Moez Khenissi

Julien Royer

LOCAL ENERGY DECAY AND SMOOTHING EFFECT FOR THE DAMPED SCHRÖDINGER EQUATION

We prove the local energy decay and the smoothing effect for the damped Schrödinger equation on R d . The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform resolvent estimates obtained by the dissipative Mourre method. All the results depend on the strength of the dissipation which we consider.

1

Introduction

Let d 3. Our purpose in this paper is to study on R d the local energy decay and the Kato smoothing effect for the damped Schrödinger equation -i∂ t u + P uia(x) D α a(x)u = 0, u(0) = u 0 .

(1.1)

The operator P is a Laplacian in divergence form (or a Laplace-Beltrami operator) associated to a metric which is a long-range perturbation of the usual flat metric (see (1.2) below). For the dissipative part we have denoted by

• the function (1 + |•| 2 )
1 2 and by D the square root of the free Laplacian, so that D α stands for (1 -∆) α 2 . The parameter α belongs to [0, 2[. The non-negative valued function a will be assumed to be of short range (see (1.7)), so that in terms of spacial decay, we have an absorption index a(x) 2 which decays at least like x -2-2ρ for some ρ > 0.

It is known that the free Schrödinger equation ((1.1) with P = -∆ and a = 0) preserves the L 2 -norm but satisfies the local energy decay: if u 0 is supported in the ball B(R) = {|x| R} of R d for some R > 0 we have

e it∆ u 0 L 2 (B(R)) C R t -d 2 u 0 L 2 (R d ) .
This means that the "mass" of the solution escapes at infinity. On the other hand, the Schrödinger equation has a regularizing effect. The solution e it∆ u 0 belongs to C ∞ for t = 0 and

R (1 -∆) 1 4 e it∆ u 0 2 L 2 (B(R)) dt C R u 0 2 L 2 (R d ) .
There are many papers dealing with these properties for more and more general Schrödinger equations. Concerning the local energy decay for a self-adjoint Schrödinger equation, we only refer to [START_REF] Rauch | Local decay of scattering solutions to Schrödinger's equation[END_REF] for the Schrödinger operator with an exponentially decaying potential, to [START_REF] Tsutsumi | Local energy decay of solutions to the free Schrödinger equation in exterior domains[END_REF] for the free Schrödinger equation on an exterior demain, and to [START_REF] Bouclet | Low frequency estimates and local energy decay for asymptotically Euclidean laplacians[END_REF][START_REF] Bony | Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds[END_REF] for a laplacian associated to a long-range perturbation of the flat metric. For all these papers, the local energy decays like t -d 2 or like t -d 2 +ε under a non-trapping assumption. There is also a huge litterature for the closely related problem of the local energy decay for the wave equation (see [START_REF] Lax | Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle[END_REF][START_REF] Ralston | Solution of the wave equation with localized energy[END_REF][START_REF] Morawetz | Decay of the solution of the wave equation outside non-trapping obstacles[END_REF][START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF][START_REF] Tataru | Local decay of waves on asymptotically flat stationary space-times[END_REF][START_REF] Guillarmou | Resolvent at low energy III: the spectral measure[END_REF] and references therein).

Concerning the smoothing effect we mention [START_REF] Constantin | Local smoothing properties of dispersive equations[END_REF][START_REF] Sjölin | Regularity of solutions to the Schrödinger equation[END_REF] for the laplacian on R d , [START_REF] Ben-Artzi | Decay and regularity for the Schrödinger equation[END_REF] for the Schrödinger operator with a potential and [START_REF] Burq | On nonlinear Schrödinger equations in exterior domains[END_REF] for the problem on an exterior domain. We also refer to [START_REF] Doi | Smoothing effects of Schrödinger evolution groups on Riemannian manifolds[END_REF][START_REF] Doi | Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow[END_REF][START_REF] Burq | Smoothing effect for Schrödinger boundary value problems[END_REF] for the necessity of the non-trapping condition.

In the dissipative context, the local energy decay for the damped Schrödinger equation in an exterior domain has been proved in [START_REF] Aloui | Stabilization of Schrödinger equation in exterior domains[END_REF]. In this context, the non-trapping condition can be replaced by the geometric control condition (see [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF][START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]): there can be bounded classical trajectories but they have to go through the damping region. Then the local energy decays like t -d 2 , as in the self-adjoint case under the non-trapping condition. A similar result has been obtained in [START_REF] Aloui | Boundary stabilization of the wave and Schrödinger equations in exterior domains[END_REF] for the free Schrödinger equation on an exterior domain with dissipation at the boundary, and in [START_REF] Royer | Exponential decay for the Schrödinger equation on a dissipative wave guide[END_REF] for the similar problem on a wave guide. In the latter case the global energy decays exponentially and we have a smoothing effect in the unbounded directions. We also mention [START_REF] Bortot | Asympotic stability for the damped Schrödinger equation on noncompact Riemannian manifolds and exterior domains[END_REF], where an exponential decay for the global energy is proved for the solution of the Schrödinger equation with a dissipation effective on a neighborhood of the infinity.

The dissipation by a potential (α = 0 in our setting) is not strong enough to recover under the damping condition the same smoothing effect as under the non-trapping condition. However it is known that this is the case for the so-called regularized Schrödinger equation (α = 1). See [START_REF] Aloui | Smoothing effect for regularized Schrödinger equation on compact manifolds[END_REF][START_REF] Aloui | Smoothing effect for regularized Schrödinger equation on bounded domains[END_REF] for the problem on a compact manifold and [AKR] for the problem on an exterior domain. As in the self-adjoint case (see [START_REF] Burq | Smoothing effect for Schrödinger boundary value problems[END_REF]), we can recover a H 1 2 -ε smoothing effect if only a few classical trajectories fail to satisfy the assumption (see [START_REF] Aloui | Smoothing effect for the regularized Schrödinger equation with non-controlled orbits[END_REF]).

In these works, the problem is a compact perturbation of the free Schrödinger equation. Our purpose in this paper is to prove the local energy decay and the Kato smoothing effect for an asymptotically vanishing perturbation. In a similar context, the local energy decay has been studied for the dissipative wave equation in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF].

We now describe more precisely the setting of our paper. We consider on R d a metric G(x) which is a long range perturbation of the identity: for some ρ > 0 there exist constants

C β for β ∈ N d such that ∂ β G(x) -I d C β x -ρ-|β| . (1.2) Let w ∈ C ∞ (R d ) be such that C -1 w(x) C (1.
3) for some C 1. We consider b 1 , . . . , b d ∈ C ∞ 0 (R d ) such that the operator

P = -div G(x)∇ + W, where W = d j=1 b j (x)D j , (1.4)
is self-adjoint and non-negative on L 2 w := L 2 (R d , w(x) dx) (with domain H 2 w = H 2 (R d , w(x) dx)). Here and everywhere below, D j stands for -i∂ xj . For a Laplacian in divergence form we only have to set

w = 1, b 1 = • • • = b n = 0.
(1.5) We now turn to the Laplace-Beltrami operator associated to a metric g. It is defined as

-∆ g = - d j,k=1 |g(x)| -1 2 ∂ ∂x j |g(x)| 1 2 G j,k (x) ∂ ∂x k ,
with |g(x)| = det(g j,k (x)) and (G j,k ) = (g j,k ) -1 . If g is a long-range perturbation of the flat metric, then so is G = g -1 . We recall from [START_REF] Bouclet | Low frequency estimates and local energy decay for asymptotically Euclidean laplacians[END_REF] that we can assume without loss of generality that |g(

x)| = 1 outside a compact set of R d . Thus there exist b 1 , . . . b d ∈ C ∞ 0 (R d ) such that -∆ g is as in (1.4) with w = |g(x)| 1 2
and G = g -1 .

(1.6) Concerning the dissipative term, a is a smooth and non-negative valued function on R d . As already mentioned, it is of short range:

∂ β a(x) C β x -1-ρ-|β| .
(1.7)

We will use the following notation:

B α = a(x) D α a(x) and H = P -iB α . (1.8)
We also set

α = min(1, α) and κ = d 2 if d is even, d+1 2 if d is odd.
(1.9)

We will see that H is a maximal dissipative operator on L 2 w . In particular, for u 0 ∈ D(H) = H 2 w the problem (1.1) has a unique solution t → e -itH u 0 . The main purpose of this paper is to prove that this solution satisfies the local energy decay and the Kato smoothing effect as stated in the following two theorems:

Theorem 1.1 (Local energy decay). Let ε > 0. Let δ > κ + 1 2 , N ∈ N and σ ∈ [0, 2].
Assume that (i) there are no bounded geodesics (see the non-trapping condition (1.12) below) (ii) or the bounded geodesics go through the damping region (see (1.13)), N α + σ 2 and δ > N -1 2 . Then there exists C 0 such that for u 0 ∈ H σ,δ and t 0 we have

e -itH u 0 L 2,-δ Ct -d 2 +ε u 0 H σ,δ .
In this statement L 2,-δ denotes the weighted space L 2 x -2δ dx , while u 0 H σ,δ is the L 2 -norm of x δ D σ u 0 . We remark that we have to take σ = 2 in the second case if α = 0. This means that we have a loss of two derivatives. If α > 0 we can take σ = 0 (no loss of derivative) as long as we choose δ large enough (if α 1 then we can take N = 2 and in this case the condition δ > N -1 2 is weaker than δ > κ + 1 2 ). Under the non-trapping condition we can always take σ = 0. Theorem 1.2 (Kato smoothing effect). There exists C 0 such that for all u 0 ∈ L 2 we have

+∞ 0 x -1 D α/2 e -itH u 0 2 L 2 dt C u 0 2 L 2 .
Notice that according to (1.3) we have L 2,δ = L 2,δ w and H σ,δ w = H σ,δ with equivalent norms, so we can state these estimates in L 2 and H σ,δ even when w ≡ 1.

The proofs of Theorems 1.1 and 1.2 are based on uniform resolvent estimates. According to Proposition 2.2 below, the operator H is maximal dissipative, so for all z in

C + := {z ∈ C : Im z > 0} we can consider the resolvent R(z) = H -z -1 ∈ L(L 2 w ) = L(L 2 ).
(1.10)

Here we have denoted by L(L 2 ) the space of bounded operators on L 2 . After a Fourier transform, the solution u of (1.1) can be written as the integral over frequencies Re(z) of this resolvent when Im(z) goes to 0 (see Section 6). Thus the problem will be reduced to proving uniform estimates for R(z) and its derivatives for Im(z) small, and then to control the dependance of these estimates with respect to Re(z). Since the self-adjoint part P of H is a non-negative operator, the estimates for Re(z) < 0 are easy: for n ∈ N and z ∈ C + with Re(z) -c 0 < 0 we have

R n+1 (z) L(L 2 ) C w |Re(z)| n+1 .
(1.11) Thus we will focus on z ∈ C + with Re(z) -c 0 where 0 < c 0 ≪ 1. As usual, the difficulties will arise for low frequencies (Re(z) close to 0) and high frequencies (Re(z) ≫ 1). We first state the uniform resolvent estimates for intermediate frequencies:

Theorem 1.3 (Intermediate frequency estimates). Let K be a compact subset of C \ {0}. Let n ∈ N and δ > n + 1 2 .
Then there exists C 0 such that for all z ∈ K ∩ C + we have

x -δ R n+1 (z) x -δ L(L 2 )
C.

We remark that compared to the resolvent for the dissipative wave equation (see [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF]), the derivatives of the resolvent correspond to its powers:

R (n) (z) = n! R n+1 (z).
This will significantly simplify the discussion.

