Nalini Anantharaman
email: nalini.anantharaman@math.u-psud.fr

AND Clotilde Fermanian-Kammerer

Fabricio Macià
email: fabricio.macia@upm.es

SEMICLASSICAL COMPLETELY INTEGRABLE SYSTEMS : LONG-TIME DYNAMICS AND OBSERVABILITY VIA TWO-MICROLOCAL WIGNER MEASURES

We look at the long-time behaviour of solutions to a semi-classical Schrödinger equation on the torus. We consider time scales which go to infinity when the semi-classical parameter goes to zero and we associate with each time-scale the set of semi-classical measures associated with all possible choices of initial data. On each classical invariant torus, the structure of semi-classical measures is described in terms of two-microlocal measures, obeying explicit propagation laws.

We apply this construction in two directions. We first analyse the regularity of semiclassical measures, and we emphasize the existence of a threshold : for time-scales below this threshold, the set of semi-classical measures contains measures which are singular with respect to Lebesgue measure in the "position" variable, while at (and beyond) the threshold, all the semi-classical measures are absolutely continuous in the "position" variable, reflecting the dispersive properties of the equation. Second, the techniques of twomicrolocal analysis introduced in the paper are used to prove semiclassical observability estimates. The results apply as well to general quantum completely integrable systems.

1. Introduction 1.1. The Schrödinger equation in the large time and high frequency régime. This article is concerned with the dynamics of the linear equation [START_REF] Aïssiou | Semiclassical limits of eigenfunctions on flat n-dimensional tori[END_REF] ih∂ t ψ h (t, x) = (H(hD x ) + h 2 V h (t)) ψ h (t, x) , (t, x) ∈ R × T d , ψ h|t=0 = u h , on the torus T d := (R/2πZ) d , with H a smooth, real-valued function on (R d ) * (the dual of R d ), and h > 0. In other words, H is a function on the cotangent bundle T * T d = T d ×(R d ) * that does not depend on the d first variables, and thus gives rise to a completely integrable Hamiltonian flow. For the sake of simplicity, we shall assume that H ∈ C ∞ R d . However the smoothness assumption on H can be relaxed to C k , where k large enough, in most results of this article. The lower order term V h (t) is a bounded self-adjoint operator (possibly depending on t and h). We assume that the map t →

V h (t) L(L 2 (T d )) is in L 1 loc (R) ∩ L ∞ (R)
, uniformly with respect to h. This condition ensures the existence of a semi-group associated with the operator H(hD x ) + h 2 V h (t) (see Appendice B in [START_REF] Derezinski | Scattering theory of classical and quantum N-particle systems[END_REF], Proposition B. 3.6).

N. Anantharaman wishes to acknowledge the support of Agence Nationale de la Recherche, under the grant ANR-09-JCJC-0099-01. F. Macià takes part into the visiting faculty program of ICMAT and is partially supported by grants MTM2010-16467 (MEC), ERC Starting Grant 277778.

We are interested in the simultaneous limits h → 0 + (high frequency limit) and t → +∞ (large time evolution). Our results give a description of the limits of sequences of "position densities" |ψ h (t h , x)| 2 at times t h that tend to infinity as h → 0 + . Remark 1.1. In future applications, it will be interesting to note as of now that we may allow H = H h to depend on the parameter h, in such a way that H h converges to some limit H 0 in the C k topology on compact sets, for k sufficiently large. For instance, we allow H h (ξ) = H(ξ + hω), where ω ∈ (R d ) * is a fixed vector.

To be more specific, let us denote by S h (t, s) the semigroup associated with the operator H(hD x ) + h 2 V h (t) and set S t h = S h (t, 0). Fix a time scale, that is, a function

τ : R * + -→ R * + h -→ τ h ,
such that lim inf h→0 + τ h > 0 (actually, we shall be mainly concerned in scales that go to +∞ as h → 0 + ). Consider a family of initial conditions (u h ), normalised in L 2 (T d ): u h L 2 (T d ) = 1 for h > 0, and h-oscillating in the terminology of [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch[END_REF][START_REF] Gérard | Ergodic properties of eigenfunctions for the Dirichlet problem[END_REF], i.e.:

(2) lim sup

h→0 + 1 [R,+∞[ -h 2 ∆ u h L 2 (T d ) -→ R-→∞ 0,
where 1 [R,+∞[ is the characteristic function of the interval [R, +∞[. Our main object of interest is the density |S t h u h | 2 , and we introduce the probability measures on T d :

ν h (t, dx) := S t h u h (x) 2 dx;
the unitary character of S t h implies that ν h ∈ C R; P T d (in what follows, P (X) stands for the set of probability measures on a Polish space X).

To study the long-time behavior of the dynamics, we rescale time by τ h and look at the time-scaled probability densities: [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] ν h (τ h t, dx) .

When t = 0 is fixed and τ h grows too rapidly, it is a notoriously difficult problem to obtain a description of the limit points (in the weak- * topology) of these probability measures as h → 0 + , for rich enough families of initial data u h . See for instance [START_REF] Schubert | Semiclassical wave propagation for large times[END_REF][START_REF] Paul | Semiclassical approximation and noncommutative geometry[END_REF] in the case where the underlying classical dynamics is chaotic, the u h are a family of lagrangian states, and τ h = h -2+ǫ . In completely integrable situations, such as the one we consider here, the problem is of a different nature, but rapidly leads to intricate number theoretical issues [START_REF] Marklof | Pair correlation densities of inhomogeneous quadratic forms[END_REF][START_REF] Marklof | Pair correlation densities of inhomogeneous quadratic forms[END_REF][START_REF] Marklof | Mean square value of exponential sums related to the representation of integers as sums of squares[END_REF].

We soften the problem by considering the family of probability measures (3) as elements of L ∞ R; P T d . Our goal will be to give a precise description of the set M (τ ) of their accumulation points in the weak- * topology for L ∞ R; P T d , obtained as (u h ) varies among all possible sequences of initial data h-oscillating and normalised in L 2 T d .

The compactness of T d ensures that M (τ ) is non-empty. Having ν ∈ M (τ ) is equivalent to the existence of a sequence (h n ) going to 0 and of a normalised, h n -oscillating sequence (u hn ) in L 2 T d such that: for every real numbers a < b and every χ ∈ C T d . In other words, we are averaging the densities |S t h u h (x)| 2 over time intervals of size τ h . This averaging, as we shall see, makes the study more tractable.

If case (4) occurs, we shall say that ν is obtained through the sequence (u hn ). To simplify the notation, when no confusion can arise, we shall simply write that h -→ 0 + to mean that we are considering a discrete sequence h n going to 0 + , and we shall denote by (u h ) (instead of (u hn )) the corresponding family of functions.

Remark 1.2. When the function τ is bounded, the convergence of ν h (τ h t, •) to an accumulation point ν (t, •) is locally uniform in t. According to Egorov's theorem (see, for instance, [START_REF] Zworski | Semiclassical analysis[END_REF]), ν can be completely described in terms of semiclassical defect measures of the corresponding sequence of initial data (u h ), transported by the classical Hamiltonian flow φ s : T * T d -→ T * T d generated by H, which in this case is completely integrable :

(5) φ s (x, ξ) := (x + sdH(ξ), ξ).

As an example, take τ h = 1 and consider the case where the initial data u h are coherent states :

fix ρ ∈ C ∞ c R d with ρ L 2 (R d ) = 1, fix (x 0 , ξ 0 ) ∈ R d × R d
, and let u h (x) be the 2πZ d -periodization of the following coherent state:

1 h d/4 ρ x -x 0 √ h e i ξ 0 h •x .
Then ν h (t, •) converges, for every t ∈ R, to:

δ x 0 +tdH(ξ 0 ) (x) .
When the time scale τ h is unbounded, the t-dependence of elements ν ∈ M (τ ) is not described by such a simple propagation law. From now on we shall only consider the case where τ h -→ h-→0

+∞.

The problem of describing the elements in M (τ ) for some time scale (τ h ) is related to several aspects of the dynamics of the flow S t h such as dispersive effects and unique continuation. In [START_REF] Anantharaman | The dynamics of the Schrödinger flow from the point of view of semiclassical measures[END_REF][START_REF] Macià | The Schrödinger flow in a compact manifold: high-frequency dynamics and dispersion[END_REF] the reader will find a description of these issues in the case where the operator S t h is the semiclassical Schrödinger propagator e iht∆ corresponding to the Laplacian on an arbitrary compact Riemannian manifold. In that setting, the time scale τ h = 1/h appears in a natural way, since it transforms the semiclassical propagator into the non-scaled flow e ihτ h t∆ = e it∆ . The possible accumulation points of sequences of probability densities of the form |e it∆ u h | 2 depend on the nature of the dynamics of the geodesic flow. When the geodesic flow has the Anosov property (a very strong form of chaos, which holds on negatively curved manifolds), the results in [START_REF] Anantharaman | Dispersion and controllability for the Schrödinger equation on negatively curved manifolds[END_REF] rule out concentration on sets of small dimensions, by proving lower bounds on the Kolmogorov-Sinai entropy of semiclassical defect measures. Even in the apparently simpler case that the geodesic flow is completely integrable, different type of concentration phenomena may occur, depending on fine geometrical issues (compare the situation in Zoll manifolds [START_REF] Macià | Semiclassical measures and the Schrödinger flow on Riemannian manifolds[END_REF] and on flat tori [START_REF] Macià | High-frequency propagation for the Schrödinger equation on the torus[END_REF][START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF]).

1.2. Semiclassical defect measures. Our results are more naturally described in terms of Wigner distributions and semiclassical measures (these are the semiclassical version of the microlocal defect measures [START_REF] Gérard | Microlocal defect measures[END_REF][START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF], and have also been called microlocal lifts in the recent literature about quantum unique ergodicity, see for instance the celebrated paper [START_REF] Lindenstrauss | Invariant measures and arithmetic quantum unique ergodicity[END_REF]). The Wigner distribution associated to u h (at scale h) is a distribution on the cotangent bundle T * T d , defined by ( 6)

T * T d a(x, ξ)w h u h (dx, dξ) = u h , Op h (a)u h L 2 (T d ) , for all a ∈ C ∞ c (T * T d ),
where Op h (a) is the operator on L 2 (T d ) associated to a by the Weyl quantization. The reader not familiar with these objects can consult the appendix of this article or the book [START_REF] Zworski | Semiclassical analysis[END_REF]. For the moment, just recall that w h u h extends naturally to smooth functions χ on T * T d = T d × (R d ) * that depend only on the first coordinate, and in this case we have ( 7)

T * T d χ(x)w h u h (dx, dξ) = T d χ(x)|u h (x)| 2 dx.
The main object of our study will be the (time-scaled) Wigner distributions corresponding to solutions to [START_REF] Aïssiou | Semiclassical limits of eigenfunctions on flat n-dimensional tori[END_REF]:

w h (t, •) := w h S τ h t h u h
The map t -→ w h (t, •) belongs to L ∞ (R; D ′ T * T d ), and is uniformly bounded in that space as h -→ 0 + whenever (u h ) is normalised in L 2 T d . Thus, one can extract subsequences that converge in the weak- * topology on L ∞ (R; D ′ T * T d ). In other words, after possibly extracting a subsequence, we have R T * T d ϕ(t)a(x, ξ)w h (t, dx, dξ)dt -→ h-→0 R T * T d ϕ(t)a(x, ξ)µ(t, dx, dξ)dt for all ϕ ∈ L 1 (R) and a ∈ C ∞ c (T * T d ), and the limit µ belongs to L ∞ R; M + T * T d (here M + (X) denotes the set of positive Radon measures on a Polish space X).

The set of limit points thus obtained, as (u h ) varies among normalised sequences, will be denoted by M (τ ). We shall refer to its elements as (time-dependent) semiclassical measures.

Moreover, if (u h ) is h-oscillating (see [START_REF] Aïssiou | Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures[END_REF]), it follows that µ ∈ L ∞ R; P T * T d and identity [START_REF] Bony | Quantification asymptotique et microlocalisation d'ordre supérieure[END_REF] is also verified in the limit :

b a T d χ (x) |S τ h t h u h (x) | 2 dxdt -→ h-→0 b a T * T d χ (x) µ (t, dx, dξ) dt,
for every a < b and every χ ∈ C ∞ T d . Therefore, M (τ ) coincides with the set of projections onto x of semiclassical measures in M (τ ) corresponding to h-oscillating sequences [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch[END_REF][START_REF] Gérard | Ergodic properties of eigenfunctions for the Dirichlet problem[END_REF].

It is also shown in the appendix that the elements of M (τ ) are measures that are Hinvariant, by which we mean that they are invariant under the action of the hamiltonian flow φ s defined in [START_REF] Anantharaman | Dispersion and controllability for the Schrödinger equation on negatively curved manifolds[END_REF].

1.3. Results on the regularity of semiclassical measures. The main results in this article are aimed at obtaining a precise description of the elements in M (τ ) (and, as a consequence, of those of M (τ )). We first present a regularity result which emphasises the critical character of the time scale τ h = 1/h in situations in which the Hessian of H is non-degenerate, definite (positive or negative).

Theorem 1.3. (1) If τ h ≪ 1/h then M (τ ) contains elements that are singular with respect to the Lebesgue measure dtdx. Actually, M (τ ) contains all measures invariant by the flow φ s defined in [START_REF] Anantharaman | Dispersion and controllability for the Schrödinger equation on negatively curved manifolds[END_REF].

(

) Suppose τ h ∼ 1/h or τ h ≫ 1/h. Assume that the Hessian d 2 H(ξ) is definite for all ξ. Then M (τ ) ⊆ L ∞ R; L 1 T d 2
, in other words the elements of M (τ ) are absolutely continuous with respect to dtdx.

The proof of (1) in Theorem 1.3 relies on the construction of examples, while the proof of ( 2) is based on the forthcoming Theorem 1.10, which contains a careful analysis of the case τ h = 1/h (see section 1.4). A comparison argument between different time-scales allows to treat the case τ h ≫ 1/h (see section 1.5).

Note also that the construction leading to Theorem 1.3 (2) also yields observability results : see section 7 below. Finally, we point out in Section 1.7 that Theorem 1.3 extends to general quantum completely integrable systems. An interesting and immediate by-product of Theorem 1.3 is the following corollary.

Corollary 1.4. Theorem 1.3(2) applies in particular when the data (u h ) are eigenfunctions of H(hD x ), and shows (assuming the Hessian of H is definite) that the weak limits of the probability measures |u h (x)| 2 dx are absolutely continuous.

Note that statement (2) of Theorem 1.3 has already been proved in the case H(ξ) = |ξ| 2 in [START_REF] Bourgain | Analysis results and problems related to lattice points on surfaces[END_REF] and [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] with different proofs (the proof in the second reference extends to the xdependent Hamiltonian |ξ| 2 + h 2 V (x)). However, the extension to more general H of the method in [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] is not straightforward, even in the case where H(ξ) = ξ • Aξ, where A is a symmetric linear map : (R d ) * -→ R d (i.e. the Hessian of H is constant), the difficulty arising when A has irrational coefficients.

Let us now comment on the assumptions of the theorem. We first want to emphasize that the conclusion of Theorem 1.3(2) may fail if the condition on the Hessian of H is not satisfied.

Example 1.5. Fix ω ∈ R d and take H (ξ) = ξ•ω and V h (t) = 0. Let µ 0 be an accumulation point in D ′ T * T d of the Wigner distributions w h u h defined in [START_REF] Bony | Second microlocalisation and propagation of singularities for semi-linear hyperbolic equations[END_REF], associated to the initial data (u h ). Let µ ∈ M (τ ) be the limit of

w h S τ h t h u h in L ∞ (R; D ′ T * T d ).
Then an application of Egorov's theorem (actually, a particularly simple adaptation of the proof of Theorem 4 in [START_REF] Macià | Semiclassical measures and the Schrödinger flow on Riemannian manifolds[END_REF]) gives the relation, valid for any time scale (τ h ) :

T * T d a (x, ξ) µ (t, dx, dξ) = T * T d a (x, ξ) µ 0 (dx, dξ) ,
for any a ∈ C ∞ c T * T d and a.e. t ∈ R. Here a stands for the average of a along the Hamiltonian flow φ s , that is in our case

a (x, ξ) = lim T →∞ 1 T T 0 a (x + sω, ξ) ds.
Hence, as soon as ω is resonant (in the sense of §2.1) and µ 0 = δ x 0 ⊗ δ ξ 0 for some (x 0 , ξ 0 ) ∈ T * T d , the measure µ will be singular with respect to dtdx.

It is also easy to provide counter-examples where the Hessian of H is non-degenerate, but not definite.

Example 1.6. On the two-dimensional torus T 2 , consider

H(ξ) = ξ 2 1 -ξ 2 2
, where ξ = (ξ 1 , ξ 2 ). Take for (u h (x 1 , x 2 )) the periodization of

1 (2πh) 1/2 ρ x 1 -x 2 h where ρ ∈ C ∞ c (R) satisfies ρ L 2 (R) = 1.
Then the functions u h are eigenfunctions of H(hD x ) for the eigenvalue 0 and the measures

|u h (x 1 , x 2 )| 2 dx 1 dx 2 obviously concentrate on the diagonal {x 1 = x 2 }.
Note however that in this example the system is isoenergetically degenerate at ξ = 0. Recall the definition of isoenergetic non-degeneracy : the Hamiltonian H is isoenergetically nondegenerate at ξ if for all η ∈ (R d ) * , and λ ∈ R,

dH(ξ) • η = 0 and d 2 H(ξ) • η = λdH(ξ) =⇒ (η, λ) = (0, 0).
Definiteness of the Hessian implies isoenergetic non-degeneracy at all ξ such that dH(ξ) = 0. In view of the previous example, one may wonder whether isoenergetic non-degeneracy is a sufficient assumption for our results. In Section 4.5 we give a sufficient set of assumptions for our results which is weaker than definiteness, but is not implied by isoenergetic nondegeneracy except in dimension d = 2. As a conclusion, isoenergetic non-degeneracy is sufficient for all our results in dimension d = 2, but not in dimensions d ≥ 3, as is finally shown by the following counter-example :

Example 1.7. Take d = 3. On (R 3 ) * consider H(ξ) = ξ 2 1 + ξ 2 2 -ξ 3 3 , and let u h (x 1 , x 2 , x 3 ) be the periodization of 1 (2πǫ) 1/2 ρ x 2 + x 3 ǫ e i αx 1 +x 2 +x 3 h , where ρ ∈ C ∞ c (R) , ρ L 2 (R) = 1
, and ǫ = ǫ(h) tends to 0 with ǫ(h) ≫ h. Note that u h is an eigenfunction of H(hD x ). The Wigner measures of (u h ) concentrate on the set {ξ 1 = α, ξ 2 = ξ 3 = 1} where the system is isoenergetically non-degenerate if α = 0. Its projection on T 3 is supported on the hyperplane {x 2 + x 3 = 0}.

In Section 5 we present an example communicated to us by J. Wunsch showing that absolute continuity of the elements of M (1/h) may fail in the presence of a subprincipal symbol of order h β with β ∈ (0, 2) even in the case H (ξ) = |ξ| 2 . We also show in Section 5 that absolute continuity may fail for the elements of M (1/h) when H (ξ) = |ξ| 2k , k ∈ N and k > 1; a situation where the Hessian is degenerate at ξ = 0.

