Julien Royer

Semiclassical measure for the solution of the dissipative Helmholtz equation

Keywords: 2010 Mathematical Subject Classification. 35J10, 35Q40, 35S30, 47B44, 47G30, 81Q20. Keywords. Dissipative operators, Helmholtz equation, Semiclassical measures

We study the semiclassical measures for the solution of a dissipative Helmholtz equation with a source term concentrated on a bounded submanifold. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption coefficient is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagragian distributions. Moreover we prove and use the fact that the outgoing solution of the dissipative Helmholtz equation is microlocally zero in the incoming region.

Introduction and statement of the result

We consider on L 2 (R n ) the dissipative semiclassical Helmholtz equation:

(-h 2 ∆ + V h -E h )u h = S h
(1.1) in the high frequency limit, that is when the semiclassical parameter h > 0 goes to 0. Here the potential V h = V 1 -ihV 2 has a nonpositive imaginary part of size h. We recall (see [START_REF] Benamou | A geometrical optics-based numerical method for high frequency electromagnetic fields computations near fold caustics[END_REF]) that this equation modelizes for instance the propagation of the electromagnetic field of a laser in material medium. In this setting the parameter h is the wave length of the laser, Re(E h -V h ) is linked to the electronic density of the material medium (and plays the role of the refraction index for the corresponding hamiltonian problem) while h -1 Im(E h -V h ) is the absorption coefficient of the laser energy by the material. Thus, in order to consider the case of a non-constant absorption coefficient we have to allow non-real potentials. We proved in [Roy] that if the potential has non-positive imaginary part then (with decay and regularity assumptions on V ) the resolvent (-h 2 ∆ + V hz) -1 is well-defined for Im z > 0 and is of size O(h -1 ) uniformly for z close to E ∈ R * + on condition E satisfies an assumption on classical trajectories for the corresponding hamiltonian problem. In this case, the resolvent has a limit for z → E in the space of bounded operators in some weighted spaces, and this limit operator gives the (outgoing) solution for (1.1) (see below).

Given a source term S h and such an energy E > 0, our purpose in this paper is to study the asymptotic when h → 0 for the outgoing solution u h of (1.1). More precisely we are interested in the semiclassical measures (or Wigner measures) of u h . The first work in this direction seems to be the paper of J.-D. Benamou, F. Castella, T. Katsaounis and B. Perthame ([BCKP02]). In their paper S h = S(x/h)/h concentrates on 0 and Im E h = hα h with α h → α 0. They consider the family of Wigner transforms f h of the solutions u h and prove that after extracting a subsequence, this family of Wigner transforms converges to a measure f which is the (outgoing) solution of the transport equation 1 :

αf + ξ.∂ x f (x, ξ) - 1 2 ∂ x V 1 (x).∂ ξ f (x, ξ) = 1 (4π) 2 δ(x) Ŝ(ξ) 2 δ(|ξ| = 1) (1.2)
Note that the solution is estimated by Morrey-Companato-type estimates (see [START_REF] Perthame | Morrey-Campanato estimates for Helmholtz equations[END_REF]) and that part of the result is left as a conjecture and proved in [START_REF] Castella | The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave-packet approach[END_REF]. F. Castella, B. Perthame and O. Runborg study in [START_REF] Castella | High frequency limit of the Helmholtz equation. II: Source on a general smooth manifold[END_REF] the similar problem with a source term which concentrates on an unbounded submanifold of R n . As a consequence there is a lack of decay of the source and Morrey-Companato estimates cannot be used. Actually only a formal description of the asymptotics is given and the proof concerns the case where the refraction index is constant, that is V 1 = 0, and the submanifold is an affine subspace. X.-P. Wang and P. Zhang give a proof for V 1 = 0 (variable refraction index) in [START_REF] Wang | High-frequency limit of the Helmholtz equation with variable refraction index[END_REF] using uniform estimates given by Mourre method. We also mention the work of E. Fouassier who considered the case of a source which concentrates on two points (see [START_REF] Fouassier | High frequency analysis of Helmholtz equations: case of two point sources[END_REF], V 1 = 0 in this case) and the case of a potential discontinuous along an affine hyperplane in [START_REF] Fouassier | High frequency limit of helmholtz equations: refraction by sharp interfaces[END_REF] (the source concentrates on 0 in this case). All this papers use a priori estimates of the solution in Besov spaces (we have already mentionned [START_REF] Perthame | Morrey-Campanato estimates for Helmholtz equations[END_REF], see also [START_REF] Castella | Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and C 2 potential[END_REF][START_REF] Wang | High-frequency limit of the Helmholtz equation with variable refraction index[END_REF][START_REF] Wang | Microlocal estimates of the stationnary Schrödinger equation in semiclassical limit[END_REF][START_REF] Castella | Semiclassical resolvent estimates for Schrödinger operators with Coulomb singularities[END_REF] for further results about these estimates).

Here we are going to use the point of view of J.-F. Bony (see [Bon]). He considers the case of a source which concentrates on one or two points (with V 1 = 0) using a time-dependant method based on a BKW approximation of the propagator to prove that, microlocally, the solution of the Helmholtz equation is a finite sum of lagrangian distributions. In particular, abstract estimates of the solution are only used for the large times control, and this part of the solution has no contribution for the semiclassical measure, so the measure is actually constructed explicitely. Moreover, this method requires a geometrical assumption weaker than the Virial hypothesis used in the previous works.

In this paper we consider the case where not only the refraction index but also the absorption coefficient can be non-constant, and hence we have to work with a non-selfadjoint Schrödinger operator. But, as already mentionned, we know that the resolvent is well-defined for a spectral parameter z with Im z > 0. For the selfadjoint semiclassical Schrödinger, we need a non-trapping condition on classical trajectories of energy E > 0 to have uniform estimates of the resolvent and the limiting absorption principle around E (see [START_REF] Robert | Semi-classical estimates for resolvents and asymptotics for total scattering cross-sections[END_REF][START_REF] Wang | Time-decay of scattering solutions and classical trajectories[END_REF]). In the dissipative case, this assumption can be weakened as follows: any trajectory should either go to infinity or meet the region where V 2 > 0. This is the assumption we are going to use, and as as consequence, even if we can show that the outgoing solution u h of (1.1) is microlocally zero in the incoming region, the contribution of large times in u h does not vanishes when h → 0 as is the case in [Bon], and in particular the solution can be an infinite sum of lagrangian distributions around some points of the phase space. However, the assumption that bounded trajectories should meet the region where there is absorption will make the series of amplitudes of these distributions convergent, which is the key argument in order to have a well-defined semiclassical measure in our case.

Concerning the source term, S h is allowed to concentrate on any bounded submanifold of R n . We do not have problem like in [START_REF] Castella | High frequency limit of the Helmholtz equation. II: Source on a general smooth manifold[END_REF][START_REF] Wang | High-frequency limit of the Helmholtz equation with variable refraction index[END_REF] with decay assumptions, but this allows us to see what happens when the source concentrates on a non-flat submanifold. Note that we do not have phase factor in our source term (see below) so we are in the propagative regime described in [START_REF] Castella | High frequency limit of the Helmholtz equation. II: Source on a general smooth manifold[END_REF].

Let us now state the assumptions we are going to use in this work. We denote the free laplacian -h 2 ∆ by H h 0 and H h is the dissipative Schrödinger operator on L 2 (R n ) (n 1):

H h = -h 2 ∆ + V 1 (x) -ihV 2 (x)
We also denote by

H h 1 = -h 2 ∆ + V 1 (x) the selfadjoint part of H h . V 1 , V 2 are smooth real functions on R n , V 2 is nonnegative and for j ∈ {1, 2}, α ∈ R n : |∂ α V j (x)| C α x -ρ-|α| (1.3)
for some ρ > 0. Here • denotes the function x → (1 + |x| 2 ) 1 2 . Let p : (x, ξ) → ξ 2 + V 1 (x) be the symbol on R 2n ≃ T * R n of the selfadjoint part H h 1 . The classical trajectories for this problem are the solutions φ t (w) = (x(t, w), ξ(t, w)) for w ∈ R 2n of the hamiltonian problem:

    
∂ t x(t, w) = 2ξ(t, w) ∂ t ξ(t, w) = -∇V 1 (x(t, w)) φ 0 (w) = w We recall from [Roy] that the exact hypothesis we need on an energy E > 0 to have the limiting absorption principle around E is the following: if we set

O = {x ∈ R n : V 2 (x) > 0}
then for all w ∈ R 2n such that p(w) = E we have:

φ t (w), t ∈ R is unbounded in R 2n or φ t (w), t ∈ R ∩ O = ∅ (1.4)
which means that any trapped trajectories should meet the set where there is absorption.

For further use we also set, for γ > 0:

O γ = {x ∈ R n : V 2 (x) > γ}
With this condition (which is actually necessary), for any α > 1 2 there exist ε > 0 and c 0 such that: sup |Re z-E| ε,Im z>0

x -α (H hz) -1 x -α c h and for all λ ∈ [Eε, E + ε] the limit:

(H h -(E + i0)) -1 := lim µ→0 + (H h -(E + iµ)) -1
exists (and is a continuous function of λ) in the space of bounded operators from L 2,α (R n ) to L 2,-α (R n ), where L 2,δ (R n ) stands for L 2 ( x 2δ dx). Then for all S h ∈ L 2,α (R n ), u h = (H h -(E + i0)) -1 S h ∈ L 2,-α (R n ) is the outgoing solution for (1.1). About the classical hamiltonian problem, we use the following notations:

Ω ± b (J) = w ∈ R 2n : {x(t, w), ±t 0} is bounded Ω ± ∞ (J) = w ∈ R 2n : |x(t, w)| ----→ t→±∞ +∞ Note that Ω ± ∞ (J) is open if J is open and Ω ± b (J) is closed if J is closed.
Let us now introduce the source term we consider. Given a (bounded) submanifold Γ 2 of dimension d ∈ 0, n -1 in R n with the measure σ induced by the Lebesgue measure on R n , a smooth function A of compact support on Γ 2 and a Schwartz function S ∈ S(R n ), we note for x ∈ R n :

S h (x) = h 1-n-d 2 z∈Γ A(z)S x -z h dσ(z) (1.5) We can choose Γ and Γ 1 open in Γ 2 such that Γ 0 := supp A ⊂ Γ, Γ ⊂ Γ 1 and Γ 1 ⊂ Γ 2 (if Γ 2 is compact we can have Γ 0 = Γ = Γ 1 = Γ 2 ).
As usual, for z ∈ Γ 2 and ζ ∈ T z Γ 2 small enough (where T z Γ 2 is the tangent space to Γ 2 at z), we denote by exp

z (ζ) the point c ζ (1) where t → c ζ (t) is the unique geodesic on Γ 2 with initial conditions c ζ (0) = z and c ′ ζ (0) = ζ (see [GHL90, §2.86]).
On Γ 2 we define the distance d Γ as usual: for x, y ∈ Γ 2 , d Γ (x, y) is the infimum of the length of all piecewise C 1 curves from x to y. For z ∈ Γ 2 , there exists a neighborhood U of z in Γ 2 and ε > 0 such that for x, y ∈ U there is a unique geodesic c from x to y of length less than ε. And the length of

c is d Γ (x, y) (see [GHL90, §2.C.3]).
We consider a family of energies E h ∈ C for h ∈]0, 1]. We assume that Im E h 0 and:

E h = E 0 + hE 1 + o h→0 (h) (1.6)
where E 0 > 0 satisfies (1.4) and:

∀z ∈ Γ, V 1 (z) < E 0 (1.7) We set N Γ = {(z, ξ) ∈ Γ × R n : ξ⊥T z Γ}, N E Γ = (z, ξ) ∈ N Γ : |ξ| = E 0 -V 1 (z) and: Λ = φ t (z, ξ); t > 0, (z, ξ) ∈ N E Γ We similarly define N E Γ 0 and N E Γ 1 . For (z, ξ) ∈ N E Γ and (Z, Ξ) ∈ T (z,ξ) N E Γ we have Z ∈ T z Γ and Ξ ∈ R n decomposes as Ξ = Ξ T + Ξ + Ξ ⊥ with Ξ T ∈ T z Γ, Ξ ∈ Rξ and Ξ ⊥ ∈ (T z Γ ⊕ Rξ) ⊥ .
Then N E Γ is endowed with the metric g defined by:

g (z,ξ) (Z 1 , Ξ 1 ), (Z 2 , Ξ 2 ) = Z 1 , Z 2 R n + Ξ 1 ⊥ , Ξ 2 ⊥ R n for all (Z 1 , Ξ 1 ), (Z 2 , Ξ 2 ) ∈ T (z,ξ) N E Γ.
This means that we do not take into account the part of Ξ colinear to ξ and T z Γ, which is allowed since (Z, Ξ) never reduces to (0, Ξ T + Ξ ) unless (Z, Ξ) = (0, 0). Indeed, if Z = 0 then Ξ ∈ T (z,ξ) (N E Γ ∩ N z Γ) and hence Ξ = Ξ ⊥ . Now we denote by σ the canonical measure on N E Γ given by the metric g. This means that for any smooth map ψ : U → V (where

U is an open set in R n-1 and V is an open set in N E Γ) and any function f on V we have (see [GHL90, §3.H]): V f (v) dσ(v) = U f (ψ(u)) det(g ψ(u) (∂ i ψ(u), ∂ j ψ(u))) 1 i,j n-1 1 2 du
Finally we set:

Φ 0 = (z, ξ) ∈ N E Γ : ∃t > 0, φ t (z, ξ) ∈ N E Γ
The last assumption we need is:

σ(Φ 0 ) = 0 (1.8)
In [Bon, section 4] is given an example of what can happen without an hypothesis of this kind. Note that when Γ = {0}, this assumption is weaker than the assumption ν 0 (E 0 -V 1 (x))x.∇V 1 (x) c 0 > 0 for some ν 0 ∈]0, 2] which is used for instance in [START_REF] Wang | Microlocal estimates of the stationnary Schrödinger equation in semiclassical limit[END_REF]. This is no longer true in general (for instance we can take V 1 = 0, E 0 = 1 and any circle in R 2 for Γ).

To study semiclassical measures of u h , we choose the point of view of pseudo-differential operators. Let us recall that the Weyl quantization of an observable a : R 2n → C is the operator:

Op w h (a)u(x) = 1 (2πh) n R n R n e i h x-y,ξ a x + y 2 , ξ u(y) dy dξ
We also use the standard quantization: [START_REF] Martinez | An Introduction to semiclassical and microlocal analysis[END_REF][START_REF] Evans | Lectures on semiclassical analysis[END_REF] for more details about semiclassical pseudo-differential operators, [START_REF] Gérard | Mesures semi-classiques et ondes de bloch[END_REF] for semiclassical measures. We are going to use the following classes of symbols. For δ ∈ R we set:

Op h (a)u(x) = 1 (2πh) n R n R n e i h x-y,ξ a(x, ξ)u(y) dy dξ See [Rob87,
S δ = a ∈ C ∞ (R 2n ) : ∀α, β ∈ N n , ∃c α,β , ∀(x, ξ) ∈ R 2n , ∂ α x ∂ β ξ a(x, ξ) c α,β x δ-|α| while S b is the set of C ∞ (R 2n
) functions whose derivatives up to any order are in L ∞ (R 2n ).

We can now state the main theorem of this paper:

Theorem 1.1. There exists a Radon measure µ on R 2n such that for all q ∈ C ∞ 0 (R 2n ):

Op w h (q)u h , u h ---→ h→0 R 2n q dµ
Moreover µ is characterized by the following three properties:

(i) µ is supported on the hypersurface of energy E 0 :

supp µ ⊂ p -1 ({E 0 }) (ii) µ vanishes in the incoming region: let σ ∈]0, 1[, then there exists R 0 such that for q ∈ C ∞ 0 (R 2n
) supported in the incoming region Γ -(R, -σ) (see definition in section 5.1) we have: q dµ = 0 (iii) µ satisfies the Liouville equation:

(H p + 2 Im E 1 + 2V 2 )µ = π(2π) d-n A(z) 2 |ξ| -1 Ŝ(ξ) 2 σ (1.9)
where H p = {p, •} = 2ξ.∂ x -∇V 1 (x).∂ ξ and σ is extended by 0 on R 2n \ N E Γ. This means that for any q ∈ C ∞ 0 (R 2n ) we have:

R 2n (-H p + 2 Im E 1 + 2V 2 )q dµ = π(2π) d-n NEΓ q(z, ξ)A(z) 2 |ξ| -1 Ŝ(ξ) 2 dσ(z, ξ)
We first remark that this theorem gives not only existence of a semiclassical measure but also uniqueness, since we do not need to extract a subsequence to have convergence of Op w h (q)u h , u h when h → 0. Moreover, we see that in the Liouville equation the absorption coefficient α of (1.2) is replaced by our full non-constant absorption coefficient Im E 1 + V 2 , as one could expect.

