Stéphane Nonnenmacher
email: snonnenmacher@cea.fr

QUANTUM TRANSFER OPERATORS AND QUANTUM SCATTERING

Introduction and statement of the result

These notes present a new method, developed in collaboration with Johannes Sjöstrand and Maciej Zworski, the aim of which is a better understanding of quantum scattering systems, in situations where the set of classically trapped trajectories at some energy E > 0 is bounded, but can be a complicated fractal set. In particular, we are interested in the situations where this trapped set is a "chaotic repeller" hosting a hyperbolic (Axiom A) flow. Such a scattering system belongs to the realm of quantum chaos, namely the study of wave or quantum systems, the classical limit of which enjoy chaotic properties. This type of dynamics occurs for instance in the scattering by 3 or more disks in the Euclidean plane [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of several convex bodies[END_REF], but also in scattering by a smooth potential (see fig 1). Chaotic scattering systems are physically relevant: for instance, mesoscopic quantum dots are often modelled by open chaotic billiards [START_REF] Nakamura | Quantum Chaos and Quantum Dots[END_REF]; the ionization of atoms or molecules in presence of external electric and/or magnetic fields also involves classical chaotic trajectories [START_REF] Blümel | Chaos in Atomic Physics[END_REF]; Open quantum billiards can also be realized in microwave billiard expermiments [START_REF] Stöckmann | Scattering Properties of Chaotic Microwave Billiards[END_REF].

The method we propose is a quantum version of the Poincaré section/Poincaré map construction used to analyze the classical flow (see §1). Namely, around some scattering energy E > 0 we will construct a quantum transfer operator (or quantum monodromy operator), which contains the relevant information of the quantum dynamics at this energy, in a much reduced form: this operator has finite rank (which increases in the semiclassical limit), it allows to characterize the quantum resonances of the scattering system in the vicinity of the energy E. The quantum transfer operator is very similar with the open quantum maps studied as toy models for chaotic scattering [START_REF] Borgonovi | Statistics of quantum lifetimes in a classically chaotic system[END_REF][START_REF] Nonnenmacher | Distribution of resonances for open quantum maps[END_REF][START_REF] Schomerus | Quantum-to-classical crossover of quasi-bound states in open quantum systems[END_REF].

Our main result (Theorem 1) will be stated in §1.2. In §2 and §3 we sketch the proof of this result. We defer the details of the proofs, as well as some applications of the method, to a forthcoming publication [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF].

From flows to maps, and back. Let us recall some facts from classical dynamics. In the theory of dynamical systems, the study of a flow Φ t : Y → Y generated by some vector field (or ODE) on a phase space Y (say, a smooth manifold) can obten be facilitated by considering a Poincaré section of that flow, namely a family Σ = {Σ i , i = 1, . . . , J} of hypersurfaces of Y , which intersect the flow transversely. The successive intersections of the flow with Σ define a first return (or Poincaré) map κ : Σ → Σ (see fig. 1). This map, defined on a phase space Σ of codimension 1, conveniently represents the flow on Y . Long time properties of κ are often easier to analyze than the corresponding properties of the Figure 1. Left: a 3-bump potential, which admits a fractal hyperbolic trapped set at intermediate energies [START_REF] Sjöstrand | Geometric bounds on the density of resonances for semiclassical problems[END_REF]Appendix]. Right: schematic representation of a Poincaré section. flow. One can reconstruct the flow Φ t from the knowledge of κ together with the return time function τ : Σ → R + , which measures the time spanned between the intersections ρ and κ(ρ). Below we will explain how transfer operators associated with κ can also help to compute long time properties of the flow. 1.0.1. Hamiltonian scattering. The flows we consider are Hamiltonian flows defined on the cotangent space T * R n . A Hamiltonian (function) p ∈ C ∞ (T * R n ) defines a Hamilton vector field H p on phase space, which generates the flow Φ t = exp(tH p ). For our specific choice (1.1), the flow is complete. It preserves the symplectic structure on T * R n , and leaves invariant each energy shell p -1 (E), so it makes sense to study the dynamics on each individual shell. A Poincaré section Σ ⊂ p -1 (E) naturally inherits a symplectic structure, which is preserved by the Poincaré map κ. Hence, the Poincaré maps we consider are (local) symplectomorphisms on Σ.

We will specifically consider Hamiltonians of the form

(1.1) p(x, ξ) = |ξ| 2 2 + V (x), with a potential V ∈ C ∞ c (R n ) (say, supported in a ball B(0, R 0 ) ⊂ R n ).
Such a Hamiltonian generates a scattering system: for any energy E > 0, particles can come from infinity, scatter on the the potential, and be sent back towards infinity. Depending on the shape of V and of the energy, some trajectories can also be trapped forever (in the past and/or in the future) inside the ball B(0, R 0 ). This leads to the definition of the trapped set at energy E:

(1.2) K E def = {ρ ∈ p -1 (E) : exp(RH p )(ρ) is bounded},
which is a compact, flow-invariant subset of p -1 (E). The interesting long time dynamics takes place in the vicinity of K E , so the Poincaré section Σ need only represent correctly the flow Φ t restricted on K E , or on some neighbourhood of it. The Poincaré map κ will

Stéphane Nonnenmacher

VII-2 also be defined in some neighbourhood of the reduced trapped set T E def = K E ∩ Σ. We will give a more precise description of Σ in §2.1.

1.0.2. Chaotic dynamics and transfer operators. Our Theorem 1 will be relevant to the case where the flow on K E is uniformly hyperbolic (and satisfies Smale's Axiom A). Such a flow is, in a sense "maximally chaotic". Hyperbolicity means that at each point ρ ∈ K E the tangent space T ρ p -1 (E) can be split between the flow direction, an unstable and a stable subspaces:

(1.3) T ρ p -1 (E) = RH p ⊕ E + (ρ) ⊕ E -(ρ) ,
where the (un)stable subspaces are defined by the long time properties of the tangent map: there exist C, λ > 0 such that, for any

ρ ∈ K E , v ∈ E ∓ (ρ) ⇐⇒ dΦ ±t v ≤ C e -λt , t > 0 .
The Poincaré map κ then inherits the Axiom A property.

To study the long time properties of such chaotic flow, it has proved convenient to use transfer operators associated with κ [START_REF] Baladi | Positive Transfer Operators and Decay of Correlations[END_REF]. Let us give an example of such operators. Given any weight function f ∈ C(Σ, R), one define the transfer operator L f by a weighted pushforward on functions ϕ : Σ → R:

L f ϕ(ρ) def = ρ ′ :κ(ρ ′ )=ρ e f (ρ ′ ) ϕ(ρ ′ ) .
Provided L f is applied to some appropriate functional space 1 , its spectrum can deliver relevant information about the long time dynamics of κ. For instance, the spectral radius r sp (L f ) determines the topological pressure of κ associated with the weight f , which provides statistical information on the long periodic orbits of κ:

log r sp (f ) = P(κ, f ) def = lim T →∞ 1 T log |γ|≤T e R γ f .
(here γ f is the sum of values of f (ρ) along the periodic orbit γ). The topological pressure of the flow Φ t , associated with a weight F ∈ C(X), can also be computed through transfer operators. One defines on Σ the function

f (ρ) = τ (ρ) 0 F (Φ t (ρ))
, that is the accumulated weight from ρ ∈ Σ to its next return κ(ρ), and considers the family {L f -sτ , s ∈ R} of transfer operators. The following relation then relates the pressures of κ and Φ t :

s = P(Φ t , F ) ⇐⇒ P(κ, f -sτ ) = 0 ⇐⇒ r sp (L f -sτ ) = 1 .
The decay of correlations for the Axiom A flow Φ t is encoded in the Ruelle-Pollicott resonances, which are the poles of the Fourier transform of the correlation function [START_REF] Pollicott | On the rate of mixing of Axiom A flows[END_REF]. Within some strip S ⊂ C, these resonances {z i } can be characterized by using the family {L f -zτ , z ∈ C} of complex weighted transfer operators: z i ∈ S is a resonance iff L f -z i τ 1 The functional space can be rather complicated, see e.g. [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] for the case of Anosov diffeomorphisms. has an eigenvalue equal to 1. This property can be written (abusively, because transfer operators are usually not trace class) as

(1.4) z ∈ S is a Ruelle-Pollicott resonance ⇐⇒ det(1 -L f -zτ ) = 0 .
1.1. A quantum scattering problem. Let us now introduce the quantum dynamics we are interested in. The operator

P = P (h) = - h 2 ∆ 2 + V (x) , V ∈ C ∞ c (R n ) ,
generates the Schrödinger dynamics U(t) = exp(-itP (h)/h) on L 2 (R n ). P (h) is the hquantization of the classical Hamiltonian (1.1), so the semiclassical behaviour of the quantum dynamics will be strongly influenced by the flow exp(tH p ). We focus on the dynamics around some positive energy E > 0, so the flow we need to understand is Φ t ↾ p -1 (E). We will assume that

• the flow on p -1 (E) has no fixed point: dp↾ p -1 (E) = 0.

• the trapped set K E has topological dimension 1. Equivalently, the reduced trapped set

T E = K E ∩ Σ is totally disconnected.
These conditions are satisfied, for example, for a 3-bump potential at intermediate energies (see fig. 1). The second condition was absent in previous studies of such systems [START_REF] Sjöstrand | Fractal upper bounds on the density of semiclassical resonances[END_REF][START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF], it is a technical constraint specific to the approach we develop below (as we explain after Thm 1, the condition required for the method to work is actually weaker). We are interested in the long time Schrödinger dynamics near energy E, so it is natural to investigate the spectrum of P (h) near E. That operator is self-adjoint on L 2 (R n ), with domain H 2 (R n ); but, due to the bounded support of V (x), the spectrum of P (h) is absolutely continuous on R + , without any embedded eigenvalue. Nevertheless, the truncated resolvent ψ(P (h)z) -1 ψ (with ψ ∈ C ∞ c (R n )), well-defined in the quadrant {Re z >, Im z > 0}, can be meromorphically extended across the real axis into {Re z > 0, Im z < 0}. The finite rank poles {z i } in this region are called its resonances (they do not depend on the specific cutoff ψ). Resonances are often understood as "generalized eigenvalues": they are associated with metastable modes u i (x) which are not square-integrable, but satisfy the differential equation P (h)u i = z i u i , so that they decay expontially in time, at a rate given by

| Im z i |/h.
One of our objectives is to better understand the distribution of these resonances in the h-neighbourhood of the energy E, that is in disks D(E, Ch) (see fig. 2). More precisely, we want to investigate:

• the number of resonances in D(E, Ch). So far fractal upper bounds have been proven [START_REF] Sjöstrand | Fractal upper bounds on the density of semiclassical resonances[END_REF]. We wish to investigate whether similar lower bounds can be obtained, at least for a generic system. • the width of the resonance free strip in D(E, Ch). A lower bound for such a strip has been expressed in terms of some topological pressure [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of several convex bodies[END_REF][START_REF] Gaspard | Semiclassical quantization of the scattering from a classical chaotic repeller[END_REF][START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF], but a recent result of Petkov-Stoyanov (for obstacle scattering) shows that this lower bound is in general not sharp [START_REF] Petkov | Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function[END_REF].

1.2. Our result. Our main result is a "quantization" of the Poincaré section method presented above.

Theorem 1. Assume that, for some energy E > 0, the trapped set K E for the flow exp(tH p ) is topologically one dimensional, and contains no fixed point.

Then, for h > 0 small enough, there exists a family of matrices {M(z, h), z ∈ D(0, Ch)} holomorphic w.r.to z, such that the zeros of the function

(1.5) ζ(z, h) def = det(I -M(z, h))
give the resonances of (P (h) -E) in D(0, Ch), with correct multiplicities.

The matrices M(z, h) have the following structure. There exists a Poincaré section

Σ = ⊔ J i=1 Σ i and map κ : Σ → Σ, an h-Fourier integral operator M(z, h) : L 2 (R n-1 ) J quantizing κ, and a projector Π h of rank r(h) ≍ h -n+1 , such that M(z, h) = Π h M(z, h) Π h + O(h N ) .
The remainder estimate holds in the operator norm on C r(h) . The exponent N can be assumed arbitrary large.

