Effective Mass Theorems with Bloch Modes Crossings

Abstract

We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.

1 Introduction

1.1 Description of the problem

We consider the dynamics of an electron in a crystal in the regime of small wave-length comparable to the characteristic scale of the crystal. After a suitable rescaling (see for instance [52]), such an analysis leads to an $\varepsilon$-dependent Schrödinger equation where $\varepsilon$ is a small parameter $\varepsilon \ll 1$

$$\begin{aligned} \left\{ \begin{array}{l} i\partial _t \psi ^\varepsilon (t,x)+\dfrac{1}{2} \Delta _x \psi ^\varepsilon (t,x) - \dfrac{1}{\varepsilon ^{2}} V_{\mathrm{per}}\left( \dfrac{x}{\varepsilon }\right) \psi ^\varepsilon (t,x) - V_{\mathrm{ext}}(t,x)\psi ^\varepsilon (t,x) =0, \\ \psi ^\varepsilon |_{t=0}=\psi ^\varepsilon _0. \end{array}\right. \nonumber \\ \end{aligned}$$

(1.1)

The potential $V_{\mathrm{per}}$ is supposed to be smooth, real-valued and ${{\mathbb {Z}}}^d$-periodic; it models the interactions due to the crystalline structure. The external potential $V_{\mathrm{ext}}$, takes into accounts the impurities; we assume that $t\mapsto V_{\mathrm{ext}}(t,\cdot )$ is a bounded map from ${{\mathbb {R}}}$ into the set of smooth, real-valued functions on ${{\mathbb {R}}}^d _x$ with bounded derivatives. The times-scales of the equation 1.1 are characteristic of the analysis of the obstructions to the dispersion of the energy. It is the long time scaling studied in [1, 2, 7, 9, 37, 52, 54], by contrast to the short time analysis that allows to analyze transport effects and is performed for example in [8, 13, 38, 51, 55] (the Schrödinger equation therein is obtained from (1.1) by changing t into $\tau =\varepsilon t$).

1.1.1 The Wave Function, Observables and Quadratic Quantities

We are interested in the asymptotic behavior of the time-averaged position densities $|\psi ^\varepsilon (t,x)|^2$ as $\varepsilon$ goes to 0. The wave function itself cannot be directly measured, but these densities that allow to compute the probability P(A) of finding the position of the electron in a set A at time t according to

$$\begin{aligned} P(A) = \int _{A} |\psi ^\varepsilon (t,x)|^2 \mathrm{d}x. \end{aligned}$$

We are interested in describing at leading order in $\varepsilon$ the value of this probability for bounded sets A. In other words, we would like to characterize, in the set of time-dependent probability measures, the time-averaged limit as $\varepsilon \rightarrow 0^+$ of the position densities $|\psi ^\varepsilon (t,x)|^2dx$, i.e. the limits as $\varepsilon \rightarrow 0^+$ of the quantities

$$\begin{aligned} \int _a^b\int _{{{\mathbb {R}}}^d} \phi (x)|\psi ^\varepsilon (t,x)|^2 \mathrm{d}x\,\mathrm{d}t,\;\; \phi \in {{\mathcal {C}}}_0({{\mathbb {R}}}^d),\;\; a<b, \end{aligned}$$

(1.2)

where ${\mathcal {C}}_0({{\mathbb {R}}}^d)$ stands for the space of continuous compactly supported functions on ${{\mathbb {R}}}^d$. Note that these objects do not capture the fraction of the mass of the measure that goes to infinity in x as $\varepsilon$ goes to 0.

We will derive representations of these limits in terms of Effective mass equations. In its full generality, our result also describes the evolution of the action of observables on the wave functions. Indeed, the wave function itself has no physical meaning and it is the evolution of quadratic quantities such as those of Section 3.1 that carries information like the average momenta or the average energy. In (1.2), the averaging in time takes into account the fact that a physical observation is not instantaneous and, though small, its duration is not negligible. However, we will discuss situations where local in time point-wise information can be derived about the evolution of the energy densities (see Section 1.4.3).

1.1.2 Floquet-Bloch Theory

It is classical in this context to use Floquet-Bloch theory in order to diagonalize $-\frac{1}{2}\Delta _x+V_{\mathrm{per}}$. To this end, one introduces, for $\xi \in {{\mathbb {R}}}^d$, the operator

$$\begin{aligned} P(\xi ):= \frac{1}{2} |\xi +D_y|^2 + V_{\mathrm{per}}(y),\;\; y\in {{\mathbb {T}}}^d, \end{aligned}$$

where ${{\mathbb {T}}}^d={{\mathbb {R}}}^d\backslash {{\mathbb {Z}}}^d$ is a flat torus. It is well known that this operator is essentially self-adjoint on $L^2({{\mathbb {T}}}^d)$ with domain $H^2({{\mathbb {T}}}^d)$, and has a compact resolvent, hence a non-decreasing sequence of eigenvalues counted with their multiplicities, which are called Bloch energies or band functions

$$\begin{aligned} \varrho _1(\xi )\le \varrho _2(\xi )\le \cdots \le \varrho _n(\xi )\longrightarrow +\infty , \end{aligned}$$

and an orthonormal basis of eigenfunctions $\left( \varphi _n(\cdot ,\xi )\right) _{n\in {{\mathbb {N}}}^*}$ called Bloch waves or Bloch modes, satisfying for all $\xi \in {{\mathbb {R}}}^d$ and $n\in {{\mathbb {N}}}^*$:

$$\begin{aligned} P(\xi )\varphi _n(\cdot ,\xi )=\varrho _n(\xi )\varphi _n(\cdot ,\xi ). \end{aligned}$$

(1.3)

Both Bloch waves and Bloch energies are continuous functions of the $\xi$-variable. Besides, for all $k\in 2\pi {{\mathbb {Z}}}^d$, the operator $P(\xi +k)$ is unitarily equivalent to $P(\xi )$ through multiplication by $y\mapsto \mathrm{e}^{ik\cdot y}$, which implies that for all $n\in {{\mathbb {N}}}^*$, the maps $\xi \mapsto \varrho _n(\xi )$ are $2\pi {{\mathbb {Z}}}^d$-periodic and the map $\xi \mapsto \varphi _n (\cdot ,\xi )$ belongs to ${\mathcal {C}}({{\mathbb {R}}}^d_\xi , L^2({{\mathbb {T}}}^d_y))$. The spectrum of $-\frac{1}{2}\Delta _x+V_{\mathrm{per}}$ is then the union of the Bloch bands $B_n:=\varrho _n([0,2\pi ]^d)$, which are closed intervals:

$$\begin{aligned} {{\,\mathrm{Sp}\,}}\left( -\frac{1}{2}\Delta _x+V_{\mathrm{per}}\right) =\bigcup _{n\in {{\mathbb {N}}}^*}B_n. \end{aligned}$$

The Bloch energies $\xi \mapsto \varrho _n(\xi )$ are Lipschitz functions which are analytic outside a set of zero Lebesgue measure (see [56]). In particular Bloch energies that are of constant multiplicity as $\xi$ varies are always analytic functions of $\xi$. These energies are then called isolated. The opposite situation, that is, when two, otherwise distinct, Bloch energies coincide at some point $\xi$ is referred to as a crossing. At those points, the multiplicity is greater than one and the corresponding Bloch bands have non-empty intersection. When the space dimension is one, two Bloch bands can touch at one edge and their crossing set consists on isolated points (see “Appendix A” and the references therein); in higher dimensions more complicated situations can occur: most bands overlap (in fact as soon as $d\ge 2$ only a finite number of gaps exist) and the crossing set may be a higher dimensional manifold (in fact, the union of the graphs of the band functions form a real analytic variety). The survey article [39] provides additional details on these issues.

1.1.3 Effective Mass Theory

Sometimes also called effective Hamiltonian theory, effective mass theory consists in showing that, under suitable assumptions on the initial data $\psi ^\varepsilon _0$, the energy density associated with the solutions of (1.1) can be approximated for $\varepsilon$ small by those of a simpler Schrödinger equation, the Effective mass equation, which does not depend on $\varepsilon$ and involves quantities related to the Bloch energies.

Effective mass equations have then been derived in various contributions [1, 2, 7, 9, 37, 52, 54] under the assumptions that the orthogonal projection of the initial datum $\psi ^\varepsilon _0$ on spectral subspaces corresponding to the non-simple Bloch energies is negligible, and that critical points of this band functions are non-degenerate.

All these contributions emphasize the important role played by the set of critical points of the Bloch energies. Indeed, the group velocity of the n-th mode is $\varepsilon ^{-1} \nabla \varrho _n(\varepsilon \xi )$, which becomes infinite in the limit $\varepsilon \rightarrow 0$; this implies that the obstructions to the dispersive effects created by the n-th band, $n\in {{\mathbb {N}}}^*$, have to be found above the set $\Lambda _n$ of critical points of the function $\varrho _n$:

$$\begin{aligned} \Lambda _n:=\{\xi \in {{\mathbb {R}}}^d\,:\,\nabla \varrho _n(\xi )=0\}. \end{aligned}$$

(1.4)

Quantifying the loss of mass at infinity due to dispersion can be done for instance as in in [2], where the authors introduce a $\varepsilon$-dependent drift in order to take into account the displacement to infinity of part of the mass.

