Wigner measures and effective mass theorems

We study a semi-classical Schr{\"o}dinger equation which describes the dynamics of an electron in a crystal in the presence of impurities. It is well-known that under suitable assumptions on the initial data, the wave function can be approximated in the semi-classical limit by the solution of a simpler equation, the effective mass equation. Using Floquet-Bloch decomposition and with a non-degeneracy condition on the critical points of the Bloch bands, as it is classical in this subject, we establish effective mass equations for more general initial data. Then, when the critical points are degenerated (which may occur in dimension strictly larger than one), we prove that a similar analysis can be performed, leading to a new type of effective mass equations which are operator-valued and of Heisenberg form. Our analysis relies on Wigner measure theory and, more precisely, to its applications to the analysis of dispersion effects.


The dynamics of an electron in a crystal and the effective mass equation
The dynamics of an electron in a crystal in the presence of impurities is described by a wave function Ψ(t, x) that solves the Schrödinger equation: The potential Q per is periodic with respect to some lattice in R d and describes the interactions between the electron and the crystal. The external potential Q ext takes into account the effects of impurities on the otherwise perfect crystal. Here denotes the Planck constant and m is the mass of the electrons. In many cases of physical interest, the ratio ε between the mean spacing of the lattice and the characteristic length scale of variation of Q ext is very small. After performing a suitable change of units, and rescaling the external potential and the wave function (see for instance [PR96]) the Schrödinger equation becomes: The potential V per is periodic with respect to a fixed lattice in R d , which, for the sake of definiteness will be assumed to be Z d .
Effective Mass Theory consists in showing that, under suitable assumptions on the initial data ψ ε 0 , the solutions of (1.2) can be approximated for ε small by those of a simpler Schrödinger equation, the effective mass equation, which is of the form: where, as usual, D x = 1 i ∂ x . The approximation has to be understood in the sense that any weak limits of the density |ψ ε (t, x)| 2 dxdt is the density |φ(t, x)| 2 dxdt as ε goes to 0. In the equation (1.3), B is a d × d matrix called the effective mass tensor, it generates the effective Hamiltonian The effective mass tensor is an experimentally accessible quantity that can be used to study the effect of the impurities on the dynamics of the electrons. Both the question of finding those initial conditions for which the corresponding solutions of (1.2) converge (in a suitable sense) to solutions to the effective mass equation and that of clarifying the dependence of B on the sequence of initial data have been extensively studied in the literature [AP05,BBA11,BLP78,HW11,PR96]. The effective mass tensor is related to the critical points of the Bloch modes. These are the eigenvalues of the operator P (ξ) on L 2 (T d ) which is canonically associated with the equation (1.2), We focus here on initial data which are structurally related with one of the Bloch mode in a sense that we will make precise later, we assume that this Bloch mode is of constant multiplicity and we introduce a new method for deriving rigorously the equation (1.3). The advantage of this method is that it allows to treat the case where the critical points of the considered Bloch modes are degenerate, leading to the introduction of a new family of Effective mass equations which are of Heisenberg type. Our strategy is based on the analysis of the dispersion of PDEs by a Wigner measure approach which has led us to develop global two microlocal Wigner measures in this specific context, while they are only defined locally in general ( [FK00,FK05]).

Floquet-Bloch decomposition
The analysis of Schrödinger operators with periodic potentials has a long history that has its origins in the seminal works by Floquet [Flo83] on ordinary differential equations with periodic coefficients, and by Bloch [Blo28], who developed a spectral theory of periodic Schrödinger operators in the context of solid state physics. Floquet-Bloch theory can be used to study the spectrum of the perturbed periodic Schrödinger operator: see for instance [Kuc01,Kuc04,Kuc16,RS78] and the references therein, and [GMS91,HW11,Out87] for results in the semiclassical context. The Floquet-Bloch decomposition gives as a result that the corresponding Schrödinger evolution can be decoupled in an infinite family of dispersive-type equations for the so-called Bloch modes. We briefly recall the basic facts that we shall need by following the approach in [Gér91a,GMS91].
The Floquet-Bloch decomposition is based on assuming that the solutions to (1.2) depend on both the "slow" x and the "fast" x/ε variables. The fast variables should moreover respect the symmetries of the lattice. This leads to the following Ansatz on the form of the solutions ψ ε of (1.2): where U ε (t, x, y) is assumed to be Z d -periodic with respect to the variable y (and, therefore, that it can be identified to a function defined on R × R d × T d , where T d denotes the torus R d /Z d ). The function U ε then satisfies the equation: (1.6) where the operator L ε maps functions F defined on R d × T d on functions on R d according to: and P (εD x ) denotes the operator-valued Fourier multiplier associated with the symbol ξ → P (εξ) defined in (1.4). The initial condition in (1.6) can be interpreted in terms of the natural embedding L 2 (R d x ) → L 2 (R d x × T d y ) by taking U ε 0 (x, y) = ψ ε 0 (x) ⊗ 1(y). One can also have more elaborated identifications depending on the structure of the initial data, as we shall see later. Identity (1.5) makes sense, since one can check that, under suitable assumptions on the initial datum, U ε (t, x, · ) has enough regularity with respect to the variable y (the fact that ψ ε must be given by (1.5) following from the uniqueness of solutions to the initial value problem (1.2)).
Assuming that the function y → V per (y) is smooth is enough for proving that the operator P (ξ) is self-adjoint on L 2 (T d ) (with domain H 2 (R d )) and has a compact resolvent. For the sake of simplicity, we shall make here this assumption, even though it can be relaxed into assuming V per ∈ L p (T d ) for some convenient set of indices p which authorizes Coulombian singularity in dimension 3 (see [Lew17]). As a consequence of the fact that P (ξ) has compact resolvent, there exist a non-decreasing sequence of eigenvalues (the so-called Bloch energies): the multiplicity of the eigenvalue n (ξ) is equal to the same constant for all ξ ∈ R d , then n and the eigenprojector Π n on this mode are globally analytic functions of ξ. The reader can refer to [Kuc16] for a survey on the subject.
Observing that, via the decomposition in Fourier series, any U ∈ L 2 (R d x × T d y ) can be written as:

