22 and Nt a normalization constant, and an effective charge Zef f > 0 such that (ψt , hBR ψt ) < 0

for m = 0.

(3.6)

Thus σd (h ) = ∅ which assures the existence of a ground state for a given nuclear charge Z0 . A fixed trial function for Z0 satisfying (3.6) then also satisfies this inequality for all Z > Z0 . This follows from the fact that the negative part −(V1 + V2 ) of hBR (V1 and V2 are symmetric operators and hence are ≤ 0 as is V ) increases linearly with γ (i.e. with Z) while the positive part EA is independent of Z. We remark that a variational solution to (3.6) can also be found for Z < 20, however, the convergence of the respective integrals in (ψt , hBR ψt ) gets increasingly poor when Z becomes smaller. BR

(b) Case m > 0 BR − m, where the subtraction of the rest We consider the operator hBR − =h BR energy m has the advantage that σess (h− ) = [0, ∞) implying that the groundstate energy is below zero for all m. We show that (ψt , hBR − ψt ) with the fixed ψt from (3.5) is monotonically decreasing with m for m < mc when mc is sufficiently small. For the kinetic part we have, using that EA ψt = p23 + m2 ψt (see also Sect. 4) (ψt , (EA − m) ψt ) = (φˆt , ( k 2 + m2 − m) φˆt ), (3.7) √ 2 2 / eB ) K1 ( Zef + k 0 f ˜ , φˆt (k) = N 2 2 Zef f +k where φˆt ∈ L2 (R) is (up to a constant) the Fourier transform of exp(−Zef f ˜ > 0 a normalization constant x23 + 1/eB0 ), K1 a modified Bessel function and N [9, p.482],[12]. With the mean value theorem applied at m = 0 we get

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(φˆt , ( k 2 + m2 − m) φˆt ) = (φˆt , |k| φˆt ) − m · Dt (ξ) with ξ ∈ [0, m] and ˜2 Dt (ξ) := 2N

∞ dk 0

2 2 K12 ( Zef f + k /eB0 ) 2 Zef f

+ k2

·

(ξ +

k2

. k2 + ξ 2 ) k2 + ξ 2

(3.8)

(3.9)

Since the second factor in (3.9) is nonnegative, bounded by 1 and decreasing with ξ, the integrand is positive except for k = 0 leading to 0 < Dt (m1 ) < Dt (ξ) ≤ 1 for all ξ and m below some fixed value m1 . Consequently, (3.8) is decreasing with m (linearly for m → 0). Fourier transforming the expectation value of the potential part of hBR , we have ∞ ∞ (ψt , (V1 + V2 ) ψt ) = −γ c0 dk dk φˆt (k) Vˆ0 (k − k ) fkk (m) φˆt (k ) −∞

−∞

(3.10) fkk (m) := AE (k)AE (k ) + AE (k) where c0 is a constant, EA (k) =

√

k k AE (k ) EA (k) + m EA (k ) + m

A (k)+m k 2 + m2 , AE (k) = ( E2E ) 2 , and Vˆ0 (q) is the A (k) 1

2

Fourier transformed expectation value of x1 with the transverse function e−eB0 /4 [for details see Sect. 4, in particular (4.6)]. One can split F (m) := (ψt , (V1 + V2 ) ψt ) into F+ (m) + F− (m), where F+ (m) results from kk ≥ 0 in the integrand of (3.10) while F− (m) is assigned to kk ≤ 0. The function fkk is positive and analytic for m ≥ 0 if (k, k ) ∈ R2 \Sf with Sf := (0, R) ∪ (R, 0). Concerning F+ (m), i.e. kk > 0, one gets by elementary 2

fkk kk computation dfdm |m=0 = 0 and ddm (except for k = k ). Thus fkk 2 |m=0 = 0 has (almost everywhere) a local extremum in m = 0. Since the remaining factors in the integrand of (3.10) are independent of m and positive (see Lemma 2 for Vˆ0 ), F+ (m) has also a local extremum in m = 0. Thus, there is an mc such that the variation of F+ (m) in [0, mc ) is considerably weaker than the linear dependence (3.8) of the kinetic term. The other contribution, F− (m), is monotonically decreasing like the kinetic term. In fact, if |kk | is kk < 0, fkk is monotonically increasing with m (as 1 − (EA (k)+m)(E A (k )+m) BR increasing with m). As a result, (ψt , (h − m) ψt ) is decreasing with m in [0, mc ). This feature is confirmed numerically, with a quite large mc . Consequently, for a given nuclear charge Z0 , once (ψt , hBR − ψt ) < 0 is established at m = 0 it follows that (ψt , hBR ψ ) < 0 for m ∈ [0, mc ). On the t − other hand, as discussed in the context of m = 0 (but remains true for m > 0), (ψt , hBR − ψt ) < 0 for all charges greater than Z0 . Since Z0 10 is arbitrary, this guarantees the existence of a ground state of hBR − below 0 for all m < mc .

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Alternatively, the existence of a discrete ground state (and, in addition, infinitely many bound states) of hBR below m (for m > 0) may be based on a theorem by Matte and Stockmeyer [15]. However, its applicability requires the fulfilment of certain nontrivial conditions on the Weyl sequences for the essential spectrum of the free Dirac operator DA . In the remaining part of the present work, we prove the convergence of the ˜g (m) ˜g (0). We start by restricting the sequence of eigenvalues E ˜ for m ˜ → 0 to E ground-state function to the lowest Landau level and show continuity of the expec˜ BR at m ˜ = 0 for a certain class of functions. Then, we allow for tation value of h ˜ BR the the presence of higher Landau states. From the continuity property of h ˜ ˜ convergence of Eg (m) ˜ to Eg (0) is deduced.

4. Reduction to a One-Dimensional Problem We want to gain information on the ground-state wavefunction of hBR for arbitrary mass m ≥ 0. To this aim we reduce the three-dimensional problem to a one-dimensional one by invoking the eigenfunctions of the Pauli operator. The resulting eigenvalue equation is then transformed to an integral equation in Fourier space from which the basic properties of the ground-state function can be extracted. The eigenfunctions ψnlds (x1 , x2 ) = ϕnld (x1 , x2 ) χs of the two-dimensional Pauli operator obey [18, p. 196] 2

2 − p23 − m2 )ψnlds = (pi − eAi )2 − eσ3 B0 ψnlds = (2n + 1 − s)eB0 ψnlds (EA i=1

1, n = 0 , (4.1) ±1, n ≥ 1 1 0 where the spin functions are χ+1 = , χ−1 = . These eigenfunctions form 0 1 a complete set of orthonormal functions (see e.g. [8],[13, §111]) such that an eigenstate ψ ∈ L2 (R3 ) ⊗ C2 of hBR can be expanded in terms of these functions. When B strongly dominates the Coulomb field the ground state of hBR is approximately characterized by the ground state of the Pauli operator which is determined by the quantum numbers n = 0 and s = +1 (where the r.h.s of (4.1) attains its minimum zero). Indeed it was shown for the Dirac operator in a homogeneous magnetic field [5] that the n = 0 approximation becomes exact when the nuclear charge tends to zero or equivalently, when B → ∞. With the restriction to n = 0 and s = +1 the (normalized) ground state ψ of hBR can be written as ∞ 2 ψn=0 (x) = al Nl e−eB0 /4 l eildϕ φl (x3 ) χ+1 , l ∈ N0 ,

n ∈ N0 ,

s = ±1,

l=0

d=

1214


Nl =

(eB0 )l+1 2l+1 l!π


12 ,

(4.2)

where al is an expansion coefficient, = x21 + x22 , φl (x3 ) a normalized function yet to be determined, and we have used the explicit form of ϕ0l [18, p.196], [16, the negative sign of d in that work relates to a positive coupling in (4.1), (pi + eAi )]. When n = 0 and the spin is fixed in ψ one can show by using σpA x1 σpA = pA x1 pA + iσ(pA × x1 pA ), that not only EA and V1 , but also V2 is diagonal in l. For s = +1 we have from (4.1) for the kinetic part (ϕ0l , EA ϕ0l ) = p23 + m2 =: EA (p3 ) and it is straightforward to verify for the potential part 1 Vm (x3 , l) := (χ+1 ϕ0l , − (V1 + V2 ) ϕ0l χ+1 ) γ ∞ 2 (eB0 )l+1 = AE (p3 ) d 2l+1 e−eB0 /2 2l l! 0

1 p3 1 p3 AE (p3 ) (4.3) · + EA (p3 ) + m 2 + x23 EA (p3 ) + m

2 + x23 where the functional dependence of AE (p3 ) on EA (p3 ) is given by (1.3). The smallest value of (ψ, hBR ψ) is given by l = 0. In fact, EA (p3 ) is independent of l. More2 2 − 12 in the integral in (4.3) is peaked at max = over, the weight factor of ( + x3 ) (2l + 1)/eB0 , its normalized value at max slightly decreasing with l. Thus Vm (x3 , l) attains its maximum at l = 0, where max is smallest. For the consideration of the ground state we can therefore restrict ourselves to l = 0. Thus, drop 1 ping the sum in (4.2) and the index on φ, we have ψn=0 (x) = ϕ00 (x1 , x2 )φ(x3 ) . 0 Disregarding the coupling to higher Landau states, the eigenvalue equation for hBR − reduces to 2 2 p3 + m − m − γ Vm (x3 , 0) φ(x3 ) = Em− φ(x3 ). (4.4) where Em− is the ground-state energy of hBR − m under the restriction n = 0 (and s = +1). When Fourier transforming φ in (4.4) and projecting with an eigenstate to p3 , exp(ipx3 ), the momentum operators in Vm turn into functions of k and p, respectively. As a result, we obtain an eigenvalue equation for the momentumˆ space function φ,

p2

+

m2

γ ˆ − m φ(p) − √ 2π

∞ dkAE (p) −∞

p k ˆ ˆ ˆ ˆ V0 (p − k) = Em− φ(p) (4.5) · V0 (p − k) + AE (k)φ(k) EA (p) + m EA (k) + m

Vol. 10 (2009)


1215

for p ∈ R, where we have introduced the Fourier transformed expectation value V0 (x3 ) of x1 [9, p.419], ∞ ∞ 2 1 eB0 −iqx3 ˆ V0 (q) := √ dx3 e

d e−eB0 /2 2 2π

+ x23 −∞ ∞ 2eB0

= √

2π

0

2

d e−eB0

/2

K0 (|q| ).

(4.6)

0

Lemma 2. The momentum-space potential Vˆ0 from (4.6) obeys ⎧ ⎪ ⎨ −c1 ln |q|, q → 0 ˆ V0 (q) ∼ c ⎪ ⎩ 2, |q| → ∞ q2 with c1 > 0 and c2 ≥ 0. Vˆ0 (q) ≥ 0 is monotonically decreasing with |q|.

(4.7)

Proof. For q → 0 the modified Bessel function diverges logarithmically, K0 (|q| ) ∼ − ln(|q| ) = − ln |q| − ln , and in this limit the integral over is convergent. For large |q| we make the substitution |q| = z and obtain ∞ 2 2 2eB0 2eB0 Vˆ0 (q) = √ z dz e−eB0 z /2q K0 (z) ≤ √ , (4.8) 2π q 2 2π q 2 0

where we estimated the exponential by unity and used [9, p.684]. Since in (4.6) K0 is monotonically decreasing and the integrand ≥ 0, so is Vˆ0 (q). ˆ With the help of Lemma 2 some properties of φ(k) can be derived. We have ˆ Lemma 3. For m ≥ 0 the restricted ground-state function in momentum space, φ, has the following properties ˆ (a) φ(k) is uniformly bounded for k ∈ R. c ˆ , |k| → ∞, where c ≥ 0 is some constant. (b) |φ(k)| ≤ |k| Proof. For the proof of boundedness we use that p2 + m2 −m+ |Em− | ≥ |Em− | and estimate from (4.5), using that AE ≤ 1 and k/(EA (k) + m) ≤ 1 (for m ≥ 0), ∞ 2γ 1 ˆ ˆ √ |φ(p)| ≤ dk Vˆ0 (p − k) |φ(k)| 2π |Em− | −∞

⎛

1 ⎝ 2γ ≤ √ |E 2π m− |

∞

dq

⎞1 2 2 ˆ Vˆ0 (q) ⎠ φ,

(4.9)

−∞

where the Schwartz inequality was applied. Due to Lemma 2 the integral is c ˆ = 1, such that |φ(p)| ˆ bounded, and φ ≤ |Em− | where the constant c does

1216



neither depend on φˆ nor on m. Recall that Em− < 0 exists for all m ≥ 0 since the trial function ψt is an n = 0 (spin-up) state. ˆ It is now easy to derive the behaviour of φ(p) at |p| → ∞. Without restriction 2 ≥ √|p| . we can assume |p| > m such that p2 + m2 − m + |Em− | ≥ √ 2 p 2 2+1 p +m +m

For the estimate of the integral in (4.5) we can use (4.9). Then, with the same constant as above, √ 2+1 ˆ · c for |p| → ∞. (4.10) |φ(p)| ≤ |p|

5. Continuity Property of hBR in m = 0 Let Mpos := {ϕ ∈ L2 (R) : ϕ = 1, ϕˆ uniformly bounded near 0}

(5.1)

be the set of square integrable functions for which there exists C > 0, 0 < δ < 1, such that |ϕ(k)| ˆ ≤ C for k ∈ [−δ, δ]. Then for any 0 > 0 there exists 0 < δ0 ≤ δ with 2 ( sup |ϕ|) ˆ 2 · 2δ0 ≤ 4C 2 δ0 ≤ 0 .

(5.2)

[−δ,δ]

0 We can choose δ0 = min{ 4C 2 , δ} for all ϕ ∈ Mpos .

Lemma 4. Let hBR (m) be the Brown–Ravenhall operator for a particle with mass 2 m and let ψ(x) = N0 e−eB0 /4 φ(x3 ) χ+1 with φ ∈ Mpos . Then (ψ, hBR (m) ψ) is continuous in m = 0, i.e. for any > 0 ∃ m0 > 0 : |(ψ, hBR (m) − hBR (0) ψ)| < for all m < m0 . (5.3) As a consequence, the expectation value of hBR − (m) is also continuous in 0. Proof. Denoting the dependence on m by an additional subscript we have for the kinetic part of hBR (m) from (1.3), |(ψ, (EA,m − EA,0 ) ψ)| ≤ ψ

m2 ψ ≤ m ψ2 , EA,m + EA,0

(5.4)

which shows the continuity in m = 0. Note that (5.4) holds for all ψ ∈ L2 (R3 )⊗C2 . The potential part can be decomposed in the following way, 1 1 V1,m − V1,0 = −γ AE,m (AE,m − AE,0 ) − γ (AE,m − AE,0 ) AE,0 (5.5) x x with a similar formula for V2 . The expectation value of the second summand is the complex conjugate one of the first term (except for the replacement of AE,m by AE,0 ) and need not be considered separately. Transforming into Fourier space we have from (4.6)

Vol. 10 (2009)


|(ψ, (AE,m − AE,0 )

1217

1 AE,m ψ)| = |(φ, (AE,m (p3 ) − AE,0 (p3 )) V0 AE,m (p3 ) φ)| x ∞ ∞ ˆ ˆ )| ≤ dk |φ(k)| dk K(k, k ) |φ(k (5.6) −∞

−∞

with the kernel 1 K(k, k ) := √ |AE,m (k) − AE,0 (k)| Vˆ0 (k − k ) AE,m (k ). 2π

(5.7)

We estimate (5.6) further by applying the Schwarz inequality, ⎛ ∞ ⎞ 12 ⎛ ∞ ⎞ 12 1 2 ˆ ˆ )|2 J(k )⎠ |(ψ, (AE,m − AE,0 ) AE,m ψ)| ≤ ⎝ dk|φ(k)| I(k)⎠ ⎝ dk |φ(k x ∞ I(k) :=

−∞

dk K(k, k ),

J(k ) :=

−∞

∞

−∞

dk K(k, k ).

(5.8)

−∞

We will show that each of the two factors can be bounded by an arbitrarily small 1 2 , provided m is sufficiently small. We have, substituting ξ = k − k and using that AE,m (k ) is bounded by 1, 1 |AE,m (k) − AE,0 (k)| I(k) ≤ √ 2π

∞ dξ Vˆ0 (ξ),

(5.9)

−∞

where by Lemma 2 the integral is equal to some finite constant cI . We make use of the fact that AE,m (k) is continuous at m = 0 if k = 0, choose 0 > 0 and take δ, δ0 from the definition (5.1) of Mpos . Then for |k| ≥ δ0 there is m00 > 0 such that |AE,m (k) − AE,0 (k)| < 0 for all m < m00 . We decompose the integration interval according to (−∞, ∞) = (−∞, −δ0 ) ∪ [−δ0 , δ0 ] ∪ (δ0 , ∞) and obtain ⎧ ⎛ ⎞ −δ0 ∞ ∞ ⎨ c I 2 2 ˆ ˆ dk |φ(k)| I(k) ≤ √ + ⎠ dk |φ(k)| 0 ⎝ 2π ⎩ −∞ −∞

δ0

δ0 + −δ0

⎫ ⎬ 2 ˆ dk |φ(k)| |AE,m (k) − AE,0 (k)| . ⎭

(5.10)

We have |AE,m (k) − AE,0 (k)| ≤ 2, and since φˆ ∈ Mpos , by (5.2) the last term ˆ )2 · 2δ0 ≤ 0 . The other two in the curly brackets is estimated by 2 (sup[−δ0 ,δ0 ] |φ| 2 ˆ = 0 . This leads to terms can be estimated by 0 φ

1218


∞ −∞

2cI 2 ˆ 0 dk |φ(k)| I(k) ≤ √ 2π


for m < m00 .

(5.11)

For J(k ) we treat the case |k| ≥ δ1 with δ1 < 1 by estimating |AE,m (k) − AE,0 (k)| < 0 for, say, m < m01 . Then we obtain, substituting ξ = k − k for k, ⎧ ⎛ ⎞ −δ ⎪ 1 −k ∞ ⎨ 1 ⎜ ⎟ J(k ) ≤ √ AE,m (k ) 0 ⎝ + ⎠ dξ Vˆ0 (ξ) ⎪ 2π ⎩ −∞ δ1 −k ⎫ δ 1 −k ⎪ ⎬ ˆ (5.12) + dξ |AE,m (ξ + k ) − AE,0 (ξ + k )| V0 (ξ) . ⎪ ⎭ −δ1 −k

The first term in the curly brackets is estimated by cI 0 . As concerns the second term we profit from the fact that Vˆ0 (ξ) is symmetric and monotonically decreasing with |ξ|. Thus, estimating the difference between AE,m and AE,0 by 2, the remaining integral over Vˆ0 (ξ) has its maximum value for k = 0. With (4.7) we have δ 1 −k

δ1 dξ Vˆ0 (ξ) ≤

−δ1 −k

dξ Vˆ0 (ξ) −δ1

δ1 ln ξ dξ = 2c1 δ1 (1 − ln δ1 ).

≤ −2c1

(5.13)

0

The r.h.s tends to zero as δ1 → 0 and thus can be made smaller than 0 for sufficiently small δ1 . Then, with AE,m ≤ 1, ∞ −∞

ˆ )|2 J(k ) ≤ √1 {cI 0 + 20 } dk |φ(k 2π

for m < m01 .

(5.14)

The estimate of the second potential part, V2,m − V2,0 , proceeds in the same way. Only one has to replace throughout AE,m (k) by A˜E,m (k) := AE,m (k) · k √ which is also bounded by 1 and continuous at m = 0 for k = 0. This k2 +m2 +m proves that the potential part is bounded by for m < min{m00 , m01 } =: m0 . We continue by showing that the state φm , defining the ground state ψ0,m of both hBR (m) and hBR − (m) (under the restriction n = 0, s = +1; the subscript m is added for clarity), is a member of Mpos for m sufficiently small. For m = 0 one has φ0 ∈ Mpos because from (4.9), |φˆ0 (p)| ≤ |Ec0 | for all p ∈ R (where E0 is the ground-state energy of the restricted hBR (0)), and we may choose any bound C ≥ |Ec0 | . Therefore, from Lemma 4 (with possibly a slightly smaller m0 ),

Vol. 10 (2009)


BR |(ψ0,0 , hBR (0) ψ0,0 )| < − (m) ψ0,0 ) − (ψ0,0 , h

for m < m0 .

1219

(5.15)

As a consequence, (ψ0,0 , hBR − (m) ψ0,0 ) < E0 + . On the other hand, Em− ≤ (ψ0,0 , hBR − (m) ψ0,0 ) since the expectation value BR of (the restricted) h− (m) taken with an arbitrary function leads to an upper bound of the ground-state energy Em− . Combining these two inequalities we get Em− < E0 +

for m < m0 .

(5.16)

Since E0 < 0 there exists δ2 < 0 such that E0 < δ2 < 0. Let be so small that E0 + < δ2 . Then, from (5.16), Em− < δ2 for all m < m0 . This leads to an m-independent bound of φˆm from (4.9), |φˆm (p)| ≤

c |Em− |

|Ec0 | , we have found a universal bound on φˆm for all m < m0 . As a consequence, Lemma 4 holds for all eigenstates ψ0,m with m < m0 . Let us now generalize the ansatz for the ground-state function ψ of hBR by including the higher Landau states (n > 0 and s = −1) in the expansion (4.2). This allows for the coupling of the potential V1 + V2 to n ≥ 1 states as well as for spin-flip induced by V2 . However, for high magnetic fields the ground state is not much affected. For the Dirac operator, it was shown numerically [6] that for e.g. Z = 68 and B = 2.3×1016 G (where Z 2 m2 e3 = 1.1×1013 G is the particular field for which Coulombic and magnetic forces on the electron are equally important) the effect of considering the n > 0 contributions is about 3%. With ψn=0 being the ground state of hBR with the coupling to higher Landau states switched off, we now add a spin-flip term ψ−1 and a remainder ψr , ψ(x) = a0 ψn=0 (x) + a1 ψ−1 (x) + a2 ψr (x).

(5.18)

ψ−1 is characterized by n = 0, spin s = −1 and l = 1 since V2 changes l by one unit simultaneously with the spin-flip. ψr is composed of Landau states with n ≥ 1. All functions in (5.18) are normalized and mutually orthogonal. The weight factors obey |a0 |2 + |a1 |2 + |a2 |2 = 1, guaranteeing that ψ is normalized too. SinceEA , acting on a Landau state with√n ≥ 1, is according to (4.1) given by EA = 2neB0 + eB0 (1 − s) + p23 + m2 ≥ 2eB0 + m2 it is strictly positive at m = 0. The same is true when √ EA acts on an n = 0 spin-down state, resulting in EA = 2eB0 + p23 + m2 ≥ 2eB0 + m2 . Hence, the m-dependent factors in V1 and V2 , AE and σpA /(EA + m), are analytic in m = 0. As a consequence, (ψr , hBR ψr ) and (ψ−1 , hBR ψ−1 ) are continuous in m = 0 (in the sense that ψr , respectively ψ−1 , is kept fixed when performing the limit m → 0). When proving Lemma 4 for ψ from (5.18) it thus remains to show that the off-diagonal matrix elements are also bounded by for m sufficiently small. Considering the potential V1 we have, using the decomposition (5.5),

1220


1 − (ψn=0 , (V1,m − V1,0 ) ψr ) = γ

1


1

1 AE,m ψn=0 , 1 (AE,m − AE,0 ) ψr x2 x2 1 1 + . 1 (AE,m − AE,0 ) ψn=0 , 1 AE,0 ψr x2 x2 (5.19)

The r.h.s of (5.19) can be estimated from above by

1 1

x2

AE,m ψn=0 ·

1 1

x2

(AE,m − AE,0 ) ψr

12 1 1 ˜ 1 + ψn=0 , (AE,m − AE,0 ) AE ψn=0 · (ψr , − V1,0 ψr ) 2 x γ

(5.20)

with the bounded operator A˜E := AE,m − AE,0 . Since all Landau states, and 2 thus ψr , have a Gaussian decay (∼ e−eB0 /4 [8]), (ψr , V1,0 ψr ) is bounded. The 1 multiplication factor is bounded by 2 according to the proof of Lemma 4. The two factors in the first term of (5.20) are bounded by since (ψn=0 , V1,m ψn=0 ) is bounded and AE,m continuous in m = 0 when acting on ψr . The corresponding estimates for the potential V2 are done in the same way. All these estimates also hold when ψr in (5.18) is replaced by ψ−1 or when ψn=0 is replaced by ψ−1 . Together with (5.4) this establishes the continuity property of hBR , respectively hBR − , BR (0)) ψ)| < |(ψ, (hBR − (m) − h

for m < m0

(5.21)

and m0 sufficiently small, where ψ from (5.18) is taken as eigenstate to hBR (m ) for any fixed m with 0 ≤ m < m0 .

6. Convergence of the Sequence of Eigenvalues With (5.21) at hand it is easy to prove BR Lemma 5. Let Eg− (m) be the ground-state energy of hBR (m) − m, and − (m) = h BR let Eg (0) be that of h (0). Then for a given > 0 there is an m0 > 0 such that

|Eg− (m) − Eg (0)| <

for all m < m0 .

(6.1)

Proof. Let us choose ψ in (5.21) as eigenstate ψm to hBR (m). Then BR (0) ψm )| < |(ψm , hBR − (m) ψm ) − (ψm , h

for m < m0 ,

(6.2)

such that (ψm , hBR (0) ψm ) < Eg− (m) + . Moreover, one has Eg (0) ≤ (ψm , hBR (0) ψm ) since ψm will differ from the ground-state function ψ0 of hBR (0). The combination of these two inequalities leads to Eg− (m) > Eg (0) −

for m < m0 .

(6.3)

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1221

On the other hand, with ψ := ψ0 , the step from (5.15) to (5.16) can be mimicked. Then (5.16) turns into Eg− (m) < Eg (0) + which gives, combined with (6.3), |Eg− (m) − Eg (0)| <

for m < m0 .

This proves the continuity of Eg− (m) (and of Eg (m) as well) at m = 0.

(6.4)

Acknowledgement It is a pleasure to thank P. M¨ uller for enlightening discussions.

References [1] Avron, J., Herbst, I., Simon, B.: Schr¨ odinger operators with magnetic fields. I. General interactions. Duke Math. J. 45, 847–883 (1978) [2] Avron, J., Herbst, I., Simon, B.: III. Atoms in homogeneous magnetic fields. Commun. Math. Phys. 79, 529–572 (1981) [3] Brown, G.E., Ravenhall, D.G.: On the interaction of two electrons. Proc. R. Soc. Lond. A 208, 552–559 (1951) [4] De Vries, E.: Foldy–Wouthuysen transformations and related problems. Fortschr. Phys. 18, 149–182 (1970) [5] Dolbeault, J., Esteban, M.J., Loss, M.: Relativistic hydrogenic atoms in strong magnetic fields. Ann. Henri Poincaré 8, 749–779 (2007) [6] Dolbeault, J., Esteban, M.J., Loss, M.: Characterization of the critical magnetic field in the Dirac-Coulomb equation. J. Phys. A 41(185303), 1–13 (2008) [7] Evans, W.D., Perry, P., Siedentop, H.: The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Commun. Math. Phys. 178, 733–746 (1996) [8] Garstang, R.H.: Atoms in high magnetic fields. Rep. Prog. Phys. 40, 105–154 (1977) [9] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic, New York (1965) [10] Jakubassa-Amundsen, D.H.: The single-particle pseudorelativistic Jansen-Hess operator with magnetic field. J. Phys. A 39, 7501–7516 (2006) [11] Jakubassa-Amundsen, D.H.: Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field. J. Math. Phys. 49(032305), 1–22 (2008) [12] Jakubassa-Amundsen, D.H.: Variational ground state for relativistic ions in strong magnetic fields. Phys. Rev. A 78(062103), 1–9 (2008) [13] Landau, L.D., Lifschitz, E.M.: Lehrbuch der Theoretischen Physik. III Quantenmechanik. Akademie-Verlag, Berlin (1974) [14] Lieb, E.H., Siedentop, H., Solovej, J.P.: Stability and instability of relativistic electrons in classical electromagnetic fields. J. Stat. Phys. 89, 37–59 (1997) [15] Matte, O., Stockmeyer, E.: On the eigenfunctions of no-pair operators in classical magnetic fields. Integr. Equ. Oper. Theory (2008, to appear). arXiv:mathph/0810.4897v1

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[16] Rau, A.R.P., Mueller, R.O., Spruch, L.: Simple model and wave function for atoms in intense magnetic fields. Phys. Rev. A 11, 1865–1879 (1975) [17] Rau, A.R.P., Spruch, L.: Energy levels of hydrogen in magnetic fields of arbitrary strength. Astrophys. J. 207, 671–679 (1976) [18] Thaller, B.: The Dirac Equation. Springer, Berlin (1992) Doris H. Jakubassa-Amundsen Mathematics Institute University of Munich Theresienstr. 39 80333 Munich Germany e-mail: [email protected] Communicated by Rafael D. Benguria. Received: March 25, 2009. Accepted: July 7, 2009.

Ann. Henri Poincaré 10 (2010), 1223–1249 c 2010 Birkh¨ auser Verlag AG, Basel/Switzerland 1424-0637/10/071223-27, published online January 30, 2010 DOI 10.1007/s00023-010-0021-z


Dynamical Phase Transition for a Quantum Particle Source Maximilian Butz and Herbert Spohn Abstract. We analyze the time evolution describing a quantum source for non-interacting particles, either bosons or fermions. The growth behavior of the particle number (trace of the density matrix) is investigated, leading to spectral criteria for sublinear or linear growth in the fermionic case, but also establishing the possibility of exponential growth for bosons. We further study the local convergence of the density matrix in the long time limit and prove the semi-classical limit.

1. Introduction Particle sources are an indispensable part of any scattering experiment. Nevertheless in the theoretical description they are mostly disregarded on the basis that a “suitable” wave function has been prepared. Of course, a fully realistic modeling of a particle source will be difficult and possibly of marginal interest. However, on intermediate grounds, having a simple model source could be of use. The purpose of our paper is to study a, in a certain sense, minimal model. Surprisingly enough, at least to us, we will find that for bosons there is a dynamical phase transition. On the classical level a particle source is easily modeled and, in variation, used widely without further questioning. To explain the principle let us discuss particles in one dimension with position xj ∈ R and velocity vj ∈ R, j = 1, 2, . . .. The source is located at the origin and turned on at time t = 0. Particles are created at times 0 < t1 < t2 < · · ·. Once the j-th particle is created at time tj , it moves freely as xj (t) = vj (t − tj ), t ≥ tj . To have a Markov process it is assumed that (tj+1 − tj ) are independent and exponentially distributed with rate λ. Also at the moment of creation the velocity is distributed according to h(v)dv independently of all the other particles. The average density f on the one-particle phase space is then governed by ∂ ∂ f (x, v, t) + v f (x, v, t) = λδ(x)h(v). (1.1) ∂t ∂x

1224

M. Butz and H. Spohn


Of course, one could imagine some statistical dependence. But then the simplicity of Eq. (1.1) is lost. In applications, the left hand side of the transport equation may contain further items as an external potential, a nonlinear collision operator, and the like, to which the source term on the right hand side is simply added. The underlying reasoning for the source term still follows from the statistical assumptions stated above. On the basis of (1.1) one concludes that, provided f (x, v, 0) = 0, the particle number increases linearly in time, f (x, v, t)dx dv = tλ h(v)dv. (1.2) R

R2

One can also show, that f (·, t) converges to a steady state as t → ∞. In the quantum case the Markov process on the many particle level is replaced by a quantum dynamical semigroup on Fock space. As in (1.1), the crucial constraint comes from the condition to have a closed equation on the one-particle level, a condition which essentially determines the model uniquely. Particles are created in the pure state φ ∈ H = L2 Rd . Once created they move in Rd according to the Schr¨ odinger equation d (1.3) i ψ = Hψ. dt The one-particle Hamiltonian H is a self-adjoint operator with domain D(H) ⊂ H. For later purpose, Ppp , resp. Pac , is the spectral projection of H onto the pure point, resp. absolutely continuous, part of the spectrum. Primarily, we think of the free Schr¨ odinger evolution, in which case H = −∆ (we use units such that the Planck constant equals 1 and the particle mass m equals 1/2). But some of our results also hold abstractly, in particular for the Schr¨ odinger operator with a potential, H = −∆ + V (x). We use the shorthand H0 = −∆. Our techniques generalize without great efforts to the case of particles being created in a mixed state. Hence we decided to stay with a pure state as minimal model. To be more precise, we introduce the Fock space ∞ F± = S± H⊗n , (1.4) n=0 ⊗n

denotes the (anti)symmetrized n-fold tensor product of H with where S± H itself, i.e. the either bosonic (+) or fermionic (−) n-particle subspace. The bosonic or fermionic creation and annihilation operators are denoted by a± (φ)∗ and a± (φ). They satisfy the (anti)commutation relations [a± (f ), a∗± (g)]∓ = f, g 1F± ,

[a± (f ), a± (g)]∓ = 0 = [a∗± (f ), a∗± (g)]∓

(1.5)

with ·, · the scalar product of H and [A, B]− = [A, B] = AB − BA, [A, B]+ = {A, B} = AB + BA for operators A, B on F± . We postulate an evolution equation of Lindblad type

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A Quantum Particle Source

1225

d A(t) = LA(t) (1.6) dt with bounded A(t) on F± , where L is the generator of a completely positive dynamical semigroup. To have a closed equation on the one-particle space, L has to be quadratic in a± (φ)∗ , a± (φ), in other words L has to generate a quasi-free dynamical semigroup. Clearly L is the sum of the Hamiltonian part L0 and the source part Ls . L0 is obviously quasi-free and completely positive. If particles are created in the state φ ∈ H, φ = 1 with rate |λ| ≥ 0, then the only possible source term has to be of the form Ls A = |λ| (2a± (φ)Aa± (φ)∗ − a± (φ)a± (φ)∗ A − Aa± (φ)a± (φ)∗ )

(1.7)

Let ω0 be the initial state as density matrix on F± . Then ωt (A) = ω0 (eLt A) defines the state at time t. Its one-particle density matrix is given by g, ρ(t)f = ωt (a∗ (f )a(g)).

(1.8)

∗

Here ρ(t) = ρ(t), ρ(t) ≥ 0, tr (ρ(t)) < ∞ and ρ(t) ≤ 1 in the case of fermions. If ω0 is quasi-free, then ωt is also quasi-free. In particular, ωt is uniquely determined by ρ(t). From (1.7), (1.8) one readily obtains the evolution equation for the one-particle density matrix ρ(t), d ρ(t) = −i[H, ρ(t)] + 2|λ|Pφ + λ (Pφ ρ(t) + ρ(t)Pφ ) , (1.9) dt where Pφ = |φ φ| is the orthogonal projection onto φ. We regard λ as real parameter, λ ∈ R. Then λ > 0 in (1.9) is the evolution equation for bosons, while λ < 0 refers to fermions, and we maintain this convention throughout. (1.9) holds for t ≥ 0 and is supplemented by the initial ρ(0) = ρ0 . It is the quantum analogue of the classical equation (1.1). The goal of our paper is a detailed analysis of Eq. (1.9). While precise conditions will be given in the main part, let us explain already now the rough overall picture which emerges from our study. We take H = H0 = −∆, a “reasonable” wave function φ, and start with ρ(0) = 0, which corresponds to the Fock vacuum. Hence ωt is quasi-free and gauge invariant. Basically there is a competition between the speed of transport through −∆ and the rate |λ| at which new particles are supplied. For small |λ| the Laplacian dominates. At time t the front particles have traveled a distance of order t from the origin. In essence, the state ωt is an incoherent mixture of single particle wave functions, similar to the classical set-up described by (1.1). If λ → −∞, i.e. per unit time a large number of fermions are created, then the replenishment becomes constrained because of the exclusion principle and, while the number of particles still increases linearly in time, the particle current should level off. In fact, we will show that the current vanishes in the limit of large production rate. On the other side for bosons, beyond some critical value λc , the operator iH0 + λPφ attains an isolated, non-degenerate eigenvalue satisfying (iH0 + λPφ )φλ =

1226



α(λ)φλ , φλ ∈ H, with Re α(λ) > 0. Therefore in the long time limit ρ(t) ∼ = Pφ exp[2 Re α(λ)t]. λ

(1.10)

The particle number increases exponentially and a pure Bose condensate with condensate wave function φλ is generated. φλ → φ as λ → ∞. We will prove this scenario for sufficiently large λ. The critical regime, which in principle could be more complicated, remains to be explored. Quasi-free dynamical semigroups were introduced in [5,4]. A very readable account is the review by Alicki in [2]. He writes down Eqs. (1.7) and (1.9) for the case of a general source and sink. He also discusses coupled quantum fields, for which such kind of equations arise in a weak coupling limit. The case of a sink only is considered in [1]. A discrete time model is studied, which however, in a continuous time limit converges to a quasi-free dynamical semigroup of the type (1.7) with creation and annihilation operators interchanged. We also refer to [3], where more recent mathematical contributions are listed. To give a brief outline: In the following two sections we properly define the solution to (1.9) and list the main results. In Sects. 4–6 we study the particle number, N (t) = tr (ρ(t)), when starting with an empty space ρ0 = 0. In particular we establish both asymptotically linear and exponential growth depending on the parameters. In Sect. 7 the convergence of the local density matrix as t → ∞ is investigated. Finally we prove that in the semi-classical limit, |λ| = O( ) and time, space O( −1 ) the solution of Eq. (1.9) converges to the solution of the classical 2 ˆ . source equation (1.1) with h(v) ∼ |φ(v)|

2. Existence of Solutions The formal solution of Eq. (1.9) for t ≥ 0 reads ρ(t) = sgn(λ) e(−iH+λPφ )t e(iH+λPφ )t − 1 + e(−iH+λPφ )t ρ0 e(iH+λPφ )t , (2.1) where 1 = 1H is the identity map on H. Since H is self-adjoint, iH generates a strongly continuous unitary group on H and, considering ±λPφ as a bounded perturbation of this generator, we can apply Theorem 2.1 of [6], Chapter IX., Sect. 2 to conclude that T = iH ± λPφ : D(H) → H still is the generator of a strongly continuous group which is norm-bounded as etT ≤ e|λt| . Therefore, the (semi)groups occurring in (2.1) are well-defined. Furthermore, the well-known formula −n (±iH + λPφ )t (2.2) e(±iH+λPφ )t = s − lim 1 − n→∞ n implies

for all t ∈ R.

∗ e(−iH+λPφ )t = e(iH+λPφ )t

(2.3)

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Next we have to ensure that ρ(t) is actually a density matrix, i.e. a positive trace class operator for all t ≥ 0. From (2.1) one obtains t ρ(t) = 2|λ|

e(−iH+λPφ )s Pφ e(iH+λPφ )s ds + e(−iH+λPφ )t ρ0 e(iH+λPφ )t

(2.4)

0

in which the integral is a trace-class valued Riemann integral (recall that the semigroups are strongly continuous). Since ρ0 and Pφ are positive trace class operators, (2.3) implies the same property for ρ(t). In case λ < 0, we have to check the fermionic property ρ(t) ≤ 1. But if the system starts in a fermionic state, ρ0 ≤ 1, (2.1) can be reordered to 1 − ρ(t) = e(−iH+λPφ )t (1 − ρ0 ) e(iH+λPφ )t ,

(2.5)

which stays positive for all t. Thus the fermionic property of ρ0 is preserved in time.

3. Main Results 3.1. Asymptotics of the Particle Number Most basically one would like to know the number of particles, N (t), produced by the source. Na¨ıvely one would expect N (t) to grow linearly. However, the statistics of the particles induces an effective attraction, respectively repulsion, which might change such simplistic picture. From (2.1) one easily calculates

d tr(ρ(t)) = 2|λ|e(−iH+λPφ )t φ2 + 2λ φ, e(−iH+λPφ )t ρ0 e(iH+λPφ )t φ . (3.1) dt Together with the observation that for all ψ ∈ H,

2 d (−iH+λPφ )t 2 e ψ = 2λ φ, e(−iH+λPφ )t ψ , (3.2) dt equation (3.1) immediately implies that for any Hamiltonian H, the number of particles grows at least linearly in time for λ > 0, and at most linearly for λ < 0. The number of fermions is monotonically increasing, which is an easy consequence of (3.1) together with the fermionic property ρ0 ≤ 1 and the fact that e(−iH+λPφ )t φ2 = e(+iH+λPφ )t φ2 (their derivatives with respect to t are equal). Concerning the asymptotic growth of N (t), the initial density ρ0 does not change the qualitative behaviour, and is therefore set to ρ0 = 0. In addition to the upper bound N (t) ≤ 2|λ|t for the number of fermions, there is an easy characterization of those source states for which the number of fermions stays bounded. Theorem 3.1. Let λ < 0. N (t) stays bounded as t → ∞ if and only if the source state φ is a finite linear combination of eigenvectors of H. In this case lim N (t) = N.

t→∞

(3.3)

with N the number of different eigenvalues corresponding to these eigenvectors.

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Concerning the distinction of source states generating linear or sublinear growth of N (t) in the fermionic case we have the following result. Theorem 3.2. Let λ < 0. If Ppp φ = φ, then the limit rate of particle production vanishes and N (t) grows sublinearly. On the other hand, if Pac φ = 0, then N (t) increases linearly. In particular, for H = H0 , and arbitrary φ, the number of fermions increases linearly in time. For bosons, linear growth is not an upper, but a lower bound for N (t). But also in this case one can characterize explicitly a class of source states which yield linear growth for small λ. ∞ Theorem 3.3. Let φ ∈ H, φ = 1 and assume that τ := 0 | φ, e−iHt φ |dt < ∞. d N (t) converges to a non-zero limit as Then for all λ < 0 or 0 < λ < τ −1 , dt t → ∞, which satisfies 2|λ| d N (t) ≤ (λ > 0), 2|λ| ≤ lim 2 t→∞ dt (1 − λτ ) (3.4) d 2|λ| N (t) ≤ 2|λ| (λ < 0). lim 2 ≤ t→∞ dt (1 − λτ )

The quantity φ, e−iHt φ is the overlap between the source state at time 0 and at time t. Thus τ should be regarded as a measure for how long it takes the time evolution to transport an emitted particle away from its source. In this context, |λ| < τ −1 means, that the strength of the source is smaller than the “transport capacity” of the time evolution. Therefore, the emitted particles hardly influence each other, and in essence the bosonic or fermionic character does not show in the evolution of the density matrix. The limit particle production is constant as in the classical case. An example for such φ in the physically relevant case H = L2 R3 ,

H0 = −∆ is φ ∈ L1 ∩ L2 R3 , since φ, e−iHt φ has a t−3/2 decay, as can be deduced from the free propagator −iH0 t i|x−y|2 −d e φ (x) = (4πit) 2 e 4t φ(y)dy (3.5) 1

2

Rd

for φ ∈ L ∩ L R (compare [10], Chapter IX.7). For fermions, this is also an example where one has an explicit estimate for the speed of particle production. However, the form of the lower bound in (3.4) already suggests that the limit growth rate might decrease to 0 as λ → −∞. In fact, we have d

Theorem 3.4. For φ ∈ D(H), we have d Nλ (t) = 0. lim lim λ→−∞ t→∞ dt

(3.6)

For bosons, the behavior at large values of λ is qualitatively very different. In this regime, exponential growth occurs for arbitrary choices of the Hamiltonian H and the source state φ.

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Theorem 3.5. For sufficiently large λ > 0 the operator iH + λPφ has an eigenvalue α(λ) with positive real part, and limλ→∞ λ−1 α(λ) = 1. If such an eigenvalue α(λ) with normalized eigenvector ψ exists, the number of bosons can be estimated by tr(ρ(t)) ≥ ψ, ρ(t)ψ ≥ e2 Re α(λ)t − 1.

(3.7)

Hence Theorem 3.5 implies the existence of a critical strength λc ≥ 0 such that the number of particles grows exponentially in time for all λ > λc . For source states φ as in Theorem 3.3, our result implies a dynamical phase transition from linear (0 < λ < τ −1 ) to exponential (λ > λc ) growth. The general picture is more complicated, however. If H has point spectrum, it is obvious that choosing φ as an eigenvector of H will generate exponential growth for all λ > 0: N (t) = e2λt − 1. But also for H = H0 there are source states φ which generate exponential growth for all λ > 0. Theorem 3.6. Let φ ∈ D(H0 ) = H 2 Rd with Fourier transform 3 ˆ . (3.8) φ(p) = |S d−1 |−1/2 |p|(1−d)/2 π(|p|6 + 1) Then iH0 + λPφ has an eigenvalue α(λ) with positive real part for all λ > 0. In the proof of this theorem, the eigenvalue will not be computed explicitly. With a view towards the semi-classical limit, one can infer that Re α(λ) = o(λ) has to hold. In this limit one considers a source with activity λ = c on a time scale −1 t and thus the exponent in (3.7), 2 Re α(c ) −1 t,

(3.9)

has to vanish in the limit → 0 so to yield the linear classical growth behaviour. √ In contrast to Theorem 3.3 the source state (3.8) has an overlap decay as 1/ t, thus τ = ∞. As a consequence, the emitted particles stay close to the source for a long time and, being bosonic, they pull further particles out of the source, which leads to an exponential growth of the particle number regardless of how small λ. 3.2. Convergence of ρ(t) Having studied N (t) one may ask whether ρ(t) has a limit as t → ∞. Since mostly N (t) → ∞ for large t, the natural notion is to study the local limit of ρ(t). Let Ω ⊂ Rd be a bounded region and let PΩ denote the orthogonal projection of H onto the subspace L2 (Ω). We consider the number of particles in Ω, i.e. NΩ (t) = tr(PΩ ρ(t)PΩ ).

(3.10)

If this quantity stays bounded as t → ∞, one can use that PΩ ρ(t)PΩ has a positive time derivative (at least for ρ0 = 0) to infer that the restricted density matrix PΩ ρ(t)PΩ has even a trace class limit.

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A first result shows that for fermions and H = H0 this limit, as far as it exists, does not depend on the initial condition ρ0 . Theorem 3.7. For Ω ⊂ Rd , with finite Lebesgue measure |Ω| < ∞, λ < 0 and ρ0 an arbitrary trace class operator, it holds PΩ e(−iH0 +λPφ )t ρ0 e(iH0 +λPφ )t PΩ tr → 0

as t → ∞.

(3.11)

For all remaining results, we return to ρ0 = 0 again. The next result is rather obvious: If particles are generated at most linearly in time, and if they move away from a certain region Ω in space within finite time, the number of particles in Ω will not diverge: Theorem 3.8. Let us choose φ and λ such that N (t) has a linear bound and assume that ∞ PΩ e−iHt φdt < ∞. (3.12) 0

Then tr(PΩ ρ(t)PΩ ) approaches a finite limit as t → ∞. In particular, for H = H0 this theorem applies to Ω with |Ω| < ∞ for all φ ∈ L1 ∩ L2 R3 and all λ < τ −1 : (Sub)linearity follows from Theorem 3.3, and (3.12) is deduced from the explicit form of the integral kernel (3.5). The next result is characteristic for the behaviour of a fermionic particle source. Consider the fact that in a region Ω of finite measure, there are only finitely many states with kinetic energy below a certain bound (cf. [9], p. 27). If the source state has no high-energy contributions, it should only be able to charge a finite number of states, so that the particle number in Ω stays finite. Theorem 3.9. Let H = H0 , φ ∈ H, with Fourier transform φˆ supported in K ⊂ Rd where |K| < ∞, and let Ω ⊂ Rd with |Ω| < ∞. Then, for λ < 0, lim tr(PΩ ρ(t)PΩ ) ≤ (2π)−d |Ω||K| < ∞.

t→∞

(3.13)

The upper bound corresponds to the phase space volume of fermions, i.e. to one fermion per unit cell. 3.3. Semi-classical Limit To pass from the Hilbert space formulation of quantum mechanics to phase space, one convenient tool is the Wigner transform. One defines y y dy, (3.14) W [κ](x, p) = (2π)−d e−ip·y κ x + , x − 2 2 Rd

where κ ∈ L2 Rdx × Rdy is the integral kernel of a Hilbert-Schmidt operator. In particular, since all trace-class operators are Hilbert-Schmidt, we can consider the

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Wigner transform for any density matrix. Denoting, as usual, the semi-classical parameter by , the semi-classical limit corresponds to the small behaviour of −1 f (X, P, T ) = −d W ρ −1 T X, P , (X, P, T ) ∈ Rd × Rd × R+ , (3.15) where ρ (t) is the solution of d ρ (t) = −i [H0 , ρ (t)] + 2|c| Pφ + c (Pφ ρ (t) + ρ (t)Pφ ) (3.16) dt for all t ≥ 0, with the convention H0 = −∆/2. The initial densities ρ0 are chosen such that ρ0 tr is bounded uniformly in > 0 and (3.17) −d W [ρ0 ] −1 X, P → g(X, P ) for some distribution g ∈ D RdX × RdP . The Eqs. (3.15)–(3.17) describe a quantum particle source producing particles on a microscopic scale (x, p, t) correlated to the macroscopic scale (X, P, T ) by (X, P, T ) = ( x, p, t), the standard semiclassical scaling, see e.g. [8]. To have a bounded rate |c| on the macroscopic time scale, we have set λ = c . In this case, we have the following convergence result for the phase space density f . Theorem 3.10. For all g ∈ D Rdx × Rdp , φ ∈ H, φ = 1, and for all T ≥ 0, the limit T 0 ˆ )|2 ds (3.18) lim f (X, P, T ) = f (X, P, T ) = g(X −P T, P )+2|c| δ(X −P s)|φ(P →0

holds in the topology of D

RdX

×

RdP

0

. This limit solves

∂ 0 ˆ )|2 , f (X, P, T ) + P · ∇X f 0 (X, P, T ) = 2|c|δ(X)|φ(P ∂T f 0 (X, P, 0) = g(X, P )

(3.19)

in the sense of distributions on phase space. (3.19) is the weak form of Eq. (1.1) with a source term defined through the semi-classical limit of φ. A natural step would be to include an external potential varying on the macroscopic scale. We leave this as an open problem.

4. Particle Production in the Fermionic Case 4.1. Proof of Theorem 3.1 Proof. Since we are in the case ρ0 = 0, ρ(t) is differentiable in trace class with a positive operator as derivative, d ρ(t) = 2|λ|e(−iH+λPφ )t Pφ e(iH+λPφ )t . dt

(4.1)

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Thus ρ(t)−ρ(s)tr = | tr(ρ(t))−tr(ρ(s))|, so if the particle number stays bounded and tr (ρ(t)) approaches a finite limit from below, ρ(t) converges in trace class as t tends to infinity. As a fermionic density matrix, the limit ρ∞ obeys 0 ≤ ρ∞ ≤ 1 and it stays invariant under the time evolution given by (2.1), ρ∞ = 1 − e(−iH+λPφ )t e(iH+λPφ )t + e(−iH+λPφ )t ρ∞ e(iH+λPφ )t .

(4.2)

Furthermore, since tr(ρ(t)) converges to a finite limit, its (monotonously decreasing) time derivative tends to zero, and therefore, φ, (1 − ρ∞ )φ = lim φ, (1 − ρ(t))φ = lim e(iH+λPφ )t φ2 t→∞

= lim e

t→∞

(−iH+λPφ )t

t→∞

2

φ = lim (2|λ|)−1 t→∞

d tr(ρ(t)) = 0. dt

(4.3)

Since 1 − ρ∞ is positive, 0 = φ, (1 − ρ∞ )φ can only hold if (1 − ρ∞ )φ = 0. Together with (4.2) this yields: d iHt e ρ∞ e−iHt = |λ|eiHt Pφ e(−iH+λPφ )t (1 − ρ∞ )e(iH+λPφ )t e−iHt dt + |λ|eiHt e(−iH+λPφ )t (1 − ρ∞ )e(iH+λPφ )t Pφ e−iHt = |λ|eiHt (Pφ (1 − ρ∞ ) + (1 − ρ∞ )Pφ ) e−iHt = 0.

(4.4)

Therefore, we have for t ∈ R eiHt ρ∞ e−iHt = ρ∞ .

(4.5)

Now consider the eigenspace V of ρ∞ corresponding to the eigenvalue 1. V has a positive, but finite dimension because φ ∈ V and ρ∞ is trace class, and due to (4.5), V is invariant under the action of the group e−iHt . So since V is finitedimensional, we have V ⊂ D(H) and V has a orthonormal basis of eigenvectors of H, and φ can be written as a linear combination of those. N Conversely, assume that φ = n=1 cn ψn with ψn = 1, Hψn = ωn ψn , with all cn = 0 and all ωn different. −iH + λPφ is a bounded operator on W := span(ψ1 , . . . , ψn ), so e(−iH+λPφ )t φ can be calculated by the power series, to see that t ρ(t) = 2|λ|

e(−iH+λPφ )s Pφ e(iH+λPφ )s ds.

(4.6)

0

contains only states from W , which, together with the fermionic property, yields the bound tr(ρ(t)) ≤ N . Thus ρ∞ exists and, by (4.5) commutes with H on W , so that they have a common basis of eigenvectors, which necessarily means ρ∞ =

N n=1

bn |ψn ψn |.

(4.7)

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Now it only remains to check that ρ∞ = 1W , which follows from N

cn ψn = φ = ρ∞ φ =

n=1

N

cn bn ψn .

(4.8)

n=1

and the fact that all cn = 0. So tr(ρ∞ ) = dim(W ) = N .

4.2. Proof of Theorem 3.2 Proof. For a state φ ∈ Hpp we have to show that d tr(ρ(t)) = lim 2|λ|e(−iH+λPφ )t φ2 = 0. t→∞ dt To this end, expand φ in a basis of eigenvectors of H, lim

t→∞

φ=

∞

cn ψn

(4.9)

(4.10)

n=1

with ψn = 1, Hψn = ωn ψn . Since e(iH+λPφ )t e(−iH+λPφ )t ≥ 0, d (iH+λPφ )t (−iH+λPφ )t e e = −2|λ|e(iH+λPφ )t Pφ e(−iH+λPφ )t ≤ 0, dt

(4.11)

polarization implies that e(iH+λPφ )t e(−iH+λPφ )t converges weakly to a bounded, positive, self-adjoint operator A with A = e(iH+λPφ )t Ae(−iH+λPφ )t . As the ψn are eigenvectors of H,

ψn , e−iHt AeiHt ψn = ψn , Aψn .

(4.12)

(4.13)

The last two equations prove d 0= ψn , e−iHt AeiHt ψn dt = λ ψn , e−iHt (Pφ A + APφ )eiHt ψn = λ ψn , (Pφ A + APφ )ψn .

(4.14)

Taking the sum over all n, one obtains φ, Aφ = 0, (even Aφ = 0). By the definition of A, this means

lim e(−iH+λPφ )t φ2 = lim φ, e(iH+λPφ )t e(−iH+λPφ )t φ = φ, Aφ = 0, (4.15) t→0

t→0

which proves the first assertion. For the proof of the second part of the theorem, assume that there is a source state φ with Pac φ = 0 and a λ < 0 such that the number of particles grows sublinearly in time, i.e. (4.16) h(t) = e(−iH+λPφ )t φ = e(iH+λPφ )t φ → 0 (t → ∞).

1234



By the same arguments as in (4.11), we have the weak convergence e(iH+λPφ )t e(−iH+λPφ )t → A

(4.17)

with A having the invariance property (4.12), and, by assumption, Aφ = 0.

(4.18)

Following the proof of Theorem 3.1, one can deduce from (4.12) and Aφ = 0 that A = eiHt Ae−iHt

(4.19)

for all t. Thus A and H commute, and so do A and all operators given by the functional calculus for H, Af (H) = f (H)A,

(4.20)

for all bounded Borel functions f on R. We will now apply this fact to a certain choice of f . For ψ = Pac φ = 0, we have µψ (dE) = ρ(E)dE with a positive L1 (R) function ρ. Defining the Borel function fc = 1{ρ ≤ c} · 1 (R \ supp(µpp )) · 1 (R \ supp(µsing ))

(4.21)

for c > 0, we have ψc = fc (H)φ → ψ

(c → ∞)

(4.22)

and Aψc = Afc (H)φ = fc (H)Aφ = 0.

(4.23)

Thus for c fixed large enough, ψc = 0 and 2 lim e(−iH+λPφ )t ψc = ψc , Aψc = 0.

(4.24)

t→∞

Considering the semigroup as a perturbation of the unitary group, we can write e

(−iH+λPφ )t

ψc = e

−iHt

t ψc + λ

e(−iH+λPφ )(t−s) φ φ, e−iHs ψc ds.

(4.25)

0

By the choice of ψc , the scalar product reads

−iHs φ, e ψc = 1{ρ ≤ c}ρ(E)eiEs dE, R

and therefore is an L2 (Rs ) function with norm not larger than > 0 and T > 0 such, that ∞ T

| φ, e−iHs ψc |2 ds <

(4.26) √

2πc. Choosing

(4.27)

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we can split the integral in two parts to see for all t ≥ T T (−iH+λPφ )t ψc ≥ ψc − |λ| h(t − s) φ, e−iHs ψc ds e 0

−|λ|

sup χ∈H,ξ=1

t

χ, e(−iH+λPφ )(t−s) φ φ, e−iHs ψc ds. T

(4.28) 2

Now actually both scalar products in the second integral are L functions, since

2 d (iH+λPφ )s 2 e χ = 2λ φ, e(iH+λPφ )s χ (4.29) ds and thus ∞

2 χ2 χ2 − limt→∞ e(iH+λPφ )t χ2 ≤ . (4.30) φ, e(iH+λPφ )s χ ds = 2|λ| |2λ| 0

Therefore an application of the Cauchy-Schwarz inequality to (4.28) shows T −iHs |λ| (−iH+λPφ )t . (4.31) ψc ≥ ψc − |λ| h(t − s) φ, e ψc ds − e 2 0

For fixed T , the integral in the last line tends to zero as t → ∞ by dominated convergence and the assumption that h(t) → 0. Since one can take arbitrarily small, this would imply lim e(−iH+λPφ )t ψc ≥ ψc > 0, (4.32) t→∞

contradicting (4.24). Thus sublinear growth is not possible for source states φ with Pac φ = 0.

5. Linear Growth 5.1. Proof of Theorem 3.3 Proof. For t > 0 define h(t) = e(−iH+λPφ )t φ. Considering the non-unitary time evolution as a perturbation of the unitary group, one can write e

(−iH+λPφ )t

φ=e

−iHt

t φ+λ

e(−iH+λPφ )(t−s) φ φ, e−iHs φ ds.

(5.1)

0

For 0 < λ < τ −1 , the monotonicity of h(t) implies h(t) ≤ 1 + λτ h(t) h(t) ≤ (1 − λτ )−1 ,

(5.2)

1236



so limt→∞ h(t) exists and is bounded by (1 − λτ )−1 . For λ < 0, limt→∞ h(t) = h∞ exists since h(t) decreases monotonically, and dominated convergence implies the estimate h∞ ≥ 1 − |λ|τ h∞

(5.3)

h∞ ≥ (1 − λτ )−1 .

Setting ρ0 = 0 in (3.1) and using the unitarity of eiHt , this implies the existence d of limt→∞ dt N (t) and the estimates lim

2|λ| d N (t) ≤ 2 dt (1 − λτ )

(λ > 0)

(5.4)

lim

2|λ| d N (t) ≥ 2 dt (1 − λτ )

(λ < 0).

(5.5)

t→∞

and t→∞

The other bounds follow from (3.1) and (3.2). 5.2. Proof of Theorem 3.4 For λ < 0, φ ∈ D(H), one has d (iH−λPφ )t λPφ t e e φ = ieλt Hφ, dt

(5.6)

and thus e

(−iH+λPφ )t

t φ=e φ−i λt

eλs e(−iH+λPφ )(t−s) Hφ ds.

(5.7)

0

Therefore, 1 1 (−iH+λPφ )t → φ ≤ e−|λ|t + e |λ| |λ| so that

lim

t→∞

(t → ∞),

d 2 Nλ (t) ≤ , dt |λ|

(5.8)

(5.9)

which is exactly the |λ|−1 suggested by (3.4). To see why this theorem does not hold for all φ ∈ / D(H), consider the Hilbert space H = L2 (Rx ) with the multiplication operator H = x and the source state φ with φ(x) = √ 1 2 . In this case, the simple form of the overlap φ, e−iHt φ = π(1+x )

e−|t| allows for an explicit representation (λ−1+ix)t −1 e (−iH+λPφ )t −ixt e +1 φ (x) = e φ(x) λ λ − 1 + ix

(5.10)

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for all t ≥ 0. As t → ∞, for λ < τ = 1, eiHt φ strongly converges to the limit function −1 + ix (5.11) φ(x) · λ − 1 + ix √ with norm 1/ 1 − λ, yielding the limit d 2|λ| Nλ (t) = lim = 2, (5.12) lim lim λ→−∞ t→∞ dt λ→−∞ |1 − λ| which is a positive saturation value for the particle flux. Also note that for the critical coupling constant λ = 1, eixt − 1 (−iH+Pφ )t −ixt e φ (x) = e φ(x) 1 + (5.13) ix with

(−iH+Pφ )t 2 φ = 1 + 2t e

(5.14)

and thus N1 (t) = t2 + t. So in this case, the transition region implied by the Theorems 3.3 and 3.5 really consists only of the critical point λ = τ = 1, with quadratical growth of the particle number at the transition point itself, and exponential growth behaviour of the form e(λ−1)t for larger values of λ.

6. Exponential Growth 6.1. Proof of Theorem 3.5 Proof. It suffices to show that for any choice H, φ and 0 <
0 with φ − φE < /8. For this choice of E, take HE = QE H and λ > 4E/ . Writing PφE = |φE φE |, we first consider the operator iH + PφE (6.5) λ restricted to HE . PφE has exactly one eigenvalue in B (1), which is φE 2 ∈ [1 − /4, 1], and thus we have the estimate E 4 1 −1 max iH (PφE − z) /λ ≤ · < . (6.6) z∈Γ λ 3 3 Therefore, the above-mentioned perturbation result applies, and the operator from (6.5) has one single eigenvalue in B (1). Furthermore, one can estimate its resolvent by the Neumann series to obtain −1 −1 −1 1 + iH (P − z) /λ max (iH/λ + PφE − z) ≤ max HE φE z∈Γ z∈Γ −1 · (PφE − z) < (1 − 1/3)−1 · 4/(3 ) = 2/

(6.7)

⊥ on the restricted space HE . On the orthogonal complement HE , we have iH/λ + PφE = iH/λ, so this operator has purely imaginary spectrum, and its resolvent can easily be estimated by 1 −1 −1 < 2. (6.8) max (iH/λ + PφE − z) = max (iH/λ − z) ≤ z∈Γ z∈Γ 1− ⊥ Since the operator iH/λ + PφE on H is decomposed by the pair HE , HE as described in [6], Chapter III., §5.6, one can simply combine the results concerning the subspaces, to obtain that the operator on the whole space still has one eigenvalue in B (1) and obeys the estimate −1 (6.9) max (iH/λ + PφE − z) < 2/ . z∈Γ

Now we are ready for the second application of the perturbation result. As Pφ − PφE ≤ 2φ − φE < /4, we have 2 1 −1 max (Pφ − PφE ) (iH/λ + PφE − z) < · = . (6.10) z∈Γ 4 2

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As a consequence, iH/λ + PφE + (Pφ − PφE ) = iH/λ + Pφ has one eigenvalue of multiplicity one in B (1). 6.2. Proof of Theorem 3.6 Proof. The polynomial p(z) := z 4 + 2iz 3 + (λi − 2)z 2 − (2λ + i)z − (3/2)λi

(6.11)

has a root with positive imaginary part for all λ > 0. To show this let zk , k = 1, . . . , 4 be its roots, counting multiplicities. For all z with p(z) = 0, the product rule implies 4

p(z)p (z) p (z) 1 = = |p(z)|2 p(z) z − zk

(6.12)

k=1

so if all Im zk ≤ 0, the right side, and therefore also p(z)p (z) would never have a positive imaginary part for z ∈ R. But Im p( λ/2)p ( λ/2) = λ/2 + o(λ), (λ → 0), (6.13) so p must have a root with positive imaginary part for small positive λ. Furthermore, p does not have a real root for any positive λ, because one can easily check that the real and imaginary part of p(t) · (t3 − 2it2 − 2t + i) = (t6 + 1) · t + 3λ/2 + λ(2it5 + it3 + 2it)/2 (6.14) cannot equal zero for the same t ∈ R. Since the roots of p depend continuously on λ, this means that there is a z0 with Im z0 > 0 and p(z0 ) = 0 for all λ > 0. The vector ψ ∈ D(H0 ) which we will prove to be a (non-normalized) eigenvector is −1 φ. (6.15) ψ := H0 − z02 Taking CR = {|z| = R, Im z ≥ 0} ∪ (−R, R) as contour of integration, one has by Cauchy’s Integral theorem and by the fact that Im z0 > 0 1 3 dk φ, ψ = 2π (k 6 + 1) (k 2 − z02 ) R 3 1 = lim dz 6 R→∞ 2π (z + 1) (z 2 − z02 ) CR

iz 2 − 2z0 − 3i/2 =i 4 0 3 z0 + 2iz0 − 2z02 − iz0 i (6.16) =− , λ where we have used p(z0 ) = 0 in the last line. Therefore, ψ is an eigenvector: −1 z02 z2 (iH0 +λPφ ) ψ = iH0 H0 − φ+λ φ, ψ φ = iφ+i 0 ψ−iφ = α(λ)ψ. (6.17) 2 2

1240



Since H0 has no eigenvectors, we have Pφ ψ = 0, and thus ψ2 Re α(λ) = Re ψ, (iH0 + λPφ ) ψ = λ| φ, ψ |2 > 0.

(6.18)

7. Restricted Limits 7.1. Independence (Proof of Theorem 3.7) For the proof of theorem 3.7, we start with two lemmas: Lemma 7.1. For λ < 0, φ an arbitrary normalized source state, e(−iH0 +λPφ )t → 0

(7.1)

in the weak operator topology as t → ∞.

Proof. Let ψ, χ ∈ H. We have to show that ψ, e(−iH0 +λPφ )t χ → 0 as t → ∞. To do so, we write the scalar product as t

(−iH0 +λPφ )t −iH0 t ψ, eiH0 (t−s) φ φ, e(−iH0 +λPφ )s χ ds ψ, e χ = ψ, e χ +λ 0

(7.2) The first term on the right-hand side tends to 0 as t → ∞ by the RiemannLebesgue Lemma (cf. [9], p. 18, Example 1.1, note that Hac (H0 ) = H, i.e. H0 has only absolutely continuous spectrum). The second term will only be analyzed for all ψ from a dense subset of H. Since ψ, e(−iH0 +λPφ )t χ depends continuously on ψ uniformly in t > 0, this is enough to prove the assertion. According to Theorem 1.3 in [9] (p. 20), there is a subset K(H0 ) which is dense in Ha.c. (H0 ) = H such that

ψ, eiH0 t φ ∈ L2 (Rt ) (7.3) for all ψ ∈ K(H0 ). Thus, by (4.30), the integral on the right side of (7.2) can be estimated by the convolution of two L2 (R) functions, and therefore tends to 0 as t → ∞. Lemma 7.2. For a set Ω ⊂ Rd with finite Lebesgue measure µ(Ω) < ∞, and λ < 0 PΩ e(−iH0 +λPφ )t → 0 in strong operator topology as t → ∞. Proof. If q > d2 , q ≥ 2, one has 1Ω (x) ∈ Lq (Rdx ) and

(7.4)

p2 2

−1 +i ∈ Lq (Rdp ), and by

Theorem XI.20 in [11], (p. 47), −1 (−i∇)2 −1 +i ∈ Jq = {A ∈ B(H) : tr (|A|q ) < ∞.} PΩ (H0 + i) = 1Ω (x) 2 (7.5)

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1241

−1

Thus PΩ (H0 + i) is compact. First, take ψ ∈ D(H0 ) = H 2 (Rd ) and apply the fact that a strongly continuous group commutes with its generator, PΩ e(−iH0 +λPφ )t ψ = PΩ (H0 + i)−1 (H0 + iλPφ + i − iλPφ )e(−iH0 +λPφ )t ψ = PΩ (H0 + i)−1 e(−iH0 +λPφ )t (H0 + iλPφ + i)ψ − iλPΩ (H0 + i)−1 Pφ e(−iH0 +λPφ )t ψ.

(7.6)

Both terms on the right-hand side tend to 0 in · H as t → ∞. The first one −1 because of Lemma 7.1 and the fact that the compact operator PΩ (H0 + i) maps weakly convergent sequences to convergent ones, and the second one directly by Lemma 7.1. Since D(H0 ) is dense in H, this already implies the claim by continuity. Now one can easily show that the initial density ρ0 does not contribute to the value of limt→∞ tr(PΩ ρF (t)PΩ ) in the fermionic case. Proof. (Theorem 3.7) Since s − limt→∞ PΩ e(−iH0 +λPφ )t = 0 and ρ0 ∈ J1 (the trace class), Lemma 3.1 from [8] applies, and one has: PΩ e(−iH0 +λPφ )t ρ0 e(iH0 +λPφ )t PΩ tr ≤ PΩ e(−iH0 +λPφ )t ρ0 tr → 0 as t → ∞.

(7.7)

7.2. Existence (Proof of Theorems 3.8, 3.9) The main idea for the proofs of theorems 3.8 and 3.9 is the formula d tr(PΩ ρ(t)PΩ ) = 2|λ|PΩ e(−iH0 +λPφ )t φ2 , dt which is obtained from (2.1) when setting ρ0 = 0.

(7.8)

Proof. (Theorem 3.8) The assumption of sublinear particle production (which is always the case for fermions) implies by (3.1) and (3.2), that ∞

| φ, e(−iH0 +λPφ )s φ |2 ds < ∞. (7.9) 0

Furthermore, PΩ e(−iH0 +λPφ )t φ t

−iH0 t = φ+λ PΩ e−iH0 (t−s) φ φ, e(−iH0 +λPφ )s φ ds PΩ e 0

≤ PΩ e

−iH0 t

t φ + |λ|

PΩ e−iH0 (t−s) φ φ, e(−iH0 +λPφ )s φ ds

(7.10)

0

By assumption (3.12), PΩ e−iH0 t φ ∈ L1 ∩ L∞ (R+ t ). Thus, the first term in (7.10) obviously is in L2 (R+ ), and by (7.9), the second one is a convolution of an L1 (R+ ) t

1242



function with an L2 (R+ ) function, and therefore also contained in L2 (R+ t ). By (7.8), this proves the assertion. ˆ be the support of φ. ˆ Since Fourier transProof. (Theorem 3.9) Let K := supp(φ) form diagonalizes H0 , e

(−iH0 +λPφ )t

implies supp Fe

φ=e

−iH0 t

(−iH0 +λPφ )t

+ λe

−iH0 t

t

eiH0 s φ φ, e(−iH0 +λPφ )s φ ds

(7.11)

0

φ ⊂ K for all times t ≥ 0. This means

PΩ e(−iH0 +λPφ )t φ = PΩ 1K (−i∇)e(−iH0 +λPφ )t φ,

(7.12)

where PΩ 1K (−i∇) is a Hilbert-Schmidt operator with corresponding norm PΩ 1K (−i∇)J2 ≤ (2π)−d/2 (|Ω||K|)

1/2

(7.13)

by theorem XI.20 in [11], p. 47. Applying (4.30) to a singular value decomposition of PΩ 1K (−i∇) one obtains ∞ lim tr(PΩ ρF (t)PΩ ) = 2|λ|

t→∞

PΩ e(−iH0 +λPφ )t φ2 dt

0

∞ = 2|λ|

PΩ 1K (−i∇)e(−iH0 +λPφ )t φ2 dt

0

≤ PΩ 1K (−i∇)2J2 ≤ (2π)−d |Ω||K|.

(7.14)

8. Semi-classical Limit (Proof of Theorem 3.10) To obtain a semi-classical limit, it is important to observe that the distinction between the bosonic and fermionic character of the semigroups disappears in the limit → 0. Lemma 8.1. For any φ, ψ ∈ H, φ = 1 one has for all t, c ∈ R: lim exp(−iH0 t/ + cPφ t)ψ − exp(−iH0 t/ )ψ = 0.

→0

(8.1)

Proof. One can write this difference as exp(−iH0 t/ + cPφ t)ψ − exp(−iH0 t/ )ψ t = c φ, exp(−iH0 s/ + cPφ s)ψ exp(−iH0 (t − s)/ )φ ds. 0

(8.2)

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For the scalar product in the integral of (8.2), one has φ, exp(−iH0 s/ + cPφ s)ψ = φ, exp(−iH0 s/ )ψ s + c φ, exp(−iH0 r/ + cPφ r)ψ φ, exp(−iH0 (s − r)/ )φ dr.

(8.3)

0

The first term in (8.3) tends to zero for all s = 0 as → 0 by the RiemannLebesgue Lemma. By the same argument, the integrand of the second term tends to zero almost everywhere, and it is bounded by e|cs| . By dominated convergence this implies lim φ, exp(−iH0 s/ + cPφ s)ψ = 0.

→0

(8.4)

for all s = 0. But dominated convergence also applies to (8.2) and the assertion is proven. To show Theorem 3.10, it is useful to have the inverse for the Wigner transform, which is given by the Weyl quantization. For a function a ∈ L2 Rdx × Rdp one can define an operator on H by x+y 1 (Op[a]ψ)(x) = , p eip·(x−y) ψ(y) dy dp a (8.5) (2π)d 2 d Rd p Ry

Up to a factor, this is a unitary map from L2 Rd × Rd to the space J2 of Hilbert-Schmidt operators on H (cf. Equation (3.3) in [8]): a(x, p)b(x, p)dx dp = (2π)d tr (Op[a]∗ Op[b]) . (8.6) R2d

With this definition, one has for all Hilbert-Schmidt operators κ, Op [W [κ]] =

1 κ, (2π)d

(8.7)

where we identified operators and kernels. This is the main tool for the proof of Theorem 3.10. Proof. Let θ ∈ D RdX × RdP = C0∞ RdX × RdP be a test function. We have to establish the existence of f (X, P, T )θ(X, P ) dXdP. (8.8) lim →0 R2d

1244



By (8.6) one can transform this integral to f (X, P, T )θ(X, P ) dX dP = −d W [ρ (T / )] (X/ , P ) θ(X, P ) dX dP R2d

R2d

W [ρ (T / )] (x, p) θ( x, p)dx dp

= R2d

= tr (ρ (T / ) Op[θ ]), ∗

(8.9)

where Op[a] = Op[a] is used, and θ (x, p) = θ( x, p). By (2.1), the following expression holds, ρ (T / ) = sgn(c) (exp(−iH0 T / + cPφ T ) exp(iH0 T / + cPφ T ) − 1) + exp(−iH0 T / + cPφ T )ρ0 exp(iH0 T / + cPφ T ).

(8.10)

To keep notation simple, we first assume ρ0 = 0. The estimate (|ψ ψ| − |χ χ|) tr ≤ ψ − χ · (ψ + χ) yields, together with Lemma 8.1 and the dominated convergence theorem, T ρ (T / ) − 2|c|

e−iH0 S/ Pφ eiH0 S/ dStr

0

T ≤ 2|c|

e−iH0 S/+cPφ S Pφ eiH0 S/+cPφ S − e−iH0 S/ Pφ e+iH0 S/ tr dS

0

|cT |

≤ 2|c| e

T +1 e−iH0 S/+cPφ S φ−e−iH0 S/ φ dS → 0

( → 0)

(8.11)

0

Furthermore, by the theorem of Calder´ on-Vaillancourt (Theorem 2.8.1 in [7]), the operators Op[θ ] are uniformly bounded in B(H) as tends to zero, and thus, by the linearity and cyclicity of the trace, T lim tr (ρ (T / ) Op[θ ]) − 2|c| tr eiH0 S/ Op[θ ]e−iH0 S/ Pφ dS = 0. (8.12) →0 0

The Heisenberg time evolution applied to Op[θ ] can be carried over to phase space by defining ηS (x, p) := θ (x + pS/ , p) = θ( x + pS, p)

(8.13)

eiH0 S/ Op[θ ]e−iH0 S/ = Op[ηS ].

(8.14)

so that (cf. [8])

Now Lemma 3.2 from [8] applies to ηS , stating that the operators Op[ηS ]≥0 are uniformly bounded in B(H) and converge strongly to Op[ηS0 ] as tends to zero.

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1245

Thus also tr (Op[ηS ]Pφ ) converges and, by dominated convergence, the desired limit exists, lim

→0 R2d

T f (X, P, T )θ(X, P ) dX dP = lim 2|c| →0

tr (Op[ηS ]Pφ ) dS 0

T = 2|c|

tr Op[ηS0 ]Pφ dS.

(8.15)

0

By (8.13), ηS0 does not depend on X. Op[ηS0 ] is a multiplication operator in momentum space and thus T 2|c|

tr Op[ηS0 ]Pφ dS = 2|c|

0

T

ˆ )|2 dS. θ(P S, P )|φ(P

(8.16)

0 Rd

At this point of the proof, one should remark that ηS0 is no longer in L2 Rd × Rd so Op[ηs0 ] is not given by our definition of the Weyl quantization (8.5). But it can be obtained as the quantization of a C ∞ Rd × Rd symbol with all derivatives bounded. This construction is described, for example, in Chapter 2 of [7]. Next, we allow a general initial condition, i.e. ρ0 is an arbitrary trace-norm bounded sequence of density matrices such that (3.17) holds for some distribution g on macroscopic phase space. Concentrating on the last term in (8.10), one has to evaluate tr e−iH0 T /+cPφ T ρ0 eiH0 T /+cPφ T Op[θ ] .

(8.17)

As before, one first has to check that the action of the semigroups can be substituted by the free evolution, e−iH0 T /+cPφ T ρ0 eiH0 T /+cPφ T Op[θ ] − e−iH0 T / ρ0 eiH0 T / Op[θ ]tr ≤ eiH0 T / e−iH0 T /+cPφ T ρ0 eiH0 T /+cPφ T e−iH0 T / − ρ0 tr · Op[θ ]B(H) ≤ 2|c|e|cT |

T

Pφ e−iH0 S/+cPφ S ρ0 tr dS · Op[θ ]B(H) .

(8.18)

0

The operator norm in the last line is bounded uniformly by the Calder´ on– Vaillancourt theorem. For the integral, Lemma 8.1 and the uniform boundedness of ρ0 tr allow an application of dominated convergence,

1246


T lim sup →0

Pφ e−iH0 S/+cPφ S ρ0 tr dS = lim sup →0

0

T


Pφ e−iH0 S/ ρ0 tr dS

0

T ρ0 eiH0 S/ φ dS = 0. (8.19)

= lim sup →0

0

The last equality is seen as follows: Since the spectrum of H0 is absolutely continuous, we first can consider a state φ with bounded spectral density, i.e. dµφ (E) ≤ C2 (8.20) dE

for some C > 0. Then for any ψ ∈ H, g(t) = ψ, eiH0 t φ is the Fourier transform of an L2 (R) function, satisfying gL2 ≤

√

2πCψ.

(8.21)

Therefore, the Cauchy-Schwarz inequality yields T

√ √ ψ, eiH0 S/ φ dS ≤ 2πT Cψ .

(8.22)

0

Now we can write ρ0 =

an |ψn ψn |

(8.23)

n∈N

with (ψn ) an -dependent orthonormal basis and Then (8.22) implies T ρ0 eiH0 S/ φdS

≤

n∈N

0

|an |

n∈N

|an | ≤ K uniformly in .

T

iH0 S/ φ dS ψn , e 0

√ √ ≤ K 2πT C → 0

( → 0).

(8.24)

Approximating general φ with states φC of bounded spectral density, one can use the uniform boundedness of the operators ρ0 eiH0 S/ to show T ρ0 eiH0 S/ φ dS ≤ T Kφ − φC ,

lim sup →0

0

(8.25)

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1247

where the right hand side can be made arbitrarily small. Thus one can in fact replace the semigroups by the unitary group, and then apply (8.6), lim tr e−iH0 T /+cPφ T ρ0 eiH0 T /+cPφ T Op[θ ] →0 = lim tr e−iH0 T / ρ0 eiH0 T / Op[θ ] →0

= lim tr (ρ0 Op[ηT ]) →0 W [ρ0 ] (x, p) θ( x + pT, p) dx dp = lim →0 R2d

= lim

→0 R2d

−d W [ρ0 ] (X/ , P ) θ(X + P T, P ) dX dP

=

g(X, P )θ(X + P T, P ) dX dP R2d

g(X − P T, P )θ(X, P ) dX dP.

=

(8.26)

R2d

Adding (8.16) and (8.26), one obtains the limit for general initial conditions, lim

→0 R2d

f (X, P, T )θ(X, P ) dX dP

T g(X − P T, P )θ(X, P ) dX dP + 2|c|

= R2d

0

= f (T ), θ

ˆ )|2 dP dS θ(P S, P )|φ(P

0 Rd

(8.27)

D ,D

The distribution in the last line is given by T

0

f (X, P, T ) = g(X − P T, P ) + 2|c|

ˆ )|2 dS. δ(X − P S)|φ(P

(8.28)

0

It remains to show (3.19). By (3.17), the initial value is f 0 (X, P, 0) = lim −d W [ρ0 ] (X/ , P ) = g(X, P ). →0

(8.29)

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By testing with an arbitrary θ ∈ D RdX × RdP , one can check that f 0 also solves the differential equation, d 0 f (T ), θ D ,D dT ⎛ ⎞ d ⎝ = g(X, P )θ(X + P T, P ) dX dP ⎠ dT R2d

⎛

⎞ T d ⎝ ˆ )|2 dP dS ⎠ + 2|c| θ(P (T − S), P )|φ(P dT 0 Rd = g(X, P )P · ∇X θ(X + P T, P ) dX dP R2d

T +2|c|

ˆ )|2 dP dS P · ∇X θ(P (T − S), P )|φ(P

0 Rd

+2|c|

ˆ )|2 dP θ(0, P )|φ(P

Rd

ˆ )|2 , θ = f 0 (T ), P · ∇X θ D ,D + 2|c|δ(X)|φ(P D ,D

ˆ )|2 , θ = −P · ∇X f 0 (T ) + 2|c|δ(X)|φ(P . D ,D

(8.30)

Acknowledgements We would like to thank V. S. Buslaev and A. Komech for instructive and encouraging discussions. M. Butz acknowledges support from the ENB graduate program TopMath and a grant sponsored by Max Weber-Programm Bayern.

References [1] Alicki, R., Fannes, M., Haegeman, B., Vanpeteghem, D.: Coherent transport and dynamical entropy for fermionic systems. J. Stat. Phys. 113, 549–574 (2003) [2] Alicki, R., Lendi, K.: Quantum dynamical semigroups and applications, Lecture Notes in Physics, vol. 717. Springer, Berlin (2007) [3] Attal, S., Joye, A., Pillet, C.-A.: Open Quantum Systems III: Recent Developments. Lecture Notes in Mathematics, vol. 1882. Springer, Berlin (2006) [4] Damoen, B., Vanheuverzwijn, P., Verbeure, A.: Completely positive maps on the CCR. Lett. Math. Phys. 2, 161–166 (1978) [5] Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London (1976)

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[6] Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976) [7] Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Springer, New York (2002) [8] Nier, F.: Asymptotic analysis of a scaled Wigner equation and quantum scattering. Transp. Theory Stat. Phys. 24, 591–628 (1995) [9] Perry, P.A.: Scattering Theory by the Enss Method. Harwood Academic Publishers, New York (1983) [10] Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Academic Press, London (1975) [11] Reed, M., Simon, B.: Methods of Modern Mathematical Physics III. Academic Press, London (1979) Maximilian Butz, Herbert Spohn Technische Universit¨ at M¨ unchen Zentrum Mathematik Boltzmannstraße 3 85748 Garching Germany e-mail: [email protected]; [email protected] Communicated by Jean Bellissard. Received: August 31, 2009. Accepted: December 2, 2009.

Ann. Henri Poincaré 10 (2010), 1251–1284 c 2009 Birkh¨ auser Verlag AG, Basel/Switzerland 1424-0637/10/071251-34, published online December 16, 2009 DOI 10.1007/s00023-009-0017-8


Repeated and Continuous Interactions in Open Quantum Systems Laurent Bruneau, Alain Joye and Marco Merkli Abstract. We consider a finite quantum system S coupled to two environments of different nature. One is a heat reservoir R (continuous interaction) and the other one is a chain C of independent quantum systems E (repeated interaction). The interactions of S with R and C lead to two simultaneous dynamical processes. We show that for generic such systems, any initial state approaches an asymptotic state in the limit of large times. We express the latter in terms of the resonance data of a reduced propagator of S + R and show that it satisfies a second law of thermodynamics. We analyze a model where both S and E are two-level systems and obtain the asymptotic state explicitly (at lowest order in the interaction strength). Even though R and C are not directly coupled, we show that they exchange energy, and we find the dependence of this exchange in terms of the thermodynamic parameters. We formulate the problem in the framework of W ∗ -dynamical systems and base the analysis on a combination of spectral deformation methods and repeated interaction model techniques. We analyze the full system via rigorous perturbation theory in the coupling strength, and do not resort to any scaling limit, like e.g. weak coupling limits, or any other approximations in order to derive some master equation.

A. Joye Supported partially by Institute for Mathematical Sciences, National University of Singapore, through the program “Mathematical Horizons for Quantum Physics”, during which parts of this work have been performed. Partially supported by the Ministère Fran¸cais des Affaires ´ Etrang` eres through a séjour scientifique haut niveau. M. Merkli Supported partially by Institute for Mathematical Sciences, National University of Singapore, through the program “Mathematical Horizons for Quantum Physics”, during which parts of this work have been performed. Partially supported by the Ministère Fran¸cais des Affaires ´ Etrang` eres through a séjour scientifique haut niveau. Supported by NSERC under Discovery Grant 205247.

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1. Introduction Over the last years, the rigorous study of equilibrium and non-equilibrium quantum systems has received much and renewed attention. While this topic of fundamental interest has a long tradition in physics and mathematics, conventionally explored via master equations [6,9], dynamical semi-groups [3,6] and algebraic scattering theory [16,33], many recent works focus on a quantum resonance theory approach. The latter has been applied successfully to systems close to equilibrium [18,27–29] and far from equilibrium [19,26]. In both situations, one of the main questions is the (time-) asymptotic behaviour of a quantum system consisting of a subsystem S interacting with one or several other subsystems, given by thermal reservoirs R1 , . . . , Rn . It has been shown that if S + R starts in a state in which the reservoir is in a thermal state at temperature T > 0 far away from the system S, then S + R converges to the joint equilibrium state at temperature T , as time t → ∞. This phenomenon is called return to equilibrium (see also [23] for the situation where several equilibrium states at a fixed temperature coexist). In case S is in contact with several reservoirs having different temperatures (or different other macroscopic properties), the whole system converges to a non-equilibrium stationary state (NESS). The success of the resonance approach is measured not only by the fact that the above-mentioned phenomena can be described rigorously and quantitatively (convergence rates), but also by that the asymptotic states can be constructed (via perturbation theory in the interaction) and their physical and mathematical structure can be examined explicitly (entropy production, heat- and matter fluxes). One of the main advantages of this method over the usual master equation approach (and the related van Hove limit) is that it gives a perturbation theory of the dynamics which is uniform in time t ≥ 0. While the initial motivation for the development of the dynamical resonance theory was the investigation of the time-asymptotics, the method is becoming increasingly refined. It has been extended to give a precise picture of the dynamics of open quantum systems for all times t ≥ 0, with applications to the phenomena of decoherence, disentanglement, and their relation to thermalization [24,27–29]. An extension to systems with rather arbitrary time-dependent Hamiltonians has been presented in [30] (see also [2] for time-periodic systems). A further direction of development is a quantum theory of linear response and of fluctuations [21]. In certain physical setups, the reservoir has a structure of a chain of independent elements, C = E1 + E2 + · · ·. An example of such a system is the so-called “one-atom maser” [25], where S describes the modes of the electromagnetic field in a cavity, interacting with a beam C of atoms Ej , shot one by one into the cavity and interacting for a duration τj > 0 with it. A mathematical treatment of the oneatom maser is provided in [14]. Another instance of the use of such systems is the construction of reservoirs made of “quantum noises” by means of adequate scaling limits of the characteristics of the chain C and its coupling with S, which lead to certain types of master equations as well as Quantum Langevin equations [1,4–7].

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The central feature of such systems is that S interacts successively with independent elements Ej constituting a reservoir. This independence implies a markovian property which simplifies the mathematical treatment considerably. In essence it enables one to express the dynamics of S at time t = τ1 + · · · + τN by a propagator of product form M1 (τ1 ) · · · MN (τN ), where each Mj (τj ) encodes the dynamics of S with a fixed element Ej . In case each element Ej is physically the same and each interaction is governed by a fixed duration τ (and a fixed interaction operator), the dynamics is given by M (τ )N and the asymptotics is encoded in the spectrum of the reduced dynamics operator M (τ ) [11]. An analysis for non-constant interactions is more involved. It has been carried out in [12,13] for systems with random characteristics (e.g. random interaction times). See also [31] for related issues. In both the deterministic and the random settings, the system approaches a limit state as t → ∞, called a repeated interaction asymptotic state (RIAS), whose physical and mathematical properties have been investigated explicitly. In the present work we make the synthesis of the above two situations. We consider a system S interacting with two environments of distinct nature (we are thus in a non-equilibrium situation). On the one hand, S is coupled in the repeated interaction way to a chain C = E + E + · · ·, and on the other hand, S is in continuous contact with a heat reservoir R. Such a system describes for example a “one-atom maser” in which one also takes into account some losses in the cavity, the latter being not completely isolated from the exterior world, e.g. from the laboratory [15]. It is assumed that C and R do not interact directly. This assumption is physically reasonable. Indeed, again for the “one-atom maser” experiment, the idea is that the atoms are ejected from an oven one by one just before they interact with the cavity and moreover the atom-field interaction time τ is typically much smaller than the damping time due to the presence of the heat reservoir. Therefore, the atoms do not have enough time to feel the effects of the reservoir before and during their interaction with the field. Our goal is to construct the asymptotic state of the system and to analyze its physical properties. The paper is organized as follows. We present in Sect. 1.2 our results on the convergence to, and form of the asymptotic state, in Sect. 1.3 the thermodynamic properties of it, and in Sect. 1.4 we present the analysis of an explicit model. The proofs are given in the next sections. Namely, in Sect. 2 we prove the results of Sect. 1.2, i.e. Theorem 1.3. In Sect. 3 we show how to reduce the analysis of the Fermi Golden Rule (one of the main assumptions of Theorem 1.3) to standard perturbation theory of discrete eigenvalues. In Sect. 4 we prove the results on thermodynamic properties of the asymptotic state. Finally, in Sect. 5 we give some details about the explicit model presented in Sect. 1.4. 1.1. Description of the System The following is a unified description of S, R, C in the language of algebraic quantum statistical mechanics (we refer the reader to e.g. [32] for a more detailed exposition). For the reader’s convenience, we start from the C ∗ -dynamical systems formalism. A C ∗ -dynamical system is a pair (A, α) where A is a C ∗ -algebra

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(describing the observables of the physical system under consideration) and t → αt is a strongly continuous group of ∗-automorphisms of A (describing the evolution of the observables). A state of the system is described by a positive linear functional ω on A satisfying ω(1l) = 1. Following [19], a triple (A, α, ω), where ω is an invariant state (i.e. ω ◦ αt ≡ ω), is called a quantum dynamical system. Concrete examples of such quantum dynamical systems are given in Sect. 1.4. Each subsystem # = S, R, E is described by a quantum dynamical system (A# , α# , ω# ). The “reference” states ω# determine the macroscopic properties of the systems,1 e.g. they are KMS states at some inverse temperature β# . We also assume that they are faithful states, i.e. for any A ∈ A# , ω# (A∗ A) = 0 ⇒ A = 0. In our paper, we will study the (time) asymptotic behaviour of the system using a spectral approach. For that purpose, it is convenient to have a “Hilbert space description” of the system. Such a description is easy to obtain via the GNSrepresentation (H# , π# , Ψ# ) of the algebras A# associated to the states ω# . Since the ω# are faithful, the π# are injections and we can identify A# and π# (A# ) (in the rest of the paper we will therefore simply write A for π(A)). We set M# = π# (A# ) ⊂ B(H# ). The M# form the von Neumann algebras of observables. Finally, by construction the representative vectors Ψ# are cyclic for M# [10], and we assume that they are also separating vectors for M# , i.e. AΨ# = 0 ⇒ A = 0 for any A ∈ M# (note that since ω# is faithful, this is automatic when A ∈ π# (A# )). Typically, the Ψ# describe the equilibrium states at any fixed temperature T# > 0. We assume that dim HS < ∞ (i.e. AS was a matrix algebra Mn (C)) and dim HE may be finite or infinite. R being a reservoir, its Hilbert space is assumed to be infinite-dimensional, dim HR = ∞. The free dynamics α# of each constituent is implemented in the GNS-representation by the so-called Liouville operators L# , i.e., the Heisenberg evolution of an observable A ∈ M# at time t is given t (A)) = eitL# π# (A)e−itL# . Since by eitL# Ae−itL# . In other words we have π# (α# the ω# were invariant states, we can also chose the Liouville operators L# so that L# Ψ# = 0 (actually such an L# is unique, see e.g. [10]). The Hilbert space HC of the chain is the infinite tensor product of factors HE , taken with respect to the stabilizing sequence ΨC = ⊗j≥1 ΨE , i.e. HC is obtained by taking the completion of the vector space of finite linear combinations of the form ⊗j≥1 ψj , where ψj ∈ HE , ψj = ΨE except for finitely many indices, in the norm induced by the inner product ⊗j ψj , ⊗j φj = ψj , φj HE . j

The algebra of observables MC of the chain is the von Neumann algebra MC = ⊗m≥1 ME 1

In other words, it determines the folium of normal states. If the Hilbert space is finite-dimensional then the set of normal states is unique, but for infinite systems different classes of normal states are determined by different macroscopic parameters, such as the temperature.

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acting on HC , which is obtained by taking the weak closure of finite linear combinations of operators ⊗j≥1 Aj , where Aj ∈ ME and Aj = 1lHE except for finitely many indices. In summary, the non-interacting system is given by a von Neumann algebra M = MS ⊗ MR ⊗ MC , acting on the Hilbert space H = HS ⊗ HR ⊗ HC , and its dynamics is generated by the Liouvillian LEk . L0 = LS + LR +

(1.1)

k≥1

Here we understand that LEk acts as the fixed operator LE on the kth factor of HC , and we do not display obvious factors 1l. The operators governing the couplings between S and E and S and R are given by VSE ∈ MS ⊗ ME

and VSR ∈ MS ⊗ MR

respectively, and the total interaction is V (λ) = λ1 VSR + λ2 VSE ∈ MS ⊗ MR ⊗ ME ,

(1.2)

odinger) dynamwhere λ1 , λ2 are coupling constants (λ = (λ1 , λ2 )). The full (Schr¨ ics is ψ → U (m)ψ,

(1.3)

U (m) = e−iτ (L0 +Vm ) e−iτ (L0 +Vm−1 ) · · · e−iτ (L0 +V1 ) ,

(1.4)

where U (m) is the unitary map

τ > 0 being the time-scale of the repeated interaction and Vk being the operator V (λ), (1.2), acting nontrivially on HS , HR and the kth factor HE of HC (we will also write Lm = L0 +Vm ). We discuss here the dynamics (1.3) at discrete time steps mτ only, a discussion for arbitrary continuous times follows in a straightforward manner by decomposing t = mτ + s, s ∈ [0, τ ), see [11]. Explicit form of finite systems and thermal reservoirs. (A)

Finite systems. We take S (and possibly E) to be finite, i.e. hS = Cn for some n. The Hamiltonian of S is given by hS , acting on hS . In other words, AS = Mn (C) and αSt (A) = eithS Ae−ithS . The (Gelfand–Naimark–Segal) Hilbert space, the observable algebra and the Liouville operator are given by HS = hS ⊗ hS ,

MS = B(HS ) ⊗ 1l,

LS = hS ⊗ 1l − 1l ⊗ hS .

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The reference state is chosen to be the trace state, represented by dim HS 1 ΨS = √ ϕj ⊗ ϕ j , dim HS j=1

(B)

where {ϕj } is an orthonormal basis of hS diagonalizing hS . Thermal reservoirs. We take R (and possibly E) to be a thermal reservoir of free Fermi particles at a temperature T > 0, in the thermodynamic limit. Its description was originally given in the work by Araki and Wyss [8]; see also [18] and [30], Appendix A, for an exposition close to ours. We give directly the description in the GNS-representation, we provide a precise derivation of this formalism starting from the usual description of a reservoir of non-interacting and non-relativistic fermions via C ∗ -dynamical systems in Sect. 1.4. The Hilbert space is the anti-symmetric Fock space n P− [h⊗j=1 ] HR = Γ− (h) := n≥0

over the one-particle space h = L2 (R, G),

(1.5)

where P− is the orthonormal projection onto the subspace of anti-symmetric functions, and G is an ‘auxiliary space’ (typically an angular part like L2 (S 2 )). In this representation, the one-particle Hamiltonian h is the operator of multiplication by the radial variable (extended to negative values) s ∈ R of (1.5), i.e. for ϕ ∈ L2 (R, G) (hϕ)(s) = sϕ(s). The Liouville operator is the second quantization of h, LR = dΓ(h) :=

n

hj ,

(1.6)

n≥0 j=1 n

where hj is understood to act as h on the jth factor of P− [L2 (h)]⊗j=1 and trivially on the other ones. The von Neumann algebra MR is the subalgebra of B(HR ) generated by the thermal fermionic field operators (at inverse temperature β), represented on HR by 1 ϕ(gβ ) = √ [a∗ (gβ ) + a(gβ )] . 2 Here, we define for g ∈ L2 (R+ , G) gβ (s) =

1

e−βs

+1

g(s) g(−s)

if s ≥ 0 if s < 0.

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We choose the reference state to be thermal equilibrium state, represented by the vacuum vector of HR , ΨR = Ω. 1.2. Convergence to Asymptotic State One of our main interests is the behaviour of averages ρ(U (m)∗ Om U (m)) as m → ∞, where ρ is any (normal) initial state of the total system, and where Om is a so-called instantaneous observable [11–13]. Definition 1.1. An observable Om is called an instantaneous observable if there exist ASR ∈ MS ⊗ MR and Bj ∈ ME , j = −l, . . . , r, where l, r ≥ 0 are integers such that Om = ASR ⊗m+r j=m−l ϑj (Bj−m ) ∈ M,

(1.7)

where ϑj (B) is the observable of M which acts as B on the jth factor of HC , and trivially everywhere else (ϑj is the translation to the jth factor). Note that an instantaneous observable is a time-dependent one. It may be viewed as a train of fixed observables moving with time along the chain C so that at time m it is “centered” at the m-th factor HE of HC , on which it acts as B0 . If O acts trivially on the elements of the chain, then the corresponding instantaneous observable is constant and Om = O. However, in order to be able to reveal interesting physical properties of the system, instantaneous observables are needed. For instance observables measuring fluxes of physical quantities (like energy, entropy) between S and the chain involve instantaneous observables acting non-trivially on Em and on Em+1 , which corresponds to nontrivial ASR and B0 , B1 . We denote the Heisenberg dynamics of observables by (see (1.4)) αm (Om ) = U (m)∗ Om U (m).