A second feature that is assumed in the aforementioned references is the simplicity of the band functions, which is an important technical ingredient in the proofs. Simple band functions are smooth, and therefore group velocity is well defined everywhere. This property may fail in the presence of band crossings which produce at worst a loss of regularity at the crossing points. In that case, the group velocity $\varepsilon ^{-1}\nabla _\xi \varrho _n (\varepsilon \xi )$ is no longer defined at the crossing points, even though it may have directional limits, the archetype being the conical singularity $\varrho _n (\xi )\sim \xi /|\xi |$ close to $\xi =0$. Our aim here is to deal with this difficulty: we will not take into account the loss of mass at infinity and focus instead on the limit (1.2); note, however, that our results take into account situations more general than the ones considered in references quoted above, since we allow for band crossings and the singularities they infer.

This motivates the introduction of the crossing set of two distinct Bloch energies:

$$\begin{aligned} \Sigma _{n,n'}:=\{\xi \in {{\mathbb {R}}}^d\,:\,\varrho _n(\xi )=\varrho _{n'}(\xi )\},\quad n,n'\in {{\mathbb {N}}}^*, \;\varrho _n\ne \varrho _{n'}. \end{aligned}$$

(1.5)

The band functions $\varrho _n$, $n\in {{\mathbb {N}}}^*$, are piece-wise real analytic; their non-smoothness points lie in the union of crossing sets $\bigcup _{\varrho _n\ne \varrho _{n'}}\Sigma _{n,n'}$. We will also consider the sets

$$\begin{aligned} \Sigma _n:=\Sigma _{n,n+1},\quad n\in {{\mathbb {N}}}^*. \end{aligned}$$

(1.6)

The crossing problematic has been addressed since long for equations that are scaled differently in the small parameter, in particular by George Hagedorn in the 90s [36]. Since then, different approaches have been devoted to understand propagation through crossings, from the use of normal forms [16, 17], the analysis in terms of Wigner measures [23, 26, 27] and Wigner functions [29,30,31], up to, more recently, the analysis in terms of wave packets [55] in the case of non-singular crossings. Indeed, the growing interest in crossings, especially conical ones, is linked with the technological interest of new materials that are topological insulators (see [19, 20] and references therein). However, this question has never been addressed in the context of the particular scaling in $\varepsilon$ of equation (1.1).

In [14, 15], the range of validity of the Effective Mass Theory has been extended to include degenerate critical points, through the introduction of a new class of Effective mass equation which are of von Neumann type. However, the Bloch modes involved in the description of the initial data are still assumed to be of constant multiplicity. Our aim here is to consider situations where different Bloch energies may have non-empty intersections inducing singularities and to treat rather general initial data. Our result gives a a complete description of the weak limits of the densities $|\psi ^\varepsilon (t,x)|^2$ as $\varepsilon$ goes to 0 when the crossings are conical, in a sense that we shall make precise later. This is done through the analysis of the weak limits of the Wigner function associated with $\psi ^\varepsilon (t,x)$. Indeed, the Wigner function introduced in Section 3 below, plays the role of a generalized energy-density in the phase space $T^*{{\mathbb {R}}}^d= {{\mathbb {R}}}^d_x\times {{\mathbb {R}}}^d_\xi$, the density $|\psi ^\varepsilon (t,x)|^2$ being its projection in the configuration space ${{\mathbb {R}}}^d_x$. Our result covers all possible cases when $d=1$ and generic situations in higher dimensions. We also complete the description of the picture by providing a characterisation of these limits when crossings are degenerate, exhibiting the persistence of terms due to interactions between the Bloch energies that cross. Our results rely on the use of a two-microlocal analysis in the spirit of [3, 4, 6, 42, 45], using two-scale Wigner distributions [22, 24, 49, 50].

1.2 General Assumption on the Initial Data

We denote by $A(\varepsilon D_x)$, for $\varepsilon >0$, the scaled Fourier multiplier associated with the function $A(\xi )$, i.e. the operator satisfying

$$\begin{aligned} \forall f\in {{\mathcal {S}}}({{\mathbb {R}}}^d),\quad \widehat{A(\varepsilon D_x) f}(\xi )= A(\varepsilon \xi ) {{\widehat{f}}}(\xi ), \end{aligned}$$

where the following normalization has been used for the Fourier transform

$$\begin{aligned} {{\widehat{f}}}(\xi )=\int _{{{\mathbb {R}}}^d} \text {e}^{-i\xi \cdot x}f(x)\mathrm{d}x. \end{aligned}$$

Along the paper, we will consider the functions spaces $H^s_\varepsilon ({{\mathbb {R}}}^d)$, defined for $s\ge 0$, that are the Sobolev spaces equipped with the norms

$$\begin{aligned} \Vert f\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d)}:=\Vert \left\langle \varepsilon D_x\right\rangle ^s f\Vert _{L^2({{\mathbb {R}}}^d)}, \end{aligned}$$

where $\left\langle \xi \right\rangle :=(1+|\xi |^2)^{1/2}$.

Any function $U\in L^2({{\mathbb {R}}}^d_x\times {{\mathbb {T}}}^d_y)$ can be written in terms of Fourier series as

$$\begin{aligned} U(x,y)=\sum _{k\in {{\mathbb {Z}}}^d}U_k(x)\mathrm{e}^{i2\pi k\cdot y}\;\;\mathrm{with}\;\; \Vert U\Vert _{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}^2=\sum _{k\in {{\mathbb {Z}}}^d}\Vert U_k\Vert _{L^2({{\mathbb {R}}}^d)}^2. \end{aligned}$$

We denote by $H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$, for $s\ge 0$, the Sobolev space consisting of those functions $U \in L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$ such that there exists $\varepsilon _0,C>0$ for which we have

$$\begin{aligned} \forall \varepsilon \in (0,\varepsilon _0),\;\;\Vert U\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}^2:=\sum _{k\in {{\mathbb {Z}}}^d}\int _{{{\mathbb {R}}}^d}(1+|\varepsilon \xi |^2+|k|^2)^s|\widehat{U_k}(\xi )|^2\mathrm{d}\xi \le C.\nonumber \\ \end{aligned}$$

(1.7)

These functions can be projected on the bands as follows. For every $n\in {{\mathbb {N}}}^*$ and $\xi \in {{\mathbb {R}}}^d$, we denote by $\Pi _n(\xi )$ the projector from $L^2({{\mathbb {T}}}^d)$ onto the eigenspace corresponding to $\varrho _n(\xi )$. The corresponding Fourier multiplier $\Pi _n(\varepsilon D_x)$ acts on $L^2({{\mathbb {R}}}_x^d\times {{\mathbb {T}}}_y^d)$, since band functions are bounded. Finally, we define the operator $L^\varepsilon$ acting on functions $F\in H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$, $s>d/2$, by

$$\begin{aligned} (L^\varepsilon F)(x):=F\left( x,\frac{x}{\varepsilon }\right) . \end{aligned}$$

Then there exists $C_s>0$ such that, for every $F\in H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$,

$$\begin{aligned} \Vert L^\varepsilon F\Vert _{L^2({{\mathbb {R}}}^d)}\le C_s \Vert F\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}, \end{aligned}$$

(1.8)

uniformly in $\varepsilon >0$. See [15, Lemma 6.2].

We will make the following assumption on the family of initial data in (1.1).

• ${{\textbf {H}}}{{\textbf {0}}}$  There exists a bounded family $(U^\varepsilon _0)$ in $H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$ for some $s>d/2$ such that

  $$\begin{aligned} \psi ^\varepsilon _0=L^\varepsilon U^\varepsilon _0. \end{aligned}$$

Note that if $(\psi ^\varepsilon _0)_{\varepsilon >0}$ is bounded in $H^s_\varepsilon ({{\mathbb {R}}}^d)$ with $s>d/2$, H0 holds with $U^\varepsilon _0(y,x)= \psi ^\varepsilon _0(x)\otimes \mathbf{1}_{{{\mathbb {T}}}^d}(y)$.

1.3 The Case of the Dimension One

Let us first state our results in dimension $d=1$. When $d=1$, one can prove that $\Lambda _n$ is contained in $\pi {{\mathbb {Z}}}$, and consists only on non-degenerate critical points. In addition, when $|n-n'|>1$, $\Sigma _{n,n'}= \emptyset$ and $\Sigma _n\cap \Lambda _n=\emptyset$

(see Lemma A.1). In this specific case, we are able to give a complete description of the limit of the energy density of families of solutions to (1.1) with initial data of the form stated in H0.

Theorem 1.1

Assume $(\psi ^\varepsilon _0 )$ satisfies H0. Then there exists a subsequence $(\psi ^{\varepsilon _\ell }_0)_{\ell \in {{\mathbb {N}}}}$ of the initial data, such that for every $a<b$ and every $\phi \in {\mathcal {C}}_0({{\mathbb {R}}})$,

$$\begin{aligned} \lim _{\ell \rightarrow \infty }\int _a^b\int _{{{\mathbb {R}}}}\phi (x)|\psi ^{\varepsilon _\ell }(t,x)|^2\mathrm{d}x\mathrm{d}t= \sum _{n\in {{\mathbb {N}}}^*}\; \sum _{\xi \in \Lambda _n}\int _a^b\int _{{{\mathbb {R}}}}\phi (x)|\psi _{\xi }^{(n)} (t,x)|^2\mathrm{d}x\mathrm{d}t, \end{aligned}$$

where, for every $n\in {{\mathbb {N}}}^*$ and $\xi \in \Lambda _n$, $\psi _\xi ^{(n)}$ solves the effective mass Schrödinger equation

$$\begin{aligned} i\partial _t \psi _{\xi }^{(n)}(t,x) ={1\over 2}\partial ^2_\xi \varrho _n(\xi )\partial _x^2 \psi _{\xi }^{(n)}(t,x)+V_{\mathrm{ext}}(t,x)\psi _{\xi }^{(n)}(t,x) \end{aligned}$$

(1.9)

with initial datum

$\psi _{\xi }^{(n)}|_{t=0}$ is the weak limit in $L^2({{\mathbb {R}}})$ of the sequence $\left( \mathrm{e}^{-\frac{i}{\varepsilon _\ell } \xi x} L^{\varepsilon _\ell } \Pi _n(\varepsilon _\ell D_x)U^{\varepsilon _\ell }_0\right)$.