And finally we will prove that the three properties of the theorem implies that the measure µ is given, for q ∈ C ∞ 0 (R 2n ), by:

R 2n q dµ = π(2π) d-n R+ NE Γ A(z) 2 |ξ| -1 Ŝ(ξ) 2 q(φ t (z, ξ))e -2t Im E1-2 R t 0 V2(x(s,z,ξ)) ds dσ(z, ξ) dt
(1.10) To prove this theorem we write as in [Bon] the resolvent as the integral over positive times of the propagator, the main difference being the large times contribution. Let:

U h (t) = e -it h H h , U h 0 (t) = e -it h H h 0 , and U E h (t) = e -it h (H h -E h )
Then:

u h = (H h -(E h + i0)) -1 S h = i h +∞ 0 U E h (t)S h dt (1.11)
and for T 0 we set:

u T h = (H h -(E h + i0)) -1 S h -(H h -(E h + i0)) -1 U E h (T )S h = i h T 0 U E h (t)S h dt (1.12)
Our purpose is to study the quantity:

lim h→0 lim T →+∞ Op h (q)u T h , u T h
which we cannot do directly. Around w ∈ R 2n , troubles appear when proving that relevant parts of integral (1.11) are around times t for which we can find (z, ξ) ∈ N E Γ such that φ t (z, ξ) = w (see proposition 4.1). Indeed, far from these times we can find t such that φ t (N E Γ) is close to w, giving contribution for the semiclassical measure in any neighborhood of w. Moreover, the Egorov theorem we use gives estimates uniform in h but not in time (see [START_REF] Bouzouina | Uniform semiclassical estimates for the propagation of quantum observables[END_REF] for a discussion of this problem). The key of our proof is to check that even if the contribution of large times is not zero as for the non-trapping case, the damping term V 2 makes it so small that the semiclassical measure is also given by: lim

T →+∞ lim h→0 Op h (q)u T h , u T h
which is much easier to study. Indeed, this means that we study the semiclassical measure for the family (u T h ). This can be done as for the non-trapping case since we do not have to worry about large times behavior. This gives a family of measures on R 2n , and then we can take the limit T → +∞, since we no longer have problems with the parameter h. It only remains to check this gives the measure we are looking for.

We begin the proof by a few preliminary results: we show to what extent the damping term V 2 implies a decay of U h (t), we look at the classical trajectories around the submanifold Γ and give more details about the assumption on Φ 0 . Finally we show that the solution u h concentrates on the hypersurface of energy E 0 . In section 3 we give an estimate of the solution near Γ, since we cannot give a precise description of u h there. This part is close to section 3.3 of [Bon] but we give a complete proof in order to see how to deal with the general case dim Γ 1. In section 4 we study the finite times contribution and give the semiclassical measure for u T h , and then in section 5 we prove that taking the limit T → +∞ for this family of measures gives a semiclassical measure for the solution u h . We also show that this limit is the solution of the Liouville equation (1.9) where V 2 naturally appears as a damping factor. Finally in section 6 we give the proof of the estimate in the incoming region we use in section 5. Indeed if we no longer assume that all the classical trajectories of energy E 0 go to infinity, there still are some non-trapped trajectories. So we still need the estimate of the outgoing solution in the incoming region used in the non-trapping case. For the self-adjoint Schrödinger operator, this is proved in [START_REF] Robert | Asymptotic behavior of scattering amplitudes in semiclassical and low energy limits[END_REF] but here we need to show that this remains true in our dissipative setting.

2 Some preliminary results

Damping effect of the absorption coefficient on the semigroup generated by H h

We saw in [Roy] that assumption (1.4) is actually satisfied for any energy close enough to E 0 , hence we can consider two closed intervals I and J such that E 0 ∈ I, I ⊂ J and any trapped trajectory of energy in J meets O.

The main tool we need in this section is the dissipative version of Egorov theorem. We already stated this theorem in [Roy] but we give here a more precise version we are going to use in the proof of proposition 4.1.

Proposition 2.1. Let a ∈ S b .

(i) There exists a family of symbols α j (t) for j ∈ N and t 0 such that for any N ∈ N and t 0 the symbol A N (t, h) = N j=0 h j α j (t) satisfies:

U h (t) * Op w h (a)U h (t) = Op w h (A N (t, h)) + O h→0 (h N +1 )
where the rest is bounded as an operator on L 2 (R n ) uniformly in t ∈ [0, T ] for any T 0.

(ii) α 0 (t) = (a • φ t ) exp -2 t 0 V 2 • φ s ds where for (x, ξ) ∈ R 2n , V 2 (x, ξ) means V 2 (x).
(iii) If a vanishes on the open set W ⊂ R 2n then for all j ∈ N the symbol α j (t) vanishes on φ -t (W).

Proof. In [Roy] we proved (i) for N = 0 and (ii). Moreover (iii) is a direct consequence of (ii) for j = 0. What remains can be proved as in the selfadjoint case (see [START_REF] Robert | Autour de l'appoximation semi-classique[END_REF]) so we only recall the ideas. (i) is proved by induction. More precisely, we show that for any N ∈ N:

U h (t) * Op w h (a)U h (t) = N j=0 h j Op w h (α j (t)) +h N +1 t τ1=0 τ1 τ2=0 . . . τN τN+1=0 U h (τ N +1 ) * Op w h (b N (τ 1 , . . . , τ N +1 , h))U h (τ N +1 ) dτ N +1 . . . dτ 1
for some symbol b N . The case N + 1 is obtained by applying the case N = 0 to the principal symbol of b N .

To prove (iii) we take the derivative of U h (t) * Op w h (a)U h (t) with report to t. This gives, for j ∈ N:

∂ t α j (t) = H p (α j ) -2V 2 α j (t) + j-1 q=0 C j,q D * j,q α q
where C j,q is a function with bounded derivatives and D * j,q is a differential operator. Then if αj (t) = (α j (t) • φ -t ) exp 2 t 0 V 2 • φ -s ds we have:

∂ t αj (t) = j-1 q=0 C j,q D * j,q (α q (t) • φ -t ) exp 2 t 0 V 2 • φ -s ds
and it is easy to check by induction on j 1: αj (0) = 0, ∂ t αj (t) = 0 on W, and hence α j (t) = 0 on φ -t (W)

Lemma 2.2. Let K be a compact subset of Ω + b (J). There is C 0 and δ > 0 such that:

∀w ∈ K, exp - t s=0 V 2 (φ s (w)) ds Ce -δt
Proof. 1. We first recall that if w ∈ Ω + b (J) then there exists T 0 such that φ T (w) ∈ O (this is slightly stronger than assumption (1.4)). Indeed, the set K w = {φ t (w), t 0} is compact, so there is an increasing sequence (t m ) m∈N with t m → +∞ and

w ∞ ∈ K w such that φ tm (w) → w ∞ . Since Ω + b ({p(w)}) is closed, w ∞ ∈ Ω + b ({p(w)}). Moreover, for M ∈ N and m M we have φ -tM (φ tm (w)) ∈ K w and hence φ -tM (w ∞ ) ∈ K w , which proves that w ∞ ∈ Ω - b (R). By assymption (1.4), there is T ∈ R such that φ T (w ∞ ) ∈ O.
Hence φ T +tm (w) lies in O for large m. Since T + t m 0 when m is large enough, the claim is proved.

2. We set: K = {φ t (w), t 0, w ∈ K} By definition of K, K is compact in R 2n . Let w ∈ K. There are T w 0 and γ w > 0 such that φ Tw (w) ∈ O 2γw , so we can find τ w > 0 and a neighborhood V w of w in R 2n such that for all v ∈ V w and t ∈ [T wτ w , T w ] we have: φ t (v) ∈ O γw . As K is compact we can find w 1 , . . . , w k such that K ⊂ ∪ k i=1 V wi . Then we take T = max{T wi , 1 i k}, τ = min{τ i , 1 i k} and γ = min{γ wi , 1 i k}. For all w ∈ K and t 0, φ t (w) is in K and hence in [t, t + T ] there is a subinterval I w,t of length at least τ such that φ s (w) ∈ O γ for s ∈ I w,t . Thus:

exp t+T s=t V 2 (φ s (w)) ds e -τ γ
We apply this for t n = nT with n t/T and this gives:

exp t 0 V 2 (φ s (w)) ds e -t-T T τ γ
e τ γ e -t τ γ T so the result follows with C = e τ γ et δ = τ γ T . Proposition 2.3. Let q, q ′ ∈ C ∞ 0 (R 2n ) supported in p -1 (J) and ε > 0. Then there exists T 0 0 such that for all T T 0 we can find h T > 0 which satisfies:

∀h ∈]0, h T ], Op w h (q)U h (T )Op w h (q ′ ) ε Proof. We set K = supp q ′ ∩ Ω + b (R).
As K is a compact subset of Ω + b (J), lemma 2.2 shows that there is T 0 0 such that:

sup w∈K q ∞ q ′ ∞ exp - T s=0 (V 2 • φ s )(w) ds ε 4
As the left-hand side is a continuous function of w, we can find a neighborhood V of K in R 2n such that this holds for w ∈ V after having replaced ε/4 by ε/2. Let now

K ∞ = supp q ′ \ V. K ∞ is a compact subset of Ω ∞ + .
Therefore, if T 0 is large enough, we can assume that for T T 0 and w ∈ K ∞ we have φ T (w) / ∈ supp q. Hence by Egorov theorem (see also remark 4.4 in [Roy]), for any T T 0 we have:

Op w h (q)U h (T )Op w h (q ′ ) = U h 1 (-T )Op w h (q)U h (T )Op w h (q ′ ) = Op w h (q • φ T )e - R T s=0 V2•φ s ds Op w h (q ′ ) + O h→0 (h) sup w∈R 2n q ′ (w)(q(φ T (w)))e - R T s=0 V2(φ s (w)) ds + C(T ) √ h ε 2 + C(T ) √ h (2.1)
and hence for any fixed T T 0 we can find h T > 0 small enough to conclude.

Classical trajectories around Γ

In this section we assume that assumptions (1.3), (1.4) and (1.7) are satisfied.

Proposition 2.4. There exists τ 0 > 0 such that:

T : ]0, 3τ 0 ] × N E Γ 1 → R n (t, w) → x(t, w) (2.2)
is one-to-one and Ran(T ) ∪ Γ 1 is a neighborhood of Γ in R n . Furthermore:

(i) We can choose τ 0 to have:

∀t ∈]0, 3τ 0 ], ∀w ∈ N E Γ 1 , 2γ m t d(x(t, w), Γ 2 ) 2γ M t (2.3)
for some γ M γ m > 0.

(ii) If f is a continuous function with support in T (]0, 3τ 0 [×N E Γ) then:

x∈R n f (x) dx = 2 n-d 3τ0 0 NE Γ f (x(t, z, ξ))t n-d-1 |ξ| 1 + O t→0 (t) dσ(z, ξ) dt (2.4)
For 0 r 1 r 2 3τ 0 we set:

Γ(r 2 ) = T ([0, r 2 ] × N E Γ) and Γ(r 1 , r 2 ) = T (]r 1 , r 2 ] × N E Γ) When x ∈ Γ(0, 3τ 0 ) we write (t x , z x , ξ x ) = T -1 (x).
Proof. For τ > 0, let :

N (τ ) = (z, ξ) ∈ N Γ 1 : |ξ| τ E 0 -V 1 (z)
We consider the function T from N (1) to R n defined by:

T (z, ξ) =    x |ξ| √ E0-V1(z) , z, ξ √ E0-V1(z) |ξ| if ξ = 0 z if ξ = 0
We have:

T (z, ξ) = z + 2ξ + o(|ξ|)
Hence for τ 0 > 0 small enough, T is a diffeomorphism from N (3τ 0 ) to a tubular neighborhood of Γ 1 (we can follow the proof for the function (x, ξ) → z + 2ξ, see for instance theorem 2.7.12 in [START_REF] Berger | Géométrie différentielle : variétés, courbes et surfaces[END_REF]). In particular T and hence T : (t, z, ξ) → T (z, tξ) are one-to-one and

Ran T ∪ Γ 1 = Ran T N (3τ0)
∪ Γ 1 is a neighborhood of Γ 0 .

(i) We have:

x(t, z, ξ) -z = t 0 2ξ(s, z, ξ) ds = 2tξ -2 t 0 s 0 ∇V 1 (u, z, ξ) du ds Hence, if M = sup x∈R n |∇V 1 (x)| this gives: |x(t, z, ξ) -z -2tξ| 2t 2 M Denote ξ min = min{|ξ| , ξ ∈ N E Γ 1 } > 0 and ξ max = max{|ξ| , ξ ∈ N E Γ 1 }. We recall from [BG87] that for (z, ξ) ∈ N E Γ 1 and t small enough we have d(z + tξ, Γ 2 ) = t |ξ|.
Then for τ 0 small enough we have 2τ 0 M ξ min so:

d(x(t, z, ξ), Γ 2 ) d(z + 2tξ, Γ 2 ) -|x(t, z, ξ) -z -2tξ| 2t |ξ| -tξ min tξ min and: d(x(t, z, ξ), Γ 2 ) d(z + 2tξ, Γ 2 ) + |x(t, z, ξ) -z -2tξ| 2t |ξ| + tξ min t(2ξ max + ξ min ) (ii) Let (t, z, ξ) ∈]0, 3τ 0 [×N E Γ. For (T 1 , Z 1 , Ξ 1 ), (T 2 , Z 2 , Ξ 2 ) ∈ T (t,z,ξ) (]0, 3τ 0 [×N E Γ) we set: g(t,z,ξ) ((T 1 , Z 1 , Ξ 1 ), (T 2 , Z 2 , Ξ 2 )) = T 1 T 2 + g (z,ξ) ((Z 1 , Ξ 1 ), (Z 2 , Ξ 2 ))
We first look for good orthonormal bases of T (t,z,ξ) (]0, 3τ 0 [×N E Γ) (for the metric g) and R n (for the usual metric) to compute the jacobian of T .

N E Γ ∩ ({z} × R n ) is a submanifold of dimension n -d -1 in N E Γ,
so we can consider an orthonormal basis ((0, Ξ j )) d+2 j n of its tangent space at (z, ξ). We now choose an orthonormal basis (Z j ) 2 j d+1 of T z Γ. We can find Ξ 2 , . . . , Ξ d+1 ∈ R n such that (Z j , Ξ j ) ∈ T (z,ξ) N E Γ for j ∈ 2, d + 1 and since linear combinations of (0, Ξ d+2 ), . . . , (0, Ξ n ) can be added, we may assume that Ξ j ∈ T z Γ ⊕ Rξ for all j ∈ 2, d + 1 . These n -1 vectors form an orthonormal family of T (z,ξ) N E Γ to which we add the canonical unit vector of R for the time component. This gives an orthonormal basis B (t,z,ξ) of T (t,z,ξ) (]0, 3τ 0 [×N E Γ). In R n we consider the orthonormal basis:

BT (t,z,ξ) = (ξ/ |ξ| , Z n-d , . . . , Z n-1 , Ξ 1 , . . . , Ξ n-d-1 )
Since T (t, z, ξ) = z + 2tξ + O(t 2 ), the jacobian matrix of T in these two bases is:

Mat B (t,z,ξ) → BT (t,z,ξ) D (t,z,ξ) T =   2 |ξ| 0 0 0 I d 0 0 0 2tI n-d-1   1 + O t→0 (t)
On the other hand, since basis B (t,z,ξ) and BT (t,z,ξ) are orthonormal, we have, for x ∈ Γ(0, 3τ 0 ):

det(g T -1 (x) (∂ i T -1 (x), ∂ j T -1 (x))) 1 i,j n 1 2 = det Mat Bx→B T -1 (x) D x T -1
Thus, using the definition of the measure dt dσ on ]0, 3τ 0 [×N E Γ and the fact that T -1 : Γ(0, 3τ 0 ) →]0, 3τ 0 [×N E Γ can be seen as a map for the manifold ]0, 3τ 0 [×N E Γ, we obtain:

x∈R n f (x) dx = x∈R n (f • T )(T -1 x) det Mat Bx→B T -1 (x) D x T -1 det Mat B T -1 (x) → Bx D T -1 (x) T dx = 3τ0 t=0 (z,ξ)∈NEΓ (f • T )(t, z, ξ) det Mat B (t,z,ξ) → BT (t,z,ξ) D (t,z,ξ) T dσ(z, ξ) dt = 2 n-d 3τ0 0 NE Γ f (T (t, z, ξ))t n-d-1 |ξ| 1 + O t→0 (t) dσ(z, ξ) dt Corollary 2.5. Let (t, z, ξ) = (s, ζ, η) ∈ R * + × N E Γ such that φ t (z, ξ) = φ s (ζ, η).
Then |t -s| 3τ 0 where τ 0 is given by proposition 2.4.