It is known that even for the free Laplacian the estimates of Theorem 1.3 fail to hold uniformly when z goes to 0 if n is too large. This explains the restriction in the rate of decay in Theorem 1.1. For low frequencies we prove the following result: Theorem 1.4 (Low frequency estimates). Let ε > 0. Let n ∈ N and let δ be such that

δ > n + 1 2 if 2n + 1 d, n + 1 if 2n + 1 < d.
Then there exist C 0 and a neighborhood U of 0 in C such that for all z ∈ U ∩ C + we have

x -δ R n+1 (z) x -δ L(L 2 ) C 1 + |z| d 2 -ε-1-n .
In the self-adjoint case we can improve the estimate for a single resolvent. More precisely we can replace the weight x -δ for δ > 1 by x -1 . See [BR]. This is particularly interesting for Theorem 1.2 which does not require estimates for the derivatives of the resolvent. This sharp resolvent estimates is also valid in our dissipative context: Theorem 1.5 (Sharp low frequency estimate). There exist C 0 and a neighborhood U of 0 in C such that for all z ∈ U ∩ C + we have

x -1 R(z) x -1 L(L 2 )
C.

The high frequency properties of the problem are closely related to the corresponding classical problem. Here the classical flow is the geodesic flow on R 2d ≃ T * R d for the metric G(x) -1 (that is the geodesic flow of the metric g when P = -∆ g ). It is the Hamiltonian flow corresponding to the symbol p(x, ξ) = G(x)ξ, ξ . We denote by φ t = (X(t), Ξ(t)) this flow. Let

Ω b = w ∈ p -1 ({1}) : sup t∈R |X(t, w)| < +∞ .
We say that the classical flow is non-trapping if there is no bounded geodesic:

Ω b = ∅.
(1.12)

We say that the damping condition on bounded geodesics (or Geometric Control Condition) is satisfied if every bounded geodesic goes through the damping region {a(x) > 0}:

∀w ∈ Ω b , ∃T ∈ R, a X(T, w) > 0.
(1.13) Theorem 1.6 (High frequency estimates). Let n ∈ N and δ > n + 1 2 . (i) Assume that the non-trapping assumption (1.12) holds. Then there exists C 0 such that for z ∈ C + with Re(z) C we have

x -δ R n+1 (z) x -δ L(L 2 ) C |z| -n+1 2 .
(ii) Assume that the damping condition (1.13) holds. Then there exists C 0 such that for z ∈ C + with Re(z) C we have

x -δ R n+1 (z) x -δ L(L 2 ) C |z| -(n+1) α 2 .
In order to prove the uniform estimates of Theorems 1.3, 1.4 and 1.6 we use the commutators method of Mourre (see [START_REF] Mourre | Absence of singular continuous spectrum for certain self-adjoint operators[END_REF], see also [START_REF] Amrein | C 0 -groups, Commutator Methods and Spectral theory of N -body Hamiltonians[END_REF] and references therein for an overview of the subject). The method has been generalized to the dissipative setting in [START_REF] Royer | Limiting absorption principle for the dissipative Helmholtz equation[END_REF], then in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF] for the estimates of the derivatives of the resolvent and finally in [Roy] for a dissipative perturbation in the sense of forms. Here the dissipative perturbation B α is well defined as an operator on L 2 relatively bounded with respect to the self-adjoint part P . However, for d ∈ {3, 4} the rescaled version of the dissipative part which we are going to use for low frequencies will be uniformly bounded as an operator in L(H 1 , H -1 ) but not in L(H 2 , L 2 ), so we will have to see H as a dissipative perturbation of P in the sense of forms. See Remark 4.7.

Let us come back to the statement of Theorem 1.2. To prove this theorem we will use in particular the resolvent estimates of Theorem 1.6, which in turn rely on the damping assumption (1.13). These estimates and hence the smoothing effect we obtain are optimal (in the sense that they are as good as in the self-adjoint case with the non-trapping condition) when α 1. However with a weaker dissipation (α < 1) we can obtain (weaker) resolvent estimates and a (weaker) smoothing effect. Similarly, it is possible to prove high frequency resolvent estimates weaker than those of Theorem 1.6 without the damping condition. We have already mentioned [START_REF] Burq | Smoothing effect for Schrödinger boundary value problems[END_REF] in the self-adjoint case and [START_REF] Aloui | Smoothing effect for the regularized Schrödinger equation with non-controlled orbits[END_REF] in the dissipative setting, where only a few hyperbolic classical trajectories deny the assumption (in these cases the high-frequency resolvent estimates are of size ln |z| / |z|, which gives a gain of 1 2ε derivative). We do not prove resolvent estimates without damping condition in this paper, but we emphasize this fact with a more general version of Theorem 1.2 (for self-adjoint operators, we mention the result of [START_REF] Thomann | A remark on the Schrödinger smoothing effect[END_REF] which give a relation between the smoothing effect and the decay of the spectral projections).

Theorem 1.7. Let γ ∈ [0, 2]. Assume that there exists C 0 such that for all z ∈ C + we have

x -1 R(z) x -1 L(L 2 ) C z -γ 2 . (1.14)
Then for all u 0 ∈ L 2 we have

+∞ 0 x -1 D γ 2 e -itH u 0 2 L 2 dt C u 0 2 L 2 .
It is classical in the self-adjoint setting to prove the smoothing effect from resolvent estimates by means of the theory of relatively smooth operators in the sense of Kato (see [START_REF] Kato | Wave operators and similarity for some non-selfadjoint operators[END_REF][START_REF] Reed | Method of Modern Mathematical Physics, volume IV, Analysis of Operator[END_REF]). Other ideas have been used for dissipative operators (see [START_REF] Aloui | Smoothing effect for the regularized Schrödinger equation with non-controlled orbits[END_REF] and [AKR]). However the theory of Kato can also be used in this context (see [START_REF] Royer | Limiting absorption principle for the dissipative Helmholtz equation[END_REF] and [START_REF] Royer | Exponential decay for the Schrödinger equation on a dissipative wave guide[END_REF]). We will follow this idea to prove Theorem 1.7 and hence Theorem 1.2. This paper is organized as follows. In Section 2 we recall all the abstract properties we need concerning dissipative operators (including the statement of the Mourre method). In Section 3 we prove Theorem 1.3. In Section 4 we deal with low frequencies. We first prove Theorem 1.4 for a small perturbation of the free laplacian in Section 4.1 and then in the general setting in Section 4.2. Theorem 1.5 is proved in Section 4.3. In Section 5 we prove Theorem 1.6 concerning the high-frequency resolvent estimates. Finally we turn to the time dependant problem: we prove Theorem 1.1 in Section 6 and Theorems 1.7 and 1.2 in Section 7.

Abstract properties for dissipative operators

In this section we recall some general properties about dissipative operators. In particular we give the version of the Mourre's method which we use in this paper.

Let H be a Hilbert space. An operator H with domain D(H) on H is said to be dissipative (respectively accretive) if ∀ϕ ∈ D(H), Im Hϕ, ϕ H 0 respectively Re Hϕ, ϕ H 0 .

Moreover H is said to be maximal dissipative (respectively maximal accretive) if it has no other dissipative (respectively accretive) extension than itself. Notice that H is (maximal) dissipative if and only if iH is (maximal) accretive. We recall that a dissipative operator H is maximal dissipative if and only if there exists z ∈ C + such that the operator (Hz) has a bounded inverse on H. In this case any z ∈ C + belongs to the resolvent set of H and

(H -z) -1 L(H) 1 Im(z) . (2.1)
According to the Hille-Yosida theorem this implies in particular that -iH generates a contractions semi-group, and then for all u 0 ∈ D(H) the function u : t → e -itH u 0 belongs to

C 0 (R + , D(H)) ∩ C 1 (R + , H
) and is the unique solution for the problem

-i∂ t u + Hu = 0, ∀t > 0, u(0) = u 0 .
Moreover we have ∀t 0, u(t) H u 0 H .

Remark 2.1. Assume that H is both dissipative and accretive. Then it is maximal dissipative if and only if it is maximal accretive. Indeed both properties are equivalent to the fact that (H -(-1 + i)) has a bounded inverse on H. Moreover in this case for z ∈ C with Im(z) > 0 or Re(z) < 0 we have

(H -z) -1 L(H) 1 max Im(z), -Re(z)
.

Proposition 2.2. The operator H defined by (1.8) is maximal dissipative and maximal accretive on L 2 w . Proof. The operator P and B α are self-adjoint and non-negative, so that H = P -iB α is dissipative and accretive. Let ϕ ∈ H 2 w = D(P ). By interpolation there exists C 0 (which only depends on a and α) such that for any ε > 0

B α ϕ L 2 w C ϕ H α w C ϕ α 2 H 2 w ϕ 1-α 2 L 2 w αεC 2 ϕ H 2 w + C 1 - α 2 ε -α 2-α ϕ L 2 w .
With ε > 0 small enough we obtain that the dissipative operator -iB α is relatively bounded with respect to P with relative bound less that 1. According to [Roy10, Lemma 2.1] this proves that H is maximal dissipative in L 2 w . By Remark 2.1, H is also maximal accretive.

According to Proposition 2.2 the estimate of Remark 2.1 holds for H in L(L 2 w ), and hence in L(L 2 ) up to a multiplicative constant. As already mentioned, the difficulties in Theorems 1.3, 1.4 and 1.6 come from the behavior of the resolvent R(z) when the spectral parameter z ∈ C + approaches the non-negative real axis. For this we are going to use a dissipative version of the Mourre method, which we recall now.

Let q 0 be a quadratic form closed, densely defined, symmetric and bounded from below. We set K = D(q 0 ). Let q Θ be another symmetric form on H, non-negative and q 0 -bounded. Let q = q 0iq Θ and let H be the corresponding maximal dissipative operator (see Proposition 2.2 in [Roy]). We denote by H : K → K * the operator which satisfies q(ϕ, ψ) = Hϕ, ψ

K * ,K
for all ϕ, ψ ∈ K. Similarly we denote by H0 and Θ the operators in L(K, K * ) which correspond to the forms q 0 and q Θ , respectively. By the Lax-Milgram Theorem, the operator

( H -z) has a bounded inverse in L(K * , K) for all z ∈ C + . Moreover for ϕ ∈ H we have (H -z) -1 ϕ = ( H -z) -1 ϕ.
Definition 2.3. Let A be a self-adjoint operator on H and N ∈ N * . We say that A is a conjugate operator (in the sense of forms) to H on the interval J, up to order N , and with bounds α 0 ∈]0, 1], β 0 and Υ N 0 if the following conditions are satisfied:

(i) The form domain K is left invariant by e -itA for all t ∈ R. We denote by E the domain of the generator of e -itA K . (ii) The commutators B 0 = [ H0 , iA] and B 1 = [ H, iA], a priori defined as operators in L(E, E * ), extend to operators in L(K, K * ). Then for all n ∈ 1, N the operator [B n , iA] defined (inductively) in L(E, E * ) extends to an operator in L(K, K * ), which we denote by B n+1 . (iii) We have

B √ α 0 Υ N , B + βΘ B 0 α 0 Υ N , [B, A] + β [Θ, A] α 0 Υ N and N +1 n=2 B n L(K,K * ) α 0 Υ N ,
where all the norms are in L(K, K * ). (iv) We have

1 J (H 0 )(B 0 + βΘ)1 J (H 0 ) α 0 1 J (H 0 ).
(2.2) Theorem 5.5 of [Roy] in the particular case where all the inserted factors are equal to Id H gives the following abstract resolvent estimates: Theorem 2.4. Suppose that the self-adjoint operator A is conjugate to the maximal dissipative operator H on J up to order N 2 with bounds (α, β, Υ N ). Let n ∈ 1, N . Let I ⊂ J be a compact interval. Let δ > n -1 2 . Then there exists c 0 which only depends on J, I, δ, β and Υ N and such that for all z ∈ C I,+ we have

A -δ (H -z) -n A -δ L(H) c α n 0 .
We finish this general section with the so-called quadratic estimates. The following result is a consequence of Proposition 4.4 in [Roy]:

Proposition 2.5. Let T ∈ L(K, H) be such that T * T q Θ in the sense of forms on K. Let Q ∈ L(H, K * ). Then for all z ∈ C + we have T ( H -z) -1 Q L(H) Q * ( H -z) -1 Q 1 2 L(H)
.