(1.8)

The total reference vector Ψ0 = ΨS ⊗ ΨR ⊗ ΨC

(1.9)

is cyclic and separating for the von Neumann algebra M. We also introduce, for later purposes, the projector PSR = 1lSR ⊗|ΨC ΨC |, which range we often identify with HSR = HS ⊗ HR . Let J and ∆ be the modular conjugation and the modular operator associated to the pair (M, Ψ0 ) [10]. In order to represent the dynamics in a convenient way (using a so-called C-Liouville operator), we make the following assumption. H1 The interaction operator V (λ), (1.2), satisfies ∆1/2 V (λ)∆−1/2 ∈ MS ⊗ MR ⊗ ME . Since we will be using analytic spectral deformation methods on the factor HR of H, we need to make a regularity assumption on the interaction. Let R θ → T (θ) ∈ B(HR ) be the unitary group defined by T (θ) = Γ(e−θ∂s )

on Γ− (L2 (R, G)),

(1.10)

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where for any f ∈ L2 (R, G), (e−θ∂s f )(s) = f (s − θ). In the following, we will use the notation T (θ) = 1lS ⊗ T (θ) ⊗ 1lE for simplicity. Note that T (θ) commutes with all observables acting trivially on HR , in particular with PSR . Also, we have T (θ)ΨR = ΨR for all θ. The spectral deformation technique relies on making the parameter θ complex. H2 The coupling operator WSR := VSR − J∆1/2 VSR ∆−1/2 J is translation analytic in a strip κθ0 = {z : 0 < z < θ0 } and strongly continuous on the real axis. More precisely, there is a θ0 > 0 such that the map R θ → T −1 (θ)WSR T (θ) = WSR (θ) ∈ MS ⊗ MR , admits an analytic continuation into θ ∈ κθ0 which is strongly continuous as θ ↓ 0, and which satisfies sup 0≤θ 0 (depending on θ1 in general) such that for all λ with 0 < |λ| < λ0 , σ(Mθ1 (λ)) (spectrum) lies inside the complex unit disk, and σ(Mθ1 (λ)) ∩ S = {1}, the eigenvalue 1 being simple and isolated. (S is the complex unit circle.) This condition is verified in practice by perturbation theory (small λ). It is also possible to prove that if the spectral radius of Mθ1 (λ) is determined by discrete eigenvalues only, then the spectrum of Mθ1 (λ) is automatically inside the unit disk (see Proposition A.3). Since the spectrum is a closed set the FGR condition implies that apart from the eigenvalue 1 the spectrum is contained in a disk of radius e−γ < 1. FGR

Theorem 1.3 (Convergence to asymptotic state). Assume that assumptions H1, H2 and FGR are satisfied. Then there is a λ0 > 0 s.t. if 0 < |λ| < λ0 , the following holds. Let ρ be any normal initial state on M, and let Om be an analytic instantaneous observable of the form (1.7). Then r ΨE |Bj ΨE , lim ρ (αm (Om )) = ρ+,λ PSR αl+1 ASR ⊗0j=−l Bj PSR

m→∞

j=1

(1.12) where ρ+,λ is a state on MS ⊗ MR , PSR is the orthogonal projection onto the subspace HS ⊗ HR , and where αl is the dynamics (1.8). Moreover, for analytic A ∈ M, we have the representation ρ+,λ (PSR APSR ) = ψθ∗1 (λ)|T (θ1 )−1 PSR APSR ΨS ⊗ ΨR ,

(1.13)

where ψθ∗1 (λ) is the unique invariant vector of the adjoint operator [Mθ1 (λ)]∗ , normalized as ψθ∗1 (λ)|Ψ0 = 1. Remark. The operators Bj with j ≥ 1 measure quantities on elements Em+j which, at time m, have not yet interacted with the system S. Therefore, they evolve independently simply under the evolution of Em+j . For large times m → ∞, the elements of the chain approach the reference state ΨE (because the initial state is normal), and the latter is stationary w.r.t. the uncoupled evolution. This explains the factorization in (1.12). As a special case of Theorem 1.3 we obtain the reduced evolution of S + R. Corollary 1.4. Assume the setting of Theorem 1.3. Then lim ρ (αm (ASR )) = ρ+,λ (ASR ).

m→∞

1.3. Thermodynamic Properties of Asymptotic State The total energy of the system is not defined, since R and C are reservoirs (and typically have infinite total energy). However, the energy variation is well defined. In order to quantify these energy jumps, let us assume for a moment that the various components, i.e. S, R and the elements E, are described via the usual Hamiltonian framework. We denote by hS and hS the Hilbert space and Hamiltonian describing

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the system S, by hR and hR those for the reservoir and by hE and hE those for an element E, and let vSR , resp. vSE , be a selfadjoint operator on hS ⊗ hR , resp. hS ⊗ hE , describing the interaction between S and R, resp. S and E. During each time interval [(m − 1)τ, mτ ), the hamiltonian of the total system writes hE,k + λ1 vSR + λ2 θm (vSE ), hm = hS + h R + k

where hE,k acts non trivially on the kth element of the chain on which it equals hE , and θm (v) = v acting on S and the mth element of the chain. It is then clear that energy change from time t1 to time t2 writes ∆E(t2 , t1 ) = u(t2 )∗ hm(t2 )+1 u(t2 ) − u(t1 )∗ hm(t1 )+1 u(t1 ),

(1.14)

where u(t) = e−is(t)hm(t)+1 e−iτ hm(t) · · · e−iτ h1 , and where we decomposed t1 and t2 as t = m(t)τ +s(t), m(t) ∈ N and s(t) ∈ [0, τ ). Now, for (m−1)τ ≤ t1 < mτ ≤ t2 < (m+1)τ , it is easy to see that (1.14) simplifies to ∆E(t2 , t1 ) = λ2 u(mτ )∗ (θm+1 (vSE ) − θm (vSE ))u(mτ ) =: δE(m),

(1.15)

which we interpret as the energy jump observable as time passes the moment mτ (note that during each interaction, i.e. in each time interval of the form [mτ, (m + 1)τ ), the full system is autonomous so that there is no energy variation in it). Starting from a Hamiltonian description of the system and given reference states ωS (resp. ωR and ωE ) of S (resp. R and E), one then performs the GNS representation (HS , πS , ψS ) of (B(hS ), ωS ), and similarly for R and E. The interaction operator VSR and VSE are then given by VSR = πS ⊗ πR (vSR ) and VSE = πS ⊗ πE (vSE ). Now, for any observable O one has π (u(mτ )∗ Ou(mτ )) = αm (π(O)) so that ∆E tot (m) := π(δE(m)) = λ2 αm (ϑm+1 (VSE ) − ϑm (VSE )) . In view of the above (formal) discussion, we therefore define the energy jump observable as time passes moment mτ by ∆E tot (m) = λ2 αm (ϑm+1 (VSE ) − ϑm (VSE )).

(1.16)

The variation ∆E tot (m) is thus an instantaneous observable. In applications this observable is analytic and hence we obtain, under the conditions of Theorem 1.3 (see also [11]), that 1 tot tot ρ ∆E tot (m) = ρ+,λ (j+ := lim ), dE+ m→∞ m tot tot where j+ = V − ατ (V ) is the total energy flux observable. The quantity dE+ represents the asymptotic energy change per unit time τ of the entire system.

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In the same way we define the variation of energy within the system S, the reservoir R and the chain C between times m and m + 1 by ∆E S (m) = αm+1 (LS ) − αm (LS ),

(1.17)

R

m+1

(LR ) − α (LR ),

(1.18)

C

m+1

(LEm+1 ) − α (LEm+1 ).

(1.19)

∆E (m) = α ∆E (m) = α

m

m

Remark. One would a priori define the various energy variations using the hamiltonians h# instead of the Liouvilleans L# , i.e. in the same way as for the total energy variation and with the same notation, ∆E # (m) = π(δE # (m)) where δE # (m) := u(m + 1)∗ h# u(m + 1) − u(m)∗ h# u(m). It is easy to see that (at least formally) this leads to the same expression as those of (1.17)–(1.19). The energy variations (1.17)–(1.19) can be expressed in terms of commutators [VSE , L# ] and [VSR , L# ], where # = S, E, R (see Sect. 4). Since [VSE , L# ] acts on S + E only, it is an analytic observable (see sentence after (1.2)). We make the following Assumption. H3 The commutators [VSR , L# ], where # = S, E, R, are analytic observables in M. We can thus apply Theorem 1.3 to obtain (see Sect. 4) # := lim dE+

m→∞

1 # ρ(∆E # (m)) = ρ+,λ (j+ ), m

# = S, R, C,

(1.20)

where j # are explicit ‘flux observables’ (c.f. (4.3)–(4.5)). We show in Propositot S R C S = j+ + j+ + j+ , and that ρ+,λ (j+ ) = 0. It follows immediately tion 4.1 that j+ that tot R C dE+ = dE+ + dE+ .

(1.21)

The total energy variation is thus the sum of the variations in the energy of C and R. The details of how the energy variations are shared between the subsystems depend on the particulars of the model considered; see below for an explicit example. Next, we consider the entropy production. Given two normal states ρ and ρ0 on M, the relative entropy of ρ with respect to ρ0 is denoted by Ent(ρ|ρ0 ). (This definition coincides with the one in [11] and differs from certain other works by a sign; here Ent(ρ|ρ0 ) ≥ 0). We examine the change of relative entropy of the state of the system as time evolves, relative to the reference state ρ0 represented by the reference vector Ψ0 , see (1.9). For a thermodynamic interpretation of the entropy, we take the vectors Ψ# , # = S, E, R to represent equilibrium states of respective temperatures βS , βE , βR . We analyze the change of relative entropy ∆S(m) = Ent(ρ ◦ αm |ρ0 ) − Ent(ρ|ρ0 )

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proceeding as in [11]. We show in Sect. 4 (see (4.6)) that ∆S(m) R tot = (βR − βE )dE+ + βE dE+ . m→∞ m Combining this result with (1.21), we arrive at dS+ := lim

Corollary 1.5. The system satisfies the following asymptotic 2nd law of thermodynamics, C R + βR dE+ . dS+ = βE dE+

1.4. An Explicit Example We consider S and E to be two-level systems. The observable algebra for S and for E is AS = AE = M2 (C). Let ES , EE > 0 be the “excited” energy level of S and of E, respectively. Accordingly, the Hamiltonians are given by 0 0 0 0 hS = and hE = . 0 ES 0 EE The dynamics are αSt (A) = eithS Ae−ithS and αEt (A) = eithE Ae−ithE . We choose the reference state of E to be the Gibbs state at inverse temperature βE , i.e. ρβE ,E (A) =

Tr(e−βE hE A) , ZβE ,E

where ZβE ,E = Tr(e−βE hE ),

and we choose (for computational convenience) the reference state for S to be the tracial state, ρS (A) = 12 Tr(A). The interaction operator between S and an element E of the chain is defined by λ2 vSE , where λ2 is a coupling constant, and vSE := aS ⊗ a∗E + a∗S ⊗ aE . The above creation and annihilation operators are represented by the matrices 0 1 0 0 a# = and a∗# = . 0 0 1 0 To get a Hilbert space description of the system, one performs the Gelfand– Naimark–Segal (GNS) construction of (AS , ρS ) and (AE , ρβE ,E ) as in Sect. 1.1, see, e.g. [10,11]. In this representation, the Hilbert spaces are given by HS = HE = C2 ⊗ C2 , the Von Neumann algebras by MS = ME = M2 (C) ⊗ 1lC2 ⊂ B(C2 ⊗ C2 ), and the vectors representing ρS and ρβE ,E are 1 ΨS = √ (|0 ⊗ |0 + |1 ⊗ |1 ), 2

1 |0 ⊗ |0 + e−βE EE /2 |1 ⊗ |1 . ΨE = √ Tr e−βE hE In other words, ρS (A) = ψS , (A ⊗ 1l)ψS and ρβE ,E (A) = ψE , (A ⊗ 1l)ψE . Above, |0 (resp. |1 ) denotes the ground (resp. excited) state of hS and hE . For shortness,

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in the following we will denote |ij for |i ⊗ |j , i, j = 0, 1. The free Liouvilleans LS and LE are given by LS = hS ⊗ 1lC2 − 1lC2 ⊗ hS ,

LE = hE ⊗ 1lC2 − 1lC2 ⊗ hE

and the interaction operator VSE is VSE = (aS ⊗ 1lC2 ) ⊗ (a∗E ⊗ 1lC2 ) + (a∗S ⊗ 1lC2 ) ⊗ (aE ⊗ 1lC2 ). For the reservoir, we consider a bath of non-interacting and non-relativistic fermions. The one particle space is hR = L2 (R3 , d3 k) and the one-particle energy operator hR is the multiplication operator by |k|2 . The Hilbert space for the reservoir is thus Γ− (hR ) and the Hamiltonian is the second quantization dΓ(hR ) of hR (see (1.6)). The algebra of observables is the C ∗ -algebra of operators A generated by {a# (f ) | f ∈ hR } where a/a∗ denote the usual annihilation/creation operators on Γ− (hR ). The dynamics is given by τft (a# (f )) = a# (eith f ), where h is the Hamiltonian of a single particle, acting on h. It is well known (see e.g. [10]) that for any βR > 0 there is a unique (τf , β)-KMS state ρβR on A, determined by the two point function ρβR (a∗ (f )a(f )) = f, (1 + eβR hR )−1 f , and which we choose to be the reference state of the reservoir. Finally, the interaction between the small system S and the reservoir is chosen of electric dipole type, i.e. of the form vSR = (aS + a∗S ) ⊗ ϕR (f ) where f ∈ hR is a form factor and ϕ(f ) = √12 (a(f ) + a∗ (f )). We know explain how to get a description of the reservoir similar to the one given in Sect. 1.1. As for S and E, the first point is to perform the GNS representation of (A, ρβR ), so called Araki-Wyss representation [8]. Namely, if Ω denotes the Fock vacuum and N the number operator of Γ− (hR ), the Hilbert space is given by ˜ R = Γ− (L2 (R3 , d3 k)) ⊗ Γ− (L2 (R3 , d3 k)), H the Von-Neumann algebra of observables is ˜ R = πβ (A) M where

eβh/2 1 f ⊗ 1l + (−1)N ⊗ a∗ √ f¯ =: aβ (f ), 1 + eβh 1 + eβh βh/2 1 e ∗ ∗ N ¯ √ √ f ⊗ 1l + (−1) ⊗ a πβ (a (f )) = a f =: a∗β (f ), βh βh 1+e 1+e πβ (a(f )) = a

√

the reference vector is ˜ R = Ω ⊗ Ω, Ψ and the Liouvillean is ˜ R = dΓ(hR ) ⊗ 1l − 1l ⊗ dΓ(hR ). L

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Ann. Henri Poincaré √

We then consider the isomorphism between L2 (R3 , d3 k) and L2 (R+ × S 2 , 2r drdσ) √ L2 (R+ , 2r dr; G), where G = L2 (S 2 , dσ), so that the operator hR (the multiplication by |k|2 ) becomes multiplication by r ∈ R+ (i.e. we have r = |k|2 ). The ˜ R is thus isomorphic to Hilbert space H √ √ r r 2 + 2 + dr; G ⊗ Γ− L R , dr; G . (1.22) Γ− L R , 2 2 Next, we make use of the maps a# (f ) ⊗ 1l → a# (f ⊕ 0),

(−1)N ⊗ a# (f ) → a# (0 ⊕ f )

to define an isometric isomorphism between (1.22) and √ √ r r dr; G ⊕ L2 R+ , dr; G . Γ− L2 R+ , 2 2 A last isometric isomorphism between the above Hilbert space and HR := Γ− L2 (R, ds; G) is induced by the following isomorphism between the one-particle spaces L2 (R+ , √ √ r r 2 + 2 2 dr; G) ⊕ L (R , 2 dr; G) and L (R, ds; G) =: h |s|1/4 f (s) if s ≥ 0, f ⊕ g → h, where h(s) = √ g(−s) if s < 0. 2 Using the above isomorphisms, one gets a description of the form given in Sect. 1.1 for the reservoir R. In this representation, the interaction operator vSR becomes VSR = (σx ⊗ 1lC2 ) ⊗ ϕ(fβR ) ∈ MS ⊗ MR , where σx = aS + a∗S is the Pauli matrix and fβR ∈ h = L2 (R, ds; L2 (S 2 , dσ)) is related to the initial form factor f ∈ L2 (R3 , d3 k) as follows √ if s ≥ 0, f ( s σ) |s|1/4 1 (1.23) (fβR (s)) (σ) = √ √ √ 2 1 + e−βR s f¯( −s σ) if s < 0. As mentioned at the beginning of the introduction, the situation where S is interacting with R or C alone has been treated in previous works [18,30] and [11]. If S is coupled to R alone, then a normal initial state approaches the joint equilibrium state, i.e. the equilibrium state of the coupled system S +R at temper−1 , with speed e−mτ γth (we consider discrete moments in time, t = mτ to ature βR compare with the repeated interaction situation). If S is coupled to C alone, initial normal states approach a repeated interaction asymptotic state, which turns out to be the equilibrium state of S at inverse temperature βE where βE = βE

EE , ES

(1.24)

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and with speed e−mτ γri . The convergence rates are given by

π (2) (2) ES f ( ES )2G (1.25) γth = λ21 γth + O(λ41 ), with γth = 2 τ (EE − ES ) (2) (2) γri = λ22 γri + O(λ42 ), with γri = τ sinc2 , (1.26) 2 √ √ where sinc(x) = sin(x)/x and f ( ES )2G := S 2 |f ( ES σ)|2 dσ. In order to satisfy the translation analyticity requirement H2, we need to make some assumption on the form factor f . Let I(δ) ≡ {z ∈ C, |(z)| < δ}. We denote by H 2 (δ) the Hardy class of analytic functions h : I(δ) → G which satisfy hH 2 (δ) := sup h(s + iθ)2G ds < ∞. |θ| 0 s.t. e−βR s/2 f0 (s) ∈ H 2 (δ). √ Proposition 1.6 (Asymptotic state of S). Assume f satisfies H4, f ( ES )G = 0 / 2πZ∗ . Then the asymptotic state ρ+,λ is given by and τ (EE − ES ) ∈ ρ+,λ = γρβR ,S + (1 − γ)ρβE ,S ⊗ ρβR ,R + O(λ),

H4

where ρβ,# is the Gibbs state of #, # = S, R, at inverse temperature β and where γ is given by (2)

γ=

λ21 γth (2)

(2)

λ21 γth + λ22 γri

.

Remark. The fact that the asymptotic state ρ+,λ is a convex combination of the two asymptotic states ρβR ,S and ρβE ,S holds only because the system S is a twolevel system and is not true in general. Using (4.2)–(4.5), Corollary 1.5 and Proposition 1.6, an explicit calculation of the energy fluxes and the entropy production for this concrete model reveals the following result. √ / 2πZ∗ . Then Proposition 1.7. Assume that f ( ES )G = 0 and τ (EE − ES ) ∈

C dE+ = κEE e−βR ES − e−βE ES + O(λ3 ),

R dE+ = κES e−βE ES − e−βR ES + O(λ3 ),

tot dE+ = κ(EE − ES ) e−βR ES − e−βE ES + O(λ3 ),

dS+ = κ(βE ES − βR ES ) e−βR ES − e−βE ES + O(λ3 ), where κ = Zβ−1 Zβ−1 ,S R ,S E

(2)

(2)

λ21 γth λ22 γri (2)

(2)

λ21 γth + λ22 γri

.

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Remarks. 1. The constant κ is positive and of order λ2 . Moreover it is zero if at least one of the two coupling constants vanishes (we are then in an equilibrium situation and there is no energy flux neither entropy production). C is positive (energy flows into chain) if and only if the 2. The energy flux dE+ −1 is greater than the renormalized temperature reservoir temperature TR = βR −1 TE = βE of the chain, i.e. if and only if the reservoir is “hotter”. A similar R statement holds for the energy flux dE+ of the reservoir. Note that it is not the temperature of the chain which plays a role but its renormalized value (1.24). 3. When both the reservoir and the chain are coupled to the system S (λ1 λ2 = 0) the entropy production vanishes (at the main order) if and only if the two temperatures TR and TE are equal, i.e. if and only if we are in an equilibrium situation. Once again, it is not the initial temperature of the chain which plays a role but the renormalized one. 4. The total energy variation can be either positive or negative depending on the parameters of the model. This is different from the situation considered in [11], where that variation was always non-negative.

2. Proof of Theorem 1.3 2.1. Generator of Dynamics Km We recall the definition of the so-called ‘C-Liouvillean’ introduced for the study of open systems out of equilibrium in [19], and further developed in [11–13,27,28,30] (see also references in the latter papers). Let (J# , ∆# ) denote the modular data associated with (M# , Ψ# ), with # given by S, R or E. Then (J, ∆) = (JS ⊗ JR ⊗ JE , ∆S ⊗ ∆R ⊗ ∆E ) are the modular data associated with (MS ⊗ MR ⊗ ME , ΨS ⊗ ΨR ⊗ ΨE ). We will write Jm and ∆m to mean that these operators are considered on the m-th copy of the infinite tensor product HC . We define the C-Liouville operator 1/2 Vm (λ)∆−1/2 Jm ≡ L0 + Wm (λ), Km = Lm − Jm ∆m m

m ≥ 1, where Wm (λ) ∈ MS ⊗ MR ⊗ ME is given by

1/2 1/2 1/2 1/2 Wm (λ) = λ1 VSR − (JS ∆S ⊗ JR ∆R )VSR (JS ∆S ⊗ JR ∆R )

1/2 1/2 1/2 1/2 )VSE,m (JS ∆S ⊗ Jm ∆m ) +λ2 VSE,m − (JS ∆S ⊗ Jm ∆m ≡ λ1 WSR + λ2 WSE,m .

(2.1)

Of course, WSE,m is the operator acting as WSE on the subspace HS ⊗ HEm of H, and trivially on its orthogonal complement.

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The operators Km have two crucial properties [11,19,27]. The first one is that they implement the same dynamics as the Lm : eitLm Ae−itLm = eitKm Ae−itKm ,

∀t ≥ 0, ∀A ∈ MS ⊗ MR ⊗ MC .

The second crucial property is that the reference state Ψ0 , (1.9), is left invariant under the evolution eitKm , Km Ψ0 = 0,

∀m.

(2.2)

2.2. Reduced Dynamics Operator We follow the strategy of [11] to reduce the problem to the study of the high powers of an effective dynamics operator. The main difference w.r.t. [11] is that in the present setup, the effective dynamics operator acts now on the infinite dimensional Hilbert space HS ⊗ HR . We first split off the free dynamics of elements not interacting with S by writing the product of exponentials in U (m), (1.4), as − −iτ Lm −iτ Lm−1 + e e · · · e−iτ L1 Um , U (m) = Um

where Lj = LS + LR + LE + V (λ) acts nontrivially on the subspace HS ⊗ HR ⊗ HEj and m − Um = exp −i (m − k)τ LEk , + = exp −i Um

k=1 m+1 k=2

(k − 1)τ LEk − imτ

LEk .

k>m+1

Let Om be an instantaneous observable (see (1.7)). A straightforward computation shows that (see also [13, equation (2.19)])

with

+ ∗ iτ L1 + ) e · · · eiτ Lm N (Om )e−iτ Lm · · · e−iτ L1 Um , αm (Om ) = (Um

(2.3)

iτ |j|LE −iτ |j|LE e ⊗rj=0 ϑm+j (Bj ). N (Om ) = ASR ⊗−1 ϑ B e m+j j j=−l

(2.4)

As normal states are convex combinations of vector states, it is sufficient to consider the latter. Let Ψρ be the GNS vector representing the initial state ρ, i.e., ρ(·) = Ψρ | · Ψρ . Since every Φ ∈ H is approximated in the norm of H by finite linear combinations of vectors of the form ΦS ⊗ ΦR ⊗m≥1 Φm , where ΦS ∈ HS , ΦR ∈ HR , and Φm = ΨE if m > N , for some N < ∞, it suffices to prove (1.12) for vector states determined by vectors Ψρ of the form Ψρ = ΦS ⊗ ΦR ⊗N m=1 Φm ⊗m>N ΨE ,

(2.5)

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for some arbitrary N < ∞. Finally, since the vectors ΨS , ΨR , ΨE are cyclic for the commutants MS , MR , ME , any vector of the form (2.5) is approximated by a Ψ = B Ψ0 ,

(2.6)

⊗N B = BS ⊗ BR m=1 Bm ⊗m>N 1lE ∈ M ,

(2.7)

for some ∈ MR , Bm ∈ ME . It is therefore sufficient to show (1.12), for with BS ∈ MS , BR vector states with vectors Ψρ of the form (2.6) and (2.7). Let Om be an instantaneous observable and let us consider the expression

Ψρ |αm (Om )Ψρ = B Ψ0 |αm (Om )B Ψ0 = Ψ0 |(B )∗ B αm (Om )Ψ0 . We use expression (2.3) and the properties of the generators Kn to obtain + ∗ iτ K1 Ψ0 |(B )∗ B αm (Om )Ψ0 = Ψ0 |(B )∗ B (Um ) e · · · eiτ Km N (Om )Ψ0 . + Ψ0 = Ψ0 . Let Note that Um

PN = 1lS ⊗ 1lR ⊗ 1lE1 ⊗ · · · 1lEN ⊗ PΨEN +1 ⊗ PΨEN +2 ⊗ · · · , where PΨEk = |ΨEk ΨEk |. Since (B )∗ B acts non-trivially only on the factors of the chain Hilbert space having index ≤ N , we have Ψ0 |(B )∗ B αm (Om )Ψ0 + )∗ eiτ K1 · · · eiτ KN PN eiτ KN +1 · · · eiτ Km N (Om )Ψ0 , = Ψ0 |(B )∗ B (U N

(2.8)

where (for m > N ; we have the limit m → ∞ in mind) N + + U = PN Um = PN exp −i (k − 1)τ LE k

N

k=2

Recall PSR = 1lS ⊗ 1lR ⊗ |ΨC ΨC |. We have for m > N + l PN eiτ KN +1 · · · eiτ Km N (Om )Ψ0 = PSR eiτ KN +1 · · · eiτ Km N (Om )Ψ0 = PSR M m−l−N −1 eiτ Km−l · · · eiτ Km N (Om )Ψ0 , (2.9) where we have introduced the following reduced dynamics operator (RDO), see (2.21) in [13] PSR eiτ K PSR = M ⊗ |ΨC ΨC | M acting on HS ⊗ HR .

(2.10)

In the last step of (2.9), we use the property PSR eiτ Ks eiτ Ks+1 · · · eiτ Kt PSR = PSR eiτ Ks PSR eiτ Ks+1 PSR · · · PSR eiτ Kt PSR , which holds for any 1 ≤ s < t. This property follows from the independence of the Ej for different j, see [11, Proposition 4.1].

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Combining (2.8) with (2.9) we obtain Ψ0 |(B )∗ B αm (Om )Ψ0 + )∗ eiτ K1 · · · eiτ KN PSR M m−l−N −1 eiτ Km−l · · · eiτ Km N (Om )Ψ0 . = Ψ0 |(B )∗ B (U N

(2.11) In order to emphasize the dependence on the coupling constants λ = (λ1 , λ2 ), we write K(λ) and M (λ). The following are general properties of the RDO. Proposition 2.1. Let λ ∈ R2 be arbitrary. We have (i) M (λ) ∈ B(HSR ) (ii) M (λ)ΨSR = ΨSR , where ΨSR := ΨS ⊗ ΨR (iii) For any ϕ in the dense set D = {ASR ΨSR , ASR ∈ MSR }, there exists a constant C(ϕ) < ∞ s.t. sup M (λ)n ϕ ≤ C(ϕ).

(2.12)

n∈N

Proof. (i) follows from the fact that K is a bounded perturbation of a self-adjoint operator and (ii) is a consequence of (2.2). To prove (iii), first note that D is dense since ΨSR is cyclic for MSR . Then note that the following identity holds for all BSR ∈ MSR BSR Ψ0 |αn (ASR )Ψ0 = BSR ΨSR |M (λ)n ASR ΨSR . Statement (iii) of the proposition follows from the density of D and unitarity of the Heisenberg evolution, with C(ASR ΨSR ) = ASR . Remark. Contrarily to the cases dealt with in [11,12] and [13], where the underlying Hilbert space is finite dimensional, we cannot conclude from (2.12) that M (λ) is power bounded. Hence we do not know a priori that σ(M (λ)) ⊂ {z : |z| ≤ 1}. 2.3. Translation Analyticity To separate the eigenvalues from the continuous spectrum, we use analytic spectral deformation theory acting on the (radial) variable s of the reservoir R. Recall the definition (1.10) of the translation. It is not difficult to see that Kθ := T (θ)−1 KT (θ) = L0 + θN + λ1 WSR (θ) + λ2 WSE ,

(2.13)

where N is the number operator, and that the right side of (2.13) admits an analytic continuation into θ ∈ κθ0 , strongly on the dense domain D(L0 ) ∩ D(N ), defining a family of closed operators (see [18]). Theorem 2.2 (Analyticity of propagator). Assume that H1 and H2 hold. Then 1. T (θ)−1 eiτ K T (θ) has an analytic continuation from θ ∈ R into the upper strip κθ0 , and this continuation is strongly continuous as θ ↓ 0. 2. For each θ ∈ κθ0 ∪ R, the analytic continuation of T (θ)−1 eiτ K T (θ) is given by eiτ Kθ , which is understood as an operator-norm convergent Dyson series (with ‘free part’ eiτ (L0 +θN ) ).

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For each θ ∈ κθ0 , λj → T (θ)−1 eiτ K T (θ), j = 1, 2, are analytic entire functions.

3.

Remarks. 1.

The proof of this result yields the following bound for all θ ∈ κθ0 ∪R, T (θ)−1 eiτ K T (θ) ≤ eτ sup0≤θ 0 (see proof of Lemma 2.4) and where ˜ + (B )∗ B Ψ0 × χσ T (θ1 ) × R2 (Om ) ≤ U N

N

eiτ Kj,θ1

j=1

×T (θ1 )−1 eiτ Km−l · · · eiτ Km N (Om )Ψ0 eN γ ≤ C(θ0 , N )T (θ1 )−1 eiτ Km−l · · · eiτ Km N (Om )Ψ0 eθ1 σ . The latter quantity is bounded uniformly in m. To arrive at the second line in (2.20) we made use of the fact that Ψ0 is invariant under the action of all of χσ , T (θ) and eiτ Kj,θ and that B Ψ0 2 = 1 (B Ψ0 is the initial vector state). We combine estimates (2.18) and (2.20) to arrive at ρ(αm (Om )) − ψθ∗ |T (θ1 )−1 eiτ Km−l · · · eiτ Km N (Om )Ψ0 1

≤ R1 (Om ) + R2 (Om )e−mγ . Hence,

lim sup ρ(αm (Om )) − ψθ∗1 |T (θ1 )−1 eiτ Km−l · · · eiτ Km N (Om )Ψ0 m→∞

≤ lim sup |R1 (Om )|.

(2.21)

m→∞

Finally, by taking σ → ∞ and since limσ→∞ R1 = 0 uniformly in m, this completes the proof of Theorem 1.3.

3. Analysis of M (λ) An important issue in the analysis of concrete models is the verification of the Fermi Golden Rule assumption FGR (see before Theorem 1.3). We have introduced the description of the two types of systems, ‘small’ and ‘reservoir’ in Sect. 1.1. For a more detailed analysis, we need to complement that description. We denote the eigenvalues and associated eigenvectors of HS by E1 , . . . , Ed ,

and

ϕ1 , . . . , ϕd .

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Before analyzing the spectrum of Mθ (λ) in general, we mention some easier special cases. • In the unperturbed case (λ = 0), we have M (0) = eiτ LS ⊗ eiτ LR , with σ(M (0)) = eiτ (Ej −Ek )

(j,k)∈{1,2,··· ,d}2

•

∪ eil , l ∈ R ,

where the eigenvalues eiτ (Ej −Ek ) are embedded and have corresponding eigenvectors ϕj ⊗ ϕk ⊗ ΨR . The eigenvalue 1 is at least d-fold degenerate. In case the coupling λ1 between the small system and the reservoir is zero, we have ˜ (λ2 ) ⊗ eiτ LR on HS ⊗ HR , M (0, λ2 ) = M where ˜ (λ2 ) PS eiτ (LS +LE +λ2 VSE ) PS and PS = 1lS ⊗ |ΨE ΨE |. M ˜ (λ2 ), which is nothing but the RDO correspondThe results of [11] apply to M ing to the repeated interaction quantum system formed by S and C only. In ˜ (λ2 ) is and eiτ LR is particular, we get that M (0, λ2 ) is power bounded, as M unitary. Moreover, assuming the interaction VSE “effectively” couples S and C, ˜ (λ2 ) satisfies hypothesis (E) in [11], we know that the spectrum of M ˜ (λ2 )) = {µj (λ2 )} σ(M

j=1,2,...,d2 ,

with µ1 (λ2 ) = 1 a simple eigenvalue with eigenvector ΨS and µj (λ2 ) ∈ {z||z| < 1}. Hence, ˜ (λ2 )) ∪ {|µj (λ2 )|eil , l ∈ R}j=1,...,d2 , σ(M (0, λ2 )) = σ(M

•

where the eigenvalues are embedded in the absolutely continuous spectrum again. In case the chain is decoupled, i.e. if λ2 = 0, we get M (λ1 , 0) = eiτ (LS +LR +λ1 WSR )

on HS ⊗ HR ,

whose spectral analysis already requires the tools we will use for general λ. We now turn to a perturbative analysis of Mθ (λ) (small λ). Take θ ∈ κθ0 and let λ1 = λ2 = 0. Then Mθ (0) = eiτ (LS +LR +θN ) = eiτ LS ⊗ eiτ LR eiτ θN and σ(Mθ (0)) = {eiτ (Ej −Ek ) }j,k∈{1,...,d} ∪ {eil e−τ jθ , l ∈ R}j∈N∗ . The effect of the analytic translation is to push the continuous spectrum of Mθ (0) onto circles with radii e−τ jθ , j = 1, 2, . . ., centered at the origin. Hence the discrete spectrum of Mθ (0), lying on the unit circle, is separated from the continuous spectrum by a distance 1 − e−τ θ . Analytic perturbation theory in the parameters

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λ1 , λ2 guarantees that the discrete and continuous spectra stay separated for small coupling. The following result quantifies this. Proposition 3.1. Let C0 (λ) := supθ∈κθ0 ∪R Wθ . Take θ ∈ κθ0 and suppose that τ C0 (λ)eτ C0 (λ)
0. For λ small enough Eq. (3.1) is therefore satisfied, so that we can verify the (FGR) hypotheses using perturbation theory for a finite set of eigenvalues, those four (0) eigenvalues which are located in σθ (λ) (see (3.2)). When the coupling constants are turned off, we have (0) σθ (0) = σ(eiτ LS ) = 1, eiτ ES , e−iτ ES where the eigenvalue 1 has multiplicity 2. In order to make the computation in perturbation theory as simple as possible, we will assume that these eigenvalues do / πN. However, this assumption is certainly not necessary. not coincide, i.e. τ ES ∈ Using a Dyson expansion for Mθ (λ) as in the proof of Theorem 2.2 and regu(0) lar perturbation theory (see e.g. [17,22]) we compute the four elements of σθ (λ). (0) We know that 1 always belongs to σθ (λ). The other ones respectively write √ τ (EE − ES ) π√ 2 2 2 2 e0 (λ) = 1 − ES f ( ES )G − λ2 τ sinc + O(λ3 ), 2 2 √ τ (EE − ES ) τ2 π√ e+ (λ) = eiτ ES 1 − λ21 τ ES f ( ES )2G − λ22 sinc2 4 2 2 ⎛ ⎞⎤

2 |s|f ( |s|) τ 1 − sinc(τ (E − E )) E S G ⎠⎦ + O(λ3 ), − i ⎝λ21 PV ds + λ22 τ 2 4 s − ES τ (EE − ES ) R √ τ (EE − ES ) τ2 π√ ES f ( ES )2G − λ22 sinc2 e− (λ) = e−iτ ES 1 − λ21 τ 4 2 2 ⎛ ⎞⎤

2 |s|f ( |s|) τ 1 − sinc(τ (E − E )) E S G 2 2 2 ⎠⎦ + O(λ3 ), + i ⎝λ1 PV ds + λ2 τ 4 s − ES τ (EE − ES ) λ21 τ

R

where sinc(x) = the following

sin(x) x

and PV stands for Cauchy’s principal value. We thus get

√ Lemma 5.1. Assume that f ( ES )G = 0 and τ (EE − ES ) ∈ / 2πZ∗ , then (FGR) is satisfied. In order to compute the asymptotic state ρ+,λ , we compute the (unique) invariant vector ψθ∗ (λ) of Mθ (λ)∗ (see (1.12)). Once again, standard perturbation theory shows that ψθ∗ (λ) = ψS∗ (λ) ⊗ ΨR + Oθ (λ) with + λ22 γri Zβ−1 √ λ21 γth Zβ−1 R ,S E ,S = 2 |00 (2) (2) λ21 γth + λ22 γri (2)

ψS∗ (λ)

(2)

+ λ22 γri e−βE ES Zβ−1 √ λ21 γth e−βR ES Zβ−1 R ,S E ,S + 2 |11 , (2) (2) λ21 γth + λ22 γri (2)

(2)

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(2)

where γth and γri are defined in (1.25) and (1.26). Inserting the above expression in (1.13), this proves Proposition 1.6.

Appendix A. Some Operator Theory Our analysis of the spectrum of Mθ (λ) makes use of a translated version of (2.12), which replaces the powerboundedness of Mθ (λ) in our setup. Lemma A.1. Assume ASR and BSR are translation analytic in κθ0 . Then sup |BSR ΨSR |Mθm (λ)ASR Ψ0 | ≤ ASR (θ)BSR (θ).

m∈N

Proof.

Consider

BSR (θ)ΨSR |αm (ASR (θ))ΨSR = BSR ΨSR |T (θ)M (λ)m T −1 (θ)ASR ψSR = BSR ΨSR |Mθ (λ)m ASR ψSR We can use this property to bound the spectral radius of Mθ (λ) when it is determined by discrete eigenvalues only. This means that there are finitely many eigenvalues αj , j = 1, . . . , N , all of equal modulus α, such that sup{|z| : z ∈ σ(Mθ (λ))} = α and σess (Mθ (λ)) ∩ {|z| = α} = ∅. Lemma A.2. Assume that for some θ ∈ κθ0 , spr (Mθ (λ)) is determined by discrete eigenvalues only. Then spr (Mθ (λ)) = 1. This is an application of the following result stated in a more abstract setting. Proposition A.3. Let M be a bounded operator on a Hilbert space H such that: i) there exists a dense set of vectors C ⊂ H satisfying sup |ϕ|M n ψ | ≤ C(ϕ, ψ),

n∈N

ii)

∀ϕ, ψ ∈ C,

spr (M ) is determined by discrete eigenvalues only, i.e. σ(M ) ∩ {z ∈ C | |z| = spr (M )} ⊂ σd (M ).

Then, spr (M ) ≤ 1 and the eigenvalues of modulus one, if any, are semisimple. Proof. Let {αj }j=1,...,N , be the discrete eigenvalues such that |αj | = α = spr (M ) and let Pj and Dj be the corresponding eigenprojectors and eigennilpotents. Recall that [Dj , Pj ] = 0 and Pj Pk = δjk Pj , for all j, k ∈ {1, . . . , N }. Setting Q = 1l − "N j=1 Pj , we can write, by assumption (ii) M=

N j=1

αj Pj + Dj + QM Q,

(A.1)

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where (QM Q)n ≤ eβn ∗

with β < ln α.

Let K ∈ N be such that = 0 for all j ∈ {1, . . . , N } and DjK0 = 0, for some j0 ∈ {1, . . . , N }. If all eigennilpotents are zero, we set K = 0. Using the properties of the spectral decomposition (A.1), we get for any n ∈ N large enough N K n n−k k n n αj Pj + αj Dj + (QM Q)n . M = k DjK+1

j=1

k=1

Consider first the case K = 0, where all Dj = 0. Assume that α > 1 and consider ϕ ∈ H such that Pj0 ϕ = 0, for some j0 ∈ {1, . . . , N }. We define ϕ0 = Pj0 ϕ/Pj0 ϕ such that M n ϕ0 = αjn0 ϕ0 . Now, C being dense, for any > 0, there exists ϕ˜0 ∈ C with ϕ˜0 − ϕ0 ≤ so that M n ϕ˜0 = M n ϕ0 +

N

αjn Pj (ϕ˜0 − ϕ0 ) + (QM Q)n (ϕ˜0 − ϕ0 ),

j=1

"N where the norm of the last two terms is bounded by αn ( j=1 Pj + e(β−ln α)n ). Hence, ϕ˜0 |M n ϕ˜0 = αjn0 (ϕ˜0 |ϕ0 + O()),

with O() uniform in n,

and ϕ˜0 |ϕ0 = 1 + O(). Thus the modulus of the RHS goes to infinity exponentially fast with n (since |αj0 | = α > 1), whereas the LHS should be uniformly bounded in n by assumption i). Consider now K > 0 and let ϕ ∈ H be such that DjK0 ϕ = 0. Assume α ≥ 1 and set, as above, ϕ0 = Pj0 ϕ/Pj0 ϕ. We have for n large enough K n M n ϕ0 = αjn0 ϕ0 + Djk ϕ0 , αj−k 0 k k=1

where, for 1 ≤ k ≤ K and n large, n n < nK /K!. k K Let ψ0 = DjK0 ϕ0 /DjK0 ϕ0 2 , and, for any > 0, ϕ˜0 , ψ˜0 in C such that ϕ˜0 −ϕ0 < and ψ˜0 − ψ0 < . Then, as n → ∞, K−1 n n −K −k n n k αj0 + ψ0 |M ϕ0 = αj0 αj0 ψ0 |Dj ϕ0 + ψ0 |ϕ0 K k k=1 n = αjn0 αj−K (1 + O(1/n)). 0 K Thus ψ˜0 |M n ϕ˜0 = ψ0 |M n ϕ0 + ψ˜0 − ψ0 |M n ϕ˜0 + ψ0 |M n (ϕ˜0 − ϕ0 ) ,

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where the vector M n ϕ˜0 =

N j=1

αjn Pj +

K n k=1

k


αjn−k Djk

ϕ˜0 + (QM Q)n ϕ˜0

satisfies for n large enough and some constant C uniform in n, n n n n n M ϕ˜0 ≤ Cα ϕ˜0 ≤ Cα (1 + ), K K ˜ and a similar estimate holds for (M n )∗ ψ0 . We finally get, for some constant C, uniform in n and , n |ψ˜0 |M n ϕ˜0 | ≥ |ψ0 |M n ϕ0 | − Cαn (ϕ˜0 + ψ0 ) K n ˜ (1 − C(1/n + )). = αn−K K Again, if α ≥ 1, the RHS diverges as n → ∞ whereas the LHS should be bounded by (ii), and the result follows. Remark. To get Lemma A.2, from this Proposition, note that ΨSR is cyclic for MS ⊗MR and the set of analytic observables ASR is (strongly) dense in MS ⊗MR . Moreover, Lemma A.1 shows that the dense set of analytic vectors of the form {ASR ΨSR } satisfies assumption (i). Finally, as ΨSR is invariant by Mθ (λ), the spectral radius is equal to 1.

References [1] Attal, S.: Extensions of the quantum stochastic calculus. Quantum Probability Communications vol. XI, pp. 1–38. World Scientific, Singapore (2003) [2] Abou Salem, W.K., Fr¨ ohlich, J.: Cyclic thermodynamic processes and entropy production. J. Stat. Phys. 126(3), 431–466 (2007) [3] Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Lecture Notes in Physics, vol. 717. Springer, Berlin (2007) [4] Attal, S., Joye, A.: Weak coupling and continuous limits for repeated quantum interaction. J. Stat. Phys. 126, 1241–1283 (2007) [5] Attal S., Joye A.: The Langevin equation for a quantum heat bath. J. Funct. Anal. 247, 253–288 (2007) [6] Attal, S., Joye, A., Pillet, C.-A. (eds.): Open Quantum Systems I–III. Lecture Notes in Mathematics, vol. 1880–1882. Springer, Berlin (2006) [7] Attal, S., Pautrat, Y.: From repeated to continuous quantum interactions. Ann. Henri Poincaré 7, 59–104 (2006) [8] Araki, H., Wyss, W.: Representations of canonical anticommutation relations. Helv. Phys. Acta 37, 136–159 (1964) [9] Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)

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[10] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vols. 1 and 2. Texts and Monographs in Physics. Springer, Berlin (1996) [11] Bruneau, L., Joye, A., Merkli, M.: Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 239, 310–344 (2006) [12] Bruneau, L., Joye, A., Merkli, M.: Infinite Products of Random Matrices and Repeated Interaction Dynamics. preprint arxive:math.PR/0703625. Annales de l’Inst. Henri Poincaré Probabilité et Statistique (to appear) [13] Bruneau, L., Joye, A., Merkli, M.: Random repeated interaction quantum systems. Commun. Math. Phys. 284, 553–581 (2008) [14] Bruneau, L., Pillet, C.-A.: Thermal relaxation of a QED cavity. J. Stat. Phys. 134 (5–6), 1071–1095 (2009) [15] Filipowicz, P., Javanainen, J., Meystre, P.: Theory of a microscopic maser. Phys. Rev. A 34, 3077–3087 (1986) [16] Fr¨ ohlich, J., Merkli, M., Ueltschi, D.: Dissipative transport: thermal contacts and tunnelling junctions. Ann. Henri Poincaré 4(5), 897–945 (2003) [17] Hunziker, W., Pillet, C.A.: Degenerate asymptotic perturbation theory. Commun. Math. Phys. 90, 219–233 (1983) [18] Jak˘sić, V., Pillet, C.-A.: On a model for quantum friction. III. Ergodic properties of the spin-boson system. Commun. Math. Phys. 178(3), 627–651 (1996) [19] Jak˘sić, V., Pillet, C.-A.: Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Commun. Math. Phys. 226, 131–162 (2002) [20] Jak˘sić, V., Pillet, C.-A.: A note on the entropy production formula. Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002), pp. 175–180. Contemp. Math., vol. 327. American Mathematical Society, Providence (2003) [21] Jak˘sić, V., Ogata, Y., Pillet, C.-A.: The Green–Kubo formula for the spin-fermion system. Commun. Math. Phys. 268(2), 369–401 (2006) [22] Kato, K.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976) [23] Merkli, M.: Stability of equilibria with a condensate. Commun. Math. Phys. 257(3), 621–640 (2005) [24] Merkli, M.: Evolution of entanglement and coherence via quantum resonances. (Preprint 2009) [25] Meschede, D., Walther, H., M¨ uller, G.: One-atom maser. Phys. Rev. Lett. 54, 551–554 (1993) [26] Merkli, M., M¨ uck, M., Sigal, I.M.: Instability of equilibrium states for coupled heat reservoirs at different temperatures. J. Funct. Anal. 243, 87–120 (2007) [27] Merkli, M., Sigal, I.M., Berman, G.P.: Decoherence and thermalization. Phys. Rev. Lett. 98(13), 130401, 4 pp. (2007) [28] Merkli, M., Sigal, I.M., Berman, G.P.: Resonance theory of decoherence and thermalization. Ann. Phys. 323(2), 373–412 (2008) [29] Merkli, M., Berman, G.P., Sigal, I.M.: Dynamics of collective decoherence and thermalization. Ann. Phys. 323(12), 3091–3112 (2008) [30] Merkli, M., Starr, S.: A resonance theory for open quantum systems with timedependent dynamics. J. Stat. Phys. 134(5–6), 871–898 (2009)

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[31] Nechita, I., Pellegrini, C.: Random repeated quantum interactions and random invariant states. Preprint, http://arxiv.org/abs/0902.2634 [32] Pillet, C.A.: Quantum dynamical systems. In: Attal, S., Joye, A., Pillet, C.A. (eds.) Open Quantum Systems. The Hamiltonian Approach, vol. I. Springer Lecture Notes in Mathematics, vol. 1880, pp. 107–182 (2006) [33] Ruelle, D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Phys. 98, 57 (2000) Laurent Bruneau Université de Cergy-Pontoise Département de Mathématiques CNRS (UMR 8088) 95000 Cergy-Pontoise, France e-mail: [email protected] Alain Joye Institut Fourier UMR 5582 CNRS-Université de Grenoble I BP 74, 38402 Saint-Martin d’Hères, France e-mail: [email protected] Marco Merkli Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, Canada e-mail: [email protected] Communicated by Claude Alain Pillet. Received: May 13, 2009. Accepted: September 21, 2009.

Ann. Henri Poincaré 10 (2010), 1285–1309 c 2010 Birkh¨ auser Verlag AG, Basel/Switzerland 1424-0637/10/071285-25, published online January 20, 2010 DOI 10.1007/s00023-009-0018-7


Gell-Mann and Low Formula for Degenerate Unperturbed States Christian Brouder, Gianluca Panati and Gabriel Stoltz Abstract. The Gell-Mann and Low switching allows to transform eigenstates of an unperturbed Hamiltonian H0 into eigenstates of the modified Hamiltonian H0 + V . This switching can be performed when the initial eigenstate is not degenerate, under some gap conditions with the remainder of the spectrum. We show here how to extend this approach to the case when the ground state of the unperturbed Hamiltonian is degenerate. More precisely, we prove that the switching procedure can still be performed when the initial states are eigenstates of the finite rank self-adjoint operator P0 V P0 , where P0 is the projection onto a degenerate eigenspace of H0 .

1. Introduction Adiabatic switching is a crucial ingredient of many-body theory. It provides a way to express the eigenstates of a Hamiltonian H0 + V in terms of the eigenstates of H0 . Its basic idea is to switch very slowly the interaction V , i.e., to transform H0 + V into a time-dependent Hamiltonian of the typical form H0 + e−ε|t| V , where the small parameter ε > 0 eventually vanishes. It may be expected that an eigenstate of H0 + V is obtained by taking the limit of an eigenstate of H0 , evolved according to the time-dependent Hamiltonian H0 + e−ε|t| V when ε tends to zero. It turns out that this naive expectation is not justified since the time-dependent eigenstate has no limit when ε → 0 because of some non-convergent phase factor. When the initial state belongs to a non-degenerate eigenspace, Gell-Mann and Low solved the problem by dividing out the oscillations by a suitable factor [7]. The ratio becomes, in the limit ε → 0, the Gell-Mann and Low wavefunction. Mathematically, the convergence of this procedure has been proved in 1989 by Nenciu and Rasche [16], elaborating on the adiabatic theorem [3,6,12]. On the other side, the physics community realized about 50 years ago [2] that a generalization of the Gell-Mann and Low formula is needed in the case of a degenerate eigenvalue of H0 . This happens in many practical situations, for instance,

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when the system contains unfilled shells. This problem has been discussed in several fields, including nuclear physics, solid state physics, quantum chemistry, and atomic physics (see the references in [4,5]). In most cases, it is assumed that there is some eigenstate in the degenerate eigenspace E0 of H0 for which the Gell-Mann and Low formula holds. In general, however, the Gell-Mann and Low formula is not applicable when this state is chosen at random in the degenerate subspace, as illustrated in the simple model analytically studied in [4]. We show in this paper that the switching can be performed provided the initial eigenstates are also eigenstates of P0 V P0 E , the perturbation restricted 0 to act on the degenerate eigenspace. If the latter operator has itself degenerate eigenvalues, a further analysis is required, as discussed in Sect. 3.4. The result is based on the recent progress in the mathematical analysis of adiabatic problems (see [1,8–10,14,15,17,21,22] and references therein). The physical consequences of our result are discussed in the companion physics paper [5], where we also comment on the formal relation with different types of Green functions.

2. Statement of the Results 2.1. Spectral Structure of the Problem Consider a Hilbert space H, a self-adjoint operator H0 , with dense domain D(H0 ) ⊂ H, and a symmetric perturbation V , H0 -bounded with relative bound a < 1. Then, according to the Kato–Rellich theorem (Theorem X.12 in [18]), H0 + λV is selfadjoint on D(H0 ) for any 0 ≤ λ ≤ 1. We denote1 H(λ) = H0 + λV, with λ ∈ [0, 1]. In all this study, we assume that the spectrum has the following structure: Assumption 1. (Structure of the spectrum) The spectrum of H(λ) = H0 + λV , λ ∈ [0, 1], consists of two disconnected pieces \σN (λ) , σ H(λ) = σN (λ) ∪ σ H(λ) where σN (λ) is a finite subset of the discrete spectrum: j (λ), j = 1, . . . , N ⊂ σdisc H(λ) , σN (λ) = E j (0) = E k (0) for all 1 ≤ j, k ≤ N . and the initial state is degenerate: E In order to apply results and techniques from adiabatic theory [1,3,12,15], we make the following standard assumption on the existence of a gap in the spectrum. 1 For reasons that will become clear once a time variable is introduced, we will always denote with a functions of the variable λ ∈ [0, 1]. Untilded functions will have time as an argument.

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Assumption 2. (Gap condition) There is a gap between the two parts of the spectrum, in the sense that min E , ∆(λ) = min j (λ) − E , E ∈ σ(H(λ))\ E1 (λ), . . . , EN (λ) j=1,...,N

is bounded from below by a positive constant: inf ∆(λ) = ∆∗ > 0.

λ∈[0,1]

j (λ) (counted with their The projectors associated with the N eigenvalues E multiplicities) are denoted by Pj (λ), for 1 ≤ j ≤ M with M ≤ N . The projector onto the subspace orthogonal to the eigenspace spanned by the N eigenvectors is M PN +1 (λ) = I − j=1 Pj (λ). We denote in the sequel P0 =

M

Pj (0)

j=1

the projector onto the eigenspace E0 = Ran(P0 ) spanned by the N degenerate eigenstates of H0 . For simplicity, we assume that the perturbation V is sufficient to split the degeneracy (so that M = N ), in the sense that the following assumption holds true: Assumption 3. (Degeneracy splitting) The finite rank self-adjoint operator P0 V P0 : E0 → E0 has non-degenerate eigenvalues, and there is a gap between the N first levels in the interval (0, 1]: for any λ∗ > 0, there exists α (depending on λ∗ ) such that l (λ) ≥ α > 0. E (2.1) inf min (λ) − E k ∗ λ ≤λ≤1 k=l

This implies that the projectors Pj (λ) are rank-1 projectors for any λ > 0 (since it can be proved that the perturbation V is enough to split the eigensubspaces, and the gap condition on (0, 1] ensures that no crossing can happen (see Sect. 3.1 for more details). Remark 4. Assumption 3 may be relaxed in several ways. First, the operator P0 V P0 can have degenerate eigenvalues, but then higher-order terms should be considered in the perturbative expansion of the eigenvalues. The gap assumption can be relaxed as well, and some crossings could be allowed. Besides, the general case of M < N projectors of ranks greater or equal to 1 can be treated similarly upon modifying the condition Pj (1) − Pj (0) < 1 required in Theorem 7. All these extensions are discussed in Sect. 3.4. 2.2. Switching Procedure Consider, for τ ∈ (−∞, 0], (τ )) = H0 + f (τ ) V, H(τ ) = H(f

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where the switching function f has values in [0, 1] (in order for the operator H(τ ) to be well-defined as a self-adjoint operator on D(H0 )). We denote by Pj (τ ) the eigenprojectors and eigenvalues corresponding to the first N eigenvalues Ej (τ ) of N H(τ ); also, PN +1 (τ ) = I − k=1 Pk (τ ). Of course, Pj (τ ) = Pj (f (τ )),

j (f (τ )). Ej (τ ) = E

For the subsequent analysis, we assume that Assumption 5. The switching function f : (−∞, 0] → [0, 1] is a C2 function such that (i) f is non-decreasing; (ii) f (0) = 1 and limτ →−∞ f (τ ) = 0; (iii) f, f ∈ L1 ((−∞, 0]). The most common choice in practice is f (τ ) = eτ . Notice, however, that any C non-decreasing compactly supported function with f (0) = 1 satisfies the above assumptions. In the latter case, the monotonicity of f implies that the support of f is a compact interval [Rf , 0], and f (t) > 0 for t ∈ (Rf , 0]. The assumption f ∈ C2 ensures that the adiabatic evolution (see (3.13)) is well defined. As a consequence of these assumptions, f ≥ 0 and f ∈ L1 ((−∞, 0]) ∩ ∞ L ((−∞, 0]); hence f ∈ L2 ((−∞, 0]). Indeed, the boundedness of f is a consequence of the fundamental theorem of calculus and the fact that f ∈ L1 ((−∞, 0]).

0 Besides, t f = f (0) − f (t) ≤ 1, and f ≥ 0; hence f ∈ L1 ((−∞, 0]). 2

Remark 6. It can be shown that eigenprojectors and eigenvectors are analytic with respect to λ = f (τ ) (see Sect. 3.1). When the switching function f is analytic, the eigenvalues Ej (τ ) (and the associated eigenvectors and eigenprojectors) are also analytic with respect to τ . We denote by Uε (s, s0 ) the unitary evolution generated by H(εs), i.e., the unique solution (which is well defined by Theorem X.70 in [18]) of the problem: dUε (s, s0 ) = H(εs) Uε (s, s0 ), Uε (s0 , s0 ) = I. ds In order to remove divergent phase factors (see the proof in Sect. 3.3.1), it is convenient to consider evolution operators in the interaction picture: i

Uε,int (s, s0 ) = eisH0 Uε (s, s0 ) e−is0 H0 . It is actually more convenient to rescale the time and to consider a macroscopic time t = εs. The unitary evolution U ε (t, t0 ) in terms of the macroscopic time is the solution of dU ε (t, t0 ) iε = H(t)U ε (t, t0 ), U ε (t0 , t0 ) = I, dt and, in the interaction picture, ε Uint (t, t0 ) = eitH0 /ε U ε (t, t0 ) e−it0 H0 /ε .

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ε ε Standard results show that Uint (t, −∞)ψ = limt0 →−∞ Uint (t, t0 )ψ exists for ψ ∈ D(H0 ) (for instance, by using a Cook’s type argument and rewriting this operator as the integral of its derivative with respect to t0 ).

2.3. Main Results We are now in position to state our main results. Theorem 7. Suppose that the gap conditions on H0 and V (Assumptions 1 and 2) are satisfied, and that the perturbation term V lifts the degeneracy (Assumption 3). Consider a switching function verifying Assumption 5. Let (ψ1 , . . . , ψN ) be an orthonormal basis of E0 which diagonalizes the bounded operator P0 V P0 E . Then, 0 if Pj (−∞) − Pj (0) < 1, (2.2) the limit ε Uint (0, −∞)ψj ε→0 ψj | U ε (0, −∞)ψj int

(2.3)

Ψj = lim

exists and is an eigenstate of H0 + V . Notice that, for a generic state ψ ∈ Ran P0 which is not an eigenvector of P0 V P0 E the above limit generically does not exist, as showed in [4] by using a 0 simple toy model. It is therefore crucial to select the appropriate initial states, so that the Gell-Mann & Low limit (2.3) does exist. As an intermediate step, the eigenprojector Pj (0) and a corresponding eigenfunction Ψj can be recovered by Kato’s geometric evolution [12]. Definition 8. The Kato evolution operator A(s, s0 ), for s, s0 ∈ R is the unique solution of the problem dA(s, s0 ) = K(s) A(s, s0 ), A(s0 , s0 ) = I, (2.4) ds with N +1 dPj (s). Pj (s) K(s) = − ds j=1 By our assumptions, the operator K(s) is uniformly bounded (see the comment after Definition 11). The Kato evolution operator is a unitary operator which intertwines the spectral subspaces of H(s) and H(s0 ), in the sense that A(s, s0 )Pj (s0 ) = Pj (s)A(s, s0 ). Equipped with this notation, we have the following result, where no condition analogous to (2.2) is assumed: Proposition 9. Let Assumptions 1, 2, 3 and 5 be satisfied. Let (ψ 1 , . . . , ψN ) be an orthonormal basis of E0 which diagonalizes the operator P0 V P0 E . Then 0

Ψj := A(0, −∞)ψj is an eigenvector of H0 + V .

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It is actually much simpler to consider the geometric evolution operator A ε since less conditions on V and H0 are rather than the evolution operator Uint required. Indeed, there is no denominator which needs to be considered in order to remove a divergent phase. However, the many-body theory used in physics is ε and not in terms of A. defined in terms of Uint We sketch shortly the structure of the proof, which is done in four steps: (i)

first, we use the Kato geometric evolution backward in time, in order to identify, though in a non-explicit manner, the initial subspaces of P0 whose vectors can be considered as convenient initial states; (ii) in a second step (Sect. 3.2), we give an explicit description of these initial subspaces, in terms of the eigenvectors of P0 V P0 E . At this stage, we are 0 already in position to prove Proposition 9; ε (iii) then, we show how the limit of the full evolution Uint can be related to the geometric evolution as ε → 0 (Sect. 3.3). A first step is to introduce an intermediate concept, the adiabatic evolution, which takes some dynamics into account (arising from the Hamiltonian operator). The adiabatic evolution is also an intertwiner. Since intertwiners differ only by a phase (in a sense to be made precise), and, provided this phase can be removed, the adiabatic evolution can be reduced to the geometric one (see Sect. 3.3.1); (iv) the last point is to show that the limit as ε → 0 of the full evolution is the adiabatic evolution (see Sect. 3.3.2). Steps (iii) and (iv) are straightforward extensions of previous results in adiabatic theory, and we heavily relied on the paper by Nenciu and Rasche [16] for Sect. 3.3.1 and the book by Teufel [22] for Sect. 3.3.2.

3. Proof of the Results 3.1. Geometric Evolution and Definition of the Initial States In view of the local gap assumption, the projectors and eigenvalues of H(λ) are real analytic functions of λ ∈ (0, 1]. Besides, Theorem II.6.1 in [13] shows that the j and projectors Pj can be analytically continued in the limit λ → 0. eigenvalues E The Kato construction of unitary operators A intertwining projectors can then be performed, see for instance Theorem XII.12 in [19] or Sections II.4 and II.6.2 in [13]. Consider the operator K(λ) =−

N +1 j=1

dPj Pj (λ) (λ), dλ

first proposed in [12], and the unique solution of λ0 ) dA(λ, λ0 ), = K(λ) A(λ, dλ

0 , λ0 ) = I. A(λ

(3.1)

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λ0 ) is well-defined and strongly Since K(λ) is uniformly bounded, the operator A(λ, λ0 ) is unitary, and intertwines continuous (see Theorem X.69 in [18]). Besides, A(λ, the spectral subspaces: λ0 )Pj (λ0 )A(λ, λ0 )∗ . Pj (λ) = A(λ, 2 , λ1 )A(λ 1 , λ0 ) = A(λ 2 , λ0 ), for instance, by comIt is also easily shown that A(λ puting the derivative of both expressions with respect to λ2 and using the uniqueness of the solution of (3.1). We define the initial subspaces by evolving backwards eigenstates of the Ham iltonian H(λ) for which the perturbation has split the degeneracy: the corresponding eigenprojector is defined as λ)Pj (λ)A(λ, 0), Pjinit := A(0,

(3.2)

the definition being independent of λ > 0. Eigenstates of H(1) = H0 + V are then obtained by evolving initial states belonging to the range of Pjinit according to the Kato evolution operator. Indeed, 0)P init = A(1, 0)A(0, λ)Pj (λ) = A(1, λ)Pj (λ). Thanks to the intertwining A(1, j property of A, it holds 1). 0)Pjinit A(0, Pj (1) = A(1,

(3.3)

3.2. Characterization of the Initial States The above paragraph shows that it is crucial to identify Ran(Pjinit ). We now characterize these spaces by an explicit condition. General expressions of the eigenvalues and eigenvectors. Since the eigenvalues and eigenprojectors of H(λ) are analytic in λ ∈ [0, 1], the following expansions are valid for 1 ≤ j ≤ N : j (λ) = E

+∞

λn Ej,n ,

(3.4)

n=0

and Pj (λ) =

+∞

λn Pj,n .

n=0

j (0), the common value of the energy in the degenerate Of course, Ej,0 = E0 = E ground-state. Notice also that the operators Pj,n are not necessarily orthogonal projectors. To define Pj (λ), it is more convenient to consider an eigenvector φj (λ) assoj (λ), i.e., a non-zero element of H satisfying ciated with E H(λ)φ j (λ) = Ej (λ) φj (λ).

(3.5)

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Such an eigenvector can be chosen to be analytic, by the same results which allow to conclude to the analyticity of the eigenprojectors. We therefore write φj (λ) =

+∞

λn ϕj,n .

(3.6)

n=0

Once such an eigenvector is known, the analytic eigenprojector can be constructed as φj (λ) φj (λ) . Pj (λ) = φj (λ) φj (λ) The aim of this section is to provide an explicit expression of the leading terms of the above expansions, in order to have a more explicit definition of Pjinit . To this end, we first construct a basis of E0 , which will turn out to be particularly useful to characterize the terms in the expansions (3.4) and (3.6). Diagonalization of P0 V P0 . Since P0 V P0 and P0 commute, it is possible to construct an orthonormal basis (ϕ1,0 , . . . , ϕN,0 ) of E0 such that P0 V P0 ϕj,0 = αj ϕj,0

(3.7)

for some real numbers α1 , . . . , αN , and ∀j = k,

ϕk,0 | P0 V P0 | ϕj,0 = 0.

(3.8)

Expressions for the terms in the expansions (3.4)–(3.6) at order 1. We identify the terms associated with the same powers of λ in (3.5). An additional normalization condition should be added in order to uniquely define the solution, so we impose ∀λ ∈ [0, 1],

ϕj,0 | φj (λ) = 1,

(3.9)

as is done in [20]. As will be seen below, this condition is simpler to work with than the standard condition φj (λ) = 1. The identification of the terms in (3.5) gives, for 1 ≤ j ≤ N , the following hierarchy of equations: (H0 − E0 )ϕj,0 = 0, (H0 − E0 )ϕj,1 = (Ej,1 − V )ϕj,0 , (H0 − E0 )ϕj,2 = (Ej,1 − V )ϕj,1 + Ej,2 ϕj,0 , and, for n ≥ 2, (H0 − E0 )ϕj,n+1 = (Ej,1 − V )ϕj,n +

n−1

Ej,n+1−m ϕj,m .

(3.10)

m=0

The equation on the term of order zero does not give any information on the choice of the initial states ϕj,0 . This information can be obtained from the first order condition: (H0 − E0 )ϕj,1 = (Ej,1 − V )ϕj,0 .

(3.11)

A necessary condition for this equation to have a solution is that the righthand side belongs to E0⊥ (since the left-hand side does):

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ϕk,0 , (Ej,1 − V )ϕj,0 = 0.

(3.12)

This requires Ej,1 = ϕj,0 , V ϕj,0 , and ∀k = j,

ϕk,0 , V ϕj,0 = 0.

Therefore, the conditions (3.12) for k = j cannot be fulfilled for a general basis. A necessary condition is that the basis {ϕk,0 }k=1,...,N of E0 diagonalizes P0 V P0 . Besides, the first-order term in the energy shifts are exactly the eigenvalues of P0 V P0 . This condition determines uniquely the basis when P0 V P0 has non-degenerate eigenvalues. If this is not the case, information about the higher-order equations in the hierarchy is needed (see Sect. 3.4). Remark 10. Assuming that the bands do not recross after the initial splitting, and if the degenerate state is the ground state of H0 , then the ground state of H0 + V is obtained by following the eigenstate associated with the lowest Ej,1 . Once the initial basis and the first energy shifts have been defined, the firstorder term in the variation of the eigenstates can be obtained from (3.11) as the sum of the reduced resolvent applied to the right-hand side, and some solution of the homogeneous equation (H0 − E0 )ψ = 0: ϕj,1 =

N

c1j,k ϕk,0 + (H0 − E0 )

k=1

=

−1

E0⊥

(Ej,1 − V )ϕj,0

c1j,k ϕk,0 − R0 V ϕj,0 ,

k=j

where

R0 = (H0 − E0 )−1

E0⊥

= (I − P0 ) (H0 − E0 )

−1

(I − P0 )

is a bounded operator from E0⊥ to E0⊥ ∩ D(H0 ), and c1j,j = 0 in view of the normalization condition (3.9). The coefficients c1k,j (for k = j) are undetermined at this stage. They have to be chosen so that the right-hand side of the next equation in the hierarchy is in E0⊥ . Conclusion: characterization of the initial subspaces. The above computations λ) − I = show that Pj (λ) = Pj,0 + O(λ), with Pj,0 = |ϕj,0 ϕj,0 |. Besides, A(0, O(λ) in view of the differential equation (3.1) satisfied by A. The initial subspace (3.2) is therefore λ) [Pj,0 + O(λ)] = Pj,0 . λ)Pj (λ) = lim A(0, Pjinit = A(0, λ→0

Proof of Proposition 9. Let ψ ∈ E0 be an eigenvector of P0 V P0 . Then, there exists j ∈ {1, . . . , N } such that ψ ∈ Ran(Pj,0 ) = Ran(Pjinit ). Using (3.3), it follows

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˜ 0)ψj ∈ Ran Pj (1) , A(0, −∞)ψj = A(1, which proves the claim. 3.3. Adiabatic Evolution and Limit of the Full Evolution Definition 11. The adiabatic evolution operator UA (s, s0 ) is defined for (s, s0 ) ∈ R2 as the unique solution of the problem dUA (s, s0 ) = HA (s)UA (s, s0 ), ds where the adiabatic Hamiltonian is i

UA (s0 , s0 ) = I,

(3.13)

HA (s) = H(s) + iK(s), with K(s) = −

N +1

Pj (s)

j=1

dPj (s). ds

(s)) so that Notice that K(s) = f (s) K(f K(s) ≤ Cf (s)

(3.14)

for some constant C > 0. Therefore, K(s) is uniformly bounded since f is bounded by our assumptions on the switching function. Compared to the geometric evolution (3.1), a Hamiltonian term has been added, which is at the origin of some dynamical phase factor in the dynamics. The adiabatic dynamics is well defined in view of the assumptions made on H0 , V , and f (see Theorem X.70 in [18]). A simple computation shows that it intertwines the spectral subspaces: Pj (s) = UA (s, s0 )Pj (s0 )UA (s, s0 )∗ . Switching to the interaction picture, we define UA,int (s, s0 ) = eisH0 UA (s, s0 ) e−is0 H0 . The factor ε is introduced by slowing down the switching as dUε,A (s, s0 ) = HA (εs)Uε,A (s, s0 ), Uε,A (s0 , s0 ) = I, (3.15) ds and the corresponding operator in the interaction picture is eisH0 Uε,A (s, s0 ) e−is0 H0 . It is convenient to rewrite the evolution (3.15) in the rescaled time variable t = εs: dU ε (t, t0 ) ε = HA (t)UAε (t, t0 ), UAε (t0 , t0 ) = I, (3.16) iε A dt ε with HA (t) = H(t) + iεK(t). The associated operator in the interaction picture is i

ε (t, t0 ) = eitH0 /ε UAε (t, t0 ) e−it0 H0 /ε . UA,int

Theorem 7 is then a consequence of the following results.

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Lemma 12. Let ψj ∈ Pjinit (defined by (3.2)). Then, under the assumptions of Theorem 7, the vector ε UA,int (0, −∞)ψj UA,int (0, −∞)ψj = ε

ψj | UA,int (0, −∞)ψj

ψj | UA,int (0, −∞)ψj

(3.17)

is an eigenstate of H0 . Lemma 13. Let ψj ∈ Pjinit . Then, under the assumptions of Theorem 7,

ε ε UA,int (0, −∞)ψj (0, −∞)ψj Uint lim = 0. ε (0, −∞)ψ − ψ | U ε ε→0

ψj | Uint j j A,int (0, −∞)ψj 3.3.1. Proof of Lemma 12. We show first in this section that ψj can be trans formed into an eigenstate of H(0) = H(1) using the adiabatic evolution defined from (3.13), and then the equality of the ratios (3.17). The proof presented here reproduces the argument of Nenciu and Rasche [16], which was given in the case N = 1 with our notation, but can be applied mutatis mutandis to the case considered here. We, however, present the proof for completeness. Evolution in the case ε = 1. Since both UA and A are intertwiners, they differ only by a phase which commutes with the spectral projectors. Indeed, define Φ(s, s0 ) = A(s, s0 )∗ UA (s, s0 ), so that UA (s, s0 ) = A(s, s0 ) Φ(s, s0 ). Then, [Φ(s, s0 ), Pj (s0 )] = 0, as can be seen using the intertwining properties: [Φ(s, s0 ), Pj (s0 )] = A(s, s0 )∗ UA (s, s0 )Pj (s0 ) − Pj (s0 )A(s, s0 )∗ UA (s, s0 ) = A(s, s0 )∗ Pj (s)UA (s, s0 ) − A(s, s0 )∗ Pj (s)UA (s, s0 ) = 0. The time-evolution of the phase matrix can be simplified due to this commutation property. First, dΦ(s, s0 ) = −iA(s, s0 )∗ H(s)UA (s, s0 ), ds since K(s)∗ = −K(s). Besides,

N +1

N +1 N +1 Pk (s0 ) Φ(s, s0 ) Pk (s0 ) = Φk (s, s0 ), Φ(s, s0 ) = k=1

k=1

k=1

where Φk (s, s0 ) = Pk (s0 )Φ(s, s0 )Pk (s0 ). The time evolution of the projected phasematrix is a scalar phase since d Φk (s, s0 ) = −iEk (s)Φk (s, s0 ); ds

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hence

⎛ Φ(s, s0 )Pj (s0 ) = exp ⎝−i

s


⎞ Ej (r) dr⎠ Pj (s0 ).

s0

The geometric evolution and the adiabatic evolution are therefore related through some global dynamical phase: ⎛ ⎞ s UA (s, s0 )Pj (s0 ) = A(s, s0 )Φ(s, s0 )Pj (s0 ) = exp ⎝−i Ej (r) dr⎠ A(s, s0 )Pj (s0 ). s0

To describe the asymptotic evolution, we follow closely the approach of [16]. In order for UA (s, s0 )Pj (s0 ) to be defined in the limit s0 → −∞, it is important to work in the interaction picture. Then, UA,int (s, s0 )Pj (−∞) = eisH0 A(s, s0 )Φ(s, s0 )e−is0 H0 Pj (−∞) = e−is0 E0 eisH0 A(s, s0 )e−isH0 eisH0 Φ(s, s0 )Pj (−∞). Using

⎛

Φ(s, s0 )Pj (s0 ) = Pj (s0 )Φ(s, s0 )Pj (s0 ) = exp ⎝−i

s

⎞ Ej (r) dr⎠ Pj (s0 ),

s0

it holds e−is0 E0 eisH0 Φ(s, s0 )Pj (−∞) = e−is0 E0 eisH0 Φ(s, s0 )Pj (s0 ) + e−is0 E0 eisH0 Φ(s, s0 )(Pj (−∞) − Pj (s0 )) ⎛ ⎞ s = exp ⎝−i Ej (r) dr − is0 E0 ⎠ eisH0 Pj (s0 ) s0

+e

−is0 E0 isH0

⎛

= exp ⎝−i

e s

Φ(s, s0 )(Pj (−∞) − Pj (s0 )) ⎞ Ej (r) dr − is0 E0 ⎠ eisH0 Pj (−∞) + eisH0 (Pj (s0 ) − Pj (−∞))

s0

+e

−is0 E0 isH0

⎛

= exp ⎝−i

e s

Φ(s, s0 )(Pj (−∞) − Pj (s0 )) ⎞

Ej (r) − E0 dr⎠ Pj (−∞) + W (s, s0 )(Pj (s0 ) − Pj (−∞)),

s0

where W ≤ 2. Since λ → Ej (λ) is C 1 on the compact interval [0, 1], there exists a constant C > 0 such that |Ej (r) − E0 | = E j (f (r)) − Ej (0) ≤ Cf (r).

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Since f ∈ L1 ((−∞, 0]), this shows that the function r → Ej (r) − E0 is integrable on (−∞, 0]. Besides, P (s0 ) → Pj (−∞) when s0 → −∞. The limit s0 → −∞ of UA,int (s, s0 )Pj (−∞) is therefore well-defined: ⎛ s ⎞ UA,int (s, −∞)Pj (−∞) = exp⎝−i Ej (r) − E0 dr⎠ eisH0 A(s, −∞)e−isH0 Pj (−∞). −∞

(3.18) The above equality reads, for s = 0, ⎛ UA,int (0, −∞)Pj (−∞) = exp ⎝−i

0

⎞ Ej (r) − E0 dr⎠ A(0, −∞)Pj (−∞).

−∞

Since Pj (0)A(0, −∞) = A(0, −∞)Pj (−∞), it holds, for ψj ∈ Pjinit = Pj (−∞) = Ran(ϕj,0 ), Pj (0)ψj = A(0, −∞)Pj (−∞)A(0, −∞)∗ ψj = ψj | A(0, −∞)∗ ψj A(0, −∞)ψj . (3.19) Finally, A(0, −∞)ψj UA,int (0, −∞)ψj Pj (0)ψj Pj (0)ψj = = , = Pj (0)ψj 2

ψj | Pj (0)ψj

ψj | A(0, −∞)ψj

ψj | UA,int (0, −∞)ψj which shows that the adiabatic evolution transforms the initial eigenstate into an eigenstate of H(1) provided Pj (0)ψj = 0, which is the case when Pj (0) − Pj (−∞) < 1. Evolution in the case ε > 0. Let us conclude this section by proving the equality (3.17). Computations similar to the ones performed in the case ε = 1 lead to ⎛ ⎞ 0 i ε (0, −∞)Pj (−∞) = exp ⎝− Ej (τ ) − E0 dτ ⎠ A(0, −∞)Pj (−∞). UA,int ε −∞

This can be seen for instance by noticing that (3.16) can be rewritten in the form ε (0, −∞) (3.13), upon considering the Hamiltonian H/ε in (3.13). Therefore, UA,int Pj (−∞) is equal, up to the ε-dependence in the phase factor, to UA,int (0, −∞) Pj (−∞). The non-convergent phase factor can be eliminated precisely by considering the Gell-Mann and Low ratio (3.17). 3.3.2. Proof of Lemma 13. It is sufficient to prove that lim U ε (0, −∞) − UAε (0, −∞) = 0,

ε→0

which indeed gives the result since ε ε (t, t0 ) − UA,int (t, t0 ) = U ε (t, t0 ) − UAε (t, t0 ) . Uint

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Notice that, although none of the operators U ε (0, −∞), UAε (0, −∞) has a limit when ε → 0, the difference nonetheless vanishes in this limit. The proof is based on the proofs of Theorem 2.2 and Corollary 2.5 in the book by Teufel [22], which are extended to the case of non-compactly supported switching functions and N > 1 with our notation. In this section, C and C denote constants, which may change from line to line, but are always independent of t, ε, etc., and depend only on the relative H0 -bound of V , on N , on ∆∗ and on bounds on the functions Pj and their derivatives on [0, 1]. We denote by δj (t) ≥ 0 the local gap around Ej (t): δj (t) = min {|Ej (t) − E|, E ∈ σ(H(t))\{Ej (t)}} . Notice that δj (t) > 0 when f (t) > 0, but δj (t) → 0 when f (t) → 0 since the initial eigenvalue is N -fold degenerate (see Assumption 3). In fact, the analysis of Sect. 3.2 shows that there exist α1 , α2 > 0 such that δj (t) ≤ α2 α1 ≤ (3.20) f (t) when f (t) > 0. Rewriting the difference as an integral. The difference between the two unitary evolutions is rewritten as the integral of the derivative, as ε

U (t, t0 ) −

UAε (t, t0 )

ε

t

d (U ε (t0 , t )UAε (t , t0 )) dt dt

= −U (t, t0 ) t0

i = − U ε (t, t0 ) ε = −U ε (t, t0 )

t

U ε (t0 , t ) [H(t ) − HA (t )] UAε (t , t0 ) dt

t0

t

U ε (t0 , t )K(t )UAε (t , t0 ) dt .

t0

The idea is to rewrite K(t) as a commutator, so that t → U ε (t0 , t)K(t)UAε (t, t0 ) is the derivative of a function (up to negligible terms), and an integration by parts gives the required estimates. The proof proposed here is an extension of the proof presented in [22, Chapter 2] in the case when several pieces of the discrete spectrum are considered independently. It would also have been possible to use the twiddle operation introduced in [1], which is, in some sense, the inverse operation of the commutator with the Hamiltonian. Construction of the function used in the commutator. Consider −∞ < t ≤ 0 such that f (t) > 0 (for compactly supported switching functions, this means that t is

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in the interior of the support). Define ⎛ ⎞ N +1 1 ⎝ F (t) = − Fj (t) + Gj (t)⎠ , 2 j=1 with, for 1 ≤ j ≤ N , Fj (t) =

1 2iπ

˙ Pj⊥ (t)R(z, t)R(z, t) dz,

(3.21)

˙ R(z, t)R(z, t)Pj⊥ (t) dz,

(3.22)

Γj (t)

1 Gj (t) = 2iπ

Γj (t)

where dH(t) d ˙ (H(t) − z)−1 = −R(z, t) R(z, t), R(z, t) = dt dt and Γj (t) is a contour enclosing Ej (t) and no other element of the spectrum (such a contour exists in view of Assumption 3). For j = N + 1, we denote by ΓN +1 (t) a contour enclosing all the first N eigenvalues Ek (t), k = 1, . . . , N , and separated from the remainder of the spectrum (such a contour exists in view of Assumption 2), and define ⊥

N 1 ˙ FN +1 (t) = − Pk (t) R(z, t)R(z, t) dz, (3.23) 2iπ R(z, t) = (H(t) − z)−1 ,

ΓN +1 (t)

1 GN +1 (t) = − 2iπ

k=1

˙ R(z, t)R(z, t)

N

⊥ Pk (t)

dz.

(3.24)

k=1

ΓN +1 (t)

By definition of the contours, 1 − R(z, t) dz = Pj (t), 2iπ

1 ≤ j ≤ N,

Γj (t)

and 1 − 2iπ

R(z, t) dz = ΓN +1 (t)

N

Pk (t) = PN⊥+1 (t).

k=1

Besides, in view of the continuity of t → Ej (t) for all 1 ≤ j ≤ N , it is possible to use contours which are locally constant in time, i.e., for a given t > −∞ such that f (t) > 0, there exists a (small) time interval (t − τ, t + τ ) and a contour Γtj such that 1 ∀s ∈ (t − τ, t + τ ), − R(z, s) dz = Pj (s) 2iπ Γtj

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for 1 ≤ j ≤ N , a similar result holding for j = N + 1. Using such locally constant contours, the time derivative of the contour integral defining the projector can be restated as a contour integral of the time derivative of the resolvent: dPj (t) 1 ˙ R(z, t) dz = , 1 ≤ j ≤ N, − 2iπ dt Γj (t)

and

1 − 2iπ

˙ R(z, t) dz =

N dPk (t) k=1

ΓN +1 (t)

dt

=−

dPN +1 (t) . dt

Boundedness of F . The operator F (t) is bounded. To see this, we first rewrite Fj (1 ≤ j ≤ N ) as Fj (t) = Pj⊥ (t)R(Ej (t), t)Pj⊥ (t)

dPj (t) . dt

(3.25)

Indeed, using the expression (3.21) of Fj , dPj (t) Fj (t) − Pj⊥ (t)R(Ej (t), t)Pj⊥ (t) dt 1 ⊥ ˙ Pj (t)(R(z, t) − R(Ej (t), t))Pj⊥ (t)R(z, t) dz = 2iπ Γj (t)

1 − 2iπ

˙ Pj⊥ (t)(R(z, t) − R(Ej (t), t))Pj⊥ (t)R(z, t)H(t)R(z, t) dz.

Γj (t)

When the contour encircles closely enough Ej (t), R(z, t) ≤

2 . δj (t)

Using the resolvent identity, it follows Pj⊥ (t)(R(z, t) − R(Ej (t), t))Pj⊥ (t)R(z, t) = |z − Ej (t)| · R(z, t)Pj⊥ (t)R(Ej (t), t)Pj⊥ (t)R(z, t) ≤

8|z − Ej (t)| . δj (t)3

Then, the difference f (t) ⊥ ⊥ ˙ Pj (t)(R(z, t) − R(Ej (t), t))R(z, t)Pj (t)H(t)R(z, t) dz ≤ C |Γj (t)| δj (t)3 Γj (t) can be made arbitrarily small by decreasing the radius of the contour Γj (t), with a constant C depending on the relative H0 -bound of V .

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From the expression (3.25), and the bound Pj⊥ (t)R(Ej (t), t)Pj⊥ (t) ≤ δj (t) , it holds finally −1

Fj (t) ≤

f (t) P˙j (t) ≤C , δj (t) f (t)

where we recall that both f and f are non-negative. This shows that Fj (t) is a bounded operator when f (t) > 0. A similar bound holds for Gj . The terms FN +1 (t), GN +1 (t) require a different treatment. In this case, the uniformity of the gap between the N eigenvalues encircled by ΓN +1 (t) and the remainder of the spectrum may be used to construct a contour ΓN +1 (t) such that 4 . ∀z ∈ ΓN +1 (t), R(z, t) ≤ ∆(t) This can be done by ensuring that the contour remains far away enough from the remainder of the spectrum, while still being at a finite distance of the first N eigenvalues. In particular, it is possible to construct a contour intersecting the real axis at a point γ such that |γ − EN (t)| ≥ ∆(t)/4 and inf {|γ − E|, E ∈ σ(H(t))\{E1 (t), . . . , EN (t)}} ≥ ∆(t)/4. Then,

⊥

N f (t) f (t) 2 FN +1 (t) = Pk (t) R(z, t) V R(z, t) dz ≤ C , 2iπ ∆(t)3 k=1 ΓN +1 (t)

(3.26)

and so FN +1 is bounded since ∆(t) ≥ ∆∗ > 0 and f is bounded. A similar bound holds for GN +1 . In conclusion, F (t) ≤ CF

f (t) , f (t)

(3.27)

for some constant CF independent of t. Computation of the commutator. It is easily seen that F (t) maps the Hilbert space H to D(H0 ). The commutator [H(t), F (t)] can then be defined as an unbounded operator with domain D(H(t)) = D(H0 ). For a given 1 ≤ j ≤ N , it holds, using the commutation property Pj⊥ (t)H(t) = H(t)Pj⊥ (t), 1 ˙ [H(t), Fj (t)] = [H(t), Pj⊥ (t)R(z, t)R(z, t)] dz 2iπ Γj (t)

1 = 2iπ

˙ [H(t) − z, Pj⊥ (t)R(z, t)R(z, t)] dz

Γj (t)

1 = 2iπ

Γj (t)

˙ ˙ Pj⊥ (t)R(z, t) − Pj⊥ (t)R(z, t)R(z, t)(H(t) − z) dz

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⎛ = −Pj⊥ (t)

dPj (t) ⎜ 1 + Pj⊥ (t) ⎝ dt 2iπ

⎞ ⎟ ˙ R(z, t)2 dz ⎠ H(t)

Γj (t)

dPj (t) = −(I − Pj (t)) , dt following the proof of Theorem 2.2 in [22]. Similar computations show [H(t), Gj (t)] =

dPj (t) (I − Pj (t)). dt

Finally, for 1 ≤ j ≤ N ,

dPj (t) [H(t), Fj (t) + Gj (t)] = Pj (t), . dt

In the same way,

dPN⊥+1 (t) dPN +1 (t) = PN +1 (t), [H(t), FN +1 (t) + GN +1 (t)] = − PN +1 (t), . dt dt

Since K(t) = −

N +1 j=1

Pj (t)

N +1 1 dPj (t) dPj (t) =− Pj (t), , dt 2 j=1 dt

it holds [H(t), F (t)] = K(t).

(3.28)

Integration by parts. Consider now −∞ < t0 < t ≤ 0 such that f (t0 ) > 0 (hence f (t) > 0 since f is non-decreasing). Define K(t) = −iε U ε (t0 , t)F (t)U ε (t, t0 ). Then K (t) = U ε (t0 , t)[H(t), F (t)]U ε (t, t0 ) − iεU ε (t0 , t)F (t)U ε (t, t0 ). In view of (3.28), the difference between the evolution operators is rewritten as U ε (t, t0 ) − UAε (t, t0 ) t ε = −U (t, t0 ) U ε (t0 , t )K(t )UAε (t , t0 ) dt t0 ε

t

= −U (t, t0 ) t0

dK(t ) ε dF (t ) ε + iεU (t , t ) U (t , t ) U ε (t0 , t )UAε (t , t0 ) dt , 0 0 dt dt (3.29)

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so that, after an integration by parts for the term associated with K , t ε ε U ε (t, t0 ) − UAε (t, t0 ) = U (t , t )K(t )U (t , t ) dt 0 0 A

(3.30)

t0

t

F

≤ K(t) + K(t0 ) + ε t0

t d ε ε + K(t ) (U (t , t )U (t , t )) dt 0 0 A dt t0 ⎛ t ≤ ε ⎝ F (t) + F (t0 ) + F (t ) dt t0

t +

⎞

F (t ) K(t ) dt ⎠ .

(3.31)

t0

The first two terms in the above equality are bounded with the bound (3.27) on F . For the last one, we use again the bound (3.27) on F , and the fact that K is uniformly bounded (see (3.14)), so that t

t

F (t ) K(t ) dt ≤ C t0

t0

C (f )2 ≤ f f (t0 )

t

(f )2 .

(3.32)

t0

We now turn to the central term. For 1 ≤ j ≤ N , and using (3.25), t

Fj (t ) dt

t ≤

t0

t0

t ≤ t0

P¨j (t ) dt + δj (t ) P¨j (t ) dt + δj (t ) t

+

t t0

t t0

d ⊥ ⊥ P˙j (t ) dt Pj (t )R(Ej (t ), t )Pj (t ) 2 P˙j (t ) 2 dt δj (t )

P˙j (t ) Pj⊥ (t )R(Ej (t ), t )V R(Ej (t ), t )Pj⊥ (t ) f (t ) dt

t0

t ≤ t0

dt

P¨j (t ) 2 P˙j (t ) 2 P˙j (t ) + + Cf (t) dt δj (t ) δj (t ) δj (t )2

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2 t 2 f (t ) + 3 f (t ) + f (t ) ≤C dt f (t ) f (t ) f (t ) t0 ⎛ ⎞ t t 1 1 |f | + 3(f )2 + ≤ C ⎝ (f )2 ⎠ , f (t0 ) f (t0 )2

t0

t0

for some constants C, C > 0 (related to the relative H0 -bound of V ). Similar expressions can be obtained for Gj (1 ≤ j ≤ N ). Straightforward estimates can be used for FN +1 , GN +1 , following a treatment similar to what was done to obtain (3.26), upon deriving the terms appearing in the contour integral:

N |f (t)| f (t) ˙ f (t)2 + Pk (t) + , FN +1 (t) ≤ C ∆(t)3 ∆(t)3 ∆(t)4 k=1

with

P˙k (t) = f (t) ∂λ P(f (t)) ≤ C f (t).

In conclusion, t t0

⎛

1 F (t ) dt ≤ C ⎝ f (t0 )

t

1 + f (t0 )2

|f | + (f )2

t0

t

⎞ (f )2 ⎠ ,

(3.33)

t0

for some constant C > 0. Decomposition of the integral close to the degeneracy. In order to avoid the singularities when f (t0 ) → 0, the difference of the unitary operators is separated into two contributions as ε

U (0, t0 ) −

UAε (0, t0 )

ε

T

= −U (0, t0 )

U ε (t0 , t)K(t)UAε (t, t0 ) dt

t0 ε

0

−U (0, t0 )

U ε (t0 , t)K(t)UAε (t, t0 ) dt,

T

where T is chosen such that f (T ) > 0. The first term is bounded using the straightforward estimate T T N ε U (0, t0 ) U ε (t0 , t)K(t)UAε (t, t0 ) dt ≤ C P˙k (t) dt k=1 t0

t0

≤C

T t0

f (t) dt ≤ C f (T ). (3.34)

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For t ∈ [T, 0], f (t) ≥ f (T ) > 0 and there is a gap proportional to f (T ) between the eigenvalues: ∀1 ≤ j ≤ N,

∀t ∈ [0, T ],

δj (t) ≥ αf (T ),

for some α > 0. The inequality (3.30), combined with (3.27), (3.32) and (3.33), allows to bound the second term as 0 0 ε U (0, t0 ) U ε (t0 , t)K(t)UAε (t, t0 ) dt = U ε (T, t)K(t)UAε (t, T ) dt T T ⎛ ⎞ 0 0 f (0) f (T ) 1 1 2 ⎝ ≤ Cε + + |f | + (f ) + (3.35) (f )2 ⎠ . f (0) f (T ) f (T ) f (T )2 T

T

The limit t0 → −∞ can then be taken in the above expressions. Moreover, upon choosing T small enough so that f (T ) = ε1/3 1, it follows, adding (3.34) and (3.35), and using the fact that f ∈ L1 ((−∞, 0]) ∩ L∞ ((−∞, 0]) and f ∈ L1 ((−∞, 0]), 1 U ε (0, −∞) − UAε (0, −∞) ≤ C f (T ) + ε 1 + (3.36) ≤ 3Cε1/3 . f (T )2 This concludes the proof. 3.4. Extensions The above proofs can be straightforwardly extended to the following cases (see Sect. 2 for the notation). Definition of the initial states when P0 V P0 has degenerate eigenvalues. Two changes should be made in the proofs presented in this paper: (i) the estimate obtained in the adiabatic limit degrades; (ii) more conditions are required to define the initial states. Denote by E0,i the M < N eigenspaces associated with the eigenvalues of P0 V P0 , set ni = dim(E0,i ), and define Ni = k ∈ {1, . . . , N } ϕk,0 ∈ E0,i , the set of indices corresponding to the ith eigenspace of P0 V P0 . Of course, M

ni = N,

Card(Ni ) = ni .

i=1

In view of Assumption 3, for any (k, l) ∈ Ni2 , k = l, there exists an integer pk,l ≥ 2 and an analytic function ekl (λ) such that Ek (λ) − El (λ) = λpk,l ekl (λ),

ek,l (0) = 0.

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Denote by p∗ the maximal integer for all couples 1 ≤ k, l ≤ N . Then, the final estimate (3.36) in the proof of the adiabatic limit reads 1 ≤ 3Cε1/(2p∗ +1) , U ε (0, −∞) − UAε (0, −∞) ≤ C f (T ) + ε 1 + f (T )2p∗ which is indeed larger than the ε1/3 bound found in the case p = 1 (no degeneracy of the perturbation restricted to E0 ). We now describe an iterative procedure which determines the initial states in a unique manner, using the higher-order equations in the hierarchy (3.10). We start with the conditions of order 2. A necessary condition for (3.10) to have a solution is that its right-hand side belongs to E0⊥ . This requires, for 1 ≤ j, k ≤ N ,

ϕk,0 , V R0 V ϕj,0 + Ej,2 δj,k + (Ej,1 − Ek,1 )c1j,k = 0,

(3.37)

where δa,b is the Kronecker symbol. In particular, ∀i ∈ {1, . . . , M },

∀(j, k) ∈ Ni2 ,

ϕk,0 , V R0 V ϕj,0 + Ej,2 δj,k = 0.

Therefore, {ϕj,0 }j∈Ni has to be an eigenbasis of P0,i V R0 V P0,i where P0,i denotes the projector onto E0,i . If P0,i V R0 V P0,i has non-degenerate eigenvalues, the initial eigenfunctions {ϕk,0 }k∈Ni are uniquely defined. Otherwise, the procedure must be repeated. Recall that there exists an integer p∗ such that after p∗ steps the degeneracy has no further split (see the discussion at the beginning of this paragraph). When the degeneracy is not permanent (see below for this case), this allows determining the initial states in a unique manner. See for instance [11]. In many practical cases, however, degeneracy is never totally split because V shares some symmetries with H0 . In this case, permanent degeneracy has to be taken into account (see below). Decomposition of the switching. In the case when (2.2) is not satisfied, i.e., Pj (0) − P (−∞) = 1 or equivalently Pj (0)ψj = 0 (since the eigenspaces are assumed to be one-dimensional), the switching should be done in several steps. The intermediate steps can be chosen by finding a finite number of values λk ∈ [0, 1] (k = 1, . . . , N − 1), with λ0 = 0 and λN = 1, such that Pj (λk+1 ) − Pj (λk ) < 1. This is possible since Pj is continuous on the compact interval [0, 1]. The initial state ψ0 is evolved into a state ψ1 by switching from H0 to H0 + λ1 V as ε1 Uint,λ (0, −∞)ψ0 1 ! ", ψ1 = lim ε1 ε1 →0 ψ0 Uint,λ (0, −∞)ψ 0 1

where the evolution operator ε Uint,λ (t, t0 ) = eitH0 /ε Uλε1 (t, t0 ) e−it0 H0 /ε 1

is the following operator in the interaction picture: iε

dUλε1 (t, t0 ) = (H0 + λ1 f (t)V ) Uλε1 (t, t0 ), dt

Uλε1 (t0 , t0 ) = I.

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The state ψ1 is then evolved into a state ψ2 by switching H0 + λ1 V to H0 + λ2 V as ε2 Uint,λ (0, −∞)ψ1 2 ,λ1 ! ", ψ2 = lim ε2 ε2 →0 ψ0 Uint,λ2 ,λ1 (0, −∞)ψ0

where the evolution operator ε (t, t0 ) = eitH0 /ε Uλε2 ,λ1 (t, t0 ) e−it0 H0 /ε Uint,λ 2 ,λ1

is defined as the following operator in the interaction picture: iε

dUλε2 ,λ1 (t, t0 ) = (H0 + λ1 V + (λ2 − λ1 )f (t)V ) Uλε2 ,λ1 (t, t0 ), dt

Uλε2 ,λ1 (t0 , t0 ) = I.

This construction is repeated until an eigenstate ψN of H0 + V = H0 + λN V is obtained. Notice that it is important to do the procedure sequentially. Permanently degenerate eigenspaces. When there are permanently degenerate j (λ) or Ej (t), the determieigenspaces associated with one of the eigenvalues E nation of the initial basis can still be performed as it is presented in Sect. 3.2. However, the argument leading to (3.19) in Sect. 3.3.1 cannot be extended as such to the case when Ran Pj (0) is of dimension larger or equal to 2. This is not a problem since A(0, −∞)ψj is still an eigenvector of Pj (0), and its phase can be removed upon considering ε UA,int (0, −∞)ψj A(0, −∞)ψj = ε

φ | UA,int (0, −∞)ψj

φ | A(0, −∞)ψj

for some fixed state φ, provided the denominator is non-zero. In Theorem 7, the choice φ = ψj is done, together with the assumption φ | A(0, −∞)ψj = 0. This assumption could in this specific case be translated into an assumption on Pj (0) − Pj (−∞) , but in general it should then be assumed that there exists φ ∈ H such that φ | A(0, −∞)ψj = 0. Existence of finitely many crossings. The projectors being analytic, the Kato operator can still be defined when there are eigenvalue crossings. The main issue in extending the Gell-Mann and Low formula to this case is therefore the proof of the adiabatic limit, which can, however, still be handled with [22, Corollary 2.5] since the crossings are regular (again, because the eigenvalues are analytic). Initial subspace composed of several degenerate spaces E0 , E1 , . . .. In this case, the operator V should be diagonalized in each subspaces, i.e., the self-adjoint finiterank operators Pj V Pj E are diagonalized in order to construct a basis of Ej . A j global basis is then obtained by concatenation (direct sum).

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Acknowledgements G.P. is grateful to S. Teufel and J. Wachsmuth for a useful discussion in a preliminary stage of this work. We gratefully thank an anonymous referee for useful comments and remarks, which encouraged us to generalize the result appearing in the first version of the paper.

References [1] Avron, J.E., Seiler, R., Yaffe, L.G.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys. 110, 33–49 (1987) [2] Bloch, C., Horowitz, J.: Sur la détermination des premiers états d’un système de fermions dans le cas dégénéré. Nucl. Phys. 8, 91–97 (1958) [3] Born, M., Fock, V.: Beweis des Adiabatensatzes. Z. Phys. 51, 165–180 (1928) [4] Brouder, C., Stoltz, G., Panati, G.: Adiabatic approximation, Gell-Mann and Low theorem and degeneracies: a pedagogical example. Phys. Rev. A 78, 042102 (2008) [5] Brouder, C., Stoltz, G., Panati, G.: Many-body Green function of degenerate systems. Phys. Rev. Lett. 103, 230401 (2009) [6] Garrido, L.M.: Generalized adiabatic invariance. J. Math. Phys. 5, 335–362 (1964) [7] Gell-Mann, M., Low, F.: Bound states in quantum field theory. Phys. Rev. 84(2), 350–354 (1951) [8] Hagedorn, G.A.: Adiabatic expansions near eigenvalue crossings. Ann. Phys. 196, 278–295 (1989) [9] Hagedorn, G.A., Joye, A.: A time-dependent Born–Oppenheimer approximation with exponentially small error estimates. Commun. Math. Phys. 223(3), 583–626 (2001) [10] Hagedorn, G.A., Joye, A.: Mathematical analysis of Born–Oppenheimer approximations. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proceedings of the Symposium in Pure Mathematics, vol. 76, pp. 203–226. American Mathematical Society, Providence (2007) [11] Hirschfelder, J.O.: Formal Rayleigh–Schr¨ odinger perturbation theory for both degenerate and non-degenerate energy states. Int. J. Quantum Chem. 3, 731–748 (1969) [12] Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn. 5, 435– 439 (1950) [13] Kato, T.: Perturbation theory for linear operators. Grundlehren der mathematischen Wissenschaften, vol. 132. Springer, Berlin (1976) [14] Martinez, A., Sordoni, V.: A general reduction scheme for the time-dependent Born– Oppenheimer approximation. C. R. Math. Acad. Sci. Paris 334(3), 185–188 (2002) [15] Nenciu, G.: On the adiabatic theorem of quantum mechanics. J. Phys. A: Math. Gen. 13, L15–L18 (1980) [16] Nenciu, G., Rasche, G.: Adiabatic theorem and Gell-Mann & Low formula. Helvetica Phys. Acta 62, 372–388 (1989) [17] Panati, G., Spohn, H., Teufel, S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7(1), 145–204 (2003)

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[18] Reed, M., Simon, B.: Fourier analysis and self-adjointness. In: Methods of Modern Mathematical Physics, vol. II. Academic Press, New York (1975) [19] Reed, M., Simon, B.: Analysis of operators. In: Methods of Modern Mathematical Physics, vol. IV. Academic Press, New York (1978) [20] Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, Reading (1994) [21] Sj¨ ostrand, J.: Projecteurs adiabatiques du point de vue pseudodifférentiel. C. R. Acad. Sci. Paris Sér. I Math. 317(2), 217–220 (1993) [22] Teufel, S.: Adiabatic perturbation theory in quantum dynamics. In: Lecture Notes in Mathematics, vol. 1821. Springer, Berlin (2003) Christian Brouder Institut de Minéralogie et de Physique des Milieux Condensés CNRS UMR 7590 Universités Paris 6 et 7, IPGP 140, rue de Lourmel 75015 Paris, France e-mail: [email protected] Gianluca Panati Dipartimento di Matematica Universit` a di Roma La Sapienza Rome, Italy e-mail: [email protected] Gabriel Stoltz Université Paris Est CERMICS Projet MICMAC ENPC, INRIA 6 & 8 Av. Pascal 77455 Marne-la-Vallée Cedex 2, France e-mail: [email protected] Communicated by Claude Alain Pillet. Received: June 12, 2009. Accepted: November 23, 2009.