We point out that Theorem 1.3(2) admits a microlocal refinement, which allows us to deal with more general Hamiltonians H whose Hessian is not necessarily definite at every ξ ∈ R d . Given µ ∈ M (τ ) we shall denote by μ the image of µ under the map

π 2 : (x, ξ) -→ ξ. For V h (t) = Op h (V (t, x, ξ)) with V ∈ C ∞ (R × T * T d ), it is shown in the appendix that μ does not depend on t if τ h ≪ h -2 : in this case we have μ = (π 2 ) * µ 0 ,
where the measure µ 0 is an accumulation point in D ′ T * T d of the sequence w h u h . For simplicity we restrict our attention to that case in the following theorem :

Theorem 1.8. Assume that V h (t) = Op h (V (t, •)) with V ∈ C ∞ (R × T * T d ) bounded.
Let µ ∈ M (1/h) and denote by µ ξ (t, •) the disintegration of µ(t, •) with respect to the variable ξ, i.e. for every θ ∈ L 1 (R) and every bounded measurable function f : R θ(t)

T d ×R d f (x, ξ)µ(t, dx, dξ)dt = R θ (t) R d T d f (x, ξ)µ ξ (t, dx) μ(dξ)dt.
Then for μ-almost every ξ where d 2 H(ξ) is definite, the measure µ ξ (t, •) is absolutely continuous.

Let us introduce the closed set

C H := ξ ∈ R d : d 2 H(ξ) is not definite .
The following consequence of Theorem 1.8 provides a refinement on Theorem 1.3 [START_REF] Aïssiou | Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures[END_REF], in which the global hypothesis on the Hessian of H is replaced by a hypothesis on the sequence of initial data.

Corollary 1.9. Suppose ν ∈ M (1/h) is obtained through an h-oscillating sequence (u h ) having a semiclassical measure µ 0 such that µ 0 T d × C H = 0. Then ν is absolutely continuous with respect to dtdx.

1.4. Second-microlocal structure of the semiclassical measures. Theorem 1.8 is a consequence of a more detailed result on the structure of the elements of M (1/h) on which we focus in this paragraph. We follow here the strategy of [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] that we adapt to a general Hamiltonian H(ξ). The proof relies on a decomposition of the measure associated with the primitive submodules of (Z d ) * . Before stating it, we must introduce some notation.

Recall that (R d ) * is the dual of R d . Later in the paper, we will sometimes identify both by working in the canonical basis of R d . We will denote by

(Z d ) * the lattice in (R d ) * defined by (Z d ) * = {ξ ∈ (R d ) * , ξ.n ∈ Z, ∀n ∈ Z d }. We call a submodule Λ ⊂ (Z d ) * primitive if Λ ∩ (Z d ) * = Λ (
here Λ denotes the linear subspace of (R d ) * spanned by Λ). Given such a submodule we define: (8)

I Λ := ξ ∈ (R d ) * : dH (ξ) • k = 0, ∀k ∈ Λ .
We note that I Λ \ C H is a smooth submanifold. We define also

L p T d , Λ for p ∈ [1, ∞] to be the subspace of L p T d consisting of the functions u such that u (k) = 0 if k ∈ (Z d ) * \ Λ (here u (k) stand for the Fourier coefficients of u). Given a ∈ C ∞ c T * T d and ξ ∈ R d , denote by a Λ (•, ξ) the orthogonal projection of a (•, ξ) on L 2 T d , Λ : (9) a Λ (x, ξ) = k∈Λ a k (ξ) e ikx (2π) d
Note that if a only has frequencies in Λ, then a Λ = a.

For ω in the torus Λ /Λ, we denote by

L 2 ω (R d , Λ) the subspace of L 2 loc (R d ) ∩ S ′ (R d ) formed by the functions whose Fourier transform is supported in Λ -ω. Each L 2 ω (R d , Λ) has a natural Hilbert space structure.
We denote by m a Λ (ξ) the operator acting on each L 2 ω R d , Λ by multiplication by a Λ (•, ξ).

(10) R θ (t) T * T d a (x, ξ) µ (t, dx, dξ) dt = Λ⊆Z d R θ (t) T d ×I Λ a (x, ξ) µ final Λ (t, dx, dξ) dt,
the sum being taken over all primitive submodules of (Z d ) * .

In addition, there exists a measure μΛ (t) on ( Λ /Λ) × I Λ and a measurable family {N Λ (t, ω, ξ)} t∈R,ω∈ Λ /Λ,ξ∈I Λ of non-negative, symmetric, trace-class operators acting on L 2 ω R d , Λ , such that the following holds:

(11)

T d ×I Λ a (x, ξ) µ final Λ (t, dx, dξ) = ( Λ /Λ)×I Λ Tr m a Λ (ξ) N Λ (t, ω, ξ) μΛ (t, dω, dξ).
(

) If V h (t) = Op h (V (t, •)) with V ∈ C ∞ (R × T * T d 2
), then μΛ does not depend on t, and N Λ (t, ω, ξ) depends continuously on t, and solves the Heisenberg equation labelled below as ( Heis Λ,ω,ξ ).

When the Hessian of H is definite, formula [START_REF] Burq | Control for Schrödinger operators on tori[END_REF] holds for every a ∈ C ∞ c T * T d and therefore completely describes µ.

Remark 1.11. The arguments in Section 6.1 of [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] show that Theorem 1.8 is a consequence of Theorem 1.10. Therefore, in this article only the proof of Theorem 1.10 will be presented. Theorem 1.10 has been proved for H (ξ) = |ξ| 2 in [START_REF] Macià | High-frequency propagation for the Schrödinger equation on the torus[END_REF] for d = 2 and in [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] for the x-dependent Hamiltonian |ξ| 2 + h 2 V (x) in arbitrary dimension (in these papers the parameter ω does not appear, and all measures have Dirac masses at ω = 0).

The measures µ final Λ in equation [START_REF] Burq | Control for Schrödinger operators on tori[END_REF] are obtained as the final step of an interative procedure that involves a process of successive microlocalizations along nested sequences of submanifolds in frequency space.

Theorem 1.10(2) allows to describe the dependence of µ on the parameter t. This is a subtle issue since, as was noticed in [START_REF] Macià | Semiclassical measures and the Schrödinger flow on Riemannian manifolds[END_REF][START_REF] Macià | High-frequency propagation for the Schrödinger equation on the torus[END_REF], the semiclassical measures of the sequence of initial data (u h ) do not determine uniquely the time dependent semiclassical measure µ. Thus, when

V h (t) = Op h (V (t, •)) with V ∈ C ∞ (R × T * T d ), the measure µ final Λ (t, dx, dξ)
is fully determined by the measures μΛ and the family of operators N Λ (0, ω, ξ), which are objects determined by the initial data (u h ). The N Λ (t, ω, ξ) are obtained from N Λ (0, ω, ξ) by propagation along a Heisenberg equation (Heis Λ,ω,ξ ), written in Theorem 3.2, which is the evolution equation of operators that comes from the following Schrödinger equation in

L 2 ω (R d , Λ) : (S Λ,ω,ξ ) i∂ t v = 1 2 d 2 H(ξ)D y • D y + V (•, ξ) Λ v.
This process gives an explicit construction of µ in terms of the initial data. Full details on the structure of these objects are provided in Sections 3 and 4. Theorem 1.10 is stated for the time scale τ h = 1/h; if τ h ≪ 1/h, the elements of M (τ ) can also be described by a similar result (see Section 4.3) involving expression [START_REF] Burq | Control for Schrödinger operators on tori[END_REF]. However, in that case, the propagation law involves classical transport rather than propagation along a Schrödinger flow, and as a result Theorem 1.3(2) does not hold for τ h ≪ 1/h.

Second microlocalisation has been used in the 80's for studying propagation of singularities (see [START_REF] Bony | Second microlocalisation and propagation of singularities for semi-linear hyperbolic equations[END_REF][START_REF] Bony | Quantification asymptotique et microlocalisation d'ordre supérieure[END_REF][START_REF] Delort | transformation, Second microlocalization and semilinear caustics[END_REF][START_REF] Lebeau | Deuxième microlocalisation sur les sous-variétés isotropes[END_REF]). The two-microlocal construction performed here is in the spirit of that done in [START_REF] Nier | A semi-classical picture of quantum scattering[END_REF][START_REF] Fermanian-Kammerer | Mesures semi-classiques 2-microlocales[END_REF][START_REF] Kammerer | Propagation and absorption of concentration effects near shock hypersurfaces for the heat equation[END_REF] in Euclidean space in the context of semi-classical measures. We also refer the reader to the articles [START_REF] Vasy | Semiclassical second microlocal propagation of regularity and integrable systems[END_REF][START_REF] Vasy | Erratum to: "Semiclassical second microlocal propagation of regularity and integrable systems[END_REF][START_REF] Wunsch | Non-concentration of quasimodes for integrable systems[END_REF] for related work regarding the study of the wave-front set of solutions to semiclassical integrable systems.

When the Hessian of H is constant Theorem 1.10 gives a complement to the results announced in [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] (where the argument was only valid when the Hessian has rational coefficients).

1.5. Hierarchy of time scales. In this section, we discuss the dependence of the set M (τ ) on the time scale τ . The following proposition allows to derive Theorem 1.3(2) for τ h ≫ 1/h from the result about τ h = 1/h. Denote by M av (τ ) the subset of P T d consisting of measures of the form:

1 0 ν (t, •) dt, where ν ∈ Conv M (τ ) .
where Conv X stands for the convex hull of a set X ⊂ L ∞ R; P T d with respect to the weak- * topology. We have the following result. Proposition 1.12. Suppose (τ h ) and (τ ′ h ) are time scales tending to infinity and such that τ ′ h ≪ τ h . Then:

M (τ ) ⊆ L ∞ (R; M av (τ ′ )) .
It is also important to clarify the link between the time-dependent Wigner distributions and those associated with eigenfunctions. Eigenfunctions are the most commonly studied objects in the field of quantum chaos, however, we shall see that they do not necessarily give full information about the time-dependent Wigner distributions. For the sake of simplicity, we state the results that follow in the case V h (t) = 0, although they easily generalise to the case in which V h (t) does not depend on t. Start noting that the spectrum of H (hD x ) coincides with H hZ d ; given E h ∈ sp(H (hD x )) the corresponding normalised eigenfunctions are of the form:

(12) u h (x) = H(hk)=E h c h k e ik•x , with k∈Z d c h k 2 = 1 (2π) d .
In addition, one has:

ν h (τ h t, •) = S τ h t h u h 2 = |u h | 2 ,
independently of (τ h ) and t. Let us denote by M (∞) the set of accumulation points in P T d of sequences |u h | 2 where (u h ) varies among all possible h-oscillating sequences of normalised eigenfunctions [START_REF] Verdière | Méthodes Semi-Classiques et Théorie Spectrale[END_REF], we have

M (∞) ⊆ M (τ ) .
As a consequence of Theorem 1.3, we obtain the following result.

Corollary 1.13. All eigenfunction limits M (∞) are absolutely continuous under the definiteness assumption on the Hessian of H.

A time scale of special importance is the one related to the minimal spacing of eigenvalues : define

(13) τ H h := h sup E 1 h -E 2 h -1 : E 1 h = E 2 h , E 1 h , E 2 h ∈ H hZ d .
It is possible to have τ H h = ∞: for instance, if H (ξ) = |ξ| α with 0 < α < 1 or H (ξ) = ξ • Aξ with A a real symmetric matrix that is not proportional to a matrix with rational entries (this is the content of the Oppenheim conjecture, settled by Margulis [START_REF] Shrikrishna | Values of quadratic forms at primitive integral points[END_REF][START_REF] Gregori | Discrete subgroups and ergodic theory[END_REF]). In some other situations, such as

H (ξ) = |ξ| α with α > 1, (13) is finite : τ H h = h 1-α . Proposition 1.14. If τ h ≫ τ H h one has: M (τ ) = Conv M (∞) .
This result is a consequence of the more general results presented in Section 6. Note that Proposition 1.14 allows to complete the description of M (τ ) in the case H (ξ) = |ξ| 2 as the time scale varies.

Remark 1.15. Suppose H (ξ) = |ξ| 2 , or more generally, that τ H h ∼ 1/h and the Hessian of H is definite. Then:

if τ h ≪ 1/h, ∃ν ∈ M (τ ) such that ν ⊥ dtdx; if τ h ∼ 1/h, M (τ ) ⊆ L ∞ R; L 1 T d ; if τ h ≫ 1/h M (τ ) = Conv M (∞) .
Finally, we point out that in this case the regularity of semiclassical measures can be precised. The elements in M (∞) are trigonometric polynomials when d = 2, as shown in [START_REF] Jakobson | Quantum limits on flat tori[END_REF]; and in general they are more regular than merely absolutely continuous, see [START_REF] Aïssiou | Semiclassical limits of eigenfunctions on flat n-dimensional tori[END_REF][START_REF] Jakobson | Quantum limits on flat tori[END_REF][START_REF] Nadirashvili | Geometric properties of eigenfunctions[END_REF]. The same phenomenon occurs with those elements in M (1/h) that are obtained through sequences whose corresponding semiclassical measures do not charge {ξ = 0}, see [START_REF] Aïssiou | Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures[END_REF].

1.6. Application to semiclassical and non-semiclassical observability estimates. As was already shown in [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] for the case H (ξ) = |ξ| 2 the characterization of the structure of the elements in M (1/h) implies quantitative, unique continuation-type estimates for the solutions of the Schrödinger equation (1) known as observability inequalities. This is the case again in this setting; here we shall prove the following result.

∈ C ∞ c (R d ) such that supp χ ∩ C H = ∅. Assume that V h (t) = Op h (V (t, •)) with V ∈ C ∞ (R × T * T d ) bounded.
Then the following are equivalent: i) Semiclassical observability estimate. There exists C = C(U, T, χ) > 0 and h 0 > 0 such that:

(14) χ (hD x ) u 2 L 2 (T d ) ≤ C T 0 U S t/h h χ (hD x ) u (x) 2 dxdt,
for every u ∈ L 2 (T d ) and h ∈ (0, h 0 ]. ii) Unique continuation in M (1/h). For every µ ∈ M (1/h) with µ(supp χ) = 0 and µ(C H ) = 0 (recall that µ is the image of µ under the projection π 2 ) one has:

T 0 µ (t, U × supp χ) dt = 0.
Besides, any of i) or ii) is implied by the following statement.

iii) Unique continuation for the family of Schrödinger equations (S Λ,ω,ξ ). For every Λ ⊂ Z d , every ξ ∈ supp χ with Λ ⊆ dH(ξ) ⊥ and every ω ∈ Λ /Λ, one has the following unique continuation property: if v ∈ C R; L 2 ω (R d , Λ) solves the Schrödinger equation (S Λ,ω,ξ ) and v| (0,T )×U = 0 then v = 0. This result will be proved as a consequence of the structure Theorem 1.10.

Remark 1.17. The unique continuation property for (S Λ,ω,ξ ) stated in Theorem 1.16, iii) is known to hold in any of the following two cases: i) V (•, ξ) is analytic in (t, x) for every ξ. This is a consequence of Holmgren's uniqueness theorem (see [START_REF] Tataru | Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem[END_REF])

ii) V (•, ξ) is smooth (or even continuous outside of a set of null Lebesgue measure) for every ξ and does not depend on t (see Theorem 1.20 below).

Corollary 1.18. Let U, T , χ, and V h (t) be as in Theorem 1.16, and suppose that V satisfies any of the two conditions in Remark 1.17. Then the semiclassical observability estimate ( 14) holds.

The nature of the observability estimate ( 14) is better appreciated when H is itself quadratic. Suppose:

H A,θ (ξ) = 1 2 A (ξ + θ) • (ξ + θ) , where θ ∈ R d ,
T > 0. Let V h (t) = Op h (V (t, •)) with V ∈ C ∞ (R × T * T d ) bounded.
Suppose that the following unique continuation result holds:

For every Λ ⊂ Z d and every ξ ∈ R d with Λ ⊆ dH(ξ) ⊥ , if v ∈ C R; L 2 ω (R d , Λ) , ω ∈ Λ /Λ, solves: (15) i∂ t v = (H A,θ (D y ) + V (•, ξ) Λ ) v and v| (0,T )×U = 0 then v = 0.
Then there exist C > 0 such that for every u ∈ L 2 (T d ) one has:

(16) u 2 L 2 (T d ) ≤ C T 0 U S t u (x) 2 dxdt.
Note that an estimate such as ( 16) implies a unique continuation result for solutions to (1): S t u| U = 0 for t ∈ (0, T ) =⇒ u = 0. Corollary 1. [START_REF] Fermanian | Mesures semi-classiques et croisement de modes[END_REF] shows in particular that this (weaker) unique continuation property for family of quadratic Hamiltonians in equations [START_REF] Derezinski | Scattering theory of classical and quantum N-particle systems[END_REF] actually implies the stronger estimate [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]. We also want to stress the fact that Corollary 1.19 establishes the unique continuation property for perturbations of pseudodifferential type from the analogous property for perturbations that are merely multiplication by a potential. It should be also mentioned that the proof of Theorem 4 in [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] can be adapted almost word by word to prove estimate [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] in the case when V does not depend on t, without relying in any a priori unique continuation result except those for eigenfunctions. In fact, the function V can be supposed less regular than smooth: it suffices that it is continuous outside of a set of null Lebesgue measure.

Theorem 1.20. Suppose V only depends on x; let U ⊂ T d a nonempty open set, and let T > 0. Then (16) holds; in particular, any solution S t u that vanishes identically on (0, T ) × U must vanish everywhere.

In the proof of Theorem 1.20, unique continuation for solutions to the time dependent Schrödinger equation is replaced by a unique continuation result for eigenfunctions of H A,θ (D x ) + V (t, •). This allows to reduce the proof of ( 16) to that of a semiclassical observability estimate ( 14) with a cut-off χ vanishing close to ξ = 0. At this point, the validity of ( 16) is reduced to the validity of the corresponding estimate on T d-1 . The proof of ( 16) is completed by applying an argument of induction on the dimension d. We refer the reader to the proof of [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF], Theorem 4 for additional details (see also [START_REF] Macià | The Schrödinger flow in a compact manifold: high-frequency dynamics and dispersion[END_REF] for a proof in a simpler case in d = 2).

Let us finally mention that Theorem 1.20 was first proved in the case H (ξ) = |ξ| 2 in [24] for V = 0, in [START_REF] Burq | Control for Schrödinger operators on tori[END_REF] for d = 2 and in [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] for general d, the three results having rather different proofs. We also refer the reader to [START_REF] Anantharaman | The dynamics of the Schrödinger flow from the point of view of semiclassical measures[END_REF][START_REF] Burq | Geometric control in the presence of a black box[END_REF][START_REF] Laurent | Internal control of the Schrödinger equation[END_REF][START_REF] Macià | The Schrödinger flow in a compact manifold: high-frequency dynamics and dispersion[END_REF] for additional results and references concerning observability inequalities in the context of Schrödinger-type equations.