Main result
We consider the following set of assumptions.
(H1) Assume V per is smooth and real-valued and that V ext is a continuous function in time taking values in the set of smooth, real-valued, bounded functions on R d with bounded derivatives. (H2) Assume that n is a Bloch mode of constant multiplicity and that the set of critical points of n is a submanifold of R d . (H3) Assume that the Hessian d 2 n (ξ) is of maximal rank above each point ξ ∈ Λ n (or equivalently that Ker d 2 n (ξ) = T ξ Λ n for all ξ ∈ Λ n ), (H4) Assume that the initial data ψ ε 0 (x) satisfies for some s > d/2. It will be convenient to identify n to a function defined on (R d ) * rather than R d (via the standard identification by duality). Then we define the cotangent bundle of Λ n as the union of all cotangent spaces to Λ n (1.9) Note that this is welldefined, since T * Λ n ⊂ (R d ) * * = R d . We shall denote by M + (T * Λ n ) the set of positive Radon measures on T * Λ n . We also define the normal bundle of Λ n which is the union of those linear subspaces of R d that are normal to Λ n : (1.10) to denote the operator acting on L 2 (N ξ Λ n ) by multiplication by φ(v + · ). Note that assumption (H3) implies that the Hessian of n defines an operator d 2 n (ξ)D z · D z acting on N ξ Λ n for any ξ ∈ Λ n . In the statement below, the weak limit of the energy density are described by means of a time-dependent family M n of trace-class operators acting on a certain L 2space. More precisely, the operators M n depend on t ∈ R and on ξ ∈ Λ n , v ∈ T * ξ Λ n ; for every choice of these parameters, M n (t, v, ξ) is a trace-class operator acting on L 2 functions of the vector space N ξ Λ. Note that M n (t, · ) can also be viewed as a section of a vector bundle over T * Λ n , namely: (v,ξ)∈T * Λ L 1 + (L 2 (N ξ Λ n )).
Theorem 1.1. -Assume the hypotheses (H1) to (H4). Then, there exist a subsequence (ε k ) k∈N , a positive measure ν n ∈ M + (T * Λ n ), and a measurable family of self-adjoint, positive, trace-class operator = 1, such that for every for every a < b and every φ ∈ C c (R d ) one has: where M n ( · , v, ξ) ∈ C(R; L 1 + (L 2 (N ξ Λ n )) solves the Heisenberg equation: (1.11) Remark 1.2. -We point out that the measure ν n and the family of operators M 0,n only depend on the subsequence ψ ε k 0 of initial data. The way of computing them will be made clear in Section 5.
When the critical points of n (ξ) are all non degenerate, then the set Λ n is discrete and 2πZ d -periodic, T * Λ n = Λ n × {0} and N Λ n = R d . We then have the following corollary.
Corollary 1.3. -Assume we have assumptions (H1) to (H4) and that the critical points of n (ξ) are all non degenerate. Then the measure ν n and the operator M n of Theorem 1.1 above satisfy: (1) The operator M n (t, ξ) is the orthogonal projection on ψ ξ which solves the effective mass equation: with initial data:

ANNALES HENRI LEBESGUE
(2) The measure ν n is given by This corollary is well known and we refer to the work by Allaire and Piatnitski [AP05] or to [AP06] for similar results in a related problem; in that work homogenization and two-scale convergence techniques are used to obtain a precise description of the solution profile for similar data than ours and for Bloch mode having non-degenerated critical points. In [BBA11], Barletti and Ben Abdallah obtained a result similar to Corollary 1.3 by following the approach initiated by Kohn and Luttinger in [LK55] consisting in introducing a (non-canonical) basis of modified Bloch functions.
The starting point in our approach is conceptually closer to that in [PR96], in the sense that we analyse the structure of Wigner measures associated to sequences of solutions. The main novelty here is the use of two-microlocal Wigner measures, that give a more explicit geometric description of the mechanism that underlies the Effective Mass Approximation, showing that it is a result of the dispersive effects associated to high-frequency solutions to the semiclassical Bloch band equations. Moreover, we are able to deal with the presence of non-isolated critical points on the Bloch energies and to prove Theorem 1.1. We believe our approach is sufficiently robust to be implemented on a Bloch band, isolated from the remainder of the spectrum, and consisting of several Bloch modes which may present crossings. We will devote further works to this specific problem. It is also interesting to notice that our result generalizes to initial data which are a finite sum of data satisfying Assumption (H4). The weak limit of the energy density associated with the solution corresponding to this new data is the sum of weak limits of the energy densities of the solution associated with each term of the data, without any interference (see Section 6.5 for a precise statement).

Strategy of the proof
The proof of Theorem 1.1 relies on the analysis of the solution U ε to equation (1.6) with initial data U ε 0 as introduced in Assumption (H4), and more precisely on its component U ε n on the n th Bloch mode and its restriction ψ ε n by L ε : It is shown in Section 6.3 that the family (ψ ε n ) solves the equation There, we prove that (1.14) (1.14) shows that no other Bloch modes is concerned in the decomposition of U ε and ψ ε : the mass of ψ ε remains above the specific mode n because it is separated from the other ones. Therefore, a crucial step in this strategy consists in performing a detailed analysis of the dispersive equation (1.13).

Structure of the article
Sections 2 to 5 are devoted to the analysis of a dispersive equation of the form (1.13) in a more general setting. For this, we use pseudodifferential operators and semiclassical measures (Section 3) and we introduce two-microlocal tools (Section 4) that allow us to prove the main results of Section 2 in Section 5. Finally, in Section 6 we come back to the effective mass equations and prove Theorem 1.1, which requires additional results on the restriction operator L ε , the projector Π n (ξ) and energy estimates for solutions to (1.6). Some Appendices are devoted to basic results about pseudodifferential calculus and trace-class operator-valued measures, and to the proof of technical lemma.