Applied with Q = T * , this proposition gives the following particular case:

Corollary 2.6. Let T be as in Proposition 2.5. Then for all z ∈ C + we have

T ( H -z) -1 T * L(H) 1.
We are going to use all these results with the forms q 0 : ϕ → P ϕ, ϕ and q Θ : ϕ → B α ϕ, ϕ defined on K = H 1 (R d ).

Intermediate frequency estimates

In this section we prove Theorem 1.3. For this we will apply Theorem 2.4 with the generator of dilations as the conjugate operator. Let

A = - i 2 (x • ∇ + ∇ • x) = -i (x • ∇) - id 2 .
We recall in the following proposition the main properties of A we are going to use in this paper:

Proposition 3.1. (i) For θ ∈ R, u ∈ S and x ∈ R d we have (e iθA u)(x) = e dθ 2 u(e θ x). (ii) For j ∈ 1, d and γ ∈ C ∞ (R d ) we have on S: [∂ j , iA] = ∂ j and [γ, iA] = -(x • ∇)γ. (iii) For p ∈ [1, +∞], θ ∈ R and u ∈ S we have e iθA u L p = e θ( d 2 -d p ) u L p .

Now we give a proof of Theorem 1.3:

Proof of Theorem 1.3. Let E > 0. We check that the generator of dilations A is a conjugate operator for H on a neighborhood J of E in the sense of Definition 2.3. The form domain of H is the Sobolev space H 1 (R d ). According to Proposition 3.1, it is left invariant by the dilation e -itA for any t ∈ R. By pseudo-differential calculus we can see that the commutators [P, iA],

[[P, iA], iA], [B α , iA] and [[B α , iA], iA] define operators in L(H 2 , L 2 ), hence in L(L 2 , H -2
) by duality, and in L(H 1 , H -1 ) by interpolation 1 . Since we need estimates for a single operator, we do not have to worry about the estimates of the third assumption. We only have to choose Υ N large enough. Finally we use the usual trick for the main assumption. For σ > 0 we set

J σ = [E -σ, E + σ].
We have

1 Jσ (P )[P, iA]1 Jσ (P ) = 1 Jσ (P ) 2P 1 Jσ (P ) + W 1 Jσ (P ) 2(E -σ)1 Jσ (P ) + W 1 Jσ (P )
where

W = 1 Jσ (P )   div (x • ∇)G(x) ∇ - d j=1 (x • ∇)b j D j - d j=1 b j D j  
is a compact operator. Since E > 0 is not an eigenvalue of P (see [START_REF] Koch | Carleman estimates and absence of embedded eigenvalues[END_REF]) the operator 1 Jσ (P ) goes strongly to 0 when σ goes to 0. Then for σ small enough we have 1 Jσ (P )[P, iA]1 Jσ (P ) E1 Jσ (P ).

Thus we can apply Theorem 2.4, which gives Theorem 1.3 for Re(z) ∈ J σ and with weights A -δ . By compactness of K ⊂ C * and the easy estimate of Remark 2.1, we have a uniform

1 In fact we can also compute these commutators explicitely with Proposition 3.1, except for the commutators of D α with A: for this we can write

D α = (1 -∆) 2 × (1 -∆) -2 -∆ α 2
and use the Helffer-Sjöstrand formula for the second factor (see [START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF][START_REF] Davies | The functional calculus[END_REF]).

estimate for all z ∈ C + ∩ K. It remains to replace A -δ by x -δ . For this we use the resolvent identity

R(z) = R(i) + (z -i)R(i)R(z) = R(i) + (z -i)R(z)R(i)
to prove by induction on m ∈ N * that R n+1 (z) can be written as a sum of terms of the form

(z -i) β R n+1+β (i) with β ∈ N or (z -i) 2m-n-1+ν R m (i)R ν (z)R m (i) with max(1, n + 1 -2m) ν n + 1. On the one hand R n+1+β (i) is uniformly bounded in L(L 2
) and on the other hand

x -δ R m (i)R ν (z)R m (i) x -δ x -δ R m (i) A δ A -δ R ν (z) A -δ A δ R m (i) x -δ .
The first and third factors are bounded by pseudo-differential calculus if m is large enough and the second has been estimated uniformly by the Mourre method. This concludes the proof of Theorem 1.3.

Low frequency estimates

In this section we prove Theorems 1.4 and 1.5. As in [START_REF] Bouclet | Low frequency estimates and local energy decay for asymptotically Euclidean laplacians[END_REF][START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF], the proof of Theorem 1.4 is based on a scaling argument for a small perturbation of the free Laplacian (see Section 4.1), and then on a perturbation argument to deal with the general case (see Section 4.2). Theorem 1.5 is proved in Section 4.3.

Let χ ∈ C ∞ 0 (R d ) be equal to 1 on a neighborhood of 0. For η ∈]0, 1] we set χ η : x → χ(ηx). Then for η 1 ∈]0, 1] we set G η1 (x) = χ η1 (x)I d + (1 -χ η1 (x))G(x), P η1 = -div G η1 (x)∇ and P η1,c = P -P η1 = -div χ η1 (x)(G(x) -I d ) ∇. (4.1)
For the dissipative part we set

B α η2 = a(1 -χ η2 ) D α a + aχ η2 D α a(1 -χ η2 ) (4.2) and B α η2,c = B α -B α η2 = aχ η2 D α aχ η2 ,
where η 2 ∈]0, 1]. Finally, for the full operator we define

H η = P η1 -iB α η2 and R η (z) = (H η -z) -1 , where η = (η 1 , η 2 ) ∈]0, 1] 2 .
4.1. Low frequency estimates for a small perturbation of the Laplacian. In this paragraph we prove Theorem 1.4 with R(z) replaced by R η (z). Then in Section 4.2 we will add the contributions of P η1,c , W and B α η2,c .

The proof relies on a scaling argument. To this purpose we use for z ∈ C * the operator

Θ z = exp i ln |z| 2 A .
For u ∈ S and x ∈ R d we have

(Θ z u)(x) = |z| d/4 u |z| 1/2 x . According to Proposition 3.1 we have for p ∈ [1, +∞] Θ z L(L p ) = |z| d 4 -d 2p . (4.3)
For a function u on R d and z ∈ C * we denote by u z the function

u z : x → u x |z| .
Compared to the scaling for the wave equation we are using the parameter |z| instead of |z|.

Now we introduce the rescaled versions of our operators:

H η,z = 1 |z| Θ -1 z H η Θ z = P η1,z -iB α η2,z
where P η1,z =div G η1,z (x)∇ and

B α η2,z = 1 |z| (1 -χ η2 )a z 1 -|z| ∆ α 2 a z + 1 |z| (χ η2 a) z 1 -|z| ∆ α 2 (1 -χ η2 )a z . Then for ζ ∈ C + we set R η,z (ζ) = H η,z -ζ -1
, so that with the notation ẑ = z/ |z| we have

for z ∈ C + R η (z) = 1 |z| Θ z R η,z (ẑ)Θ -1 z .
Our analysis of the rescaled operators is based on the fact that if a function φ decays like x -ν-ρ 2 (recall that ρ > 0 is fixed by (1.2) and (1.7)) then the multiplication by the rescaled function φ λ behaves like an differential operator of order ν for low frequencies, in the sense that it is of size λ ν as an operator from H s to H s-ν . Since this observation relies on the Sobolev embeddings, there is however a restriction in the choice of ν and s.

For σ ∈ R, let S -σ (R d ) be the set of functions φ ∈ C ∞ (R d ) such that ∂ β φ(x)
x -σ-|β| .

For ν 0, N ∈ N and φ ∈ S -ν-ρ 2 (R d ) we set φ ν,N = sup |β| κ+1 0 m N sup x∈R d x ν+ ρ 2 +|β| ∂ β (x • ∇) m φ (x) .
We recall that the integer κ was defined in (1.9). The following result is Proposition 7.2 in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF]:

Proposition 4.1. Let ν ∈ 0, d 2 and s ∈ -d 2 , d 2 be such that s -ν ∈ -d 2 , d 2 . Then there exists C 0 such that for φ ∈ S -ν-ρ 2 (R d ), u ∈ H s and λ > 0 we have φ λ u Ḣs-ν Cλ ν φ ν,0 u Ḣs and φ λ u H s-ν Cλ ν φ ν,0 u H s .
The interest of replacing G(x) by G η1 (x) and a by a(1χ η2 ) in the definition of H η is that for all N ∈ N we have

N η,N := d j,k=1 G η1,j,k (x) -δ j,k 0,N + (1 -χ η2 )a 1,N a 1,N + χ η2 a 1,N = O η→0 |η| ρ/2 . (4.4)
Thus this quantity is as small as we wish if we choose η 1 and η 2 small enough. x d . At the beginning of the section we said that H η has to be close to the free Laplacian. What we need precisely is the following result:

Given two operators

Proposition 4.2. Let µ ∈ N d , m ∈ N, ε 0 > 0 and s ∈ R. There exists η 0 ∈]0, 1] such that for η = (η 1 , η 2 ) ∈]0, η 0 ] 2 the following statements hold: (i) If s ∈ -d 2 , d 2 then for z ∈ C + with |z| 1 we have ad µ x ad m A P η1,z + ∆) L(H s+1 ,H s-1 ) ε 0 . (ii) If s ∈ -d 2 + 1, d 2 -1 then we also have ad µ x ad m A B α η2,z L(H s+1 ,H s-1 ) ε 0 . (iii) For u ∈ H 2 we have 1 2 u Ḣ2 P η1 u L 2 2 u Ḣ2 .
Proof. The first statement is the same as for the wave equation. See Proposition 7.6 in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF]. In particular with s = 1, |z| = 1 and ε 0 = 1 2 we obtain the last statement. It remains to prove (ii). Let D z = |z|D. We write

((1 -χ η2 )a) z D z α a z = ((1 -χ η2 )a) z (-|z| ∆ + 1) D z α-2 a z .
Then ad µ x ad m A ((1χ η2 )a) z D z α a z can be written as a sum of terms of the form

ad µ1 x ad m1 A ((1 -χ η2 )a) z ad µ2 x ad m2 A -|z| ∆ + 1 ad µ3 x ad m3 A D z α-2 ad µ4 x ad m4 A (a z ), where µ 1 , µ 2 , µ 3 , µ 4 ∈ N d and m 1 , m 2 , m 3 , m 4 ∈ 0, m are such that µ 1 + µ 2 + µ 3 + µ 4 = µ and m 1 + m 2 + m 3 + m 4 = m. Let γ ∈ [0, 1]. According to Proposition 4.1 we have for z ∈ C + ad µ1 x ad m1 A (1 -χ η2 )a z L(H s-1+γ ,H s-1 ) η 1+ ρ 2 -γ 2 |z| γ 2 (4.5) and ad µ4 x ad m4 A a z L(H s+1 ,H s+1-γ ) |z| γ 2 . (4.6)
To estimate ad µ3 x ad m3 A D z α-2 we use the Helffer-Sjöstrand formula (see [START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF][START_REF] Davies | The functional calculus[END_REF]). We can check that for ζ ∈ C \ R we have

ad µ3 x ad m3 A -|z| ∆ -ζ -1 L(H s+1-γ ) ζ |µ3|+m3 |Im(ζ)| |µ3|+m3+1 . Let f : τ → (τ + 1) α-2 2 . Let φ ∈ C ∞ 0 (R, [0, 1]) be supported in [-2, 2] and equal to 1 on [-1, 1]. For m > |µ 3 | + m 3 + 1 and ζ = x + iy we set fm (ζ) = φ y x m k=0 f (k) (x) (iy) k k! .
We have

∂ fm ∂ζ (ζ) 1 { x |y| 2 x } (ζ) x -1+ α-2 2 + 1 {|y| 2 x } (ζ) |y| m x -m-1+ α-2 2 ,
so we can write

(-|z| ∆ + 1) α-2 2 = 1 π ζ=x+iy∈C ∂ fm ∂z (ζ)(-|z| ∆ -ζ) -1 dx dy.
Then we can check that

ad µ3 x ad m3 A D z α-2 L(H s+1-γ )
1.