1.7. Generalization to quantum completely integrable systems. Our results may be transferred to more general completely integrable systems as follows. Let (M, dx) be a compact manifold of dimension d, equipped with a density dx. Assume we have a family ( Â1 , . . . , Âd ) of d commuting self-adjoint h-pseudodifferential operators of order 0 in h. By this, we mean an operator a(x, hD x ) where a is in some classical symbol class S l , or may even have an asymptotic expansion a ∼ +∞ k=0 h k a k in this S l (the term a 0 will then be called the principal symbol). Let H = f ( Â1 , . . . , Âd ) where

f : R d -→ R is smooth. Let A = (A 1 , . . . A d ) : T * M -→ R d be the principal symbols of the operators Âi . Note that the commutation [ Âi , Âj ] = 0 implies the Poisson commutation {A i , A j } = 0. Assume that there is an open subset W of R d and a symplectomorphism T : T d × W -→ A -1 (W ) with A i • T = ξ j (
note that, by Arnold-Liouville Theorem, this situation occurs locally where the differentials of the A i are linearly independent). Then, there exists a Fourier integral operator Û : L 2 (T d ) -→ L 2 (M) associated with T , such that Û Û * = I + O(h ∞ ) microlocally on A -1 (W ), and such that [START_REF] Verdière | Méthodes Semi-Classiques et Théorie Spectrale[END_REF], Theorem 78 ( 1)).

Û * Âj Û = hD x j + k≥1 h k S j,k (hD x ) on T d × W with S j,k ∈ C ∞ (R d ) (see
This may be used to generalize our results to the equation ( 17)

ih∂ t ψ h (t, x) = Ĥ + h 2 V ψ h (t, x) , (t, x) ∈ R × M, ψ h |t=0 = u h ,
where V is a pseudodifferential operator of order 0.

If a is smooth, compactly supported inside A -1 (W ), if χ is supported in A -1 (W ) taking the value 1 on the support of a, and if t stays in a compact set of R, we have

Op h (a)S τ h t u h = Op h (a)S τ h t Op h (χ)u h + o(1) as long as τ h ≪ h -2 (or for all τ h if V = 0) and Op h (a)S τ h t Op h (χ)u h = Op h (a) Û Û * S τ h t Û Û * Op h (χ)u h + O(h ∞ ). Note that Û * S τ h t Û Û * Op h (χ)u h coincides modulo o(1) with Sτ h t Û * Op h (χ)u h where Sτ h t is the propagator associated to (18) ih∂ t ψ h (t, x) = f h (hD x ) + h 2 Û * V Û ψ h (t, x) , (t, x) ∈ R × T d .
where

f h (ξ) = f (ξ + k≥1 h k S k (ξ)).
The semiclassical measures associated with equation ( 17), when restricted to A -1 (W ), are exactly the images under T of the semiclassical measures coming from ( 18) supported on T d × W . Applying Theorem 1.3 to the solutions of ( 18) and transporting the statement by the symplectomorphism T , we obtain the following result :

Theorem 1.21. If τ h ≥ h -1
then the semiclassical measures associated with solutions of (18) are absolutely continuous measures of the lagrangian tori

A -1 (ξ), for μ-almost every ξ ∈ V such that d 2 f (ξ) is definite.
The observability results could also be rephrased in this more general setting.

1.8. Organisation of the paper. When τ h ≤ 1/h, the key argument of this article is a second microlocalisation on primitive submodules which is the subject of Section 2 and leads to Theorems 2.5 and 2.6. Sections 3 and 4 are devoted to the proof of these two theorems. At that stage of the paper, the proofs of Theorem 1.10 and Theorem 1.3(2) when τ h ∼ 1/h are then achieved. Examples are developed in Section 5 in order to prove Theorem 1.3 [START_REF] Aïssiou | Semiclassical limits of eigenfunctions on flat n-dimensional tori[END_REF]. Finally, the results concerning hierarchy of time-scales are proved in Section 6 (and lead to Theorem 1.3 for τ h ≫ 1/h), whereas the proof of Theorem 1.16 is given in Section 7.

Two-microlocal analysis of integrable systems on T d

In this section, we develop the two-microlocal analysis of the elements of M(τ ) that will be at the core of the proof of Theorems 1.3, 1.8 and 1.10 in the case where τ h ≤ 1/h. From now on, we shall assume that the time scale (τ h ) satisfies: [START_REF] Fermanian | Mesures semi-classiques et croisement de modes[END_REF] (hτ h ) is a bounded sequence.

Note however that the discussion of section 2.1 does not require this assumption.

2.1. Invariant measures and a resonant partition of phase-space. As in [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF], the first step in our strategy to characterise the elements in M (τ ) consists in introducing a partition of phase-space T * T d according to the order of "resonance" of its elements, that induces a decomposition of the measures µ ∈ M (τ ).

Using the duality

((R d ) * , R d ), we denote by A ⊥ ⊂ R d the orthogonal of a set A ∈ (R d ) * and by B ⊥ ⊂ (R d ) * the orthogonal of a set B ∈ R d .
Recall that L is the family of all primitive submodules of (Z d ) * and that with each Λ ∈ L, we associate the set I Λ defined in [START_REF] Bourgain | Analysis results and problems related to lattice points on surfaces[END_REF]:

I Λ = dH -1 (Λ ⊥ ).
Denote by Ω j ⊂ R d , for j = 0, ..., d, the set of resonant vectors of order exactly j, that is:

Ω j := ξ ∈ (R d ) * : rk Λ ξ = d -j , where Λ ξ := k ∈ (Z d ) * : k • dH(ξ) = 0 = dH(ξ) ⊥ ∩ Z d .
Note that the sets Ω j form a partition of (R d ) * , and that Ω 0 = dH -1 ({0}); more generally, ξ ∈ Ω j if and only if the Hamiltonian orbit {φ s (x, ξ) : s ∈ R} issued from any x ∈ T d in the direction ξ is dense in a subtorus of T d of dimension j. The set Ω := d-1 j=0 Ω j is usually called the set of resonant momenta, whereas

Ω d = (R d ) * \ Ω is referred to as the set of non-resonant momenta. Finally, write (20) R Λ := I Λ ∩ Ω d-rk Λ .
Saying that ξ ∈ R Λ is equivalent to any of the following statements:

(i) for any x 0 ∈ T d the time-average 1 T T 0 δ x 0 +tdH(ξ) (x) dt converges weakly, as T → ∞, to the Haar measure on the torus x 0 + T Λ ⊥ . Here, we have used the notation

T Λ ⊥ := Λ ⊥ / 2πZ d ∩ Λ ⊥ , which is a torus embedded in T d ; (ii) Λ ξ = Λ. Moreover, if rk Λ = d -1 then R Λ = dH -1 Λ ⊥ \ {0} = I Λ \ Ω 0 . Note that, (21) (R d ) * = Λ∈L R Λ , that is, the sets R Λ form a partition of (R d ) * . As a consequence, any measure µ ∈ M + (T * R d ) decomposes as (22) µ = Λ∈L µ⌉ T d ×R Λ .
Therefore, the analysis of a measure µ reduces to that of µ⌉ T d ×R Λ for all primitive submodule Λ. Given an H-invariant measure µ, it turns out that µ⌉ T d ×R Λ are utterly determined by the Fourier coefficients of µ in Λ. Indeed, define the complex measures on R d :

µ k := T d e -ik•x (2π) d/2 µ (dx, •) , k ∈ Z d , so that, in the sense of distributions, µ (x, ξ) = k∈Z d µ k (ξ) e ik•x (2π) d/2 .
Then, the following proposition holds.

Proposition 2.1. Let µ ∈ M + T * T d and Λ ∈ L. The distribution:

µ Λ (x, ξ) := k∈Λ µ k (ξ) e ik•x (2π) d/2
is a finite, positive Radon measure on T * T d . Moreover, if µ is a positive H-invariant measure on T * T d , then every term in the decomposition ( 22) is a positive H-invariant measure, and

(23) µ⌉ T d ×R Λ = µ Λ ⌉ T d ×R Λ .
Besides, identity ( 23) is equivalent to the fact that µ⌉ T d ×R Λ is invariant by the translations

(x, ξ) -→ (x + v, ξ) , for every v ∈ Λ ⊥ .
The proof of Proposition 2.1 follows the lines of those of Lemmas 6 and 7 of [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF]. We also point out that this decomposition depends on the function H through the definition of I Λ . In the following, our aim is to determine µ restricted to T d × R Λ for any Λ ∈ L.

2.2.

Second microlocalization on a resonant submanifold. Let (u h ) be a bounded sequence in L 2 T d and suppose (after extraction of a subsequence) that its Wigner distributions

w h (t) = w h S tτ h h u h converge to a semiclassical measure µ ∈ L ∞ R; M + T * T d in the weak- * topology of L ∞ R; D ′ T * T d .
Given Λ ∈ L, the purpose of this section is to study the measure µ⌉ T d ×R Λ by performing a second microlocalization along I Λ in the spirit of [START_REF] Fermanian-Kammerer | Mesures semi-classiques 2-microlocales[END_REF][START_REF] Kammerer | Propagation and absorption of concentration effects near shock hypersurfaces for the heat equation[END_REF][START_REF] Fermanian | Mesures semi-classiques et croisement de modes[END_REF][START_REF] Nier | A semi-classical picture of quantum scattering[END_REF][START_REF] Miller | Propagation d'ondes semi-classiques à travers une interface et mesures 2-microlocales[END_REF] and [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF][START_REF] Macià | High-frequency propagation for the Schrödinger equation on the torus[END_REF]. By Proposition 2.1, it suffices to characterize the action of µ⌉ T d ×R Λ on test functions having only x-Fourier modes in Λ. With this in mind, we shall introduce two auxiliary "distributions" which describe more precisely how w h (t) concentrates along T d × I Λ . They are actually not mere distributions, but lie in the dual of the class of symbols S 1 Λ that we define below.

In what follows, we fix ξ 0 ∈ R Λ such that d 2 H(ξ 0 ) is definite and, by applying a cut-off in frequencies to the data, we restrict our discussion to normalised sequences of initial data (u h ) that satisfy:

u h (k) = 0, for hk ∈ R d \ B(ξ 0 ; ǫ/2)
, where B(ξ 0 , ǫ/2) is the ball of radius ǫ/2 centered at ξ 0 . The parameter ǫ > 0 is taken small enough, in order that

d 2 H(ξ) is definite for all ξ ∈ B(ξ 0 , ǫ);
this implies that I Λ ∩B(ξ 0 , ǫ) is a submanifold of dimension d -rk Λ, everywhere transverse to Λ , the vector subspace of (R d ) * generated by Λ. Note that this is actually achieved under the weaker hypothesis that d 2 H(ξ) is non-singular and defines a definite bilinear form on Λ × Λ (Section 4.5 gives a set of assumptions which is weaker than definiteness but sufficient for our results). By eventually reducing ǫ, we have

B(ξ 0 , ǫ/2) ⊂ (I Λ ∩ B(ξ 0 , ǫ)) ⊕ Λ ,
by which we mean that any element ξ ∈ B(ξ 0 , ǫ/2) can be decomposed in a unique way as ξ = σ + η with σ ∈ I Λ ∩ B(ξ 0 , ǫ) and η ∈ Λ . We thus get a map

F : B(ξ 0 , ǫ/2) -→ (I Λ ∩ B(ξ 0 , ǫ)) × Λ (24) ξ -→ (σ(ξ), η(ξ))
With this decomposition of the space of frequencies, we associate two-microlocal testsymbols.

Definition 2.2. We denote by S 1 Λ the class of smooth functions a (x, ξ, η) on T * T d × Λ that are:

(i) compactly supported on (x, ξ) ∈ T * T d , ξ ∈ B(ξ 0 , ǫ/2),
(ii) homogeneous of degree zero at infinity w.r.t. η ∈ Λ , i.e. such that there exist

R 0 > 0 and a hom ∈ C ∞ c T * T d × S Λ with a (x, ξ, η) = a hom x, ξ, η |η| , for |η| > R 0 and (x, ξ) ∈ T * T d
(we have denoted by S Λ the unit sphere in Λ ⊆ (R d ) * , identified later on with the sphere at infinity);

(iii) such that their non vanishing Fourier coefficients (in the x variable) correspond to frequencies k ∈ Λ:

a (x, ξ, η) = k∈Λ a k (ξ, η) e ik•x (2π) d/2 .
We will also express this fact by saying that a has only x-Fourier modes in Λ.

The index 1 in the notation S 1 Λ refers to the fact that we have added one variable (η) to the standard class of symbols corresponding to the second microlocalisation. In Section 4, we will perform successive higher order microlocalisations corresponding to the addition of k ≥ 1 variables and we will consider spaces denoted S k Λ . For a ∈ S 1 Λ , we introduce the notation

Op Λ h (a(x, ξ, η)) := Op h (a (x, ξ, τ h η(ξ))) . Notice that, for all β ∈ N d , (25) ∂ β ξ (a (x, hξ, τ h η(hξ))) L ∞ ≤ C β (τ h h) |β| .
The Calderón-Vaillancourt theorem (see [START_REF] Calderón | On the boundedness of pseudo-differential operators[END_REF] or the appendix of [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF] for a precise statement) therefore ensures that there exist N ∈ N and

C N > 0 such that (26) ∀a ∈ S 1 Λ , Op Λ h (a) L(L 2 (R d )) ≤ C N |α|≤N ∂ α x,ξ,η a L ∞ ,
since (hτ h ) is assumed to be bounded. Therefore the family of operators Op Λ h (a) is a bounded family of L 2 (T d ).

We are going to use this formalism to decompose the Wigner transform w h (t). Let χ ∈ C ∞ c ( Λ ) be a nonnegative cut-off function that is identically equal to one near the origin. For a ∈ S 1 Λ , R > 1, δ < 1, we decompose a into: a(x, ξ, η) = 3 j=1 a j (x, ξ, η) with

a 1 (x, ξ, η) := a(x, ξ, η) 1 -χ η R 1 -χ η(ξ) δ , a 2 (x, ξ, η) := a(x, ξ, η) 1 -χ η R χ η(ξ) δ , (27)
a 3 (x, ξ, η) := a(x, ξ, η)χ η R . ( 28
)
Since any smooth compactly supported function with Fourier modes in Λ can be viewed as an element of S 1 Λ (which is constant in the variable η), this induces a decomposition of the Wigner distribution:

w h (t) = w I Λ h,R,δ (t) + w I Λ ,h,R (t) + w I c Λ h,R,δ (t)
, where:

w I Λ h,R,δ (t) , a := T * T d a 2 (x, ξ, τ h η(ξ)) w h (t) (dx, dξ) , (29)
w I Λ ,h,R (t) , a := T * T d a 3 (x, ξ, τ h η(ξ)) w h (t) (dx, dξ) , and
w I c Λ h,R,δ (t) , a := T * T d a 1 (x, ξ, τ h η(ξ)) w h (t) (dx, dξ) ,
that we shall analyse in the limits h -→ 0 + , R -→ +∞ and δ -→ 0 (taken in that order).

The distributions w I Λ h,R,δ (t) and w I Λ ,h,R (t) can be expressed for all t ∈ R by

w I Λ h,R,δ (t), a = u h , S τ h t * h Op Λ h (a 2 )S τ h t h u h L 2 (T d ) , (30)
w Λ I Λ ,h,R (t), a = u h , S τ h t * h Op Λ h (a 3 )S τ h t h u h L 2 (T d ) . ( 31
)
As a consequence of (26), both w I Λ h,R,δ and w

I Λ ,h,R are bounded in L ∞ R; (S 1 Λ ) ′ .
The first observation is that

lim δ-→0 lim R→∞ lim h→0 R θ(t) w I c Λ h,R,δ (t) , a dt = R T * T d θ(t)a hom x, ξ, η(ξ) |η(ξ)| µ(t, dx, dξ)⌉ T d ×I c Λ dt
where µ ∈ M (τ h ) is the semiclassical measure obtained through the sequence (u h ). Since R Λ ⊂ I Λ , the restriction of the measure thus obtained to T d × R Λ vanishes, and we do not need to further analyse the term involving the distribution w

I c Λ h,R,δ (t).
Then, after possibly extracting subsequences, one defines limiting objects μΛ and μΛ such that for every

ϕ ∈ L 1 (R) and a ∈ S 1 Λ , R ϕ (t) μΛ (t, •) , a dt := lim δ-→0 lim R→∞ lim h→0 + R ϕ (t) w I Λ h,R,δ (t) , a dt, and (32)
R ϕ (t) μΛ (t, •) , a dt := lim R→∞ lim h→0 + R ϕ (t) w I Λ ,h,R (t) , a dt.

From the decomposition w

h (t) = w I Λ h,R,δ (t) + w I Λ ,h,R (t) + w I c Λ h,R,δ (
t) (when testing against symbols having Fourier modes in Λ), it is immediate that the measure µ (t, •)⌉ T d ×R Λ is related to μΛ and μΛ according to the following Proposition.

Proposition 2.3. Let µ Λ (t, •) := Λ μΛ (t, •, dη)⌉ T d ×R Λ , µ Λ (t, •) := Λ μΛ (t, •, dη)⌉ T d ×R Λ .
Then both µ Λ (t, •) and µ Λ (t, •) are H-invariant positive measures on T * T d and satisfy:

(33) µ (t, •)⌉ T d ×R Λ = µ Λ (t, •) + µ Λ (t, •) .
This proposition motivates the analysis of the structure of the accumulation points μΛ (t, •) and μΛ (t, •). It turns out that both μΛ and μΛ have some extra regularity in the variable x, although for two different reasons. Our next two results form one of the key steps towards the proof of Theorem 1.3.

Remark 2.4. All the results in this section remain valid if the Hamiltonian H h depends on h as stated in Remark 1.1. The proofs are completely analogous, the only difference being that the the resonant manifolds I Λ and the coordinate system F = (σ, η) now also vary with h. Therefore, the definitions of w I Λ h,R,δ and w I Λ ,h,R should be modified accordingly. 2.3. Properties of two-microlocal semiclassical measures. We define, for

(x, ξ, η) ∈ T * T d × ( Λ \ {0}) and s ∈ R, φ 0 s (x, ξ, η) := (x + sdH(ξ), ξ, η) , φ 1 s (x, ξ, η) := x + sd 2 H(σ(ξ)) • η |η| , ξ, η .
This second definition extends in an obvious way to η ∈ S Λ (the sphere at infinity). On the other hand, the map (x, ξ, η) → φ 1 s|η| (x, ξ, η) extends to η = 0. We first focus on the measure μΛ . We point out that because of the existence of R 0 > 0 and of

a hom ∈ C ∞ c T * T d × S Λ such that a (x, ξ, η) = a hom x, ξ, η |η| , for |η| ≥ R 0 ,
the value w I Λ h,R,δ (t) , a only depends on a hom . Therefore, the limiting object μΛ (t,

•) ∈ (S 1 Λ )
′ is zero-homogeneous in the last variable η ∈ R d , supported at infinity, and, by construction, it is supported on ξ ∈ I Λ . This can be also expressed as the fact that μΛ is a "distribution" on T d × I Λ × Λ (where Λ is the compactification of Λ by adding the sphere S Λ at infinity) supported on {η ∈ S Λ }. Moreover, we have the following result.

Theorem 2.5. μΛ (t, •) is a positive measure on T d × I Λ × Λ supported on the sphere at infinity S Λ in the variable η. Besides, for a.e. t ∈ R, the measure μΛ (t, •) satisfies the invariance properties:

(34) φ 0 s * μΛ (t, •) = μΛ (t, •) , φ 1 s * μΛ (t, •) = μΛ (t, •) , s ∈ R.
Note that this result holds whenever τ h ≪ 1/h or τ h = 1/h. This is in contrast with the situation we encounter when dealing with μΛ (t, •). The regularity of this object indeed depends on the properties of the scale. (3) If τ h = 1/h, the projection of μΛ (t, •) on T * T d is a positive measure, whose projection on T d is absolutely continuous with respect to the Lebesgue measure. (4) If τ h ≪ 1/h, then μΛ satisfy the following propagation law: [START_REF] Miller | Propagation d'ondes semi-classiques à travers une interface et mesures 2-microlocales[END_REF] ∀t ∈ R, μΛ (t, x, ξ, η) = (φ 1 t|η| ) * μΛ (0, x, ξ, η). Note that (4) implies the continuous dependence of μΛ (t, •) with respect to t in the case τ h ≪ 1/h. For τ h = 1/h the dependence of μΛ (t, x, ξ, η) on t will be investigated in Section 3.