Quantifying the lack of dispersion
As emphasized in the introduction, understanding the limiting behavior as ε → 0 of the position densities of solutions to the Schrödinger equation (1.2) relies on a careful analysis of the solutions of equations of the form: This equation ceases to be dispersive as soon as λ(ξ) has critical points ξ = 0, and this is always the case if λ is a Bloch energy. Heuristically, one can think that one of the consequences of a dispersive time evolution is a regularization of the high-frequency effects (that is associated to frequencies εξ = c = 0) caused by the sequence of initial data. These heuristics have been made precise in many cases; a presentation of our results from this point of view can be found in [CFKM19]. The reader can also find there a detailed account on the literature on the subject.
Here we show that, in the presence of critical points of λ, some of the highfrequency effects exhibited by the sequence of initial data persist after applying the time evolution (2.1). We provide a quantitative picture of this persistence by giving a complete description of the asymptotic behavior of the densities |u ε (t, x)| 2 dxdt associated to a bounded sequence (u ε ) of solutions to (2.1). We give an explicit procedure to compute all weak-accumulation points of the sequence of positive measures (|u ε (t, x)| 2 dxdt) in terms of quantities that can be obtained from the sequence of initial data (u ε 0 ). These results are of independent interest; we have thus chosen to present them in a more general framework than what is necessary in our applications to Effective Mass Theory.
In order to obtain a non trivial result we must make sure that the characteristic length-scale of the oscillations carried by the sequence of initial data is of the order of ε. The following assumption is sufficient for our purposes: (H0) The sequence (u ε 0 ) is uniformly bounded in L 2 (R d ) and ε-oscillating, in the sense that its energy is concentrated on frequencies smaller or equal than 1/ε: We shall assume that λ is smooth and grows at most polynomially, and that its set of critical points is a submanifold of R d . More precisely, we impose the following hypotheses on λ and V : ( together with its derivatives, grows at most polynomially; i.e. there exists N > 0 such that, for every α ∈ N d + , one has: sup is a connected, closed embedded submanifold of R d of codimension 0 < p d and the Hessian d 2 λ is of maximal rank over Λ. The hypothesis (H2) implies the existence of tubular coordinates in a neighborhood of Λ. A stronger version of (H2) is to suppose that all critical points of λ are nondegenerate (that is, the Hessian of λ, d 2 λ(ξ) is a non-degenerate quadratic form for every ξ ∈ Λ). This implies that p = d and Λ is a discrete set in R d ; if moreover one has that λ is Z d -periodic, which is the situation when λ is a Bloch energy, this set is finite modulo Z d . We first state the main result of this section under this stronger hypothesis.
Theorem 2.1. -Suppose that the sequence of initial data (u ε 0 ) verifies (H0), denote by (u ε ) the corresponding sequence of solutions to (2.1). Suppose in addition that (H1) is satisfied and all critical points of λ are non-degenerate. Then there exists a subsequence (u ε k 0 ) such that for every a < b and every φ ∈ C c (R d ) the following holds: where u ξ solves the following Schrödinger equation: with initial data: If Λ = ∅ then the right-hand side of (2.3) is equal to zero.
Note that u ξ may be identically equal to zero even if the sequence (u ε 0 ) oscillates in the direction ξ. For instance, if the sequence of initial data is a coherent state: Theorem 2.1 allows us to conclude that the corresponding solutions (u ε ) converge to zero in L 2 loc (R × R d ). Theorem 2.1 can be interpreted as a description of the obstructions to the validity of smoothing-type estimates for the solutions to equation (2.1) in the presence of critical points of the symbol of the Fourier multiplier. We refer the reader to [CFKM19] for additional details concerning this issue and a simple proof of Theorem 2.1. Here, we obtain Theorem 2.1 as a particular case of a more general result which requires some geometric preliminaries.
As for the mode Bloch n in the Introduction, we identify λ to a function defined on (R d ) * rather than R d , and we associate with Λ its cotangent bundle T * Λ and its normal bundle N Λ. In the analogue of Theorem 2.1 in this context, the sum over critical points is replaced by an integral with respect to a measure over T * Λ, and the Schrödinger equation (2.4) becomes a Heisenberg equation for a time-dependent family M of trace-class operators of (v,ξ)∈T * Λ L 1 + (L 2 (N ξ Λ)).
Theorem 2.2. -Let (u ε 0 ) be a sequence of initial data satisfying (H0), and denote by (u ε ) the corresponding sequence of solutions to (2.1). If (H1) and (H2) hold, then there exist a subsequence (u ε k 0 ), a positive measure ν ∈ M + (T * Λ) and a measurable family of self-adjoint, positive, trace-class operators such that for every a < b and every φ ∈ C c (R d ) one has: ) solves the following Heisenberg equation: Remark 2.3. -When the (H2) hypothesis about the rank of the Hessian d 2 λ is dropped, then an additional term appears in (2.5) (see [CFKM19]).
When Λ consists of a set of isolated critical points, Theorems 2.1 and 2.2 are completely equivalent. Note that in this case, T * Λ = {0} × Λ and the measure ν (which in this case is a measure depending on ξ ∈ R d only) is simply In addition, N ξ Λ = R d and the operator M t (ξ) (which again does not depend on z) is the orthogonal projection onto u ξ (t, · ) in L 2 (R d ) (recall that u ξ solves the Schrödinger equation (1.12)). These orthogonal projections satisfy the Heisenberg equation (2.6).
The proof of Theorem 2.2 follows a strategy developed in the references [AFKM15, AM12, Mac10] in a different (though related) context. As in those references, the measure ν and the family of operators M 0 only depend on the subsequence of initial data (u ε k 0 ); we will see in Section 3 that they are defined as two microlocal Wigner measures of (u ε k 0 ) in the sense of [FK95,FK00,FK05,Mac10]. At this point, it might be useful to stress out that in this regime the limiting objects M, ν cannot be computed in terms of the Wigner/semiclassical measure of the sequence of initial data, as it is the case when dealing with the semiclassical limit. In [CFKM19], we have explicitly constructed sequences of initial data having the same semiclassical measure but such that their time dependent measures differ. This type of behavior was first remarked in this context in the case of the Schrödinger equation on the torus, see [Mac09,Mac10].
We also emphasize that the original definition of two-microlocal Wigner measures performed in [FK00] and their extension to more general geometric setting [FK05] were only defined locally. We prove here that they extend to global objects in the geometric context of closed simply connected embedded submanifolds of R d ; related constructions were performed in the torus [AFKM15, AM12, Mac10, MR18] and the disk [ALM16].
See also, that as soon as Λ has strictly positive dimension (i.e. it is not a union of isolated critical points), the measure ν may be singular with respect to the z variable, while when Λ consists in isolated points, the weak limit of the densities |ψ ε (t, x)| 2 dx are proved to be uniformly continuous with respect to the measure dx. See [CFKM19] for specific examples exhibiting this type of behavior; see also that reference for examples proving the necessity of Hypothesis (H2); it is shown there that different types of behavior can happen whenever the Hessian of λ is not of full rank on Λ.
The main idea of the proof comes from the following remark.
, which means that, in the preceding analysis, we performed the semiclassical limit ε → 0 in (2.7) simultaneously with the limit t/ε → +∞. Such analysis, combining high-frequencies (ε → 0) and long times (t ∼ t ε → +∞) is relevant if one wants to understand the behavior of solutions of (2.7) beyond the Ehrenfest time. This approach was followed in the case of confined geometries in the references [AFKM15,Mac09,MR16]. Note also that in the particular case when λ(ξ) is homogeneous of degree two, this change of time scale transforms the semiclassical equation (2.7) into the non-semiclassical one (that is, the one corresponding to ε = 1). Therefore, it is possible to derive results on the dynamics of the Schrödinger equation via this scaling limit, see [ALM16,AM14,AR12,Mac10]. The reader can consult the survey articles [AM12,Mac11] and the introductory lecture notes [Mac15] for additional details and references on this approach.