(4.7)

It remains to estimate

ad µ2 x ad m2 A D z 2 = -|z| ad µ2 x ad m2 A ∆ + ad µ2 x ad m2 A (1). (4.8)
We have |z| ad µ2 x ad m2 A ∆ |z| in L(H s+1 , H s-1 ) so with (4.5), (4.6) and (4.7) applied with γ = 0 we obtain in L(H s+1 , H s-1 )

ad µ1 x ad m1 A ((1 -χ η2 )a) z ad µ2 x ad m2 A -|z| ∆ ad µ3 x ad m3 A D z α-2 ad µ4 x ad m4 A (a z ) |z| η 1+ ρ 2 2
. (4.9)

If |µ 2 | = m 2 = 0 we also have to consider the second term in (4.8). For this we apply (4.5), (4.6) and (4.7) with γ = 1, which gives

ad µ1 x ad m1 A ((1 -χ η2 )a) z ad µ3 x ad m3 A D z α-2 ad µ4 x ad m4 A (a z ) L(H s+1 ,H s-1 ) |z| η ρ 2
2 .

Thus we have proved that

ad µ x ad m A ((1-χ η2 )a) z D z α a z is of size O(|z| η ρ 2
2 ) in L(H s+1 , H s-1 ). We proceed similarly for ad µ x ad m A (χ η2 a) z D z α ((1χ η2 )a) z , and the statement follows.

Remark 4.3. If d 5 we can replace P η1 by H η in the last statement of Proposition 4.2. This is not the case for d ∈ {3, 4}. This is due to the fact that s = 1 does not belong to

-d 2 + 1, d 2 -1 and hence B α η2,z is not small in L( Ḣ2 , L 2 ) in these cases. Proposition 4.4. Let µ ∈ N d , m ∈ N and s ∈ -d 2 + 1, d 2 -1 . There exists η 0 ∈]0, 1] such that the operator ad µ x ad m A R η,z (-1) is bounded as an operator from H s-1 to H s+1 uniformly in z ∈ C + with |z| 1 and η = (η 1 , η 2 ) ∈]0, η 0 ] 2 .
Proof. The idea of the proof is the same as the proof of Proposition 7.9 in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF]. We only have to be careful with the fact that the dissipative term has to be seen as an operator of order 2. However, with the smallness assumption on a(1χ η2 ), it is still a small perturbation of -∆, and we can proceed as for the wave resolvent. We also have to be careful with the restriction on s which is stronger than for the wave equation. This is due to the analogous restriction in the second statement of Proposition 4.2. We omit the details.

Proposition 4.5. (i) Let s ∈ 0, d 2 , δ > s and m ∈ N be such that m s. Then there exist η 0 ∈]0, 1] and C 0 such that for z ∈ C + with |z| 1 and η ∈]0, η 0 ] 2 we have

x -δ Θ z R m η,z (-1)Θ -1 z x -δ L(L 2 ) C |z| s .
(ii) Let s ∈ 0, d 2 , δ > s, and m ∈ N large enough (say m δ + s 2 + 1). Then there exist η 0 ∈]0, 1] and C 0 such that for z ∈ C + with |z| 1 and η ∈]0, η 0 ] 2 we have

x -δ Θ z R m η,z (-1) A δ L(L 2 ) C |z| s 2 and A δ R m η,z (-1)Θ -1 z x -δ L(L 2 ) C |z| s 2 .
Proof. According to Proposition 4.4 the operator R m η,z (-1) is bounded in L(H -s , H s ) uniformly in z ∈ C + with |z| 1 and η close to (0,0). On the other hand, according to the Sobolev embedding H s ⊂ L p for p = 2d d-2s , the fact that x -δ belongs to L(L p , L 2 ) and (4.3) we have

x -δ Θ z L(H s ,L 2 ) Θ z L(L p ) |z| s 2 .
We similarly have

Θ -1 z x -δ L(L 2 ,H -s ) |z| s 2 ,
and the first statement follows. For the second statement we use the same idea as in the proof of Proposition 7.11 in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF]. We only prove the first estimate. For this we first remark that

x -δ Θ z 1 + |x| δ L(H s ,L 2 ) x -δ Θ z L(H s ,L 2 ) + x -δ Θ z |x| δ L(L 2 ) Θ z L(L p ) + |z| δ 2 x -δ |x| δ Θ z L(L 2 ) |z| s 2
where, again, p stands for 2d d-2s . Then it remains to prove that for all δ 0 (we no longer need the assumption that δ > s), m δ + s 2 + 1 and µ ∈ N d the operator

x -δ ad µ x R m η,z (-1) A δ is bounded in L(L 2 , H s ) uniformly in z ∈ C + .
With µ = 0 this will conclude the proof. By interpolation it is enough to consider the case where δ is an integer and m δ + s 2 (we do not mean to be sharp with this assumption). We proceed by induction. The statement for δ = 0 is given by Proposition 4.4. Now let δ ∈ N * . We have

R m η,z (-1)A δ = δ k=0 C k δ R m-1 η,z (-1)A δ-k ad k A R η,z (-1) .
When k = 0 we can apply the inductive assumption to R m-1 η,z (-1)A δ-k . With Proposition 4.4 we obtain that the contributions of the corresponding terms are uniformly bounded in L(L 2 , H s ) as expected. It remains to consider the term corresponding to k = 0. It is enough to consider

R m-1 η,z (-1)A δ-1 x j D j R η,z (-1)
for some j ∈ 1, d . The operator D j R η,z (-1) and its commutators with powers of x are uniformly bounded operators on L 2 , and R m-1 η,z (-1)A δ-1 x j = x j R m-1 η,z (-1)A δ-1 + ad xj R m-1 η,z (-1)A δ-1 . We conclude with the inductive assumption.

Proposition 4.6. Let k ∈ N and δ > k + 1 2 . Then there exist η 0 ∈]0, 1] and C 0 such that for z ∈ C + with |z| 1 and η = (η 1 , η 2 ) ∈]0, η 0 ] 2 we have

A -δ R k+1 η,z (ẑ) A -δ L(L 2 ) C.
Proof. The estimate is clear when ẑ is outside some neighborhood of 1. For ẑ close to 1 we apply Theorem 2.4 uniformly in z with A as a conjugate operator. We have already said that e -itA leaves H 1 invariant for all t ∈ R. The assumptions (ii) and (iii) of Definition 2.3 with α 0 = 1/2 and β = 0 are consequences of Proposition 4.2 applied with s = 0, µ = 0 and m ∈ N * . For m ∈ {0, 1}, z ∈ C + and u ∈ S we have

ad m iA (P η1,z )u, u L 2 -2 m -∆u, u L 2 d j,k=1 (2 -x • ∇) m (G η1,z,j,k -δ j,k )D j u, D k u L 2 O(η ρ/2 1 ) ∇u 2 L 2 , and hence [P η1,z , iA] 2 -O η ρ/2 1 (-∆) 2 -O η ρ/2 1 P η1,z .
Let J = 1 2 , 3 2 . After conjugation by 1 J (P η1,z ) we obtain that if η 0 is small enough then for all η 1 ∈]0, η 0 ] and z ∈ C + we have 1 J (P η1,z )[P η1,z , iA]1 J (P η1,z ) 1 2 1 J (P η1,z ).

Then Proposition 4.6 follows from Theorem 2.4.

Remark 4.7. It is important to notice that we have estimated [B α η2 , iA] and [[B α η2 , iA], iA] in L(H 1 , H -1 ) and not in L(H 2 , L 2 ). By pseudo-differential calculus, these two commutators define operators in L(H 2 , L 2 ). But in low dimensions (d ∈ {3.4}) they can be estimated uniformly by Proposition 4.2 only in the sense of forms. This is why we need a form version of the dissipative Mourre method here. Proposition 4.8. Let ε > 0 and n ∈ N. Let δ be as in the statement of Theorem 1.4. Then there exist η 0 ∈]0, 1], C 0 and a neighborhood U of 0 in C such that for η = (η 1 , η 2 ) ∈]0, η 0 ] 2 and β l , β r ∈ R + with β l + β r 2 we have

x -δ D β l R n+1 η (z) D βr x -δ L(L 2 ) C 1 + |z| d 2 -ε-1-n .
Remark 4.9. Compared to the analogous result for the wave equation (see Theorem 1.3 in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF]) there is no gain when we add a derivative. This is a consequence of the restriction on the Sobolev index s in Proposition 4.4, which is stronger than in Proposition 7.9 in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF].

Proof of Proposition 4.8. First assume that n 1. By the resolvent identity we have

x -δ D β l R n+1 η (z) D βr x -δ = x -δ D β l R η (-1) x δ × x -δ R n-1 η (z) + 2(1 + z)R n η (z) + (1 + z) 2 R n+1 (z) x -δ × x δ R η (-1) D βr x -δ .
The first and last factors are bounded on L 2 uniformly in η ∈]0, 1] 2 by pseudo-differential calculus, so it is enough to prove the statement without additionnal derivatives if n 1. Since β l + β r 2 we have a similar argument for n = 0.

We have

x -δ R n+1 η (z) x -δ = |z| -(n+1) x -δ Θ z R n+1 η,z (ẑ)Θ -1 z x -δ .
As in the proof of Theorem 1.3 in Section 3 we can prove by induction on m ∈ N * that R n+1 η,z (ẑ) can be written as a sum of terms of the form

(1 + ẑ) β R n+1+β η,z (-1) or (1 + ẑ) 2m-n-1+ν R m η,z (-1)R ν η,z (ẑ)R m η,z (-1), (4.10)
where max(1, n + 1 -2m) ν n + 1 and β ∈ N. Let s = min n + 1, d 2ε . For β ∈ N we have s ∈ 0, d 2 , n + 1 + β s and δ > s so according to the first statement of Proposition 4.5 we have

|z| -(n+1) x -δ R n+1+β η,z (-1) x -δ L(L 2 ) |z| s-(n+1) 1 + |z| d 2 -ε-n-1 .
Now we consider the contributions of terms of the second kind in (4.10). We can assume that m is large enough to apply the second statement of Proposition 4.5. We have δ > ν -1 2 so with proposition 4.6 we get |z|

-(n+1) x -δ Θ z R m η,z (-1)R ν η,z (ẑ)R m η,z (-1)Θ -1 z x -δ |z| -(n+1) x -δ Θ z R m η,z (-1) A δ A -δ R ν η,z (ẑ) A -δ A δ R m η,z (-1)Θ -1 z x -δ |z| s-(n+1) 1 + |z| d 2 -ε-n-1 .
This concludes the proof.

4.2.

Low frequency estimates for a general perturbation of the Laplacian. In this paragraph we use the estimates on R η (z) to prove the same estimates for R(z). To this purpose we have to add the contributions of P η1,c and W in the self-adjoint part, and the contribution of B α η2,c in the dissipative part.

For η 0 , η 2 ∈]0, 1] and η = (η 0 , η 2 ) we set

K η0 = P η0,c + W and, for ψ ∈ C ∞ 0 (R), S ψ,η (z) = K η0 R η (z)ψ(P ).
From Proposition 4.8 we obtain the following result:

Proposition 4.10. Let ε > 0, n ∈ N and M ∈ R. Let ψ ∈ C ∞ 0 (R).
Let δ be as in the statement of Theorem 1.4. Let η 0 ∈]0, 1] be given by Proposition 4.8. Then there exists C 0 and a neighborhood U of 0 in C such that for η 2 ∈]0, η 0 ] and z ∈ U ∩ C + we have

x M S (n) ψ,η (z) x -δ L(L 2 ) C 1 + |z| d 2 -ε-1-n ,
where η = (η 0 , η 2 ).