Remark 1.1. The 1-dimensional condition we impose on K E is not strictly necessary. What one needs is the existence of a Poincaré section Σ intersecting Φ t ↾ K E , such that ∂Σ ∩ K E = ∅; in particular, we don't need the flow to be hyperbolic on K E . Still, Axiom A flows provide the most obvious example for which this condition is satisfied [START_REF] Bowen | One-dimensional hyperbolic sets for flows[END_REF]; it holds as well for the broken geodesic flow in the scattering by 3 disks satisfying a no-eclipse condition [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of several convex bodies[END_REF]. This theorem shows that the dynamics generated by the Hamiltonian P (h) near E can be "summarized" in the family of quantum transfer operators {M(z, h), z ∈ D(0, Ch)}. One reason for this terminology is that M(z, h) bears some resemblance with the transfer operators L f -zτ briefly described in §1.0.2. The equation (1.5) characterizing quantum resonances is obviously the quantum analogue of the (generally formal) equation (1.4) defining Ruelle-Pollicott resonances. Also, the notation ζ(z, h) in (1.5) hints at an analogy, or relationship, between this spectral determinant and some form of semiclassical zeta function (such functions have been mostly studied in the physics literature, see e.g. [START_REF] Cvitanović | A Fredholm determinant for semiclassical quantization[END_REF]).

The operators M(z, h) have the same semiclassical structure as open quantum maps studied in the (mostly physical) literature as toy models of quantum scattering systems. For instance, the distribution of resonances and resonant modes has proven to be much easier to study numerically for open quantum maps, than for realistic flows [START_REF] Borgonovi | Statistics of quantum lifetimes in a classically chaotic system[END_REF][START_REF] Schomerus | Quantum-to-classical crossover of quasi-bound states in open quantum systems[END_REF][START_REF] Nonnenmacher | Distribution of resonances for open quantum maps[END_REF][START_REF] Keating | Semiclassical structure of quantum fractal eigenstates[END_REF]. The novelty here, is that the operators M(z, h) allow to characterize a "physical" resonance spectrum.

1.2.1. Historical remarks. Actually, a similar method has been introduced in the theoretical physics literature devoted to "quantum chaos". To the author's knowledge, the first such construction appeared in Bogomolny's work [START_REF] Bogomolny | Semiclassical quantization of multidimensional systems[END_REF] on multidimensional closed quantum systems. In that work, a family of quantum transfer operators T (E) is constructed, which are integral operators defined on a hypersurface in configuration space. The eigenvalues of the bound Hamiltonian are then obtained, in the semiclassical limit, as roots of the equation det(1 -T (E)) = 0. This work generated a lot of interest in the quantum chaos community. Smilansky and co-workers derived a similar quantization condition for closed Euclidean 2-dimensional billiards [START_REF] Doron | Semiclassical quantization of chaotic billiards: a scattering theory approach[END_REF], replacing T (E) by a scattering matrix S(E) associated with the dual scattering problem. Bogomolny's method was also extended to study quantum scattering situations [START_REF] Georgeot | Fredholm theory for quasiclassical scattering[END_REF]. On the other hand, Prosen developed an "exact" (that is, not necessarily semiclassical) quantum surface of section method to study certain closed Hamiltonian systems [START_REF] Prosen | General quantum surface-of-section method[END_REF].

In the mathematics literature similar operators appeared in the framework of obstacle scattering [START_REF] Gérard | Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes[END_REF][START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of several convex bodies[END_REF]: the scattering problem was analyzed through integral operators defined on the obstacle boundaries, which also have the structure of Fourier integral operators associated with the bounce map. More recently, a monodromy operator formalism has been introduced in [START_REF] Sjöstrand | Quantum monodromy and semiclassical trace formulae[END_REF] to study the Schrödinger dynamics in the vicinity of a single isolated periodic orbit. This approach has then been used to investigate concentration properties of eigenmodes in the vicinity of such an orbit [START_REF] Christianson | Quantum monodromy and non-concentration near a closed semi-hyperbolic orbit[END_REF]. The construction we present below heavily borrows from the techniques developed in [START_REF] Sjöstrand | Quantum monodromy and semiclassical trace formulae[END_REF]. It improves them on two aspects: first, our invariant set K E is more complex than a single periodic orbit. Second, the connection we establish between the operators (P (h) -Ez) and M(z, h) is deeper than previously.

Formal construction of the quantum transfer operator

The proof of Thm 1 proceeds in several steps. It uses many tools of h-pseudodifferential calculus (we will use the notations of [START_REF] Dimassi | Spectral Asymptotics in the semi-classical limit[END_REF]10]). We just recall a few of them:

Stéphane Nonnenmacher VII-6 • a state u = u(h) ∈ L 2 is microlocalized in a domain U ⋐ T * R n iff, for any function χ ∈ C ∞ c (T * R n ) with supp χ ∩ Ū = ∅, one has χ w u L 2 = O(h ∞ ) u L 2 . (here χ w = χ w (x, hD x ) denotes the h-Weyl quantization of χ). • two states u, v are said microlocally equivalent in U ⋐ T * R n iff, for any cutoff χ ∈ C ∞ c (U), one has χ w (u -v) L 2 = O(h ∞ ). • similarly, two operators A, B are said microlocally equivalent in V ×U (with U, V ⋐ T * R n ) iff, for any cutoffs χ 1 ∈ C ∞ c (U), χ 2 ∈ C ∞ c (V ), one has χ w 2 (A-B)χ w 1 L 2 →L 2 = O(h ∞ ).
• an operator A is microlocally defined in V × U iff it is microlocally equivalent in V × U to some globally defined operator. "Microlocally defined in U" will mean "microlocally defined in U × U".

The present section constructs the quantum transfer operators microlocally in a neighbourhood of the trapped set K E , without paying attention to the rest of the phase space. The arguments making the construction globally well-defined will be presented in §3.

The microlocal construction being strongly tied to a Poincaré section, we start by describing the latter in some detail.

2.1. Description of the Poincaré section. According to the assumptions of the theorem, the trapped set K E is a compact set of topological dimension unity. It is then possible to construct a Poincaré section Σ = ⊔ J i=1 Σ i ⊂ p -1 (E) with the following properties: • each Σ i is a (2n -2)-dimensional topological disk, transverse to the flow.

• the maximal diameter of the Σ i can be chosen arbitrary small.

• there exists a time τ max > 0 such that, for any ρ ∈ K E , the trajectory Φ t (ρ) intersects Σ at some time 0 < t ≤ τ max .

• the boundary ∂Σ = ⊔ i Σ i does not intersect K E .
If we restrict ourselves to points in the reduced trapped set T E def = K E ∩ Σ, the map ρ → ρ + (ρ) defines a bicontinuous bijection κ : T → T .

Each component of the reduced trapped set, T i def = K E ∩ Σ i , splits in two different ways:

T i = ⊔ j D ji , where D ji = {ρ ∈ T i , κ(ρ) ∈ T j } (2.1)
T i = ⊔ j A ij , where A ij = {ρ ∈ T i , κ -1 (ρ) ∈ T j } (2.2)
We will denote by J + (i) (resp. J -(i)) the set of indices in the "outflow" (resp. "inflow") of T i , that is such that D ji and A ji (resp. D ij and A ij ) are not empty. The map κ is the union of components κ ij , which relate bijectively D ij with A ij . Since T i lies in the interior of Σ i , the components D ji (resp. A ij ) are disconnected from one another. Hence, each κ ij can be extended to be a bijection 3 for a sketch).

κ ij : D ij → A ij , where D ij , A ij are open neighborhoods of D ij and A ij , respectively in Σ j and Σ i . The extended map κ ij : D ij → A ij is a symplectomorphism (see fig.

Microlocal solutions.

In this section we show that, for z ∈ D(0, Ch), any solution to the equation (P (h) -Ez)u = 0, microlocally near some part of K E , can be "encoded" by a transversal function w ∈ L 2 (R n-1 ), which "lives" on one component Σ i of the Poincaré section.

Take such a component Σ i . From the assumption dp ↾ p -1 (E) = 0, there exists an open neighbourhood V i of Σ i , and a set of symplectic coordinates (y 1 , . . . , y n ; , η 1 , . . . , η n ) on V i , such that

• the Hamiltonian p(ρ) = E + η 1 for any ρ ∈ V i • the section Σ i is locally defined by {y 1 = η 1 = 0}, and the origin y = η = 0 corresponds to a point in T i .

Equivalently, there exists a neighbourhood (0, 0) ∈ Ṽi ∈ T * R n and a symplectomorphism κi : Ṽi → V i , such that p • κi (y, η) = E + η 1 , etc. The change of coordinates κi can be h-quantized into an h-Fourier integral operator

U i : L 2 (R n ) → L 2 (R n ), microlocally defined and unitary near V i × Ṽi , such that (2.3)
U * i (P (h) -E) U i is microlocally equivalent with hD y 1 in Ṽi × Ṽi . This "quantum change of coordinates" allows one to easily characterize, for z ∈ D(0, Ch), the microlocal solutions to the equation

(2.4) (P (h) -E -z)u = 0 microlocally in V i .
Indeed, the equation

(2.5) (hD y 1 -z)v = 0 is obviously solved by (2.6) v(y 1 , y ′ ) = e izy 1 /h w(y ′ ), w ∈ L 2 (R n-1 ) , Stéphane Nonnenmacher VII-8
that is by extending some "transversal data" w. We denote this extension by v = K(z)w.

Conversely, the solution v can easily be "projected" onto the data w: consider some monotone χ ∈ C ∞ (R n ) such that χ(y) = 0 for y 1 < ǫ, χ(y) = 1 for y 1 > ǫ. Then, w can be recovered from v through

w(y ′ ) = R e -izy 1 /h ∂ y 1 χ(y) v(y) dy 1 .
In a more compact form, we write

w = K(z) * χ ′ v, with χ ′ = i h [hD y 1 , χ].
The solutions of (2.4) are then given by selecting some w ∈ L 2 (R n-1 ) (microlocalized near the origin), and take

(2.7) u = U i K(z)w def = K i (z) w .
That is, the operator K i (z) builds a microlocal solution of (2.4) near Σ i , starting from "transversal data" w ∈ L 2 (R n-1 ). The latter can be interpreted as a quantum state living in the reduced phase space Σ i . The converse "projection" is given by

(2.8) w = K(z) * χ ′ U * i u = K i (z) * χ ′ i u def = R +i (z) u .
Here χ i is the cutoff corresponding to χ near the section Σ i . To get a consistent definition for χ i , we must assume that it jumps back down to 0 a little further along the flow, but the precise position will be irrelevant. Indeed, the commutator i h [P (h), χ w i (x, hD x )] is equal (microlocally near K E ) to the sum of two pseudodifferential operators with disjoint wavefront sets. The first one is microlocalized near Σ i (in the region where χ i jumps from 0 to 1), we will denote it by χ ′ i = i h [P (h), χ w i (x, hD x )] i ; the second component "lives" in the region where χ i decreases from 1 to 0, and will not play any role.

The same construction can be performed independently near each Σ j , j = 1, . . . , J. We will call w j the transversal data associated with the section Σ j , and K j (z), R +j (z) the corresponding operators.

2.3.

From one transversal parametrization the next. The the solution (2.7) is microlocalized in V i , since U i is only defined microlocally in V i × Ṽi . However, this solution can be extended in a forward cylinder ∪ 0≤t≤T Φ t Σ i by using the propagator e -it(P -E-z)/h : if u is a solution near ρ ∈ Σ i , then e -it(P -E-z)/h u is the extension of this solution near Φ t (ρ). This way, we can extend u up to the vicinity of the sections Σ j in the outflow of Σ i . This extended solution will still be denoted by u = K i (z)w i . Near Σ j , this solution can also be parametrized by the "transversal" function w j = R +j u ∈ L 2 (R n-1 ). The map w i → w j , which amounts to changing the transversal parametrization for a single solution u, defines our quantum Poincaré map:

(2.9)

M ji (z, h) def = R +j (z)K i (z) .
This operator is a Fourier integral operator quantizing the Poincaré map κ ji ; it is microlocally defined, and microlocally unitary, on D ji × A ji .