Note some of the accumulation points of $\mathrm{e}^{-\frac{i}{\varepsilon _\ell } \xi x} L^{\varepsilon _\ell } \Pi _n(\varepsilon _\ell D_x)U^{\varepsilon _\ell }_0$ may just be 0. For example if one has $V_{\mathrm{per}}=0$, only the first Bloch energy $\varrho _1$ has critical points and they are precisely $\Lambda _1=2\pi {{\mathbb {Z}}}$. The addition, the associated projector $\Pi _1(\xi )$ coincides with the orthogonal projection onto ${{\mathbb {C}}}\mathrm{e}^{i ky}$ whenever $\xi \in (k-\pi ,k+\pi )$ and $k\in 2\pi {{\mathbb {Z}}}$. Therefore, if one takes $U^\varepsilon _0(x,y)=\psi ^{\varepsilon }_0(x)\otimes \mathbf{1}_{y\in {{\mathbb {T}}}^d}$, then

$$\begin{aligned} \Pi _1(\varepsilon \xi )\widehat{U^{\varepsilon _\ell }_0}(\xi ,\cdot )={\mathbf {1}}_{(-\pi ,\pi )}(\varepsilon \xi )\widehat{\psi ^{\varepsilon _\ell }_0}(\xi ), \end{aligned}$$

and $\mathrm{e}^{-\frac{i}{\varepsilon _\ell } 2\pi k x} \Pi _1(\varepsilon D_x)(\psi ^{\varepsilon _\ell }_0\otimes \mathbf{1}_{y\in {{\mathbb {T}}}})$ weakly converges to zero when $k\ne 0$.

Theorem 1.1 is derived as a consequence of a more general analysis that is valid in any dimension under assumptions that are satisfied for all Bloch energies when $d=1$, and that is presented in the next section.

1.4 The Generic case with $d\ge 1$-Conical Crossings

We now present results that, under a set of assumptions that always hold when $d=1$, give a description of effective mass equations in higher dimension under the presence of generic crossings for data satisfying H0.

1.4.1 Assumptions

Our first assumption concern the multiplicity of Bloch bands.

• ${{\textbf {H}}}{{\textbf {1}}}$  For $n\in {{\mathbb {N}}}^*$, the multiplicity of the Bloch energy $\varrho _n$ is one, except at crossing points, where it is two. This implies that a global labeling of the band functions exists such that $\Sigma _{n,n'}\ne \emptyset$ implies $|n-n'|=1$.

Remark 1.2

Hypothesis H 1 is thought to be generic, as follows from the variational characterization of eigenvalues of Schrödinger operators with Bloch periodicity conditions. We make it in order to avoid having statements that are unnecessarily involved. As we stated it, it prevents from having simultaneous crossings of more than two Bloch energies, and higher multiplicities (both scenarios are non-generic). The proofs we provide can be adapted in order to deal with these situations.

We also consider a generic assumption on the set of critical points $\Lambda _n$ defined in (1.4).

• ${{\textbf {H}}}{{\textbf {2}}}$  For $n\in {{\mathbb {N}}}^*$, we assume that ${{\,\mathrm{Hess}\,}}\varrho _n$ is of constant rank in a neighborhood of each connected component of $\Lambda _n$.

Remark 1.3

Let $X\subseteq \Lambda _n$ be a connected component of $\Lambda _n$. By the constant rank level set theorem, this hypothesis implies that each connected component $X\subseteq \Lambda _n$ is a closed submanifold of ${{\mathbb {R}}}^d$ of dimension $d-{{\,\mathrm{rk}\,}}{{\,\mathrm{Hess}\,}}\varrho _n|_X$.

Finally, our third set of hypothesis concerns the geometry of the crossing sets $\Sigma _n$. For stating this assumption, we introduce geometric objects associated with a submanifold X of $({{\mathbb {R}}}^d)^*$: we consider its tangent spaces $T_\xi X$ and define the fibre of the normal bundle NX of X above $\xi \in X$ as the vector space $N_\xi X$ consisting of those $\eta \in ({{\mathbb {R}}}^d)^{**}={{\mathbb {R}}}^d$ that annihilate $T_\xi X$

$$\begin{aligned} NX:=\{(\xi ,\eta )\in X\times {{\mathbb {R}}}^d \, : \, \eta \cdot \zeta =0 ,\;\;\forall \zeta \in T_\xi X \}. \end{aligned}$$

(1.10)

With a Bloch mode $\varrho _n$ presenting crossings on a manifold $\Sigma _n$, we associate the function $g_n$ defined on $N\Sigma _n$ by

$$\begin{aligned} (\xi ,\eta )\mapsto g_n(\xi ,\eta ):=\frac{1}{2}\left( \varrho _{n+1}(\xi +\eta ) -\varrho _n(\xi +\eta )\right) ,\;\;\xi \in \Sigma _n,\;\;\eta \in N_\xi \Sigma _n.\nonumber \\ \end{aligned}$$

(1.11)

Note that $g_n(\xi ,\eta )\ge 0$ and $g_n(\xi ,\eta )=0$ if and only if $\eta =0$. Besides, for any $\xi \in \Sigma _n$, $\eta \mapsto g_n(\xi ,\eta )$ is differentiable in all $\eta \not =0$ (see “Appendix B”). We denote by $\nabla _\eta g_n(\xi ,\eta )$ this differential, which can be identified with a vector of $N_\xi \Sigma _n\subset T_\xi ({{\mathbb {R}}}^d)^*$.

Definition 1.4

We say that the crossings of $\Sigma _n$ are conic if and only if there exists a neighborhood U of $\Sigma _n$ such that $\varrho _n$ and $\varrho _{n+1}$ are of multiplicity 1 outside $\Sigma _n$ in U and there exists $c>0$ such that

$$\begin{aligned}\forall (\xi ,\eta )\in N \Sigma _n,\;\; |g_n(\xi ,\eta )|\ge c |\eta |. \end{aligned}$$

Note that the critical sets $\Lambda _n$ contain no conical crossing point. Besides, one can prove (see “Appendix B”) that, generically, as soon as the crossing set $\Sigma _n$ is a closed submanifold of ${{\mathbb {R}}}^d$, either $\varrho _n$ has a conical singularity along $\Sigma _n$, either $\varrho _n$ is in ${\mathcal {C}}^{1,1}$. We set, for $n\in {{\mathbb {N}}}^*$,

$$\begin{aligned} \lambda _n(\xi )=\frac{1}{2} \left( \varrho _n(\xi )+\varrho _{n+1}(\xi )\right) ,\;\;\xi \in {{\mathbb {R}}}^d,\;\; n\in {{\mathbb {N}}}^*, \end{aligned}$$

(1.12)

and we introduce the following last assumption:

• ${{\textbf {H}}}{{\textbf {3}}}$  For $n\in {{\mathbb {N}}}^*$, we assume that the crossing set $\Sigma _n$ is a smooth closed submanifold of ${{\mathbb {R}}}^d$. Moreover, the crossing is of conic type in the sense of Definition 1.4 and for all $\xi \in \Sigma _n$, $\eta \in N_\xi \Sigma _n$ with $\eta \not =0$,

  $$\begin{aligned} \nabla _\xi \lambda _{n} (\xi ) \pm \nabla _{\eta } g_n(\xi ,\eta ) \not =0. \end{aligned}$$

1.4.2 The Result: Dispersion Above Conical Crossings

For stating the result, we need to introduce other geometric objects associated with a submanifold X of $({{\mathbb {R}}}^d)^*$. We define its cotangent bundle as the union of all cotangent spaces to X

$$\begin{aligned} T^*X:=\{(\xi ,x)\in X\times {{\mathbb {R}}}^d \, : \, x \in T_\xi ^* X \}, \end{aligned}$$

(1.13)

each fibre $T_\xi ^*X$ is the dual space of the tangent space $T_\xi X$. We shall denote by ${\mathcal {M}}_+(T^*X)$ the set of non-negative Radon measures on $T^*X$. We observe that every point $x\in {{\mathbb {R}}}^d$ can be uniquely written as

$$\begin{aligned} x=v+z\;\; \text{ where } \;\; v\in T^*_\xi X\;\; \text{ and }\;\; z\in N_\xi X. \end{aligned}$$

Then, given a function $\phi \in L^\infty ({{\mathbb {R}}}^d)$ and a point $(\xi ,v)\in T^*X$, we denote by $m^X_\phi (\xi ,v)$ the operator acting on $L^2(N_\xi X)$ by multiplication by $\phi (v+\cdot )$. We shall denote by ${\mathcal {L}}(L^2(N_\xi X))$ the set of bounded operators acting on $L^2(N_\xi X)$ and by ${\mathcal {L}}^1_+(L^2(N_\xi X))$ the set of operators that are non-negative and trace-class. When $X=\Lambda _n$ and assumption H2 holds, we will consider the operator $\mathrm{Hess}\, \varrho _n(\xi )D_z\cdot D_z$ acting on $N_\xi \Lambda _n$ for any $\xi \in \Lambda _n$.