Let w ∈ R 2n and denote:

((t w,k , z w,k , ξ w,k )) 1 k Kw = (t, z, ξ) ∈ R * + × N E Γ : φ t (z, ξ) = w with t w,1 < t w,2 < . . . and K w ∈ N ∪ {∞} ( 1, K w is to be understood as N * if K w = ∞ and K w = 0 if w / ∈ Λ). We also define K T w = sup {k ∈ 1, K w : t w,k T } ∈ N. For w ∈ R 2n and k ∈ 1, K w we write: Λ w,k = φ t (z, ξ), |t -t w,k | < τ 0 , |(z, ξ) -(z k , ξ k )| < τ 0 and if w ∈ N E Γ: Λ w,0 = φ t (z, ξ), |t| < τ 0 , |(z, ξ) -w| < τ 0 Proposition 2.6. Let w = (x, ξ) ∈ R 2n and j, k ∈ 1, K w ( 0, K w if w ∈ N E Γ). Then (i) Λ w,j ∩ Λ w,k is of measure zero in Λ w,j is and only if it is of measure zero in Λ w,k . (ii) Assumption (1.8) is equivalent to: ∀w ∈ R 2n , ∀j, k ∈ 1, K w (or 0, K w ), Λ j ∩ Λ k is of measure 0 in Λ j (2.5)
This proposition is proved in section 6 of [Bon].

Localization around E 0 -energy hypersurface

Proposition 2.7. For any δ ∈ R we have:

S h L 2,δ (R n ) = O h→0 √ h (2.6)
Proof. 1. There exists C 0 such that for all x ∈ R n and r > 0, the measure of B(x, r) ∩ Γ in Γ is less than Cr d . Otherwise for all m ∈ N we can find x m ∈ R n and r m > 0 such that the measure of the ball B(x m , r m ) ∩ Γ in Γ is greater than mr d m . As Γ is of finite measure, r m necessarily goes to 0 as m → +∞. On the other hand x m has to stay close to Γ, hence in a compact subset of R n , so taking a subsequence we can assume that

x m → x ∞ ∈ Γ. But the part of Γ close to x ∞ is diffeomorphic to a subset of R d ⊂ R n , hence the measure of B(x ∞ , r) ∩ Γ in Γ is less than Cr d for some C 0. 2. Let x ∈ R n . We have: S h (x) 2 = h 1-n-d m∈N mh |x-z|<(m+1)h A(z)S x -z h dσ(z) 2 c h 1-n-d m∈N m 2 mh |x-z|<(m+1)h A(z)S x -z h dσ(z) 2 c h 1-n m∈N m 2+d mh |x-z|<(m+1)h S x -z h 2 dσ(z)
and hence:

S h 2 L 2,δ (R n ) c h 1-n x∈R n m∈N m 2+d mh |x-z|<(m+1)h x 2δ S x -z h 2 dσ(z) dx c h m∈N m 2+d z∈Γ m |y|<(m+1) z + hy 2δ S(y) 2 dy dσ(z) c h m∈N m 2+d z∈Γ m |y|<(m+1) y 2δ S(y) 2 dy dσ(z)
for h ∈]0, 1], since Γ is bounded. As S decays faster than y -n+2δ+4+d 2 we have:

S h 2 L 2,δ (R n ) c h m∈N m 2 m -4-d c h Since (H h -(E h + i0)) -1 = O(h -1 ) as an operator from L 2,α (R n ) to L 2,-α (R n ) we get: Corollary 2.8. u h = O h→0 (h -1 2 ) in L 2,-α (R n ).
The same applies to u T h for all T 0.

Proposition 2.9.

S h is microlocalized in N Γ 0 . Proof. Let q ∈ C ∞ 0 (R 2n ) supported outside N Γ 0 .
We have:

Op w h (q)S h (x) = 1 (2πh) n Γ R n R n e i h x-y,ξ q(x, ξ)A(z)S y -z h dy dξ dσ(z) = 1 (2π) n Γ R n R n e i h x-z,ξ e -i v,ξ q(x, ξ)A(z)S(v) dv dξ dσ(z) If ∂ z x -z, ξ = 0 and ∂ ξ x -z, ξ = 0 then x = z and ξ ∈ N z Γ so A(z)q(x, ξ) = 0.
According to the non-stationnary phase theorem, we have

Op w h (q)S h = O(h ∞ ) in L 2 (R n ). Proposition 2.10. (i) Let g ∈ S b equal to 1 in a neighborhood of p -1 ({E 0 }).
We have:

Op w h (1 -g)(H h -(E h + i0)) -1 L 2,α (R n )→L 2-α (R n ) = O h→0 (1) (2.7) (ii) Let f ∈ S b equal to 1 in a neighborhood of N E Γ 0 , then in L 2,-α (R n ): u h = (H h -(E h + i0)) -1 Op h (f )S h + O h→0 ( √ h) (2.8) (iii) Moreover there exists g ∈ C ∞ 0 (R) equal to 1 in a neighborhood of E 0 such that in L 2,-α (R n ): (H h -(E h +i0)) -1 Op h (1-f )S h = (1-g)(H h 1 )(H h -(E h +i0)) -1 Op h (1-f )S h + O h→0 h 3 2
(2.9) Similar results hold for u T h , T 0. Proof. (i) For Im z > 0 we have:

Op h (1 -g)(H h -z) -1 = Op h (1 -g)(H h 1 -z) -1 (1 + hV 2 (H h -z) -1 )
According to [START_REF] Helffer | Calcul fonctionnel par la transformation de melin et opérateurs admissibles[END_REF] we have:

(H h 1 -z) -1 = Op w h (p(x, ξ) -z) -1 + O h→0 (h) Since (p(x, ξ) -z) -1 is bounded on supp(1 -g) uniformly for z close to E 0 , Im z > 0, the operator Op w h (1 -g)(H h 1 -z) -1 is uniformly bounded in h > 0 and z close to E 0 , Im z > 0. Moreover (1 + hV 2 (H h -z) -1 ) is uniformly bounded as an operator from L 2,α (R n ) to L 2,-α (R n ) so: Op w h (1 -g)(H h -z) -1 L 2,α (R n )→L 2-α (R n ) = O h→0 (1)
uniformly in z. Taking the limit z → E h + i0 gives (2.7).

(ii) Let U be a neighborhood of

N E Γ 0 in R 2n such that f = 1 on U. We can find ε > 0 such that p -1 ([E 0 -2ε, E 0 + 2ε]) \ U does not intersect N Γ 0 . Let χ ∈ C ∞ 0 (R) supported in ]E 0 -2ε, E 0 +2ε[ and equal to 1 on ]E 0 -ε, E 0 +ε[. Since modulo O(h ∞ ) the operator χ(H 1 ) is a pseudo-differential operator with symbol supported in supp(χ • p) and S h is microlocalized on N Γ 0 we have in L 2,α (R n ): (H h -(E h + i0)) -1 Op h (1 -f )χ(H 1 )S h = O h→0 (h ∞ )
On the other hand, as we proved (2.7) we see that:

(H h -(E h + i0)) -1 (1 -χ)(H h 1 ) = O h→0 (1) so (2.8) follows since Op w h (1 -f )S h = O( √ h). (iii) Let us refine this last estimate. Let g ∈ C ∞ 0 (R) supported in [E 0 -ε, E 0 + ε] and equal to 1 in a neighborhood of E 0 . Since (1 -χ)g = 0, we have: g(H h 1 )(H h -z) -1 (1 -χ)(H h 1 ) = g(H h 1 )(H h -z) -1 (1 -χ)(H h 1 )(1 -g)(H h 1 ) = g(H h 1 )(1 + h(H h -z) -1 V 2 )(H h 1 -z) -1 (1 -χ)(H h 1 )(1 -g)(H h 1 ) = hg(H h 1 ) (H h -z) -1 V 2 (1 -χ)(H h 1 ) (1 -g)(H h 1 ) (H h 1 -z) -1

It only remains to see that the operators (H

h -z) -1 V 2 (1-χ)(H h 1 ) and (1-g)(H h 1 )(H h 1 -z) -1 are bounded uniformly in h ∈]0, 1] and z close to E 0 with Im z > 0.
As a first consequence of this proposition we see that the solution u h consentrates on p -1 ({E 0 }):

Corollary 2.11. If q ∈ C ∞ 0 (R n ) has support outside p -1 ({E 0 }) then: Op w h (q)u h , u h ---→ h→0 0 Proof. Let q ∈ C ∞ 0 (R 2n
) supported outside p -1 ({E 0 }) and equal to 1 on supp q. We have:

Op w h (q)u h , u h = Op w h (q)u h , Op w h (q)u h + O h→0 (h ∞ ) = O h→0 (h)
3 Around Γ

WKB method

According to proposition IV.14 in [START_REF] Robert | Autour de l'appoximation semi-classique[END_REF] or lemma 10.10 in [EZ] applied with the symbol

p E : (x, ξ) → ξ 2 + V 1 (x) -E 0 we know that if τ 0 is small enough, then there exists a function ϕ ∈ C ∞ ([-3τ 0 , 3τ 0 ] × R 2n
) such that:

∂ t ϕ(t, x, ξ) + |∂ x ϕ(t, x, ξ)| 2 + V 1 (x) -E 0 = 0 ϕ(0, x, ξ) = x, ξ (3.1)
Moreover ϕ is unique and:

ϕ(t, x, ξ) = y(t, x, ξ), ξ + 2 t 0 ξ(s, t, x, ξ) 2 ds -tp E (x, ξ) = x, ξ -2 t 0 ξ(s, t, x, ξ), ξ ds + 2 t 0 ξ(s, t, x, ξ) 2 ds -tp E (x, ξ) = x, ξ -tp E (x, ξ) + t 2 r(t, x, ξ) (3.2)
where y(t, x, ξ) is the unique point in R n such that x(t, y(t, x, ξ), ξ) = x (note that y(t, x, ξ) is well-defined for t small enough, see [START_REF] Robert | Autour de l'appoximation semi-classique[END_REF]) and:

r(t, x, ξ) = 2 t 2 t s=0 t τ =s ξ(s, t, x, ξ), ∇V 1 (x(τ, t, x, ξ)) dτ ds = ξ, ∇V 1 (x) + O t→0 (t) Proposition 3.1. Let f ∈ C ∞ 0 (R 2n , R). We can find a function a(h) ∈ C ∞ 0 ([0, 3τ 0 ] × R 2n ) such that: a(0, x, ξ, h) = f (x, ξ) (3.3)
and:

sup t∈[0,3τ0] a(t, x, ξ, h)e i h ϕ(t,x,ξ) -e -it h (H h -E h ) f (x, ξ)e i h x,ξ L 2 (R 2n ) ---→ h→0 0 (3.4)
Proof. We define:

η(s, t, x, ξ) = exp t s (iE 1 -V 2 (x(τ, t, x, ξ) -∆ x ϕ(τ, x(τ, t, x, ξ), ξ)) dτ
Then:

a 0 (t, x, ξ) = f (y(t, x, ξ), ξ)η(0, t, x, ξ)
and:

a 1 (t, y, ξ) = i t 0 ∆ x a 0 (s, x(s, t, x, ξ), ξ)η(s, t, x, ξ) ds
where for 0 s t τ 0 we have set x(s, t, x, ξ) = x(s, y(t, x, ξ), ξ). Then we set a(h) = a 0 + ha 1 . Initial condition (3.3) is true and we can check that:

(∂ t + 2∂ x ϕ.∂ x + ∆ x ϕ + V 2 -iE 1 ) a 0 (t, x, ξ) = 0 and: (∂ t + 2∂ x ϕ.∂ x + ∆ x ϕ + V 2 -iE 1 ) a 1 (t, x, ξ) = i∆ x a 0 (t, x, ξ)
which, with (3.1), give (3.4). Note that the function a(h) is of compact support and the absorption coefficient V 2 does not change the phase ϕ. Only a depends on V 2 and the bigger V 2 is the faster a decays with time.

Remark. If (1.6) is replaced by:

E h = N j=0 h j E j + O(h N +1 ) for all N ∈ N (3.5)
then we can define:

a j (t, y, ξ) = i t 0 ∆ x a j-1 (s, x(s, t, x, ξ), ξ) + j-2 k=0 E j-k a k (x, x(x, t, x, ξ), ξ) η(s, t, x, ξ) ds
for all j 2 and a ∼ h j a j by Borel theorem (see [START_REF] Evans | Lectures on semiclassical analysis[END_REF]th. 4.16]). Then the rest is of size O(h ∞ ) instead of o(1) in (3.4) and hence in (3.16) and (3.26) below.

Critical points of the phase function

For t ∈ [0, 3τ 0 ], x, ξ ∈ R n and z ∈ Γ 1 we write:

ψ(t, x, z, ξ) = ϕ(t, x, ξ) -z, ξ
In this section we study the critical points of ψ with report to t, ξ and z with t ∈]0, 3τ 0 ], that is the solutions of the system:

         ∂ t ψ(t, x, z, ξ) = 0 ∂ z ψ(t, x, z, ξ) = 0 ∂ ξ ψ(t, x, z, ξ) = 0 t ∈]0, 3τ 0 ] ⇐⇒          ∂ t ϕ(t, x, ξ) = 0 ξ ∈ N z Γ 1 ∂ ξ ϕ(t, x, ξ) = z t ∈]0, 3τ 0 ] (3.6) Proposition 3.2. Let t ∈]0, 3τ 0 ], x, ξ ∈ R n and z ∈ Γ. If (t, x, ξ, z) is a solution of (3.6) then (z, ξ) ∈ N E Γ 1 and x = x(t, z, ξ).
Proof. Assume that (t, x, ξ, z) is such a solution. We already know that ξ ∈ N z Γ 1 . By proposition IV.14 in [START_REF] Robert | Autour de l'appoximation semi-classique[END_REF] we have:

(x, ∂ x ϕ(t, x, ξ)) = φ t (∂ ξ ϕ(t, x, ξ), ξ) = φ t (z, ξ) (3.7)
and in particular: x = x(t, z, ξ). Moreover, since ϕ is a solution of (3.1) we also have:

p(z, ξ) = p(x, ∂ x ϕ(t, x, ξ)) = |∂ x ϕ(t, x, ξ)| 2 + V 1 (x) = E 0 -∂ t ϕ(t, x, ξ) = E 0 which proves that |ξ| 2 = E 0 -V 1 (z).
We prove that for x close to Γ (but not on Γ 1 ), there is a solution (t, x, ξ, z) for (3.6). By proposition 3.2, this solution must be (t x , x, z x , ξ x ) (defined in proposition 2.4), so we already have uniqueness.