Ann. Henri Poincaré 10 (2010), 1311–1333 c 2009 Birkh¨ auser Verlag AG, Basel/Switzerland 1424-0637/10/071311-23, published online December 19, 2009 DOI 10.1007/s00023-009-0016-9


Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations Jean Dolbeault and Robert Sta´ nczy Abstract. Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich–Pohoˇzaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

1. Introduction This paper is devoted to non-existence and uniqueness results for various supercritical semilinear elliptic equations depending on a bifurcation parameter, in a star-shaped domain in Rd . We shall distinguish the multiplicative case when the equation can be written as ∆u + λ f (u) = 0

(1)

and the additive case for which the equation is ∆u + f (u + µ) = 0.

(2)

We shall also distinguish two sub cases for each equation. The local case when λ and µ are the bifurcation parameters, and the non-local case when λ and µ are determined by a non-local condition, respectively,

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J. Dolbeault and R. Sta´ nczy


λ

f (u) dx = κ Ω

and

f (u + µ) dx = M. Ω

In the multiplicative non-local case, the equation is f (u) =0. (3) ∆u + κ f (u) dx Ω In many applications, the term f (u)/ Ω f (u) dx is interpreted as a probability measure and κ is a coupling parameter. Such a parameter arises from physical constants after a proper adimensionalization. In the additive non-local case (cf. [18]), the problem to solve is ∆u + f (u + µ) = 0, M = f (u + µ)dx. (4) Ω

The parameter M is typically mass and, in a variational setting, µ can be interpreted as a Lagrange multiplier associated with mass constraint, i.e., a chemical potential from the point of view of physics. We shall consider the four problems, (1)–(4), and prove that if the domain Ω is star-shaped, with boundary ∂Ω in C 2,γ , γ ∈ (0, 1), and if f is a non-decreasing non-linearity with supercritical growth at infinity, such that f (0) > 0 in the case of (1) or (3), or such that f > 0 on (¯ µ, ∞) ¯ ∈ [−∞, ∞) in the case of (2) or (4), then solutions and limµ→¯µ f (µ) = 0 for some µ are unique in L∞ ∩ H01 (Ω) in a certain range of the parameters λ, µ, κ or M , while no solution exists for large enough values of the same parameters. Typical nonlinearities are the exponential function f (u) = eu and the power law non-linearity f (u) = (1 + u)p , for some p > (d + 2)/(d − 2), d ≥ 3. In the exponential case, (1) is the well-known Gelfand equation, cf. [36]. Our approach is based on Pohoˇzaev’s estimate, see [55], which is obtained by multiplying the equations by (x · ∇u), integrating over Ω and then integrating by parts. Also see [63] for an earlier result based on the local dilation invariance in a linear setting. In this paper, we shall only consider solutions in L∞ ∩ H01 (Ω), which are, therefore, classical solutions, so that multiplying the equation by u or by (x · ∇u) is allowed. Some results can be extended to the H01 (Ω) framework, but some care is then required. This paper is organized as follows. In Sect. 2, we consider the multiplicative local and non-local bifurcation problems, respectively, (1) and (3). In Sect. 3, we study the additive local and non-local bifurcation problems, respectively, (2) and (4). In all cases, we establish non-existence and uniqueness results, and give some indications on how to construct the branches of solutions, although this is not our main purpose. Before giving the details of our results, let us give a brief review of the literature. Concerning (1), we primarily refer to the contributions of Schaaf [64] and

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Schmitt [65], which cover even more general cases than ours and will be discussed more thoroughly later in this section. The parameter λ in (1) can be seen as a bifurcation parameter. Equation (1) is sometimes called a non-linear eigenvalue problem. It is well known that for certain values of λ, multiplicity of solutions can occur, see for instance [40]. In some cases, there are infinitely many positive solutions, even in the radial case, when Ω is a ball. Radial solutions have been intensively studied. We refer for instance to [19] for a review of problems with positone structure, i.e. for which f (0) < 0 and f changes sign once on R+ . A detailed analysis of bifurcation diagrams can be found in [52,53]. Also see [43] for earlier and more qualitative results. Positive bounded solutions of such a non-linear scalar field equation are often called ground states and can be characterized in many problems as minimizers of a semi-bounded coercive energy functional. They are relevant in many cases of practical interest in physics, chemistry, mathematical biology, etc. When Ω is a ball, all bounded positive solutions are radial under rather weak conditions on the non-linearity f , according to [37] and subsequent papers. Lots of efforts have been devoted to uniqueness issues for the solutions of the corresponding ODE and slightly more general problems like quasilinear elliptic ones, see, e.g., [32]. Several other results also cover the case Ω = Rd , see [67]. There are also numerous papers in the case of more general non-linearities, including, for instance, functions of x, u, and ∇u (see [42]), or more general bifurcation problems than the ones considered in this paper. It is out of the scope of this introduction to review all of them. In a ball, the set of bounded solutions can often be parametrized. The corresponding bifurcation diagrams have the following properties. For nonlinearities with subcritical growth, for instance for f (u) = (1 + u)p , p < (d + 2)/ (d−2), d ≥ 3, multiple positive solutions may exist when λ is positive, small, while for supercritical growths, for example f (u) = (1 + u)p with p > (d + 2)/(d − 2), d ≥ 3, or f (u) = eu and d = 3, there is one branch of positive solutions which oscillates around some positive, limiting value of λ and solutions are unique only for λ positive, small. See [4,27,29,33,43,52,53,75] for more details. Another well-known fact is that, at least for star-shaped domains, Pohoˇzaev’s method allows to discriminate between super- and subcritical regimes. This approach has been used mostly to prove the non-existence of non-trivial solutions, see [14,24,57,59], and [55,63] for historical references. Such a method is for instance at the basis of the result of [14] on the Brezis-Nirenberg problem. Also see [4] and references therein for more details. The identity in Pohoˇzaev’s method amounts to consider the effect of a dilation on an energy associated with the solution and therefore carries some important information on the problem, see, e.g., [28,60]. In this context, stereographic projection and connections between Euclidean spaces and spheres are natural, as was already noted in Bandle and Benguria [2]. In this paper we are going to study first the regime corresponding to λ small and show that Pohoˇzaev’s method provides a uniqueness result also in cases for which a non-trivial solution exists. The existence of a branch of positive solutions of (1) is a widely studied issue, see for instance [25,58]. Also see [65] for a review,

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and references therein. As already said, our two basic examples are based on the power law case, f (u) = (1 + u)p , and the exponential non-linearity, f (u) = eu , for which useful informations and additional references can be found in [27,40,50,66, 77]. We shall also consider a third example, with a non-linearity corresponding to the case of Fermi-Dirac statistics, which behaves like a power law for large, positive values of u, and like an exponential function for large, negative values of u. The functional framework of bounded solutions and a bootstrap argument imply that we work with classical solutions. Apart from the condition that the domain is star-shaped and satisfies the Poincaré inequality, e.g., is bounded in one direction, with some compactness properties, we will assume no other geometrical condition. In the local multiplicative case, several uniqueness results are known for small λ > 0, including in the case of Gelfand’s equation, see [46,64,65]. One should note that in the framework of the larger space H01 (Ω), if the boundedness assumption is relaxed, it is not even known if all solutions are radial when Ω is a ball. The results of [37] and subsequent papers almost always rely on the assumption that the solutions are continuous or at least bounded on Ω. Notice that, according to [44,62], even for a ball, it is possible to prescribe a given isolated singularity which is not centered. In [62], the case of our two basic examples, f (u) = eu and d+2 d+1 < p < d−3 , d > 3, has been studied and then generalized f (u) = (1+u)p , with d−2 to several singularities in [61]. Also see [45,54] for an earlier result. These singularities are in H01 (Ω) and, for a given value of a parameter λ set apart from zero, they are located at an a priori given set of points. Similar problems on manifolds were considered in [6]. We refer to [3,35] for bounds on the solutions to Gelfand’s problem, which have been established earlier than uniqueness results but are actually a key tool. Also see [48] for a more recent contribution. Concerning the uniqueness of the solutions to Gelfand’s problem for d ≥ 3 and λ > 0, small, we refer to [46,64,65]. In the case of a ball, the result goes back to the paper of Joseph and Lundgren [40], when combined with the symmetry result of [37]. The local multiplicative case corresponding to Problem (1) is the subject of Sect. 2.1. The literature on such semilinear elliptic problems and associated bifurcation problems is huge. The results of non-existence of non-trivial solutions are well known, see [26,57,64] and references therein. Also see [49] for an extension to systems. Concerning the uniqueness result on non-trivial solutions, the method was apparently discovered independently by several people including Mignot and Puel [35] and Cabré and Majer [15], but it seems that the first published reference on uniqueness results by Rellich–Pohoˇzaev type estimates is due to Schmitt [65] and later, to Schaaf [64]. A more general result for the multiplicative case has been obtained in [13] to the price of more intricate reasonings. Numerous papers have been devoted to the understanding of the role of the geometry and they extend the standard results, mostly the non-existence results, to the case of non-strictly starshaped domains, see for instance [26,57,64] and several papers of McGough et al. see [46–48], which are, as far as we know, the most up-to-date results on such issues.

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As already mentioned above, Problem (1) has been studied by Schmitt [65] and Schaaf [64]. In [65, Theorem 2.6.7], it is proved that if one replaces f (u) in (1) by a more general function f (x, u) in C 2 (Ω × R+ ) satisfying f (x, u) > 0, fu (x, u) > 0, u ≥ 0, x ∈ Ω, 2 d F (x, u) < 1, (ii) lim sup sup u→∞ x∈Ω (d − 2) u f (x, u) (i)

(iii)

[∇x F (x, u + 1) − ∇x F (x, u) − u ∇x f (x, u)] · x ≤ 0 for u 1; x ∈ Ω,

then uniqueness holds for a star-shaped domain Ω. A survey on the existence and continuation results for linear and superlinear (sub- and supercritical) growth of the non-linear term f in (1) can also be found in [65], as well as a study of the influence of the geometry, topology and dimension of the domain, which is of interest for our purpose. In [64], Schaaf studies uniqueness results for the semilinear elliptic prob1 lem (1) under the asymptotic condition lim supu→∞ uFf(u) (u) < 2 − M (Ω), where M (Ω) = 1/d for star-shaped domains. In general M (Ω) is some number in the interval (0, 1/d]. In the autonomous case, the above asymptotic condition is equivalent to the Assumption (ii) made by Schmitt [65] or to our Assumption (8), to be found below. Our contribution to the question of the uniqueness for (1) relies on a simplification of the proof in [64,65]. Imposing a non-local constraint dramatically changes the picture. For instance, in case of Maxwell–Boltzmann statistics, f (u) = eu , in a ball of R2 , the solution of (1) has two solutions for any λ ∈ (0, λ∗ ) and no solution for λ > λ∗ , while uniqueness holds in (3) in terms of M , for any M for which a solution exists, see [7,40]. Non-local constraints are motivated by considerations arising from physics. Also see [38] for the case of negative values of λ. In the case of the exponential non-linearity with a mass normalization constraint, a considerable effort has been done in the 2D case for understanding the statistical properties of the so-called Onsager solutions of the Euler equation, see [16,17,51]. The same model, but rather in dimension d = 3, is relevant in astrophysical models for systems of gravitating particles, see [13]. Other standard examples are the polytropic distributions, with f (u) = up , and Bose–Einstein or Fermi-Dirac distributions which result in non-linearities involving special functions. Existence and non-existence results were obtained for instance in [7] and [71,72], respectively, for Maxwell–Boltzmann and Fermi-Dirac statistics. An evolution model compatible with Fermi-Dirac statistics and the convergence of its solutions towards steady states has been thoroughly examined in [9], while the steady state problem was considered by Sta´ nczy [71,72,74]. See [22] and references therein for a model improved with respect to thermodynamics, [72] and references therein for more elaborate models, and [23] for a derivation of an evolution equation involving a mean field term, which also provides a relevant, stationary model studied in [13,73]. Also see [21,30] for an alternative, phenomenological derivation of drift-diffusion equations and their stationary counterparts, and [74]

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for the existence of radial solutions by fixed point methods in weighted function spaces, under non-local constraints. The case of a decoupled, external potential goes back to the work of Smoluchowski, see [20,68]. For this reason, the evolution model is often referred to as the Smoluchowski–Poisson equation. Our purpose is not to study the above mentioned evolution equations, but only to emphasize that for the corresponding steady states, non-local constraints are very natural, since they correspond to quantities which are conserved along evolution. Hence, to identify the asymptotic state of the solutions to the evolution equation, we have to solve a semilinear elliptic equation with a non-local constraint, which corresponds, for instance, to mass conservation.

2. The Multiplicative Case 2.1. The Local Bifurcation Problem We consider Problem (1) on a domain Ω in Rd . Our first assumption is the geometrical condition that a Poincaré inequality holds: |u|2 dx ≤ CP |∇u|2 dx (5) Ω

Ω

H01 (Ω)

and some positive constant CP > 0. Such an inequality holds for any u ∈ for instance if Ω is bounded in one direction. See [69, Proposition 2.1] for more details, and also [70]. Inequality (5) is called Friedrichs’ inequality in some areas of analysis (see [34,41,56] for historical references; we also refer to [39]). We shall further require that |u|2 dx = CP |∇u|2 dx . (6) ∃ u ∈ H01 (Ω) such that u > 0 and Ω

Ω

Such a property arises for instance as a consequence of the compactness of the embedding H01 (Ω) → L2 (Ω), if Ω is connected. The compactness is granted if the volume of Ω is finite. If Ω is unbounded, we refer to [5, Theorem 2.8] and [1, Theorems 6.16 and 6.19] for compactness issues. The goal of this section is to state a non-existence result for large values of λ and give sufficient conditions on f ≥ 0 such that, for some λ0 > 0, Equation (1) has a unique solution in L∞ ∩ H01 (Ω) for any λ ∈ (0, λ0 ). We assume that f is of class C 2 . By standard elliptic bootstraping arguments, a bounded solution is then a classical one. Next we assume that for some λ∗ > 0, there exists a branch of positive minimal solutions (λ, uλ )λ∈(0,λ∗ ) originating from (0, 0) and such that (7) lim uλ L∞ (Ω) + ∇uλ L∞ (Ω) = 0 . λ→0+

Sufficient conditions for such a property to hold can be found in various papers. We can for instance quote the following result.

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Lemma 1. Assume that Ω is bounded with smooth, i.e. C 2,γ for some γ ∈ (0, 1), boundary, f ∈ C 2 is positive on [0, ∞) and inf u>0 f (u)/u > 0. Then (7) holds. We refer for instance to [65] for a proof. The solutions satisfying (7) can be characterized as a branch of minimal solutions, using sub- and super-solutions. Although this is standard, for the sake of completeness let us state a non-existence result for values of the parameter λ large enough. Proposition 2. Assume that (5) and (6) hold. If Λ := inf u>0 f (u)/u > 0, then there exists λ∗ > 0 such that (1) has no non-trivial non-negative solution in H01 (Ω) if λ > λ∗ . The lowest possible value of λ∗ is usually called the critical explosion parameter. Proof. Let ϕ1 be a positive eigenfunction associated with the first eigenvalue λ1 = 1/CP of −∆ in H01 (Ω): −∆ϕ1 = λ1 ϕ1 . By multiplying this equation by u and (1) by ϕ1 , we get λ1 u ϕ1 dx = ∇u · ∇ϕ1 dx = λ f (u) ϕ1 dx ≥ Λ λ u ϕ1 dx, Ω

Ω

Ω

Ω

thus proving that there are no non-trivial non-negative solutions if λ > λ1 /Λ.

Next we present a simplified version of the proof of a uniqueness result stated in [64], under slightly more restrictive hypotheses. We assume that d ≥ 3 and f has a supercritical growth at infinity, i.e., f is such that lim sup u→∞

d−2 F (u) =η< , u f (u) 2d

(8)

u where F (u) := 0 f (s) ds. Notice that, in Proposition 2, Λ > 0 if (8) holds and if we assume that f is positive. Theorem 3. Assume that Ω is a bounded star-shaped domain in Rd , d ≥ 3, with C 2,γ boundary, such that (5) holds for some CP > 0. If f (z) is positive for large values of z, of class C 2 and satisfies (7) and (8), then there exists a positive constant λ0 such that Eq. (1) has at most one solution in L∞ ∩ H01 (Ω) for any λ ∈ (0, λ0 ). Proof. We follow the lines of the proof of [64] with some minor simplifications. Up to a translation, we can assume that Ω is star-shaped with respect to the origin. Assume that (1) has two solutions, u and u+v. With no restriction, we can assume that u is a minimal solution and satisfies (7). As a consequence, v is non-negative and satisfies ∆v + λ [f (u + v) − f (u)] = 0.

(9)

1318



If we multiply (9) by v and integrate with respect to x ∈ Ω, we get 2 |∇v| dx = λ v [f (u + v) − f (u)] dx. Ω

(10)

Ω

Multiply (9) by x · ∇v and integrate with respect to x ∈ Ω to get d−2 1 |∇v|2 dx + |∇v|2 (x · ν(x)) dσ 2 2 Ω ∂Ω = d λ [F (u + v) − F (u) − F (u) v] dx Ω

+λ

(x · ∇u) [f (u + v) − f (u) − f (u) v] dx

(11)

Ω

where dσ is the measure induced by Lebesgue’s measure on ∂Ω. Recall that F is a primitive of f such that F (0) = 0. Take η1 ∈ (η, (d − 2)/(2 d)) where η is defined in Assumption (8). Since u = uλ is a minimal solution and, therefore, uniformly small as λ → 0+ , for any ε > 0, we obtain |x · ∇u| ≤ ε for any x ∈ Ω, provided λ > 0 is small enough. Define hε by hε (u, v) := d [F (u + v) − F (u) − F (u) v] + ε |f (u + v) − f (u) − f (u) v| − d η1 v [f (u + v) − f (u)] . Because of the smoothness of f and by Assumption (8), the function hε (u, v)/v 2 is bounded from above by some constant H, uniformly in ε > 0, small enough. By the assumption of star-shapedeness of the domain Ω, x · ν(x) ≥ 0 for any x ∈ ∂Ω. From (10) and (11), it follows that d−2 |∇v|2 dx ≤ d λ H |v|2 dx + d η1 |∇v|2 dx. 2 Ω

Ω

Ω

Due to the Poincaré inequality (5), the condition d−2 1 − η1 λ< CP H 2d implies v = 0 and the uniqueness follows.

Examples. If f (u) = eu , Condition (8) is always satisfied. Notice that if d = 2 and Ω is a ball, the uniqueness result is not true, see [40]. d+2 . Also see 2. If f (u) = (1 + u)p , d ≥ 3, Condition (8) holds if and only if p > d−2 [40] for more details. Similarly in the same range of parameters for f (u) = up we only get the trivial, zero solution. 1.

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The Fermi-Dirac distribution ∞ f (u) = fδ (u) := 0

tδ dt 1 + et−u

(12)

1 δ+1 δ+1 u

behaves like as u → ∞. Condition (8) holds if and only if δ + 1 > (d + 2)/(d − √2). The physically relevant examples require that δ = d/2 − 1, i.e., d > 2 (1 + 2) ≈ 4.83. For more properties of these functions see, e.g., [9,12]. 2.2. The Non-Local Bifurcation Problem In this section we address, in L∞ ∩ H01 (Ω), the non-local boundary value problem (3) with parameter κ > 0. Here Ω is a bounded domain in Rd , d ≥ 3, with C 1 boundary. We start with a non-existence result. Computations are similar to the ones of Sect. 2.1 and rely on Pohoˇzaev’s method. First multiply (3) by u to get u f (u) dx |∇u|2 dx = κ Ω . (13) f (u) dx Ω Ω

Multiplying (3) by (x · ∇u), we also get F (u) dx d−2 1 2 2 |∇u| dx + |∇u| (x · ν) dσ = d κ Ω 2 2 f (u) dx Ω Ω

(14)

∂Ω

where F is the primitive of f chosen so that F (0) = 0 and dσ is the measure induced by Lebesgue’s measure on ∂Ω. A simple integration of (3) gives κ = − ∆u dx = − ∇u · ν dσ. Ω

∂Ω

By the Cauchy–Schwarz inequality, ⎛ ⎞2 κ2 = ⎝ ∇u · ν dσ ⎠ ≤ |∂Ω| |∇u · ν|2 dσ = |∂Ω| |∇u|2 dσ, ∂Ω

∂Ω

∂Ω

where the last equality holds because of the boundary conditions. Assume that Ω is strictly star-shaped with respect to the origin α := inf (x · ν(x)) > 0. x∈∂Ω

(15)

Because of the invariance by translation of the problem, this is equivalent to assume that Ω is strictly star-shaped with respect to any other point in Rd . Hence α κ2 . |∇u|2 (x · ν) dσ ≥ α |∇u|2 dσ ≥ |∂Ω| ∂Ω

∂Ω

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Collecting this estimate with (13) and (14), we obtain ακ [2 d F (u) − (d − 2) u f (u)] dx ≥ f (u) dx. |∂Ω| Ω

Ω

As a straightforward consequence, we obtain the following result. Theorem 4. Assume that Ω is a bounded domain in Rd , d ≥ 3, with C 1 boundary satisfying (15) for some α > 0. If f is a C 1 function such that for some C > 0, 2 d F (u) ≤ (d − 2) u f (u) + C f (u) ∞

for any u ≥ 0, then (3) has no solution in L 1.

∩

H01 (Ω)

(16)

if κ > C |∂Ω|/α.

Standard examples, for which Condition (16) is satisfied, are: Exponential case: f (u) = eu with C = 2 d, cf. [7]. A sharper estimate can be easily achieved as follows. The function h(u) := C eu +(d−2) u eu −2 d (eu −1) is non-negative if C is such that 0 = h (u) = h(u) for some u ≥ 0. After eliminating u, we find d−2 C = d + 2 + (d − 2) log . (17) 2d

d+2 and C = 0, Pure power law case: If f (u) = up , the result holds with p ≥ d−2 cf. [36,76]. There are no non-trivial solutions. d+2 , then (16) holds with 3. Power law case: If f (u) = (1 + u)p with p ≥ d−2 C = d − 2. Uniqueness results in the non-local case follow from Sect. 2.1, when the coupling constant κ is positive, small. In case of non-linearities of exponential type, as far as we know, uniqueness results were guaranteed only under some additional assumptions, see [10,11]. We are now going to extend such uniqueness results to more general non-linearities satisfying (7) and (8) by comparing Problems (1) and (3). Denote by uλ the solutions of (1). For λ > 0, small, a branch of solutions of (3) can be parametrized by λ → κ(λ) := λ Ω f (uλ ) dx, uλ . Reciprocally, if Ω is bounded and

2.

0 < β := inf f (u), u≥0

∞

then any solution u ∈ L

∩

H01 (Ω)

of (3) is also a solution of (1) with κ κ λ= . ≤ β |Ω| f (u) dx Ω

This implies that λ is small for small κ and, as a consequence, for small values of κ, all solutions to (3) are located somewhere on the local branch originating from (0, 0). Moreover, as κ → 0+ , the solution of (3) also converges to (0, 0). To prove the uniqueness in L∞ ∩ H01 (Ω) of the solutions of (3), it is, therefore, sufficient to establish the monotonicity of λ → κ(λ) for small values of λ. Assume that

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f (0) > 0 and f is monotone non-decreasing on R+ .

(18)

Under this assumption, we observe that β = f (0). Let u1 and u2 be two solutions of (1) with λ1 < λ2 and let v := u2 − u1 . Then for some function θ on Ω, with values in [0, 1], we have −∆v − λ1 f (u1 + θ v) v = (λ2 − λ1 ) f (u2 ) ≥ 0, so that, by the Maximum Principle, v is non-negative. Notice indeed that for λ2 small enough, u1 and u2 are uniformly small since they lie on the local branch, close to the point (0, 0) and, therefore, λ1 f (u1 + θ v) < 1/CP . It follows that f (u2 ) dx = f (u1 + v) dx ≥ f (u1 ) dx, Ω

thus proving that κ(λ2 ) = λ2

Ω

Ω

f (u2 ) dx > λ1

Ω

Ω

f (u1 ) dx = κ(λ1 ).

Corollary 5. Under the assumptions of Theorem 3, if moreover f satisfies (18), then there exists a positive constant κ0 such that Equation (3) has at most one solution in L∞ ∩ H01 (Ω) for any κ ∈ (0, κ0 ).

3. The Additive Case 3.1. The Local Bifurcation Problem Consider in L∞ ∩ H01 (Ω) (2). In the two standard examples of this paper the problem can be reduced to (1) as follows. 1. Exponential case: If f (u) = eu , (2) is equivalent to (1) with λ = eµ and the limit λ → 0+ corresponds to µ → −∞. 2. Power law case: If f (u) = (1+u)p , (2) is equivalent to (1) with λ = (1+µ)p−1 and the limit λ → 0+ corresponds to µ → −1+ . If u is a solution of ∆u + (1 + u + µ)p = 0, one can indeed observe that v such that 1 + u + µ = (1 + µ)(1 + v) solves ∆v + λ (1 + v)p = 0 with λ = (1 + µ)p−1 . Equation (2) is however not completely equivalent to (1). To obtain a nonexistence result for large values of µ, we impose the assumption that reads lim

u→∞

f (u) = +∞. u

(19)

Proposition 6. Assume that (5), (6) and (19) hold. There exists µ∗ > 0 such that (2) has no positive, bounded solution in H01 (Ω) if µ > µ∗ . Proof. The proof is similar to the one of Proposition 2. Let ϕ1 be a positive eigenfunction associated with the first eigenvalue λ1 = 1/CP of −∆ in H01 (Ω). For any µ ≥ 0, λ1 u ϕ1 dx = f (u + µ) ϕ1 dx ≥ Λ(µ) (u + µ) ϕ1 dx ≥ Λ(µ) u ϕ1 dx, Ω

Ω

Ω

Ω

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where Λ(µ) := inf s≥µ f (s)/s, thus proving that there are no non-negative solutions if Λ(µ) > λ1 . Let us make a few comments on the existence of a branch of solutions, although this is out of the main scope of this paper. Let f be a positive function of class C 2 on (¯ µ, ∞), for some µ ¯ ∈ [−∞, ∞), with limµ→¯µ+ f (µ) = 0. We shall assume that there is a branch of minimal solutions (µ, uµ ) originating from (¯ µ, 0) and such that lim uµ L∞ (Ω) + ∇uµ L∞ (Ω) = 0. (20) µ→¯ µ

This can be guaranteed if Ω is bounded and if we additionally require that the function f is increasing, as in [71] for the Fermi-Dirac model. This is also true for exponential and power-like non-linearities. At least at a formal level, this can easily be understood by taking ζ = f (µ) as a bifurcation parameter. A solution −1 of (2) is then a zero of F (ζ, u) = u − (−∆) f (u + (f )−1 (ζ)) and it is therefore easy to find a branch issued from (ζ, u) = (0, 0) by applying the implicit function theorem at (ζ, u) = (0, 0) with F (0, 0) = 0, even if µ ¯ = −∞. Using comparison arguments, one can prove that this branch is a branch of minimal solutions. We shall now address the uniqueness issues. We assume that (8) holds: F (u) − η1 u f (u) d−2 = η − η1 < 0 . ∀ η1 ∈ η, , lim sup 2d u f (u) u→∞ As a consequence, for any µ > µ ¯, F (v + µ) − F (µ) − F (µ) v − η1 v [f (v + µ) − f (µ)] is negative for large v, and the function H(v, µ, η1 ) defined by v 2 H(v, µ, η1 ) = F (v + µ) − F (µ) − F (µ) v − η1 v [f (v + µ) − f (µ)] achieves a maximum for some finite value of v. With H(µ, η1 ) = supv>0 H(v, µ, η1 ), we have F (v + µ) − F (µ) − F (µ) v − η1 v [f (v + µ) − f (µ)] ≤ H(µ, η1 ) v 2 . Next we assume that, for some η1 ∈ η, d−2 2 d , we have

(21)

d−2 − η1 , (22) 2d where CP is the Poincaré constant. This condition is non-trivial. It relates H(µ, η1 ), a quantity attached to the non-linearity, to CP which has to do only with Ω. It is satisfied for all our basic examples. 1. Exponential case: If f (u) = eu , we take µ negative, with |µ| big enough. Indeed, using the homogeneity, one obtains H(v, µ, η1 ) = eµ H(v, 0, η1 ). Since limv→0+ H(v, 0, η1 ) = (1 − 2 η1 )/2 and H(v, 0, η1 ) becomes negative as v → +∞, as a function of v ∈ R+ , H(v, 0, η1 ) admits a maximum value. To get a more explicit bound, we take a Taylor expansion at second order, namely eθ v (1 − 2 η1 − η1 θ v)/2 for some intermediate number θ ∈ (0, 1). An upper CP H(µ, η1 )
2 (1 + 2), which is stronger than Assumption (8), as can easily be recovered by integrating f (u) − η [u f (u) + 2 f (u)] twice, for large values of u. Take η1 ∈ (η, (d − 2)/(2 d)). A Taylor expansion shows that H(v, µ, η1 ) = f (u) − η1 (u f (u) + 2 f (u)) + µ η1 f (u) η + η1 = a f (u) − (u f (u) + 2 f (u)) + (µ − b u) η1 f (u) 2 1−2 η1 η1 −η , b = 2 η1 (1−η−η and u = µ + θ v for some θ ∈ (0, 1). Both with a = 1−η−η 1 1) terms in the above right-hand side are negative for u large enough, which proves the existence of a constant H(µ, η1 ) such that (21) holds. Notice that by [12, Appendix], f and its derivatives behave like exponentials for u < 0, |u| large. Under the additional assumption d ≥ 6, a tedious but elementary computation shows that, as µ → −∞, the maximum of

1 (u f (u) + 2 f (u)) + (µ − b u) η1 f (u) u → a f (u) − η+η 2

is achieved at some u = o(µ), √ which proves that limµ→−∞ H(µ, η1 ) = 0. Moreover, for any d > 2(1 + 2) one can still show that this maximum value behaves like exp(µ) and thus can be made arbitrarily small for negative µ with |µ| large enough. Assume that (2) has two solutions, u and u + v, with v ≥ 0, and let us write the equation for the difference v as ∆v + f (u + v + µ) − f (u + µ) = 0.

(24)

The method is the same as in Sect. 2. Multiply (24) by x · ∇v and integrate with respect to x ∈ Ω. If F is a primitive of f such that F (¯ µ) = 0, then 1 d−2 |∇v|2 dx + |∇v|2 (x · ν(x)) dσ 2 2 Ω ∂Ω =d [F (u + v + µ) − F (u + µ) − F (u + µ) v] dx Ω

+ Ω

(x · ∇u) [f (u + v + µ) − f (u + µ) − f (u + µ) v] dx.

1324



Assume that (22) holds for some η1 . If Ω is bounded, |x · ∇u| is uniformly small as µ → µ ¯+ , and we may assume that for any ε > 0, arbitrarily small, there exists ¯, sufficiently close to µ ¯ (i.e., µ0 − µ ¯ > 0, small if µ ¯ > −∞, or µ0 < 0, |µ0 | µ0 > µ big enough if µ ¯ = −∞), such that |x · ∇u| ≤ ε for any x ∈ Ω if µ ∈ (¯ µ, µ0 ). Next we define hε (v) := d [F (z + v) − F (z) − F (z) v] + ε | f (z + v) − f (z) − f (z) v | − d η1 v [f (z + v) − f (z)] . where z = u + µ. Using the star-shapedeness of the domain Ω, we have d−2 |∇v|2 dx ≤ hε (v)dx + d η1 v [f (z + v) − f (z)] dx. 2 Ω

Ω

Ω

Up to a small change of η1 , so that Condition (22) still holds, for ε > 0, small enough, we get 1 hε (v) ≤ F (z + v) − F (z) − F (z) v − η1 v [f (z + v) − f (z)] . d As ε → 0+ , z converges to µ uniformly and the above right-hand side is equivalent to F (v + µ) − F (µ) − F (µ) v − η1 v [f (v + µ) − f (µ)]. For some δ > 0, arbitrarily small, we obtain 1 hε (v) ≤ (H(µ, η1 ) + δ) v 2 . d From (24) multiplied by v, after an integration by parts we obtain |∇v|2 dx = v [f (z + v) − f (z)] dx. Ω

Ω

Hence we have shown that d−2 − η1 |∇v|2 dx ≤ (H(µ, η1 ) + δ) |v|2 dx. 2d Ω

Ω

By the Poincaré inequality (5), the left-hand side is bounded from below by d−2 d−2 1 − η1 − η1 |∇v|2 dx ≥ |v|2 dx. 2d CP 2d Ω

Ω

2

Summarizing, we have proved that, if Ω |v| dx = 0, then, for an arbitrarily small δ > 0, d−2 1 − η1 ≤ H(µ, η1 ) + δ CP 2d if µ − µ ¯ > 0 is small if µ ¯ > −∞, or µ < 0, |µ| big enough if µ ¯ = −∞. This contradicts (22) unless v ≡ 0.

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Theorem 7. Assume that Ω is a bounded star-shaped domain in Rd , with C 2,γ boundary, γ ∈ (0, 1), such that (5) holds. If f ∈ C 2 satisfies (8) and (22), if µ, ∞) such that Equation (2) has at limµ→¯µ f (µ) = 0, then there exists a µ0 ∈ (¯ µ, µ0 ). most one solution in L∞ ∩ H01 (Ω) for any µ ∈ (¯ In cases of practical interest for applications, one often has to deal with the equation ∆u + f (x, u + µ) = 0. Our method can be adapted in many cases, that we omit here for simplicity. The necessary adaptations are left to the reader. 3.2. The Non-Local Bifurcation Problem In this section we address problem (4) with parameter M > 0, in a bounded starshaped domain Ω in Rd . Consider in L∞ ∩ H01 (Ω) the positive solutions of (4), i.e., of ∆u + f (u + µ) = 0

(25)

where µ is determined by the non-local normalization condition M = f (u + µ) dx.

(26)

Ω

We observe that in the exponential case, f (u) = eu , (4) is equivalent to the non-local multiplicative case (3). Condition (26) is indeed explicitly solved by eµ Ω eu dx = M = κ. Non-existence results for large values of M can be achieved by the same method as in the multiplicative non-local case. If we multiply (25) by u and (x·∇u), we get |∇u|2 dx = u f (u + µ)dx, d−2 2

Ω

1 |∇u|2 dx + 2

Ω

Ω 2

|∇u| (x · ν)dσ = d

(F (u + µ) − F (µ)) dx. Ω

∂Ω

The elimination of Ω |∇u|2 dx gives [2 d (F (u + µ) − F (µ)) − (d − 2) u f (u + µ)] dx ≥ |∇u|2 (x · ν)dσ. Ω

∂Ω

By the Cauchy–Schwarz inequality, we know that ⎛ ⎞2 M 2 = ⎝ ∇u · ν dσ ⎠ ≤ |∂Ω| |∇u|2 dσ. ∂Ω

∂Ω

If (15) holds, then, as in Sect. 2.2, α M 2 ≤ |∂Ω|

∂Ω

|∇u|2 (x · ν)dσ.

1326



Summarizing, we have found that α M2 . [2 d (F (u + µ) − F (µ)) − (d − 2) u f (u + µ)] dx ≥ |∂Ω|

(27)

Ω

This suggests a condition similar to the one in the multiplicative case, (16). Define G(µ) := sup [2 d (F (z) − F (µ)) − (d − 2) f (z) (z − µ)] /f (z). z>µ

If f is supercritical in the sense of (8), G is well defined, but in some cases, it also makes sense for d = 2. For simplicity, we shall assume that G is a non-decreasing function of µ. As a consequence, we can state the following theorem, which generalizes known results on exponential and Fermi-Dirac distributions, cf. [7] and [71,72], respectively. Theorem 8. Assume that Ω is a bounded domain in Rd , d ≥ 2, with C 1 boundary satisfying (15) for some α > 0. If f is a C 1 positive, non-decreasing function such that (8) holds and if G is non-decreasing, then (4) has no solution in L∞ ∩H01 (Ω) if M |∂Ω| −1 (G ◦ f ) M> . α |Ω| Here by f −1 one has to understand the generalized inverse given by f −1 (t) := sup {s ∈ R : f (s) ≤ t}. Proof. From the above definitions and computations, we have α M2 ≤ G(µ) M. |∂Ω| Since f is non-decreasing and the solution u of (25) is positive, while M = µ) dx ≥ f (µ) |Ω|, this completes the proof. 1.

2.

Ω

f (u+

Theorem 8 can be illustrated by the following examples. Exponential case: if f (u) = eu and d ≥ 3, then G(µ) ≡ d + 2 + (d − 2) log( d−2 2d ) does not depend on µ. If d = 2, G(µ) ≡ 4. In both cases (4) has no bounded solution if M > |∂Ω| G/α. We recover here the condition corresponding to (17) and Theorem 4. d+2 , then G(µ) = µ G(1). Using Power law case: if f (u) = up with p ≥ d−2 1/p µ ≤ (M/|Ω|) , it follows that (4) has no bounded solution if M

3.

p−1 p

>

G(1) |∂Ω| . α |Ω|1/p

Fermi-Dirac distribution case: If f (u) = fδ (u) where fδ is the √ Fermi-Dirac distribution defined by (12) with δ = d/2 − 1 and d > 2 (1 + 2), then f is increasing, F = d2 fd/2 is the primitive of f such that limu→−∞ F (u) = 0,

Gd := sup 4 fd/2 (z) − (d − 2) z fd/2−1 (z) = sup [2 d F (z) − (d − 2) z f (z)] z∈R

z∈R

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is finite according to [12, Appendix] and depends only on the dimension d. It is indeed known that fδ = δ fδ−1 , fδ (z) ∼ Γ(δ + 1) ez as z → −∞ and fδ (z) ∼ uδ+1 /(δ + 1) as z → +∞. From (27), we deduce that α M2 ≤ [2 d (F (z) − F (µ)) − (d − 2) z f (z)] dx + (d − 2) µ f (z) dx |∂Ω| Ω

Ω

with z := u + µ. By dropping the term F (µ), we see that the first integral in the right-hand side is bounded by Gd |Ω|, and the second one by (d − 2)µ M . Since f is increasing and u positive, f (µ) |Ω| ≤ Ω f (z) dx = M and therefore µ ≤ f −1 (M/|Ω|)). As a consequence, (4) has no bounded solution if M . α M 2 > |∂Ω| Gd |Ω| + (d − 2) M f −1 |Ω| For a similar approach, one can refer to [72]. Denote by uµ a branch of solutions of (2) satisfying (20). For µ − µ ¯ > 0, small if µ ¯ > −∞, or µ < 0, |µ| big enough if µ ¯ = −∞, a branch of solutions of (4) can be parametrized by µ → M (µ) := Ω f (uµ + µ) dx , uµ . Reciprocally, if Ω is bounded, then any solution u ∈ L∞ ∩ H01 (Ω) of (4) is of course a solution of (2) with µ = µ(M ) determined by (26). If f is monotone increasing, we additionally know that µ ¯ < µ < f −1 (M/|Ω|). To prove the uniqueness in L∞ ∩ H01 (Ω) of the solutions of (4), it is therefore sufficient to establish the monotonicity of µ → M (µ). Assume that lim f (µ) = lim f (µ) = 0

µ→¯ µ

µ→¯ µ

and

f is monotone increasing on (¯ µ, ∞). (28)

The function v := duµ /dµ is a solution in H01 (Ω) of ∆v + f (uµ + µ) (1 + v) = 0. As in the proof of Corollary 5, by the Maximum Principle, v is non-negative when µ is in a right neighborhood of µ ¯, thus proving that dM = f (uµ + µ) (1 + v) dx dµ Ω

is non-negative. Using Theorem 7, we obtain the following result. Theorem 9. Assume that Ω is a bounded star-shaped domain in Rd with C 2,γ boundary. If f ∈ C 2 is non-negative, increasing, satisfies (5), (8), (22), and (28), then there exists M0 > 0 such that (4) has at most one solution in L∞ ∩ H01 (Ω) for any M ∈ (0, M0 ).

4. Concluding Remarks Uniqueness issues in non-linear elliptic problems are difficult questions when no symmetry assumption is made on the domain. In this paper, we have considered only a few simple cases, which illustrate the efficiency of the approach based on

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Pohoˇzaev’s method when dealing with bifurcation problems. Our main contribution is to extend what has been done in the local multiplicative case to the additive case, and then to problems with non-local terms or constraints. The key point is that Pohoˇzaev’s method, which is well known to provide nonexistence results in supercritical problems, also gives uniqueness results. One can incidentally notice that non-existence results in many cases, for instance supercritical pure power law, are more precisely non-existence results of non-trivial solutions. The trivial solution is then the unique solution. The strength of the method is that minimal geometrical assumptions have to be done, and the result holds true even if no symmetry can be expected. As a non-trivial byproduct of our results, when the domain Ω presents some special symmetry, for instance with respect to a hyperplane, then it follows from the uniqueness result that the solution also has the corresponding symmetry.

Acknowledgements J. Dolbeault thanks J.-P. Puel for explaining him the method in the local, multiplicative case, for the exponential non-linearity, f (u) = eu , and X. Cabré for pointing to him [65]. The authors thank a referee for pointing them an important missing assumption and M. Jakszto for pointing them some references on the Poincaré/Friedrichs inequality. This work has been partially supported by the EU financed network HPRN-CT-2002-00282 and the Polonium contract no. 13886SG. It was also partially supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by the Polish Ministry of Science project N201 022 32/0902.

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[44] Matano, H.: Solutions of a nonlinear elliptic equation with a prescribed singularity. Personal Communication [45] Mazzeo, R., Pacard, F.: A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. J. Differ. Geom. 44, 331–370 (1996) [46] McGough, J., Mortensen, J.: Pohozaev obstructions on non-starlike domains. Calc. Var. Partial Differ. Equ. 18, 189–205 (2003) [47] McGough, J., Mortensen, J., Rickett, C., Stubbendieck, G.: Domain geometry and the Pohozaev identity. Electron. J. Differ. Equ. 16 (2005) [48] McGough, J., Schmitt, K.: Applications of variational identities to quasilinear elliptic differential equations. Math. Comput. Model. 32, 661–673 (2000) [49] McGough, J.S.: Nonexistence and uniqueness in semilinear elliptic systems. Differ. Equ. Dyn. Syst. 4, 157–176 (1996) [50] Mignot, F., Puel, J.-P.: Quelques résultats sur un problème elliptique avec non ´ linéarité exponentielle. In: Gauthier-Villars (ed.) Equations aux dérivées partielles et applications. Sci. Méd, pp. 683–704. Elsevier, Paris (1998) [51] Neri, C.: Statistical mechanics of the N -point vortex system with random intensities on a bounded domain. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 381–399 (2004) [52] Ouyang, T., Shi, J.: Exact multiplicity of positive solutions for a class of semilinear problems. J. Differ. Equ. 146, 121–156 (1998) [53] Ouyang, T., Shi, J.: Exact multiplicity of positive solutions for a class of semilinear problem. II. J. Differ. Equ. 158, 94–151 (1999) [54] Pacard, F.: Solutions de ∆u = −λeu ayant des singularités ponctuelles prescrites. C. R. Acad. Sci. Paris Sér. I Math. 311, 317–320 (1990) [55] Pohoˇzaev, S.I.: On the eigenfunctions of the equation ∆u + λf (u) = 0. Dokl. Akad. Nauk SSSR 165, 36–39 (1965) [56] Poincaré, H.: Sur la théorie analytique de la chaleur. C. R. Acad. Sci. 104, 1753–1759 (1887) [57] Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35, 681–703 (1986) [58] Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971) [59] Rabinowitz, P.H.: Variational methods for nonlinear elliptic eigenvalue problems. Indiana Univ. Math. J. 23, 729–754 (1973, 1974) [60] Ramos, M., Terracini, S., Troestler, C.: Superlinear indefinite elliptic problems and Pohoˇzaev type identities. J. Funct. Anal. 159, 596–628 (1998) [61] Réba¨ı, Y.: Solutions of semilinear elliptic equations with many isolated singularities: the unstable case. Nonlinear Anal. 38, 173–191 (1999) [62] Réba¨ı, Y.: Solutions of semilinear elliptic equations with one isolated singularity. Differ. Integr. Equ. 12, 563–581 (1999) [63] Rellich, F.: Darstellung der Eigenwerte von ∆u + λu = 0 durch ein Randintegral. Math. Z. 46, 635–636 (1940) [64] Schaaf, R.: Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry. Adv. Differ. Equ. 5, 1201–1220 (2000)

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[65] Schmitt, K.: Positive solutions of semilinear elliptic boundary value problems. In: Topological methods in differential equations and inclusions (Montreal, PQ, 1994). NATO Advanced Science Institutes Series C Mathematical and Physical Sciences, vol. 472, pp. 447–500. Kluwer, Dordrecht (1995) [66] Senba, T., Suzuki, T.: Applied analysis. Mathematical methods in natural science. Imperial College Press, London (2004) [67] Serrin, J., Tang, M.: Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J. 49, 897–923 (2000) [68] Smoluchowski, M.: Drei vortrage u ¨ ber diffusion, brownsche molekularbewegung und koagulation von kolloidteilchen. Phys. Zeit. 17, 557–571, 585–599 (1916) [69] Souplet, P.: Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations. Commun. Partial Differ. Equ. 24, 951–973 (1999) [70] Souplet, P., Weissler, F.B.: Poincaré’s inequality and global solutions of a nonlinear parabolic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 335– 371 (1999) [71] Sta´ nczy, R.: Self-attracting Fermi-Dirac particles in canonical and microcanonical setting. Math. Methods Appl. Sci. 28, 975–990 (2005) [72] Sta´ nczy, R.: Steady states for a system describing self-gravitating Fermi-Dirac particles. Differ. Int. Equ. 18, 567–582 (2005) [73] Sta´ nczy, R.: On some parabolic–elliptic system with self-similar pressure term. In: Self-Similar Solutions of Nonlinear PDE, vol. 74, pp. 205–215. Banach Center Publications, Warsaw (2006) [74] Sta´ nczy, R.: The existence of equilibria of many-particle systems. Proc. R. Soc. Edinb. 139A, 623–631 (2009) [75] Suzuki, T.: Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 367– 397 (1992) [76] Suzuki, T.: Semilinear elliptic equations. GAKUTO International Series. In: Mathematical sciences and applications, vol. 3. Gakk¯ otosho Co. Ltd, Tokyo (1994) [77] Suzuki, T.: Free energy and self-interacting particles. In: Progress in nonlinear differential equations and their applications, vol. 62. Birkh¨ auser, Boston (2005) Jean Dolbeault Ceremade (UMR CNRS no. 7534) Université Paris-Dauphine Place de Lattre de Tassigny 75775 Paris Cedex 16, France e-mail: [email protected] Robert Sta´ nczy Instytut Matematyczny Uniwersytet Wroclawski pl. Grunwaldzki 2/4 50-384 Wroclaw, Poland e-mail: [email protected]

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Communicated by Rafael D. Benguria. Received: January 6, 2009. Accepted: July 7, 2009.

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Hilbert Lattice Equations Norman D. Megill and Mladen Paviˇcić Abstract. There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet’s EA , Godowski, Mayet–Godowski, and Mayet’s E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet–Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.

1. Introduction In 1995, Solèr [1,2] proved that an infinite-dimensional Hilbert space can be recovered from an orthomodular lattice (OML) together with a small number of additional conditions, with the only ambiguity being that its field may be real, complex, or quaternionic. Specifically, any OML that is complete, is atomic, satisfies a superposition principle, has height at least 4, and has an infinite set of mutually orthogonal atoms, which completely determines such a Hilbert space. This provides us with a dual, purely lattice-theoretical way to work with the Hilbert spaces of quantum mechanics. In addition to offering the potential for new insights, the lattice-theoretical approach may be computationally efficient for certain kinds of problems in quantum mechanics, particularly if, in the future, we are able to exploit what may be a “natural” fit with quantum computation. However, the approach cannot be applied straightforwardly because unlike the equations (identities) defining OML, the additional conditions needed to recover Hilbert space are first- and second-order quantified conditions. Quantified conditions can complicate computational work: trivially, a computer cannot scan infinite lattices or an infinite number of lattices to determine if “there exists” and/or “for all” conditions are satisfied; more generally, quantified theorem-proving algorithms may be needed to achieve rigorous results. Thus, it is desirable to find

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N. D. Megill and M. Paviˇcić


equations that can partially express some of these quantified conditions, allowing them to be weakened or possibly even replaced. The goal is to get as close as possible to a purely equational description of C(H) (the lattice of closed subspaces of a Hilbert space H), in other words to find smaller and smaller equational varieties that contain it. Until 1975, the only lattice equations known to hold in C(H) were those defining OML itself. Then Alan Day discovered that a stronger equation, the orthoarguesian law, also holds. There have been several advances since then. In 2000, Megill and Paviˇcić [3] discovered an infinite family of equations that generalised the orthoarguesian law and called them generalised orthoarguesian laws. In 2006, Mayet [4] described a family of equations EA , obtained with a technique similar to that used to derive the generalised orthoarguesian laws, that may further generalise these laws. In this paper, we obtain a result that opens a possibility that the latter class coincide with the former. While the previous equations are not related to the states lattices admit, the other equations are. In 1981, Godowski [5] discovered an infinite family of equations derived by considering states on the lattice. In 1986, Mayet [6] generalised (strengthened) Godowski’s equations with a new family, but the examples he gave were shown by Megill and Paviˇcić [3] to actually be instances of Godowski’s equations. In 2006, Megill and Paviˇcić [7] showed the Mayet–Godowski class to be independent from the Godowski class. We also present a new algorithm for generating Mayet–Godowski equations (MGEs) that differs considerably from other methods described by Mayet [6,8]. In 2006, Mayet [4] discovered several new series of equations that hold provided the underlying field of H is real, complex, or quaternionic, which are also the ones of interest for quantum mechanics. Mayet found these by considering vector-valued states on C(H) and showed that they were independent of any of the other equations found so far. In this paper, we obtain several new results on these equations. To achieve our results cited above, we developed several new algorithms. The main part of this paper describes the two most important ones, which are incorporated into the computer programs that found these results. The first algorithm (Sect. 4) determines whether a finite OML admits a “strong set of states” (defined elsewhere) and if not, an extension to the algorithm (Sect. 5) generates MGE that fails in the input OML but holds in every Hilbert lattice. The second algorithm (Sect. 6) enables us to prove whether or not this generated equation is independent from every equation in the infinite family found by Godowski. This second algorithm also enabled us to find Godowski lattices of much higher order than before and to show that it is possible to reduce their original size, therefore speeding up calculations that make use of them; some of these results are presented at the end of Sect. 6. The last part of the paper presents two new results that were partly assisted by our programs. In Sect. 7, we show that an example provided by Mayet from his new family of orthoarguesian-related equations in fact can be derived from the

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generalised orthoarguesian laws, leaving open the problem of whether this new family has members that are strictly stronger than these laws. In Sect. 8, we show the solution to an open problem posed by Mayet [4] concerning his new families of equations related to strong sets of Hilbert space-valued states [8].

2. Definitions for Lattice Structures We briefly recall the definitions we will need. For further information, see Refs. [3, 7,9,10]. Definition 2.1. [11] A lattice is an algebra L = LO , ∩, ∪ such that the following conditions are satisfied for any a, b, c ∈ LO : a ∪ b = b ∪ a, a ∩ b = b ∩ a, (a ∪ b) ∪ c = a ∪ (b ∪ c), (a ∩ b) ∩ c = a ∩ (b ∩ c), a ∩ (a ∪ b) = a, a ∪ (a ∩ b) = a. def

Theorem 2.2. [11] The binary relation ≤ defined on L as a ≤ b ⇐⇒ a = a ∩ b is a partial ordering. Definition 2.3. [12] An ortholattice (OL) is an algebra LO , , ∩, ∪, 0, 1 such that LO , ∩, ∪ is a lattice with unary operation called orthocomplementation which satisfies the following conditions for a, b ∈ LO (a is called the orthocomplement of a): a ∪ a = 1, a ∩ a = 0, a ≤ b ⇒ b ≤ a , a = a Definition 2.4. [13,14] An OML is an OL in which the following condition holds: def

a ↔ b = 1 ⇔ a = b, where a ↔ b = 1 ⇐⇒ a → b = 1 & b → a = 1, where def a → b = a ∪ (a ∩ b). Definition 2.5. [15] We say that a and b commute in OML, and write aCb, when the following equation holds: a ∩ (a ∪ b) ≤ b. Definition 2.6. An OML which satisfies the following conditions is a Hilbert lattice, HL.1 1. 2.

Completeness: The meet and join of any subset of an HL exist. Atomicity: Every non-zero element in an HL is greater than or equal to an atom. (An atom a is a non-zero lattice element with 0 < b ≤ a only if b = a.) 3. Superposition principle: (The atom c is a superposition of the atoms a and b if c = a, c = b, and c ≤ a ∪ b.) (a) Given two different atoms a and b, there is at least one other atom c, c = a, and c = b, that is a superposition of a and b. (b) If the atom c is a superposition of distinct atoms a and b, then atom a is a superposition of atoms b and c. 4. Minimum height: The lattice contains at least two elements a, b satisfying: 0 < a < b < 1.

1

For additional definitions of the terms used in this section see Refs. [2, 3, 16, 17].

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Note that atoms correspond to pure states when defined on the lattice. We recall that the irreducibility and the covering property follow from the superposition principle [16, pp. 166, 167]. We also recall that any Hilbert lattice must contain a countably infinite number of atoms [18]. By Birkhoff’s HSP theorem [19, p. 2], the family HL is not an equational variety, since a finite sublattice is not an HL. A goal of studying equations that hold in HL is to find the smallest variety that includes HL, so that the fewest number of non-equational (quantified) conditions such as the above will be needed to complete the specification of HL. Definition 2.7. A state (also called probability measures or simply probabilities [17,20–22]) on a lattice L is a function m : L −→ [0, 1] such that m(1) = 1 and a ⊥ b ⇒ m(a ∪ b) = m(a) + m(b), where a ⊥ b means a ≤ b . Lemma 2.8. The following properties hold for any state m: m(a) + m(a ) = 1,

(2.1)

a ≤ b ⇒ m(a) ≤ m(b), 0 ≤ m(a) ≤ 1,

(2.2) (2.3)

m(a1 ) = · · · = m(an ) = 1 ⇔ m(a1 ) + · · · + m(an ) = n, m(a1 ∩ · · · ∩ an ) = 1 ⇒ m(a1 ) = · · · = m(an ) = 1.

(2.4) (2.5)

Definition 2.9. A set S of states on L is called a strong2 set of states if (∀a, b ∈ L)([(∀m ∈ S)(m(a) = 1 ⇒ m(b) = 1)] ⇒ a ≤ b).

(2.6)

Theorem 2.10. [3] Every Hilbert lattice admits a strong set of states.

3. Definitions of Equational Families Related to States First, we will define the family of equations found by Godowski, introducing a special notation for them. These equations hold in any lattice admitting a strong set of states and thus, in particular, any Hilbert lattice [3]. Definition 3.1. Let us call the following expression the Godowski identity: γ

def

a1 ≡ an = (a1 → a2 ) ∩ (a2 → a3 ) ∩ · · · ∩ (an−1 → an ) ∩ (an → a1 ), γ

n = 3, 4, . . . (3.1)

We define an ≡ a1 in the same way with variables ai and an−i+1 swapped.

2

Some authors use the term rich instead of strong, e.g. Ref. [23, p. 21].

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Theorem 3.2. Godowski’s equations [5] γ

γ

γ

γ

γ

γ

a1 ≡ a3 = a3 ≡ a1 ,

(3.2)

a1 ≡ a4 = a4 ≡ a1 ,

(3.3)

a1 ≡ a5 = a5 ≡ a1 ,

(3.4)

... hold in all ortholattices, OLs, with strong sets of states. An OL to which these equations are added is a variety smaller than OML. We shall call these equations n-Go (3-Go, 4-Go, etc.). We also denote by nGO (3GO, 4GO, etc.) the OL variety determined by n-Go, and we call equation n-Go the nGO law.3 Next, we define a generalisation of this family, first described by Mayet [24]. These equations also hold in all lattices admitting a strong set of states, and in particular in all HLs. Definition 3.3. An MGE is an equality with n ≥ 2 conjuncts on each side: t1 ∩ · · · ∩ tn = u 1 ∩ · · · ∩ u n ,

(3.5)

where each conjunct ti (or ui ) is a term consisting of either a variable or a disjunction of two or more distinct variables: ti = ai,1 ∪ · · · ∪ ai,pi , i.e. pi disjuncts, ui = bi,1 ∪ · · · ∪ bi,qi , i.e. qi disjuncts

(3.6) (3.7)

and where the following conditions are imposed on the set of variables in the equation: 1. All variables in a given term ti or ui are mutually orthogonal. 2. Each variable occurs the same number of times on each side of the equality. We will call a lattice in which all MGEs hold an MGO; i.e. MGO is the largest class of lattices (equational variety) in which all MGEs hold. The following three theorems about MGEs and MGOs are proved in Ref. [7]. Theorem 3.4. Every MGE holds in any ortholattice L admitting a strong set of states and thus, in particular, in any Hilbert lattice. Theorem 3.5. The family of all MGEs includes, in particular, the Godowski equations [Eqs. (3.2), (3.3),. . . ]; in other words, the class MGO is included in nGO for all n. 3

The equation n-Go can also be expressed with 2n variables: a1 ⊥ b1 ⊥ a2 ⊥ b2 ⊥ · · · an ⊥ bn ⊥ a1 ⇒ (a1 ∪ b1 ) ∩ · · · (an ∪ bn ) ≤ b1 ∪ a2 , where n ≥ 3. We remark that if we set n = 2, this equation holds in all OMLs, answering a question in Ref. [8, p. 536]. This can be seen as follows. The equation that results from setting n = 2 in the equation series of Theorem 3.2 has two variables and is easily shown to hold in all OMLs. The proof of Th. 3.19 of Ref. [3], which converts it to the 2n-variable form, involves only OML manipulations.

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Theorem 3.6. The class MGO is properly included in all nGOs, i.e. not all MGEs can be deduced from the equations n-Go. Definition 3.7. A condensed state equation is an abbreviated representation of an MGE constructed as follows: all (orthogonality) hypotheses are discarded, all meet symbols, ∩, are changed to +, and all join symbols, ∪, are changed to juxtaposition. For example, the 3-Go equation can be expressed as [7]: a⊥d⊥b⊥e⊥c⊥f ⊥a ⇒ (a ∪ d) ∩ (b ∪ e) ∩ (c ∪ f ) = (d ∪ b) ∩ (e ∪ c) ∩ (f ∪ a),

(3.8)

which, in turn, can be expressed by the condensed state equation ad + be + cf = db + ec + f a.

(3.9)

The one-to-one correspondence between these two representations of an MGE should be obvious.

4. Finding States on Finite Lattices It is possible to express the set of constraints imposed by states as a linear programming (LP) problem. LP is used by industry to minimise cost, labour, etc., and many efficient programs have been developed to solve these problems, most of them based on the simplex algorithm. We will examine a particular example in detail to illustrate how the problem is expressed. For this example, we will consider a Greechie diagram with 3-atom blocks, although the principle is easily extended to any number of blocks. If m is a state, then each 3-atom block with atoms (a, b, c) and complements (a , b , c ) imposes the following constraints: m(a) + m(b) + m(c) = 1 m(a ) + m(a) = 1, m(b ) + m(b) = 1, m(c ) + m(c) = 1, m(x) ≥ 0,

(4.1) x = a, b, c, a , b , c .

To obtain Eq. (4.1), note that in any Boolean block, a ⊥ b ⊥ c ⊥ a, so m(a) = 1 − m(a ) = 1 − m(b ∪ c) = 1 − m(b) − m(c). Let us take the specific example of the Peterson lattice, which we know does not admit a set of strong states. The Greechie diagram for this lattice, shown in Fig. 1, can be expressed with the textual notation 123,345,567,789,9AB,BC1,2E8,4FA,6DC,DEF.

(see Ref. [7]), where each digit or letter represents an atom, and groups of them represent blocks (edges of the Greechie diagram).

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8

F

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7

E

D

6 5

B 4

C 1

2

3

Figure 1. Greechie diagram for the Peterson lattice. Referring to the textual notation, we designate the atoms by 1, 2, . . . , F and their orthocomplements by 1 , 2 , . . . , F . We will represent the values of state m on the atoms by m(1), m(2), . . . , m(F ). This gives us the following constraints for the 10 blocks: m(1) + m(2) + m(3) = 1, m(3) + m(4) + m(5) = 1, m(5) + m(6) + m(7) = 1, m(7) + m(8) + m(9) = 1, m(9) + m(A) + m(B) = 1, m(B) + m(C) + m(1) = 1, m(2) + m(E) + m(8) = 1, m(4) + m(F ) + m(A) = 1, m(6) + m(D) + m(C) = 1, m(D) + m(E) + m(F ) = 1.

In addition, we have m(a ) + m(a) = 1, m(a) ≥ 0, and m(a ) ≥ 0 for each atom a, adding potentially an additional 15 × 3 = 45 constraints. However, we can omit all but one of these since most orthocomplemented atoms are not involved in this problem, the given constraints are sufficient to ensure that the state values for atoms are less than 1, and the particular LP algorithm we used assumes all variables are nonnegative. This speeds up the computation considerably. The only one we will need is m(7) + m(7 ) = 1 because, as we will see, the orthocomplemented atom 7 will be part of the full problem statement. We pick two incomparable nodes, 1 and 7 , which are on opposite sides of the Peterson lattice. (The program will try all possible pairs of incomparable nodes,

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but for this example, we have selected a priori a pair that will provide us with the answer). Therefore, it is the case that ∼ 1 ≤ 7 . If the Peterson lattice admitted a strong set of states, for any state m we would have: (m(1) = 1 ⇒ m(7 ) = 1) ⇒ 1 ≤ 7 . Since the conclusion is false, for some m we must have ∼ (m(1) = 1 ⇒ m(7 ) = 1), i.e. ∼ (∼ m(1) = 1 ∨ m(7 ) = 1), i.e. m(1) = 1 & ∼ m(7 ) = 1. So this gives us another constraint: m(1) = 1; and for a set of strong states to exist, there must be some m such that m(7 ) < 1. So, our final LP problem becomes (expressed in the notation of the publicly available program lp solve4 ): min: m1 = m7 + m1 + m3 + m5 + m7 + m9 + mB + m2 + m4 + m6 + mD +

m7 ; 1; m7 = 1; m2 + m3 = m4 + m5 = m6 + m7 = m8 + m9 = mA + mB = mC + m1 = mE + m8 = mF + mA = mD + mC = mE + mF =

1; 1; 1; 1; 1; 1; 1; 1; 1; 1;

which means “minimise m(7 ), subject to constraints m(1) = 1, m(7) + m(7 ) = 1, . . ..” The variable to be minimised, m(7 ), is called the objective function (or “cost function”). When this problem is given to lp solve, it returns an objective function value of 1. This means that regardless of m, the other constraints impose a minimum value of 1 on m(7 ), contradicting the requirement that m(7 ) < 1. Therefore, we have a proof that the Peterson lattice does not admit a set of strong states. The program states.c that we use reads a list of Greechie diagrams and, for each one, indicates whether or not it admits a strong set of states. The program embeds the lp solve algorithm, wrapping around it an interface that translates, internally, each Greechie diagram into the corresponding LP problem. 4

Version 3.2, available at http://m3k.grad.hr/lp solve 32/ (as of January 2010).

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5. Generation of MGEs from Finite Lattices When the LP problem in the previous section finds a pair of incomparable nodes that prove that the lattice admits no strong set of states, the information in the problem can be used to find an equation that holds in any OML admitting a strong set of states, and in particular in HL, but fails in the OML under test. Typically, an OML to be tested was chosen because it does not violate any other known HL equation. Thus, by showing an HL equation that fails in the OML under test, we would have found a new equation that holds in HL and is independent from other known equations. The set of constraints that lead to the objective function value of 1 in our LP problem turns out to be redundant. Our algorithm will try to find a minimal set of hypotheses (constraints) that are needed. The equation-finding mode of the states.c program incorporates this algorithm, which will try to weaken the constraints of the LP problem one at a time, as long as the objective function value remains 1 (as in the problem in the previous section). The equation will be constructed based on a minimal set of unweakened constraints that results. The theoretical basis for the construction is described in the proof of Theorem 30 of Ref. [7]. Here, we will describe the algorithm by working through a detailed example. Continuing from the final LP problem of the previous section, the program will test each constraint corresponding to a Greechie diagram block, i.e. each equation with three terms, as follows. It will change the right-hand side (r.h.s.) of the constraint equation from = 1 to ≤ 1, thus weakening it, then it will run the LP algorithm again. If the weakened constraint results in an objective function value m(7 ) < 1, it means that the constraint is needed to prove that the lattice does not admit a strong set of states, so we restore the r.h.s. of that constraint equation back to 1. On the other hand, if the objective function value remains m(7 ) = 1 (as in the original problem), a tight constraint on that block is not needed for the proof that the lattice does not admit a strong set of states, so we leave the r.h.s. of that constraint equation at ≤ 1. After the program completes this process, the LP problem for this example will look like this: min: m1 = m7 + m1 + m3 + m5 + m7 + m9 + mB + m2 + m4 +

m7 ; 1; m7 = 1; m2 + m3 0, m > 0 and large enough positive λ, V (r) is positive between two zeroes, corresponding to a black-hole event horizon and a cosmological event horizon. The solution generalises the asymptotically dS Kottler solution. With Λ < 0 and m > 0, V again has a single zero, corresponding to an event horizon, and the solution generalises the asymptotically-AdS Kottler solution. The solutions in the previous class with λ ≤ 0 may have no global symmetries except the staticity Killing vector. This is because compact, negative scalar curvature Einstein manifolds have no global symmetries, nor does, for example, the Ricci-flat metric on K3 (an example with λ = 0 and n = 4). rn−1 =

•

• • •

2.2. Possible Interiors 2.2.1. Some Einstein Metrics. We construct some Einstein (n + 1)-metrics in the familiar way as cones on Einstein n-metrics. Proposition 2.7. If I ⊂ R is an open interval, f : I → R is a positive smooth function and (N, dσ 2 ) is as before, then the Riemannian metric defined on the (n + 1)-dimensional manifold I × N by dρ2 + f (ρ)2 dσ 2

(5)

is Einstein with Ricci scalar k(n + 1) precisely in the following cases: 1. If k > 0 then k = ν 2 n, 2.

λ = ν 2 (n − 1),

f = sin(νρ)

for some ν > 0. If k = 0 then λ = n − 1,

f = ρ.

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If k < 0 then k = −ν 2 n,

λ = ν 2 (n − 1),

f = sinh(νρ),

or k = −ν 2 n,

λ = 0,

f = e±νρ ,

or k = −ν 2 n,

λ = −ν 2 (n − 1),

f = cosh(νρ),

for some ν > 0, according as λ > 0, λ = 0 or λ < 0. These metrics typically have singularities at the origin ρ = 0. Proposition 2.8. The Kretschmann scalar K of the (n + 1)-metric above (that is the trace of the square of the Riemann tensor) is related to the square C 2 of the Weyl tensor of the base n-metric by K=

C2 + const. f4

(6)

Therefore, the (n + 1)-metric is singular anywhere f vanishes, unless the n-metric is conformally flat (like an n-sphere with the standard metric).3 This can be avoided in case 3 with f = e±νρ when the metric has an internal infinity (or a cusp) or f = cosh(νρ) when the metric has a minimal surface and a second asymptotic region, which will correspond to a space–time wormhole. Otherwise, if f has a zero at which the metric is singular, we shall need to check whether this singularity is visible from infinity in the resulting space–time. 2.2.2. Some FLRW-Like Metrics. The previous subsection suggests a family of (n + 2)-dimensional FLRW-like metrics. Proposition 2.9. The (n + 2)-dimensional Lorentzian metric ds2 = −dτ 2 + R2 (τ )(dρ2 + f 2 (ρ)dσ 2 )

(7)

is a solution of the Einstein equations with cosmological constant Λ and energymomentum tensor Tab = µ ua ub , corresponding to a dust fluid with density µ and velocity ua dxa = dτ , if and only if R(τ ) and µ(τ ) satisfy the conservation equation µRn+1 = µ0 ,

(8)

for constant µ0 , and the Friedman-like equation R˙ 2 Λ k 2κµ + . + = R2 nR2 n(n + 1) n + 1 3

(9)

Notice that there exist Einstein metrics on certain spheres which are not conformally flat [5], which could be used here and elsewhere in this article.

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Remark 2.10. There are other solutions of this type which we shall exploit below in Sect. 3, namely ds2 = −dτ 2 + R2 (τ )hij dxi dxj , i

(10)

j

where hij dx dx is chosen to be any Einstein (n + 1)-metric with Ricci scalar k(n + 1). Explicitly we shall take the Einstein metric to be one of Eguchi–Hansen [8], k-Eguchi–Hansen [18] or k-Taub-NUT [6]. FLRW-like dust cosmologies are again given by solutions of (8) and (9). 2.2.3. Some Lemaˆıtre–Tolman–Bondi-Like Solutions. We shall now introduce Lemaˆıtre–Tolman–Bondi-like (LTB-like) solutions, generalising those of [10], related to the metrics of Sect. 2.2.2. Proposition 2.11. The (n + 2)-dimensional Lorentzian metric ds2 = −dτ 2 + A(τ, ρ)2 dρ2 + B(τ, ρ)2 dσ 2

(11)

is a solution to the Einstein equations with cosmological constant Λ and energymomentum tensor Tab = µ ua ub , corresponding to a dust with density µ and velocity ua dxa = dτ , if and only if A(τ, ρ), B(τ, ρ) and µ(τ, ρ) satisfy A = B (1 + w(ρ)), n

µAB = M (ρ)(1 + w(ρ)), for some functions w(ρ) and M (ρ), and λ 1 Λ 2κM (ρ) 2 n−1 2 ˙ B B − B B n−1 = + − 2 n − 1 (1 + w(ρ)) n+1 n

(12) (13)

(14)

(where dot and prime denote differentiation with respect to τ and ρ). Remark 2.12. This metric has three free functions of ρ, namely w(ρ), M (ρ) and B(0, ρ), one of which can be removed by coordinate freedom. 2.3. Matching In this subsection, we seek to match an interior represented by the metric (7) to a static exterior represented by the metric (2) at a surface Ω ruled by radial time-like geodesics in (2) which is comoving, i.e. a surface of constant ρ (say ρ = ρ0 ) in (7). We find that this can be done, subject to conditions found below. Proposition 2.13. The metric (2) can be matched to the FLRW-like metric (7) at f (ρ0 )n+1 ρ = ρ0 provided that f (ρ0 ) > 0 and m = κµ0n(n+1) . Proof. The interior metric on the matching surface Ω is −dτ 2 + R(τ )2 f (ρ0 )2 dσ 2 , while the geodesic in the exterior, parameterised by proper time τ , has E t˙ = , V

r˙ 2 = E 2 − V,


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and the exterior metric on Ω becomes −dτ 2 + r(τ )2 dσ 2 . Ω

Introducing = to mean equal at Ω we must then have Ω

r = R(τ )f (ρ0 ).

(15)

For the second fundamental form, the matching reduces to a calculation already done in [17], and is V t˙ Ω f , (16) = r Rf which, with (15) and the geodesic equation, reduces to E = f (ρ0 ).

(17)

Since we need E > 0, this constrains the matching to a region where f > 0. The other geodesic equation, with the dot of (15), is r˙ 2 = E 2 − V = R˙ 2 f (ρ0 )2 , which, with (1), reduces to the Friedman equation (9) if we make the identifications m=

κµ0 f (ρ0 )n+1 , n(n + 1)

(18)

and kf (ρ0 )2 λ − . n−1 n The first of these determines the mass m of the exterior from the density and size of the interior. The second is an identity, as can be checked from the formulae in Sect. 2.2.1. E2 =

Remark 2.14. We shall use the term FLRW-Kottler space–times for these matched solutions. Since the matching requires m > 0 in the exterior and V > 0 at Ω, we can have FLRW-Kottler space–times with any sign on Λ for λ > 0, but if λ ≤ 0 then the matching requires Λ < 0. Proposition 2.15. The metric (2) can be matched to the LTB-like metric (11) at ρ = ρ0 provided that 1 + w(ρ0 ) > 0 and m = nκ M (ρ0 ). Proof. The proof is analogous to the previous one, and we obtain Ω

r = B(τ, ρ0 ), E = (1 + w(ρ0 ))−1 , κ m = M (ρ0 ), n in place of (15), (17), and (18), respectively.

Remark 2.16. We shall use the term LTB-Kottler space–times for these matched solutions.

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(a)


(b)

Figure 1. Penrose diagram for Λ = 0 and a k ≤ 0; b k > 0, showing the matching surfaces and the horizons. 2.4. Global Properties We will now analyse in detail the global properties of the FLRW-Kottler spacetime in the three cases Λ = 0, Λ > 0 and Λ < 0, and make some remarks about the global properties of the LTB-Kottler spacetime. Proposition 2.17. If Λ = 0 (hence λ > 0) and (N, dσ 2 ) is not an n-sphere then the locally naked singularity of the FLRW-Kottler spacetime at ρ = 0 is always visible from I + for k ≤ 0, but can be hidden if k > 0 and n ≥ 4 (space–time dimension n + 2 ≥ 6). Proof. The first statement is clear from the Penrose diagram depicted in Fig. 1. For the second statement we must compare the conformal lifetime of the FLRW universe R max dR 2π ∆T = 2 = ˙ ν(n − 1) RR 0

π . For the singularity with the supremum of the possible values of ρ0 , which is 2ν to be hidden it is necessary that the radial light ray emanating from ρ = 0 at the Big Bang is to the future of the future event horizon, and it is clear that in this π π case one will have ρ0 > ∆T 2 . This is only possible if 2ν > ν(n−1) , i.e. n > 3.

Remark 2.18. A special role for space–time dimension n + 2 = 6 in dust collapse was also found in [13]. Proposition 2.19. If Λ > 0 (hence λ > 0) and (N, dσ 2 ) is not an n-sphere then the locally naked singularity of the FLRW-Kottler spacetime at ρ = 0 can be always be hidden except if the FLRW universe is recollapsing (hence k > 0) and n < 4. Proof. If the FLRW universe is recollapsing then one can show that its conformal 2π as Λ → 0. Therelifetime is an increasing function of Λ, and approaches ν(n−1) fore, the singularity can be hidden for sufficiently small Λ if n ≥ 4, but not if n < 4. If the FLRW universe is not recollapsing then one can show that its conformal lifetime is a decreasing function of Λ which approaches zero as Λ → +∞. Therefore, the singularity can be hidden for sufficiently large Λ (Fig. 2).


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(b)

Figure 2. Penrose diagram for Λ > 0 with the FLRW universe a recollapsing; b non-recollapsing, showing the matching surfaces and the horizons.

(a)

(b)

(c)

Figure 3. Penrose diagram for Λ < 0 and a λ > 0; b λ = 0; c λ < 0, showing the matching surfaces and the horizons. Proposition 2.20. For Λ < 0 the FLRW-Kottler spacetime satisfies the following: 1. If λ > 0 and (N, dσ 2 ) is not an n-sphere then the locally naked singularity of the FLRW-Kottler spacetime at ρ = 0 can always be hidden. 2. If λ = 0 then the cusp singularity is not locally naked. 3. If λ < 0 then no causal curve can cross the wormhole from one I to the other. Proof. To prove the first statement one just has to check that the conformal lifetime of the FLRW universe goes to zero as Λ → −∞. Therefore, one can always hide the singularity by taking Λ small enough (Fig. 3). The second statement follows from the fact that for λ = 0 one must have f (ρ) = eνρ , and hence the cusp singularity is at ρ = −∞. To prove the third statement (which can be seen, essentially, as a corollary of a result of Galloway [9]) one notices that the future horizons hit the matching surfaces at marginally outer trapped surfaces. The set of all these surfaces forms the curve R˙ + ν tanh(νρ) = 0, which can be seen to be spacelike with the help of the Friedman-like equation (9). A similar argument shows that the past horizons are connected by the spacelike curve of marginally anti-trapped surfaces. The statement now follows from the observation that these two curves touch at R˙ = ρ = 0.

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Remark 2.21. The global properties of the LTB-Kottler spacetime obtained in Proposition 2.15 are much more diverse. For instance, one can easily find examples of black hole formation with wormholes inside the matter with positive λ and Λ = 0 (similar results in 4-dimensions are in [14]). Indeed, take data B(0, ρ) = a2 + ρ2 ,

˙ B(0, ρ) = 0,

A(0, ρ) = 1.

Then τ = 0 is a surface of time symmetry, the metric on τ = 0 has a minimal surface at ρ = 0, and (1 + w)−1 = 2ρ. Equation (14) becomes λ 2κM (ρ) 1−n 2 2 ˙ B =− − 4ρ + B . n−1 n We restrict ρ so that the first term is strictly negative, ρ2
0 with Λ = 0 and a wormhole. To rule out shell-crossing, which would occur at a zero of A in (11), we consider the (ρ, ρ) component of the Einstein equations. This is A˙ B˙ A¨ n B κµ +n − , = A AB AB A n ρ2
0 for τ > 0 and so A is never zero, i.e. there is no shell-crossing, for τ ≥ 0. Since A is an even function of τ this shows that there is no shell-crossing at all for this example.


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3. Collapse to Black Hole with Gravitational Wave Emission The matchings in the previous section involved static exteriors. In this section, we shall consider gravitational collapse with a gravitational wave exterior, so that the exterior metrics will be time-dependent generalisations of those in (2). For simplicity, we confine our attention to one example, the Bizo´ n–Chmaj–Schmidt metric in (4 + 1)-dimensions [3], though a similar ansatz can be made in other dimensions and with other symmetries (see [4]). We shall consider three different interiors with this exterior, built on Riemannian Bianchi type IX spatial metrics. 3.1. The Exterior: Bizo´ n–Chmaj–Schmidt Metric Consider the metric [3] r2 2B 2 r2 e (σ1 + σ22 ) + e−4B σ32 , (21) 4 4 where A, δ and B are functions of t and r. The one-forms σi are left-invariant for the standard Lie group structure on S 3 , satisfy the differential relations dσi = 1 2 ijk σj ∧ σk , and can be taken to be ds2+ = −Ae−2δ dt2 + A−1 dr2 +

σ1 = cos ψdθ + sin θ sin ψdφ, σ2 = sin ψdθ − sin θ cos ψdφ,

(22)

σ3 = dψ + cos θdφ, where θ, ψ, φ are Euler angles on S 3 with 0 < θ < π, 0 < φ < 2π and 0 < ψ < 4π. The Schwarzschild limit of (21) is obtained by setting B = 0. The space–time with B = 0 is interpreted as containing pure gravitational waves with radial symmetry [3]. Note that there is a residual coordinate freedom t → tˆ = f (t);

δ → δˆ = δ + log f˙

(23)

in the metric (21), which one can exploit to choose δ arbitrarily along a timelike curve. The (4 + 1)-dimensional vacuum EFEs give 1 2A + (8e−2B − 2e−8B ) − 2r(e2δ A−1 (∂t B)2 + A(∂r B)2 ), r 3r ∂t A = −4rA(∂t B)(∂r B),

∂r A = −

2δ

−2

∂r δ = −2r(e A

2

2

(∂t B) + (∂r B) ),

(24) (25) (26)

together with the quasi-linear wave equation for B 4 ∂t (eδ A−1 r3 (∂t B)) − ∂r (e−δ Ar3 (∂r B)) + e−δ r(e−2B − e−8B ) = 0. (27) 3 In [3] the authors solve this system by giving B and ∂t B at t = 0 with A(0, 0) = 1 and δ(t, 0) = 0. We shall be interested in giving data A, B and the normal derivative ∇n B at the timelike boundary Ω of the collapsing interior, which is noncharacΩ teristic for this system, with the gauge choice δ = 0. Uniqueness and local existence follow as standard. From [7,15] one knows that the 5-dimensional Schwarzschild

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metric is stable among the BCS solutions, so that if data close to that for Schwarzschild is given on an asymptotically-flat hypersurface then the solution will exist forever and stay close to the Schwarzschild solution. As we shall see, data on Ω can be chosen to be close to data for Schwarzschild. This is not sufficient to deduce that the solution exists forever and is asymptotically flat in the exterior, but it makes it rather plausible. The matching surface is parameterised by Ω+ = {t(τ ), r(τ )}, and the first fundamental form on Ω+ is r2 2B 2 r2 e (σ1 + σ22 ) + e−4B σ32 . 4 4 The normal vector to the matching surface is ds2+ |Ω+ = −dτ 2 +

eδ ˙ r, ∂t + Ae−δ t∂ A where the dot denotes differentiation with respect to the parameter τ . The boundary as seen from the exterior is ruled by geodesics which obey na+ = r˙

Ae−2δ t˙2 − A−1 r˙ 2 = 1.

(28)

+

The second fundamental form on Ω reads r2 2B r ˙ e ∇n B + e2B Ae−δ t, 4 4 r2 e−4B r ˙ ∇n B + e−4B Ae−δ t. =− 2 4

+ + K11 = K22 = + K33

3.2. The Interiors As interior metrics, we shall consider three classes of FLRW-like solutions based on Riemannian Bianchi-IX spatial metrics which are respectively the Eguchi–Hanson metric (with Rij = 0), the k-Eguchi–Hanson metric (with Rij = kgij excluding the case k = 0) and the k-Taub-NUT metric (with Rij = kgij , including k = 0 as a particular case). We summarize our results as follows: Theorem 3.1. In each case, the interior metric gives consistent data for the metric (21) at a comoving time-like hypersurface. Local existence of the radiating exterior in the neighbourhood of the matching surface is then guaranteed. In the case of Eguchi–Hanson and k-Taub-NUT with k < 0, the data can be chosen to be close to the data for the Schwarzschild solution. 3.2.1. The Eguchi–Hanson Metric. Eguchi and Hanson found a class of self-dual solutions to the Euclidean Einstein equations with metric given by [8] −1 a4 ρ2 2 ρ2 a4 2 2 hEH = 1 − 4 dρ + (σ1 + σ2 ) + (29) 1 − 4 σ32 ρ 4 4 ρ


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with σi given by (22) and a is a real constant. The level sets of ρ are topologically S 3 /Z2 , rather than S 3 , but the corresponding quotient can also be taken on the metric (21). The FLRW-like metric built on this is ds2− = −dτ 2 + R2 (τ )hEH , with the Einstein equations for a dust source reducing to κµ0 . µR4 = µ0 , R˙ 2 = 6R2 We shall match at ρ = ρ0 so that Ω− is parameterised by

(30)

Ω− = {τ, ρ = ρ0 }. The corresponding first fundamental form on Ω− is 2 ρ ρ2 2 a4 2 2 2 2 2 − ds |Ω = −dτ + R (τ ) 1 − 4 σ3 + (σ1 + σ2 ) , 4 ρ 4 and the equality of the first fundamental forms then gives Ω

r = Rρe−B , Ω

e−6B = 1 −

a4 . ρ4

(31)

The normal vector to the matching surface is 1 1 a4 2 n− = ∂ρ , 1− 4 R ρ and the associated non-zero components of the second fundamental form on Ω− are 1 a4 2 − − Ω ρR K11 = K22 = , 1− 4 4 ρ 1 a4 2 a4 − Ω R K33 = . 1− 4 ρ+ 3 4 ρ ρ The equality of the second fundamental forms gives − 12 2a4 a4 Ω ∇n B = − , 1− 4 3Rρ5 ρ a4 Ω Ae−δ t˙ = e2B 1 − 4 . 3ρ Then from (28), (30) and (31) we calculate 2 κµ0 a4 Ω 4B A=e − 2 ρ4 e−4B . 1− 4 3ρ 6r

(32)

(33)

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From the EFEs (24)–(26) on Ω we get (using the matching conditions) Ω

∂t B = −∇n B re ˙ −δ ,

Ω

˙ −δ , ∂r B = ∇n B te

and then it is straightforward to check that the expression (33) for A is consistent with A˙ calculated from (24) and (25). At this point, we have B, ∇n B and A on Ω, so that (32) gives the combina˙ We cannot expect to obtain the two factors separately because of the tion e−δ t. gauge freedom, which we can use to set δ = 0 on Ω. Ω Ω By (31) we have B = O(a4 /ρ4 ) and by (32) ∇n B = (ρR)−1 O(a4 /ρ4 ) and, to say that the data are close to Schwarzschild data, we want these to be small. The first term is small if ρ a. The second term has dimension (length)−1 and will increase without bound as R decreases to zero in the contracting direction. This will only happen inside the horizon. If we restrict R by its value when a marginally outer-trapped surface forms on Ω then, from the Friedman equation and with ρ a, this happens when R2 ρ2 ∼ κµ0 ρ4 , so that we control ∇n B on Ω by controlling µ0 . Now by choice of the location of Ω, at ρ = ρ0 , and choice of µ0 we can choose data close to Schwarzschild. 3.2.2. The k-Eguchi–Hanson metric. By this we mean the metric of Pedersen [18], which can be regarded as the Eguchi–Hanson metric with a cosmological constant (k rather than Λ, with our conventions), given by hkEH = ∆−1 dρ2 + where ∆ = 1 −

a4 ρ4

ρ2 2 ρ2 (σ1 + σ22 ) + ∆σ32 , 4 4

(34)

− k6 ρ2 . This metric is complete for k < 0 and

a4 =

4 (p − 2)2 (p + 1), 3k 2

ρ>

−

2(p − 2) k

12 ,

where p ≥ 3 in an integer. Then the singularity at ∆ = 0 is a removable bolt and the level sets of ρ are topologically S 3 /Zp . Since k is related to a for a complete solution, we cannot obtain the previous case from this case by taking k → 0. However, the matching formulae do formally allow this limit, as we shall see. Now the Einstein equations for dust source reduce to µR4 = µ0 ,

κµ0 k R˙ 2 + = . 3 6R2

(35)

We again take the matching surface at constant ρ so that the first fundamental form on Ω− is 2 ρ ρ2 2− 2 2 2 2 2 (σ + σ2 ) + ∆σ3 . ds |Ω− = −dτ + R (τ ) 4 1 4


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The equality of the first fundamental forms on Ω gives Ω

r = Rρe−B , Ω

e−6B = ∆.

(36)

The normal vector to the matching surface is 1 1 n− = ∆ 2 ∂ρ , R and the associated non-zero components of the second fundamental form on Ω− are 1 − − Ω ρR ∆2 , = K22 = K11 4 1 k 3 a4 − Ω R K33 = ρ + 3 − ρ ∆2 . 4 ρ 3 The equality of the second fundamental forms gives 1 ∆− 2 2a4 kρ2 Ω ∇n B = − − , 3ρR ρ4 6 2kρ2 a4 Ω Ae−δ t˙ = e2B 1 − 4 − . 3ρ 9 We can calculate A as before, to find 2 κµ0 ρ4 e−4B 2kρ2 kρ2 a4 Ω 4B ∆ − + , 1− 4 − A=e 3ρ 9 3 6r2

(37)

(38)

and as before check that this is consistent with A˙ calculated from (24) and (25). It is not so clear that we may choose data close to Schwarzschild data in this Ω case. We can take B = 0, but then the normal derivative is kρ , 6R so that, for this to be small, we would require R to be large on Ω outside the marginally trapped surface. It is hard to see how to arrange this and so, although the solution in the exterior exists locally, we do not have a good reason to think that it will settle down to Schwarzschild. Ω

∇n B =

3.2.3. k-Taub-NUT. We take the Riemannian Taub-NUT metric with a cosmological constant (k rather than Λ with our conventions) [1,6] hTN =

1 −1 2 1 2 Σ dρ + (ρ − L2 )(σ12 + σ22 ) + L2 Σσ32 , 4 4

where Σ=

(ρ − L)(1 −

k 12 (ρ

− L)(ρ + 3L)) , ρ+L

(39)

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and use it to construct the interior: ds2− = −dτ 2 + R2 (τ )hTN . The Einstein equations for a dust source are again (35). At the matching surface ρ = ρ0 the first fundamental form is 1 2 2− 2 2 2 2 2 2 2 (ρ − L )(σ1 + σ2 ) + L Σσ3 . ds |Ω− = −dτ + R (τ ) 4 From matching the first fundamental forms we get Ω

1

r = R(ρ2 − L2 ) 2 e−B , Ω

e−6B =

4L2 Σ . ρ2 − L2

(40)

The normal vector to Ω− is taken to be 2 1 n− = Σ 2 ∂ρ , R and the non-zero components of the second fundamental form in this case are 1 − − Ω 1 K11 = K22 = RΣ 2 ρ, 2 (41) 1 2ΣL k(ρ − L) − Ω − K33 = RL2 Σ 2 . ρ2 − L2 6 The second matching conditions read 2 2L (2ρ + L)Σ k 2 −δ ˙ Ω 4R 12 4B Σ e − L (ρ − L) , Ae t = 3r ρ2 − L2 6 2 1 2L Σ k 2 Ω 4R 4B 2 + L (ρ − L) ∇n B = 2 Σ e 3r ρ+L 6 1/2 2Σ k(ρ − L) Ω 1 + = . 3R ρ + L 6Σ1/2

(42) (43)

We calculate A on Ω as before and obtain an expression of the form A = c1 (ρ) +

c2 (ρ) r2

and, as before, we can check that this is consistent with A˙ calculated from (24) and (25). Now note that if kL2 = −3 then the metric (39) is precisely the 4-dimensional hyperbolic metric. In this case, B and ∇n B vanish on Ω whatever the value of ρ0 , so that the exterior metric is precisely Schwarzschild: this is a case from Sect. 2 as the interior is now a standard FLRW cosmology. Consequently, if we take kL2 close to −3 we expect to get data close to Schwarzschild data. To see that this is the case, set kL2 = −3(1 + ).


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Then Σ=

(ρ2 − L2 ) (1 + O( )) , 4L2

so that Ω

e−6B = 1 + O( ), and 1 O( ). LR Now, clearly the data (B, ∇n B) can be chosen as small as desired by choosing large ρ0 and small . Ω

∇n B =

Acknowledgements We thank CRUP/British Council for Treaty of Windsor grant B-29/08. FM was supported by CMAT, University of Minho. JN was supported by FCT (Portugal) through program POCI 2010/FEDER and grant POCI/MAT/ 58549/2004. FM and JN thank the EPSRC and the Oxford Centre for Nonlinear PDE (EP/E035027/1), where this work was initiated, for hospitality. FM and PT thank Dep. Matemática, Instituto Superior Técnico, for hospitality.

References [1] Akbar, M.M.: Classical boundary-value problem in Riemannian quantum gravity and Taub–Bolt-anti-de Sitter geometries. Nucl. Phys. B 663, 215–230 (2003) [2] Birmingham, D.: Topological black holes in anti-de Sitter space. Class. Quantum Grav. 16, 1197–1205 (1999) [3] Bizo´ n, P., Chmaj, T., Schmidt, B.G.: Critical behaviour in vacuum gravitational collapse in 4 + 1-dimensions. Phys. Rev. Lett. 95, 071102 (2005) [4] Bizo´ n, P., Chmaj, T., Rostworowski, A., Schmidt, B.G., Tabor, Z.: Vacuum gravitational collapse in nine dimensions. Phys. Rev. D 72, 121502 (2005) [5] B¨ ohm, C.: Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Invent. Math. 134, 145 (1998) [6] Boutaleb-Joutei, H.: The general Taub-NUT de Sitter metric as a self-dual YangMills solution of gravity. Phys. Lett. B 90, 181–184 (1980) [7] Dafermos, M., Holzegel, G.: On the nonlinear stability of higher dimensional triaxial Bianchi-IX black holes. Adv. Theor. Math. Phys. 10, 503–523 (2006) [8] Eguchi, T., Hanson, A.J.: Asymptotically flat self-dual solutions to Euclidean gravity. Phys. Lett. B 74, 249–251 (1978) [9] Galloway, G.J.: A ‘finite infinity’ version of topological censorship. Class. Quantum Grav. 13, 1471–1478 (1996) [10] Ghosh, S.G., Beesham, A.: Higher dimensional inhomogeneous dust collapse and cosmic censorship. Phys. Rev. D 64, 124005 (2001)

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[11] Gibbons, G.W., Hartnoll, S.A.: Gravitational instability in higher dimensions. Phys. Rev. D 66, 064024 (2002) [12] Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness and non-uniqueness of static vacuum black holes in higher dimensions. Prog. Theor. Phys. Suppl. 148, 284–290 (2003) [13] Goswami, R., Joshi, P.: Cosmic censorship in higher dimensions. Phys. Rev. D 69, sss104002 (2004) [14] Hellaby, C.: A Kruskal-like model with finite density. Class. Quantum Grav. 4, 635– 650 (1987) [15] Holzegel, G.: Stability and decay-rates for the five-dimensional Schwarzschild metric under biaxial perturbations. Preprint, arXiv:0808.3246 (2008) [16] Lemos, J.P.S.: Gravitational collapse to toroidal, cylindrical and planar black holes. Phys. Rev. D 57, 4600–4605 (1998) [17] Mena, F.C., Nat´ ario, J., Tod, P.: Gravitational collapse to toroidal and higher genus asymptotically AdS black holes. Adv. Theor. Math. Phys. 12, 1163–1181 (2008) [18] Pederson, H.: Eguchi–Hanson metrics with a cosmological constant. Class. Quantum Grav. 2, 579–587 (1985) [19] Smith, W.L., Mann, R.B.: Formation of topological black holes from gravitational collapse. Phys. Rev. D 56, 4942–4947 (1997) Filipe C. Mena Departamento de Matem´ atica Universidade do Minho 4710-057 Braga, Portugal e-mail: [email protected] José Nat´ ario Departamento de Matem´ atica Instituto Superior Técnico 1049-001 Lisbon, Portugal e-mail: [email protected] Paul Tod Mathematical Institute University of Oxford St Giles’ 24-29 Oxford OX1 3LB, UK e-mail: [email protected] Communicated by Piotr T. Chru´sciel. Received: June 25, 2008. Accepted: September 14, 2009.

Ann. Henri Poincaré 10 (2010), 1377–1393 c 2009 Birkh¨ auser Verlag AG, Basel/Switzerland 1424-0637/10/071377-17, published online December 16, 2009 DOI 10.1007/s00023-009-0015-x


Perturbation Method for Particle-like Solutions of the Einstein–Dirac Equations Simona Rota Nodari Abstract. The aim of this work is to prove by a perturbation method the existence of solutions of the coupled Einstein–Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. We relate the solutions of our equations to those of the nonlinear Choquard equation and we show that the nondegenerate solution of Choquard’s equation generates solutions of the Einstein–Dirac equations.

1. Introduction In this paper, we study the coupled Einstein–Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Using numerical methods, Finster et al. found, in [1], particle-like solutions; our goal is to give a rigorous proof of their existence by a perturbation method.1 The Einstein–Dirac equations take the form (D − m)ψ = 0 (1.1) 1 Rji − Rδji = −8πTji (1.2) 2 where D denotes the Dirac operator, ψ is the wave function of a fermion of mass m, Rji is the Ricci curvature tensor, R indicates the scalar curvature and, finally, Tji is the energy-momentum tensor of the Dirac particle. In [1], Finster et al. work with the Dirac operator into a static, spherically symmetric space–time where the metric, in polar coordinates (t, r, ϑ, ϕ), is given by 1

After completing this work, we learned from professor Joel Smoller that Erik J. Bird had proved the existence of small solutions of the Einstein–Dirac equations in his doctoral thesis in 2005 [2]. His method is quite different from ours: he uses Schauder’s fixed point theorem.

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gij g ij

1 1 2 2 2 = diag , − , −r , −r sin ϑ T2 A 1 1 2 = diag T , −A, − 2 , − 2 2 r r sin ϑ

(1.3) (1.4)

with A = A(r), T = T (r) positive functions; so, the Dirac operator can be written as i T i t r −1/2 − 1−A D = iγ ∂t + γ i∂r + (1.5) + iγ ϑ ∂ϑ + iγ ϕ ∂ϕ r 2T with γ t = T γ¯ 0 √ γ r = A γ¯ 1 cos ϑ + γ¯ 2 sin ϑ cos ϕ + γ¯ 3 sin ϑ sin ϕ 1 1 γϑ = −¯ γ sin ϑ + γ¯ 2 cos ϑ cos ϕ + γ¯ 3 cos ϑ sin ϕ r 1 2 ϕ −¯ γ sin ϕ + γ¯ 3 cos ϕ γ = r sin ϑ

(1.6) (1.7) (1.8) (1.9)

where γ¯ i are the Dirac matrices in Minkowski space (see [1]). Moreover, Finster, Smoller and Yau are looking for solutions taking the form ⎞ 1 ⎟ ⎜ Φ1 0 −iωt −1 1/2 ⎜ ⎟, ψ=e r T ⎝ 1 ⎠ iΦ2 σ r 0 ⎛

(1.10)

1 where σ r = σ ¯ cos ϑ + σ ¯ 2 sin ϑ cos ϕ + σ ¯ 3 sin ϑ sin ϕ is a linear combination of the Pauli matrices σ ¯ i and Φ1 (r), Φ2 (r) are radial real functions. We remind also that the energy-momentum tensor is obtained as the variation of the classical Dirac action

¯ − m)ψ |g| d4 x S = ψ(D and takes the form Tji

1 1 = 2 diag 2ωT 2 |Φ|2 , −2ωT 2 |Φ|2 + 4T Φ1 Φ2 + 2mT Φ21 − Φ22 , r r 1 1 −2T Φ1 Φ2 , −2T Φ1 Φ2 r r

(see [1] for more details).

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In this case, the coupled Einstein–Dirac equations can be written as √ 1 AΦ1 = Φ1 − (ωT + m)Φ2 r √ 1 AΦ2 = (ωT − m)Φ1 − Φ2 r rA = 1 − A − 16πωT 2 Φ21 + Φ22 T 1 2rA = A − 1 − 16πωT 2 Φ21 + Φ22 + 32π T Φ1 Φ2 T r + 16πmT Φ21 − Φ22

(1.11) (1.12) (1.13)

(1.14)

with the normalization condition

∞ 0

1 T . |Φ|2 √ dr = 4π A

(1.15)

In order that the metric be asymptotically Minkowskian, Finster, Smoller and Yau assume that lim T (r) = 1.

r→∞

Finally, they also require that the solutions have finite (ADM) mass; namely lim

r→∞

r (1 − A(r)) < ∞. 2

In this paper, we will prove the existence of solutions of (1.1) and (1.2) in the form (1.10) by a perturbation method. In particular, we follow the idea described by Ounaies in [3] (see also [4] for a rigorous existence proof of nonlinear Dirac solitons based on Ounaies’ approach). Ounaies, by a perturbation parameter, relates the solutions of a nonlinear Dirac equation to those of a nonlinear Schr¨ odinger equation. Imitating the idea of Ounaies, we relate the solutions of our equations to those of nonlinear Choquard’s equation (see [5,6] for more details on Choquard’s equation) and we obtain the following result. Theorem 1.1. Given 0 < ω < m such that m − ω is sufficiently small, there exists a non trivial solution of (1.11)–(1.14). In Sect. 2, we solve the Einstein–Dirac equations by means of the perturbation method suggested by Ounaies; in particular in the first subsection we describe a useful rescaling and some properties of the operators involved, whereas in the second subsection we prove the existence of solutions generated by the solution of the Choquard equation.