Remark 2.7. Consider the decomposition µ (t, •) = Λ∈L µ Λ (t, •) + Λ∈L µ Λ (t, •) .
given by Proposition 2.3. When τ h = 1/h, Theorem 2.6(3) implies that the second term defines a positive measure whose projection on T d is absolutely continuous with respect to the Lebesgue measure.

Theorem 2.6 calls for a few comments.

The fact that the distribution μΛ is supported on T d ×I Λ × Λ is straightforward. Indeed, we have for all t, [START_REF] Nadirashvili | Geometric properties of eigenfunctions[END_REF] w

I Λ ,h,R (t), a(x, ξ, η) = w I Λ ,h,R (t), a(x, σ(ξ), η) + O(τ -1 h )
since, by [START_REF] Lebeau | Deuxième microlocalisation sur les sous-variétés isotropes[END_REF],

Op Λ h (a 3 (x, ξ, η)) = Op Λ h (a(x, σ(ξ) + τ -1 h η, η)χ(η/R)) = Op Λ h (a(x, σ(ξ), η)χ(η/R)) + O(τ -1 h ) where the O(τ -1
h ) term is understood in the sense of the operator norm of L(L 2 (R d )) and depends on R (the fact that we first let h go to 0 + is crucial here).

When τ h ≪ 1/h the quantization of our symbols generates a semi-classical pseudodifferential calculus with gain hτ h . The operators Op Λ h (a) are semiclassical both in ξ and η. This implies that the accumulation points μΛ and μΛ are positive measures (see for instance [START_REF] Miller | Propagation d'ondes semi-classiques à travers une interface et mesures 2-microlocales[END_REF] or [START_REF] Fermanian | Mesures semi-classiques et croisement de modes[END_REF]).

When τ h = 1/h, we will see in Theorem 3.2 in Section 3 that the distributions μΛ (t, •) satisfy an invariance law that can be interpreted in terms of a Schrödinger flow type propagator.

Let us now comment on the invariance by the flows. Note first that of major importance is the observation that for all ξ ∈ R d * \ C H (recall that C H stands for the points where the Hessian d 2 H (ξ) is not definite) we have the decomposition

R d = Λ ⊥ ⊕ d 2 H (ξ) Λ .
Therefore, the flows φ 0 s and φ 1 s are independent on

T d × (R Λ \ C H ) × Λ .
Then, the following remark holds: Remark 2.8. In the case where rk Λ = 1 then (34) implies that, for a.e. t ∈ R, and for any ν in the 1-dimensional space Λ , the measure μΛ (t,

•)⌉ T d ×R Λ × Λ is invariant under (x, σ, η) -→ (x + d 2 H(σ) • ν, σ, η).
On the other hand, the invariance by the Hamiltonian flow and Proposition 2.1, imply that μΛ (t, •)⌉ T d ×R Λ × Λ is also invariant under

(x, σ, η) -→ (x + v, σ, η)
for every v in the hyperplane Λ ⊥ . Using the independence of the different flows and the fact that the Hessian d 2 H (σ) is definite on the support of μΛ (t, •)⌉ T d ×R Λ × Λ , we conclude that the measure μΛ (t, •)⌉ T d ×R Λ × Λ is constant in x ∈ T d in this case. For rk Λ > 1, we will develop a similar argument thanks to successive microlocalisations (see section 4).

In the next subsection, we prove the invariance properties stated in Theorems 2.5 and 2.6 (2) and (4). For τ h = h -1 the detailed analysis of the measure μΛ is performed in section 3 and the proof of the absolute continuity of its projection on T * T d is done in section 4.

Invariance properties of two-microlocal semiclassical measures.

Proof of Theorem 2.5. The positivity of μΛ (t, •) can be deduced following the lines of [START_REF] Fermanian | Mesures semi-classiques et croisement de modes[END_REF] §2.1, or those of the proof of Theorem 1 in [START_REF] Gérard | Microlocal defect measures[END_REF]; see also the appendix of [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF]. The proof of invariance of μΛ (t, •) under φ 0 s is similar to the proof of invariance of µ under φ s done in the appendix.

Let us now check the invariance property [START_REF] Marklof | Mean square value of exponential sums related to the representation of integers as sums of squares[END_REF]. Using [START_REF] Lindenstrauss | Invariant measures and arithmetic quantum unique ergodicity[END_REF], we have (along any convergent subsequence)

R ϕ(t) μΛ (t, •), a dt = lim δ→0 lim R→+∞ lim h→0 R ϕ(t) w I Λ h,R,δ (t) , a dt (37) = lim δ→0 lim R→+∞ lim h→0 R ϕ(t) w h (t), a 2 (x, ξ, τ h η(ξ)) dt.
Notice that the symbol

a 2 • φ 1 s (x, ξ, η) = a 2 x + sd 2 H(σ(ξ)) η |η| , ξ, η ,
is a well-defined element of S 1 Λ , since, for fixed R, a 2 is identically equal to zero near η = 0; moreover

∀ω ∈ S Λ , (a 2 • φ 1 s ) hom (x, ξ, ω) = a hom (x + sd 2 H(σ(ξ))ω, ξ, ω). We write d ds | s=0 a 2 • φ 1 s (x, ξ, τ h η(ξ)) = d 2 H(σ(ξ)) η(ξ) |η(ξ)| • ∂ x a 2 (x, ξ, τ h η(ξ)) .
Using the Taylor expansion

d 2 H(σ(ξ))η(ξ) + G(ξ)[η(ξ), η(ξ)] = dH(ξ) -dH(σ(ξ))
where ( 38)

G(ξ) = 1 0 d 3 H(σ(ξ) + tη(ξ))(1 -t)dt,
is uniformly bounded, and taking into account the fact that η(ξ) = O(δ) on the support of a 2 , we have

d 2 H(σ(ξ)) η(ξ) |η(ξ)| • ∂ x a 2 (x, ξ, τ h η(ξ)) = dH(ξ) -dH(σ(ξ)) |η(ξ)| • ∂ x a 2 (x, ξ, τ h η(ξ)) + O(δ).
Because a 2 has only x-Fourier coefficients in Λ and dH(σ(ξ)) ∈ Λ ⊥ , we can write

dH(ξ) -dH(σ(ξ)) |η(ξ)| • ∂ x a 2 (x, ξ, τ h η(ξ)) = dH(ξ) |η(ξ)| • ∂ x a 2 (x, ξ, τ h η(ξ)) .

Note now that

Op h dH(ξ) |η(ξ)| • ∂ x a 2 (x, ξ, τ h η(ξ)) = i h H(hD x ) + h 2 V h (t), Op h a 2 (x, ξ, τ h η(ξ)) |η(ξ)| + O(h) + O hτ h R .
For the last term, we have only used that V h (t) is a bounded operator on L 2 and

Op h a 2 (x, ξ, τ h η(ξ)) |η(ξ)| L 2 -→L 2 = O τ h R since 1 |η(ξ)| = O τ h R on the support of a 2 . To conclude, take a test function ϕ(t) ∈ C ∞ c (R) (those are dense in L 1 (R)). R ϕ(t) w I Λ h,R,δ (t) , d ds | s=0 a 2 • φ 1 s (x, ξ, τ h η(ξ)) dt = R ϕ(t) S τ h t h u h , i h H(hD x ) + h 2 V h (t), Op h a 2 (x, ξ, τ h η(ξ)) |η(ξ)| S τ h t h u h dt +O(h) + O hτ h R + O(δ) = 1 τ h R ϕ(t) d dt S τ h t h u h , Op h a 2 (x, ξ, τ h η(ξ)) |η(ξ)| S τ h t h u h dt + O(h) + O hτ h R + O(δ) = - 1 τ h R ϕ ′ (t) S τ h t h u h , Op h a 2 (x, ξ, τ h η(ξ)) |η(ξ)| S τ h t h u h dt + O(h) + O hτ h R + O(δ) = O(τ -1 h ) + O(h) + O hτ h R + O(δ).
Taking the limit h -→ 0 followed by R -→ +∞ and δ -→ 0, we obtain

R ϕ(t) μΛ (t), d ds | s=0 a • φ 1 s dt = 0
for any ϕ and a, which ends the proof of Theorem 2.5.

Proof of ( 1), ( 2) and (4) of Theorem 2.6 for hτ h -→ 0. The statement on the support of the measure μΛ has already been discussed in Section 2.2 (after Remark 2.7). The positivity of μΛ is standard once we notice that the two-scale quantization admits the gain hτ h (in view of ( 25)). Note also that (4) implies the continuous dependence with respect to t. Thus let us prove part (4) of the theorem. The propagation law (and hence, the continuity with respect to t) is proved as follows. Let

φ1 t (x, ξ, η) = φ 1 t|η| (x, ξ, η) = (x + td 2 H(σ(ξ))η, ξ, η). We write d dt|t=0 a 3 • φ1 t (x, ξ, τ h η(ξ)) = τ h d 2 H(σ(ξ))η(ξ) • ∂ x a 3 (x, ξ, τ h η(ξ))
and the same argument as in the previous proof now yields

τ h d 2 H(σ(ξ))η(ξ) • ∂ x a 3 (x, ξ, τ h η(ξ)) = τ h dH(ξ) • ∂ x a 3 (x, ξ, τ h η(ξ)) + O R 2 τ h
where we now use that |η(ξ

)| = O R τ h
on the support of a 3 . Note now that

τ h Op h (dH(ξ) • ∂ x a 3 (x, ξ, τ h η(ξ))) = iτ h h H(hD x ) + h 2 V h (t), Op h (a 3 (x, ξ, τ h η(ξ)) +O (hτ h ) ,
using only the fact that V h (t) is a bounded operator.

To conclude, take a test function

ϕ(t) ∈ C ∞ c (R). R ϕ(t) w I Λ ,h,R (t) , d dt|t=0 a 3 • φ1 t (x, ξ, τ h η(ξ)) dt = R ϕ(t) S τ h t h u h , iτ h h H(hD x ) + h 2 V h (t), Op h (a 3 (x, ξ, τ h η(ξ))) S τ h t h u h dt +O (hτ h ) + O R 2 τ h = R ϕ(t) d dt S τ h t h u h , Op h (a 3 (x, ξ, τ h η(ξ))) S τ h t h u h dt + O (hτ h ) + O R 2 τ h = - R ϕ ′ (t) S τ h t h u h , Op h (a 3 (x, ξ, τ h η(ξ))) S τ h t h u h dt + O (hτ h ) + O R 2 τ h Taking the limit h -→ 0 followed by R -→ +∞, we obtain R ϕ(t) μΛ (t), d dt | t=0 a • φ1 t dt = - R ϕ ′ (t) μΛ (t), a dt
for any ϕ and a, which is the announced result.

3. Regularity and transport of μΛ .

In this section, we suppose τ h = 1/h and we prove statement (3) of Theorem 2.6. This constitutes a first step towards the proof of Theorem 1.10 (and Theorem 1.3( 2)) which will be achieved in Section 4. In Theorem 3.2 below, we give a description of the measure μΛ . The first part of our result implies in particular that the projection of μΛ onto T d is absolutely continuous. For this result to hold we only assume that V h (t) is a general bounded self-adjoint perturbation as described in the Introduction. The second part of Theorem 3.2 shows that μΛ satisfies a propagation law that involves a Heisenberg equation. For that part we need to assume that V h (t) = Op h (V (t, •)) for some smooth bounded symbol V .

Recall that for ω in the torus Λ /Λ, we denote by

L 2 ω (R d , Λ) the subspace of L 2 loc (R d ) ∩ S ′ (R d ) formed by the functions whose Fourier transform is supported in Λ -ω. In other words, f ∈ L 2 ω (R d , Λ) if and only if f ∈ L 2 loc (R d ) and: f (• + v) = f, ∀v ∈ Λ ⊥ , (39) f (• + k) = e -iω•k f, ∀k ∈ 2πZ d ,
where, recall, Λ ⊥ stands for the orthogonal of Λ in the duality sense. Clearly,

f ∈ L 2 ω (R d , Λ) if and only if there exists g ∈ L 2 (T d , Λ) (the set of L 2 function on T d which have Fourier modes in Λ) such that f (x) = e -iω•x g (x); this characterization induces a natural Hilbert structure on L 2 ω (R d , Λ). We introduce an auxiliary lattice Λ ⊂ 2πZ d such that 2πZ d ⊂ Λ ⊥ ⊕ Λ. Let D Λ ⊂ Λ be a fundamental domain of the action of Λ on Λ . Each space L 2 ω (R d , Λ) is naturally isomorphic to L 2 (D Λ)
, (simply by extending by continuity the restriction of functions in

C ∞ (R d ) ∩ L 2 ω (R d , Λ)). Under this isomorphism, the norm in L 2 (D Λ) equals a factor |D Λ| 1/2 / (2π) d/2 times the norm in L 2 ω (R d , Λ). Introduce a vector bundle F over ( Λ /Λ) × I Λ , formed of pairs (ω, σ, f ) where (ω, σ) ∈ ( Λ /Λ) × I Λ and f ∈ L 2 ω (R d , Λ)
. This vector bundle is trivial, it may be identified with ( Λ /Λ) × I Λ × L 2 (D Λ) just by taking the above isomorphism (note, however, that the subbundle formed of triples (ω, σ, f ) such that f is smooth does not admit a smooth trivialisation).

We denote by L(F) (resp. K(F), L 1 (F)) the vector bundles over

( Λ /Λ) × I Λ formed of pairs (ω, σ, Q) where Q ∈ L(L 2 (R d , Λ, ω)) (resp. K(L 2 (R d , Λ, ω)), L 1 (L 2 (R d , Λ, ω))).
Again, all these bundles are trivial. Recall that, given a Hilbert space H, L(H), K(H) and L 1 (H) stands respectively for the space of bounded, compact and trace-class operators acting on H.

Remark 3.1. The Bloch-Floquet spectral decomposition shows that L ∞ sections of L(F) are in one-to-one correspondence with

L ∞ maps I Λ ∋ σ -→ Q(σ) ∈ L(L 2 (R d ))
, where, in addition, a.e. Q(σ) commutes with all translations by vectors k ∈ 2πZ d + Λ ⊥ .

We denote by Γ(K(F)) the space of continuous sections of K(F). Using the previous trivialisation, it is isomorphic to C((

Λ /Λ) × I Λ , K(L 2 (D Λ))), the space of continuous functions on ( Λ /Λ)×I Λ taking values in K(L 2 (D Λ)). The dual space Γ(K(F)) ′ to Γ(K(F)) is isomorphic to M(( Λ /Λ)×I Λ , L 1 (L 2 (D Λ))), the space of measures on ( Λ /Λ)×I Λ taking values in L 1 (L 2 (D Λ)).
We denote by Γ + (K(F)) the subset of positive sections, and Γ(K(F)) ′ + the positive elements of the dual, which correspond to elements of M + (( Λ /Λ) × I Λ , L 1 (L 2 (D Λ))) (the space of measures taking values in positive trace-class operators). Note that, by the Radon-Nikodym theorem (see for instance the appendix in [START_REF] Gérard | Microlocal defect measures[END_REF]), an element ρ ∈ Γ + (K(F)) can be written as ρ = Mν where ν = Tr ρ is a positive measure on ( Λ /Λ) × I Λ and M is a ν-integrable section of L 1 (F). We shall denote by Γ 1 (L 1 (F); ν) the set of such sections.

In order to state the propagation law obeyed by μΛ when V h (t) = Op h (V (t, x, ξ)), let us introduce one more notation. Write x = s + y ∈ R d with (s, y) ∈ Λ ⊥ × Λ and let σ ∈ I Λ ; we denote by V (t, y, σ) Λ the average of V (t, s + y, σ) w.r.t. s, thus getting a function that does not depend on s. We denote by V (t) Λ,σ the multiplication operator on L 2 ω (R d , Λ) associated to the multiplication by the σ-depending function V (t, y, σ) Λ . In the trivialisation introduced above, this operator does not depend on ω. In the case of a function a(x, ξ, η) ∈ S 1 Λ , the function a(s + y, ξ, η) does not depend on s so that a(y, σ, η) Λ = a(y, σ, η). We denote by a σ the section of L(F) that associates to (ω, σ) the operator acting on L 2 ω (R d , Λ) by a(y, σ, D y ) (Weyl quantization). In the case of a function a(x, ξ) ∈ C ∞ 0 (T * T d ) with Fourier modes in Λ (and independent of η), the function a(s+y, ξ) does not depend on s so that a(y, σ) Λ = a(y, σ). In this case a σ is the section of L(F) that associates to (ω, σ) the operator acting on L 2 ω (R d , Λ) by multiplication by a(y, σ). Finally, (d 2 H(σ)D y • D y ) ω will be used to denote the operator d 2 H(σ)D y • D y acting on L 2 ω (R d , Λ).

Theorem 3.2.

There exists m Λ ∈ M + (( Λ /Λ) × I Λ ) and M 0 Λ a m Λ -integrable section of L 1 (F), which only depend on the sequence of initial data, such that for all a ∈ S 1 Λ and all t ∈ R:

(40) μΛ (t, •) , a = ( Λ /Λ)×I Λ Tr L 2 ω (R d ,Λ) (a σ M Λ (t, ω, σ)) m Λ (dω, dσ).
where M Λ (t, ω, σ) solves, m Λ -a.e. (ω, σ)

∈ ( Λ /Λ) × R Λ , (Heis Λ,ω,σ ) i∂ t M Λ (t, ω, σ) = 1 2 d 2 H(σ)D y • D y + V (t) Λ,σ , M Λ (t, ω, σ) , with M Λ (0, •) = M 0 Λ . In other words, (41) M Λ (t, ω, σ) = U Λ,ω,σ (t) M 0 Λ (ω, σ)U * Λ,ω,σ (t)
, where U Λ,ω,σ (t) is the propagator starting at t = 0 of the unitary evolution associated to the operator Λ). Remark 3.3. i) Note that the fact that M Λ (t, •) is given by (Heis Λ,ω,σ ) and ( 41) implies that it is a continuous function of t. Therefore, μΛ (t, •) itself can be identified to a family of positive measures depending continuously on time .

1 2 (d 2 H(σ)D y • D y ) + V (t) Λ,σ on L 2 ω (R d ,
ii) The proof of Theorem 3.2 is carried out using the trivialisation obtained by identifying

L 2 ω (R d , Λ) with L 2 (D Λ)
and the final result does not depend on the choice of Λ and D Λ. iii) Identity ( 40) holds when V h (t) is a bounded family of perturbations as described in the introduction. In that case, the measure m Λ may also depend on time and equation ( 41) is not available.