We consider the function Φ defined as follows: for y ∈ Γ1 (0, 3τ 0 ), ξ ∈ R n , ζ ∈ T zy Γ 1 of norm less than 1, δ ∈ [0, γ 1 ] (where γ 1 ∈]0, 1] is chosen small enough for exp z (δζ) being defined in Γ 2 for all z ∈ Γ 1 and ζ of norm less than 1) and θ ∈]0, 3τ 0 /γ 1 ] then:

Φ(θ, y, ζ, ξ, δ) = 1 δ ϕ(δθ, x(δt y , z y , ξ y ), ξ) -exp zy (δζ), ξ if δ = 0 ξ y -ζ, ξ -θ(ξ 2 + V 1 (z y ) -E 0 ) if δ = 0 (3.8)
For δ ∈]0, γ 1 ], t ∈ 0, 3τ0δ γ1 , x ∈ Γ1 (0, δτ 0 ), z such that d Γ (z x , z) δ and ξ ∈ R n we have:

ψ(t, x, ξ, z) = δΦ t δ , x t x δ , z x , ξ x , 1 δ (exp zx ) -1 (z), ξ, δ
Thus:

∂ t ψ(t, x, z, ξ) = 0 ⇐⇒ ∂ θ Φ t δ , x t x δ , z x , ξ x , 1 δ (exp zx ) -1 (z), ξ = 0 (3.9a) ∂ ξ ψ(t, x, z, ξ) = 0 ⇐⇒ ∂ ξ Φ t δ , x t x δ , z x , ξ x , 1 δ (exp zx ) -1 (z), ξ = 0 (3.9b) ∂ z ψ(t, x, z, ξ) = 0 ⇐⇒ ∂ ζ Φ t δ , x t x δ , z x , ξ x , 1 δ (exp zx ) -1 (z), ξ = 0 (3.9c) Proposition 3.3. Let K = T τ0 2 , 3τ 0 × N E Γ .
There exists δ 0 ∈]0, γ 1 ] such that for all y ∈ K and δ ∈ [0, δ 0 ] the system:

∂ θ,ξ,ζ Φ(θ, y, ζ, ξ, δ) = 0 θ ∈ 0, 3τ0 γ1 (3.10) has a solution (θ, ξ, ζ) ∈]0, τ 0 /γ 1 ] × R n × T zy Γ. Proof. For δ ∈]0, γ 1 ] we compute: Φ(θ, y, ζ, ξ, δ) = 1 δ ϕ(δθ, x(δt y , z y , ξ y ), ξ) -exp zy (δζ), ξ = 1 δ x(δt y , z y , ξ y ), ξ -δθ(ξ 2 + V 1 (x(δt y , z y , ξ y )) -E 0 ) + δ 2 θ 2 r(δθ, x(δt y , z y , ξ y ), ξ) -exp zy (δζ), ξ = 2t y ξ y -ζ, ξ -θ ξ 2 + V 1 (z y ) -E 0 + θ(V 1 (x(δt y , z y , ξ y )) -V 1 (z y )) + δθ 2 r(δθ, x(δt y , z y , ξ y ), ξ, h) - 1 δ exp zy (δζ) -z y -δζ, ξ = 2t y ξ y -ζ, ξ -θ ξ 2 + V 1 (z y ) -E 0 + δR(θ, y, ξ, ζ, δ)
where R is of class C 1 . This proves that Φ is of class C 1 . The point (θ, y, ζ, ξ, 0) is a solution of (3.10) if and only if:

         |ξ| = E 0 -V 1 (z y ) ξ ∈ N * zy Γ 2t y ξ y -ζ = 2θξ θ ∈ 0, τ0 γ1
Let y ∈ K. This system has a unique solution which we denote ( θy,0 , ζy,0 , ξy,0 ). It is given by: θy,0 = t y ; ζy,0 = 0; ξy,0 = ξ y (3.11)

For z ∈ Γ and ξ ∈ R n we denote by ξ z the orthogonal projection of ξ on T z Γ and ξ ⊥ z = ξξ z . Then we have:

Hess θ,ζ,ξ Φ(θ, y, ζ, ξ, δ) =      0 0 -2 t ξ z -2 t ξ ⊥ z 0 0 -I d 0 -2ξ z -I d -2θI d 0 -2ξ ⊥ z 0 0 -2θI n-d      + O δ→0 (δ)
and in particular:

det Hess θ,ζ,ξ Φ( θy,0 , y, ζy,0 , ξy,0

, 0) = 2 n-d+1 (-1) n-d t n-d-1 y |ξ z | 2
The derivative of the function:

(θ, y, ζ, ξ, δ) → ∂ θ,ζ,ξ Φ(θ, y, ζ, ξ, δ) ∈ R n+d+1
with report to θ, ζ and ξ at the point ( θy,0 , 0, ζy,0 , ξy,0 , 0) is:

Hess θ,ζ,ξ Φ(( θy,0 , 0, ζy,0 , ξy,0 , 0)) ∈ GL n+d+1 (R)
so we can apply the implicit function theorem around ( θy,0 , y, ζy,0 , ξy,0 , 0). We obtain that there exists δ y > 0, a neighborhood V y of y in R n and a function ϕ y which maps V y × [0, δ y ] into a neighborhood U y of ( θy,0 , ζy,0 , ξy,0 ) in ]0,

τ 0 /γ 1 ] × T zy Γ × R n such that: ∀(v, δ) ∈ V y × [0, δ y ], ∀(θ, ζ, ξ) ∈ U y , ∂ θ,ζ,ξ Φ(θ, v, ζ, ξ, δ) = 0 ⇐⇒ (θ, ζ, ξ) = ϕ y (v, δ)
K is covered by a finite number of such neighborhoods V y . We get the result if we take for δ 0 the minimum of the corresponding δ y .

Corollary 3.4. For all x ∈ Γ(0, 2δ 0 τ 0 ) there is a unique

(t, z, ξ) ∈]0, τ 0 ] × Γ × R n such that (t, x, z, ξ
) is a solution of the system (3.6). Moreover this solution is given by (t x , x, z x , ξ x ).

Proof. After proposition 3.2, there only remains to prove existence. Let x ∈ Γ(0, 2δ 0 τ 0 ). There is δ ∈]0, δ 0 ] such that y = x tx δ , z x , ξ x ∈ Γ(τ 0 , 2τ 0 ). Proposition 3.3 and equations (3.9) give the result.

Small times control

We can find a neighborhood G of N E Γ 0 such that for all t ∈ [0, τ 0 ] and (x, ξ) ∈ G we have 0 < d 1 |ξ| d 2 and x(t, x, ξ) ∈ Γ(2τ 0 ). We choose a function χ ∈ C ∞ 0 (R) supported in ] -1, τ 0 [ and equal to 1 in a neighborhood of 0. For f ∈ C ∞ 0 (R 2n ) supported in G, we set:

B 0 (h) = i h ∞ 0 χ(t)e -it h (H h -E h ) Op h (f )S h dt (3.12)
Egorov theorem (see proposition 2.1) yields:

½ R n \ Γ(2τ0) B 0 (h) L 2 (R n ) = O h→0 (h ∞ ) (3.13)
Proposition 3.5. If τ 0 > 0 is small enough, then for all ε > 0, there exists τ 1 ∈]0, τ 0 ] and

h 0 > 0 such that for all f ∈ C ∞ 0 (R 2n
) supported in G we have:

∀h ∈]0, h 0 ], ½ Γ(τ1) B 0 (h) L 2 (R n ) ε (3.14)
Proof. 1. If F h denotes the semiclassical Fourier transform we have:

F h S h (ξ) = h 1-n-d 2 R n Γ e -i h x,ξ A(z)S x -z h dσ(z) dx = h 1+n-d 2 Γ A(z)e -i h z,ξ R n e -i y,ξ S(y) dy dσ(z) = h 1+n-d 2 Ŝ(ξ) Γ A(z)e -i h z,ξ dσ(z)
where Ŝ is the usual Fourier transform of S, and then:

Op h (f )S h (x) = 1 (2πh) n R n e i h x,ξ f (x, ξ)F h S h (ξ) dξ = h 1+n-d 2 (2πh) n Γ R n e i h x-z,ξ A(z)f (x, ξ) Ŝ(ξ) dξ dσ(z) so: B 0 (h) = ih -1+n+d 2 (2π) n +∞ 0 Γ R n χ(t)A(z)e -i h z,ξ e -it h (H h -E h ) e i h •,ξ f (•, ξ) Ŝ(ξ) dξ dσ(z) dt
(3.15) Let a and ϕ given by WKB method (see section 3.1). We define:

J(x, h) = ∞ 0 Γ R n χ(t)e i h (ϕ(t,x,ξ)-z,ξ ) a(t, x, ξ, h)A(z) Ŝ(ξ) dξ dσ(z) dt
so that by (3.4):

B 0 (h) = ih -1+n+d 2 (2π) n J(h) 1 + o h→0 (1) in L 2 (R n ) (3.16) Let: κ(t, x, z, ξ, h) = χ(t)a(t, x, ξ, h)A(z) Ŝ(ξ)
κ is smooth and of compact support in t, x, z, ξ so all its derivatives are bounded. We recall that we wrote ψ(t, x, ξ, z) = ϕ(t, x, ξ)z, ξ .

2.

Let N ∈ N. To estimate J, we define, for all δ ∈]0, δ 0 ]:

J δ (x) = ½ Γ(δτ0,2δτ0) (x) R Γ R n e i h ψ(t,x,z,ξ) κ(t, x, z, ξ, h) dξ dσ(z) dt
Let:

J δ (x) = ½ Γ(δτ0,2δτ0) (x) R Γ ˛ξ z ˛>d1δ e i h ψ(t,x,z,ξ) κ(t, x, z, ξ, h) dξ dσ(z) dt
Since ∂ z ψ(t, x, z, ξ) = ξ z , N partial integrations in z show that:

J δ (x) c ½ Γ(δτ0,2δτ0) (x)
h δ N and hence:

J δ L 2 (R n ) c h N δ n-d 2 -N
(3.17)

3. By (3.2) we have:

∂ ξ ψ(t, x, z, ξ) = x -(z + 2tξ) + t 2 ∂ ξ r(t, x, ξ)
and hence:

[x -(z + 2tξ)] ∧ .∂ ξ ψ(t, x, z, ξ) = |x -(z + 2tξ)| + t 2 [x -(z + 2tξ)] ∧ .∂ ξ r(t, x, ξ)
where x stands for x |x| . For t δτ 0 min 1, γm 4d2 (γ m is defined in proposition 2.4) and

x ∈ Γ(δτ 0 , 2δτ 0 ) we have:

|x -(z + 2tξ)| |x -z| -2t |ξ| δτ 0 γ m -2td 2 δτ 0 γ m 2
and hence:

|x -(z + 2tξ)| + t 2 [x -(z + 2tξ)] ∧ .∂ ξ r δ τ 0 γ m 2 -M τ 2 0 (3.18)
where M = ∂ ξ r L ∞ ([0,τ0]×R 2n ) . Taking τ 0 smaller we may assume that the quantity in brackets is positive.

On the other hand if

t ∈ δ 2τ0(2d1+γM )+1 d1 , τ 0 , z xx is a point of Γ 1 for which |x -z xx | = d(x, Γ 1 ) and ξ z δd 1 , then: |x -(z + 2tξ)| |z + 2tξ -z xx | -|x -z xx | z + 2tξ ⊥ z -z xx -2δτ 0 d 1 -2δτ 0 γ M 2td 1 -2δτ 0 (2d 1 + γ M ) since for t small enough d(z + 2tξ ⊥ z , Γ) = 2tξ ⊥ z 2t |ξ| -2t ξ z . Thus: |x -(z + 2tξ)| + t 2 [x -(z + 2tξ)] ∧ .∂ ξ r t(d 1 -τ 0 M ) + td 1 -2δτ 0 (2d 1 + γ M ) δ + t d 1 2 (3.19)
if d 1 2τ 0 M , which may be assumed. In particular we have proved that there exists C, c 0 > 0 such that:

∀δ ∈]0, δ 0 ], ∀t ∈ 0, δ C ∪ [Cδ, τ 0 ], |x -(z + 2tξ)| + t 2 [x -(z + 2tξ)] ∧ .∂ ξ r c 0 δ
on the support of ½ Γ(δτ0,2δτ0) (x)κ(t, x, z, ξ, h). We get:

∂ α ξ [x -(z + 2tξ)] ∧ |x -(z + 2tξ)| + t 2 [x -(z + 2tξ)] ∧ .∂ ξ r(t, x, ξ) c α δ -|α|
on this support, since the derivatives of [x -(z + 2tξ)] ∧ with report to ξ are bounded for t ∈ [0, δ/C] ∪ [Cδ, τ 0 ] according to (3.18) and (3.19). We choose a function χ 1 ∈ C ∞ (R) equal to 1 in a neighborhood of -∞, 1 2C ∪ [2C, +∞] and zero on 1 C , C and χ 0 = 1χ 1 . Then we have J δ = J 1 δ + J 0 δ + J δ where, for j ∈ {0, 1}:

J j δ (x) = ½ Γ(δτ0,2δτ0) (x) ∞ 0 Γ ˛ξ z ˛ δd1 χ j t δ e i h ψ(t,x,z,ξ) κ(t, x, z, ξ, h) dξ dσ(z) dt
We consider the operator:

L : u → (t, x, z, ξ, h) → -ih [x -(z + 2tξ)] ∧ .∂ ξ u |x -(z + 2tξ)| + t 2 [x -(z + 2tξ)] ∧ .∂ ξ r
The function (t, x, z, ξ, h) → exp i h ψ(t, x, z, ξ) is invariant by L and the adjoint L * is given by:

L * : v → (t, x, z, ξ) → ih div ξ [x -(z + 2tξ)] ∧ .v |x -(z + 2tξ)| + t 2 [x -(z + 2tξ)] ∧ .∂ ξ r
N partial integrations with L prove:

J 1 δ (x) C N h δ N ½ Γ(δτ0,2δτ0) (x)
and hence:

J 1 δ C N h N δ n-d 2 -N (3.20)
4. We now turn to J 0 δ . We recall that for all z ∈ Γ 1 and ζ ∈ T z Γ 1 of norm less than γ 1 then exp z (ζ) is well-defined (on Γ 2 ) and d Γ1 (z, exp z (ζ)) = |ζ|. For τ 0 small enough, if x ∈ Γ(δτ 0 , 2δτ 0 ) and d Γ1 (z, z x ) γ 1 δ then |x -z| . As a result we can do partial integrations with L as before and see that modulo O((h/δ) N ), J 0 δ (x) is given by integration over z in a neighborhood of radius δ around z x :

J 0 δ (x) = ½ Γ(δτ0,2δτ0) (x) ∞ 0 BΓ(zx,γ1δ) ˛ξ z ˛ δd1 χ 0 t δ e i h ψ(t,x,z,ξ) κ(t, x, z, ξ, h) dξdσ(z) dt + O (h/δ) N
After the change of variables t = θδ and z = exp zx (δζ), ζ ∈ T zx Γ, we get for y ∈ R n :

J 0 δ (x(δt y , z y , ξ y )) = δ 1+d ½ Γ(τ0,2τ0) (y) R n ∞ 0 χ 0 (θ)κ(θ, y, ξ, ζ, h)e i h δΦ(θ,y,ξ,ζ,δ) dθ dξ dζ + O (h/δ) N
where integral in ζ is over the ball or radius γ 1 in T zy Γ and:

κ(θ, y, ξ, ζ, h, δ) = χ(y)κ(δθ, δy, ξ, exp zx (δζ), h)∂ ζ exp zx (δζ) with χ ∈ C ∞ 0 (R 2n ) supported in {τ 0 /2 t y 3τ 0 } and equal to 1 on {τ 0 t y 2τ 0 }. κ(h, δ) is of compact support in ]0, +∞[×(R n \ Γ) × (R n \ {0}) × T zy Γ. Φ is defined in (3.8).
For y such that τ 0 /2 t y 3τ 0 and δ ∈]0, δ 0 ], there is by proposition 3.3 a unique ( θy,δ , ξy,δ , ζy,δ ) such that ( θy,δ , y, ξy,δ , ζy,δ , δ) is a critical point of φ and θ > 0. Moreover:

∂ θ,ξ,z Φ(θ,
|∂ θ,ζ,ξ Φ(θ, y, z, ξ, δ)|
is uniformly bounded. So we can use theorems 7.7.5 and 7.7.6 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators, I[END_REF], which give:

J 0 δ (x) cδ 1+d h δ n+d+1 2 ½ Γ(δτ0,2δτ0) (x) + c h δ N ½ Γ(δτ0,2δτ0) (x)
and thus:

J 0 δ cδ 1 2 h n+d+1 2 + c h N δ n-d 2 -N (3.21)
5. For γ ∈]0, 1] we define :

Jγ (x) = ½ Γ(2γτ0) (x) R Γ R n e i h ψ(t,x,z,ξ) κ(t, x, z, ξ, h) dξ dσ(z) dt
J γ is defined as J δ with ½ Γ(δτ0,2δτ0) replaced by ½ Γ(2γτ0) . An estimate analog to (3.17) holds for J γ . We now note χ

+ = ½ [C,+∞[ χ 1 , χ -= 1 -χ + , and: J± γ (x) = ½ Γ(2γτ0) (x) R Γ ˛ξ z ˛ γd1 χ ± t γ e i h ψ(t,x,z,ξ) κ(t, x, z, ξ, h) dσ(z) dξ dt
As we did for J 1 δ we see that:

J+ γ C N h N γ n-d 2 -N (3.22)
To estimate J - γ , we remark that we are integrating a bounded function over a set of size O(γ) in t and over {(z, ξ), ξ z γd 1 } whose volume is of size O(γ d ), so:

J- γ (x) cγ 1+d ½ Γ(2γτ0) (x)
Taking the L 2 (R n ) norm in x gives:

J- γ L 2 (R n ) cγ 1+ n+d 2 (3.23)
6. Estimates (3.17), (3.20), (3.21), (3.22) and (3.23) allow to conclude: let τ 1 ∈]0, δ 0 τ 0 ] and µ ∈]0, 1[, we use a dyadic decomposition δ = 2 -m with h 1-µ < δ < τ 1 /τ 0 , that is ln 2 (τ 0 )ln 2 (τ 1 ) < m < -(1µ) ln 2 h. We write m -= ln 2 (τ 0 )ln 2 (τ 1 ) and m + = -(1µ) ln 2 h. Then:

½ Γ(τ1) J Jh 1-µ + m-<m<m+ J 2 -m with: Jh 1-µ J h 1-µ + J- h 1-µ + J+ h 1-µ c N h (1-µ)( n+d 2 +1) + h (1-µ) n-d 2 +µN c N h n+d+1 2 h 1 2 -µ( n+d 2 +1) + h µN -1 2 -d-µ n-d 2 and: m-<m<m+ J 2 -m m-<m<m+ J 1 2 -m + J 0 2 -m + J 2 -m c N   h N m m+ 2 N -n-d 2 m + h n+d+1 2 m-m 2 -m 2   c N h N -(1-µ)(N -n-d 2 ) + h n+d+1 2 √ τ 1 c N h n+d+1 2 h µN -1 2 -d-µ n-d 2 + √ τ 1
We now take µ > 0 small enough to have ν := 1 2µ n+d 2 + 1 > 0 and then N big enough to have µN -1 2dµ n-d 2 0. This gives:

½ Γ(τ1) J L 2 (R n ) c h n+d+1 2 ( √ τ 1 + h ν )
If τ 1 and h 0 are small enough we have c(

√ τ 1 + h ν ) ε 2 for all h ∈]0, h 0 ]
. By (3.16), if h 0 is small enough we finally reach the result: This means that B 0 is of the form B 0 (x) = e i h ψ(x) b 0 (x) + o(1). Note that if (3.5) holds we can have B 0 (x) = e i h ψ(x) b(x, h) + O(h ∞ ) where b(x, h) ∼ h j b j (x) for some functions b j , j 1. See [START_REF] Sogge | Fourier Integrals in Classical Analysis[END_REF] for more details about lagrangian distributions (in the microlocal setting).