Proof. The Proposition is a consequence of Proposition 4.8, the boundedness of ψ(P ) in L 2,δ and the boundedness of

x M K η0 (1 -∆) -1 x δ .
Remark 4.11. Until now we had not used the distinction between η 1 and η 2 . However the size of x M K η1 depends on η 1 , so η 1 has to be fixed in order to obtain uniform estimates in Proposition 4.10 and in Proposition 4.12 below. On the other hand we have to keep the possibility to take η 2 small. More precisely the choice of the cut-off function ψ in Proposition 4.12 (and hence in the proof of Proposition 4.13) will depend on η 1 , and then the choice of η 2 will in turn depend on ψ. This is why we could not simply take η 1 = η 2 in the definition of H η . Proposition 4.12. Let η 0 ∈]0, 1] be given by Proposition 4.8. Let ε 1 > 0, σ > 2 and M 0. Then there exist a bounded neighborhood U of 0 in C, ψ ∈ C ∞ 0 (R) equal to 1 on a neighborhood of 0 and η ∈]0, η 0 ] such that for η 2 ∈]0, η] and z ∈ U ∩ C + we have

x M S ψ,η (z) x -σ L(L 2 ) ε 1 , where η = (η 0 , η 2 ).
Proof. According to the Hardy inequality we have for u ∈ S

x M P η0,c u L 2 d j,k=1 x M D j (χ η0 G j,k ) D k u L 2 + d j,k=1 x M χ η0 G j,k -δ j,k D j D k u L 2 u Ḣ2 and x M W u L 2 d j=1 x M b j D j u L 2 u Ḣ2 .
According to the third statement of Proposition 4.2 we obtain for µ > 0

x M K η0 R η (iµ)ψ(P ) x -σ u L 2 R η (iµ)ψ(P ) x -σ u Ḣ2 P η0 R η (iµ)ψ(P ) x -σ u L 2 ψ(P ) x -σ u L 2 + µ R η (iµ)ψ(P ) x -σ u L 2 + B α η2 R η (iµ)ψ(P ) x -σ u ψ(P ) x -σ u L 2 + O(η 1+ρ 2 ) x -1-ρ (-∆ + 1)R η (iµ)ψ(P ) x -σ u L 2
.

The term with the factor µ is estimated by the analog of (2.1) for H η . For the term involving B α η2 we have used the fact that

B α η2 (-∆ + 1) -1 x 1+ρ L(L 2 ) = O η2→0 (η 1+ρ 2 ).
Let

ψ 1 ∈ C ∞ 0 (R) be equal to 1 on [-1,1]. For ψ ∈ C ∞ 0 (R) supported in ] -1, 1[ we have ψ(P ) x -σ = ψ(P )ψ 1 (P ) x -σ
. The operator ψ 1 (P ) x -σ is compact. On the other hand, since 0 is not an eigenvalue of P the operator ψ(P ) goes weakly to 0 when the support of ψ shrinks to {0}. Thus we can find ψ equal to 1 on a neighborhood of 0 such that for µ > 0 and η 2 small enough we have

x M S ψ,η (iµ) x -σ L(L 2 ) ε 1 2 .
Now let τ ∈ R and µ > 0. We have

x M K η0 R η (τ + iµ) -R η (iµ) ψ(P ) x -σ x M K η0 (-∆ + 1) -1 x σ × τ 0 x -σ (-∆ + 1)R 2 η (θ + iµ) x -σ dθ.
The first factor is bounded by pseudo-differential calculus, and the second factor is of size O(|τ |) according to Proposition 4.8. Thus this norm is not greater that ε1 2 if τ is small enough, and the proposition is proved.

For z ∈ C + and η 2 ∈]0, 1] we set R 0 (z) = P -z -1 (4.11) and Rη2 (z) = P -iB α η2 -z -1 .
In the following proposition we prove the resolvent estimates for Rη2 (z). Then we will add the contribution of B α η2,c in the dissipative part to conclude the proof of Theorem 1.4. Proposition 4.13. Let ε > 0 and n ∈ N. Let δ be as in the statement of Theorem 1.4. Then there exist η 2 , C 0 and a neighborhood U of 0 in C such that for z ∈ U ∩ C + and β l , β r ∈ R + with β l + β r 2 we have

x -δ D β l Rn+1 η2 (z) D βr x -δ L(L 2 ) C 1 + |z| d 2 -ε-1-n .
Proof. As for Proposition 4.8 we see that it is enough to consider the case β l = β r = 0. Let σ = max(δ, 3). Let ε 1 ∈ 0, 1 4 and consider ψ ∈ C ∞ 0 (R) as given by Proposition 4.12 for M = σ. We set B ψ (z) = R 0 (z)(1ψ)(P ). For any γ ∈ R, this operator and its derivatives are uniformly bounded on L 2,γ for z ∈ C + close to 0. Let η 0 be given by proposition 4.8. For η 2 ∈]0, η 0 ] we write η for (η 0 , η 2 ). We have

Rη2 (z) = R η (z)ψ(P ) -Rη2 (z)S ψ,η (z) + B ψ (z) + i Rη2 (z)B α η2 B ψ (z),
and hence for n ∈ N

R(n) η2 (z) = R (n) η (z)ψ(P ) + R(n) η2 (z) -S ψ,η (z) + iB α η2 B ψ (z) + B (n) ψ (z) + i n-1 j=0 C n j R(j) η2 (z) -S (n-j) ψ,η (z) + iB α η2 B (n-j) ψ (z) . (4.12)
We prove by induction on n ∈ N that

x -δ R(n) η2 (z) x -σ 1 + |z| d 2 -ε-n-1 . (4.13)
According to Propositions 4.8, 4.10 and 4.12, the fact that ψ(P ) is uniformly bounded on L 2,σ and the inductive induction for the sum in (4.12) (it vanishes if n = 0), there exists C 0 such that for z ∈ C + close to 0 we have

x -δ R(n) η2 (z) x -σ 1 -ε 1 -x σ B α η2 B ψ (z) x -σ C 1 + |z| d 2 -ε-n-1 .
By pseudo-differential calculus we see that the norm of x σ B α η2 B ψ (z) x -σ goes to 0 when η 2 goes to 0. Thus if η 2 is small enough we have

1 -ε 1 -x σ B α η2 B ψ (z) x -σ 1 2 ,
which concludes the proof of (4.13). In order to replace σ by δ we use (4.12) again and, estimating the second term with (4.13) and Proposition 4.10 instead of Proposition 4.12 we obtain

x -δ R(n) η2 (z) x -δ 1 -x δ B α η2 B ψ (z) x -δ C 1 + |z| d 2 -ε-n-1 ,
and we conclude similarly.

In order to prove Theorem 1.4 it remains to add the dissipative part with compactly supported absorption index. We begin with a lemma: Lemma 4.14. Let H be a Hilbert space. Let R 0 , R 1 ∈ L(H) and let B be such that

R 1 = R 0 -R 0 BR 1 = R 0 -R 1 BR 0 .
Then for all m ∈ N we can write R m+1 1 as a linear combination of terms of the form

R m1+1 0 BR m2+1 j2 B . . . BR m k +1 j k-1 BR m k +1 0 (4.14)
where k ∈ N * , j 1 , . . . , j k-1 ∈ {0, 1} and m 1 , . . . , m k ∈ N are such that k l=1 m l m and m l = 0 if j l = 1.

Proof. Using both of the identities between R 1 and R 0 we obtain

R 1 (z) = R 0 (z) -R 0 (z)BR 0 (z) + R 0 (z)BR 1 (z)BR 0 (z).
Then the result is proved by induction on m.

Now we can finish the proof of Theorem 1.4:

Proof of Theorem 1.4. Let η 2 be given by Proposition 4.13. Let T = D α 2 a ∈ L(H 1 , L 2 ). We have T * T = B α so according to Corollary 2.6 we have

T R(z)T * L(L 2 ) 1. Let M 0 and T M = x -M D α 2 . We can write B α η2,c = T * M B 1 T = T * B 2 T M = T * M B 3 T M where B 1 , B 2
and B 3 are bounded on L 2 . According to Lemma 4.14 applied with

R 0 = Rη2 (z), R 1 = R(z) and B = B α
η2,c , we can write R m+1 (z) as a sum of terms of the form

T = R m1+1 0 (z)BR m2+1 j2 (z)B . . . BR m k +1 j k-1 (z)BR m k +1 0 (z), (4.15)
where k ∈ N * , j 1 , . . . , j k-1 ∈ {0, 1} and m 1 , . . . , m k ∈ N are such that k l=1 m l m and m l = 0 if j l = 1. If M is large enough we obtain for such a term

x -δ T x -δ x -δ Rm1+1 η2 (z)T * M × k-1 l=2 j l =0 T M Rm l +1 η2 (z)T * M × k-1 l=2 j l =1 T R(z)T * × T M Rm k +1 η2 (z) x -δ k l=1 1 + |z| d 2 -1-m l -ε 1 + |z| d 2 -1-m-ε .
This concludes the proof of Theorem 1.4.

4.3. Sharp low frequency resolvent estimate. We finish this section with the proof of Theorem 1.5. The result follows from the self-adjoint analog by a simple perturbation argument, using the quadratic estimates and the spatial decay of the dissipative term:

Proof of Theorem 1.5. According to the resolvent identity, Proposition 2.5 and Theorem 1.1 in [BR] we have

x -1 R(z) x -1 = x -1 R 0 (z) x -1 + x -1 R 0 (z) B α B α R(z) x -1 1 + x -1 R 0 (z) B α x -1 R(z) x -1 1 2 .
Moreover,

x -1 R 0 (z) B α x -1 R 0 (i) B α + |z -i| x -1 R 0 (z) x -1 x R 0 (i) B α 1.
For the norms involving R 0 (i) we have used the fact that x σ R 0 (i) √ Ba extends to a bounded operator since for σ 1 and u ∈ S we have by pseudo-differential calculus

B α R 0 (i) x σ u 2 L 2 x σ R 0 (-i)B α R 0 (i) x σ u, u u 2 L 2 . This gives x -1 R(z) x -1 1 + x -1 R(z) x -1 1 2 ,
from which the conclusion follows.

High frequency estimates

In this section we prove Theorem 1.6. To this purpose we use semiclassical analysis (see for instance [START_REF] Zworski | Semiclassical Analysis[END_REF]). For h > 0 and ζ ∈ C + we set

H h = h 2 H, P h = h 2 P and R h (ζ) = (H h -ζ) -1 . Then for n ∈ N, z ∈ C + and h = |z| -1 2 we have R(z) n+1 = 1 |z| n+1 R n+1 h (ẑ) = h 2(n+1) R n+1 h (ẑ) (5.1)
(we recall that ẑ = z/ |z|).

In order to prove uniform estimates for the resolvent R h (z) we use again the Mourre method. For high frequencies and in a dissipative context we follow [START_REF] Royer | Limiting absorption principle for the dissipative Helmholtz equation[END_REF][START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF]. Here we have to be careful with the form of the dissipative part h 2 B α .

Let χ α ∈ C ∞ 0 (R) be positive in a neighborhood of 1 and such that 0

χ α (r) r α 2 for all r ∈ R + . For h ∈]0, 1] we set B α h = a(x)χ α -h 2 ∆ a(x). Then we have 0 h 2-αB α h h 2-α B α h h 2 (-∆) α 2 h 2 B α , (5.2)
in the sense that for all ϕ ∈ H α/2 (R d ) we have

0 h 2-α B α h ϕ, ϕ L 2 (R d ) h 2 B α ϕ, ϕ L 2 (R d ) .
(5.3)

The operator B α h is a bounded pseudo-differential operator on L 2 . Its principal symbol is b(x, ξ) = a(x) 2 χ α (|ξ| 2 ).

