Ivana Alexandrova
email: alexandrovai@ecu.edu

Jean-Franc ¸ois

Thierry Ramond
email: thierry.ramond@math.u-psud.fr

Jean-Franc ¸ois Bony
email: bony@math.u-bordeaux1.fr

SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM OF THE POTENTIAL

Keywords: April 2, 2008. 2000 Mathematics Subject Classification. 81U20, 35P25, 35B38, 35C20 Scattering amplitude, critical energy, Schrödinger equation

 to analyze the contributions of the trapped trajectories. We prove a semiclassical expansion of the scattering amplitude and compute its leading term. We show that it has different orders of magnitude in specific regions of phase space. We also prove upper and lower bounds for the resolvent in this setting.

Introduction

We consider the semiclassical behavior of the scattering amplitude at energy E > 0 for Schrödinger operators (1.1)

P (x, hD) = - h 2 2 ∆ + V (x)
where V is a real valued C ∞ function on R n , which vanishes at infinity. We suppose that E is close to a critical energy level E 0 for P , which corresponds to a non-degenerate global maximum of the potential. Here, we address the case where this maximum is unique.

Let us recall that, if V (x) = O( x -ρ ) for some ρ > (n + 1)/2, then for any ω = θ ∈ S n-1 and E > 0, the problem

     P (x, hD)u = Eu, u(x, h) = e i √ 2Ex•ω/h + A(ω, θ, E, h) e i √ 2Ex•θ/h
|x| (n-1)/2 + o(|x| (1-n)/2 ) as x → +∞,

x |x| = θ,
has a unique solution in L 2 loc (R n ). The scattering amplitude at energy E for the incoming direction ω and the outgoing direction θ is the real number A(ω, θ, E, h).

For potentials that are not decaying that fast at infinity, the scattering amplitude cannot be so easily defined through a stationary approach: If V (x) = O( x -ρ ) for some ρ > 1, the scattering matrix S(E, h) at energy E can be given in terms of the wave operators (see Section 4 below). Then, writing (1.2) S(E, h) = Id -2iπT (E, h), one can see that T (E, h) is a compact operator on L 2 (S n-1 ), whose kernel T (ω, θ, E, h) is smooth away from the diagonal in S n-1 × S n-1 . Then, the scattering amplitude is given for θ = ω, by (1.3) A(ω, θ, E, h) = c(E)h (n-1)/2 T (ω, θ, E, h),

where

(1.4) c(E) = -2π(2E) -n-1 4 (2π)
n-1 2 e -i (n-3)π 4 .

We proceed here as in [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF], where D. Robert and H. Tamura have studied the semiclassical behavior of the scattering amplitude for short range potentials at a non-trapping energy E . An energy E is said to be non-trapping when K(E), the trapped set at energy E, is empty. This trapped set is defined as (1.5) K(E) = (x, ξ) ∈ p -1 (E); exp(tH p )(x, ξ) → ∞ as t → ±∞ , where H p is the Hamiltonian vector field associated to the principal symbol p(x, ξ) = 1 2 ξ 2 + V (x) of the operator P . Notice that the scattering amplitude has been first studied, in the semiclassical regime, by B. Vainberg [START_REF] Vaȋnberg | Quasiclassical approximation in stationary scattering problems[END_REF] and Y. Protas [START_REF] Protas | Quasiclassical asymptotic behavior of the scattering amplitude of a plane wave on the inhomogeneities of a medium[END_REF] in the case of compactly supported potential and for non-trapping energies, where they obtained the same type of result.

Under the non-trapping assumption, and some other non-degeneracy condition (in fact our assumption (A4) below), D. Robert and H. Tamura have shown that the scattering amplitude has an asymptotic expansion with respect to h. This non-degeneracy assumption implies in particular that there is a finite number N ∞ of classical trajectories for the Hamiltonian p, with asymptotic direction ω for t → -∞ and asymptotic direction θ as t → +∞. Robert and Tamura's result is the following asymptotic expansion for the scattering amplitude:

(1.6)

A(ω, θ, E, h)

= N∞ j=1 e iS ∞ j /h m≥0 a j,m (ω, θ, E)h m + O(h ∞ ), h → 0,
where S ∞ j is the classical action along the corresponding trajectory. Also, they have computed the first term in this expansion, showing that it can be given in terms of quantities attached to the corresponding classical trajectory only.

V. Guillemin [START_REF] Guillemin | Sojourn Times and Asymptotic Properties of the Scattering Matrix[END_REF] has established a similar asymptotic expansion in the setting of smooth compactly-supported metric perturbations of the Laplacian. For short-range potentials, K. Yajima has proved in [START_REF] Yajima | The Quasiclassical Limit of Scattering Amplitude. L 2 Approach for Short Range Potentials[END_REF] an asymptotic expansion of the form (1.6) of the scattering amplitude in the L 2 sense. Most recently, A. Hassell and J. Wunsch [START_REF] Hassell | The Semiclassical Resolvent and the Propagator for Nontrapping Scattering Metrics[END_REF] have shown that the scattering matrix at non-trapping energies on a compact manifold with boundary with a scattering matric is a Legendrian-Lagrangian distribution associated to the total sojourn relation.

There is also a small number of results concerning the scattering amplitude when the nontrapping assumption is not fulfilled. In [START_REF] Michel | Semi-classical behavior of the scattering amplitude for trapping perturbations at fixed energy[END_REF] L. Michel has shown that, if there is no trapped trajectory with incoming direction ω, and θ is ω-regular (see the discussion after (2.7) below), and if there is a resonance free complex neighborhood of E of size ∼ h N for some N ∈ N, then A(ω, θ, E, h) is still given by (1.6). The potential is also supposed to be analytic in a sector out of a compact set, and the assumption on the resonance free domain near E amounts to an estimate on the boundary values of the meromorphic extension of the truncated resolvent of the form (1.7) χ(P -

(E ± i0)) -1 χ = O(h -N ), χ ∈ C ∞ 0 (R n ).
Note that, these assumptions allow the existence of a non-empty trapped set.

In [START_REF] Alexandrova | Structure of the semi-classical amplitude for general scattering relations[END_REF] and [START_REF]Structure of the short range amplitude for general scattering relations[END_REF], the first author has shown that at non-trapping energies or in Michel's setting, the scattering amplitude is an h-Fourier integral operator associated to a natural scattering relation. These results imply that the scattering amplitude admits an asymptotic expansion, in the sense of oscillatory integrals, even without the non-degeneracy assumption. In particular, the expansion (1.6) is recovered under the non-degeneracy assumption.

In [START_REF] Lahmar-Benbernou | Semiclassical asymptotics of the residues of the scattering matrix for shape resonances[END_REF], A. Lahmar-Benbernou and A. Martinez have computed the scattering amplitude at energy E ∼ E 0 , in the case where the trapped set K(E 0 ) consists in one single point corresponding to a local minimum of the potential (a well in the island situation). In that case, the estimate (1.7) is not true, and their result is obtained through a construction of the resonant states.

In the present work, we compute the scattering amplitude at energy E ∼ E 0 in the case where the trapped set K(E 0 ) corresponds to the unique global maximum of the potential. The one-dimensional case has been studied in [START_REF] Ramond | Semiclassical study of quantum scattering on the line[END_REF][START_REF] Fujiié | Matrice de scattering et résonances associées à une orbite hétérocline[END_REF][START_REF]Breit-Wigner formula at barrier tops[END_REF], with specific techniques, and we consider here the general n > 1 dimensional case. Notice that J. Sjöstrand in [START_REF] Sjöstrand | Semiclassical resonances generated by nondegenerate critical points[END_REF], and P. Briet, J.-M. Combes and P. Duclos in [START_REF] Briet | On the location of resonances for Schrödinger operators in the semiclassical limit. I. Resonances free domains[END_REF][START_REF]On the location of resonances for Schrödinger operators in the semiclassical limit II: Barrier top resonances[END_REF] have described the resonances close to E 0 in the case where V is analytic in a sector around R n . From their result, it follows that Michel's assumption on the existence of a not too small resonance-free neighborhood of E 0 is satisfied. However, we show below (see Proposition 2.5) that for any ω ∈ S n-1 , there is at least one half-trapped trajectory with incoming direction ω, so that Michel's result never applies here.

Here, we do not assume analyticity for V . We compute the contributions to the scattering amplitude arising from the classical trajectories reaching the unstable equilibrium point, which corresponds to the top of the potential barrier. At the quantum level, tunnel effect occurs, which permits the particle to pass through this point. Our computation here relies heavily on [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF], where a precise description of this phenomena has been obtained. In a forthcoming paper, we shall show that in this case also, the scattering amplitude is an h-Fourier integral operator.

This paper is organized in the following way. In Section 2, we describe our assumptions, and state our main results: a resolvent estimate, and the asymptotic expansion of the scattering amplitude in the semiclassical regime. Section 3 is devoted to the proof of the resolvent estimate, from which we deduce in Section 4 estimates similar to those in [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF]. In that section, we also recall briefly the representation formula for the scattering amplitude proved by H. Isozaki and H. Kitada, and introduce notations from [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF]. The computation of the asymptotic expansion of the scattering amplitude is conducted in sections 5, 6 and 7, following the classical trajectories. Eventually, we have put in four appendices the proofs of some side results or technicalities.

Assumptions and main results

We suppose that the potential V satisfies the following assumptions (A1) V is a C ∞ function on R n , and, for some ρ > 1,

∂ α V (x) = O( x -ρ-|α| ).
(A2) V has a non-degenerate maximum point at x = 0, with E 0 = V (0) > 0 and

∇ 2 V (0) =    -λ 2 1 . . . -λ 2 n    , 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ n .
(A3) The trapped set at energy E 0 is K(E 0 ) = {(0, 0)}.

Notice that the assumptions (A1)-(A3) imply that V has an absolute global maximum at x = 0. Indeed, if L = {x = 0; V (x) ≥ E 0 } was non empty, the geodesic, for the Agmon distance (E 0 -V (x))

1/2 + dx, between 0 and L would be the projection of a trapped bicharacteristic (see [START_REF] Abraham | Foundations of mechanics[END_REF]Theorem 3.7.7]).

As in D. Robert and H. Tamura in [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF], one of the key ingredient for the study of the scattering amplitude is a suitable estimate for the resolvent. Using the ideas in [5, Section 4], we have obtained the following result, that we think to be of independent interest. Theorem 2.1. Suppose assumptions (A1), (A2) and (A3) hold, and let α > 1 2 be a fixed real number. We have

(2.1) (P -(E ± i0)) -1 α,-α h -1 | ln h|,
uniformly for |E -E 0 | ≤ δ, with δ > 0 small enough. Here Q α,β denotes the norm of the bounded operator Q from L 2 ( x α dx) to L 2 ( x β dx).

Moreover, we prove in the Appendix B that our estimate is not far from optimal. Indeed, we have the Proposition 2.2. Let ψ ∈ C ∞ 0 (R n ) with ψ(0) = 0. Under the assumptions (A1) and (A2), we have

(2.2) ψ(P -(E 0 ± i0)) -1 ψ h -1 | ln h|.
In particular,

(2.3) (P -(E ± i0)) -1 α,-α h -1 | ln h|,
for all α > 1 2 . We would like to mention that in the case of a closed hyperbolic orbit, the same upper bound has been obtained by N. Burq [START_REF] Burq | Smoothing effect for Schrödinger boundary value problems[END_REF] in the analytic category, and in a recent paper [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF] by H. Christianson in the C ∞ setting.

As a matter of fact, in the present setting, S. Nakamura has proved in [START_REF] Nakamura | Semiclassical resolvent estimates for the barrier top energy[END_REF] an O(h -2 ) bound for the resolvent. Nakamura's estimate would be sufficient for our proof of Theorem 2.6, but it is not sharp enough for the computation of the total scattering cross section along the lines of D. Robert and H. Tamura in [START_REF] Robert | Semiclassical estimates for resolvents and asymptotics for total scattering cross-sections[END_REF]. In that paper, the proof relies on a bound O(h -1 ) for the resolvent, but it is easy to see that an estimate like O(h -1-ε ) for any small enough ε > 0 is sufficient. If we denote 1) . Now we state our assumptions concerning the classical trajectories associated with the Hamiltonian p, that is curves t → γ(t, x, ξ) = exp(tH p )(x, ξ) for some initial data (x, ξ) ∈ T * R n . Let us recall that, thanks to the decay of V at infinity, for given α ∈ S n-1 and z ∈ α ⊥ ∼ R n-1 (the impact plane), there is a unique bicharacteristic curve

(2.4) σ(ω, E, h) = S n-1 |A(ω, θ, E, h)|
, n ≥ 2. If |E -E 0 | < δ for some δ > 0 small enough, then (2.5) σ(ω, E, h) = 4 ω ⊥ sin 2 2 -1 (2E) -1/2 h -1 R V (y + sω)ds dy + O h -(n-1)/(ρ-
(2.6) γ ± (t, z, α, E) = (x ± (t, z, α, E), ξ ± (t, z, α, E)) such that (2.7) lim t→±∞ |x ± (t, z, α, E) - √ 2Eαt -z| = 0, lim t→±∞ |ξ ± (t, z, α, E) - √ 2Eα| = 0.
We shall denote by Λ - ω the set of points in T * R n lying on trajectories going to infinity with direction ω as t → -∞, and Λ + θ the set of those which lie on trajectories going to infinity with direction θ as t → +∞:

(2.8) Λ - ω = γ -(t, z, ω, E 0 ) ∈ T * R n ; z ∈ ω ⊥ , t ∈ R , Λ + θ = γ + (t, z, θ, E 0 ) ∈ T * R n ; z ∈ θ ⊥ , t ∈ R .
From the discussion of Section 4 one can see that Λ - ω and Λ + θ are Lagrangian submanifolds of T * R n .

Under the assumptions (A1), (A2) and (A3) there are only two possible behaviors for x ± (t, z, α, E 0 ) as t → ∓∞: either it escapes to ∞, or it goes to 0.

First we state our assumptions for the first kind of trajectories. For these, we also have, for some (r

∞ (z, ω, E 0 ), ξ ∞ (z, ω, E 0 )), (2.9) lim t→+∞ ξ -(t, z, ω, E 0 ) = ξ ∞ (z, ω, E 0 ), lim t→+∞ x -(t, z, ω, E 0 ) -ξ ∞ (z, ω, E 0 )t = r ∞ (z, ω, E 0 ),
and we shall say that the trajectory γ -(t, z, ω, E 0 ) has initial direction ω and final direction

θ = ξ ∞ (z, ω, E 0 )/ √ 2E 0 .
As in [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF] we shall make some non-degeneracy assumption on the trajectories with initial direction ω. This assumption can be given in terms of the angular density

(2.10) σ(z) = | det(ξ ∞ (z, ω, E 0 ), ∂ z 1 ξ ∞ (z, ω, E 0 ), . . . , ∂ z n-1 ξ ∞ (z, ω, E 0 ))|.
Definition 2.4. The outgoing direction θ ∈ S n-1 is called regular for the incoming direction ω ∈ S n-1 , or ω-regular, if θ = ω and, for all

z ′ ∈ ω ⊥ with ξ ∞ (z ′ , ω, E 0 ) = √ 2E 0 θ, the map ω ⊥ ∋ z → ξ ∞ (z, ω, E 0 ) ∈ S n-1 is non-degenerate at z ′ , i.e. σ(z ′ ) = 0.
We fix the incoming direction ω ∈ S n-1 , and we assume that

(A4) Λ - ω ∩ Λ + θ is a finite set of Hamiltonian trajectories (γ ∞ j ) 1≤j≤N∞ , and the direction θ ∈ S n-1 is ω-regular. We denote γ ∞ j (t) = γ ∞ (t, z ∞ j ) = (x ∞ j (t), ξ ∞ j (t)).
Then one can show that Λ - ω and Λ + θ intersect transversely along each of these trajectories.

We now turn to trapped bicharacteristics. Let us notice that the linearization F p at (0, 0) of the Hamilton vector field H p has eigenvalues -λ n , . . . , -λ 1 , λ 1 , . . . , λ n . Thus (0, 0) is a hyperbolic fixed point for H p , and the Stable Manifold Theorem gives the existence of a stable incoming Lagrangian manifold Λ -and a stable outgoing Lagrangian manifold Λ + characterized by (2.11)

Λ ± = {(x, ξ) ∈ T * R n ; exp(tH p )(x, ξ) → 0 as t → ∓∞} .
In this paper, we shall describe the contribution to the scattering amplitude of the trapped trajectories, that is those going from infinity to the fixed point (0, 0). We have proved in Appendix A the following result, which shows that there are always such trajectories. Proposition 2.5. For every ω, θ ∈ S n-1 , we have

(2.12) Λ - ω ∩ Λ -= ∅ and Λ + θ ∩ Λ + = ∅.
We suppose that (A5) Λ - ω and Λ -(resp. Λ + θ and Λ + ) intersect in a finite number N -(resp N + ) of bicharacteristic curves, with each intersection transverse.

We denote these curves, respectively,

(2.13) γ - k : t → γ -(t, z - k ) = (x - k (t), ξ - k (t)), 1 ≤ k ≤ N -, and
(2.14) γ + ℓ : t → γ + (t, z + ℓ ) = (x + ℓ (t), ξ + ℓ (t)), 1 ≤ ℓ ≤ N + .
Here, the z - k (resp. the z + ℓ ) belong to ω ⊥ (resp. θ ⊥ ) and determine the corresponding curve by (2.7).

We recall from [20, Section 3] (see also [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]Section 5]), that each integral curve γ ± (t) = (x ± (t), ξ ± (t)) ∈ Λ ± satisfies, in the sense of expandible functions (see Definition 6.1 below), (2.15) γ ± (t) ∼ j≥1 γ ± j (t)e ±µ j t , as t → ∓∞,

where µ 1 = λ 1 < µ 2 < . . . is the strictly increasing sequence of linear combinations over N of the λ j 's. Here, the functions γ ± j : R → R 2n are polynomials, that we write

(2.16) γ ± j (t) = M ′ j m=0 γ ± j,m t m .
Considering the base space projection of these trajectories, we denote (2.17)

x ± (t) ∼ +∞ j=1 g ± j (t)e ±µ j t , as t → ∓∞, g ± j (t) = M ′ j m=0 g ± j,m t m .
Let us denote by  the (only) integer such that µ  = 2λ 1 . We prove in Proposition 6.11 below that if j < , then M ′ j = 0, or more precisely, that γ ± j (t) = γ ± j is a constant vector in Ker(F p ∓ λ j ). We also have M ′  ≤ 1, and g - ,1 can be computed in terms of g - 1 . In this paper, we will denote the objects associated to the k-th incoming or ℓ-th outgoing trajectory by attaching z - k or z + ℓ to the notation. Concerning the incoming trajectories, we shall assume that

(A6) For each k ∈ {1, . . . , N -}, g - 1 (z - k ) = 0.
Finally, we state our assumptions for the outgoing trajectories

γ + ℓ ⊂ Λ + ∩ Λ + θ .
First of all, it is easy to see, using Hartman's linearization theorem, that, for all ℓ, there always exists m ∈ N such that g + m (z + ℓ ) = 0. We let

(2.18) ℓ ℓ ℓ = ℓ ℓ ℓ(ℓ) = min{m; g + m (z + ℓ ) = 0}
be the smallest of these m's. We know that µ ℓ ℓ ℓ is one of the λ j 's, and that M ′ ℓ ℓ ℓ = 0. In [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF], we have been able to describe the branching process between an incoming curve γ -⊂ Λ -and an outgoing curve γ + ⊂ Λ + provided g - 1 |g + 1 = 0 (see the definition for Λ + (ρ -) before [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]Theorem 2.6]). Here, for the computation of the scattering amplitude, we can relax this assumption a lot, and analyze the branching in other cases which we now describe. Let us denote, for a given pair of paths (γ

-(z - k ), γ + (z + ℓ )) in (Λ - ω ∩ Λ -) × (Λ + θ ∩ Λ + ), (2.19) M 2 (k, ℓ) = - 1 8λ 1 j∈I 1 (2λ 1 )
α,β∈I 2 (λ 1 )

∂ j ∂ β V (0) (g - 1 (z - k )) β β! ∂ j ∂ α V (0) (g + 1 (z + ℓ )) α α! ,
and

M 1 (k, ℓ) = - j∈I 1 α∈I 2 (λ 1 ) ∂ j ∂ α V (0) α! (g - 1 (z - k )) α (g + ,0 (z + ℓ )) j + (g - ,0 (z - k )) j (g + 1 (z + ℓ )) α + α,β∈I 2 (λ 1 ) (g - 1 (z - k )) α α! (g + 1 (z + ℓ )) β β! C α,β , (2.20)
where

C α,β = -∂ α+β V (0) + j∈I 1 \I 1 (2λ 1 ) 4λ 2 1 λ 2 j (4λ 2 1 -λ 2 j ) ∂ j ∂ α V (0)∂ j ∂ β V (0) - j∈I 1 γ,δ∈I 2 (λ 1 ) γ+δ=α+β (γ + δ)! γ! δ! 1 2λ 2 j ∂ j ∂ γ V (0)∂ j ∂ δ V (0). (2.21)
Here, we have set I 1 = {1, . . . , n}, that we sometimes identify with {1 j , j = 1 . . . n},

1 j = (δ ij ) i=1,...,n ∈ N n and (2.22) I m (µ) = {β ∈ N n ; β = 1 k 1 + • • • + 1 km with λ k 1 = • • • = λ km = µ},
the set of multi-indices β of length |β| = m with each index of its non-vanishing components in the set {j ∈ N; λ j = µ}. We also denote I m ⊂ N n the set of all multi-indices of length m.