Pseudodifferential operators and semiclassical measurespreliminaries
In this section we recall some basic facts on Wigner distributions and semiclassical measures, which are the tools we are going to use to prove Theorem 2.2, and derive preliminary results about Wigner measures associated with families of solutions of equations of the form (2.1).

Wigner transform and Wigner measures
Wigner distributions provide a useful way for computing weak-accumulation points of a sequence of densities |f ε (x)| 2 dx issued from a L 2 -bounded sequence (f ε ) of solutions of a semiclassical (pseudo) differential equation. They provide a joint physical/Fourier space description of the energy distribution of functions in R d . The Wigner distribution of a function f ∈ L 2 (R d ) is defined as: and has several interesting properties (see, for instance, [Fol89]). • f on x or ξ gives the position or momentum densities of f respectively: where op ε (a) is the semiclassical pseudodifferential operator of symbol a obtained through the Weyl quantization rule: . This is proved using identity (3.1) combined with the fact that the operators op ε (a) are uniformly bounded by a suitable seminorm in S(R d × R d ), see (A.1). Appendix A contains additional facts on the theory of pseudodifferential operators, as well as references to the literature.
In addition, every accumulation point of ( is an accumulation point of (W ε f ε ) along some subsequence (ε k ) and (|f ε k | 2 ) converges weakly-towards a measure ν ∈ M + (R d ) then one has: Equality holds if and only if (f ε ) is ε-oscillating: GL93,GMMP97]. The Hypothesis (H0) that we made on the initial data for equation (2.1), is this ε-oscillating property. Note also that (3.2) implies that µ is always a finite measure of total mass bounded by sup ε f ε 2 is uniformly bounded for some constant s > 0, then the family f ε is ε-oscillating.

Wigner measure and family of solutions of dispersive equations
We will now consider Wigner distributions associated to solutions of the evolution equation (2.1) where V ext and λ satisfy hypothesis H1 and (g ε (t, · )) is locally uniformly bounded with respect to t in L 2 (R d ).
When the sequence (u ε 0 ) of initial data is uniformly bounded in L 2 (R d ), so is the corresponding sequence (u ε (t, · )) of solutions to (2.1) for every t ∈ R. Therefore the sequence of . Nevertheless, its time derivatives are unbounded and, in general, one cannot hope to find a subsequence that converges pointwise (or even almost everywhere) in t (see Proposition 3.4 below). This difficulty can be overcome if one considers the time-average of the Wigner distributions.
Proposition 3.2. -Let (u ε ) be a sequence of solutions to (2.1) issued from an L 2 (R d )-bounded family of initial data (u ε 0 ). Then there exist a subsequence (ε k ) tending to zero as k → ∞ and a t-measurable family µ t ∈ M + (R d × R d ) of finite measures, with total mass essentially uniformly bounded in t ∈ R, such that, for every θ ∈ L 1 (R) and a ∈ C ∞ c (R d × R d ): If moreover, the families (u ε 0 ) and g ε (t, · ) are ε-oscillating, then for every θ ∈ L 1 (R) and φ ∈ C ∞ c (R d ): This result is proved in [Mac09, Theorem 1]; see also [MR16,Appendix B]. Note that its proof uses the following observation.