The damping assumption (1.13) on bounded trajectories is satisfied with b instead of a:

∀w ∈ Ω b , ∃T ∈ R, b φ T (w) > 0. Set f 0 (x, ξ) = x • ξ.
As in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF] (see Proposition 8.1), we can prove that there exist an open neighborhood J of 1,

f c ∈ C ∞ 0 (R 2n , R), β 0 and c 0 > 0 such that on p -1 ( J) we have {p, f 0 + f c } + βb 3c 0 , (5.4)
where {p, q} is the Poisson bracket ∇ ξ p • ∇ x q -∇ x p • ∇ ξ q. The fact that the symbol of the dissipative part depends on ξ does not change anything in the proof of this statement. We set

F h = Op w h (f 0 + f c ),
where Op w h is the Weyl quantization:

Op w h (q)u(x) = 1 (2πh) n R n R n e i h x-y,ξ q
x + y 2 , ξ u(y) dy dξ.

Let J be a neighborhood of 1 and a compact subset of J. Let χ ∈ C ∞ 0 ( J, [0, 1]) be equal to 1 on a neighborhood of J. After multiplication by (χ • p) 2 , the (easy) Gårding inequality (Theorem 4.26 in [START_REF] Zworski | Semiclassical Analysis[END_REF]) gives for h > 0 small enough

Op h (χ • p) 2 {p, f 0 + f c } + βb(χ • p) 2 + 3c 0 1 -(χ • p) 2 3c 0 -O(h) 2c 0 .
After multiplication by h 2-α we obtain χ(P h ) [P h , ih 1-αF h ] + βh 2-αB α h χ(P h ) + 3c 0 (1χ 2 )(P h ) 2c 0 h 2-α -O(h 3-α). After conjugation by 1 J (P h ) we obtain for h small enough 1 J (P h ) [P h , ih 1-αF h ] + βh 2-αB α h 1 J (P h ) c 0 h 2-α1 J (P h ). According to (5.2) this finally gives

1 J (P h ) [P h , ih 1-αF h ] + βh 2 B α 1 J (P h ) c 0 h 2-α1 J (P h ), (5.5)
which is the main assumption of Definition 2.3 with βh 2 instead of β and α = c 0 h 2-α.

It remains to check the other assumptions of Definition 2.3. The first is proved as in [START_REF] Bouclet | Local energy decay for the damped wave equation[END_REF] (except that we look at the norm in the form domain H 1 instead of the domain H 2 ), and the commutator properties are proved using (standard) pseudo-differential calculus, considering h as a parameter (for the dissipative part we cannot use h 2-α B α h as above, so we have to control directly the commutators of h 2 B α with h 1-αF h ) .

Thus we have proved that for h ∈]0, h 0 ] the operator h 1-αF h is a conjugate operator to H h on a neighborhood J of 1 with lower bounds h 2-αc 0 for some c 0 > 0. According to Theorem 2.4 we have proved the following result with F h -δ instead of x -δ :

Proposition 5.1. Let n ∈ N and δ > n + 1 2 . There exists a neighborhood J of 1, h 0 > 0 and C 0 such that for all ζ ∈ C + with Re(ζ) ∈ J we have

x -δ R n+1 h (ζ) x -δ L(L 2 ) C h (2-α)(n+1) .
In order to have the estimate with x -δ we proceed as usual (see the end of Section 3 for intermediate frequencies or [START_REF] Royer | Limiting absorption principle for the dissipative Helmholtz equation[END_REF] in the semi-classical context). With (5.1) and Proposition 5.1 we obtain the second statement of Theorem 1.6. For the first statement, we observe that under the non-trapping condition we can proceed as above with β = 0 and with α replaced by 1 in (5.5).

Local energy decay

In this section we use Theorems 1.3, 1.4 and 1.6 to prove Theorem 1.1. Let u 0 ∈ S. We denote by u the solution of (1.1). Let µ > 0. For t ∈ R we set

u µ (t) = 1 R+ (t)u(t)e -tµ .
Then for τ ∈ R we set ǔµ (τ ) = R e itτ u µ (t) dt = +∞ 0 e it(τ +iµ) u(t) dt, (6.1) so that for all n ∈ N and τ ∈ R we have

ǔ(n) µ (τ ) = R (it) n e itτ u µ (t) dt. (6.2)
We multiply (1.1) by e it(τ +iµ) and integrate over R + . This yields

H -(τ + iµ) ǔµ (τ ) = -iu 0
and hence, for all n ∈ N:

ǔ(n) µ (τ ) = -i n! R n+1 (τ + iµ)u 0 . (6.3) Lemma 6.1. For all n ∈ N * and µ > 0 the map τ → R n+1 (τ + iµ)u 0 belongs to L 1 (R, L 2 (R d )).
Proof. Let χ 0 ∈ C ∞ 0 (R, [0, 1]) be equal to 1 on a neighborhood of 0. According to (1.11) the map τ → R n+1 (τ + iµ)u 0 is bounded, so it is enough to prove that τ → (1

-χ 0 )(τ )R n+1 (τ + iµ)u 0 belongs to L 1 (R). Let z ∈ C + . Using twice the identity R(z) = R(z)(H + 1) -1 z + 1 we get R(z)u 0 = 1 (z + 1) 2 R(z)(H + 1) 2 u 0 - 1 (z + 1) 2 (H + 1)u 0 - 1 z + 1 u 0 .
The result follows after at least one differentiation with respect to z.

This lemma does not provide any uniform estimate, but now we can take the Fourier transform of (6.2). With (6.3) this gives for all t 0: (it) n e -tµ u(t) = -in! 2π τ ∈R e -itτ R n+1 (τ + iµ)u 0 dτ. (6.4)

We consider χ -, χ 0 , χ ∈ C ∞ (R, [0, 1]) such that χ -is supported in ] -∞, 0[, χ 0 is compactly supported and equal to 1 on a neighborhood of 0, χ is compactly supported in ]0, +∞[ and

χ -+ χ 0 + j∈N * χ j = 1 on R,
where for j ∈ N * and τ ∈ R we have set χ j (τ ) = χ(τ /2 j-1 ). We set χ + = j∈N * χ j . Starting from (6.4) applied with n = κ -1 (κ was defined in (1.9)) we can write

u µ (t) = - in! 2π(it) κ-1 v -(t) + v 0 (t) + v + (t) (6.5)
where for * ∈ {-, 0, +} we have set

v * (t) = τ ∈R χ * (τ )e -itτ R κ (τ + iµ)u 0 dτ. (6.6)
To simplify the notation we forget the dependance in µ. From now on, all the quantities depend on µ > 0 but the estimates are uniform in µ. Proposition 6.2. Let k ∈ N. There exists C 0 which does not depend on u 0 ∈ S such that for all µ > 0 and t 0 we have

v -(t) L 2 C t -k u 0 L 2 .
This implies that the corresponding contribution for u(t) decays like any power of t in L 2 .

Proof. After k partial integrations in (6.6) we get

(it) k v -(t) = R e -itτ d k dτ k χ -(τ )R(τ + iµ) κ u 0 dτ. According to Remark 2.1 we have d k dτ k χ -(τ )R(τ + iµ) κ L(L 2 ) τ -(κ+k) ,
and the result follows.

We now deal with v 0 . The following result is (a slightly modified version of) Lemma 4.3 in [BR14]: Lemma 6.3. Let H be a Hilbert space. Let f ∈ C 1 (R * , H) be equal to 0 outside a compact subset of R. Assume that for some γ ∈]0, 1[ and M f 0 we have

∀τ ∈ R * , f (τ ) H M f |τ | -γ and f ′ (τ ) H M f |τ | -1-γ .
Let β ∈ [0, 1[. Then there exists C 0 which does not depend on f and such that for all t ∈ R we have

f (t) H C M f t β(γ-1) . Proof. Following the proof of [BR14] we set f t (τ ) = 1 -1 f τ -s t ds where φ ∈ C ∞ 0 (] -1, 1[, R) satisfies R φ = 1 and we write for |t| 1 f (t) |τ | t -β f (τ ) dτ + |τ | t -β f (τ ) -f t (τ ) dτ + |τ | t -β e -itτ f t (τ ) dτ |t| β(1-γ) + |t| γβ-1 + 1 t f t (t -β ) + f t (-t -β ) + |τ | t -β e -itτ f ′ t (τ ) dτ |t| β(1-γ) .
We omit the details.

Proposition 6.4. Let ε ∈ 0, 1 2 and δ > κ + 1 2 . Then there exists C 0 which does not depend on u 0 ∈ S and such that for all µ > 0 and t 0 we have

v 0 (t) L 2,-δ t κ-1-d 2 +ε u 0 L 2,δ .
Proof. According to Theorem 1.4 applied with ε/2 instead of ε and Theorem 1.3 there exists C 0 (which does not depend on u 0 ) such that for µ > 0, τ ∈ R and z = τ + iµ we have

χ 0 (τ )R κ (z)u 0 L 2,-δ C |z| d 2 -κ-ε 2 u 0 L 2,δ and d dτ χ 0 (τ )R κ (z) u 0 L 2,-δ C |z| d 2 -κ-1-ε 2 u 0 L 2,δ .
Then the statement follows from Lemma 6.3 applied with β ∈]0, 1[ so close to 1 that

β κ - d 2 -1 + ε 2 κ - d 2 -1 + ε.
To finish the proof of Theorem 1.1 we have to estimate v + (t). As for v -(t) above, k partial integrations yield

(it) k v + (t) = R e -itτ k j=1 C j k χ (j) + (τ )R κ+k-j (τ + iµ)u 0 dτ + R e -itτ χ + (τ )R κ+k (τ + iµ)u 0 dτ =: v 0 +,k (t) + w k (t)
The following proposition proves that the contribution of v + (t) in (6.5) decays like any power of t. However there may be a loss of two derivatives when α = 0 if the non-trapping assumption does not hold. Proposition 6.5. Let k ∈ N * and δ > κ + k -1 2 . Let σ ∈ [0, 2]. (i) There exists C 0 which does not depend on u 0 and such that for all µ > 0 and t 1 we have

x -δ v 0 +,k (t) L 2 C u 0 L 2,δ .
(ii) Assume that the non-trapping assumption (1.12) holds or that we have the damping condition (1.13) together with (κ + k)α + σ 2. Then there exists C 0 which does not depend on u 0 such that for all µ > 0 and t 1 we have

x -δ w k (t) L 2 C u 0 H σ,δ .
Proof. • Statement (i) follows from Theorem 1.3 and the fact that χ

+ is compactly supported in ]0, +∞[ for all j 1.

• Assume that α 1 (and hence α = 1) or that the non-trapping condition holds. Then according to Theorem 1.6 we have for τ ∈ supp(χ + )

x -δ R κ+k (τ + iµ) x -δ L(L 2 ) |τ | -κ+k 2 .
Since κ + k 3 this gives the second statement with σ = 0.