The proof of Proposition 3.2 is divided in three steps:

(1) We first define an operator K h which maps functions on R d to distributions with Fourier frequencies only in Λ ; in addition, this operator maps (2πZ d )-periodic functions to distributions on I Λ taking values in functions satisfying a Bloch-Floquet periodicity condition. (2) Then, we express w I Λ ,h,R (t) in terms of K h and take limits, first h → 0 + then followed by R → +∞. This defines an element ρ Λ ∈ L ∞ (R; Γ + (K(F))).

(3) We study the dependence in t of the limit object ρ Λ and show that it obeys a Heisenberg equation similar to (Heis Λ,ω,σ ). Note that the latter is of lower dimension than the original one (1) as soon as rk Λ < d. Each of the next subsections is devoted to one of the steps of the proof.

First Step: Construction of the operator K

h . Take m ∈ C ∞ 0 ((R d ) *
) supported in the ball B(ξ 0 , ǫ) ⊂ (R d ) * , and identically equal to 1 on B(ξ 0 , ǫ/2). For f a tempered distribution, we let

F f (ξ) := R d f (x)e -iξ•x dx (2π) d/2 .
In particular, if f is a 2πZ d -periodic function, we have

F f = k∈Z d f (k)δ k .
In what follows we shall denote d Λ := rk Λ and d Λ ⊥ := rk Λ ⊥ .

The operator K h maps a tempered distribution f to a distribution on I Λ × R d as follows:

K h f (σ, y) := x∈R d f (x) η∈ Λ m(σ)e i h η•y e -i h (σ+η)•x dη (2πh) d Λ /2 dx (2πh) d/2 = m(σ) h d/2 Λ F f σ + η h e i h η•y dη (2πh) d Λ /2 .
In order to get more insight on the properties of K h f it is useful to introduce some notations. Let π Λ be the projection on Λ , in the direction of Λ ⊥ . We have π Λ(2πZ d ) = Λ ⊂ Z d . For ξ ∈ (R d ) * , we shall denote by ξ Λ ∈ Λ the linear form ξ Λ (y) := ξ • π Λ(y) (in other words, the projection of ξ on Λ , in the direction Λ⊥ ). Note that for ξ ∈ Z d one has ξ Λ ∈ Λ. We fix a bounded fundamental domain D Λ for the action of Λ on Λ . For η ∈ Λ , there is a unique {η} ∈ D Λ (the "fractional part" of η) such that η -{η} ∈ Λ.

Sometimes we shall use the decomposition (R d ) * = Λ⊥ ⊕ Λ . This decomposition is related to the one given by the local coordinate system F (ξ) = (σ (ξ) , η (ξ)) defined in [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF] as follows. Let (ξ 1 , ξ 2 ) ∈ Λ⊥ × Λ such that F is defined on ξ = ξ 1 + ξ 2 (and therefore ξ = σ (ξ) + η (ξ)). Then σ (ξ) = σ (ξ 1 ) does not depend on ξ 2 and

ξ 2 = η (ξ) + σ (ξ 1 ) Λ .
In other words, (ξ 1 , ξ 2 ) ∈ Λ⊥ × Λ corresponds through F (whenever F (ξ) is defined) to a pair (σ, η) ∈ I Λ × Λ given by:

(42) σ = σ (ξ 1 ) , η = ξ 2 -σ (ξ 1 ) Λ .
These relations imply that for every ξ 1 ∈ Λ⊥ and y ∈ R d the following holds:

(43) K h f (σ (ξ 1 ) , y) = e -i σ(ξ 1 ) Λ h

•y m(σ(ξ 1 ))

Λ ⊥ f (s + π Λ (y))e -i ξ 1 h •s ds (2πh) d Λ ⊥ /2 .
In the above formula, ds is the dual density of dξ 1 , which in turn is defined to have dξ = dξ 1 dξ 2 where dξ 2 stands for the natural density on Λ . Note that dξ 1 is a constant multiple of the natural density on Λ⊥ .

If f is a 2πZ d -periodic function then:

K h f (σ, y) = h d Λ ⊥ /2 (2π) d Λ /2 kσ∈σ(hZ d ) δ kσ (σ) kη∈ Λ , (kσ,kη)∈F (hZ d ) m(k σ ) f k σ + k η h e i h kη•y .
It is clear from the above formula that for every y ∈ R d , the distribution K h f (•, y) is supported on the set

I h Λ := σ ∈ I Λ : σ h ∈ Z d -Λ .
We gather the properties of the operator K h that will be used in the sequel in the following lemma.

Lemma 3.4. (i) The Fourier transform of K h f (σ, •) w.r.t. the second variable is:

(44) F K h f (σ, η) = 2π h d Λ ⊥ /2 m (σ) F u σ h + η δ Λ (η) , in particular it is supported in Λ . (ii) The support of K h f (σ, •) is included in supp f + Λ ⊥ . (iii) If f is (2πZ) d -periodic, then supp K h f (•, y) ⊂ I h Λ and K h f (σ,
•) satisfies a Floquet periodicity condition:

(45) K h f (σ, y + k) = K h f (σ, y)e -iω h (σ)•k
for all k ∈ 2πZ d , where

ω h : I h Λ -→ Λ /Λ : σ -→ σ Λ h .
Statement ( 45) is equivalent to the fact that K h f (σ, •) has only frequencies in Λω h (σ).

(iv) Let f be a 2πZ d -periodic function, and let χ be a compactly supported function on

R d such that k∈2πZ d χ(• + k) ≡ 1. Then (46) k∈ Λ K h (χf )(σ, y + k)e iω h (σ)•k = K h f (σ, y). (v) If f ∈ L 2 (T d
) then the following Plancherel-type formula holds:

(47)

k∈Z d | f (k)m(hk)| 2 = σ∈I h Λ T d |K h f (σ, y)| 2 dy.
Proof. All points are quite obvious except (iii), which we prove below. Formula [START_REF] Vasy | Erratum to: "Semiclassical second microlocal propagation of regularity and integrable systems[END_REF] shows that the Fourier transform of K h f (σ, •) is supported in Λ . On the other hand, if f is 2πZ d -periodic then its Fourier transform is supported in Z d . Therefore, because of identity [START_REF] Vasy | Erratum to: "Semiclassical second microlocal propagation of regularity and integrable systems[END_REF], on the support of F K h f (σ, η) one must have:

σ h + η ∈ Z d , η ∈ Λ .
In other words, σ ∈ I h Λ and η ∈ Λ . Taking the projection on Λ in the direction Λ⊥ yields η ∈ Λ -σ Λ h , which is equivalent to η ∈ Λ -{ σ Λ h }. Note that any other choice of the auxiliary lattice used to define the projection onto Λ would lead to a σ ′ Λ ∈ Λ that differs from σ Λ on an element of hΛ. This shows, in particular, that the mapping ω h is well-defined on I h Λ . Remark 3.5. Let f be 2πZ d -periodic and let θ ∈ (R d ) * /(Z d ) * . Let g θ (y) := e -iθ•y f (y); then the proof of Lemma 3.4 (iii) shows that K h g θ satisfies the following Bloch-Floquet periodicity condition:

K h g θ (σ, y + k) = K h g θ (σ, y)e -i(ω h (σ)+θ Λ )•k , for every k ∈ 2πZ d .

Second step: Link between w I Λ ,h,R and K h . Now we show how the two-microlocal

Wigner distributions (w I Λ ,h,R ) of the sequence S t/h h u h can be expressed in terms of

K h S t/h h u h . Let χ ∈ C ∞ c (R d ) be such that k∈2πZ d χ(• + k) ≡ 1.
All the following identities hold independently of the choice of such χ. We start expressing the standard Wigner distributions w h u h in terms of the decomposition ξ

= (ξ 1 , ξ 2 ) ∈ Λ⊥ × Λ of (R d ) * . Let b ∈ C ∞ c (T d × R d ),
possibly depending on h. The following holds:

I(b, h) := T * T d b (x, ξ) w h u h (dx, dξ) = 1 (2π) d/2 (R d ) * ×(R d ) * F (χu h ) (ξ)F u h (ξ ′ )F b ξ ′ -ξ, h ξ + ξ ′ 2 dξdξ ′ = 1 (2π) d/2 h d Λ ⊥ ξ 1 ,ξ ′ 1 ∈ Λ⊥ ,ξ 2 ,ξ ′ 2 ∈ Λ F (χu h ) (ξ 1 , ξ 2 )F u h (ξ ′ 1 , ξ ′ 2 ) F b ξ ′ 1 -ξ 1 h + ξ ′ 2 -ξ 2 , ξ 1 + ξ ′ 1 2 + h ξ 2 + ξ ′ 2 2 dξ 1 dξ ′ 1 dξ 2 dξ ′ 2 .
Identity [START_REF] Tataru | Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem[END_REF] shows that:

ξ = ξ 1 h + ξ 2 = σ h + η,
where σ = σ(ξ 1 ) and η = ξ 2 -σ(ξ 1 ) Λ h . For every σ ∈ I Λ , we denote by ξ 1 (σ) the element in Λ⊥ characterised by ξ 1 (σ)σ ∈ Λ . The density dξ 1 on Λ⊥ is transferred to a density dσ on I Λ (note that dξ = dξ 1 dξ 2 = dσdη).

With this in mind, we obtain using [START_REF] Vasy | Erratum to: "Semiclassical second microlocal propagation of regularity and integrable systems[END_REF], provided we assume that u h has frequencies in B(ξ 0 , ǫ/2):

(48) I(b, h) = (2πh) -d Λ ⊥ (2π) d/2 σ,σ ′ ∈I Λ ,η,η ′ ∈R d F K h χu h (σ, η)F K h u h (σ ′ , η ′ ) F b ξ 1 (σ ′ ) -ξ 1 (σ) h + σ ′ Λ h + η ′ - σ Λ h -η, σ ⊕ σ ′ 2 - σ ⊕ σ ′ 2 Λ + σ Λ + σ ′ Λ 2 + h η + η ′ 2 dσdσ ′ dηdη ′ ,
where σ⊕σ ′ 2 is a notation for the image under the coordinate map σ (ξ) of σ+σ ′ 2 . If b is invariant in the direction Λ ⊥ then the previous integral reduces to an integral over σ = σ ′ and takes the simpler form

I(b, h) = (2πh) -d Λ ⊥ (2π) d Λ /2 σ∈I Λ ,η,η ′ ∈R d F K h χu h (σ, η)F K h u h (σ, η ′ )F b η ′ -η, σ + h η + η ′ 2 dσdηdη ′ ,
where now F b should just be interpreted as a partial Fourier transform in the direction Λ .

In (48), the function K h χu h may be replaced by any other function satisfying the identity [START_REF] Zworski | Semiclassical analysis[END_REF]. In particular, we may replace K h χu h by χ 0 K h u h , where χ 0 is a function that satisfies k∈ Λ χ 0 (• + k) ≡ 1 and χ 0 is constant in the direction Λ ⊥ . In what follows, we take χ 0 to be the characteristic functions of D Λ, a fundamental domain for the action of Λ on R d .

By the changes of variables

η -→ η -σ Λ h + 1 h σ⊕σ ′ 2 Λ and η ′ -→ η ′ - σ ′ Λ h + 1 h σ⊕σ ′ 2
Λ , (48) may be transformed into

(49) I(b, h) = (2πh) -d Λ ⊥ (2π) d/2 σ,σ ′ ∈I Λ ,η,η ′ ∈R d F χ 0 K h u h σ, η - σ Λ h + 1 h σ ⊕ σ ′ 2 Λ × F K h u h σ ′ , η ′ - σ ′ Λ h + 1 h σ ⊕ σ ′ 2 Λ × F b ξ 1 (σ ′ ) -ξ 1 (σ) h + η ′ -η, σ ⊕ σ ′ 2 + h η + η ′ 2 dσdσ ′ dηdη ′ .

Next we write

K h u h (σ, y) = k∈ Λ χ 0 (y + k)K h u h (σ, y) = k∈ Λ(χ 0 K h u h )(σ, y + k)e iω h (σ)•k , so that F K h u h (σ, η) = F χ 0 K h u h (σ, η)δ Λ-ω h (σ)+ Λ⊥ (η) ,
Thus (49) is also

(50) I(b, h) = (2πh) -d Λ ⊥ (2π) d/2 σ,σ ′ ∈I Λ ,η,η ′ ∈R d F χ 0 K h u h σ, η - σ Λ h + 1 h σ ⊕ σ ′ 2 Λ F χ 0 K h u h σ ′ , η ′ - σ ′ Λ h + 1 h σ ⊕ σ ′ 2 Λ F b ξ 1 (σ ′ ) -ξ 1 (σ) h + η ′ -η, σ ⊕ σ ′ 2 + h η + η ′ 2 dσdσ ′ dηdη ′ δ η ′ ∈ Λ⊥ +ω h ( σ⊕σ ′ 2 )
. This motivates the following definition : Definition 3.6. If Q(ω, s, σ) is a smooth compactly supported function on Λ /Λ×Λ ⊥ ×I Λ , taking values in L(L 2 (D Λ)), we define:

P h Q (s, σ) := e i σ Λ • h Q (ω h (σ) , s, σ) e -i σ Λ • h , v h (σ, y) := χ 0 (y) e i σ Λ h •y K h u h (σ, y)
,

where e i σ Λ • h denotes multiplication by e i σ Λ h •y . Define (51) ρ h u h , Q := v h , P h Q (hD σ , •) v h L 2 (IΛ;L 2 (D Λ)) ,
where

P h Q (hD σ , σ) is obtained from P h Q by Weyl quantization.
More explicitly, we have:

(52) ρ h u h , Q = 1 (2πh) d Λ ⊥ σ,σ ′ ∈I Λ s∈Λ ⊥ e -i ξ 1 (σ ′ )-ξ 1 (σ) h •s v h (σ ′ , •), P h Q s, σ ⊕ σ ′ 2 v h (σ ′ , •) L 2 (D Λ) dσdσ ′ ds.
The interest of this definition becomes clearer if we realize the following identity. Let b ∈ C ∞ c (T * T d ), then formula (50) is equivalent to the identity (53)

u h , Op h (b)u h L 2 (T d ) = ρ h u h , Q h b , where Q h b (ω, s, σ) is the operator on L 2 (D Λ)
given by the kernel

(54) Q h b (ω, s, σ)(ỹ ′ , ỹ) = 1 (2π) d Λ k∈ Λ e iωk η∈ Λ b s + ỹ + ỹ′ 2 , σ + hη e iη•(ỹ ′ -ỹ+k) dη,
where, recall we have written

x = s + ỹ ∈ Λ ⊥ ⊕ Λ . Note that if one identifies L 2 (D Λ) with L 2 ω (R d , Λ
) as we did before, the operator Q h b (ω, s, σ) then corresponds to the (ωindependent) Weyl pseudodifferential operator b(y, σ + hD y ) acting on L 2 ω (R d , Λ). Let us now consider a ∈ S 1 Λ and let b h,R (x, ξ) := a(x, ξ, η(ξ)/h)χ(η(ξ)/hR) . We then have

S t/h h u h , Op h (b h )S t/h h u h L 2 (T d ) = w I Λ ,h,R (t) , a = ρ h (t), Q h a,R + O R (h) (55) where ρ h (t) := ρ h S t/h h u h and Q h a,R := Q h b h,R . Note that Q h a,R
does not depend on s, since as a has only frequencies in Λ it is a function independent on s.

We now take limits as h tends to zero : Proposition 3.7. After extraction of a subsequence, there exist

ρ Λ ∈ L ∞ R t , D ′ Λ /Λ × Λ ⊥ × I Λ ; L 1 L 2 (D Λ) such that for every Q ∈ C ∞ c Λ /Λ × Λ ⊥ × I Λ ; K (L 2 (D Λ)) and every φ ∈ L 1 (R): R φ (t) ρ h (t), Q dt -→ R φ (t) ρ Λ (t), Q dt.
In addition, ρ Λ is positive when restricted to symbols Q(ω, s, σ) that do not depend on s.

Proof. Note that Lemma 3.4, v) implies that (v h ) is bounded in L 2 (I Λ ; L 2 (D Λ))
and that the Calderón-Vaillancout theorem gives that the operators P h Q (hD σ , σ) are uniformly bounded with respect to h. Therefore, the linear map

L h : Q → R ρ h (t), Q (t) dt
is uniformly bounded as h -→ 0. Therefore, for any Q, up to extraction of a subsequence, it has a limit l(Q).

Considering a countable dense subset of

L 1 R; C ∞ c Λ /Λ × Λ ⊥ × I Λ ; K (L 2 (D Λ))
, and using a diagonal extraction process, one finds a sequence (h n ) tending to 0 as n goes to +∞ such that for any

Q ∈ L 1 R; C ∞ c Λ /Λ × Λ ⊥ × I Λ ; K (L 2 (D Λ))
, the sequence L hn (Q) has a limit as n goes to +∞. The limit is a linear form on

L 1 R; C ∞ c Λ /Λ × Λ ⊥ × I Λ ; K (L 2 (D Λ)) , characterized by an element ρ Λ of the dual bundle L ∞ R t , D ′ Λ /Λ × Λ ⊥ × I Λ ; L 1 (L 2 (D Λ))
. Finally, note that if Q(t, ω, σ) is a positive operator independent of s, equation (51) gives L h (Q) ≥ 0, whence the positivity of ρ Λ when restricted to symbols that do not depend on s.

As a consequence and in view of (55), letting h going to 0, then R to +∞, we have (possibly along a subsequence) for every a ∈ S 1 Λ and φ ∈ L 1 (R):

lim R→+∞ lim h→0 + R φ (t) w I Λ ,h,R (t), a dt = lim R→+∞ lim h→0 + R φ (t) ρ h (t), Q a,R dt = lim R→+∞ R φ (t) ρ Λ (t), Q a,R dt = R φ (t) ρ Λ (t), Q a,∞ dt,
where Q a,∞ (ω, s, σ) is the bounded operator on L 2 (D Λ) given by the kernel

(56) Q a,∞ (ω, s, σ)(ỹ ′ , ỹ) = 1 (2π) d Λ k∈ Λ e iωk η∈ Λ a s + ỹ + ỹ′ 2 , σ, η e iη•(ỹ ′ -ỹ+k) dη.
As discussed before, the operator Q a,∞ corresponds to the Weyl operator a(s + y, σ, D y ) acting on L 2 ω (R d Λ ). In particular, when a ∈ C ∞ c (T d ) has only frequences in Λ, the operator Q a,∞ is the multiplication operator a σ appearing in identity (40) of Theorem 3.2.

At this stage of the analysis, we have completed the proof of the first part of Theorem 3.2 (equation ( 40)), using only the fact that V h (t) is a bounded perturbation : we let m Λ (t, ω, s, σ)

:= Tr L 2 ω (R d ,Λ) ρ Λ (t, ω, s, σ). We have ρ Λ = M Λ m Λ where TrM Λ = 1.
As already noted, equation [START_REF] Schubert | Semiclassical wave propagation for large times[END_REF] implies the absolute continuity result, Theorem 1.3 (2).

3.3.

Step 3: Showing a propagation law. From now on we shall assume V h (t) = Op h (V (t, •)) and prove the propagation law (Heis Λ,ω,σ ). The first crucial observation is the following lemma.

Lemma 3.8. The measure ρ Λ is invariant by the Hamiltonian flow. More precisely, for every

Q ∈ C ∞ c ( Λ /Λ × Λ ⊥ × I Λ , K(L 2 (D Λ))) and a.e. t, ρ Λ (t) , dH(σ) • ∂ s Q = 0.
In particular, the restriction of ρ Λ (t) to σ ∈ R Λ is invariant under the action of Λ ⊥ by translation on the parameter s.