We will suppose that

(A7) For each pair of paths (γ -(z - k ), γ + (z + ℓ )), k ∈ {1, . . . , N -}, ℓ ∈ {1, .
. . , N + }, one of the three following cases occurs:

(a) The set m <  ; g - m (z - k )|g + m (z + ℓ ) = 0 is not empty. Then we denote k = min m <  ; g - m (z - k )|g + m (z + ℓ ) = 0 . (b) For all m < , we have g - m (z - k )|g + m (z + ℓ ) = 0, and M 2 (k, ℓ) = 0. (c) For all m < , we have g - m (z - k )|g + m (z + ℓ ) = 0, M 2 (k, ℓ) = 0 and M 1 (k, ℓ) = 0.
As one could expect (see [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF], [START_REF] Ramond | Semiclassical study of quantum scattering on the line[END_REF] or [START_REF]Breit-Wigner formula at barrier tops[END_REF]), action integrals appear in our formula for the scattering amplitude. We shall denote

S ∞ j = +∞ -∞ (|ξ ∞ j (t)| 2 -2E 0 )dt -r ∞ (z ∞ j , ω, E 0 )| 2E 0 θ , j ∈ {1, . . . , N ∞ }, (2.23) S - k = +∞ -∞ |ξ - k (t)| 2 -2E 0 1 t<0 dt, k ∈ {1, . . . , N -}, (2.24)
S + ℓ = +∞ -∞ |ξ + ℓ (t)| 2 -2E 0 1 t>0 dt, ℓ ∈ {1, . . . , N + }, (2.25) and ν ∞ j , ν + ℓ , ν - k the Maslov indexes of the curves γ ∞ j , γ + ℓ , γ - k respectively. Let also D - k = lim t→+∞ det ∂x -(t, z, ω, E 0 ) ∂(t, z) | z=z - k e -(Σ j λ j -2λ 1 )t , (2.26) D + ℓ = lim t→-∞ det ∂x + (t, z, ω, E 0 ) ∂(t, z) | z=z + ℓ e (Σ j λ j -2λ ℓ ℓ ℓ )t , (2.27)
be the Maslov determinants for γ - k , and γ + ℓ respectively. We show below that 0

< D - k , D + ℓ < +∞. Eventually we set (2.28) Σ(E, h) = n j=1 λ j 2 -i E -E 0 h •
Then, the main result of this paper is the Theorem 2.6. Suppose assumptions (A1) to (A7) hold, and that

E ∈ R is such that E -E 0 = O(h). Then A(ω, θ, E, h) = N∞ j=1 A reg j (ω, θ, E, h) + N - k=1 N + ℓ=1 A sing k,ℓ (ω, θ, E, h) + O(h ∞ ), (2.29)
where (2.30)

A reg j (ω, θ, E, h) ∼ e iS ∞ j /h m≥0 a reg j,m (ω, θ, E)h m , with (2.31) a reg j,0 (ω, θ, E) = e -iν ∞ j π/2 σ(z ∞ j ) 1/2 e -r∞(z ∞ j ,ω,E 0 )| √ 2E 0 -1 θ E-E 0 h • Moreover we have • In case (a) A sing k,ℓ (ω, θ, E, h) ∼ e i(S - k +S + ℓ )/h m≥0 a sing k,ℓ,m (ω, θ, E, ln h)h (Σ(E)+ µm)/µ k -1/2 , (2.32)
where the a sing k,ℓ,m (ω, θ, E, ln h) are polynomials with respect to ln h, and

a sing k,ℓ,0 (ω, θ, E, ln h) = e iπ/4 E (n-1)/4 2 (n+1)/4 √ π n j=1 λ j -1/2 Γ Σ(E) µ k (2λ 1 λ ℓ ℓ ℓ ) 3/2 µ k × e -iν + ℓ π/2 e -iν - k π/2 (D - k D + ℓ ) -1/2 × |g - 1 (z - k )| |g + ℓ ℓ ℓ (z + ℓ )| 2iµ k g - k (z - k ) g + k (z + ℓ ) -Σ(E)/µ k . (2.33) • In case (b) (2.34) A sing k,ℓ (ω, θ, E, h) = e i(S + ℓ +S - k )/h a sing k,ℓ (ω, θ, E) h Σ(E)/2λ 1 -1/2 | ln h| Σ(E)/λ 1 (1 + o(1)),
where

a sing k,ℓ (ω, θ, E) = e iπ/4 E (n-1)/4 2 (n+1)/4 √ π n j=1 λ j -1/2 Γ Σ(E) 2λ 1 (2λ 1 λ ℓ ℓ ℓ ) 3/2 (2λ 1 ) Σ(E)/λ 1 -1 × e -iν + ℓ π/2 e -iν - k π/2 (D - k D + ℓ ) -1/2 × |g - 1 (z - k )| |g + ℓ ℓ ℓ (z + ℓ )| -iM 2 (k, ℓ) -Σ(E)/2λ 1 . (2.35) • In case (c) (2.36) A sing k,ℓ (ω, θ, E, h) = e i(S + ℓ +S - k )/h a sing k,ℓ (ω, θ, E) h Σ(E)/2λ 1 -1/2 | ln h| Σ(E)/2λ 1 (1 + o(1)),
where

a sing k,ℓ (ω, θ, E) = e iπ/4 E (n-1)/4 2 (n+1)/4 √ π n j=1 λ j -1/2 Γ Σ(E) 2λ 1 (2λ 1 λ ℓ ℓ ℓ ) 3/2 (2λ 1 ) Σ(E)/2λ 1 -1 × e -iν + ℓ π/2 e -iν - k π/2 (D - k D + ℓ ) -1/2 × |g - 1 (z - k )| |g + ℓ ℓ ℓ (z + ℓ )| -iM 1 (k, ℓ) -Σ(E)/2λ 1 . (2.37)
Here, the µ j are the linear combinations over N of the λ k 's and µ kµ k 's for k ≥ k, and the function z → z -Σ(E)/µ k is defined on C\] -∞, 0] and real positive on ]0, +∞[. Of course the assumption that g - 1 |g + 1 = 0 (a subcase of (a)) is generic. Without the assumption (A4), the regular part A reg of the scattering amplitude has an integral representation as in [START_REF]Structure of the short range amplitude for general scattering relations[END_REF]. When the assumption (A7) is not fulfilled, that is when the terms corresponding to the µ j with j ≤  do not contribute, we don't know if the scattering amplitude can be given only in terms of the g ± j 's and of the derivatives of the potential at the critical point.

Proof of the main resolvent estimate

Here we prove Theorem 2.1 using Mourre's Theory. We start with the construction of an escape function close to the stationary point (0, 0) in the spirit of [START_REF] Burq | Geometric control in the presence of a black box[END_REF] and [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]. Since Λ + and Λ -are Lagrangian manifolds, one can find a local symplectic map κ : (x, ξ) → (y, η) such that

(3.1) p(x, ξ) -E 0 = B(y, η)y • η,
where (y, η) → B(y, η) is a C ∞ mapping from a neighborhood of (0,0) in T * R n to the space M n (R) of n × n matrices with real entries, such that,

(3.2) B(0, 0) =    λ 1 /2 . . . λ n /2    .
We denote by U a unitary Fourier integral operator (FIO) microlocally defined in a neighborhood of (0, 0), whose canonical transformation is κ, and we set

(3.3) P = U (P -E 0 )U * .
Here the FIO U * is the adjoint of U , and we have

U U * = Id + O(h ∞ ) and U * U = Id + O(h ∞ )
microlocally near (0, 0). Then P is a pseudodifferential operator, with a real (modulo O(h ∞ )) symbol p(y, η) = j p j (y, η)h j , such that

(3.4) p 0 = B(y, η)y • η. We set B 1 = Op h (b 1 ), (3.5) b 1 (y, η) = ln y √ hM -ln η √ hM χ 2 (y, η),
where M > 1 will be fixed later and

χ 1 ≺ χ 2 ∈ C ∞ 0 (T * R n ) with χ 1 = 1 near (0, 0).
In what follows, we will assume that hM < 1. In particular, b 1 ∈ S 1/2 (| ln h|). Here and in what follows, we use the usual notation for classes of symbols. For m an order function, a function a(x, ξ, h) ∈ C ∞ (T * R n ) belongs to S δ (m) when (3.6) ∀α ∈ N 2n , ∃C α > 0, ∀h ∈]0, 1], |∂ α x,ξ a(x, ξ, h)| ≤ C α h -δ|α| m(x, ξ). We also recall that, with Op h (a) denoting the Weyl quantization, if a ∈ S α (1) and b ∈ S β (1), with α, β < 1/2, we have

(3.7) Op h (a), Op h (b) = Op h ih{b, a} + h 3(1-α-β) Op h (r),
with r ∈ S min(α,β) (1): In particular the term of order 2 vanishes.

Hence, we have here

(3.8) [B 1 , P ] = Op h ih{ p 0 , b 1 } + | ln h| h 3/2 Op h (r M ),
with r M ∈ S 1/2 (1). The semi-norms of r M may depend on M . We have

(3.9) { p 0 , b 1 } = c 1 + c 2 ,
with

c 1 = ln y √ hM -ln η √ hM { p 0 , χ 2 } (3.10) c 2 = p 0 , ln y √ hM -ln η √ hM χ 2 = By + (∂ η B)y • η • y hM + y 2 + Bη + (∂ y B)y • η • η hM + η 2 χ 2 . (3.11) The symbols c 1 ∈ S 1/2 (| ln h|), c 2 ∈ S 1/2 (1) satisfy supp(c 1 ) ⊂ supp(∇ χ 2 ). Let ϕ ∈ C ∞ 0 (T * R n
) be a function such that ϕ = 0 near (0, 0) and ϕ = 1 near the support of ∇ χ 2 . We have

Op h (c 1 ) = Op h ( ϕ) Op h (c 1 ) Op h ( ϕ) + O(h ∞ ) ≥ -C 1 h| ln h| Op h ( ϕ) Op h ( ϕ) + O(h ∞ ) ≥ -C 1 h| ln h| Op h ( ϕ 2 ) + O(h 2 | ln h|), (3.12)
for some C 1 > 0. On the other hand, using [5, (4.96)-(4.97)], we get

(3.13) Op h (c 2 ) ≥ εM -1 Op h ( χ 1 ) + O(M -2 ),
for some ε > 0. With the notation A 1 = U * B 1 U , the formulas (3.8), (3.9), (3.12) and (3.13) imply

-i[A 1 , P ] = -iU * [B 1 , P ]U + O(h ∞ ) ≥εhM -1 U * Op h ( χ 1 )U -C 1 h| ln h|U * Op h ( ϕ 2 )U + O(hM -2 ) + O M (h 3/2 | ln h|). (3.14) Here, χ j = χ j • κ, j = 1, 2 and ϕ = ϕ • κ are C ∞ 0 (T * R n , [0, 1]
) functions which satisfy χ 1 = 1 near (0, 0) and ϕ = 0 near (0, 0). Using Egorov's Theorem, (3.14) becomes

(3.15) -i[A 1 , P ] ≥ εhM -1 Op h (χ 1 ) -C 1 h| ln h| Op h (ϕ 2 ) + O(hM -2 ) + O M (h 3/2 | ln h|).
Now, we build an escape function outside of supp(χ 1 ) as in [START_REF] Martinez | Resonance free domains for non globally analytic potentials[END_REF]. Let

1 (0,0) ≺ χ 0 ≺ χ 1 ≺ χ 2 ≺ χ 3 ≺ χ 4 ≺ χ 5 be C ∞ 0 (T * R n , [0, 1]) functions with ϕ ≺ χ 4 . We define a 3 = g(ξ)(1 -χ 3 (x, ξ))x • ξ where g ∈ C ∞ 0 (R n ) satisfies 1 p -1 ([E 0 -δ,E 0 +δ]) ≺ g. Using [6, Lemma 3.1], we can find a bounded, C ∞ function a 2 (x, ξ) such that (3.16) H p a 2 ≥ 0 for all (x, ξ) ∈ p -1 ([E 0 -δ, E 0 + δ]), 1 for all (x, ξ) ∈ supp(χ 4 -χ 0 ) ∩ p -1 ([E 0 -δ, E 0 + δ]),
and we set A 2 = Op h (a 2 χ 5 ), A 3 = Op h (a 3 ). We denote (3.17)

A = A 1 + C 2 | ln h|A 2 + | ln h|A 3 ,
where C 2 > 1 will be fixed later. Now let ψ ∈ C ∞ 0 ([E 0δ, E 0 + δ], [0, 1]) with ψ = 1 near E 0 . We recall that ψ(P ) is a classical pseudodifferential operator of class Ψ 0 ( ξ -∞ ) with principal symbol ψ(p). Then, from (3.15), we obtain

-i ψ(P )[A, P ] ψ(P ) ≥εhM -1 ψ(P ) Op h (χ 1 ) ψ(P ) -C 1 h| ln h| ψ(P ) Op h (ϕ 2 ) ψ(P ) + C 2 h| ln h| Op h ψ 2 (p)(χ 4 -χ 0 ) + C 2 h| ln h| Op h ψ 2 (p)a 2 H p χ 5 + h| ln h| Op h ψ 2 (p)(ξ 2 -x • ∇V )(1 -χ 3 ) + h| ln h| Op h ψ 2 (p)x • ξH p (gχ 3 ) + O(hM -2 ) + O M (h 3/2 | ln h|). (3.18)
From (A1), we have x•∇V (x) → 0 as x → ∞. In particular, if χ 3 is equal to 1 in a sufficiently large zone, we have

(3.19) ψ 2 (p)(ξ 2 -x • ∇V )(1 -χ 3 ) ≥ E 0 ψ 2 (p)(1 -χ 3 ). If C 2 > 0 is large enough, the Gårding inequality implies (3.20) C 2 Op h ψ 2 (p)(χ 4 -χ 0 ) -C 1 Op h ψ 2 (p)ϕ 2 + Op h ψ 2 (p)x • ξH p (gχ 3 ) ≥ Op h ψ 2 (p)(χ 4 -χ 0 ) + O(h).
As in [START_REF] Martinez | Resonance free domains for non globally analytic potentials[END_REF], we take χ 5 (x, ξ) = χ 5 (µx)g(ξ) with µ small and χ 5 ∈ C ∞ 0 (R n , [0, 1]), χ 5 = 1 near 0. Since a 2 is bounded, we get

(3.21) C 2 ψ 2 (p)a 2 H p χ 5 ≤ µC 2 a 2 L ∞ H p χ 5 L ∞ µ.
Therefore, if µ is small enough, (3.19) implies

(3.22) Op h ψ 2 (p)(ξ 2 -x • ∇V )(1 -χ 3 ) + C 2 Op h ψ 2 (p)a 2 H p χ 5 ≥ E 0 2 Op h ψ 2 (p)(1 -χ 3 ) .
Then (3.18), (3.20), (3.22) and the Gårding inequality give, for some ε > 0,

-i ψ(P )[A, P ] ψ(P ) ≥εhM -1 Op h ψ 2 (p)χ 1 + h| ln h| Op h ψ 2 (p)(χ 4 -χ 0 ) + E 0 2 h| ln h| Op h ψ 2 (p)(1 -χ 3 ) + O(hM -2 ) + O M (h 3/2 | ln h|) ≥εhM -1 Op h ψ 2 (p) + O(hM -2 ) + O M (h 3/2 | ln h|). (3.23)
Choosing M large enough and 1 E 0 ≺ ψ ≺ ψ, we have proved the Lemma 3.1. Let M be large enough and ψ ∈ C ∞ 0 ([E 0δ, E 0 + δ]), δ > 0 small enough, with ψ = 1 near E 0 . Then, we have From the properties of the support of the χ j , we have

[[P, A], A] =[[P, A 1 ], A 1 ] + C 2 | ln h|[[P, A 1 ], A 2 ] + C 2 | ln h|[[P, A 2 ], A 1 ] + C 2 2 | ln h| 2 [[P, A 2 ], A 2 ] + C 2 | ln h| 2 [[P, A 2 ], A 3 ] + C 2 | ln h| 2 [[P, A 3 ], A 2 ] + | ln h| 2 [[P, A 3 ], A 3 ] + O(h ∞ ). (3.26)
We also know that

P ∈ Ψ 0 ( ξ 2 ), A 2 ∈ Ψ 0 ( ξ -∞ ) and A 3 ∈ Ψ 0 ( x ξ -∞
). Then, we can show that all the terms in (3.26) with j, k = 2, 3 satisfy

(3.27) [[P, A j ], A k ] ∈ Ψ 0 (h 2 ).
On the other hand,

(3.28) [[P, A 1 ], A 2 ] = U * [[ P , B 1 ], U A 2 U * ]U + O(h ∞ ), with U A 2 U * ∈ Ψ 0 (1). From (3.8) -(3.11), we have [ P , B 1 ] ∈ Ψ 1/2 (h| ln h|) and then (3.29) [[P, A 1 ], A 2 ] = O(h 3/2 | ln h|).
The term [[P, A 2 ], A 1 ] gives the same type of contribution. It remains to study

(3.30) [[P, A 1 ], A 1 ] = U * [[ P , B 1 ], B 1 ]U + O(h ∞ ). Let χ 3 ∈ C ∞ 0 (T * R n ), [0, 1]) with χ 2 ≺ χ 3 and (3.31) f = ln y √ hM -ln η √ hM χ 3 (y, η) ∈ S 1/2 (| ln h|).
Then, with a remainder r M ∈ S 1/2 (1) which differs from line to line,

i[ P , B 1 ] =h Op h f { χ 2 , p 0 } + c 2 -h 3/2 | ln h| Op h (r M ) =h Op h (f ) Op h ({ χ 2 , p 0 }) + h Op h (c 2 ) + h 3/2 | ln h| Op h (r M ). (3.32) In particular, since [ P , B 1 ] ∈ Ψ 1/2 (h| ln h|), c 2 ∈ S 1/2 (1) and f ∈ S 1/2 (| ln h|), [[ P , B 1 ], B 1 ] =[[ P , B 1 ], Op h (f χ 2 )] = -ih[Op h (f ) Op h ({ χ 2 , p 0 }), Op h (f χ 2 )] -ih[Op h (c 2 ), Op h (f χ 2 )] + O(h 3/2 | ln h| 2 ) = -ih[Op h (f ) Op h ({ χ 2 , p 0 }), Op h (f ) Op h ( χ 2 )] + O(h| ln h|) = -ih Op h (f )[Op h ({ χ 2 , p 0 }), Op h (f )] Op h ( χ 2 ) -ih Op h (f ) Op h (f )[Op h ({ χ 2 , p 0 }), Op h ( χ 2 )] -ih Op h (f )[Op h (f ), Op h ( χ 2 )] Op h ({ χ 2 , p 0 }) + O(h| ln h|)
=O(h| ln h|). As a matter of fact, using [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF], one can show that [[P, A], A] = O(h). Now we can use the following proposition which is an adaptation of the limiting absorption principle of Mourre [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF] (see also [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]Theorem 4.9], [21, Proposition 2.1] and [4, Theorem 7.4.1]).

Proposition 3.2. Let (P, D(P )) and (A, D(A)) be self-adjoint operators on a separable Hilbert space H. Assume the following assumptions:

i) P is of class C 2 (A). Recall that P is of class C r (A) if there exists z ∈ C \ σ(P ) such that (3.35) R ∋ t → e itA (P -z) -1 e -itA ,
is C r for the strong topology of L(H). ii) The form [P, A] defined on D(A) ∩ D(P ) extends to a bounded operator on H and

(3.36) [P, A] β.
iii) The form [[P, A], A] defined on D(A) extends to a bounded operator on H and

(3.37) [[P, A], A] γ.
iv) There exist a compact interval

I ⊂ R and g ∈ C ∞ 0 (R) with 1 I ≺ g such that (3.38) ig(P )[P, A]g(P ) γg 2 (P ). v) β 2 γ 1.
Then, for all α > 1/2, lim ε→0 A -α (P -E ± iε) -1 A -α exists and

(3.39) A -α (P -E ± i0) -1 A -α γ -1 ,
uniformly for E ∈ I.