Exp. n o VII-Quantum transfer operators and quantum scattering

VII-9

Let us see how the operators M ji (z) can be used. Assume E + z is a resonance of P (h), with z ∈ D(0, Ch). Then, there exists a metastable state u ∈ L 2 loc , global solution to the equation (P -Ez)u = 0. The above procedure associates to this solution J parametrizations w i = R +i (z)u, microlocally defined near T j . For any i and j ∈ J + (i), these parametrizations satisfy w j = M ji (z)w i : can be written in the compact form (2.10) w = M(z) w , where w = (w i ) i is the column vector of all J local parametrizations, and M(z) is the operator valued matrix M ij (z) .

In the next subsections we prove that the converse statement holds as well: the existence of a solution of (M(z) -Id)w = 0 microlocally near T E implies the existence of a solution of (P -Ez)u = 0 microlocally near K E . To prove this we will set up a formal Grushin problem, in which the operator (M(z) -I) will appear as the "effective Hamiltonian" for the original operator (P -Ez).

Grushin problems.

A Grushin problem for the family of operators 2 {(P -Ez) :

H 2 h (R n ) → L 2 (R n ), z ∈ D(0, Ch)
} consists in the insertion of that operator inside an operator valued matrix (2.11)

P(z) = i h (P -E -z) R -(z) R + (z) 0 : H 2 h (R n ) × H → L 2 (R n ) × H ,
in a way such that P(z) is invertible (see e.g. [START_REF] Sjöstrand | Elementary linear algebra for advanced spectral problems[END_REF] or [10,Appendix] for a general presentation of this method). Ideally, the auxiliary space H is "much smaller" than L 2 or H 2 h (in our final version, H will be finite dimensional). The inverse of P(z) is traditionally written in the form

P(z) -1 = E(z) E + (z) E -(z) E -+ (z) .
The invertibility of (P -E -z) is then equivalent with that of the operator E -+ (z): Schur's complement formula shows that (2.12)

h i (P -E-z) -1 = E(z)-E + (z)E -+ (z) -1 E -(z) , E -+ (z) -1 = - h i R + (z)(P -E-z) -1 R -(z) ,
so that dim ker(P -Ez) = dim ker E -+ (z). For this reason, E -+ (z) is called an effective Hamiltonian associated with (P (h) -Ez). It has a smaller rank than P (h), but its dependence in the spectral parameter z is nonlinear.

2 H 2 h (R n ) is the semiclassical Sobolev space of norm u H 2 h = |ũ(ξ)| 2 (1 + |hξ| 2 )
2 dξ, with ũ the Fourier transform of u.

Stéphane Nonnenmacher

VII-10 2.5. Our formal Grushin problem. We will first build our Grushin problem microlocally near K E (so we can identify H 2 h with L 2 ). Our auxiliary space H will contain local "transversal data" w i ∈ L 2 (R n-1 ), one for each section Σ i , so we have formally H = L 2 (R n-1 ) J . The auxiliary operators are then vectors of operators: R + (z) = (R +1 , . . . , R +J ), R -(z) = t (R -1 , . . . , R -J ), which will for now be defined only microlocally:

• R +i (z) is the "projector" (2.8) of L 2 (R n ) onto the parametrization w i living on Σ i .

We will rebaptize χ i def = χ f i (for forward) the cutoff used in the definition of R +i . • on the opposite, R -i (z) takes the data w i ∈ L 2 (R n-1 ) to produce a microlocal solution, and cuts off this solution by applying the derivative of another cutoff χ b i : (2.13)

R -i (z) = χ b′ i K i (z) . The cutoff χ b
i (for backward) is similar with χ f i , and χ b′ i is, as before, the component of [ i h P (h), (χ b i ) w ] microlocalized near Σ i . We require that the jump of χ b i occurs before that of χ f i , and that the whole family {χ b i , i = 1, . . . , J} satisfies a local resolution of identity near K E :

(2.14) i χ b i = 1 , in some neighbourhood of K E .

Homogeneous problem.

Let us now try to invert the matrix P(z) we have just defined, at least microlocally near K E × i T i . First we consider arbitrary transversal data w = (w i ), and try to solve (in

u ∈ L 2 (R n ), u -∈ L 2 (R n-1 ) J ) the system i h (P -E -z)u + J i=1 R -i (z)u -i = 0 (2.15) R +i (z)u = w i , i = 1, . . . , J . (2.16)
Eq. (2.16) suggests that u could be a microlocal solution parametrized by w i , at least in the region where χ f j jumps from 0 to 1. Since χ b i ≡ 1 in this region, we take the Ansatz (2.17)

u = J i=1 (χ b i ) w K i (z) w i def = i E +i (z) w i .
Injecting this Ansatz in (2.15), we obtain (2.18)

J i=1 i h [P, (χ b i ) w ]K i (z) w i + J i=1 R -i (z)u -i = 0 , which we want to solve in (u -i ). Each commutator i h [P, (χ b i ) w ] is the sum of a component χ b′ i = i h [P, (χ b i ) w ] i microlocalized near Σ i ,
and of components i h [P, (χ b i ) w ] j microlocalized near A ji ⊂ Σ j , for each index j ∈ J + (i). The resolution of identity (2.14) shows that near we Exp. n o VII-Quantum transfer operators and quantum scattering VII-11 have H p χ b′ i + H p χ b′ j = 0, the quantum version of which reads i h [P, (χ b i ) w ] j + i h [P, (χ b j ) w ] j = 0 microlocally near A ji . As a result (2.18) can be rewritten as

J i=1 χ b′ i K i (z) w i - J i=1 j∈J + (i) χ b′ j K i (z) w i + J i=1 R -i (z)u -i = 0 . Near each Σ j , j ∈ J + (i) we have K i (z)w i = K j (z)M ji (z) w i .
For each i we can group together the terms localized near Σ i , and get:

R -i (z) w i - i∈J + (j) R -i (z)M ij (z) w j + R -i (z)u -i = 0 .
This leads to the microlocal solution

(2.19) u -i = -w i + i∈J + (j) M ij (z) w j def = j E -+ij (z)w j .
We have thus solved the system (2.15,2.16) microlocally near K E × i T i , and provided explicit expressions for the operators E + (z) and E -+ (z) = M(z) -Id, microlocally near the trapped set.

Nonhomogeneous problem.

To complete the microlocal inversion of P(z), we now take v ∈ L 2 (R n ) microlocalized near K E , and try to solve (in u, u -, microlocally near

K E ) (2.20) i h (P -E -z)u + J i=1 R -i (z)u -i = v .
Let us first assume that v is microlocalized inside the region {χ b i (ρ) = 1} for some index i. We then take the truncated parametrix E(z) = T 0 e -it(P -E-z)/h dt, with T large enough so that e -iT P/h v is microlocalized beyond supp χ b i , and define the Ansatz

u = (χ b i ) w E(z) v. The latter satisfies i h (P -E -z)u = v + i h [P, (χ b i ) w ] E(z) v dt (2.21) = v + χ b′ i E(z)v - j∈J + (i) χ b′ j E(z)v . (2.22)
(we have used the splitting of the commutator explained above). Now, provided T is not too large, the state E(z)v is microlocalized away from Σ i so that χ b′ i E(z)v = O(h ∞ ) On the opposite, for each j ∈ J + (i) that state is a microlocal solution of (P -Ez)u -0 near A ji , which can then be written as K j (z)u -j with u -j = R +j E(z)v. The above equality becomes i h

(P -E -z) u = v - j∈J + (i) R -j u -j ,
and solves (2.20) microlocally.

Stéphane Nonnenmacher

VII-12

If v is microlocalized near Σ i , we cutoff E(z)v by ( j∈J -(i) χ b j + χ b i ) w , which is equivalent to identity near T i , and take as above u -j = R +j E(z)v, j ∈ J + (i).

We have now fully inverted P(z) microlocally near K E × i T i , and the norm of the inverse can be shown to be of order unity. The effective Hamiltonian reads E -+ (z) = M(z) -Id. Hence, as anticipated above, the existence of a nontrivial state w satisfying w = M(z) w is equivalent with that of a microlocal solution to (P -Ez)u = 0 near K E .

To prove Thm 1, we must define our Grushin problem globally, that is properly define the auxiliary space H and the operators R ± , in such a way that P(z) is invertible. One then says that the Grushin problem is well-posed.

From the formal to the well-posed Grushin problem

In order to make our Grushin problem well-posed, we will first "deform" the original Schrödinger operator P (h) in order to transform its resonances z i into bona fide L 2 eigenfunctions of the deformed operator P θ (h). This deformation is performed through a "complex scaling" of P (h) far away from the scattering region. The operator (P θ (h) -E) will now be elliptic outside a large ball B(0, R). In order to enlarge this zone of ellipticity to the complement of a smaller neighbourhood of K E , we will then modify the Hilbert structure of our auxiliary states, using an appropriate escape function G(x, ξ). After these two modifications, we will be able to complete the construction of a well-posed Grushin problem, with finite dimensional auxiliary spaces.

Γ θ ∩ B C n (0, R 0 ) = B R n (0, R 0 ) , Γ θ ∩ C n \ B C n (0, 2R 0 ) = e iθ R n ∩ C n \ B C n (0, 2R 0 ) .
We then define the operator P θ (h) acting on u ∈ C ∞ c (Γ θ ), by P θ u = P (ũ)↾ Γ θ , where ũ is an almost analytic extension of u. Through the identification Γ θ ∋ x ←→ (sin θ) -1 Re x ∈ R n , the operator P θ can considered as acting on functions in C ∞ c (R n ), with the action -e -2iθ h 2 ∆ 2 outside B(0, 2R 0 ). One can then show [START_REF] Sjöstrand | A trace formula and review of some estimates for resonances[END_REF] that the resolvent (P θz) -1 : L 2 → H 2 h is meromorphic in the region {arg(z) > -2θ}. The L 2 spectrum of P θ (h) in that region is discrete, independent of θ and R, and consists of the resonances of the initial operator P (h).

Since P θ (h) = P (h) inside the ball B(0, R 0 ) ⊃ πK E , our formal Grushin problem remains unchanged if we replace P (h) by P θ (h).

Below we will take values of θ of the form θ ∼ C h log(1/h), C > 0 fixed.

Exp. n o VII-Quantum transfer operators and quantum scattering VII-13

3.2. Finite dimensional auxiliary spaces. We have built in §2.5 a Grushin problem which is invertible microlocally near the trapped set. To make that Grushin problem wellposed, we need to make definite choices for the auxiliary spaces, that is for each i = 1, . . . , J define a subspace of H i ⊂ L 2 (R n-1 ) containing the transversal data. This subspace should contain states microlocalized in some neighbourhood S i of T i , small enough to lie in the domain ∪ j∈J + (i) A ji where κ is defined. To construct this subspace explicitly, we may define the neighbourhood S i as S i = {q i (ρ) < 0}, for a well-chosen q i ∈ C ∞ (R n-1 ) satisfying lim inf ρ→∞ q i (ρ) > 0. The subspace H i can then be defined as the range of the spectral projector

(3.1) Π i def = 1l R -(q w i (y, hD y ))
. According to Weyl's law, for h small enough the space H i has a finite dimension of the order of vol(S i ) h -n+1 .

One can then consider the Grushin problem (2.11), with P replaced by P θ , the auxiliary space H = i H i and the operators

(3.2) R +i (z) def = Π i K * i (z)χ b′ i , R -i def = χ b′ i K i (z)Π i .
Unfortunately, when trying to solve this new Grushin problem, that is invert P(z), one encouters difficulties. Some of them are due to the fact that κ does not leave the neighbourhoods S i invariant (see fig. 3). As a result, an initial datum w i ∈ H i is propagated through M ji (z) into a state M ji (z)w i which, in general, is not microlocalized in S j , and thus cannot belong to H j . Brutally applying the projector Π j to M ji (z)w i produces an extra term (1 -Π j )M ji w i , which is difficult to solve away. Another difficulty arises when trying to solve the unhomogeneous problem (2.20) for data v microlocalized at some distance from K E .

3.3. Escape functions and modified norms. These difficulties can be tackled by modifying the Hilbert norms on H 2 h (R n ) and the auxiliary space L 2 (R n-1 ) J . The new norms will be defined in terms of well-chosen escape functions

G ∈ C ∞ c (T * R n ), G i ∈ C ∞ c (T * R n-1
). Using these new norms, the problems mentioned above will disappear, because the states microlocalized away from K E will become easily solvable.