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2. Perturbation Method for the Einstein–Dirac Equations First of all, we observe that writing T (r) = 1 + t(r) and using Eq. (1.13), the coupled Einstein–Dirac equations become √

1 Φ1 − (ω + m)Φ2 − ωtΦ2 r √ 1 AΦ2 = (ω − m)Φ1 + ωtΦ1 − Φ2 r AΦ1 =

(2.1) (2.2)

1 2rAt = (A − 1)(1 + t) − 16πω(1 + t)3 Φ21 + Φ22 + 32π (1 + t)2 Φ1 Φ2 r (2.3) +16πm(1 + t)2 Φ21 − Φ22

where 16πω A(r) = 1 − r

r

16πω Q(r). (1 + t(s))2 Φ1 (s)2 + Φ2 (s)2 ds := 1 − r

(2.4)

0

Furthermore, because we want A(r) > 0, we have that the following condition must be satisfied 0≤

Q(r) 1 < r 16πω

(2.5)

for all r ∈ (0, ∞). Now, to find a solution of Eqs. (2.1)–(2.3), we exploit the idea used by Ounaies in [3]. In particular, we proceed as follow: in a first step we use a rescaling argument to transform (2.1)–(2.3) in a perturbed system of the form ⎧ d 1 ⎪ ⎨ A (ε, ϕ, χ, τ ) dr ϕ − r ϕ + 2mχ + K1 (ε, ϕ, χ, τ ) = 0 d (2.6) A (ε, ϕ, χ, τ ) dr χ + 1r χ + ϕ − mϕτ + K2 (ε, ϕ, χ, τ ) = 0 ⎪ ⎩ d 8πm r 2 A (ε, ϕ, χ, τ ) dr τ + r2 0 ϕ ds + K3 (ε, ϕ, χ, τ ) = 0 where ϕ, χ, τ : (0, ∞) → R. Second, we relate the solutions of (2.6) to those of the nonlinear system ⎧ ∞ ϕ2 d2 ⎪ − 16πm3 0 max(r,s) ds ϕ = 0 ⎨ − dr2 ϕ + 2mϕ 1 1 d (2.7) χ(r) = 2m ϕ − dr ϕ r ⎪ ∞ ϕ2 ⎩ τ (r) = 8πm 0 max(r,s) ds. We remark that ϕ is a solution of (2.7) if and only if u(x) = ϕ(|x|) solves the |x| nonlinear Choquard equation ⎞ ⎛

2 |u(y)| (2.8) − u + 2mu − 4m3 ⎝ dy ⎠ u = 0 in H 1 R3 . |x − y| R3

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1381

To prove this fact it is enough to remind that for a radial function ρ, 1 d 2 d ρ = 2 ρ r r dr dr and

⎞ ⎛ ∞

2 −1 s ρ(s) | · | ρ (x) = 4π ⎝ ds⎠ max(r, s) 0

with r = |x|. We observe also that if we write

u(x) v(x)

⎞ 1 ϕ(r) ⎜ 0 ⎟ ⎟ = r−1 ⎜ ⎝ 1 ⎠ r iχ(r)σ 0 ⎛

with r = |x|, (ϕ, χ) is a solution of (2.7) if and only if (u(x), v(x)) solves ⎞ ⎛

2 −i¯ σ ∇u |u(y)| − u + 2mu − 4m3 ⎝ dy ⎠ u = 0 v = |x − y| 2m

(2.9)

R3

3 in H 1 (R3 ) where σ ¯ ∇ = i=1 σ ¯ i ∂i . It is well known that Choquard’s equation (2.8) has a unique radial, positive solution u0 with |u0 |2 = N for some N > 0 given. Furthermore, u0 is infinitely differentiable and goes to zero at infinity; more precisely there exist some positive constants Cδ,η such that |Dη (u0 )| ≤ Cδ,η exp(−δ|x|) for x ∈ R3 . At last, u0 ∈ H 1 (R3 ) is a radial nondegenerate solution; by this we mean that the linearization of (2.8) around u0 has a trivial nullspace in L2r (R3 ). In particular, the linear operator L given by ⎞ ⎞ ⎛ ⎛

2 |u0 (y)| ξ(y)u0 (y) ⎠ dy ⎠ ξ − 8m3 ⎝ dy u0 Lξ = −ξ + 2mξ − 4m3 ⎝ |x − y| |x − y| R3

R3

satisfies ker L = {0} when L is restricted to L2r (R3 ) (see [5–7] for more details). The main idea is that the solutions of (2.6) are the zeros of a C 1 operator D : R × Xϕ × Xχ × Xτ → Yϕ × Yχ × Yτ . If we denote by Dϕ,χ,τ (ε, ϕ, χ, τ ) the derivative of D(ε, ·, ·, ·), by (ϕ0 , χ0 , τ0 ) the ground state solution of (2.7) and we observe that Dϕ,χ,τ (0, ϕ0 , χ0 , τ0 ) is an isomorphism, the application of the implicit function theorem (see [8]) yields the following result, which is equivalent to Theorem 1.1. Theorem 2.1. Let (ϕ0 , χ0 , τ0 ) be the ground state solution of (2.7), then there exists δ > 0 and a function η ∈ C((0, δ), Xϕ × Xχ × Xτ ) such that η(0) = (ϕ0 , χ0 , τ0 ) and (ε, η(ε)) is a solution of (2.6), for 0 ≤ ε < δ.

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2.1. Rescaling In this subsection we are going to introduce the new variable (ϕ, χ, τ ) such that Φ1 (r) = αϕ(λr), Φ2 (r) = βχ(λr) and t(r) = γτ (λr), where Φ1 , Φ2 , t satisfy (2.1)– (2.3) and α, β, γ, λ > 0 are constants to be chosen later. Using the explicit expression of A, given in (2.4), we have 2

r β 16πωα2 2 2 χ (1 + γτ ) ϕ + A(Φ1 , Φ2 , t) = 1 − ds r α 0

:= Aα,β,γ (ϕ, χ, τ )

(2.10)

It is now clear that if Φ1 , Φ2 , t satisfy (2.1)–(2.3), then ϕ, χ, τ satisfy the system ⎧ αλ 1 αλ d ⎪ ⎨ Aα,β,γ β dr ϕ − β r ϕ + (m + ω)χ + ωγτ χ = 0 α d (2.11) Aα,β,γ dr χ + 1r χ + βλ (m − ω)ϕ − αγ βλ ωτ ϕ = 0 ⎪ ⎩ d 2Aα,β,γ r dr τ + Kα,β,γ = 0 with

⎛ r ⎞ 2

β 16πω α2 ⎝ χ Kα,β,γ (ϕ, χ, τ ) = (1 + γτ )2 ϕ2 + ds⎠ (1 + γτ ) r γ α 0 2 2 β α λαβ 1 3 2 + 16πω (1 + γτ ) ϕ + χ (1 + γτ )2 ϕχ −32π γ α γ r 2 β α2 2 2 χ − 16πm (1 + γτ ) ϕ − . γ α 2

αγ α α By adding the conditions βλ (m − ω) = 1, αλ β = 1, βλ = 1, γ = 1 and m − ω ≥ 0, we obtain α = (m − ω)1/2 , λ = (m − ω)1/2 , β = m − ω and γ = m − ω. Denoting ε = m − ω, (2.11) is equivalent to ⎧ d 1 ⎪ ⎨ A (ε, ϕ, χ, τ ) dr ϕ − r ϕ + 2mχ + K1 (ε, ϕ, χ, τ ) = 0 d (2.12) A (ε, ϕ, χ, τ ) dr χ + 1r χ + ϕ − mϕτ + K2 (ε, ϕ, χ, τ ) = 0 ⎪ ⎩ d 8πm r 2 A (ε, ϕ, χ, τ ) dr τ + r2 0 ϕ ds + K3 (ε, ϕ, χ, τ ) = 0

where A(ε, ϕ, χ, τ ), K1 (ε, ϕ, χ, τ ), K2 (ε, ϕ, χ, τ ) and K3 (ε, ϕ, χ, τ ) are defined by 16π(m − ε)ε A (ε, ϕ, χ, τ ) = 1 − r

r

(1 + ετ )2 ϕ2 + εχ2 ds;

(2.13)

0

K1 (ε, ϕ, χ, τ ) = −εχ + ε(m − ε)τ χ;

(2.14)

K2 (ε, ϕ, χ, τ ) = ετ ϕ;

(2.15)

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and 8πmε K3 (ε, ϕ, χ, τ ) = r2

r

16πmε χ ds + r2

0

+

r

2

8πmε2 r2 ⎛

8πε − 2 ⎝ r

τ ϕ2 + εχ2 ds

0

r 0

r

τ 2 ϕ2 + εχ2 ds

⎞ (1 + ετ )2 ϕ2 + εχ2 ds⎠

0

⎞ ⎛ r

8π(m − ε)ε ⎝ + (1 + ετ )2 ϕ2 + εχ2 ds⎠ τ r2 0

2 2 χ2 2 2 3 ϕ + εχ +16πmε + 8πmε(3τ + 3ετ + ε τ ) r r 2 ϕ + εχ2 ϕχ −8πε(1 + ετ )3 − 16πε(1 + ετ )2 2 r r 2 2 ϕ − εχ . (2.16) −8πmε(2τ + ετ 2 ) r For ε = 0, (2.12) becomes ⎧ d 1 ⎪ ⎨ dr ϕ − r ϕ + 2mχ = 0 d 1 dr χ + r χ + ϕ − mϕτ = 0 ⎪ ⎩ d 8πm r 2 ϕ ds = 0 dr τ + r 2 0 that is equivalent to ⎧ ∞ d2 3 ⎪ − ϕ + 2mϕ − 16πm ⎪ 2 0 ⎪ ⎨ dr 1 1 d χ(r) = 2m r ϕ − dr ϕ ⎪ ⎪ ⎪ ⎩ τ (r) = 8πm ∞ ϕ2 ds. 0 max(r,s)

ϕ2 max(r,s)

(2.17)

ds ϕ = 0

Then, we denote by (ϕ0 , χ0 , τ0 ) a solution of (2.18); in particular r d ϕ0 χ0 (r) = − 2m dr r

∞ ϕ20 ds. τ0 (r) = 8πm max(r, s) 0

(2.18)

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Now, to obtain a solution of (2.12) from (ϕ0 , χ0 , τ0 ), we define the operators L1 : R×Xϕ ×Xχ ×Xτ → Yϕ , L2 : R×Xϕ ×Xχ ×Xτ → Yχ , L3 : R×Xϕ ×Xχ ×Xτ → Yτ and D : R × Xϕ × Xχ × Xτ → Yϕ × Yχ × Yτ by 1 d ϕ− A (ε, ϕ, χ, τ ) r dr 1 d χ+ L2 (ε, ϕ, χ, τ ) = A (ε, ϕ, χ, τ ) r dr

ϕ χ 1 + 2m + K1 (ε, ϕ, χ, τ ) 2 r r r χ ϕ 1 ϕ + − m τ + K2 (ε, ϕ, χ, τ ) 2 r r r r

r 8πm d L3 (ε, ϕ, χ, τ ) = A (ε, ϕ, χ, τ ) τ + 2 ϕ2 ds + K3 (ε, ϕ, χ, τ ) dr r

L1 (ε, ϕ, χ, τ ) =

0

and D(ε, ϕ, χ, τ ) = (L1 (ε, ϕ, χ, τ ), L2 (ε, ϕ, χ, τ ), L3 (ε, ϕ, χ, τ )) where Xϕ = Xχ = Xτ = Yϕ =

ϕ(|x|) 1 3 2 1 ϕ : (0, ∞) → R ∈ H R ,R 0 |x| χ(|x|) r 1 3 2 1 σ χ : (0, ∞) → R ∈ H R ,C 0 |x| d 1 τ : (0, ∞) → R lim τ (r) = 0, τ ∈ L ((0, ∞), dr) r→∞ dr 3 2 Yχ = Lr R

Yτ = L1 ((0, ∞), dr). Furthermore we define the following norms: ϕ(|x|) ϕ Xϕ = |x| 1 3 , H (R ) χ(|x|) r 1 σ χ Xχ = , 0 H 1 (R3 ) |x| d τ τ Xτ = . dr 1 L ((0,∞),dr) It is well known that

H 1 R3 → Lq R3 ∞

Xτ → L

2≤q≤6

((0, ∞), dr) .

Moreover, using Hardy’s inequality

|f |2 dx ≤ 4 |∇f |2 dx, |x|2 R3

R3

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we get the following properties: ρ ∈ H 1 ((0, ∞), dr) → L∞ ((0, ∞), dr) ρ ∈ L2 ((0, ∞), dr) r

(2.19)

∀ρ ∈ Xϕ , ∀ρ ∈ Xχ . Since the operator A(ε, ϕ, χ, τ ) must be strictly positive, we consider Bϕ , Bχ , Bτ , defined as the balls of the spaces Xϕ , Xχ , Xτ , and ε1 , ε2 , depending on m and on the radius of Bϕ , Bχ , Bτ , such that 16π(m − ε)ε 1− r

r

(1 + ετ )2 ϕ2 + εχ2 ds ≥ δ > 0

0

for all (ε, ϕ, χ, τ ) ∈ (−ε1 , ε2 ) × Bϕ × Bχ × Bτ . The existence of ε1 , ε2 is assured by the fact that ϕ, χ, τ are bounded; in particular, if ε ≥ 0, 16π(m − ε)ε 1− r

r

(1 + ετ )2 ϕ2 + εχ2 ds

0

2 2 2 ≥ 1 − 20mε ϕ Xϕ − 8mε2 5 τ Xτ ϕ Xϕ + χ Xχ 2 2 2 2 − 4mε3 τ Xτ 5 τ Xτ ϕ Xϕ + 4 χ Xχ − 8mε4 τ Xτ χ Xχ , then there exists ε2 > 0 such that A(ε, ϕ, χ, τ ) > 0 for all ε ∈ [0, ε2 ). In the same way, if ε < 0, 16π(m − ε)ε 1− r ≥

r

(1 + ετ )2 ϕ2 + εχ2 ds

0 2 2 1 − 8mε χ Xχ − 8m|ε|3 χ Xχ 1 + 2 τ Xτ 2 2 −8mε4 χ Xχ τ Xτ 2 + 1 τ Xτ − 8m|ε|5 τ Xτ 2

2

χ Xχ ,

then there exists ε1 > 0 such that A(ε, ϕ, χ, τ ) > 0 for all ε ∈ (−ε1 , 0). Lemma 2.2. The operators L1 , L2 ∈ C 1 ((−ε1 , ε2 ) × Bϕ × Bχ × Bτ , Yϕ ) and L3 ∈ C 1 ((−ε1 , ε2 ) × Bϕ × Bχ × Bτ , Yτ ). Before startingthe proof of the lemma we observe that for a radial function ρ such that ρr ∈ Hr1 R3 we have ρ(r) 1/2 d . |ρ(r)| ≤ r (2.20) 2 dr r L r

We remind that

Hr1 (R3 )

= {u ∈ H (R )| u is radial}. 1

3

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Proof. We begin with L3 ; first, we have to prove that it is well defined in Yτ = L1 ((0, ∞), dr). We remark that

r d C2 2 ϕ + εχ2 ds + C3 ϕ2 + εχ2 |L3 (ε, ϕ, χ, τ )| ≤ C1 τ + 2 dr r r 0

C4 C5 2 + 2 |ϕχ| + ϕ − εχ2 r r where C1 , C2 , C3 , C4 , C5 are positive constants and, by definition, we have that d 1 dr τ ∈ L ((0, ∞), dr). Next, we have

∞ r

∞ 2 2 ϕ + εχ2 1 ϕ + εχ2 ds dr = ds < +∞, r2 s 0

0

0

r 2 1 2 using H¨ older’s inequality, then r 2 0 ϕ + εχ conclude that 1r ϕ2 + εχ2 , 1r ϕ2 − εχ2 ∈ Yτ .

ds ∈ Yτ . In the same way, we can

Finally,

∞

ϕ χ |ϕ| |χ| dr ≤ C < +∞ r r r L2 ((0,∞)) r L2 ((0,∞))

0

thanks to (2.19), then r12 ϕχ ∈ Yτ . Now, we have to prove that L3 (ε, ϕ, χ, τ ) is C 1 ; by classical arguments, it is enough to show that for (h1 , h2 , h3 ) ∈ Bϕ × Bχ × Bτ ∂ (L3 (ε, ϕ, χ, τ )) h1 ∈ Yτ , ∂ϕ ∂ (L3 (ε, ϕ, χ, τ )) h2 ∈ Yτ , ∂χ ∂ (L3 (ε, ϕ, χ, τ )) h3 ∈ Yτ . ∂τ ∂ ∂ϕ

(L3 (ε, ϕ, χ, τ )), ∂ d ∂ (L3 (ε, ϕ, χ, τ )) h1 = (A (ε, ϕ, χ, τ )) h1 τ ∂ϕ ∂ϕ dr ⎛ r ⎞

16π(m − ε) ⎝ + (1 + ετ )2 ϕh1 ds⎠ (1 + ετ ) r2

We begin with

0

+16π(m − ε)(1 + ετ )3 −16πm(1 + ετ )2

ϕh1 ; r

ϕh1 h1 χ − 16πε(1 + ετ )2 2 r r

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∂ ∂χ


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(L3 (ε, ϕ, χ, τ )),

∂ (L3 (ε, ϕ, χ, τ )) h2 = ∂χ

∂ d (A (ε, ϕ, χ, τ )) h2 τ ∂χ dr ⎛ r ⎞

16π(m − ε)ε ⎝ + (1 + ετ )2 χh2 ds⎠ (1 + ετ ) r2 0

+16π(m − ε)ε(1 + ετ )3 −16πmε(1 + ετ )2

χh2 ϕh2 − 16πε(1 + ετ )2 2 r r

χh2 r

and, finally, ∂ (L3 (ε, ϕ, χ, τ )) h3 = ∂τ

∂ d (A (ε, ϕ, χ, τ )) h3 τ ∂τ dr d +A (ε, ϕ, χ, τ ) h3 dr ⎞ ⎛ r

2 16π(m − ε)ε ⎝ + (1 + ετ )h3 ϕ + εχ2 ds⎠ (1 + ετ ) r2 0 ⎛ r ⎞

8π(m − ε)ε ⎝ + (1 + ετ )2 ϕ2 + εχ2 ds⎠h3 r2 0 2 2 2 ϕ + εχ h3 + 24π(m − ε)ε(1 + ετ ) r ϕχ − 32πε2 (1 + ετ ) 2 h3 r ϕ2 − εχ2 h3 . − 16πmε(1 + ετ ) r

First of all, we remark that if ϕ, h1 ∈ Bϕ , χ, h2 ∈ Bχ and τ, h3 ∈ Bτ , then ∂ ∂ ∂ ∂ϕ (A (ε, ϕ, χ, τ )) h1 , ∂χ (A (ε, ϕ, χ, τ )) h2 and ∂τ (A (ε, ϕ, χ, τ )) h3 are bounded. So, we have that ⎛ r ⎞

d C2 ∂L3 ⎝ |ϕh1 | ds⎠ + C3 |ϕh1 | + C4 |h1 χ| ∂ϕ h1 ≤ C1 dr τ + r2 r r2 0 ⎛ r ⎞

∂L3 d C6 ⎝ |χh2 | ds⎠ + C7 |χh2 | + C8 |ϕh2 | ∂χ h2 ≤ C5 dr τ + r2 r r2 0

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2

r ϕ + εχ2 ∂L3 d d C11 2 2 ϕ + εχ ds + C12 ∂τ h3 ≤ C9 dr τ + C10 dr h3 + r2 r 0 2 ϕ − εχ2 |ϕχ| + C13 2 + C14 r r with Ci positive constants. With exactly the same arguments used above, we conclude that

∞ ∂ ∂ϕ (L3 (ε, ϕ, χ, τ )) h1 dr < +∞ 0

∞ ∂ (L (ε, ϕ, χ, τ )) h 2 dr < +∞ ∂χ 3 0

∞ ∂ (L (ε, ϕ, χ, τ )) h 3 dr < +∞ ∂τ 3 0

if (ε, ϕ, χ, τ ) ∈ (−ε1 , ε2 ) × Bϕ × Bχ × Bτ , and (h1 , h2 , h3 ) ∈ Bϕ × Bχ × Bτ . ∂L3 3 ∂L3 Furthermore ∂L ∂ϕ , ∂χ and ∂τ are continuous; thus the proof for L3 . We consider now L1 ; first, we have to prove that it is well defined in Yϕ . We observe that χ 1 d ϕ ϕ + 2 + C2 |L1 (ε, ϕ, χ, τ )| ≤ C1 r dr r r with C1 , C2 positive constants then L1 (ε, ϕ, χ, τ ) ∈ L2 R3 , thanks to conditions (2.19). Now, we have to prove that L1 (ε, ϕ, χ, τ ) is C 1 ; by classical arguments, it is enough to show that for (h1 , h2 , h3 ) ∈ Bϕ × Bχ × Bτ ∂ (L1 (ε, ϕ, χ, τ )) h1 ∈ Yϕ , ∂ϕ ∂ (L1 (ε, ϕ, χ, τ )) h2 ∈ Yϕ , ∂χ ∂ (L1 (ε, ϕ, χ, τ )) h3 ∈ Yϕ . ∂τ By a straightforward computation, we find out 1 d ∂L1 1 d 1 −1/2 ∂A h1 h1 = A h1 ϕ + A1/2 h1 − 2 , ∂ϕ 2 ∂ϕ r dr r dr r 1 d ∂L1 h2 h2 1 −1/2 ∂A h2 = A h2 ϕ + (2m − ε) + ε(m − ε)τ , ∂χ 2 ∂χ r dr r r ∂L1 1 d χ 1 −1/2 ∂A h3 = A h3 ϕ + ε(m − ε)h3 ; ∂τ 2 ∂τ r dr r

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and, using the positivity of A, ∂L1 1 d 1 d h1 ∂ϕ h1 ≤ C1 r dr ϕ + C2 r dr h1 + r2 ∂L1 1 d h2 ∂χ h2 ≤ C3 r dr ϕ + C4 r ∂L1 1 d + C6 χ ≤ C h ϕ 3 5 ∂τ r dr r with Ci positive constants. Then, we can conclude that 2

∂ ∂ϕ (L1 (ε, ϕ, χ, τ )) h1 dx < +∞ R3

2

∂ ∂χ (L1 (ε, ϕ, χ, τ )) h2 dx < +∞

R3

2

∂ ∂τ (L1 (ε, ϕ, χ, τ )) h3 dx < +∞

R3

if (ε, ϕ, χ, τ ) ∈ (−ε1 , ε2 ) × Bϕ × Bχ × Bτ , and (h1 , h2 , h3 ) ∈ Bϕ × Bχ × Bτ . ∂L1 1 ∂L1 Furthermore ∂L ∂ϕ , ∂χ and ∂τ are continuous; thus the proof for L1 and with the same arguments for L2 . 2.2. Branches Generated by the Solution of Choquard’s Equation In this subsection, we show that a solution φ0 = (ϕ0 , χ0 , τ0 ) of (2.7) can generate a local branch of solutions of (2.6). First, we linearize the operator D on (ϕ, χ, τ ) around (0, φ0 ) ⎞ ⎛ 1 d h k r dr h − r 2 + 2m r ⎟ ⎜ d Dϕ,χ,τ (0, φ0 )(h, k, l) = ⎝ 1r dr k + rk2 + hr − m hr τ0 − m ϕr0 l ⎠ . d 16πm r ϕ0 h ds dr l + r 2 0 Now, if we prove that Dϕ,χ,τ (0, φ0 ) is an isomorphism, the implicit function theorem can be applied and we can find solutions of (2.6) near the ground state φ0 . Lemma 2.3. We define the operator V : Xϕ × Xχ → Yϕ × Yχ , by 1 d 1 1 r dr ϕ − r 2 ϕ + 2m r χ V (ϕ, χ) = , 1 d 1 1 r dr χ + r 2 χ + r ϕ then V is an isomorphism of Xϕ × Xχ onto Yϕ × Yχ . This lemma is obvious if we remind that L2 (R3 , C4 ) can be written as the direct sum of partial wave subspaces and that the Dirac operator leaves invariant all these subspaces (see [9]). So, thanks to Lemma 2.3 of [3], we know that

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V : H 1 (R3 , C2 ) × H 1 (R3 , C2 ) → L2 (R3 , C2 ) × L2 (R3 , C2 ) defined by i¯ σ ∇u + 2mv V (u, v) = −i¯ σ ∇v + u is an isomorphism of H 1 (R3 , C2 ) × H 1 (R3 , C2 ) onto L2 (R3 , C2 ) × L2 (R3 , C2 ) and then V is an isomorphism of each partial wave subspace. In particular, V coincide with V on the partial wave subspace Xϕ × Xχ . Lemma 2.4. We define the operator W : Xϕ × Xχ × Xτ → Yϕ × Yχ × Yτ , by ⎛ 1 d ⎞ h k r dr h − r 2 + 2m r ⎜ d ⎟ W (h, k, l) = ⎝ 1r dr k + rk2 + hr − m ϕr0 l ⎠ , d dr l

then W is an isomorphism of Xϕ × Xχ × Xτ onto Yϕ × Yχ × Yτ . Proof. First we prove that W is one to one. We observe that W (h, k, l) = 0 if and only if (h, k, l) satisfies ⎧1 d h k ⎪ ⎨ r dr h − r2 + 2m r = 0 ϕ0 1 d k h r dr k + r 2 + r − m r l = 0 ⎪ ⎩ d dr l = 0 in Yϕ × Yχ × Yτ . In particular, we must have l ≡ 0 and (h, k) solution of 1 d h k r dr h − r 2 + 2m r = 0 1 d r dr k

+

k r2

+

h r

=0

that is equivalent to V (h, k) = 0. So, thanks to Lemma 2.3, h ≡ k ≡ 0 and W is one to one in Yϕ × Yχ × Yτ . Secondly, we have to prove that for f = (f1 , f2 , f3 ) ∈ Yϕ × Yχ × Yτ , there exists (h, k, l) ∈ Xϕ × Xχ × Xτ such that W (h, k, l) = f . This means that the system ⎧1 d h k ⎪ ⎨ r dr h − r2 + 2m r = f1 ϕ0 1 d k h r dr k + r 2 + r − m r l = f2 ⎪ ⎩ d dr l = f3 has a solution in Xϕ × Xχ × Xτ for all (f1 , f2 , f3) ∈ Yϕ × Yχ × Yτ . We observe ∞ d ∗ that ∀f3 ∈ L1 ((0, ∞), dr) there exists l∗ (r) = − r f3 ds such that dr l = f3 ; ∗ furthermore l ∈ Xτ . So, we have to show that 1 d h k r dr h − r 2 + 2m r = f1 (2.21) ϕ0 ∗ 1 d k h r dr k + r 2 + r = f2 + m r l has a solution in Xϕ × Xχ for all (f1 , f2 ) ∈ Yϕ × Yχ .

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Now, we remark that ϕr0 l∗ ∈ L2 (R3 ) and then, thanks to Lemma 2.3, (2.21) has a solution in Xϕ × Xχ for all (f1 , f2 ) ∈ Yϕ × Yχ . In conclusion W is an isomorphism of Xϕ × Xχ × Xτ onto Yϕ × Yχ × Yτ . Finally, we observe that Dϕ,χ,τ (0, φ0 )(h, k, l) can be written as Dϕ,χ,τ (0, φ0 )(h, k, l) = W (h, k, l) + S(h) with

⎛ ⎜ S(h) = ⎝

0

(2.22)

⎞

⎟ −m hr τ0 ⎠. 16πm r ϕ0 h ds r2 0

(2.23)

Theorem 2.5. Let φ0 be the ground state solution of (2.7), then there exists δ > 0 and a function η ∈ C((0, δ), Xϕ × Xχ × Xτ ) such that η(0) = φ0 and D(ε, η(ε)) = 0 for 0 ≤ ε < δ. Proof. Since D(0, φ0 ) = 0 and D is continuously differentiable in a neighborhood of (0, φ0 ), to apply the implicit function theorem we have to prove that Dϕ,χ,τ (0, φ0 ) is an isomorphism of Xϕ × Xχ × Xτ onto Yϕ × Yχ × Yτ . We observe that Dϕ,χ,τ (0, φ0 )(h, k, l) = 0 if and only if (h, k, l) satisfies ⎧ d h ⎪ ⎨ dr h − r + 2mk = 0 d k (2.24) dr k + r + h − mhτ0 − mϕ0 l = 0 ⎪ ⎩ d 16πm r ϕ0 h ds = 0 dr l + r 2 0 that means ⎧ ∞ d2 3 ⎪ − h+2mh−16πm ⎪ 0 ⎨ dr2 d 1 dr h − r h + 2mk = 0 ⎪ ⎪ ⎩ l = 16πm ∞ ϕ0 h ds 0 max(r,s)

ϕ20 max(r,s)

∞ ds h−32πm3 0

ϕ0 h max(r,s)

ds ϕ0 = 0 (2.25)

Now, if we write ξ(x) = h(|x|) and we remind that ϕ0 (|x|) = |x|u0 (x) with u0 |x| solution of (2.8), we have that (h, k, l) is a solution of (2.25) if ⎛ ⎞ 1 h(r) ⎜ 0 ⎟ ξ(x) ⎟ = r−1 ⎜ ⎝ 1 ⎠ ζ(x) r ik(r)σ 0 satisfies −ξ + 2mξ − 4m3 R3 ζ=

−i¯ σ ∇ξ 2m

|u0 (y)|2 |x−y|

dy ξ − 8m3 R3

and

l(x) = 4m R3

ξ(y)u0 (y) dy. |x − y|

ξ(y)u0 (y) |x−y|

dy u0 = 0

(2.26)

(2.27)

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It is well known that the unique solution of the first equation of (2.26) in Hr1 (R3 ) is ξ ≡ 0 (see [7] for more details) and that implies ζ ≡ l ≡ 0. So the unique solution of (2.24) is h ≡ k ≡ l ≡ 0 and Dϕ,χ,τ (0, φ0 ) is one to one in Xϕ × Xχ × Xτ . Next, if we show that S(h) is a compact operator, we have that Dϕ,χ,τ (0, φ0 ) is a one to one operator that can be written as a sum of an isomorphism and a compact operator and then it is an isomorphism. r First, we can easily see that T (h) = r12 ( 0 ϕ0 h ds) is a compact operator from Xϕ on Yτ ; in particular, we use the fact that Hr1 (R3 ) is compactly embedded in Lq (R3 ), for 2 < q < 6, to prove that for any bounded sequence {hn } ⊂ Xϕ , the sequence {T (hn )} ⊂ Yτ contains a Cauchy subsequence. Second, we have to show that the operator hr τ0 from Xϕ to L2 (R3 ) is compact. hn If { r } is a bounded sequence in H 1 (R3 ) then { hrn τ0 } is precompact on L2loc (R3 ), thanks to compact Sobolev embedding and, since τ0 (r) → 0 when r → +∞, we can conclude that { hrn τ0 } is precompact on L2 (R3 ). So S(h) is a compact operator from Xϕ on Yϕ × Yχ × Yτ and Dϕ,χ,τ (0, φ0 ) is an isomorphism of Xϕ × Xχ × Xτ onto Yϕ × Yχ × Yτ . In conclusion, we can apply the implicit function theorem to find that there exists δ > 0 and a function η ∈ C((0, δ), Xϕ × Xχ × Xτ ) such that η(0) = φ0 and D(ε, η(ε)) = 0 for 0 ≤ ε < δ.

Acknowledgement The author would like to thank professor Eric Séré for helpful discussions and useful comments.

References [1] Finster, F., Smoller, J., Yau, S.T.: Particlelike solutions of the Einstein–Dirac equations. Phys. Rev. D. Particles Fields, Third Series 59 (1999) [2] Bird, E.J.: A proof of existence of particle-like solutions of Einstein Dirac Equations. Ph.D. thesis, University of Michigan (2005) [3] Ounaies, H.: Perturbation method for a class of non linear Dirac equations. Differ Integral Equ. 13(4–6), 707–720 (2000) [4] Guan, M.: Solitary wave solutions for the nonlinear Dirac equations. Preprint (2008) [5] Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977) [6] Lions, P.L.: The Choquard equation and related questions. Nonlinear Anal. Theory Methods Appl. 4(6), 1063–1073 (1980) [7] Lenzmann, E.: Uniqueness of ground states for pseudo-relativistic Hartree equations. Preprint (2008) [8] Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, pp. 337– 338. Springer, Berlin (1993) [9] Thaller, B.: The Dirac Equation. Springer, Berlin (1992)

Vol. 10 (2010)


Simona Rota Nodari Ceremade (UMR CNRS 7534) Université Paris-Dauphine Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France e-mail: [email protected] Communicated by Piotr T. Chrusciel. Received: July 8, 2009. Accepted: September 29, 2009.

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On Heisenberg-like Super Group Structures Ramon Peniche and O. A. Sánchez-Valenzuela Abstract. The ring-like structures that can be defined on the ground supermanifolds R1|1 and C1|1 are classified up to equivalence in the category of smooth and complex Berezin-Kostant-Leites-Manin supermanifolds. It is proved that there are three different such equivalence classes in the real case, whereas there are two for the complex field. The corresponding module structures—defined componentwise on the product of k copies of R1|1 or C1|1 —are also classified up to equivalence. The notions of linearity and bilinearity are reviewed and used to define Heisenberg-like super group structures. It turns out that there are three non-isomorphic real such super groups, whereas only two over the complex field. The use of the appropriate exponential maps introduces the possibility of defining Heisenberg-like super group structures on the product of k copies of the ground supermanifold, with an appropriate super circle. The corresponding classification is also obtained.

1. Introduction Let F be either the real or the complex numbers field R or C, respectively. Let F1|1 = (F, F) be the (1, 1)-dimensional real or complex supermanifold defined by the sheaf F = CF∞ ⊗ ∧F∗ , where F∗ stands for the dual vector space to F. We shall follow the standard references for the basics of what is usually referred to as the category of Berezin-Kostant-Leites-Manin supermanifolds (e.g., [2,3]). We study Heisenberg-like super group structures Λ defined on (V, V) × F1|1 , where (V, V) = F1|1 ×· · ·×F1|1 (k factors) is further endowed with vector space-like maps of sum (V, V) × (V, V) → (V, V), and scalar multiplication F1|1 × (V, V) → (V, V), and we are given a bilinear-like morphism B : (V, V) × (V, V) → F1|1 , so that Λ ((u1 + ξ1 , z1 + ζ1 ), (u2 + ξ2 , z2 + ζ2 )) = ((u1 + u2 ) + (ξ1 + ξ2 ), (z1 + z2 + B(u1 + ξ1 , u2 + ξ2 )) + (ζ1 + ζ2 )) , where the ui ’s denote the even vector coordinates on V , the ξi ’s are their odd counterparts, the zi ’s are the even point coordinates on the ground field F, and ζi

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are their odd counterparts. They can all be rigorously defined as even (resp., odd) sections of the corresponding sheaves. One also has to define the nature of the “+” operations involved, the actual meaning of a bilinear B : (V, V) × (V, V) → F1|1 in the category, and finally check the associative property of Λ. Actually, in this paper we carefully define and construct all what is needed, and check or prove that the algebraic properties and relations do work out fine in order to make sense of the composition law just displayed. But assuming at this stage that everything is indeed well defined, we may further ask ourselves for the Lie superalgebra g = g0 ⊕ g1 of left-invariant super vector fields on (V, V) × F1|1 . Now, B defines an ˜ : V × V → F. It turns out that g is (k + 1, k + 1)-dimenordinary bilinear map B sional and can be obtained from V and a 1-dimensional even central element Z0 , ˜ namely, let g0 = V ⊕ Z0 , and using the super skew-symmetric envelope of B; think of g1 as the vector space ΠV ⊕ ΠZ0 , where Π is thought of as the identity map, but having the effect of changing the parity of the even elements, to produce their odd counterparts. Then, for any pair of elements u and v from V we have ˜ v) − B(v, ˜ u) Z0 , [u, v] = B(u, ˜ v) − B(v, ˜ u) ΠZ0 , [u, Πv] = B(u, ˜ v) + B(v, ˜ u) Z0 , [Πu, Πv] = −ε B(u, where, ε ∈ F is a scalar parameter whose role is to be determined, and the commutation relations are completed by demanding, [Z0 , · ] = 0, and [ΠZ0 , · ] = 0. This is clearly a particular generalization of Heisenberg’s Lie algebra. First of all, it is a Lie superalgebra, and therefore [Πu, Πv] is really an anticommutator. There are some others generalizations in the literature (e.g., [1,6,7] or [9]). The main difference is that our construction introduces, not a 1-dimensional, but a (1, 1)˜ does not need to have a specific dimensional center; besides, the bilinear form B type of symmetry or skew-symmetry, nor need it be non-degenerate. It is clear, ˜ is skew-symmetric and nondegenerate, the Lie superalgehowever, that when B bra just described restricts its even subalgebra to the well known Heisenberg Lie algebra. Put this way, all what is needed is to define a vector space-like structure on (V, V) = (F1|1 )k = Fk|k and in order to do that with componentwise operations, all what is needed is to give a ring-like structure on F1|1 . Once this is done, an appropriate notion of bilinearity can be defined for a given B. These preliminaries, however, are not entirely trivial to achieve in the category of supermanifolds, for there is no unique way to define ring-like structures on F1|1 , and ‘taking out scalars’ from the second argument of a bilinear B is a rather technical condition. Thus we shall proceed according to the following program: 1.

Classification of ring-like structures on F1|1 . We shall prove in Sect. 2 below that up to isomorphism there are three (resp., two) different ring-like structures on F1|1 , when F is the real (resp., complex) numbers field, defined by the

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On Heisenberg-like Super Group Structures

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sum and multiplication maps s : F1|1 ×F1|1 → F1|1 , and m : F1|1 ×F1|1 → F1|1 , respectively, given by s∗ t = t1 + t2 , ∗

s ζ = ζ 1 + ζ 2,

m∗ t = t1 t2 + εζ 1 ζ 2 , m∗ ζ = t2 ζ 1 + t1 ζ 2 .

Here {t, ζ} is a local coordinates system for F1|1 whereas {t1 , t2 , ζ 1 , ζ 2 } is the natural set of local coordinates for F1|1 × F1|1 arising from the pull-back of the projections πi : F1|1 × F1|1 → F1|1 (i = 1, 2); that is, ti = πi∗ t and ζ i = πi∗ ζ. The three (resp., two) different isomorphism classes correspond to the values ε = −1, 0, 1 (resp., ε = 0, 1). 2. Left F1|1 -module structures on Fk|k ; F1|1 -linearity and bilinearity. In Sect. 3 we use the different ring-like structures on F1|1 to define three (resp., two) different left F1|1 -module structures on Fk|k in a componentwise way. It turns out that under the appropriate notion of isomorphism, the obtained module structures are non-isomorphic. We can conveniently refer to them as defined through the sum and scalar multiplication maps σ : Fk|k × Fk|k → Fk|k , and µ : F1|1 × Fk|k → Fk|k , and the three (resp., two) different isomorphism classes correspond to the values ε = −1, 0, 1 (resp., ε = 0, 1) inherited form the ring-like structure on F1|1 . Thus, σ ∗ ti = t1i + t2i ,

µ∗ ti = t0 ti + εζ0 ζi

σ ∗ ζi = ζi1 + ζi2 , µ∗ ζi = ti ζ0 + t0 ζi

3.

where {ti , ζi | i = 1, 2, . . . , k} is a local coordinate system in Fk|k , and {ti , ζi | i = 1, 2, . . . , k and = 1, 2} and {tj , ζj | j = 0, 1, . . . , k} are the the natural local coordinate systems for Fk|k × Fk|k and F1|1 × Fk|k , respectively, arising from the pull-back of the projections. We also generalize the work in [8] in order to obtain the appropriate notions of linearity and bilinearity with respect to the three (resp., two) different vector space structures that can be defined on Fk|k when F = R (resp., = C). In [8] only the case ε = 1 was studied. It turns out, however, that the results obtained there are still valid for the other parameter values of ε; namely, that linear and bilinear maps in the category are completely characterized by their underlying linear and bilinear maps, regardless of the super vector space structure defined; the only restriction is to use the same type of structure in the domain and the codomain. Heisenberg-like super group structures and their corresponding left-invariant super vector fields. We look in Sect. 4 for an associative map Λ : H × H → H, having the identity and inverse elements properties in order to define a super group on H = Fk|k × F1|1 . This is achieved as follows: Both factors, have sum maps defined on them; namely, σ : Fk|k ×Fk|k → Fk|k and s : F1|1 ×F1|1 → F1|1 , respectively. Furthermore, an F1|1 -bilinear B : Fk|k ×Fk|k → F1|1 can be added to the result of the sum map s in the second factor, in such a way that the

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˜ : (Fk × F) × (Fk × F) → Fk × F is underlying Lie group composition map Λ given by, ˜ 1 , h1 ), (v2 , h2 )) = (v1 + v2 , h1 + h2 + B ˜ (v1 , v2 )), Λ((v ˜ : Fk × Fk → F is the underlying bilinear map with which we produce where B a super bilinear B on Fk|k . It is shown that up to isomorphism there are three (resp., two) different possibilities to achieve this over F = R (resp., C) which are inherited precisely from the corresponding F1|1 ring-like structures and left F1|1 -module structures used on H. Once the Lie super group structure is set, we determine its Lie superalgebra of left-invariant super vector fields g = g0 ⊕ g1 . It turns out that it is (k + 1, k + 1)-dimensional and spanned over F by the even (resp., odd) elements {X1 , . . . , Xk , Z0 } (resp., {Y1 , . . . , Yk , Z1 }) satisfying the following super commutation relations: [Zα , Xi ], [Zα , Yi ], and [Zα , Zβ ] are identically zero, and, [Xi , Xj ] = (Bij − Bji )Z0 [Xi , Yj ] = (Bij − Bji )Z1 [Yi , Yj ] = −ε(Bij + Bji )Z0

4.

˜ In The coefficients Bij ∈ F are the entries of the underlying bilinear map B. particular, this result is already interesting in its own. We could have started ˜ with a real or complex k-dimensional vector space V , and a bilinear map B defined on it, and then consider the (k + 1, k + 1)-dimensional super vector space (V ⊕ ΠV ) ⊕ (Z0 ⊕ ΠZ0 ), where we think of Π as the identity map, but changing the parity of the Z2 -homogeneous elements on which it acts. We can then turn this super vector space into a Lie superalgebra by defining the Lie brackets on generators as above with the understanding that Yi = ΠXi , and Z1 = ΠZ0 . It is easy to see that Jacobi identity holds true. Furthermore, this Lie superalgebra integrates to a unique simply connected Heisenberg-like super group (e.g., as in [4,5]). Heisenberg-like super group structures on G := Fk|k × S 1|dF . (Here, dF = 1 or dF = 2, depending on whether F is either R or C, respectively). It follows from the results in [4] that for each additive structure on F1|1 one obtains an ˙ C ∞ | ˙ × ∧F∗ ), associated multiplicative super group structure on F˙ 1|1 = (F, F F where F˙ = F − {0}. This super group structure projects onto the multiplicative group structure on F˙ defined by the ordinary product on the ground field F. When F = C, it can be shown that each of the three super groups restrict over the unit circle S 1 = {z ∈ C | |z| = 1 } to a real (1, 2)-dimensional super group. The corresponding supermanifold sheaf is CS∞1 ⊗ ∧C∗ . On the other hand, when F = R, the multiplicative super group structure on R˙ 1|1 can be pushed forward to a super group structure on a (1, 1)-dimensional supermanifold over the unit circle S 1 whose supermanifold sheaf is CS∞1 ⊗ ∧R∗ . In either case, the corresponding super groups can be suggestively denoted by S 1|dF and can be naturally called super circles. It also follows from the results

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in [4] that in either case, there are well defined super group epimorphisms E : F1|1 → S 1|dF mapping each super sum into its corresponding super multiplication. Fixing a super group structure on S 1|dF , an F1|1 -module structure on Fk|k , and a bilinear B : Fk|k × Fk|k → F1|1 , we may look for a super group structure Ω on G = Fk|k × S 1|dF , using the sum map Fk|k × Fk|k → Fk|k on the first factor, and then the product map on S 1|dF together with an F1|1 -bilinear B : Fk|k × Fk|k → F1|1 pushed into S 1|dF with the group epimorphism E on ˜ on Fk × S 1 is given by the second, in such a way that its underlying Ω ˜ 1 , h1 ), (v2 , h2 )) = (v1 + v2 , h1 h2 E( B ˜ (v1 , v2 ))), Ω((v It is shown in Sect. 5 below that only for one of the super group structures on S 1|dF it is possible to define a super group structure Ω on G with the required conditions. Furthermore, in this case there are three (resp., two) different such Ω’s when F = R (resp., C) essentially arising from the conditions on B and the F1|1 -module structures on Fk|k . Finally, there is a natural correspondence between the Heisenberg-like structures Λ on H, and the super group structures Ω on G. Each Heisenberg Lie algebra encodes important phase space information of a given dynamical system through its symplectic form. Quantization involves looking at their representations, and the celebrated Stone-von Neumann theorem states that their unitary representations are essentially unique. Our approach incorporates fermion-boson ‘interactions’ through the supersymmetry transformations to the canonical commutation relations that are explicitly exhibited precisely through the super group composition law Λ; their infinitesimal versions build up the Lie superalgebra just described in this introduction. Moreover, our approach also allows the possibility of dealing with constraints as the skew-symmetric part of the ˜ may degenerate. The question of determining their unitary repbilinear map B, resentations rigorously within the ‘super’ category, where care must be taken on how to realize the super unitary group of the corresponding representation Hilbert superspace, and the role to be played by the odd component of the center, are at this point, and as far as the authors’ know, questions that remain to be settled.

2. Classification of Ring-like Structures on F1|1 Our first goal is to define a ring structure on F1|1 . Thus, we need two associative operations defined through morphisms of the form F : F1|1 × F1|1 → F1|1 , that satisfy a distributive property between them. Note first that the associativity condition (A) on F can be stated as follows: F ◦ (F ◦ (π1 × π2 ) × π3 ) = F ◦ (π1 × F ◦ (π2 × π3 ))

(A)

where πi : F1|1 × F1|1 × F1|1 → F1|1 is the natural projection onto the ith factor (i = 1, 2, 3). Let {z, ζ} be a set of local coordinates on F1|1 over an open subset

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U ⊂ F. We then write F ∗ z = F˜ + F12 ζ1 ζ2 F ∗ ζ = F1 ζ1 + F2 ζ2 where F˜ , F12 , F1 , F2 ∈ CF∞2 (F −1 (U )). We are particularly interested in the cases where F˜ : F × F → F is either the ordinary sum or the ordinary product on the field. Proposition 2.1. (a) The associative morphisms s : F1|1 × F1|1 → F1|1 whose underlying morphism s˜ : F × F → F is s˜(z1 , z2 ) = z1 + z2 are determined by a pair of constants a and b in F satisfying ab = 0, and a non vanishing smooth function f : F → F such that f (0) = 1 in such a way that b ζ1 ζ2 f (z1 )f (z2 ) f (z1 + z2 ) f (z1 + z2 ) ζ1 + ζ2 . f (z1 ) f (z2 )

s∗ z = z1 + z2 + s∗ ζ = eaz2

(b) The associative morphisms m : F1|1 ×F1|1 → F1|1 whose underlying morphism m ˜ : F × F → F is m(z ˜ 1 , z2 ) = z1 z2 are determined by a pair of constants c and ε in F satisfying cε = 0, and a non vanishing smooth function g : F → F such that g(1) = 1 in such a way that εz1 z2 ζ1 ζ2 m∗ z = z 1 z 2 + g(z1 )g(z2 ) g(z1 z2 ) g(z1 z2 ) ζ1 + ζ2 . m∗ ζ = z2c g(z1 ) g(z2 ) Proof. Through a straightforward computation it is easy to see that (A) is equivalent to the following set of conditions: F˜ F˜ (z1 , z2 ), z3 = F˜ z1 , F˜ (z2 , z3 ) (2.1) ∂ F˜ ˜ F (z1 , z2 ), z3 F12 (z1 , z2 ) = F12 z1 , F˜ (z2 , z3 ) F1 (z2 , z3 ) ∂x F12 F˜ (z1 , z2 ), z3 F1 (z1 , z2 ) = F12 z1 , F˜ (z2 , z3 ) F2 (z2 , z3 ) ∂ F˜ z1 , F˜ (z2 , z3 ) F12 (z2 , z3 ) F12 F˜ (z1 , z2 ), z3 F2 (z1 , z2 ) = ∂y ˜ F1 F (z1 , z2 ), z3 F1 (z1 , z2 ) = F1 z1 , F˜ (z2 , z3 ) F1 F˜ (z1 , z2 ), z3 F2 (z1 , z2 ) = F2 z1 , F˜ (z2 , z3 ) F1 (z2 , z3 ) F2 F˜ (z1 , z2 ), z3 = F2 z1 , F˜ (z2 , z3 ) F2 (z2 , z3 ) ∂F2 ˜ ∂F1 F (z1 , z2 ), z3 F12 (z1 , z2 ) = z1 , F˜ (z2 , z3 ) F12 (z2 , z3 ) ∂x ∂y

(2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8)

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If F˜ is either the sum or the product then (2.1) is trivially satisfied. Actually, when F˜ is the sum, a simple analysis of (2.5)–(2.7) shows that the Fi ’s are totally determined by two functions fi : F → F through F1 (x, y) =

f1 (x + y) f1 (x)

and

F2 (x, y) =

f2 (x + y) , f2 (y)

where f1 (x) = F1 (0, x) and f2 (x) = F2 (x, 0). Notice that (2.5) and (2.7) imply that F1 (0, 0) = F2 (0, 0) = 1, so that f1 (0) = f2 (0) = 1. From (2.6), f1 (x + y + z) f2 (x + y) f2 (x + y + z) f1 (y + z) = . f1 (x + y) f2 (y) f2 (y + z) f1 (y) Letting h(t) =

f1 (t) f2 (t) ,

the above condition implies that h(x + y + z)h(y) = h(x + y)h(y + z).

In particular h(x + y)h(0) = h(x)h(y), with h(0) = 1. Therefore h(t) = eat , where a = h (0), and f1 (t) = eat f2 (t). Thus, F1 (x, y) = eay F2 (y, x)

and

F2 (x, y) =

f2 (x + y) . f2 (y)

In what follows we shall write f instead of f2 . Now, the conditions (2.2), (2.3) and (2.4) can be written as F12 (x, y) = F12 (x, y + z) eaz F2 (z, y) F12 (x, y + z)F2 (y, z) = F12 (x + y, z) eay F2 (y, x) F12 (y, z) = F12 (x + y, z)F2 (x, y)

(2.9) (2.10) (2.11)

whereas (2.8) is transformed into ∂F2 (x + y, z)F12 (x, y) ∂x =e

a(y+z)

∂F2 (y + z, x) F12 (y, z) aF2 (y + z, x) + ∂x

(2.12)

Choosing z = −y in (2.9) we obtain F12 (x, y) = e−ay F12 (x, 0)F2 (−y, y) = e−ay F12 (x, 0)

1 , f (y)

and choosing x = −y and z = 0 in (2.11), we find, F12 (y, 0) = F12 (0, 0)F2 (−y, y) =

b , f (y)

where b = F12 (0, 0). Therefore, F12 (x, y) = b

e−ay , f (x)f (y)

but now, (2.10) implies b e2ay −1 = 0, so that ab = 0. In any case, and due to the fact that

∂F2 ∂x (x, y)

=

f (x+y) f (y) ,

(2.12) is trivially satisfied.

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Now, when F˜ (x, y) = xy we see from (2.5), (2.6) and (2.7) that F1 (1, 1) = F2 (1, 1) = 1, and as before, F2 is determined by a function g such that g(1) = 1. Similarly F1 is determined by a scalar c and the function g as follows: F1 (x, y) = y c F2 (y, x)

and F2 (x, y) =

g(xy) . g(y)

On the other hand, (2.2), (2.3), (2.4) and (2.8) imply z F12 (x, y) = F12 (x, yz)F1 (y, z) F12 (xy, z)F1 (x, y) = F12 (x, yz)F2 (y, z) x F12 (y, z) = F12 (xy, z)F2 (x, y) ∂F2 ∂F2 c−1 y (xy, z) F12 (x, y) = (yz) (yz, x) F12 (y, z). c F2 (yz, x) + yz ∂x ∂x The first equation implies F12 (x, y) = yF12 (x, 1)F1 (y, y −1 ) =

y 1−c F12 (x, 1) g(y)

and the third equation implies F12 (y, 1) = yF12 (1, 1)F2 (y −1 , y) = ε

y , g(y)

where ε = F12 (1, 1), so that F12 (x, y) = ε

x y 1−c . g(x)g(y)

The second equation imposes further restrictions: ε [y c − 1] = 0, and in any case, the last equation is trivially satisfied. We now explore the conditions imposed on a given sum s and a product m by the distributive properties (D.1) and (D.2) that follow: m ◦ (π1 × (s ◦ (π2 × π3 ))) = s ◦ [(m ◦ (π1 × π2 )) × (m ◦ (π1 × π3 ))]

(D.1)

m ◦ ((s ◦ (π1 × π2 )) × π3 ) = s ◦ [(m ◦ (π1 × π3 )) × (m ◦ (π2 × π3 ))]

(D.2)

where, again, πi : F1|1 × F1|1 × F1|1 → F1|1 denotes the projection to the ith factor (i = 1, 2, 3). Proposition 2.2. Let be given a pair of morphisms s and m as in Proposition 2.1 (a) and (b), respectively. Then s and m satisfy (D.1) and (D.2) if and only if (x) a = b = c = 0 and g(x) = xf f (1) . In order to prove the statement it is convenient to realize that for any function ϕ ∈ C ∞ (F2 ) and any pair of morphisms s and m as in Proposition 2.1, we have

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∗ (π1 × (s ◦ (π2 × π3 ))) ϕ = ϕ ◦ (π1 × (s ◦ (π2 × π3 )))

+

b ∂ϕ ◦ (π2 × π3 )))ζ2 ζ3 (π1 × (s (f ◦ π2 )(f ◦ π3 ) ∂y

∗ ((s ◦ (π1 × π2 )) × π3 ) ϕ = ϕ ◦ ((s ◦ (π1 × π2 )) × π3 )

+

b ∂ϕ × π2 )) × π3 )ζ1 ζ2 ((s ◦ (π1 (f ◦ π1 )(f ◦ π2 ) ∂x

[(m ◦ (π1 × π2 )) × (m ◦ (π1 × π3 ))] ϕ = ϕ ◦ [(m ◦ (π1 × π2 )) × (m ◦ (π1 × π3 ))] ∗

+

ε · π1 · π2 ∂ϕ × (m ◦ (π1 × π3 ))]ζ1 ζ2 [(m ◦ (π1 × π2 )) (g ◦ π1 )(g ◦ π2 ) ∂x

+

∂ϕ ε · π1 · π3 × (m ◦ (π1 × π3 ))]ζ1 ζ3 [(m ◦ (π1 × π2 )) (g ◦ π1 )(g ◦ π3 ) ∂y

∗ [(m ◦ (π1 × π3 )) × (m ◦ (π2 × π3 ))] ϕ = ϕ ◦ [(m ◦ (π1 × π3 )) × (m ◦ (π2 × π3 ))]

+

ε · π1 · π3 ∂ϕ × (m ◦ (π2 × π3 ))]ζ1 ζ3 [(m ◦ (π1 × π3 )) (g ◦ π1 )(g ◦ π3 ) ∂x

+

∂ϕ ε · π2 · π3 × (m ◦ (π2 × π3 ))]ζ2 ζ3 [(m ◦ (π1 × π3 )) (g ◦ π2 )(g ◦ π3 ) ∂y

These can be checked by using Taylor’s theorem. Proof. Note that the pull-back of (D.1) applied to z implies ε·

z1 (z2 + z3 ) eaz3 f (z2 + z3 ) z1 z2 z3c g(z1 z2 )g(z1 z3 ) = ε· −b· g(z1 )g(z2 + z3 )f (z2 ) g(z1 )g(z2 ) f (z1 z2 )f (z1 z3 )g(z1 )g(z2 ) ε·

z1 z 3 z2c g(z1 z2 )g(z1 z3 ) z1 (z2 + z3 )f (z2 + z3 ) = ε· +b· g(z1 )g(z2 + z3 )f (z3 ) g(z1 )g(z3 ) f (z1 z2 )f (z1 z3 )g(z1 )g(z3 ) b·

g(z1 z2 )g(z1 z3 ) z1 = b· f (z2 )f (z3 ) f (z1 z2 )f (z1 z3 )g(z2 )g(z3 )

Assuming ε = 0 the first equation says that b = 0. On the other hand, if ε = 0 and b = 0 then a = 0. Moreover, after the exchange of z2 and z3 in the second equation, and comparing the result with the first, we obtain a contradiction. Therefore (D.1) applied to z implies b = 0 and ε·

z1 z2 z1 (z2 + z3 ) eaz3 f (z2 + z3 ) = ε· g(z1 )g(z2 + z3 )f (z2 ) g(z1 )g(z2 ) ε·

z1 z 3 z1 (z2 + z3 )f (z2 + z3 ) = ε· g(z1 )g(z2 + z3 )f (z3 ) g(z1 )g(z3 )

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Now, the pull-back of (D.1) applied to ζ implies (z2 + z3 )c g(z1 (z2 + z3 )) f (z1 z2 + z1 z3 ) eaz1 z3 z2c g(z1 z2 ) = g(z1 ) f (z1 z2 )g(z1 ) +

f (z1 z2 + z1 z3 )z3c g(z1 z3 ) f (z1 z3 )g(z1 )

g(z1 (z2 + z3 )) eaz3 f (z2 + z3 ) f (z1 z2 + z1 z3 ) eaz1 z3 g(z1 z2 ) = g(z2 + z3 )f (z2 ) f (z1 z2 )g(z2 ) g(z1 (z2 + z3 ))f (z2 + z3 ) f (z1 z2 + z1 z3 )g(z1 z3 ) = g(z2 + z3 )f (z3 ) f (z1 z3 )g(z3 ) ε·

z1 z3 eaz1 z3 f (z1 z2 + z1 z3 )g(z1 z2 ) z1 z2 g(z1 z3 )f (z1 z2 + z1 z3 ) = aε · g(z1 )g(z2 )f (z1 z3 )g(z3 ) g(z1 )g(z3 )f (z1 z2 )g(z2 )

z1 z3 eaz1 z3 f (z1 z2 + z1 z3 )g(z1 z2 ) g(z1 )g(z3 )f (z1 z2 )g(z2 ) Just as before, after the exchange of z2 and z3 in the third equation and comparing the result with the second, we see that a = 0. Also, from third equation g(y + z)g(xz) g(x(y + z))g(z) = f (x(y + z))f (z) f (y + z)f (xz) +ε·

and, if we write k(t) =

g(t) f (t) ,

then k(x(y + z))k(z) = k(y + z)k(xz); and through

a straightforward computation one is led to k(x) = k(1)xν , where ν = xν f (x) f (1)

k (1) k(1) .

It

and, in order to fulfill all the equations arising from follows that g(x) = (D.1) (both, for z and ζ), we must have ε · (z2 + z3 )1−ν = ε · z21−ν ε · (z2 + z3 )1−ν = ε · z31−ν (z2 + z3 )c+ν = z2c+ν + z3c+ν ε · z21−ν f (z1 z2 + z1 z3 ) = ε · z31−ν f (z1 z2 + z1 z3 ). Thus, if ε = 0 it follows that ν = 1 and c = 0. Otherwise, if ε = 0, then c + ν = 1. One notes that the pull-back of (D.2) applied to z gives no new conditions. When applied to ζ, however, it yields (z1 + z2 )1−c = z11−c + z21−c , so that c = 0. Therefore ν = 1. We now want to determine the isomorphism classes of ring-like structures on F1|1 , under the obvious notion of isomorphism. Theorem 2.3. The isomorphism classes of ring-like structures defined on F1|1 are parametrized by the different values of the scalar ε appearing in s∗ z = z1 + z2 s∗ ζ = ζ1 + ζ2

m∗ z = z1 z2 + εζ1 ζ2 m∗ ζ = z2 ζ1 + z1 ζ2

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according to the following scheme: If F = R, the classes correspond to the values ε = −1, 0, 1. If F = C, they correspond to ε = 0, 1. Proof. Assuming that ε and f define the two morphisms s and m as in Proposition 2.2 and ε and f define another pair of morphisms s and m, we want to find all the isomorphism classes. Now, let Φ : F1|1 → F1|1 be an isomorphism in the category of supermanifolds, satisfying, Φ ◦ s = s ◦ (Φ ◦ π1 × Φ ◦ π2 ) and

Φ◦m=m ◦ (Φ ◦ π1 × Φ ◦ π2 ).

Writing Φ∗ z = ϕ and Φ∗ ζ = ψζ, the first condition implies: ϕ(z1 + z2 ) = ϕ(z1 ) + ϕ(z2 ); ψ(z1 + z2 )

f (z1 + z2 ) f(ϕ(z1 ) + ϕ(z2 )) = ψ(z1 ) ; f (z1 ) f(ϕ(z1 ))

ψ(z1 + z2 )

f (z1 + z2 ) f(ϕ(z1 ) + ϕ(z2 )) = ψ(z2 ) ; f (z2 ) f(ϕ(z2 ))

and second condition implies ϕ(z1 z2 ) = ϕ(z1 )ϕ(z2 ) ; εf (1)2 ϕ (z1 z2 ) εf(1)2 ψ(z1 )ψ(z2 ) = ; f (z1 )f (z2 ) f(ϕ(z1 ))f(ϕ(z2 )) z2 ψ(z1 z2 )f (z1 z2 ) ϕ(z2 ) ψ(z1 )f(ϕ(z1 )ϕ(z2 )) = ; f (z1 ) f(ϕ(z1 )) z1 ψ(z1 z2 )f (z1 z2 ) ϕ(z1 ) ψ(z2 )f(ϕ(z1 )ϕ(z2 )) = ; f (z2 ) f(ϕ(z2 )) It follows that ϕ(x) = x and the other equations are ψ(x)f (x) = ψ(0)f(x) and εf (1)2 = εf(1)2 ψ(0)2 . Thus, for a given f and ε, we can choose ψ(y) = ψ(0)f(y) so that f (y) = 1 and ε = εf(1)2 ψ(0)2 . If ε = 0 then ε = 0, otherwise, in the real case, depending on the sign of ε, we can always choose ψ(0) so that ε = ±1; in the complex case we may we may choose ψ(0)2 so that ε = 1.

3. Left F1|1 -Module Structures on Fk|k ; F1|1 -Linearity and Bilinearity In the previous section we found all possible ring-like structures that can be given to F1|1 by finding pairs of the associative morphisms s and m which also satisfy distributive conditions. Following the idea given in [8], we may also construct free F1|1 -modules as usual by taking a direct product of copies of F1|1 with itself and endowing the resulting supermanifold with two morphisms; namely, σ the ‘sum

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of vectors’ and µ the ‘scalar multiplication’, defined componentwise. We shall use the following notation (see [8]) ˆ τj : Fk|k × Fk|k → F1|1 ; E 1|1

ρi : F

×F

k|k

1|1

→F

ˆ ;E

for the projection onto the jth factor, 1 ≤ j ≤ 2k for the projection onto the (i + 1)th factor, 0 ≤ i ≤ k.

Therefore, the morphisms σ : Fk|k × Fk|k → Fk|k and µ : F1|1 × Fk|k → Fk|k are defined by σ = s ◦ (τ1 × τk ) × s ◦ (τ2 × τk+2 ) × · · · × s ◦ (τk × τ2k ) µ = m ◦ (ρ0 × ρ1 ) × m ◦ (ρ0 × ρ2 ) × · · · × m ◦ (ρ0 × ρk ). Both morphisms can be written in terms of the local coordinates induced from the coordinates on F1|1 via pull-back. Thus, if we consider the projections pi : Fk|k → F1|1 , π : F

k|k

×F

k|k

for i = 1, 2, . . . k,

→F

k|k

,

for = 1, 2,

then {ti , ζi | i = 1, . . . , k} are coordinates for Fk|k and {ti , ζi | i = 1, . . . , k; = 1, 2} are coordinates for Fk|k × Fk|k , where ti = p∗i t, ζi = p∗i ζ and ti = π∗ ti , ζi = π∗ ζi . It follows that σ ∗ ti = t1i + t2i σ ∗ ζi = ζi1 + ζi2 µ∗ ti = t0 ti + εζ0 ζi µ∗ ζi = ti ζ0 + t0 ζi . Notice that pi ◦ π1 = τi and pi ◦ π2 = τk+i for i = 1, 2, . . . k. The linear and bilinear transformations in the category of supermanifolds are well understood in [8] in the particular case when ε = 1. Here we are also interested in the other cases, and with the explicit expressions for σ and µ above we shall be able to deal with any of the F1|1 -module structures on Fk|k simultaneously. Let us fix a ring-like structure on F1|1 . Following [8] we say that L : Fk|k → 1|1 F is F1|1 -linear if it satisfies the conditions (L.1) and (L.2) that follow: L ◦ σ = s ◦ (L ◦ (τ1 × τ2 × · · · × τk ) × L ◦ (τk+1 × τk+2 × · · · × τ2k ))

(L.1)

L ◦ µ = m ◦ (ρ0 × L ◦ (ρ1 × · · · × ρk ))

(L.2)

It was proved in [8] for the case ε = 1 that any linear morphism in the category is determined by its underlying linear map. The proof of the following proposition is similar to the proof given in [8] but takes into account the cases ε = 0 and ε = −1 as well. Proposition 3.1. If L : Fk|k → F1|1 satisfies (L.1) and (L.2), then the coordinate description of L is L∗ t =

k i=1

where Li ∈ F.

Li ti ,

L∗ ζ =

k i=1

Li ζi

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Proof. One writes L in the given coordinate systems as ˜ + Lij ζi ζj + · · · L∗ t = L ∗

L ζ=

i<j

Li ζi +

i=1

Lijr ζi ζj ζr + · · ·

i<j 0 sufficiently small, a real analytic Hamiltonian of the form: H(x, y, ε) = h0 (y n0 ) + εm1 h1 (y n1 ) + · · · + εma ha (y na ) + εma +1 p(x, y, ε),

(1.1)

Y. Han was partially supported by NSFC Grant 10601019 and Chinese Postdoctoral Science Foundation. Y. Li was partially supported by NSFC Grant 10531050, National 973 project of China 2006CD805903, SRFDP Grant 20040183030, and 985 Program of Jilin University. Y. Yi was partially supported by NSF grant DMS0708331, NSFC Grant 10428101, and a Changjiang Scholarship from Jilin University.

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Y. Han, Y. Li and Y. Yi


where a, m, and l, ni [i = 0, 1, . . . , a], mj [j = 1, 2, . . . , a], are positive integers satisfying n0 ≤ n1 ≤ · · · ≤ na =: n, m1 ≤ m2 ≤ · · · ≤ ma =: m, y ni = (y1 , . . . , yni ) , for i = 1, 2, . . . , a, and p depends on ε smoothly. The Hamiltonian (1.1) is clearly nearly integrable when the parameter ε is sufficiently small. Hamiltonians of the form (1.1) arise frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and nonrestricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales (see [29] and references therein). After certain regularization and normalization (see e.g., [9,14,21]), the Hamiltonians typically have the form (1.1), for which proper degeneracies, mainly due to the super-integrability of the Keplerian, usually occur in a way that for each time scale of order εmj , the normalized Hamiltonian hj is only a function of the first nj action variables for some positive integers mj , nj . Hence they are properly degenerate in the sense of Arnold ([1,2]). The existence of quasi-periodic motions for properly degenerate Hamiltonian (1.1) was first shown by Arnold ([1]) for the case a = m = 1 under a so-called degeneracy-removing condition that h0 + εh1 satisfies either the Kolmogorov or iso-energetic non-degenerate condition. Such a degeneracy-removing condition is known to be satisfied in many planar or restricted 3-body problems and n-body problems, leading to the existence of quasi-periodic invariant tori (see [2–4,6,10,11,14,15,19,20] and references therein). However, it is also known that in many cases of perturbed Kepler problems, the leading order of the perturbed Hamiltonian is insufficient to remove the degeneracy. For instance, the normalized Hamiltonian associated with the spatial lunar problem considered by Sommer ([29]) has the form: H = J1 + εh1 (J1 , J2 ) + ε2 h2 (J1 , J2 , J3 , ε) + εl P (J, φ, ε), J = (J1 , J2 , J3 ) ∈ R3 ,

(1.2)

3

φ∈T , where l > 3 is a real number. This Hamiltonian actually involves three time scales [i.e. a = m = 2 in (1.1)]. As shown in [29], besides Arnold’s singularity-removing condition imposed on the O(ε) order term h1 , the existence of quasi-periodic invariant tori for (1.2) requires a further singularity-removing condition of Kolmogorov type imposing on the O(ε2 ) order term h2 . Motivated by applications arising in a broader class of perturbed Kepler problems, the goal of this work is to present a KAM type of result for Hamiltonians of type (1.1) by taking into account of higher order singularity-removing conditions. To set up the problem, we consider the Hamiltonian (1.1) in a bounded closed region G × T n ⊂ Rn × T n . It is clear that for each ε the integrable part of (1.1): Nε (y) = h0 (y n0 ) + εm1 h1 (y n1 ) + · · · + εma ha (y na ),

(1.3)

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admits a family of invariant n-tori Tξε = {ξ} × T n , with linear flows {x0 + ω ε (ξ)t}, ξ ∈ G, where for each ξ ∈ G, ω ε (ξ) = ∇Nε (ξ) is the frequency vector of the n-torus Tξε . When ω ε (ξ) is non-resonant, the n-torus Tξε becomes quasi-periodic with slow and fast frequencies of different scales. Adopting the terminology of Arnold ([1,2]), we refer the integrable part Nε and its associated tori {Tξε } to as the intermediate Hamiltonian and intermediate tori, respectively. Let yˆni = (yni−1 +1 , . . . , yni ) , i = 0, 1, . . . , a, where n−1 = 0 (hence yˆn0 = y n0 ), and define: Ω = (∇yˆn0 h0 (y n0 ), . . . , ∇yˆna ha (y na )),

(1.4)

where for each i = 0, 1, . . . , a, ∇yˆni denotes the gradient with respect to yˆni . We assume the following high order, degeneracy-removing condition of Bruno– R¨ ussmann type: (A)

There is a positive integer N such that: Rank{∂yα Ω(y); 0 ≤ |α| ≤ N } = n,

∀ y ∈ G.

We note that the condition (A) above is equivalent to the following condition: (A ) For each i = 0, 1, . . . , a, there is a positive integer Ni such that α ∂ hi (y ni ) Rank ; 1 ≤ |α| ≤ Ni = ni − ni−1 , ∀ y ∈ G. ∂(ˆ y ni )α We will prove the following: Theorem (Main result). Assume the condition (A) and let 0 < δ < 15 be given. Then there exists an ε0 > 0 and a family of Cantor sets Gε ⊂ G, 0 < ε < ε0 , with δ |G \ Gε | = O(ε N ), such that each ξ ∈ Gε corresponds to a real analytic, invariant, quasi-periodic n-torus Tˆξε of the Hamiltonian (1.1) which is slightly deformed from the intermediate n-torus Tξε . Moreover, the family {Tˆξε : ξ ∈ Gε , 0 < ε < ε0 } varies Whitney smoothly. Remark. (1) Using arguments in [8], the above theorem also holds on a submanifold M of Rn if the condition (A) is only assumed for ξ ∈ M (e.g., M is a fixed energy surface). This, in particular, leads to an iso-energetic version of the theorem (see [8] for detail). One can further show the partial preservation of frequency components for the perturbed tori in the above theorem. More precisely, let i1 , . . . , in∗ be the row indexes of a non-singular principal minor of the matrix ∂Ω on G. Then the i1 , . . . , in∗ components of each perturbed toral frequency remain the same as the corresponding ones of the associated unperturbed toral frequency.

1422

(2)



Differing from the case for a usual nearly integrable Hamiltonian system, the excluding measure for the existence of quasi-periodic invariant tori in the δ properly degenerate case is of a fairly large order of ε N for a pre-fixed small positive constant δ, as shown in the theorem above. This is mainly caused by a normal form reduction which pushes the perturbation to an order higher than εN b+δ for a b= mi (ni − ni−1 ), (1.5) i=1 δ

(3)

for which the domain G needs to shrink by an order of ε N in measure. This is necessary for general properly degenerate Hamiltonian systems like (1.1) in order for the standard KAM iterations to apply (see the discussion below). However, if the perturbation in (1.1) is already in an order of O(εN b+δ ), then a normal form reduction will not be necessary, and the excluding measure for the existence of quasi-periodic invariant tori can be improved to an order of εb (see the measure estimate in Sect. 3). Indeed, this is the case for (1.2) because l > b = 3 and N = 1 there. We note that in the case l = 3 in (1.2), direct KAM iterations are not applicable. Instead, one can apply the theorem above to obtain a nearly full measure set of KAM tori with the excluding measure in an order of εδ for some pre-fixed small positive constant δ. In applications, verification of the condition (A) should rely on certain a priori regularization or normalization procedures which add higher order averaged terms to the properly degenerate part until the degeneracy-removing condition (A) is satisfied. Such an averaging procedure can be made general if lower dimensional tori are considered (see [13]) but it can be very delicate for the case of full dimensional tori(see [29] for a complete treatment of the spatial lunar problem). For a usual nearly integrable Hamiltonian system: H(x, y) = N (y) + εP (x, y), (y, x) ∈ G × T n ⊂ Rn × T n ,

the majority existence of invariant, quasi-periodic n-tori is asserted by the classical KAM theorem under the Kolmogorov non-degenerate condition that ∂ω(y), where ω(y) = ∇N (y), is non-singular over G. The same was shown to hold by Bruno ([5]) under the Bruno non-degenerate condition that: Rank(ω(y), ∂ω(y)) = n, y ∈ G. The weakest condition guaranteeing such persistence was given by R¨ ussmann [25] under the R¨ ussmann non-degenerate condition that ω(G) should not lie in any n − 1 dimensional subspace (see also [7] for a similar geometric condition). KAM type of theorems under the R¨ ussmann non-degenerate condition were shown in [27,31]. In particular, it was shown in [31] (see also [30] that the R¨ ussmann nondegenerate condition is equivalent to the condition A) above with respect to the present frequency map ω. We refer the readers to [8,12,16–18,24,28] for more KAM type of results under R¨ ussmann non-degenerate conditions.

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Unfortunately, these results as well as their proofs do not apply to the properly degenerate Hamiltonian system (1.1) directly, simply because the order of its non-integrable perturbation is not high enough for the usual KAM iterations to carry over. Due to the nature of the proper degeneracy in (1.1), it is not hard to see that a possible KAM iteration for the Hamiltonian would have to be carried out over a frequency domain containing Diophantine frequencies of type (γ, τ ), for τ > max{(N + 1)N − 1, (n + 1)n − 1} and γ ∼ εN b , where b is as in (1.5). This would automatically require a perturbation order that is higher than εN b . To overcome this obstacle, a crucial idea in the proof of our main result is to first obtain a normal form for (1.1) by conducting finitely many steps of KAM iterations on relatively small domains so that the non-integrable perturbation is pushed into a sufficiently high order. We will do so in Sect. 2 by adopting a quasilinear KAM iterative scheme introduced in [16] which involves solving a system of quasi-linear homological equations at each KAM step instead of linear ones. Our main result will be proved in Sect. 3 by performing a linear KAM scheme for infinite steps. Throughout the paper, unless specified otherwise, we will use the same symbol | · | to denote an equivalent (finite dimensional) vector norm and its induced matrix norm, absolute value of functions, and measure of sets, etc., and use | · |D to denote the sup-norm of functions on a domain D. For any r˜, s˜ > 0, we let: D(˜ r, s˜) = {(x, y) : |Imx| < r˜, |y| < s˜} be the (˜ r, s˜)-complex neighborhood of T n × {0} ⊂ T n × Rn , and D(˜ s) = {y : |y| < s˜} be the s˜-complex neighborhood of {0} ⊂ Rn .

2. Reduction to Normal Form As usual, the translations y → y + ξ, x → x, ξ ∈ G =: G0 , transform (1.1) into a smooth family of real analytic Hamiltonians: H0 = N 0 (y, ξ, ε) + εma P 0 (x, y, ξ, ε), N 0 = Nε (y + ξ) = e0 (ξ, ε) + ω 0 (ξ, ε), y + h0 (y, ξ, ε)

(2.1)

parametrized by ξ ∈ G0 , where ω 0 = ωε , h0 = O(|y n0 |2 + εm1 |y n1 |2 + · · · + εma |y na |2 ), and P 0 = εp(x, y + ξ, ε). It is clear that ω 0 has the form: ω 0 = (ω00 , εm1 ω10 , . . . , εma ωa0 ), where ωi0 = ∇yˆni hi (ξ ni ) + O(εmi+1 −mi ) for i = 0, 1, . . . , a − 1 and ωa0 = ∇yˆna ha (ξ na ). We denote: Ω0 =: Ω = (ω00 , ω10 , . . . , ωa0 ). We will derive a desired normal form for the Hamiltonian (2.1) via finite steps of KAM iterations using the quasi-linear iterative scheme introduced in [16].

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As to be seen later, the term εma in the perturbation plays an important role during the iterations in controlling derivatives of the transformations. Hence the Hamiltonian (2.1) cannot be rescaled to include the term h0 into the perturbation, which requires that each KAM iteration keeps a similar term in the integrable part. This is indeed one of the advantages of the quasi-linear scheme. For the remaining part of the paper, all derivatives with respect to the parameter ξ should be understood in the sense of Whitney. For the fixed 0 < δ < 15 prescribed in the main result, we let γ0 = εδ , s0 = ε2δ , μ0 = ε1−5δ . Also, let 0 < r0 < 1 be given such that the Hamiltonian (2.1) is real analytic in D(r0 , s0 ). Then it is easy to see that: |∂ξl P 0 |D(r0 ,s0 )×G0 < γ0 s20 μ0 , |l| ≤ N. Our normal form theorem states as follows: Normal Form Theorem. Consider the Hamiltonian (2.1) under the condition (A). Then as ε > 0 sufficiently small, there exist a subsets G∗ ⊂ G0 , with δ |G0 \ G∗ | = O(ε N ), and a smooth family of canonical, real analytic transfor1 6δ mations Φ∗ : D(r0 , s0 ) → D(r∗ , s∗ ), where r∗ = r20 and s∗ = O(ε 5 + 5 ), such that the transformed Hamiltonian reads: H∗ = H0 ◦ Φ∗ = e∗ (ξ, ε) + ω ∗ (ξ, ε), y + h∗ (y, ξ, ε) + P ∗ (x, y, ξ, ε), ∗

2

∗

∗

(2.2)

(ω0∗ , εm1 ω1∗ , . . . , εma ωa∗ )

with ωi being an where h = O(|y| ), ω has the form ω = ni − ni−1 dimensional vector for each i = 0, 1, . . . , a, respectively, and P ∗ satisfies: 2(N +6)

|∂ξl P ∗ |D(r∗ ,s∗ )×G∗ ≤ εγ∗ with γ∗ = εb , μ∗ = ε

ma +δ 2

s∗ μ2∗ , |l| ≤ N

(2.3)

. Moreover, if we denote Ω∗ = (ω0∗ , ω1∗ , . . . , ωa∗ ), then: δ

|∂ξl Ω∗ − ∂ξl Ω0 |G∗ ≤ ε1− 2 , |l| ≤ N.

(2.4)

We will prove the normal form theorem inductively via a finite sequence of quasi-linear iterations. Suppose that at a νth-step, we have obtained the following smooth family of real analytic Hamiltonians: H = N + εma P, (2.5) N = e(ξ, ε) + ω(ξ, ε), y + h(y, ξ, ε), where (x, y) ∈ D(r, s) for some 0 < r = r < r0 , 0 < s = s < s0 , ξ ∈ G with G ⊂ Rn being a bounded region, ω has the form: ω(ξ, ε) = (ω0 , εm1 ω1 , . . . , εma ωa ) with ωi ’s being an ni − ni−1 dimensional vectors for each i = 0, 1, . . . , a, respectively, h has the form: h(y, ξ, ε) = O(|y n0 |2 + εm1 |y n1 |2 + · · · + εma |y na |2 ), and |∂ξl P|D(r,s)×G ≤ γ0 s2 μ, |l| ≤ N, for some 0 < μ ≤ μ0 .

(2.6)

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For “+”=: ν + 1, we will find a symplectic transformation Φ+ , which, on a small phase domain D(r+ , s+ ) and a smaller parameter domain G+ , transforms (2.5) into a family of Hamiltonians: H+ = H ◦ Φ+ = N + + εma P + in the (ν+1)th-step which enjoy similar properties as (2.5) but with a much smaller non-integrable perturbation P + . All constants c1 − c5 below are independent of iteration process. For simplicity, we will use c to denote any intermediate positive constant which is independent of the iteration process. Define: Ω = (ω0 , ω1 , . . . , ωa ) and let: r0 r + , 2 4 1 1 = αs, α = μ 3 , 4 3 1 = log +1 , μ i−1 i (r − r+ ), αs , i = 1, 2, 3, 4, = D r+ + 4 4 3 = D r+ + (r − r+ ), s , 4 γ0 = ξ ∈ G : | k, Ω(ξ, ε)| > , 0 < |k| ≤ K+ . |k|τ

r+ = s+ K+ D 4i α ˆ D(s) G+

Hereafter, we let τ > max{(N + 1)N − 1, (n + 1)n − 1} be fixed. We consider the truncation: √ R= pkj y j e −1k,x |k|≤K+ ,|j|≤2

of the Taylor–Fourier series: P=

√ −1k,x

pkj y j e

.

n k∈Z n ,j∈Z+

Lemma 2.1. Assume: r−r+ (H1) e−K+ 4 = o(μ(r − r+ )n ). Then there is a constant c1 such that: |∂ξl (P − R)|D 3 α ×G ≤ c1 γ0 s2 μ2 , 4

|∂ξl R|D 3 α ×G 4

for all |l| ≤ N .

≤ c1 γ0 s2 μ

(2.7)

1426



Proof. Write: P = R + I + II, where

I=

√ −1k,x

pkj y j e

n |k|>K+ ,j∈Z+

II =

√ −1k,x

pkj y j e

,

.

|k|≤K+ ,|j|≥3

The standard Cauchy estimate yields that:

|∂ξl I|D(s)×G ≤ ˆ

|∂ξl P|D(r,s)×G e−|k|

r−r+ 4

≤ γ0 s2 μ

≤ γ0 s2 μ

κn e−κ

r−r+ 4

κ=K+

|k|>K+

∞

∞ K+

tn e−t

r−r+ 4

dt ≤ γ0 s2 μ

n

r−r+ 4 n! e−K+ 4 ≤ γ0 s2 μ2 . (r − r+ )n

It follows that: Let

≤ |∂ξl P|D(r,s)×G + |∂ξl I|D(s)×G ≤ 2γ0 s2 μ. |∂ξl (P − I)|D(s)×G ˆ ˆ be the obvious anti-derivative of

∂3 . We have by Cauchy estimate that: ∂y 3

|∂ξl II|D 3 α ×G 4

c

l c

≤ 3 ∂ξ (P − I − R) D(s)×G dy

= 3 ˆ s s D3α 4

c

≤ 3 γ0 s2 μdy

≤ cγ0 s2 μ2 . s D3 4

l

∂ξ (P − I) ˆ dy

D(s)×G

D3α 4

(2.8)

α

Thus, |∂ξl (P − R)|D 3 α ×G ≤ cγ0 s2 μ2 ,

(2.9)

4

and,

l

∂ξ R

D 3 α ×G 4

≤ ∂ξl (P − R) D

3 α ×G 4

+ ∂ξl P D(r,s)×G ≤ cγ0 s2 μ.

We wish to average out all coefficients of R by constructing a symplectic transformation as the time-1 map φ1F of the flow generated by a Hamiltonian F of the form: √ fkj y j e −1k,x . F = 0 log log 5−log 4 is fixed. The following iteration lemma and convergence result are special cases of those contained in [8].

Lemma 3.1. Let ε be sufficiently small. Then the following holds for all ν = 1, 2, . . .. (1) There is a sequence of smooth families of symplectic, real analytic, near identity transformations Φξν : D(rν , sν ) → D(rν−1 , sν−1 );

ξ ∈ Λν

such that Hν = Hν−1 ◦ Φξν =: Nν + Pν , Nν = eν + ων , y,

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where:

γν−1 , 0 < |k| ≤ Kν Λν = ξ ∈ Λν−1 : | k, ων−1 | > |k|τ γν−1 = ξ ∈ Λν−1 : | k, ων−1 (ξ)| > , Kν−1 < |k| ≤ Kν , |k|τ |∂ξl ων − ∂ξl ω0 |Λν ≤ γ0 μ0 , |l| ≤ N,

|∂ξl Pν |Dν ×Λν (2)

(3.1)

≤ γν sν μν , |l| ≤ N.

The Whitney extensions of: Ψν =: Φξ1 ◦ Φξ2 ◦ · · · ◦ Φξν converge C N uniformly to a smooth family of symplectic maps, say, Ψ∞ , on D( r20 , s20 ) × Λ∗ , where: Λν , Λ∗ = ν≥0

such that: Hν = H0 ◦ Ψν−1 → H∞ =: H0 ◦ Ψ∞ = e∞ + ω∞ , y + P∞ with e∞ = limν→∞ eν , ω∞ = limν→∞ ων , P∞ = limν→∞ Pν , and moreover, ∂yj P∞ |D( r20 ,0)×Λ∗ = 0, |j| ≤ 2. Hence for each ξ ∈ Λ∗ , T n × {0} is an analytic invariant torus of H∞ with Diophantine frequency ω∞ (ξ) of type (γ∗ , τ ) for γ∗ = limν→∞ γν . We now estimate the measure |Λ0 \Λ∗ |. For each k ∈ Z n \{0} and ν = 0, 1, . . ., we consider the set: γν ν+1 ν , Rk = ξ ∈ Λν : |gk (ξ)| ≤ |k|τ +1 where gkν (ξ) =

k , ων |k|

.

Then: Λ0 \ Λ∗ =

∞

ν=0 Kν 0 such that

N ν

∂ gk

b

∂ξ N ≥ cε . Λν

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It then follows from Lemma 2.1 in [31] that: (N +6)b N1 ε ν+1 , k ∈ Z n \ {0}, |Rk | ≤ c(ε) |k|τ +1 where

ν = 0, 1, . . . ,

1 1 c(ε) = 2 2 + 3 + · · · + + b n − 1 cε

.

Hence: |Λ0 \ Λ∗ | ≤

∞

6b

ν=0 Kν 0 as the mass of the isolated body whose center of mass is at r = 0 and the event horizon is at r = 2M . Scaling F0 by a factor of ε shrinks the mass of the body down to M ε and “brings the asymptotia and the event horizon closer in”. Thus, one may interpret the family of functions εF0 , ε → 0+ , as modeling a body which “shrinks to zero size”. One particular solution of (1.1) is F (r) = Λ3 r2 , a function corresponding to the de Sitter data. Adding the “shrinking” Schwarzschild body to the de Sitter

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I. Stavrov Allen


background amounts to considering the general solution of (1.1): Λ 2M ε . Fε (r) = F (r) + εF0 (r) = r2 + 3 r The solution Fε corresponds to the one-parameter family of metrics gε for Schwarzschild–de Sitter data: −1 Λ 2M ε ds2 = 1 − r2 − dr2 + r2 dω 2 . (1.2) 3 r Let us proceed by examining what happens if we take ε → 0+ , that is, if we take a “point-particle limit” of gε . First of all, we note that the 2M ε-term in (1.2) converges to zero (away from r = 0) and that gε approach the de Sitter metric −1 2 ds2 = 1 − Λ3 r2 dr + r2 dω 2 . This is to be expected as the contribution of a “point-particle” to a large-scale geometry ought to be negligible. What is perhaps more interesting is “zooming in” and paying attention to small scale behavior near r = 0. To that end we consider the metric ε12 gε in scaled coordinates (ρ, ω) where ρ = rε : −1 1 2M 2 2Λ 2 ds = 2 gε = 1 − ε ρ − dρ2 + ρ2 dω 2 . ε 3 ρ In the limit as ε → 0+ , we recover the original Schwarzschild body! 1.4. The Two Point-Particle Limit Properties Let us now precisely state the two limit properties illustrated above. The terminology we use is motivated by that of [7]. Let (M, g, K) be large-scale initial data, let S ∈ M and let (M0 , g0 , K0 ) be AE initial data. In the example above (M, g, K) corresponds to de Sitter data with S as the origin in R3 , and (M0 , g0 , K0 ) the Schwarzschild data. A family (Mε , gε , Kε ) of initial data (Schwarzschild–de Sitter in the example) obeys pointparticle limit properties with respect to (M, g, K), (M0 , g0 , K0 ) and S ∈ M if the following hold. 1. The ordinary point-particle limit property. Let K ⊆ M {S} be a compact set. For small enough ε > 0 there exist embeddings iε : K → Mε such that for all k ∈ N ∪ {0} we have in the limit ε → 0+ that (iε )∗ gε − g C k (K,g) → 0, (iε )∗ Kε − K C k (K,g) → 0. 2. The scaled point-particle limit property. Let K ⊆ M0 be a compact set. For small enough ε > 0 there exist embeddings ιε : K → Mε such that for all k ∈ N ∪ {0} we have in the limit ε → 0+ that 1 1 (ιε )∗ gε − g0 (ιε )∗ Kε − K0 → 0, → 0. k k ε2 ε C (K,g0 )

C (K,g0 )

From this point on every use of the word “shrinking” should be interpreted in terms of the scaled point-particle limit property. Unless specifically stated otherwise, the parameter ε should be assumed to be positive.

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1.5. The Results Our main result is a gluing construction which adds scaled and truncated AE initial data to compact cosmological initial data. The outcome of the construction is a one-parameter family of initial data which has ordinary and scaled point-particle limits as described above. Our analytical work is based upon the IMP-gluing procedure [8] and, in particular, the conformal method. We work with data (M, g, K) and (M0 , g0 , K0 ) for which τ := Trg K and Trg0 K0 are constant. The advantage of considering such constant mean curvature (CMC) data is that the conformal method becomes particularly user-friendly. The IMP-gluing techniques (e.g. [5,8]) require certain non-degeneracy conditions. The extent to which these conditions are generically satisfied is addressed in [2] (see Theorem 1.3). For our results we need to assume the following conditions on the large-scale initial data: CKVF assumption. (M, g) has no conformal Killing vector fields; Injectivity assumption. The operator Δg − |K|2g − Λ on (M, g) has trivial kernel. We address these assumptions in more detail in an appendix to this paper. Note that the metrics g which satisfy the CKVF assumption are in some sense generic (see Sect. 4 of [2]). Also, if the cosmological constant Λ is relatively small, that is Λ < |K|2g , the Injectivity Assumption is unnecessary due to the Maximum Principle. The following is our main result. Theorem 1.1. Let (M, g, K) be a (not necessarily connected) compact and smooth CMC solution of the vacuum constraints corresponding to a cosmological constant Λ such that the CKVF and the Injectivity assumption are satisfied. Furthermore, let (M0 , g0 , K0 ) be a smooth AE CMC solution of the vacuum constraints with no cosmological constant. Let n be the number of asymptotic ends of (M0 , g0 ), let S1 , S2 , . . . , Sn ∈ M and let I1 , I2 , . . . , In ∈ M 0 be the asymptotic endpoints of M0 . Finally, let ν ∈ 32 , 2 . There exists ε0 > 0 such that for each ε ∈ (0, ε0 ) there is a compact and smooth CMC solution (Mε , gε , Kε ) of the vacuum constraints corresponding to the cosmological constant Λ with the following properties. 1. Mε is diffeomorphic to an n-fold connected sum of M and M 0 obtained by excising small balls around S1 , . . . , Sn and I1 , . . . , In and identifying the boundaries of the balls at S1 and I1 , S2 and I2 ,. . . , Sn and In . 2. For K ⊆ M {S1 , S2 , . . . , Sn } a compact set and ε small enough, depending on K, there exist embeddings iε : K → Mε such that for all k ∈ N ∪ {0} we have as ε → 0 that (iε )∗ gε − g C k (K,g) = O(εν/2 ),

(iε )∗ Kε − K C k (K,g) = O(εν/2 ).

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3. For K ⊆ M0 a compact set and ε small enough, depending on K, there exist embeddings ιε : K → Mε such that for all k ∈ N ∪ {0} we have as ε → 0 that 1 (ιε )∗ gε − g0 = O(ε1−ν/2 ), k ε2 C (K,g0 ) 1 (ιε )∗ Kε − K0 = O(ε1−ν/2 ). k ε C (K,g0 ) Our result provides examples of initial data which are generalizations of Schwarzschild–de Sitter initial data. In future work we intend to investigate if one could use these examples to generate one-parameter families of space-times required in [7]. Another consequence of our main theorem is that one can use any AE initial data as a “shrinking prefabricated bridge” connecting several compact initial data so long as the compact data satisfy CKVF and Injectivity assumptions. This generalizes IMP-type results [8] in which the geometry of the gluing region is, essentially, a shrinking Cauchy slice in the Kruskal extension of Schwarzschild space-time. The idea behind this application of our result is related to the work of Joyce in [9]. 1.6. The Gluing Method Here, we review the steps of a typical initial data gluing procedure. • The topological gluing. For a connected sum of compact initial data one excises balls of radius “ε” and identifies the surrounding small annular regions, A1,ε and A2,ε , using inversion.1 The main difference between our work and the gluing procedures in the literature thus far is that we do not use inversion but directly identify the nearly Euclidean rescaled asymptotia (truncated to an annular shape) with an “almost” Euclidean annular region near the center of a geodesic normal coordinate patch in the large-scale data. • The approximate initial data. One uses cut-offs with differentials supported in the annular regions A1,ε , A2,ε to combine the Riemannian structures and the second fundamental forms. The derivatives of the cut-offs make the resulting “data” (Mε , gε , Kε ) violate the constraints in such a way that the “size” of the violation approaches zero as ε → 0. We note that (Mε , gε , Kε ) can at least be made CMC by killing off the trace-free part μ := K − τ3 g in the region where the metric gε transitions from g A1,ε to g A2,ε . • Repairing the momentum constraint. The idea here is to perturb Kε or, rather, its trace-free part με := Kε − τ3 gε so that the momentum constraint 1

IMP-construction does not literally involve inversion. Small neighborhoods of two points, S1 and S2 are first conformally blown up to become asymptotic cylinders, which are then truncated and glued to each other along what used to be the geodesic polar coordinate r. Up to an additive constant IMP-procedure identifies the logarithm of the geodesic distance from S1 , i.e. ln(r1 ), with the negative of the logarithm of the geodesic distance from S2 , − ln(r2 ). In effect, r1 is identified with (a small constant multiple of) r1 . This is nothing but inversion. 2

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is satisfied. This is done using the conformal Killing operator Dε and the ε is given by vector Laplacian Lε = Dε∗ Dε on (Mε , gε ); the perturbation μ μ ε := με + Dε Xε , where Lε Xε = (divgε με ) . • The Lichnerowicz equation. The Hamiltonian constraint can be repaired using conformal changes which do not break the momentum constraint we just fixed; see [1] for details. The conformal change we make in our paper is of the form ε + τ3 φ4ε gε , where φε is a positive solution of the gε → φ4ε gε , με + τ3 gε → φ−2 ε μ Lichnerowicz equation 1 1 με |2gε φ−7 Δgε φε − R(gε )φε + | ε + 8 8

Λ τ2 − 4 12

φ5ε = 0.

(1.3)

By making careful estimates in geometrically adapted weighted H¨ older spaces one shows that the resulting one-parameter family of glued data obeys any desired limit properties.

1.7. The Organization of the Paper The presentation of our gluing procedure begins with A List of Notational Conventions. The details of the topological gluing and the creation of the approximate data can be found in the following section, The Approximate Data. We proceed in the Sect. 4 to give a detailed description of several weighted H¨ older spaces used in our construction. Not only do these spaces permit a unified approach to showing that both point-particle limit properties hold, they also provide a context in which many relevant facts (e.g. Theorem 4.7 and its consequence (5.6)) can be most easily demonstrated and understood. It is our opinion that their importance warrants a clear, albeit somewhat lengthy, exposition. The remainder of the paper is dedicated to the analysis of the PDE’s needed to repair the momentum and the Hamiltonian constraints. This material is organized in Sects. 5 and 6. Each of these two sections ends with an examination of the point-particle limit properties. Our paper has two appendices, the first of which is On the CKVF and Injectivity assumptions. The second appendix concerns (the kernel of) the Euclidean vector Laplacian Lδ which is involved in repairing the momentum constraint. We have found two ways of executing this step of the proof, one being the direct computation of the kernel. The approach included in the main body of our paper is more elegant and more in the spirit of our other arguments. The downside of the indirect approach is that it only works for a narrower set of weights (which turns out to be necessary elsewhere in the paper). In this respect, the direct computation of ker(Lδ ) is more optimal. Having found no resource in the literature with an explicit treatment of Lδ and its kernel we have decided to append our computation of Lδ to this article. Thus, our paper ends with Appendix B. The Euclidean Vector Laplacian in Spherical Coordinates.

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2. A List of Notational Conventions • We let f (ε, x) g(ε, x) mean that there is some constant C > 0 such that for all points x on a specified domain, and all sufficiently small ε > 0 we have f (ε, x) ≤ Cg(ε, x). In particular, the symbol is to be interpreted as being uniform in ε. We use f (ε, x) ∼ g(ε, x) for g(ε, x) f (ε, x) g(ε, x). In case of additional parameters we let f (ε, x, Φ) g(ε, x, Φ) independently of Φ

• •

• •

•

•

mean that the constant C > 0 can be chosen independently of Φ; the same comment applies to ∼. We use letters k, α, ν in the context of weighted function spaces. Unless specifically stated otherwise, the reader should assume that k ∈ N ∪ {0}, α ∈ [0, 1) and ν ∈ R. We use p, q ∈ N ∪ {0} in denoting the type of a tensor. Unless specifically i ...i stated otherwise, a tensor T should be assumed to be of type Tj11...jqp . If the tensor type is clear from context p, q may not be emphasized. We use δ to denote the Euclidean metric. ¯r ) denotes an open (resp. closed) ball of radius r in In principle, Br (resp. B an ambient space which should be clear from the context. Unless explicitly stated otherwise, the reader should assume that the ball is centered at the origin. We use | · | to denote the point-wise norm of tensors. This symbol is typically decorated by an index which reveals which metric has been used to compute the norm. If the symbol | · | is left undecorated the reader should assume that we are discussing the Euclidean norm or the absolute value. The symbols for geometric differential operators are decorated by a superscript or a subscript which indicates the metric with respect to which the g operators are defined. For example, ∇ denotes the covariant differentiation with respect to some metric g, while Δδ denotes the Euclidean scalar Laplacian.

3. The Approximate Data Let (M, g, K) be a smooth compact CMC solution of the vacuum constraints with cosmological constant Λ. Assume that (M, g, K) satisfy CKVF and Injectivity assumptions. Let τ := Trg K and μ := K − τ3 g. The 2-tensor μ is trace-free and, by virtue of the momentum constraint, also divergence-free. Let (M0 , g0 , K0 ) be a smooth asymptotically Euclidean CMC vacuum initial data with no cosmological constant. We adopt the notation used in Definition 1. For simplicity we outline the gluing procedure assuming (M0, g0 ) has one asymp ¯R . The AE-estimates totic end. For sufficiently large R we set 0ΩR := 0σ R3 B of Definition 1 imply Trg0 K0 = 0. We also let μ0 := K0 .