Remark 3.3. From Theorem 6.2.10 of [START_REF] Amrein | C0-groups, commutator methods and spectral theory of N -body Hamiltonians[END_REF], we have the following useful characterization of the regularity C 2 (A). Assume that (ii) and (iv) hold. Then, P is of class C 2 (A) if and only if, for some z ∈ C \ σ(P ), the set {u ∈ D(A);

(P -z) -1 u ∈ D(A) and (P -z) -1 u ∈ D(A)} is a core for A.
Proof. The proof follows the work of Hislop and Nakamura [START_REF] Hislop | Semiclassical resolvent estimates[END_REF]. For ε > 0, we define M 2 = ig(P )[P, A]g(P ) and G ε (z) = (P -iεM 2z) -1 which is analytic for Re z ∈ I and Im z > 0.

Following [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]Lemma 4.14] with (3.35), we get

(3.40) g(P )G ε (z)ϕ (εγ) -1/2 |(ϕ, G ε (z)ϕ)| 1/2 , (3.41) (1 -g(P ))G ε (z) 1 + εβ G ε (z) ,
and then

(3.42) G ε (z) (εγ) -1 ,
for ε < ε 0 with ε 0 small enough, but independent of β, γ.

As in [START_REF] Hislop | Semiclassical resolvent estimates[END_REF], let

D ε = (1 + |A|) -α (1 + ε|A|) α-1 for α ∈]1/2, 1] and F ε (z) = D ε G ε (z)D ε . Of course, from (3.42), (3.43) F ε (z) (εγ) -1 ,
and (3.40) and (3.41) with ϕ = D ε ψ give

(3.44) G ε (z)D ε 1 + (εγ) -1/2 F ε 1/2 .
The derivative of F ε (z) is given by (see [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]Lemma 4.15])

(3.45) ∂ ε F ε (z) = iD ε G ε M 2 G ε D ε = Q 0 + Q 1 + Q 2 + Q 3 , with Q 0 =(α -1)|A|(1 + |A|) -α (1 + ε|A|) α-2 G ε (z)D ε + (α -1)D ε G ε (z)|A|(1 + |A|) -α (1 + ε|A|) α-2 (3.46) Q 1 =D ε G ε (1 -g(P ))[P, A](1 -g(P ))G ε D ε (3.47) Q 2 =D ε G ε (1 -g(P ))[P, A]g(P )G ε D ε + D ε G ε g(P )[P, A](1 -g(P ))G ε D ε (3.48) Q 3 = -D ε G ε [P, A]G ε D ε . (3.49) From (3.44), we obtain (3.50) Q 0 ε α-1 1 + (εγ) -1/2 F ε 1/2 ,
and from (3.36), v) of Proposition 3.2, (3.41), and (3.42), we have

(3.51) Q 1 γ -1 .
Using in addition (3.44), we obtain

(3.52) Q 2 1 + (εγ) -1/2 F ε 1/2 .

Now we write

Q 3 = Q 4 + Q 5 with Q 4 = -D ε G ε [P -iεM 2 -z, A]G ε D ε (3.53) Q 5 = -iεD ε G ε [M 2 , A]G ε D ε . (3.54)
For Q 4 , we have the estimate

(3.55) Q 4 ε α-1 1 + (εγ) -1/2 F ε 1/2
On the other hand, (3.36), (3.37) and v) imply

(3.56) [M 2 , A] γ.
Then (3.44) gives

(3.57) Q 5 1 + F ε .
Using the estimates on the Q j , we get

(3.58) ∂ ε F ε ε α-1 γ -1 + (εγ) -1/2 F ε 1/2 + F ε .
Using (3.43) and integrating (3.37) N times with respect to ε, we get

(3.59) F ε γ -1 1 + ε 2α(1-2 -N )-1 , so that, for N large enough, (3.60) lim sup δ→0 sup E∈I A -α (P -E ± iδ) -1 A -α γ -1 .
Using, as in [START_REF] Hislop | Semiclassical resolvent estimates[END_REF], the fact that z → F 0 (z) is Hölder continuous, we prove the existence of the limit lim Im z→0 F 0 (z) for Re z ∈ I and the proposition follows from (3.60).

From Lemma 3.1 and (3.34), we can apply Proposition 3.2 with A = A/| ln h|, β = h and γ = h/| ln h|. Therefore we have the estimate

(3.61) A -α (P -E ± i0) -1 A -α h -1 | ln h|, for E ∈ [E 0 -δ, E 0 + δ].
As usual, we have

(3.62) x -α A α = O(1),
for α ≥ 0. Indeed, (3.62) is clear for α ∈ 2N, and the general case follows by complex interpolation. Then, (3.61) and (3.26) imply Theorem 2.1.

Representation of the scattering amplitude

As in [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF], our starting point for the computation of the scattering amplitude is the representation given by Isozaki and Kitada in [START_REF] Isozaki | Modified wave operators with time-independent modifiers[END_REF]. We recall briefly their formula, that they obtained writing parametrices for the wave operators W ± as Fourier integral operators, taking advantage of the well-known intertwining property W ± P = P 0 W ± , P = P 0 + V , with P 0 = -h 2 2 ∆. The wave operators are defined by (4.1)

W ± = s-lim t→±∞ e itP/h e -itP 0 /h ,
where the limits exist in L 2 (R n ) thanks to the short-range assumption (A1). The scattering operator is by definition S = (W + ) * W -, and the scattering matrix S(E, h) is then given by the decompostion of S with respect to the spectral measure of P 0 . Now we recall briefly the discussion in [32, Section 1,2] (see also [START_REF]Structure of the short range amplitude for general scattering relations[END_REF]), and we start with some notations.

If Ω is an open subset of T * R n , we denote by A m (Ω) the class of symbols a such that (x, ξ) → a(x, ξ, h) belongs to C ∞ (Ω) and (4.2) ∂ α x ∂ β ξ a(x, ξ) ≤ C αβ x m-|α| ξ -L , for all L > 0, (x, ξ) ∈ Ω, (α, β) ∈ N d × N d .
We also denote by

(4.3) Γ ± (R, d, σ) = (x, ξ) ∈ R n × R n ; |x| > R, 1 d < |ξ| < d, ± cos(x, ξ) > ±σ with R > 1, d > 1, σ ∈ (-1, 1
), and cos(x, ξ) = x•ξ |x| |ξ| , the outgoing and incoming subsets of T * R n , respectively. Eventually, for α > 1 2 , we denote the bounded operator F 0 (E, h) :

L 2 α (R n ) → L 2 (S n-1 ) given by (4.4) (F 0 (E, h)f ) (ω) = (2πh) -n 2 (2E) n-2 4 R n e -i h √ 2Eω•x f (x)dx, E > 0.
Isozaki and Kitada have constructed phase functions Φ ± and symbols a ± and b ± such that, for some

R 0 >> 0, 1 < d 4 < d 3 < d 2 < d 1 < d 0 , and 0 < σ 4 < σ 3 < σ 2 < σ 1 < σ 0 < 1: i) Φ ± ∈ C ∞ (T * R n ) solve the eikonal equation (4.5) 1 2 |∇ x Φ ± (x, ξ)| 2 + V (x) = 1 2 |ξ| 2 in (x, ξ) ∈ Γ ± (R 0 , d 0 , ±σ 0 ), respectively. ii) (x, ξ) → Φ ± (x, ξ) -x • ξ ∈ A 0 (Γ ± (R 0 , d 0 , ±σ 0 )) . iii) For all (x, ξ) ∈ T * R n (4.6) ∂ 2 Φ ± ∂ x j ∂ ξ k (x, ξ) -δ jk < ε(R 0 ),
where δ jk is the Kronecker delta and ε(R

0 ) → 0 as R 0 → +∞. iv) a ± ∼ ∞ j=0 h j a ±j , where a ±j ∈ A -j (Γ ± (3R 0 , d 1 , ∓σ 1 )), supp a ±j ⊂ Γ ± (3R 0 , d 1 , ∓σ 1 ), a ±j solve (4.7) ∇ x Φ ± |∇ x a ±0 + 1 2 (∆ x Φ ± ) a ±0 = 0 (4.8) ∇ x Φ ± |∇ x a ±j + 1 2 (∆ x Φ ± ) a ±j = i 2 ∆ x a ±j-1 , j ≥ 1,
with the conditions at infinity (4.9)

a ±0 → 1, a ±j → 0, j ≥ 1, as |x| → ∞. in Γ ± (2R 0 , d 2 , ∓σ 2 )
, and solve (4.7) and (4.8

) in Γ ± (4R 0 , d 1 , ∓σ 2 ). v) b ± ∼ ∞ j=0 h j b ±j , where b ±j ∈ A -j (Γ ± (5R 0 , d 3 , ±σ 4 )), supp b ±j ⊂ Γ ± (5R 0 , d 3 , ±σ 4
), b ±j solve (4.7) and (4.8) with the conditions at infinity (4.9) in Γ ± (6R 0 , d 4 , ±σ 3 ), and solve (4.7) and (4.8) in Γ ± (6R 0 , d 3 , ±σ 3 ).

For a symbol c and a phase function ϕ, we denote by I h (c, ϕ) the oscillatory integral (4.10)

I h (c, ϕ) = 1 (2πh) n R n e i h (ϕ(x,ξ)-y•ξ) c(x, ξ)dξ and we set (4.11) K ±a (h) = P (h)I h (a ± , Φ ± ) -I h (a ± , Φ ± )P 0 (h), K ±b (h) = P (h)I h (b ± , Φ ± ) -I h (b ± , Φ ± )P 0 (h).
The operator

T (E, h) for E ∈] 1 2d 2 4 , d 2
4
2 [ is then given by (see [START_REF] Isozaki | Modified wave operators with time-independent modifiers[END_REF]Theorem 3.3]) (4.12)

T (E, h) = T +1 (E, h) + T -1 (E, h) -T 2 (E, h),
where (4.13)

T ±1 (E, h) = F 0 (E, h)I h (a ± , Φ ± ) * K ±b (h)F * 0 (E, h) and (4.14) T 2 (E, h) = F 0 (E, h)K * +a (h)R(E + i0, h) (K +b (h) + K -b (h)) F * 0 (E, h), where we denote from now on R(E ± i0, h) = (P -(E ± i0)) -1 .
Writing explicitly their kernel, it is easy to see, by a non-stationary phase argument, that the operators

T ±1 are O(h ∞ ) when θ = ω. Therefore we have (4.15) A(ω, θ, E, h) = -c(E)h (n-1)/2 T 2 (ω, θ, E, h) + O(h ∞ ),
where c(E) is given in (1.4).

As in [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF], we shall use our resolvent estimate (Theorem 2.1) in a particular form. It was noticed by L. Michel in [26, Proposition 3.1] that, in the present trapping case, the following proposition follows easily from the corresponding one in the non-trapping setting. Indeed, if ϕ is a compactly supported smooth function, it is clear that

P = -h 2 ∆ + (1 -ϕ(x/R))V (x)
satisfies the non-trapping assumption for R large enough, thanks to the decay of V at ∞. Writing [32, Lemma 2.3] for P , one gets the

Proposition 4.1. Let ω ± ∈ A 0 have support in Γ ± (R, d, σ ± ) for R > R 0 . For E ∈ [E 0 - δ, E 0 + δ], we have (i) For any α > 1/2 and M > 1, then, for any ε > 0, (4.16) R(E ± i0, h)ω ± (x, hD x ) -α+M,-α = O(h -3-ε ). (ii) If σ + > σ -, then for any α ≫ 1, (4.17) ω ∓ (x, hD x )R(E ± i0, h)ω ± (x, hD x ) -α,α = O(h ∞ ). (iii) If ω(x, ξ) ∈ A 0 has support in |x| < (9/10)R, then for any α ≫ 1 (4.18) ω(x, hD x )R(E ± i0, h)ω ± (x, hD x ) -α,α = O(h ∞ ).
Then we can follow line by line the discussion after Lemma 2.1 of D. Robert and H. Tamura, and we obtain (see Equations 2.2-2.4 there):

(4.19) A(ω, θ, E, h) = c(E)h -(n+1)/2 R(E + i0, h)g -e iψ -/h , g + e iψ + /h + O(h ∞ ), where c(E) = (2π) (1-n)/2 (2E) (n-3)/4 e -i (n-3)π 4 , (4.20)
g ± = e -iψ ± /h [χ ± , P ]a ± (x, h)e iψ ± /h , and
(4.21) ψ + (x) = Φ + (x, √ 2Eθ), ψ -(x) = Φ -(x, √ 2Eω).
Moreover the functions χ ± are C ∞ 0 (R n ) functions such that χ ± = 1 near some ball B(0, R ± ), with support in B(0, R ± + 1).

Eventually, we shall need the following version of Egorov's Theorem, which is also used in Robert and Tamura's paper. Proposition 4.2 ([32, Proposition 3.1]). Let ω(x, ξ) ∈ A 0 be of compact support. Assume that, for some fixed t ∈ R, ω t is a function in A 0 which vanishes in a small neighborhood of

{(x, ξ); (x, ξ) = exp(tH p )(y, η), (y, η) ∈ supp ω}. Then Op h (ω t )e -itP/h Op h (ω) -α,α = O(h ∞ ),
for any α ≫ 1. Moreover, the order relation is uniform in t when t ranges over a compact interval of R.

In the three next sections, we prove Theorem 2.6 using (4.19). We set

(4.22) u -= u h -= R(E + i0, h)g -e iψ -/
h , and our proof consists in the computation of u -in different region of the phase space, following the classical trajectories γ ∞ j , or γ - k and γ + ℓ . It is important to notice that we have (P -E)u -= 0 out of the support of g -.

5.

Computations before the critical point 5.1. Computation of u -in the incoming region.

We start with the computation of u -in an incoming region which contains the microsupport of g -. Notice that, thanks to Theorem 2.1, x -α u -(x) is a semiclassical family of distributions for α > 1/2. Lemma 5.1. Let P = -h 2 2 ∆ + V , where V satisfies assumption (A1) with ρ > 0. Let also I be a compact interval of ]0, +∞[, and d > 0 such that I ⊂] 1 2d 2 , d 2 2 [. For any 0 < σ + < 1, there exists R(σ + ) > 0 such that, for all R > R(σ + ) and any compact subset K ⊂ T * R n of p -1 (I), there exists T > 0 such that, if ρ ∈ K and t > T ,

(5.1) exp(tH p )(ρ) ∈ Γ + (R/2, d, σ + ) ∪ (B(0, R/2) × R n ).
Proof. We recall from the constructions of C. Gérard and J. Sjöstrand in [START_REF] Gérard | Semiclassical resonances generated by a closed trajectory of hyperbolic type[END_REF] that for any δ > 0, there exist R δ > 0 and a function

G(x, ξ) ∈ C ∞ (R 2n ) such that, (H p G)(x, ξ) ≥ 0 for all (x, ξ) ∈ p -1 (] 1 2d 2 , d 2 2 [), (5.2) (H p G)(x, ξ) > 2E(1 -δ) for |x| > R δ and p(x, ξ) = E ∈] 1 2d 2 , d 2 2 [, (5.3) G(x, ξ) = x • ξ for |x| > R δ . (5.4)
We choose δ > 0 such that 1 -3δ > σ + . We can assume that

(5.5) |ξ| ≤ √ 2E(1 + δ), for |x| ≥ R δ , (x, ξ) ∈ p -1 (] 1 2d 2 , d 2 2 [).
We first assume that R > 4R δ , and that K is a compact subset of p -1 (I). For ρ ∈ K and γ(t) = (x(t), ξ(t)) = exp(tH p )(ρ), the corresponding Hamiltonian curve, we distinguish between 2 cases: 1) For all t > 0, we have |x(t)| > R δ .

Then G(γ(t)) > 2E(1δ)t + G(ρ) and, for t > T 1 with T 1 large enough,

(5.6) G(γ(t)) > 2 sup x∈B(0,R δ ) p(x,ξ)∈I G(x, ξ).
By continuity, there exists a neighborhood U of ρ such that, for all ρ ∈ U, we have

(5.7) G( γ(T 1 )) > sup x∈B(0,R δ ) p(x,ξ)∈I G(x, ξ).
Since G is non-decreasing along γ(t), we have | x(t)| > R δ for all t > T 1 , and then

(5.8) G( γ(t)) > 2E(1 -δ)(t -T 1 ) + G( γ(T 1 )) > 2E(1 -δ)t -C.
From (5.5) and (5.8), we get [START_REF] Abraham | Foundations of mechanics[END_REF]. On the other hand using (5.5) we have | x(t)| ≤ √ 2E(1 + δ)t + C, for some C > 0 independent of ρ ∈ U. In particular, the previous estimates give, for t > T U with T U large enough but independent of ρ ∈ U (5.9)

| x(t)| > 1 C t -C for all ρ ∈ U, and then | ξ(t)| = √ 2E + o t→∞
| x(t)| > R/2, (5.10) cos x, ξ (t) > 2E(1 -δ)t -C ( √ 2E(1 + δ)t + C)( √ 2E + o t→∞ (1)) = 1 -δ 1 + δ + o t→∞ (1) > 1 -3δ,
Thus, for t > T U and ρ ∈ U, we have

(5.11) γ(t) ∈ Γ + (R/2, d, σ + ),
where we recall that σ + < 1 -3δ.

2) There exists

T 2 > 0 such that |x(T 2 )| = R δ .
Then there exists a neighborhood V of ρ such that for all ρ ∈ V we have

| x(T 2 )| < 2R δ , where ( x(t), ξ(t)) = exp tH p ( ρ). Now let t > T 2 . a) If | x(t)| ≤ R/2, then γ(t) ∈ B(0, R/2) × R n . b) Assume now | x(t)| > R/2. Denote by T 3 (> T 2 ) the last time (before t) such that | x(T 3 )| = 2R δ . Then G( γ(t)) >2E(1 -δ)(t -T 3 ) + G( γ(T 3 )) >2E(1 -δ)(t -T 3 ) -C, (5.12)
where C depends only on R δ . On the other hand, the have

| x(t)| < √ 2E(1 + δ)(t -T 3 ) + C, where the constant C depends only on R δ . Then, (5.13) t -T 3 > | x(t)| √ 2E(1 + δ) - C √ 2E(1 + δ) , (5.14) | ξ(t)| = √ 2E + o R→∞ (1), cos x, ξ (t) > 2E(1 -δ)| x(t)| | x(t)|( √ 2E(1 + δ))( √ 2E + o R→∞ (1)) + O(R -1 ) > 1 -δ 1 + δ + o R→∞ (1) > 1 -2δ + o R→∞ (1). (5.15) So, if R is large enough, γ(t) ∈ Γ + (R/2, d, σ + ).
Then a) and b) imply that, for all ρ ∈ V and t > T := T 2 , we have (5.16)

γ(t) ∈ Γ + (R/2, d, σ + ) ∪ (B(0, R/2) × R n ).
The lemma follows from (5.11), (5.16) and a compactness argument.

Recall that the microsupport of g

-(x)e iψ -(x)/h ∈ C ∞ 0 (R n ) is contained in Γ -(R -, d 1 , σ 1 ). Let ω -(x, ξ) ∈ A 0 with ω -= 1 near Γ -(R -/2, d 1 , σ 1 ) and supp(ω -) ⊂ Γ -(R -/3, d 0 , σ 0 ). Using the identity (5.17) u -= i h T 0 e -it(P -E)/h (g -e iψ -/h )dt + R(E + i0, h)e -iT (P -E)/h (g -e iψ -/h ),
and Proposition 4.1, Proposition 4.2 and Lemma 5.1, we get

(5.18) Op h (ω -)u -= Op h (ω -) i h T 0 e -it(P -E)/h (g -e iψ -/h )dt + O(h ∞ ),
for some T > 0 large enough. In particular,

(5.19) MS(Op h (ω -)u -) ⊂ Λ - ω ∩ (B(0, R -+ 1) × R n ). 5.2. Computation of u -along γ - k .
Here we compute u -microlocally along a trajectory γ - k . We recall that

γ - k is a bicharac- teristic curve (x - k (t), ξ - k (t)) such that (x - k (t), ξ - k (t)
) → (0, 0) as t → +∞, and such that, as t → -∞,

(5.20) |x - k (t) - √ 2E 0 ωt -z - k | → 0, |ξ - k (t) - √ 2E 0 ω| → 0.
The symbol a -solves (4.7) and (4.8

) near γ - k ∩ MS(g -e iψ -/h ). In particular, if R -is large enough, microlocally near γ - k ∩ Γ -(R -/2, d 1 , σ 1 ) ∩ (B(0, R -) × R n ), u -is
given by (5.18) and

u -= i h T 0 e -it(P -E)/h ([χ -, P ]a -e iψ -/h )dt + O(h ∞ ) = i h T 0 e -it(P -E)/h (χ -(P -E)a -e iψ -/h )dt - i h T 0 e -it(P -E)/h ((P -E)χ -a -e iψ -/h )dt + O(h ∞ ) = - i h T 0 (P -E)e -it(P -E)/h (χ -a -e iψ -/h )dt + O(h ∞ ) = -(P -E)R(E + i0, h)a -e iψ -/h + O(h ∞ ) = -a -e iψ -/h + O(h ∞ ). (5.21)
Now, using (5.21), and the fact that u -is a semiclassical distribution satisfying (5.22) (P -E)u -= 0, near B(0, R -), we can compute u -microlocally near γ - k ∩ B(0, R -) using Maslov's theory (see [START_REF] Maslov | Semiclassical approximation in quantum mechanics[END_REF] for more details). Moreover, it is proved in Proposition C.1 (see also [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]Lemma 5.8]) that the Lagrangian manifold Λ - ω has a nice projection with respect to x in a neighborhood of γ - k close to (0, 0). Then, in such a neighborhood, u -can be written as

(5.23) u -(x) = -a -(x, h)e -iν - k π/2 e iψ -(x)/h ,
where ν - k denotes the Maslov index of γ - k . The phase ψ -satisfies the usual eikonal equation (5. [START_REF] Martinez | Resonance free domains for non globally analytic potentials[END_REF] p(x, ∇ψ -) = E 0 .