Theorem 1.5

Assume H1, H2 and H3 are satisfied for all $n\in {{\mathbb {N}}}^*$ and consider $(\psi ^\varepsilon )_{\varepsilon >0}$ a family of solutions to equation (1.1) with an initial data $(\psi ^\varepsilon _0)_{\varepsilon >0}$ that satisfies H0. Then, there exist a subsequence $(\psi ^{\varepsilon _\ell }_0)_{\ell \in {{\mathbb {N}}}}$ of the initial data, a sequence of non negative measures $(\nu _n)_{n\in {{\mathbb {N}}}}$ on $T^*\Lambda _n$, and a sequence of measurable non negative trace-class operators $(M_n)_{n\in {{\mathbb {N}}}}$

$$\begin{aligned} M_n:T^*_\xi \Lambda _n\ni (\xi ,v) \mapsto M_n(\xi ,v)\in {\mathcal {L}}^1_+(L^2(N_\xi \Lambda _n)),\;\;\mathrm{Tr} _{L^2(N_\xi \Lambda _n)}M_n(\xi ,v) =1, \end{aligned}$$

both depending only on $(\psi ^{\varepsilon _\ell }_0)_{\ell \in {{\mathbb {N}}}}$, such that for every $a<b$ and every $\phi \in {{\mathcal {C}}}_0({{\mathbb {R}}}^d)$ one has

$$\begin{aligned}&\lim _{\ell \rightarrow +\infty } \int _a^b\int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\varepsilon _\ell } (t,x)|^2 \mathrm{d}x \mathrm{d}t \nonumber \\&\quad = \sum _{n\in {{\mathbb {N}}}} \int _a^b \int _{T^*\Lambda _n}\mathrm{Tr} _{L^2(N_\xi \Lambda _n)} \left( m^{\Lambda _n}_\phi (\xi ,v)M^t_n(\xi ,v)\right) \nu _n(d\xi ,dv)\mathrm{d}t, \end{aligned}$$

(1.14)

where $t\mapsto M^t_n(\xi ,v) \in {\mathcal {C}}({{\mathbb {R}}}, {\mathcal {L}}^1_+(L^2(N_\xi \Lambda _n))$ solves the von Neumann equation

$$\begin{aligned} \left\{ \begin{array}{l} i\partial _t M^t_n (\xi ,v) =\left[ \frac{1}{2}\mathrm{Hess} \varrho _n(\xi ) D_z\cdot D_z + m_{V_{\mathrm{ext}}}^{\Lambda _n} (\xi ,v) \;,\; M^t_n(\xi ,v) )\right] \\ M_n^0= M_n. \end{array}\right. \end{aligned}$$

(1.15)

(recall that $m^{\Lambda _n}_\phi (\xi ,v)$ (resp. $m^{\Lambda _n}_{V_{\mathrm{ext}}}(\xi ,v)$) denotes the operator acting on $L^2(N_\xi \Lambda _n)$ by multiplication by $\phi (v+\cdot )$ (resp. $V_{\mathrm{ext}}(v+\cdot )$)).

Above and throughout this article, when A and B are two operators acting on the same Hilbert space, the notation [A, B] denotes the commutator $AB-BA$. Several remarks are in order. First, note that the n-th term of the sum in (1.14) measures how much the critical points of the n-th Bloch mode trap the energy and prevent the dispersion effects. Theorem 1.5 also tells that conical crossings do not trap energy. This Theorem 1.5 has exactly the same form than Theorem 2.2 in [15] while the assumptions are quite different since crossings between Bloch energies are authorized as long as they are conical. We shall see in the next subsection that crossing points may trap energy when they also are critical points of the Bloch energies (and thus they are no longer conical).

Secondly, we emphasize that Remark 4.2 (1) comments the determination of $(M_n)_{n\in {{\mathbb {N}}}^*}$ and $(\nu _n)_{n\in {{\mathbb {N}}}^*}$ from the initial data. The special case where $\Lambda _n$ is a point is discussed in the next subsection.

Thirdly, recall that when $d=1$, the assumptions H1, H2 and H3 are automatically satisfied (see “Appendix A”). Therefore, Theorem 1.1 is a consequence of Theorem 1.5 in the case where critical points are isolated.

Finally, we emphasize that Theorem 1.5 extends to situations where the Fourier transform of the initial data is localized on a set of the form $\{\varepsilon \xi \in \Omega +2\pi {{\mathbb {Z}}}^d\}$ for some open subset $\Omega$ of a unit cell of $2\pi {{\mathbb {Z}}}^d$, provided the assumptions H2 and H3 are satisfied for all $n\in {{\mathbb {N}}}^*$ above points of $\Omega +2\pi {{\mathbb {Z}}}^d$. We esquiss this approach in Section 7.4 and explain how the arguments of the proofs detailed below can be adapted to this setting by localisation (see Lemma 7.3).

1.4.3 The Special Case of Isolated Critical Points

When H1, H2 and H3 are satisfied for all $n\in {{\mathbb {N}}}^*$ and, moreover, all the sets $\Lambda _n$ consist in a family of isolated critical points then $T^*\Lambda _n=\Lambda _n\times \{0\}$, $N\Lambda _n={{\mathbb {R}}}^d$ so that the operators $M_n^t$ only depend on the parameter $\xi \in \Lambda _n$ and the operator $m_\phi ^{\Lambda _n}(\xi ,v)$ simply is the operator of multiplication by $\phi$. One can prove a statement very similar to Theorem 1.1 (see also Corollary 1.3 in [15] and Remark 4.2 (2)): the measures $\nu ^t_n$ are linear combinations of Dirac masses at the points $\xi _n\in \Lambda _n$ and $M_n^t(\xi _n)$ are orthogonal projectors on ${{\mathbb {C}}}\psi ^{\xi _n} (t)$, the solution to

$$\begin{aligned} i\partial _t \psi ^{\xi _n}(t,x) = \frac{1}{2} \mathrm{Hess} \varrho _n(\xi _n) D_x\cdot D_x\psi ^{\xi _n}(t,x) + V_{\mathrm{ext}} \psi ^{\xi _n}(t,x) \end{aligned}$$

(1.16)

with initial data $\psi ^{\xi _n}(0)$ that is a weak limit of $\left( \mathrm{e}^{-\frac{i}{\varepsilon _\ell } x\cdot \xi _{n}} L^{\varepsilon _\ell } \Pi _{n}(\varepsilon _\ell D_x) U^{\varepsilon _\ell }_0\right) _{\varepsilon _\ell >0}$ in $L^2({{\mathbb {R}}}^d)$.

In order to illustrate this type of result in the multidimensional setting, we state it in the particular case of well-prepared data that satisfy $U^\varepsilon _0(y,x)=\varphi _{n_0}(y,\varepsilon D_x) u_{n_0}^\varepsilon (x)$ for some $n_0\in {{\mathbb {N}}}^*$ (and therefore $\Pi _n(\varepsilon D_x) U^\varepsilon _0=0$ for $n\not =n_0$). More specifically, the class of well-prepared initial data that we will consider is those of the form

$$\begin{aligned} \psi ^\varepsilon _{0,n}(x):= \varphi _{n}\left( \frac{x}{\varepsilon },\varepsilon D_x\right) u^\varepsilon _{n}(x),\quad u^\varepsilon _{n}(x)=\mathrm{e}^{ \frac{i}{\varepsilon }x\cdot \xi _{n}} v_{n}^\varepsilon (x),\;\;n\in {{\mathbb {N}}}^* \end{aligned}$$

(1.17)

with $\xi _{n}\in \Lambda _{n}$, $(v_{n}^\varepsilon )_{\varepsilon >0}$ bounded in $H^s({{\mathbb {R}}}^d)$, $s>d/2$ or $s>1$ when $d=1$, and such that

$$\begin{aligned} v_{n}^\varepsilon \rightharpoonup v_{n},\quad \varepsilon \rightarrow 0^+,\quad \text { in }L^2({{\mathbb {R}}}^d). \end{aligned}$$

These data are closely related with those considered in [1, 2] for example (this connection is explained in detail in Lemma E.1). Their main properties are studied in “Appendix E”).