Localisation of Wigner measures on the critical set
The fact that (u ε (t, · )) is a sequence of solutions to (2.1) imposes restrictions on the measures µ t that can be attained as a limit of their Wigner functions. In the region in the phase space R d x × R d ξ where equation (2.1) is dispersive (i.e. away from the critical points of λ) the energy of the sequence (u ε (t, · )) is dispersed at infinite speed to infinity. These heuristics are made precise in the following result.
Proposition 3.4. -Let (u ε (t, · )) be a sequence of solutions to (2.1) issued from an L 2 (R d )-bounded and ε-oscillating sequence of initial data (u ε 0 ), and suppose that the measures µ t are given by Proposition 3.2. Then, for almost every t ∈ R the measure µ t is supported above the set of critical points of λ: The result of Proposition 3.4 follows from a geometric argument : the fact that u ε are solutions to (2.1) translates in an invariance property of the measures µ t .
Lemma 3.5. -For almost every t ∈ R, the measure µ t is invariant by the flow This means that for every function a on R d × R d that is Borel measurable one has: This result is part of [Mac09, Theorem 2]. We reproduce the argument here for the reader's convenience, since we are going to use similar techniques in the sequel.
Proof of Lemma 3.5. -It is enough to show that, for all a ∈ C ∞ c (R d × R d ) and θ ∈ C ∞ c (R), the quantity and, using the fact that u ε solves (2.1): This estimate together with identity (3.1) show that R ε (θ, a) = O(ε), which gives the result that we wanted to prove. Proposition 3.4 follows easily from Lemma 3.5 and the following elementary fact.
Lemma 3.6. -Let Ω ⊂ R d and Φ s : R d × Ω −→ R d × Ω a flow satisfying: for every compact K ⊂ R d × Ω such that K contains no stationary points of Φ there exist constants α, β > 0 such that: Let µ be a finite, positive Radon measure on R d × Ω that is invariant by the flow Φ s . Then µ is supported on the set of stationary points of Φ s .
Proof. -It suffices to show that µ(K) = 0 for every compact set K ⊂ R d ×Ω as in the statement of the Lemma 3.6. By the assumption made on Φ s , it is possible to find a sequence s k → +∞ such that Φ s k (K), k ∈ N, are mutually disjoint. The invariance property of µ implies that µ(Φ s k (K)) = µ(K) and therefore, for every N > 0: Since µ is finite, we must have µ(K) = 0.

Two-microlocal Wigner distributions
The localization result for semiclassical measures that we obtained in the preceding section is still very far from the conclusions of Theorems 2.1 and 2.2. In particular, Proposition 3.4 does not explain how the measures µ t depend on the sequence of initial data of the sequence of solutions (u ε (t, · )). For obtaining more information, we use two-microlocal tools that we introduce in a rather general framework in this section.
From now on, we assume that X is a connected, closed embedded submanifold of (R d ) * with codimension p > 0. Given any σ ∈ X, T σ X and N σ X will stand for the cotangent and normal spaces of X at σ respectively (as defined in (1.9) and (1.10)).
The tubular neighborhood theorem (see for instance [Hir94]) ensures that there exists an open neighborhood U of {(σ, 0) : σ ∈ X} ⊆ N X such that the map: is a diffeomorphism onto its image V . Its inverse is given by: for some smooth map σ : V −→ X. When X = {ξ 0 } consists of a single point, the function σ is constant, identically equal to ξ 0 .
We extend the phase space The test functions associated with this extended phase space are those functions a ∈ C ∞ (T * R d x,ξ × R d η ) which satisfy the two following properties: (1) There exists a compact K ⊂ T * R d such that, for all η ∈ R p , the map (x, ξ) → a(x, ξ, η) is a smooth function compactly supported in K.
(2) There exists a smooth function a ∞ defined on T * R d × S d−1 and R 0 > 0 such that, if |η| > R 0 , then a(x, ξ, η) = a ∞ (x, ξ, η/|η|). We denote by A the set of such functions and for a ∈ A we write: Since a ε (x, εξ) = a x, εξ, εξ−σ(εξ) has derivatives that are uniformly bounded in ε, the Calderón-Vaillancourt Theorem (see Appendix A) gives the uniform boundedness of the family of operators (op ε (a ε )) ε>0 in L 2 (R d ). In addition, any a ∈ C ∞ c (R d × V ) can be naturally identified to a function in A which does not depend on the last variable. For such a, one clearly has Putting the above remarks together, one obtains the following.
-Let (f ε ) ε>0 be bounded in L 2 (R d ); suppose in addition that this sequence has a semiclassical measure µ. Then, (W X,ε f ε ) ε>0 is a bounded sequence in D (R d × V × R d ) whose accumulation points µ X satisfy: The distributions µ X turn out to have additional structure (they are not positive measures on R d × V × R d , though) and can be used to give a more precise description of the restriction µ R d ×X of semiclassical measures. The measure µ X decomposes into two parts: a compact part, which is essentially the restriction of µ X to the and a part at infinity, which corresponds to the restriction to the sphere at infinity R d × V × S d−1 .

The compact part
For σ ∈ X, we define functions of L 2 (N σ X) as functions where z is the parameter of a parametrization of N σ X. These parametrizations depend on the system of equations of X that we choose in a neighborhood of the point σ. Let ϕ(ξ) = 0 be such a system in an open set Ω that we can assume included in the set V where the map σ is defined. Then, a parametrization of N σ X associated to this system of equations is Besides, one associate with the system ϕ(ξ) = 0 a smooth map ξ → B(ξ) from the neighborhood Ω of σ into the set of d × p matrices such that Given a function a ∈ C ∞ c (R d × Ω × R d ) and a point (σ, v) ∈ T X, we can use the system of coordinates ϕ(ξ) = 0 to define an operator acting on f ∈ L 2 (N σ X) given by: In other words, Q ϕ a (σ, v) is obtained from a by applying the non-semiclassical Weyl quantization to the symbol If one changes the system of coordinates into ϕ(ξ) = 0 on some open neighborhood Ω of σ, then, there exists a smooth map R(ξ) defined on the open set Ω ∩ Ω (where both system of coordinates can be used), and valued in the set of invertible p × p matrices, such that ϕ(ξ) = R(ξ)ϕ(ξ). One then observe that the matrix B(ξ) associated with the choice of ϕ is given by where U (σ) is the unitary operator of L 2 (N σ X) ∼ L 2 (R p ) associated with the linear map from R p into itself : z → t R(σ)z. More precisely, This map is the one associated with the change of parametrization on N σ X induced by turning ϕ into ϕ, and the map (z, ζ) → ( t R(σ)z, R(σ) −1 ζ) is a symplectic transform of the cotangent of R p . This is the standard rule of transformation of pseudodifferential operators through linear change of variables (see [AG07] for an example or any textbook about pseudodifferential calculus). Because of this invariance property with respect to the change of system of coordinates, we shall say that a defines an operator Q a on L 2 (N σ X). Clearly, Q a (σ, v) is smooth and compactly supported in (σ, v); moreover, Q a (σ, v) ∈ K(L 2 (N σ X)), for every (σ, v) ∈ T X, where K(L 2 (N σ X)) stands for the space of compact operators on L 2 (N σ X).
The proof of this lemma is in the Appendix C. This lemma reduces the problem to the analysis of the concentration of the bounded family f ε = (U ε f ) on the submanifold Λ 0 = {ξ = 0} which has the additional property to be a vector space. This special case has been studied in [CFKM19, p. 96-97, Proposition 2] where it is proved that up to a subsequence, there exist a positive measure ν 0 on T * R d−p and a measurable family of trace 1 operators: The reader will find in Appendix B comments on the operator-valued families. Therefore, for compactly supported a ∈ A, and choosing Note that the map θ → σ = Φ(0, θ ) is a parametrization of X with associated parametrization of T * X, Since the Jacobian of this mapping is 1, after a change of variable, we obtain an operator valued measurable family M on T * X and a measure ν on T * X such that We now take advantage of the fact that ϕ(Φ(ζ)) = ζ for all ζ ∈ R d in order to write dϕ(Φ(ζ))dΦ(ζ) = (Id, 0).