Here, to the contrary of (4.21), we have written 1), and we choose to work with E 1 in the amplitudes instead of the phases. As usual, we have

E = E 0 + E 1 h with E 1 = O(
(5.25) ∂ t (ψ -(x - k (t))) = ∇ψ -(x - k (t)) • ∂ t x - k (t) = ∇ψ -(x - k (t)) • ξ - k (t) = |ξ - k (t)| 2 , so that (5.26) ψ -(x - k (t)) = ψ -(x - k (s)) + t s |ξ - k (u)| 2 du
We also have ψ

-(x - k (s)) = ( √ 2E 0 ωs + z - k ) • √ 2E 0 ω + o(1)
as s → -∞, and then

(5.27) ψ -(x - k (t)) = 2E 0 s + t s |ξ - k (u)| 2 du + o(1), s → -∞.
We have obtained in particular that (5.28)

ψ -(x - k (t)) = t -∞ |ξ - k (u)| 2 -2E 0 1 u<0 du = t -∞ 1 2 |ξ - k (u)| 2 -V (x - k (u)) + E 0 sgn(u) du.
We turn to the computation of the symbol. The function a -(x, h) ∼ ∞ k=0 a -,k (x)h k satisfies the usual transport equations:

(5.29)

     ∇ψ -• ∇a -,0 + 1 2 (∆ψ --2iE 1 )a -,0 = 0, ∇ψ -• ∇a -,k + 1 2 (∆ψ --2iE 1 )a -,k = i 1 2 ∆a -,k-1 , k ≥ 1,
In particular, we get for the principal symbol

(5.30) ∂ t (a -,0 (x - k (t))) = ∇a -,0 (x - k (t)) • ξ - k (t) = ∇a -,0 (x - k (t)) • ∇ψ -(x - k (t)), so that, (5.31) ∂ t (a -,0 (x - k (t))) = - 1 2 ∆ψ -(x - k (t)) -2iE 1 a -,0 (x - k (t)) and then (5.32) a -,0 (x - k (t)) = a -,0 (x - k (s)) exp - 1 2 t s ∆ψ -(x -(u)) du + i(t -s)E 1 .
On the other hand, from [32, Lemma 4.3], based on Maslov theory, we have

(5.33) a -,0 (x - k (t)) = (2E 0 ) 1/4 D - k (t) -1/2 e itE 1
, where (5.34)

D - k (t) = det ∂x -(t, z, ω, E 0 ) ∂(t, z) | z=z - k .

Computation of u -at the critical point

In this section we use the results of [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF] to obtain a representation of u -in a whole neighborhood of the critical point. Indeed we saw already that (P -E)u -= 0 outside the support of g -, in particular in a neighborhood of the critical point. First, we need to recall some terminology from [START_REF] Helffer | Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells[END_REF] and [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF].

We recall from Section 2 that (µ j ) j≥0 is the strictly increasing sequence of linear combinations over N of the λ j 's, with µ 0 = 0. Let u(t, x) be a function defined on [0, +∞[×U , U ⊂ R m . Definition 6.1. We say that u : [0, +∞[×U → R, a smooth function, is expandible, if, for any N ∈ N, ε > 0, (α, β) ∈ N 1+m , (6.1)

∂ α t ∂ β x u(t, x) - N j=1 u j (t, x)e -µ j t = O e -(µ N +1 -ε)t ,
for a sequence (u j ) j of smooth functions, which are polynomials in t. We shall write u(t, x) ∼ j≥1 u j (t, x)e -µ j t , when (6.1) holds.

We say that f (t, x) = O(e -µt ) if for all (α, β) ∈ N 1+m and ε > 0 we have (6.2)

∂ α t ∂ β x f (t, x) = O(e -(µ-ε)t
). Definition 6.2. We say that u(t, x, h), a smooth function, is of class S A,B if, for any ε > 0, (α, β) ∈ N 1+m , (6.3)

∂ α t ∂ β x u(t, x, h) = O h A e -(B-ε)t . Let S ∞,B = A∈R S A,B
. We shall say that u(t, x, h) is a classical expandible function of order (A, B), if, for any K ∈ N,

(6.4) u(t, x, h) - K k=A u k (t, x)h k ∈ S K+1,B ,
for a sequence (u k ) k of expandible functions. We shall write

u(t, x, h) ∼ k≥A u k (t, x)h k , in that case.
Now, since the intersection between Λ - ω and Λ -is transverse along the trajectories γ - k (z - k ), and since g - 1 (z - k ) = 0, Theorem 2.1 and Theorem 5.4 of [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF] imply that one can write, microlocally near (0, 0), (6.5)

u -= 1 √ 2πh N - k=1 α k (t, x, h)e iϕ k (t,x)/h dt,
where the α k (t, x, h)'s are classical expandible functions in S 0,2 Re Σ(E) : (6.6)

α k (t, x, h) ∼ m≥0 α k m (t, x)h m , α k m (t, x) ∼ j≥0 α k m,j (t, x)e -2(Σ(E)+µ j )t ,
and where the α k m,j (t, x)'s are polynomial with respect to t. We recall from (2.28) that, for

E = E 0 + hE 1 , (6.7) Σ(E) = n j=1 λ j 2 -iE 1 .
Following line by line [5, Section 6], we obtain (see [5, (6.26)])

α k 0,0 (0) = -e iπ/4 (2λ 1 ) 3/2 e -iν - k π/2 |g(γ - k )|(D - k ) -1/2 (2E 0 ) 1/4 . (6.8)
Notice that from (5.32) and Proposition C.1, we have 0 < D - k < +∞.

From [5, Section 5], we recall that the phases ϕ k (t, x) satisfy the eikonal equation (6.9)

∂ t ϕ k + p(x, ∇ x ϕ k ) = E 0 ,
and that they have the asymptotic expansions (6.10)

ϕ k (t, x) ∼ +∞ j=0 M k j m=0 ϕ k j,m (x)t m e -µ j t ,
with M k j < +∞. In the following, we set (6.11)

ϕ k j (t, x) = M k j m=0 ϕ k j,m (x)t m ,
and, still from [5, Section 5], we have that the first ϕ k j 's are of the form

ϕ k 0 (t, x) =ϕ + (x) + c k (6.12) ϕ k 1 (t, x) = -2λ 1 g - 1 (z - k ) • x + O(x 2 ), (6.13)
where c k ∈ R is the constant depending on k given by (6.14)

c k = "ψ -(0)" = lim t→+∞ ψ -(x - k (t)) = S - k ,
thanks to (5.28) (see also [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]Lemma 5.10]). Moreover ϕ + is the generating function of the outgoing stable Lagrangian submanifold Λ + with ϕ + (0) = 0, and

(6.15) ϕ + (x) = j λ j 2 x 2 j + O(x 3 ).
The fact that ϕ k 1 (t, x) does not depend on t and the expression (6.13) follows also from Corollary 6.6 and (6.109).

6.1. Study of the transport equations for the phases. Now, we examine the equations satisfied by the functions ϕ k j (t, x), defined in (6.10) and (6.11), for the integers j ≤  (recall that  is defined by µ  = 2λ 1 ). For clearer notation, we omit the superscript k until further notice.

Recall that ϕ(t, x) satisfies the eikonal equation (6.9), which implies (see (6.10)) (6.16)

j M j m=0 e -µ j t ϕ j,m (x)(-µ j t m + mt m-1 ) + 1 2 j M j m=0 ∇ϕ j,m (x)t m e -µ j t 2 + V (x) ∼ E 0 ,
and then

j M j m=0 e -µ j t ϕ j,m (x)(-µ j t m + mt m-1 ) + 1 2 j,  M j m=0 M  m=0 ∇ϕ j,m ∇ϕ , m (x)e -(µ j +µ  )t t m+ m
+V (x) ∼ E 0 . (6.17) When µ j < 2λ 1 , the cross product in the previous formula provides a term of the form e -µ j t if and only if µ j = 0 or µ  = 0. In particular, the term of order e -µ j t in (6.17) gives (6.18)

M j m=0 ϕ j,m (x)(-µ j t m + mt m-1 ) + ∇ϕ + (x) • M j m=0 ∇ϕ j,m (x)t m = 0.
When µ j = 2λ 1 , one gets also a term of order e -2λ 1 t for µ j = µ  = λ 1 and then

M j m=0 ϕ j,m (x)(-µ j t m + mt m-1 ) + ∇ϕ + (x) • M j m=0 ∇ϕ j,m (x)t m + 1 2 M 1 m=0 M 1 m=0 t m+ m ∇ϕ 1,m (x)∇ϕ 1, m (x) = 0. (6.19)
To study these equations, we denote by (6.20)

L = ∇ϕ + (x) • ∇
the vector field that appears in (6.18) and (6.19). We set also L 0 = j λ j x j ∂ j its linear part at x = 0, and we begin with the study of the solution of

(6.21) (L -µ)f = g, with µ ∈ R and f , g ∈ C ∞ (R n ).
First of all, we show that it is sufficient to solve (6.21) for formal series.

Proposition 6.3. Let g ∈ C ∞ (R n
) and g 0 be the Taylor series of g at 0. For each formal series f 0 such that (Lµ)f 0 = g 0 , there exists a unique function f ∈ C ∞ (R n ) defined near 0 such that f has Taylor series f 0 at 0 and

(6.22) (L -µ)f = g, near 0.
Proof. Let f 0 be a C ∞ function having f 0 has Taylor series at 0. With the notation f = f 0 +r, the problem (6.22) is equivalent to finding r = O(x ∞ ) with

(6.23) (L -µ)r = g -(L -µ) f 0 = r,
where r ∈ C ∞ has g 0 -(Lµ)f 0 = 0 as Taylor series at 0. Let y(t, x) be the solution of (6.24)

∂ t y(t, x) = ∇ϕ + (y(t, x)), y(0, x) = x.
Thus, (6.23) is equivalent to (6.25) r(x) = 0 t e -µs r(y(s, x))ds + e -µt r(y(t, x)).

Since r(x), r(x) = O(x ∞ ) and y(s, x) = O(e λ 1 t |x|) for t < 0, the functions e -µt r(y(t, x)), e -µt r(y(t, x)) are O(e N t ) as t → -∞ for all N > 0. Then (6.26)

r(x) = 0 -∞
e -µs r(y(s, x))ds, and r(x) = O(x ∞ ). The uniqueness follows and it is enough to prove that r given by (6.26) is C ∞ . We have (6.27) ∂ t (∇ x y) = (∇ 2 x ϕ + (y))(∇ x y), and since ∇ 2

x ϕ + is bounded, there exists C > 0 such that (6.28)

|∇ x y(t, x)| e -Ct , has t → -∞.
Then, e -µs (∇ r)(y(s, x))(∂ j y(t, x)) = O(e N t ) as t → -∞ for all N > 0 and ∂ j r(x) = 0 -∞ e -µs (∇ r)(y(s, x))(∂ j y(t, x))ds. The derivatives of order greater than 1 can be treated in the same way.

We let (6.29)

L µ = L -µ : C x → C x ,
where we use the standard notation C x for formal series, and C p x for formal series of degree ≥ p. We notice that (6.30)

L µ x α = (L 0 -µ)x α + C |α|+1 x = (λ • α -µ)x α + C |α|+1 x .
Recall that I ℓ (µ) has been defined in (2.22). The number of elements in I ℓ (µ) will be denoted (6.31) n ℓ (µ) = #I ℓ (µ).

One has for example n 2 (µ) = n 1 (µ)(n 1 (µ)+1) i) The kernel of L µ has dimension n 1 (µ), and one can find a basis (E j 1 , . . . , E j n 1 (µ) ) of Ker L µ such that E j (x) = x j + C 2 x , j ∈ I 1 (µ).

ii) A formal series F = F 0 + n j=1 F j x j + C 2 x belongs to Im L µ if and only if F j = 0 for all j ∈ I 1 (µ).

Remark 6.5. Thanks to Propostion 6.3, the same result is true for germs of C ∞ functions at 0. Notice that when µ = µ j for all j, L µ is invertible.

Proof. For a given F = α F α x α ∈ C x , we look for solutions

E = α E α x α ∈ C x to the equation (6.32) L µ α E α x α = α F α x α .
The calculus of the term of order x 0 in (6.32) leads to the equation (6.33)

E 0 = - F 0 µ .
With this value for E 0 , (6.32) becomes, using again (6.30), (6.34)

|α|=1 (λ • α -µ)E α x α = |α|=1 F α x α + C 2 x .
We have two cases:

If α / ∈ I 1 (µ), one should have (6.35) E α = F α λ • α -µ .
If α ∈ I 1 (µ), the formula (6.34) gives F α = 0. In that case, the corresponding E α can be chosen arbitrarily. Now suppose that the E α are fixed for all |α| ≤ p -1 (with p ≥ 2), and such that (6.36)

L µ |α|≤p-1 E α x α = α F α x α + C p x .
We can write (6.32) as (6.37)

L µ |α|=p E α x α = α F α x α -L µ |α|≤p-1 E α x α + C p+1 x ,
or, using again (6.30), (6.38)

|α|=p (λ • α -µ)E α x α = |α|≤p F α x α -L µ |α|≤p-1 E α x α + C p+1 x .
Since |α| ≥ 2, one has λ • α ≥ 2λ 1 > µ, so that (6.38) determines by induction all the E α 's for |α| = p in a unique way.

Corollary 6.6. If j < , the function ϕ j (t, x) does not depend on t, i.e. we have M j = 0.

Proof. Suppose that M j ≥ 1, then (6.18) gives the system (6.39) (Lµ j )ϕ j,M j = 0, (Lµ j )ϕ j,M j -1 = -M j ϕ j,M j , with ϕ j,M j = 0. But this would imply that ϕ j,M j ∈ Ker L µ ∩ Im L µ , a contradiction.

As a consequence, for j < , the equation (6.18) on ϕ j reduces to (6.40) (Lµ j )ϕ j,0 = 0, and, from Proposition 6.4, we get that (6.41)

ϕ j (t, x) = ϕ j,0 (x) = k∈I 1 (µ j ) d j,k x k + O(x 2 ).
We now consider the case j = , and we study (6.19). We have already seen that ϕ 1 does not depend on t, so that this equation can be written (6.42)

M j m=0 ϕ j,m (x)(-µ j t m + mt m-1 ) + ∇ϕ + (x) • M j m=0 ∇ϕ j,m (x)t m + 1 2 ∇ϕ 1 (x) 2 = 0.
As for the study of (6.18), we begin with that of (6.21), with µ = 2λ 1 . We denote by Ψ : R n 1 (2λ 1 ) -→ R n 2 (λ 1 ) the linear map (6.43)

Ψ(E β 1 , . . . , E β n 1 (2λ 1 ) ) = β∈I 1 (2λ 1 ) E β 1 α! ∂ α (L -µ)x β | x=0 α∈I 2 (λ 1 ) ,
and we set (6.44) n(Ψ) = dim Ker Ψ.

Recalling that L = ∇ϕ + (x) • ∇, we see that

(6.45) Ψ(E β 1 , . . . , E β n 1 (2λ 1 ) ) = β∈I 1 (2λ 1 ) E β ∂ α ∂ β ϕ + (0) α! α∈I 2 (λ 1 )
.

More generally, for any |α| = 2, we denote

(6.46) Ψ α ((E β ) β∈I 1 (2λ 1 ) ) = β∈I 1 (2λ 1 ) E β ∂ α ∂ β ϕ + (0) α! •
Then, at the level of formal series, we have the Proposition 6.7. Suppose µ = 2λ 1 . Then i) Ker L µ has dimension n 2 (λ 1 ) + n(Ψ).

ii) A formal series F = α F α x α belongs to Im L µ if and only if ∀α ∈ I 1 (2λ 1 ), F α = 0, (6.47)

|β|=1 β / ∈I 1 (2λ 1 ) ∂ β ∂ α ϕ + (0) α! F β 2λ 1 -λ • β + F α α∈I 2 (λ 1 ) ∈ Im Ψ. (6.48) iii) If F ∈ Im L µ , any formal series E = α E α x α with L µ E = F satisfies E 0 = 1 -2λ 1 F 0 , (6.49) E α = 1 λ • α -2λ 1 F α , for α ∈ I 1 \ I 1 (2λ 1 ), (6.50) Ψ (E β β∈I 1 (2λ 1 ) ) = |β|=1 β / ∈I 1 (2λ 1 ) ∂ β ∂ α ϕ + (0) α! F β 2λ 1 -λ • β + F α α∈I 2 (λ 1 ) . (6.51) Moreover for α ∈ I 2 \ I 2 (λ 1 ), one has (6.52) E α = 1 λ • α -2λ 1 F α -Ψ α ((E β ) β∈I 1 (2λ 1 ) ) + |β|=1 β / ∈I 1 (2λ 1 ) F β 2λ 1 -λ • β ∂ α+β ϕ + (0) α! .
Lastly, E is completely determined by F and a choice of the E α for |α| ≤ 2 such that (6.49)-(6.52) are satisfied. iv) Ker L µ ∩ Im(L µ ) 2 = {0}.

Proof. For a given F = α F α x α we look for a E = α E α x α such that L 2λ 1 E = F . First of all, we must have (6.53)

E 0 = - F 0 2λ 1 .
When this is true, we get (6.54)

|α|=1 E α (L 0 -2λ 1 )x α = |α|=1 F α (L -2λ 1 )x α + C 2 x ,
and we obtain as necessary condition that F α = 0 for any α ∈ I 1 (2λ 1 ). So far, the E α for α ∈ I 1 (2λ 1 ) can be chosen arbitrarily, and we must have (6.55)

E α = F α λ • α -2λ 1 , α ∈ I 1 \ I 1 (2λ 1 ).
We suppose that (6.53) and (6.55) hold. Then we have (6.56)

|α|=2 E α (L 0 -2λ 1 )x α = |α|=2 F α x α + |α|=1 α / ∈I 1 (2λ 1 ) F α x α - |α|=1 E α (L -2λ 1 )x α + C 3 x .
Notice that the second term in the R.H.S of (6.56) belongs to C 2 x thanks to (6.55). Again, we have two cases:

• When α ∈ I 2 (λ 1 ), the corresponding E α can be chosen arbitrarily, but one must have

F α = |β|=1 E β 1 α! ∂ α (L -2λ 1 )x β | x=0 (6.57) =Ψ α ((E β ) β∈I 1 (2λ 1 ) ) + |β|=1 β / ∈I 1 (2λ 1 ) E β ∂ α+β ϕ + (0) α! , (6.58)
and this, with (6.55), gives (6.51).

• When |α| = 2, α / ∈ I 2 (λ 1 ), one obtains

E α = 1 λ • α -2λ 1 F α - |β|=1 E β 1 α! ∂ α (L -2λ 1 )x β | x=0 = 1 λ • α -2λ 1 F α -Ψ α ((E β ) β∈I 1 (2λ 1 ) ) - |β|=1 β / ∈I 1 (2λ 1 ) E β ∂ α+β ϕ + (0) α! , (6.59)
and this, with (6.55), gives (6.52). Now suppose that (6.53), (6.55), (6.57) and (6.59) hold, and that we have chosen a value for the free variables E α for α ∈ I 1 (2λ 1 ) ∪ I 2 (λ 1 ). Thanks to the fact that λ • α = 2λ 1 for any α ∈ N n with |α| = 3, we see as in the proof of Propostion 6.4, that the equation ( 6.54) has a unique solution, and the points (i), (ii) and (iii) follow easily.