Proposition 1.6

Assume H1, H2 and H3 are satisfied for all $n\in {{\mathbb {N}}}^*$ and that, for some $n_0\in {{\mathbb {N}}}^*$, the set $\Lambda _{n_0}$ is discrete. Consider $(\psi ^\varepsilon )_{\varepsilon >0}$ a family of solutions to equation (1.1) with initial data $\psi ^\varepsilon _0$ satisfying (1.17) for some $\xi _{n_0}\in \Lambda _{n_0}$ and $v^\varepsilon _{n_0}$, $v_{n_0}$ in $L^2({{\mathbb {R}}}^d)$. Then, there exists a subsequence $(\psi ^{\varepsilon _\ell }_0)_{\ell \in {{\mathbb {N}}}}$ of the initial data such that for every $a<b$ and every $\phi \in {{\mathcal {C}}}_0({{\mathbb {R}}}^d)$ one has

$$\begin{aligned} \lim _{\ell \rightarrow +\infty } \int _a^b\int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\varepsilon _\ell } (t,x)|^2 \mathrm{d}x \mathrm{d}t = \int _a^b \int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\xi _{n_0}}(t,x)|^2 \mathrm{d}t, \end{aligned}$$

where $\psi ^{\xi _{n_0}}(t)$ solves (1.16) with initial data $\psi ^{\xi _{n_0}}(0) = v_{n_0}$

If $v^\varepsilon _0\rightarrow v_{n_0}$ as $\varepsilon \rightarrow 0^+$ in $L^2({{\mathbb {R}}}^d)$, then one has for all $t\in {{\mathbb {R}}}$, $\Vert \psi ^\varepsilon _0\Vert _{L^2}\rightarrow \Vert \psi ^{\xi _{n_0}}(t)\Vert _{L^2}$, which implies that no energy is dispersed. As a consequence of Remark 7.2, one obtains that the result of Proposition 1.6 holds locally in time: for all $T>0$, there exists a subsequence $(\psi ^{\varepsilon _\ell }_0)$ of the initial data such that for every $\phi \in {{\mathcal {C}}}_0({{\mathbb {R}}}^d)$ one has for all $t\in [0,T]$

$$\begin{aligned} \lim _{\ell \rightarrow +\infty } \int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\varepsilon _\ell } (t,x)|^2 \mathrm{d}x \mathrm{d}t = \int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\xi _{n_0}}(t,x)|^2. \end{aligned}$$

Proposition 1.6 is an improvement of Theorem 3.2 in [2] since it holds without any simplicity assumption on the Bloch energy $\varrho _{n_0}$. However, in the case where $\xi _{n_0}\notin \Lambda _{n_0}$ as in Theorem 4.5 of [2] where a drift is taken into account, the application of Theorem 1.1 does not bring any improvement. In particular, if $\xi _{n_0}\in \Sigma _{n_0}$, Theorem 1.1 gives 0 at the limit and Theorem 4.5 of [2] does not apply (the drift $\nabla \varrho _{n_0}(\xi _{n_0})$ is not defined). Studying how the mass goes to infinity in that special case would be an interesting follow-up of the results presented here.

1.5 A Non-generic Case with Interactions of Bloch Bands Above Non-conical Crossing Points

In that section, we consider degenerate crossing points that we define as follows:

Definition 1.7

We say that the crossing set $\Sigma _n$ is degenerate if some $q\ge 2$ exists such that the function $g_n$ defined in (1.11), satisfies

$$\begin{aligned} \exists c>0,\;\; \forall (\xi ,\eta )\in N\Sigma _n ,\;|\eta |<1,\;g_n(\xi , \eta ) \le c|\eta | ^q. \end{aligned}$$

If the above inequality fails for any $q'>q$, we say the crossing set has degeneracy of order q.

1.5.1 Assumptions

We consider two Bloch energies that cross in a degenerate manner, though isolated from the remainder of the spectrum.

• H1’ $\varrho _n(\xi )$ and $\varrho _{n+1}(\xi )$ are two Bloch energies that cross on $\Sigma _n$ and are of multiplicity 1 outside $\Sigma _n$.

• H2’ ${{\,\mathrm{Hess}\,}}\varrho _n$ (resp. ${{\,\mathrm{Hess}\,}}\varrho _{n+1}$) is of constant rank in a neighborhood of each connected component of $\Lambda _n$ (resp. $\Lambda _{n+1}$).

• H3’ The crossing set $\xi _n$ is a smooth closed submanifold of ${{\mathbb {R}}}^d$ included in $\Lambda _n \cap \Lambda _{n+1}$ and is degenerated of order q. Besides, $\Sigma _{n-1}=\Sigma _{n+1}=\emptyset$ and

  • if $q> 2$, for all $\xi \in \Sigma _n$, the Hessian of $\lambda _n$ is of rank $d-\mathrm{Rank} \Sigma _n$ in a neighborhood of $\Sigma _n$,

  • if $q=2$, for all $(\xi ,\eta )\in N\Sigma _n$ with $|\eta |=1$,

    $$\begin{aligned} \mathrm{Hess}\, \lambda _n (\xi )\eta \pm \nabla _\eta g_n(\xi ,\eta )\not =0 . \end{aligned}$$

Note that the latter assumption can be considered as a maximal rank assumption. Indeed, if $q> 2$, the Bloch energies are ${{\mathcal {C}}}^{2}$ and $\mathrm{Hess} \, \varrho _n(\xi )= \mathrm{Hess} \, \varrho _{n+1}(\xi )= \mathrm{Hess} \, \lambda _n(\xi )$ above points $\xi \in \Sigma _n$. Moreover, if $q=2$ and $(\xi _k,\omega _k)_{k\in {{\mathbb {N}}}}$ is a sequence of points satisfying for all $k\in {{\mathbb {N}}}$, $\omega _k\in N_{\xi _k} \Sigma _n$ with $|\omega _k|=1$, we have the following property: if $(\xi _k,\omega _k)\mathop {\longrightarrow }\limits _{\ell \rightarrow +\infty }(\xi ,\omega )\in N \Sigma _n$ then

$$\begin{aligned} \mathrm{Hess} \, \varrho _n (\xi _k) \omega _k \mathop {\longrightarrow }\limits _{(\xi _k,\omega _k)\rightarrow (\xi ,\omega )} \mathrm{Hess}\, \lambda _n(\xi ) \omega \pm \nabla _\eta g_n(\xi ,\omega ). \end{aligned}$$

This property shows the link between $\mathrm{Hess}\, \lambda _n (\xi )\eta \pm \nabla _\eta g_n(\xi ,\eta )$ and $\mathrm{Hess} \, \varrho _n (\xi )\eta$.

The assumption H3’ implies that $\nabla \lambda _n=0$ on $\Sigma _n$. We shall see in the proof that if $\nabla \lambda _n$ does not vanish on $\Sigma _n$, then these degenerate crossing points do not contribute. The result then is comparable to the one of Theorem 1.5. Notice that one always has

$$\begin{aligned} \Lambda _n\cap \Lambda _{n+1} = \Sigma _n \cap \Lambda _n= \Sigma _n\cap \Lambda _{n+1}= \Sigma _n \cap \{ \nabla \lambda _n(\xi )=0\}. \end{aligned}$$

At the difference with the results of the preceding section that were obtained under the assumption that H1, H2 and H3 were satisfied for all $n\in {{\mathbb {N}}}^*$, we will assume in this section that we have H1’, H2’ and H3’ for some single $n\in {{\mathbb {N}}}^*$ and we will choose well-prepared data concentrating on the bands $\varrho _n$ and $\varrho _{n+1}$ involved in the assumptions.

We assume that

$$\begin{aligned} \psi ^\varepsilon _0(x)= \varphi _n\left( \frac{x}{\varepsilon },\varepsilon D\right) u^\varepsilon _{n}(x) + \varphi _{n+1}\left( \frac{x}{\varepsilon },\varepsilon D\right) u^\varepsilon _{n+1}(x), \end{aligned}$$

(1.18)

where $\varphi _n$ (resp. $\varphi _{n+1}$) is the Bloch wave associated with $\varrho _n$ (resp. $\varrho _{n+1}$), and $(u^\varepsilon _{n})_{\varepsilon >0}$ and $(u^\varepsilon _{n+1})_{\varepsilon >0}$ are bounded families in $H^s_\varepsilon ({{\mathbb {R}}}^d)$, $s>d/2$. These data are somehow more general than those of (1.17) since no assumption is made on $u^\varepsilon _{n}$ ans $u^\varepsilon _{n+1}$.

1.5.2 The Result: Concentration Above Degenerate Crossings

We prove that for data as in (1.18), the way their components interact above the crossing set plays a role in the determination of the weak limits of the time-averaged energy density. We associate with the function $g_n$ defined on $N\Sigma _n$ (see (1.11)) the operator $Q_{g_n}^{\Sigma _n} (\xi )$ acting on $L^2(N_\xi \Sigma _n)$ as a Fourier multiplier.

Theorem 1.8

Let $(\psi ^\varepsilon )_{\varepsilon >0}$ be a family of solutions to equation (1.1) with initial data satisfying (1.18). Assume the Bloch energies $\varrho _n$ and $\varrho _{n+1}$ satisfies H1’, H2’ and H3’ with $\Lambda _n=\Lambda _{n+1}=\Sigma _n$. Then, there exists a subsequence $(\psi ^{\varepsilon _\ell }_0)_{\ell \in {{\mathbb {N}}}}$ of the initial data, a non negative measure $\nu ^0\in {\mathcal {M}}^+(\Sigma _n)$ and a matrix M of measurable trace-class operators

$$\begin{aligned}&M:\,T^*_\xi \Sigma _n\ni (\xi ,v) \mapsto M(\xi ,v) \in {\mathcal {L}}^1_+(L^2(N_\xi \Sigma _n,{{\mathbb {C}}}^2)),\\&\mathrm{Tr} _{L^2(N_\xi \Sigma _n,{{\mathbb {C}}}^2)}M (\xi ,v) =1\; d\nu ^0\; a.e., \end{aligned}$$

both depending only on $(\psi ^{\varepsilon _\ell }_0)_{\ell \in {{\mathbb {N}}}}$, such that for every $a<b$ and every $\phi \in {{\mathcal {C}}}_0({{\mathbb {R}}}^d)$ one has

$$\begin{aligned}&\lim _{\ell \rightarrow +\infty } \int _a^b\int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\varepsilon _\ell } (t,x)|^2 \mathrm{d}x \mathrm{d}t\\&\quad = \int _a^b \int _{T^*\Sigma _n}\mathrm{Tr} _{L^2(N_\xi \Sigma _n, {{\mathbb {C}}})}\\&\qquad \qquad \qquad \qquad [m^{\Sigma _n}_\phi (\xi ,v) (m_n^t + m_{n+1}^t+2 \mathrm{Re} (m_{n,n+1}^t ))(\xi ,v) ] \nu ^0(d\xi , dv)\mathrm{d}t , \end{aligned}$$

where

$$\begin{aligned} M^t(\xi ,v) = \begin{pmatrix} m_n^t(\xi ,v) &{} m_{n,n+1}^t(\xi ,v) \\ m_{n,n+1}^t(\xi ,v)^* &{} m_{n+1}^t(\xi ,v)\end{pmatrix} \end{aligned}$$

is a non negative trace class operator on $L^2( N_\xi \Sigma _n,{{\mathbb {C}}}^2)$. In addition, the map

$$\begin{aligned} t\mapsto M^t (\xi ,v)\in {\mathcal {C}}({{\mathbb {R}}}, {\mathcal {L}}^1_+(L^2(N_\xi \Sigma _n, {{\mathbb {C}}}^2)) \end{aligned}$$

solves a von Neumann equation that depends on the value of q:

• If $q>2$, it solves

  $$\begin{aligned}&i\partial _t M^t (\xi ,v) =\left[ \left( \frac{1}{2}\mathrm{Hess} \lambda _n(\xi ) D_z\cdot D_z + m^{\Sigma _n}_{V_{\mathrm{ext}}}(\xi ,v)\right) \mathrm{Id_{{{\mathbb {C}}}^2}} \;,\; M^t(\xi ,v) \right] ,\nonumber \\&M^0=M. \end{aligned}$$

  (1.19)

• If $q=2$, it solves

  $$\begin{aligned}&i\partial _ t M^t (\xi ,v) = \left[ \left( \frac{1}{2} \mathrm{Hess}\, \lambda _n(\xi ) D_z\cdot D_z + m^{\Sigma _n}_{V_{\mathrm{ext}}(t,\cdot )}(\xi ,v) \right) \mathrm{Id}_{{{\mathbb {C}}}^{2}} - Q^{\Sigma _n} _{g_n}(\xi ) \, J \;,\; M^t(\xi ,v) \right] ,\;\nonumber \\&M^0=M \end{aligned}$$

  (1.20)

  with $J=\begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix}$.

(recall that $m^{\Sigma _n}_\phi (\xi ,v)$ (resp. $m^{\Sigma _n}_{V_{\mathrm{ext}}}(\xi ,v)$) denotes the operator acting on $L^2(N_\xi \Sigma _n)$ by multiplication by $\phi (v+\cdot )$ (resp. $V_{\mathrm{ext}}(v+\cdot )$), and $Q^{\Sigma _n} _{g_n}(\xi )$ is the Fourier multiplier on $L^2(N_\xi \Sigma _n)$ associated with the function $g_n(\xi , \cdot )$).

In the latter statement, we have assumed for simplicity that the critical sets of the two Bloch energies $\varrho _n$ and $\varrho _{n+1}$ coincide with the crossing set $\Sigma _n$. We indeed prove this result without this assumption; however, the resulting statement is more involved (see Theorem 7.1).

1.5.3 The Special Case of Isolated Degenerate Crossing Points

When $\Sigma _n$ consists in a family of isolated degenerate crossing points, the preceding statement admits a simpler, more transparent formulation. Indeed, as in Section 1.4.3, the operator $M^t$ only depend on the parameter $\xi \in \Sigma _n$ and the operator $m_\phi ^{\Lambda _n}(\xi ,v)$ simply is the operator of multiplication by $\phi$ and $Q^{\Sigma _n}_{g_n}= g_n(\xi , D_x)$. We now assume that $\psi ^\varepsilon _0$ satisfies (1.18) with

$$\begin{aligned} u^\varepsilon _{n}(x)=\mathrm{e}^{ \frac{i}{\varepsilon }x\cdot \xi _{n}} v_{n}^\varepsilon (x)\;\;\text{ and } \;\;u^\varepsilon _{n+1}(x)=\mathrm{e}^{ \frac{i}{\varepsilon }x\cdot \xi _{n}} v_{n+1}^\varepsilon (x) \end{aligned}$$

with $\xi _{n}\in \Sigma _{n}$, and for $j\in \{n,n+1\}$, $(v_{j}^\varepsilon )_{\varepsilon >0}$ bounded in $H^s({{\mathbb {R}}}^d)$, $s>d/2$ or $s>1$ when $d=1$, and such that:

$$\begin{aligned} v_{j}^\varepsilon \rightharpoonup v_{j},\quad \varepsilon \rightarrow 0^+,\quad \text { in }L^2({{\mathbb {R}}}^d). \end{aligned}$$

Proposition 1.9

Let $(\psi ^\varepsilon )_{\varepsilon >0}$ be a family of solutions to equation (1.1) with initial data satisfying (1.18). Assume the Bloch energies $\varrho _n$ and $\varrho _{n+1}$ satisfy H1’, H2’ and H3’. Assume moreover $\Lambda _n=\Lambda _{n+1}=\Sigma _n$ consists in a family of isolated critical crossing points. Then, there exists a subsequence $(\psi ^{\varepsilon _\ell }_0)_{\ell \in {{\mathbb {N}}}}$ of the initial data such that for every $a<b$ and every $\phi \in {{\mathcal {C}}}_0({{\mathbb {R}}}^d)$ one has

$$\begin{aligned} \lim _{\ell \rightarrow +\infty } \int _a^b\int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\varepsilon _\ell } (t,x)|^2 \mathrm{d}x \mathrm{d}t= & {} \sum _{\xi \in \Sigma _n} \int _a^b \int _{{{\mathbb {R}}}^d} \phi (x)\qquad \qquad \\&\qquad \qquad |\psi ^{\xi _n}_n(t,x)+\psi ^{\xi _n}_{n+1}(t,x)|^2 \mathrm{d}x \mathrm{d}t, \end{aligned}$$

where

• if $q>2$, for $j\in \{n,n+1\}$, $\psi ^{\xi }_{j}$ solves the effective mass Schrödinger equation

  $$\begin{aligned} i\partial _t \psi ^{\xi _n}_j = \frac{1}{2} \mathrm{Hess} \lambda _n(\xi _n) D_x\cdot D_x\psi ^{\xi _n}_j+ V_{\mathrm{ext}} \psi ^{\xi _n}_j \end{aligned}$$

  (1.21)

• if $q=2$, for $j\in \{n,n+1\}$, $\psi ^{\xi }_{j}$ solves the effective mass Schrödinger equation

  $$\begin{aligned} i\partial _t \psi ^{\xi _n}_j = \frac{1}{2} \mathrm{Hess} \lambda _n(\xi _n) D_x\cdot D_x\psi ^{\xi _n}_j +\eta _j g_n(\xi _n, D_x)\psi ^{\xi _n}_j + V_{\mathrm{ext}} \psi ^{\xi _n}_j, \end{aligned}$$

  (1.22)

  with $\eta _n=1 \;\text{ and }\; \eta _{n+1}=-1$.

Besides, for $j\in \{n,n+1\}$, the initial data $\psi ^{\xi _n}_{j}(0)=v_j$.

We stress the fact that, in contrast to what happened in the presence of a singular crossing, the description of the limiting position density limit involves a term of the form $2\mathrm{Re} (\psi ^{\xi _n} \overline{\psi ^{\xi _{n+1}}})$ which takes into account the coupling between the modes corresponding to the two Bloch bands.

1.6 Ideas of the Proofs and Organisation of the Paper

We follow the semi-classical approach developed in [14, 15] which is based on semi-classical analysis. In these references, the Bloch energies in consideration are smooth, and we have exhibited the role of the critical points of the Bloch energies as principal contributors to the weak limits of the time-averaged energy densities. We have also explained how a second microlocalisation allows to compute quantitatively this contribution. We follow here this scheme of thoughts with additional difficulties that are two-fold.

Firstly, in order to consider general initial data as in Theorem 1.1 and 1.5, and to decompose them on the Bloch energies, we shall need to treat infinite series. The assumption that the data satisfy H0 is the key point that we use technically for treating this issue. We explain in Section 2 how we perform the decomposition and which properties of the solution we use.

The second difficulty comes from the lack of regularity of the Bloch energies close to the crossing sets, which requires to perform semi-classical calculus with symbols of low regularity, what we do by using and developing ideas from [28]. We explain and construct in Sections 3 and 4 the semi-classical and two-microlocal analysis of our problem.

This led us to the statement of two theorems that are interesting by themselves: in Theorems 4.5 and 4.6, we describe the evolution of two-microlocal semi-classical measures associated to the concentration of the solutions of (1.1) on one of the sets of critical points $\Lambda _n$, $n\in {{\mathbb {N}}}^*_0$, and in the context given by hypothesis H 1, H 2 and H 3 for the first one and H1’, H2’ and H3’ for the second one. These two theorems are proved in Sections 5 and 6 respectively; they are the core of the proofs of Theorems 1.5 and 1.8, which are performed themselves in Section 7, together with the proof of Theorem 1.1, Propositions 1.6 and 1.9.

Finally, some appendices are devoted to technical elements that we use in the proofs of this paper: special features of the Bloch decomposition in dimension 1 (“Appendix A”), properties of the Bloch energies at a crossing (“Appendix B”), elements of matrix-valued pseudo-differential calculus, in particular with low regularity (“Appendix C”), two scale pseudodifferential calculus (“Appendix D”) and various remarks above well-prepared data (“Appendix E”).