• Now assume that α ∈ [0, 1[. For j ∈ N * we set

w k,j (t) = τ ∈R χ j (τ )e -itτ R κ+k (τ + iµ)u 0 dτ. Let χ ∈ C ∞ 0 (R * + , [0, 1]) be equal to 1 on a neighborhood of supp χ. For τ ∈ R and j ∈ N * we set χj (τ ) = χ(τ /2 j-1 ). Let I k,j (t) = τ ∈R χ j (τ )e -itτ x -δ R κ+k (τ + iµ) x -δ dτ ∈ L(L 2 ).
We have x -δ w k,j (t) = w 1 k,j (t) + w 2 k,j (t) + w 3 k,j (t) where w 1 k,j (t) = χj (P )I k,j (t) χj (P ) x δ u 0 , w 2 k,j (t) = (1χj )(P )I k,j (t) χj (P ) x δ u 0 and w 3 k,j (t) = I k,j (t)(1χj )(P ) x δ u 0 . • By almost orthogonality, Theorem 1.6 and almost orthogonality again we have

j∈N * w 1 k,j (t) 2 j∈N * w 1 k,j (t) 2 sup j∈N * τ ∈R χ j (τ ) x -δ R κ+k (τ + iµ) x -δ dτ 2 × j∈N * χj (P ) x δ u 0 2 sup j∈N * 2 2j 2 -j(κ+k)α 2 -jσ P σ 2 x δ u 0 2 u 0 2 H σ,δ . It remains to prove that w 2 k,j (t) + w 3 k,j (t) 2 -j u 0 L 2,δ . (6.7)
• For the contribution of w 2 k,j (t) we prove that there exists C 0 such that for j ∈ N * and τ ∈ supp(χ j ) we have

(1 -χj )(P ) x -δ R κ+k (τ + iµ) x -δ L(L 2 ) C2 -2j . (6.8) Let ζ ∈ C ∞ 0 (R * + , [0, 1
]) be equal to 1 on a neighborhood of supp χ and such that χ = 1 on a neighborhood of supp ζ. For j ∈ N * and τ ∈ R we set ζ j (τ ) = ζ(τ /2 j-1 ). According to Theorem 8.7 in [START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF] about functions of a self-adjoint semiclassical pseudo-differential operator (with h = 2 -j-1

2 ) we see that for any M 0 we have

(1 -χj )(P ) x -δ ζ j (P ) x δ L(L 2 ) 2 -jM .
Since (1χj )(P ) is uniformly bounded, it is remains to prove

x -δ (1 -ζ j )(P )R κ+k (τ + iµ) x -δ L(L 2 ) 2 -2j .
We recall that for z ∈ C + we have set R 0 (z) = (Pz) -1 . By the resolvent identity we have

R κ+k (z) = R 0 (z)R κ+k-1 (z) + iR 0 (z)B α R κ+k (z) = R 0 (z) 2 R κ+k-2 (z) + iR 2 0 (z)B α R κ+k-1 (z) + iR 0 (z)B α R κ+k (z). For Re(z) ∈ supp(χ j ) we have x -δ (1 -ζ j )(P )R 2 0 (z)R κ+k-2 (z) x -δ x -δ (1 -ζ j )(P )R 2 0 (z) x δ x -δ R κ+k-2 (z) x -δ 2 -2j .
We have used the Spectral Theorem and pseudo-differential calculus to estimate the first factor, and Theorem 1.6 for the second. Similarly

x -δ (1 -ζ j )(P )R 2 0 (z)B α R(z) κ+k-1 x -δ x -δ (1 -ζ j )(P )R 2 0 (z)B α x δ x -δ R(z) κ+k-1 x -δ 2 j(-2+ α 2 )-αj 2 (κ+k-1) 2 -2j .
For the last term we have to prove

x -δ (1ζ j )(P )R 0 (z)B α R κ+k (z) x -δ 2 -2j . (6.9)

We proceed as above. We consider φ ∈ C ∞ 0 (R * + , [0, 1]) equal to 1 on a neighborhood of supp(χ) and such that ζ = 1 on a neighborhood of supp(φ), and then we set φ j = φ(•/2 j-1 ) for j ∈ N * . We write

x -δ (1ζ j )(P )R 0 (z)B α R κ+k (z) x -δ

x -δ (1ζ j )(P )R 0 (z)B α φ j (P )

x δ x -δ R κ+k (z) x -δ + x -δ (1 -ζ j )(P )R 0 (z)B α x δ x -δ (1 -φ j )(P )R κ+k (z) x -δ 2 -jM + 2 j( α 2 -1)
x -δ (1φ j )(P )R κ+k (z) x -δ , for any M 0. Then we use again the resolvent identity

x -δ (1 -φ j )(P )R κ+k (z) x -δ = x -δ (1 -φ j )(P )R 0 (z)R κ+k-1 (z) x -δ + x -δ (1 -φ j )(P )R 0 (z)B α R κ+k (z) x -δ
and we conclude as above. We obtain (6.9), then (6.8), and finally the contribution of w 2 k,j in (6.7) after integration over supp(χ j ). The contribution of w 3 k,j is estimated similarly. Then it remains to sum over j ∈ N * to conclude the proof of the proposition and hence the proof of Theorem 1.1.

Smoothing effect

In this section we prove Theorem 1.7. With Theorems 1.3, 1.5 and 1.6 it implies Theorem 1.2. For this we use a dissipative version of the theory of relatively smooth operators in the sense of Kato.

Proposition 7.1. Under the assumption of Theorem 1.7 there exists C 0 such that for all z ∈ C + we have x -1

1 + x -1 P γ 4 R(i) x x -1 R(z) x -1
x R(i) P γ 4

x -1 .

By pseudo-differential calculus the operators P is also bounded on L 2 uniformly in z with |z| 1, and we have similar estimates for the adjoint operators of (7.1) and (7.2). Thus

x -1 P γ 4 χ z (P )R(z)χ z (P ) P γ 4

x -1 |z| γ 2

x -1 |z| -1 P γ 4 χ z (P )R(z)χ z (P ) |z| -1 P γ 4

x -1 |z| γ 2

x -1 R(z) x -1

1.

• With R 0 (z) = (Pz) -1 we have the resolvent identity R(z) = R 0 (z) + iR(z)B α R 0 (z).

We have

x -1 P γ 4 χ z (P )R 0 (z)(1χ z )(P ) P γ 4

x -1

x -1 P γ 4 χ z (P ) R 0 (z)(1χ z )(P ) P We have estimated the first factor as above and the second by the Spectral Theorem. On the other hand, since the operator √ B α P -1 2 is bounded we also have by Proposition 2.5

x -1 P γ 4 χ z (P )R(z)B α R 0 (z)(1χ z )(P ) P γ 4

x -1

x -1 P γ 4 χ z (P ) x x -1 R(z) B α P This proves that x -1 P γ 4 χ z (P )R(z)(1χ z )(P ) P γ 4

x -1

1.

• The operator x -1 P γ 4 (1χ z )(P )R(z)χ z (P ) P γ 4

x is estimated similarly. Finally for x -1 P γ 4 (1χ z )(P )R(z)(1χ z )(P ) P γ 4

x -1

we only have to use twice the resolvent identity: R(z) = R 0 (z) + iR 0 (z)B α R 0 (z) -R 0 (z)B α R(z)B α R 0 (z).

Then we apply the same idea as above, using Corollary 2.6 to estimate √ B α R(z) √ B α . This concludes the proof.

Taking the adjoint in the estimate of proposition 7.1 we obtain the same estimate with R(z) replaced by R(z) * = (P + iB αz) -1 (the same is true for the estimates of Theorems 1.3, 1.5 and 1.6). In particular we obtain the following result: Corollary 7.2. Then there exists C 0 such that for all z ∈ C + and ϕ ∈ S we have

(H -z) -1 -(H * -z) -1 P γ 4
x -1 ϕ, P γ 4

x -1 ϕ

L 2 C ϕ 2 L 2 .
It is known that such an estimate on the resolvent implies Theorem 1.7. This comes from the dissipative version of the theory of relatively smooth operators. The self-adjoint theory can be found in [START_REF] Reed | Method of Modern Mathematical Physics, volume IV, Analysis of Operator[END_REF]§XIII.7]. The dissipative version uses the theory of self-adjoint dilations for a dissipative operator described in [START_REF] Nagy | Harmonic analysis of operators on Hilbert spaces[END_REF]. All this has been combined in Proposition 6.2 in [Roy], according to which Theorem 1.7 follows.

Moez

  T and S we set ad 0 T (S) = S, ad T (S) = ad 1 T (S) = [S, T ] and then for m 2: ad m T (S) = [ad m-1 T (S), S]. For µ = (µ 1 , . . . , µ d ) ∈ N d we set ad µ x := ad µ1 x1 . . . ad µ d

  Proof. • Let K be a compact subset of C. Using the resolvent identityR(z) = R(i) + (zi)R(i) 2 + (zi) 2 R(i)R(z)R(i),we obtain for z ∈ C + ∩ K x

γ 4 R

 4 (i) and R(i) P γ 4 are bounded on L 2,-1 and L 2,1 , respectively. For the second factor in the right-hand side we use (1.14), and the conclusion follows for z ∈ C + ∩ K. • It remains to prove the result for |z| ≫ 1. Let χ ∈ C ∞ 0 (R, [0, 1]) be supported on [-3,3] and equal to 1 on [-2,2]. For z ∈ C + we define χ z : λ → χ(λ/ |z|). The operator ε γ 4 P γ 4 εP -γ 4 is a pseudo-differential operator whose symbol has bounded derivatives uniformly in ε ∈]0, 1], so the operator |z| bounded operator on L 2 uniformly in z with |z| 1. The operator x

  KHENISSI École Supérieure des Sciences et de Technologie de Hammam Sousse Rue Lamine El Abbessi 4011 Hammam Sousse Tunisia moez.khenissi@fsg.rnu.tn

	Julien ROYER
	Institut	de	mathématiques	de
	Toulouse
	118, route de Narbonne
	31062 Toulouse Cédex 9
	France
	julien.royer@math.univ-toulouse.fr


Jérémy Michaud

Pamela Del Vecchio

Laurent Béchou

David Veyrié

Mauro Andrea Bettiati

Francois Laruelle

Stéphane Grauby

Precise Facet Temperature Distribution of High-Power Laser Diodes: Unpumped Window Effect

Keywords: Catastrophic optical degradation, facet temperature variation, laser diode, unpumped window

A thermoreflectance technique is used to evaluate the temperature variations at the output facet of high-power GaAs-based laser diodes emitting at 980 nm. Two kinds of diodes with different unpumped windows (UPWs) are studied to determine the influence of UPW length on the temperature variation. We show that in the vicinity of the active region, where a catastrophic optical damage is most susceptible to occur, the short UPW diode heats much more (up to 40%) than the long UPW one.

I. INTRODUCTION

F ACET heating, mainly due to laser light absorption and surface carrier recombination, plays a major role concerning laser diode reliability [START_REF] Fukuda | Degradation behavior of 0.98-μm strained quantum well InGaAs/AlGaAs lasers under high-power operation[END_REF]. The heating reduction, in particular near the active region, can increase the current threshold of catastrophic optical degradation (COD) and consequently the maximum output optical power. Introducing an unpumped window (UPW) architecture is one of the possible strategies to reduce the facet temperature and hence fabricate highly reliable devices with high power operation [START_REF] Horie | Thermal rollover characteristics up to 150 °C of buried-stripe type 980-nm laser diodes with a current injection window delineated by a SiN x layer[END_REF], [START_REF] Rinner | Facet temperature reduction by a current blocking layer at the front facets of high-power InGaAs/AlGaAs lasers[END_REF]. The UPW consists in a region located at the diode front facet where current injection is reduced.

In this letter, we study 980nm emitting laser diodes which constitute performing systems in terms of output power, conversion efficiency and reliability. Two kinds of laser diodes are compared, considering either a short or a long UPW. Very few studies of the UPW impact on the facet temperature are reported in literature and most of them are simulations [START_REF] Liu | Thermal management strategies for high power semiconductor pump lasers[END_REF] or measurements of the ridge temperature [START_REF] Raad | Thermoreflectance measurements for optically emitting devices[END_REF]. μ-Raman facet measurements [START_REF] Hempel | High singlespatial-mode pulsed power from 980 nm emitting diode lasers[END_REF] can also be found but with large uncertainties and long acquisition times, which prevent this This work was supported by the Centre National d'Etudes Spatiales. J. Michaud and S. Grauby are with the Université de Bordeaux, Laboratoire Ondes et Matière d'Aquitaine, Talence 33400, France (e-mail: jmichaud25@gmail.com; stephane.grauby@u-bordeaux.fr). P. Del Vecchio, M. A. Bettiati, and F. Laruelle are with 3S Photonics, Nozay 91625, France (e-mail: pamela.delvecchio@ims-bordeaux.fr; mbettiati@3spgroup.com; flaruelle@3spgroup.com).

L. Béchou is with the Intégration du Matériau au Système, Université de Bordeaux, Talence 33405, France (e-mail: laurent.bechou@ims-bordeaux.fr). D. Veyrié is with the Centre National d'Études Spatiales, Toulouse 31400, France (e-mail: david.veyrie@cnes.fr). technique to be used for temperature mapping. Then, we use thermoreflectance (TR), which is a fast and very accurate technique, to compare the temperature variation for both structures in different locations of the output facet with a micrometric spatial resolution. Hence, for the first time, we can detail the influence of the UPW length on the temperature both far from the active region and close to it, where a COD is most likely to occur.