Let us describe the escape function. For some small δ > 0, we consider the thickened energy shell p -1 (E) = |s|≤δ p -1 (E + s) and trapped set K E = |s|≤δ K E+s . It is shown in [28, § §4.1,4.2,7.3] and [20, §6.1] that, for any small δ 0 > 0 and large R > 0, and any neighbourhoods

U ⊂ U ⊂ V of K E , one can construct a function G 0 ∈ C ∞ c (T * R n ) such that G 0 = 0 on U, H p G 0 ≥ 0 on T * B(0,3R) R n , (3.3) H p G 0 ≥ 1 on T * B(0,3R) R n ∩ ( p -1 (E) \ V ), H p G 0 ≥ -δ 0 on T * R n . (3.4) Stéphane Nonnenmacher

VII-14

It is convenient 3 to slightly modify this function in the neighbourhood of the sets S i ⊂ Σ i . Namely, we consider open neighbourhoods Wi ⋐ W i of S i in T * R n , and modify

G 0 into a function G 1 , such that H p G 1 = 0 in Wi while H p G 1 ≥ 1 on T * B(0,3R) R n ∩( p -1 (E)\(V ∪ i W i ).
We then set G def = Nh log(1/h)G 1 , with N > 0 fixed but arbitrary large. The exponential exp(G w (x, hD)/h) is a pseudodifferential operator in some mildly exotic class, bounded and of bounded inverse on L 2 , with norms O(h -CN ). We call H G the vector space H 2 h (R n ) equipped with the Hilbert norm

(3.5) u H G def = exp(-G w (x, hD x )/h)u H 2 h . Similarly, we consider functions G i ∈ C ∞ c (T * R n-1
) such that (using the coordinate change κi near Σ i ) G i (y ′ , η ′ ) = G • κi (0, y ′ ; 0, η ′ ) in some neighbourhood of S i , and modify the Hilbert norms on the space L 2 (R n-1 ) attached to the section Σ i by (3.6) w

i H G i def = e -(G i ) w (y ′ ,hD y ′ )/h w i L 2 (R n-1 ) .
3.4. How these norms resolve our problems. Let us explain how this change of norm helps us. The action of P θ (h) on the Hilbert space H G is equivalent to the action of This shows that, for any z ∈ D(0, Ch), the symbol (p θ,G -Ez) is invertible outside V ∪ i W i , with inverse of order (h log h -1 ) -1 . Hence, for any v ∈ L 2 microlocalized outside V ∪ i W i , the equation (P θ,G -Ez)u = v can be solved up to O(h ∞ ), with a solution u microlocalized outside V ∪ i W i . This remark basically tackles the second problem mentioned at the end of §3.2.

P θ,G (h) def = e -G w /h P θ (h) e G w /h on H 2 h (R n ),
The first problem (the fact that κ ji (S i ) is not contained in S j ) is also resolved through this change of norms. Indeed, the escape function G 1 can be chosen such that it uniformly increases (say, by some 2C > 0) along all trajectories of the form ρ ∈ S i → κ ji (ρ) ∈ Σ j \ ∈ S j , so that e -G(κ ji (ρ))/h e -G(ρ)/h ≤ h 2N C . This implies that, for any state w i microlocalized near such a point ρ, one gets (taking the definition (2.8) for R +j )

(3.8) R +j (z)K i (z)w i H G j = O(h N C ) w i H G i .
3 The role of this modification is to ultimately keep the norms R +i (z) HG→Hi , R -i (z) Hj→HG , M ji (z) Hi→Hj uniformly bounded We then need to modify the finite rank projectors (3.1) defining our auxiliary states, such as to make them orthogonal w.r.to the new norms (otherwise Π i H G i →H G i could be very large). This modification only amounts to adding a subprincipal (complex valued) term to the function q i , such that q w i becomes selfadjoint on H G i , and the spectral projector (3.1) orthogonal. The space H i def = Π i H G i is still made of states microlocalized in S i , and has dimension ∼ vol(S i )h -n+1 . Our operators R ±i will be defined by (3.2).

Let us reconsider the homogeneous problem §2.5.1 with data w i ∈ H i in our new Grushin problem. For j ∈ J + (i), the state M ji (z) w i does not a priori belong to H j . However,, the estimate (3.8) shows that the component of M ji (z) w i microlocalized outside S j has an H G j -norm of order O(h N C ). As a result, defining the finite rank operators M ij (z)w j ∈ H i provides a solution to the homogeneous problem, up to an error O(h N C )( i w i H i ).

The nonhomogeneous problem (2.20) can be solved as well, up to a comparable error (see [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF] for details).

To summarize, our globally defined Grushin problem has an approximate inverse E(z):

P(z)E(z) = I + R(z), R(z) L 2 ×H→L 2 ×H = O(h N C ) ,
where we insist on the fact that N can be chosen arbitrary large (it comes from the factor in front of the escape function G). Hence, for h small enough this operator has the exact inverse Ẽ(z) = E(z)(I +R(z)) -1 = E(z)+O L 2 ×H→H 2 h ×H (h N C ). In particular, the lower-right entry of Ẽ(z) (that is, the exact effective Hamiltonian) reads

E -+ (z) = I -M(z) + O H→H (h N C ) ,
where M(z) is the matrix composed of the finite dimensional operators (3.9).

As explained in §2.4, this exact inversion implies that the eigenvalues {z i } of (P θ -E) in D(0, Ch) coincide (with multiplicities) with the zeros of det(E -+ (z)).

Séminaire 18 VII- 1 -

 181 Équations aux dérivées partielles Centre de mathématiques Laurent Schwartz, 2009-2010 Exposé n o VII, 1-

  Exp. n o VII-Quantum transfer operators and quantum scattering VII-3

Figure 2 .

 2 Figure 2. Schematic representation of the spectrum of P (h) and its resonances near the energy E.

  Exp. n o VII-Quantum transfer operators and quantum scattering VII-5

Figure 3 .

 3 Figure 3. Schematic representation of a hyperbolic Poincaré map. The light blue (resp. pink) regions represent D 31 and A 31 (resp. D 21 and A 21 ). The (un)stable directions are represented by the dashed horizontal and vertical lines. The black squares show a coarse-graining of T E .

3. 1 . 2 i.

 12 Complex scaling. Here we use the fact that outside a ball B(0, R 0 ) ⋑ supp V , one has P (h) = -h 2 In that region that operator can be holomorphically exended into P = -h 2 functions on C n . For θ > 0 small, we deform R n ⊂ C n into a smooth contour Γ θ ⊂ C n :

  which is a pseudodifferential operator with symbolp θ,G (ρ) = p(ρ) -iNh log(1/h)H p G 1 (ρ) + O(h 2 log 2 (1/h)) , ρ ∈ T * B(0,R) R n .Provided we have chosen a dilation angle θ ≪ δ 0 Nh log(1/h), the properties of G 1 show that(3.7) ∀ρ ∈ (V ∪ i W i ), | Re p θ,G (ρ) -E| ≤ δ/2 =⇒ Im p θ,G ≤ -θ/C ,

  Exp. n o VII-Quantum transfer operators and quantum scattering VII-15

(3. 9 )

 9 M ji (z) def = Π j M ji (z) : H i → H j , we find that u -i def = -w i + j∈J + (i)

Acknowledgements. The author has been partially supported by the Agence Nationale de la Recherche through the grant ANR-09-JCJC-0099-01. These notes were written while the author was visiting the Institute of Advanced Study in Princeton, supported by the National Science Foundation under agreement No. DMS-0635607.


Stéphane Nonnenmacher

Spectral theory of damped quantum chaotic systems

We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on X and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.

Introduction

Spectrum of the damped wave equation

Given a Riemannian manifold (X, g) and a damping function a ∈ C ∞ (X, R + ), we are interested in the solutions of the damped wave equation (DWE)

(∂ 2 t -∆ + 2a(x)∂ t )v(x, t) = 0, v(x, 0) = v 0 , ∂ t v(x, 0) = v 1 . (1.1)
The natural setting is to take intial data (v 0 , v 1 ) in the space H def = H 1 (X) × L 2 (X). This equation is equivalent with the system

(i∂ t + A) v(t) = 0, A def = 0 I -∆ -2ia , v(0) ∈ H, (1.2)
with the correspondence v(t) = (v(t), i∂v(t)). A generates a strongly continuous semigroup on H, so the solution to (1.1,1.2) reads v(t) = e -itA v(0).

(1.3)

We will always assume that the damping is nontrivial, a ≡ 0. Apart from the constant solution v(t) = (1, 0), all solutions then decay. Physically, this is expressed by the fact that the energy of the waves,

E(v(t)) = 1 2 ∇v(t) 2 + ∂ t v(t) 2 ,
(1.4) will decay to zero when t → ∞ for any v(0) ∈ H. In some sense, the waves are stabilized by the damping.

To analyze this decay, it it natural to try to expand the solution in terms of the spectrum of A. This spectrum is discrete, consisting of countably many complex eigenvalues {τ n } with Re τ n → ±∞. It can be obtained by solving the generalized eigenvalue equation -∆ -τ 2 -2iaτ u = 0.

(1.5)

The following properties are easily shown (a min def = min x∈X a(x), similarly for a max ).

Proposition 1. [START_REF] Lebeau | Equation des ondes amorties, Algebraie and geometric methods in mathematical physics[END_REF]All eigenvalues except τ 0 = 0 satisfy Im τ n < 0.

If Re τ n = 0 then -Im τ n ∈ [a min , a max ].
The spectrum is symmetric w.r.to the imaginary axis.

To each eigenvalue τ n corresponds a quasi-stationary mode u n (x), an eigenstate u n = (u n , τ n u n ) of A reads, and a solution v n (t, x) = e -itτn u n (x) of the DWE. Hence Im τ n represents the quantum decay rate of the mode u n .

If a min > 0, then the energy decays exponentially, uniformly for initial data v(0) ∈ H. Precisely, there exists γ > 0, C > 0 such that

E(v(t)) ≤ C e -2γt E(v(0)), ∀v (0)
∈ H, ∀t ≥ 0.

(1.6)

The Geometric Control Condition

The condition a min > 0 is not necessary to ensure such a uniform exponential decay. Since all eigenvalues except τ 0 = 0 satisfy Im τ n < 0, for any C > 0 the subspace H C ⊂ H spanned by the eigenstates {u n , |Re τ n | ≤ C} is finite dimensional, and for any initial data v(0) ⊂ H C the energy will decay exponentially. Hence, the failure of exponential decay can only come from the behaviour of waves at high frequency.

In this high frequency limit, a natural connection can be made with the classical ray dynamics on X, equivalently the geodesic flow Φ t on the unit cotangent bundle S * X (see § 1.4). Using this connection, Rauch and Taylor [START_REF] Rauch | Decay of solutions to nondissipative hyperbolic systems on compact manifolds[END_REF] showed that the uniform exponential decay (1.6) is equivalent with the Geometric Control Condition (GCC), which states that every geodesic meets the damping region {x ∈ X, a(x) > 0} (due to the compactness of S * X each geodesic does it within some time T 0 > 0). This condition can be expressed in terms of the time averages of the damping, namely the functions

a t (ρ) = 1 t t 0 a • Φ s (ρ) ds, ρ ∈ T * X, t > 0.
(1.7)

GCC is equivalent to the fact that, for t > 0 large enough (say, t > 2T 0 ), the function a t is strictly positive on S * X. Lebeau [Leb93] generalized this result to the case of manifolds with boundaries. He also showed that the optimal decay rate γ is given by min (G, a -), where G = inf {-Im τ n , τ n = 0} is the spectral gap, while

a - def = lim t→∞ min S * X a t (1.8) IX-2 X a(x)>0 Figure 1.1: A damped geodesic.
is the minimal asymptotic damping. We will explain the relevance of a -in §2.2. Koch and Tataru [START_REF] Koch | On the spectrum of hyperbolic semigroups[END_REF] studied the same question in a more general context (case of manifolds with boundaries, and of a damping taking place both in the "bulk" and on the boundary). They showed that the averages a t govern the decay of the semigroup, up to a compact subspace.