2 Separation of Scales and Control of the Oscillations

Here we present the first steps of the strategy that will lead to the proof of Theorems 1.1, 1.5 and 1.8. Our starting point is the following Ansatz that is widely used in this context and consists in separating the slow and fast scales of oscillation. We look for a solution to (1.1) of the form

$$\begin{aligned} \psi ^\varepsilon (t,x)=U^\varepsilon \left( t,x,{x\over \varepsilon }\right) ,\;\;(t,x)\in {{\mathbb {R}}}\times {{\mathbb {R}}}^d, \end{aligned}$$

(2.1)

where $U^\varepsilon (t,\cdot ,\cdot )$ is a function on ${{\mathbb {R}}}^d\times {{\mathbb {T}}}^d$. In order to make sense of this, some regularity on the solutions $\psi ^\varepsilon$ is required, and this partly justifies our assumption H0. Uniqueness for solutions to the initial value problem for the Schrödinger equation (1.1) with initial data satisfying H0 implies that (2.1) holds provided $U^\varepsilon$ is a solution to the system

$$\begin{aligned} \left\{ \begin{array}{l} i\varepsilon ^2\partial _t U^\varepsilon (t,x,y)=P(\varepsilon D_x)U^\varepsilon (t,x,y)+\varepsilon ^2 V_{\mathrm{ext}}(t,x) U^\varepsilon (t,x,y), \quad \\ U^\varepsilon _{|t=0}= U^\varepsilon _0,\end{array}\right. \end{aligned}$$

(2.2)

By Lemma 6.5 in [15], this equation is well-posed in $H^s_\varepsilon ({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d)$: there exists $C_s>0$ such that for every $t\in {{\mathbb {R}}}$ and $U^\varepsilon _0\in H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$,

$$\begin{aligned} \Vert U^\varepsilon (t,\cdot )\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\le \Vert U^\varepsilon _0\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}+C_s\varepsilon |t| \end{aligned}$$

(2.3)

uniformly in $\varepsilon >0$.

We shall use a decomposition of the solution $U^\varepsilon$ in the basis of Bloch modes. We set, for $j\in {{\mathbb {N}}}_0$,

$$\begin{aligned} P^\varepsilon _{\varphi _j} W(x,y):=\varphi _j\left( y,\varepsilon D_x\right) \int _{{{\mathbb {T}}}^d}\overline{\varphi _j}(z,\varepsilon D_x) W(x,z)\mathrm{d}z,\;\;\forall W\in L^2({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d).\nonumber \\ \end{aligned}$$

(2.4)

Lemma 2.1

If $(\psi ^\varepsilon _0)_{\varepsilon >0}$ satisfies H0, then for all $t\in {{\mathbb {R}}}$, the solution of (1.1) is given by

$$\begin{aligned} \psi ^\varepsilon (t,\cdot )=\sum _{n\in {{\mathbb {N}}}^*} \psi ^\varepsilon _n(t,\cdot ), \end{aligned}$$

(2.5)

where the convergence of the series takes place in $L^2({{\mathbb {R}}}^d)$ and

$$\begin{aligned} \psi ^\varepsilon _n(t,x):=L^\varepsilon P^\varepsilon _{\varphi _n} U^\varepsilon (t,x)= \varphi _n\left( \frac{x}{\varepsilon }, \varepsilon D_x\right) \int _{{{\mathbb {T}}}^d}\overline{\varphi _n}(y,\varepsilon D_x) U^\varepsilon (t,x,y) \mathrm{d}y.\nonumber \\ \end{aligned}$$

(2.6)

Moreover, for every $t\in {{\mathbb {R}}}$,

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+} \left\| \sum _{n>N} \psi ^\varepsilon _n(t,\cdot )\right\| _{L^2({{\mathbb {R}}}^d)} \mathop {\longrightarrow }\limits _{N\rightarrow \infty }0. \end{aligned}$$

(2.7)

The proof of this result requires two important technical facts that we gather in the next two remarks.

Remark 2.2

Modulo the addition of a positive constant to equation (1.1), we may assume that $P(\varepsilon D_x)$ is a positive operator (this will modify the solutions only by a constant phase in time). In that case there exists constants $\varepsilon _0,c>0$ such that

$$\begin{aligned} c^{-1}\Vert U\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\le & {} \ \Vert \left\langle \varepsilon D_x\right\rangle ^s U\Vert _{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}+\Vert P(\varepsilon D_x)^{s/2}U\Vert _{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)} \\\le & {} c\Vert U\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)} \end{aligned}$$

for every $U\in H^s({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$ and $0<\varepsilon <\varepsilon _0$.

Remark 2.3

In view of (1.3), and the fact that $P(\xi )$ depends analytically on $\xi$, follows that $\xi \mapsto \varphi _j(\cdot ,\xi )$, $j\in {{\mathbb {N}}}^*$, are continuous functions from ${{\mathbb {R}}}^d$ to $L^2({{\mathbb {T}}}^d)$ (see also [56]) and that, for every $s>0$, the family $P(\xi )^s$ depends continuously on $\xi$ (with $P(\xi )$ positive).

These remarks imply the boundedness of the operators $P_{\varphi _j}$ in $H^s_\varepsilon ({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d)$ for all $j\in {{\mathbb {N}}}_0$. To see this, note that Remark 2.3 implies that formula 2.4 defines a bounded operator on $L^2({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d)$. Moreover, we have

$$\begin{aligned}{}[P(\varepsilon D_x)^{s/2},P^\varepsilon _{\varphi _j}]=[\left\langle \varepsilon D_x\right\rangle ^s,P^\varepsilon _{\varphi _j}]=0. \end{aligned}$$

If follows from Remark 2.2 that there exists $c_1>0$ such that, for all $W\in H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$,

$$\begin{aligned} \Vert P^\varepsilon _{\varphi _j}W\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\le c_1\Vert W\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}, \end{aligned}$$

and, more generally, that every $W\in H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$ can be expressed in the topology of $H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$ as

$$\begin{aligned} W=\sum _{n\in {{\mathbb {N}}}^*}P^\varepsilon _{\varphi _n} W. \end{aligned}$$

Proof of Lemma 2.1

The boundedness in $H^s_\varepsilon ({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d)$ of the operator $P_{\varphi _j}$ and the boundedness of $L^\varepsilon$ from $H^s_\varepsilon ({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d)$ to $L^2({{\mathbb {R}}}^d)$ for $s>d/2$ imply that that  (2.5) holds in $L^2({{\mathbb {R}}}^d)$.

It remains to prove (2.7). In view of (2.3), (1.8), it is enough to show that if $(V^\varepsilon )_{\varepsilon >0}$ is a bounded family in $H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d )$, $s>d/2$, we have, for $d/2<r<s$,

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+}\left\| \sum _{n>N} P^\varepsilon _{\varphi _n} V^\varepsilon \right\| _{H^r_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\mathop {\longrightarrow }\limits _{N\rightarrow \infty }0. \end{aligned}$$

Remark 2.2 implies that we only have to prove

$$\begin{aligned}&\limsup _{\varepsilon \rightarrow 0^+}\left\| \sum _{n>N} P(\varepsilon D_x)^{r/2}P^\varepsilon _{\varphi _n} V^\varepsilon \right\| ^2_{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\nonumber \\&\quad + \limsup _{\varepsilon \rightarrow 0^+}\left\| \sum _{n>N} \langle \varepsilon D_x\rangle ^{r} P^\varepsilon _{\varphi _n} V^\varepsilon \right\| ^2_{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\mathop {\longrightarrow }\limits _{N\rightarrow \infty }0. \end{aligned}$$

(2.8)

We thus focus on proving (2.8).

Let us consider the series $\sum _{n>N} P(\varepsilon D_x)^{r/2} P^\varepsilon _{\varphi _n} V^\varepsilon$ (the proof for $\sum _{n>N}\langle \varepsilon D_x\rangle ^{r} P^\varepsilon _{\varphi _n} V^\varepsilon$ is similar). In view of (2.4),

$$\begin{aligned} P(\varepsilon D_x) P^\varepsilon _{\varphi _n} V^\varepsilon (x,y)&=\varphi _n(y,\varepsilon D_x) \varrho _n(\varepsilon D_x) \,\int _{{{\mathbb {T}}}^d} \overline{\varphi _n}(z,\varepsilon D_x) V^\varepsilon (x,z) \mathrm{d}z, \end{aligned}$$

This implies

$$\begin{aligned} \left\| \sum _{n>N} P(\varepsilon D_x)^{r/2}P^\varepsilon _{\varphi _n} V^\varepsilon \right\| ^2_{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}= \sum _{n>N} \left\| P(\varepsilon D_x)^{r/2}P^\varepsilon _{\varphi _n} V^\varepsilon \right\| ^2_{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}. \end{aligned}$$

We decompose $V^\varepsilon$ in Fourier series and write $V^\varepsilon (x,y) =\sum _{j\in {{\mathbb {Z}}}^d} V^\varepsilon _j(x) \mathrm{e}^{2i\pi j\cdot y}$, whence

$$\begin{aligned}&P(\varepsilon D_x) P^\varepsilon _{\varphi _n} V^\varepsilon (x,y)\\&\quad =\varphi _n(y,\varepsilon D_x) \sum _{j\in {{\mathbb {Z}}}^d} \varrho _n(\varepsilon D_x) \,\left( \int _{{{\mathbb {T}}}^d} \overline{\varphi _n}(z,\varepsilon D_x) \mathrm{e}^{2i\pi j\cdot z} \mathrm{d}z \right) V^\varepsilon _j(x), \end{aligned}$$

and by functional calculus,

$$\begin{aligned} P(\varepsilon D_x)^{r/2} P^\varepsilon _{\varphi _n} V^\varepsilon (x,y)&=\varphi _n(y,\varepsilon D_x) \sum _{j\in {{\mathbb {Z}}}^d} d_{n} (\varepsilon D_x,j) V^\varepsilon _j(x) \end{aligned}$$

with

$$\begin{aligned} d_{n} (\xi ,j)= \varrho _n(\xi )^{r/2} \,\left( \int _{{{\mathbb {T}}}^d} \overline{\varphi _n}(z,\varepsilon D_x) \mathrm{e}^{2i\pi j\cdot z} \mathrm{d}z \right) \end{aligned}$$

We use three observations.