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In this section, we assume small values of the parameter ε > 0. To be precise, ε needs to satisfy 4Cε < (4C)−1 where C is the constant involved in Definition 1. 3.1. Scaling Consider (M0 , ε2 g0 , εK0 ); these data also satisfy the constraints. By precomposing σ with x → xε one obtains “scaled” coordinates εσ for the asymptotia of M0 : ε ¯Cε → M0 . σ : R3 B

0

We let ρ(X) := |εσ −1 (X)| = ε|0σ −1 (X)| denote our new radial function in the asymptotia. Note the following AE-estimates: ε ε ∗ 2 ( σ ε g0 )ij − δij , ρ (3.1) 1 1 ε β ε ∗ 2 ∂ ( σ ε g0 )ij ρ |β|+1 · |β| |β|+1 if |β| ≥ 1, (ε) ε ρ 1 1 1 ε β ε ∗ (3.2) ∂ ( σ εK0 )ij · ρ |β|+2 · |β| |β|+2 if |β| ≥ 0. ε (ε) ε ρ By truncating our scaled AE data we mean considering M0 εΩC −1 where

ε ΩC −1 := εσ x ∈ R3 |x| > C −1 = 0Ω(εC)−1 . 3.2. The Topological Gluing Consider S ∈ M and geodesic normal coordinates Mσ with respect to (M, g) centered at S. By increasing the value of C if necessary we may assume that these coordinates are defined on BC −1 . We have ( Mσ ∗ g)ij = δij + lij (x)

(3.3)

where lij (x) = O(|x|2 ) as |x| → 0. Moreover, the first derivatives of lij satisfy ∂lij (x) = O(|x|) while the higher derivatives satisfy ∂ β lij (x) = O(1) as |x| → 0. M := Mσ (BCε ). We proceed by excising a small geodesic ball BCε ε := M B M ∪ (M0 εΩC −1 ) and the equivaDefinition 2. Consider M Cε ε generated by the requirement that P ∼ε Q whenever lence relation ∼ε on M ε −1 M −1 σ (P ), σ (Q) are both defined and satisfy Mσ −1 (P ) = εσ −1 (Q). We define

ε . the family of manifolds Mε by Mε := M ∼ε Thus, loosely speaking, the manifolds Mε are created by identifying a large annular region in the asymptotia of M0 (characterized by |0σ −1 | ∈ [C, (εC)−1 ]) with a small annular region centered at S ∈ M (given by | Mσ −1 | ∈ [Cε, C −1 ]). M . The (truncated) geometry of M0 is scaled in order to “fit” within BCε There are natural quotient maps M iε : M BCε → Mε and ιε : (M0 εΩC −1 ) → Mε

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which we may use to put coordinates on Mε . Note that σ := iε ◦ Mσ = ιε ◦ εσ maps the annular region Aε := {x ∈ R3 | Cε ≤ |x| ≤ C −1 } to the gluing region [εσ (Aε )] = [ Mσ(Aε )] ⊆ Mε . ∼ε

∼ε

The map σ defines our preferred coordinates for the gluing region. For the purposes of dealing with point-particle limits note that for any compact subset K ⊆ M {S} and sufficiently small ε the quotient map iε gives rise to an embedding of K into Mε . Likewise, if K ⊆ M0 is a compact subset then for ε small enough the quotient map ιε defines an embedding of K into Mε . 3.3. Combining the Metrics We use partition of unity to combine the metrics g and ε2 g0 . Let χ be a decreasing smooth cut-off function on R such that χ ≡ 1 on (−∞, 1]

and

χ ≡ 0 on [4, +∞).

We define gε on Mε to be the unique metric for which i∗ε gε = g and ι∗ε gε = ε2 g0 away from the gluing region while in the gluing region itself we have |x| ε ∗ 2 |x| σ ε g0 + (1 − χ) √ [ Mσ ∗ g] . (3.4) σ ∗ gε = χ √ ε ε x x x part of the gluing region Loosely speaking, √ our metric gε “matches” with g on the √ given by |x| ≥ 4 ε and it “matches” with ε2 g0 on |x| ≤ ε. 3.4. Combining the Second Fundamental Forms The cut-off χ is also involved in combining the trace-free parts, μ and εμ0 , of K and εK0 . We define με to be the unique symmetric 2-tensor for which i∗ε με = μ and ι∗ε με = εμ0 away from the gluing region while in the gluing region itself we have 6 3 √ |x| − 2 [ Mσ ∗ μ] . (3.5) σ ∗ με = χ √ |x| − 2 [εσ ∗ εμ0 ] + (1 − χ) ε 4 ε x x x It follows from (3.5) that on the part of the gluing region given by √ √ ε ≤ |x| • √ √ ≤ 4 ε we have με ≡ 0; • |x| ≥ 4 ε we have that με is a multiple of (the pullback of) μ; on |x| ≥ 8 ε the tensor √ √ με exactly “matches” μ; • |x| ≤ ε we have that με is a multiple of (the pullback of) εμ0 ; on |x| ≤ 12 ε the tensor με exactly “matches” εμ0 . Since μ and μ0 are traceless with respect to g and g0 , respectively, we see that Trgε με = 0. It possible that divgε με = 0, with its support in the part of the gluing √ √ region given by 12 ε ≤ |x| ≤ 8 ε. In other words, the data Mε , gε , με + 13 τ gε where τ = Trg K is only an approximate solution of the constraints.

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4. Weighted Function Spaces Our analysis relies on weighted H¨ older spaces described below. A reader may find it useful to examine the geometric relationship between the four types of spaces. 1. Weighted H¨ older spaces on (Mε , gε ) which in some sense measure the (pointwise) size of tensors relative to the proximity to the gluing region; 2. Weighted H¨ older spaces on (M {S}, g) which keep track of (point-wise) decay and/or growth of tensors in terms of the geodesic distance from S; 3. Weighted H¨ older spaces on (M0 , g0 ) which keep track of the decay of tensors with respect to the radial function of the “asymptotia”. 4. Weighted H¨ older spaces on the Euclidean R3 {0}, δ which keep track of the decay and/or growth of tensors near 0 and ∞ in terms of the radial function r : x → |x|. All four of these function spaces are defined using preferred atlases for the underlying manifolds. The charts in these atlases are in turn determined by the respective weight functions. The weight functions are in no way “canonical”—we choose them so that they are conducive to obtaining the desired results. We only need the weight functions to obey properties such as those stated in our Proposition 4.1. This approach to weighted H¨ older spaces has been highly influenced by [4] (see the appendix on “scaling properties”) and [10], and we thank the authors of these two papers for addressing the function spaces in detail. Since the weighted spaces on (Mε , gε ) are the most delicate of all we present their construction in detail; the remaining weighted spaces are discussed only briefly. 4.1. The Weight Function for (Mε , gε ) Consider smooth, increasing, positive functions w0,R and wM,R on (0, ∞) for which s if s ≥ 2C C −1 if s ≥ ( 32 C)−1 w0,R (s) = (s) = and w M,R C if s ≤ 32 C, s if s ≤ (2C)−1 . We define the weight function wε to be the unique function on Mε which satisfies if |x| ≥ 2Cε wM,R (|x|) (wε ◦ σ) (x) = |x| if |x| ≤ (2C)−1 ε · w0,R ε for x ∈ Aε and which is constant away from σ(Aε ). In other words, • wε ≡ C −1 on the component of Mε σ(Aε ) arising from M ; • wε on σ(Aε ) in some sense describes the distance in (M, g) away from S. Equivalently, one may think of wε on σ(Aε ) as being the radial function in the asymptotia of (M0 , ε2 g0 ); • wε ≡ Cε on the connected component of Mε σ(Aε ) arising from M0 . 4.2. Special Atlases for Mε For each Mε , we construct two atlases, one a refinement of the other. Within these atlases we distinguish three types of charts, depending on which region of Mε they

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cover. The charts are described in detail below, but we take the opportunity to outline the three types first. One type (Type G) are the charts whose images are well within the gluing region. To be more precise, these charts cover Gε ⊆ σ(Aε ) where Gε is determined by

σ −1 (Gε ) = x ∈ R3 4Cε < |x| < (4C)−1 . There are two connected components of Mε Gε : one arising from M and one arising from M0 . By Type M we mean the charts (described below) which cover the component arising from M and by Type M0 we mean the charts which cover the component arising from M0 . Type G. To a point P = σ(xP ) ∈ Gε we associate the chart σP = σ ◦ HP where |xP | |xP | · x. HP : B1 → B xP ; , HP (x) = xP + 2 2 This chart is well defined since 3 3 |xP | < HP (x) < |xP | < C −1 < C −1 for all x ∈ B1 . Cε < 2Cε < 2 2 8 It is important to note that wε (σP (x)) = |HP (x)| for all x ∈ B1 . To a chart σP we associate the scaled pullback gP :=

4 σ ∗ gε . |xP |2 P

(4.1)

Finally, we also consider the restriction σP of σP to B 21 . M Type M . Consider finitely many charts Mσ 1 , . . . , Mσ N which cover M B(4C) −1 M whose domains are B1 and whose images are contained in M B(8C) . More−1 M over, assume that their restrictions to B 14 cover M B(4C) −1 . By composing M σ 1 , . . . , Mσ N with iε we obtain charts σn := iε ◦ Mσ n ,

1≤n≤N

which cover the component of Mε Gε arising from M . To σn we associate the pullback gn := σn∗ gε = ( Mσ n )∗ g. The restrictions of σn to B 12 will be denoted by σn . Type M0 . Consider finitely many2 charts 0σ 1 , . . . , 0σ N covering M0 0Ω4C = M0 εΩ4Cε whose domains are B1 , whose images are contained in M0 0Ω8C , and whose restrictions to balls B 41 cover M0 0Ω4C . By composing with ιε we obtain σ−n := ιε ◦ 0σ n ,

1 ≤ n ≤ N,

2 Note that, without loss of generality, we may assume that the number N of charts of Type M is the same as the number of charts of Type M0 .

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charts which cover the component of Mε Gε arising from M0 . To a chart σ−n we associate the scaled pullback 1 ∗ g−n := 2 σ−n gε = (0σ n )∗ g0 . ε Note that g−n are independent of ε. The restrictions of σ−n to B 21 will be denoted by σ−n . The collection of charts σP for P ∈ Gε and σn , σ−n for 1 ≤ n ≤ N will be denoted by Cε . The collection of their restrictions to B 12 will be denoted by Cε . We list some properties of Cε and Cε ; the reader may find it useful to compare these with the scaling properties of [4]. Proposition 4.1. 1. We have wε (Φ(x)) ∼ wε (Φ(0)) independently of Φ ∈ Cε . 2. The metrics gP for P ∈ Gε are uniformly close to δ in the sense that (for each k) gP − δ C k (B1 ,δ) 1 independently of P ∈ Gε . 3. The transition functions for the charts in Cε and Cε have uniformly bounded Euclidean H¨ older norms. In particular, |F (x) − F (y)| ∼1 |x − y|

(4.2)

independently of the transition function F . δ ν 4. For a fixed ν ∈ R we have ∇k (wε ◦ Φ)ν (wε ◦ Φ) independently of Φ ∈ Cε . Proof. It is immediate that for all x ∈ B1 and all charts Φ of Type Gε we have 1 3 wε (Φ(0)) ≤ wε (Φ(x)) ≤ wε (Φ(0)) . 2 2 For the remaining charts the claim (1) follows from the fact that wε ∼ 1 on iε M B(8C)−1 and wε ∼ ε on ιε (M0 εΩ8Cε ). To prove part (2) of our proposition let P ∈ Gε and x ∈ B1 . It follows from the definition (4.1) of gP , the definition (3.4) of gε , the estimate (3.1), and expansion (3.3) that the components of gP satisfy (gP )ij − δij = (σ ∗ gε )ij − δij x HP (x) ε ∗ 2 ≤ ( σ ε g0 )ij − δij + ( Mσ ∗ g)ij − δij HP (x) HP (x) 2 ε + HP (x) . (4.3) HP (x) Since ε HP (x) 1 independently of P ∈ Gε we see that (gP )ij − δij is uniformly bounded from above. In fact, the same argument can be applied to the derivatives of (gP )ij . The only complication here is the presence of the derivatives

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P (x)| . However, each derivative of this function is of the cut-off function χ |H√ ε uniformly bounded on B1 . To see this, decompose χ

|HP (x)| 1 |xP | √ (y) = χ(|y|) and H =χ ◦H P (x), for χ P (x) = √ · xP + √ · x. ε ε 2 ε

All derivatives of χ are bounded on R3 , and are supported in the annulus 1 ≤ |y| ≤ 4. Since |x | |β| √P ◦H ∂β χ · (∂ β χ ) ◦ H P = P 2 ε

(4.4)

◦H is supported on the set {x ∈ for all multi-indices β, we see that ∂ β χ P |x | 3|xP | P B1 1 ≤ H P (x) ≤ 4}. Moreover, 2√ε ≤ HP (x) ≤ 2√ε for all x ∈ B1 and so ◦H ∂β χ P = 0 only if 1 |xP | ≤ √ ≤ 4. 3 2 ε P (x)| for each individual multi-index β It now follows from (4.4) that ∂xβ χ |H√ ε is bounded independently of ε, P and x. Arguing as in (4.3) completes the proof of part (2) of our proposition. To prove part (3) it suffices to consider the transition functions between two charts one of which is of Type G. A transition function σP−1 ◦ σQ for P, Q ∈ Gε is ε (Q) a composition of a translation and a dilation with coefficient w wε (P ) . In order for

1 ε (Q) Im(σP ) ∩ Im(σQ ) = ∅ we need to have w wε (P ) ∈ [ 3 , 3]. Therefore, all the derivatives of the transition functions of the form σP−1 ◦ σQ are bounded from above by 3. To understand the transition functions of type(s) σn−1 ◦σP and σP−1 ◦σn we inspect the condition Im(σn ) ∩ Im(σP ) = ∅. This condition implies wε (P ) ∼ 1 independently of σn−1 ◦ σP (or its inverse). Since σn−1 ◦ σP = σn−1 ◦ σ ◦ HP , and since there are only finitely many transition functions of the form σn−1 ◦ σ, it follows that all the derivatives of σn−1 ◦ σP and σP−1 ◦ σn are bounded uniformly in ε. Similar consid−1 ◦ σP and erations prove the boundedness in the last remaining cases: those of σ−n −1 ε σP ◦ σ−n . Roughly speaking, here we have wε (P ) = O(ε) while σ = ιε ◦ σ features a scaling by ε such that the resulting net contribution of σP and σP−1 in terms of ε is simply O(1). The explicit details are left to the reader. Part (4) of the proposition is immediate for of Type M due to the charts their finite number. Since wε ◦ ιε (X) = ε w0,R 0σ −1 (X) for all X ∈ 0ΩC , the same is true for charts of Type M0 . Thus, it remains to understand the charts of Type G.

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In general, there exist constants cj = cj (ν) such that δ δ k ∇ δ ∇i1 (wε ◦ Φ) ij (wε ◦ Φ) k ν ν cj (wε ◦ Φ) ∇ ((wε ◦ Φ) ) ≤ ··· . wε ◦ Φ wε ◦ Φ j=1 Therefore, it suffices to prove that δ k ∇ (wε ◦ Φ) (wε ◦ Φ)

(4.5)

independently of Φ = σP , P = σ(xP ) ∈ Gε . For k = 1 one can directly compute: d (w ◦ Φ) 3 (xa + |xP | xa ) · |xP | a=1 P ε a 2 2 (x) = dx wε ◦ Φ |xP + |x2P | x|2 ≤

|xP | 2 |xP + |x2P | x|

·

3 |xaP +

|xP | a 2 x | |xP | a=1 |xP + 2 x|

≤ 3.

The remainder of the proof of (4.5) is easily done by induction. To avoid notational complications we only do the case of k = 2 and leave the general induction step to the reader. We have δ δ ∇2 (wε ◦ Φ)2 = 2d (wε ◦ Φ) ⊗ d (wε ◦ Φ) + 2(wε ◦ Φ)∇2 (wε ◦ Φ), δ δ 3 |xP | a 2 a , viewed as a matrix, is a while ∇2 (wε ◦ Φ)2 (x) = ∇2 (x + x ) a=1 P 2 2 product of |x2P | and the identity. Dividing by (wε ◦ Φ)2 , using the boundedness |xP | 2 ε ◦Φ) 2 of (wε ◦Φ)2 and the already established estimate for d(w wε ◦Φ we obtain δ ∇ δ2 2 2 (wε ◦ Φ) ∇ (wε ◦ Φ) d (wε ◦ Φ) 2 + ≤ 1. wε ◦ Φ (wε ◦ Φ)2 wε ◦ Φ 4.3. H¨ older norms on Mε We start by defining the norms. The reader may find it instructive to compare our approach to that of [10]. Definition 3. If T is a tensor field with locally C k,α components, then set T k,α := sup Φ∗ T C k,α (B1 ,δ) Φ∈Cε

and T k,α := sup Φ∗ T C k,α (B 1 ,δ) . Φ∈Cε

2

The norms defined above are referred to as “H¨ older norms” or “C k,α -norms”. To develop an intuition about them we work out an equivalent form of the C k,0 -norm.

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We start by analyzing (σP )∗ T C k,0 for a chart σP of Type G and a tensor i i ...i T = Tj11j22...jpq on Mε . We have −p+q+k g gP gP k ∗ wε (P ) εk ∇ (σP T ) = ∇ k (σP∗ T ) 4 = · ∇ T ◦ σP . gP σ∗ g 2 gε wε (P )2 P ε Part 1 of Proposition 4.1 implies that wε ◦ σP ∼ wε (P ) independently of P , while Part 2 implies that for each k, p, q there exists Cp,q,k > 0 such that δ ∗ gε k ∗ −p+q+k k ∇ T g ◦ σP ≤ Cp,q,k ∇ σP T + · · · + σP T wε ε

gε δk ∗ ∇ σP T ≤ Cp,q,k wε−p+q+k ∇ k T

gε

+ · · · + wε−p+q T gε ◦ σP ,

point-wise. Analogues of these estimates also hold for charts of Type M and Type M0 . Indeed, one may apply the reasoning outlined above to metrics g±n and take advantage of the fact that wε ◦ σn ∼ 1, while wε ◦ σ−n ∼ ε. It follows that the i ...i · k,0 -norm on the space of tensors T = Tj11...jqp is equivalent to the norm k

gε sup wε−p+q+j ∇ j T gε ,

j=0 Mε

with equivalence constants depending on p, q, k but not on ε. As the same argument applies to the · k,0 -norm we have the uniform equivalence · k,0 ∼ · k,0 . In fact, more is true: Proposition 4.2. The norms · k,α and · k,α are equivalent uniformly in ε. Proof. Since T k,α ≤ T k,α it remains to show that for all Φ ∈ Cε and all tensors T of a given type we can bound δ δ k ∗ ∇ Φ T (x) − ∇k Φ∗ T (y) |x − y|α ˜ ∗ T C k,α (B ,δ) for some Φ ˜ ∈ Cε ; the word from above by a uniform multiple of Φ 1 2

“uniform” here should be interpreted to mean “independent of ε, Φ, T , x, y, and ˜ To this end, let m ∈ N be a uniform upper bound implied by (4.2). For a given Φ”. Φ ∈ Cε , tensor T and x, y ∈ B1 let x = x0 , x1 , x2 , . . . , x8m−1 , x8m = y be the division points of the line segment xy into 8m congruent subsegments. Note 1 for all 0 ≤ a ≤ 8m − 1. Since by assumpthat |x − y| < 2 implies |xa − xa+1 | < 4m tion the restrictions of elements of Cε to B 41 cover Mε we know that for each a there is Φa such that Φ(xa ) ∈ Φa B 14 . Observe that Φ(xa ), Φ(xa+1 ) ∈ Φa B 21 , 0 ≤ a ≤ 8m − 1.

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Indeed, if this there would exist a point x∗ of the segment xa xa+1 −1were not the case 1 −1 ∗ such that Φa ◦ Φ(xa ) − Φa ◦ Φ(x ) > 4 which then by definition of m would 1 imply |x∗ − xa | > 4m . Applying the triangle inequality we see that δ δ δ δ k ∗ k ∗ ∇ Φ T (xa¯ ) − ∇k Φ∗ T (xa¯+1 ) ∇ Φ T (x) − ∇k Φ∗ T (y) 1−α ≤ (8m) . |x − y|α |xa¯ − xa¯+1 |α δ δ ∗ ∗ Φa¯ T for some 0 ≤ a ¯ ≤ 8m − 1. The components of ∇k Φ∗ T = ∇k Φ−1 a ¯ ◦Φ at some point ξ can be written as sums of products of the components of the −1 −1 covariant derivatives of Φ∗a¯ T at Φ−1 at a ¯ ◦ Φ(ξ), the derivatives of Φa ¯ ◦Φ −1 −1 Φa¯ ◦ Φ(ξ) and the derivatives of Φa¯ ◦ Φ at ξ. The uniform bounds on the derivatives of transition functions and allow us to control expressions discussed −1 in (4.2) β Φa¯ ◦ Φ , which then the C 0,α (B 21 , δ)-norms of ∂ β Φ−1 ◦ Φa¯ ◦ Φ−1 a ¯ ◦ Φ and ∂ leaves us with δ δ k ∗ ∇ Φ T (xa¯ ) − ∇k Φ∗ T (xa¯+1 ) Φ∗a¯ T C k,α (B 1 ,δ) |xa¯ − xa¯+1 |α 2 independently of a ¯, Φ and T . This completes our proof. There is an important feature of the previous proof: it shows that the atlases Cε and Cε in Definition 3 can be replaced by any finite sub-atlases whose charts after restriction to B 41 still cover Mε . Indeed, the only effect such a change of atlases has is that · k,α (and · k,α ) are being replaced by equivalent norms. Consequently, compactness of Mε implies that any tensor field T on Mε with locally C k,α components satisfies T k,α < ∞ and equivalently, T k,α < ∞ . A natural question to investigate at this point is whether the space of all i ...i tensors Tj11...jqp on Mε with locally C k,α components is complete with respect to the H¨ older norm. A purely formal exercise which solely uses Definition 3 and completeness of C k,α (B1 ) gives a positive answer to the question. i ...i

Definition 4. The Hölder norm · k,α gives the set of all tensors Tj11...jqp on Mε with locally C k,α components the structure of a Banach space, which we denote k,α (Mε ); this space is also referred to as a Hölder space. For simplicity we by Cp,q k,α (Mε ). often write C k,α (Mε ) in place of Cp,q 4.4. Weighted Hölder Spaces on Mε i ...i

Definition 5. The Banach space of all tensors Tj11...jqp with locally C k,α components and norm(s) T k,α,ν := wεν T k,α

(and /or T k,α,ν := wεν T k,α )

k,α,ν is denoted by Cp,q (Mε ). The norm · k,α,ν is referred to as a weighted Hölder k,α,ν norm, and Cp,q (Mε ) is referred to as a weighted Hölder space.

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We point out that the two weighted H¨ older norms ( · and · ) are equivalent uniformly in ε (Proposition 4.2). We proceed by discussing some equivalent representations of our weighted H¨ older norm. For Φ ∈ Cε (or Cε ) set w ε,Φ := wε (Φ(0)); the reader should think of w ε,Φ as “a sample value” of the weight function wε on the chart Φ. Proposition 4.3. The weighted H¨ older norms · k,α,ν and · k,α,ν are equivalent to the norms ∗ T → sup w ε,Φ Φ T C k,α (B1 ,δ) ν

Φ∈Cε

and

∗ T → sup w ε,Φ Φ T C k,α (B 1 ,δ) . ν

Φ∈Cε

2

This equivalence is uniform in ε. Proof. We focus on the norms arising from B1 ; the norms arising from B 21 can be dealt with analogously. The main ingredient of our proof is showing the uniform estimate ∗ ν ν ∗ w (4.6) Φ (wε T ) ε,Φ Φ T C k,α (B1 ,δ) , C k,α (B1 ,δ)

δ δ j δ independently of Φ. Since ∇j (Φ∗ (wεν T )) = i=0 ∇i ((wε ◦ Φ)ν ) ⊗ ∇j−i (Φ∗ T ) for all 0 ≤ j ≤ k, Proposition 4.1 implies the point-wise estimate j δ δ j−i ∗ j ∗ ν ∇ (Φ T ) ∇ (Φ (wε T )) (wε ◦ Φ)ν

i=0 ∗ (wε ◦ Φ)ν Φ∗ T C k,α (B1 ,δ) w ε,Φ Φ T C k,α (B1 ,δ) . ν

The same line of reasoning gives us δ δ k ∗ ν ∇ (Φ (wε T )) (x) − ∇k (Φ∗ (wεν T )) (y) |x − y|α δ δ k ∇j ((w ◦ Φ)ν ) (x) − ∇j ((w ◦ Φ)ν ) (y) ε ε δ ≤ · ∇k−j (Φ∗ T ) (x) α |x − y| j=0 δ δ k−j ∗ k (Φ T ) (x) − ∇k−j (Φ∗ T ) (y) ∇ δj ν ∇ ((wε ◦ Φ) ) (y) · + |x − y|α j=0 ∗ (wε ◦ Φ)ν C k+1 (B1 ,δ) · Φ∗ T C k,α (B1 ,δ) w ε,Φ Φ T C k,α (B1 ,δ) ν

independently of Φ. Estimate (4.6) is now immediate. Using −ν in place of ν in (4.6) we obtain −ν ∗ Φ T C k,α (B1 ,δ) = Φ∗ wε−ν (wεν T ) C k,α (B1 ,δ) w Φ∗ (wεν T ) C k,α (B1 ,δ) , ε,Φ ∗ ∗ ν i.e. w ε,Φ Φ T C k,α (B1 ,δ) Φ (wε T ) C k,α (B1 ,δ) . Overall, we have ν

∗ ∗ ν ∗ w ε,Φ Φ T C k,α (B1 ,δ) Φ (wε T ) C k,α (B1 ,δ) w ε,Φ Φ T C k,α (B1 ,δ) , ν

which yields the claimed equivalence of norms.

ν

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As a consequence of Proposition 4.3 and the description of the (unweighted) k,0,ν (Mε ) is equivalent to the norm C k,0 -norm we have that the · k,0,ν -norm on Cp,q k

gε sup wε−p+q+ν+j ∇ j T g , ε

j=0 Mε

(4.7)

with equivalence constants possibly depending on p, q, k, ν but not on ε. older Proposition 4.3 also implies that for any fixed ε and any ν1 , ν2 ∈ R the H¨ norms · k,α,ν1 and · k,α,ν2 are equivalent. However, it also shows that the norms are not equivalent uniformly in ε. It is this non-uniformity that makes our weighted H¨ older spaces useful. 4.5. Restrictions of (Weighted) H¨ older Norms to Open Subsets of Mε k,α,ν We now define function spaces Cp,q (U ) for open subsets U ⊆ Mε . These restrictions of (weighted) H¨ older spaces are avoidable, but we have decided to keep them in our work as an efficient book-keeping tool.

Definition 6. Let U ⊆ Mε be an open subset and let Cε;U := ΦΦ−1 (U ) Φ ∈ Cε . For tensor fields T on U with locally C k,α components define T k,α;U := sup Φ∗ T C k,α (Φ−1 (U ),δ) Φ∈Cε;U

and T k,α,ν;U := wεν T k,α;U .

i ...i

The Banach space of all tensors Tj11...jqp on U for which T k,α,ν;U < ∞ is denoted k,α,ν by Cp,q (U ). It is important to notice that the proof of Proposition 4.3 carries over to our new set-up and that we have the norm equivalences ∗ sup w ε,Φ Φ T C k,α (Φ−1 (U ),δ) ν

Φ∈Cε;U

∗ T k,α,ν;U sup w ε,Φ Φ T C k,α (Φ−1 (U ),δ) , ν

Φ∈Cε;U

T k,0,ν ∼

k

gε sup wε−p+q+j+ν ∇ j T |gε ,

(4.8) (4.9)

j=0 U

with equivalence constants which are independent of U ⊆ Mε . A careful reader has surely noticed that we have not defined the norm · k,α,ν;U . The reason for this is twofold: the norm · k,α,ν is only needed to obtain elliptic estimates on the entire Mε (e.g. Theorem 4.6), and, the proof of Proposition 4.2 does not carry over as it highly depends on the “convexity” of U . Given this situation we have decided it is best to avoid · k,α,ν;U -norm altogether. It is clear from Definition 6 that U ⊆ V ⊆ Mε =⇒ T k,α,ν;U ≤ T k,α,ν;V

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for all choices of k, α, ν and T . Other properties of the restricted H¨ older norms which we need are listed below; they follow directly from (4.8) and their proofs are left to the reader. Proposition 4.4. Let U ⊆ Mε be an open set. 1. If ν1 < ν2 then (inf wε )ν2 −ν1 T k,α,ν1 ;U T k,α,ν2 ;U (sup wε )ν2 −ν1 T k,α,ν1 ;U U

U

and in particular εν2 −ν1 T k,α,ν1 ;U T k,α,ν2 ;U T k,α,ν1 ;U independently of U ⊆ Mε and C k,α -tensor fields T . 2. The point-wise tensor product satisfies T1 ⊗ T2 k,α,ν1 +ν2 ;U T1 k,α,ν1 ;U T2 k,α,ν2 ;U independently of U ⊆ Mε and C k,α -tensor fields T1 , T2 of a particular type. 3. Contraction of a tensor field gives rise to a continuous linear map k,α,ν k,α,ν C : Cp+1,q+1 (U ) → Cp,q (U )

whose norm is bounded uniformly in ε and independently of U ⊆ Mε . 4. Raising an index of a tensor field on (U, gε ) gives rise to a continuous linear map

k,α,ν k,α,ν+2 : Cp,q+1 (U ) → Cp+1,q (U )

whose norm is bounded uniformly in ε and independently of U ⊆ Mε . 4.6. Differential Operators and Uniform Elliptic Estimates on Mε Here are several differential operators we use in our analysis: • The conformal Killing operator Dε , whose action on vector fields is given by (Dε X)ab =

1 1 (Xa;b + Xb;a ) − (divgε X) (gε )ab . 2 3

k+1,α,ν k,α,ν−2 (Mε ) → C0,2 (Mε ) has uniformly bounded The operator Dε : C1,0 norm (see Proposition 4.5). Intuitively, the reason for the change in weights (from ν to ν − 2) is that Dε lowers the index on X and hence reduces the weight by 2 (cf. part 4 of Proposition 4.4). It is easy to see that Im(Dε ) is contained in the subspace of trace-free symmetric 2-tensors and that the formal adjoint of Dε is k+1,α,ν−2 k,α,ν+2 Dε∗ : C0,2 (Mε ) → C1,0 (Mε ) with Dε∗ (ω) = −(divgε ω) ;

the reason for the change in weights (from ν − 2 to ν + 2) can be traced back to the fact that ω → −(divgε ω) requires raising indices twice.

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• The vector Laplacian Lε , whose action on vector fields is given by Lε X = Dε∗ Dε X = −divgε (Dε X) .

k+2,α,ν The operators Lε are self-adjoint and elliptic. They map C1,0 (Mε ) to k,α,ν+2 (Mε ) and, as we see in Theorem 4.6 below, they allow uniform ellipC1,0 tic estimates. • The linearized Lichnerowicz operators Lε , whose action on functions can schematically be represented as 1 Lε f = Δgε f − R(gε )f + hε f, 8 where hε : Mε → R is a family of functions of uniformly bounded C k,α,2 norms: hε k,α,2 1. By Theorem 4.6 below, the family of second order (selfk+2,α,ν k,α,ν+2 (Mε ) → C0,0 (Mε ) adjoint, elliptic) differential operators Lε : C0,0 allows uniform elliptic estimates. We now prove the above-mentioned properties of Dε , Lε and Lε . k+1,α,ν k,α,ν−2 Proposition 4.5. If X ∈ C1,0 (Mε ) then Dε X ∈ C0,2 (Mε ) and

Dε X k,α,ν−2 X k+1,α,ν . Proof. We see from Proposition 4.3 that it suffices to show 2

∗ Φ∗ (Dε X) C k,α (B1 ,δ) w ε,Φ Φ X C k+1,α (B1 ,δ)

independently of Φ ∈ Cε . Our proof splits into three cases depending on the type of chart Φ ∈ Cε . In the interest of brevity we focus on the most interesting of the three cases, Type G. Let Φ = σP with P ∈ Gε . We compute: wε (P )2 DgP σP∗ X k,α σP∗ (Dε X) C k,α (B1 ,δ) = DσP∗ gε σP∗ X C k,α (B1 ,δ) = C (B1 ,δ) 4 2

gP

∗ w ε,σP ∇ (σP X) C k,α (B1 ,δ) . gP

δ

Recall that by Proposition 4.1 the linear operator ∇ − ∇ is bounded uniformly in P ∈ Gε and ε, together with all of its derivatives; thus δ 2 ∗ ∗ ∗ σP (Dε X) C k,α (B1 ,δ) w ∇(σP X) C k,α (B1 ,δ) + σP X C k,α (B1 ,δ) ε,σP 2

∗ w ε,σP σP X C k+1,α (B1 ,δ) .

The cases of Type M and Type M0 are handled in the same manner, using w ε,σn ∼ 1, w ε,σ−n ∼ ε, and the uniformity of g±n in ε.

(4.10)

Theorem 4.6. Let α ∈ (0, 1). If vector field X ∈ C 0,0,ν (Mε ) is such that Lε X ∈ C k,α,ν+2 (Mε ) then X ∈ C k+2,α,ν (Mε ) with: X k+2,α,ν Lε X k,α,ν+2 + X 0,0,ν .

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Likewise, if function f ∈ C 0,0,ν (Mε ) satisfies Lε f ∈ C k,α,ν+2 (Mε ) then f ∈ C k+2,α,ν (Mε ) with f k+2,α,ν Lε f k,α,ν+2 + f 0,0,ν . Proof. Our argument relies on the scaling properties of the vector Laplacian Lε and the linearized Licherowicz operator Lε . The reader may find it instructive to compare the following proof with that of Proposition 4.5. We analyze the vector Laplacian first; it scales as follows: LσP∗ gε = L wε (P )2 g = 4 wε,σP 4

P

−2

LgP for P ∈ Gε , and

−2 ∗ g = L 2 Lg−n for 1 ≤ n ≤ N. Lσ−n ε g−n = ε ε

(4.11)

By assumption we have Lh Φ∗ X ∈ C k,α (B1 , δ) for all Φ ∈ Cε and corresponding metrics h ∈ {gP |P ∈ Gε } ∪ {g±n |1 ≤ n ≤ N }. We now apply the basic Schauder interior regularity [6] to Lh and open sets B 12 ⊂ B1 . Recall that the constant in the interior regularity estimates only depends on k,α, the (Euclidean) distance between B 21 and B1 , a lower bound on the eigenvalues of the principal symbol of Lh , and an upper bound on the C k,α -norm of the coefficients of Lh . Since our metrics h are uniformly close to the Euclidean metric δ by Proposition 4.1, the constants in the interior regularity estimates can be chosen independently of h and ε. We now have Φ∗ X ∈ C k+2,α (B 12 , δ) with Φ∗ X C k+2,α (B 1 ,δ) Lh Φ∗ X C k,α (B1 ,δ) + Φ∗ X C 0,0 (B1 ,δ) 2

independently of h, Φ and X. The scaling properties (4.10) and (4.11) further imply ∗ w ε,Φ Φ X C k+2,α (B 1 ,δ) w ε,Φ ν

2

ν+2

∗ Φ∗ Lε X C k,α (B1 ,δ) + w ε,Φ Φ X C 0,0 (B1 ,δ) ν

independently of h, Φ and X. Taking supremum over Φ yields X k+2,α,ν Lε X k,α,ν+2 + X 0,0,ν < ∞. The claimed uniform elliptic regularity follows from the (uniform) equivalence of · and · norms (Proposition 4.2). In the case of the linearized Lichnerowicz operator Lε we rely on the properties of the differential operators LΦ on B1 defined by 2

∗ LΦ = w ε,Φ Φ Lε .

It is clear from the above that it suffices to show the existence of a uniform lower bound (denoted λ) on the eigenvalues of the principal symbol of operators LΦ , and a uniform upper bound (denoted λ) on the C k,α -norm of the coefficients of LΦ ; the word “uniform” here should be interpreted to mean “independent of Φ ∈ Cε and

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ε”. We only present the argument in the case of Φ = σ−n and leave the remaining cases (Φ = σP , Φ = σn ) to an interested reader. Observe that 1 2 ∗ ∗ Lσ−n = w R(σ − g ) + h ◦ σ Δ ε,σ−n σ−n gε ε −n −n ε 8 2 w 1 2 ε,σ−n = Δg−n − R(g−n ) + w ε,σ−n hε ◦ σ−n . 2 ε 8 The existence of the uniform lower bound λ and the uniform upper bound λ follows from the independence of g−n in ε, the scaling property (4.10), and the assumption that hε k,α,2 1. 4.7. Weighted Function Spaces on (M {S}, g) In this paper, we also need function spaces which measure decay/growth of tensors in terms of the geodesic distance from S. The construction of these spaces is analogous to that used for the spaces C k,α,ν (Mε ). The weight function wM : M {S} → R we utilize here can roughly be described as the geodesic distance from S. We define wM as the unique continuous M function which is constant away from BC −1 and satisfies (wM ◦ Mσ) (x) = wM,R (|x|) M for the function wM,R of Sect. 4.1. Note that wM = wε ◦ iε on M B2Cε . Next, we introduce the special atlases CS and CS for M {S}. These contain two types of charts:

−1 ⊆ M {S} and let 1. Charts near S. Let G = Mσ x | 0 < |x| < C4 M M P = σ(xP ) ∈ G. To point P we associate the chart σ P := Mσ ◦ HP and the metric 4 M ∗ M gP = σ g, |xP |2 P

where HP : B1 → R3 is defined by HP (x) = xP + |x2P | x (see Sect. 4.2). We also consider the restriction Mσ P of Mσ P to B 12 . Note that iε ◦ Mσ P = σP for |xP | ≥ 4Cε and that (wM ◦ Mσ P ) (x) = HP (x) for all x ∈ B1 . 2. Charts away from S. To cover M (G ∪ {S}) we use the charts M σ 1 , . . . , Mσ N introduced in the Sect. 4.2. In addition, we use the restrictions M σ 1 , . . . , Mσ N of these charts to the ball B 12 , and the metrics gn = Mσ ∗n g, 1 ≤ n ≤ N discussed before. Define the atlas CS as the collection of charts Mσ P for P ∈ G and charts Mσ n , 1 ≤ n ≤ N ; the atlas CS consists of the corresponding restrictions. It is important to notice that Proposition 4.1 can easily be modified to yield a result regarding wM and CS (in place of wε and Cε ). The Hölder norms in this setting are defined as T k,α := sup Φ∗ T C k,α (B1 ,δ) , Φ∈CS

ν T k,α,ν := wM T k,α ;

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analogous expressions yield · . Using scaling properties analogous to Proposition 4.1 one can prove the norm equivalences ∗ ∗ T k,α,ν ∼ sup w M,Φ Φ T C k,α (B1 ,δ) ∼ T k,α,ν ∼ sup w M,Φ Φ T C k,α (B 1 ,δ) , ν

ν

Φ∈CS

Φ∈CS

2

where w M,Φ := wM (Φ(0)) are sample values of the weight function, as well as T k,0,ν ∼ T k,0,ν ∼ We let

k

sup

j=0 M {S}

i ...i k,α (M ; S) := Tj11...jqp Cp,q i ...i k,α,ν Cp,q (M ; S) := Tj11...jqp

g −p+q+j+ν j ∇ T |g . wM

(4.12)

T k,α < ∞}, and T k,α,ν < ∞

be the corresponding (weighted) function spaces. We have the following weighted elliptic estimates for the vector Laplacian Lg and the linearized Lichnerowicz operator Lg . The latter can schematically be represented by 1 Lg f = Δg f − R(g) f + h f 8 with h : M → R smooth. Theorem 4.7. Let α ∈ (0, 1). If vector field X ∈ C 0,0,ν (M ; S) is such that Lg X ∈ C k,α,ν+2 (M ; S) then X ∈ C k+2,α,ν (M ; S) and X k+2,α,ν Lg X k,α,ν+2 + X 0,0,ν . Likewise, if function f ∈ C 0,0,ν (M ; S) satisfies Lg f ∈ C k,α,ν+2 (M ; S) then f ∈ C k+2,α,ν (M ; S) and f k+2,α,ν Lg f k,α,ν+2 + f 0,0,ν . The proof of this theorem is analogous to the proof of Theorem 4.6: one uses the scaling properties of the vector Laplacian and the linearized Licherowicz operator together with the basic interior elliptic estimates and various norm equivalences. We omit the details, but point out that in the case of the linearized Licherowicz operator we use the fact that h ∈ C k,α,2 (M ; S). 4.8. Weighted Function Spaces on (M0 , g0 ) We also need function spaces which keep track of the decay of tensors on (M0 , g0 ) with respect to the radial function of the asymptotia. The construction of these spaces is analogous to that used for the spaces C k,α,ν (Mε ) and C k,α,ν (M ; S). The weight function w0 : M0 → R below can be viewed as a smooth extension of the radial function |0σ −1 (.)|. We define w0 to be the unique continuous function ¯C and satisfies which is constant away from 0σ R3 B

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(w0 ) ◦ (0σ) : x → w0,R (|x|) for the function w0,R of Sect. 4.1. We note that wε ◦ ιε = εw0 on M0 0Ω(εC)−1 . Next we introduce the special atlases C0 and C0 for M0 . These contain two types of charts: 1. Charts in the asymptotia. Let P = 0σ(xP ) ∈ 0Ω4C . To this point we associate the chart 0σ P := 0σ ◦ HP , where HP : B1 → R3 is defined by HP (x) = xP + |x2P | x. By construction (w0 ◦ 0σ P ) (x) = HP (x) for all x ∈ B1 . The methods used in the proof of Proposition 4.1 show that the metrics 0

gP =

4 |xP |2

σ ∗P g0 ,

0

P ∈ 0Ω4C ,

are uniformly close to the Euclidean metric δ. We also consider the restrictions σ P of 0σ P to the ball B 12 . 2. Charts away from the asymptotia. We cover M0 0Ω4C with the charts 0 σ n , 1 ≤ n ≤ N , introduced in Sect. 4.2. We also consider the restrictions 0σ n of these charts to B 21 , and the metrics g−n = 0σ ∗n g0 discussed before. 0

Define the atlas C0 to be the collection of charts 0σ P for P ∈ 0Ω4C and charts 0σ n , 1 ≤ n ≤ N . Furthermore, define the H¨ older norm T k,α,ν by T k,α := sup Φ∗ T C k,α (B1 ,δ) , Φ∈C0

T k,α,ν := w0ν T k,α .

The atlas C0 and the norm T k,α,ν are defined analogously using restrictions of Φ ∈ C0 to B 12 . One can easily extend the results of Proposition 4.1 to prove the following norm equivalences: ∗ ∗ T k,α,ν ∼ sup w 0,Φ Φ T C k,α (B1 ,δ) ∼ T k,α,ν ∼ sup w 0,Φ Φ T C k,α (B 1 ,δ) , ν

ν

Φ∈C0

Φ∈C0

T k,0,ν ∼ T k,0,ν ∼

k

2

g0 sup w0−p+q+j+ν ∇ j T |g0 ;

j=0 M0

here w 0,Φ := w0 (Φ(0)) are sample values of the weight function. Let i ...i k,α Cp,q (M0 ; ∞) := Tj11...jqp T k,α < ∞} and i ...i k,α,ν Cp,q (M0 ; ∞) := Tj11...jqp T k,α,ν < ∞ denote the corresponding (weighted) function spaces. We use these function spaces when studying the vector Laplacian Lg0 . The following theorem is analogous to Theorem 4.6. 0,0,ν k,α,ν+2 (M0 ; ∞) with Lg0 X ∈ C1,0 (M0 ; ∞) Theorem 4.8. Let α ∈ (0, 1). If X ∈ C1,0 k+2,α,ν (M0 ; ∞) and then X ∈ C1,0

X k+2,α,ν Lg0 X k,α,ν+2 + X 0,0,ν .

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To prove the theorem one uses the scaling properties of the vector Laplacian together with the basic interior elliptic estimates and various norm equivalences. The details are left to the reader. 4.9. Weighted Function Spaces on (R3 {0}, δ) It is avoidable, but highly convenient to use function spaces on (R3 {0}, δ) with weight function r : x → |x|. In a sense these function spaces blend the usefulness of C k,α,ν (M ; S) near the origin with that of C k,α,ν (M0 ; ∞) “near” ∞. The special atlas C consists of charts of the form HP : B1 → R3 {0}, HP (x) = P +

|P | x, where P ∈ R3 {0}. 2

We define the (weighted) H¨ older norm in the following manner: T k,α := sup HP∗ T C k,α (B1 ,δ) , HP ∈C

T k,α,ν := rν T k,α .

The atlas C and the norm T k,α,ν are defined analogously using the restrictions of HP ∈ C to B 21 . One can easily extend the results of Proposition 4.1 and prove the norm equivalences T k,α,ν ∼ sup |P |ν HP∗ T C k,α (B1 ,δ) ∼ T k,α,ν ∼ sup |P |ν HP∗ T C k,α (B 1 ,δ) , HP ∈C

T k,0,ν ∼ T k,0,ν ∼

k

sup

3 j=0 R {0}

HP ∈C

2

δ r−p+q+j+ν ∇j T |δ .

The corresponding weighted function spaces are i ...i k,α,ν (R3 ; 0, ∞) := Tj11...jqp T k,α,ν < ∞ . Cp,q We use these function spaces when studying the vector Laplacian Lδ and the scalar Laplacian Δδ . The following theorem is analogous to Theorem 4.6. 0,0,ν k,α,ν+2 (R3 ; 0, ∞) with Lδ X ∈ C1,0 Theorem 4.9. Let α ∈ (0, 1). If X ∈ C1,0 k+2,α,ν (R3 ; 0, ∞) then X ∈ C1,0 (R3 ; 0, ∞) and

X k+2,α,ν Lδ X k,α,ν+2 + X 0,0,ν . If function f ∈ C 0,0,ν (R3 ; 0, ∞) satisfies Δδ f ∈ C k,α,ν+2 (R3 ; 0, ∞) then f ∈ C k+2,α,ν (R3 ; 0, ∞) and f k+2,α,ν Δδ f k,α,ν+2 + f 0,0,ν . The details of the proof are left to the reader.

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5. Repairing the Momentum Constraint We start by estimating the extent to which our approximate data (Mε , gε , Kε ) fail to satisfy the momentum constraint. With the notational conventions of Sect. 3.4 this constraint reduces to divgε με = 0. To accommodate the needs of our discussion later on, we not only prove an estimate for (divgε με ) but also a more general result regarding με . Proposition 5.1. 1. We have με k,α,−1 √ 1. ε −1 2. Let Uε := wε 6 , +∞ ⊆ Mε . We have με k,α,−2;Uε 1. √ 3. Let Vε := wε−1 (0, 24 ε) ⊆ Mε . We have με k,α,0;Vε ε. 4. Finally, (divgε με ) k,α,ν ε(ν/2)−1 . Proof. We use the definition of the H¨ older norms and study the pullback of με along Φ ∈ Cε . There are three cases to consider, based on the type of Φ. We begin with the most interesting, Type G. Let Φ = σP , P = σ(xP ) ∈ Gε . To simplify our computation, we set 6 3 √ |HP (x)| − 2 , χ1 (x) := χ √ |HP (x)| − 2 and χ2 (x) := (1 − χ) ε 4 ε so that σP∗ με = εχ1 · [HP∗ εσ ∗ μ0 ] + χ2 · [HP∗ Mσ ∗ μ] . While the cut-offs χ1 , χ2 have a potential to contribute a significant amount to the derivatives of με they do not cause any actual trouble: the methods used in the proof of Proposition 4.1, along with √ |dχ1 |2 + |dχ2 |2 = 0 =⇒ |xP | ∼ ε, show that for each multi-index β the derivatives ∂ β χ1 and ∂ β χ2 are bounded on B1 independently of ε and P . We estimate the norm of the term involving μ0 = K0 by using (3.2). As HP dilates by a factor of |x2P | it follows that (for each multi-index β) 1 β ∗ε ∗ |xP ||β| 1 (5.1) ∂ (HP σ μ0 )ij |xP |2 |xP ||β|+2 independently of P . Likewise, β ∗M ∗ ∂ (HP σ μ)ij |xP |2 ∂ β ( Mσ ∗ μ)ij ◦ HP |xP ||β| |xP |2+|β| |xP |2 . (5.2) These estimates yield σP∗ με C k,α (B1 ,δ) ε + w ε,σP

2

independently of P ∈ Gε . 2

For charts Φ of Type M , we have Φ∗ με C k,α (B1 ,δ) 1 ε + w ε,Φ , while 2

for the charts of type Type M0 we have Φ∗ με C k,α (B1 ,δ) ε ε + w ε,Φ .

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The first claim of our proposition now follows from ε + w ε,Φ w ε,Φ and w ε,Φ

−1

Φ∗ με C k,α (B1 ,δ) 1 independently of Φ ∈ Cε . 2

2

The second (respectively, third) claim follows from ε+ w ε,Φ w ε,Φ (respectively, 2 −2 ∗ ∗ ε + w ε) and w Φ μ 1 (respectively, Φ με C k,α (B1 ,δ) ε,Φ ε,Φ ε C k,α (B1 ,δ) ε) which hold independently of Φ ∈ Cε;Uε (respectively, Φ ∈ Cε;Vε ). A similar method is used to estimate the size of (divgε με ) . Note that √ in order √ ∗ for Φ (divgε με ) = 0 the chart Φ needs to be of Type G with |xP | ∈ 3ε , 16 ε . It follows from gradσP∗ gε = σP∗ divgε με = ε

4 |xP

4 |xP |2 gradgP , divg μ

|2

gradgP χ1

= 0 and divg0 μ0 = 0 that

(HP∗ εσ ∗ μ0 ) +

4 |xP

|2

gradgP χ2

(HP∗ Mσ ∗ μ)

where V ω denotes the 1-form ω(·, V ). Given that the metrics gP are uniformly close to the Euclidean metric δ (see Proposition 4.1), relations (5.1), and (5.2) imply ∗ σP (divgε με ) k,α (B1 ,δ)

C

1 1 1 HP∗ εσ ∗ μ0 C k,α (B1 ,δ) + 2 HP∗ Mσ ∗ μ C k,α (B1 ,δ) ε ε ε √ √ ε independently of P with |xP | ∈ 3 , 16 ε . In conclusion, we have ν σP∗ (divg με ) k,α w ε(ν/2)−1 ε,σP ε C (B ,δ) 1

and (divgε με ) k,α,ν ε

(ν/2)−1

.

To repair the momentum constraint we perturb με so that the resulting tracefree symmetric 2-tensor is divergence-free. One classic way of doing this [8] involves solving the linear PDE Lε Xε = (divgε με ) .

(5.3)

k+2,α,ν C1,0 (Mε )

Observe that (5.3) has a solution in for each k, α, ν. Indeed Lε : H k+2 (Mε ) → H k (Mε ) (viewed as an operator between ordinary Sobolev spaces) is self-adjoint and elliptic; in particular, Im(Lε ) = Ker(Lε )⊥ with respect to the L2 -pairing. If Y ∈ Ker(Lε ) then integration by parts yields Y ∈ Ker(Dε ) and gε (divgε με ) , Y = − gε (Dε∗ με , Y ) = − gε (με , Dε Y ) = 0. Mε

Mε

Mε

It follows that (divgε με ) ∈ Im(Lε ) ⊆ H (Mε ) and that there exists a solution Xε ∈ H k+2 (Mε ) of (5.3). By the Sobolev Embedding Theorem, we see that Xε ∈ 0,0,ν k+2,α,ν C1,0 (Mε ). Elliptic regularity, Theorem 4.6, shows that Xε ∈ C1,0 (Mε ).

k

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We use Xε to make a small perturbation of με and repair the momentum constraint. Thus, it is crucial that we have a control on the size of Xε . We achieve this by proving the following uniformity property of the family of operators Lε . Proposition 5.2. Let ν ∈ 32 , 2 and let ε > 0 be sufficiently small. We have X 0,0,ν Lε X 0,0,ν+2 independently of smooth vector fields X on Mε . Proof. We assume opposite: that there exist εj ↓ 0 (j ∈ N) and vector fields Xj on Mεj such that Xj 0,0,ν = 1 and Lεj Xj 0,0,ν+2 → 0.

(5.4)

Equivalently, the first property of Xj can be written as max wε−1+ν |Xj |gε = 1. j Mεj

j

Let Pj ∈ Mεj be the points at which these maxima are reached. Consider the sequence wεj (Pj ) j∈N . One of the following holds: Case M {S}. There exists a subsequence of (Pj )j∈N , which we may without loss of generality assume is (Pj )j∈N itself, and a number cM > 0 such that wεj (Pj ) ≥ cM for all j ∈ N. 3

Case R {0}. There exists a subsequence of (Pj )j∈N , which we may without loss of generality assume is (Pj )j∈N itself, such that εj → 0. wεj (Pj ) → 0, wεj (Pj ) Case M0 . There exists a subsequence of (Pj )j∈N , which we may without loss of generality assume is (Pj )j∈N itself, and a number cM0 > 0 such that wεj (Pj ) ≤ cM0 εj for all j ∈ N. In each of the three cases, we use the sequence (Xj )j∈N to construct a non-trivial vector field on the indicated manifold. By construction the vector field is in a particular weighted H¨ older space and is in this kernel of a vector Laplacian. We obtain a contradiction by arguing that there is no such vector field. The reasoning is similar in each of the three cases. In the interest of brevity, we present only one of the cases in full detail, namely Case M {S}. Obtaining the contradiction in Case M {S}. Our first step is the construction of a vector field on M {S} with peculiar properties; we refer to this step as the Exhaustion Argument. ! 2 ⊂ D2 ⊂ · · · ⊂ M {S} be a sequence of compact subsets ! 1 ⊂ D1 ⊂ D Let D of M {S} such that ∞ " n=1

Int(Dn ) = M {S}.

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Without loss of generality we may assume that wM D! ≥ cM and that the restric1 tion of the quotient map iεj to D1 , iεj : D1 → Mεj , is an embedding with i∗εj gεj = g for all j ∈ N. The restrictions of vector fields Xj := i∗εj Xj to D1 satisfy sup |Xj |g ≥ c, !1 D

1−ν |Xj |g ≤ wM ,

−1−ν |Lg Xj |g ≤ cj · wM and

lim cj = 0, (5.5)

j→∞

where c > 0 is a constant independent of j ∈ N, and where cj := Lεj Xj 0,0,ν+2 . Now consider the interior elliptic estimate Xj H 2 (D! 1 ,g) ≤ C1 Lg Xj L2 (D1 ,g) + Xj L2 (D1 ,g) in Sobolev spaces with respect to the metric g. It follows from (5.5) and the uniform boundedness of wM = wεj ◦ iεj on D1 away from 0 that the sequences (Xj )j∈N and (Lg Xj )j∈N are bounded in L2 (D1 , g). Consequently, (Xj )j∈N is bounded as ! 1 , g). From the Rellich Lemma and the Sobolev Embedding a sequence in H 2 (D Theorem, we see that there is a subsequence of (Xj )j∈N which is convergent in ! 1 , g). We extract and relabel the subsequences (εj )j∈N , (Xj )j∈N to get C 0 (D Xj → Y1

! 1 , g). in C 0 (D

1−ν ! 1 due to (5.5). on D Note that Y1 = 0 and |Y1 |g ≤ wM We now repeat the process: eliminating finitely many terms of (εj )j∈N and (Xj )j∈N we ensure that i∗εj gεj = g on D2 . The interior elliptic estimate Xj H 2 (D! 2 ,g) ≤ C2 Lg Xj L2 (D2 ,g) + Xj L2 (D2 ,g)

implies the existence of a subsequence of (Xj )j∈N whose pullback is convergent in ! 2 , g). As above, we extract and relabel this subsequence so to have C 0 (D ! 2 , g). in C 0 (D

Xj → Y2

Since Y2 is a subsequential limit of the sequence which defines Y1 we have Y2 D! 1 = 1−ν Y1 . We also note that |Y2 |g ≤ wM and that (5.5) holds on D2 . The process described above gives rise to an iterative construction of vector ! n , g), n ∈ N such that fields Yn ∈ C 0 (D 1−ν = Yn−1 , |Yn |g ≤ wM . Yn D! n−1

Define the vector field Y on M {S} by Y D! = Yn . n

0,0,ν (M ; S), C1,0

Y 0,0,ν ≤ 1 and Y = 0. We have Y ∈ The punch-line of the Exhaustion Argument is that Lg Y = 0 on M {S}. By elliptic regularity it suffices to show that Lg Y = 0 weakly. To that end, let

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ξ be a compactly supported vector field on M {S}. Let m ∈ N be such that ! m . Since Y ! = Ym and since Lg is formally self adjoint we have supp(ξ) ⊆ D D m

g (Y, Lg ξ) dvolg = M {S} D

g (Ym , Lg ξ) dvolg m g (Xj , Lg ξ) dvolg = lim = lim j→∞ j→∞ D D m

m

g (Lg Xj , ξ) dvolg

! ≤ ξC 0 (D m ,g) volg (Dm ) · lim Lg Xj C 0 (D m ,g) . j→∞

−1−ν ! m there is a constant c(D ! m ) such that Since wM is bounded on D

! m ) for all j ∈ N. Lg Xj C 0 (D! m ,g) ≤ cj c(D In particular, we have limj→∞ Lg Xj C 0 (D! m ,g) = 0 and Lg Y = 0 on M {S}.

2,0,ν It is now important to notice that Theorem 4.7 implies Y ∈ C1,0 (M ; S). Furthermore, it follows from (4.12) that there is a constant c˜ with −ν . |∇Y |g ≤ c˜ · wM

(5.6)

We now show that the existence of the vector field Y described above is a contradiction. We start by showing that Lg Y = 0 weakly on M . Let ξ be a vector field on M and let BrM be a geodesic ball #of (small) radius r centered # at S. To understand # g(Y, L ξ) dvol we estimate g(Y, L ξ) dvol and g(Y, Lg ξ) dvolg g g g g M BrM M BrM individually. 1−ν For the first integral we take the advantage of |Y |g ≤ wM to see that for some constant c1 (ξ) (independent of r) we have g(Y, Lg ξ) dvolg ≤ c1 (ξ) · r4−ν . B M r

On the other hand, integration by parts and the fact that Lg Y = 0 on M {S} imply g(Y, Lg ξ) dvolg ≤ |Dg ξ (Y, n) + Dg Y (ξ, n) | dvolg , ∂B M M B M r

r

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where n denotes a unit normal to the geodesic sphere ∂BrM . Point-wise estimates for |Y |g and |∇Y |g (see (5.6)) yield c2 (ξ)r1−ν dvolg ≤ c3 (ξ)r3−ν ,

|Dg ξ(Y, n)| dvolg ≤ ∂BrM

∂BrM

|Dg Y (ξ, n)| dvolg ≤ c4 (ξ)r2−ν ∂BrM

for some constants c2 (ξ), c3 (ξ), c4 (ξ) independent of r. Combining all of the above we obtain g(Y, Lg ξ) dvolg ≤ c5 (ξ)r2−ν M

for some constant c5 (ξ) independent of r. Since r is arbitrary we may take the limit as r → 0; as a result we obtain g(Y, Lg ξ) dvolg = 0. M

In other words, we have that Lg Y = 0 weakly on M . By elliptic regularity Lg Y = 0 strongly and Y is smooth on all of M . Integrating by parts, we further see that 2

|Dg Y |g dvolg ,

g(Lg Y, Y ) dvolg =

0= M

M

i.e. that Dg Y = 0 on M . This is a contradiction since Y = 0 and there are no non-trivial conformal Killing vector fields on M . Obtaining the contradiction in Case R3 {0}. In this case, we have Pj ∈ Gεj for all but finitely many j ∈ N. To be able to employ the Exhaustion Argument we need to do some re-scaling. More precisely, we blow up the gluing region Gεj by a factor of wj := wεj (Pj ) and re-scale the metrics and vector fields correspondingly. Consider the dilation Hj : x → wj · x of R3 and define $ % εj C −1 Ωj := (σ ◦ Hj )−1 Gεj = x ∈ R3 4C < |x| < . wj 4wj This choice is motivated by the fact that the points Qj ∈ Ωj with Pj = σ ◦ Hj (Qj ) ε satisfy |Qj | = 1. Since wj → 0 and wjj → 0 as j → ∞ each compact subset D ⊆ R3 {0} is contained in all but finitely many Ωj . Next consider the metrics 1 gjΩ := 2 · (σ ◦ Hj )∗ gεj wj

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on Ωj . The methods used in the proof of Proposition 4.1 (see (4.3) for details) show that for each compact subset D ⊆ R3 {0} there exists a constant c(D) such that εj + wj2 |gjΩ − δ| ≤ c(D) wj for all but finitely many j. It follows from wj → 0 and wjj → 0 that gjΩ converges to the Euclidean metric δ uniformly on D as j → ∞. A similar line of reasoning shows that gjΩ converges to δ in the C k (D, δ)-norm for any compact subset D ⊆ R3 {0}. ∗ Define Xj := wjν (σ ◦ Hj ) Xj . We claim there exists a sequence (cj )j∈N which converges to zero and a constant c > 0 with the following properties. ε

• The supremum over the unit sphere S 2 ⊆ R3 {0} satisfies supS 2 |Xj |δ ≥ c for all j ∈ N. • For each compact subset D ⊆ R3 {0} there is j0 ∈ N such that for all j ≥ j0 and all Q ∈ D we have |Xj (Q)|δ ≤ 1c |Q|1−ν . • For each compact subset D ⊆ R3 {0} there exist j0 ∈ N and a constant c(D) such that for all j ≥ j0 we have |LgjΩ Xj |gjΩ ≤ cj · c(D) on D. These properties are consequences of the normalization Qj ∈ S 2 for all j ∈ N, the scaling identities |Xj |gjΩ = wjν−1 |Xj |gεj ◦ Hj ,

wεj ◦ σ ◦ Hj (Q) = wj |Q|,

LgjΩ = wj2 L(σ◦Hj )∗ gεj

and the convergence gjΩ → δ on compact subsets. To illustrate the proofs of these properties we verify the last inequality. Let cj := Lεj Xj 0,0,ν+2 ; by assumption cj → 0 as j → ∞. Further, let D ⊆ R3 {0} be a compact set and let Q ∈ D. Since gjΩ → δ on D there is j0 ∈ N such that Lεj Xj 0,0,ν+2 = cj |Q|−1−ν LgjΩ Xj (Q) Ω = wj1+ν Lεj Xj (Q)gε ≤ wj1+ν · j gj (σ ◦ Hj )∗ wε1+ν (Q) j −1−ν

for all j ≥ j0 . The constant c(D) can be chosen to be (inf Q∈D |Q|) . We now apply the Exhaustion Argument to the vector fields Xj and obtain a vector field Y = 0 on R3 {0} such that 0,0,ν (R3 ; 0, ∞) and Lδ Y = 0. Y ∈ C1,0

(5.7)

The Exhaustion Argument here is essentially identical to the one presented in Case M {S}, except in the integration by parts step which proves that Lδ Y = 0 weakly on R3 {0}. The key difference in the step is that, for a given test vector ! m ⊇ supp(ξ), we use g Ω − δ 2 ! field ξ and compact set D j C (Dm ,δ) → 0 and its consequence = 0. (5.8) lim LgjΩ ξ − Lδ ξ j→∞

! m ,δ) C 0 (D

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For clarity, we do the integration explicitly here: δ (Y, Lδ ξ) dvolδ = δ (Ym , Lδ ξ) dvolδ R3 {0} D! m Ω = lim gj Xj , LgjΩ ξ dvolgjΩ j→∞ D! m Ω = lim gj LgjΩ Xj , ξ dvolgjΩ j→∞ D! m !m) ≤ lim ξ C 0 (D! m ,gΩ ) LgjΩ Xj C 0 (D! m ,gΩ ) volgjΩ (D j j j→∞ ! m ) · lim cj c(D ! m ) = 0. ≤ ξ C 0 (D! m ,δ) volδ (D j→∞

The next step is to show that the existence of Y described above is a contradiction. We have discovered two ways to do this. One way is to use spherical coordinates and spherical harmonics to explicitly compute the kernel of Lδ ; this approach is addressed in the Appendix B. The approach we take here is to show that Y is a conformal Killing vector field on (R3 , δ) which decays at ∞; the explicit knowledge of all the Euclidean conformal Killing vector fields shows that the existence of such a Y is impossible. We first observe that Lδ Y = 0 weakly (and hence strongly) on R3 . Indeed, 2,0,ν (R3 ; 0, ∞) and, in particular, Theorem 4.9 shows that Y ∈ C1,0 |Y | ≤ c1 r1−ν ,

|∇Y | ≤ c1 r−ν

(5.9)

for some constant c1 > 0. These estimates ensure that the integration-by-parts argument of Case M {S} carries over with no changes. Next let Bρ (resp. Sρ2 ) denote the Euclidean ball (resp. sphere) of radius ρ centered at the origin. Consider the integration by parts formula 0=

Dδ Y (Y, n) dvolδ +

δ (Lδ Y, Y ) dvolδ = Bρ

Sρ2

2

|Dδ Y | dvolδ

Bρ

in which Dδ denotes the Euclidean conformal Killing operator and in which n denotes the appropriately oriented unit normal to Sρ2 . It follows from (5.9) that Dδ Y (Y, n) dvolδ ≤ c2 ρ3−2ν Sρ2

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for some constant c2 . By assumption ν >

3 2

2

|Dδ Y | dvolδ = − lim

ρ→+∞ Sρ2

R3

1471

and so Dδ Y (Y, n) dvolδ = 0.

We conclude that Dδ Y = 0 on R3 . In other words, we have that Y ∈ 2,0,ν (R3 ; 0, ∞) is a non-zero conformal Killing vector field on R3 which decays C1,0 at ∞. To see that this is a contradiction one can appeal to a generalization of a theorem of Christodoulou [8, Prop. 13] or simply recall that the space of conformal vector fields ei , Killing vector fields on R3 is spanned by coordinate (translation) i i j e − x e , dilation vector field x e , and the special vecrotation vector fields x j i i i i 2 x ei − (x ) ej , none of which decay at ∞. This completes tor fields 2xj the proof in Case R3 {0}. Obtaining the contradiction in Case M0 . To apply the Exhaustion Argument we need to re-scale M0 “back to its original size”. We consider the regions Ωj := M0 0Ω(2√ε )−1 , j which ιεj map diffeomorphically3 into Mεj . Since εj → 0 each compact subset D ⊆ M0 is contained in all but finitely many Ωj . If necessary, we eliminate finitely many terms of (εj )j∈N so that Pj ∈ ιεj (Ωj ) and ι∗εj gεj = ε2j g0 for all j. Let Qj ∈ M0 be such that Pj = ιεj (Qj ). Finally, consider the vector fields ∗ Xj := ενj ιεj Xj on Ωj . Using wεj ◦ ιεj = εj w0 on the domain of ιεj one can easily verify the following properties of the vector fields Xj : 1. There is a constant c1 independent of j such that |Xj |g0 ≤ c1 point-wise on Ωj , 2. |Xj |g0 ≤ w01−ν point-wise on the “asymptotic” region Ωj ∩ 0Ω2C , 3. |Xj (Qj )|g0 ≥ c1−ν M0 and 4. There is a constant c2 independent of j with |Lg0 Xj |g0 ≤ c2 Lεj Xj 0,0,ν+2 point-wise on Ωj . To illustrate the proofs of these properties, we verify the second inequality: 1−ν ∗ ∗ ν−1 w ≤ w01−ν Xj 0,0,ν = w01−ν . |Xj |g0 = εν−1 ι |X | = w ι |X | j gε j gεj εj εj εj 0 j j

√ −1 Since our proof here does not rely on ι∗εj μεj = εj μ0 the choice of 2 εj may appear unmotivated. However, the blow-up argument we use later on to prove Proposition 6.2 does rely on √ −1 throughout. ι∗εj μεj = εj μ0 . For the purposes of unified exposition we use 2 εj

3

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Applying the Exhaustion Argument to the vector fields (Xj )j∈N we obtain a non0,0,ν zero vector field Y ∈ C1,0 (M0 ; ∞) for which Lg0 Y = 0. It follows from Theo2,0,ν rem 4.8 that Y ∈ C1,0 (M0 ; ∞) and |Y |g0 ≤ c˜ w01−ν , |∇Y |g0 ≤ c˜ w0−ν

(5.10)

for some constant c˜. With these estimates in hand we apply the integrationby-parts argument used in Case R3 {0} to show that Y is a conformal Killing vector field on (M0 , g0 ). In addition, we see from (5.10) that Y = 0 must “decay at infinity”. This situation is impossible according to a generalization of a theorem of Christodoulou (see [8, Prop. 13]). Thus, we have reached our final contradiction! The following important consequence of Proposition 5.2 is immediate from Theorem 4.6. Theorem 5.3. Let α ∈ (0, 1) and ν ∈ 32 , 2 . If ε > 0 is sufficiently small, then X k+2,α,ν Lε X k,α,ν+2 independently of smooth vector fields X on Mε . Recall that our strategy for repairing the momentum constraint involves solving the equation (5.3), for which there always exists a solution Xε . Proposition 5.1 and the previous theorem now provide a weighted H¨ older estimate for Xε . More precisely, if ν ∈ 32 , 2 and ε is small then Xε k+2,α,ν εν/2 .

(5.11)

Consider the perturbation μ ε of με defined by μ ε := με + Dε Xε ,

where

Lε Xε = (divgε με ) .

Since Dε maps into the subspace of symmetric and trace-free 2-tensors, the tensor μ ε is itself symmetric and trace-free with respect to gε . Furthermore, the choice ε is also divergence-free: of Xε ensures that μ ε ) = (divgε με ) − Dε∗ Dε Xε = Lε Xε − Lε Xε = 0. (divgε μ

ε + τ3 gε ) satisfies the momentum constraint. The Thus, the pair of tensors (gε , μ ε is indeed a small following proposition compares μ ε with με and shows that μ perturbation of με . Proposition 5.4. If ν ∈ 32 , 2 then: 1. μ − με k,α,ν−2 εν/2 . ε 2 2 με |gε − |με |gε 2. | εν/2 . k,α,ν+1 √ 2 2 με |gε − |με |gε 3. | εν/2 , where Uε := wε−1 6ε , +∞ ⊆ Mε . k,α,ν;Uε

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Proof. The claim regarding the size of μ ε − με = Dε Xε follows immediately from (5.11) and Proposition 4.5. The second estimate is a consequence of Proposition 4.4, the earlier estimate for Dε Xε , and the first claim of Proposition 5.1. Indeed 2 2 με |gε − |με |gε | k,α,ν+1

gε (Dε Xε , με + Dε Xε ) k,α,ν+1 Dε Xε ⊗ (με + Dε Xε ) k,α,ν−3 Dε Xε k,α,ν−2 με k,α,−1 + Dε Xε k,α,−1 εν/2 1 + ε1−ν Dε Xε k,α,ν−2 εν/2 1 + ε1−ν/2 εν/2 ; note that the above relies on the fact that −1 < ν − 2 and 1 − estimate is proved similarly using

ν 2

> 0. The third

Dε Xε k,α,−2;Uε ε−ν/2 Dε Xε k,α,ν−2;Uε 1

and the second claim of Proposition 5.1.

With little additional work one can prove the following estimates for | με |2gε itself; these estimates are crucial in our analysis of the Lichnerowicz equation. Proposition 5.5. Let ν ∈ 32 , 2 . 2 με |gε 1. We have | 1. √ k,α,2 2 με |gε 2. If Uε := wε−1 6ε , +∞ ⊆ Mε , then | 1. k,α,0;Uε √ 2 με |gε 3. If Vε := wε−1 (0, 24 ε) ⊆ Mε , then | εν−1 . k,α,ν+1;V ε √ √ 2 με |gε 4. If Wε := wε−1 6ε , 24 ε ⊆ Mε , then | εν/2 . k,α,ν;Wε

Proof. The proofs of all four claims are simple manipulations of estimates in Propositions 5.1 and 5.4, using properties of Proposition 4.4. To illustrate the arguments we prove the last claim of our proposition. 2 2 2 2 με |gε με |gε − |με |gε ε(ν−4)/2 |με |gε + | | k,α,ν;Wε

k,α,4;Wε

ε

(ν−4)/2

με ⊗ με k,α,0;Wε + ε

k,α,ν;Wε

ν/2

ε

(ν−4)/2

· ε2 + εν/2 εν/2 .

The remaining proofs are left to the reader.

We conclude this section with an interpretation of με − με k,α,ν−2 εν/2 in terms of the ordinary and the scaled point-particle limit properties. Consider a compact subset K ⊆ M {S} and the corresponding embedding iε : K → Mε . Since 1 wε on iε (K) the norm equivalence (4.7) yields (iε )∗ μ ε − μ C k (K,g) = O(εν/2 )

as ε → 0.

(5.12)

On the other hand, consider a compact set K ⊆ M0 and the corresponding embedding ιε : K → Mε . Note that wε ∼ ε on ιε (K). Proposition 5.4 and the norm

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equivalence (4.7) imply gε ε(2+k)+(ν−2) ∇ k ( με − με )

gε

g0 = εν−2 ∇ k ( με − με )

∗ Since (ιε ) τ3 gε C k (K,g ) = τ ε2 we have 0 1 τ 1−ν/2 (ιε )∗ μ g + = O ε − K ε ε 0 ε k 3 C (K,g0 )


g0

εν/2 .

as ε → 0.

(5.13)

ε + τ3 gε satisfy The limits (5.12) and (5.13) show that the approximate data gε , μ the ordinary and the scaled point-particle limit properties.

6. The Lichnerowicz Equation

While the data gε , μ ε + τ3 gε satisfy the momentum constraint, they need not satisfy the Hamiltonian constraint. To address the situation we apply the conformal method; in other words, we make a suitable conformal change of the data which repairs the Hamiltonian constraint and yet preserves the momentum constraint. One such change is τ τ ε + gε → φ−2 ε + φ4ε gε , (6.1) gε → φ4ε gε , μ ε μ 3 3 where φε is a positive solution of the Lichnerowicz equation (1.3). In light of (5.12) and (5.13) we see that the ordinary and the scaled point-particle limit properties hold for the resulting data provided the solution of the Lichnerowicz equation satisfies φε ≈ 1 (in some sense of the word). To avoid notational confusion we let φ0 be the constant function φ0 ≡ 1. Most of the work in this section is dedicated to establishing the existence of a solution φε of (1.3) such that φε − φ0 satisfies desirable Hölder estimates. The first step in this analysis is to understand the extent to which the function φ0 fails to be a solution of (1.3). More specifically, we need H¨ older estimates of Nε (φ0 ) where Nε denotes the (non-linear) Lichnerowicz operator Λ τ2 1 1 2 −7 με |gε φ + − Nε (φ) := Δgε φ − R(gε )φ + | φ5 . 8 8 4 12 Proposition 6.1. If ν ∈ 32 , 2 then Nε (φ0 ) k,α,ν+1 εν/2 . Proof. We show that w ε,Φ

ν+1

Φ∗ Nε (φ0 ) C k,α (B1 ,δ) εν/2

(6.2)

independently of Φ ∈ Cε . We distinguish three cases, based on whether a suitable pullback of (gε , με ) matches with (g, μ), with (ε2 g0 , εμ0 ), or with neither. √ The case of Im(wε ◦ Φ) ⊆ [8 ε, ∞). Here we use the fact that (g, K) satisfies the Hamiltonian constraint with cosmological constant Λ. Since μ is trace-free

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we have |K|2g =

τ2 3

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+ |μ|2g and

2 R(g) − |μ|2g + τ 2 − 2Λ = 0. 3 Consequently, the expression for Φ∗ Nε (1) simplifies to: 1 2 1 2 2 με |gε − |με |gε Φ∗ Nε (φ0 ) = − R(Φ∗ gε ) − |Φ∗ με |2Φ∗ gε + τ 2 − 2Λ + Φ∗ | 8 3 8 1 ∗ 2 2 με |gε − |με |gε . = Φ | 8 The claim (6.2) is now immediate from 5.4. Proposition The case of Im(wε ◦ Φ) ⊆

∗

0,

2

√ ε 2

. We first simplify the expression for

Φ Nε (1) using the fact that (ε g0 , εK0 ) satisfies the Hamiltonian constraint with no cosmological constant: Λ τ2 1 2 2 με |gε − |με |gε + − Φ∗ Nε (φ0 ) = Φ∗ | . 8 4 12 √ By assumption we have wε ◦ Φ ε and therefore ν+1 ν Λ τ2 ν+1 w ε 2 ε2 . (6.3) ε,Φ 4 − 12 k,α C

(B1 ,δ)

Once again, the claim (6.2) follows Proposition 5.4. √ from √ ε The case of Im(wε ◦Φ)∩ 2 , 8 ε = ∅. More specifically, we have Φ = σP √ √ for some P = σ(xP ) with 3ε < |xP | < 16 ε. Our strategy is to estimate each individual term of Φ∗ Nε (φ0 ). Recall from the proof of Proposition 4.1 that ε + |xP |2 . gP − δ C k+3 |xP | √ This estimate implies R(gP ) C k,α (B1 ,δ) ε and w ε,Φ

ν+1

Φ∗ R(gε ) C k,α (B1 ,δ) = |xP |ν+1

4

R(gP ) C k,α (B1 ,δ) εν/2 |xP |2 √ √ independently of P such that 3ε < |xP | < 16 ε. To estimate | μ |2gε we use the √ ε √ last claim of Proposition 5.5 which, by virtue of ImΦ ⊆ wε−1 6ε , 24 ε , implies w ε,Φ

ν+1

Φ∗ | με |2gε C k,α (B1 ,δ) εν/2 .

Since (6.3) continues to hold in this case, our proof is complete.

The main ingredient in our study of the Lichnerowicz equation is the uniform invertibility of the linearizations Λ τ2 1 7 2 με |gε + 5 − Lε := Δgε − R(gε ) − | 8 8 4 12

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of Nε at φ0 . To match the notation here with that used in Sect. 4.6 we let Λ τ2 7 με |2gε + 5 − hε := − | . 8 4 12 of the form hε k,α,2 1. It was claimed in Sect. 4.6 that hε satisfy an estimate Λ τ2 This fact is a consequence of Proposition 5.5 and 4 − 12 1. k,α,2

Our approach to proving the uniform invertibility of Lε is analogous to the approach taken in our analysis of the vector Laplacian. As the proofs of the main technical propositions, Proposition 5.2 and Proposition 6.2, share many common features we avoid repetition whenever possible and refer the reader to the proof of Proposition 5.2. Proposition 6.2. Let ν ∈ 32 , 2 and let ε > 0 be sufficiently small. We have φ 0,0,ν−1 Lε φ 0,0,ν+1 independently of smooth functions φ on Mε . Proof. We assume opposite: that there are εj ↓ 0 (j ∈ N) and (smooth) functions φj on Mεj with φj 0,0,ν−1 = 1 and Lεj φj 0,0,ν+1 → 0. Let Pj ∈ Mεj be the points at which wεj (Pj )−1+ν |φj (Pj )| = 1. Depending on the nature of the sequence wεj (Pj ) j∈N we distinguish three cases: Case M {S}, Case R3 {0} and Case M0 ; this is done exactly as in the proof of Proposition 5.2. Obtaining the contradiction in Case M {S}. The strategy in this case is to use the sequence (φj )j∈N to construct a non-zero function ψ : M → R which is in the kernel of Λ τ2 7 1 Lg := Δg − R(g) − |μ|2g + 5 − ; 8 8 4 12 the existence of such a function contradicts the Injectivity assumption on (M, g, K). −1 ([cM , +∞)). By Let D ⊆ M {S} be a compact subset which contains wM eliminating finitely many φj from consideration we may assume that the functions ψj : D → R,

ψj := φj ◦ iεj

are well-defined and satisfy 1−ν sup |ψj | ≥ c1 and |ψj | ≤ wM D

(6.4)

for some constant c1 > 0 independent of D and j. Since wM D is bounded away from zero the sequence (ψj )j∈N is bounded in C 0 (D, g) and i∗εj Lεj φj C 0 (D,g) → 0.

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Let j be large enough so that on D we have i∗εj gεj = g, i∗εj μεj = μ. One easily computes 7 |ψ | 2 ∗ j 2 μ ◦ i − |μ| iεj Lεj φj − Lg ψj = εj g εj g . εj 8 Proposition 5.4 implies ∗ iεj Lεj φj − Lg ψj

ν/2

C 0 (D,g)

≤ c2 (D)εj , j 1

for some constant c2 (D) which potentially depends on D. It now follows that lim Lg ψj C 0 (D,g) = 0.

j→+∞

(6.5)

The properties of (6.4) and (6.5) allow us to apply the Exhaustion Argument (see the proof of Proposition 5.2) to functions ψj . We conclude that there exists a non-zero function ψ ∈ C 0,0,ν−1 (M ; S) such that Lg ψ = 0 We see from Theorem 4.7 that ψ ∈ C

on M {S}. 2,0,ν−1

(M ; S) and that, by (4.12),

−ν |∇ψ|g ≤ c3 · wM

for some constant c3 . This knowledge of the “blow-up” rate of ψ at S allows us to show that ψ satisfies Lg ψ = 0 weakly on M . The computation is analogous to that in the proof of Proposition 5.2. The conclusion is that ψ is a non-zero smooth function on M which is in the kernel of Lg . On the other hand, we see from the Hamiltonian constraint that R(g)−|μ|2g + 2 2 3 τ − 2Λ = 0 and 1 Lg = Δg − |μ|2g + τ 2 − Λ = Δg − |K|2g − Λ . 3 It follows from the Injectivity assumption that Lg has trivial kernel. We have reached a contradiction which completes the argument in Case M {S}. Obtaining the contradiction in Case R3 {0}. The strategy in this case is to use the sequence (φj )j∈N to construct a non-zero harmonic function ψ : R3 → R which is in C 0,0,ν−1 (R3 ; 0, ∞). The existence of such a function is, by the Maximum Principle, a contradiction. Adopt the notation used in the proof of Proposition 5.2, Case R3 {0}. Define ψj : Ωj → R ψj := wjν−1 (φj ◦ σ ◦ Hj ) . The scaling identity wεj ◦ σ ◦ Hj (Q) = wj |Q| implies the following properties of the functions ψj : • The supremum over the unit sphere S 2 ⊆ R3 {0} satisfies supS 2 |ψj | ≥ 1. • If Q ∈ Ωj then |ψj (Q)| ≤ |Q|1−ν .

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Consider the operators

2 Λ τ2 1 7 2 − Lj := ΔgjΩ − R(gjΩ ) − wj2 μ ; εj g ◦ σ ◦ Hj + 5 wj εj 8 8 4 12 these operators are of interest since Lj ψj = wjν+1 (Lεj φj ) ◦ σ ◦ Hj

for all j ∈ N. Using our assumption on Lεj φj one can easily show that lim Lj ψj C 0 (D,gjΩ ) = 0

j→+∞

on each compact subset D ⊆ R3 {0}. Notice that, in some sense, the sequence of operators (Lj )j∈N itself converges. More precisely, we claim that if η is a test function on R3 {0} then lim Lj η − Δδ η C 0 (R3 ,δ) = 0.

(6.6)

j→∞

Before we prove (6.6) we point out that this identity plays the same role in the overall proof of Proposition 6.2 as identity (5.8) plays in the proof of Proposition 5.2. To prove (6.6) let η be a test function on R3 {0}. We compute: 1 |Lj η − Δδ η| ≤ |ΔgjΩ η − Δδ η| + R(gjΩ )η 8 7 2 2 τ2 2 Λ + wj μ − ◦ σ ◦ H |η| + 5 w η . εj g j j εj 8 4 12 Since gjΩ → δ on supp(η) uniformly with all the derivatives, we have that both ΔgjΩ η − Δδ η C 0 (R3 ,δ) and R(gjΩ )η C 0 (R3 ,δ) converge to 0 as j → ∞. The second and the third estimate of Proposition 5.5, together with (4.9), imply that for some constant c˜ > 0 and all j ∈ N at least one of the following two inequalities holds: 2 |μ ˜, εj |gε ≤ c

2 wεν+1 |μ ˜ εν−1 . εj |gε ≤ c j j

j

j

Therefore if Q ∈ supp(η) then wεj ◦ σ ◦ Hj (Q) = wj |Q| and at least one of the estimates 2 2 ˜(η), wjν+1 |μ ˜(η) εν−1 , |μ εj |gε ◦ σ ◦ Hj (Q) ≤ c εj |gε ◦ σ ◦ Hj (Q) ≤ c j j

j

where c˜(η) is a constant depending only on η. We can re-write these estimates jointly as ν−1 & 2 εj 2 2 wj μ |η| · max wj , . ˜(η) · max εj gε ◦ σ ◦ Hj |η| ≤ c 3 j wj R Since both wj , wjj → 0 in Case R3 {0}, we conclude that 2 2 lim wj μ = 0. εj g ◦ σ ◦ Hj η ε

j→∞

εj

C 0 (R3 ,δ)

The convergence (6.6) is now immediate from limj→∞ wj2 Λ4 −

η

τ2 12

C 0 (R3 ,δ)

= 0.

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We now apply the Exhaustion Argument to (ψj )j∈N ; we get a non-zero function ψ ∈ C 0,0,ν−1 (R3 ; 0, ∞) such that Δδ ψ = 0 on R3 {0}. Note that by Theorem 4.9 we in fact have ψ ∈ C 2,0,ν−1 (R3 ; 0, ∞). Consequently there is a constant c such that |ψ| < c r1−ν ,

δ

|∇ψ| < c r−ν

on R3 {0}. This control on the “blow-up” of ψ at the origin allows us to use an integration-by-parts argument to show that Δδ ψ = 0 weakly on R3 . We omit the integration details and refer the reader to the corresponding part of the proof of Proposition 5.2. The overall conclusion is that ψ is a non-zero harmonic function which, by virtue of ψ ∈ C 0,0,ν−1 (R3 ; 0, ∞), decays at ∞. This is a contradiction to the Maximum Principle. Our proof in Case R3 {0} is now complete. Obtaining the contradiction in Case M0 . The strategy in this case is to construct a non-zero function ψ ∈ C 0,0,ν−1 (M0 ; ∞) which is in the kernel of the operator Δg0 − |μ0 |2g0 on M0 ; the existence of such a function is a contradiction to the Maximum Principle. Adopt the notation used in the proof of Proposition 5.2, Case M0 . Define the functions φj ◦ ιεj ψj := εν−1 ψj : Ωj → R, j and the operator Lj := Δg0

Λ τ2 1 7 2 2 2 − − R(g0 ) − εj μ + 5εj . εj g ◦ ιεj εj 8 8 4 12

One easily verifies the following properties of the sequence (ψj )j∈N . • • • •

There is a constant c1 independent of j such that |ψj | ≤ c1 point-wise on Ωj , |ψj | ≤ w01−ν point-wise on the “asymptotic” region Ωj ∩ 0Ω2C , |ψj (Qj )| ≥ c1−ν M0 and There is a constant c2 independent of j such that |Lj ψj | ≤ c2 Lεj φj 0,0,ν+1 on Ωj . We now establish a property of the sequence of operators (Lj )j∈N which is analogous to (6.6). Let η be a test function on M0 , and let 1 7 2 L0 := Δg0 − R(g0 ) − |μ0 |g0 . 8 8 To compare the actions of Lj and L0 on η we use Proposition 5.4, from which we see that 2 −1−ν/2 μ ◦ ι μ | − |ε εj g εj j 0 ε2 g0 ≤ c3 εj εj j for some constant c3 . This estimate implies 2 2 2 εj μ − |μ0 |g0 → 0 εj g ◦ ιεj εj

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uniformly on supp(η). The convergence lim Lj η − L0 η C 0 (M0 ,g0 ) = 0

j→∞

is now immediate. The Exhaustion Argument applied to (ψj )j∈N yields a non-zero function ψ in C 0,0,ν−1 (M0 ; ∞) such that L0 ψ = 0. To understand the operator L0 better we use the Hamiltonian constraint R(g0 ) − |μ0 |2g0 = 0 and re-write L0 as L0 = Δg0 − |μ0 |2g0 . Since the Maximum Principle applies to such operators, there are no non-trivial functions in the kernel of L0 which decay at infinity. The existence of ψ ∈ C 0,0,ν−1 (M0 ; ∞) is therefore a contradiction. Our proof is now complete. 3 For α ∈ (0, 1), ν ∈ 2 , 2 and smooth φ we now have φ k+2,α,ν−1 Lε φ k,α,ν+1 by virtues of Proposition 6.2 and Theorem 4.6. Hence Lε : C k+2,α,ν−1 (Mε ) → C k,α,ν+1 (Mε ) is injective. In fact, we have a stronger result. Theorem 6.3. Let α ∈ (0, 1), ν ∈ 32 , 2 and let ε be sufficiently small. The linearized Lichnerowicz operator Lε : C k+2,α,ν−1 (Mε ) → C k,α,ν+1 (Mε ) is invertible, and the norm of its inverse is bounded uniformly in ε. Proof. It remains to verify that Lε : C k+2,α,ν−1 (Mε ) → C k,α,ν+1 (Mε ) is surjective. As a self-adjoint elliptic operator on Sobolev spaces, Lε : H k+2 (Mε ) → H k (Mε ) is Fredholm of index zero. The kernel of this operator consists of smooth functions and is therefore the same as the kernel of the operator Lε : C k+2,α,ν−1 (Mε ) → C k,α,ν+1 (Mε ), which is trivial. Thus Lε acting between Sobolev spaces is injective and, by Fredholm theory, surjective. It now follows from C k,α,ν+1 (Mε ) ⊆ H k (Mε ), H k+2 (Mε ) ⊆ C 0,0,ν−1 (Mε ), and elliptic regularity (Theorem 4.6) that Lε : C k+2,α,ν−1 (Mε ) → C k,α,ν+1 (Mε ) is also surjective. We solve the Lichnerowicz equation (1.3) by interpreting it as a fixed point problem. A formal computation shows that the difference ηε := φε − φ0 between a solution φε of the Lichnerowicz equation and the constant function φ0 ≡ 1 satisfies ηε = −(Lε )−1 (Nε (φ0 ) + Qε (ηε )) , where Qε (η) := Nε (φ0 + η) − Nε (φ0 ) − Lε η Λ τ2 1 2 −7 με | (1 + η) − 1 + 7η + − (1 + η)5 − 1 − 5η = | 8 4 12

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is a “quadratic error term”. Our strategy now is to show that the map Pε : η → −(Lε )−1 (Nε (φ0 ) + Qε (η)) is a contraction mapping from a small ball in C this approach we need some estimates for Qε .

k,α,ν−1

(6.7)

(Mε ) to itself. To execute

Proposition 6.4. For a given c > 0 and sufficiently small ε there exists c > 0 independent of ε such that for all η1 , η2 ∈ C k,α,ν−1 (Mε ) with η1 k,α,ν−1 , η2 k,α,ν−1 ≤ c εν/2 we also have Qε (η1 ) − Qε (η2 ) k,α,ν+1 ≤ c εν/2 η1 − η2 k,α,ν−1 . Proof. We start by proving some algebraic results. First note that our assumption on the weighted H¨ older norms of η1 , η2 implies η1 k,α , η2 k,α ≤ c˜ ε1−ν/2

(6.8)

for some constant c˜ > 1 independent of ε. By increasing the value of c˜ if necessary (and by Proposition 4.4) we get b c˜ a+b−1 η1 k,α,ν−1 η1 a−1 if a ≥ 1, a b k,α η2 k,α η1 η2 k,α,ν−1 ≤ b−1 c˜ a+b−1 η2 k,α,ν−1 η1 ak,α η2 k,α if b ≥ 1; that is,

a+b−1 η1a η2b k,α,ν−1 ≤ c εν/2 c˜2 ε1−ν/2

for all a, b ∈ N ∪ {0} with a + b ≥ 1. In light of η1j − η2j = (η1 − η2 ) η1j−1 + η1j−2 η2 + · · · + η2j−1 for j ∈ N and 2(ν − 1) < ν + 1 we further obtain j j η1 − η2 k,α,ν−1 · η1 − η2 k,α,ν+1

η1a η2b k,α,ν−1

a+b=j−1

jεν/2 c˜2 ε1−ν/2

j−2

η1 − η2 k,α,ν−1

(6.9)

independently of j ∈ N. For the rest of the proof assume ε is small enough so that c˜2 ε1−ν/2 < 12 . Under this assumption estimates (6.8) imply Im(η1 ), Im(η2 ) ⊆ (− 12 , 12 ). Thus we may expand the algebraic terms in Qε (η1 ) and Qε (η2 ) into binomial series. Together with (6.9) this process results in ∞ aj · η1j − η2j k,α,ν+1 Qε (η1 ) − Qε (η2 ) k,α,ν+1 ≤ j=2

εν/2

∞ jaj η1 − η2 k,α,ν−1 , j−2 2 j=2

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where aj are some positive numbers (absolute values of linear combinations of jaj binomial coefficients). Our claim now follows from the convergence of 2j−2 . We now prove the map (6.7) is a contraction of a small ball in C k,α,ν−1 (Mε ) to itself. Proposition 6.5. Let α ∈ (0, 1) and ν ∈ 32 , 2 . For sufficiently large c and sufficiently small ε, the map Pε is a contraction of the closed ball of radius c εν/2 around 0 in C k,α,ν−1 (Mε ). Proof. Let η k,α,ν−1 ≤ c εν/2 for some c which is determined below. Proposition 6.4 and the fact that Qε (0) = 0 show that Qε (η) k,α,ν+1 ≤ c εν for some c . We now use Proposition 6.1 to see that, for sufficiently small ε, there is a constant c˜ independent of c and c such that Nε (φ0 ) + Qε (η) k,α,ν+1 ≤ c˜ εν/2 . This means that for sufficiently small ε we have Pε (η) k,α,ν−1 = (Lε )−1 (Nε (φ0 ) + Qε (η)) k,α,ν−1 ≤ c˜ lεν/2 , where l is the (uniform) upper bound on the inverses of linearized Lichnerowicz operators discussed in Theorem 6.3. By choosing c ≥ c˜ l we ensure that ¯c εν/2 → B ¯c εν/2 . Pε : B ¯c εν/2 , assume ε To examine if this restriction of Pε is a contraction let η1 , η2 ∈ B is small and compute Pε (η1 ) − Pε (η2 ) k,α,ν−1 = (Lε )−1 (Qε (η1 ) − Qε (η2 )) k,α,ν−1 ≤ c l εν/2 η1 − η2 k,α,ν−1 ; the constant c used here is the one discussed in Proposition 6.4. The last estimate ¯c εν/2 → B ¯c εν/2 is a contracimplies that, for ε small enough, the map Pε : B tion. ¯c εν/2 → B ¯c εν/2 we Applying the Banach Fixed Point Theorem to Pε : B obtain our main result regarding the Lichnerowicz equation (1.3). Proposition 6.6. Let α ∈ (0, 1) and ν ∈ 32 , 2 . If ε is sufficiently small, there exists a function φε on Mε which solves the Lichnerowicz equation Λ τ2 1 1 με |2gε φ−7 − Δgε φε − R(gε )φε + | + φ5ε = 0. ε 8 8 4 12 The function φε is a small perturbation of the constant function φ0 ≡ 1; that is φε − φ0 k,α,ν−1 εν/2 .

(6.10)

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The outcome of our work is (the family of) data φ4ε gε , φ−2 ε + τ3 φ4ε gε on ε μ Mε which by construction satisfy the momentum and the Hamiltonian constraints with the cosmological constant Λ. What remains to be discussed is whether these initial data obey the point-particle limit properties discussed in Theorem 1.1. First consider a compact subset K ⊆ M {S}; let iε : K → Mε be the corresponding embedding. Since 1 wε on iε (K) the estimate (6.10) yields 4 φε ◦ iε − 1 k εν/2 . C (K,g)

It now follows from the definition of gε that (iε )∗ φ4ε gε − g k = O(εν/2 ) as ε → 0. (6.11) C (K,g) −2 Likewise, the estimate φε ◦ iε − 1C k (K,g) εν/2 together with (5.12) and (6.11) implies τ 4 − K φ μ + g = O(εν/2 ) as ε → 0. k (iε )∗ φ−2 ε ε ε 3 ε C (K,g) On the other hand, consider a compact set K ⊆ M0 and the corresponding embedding ιε : K → Mε . Note that wε ∼ ε on ιε (K) and that, therefore, for each given j with 0 ≤ j ≤ k we have gε g0 εj+(ν−1) ∇ j φ4ε ◦ ιε − 1 = εν−1 ∇ j φ4ε ◦ ιε − 1 εν/2 . gε

We now see that 4 φε ◦ ιε − 1 k

C (K,g)

g0

1−ν/2 ε1−ν/2 and similarly φ−2 . ε ◦ ιε − 1 C k (K,g) ε

Using the definition of gε one readily verifies 1 (ιε )∗ φ4ε gε − g0 = O ε1−ν/2 as ε → 0. k ε2 C (K,g)

Combining this with (5.13) further gives 1 τ 4 1−ν/2 (ιε )∗ φ−2 − K as ε → 0. φ μ + g = O ε ε ε 0 ε ε k 3 ε C (K,g0 ) Thus, the initial data φ4ε gε , φ−2 μ ε + τ3 φ4ε gε obey the point-particle limit properties discussed in the statement of Theorem 1.1.

Acknowledgments I use this opportunity to thank Paul T. Allen and James Isenberg for encouragement during my transition into this area of mathematical general relativity. The majority of the work for this particular project was done during a Junior Sabbatical Leave from Lewis and Clark College in 2008, while visiting the Geometric Analysis Group of The Albert Einstein Institute in Golm, Germany. I am particularly grateful to Prof. Gerhard Huiskin for his generous support of my visit. I also thank Vincent Moncrief for helpful discussions regarding the material in

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Appendix A, and the anonymous referee for his/her extremely thorough review and helpful comments.

Appendix A. On the CKVF and Injectivity Assumptions Moncrief pointed to us that these assumptions are related to non-existence of Killing vector fields in a space-time development of (M, g, K). Subsequently, we examined if the computations of [11] can be modified to accommodate for the presence of a cosmological constant and concluded that if M is orientable and the initial data (M, g, K) satisfy CKVF and Injectivity assumptions then no space-time development of (M, g, K) has a Killing vector field. We outline the argument below. Suppose there exists a Killing vector field X on a space-time development of (M, g, K). Let n denote a unit normal vector field to the Cauchy slice (M, g, K) within this space-time development. We may write X = Cn + X where X is tangent to M and C is some function on M . Note that the CKVF assumption implies C ≡ 0. A lengthy computation which follows [11] shows that 0 = −2CK + LX g (A.1) 0 = −Hessg (C) + Riccig − 2K 2 + τ K − Λg C + LX K, 2 := Kac Kcb and where LX denotes the Lie derivative. Tracing the second where Kab equation, using the first equation to eliminate the Lie derivative term, and using the Hamiltonian constraint to eliminate the scalar curvature term yields 0 = −Δg C + |K|2g − Λ C.

By Injectivity assumption we must have C ≡ 0. This produces a contradiction. Readers familiar with the notion of Killing Initial Data have surely noticed that (A.1) are the defining equations for KIDs on (M, g, K) (see [2]) and that our assumptions imply (M, g, K) has no KIDs.

Appendix B. The Euclidean Vector Laplacian in Spherical Coordinates The Euclidean vector Laplacian Lδ can be understood explicitly using spherical harmonics. Let r : x → |x| and let φl,m l ∈ N ∪ {0}, m ∈ Z ∩ [−l, l] ⊆ L2 (S 2 ) be the complete orthonormal basis of eigenfunctions for the Hodge Laplacian ΔH on S 2 : ΔH φl,m = δdφl,m = l(l + 1)φl,m . 1 √ 2 π

We note that φ0,0 = is a constant function. The set of 1-forms & & δ φl,m dφl,m ' l ∈ N, m ∈ Z ∩ [−l, l] ∪ ' l ∈ N, m ∈ Z ∩ [−l, l] l(l + 1) l(l + 1)

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is a complete orthonormal basis for the L2 -space of 1-forms on S 2 . For notational simplicity we set dφl,m δ φl,m √ and Wl,m := , λl := l(l + 1), Vl,m := √ λl λl where denotes the vector field dual with respect to the standard round metric on S 2 . In what follows we abuse notation and let Vl,m , Wl,m denote the trivial (r-independent) extensions of the vector fields to R3 {0}. We point out that Vl,m and Wl,m are only defined for l = 0. A vector field Y on R3 {0} ≈ (0, +∞) × S 2 can be decomposed as Y = fr · ∂r + Zr , where fr : S 2 → R is a one-parameter family of functions and where Zr is a one-parameter family of vector fields on S 2 . Using spherical harmonics we may decompose ul,m (r)φl,m , Zr = [vl,m (r)Vl,m + wl,m (r)Wl,m ] . fr = L2 (S 2 )

L2 (S 2 )

A lengthy computation involving Weitzenb¨ ock formulae (see [8, Sect. 3.1]) leads to: 2 4 u0,0 4 u0,0 u + − Lδ Y = − φ0,0 ∂r 3 0,0 3 r 3 r2 √ √ 2 4 ul,m λl λl 4 ul,m λl ul,m u + − vl,m − v − − + φl,m ∂r 3 l,m 3 r 3 r2 2 r2 r 6 l,m L2 (S 2 ) √ √ λl 4 λl ul,m 2λl r2 u + + 2rvl,m + 1 − − vl,m + vl,m Vl,m 6 l,m 3 r 3 2 L2 (S 2 ) r2 λl wl,m − + 2rwl,m + 1− wl,m Wl,m . 2 2 2 2 L (S )

This decoupling allows us to find the kernel of Lδ explicitly: it is spanned by • Basis vector fields which blow up at ∞. The vector field r∂r along with three families indexed by l ∈ N and m ∈ Z ∩ [−l, l]: ' {(l − 6) λl · rl+1 φl,m ∂r + l(l + 9)rl Vl,m }, ' { λl · rl−1 φl,m ∂r + (l + 1)rl−2 Vl,m } and {rl−1 Wl,m }. • Basis vector fields which blow up at the origin. The vector field r−2 ∂r and three families indexed by l ∈ N and m ∈ Z ∩ [−l, l]: ' (l + 7) λl · r−l φl,m ∂r − (l + 1)(l − 8)r−l−1 Vl,m , ' λl · r−l−2 φl,m ∂r − lr−l−3 Vl,m and {r−l−2 Wl,m }.