Measure structure of the part at infinity
To analyze the part at infinity, we use a cut-off function χ ∈ C ∞ c (R d ) such that 0 χ 1, χ(η) = 1 for |η| 1 and χ(η) = 0 for |η| 2, Observe that a R is compactly supported in all variables. We thus focus on the second part, and more precisely on the quantity We denote by SΛ the compactified normal bundle to Λ, viewed as a submanifold of R d × R d , the fiber of which is T * σ R d × S σ Λ above σ with S σ Λ being obtained by taking the quotient of N σ Λ by the action of R * + by homotheties. Proposition 4.4. -Let (f ε ) be a bounded family of L 2 (R d ). There exists a subsequence ε k and a measure γ on SΛ such that for all a ∈ A, where X c denotes the complement of the set X in R d .
Proof. -We begin by recalling the arguments that prove the existence of the measure γ, which are the same that the one developed in the vector case in [CFKM19]. Since a = a ∞ for |η| large enough, we have a R = a R ∞ as soon as R is large enough and the quantity lim sup will only depend on a ∞ . Therefore, by considering a dense subset of C c (T * R d × S d−1 ), we can find a subsequence (ε k ) by a diagonal extraction process such that the following linear form on This implies that the symbolic calculus on symbols (a R ) ε is semiclassical with respect to the small parameter √ ε 2 + R −2 . To be precise, one has the following weak Gårding inequality: if a 0, then, for all κ > 0, there exists a constant C κ such that

ANNALES HENRI LEBESGUE
We then conclude that the linear form defined above is positive and defines a positive Radon measure ρ. It remains to compute ρ outside X. In this purpose, we set and we observe that, by the definition of µ: which concludes the proof of the existence of the measure γ.
Let us now analyze the geometric properties of this measure. We choose a system of local coordinates of Λ and introduce the matrix B as in (4.3). By Lemma 4.3 and the result of [CFKM19] for vector spaces: up to a subsequence, there exists a The mapping ξ → Φ(0, ξ ) is a parametrization of X and the mapping is the associated mapping of T * X R d . Therefore, this relation defines a measure γ on the bundle T * X × S p−1 such that Besides, using that for any σ 0 ∈ X, we deduce that for any ζ ∈ T σ 0 R d , we have the decomposition Now, since dϕ is of rank p, one can write any ω ∈ S p−1 as ω = dϕ(σ 0 )ζ and the points B(σ 0 )ω are in N σ 0 X. By identification of γ in (4.5), we deduce that γ(x, σ, · ) is a measure on the set which completes the proof of the Proposition 4.4. TOME 3 (2020)

Two microlocal Wigner measures and families of solutions to dispersive equations
We now consider families of solutions to equation (2.1). As proved in Proposition 3.4, the Wigner measure of the family (u ε (t, · )) concentrates on the critical set Λ = {∇λ(ξ) = 0}. In order to analyze µ t above Λ, we perform a second microlocalization above the set X = Λ, with average in time. We consider for θ ∈ L 1 (R) the quantities R θ(t) W Λ,ε u ε (t,·) , a dt for symbols a ∈ A. Up to extracting a subsequence ε k , we construct L ∞ maps valued respectively on the set of positive Radon measures on R d × Λ × S d−1 , on the set of positive Radon measures on T * Λ and finally on the set of measurable families from T * Λ onto the set of positive trace class operators on L 2 (N Λ), such that for θ ∈ L 1 (R) and a ∈ A: The measures γ t and ν t , and the map M t satisfy additional properties coming from the fact that the family (u ε (t, · )) solves a time-dependent equation. These properties are discussed in the next two sections. We shall see that the measures γ t are invariant under a linear flow and that we can choose the sequence ε k such that the map t → M t is continuous (and even C 1 ).

Transport properties of the compact part
Since Λ is the set of critical points of λ, the matrix d 2 λ is intrinsically defined above points of Λ. Thus, using the formalism of the preceding sections, Proposition 5.1. -The map t → ν t is constant and the map t → M t (σ, v) ∈ C(R; L 1 + (L 2 (N σ Λ)) solves the Heisenberg Equation (2.6).
Proof. -We analyze for a ∈ C ∞ c (R 3d ) the time evolution of the quantity W Λ,ε u ε (t,·) , a . We have d dt By standard symbolic calculus for Weyl quantization, we have in Besides, by Taylor formula and by use of ∇λ(σ(ξ)) = 0, we have where Γ is a smooth matrix. This yields a(x, ξ, η).
At this stage of the proof, we see that d dt W Λ,ε u ε (t,·) , a is uniformly bounded in ε, thus using a suitable version of Ascoli's theorem and a standard diagonal extraction argument, we can find a sequence (ε k ) such that the limit exists for all a ∈ C ∞ c (R 3d ) and all time t ∈ [0, T ] (for some T > 0 fixed) with a limit that is a continuous map in time. The transport equation that we are now going to prove shall guarantee the independence of the limit from T > 0.
We observe that for any local system of equations of Λ, ϕ(ξ) = 0, the operator Q ϕ b satisfies for (σ, v) ∈ T Λ, On the other hand, we observe that, setting we have and we now focus on the matrix t dϕ(σ) t B(σ)d 2 λ(σ)B(σ), and thus on the properties of the hessian d 2 λ(σ).
For ξ ∈ Λ, the bilinear form d 2 λ(ξ) is defined intrinsically on T ξ R d and d 2 λ(ξ) = 0 on T ξ Λ. We deduce from (4.6) that any ζ ∈ T ξ R d satisfies . Taking into account this information, Equation (5.2) becomes Taking the trace, we get ∂ t ν t = 0, thus ν t is equal to some constant measure ν and M t satisfies Equation (2.6), which proves the Proposition 5.1.