Theorem 2. [KoTa94, Thm 2]For each > 0 and each t > 0 there exists a subspace

H ,t ⊂ H of finite codimension such that e -itA H ,t→H ≤ exp -t min S * X a t + . For any subspace H 1 ⊂ H of finite codimension, exp -t min S * X a t ≤ e -itA H 1 →H .
A consequence of this result is the characterization of the Fredholm spectrum of the semigroup. In the present situation of damped waves on a compact X withough boundary, their result states1 that this Fredholm spectrum is given by the annulus {z ∈ C, e -ta + ≤ |z| ≤ e -ta -}.

Notice that it is possible to have a positive gap G > 0 and at the same time a -= 0 (failure of the GCC), see e.g. an example in [START_REF] Renardy | On the linear stability of hyperbolic PDEs and viscoelastic flows[END_REF]. This reflects the fact that the spectrum of the semigroup e -itA is not controlled by the spectrum of generator A, a frequent problem for nonnormal generators like A.

Beyond Geometric control: a few cases of uniform energy decay for regular data

If GCC fails, one can construct initial data v(0) ∈ H such that E(v(t)) decays arbitrarily slowly [START_REF] Lebeau | Equation des ondes amorties, Algebraie and geometric methods in mathematical physics[END_REF]. Yet, it is possible to show that more regular data, say in some Sobolev space H s = H s+1 × H s , s > 0, decay in some uniform way. In the most general situation, Lebeau showed a logarithmic decay of the energy2 [Leb93, Thm 1],

E(v(t)) 1/2 ≤ C s log(2 + t) -s v H s , ∀v = v(0) ∈ H s , ∀t ≥ 0.
The proof uses Carleman estimates, which imply some control of the resolvent (τ

-A) -1 in a strip {Re τ ≥ C, | Im τ | ≤ f (Re τ )} for some function f (r) which
decays exponentially as r → ∞. By interpolation, proving such an estimate for some s 0 > 0 implies a similar estimate for all s > 0.

In specific dynamical situations, one can show resolvent estimates in a larger strip, e.g. for some function f (r) decaying algebraically: one then gets an algebraic decay O(t -γs v H s ) for the energy of regular data. Burq and Hitrik showed that this is the case for damped waves in the (chaotic) stadium billiard, if the damping function vanishes only in some part of the central rectangle, so that the set of undamped trajectories,

K def = ρ ∈ S * X, ∀t ∈ R, a(Φ t (ρ)) = 0 , (1.9)
consists in a collection of (neutrally stable) "bouncing ball" trajectories [START_REF] Burq | Energy decay for damped wave equations on partially rectangular domains[END_REF].

Christianson studied the case where the set K consists in a single hyperbolic closed geodesic: the energy decay is bounded by a stretched exponential O(e -γst 1/2 v H s ) [Chris07, Thm 5'] [START_REF] Christianson | Quantum Monodromy and Non-concentration Near a Closed Semi-hyperbolic Orbit[END_REF]. The hyperbolicity of the geodesic induces a strong dispersion of the waves, responsible for this fast decay.

In §3.3 and 3.4 we will present situations, with (X, g) a manifold of negative curvature, for which the energy of regular data decays exponentially:

E(v(t)) 1/2 ≤ C s e -γst v H s , ∀v ∈ H s , ∀t ≥ 0.
(1.10)

The proof proceeds by controlling the resolvent (τ

-A) -1 in a strip {Re τ > C, | Im τ | ≤ c},
and then uses standard arguments [START_REF] Christianson | Applications of Cutoff Resolvent Estimates to the Wave Equation[END_REF]. Here as well, the resolvent bounds are based on certain hyperbolic dispersion estimates. See the Corollaries 16 and 19 for the precise dynamical conditions, which involve the interplay between the flow and the damping function.

High frequency limit -semiclassical formulation and generalization

As explained above, the decay of the energy is mainly governed by the high frequencies Re τ 1. A semiclassical formulation of the problem was used in [START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF], allowing to use -pseudodifferential techniques. Take an effective Planck's "constant" 0 < 1. In order to study the frequencies Re τ = -1 + O(1), we perform the rescaling

τ = √ 2z with z in the disk D(1/2, C ).
(1.11)

The equation (1.5) becomes

(P ( , z) -z)u = 0, P ( , z) = - 2 ∆ 2 -i √ 2z a = - 2 ∆ 2 -i a + O( 2 ). (1.12)
More generally, we consider operators of the type

P ( , z) = - 2 ∆ 2 + i Op (q z ). (1.13)
where the symbol3 q z ∈ S 0 (T * X) is now a function on the phase space T * X, and depends holomorphically on z ∈ D(1/2, C ), with q z = q 1/2 + O( ) [START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF]. With a slight abuse, will call q = q 1/2 (x, ξ) "the damping function" (a damping effectively occurs only at phase space points where q(x, ξ) < 0, while positive values of q rather correspond to a "gain" or "pumping").

In this framework, we are interested in the semiclassical distribution of the generalized eigenvalues z n ( ) of the operator P ( , z) in the disk D(1/2, C ). Because we will specify this spectrum at the scale , we may for convenience drop the term O( 2 ) in (1.13), and rather study the spectrum of the z-independent operator P ( ) = P 0 ( ) + i Op (q), P 0 ( )

def = - 2 ∆ 2 . (1.14)
The principal symbol of this operator is p 0 (x, ξ) = |ξ| 2 2 , it generates the geodesic flow Φ t = exp(tH p 0 ) on T * X, with unit speed on the energy shell p -1 0 (1/2) = S * X.

Semiclassical spectral distribution of P ( ). General results

The results we present in this section hold for arbitrary Riemannian manifolds (X, g). We skip the proofs.

Weyl law for the real parts

In the semiclassical limit, the number of eigenvalues of the selfadjoint operator

P 0 ( ) = - 2 ∆ 2 in the interval 1 2 ± def = 1 2 -, 1 2
+ is approximately given by Weyl's law. When applying the perturbation i Op (q) = O( ), it seems natural to expect that most eigenvalues of P 0 ( ) in this interval will remain inside a (complex) neighbourhood 1 2 ± +O( ). One can indeed show that the "macroscopic" spectral distribution is given by the same Weyl's law as in the undamped case.

Theorem 3. [START_REF] Markus | Comparison theorems for spectra of linear operators, and spectral asymptotics[END_REF][START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF] For any > 0 small enough (possibly depening on ), the number of eigenvalues of P ( ) in the strip S = 1 2 -≤ Re z ≤ 1 2 + is semiclassically given by

# {Spec P ( ) ∩ S } = (2π ) -d Vol {ρ ∈ T * X, |p 0 (ρ) -1/2| ≤ } + O -d+1 .
Here Vol corresponds to the symplectic volume in T * X. This estimate implies that, for C > 0 large enough,

# {Spec P ( ) ∩ D(1/2, C )} -d+1 . (2.1)
This result is actually not so easy to prove. It uses some subtle analytic Fredholm theory in order to relate the spectrum of P ( ) with that of a simpler "comparison operator".

By Op (q) we refer to a semiclassical quantization on X, with the property that Op (q) is symmetric if q is real (see e.g [EvZw, App. E]). The operators Op (q) form the class Ψ k (X) of pseudodifferential operators (PDO).

Restrictions for the imaginary parts (quantum decay rates)

Eventhough the operator P ( ) was inspired by the wave equation, it is convenient for us to consider it instead as the generator of a Schrödinger equation

i ∂ t v -P ( )v = 0. (2.2)
Any eigenvalue/vector (z n , u n ) of P ( ) is thus associated with a solution v n (x, t) = e -iznt/ u n (x) of this equation, and we will also call Im z n / the quantum decay rate4 associated with u n . First we remark that an eigenmode u n associated to an eigenvalue

z n ∈ D(1/2, C ) is semiclassically microlocalized on p -1 0 (1/2) = S * X.
To state this property we need to introduce energy cutoffs.

Definition 4. An energy cutoff will be function

χ ∈ C ∞ c ([1/2 ± δ] , [0, 1]
) for some δ > 0, and satisfies χ(s) = 1 near s = 1/2. The cutoff χ 0 is said to be embedded in χ 1 , and we note χ 1 χ 0 , iff χ 1 ≡ 1 near supp χ 0 . From the cutoff χ we construct the (pseudodifferential) cutoff operator χ(P 0 ( )) ∈ Ψ -∞ (X).

Proposition 5. Take an energy cutoff χ. Let u = u( ) be an eigenstate of P ( )

with eigenvalue z( ) ∈ D(1/2, C ). Then, (I -χ(P 0 )) u = O( ∞ ) u .
This localization property explains why only the neighbourhood of S * X will be important for us.

A factorization of the Schrödinger propagator

An easy way to realize how the skew-adjoint subprincipal term i Op (q) implies a "damping" is to analyze the propagator for the Schrödinger equation (2.2). The following Proposition adapts a more general result due to Rauch and Taylor [START_REF] Rauch | Decay of solutions to nondissipative hyperbolic systems on compact manifolds[END_REF]. Proposition 6. Assume q ∈ C ∞ c (T * X), and take P ( ) as in (1.14). For any fixed t ∈ R, decompose the Schrödinger propagator V t = e -itP/ into V t = U t B(t) , where U t = e -itP 0 / is the undamped propagator.

(2.3)

The operator B(t) is a PDO in Ψ 0 (X) of principal symbol b(t, ρ) = exp t 0 q • Φ s (ρ) ds = exp {t q t (ρ)} ,
where the time averaged damping q t generalizes (1.7).

The factor b(t, ρ) is the accumulated damping along the orbit {Φ s (ρ), 0 ≤ s ≤ t}. If one starts from a Gaussian wavepacket u 0 microlocalized at a point ρ, then the state V t u 0 is a wavepacket microlocalized at Φ t (ρ), and the above statement shows that the L 2 norms are related through this factor:

u t = u 0 (b(t, ρ) + O( )) .
Applying an energy cutoff allows to extend the factorization (2.3) damping functions q ∈ S 0 (T * X) of noncompact support.

Lemma 7. Consider P ( ) with a damping function q ∈ S 0 (T * X). Take two embedded cutoffs χ 1 χ 0 supported in [1/2 ± δ], and the truncated damping q def = χ 1 (p 0 ) q. Then, for any t ≥ 0 fixed, one has

V t χ 0 (P 0 ) = Ṽ t χ 0 (P 0 ) + O L 2 →L 2 ( ∞ ) , (2.4)
where Ṽ t is the propagator corresponding to P = P 0 + i Op (q). This identity uses the fact that the propagation V t does not modify the energy localization properties.

From propagator factorization to a resolvent bound

We can now easily obtain a first constraint on the quantum decay rates. Applying Proposition 6 to the function q = χ 1 (p 0 )q of Lemma 7, we get for any fixed t ≥ 0:

V t χ 0 (P 0 ) L 2 →L 2 = B(t)χ 0 (P 0 ) L 2 →L 2 + O( ∞ ) = max T * X b(t)χ 0 (p 0 ) + O t ( ) (2.5) ≤ exp t max supp χ 0 •p 0 q t + O t ( ).
On the second line we used the sharp Gårding (in)equality for the operator B(t) ∈ Ψ 0 (X). The maximum max p -1 0 ([1/2±δ]) q t decreases when t → ∞ or when δ → 0, and converges to the asymptotic maximum on S * X,

q + def = lim t→∞ max ρ∈S * X q t (ρ).
(2.6)

One similarly defines an asymptotic minimum q -. We then get the following norm bounds for the propagator.

Proposition 8. Fix > 0. If the energy cutoff χ 0 has a small enough support, and T > 0 is large enough, then for any (fixed) t ≥ T and any small enough > 0, the following bounds holds:

V t χ 0 (P 0 ) L 2 →L 2 ≤ e (q + + )t , V -t χ 0 (P 0 ) L 2 →L 2 ≤ e (-q -+ )t .
(2.7)

From there one easily obtains the following resolvent and spectral bounds.

Theorem 9. [START_REF] Lebeau | Equation des ondes amorties, Algebraie and geometric methods in mathematical physics[END_REF][START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF]Take any > 0. Then, there exists , C > 0 such that for < , the following resolvent estimate holds:

∀z ∈ D(1/2, C ) \ {z, Im z/ ∈ [q --, q + + ]} , (P ( ) -z) -1 ≤ C . (2.8) As a consequence, for < all eigenvalues z n ∈ D(1/2, C ) of P ( ) satisfy Im z n ∈ [q --, q + + ] .
In the case of the damped wave equation (q(x, ξ) = -a(x)) and in a situation of geometric control (a -= -q + > 0), the resolvent estimate (2.8) can be used to show the uniform exponential energy decay (1.6) [START_REF] Hitrik | Eigenfrequencies and expansions for damped wave equations[END_REF].