½ Γ(τ1) B 0 (h) L 2 (R n ) ε For z ∈ Γ and x ∈ R n we set: ψx,z : (t, ζ, ξ) → ψ(t, x, exp z (ζ), ξ) (3.24) This is defined for t ∈]0, τ 0 ], ξ ∈ R n and ζ in a neighborhood U z of 0 in T z Γ. Now for x ∈ Γ(0, 2τ 0 ) we let ψ(x) = ψ(t x , x, z x , ξ x ) = ϕ(t x , x, ξ x ) -z x , ξ
Proof. Everything we need is already in the proof of proposition 3.5. By Egorov theorem there exists τ 2 ∈]0, τ 0 ] such that:

½ Γ(τ0)\U B 0 = ½ Γ(τ2,τ0) B 0 + O h→0 (h ∞ )
Let us come back to the proof of (3.18) with δ = τ 2 . We see

that if χ ∈ C ∞ 0 (R * + ) is such that χ(t) = χ(t) for t γmτ2τ0 4d2 then in L 2 ( Γ(τ 2 , τ 0 )): B 0 (x) = ih -1+n+d 2 (2π) n ∞ 0 Γ R n χ(t)e i h ψ(t,x,z,ξ) a(t, x, ξ, h)A(z) Ŝ(ξ) dξ dσ(z) dt 1 + o h→0 (1)
Moreover as we explained for J 0 δ the only relevant part of integration on z is around z x , so:

B 0 (x, h) = ih -1+n+d 2 (2π) n ∞ 0 Uz x R n χ(t)e i h ψx,zx (t,ζ,ξ) a(t, x, ξ, h)A(z) Ŝ(ξ) Jac(exp zx )(ζ) dξ dζ dt × 1 + o h→0 (1)
(3.26)

Then, as we did to study J 0 δ , we use the results of section 3.2 and stationnary phase method to get the result (in particular the only stationnary point for ψx,zx is (t x , 0, ξ x ).

Proposition 3.7. Let x ∈ Γ(τ 0 ). We have:

det Hess ψx,zx (t x , 0, ξ x ) = 2 n-d+1 t n-d-1 x |ξ x | 2 + O tx→0 (t n-d x ) (3.27)
where the size of the rest is uniform in x.

Proof. (ii). By (3.2) we have:

det Hess ψx,z (t, 0, ξ) =

∂ 2 t ϕ(t, x, ξ) 0 -2 t ξ z -2 t ξ ⊥ z 0 A -I d 0 -2ξ z -I d -2tI d 0 -2ξ ⊥ z 0 0 -2tI n-d 1 + O t→0 (t) = (-1) n-d 2 n-d+1 t n-d-1 ξ ⊥ z 2 + O t→0 (t n-d )
where for 1 i, j d:

A ij = -∂ 2 ζiζj exp z (ζ)
, ξ only appears in the rest, and (ξ

x ) ⊥ zx = ξ x since (z x , ξ x ) ∈ N E Γ.
4 Partial result for finite times

Intermediate times contribution

We begin with a proposition which proves that for w ∈ R 2n and q ∈ C ∞ 0 (R 2n ) supported close to w, then in the integral:

u T h = i h T 0 U E h (t)S h dt
only times around t w,k for 1 k K T w (and on a neighborhood of 0 if w ∈ N E Γ) give a relevant contribution.

Proposition 4.1. Let w ∈ R 2n , T > 0 and χ ∈ C ∞ 0 (R) a function which is zero near t w,k for k ∈ 1, K w (and 0 if w ∈ N E Γ). Then there exists a neighborhood V w,T of w in R 2n and a neighborhood G w,T ⊂ G of N E Γ (G was defined in section 3.3) such that for all q ∈ C ∞ 0 (R 2n ) supported in V w,T and f ∈ C ∞ 0 (R 2n ) supported in G w,T , we have in L 2 (R n ):

Op w h (q) i h T 0 χ(t)U E h (t)Op h (f )S h dt = O h→0 (h ∞ )
Proof. There exists a neighborhood G w,T ⊂ G of N E Γ in R 2n and ρ > 0 such that for all w ∈ G and t ∈ supp χ we have:

φ t ( w) -w 2ρ
Otherwise for all m ∈ N * we can find t m ∈ supp χ and w m ∈ R 2n with d(w m ), N E Γ)

1 m such that |φ tm (w m ) -w| 1 m . We can extract a subsequence so that t m k → t ∈ supp χ and w m k → w ∞ ∈ N E Γ. Then we have φ t (w ∞ ) = w, which is impossible since t / ∈ {t w,1 , . . . , t w,Kw } (∪ {0} if w ∈ N E Γ).
Let V w,T be the ball B(w, ρ) and q ∈ C ∞ 0 (R 2n ) supported in V w,T . By Egorov theorem, we have for all t ∈ [0, T ]:

Op w h (q)U E h (t)Op h (f ) = O h→0 (h ∞ )
where the remainder is uniform in t ∈ [0, T ]. An integration over t gives the result.

Remark. Note that neither the neighborhoods G w,T and V w,T nor the size of the remainder can be uniform in T . That is the main reason why we cannot deal directly with u h and have to begin with a study of u T h . Let w ∈ Λ and τ w = min(t w,1 , τ 0 ). We consider χ w ∈ C ∞ 0 (R) supported in ]0, 2τ w [ and equal to 1 in a neighborhood of τ w , and set:

B w (h) = i h ∞ t=0 χ w (t)U E h (t)Op h (f )S h dt
Moreover, for k ∈ 1, K w we denote:

B w,k (h) = i h ∞ t=0 χ w (t -t w,k + τ w )U E h (t)Op h (f )S h dt (4.1)
As in proposition 3.6 (and we do not even have to worry about very small times since χ w vanishes around 0) we see that B w (h) is a lagrangian distribution of submanifold det Hess ψzx (t x , 0, ξ x )

Λ 0 = (x, ∂ x ψ), x ∈ Γ(0, 2τ 0 ) = φ tx (z x , ξ x ), x ∈ Γ(0, 2τ 0 ) = φ t (z, ξ), t ∈]0, 2τ 0 ], (z, ξ) ∈ N E Γ and of principal symbol b w (x) = i(2π)
1 2 A(z x )a 0 (t x , x, ξ x ) Ŝ(ξ x )χ w (t x )
Proposition 4.2. For all w ∈ Λ and k ∈ 1, K w , B w,k (h) is a lagrangian distribution of lagrangian submanifold Λ w,k := φ t w,k Λ 0 . We denote by b w,k and ψ w,k the principal symbol and the phase of this distribution.

Remark. Again, with (1.6) this means that B w,k (h) = e Proof. We have:

B w,k (h) = i h ∞ t=0 χ w (t -t w,k + τ w )U E h (t)Op h (f )S h dt = i h ∞ t=-t w,k +τw χ w (t)U E h (t + t w,k -τ w )Op h (f )S h dt = U E h (t w,k -τ w )B w (h) It is known that e -i(t w,k -τw ) h (H h 1 -E h )
turns a lagrangian distribution of submanifold Λ 0 into a lagrangian distribution of submanifold φ t w,k -τw Λ 0 (see [START_REF] Sogge | Fourier Integrals in Classical Analysis[END_REF][START_REF] Evans | Lectures on semiclassical analysis[END_REF]). We can similarly see that this also applies to U E h (t w,kτ w ). Computations are actually close to what is done for WKB method, where we see that the imaginary part does not affect the phase factor but only the amplitude. Here again V 2 only appears in the symbol b w,k of the lagrangian distribution.

We give another property of B w,k we are going to use in section 5.3: Proposition 4.3. Let w ∈ Λ. For all k ∈ 1, K w we have:

(H h -E h )B w,k (h) = 0 microlocally near w Proof. We have: (H h -E h )B w,k (h) = (H h -E h ) i h +∞ 0 χ w (t -t k + τ w )U E h (t)Op h (f )S h dt = - +∞ 0 χ w (t -t k + τ w )∂ t U E h (t)Op h (f )S h dt = +∞ 0 χ ′ w (t -t k + τ w )U E h (t)Op h (f )S h dt As ∂ t χ w (t -t k + τ w
) is zero near t = t j for j ∈ 1, K w (and t = 0), the result is given by Egorov theorem as in the proof of theorem 4.1.

Convergence toward a partial semiclassical measure

We are now ready to give the semiclassical measure for u T h .

Theorem 4.4. Let T 0. There exists a nonnegative Radon measure µ T on R 2n such that for all q ∈ C ∞ 0 (R 2n ) we have:

Op w h (q)u T h , u T h ---→ h→0 q dµ T (4.2)
Proof. 1. Localization around a point w ∈ R 2n . We are going to show that for any w ∈ R 2n and T 0, there is a neighborhood V w,T ⊂ R 2n such that for all q ∈ C ∞ 0 (R 2n ) supported in V w,T we have:

Op w h (q)u T h , u T h ---→ h→0 q dµ w,T (4.3)
where µ w,T is a Radon measure on V w,T . If w 1 , w 2 ∈ R 2n are such that V w1,T ∩ V w2,T = ∅, then the two measures µ w1,T and µ w2,T coincide on V w1,T ∩ V w2,T (we only have to consider the two versions of (4.3) for q ∈ C ∞ 0 (R 2n ) supported in V w1,T ∩V w2,T ). Thus we can define the measure µ T on R 2n as the only measure which coincides with µ w,T on V w,T for all w ∈ R 2n . Then for all q ∈ C ∞ 0 (R 2n ) a partition of unity and a finite numbers of applications of (4.3) give (4.2).

So let w ∈ R 2n . If w / ∈ (N E Γ ∪ Λ) we can choose a neighborhood V w of w which does not intersect N E Γ ∪ Λ. Proposition 4.1 with χ = 1 on [0, T ] shows:

Op w h (q)u T h , u T h ---→ h→0 0 for all q ∈ C ∞ 0 (R 2n ) supported in V w .
Hence we set µ w,T = 0 on V w,T . This proves that if µ T exists then we must have:

µ T = 0 outside N E Γ ∪ Λ (4.4)
We now assume that w ∈ N E Γ ∪ Λ.

2. Localization around relevant times. Let δ w = 1 if w ∈ N E Γ and δ w = 0 otherwise. We recall that χ and χ w have been chosen in sections 3.3 and 4.1. By corollary 2.5, if w ∈ N E Γ then t w,1 3τ 0 so for all w ∈ N E Γ ∪ Λ supports of functions δ w χ and χ w (•t w,k + τ w ) for 1 k K T w are pairwise disjoint, so we can consider a function χ ∈ C ∞ 0 (R, [0, 1]) such that:

∀t ∈ [0, T ], δ w χ(t) + K T w k=1 χ w (t -t k + τ w ) + χ(t) = 1
By proposition 4.1 there exists a function f w,T ∈ C ∞ 0 (R 2n ) equal to 1 around N E Γ and a neighborhood V w,T of w in R 2n such that for q supported in V w,T we have in L 2 (R n ):

Op w h (q)v T h = Op w h (q)ũ T h + O h→0 (h ∞ )
where:

v T h = i h T 0 U E h (t)Op h (f w,T )S h dt and ũT h = δ w B T w,0 + K T w k=1 B T w,k
with B T w,0 defined in (3.12) and the B T w,k given by (4.1) with f replaced by f w,T . Let g be given by proposition 2.10. We have:

Op w h (q)ũ T h , ũT h (4.5) = Op w h (q) v T h + (1 -g)(H h 1 )(u T h -v T h ) + O(h) , v T h + (1 -g)(H h 1 )(u T h -v T h ) + O(h) = Op w h (q)v T h , v T h + Op w h (q)(u T h -v T h ), (1 -g)(H h 1 )v T h + Op w h (q)(1 -g)(H h 1 )v T h , u T h -v T h + O h→0 ( √ h) = Op h (q)ũ T h , ũT h + O h→0 ( √ h)
3. Definition of the measure µ w,T . For k ∈ 1, K T w and Ω a borelian set in V w,T we define:

µ w,T,k (Ω) = R n ½ Ω (x, ∂ψ w,k (x)) |b w,k (x)| 2 dx ; µ w,T,0 (Ω) = δ w R n ½ Ω (x, ∂ψ(x)) |b 0 (x)| 2 dx
and finally:

µ w,T (Ω) = K T w k=0 µ w,T,k
which defines a measure on V w,T . Note that all these measures are nonnegative. V w,T and µ w,T are now fixed, and we have to prove that for any ε > 0 and q ∈ C ∞ 0 (R 2n ) supported in V w,T , there is h 0 > 0 such that for all h ∈]0, h 0 ]:

Op w h (q)u T h , u T h -q dµ w,T ε (4.6)
Let ε > 0 and q supported in V w,T . (4.5) yields:

Op w h (q)u T h , u T h -Op w h (q)ũ T h , ũT h ε 9 (4.7) with h ∈]0, h 0 ] for some h 0 > 0. 4. Self-intersections of Λ. Let j, k ∈ 1, K w with j = k (j, k ∈ 0, K w if w ∈ N E Γ).
Λ w,j ∩ Λ w,k is a closed set of measure 0 in the smooth manifold Λ w,j , hence by regularity of the measure on Λ w,j , for all m ∈ N we can find an open subset U m j of Λ w,j of measure less than 1 m such that Λ w,j ∩ Λ w,k ⊂ U m j . We can find an open sett V m j in R 2n of measure less than 1 m such that U m j = V m j ∩ Λ w,j , and by Uryshon lemma there exists a function γ m j ∈ C ∞ 0 (R 2n , [0, 1]) equal to 1 outside V m j and zero in a neighborhood of Λ w,j ∩ Λ w,k . We construct similarly a function γ m k interverting j and k, we set γ m j,k = γ m j γ m k and finally:

γ m = 1 j<k K T w γ m j,k   or 0 j<k K T w γ m j,k if w ∈ N E Γ   (4.8)
so that the sets Λ w,k ∩ V w,T for 1 k K T w (or 0 k K T w ) do not intersect on the support of γ m and:

mes Λ supp(1 -γ m ) ∩ ∪ K T w j=0 Λ w,k 1 m (4.9)
For all k ∈ 0, K T w , the support of the function x → (1γ m )(x, ∂ψ k (x)) is of measure less than C m in R n where C only depends on Γ. Op w h (γ m )B T w,k is a lagrangian distribution microlocally supported in Λ w,k ∩ supp(γ m ) with symbols uniformly bounded in h and k, so there is c 0 such that for all h ∈]0, h 0 ]:

ũT h -Op w h (γ m )ũ T h c m (4.10)
Moreover, for j = k ∈ 0, K w the distributions Op w h (qγ m )B T w,j and Op w h (qγ m )B T w,k have disjoint microsupports, so we have:

Op w h (qγ m )B T w,j , Op w h (qγ m )B T w,k = O h→0 (h ∞ ) (4.11)
Taking m ∈ N large enough and using (4.7), (4.10) et (4.11), we obtain for all h ∈]0, h 0 ]:

Op w h (q)u h , u h -δ w Op w h (qγ m )B T w,0 , B T w,0 - K T w k=1 Op w h (qγ m )B T w,k , B T w,k ε 3 (4.12)

Convergence for intermediate times.

Let k ∈ 1, K T w . We know that B T w,k is a lagrangian distribution of phase ψ w,k and of principal symbol b w,k , hence we have:

Op w h (q)Op w h (γ m )B T w,k , B T k,w = R n q(x, ∂ψ w,k (x))γ m (x, ∂ψ w,k (x)) |b w,k (x)| 2 dx + o h→0 (1)
If m is large enough and h 0 small enough, we have for all h ∈]0, h 0 ]:

Op w h (q)Op w h (γ m )B T w,k , B T w,k - R n q(x, ∂ψ w,k (x)) |b w,k (x)| 2 dx ε 3K T w (4.13)
6. Convergence for small times.

It only remains to consider the term δ w Op w h (q)Op w h (γ m )B T w,0 , B T w,0 . We assume that w belongs to N E Γ.