To prove the last point of the proposition, suppose that (6.60)

E = α∈N n E α x α ∈ Ker L µ ∩ Im(L µ ) 2 .
First, we have E ∈ Ker L µ ∩ Im L µ . Thus, E 0 = 0 by (6.49), E α = 0 for α ∈ I 1 (2λ 1 ) by (6.47), and E α = 0 for α ∈ I 1 \ I 1 (2λ 1 ) by (6.50). Last, since L µ E = 0, we also have E α = 0 for α ∈ I 2 \ I 2 (λ 1 ), and finally, (6.61)

E = α∈I 2 (λ 1 ) E α x α + C 3 x .
Moreover, one can write E = L µ G for some G ∈ Im L µ . Since E 0 = 0, we must have G 0 = 0. Since G ∈ Im L µ , by (6.47), we have G α = 0 for α ∈ I 1 (2λ 1 ). Finally, since E α = 0 for |α| = 1, α / ∈ I 1 (2λ 1 ), the same is true for the corresponding G α , and

(6.62) G = |α|≥2 G α x α .
Then, since L µ x α = 0 + C 3 [x] for α ∈ I 2 (λ 1 ), we obtain E α = 0 for α ∈ I 2 (λ 1 ). As above, we then get that, for |α| ≥ 3, E α = 0, and this ends the proof.

Corollary 6.8. We have M  ≤ 2. If, in addition, λ k = 2λ 1 for all k ∈ {1, . . . , n}, then M  ≤ 1.

Proof. Suppose that M  ≥ 3. Then (6.42) gives

(L -µ  )ϕ ,M  = 0 (6.63) (L -µ  )ϕ ,M  -1 = -M  ϕ ,M  (6.64) (L -µ  )ϕ ,M  -2 = -(M  -1)ϕ ,M  -1 , (6.65)
with ϕ ,M  = 0. Notice that we have used the fact that M  -2 > 0 in (6.65). But this gives ϕ

,M  ∈ Ker(L -µ  ) and (L -µ  ) 2 ϕ ,M  -2 = M  (M  -1)ϕ ,M  , so that ϕ ,M  ∈ Im(L -µ  ) 2 .
This contradicts point (iv) of Proposition 6.7. Now we suppose that λ k = 2λ 1 for all k ∈ {1, . . . n}, that is I 1 (2λ 1 ) = ∅, and that M  = 2. Then (6.42) gives

(L -µ  )ϕ ,M  = 0 (6.66) (L -µ  )ϕ ,M  -1 = -M  ϕ ,M  (6.67) with ϕ ,M  = 0. Therefore we have ϕ ,M  ∈ Ker L µ  ∩ Im L µ  ,
and we get the same conclusion as in (6.61): ϕ ,M  (x) = O(x 2 ). Then, we write (6.68) ϕ ,M  = (Lµ  )g, and we see, as in (6.62), that g = O(x 2 ), here because I 1 (2λ 1 ) = ∅. Finally, we conclude also that ϕ ,M  = 0, a contradiction.

6.2.

Taylor expansions of ϕ + and ϕ k 1 . Now we compute the Taylor expansions of the leading terms with respect to t, of the phase functions ϕ(t, x) = ϕ k (t, x). Lemma 6.9. The smooth function ϕ

+ (x) = n j=1 λ j 2 x 2 j + O(x 3 ) satisfies (6.69) ∂ α ϕ + (0) = - 1 λ • α ∂ α V (0),
for |α| = 3, and (6.70)

∂ α ϕ + (0) = - 1 2(λ • α) n j=1 β,γ∈I 2 α=β+γ α! β! γ! ∂ j ∂ β V (0) λ j + λ • β ∂ j ∂ γ V (0) λ j + λ • γ - 1 λ • α ∂ α V (0), for |α| = 4, where α, β, γ ∈ N n .
Proof. The smooth function x → ϕ + (x) is defined in a neighborhood of 0, and it is characterized (up to a constant: we have chosen ϕ + (0) = 0) by (6.71)

     p(x, ∇ϕ + (x)) = 1 2 |∇ϕ + (x)| 2 + V (x) = E 0 ∇ϕ + (x) = (λ j x j ) j=1,...,n + O(x 2 )
The Taylor expansion of ϕ + at x = 0 is (6.72)

ϕ + (x) = n j=1 λ j 2 x 2 j + |α|=3,4 ∂ α ϕ + (0) α! x α + O(x 5 ),
and we have (6.73)

∂ j ϕ + (x) = λ j x j + |α|=3,4 α j ∂ α ϕ + (0) α! x α-1 j + O(x 4 ). Therefore |∇ϕ + (x)| 2 = n j=1 λ 2 j x 2 j + 2 |α|=3 n j=1 λ j α j ∂ α ϕ + (0) α! x α + 2 |α|=4 n j=1 λ j α j ∂ α ϕ + (0) α! x α + n j=1 |α|=3 α j ∂ α ϕ + (0) α! x α-1 j 2 + O(x 5
). (6.74) Let us compute further the last term in (6.74):

n j=1 |α|=3 α j ∂ α ϕ + (0) α! x α-1 j 2 = n j=1 |β|,|γ|=3 β j γ j ∂ β ϕ + (0) β! ∂ γ ϕ + (0) γ! x β+γ-21 j = n j=1 |α|=4 x α α=β+γ |β|,|γ|=2 ∂ j ∂ β ϕ + (0) β! ∂ j ∂ γ ϕ + (0) γ! • (6.75)
Writing the Taylor expansion of V at x = 0 as (6.76)

V (x) = E 0 - n j=1 λ 2 j 2 x 2 j + |α|=3,4 ∂ α V (0) α! x α + O(x 5 ),
and using the eikonal equation (6.71), we obtain first, for any α ∈ N n with |α| = 3, (6.77)

∂ α ϕ + (0) = - 1 λ • α ∂ α V (0).
Then, (6.74) and (6.75) give (6.78)

∂ α ϕ + (0) = - 1 λ • α ∂ α V (0) - 1 2(λ • α) n j=1 β,γ∈I 2 α=β+γ α! β!γ! ∂ j ∂ β V (0) λ j + λ • β ∂ j ∂ γ V (0) λ j + λ • γ , for |α| = 4.
Now we turn to the function ϕ 1 . This function is a solution, in a neighborhood of x = 0, of the transport equation (6.79)

Lϕ 1 (x) = λ 1 ϕ 1 (x),
where L is given in (6.20).

Lemma 6.10.

The C ∞ function ϕ 1 (x) = -2λ 1 g - 1 (z -) • x + O(x 2 ) satisfies (6.80) ∂ α ϕ 1 (0) = 2λ 1 α! (λ 1 -λ • α)(λ 1 + λ • α) n j=1 ∂ j ∂ α V (0) α! g - 1 (z -) j ,
for |α| = 2, and

∂ α ϕ 1 (0) = - 2λ 1 λ 1 -λ • α 1 k ∈I 1 (λ 1 ),j∈I 1 β,γ∈I 2 α+1 j =β+γ α!γ j β!γ! ∂ j ∂ β V (0) λ j + λ • β ∂ k ∂ γ V (0) (λ 1 -λ • γ)(λ 1 + λ • γ) g - 1 (z -) k + λ 1 (λ 1 -λ • α)(λ 1 + λ • α) k∈I 1 , j ∈I 1 (λ 1 ) β,γ∈I 2 1 j +α=β+γ (α + 1 j )! β!γ! ∂ k ∂ β V (0) λ k + λ • β ∂ k ∂ γ V (0) λ k + λ • γ g - 1 (z -) j + 2λ 1 (λ 1 -λ • α)(λ 1 + λ • α) 1 j ∈I 1 (λ 1 ) ∂ j ∂ α V (0) g - 1 (z -) j . (6.81) for |α| = 3.
Proof. We write (6.82)

ϕ 1 (x) = n j=1 a j x j + |α|=2,3 a α x α + O(x 4 ),
and Lemma 6.9 together with (6.73) give all the coefficients in the expansion (6.83)

∇ϕ + (x) = λ j x j + |α|=2,3 A j,α x α + O(x 4 ) j=1,...,n
.

In fact, we have (6.84)

A j,α = ∂ α+1 j ϕ + (0) α! and a α = ∂ α ϕ 1 (0) α! .
We get

Lϕ 1 (x) = n j=1 ∂ j ϕ + (x)∂ j ϕ 1 (x) = n j=1 a j λ j x j + |α|=2 α j λ j a α + a j A j,α x α + |α|=3 α j λ j a α x α + |β|=|γ|=2 A j,β γ j a γ x β+γ-1 j + |α|=3 a j A j,α x α + O(x 4 ) = n j=1 a j λ j x j + |α|=2 λ • α a α + n j=1 A j,α a j x α + |α|=3 λ • α a α + n j=1 α=β+γ-1 j |β|,|γ|=2
A j,β γ j a γ + a j A j,α x α + O(x 4 ). (6.85) Thus, (6.79) gives, for all α ∈ N n with |α| = 2, (6.86)

a α = 1 λ 1 -λ • α n j=1 A j,α a j ,
and, for all α ∈ N n with |α| = 3, (6.87)

a α = 1 λ 1 -λ • α n j=1 β,γ∈I 2 α+1 j =β+γ γ j A j,β a γ + a j A j,α .
Then, the lemma follows from (6.84).

Asymptotics near the critical point for the trajectories.

The knowledge obtained so far is not sufficient for the computation of the ϕ j 's. We shall obtain here some more information by studying the behavior of the incoming trajectory γ -(t) as t → +∞. We recall from [20, Section 3] (see also [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]Section 5]), that the curve

γ -(t) = (x -(t), ξ -(t)) ∈ Λ -∩ Λ -
ω satisfies, in the sense of expandible functions, (6.88)

γ -(t) = j≥1 M ′ j m=0 γ - j,m t m e -µ j t ,
Notice that we continue to omit the subscript k for γ

- k = (x - k , ξ - k ), z - k , . . . Writing also (6.89) x -(t) ∼ +∞ j=1 g - j (t, z -)e -µ j t , g - j (t, z -) = M ′ j m=0 g - j,m (z -)t m ,
for some integers M ′ j , we know that g -

1 (t, z -) = g - 1,0 (z -) = 0. Since ξ -(t) = ∂ t x -(t), we have (6.90) ξ -(t) ∼ +∞ j=1 M ′ j m=0 g - j,m (z -)(-µ j t m + mt m-1 )e -µ j t .
Proposition 6.11. If j < , then M ′ j = 0. We also have M ′  ≤ 1, and M ′  = 0 when I 1 (2λ 1 ) = ∅. Moreover

(6.91) (g - ,1 (z -)) β =      1 4λ 1 |α|=2 ∂ α+β V (0) α! (g - 1 (z -)) α for β ∈ I 1 (2λ 1 ), 0 for β / ∈ I 1 (2λ 1 ),
and, for |β| = 1, β / ∈ I 1 (2λ 1 ),

(6.92) (g - ,0 (z -)) β = 1 (2λ 1 + λ • β)(2λ 1 -λ • β) |α|=2 ∂ α+β V (0) α! (g - 1 (z -)) α .
Proof. First of all, since ∂ t γ -(t) = H p (γ -(t)), we can write (6.93)

∂ t γ -(t) = F p (γ -(t)) + O(t 2M ′ 1 e -2λ 1 t ),
where (6.94)

F p = d (0,0) H p = 0 I Λ 2 0 , Λ 2 = diag(λ 2 1 , . . . , λ 2 n ).
We obtain (6.95)

1≤j<  M ′ j m=0 (F p + µ j )γ - j,m t m e -µ j t = 1≤j<  M ′ j m=0 γ - j,m mt m-1 e -µ j t .
Now suppose j <  and M ′ j ≥ 1. We get, for this j, for some γ - j,M ′ j = 0, (6.96)

   (F p + µ j )γ - j,M ′ j = 0, (F p + µ j )γ - j,M ′ j -1 = M ′ j γ - j,M ′ j ,
so that Ker(F p + µ j ) ∩ Im(F p + µ j ) = {0}. Since F p is a diagonizable matrix, this can easily be seen to be a contradiction.

Now we study M ′  . So far we have obtained that (6.97)

γ -(t) = 1≤j<  γ - j e -µ j t + M ′  m=0 γ - ,m t m e -2λ 1 t + O(t C e -µ +1 t ),
and we can write (6.98)

H p (x, ξ) =     ξ Λ 2 x - |α|=2 ∂ α ∇V (0) α! x α + O(x 3 )     .
Thus we have (6.99)

H p (γ -(t)) = F p j<  γ - j e -µ j t + M ′  m=0 γ - ,m t m e -2λ 1 t + e -2λ 1 t A(γ - 1 ) + O(e -(2λ 1 +ε)t ),
where, noticing that µ j + µ j ′ = 2λ 1 if and only if j = j ′ = 1, (6.100)

A(γ - 1 ) =     0 - |α|=2 ∂ α ∇V (0) α! (g - 1 ) α     .
For the terms of order e -2λ 1 t , we have, since

∂ t γ -(t) = H p (γ -(t)), (6.101) (F p + 2λ 1 ) M ′  m=0 γ - ,m t m = M ′  m=0 γ - ,m mt m-1 -A(γ - 1 ).
Thus, if we suppose that M ′  ≥ 2, we obtain (6.102)

   (F p + 2λ 1 )γ - ,M ′  = 0, (F p + 2λ 1 )γ - ,M ′ -1 = M ′  γ - ,M ′  .
Then again we have γ -

,M ′  ∈ Ker(F p + 2λ 1 ) ∩ Im(F p + 2λ 1 ), a contradiction.
Finally, if λ j = 2λ 1 for all j, then Ker(F p + 2λ 1 ) = {0}. Therefore, if we suppose that M ′  = 1, we see that γ ,1 = 0 satisfies the first equation in (6.102) and we obtain a contradiction.

Now we compute γ

-  (t) = γ - ,1 t + γ - ,0 . We have (6.103) (F p + 2λ 1 )γ - ,1 = 0, (F p + 2λ 1 )γ - ,0 = γ - ,1 -A(γ - 1 ),
and we see that

γ - ,1 = Πγ - ,1 = ΠA(γ - 1 )
, where Π is the projection on the eigenspace of F p associated to -2λ 1 . We denote by e j = (δ i,j ⊗ 0) i=1,...,n and ε j = (0 ⊗ δ i,j ) i=1,...,n for j = 1, . . . , n, so that (e 1 , . . . e n , ε 1 , . . . , ε n ) is the canonical basis of R 2n = T (0,0) T * R n . Then it is easy to check that, for all j, v ± j = e j ± λ j ε j is an eigenvector of F p for the eigenvalue ±λ j . In the basis {e 1 , ε 1 , . . . , e n , ε n } the projector Π is block diagonal and, if K j = span(e j , ε j ), we have

(6.104) Π | K j =    1/2 -1/4λ 1 -λ 1 1/2 for j ∈ I 1 (2λ 1 ), 0 for j / ∈ I 1 (2λ 1 ).
Therefore, we obtain

(6.105) (g - ,1 ) β =      - 1 4λ 1 |α|=2 ∂ β ∂ α V (0) α! (g - 1 (z -)) α for β ∈ I 1 (2λ 1 ), 0 for β / ∈ I 1 (2λ 1 ).
Now suppose that k / ∈ I 1 (2λ 1 ). Then the second equality in (6.103) restricted to K k gives (6.106)

2λ 1 1 λ 2 k 2λ 1 Π k γ ,0 = -Π k A(γ - 1 ),
where Π k denotes the projection onto K k . Solving this system, one gets

(6.107) (g - ,0 ) k = 1 4λ 2 1 -λ 2 k Π x Π k A(γ - 1 ),
and, together with (6.100), this ends the proof of Proposition 6.11.

6.4. Computation of the ϕ k j 's.

Here we compute the ϕ k j 's for j ≤ . We continue to omit the superscript k. From [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF], we know that ξ -(t) = ∇ x ϕ t, x -(t) , so that, using Corollary 6.6, and Corollary 6.8,

ξ -(t) =∇ϕ + (x -(t)) + ∇ϕ 1 (x -(t))e -λ 1 t + 2≤j<  ∇ϕ j (0)e -µ j t + ∇ϕ ,2 (0)t 2 e -2λ 1 t + ∇ϕ ,1 (0)te -2λ 1 t + ∇ϕ ,0 (0)e -2λ 1 t + O(e -µ +1 t ). (6.108) Since ϕ + = -ϕ -and (x -, ξ -) ∈ Λ -, we have ∇ϕ + (x -(t)) = -ξ -(t)
, and we obtain first, by (6.90), (6.109)

∇ϕ j (0) = -2µ j g - j (z -), for 1 ≤ j < . Now we study ϕ  (t, x) = ϕ ,0 (x) + tϕ ,1 (x) + t 2 ϕ ,2 (x) when I 1 (2λ 1 ) = ∅. It follows from (6.108) that we have (6.110) -4λ 1 g - ,1 (z -) = ∇ϕ ,1 (0), -4λ 1 g - ,0 (z -) + 2g - ,1 (z -) = ∇ϕ ,0 (0) + ∇ 2 ϕ 1 (0)g - 1 (z -).
On the other hand, we have seen that, by (6.42), the functions ϕ ,2 , ϕ ,1 and ϕ ,0 satisfy (6.111)

       (L -2λ 1 )ϕ ,2 = 0, (L -2λ 1 )ϕ ,1 = -2ϕ ,2 , (L -2λ 1 )ϕ ,0 = -ϕ ,1 - 1 2 |∇ϕ 1 (0)| 2 .
In particular ϕ ,2 ∈ Ker(L -2λ 1 ) ∩ Im(L -2λ 1 ) so that (see (6.61)),

(6.112) ϕ ,2 (x) = α∈I 2 (λ 1 ) c 2,α x α + O(x 3 ).
Going back to (6.108), we now obtain (6.113) and this equality is consistent with Proposition 6.11.

ξ -(t) =∇ϕ + (x -(t)) + ∇ϕ 1 (x -(t))e -λ 1 t + 2≤j<  ∇ϕ j (0)e -µ j t ∇ϕ ,1 (0)te -2λ 1 t + ∇ϕ ,0 (0)e -2λ 1 t + O(e -µ +1 t ),
Then, (6.49) and (6.50) give

(6.114) ϕ ,1 (x) = α∈I 1 (2λ 1 ) c 1,α x α + |α|=2 c 1,α x α + O(x 3 ),
and, by (6.51), we have

(6.115) Ψ((c 1,β ) β∈I 1 (2λ 1 ) ) = (-2c 2,α ) α∈I 2 (λ 1 )
.

By (6.52), we also have for |α| = 2, α / ∈ I 2 (λ 1 ), (6.116)

c 1,α = 1 2λ 1 -λ • α β∈I 1 (2λ 1 ) ∂ α+β ϕ + (0) α! c 1,β .
The function ϕ ,0 (x) = |α|≤2 c 0,α x α + O(x 3 ) satisfies (see (6.42))

(6.117) (L -2λ 1 )ϕ ,0 = -ϕ ,1 - 1 2 ∇ϕ 1 (x) 2 .
First of all, the compatibility condition (6.47) gives (6.118)

∀α ∈ I 1 (2λ 1 ), c 1,α = -∇ϕ 1 (0) • ∂ α ∇ϕ 1 (0),
so that in particular, by (6.115), the function ϕ ,2 is known up to O(x 3 ) terms:

(6.119) ∀α ∈ I 2 (λ 1 ), c 2,α = 1 2 β∈I 1 (2λ ) ∂ α+β ϕ + (0) α! ∇ϕ 1 (0) • ∂ β ∇ϕ 1 (0), and
(6.120) ∀α / ∈ I 2 (λ 1 ), |α| = 2, c 1,α = - 1 2λ 1 -λ • α β∈I 1 (2λ 1 ) ∂ α+β ϕ + (0) α! ∇ϕ 1 (0) • ∂ β ∇ϕ 1 (0).
Now (6.49) and (6.50) give (6.121)

c 0,0 = ϕ ,0 (0) = 1 4λ 1 |∇ϕ 1 (0)| 2 ,
and

(6.122) ∀α / ∈ I 1 (2λ 1 ), |α| = 1, c 0,α = 1 2λ 1 -λ • α ∇ϕ 1 (0) • ∂ α ∇ϕ 1 (0).
From the other compatibility condition (6.48), we know that

c 1,α + 1 α! ∇ϕ 1 (0) • ∂ α ∇ϕ 1 (0) + 1 2 β,γ∈I 1 (λ 1 ) β+γ=α ∂ β ∇ϕ 1 (0) • ∂ γ ∇ϕ 1 (0) + |β|=1 β / ∈I 1 (2λ 1 ) ∂ α+β ϕ + (0) α! ∇ϕ 1 (0) • ∂ β ∇ϕ 1 (0) 2λ 1 -λ • β α∈I 2 (λ 1 )
∈ Im Ψ, (6.123) and, from (6.51), we obtain the following relation between the (c 0,β ) β∈I 1 (2λ 1 ) and the (c

1,α ) α∈I 2 (λ 1 ) ∀α ∈ I 2 (λ 1 ), c 1,α = - 1 α! ∂ α ∇ϕ 1 (0) • ∇ϕ 1 (0) - 1 2 β,γ∈I 1 (λ 1 ) β+γ=α ∂ β ∇ϕ 1 (0) • ∂ γ ∇ϕ 1 (0) - β∈I 1 (2λ 1 ) ∂ α+β ϕ + (0) α! c 0,β - |β|=1 β / ∈I 1 (2λ 1 ) ∂ α+β ϕ + (0) α! ∇ϕ 1 (0) • ∂ β ∇ϕ 1 (0) 2λ 1 -λ • β • (6.124)
Using the second equation in (6.110), we obtain, for |β| = 1, (6.125)

c 0,β = -4λ 1 (g - ,0 (z -)) β + 2(g - ,1 (z -)) β -∂ β ∇ϕ 1 (0) • g - 1 (z -).
So far, we have computed the functions ϕ ,1 (x) and ϕ ,2 (x) up to O(x 3 ), in terms of derivatives of ϕ + and ϕ 1 , and of the g - ,m (z -). We shall now use the expressions we have obtained in Section 6.2 and in Section 6.3 to give these functions in terms of g - 1 and of derivatives of V only.