Proof. This lemma may be understood as a consequence of the invariance by the classical flow of semi-classical measures. Indeed, the same arguments than those of Appendix 8 give that for all ℓ ∈ R, we have 1) as h goes to 0 (R fixed). As a consequence, we deduce that for all ℓ ∈ R, 1) as h goes to 0 and R to +∞. Recall that φ 0 ℓ (x, ξ) = (x + ℓdH(ξ), ξ) and that for σ ∈ I Λ , dH(σ) ∈ Λ ⊥ . As a consequence, if x = s + y ∈ Λ ⊥ ⊕ Λ , then for σ ∈ I Λ , the vector x + ℓdH(σ) decomposes as

w I Λ ,h,R (t), a = w I Λ ,h,R (t), a • φ 0 ℓ + o(
ρ h (t), Q a,R = ρ h (t), Q a•φ 0 ℓ ,R + o(
x + ℓdH(σ) = (s + ℓdH(σ)) + y ∈ Λ ⊥ ⊕ Λ ,
and the kernel of the operator Q a•φ 0 ℓ ,R (ω, s, σ) is the function

Q a•φ 0 ℓ ,R (ω, s, σ)(ỹ ′ , ỹ) = 1 (2π) d Λ k∈ Λ e iωk a s + ℓdH(σ) + ỹ + ỹ′ 2 , σ, η × χ (η/R) e iη•(ỹ ′ -ỹ+k) dη.
The result follows if we note that compact pseudodifferential operators (e.g. operators of the form Q a,R (ω, s, σ)) are dense in K(L 2 (D Λ)) for the weak topology of operators .

We finally show that ρ Λ , restricted to σ ∈ R Λ obeys the propagation law (Heis Λ,ω,σ ). From now on, we only consider test symbols Q(ω, σ) that do not depend on the parameter s ∈ Λ ⊥ . We recall that we write V (t, x, ξ) as V (t, s + ỹ, σ + η) and that we use the notation V (t, ỹ, σ + η) Λ to mean that we are averaging V (t, s + ỹ, σ + η) w.r.t. s, thus getting a function that does not depend on s (nor η when ξ ∈ I Λ ). In the notation of (54), we note that Q h=0 V (t,•,σ) Λ = V (t) Λ,σ defines a multiplication operator on L 2 (D Λ) (for which the Floquet-Bloch periodicity conditions are transparent). To simplify the notation, we set in what follows:

A (σ, η) := 1 2 d 2 H(σ)η • η.
To prove that ρ Λ satisfies a Schrödinger-type equation, we note that

K h H(hD x )f (σ, y) = 1 h d/2 m(σ) η∈ Λ H(σ + η)F f σ + η h e i h η•y dη (2πh) d Λ /2 = H(σ + hD y )K h f (σ, y),
and that

K h Op h (V (t, •)) f (σ, y) = P h Q h V (hD σ , σ)K h f (σ, y)
(we used the notation of Definition 3.6). Therefore, w h (t,

•) := K h S t/h h u h solves: i∂ t w h (t, σ, y) = h -2 H (σ + hD y ) + P h Q h V (hD σ , σ) w h (t, σ, y) .
Note that if Q(ω, s, σ) does not depend on s, we have

ρ h (t), [h -2 H(σ + hD y ), Q] = χ 0 e i σ Λ h • h -2 H (σ + hD y ) w h (t), P h Q (hD σ , •) χ 0 e i σ Λ h • w h (t) -χ 0 e i σ Λ h • w h (t), P h Q (hD σ , •) χ 0 e i σ Λ h • h -2 H (σ + hD y ) w h (t) L 2 (IΛ;L 2 (D Λ))
Hence, passing to the limit h -→ 0,

R φ ′ (t) ρ h (t), Q dt = i R φ (t) ρ h (t), [h -2 H(σ + hD y ) ω + Q h V (s, σ), Q] dt + o(1).
Above, the index ω in H(σ + hD y ) ω indicates that the operator acts on L 2 ( D Λ ) with Floquet periodicity conditions [START_REF] Wunsch | Non-concentration of quasimodes for integrable systems[END_REF]. We perform a Taylor expansion of H(σ + hD y ) and write, in

L(L 2 ( D Λ )), for any Q ∈ K (L 2 ( D Λ )), H(σ + hD y )Q = H(σ)Q + hdH(σ)D y Q + h 2 A(σ, D y )Q + O(h 3 ).
At this point, note that dH(σ)D y = 0 (since

σ ∈ I Λ one has dH(σ) ∈ Λ ⊥ ). Therefore, for Q ∈ C ∞ c ( Λ /Λ × Λ ⊥ × I Λ , K(L 2 (D Λ))), h -2 H(σ + hD y ) ω , Q(ω, hD σ , σ) = [A(σ, D y ) ω , Q(ω, hD σ , σ)] + O(h).
As a consequence, we obtain:

R φ ′ (t) ρ h (t), Q dt = i R φ (t) ρ h (t), [A(σ, D y ) ω + Q h V (hD σ , σ), Q (ω, hD σ , σ)] dt + o(1).
Taking limits, we obtain

R φ ′ (t) ρ Λ (t), Q dt = i R φ (t) ρ Λ (t), [A(σ, D y ) ω + Q 0 V (s, σ), Q] dt = i R ρ Λ (t), [A(σ, D y ) ω + Q 0 V (s, σ), Q] dt
where the potential V (s, σ) is averaged along the flow s → s + tdH(σ) (because of Lemma 3.8). But V does not depend on s for σ ∈ R Λ , and it is simply the average of V w.r.t. s. Hence Q 0 V = V (t) Λ,σ and ρ Λ satisfies the following Heisenberg equation for

σ ∈ R Λ : i∂ t ρ Λ (t, ω, σ) = [A(σ, D y ) ω + V (t) Λ,σ , ρ Λ (t, ω, σ)] (note that ρ Λ (t, ω, σ) does not depend on s for σ ∈ R Λ ). Let m Λ (t, ω, s, σ) := Tr L 2 ω (R d ,Λ) ρ Λ (t, ω, s, σ);
the propagation law (57) implies that m Λ does not depend on t. Therefore, ρ Λ = M Λ m Λ where M Λ (•, ω, σ) solves (Heis Λ,ω,σ ) for σ ∈ R Λ and TrM Λ = 1. This concludes the proof of Theorem 3.2 (in the statement, the parameter s disappeared since all test functions are independent of s).

An iterative procedure for computing µ

In this section, we develop the iterative procedure which leads to the proof of Theorem 1.10 4.1. First step of the construction. What was done in the previous section can be considered as the first step of an iterative procedure that allows to effectively compute the measure µ(t, •) solely in terms of the sequence of initial data (u h ). Recall that we assumed in §2.2, without loss of generality, that the projection on ξ of µ (t, •) was supported in a ball contained in R d \ C H . We have decomposed this measure as a sum

µ(t, •) = Λ∈L µ Λ (t, •) + Λ∈L µ Λ (t, •),
where Λ runs over the set of primitive submodules of Z d , and where

µ Λ (t, •) = Λ μΛ (t, •, dη)⌉ T d ×R Λ , µ Λ (t, .) = Λ μΛ (t, •, dη)⌉ T d ×R Λ .
From Theorem 2.6, the distributions μΛ have the following properties :

(1) μΛ (t, dx, dξ, dη) is in C (R; (S 1 Λ ) ′ ) and all its x-Fourier modes are in Λ; with respect to the variable ξ, μΛ (t, dx, dξ, dη) is supported in I Λ ;

(2) if τ h ≪ 1/h then for every t ∈ R, μΛ (t, •) is a positive measure and:

μΛ (t, •) = φ1 t * μΛ (0, •) , where: φ1 s : (x, ξ, η) -→ (x + sd 2 H(σ(ξ))η, ξ, η); (3) if τ h = 1/h then Λ μΛ (t, •, dη) is in C(R; M + (T * T d )) and R d × Λ μΛ (t, •, dξ, dη) is
an absolutely continuous measure on T d . In fact, with the notations of Section 2.4, we have, for every a ∈ C ∞ c T * T d with Fourier modes in Λ,

T d ×I Λ × Λ a(x, ξ)μ Λ (t, dx, dξ, dη) = ( Λ /Λ)×I Λ Tr (a σ ρ Λ (t, dω, dσ)) where ρ Λ ∈ L ∞ R t , Γ (K(F)) ′ +
and a σ is the section of L(F) defined by the map (ω, σ) → multiplication by a(y, σ). In addition, if

V h (t) = Op h (V (t, •)) then ρ Λ = M Λ m Λ where m Λ ∈ M + (( Λ /Λ) × I Λ ), M Λ is a section of L 1 (F) integrable with respect to m Λ . Moreover, Tr L 2 ω (R d ,Λ) M Λ (t, ω, σ) = 1 and M Λ (•, ω, σ) satisfies a Heisenberg equation (Heis Λ,ω,σ ).
On the other hand, the measures μΛ satisfy:

(1) for a ∈ S 1 Λ , μΛ (t, dx, dξ, dη), a(x, ξ, η) is obtained as the limit of

w I Λ h,R,δ (t) , a = T * T d χ η (ξ) δ 1 -χ τ h η(ξ) R a (x, ξ, τ h η(ξ)) w h (t) (dx, dξ) ,
in the weak- * topology of L ∞ (R, (S 1 Λ ) ′ ), as h -→ 0 + , R -→ +∞ and then δ -→ 0 + (possibly along subsequences);

(2) μΛ (t, dx, dξ, dη) is in L ∞ (R, M + (T * T d × Λ )) and all its x-Fourier modes are in Λ.

With respect to the variable η, the measure μΛ (t, dx, dξ, dη) is 0-homogeneous and supported at infinity : we see it as a measure on the sphere at infinity S Λ . With respect to the variable ξ, it is supported on {ξ ∈ I Λ }; (3) μΛ is invariant by the two flows, φ 0 s : (x, ξ, η) -→ (x + sdH(ξ), ξ, η), and

φ 1 s : (x, ξ, η) -→ (x + sd 2 H(σ(ξ)) η |η| , ξ, η).
This is the first step of an iterative procedure; the next step is to decompose the measure µ Λ (t, •) according to primitive submodules of Λ. We need to adapt the discussion of [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF]; to this aim, we introduce some additional notation.

Fix a primitive submodule Λ ⊆ Z d and σ ∈ I Λ \ C H . For Λ 2 ⊆ Λ 1 ⊆ Λ primitive submodules of (Z d ) * , for η ∈ Λ 1 , we denote Λ η (σ, Λ 1 ) := Λ ⊥ 1 ⊕ R d 2 H(σ).η ⊥ ∩ (Z d ) * = R d 2 H(σ).η ⊥ ∩ Λ 1 ,
where the orthogonal is always taken in the sense of duality. We note that Λ η (σ, Λ 1 ) is a primitive submodule of Λ 1 , and that the inclusion

Λ η (σ, Λ 1 ) ⊂ Λ 1 is strict if η = 0 since d 2 H(σ) is definite. We define: R Λ 1 Λ 2 (σ) := {η ∈ Λ 1 , Λ η (σ, Λ 1 ) = Λ 2 }. Because d 2 H(σ) is definite, we have the decomposition (R d ) * = (d 2 H(σ).Λ 2 ) ⊥ ⊕ Λ 2 .
We define P σ Λ 2 to be the projection onto Λ 2 with respect to this decomposition.

4.2.

Step k of the construction. In the following, we set Λ = Λ 1 , corresponding to step k = 1. We now describe the outcome of our decomposition at step k (k ≥ 1); we will indicate in §4.3 how to go from step k to k + 1, for k ≥ 1.

At step k, we have decomposed µ(t, •) as a sum

µ(t, •) = 1≤l≤k Λ 1 ⊃Λ 2 ⊃...⊃Λ l µ Λ 1 Λ 2 ...Λ l-1 Λ l (t, •) + Λ 1 ⊃Λ 2 ⊃...⊃Λ k µ Λ 1 Λ 2 ...Λ k (t, •),
where the sums run over the strictly decreasing sequences of primitive submodules of (Z d ) * (of lengths l ≤ k in the first term, of length k in the second term). We have

µ Λ 1 Λ 2 ...Λ l-1 Λ l (t, x, ξ) = R Λ 1 Λ 2 (ξ)×...×R Λ l-1 Λ l (ξ)× Λ l μΛ 1 Λ 2 ...Λ l-1 Λ l (t, x, ξ, dη 1 , . . . , dη l )⌉ T d ×R Λ 1 , µ Λ 1 Λ 2 ...Λ k (t, x, ξ) = R Λ 1 Λ 2 (ξ)×...×R Λ k-1 Λ k (ξ)×S Λ k μΛ 1 Λ 2 ...Λ k (t, x, ξ, dη 1 , . . . , dη k )⌉ T d ×R Λ 1 . The distributions μΛ 1 Λ 2 ...Λ l-1 Λ l
have the following properties :

(1)

μΛ 1 Λ 2 ...Λ l-1 Λ l ∈ C R, D ′ T * T d × S Λ 1 × . . . × S Λ l-1 × Λ l
and all its x-Fourier modes are in Λ l ; with respect to ξ it is supported in I Λ 1 ;

(2) for every t ∈ R,

μΛ 1 Λ 2 ...Λ l-1 Λ l (t,
•) is invariant under the flows φ j s (j = 0, 1, . . . , l -1) defined by φ 0 s (x, ξ, η 1 , ..., η l ) = (x + sdH(ξ), ξ, η 1 , ..., η l-1 , η l );

φ j s (x, ξ, η 1 , ..., η l ) = (x + sd 2 H(ξ) η j |η j | , ξ, η 1 , ..., η l ); (3) if τ h ≪ 1/h then for every t ∈ R, μΛ 1 Λ 2 ...Λ l-1 Λ l (t,
•) is a positive measure and

μΛ 1 Λ 2 ...Λ l-1 Λ l (t, •) = φl t * μΛ 1 Λ 2 ...Λ l-1 Λ l (0, •),
where, for

(x, ξ, η 1 , .., η l ) ∈ T * T d × S Λ 1 × . . . × S Λ l-1 × Λ l we define: φl s : (x, ξ, η 1 , ..., η l ) -→ (x + sd 2 H(ξ)η l , ξ, η 1 , ..., η l ); (4) if τ h = 1/h then Λ l μΛ 1 Λ 2 ...Λ l-1 Λ l (t, •, dη l ) is in C(R, M + (T * T d ×S Λ 1 ×. . .×S Λ l-1 ))
and the measure

(R d ) * ×S Λ 1 ×...×S Λ l-1 × Λ l μΛ 1 Λ 2 ...Λ l-1 Λ l (t, •, dξ, dη 1 , . . . , dη l ) is an ab- solutely continuous measure on T d . Besides, if a ∈ C ∞ c T * T d has only Fourier modes in Λ l , then, define L(F l ) the bundle over ( Λ l /Λ l )×I Λ 1 ×S Λ 1 ×. . .×S Λ l-1 formed of elements (ω, σ, η 1 , • • • , η l-1 , Q) where Q ∈ L(L 2 ω (R d , Λ l )), define similarly K(F l ) and L 1 (F l ), then T * T d ×S Λ 1 ×...×S Λ l-1 × Λ l a(x, ξ)μ Λ 1 Λ 2 ...Λ l-1 Λ l (t, dx, dξ, dη 1 , . . . , dη l ) = ( Λ l /Λ l )×I Λ 1 ×S Λ 1 ×...×S Λ l-1 Tr a σ ρ Λ 1 Λ 2 •••Λ l-1 Λ l (t, dσ, dη 1 , • • • , dη ℓ-1 ) , where ρ Λ 1 Λ 2 •••Λ l-1 Λ l is L ∞ in t,
a positive section element of Γ(K(F l )) ′ and where a σ is the section of L(F l ) defined by multiplication by a(σ, y).

When V h (t) = Op h (V (t, •)) then ρ Λ 1 Λ 2 •••Λ l-1 Λ l = M Λ 1 Λ 2 •••Λ l-1 Λ l m Λ 1 Λ 2 •••Λ l-1 Λ l where m Λ 1 Λ 2 •••Λ l-1 Λ l ∈ M + (( Λ l /Λ l ) × I Λ 1 × S Λ 1 × . . . × S Λ l-1 ), M Λ 1 Λ 2 •••Λ l-1 Λ l is a section of L 1 (F l ) integrable with respect to m Λ 1 Λ 2 •••Λ l-1 Λ l . More- over, Tr L 2 ω (R d ,Λ l ) M Λ 1 Λ 2 •••Λ l-1 Λ l = 1 and M Λ 1 Λ 2 •••Λ l-1 Λ l satisfies a Heisenberg equation (Heis Λ,ω,σ ) with Λ = Λ l . On the other hand μΛ 1 Λ 2 ...Λ k satisfy: (1) μΛ 1 Λ 2 ...Λ k is in L ∞ (R, M + (T * T d × S Λ 1 × . . . × S Λ k )
) and all its x-Fourier modes are in Λ k ; (2) μΛ 1 Λ 2 ...Λ k is invariant by the k + 1 flows, φ 0 s : (x, ξ, η) → (x + sdH(ξ), ξ, η 1 , . . . , η k ), and φ l s : (x, ξ, η 1 , . . . , η k ) -→ (x + sd 2 H(σ(ξ)) η l |η l | , ξ, η 1 , . . . , η k ) (where l = 1, . . . , k).

Finally, we define the space S k Λ k which is the class of smooth functions a(x, ξ, η 1 , . . . , η k ) on T * T d × Λ 1 × . . . × Λ k that are (i) smooth and compactly supported in (x, ξ) ∈ T * T d ; (ii) homogeneous of degree 0 at infinity in each variable η 1 , . . . , η k ;

(iii) such that their non-vanishing x-Fourier coefficients correspond to frequencies in Λ k .