Questions on the spectral distribution

So far we showed that the quantum decay rates are bounded by the asymptotic extrema of the time-averaged damping. The following questions were raised in [Sjo00, AschLeb, Anan10] concerning their semiclassical distribution.

1. What are the possible accumulation points of the quantum decay rates when → 0? In particular, are there sequences of decay rates (Im z( )/ ) →0 converging to the extremal values q ± ? 2. Do the quantum decay rates admit an asymptotic distribution when → 0?

Namely, for a given interval I ⊂ [q -, q + ] and 1 ( ) , does the ratio

# z ∈ Spec P ( ), | Re z -1/2| ≤ , Im z ∈ I # {z ∈ Spec P ( ), | Re z -1/2|
≤ } have a limit when → 0? Is this limit distribution related with the value distributions of the averages q t ?

Spectral estimates on Anosov manifolds

The above questions are open in general. In order to get more precise informations on the spectrum of P ( ), one needs to make specific assumptions on the geodesic flow on X. For instance, the case of a completely integrable dynamics has been considered by Hitrik-Sjöstrand in a sequence of papers (see e.g. [START_REF] Hitrik | Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2[END_REF]and reference therein). The case of nearly-integrable dynamics including KAM invariant tori has been studied by Hitrik-Sjöstrand-Vũ Ngo . c [START_REF] Hitrik | Diophantine tori and spectral asymptotics for non-selfadjoint operators[END_REF]. In these cases, one can transform the Hamiltonian flow into a normal form near each invariant torus, which leads to a precise description of the spectrum "generated" by this torus. A Weyl law for the quantum decay rates was recently obtained in [START_REF] Hitrik | Diophantine tori and Weyl laws for nonselfadjoint operators in dimension two[END_REF] (for skew-adjoint perturbations iθ( ) Op (q), with θ( ) ). On the other hand, Asch-Lebeau [AschLeb] addressed Question 1 for the case of the 2-dimensional standard sphere. They show that, if the damping function q has real analytic real and imaginary parts, there is generically a spectral gap γ > 0: for small enough, all eigenvalues z n ∈ D(1/2, C ) have quantum decay rates

Im z n / ∈ [q -+ γ, q + -γ] .
The proof proceeds by applying a complex canonical transformation, such that the real part of the pulled-back damping function takes values in the above interval. In this case, the range of the quantum decay rates is strictly smaller than the range of classical decay rates.

We will see below that such a spectral gap may also occur in the case of Anosov geodesic flows, thanks to a different mechanism, namely a hyperbolic dispersion property due to the instability of the classical flow (see Thms 15 and 18 below).

In the next section we recall the definition and properties of Anosov manifolds.

A short reminder on Anosov manifolds [KatHas95]

At the "antipode" of the completely integrable case, one finds the "strongly chaotic" flows, namely the Anosov (or uniformly hyperbolic) flows. Uniform hyperbolicity IX-8 means that at each point ρ ∈ S * X there exists a splitting of the tangent space,

E + E t t + ( ) J
T ρ S * X = RH p (ρ) ⊕ E + (ρ) ⊕ E -(ρ),
where H p (ρ) is the Hamiltonian vector field, E ± (ρ) are the unstable and stable subspaces at the point ρ; they both have dimension d -1, and are uniformly transverse to e.o.. The families {E ± (ρ), ρ ∈ S * X} form the unstable/stable distributions, they are invariant w.r.to the flow, Hölder continuous, and are characterized by the following property: there exists C, λ > 0 such that

∀ρ ∈ S * X, ∀v ∈ E ∓ (ρ), ∀t > 0, dΦ ±t ρ v ≤ C e -λt v . (3.1)
As was shown by Hadamard, such a geodesic flow is obtained if the manifold (X, g) has a negative sectional curvature. The long time properties of such a strongly chaotic flow are well understood. In particular, Hopf proved that this geodesic flow is ergodic w.r.to the Liouville measure on S * X.

Definition 10. Call µ L the normalized Liouville measure on S * X. Then, the geodesic flow Φ t on S * X is ergodic (w.r.to µ L ) iff, for any continuous function q on S * X, for µ L -almost every ρ ∈ S * X, lim

t→∞ q t (ρ) = q def = µ L (q).
The following quantities provide quantitative measures of the hyperbolicity:

the maximal expansion rate

λ max def = lim t→∞ sup ρ∈S * X 1 t log dΦ t ρ , (3.2) the unstable Jacobian J + (ρ, t) = det dΦ t E + (ρ) , t > 0,
and its infinitesimal version ϕ + (ρ)

def = d dt log J + (ρ, t) t=0 , (3.3) the minimal expansion rate ν min def = 1 d -1 lim t→∞ min ρ∈S * X 1 t log J + (ρ, t) . (3.4)
The Jacobian depends on the choice of norms on the spaces T ρ S * X. However, its long time asymptotics is independent of this choice. In variable curvature these rates are related by λ max ≥ ν min ≥ λ > 0. In particular, if the positive Lyapunov exponents are not all equal, one has ν min < λ max .

A particular class of Anosov manifolds consists in quotients of the d-dimensional hyperbolic space H d by co-compact subgroups of its isometry group. These manifolds IX-9 have a constant curvature5 -Λ 2 . Using the natural norms on T ρ S * X, one then has

J + (ρ, t) = e t(d-1)Λ , ϕ + (ρ) = Λ(d -1), λ max = ν min = Λ.
In this case the contraction/expansion are both homogeneous (independent of the point ρ) and isotropic (independent in the direction).

Let us finally notice that the study of the operator P ( ) on an Anosov manifold belongs to the field of "quantum chaos". The methods we will use below occur in various problems of this field, e.g. the study of eigenstates of the Laplacian P 0 ( ) on such manifolds [START_REF] Zelditch | Recent developments in mathematical quantum chaos[END_REF].

Fractal Weyl upper bounds for the quantum decay rates

In this section we address Question 2, that is the asymptotic distribution of the quantum decay rates, for the case of Anosov manifolds.

Typical quantum decay rates for ergodic flows

A basic property of the geodesic flow on any Anosov manifold is ergodicity. Sjöstrand has studied the asymptotic disribution of the quantum decay rates for any ergodic geodesic flow [START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF], and obtained the following result.

Theorem 11. [START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF] Assume the geodesic flow on S * X is ergodic w.r.to the Liouville measure. Then, For any C > 0, > 0, one has as → 0,

# z ∈ Spec P ( ) ∩ D(1/2, C ), Im z ∈ [q -, q + ] = o( -d+1 ).
Comparing this bound with the Weyl law (2.1) shows that, in the semiclassical limit, almost all the quantum decay rates are close to the phase space average q. The latter could be called the typical value for quantum decay rates, while values at finite distance from q are atypical. The asymptotic distribution is simply the delta measure δ q.

Remark 12. As was noticed in [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF], this concentration result could be seen as a "nonperturbative version" of quantum ergodicity. The latter property states that almost all the eigenstates u 0 n of P 0 ( ) with eigenvalues z 0 n ∈ D(1/2, C ) satisfy u 0 n , Op (q)u 0 n = q + o →0 (1). A naive perturbation theory argument would predict that, when switching on i Op (q), the eigenvalues z 0 n move to z n = z 0 n + i u 0 n , Op (q)u 0 n +O( 2 ), so that the quantum decay rates Im z n / = u 0 n , Op (q)u 0 n + O( ), the RHS being equal to q + o(1) for almost all n. Of course, this argument does not apply because the perturbation i Op (q) is much stronger than the mean spacing between successive eigenvalues. To prove the above theorem one relates the counting of quantum decay rates with the value distribution of the quantum averages q t on S * X, that is the volumes

µ L {ρ ∈ S * X, q t (ρ) ≥ α} , α ∈ R, t > 0.
(3.5) Indeed, the main intermediate result in the proof is the bound

# z ∈ Spec P ( ) ∩ D(1/2, C ), Im z ≥ α ≤ C t,δ -d+1 µ L {ρ ∈ S * X, q t (ρ) ≥ α -δ} ,
(3.6) which holds for any fixed t > 0 and δ > 0 (this bound holds independently of the ergodicity assumption). Ergodicity then implies that the value distribution of q t converges to δ q when t → ∞; in particular, for any α > q the volume (3.5) decays to zero when t → ∞. We have then obtained the bound o( -d+1 ) for the quantum decay rates ≥ α > q. The case of values ≤ α < q is treated analogously.

Large deviation estimates for Anosov flows

For an Anosov flow, one has more precise estimates for the volumes (3.5), which can then induce sharper bounds on the number of the atypical quantum decay rates. These volume estimate take the form of large deviation estimates. Let us introduce some notations. We call M the set of Φ t -invariant probability measures on S * X. For each such measure µ, we denote by h KS (µ) its Kolmogorov-Sinai entropy: this is a nonnegative number, which measures the complexity of a µ-typical trajectory [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]. Then, we define the rate function H : R → R as follows:

∀s ∈ R, H(s) def = sup h KS (µ) -µ(ϕ + ), µ ∈ M, µ(q) = s , ( 3.7)
where we recall that ϕ + is the infinitesimal unstable Jacobian (3.3). We are now ready to state our large deviation result.

Theorem 13. [START_REF] Kifer | Large deviations in dynamical systems and stochastic processes[END_REF]Assume the geodesic flow on S * X is Anosov. Then, for any closed interval I ⊂ R and for any q ∈ C ∞ (S * X), the time averages q t satisfy lim sup

t→∞ 1 t log µ L {ρ ∈ S * X, q t (ρ) ∈ I} ≤ sup s∈I H(s). (3.8)
The rate function H is continuous on [q -, q + ], smooth and strictly concave on (q -, q + ), negative except at the point H(q) = 0, satisfies H(s) ≥ -sup µ∈M µ(ϕ + ) for s ∈ [q -, q + ], and is equal to -∞ outside [q -, q + ]. As a consequence of (3.8), for any α ≥ q and > 0 arbitrary small, there exists T α, > 0 such that ∀t ≥ T α, , µ L {ρ ∈ S * X, q t (ρ) ≥ α} ≤ e t( H(α)+ ) .

(3.9)

Due to the negativity of H(α) for α = q, we see that the probability of q t taking atypical values decays exponentially when t → ∞.

Fractal Weyl upper bounds on Anosov manifolds

Using these large deviation estimates, Anantharaman [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF] improved Thm 11 by letting the averaging time grow with in a controlled way. The optimal time is the Ehrenfest time

T = T Ehr = (1 -2 ) log 1/ λ max , (3.10)
where λ max is the largest expansion rate (3.2) and > 0 arbitrary small. What is the signification of this time? For any f ∈ S -∞ (T * X) supported in an -neighbourhood of S * X, the classically evolved observable f •Φ t remains in the "good" symbol class6 S -∞ 1/2-(T * X) uniformly for times |t| ≤ T Ehr /2. In turn, the symmetric averages

f t,sym = 1 t t/2 -t/2 f • Φ s ds (3.11) belong to S -∞ 1/2-(T * X)
for |t| ≤ T Ehr , and this time is sharp. Let us insist on the fact that controlling the time evolution up to times t log 1/ is a crucial ingredient in order to obtain refined spectral estimates on Anosov manifolds. This will also be the case in Sections 3.3 and 3.4 when proving hyperbolic dispersion estimates and spectral gaps.

The pseudodifferential calculus on Ψ 2 1/2-(X) allows to extend the validity of the bound (3.6) up to the Ehrenfest time. The large deviation estimate (3.9) then leads to the following fractal Weyl upper bound for the number of atypical quantum decay rates.

Theorem 14. [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF]Assume the geodesic flow on S * X is Anosov. Then, for any α ≥ q and > 0, one has for small enough

# z ∈ Spec P ( ) ∩ D(1/2, C ), Im z ≥ α ≤ -H(α) λmax +(1-d)-, (3.12)
where H is the rate function (3.7). A similar expression holds when counting quantum decay rates smaller than α ≤ q.