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An alternative way of obtaining the contradiction in Case R3 {0} of Proposition 5.2 is to show 0,0,ν (R3 ; 0, ∞) ∩ ker(Lδ ) = {0} C1,0

for ν ∈ (1, 2). This identity is easy to prove from the above description of ker(Lδ ).

References [1] Bartnik, R., Isenberg, J.: Constraint Equations. In: Chru´sciel, P., Friedrich, H. (eds.) The Einstein Equations and Large Scale Behavior of Gravitational Fields, pp. 1–38. Birh¨ auser, Berlin (2004) [2] Beig, R., Chru´sciel, P.T., Schoen, R.: KIDs are non-generic. Ann. Henri Poincaré 6(1), 155–194 (2005) [3] Choquet-Bruhat, Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partialles non linéaires. Acta. Math. 88, 141–225 (1952) [4] Chru´sciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. Fr. (N.S.) 94 (2003) [5] Chru´sciel, P.T., Isenberg, J., Pollack, D.: Initial data engineering. Comm. Math. Phys. 257(1), 29–42 (2005) [6] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of the Second Order. Springer, Berlin (1983) [7] Gralla, S., Wald, R.: A rigorous derivation of gravitational self-force. Class. Quantum Grav. 25, 205009 (2008) [8] Isenberg, J., Mazzeo, R., Pollack, D.: Gluing and wormholes for the Einstein constraint equations. Comm. Math. Phys. 231, 529–568 (2002) [9] Joyce, D.: Constant scalar curvature metrics on connected sums. Int. J. Math. Sci. 7, 405–450 (2003) [10] Lee, J.: Fredholm operators and Einstein metrics on conformally compact manifolds. Mem. Am. Math. Soc. 183 (2006) [11] Moncrief, V.: Spacetime symmetries and linearization stability of the Einstein equations I. J. Math. Phys. 16, 493–498 (1975) Iva Stavrov Allen Department of Mathematical Sciences Lewis & Clark College Portland, OR 97219, USA e-mail: [email protected] Communicated by Piotr T. Chrusciel. Received: August 21, 2009. Accepted: December 31, 2009.

Ann. Henri Poincaré 10 (2010), 1487–1535 c 2010 Birkh¨ auser / Springer Basel AG 1424-0637/10/081487-49, published online March 10, 2010 DOI 10.1007/s00023-010-0024-9


A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends Almir Silva Santos Abstract. It has been showed by Byde (Indiana Univ. Math. J. 52(5):1147– 1199, 2003) that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a oneparameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.

1. Introduction In 1960, Yamabe [45] claimed that every n-dimensional compact Riemannian manifold M , n ≥ 3, has a conformal metric of constant scalar curvature. Unfortunately, in 1968, Trudinger discovered an error in the proof. In 1984, Schoen [38], after the works of Yamabe [45], Trudinger [44], and Aubin [4], was able to complete the proof of The Yamabe Problem: Let (M n , g0 ) be an n-dimensional compact Riemannian manifold (without boundary) of dimension n ≥ 3. Find a metric conformal to g0 with constant scalar curvature. See [20,42] for excellent reviews of the problem. It is then natural to ask whether every noncompact Riemannian manifold of dimension n ≥ 3 is conformally equivalent to a complete manifold with constant scalar curvature. For noncompact manifolds with a simple structure at infinity, this question may be studied by solving the so-called singular Yamabe problem:

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Given (M, g0 ) an n-dimensional compact Riemannian manifold of dimension n ≥ 3 and a nonempty closed set X in M , find a complete metric g on M \X conformal to g0 with constant scalar curvature. In analytical terms, since we may write g = u4/(n−2) g0 , this problem is equivalent to finding a positive function u satisfying n+2 n−2 n−2 Δg0 u − 4(n−1) Rg0 u + 4(n−1) Ku n−2 = 0 on M \X (1.1) u(x) → ∞ as x → X where Δg0 is the Laplace–Beltrami operator associated with the metric g0 , Rg0 denotes the scalar curvature of the metric g0 , and K is a constant. We remark that the metric g will be complete if u tends to infinity with a sufficiently fast rate. The singular Yamabe problem has been extensively studied in recent years, and many existence results as well as obstructions to existence are known. This problem was considered initially in the negative case by Loewner and Nirenberg [23], when M is the sphere Sn with its standard metric. In the series of papers [1–3] Aviles and McOwen have studied the case when M is arbitrary. For a solution to exist on a general n-dimensional compact Riemannian manifold (M, g0 ), the size of X and the sign of Rg must be related to one another: it is known that if a solution exists with Rg < 0, then dim X > (n − 2)/2, while if a solution exists with Rg ≥ 0, then dim X ≤ (n − 2)/2, and in addition the first eigenvalue of the conformal Laplacian of g0 must be nonnegative. Here the dimension is to be interpreted as Hausdorff dimension. Unfortunately, only partial converses to these statements are known. For example, Aviles and McOwen [2] proved that when X is a closed smooth submanifold of dimension k, a solution for (1.1) exists with Rg < 0 if and only if k > (n − 2)/2. We direct the reader to the papers [1–3,12,13,23,28–30,33–35,37,40,41] and the references contained therein. In the constant negative scalar curvature case, it is possible to use the maximum principle, and solutions are constructed using barriers regardless of the dimension of X. See [1–3,12,13] for more details. Much is known about the constant positive scalar curvature case. When M is the round sphere Sn and X is a single point, by a result of Caffarelli et al. [9], it is known that there is no solution of (1.1). See [33] for a different proof. In the case where M is the sphere with its standard metric, in 1988, Schoen [40] constructed solutions with Rg > 0 on the complement of certain sets of Hausdorff dimension less than (n − 2)/2. In particular, he produced solutions to (1.1) when X is a finite set of points of at least two elements. Using a different method, later in 1999, Mazzeo and Pacard proved the following existence result: Theorem 1.1 [30]. Suppose that X = X ∪ X is a disjoint union of submanifolds in Sn , where X = {p1 , . . . , pk } is a collection of points, and X = ∪m j=1 Xj where dim Xj = kj . Suppose further that 0 < kj ≤ (n − 2)/2 for each j, and either

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k = 0 or k ≥ 2. Then there exists a complete metric g on Sn \X conformal to the standard metric on Sn , which has constant positive scalar curvature n(n − 1). Also, it is known that if X is a finite set of at least two elements, and M = Sn , the moduli space of solutions has dimension equal to the cardinality of X (see [33]). The first result for arbitrary compact Riemannian manifolds in the positive case appeared in 1996. Mazzeo and Pacard [28] established the following result: Theorem 1.2 [28]. Let (M, g0 ) be any n-dimensional compact Riemannian manifold with constant nonnegative scalar curvature. Let X ⊂ M be any finite disjoint union of smooth submanifolds Xi of dimensions ki with 0 < ki ≤ (n − 2)/2. Then there is an infinite dimensional family of complete metrics on M \X conformal to g0 with constant positive scalar curvature. Their method does not apply to the case in which X contains isolated points. If X = {p}, an existence result was obtained by Byde in 2003 under an extra assumption, which can be stated as follows: Theorem 1.3 [5]. Let (M, g0 ) be any n-dimensional compact Riemannian manifold of constant scalar curvature n(n − 1), nondegenerate about 1, and let p ∈ M be a point in a neighborhood of which g0 is conformally flat. There is a constant ε0 > 0 and a one-parameter family of complete metrics gε on M \{p} defined for ε ∈ (0, ε0 ), conformal to g0 , with constant scalar curvature n(n − 1). Moreover, gε → g0 uniformly on compact sets in M \{p} as ε → 0. See [5,27,30,33,35] for more details about the positive singular Yamabe problem. This work is concerned with the positive singular Yamabe problem in the case X is a single point (or when X is finite, more generally). Our main result is the construction of solutions to the singular Yamabe problem under a condition on the Weyl tensor. If the dimension is at most 5, no condition on the Weyl tensor is needed, as we will see below. We will use the gluing method, similar to that employed by Byde [5], Jleli [14], Jleli and Pacard [15], Kaabachi and Pacard [16], Kapouleas [17], Mazzeo and Pacard [29,30], Mazzeo, Pacard and Pollack [31,32], and other authors. Our result generalizes the result of Byde, Theorem 1.3, and it reads as follows: Main Theorem. Let (M n , g0 ) be an n-dimensional compact Riemannian manifold of scalar curvature n(n − 1), nondegenerate about 1, and let p ∈ M with ∇kg0 Wg0 (p) = 0 for k = 0, . . . , [ n−6 2 ], where Wg0 is the Weyl tensor of the metric g0 . Then, there exist a constant ε0 > 0 and a one-parameter family of complete metrics gε on M \{p} defined for ε ∈ (0, ε0 ), conformal to g0 , with scalar curvature n(n − 1). Moreover, each gε is asymptotically Delaunay and gε → g0 uniformly on compact sets in M \{p} as ε → 0. For the gluing procedure to work, there are two restrictions on the data (M, g0 , X): non-degeneracy and the Weyl vanishing condition. The non-degeneracy is defined as follows (see [5,18] and [34]):

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Definition 1.4. A metric g is nondegenerate at u ∈ C 2,α (M ) if the operator Lug : 2,α (M ) → C 0,α (M ) is surjective for some α ∈ (0, 1), where CD Lug (v) = Δg v −

4 n−2 n(n + 2) n−2 Rg v + u v, 4(n − 1) 4

Δg is the Laplace operator of the metric g and Rg is the scalar curvature of g. Here, C k,α (M ) are the standard H¨ older spaces on M , and the D subscript indicates the restriction to functions vanishing on the boundary of M (if there is one). Although it is the surjectivity that is used in the nonlinear analysis, it is usually easier to check injectivity. This is a corollary of the non-degeneracy condition on M in conjunction with self-adjointness. For example, it is clear that the round sphere Sn is degenerate because L1g0 = Δg0 +n annihilates the restrictions of linear functions on Rn+1 to Sn . As it was already expected by Chru´sciel and Pollack [11] (see also [10]), when 3 ≤ n ≤ 5 we do not need any hypothesis about the Weyl tensor, that is, in this case, (1.1) has a solution for any nondegenerate compact manifold M and X = {p} with p ∈ M arbitrary. We will show in Sect. 5 that the product manifolds S2 (k1 ) × S2 (k2 ) and S2 (k3 ) × S3 (k4 ) are nondegenerate except for countably many values of k1 /k2 and k3 /k4 . Therefore, our Main Theorem applies to these manifolds. We notice that they are not locally conformally flat. Byde proved his theorem assuming that M is conformally flat in a neighborhood of p. With this assumption, the problem gets simplified since in the neighborhood of p the metric is conformal to the standard metric of Rn , and in this case it is possible to transfer the metric on M \{p} to cylindrical coordinates, where there is a family of well-known Delaunay-type solutions. In our case we only have that the Weyl tensor vanishes to sufficiently high order at p. Since the singular Yamabe problem is conformally invariant, we can work in conformal normal coordinates. In such coordinates it is more convenient to work with the Taylor expansion of the metric, instead of dealing with derivatives of the Weyl tensor. As indicated in [18], we get some simplifications. In fact, this assumption will be fundamental to solve the problem locally in Sect. 3. We will exploit the fact that the first term in the expansion of the scalar curvature, in conformal normal coordinate, is orthogonal to the low eigenmodes. Pollack [37] has indicated that it would be possible to find solutions with one singular point with some Weyl vanishing condition, as opposed to the case of the round metric on Sn . in the Main Theorem comes from the Weyl VanThe motivation for n−6 2 ishing Conjecture (see [39]). It states that if a sequence vi of solutions to the equation Δg vi −

n+2 n−2 Rg vi + vin−2 = 0 4(n − 1)

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in a compact Riemannian manifold (M, g), blows-up at p ∈ M , then one should have n−6 k ∇g Wg (p) = 0 for every 0 ≤ k ≤ . 2 Here, Wg denotes the Weyl tensor of the metric g. This conjecture has been proved by Marques for n ≤ 7 in [24], Li and Zhang for n ≤ 9 in [21] and for n ≤ 11 in [22], and by Khuri et al. for n ≤ 24 in [18]. In [25], based on the works of Brendle [6] and Brendle and Marques [8], Marques constructs counterexamples for any n ≥ 25. The Weyl Vanishing Conjecture was in fact one of the essential pieces of the program proposed by Schoen in [39] to establish the Schoen’s compactness conjecture in high dimensions (see [18]). This last conjecture asserts that the set of solutions to the Yamabe equation in a compact Riemannian manifold is compact except on the standard Sn . The first counterexamples to the Schoen’s compactness conjecture were given in [6] (for n ≥ 52), and subsequently extended in [8] (for 25 ≤ n ≤ 51); the conjecture was later confirmed in dimension n ≤ 24 in [18]. n−6 The order 2 comes up naturally in our method, but we do not know if it is the optimal one (see Remark 3.5.) The Delaunay metrics form the local asymptotic models for isolated singularities of locally conformally flat constant positive scalar curvature metrics (see [9,19]). In dimensions 3 ≤ n ≤ 5 this also holds in the non-conformally flat setting. In [26], Marques proved that if 3 ≤ n ≤ 5, then every solution of the Eq. (1.1) with a nonremovable isolated singularity is asymptotic to a Delaunay-type solutions. This motivates us to seek solutions that are asymptotic to Delaunay. We use a perturbation argument together with the fixed point method to find solutions close to a Delaunay-type solution in a small ball centered at p with radius r. We also construct solutions in the complement of this ball. After that, we show that for small enough r the two metrics can be made to have exactly matching Cauchy data. Therefore (via elliptic regularity theory) they match up to all orders. See [14,15] for an application of the method. We will indicate in the end of this paper how to handle the case of more than one point. We prove Theorem 1.5. Let (M n , g0 ) be an n-dimensional compact Riemannian manifold of scalar curvature n(n − 1), nondegenerate about 1. Let {p1 , . . . , pk } a set of points in M with ∇jg0 Wg0 (pi ) = 0 for j = 0, . . . , [ n−6 2 ] and i = 1, . . . , k, where Wg0 is the Weyl tensor of the metric g0 . There exists a complete metric g on M \{p1 , . . . , pk } conformal to g0 , with constant scalar curvature n(n − 1), obtained by attaching Delaunay-type ends to the points p1 , . . . , pk . The organization of this paper is as follows. In Sect. 2 we record some notation that will be used throughout the paper. We review some results concerning the Delaunay-type solutions, as well as the function spaces on which the linearized operator will be defined. We will recall some results about the Poisson operator for the Laplace operator Δ defined in

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Br (0)\{0} ⊂ Rn and in Rn \Br (0). Finally, we will review some results concerning conformal normal coordinates and scalar curvature in these coordinates. In Sect. 3, with the assumption of the Weyl tensor and using a fixed point argument we construct a family of constant scalar curvature metrics in a small ball centered at p ∈ M , which depends on n + 2 parameters with prescribed Dirichlet data. Moreover, each element of this family is asymptotically Delaunay. In Sect. 4, we use the non-degeneracy of the metric g0 to find a right inverse for the operator L1g0 in a suitable function space. After that, we use a fixed point argument to construct a family of constant scalar curvature metrics in the complement of a small ball centered at p ∈ M , which also depends on n + 2 parameters with prescribed Dirichlet data. Each element of this family is a perturbation of the metric g0 . In Sect. 5, we put the results obtained in previous sections together to find a solution for the positive singular Yamabe problem with only one singular point. Using a fixed point argument, we examine suitable choices of the parameter sets on each piece so that the Cauchy data can be made to match up to be C 1 at the boundary of the ball. The ellipticity of the constant scalar curvature equation then immediately implies that the glued solutions are smooth. Finally, in Sect. 6, we briefly explain the changes that need to be made in order to deal with more than one singular point.

2. Preliminaries In this section, we record some notation and results that will be used frequently, throughout the rest of the work and sometimes without comment. 2.1. Notation Let us denote by θ → ej (θ), for j ∈ N, the eigenfunction of the Laplace operator on Sn−1 with corresponding eigenvalue λj . That is, ΔSn−1 ej + λj ej = 0. These eigenfunctions are restrictions to Sn−1 ⊂ Rn of homogeneous harmonic polynomials in Rn . We further assume that these eigenvalues are counted with multiplicity, namely λ0 = 0,

λ1 = · · · = λn = n − 1,

λn+1 = 2n, . . .

2

and

λj ≤ λj+1 ,

and that the eigenfunctions have L −norm equal to 1. The ith eigenvalue counted without multiplicity is i(i + n − 2). It will be necessary to divide the function space defined on Sn−1 , the sphere r with radius r > 0, into high and low eigenmode components. If the eigenfunction decomposition of the function φ ∈ L2 (Sn−1 ) is given by r ∞ φj (r)ej (θ) where φj (r) = φ(r·)ej , φ(rθ) = j=0

Sn−1

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then we define the projection πr onto the high eigenmode by the formula ∞ πr (φ)(rθ) := φj (r)ej (θ). j=n+1

Sn−1 r

is spanned by the constant functions and the restricThe low eigenmode on of linear functions on Rn . We always will use the variable θ for points tions to Sn−1 r in Sn−1 , and use the expression a · θ to denote the dot-product of a vector a ∈ Rn with θ considered as a unit vector in Rn . We will use the symbols c, C, with or without subscript, to denote various positive constants. 2.2. Constant Scalar Curvature Equation It is well known that if the metric g0 has scalar curvature Rg0 , and the metric g = u4/(n−2) g0 has scalar curvature Rg , then u satisfies the equation n+2 n−2 n−2 Δg0 u − Rg u + Rg u n−2 = 0, (2.1) 4(n − 1) 0 4(n − 1) see [20,42]. In this work we seek solutions to the singular Yamabe problem (1.1) when (M n , g0 ) is an n-dimensional compact nondegenerate Riemannian manifold with constant scalar curvature n(n − 1), X is a single point {p}, by using a method employed by [5,14,15,17,29–32,35] and others. Thus, we need to find a solution u for the Eq. (2.1) with Rg constant, requiring that u tends to infinity on approach to p. We introduce the quasi-linear mapping Hg , Hg (u) = Δg u −

4 n−2 n(n − 2) n−2 Rg u + |u| u, 4(n − 1) 4

(2.2)

and seek functions u that are close to a function u0 , so that Hg (u0 + u) = 0, u0 + u > 0 and (u + u0 )(x) → +∞ as x → p. This is done by considering the linearization of Hg about u0 , 4 ∂ n(n + 2) n−2 u0 Lg (u) = Hg (u0 + tu) u0 u, = Lg u + (2.3) ∂t 4 t=0

where n−2 Rg u 4(n − 1) is the Conformal Laplacian. The operator Lg obeys the following relation concerning conformal changes of the metric Lg u = Δg u −

Lv4/(n−2) g u = v − n−2 Lg (vu). n+2

The method of finding solutions to (1.1) used in this work is to linearize about a function u0 , not necessarily a solution. Expanding Hg about u0 gives Hg (u0 + u) = Hg (u0 ) + Lug 0 (u) + Qu0 (u),

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where the non-linear remainder term Qu0 (u) is independent of the metric, and given by Qu0 (u) =

n(n + 2) u 4

1

4

4

|u0 + tu| n−2 − u0n−2

dt.

(2.4)

0

It is important to emphasize here that in this work (M n , g0 ) always will be a compact Riemannian manifold of dimension n ≥ 3 with constant scalar curvature n(n − 1) and nondegenerate about the constant function 1. This implies that (2.2) is equal to Hg (u) = Δg u −

4 n(n − 2) n−2 n(n − 2) u+ |u| u 4 4

and the operator L1g0 : C 2,α (M ) → C 0,α (M ) given by L1g0 (v) = Δg0 v + nv,

(2.5)

is surjective for some α ∈ (0, 1), see Definition 1.4. 2.3. Delaunay-type Solutions In Sect. 3 we will construct a family of singular solutions to the Yamabe Problem in the punctured ball of radius r centered at p, Br (p)\{p} ⊂ M , conformal to the metric g0 , with prescribed high eigenmode boundary data at ∂Br (p). It is natural to require that the solution is asymptotic to a Delaunay-type solution, called by some authors Fowler solutions. We recall some well-known facts about the Delaunay-type solutions that will be used extensively in the rest of the work. See [30,33] for facts not proved here. 4 If g = u n−2 δ is a complete metric in Rn \{0} with constant scalar curvature Rg = n(n−1) conformal to the Euclidean standard metric δ on Rn , then u(x) → ∞ when x → 0 and u is a solution of the equation Hδ (u) = Δu +

n+2 n(n − 2) n−2 u =0 4

(2.6)

in Rn \{0}. It is well known that u is rotationally invariant (see [9], Theorem 8.1), and thus the equation it satisfies may be reduced to an ordinary differential equation. 2−n n−2 Therefore, if we define v(t) := e 2 t u(e−t θ) = |x| 2 u(x), where t = − log |x| x and θ = |x| , then we get that v −

n+2 n(n − 2) n−2 (n − 2)2 v+ v = 0. 4 4

(2.7)

Because of their similarity with the CMC surfaces of revolution discovered by Delaunay a solution of this ODE is called Delaunay-type solution.

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Setting w := v this equation is transformed into a first-order Hamiltonian system v = w 2 n+2 , v − n(n−2) v n−2 w = (n−2) 4 4 whose Hamiltonian energy, given by 2n (n − 2)2 2 (n − 2)2 n−2 v + v , (2.8) 4 4 is constant along solutions of (2.7). We summarize the basic properties of this solutions in the next proposition (see Proposition 1 in [30]).

H(v, w) = w2 −

Proposition 2.1. For any H0 ∈ (−((n − 2)/n)n/2 (n − 2)/2, 0), there exists a unique bounded solution of (2.7) satisfying H(v, v ) = H0 , v (0) = 0 and v (0) > 0. This solution is periodic, and for all t ∈ R we have v(t) ∈ (0, 1). This solution can be indexed by the parameter ε = v(0) ∈ (0, ((n−2)/n)(n−2)/4 ), which is the smaller of the two values v assumes when v (0) = 0. When H0 = −((n − 2)/n)n/2 (n − 2)/2, there is a unique bounded solution of (2.7), given by v(t) = ((n − 2)/n)(n−2)/4 . Finally, if v is a solution with H0 = 0 then either v(t) = (cosh(t − t0 ))(2−n)/2 for some t0 ∈ R or v(t) = 0. We will write the solution of (2.7) given by Proposition 2.1 as vε , where vε (0) = min vε = ε ∈ (0, ((n − 2)/n)(n−2)/4 ) and the corresponding solution of (2.6) as uε (x) = |x|(2−n)/2 vε (− log |x|). Although we do not know them explicitly, the next proposition gives sufficient information about their behavior as ε tends to zero for our purposes (see [30]). Proposition 2.2. For any ε ∈ (0, ((n − 2)/n)(n−2)/4 ) and any x ∈ Rn \{0} with |x| ≤ 1, the Delaunay-type solution uε (x) satisfies the estimates n+2 ε uε (x) − (1 + |x|2−n ) ≤ cn ε n−2 |x|−n , 2 n+2 |x|∂r uε (x) + n − 2 ε|x|2−n ≤ cn ε n−2 |x|−n 2 and

2 2 2 n+2 |x| ∂r uε (x) − (n − 2) ε|x|2−n ≤ cn ε n−2 |x|−n , 2

for some positive constant cn that depends only on n. As indicated by Mazzeo–Pacard [30], there are some important variations of these solutions, leading to a (2n + 2)-dimensional family of Delaunay-type solutions. For our purpose, it is enough to consider the family of solutions where only translations along Delaunay axis and of the “point at infinity” are allowed. Therefore, we will work with the following family of solutions of (2.6)

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uε,R,a (x) := |x − a|x|2 |

2−n 2


vε (−2 log |x| + log |x − a|x|2 | + log R).

(2.9)

See [30] for details. In Sect. 3 we will find solutions to the singular Yamabe problem in the punctured ball Br (p)\{p} only with prescribed high eigenmode Dirichlet data, so we need other parameters to control the low eigenmode. The parameters a ∈ Rn and R ∈ R+ in (2.9) will allow us to have control over the low eigenmode. The first corollary is a direct consequence of (2.9), and it will control the space spanned by the coordinates functions, and the second one follows from Proposition 2.2, and it will control the space spanned by the constant functions in the sphere. Notation. We write f = O (Krk ) to mean f = O(Krk ) and ∇f = O(Krk−1 ), for K > 0 constant. O is defined similarly. Corollary 2.1. There exists a constant r0 ∈ (0, 1), such that for any x and a in Rn with |x| ≤ 1, |a||x| < r0 , R ∈ R+ , and ε ∈ (0, ((n − 2)/n)(n−2)/4 ) the solution uε,R,a satisfies the estimates uε,R,a (x) = uε,R (x) + ((n − 2)uε,R (x) + |x|∂r uε,R (x))a · x + O (|a|2 |x|

6−n 2

) (2.10)

and

2−n uε,R,a (x) = uε,R (x)+((n − 2)uε,R (x)+|x|∂r uε,R (x))a · x+O |a|2 εR 2 |x|2 (2.11)

if R ≤ |x|. Proof. Using the Taylor’s expansion we obtain that

x − a|x| + log R = vε (− log |x| + log R) vε − log |x| + log |x| − vε (− log |x| + log R)a · x + vε (− log |x| + log R)O (|a|2 |x|2 ) + vε (− log |x| + log R + ta,x )O (|a|2 |x|2 ) x for some ta,x ∈ R with 0 < |ta,x | < | log | |x| − a|x|||, since x − a|x| = −a · x + O (|a|2 |x|2 ). log |x|

for |a||x| < r0 and some r0 ∈ (0, 1). Observe that ta,x → 0 as |a||x| → 0. Now, by the Eq. (2.7) and the fact that H(vε , vε ) =

n+2 (n − 2)2 2 n−2 ε (ε − 1), 4

where H is defined in (2.8), it follows that |vε | ≤ cn vε , |vε | ≤ cn vε , for some constant cn that depends only on n.

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Since − log |x| + log R ≤ 0 if R ≤ |x|, and vε (t) ≤ εe [30]), we obtain that vε (− log |x| + log R) ≤ εR

2−n 2

|x|

n−2 2 |t|

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, for all t ∈ R (see

n−2 2

and vε (− log |x| + log R + ta,x ) ≤ cεR

2−n 2

|x|

n−2 2

,

for some constant c > 0 that does not depend on x, ε, R and a. 2−n 2−n Therefore, from (2.9), 0 < vε (t) ≤ 1 and |x − a|x|2 | 2 = x| 2 + n−2 2 a· 2−n 6−n 2 2 2 + O (|a| |x| ), for |a||x| < r0 and some r0 ∈ (0, 1), we deduce the x|x| result. Corollary 2.2. For any ε ∈ (0, ((n − 2)/n)(n−2)/4 ) and any x in Rn with |x| ≤ 1, the function uε,R := uε,R,0 satisfies the estimates

n+2 n−2 n+2 ε 2−n uε,R (x) = R 2 + R 2 |x|2−n + O (R 2 ε n−2 |x|−n ), 2 n+2 n+2 2 − n n−2 2−n εR 2 |x| |x|∂r uε,R (x) = + O (R 2 ε n−2 |x|−n ) 2 and |x|2 ∂r2 uε,R (x) =

n+2 n−2 n+2 (n − 2)2 εR 2 |x|2−n + O(R 2 ε n−2 |x|−n ). 2

Proof. Use Proposition 2.2 and the fact that uε,R (x) = R

2−n 2

uε (R−1 x).

2.4. Function Spaces Now, we will define some function spaces that we will use in this work. The first one is the weighted H¨ older spaces in the punctured ball. They are the most convenient spaces to define the linearized operator. The second one appears so naturally in our results that it is more helpful to define it here. Finally, the third one is the weighted H¨ older spaces in which the exterior analysis will be carried out. These are essentially the same weighted spaces as in [14,15,30]. Definition 2.1. For each k ∈ N, r > 0, 0 < α < 1 and σ ∈ (0, r/2), let u ∈ C k (Br (0)\{0}), set ⎛ ⎞ k |∇k u(x) − ∇k u(y)| σ j |∇j u(x)|⎠ +σ k+α sup . u(k,α),[σ,2σ] = sup ⎝ |x − y|α |x|∈[σ,2σ] |x|,|y|∈[σ,2σ] j=0 Then, for any μ ∈ R, the space Cμk,α (Br (0)\{0}) is the collection of functions u that are locally in C k,α (Br (0)\{0}) and for which the norm u(k,α),μ,r = sup σ −μ u(k,α),[σ,2σ] 0 0. Let φ ∈ C k (Sn−1 ), set r φ(k,α),r := φ(r·)C k,α (Sn−1 ) . Then, the space C k,α (Sn−1 ) is the collection of functions φ ∈ C k (Sn−1 ) for which r r the norm φ(k,α),r is finite. The next lemma shows a relation between the norms of Definitions 2.1 and 2.2. To prove it, we use the decomposition of the function spaces in the sphere. Lemma 2.3. Let α ∈ (0, 1) and r > 0 be constants. Then, there exists a constant c > 0 that does not depend on r, such that πr (ur )(2,α),r ≤ cK

(2.12)

r∂r πr (ur )(1,α),r ≤ cK,

(2.13)

and for all function u : {x ∈ R ; r/2 ≤ |x| ≤ r} → R satisfying u(2,α),[r/2,r] ≤ K, for some constant K > 0. Here, ur is the restriction of u to the sphere of radius r, ⊂ Rn . Sn−1 r n

)) and π (Cμk,α (Br (0)\{0})) for Remark 2.4. We often will write π (C k,α (Sn−1 r {φ ∈ C k,α (Sn−1 ); πr (φ) = φ} r and

, u ∈ Cμk,α (Br (0)\{0}); πs (u(s·))(θ) = u(sθ), ∀s ∈ (0, r) and ∀θ ∈ Sn−1 r

respectively. Next, consider (M, g) an n-dimensional compact Riemannian manifold and Ψ : Br1 (0) → M some coordinate system on M centered at some point p ∈ M , where Br1 (0) ⊂ Rn is the ball of radius r1 > 0. For 0 < r < s ≤ r1 define Mr := M \Ψ(Br (0))

and

Ωr,s := Ψ(Ar,s ),

where Ar,s := {x ∈ R ; r ≤ |x| ≤ s}. n

Definition 2.5. For all k ∈ N, α ∈ (0, 1) and ν ∈ R, the space Cνk,α (M \{p}) is the k,α space of functions v ∈ Cloc (M \{p}) for which the following norm is finite: vCνk,α (M \{p}) := vC k,α (M 1 r

2 1

)

+ v ◦ Ψ(k,α),ν,r1 ,

where the norm · (k,α),ν,r1 is the one defined in Definition 2.1.

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For all 0 < r < s ≤ r1 , we can also define the spaces Cμk,α (Ωr,s ) and Cμk,α (Mr ) to be the space of restriction of elements of Cμk,α (M \{p}) to Mr and Ωr,s , respectively. These spaces are endowed with the following norm: f Cμk,α (Ωr,s ) := sup σ −μ f ◦ Ψ(k,α),[σ,2σ] r≤σ≤ 2s

and hCμk,α (Mr ) := hC k,α (M 1 r

2 1

)

+ hCμk,α (Ωr,r ) . 1

2.5. The Linearized Operator Let us fix one of the solutions of (2.6), uε,R,a given by (2.9). Hence, uε,R,a satisfies Hδ (uε,R,a ) = 0. The linearization of Hδ at uε,R,a is defined by u

Lε,R,a (v) := Lδ ε,R,a (v) = Δv +

4 n(n + 2) n−2 uε,R,a v, 4

(2.14)

u

where Lδ ε,R,a is given by (2.3). In [30], Mazzeo and Pacard studied the operator Lε,R := Lε,R,0 defined in weighted H¨ older spaces. They showed that there exists a suitable right inverse with two important features; the corresponding right inverse has norm bounded independently of ε and R when the weight is chosen carefully, and the weight can be improved if the right inverse is defined in the high eigenmode. These properties will be fundamental in Sect. 3. To summarize, they establish the following result: Proposition 2.3. (Mazzeo–Pacard, [30]) Let R ∈ R+ , α ∈ (0, 1) and μ ∈ (1, 2). Then there exists ε0 > 0 such that, for all ε ∈ (0, ε0 ], there is an operator 0,α (B1 (0)\{0}) → Cμ2,α (B1 (0)\{0}) Gε,R : Cμ−2 0,α with the norm bounded independently of ε and R, such that for f ∈ Cμ−2 (B1 (0)\{0}), the function w := Gε,R (f ) solves the equation in B1 (0)\{0} Lε,R (w) = f . (2.15) π1 (w|Sn−1 ) = 0 on ∂B1 (0) 0,α Moreover, if f ∈ π (Cμ−2 (B1 (0)\{0})), then w ∈ π (Cμ2,α (B1 (0)\{0})) and we may take μ ∈ (−n, 2).

Proof. The statement in [30] is that for each fixed R the norm of Gε,R is independent of ε, but this bound might depend on R. In [5], Byde observed that the norm of Gε,R also does not depend on R. We will work in Br (0)\{0} with 0 < r ≤ 1; then it is convenient to study the operator Lε,R in function spaces defined in Br (0)\{0}.

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0,α Let f ∈ Cμ−2 (B1 (0)\{0}) and w ∈ Cμ2,α (B1 (0)\{0}) be solution of (2.15). Considering g(x) = r−2 f (r−1 x) and wr (x) = w(r−1 x) we get that (2.15) is equivalent to in Br (0)\{0} Lε,rR (wr ) = g . ) = 0 on ∂Br (0) πr (wr |Sn−1 r

Furthermore, since ∇j wr (x) = r−j ∇j w(r−1 x) and ∇j g(x) = r−2−j ∇j f (r−1 x), we get wr (2,α),μ,r ≤ cg(0,α),μ−2,r , where c > 0 is a constant that does not depend on ε, r and R. Thus, we obtain the following corollary. Corollary 2.3. Let μ ∈ (1, 2), α ∈ (0, 1), ε0 > 0 given by Proposition 2.3. Then for all ε ∈ (0, ε0 ), R ∈ R+ and 0 < r ≤ 1 there is an operator 0,α Gε,R,r : Cμ−2 (Br (0)\{0}) → Cμ2,α (Br (0)\{0}) 0,α with norm bounded independently of ε, R and r, such that for each f ∈ Cμ−2 (Br (0)\{0}), the function w := Gε,R,r (f ) solves the equation in Br (0)\{0} Lε,R (w) = f . ) = 0 on ∂Br (0) πr (w|Sn−1 r 0,α Moreover, if f ∈ π (Cμ−2 (Br (0)\{0})), then w ∈ π (Cμ2,α (Br (0)\{0})) and we may take μ ∈ (−n, 2).

In fact, we will work with the solution uε,R,a , and so, we need to find an inverse to Lε,R,a with norm bounded independently of ε, R, a and r. But this is the content of the next corollary, whose proof is a perturbation argument. Corollary 2.4. Let μ ∈ (1, 2), α ∈ (0, 1), ε0 > 0 given by Proposition 2.3. Then for all ε ∈ (0, ε0 ), R ∈ R+ , a ∈ Rn and 0 < r ≤ 1 with |a|r ≤ r0 for some r0 ∈ (0, 1), there is an operator 0,α Gε,R,r,a : Cμ−2 (Br (0)\{0}) → Cμ2,α (Br (0)\{0}), 0,α with norm bounded independently of ε, R, r, and a, such that for each f ∈ Cμ−2 (Br (0)\{0}), the function w := Gε,R,r,a (f ) solves the equation in Br (0)\{0} Lε,R,a (w) = f . ) = 0 on ∂Br (0) πr (w|Sn−1 r

Proof. We will use a perturbation argument. Thus,

4 4 n(n + 2) n−2 n−2 (Lε,R,a − Lε,R )v = uε,R,a − uε,R v 4

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implies

4 4 n−2 n−2 ≤ c uε,R,a − uε,R

(Lε,R,a − Lε,R )v(0,α),[σ,2σ]

(0,α),[σ,2σ]

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v(0,α),[σ,2σ] ,

where c > 0 does not depend on ε, R, a, and r. Note that

4 4 x n−2 − a|x| + log R = vεn−2 (− log |x| + log R) vε − log |x| + log |x| x log| |x| −a|x||

4 + n−2

6−n n−2

vε

vε

(− log |x|+log R+t) dt.

0

Therefore, from (2.9) and the expansion |x − a|x|2 |−2 = |x|−2 + O(|a||x|−1 ), we get 4 n−2

4 n−2

uε,R,a (x) = uε,R (x) +

−2

4|x| n−2

4 n−2

+ O(|a||x|−1 )vε

x log| |x| −a|x||

6−n vεn−2 vε (− log |x| + log R + t) dt

0

x − a|x| + log R . − log |x| + log |x|

From the proof of Corollary 2.1 we know that |vε | ≤ cn vε . Hence, 4 4 n−2 n−2 u ≤ cn |x|−2 (x)−u (x) ε,R ε,R,a

O(|a||x|)

4

vεn−2 (− log |x|+log R + t) dt + O(|a||x|−1 ),

0 −1

since log |x|x|

− a|x|| = O(a|x|) and 0 < ε ≤ vε ≤ 1. Thus 4 4 n−2 n−2 u ≤ cn |a||x|−1 , (x) − u (x) ε,R ε,R,a

(2.16)

where the constant c > 0 does not depend on ε, R and a. The estimate for the full H¨ older norm is similar. Hence 4 4 n−2 n−2 u − u ≤ c|a|σ −1 ε,R ε,R,a (0,α),[σ,2σ]

and then (Lε,R,a − Lε,R )v(0,α),μ−2,r ≤ c|a|rv(2,α),μ,r , where c > 0 is a constant that does not depend on ε, R, a, and r. Therefore, Lε,R,a has a bounded right inverse for small enough |a|r, and this inverse has norm bounded independently of ε, R, a, and r. In fact, if we choose

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r0 so that r0 ≤ 12 K −1 , where the constant K > 0 satisfies Gε,R,r ≤ K for all ε ∈ (0, ε0 ), R ∈ R+ and r ∈ (0, 1), then 1 . 2 has a bounded right inverse given by

Lε,R,a ◦ Gε,R,r − I ≤ Lε,R,a − Lε,R Gε,R,r ≤ This implies that Lε,R,a ◦ Gε,R,r

(Lε,R,a ◦ Gε,R,r )−1 :=

∞

(I − Lε,R,a ◦ Gε,R,r )i ,

i=0

and it has norm bounded independently of ε, R, a, and r, in fact < 1. Therefore, we define a right inverse of Lε,R,a as Gε,R,r,a := Gε,R,r ◦ (Lε,R,a ◦ Gε,R,r )−1 . 2.6. Poisson Operator Associated with the Laplacian Δ 2.6.1. Laplacian Δ in Br (0)\{0} ⊂ Rn . Since πr (Gε,R,r,a (f )|Sn−1 ) = 0 on r ∂Br (0), we need to find some way to prescribe the high eigenmode boundary data at ∂Br (0). This is done using the Poisson operator associated with the Laplacian Δ. Proposition 2.4. Given α ∈ (0, 1), there is a bounded operator

P1 : π1 (C 2,α (Sn−1 )) −→ π1 C22,α (B1 (0)\{0}) , so that

Δ(P1 (φ)) = 0 π1 (P1 (φ)|Sn−1 ) = φ

in B1 (0) . on ∂B1 (0)

Proof. See Proposition 2.2 in [5], Proposition 11.25 in [14] and Lemma 6.2 in [36]. For μ ≤ 2 and 0 < r ≤ 1 we can define an analogous operator, −→ πr Cμ2,α (Br (0)\{0}) Pr : πr C 2,α Sn−1 r as Pr (φr )(x) = P1 (φ)(r−1 x),

(2.17)

where φ(θ) := φr (rθ). By Proposition 2.4 we deduce that Δ(Pr (φr )) = 0 in Br (0)\{0} ) = φ on ∂Br (0) πr (Pr (φr )|Sn−1 r r and Pr (φr )(2,α),μ,r ≤ Cr−μ φr (2,α),r ,

(2.18)

where the constant C > 0 does not depend on r and the norm φr (2,α),r is defined in Definition 2.2.

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2.6.2. Laplacian Δ in Rn \Br (0). For the same reason as before we will need a Poisson operator associated to the Laplacian Δ defined in Rn \Br (0). Proposition 2.5. Assume that ϕ ∈ C 2,α (Sn−1 ) and let Q1 (ϕ) be the only solution of Δv = 0 in Rn \B1 (0) v=ϕ on ∂B1 (0) which tends to 0 at ∞. Then Q1 (ϕ)C 2,α (Rn \B1 (0)) ≤ Cϕ(2,α),1 , 1−n

if ϕ is L2 -orthogonal to the constant function. Proof. See Lemma 13.25 in [14] and also [16].

Here, the space Cμk,α (Rn \Br (0)) is the collection of functions u that are locally in C k,α (Rn \Br (0)) and for which the norm uCμk,α (Rn \Br (0)) := sup σ −μ u(k,α),[σ,2σ] σ≥r

is finite. Remark 2.6. In this case, it is very useful to know an explicit expression for Q1 , since it has a component in the space spanned by the coordinate functions and this ∞ will be important to control this space in Sect. 5. Hence, if we write ϕ = i=2 ϕi , with ϕ belonging to the eigenspace associated with the eigenvalue i(i + n − 2), then Q1 (ϕ)(x) =

∞

|x|2−n−j ϕi .

i=1

Now, define Qr (ϕr )(x) := Q1 (ϕ)(r−1 x),

(2.19)

where ϕr (x) := ϕ(r−1 x). From Proposition 2.5, we deduce that ΔQr (ϕr ) = 0 in Rn \Br (0) Qr (ϕr ) = ϕr on ∂Br (0) and Qr (ϕr )C 2,α (Rn \Br (0)) ≤ Crn−1 ϕr (2,α),r , 1−n

where C > 0 is a constant that does not depend on r.

(2.20)

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2.7. Conformal Normal Coordinates Since our problem is conformally invariant, in Sect. 3 we will work in conformal normal coordinates. In this section, we introduce some notation and an asymptotic expansion for the scalar curvature in conformal normal coordinates, which will be essential in the interior analysis of Sect. 3. Theorem 2.7 [20]. Let M n be an n-dimensional Riemannian manifold and P ∈ M . For each N ≥ 2 there is a conformal metric g on M such that det gij = 1 + O(rN ), where r = |x| in g-normal coordinates at P. In these coordinates, if N ≥ 5, the scalar curvature of g satisfies Rg = O(r2 ). In conformal normal coordinates it is more convenient to work with the Taylor expansion of the metric. In such coordinates, we will always write gij = exp(hij ), where hij is a symmetric two-tensor satisfying hij (x) = O(|x|2 ) and trhij (x) = O(|x|N ). Here N is a large number. n ∂i ∂j hij . In what follows, we write ∂i ∂j hij instead of i,j=1

Lemma 2.8. The functions hij satisfy the following properties: a) Sn−1 ∂i ∂j hij = O(rN ); r b) Sn−1 xk ∂i ∂j hij = O(rN ) for every 1 ≤ k ≤ n, r where N is as big as we want. This lemma plays a central role in our argument for n ≥ 8 in Sect. 3. Using this notation we obtain the following proposition whose proof can be found in [6,18]: Proposition 2.6. There exists a constant C > 0 such that |Rg − ∂i ∂j hij | ≤ C

d

|hijα |2 |x|2|α|−2 + C|x|n−3 ,

|α|=2 i,j

if |x| ≤ r ≤ 1, where hij (x) =

hijα xα + O(|x|n−3 )

2≤|α|≤n−4

and C depends only on n and |h|C N (Br (0)) .

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3. Interior Analysis Now that we have a right inverse for the operator Lε,R,a and a Poisson operator associated with the Laplacian Δ, we are ready to show the existence of solutions with prescribed boundary data for the equation Hg0 (v) = 0 in a small punctured ball Br (p)\{p} ⊂ M . The point p is a nonremovable singularity, that is, u blows-up at p. In fact, the hypothesis on the Weyl tensor is fundamental to our construction if n ≥ 6. But, if 3 ≤ n ≤ 5 we do not need any additional hypothesis on the point p. We do not know whether it is possible to show the Main Theorem assuming the . This should be an interesting Weyl tensor vanishes up to order less than n−6 2 question. First, we will explain how to use the assumption on the Weyl tensor to reduce the problem to a problem of finding a fixed point of a map, (3.8) and (3.12). After that, we will show that these maps have a fixed point for suitable parameters. 3.1. Analysis in Br (p)\{p} ⊂ M , and g will be a smooth conformal Throughout the rest of this work d = n−2 2 metric to g0 in M given by Theorem 2.7, with N a large number. Hence, by the proof of Theorem teo07 in [20], we can find some smooth function F ∈ C ∞ (M ) 4 such that g = F n−2 g0 and F(x) = 1 + O(|x|2 ) in g-normal coordinates at p. In this section we will work in these coordinates around p, in the ball Br1 (p) with 0 < r1 ≤ 1 fixed. Recall that (M, g0 ) is an n-dimensional compact Riemannian manifold with Rg0 = n(n − 1), n ≥ 3, and the Weyl tensor Wg0 at p satisfies the condition ∇l Wg0 (p) = 0,

l = 0, 1, . . . , d − 2.

(3.1)

Since the Weyl tensor is conformally invariant, it follows that Wg , the Weyl tensor of the metric g, satisfies the same condition. Note that if 3 ≤ n ≤ 5, then the condition on Wg does not exist. From Theorem 2.7 the scalar curvature satisfies Rg = O(|x|2 ), but for n ≥ 8 we can improve this decay, using the assumption of the Weyl tensor. This assumption implies hij = O(|x|d+1 ) (see [7]), and it follows from Proposition 2.6 that Rg = ∂i ∂j hij + O(|x|n−3 ).

(3.2)

d−1

We conclude that Rg = O(|x| ). On the other hand, for n = 6 and 7 we have d = 2 and in this case, we will consider Rg = O(|x|2 ), given directly by Theorem 2.7. The main goal of this section is to solve the PDE Hg (uε,R,a + v) = 0

(3.3)

in Br (0)\{0} ⊂ R for some 0 < r ≤ r1 , ε > 0, R > 0 and a ∈ R , with uε,R,a + v > 0 and prescribed Dirichlet data, where the operator Hg is defined in (2.2) and uε,R,a in (2.9). To solve this equation, we will use the method used by Byde and others, the fixed point method on Banach spaces. In [5], Byde solves an equation like n

n

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this assuming that g is conformally flat in a neighborhood of p, and thus he uses directly the right inverse of Lε,R given by Corollary 2.4, to reduce the problem to a problem of fixed point. The main difference here is that we work with metrics not necessarily conformally flat, so we need to rearrange the terms of the Eq. (3.3) in such a way that we can apply the right inverse of Lε,R,a . For each φ ∈ π (C 2,α (Sn−1 )) define vφ := Pr (φ) ∈ π (C22,α (Br (0)\{0})) as r in Proposition 2.4. It is easy to see that the Eq. (3.3) is equivalent to Lε,R,a (v) = (Δ − Δg )(uε,R,a + vφ + v) +

n−2 Rg (uε,R,a + vφ + v) 4(n − 1)

4 n(n + 2) n−2 uε,R,a vφ , 4 solves the Eq. (2.6). Here Lε,R,a is defined as in (2.14),

− Qε,R,a (vφ + v) −

since uε,R,a

Qε,R,a (v) := Quε,R,a (v) uε,R,a

and Q

(3.4)

(3.5)

is defined in (2.4).

Remark 3.1. Throughout this work we will consider |a|rε ≤ 1/2 with rε = εs , s restricted to (d + 1 − δ1 )−1 < s < 4(d − 2 + 3n/2)−1 and δ1 ∈ (0, (8n − 16)−1 ). From this and (2.9) it follows that there are constants C1 > 0 and C2 > 0 that do not depend on ε, R and a, so that C1 ε|x|

2−n 2

≤ uε,R,a (x) ≤ C2 |x|

2−n 2

,

(3.6)

for every x in Brε (0)\{0}. These restrictions are made to ensure some conditions that we need in the next lemma and in Sect. 5. Lemma 3.2. Let μ ∈ (1, 3/2). There exists ε0 ∈ (0, 1) such that for each ε ∈ (0, ε0 ), a ∈ Rn with |a|rε ≤ 1, and for all vi ∈ Cμ2,α (Brε (0)\{0}), i = 0, 1, and w ∈ 2+d−μ− n −δ

1 2,α 2 C2+d− and w(2,α),2+d− n2 ,rε ≤ n (Brε (0)\{0}) with vi (2,α),μ,rε ≤ crε 2 c, for some constant c > 0 independent of ε, we have that Qε,R,a given by (3.5) satisfies the inequalities

Qε,R,a (w + v1 ) − Qε,R,a (w + v0 )(0,α),μ−2,rε ≤ Cελn rεd+1 v1 − v0 (2,α),μ,rε w(2,α),2+d− n2 ,rε

+ v1 (2,α),μ,rε + v0 (2,α),μ,rε , and 3+2d− n 2 −μ

Qε,R,a (w)(0,α),μ−2,rε ≤ Cελn rε Here, λn = 0 for 3 ≤ n ≤ 6, λn = not depend on ε, R and a.

6−n n−2

w2(2,α),2+d− n ,rε . 2

for n ≥ 7, and the constant C > 0 does

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Proof. By the hypothesis, we conclude that 2+d− n 2 −δ1

|vi (x)| ≤ crε and

2+d− n 2

|w(x)| ≤ crε for all x ∈ Brε (0)\{0}. Using (3.6), we get uε,R,a (x) + w + vi (x) ≥ ε|x|

2−n 2

(C1 − c(|x|rε−1 )

n−2 2

εs(d+1−δ1 )−1 ),

with s(d + 1 − δ1 ) − 1 > 0, since s > (d + 1 − δ1 )−1 . Therefore, 0 < C3 ε|x|

2−n 2

≤ uε,R,a (x) + w(x) + vi (x) ≤ C4 |x|

2−n 2

(3.7)

for small enough ε > 0, since |x| ≤ rε . Thus, by (2.4), we can write Qε,R,a (w + v1 ) − Qε,R,a (w + v0 ) 1 1 6−n n(n + 2) (v1 − v0 ) (uε,R,a + szt ) n−2 zt dt ds = n−2 0

0

and Qε,R,a (w) =

n(n + 2) 2 w n−2

1 1

6−n

(uε,R,a + stw) n−2 t dt ds, 0

0

where zt = w + tv1 + (1 − t)v0 . From this we obtain Qε,R,a (w + v1 ) − Qε,R,a (w + v0 )(0,α),[σ,2σ] ≤ Cv1 − v0 (0,α),[σ,2σ] w(0,α),[σ,2σ] 6−n +v1 (0,α),[σ,2σ] + v0 (0,α),[σ,2σ] max (uε,R,a + szt ) n−2 (0,α),[σ,2σ] 0≤s,t≤1

and 6−n

Qε,R,a (w)(0,α),[σ,2σ] ≤ Cw2(0,α),[σ,2σ] max (uε,R,a + stw) n−2 (0,α),[σ,2σ] . 0≤s,t≤1

From (3.7) we deduce that 6−n

n−6 2

6−n

n−6 2

|(uε,R,a + szt ) n−2 (x)| ≤ Cελn |x| and |(uε,R,a + stw) n−2 (x)| ≤ Cελn |x|

,

for some constant C > 0 independent of ε, a and R. The estimate for the full H¨ older norm is similar. Hence, we conclude that max (uε,R,a + szt ) n−2 (0,α),[σ,2σ] ≤ Cελn σ

6−n

n−6 2

6−n

n−6 2

0≤s,t≤1

and max (uε,R,a + stw) n−2 (0,α),[σ,2σ] ≤ Cελn σ

0≤s,t≤1

.

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Therefore, σ

2−μ

Qε,R,a (w + v1 ) − Qε,R,a (w + v0 )(0,α),[σ,2σ]

≤ Cελn rεd+1 v1 −v0 (2,α),μ,rε w(2,α),2+d− n2 ,rε +v1 (2,α),μ,rε + v0 (2,α),μ,rε

and 3+2d− n 2 −μ

σ 2−μ Qε,R,a (w)(0,α),[σ,2σ] ≤ Cελn rε

w2(2,α),2+d− n ,rε , 2

since 1 < μ < 3/2 implies 2 + d − n/2 < μ and 3 + 2d − n/2 − μ > 0.

Now to use the right inverse of Lε,R,a , given by Gε,R,rε ,a , all terms of the right-hand side of the Eq. (3.4) have to belong to the domain of Gε,R,rε ,a . But this does not happen with the term Rg uε,R,a if n ≥ 8, since Rg = O(|x|d−1 ) implies n 0,α Rg uε,R,a = O(|x|d− 2 ) and so Rg uε,R,a ∈ Cμ−2 (Brε (0)\{0}) for every μ > 1. However, when 3 ≤ n ≤ 7 we get the following lemma: Lemma 3.3. Let 3 ≤ n ≤ 7, μ ∈ (1, 3/2), κ > 0 and c > 0 be fixed constants. There exists ε0 ∈ (0, 1) such that for each ε ∈ (0, ε0 ), for all v ∈ 2+d−μ− n 2 −δ1 Cμ2,α (Brε (0)\{0}) and φ ∈ π (C 2,α (Sn−1 and rε )) with v(2,α),μ,rε ≤ crε 2+d− n 2 −δ1

φ(2,α),rε ≤ κrε 0,α (Brε (0)\{0}). Cμ−2

, we have that the right-hand side of (3.4) belongs to

Proof. Initially, note that by (2.18) we obtain 2+d−μ− n 2 −δ1

vφ + v(2,α),μ,rε ≤ (c + κ)rε

,

0,α (Brε (0)\{0}). by Lemma 3.2 we get that Qε,R,a (vφ + v) ∈ Cμ−2 Now it is enough to show that the other terms have the decay O(|x|μ−2 ). Using the expansion (2.10), it follows that

(Δ − Δg )uε,R,a = (Δ − Δg )uε,R + (Δ − Δg )(uε,R,a − uε,R ), 4−n

with uε,R,a − uε,R = O (|a||x| 2 ). Moreover, since in conformal normal coordinates Δg = Δ + O(|x|N ) when applied to functions that depend only on |x|, where N can be any big number (see proof of Theorem 3.5 in [42], for example), we get

(Δ − Δg )uε,R = O(|x|N ), where N is big for N big. Since gij = δij + O(|x|d+1 ), we get

2−n = O(|x|μ−2 ) (Δ − Δg )(uε,R,a − uε,R ) = O |x|d+ 2

when μ ≤ 3 + d − 3/2. Since vφ = O(|x|2 ), gij = δij + O(|x|d+1 ), Rg = O(|x|2 ), using (3.6) we get the same decay for the remaining terms. Hence the assertion follows.

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Now this lemma allows us to use the map Gε,R,rε ,a . Let μ ∈ (1, 3/2) and c > 0 be fixed constants. To solve the Eq. (3.3) we need to show that the map Nε (R, a, φ, ·) : Bε,c,δ1 → Cμ2,α (Brε (0)\{0}) has a fixed point for suitable parameters 2+d−μ− n −δ

1 2 ε, R, a, and φ, where Bε,c,δ1 is the ball in Cμ2,α (Brε (0)\{0}) of radius crε and Nε (R, a, φ, ·) is defined by

n−2 Nε (R, a, φ, v) = Gε,R,r,a (Δ − Δg )v + Rg v − Qε,R,a (vφ + v) 4(n − 1) 4 n−2 n(n + 2) n−2 Rg (uε,R,a + vφ ) − uε,R,a vφ . + (Δ − Δg )(uε,R,a + vφ ) + 4(n − 1) 4 (3.8)

Let us now consider n ≥ 8. Since Rg = O(|x|d−1 ), we have Rg uε,R,a = n 0,α O(|x|d− 2 ), and this implies that Rg uε,R,a ∈ Cμ−2 (Brε (0)\{0}) for μ > 1. Hence, we cannot use Gε,R,rε ,a directly. To overcome this difficulty we will consider the expansion (2.10), the expansion (3.2) and use the fact that ∂i ∂j hij is orthogonal to {1, x1 , . . . , xn } modulo a term of order O(|x|N ) with N as big as we want (see Lemma 2.8.) It follows from this fact and Corollary 2.3, that there exists wε,R in the space 2,α C2+d− n (Brε (0)\{0}) such that 2

Lε,R (wε,R ) =

n − 2 π (∂i ∂j hij )uε,R . 4(n − 1)

(3.9)

This is because uε,R depends only on |x|. Again by Corollary 2.3 wε,R (2,α),2+d− n2 ,rε ≤ cπ (∂i ∂j hij )uε,R (0,α),d− n2 ,rε ≤ c,

(3.10)

for some constant c > 0 that does not depend on ε and R, since ∂i ∂j hij uε,R = n O(|x|d− 2 ). Considering the expansion (2.10) and substituting v for wε,R + v in the Eq. (3.4), we obtain n−2 Rg (wε,R + vφ + v) Lε,R,a (v) = (Δ − Δg )(uε,R,a + wε,R + vφ + v) + 4(n − 1) n−2 − Qε,R,a (wε,R + vφ + v) + ∂i ∂j hij (uε,R,a − uε,R ) 4(n − 1)

4 4 n−2 n(n + 2) n−2 n−2 (Rg − ∂i ∂j hij )uε,R,a + + uε,R − uε,R,a wε,R 4(n − 1) 2 4 n(n + 2) n−2 n−2 uε,R,a vφ + − huε,R (3.11) 4 4(n − 1) 4−n

4

4

n−2 n−2 − uε,R = where Rg − ∂i ∂j hij = O(|x|n−3 ), uε,R,a − uε,R = O(|a||x| 2 ), uε,R,a O(|a||x|−1 ) by the proof of Corollary 2.4, and h = ∂i ∂j hij −π (∂i ∂j hij ) = O(|x|N ) with N large. Hence, we obtain the following lemma

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Lemma 3.4. Let n ≥ 8, μ ∈ (1, 3/2), κ > 0 and c > 0 be fixed constants. There exists ε0 ∈ (0, 1) such that for each ε ∈ (0, 1), for all v ∈ 2+d−μ− n 2 −δ1 , Cμ2,α (Brε (0)\{0}) and φ ∈ π (C 2,α (Sn−1 rε )) with v(2,α),μ,rε ≤ crε 2+d− n 2 −δ1

and φ(2,α),rε ≤ κrε 0,α (Brε (0)\{0}). to Cμ−2

, we have that the right-hand side of (3.11) belongs

Proof. As before in Lemma 3.3, we obtain 0,α Qε,R,a (wε,R + vφ + v) ∈ Cμ−2 (Brε (0)\{0})

and

n (Δ − Δg )uε,R,a = O |x|1+d− 2 = O(|x|μ−2 ).

Therefore, the assertion follows, since for the remaining terms we obtain the same estimate. Let μ ∈ (1, 3/2) and c > 0 be fixed constants. It is enough to show that the map Nε (R, a, φ, ·) : Bε,c,δ1 → Cμ2,α (Br (0)\{0}) has a fixed point for suitable parameters ε, R, a and φ, where Bε,c,δ1 is the ball in Cμ2,α (Br (0)\{0}) of radius 2+d−μ− n 2 −δ1

and Nε (R, a, φ, ·) is defined by

n−2 Rg v − Qε,R,a (vφ + wε,R + v) Nε (R, a, φ, v) = Gε,R,r,a (Δ − Δg )v + 4(n − 1) n−2 Rg (vφ + wε,R ) + (Δ − Δg )(uε,R,a + vφ + wε,R ) + 4(n − 1) 4 n−2 n(n + 2) n−2 (Rg − ∂i ∂j hij )uε,R,a − uε,R,a vφ + 4(n − 1) 4

4 4 n(n + 2) n−2 n−2 n−2 + − uε,R,a huε,R wε,R + uε,R 2 4(n − 1) n−2 ∂i ∂j hij (uε,R,a − uε,R ) . + (3.12) 4(n − 1)

crε

In fact, we will show that the map Nε (R, a, φ, ·) is a contraction for small enough ε > 0, and as a consequence of this we will get that the fixed point is continuous with respect to the parameters ε, R, a, and φ. Remark 3.5. The vanishing of the Weyl tensor up to the order d−2 is sharp, in the following sense: if ∇l Wg (0) = 0, l = 0, 1, . . . , d−3, then for n ≥ 6, gij = δij +O(|x|d ) and n (Δ − Δg )uε,R,a = O |x|d− 2 . 0,α (Brε (0)\{0}), with μ > 1. This implies (Δ − Δg )uε,R,a ∈ Cμ−2

The next lemma will be very useful to show Proposition 3.1. To prove it use the Laplacian in local coordinates.

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Lemma 3.6. Let g be a metric in Br (0) ⊂ Rn in conformal normal coordinates with the Weyl tensor satisfying the assumption (3.1). Then, for all μ ∈ R and v ∈ Cμ2,α (Br (0)\{0}) there is a constant c > 0 that does not depend on r and μ such that (Δ − Δg )(v)(0,α),μ−2,r ≤ crd+1 v(2,α),μ,r . 3.2. Complete Delaunay-type Ends The previous discussion tells us that to solve the Eq. (3.3) with prescribed boundary data on a small sphere centered at 0, we have to show that the map Nε (R, a, φ, ·), defined in (3.8) for 3 ≤ n ≤ 7 and in (3.12) for n ≥ 8, has a fixed point. To do this, we will show that this map is a contraction using the fact that the right inverse Gε,R,rε ,a of Lε,R,a in the punctured ball Brε (0)\{0}, given by Corollary 2.4, has norm bounded independently of ε, R, a, and rε . Next we will prove the main result of this section. This will solve the singular Yamabe problem locally. Remark 3.7. To ensure some estimates that we will need, from now on, we will 2−n consider R 2 = 2(1 + b)ε−1 , with |b| ≤ 1/2. Proposition 3.1. Let μ ∈ (1, 5/4), τ > 0, κ > 0 and δ2 > δ1 be fixed constants. There exists a constant ε0 ∈ (0, 1) such that for each ε ∈ (0, ε0 ], |b| ≤ 1/2, a ∈ Rn 2+d− n 2 −δ1 with |a|rε1−δ2 ≤ 1, and φ ∈ π (C 2,α (Sn−1 , there rε )) with φ(2,α),rε ≤ κrε 2+d−μ− n 2

exists a fixed point of the map Nε (R, a, φ, ·) in the ball of radius τ rε Cμ2,α (Brε (0)\{0}).

in

Proof. First, note that |a|rε ≤ rεδ2 → 0 when ε tends to zero. It follows from Corollary 2.4, Lemmas 3.3 and 3.4 that the map Nε (R, a, φ, ·) is well defined in the 2+d−μ− n 2 ball of radius τ rε in Cμ2,α (Brε (0)\{0}) for small ε > 0. Following [5] we will show that 1 2+d−μ− n2 Nε (R, a, φ, 0)(2,α),μ,rε < τ rε , (3.13) 2 2+d−μ− n

2 and for all vi ∈ Cμ2,α (Brε (0)\{0}) with vi (2,α),μ,rε ≤ τ rε , i = 1, 2, we will have 1 Nε (R, a, φ, v1 ) − Nε (R, a, φ, v2 )(2,α),μ,rε < v1 − v2 (2,α),μ,rε . (3.14) 2 It will follow from this that for all v ∈ Cμ2,α (Brε (0)\{0}) in the ball of radius

2+d−μ− n 2

τ rε

we will get

Nε (R, a, φ, v)(2,α),μ,rε ≤ Nε (R, a, φ, v) − Nε (R, a, φ, 0)(2,α),μ,rε + Nε (R, a, φ, 0)(2,α),μ,rε . Hence, we conclude that the map Nε (R, a, φ, ·) will have a fixed point belonging 2+d−μ− n 2 to the ball of radius τ rε in Cμ2,α (Brε (0)\{0}). Consider 3 ≤ n ≤ 7.

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Since Gε,R,rε ,a is bounded independently of ε, R and a, it follows that Nε (R, a, φ, 0)(2,α),μ,rε ≤ c (Δ − Δg )(uε,R,a + vφ )(0,α),μ−2,rε + Rg (uε,R,a + vφ )(0,α),μ−2,rε + Qε,R,a (vφ )(0,α),μ−2,rε 4 n−2 + , uε,R,a vφ (0,α),μ−2,rε

where c > 0 is a constant that does not depend on ε, R and a. Using local coordinates we obtain that σ 2−μ (Δ − Δg )(uε,R,a − uε,R )(0,α),[σ,2σ] ≤ cσ 1+d−μ uε,R,a − uε,R (2,α),[σ,2σ] ≤ c|a|σ 3+d−μ− 2 , n

since uε,R,a = uε,R + O (|a||x|

4−n 2

), by (2.10). The condition μ < 3/2 implies 3+d−μ− n 2

(Δ − Δg )(uε,R,a − uε,R )(0,α),μ−2,rε ≤ c|a|rε

.

(3.15)

As in the proof of Lemma 3.3 we have that (Δ − Δg )uε,R = O(|x| ), and from this we obtain N

(Δ − Δg )uε,R (0,α),μ−2,rε ≤ crεN ,

(3.16)

where N is as big as we want. Hence, from (3.15) and (3.16), we get 2+d−μ− n 2

(Δ − Δg )uε,R,a (0,α),μ−2,rε ≤ crεδ2 rε since

,

(3.17)

|a|rε1−δ2

≤ 1, with δ2 > 0. From Lemma 3.6 and (2.18), we conclude that 3+2d−μ− n 2 −δ1

(Δ − Δg )vφ (0,α)μ−2,rε ≤ crε1+d−μ φ(2,α),rε ≤ cκrε and then

2+d−μ− n 2

(Δ − Δg )vφ (0,α)μ−2,rε ≤ cκrε1+d−δ1 rε

.

(3.18)

Furthermore, since 5 − μ − n/2 ≥ 3 + d − μ − n/2, Rg = O(|x|2 ) and we have (3.6), we get that 5−μ− n 2

Rg uε,R,a (0,α),μ−2,rε ≤ crε

2+d−μ− n 2

≤ crε rε

.

(3.19)

Using (2.18), we also get 2+d−μ− n 2

Rg vφ (0,α),μ−2,rε ≤ crε4−μ φ(2,α),rε ≤ cκrε4−δ1 rε

,

(3.20)

with 4 − δ1 > 0. By Lemma 3.2 and (2.18), we obtain

2+d−μ− n 2

Qε,R,a (vφ )(0,α),μ−2,rε ≤ cελn rε1+d−2μ φ2(2,α),rε ≤ cκ2 εδ rε

,

(3.21)

with δ = λn + s(3 + 2d − μ − n/2 − 2δ1 ) > 0, since μ < 5/4, s > (d + 1 − δ1 )−1 and 0 < δ1 < (8n − 16)−1 .

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1513

4

n−2 Let us estimate the norm uε,R,a vφ (0,α),μ−2,rε . 4

4

n−2 n−2 = uε,R + O(|a||x|−1 ). Hence, using (2.18), we First, (2.16) implies uε,R,a deduce that

4 n−42 2+d−μ− n n−2 2−μ 2 σ uε,R,a −uε,R vφ ≤ C|a|rε1−μ φ(2,α),rε ≤ Cκrεδ2−δ1 rε ,

(0,α),[σ,2σ]

(3.22) since |a|rε1−δ2 ≤ 1, with δ2 − δ1 > 0. If rε1+λ ≤ |x| ≤ rε with λ > 0, then −s log ε ≤ − log |x| ≤ −s(1 + λ) log ε, 2−n

and by the choice of R, R 2 = 2(1 + b)ε−1 with |b| < 1/2, see Remark 3.7,we obtain

2 2 − s log ε + log(2 + 2b) 2−n ≤ − log |x| + log R n−2

2 2 −s(1+λ) log ε+log(2 + 2b) 2−n , ≤ n−2 with

2 n−2

− s > 0, since s < 4(d − 2 + 3n/2)−1 < 2(n − 2)−1 . We also have

vε (− log |x| + log R) ≤ εe(

n−2 2 s−1

) log ε+log(2+2b) = (2 + 2b)ε n−2 2 s

for small enough λ > 0. This follows from the estimate vε (t) ≤ εe Hence 4

2 s(n−2)

− 1 fixed, then

4 n−2 u ε,R

(0,α),[σ,2σ]

for

rε1+λ

, ∀t ∈ R.

4

n−2 uε,R (x) = |x|−2 vεn−2 (− log |x| + log R) ≤ Cn |x|−2 rε2 .

If we take 0 < λ < we get

n−2 2 |t|

2 n−2

(3.23)

− s(1 + λ) > 0 and from (3.23)

≤ Cσ −2 rε2 ,

−1

≤σ≤2

rε , and then 4 2−μ n−2 σ uε,R vφ

(0,α),[σ,2σ]

2+d−μ− n 2

≤ Cκrε2−δ1 rε

,

(3.24)

with 2 − δ1 > 0. For 0 ≤ σ ≤ rε1+λ , we have 4 2+d−μ− n 2−μ n−2 2 σ uε,R vφ ≤ Crε(2−μ)(1+λ)−2 φ(2,α),rε ≤ Cκrε(2−μ)λ−δ1 rε , (0,α),[σ,2σ]

(3.25) 1 2 Since s < 4(d − 2 + 3n/2)−1 , we can take λ such that 4n−8 < λ < s(n−2) − 1. This, −1 together with μ < 5/4 and 0 < δ1 < (8n − 16) implies (2 − μ)λ − δ1 > 0.

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Therefore, by (3.22), (3.24) and (3.25) we obtain 4 n−2 2+d−μ− n 2 u ≤ crεδ −μ φ ≤ cκrεδ −δ1 rε , ε,R,a vφ

(3.26)

(0,α),μ−2,rε

for some δ > δ1 fixed independent of ε. Therefore, from (3.17), (3.18), (3.19), (3.20), (3.21) and (3.26) it follows (3.13) for small enough ε > 0. For the same reason as before, Nε (R, a, φ, v1 ) − Nε (R, a, φ, v2 )(2,α),μ,rε ≤ c (Δg − Δ)(v1 − v2 )(0,α),μ−2,rε + Rg (v1 − v2 )(0,α),μ−2,rε + Qε,R,a (vφ + v1 ) − Qε,R,a (vφ + v2 )(0,α),μ−2,rε , where c > 0 is a constant independent of ε, R and a. From Lemma 3.6 and Rg = O(|x|2 ) we obtain (Δ − Δg )(v1 − v2 )(0,α),μ−2,rε ≤ crεd+1 v1 − v2 (2,α),μ,rε

(3.27)

Rg (v1 − v2 )(0,α),μ−2,rε ≤ crε4 v1 − v2 (2,α),μ,rε .

(3.28)

and

As before, Lemma 3.2 and (2.18) imply Qε,R,a (vφ + v1 ) − Qε,R,a (vφ + v2 )(0,α),μ−2,rε ≤ cκ ελn +s(3+2d−μ− 2 −δ1 ) v1 − v2 (2,α),μ,rε n

(3.29)

with λn + s(3 + 2d − μ − n/2 − δ1 ) > 0 as in (3.21). Therefore, from (3.27), (3.28) and (3.29), we deduce (3.14) provided v1 , v2 2+d−μ− n 2 belong to the ball of radius τ rε in Cμ2,α (Brε (0)\{0})) for ε > 0 chosen small enough. Consider n ≥ 8. Similarly, Nε (R, a, φ, 0)(2,α),μ,rε ≤ c (Δ − Δg )(uε,R,a + vφ + wε,R )(0,α),μ−2,rε + Rg (vφ + wε,R )(0,α),μ−2,rε + Qε,R,a (vφ + wε,R )(0,α),μ−2,rε + (Rg − ∂i ∂j hij )uε,R,a (0,α),μ−2,rε + ∂i ∂j hij (uε,R,a − uε,R )(0,α),μ−2,rε 4

4

4

n−2 n−2 n−2 + uε,R,a vφ (0,α),μ−2,rε + (uε,R − uε,R,a )wε,R (0,α),μ−2,rε + huε,R (0,α),μ−2,rε ,

where c > 0 does not depend on ε, R and a. From (2.18), (3.10), Lemma 3.2 and the fact that Rg = O(|x|d−1 ), we get 2+d−μ− n 2

(Δ − Δg )wε,R (0,α),μ−2,rε ≤ crεd+1 rε Rg (vφ + wε,R )(0,α),μ−2,rε ≤

,

2+d−μ− n 2 cκrε1+d−δ1 rε ,

(3.30) (3.31)

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1515

and 2+d−μ− n 2

Qε,R,a (vφ + wε,R )(0,α),μ−2,rε ≤ crεδ rε

,

(3.32)

for some δ > 0. Note that and

n (Rg − ∂i ∂j hij )uε,R,a = O |x| 2 −2 n ∂i ∂j hij (uε,R,a − uε,R ) = O |a||x|1+d− 2 ,

by Corollary 2.1. This implies 2+d−μ− n 2

(Rg − ∂i ∂j hij )uε,R,a (0,α),μ−2,rε ≤ crε rε

(3.33)

and 2+d−μ− n 2

∂i ∂j hij (uε,R,a − uε,R )(0,α),μ−2,rε ≤ c|a|rε rε 4 n−2

.

(3.34)

4 n−2

Finally, by the proof of Corollary 2.4 we have uε,R,a − uε,R = O(|a||x|−1 ). Hence, 4 4 n−2 2+d−μ− n n−2 2 (u ≤ c|a|rε rε . (3.35) ε,R − uε,R,a )wε,R (0,α),μ−2,rε

Since h = O(|x|N ), where N is as big as we want, by (3.17), (3.18), (3.26), (3.30), (3.31), (3.32), (3.33), (3.34) and (3.35), we deduce (3.14) for ε > 0 small enough. Now, we have Nε (R, a, φ, v1 ) − Nε (R, a, φ, v2 )(2,α),μ,rε ≤ c (Δg − Δ)(v1 − v2 )(0,α),μ−2,rε + Qε,R,a (vφ + wε,R,a + v1 ) − Qε,R,a (vφ + wε,R,a + v2 )(0,α),μ−2 , rε + Rg (v1 − v2 )(0,α),μ−2,rε . As before, we obtain (Δ − Δg )(v1 − v2 )(0,α),μ−2,rε ≤ crεd+1 v1 − v2 (2,α),μ,rε

(3.36)

Rg (v1 − v2 )(0,α),μ−2,rε ≤ crεd+1 v1 − v2 (0,α),μ,rε .

(3.37)

and By Lemma 3.2 and (2.18), we obtain Qε,R,a (vφ + wε,R,a + v1 ) − Qε,R,a (vφ + wε,R,a + v2 )(0,α),μ−2,rε ≤ cκ ελn +s(3+2d−μ− 2 −δ1 ) v1 − v2 (2,α),μ,rε , n

(3.38)

with λn + s(3 + 2d − μ − n/2 − δ1 ) > 0. Therefore, from (3.36), (3.37) and (3.38), we deduce (3.14) provided v1 , v2 2+d−μ− n 2 belong to the ball of radius τ rε in Cμ2,α (Brε (0)\{0})) for ε > 0 chosen small enough.

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We summarize the main result of this section in the next theorem. Theorem 3.8. Let μ ∈ (1, 5/4), τ > 0, κ > 0 and δ2 > δ1 be fixed constants. There exists a constant ε0 ∈ (0, 1) such that for each ε ∈ (0, ε0 ], |b| ≤ 1/2, a ∈ Rn with 2+d− n 2 −δ1 |a|rε1−δ2 ≤ 1 and φ ∈ π (C 2,α (Sn−1 , there exists rε )) with φ(2,α),rε ≤ κrε a solution Uε,R,a,φ ∈ Cμ2,α (Brε (0)\{0}) for the equation Hg (uε,R,a + wε,R + vφ + Uε,R,a,φ ) = 0 in Brε (0)\{0} on ∂Brε (0) πrε ((vφ + Uε,R,a,φ )|∂Brε (0) ) = φ 2,α where wε,R ≡ 0 for 3 ≤ n ≤ 7 and wε,R ∈ π (C2+d− n (Brε (0)\{0})) is solution of 2 the Eq. (3.9) for n ≥ 8. Moreover, 2+d−μ− n 2

Uε,R,a,φ (2,α),μ,rε ≤ τ rε

(3.39)

and Uε,R,a,φ1 − Uε,R,a,φ2 (2,α),μ,rε ≤ Crεδ3 −μ φ1 − φ2 (2,α),rε ,

(3.40)

for some constants δ3 > 0 that do not depend on ε, R, a and φi , i = 1, 2. Proof. The solution Uε,R,a,φ is the fixed point of the map Nε (R, a, φ, ·) given by Proposition 3.1 with the estimate (3.39). Use the fact that Uε,R,a,φ is a fixed point of the map Nε (R, a, φ, ·) to show that Uε,R,a,φ1 − Uε,R,a,φ2 (2,α),μ,rε ≤ 2Nε (R, a, φ1 , Uε,R,a,φ2 ) − Nε (R, a, φ2 , Uε,R,a,φ2 )(2,α),μ,rε .

From this we obtain the inequality (3.40).

We will write the full conformal factor of the resulting constant scalar curvature metric with respect to the metric g as Aε (R, a, φ) := uε,R,a + wε,R + vφ + Uε,R,a,φ , in conformal normal coordinates. More precisely, the previous analysis says that 4 the metric gˆ = Aε (R, a, φ) n−2 g is defined in Brε (p)\{p} ⊂ M ; it is complete and has constant scalar curvature Rgˆ = n(n − 1). The completeness follows from the estimate Aε (R, a, φ) ≥ c|x| for some constant c > 0.

2−n 2

,

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4. Exterior Analysis In Sect. 3 we have found a family of constant scalar curvature metrics on Brε (p)\{p} ⊂ M , conformal to g0 and with prescribed high eigenmode data. Now we will use the same method of the previous section to perturb the metric g0 and build a family of constant scalar curvature metrics on the complement of some suitable ball centered at p in M . First, using the non-degeneracy we find a right inverse for the operator L1g0 [see (2.5)], in the complement of the ball Br (p) ⊂ M for small enough r, with bounded norm independently of r. In contrast with the previous section, in which we worked with conformal normal coordinates, in this section it is better to work with the constant scalar curvature metric, since in this case the constant function 1 satisfies Hg0 (1) = 0. Hence, in this section, (M n , g0 ) is an n-dimensional nondegenerate compact Riemannian manifold of constant scalar curvature Rg0 = n(n − 1). 4.1. Analysis in M \Br (p) Let r1 ∈ (0, 1) and Ψ : Br1 (0) → M be a normal coordinate system with respect to 4 g = F n−2 g0 on M centered at p, where F is defined in Sect. 3. We denote by Gp (x) the Green’s function for L1g0 = Δg0 +n, the linearization of Hg0 about the constant function 1, with pole at p (the origin in our coordinate system). We assume that Gp (x) is normalized such that in the coordinates Ψ we have limx→0 |x|n−2 Gp (x) = 1. This implies that |Gp ◦ Ψ(x)| ≤ C|x|2−n , for all x ∈ Br1 (0). In these coordinates we have that (g0 )ij = δij + O(|x|2 ), since gij = δij + O(|x|2 ) and F = 1 + O(|x|2 ). Our goal in this section is to solve the equation Hg0 (1 + λGp + u) = 0 on M \Br (p)

(4.1)

with λ ∈ R, r ∈ (0, r1 ) and prescribed boundary data on ∂Br (p). In fact, we will get a solution with prescribed boundary data, except in the space spanned by the constant functions. To solve this equation we will use basically the same techniques that were used in Proposition 3.1. We linearize Hg0 about 1 to get Hg0 (1 + λGp + u) = L1g0 (u) + Q1 (λGp + u), since Hg0 (1) = 0 and L1g0 (Gp ) = 0, where Q1 is given by (2.4). Next, we will find a right inverse for L1g0 in a suitable space and so we will reduce the Eq. (4.1) to the problem of fixed point as in the previous section. 4.2. Inverse for L1g0 in M \Ψ(Br (0)) To find a right inverse for L1g0 , we will follow the method of Jleli in [14]. This problem is approached by decomposing f as the sum of two functions, one of them with support contained in an annulus inside Ψ(Br1 (0)). Inside the annulus we transfer the problem to normal coordinates and solve it. For the remainder term we use the right invertibility of L1g0 on M which is a consequence of the non-degeneracy.

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The next two lemmas allow us to use a perturbation argument in the annulus contained in Ψ(Br1 (0)). Lemma 4.1. Fix any ν ∈ R. There exists C > 0 independent of r and s such that if 0 < 2r < s ≤ r1 , then 1 (Lg − Δ)(v) 0,α ≤ Cs2 vCν2,α (Ωr,s ) , 0 C (Ω ) ν−2

r,s

for all v ∈ Cν2,α (Ωr,s ).

Proof. Use the Laplacian in local coordinates.

Lemma 4.2. Assume that ν ∈ (1 − n, 2 − n) is fixed and that 0 < 2r < s ≤ r1 . Then there exists an operator ˜ r,s : C 0,α (Ωr,s ) → Cν2,α (Ωr,s ) G ν−2 such that, for all f ∈ Cν0,α (Ωr,s ), the ⎧ ⎨ Δw = f w=0 ⎩ w∈R

˜ r,s (f ) is a solution of function w = G in Bs (0)\Br (0) on ∂Bs (0) . on ∂Br (0)

In addition, ˜ r,s (f ) 2,α G Cν (Ωr,s ) ≤ Cf C 0,α (Ωr,s ) , ν−2

for some constant C > 0 that does not depend on s and r.

Proof. See Lemma 13.23 in [14,15]. Proposition 4.1. Fix ν ∈ (1 − n, 2 − n). There exists r2 < r ∈ (0, r2 ) we can define an operator

1 4 r1

such that, for all

0,α (Mr ) → Cν2,α (Mr ), Gr,g0 : Cν−2 0,α with the property that, for all f ∈ Cν−2 (Mr ) the function w = Gr,g0 (f ) solves

L1g0 (w) = f, in Mr with w ∈ R constant on ∂Br (p). In addition, Gr,g0 (f )Cν2,α (Mr ) ≤ Cf C 0,α (Mr ) , ν−2

where C > 0 does not depend on r. Proof. From Lemma 4.1, a perturbation argument follows that the result of Lemma 4.2 holds for s = r1 small enough when Δ is replaced by L1g0 . We denote by Gr,r1 the corresponding operator. 0,α Let f ∈ Cν−2 (Mr ) and define a function w0 ∈ Cν2,α (Mr ) by w0 := ηGr,r1 (f |Ωr,r1 )

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where η is a smooth, radial function equal to 1 in B 21 r1 (p), vanishing in Mr1 and satisfying |∂r η(x)| ≤ c|x|−1 and |∂r2 η(x)| ≤ c|x|−2 for all x ∈ Br1 (0). From this it follows that η(2,α),[σ,2σ] is uniformly bounded in σ, for every r ≤ σ ≤ 12 r1 . Thus, w0 Cν2,α (Mr ) ≤ Cf C 0,α (Mr ) ,

(4.2)

ν−2

where the constant C > 0 is independent of r and r1 . Since w0 = Gr,r1 (f |Ωr,r1 ) in Ωr, 12 r1 , the function h := f − L1g0 (w0 ) is supported in M 12 r1 . We can consider that h is defined on the whole M with h ≡ 0 in B 21 r1 (p). By (4.2) we get hC 0,α (M ) ≤ Cr1 f C 0,α (Mr ) ,

(4.3)

ν−2

with the constant Cr1 > 0 independent of r. Since L1g0 : C 2,α (M ) → C 0,α (M ) has a bounded inverse, we can define the function w1 := χ(L1g0 )−1 (h), where χ is a smooth, radial function equal to 1 in M2r2 , vanishing in Br2 (p) and satisfying |∂r χ(x)| ≤ c|x|−1 and |∂r2 χ(x)| ≤ c|x|−2 for all x ∈ B2r2 (0), and some r2 ∈ (r, 14 r1 ) to be chosen later. This implies that χ(2,α),[σ,2σ] is uniformly bounded for r ≤ σ ≤ 12 r1 . Hence, from (4.3) w1 Cν2,α (Mr ) ≤ Cr1 (L1g0 )−1 (h)C 2,α (M ) ≤ Cr1 hC 0,α (M ) ≤ Cr1 f C 0,α (Mr ) , ν−2

(4.4) since ν < 0, where the constant Cr1 > 0 is independent of r and r2 . 0,α Define an application Fr,g0 : Cν−2 (Mr ) → Cν2,α (Mr ) as Fr,g0 (f ) = w0 + w1 . From (4.2), (4.3) and (4.4) we obtain Fr,g0 (f )Cν2,α (Mr ) ≤ Cf C 0,α (Mr ) ,

(4.5)

ν−2

and L1g0 (Fr,g0 (f )) − f C 0,α (Mr ) ≤ Cr2−1−ν f C 0,α (Mr ) ν−2

ν−2

(4.6)

since 1 − n < ν < 2 − n implies that 2 − ν > 0 and −1 − ν > 0, for some constant C > 0 independent of r and r2 . The assertion follows from a perturbation argument by (4.5) and (4.6), as in the proof of Corollary 2.4.

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4.3. Constant Scalar Curvature Metrics on M \Br (p) In this section we will solve the Eq. (4.1) using the method employed in the interior analysis, the fixed point method. In fact we will find a family of metrics with parameters λ ∈ R, 0 < r < r1 and some boundary data. For each ϕ ∈ C 2,α (Sn−1 ) L2 -orthogonal to the constant functions, let uϕ ∈ r 2,α Cν (Mr ) be such that uϕ ≡ 0 in Mr1 and uϕ ◦ Ψ = ηQr (ϕ), where Qr is defined in Sect. 2.6.2, η is a smooth, radial function equal to 1 in B 12 r1 (0), vanishing in Rn \Br1 (0), and satisfying |∂r η(x)| ≤ c|x|−1 and |∂r2 η(x)| ≤ c|x|−2 for all x ∈ Br1 (0). As before, we have η(2,α),[σ,2σ] ≤ c, for every r ≤ σ ≤ 12 r1 . Hence, using (2.20) we conclude that uϕ Cν2,α (Mr ) ≤ cr−ν ϕ(2,α),r ,

(4.7)

for all ν ≥ 1 − n. Finally, substituting u := uϕ + v in Eq. (4.1), we have that to show the existence of a solution of the Eq. (4.1) it is enough to show that for suitable λ ∈ R, ) the map Mr (λ, ϕ, ·) : Cν2,α (Mr ) → Cν2,α (Mr ), given by and ϕ ∈ C 2,α (Sn−1 r Mr (λ, ϕ, v) = −Gr,g0 (Q1 (λGp + uϕ + v) + L1g0 (uϕ )),

(4.8)

has a fixed point for small enough r > 0. We will show that Mr (λ, ϕ, ·) is a contraction, and as a consequence the fixed point will depend continuously on the parameters r, λ and ϕ. Proposition 4.2. Let ν ∈ (3/2 − n, 2 − n), δ4 ∈ (0, 1/2), β > 0 and γ > 0 be fixed constants. There exists r2 ∈ (0, r1 /4) such that if r ∈ (0, r2 ), λ ∈ R with 3n ) is L2 -orthogonal to the constant functions |λ|2 ≤ rd−2+ 2 , and ϕ ∈ C 2,α (Sn−1 r 2+d− n −δ 4 2 , then there is a fixed point of the map Mr (λ, ϕ, ·) with ϕ(2,α),r ≤ βr n in the ball of radius γr2+d−ν− 2 in Cν2,α (Mr ). Proof. As in Proposition 3.1 we will show that Mr (λ, ϕ, 0)Cν2,α (Mr ) ≤

1 2+d−ν− n2 γrε 2

(4.9)

and Mr (λ, ϕ, v1 ) − Mr (λ, ϕ, v2 )Cν2,α (Mr ) ≤

1 v1 − v2 Cν2,α (Mr ) , 2

(4.10)

2+d−ν− n

2 , i = 1 and 2. for all vi ∈ Cν2,α (Mr ) with vi Cν2,α (Mr ) ≤ γrε From (4.8) and Proposition 4.1 it follows that Mr (λ, ϕ, 0)Cν2,α (Mr ) ≤ c Q1 (λGp + uϕ )C 0,α (Mr ) + L1g0 (uϕ )C 0,α (Ω ν−2

for some constant c > 0 independent of r.

ν−2

r,r1 )

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Note that |λGp | ≤ cr1+ 2 − 4 , with 1 + d/2 − n/4 > 0 and c > 0 independent of r, and from (2.4) d

n

n(n + 2) 2 u Q (u) = n−2 1

1 1

6−n

(1 + stu) n−2 s ds dt 0

(4.11)

0

for 1 + stu > 0. Since 0 < c < 1 + stλGp < C in Mr1 for small enough r, then 6−n

C 0,α M 1 r

max (1 + stλGp ) n−2

t∈[0,1]

≤ c,

2 1

and Gp C 0,α (M 1 r

2 1

)

≤ c,

(4.12)

where c > 0 is a constant independent of r. Thus, by (4.11) and (4.12) we have Q1 (λGp )

C 0,α M 1 r

≤ C|λ|2 ≤ Crδ r2+d−ν− 2 , n

(4.13)

2 1

where the constant C > 0 does not depend on r and δ = 2n − 4 + ν > 0 since ν > 3/2 − n. n Now, observe that (4.7) implies |uϕ (x)| ≤ cβr2+d− 2 −δ4 , ∀x ∈ Mr , with d n 2 + d − n/2 − δ4 > 0. From this and |λGp (x)| ≤ cr1+ 2 − 4 for all x ∈ Ωr,r1 , we get 0 < c < 1 + t(λGp + uϕ ) < C for every 0 ≤ t ≤ 1. Again, using (4.7) and 6−n d n |λ∇Gp | ≤ cr 2 − 4 , we conclude that the H¨ older norm of (1 + t(λGp + uϕ )) n−2 is bounded independently of r and t. Therefore, 6−n

max (1 + t(λGp + uϕ )) n−2 (0,α),[σ,2σ] ≤ C.

0≤t≤1

From (4.11) we obtain σ

2−ν

5

Q1 (λGp + uϕ )(0,α),[σ,2σ] ≤ Cσ 2−ν λGp + uϕ 2(0,α),[σ,2σ] ≤ Cβ r 2 +d−ν− 2 , n

since n ≥ 3, δ4 < 1/2, r ≤ σ and ν > 3/2 − n implies that 6 + 2d − ν − n − 2δ4 > 5/2 + d − ν − n/2 and 9/2 − ν − 2n < 0. Therefore, Q1 (λGp + uϕ )C 0,α (Ωr,r ν−2

1

1

)

≤ Cβ r 2 r2+d−ν− 2 , n

(4.14)

and from (4.13) and (4.14), we get

Q1 (λGp + uϕ )C 0,α (Mr ) ≤ Cβ rδ r2+d−ν− 2 , n

ν−2

(4.15)

for some constant δ > 0 independent of r. From ΔQr = δ ij ∂i ∂j Qr = 0, (g0 )ij = δij + O(|x|2 ) and det g0 = 1 + O(|x|2 ), we obtain Δg0 (uϕ )(0,α),[σ,2σ] ≤ C Qr (ϕ)(2,α),[σ,2σ] + σ −2 η(0,α),[σ,2σ] Qr (ϕ)(2,α),[σ,2σ] , where the term with σ −2 appears only for σ > 14 r1 , since ∂i η ≡ 0 in B 12 r1 (0).

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Therefore, using that 3 − n − ν > 0 and δ4 ∈ (0, 1/2) we get σ 2−ν L1g0 (uϕ )(0,α),[σ,2σ] ≤ Cr1 rn−1 ϕ(2,α),r = Cr1 βrn−1+ν−δ4 r2+d−ν− 2 , n

with n − 1 + ν − δ4 > 0. This implies 1 Lg (uϕ ) 0,α 0 C (Ω ν−2

≤ Cr1 βrn−1+ν−δ4 r2+d−ν− 2 , n

r,r1 )

(4.16)

with n − 1 + ν − δ4 > 0. Therefore, by (4.15) and (4.16) we obtain (4.9) for r > 0 small enough. For the same reason as before, we have Mr (λ, ϕ, v1 ) − Mr (λ, ϕ, v0 )Cν2,α (Mr ) ≤ cQ1 (λGp + uϕ + v1 ) − Q1 (λGp + uϕ + v0 )C 0,α (Mr ) . ν−2

Furthermore, Q1 (λGp + uϕ + v1 ) − Q1 (λGp + uϕ + v0 ) 1 1 6−n n(n + 2) = (v1 − v0 ) (1 + szt ) n−2 zt ds dt, n−2 0

0

where zt = λGp + uϕ + v0 + t(v1 − v0 ), since for small enough r > 0 we have 0 < c < 1 + szt < C. This implies 6−n

(1 + szt ) n−2 C 0,α (M 1 r

2 1

)

≤C

and

6−n

(1 + szt ) n−2 (0,α),[σ,2σ] ≤ C,

with the constant C > 0 independent of r. Then, by (4.12), we have Q1 (λGp + v1 ) − Q1 (λGp + v0 )C 0,α (Mr1 )

d 3n n ≤ C r 2 −1+ 4 + r2+d−ν− 2 v1 − v0 Cν2,α (Mr ) and σ 2−ν Q1 (λGp + uϕ + v1 ) − Q1 (λGp + uϕ + v0 )(0,α),[σ,2σ] ≤ C |λ|σ 4−n + σ 2 uϕ (2,α),[σ,2σ] + σ 2 v1 (2,α),[σ,2σ] + σ 2 v0 (2,α),[σ,2σ] σ −ν v1 − v0 (0,α),[σ,2σ] ≤ Cr1 ,β r2+ 2 − 4 v1 − v0 Cν2,α (Mr ) d

n

since 1 + ν < 0, 2 + d/2 − n/4 < 3 + d − n/2 < 4 + d − n/2 − δ4 and 0 < δ4 < 1/2. Notice that 2 + d/2 − n/4 > 0. Therefore, we deduce (4.10) for small enough r > 0. From Proposition 4.2 we get the main result of this section.

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Theorem 4.3. Let ν ∈ (3/2 − n, 2 − n), δ4 ∈ (0, 1/2), β > 0 and γ > 0 be fixed con3n stants. There is r2 ∈ (0, r1 /2) such that if r ∈ (0, r2 ), λ ∈ R with |λ|2 ≤ rd−2+ 2 , ) is L2 -orthogonal to the constant functions with ϕ(2,α),r ≤ and ϕ ∈ C 2,α (Sn−1 r 2+d− n −δ 4 2 , then there is a solution Vλ,ϕ ∈ Cν2,α (Mr ) to the problem βr Hg0 (1 + λGp + uϕ + Vλ,ϕ ) = 0 in Mr . (uϕ + Vλ,ϕ ) ◦ Ψ|∂Br (0) − ϕ ∈ R on ∂Mr Moreover, Vλ,ϕ Cν2,α (Mr ) ≤ γr2+d−ν− 2 ,

(4.17)

Vλ,ϕ1 − Vλ,ϕ2 Cν2,α (Mr ) ≤ Crδ5 −ν ϕ1 − ϕ2 (2,α),r ,

(4.18)

n

and

for some constant δ5 > 0 small enough independent of r. Proof. The solution Vλ,ϕ is the fixed point of Mr (λ, ϕ, ·) given by Proposition 4.2 with the estimate (4.17). The inequality (4.18) follows similarly to (3.40). Define f := 1/F, where F is the function defined in Sect. 3.1. We have 4 g0 = f n−2 g with f = 1 + O(|x|2 ) in conformal normal coordinates centered at p. We will denote the full conformal factor of the resulting constant scalar curvature metric in Mr with respect to the metric g as Br (λ, ϕ), that is, the metric 4

g˜ = Br (λ, ϕ) n−2 g has constant scalar curvature Rg˜ = n(n − 1), where Br (λ, ϕ) := f + λf Gp + f uϕ + f Vλ,ϕ .

5. Constant Scalar Curvature on M \{p} The main task of this section is to prove the following theorem: Theorem 5.1. Let (M n , g0 ) be an n-dimensional compact Riemannian manifold of scalar curvature Rg0 = n(n − 1), nondegenerate about 1, and let p ∈ M be such that ∇k Wg0 (p) = 0 for k = 0, . . . , d − 2, where Wg0 is the Weyl tensor. Then there exist a constant ε0 and a one-parameter family of complete metrics gε on M \{p} defined for ε ∈ (0, ε0 ) such that i) ii) iii)

each gε is conformal to g0 and has constant scalar curvature Rgε = n(n − 1); gε is asymptotically Delaunay; gε → g0 uniformly on compact sets in M \{p} as ε → 0.

If the dimension is at most 5, no condition on the Weyl tensor is needed. Let us give some examples of non-locally conformally flat manifolds for which the theorem applies.

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Example. The spectrum of the Laplacian on the n-sphere Sn (k) of constant curvature k > 0 is given by Spec(Δg ) = {i(n + i − 1)k; i = 0, 1, . . .}. Consider the product manifolds S2 (k1 ) × S2 (k2 ) and S2 (k3 ) × S3 (k4 ). If we normalize so that the curvatures satisfy the conditions k1 + k2 = 6 and k3 + 3k4 = 10, then the operator given in Definition 1.4 with u = 1 is equal to L1g12 = Δg12 + 4 and L1g34 = Δg34 + 5, where g12 and g34 are the standard metrics on S2 (k1 ) × S2 (k2 ) and S2 (k3 ) × S3 (k4 ), respectively. Notice that we have Rg12 = 12 and Rg23 = 20. It is not difficult to show that the spectra satisfy Spec L1g12 ⊆ {i(i + 1)km − 4; m = 1, 2 and i = 0, 1, . . .} ∪ [8, ∞) and

Spec L1g34 ⊆ {i(i + 1)k3 − 4, i(i + 2)k4 − 4; i = 0, 1, . . .} ∪ [6, ∞).

The product S2 (k1 ) × S2 (k2 ) with normalized constant scalar curvature equal to 12, is degenerate if and only if k1 = 4/(i(i + 1)) or k2 = 4/(i(i + 1)) for some i = 1, 2, . . . For the product S2 (k3 ) × S3 (k4 ) with normalized constant scalar curvature equal to 20, we conclude that it is degenerate if and only if k3 = 4/(i(i + 1)) or k4 = 4/(i(i + 2)), for some i = 1, 2, . . . Therefore, we conclude that only countably many of these products are degenerate. In previous sections we have constructed a family of constant scalar curvature metrics on Brε (p), conformal to g0 and singular at p, with parameters ε ∈ (0, ε0 ) for some ε0 > 0, R > 0, a ∈ Rn and high eigenmode boundary data φ. We have also constructed a family of constant scalar curvature metrics on Mr = M \Br (p) conformal to g0 with parameters r ∈ (0, r2 ) for some r2 > 0, λ ∈ R and boundary data ϕ L2 -orthogonal to the constant functions. In this section, we examine suitable choices of the parameter sets on each piece so that the Cauchy data can be made to match up to be C 1 at the boundary of Brε (p). In this way we obtain a weak solution to Hg0 (u) = 0 on M \{p}. It follows from elliptic regularity theory and the ellipticity of Hg0 that the glued solutions are smooth metric. To do this we will split the equation that the Cauchy data must satisfy in an equation corresponding to the high eigenmode, another one corresponding to the space spanned by the constant functions, and n equations corresponding to the space spanned by the coordinate functions. 5.1. Matching the Cauchy Data From Theorem 3.8 there is a family of constant scalar curvature metrics in Brε (p)\{p}, for small enough ε > 0, satisfying the following: 4

gˆ = Aε (R, a, φ) n−2 g, with Rgˆ = n(n − 1), Aε (R, a, φ) = uε,R,a + wε,R + vφ + Uε,R,a,φ ,

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in conformal normal coordinates centered at p, and with I1) I2) I3) I4) I5)

2−n

R 2 = 2(1 + b)ε−1 and |b| ≤ 1/2; 2+d− n 2 −δ1 φ ∈ π (C 2,α (Sn−1 , δ1 ∈ (0, (8n − 16)−1 ) rε )) with φ(2,α),rε ≤ κrε and κ > 0 is some constant to be chosen later; |a|rε1−δ2 ≤ 1 with δ2 > δ1 ; 2,α wε,R ≡ 0 for 3 ≤ n ≤ 7, wε,R ∈ π (C2+d− n (Brε (0)\{0})) is the solution of 2 the Eq. (3.9) for n ≥ 8; Uε,R,a,φ ∈ Cμ2,α (Brε (0)\{0}) with πrε (Uε,R,a,φ |∂Brε (0) ) = 0, satisfies the 2+d−μ− n 2

inequality (3.40) and has norm bounded by τ rε and τ > 0 is independent of ε and κ.