Let

τ 1 ∈]0, τ 0 ] and v ∈ C ∞ 0 (R 2n , [0, 1]) such that supp v ⊂ Γ(τ 1
) and v is equal to 1 in a neighborhood of supp A. By proposition 3.5, if τ 1 > 0 is small enough we have:

vB T w,0 L 2 (R n ) ε 6 (4.14)
On the other hand, since (1v) vanishes around supp A, we can write (1v(x))B T w,0 as a lagrangian distribution (see proposition 3.6):

Op w h (q)Op w h (1 -v)Op w h (γ m )B T w,0 , B T w,0 = R n (qγ m )(x, ∂ψ(x))(1 -v(x)) |b 0 (x)| 2 dx + o h→0 (1)
Thus, if τ 1 and h 0 are small enough, then for all h ∈]0, h 0 ]:

Op w h (q)Op w h (1 -v)Op w h (γ m )B T w,0 , B T w,0 -q dµ w,0 ε 6 (4.15)
7. Conclusion. According to (4.12), (4.13) and (4.15), we can conclude that (4.6) holds.

5 Convergence toward a semiclassical measure

Large times control

For R > 0, d > 0 and σ ∈] -1, 1[ we note:

Γ ± (R, d, σ) = (x, ξ) ∈ R 2n : |x| R, |ξ| d and x, ξ ≷ σ |x| |ξ| Γ ± (d, σ) = (x, ξ) ∈ R 2n : |ξ| d and x, ξ ≷ σ |x| |ξ|
As mentionned in the introduction, the following proposition states that the outgoing solution u h is microlocally zero in the incoming region. The proof of this proposition is postponed to section 6.

Proposition 5.1. Let d > 0, σ ∈]0, 1[ and E h such that Im E h > 0 or E h is positive and satisfies (1.4). Then there exists R > 0 such that if ω -, ω ∈ S 0 are supported in Γ -(R, d, -σ) (respectively outside Γ -(R 1 , d 1 , -σ 1 ) for some R 1 < R, d 1 < d and σ 1 < σ) then: Op h (ω -)(H h -(E 0 + i0)) -1 Op h (ω) = O h→0 (h ∞ )
We now use this proposition to show that for T large enough, Op w h (q)u T h , u T h is a good approximation of Op w h (q)u h , u h .

Proposition 5.2. Let q ∈ C ∞ 0 (R 2n ) be supported in p -1 (I) and ε > 0. Then there exists T 0 0 such that for all T T 0 we can find h T > 0 which satisfies:

∀h ∈]0, h T ], Op w h (q)u h , u h -Op w h (q)u T h , u T h ε Proof. 1. Let R b 0 such that Γ ⊂ B R n (R b ), supp q ⊂ B x (R b ) = (x, ξ) ∈ R 2n : |x| < R b
and any trajectory of energy in J which leaves B x (R b ) never comes back (and goes to infinity

). Let χ ∈ C ∞ 0 (R n ) supported in B(2R b ) and equal to 1 on B(R b ). Let Q ∈ C ∞ 0 (R 2n
) supported in p -1 (J) and equal to 1 in a neighborhood of p -1 (I) ∩ B x (2R b ) and of supp q. Let T 0 and ω -equal to 1 in the incoming region Γ -(R b , -1/2) and zero outside Γ -(R b /2, -1/4). We have:

Op w h (Q)u h = i h T t=0 Op w h (Q)U E h (t)S h dt + Op w h (Q)U E h (T )u h = Op w h (Q)u T h + Op w h (Q)U E h (T )Op w h (Q)u h + Op w h (Q)U E h (T )Op w h (1 -Q)χ(x)u h + Op w h (Q)U E h (T )Op w h (1 -Q)(1 -χ(x))Op h (ω -)u h + Op w h (Q)U E h (T )Op w h (1 -Q)(1 -χ(x))Op h (1 -ω -)u h (5.1)
For T large enough the last three terms are O h→0 ( √ h) respectively by the localization close to the E 0 -energy hypersurface (proposition 2.10, which implies that Op w h (1 -Q)χ(x)u h is small), estimates on the incoming region (Op w h (ω -)u h is small by proposition 5.1, changing quantization is harmless here) and Egorov theorem (Op

w h (Q)U E h (T )Op h (1 -ω -)(1 -χ(x)
) is small). Hence we have:

1 -Op w h (Q)U E h (T )Op w h ( Q) Op w h (Q)u h = Op w h (Q)u T h + O h→0 ( √ h) (5.2) where Q ∈ C ∞ 0 (R 2n
) is supported in p -1 (J) and equal to 1 on the support of Q. Furthermore:

Op w h (Q)u T h 2 = Op w h (Q) 2 u T h , u T h ---→ h→0 Q 2 dµ T < +∞
Hence for any (large enough) fixed T , the right-hand side of (5.2) is uniformly bounded in h. Moreover, by proposition 2.3, there exists T 0 such that for all T T 0 there is h T > 0 which satisfies:

∀h ∈]0, h T ], Op w h (Q)U E h (T )Op w h ( Q) 1 2
As a consequence, the operator (1

-Op w h (Q)U E h (T )Op w h ( Q)
) is invertible and its inverse is bounded uniformly in T T 0 and h ∈]0, h T ]. This proves that the quantity:

Op w h (Q)u h = 1 -Op w h (Q)U E h (T )Op w h ( Q) -1 Op w h (Q)u T h + O h→0 ( √ h)
is bounded uniformly in h ∈]0, h T ] for fixed T T 0 and hence is bounded uniformly for h small enough since the left hand side does not depend on T .

2. As for (5.1) we see that:

Op w h (q)u h = Op w h (q)u T h + Op w h (q)U E h (T )Op w h (Q)u h + Op w h (q)U E h (T )Op w h (1 -Q)χ(x)u h + Op w h (q)U E h (T )Op w h (1 -Q)(1 -χ(x))Op h (ω -)u h + Op w h (q)U E h (T )Op w h (1 -Q)(1 -χ(x))Op h (1 -ω -)u h (5.3)
And as for (5.1) the last three terms are O h→0 ( √ h) by localization close to E 0 -energy hypersurface, estimates in the incoming region and Egorov theorem. Moreover the second term is:

Op w h (q)U E h (T )Op w h (Q)u h = Op w h (q)U E h (T )Op w h ( Q) (Op w h (Q)u h ) + O h→0 ( √ h)
But Op w h (Q)u h is bounded uniformly in h and the operator Op w h (q)U h (T )Op h ( Q) is of norm less than any δ > 0 for T big enough and h small enough (depending of the chosen T ). Hence we have proved:

∀δ > 0, ∃T 0 0, ∀T T 0 , ∃h T > 0, ∀h ∈]0, h T ], Op w h (q)(u h -u T h ) δ (5.4)
and in particular:

∃C 0, ∀T T 0 , ∀h ∈]0, h T ], Op w h (q)u T h C (5.5)
We consider q ∈ C ∞ 0 (R 2n ) supported in p -1 (I), equal to 1 on supp q and such that Q = 1 on a neighborhood of supp q. We can assume that (5.4)-(5.5) hold for q and q. Let δ ∈ 0, ε 4C and then T and h T given by (5.4). For all h ∈]0, h T ] we have:

Op w h (q)u h , u h -Op w h (q)u T h , u T h = Op w h (q)u h , Op w h (q)u h -Op w h (q)u T h , Op w h (q)u T h + O h→0 (h ∞ ) Op w h (q)(u h -u T h ), Op w h (q)u T h + Op w h (q)u h , Op w h (q)(u h -u T h ) + O h→0 (h ∞ ) δ Op w h (q)u T h + Op w h (q)u T h + O h→0 (h ∞ ) ε 2 + O h→0 ( √ h)
and this last quantity is less than ε if we choose h small enough.

Convergence of the partial semiclassical measure

Proposition 5.3. There exists a Radon measure µ on R 2n such that for all q ∈ C ∞ 0 (R 2n ):

q dµ T -----→ T →+∞
q dµ and we have:

Op w h (q)u h , u h ---→ h→0 q dµ
Proof. 1. We can assume that for any w ∈ R 2n , the family of neighborhoods V w,T , T 0, decreases when T increases. Let T 1 T 2 ∈ R + . For w ∈ R 2n and q ∈ C ∞ 0 (R 2n ) supported in V w,T2 ⊂ V w,T1 we have:

q dµ T1 = q dµ w,T1 = K T 1 w k=0 q dµ w,T1,k K T 2 w k=0 q dµ w,T2,k = q dµ T2 Since any q ∈ C ∞ 0 (R 2n
) can be written as a finite sum q i where q i is supported in V wi,T2 for some w i , the same applies for all q ∈ C ∞ 0 (R 2n ). This proves that q dµ T grows with T , and hence has a limit in R + ∪ {+∞} when T goes to +∞.

2. If supp q ∩ p -1 ({E 0 }) = ∅, then q dµ T = 0 -----→ T →+∞ 0
This is consistent with corollary 2.11.

3. Now let q ∈ C ∞ 0 (R 2n
) supported in p -1 (I), q and C as in the proof of proposition 5.2 (see (5.5)). We have:

q dµ T = lim h→0 Op w h (q)u T h , u T h = lim h→0 Op w h (q)u T h , Op w h (q)u T h C 2
As a result, q dµ T as a finite limit when T goes to +∞. This limit defines a nonnegative (each µ T is a nonnegative measure) linear form on

C ∞ 0 (R 2n ). Let K be compact in R 2n and Q ∈ C ∞ 0 (R 2n
) equal to 1 on K. Then for all q ∈ C ∞ 0 (R 2n ) supported in K we have:

q dµ lim T →∞ |q| dµ T q ∞ lim T →∞ Q dµ T c q ∞
and hence this limit is a continuous function of q (is the space of compactly supported continuous functions). Thus the application q → lim T →+∞ q dµ T can be extended to a nonnegative continuous linear form on the space of compactly supported continuous functions so, by Riesz theorem, there is a nonnegative Radon measure µ on R 2n such that: lim

T →∞ q dµ T = q dµ 4. For q ∈ C ∞ 0 (R 2n , [0, 1]
) there exists T 0 such that: 0 q dµq dµ T ε 3

According to proposition 5.2, if T is chosen large enough there is h T > 0 such that:

∀h ∈]0, h T ], Op w h (q)u h , u h -Op w h (q)u T h , u T h ε 3
and by theorem 4.4, there is h 0 ∈]0, h T ] such that for all h ∈]0, h 0 ] we have:

Op w h (q)u T h , u T h -q dµ T ε 3
Hence we get:

∀h ∈]0, h 0 ], Op w h (q)u h , u hq dµ ε which proves the proposition.

Characterization of the semiclassical measure

We now finish the proof of theorem 1.1:

Proof. 1. Statement (i) is already proved and similarly, (ii) is a consequence of the estimate in the incoming region (see proposition 5.1).

2. Let q ∈ C ∞ 0 (R 2n ) such that supp q ∩ (N E Γ ∪ Λ) = ∅.
We have:

q (H p + 2 Im E 1 + 2V 2 )dµ = (-H p + 2 Im E 1 + 2V 2 )q dµ = lim T →∞ (-H p + 2 Im E 1 + 2V 2 )q dµ T = 0
according to (4.4) since the support of (-

H p + 2 Im E 1 + 2V 2 )q does not meet N E Γ ∪ Λ. 3. Let w ∈ Λ, T 0 and q ∈ C ∞ 0 (R 2n ) such that supp q ⊂ V w,T . Since 2ih Im E 1 = E h -E h + o h→0 (h) and H p (q) = {p, q} is the principal symbol of the operator i h [H h 1 , Op w h (q)],
we have:

Op w h (H p (q)) = i h [H h 1 , Op w h (q)] + hOp w h (r 1 ) + O h→0 (h 2 )
for some symbol r 1 ∈ C ∞ 0 (R 2n ). But Op w h (r 1 )B T w,k , B T w,k as a limit as h goes to 0 (which is r 1 dµ w,T,k , see step 5 in the proof of theorem 4.4) and B T w,k = O(h -1 2 ), so:

(-H p + 2 Im E 1 + 2V 2 )q dµ w,T,k (5.6) = lim h→0 Op w h (-H p (q) + 2 Im E 1 q + 2V 2 q)B T w,k , B T w,k = lim h→0 - i h [H h 1 , Op w h (q)] + 2 Im E 1 Op w h (q) + 2V 2 Op w h (q)B T w,k , B T w,k = -lim h→0 i h ((H h -E h ) * Op w h (q) -Op w h (q)(H h -E h ))B T w,k , B T w,k = -lim h→0 i h Op w h (q)B T w,k , (H h -E h )B T w,k -(H h -E h )B T w,k , Op w h (q)B T w,k = 0
according to proposition 4.3.

4. Let q ∈ C ∞ 0 (R 2n
) and ε > 0. There exists T 0 such that:

q dµ T q dµ - ε 2
We can find a finite number of w i ∈ R 2n such that supp q ⊂ ∪V wi,T and either

w i ∈ N E Γ∪Λ or V wi,T ∩ (N E Γ∪ Λ) = ∅.
With a partition of unity, we can write q = q i with supp q i ⊂ V wi,T and show the result for each q i . So without loss of generality we can assume that supp q ⊂ V w,T for some w ∈ N E Γ ∪ Λ. According to (5.6) we have:

(-H p + 2 Im E 1 + 2V 2 )q dµ T = K T w j=0 (-H p + 2 Im E 1 + 2V 2 )q dµ w,T,k = (-H p + 2 Im E 1 + 2V 2 )q dµ w,T,0
This is zero unless w ∈ N E Γ, which we now assume. Let g ∈ C ∞ 0 (R) supported in ]-∞, 1] with g = 1 near 0. For m ∈ N and (x, ξ) ∈ Γ(τ 0 )× R n we set g m (x, ξ) = g(mt x ). In particular the function (1g m )q vanishes near N E Γ, so:

(-H p + 2V 2 + 2 Im E 1 )(1 -g m )q dµ = 0
Then since g m is supported in Γ(0, τ 0 ) for all m ∈ N, we can use (2.4) to have:

R 2n (-H p + 2 Im E 1 + 2V 2 )q dµ w,T,0 = R 2n (-H p + 2 Im E 1 + 2V 2 )qg m dµ w,T,0 = Γ(0,τ0) (-H p + 2 Im E 1 + 2V 2 )(qg m )(x, ∂ψ(x)) |b 0 (x)| 2 dx = 2 n-d τ0 0 NE Γ t n-d-1 |ξ| 1 + O t→0 (t) |b 0 (x(t, z, ξ))| 2 ×(-H p + 2 Im E 1 + 2V 2 )(qg m )(x(t, z, ξ), ∂ψ(x(t, z, ξ))) dσ(z, ξ) dt
According to (3.7) we have (x, ∂ψ(x)) = φ tx (z x , ξ x ). On the other hand, by (3.25) and (3.27) we have:

2 n-d t n-d-1 |ξ| |b 0 (x(t, z, ξ))| 2 ---→ t→0 π(2π) d-n A(z) 2 |ξ| -1 Ŝ(ξ) 2 =: c(z, ξ) (5.7) so: R 2n (-H p + 2 Im E 1 + 2V 2 )q dµ w,T,0 = - τ0 0 NE Γ (∂ t -2 Im E 1 -2V 2 )(q(φ t (z, ξ))g(mt))c(z, ξ) 1 + O t→0 (t) dσ(z, ξ) dt = - τ0 0 NE Γ g(tm)(∂ t -2 Im E 1 -2V 2 )(q(φ t (z, ξ)))c(z, ξ) 1 + O t→0 (t) dσ(z, ξ) dt - τ0 0 NE Γ mg ′ (tm)q(φ t (z, ξ))c(z, ξ) 1 + O t→0 (t) dσ(z, ξ) dt
and hence:

(-H p + 2 Im E 1 + 2V 2 )q dµ w,T,0 - NE Γ q(z, ξ)c(z, ξ) dσ(z, ξ) O 1 m + τ0 0 NE Γ mg ′ (tm) q(z, ξ) -q(φ t (z, ξ)) c(z, ξ) dσ(z, ξ) dt O 1 m + τ0 0 NE Γ m |g ′ (tm)| sup 0 t 1 m q(z, ξ) -q(φ t (z, ξ)) c(z, ξ) dσ(z, ξ) dt = O 1 m
It only remains to choose m so large that the rest is less than ε 2 . As said in the introduction, µ is actually characterized by the three properties of theorem 1.1 and is given by (1.10):

Proposition 5.4. Let ν be a Radon measure on R 2n which satisfies the three properties of theorem 1.1. Then for all q ∈ C ∞ 0 (R 2n ) we have:

R 2n q dν = +∞ 0 NEΓ c(z, ξ)q(φ t (z, ξ))e -2t Im E1-2 R t 0 V2(x(s,z,ξ)) ds dσ(z, ξ) dt (5.8)
where the function c is defined in (5.7).

Proof. Let I 1 be an open interval such that I ⊂ I 1 ⊂ I 1 ⊂ J. Let q ∈ C ∞ 0 (R 2n ). According to property (i), if supp q ⊂ p -1 (R \ I) then q dν = 0 which is consistent with (5.8), since both sides are zero. So we can assume that supp q ⊂ p -1 (I 1 ).