II. EXPERIMENTAL SET UP AND DEVICES UNDER TEST

The devices under test (fig. 1) are single AlGaAs/InGaAs quantum well laser diodes emitting at 980nm used for amplification in optical communication transmission systems [START_REF] Bettiati | High optical strength GaAs-based laser structures[END_REF]. The 3.9mm long ridge structure, covered by a gold layer, allows to confine the current and the light emission in the quantum well of the active region at the p-n junction.

We studied two kinds of devices, differing by their UPW length, either short (noted S-UPW) or long (noted L-UPW). The S-UPW is much smaller than the 30μm long UPW studied in [START_REF] Rinner | Facet temperature reduction by a current blocking layer at the front facets of high-power InGaAs/AlGaAs lasers[END_REF] whereas the L-UPW has been lengthened with a diode power reduction limited to 1% at maximum. The diodes are biased using a ITC510 current source, to adjust the operating point, connected with a 33250A voltage generator to modulate the driving current which is a 50% duty cycle square current (f=442Hz) varying from 0.1mA to I max ; I max can be adjusted from 6mA to 1A. This induces a temperature variation T, hence a reflectivity relative variation R/R:

R R = 1 R ∂ R ∂ T T = κ T ( 1
)
where R is the sample mean reflectivity and κ is the thermoreflectance coefficient mainly depending on the nature of the material and on the probe light wavelength. The total reflectivity can be written as:

R = R 0 + R (2)
where R 0 is the reflectivity signal at room temperature and R is the reflectivity variation due to temperature variation:

R = R 0 + R f cos(2π f t + ϕ) + higher frequency terms (3)
with R 0 the DC reflectivity variation due to the DC dissipated power, R f the reflectivity variation due to the dissipated power at frequency f, ϕ the thermal phase shift at frequency f. "higher frequency terms" includes all the reflectivity variations due to the dissipated power at harmonics 2f, 3f… Since the power is mainly dissipated at frequency f, we measure R f /R to evaluate the T f temperature variation at frequency f. Among thermal mapping methods, thermoreflectance [START_REF] Dilhaire | Thermoreflectance calibration procedure on a laser diode: Application to catastrophic optical facet damage analysis[END_REF], [START_REF] Epperlein | Mapping of local temperatures on mirrors of GaAs/AlGaAs laser diodes[END_REF] is an accurate, spatially resolved, non-contact and non-invasive method. The reflectivity variation implies an intensity variation of the light reflected to the photodetector. Thus, measuring the relative variation of detector photocurrent I/I, we deduce R f /R and then T f if κ is known. The set-up is presented in fig. 2. The probe laser beam first crosses a Faraday isolator coupled to a half wave plate to eliminate a possible probe laser signal coming back into the laser cavity and to adjust the laser intensity. Then, it is separated into a reference beam and a probe beam using a polarizing beam splitter (PBS). The probe beam is then focused on the sample surface through a ×40 magnitude microscope objective. After reflection, the probe beam is transmitted towards the detection arm by a second PBS and focused on the differential detector. A removable mirror sends the reflected probe and the light coming from a white lamp to a CCD camera to visualize the probe position on the sample surface. A dichroic mirror prevents the diode signal from back reflections inside the probe laser cavity. In addition, a 980nm notch filter and two 632nm interferential filters are used so that only the probe wavelength could be transmitted to the differential detector. This detector eliminates the DC component dominating the reflected signal, the reference signal input being adjusted by a half wave plate to compensate the DC reflected signal. Then the differential signal is measured by a lock-in amplifier locked at f to extract R f . We also measure the total reflectivity R to deduce R f /R = κ T f . The differential detector also enables to avoid a possible lock-in amplifier input saturation and to eliminate possible experimental noise sources affecting both reference and reflected beams. Subsequently, R f /R is simply noted R/R. Since κ is wavelength dependent, we had first compared the TR signal measured on the diode facet for different laser probe wavelengths, namely 408, 532, 632 and 835nm. The maximum signal had been obtained for the 632nm wavelength. Subsequently, a He-Ne laser was chosen as a probe for all the measurements presented. The spot diameter was evaluated to 3μm approximately and the R/R set-up sensitivity was measured to be as low as 2×10 -6 .

III. TEMPERATURE VARIATION MEASUREMENTS

The measurements are done on 3 laser diodes of each type, and for each one, they are repeated three times. The results are the mean values obtained on each kind of diodes. To evaluate the UPW length influence on the output facet temperature variation, we first measured the TR signal along the y axis when I max =800mA (fig. 3). The first point (taken as y=0) is just under the emitting source towards the AlN heat sink. The displacement step is 2μm on the first 10μm and then 5μm between y=10μm and 80μm. Close to the source (from y=0 to 5μm), the S-UPW diode is heating more than the L-UPW one, with a difference reaching almost 40% for y=0. Then, beyond 5μm, the L-UPW diode heats more than the S-UPW one from 20% at y=6μm down to less than 10% at 80μm. The maximum R/R measured at y=0 was (4.160×10 -3 ±2×10 -6 ) and (2.680×10 -3 ±2×10 -6 ) for the S-UPW and L-UPW diodes respectively. The uncertainty corresponds to the repeatability error. Other measurements, made along the black horizontal full line, for y=5μm and between x=-20μm and x=20μm, confirmed that on the active region and in its close vicinity (closer than 5 μm), the S-UPW laser diode heats more by 20% than the L-UPW laser diode, in accordance with simulation results [START_REF] Liu | Thermal management strategies for high power semiconductor pump lasers[END_REF]. On the contrary, when moving away from the active zone, the S-UPW diode signal becomes lower than the L-UPW diode one with a 10% difference. The thermal power seems then to be dissipated differently: for the S-UPW diodes, the heating is more concentrated on the active region whereas, for the L-UPW diodes, it spreads on the whole facet, being more homogeneously distributed.

Then, we particularly focused on two particular positions: one close (10μm) to the emitting source (M 1 ) and one 60μm far from it (M 2 ) as indicated in fig. 1. For each position, we measured the TR signal as a function of the current, I max varying from 6mA to 800mA with a 3mA step (fig. 4). We distinguish three zones. The first one (zone A), under the threshold, has a linear behavior with a slope α 1 . Then, between the threshold and a transition current I t (zone B), the behavior is still linear but with a different slope α 2 . Finally, in the last zone C beyond I t , the behavior appears non-linear. This can result in a fast and strong temperature increase and then in a COD at very high current.

From a thermal point of view, three contributions can be considered in the facet temperature variation [START_REF] Dilhaire | Laser diode COFD analysis by thermoreflectance microscopy[END_REF]. The first one is due to the conversion of the electrical power into heat and is the only active contribution under the threshold. Starting above the laser threshold, part of the input electrical power is extracted as optical output thus reducing the temperature slope: this is the second contribution. The last contribution comes from the surface heat source which creates a new temperature increase and can lead to a huge thermal runaway rapidly inducing a COD since it is concentrated on the emission area and its amplitude strongly increases with the current. This last contribution is then negligible at low current (A and B zones) and far from the active region (at M 2 ). But when approaching the active region at M 1 , it becomes dominant, even at relatively low current (zone B), and can compensate the second contribution, finally leading to a temperature increase in zone B. The α 2 /α 1 ratio is then an indicator of the facet heating.

For position M 2 , as expected, α 2 /α 1 is smaller than 1 since a part of the electrical power injected is converted into light and hence the proportion of the electrical power converted into heat decreases. Then, in zone C, the TR signal increases faster because of the surface source whose amplitude strongly increases with the supplied current. At this position, the L-UPW diode heats more than the S-UPW one, in agreement with fig. 3. For position M 1 , the L-UPW diode still heats more than the S-UPW one. When moving from M 2 to M 1 , the α 2 /α 1 ratio increases from 0.54 to 0.7 for the S-UPW diode and from 0.67 to 1.18 for the L-UPW one. Hence, for both kinds of diodes, when approaching the emitting zone, the surface source contribution increases but, in the last case (L-UPW at position M 1 ), the surface source contribution is clearly dominating, which is consistent with the previous results showing that beyond 5μm from the source, the L-UPW diode heats more than the other one.

To determine which UPW will be more sensitive to COD, the non-linear behavior needs to be analyzed in detail since COD will occur at very high current well beyond 800mA, i.e. far away in the non-linear zone C. The I t current, indicating the onset of the non-linear behavior (which leads to thermal runaway), is evaluated to 500mA for the L-UPW diode but only 400mA for the S-UPW diode (fig. 4), hence 20% lower. In addition, we fitted the curves by a polynomial fit in the I t -0.8A range and extrapolated them at very high current up to 2.75A (inset in fig. 4). We then observe that the non-linearity is higher for the S-UPW: the quadratic term is 4.2×10 -4 and 3.3×10 -4 for the S-UPW and L-UPW respectively. As a result, for very high current beyond 2.16A, the S-UPW diode heats more than the L-UPW counterpart. The thermal runaway will then be triggered at lower current. We confirm it measuring the optical power as a function of the current until the COD arises. It occurred at 2.65A for the S-UPW diode whereas for the L-UPW one, the COD was not yet reached at 3A.

All the measurements lead to the conclusion that the L-UPW diode heats a little bit more than the S-UPW one for low currents and far from the source (beyond 5μm from it). However, the temperature variation is much higher (up to 40%) for the S-UPW diodes in the source close vicinity and at high currents. In addition, for currents above I t , the temperature variation becomes non-linear for both kinds of diodes but it increases more rapidly for the short UPW diode. These S-UPW diodes may then be less reliable at very high current and should be more sensitive to COD.

Other papers have studied the influence of unpumped windows on laser diode reliability, either by simulation [START_REF] Liu | Thermal management strategies for high power semiconductor pump lasers[END_REF] or experimentally [START_REF] Rinner | Facet temperature reduction by a current blocking layer at the front facets of high-power InGaAs/AlGaAs lasers[END_REF], [START_REF] Ohgoh | Technologies and applications of Al-free high-power laser diodes[END_REF]. In [START_REF] Ohgoh | Technologies and applications of Al-free high-power laser diodes[END_REF], Ohgog and al studied the effect of a current blocking region on the light output power and on the driving current. In [START_REF] Rinner | Facet temperature reduction by a current blocking layer at the front facets of high-power InGaAs/AlGaAs lasers[END_REF], Rinner uses μ-Raman spectroscopy to evaluate the facet and averaged waveguide temperatures. It is quite difficult to compare their results with ours because, concerning Ohgoh, the laser diode structure he studied is very different. As for Rinner, he shows a reduction of the facet overheating due to the unpumped window much larger than the one we have measured. One of the possible explanations might be that the measurements are not performed at the same position since we have measured under the emitting zone while he has investigated directly on it. Moreover, μ-Raman measurements reported large uncertainties (up to ±50%) [START_REF] Rinner | Facet temperature reduction by a current blocking layer at the front facets of high-power InGaAs/AlGaAs lasers[END_REF], [START_REF] Hempel | High singlespatial-mode pulsed power from 980 nm emitting diode lasers[END_REF]. Nevertheless, all these studies lead to the same conclusion that a long unpumped window improves the laser diode reliability, especially by increasing the COD threshold. Our study confirms these results but also shows for the first time and with a very good accuracy, how the thermoreflectance signal, hence the temperature variation, is spatially distributed on the output facet.

Fig. 1 .

 1 Fig. 1. Generic facet view of the laser diode.

Fig. 2 .

 2 Fig. 2. Thermoreflectance set-up.

Fig. 3 .

 3 Fig. 3. R/R signal along the vertical y-axis from source to base at 800mA.

Fig. 4 .

 4 Fig. 4. R/R signal for two positions (M 1 and M 2 ). Zones B and C correspond to the short UPW diode. Dashed lines correspond to the linear fit between I th and I t . Inset: extrapolation for high currents.