, with μ ∈ (1, 5/4)

Also, from Theorem 4.3 there is a family of constant scalar curvature metrics in Mrε = M \Brε (p), for small enough ε > 0, satisfying the following: 4

g˜ = Brε (λ, ϕ) n−2 g, with Rg˜ = n(n − 1), Brε (λ, ϕ) = f + λf Gp + f uϕ + f Vλ,ϕ , in conformal normal coordinates centered at p, with E1)

f = 1 + f with f = O(|x|2 );

E2) E3)

2 λ ∈ R with |λ|2 ≤ rε ; 2,α n−1 2 ϕ ∈ C (Srε ) is L -orthogonal to the constant functions and belongs to

E4)

4 2 , δ4 ∈ (0, 1/2) and β > 0 is a constant to be the ball of radius βrε chosen later; Vλ,ϕ ∈ Cν2,α (Mrε ) is constant on ∂Mrε , satisfies the inequality (4.18) and 2+d−ν− n 2 has norm bounded by γrε , with ν ∈ (3/2 − n, 2 − n) and γ > 0 is a constant independent of ε and β.

d−2+ 3n

2+d− n −δ

Recall that rε = εs with (d+1−δ1 )−1 < s < 4(d−2+3n/2)−1 , see Remark 3.1. For example, we can choose δ1 = 1/8n and s = 2(n − 1 − 1/2n)−1 . We want to show that there are parameters, R ∈ R+ , a ∈ Rn , λ ∈ R, and ϕ, φ ∈ C 2,α (Sn−1 rε ) such that Aε (R, a, φ) = Brε (λ, ϕ) (5.1) ∂r Aε (R, a, φ) = ∂r Brε (λ, ϕ) on ∂Brε (p). First, let δ1 ∈ (0, (8n − 16)−1 ) be fixed. If we take ω and ϑ in the ball of 2+d− n 2 −δ1 radius rε in C 2,α (Sn−1 rε ), with ω belonging to the space spanned by the coordinate functions, ϑ belonging to the high eigenmode, and we define ϕ := ω +ϑ, then we can apply Theorem 4.3 with β = 2 and δ4 = δ1 , to define Brε (λ, ω + ϑ), 2+d− n 2 −δ1 since ϕ(2,α),rε ≤ 2rε .

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Now define ) φϑ := πrε ((Brε (λ, ω + ϑ) − uε,R,a − wε,R )|Sn−1 rε = πrε ((f + λf Gp + f uω+ϑ + f Vλ,ω+ϑ − uε,R,a − wε,R )|Sn−1 ) + ϑ, rε

(5.2)

where in the second equality we use that πrε (uω+ϑ |Sn−1 ) = ϑ, πrε (Vλ,ω+ϑ |Sn−1 )=0 rε rε 2 and f = 1 + f , with f = O(|x| ). We have to derive an estimate for φϑ (2,α),rε . To do this, we will use the inequality (2.12) in Lemma 2.3. But before that, from (2.11) in Corollary 2.1, we obtain

= O |a|2 rε2 , (5.3) πrε uε,R,a |Sn−1 r ε

2−n 2

since rε = εs and R = 2(1 + b)ε−1 with s < 4(d − 2 + 3n/2)−1 < 2(n − 2)−1 and |b| ≤ 1/2 implies that R < rε for small enough ε > 0. d− n Let 1 + d/2 − n/4 > δ2 > δ1 and let a ∈ Rn with |a|2 ≤ rε 2 (δ2 = 1/8, 1+ d − n −δ

for example). Hence, we have that |a|rε1−δ2 ≤ rε 2 4 2 tends to zero when ε goes to zero, and I3) is satisfied for ε > 0 small enough. Furthermore, since 2+d− n 2 |a|2 rε2 ≤ rε , we can show that 2+d− n 2

πrε (uε,R,a |Sn−1 )(2,α),rε ≤ Crε rε

,

(5.4)

for some constant C > 0 independent of ε, R and a.

d−2+ 3n 2

Observe that (f Gp )(x) = |x|2−n + O(|x|3−n ) and |λ|2 ≤ rε

2+ d − n πrε (λ(f Gp )|Sn−1 ) = O rε 2 4 , rε

imply

with 2 + d/2 − n/4 > 2 + d − n/2. Thus 2+d− n 2

πrε (λ(f Gp )|Sn−1 )(2,α),rε ≤ Crε r ε

.

(5.5)

Now, using (2.20), (3.10), (4.17), (5.2), Lemma 2.3 and the fact that f = O(|x|2 ), we deduce that 2+d− n 2

φϑ − ϑ(2,α),rε ≤ crε

,

(5.6)

and 2+d− n 2 −δ1

φϑ (2,α),rε ≤ crε

, 2+d− n −δ

1 2 for every ϑ ∈ π (C 2,α (Sn−1 , for some constant rε )) in the ball of radius rε c > 0 that does not depend on ε. Therefore, we can apply Theorem 3.8 with κ equal to this constant c and Aε (R, a, φϑ ) is well defined. The definition (5.2) immediately yields

n−1 = π B . (λ, ω + ϑ)| πrε Aε (R, a, φϑ )|Sn−1 r r ε S ε rε rε

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We project the second equation of the system (5.1) on the high eigenmode, the space of functions which are L2 (Sn−1 )-orthogonal to e0 , . . . , en . This yields a nonlinear equation which can be written as rε ∂r (vϑ − uϑ ) + Sε (a, b, λ, ω, ϑ) = 0,

(5.7)

on ∂r Brε (0), where

+ rε ∂r wε,R Sε (a, b, λ, ω, ϑ) = rε ∂r vφϑ −ϑ + rε ∂r πrε uε,R,a |Sn−1 rε

n−1 . − r (f V +rε ∂r πrε (Uε,R,a,φϑ − f − λf Gp − f uω+ϑ )|Sn−1 ∂ π )| ε r λ,ω+ϑ rε Srε rε

Since vϑ = Pr (ϑ) and uϑ = Qr (ϑ) in Ωrε , 12 r1 ⊂ Mrε for some r1 > 0, see Sect. 4.3, from (2.17) and (2.19), we conclude that rε ∂r (vϑ − uϑ )(rε ·) = ∂r (P1 (ϑ1 ) − Q1 (ϑ1 )), where ϑ1 ∈ C 2,α (Sn−1 ) is defined by ϑ1 (θ) := ϑ(rθ). Define an isomorphism Z : π (C 2,α (S n−1 )) → π (C 1,α (S n−1 )) by Z(ϑ) := ∂r (P1 (ϑ) − Q1 (ϑ)), (see [14], proof of Proposition 8 in [30] and proof of Proposition 2.6 in [36]). To solve the Eq. (5.7) it is enough to show that the map Hε (a, b, λ, ω, ·) : Dε → π (C 2,α (S n−1 )) given by Hε (a, b, λ, ω, ϑ) = −Z −1 (Sε (a, b, λ, ω, ϑrε )(rε ·)), 2+d− n 2 −δ1

has a fixed point, where Dε := {ϑ ∈ π (C 2,α (Sn−1 )); ϑ(2,α),1 ≤ rε ϑrε (x) := ϑ(rε−1 x).

} and

Lemma 5.2. There is a constant ε0 > 0 such that if ε ∈ (0, ε0 ), a ∈ Rn with d−2+ 3n

d− n

2 |a|2 ≤ rε 2 , b and λ in R with |b| ≤ 1/2 and |λ|2 ≤ rε , and ω ∈ C 2,α (Sn−1 rε ) belongs to the space spanned by the coordinate functions and with norm bounded 2+d− n 2 −δ1 , then the map Hε (a, b, λ, ω, ·) has a fixed point in Dε . by rε

Proof. As before, in Propositions 3.1 and 4.2 it is enough to show that 1 2+d− n2 −δ1 Hε (a, b, λ, ω, 0)(2,α),1 ≤ rε 2 and 1 Hε (a, b, λ, ω, ϑ1 ) − Hε (a, b, λ, ω, ϑ2 )(2,α),1 ≤ ϑ1 − ϑ2 (2,α),1 , 2 for all ϑ1 , ϑ2 ∈ Dε . Since Z is an isomorphism, we have that Hε (a, b, λ, ω, 0)(2,α),1 ≤ CSε (a, b, λ, ω, 0)(1,α),rε where by (5.6), φ0 satisfies 2+d− n 2

φ0 (2,α),rε ≤ crε

,

where the constant C > 0 and c > 0 are independent of ε.

(5.8)

(5.9)

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From (2.13), (2.18), (2.20), (3.10), (3.39), (4.17), (5.4) and (5.5) and the fact that f = O(|x|2 ) we obtain 2+d− n 2

Sε (a, b, λ, ω, 0)(1,α),rε ≤ crε

.

for some constant c > 0 independent of ε. Therefore, we get (5.8) for small enough ε. Now, we have Hε (a, b, λ, ω, ϑ1 ) − Hε (a, b, λ, ω, ϑ2 )(2,α),1 ≤ C rε ∂r vφϑr ,1 −ϑrε ,1 −(φϑr ,2 −ϑrε ,2 ) (1,α),rε ε ε

(1,α),rε + rε ∂r πrε (Uε,R,a,φϑr ,1 − Uε,R,a,φϑr ,2 )|Sn−1 rε ε ε

(1,α),rε + rε ∂r πrε f (Vλ,ω+ϑrε ,1 − Vλ,ω+ϑrε ,2 ) |Sn−1 rε

+ rε ∂r πrε ((f uϑrε ,1 −ϑrε ,2 )|Sn−1 )(1,α),rε , rε where, by (5.2) φϑrε ,1 − ϑrε ,1 − (φϑrε ,2 − ϑrε ,2 )

. = πrε f uϑrε ,1 −ϑrε ,2 + f (Vλ,ω+ϑrε ,1 − Vλ,ω+ϑrε ,2 ) |Sn−1 rε Using the inequality (2.12) of Lemma 2.3, (2.20), (4.18) and the fact that f = O(|x|2 ), we obtain φϑrε ,1 − ϑrε ,1 − (φϑrε ,2 − ϑrε ,2 )(2,α),rε ≤ crεδ6 ϑrε ,1 − ϑrε ,2 (2,α),rε , for some constants δ6 > 0 and c > 0 that does not depend on ε. This implies ≤ crεδ6 ϑ1 − ϑ2 (2,α),1 . (5.10) rε ∂r vφϑr ,1 −ϑrε ,1 −(φϑr ,2 −ϑrε ,2 ) ε

ε

(1,α),rε

From (3.40) and (4.18) we conclude that ≤ Crεδ1 ϑrε ,1 − ϑrε ,2 (2,α),rε Uε,R,a,φϑrε ,1 − Uε,R,a,φϑrε ,2 1 (2,α),[ 2 rε ,rε ]

and Vλ,ω+ϑrε ,1 − Vλ,ω+ϑrε ,2 (2,α),[rε ,2rε ] ≤ Crεδ5 ϑrε ,1 − ϑrε ,2 (2,α),rε , for some δ1 > 0 and δ5 > 0 independent of ε. From this, (2.20) and the fact that f = 1 + f , we derive an estimate as (5.10) for the other terms, and from this the inequality (5.9) follows, since ε is small enough. Therefore, there exists a unique solution of (5.7) in the ball of radius 2+d− n 2 −δ1 rε in C 2,α (Sn−1 rε ). We denote by ϑε,a,b,λ,ω this solution given by Lemma 5.2. Since this solution is obtained through the application of fixed point theorems for contraction mappings, it is continuous with respect to the parameters ε, a, b, λ, and ω.

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2−n

Recall that R 2 = 2(1 + b)ε−1 with |b| ≤ 1/2. Hence, using (5.3) and Corollaries 2.1 and 2.2 we show that uε,R,a (rε θ) = 1 + b +

ε2 rε 2−n + ((n − 2)uε,R (rε θ) + r∂r uε,R (rε θ))a · x 4(1 + b)

+ O(|a|2 rε2 ) + O(ε2 n−2 rε −n ), n+2

where the last term, O(ε2 n−2 rε−n ), does not depend on θ. Hence, we have n+2

ε2 rε 2−n +vφϑε,a,b,λ,ω (rε θ) + wε,R (rε θ) 4(1 + b) + ((n − 2)uε,R (rε θ) + rε ∂r uε,R (rε θ))rε a · θ

n+2 + Uε,R,a,φϑε,a,b,λ,ω (rε θ)+O |a|2 rε2 +O ε2 n−2 rε−n .

Aε (R, a, φϑε,a,b,λ,ω )(rε θ) = 1+ b +

In the exterior manifold Mrε , in conformal normal coordinate system in the neighborhood of ∂Mrε , namely Ωrε , 12 r1 , we have Brε (λ, ω + ϑε,a,b,λ,ω )(rε θ) = 1 + λrε2−n + uω+ϑε,a,b,λ,ω (rε θ) + f (rε θ) + (f uω+ϑε,a,b,λ,ω )(rε θ) + (f Vλ,ω+ϑε,a,b,λ,ω )(rε θ) + O |λ|rε3−n . 2,α Using that wε,R ∈ π (C2+d− n (Brε (0)\{0})), we now project the system (5.1) on 2 the set of functions spanned by the constant function. This yields the equations ⎧ 2

ε ⎨b + − λ rε2−n = H0,ε (a, b, λ, ω) 4(1+b) 2

, (5.11) ε ⎩ (2 − n) − λ rε2−n = rε ∂r H0,ε (a, b, λ, ω) 4(1+b)

where H0,ε and ∂r H0,ε are continuous maps and satisfy

2+d− n 2+d− n 2 2 and rε ∂r H0,ε (a, b, λ, ω) = O rε . H0,ε (a, b, λ, ω) = O rε

(5.12)

Lemma 5.3. There is a constant ε1 > 0 such that if ε ∈ (0, ε1 ), a ∈ Rn with d− n |a|2 ≤ rε 2 and ω ∈ C 2,α (Sn−1 rε ) belongs to the space spanned by the coordinate 2+d− n 2 −δ1

functions and has norm bounded by rε 2

2

tion (b, λ) ∈ R , with |b| ≤ 1/2 and |λ| ≤

, then the system (5.11) has a solu-

d−2+ 3n 2 rε .

Proof. Define a continuous map Gε,a,ω : D0,ε → R2 by

rε ∂r H0,ε (a, b, λ, ω) + H0,ε (a, b, λ, ω), Gε,a,ω (b, λ) := n−2 rεn−1 ε2 + ∂r H0,ε (a, b, λ, ω) , 4(1 + b) n − 2 d

where D0,ε := {(b, λ) ∈ R2 ; |b| ≤ 1/2 and |λ| ≤ rε2

−1+ 3n 4

}.

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Then, using (5.12) and the fact that 2 > s(d/2 − 1 + 3n/4), we can show that Gε,a,ω (D0,ε ) ⊂ D0,ε , for small enough ε > 0. By the Brouwer’s fixed point theorem it follows that there exists a fixed point of the map Gε,a,ω . Obviously, this fixed point is a solution of the system (5.11). With further work, one can also show that the mapping is a contraction, and hence that the fixed point is unique and depends continuously on the parameter ε, a and ω. From now on we will work with the fixed point given by Lemma 5.3 and we will write simply as (b, λ). Finally, we project the system (5.1) over the space of functions spanned by the coordinate functions. It will be convenient to decompose ω in n ω= ωi ei , where ωi = ω(rε ·)ei . (5.13) i=1

Sn−1

Hence, |ωi | ≤ cn supSn−1 |ω|. From this and Remark 2.6 we get the system rε F (rε )rε ai − ωi = Hi,ε (a, ω) (5.14) G(rε )rε ai − (1 − n)ωi = rε ∂r Hi,ε (a, ω), i = 1, . . . , n, where F (rε ) := (n − 2)uε,R (rε θ) + rε ∂r uε,R (rε θ), G(rε ) := (n − 2)uε,R (rε θ) + nrε ∂r uε,R (rε θ) + rε2 ∂r2 uε,R (rε θ),

2+d− n 2+d− n 2 2 and rε ∂r Hi,ε (a, ω) = O rε . Hi,ε (a, ω) = O rε

(5.15)

The maps Hi,ε and ∂r Hi,ε are continuous. Lemma 5.4. There is a constant ε2 > 0 such that if ε ∈ (0, ε2 ), then the system d− n 2 2 (5.14) has a solution (a, ω) ∈ Rn × C 2,α (Sn−1 and ω given by rε ) with |a| ≤ rε 2+d− n 2 −δ1

(5.13) of norm bounded by rε

.

Proof. Define a continuous map Ki,ε : Di,ε → R2 by Ki,ε (ai , ωi ) := (G(rε ) + (n − 1)F (rε ))−1 rε−1 (rε ∂r Hi,ε (a, ω) + (n − 1)Hi,ε ), (G(rε )+(n − 1)F (rε ))−1 F (rε )(rε ∂r Hi,ε (a, ω)+(n − 1)Hi,ε ) − Hi,ε ) , d− n

2+d− n −δ

1 −1 2 rε }, where Di,ε := {(ai , ωi ) ∈ R2 ; |ai |2 ≤ n−1 rε 2 and |ωi | ≤ n−1 ki,n ki,n = ei (2,α),1 , F (rε ) = (n−2)(1+b)+O(ε2−s(n−2) ) and G(rε )+(n−1)F (rε ) = n(n − 2)(1 + b) + O(ε2−s(n−2) ) with 2 − s(n − 2) > 0. From (5.15) we obtain that Ki,ε (Di,ε ) ⊂ Di,ε , for small enough ε > 0. Again, by the Brouwer’s fixed point theorem there exists a fixed point of the map Ki,ε , and this fixed point is a solution of the system (5.14).

Now we are ready to prove the main theorem of this paper.

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Proof of Theorem 5.1. We keep the previous notations. From Lemmas 5.2, 5.3 and 5.4 we conclude that there is ε0 > 0 such that for all ε ∈ (0, ε0 ) there are parameters Rε , aε , φε , λε and ϕε for which the functions Aε (Rε , aε , φε ) and Brε (λε , ϕε ) coincide up to order one in ∂Brε (p). Hence using elliptic regularity we show that the function Uε defined by Uε := Aε (Rε , aε , φε ) in Brε (p)\{p} and Uε := Brε (λε , ϕε ) in M \Brε (p) is a positive smooth function in M \{p}. Moreover, Uε tends to infinity on approaching p. 4

Therefore, the metric gε := Uεn−2 g is a complete smooth metric defined in M \{p} and by Theorems 3.8 and 4.3 it satisfies i), ii), and iii).

6. Multiple Point Gluing In this final section. we discuss the minor changes that need to be made in order to deal with more than one singular point. Let X = {p1 , . . . , pk } so that at each point we have ∇l Wg0 (pi ) = 0, for l = 0, . . . , d − 2. As in the previous case, there are three steps. In Sect. 3 we do not need to make any changes, since the analysis is done at each point pi . Here, we find a family of metrics defined in Brεi (p)\{p}, with εi = ti ε, ε > 0, ti ∈ (δ, δ −1 ) and δ > 0 fixed, i = 1, . . . , k. In order to get a family of metrics as in Sect. 4 we need to make some changes. 4

Let Ψi : B2r0 (0) → M be a normal coordinate system with respect to gi = Fin−2 g0 on M centered at pi . Here, Fi is such that as in Sect. 4. Therefore, each metric gi gives us conformal normal coordinates centered at pi . Recall that Fi = 1 + O(|x|2 ) in the coordinate system Ψi . Denote by Gpi the Green’s function for L1g0 with pole at pi and assume that lim |x|n−2 Gpi (x) = 1 in the coordinate system Ψi . Let x→0

Gp1 ,...,pk ∈ C ∞ (M \{p1 , . . . , pk }) be such that Gp1 ,...,pk =

k

λi Gpi ,

i=1

where λi ∈ R. Let r = (rε1 , . . . , rεk ). Denote by Mr the complement in M of the union of Ψi (Brεi (0)) and define the space Cνl,α (M \{p1 , . . . , pk }) as in Definition 2.5, with the following norm vCνl,α (M \{p}) := vC l,α (M 1 r

2 0

)+

k

v ◦ Ψi (l,α),ν,r0 .

i=1

The space Cνl,α (Mr ) is defined similarly. It is possible to show an analog of Proposition 4.1 in this context, with w ∈ R constant on any component of ∂Mr . Let ϕ = (ϕ1 , . . . , ϕk ), with ϕi ∈ C 2,α (Sn−1 )L2 -orthogonal to the constant r 2,α functions. Let uϕ ∈ Cν (Mr ) be such that uϕ ◦ Ψi = ηQrεi (ϕi ), where η is a

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smooth, radial function equal to 1 in Br0 (0), vanishing in Rn \B2r0 (0), and satisfying |∂r η(x)| ≤ c|x|−1 , and |∂r2 η(x)| ≤ c|x|−2 for all x ∈ B2r0 (0). Finally, in the same way that we showed the existence of solutions to the Eq. (4.1), we solve the equation Hg0 (1 + Gp1 ,...,pk + uϕ + u) = 0. The result reads as follows: Theorem 6.1. Let (M n , g0 ) be an n-dimensional compact Riemannian manifold of scalar curvature n(n−1), nondegenerate about 1. Let {p1 , . . . , pk } a set of points in M so that ∇jg0 Wg0 (pi ) = 0 for j = 0, . . . , [ n−6 2 ] and i = 1, . . . , k, where Wg0 is the Weyl tensor of the metric g0 . There exists a complete metric g on M \{p1 , . . . , pk } conformal to g0 , with constant scalar curvature n(n − 1), obtained by attaching Delaunay-type ends to the points p1 , . . . , pk .

Acknowledgements The content of this paper is from the author’s doctoral thesis at IMPA [43]. The author is specially grateful to his advisor Prof. Fernando C. Marques for numerous mathematical conversations and constant encouragement. The author was partially supported by CNPq-Brazil.

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[9] Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth. Comm. Pure Appl. Math. 42(3), 271–297 (1989) [10] Chru´sciel, P.T., Pacard, F., Pollack, D.: Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends II. Generic metrics. Math. Res. Lett. 16(1), 157–164 (2009) [11] Chru´sciel, P.T., Pollack, D.: Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends. Ann. H. Poincaré 9(4), 639–654 (2008) [12] Finn, D.: On the negative case of the singular Yamabe problem. J. Geom. Anal. 9(1), 73–92 (1995) [13] Finn, D., McOwen, R.: Singularities and asymptotics for the equation Δg u − uq = Su. Indiana Univ. Math. J. 42(4), 1487–1523 (1993) [14] Jleli, M.: Constant mean curvature hypersurfaces, PhD Thesis, University of Paris 12 (2004) [15] Jleli, M., Pacard, F.: An end-to-end construction for compact constant mean curvature surfaces. Pac. J. Math. 221(1), 81–108 (2005) [16] Kaabachi, S., Pacard, F.: Riemann minimal surfaces in higher dimensions. J. Inst. Math. Jussieu 6(2), 613–637 (2007) [17] Kapouleas, N.: Complete constant mean curvature surfaces in Euclidean threespace. Ann. Math. (2) 131(2), 239–330 (1990) [18] Khuri, M.A., Marques, F.C., Schoen, R.: A compactness theorem for the Yamabe problem. J. Diff. Geom. 81(1), 143–196 (2009) [19] Korevaar, N.A., Mazzeo, R., Pacard, F., Schoen, R.: Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135(2), 233– 272 (1999) [20] Lee, J., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17(1), 37–91 (1987) [21] Li, Y.Y., Zhang, L.: Compactness of solutions to the Yamabe problem II. Calc. Var. PDE 24(2), 185–237 (2005) [22] Li, Y.Y., Zhang, L.: Compactness of solutions to the Yamabe problem III. J. Funct. Anal. 245(2), 438–474 (2007) [23] Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations. In: Contributions to Analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York (1974) [24] Marques, F.C.: A priori estimates for the Yamabe problem in the non-locally conformally flat case. J. Diff. Geom. 71(2), 315–346 (2005) [25] Marques, F.C.: Blow-up examples for the Yamabe problem. Calc. Var. PDE 36(2), 377–397 (2009) [26] Marques, F.C.: Isolated singularities of solutions to the Yamabe equation. Calc. Var. PDE 32(3), 349–371 (2008) [27] Mazzeo, R.: Regularity for the singular Yamabe equation. Indiana Univ. Math. J. 40(4), 1277–1299 (1991) [28] Mazzeo, R., Pacard, F.: A construction of singular solutions for a semilinear elliptic equation Using asymptotic analysis. J. Diff. Geom. 44(2), 331–370 (1996)

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[29] Mazzeo, R., Pacard, F.: Constant mean curvature surfaces with delaunay ends. Comm. Anal. Geom. 9(1), 169–237 (2001) [30] Mazzeo, R., Pacard, F.: Constant scalar curvature metrics with isolated singularities. Duke Math. J. 99(3), 353–418 (1999) [31] Mazzeo, R., Pacard, F., Pollack, D.: The conformal theory of Alexandrov embedded constant mean curvature surfaces in R3 . In: Global Theory of Minimal Surfaces. Clay Math. Proc., vol. 2, pp. 525–559. Amer. Math. Soc., Providence (2005) [32] Mazzeo, R., Pacard, F., Pollack, D.: Connected sum of constant mean curvature surfaces in Euclidean 3 space. J. Reine Angew. Math. 536, 115–165 (2001) [33] Mazzeo, R., Pollack, D., Uhlenbeck, K.: Moduli spaces of singular Yamabe metrics. J. Am. Math. Soc. 9(2), 303–344 (1996) [34] Mazzeo, R., Pollack, D., Uhlenbeck, K.: Connected sum constructions for constant scalar curvature metrics. Topol. Methods Nonlinear Anal. 6(2), 207–233 (1995) [35] Mazzeo, R., Smale, N.: Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere. J. Diff. Geom. 34(3), 581–621 (1991) [36] Pacard, F., Rivière, T.: Linear and Nolinear Aspects of Vortices: the GinzburgLandau Model. In: Progress in Nonlinear Differential Equations and their Applications, vol. 39. Birkh¨ auser, Boston [37] Pollack, D.: Compactness Results for Complete Metrics of Constant Positive Scalar Curvature on Subdomains of Sn . Indiana Univ. Math. J. 42(4), 1441–1456 (1993) [38] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom. 20(2), 479–495 (1984) [39] Schoen, R.: A report on some recent progress on nonlinear problems in geometry. In: Surveys in Differential Geometry (Cambridge, MA, 1990), pp. 201–241. Lehigh Univ., Bethlehem (1991) [40] Schoen, R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Comm. Pure Appl. Math. 41(3), 317–392 (1988) [41] Schoen, R., Yau, S.-T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92(1), 47–71 (1988) [42] Schoen, R., Yau, S.-T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology. International Press Inc. (1994) [43] Silva Santos, A.: A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends. Doctoral thesis, IMPA, Brazil (2009) [44] Trudinger, N.: Remarks concerning the conformal deformation of a Riemannian structure on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, 265–274 (1968) [45] Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)

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Almir Silva Santos Present Address: Departamento de Matem´ atica Centro de Ciências Exatas e Tecnologia Universidade Federal de Sergipe Av. Marechal Rondon s/n S˜ ao Crist´ ov˜ ao, SE 49100-000, Brazil and Instituto de Matem´ atica Pura e Aplicada (IMPA) Estrada Dona Castorina 110 Rio de Janeiro, RJ 22460-320, Brazil e-mail: [email protected] Communicated by Piotr T. Chrusciel. Received: December 14, 2009. Accepted: January 7, 2010.

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Ann. Henri Poincaré 10 (2010), 1537–1557 c 2010 Birkh¨ auser / Springer Basel AG 1424-0637/10/081537-21, published online March 12, 2010 DOI 10.1007/s00023-010-0022-y


All Vacuum Near Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries Stefan Hollands and Akihiro Ishibashi Abstract. We explicitly construct all stationary, non-static, extremal near horizon geometries in D dimensions that satisfy the vacuum Einstein equations, and that have D −3 commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in D = 4, 5. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology S 2 × T D−4 , or S 3 × T D−5 , or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as (D − 2)(D − 3)/2 continuous parameters. Not all of our metrics in D ≥ 6 seem to arise as the near-horizon limits of known black hole solutions.

1. Introduction Many known families of black hole solutions possess a limit wherein the black hole horizon becomes degenerate, i.e., where the surface gravity tends to zero; such black holes are called extremal. While extremal black holes are not believed to be physically realized as macroscopic objects in nature, they are nevertheless highly interesting from the theoretical viewpoint. Due to the limiting procedure, they are in some sense at the fringe of the space of all black holes, and therefore possess special properties which make them easier to study in various respects. For example, in string theory, the derivation of the Bekenstein–Hawking entropy of black holes from counting microstates (see e.g., [13] for a review) is best understood for extremal black holes. Furthermore, many black hole solutions that have been constructed in the context of supergravity theories (see e.g., [17,18]) have supersymmetries, and are thus automatically extremal.

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Many of the arguments related to the derivation of the black hole entropy—especially in the context of the “Kerr-CFT correspondence” [1,3,10,19, 21,34]—actually only involve the spacetime geometry in the immediate (actually infinitesimal) neighborhood of the black hole horizon. More precisely, by applying a suitable scaling process to the spacetime metric which in effect blows up this neighborhood, one can obtain in the limit a new spacetime metric, called a “near horizon geometry.” It is the near-horizon geometry which enters many of the arguments pertaining to the derivation of the black hole entropy. The near-horizon limit can be defined for any spacetime (M, g) with a degenerate Killing horizon, N —not necessarily a black hole horizon. The construction runs as follows.1 First, recall that a spacetime with degenerate Killing horizon by definition has a smooth, codimension one, null hypersurface N , and a Killing vector field K whose orbits are tangent N , and which on N are tangent to affinely2 parametrized null-geodesics. Furthermore, by assumption, there is a “cross section”, H, of codimension one in N with the property that each generator of K on N is isomorphic to R and intersects H precisely once. In the vicinity of N , one can then introduce “Gaussian null coordinates” u, v, y a as follows, see, e.g., [36]. First, we choose arbitrarily local3 coordinates y a on H, and we Lie-transport them along the flow of K to other places on N , denoting by v the flow parameter. Then, at each point of N we shoot off affinely parametrized null-geodesics and take u to be the affine parameter along these null geodesics. The tangent vector ∂/∂u to these null geodesics is required to have unit inner product with K = ∂/∂v on H, and to be orthogonal to the Lie-transported cross-section H. It can be shown that the metric then takes the Gaussian null form g = 2dv(du + u2 αdv + uβa dy a ) + γab dy a dy b ,

(1.1)

where the function α, the one-form β = βa dy a , and the tensor field γ = γab dy a dy b do not depend on v. The Killing horizon N is located at u = 0, and the cross section H at u = v = 0. The near horizon limit is now taken by applying to g the diffeomorphism v → v/, u → u (leaving the other coordinates y a unchanged), and then taking → 0. The so-obtained metric looks exactly like Eq. (1.1), but with new metric functions obtained from the old ones by evaluating them at u = 0. Thus, the fields α, β, γ of the near horizon metric neither depend on v nor u and can therefore be viewed as fields on H. If the original spacetime with degenerate Killing horizon satisfies the vacuum Einstein equation or the Einstein equation with a cosmological constant, then the near-horizon limit does, too. 1 The general definition of a near-horizon limit was first considered in the context of supergravity black holes in [39], and in the context of extremal, but not supersymmetric black holes in [12] for the static case and in [30] for the general case. The concept of near-horizon geometry itself has appeared previously in the literature, e.g., [20] for 4-dimensional vacuum case (also see [33] for the isolated horizon case). 2 For a non-degenerate horizon, the orbits on N of K would not be affinely parametrized. 3 Of course, it will take more than one patch to cover H, but the fields γ, β, α on H below in Eq. (1.1) are globally defined and independent of the choice of coordinate systems.

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The near-horizon limit is simpler than the original metric in the sense that it has more symmetries. For example, if the limit procedure is applied to the extremal Kerr metric in D = 4 spacetime dimensions with symmetry group R×U (1), then— as observed4 first by [4] (see also [5,7])—the near-horizon metric has an enhanced symmetry group of O(2, 1) × U (1). The first factor of this group is related to an AdS2 -factor in the metric. A similar phenomenon occurs for stationary extremal black holes in higher dimensions with a comparable amount of symmetry: as proved in [30], if (M, g) is a D-dimensional stationary extremal black hole with isometry group5 R × U (1)D−3 and compact horizon cross section H, then the near-horizon limit has the enhanced symmetry group O(2, 1) × U (1)D−3 . In D ≥ 5 dimensions, it is not known at present what is the most general stationary extremal black hole solution with symmetry group R × U (1)D−3 , so one can neither perform explicitly their near horizon limits. Nevertheless, because the near horizon metric has an even higher degree of symmetry—the metric functions essentially only depend non-trivially on one coordinate—one can try to classify them directly. This was done for the vacuum Einstein equations in dimensions D = 4, 5 by [29], where a list of all near-horizon geometries, i.e., metrics of the form (1.1) with metric functions α, β, γ independent of u, v, was obtained. It is a priori far from obvious that all these metrics are the near horizon limits of actual globally defined black holes. Remarkably though, [29] could prove that the metrics found are indeed the limits of the extremal black ring [14], boosted Kerr string, Myers–Perry [37], and the Kaluza–Klein black holes [32,38], respectively. In this paper, we give a classification of all possible vacuum near horizon geometries with symmetry group O(2, 1) × U (1)D−3 in arbitrary dimensions D. The method of analysis used in [29] seems restricted to D = 4, 5, so we here use a different method based on a matrix formulation of the vacuum Einstein equations that works in arbitrary dimensions. The metrics that we find come in three families depending on the topology of H, which can be either S 3 × T D−5 , S 2 × T D−4 or L(p, q) × T D−5 , where L(p, q) is a Lens space. The metrics in each of these families depend on (D − 2)(D − 3)/2 real parameters; they are given explicitly in Theorem 1 below. When specialized to D = 5, our first two families of metrics must coincide with those previously found in [29], whereas the last family is shown to arise from the first one by taking quotients (this last properties generalizes to arbitrary D). In all dimensions, examples for near-horizon geometries with topology S 2 × T D−4 are provided by the near-horizon limit of the “boosted Kerr-branes” see, e.g., [15,30]. This family of metrics depends on (D − 2)(D − 3)/2 real parameters and it is conceivable that all near horizon geometries of this topology can be 4

By construction, the near horizon geometry has the Killing fields ∂/∂v and u∂/∂u − v∂/∂v, which generate a two-parameter symmetry group. The non-trivial observation by [4] is that this actually gets enhanced to the three-parameter group O(2, 1). 5 The “rigidity theorem” [23] guarantees that a stationary extremal black hole has a symmetry group that contains R × U (1), i.e., guarantees only one axial Killing field in addition to the assumed timelike Killing field. Therefore, in D ≥ 5, assuming a factor of U (1)D−3 is a non-trivial restriction, while it is actually a consequence of the rigidity theorem in D = 4.

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obtained in this way. The analogous construction is also possible when the horizon topology is S 3 × T D−5 . However, in this case, the resulting metrics depend on fewer parameters. We should also point out that there are vacuum near-horizon geometries that possess fewer symmetries than R×U (1)D−3 . For example, the near-horizon geometry of the extremal Myers–Perry black holes, constructed explicitly in [15], has the smaller symmetry group, R×U (1)[(D−1)/2] . In this paper, we are not going to classify such less symmetric vacuum near-horizon geometries. Also, we are not going to consider the case of a non-vanishing cosmological constant, since, as far as we are aware, there has appeared no successful reduction of the Einstein gravity with a cosmological constant to a suitable nonlinear sigma model, which is, however, required in our approach. The same remark would apply to other theories with different matter fields. On the other hand, we expect our approach to be applicable to theories that can be reduced to suitable sigma-models. For D = 5 minimal gauged and ungauged supergravity, the near-horizon geometries were classified in [31,39] using a method different from ours. Also for D = 4 Einstein–Maxwell theory with a cosmological constant, see, e.g., [28].

2. Geometrical Coordinates The aim of this paper is to classify the near-horizon geometries in D dimensions. As explained in the previous section, by this we mean the problem of finding all metrics g of the form (1.1) with vanishing Ricci tensor (i.e., vacuum metrics), where γ = γab dy a dy b is a smooth metric on the compact manifold H, β = βa dy a is a 1-form on H and α is a scalar function on H. These fields do not depend on u, v, and the near-horizon geometries therefore have the Killing vectors K = ∂/∂v and X = u∂/∂u − v∂/∂v. We do not assume a priori that the near horizon metrics arise from a black hole spacetime by the limiting procedure described above. Unfortunately, this problem appears to be difficult to solve in this generality, so we will make a significant further symmetry assumption. Namely, we will assume that our metrics do not only have the Killing vectors K, X, but in addition admit the symmetry group U (1)D−3 , generated by (D − 3) commuting Killing fields ψ1 , . . . , ψD−3 that are tangent to H and also commute with K, X. Thus, the full isometry group of our metric is (at least) G2 × U (1)D−3 , where G2 denotes the Lie-group that is generated by K, X. This means roughly speaking that the metric functions can nontrivially depend only on a single variable, and our metrics may hence be called “cohomogeneity-one.” As a consequence, Einstein’s equations reduce to a coupled system of non-linear ordinary differential equations in this variable. Our aim is to solve this system in the most general way and thereby to classify all near horizon geometries with the assumed symmetry. It seems that this system becomes tractable only if certain special coordinates are introduced that are adapted in an optimal way to the geometric situation under consideration. These coordinates are the well-known Weyl–Papapetrou

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coordinates up to a simple coordinate transformation. However, to introduce these coordinates in a rigorous and careful manner is more subtle in the present case than for non-extremal horizons. These technical difficulties are closely related to the fact that the usual Weyl–Papapetrou coordinates are actually singular on H, the very place we are interested in most. To circumvent this problem, we follow the elegant alternative procedure introduced in [29,30]. That procedure applies in the form presented here to non-static geometries, and we will for the rest of this paper make this assumption. The static case has been treated previously in [12,27]. We first observe that the horizon H is a compact (D − 2)-dimensional manifold with an action of U (1)D−3 . By general and rather straightforward arguments (see, e.g., [24,26]) it follows that, topologically, H can only be of the following four types: ⎧ 3 S × T D−5 , ⎪ ⎪ ⎪ ⎨S 2 × T D−4 , H∼ (2.2) = ⎪ L(p, q) × T D−5 , ⎪ ⎪ ⎩ D−2 . T Furthermore, in the first three cases, the quotient space H/U (1)D−3 is a closed interval—which we take to be [−1, 1] for definiteness—whereas in the last case, it is S 1 . We will not treat the last case in this paper,6 but we note that the topological censorship theorem [11] implies that there cannot exist any extremal, asymptotically flat or Kaluza–Klein vacuum black holes with H ∼ = T D−2 . Thus, while there D−2 ∼ , they cannot arise as the could still be near horizon geometries with H = T limit of a globally defined black hole spacetime. In this paper, we will focus on the first three topology types. In these cases, the Gram matrix fij = γ(ψi , ψj )

(2.3)

is non-singular in the interior of the interval, and it has a one-dimensional nullspace at each of the two end points [24]. In fact, there are integers ai± ∈ Z such that fij (x)ai± → 0

at boundary points ±1.

(2.4)

The integers ai± determine the topology of H (i.e., which of the first three cases we are in), as we explain more in Theorem 1 below. The first geometric coordinate, x, parametrizes the interval [−1, +1], and is introduced as follows. Consider the 1-form on H defined by Σ = (det f ) γ (ψ1 ∧ · · · ∧ ψD−3 ), where the Hodge dual is taken with respect to the metric γ on H. Using the fact that the ψi are commuting Killing fields of γ, one can show that Σ is closed, and that it is Lie-derived by all ψi . Hence, Σ may be viewed as a closed 1-form on the orbit space H/U (1)D−3 , which, as we have said, is a closed interval. 6

See, however, the note added in proof.

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It can be seen furthermore that Σ does not vanish anywhere within this closed interval, so there exists a function x, such that dx = CΣ.

(2.5)

The constant C is chosen so that x runs from −1 to +1. We take x to be our first coordinate, and we take the remaining coordinates on H to be angles ϕ1 , . . . , ϕD−3 running between 0 and 2π, chosen in such a way that ψi = ∂/∂ϕi . In these coordinates, the metric γ on H takes the form 1 dx2 + fij (x)dϕi dϕj . γ= 2 (2.6) C det f To define our next coordinate, we consider the 1-form field β on H, see Eq. (1.1). Standard results on the Laplace operator Δγ on a compact Riemannian manifold (H, γ) guarantee that there exists a smooth function λ on H such that γ d γ β = Δγ λ,

(2.7)

where γ is the Hodge star of γ. The function λ is unique up to a constant. Because β and γ are Lie-derived by all the rotational Killing fields ψi , it follows that Lψi λ = ci are harmonic functions on H, i.e., constants. Furthermore, these constants must vanish, because the ψi have periodic orbits. Thus, λ is only a function of x. We also claim that the 1-form β − dλ has no dx-part. To see this, we let h be the scalar function on H defined by h = γ (ψ1 ∧ · · · ∧ ψD−3 ∧ [β − dλ]). Using Eq. (2.7) and the fact that the ψi are commuting Killing fields of γ, it is easy to show that dh = 0, so h is constant. Furthermore, by Eq. (2.4) there exist points in H where the linear combinations ai± ψi = 0, and it immediately follows from this that h = 0 on H. This shows that β − dλ has no dx-part and hence we can write β = dλ + Ceλ ki dϕi ,

(2.8)

where we have introduced the quantities ki := C −1 e−λ ψi · β.

(2.9)

The next coordinate is defined by r := ueλ ,

(2.10) i

and we keep v as the last remaining coordinate. The coordinates ϕ , r, x, v are the desired geometrical coordinates. In these, the metric takes the form g = e−λ [2dvdr + r2 (2αe−λ − eλ ki k i )dv 2 ] dx2 + fij (dϕi + Crk i dv)(dϕj + Crk j dv). + 2 C det f

(2.11)

We have also determined that the quantities k i , fij , α, λ are functions of x only. The indices i, j, . . . are raised with the inverse f ij of the Gram matrix, e.g., k i = f ij kj . So far, we have only used the symmetries of the metric, but not the fact that it is also required to be Ricci flat. This imposes significant further restrictions [29,30]. Namely, one finds that k i are simply constants, and that (2αe−λ − eλ ki k i ) is

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a negative7 constant, which one may choose to be −C 2 after a suitable rescaling of the coordinates r, v and the constants k i , and by adding a constant to λ. Then the Einstein equations further imply that ∂x2 (e−λ det f ) = −2; hence, e−λ = −(x − x− )(x − x+ )(det f )−1 for real numbers x± . Furthermore, λ is smooth and det f vanishes only at x = ±1 by Eq. (2.4), so x± = ±1 and consequently, e−λ = (1 − x2 )(det f )−1 .

(2.12)

Thus, in summary, we have determined that the near-horizon metric is given by g=

dx2 1 − x2 (2dvdr − C 2 r2 dv 2 ) + 2 + fij (dϕi + rCk i dv)(dϕj + rCk j dv) det f C det f (2.13)

where k i , C are constants, and where fij depends only on x. In the remainder of the paper, we will work with above form of the metric (2.13). However, we will, for completeness, also give the relation to the more familiar Weyl–Papapetrou form: For r > 0 (i.e., strictly outside the horizon), we define new coordinates (t, ρ, z, φi ) by the transformation [16] z := rx ρ := r 1 − x2

(2.14) (2.15) −1

t := Cv + (Cr) i

i

φ := ϕ + C

−1 i

k log r .

(2.16) (2.17)

i

In the new coordinates (t, ρ, z, φ ), the metric then takes the Weyl–Papapetrou form e−λ ρ2 dt2 + 2 2 (dρ2 + dz 2 ) + fij (dφi + rk i dt)(dφj + rk j dt), (2.18) g=− det f C r where it is understood that r2 = ρ2 + z 2 . Note that, by contrast with the coordinate system (v, r, x, ϕi ), the Weyl–Papapetrou coordinate system does not cover the horizon itself, i.e., it is not defined for r = 0 but only for r > 0. This can be seen in several ways, for example, by noting that the coordinate transformation is singular at r = 0, i.e., on the horizon, or alternatively, by noting that the horizon corresponds in the new coordinates to the single point ρ = z = 0. This behavior is characteristic of extremal horizons and does not happen in the non-extremal case. In obtaining our form (2.13) for the near-horizon metric, we have used up all but the ij-components of the Einstein equations. The remaining Einstein equations determine the matrix of functions fij (x). As is well-known [35], a beautifully simple form of these equations can be obtained by introducing the twist potentials of the rotational Killing fields as auxiliary variables. These potentials χi are defined up to a constant by dχi = (ψ1 ∧ · · · ∧ ψD−3 ∧ dψi ). 7

Here one must use that the metric is not static, i.e., that not all ki vanish.

(2.19)

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To see that this equation makes sense, one has to prove that the right side is an exact form. Indeed, taking d of the right side and using the vanishing of the Ricci tensor together with the fact that the Killing fields all commute, one gets zero. To see that the right side is even exact, it is best to pass to the orbit space M/(G2 × U (1)D−3 ) first, which can be identified with the interval [−1, 1]. Then the χi can be defined on this orbit space and lifted back to functions on M . It also follows from this construction that χi only depends on the coordinate x parametrizing [−1, 1]. Setting −(det f )−1 χi (det f )−1 , (2.20) Φ= −(det f )−1 χi fij + (det f )−1 χi χj it is well known that the vanishing of the Ricci-tensor implies that ∂x [(1 − x2 )Φ−1 ∂x Φ] + ∂r [r2 Φ−1 ∂r Φ] = 0.

(2.21)

These equations are normally written in the Weyl–Papapetrou coordinates ρ, z (see, e.g., [24]), and the above form is obtained simply by the change of variables Eq. (2.14). Since Φ is a function of x only in our situation (but would not be, e.g., for black holes without the near-horizon limit taken) an essential further simplification occurs: The second term in the above set of matrix equations is simply zero! Hence, the content of the remaining Einstein equations is expressed in the matrix of ordinary differential equations ∂x [(1 − x2 )Φ−1 ∂x Φ] = 0.

(2.22)

In fact, this equation could be derived formally and much more directly by simply assuming the Weyl–Papapetrou form of the metric, introducing r, x as above, and then observing that, in the near-horizon limit, the dependence on r is scaled away, so that the matrix partial differential Eq. (2.21) reduce to the ordinary differential Eq. (2.22).

3. Classification To determine all near-horizon metrics (2.13), we must solve the matrix Eq. (2.22), i.e., find fij , χi . Then the constants k i are given by ki =

1 − x2 ij f ∂ x χj , det f

(3.23)

and this determines the full metric up to the choice of the remaining constant C. We must furthermore ensure that, among all such solutions, we pick only those that give rise to a smooth metric g. The Eq. (2.22) for Φ are easily integrated to L 1+x Φ(x) = Q exp [2arcth(x) · L] = Q . (3.24) 1−x

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Here, Q = Φ(0), L = 12 (1−x2 )Φ(x)−1 ∂x Φ(x) are both constant real (D−2)×(D−2) matrices, and we mean the matrix exponential etc. It follows from the definition that Φ has the following general properties: it is symmetric, det Φ = 1, and it is positive definite. It is an easy consequence of these properties that det Q = 1, TrL = 0 (taking the determinant of the equation), that Q = QT is positive definite, and that LT Q = QL. These relations allow us to write Q = S T S for some real invertible matrix S = (sIJ ) of determinant ±1, and to conclude that SLS −1 is a real symmetric matrix. By changing S to V S, where V is a suitable orthogonal transformation, we can achieve that ⎛ ⎞ σ0 0 ... 0 ⎜0 0 ⎟ σ1 . . . ⎜ ⎟ (3.25) SLS −1 = ⎜ . .. ⎟ ⎝ .. . ⎠ 0 0 . . . σD−3 is a real diagonal matrix, while leaving Q unchanged. It then follows that Φ(x) = S T exp 2arcth(x) · SLS −1 S, that is D−3 1 + x σK ΦIJ (x) = sKI sKJ . (3.26) 1−x K=0

This is the most general solution to the field equation for Φ in the near-horizon limit, and it depends on the real parameters sIJ , σI , which are subject to the constraints D−3 det(sIJ ) = ±1, σI = 0. (3.27) I=0

The near-horizon metric is completely fixed in terms of Φ. It can be obtained combining Eq. (3.26) with Eq. (2.20) to determine fij , χi , which in turn then fix the remaining constants k i , C in the near-horizon metric. In the rest of this section, we explain how this can be done. It turns out that the smoothness of the nearhorizon metric also implies certain constraints on the parameters σI , sIJ , and we will derive the form of these. Our analysis applies in principle to all dimensions D ≥ 4. The case D = 4, while being simplest, is somewhat different from the remaining cases D ≥ 5 and would require us to distinguish these cases in many of the formulae below. Therefore, to keep the discussion simple, we will stick to D ≥ 5 in the following. First, we consider the ij-component of Φ in Eq. (3.26). By Eq. (2.20) this is also equal to D−3 1 + x σI sIi sIj = Φij = fij + (det f )−1 χi χj . (3.28) 1−x I=0

Now, the coordinate x ∈ [−1, 1] parametrizes the orbit space H/U (1)D−3 of the horizon, which is topologically a finite interval. The boundary points x = ±1 correspond to points on the horizon where an integer linear combination ai± ψi of

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the rotational Killing fields vanishes. This is equivalently expressed by the condition fij (x)aj± → 0 as x → ±1. By contrast, for all values of x ∈ (−1, +1), no linear combination of the rotational fields vanishes. Therefore, det f = 0 for x ∈ (−1, +1), while det f → 0 as x → ±1. In fact, using Eq. (2.12) one sees that (det f )−1 = 2c2+ (1 − x)−1 + 2c2− (1 + x)−1 + · · ·

as x → ±1,

(3.29)

where the dots represent contributions that go to a finite limit, and where c± are non-zero constants related to λ by 4c2± = e−λ(±1) = 0. The twist potentials χi also go to a finite limit as x → ±1. By adding suitable constants to the twist potentials if necessary, we may achieve that χi →

1 μi c±

as x → ±1,

(3.30)

where μi ∈ R are constants. The upshot of this discussion is that, as one approaches the boundary points, the components Φij are dominated by the rank-1 part (det f )−1 χi χj , which diverges as 2(1 ∓ x)−1 μi μj as x → ±1. This behavior can be used to fix the possible values of the eigenvalues σI as follows. First, it is clear that at least one of the eigenvalues must be non-zero, for otherwise the right side of Eq. (3.28) would be smooth as x → ±1, which we have just argued is not the case. Let us assume without loss of generality then that σD−3 ≥ · · · ≥ σD−3−n > 0 are the n positive eigenvalues. Multiplying Eq. (3.28) by 1 − x and taking x → +1, we see that σD−3 = 1, that μi = s(D−3)i , and that all other remaining positive eigenvalues must be strictly between 0 and 1. If we now subtract (1 − x2 )−1 μi μj from both sides of the equation, then the right side of Eq. (3.28) goes to a finite limit as x → 1, and so the left side has to have that behavior, too. This is only possible if there are no other remaining positive eigenvalues besides σD−3 . A similar argument then likewise shows that there is only one negative eigenvalue, which has to be equal to −1 (without loss of generality we may take σD−4 = −1) and that μi = s(D−4)i . In summary, we have shown that ⎧ ⎪ if I ≤ D − 5, ⎨0 σI = −1 if I = D − 4, (3.31) ⎪ ⎩ 1 if I = D − 3, and we also see that μi = s(D−3)i = s(D−4)i ,

c+ = s(D−3)0 ,

c− = s(D−4)0 .

(3.32)

The condition that det S = ±1 then moreover gives ± 1 = (c+ − c− )ijk...m s0i s1j s2k · · · μm .

(3.33)

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We may now combine this information with the Eqs. (3.26) and (2.20) and solve for fij , χi . The result can be expressed as D−5

1 + x2 (μ · ξ)2 + (sI · ξ)2 2 1−x I=0 2 D−5 eλ(x) 2 2 2 − sI0 (sI · ξ) + [c+ (1 + x) + c− (1 − x) ](μ · ξ) (1 − x ) 1 − x2

fij ξ i ξ j = 2

I=0

χi ξ i = eλ(x)

(1 − x2 )

D−5

(3.34) sI0 (sI · ξ) + [c+ (1 + x)2 + c− (1 − x)2 ](μ · ξ) .

I=0

(3.35) Here, we are using shorthand notations such as μ · ξ = μi ξ i or sI · ξ = sIi ξ i , and exp[−λ(x)] = c2+ (1 + x)2 + c2− (1 − x)2 + (1 − x2 )

D−5

s2I0 ,

(3.36)

I=0

in order to have a reasonably compact notation. This function λ agrees with that previously defined in Eq. (2.7) by Eq. (2.12). From Eq. (3.34), one now finds after a short calculation that the conditions (2.4) are equivalent to sI0 μi ai+ = c+ sIi ai+ ,

sI0 μi ai− = c− sIi ai− ,

for I = 0, . . . , D − 5. (3.37)

Either of these equations “±” can be used to solve for sI0 , because8 μi ai± = 0 for both “±”. We will do this in the following: As we have explained, the constants k i in the near-horizon metric are given by (3.23). A longer calculation using Eqs. (3.34), (3.37), (3.33) and (3.36) reveals that i i a a c 2c + − + ki = + −j . (3.38) c+ − c− μj aj+ μj a− To avoid conical singularities in the near-horizon metric (2.13), we must furthermore have9 (1 − x2 )2 det f · fij ai± aj± 8

→ C2

as x → ±1,

(3.39)

Indeed, let us assume that, say μi ai+ = 0. Then, since c+ = 0, we know that also sIi ai+ = 0. It then would follow that 0 = ijk,...,m s0i s1j s2k · · · μm , which however is in contradiction with Eq. (3.33). 9 Here the constants ai ∈ Z are normalized so that the greatest common divisor of ai , i = ± + 1, . . . , D − 3 is equal to 1, and similarly for ai− .

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and this determines C. A longer calculation using Eqs. (3.34), (3.37) shows that C=

4c2+ 4c2− = . (c+ − c− )μi ai+ (c+ − c− )μi ai−

(3.40)

Thus, we have determined all quantities C, k i , fij in the near-horizon metric (2.13). We substitute these, and make the final coordinate change x = cos θ,

0 ≤ θ ≤ π.

(3.41)

Then, after performing some algebraic manipulations, we get the following result, which summarizes our entire analysis so far: Theorem 1. All non-static near horizon metrics (except topology type H ∼ = T D−2 ) are parametrized by the real parameters c± , μi , sIi , and the integers ai± where I = 0, . . . , D − 5 and i = 1, . . . , D − 3, and g.c.d.(ai± ) = 1. The explicit form of the near horizon metric in terms of these parameters is g = e−λ (2dvdr − C 2 r2 dv 2 + C −2 dθ2 ) + e+λ +(1 + cos θ)2 c2+ +

(c+ − c− )2 (sin2 θ)Ω2

2 2 sI · a+ sI · a− Ω + (1 − cos θ)2 c2− Ω ωI − ωI − μ · a+ μ · a− I I

c2± sin2 θ 2 ((sI · a± )ωJ − (sJ · a± )ωI ) (μ · a± )2

.

(3.42)

I<J

Here, the sums run over I, J from 0, . . . , D − 5, the function λ(θ) is given by exp[−λ(θ)] = c2+ (1 + cos θ)2 + c2− (1 − cos θ)2 +

c2± sin2 θ (sI · a± )2 , (μ · a± )2

(3.43)

I

C is given by C = 4c2± [(c+ − c− )(μ · a± )]−1 , and we have defined the 1-forms Ω(r) = μ · dϕ + 4Cr ωI (r) = sI · dϕ +

c+ c− dv c+ − c−

r 2 C (sI · a+ + sI · a− )dv. 2

(3.44) (3.45)

We are also using the shorthand notations such as sIi ai+ = sI · a+ , or μ · dϕ = μi dϕi , etc. The parameters are subject to the constraints μ · a± = 0 and c2+ c2− = , μ · a+ μ · a−

c+ (sI · a+ ) c− (sI · a− ) = , μ · a+ μ · a−

±1 = (c+ − c− )ijk...m s0i s1j s2k · · · μm

(3.46)

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but they are otherwise free. The coordinates ϕi are 2π-periodic, 0 ≤ θ ≤ π, and v, r are arbitrary. When writing “±”, we mean that the formulae hold for both signs. Remarks. (1) The function λ(θ) was invariantly defined in Eq. (2.7), and therefore evidently has to be a smooth function. This is manifestly true, because both c± = 0. Because also μ · a± are both non-zero, we explicitly see that the above metrics are smooth (in fact analytic). (2) The part 2dvdr − C 2 r2 dv 2 of the metric is that of AdS2 with curvature C 2 . This is the cause for the enhanced symmetry group of O(2, 1) × U (1)D−3 . Let us finally discuss the meaning of the parameters on which the nearhorizon metrics depend. The parameters ai± ∈ Z are related to the horizon topology. Up to a globally defined coordinate transformation of the form Aij ϕj mod 2π, A ∈ SL(Z, D − 3), ϕi → we have a+ = (1, 0, 0, . . . , 0),

a− = (q, p, 0, . . . , 0),

p, q ∈ Z,

g.c.d.(p, q) = 1.

(3.47)

A general analysis of compact manifolds with a cohomogeneity-1 torus action (see, e.g., [24]) implies that the topology of H is ⎧ 3 D−5 ⎪ if p = ±1, q = 0, ⎨S × T 2 ∼ H = S × T D−4 (3.48) if p = 0, q = 1, ⎪ ⎩ D−5 otherwise. L(p, q) × T The constants μi , c± , ai± are directly related to the horizon area by AH = and we also have 1 Ji := 2

(2π)D−3 (c+ − c− )2 (μ · a± )2 , 8c4±

H

(dψi ) = (2π)D−3

c+ − c− μi . 2c− c+

(3.49)

(3.50)

In an asymptotically flat or Kaluza–Klein black hole spacetime with a single horizon H, the above integral for Ji could be converted to a convergent integral over a cross section at infinity using Stokes theorem and the vanishing of the Ricci tensor. Then the Ji would be equal to the Komar expressions for the angular momentum. The near-horizon limits that we consider do not of course satisfy any such asymptotic conditions, and hence this cannot be done. Nevertheless, if the near-horizon metric under consideration arises from an asymptotically flat or asymptotically Kaluza–Klein spacetime, then the Ji are the angular momenta of that spacetime. Hence, we see that the parameters c± , μi , ai± are directly related to geometrical/topological properties of the metric. This seems to be less clear for the remaining parameters sIi .

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The number of continuous parameters on which our metric depend can be counted as follows: first, the matrix sIi has (D − 3)(D − 4) independent components, μi has (D − 3) and c± has 2 components. These parameters are subject to D−5 the (D − 2) constraints, Eq. (3.46). However, changing sIi to J=0 RJ I sJi , with RJ I an orthogonal matrix in O(D − 4), does not change the metric. Since such a matrix depends on (D − 4)(D − 5)/2 parameters, our metrics depend only on (D − 3)(D − 4) + (D − 3) + 2 − (D − 2) − (D − 4)(D − 5)/2 = (D − 2)(D − 3)/2 real continuous parameters. It is instructive to compare this number to the number of parameters of a boosted Kerr-brane. If we start from a direct product of a 4-dimensional extremal Kerr metric with a flat torus T D−4 and apply a boost in an arbitrary direction, then the resulting family of metrics has (D − 2)(D − 3)/2 parameters, and the horizon topology is S 2 × T D−4 . It is plausible that all our metrics in our Theorem 1 for this topology can be obtained by taking the near-horizon limit of these boosted Kerr-branes. By contrast, if we start with a direct product of a 5-dimensional extremal Myers–Perry black hole with a flat torus T D−5 , then we similarly get a family of metrics which depends only on (D − 3)(D − 4)/2 + 1 parameters. Therefore in this case, we get metrics depending on fewer parameters than those in Theorem 1.

4. Examples Let us first illustrate our classification in D = 5 spacetime dimensions. According to our general result, the metrics have the discrete parameters a1± , a2± as well as the six continuous parameters μ1 , μ2 , s01 , s02 , c+ , c− which are subject to three constraints. Thus, the number of free parameters is three, and we take C [given by Eq. (3.40)] as one of them for convenience. We have the following cases to consider, depending on the possible values of the discrete parameters (see Eq. (3.47)): ∼ S 1 × S 2 : This case corresponds to the choice a+ = a− = (1, 0). The Topology H = constraints (3.27) read explicitly μ s01 =1 (4.51) c2+ μ1 = c2− μ1 , c+ s01 μ1 = c− s01 μ1 , (c+ − c− ) 1 μ2 s02 in this case. We know that μ1 cannot vanish, so the first and third equation imply together that c± = ±B for some non-zero constant B. As a consequence, the second equation then gives s01 = 0, from which the third equation then gives s02 = 1/(2c+ μ1 ). Putting all this into our formula (3.42) for the near-horizon metric gives g = 2B 2 (1 + cos2 θ)(2dvdr − C 2 r2 dv 2 + C −2 dθ2 ) + +

2 8B 2 sin2 θ 1 dϕ + Adϕ2 + C 2 r dv , 2 2 C (1 + cos θ)

C2 (dϕ2 )2 16B 4 (4.52)

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where we have put A = μ2 /μ1 . We can explicitly read off from the metric that the norm of ∂/∂ϕ1 [i.e., the coefficient of (dϕ1 )2 ] vanishes at θ = 0, π, whereas the norm of ∂/∂ϕ2 [i.e., the coefficient of (dϕ2 )2 ] never vanishes. This is the characteristic feature of the action of U (1)2 on S 2 × S 1 . Topology H ∼ = S 3 : In this case, a+ = (1, 0), a− = (0, 1). The constraints (3.27) are c2+ μ2

=

c2− μ1 ,

c+ s01 μ2 = c− s02 μ1 ,

μ (c+ − c− ) 1 μ2

s01 = 1. s02

(4.53)

The constraints allow us, e.g., to express μ1 , μ2 , s01 , s02 in terms of A := c+ , B := c− and C given by Eq. (3.40). The result must then be plugged back into the equation for the near-horizon metric (3.42). After some calculation, one ends up with the result g = e−λ (2dvdr − C 2 r2 dv 2 + C −2 dθ2 ) 2 2 4 +λ sin2 θ A2 dϕ1 + B 2 dϕ2 + rABC 2 dv +e C 2 2 C + (1 + cos θ)2 A−1 dϕ2 + r(2B)−1 C 2 dv 4 2 −1 1 2 C 2 −1 2 + (1 − cos θ) B dϕ + r(2A) C dv , 4

(4.54)

where 2

2

2

2

exp[−λ(θ)] = A (1 + cos θ) + B (1 − cos θ) +

C2 16AB

2

sin2 θ. (4.55)

The quantity A − B must be non-zero on account of the third constraint. Note that exp λ(θ) = 0 for 0 ≤ θ ≤ π, so we can explicitly read off from the metric that the norm of ∂/∂ϕ2 [i.e., the coefficient of (dϕ2 )2 ] vanishes at θ = π, whereas the norm of ∂/∂ϕ1 [i.e., the coefficient of (dϕ1 )2 ] vanishes at θ = 0. This is the characteristic feature of the action of U (1)2 on the 3-sphere. Topology H ∼ = L(p, q): In this case, a+ = (1, 0), a− = (q, p), where p, q ∈ Z and p = 0. The constraints (3.27) are explicitly c2+ (qμ1 + pμ2 ) = c2− μ1 ,

c+ s01 (qμ1 + pμ2 ) = c− (qs01 + ps02 )μ1 , μ s01 = 1. (c+ − c− ) 1 μ2 s02

(4.56)

We choose as the independent parameters A := c+ /p, B := c− /p, and C given by Eq. (3.40), and solve for the remaining ones using the constraints. The result is plugged back into the equation for the near-horizon metric (3.42). After some

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calculation, one ends up with the result g = e−λ (2dvdr − C 2 r2 dv 2 + C −2 dθ2 ) 2 2 4p sin2 θ A2 (1/p)dϕ1 + B 2 (dϕ2 − (q/p)dϕ1 ) + rABC 2 dv + p2 e+λ C 2 2 C + (1 + cos θ)2 A−1 (dϕ2 − (q/p)dϕ1 ) + r(2B)−1 C 2 dv 4p 2 2 C 2 −1 1 −1 2 + (1 − cos θ) (pB) dϕ + r(2A) C dv , (4.57) 4p where

2

exp[−λ(θ)] = p

2

2

2

2

A (1 + cos θ) + B (1 − cos θ) +

C2 16p2 AB

2

2

sin θ . (4.58)

We note that at θ = π, the Killing field ∂/∂ϕ1 has vanishing norm, while at θ = 0, the Killing field q∂/∂ϕ1 + p∂/∂ϕ2 has vanishing norm. This is the characteristic feature of the action of U (1)2 on the Lens space L(p, q). The metrics with H ∼ = L(p, q) just described are closely related to those in the case H ∼ = S 3 described in the previous example. Indeed, in the case H ∼ = S3, 1 2 1 2 consider the map given by (ϕ , ϕ ) → (ϕ + 2π/p, ϕ + 2πq/p), leaving invariant the other coordinates, where ϕ1 , ϕ2 are 2π-periodic. This map is an isometry of the metric with H ∼ = S 3 , and by repeated application generates the subgroup Zp of the full isometry group. If we factor by this group, then we get a metric with H ∼ = L(p, q), and we claim that this metric is exactly the one just given. To see this more explicitly, we note that factoring by the above group Zp of isometries in effect imposes the further identifications (ϕ1 , ϕ2 ) ∼ = (ϕ1 + 2π/p, ϕ2 + 2πq/p)

(4.59)

on the angular coordinates in the metric (4.54), which were initially 2π-periodic. If we let f : (r, v, θ, ϕ1 , ϕ2 ) → (r, p2 v, θ, (1/p)ϕ1 , ϕ2 − (q/p)ϕ1 )

(4.60)

then f provides an invertible mapping from the ordinary 2π-periodic coordinates to the coordinates with the identifications (4.59). If we now take the metric (4.54) in the case H ∼ = S 3 , factor it by Zp , pull it back by f , and furthermore put C → C/p, then we get precisely the H ∼ = L(p, q) metrics (4.57). Thus, all metrics in the case H ∼ = L(p, q) arise from the case H ∼ = S 3 by taking quotients. The same statement (with similar proof) is true in all dimensions D. Let us finally briefly discuss an example of our classification in D = 6 dimensions. In this case, the metrics are classified by the discrete parameters a± [see Eq. (3.47)] and 7 real continuous parameters. An example is

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Topology S 3 × S 1 : In this case, a+ = (1, 0, 0), a− = (0, 1, 0). The constraints are explicitly c+ s01 μ2 = c− s02 μ1

c+ s11 μ2 μ1 (c+ − c− ) μ2 μ3

= c− s12 μ1 c2+ μ2 = c2− μ1 , s01 s11 s02 s12 = 1. s03 s13

(4.61)

To simplify the formulae somewhat, we consider the special case that c+ = −c− =: A/2. Then the constraints may be solved easily for the remaining parameters. To obtain a halfway simple expression, we also consider the special case s11 = s03 = 0, and we denote the remaining free parameters as B := s01 , D = μ3 , and C as usual. The resulting metric is still rather complicated and is given by g = e−λ(θ) 2dvdr − C 2 r2 dv 2 + C −2 dθ2 2 + e+λ(θ) A4 C −2 sin2 θ dϕ1 + dϕ2 + A−1 CDdϕ3 − rC 2 dv A2 B 2 (1 + cos θ)2 (2dϕ2 + A−1 CDdϕ3 − rC 2 dv)2 4 A2 B 2 C2 2 1 −1 3 2 2 (1 − cos θ) (2dϕ + A CDdϕ − rC dv) + + (dϕ3 )2 . 4 4A4 B 2 (4.62)

+

Here, we also have e−λ(θ) =

A2 B2C 2 (1 + cos2 θ) + sin2 θ. 2 4

(4.63)

This special family of metrics depends on only four parameters. It is easy to write down the general seven-parameter family of metrics.

5. Conclusion We have determined explicitly what are the possible (non-static) stationary smooth, cohomogeneity-one near horizon geometries satisfying the vacuum Einstein equations. We excluded by hand10 the case that the horizon topology is T D−2 . The solution, described in Theorem 1, is given in closed form in terms of real and discrete parameters (corresponding to the possible topology types other than T D−2 ), which are subject to certain constraints that take the form of algebraic equations. After taking into account these constraints, the metrics depend on (D −2)(D −3)/2 independent real parameters, and two discrete ones. For example, in D = 5, we initially have three real continuous parameters. We have worked out 10

See, however, the note added in proof.

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explicitly this case as did [29], but our metrics are presented in different coordinates11 for the case H ∼ = S 3 . In D ≥ 6, not all of our metrics can be obtained as the near-horizon limit of a known black hole solution, so in this sense some of our metrics are new for D ≥ 6. By contrast to D ≤ 5, not all near-horizon metrics that we have found can be obtained as the near-horizon limits of known black hole solutions in dimensions D ≥ 6. It is conceivable that there are further extremal black hole solutions—to be found—which give our metrics in the near-horizon limit, but it is also possible that some of our metrics in D ≥ 6 simply do not arise in this way. Our method as described only works for vacuum solutions. However, we expect that it can be generalized to any theory whose equations can be recast into equations of the sigma-model type that we encounter. Thus we expect our method to be applicable, e.g., to 5-dimensional minimal supergravity, see, e.g., [6,8,9,40]. By contrast, our method does not seem applicable straightforwardly to the case of a cosmological constant. In our proof, we also assumed that the metrics are not static. All static near-horizon geometries were found in [27] in D = 5 and in [12] in arbitrary dimensions. It would be interesting to see whether our classification can be used to prove a black hole uniqueness theorem in arbitrary dimensions for extremal black holes along the lines of [2,16], thereby generalizing [24,25]. It would also be interesting to investigate whether our analysis can be used to obtain new structural insights into the origin of the Bekenstein–Hawking entropy, e.g., by considering a suitably quantized version of Eq. (2.22).

Acknowledgements S.H. would like to thank the Centro de Ciencias de Benasque Pedro Pascual for its hospitality during the inspiring programme on “Gravity—New perspectives from strings and higher dimensions”, where a key part of this work was done. He would also like to thank P. Figueras, H. Kunduri and especially J. Lucietti for numerous useful discussions. We especially would like to thank the unknown referee for pointing out an error in the counting of parameters of our solutions and for suggesting a simplification of formula (4.62). This work is supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. Note added in proof: In our analysis, we excluded by hand the horizon topology T D−2 . There cannot exist any asymptotically flat or Kaluza–Klein black hole solutions with this topology by general arguments [11,24]. At any rate, these could not arise as the near-horizon limits of a black hole. After we finished this work, it was confirmed by J. Holland that there cannot be any non-static cohomogeneity-one 11

We also do not distinguish between the subcases “A” and “B” as in [29] but instead give a unified expression for the metric.

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near horizon geometries with topology H ∼ = T D−2 [22]. Hence our main theorem 1 covers all possibilities with D − 3 commuting rotational symmetries. The static case is covered by the results of [12].

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[40] Tomizawa, S., Yasui, Y., Ishibashi, A.: A uniqueness theorem for charged rotating black holes in five-dimensional minimal supergravity. Phys. Rev. D 79, 124023 (2009) Stefan Hollands School of Mathematics, Cardiff University Cardiff, UK e-mail: [email protected] Stefan Hollands and Akihiro Ishibashi Institute of Particle and Nuclear Studies KEK Theory Center High Energy Accelerator Research Organization (KEK) Tsukuba, Japan e-mail: [email protected] Communicated by Piotr T. Chrusciel. Received: October 1, 2009. Accepted: January 19, 2010.

Ann. Henri Poincaré 10 (2010), 1559–1604 c 2010 Birkh¨ auser / Springer Basel AG 1424-0637/10/081559-46, published online March 30, 2010 DOI 10.1007/s00023-010-0027-6


The Ground State and the Long-Time Evolution in the CMC Einstein Flow Martin Reiris Abstract. Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y (Σ)). As noted by 3 Volg(k) (Σ) is monoFischer and Moncrief, the reduced volume V(k) = −k 3 tonically decreasing in the expanding direction and bounded below by Vinf = −1 3 Y (Σ) 2 . Inspired by this fact we define the ground state of the mani6 fold Σ as “the limit” of any sequence of CMC states {(gi , Ki )} satisfying: (i) ki = −3, (ii) Vi ↓ Vinf , (iii) Q0 ((gi , Ki )) ≤ Λ, where Q0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, a long time and cosmologically normalized flow consider −k 2 ˜ K , where σ = − ln(−k) ∈ [a, ∞). We prove that g, (˜ g , K)(σ) = −k 3 3 ˜ ˜ if E1 = E1 ((˜ g , K)) ≤ Λ (where E1 = Q0 + Q1 , is the sum of the zero and first ˜ order Bel–Robinson energies) the flow (˜ g , K)(σ) persistently geometrizes the three-manifold Σ and the geometrization is the ground state if V ↓ Vinf .

1. Introduction Consider a cosmological space–time solution g over M = Σ × (σ0 , ∞), where Σ is a compact three-manifold having non-positive Yamabe invariant Y (Σ).1 Suppose that the foliation {Σ × {σ}} is CMC and that σ is the logarithmic time, namely suppose that each slice Σ × {σ} is of constant mean curvature k = −e−σ . Consider This work was completed while the author was a Moore Instructor at MIT. 1

The Yamabe invariant (sometimes called sigma constant) is defined as the supremum of the scalar curvatures of unit volume Yamabe metrics. A Yamabe metric is a metric minimizing the Yamabe functional in a given conformal class.

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the Einstein (CMC) flow (g, K)(σ) where g(σ) and K(σ) are the induced threemetric and second fundamental form over each slice Σ×{σ}. A natural question to ask is the following. Suppose we observe the evolution of (g, K) at the cosmological scale, then, is the long-time fate of (g, K) (at the cosmological scale) unique, and if so, how can one characterize it? If the answer is yes, one would naturally call the limit the ground state (at the cosmological scale) as any solution would decay to it. In this article we will present partial answers to this question. We elaborate on that below. It is a simple but interesting fact that (with generality) one can interpret −k 3 as equal to the Hubble parameter H of the “universe” (g, M) at the “instant of time” Σ × {σ(k)}[14]. This cosmological interpretation of the mean curvature k (or better of −k 3 ) motivates the terminology of various notions that we describe in what follows. Consider a CMC slice Σ × {σ}. At that slice the Hubble parameter −σ is thus H = e 3 . For this particular value of H scale g as H2 g. As it is easy to see, ˜ = (H2 g, HK). the state (g, K) over the slice Σ × {σ} scales to the new state (˜ g , K) 2 In this way the Hubble parameter of the new solution H g and over the same ˜ with H = 1 (or k = −3) will be called slice will be equal to one. A state (˜ g , K) ˜ a cosmologically normalized state. The flow (˜ g , K)(σ) = (H2 (σ)g(σ), H(σ)K(σ)) will be called the cosmologically normalized Einstein CMC flow2 Note that the volume of Σ relative to the metric g˜ is given by V(σ) = H3 (σ)Volg(σ) (Σ). We will call it the reduced volume. It is a crucial and central fact observed by Fischer and Moncrief [8] that V is monotonically decreasing along the expanding direction and 3 it is bounded below by the topological invariant − 61 Y (Σ) 2 . The reduced volume is a weak quantity but its relevance is greatly enhanced if we take into account at the same time the L2g˜ norm of the space–time curvature Rm relative to the ˜ 0 = Q0 ((˜ ˜ CMC slices, namely the Bel–Robinson energy Q g , K)). Our first result in ˜ 0 , the ground state Sect. 2 will be to show that, assuming a uniform bound in Q of the manifold Σ is well defined and unique. In a geometric sense the ground state is equivalent to the Thurston geometrization of Σ. Let us be more precise on the definition of ground state (under a bound in Q0 ) and its characterization. By ground state we mean “the limit” (to be described below) of any sequence of ˜ i )} with Q0 ((˜ ˜ i )) ≤ Λ (Λ is a positive gi , K cosmologically normalized states {(˜ gi , K constant) and Vi ↓ Vinf . As is shown in the appendix, for any CMC state (g, K) the L2g -norm of Ric is controlled by |k|, Q0 and V and precisely by Ric2L2g ≤ C(|k|V + Q0 ), where C is a numeric constant. It follows that the Ricci curvature of the sequence {˜ gi } is uniformly bounded in L2g˜i . Thus [1], one can extract a subsequence of {(Σ, g˜i )} converging in the weak H 2 -topology to a (non-necessarily complete) Riemannian manifold (Σ∞ , g∞ ). We prove that the limit space (Σ∞ , g∞ ) belongs 2

Cosmologically normalized flows have been considered in [5] by Andersson and Moncrief. Note however that the terminology Cosmologically normalized has been introduced in [15].

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to one among three possibilities (independently of the sequence {(gi , Ki )}). In general terms (see Sect. 2.1 for a more elaborate description of the ground state) the three cases are: 1. (Called Case Y (Σ) = 0), Σ∞ = ∅; 2. (Called Case Y (Σ) < 0 (I)), Σ∞ = H is a hyperbolic manifold and g∞ = gH (where gH is the hyperbolic metric in Σ∞ ); 3. (Called Case Y (Σ) < 0 (II)), Σ∞ = ∪i=n i=1 Hi where {Hi } is a finite set of (non-compact) complete hyperbolic metrics of finite volume. The limit metric g∞ over each Hi is equal to gH,i (where gH,i is the hyperbolic metric of Hi ). The two-tori transversal to the hyperbolic cusps of each manifold Hi embed uniquely (up to isotopy) and incompressibly (the π1 injects) in Σ. In the second and third cases Ki converges to −gH,i weakly in H 1 . One can also describe the notion of ground state in terms of geometrizations. This viewpoint will be fundamental in Sect. 3. Recall that for any Riemannian space (Σ, g) the -thick (thin) part Σ (Σ ) of Σ is defined as the set of points p in Σ where the volume g (σ)} is a continuous radius3 ν(p) is bigger (less) or equal than . Say now that {˜ (σ ∈ [σ0 , ∞)) or discrete (σ ∈ {σ0 , σ1 , . . .}) family of Riemannian metrics on Σ. We say that {(Σ, g˜(σ))} persistently geometrizes4 Σ iff there is (σ) > 0 such that (σ) Σg˜(σ) is persistently diffeomorphic to either, the empty set, or, the (σ)-thick part of a single compact hyperbolic manifold ((H, g˜H )), or, the (σ)-thick part of a finite ˜H,i )). set of (non-compact) complete hyperbolic metrics of finite volume (∪i=n i=1 (Hi , g The (σ)-thin parts Σg˜(σ),(σ) on the other hand are persistently diffeomorphic to either, the empty set, or, a single graph manifold (G), or, a finite set of graph mang (σ)} geometrizes Σ ifolds with toric boundaries (∪i=n i=1 Gi ). In quantitative terms {˜ iff either 1. ν g˜(σ) (Σ) → 0 as σ goes to infinity (in which case there is only one persistent G piece) or 2. ν g˜(σ) (Σ) ≥ ν0 > 0 as σ goes to infinity (in which case there is only one persistent H piece) and there is a continuous function ϕ : (σ0 , ∞) × H → Σ, differentiable in the second factor, such that ϕ∗ g˜(σ) − g˜H Hg˜2 → 0 as σ goes H to infinity, or 3. the volume radius collapses in some regions and remains bounded below in some others (in which case there are a set of G pieces G1 , . . . , Gj and a set of H pieces H1 , . . . , Hk ) and for any > 0 and for any H piece (Hi , g˜Hi ) there is a continuous function ϕi : (σ0 , ∞) × Hi → Σ, differentiable in the second factor such that ϕ∗i g˜(σ) − g˜Hi Hg˜2 → 0 as σ goes to infinity. Hi

It is clear that cases 1, 2 and 3 above correspond, respectively, to the three possible cases (1, 2 and 3) of ground states defined before. 3

Given a point p in Σ the volume radius ν(p) at p is defined as the supremum of all r > 0 such that Vol(B(p, r)) ≥ μr 3 for some fixed (but arbitrary) μ > 0. We define ν = inf p∈Σ ν(p) and ν = supp∈Σ ν(p). We will be using these definitions later. 4 We have taken this terminology from [11, Sect. 10].

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Figure 1. A schematic representation of the double cusp ground state. While it is easy to give examples of ground states of the type Case Y (Σ) = 0 and Case Y (Σ) < 0 (I) (see Sect. 2.2) an example of the type Case Y (Σ) < 0 (II) is more difficult to find. We dedicate Sect. 6 to describe a ground state of this type. The new ground state, that we shall call Double Cusp, consists of a fam˜ l )} that we describe in what follows. The manifold Σ is of the form ily {Σ, (˜ gl , K Σ = H1 GH2 where Hi , i = 1, 2 are (non-compact) hyperbolic manifolds with a hyperbolic cusp each5 and the manifold G is a so called torus neck G = [−1, 1]×T 2 . ˜ l )} is parametrized by the metric “length” l of the neck. As The family {(˜ gl , K l → ∞ the geometrization takes place. More precisely, as the length l of G becomes infinite, the volume radius ν(G) over G and the total volume of G collapse to zero. Over the hyperbolic sector H1 , H2 instead, the metric gl converges to gH1 and gH2 respectively and in H 2 . A schematic picture can be seen in Fig. 1. The third part of the article (Sect. 3) deals with the long-time evolution of the cosmologically normalized Einstein flow under the assumption that the zero and the first order Bel–Robinson energies remain uniformly bounded, namely ˜ + Q1 ((˜ ˜ ≤ Λ for a positive constant Λ. The main result will E˜1 = Q0 ((˜ g , K)) g , K)) ˜ be to show that a long-time flow (˜ g , K)(σ) with E˜1 ≤ Λ, persistently geometrizes the manifold Σ. Moreover, the geometrization is the ground state if V ↓ Vinf . Using the classification of ground states (Theorem 3) it is direct to show that ground states are stable in the following sense. For any Λ there is > 0 such that any ˜ long-time cosmologically normalized flow (˜ g , K)(σ) with E˜1 ≤ Λ and initial data ˜ 0 )) with V(σ0 ) − Vinf ≤ the flow converges to the ground state in the (˜ g (σ0 , K(σ long time (in the sense of geometrizations). This result is not known in general if one drops the a priori (strong) assumption of a uniform bound on E˜1 . However, it ˜ 0 )) is was proved by Andersson and Moncrief [5] that if an initial data (˜ g (σ0 ), K(σ 5 The construction can be easily generalized to include hyperbolic manifolds with any number of cusps.

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close enough in H 3 × H 2 to a ground state (H, (gH , −gH )) of type Case Y (Σ) < 0 (I), then E˜1 converges to zero when σ → ∞ and V ↓ Vinf , thus showing stability. We will give a proof of this fact in slightly more geometrical terms. The core of the proof is however the same. Finally, in Sect. 4 we present some arguments favoring the statement that a cosmologically scaled long-time CMC flow with E˜1 uniformly bounded decays necessarily to its ground state. The problem of the long-time geometrization of the Einstein flow was investigated by Anderson in the seminal work [4]. In that article, long-time geometrization (at a particular scale) was established under suitable a priori point-wise bounds on the rescaled space–time curvature (see [4] for a detailed statement). More precisely, it was shown that the flow geometrizes along (some) sequence of diverging times. The problem of whether the geometrization persists or not, and a careful analysis of the collapsed regions remained open. Further progress on these problems was done in [15], where it was possible to show that under a priori C α point-wise bounds on the cosmological normalized space–time curvature, the flow (at the cosmological scale) persistently geometrizes and the volume of the G (or thin) regions decreases to zero. An important open problem is whether the twotori, separating the H (or thick) regions and G (or thin) regions are incompressible or not. A positive answer would imply that the geometrization is unique, that coincides with the Thurston geometrization and that the reduced volume approaches its absolute infimum in the long-time (see [15] for a discussion). The present article discusses the problem of the long-time geometrization under, instead, a priori integral bounds on the cosmologically normalized space–time curvature and its time ˜ 1 ) represent ˜ 0 and Q derivative. These integral norms (the Bel–Robinson energies Q variables which go more in the spirit of general relativity, if we think (at least formally) the Einstein equations as a hyperbolic PDE. The analysis of the Einstein ˜0 + Q ˜ 1 is, in comparison with point-wise bounds flow under a priori bounds on Q ˜ ˜ on Rm and ∇T Rm, a task of much greater complexity. Finally let us note that the idea of linking the reduced volume to geometrizations was first pointed out and investigated in a series of articles by Fischer and Moncrief (see for instance [8]). The reference [15] and the present work owe much to them. 1.1. Background We summarize now some basic formulae that will be used. The reader is encouraged to read [7] from which most of the material of this section is taken (see also [16] for related material). Let us assume we have a cosmological solution6 (M, g) with generic Cauchy hypersurface diffeomorphic to Σ. Assume Σ is a compact three-manifold with non-positive Yamabe invariant Y (Σ). Assume too that there is a CMC foliation Σ×[k0 , 0) inside7 M, where k is the mean curvature. A solution 6

Following Bartnik a cosmological solution of the Einstein equations is a maximally globally hyperbolic solution having a compact space-like Cauchy hypersurface. 7 Not necessarily covering the future of M.

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having such foliation will be called a long-time CMC solution.8 With respect to the CMC foliation the metric g splits into a space-like metric g, a lapse N and a shift X. We recover the metric g from them by g = −(N 2 − |X|2 )dk 2 + X ∗ ⊗ dk + dk ⊗ X ∗ + g, where X ∗ = gab X a . The Einstein CMC equations in the CMC gauge (and for an arbitrary shift) are R = |K|2 − k 2 , ∇.K = 0,

(1) (2)

g˙ = −2N K + LX g,

(3)

K˙ = −∇∇N + N (Ric + kK − 2K ◦ K) + LX K,

(4)

2

−ΔN + |K| N = 1,

(5)

where K is the second fundamental form, (K ◦ K)ab = Kac Kcb and LX is the Lie derivative operator along the vector field X. Sometimes we will need to use these ˜ = HK formulas in terms of the cosmologically normalized quantities g˜ = H2 g, K 2 ˜ and N = H N . They will be provided without further deductions. The expressions for the derivative of the reduced volume with respect to logarithmic time will be central in Sect. 3. It is convenient to write them right away in terms of cosmological normalized quantities. They are dV ˆ˜ 2 dv . ˜ |K| ˜ dvg˜ = − N = −3 1 − 3N g ˜ g ˜ dσ Σ

Σ

tr A where the hat Aˆ of a two tensor A denotes its traceless part Aˆ = A − 3g g. The ˜ − 1 is the so called Newtonian potential and it is sometimes a expression φ = 3N better quantity to work rather than the lapse N . Let us give now the basic elements of Weyl fields and Bel–Robinson energies. Again in this case the reader is encouraged to read the reference [7] for a complete account. A Weyl field is a traceless (4, 0) space–time tensor field having the symmetries of the curvature tensor Rm. We will denote them by Wabcd or simply W. As an example, the Riemann tensor in a vacuum solution of the Einstein equations is a Weyl field that we will be denoting by Rm = W0 (we will use indistinctly either Rm or W0 ). The covariant derivative of a Weyl field ∇X W for an arbitrary vector field X is also a Weyl field. We will be using the Weyl fields W0 = Rm and W1 = ∇T Rm, where T is the future pointing unit normal field to the CMC foliation. Given a Weyl tensor W define the current J by

∇a Wabcd = Jbcd , 8

The terminology is justified by the fact that if the manifold Σ has non-positive Yamabe invariant then the range of k (which is known to be a connected interval of the real line) cannot contain zero. If Y (Σ) ≤ 0 it is conjectured that the range of k is actually (−∞, 0).

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When W is the Riemann tensor in a vacuum solution of the Einstein equations the currents J is zero due to the Bianchi identities. The L2 -norm of a Weyl field W with respect to the foliation will be introduced through the Bel–Robinson tensor which is defined by ∗ Qabcd (W) = Walcm Wbl dm + Walcm W∗ bl dm .

The Bel–Robinson tensor is symmetric and traceless in all pairs of index and for any pair of time-like vectors T1 and T2 , the quantity9 Qabcd T1a T2b T3c T4d is positive (provided W = 0). The electric and magnetic components of W are defined as Eab = Wacbd T c T d , ∗

(6)

Bab = Wacbd T T , c

d

(7) 1 lm 2 ablm W cd .

∗

where the left dual of W is defined by Wabcd = E and B are symmetric, traceless and null in the T direction. It is also the case that W can be reconstructed from them (see [p. 143] [7]). If W is the Riemann tensor in a vacuum solution we have Eab = Ricab + kKab − Kac K cb , abl Blc

(8)

= ∇a Kbc − ∇b Kac .

(9)

The components of a Weyl field with respect to the CMC foliation are given by (i, j, k, l are spatial indices) ∗

WijkT = −ij m Bmk , Wijkl = ijm kln E

mn

WijkT = ij m Emk ,

,

∗

Wijkl = ijm kln B

(10) mn

.

(11)

We also have QT T T T = |E|2 + |B|2 , QiT T T = 2(E ∧ B)i , 1 QijT T = −(E × E)ij − (B × B)ij + (|E|2 + |B|2 )gij . 3 The operations × and ∧ are provided explicitly later. The divergence of the Bel– Robinson tensor is ∇a Q(W)abcd = Wbmd n J(W)mcn + Wbmc n J(W)mdn +∗ Wbmdn J∗ (W)mcn +∗ Wbmcn J∗ (W )mcn . where J∗bcd = ∇a (∗ Wabcd ). We have therefore ∇α Q(W)αT T T = 2E ij (W)J(W)iT j + 2B ij J∗ (W)iT j . We will denote

Q(W) = Σ 9

QT T T T (W)dvg =

|E(W)|2 + |B(W)|2 dvg ,

Σ

We will later use the notation Qabcd T1a T2b T3c T4d = QT1 T2 T3 T4 .

(12)

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and in particular when W = W0 or W = W1 we will denote Q0 = Q(W0 ) and Q1 = Q(W1 ). From the Eq. (12) we get the Gauss equation ˙ Q(W) = − 2N E ij (W)J(W)iT j +2N B ij (W)J∗ (W)iT j +3N QabT T Πab dvg . Σ

(13) Πab = ∇a Tb is the deformation tensor and plays a fundamental role in the space– time tensor algebra. Its components are Πij = −Kij , ∇i N ΠT i = , N

ΠiT = 0, ΠT T = 0.

Finally, we have divE(W)a = (K ∧ B(W))a + JT aT (W),

(14)

divB(W)a = −(K ∧ E(W))a + J∗T aT (W), (15) 3 1 curlBab (W) = E(∇T W)ab + (E(W) × K)ab − kEab (W)+JaT b (W), (16) 2 2 3 1 curlEab (W) = B(∇T W)ab + (B(W) × K)ab − kBab (W)+J∗aT b (W). (17) 2 2 The operations ∧, × and the operators Div and Curl are defined through 1 1 (A × B)ab = acd bef Ace Bdf + (A ◦ B)gab − (trA)(trB)gab , 3 3 (A ∧ B)a = abc Abd Bdc , (div A)a = ∇b Aba , 1 (curl A)ab = (alm ∇l Amb + blm ∇l Ama ). 2 In what follows we describe the main results that will be used from the theory of convergence–collapse of Riemannian manifolds under L2 -bounds on the Ricci curvature and its covariant derivatives. The reader can consult (some of) the original sources [1,17,18]. Let us mention first a classical local result. Recall that in a Riemannian manifold (Σ, g) the H i -harmonic radius ri (p) of g at p, i ≥ 2, is defined as the supremum of the radius r for which there is a coordinate chart {x} covering B(p, r) and satisfying 3 4 δjk ≤ gjk ≤ δjk , (18) 3 ⎛ 4 ⎞ 2 I α=i ∂ ⎜ ⎟ r2α−3 ⎝ g (19) ∂xI jk dvx ⎠ ≤ 1, α=2 |I|=α,j,kB(o,r)

where in the sum above I is the multi-index I = (α1 , α2 , α3 ), and as usual ∂ I /∂xI = (∂x1 )α1 (∂x2 )α2 (∂x3 )α3 . Both expressions above are invariant under the ˜μ = λxμ and r˜ = λr. Observe that if j > i ≥ 2 simultaneous scaling g˜ = λ2 g, x

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then rj (p) ≤ ri (p). A chart {x} as above will be called a canonical harmonic chart. In basic terms the H i -harmonic radius marks the greatest scale λ in which i -Sobolev norm of the scaled metric λ2 g (in the ball of radius one of the the H{x} scaled metric) is bounded above by one. i We have used the notation H{x} for the H i -Sobolev space defined with respect to a chart {x}. Instead Hgi will be the Hgi -Sobolev space defined with respect to the metric g, namely for a tensor A (of any rank) we define j=i A2Hgi (Ω) = |∇(j) A|2 dvg , Ω j=0

where Ω is a given region in Σ where the Sobolev space is defined (so properly speaking we would write Hgi (Ω)). Theorem 1. Let (Σ, g) be a complete Riemannian three-manifold and let p be a point in Σ. Then 1. Ric2L2 and ν(p) control the H 2 -harmonic radius r2 (p) of g at p from below. g

2.

Ric2H 1 and ν(p) control the H 3 -harmonic radius r3 (p) of g at p from below. g

A more complex global result is10 Theorem 2. Let {(Σ, gi )} be a sequence of compact Riemannian manifolds with RicL2g + V olgi (Σ) ≤ Λ, i

where Λ is a positive constant. Then one can extract a sub-sequence (to be denoted also by {(Σ, gi )}) with one of the following behaviors. (1) (Collapse). ν i → 0 and the sub-sequence gi collapses along a sequence of F-structures. The manifold Σ is in this case a graph manifold. (2) (Convergence). ν i ≥ ν 0 > 0 and {(Σ, {gi })} converges weakly in H 2 to a H 2 Riemannian manifold (Σ∞ (= Σ), g∞ ). (3) (Convergence–Collapse). ν i → 0 and ν i ≥ ν 0 > 0 and {(Σ, gi )} converges weakly in H 2 to a (at most) countable union ∪α (Σ∞,α , g∞,α ) of H 2 (non necessarily complete) Riemannian manifolds. Moreover, for a given (sufficiently small) the manifolds Σgi , are graph manifolds with toric boundaries. The Riemannian-manifolds {(Σgi , gi )} converge weakly in H 2 to ∪α (Σ∞,α , g∞,α ) (which has only a finite number of components). The notion of convergence that we have assumed in the statement of the theorem is the following: we say that {(Σ, gi )} converges weakly in H 2 to a limit Riemannian manifold (Σ∞ , g∞ ) (as above) if for every > 0 there are (H 3 )-diffeomorphisms ϕi : Σ∞,g∞ → Σgi such that ϕ∗i gi converges to g∞ in the weak 10

For a discussion of this result see [1]. We will not elaborate on the notion of F structure as we will not need in the present article. A graph manifold is, roughly speaking a sum along two tori, of U (1)-bundles over two-surfaces. For a discussion of the relation between graph manifolds and F structures see [1].

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H 2 -topology induced by the metric g∞ over the space of H 2 (2,0)-tensors (over Σ∞ ). The use of Weyl fields to control the gravitational field would not be justified in this article if it were not for the following fundamental property. Proposition 1. Let (g, K) be a cosmologically normalized state on a three-manifold Σ with non-negative Yamabe invariant (Y (Σ) ≤ 0) and let p be a point in Σ. Then 1.

2.

ν(p), Q0 and V olg (Σ) control from below the H 2 -harmonic radius r2 (p) of g at p. Moreover if {x} is a canonical harmonic coordinate system the Sobolev 1 norm KH{x} (B(p,r2 (p)) is controlled from above.

ν(p), Q0 + Q1 and V olg (Σ) control from below the H 3 -harmonic radius r3 (p) of g at p. Moreover if {x} is a canonical harmonic coordinate system the 2 Sobolev norm KH{x} (B(p,r3 (p)) is controlled from above.

In basic words what the (say item 2 of the) Proposition claims is that the quantity 2 1 × H{x} norm 1/ν(p) + Q0 + Q1 + Volg (Σ) controls r3 (p) from below and the H{x} of (g, K) from above, in suitable canonical harmonic coordinates {x} covering B(p, r3 (p)). The proof of Proposition 1 item 1 follows easily from the bound ⎛ ⎝

⎞ 12 |Ric|2 + |∇K|2 + |K|4 dvg ⎠ ≤ C(Volg (Σ) + Q0 ),

Σ

that is proved in the appendix (C is a numeric constant) and by applying Theorem 1. The proof of the item 2 in the Proposition 1 follows by using item 1 and applying standard elliptic estimates in the elliptic system (14)–(17) with E = E(W0 ) and B = B(W0 ) on the left-hand side and then using Eqs. (8) and (9). Details of this argument can be found in [16]. The estimates of Proposition 1 will be referred later simply as “elliptic estimates”.

2. The Ground State and Examples 2.1. The Ground State Theorem 3. (The ground state) Let Σ be a compact three-manifold with Y (Σ) ≤ 0. Say {(gi , Ki )} is a sequence of states satisfying (1) (2) (3)

ki = −3;

32 1 Vi ↓ Vinf = − Y (Σ) ; 6 Q0 ((gi , Ki )) ≤ Λ,

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where Λ is a fixed constant. Then, there is a sub-sequence of {(gi , Ki )} (to be denoted also by {(gi , Ki )}) for which one and only one of the following three phenomena occurs. Case Y (Σ) = 0. 1. Σ = G is a graph manifold. 2. ν → 0 and the Riemannian spaces (Σ, gi ) collapse with bounded L2 curvature, along a sequence of F-structures. 3. Vi ↓ Vinf = 0 Case Y (Σ) < 0 (I). 1. Σ = H is a compact hyperbolic manifold (denote its hyperbolic metric by gH ). 2. (Σ, gi ) → (Σ, gH ) in the weak H 2 -topology. 3 3. Vi ↓ V olgH = − 61 Y (Σ) 2 . Case Y (Σ) < 0 (II). 1. There is a set of incompressible two-tori {Ti2 , i = 1, . . . , iT } embedded in Σ and cutting it into a set {Hi , i = 1, . . . , iH } of manifolds admitting a complete hyperbolic metric of finite volume (in its interior) and a set {Gi , i = 1, . . . , iG } of graph manifolds. The tori Ti2 are unique up to isotopy. 2 H 2. (Σ, gi ) → ∪i=i i=1 (Hi , gH,i ) in the weak H -topology. 1 32 i=iH 3. Vi ↓ i=1 V olgH,i (Hi ) = − 6 Y (Σ) . In each of the three cases above the norms Ricgi L2g , Ki Hg1 , and Ki L4g i i i ˆ i L2 , Rg + 6L1 converge to zero. remain uniformly bounded and the norms K gi

i

gi

Moreover in the regions of convergence (the hyperbolic sector in (I) and (II)) the scalar curvature Rgi converges in the strong L2 -topology to −6. Finally, two different sub-sequences of the original sequence {(gi , Ki )} as above have the same behavior. Proof. We will make use of a number of inequalities proved in the Appendix. From Proposition 10 we have ˆ 2 + |K| ˆ 4 dvg ≤ C(|k|(V − Vinf ) + Q0 ), 2|∇K| (20) Σ

and from Proposition 11 ˆ 2 dvg ≤ C(|k|(V − Vinf ) + (|k|(V − Vinf )Q0 ) 12 ). |k|2 |K|

(21)

Σ

This in particular implies the inequality 1 2 2 2 |k| Rg + k dvg ≤ C(|k|(V − Vinf ) + (|k|(V − Vinf )Q0 ) 2 ). 3

(22)

Σ

From Proposition 12 we have L2 ≤ C |k|(V − Vinf ) + (|k|(V − Vinf )Q0 ) 12 + Q0 . Ric g

(23)

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and from Proposition 13 also in the Appendix 4 |∇R| 3 + R2 dvg ≤ C(|k|V + Q0 ).