Using property (iii) we see that:

d dt R 2n (q • φ t )e -2t Im E1-2 R t 0 V2•φ t-s ds dν = R 2n (H p -2 Im E 1 -2V 2 ) (q • φ t )e -2t Im E1-2 R t 0 V2•φ t-s ds dν = - NE Γ c(z, ξ) (q • φ t )e -2t Im E1-2 R t 0 V2•φ t-s ds (z, ξ) dσ(z, ξ)
and hence, for all τ 0:

R 2n q dµ = R 2n (q • φ τ )e -2τ Im E1-2 R τ 0 V2•φ τ -s ds dν + τ 0 NE Γ c(z, ξ) (q • φ t )e -2t Im E1-2 R t 0 V2•φ t-s ds (z, ξ) dσ(z, ξ) dt
So we only have to prove that:

R 2n (q • φ τ )e -2τ Im E1-2 R τ 0 V2•φ τ -s ds dν -----→ τ →+∞ 0 For R 0 we set: K R = p -1 (I 1 ) ∩ B x (R).
According to property (ii), we can find R 0 such that ν vanishes on Γ -(R, -1 2 ) and:

t 0 supp(q • φ t ) ⊂ Γ -R, - 1 2 ∪ K R Let χ ∈ C ∞ 0 (R 2n
) supported in p -1 (J) and equal to 1 on K R . For τ 0, since ν vanishes on Γ -R, -1 2 :

R 2n

(q • φ τ )e -2t Im E1-2 R τ 0 V2•φ τ -s ds dν = R 2n χ(q • φ τ )e -2t Im E1-2 R τ 0 V2•φ τ -s ds dν
As ν is a Radon measure, there is a constant C 0 such that for all q ∈ C ∞ 0 (R 2n ) with supp q ⊂ supp χ we have:

R 2n q dν C q L ∞ (R 2n )
so we only need to prove that:

sup w∈R 2n χ(w)(q • φ τ )(w)e -2τ Im E1-2 R τ 0 (V2•φ τ -s )(w) ds -----→ τ →+∞ 0 This is clear if Im E 1 > 0.
Otherwise, this can be done with lemma 2.2 as in the proof of proposition 2.3.

6 Estimate of the outgoing solution in the incoming region

The theorem we want to prove in this section is the following: Theorem 6.1. Let N ∈ N and E h = E 0 + O(h) be an energy such that for all h ∈]0, h 0 ], Im E h > 0 or E h satisfies (1.4). Let d > 0 and σ ∈]0, 1[. Then there exits ν ∈ N and R > 0 such that if the symbols ω + , ω ∈ S 0 have supports in Γ + (R, d, σ) (respectively outside Γ + (R 1 , d 1 , σ 1 ) with R 1 < R, d 1 < d and σ 1 < σ) then for all α > 1 2 we have:

x -α Op h (ω)(H h -(E h + i0)) -1 Op h (ω + ) x -ν = O h→0 (h N ) (6.1) Similarly, if supp ω -⊂ Γ -(R, d, -σ) and supp ω ∩ Γ -(R 1 , d 1 , -σ 1 ) = ∅ then: x -α Op h (ω)(H * h -(E h -i0)) -1 Op h (ω -) x -ν = O h→0 (h N ) (6.2)
Remark. This is the analog of lemma 2.3 in [START_REF] Robert | Asymptotic behavior of scattering amplitudes in semiclassical and low energy limits[END_REF] in the dissipative case. Note that here ν is different from α and may be large.

Remark. Taking the adjoint in (6.2) gives:

x -ν Op h (ω -)(H h -(E h + i0)) -1 Op h (ω) x -α = O h→0 (h N )
which proves proposition 5.1. This theorem proves that the solution u h = (H h -(E+i0)) -1 S h is microlocally zero in the incoming region.

To prove this theorem we follow [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF]. In particular we use the following result taken from [START_REF] Isozaki | Modified wave operators with time-independant modifiers[END_REF]:

Proposition 6.2. Let d 0 ∈]0, d 1 [ and σ 0 ∈]0, σ 1 [. There exists R 0 > 0 and φ ± ∈ C ∞ (R 2n ) satisfying: ∀(x, ξ) ∈ Γ ± (R 0 , d 0 , ±σ 0 ), |∇ x φ ± (x, ξ)| 2 + V 1 (x) = |ξ| 2 (6.3)
and:

∀(x, ξ) ∈ R 2n , ∀α, β ∈ N n , ∂ α x ∂ β ξ (φ ± (x, ξ) -x, ξ ) C α,β x 1-ρ-|α| (6.4)
for some ρ > 0.

Without loss of generality we may assume that this is the same constant ρ as in (1.3).

Remark. As mentioned in [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF] (see (2.4)), we can assume that the constants C α,β in (6.4) are as small as we wish if we take R large enough. Indeed, if we take a function χ ∈ C ∞ (R n ) such that χ(x) = 0 if |x| 1 2 and χ(x) = 1 if |x| 1, and, for R > R 0 :

φ R,± : (x, ξ) → (φ ± (x, ξ) -x, ξ )χ x R + x, ξ (6.5)
Then:

∀(x, ξ) ∈ Γ ± (R, d 0 , σ 0 ), |∇ x φ R,± (x, ξ)| 2 + V 1 (x) = |ξ| 2 (6.6)
and for any ρ 1 , ρ 2 > 0 such that ρ = ρ 1 + ρ 2 :

∀(x, ξ) ∈ R 2n , ∂ α x ∂ β ξ (φ R,± (x, ξ) -x, ξ ) C α,β R -ρ1 x 1-ρ2-|α| (6.7)
where C α,β does not depend on R.

We are going to use the Fourier integral operators I h (a, φ) defined as follows:

I h (a, φ)u(x) = 1 (2πh) n R n R n e i h (φ(x,ξ)-y,ξ ) a(x, ξ)u(y) dy dξ
As in [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF], the idea of the proof is to find two symbols a and e such that: U h (t)I h (a, φ) ≈ I h (a, φ)U h 0 (t) and Op h (ω + ) ≈ I h (a, φ)I h (e, φ) * when h goes to 0. For a short range absorption coefficient V 2 , we can actually do as in [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF], but in the long range case, we have to consider a time dependant symbol a(t, h).

In this situation we have:

U h (t)I h (a(t, h), φ ± ) -I h (a(t, h), φ ± )U h 0 (t) (6.8) = t 0 U h (t) - i h H h I h (a(s, h), φ ± ) + I h (∂ t a(s, h), φ ± ) + i h I h (a(s, h), φ ± )H h 0 U h 0 (t -s) ds
Proposition 6.3. Let a(t, h) ∈ S b be a time-dependant symbol, φ = φ + or φ -given by proposition 6.2 and h ∈]0, 1]. Then we have:

- i h H h I h (a(t, h), φ) + I h (∂ t a(t, h), φ) + i h I h (a(t, h), φ)H h 0 = I h (p(t, h), φ)
where:

p(t, h) (6.9) = - i h (|∂ x φ| 2 + V 1 -ξ 2 )a(t, h) + ∂ t a(t, h) -2∂ x a(t, h).∂ x φ -a(t, h)∆ x φ -a(t, h)V 2 +ih∆ x a(t, h)
Remark. If moreover a(t, h) is of the form:

a(t, h) = N j=0 h j a j (t)
with a j ∈ S b for all j ∈ 0, N , then p(t, h) takes the form:

p(t, h) = - i h (|∂ x φ| 2 + V 1 -ξ 2 )a(t, h) + ∂ t a 0 (t, h) -2∂ x a 0 (t).∂ x φ -a 0 (t)∆ x φ -a 0 (t)V 2 + N j=1 h j ∂ t a j (t, h) -2∂ x a j (t).∂ x φ -a j (t)∆ x φ -a j (t)V 2 + i∆ x a j-1 (t) + ih N +1 ∆ x a N (t)
This gives the transport equations the symbols a j have to satisfy if we want

I h (p(t, h), φ) = O h→0 (h N +1 ).
Remark. Similarly we have:

- i h H * h I h (a(t, h), φ) + I h (∂ t a(t, h), φ) + i h I h (a(t, h), φ)H h 0 = I h (p * (t, h), φ)
where:

p * (t, h) = - i h (|∂ x φ| 2 + V 1 -ξ 2 )a(t, h) + ∂ t a(t, h) -2∂ x a(t, h).∂ x φ -a(t, h)∆ x φ + a(t, h)V 2 +ih∆ x a(t, h)
Lemma 6.4. Let φ be a function which satisfies (6.4). Then for all (x, ξ) ∈ R 2n , the Cauchy problem:

∂r ∂t (t, x, ξ) = ∂ x φ(r(t, x, ξ), ξ) r(0, x, ξ) = x
has a unique solution defined on R. Furthermore, for γ ∈]0, σ 1 [, if R is large enough, we have the following properties:

(i) For (x, ξ) ∈ Γ ± (d 1 , ±σ 1 ) and ±t 0 we have:

|r(t, x, ξ)| |x| + (σ 1 -γ)d 1 |t| (6.10) (ii) For (x, ξ) ∈ Γ ± (d 1 , ±σ 1 )
, ±t 0 and |α| + |β| 1, there is a constant c α,β such that:

∂ α x ∂ β ξ r(t, x, ξ) c α,β max(|t| , x ) x -|α| (6.11)
Proof. Let (x, ξ) ∈ R 2n . We have:

r(t, x, ξ) = x + tξ + t 0 (∂ x φ(r(s, x, ξ), ξ) -ξ) ds (6.12)
where r(•, x, ξ) is defined, that is everywhere since (∂ x φ(r(t, x, ξ), ξ)-ξ) is bounded according to (6.4).

(i) By (6.7), if R is large enough we can assume that:

∀(x, ξ) ∈ R 2n , |∂ x φ(x, ξ) -ξ| γd 1 and hence: |r(t, x, ξ) -x -tξ| |t| γd 1 If (x, ξ) ∈ Γ ± (d 1 , ±σ 1
) and ±t 0, then:

|x + tξ| 1 |x| x, x + tξ |x| + σ 1 |t| |ξ| |x| + |t| σ 1 d 1 so: |r(t, x, ξ)| |x + tξ| -γ |t| d 1 |x| + (σ 1 -γ)d 1 |t|
which proves (6.10).

(ii) We prove (6.11) by induction on |α| + |β|, beginning by the case |α| = 1, β = 0. Let ±t 0 and (x, ξ) ∈ Γ + (d 1 , σ 1 ). We have:

∂ t ∂ x r(t, x, ξ) = ∂ 2
x φ(r(t, x, ξ), ξ).∂ x r(t, x, ξ)

According to Gronwall lemma, (6.4) and (6.10), we obtain the estimate:

∂ x r(t, x, ξ) exp t 0 ∂ 2 x φ(r(s, x, ξ), ξ) ds exp t 0 c r(s, x, ξ) -1-ρ ds exp t 0 c s -1-ρ ds c c max(|t| , x ) x -1
Similarly, if α = 0 and |β| = 1 we have:

∂ t ∂ ξ r(t, x, ξ) = ∂ 2 x φ(r(t, x, ξ), ξ).∂ ξ r(t, x, ξ) + ∂ x ∂ ξ φ(r(t, x, ξ), ξ)
and then:

∂ t ∂ ξ r(t, x, ξ) t s=0 ∂ x ∂ ξ φ(r(s, x, ξ), ξ) exp t τ =s ∂ 2 x φ(r(τ, x, ξ), ξ) dτ ds c |t|
We now assume that we have proved (6.11) for 1 |α| + |β| k ∈ N * and we consider α and β such that |α| + |β| = k + 1. For j ∈ 1, n we have:

∂ t ∂ α x ∂ β ξ r j (t, x, ξ) = ∂ α x ∂ β ξ (∂ xj φ(r(t, x, ξ), ξ)) = n l=1 ∂ 2 x l ,xj φ(r(t, x, ξ), ξ) ∂ α x ∂ β ξ r l (t, x, ξ) + B j (t, x, ξ)
where B j is a sum of terms of the form:

(∂ γ x ∂ δ ξ ∂ xj φ)(r(t, x, ξ), ξ) |γ| s=1 (∂ αs x ∂ βs ξ r ls )(t, x, ξ)
with |γ| + |δ| 2 and for all s :

l s ∈ 1, n , |α s | + |β s | k, α s = α and δ + β s = β. Then B j is smaller than: r(t, x, ξ) -|γ|-ρ |γ| s=1 max(|t| , x ) x -|αs| c x -α
and finally (6.11) holds since:

∂ t ∂ α x ∂ β ξ r(t, x, ξ) t s=0 B(t, x, ξ) exp t τ =s ∂ 2 x φ(r(τ, x, ξ), ξ) dτ ds c |t| x -α
Let r ± be the functions defined in this proposition for φ = φ ± and:

F ± (t, x, ξ) = ∆ x φ ± (r ± (t, x, ξ), ξ) ± V 2 (r ± (t, x, ξ))
In particular we have:

F ± (0, x, ξ) = ∆ x φ ± (x, ξ) ± V 2 (x) and F ± (t, r ± (s, x, ξ), ξ) = F ± (t + s, x, ξ)
Proposition 6.5. The functions a j,± (t, h), j ∈ N defined by:

a 0,± (t, x, ξ) = exp - t s=0
(F ± (2s, x, ξ)) ds and for j 1:

a j,± (t, x, ξ) = i t τ =0
∆ x a j-1,± (τ, r ± (2τ, x, ξ), ξ)a 0 (τ, x, ξ) dτ are solutions of the transport equations:

∂ t a 0,± (t, h) -2∂ x a 0,± (t).∂ x φ ± -a 0,± (t)∆ x φ ± ∓ a 0,± (t)V 2 = 0 (6.13)
and for j 1:

∂ t a j,± (t, h) -2∂ x a j,± (t).∂ x φ ± -a j,± (t)∆ x φ ± ∓ a j,± (t)V 2 + i∆ x a j-1 (t) = 0 (6.14)
and satisfy estimates:

for ± t 0, (x, ξ) ∈ Γ ± (d 1 , ±σ 1 ), ∂ α x ∂ β ξ a j,± (t, x, ξ) c α,β |t| j+(|α|+|β|)(1-ρ) x -|α| (6.15)
Proof. We prove (6.15). For α, β ∈ N n , the derivative ∂ α x ∂ β ξ a 0,± (t, x, ξ, h) is a sum of terms of the form:

J k=1 ∂ µ k x ∂ ν k ξ t 0 F ± (2s, x, ξ) ds a 0,± (t, x, ξ) with µ k = α, ν k = β and for all k ∈ 1, J : |µ k | + |ν k | 1
(and in particular J |α| + |β|). We first remark that according to (6.4) and (6.10) together with nonnegativeness of V 2 the symbol a 0 is bounded uniformly in ±t 0. Hence we have to prove:

t 0 ∂ µ k x ∂ ν k ξ F ± (2s, x, ξ) ds c α,β |t| (|µ k |+|ν k |)(1-ρ) x -|µ k |
Let ±t 0, (x, ξ) ∈ Γ ± (d 1 , ±σ 1 ) and µ, ν ∈ N n . Then:

∂ µ x ∂ ν ξ t 0 F ± (2s, x, ξ) ds
is a sum of terms of the form:

t 0 ∂ δ x ∂ λ ξ (∆ x φ ± + V 2 )(r ± (2s, x, ξ), ξ) |δ| k=1 ∂ µ k x ∂ ν k ξ r ± (2s, x, ξ) ds (6.16) with |δ| j=1 µ k = µ, |δ| j=1 ν k + λ = ν and for all k ∈ 1, |δ| : |µ k | + |ν k | 1. By (1.
3), (6.4) and (6.11) we have:

∂ µ x ∂ ν ξ t 0 F ± (s, x, ξ) ds c |t| 1-ρ x -|µ|
this proves (6.15) for j = 0. We now prove the general case by induction. For α, β ∈ N n the derivative ∂ α x ∂ β ξ a j+1,± (t, x, ξ) is a sum of terms of the form:

i t τ =0 ∂ µ x ∂ ν ξ (∆ x a j,± (t, r ± (2(τ -t), x, ξ), ξ)) × ∂ α-µ x ∂ β-ν ξ a 0,± (τ, x, ξ) dτ
We already know that for τ ∈ [0, t]:

∂ α-µ x ∂ β-ν ξ a 0,± (τ, x, ξ) c |t| (1-ρ)(|α-µ|+|β-ν|) x -|α-µ|
So it remains to show:

∂ µ x ∂ ν ξ (∆ x a j,± (τ, r ± (2τ, x, ξ), ξ)) c |t| j+(1-ρ)(|µ|+|ν|) x -|µ| But ∂ µ x ∂ ν ξ (∆ x a j,± (τ, r ± (2τ, x, ξ), ξ)
) is a sum of terms of the form:

(∂ δ x ∂ λ ξ ∆ x a j,± )(t, r ± (2τ, x, ξ), ξ) |δ| k=1 (∂ µ k x ∂ ν k ξ r ± )(2τ, x ,
(∂ δ x ∂ λ ξ ∆ x a j,± )(τ, r ± (2τ, x, ξ), ξ) |δ| j=1 (∂ µj x ∂ νj ξ r ± )(2τ, x, ξ) c |τ | j+(1-ρ)(|δ|+|λ|+2) r ± (2τ, x, ξ) -|δ|-2 max(|2τ | , x ) δ x - P |δ| j=1 µj c |t| j+(1-ρ)(|δ|+|λ|) x -|µ|
which concludes the proof after integration over τ ∈ [0, t].