Invariance and localization of the measure at infinity
We are concerned with the property of the L ∞ -map t → γ t (dx, dσ, dω) valued in the set of positive Radon measures on SΛ. We now define a flow on SΛ by setting for s ∈ R φ s 2 : (x, σ, ω) → (x + s d 2 λ(σ)ω, σ, ω).
Proposition 5.2. -The measure γ t is invariant by the flow φ s 2 . Proof. -The proof essentially follows the lines of the proof of [AFKM15, Theorem 2.5]. We use the cut-off function χ introduced before and set where for all multi-index α, β ∈ N d , there exists a constant C α,β > 0 such that r R,δ ε satisfies: sup As a consequence, W Λ,ε u ε (t,·) , r R,δ ε is uniformly bounded in R, δ, ε and: uniformly with respect to R and ε, with Note that this symbol is smooth because |ξ − σ(ξ)| > R ε on the support of a R,δ . We are going to prove that for all θ ∈ C ∞ c (R),

ANNALES HENRI LEBESGUE
Indeed, by the calculus of the preceding section, we have The symbol c R,δ s is such that for all multi-index α ∈ N d , there exists C α > 0 for which: This implies in particular: By symbolic calculus, we have We deduce that for all θ ∈ C ∞ c (R), As a conclusion, which implies the Proposition 5.2.

Proofs of Theorems 2.1 and 2.2
Remind that Theorem 2.2 implies Theorem 2.1, thus we focus on Theorem 2.2. We first observe that the measure γ t is zero. Indeed, by (H2); for σ ∈ Λ, d 2 λ(σ) is one to one on N σ Λ. Therefore, since γ t is a measure on SΛ, the invariance property of Proposition 5.2 and an argument similar to the one of Lemma 3.6 yields that γ t = 0.
As a consequence, the semi-classical measure µ t is only given by the compact part and one has for any a ∈ C ∞ c (R 2d ) and θ ∈ L 1 (R), Then, taking θ = 1 [a,b] for a, b ∈ R, a < b, and in view of Proposition 3.2 and of Lemma 3.3, we deduce that for every φ ∈ C c (R d ) one has for the subsequence defining M t and ν t : where M t satisfies (2.6). This concludes the proof of Theorem 2.2. We emphasize that the measure ν and the operator valued family M 0 are utterly determined by the initial data.

Bloch projectors and semiclassical measures
In this section we prove Theorem 1.1, as a result of the analysis in Section 4. We shall use properties of the operator of restriction L ε defined in (1.7) and of the projector Π n (εD x ). Then, we prove a priori estimates for solutions of equation (1.6) and use them to reduce the dynamics of our original problem to those of equation (1.13) (Corollary 6.8).
Note that, modulo adding a positive constant to equation (1.2), we may assume that P (εD x ) is a non-negative operator. With this in mind, the following estimates, that will repeatedly used in what follows, hold.

High frequency behavior of the operator of restriction to the diagonal and of the Bloch projectors
We first focus on the properties of the operator of restriction to the diagonal L ε and prove its boundedness in appropriate functional spaces.
Lemma 6.2. -Suppose s > d/2, then the operator ) satisfies the estimate: where 1 R is the characteristic function of {|ξ| > R}, then the sequence (L ε U ε ) is bounded in L 2 (R d ) and ε-oscillating.
for some r > d/2. Then condition (6.1) is satisfied for every d/2 < s < r. This follows from the bound: and U ε 2 Then there exist constants C, C d,s > 0 such that and therefore: Let us now show that, under the hypothesis of the proposition, v ε := L ε U ε defines an ε-oscillating sequence. Given δ > 0, since s > d/2, there exists N δ > 0 such that . Therefore, it suffices to show that for any δ > 0 the sequence If R > R 0 for R 0 > 0 large enough, one has 1 R (· + 2πk) 1 R/2 for every |k| N δ . This allows us to conclude that for R > R 0 : and the conclusion follows.

TOME 3 (2020)
We shall also need information on the derivatives with respect to ξ of the operator Π n (ξ). We recall the formula where the functions χ j ∈ C ∞ (R d /2πZ d ) form a partition of unity and, for j = 1, . . . , N , the set C j is a contour in the complex plane separating n (ξ), for ξ ∈ suppχ j , form the remainder of the spectrum. The existence of such contours is guaranteed by the fact that n (ξ) is of constant multiplicity for all ξ ∈ R d and, thus, is separated from the remainder of the spectrum. As a consequence of this formula, of Lemma 6.1 and of the relation we deduce the following result.
Lemma 6.4. -The map ξ → Π n (ξ) is a smooth bounded map from R d into L(L 2 (T d )). In addition, the operator Π n (εD x ) maps the space H s ε (R d × T d ) into itself.