Since -H(α) > 0 for all α > q, this upper bound improves the bound o( -d+1 ) of Thm. 11 by a fractional power of (the name "fractal Weyl upper bound" takes its origin in the counting of resonances of chaotic scattering systems [START_REF] Sjöstrand | Fractal upper bounds on the density of semiclassical resonances[END_REF]). For any α ∈ [q, q + ] one has H(α) ≥ -sup µ µ(ϕ + ) ≥ -λ max (d -1), so the exponent of in (3.12) is negative, allowing the presence of (many) quantum decay rates arbitrary close to q + . These fractal upper bounds are not expected to be sharp in general [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF]. In the next sections we will be able, under certain conditions, to exclude the possibility of quantum decay rates near q + (or near q -).

A pressure criterium for a spectral gap

The result of Thm 9, namely the fact that all quantum decay rates belong to the interval [q --o(1), q + + o(1)], was obtained by studying the norm of the propagator V t = e -itP/ for times |t| 1 independent of . Indeed, the sequence of equalities (2.5) directly leads to the upper bound Im z n / ≤ q + + .

(3.13)

The crucial semiclassical ingredient in (2.5) is the second equality connecting the L 2 norm of the PDO B(t)χ(P 0 ) with the supremum of its (principal) symbol. Hence, one cannot hope to improve the bound (3.13) as long as B(t)χ(P 0 ) remains a "good" PDO, which is the case if the time t ≤ T Ehr /2. To improve on (3.13) we will investigate the propagator V t for "large logarithmic times", namely t ∼ K log 1/ , with K > 0 large (but independent of ).

(3.14)

For such times, the function b(t)χ(p 0 ) oscillates on scales much smaller than , so it cannot belong to any decent symbol class. As a result, the norm of B(t)χ(P 0 ) is a priori unrelated with that function. However, using the hyperbolicity of the flow one can prove an upper bound for the propagator V t in terms of a certain topological pressure depending on the damping and the hyperbolicity, and then obtain a constraint on the quantum decay rates. Before stating our bound, let us first introduce the notion of topological pressure associated with the flow Φ t on S * X. For any observable f ∈ C(S * X), the pressure P(f ) = P(f, Φ t S * X ) can be defined by

P(f ) def = sup {h KS (µ) + µ(f ), µ ∈ M} . (3.15)
Notice that the definition (3.7) of the rate function H(s) is very similar. The pressure is also given by the growth rate of weighted sums over long closed geodesics7 :

P(f ) = lim T →∞ 1 T log T ≤|γ|≤T +1 e γ f .
The pressure appearing in the next theorem is associated with the function f = q -ϕ + /2, where ϕ + is the infinitesimal unstable Jacobian (3.3), it thus mixes the damping and hyperbolicity.

Theorem 15. [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF]Let X be an Anosov manifold, and q ∈ S 0 (T * X) a damping function.

Assume the following (purely classical) inequality holds:

P(q -ϕ + /2) < q + . (3.16)
Then, for any > 0, there exists N > 0 such that, for small enough, the following resolvent bounds:

∀z ∈ D(1/2, C ) ∩ Im z/ ≥ P(q -ϕ + /2) + , (P ( ) -z) -1 ≤ -N .
(3.17) As a consequence, all quantum decay rates for z n ( ) ∈ D(1/2, C ) satisfy Im z n ( ) ≤ P(q -ϕ + /2) + .

(3.18)

in particular we have a spectral gap.

In the next subsection we present situations for which the inequality (3.16) is satisfied. The proof of this theorem was inspired by a similar result in the case of resonances in chaotic scattering [START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF][START_REF] Nonnenmacher | Semiclassical Resolvent Estimates in Chaotic Scattering[END_REF]. We give some hints of the proof in §3.3.2.

Let us now apply this bound to the damped wave equation, that is take q(x, ξ) = -a(x).

hq hq 1/2 1/2+ 1/2 P h hq + Figure 3
.2: Spectral gap in case the pressure P(q -ϕ + /2) < q + . Corollary 16. [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF] Consider the DWE on X with nontrivial damping a ≥ 0. Assume that the GCC fails (a -= 0), but that the condition P(-a -ϕ + /2) < 0 holds true.

Then the DWE has a spectral gap, and the energy decays exponentially for regular initial data, as in (1.10).

Conditions for a spectral gap

The inequality (3.16) can hold only if the damping function q varies "sufficiently" around its average value q. Indeed, one always has P(-ϕ + /2) > 0 for an Anosov flow, so by adding a constant dampint q ≡ q one gets P(q -ϕ + /2) = q + P(-ϕ + /2) > q = q + . The pressure depends continuously of q (in the C 0 topology), so if q = q + δq with δq C 0 small, the inequality (3.16) will not be satisfied.

Let us consider the case of the DWE as in Corollary 16. The failure of the GCC is equivalent to the fact that the set of undamped trajectories K (1.9) is nonempty. This set is flow-invariant and closed, so one can define the pressure P(-ϕ + /2, Φ t K ) associated with the flow Φ t restricted on K. It was shown in [START_REF] Schenck | Exponential stabilization without geometric control[END_REF] that if we multiply the damping a(x) by a large constant C > 0 (this does not modify the set K), then

lim C→∞ P(-Ca -ϕ + /2) = P(-ϕ + /2, Φ t K ).
If the set K is thin enough, the pressure P(-ϕ + /2, Φ t K ) is negative (for instance, if K consists in a single closed geodesic, the pressure is equal to the average of -ϕ + /2 along the geodesic, which is negative). In that case, the above limit shows that the pressure P(-Ca -ϕ + /2) is also negative if C 1.

Proof of the pressure bound: decomposing into "symbolic paths"

In this section we explain the strategy of proof of Thm 15. As mentioned before, we want to bound the norm of the propagator for large logarithmic times. We will prove the following Lemma 17. Choose > 0. Then, for an energy cutoff χ 0 of small enough support, and a large enough constant K > 0, for < one has the bound

V t χ 0 (P 0 ) ≤ e t(P(q-ϕ + /2)+ ) , t ∼ K log 1/ . (3.19)
From there, obtaining the resolvent estimate (3.17) is relatively straightforward.

Proof. Our aim is to bound the norm V t u 0 for a normalized state u 0 microlocalized in p -1 0 ([1/2 ± δ]), and a time t ∼ K log 1/ . We may assume that u 0 is microlocalized in an open subset W of diameter ≤ δ. We will first decompose u 0 in an adapted basis of local "momentum" modes e η , η ∈ R d , |η| ≤ δ . Here the parameter η is the "momentum coordinate" in a Darboux coordinate frame {(y, η), |y| ≤ δ, |η| ≤ δ} in W , chosen such that η 1 = p 0 -1/2 represents the energy, y 1 the time, and the "momentum" Lagrangian leaves Λ η 0 = {(y, η 0 )} are close the unstable foliation. The momentum states e η are Lagrangian states supported by the leaves Λ η , of norms O(1). The decomposition of u 0 in this "basis" reads

u 0 = |η|≤δ û0 (η) e η dη (2π ) d/2 + O( ∞ ), (3.20)
where the factor (2π ) -d/2 ensures that û0 L 1 = O(1). The reason for this decomposition is that we are able to precisely control the evolution of each individual state e η , up to times t ∼ K log 1/ . The state V t e η is a Lagrangian state supported on the Lagrangian leaf Φ t (Λ η ). Due to the hyperbolicity of the flow, when t grows the leaf Φ t (Λ η ) expands exponentially fast along the unstable directions, and converges to the unstable foliation. To analyze the state V t e η , we can view as the combination of many local Lagrangian states associated with bounded pieces of Φ t (Λ η ). A convenient bookkeeping consists in using a partition of unity in p -1 0 ([1/2 ± δ]). One first covers this region by a finite family of open subsets {W j , j = 1, . . . , J} of diameters ≤ δ, and then constructs a smooth partition of unity {π j ∈ C ∞ c (W j , [0, 1]), j = 1, . . . , J} adapted to this cover: J j=1 π j = 1 near p -1 0 ([1/2 ± δ]). This partition is quantized into Π j def = Op (π j ), j = 1, . . . , J , such that, for any energy cutoff χ any supported in [1/2 ± δ],

J j=1 Π j χ(P 0 ) = χ(P 0 ) + O L 2 →L 2 ( ∞ ). (3.21)
We use this quantum partition to split the propagator V n into "symbolic paths":

V n e η = J α 1 ,...,αn=1 V α 1 •••αn e η + O( ∞ ), V α 1 •••αn = Π αn V Π α n-1 • • • V Π α 1 V. (3.22)
For each symbolic sequence α = α 1 • • • α n , the state V α e η is a Lagrangian state supported on the Lagrangian leaf Λ α η obtained from Λ η by a sequence of n evolutions by Φ 1 and truncations on W α k . The full evolved leaf is given by Φ

t (Λ η ) = |α|=n Λ α η . If Λ α η is empty V α e η = O( ∞ )
. Otherwise (α "admissible"), Λ α η is a leaf of diameter ≤ δ, close to the unstable foliation. One can compute the amplitude of the Lagrangian state V α e η to any order in . It is governed by both the accumulated damping and accumulated instability along the "path" α, leading to the norm estimate

V α e η ≤ C b(α) J + (α) -1/2 , where (3.23) IX-15 b(α) = n k=1 b(α k ), b(j) = max ρ∈W j b(1, ρ),
and similarly for J + -1/2 . (3.24)

Applying the triangular inequality to (3.22), we obtain a bound on the norm of V n e η which can be related with the topological pressure:

V n e η ≤ C J α admis. b(α) J + (α) -1/2 ≤ exp n P(q -ϕ + /2) + O(δ) . (3.25)
Inserting this bound in the expansion (3.20) we get the same bound for V n u 0 with an extra factor -d/2 ; however, this factor is smaller than e nδ if we take n ∼ K log 1/ and K large enough, in which case the bound (3.25) also applies to V n u 0 . This ends the proof of the norm bound (3.19).

Had we inserted (3.23) in the expansion (3.20) before summing over α, we would have obtained the following hyperbolic dispersion estimate:

V α ≤ min b(α), C -d/2 b(α) J + (α) -1/2 , |α| ≤ K log 1/ .
This type of estimate was first proved in the case of the undamped operator P 0 ( ) on an Anosov manifold [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF][START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]. The adaptation to the case of the damped wave equation was written in [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF]. A similar estimate was also shown in [START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF] in the case of chaotic scattering.

A "thickness" condition for a spectral gap

In this section we present a spectral gap condition expressed only in terms of the set of undamped trajectories (1.9), but independently of the strength of the variations of the damping (as opposed to the "pressure condition" (3.16)). A proof of this result will appear in a forthcoming publication.

To state our result in the context of a damping function q(x, ξ), we introduce the set of "least damped trajectories" K = {supp µ, µ ∈ M, µ(q) = q + }, (3.26) which generalizes 8 the set (1.9). Our condition for a spectral gap depends on a topological pressure associated with the flow Φ t K . We use quantities defined in §3.1. Theorem 18. Let X be an Anosov manifold, and q ∈ S 0 (T * X) a damping function. Assume the following condition holds:

P(-ϕ + , Φ t K ) < (d -1) ν min 2 -λ max . (3.27)
Then, there exists γ > 0 ("the gap"), C > 0 and N > 0 such that, for > 0 small enough, the following resolvent bound holds:

∀z ∈ D(1/2, C ) ∩ {Im z/ ≥ q + -γ} , (P ( ) -z) -1 ≤ C -N . (3.28)
As a consequence, all eigenvalues of P ( ) in D(1/2, C ) satisfy Im z n / ≤ q + -γ, so we have a spectral gap.

8 The two definitions are not strictly equal [START_REF] Schenck | Exponential stabilization without geometric control[END_REF], but this subtlety will be irrelevant here.

In the framework of the damped wave equation with (nontrivial) damping function a(x) ≥ 0, this gives the following Corollary 19. Assume that the set of undamped trajectories (1.9) is such that the condition (3.27) holds. Then, there exists γ > 0, C > 0 and N > 0 such that

(τ -A) -1 L 2 →L 2 ≤ C τ N , ∀τ ∈ {| Re τ | ≥ C, Im τ ≥ -γ} .
As a consequence, the energy decays exponentially for regular initial data, as in (1.10).