(24)

Σ

Recall that when using the formulae above we will be dealing with a sequence ˜ i )} with ki = −3. {(˜ gi , K Case Y (Σ) = 0. From (23) we see that the L2gi norm of Ricgi remains uniformly bounded. This case then follows from Theorem 2. Case Y (Σ) < 0. First we note that there must be a constant ν 0 > 0 such that ν i ≥ ν 0 otherwise one can extract a sub-sequence of {gi } which collapses with bounded volume and curvature. Theorem 2 then implies that Σ is a graph manifold and therefore of zero Yamabe invariant which is a contradiction. Cases (I) and (II) will be distinguished according to whether there is a sub-sequence of {gi } with ν i → 0 or not. We do that below. (I). Suppose there exists ν 0 > 0 such that ν i ≥ ν 0 . Then by Theorem 2 we can extract a sub-sequence of {gi } converging in the weak H 2 -topology to a compact Riemannian manifold (Σ, g∞ ). From (22) we deduce that Rg∞ = −6. Let us see that g∞ is hyperbolic. Note that Σ Rg2∞ dvg∞ = |Y (Σ)|2 . Consider the quadratic functional R from H 2 -metrics into the reals given by 1 g → Volg (Σ) 3 Rg2 dvg . Σ

It is known [2] that if Y (Σ) < 0 the infimum of R is given by |Y (Σ)|2 . Thus it must be δR|g∞ = 0. Let us compute the variation of R at g = g∞ and for variations which preserve the local volume. Consider then an arbitrary path of metrics g(λ) with g(λ = 0) = g∞ and (dvg(λ) ) = 0 (and Frechet derivative g = h in H 2 ). From (dvg ) = 0 we get trg h = 0. Recall that the variation of the scalar curvature is given by δh Rg = Δtrg h + δδh − Ric, h. From this we get 1

δh Rg |g=g∞ = −Volg3∞ 2R∞

g , hdvg . Ric ∞ ∞

Σ

g = 0 and g∞ is hyperbolic. Therefore this case corresponds to Case Thus, Ric ∞ Y (Σ) < 0 (I). (II). Suppose lim sup ν i = 0. Consider a H 2 -weak limit of (Σ, gi ). Denote it by (Σ∞ , g∞ ). Recall that Σ∞ may have infinitely many connected components and that g∞ may not be complete on them. Note that Σ∞ is non-empty as we have ν i ≥ ν 0 > 0. For every i consider the metric gY,i in the conformal class of gi with scalar curvature RY = −6. Writing gY,i = φ4i gi , the conformal factor φi satisfies the equation RY φ5i = −8Δgi φi + Rgi φi .

(25)

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From the maximum principle we get φi ≤ 1. Thus 0 ≤ Volgi (Σ) − VolgY,i (Σ) ≤ Volgi (Σ) − Volinf . It follows from the fact that Vi ↓ Vinf that 0 ≤ 1 − φ6i dvgi → 0. and in particular

Σ

(1 − φi )6 dvgi → 0. Note that Σ 1 3 VolgY,i Rg2Y,i dvgY,i → |Y (Σ)|2 . Σ

We will exploit this fact in what follows. Pick an arbitrary point p ∈ Σ∞ . We will g |B(p,ν (p)/2) = 0. As the point p is arbitrary this would show that show that Ric ∞ g∞ g = 0 and thus g∞ is hyperbolic. First note that by (22) it is Rg = RY = −6. Ric ∞ ∞ Also by (24) and the compact embedding H 1,4/3 → L2 we see that Rgi → RY strongly in L2 on compact sets of Σ∞ . Pick a sequence {pi } of points pi ∈ Σ such that (Bgi (pi , νgi (pi )), pi , gi ) converges to (Bg∞ (p, νg∞ (p)), p, g∞ ). Let us write Eq. (25) in the form 8Δgi φi = Rgi φi − RY φ5i ,

(26) 2

and prove that the right hand side of it is converges to zero in L and over the sequence of balls Bgi (pi , νgi (pi )) (denote them by Bi ). Write |RY φ5i − Rgi φi |2 dvgi ≤ |RY φ4i − Rgi |2 dvgi Bi

Bi

≤

2|RY |2 |φ4 − 1|2 + 2|RY − Rgi |2 dvgi .

(27)

Bi

We have

|φ4i − 1|2 dvgi ≤

Bi

|φ4i − 1|dvgi → 0,

Bi

From this and the fact that Rgi converges to RY strongly in L2 over Bi we have that the right-hand side of Eq. (27) goes to zero as claimed. By elliptic regularity φi − 1Hg2 (Bg (pi , 23 νg (pi )) converges to zero. As a result (Bgi (pi , 23 νgi (pi ), pi , gY,i ) i

i

i

converges weakly in H 2 to (Bg∞ (p, 23 νg∞ (p)), p, g∞ ). As a consequence we have that g , hi g dvg → Ric Ricg∞ , hg∞ dvg∞ , Y,i Y,i Y,i Σ

Σ∞ 2

for any traceless tensor h (in H ) with support in Bg∞ (p, νg∞ (p)/2) and traceless tensors hi (in H 2 ) with support in Bgi (pi , νgi (pi )/2) converging strongly in H 2 to g = 0 in Bg (p, νg (p)/2) h. Thus, δhi R|g=gY,i → δh R|g=g∞ . Therefore, if Ric ∞ ∞ ∞

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we can lower the infimum of |Y (Σ)|2 for the functional R over the three-manifold Σ. We prove now that g∞ is complete. Let s be an incomplete geodesic in Σ∞ . Fix p ∈ s. Let S2 be a transversal geodesic-two-simplex in Σ∞ and having p in its interior. For q ∈ s (in the incomplete direction and close to p) consider the threesimplex S3 (q) formed by all geodesics joining q with a point in S2 . Observe that because (Ω, g∞ ) is hyperbolic and s has finite length (in the incomplete direction) every r ∈ ∂S3 (x) has a cone C3 (r) inside and of size bounded below.11 Now as q approaches the end of s, we can find a sequence of points qi and r(qi ) ∈ ∂S3 (qi ) with νg∞ (r(qi )) → 0 and having a cone inside S3 (r(qi )) of size bounded below. The blow up limit of the pointed space (Σ∞ , r(qi ), ν(r(q1 i ))2 g∞ ) has ν(x) = 1 and is complete, flat and having a cone of size (α, ∞) inside. It must be R3 which is a contradiction. We can conclude then that Σ∞ consist of a finite number of connected components Hi , i = 1, . . . , iH each one admitting a complete hyperbolic metric of finite volume gH,i = g∞ . Observe that it must be Volg∞ (Σ∞ ) = Volinf (and not strictly less than Volinf ) for otherwise (see [6]) one can find a sequence of metrics H g˜ ˜i in Σ and with bounded L∞ curvature, converging to ∪i=i ˜ i=1 (Hi , gH,i ) and with g ˜i Volg˜i (Σ) → Volg∞ (Σ∞ ) and thus lowering the value |Y (Σ)|2 for the infimum of R. Now, pick a transversal torus for each one of all the hyperbolic cusps of the Riemannian manifolds (Hi , gH,i ). Denote them by {Ti , i = 1, . . . , i = iT }. Each one of the tori Ti can be embedded (up to isotopy) inside Σ. As proved in [3, Theorem 2.9] if one of the tori is compressible one can again lower the infimum value for R. Thus, the tori Ti are all incompressible. As shown in [3, p. 156] the set of tori {Ti , i = 1, . . . , iT } (of a strong geometrization as this) is unique up to isotopy. The rest of the claims in the Theorem follow from Eqs. (20)–(24). 2.2. Examples Examples of ground states (namely sequences {(gi , Ki )} of cosmologically normalized states with Vi ↓ Vinf and Q0 ≤ Λ) and of the types Case Y (Σ) = 0 or Case Y (Σ) < 0 (I) (in Theorem 2.1) are easy to find. We will show that soon below. An example of a ground state of the type Case Y (Σ) < 0 (II) is more difficult to find and will be discussed in a separate section (Sect. 6). Case Y (Σ) = 0. Take any two-surface Σgen of genus greater or equal than one. Consider the three-manifold Σ = Σgen × S 1 . Denote by l2 ds2 the metric on S 1 with total length l and denote by ggen a metric on Σgen of scalar curvature −6. An example of a ground state of the type Case Y (Σ) = 0 is given by the sequence of states {(gl , −gl )} on Σ where gl = ggen × l2 ds2 and l → 0. 11

Given a point x in a Riemannian manifold (Σ, g) a cone of size (α, l) (l < injx g) in Σ is the image under the exponential map of a cone of size (α, l) (segments from x in Tx Σ having length l and forming an angle α with a given segment).

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Case Y (Σ) < 0 (I). Take any compact hyperbolic manifold Σ with hyperbolic metric gH . The constant sequence of states {(gH , −gH )} is an example of a ground state of type Case Y (Σ) < 0 (I). 2.3. The Double Cusp Say (H1 , gH1 ) and (H2 , gH2 ) are two complete hyperbolic metrics of finite volume and suppose that each one has, for the sake of concreteness, only one hyperbolic cusp. Denote the cusps as C1 and C2 . Denote by (gHi , −gHi ) the flat cone states on Hi , i = 1, 2. Recall that the metrics gHi on the cusps are of the form gHi = dx2 + e2x gT,i where gTi is a flat (and x-independent) metric on the tori T 2 transversal to the cusps (−∞, a]×T 2 . Consider now a torus-neck, namely the manifold G = [−l, l] × T 2 with a T 2 -invariant metric gG . For any x0 < a we will find a state (gG , KG ) on G which, at the boundary ∂[−l, l] × T 2 = {−l} × T 2 ∪ {l} × T 2 , approximates to any given desired order the flat cone states of H1 and H2 at x = x0 . Once this is done we will glue (gH1 , −gH1 ), (gG , KG ) and (gH2 , −gH2 ) to get an state over H1 GH2 (satisfying the constraints equations). As x0 → −∞ these “double cusp” states display the behavior of a ground state of type Case Y (Σ) < 0 (II). A schematic picture can be seen in Fig. 1. Note that the states (gG , KG ), being T 2 -symmetric, are Gowdy and therefore explicitly tractable. The construction is organized as follows. In Sect. 2.3.1 we find a (Gowdy) polarized space–time solution on R×R×T 2 . Once this is done, we find in Sect. 2.3.3 a foliation of R × R × T 2 the states of which display (when suitable normalized) a convergence–collapse behavior of the type Case Y (Σ) < 0 (II). Although the states found in this foliation are not CMC, we will see in Sect. 2.3.5 that it is possible to find a CMC foliation the CMC states of which are not far from those found before and displaying the same convergence–collapse behavior. In Sect. 2.3.4 we find (Gowdy) non-polarized space–time solutions on R × R × T 2 . One can then repeat the analysis done in Sects. 2.3.3 and 2.3.5 to find, for each space–time nonpolarized solution, a CMC foliation displaying a convergence–collapse picture of the type Case Y (Σ) < 0 (II). The family of polarized states that we will construct is sufficient to join two arbitrary flat cone cusp sates (C1 , (gH1 , −gH1 )) and (C2 , (gH2 , −gH2 )). Suppose now we have two flat cone states (Hi , (gHi , −gH2 )) having a hyperbolic cusp each that we want to join through a state in a torus-neck. Having fixed x0 and a given error , suppose we have found a state (polarized or not) (gG , KG ) in a torus-neck G, which is compatible (up to the error ) at its ends with the flat cone cusps (C1 , (gH1 , −gH1 )) and (C2 , (gH2 , −gH2 )) at x = x0 . We will perform the gluing of (H1 , (gH1 , −gH1 )), (gG , KG ) and (H2 , (gH2 , −gH2 )) as follows. First, we glue (keeping the T 2 -symmetry) the metrics gHi , i = 1, 2 and gG on an interval ([a, b] × T 2 ) of length one in each one of the necks and centered at x = x0 . Denote the new metric by g . Then we find a transverse traceless tensor ˆ T T with respect to g and equal to −gH or KG outside the intervals where the K i ˆ T T ) we appeal to a Theorem of Isenberg metrics were glued. Using the data (g , K ˆ T T ) the Lichnerowicz [12] to show that in the conformal class of the state (g , K

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equation can be solved and therefore a CMC state found. Finally, we use standard elliptic estimates to show that if the error is small enough the CMC state constructed with the conformal method is as close to the states (gHi , −gHi ) and (gG , KG ) (in their respective domains) as we like. 2.3.1. The Geometry on a Torus Neck (the Polarized Case). On R × R × T 2 we look for a (polarized) T 2 -symmetric space–time metric in the coordinates where it looks like g = e2a (−dt2 + dx2 ) + Re2W dθ12 + Re−2W dθ22 . The functions a, R, W depend on (t, x). Define the coordinates (−, +) = (t − x, t + x). Derivatives with respect to − and + will be denoted with a subscript + or −. In this representation the Einstein equations are equivalent to the system of scalar equations ∂2R ∂2R − 2 = 0, ∂x2 ∂t ∂ ∂ ∂ ∂ R W − R W = 0, ∂t ∂t ∂x ∂x 2 R± 1 R± R±± 2 a± = − + 2W±2 . R R 2 R

(28) (29) (30)

Note that Eq. (28) is decoupled from the rest. We make the choice R(x, t) = R0 (e2(t+x) + e2(t−x) ). The Eq. (29) is the Euler–Lagrange equation of the Lagrangian L(t, ∂t W, ∂x W ) = R(∂t W )2 − R(∂x W )2 dx. We make the choice W (x, t) = W1 + W0 arctan e2x . These solutions are the W -stable solutions, i.e. those W that with fixed values at the boundary (infinity in this case) minimize the potential V = R(x, 0)(∂x W )2 dx. We proceed now to find out a. Observing that 2(W± )2 = Eq. (30) can be written 1 R±± R± a± = − 2 R R 2

W02 , 2 cosh2 2x

R± R

2 +

W02 . 2 cosh2 2x

Dividing by R± /R and adding and subtracting both equations we get 1 W02 + ∂x a = − tanh 2x, 2 2 3 W2 ∂t a = + 0 , 2 2

(31)

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which after integration give a(x, t) = a(0) −

1 W02 + 2 2

1 ln cosh 2x + 2

3 W02 + 2 2

t.

In the next section we analyze these solutions along some particular space-like foliations. 2.3.2. The Evolution of States on a Torus Neck. Convene that by observes we mean a space-like slice S(t ) moving with a parametric time t . Let us analyze the solutions found in the previous section with this perspective. First, for those observers who in a forced manner move keeping their x-coordinate constant and moving uniformly forward in time t = t , the normalized three-geometry (normal ized by e

2 W0 3 2+ 2

t

), collapses along the two-tori into the one-dimensional geometry a(0)−

g∞ = e

2 W0 1 2+ 2

ln cosh 2x 2

dx2 ,

on the real line and of finite length. However, for those observers who freely fall in space–time along time-like geodesics, the normalized three-geometry will be seen to evolve into a hyperbolic cusp g∞ = dx2 + R0 e2W±∞ e2x dθ12 + R0 e−2W±∞ e2x dθ22 . There are in fact two natural sets of free-falling observers, those who move with positive x and those with negative x. Both will observe the normalized three-geometry become into hyperbolic cusps (exponentially in time). In between them the geometry is collapsing, as will be made precise in what follows. Free falling observers. We will assume a minor approximation that in no way changes the global picture, nor the precise statements that follow on the evolution of the exact geometry. Concentrate on the region x ≥ 10. On it the metric g (in the (t, x) plane) is almost like 2

e

2 W0 3 2+ 2

W2 t− 12 + 20 x

(−dt2 + dx2 ).

We will consider time-like geodesics in this region (towards the increasing direction of t). Denote by s their proper time. Then it can be calculated that, independently of the initial velocity, the coordinates (t(s), x(s)) of time-like geodesics behave according to 1 1 W02 3 W02 1 3 + W02 + + +o − t+ x = ln , 2 2 2 2 2 1 + W02 s 1 1 W02 3 W02 1 (3 + W02 )(1 + W02 ) + + +o − x+ t = ln s + ln . 2 2 2 2 2 2 s

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What these formulas tell us is that the set of coordinates 1 W02 3 W02 + + t = − x+ t, 2 2 2 2 1 W02 3 W02 + + x = − t+ x, 2 2 2 2 form the natural coordinate system prescribed by a free-falling set of observers. In 2 2 2 W W02 1 1 3 0 + 2+ 2 these new coordinates and after choosing a(0) = 2 ln − 2 + 2 we get 2t

g=e

2

2

2( π 2 W0 +W1 )

(−dt + dx ) + R0 e π

2(t +x )

e 2

· · · + R0 e−2( 2 W0 +W1 ) (e2(t +x ) + e 2+W0

2

(t −x )

+e

2 2 2+W0

(t −x )

dθ12 + · · ·

)dθ22 .

After making W+∞ = π2 W0 + W1 and normalizing by e2t we see that the local three-geometry exponentially falls into the hyperbolic cusp g = dx2 + R0 e2W+∞ e2x dθ12 + R0 e−2W+∞ e2x dθ22 2.3.3. A Convergence–Collapse Picture. Let us describe now a global foliation of Cauchy hypersurfaces (labeled with a parameter s ≥ 1) where we can see the picture of convergence–collapse. For any s the hypersurface will be defined as: (Zone I) W02 W02 1 3 {(t, x), − 2 + 2 ln s + 2 + 2 t = s, | x |≤ ln t}, (Zone II) {(t, x), s = W2 W2 t = − 12 + 20 x + 32 + 20 t, x ≥ ln s} and (Zone III) {(t, x), s = t = W02 W2 1 x + 32 + 20 t, x ≤ − ln s}. Normalize the three-metrics over the 2 + 2 slices with the factor e−2s . As s → +∞ the limit of the normalized three-metrics are: (Zone I) x2 , g∞ = d˜ which is the infinite-length one-dimensional geometry on the real line, and (Zone II) g∞ = dx2 + R0 e2W+∞ e2x dθ12 + R0 e−2W+∞ e2x dθ22 , on the whole R × T 2 , and similarly for the Zone III. A schematic picture can be seen in Fig. 2. 2.3.4. The Geometry on a Torus Neck (the Non-Polarized Case). In this section we follow the same strategy as in Sect. 2.3.1 to find (Gowdy) T 2 -symmetric space–time solutions but this time non-polarized. On R × R × T 2 we look for a non-polarized T 2 -symmetric metric in the coordinates where it looks like g = e2a (−dt2 + dx2 ) + R(e2W + q 2 e−2W )dθ12 − Rqe−2W 2dθ1 dθ2 + Re−2W dθ22 , (32)

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Figure 2. A schematic figure showing the evolution of the normalized three-geometry. and where a, R, W depend only on (t, x) or (u, v) = (−, +) = (t − x, t + x). In this representation the Einstein equations reduce to

2

R++ − R

R−− 2 − R

R+ R R− R

2 2

R+− = 0,

(33)

2 −4W + 4W+2 + q+ e − 4a+

R+ = 0, R

(34)

2 −4W + 4W−2 + q− e − 4a−

R− = 0, R

(35)

(RW− )+ + (RW+ )− + Rq+ q− e−4W = 0, −4W

(Re

−4W

q+ )− + (Re

q− )+ = 0.

(36) (37)

Again we make the choice R(x, t) = R0 e2t cosh(2x). With this choice we will solve for time-independent W and q realizing arbitrary flat metrics on the two tori at the ends, i.e. which have prescribed asymptotic q∞ , q−∞ , W∞ , W−∞ . After that we will solve for a. Solving for time independent W and q. Equation (37) forces q to satisfy q =

2ce4W . cosh(2x)

(38)

where c is an arbitrary constant. With q of this form, Eq. (36) forces W to satisfy W + 2 tanh(2x)W =

−2c2 e4W , cosh2 (2x)

(39)

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The strategy to find the solutions to (38)–(39) for W and q and having prescribed asymptotic values at the ends (i.e. when x → ±∞) is the following. Fix c first. Then find W having the prescribed asymptotic values W (∞) = W∞ and W (−∞) = W−∞ . Then vary c keeping fixed the asymptotic conditions for W and prove that we can reach at some c the prescribed asymptotic value q(∞) = q∞ if q(−∞) = q−∞ was prescribed. We will accomplish that by proving that varying c from some value c0 toward zero, the integral from −∞ to ∞ of Eq. (38) that defines q(∞) reaches (having q−∞ as the lower limit of integration prescribed) all possible values. Although Eq. (39) is highly non-linear, it can be integrated exactly. We note that Eq. (39) is equivalent (unless W is constant in which case c = 0 and q is constant) to ((cosh(2x)W )2 ) = −(c2 e4W ) ,

(40)

cosh2 (2x)W 2 = −c2 e4W + A2 ,

(41)

which gives

for A > 0, an arbitrary positive constant. Taking the square root of (41) we get a separable variables ODE. After integration we get 1 |c| W = − ln cosh(−2A arctan e2x + B), (42) 2 A with B and arbitrary constant. We need to find A and B that solve the asymptotic conditions for W i.e. |c| cosh B = e−2W−∞ , A |c| cosh(−πA + B) = e−2W∞ . A Making the change of variables A =

B−D π

we get the equivalent equations

B = D + π | c | e2W∞ cosh D, 2W−∞

D = B−π |c|e

cosh B.

(43) (44)

Now the problem is to understand the solutions B and D to (43)–(44) as functions of c, W∞ and W−∞ . If we graph B(D) (from 43) and D(B) (from 44) on the same B − D-coordinates axis, we see (observe the factor |c| in front of cosh D and cosh B) that there is some positive c0 above which there are no solutions (the graphs do not intersect), at which there is only one and below which there are only two solutions (see Fig. 3). In the following we will analyze the solutions A and B as c → 0. We will see that given a prescribed value q−∞ we get any asymptotic value for q∞ by varying c from c0 towards zero. The equation e2W−∞ cosh B = e2W∞ cosh D, gives for the each one of the two different branches (of solutions (B, D)) the following behaviors

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Figure 3. The graphs of B(D) (from 43) and D(B) (from 44) for a small c. 1.

(Branch I). Either W∞ = W−∞ for which we get (observe that A = B−D > 0)

or W∞ = W−∞

B = −D → 0, |c| → e−2W−∞ , A for which we get

B → ∞ if W∞ > W−∞ (or − ∞ if W∞ < W−∞ ), B − D → 2(W∞ − W−∞ ) (or − 2(W∞ − W−∞ )), 2 2 A → (W∞ − W−∞ ) or − (W∞ − W−∞ ) . π π 2.

(Branch II) For any W∞ , W−∞ B → ∞, D → −∞, B + D → 2(W∞ − W−∞ ), 2B − 2(W∞ − W−∞ ) . A ∼ π

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With these behaviors for A and B (as c → 0) we get (Branch I). The formula for q c q = 2 , |c| (cosh(2x)) A cosh(−2A arctan e2x + B) shows that, starting at an arbitrary q−∞ , the function q approaches (uniformly) to the constant function q = q−∞ . (Branch II). The formula for q approximates to q ∼

±e−2W−∞ (2B − 2(W∞ − W−∞ )) . π cosh B cosh(2x)(e−2W−∞ (cosh B)−1 cosh(−2A arctan e2x + B))2

Rearranged it reads q ∼

±e2W−∞ (2B − 2(W∞ − W−∞ )) cosh B . π cosh(2x) cosh(−2A arctan e2x + B))2

(45)

The factor

A cosh(−2A arctan e2x + B) = cosh B −2 arctan e2x + 1 , B

in the denominator of Eq. (45), can be bounded above in the interval −1 ≤ x ≤ 1 by cosh 2Bx. (We note that integral

A −2 B

→

1 ± −1

−4 π ,

linearize arctan e2x (x ∼ 0) and get the bound). The

e−2W−∞ (2B − 2(W∞ − W−∞ )) cosh B dx, cosh 2x(cosh 2Bx)2

is equal, after the change of variables Bx = u, to B ± −B

e2W−∞ (2B − 2(W∞ − W−∞ )) cosh B du, 2 B cosh 2u B cosh 2u

that clearly diverges to ± infinity as B goes to infinity. Solving for a. To find out the expression for a we follow the same procedure as in the polarized case. We find a˙ and a from Eqs. (34) and (35) and then integrate in time (t) and space (x). As W and q are time independent we have 2 −W e = W 2 + 4W±2 + q±

q 2 −4W e . 4

Equation (41) gives W 2 +

q 2 −4W A2 e = . 4 cosh2 2x

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This formula makes Eqs. (34) and (35) to have the same form as Eq. (31) but with 2 W02 replaced by A2 . This gives the following expression for a 1 A2 1 3 A2 a(x, t) = a(0) − + ln cosh 2x + + t. 2 4 2 2 4 The analysis of the convergence–collapse picture for these non-polarized solutions follows exactly as in the polarized case. 2.3.5. The Gluing. CMC states in a torus neck. For simplicity we will work with the polarized solution in the torus neck we have found before the computations carry over to the non-polarized case as well. We will find a CMC slice, t = s(x), of the solution g = e2a (−dt2 + dx2 ) + Re2W dθ12 + Re−2W dθ22 , 3 W02 1 W02 1 + ln cosh 2x + + a(x, t) = a(0) − t, 2 2 2 2 2 R(x, t) = R0 (e2(t+x) + e2(t−x) ), W (x, t) = W1 + W0 arctan e2x , (1+W 2 )

with k = −3 and asymptotically of the form t = s(x) ∼ t0 ± (3+W02 ) x. With this 0 asymptotic we guarantee having (almost) flat cone initial states on the ends. The way to find such CMC slice is by finding appropriate barriers. To do that we first find a general expression for the mean curvature of a general section t = s(x). We keep the discussion brief. Given a slice t = s(x) introduce a coordinate system (¯ x, t¯, θ¯1 , θ¯2 ) defined as x)t¯, x=x ¯ + s (¯ t = s(¯ x) + t¯, θ1 = θ¯1 , θ2 = θ¯2 . In these coordinates the metric g is written ¯ t¯)(d¯ ¯ t¯) + Re2W dθ¯12 + Re−2W dθ¯22 , ¯ 2 dt¯2 + g¯(d¯ x + Xd x + Xd g = −N where g¯ = e2a ((1 + s t¯)2 − s2 ), ¯ 2 = e2a (1 − s2 ). N ¯ = 0 when t¯ = 0. From this k is calculated (at the slice t = s(x)) as and X 1 s ∂t¯R k=− √ + ∂t¯a + , 1 − s2 R ea 1 − s2

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where


1 W02 3 W02 + + s tanh 2x + , 2 2 2 2 ∂x Rs + ∂t R ∂t¯R = = 2s tanh 2x + 2, R R

∂t¯a = ∂t a + s ∂x a = −

which gives

1 W02 3 W02 s 1 k(x) = − √ + + ef − tanh 2x + s 1 − s2 2 2 2 2 1 − s2 + 2s tanh 2x + 2) ,

with

f = − a(0) −

1 W02 + 2 2

1 ln cosh 2x + 2

−

2 W0 3 2+ 2

3 W02 + 2 2

s .

τ

k(s(x)). This implies in particular Remark 1. Note that k(s(x) + τ ) = e that once we have obtained a CMC slice a CMC foliation is obtained by shifting it in the (t) time direction. 1+W 2 Now, to construct the barriers, note that for the section t = s(x) = t0 + 3+W02 x, 0 k is asymptotically (i.e. as x → +∞) constant. A direct calculation shows that for the pair of sections (on the right end) t = s± (x) = t0 +

1 + W02 1 x± , 3 + W02 x

(46)

the asymptotic (to leading terms) is ∓

−k ∼ −k0 e

2 W0 3 2+ 2

1 x

1 1+O . x

The last formula shows that −k(s+ ) < −k0 < −k(s− ) asymptotically. The extension of those sections to the center of the neck can be carried as follows. Take two sections symmetric with respect to the t-axis, that (say on the right) are (i) any 1+W 2 smooth section (s+ ) from 0 to 10 with s > 0 and s+ (10)+ 3+W02 (x−10)−ln(x−9) 0 thereafter (ii) any smooth section (s− ) from 0 to 10 with s > 0 and equal to 1+W 2 s− (10) + 3+W02 (x − 10) + ln(x − 9) thereafter. It is easy to see using the Remark 0 above that by shifting the section s− upwards, at some shift the sections have disjoint range of their mean curvatures (between the points of intersection) and that 1+W 2 at the point of intersection their tangents are 3+W02 up to ∼ 1/x. Due to that, it is 0 easy to continue these two sections as was said above (in Eq. 46), starting from an x slightly less than the x where they intersect, in such a way that they have disjoint 1+W 2 range of mean curvatures but asymptotically approaching to s(x) = t0 + 3+W02 . 0

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Note that given a CMC slice as was described above, the same slice is CMC with the same mean curvature if on the metric g we replace R0 by R0 e−2δ . Also note that on the (x , t ) coordinates, for large x the metric is written approximately 2 π 2 (t −x ) g = e2t (−dt2 + dx2 ) + R0 e2( 2 W0 +W1 ) e2(t +x ) + e 2+W0 dθ12 2 π 2 (t −x ) +R0 e−2( 2 W0 +W1 ) e2(t +x ) + e 2+W0 dθ22 . Thus, changing R0 by R0 e−2δ and changing the x coordinate by x = x − δ the metric approximates to any given desired order to the flat cone state,

π

g = e2t (−dt2 + dx2 ) + R0 e2( 2 W0 +W1 ) e2(t +x ) dθ12 π + R e−2( 2 W0 +W1 ) e2(t +x ) dθ2 . 0

2

Moreover, note that the distance between standard parts on the cusps get increased by ∼ 2δ. δ therefore parametrizes the family of CMC initial states displaying a convergence–collapse picture. A traceless transverse tensor. Having now a metric on the torus neck we glue it to the hyperbolic metrics dx2 + e2x gTi (on the right (say i = 2) and the left (say i = 1) of the neck) along intervals of left one around x = x0 and preserving the T 2 symmetry. There is some freedom of course in this process. We will use it in a moment. We will look for a T 2 -symmetric transverse traceless (2, 0)-tenˆ T T with respect to the metric that resulted from the gluing. Moreover, we sor K ˆ T T to be zero except for K ˆ T T,xx , K ˆ T T,θ θ and will demand the components of K 1 1 ˆ ˆ KT T,θ2 θ2 . Finally, we demand KT T to be unchanged on the region inside the neck ˆ T T to be unchanged which is not the gluing region and, similarly, we demand K inside the bulk of the hyperbolic manifolds H1 and H2 which is not the gluing ˆ T T to be zero on the hyperbolic sector and right after the region. Thus, we want K gluing. Observing that for any T 2 -symmetric metric the connection coefficients i Γθθki θj for i, j, k equal to 0 or 1 are zero and similarly for Γxxθi and Γθxx for i = 0, 1 we have ˆ Ti T,θ = 0, j = 0, 1. ∇i K j

î For ∇i K TT

x

we compute ˆ Ti T,x = ∂x K ˆ Tx T,x + (Γθ2 − Γθ1 )K ˆ Tx T,x ∇i K xθ2 xθ1

(47)

ˆx +K ˆ θ1 ˆ θ2 where we have implicitly used that K T T,x T T,θ1 + KT T,θ2 = 0. We need to find a solution of (47) being exactly zero after an interval of length one. To do that we choose the glued metric in such a way that Γθxθ1 1 = Γθxθ2 2 (with a small difference) ˆ θ1 θ1 on an interval of length one half inside the gluing interval. Then choose K such that the solution to (47) is exactly zero right after the gluing region. One can check that this can be done using the integral formula for the solution of a first order ODE.

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Estimates. Once having (g, K) with divK = 0 and trg K = k we invoke a theorem of Isenberg [12] guaranteeing that the Lichnerowicz equation is solvable ˆ = 0 and k = 0 as is our case. To estimate the solution to the as long as K Lichnerowicz equation 1 1 ˆ 2 −7 k 2 5 Rg φ − |K| + φ , gφ 8 8 12 we use the maximum principle and the standard local elliptic estimates. From the maximum principle we get Δφ =

ˆ 2 φ(xmax )−7 + Rg φ(xmax ) − |K|

k2 φ(xmax )5 ≤ 0, 12

ˆ 2 − 2 k 2 + (x) where (x) is nonzero only on the gluing Now note that Rg = |K| 3 region. Using this in the last equation gives ˆ 2 (φ(xmax )−φ−7 (xmax ))+ 2 k 2 (φ(xmax )5 −φ(xmax ))+(xmax )φ(xmax ) ≤ 0. (48) |K| 3 Observe that KL∞ is bounded with a bound independent of . We see from g Eq. (48) that when L∞ → 0 then φ − 1L∞ → 0. Standard elliptic estimates show that in fact φ − 1C 2,α → 0.

3. Long-Time Geometrization of the Einstein Flow 3.1. The Long-Time Geometrization of the Einstein Flow In this section we prove the following Theorem. ˜ Theorem 4. Let Σ be a compact three-manifold with Y (Σ) ≤ 0. Say (˜ g , K)(σ) is a cosmologically normalized flow with E˜1 (σ) ≤ Λ where Λ is a positive constant. ˜ Then, the cosmologically normalized flow (˜ g , K)(σ) persistently geometrizes the manifold Σ. Moreover the induced geometrization is the Thurston geometrization 3 iff V(σ) ↓ Vinf = − 16 Y (Σ) 2 . We need some preliminary propositions. Proposition 2. Let Σ be a compact three-manifold. Say g0 is a H 2 -Riemannianmetric on Σ. Say p ∈ Σ and 2R < r2 (p) where r2 (p) is the H 2 -harmonic radius of the metric g0 at the point p. According to the definition of H 2 -harmonic radius we consider a harmonic coordinate system {x} covering Bg0 (p, r2 (p)) and satisfying 3 4 δjk ≤ g0,jk ≤ δjk , 4 3 ⎞ ⎛ I 2 ⎜ ∂ gjk dvx ⎟ r2 (p) ⎝ ⎠ ≤ 1. ∂xI |I|=2,j,kB (p,r (p)) g0 2

(49)

(50)

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2 Then there is (R) such that if g − g0 H{x} ¯ ≤ (R) the inclusions (Bg0 (p,R)) ≤

id : Hgi (Bg0 (p, R)) → Hgi0 (Bg0 (p, R)) and id : Hgi0 (Bg0 (p, R)) → Hgi (Bg0 (p, R)) for i = 0, 1, 2 have norms controlled by ¯ and R. Proof. Note first the Sobolev embeddings12 1 H{x} (Bg0 (p, R)) → L4{x} (Bg0 (p, R)),

(51)

2 (Bg0 (p, R)) H{x}

(52)

→

0 C{x} (Bg0 (p, R)).

0 , R) with → 0 as ¯ → 0 (and From (51) we see that g − g0 C{x} (Bg0 (p,R)) ≤ (¯ R fixed). This in particular implies that

C1 g0,ij ≤ gij ≤ C2 g0,ij , where C1 and C2 depend on ¯ and R and tend to one as ¯ → 0 (keeping R fixed). This proves the inequality C1 U L2g

0

(Bg0 (p,R))

≤ U L2g (Bg0 (p,R)) ≤ C2 U L2g

0

(Bg0 (p,R) ,

for some C1 and C2 dependent on ¯ and R, which terminates the case i = 0. In the following we will use the notation C1 , C2 to denote generic quantities depending ¯ the covarion ¯ and R. Let us prove the case i = 1 now. Denote by ∇ and ∇ ¯ ant derivatives associated to g0 and g respectively. Write ∇ = ∇ + Γ. With this notation we have ¯ |2g = |∇U + Γ ∗ U |2g ≤ C2 (|∇U |2g + |Γ|2g |U |2g ). |∇U 0

0

0

Integrating we get ¯ |2 dvg |∇U g Bg0 (p,R)

⎛

⎜ ≤ C2 ⎜ ⎝

Bg0 (p,R)

⎛ ⎜ |∇U |2g0 dvg0 + ⎝

⎞ 12 ⎛ ⎟ ⎜ |Γ|4{x} dvx ⎠ ⎝

Bg0 (p,R)

⎞ 12 ⎞ ⎟ ⎟ |U |4g0 dvg0 ⎠ ⎟ ⎠.

Bg0 (p,R)

(53) It is direct to see from the formula 1 Γkij = (∇i (gjm − g0,jm ) + ∇j (gim − g0,im ) − ∇m (gij − g0,ij ))g km , 2 1 that ΓH{x} ¯ → 0. Sobolev embeddings applied to equation (53) (Bg0 (p,R)) → 0 as give ¯ 2 2 ≤ C2 U 2 1 , ∇U Lg (Bg0 (p,R))

12

Hg (Bg0 (p,R)) 0

(B (p, R)) and not from H It is crucial that the embeddings are from H{x} g0 0,{x} (Bg0 (p, R)). This is justified by the fact that, in the coordinate system {x} the set Bg0 (p, R) has the cone property at its boundary (see [9, p. 158]).

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M. Reiris


and thus U 2Hg1 (Bg

0

(p,R))

≤ C2 U 2Hg1

0

(Bg0 (p,R)) ,

as desired. Let us prove the other inequality. Write ¯ − Γ ∗ U |2g ≤ C1 (|∇U ¯ |2g + |Γ|2g |U |2g ). |∇U |2g0 = |∇U 0 0 0 Integrating we get |∇U |2g0 dvg0 Bg0 (p,R)

⎛

⎛

⎜ ≤ C1 ⎜ ⎝

¯ |2g dvg + ⎜ |∇U ⎝

Bg0 (p,R)

⎞ 12 ⎛

⎟ ⎜ |Γ|4g0 dvg0 ⎠ ⎝

Bg0 (p,R)

Again Sobolev embeddings give ¯ 2 2 ∇U 2 2 ≤ C1 (∇U Lg (Bg0 (p,R))

Lg (Bg0 (p,R))

0

⎞ 12 ⎞ ⎟ ⎟ |U |4g0 ⎠ ⎟ ⎠,

Bg0 (p,R)

+ Γ2H 1

{x} (Bg0 (p,R))

U 2Hg1

0

(Bg0 (p,R)) ).

Moving the second term on the right hand side to the left side and choosing ¯ sufficiently small13 we have U Hg1

0

(Bg0 (p,R))

≤ C1 U Hg1 (Bg0 (p,R)) ,

as desired. The case i = 2 follows easily from the case i = 1.

We consider now the Einstein flow with zero shift, i.e. we assume we have set X = 0. Proposition 3. (Continuity of the flow) Say Σ is a compact three-manifold with Y (Σ) ≤ 0. Say (g, K)(k) is a long-time Einstein flow with domain (at least) [−3, 0). Suppose that E1 (k) ≤ Λ where Λ is a positive constant. We use the notation (g0 , K0 ) = (g(−3), K(−3)), k0 = −3 and V(−3) = V0 . Say p ∈ Σ and r2,g0 (p) ≥ 2R. Then for any > 0 there is δk(Λ, V0 , R) > 0 such that sup

{(g, K)(k) − (g, K)(k0 )Hg2

k∈[k0 ,k0 +δk]

0

(Bg0 (p,R))×Hg1 (Bg0 (p,R)) } 0

≤ .

Remark 2. i. Proposition 3 would be self evident if we have a priori control on r2,g0 over the whole manifold Σ. It is not a priori clear how is that the regions where the harmonic radius (or volume radius) is small may affect the evolution of the regions where it is not, even in the short time. What Proposition 3 shows is that under an a priori bound in E1 this influence is not noticeable in a definite interval of time t = k (depending on E1 , ν0 and R). Note however that we do not make any claim about the continuity in Hg20 (Bg0 (p, R)) of the lapse N . As we will remark later the Hg˜20 (Bg0 (p, R)) norm of N is indeed controlled but we do not know whether N satisfies a continuity of the type claimed for g and K (in their respective spaces). In particular we do not have 13

Note that C1 does not blow up as ¯ → 0.

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ii.


1587

any estimation (in any norm) of the time derivative of N on Bg0 (p, R) even for short times. This issue will appear later in Proposition 4. The Proposition 3 is evidently true if we use the cosmologically normalized ˜ σ and E˜1 instead of the variables (g, K), k and E1 . variables (˜ g , K),

Proof. The crucial fact is to note that there are δ(R, RicL2g (Σ) ) and (R, RicL2g (Σ) ) such that if g(k) − g0 Hg2 (Bg0 (p,R)) ≤ then r2,gk (∂Bg0 (p, R)) ≥ δ. 0 This result can be easily proved by contradiction or simply invoking the discussion in [1, see p. 218, p. 227]. Recall that Ric2L2g (Σ) ≤ C(|k|V + Q0 ), where C is a numeric constant. As a result the H 2 -harmonic radius of the region Bg(k) (Bg0 (p, R), 23 δ) is controlled from below by Λ, V0 and R as long as g(k) − ˆ H 2 (B (p,R)) , g0 Hg2 (Bg0 (p,R)) ≤ . Elliptic regularity shows that the norms K g0 g(k) 0 3 1 1 N Hg(k) (Bg0 (p,R)) , E0 Hg(k) (Bg0 (p,R)) and B1 Hg(k) (Bg0 (p,R)) are controlled from above by Λ, V0 and R as long as g(k) − g0 Hg2 (Bg0 (p,R)) ≤ . Under zero shift, 0 the time derivatives of g and K are g˙ = −2N K, K˙ = −∇2 N + N (E − K ◦ K). 2 Thus g˙Hg(k) (Bg0 (p,R)) and K˙Hg10 (Bg0 (p,R)) are controlled above by (say) ˜ Λ(Λ, V0 , R) as long as g(k) − g0 H 2 (B (p,R)) ≤ . Write g0

g0

k g(k) − g0 Hg2

0

(Bg0 (p,R))

≤

g˙Hg2

0

(Bg0 (p,R)) dk,

k0

k K(k) − K0 Hg1

0

(Bg0 (p,R))

≤

K˙Hg1

0

(Bg0 (p,R)) dk.

k0

By Proposition 2 we can bound g˙Hg2 Hg10 -norm

0

(Bg0 (p,R))

2 by C1 g˙Hg(k) (Bg0 (p,R)) and simi-

larly for the of K˙. Thus the length δk of the maximal interval [k0 , k0 +δk] ˜ and similarly for the where g(k) − g0 Hg2 (Bg0 (p,R)) ≤ is greater than /(C1 Λ) 0 1 Hg0 -norm of K. ˜ Proposition 4. Let Σ be a compact three-manifold with Y (Σ) ≤ 0. Assume (˜ g , K) is a cosmologically normalized long-time flow. Assume too that E˜1 ≤ Λ with Λ a positive constant. Then, for every > 0 and R > 0 there exists σ0 such that for

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any σ ≥ σ0 and p ∈ Σ with r2,˜g(σ) ≥ 4R we have ˆ ˜ K(σ) Hg˜1(σ) (Bg˜(σ) (p,R)) ≤ ,

(54)

Ric(σ) L2g˜(σ) (Bg˜(σ) (p,R)) ≤ ,

(55)

E0 (σ)2L2 (Bg˜(σ) (p,R)) g ˜(σ)

+

B0 (σ)2L2 (Bg˜(σ) (p,R)) g ˜(σ)

≤ .

(56)

Proof. The way to prove Proposition 4 is to show that for any R > 0, any sequence of points {pi }, and any divergent sequences of logarithmic-times {σi } for which r2,˜g(σi ) ≥ 4R, the norms (54), (55) and (56) (with σi instead of σ and pi instead of p) tend to zero. We will use the terminology “Case 54” for the proof of this on ˆ ˜ and similarly for Ric (Case 55) and E0 , B0 (Case 56). K Let us start by making some elementary but important observations. Observation 1. From (the proof of ) Proposition 3 we know that there are {δσi } with |δσi | controlled from below by Λ, V and R (observe that because V is monotonic along the flow we can replace the dependence on V(σi ) for the dependence only on V0 = V(σ0 ) with σ0 some initial logarithmic time) and such that the norms ˆ ˜ H 2 (B K (p ,2R)) for σ ∈ [σi , σi + δσi ], are controlled from above by Λ, V0 g ˜(σ)

g ˜(σi )

i

and R. It follows from the maximum principle applied to the lapse equation 2 ˜ + |K(σ)| ˜ ˜ −Δg˜(σ) N g ˜(σ) N = 1,

˜ (p, σ) ≥ N ˜0 (Λ, V0 , R) > 0 for p in Bg˜(σ ) (pi , 7 R) and for σ in [σi , σi +δσi ].14 that N i 4 Observation 2. Recall that dV =3 dσ

˜ − 1dvg˜ = 3 3N

Σ

φdvg˜ , Σ

˜ −1 is the Newtonian potential where (as was introduced in the background) φ = 3N and satisfies −1 ≤ φ ≤ 0. If we integrate this equation between σi and σi + δσi (where δσi will be the one in Proposition 3) we get σi+δσi

V(σi ) − V(σi + δσi ) = −3

φ(σ)dvg˜(σ) dσ σi

Σ

σi+δσi

≥ −3

φ(σ)dvg˜(σ) dσ. σi

Bg˜(σi ) (pi ,2R)

14 The argument is by contradiction. Assume there exists a sequence of states violating the inequality an obtain a convergent sub-sequence which violated the maximum principle.

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As V(Σi ) − V(σi + δσi ) → 0 when σi → ∞ (because V is monotonic and greater than zero) it follows that ⎞ ⎛ ⎟ ⎜ φ2 (σ)dvg˜(σ) ⎠ > Γ} → 0, μ{σ ∈ [σi , σi + δσi ]/ ⎝ Bg˜(σi ) (pi ,2R)

as σi → ∞, and for any fixed Γ > 0. ˆ ˜ 2 Let us prove now that K L

(Bg˜(σi ) (pi , 74 R)) → 0 as σi → ∞. Recall that dV ˆ˜ 2 dv . ˜ |K| = −3 N g ˜ g ˜ dσ g ˜(σi )

Σ

Integrating in σ we have σi+δσi

V(σi ) − V(σi + δσi ) ≥ 3 σi

ˆ˜ 2 dv . ˜ |K| N g ˜ g ˜

Bg˜(σi ) (pi , 74 R)

It follows from Proposition 3 and Observation 1 that V(σ) can get below its ˆ˜ limit V∞ = limσ→∞ V(σ) unless limσ→∞ K L2g˜(σ ) (Bg˜(σi ) (pi , 74 R)) = 0. Now as i ˆ ˜ H 2 (B is controlled from above by Λ, V and R, it follows that if K g ˜(σi ) (pi ,2R)) g ˜(σi ) ˆ ˜ K Hg˜1(σ ) (Bg˜(σi ) (pi , 74 R)) ≥ M > 0 we can extract a sub-sequence of the pointed i spaces (Bg˜(σi ) (pi , 74 R), pi , g˜(σi )) converging to a limit space (strongly in H 2 ) ˆ ˜ is not converging to zero which is a contra(Bg˜∞ p∞ , 74 R , p∞ , g˜∞ ) where K diction. This finishes the case (54). We use now this result and Observation 1 to get an improved version of Observation 1. Observation 3. Local elliptic estimates applied to the lapse equation (in the φ-variable) ˆ˜ 2 , ˜ 2g˜ φ = |K| Δg˜ φ − |K| g ˜ give μ{σ ∈ [σi , σi + δσi ]/φHg˜2(σ) (Bg˜(σ

i)

(pi , 32 R)) (σ)

≥ Γ} → 0,

as σi → ∞ and for any fixed Γ > 0. An important consequence of this is that for any space-like tensors Uk , k = 1, 2, 3 such that Uk L2g˜(σ) (Bg˜(σ ) (pi , 32 R)) ≤ M for i some M > 0 and for any k = 1, 2, 3 we have σ +δσ i i 2 → 0, ∗ φ + U ∗ ∇φ + U ∗ ∇ φ dv dσ U 0 1 3 g ˜(σ) σi Bg˜(σi ) (pi , 32 R) as σi → ∞.

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Recalling that ˜ ˆ = −B0 , Curlg˜ K we conclude that B0 L2g˜(σ ) (Bg˜(σ ) (pi ,2R)) → 0 (which is “half” the case (56)). i i To prove the case (55) we note that it is enough from ˆ˜ 2 g˜, ˆ ˆ ˆ˜ − 1 |K| g˜ = E + K ˜ +K ˜ ◦K Ric 3 and case (54), to prove that EL2g˜(σ ) (Bg˜(σ ) (pi ,R)) tends to zero as σi → ∞. This i i is however more difficult than the cases before. We will study the quantity ˆ˜ dv , E, K g ˜ g Bg˜(σi ) (pi , 32 R)

and its time derivative with respect to logarithmic time. Differentiating with respect to σ we have ⎞ ⎛ ⎜ ˆ ˆ˜ ˆ˜ + E, K ˜ g˜ dvg˜ ⎟ E, K E; K ⎠˙= ⎝ g ˜ g ˜ Bg˜(σi ) (pi , 32 R)

Bg˜(σi ) (pi , 32 R)

ˆ˜ g˜˙ + 3E, K ˆ˜ φdv . −E ◦ K, g ˜ g ˜ g ˜

(57)

To get a more convenient expression of the right hand side of the previous equation we will use the following expressions for the time derivatives of the cosmologically ˆ ˜ normalized variables g˜, E and K ˜ˆ ˜ K, g˜˙ = 2φ˜ g − 6N ˜ ˜ Curlg˜ B − ∇N ∧g˜ B − 5 E ×g˜ K ˜ − 2 E, K ˜ g˜ g˜ − 3 E, E˙ = N ˜ 2 3 2 N ˙˜ ˜ ˜ˆ ˜ˆ ˜ˆ ˆ = −K ˆ ˜ (K K − φ˜ g − ∇2 φ + φE + E − N ◦ K − 2K).

(58) (59) (60)

We now integrate Eq. (57) in σ for σ in [σi , σi ]. After integration of the left hand side we have (naturally) the expression ⎞ ⎞ ⎛ ⎛ ⎟ ⎜ ⎜ ˆ ˆ˜ dv ⎟ ˜ g˜ dvg˜ ⎟ (σi + δσi ) − ⎜ ⎟ ⎜ (61) E, K E, K g ˜ g ˜ ⎠ (σi ). ⎠ ⎝ ⎝ Bg˜(σi ) (pi , 32 R) Bg˜(σi ) (pi , 32 R) From Case (54) and the bound ˆ˜ ˆ ˜ E, Kg˜ dvg˜ (σ) ≤ E0 L2 (Bg˜(σ ) (pi , 3 R)) (σ)K L2g˜ (Bg˜(σi ) (pi , 32 R)) (σ), g ˜ 2 i 3 Bg˜(σi ) (pi , 2 R)

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we get that for any σ (in particular for σ = σi and σ = σi + δσi ) we have that (61) tends to zero as i → ∞. Similarly, using either Observation 3, Case (54) or the B0 -part of Case (56) we have that all the terms in the right hand side of the integral in σ of Eq. (57), except perhaps the term σi+δσi

σi

|E0 |2 dvg˜ dσ,

Bg˜(σi ) (pi , 32 R)

tend to zero. Thus we are lead to conclude that this term also tends to zero when i → ∞. We will see now using the Gauss equation that ( Bg˜(σ ) (pi ,R) |E0 |2 dvg˜ )(σi ) i

tends to zero as i → ∞. That would finish Case (56) and Case (55). The argument is as follows. Consider a fixed, even and positive function f of one variable x, equal to zero for |x| ≥ 32 and equal to one for |x| ≤ 1. Consider the function f (r) where r is the geodesic radius from pi and corresponding to the metric g˜(σi ) inside Bg˜(σi ) (pi , 2R). Extend f (r) to the space–time in such a way that it is time independent. Consider finally the Weyl field W = f Rm. We have EW = f E0 , BW = f B0 , and JW,bcd = (∇a f )Rmabcd . Thus, the L2g˜ (Bg˜(σi ) (pi , 32 R)) norm of EW , BW and JW are controlled by Λ, V0 and R. It follows from integrating the Gauss equation ˜ αβ dvg˜ . ˜ ˜ ˜ Q(W) ˜ Q(W)˙ = Q(W) −9 N ˜ ˜Π αβ T T

Σ

in σ and from σi to σi + δσ that Thus if ( Bg˜(σ

˜ ˜ ˜ |Q(W)(σ i + δσ) − Q(W)(σi )| ≤ Λ(Λ, V0 , R)δσ.

|E0 |2g˜ dvg˜ )(σi ) ≥ M > 0 we can choose δσ such that for all σ ˜ in [σi , σi + δσ] it is Q(W)(σ) ≥M 2 > 0. But we have ˜ Q(W) = f 2 (|E0 |2 + |B0 |2 )dvg˜ , i)

(pi ,R)

Bg˜(σi ) (pi , 32 R)

and we know from the B0 -part of Case (56) that ⎫ ⎧⎛ ⎞ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨⎜ ⎟ 2 2 ⎜ ⎟ → 0, (σ) sup |B | dv lim f 0 g ˜ ⎝ ⎠ σi →∞ σ∈[σ ,σ +δσ] ⎪ ⎪ ⎪ ⎪ i i ⎭ ⎩ B 3 g ˜(σi ) (pi , 2 R)

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when i → ∞. Therefore, if σi is big enough ⎞ ⎛ ⎟ ⎜ M ⎜ |E0 |2 dvg˜ ⎟ ⎠ (σ) ≥ 3 ⎝ Bg˜(σi ) (pi , 32 R) for any σ in [σi , σi + δσ] which would contradict that σi+δσi

σi

|E0 |2 dvg˜ dσ,

Bg˜(σi ) (pi , 32 R)

tends to zero as σi tends to infinity.

We are ready to prove Theorem 4. The proof goes essentially along the same lines as the proof of the geometrization of the flow given in [15] for long-time flows under Cg˜α curvature bounds. We repeat it here for the sake of clarity. Proof (of Theorem We provefirst there is a divergence sequence of logarithmic 4). 1 i=n ˜ g , K)(σ gH,i , −˜ gH,i )) (weakly in times {σi } with Σ i , (˜ i ) converging to ∪i=1 (Hi , (˜ H 2 ). Introduce a new variable j = 1, 2, 3, . . .. For j = 1 find a sequence {σ1,i } with (Σ1 , g˜(σ1,i )) convergent weakly in H 2 . For j = 2 find a sub-sequence {σ2,i } of {σ1,i } with (Σ1/2 , g˜(σ2,i )) convergent in the weak H 2 topology. Proceed similarly for all j to have a double sequence {σj,i }. Now, for the diagonal sequence {σi,i }, (Σ1/i , g˜(σi,i )) converges into a union of Riemannian manifolds of finite volume, ˜ i,i ) converges strongly to −˜ g∞,ν denoted as ∪ν (Mν , g˜∞,ν ). By Proposition 4, K(σ in H 1 . Also by Proposition 4 we get that each metric g˜∞,ν is hyperbolic and the convergence is in the strong H 2 -topology. Therefore, as there is a lower bound for the volume of complete hyperbolic manifolds of finite volume and the total volume of the limit space is bounded above, there must be a finite number of components, ˜H,i ). and we can write ∪ν (Mν , g˜∞,ν ) = ∪i=n i=1 (Hi , g We prove next that each component (Hj , g˜H,j ) is persistent. For simplicity assume there is only one component and therefore (Σ1/i , g˜(σi,i )) converges in the strong H 2 -topology to (H, g˜H ). There are two possibilities according to whether the component is compact or not, we discuss them separately. 1.(The compact case) Assume (H, g˜H ) is compact. Consider the space of metrics MH in H. For every metric g consider the orbit of g under the diffeomorphism group (of H 3 -diffeomorphisms). Denote such orbit by o(g). Around g˜H consider a small (smooth) section S of MH (made of Hg˜2H metrics) and transversal to the orbits generated by the action on MH of the diffeomorphism group.15 If 0 is sufficiently small every metric g in MH with g − g˜H Hg˜2 ≤ 0 can be uniquely proH jected into S by a diffeomorphism, or in other words we can consider the projection 15 Which particular section is taken is unimportant. One can use for instance S = {g/id : (H, g) → (H, g˜H )} is harmonic (see [5, 11]).

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P (g) = o(g) ∩ S. Note that one can project every flow of metrics g˜(t) starting close g (t)), until at least the first time when P (˜ g (t)) − g˜H Hg˜2 = 0 to g˜H , to a path P (˜ H or in other words until at least when the projection touches the boundary of the gH , 0 )). ball of center g˜H and radius 0 in Hg˜2H (denote such ball as B(˜ Recall Mostow rigidity.16 Mostow rigidity (the compact case). There is 1 such that if P (gH ) ∈ B(˜ gH , 1 ), where gH is a hyperbolic metric in H then P (gH ) = g˜H . Fix 2 = min{0 , 1 }. Observe that as g˜σi,i → g˜H in H 2 there is a sequence of diffeomorphisms φi such that φ∗i (g(σi,i )) converges to g˜H in Hg˜2H . Now, if the geometrization is not persistent there is ≤ 2 and i2 such that if i ≥ i2 then P (φ∗i (˜ g (σ))) is well defined for σ ≥ σi,i until a first time σi,i + Ti when g (σi,i + Ti ))) is in ∂B(˜ gH , 2 ). But we know the sequence of Riemannian P (φ∗i (˜ g (σi,i + Ti )))) converge in H 2 to g˜H , and that means by the manifolds (H, P (φ∗i (˜ definition of H 2 convergence and Mostow rigidity that there is a sequence of diffeg (σi,i + Ti )))) converges to g˜H in Hg˜2H . This omorphisms ϕi such that P (ϕ∗i (P (φ∗i (˜ g (σi,i + Ti ))) is in ∂B(˜ gH , 2 ). contradict the fact that P (φ∗i (˜ 2. (The non-compact case). The proof of this case proceeds along the same lines as in the compact case but special care must be taken at the cusps.17 Let us assume for simplicity that there is only one cusp in the piece (H, g˜H ). Given A sufficiently small there is a unique torus transversal to the cusp, to be denoted by TA2 , of constant mean curvature and area A. Denote by HA the “bulk” side of the torus TA2 in H. Consider the set MHA of metrics g˜ on HA such that g˜ = g˜H on Bg˜H (TA2 , 1). Consider the action of the diffeomorphism group (of H 3 -diffeomorphisms) on MHA and leaving Bg˜H (TA2 , 1) invariant. Again the orbit of a metric g˜ will be denoted by o(˜ g ). Consider a small (smooth) section S of MHA transversal to the orbits of the action by the diffeomorphism group mentioned above. Finally consider the projection P (˜ g ) = o(˜ g )∩S which is well defined on a ball B(˜ gH , 0 ) for 0 small enough. Observe again that a flow of metrics g˜(t) in MHA can be projected gH , 0 ). Slightly abusinto S until at least the first time when P (˜ g (t)) is in ∂Bg˜H (˜ ing the notation (as we would require a pointed sequence) consider the sequence (Σ, g˜(σi,i )) converging in H 2 to g˜H . There is a sequence of diffeomorphisms (onto g (σi,i )) − g˜H Hg˜2 converges to zero. the image) φσi,i : HA → Σ such that φ∗σi,i (˜ H Note that if we have a map φσ : HA → Σ such that φ∗σ g˜(σ) − g˜H Hg˜2 ≤ 2 for H sufficiently small then we can deform φ∗σ g˜(σ) to a metric S(φ∗σ g˜(σ)) in MHA in 16

Mostow rigidity states that any two hyperbolic metrics on a compact manifold are necessarily isometric. What we state as Mostow rigidity here is an obvious consequence of this fact. For the notions on hyperbolic three-geometry that we will need we refer the reader to the beautiful survey by Gromov [10]. Most of the treatment of hyperbolic three-geometry we will perform here goes in parallel to a similar analysis in the Ricci flow in [11]. 17 In [15] we have used CMC tori (transversal to the cusp) of a given area to compare (in a unique way) the Riemannian spaces (H, g˜H ) and (Σ, g˜(σ)) (see [15] for details). If g˜(σ) is close to g˜H only in Hg˜2H the CMC tori of a given area and transversal to the tori may be difficult to guarantee. It is for this reason that (see later in the text) we smooth out the metrics g˜(σ) near the regions where “the CMC tori of a given area would be”.

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such a way that (a) S(φ∗σ g˜(σ)) = φ∗σ g˜(σ) on He4 A , (b) inside (He2 A − He4 A ) the metric S(φ∗σ g˜(σ)) is chosen to minimize the L2S(φ∗ g˜(σ)) -norm of the traceless part σ of its Ricci tensor. Note the following elementary fact: if is chosen small enough and we have a diffeomorphism φ : (HA , g˜H ) → (Σ, g˜) with φ∗σ g˜ − g˜H Hg˜2 ≤ 2 H and φ∗ g˜ is isometric to g˜H then the new metric S(φ∗ g˜) is the deformation of g˜H by a diffeomorphism on HA (we will recall this note later as note N). We note now the following crucial facts (justified below). i. For all A > 0 but sufficiently small there exists σ0 and 0 such that for all ≤ 0 and σ1 ≥ σ0 if there exists φσ1 : He4 A → Σ with φ∗σ1 g˜(σ1 ) − g˜H Hg˜2 ≤ H then there exists φ¯σ : HA → Σ with φ¯∗ g˜(σ1 ) − g˜H H 2 ≤ 2. Note that in this 1

σ1

g ˜H

case S(φ¯∗σ g˜(σ)) is well defined. ii. From i. we conclude that if we have φσ1 : HA → Σ with satisfying: (a) φ∗σ1 g˜(σ1 ) − g˜H Hg˜2 ≤ 2, (b) the restriction of φσ1 to He4 A with φ∗σ1 g˜(σ1 ) − H g˜H Hg˜2 (He4 A ) ≤ , (c) P (S(φ∗σ1 g˜(σ1 ))) − g˜H Hg˜2 ≤ then φσ : HA → Σ with the H properties (a) and (b) exist for σ ≥ σ1 (and varying continuously) until at least the first time σ2 for which P (S(φ∗σ2 g˜(σ2 ))) − gH Hg˜2 = . H Let us justify now claim i. Recall Mostow rigidity. Mostow rigidity (the non-compact case). There is A0 such that for any A ≤ A0 there is 0 such that if (Σ , gH ) is a complete hyperbolic manifold of finite vol )− ume and φ : HA → Σ is a diffeomorphism onto the image satisfying φ∗ (gH 18 g˜H Hg˜2 ≤ 0 then (Σ , gH ) is isometric to (H, g˜H ). H The justification of i. follows straight from Mostow rigidity. Indeed pick any A such that e2 A ≤ A0 and ≤ 0 as in the Mostow rigidity statement. Suppose there exists a divergent sequence {σi } and a sequence of diffeomorphisms onto the image φσi : He4 A → Σ such that φ∗σi g˜(σi ) − g˜H Hg˜2 ≤ but such that it H cannot be extended to a diffeomorphism φ¯σ : HA → Σ with φ¯∗ − g˜H H 2 ≤ 2. σi

i

g ˜H

We can extract a (pointed) sub-sequence of {(Σ, g˜(σi )} converging to a complete hyperbolic metric of finite volume, which, by Mostow rigidity and the choice of A and must be isometric to g˜H . Therefore for σi sufficiently big the diffeomorphism φ¯σi can be defined which is a contradiction. Now from facts i. and ii. we get that, if the geometrization (H, g˜H ) is not g (σ))) is well persistent there is ≤ 0 and σ0 such that if σi,i ≥ σ0 then P (Sφ∗σ (˜ defined for σ ≥ σi,i until a first time σi,i + Ti when P (S(φ∗i g˜(σi,i + Ti ))) is in g (σi,i + Ti )) has a sub-sequence converging in H 2 ∂B(˜ gH , ). Now the sequence φ∗i (˜ to a complete hyperbolic metric of finite volume. Again as in the compact case, 18

The justification of this claim is as follows. According to the Mostow-Prasad rigidity g and g˜H will be isometric if we can prove that Σ is diffeomorphic to H. If is chosen small enough this is equivalent to show that the number of cusps of Σ and Σ are the same. This follows from the Margulis lemma [10] and the fact that if φ∗ g − g˜H H 2 ≤ then φ∗ g − g˜H 1 ≤ C g ˜H

where C is a numeric constant.

2 Cg ˜

H

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by Mostow rigidity it must be converging in H 2 to g˜H . Therefore (recall note N) P (S(φ∗σ g˜(σ))) must be converging to a metric on HA which is a diffeomorphism g (σi,i + Ti )))) is in ∂Bg˜H (˜ gH , 2 ). of g˜H contradicting the fact that P (S(φ∗i (˜ To finish the proof of the persistence of the geometrization one still needs to show that the compliment of the persistent pieces (Hi , g˜H,i ) is the G sector or in other words that for any > 0, (Σ (σ), g˜(σ)) converges to the -thick part of the persistent pieces (Hi , g˜H,i ). The proof of this fact follows by contradiction. If this is not the case one can extract a divergent sequence of logarithmic times containing an H-piece different from the pieces (Hi , g˜H,i ). One can prove again that this new piece is persistent leading into a contradiction for if persistent, the piece must be one of the pieces (Hi , g˜H,i ) by the way these pieces are defined. 3.2. Stability of the flat cone (Case Y (Σ) < 0 (I)-ground state) In this section we will prove the stability of the Case Y (Σ) < 0 (I)-ground state. ˜ over a hyperNamely, we will show that a cosmologically normalized flow (˜ g , K) ˜ bolic three-manifold Σ, with initial data (˜ g , K)(σ0 ) close (in H 3 × H 2 ) to the 3 ground state (gH , −gH ), converges (in H × H 2 ) to the ground state (gH , −gH ) when σ → ∞. This stability has been proved by Andersson and Moncrief in [5] (for rigid hyperbolic manifolds Σ).19 Theorem 5. (Stability of the flat cone). Let Σ be a compact hyperbolic threemanifold. Then, there is an > 0 such that the cosmologically normalized CMC ˜ flow (˜ g , K)(σ) of a cosmologically normalized (H 3 × H 2 ) initial state (g0 , K0 ) = ˜ (˜ g (σ0 ), K(σ0 )) with E˜1 (σ0 ) + (V − Vinf ) ≤ , converges in Hg3H × Hg2H (and for a suitable choice of the shift vector X) to (gH , −gH ) (the standard Case Y (Σ) < 0 (I)-ground state). Remark 3. As it is stated Theorem 5 gives few information about the shift vector X. This inconvenient can be remedied if, as in [5], X is chosen in such a way that for every σ the identity id : (Σ, g˜(σ)) → (Σ, gH ) is a harmonic map (the spatially harmonic gauge [5]). Full control of the evolution of the shift vector X can be obtained in this case. We begin with a preliminary Proposition. Proposition 5. Say Σ is a compact hyperbolic three-manifold. Fix ν 0 > 0 and V0 > Vinf . Then, for every > 0 there is δ(, ν 0 , V0 ) > 0 such that for every ˆ L2 + Q ˜0 ≤ δ ˜ with ν ≥ ν , V ≤ V0 and K cosmologically normalized state (˜ g , K) 0 g ˜ we have V − Vinf ≤ . ˜ (we will forget about puttProof. It is enough to prove that any sequence (˜ g , K) ˆ ˜ L2 + Q ˜ 0 → 0 has a sub-sequence converging in H 2 to gH . ing sub-index) with K g ˜

19

The rigidity condition is a somehow mild restriction. We remove it with an appropriate use of the reduced volume. The core of the proof is essentially the same as in [5].

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From Proposition 8 in the Appendix we have (for arbitrary states (g, K)) ⎞ 12 ⎛ 1 ˆ 2 + |K| ˆ 4 dvg ⎠ ≤ C |k|K ˆ L2 + Q 2 . ⎝ 2|∇K| 0 g

M

ˆ ˆ ˜ L4 → 0 as K ˜ L2 + Q ˜ 0 → 0. From this and Thus K g ˜ g ˜ ˆ˜ 2 g˜, ˆ ˆ ˆ˜ − 1 |K| g˜ = E + K ˜ +K ˜ ◦K Ric 3 g˜ L2 → 0. Moreover from the energy constraint we get we get that Ric g ˜ Rg˜ + 6L2g˜ → 0. As V is bounded above and ν bounded below, there is a sub˜ converging (in H 2 ) to gH . Thus V → Vinf . sequence of (˜ g , K) Proof (of theorem 5). Recall from Theorem 3 (and elliptic estimates) that for any ˜ it is > 0 there is δ > 0 such that if for a cosmologically normalized state (˜ g , K) ∗ ∗ ˜ ˜ E1 + (V − Vinf ) ≤ δ then there is a diffeomorphism φ such that (φ (˜ g ), φ (K)) is -close to (gH , −gH ) in Hg3H × Hg2H . One can also find δ > 0 such that in addition the L∞ g ˜ -norm of the deformation tensor Πab = ∇a Tb (with respect to the CMC foliation) is less than . It is direct to see [5] that this implies the following inequality for the evolution of E˜1 1 ∂σ E˜1 ≤ − 2 − C E˜12 E˜1 . (62) Thus, this inequality and the monotonicity of the reduced volume show that, as long as the flow is defined it will remain close in H 3 × H 2 to the ground state20 (and thus the volume radius is controlled). As there is a lower bound for the time interval for which flows are defined if the initial data is close in H 3 × H 2 to the ground state21 we conclude that the flow is a long-time flow. Note that the argument is independent of the shift X. One may well take the zero shift X = 0. Now, it is clear from Eq. (62), that E˜1 → 0 as the logarithmic ˜ converges (in time diverges. To show that (up to diffeomorphism) the flow (˜ g , K) 3 2 HgH × HgH ) to (gH , −gH ) it remains to prove that V − Vinf → 0. By Proposition 5 if V(σ) − Vinf ≥ Γ > 0 for all σ (observe that V is monotonically decreasing) it ˜ ˆ L2 (σ) ≥ M > 0 (for some M > 0) for all σ ≥ σ1 . If is chosen must be K g ˜

˜ (σ) − 1 L∞ < 1 for all σ ≥ σ0 . The equation for the small enough it must be N 3 6 evolution of the reduced volume dV ˜ˆ 2 ˜ |K| = −3 N dvg˜ , (63) dσ Σ

20

Note again, as was explained at the end of the Introduction, that here “close in H 3 × H 2 ” means close up to diffeomorphism. 21 See Theorem 1 in [16] for more details and also [5] for a continuity criteria in the harmonic gauge.

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˜ ˆ L2 (σ) ≥ M > 0 for σ ≥ σ1 then V − Vinf must go below zero shows that if K g ˜ after some time which is a contradiction.

4. Hyperbolic Rigidity, Ground States and Gravitational Waves There are several theoretical reasons that the reduced volume V should 1 to believe 32 decrease to its infimum Vinf = − 6 Y (Σ) at least for solutions in the family of long time solutions having a uniform bound on E˜1 . It may be possible (see [15]) to prove this claim for long time solutions having uniform bounds on the Cg˜α -norm of (the electric and magnetic parts of the) space–time Riemann tensor. Proving the claim for solutions in the family of long-time solutions having a uniform bound in E˜1 could be a task of much greater difficulty. In this section we present various facts and arguments pointing to the validity of this claim. According to Margulis, hyperbolic cusps are rigid in the following sense: if a complete hyperbolic metric g˜H on a manifold R × T 2 is close enough to a hyperbolic cusp metric g˜C = dx2 + e2x gT 2 over a domain Ω = [−a, ∞) × T 2 with a positive and big enough, then g˜H is isometric to g˜C . Consider the following spaces g ∼ g˜L when x → −∞}, DC = {˜ g on R×T 2 /Rg˜ ≥ −6, g˜ ∼ g˜R when x → ∞ and˜ SC = {˜ g ∈ on D × S 1 /Rg˜ ≥ −6, and g ∼ g˜S,R when x → ∞}, where DC accounts for “double cusps” and SC for “single cusp”. g˜R and g˜L are two arbitrary but fixed hyperbolic cusp metrics on the (right and left) ends of R × T 2 and g˜S,R is an arbitrary but fixed metric on the (right) end of D × S 1 (D is the unit two-dimensional disc). Consider now two cosmologically normalized flow ˜ DC ) and (˜ ˜ SC ) over R × T 2 and D × S 1 respectively and with g˜DC (˜ gDC , K gSC , K in DC and g˜SC in SC (see Fig. 4). As the states evolve one may argue that they lose “energy” (actually they lose reduced volume) by the emission of cylindrical gravitational waves22 at the ends of the cusps. According to Margulis the states ˜ DC )) or the infinite would settle into the infinite double cusp (for the flow (˜ gDC , K ˜ single cusp (for the flow (˜ gSC , KSC )) if it were the case that these configurations are V-rigid. This is indeed true for the double cusp (a ground state) but false for the single cusp (a non-ground state) in the following sense. Proposition 6. Consider the set of metrics g˜ in DC with g˜ ∼ g˜R 23 for x ∈ [aR , ∞) and g˜ ∼ g˜L for x ∈ (−∞, aL ]. Call VR the volume of g˜R on the region (−∞, aR ]×T 2 and similarly for the left cusp (VL ). Then the volume of g˜ on the region [aL , aR ]×T 2 is strictly greater than VL + VR . 22

In the definition of the sets DC and SC we can assume the metrics g˜ are T 2 -symmetric. That would justify the statement that the system emits cylindrical gravitational waves. 23 A precise meaning for ∼ can be given.

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Figure 4. The (conjectural) evolution of the Double Cusp and Single Cusp. Proposition 7. Consider the set of metrics g˜ in SC with g˜ = g˜S,R for x ∈ [aR , ∞). Call VR the volume of g˜S,R on the region (−∞, aR ] × T 2 . Then there exist metrics g˜ as described above and having volume inside the region (−∞, aR ] × T 2 less than VR . A proof of Proposition 7 and an explicit construction of such metrics is given in [13] (the metrics are indeed T 2 -symmetric). It can be seen analytically (and numerig0 ) cally) that as time evolves the evolution of the (Yamabe) initial states (˜ g0 , −˜ described in [13] actually separates from the single infinite hyperbolic cusp (as it should be).

Acknowledgments I would like to thank Michael Anderson for his constant enthusiasm and support in the present work and also for suggesting the problem of the Double-Cusp ground state and its evolution.

Appendix We begin by recalling a useful formula from [7]. Let V a symmetric traceless (2, 0) tensor with

(curl V )ab then

Σ

(div V )a = ∇b V ba = ρ, 1 = (alm ∇l Vmb + blm ∇l Vma ) = σ, 2

1 |∇V |2 + 3Ric, V ◦ V − R|V |2 = 2

Σ

1 |σ|2 + |ρ|2 . 2

(64)

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ˆ 2 2 control Proposition 8. Say Σ is a compact three-manifold. Then Q0 and |k|2 K Lg 2 4 2 ˆ ˆ ∇K 2 , K 4 and Ric 2 . More in particular we have Lg

Lg

L

⎛ ⎝

⎞ 12

1 ˆ 2 + |K| ˆ 4 dvg ⎠ ≤ C |k|K ˆ L2 + Q 2 , 2|∇K| 0 g

(65)

M

where C is a numeric constant. Observe the absence of the volume in Eq. (65) and that all norms involved are intrinsic. ˆ + k g and V = K ˆ in Eq. (64) Proof. Substituting Ric = E − kK + K ◦ K, K = K 3 we get 2 ˆ 4 −kK, ˆ 2 +3E, K ˆ 2 + 5 |K| ˆ K ˆ ◦ K ˆ − k |K| ˆ ◦ Kdv ˆ |∇K| = |B|2 dvg . g 2 3 Σ

Σ

This equation gives the bound ˆ 2 + |K| ˆ 4 dvg ≤ C (|k|2 |K| ˆ 2 + |k||K| ˆ 3 + |K| ˆ 2 |E| + |B|2 )dvg , |∇K| Σ

(66)

Σ

Observe now that the inequalities ⎛

ˆ 2 (|E|2 + |B|2 ) 2 dvg ≤ ⎝ |K| 1

Σ

⎞ 12 1

ˆ 4 dvg ⎠ Q 2 , |K| 0

Σ

⎞ 12 ⎛ ⎞ 12 ˆ 3 dvg ≤ ⎝ |K| ˆ 2 dvg ⎠ ⎝ |K| ˆ 4 dvg ⎠ , |K| ⎛

Σ

Σ

Σ

transform Eq. (66) into 1 ˆ 2 4 − C(|k|2 K ˆ 4 4 − C |k|K ˆ L2 + Q 2 K ˆ 2 2 + Q0 ) ≤ 0. ˆ 2 2 + K 2∇K 0 Lg Lg Lg Lg g Now make x2 = 2 We get

1 ˆ 2 + |K| ˆ 4 dvg , a = |k|K ˆ L2 + Q 2 in the last equation. |∇ K| 0 Σ g x2 − Cax − Ca2 ≤ 0.

Solving for x in the inequality above we get Eq. (65) which finishes the proof. ˆ L2 with V − Vinf or V depending on the The next proposition relates K g signature of the Yamabe invariant Y (Σ).

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Proposition 9. Say Σ is a compact three-manifold. Then ˆ 2 dvg ≤ C|k| 12 V 12 ( |K| ˆ 4 dvg ) 12 , i) if Y (Σ) > 0, |k|2 Σ |K| Σ 1 ˆ 2 dvg ≤ C|k| 12 (V − Vinf ) 12 ˆ 4 dvg 2 , ii) if Y(Σ) = 0, |k|2 Σ |K| | K| Σ 1 1 2 2 ˆ dvg ≤ C |k|(V −Vinf )+|k| 2 (V −Vinf ) 12 ˆ 4 dvg 2 iii) if Y(Σ) < 0, |k| Σ |K| | K| Σ where C is a numeric constant. Proof. i) and ii) (Y (Σ) > 0 or Y (Σ) = 0). This is immediate from the formula ⎛ ⎞ 12 ˆ 2 dvg ≤ |k| 12 (|k|3 V ol(Σ)) 12 ⎝ |K| ˆ 4 dvg ⎠ . |k|2 |K| Σ

Σ

iii) Y (Σ) < 0. Assume k = −3 and let gY be the unique Yamabe metric of constant scalar curvature RY = −6 in the conformal class of g. If g = φ4 gY then φ is determined by − ΔgY φ +

1 RY ˆ 2 + 1 k 2 φ5 = 0, φ − φ−3 |K| Y 8 8 12

(67)

where Δ = ∇2 . The maximum principle implies (putting the values RY = −6 and k = −3) that ˆ 2 6(φ5min − φmin ) ≥ φ−3 min |K|Y ≥ 0, which makes φ ≥ 1. Then observe that

2 −Y (Σ) ≤ −RY ( 1 dvY ) 3 , Σ

where dvY = dvgY . This gives ⎛ ⎞ 3 3 3 0 ≤ 6 2 ⎝ φ6 − 1 dvY ⎠ ≤ 6 2 φ6 dvY − (−Y (Σ)) 2 =

Σ 2 2 2 k V ol(Σ) 3 3

Therefore

32

Σ 3

− (−Y (Σ)) 2 .

(φ − 1)k dvY ≤ C(V − Vinf ), Σ

for k = 1, . . . , 6. Integrating Eq. (67), we get ˆ 2Y dvY . 6 (φ5 − φ)dvY = φ−3 |K| Σ

Σ

(68)

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Observe that ˆ 2 dvY = φφ−3 |K| ˆ 2 dvY φ−2 |K| Y Y Σ

Σ

=

ˆ 2Y dvY + φ−3 |K|

Σ

and

ˆ 2Y dvY = (φ − 1)φ−3 |K|

Σ

ˆ 2Y dvY (φ − 1)φ−3 |K|

Σ

ˆ 2Y dvY (φ − 1)φ2 φ−5 |K|

Σ

⎛

≤⎝

⎞ 12 ⎛ (φ − 1)2 φ4 dvY ⎠ ⎝

Σ

⎞ 12 ˆ 4 dvY ⎠ . (69) φ−10 |K| Y

Σ

On the other hand note that (φ − 1)2 φ4 dvY ≤ |φ6 − 1| + 2|φ5 − 1| + |φ4 − 1|dvY ≤ C(V − Vinf ). Σ

(70)

Σ

Putting together Eqs. (68),(69) and (70) we get ˆ 24 , ˆ 2 2 ≤ C (V − Vinf ) + (V − Vinf ) 12 K K Lg Lg

(71)

which after scaling finishes the proof. Combining Propositions 8 and 9 we get Proposition 10. Say Σ is a compact three-manifold. Then if Y (Σ) > 0 we have ˆ 2 + |K| ˆ 4 dvg ≤ C(|k|V + Q0 ), 2|∇K| Σ

while if Y (Σ) ≤ 0 we have ˆ 2 + |K| ˆ 4 dvg ≤ C(|k|(V − Vinf ) + Q0 ), 2|∇K| Σ

where C is a numeric constant. We also get Proposition 11. Say Σ is a compact three-manifold. If Y (Σ) > 0 we have ˆ 2 dvg ≤ C |k|V + (|k|VQ0 ) 12 , |k|2 |K| Σ

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while if Y (Σ) ≤ 0 (same for Y (Σ) = 0 than for Y (Σ) < 0) ˆ 2 dvg ≤ C |k|(V − Vinf ) + (|k|(V − Vinf )Q0 ) 12 . |k|2 |K| Σ

where C is a numeric constant. Proof. Combine equations in Proposition 9 and Eq. (65). Making x = ˆ L2 and a = |k| 12 (V − Vinf ) 12 if Y (Σ) ≤ 0 or a = |k| 12 V if Y (Σ) > 0 we |k|K 1

arrive at the inequality x2 − Cax − Ca2 − CaQ02 ≤ 0. From it we get x2 ≤ 1

C(a2 + aQ02 ).

A direct consequence of the propositions above is Proposition 12. Say Σ is a compact three-manifold, then V, |k| and Q0 control L2 . In particular we have Ric g 2 2 ≤ C(|k|V + Q0 ), Ric Lg where C is a numeric constant. ˆ +K ˆ ◦K ˆ − 1 |K| = E − kK ˆ 2 g together with the Propositions 8 Proof. Use Ric 3 3 and 9. ˆ 2 − 2 k 2 and Proposition 10 we get Using the energy constraint R = |K| 3 Proposition 13. Let Σ be a compact three-manifold. Then, V, |k| and Q0 control the scalar curvature in the following way 4 |∇R| 3 + R2 dvg ≤ C(|k|V + Q0 ), Σ

where C is a numeric constant. 4

Note that |∇R| 3 and R2 scale as a distance−4 . Proof. Squaring the energy constraint and integrating we obtain ˆ 4 + 4 k 4 dvg ≤ C(|k|V + Q0 ) R2 dvg ≤ |K| 9 Σ

Σ

where in the last inequality we have used Proposition 10. On the other hand, dif4 ˆ 43 |K| ˆ 43 . Integrating and ferentiating the energy constraint we have |∇R| 3 ≤ C|∇K| applying the H¨ older inequality we obtain ⎞ 23 ⎛ ⎞ 13 ⎛ 4 ˆ 2 dvg ⎠ ⎝ |K| ˆ 4 dvg ⎠ , |∇R| 3 dvg ≤ C ⎝ |∇K| Σ

Σ

Σ

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and if we apply Proposition 10 over each one of the factors on the RHS of the last equation we obtain 4 |∇R| 3 dvg ≤ C(|k|V + Q0 ), Σ

as desired.

References [1] Anderson, M.T.: Extrema of curvature functionals on the space of metrics on 3-manifolds. Calc. Var. Partial Differ. Equ. 5(3), 199–269 (1997) [2] Anderson, M.T.: Scalar curvature, metric degenerations, and the static vacuum Einstein equations on 3-manifolds. II. Geom. Funct. Anal. 11(2), 273–381 (2001) [3] Anderson, M.T.: Scalar curvature and the existence of geometric structures on 3-manifolds. I. J. Reine Angew. Math. 553, 125–182 (2002) [4] Anderson, M.: On long-time evolution in general relativity and geometrization of 3-manifolds. Comm. Math. Phys. 222(3), 533–567 (2001) [5] Andersson, L., Moncrief, V.: Future Complete Vacuum Space–Times. The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 299–330, Birkha¨ user, Basel (2004) [6] Cheeger, J., Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. I. J. Differ. Geom. 23(3), 309–346 (1986) [7] Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton, NJ (1993) [8] Fischer, A.E., Moncrief, V.: In: The reduced hamiltonian of general relativity and the σ-constant of conformal geometry, Mathematical and quantum aspects of relativity and cosmology (Pythagorean, 1998). Lecture Notes in Physics, vol. 537, pp. 70–101. Springer, Berlin (2000) [9] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order (Reprint of the 1998 edition). Classics in Mathematics. Springer, Berlin (2001) [10] Gromov, M.: Hyperbolic manifolds (according to Thurston and J¨ urgensen). In: Bourbaki Seminar, vol. 1979/80, pp. 40–53. Lecture Notes in Mathematics, vol. 842. Springer, Berlin-New York (1981) [11] Hamilton, R.S.: Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7(4), 695–729 (1999). In: Cao, H.D., Chow, B., Chu, S.C., Yau, S.T. (eds.) Collected Papers on Ricci flow. Series in Geometry and Topology, vol. 37. International Press, Somerville, MA (2003) [12] Isenberg, J.: Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Gravity 12(9), 2249–2274 (1995) [13] Reiris, M.: Dissertation thesis: aspects of the long time evolution in general relativity and geometrizations of three-manifolds. State University of New York at Stony Brook (2005)

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[14] Reiris, M.: General K = −1 Friedman-Lemaˆıtre models and the averaging problem in cosmology. Class. Quantum Gravity 25(8), 085001, 26 (2008) [15] Reiris, M.: On the asymptotic spectrum of the reduced volume in cosmological solutions of the Einstein equations. General Relativity and Gravitation, vol. 41, Number 5, May 2009 [16] Reiris, M.: The constant mean curvature Einstein flow and the Bel–Robinson energy. ArXiv:0705.3070 [17] Yang, D.: Convergence of Riemannian manifolds with integral bounds on curvature. I. Ann. Sci. Ecole Norm. Sup. (4) 25(1), 77–105 (1992) [18] Yang, D.: Convergence of Riemannian manifolds with integral bounds on curvature. II. Ann. Sci. Ecole Norm. Sup. (4) 25(2), 179–199 (1992) Martin Reiris Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am M¨ uhlenberg 1 14476 Golm, Germany e-mail: [email protected] Communicated by Piotr T. Chrusciel. Received: July 18, 2009. Accepted: January 12, 2010.

Ann. Henri Poincaré 10 (2010), 1605–1610 c 2010 Birkh¨ auser / Springer Basel AG 1424-0637/10/081605-6, published online March 16, 2010 DOI 10.1007/s00023-010-0023-x


On Natural Superhomogeneous Models for Minkowski Superspacetime Ramon Peniche, G. Salgado and O. A. Sánchez-Valenzuela Abstract. It is shown how, the classification of those Lie superalgebras having u2 as its underlying Lie algebra, and further restricted by the fact that the representation of u2 into its odd module is the adjoint representation, leads to at least ten different natural superhomogeneous models for Minkowski superspacetime.

1. Introduction Local homogeneous models for Minkowski spacetime lead to identify the tangent space at a given point with the real four-dimensional Lie algebra u2 of the unitary group U2 . By considering an appropriate homogeneous supermanifold—thought of as the quotient of two Lie supergroups—one might locally model Minkowski superspacetime. What is then required is that the underlying smooth manifold associated to the homogeneous supermanifold, coincides with an ordinary homogeneous model. One such example is GL2 (C)/ U2 , whose tangent space at a given point leaves a quotient isomorphic to u2 . The supertangent space at a given point of the corresponding homogeneous supermanifold should then be a Lie superalgebra of the form g = g0 ⊕ g1 having g0 = u2 , where g1 is a representation space for g0 . Indeed, to build up a supermanifold over GL2 (C)/ U2 amounts to define a vector bundle π : E → GL2 (C)/ U2 over it. In looking for a homogeneous supermanifold, the natural GL2 (C)-action g · [g ] = [gg ] on the cosets [g ] ∈ GL2 (C)/ U2 must be liftable to a GL2 (C)-action on the bundle space E for which the bundle projection becomes GL2 (C)-equivariant (see [3]); that is, π(g · v) = g · π(v). Thus the required structure is that of a homogeneous vector bundle over GL2 (C)/ U2 , and it is well known that this comes down to specifying a representation of the isotropy group of the homogeneous base into the vector space g1 to which the typical fibers of the vector bundle E are isomorphic to. In this case, the isotropy group is U2 . Thus,

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homogeneous Minkowski superspacetimes over GL2 (C)/ U2 require a Lie group representation ρ : U2 → GL(g1 ), or at the Lie algebra level, a Lie algebra representation ρ˙ : u2 → gl(g1 ). It is also well known that any Lie superalgebra is of the form g = g0 ⊕ g1 , where g0 is an ordinary Lie algebra, and g1 is a representation space for it (see [1], [2], or [4]). To link any such Lie superalgebra with a specific homogeneous vector bundle, the representation needs to be the one arising from the isotropy group on the typical fiber. Now, from among all the representation spaces, one that is natural for a given Lie algebra, is the Lie algebra itself under the adjoint representation, in the sense that no further assumptions are to be made. What is meant here by natural Minkowski superspacetimes are precisely those for which no extra assumptions are to be given about the representation g0 → gl(g1 ); whence g1 = g0 and the representation g0 → gl(g0 ) is just the adjoint representation. Therefore, what we present here is a classification, up to isomorphism, of all the Lie superalgebras having g0 = u2 = g1 associated to the adjoint representation.

2. Basic Definitions and Examples We shall follow some well-known references for the basic definitions ( eg, [1], [2], [3], and [4]). First of all a vector superspace (or supervector space) over a field F is an F-vector space V , having a distinguished direct sum decomposition V = V0 ⊕V1 , and a parity function | · | : V0 − {0} ∪ V1 − {0} → Z2 = {0, 1} defined in such a way that |v| = μ if and only if v ∈ Vμ , μ = 0, 1. Elements in the domain of | · | are called homogeneous, and vectors in the inverse image | · |−1 (0) are called even, while those in | · |−1 (1) are called odd. A Lie superalgebra is a vector superspace g = g0 ⊕g1 , together with a bilinear map [[ · , · ]] : g × g → g satisfying [[gμ , gν ]] ⊂ gμ+ν for any μ, ν ∈ Z2 , and such that for any homogeneous elements x, y and z, [[x, y]] = −(−1)|x| |y| [[y, x]] and (−1)|x| |z| [[x, [[y, z]]]] + (−1)|z| |y| [[z, [[x, y]]]] + (−1)|y| |x| [[y, [[z, x]]]] = 0

(J)

The latter is called the super-Jacobi identity. Writing down this identity for x, y and z in g0 (so that |x| = |y| = |z| = 0) we conclude that g0 is an ordinary Lie algebra with [ · , · ] = [[ · , · ]]|g0 ×g0 . Similarly, writing the super-Jacobi identity for x, y in g0 , and z in g1 , it follows that the map ρ : g0 → gl(g1 ) defined by ρ(x) = [[x, · ]]|g1 : g1 → g1 is a representation of the Lie algebra g0 into the space g1 . One also notes that [[ · , · ]]|g1 ×g1 defines a symmetric bilinear map Γ : g1 × g1 → g0 in terms of which it is easy to write the two remaining cases of (J); namely, for one element x in g0 and two elements—say u and v in g1 —we get, [x, Γ(u, v)] = Γ(ρ(x)u, v) + Γ(u, ρ(x)v)

(J1)

whereas for three elements u, v and w in g1 we get: ρ (Γ(u, v)) (w)+ρ (Γ(w, u)) (v)+ρ (Γ(v, w)) (u) = 0

(J2)

Conversely, if g0 is an ordinary Lie algebra with bracket [ · , · ] : g0 × g0 → g0 , if ρ : g0 → gl(g1 ) is a representation, and if Γ : g1 × g1 → g0 is a symmetric bilinear

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map satisfying (J1) and (J2), then g = g0 ⊕ g1 becomes a Lie superalgebra by defining [[ · , · ]] : g × g → g as follows: for any x, y in g0 , and any u, v in g1 , [[x + u, y + v]] = [x, y] + ρ(x)v − ρ(y)u + Γ(u, v). In this way, a Lie superalgebra can be viewed as a triple (g0 , ρ, Γ) satisfying (J1) and (J2). Now, under certain assumptions, (J1) implies (J2). For example, this is the case when g0 = g1 and ρ = ad, as the reader may easily verify. In what follows we will assume that g0 and g1 have been fixed, and that ρ = ad. Under these assumptions we would like to know when the Lie superalgebras defined by (g0 , ad, Γ) and (g0 , ad, Γ ) are isomorphic. Proposition 2.1. The Lie superalgebras defined by the triples (g0 , ad, Γ) and (g0 , ad, Γ ) are isomorphic, if and only if there is a pair (T, S) ∈ Aut(g0 ) × GL(g0 ) satisfying (S −1 ◦ T ) ◦ ad(x) = ad(x) ◦ (S −1 ◦ T ) for all x ∈ g0 , and T (Γ(u, v)) = Γ (S(u), S(v)) for all u and v in g0 . This follows from the basic definitions (cf, [4]) applied to our particular setup in terms of the representation ρ = ad and the symmetric bilinear map Γ : g0 ×g0 → g0 satisfying (J1).

3. The Classification Problem We shall look first at the problem of giving the most general Γ : g0 × g0 → g0 satisfying (J1) under the assumption that g0 = gl2 over the complex field, and ρ = ad. We shall use the standard basis {I, H, E, F } of gl2 , where I stands for the identity 2 × 2 matrix, H the diagonal matrix with diagonal entries +1 and −1, and E and F the nilpotent matrices that satisfy [H, E] = 2E, [H, F ] = −2F , and [E, F ] = H. Now write gl2 = C I ⊕ sl2 , with sl2 = Span {H, E, F } and let p1 : gl2 → C I, and p2 : gl2 → sl2 be the natural projections. Since I commutes with everything, (J1) with u = I says that ad(x)(Γ(I, v)) = Γ(I, ad(x)v) for any x ∈ gl2 . Using the fact that the adjoint representation of sl2 in itself is irreducible, it follows from Schur’s Lemma that p2 ◦ Γ(I, · )|sl2 must be a scalar multiple of the identity map Idsl2 . Say, for any X ∈ sl2 , p2 ◦ Γ(I, X) = μ X. It is also easy to see that for any X ∈ sl2 , Γ(I, X) ∈ sl2 , so that p2 ◦ Γ(I, X) = Γ(I, X) = μ X. Similarly, p1 ◦Γ(I, I) = Γ(I, I) = λ I for some scalar λ. Finally, p1 ◦Γ( · , · )|sl2 ×sl2 : sl2 × sl2 → C I defines a symmetric bilinear ad-equivariant form on sl2 ; therefore it must be a scalar multiple of the Cartan-Killing form κ : sl2 × sl2 → C; that is, for any X and Y in sl2 , p1 ◦ Γ(X, Y ) = ν κ(X, Y ) I. But again, it is easy to see that for any X and Y in sl2 , p1 ◦ Γ(X, Y ) = Γ(X, Y ). Summarizing: for any X and Y in sl2 , Γ(I, I) = λ I

Γ(I, X) = μ X ,

Γ(X, Y ) = ν κ(X, Y ) I.

for arbitrary parameters λ, μ and ν in the ground field C. A different symmetric bilinear map Γ : gl2 × gl2 → gl2 would yield a different set of parameters;

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say λ , μ and ν , respectively. Let us denote by gl2 (C; λ, μ, ν) the C-Lie superalgebra gl2 ⊕ gl2 defined by the parameter values (λ, μ, ν). We now have the following statement (see [5], [6]): Theorem 3.1. The Lie superalgebras gl2 (F; λ, μ, ν) and gl2 (F; λ , μ , ν ) are isomorphic if and only if there is a Lie algebra automorphism T : gl2 → gl2 and an C-linear isomorphism S : gl2 → gl2 satisfying [T (x), S(y)] = S([x, y])

and

Γ (S(x), S(y)) = T (Γ(x, y)),

for any x and y in the Lie algebra gl2 . This is the case if and only if there are nonzero constants a, b and c in the ground field C, such that, λ = λ

1 , ab2

μ = μ

1 , abc

ν = ν

a . c2

It follows that either, the three parameters λ, μ and ν are equal to zero, or exactly two of them are zero, or exactly one is zero, or none of them is zero. That is how eight different isomorphism classes over C arise. If the ground field is R, one further sees that the product λ ν is equal to λν times a positive constant. Therefore, the sign of this product must remain constant, thus giving ten real isomorphism classes. Proof of the Theorem. Use the fact that gl2 = CI ⊕ sl2 to see that, T ∈ Aut(gl2 ) if and only if T (I + X) = aI + t(X), for a ∈ F − {0} and some t ∈ Aut(sl2 ). Now, −1 −1 from Schur’s and (S ◦ T ) ◦ ad(x) = ad(x) ◦ (S ◦ T ) we conclude that Lemma b 0 S = ( a0 0t ) 0 c Idsl2 for b, c ∈ F − {0}. According to the Proposition in §2 the Lie

superalgebras defined by Γ ↔ (λ, μ, ν) and Γ ↔ (λ , μ , ν ) are isomorphic if and only if the parameters are related as in the statement. 3.1. Main Result In order to go into Minkowski spacetime u2 , we first note that u2 is a real form of gl2 ; ie, it is a real Lie subalgebra of the complex Lie algebra gl2 , such that gl2 u2 ⊕ i u2 . In order to approach the classification problem of those Lie superalgebras having g0 = u2 as their underlying Lie algebra, and ρ = ad, we perform Weyl’s unitarian trick in the following way: First give the following basis for gl2 : w0 = iI, w3 = iH, w2 = E − F and w1 = i(E + F ). Then compute Γ in terms of this basis: Γ(w0 , w0 ) = iλw0 ,

Γ(w0 , wr ) = iμwr ,

r = 1, 2, 3

Γ(wr , wr ) = 2iνw0 , r = 1, 2, 3 If the Lie superalgebra u2 (λ, μ, ν) is to be a real Lie superalgebra, λ, μ and ν have to be restricted, from taking arbitrary complex values in gl2 (C; λ, μ, ν), to take only purely imaginary values on u2 (λ, μ, ν). But then the analysis we performed before, leads to the same statement as that given in Theorem 3.1, except for the fact that all the parameters involved are now real. This gives the ten different

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Minkowski superspacetimes that can be defined naturally with no extra information on any other representation, but the one that the ground Lie algebra u2 comes defined with by means of its own Lie algebra structure. As for the significance of our results, recall that Lie superalgebras give the infinitesimal description of supersymmetry transformations. For the superalgebras just obtained, the naturality assumptions g1 = g0 , and ρ = ad : g0 → gl(g1 ) produce a g0 -equivariant change of parity map Π : g → g such that [u, Π(v)] = Π([u, v]), and Π|gµ : gμ → gμ+1 (mod 2) is just the identity map (μ ∈ Z2 = {0, 1}). In other words the even or bosonic sector produced in this case by u2 , enters in total balance with its odd or fermionic sector, so as to produce a Lie superalgebra on which only the Lie algebra that constitutes the tangent space of the given homogeneous model triggers the supersymmetry transformations.

Acknowledgements The authors would like to acknowledge the partial support received by the following grants: CONACyT Grants 106923, and CB-2007-83973; MB Grant 1411, SEP Grants 103.5/0411908, the grant PROMEP/103.5/09/4244 and PIFI-200831MSU0098J-12; and UASLP Grant C09-FAI-03-30.30. The authors would also like to thank the referee for his/her comments and questions, as they gave them the opportunity to produce a better exposition of their results.

References [1] Corwin, L., Ne’eman, Y., Sternberg, S.: Graded Lie algebras in mathematics and physics (Bose–Fermi symmetry). Rev. Mod. Phys. 47, 573–603 (1975) [2] Kac, V.G.: Lie superalgebras. Adv. Math. 26, 8–96 (1977) [3] Kostant, B.: Graded manifolds, graded Lie theory and prequantization. In: Lecture notes in mathematics, vol. 570, pp. 177–306. Springer, New York (1977) [4] Scheunert, M.: The theory of Lie superalgebras, an introduction. In: Lecture notes in mathematics, vol. 716, vi+271 pp. Springer, New York (1979) [5] Peniche, R., S´ anchez-Valenzuela, O.A.: Lie supergroups supported over GL2 and U2 associated to the adjoint representation. J. Geom. Phys. 56(6), 903–1068 (2006) [6] Salgado, G., S´ anchez-Valenzuela, O.A.: Lie superalgebras based on gln associated to the adjoint representation, and invariant geometric structures defined on them. Commun. Math. Phys. 241, 505–518 (2003) Ramon Peniche Facultad de Matem´ aticas Universidad Aut´ onoma de Yucat´ an Mérida, Yuc., México e-mail: [email protected]; [email protected]

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G. Salgado Fac. de Ciencias UASLP Av. Salvador Nava S/N Zona Universitaria San Luis Potos´ı, SLP, México e-mail: [email protected]; [email protected] O. A. S´ anchez-Valenzuela CIMAT Apdo. Postal, 402 C.P. 36000 Guanajuato, Gto., México e-mail: [email protected]; [email protected] Communicated by Raimar Wulkenhaar. Received: March 24, 2009. Accepted: January 21, 2010.


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