A priori estimates on U ε (t, · )
In order to derive the desired properties of ψ ε n (t, x), the solution to (1.13), we need to prove some a priori estimates for the solutions of equation (1.6). We will use them for reducing the analysis of ψ ε (t, · ) (the solution to our original problem (1.2)) to that of ψ ε n (t, · ).
Proof. -In view of Remark 6.1, we are first going to study the families ( εD x U ε ) and (P (εD x ) 1/2 U ε ).
Start noticing that εD x U ε satisfies the equation As a consequence, using the boundedness of ∇ x V ext on R × R d , we obtain by the symbolic calculus of semiclassical pseudodifferential operators, that the source term can be estimated by:

ANNALES HENRI LEBESGUE
for some constant C > independent of ε > 0 and t ∈ R. Using standard energy estimates, we deduce the existence of a constant C 1 > 0 such that for all t ∈ R, A completely analogous argument yields the estimate: A standard recursive argument gives, for all s ∈ N, the existence of a constant C s > 0 such that for all t ∈ R, and the result follows for any s ∈ R + by interpolation. We now focus on the case where the initial data U ε 0 belongs to a particular Bloch eigenspace: Note that by Lemma 6.4, for any t ∈ R, the family U ε (t, · ) is uniformly bounded in Lemma 6.7. -Assume U ε 0 = Π n (εD x )U ε 0 and consider U ε (t, · ) as defined above. Then, for all T > 0, there exists C T > 0 such that Let us prove now Lemma 6.7.
We have U ε (0, · ) = U ε (0, · ) and U ε solves . The symbolic calculus of semiclassical pseudodifferential operators implies that: As for εD x U ε one has: Again, the symbolic calculus gives that C ε (t) εD x U ε (t, · ) L 2 (R d ×T d ) = O(ε) locally uniformly in t. Taking into account that εD x U ε satisfies equation (6.4) and is bounded in L 2 (R d × T d ), one concludes that: Cε|t|.
An analogous reasoning holds for P (εD x ) 1/2 (U ε (t, · ) − U ε (t, · )). One concludes using an inductive argument following the lines of the end of the proof of Lemma 6.5.

Analysis of the Bloch component ψ ε n
By the definition of ψ ε n (t, x), we have ψ ε n (t, · ) = L ε U ε (t, · ); and the family is bounded in L 2 (R d ) for all t ∈ R. Moreover, as a corollary of Lemma 6.7, the following holds.
The proof is a direct consequence of Lemma 6.7, since Lemma 6.2 ensures that . We now conclude our analysis of the Bloch component ψ ε n (t, · ). The following result gathers the remaining information that we will need in order to conclude, together with Corollary 6.8, the proof of Theorem 1.1. Proposition 6.9. -The family ψ ε n solves equation (1.13) Cε for all t ∈ R, ε > 0. Proof. -Let us first prove that ψ ε n solves (1.13). We denote by J the set of the indexes of the Bloch eigenfunctions ϕ j ( · , ξ) which form an orthonormal basis of Ran Π n (ξ). Define for j ∈ J, x, y)dy, and notice that: Since U ε solves (6.5) and P (ξ)ϕ j ( · , ξ) = n (ξ)ϕ j ( · , ξ) for all ξ ∈ R d , the family u ε j solves: iε 2 ∂ t u ε j (t, x) = n (εD x )u ε j (t, x) + ε 2 V ext (t, x)u ε j (t, x) + ε 2 g ε j (t, x), where: x)]U ε (t, x, y)dy.
Summing the relations over j ∈ J, this implies (1.13) with f ε n = L ε [Π n (εD x ), V ext ]U ε . Now, Lemma 6.2 and the symbolic calculus of pseudodifferential operators gives, for any t ∈ R: which concludes the proof of Proposition 6.9.

Proofs of Theorems 1.1
The proof of Theorem 1.1 (which implies Corollary 1.3) easily follows from our results so far.

Some comments on initial data that are a finite superposition of Bloch modes
Our results also apply to initial data that are a finite linear combination of the form: with N a finite subset of N such that for all n ∈ N , P (εD x )U ε 0,n = n (εD x )U ε 0,n , for distinct n of constant multiplicity and U ε 0,n uniformly bounded in H s ε (R d × T d ) for all n ∈ N .
Proposition 6.10. -Assume we turn assumption (H4) into (6.6) in the hypotheses and that assumptions (H2), (H3) hold for every n with n ∈ N . Then, there exist a subsequence (ε k ) k∈N , positive measures ν n ∈ M + (T * Λ n ), and measurable families of self-adjoint, positive, trace-class operators M 0,n : T * ξ Λ n (v, ξ) −→ M 0,n (v, ξ) ∈ L 1 + (L 2 (N ξ Λ n )), Tr L 2 (N ξ Λn) M 0,n (v, ξ) = 1, These operators are bounded in L 2 (R d ). The Calderón-Vaillancourt Theorem [CV71] ensures the existence of a constant C d > 0 such that for every a ∈ S one has for some J 0 ∈ N depending only on d. We make use repeatedly of the following result, known as the symbolic calculus for pseudodifferential operators.
for some constant C > 0 independent of a, b and ε.

Appendix B. Trace operator-valued measures
In this appendix we recall general considerations on operator-valued measures. Let X be a complete metric space and (Y, σ) a measure space; write H := L 2 (Y, σ) and denote by L 1 (H), K(H) and L(H) the spaces of trace-class, compact and bounded operators on H respectively. A trace-operator valued Radon measure on X is a linear functional: M : C 0 (X) −→ L 1 (H) satisfying the following boundedness condition. For every compact K ⊂ X there exist a constant C K > 0 such that: Tr |M (φ)| C K sup K |φ|, ∀ φ ∈ C 0 (K).
Such an operator-valued measure is positive if for every φ 0, M (φ) is an Hermitian positive operator. Let M be a positive trace operator-valued measure on X, denote by ν ∈ M + (X) the positive real measure defined by: Clearly, both j and e j , j ∈ N, are locally ν-integrable and where, as usual, |e j (x) e j (x)| denotes the orthogonal projection in H onto e j (x). Moreover, as a consequence of the monotone convergence theorem, the following result easily follows.
Lemma B.1. -Let M be a positive trace operator-valued measure on X. Then there exist a non-negative function ρ ∈ L 1 loc (X, ν; L 1 (Y, σ)) such that, for every a ∈ C 0 (X; L ∞ (Y, σ)) one has: where m a (x) denotes the operator acting on H by multiplication by a(x, · ). The density ρ is given by: ρ(x, y) = ∞ j=1 j (x)|e j (x, y)| 2 .