First, by (6.112), (6.119), Lemma 6.9 and Lemma 6.10, we obtain

ϕ ,2 (x) = - 1 8λ 1 γ∈I 1 (2λ 1 )
α,β∈I 2 (λ 1 )

∂ β+γ V (0) (g - 1 (z -)) β β! ∂ α+γ V (0) x α α! + O(x 3 ). (6.126) Then we have (6.127) ϕ ,1 (x) = -4λ 1 g - ,1 (z -) • x + α∈I 2 (λ 1 ) c 1,α x α + |α|=2 α / ∈I 2 (λ 1 ) c 1,α x α + O(x 3 ),
where the c 1,α are given by (6.124) and (6.125) for α ∈ I 2 (λ 1 ), and by (6.120) for α / ∈ I 2 (λ 1 ).

• For |α| = 2, α / ∈ I 2 (λ 1 ), we obtain from (6.116), Lemma 6.9, and Lemma 6.10,

c 1,α = 4λ 2 1 (2λ 1 + λ • α)(2λ 1 -λ • α) × β∈I 1 (2λ 1 ) ∂ α+β V (0) α! n j=1 1 (λ 1 + λ j )(3λ 1 + λ j ) ∂ j ∂ β ∇V (0) • g - 1 (z -)(g - 1 (z -)) j . (6.128)
Since (g - 1 (z -)) j = 0 except for 1 j ∈ I 1 (λ 1 ), we get, changing notation a bit, (6.129)

c 1,α = 1 (2λ 1 + λ • α)(2λ 1 -λ • α) γ∈I 1 (2λ 1 ) β∈I 2 (λ 1 ) ∂ α+γ V (0) α! ∂ β+γ V (0) β! (g - 1 (z -)) β .
• Now we compute c 1,α for α ∈ I 2 (λ 1 ).

For the last term in the R.H.S. of (6.124), we obtain

- |β|=1 β / ∈I 1 (2λ 1 ) ∂ α+β ϕ + (0) α! ∇ϕ 1 (0) • ∂ β ∇ϕ 1 (0) 2λ 1 -λ • β = γ∈I 1 \I 1 (2λ 1 ) β∈I 2 (λ 1 ) 8λ 2 1 (2λ 1 -λ • γ)(λ • γ)(2λ 1 + λ • γ) 2 ∂ α+γ V (0) α! ∂ β+γ V (0) β! (g - 1 (z -)) β . (6.130)
Using (6.91) and (6.125), we have also

- β∈I 1 (2λ 1 ) ∂ α+β ϕ + (0) α! c 0,β = - γ∈I 1 (2λ 1 ) ∂ α+γ V (0) α! (g - ,0 (z -)) γ + 1 4λ 2 1 γ∈I 1 (2λ 1 ) β∈I 2 (λ 1 ) ∂ α+γ V (0) α! ∂ β+γ V (0) β! (g - 1 (z -)) β . (6.131) Now we compute -1 α! ∂ α ∇ϕ 1 (0) • ∇ϕ 1 (0) for α ∈ I 2 (λ 1 )
. We obtain

- 1 α! ∂ α ∇ϕ 1 (0) • ∇ϕ 1 (0) = - β∈I 2 (λ 1 ) ∂ α+β V (0) α!β! (g - 1 (z -)) β - 1 4 n j,p,k=1 β,γ∈I 2 β+γ=α+1p+1 j ((α + 1 p ) j + 1)(α p + 1) (λ k + λ • β)(λ k + λ • γ) ∂ β+1 k V (0) β! ∂ γ+1 k V (0) γ! (g - 1 (z -)) j (g - 1 (z -)) p + 2λ 1 n j,p,k=1 β,γ∈I 2 β+γ=α+1p+1 j (α p + 1)γ j (λ 1 -λ • γ)(λ 1 + λ • γ)(λ j + λ • β) × × ∂ β+1 j V (0) β! ∂ γ+1 k V (0) γ! (g - 1 (z -)) k (g - 1 (z -)) p = I + II + III. (6.132)
Writing δ = 1 j + 1 p , we get (6.133)

II = - 1 2 n k=1 β,γ,δ∈I 2 β+γ=α+δ (α + δ)! (λ k + λ • β)(λ k + λ • γ) ∂ β+1 k V (0) β! ∂ γ+1 k V (0) γ! (g - 1 (z -)) δ α! δ! • Since δ ∈ I 2 (λ 1 ) (otherwise (g - 1 (z -)) δ = 0), we have β, γ ∈ I 2 (λ 1
) and, changing notations a bit, (6.134) II = -1 2

β∈I 2 (λ 1 ) (α + β)! α! γ,δ∈I 2 (λ 1 ) γ+δ=α+β n j=1 1 (2λ 1 + λ j ) 2 ∂ j ∂ γ V (0) γ! ∂ j ∂ δ V (0) δ! (g - 1 (z -)) β β! •
In the last term III, we can suppose that γ = 1 j +1 q for some q ∈ {1, . . . , n}. Then γ j = γ! and, writing β = 1 a + 1 b we have

III = λ 1 n j,k,p=1 (α p + 1)(g - 1 (z -)) k (g - 1 (z -)) p × a,b,q∈I 1 1a+1 b +1q=α+1p (α p + 1) (λ 1 -λ j -λ q )(λ 1 + λ j + λ q )(λ j + λ a + λ b ) ∂ j,a,b V (0)∂ j,q,k V (0). (6.135)
Since α ∈ I 2 (λ 1 ) and 1 p ∈ I 1 (λ 1 ) (otherwise (g - 1 (z -)) p = 0), we have 1 a , 1 b , 1 q ∈ I 1 (λ 1 ) so that we can write (6.136)

III = - n j,k,p=1 (α p + 1) λ 1 λ j (2λ 1 + λ j ) 2 (g - 1 (z -)) k (g - 1 (z -)) p a,b,q∈I 1 1a+1 b +1q=α+1p ∂ j,a,b V (0)∂ j,q,k V (0).
Now it is easy to check, noticing that (α + 1 p ) k ∈ {0, 1, 2, 3} and examining each case, that (6.137)

a,b,q∈I 1 1a+1 b +1q=α+1p ∂ j,a,b V (0)∂ j,q,k V (0) = (α + 1 p ) k 4 a,b,c,d∈I 1 1a+1 b +1c+1 d =α+1p+1 k ∂ j,a,b V (0)∂ j,c,d V (0).
Therefore, we have

III = - 1 4 n j,k,p=1 (α
+ 1 p + 1 k )! α! λ 1 λ j (2λ 1 + λ j ) 2 (g - 1 (z -)) k (g - 1 (z -)) p × a,b,c,d∈I 1 1a+1 b d =α+1p+1 k ∂ j,a,b V (0)∂ j,c,d V (0). (6.138)
Eventually, setting β = 1 p + 1 k , γ = 1 a + 1 b and δ = 1 c + 1 d , we get (6.139)

III = - β∈I 2 (λ 1 ) (α + β)! α! γ,δ∈I 2 (λ 1 ) γ+δ=α+β n j=1 2λ 1 λ j (2λ 1 + λ j ) 2 ∂ j ∂ γ V (0) γ! ∂ j ∂ δ V (0) δ! (g - 1 (z -)) β β! •
We are left with the computation of

- 1 2 β,γ∈I 1 (λ 1 ) β+γ=α ∂ β ∇ϕ 1 (0) • ∂ γ ∇ϕ 1 (0) = - 1 2 n j=1 β,γ∈I 1 (λ 1 ) β+γ=α ∂ j ∂ β ϕ 1 (0) • ∂ j ∂ γ ϕ 1 (0) = - 1 2 n j=1 4λ 2 1 λ 2 j (2λ 1 + λ j ) 2 β,γ∈I 1 (λ 1 ) β+γ=α n k,ℓ=1 ∂ j ∂ k ∂ β V (0)(g - 1 (z -)) k ∂ j ∂ ℓ ∂ γ V (0)(g - 1 (z -)) ℓ . (6.140)
At this point, we notice that

- 1 2 α∈I 2 (λ 1 ) β,γ∈I 1 (λ 1 ) β+γ=α ∂ β ∇ϕ 1 (0) • ∂ γ ∇ϕ 1 (0)x α = - 1 2 n j=1 4λ 2 1 λ 2 j (2λ 1 + λ j ) 2 β,γ∈I 1 (λ 1 ) α∈I 2 (λ 1 ) β+γ=α n k,ℓ=1 ∂ j ∂ k ∂ β V (0)(g - 1 (z -)) k ∂ j ∂ ℓ ∂ γ V (0)(g - 1 (z -)) ℓ x α = - 1 2 n j=1 4λ 2 1 λ 2 j (2λ 1 + λ j ) 2 α,β∈I 2 (λ 1 ) (α + β)! γ,δ∈I 2 (λ 1 ) γ+δ=α+β ∂ j ∂ γ V (0) γ! ∂ j ∂ δ V (0) δ! x α α! (g - 1 (z -)) β β! -2 α,β∈I 2 (λ 1 ) ∂ j ∂ α V (0) α! ∂ j ∂ β V (0) β! x α (g - 1 (z -)) β (6.141)
From (6.124), (6.130), (6.131) (6.139), and (6.141), we finally obtain that

α∈I 2 (λ 1 ) c 1,α x α = γ∈I 1 \I 1 (2λ 1 ) α,β∈I 2 (λ 1 ) 8λ 2 1 (2λ 1 -λ • γ)(λ • γ)(2λ 1 + λ • γ) 2 ∂ α+γ V (0) α! ∂ β+γ V (0) β! (g - 1 (z -)) β x α - γ∈I 1 (2λ 1 ) α∈I 2 (λ 1 ) ∂ α+γ V (0) α! (g - ,0 (z -)) γ x α + 1 4λ 2 1 γ∈I 1 (2λ 1 )
α,β∈I 2 (λ 1 )

∂ α+γ V (0) α! ∂ β+γ V (0) β! (g - 1 (z -)) β x α - α,β∈I 2 (λ 1 ) ∂ α+β V (0) α!β! (g - 1 (z -)) β x α - 1 2 α,β∈I 2 (λ 1 ) (α + β)! γ,δ∈I 2 γ+δ=α+β n j=1 1 (2λ 1 + λ j ) 2 ∂ j ∂ γ V (0) γ! ∂ j ∂ δ V (0) δ! (g - 1 (z -)) β β! x α α! - α,β∈I 2 (λ 1 ) (α + β)! γ,δ∈I 2 (λ 1 ) γ+δ=α+β n j=1 2λ 1 λ j (2λ 1 + λ j ) 2 ∂ j ∂ γ V (0) γ! ∂ j ∂ δ V (0) δ! (g - 1 (z -)) β β! x α α! -2 α,β∈I 2 (λ 1 ) (α + β)! γ,δ∈I 2 (λ 1 ) γ+δ=α+β n j=1 λ 2 1 λ 2 j (2λ 1 + λ j ) 2 ∂ j ∂ γ V (0) γ! ∂ j ∂ δ V (0) δ! (g - 1 (z -)) β β! x α α! + 4 α,β∈I 2 (λ 1 ) n j=1 λ 2 1 λ 2 j (2λ 1 + λ j ) 2 ∂ j ∂ α V (0) α! ∂ j ∂ β V (0) β! x α (g - 1 (z -)) β , (6.142)
or, more simply, Now we compute the scattering amplitude starting from (4.19). First of all, we change the cut-off function χ + so that the support of the right hand side of the scalar product in (4. [START_REF] Hassell | The Semiclassical Resolvent and the Propagator for Nontrapping Scattering Metrics[END_REF]) is close to (0, 0).

α∈I 2 (λ 1 ) c 1,α x α = - γ∈I 1 (2λ 1 ) α∈I 2 (λ 1 ) ∂ α+γ V (0) α! (g - ,0 (z -)) γ x α + α,β∈I 2 (λ 1 ) (g - 1 (z -)) β β! x α α! × γ∈I 1 \I 1 (2λ 1 ) 8λ 2 1 (2λ 1 -λ • γ)(λ • γ)(2λ 1 + λ • γ) 2 ∂ α+γ V (0)∂ β+γ V (0) + 1 4λ 2 1 γ∈I 1 (2λ 1 ) ∂ α+γ V (0)∂ β+γ V (0) -∂ α+β V (0) - (α + β)! 2 γ,δ∈I 2 γ+δ=α+β n j=1 1 λ 2 j ∂ j ∂ γ V (0) γ! ∂ j ∂ δ V (0) δ! + 4 n j=1 λ 2 1 λ 2 j (2λ 1 + λ j ) 2 ∂ j ∂ α V (0)∂ j ∂ β V (0) .
γ + ℓ χ + = 0 supp(∇χ + ) χ + = 1 γ + ℓ χ + = 0 supp(∇ χ + ) χ + = 1 Figure 1. The support of χ + and χ + in T * R n .
Using Maslov's theory, we construct a function v + which coincides with a + (x, h)e iψ + (x)/h out of a small neighborhood of

ℓ γ + ℓ ∩ (B(0, R + + 1) × R n ) and such that v + is a solution of (P -E)v + = 0 microlocally near ℓ γ + ℓ . Let χ + (x, ξ) ∈ C ∞ (T * R n ) be such that χ + (x, ξ) = χ + (x) out of a small enough neighborhood of ℓ γ + ℓ ∩ (B(0, R + + 1) × R n ).
In particular, (P -E)v + is microlocally 0 near the support of χ +χ + . So, we have

u -, [χ + , P ]v + = u -, [Op( χ + ), P ]v + + u -, (χ + -Op( χ + ))(P -E)v + -(P -E)u -, (χ + -Op( χ + ))v + = u -, [Op( χ + ), P ]v + + O(h ∞ ) -g -e iψ -/h , (χ + -Op( χ + ))v + = u -, [Op( χ + ), P ]v + + O(h ∞ ), (7.1)
since the microsupports of g -e iψ -/h and χ +χ + are disjoint. Thus, the scattering amplitude is given by

(7.2) A(ω, θ, E, h) = c(E)h -(n+1)/2 u -, [Op( χ + ), P ]v + + O(h ∞ ).
Now we will prove that, modulo O(h ∞ ), the only contribution to the scattering amplitude in (7.2) comes from the values of the functions u -and v + microlocally on the trajectories γ + ℓ and γ ∞ j . From (5.18), the fact that u -= O(h -C ) and (P -E)u -= 0 microlocally out of the microsupport of g -e -iψ -/h , and the usual propagation of singularities theorem, we get

(7.3) MS(u -) ⊂ Λ - ω ∪ Λ + . Moreover, we have (7.4) MS(v + ) ⊂ Λ + θ . Now, let f ∞ j (resp. f + ℓ ) be C ∞ 0 (T * R n ) functions with support in a small enough neighborhood of γ ∞ j (resp. γ + ℓ ∩ MS(v + )) such that f ∞ j = 1 (resp. f + k = 1) in a neighborhood of γ ∞ j (resp. γ + ℓ ∩ MS(v + )).
In particular, we assume that all these functions have disjoint support. Since u -and v + have disjoint microsupport out of the support of the f ∞ j and the f + ℓ , we have

A(ω, θ, E, h) = c(E)h -(n+1)/2 j Op(f ∞ j )u -, Op(f ∞ j )[Op( χ + ), P ]v + + c(E)h -(n+1)/2 ℓ Op(f + ℓ )u -, Op(f + ℓ )[Op( χ + ), P ]v + + O(h ∞ ) =A reg + A sing . (7.5)
Concerning the terms which contain f ∞ j , A reg , we are in the same setting as in [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF]Section 4] with the difference that the calculus is made for any E = E 0 + hE 1 with E 1 = O(1) and not for E = E 0 .

In Equation (5.33), we have shown that the main term of the symbol appearing in the WKB expansion on u -differs, from the case E = E 0 , by a factor e it -(ρ)E 1 for ρ ∈ γ ∞ j . The time t -(ρ) is the unique time t such that γ -(t, z ∞ j , ω, E 0 ) = ρ (see (2.6) and (2.8)). The same way, the main term of the symbol in the WKB expansion on v + differs by a factor e it + (ρ)E 1 on the curve γ ∞ j . Here t

+ (ρ) = t is such that γ + (t, z ∞ j , θ, E 0 ) = ρ, where z ∞ j is the projection of r ∞ (z ∞ j , ω, E 0 ) on θ ⊥ . The bicharacteristic curves γ -(t, z ∞ j , ω, E 0 ), γ + (t, z ∞ j , θ, E 0 ) and γ ∞ j
are the same sets, and the quantity t -t + does not depend on ρ ∈ γ ∞ j . Moreover, from (2.9), we have

(7.6) t --t + = -r ∞ (z ∞ j , ω, E 0 )| 2E 0 -1 θ .
Then, following [START_REF]Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits[END_REF]Section 4], the computation of the term A reg gives (7.7)

A reg = N∞ j=1 m≥0
a reg j,m (ω, θ, E)h m e iS ∞ j /h + O(h ∞ ), with (7.8) a reg j,0 (ω, θ, E) = a reg j,0 (ω, θ, E 0 )e i(t --t + )E 1 . Here, a reg j,0 (ω, θ, E 0 ) is the term obtained by Robert and Tamura and equal to (7.9)

a reg j,0 (ω, θ, E 0 ) = e -iν ∞ j π/2 σ(z ∞ j ) 1/2 •
Now we compute A sing . Proceeding as in Section 5.2 for u -, one can show that v + can be written as (7.10) v + (x) = a + (x, h)e iν + ℓ π/2 e iψ + (x)/h , microlocally near any ρ ∈ γ + ℓ close enough to (0, 0). Here ν + ℓ is the Maslov index of γ + ℓ . The phase ψ + and the classical symbol a + satisfy the usual eikonal and transport equations. In particular, as in (5.28) and (5.33), we have (7.11)

ψ + (x + ℓ (t)) = - +∞ t |ξ + ℓ (u)| 2 -2E 0 1 u>0 du = - +∞ t 1 2 |ξ + ℓ (u)| 2 -V (x + ℓ (u)) -E 0 sgn(u) du,
and a + (x, h) ∼ m a +,m (x)h m with (7.12) We can choose χ + so that the support of the symbol of Op(f + ℓ )[Op( χ + ), P ] is contained in a vicinity of such a point ρ ∈ γ + ℓ (see Figure 1). Then, microlocally near ρ, we have (7.14) Op(f + ℓ )[Op( χ + ), P ]v + = a + (x, h)e iν + ℓ π/2 e iψ + (x)/h , with (7.15) a + (x, h) = m≥0 a +,m (x)h m+1 , and (7.16) a +,0 (x) = -i{ χ + , p}(x, ∇ψ + (x))a +,0 (x).

a +,0 (x + ℓ (t)) = (2E 0 ) 1/4 (D + ℓ (t)) -1/2 e
From [5, Section 5], the Lagrangian manifold

{(x, ∇ x ϕ k (t, x)); ∂ t ϕ k (t, x) = 0},
coincides with Λ - ω . In particular, since MS(v + ) ⊂ Λ + θ and since there is no curve γ ∞ (z ∞ j ) sufficiently close to the critical point, the finite times in (6.5) give a contribution O(h ∞ ) to the scattering amplitude (4.19). In view of the equations (6.5), (6.12) and (7.14), the principal contribution of A sing will come from the intersection of the manifolds Λ + θ and Λ + . Recall that, from (A5), the manifolds Λ + θ and Λ + intersect transversely along γ + ℓ . In particular, to compute A sing , we can apply the method of stationary phase in the directions that are transverse to γ + ℓ . For each ℓ, after a linear and orthonormal change of variables, we can assume that g + ℓ ℓ ℓ (z + ℓ ) is collinear to the x ℓ ℓ ℓ -direction, and that V (x) satisfies (A2). We denote H ℓ x ℓ ℓ ℓ = {y = (y 1 , . . . , y n ) ∈ R n ; y ℓ ℓ ℓ = x ℓ ℓ ℓ } the hyperplane orthogonal to (0, . . . , 0, x ℓ ℓ ℓ , 0, . . . , 0). We shall compute A sing in the case where there is only one incoming curve γ - k in Λ - ω and one outgoing curve γ + ℓ in Λ + θ . In the case of several but finitely many trajectories, A sing is simply given by the sum over k and ℓ of such contributions. Using (4. [START_REF] Hassell | The Semiclassical Resolvent and the Propagator for Nontrapping Scattering Metrics[END_REF]), (6.5) and (7.14), we can write

A sing = c(E)h -(n+1)/2 √ 2πh e i(ϕ k (t,x)-ψ + (x))/h α k (t, x, h) a + (x, h)e -iν + ℓ π/2 dt dx = c(E)h -(n+1)/2 √ 2πh
x ℓ ℓ ℓ y∈H ℓ x ℓ ℓ ℓ e i(ϕ k (t,x)-ψ + (x))/h α k (t, x, h) a + (x, h)e -iν + ℓ π/2 dt dy dx ℓ ℓ ℓ . (7.17) Let Φ(y) = ϕ k (t, x ℓ ℓ ℓ , y)ψ + (x ℓ ℓ ℓ , y) be the phase function in (7.17). From (6.10)-(6.13), we can write

(7.18) Φ(y) = S - k + (ϕ + -ψ + )(x ℓ ℓ ℓ , y) + ψ(t, x ℓ ℓ ℓ , y),
where ψ = O(e -λ 1 t ) is an expandible function. Since the manifolds Λ + θ and Λ + intersect transversely along γ + ℓ , the phase function y → (ϕ +ψ + )(x ℓ ℓ ℓ , y) has a non degenerate critical point y ℓ (x ℓ ℓ ℓ ) ∈ H ℓ x ℓ ℓ ℓ ∩ Π x γ + ℓ , and x ℓ ℓ ℓ → y ℓ (x ℓ ℓ ℓ ) is C ∞ for x ℓ ℓ ℓ = 0. Then, from the implicit function theorem, the function Φ has a unique critical point y ℓ (t, x ℓ ℓ ℓ ) ∈ H ℓ x ℓ ℓ ℓ for t large enough depending on x ℓ ℓ ℓ . The function (t, x ℓ ℓ ℓ ) → y ℓ (t, x ℓ ℓ ℓ ) is expandible and we have (7.19) y ℓ (t, x ℓ ℓ ℓ ) = y ℓ (x ℓ ℓ ℓ ) -Hess(ϕ +ψ + ) -1 y ℓ (x ℓ ℓ ℓ ) ∇ϕ 1 y ℓ (x ℓ ℓ ℓ ) e -µ 1 t + O e -µ 2 t .