On a manifold of constant negative curvature -Λ 2 , the condition (3.27) takes the form

h top (Φ t K ) < (d -1)Λ/2, (3.29)
where h top is the topological entropy of the flow on K. It is usually interpreted as a measure of the "complexity" of the flow on K; we rather see it as a measure of the "thickness" of K. This bound is reminiscent of the lower bound obtained by Anantharaman when describing the localization of eigenstates of the Laplacian on Anosov manifolds [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF]. In constant curvature, she shows the the support of any semiclassical measure associated with a sequence of eigenstates must have a topological entropy greater or equal to (d -1)Λ/2, thus forbidding the eigenstates from being too localized. Following the thread of Remark 12, the spectral gap condition (3.29) could be seen as a "nonperturbative analogue" of this delocalization result. How about the case of variable curvature? For certain Anosov manifolds of dimension d ≥ 3, the condition (3.27) can never be satisfied. Indeed, the variational formula (3.15) shows that, for any closed invariant set K ⊂ S * X, the pressure

P(-ϕ + , Φ t K ) ≥ -sup µ∈M µ(ϕ + ) ≥ -(d -1)λ max .
If the instability along E + is very anistropic (meaning that the largest and smallest positive Lyapunov exponents are very different), then (d -1)λ max -sup µ∈M µ(ϕ + ) can be strictly larger than (d-1)ν min 2 , in which case

P(-ϕ + , Φ t K ) > (d -1)
ν min 2 -λ max for any closed invariant set K .

This remark hints at the fact that the condition (3.27) is probably not sharp (even in 2 dimensions) in variable curvature. What could be a sharp criterion for a gap?

Rivière [START_REF] Rivière | Delocalization of slowly damped eigenmodes on Anosov manifolds[END_REF] has recently improved Anantharaman's lower bound on the semiclassical measures for eigenstates of the Laplacian, and has also extended her analysis to the case of the damped operator P ( , z) of (1.12). In particular, he shows that if the undamped set K is "thin enough" so that the pressure P(-ϕ + /2, Φ t K ) is negative, then we have a "logarithmic gap": there exists C 0 > 0 such that, for small enough, all generalized eigenvalues z n We recall that Π + is the projector associated with the region Ω +,α . Let us indicate that a similar type of dispersion estimate was used by S. Brooks, when studying the delocalization of the eigenstates of quantized hyperbolic automorphisms of the 2-dimensional torus (the so-called "quantum cat maps") [START_REF] Brooks | On the entropy of quantum limits for 2-dimensional cat maps[END_REF].

To estimate this norm, it will be useful to replace this projector by a smoothed microlocal projector obtained by quantizing a symbol χ + = χ +,α ∈ S -∞ 1/2-(T * X), such that Op (χ + ) "dominates" Π + : Π + χ 1 (P 0 ) = Op (χ + ) Π + χ 1 (P 0 ) + O( ∞ ).

The norm (3.39) can then be bounded by:

U T ++ χ 1 (P 0 ) ≤ sup u 1 = u 2 =1
Op (χ + )u 2 , U T W (T ) Op (χ + )u 1 + O( ∞ ) .

The symbol χ + can be chosen supported inside a set of the form Ω +,α-C ∩p -1 0 ([1/2± ]). This set is quite irregular, and the main information we have on it is an estimate on its volume (using large deviation estimates like (3.31)). It is then convenient to use an anti-Wick quantization scheme for Op (χ + ), that is use a family of coherent states (Gaussian wavepackets) {e ρ , ρ ∈ T * X}10 , to define Each coherent state e ρ is normalized, and is microlocalized in a "microscopic ellipse" around ρ. This ellipse is chosen to be "adapted" to the flow. Let us describe it using the local Darboux coordinates {(y, η)} as in §3.3.2. The ellipse is "short" in the energy direction, ∆η 1 ∼ 1-/2 , and "long" in the time direction, ∆y 1 ∼ /2 . The spread along the transverse directions is chosen isotropic: ∆y j = ∆η j ∼ 1/2 , j = 2, . . . , d.

The scalar product Op + (χ + )u 2 , U T W s (T ) Op + (χ + )u 1 can now be expressed as a double phase space integral

p -1 0 ([1/2± ])
dρ 1 dρ 2 (2π ) 2d u 2 , e ρ 2 e ρ 1 , u 1 χ + (ρ 2 ) χ + (ρ 1 ) U -T /2 e ρ 2 , U T /2 W (T ) e ρ 1 .

The state W (T )e ρ 1 is approximately identical to e ρ 1 . We can then precisely describe the (undamped) evolutions of the coherent states e ρ i up to the times ±T /2. The state U T /2 e ρ 1 (resp. U -T /2 e ρ 2 ) is a (microscopic) Lagrangian state along a leaf of the weak unstable manifold of volume ∼ (d-1+ )/2 J + T /2 (ρ 1 ) centered at Φ T /2 (ρ 1 ), (resp. a leaf of the weak stable manifold of volume ∼ (d-1+ )/2 J + T /2 (Φ -T /2 (ρ 2 )) centered at Φ -T /2 (ρ 2 )). Using the sharp energy localization of these states and the fact that

( )Figure 3 . 1 :

 31 Figure 3.1: Structure of the Anosov flow near an orbit Φ t (ρ)

  ( ) ∈ D(1/2, C ) of P ( , z) satisfy Im z n ( ) ≤ -C 0 | log | .Comparing Rivière's result with Thm 18 and the discussion in §3.3.1, it seems natural to expect the following3.4.2. Proof of the norm bound for U T++In this last subsection we sketch the proof of Proposition 21, that is obtain an upper bound forU T ++ χ 1 (P 0 ) L 2 →L 2 = Π + U T W (T )Π + χ 1 (P 0 ) , T = T Ehr .(3.39)

  Op (χ + ) def = dρ (2π ) d χ + (ρ) e ρ , • e ρ .

under the condition that, for any T > 0, the set of T -periodic geodesics has measure zero.

Lebeau's paper contains a log log(10 + t) factor in the numerator, which can be removed using the abstract result of[START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonances au voisinage du réel[END_REF] Thm

3]. I thank L.Robbiano for pointing out this improvement.

For any k ∈ R, the symbol class S k (T * X) denotes the functions a(x, ξ; ) satisfying the following bounds: for any multi-indices α, β ∈ N d ,∂ α x ∂ β ξ a(x, ξ; ) ≤ C α,β (1 + |ξ|) k-|β|uniformly for ∈ (0, 1),

From (1.11) we have Im z n / = Im τ n +O( ), so this denomination is (approximately) consistent with the one of §1.1.

Usually one normalizes the metrics on H d so that the curvature is -1. We prefer to keep track of the curvature in our notations.

For any k ∈ R, δ ∈ [0, 1/2), the symbol class S k δ (T * X) consists of functions g(x, ξ; ) which may become more and more singular when → 0, but in a controlled way:∀α, β ∈ N d , ∀ρ ∈ T * X, ∂ α x ∂ β ξ g(ρ) ≤ C α -δ(|α|+|β|) (1 + |ξ|) k-|β| .In this class one can still use pseudodifferential calculus, and the expansions in powers of make sense[EvZw, Sec. 4.3].

The sum over long closed geodesics is analogous to a partition function in statistical mechanics, with the time T corresponding to the volume of the system. This explains why the growth rate is called a "pressure".

Because χ + is supported inside p -1 0 ([1/2 -δ]),it is sufficient to construct a family of coherent states with ρ in this (compact) region.

This work has been partially supported by the grant ANR-09-JCJC-0099-01 of the Agence Nationale de la Recherche.

Conjecture 20. If the set K of least damped trajectories satisfies P(-ϕ + /2, Φ t K ) < 0, (3.30)

then there is a spectral gap in the spectrum of P ( ), in the sense of Thm 18.

In constant curvature the condition (3.30) is equivalent with (3.29). In variable curvature, it is weaker than (3.27), and in any metric of negative curvature, there exist invariant sets satifying this condition (e.g. any closed geodesic).

A proof of the above conjecture should make use of local expansion rates, instead of the globally defined rates λ max and ν min , like in Rivière's work on 2-dimensional Anosov manifolds [START_REF] Rivière | Entropy of semiclassical measures in dimension 2[END_REF].

In the next subsection we sketch the proof of Thm 18.

Sketch of proof for Thm 18

As was the case for Thm 15, the proof proceeds by bounding the norm of the propagator V t χ 0 (P 0 ) for some logarithmic time (now the double of the Ehrenfest time, see (3.38)). In the course of the proof we will also need to control the evolution of a certain type of "microscopic" Lagrangian states. The novelty compared to the proof of Thm 15 is that we will now distinguish between two types of phase space points, the "weakly damped" vs. "strongly damped" points. Without loss of generality 9 , we may assume that the damping function q(x, ξ) is compactly supported inside p -1 0 ([1/2 ± ]). From now on we denote the Ehrenfest time by T def = T Ehr . We fix some level α ∈ (q, q + ), and consider the set of "weakly damped points" Ω +,α def = {ρ ∈ T * X, q T,sym (q) ≥ α} (remember that q t,sym is the symmetric time average (3.11)). The large deviation estimate (3.9) provides a bound on the volume of Ω +,α ∩ S * X:

From the definition (3.7), H(q + ) is equal to the pressure P(-ϕ + , Φ t K ) appearing in (3.27). If we assume (3.27), then, by continuity of H(s) on [q -, q + ], we may choose α ∈ (q, q + ) large enough such that

We will now associate a quantum projector to the set Ω +,α . We first symmetrize the factorization (2.3), by writing

where B s (t) has principal symbol b s (t) = e t q t,sym .

(3.33) Although this symbol is positive, the operator B s (t) may not be selfadjoint; we then take its polar decomposition

is unitary.

9 Lemma 7 can be extended to logarithmic times.

IX-18

For the same reasons as in §3.2.3 (and due to the support assumption on q), the operators B s (T ), A(T ), W (T ) remain "good" PDO up to the Ehrenfest time: B s (T ), A(T ) ∈ -C Ψ 0 1/2-(X), and W (T ) ∈ Ψ 0 1/2-(X). The operators A(T ) and B s (T ) have the same leading symbol b s (T ) = e T q T,sym . Hence, to the set Ω +,α = b s (T, ρ) ≥ e αT we associate the spectral projector

and call Π -= I -Π + .

We use these projectors to decompose the propagator at time 2T : for some energy cutoff χ 0 we write

The RHS splits into four terms. Three terms contain at least one factor Π -: for them we use the obvious bound

The remaining term contains the factor

The norm of this operator will be bounded by the following hyperbolic dispersion estimate, the proof of which is sketched in §3.4.2.

Proposition 21. Assume that for some α ∈ (q, q + ) the condition (3.32) holds. Fix some > 0. Then, if the energy cutoff χ 1 has small enough support, for > 0 small enough one has

(3.36)

Since propagation does not modify the energy localization, if choose in (3.34) a cutoff χ 0 ≺ χ 1 , we then have

Inserting this identity and the bounds (2.7), (3.35) and (3.36) in the identity (3.34), we get V 2T χ 0 (P 0 ) ≤ e 2T (q + +O( )) e -T β(α) + e T (α-q + ) + e 2T (α-q + ) .

(3.37)

We may optimize this upper bound over the level α: the optimal value of the exponent is reached for the (unique) parameter α c ∈ (q, q + ) solving

For any γ > 0 satisfying q + -γ > q + +αc 2 , we get (for > 0 small enough) the following norm bound for the propagator: V 2T χ 0 (P 0 ) ≤ e 2T (q + -γ) .

(3.38)

The proof of the resolvent estimate (3.28) is then rather straightforward.

stable and unstable manifolds intersect transversely to each other, one gets the following bound: U -T /2 e ρ 2 , U T /2 W (T ) e ρ 1 ≤ θ η 1 (ρ 1 )-η 1 (ρ 2 ) 1-J + T /2 (ρ 1 )J + T /2 (Φ -T /2 (ρ 2 ))

+ O( ∞ ), for some θ ∈ C ∞ c ([-1, 1]). The denominator is bounded below by e ν min (d-1)T /2 . Inserting this bound in the above double integral and using large deviation estimates similar with (3.31) for the support of χ + , one finally gets (after some manipulations) the bound (3.36).