Remark. This is for this part of the proof that we need a time-dependant symbol. Indeed, following exactly the proof of [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF] would have led to consider:

a 0 (x, ξ) = exp ∞ 0 F (t, x, ξ) dt
which may have no sense for a long range imaginary part of the potential V 2 . For a short range potential we do not have such a problem and the sign of V 2 we have used here does not matter.

Let σ 2 and σ 3 such that σ

1 < σ 2 < σ 3 < σ, R 2 and R 3 such that R 1 < R 2 < R 3 < R and d 2 , d 3 such that d 1 < d 2 < d 3 < d. We consider functions ρ 1 ∈ C ∞ (R) such that ρ 1 (s) = 0 if s σ 2 and 1 if s σ 3 , ρ 2 ∈ C ∞ (R) such that ρ 2 (s) = 0 and s d 2 and 1 if s d 3 and ρ 3 ∈ C ∞ (R) such that ρ 3 (s) = 0 if s R 2 and ρ 3 (s) = 1 if s R 3 . Then we set: b ± (t, x, ξ, h) = ψ ± (x, ξ) N j=0 h j a j,± (t, x, ξ) where: ψ ± (x, ξ) = ρ 1 ± x, ξ |x| |ξ| ρ 2 (|ξ|)ρ 3 (|x|)
We also set:

p ± (t, h) = i h (|∂ x φ ± | 2 + V 1 -ξ 2 )b ± (t, h) + (∂ t b ± (t, h) + 2∂ x b ± (t, h).∂ x φ ± + b ± (t, h)∆ x φ ± ± b ± (t, h)V 2 ) -ih N +1 ∆ x b ± (t, h)
as given by proposition 6.3.

Proposition 6.6. The symbols b ± and p ± satisfy: Proof. (6.17) comes from (6.15). According to (6.13) and (6.14) we have:

(i) supp b ± ⊂ Γ ± (R 2 ,
p ± (t, x, ξ, h) = 2∂ x ψ ± (x, ξ).∂ x φ ± (x, ξ) N j=0 a j,± (t, x, ξ)ih N +1 ∆ x b ± (t, x, ξ, h) so (6.18) is a consequence of (6.15) and (6.17). Finally, it remains to remark that for ±t 0 and (x, ξ) ∈ Γ ± (R 3 , d 3 , ±σ 3 ) we have p ± (t, h) = -ih N +1 ∆ x b ± (t, h) to get (6.19) from (6.17 where r ± ∈ S -N uniformly in h.

Proof. This is lemma 4.5 in [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF]. Note that b ± (0, h) is just ψ ± .

Proposition 6.8. For all δ ∈ R, there is ν ∈ N such that for all l ∈ R and ±t 0 we have: and:

x l I h (p ± (t, h), φ)U h 0 (t)I h (e ± , φ) * x -1-ν-l ch N +1 t -δ (6.21)

Proof. For u ∈ S(R n ) we have:

I h (b ± (t, h), φ ± )U h 0 (t)I h (e ± (h), φ ± ) * u(x) = 1 (2πh) n y ξ e i h ζ±(t,x,y,ξ) b ± (t, x, ξ, h)e ± (y, ξ, h)u(y) dξ dy with ζ ± (t, x, y, ξ) = φ ± (x, ξ)φ ± (y, ξ)tξ 2 . If R is large enough then for (y, ξ) ∈ supp e ± we have: for some c 0 > 0.

We consider the operator L such that for u ∈ S(R 2n ):

Lu = ih (∂ ξ φ ± (y, ξ) + 2tξ).∂ ξ u |∂ ξ φ ± (y, ξ) + 2tξ| 2
Then we have:

L * v = ih div ξ . ∂ ξ φ ± (y, ξ) + 2tξ |∂ ξ φ ± (y, ξ) + 2tξ| 2 v
In particular L e -i h (φ±(y,ξ)+tξ 2 = e -i h (φ±(y,ξ)+tξ 2 ) so for ν ∈ N: for some J ν ∈ N and for all j ∈ 1, J ν we have:

I h (b ± (t, h), φ ± )U h 0 (t)I h (e ± (h), φ ± ) * u(x) = 1 (2πh) n
∂ α x ∂ β ξ b j ν,± (t, x, ξ, h) c α,β |t| N -(|α|+|β|)(1-ρ)-ρν x ν-|α|
and e 0 ∈ S 0 : Indeed, this is true for ν = 0 by (6.17) and if this is true for some ν ∈ N then for j ∈ 1, J ν we have to compute: + (∂ ξ φ ± (y, ξ) + 2tξ).∂ ξ b j ν,± (t, x, ξ, h) e j ν,± (y, ξ) + b j ν,± (t, x, ξ, h) (∂ ξ φ ± (y, ξ) + 2tξ).∂ ξ e j ν,± (y, ξ)

ih div ξ ∂ ξ φ ± (
and check each term using (6.22). Note that the factor x ν in the estimate is due to the third term. We only gain a power t -ρν at each iteration because of the fourth term and the fact that we have a bad estimate in t for the derivatives of b ν,± . Nonetheless, for all ν ∈ N we get: I h (b ± (t, h), φ ± )U h 0 (t)I h (e ± (h), φ ± ) * = Jν j=1 I h (b j ν,± (t, h), φ ± )U h 0 (t)I h (e j ν,± (h), φ ± ) * (6.23)

For any ν ∈ N, the two operators U h 0 (t) and I h (e ν,± (h), φ ± ) * are uniformly bounded in t and h from L 2,1+l+ν into itself. The norm of I h (b ν,± (t, h), φ ± ) from L 2,1+l+ν to L 2,l is estimated by a finite number of derivatives of b j ν,± , say M (see [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF]). Then we have to choose ν such that N + M (1ρ)νρ -δ to obtain (6.20).

To prove (6.21) we introduce a function χ ∈ C ∞ (R) such that χ(s) = 0 if s σ 3 and χ(s) = 1 if s σ 4 ∈]σ 3 , σ 5 [. Then we write p 2,± (t, x, ξ, h) = p ± (t, x, ξ, h)χ ± x,ξ |x||ξ| and p 1,± (t, x, ξ, h) = p ± (t, x, ξ, h)p 2,± (t, x, ξ, h). We have:

∂ α x ∂ β ξ p 2,± (t, x, ξ, h) c α,β h N +1 |t| N +(2+|α|+|β|)(1-ρ) x -2-|α|
The same argument as above proves (6.21) with p ± replaced by p 2,± .

For p 1,± , we remark that for (x, ξ) ∈ supp p 1,± ⊂ R 2n \ Γ ± (R 4 , d 4 , ±σ 4 ) and (y, ξ) ∈ supp e ± ⊂ Γ ± (R 5 , d 5 , ±σ 5 ) we have: Corollary 6.9. For all δ ∈ R, there is ν ∈ N such that for all l ∈ R and ±t 0 we have: Proof. The proof is the same as for (6.20) but instead of an estimate of I h (b j ν,± , φ) we need an estimate of Op h (ω)I h (b j ν,± , φ) . According to lemma 4.4 in [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF] if we take R large enough, then the supports of ω(x, ∂ x φ(x, ξ)) and b j ν,± are disjoint, so this norm is only the norm of the rest given in proposition A.3 of [START_REF] Wang | Time-decay of scattering solutions and resolvant estimates for semiclassical Schrödinger operators[END_REF]. This rest is of order O(h N +1 ) and the time dependance is given as for I h (b j ν,± , φ) by a finite number of derivatives of b j ν,± so we conclude the same way. Now we can prove the main theorem of this section:

Proof of theorem 6.1. Let ν ∈ N given by proposition 6.8 for δ = 2. We prove the "+" case, and we omit the + subscript for φ, b, p and r. Let t 0. According to (6.8) and proposition 6.3, we have: U h (t)I h (b(0, h), φ) = I h (b(t, h), φ)U h 0 (t) - According to the uniform estimate for the resolvent (see [Roy]) the first term is O(h N ). We use (6.24) and (6.21) for the second and third terms, which, after taking the limit z → E h if E h ∈ R, proves (6.1).

Remark. To prove (6.2) we apply the same argument with:

(H * h -z) -1 = - i h 0 -∞ e -it h (H * h -z) dt

γ1δ2

  and |x -(z + 2tξ)| γ1δ 4

  y, z, ξ, δ) = Hess θ,z,ξ Φ( θy,δ , y, ζy,δ , ξy,δ , δ)((θ, z, ξ) -( θy,δ , ζy,δ , ξy,δ )) + O (θ,ζ,ξ)→( θy,δ , ζy,δ , ξy,δ ) (|θ -θy,δ |, |ζ -ζy,δ |, |ξ -ξy,δ |) and hence: (θ, ζ, ξ) -( θy,δ , ζy,δ , ξy,δ ) = Hess θ,ζ,ξ Φ( θy,δ , y, ζy,δ , ξy,δ , δ) -1 (∂ θ,ζ,ξ Φ(θ, ζ, ξ)) + O (θ,ζ,ξ)→( θy,δ , ζy,δ , ξy,δ ) (|θ -θy,δ |, |ζ -ζy,δ |, |ξ -ξy,δ |) y and δ stay in a compact set and zero is never an eigenvalue of Hess θ,ζ,ξ Φ( θy,δ , y, ζy,δ , ζy,δ , δ), so the norm of Hess θ,ζ,ξ (θ, ζ, ξ) -1 is bounded. As a consequence the quantity: (θ, ζ, ξ) -( θy,δ , ζ y,δ , ξy,δ )

  ψx,zx (tx,0,ξx) det Hess ψzx (t x , 0, ξ x )1 2 A(z x )a 0 (t x , x, ξ x ) Ŝ(ξ x )χ(t x ) (3.25)Proposition 3.6. Let U be a neighborhood of Γ 0 in R n . Then on Γ(τ 0 ) \ U the function B 0 is a lagrangian distribution of phase ψ and principal symbol b 0 .

  ψx,zx (tx,0,ξx)

ih

  ψ w,k b w,k +o(1), but with assumption (3.5) we can write B w,k (h) = e i h ψ w,k bw,k (h) + O(h ∞ ) where bw,k (h) is a classical symbol of principal symbol b w,k .

  ξ) with µ = |δ| k=1 µ k and ν = λ + |δ| k=1 ν k , and:

  ). Proposition 6.7. Let R 5 ∈]R 3 , R[, d 5 ∈]d 3 , d[ and σ 5 ∈]σ 3 , σ[.There exists a symbol e ± (h) of the form e ± (h) = N j=0 h j f j,± with f j,± ∈ S -j and supp f j,± ⊂ Γ ± (R 5 , d 5 , ±σ 5 ) such that:I h (b ± (0, h), φ)I h (e ν,± (h), φ) * = ω ± (x, hD) + h N +1 Op h (r ± (h))

  x l I h (b ± (t, h), φ)U h 0 (t)I h (e ± , φ) * x

  |∂ ξ φ ± (y, ξ) + 2tξ| ∂ ξ φ ± (y, ξ) + 2tξ, ŷ |y|c |y| 1-ρ + 2σ 5 |t| |ξ| c 0 (|y| + |t|) (6.22)

  h (φ±(y,ξ)+tξ 2 ) (L * ) ν e i h φ±(x,ξ) b ± (t, x, ξ, h)e ± (y, ξ, h) u(y) dξ dyWe can check by induction on ν ∈ N that: (L * ) ν e i h φ±(x,ξ) b ± (t, x, ξ, h)e ± (y, ξ, h) = Jν j=1 e i h φ±(x,ξ) b j ν,± (t, x, ξ, h)e j ν,± (y, ξ, h)

  y, ξ) + 2tξ |∂ ξ φ ± (y, ξ) + 2tξ| 2 e i h φ±(x,ξ) b j ν,± (t, x, ξ, h)e j ν,± (y, ξ, h) = ih |∂ ξ φ ± (y, ξ) + 2tξ| -2 × e i h φ±(x,ξ) × (∆ ξ φ ± (y, ξ) + 2tn)b j ν,± (t, x, ξ, h)e j ν,± (y, ξ) +2 (Hess ξ φ ± (y, ξ) + 2tI n ).(∂ ξ φ ± (y, ξ) + 2tξ) 2 |∂ ξ φ ± (y, ξ) + 2tξ| 2 b j ν,± (t,x, ξ, h)e j ν,± (y, ξ)+ i h (∂ ξ φ ± (y, ξ) + 2tξ)∂ ξ φ ± (x, ξ)b j ν,± (t,x, ξ, h)e j ν,± (y, ξ)

  |∂ ξ ζ ± (x, y, ξ, t)| c 0 (|x| + |y| + |t|)for some c 0 > 0. Indeed we have:|∂ ξ ζ(x, y, ξ, t)| = |∂ x φ ± (x, ξ) -∂ ξ φ ± (y, ξ) -2tξ| |x -(y + 2tξ)| -cR -ρ But (y + 2tξ, ξ) ∈ Γ ± (R 4 , d 4 , ±σ 4 ) so if |x| γ |y + 2tξ|: |x -(y + 2tξ)| (1γ -1 ) |x| 1γ -1 2 (|x| + |y + 2tξ|) c 0 (|x| + |y| + |t|) and if |x| |y + 2tξ|: |x -(y + 2tξ)| x -(y + 2tξ), ∓ ξ = ±1 |ξ| ( y + 2tξ, ξx, ξ ) (σ 5 |y + 2tξ|σ 4 |x|) (σ 5σ 4 ) |y + 2tξ| c 0 (|x| + |y + 2tξ|) c 0 (|x| + |y| + |t|) Then we can do partial integrations with the operator L = ∂ ξ ζ.∂ ξ |∂ ξ ζ| 2 , each iteration giving a new power of h and t -ρ .

xl

  Op h (ω)I h (b ± (t, h), φ)U h 0 (t)I h (e ± , φ) * x

  t

  d 2 , ±σ 2 ) and for ±t 0, (x, ξ) ∈ Γ ± (R 2 , d 2 , ±σ 2 ) and α, β ∈ N n we have:(ii) supp p ± ⊂ Γ ± (R 2 , d 2 , ±σ 2 ) and for ±t 0, (x, ξ) ∈ Γ ± (R 2 , d 2 , ±σ 2 ) and α, β ∈ N n we have: ∂ α x ∂ β ξ p ± (t, x, ξ, h) c α,β |t| N +(2+|α|+|β|)(1-ρ) x -|α| (6.18)If furthermore (x, ξ) ∈ Γ ± (R 3 , d 3 , ±σ 3 ) then we have:

	∂ α x ∂ β ξ b(t, x, ξ, h)	c α,β |t| N +(|α|+|β|)(1-ρ) x	-|α|	(6.17)
	∂ α x ∂ β ξ p ± (t, x, ξ, h)	c α,β h N +1 |t| N +(2+|α|+|β|)(1-ρ) x	-2-|α|	(6.19)

0

  U h (ts)I h (p(s, h), φ)U h 0 (s) ds and then, by proposition 6.7:U h (t)Op h (ω + ) = h N +1 U h (t)Op h (r(h)) + I h (b(t, h), φ)U h 0 (t)I h (e(h), φ) * U h (ts)I h (p(s, h), φ)U h 0 (s)I h (e(h), φ) * ds For α > 1 2 and Im z > 0, using (H hz) -1 = i Op h (ω)(H hz) -1 Op h (ω + ) x Op h (ω)(H hz) -1 Op h (r(h))x Op h (ω)I h (b(t, h), φ)U h 0 (t)I h (e(h), φ)

							t
							-	0
	gives:						h	∞ 0 e	it h z U h (t) dt (see theorem 1.10 in [EN00])
	x	-α -ν
		= h N +1 x	-α -ν
		+	i h	x -α	t=0 ∞	e	it h z

* x -ν dt x -α Op h (ω) ∞ s=0 e is h z (H hz) -1 I h (p(s, h), φ)U h 0 (s)I h (e(h), φ) * x -ν ds

given with our notations.

This work is partially supported by the French National Research Project NONAa, No. ANR-08-BLAN-0228-01, entitled Spectral and microlocal analysis of non selfadjoint operators.


François Gerbe

Bénédicte Brulin

Leila Makrini

Catherine Legraverend

Philippe Jay

Philippe Jay Dcamkl-

DCAMKL-1 expression identifies Tuft cells