As a consequence, Φ y ℓ (t, x ℓ ℓ ℓ ) is also expandible.

Since ϕ + and ψ + satisfy the same eikonal equation, we get (see (5.25))

(7.20) ∂ t (ϕ + -ψ + )(x + ℓ (t)) = |ξ + ℓ (t)| 2 -|ξ + ℓ (t)| 2 = 0.
Thus, (ϕ +ψ + )(y ℓ (x ℓ ℓ ℓ )) does not depend of x ℓ ℓ ℓ and is equal to

(ϕ + -ψ + )(y ℓ (x ℓ ℓ ℓ )) = lim t→-∞ (ϕ + -ψ + )(x + ℓ (t)) = +∞ -∞ |ξ + ℓ (s)| 2 -2E 0 1 s>0 ds = +∞ -∞ 1 2 |ξ + ℓ (s)| 2 -V (x + ℓ (s)) -E 0 sgn(s) ds =S + ℓ , (7.21)
where we have used (7.11). Therefore, the phase function Φ at the critical point y ℓ (t, x ℓ ℓ ℓ ) is equal to

Φ y ℓ (t, x ℓ ℓ ℓ ) =S - k + S + ℓ + m∈N µm≤2λ 1 ϕ m t, y ℓ (x ℓ ℓ ℓ ) e -µmt - 1 2
Hess(ϕ +ψ + ) -1 y ℓ (x ℓ ℓ ℓ ) ∇ϕ 1 y ℓ (x ℓ ℓ ℓ ) • ∇ϕ 1 y ℓ (x ℓ ℓ ℓ ) e -2µ 1 t + O(e -µt ), (7.22) where µ is the first of the µ j 's such that µ j > 2λ 1 .

Using the method of the stationary phase for the integration with respect to y ∈ H ℓ x ℓ ℓ ℓ in (7.17), we get (7.23)

A sing = c(E)h -(n+1)/2 √ 2πh (2πh) (n-1)/2 e iΦ(y ℓ (t,x ℓ ℓ ℓ ))/h f ℓ (t, x ℓ ℓ ℓ , h) dt dx ℓ ℓ ℓ + O(h ∞ ).
The O(h ∞ ) term follows from the fact that the error term stemming from the stationary phase method can be integrated with respect to time t, since α k ∈ S 0,2 Re Σ(E) , with Re Σ(E) > 0 (see the beginning of Section 6). The symbol f ℓ (t, x ℓ ℓ ℓ , h) is a classical expandible function of order S 1,2 Re Σ(E) in the sense of Definition 6.2:

(7.24) f ℓ (t, x ℓ ℓ ℓ , h) ∼ m≥0 f ℓ m (t, x ℓ ℓ ℓ , ln h)h 1+m ,
where the f ℓ m are polynomials with respect to ln h and (7.25) Suppose that j ∈ N is such that j < . From (6.40), we have (7.28) ϕ k j (x + ℓ (s 0 )) = e -µ j (s-s 0 ) ϕ k j (x + ℓ (s)). Combining (6.41) with (6.109), we obtain

f ℓ 0 (t, x ℓ ℓ ℓ , ln h) = α k 0 t, y ℓ (t, x ℓ ℓ ℓ ) a +,0 y ℓ (t, x ℓ ℓ ℓ ) e -iν + ℓ π/2 e i sgn Φ ′′ |H ℓ x ℓ ℓ ℓ (y ℓ (t,x ℓ ℓ ℓ ))π/4 det Φ ′′ |H ℓ x ℓ ℓ ℓ y ℓ (t, x ℓ ℓ ℓ ) 1/2 .

Using

ϕ k j (x + ℓ (s 0 )) =e µ j s 0 e -µ j s -2µ j g - j (z - k )|g + j (z + ℓ )e µ j s + O(e 2λ 1 s ) = -2µ j g - j (z - k ) g + j (z + ℓ ) e µ j s 0 . (7.29)
We suppose first that we are in the case (a) of assumption (A7). Then, (7.22) becomes (7.30) Φ y ℓ (t,

x ℓ ℓ ℓ ) = S - k + S + ℓ -2µ k g - k (z - k ) g + k (z +
ℓ ) e µ k s(x ℓ ℓ ℓ ) e -µ k t + O(e -µ k+1 t ). Here s(x ℓ ℓ ℓ ) is such that x + ℓ (s(x ℓ ℓ ℓ )) = x ℓ (x ℓ ℓ ℓ ) and the O(e -µ k+1 t ) is in fact expandible, uniformly with respect to x ℓ ℓ ℓ when x ℓ ℓ ℓ varies in a compact set avoiding 0.

Suppose now that we are in the case (b) of assumption (A7). Of course, from (7.29), we have ϕ j y ℓ (x ℓ ℓ ℓ ) = 0 for all j < . On the other hand, Corollary 6.8 and (6.111) imply (7.31) ϕ k ,2 (x + ℓ (s 0 )) = e -2λ 1 (s-s 0 ) ϕ k ,2 (x + ℓ (s)). Combining this with (6.126), we get

ϕ k ,2 (x + ℓ (s 0 )) =e 2λ 1 s 0 e -2λ 1 s - 1 8λ 1 j∈I 1 (2λ 1 )
α,β∈I 2 (λ 1 )

∂ α+1 j V (0) α! ∂ β+1 j V (0) β! g - 1 (z - k ) α g + 1 (z + ℓ ) β e 2λ 1 s + O(e 3λ 1 s ) = - 1 8λ 1 j∈I 1 (2λ 1 ) α,β∈I 2 (λ 1 ) ∂ α+1 j V (0) α! ∂ β+1 j V (0) β! g - 1 (z - k ) α g + 1 (z + ℓ ) β e 2λ 1 s 0 . (7.32)
In particular, (7.22) becomes, in that case,

Φ y ℓ (t, x ℓ ℓ ℓ ) =S - k -S + ℓ - 1 8λ 1 j∈I 1 (2λ 1 ) α,β∈I 2 (λ 1 ) ∂ α+1 j V (0) α! ∂ β+1 j V (0) β! g - 1 (z - k ) α g + 1 (z + ℓ ) β e 2λ 1 s(x ℓ ℓ ℓ ) × t 2 e -2λ 1 t + O(te -2λ 1 t ) =S - k + S + ℓ + M 2 (k, ℓ)t 2 e -2λ 1 t + O(te -2λ 1 t
). (7.33) As in (7.30), the term O(te -2λ 1 t ) is in fact expandible uniformly with respect to x ℓ ℓ ℓ when x ℓ ℓ ℓ varies in a compact set avoiding 0.

we suppose that we are in the case (c) assumption (A7). Then we obtain from (7.29) and (7.32) that ϕ j y ℓ (x ℓ ℓ ℓ ) = 0 for all j <  and ϕ ,2 y ℓ (x ℓ ℓ ℓ ) = 0. With the last identity in mind, Equation (6.111) on ϕ k ,1 implies (7.34)

ϕ k ,1 (x + ℓ (s 0 )) = e -2λ 1 (s-s 0 ) ϕ k ,1 (x + ℓ (s)). In order to compute ϕ k ,1 (x + ℓ (s))
, we put the expansion (2.17) for x + ℓ (s) (with Proposition 6.11 in mind) into (6.127). The third term in (6.127) will be, at least, O(e (µ 2 +µ 1 )s ) = o(e 2λ 1 s ). Thank to (6.91) and thanks to the fact that M 2 (k, ℓ) = 0, the first term in (6.127) will give no contribution of order se 2λ 1 s and will be of the form

(7.35) -4λ 1 g - ,1 (z - k ) • x + ℓ (s) = - j∈I 1 α∈I 2 (λ 1 ) ∂ j ∂ α V (0) α! (g - 1 (z - k )) α (g + ,0 (z + ℓ )) j e 2λ 1 s + O(e µ +1 s )
It remains to study the contribution the second term in (6.127), as given in (6.143). As previously, the first term of the third line in (6.143) will give a term of order o(e 2λ 1 s ). The other terms will contribute to the order e 2λ 1 s for

- j∈I 1 α∈I 2 (λ 1 ) ∂ j ∂ α V (0) α! (g - ,0 (z - k )) j (g + 1 (z + ℓ )) α + α,β∈I 2 (λ 1 ) (g - 1 (z - k )) α α! (g + 1 (z + ℓ )) β β! × × -∂ α+β V (0) + j∈I 1 \I 1 (2λ 1 ) 4λ 2 1 λ 2 j (4λ 2 1 -λ 2 j ) ∂ j ∂ α V (0)∂ j ∂ β V (0) - j∈I 1 γ,δ∈I 2 (λ 1 ) γ+δ=α+β (γ + δ)! γ! δ! 1 2λ 2 j ∂ j ∂ γ V (0)∂ j ∂ δ V (0) . (7.36)
Thus, combining (7.35) and (7.36), the discussion above leads to

ϕ k ,1 (x + ℓ (s 0 )) =e 2λ 1 s 0 e -2λ 1 s M 1 (k, ℓ)e 2λ 1 s + o(e 2λ 1 s ) =M 1 (k, ℓ)e 2λ 1 s 0 . (7.37)
In particular, (7.22) becomes, in that case,

Φ y ℓ (t, x ℓ ℓ ℓ ) =S - k + S + ℓ + M 1 (k, ℓ)e 2λ 1 s(x ℓ ℓ ℓ ) te -2λ 1 t + O(e -2λ 1 t ). (7.38)
As above, the O(e -2λ 1 t ) is expandible uniformly with respect to the variable x ℓ ℓ ℓ when x ℓ ℓ ℓ varies in a compact set avoiding 0.

with respect to time.

Now we perform the integration with respect to time t in (7.23). We follow the ideas of [20, Section 5] and [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]Section 6]. Since y ℓ (t, x ℓ ℓ ℓ ) is expandible (see (7.19)), and since Φ is C ∞ outside of x ℓ ℓ ℓ = 0, the symbol f ℓ (t, x ℓ ℓ ℓ , h) is expandible.

We compute only the contribution of the principal symbol (with respect to h) of f ℓ , since the other terms can be treated the same way, and the remainder term will give a contribution O(h ∞ ) to the scattering amplitude. In other word, we compute (7.39)

A sing 0 = c(E)h -(n+1)/2 √ 2πh (2πh) (n-1)/2 h e iΦ(y ℓ (t,x ℓ ℓ ℓ ))/h f ℓ 0 (t, x ℓ ℓ ℓ ) dt dx ℓ ℓ ℓ + O(h ∞ ).
First, we assume that we are in the case (a) of the assumption (A7). In that case, Φ is given by (7.30). For x ℓ ℓ ℓ fixed in a compact set away from 0, we set

τ =Φ y ℓ (t, x ℓ ℓ ℓ ) -(S - k + S + ℓ ) = -2µ k g - k (z - k ) g + k (z + ℓ ) e µ k s(x ℓ ℓ ℓ ) e -µ k t + R(t, x ℓ ℓ ℓ ), (7.40)
and we perform the change of variable t → τ in (7.39). We assume for a moment that (7.41)

g - k (z - k ) g + k (z + ℓ ) < 0.
where f j (ln h) is a polynomial with respect to ln h. The function f 0 does not depend on h and we have (7.51)

f 0 = Γ(Σ(E)/µ k )(-i) -Σ(E)/µ k f ℓ 0,0 (x ℓ ℓ ℓ ) dx ℓ ℓ ℓ .
To finish the proof, it remains to perform the integration with respect to x ℓ ℓ ℓ . From (7.25) and (7.48), (7.51) becomes

f 0 =Γ(Σ(E)/µ k ) 1 µ k 2iµ k g - k (z - k ) g + k (z + ℓ ) e µ k s(x ℓ ℓ ℓ ) -Σ(E)/µ k × α 0,0 (y ℓ ℓ ℓ ℓ ) a +,0 y ℓ (x ℓ ℓ ℓ ) e -iν + ℓ π/2 e i sgn Φ ′′ |H ℓ x ℓ ℓ ℓ (y ℓ (x ℓ ℓ ℓ ))π/4 det Φ ′′ |H ℓ x ℓ ℓ ℓ y ℓ (x ℓ ℓ ℓ ) 1/2 dx ℓ ℓ ℓ . (7.52)
Now we make the change of variable x ℓ ℓ ℓ → s given by y ℓ (x ℓ ℓ ℓ ) = x + ℓ (s) (then s(x ℓ ℓ ℓ ) = s). In particular, (7.53)

dx ℓ ℓ ℓ = ∂ s (x + ℓ,ℓ ℓ ℓ (s))ds = λ ℓ ℓ ℓ |g + ℓ ℓ ℓ (z + ℓ )|e λ ℓ ℓ ℓ s (1 + o(1)
)ds, as s → -∞. In this setting, we get (7.54) α 0,0 (x + ℓ (s)) = α 0,0 (0)(1 + o(1)), as s → -∞, where α 0,0 (0) is given in (6.8). We also have, from (7.12) and (7.16),

(7.55) a +,0 (x + ℓ (s)) = -i∂ s χ + (γ + ℓ (s)) (2E 0 ) 1/4 (D + ℓ (s)) -1/2 e -isE 1 .
Then, substituting (7.26), (7.27), (7.53), (7.54) and (7.55) in (7.52), we obtain

f 0 =Γ(Σ(E)/µ k ) -i µ k 2iµ k g - k (z - k ) g + k (z + ℓ ) -Σ(E)/µ k α 0,0 (0)∂ s χ + (γ + ℓ (s)) e -iν + ℓ π/2 × e i(n-1)π/4 j =ℓ ℓ ℓ 2λ j λ ℓ ℓ ℓ |g + ℓ ℓ ℓ (z + ℓ )|(2E 0 ) 1/4 (D + ℓ (s)) -1/2 e -isE 1 e -Σ(E)s e λ ℓ ℓ ℓ s (1 + o(1)) ds = - e i(n+1)π/4 µ k j =ℓ ℓ ℓ 2λ j -1/2 λ ℓ ℓ ℓ |g + ℓ ℓ ℓ (z + ℓ )|Γ(Σ(E)/µ k ) 2iµ k g - k (z - k ) g + k (z + ℓ ) -Σ(E)/µ k × e -iν + ℓ π/2 α 0,0 (0)(2E 0 ) 1/4 (D + ℓ ) -1/2 ∂ s χ + (γ + ℓ (s)) (1 + o(1)) ds. (7.56)
Here the o(1) does not depend on χ + . Now, we choose a family of cut-off functions ( χ j + ) j∈N such that the support of ∂ t χ j + (γ + ℓ (t)) goes to -∞ as j → +∞. We also assume that ∂ t χ j + (γ + ℓ (t)) is non-positive (see Figure 1). Then )). (7.57) as j → +∞. Since f 0 is also independent of χ + , we obtain Theorem 2.6 from (7.50) and (7.51), in the case (a) and under the assumption (7.41). When g - k (z - k )|g + k (z + ℓ ) > 0, we set τ as the opposite of the R.H.S. of (7.40), and we obtain the result along the same lines (see Remark D.2). Now we assume that we are in the case (b) of the assumption (A7). In that case, the phase function Φ is given by (7.33). For x ℓ ℓ ℓ fixed in a compact set outside from 0, we set, mimicking (7.40),

f 0 = - e i(n+1)π/4 µ k j =ℓ ℓ ℓ 2λ j -1/2 λ ℓ ℓ ℓ Γ(Σ(E)/µ k )e -iν + ℓ π/2 e iπ/4 (2λ 1 ) 3/2 e -iν - k π/2 × |g - 1 (z - k )| |g + ℓ ℓ ℓ (z + ℓ )| 2iµ k g - k (z - k ) g + k (z + ℓ ) -Σ(E)/µ k × (2E 0 ) 1/2 (D - k D + ℓ ) -1/2 × (1 + o( 1
τ =Φ y ℓ (t, x ℓ ℓ ℓ ) -(S - k + S + ℓ ) =M 2 (k, ℓ)e 2λ 1 s(x ℓ ℓ ℓ ) t 2 e -2λ 1 t + R(t, x ℓ ℓ ℓ ) (7.58)
where R(t, x ℓ ℓ ℓ ) = O(te -2λ 1 t ) is expandible with respect to t. As above, we assume that M 2 (k, ℓ) is positive (the other case can be studied the same way). Following (7.42), we want to write s := e -t as a function of τ . Since t → τ (t) is expandible with respect to t, we have

(7.59) τ = M 2 (k, ℓ)e 2λ 1 s(x ℓ ℓ ℓ ) (ln s) 2 s 2λ 1 (1 + r(s, x ℓ ℓ ℓ )),
where r(s, x ℓ ℓ ℓ ) = o(1). In particular, ∂ s τ > 0 for s positive small enough and then, for ε > 0 small enough, s → τ (s) is invertible for 0 < s < ε. We denote by s(τ ) the inverse of this function. We look for s(τ ) of the form

(7.60) s(τ ) = (2λ 1 ) 1/λ 1 τ M 2 (k, ℓ)e 2λ 1 s(x ℓ ℓ ℓ ) 1/2λ 1 u(τ, x ℓ ℓ ℓ ) (-ln τ ) 1/λ 1 ,
where u(τ, x ℓ ℓ ℓ ) has to be determined. Using (7.59), the equation for u is

τ =M 2 (k, ℓ)e 2λ 1 s(x ℓ ℓ ℓ ) (ln s) 2 s 2λ 1 (1 + r(s, x ℓ ℓ ℓ )) =τ u 2λ 1 1 - ln (2λ 1 ) -2 M 2 (k, ℓ)e 2λ 1 s(x ℓ ℓ ℓ ) ln τ + 2λ 1 ln u ln τ -2 ln(-ln τ ) ln τ 2 × 1 + r (2λ 1 ) 1/λ 1 τ M 2 (k, ℓ)e 2λ 1 s(x ℓ ℓ ℓ ) 1/2λ 1 u (-ln τ ) 1/λ 1 , x ℓ ℓ ℓ
=τ F (τ, u, x ℓ ℓ ℓ ), (7.61) where F = u 2λ 1 (1 + r(τ, u, x ℓ ℓ ℓ )) and r = o(1) for u close to 1 (here (u, x ℓ ℓ ℓ ) are the variables y in Definition 7.1). In other word, to find u, we have to solve F (t, u, x ℓ ℓ ℓ ) = 1.