4.3. From step k to step k + 1 (k ≥ 1). After step k, we leave untouched the term

1≤l≤k Λ 1 ⊃Λ 2 ⊃...⊃Λ l µ Λ 1 Λ 2 ...Λ l-1 Λ l
and decompose further Λ 1 ⊃Λ 2 ⊃...⊃Λ k µ Λ 1 Λ 2 ...Λ k . Using the positivity of μΛ 1 Λ 2 ...Λ k , we use the procedure described in Section 2.1 to write (57)

μΛ 1 Λ 2 ...Λ k (σ, •) = Λ k+1 ⊂Λ k μΛ 1 Λ 2 ...Λ k ⌉ η k ∈R Λ k Λ k+1 (σ) ,
where the sum runs over all primitive submodules Λ k+1 of Λ k . Moreover, by Proposition 2.1, all the x-Fourier modes of μΛ

1 Λ 2 ...Λ k ⌉ η k ∈R Λ k Λ k+1
(σ) are in Λ k+1 . To generalize the analysis of Section 2.2, we consider test functions in S k+1 Λ k+1 . We let

w Λ 1 Λ 2 ...Λ k+1 h,R 1 ,...,R k+1 (t, x, ξ, η 1 , • • • , η k+1 ) := 1 -χ η k+1 R k+1 × w Λ 1 Λ 2 •••Λ k h,R 1 ,••• ,R k (t, x, ξ, η 1 , • • • , η k ) ⊗ δ P ξ Λ k+1 (η k ) (η k+1 ), and
w Λ 1 Λ 2 ...Λ k Λ k+1 h,R 1 ,...,R k+1 (t, x, ξ, η 1 , • • • , η k+1 ) := χ η k+1 R k+1 × w Λ 1 Λ 2 •••Λ k h,R 1 ,••• ,R k (t, x, ξ, η 1 , • • • , η k ) ⊗ δ P ξ Λ k+1 (η k ) (η k+1 ).
By the Calderón-Vaillancourt theorem, both w Λ 1 Λ 2 ...Λ k Λ k+1 ,h,R 1 ,...,R k and w

Λ 1 Λ 2 ...Λ k+1 h,R 1 ,...,R k are bounded in L ∞ (R, (S k+1
Λ k+1 ) ′ ). After possibly extracting subsequences, we can take the following limits : lim

R k+1 -→+∞ • • • lim R 1 -→+∞ lim h-→0 w Λ 1 Λ 2 ...Λ k+1 h,R 1 ,...,R k (t) , a =: μΛ 1 Λ 2 ...Λ k+1 (t), a ,
and lim

R k+1 -→+∞ • • • lim R 1 -→+∞ lim h-→0 w Λ 1 Λ 2 ...Λ k Λ k+1 ,h,R 1 ,...,R k (t) , a =: μΛ 1 Λ 2 ...Λ k Λ k+1 (t), a .
Then the properties listed in the preceding subsection are a direct generalisation of Theorems 2.5 and 2.6 (see also [START_REF] Anantharaman | Semiclassical measures for the Schrödinger equation on the torus[END_REF], Section 4) and of the identity

µ Λ 1 Λ 2 •••Λ k (t, .)⌉ η k ∈R Λ k Λ k+1 (σ) = Λ k+1 µ Λ 1 Λ 2 •••Λ k+1 (t, ., dη k+1 )⌉ η k ∈R Λ k Λ k+1 (σ) (58)
+ Λ k+1 µ Λ 1 Λ 2 •••Λ k Λ k+1 (t, ., dη k+1 )⌉ η k ∈R Λ k Λ k+1 (σ) .
Remark 4.1. By construction, if Λ k+1 = {0}, we have μΛ 1 Λ 2 ...Λ k+1 = 0, and the induction stops. Similarly to Remark 2.8, one can also see that if rk Λ k+1 = 1, the invariance properties of μΛ 1 Λ 2 ...Λ k+1 imply that it is constant in x.

Remark 4.2. Note that in the preceding definition of k-microlocal Wigner transform for k ≥ 1, we did not use a parameter δ tending to 0 as we did when k = 0 in order to isolate the part of the limiting measures supported above R Λ k Λ k+1 (σ). This comes directly from the restrictions made in (57) and (58).

4.4.

Proof of Theorem 1.10. This iterative procedure allows to decompose µ along decreasing sequences of submodules. In particular, when τ h ∼ 1/h, it implies Theorem 1.10. Indeed, to end the proof of Theorem 1.10 , we let after the final step of the induction

µ final Λ (t, •) = 0≤k≤d Λ 1 ⊃Λ 2 ⊃•••⊃Λ k ⊃Λ µ Λ 1 Λ 2 ...Λ k Λ (t, •) = 0≤k≤d Λ 1 ⊃Λ 2 ⊃•••⊃Λ k ⊃Λ R Λ 1 Λ 2 (ξ)×...×R Λ k Λ (ξ)× Λ μΛ 1 Λ 2 ...Λ k Λ (t, •, dη 1 , . . . , dη k )⌉ T d ×R Λ 1 ,
where Λ 1 , . . . , Λ k run over the set of strictly decreasing sequences of submodules ending with Λ. We know that µ Λ 1 Λ 2 ...Λ k Λ is supported on {ξ ∈ I Λ 1 }, and since Λ ⊂ Λ 1 we have

I Λ 1 ⊂ I Λ .
We also let

ρ final Λ (t, ω, σ) = 0≤k≤d Λ 1 ⊃Λ 2 ⊃•••⊃Λ k ⊃Λ R Λ 1 Λ 2 (ξ)×...×R Λ k Λ (ξ) ρΛ 1 Λ 2 ...Λ k Λ (t, ω, σ, dη 1 , . . . , dη k )⌉ σ∈R Λ 1 ,
where the ρΛ 1 Λ 2 ...Λ k Λ are the operator-valued measures appearing in §4.2. As already mentioned, Theorem 1.10 implies Theorem 1.3 in the case τ h ∼ 1/h. The proof of Theorem 1.3 in the case τ h ≪ 1/h is discussed in Section 5 and in the case τ h ≫ 1/h, in Section 6. 4.5. Sufficient assumptions. In the induction, we used the fact that (59)

Λ k = Λ k+1 ⊕ (d 2 H(ξ) • Λ k+1 ⊥ ∩ Λ k ) for all k.
Definiteness of the Hessian d 2 H(ξ) is certainly a sufficient assumption for this, but we see that we actually need less if we note we are not using this property for arbitrary Λ k , but only for the ones arising in the construction (remember for instance that for k = 1 we only need (59) for

Λ 1 = dH(ξ) ⊥ ∩ Z d ).
A careful analysis of the proof shows that a sufficient set of assumptions is the following : Assumption 4.4. For every integer k, for all ξ, η 1 , . . . , η k ∈ (R d ) * , for every strictly decreasing sequence of primitive submodules

Λ 1 ⊃ Λ 2 ⊃ • • • ⊃ Λ k ⊃ {0} such that: Λ 1 = dH(ξ) ⊥ ∩ Z d , Λ 2 = (d 2 H(ξ) • η 1 ) ⊥ ∩ Λ 1 , . . . , Λ k = (d 2 H(ξ) • η k-1 ) ⊥ ∩ Λ k-1 ,
and

η 1 ∈ Λ 1 \ {0}, η 2 ∈ (d 2 H(ξ) • η 1 ) ⊥ \ (d 2 H(ξ) • Λ 1 ) ⊥ , η k ∈ (d 2 H(ξ) • η k-1 ) ⊥ \ (d 2 H(ξ) • Λ k-1 ) ⊥ then d 2 H(ξ) • η k ∈ Λ ⊥
k . We leave it to the reader to check that Assumptions 4.4 implies (59) and thus is a sufficient assumption for all our results.

In dimension d = 2, Assumptions 4.4 is implied by isoenergetic non-degeneracy (whereas we saw that it is no longer the case for d ≥ 3). In dimension 2, what happens is that, either dH(ξ) is a vector with rationally independent entries (in which case Λ 1 = {0} and the conditions of Assumptions 4.4 are empty), or dH(ξ) is a non zero vector with rationally dependent entries : in this case (and this is very special to dimension 2), Λ ⊥ 1 is one-dimensional and coincides with RdH(ξ). Thus Assumptions 4.4 just says that

dH(ξ) • η 1 = 0, η 1 = 0 =⇒ dH 2 (ξ) • η 1 ∈ RdH(ξ)
which is isoenergetic non-degeneracy. Remark that dH(ξ) = 0 is forbidden by isoenergetic non-degeneracy.

Note, finally, that isoenergetic non-degeneracy is only a local condition at ξ (since it involves only dH(ξ), d 2 H(ξ)) whereas condition Assumptions 4.4 contains some global features, namely the relations between dH(ξ), d 2 H(ξ) and the ring Z d , which is the homology group of T d .

Some examples of singular concentration

In Subsection 5.1, assuming V h (t) = 0, we present some examples of singular concentrations for the scales τ h ≪ h and, in that manner, we conclude the proof of Theorem 1.3 by proving the only remaining point [START_REF] Aïssiou | Semiclassical limits of eigenfunctions on flat n-dimensional tori[END_REF]. Then the two other subsections are devoted to the analysis of other cases of singular concentration which arise when the assumptions of Theorem 1.3 are not satisfied. 5.1. Singular concentration for time scales τ h ≪ 1/h. Assume V h (t) = 0 and consider ρ ∈ S R d with ρ L 2 (R d ) = 1 and such that the Fourier transform ρ is compactly supported. Let (x 0 , ξ 0 ) ∈ R d × R d and (ε h ) a sequence of positive real numbers that tends to zero as h -→ 0 + . Form the wave-packet:

(60) v h (x) := 1 (ε h ) d/2 ρ x -x 0 ε h e i ξ 0 h •x .
Define u h := Pv h , where P denotes the periodization operator Pv (x) := k∈Z d v (x + 2πk).

Since ρ is rapidly decreasing, we have 1) is a consequence of our next result.

u h L 2 (T d ) -→ h-→0 1. The family (u h ) is h-oscillatory if ε h ≫ h. Theorem 1.3(
Proposition 5.1. Let (τ h ) be such that lim h→0 + hτ h = 0; suppose that ε h ≫ hτ h . Then the Wigner distributions of the solutions S τ h t h u h converge weakly- * in L ∞ R; D ′ T * T d to µ (x 0 ,ξ 0 ) , defined by: (61)

T * T d a (x, ξ) µ (x 0 ,ξ 0 ) (dx, dξ) = lim T →∞ 1 T T 0 a (x 0 + tdH (ξ 0 ) , ξ 0 ) dt, ∀a ∈ C c (T * T d ).
We call the measures µ x 0 ,ξ 0 "uniform orbit measures" for φ s (their definition and existence as a limit is specific to translation flows on the torus). They are H-invariant and the convex hull of the set of uniform orbit measures is dense in the set of H-invariant measures. Considering initial data that are linear combinations of wave packets of the form (60), we see that the convex hull of uniform orbit measures is contained in M(τ ), and since the latter is closed, it contains all measures invariant by φ s as stated in Theorem 1.3 [START_REF] Aïssiou | Semiclassical limits of eigenfunctions on flat n-dimensional tori[END_REF].

Proof. Start noticing that the sequence of initial conditions (u h ) possesses the unique semiclassical measure µ 0 = δ x 0 ⊗ δ ξ 0 . Using property (4) in the appendix, we deduce that the image µ of µ (t, •) by the projection from

T d × R d onto R d satisfies: µ = Λ∈L µ⌉ R Λ = δ ξ 0 .
Since the sets R Λ form a partition of R d , we conclude that µ⌉ R Λ = 0 unless Λ = Λ ξ 0 and therefore µ = µ⌉ T d ×R Λ ξ 0

. Therefore, in order to characterize µ it suffices to test it against symbols with Fourier coefficients in Λ ξ 0 . Let a ∈ C ∞ c T * T d be such a symbol; we can restrict our attention to the case where a is a trigonometric polynomial in x. Let ϕ ∈ L 1 (R). Recall that the Wigner distributions w

h (t) of S τ h t h u h satisfy R ϕ (t) w h (t) , a dt = R ϕ (t) w h (0) , a • φ τ h t dt + o (1) ;
moreover the Poisson summation formula ensures that the Fourier coefficients of u h are given by:

u h (k) = (ε h ) d/2 (2π) d/2 ρ ε h h (hk -ξ 0 ) e -i(k-ξ 0 /h)•x 0 .
Combining this with the explicit formula (75) for the Wigner distribution presented in the appendix we get:

R ϕ (t) w h (t) , a dt = (ε h ) d (2π) 3d/2 k-j∈Λ ξ 0 ϕ τ h dH h k + j 2 • (k -j) a j-k h k + j 2 ρ ε h h (hk -ξ 0 ) ρ ε h h (hj -ξ 0 ) e -i(k-j)•x 0 + o (1) . (62)
Now, since kj ∈ Λ ξ 0 we can write:

dH h k + j 2 • (k -j) = dH h k + j 2 -dH (ξ 0 ) • (k -j) ≤ C h k + j 2 -ξ 0 |k -j| .
By hypothesis, both ρ and k -→ a k (ξ) are compactly supported, and hence the sum (62) only involves terms satisfying:

ε h h h k 2 -ξ 0 ≤ R, ε h h h j 2 -ξ 0 ≤ R and |j -k| ≤ R
for some fixed R. This in turn implies

τ h dH h k + j 2 • (k -j) ≤ CR 2 τ h h ε h .
This shows that the limit of (62) as h -→ 0 + coincides with that of:

(ε h ) d (2π) 3d/2 k-j∈Λ ξ 0 ϕ (0) a j-k h k + j 2 ρ ε h h (hk -ξ 0 ) ρ ε h h (hj -ξ 0 ) e -i(k-j)•x 0 = ϕ (0) w h (0) , a ,
which is precisely:

ϕ (0) a (x 0 , ξ 0 ) = ϕ (0) lim T →∞ 1 T T 0 a (x 0 + tdH (ξ 0 ) , ξ 0 ) dt,
since a has only Fourier modes in Λ ξ 0 .

We next present a slight modification of the previous example in order to illustrate the two-microlocal nature of the elements of M (τ ). Define now, for η 0 ∈ R d :

u h (x) = P v h (x) e iη 0 /(hτ h ) ,
where v h was defined in (60). Proposition 5.2. Suppose that lim h→0 + hτ h = 0 and ε h ≫ hτ h . Suppose moreover that d 2 H (ξ 0 ) is definite and that η 0 ∈ Λ ξ 0 . Then the Wigner distributions of S τ h t h u h converge weakly- * in L ∞ R; D ′ T * T d to the measure:

µ (t, •) = µ (x 0 +td 2 H(ξ 0 )η 0 ,ξ 0 ) , t ∈ R,
where µ (x 0 ,ξ 0 ) is the uniform orbit measure defined in (61).

Proof. The same argument we used in the proof of Proposition 5.1 gives µ = µ⌉ T d ×R Λ ξ 0 . We claim that w I Λ ξ 0 ,h,R (0) converges to the measure:

(63) µ Λ ξ 0 (0, x, ξ, η) = µ (x 0 ,ξ 0 ) (x, ξ) δ η 0 (η) .
Assume this is the case, Theorem 2.6 (4) implies that:

µ Λ ξ 0 (t, x, ξ, η) = µ (x 0 +td 2 H(ξ 0 )η 0 ,ξ 0 ) (x, ξ) δ η 0 (η) , ∀t ∈ R,
and, since µ Λ ξ 0 (t, •) are probability measures, it follows from Proposition 2.3 that µ Λ ξ 0 = 0 and :

µ (t, •) = Λ ξ 0 µ Λ ξ 0 (t, •, dη) = µ (x 0 +td 2 H(ξ 0 )η 0 ,ξ 0 ) .
Let us now prove the claim (63). Set

ũh (x) = v h (x) e iη 0 /(hτ h ) .
Consider h 0 > 0 and χ ∈ C ∞ 0 (R d ) such that χũ h = ũh for all h ∈ (0, h 0 ) and Pχ 2 ≡ 1. We now take a ∈ S 1 Λ and denote by ã the smooth compactly supported function defined on R d by ã = χ 2 a. Using the fact that the two-scale quantization admits the gain hτ h (in view of ( 25)),

u h , Op Λ ξ 0 h (a)u h L 2 (T d ) = u h , Op Λ ξ 0 h (ã)u h L 2 (R d ) = ũh , Op Λ ξ 0 h (a)ũ h L 2 (R d ) + O(hτ h ).
Therefore, it is possible to lift the computation of the limit of w I Λ ξ 0 ,h,R (0) to T * R d × Λ ξ 0 and, in consequence, replace sums by integrals. A direct computation gives:

ũh , Op Λ ξ 0 h (a)ũ h L 2 (R d ) = (2π) -d R 3d
e iξ•(x-y) ρ(x)ρ(y)

×a x 0 + ε h x + y 2 , ξ 0 + 1 τ h η 0 + h ε h ξ, τ h η(ξ 0 + 1 τ h η 0 + h ε h ξ) dxdydξ. Note that if F (ξ) = (σ, η), then ∀k ∈ Λ, F (ξ + k) = (σ, η + k) = F (ξ) + (0, k),
which implies that dF (ξ)k = (0, k) and dη(ξ)k = k for all k ∈ Λ ξ 0 . We deduce dη(ξ 0 )η 0 = η 0 since η 0 ∈ Λ ξ 0 and, in view of η(ξ 0 ) = 0, a Taylor expansion of η(ξ) around ξ 0 gives

τ h η ξ 0 + 1 τ h η 0 + h ε h ξ = η 0 + o(1).
Therefore, as h goes to 0, ũh , Op

Λ ξ 0 h (a)ũ h → a(x 0 , ξ 0 , η 0 ) = µ Λ ξ 0 , a .

Singular concentration for

Hamiltonians with critical points. We next show by a quasimode construction that for Hamiltonians having a degenerate critical point (of order k > 2) and for time scales τ h ≪ 1/h k-1 , the set M (τ ) always contains singular measures.

Suppose ξ 0 ∈ R d is such that:

dH (ξ 0 ) , d 2 H (ξ 0 ) , ..., d k-1 H (ξ 0 ) vanish identically.
The Hamiltonian H(ξ) = |ξ| k (k an even integer -corresponding to the operator (-∆)

k

2 ) provides such an example (with ξ 0 = 0). Let u h = Pv h , where v h is defined in (60). If ε h ≫ h it is not hard to see that

H (hD x ) u h -H (ξ 0 ) u h L 2 (T d ) = O h k / (ε h ) k . Therefore, S t h u h -e -i t h H(ξ 0 ) u h L 2 (T d ) = t O h k-1 / (ε h ) k ,
and, it follows that, for compactly supported ϕ

∈ L 1 (R) and a ∈ C ∞ c T * T d , R ϕ(t) w h (t) , a dt = R ϕ(t) u h , Op h (a)u h L 2 (T d ) dt + O τ h h k-1 / (ε h ) k .
Choosing (ε h ) tending to zero and such that ε h ≫ τ h h k-1 1/k , the latter quantity converges to a(x 0 , ξ 0 ) ϕ L 1 (R) as h -→ 0 + . In other words,

dt ⊗ δ x 0 ⊗ δ ξ 0 ∈ M(τ ), whence dt ⊗ δ x 0 ∈ M(τ ).
In the special case of H(ξ) = |ξ| k (k an even integer), we know that the threshold τ H h is precisely h 1-k . From the discussion of §6 and previously known results about eigenfunctions of the laplacian, we know that the elements of M(τ ) are absolutely continuous for scales τ h ≫ 1/h k-1 . In the case of τ h = 1/h k-1 , one can still show that elements of M(τ ) are absolutely continuous. This requires some extra work which consists in checking that all our proofs still work in this case for τ h = 1/h k-1 and ξ in a neighbourhood of ξ 0 = 0, replacing the Hessian d 2 H(ξ 0 ) by d k H(ξ 0 ), and the assumption that the Hessian is definite by the assumption that d k H(ξ 0 ).ξ k = 0 =⇒ ξ = 0 .

In the general case of a Hamiltonian having a degenerate critical point, the existence of such a threshold, and its explicit determination, is by no means obvious.

The effect of the presence of a subprincipal symbol of lower order in h.