First we remark that u → F (τ, u, x ℓ ℓ ℓ ) is real-valued and continuous. Since, for δ > 0 and τ small enough,

F (τ, 1 -δ, x ℓ ℓ ℓ ) < 1 < F (τ, 1 + δ, x ℓ ℓ ℓ ), there exists u ∈ [1 -δ, 1 + δ] such that F (τ, 1 + δ, x ℓ ℓ ℓ ) = 1.
Thanks to the discussion before (7.60), the function s(τ ) is of the form (7.60) with u(τ, x ℓ ℓ ℓ ) ∈ [1δ, 1 + δ], for τ small enough.

For τ > 0, the function F is C ∞ and, since r = o(1), we have

(7.62) ∂ u F (τ, u, x ℓ ℓ ℓ ) -1 (u(τ, x ℓ ℓ ℓ )) = 2λ 1 u 2λ 1 -1 (1 + o τ (1)) > λ 1 ,
for τ small enough. The notation o τ (1) means a term which goes to 0 as τ goes to 0. Here we have used the fact that u(τ, x ℓ ℓ ℓ ) is close to 1. In particular, the implicit function theorem implies that u(τ, x ℓ ℓ ℓ ) is C ∞ .

Proof. First of all, the second estimate follow easily from the first one: we have

|x| 3 u(x) 2 = R n |x| 6 |u(x)| 2 dx h 6α u 2 ,
since u vanishes if |x| > 2h α . Thanks to the fact that u is a product of n functions of one variable, it is enough to estimate

I = ϕ 2 t h α χ 2 h β |t| 1/2 dt = 2 2h α 0 ϕ 2 t h α χ 2 h β t 1/2 dt.
We have Remark C.2. The same result holds in the outgoing region: If γ = Λ ∩ Λ + is transverse, Λ projects nicely on R n x near γ(t), t → -∞. Then Λ = Λ ψ for some function ψ satisfying ψ ′′ (x(t)) = diag(-λ 1 , . . . , -λ ν-1 , λ ν , -λ ν+1 , . . . , -λ n ) + O(e εt ).

2 h α h 2β χ 2 h β t 1/2 dt ≤ I ≤ 2 2h α h 2β χ 2 h β t 1/2 dt + 2 h 2β 0 χ 2 h β t 1/2
Proof. We follow the proof of [20, Lemma 2.1]. There exist symplectic local coordinates (y, η) centered at (0, 0) such that Λ -(resp. Λ + ) is given by y = 0 (resp. η = 0) and y j = 1 2λ j (ξ j + λ j x j ) + O((x, ξ) 2 ), (C.4) η j = 1 2λ j (ξ jλ j x j ) + O((x, ξ) 2 ). (C.5) Then, p(x, ξ) = A(y, η)y • η with A 0 := A(0, 0) = diag(λ 1 , . . . , λ n ). In particular, the tangent vectors (δ y , δ η ) to Λ at γ(t) satisfy the following evolution equation We denote by U (t, s) the linear operator such that U (t, s)δ solves (C.6) with U (s, s) = Id.

Since the intersection Λ ∩ Λ -= γ is transverse, there exists E n-1 (t 0 ) ⊂ T γ(t 0 ) Λ, a vector space of dimension n -1 disjoint from T γ(t 0 ) Λ -. For convenience, we set E n (t 0 ) = E n-1 (t 0 ) ⊕ Rv for some v / ∈ T γ(t 0 ) Λ + T γ(t 0 ) Λ -. Let E • (t) = U (t, t 0 )E • (t 0 From [20, Lemma 2.1], there exists a n × n matrix B t = O(e -λ 1 t ) such that E n (t) is given by δ η = B t δ y . Now, if δ ∈ E n-1 (t), we have σ(H p , δ) = 0 since E n-1 (t) ⊕ RH p = T γ(t) Λ and Λ is a Lagrangian manifold. From (C.1), we have It follows that δ ∈ E n-1 (t) if and only if (δ yν , δ η ) = B t δ y ′ where B t = O(e -εt ) is a (n + 1) × (n -1) matrix. Using T γ(t) Λ = E n-1 (t) ⊕ RH p , we obtain that T γ(t) Λ has a basis formed of vector f j (t) such that f j =e y j + O(e -εt ) for j = ν (C.9) f ν =e ην + O(e -εt ). (C.10)

(C.
In the (x, ξ)-coordinates, T γ(t) Λ has a basis formed of vector f j (t) of the form f j =e ξ j + λ j e x j + O(e -εt ) for j = ν (C.11) f ν =e ξνλ j e xν + O(e -εt ), (C.12) and the lemma follows. Here (-iλ) -α = e iαπ/2 λ -α and ln(-iλ) = ln λiπ/2.

Remark D.2. Notice that one obtains the behavior of these quantities as λ → -∞ by taking the complex conjugate in these expressions.

Proof. We begin with (D.2) and assume first that β = 0. Then, we can write where z -α is defined on C\] -∞, 0] and is real positive on ]0, +∞[. In particular (D.9) lim ε→0 I 1 (α, ε) = (-iλ) -α Γ(α).

Concerning I 2 (α, ε), we remark that r(t, α) = t α-1 (1χ(t)) is a symbol which satisfies (D.10) |∂ j t ∂ k α r(t, α)| t Re α-1-j ln t k , for all j, k ∈ N uniformly for t ∈ [0, +∞[ and α in a compact subset of {Re z > 0}. Then, performing integration by parts in (D.7), we obtain (D.11) I 2 (α, ε) = 1 (εiλ) j +∞ 0 e (iλ-ε)t ∂ j t r(t, α) dt, for all j ∈ N. Now, if j is large enough (j > Re α), ∂ j t r(t, α) is integrable in time and does not depend on ε. In particular, for such j, |∂ j t u(t)| t Re α-j (-ln t) β , near 0. Let ϕ ∈ C ∞ (R) be such that ϕ = 1 for t < 1 and ϕ = 0 for t > 2. For δ > 0, we have If one of the derivatives falls on 1ϕ(t/δ), the support of this contribution is contained in [δ, 2δ]. Therefore, the corresponding term will be bounded by δ Re α-N -1 (ln δ) β and will contribute like δ Re α-N (-ln δ) β to the integral.

If one of the derivatives falls on χ(t), the support of the integrand will be a compact set away from 0 and then this function will be O [START_REF] Abraham | Foundations of mechanics[END_REF]. The contribution to the integral of such a term will be like 1.

If all the derivatives fall on u(t)t -1 , the corresponding term will satisfies Taking δ = (ελ)

( 3 . 24 )

 324 -iψ(P )[A, P ]ψ(P ) ≥ εhψ 2 (P ), for some ε > 0. Moreover (3.25) [A, P ] = O(h| ln h|). Now we estimate [[P, A], A].

  26), (3.27), (3.29) and (3.33), we get (3.34) [[P, A], A] = O(h| ln h|).

2 . 6 . 4 .

 264 Proposition Suppose µ ∈]0, 2λ 1 [. With the above notations, one has Ker L µ ⊕ Im L µ = C x . More precisely:

(6.143) 7 . 7 . 1 .

 771 Computations after the critical point Stationary phase expansion in the outgoing region.

7 . 2 .

 72 Proposition C.1, we compute the Hessian of Φ, and we getψ ′′ + y ℓ (x ℓ ℓ ℓ ) = diag(-λ 1 , . . . , -λ ℓ ℓ ℓ-1 , λ ℓ ℓ ℓ , -λ ℓ ℓ ℓ+1 , . . . , -λ n ) + o(1), ϕ ′′ + y ℓ (x ℓ ℓ ℓ ) = diag(λ 1 , . . . , λ n ) + o(1). Then, for x ℓ ℓ ℓ small enough and t large enough depending on x ℓ ℓ ℓ , we havedet Φ ′′ |H ℓ x ℓ ℓ ℓ y ℓ (t, x ℓ ℓ ℓ ) 1/2 = j =ℓ ℓ ℓ 2λ j + o(1), (7.26) sgn Φ ′′ |H ℓ x ℓ ℓ ℓ y ℓ (t, x ℓ ℓ ℓ ) = n -1, (7.27)as x ℓ ℓ ℓ goes to 0. Behaviour of the phase function Φ.

(

  O(e -λ 1 t ) 0 O(e -λ 1 t ) A 0 + O(e -λ 1 t ) δ y δ η .

7 )

 7 H p (γ(t)) = γ(t) = -λ ν ( ae ην + O(e -εt ))e -λν t ,where e ην is the basis vector corresponding to η ν , a = ±|a|, and then (C.8) 0 = σ(e λν t H p , δ) = λ ν aδ yν + O(e -εt )|δ|.

Appendix D . 0 e 0 e

 .00 Asymptotic behavior of certain integralsLemma D.1. Let α ∈ C, Re α > 0, β ∈ R and χ ∈ C ∞ 0 (] -∞, 1/2[) be such that χ = 1 near 0.As λ goes to +∞, we have(D.1) ∞ iλt t α (-ln t) β χ(t) dt t = Γ(α)(ln λ) β (-iλ) -α (1 + o(1)).Moreover, if β ∈ N, we get(D.2) ∞ iλt t α (-ln t) β χ(t) dt t = (-iλ) -α β j=0C j β Γ (j) (α)(-1) j ln(-iλ) β-j + O(λ -∞ ).

Finally, if 0 e

 0 s(t) ∈ C ∞ (]0, +∞[) satisfies (D.3) |∂ j t s(t)| = o t α-j (-ln t) β , for all j ∈ N and t → 0, then (D.4) ∞ iλt s(t)χ(t) dt t = o (ln λ) β λ -α .

I 1

 1 λ+iε)t t α χ(t) dt t = lim ε→0 I 1 (α, ε) -I 2 (α, ε) , (α, ε) = (εiλ) -α Γ(α),

I 2 0 e

 20 (α, ε) = e ijπ/2 λ -j +∞ 0 e iλt ∂ j t r(t, α) dt,and then (see (D.10) or Cauchy's formula)∂ k α lim ε→0 I 2 (α, ε) =e ijπ/2 λ -j +∞ 0 e iλt ∂ j t ∂ k α r(t, α) dt =O(λ -∞ ), (D.13)for all k ∈ N. Then we obtain (D.2) for β = 0. To obtain the result for β ∈ N, it is enough to observe that∞ iλt t α (ln t) β χ(tΓ (j) (α)ln(-iλ) β-j + O(λ -∞ ), (D.[START_REF] Fujiié | Matrice de scattering et résonances associées à une orbite hétérocline[END_REF] from (D.13). Thus, (D.2) is proved.Let u ∈ C ∞ (]0, +∞[) be such that (D.[START_REF]Breit-Wigner formula at barrier tops[END_REF]

(e

  iλt u(t)χ(t) 1ϕ(t/δ) χ(t) 1ϕ(t/δ) t -1 dt, for all N .

∞ 0 et

 0 iλt ∂ N t u(t)t -1 χ(t) 1ϕ(t/δ) dt =O(1) +∞ δ Re α-1-N (-ln t) β (1χ(t))dt (-ln δ) β δ Re α-N , (D.17)for N large enough (N > Re α).

Frome

  iλt u(t)χ(t) 1ϕ(t/δ) dt t = O (-ln δ) β δ α-N λ -N .

2δ 0 t(e 2 /ε 0 e 2 /ε 0 e is s α ( 1 -

 020201 Re α-1 (-ln t) β dt =o δ→0 (1)δ Re α (-ln δ) β . (D.21)Here o δ→0 (1) stands for a term which goes to 0 as δ goes to 0. If δ = (ελ) -1 , we have(D.22) +∞ 0 e iλt s(t)χ(t)ϕ(t/δ) dt t = o λ→+∞ (1)λ -α (ln λ) β ,when λ → +∞ and ε fixed. Taking ε small enough in (D.[START_REF] Hassell | The Semiclassical Resolvent and the Propagator for Nontrapping Scattering Metrics[END_REF]), and then λ large enough in (D.22), we obtain (D.4).It remains to prove (D.1). We need to compute iλt t α (-ln t)β ϕ(t/δ) dt t •Performing the change of variable s = λt, we get I =λ -α is s α (ln λln s) β ϕ(εs) ds s =(ln λ) β λ -α ln s/ ln λ) β ϕ(εs) ds s . (D.[START_REF] Martinez | Resonance free domains for non globally analytic potentials[END_REF]

  The first estimate follows from the fact that 2βα > 0, once we have noticed that ′′k (t) + λ 2 k t 2 u k (t) = e iλ k t 2 /2h h 2 v ′′ h (t) + ihλ k (2t∂ t + 1)v h (t) ,where we have set v h (t) = ϕ t h α χ h β |t| 1/2 . Notice that the right hand side of (B.7) is an even function, so that we only have to consider t > 0. The point here, is that we have, for t > 0, On the other hand, an easy computation gives, still for t > 0, Computing the L 2 -norm of each of these terms as in Lemma B.1 and (B.9), we obtain(B.11) h 2 v ′′ h h 2+β-2α + h 2+β-2α + h 2-3β + h 2-3β ,and, eventually, from (B.6), (B.7), (B.9) and (B.11),(P -E 0 )u h -β | ln h| -1/2 h 1+β + h 2+β-2α + h 2-3β + h 3α u .Therefore we obtain Proposition 2.2 if we can find α > 0 and β > 0 such that2 -2α > 1, 2 -4β > 1, 3α > 1 and 2β > α,and one can check that α = 5/12 and β = 11/48 satisfies these four inequalities. Λ = Λ ψ := {(x, ∇ψ(x)); x ∈ R n }. Moreover, we have

	Therefore, we obtain
	(B.9) (C.3)	(2t∂ t + 1)v h  ψ ′′ (x(t)) =          	2 =2 λ 1 + 2 2h α 0 2h α ϕ 0 h 2β 0 dt + . . . λ ν-1 t h α (2t∂ t + 1) χ 2t∂ t ϕ t h α χ 2h α h α t 2 h 2α ϕ ′ t h α χ h β |t| 1/2 h β |t| 1/2 h β 2  dt 2 λ n |t| 1/2 -λ ν λ ν+1 . . .           + O(e -εt ), dt 2 dt h 2β .
	as t → +∞.	v ′′ h (t) =h -2α ϕ ′′ t h α χ	h β t 1/2 -	h β-α t 3/2 ϕ ′ t h α χ ′ h β t 1/2	dt,
	so that (B.10)		2	h α h 2β +	h 2β t 3h β 4t 5/2 ϕ dt ≤ I ≤ 2 t h α χ ′ h β 2h α h 2β t 1/2 + h 2β t h 2β dt + 2 4t 3 ϕ	0 t h α χ ′′ h β h 2β 4 dt. t 1/2 .
	Ah α h 2β dt = h On the other hand, we have h 2β t
	-k u (B.6) h 2 2 ∆u(x) -n j=1 λ 2 j 2 x 2 j u(x) = n k=1 j =k u j (x j ) -h 2 2 u ′′ k (x k ) -λ 2 k 2 x 2
	We also have
	(B.7) h 2 u (B.8) (2t∂ t + 1) χ	h β t 1/2		= -	h β t 1/2 χ ′ h β t 1/2 + χ	h β t 1/2 =	            	2 O(1) 0	if 0 < t < if h 2β 4 < t < h 2β , h 2β 4 , if h 2β < t.

2β (2βα)| ln h| + ln A . k (x k ) .

From Lemma B.1, we get

(P -E 0 )u h β(n-1) | ln h| (n-1)/2 sup 1≤k≤n h 2 u ′′ k (t) + λ 2 k t 2 u k (t) + h 3α h βn | ln h| n/2 h -β | ln h| -1/2 sup 1≤k≤n h 2 u ′′ k (t) + λ 2 k t 2 u k (t) + h 3α u .

Here R(t, x ℓ ℓ ℓ ) = O(e -µ k+1 t ) is expandible. As in [20, Section 5] and [START_REF] Bony | Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point[END_REF]Section 6], we get

τ µ j /µ k b j (-ln τ, x ℓ ℓ ℓ ) (7.43)

τ µ j /µ k b j ln τ, x ℓ ℓ ℓ ), (7.44) where the b j 's change from line to line. These expansions are valid in the following sense: Definition 7.1. Let f (τ, y) be defined on ]0, ε[×U where U ⊂ R m . We say that f = O(g(τ )) (resp. f = o(g(τ ))), where g(τ ) is a non-negative function defined in ]0, ε[ if and only if for all α ∈ N and β ∈ N m ,

Thus, an expression like f ∼ ∞ j=1 τ µ j /µ k f j (-ln τ, x ℓ ℓ ℓ ), where f j (-ln τ, x ℓ ℓ ℓ ) is a polynomial with respect to ln τ , as in (7.42)-(7.44), means that, for all J ∈ N,

We shall call that such symbols f expandible near 0.

Since f ℓ 0 (t, x ℓ ℓ ℓ ) is expandible (see Definition 6.1) with respect to t, this symbol is also expandible near 0 with respect to τ in the sense of Definition 7.1. In particular, we get (7.47)

where the f ℓ 0,j 's are polynomials with respect to ln τ . The principal symbol f ℓ 0,0 is independent of ln τ and we have (7.48)

In that case, (7.39) becomes (7.49)

Note that f ℓ 0 (τ, x ℓ ℓ ℓ ) has in fact a compact support with respect to τ . Now, using Lemma D.1, we can perform the integration with respect to t of each term in the right hand side of (7.47), modulo a term O(h ∞ ) (see (D.3)-(D.4) in Lemma D.1). Then, we get (7.50)

We write u = 1 + v(τ, x ℓ ℓ ℓ ) and we known that v ∈ C ∞ and v = o τ (1). Differentiating the equality

. Thus we have

where r(τ, x ℓ ℓ ℓ ) = o(1) change from line to line.

Since f ℓ 0 (t, x ℓ ℓ ℓ , h) is expandible with respect to t, we get, from (7.64)-(7.66), (7.67)

where r = o(1) and (7.68)

). In that case, (7.39) becomes (7.69)

Note that f ℓ 0 (τ, x ℓ ℓ ℓ ) has in fact a compact support with respect to τ . Now, using Lemma D.1, we can perform the integration with respect to t in (7.69), modulo an error term given by (D.3)-(D.4) in Lemma D.1. Then, we get 1) , (7.70) as h goes to 0. The rest of the proof follows that of (7.57).

Lastly, the proof of Theorem 2.6 in the case (c) can be obtained along the same lines, and we omit it.

Appendix A. Proof of Proposition 2.5

We prove that Λ + θ ∩ Λ + = ∅. From Assumption (A2), the Lagrangian manifold Λ + can be described, near (0, 0) ∈ T * (R n ), as

for |ξ| < 2ε, with ε > 0 small enough. For η ∈ S n-1 , let (x(t, η), ξ(t, η)) be the bicharacteristic curve with initial condition ( ϕ(εη), εη). We have

The function ξ(t, η) is continuous on R × S n-1 . From the classical scattering theory (see [13, Section 1.3]), we know that this function ξ(t, η) converges uniformly to

Then, the function

Here we used that |ξ(t, η)| = 0 for each t ∈ [0, +∞], η ∈ S n-1 . The previous properties of ξ(t, η) imply the continuity of

On the other hand,

In particular, we have

which is impossible (see [START_REF] Fulton | Algebraic topology[END_REF]Section 23] for more details).

Appendix B. A lower bound for the resolvent

Let χ ∈ C ∞ (]0, +∞[) be a non-decreasing function such that (B.1)

where 0 < α < 2β will be fixed later on. The u j 's are of course C ∞ functions, and we have

Lemma B.1. For any h small enough, we have

Appendix C. manifolds which are transverse to Λ ± Let Λ ⊂ p -1 (E 0 ) be a Lagrangian manifold such that Λ ∩ Λ -is transverse along a Hamiltonian curve γ(t) = (x(t), ξ(t)). Then, there exist a = 0 and ν ∈ {1, . . . , n} such that (C.1) γ(t) = (a + O(e -εt ))e -λν t , as t → +∞. The vector a is an eigenvector of (C.2) 0 Id V ′′ (0) 0 , for the eigenvalue λ ν . Thus, up to a linear change of variable in R n , we can always assume that Π x a is collinear to the x ν -direction. The goal of this section is to prove the following geometric result.

Proposition C.1. For t large enough, Λ projects diffeomorphically on R n x near γ(t). In particular, there exists ψ ∈ C ∞ (R n ) defined near Π x γ, unique up to a constant, such that We remark that, in the previous equation,ln s/ ln λ >ln(2/ε)/ ln λ > -1/2 for λ large enough. Using (1 + u) β = 1 + O(|u| + |u| max (1,β) ) for u > -1/2, we get