Here we present some remarks concerning how the preceding results may change when the Hamiltonian H(hD x ) is perturbed by a potential h β V h (t) with β ∈ (0, 2) and V h (t) is a multiplication operator by some smooth function V (t, x). In this case, it is possible to find potentials V (t, x) for which Theorem 1.3(2) fails, i.e. such that there exists µ ∈ M (1/h), the projection of which on x is not absolutely continuous with respect to dtdx. The following example has been communicated to us by Jared Wunsch. On the 2-dimensional torus, take H (ξ) = |ξ| 2 and V (x 1 , x 2 ) := W (x 2 ) such that W (x 2 ) = (x 2 ) 2 /2 in the set {|x 2 | < 1/2}. Take ε ∈ (0, 1) and

u h (x, y) := 1 π 1/4 h ε/4 e i x 1 h e -(x 2 ) 2 2h ε χ(y),
where χ is a smooth function that is equal to one in {|x 2 | < 1/4} and identically equal to 0 in {|x 2 | > 1/2}. One checks that

-h 2 ∆ + h 2(1-ε) V -1 u h = h 2-ε u h + O(h ∞ ). It follows that for ϕ ∈ L 1 (R) and a ∈ C ∞ c (T * T 2 ), lim h→0 + R ϕ(t) S t/h 2(1-ǫ),h u h , Op h (a)S t/h 2(1-ǫ),h u h L 2 (T 2 ) dt = lim h→0 + R ϕ(t) u h , Op h (a)u h L 2 (T 2 ) dt = R ϕ (t) dt lim h→0 + u h , Op h (a)u h L 2 (T 2 ) dt = R ϕ (t) dt T * T 2 a (x, ξ) µ (dx, dξ) ,
and it is not hard to see that µ is concentrated on {x 2 = 0, ξ 1 = 1, ξ 2 = 0}. In particular the image of µ by the projection to T 2 is supported on {x 2 = 0}.

Hierarchies of time scales

Here we prove the results announced in §1.5 of the introduction. These results make explicit the relation between the sets M (τ ) as the time scale (τ h ) varies. Proposition 6.1. Let (τ h ) and (σ h ) be time scales tending to infinity as h -→ 0 + such that lim h→0 + σ h /τ h = 0. Then for every µ ∈ M (τ ) and almost every t ∈ R there exist

µ t ∈ Conv M (σ) such that (64) µ (t, •) = 1 0 µ t (s, •) ds.
Before presenting the proof of this result, we shall need two auxiliary lemmas. Lemma 6.2. Let (σ h ) be a time scale tending to infinity as h

-→ 0 + . Let v (n) h h>0,n∈N
be a normalised family in L 2 T d and define:

w (n) h (t, •) := w h S σ h t h v (n) h . Let c (n) h ≥ 0, n ∈ N, be such that n∈N c (n) h = 1.Then, every weak- * accumulation point in L ∞ R; D ′ T * T d of (65) n∈I h c (n) h w (n) h (t, •) belongs to Conv M (σ).
Proof. Suppose (65) possesses an accumulation point μ ∈ L ∞ R; M + T * T d that does not belong to Conv M (σ). By the Hahn-Banach theorem applied to the compact convex sets {μ} and Conv M (σ) we can ensure the existence of ε > 0

, a ∈ C ∞ c T * T d and θ ∈ L 1 (R) such that: R θ (t) μ (t, •) , a dt < -ε < 0, and, (66) R θ (t) µ (t, •) , a dt ≥ - ε 3 , ∀µ ∈ Conv M (σ) .
Suppose that μ is attained through a sequence (h k ) tending to zero. For k > k 0 big enough, R θ (t)

n∈I h k c (n) h k w (n) h k (t, •) , a dt ≤ - 3 2 ε,
which implies that there exists n k ∈ N such that:

(67) R θ (t) w (n k ) h k (t, •) , a dt ≤ - 3 2 ε.
Therefore, every accumulation point of w (n k ) h k also satisfies (67) which contradicts (66).

Lemma 6.3. Let τ , σ and µ be as in Proposition 6.1. For every α < β there exists µ α,β ∈ Conv M (σ) such that

1 β -α β α µ (t, •) dt = 1 0 µ α,β (t, •) dt.
Proof. Let µ ∈ M (τ ). Then there exists an h-oscillating, normalised sequence (u h ) such that, for every θ ∈ L 1 (R) and every a Note that Conv M (σ) is sequentially compact for the weak- * topology, therefore, there exist a sequence (ε n ) tending to zero and a measure µ t ∈ Conv M (σ) such that µ t εn converges weakly- * to µ t . Identities (68) and (69) ensure that µ (t, •) = 1 0 µ t (s, •) ds. Remark 6.4. Projecting on x in identity (64) we deduce that given ν ∈ M (τ ) there exist ν t ∈ M (σ) such that:

∈ C ∞ c T * T d : lim h→0 + R θ (t) S τ h t h u h , Op h (a)S τ h t h u h dt = R θ (t) µ (t, •) , a dt. Write N h := τ h /σ h ; by hypothesis N h -→ ∞ as h -→ 0 + . Let α < β, define L := β -α and
ν (t, •) = 1 0 ν t (s, •) ds.
This, together with the fact that elements of M (1/h) are absolutely continuous imply the conclusion of Theorem 1.3(2) when τ h ≫ 1/h. We now assume that V h (t) = 0. Denote by M (∞) the set of weak- * limit points of sequences of Wigner distributions (w u h ) corresponding to sequences (u h ) consisting of normalised eigenfunctions of H (hD x ). We now focus on a family of time scales τ for which the structure of M (τ ) can be described in terms of the closed convex hull of M (∞). Given a measurable subset O ⊆ R d , we define: Proof. Since the Fourier coefficient of S τ h t h u h are e -it τ h h H(hk) u h (k) and in view of (74) and ( 75 where P E h stands for the orthogonal projector onto the eigenspace associated to the eigenvalue E h . This can be rewritten as: R θ (t) w h (t) , a dt = θ (0)

E h ∈H(hZ d )∩H(O) c E h h w h v E h h , a ,
where v E h h :=

P E h u h P E h u h L 2 (T d )
, and c E h h := P E h u h 2 L 2 (T d ) .

Note that v E h h are eigenfunctions of H (hD x ) and the fact that (u h ) is normalised implies:

E h ∈H(hZ d )∩H(O) c E h h = 1.
We conclude by applying (a straightforward adaptation of) Lemma 6.2 to v E h h and c E h h . Corollary 6.6. Suppose τ H h := τ H h R d -→ ∞ as h -→ 0 + and that (τ h ) is a time scale such that τ H h ≪ τ h . Then M (τ ) = Conv M (∞) .

Proof. The inclusion M (τ ) ⊆ Conv M (∞) is a consequence of the previous result with O = R d . The converse inclusion can be proved by reversing the steps of the proof of Proposition 6.5.

Remark 6.7. Proposition 1.14 is a direct consequence of this result.

7.

Observability and unique continuation.

In this section we prove Theorem 1.16. Start noticing that the fact that ( 14) does not hold is equivalent to the existence of a sequence (u h ) in L 2 (T d ) such that: for every j such that λ j = 0 μΛ -a.e.. Since μΛ = 0, N Λ (•, ω, σ) = 0 on a set of positive μΛ -measure. This implies that at least for one j, ϕ j (•, ω, σ) = 0 and therefore, the unique continuation property of Theorem 1.16 iii) fails for that choice of Λ, ω and σ. This shows that iii) implies ii).

χ (hD x ) u h L 2 (T d ) = 1,
(d) Let μ be the measure on R d given by the image of µ(t, •) under the projection map (x, ξ) -→ ξ. If V h (t) = Op h (V (t, x, ξ)) is a pseudodifferential operator and if τ h ≪ h -2 then μ does not depend on t. Moreover, if µ 0 stands for the image under the same projection of any semiclassical measure corresponding to the sequence of initial data (u h ) then μ = µ 0 . For the reader's convenience, we next prove statements (c) and (d) (see also [START_REF] Macià | Semiclassical measures and the Schrödinger flow on Riemannian manifolds[END_REF] for a proof of these results in the context of the Schrödinger flow e iht∆ on a general Riemannian manifold). Let us begin with the invariance through the Hamiltonian flow. We set a s (x, ξ) := a(x + sdH(ξ), ξ) = a • φ s (x, ξ).

χ

  (x) ν (t, dx) dt,

Theorem 1 .

 1 10. (1) Let µ ∈ M (1/h). For every primitive submodule Λ ⊂ (Z d ) * there exists a positive measure µ final Λ ∈ L ∞ R; M + T * T d supported on T d × I Λ and invariant by the Hamiltonian flow φ s such that : for every a ∈ C ∞ c T * T d that vanishes on T d × C H and every θ ∈ L 1 (R):

Theorem 1 . 16 .

 116 Let U ⊂ T d open and nonempty, T > 0 and χ

Theorem 2 . 6 .

 26 [START_REF] Aïssiou | Semiclassical limits of eigenfunctions on flat n-dimensional tori[END_REF] The distributions μΛ (t, •) are supported on T d × I Λ × Λ and are continuous with respect to t ∈ R.(2) If τ h ≪ 1/h then μΛ (t, •) is a positive measure.

Remark 4 . 3 .

 43 It is clear from this construction that ρ final Λ and µ final Λ (t, •) can only charge those σ ∈ (R d ) * with Λ ⊆ dH(σ) ⊥ . Moreover, if V h (t) = Op h (V (t, •)) the measure ρ final Λ admits a decomposition ρ final Λ = N Λ µ Λ where µ Λ is a measure that does not depend on t and N Λ (•, ω, σ) is a family of positive, trace-class operators on L 2 ω (R d , Λ) with TrN Λ ≡ 1, satisfying the propagation law (Heis Λ,ω,σ ).

ua 1 0

 1 h form, for each n ∈ Z, a normalised sequence indexed by h > 0. The result then follows by Lemma 6.2 and using the fact that δ h -→ 1 as h -→ 0 + . Proof of Proposition 6.1. Let µ ∈ M (τ ); an application of the Lebesgue differentiation theorem gives the existence of a countable dense set S ⊂ C ∞ c T * T d and a set N ⊂ R of measure zero such that, for a ∈ S and t∈ R \ N, d a (x, ξ) µ (s, dx, dξ) ds = T * T d a (x, ξ) µ (t, dx, dξ) . Fix t ∈ R \ N; then, for any ε > 0 there exist µ t ε ∈ Conv M (σ) such that, for every a ∈ C ∞ c T * T d , (x, ξ) µ (s, dx, dξ) ds = T * T da (x, ξ) µ t ε (s, dx, dξ) ds.

Proposition 6 . 5 .

 65 τ H h (O) := h sup |H (hk) -H (hj)| -1 : H (hk) = H (hj) , hk, hj ∈ hZ d ∩ O . Note that the scale τ H h defined in the introduction coincides with τ H h R d . The following holds. Let O ⊆ R d be an open set such that τ H h (O) tends to infinity as h -→ 0 + . Suppose (τ h ) is a time scale such that lim h→0 + τ H h (O) /τ h = 0. If V = 0 and if µ ∈ M (τ ) is obtained through a sequence whose semiclassical measure satisfies µ 0 T d × R d \ O = 0 then µ ∈ Conv M (∞).

2 dxdt = 0 . 0 µ

 200 hD x ) u (x)This in turn, is equivalent to the existence of an element µ ∈ M (1/h) such that:(70) µ(supp χ) = 1, µ(C H ) = 0,T (t, U × supp χ) dt = 0,(recall that µ is the projection on µ on the ξ-coordinate). This establishes the equivalence between statements (i) and (ii) in Theorem 1.16.Let µ ∈ M (1/h) such that µ(C H ) = 0. Theorem 1.10 implies that µ decomposes as a sum of positive measures:µ = Λ∈L µ final Λ ,such that, see Remark 4.3 and Theorem 3.2, for any b ∈ C(T * T d ),T * T d b (x, ξ) µ final Λ (t, dx, dξ) = ( Λ /Λ)×I Λ Tr m b Λ (σ) N Λ (t, ω, σ) μΛ (dω, dσ),for some μΛ ∈ M + ( Λ /Λ) × R d and where N Λ (t, ω, σ) is given by:(71) N Λ (t, ω, σ) = U Λ,ω,σ (t) N 0 Λ (ω, σ) U * Λ,ω,σ (t) ,for some positive, self-adjoint trace-class operator N 0 Λ (ω, σ) acting on L 2 ω (R d , Λ) with Tr L 2 ω (R d ,Λ) N 0 Λ (ω, σ) = 1 and where U Λ,ω,σ (t) is the unitary propagator of the equation (S Λ,ω,σ ). Therefore, the measure μΛ only charges those σ ∈ R d satisfying Λ ⊆ dH(σ) ⊥ (see Remark 4.3) and we also have:μΛ (dω, •). If ϕ 0 j (ω, σ) j∈N is an orthonormal basis in L 2 ω (R d , Λ) consisting of eigenfunctions of the operator N 0 Λ (ω, σ) then N 0 Λ (ω, σ) = ∞ j=1 λ j (ω, σ)|ϕ 0 j (ω, σ) ϕ 0 j (ω, σ) |,where ∞ j=1 λ j = 1 and λ j ≥ 0. Now (71) implies that:(72) N Λ (t, ω, σ) = ∞ j=1 λ j (ω, σ)|ϕ j (t, ω, σ) ϕ j (t, ω, σ) | where ϕ j (t, ω, σ) ∈ L 2 ω (R d , Λ) is the solution to :

2 d 2 H

 22 i∂ t ϕ j (t, ω, σ) = 1 (σ)D y • D y + V (t, σ) Λ ϕ j (t, ω, σ)with ϕ j | t=0 = ϕ 0 j . Now, suppose that Theorem 1.16 (ii) fails. Therefore there exists µ ∈ M (1/h) which satisfies condition (70). Then there exists Λ ∈ L such thatµ final Λ (t, U × supp χ) dt = 0,for every t ∈ (0, T ), but such that µ final Λ = 0. This implies that μΛ = 0 and that, for μΛ -a.e. (ω, σ) with Λ ⊆ Λ σ :(73)Tr L 2 ω (R d ,Λ) ( 1 U Λ N Λ (t, ω, σ)) = 0.Comparing with (72), we obtain T 0 U |ϕ j (t, ω, σ)| 2 (y) dydt = 0,

  αN h + nδ h ,where ⌊LN h ⌋ is the integer part of LN h . Then,S τ h t h u h , Op h (a)S τ h t h u h L 2 (T d ) dt = S σ h t h u h , Op h (a)S σ h t h u h L 2 (T d ) dt S σ h t h u h , Op h (a)S σ h t h u h L 2 (T d ) dt

	put: β α n := 1 δ h := LN h ⌊LN h ⌋ , t h L 1 LN h βN h αN h = 1 LN h ⌊LN h ⌋ n=1 = 1 LN h ⌊LN h ⌋ n=1 where the functions v	t h n t h n-1 δ h 0	S σ h t h v h , Op h (a)S σ h t (n) h v h (n)	L 2 (T d )	dt,

  ), we can write for a ∈ C ∞ c T * T d and θ ∈ L 1 (R), we write:R θ (t) w h (t) , a dt = R θ(t) u h , S τ h t * h Op h (a)S τ h t h u h L 2 (T d ) dtOur assumptions on the semiclassical measure of the initial data implies that, for a.e.t ∈ R: µ t, T d × R d \ O = 0. Suppose that µ is obtained through the normalised sequence (u h ). Suppose that a ∈ C ∞ c T d × Oand that supp θ is compact. For 0 < h < h 0 small enough,

	=	1 (2π) d/2	k,j∈Z d	u h (k) u h (j) a j-k	h 2	(k + j)	R	θ(t)e -it τ h h (H(hk)-H(hj) dt
		=	1 (2π) d/2	h,j∈Z d	θ τ h	H (hk) -H (hj) h	u h (k) u h (j) a j-k	h 2	(k + j) .
	τ θ (0) (2π) d/2		kh,hj∈O	u h (k) u h (j) a j-k	h 2	(k + j)
							H(hk)=H(hj)
						= θ (0)

h H (hk) -H (hj) h / ∈ supp θ, ∀hk, hj ∈ O such that H (hk) = H (hj) .

Therefore, for such h, a and θ,

R θ (t) w h (t) , a dt = E h ∈H(hZ d )∩H(O) P E h u h , Op h (a)P E h u h L 2 (T d ) ,

  The symbolic calculus for Weyl's quantization implies:d ds S s h Op h (a s )S s * h = S s h Op h (∂ s a s )S s * h -+ h 2 V h (t) , Op h (a s ) S s *Therefore, for fixed s, S s h Op h (a s )S s * h = Op h (a) + O(h) (note that we have only used here the boundedness of the operator V h (t)) and forθ ∈ L 1 (R), t) u h , S τ h t * h Op h (a)S τ h t h u h dt = Op h (a)S τ h s h ) u h ds = O τ h h [V h (t), Op h (a)] L(L 2 (T d )) ,(for a only depending on ξ we have Op h (a) = a(hD x ), which commutes withH(hD x )). If V h (t) = Op h (V (t, x, ξ)) then [V h (t), Op h (a)] L(L 2 (T d )) = O(h) and τ h h [V h (t), Op h (a)] L(L 2 (T d ) = O(τ h h 2 ).

	i h	S s h H(hD) t 0 d ds w h (s) , a(ξ) ds
	=	0	t	u h ,	d ds	(S τ h s * h

h = O(h). R θ(t) w h (t) , a dt = R θ(R θ(t) u h , S τ h (t-s/τ h ) * h Op h (a • φ s )S τ h (t-s/τ h ) h u h dt + O(h) = R θ(t + s/τ h ) u h , S τ h t * Op h (a • φ s )S τ h t u h dt + O(h) = R θ(t + s/τ h ) w h (t) , a • φ s dt + O(h). Since θ(• + s/τ h )θ L 1 -→ 0 (recall that we have assumed that τ h -→ ∞ as h -→ 0 + ) we obtain R θ(t) w h (t) , a dt -R θ(t) w h (t) , a • φ s dt -→ 0, as h -→ 0 + ,

whence the invariance under φ s .

Let us now prove property (d). Consider µ the image of µ by the projection (x, ξ) -→ ξ, we have for a ∈ C ∞ 0 (R d ) :

w h (t) , a (ξ)w h u h , a (ξ) =

Acknowledgements. The authors would like to thank Jared Wunsch for communicating to them the construction in example (3) in Section 5.3. They are also grateful to Luc Hillairet for helpful discussions related to some of the results in Section 6. Part of this work was done as F. Macià was visiting the Laboratoire d'Analyse et de Mathématiques Appliquées at Université de Paris-Est, Créteil. He wishes to thank this institution for its support and hospitality.

Appendix: Basic properties of Wigner distributions and semi-classical measures

In this Appendix, we review basic properties of Wigner distributions and semiclassical measures. Recall that we have defined w h u h for u h ∈ L 2 T d as:

(74)

Start noticing that (74) admits the more explicit expression:

(75)

where

x (2π) d/2 dx denote the respective Fourier coefficients of u h and a, with respect to the variable x ∈ T d . By the Calderón-Vaillancourt theorem [START_REF] Calderón | On the boundedness of pseudo-differential operators[END_REF], the norm of Op h (a) is uniformly bounded in h: indeed, there exists an integer K d , and a constant C d > 0 (depending on the dimensiond) such that, if a is a smooth function on T * T d , with uniformly bounded derivatives, then

A proof in the case of L 2 (R d ) can be found in [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]. As a consequence of this, equation (74) gives:

Therefore, if w h (t,

an accumulation point of (w h ) for the weak- * topology.

It follows from standard results on the Weyl quantization that µ enjoys the following properties :

The unitary character of S t h implies that T * T d µ(t, dx, dξ) does not depend on t; from the normalization of u h , we have T * T d µ(τ, dx, dξ) ≤ 1, the inequality coming from the fact that T * T d is not compact, and that there may be an escape of mass to infinity. Such escape does not occur if and only if (u h ) is h-oscillating, in which case µ ∈ L ∞ R; P T * T d . (c) If τ h -→ ∞ as h -→ 0 + then the measures µ(t, •) are invariant under φ s , for almost all t and all s.