• (1) First, if $\delta >0$ is fixed, there exists $J_0$ such that

  $$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+} \sum _{|j|>J_0}\int _{{{\mathbb {R}}}^d} (1+|\varepsilon \xi |^2 +|j|^2) ^{r} |\widehat{V^\varepsilon _j}(\xi )|^2\mathrm{d}\xi <\delta . \end{aligned}$$

  To see this note that:

  $$\begin{aligned} \sum _{|j|>J_0}\int _{{{\mathbb {R}}}^d} (1+|\varepsilon \xi |^2 +|j|^2) ^{r} |\widehat{V^\varepsilon _j}(\xi )|^2 \mathrm{d}\xi \le (1+|J_0|^2)^{r-s} \Vert V^\varepsilon \Vert _{H^{s}({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}^2, \end{aligned}$$

  due to the definition of the $H^{s}_\varepsilon$-norm (1.7). Since $(V^\varepsilon )_{\varepsilon >0}$ is uniformly bounded in $H^{s}_\varepsilon ({{\mathbb {R}}}^d)$, the claim follows.

• (2) Second, given $\delta >0$ and $J_0\in {{\mathbb {N}}}$, one can find $R=R(\delta , J_0) >0$ such that

  $$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+}\sum _{|j|< J_0}\int _{|\varepsilon \xi |>R} (1+|\varepsilon \xi |^2 +|j|^2) ^{r} |\widehat{V^\varepsilon _j}(\xi )|^2\mathrm{d}\xi <\delta . \end{aligned}$$

  This follows from the estimate

  $$\begin{aligned} \int _{|\varepsilon \xi |>R} (1+|\varepsilon \xi |^2 +|j|^2) ^{r} |\widehat{V^\varepsilon _j}(\xi )|^2 \mathrm{d}\xi&\le (1+R^2)^{r-s}\Vert V^\varepsilon \Vert _{H^{s}({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}^2, \end{aligned}$$

  and again from the fact that $(V^\varepsilon )_{\varepsilon >0}$ is uniformly bounded in $H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$.

• (3) Third, given $J_0, R>0$,

  $$\begin{aligned} D_N(R,J_0):=\sup _{|j|\le J_0}\sup _{|\xi |\le R}\sum _{n>N}\left| d_{n}(\xi ,j) \right| ^2\mathop {\longrightarrow }\limits _{N\rightarrow \infty }0. \end{aligned}$$

  To see why this holds, note that, for $j\in {{\mathbb {Z}}}^d$,

  $$\begin{aligned} {{\mathbb {R}}}^d\ni \xi \longmapsto \sum _{n\in {{\mathbb {N}}}^*}\left| d_{n} (\xi ,j) \right| ^2=\left\| P(\xi )^{r/2} \mathrm{e}^{2i\pi j\cdot } \right\| _{L^2({{\mathbb {T}}}^d)}^2 \in (0,\infty ) \end{aligned}$$

  (2.9)

  is a non-negative continuous function. The claim then follows from Dini’s theorem, which ensures that for every $R>0$, $j\in {{\mathbb {Z}}}^d$ one has

  $$\begin{aligned} \sup _{|\xi |\le R}\sum _{n>N}\left| d_{n}(\xi ,j) \right| ^2\mathop {\longrightarrow }\limits _{N\rightarrow \infty }0. \end{aligned}$$

We now use these observations to treat the series whose terms are

$$\begin{aligned} \left\| P(\varepsilon D_x)^{r/2} P^\varepsilon _{\varphi _n}V^\varepsilon \right\| _{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d))}^2&= \sum _{j\in {{\mathbb {Z}}}^d} \int _{{{\mathbb {R}}}^d} |d_{n}(\varepsilon \xi ,j) |^2 |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 \mathrm{d}\xi . \end{aligned}$$

Fix $\delta >0$, and consider $J_0$ given by Point (1) and $R=R(\delta ,J_0)$ given by Point (2). Decompose the sum of integrals in three terms

$$\begin{aligned} \sum _{j\in {{\mathbb {Z}}}^d} \int _{{{\mathbb {R}}}^d} = \sum _{|j|\le J_0} \int _{|\varepsilon \xi |\le R} + \sum _{|j|\le J_0} \int _{|\varepsilon \xi |> R} +\sum _{|j| > J_0} \int _{{{\mathbb {R}}}^d}. \end{aligned}$$

We start by analyzing the third term. Note that

$$\begin{aligned} \sum _{n\in {{\mathbb {N}}}^*}|d_n(\xi ,j)|^2=\left\| P(\xi )^{r/2} \mathrm{e}^{2i\pi j\cdot } \right\| _{L^2({{\mathbb {T}}}^d)}^2\le c_r(1+|\xi |^2+|j|^2)^{r} \end{aligned}$$

Therefore,

$$\begin{aligned}&\limsup _{\varepsilon \rightarrow 0^+}\sum _{n>N} \sum _{|j|> J_0} \int _{{{\mathbb {R}}}^d} |d_n (\varepsilon \xi ,j)|^2 |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 \mathrm{d}\xi \nonumber \\&\quad \le \limsup _{\varepsilon \rightarrow 0^+}\sum _{|j|> J_0} \int _{{{\mathbb {R}}}^d} \sum _{n\in {{\mathbb {N}}}^*} |d_n (\varepsilon \xi ,j)|^2 |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 d\xi \\&\le c_r \limsup _{\varepsilon \rightarrow 0^+} \sum _{|j| > J_0} \int _{{{\mathbb {R}}}^d} (1+|\varepsilon \xi |^2 +|j|^2) ^{r} |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 \mathrm{d}\xi <c_r \delta , \end{aligned}$$

using observation (1).

The second term is analyzed using observation (2):

$$\begin{aligned}&\limsup _{\varepsilon \rightarrow 0^+} \sum _{n>N} \sum _{|j|\le J_0} \int _{|\varepsilon \xi |> R} |d_n (\varepsilon \xi ,j)|^2 |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 \mathrm{d}\xi \\&\quad \le c_r \limsup _{\varepsilon \rightarrow 0^+} \sum _{|j|\le J_0} \int _{|\varepsilon \xi | > R} (1+|\varepsilon \xi |^2 +|j|^2) ^{k} |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 \mathrm{d}\xi <c_r \delta . \end{aligned}$$

Observation (3) ensures that

$$\begin{aligned} \sum _{n>N} \sum _{|j|\le J_0}&\int _{|\varepsilon \xi |\le R} |d_n (\varepsilon \xi ,j)|^2 |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 \mathrm{d}\xi \le D_N(R, J_0) \Vert V^\varepsilon \Vert _{L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}^2. \end{aligned}$$

As a consequence of this analysis,

$$\begin{aligned}&\limsup _{N\rightarrow +\infty }\; \limsup _{\varepsilon \rightarrow 0^+} \sum _{n>N} \sum _{j\in {{\mathbb {Z}}}^d} \int _{{{\mathbb {R}}}^d} \left| \int _{{{\mathbb {T}}}^d} \varrho _n(\varepsilon \xi )^{r/2} \overline{\varphi _n}(z,\varepsilon \xi ) \mathrm{e}^{2i\pi j\cdot z} \mathrm{d}z\right| ^2 |{{\widehat{V}}}^\varepsilon _j(\xi )|^2 \mathrm{d}\xi \\&\quad <2c_r\delta . \end{aligned}$$

Since $\delta$ is arbitrary, the result follows. $\quad \square$

Lemma 2.1 provides an important element of the proof of the Theorems 1.1 and 1.5 of this paper. It allows to reduce the problem to solutions consisting only of finite superposition of Bloch modes, that we are going to study with a semi-classical perspective, as explained in the next section.

3 Semi-classical Approach to the Energy Dynamics

The nature of the propagation of the asymptotic energy density for high-frequency solutions to semi-classical dispersive-type equations is better understood if the usual, physical-space, energy density is lifted to a phase-space energy density. There is no canonical lifting procedure, roughly speaking these choices correspond to different quantization procedures. Here we will work with the lifting procedure that corresponds to the Weyl quantization, from which the Wigner functions are obtained (see the definition in (3.1)).

It should be noted that although the asymptotic limit in equation (1.1) we are interested might not appear to fit in the semi-classical regime one can indeed place it in that context. One can check that any solution $\psi ^\varepsilon$ of (1.1) becomes, after rescaling in time as $u^\varepsilon (t,\cdot ):=\psi ^\varepsilon (\varepsilon t,\cdot )$, a solution to a semi-classical Schrödinger equation with highly oscillating potential:

$$\begin{aligned} i\varepsilon \partial _t u^\varepsilon (t,x)+\frac{\varepsilon ^2}{2}\Delta _x u^\varepsilon (t,x)-V_{\mathrm{per}}\left( \dfrac{x}{\varepsilon }\right) u^\varepsilon (t,x) - \varepsilon ^2V_{\mathrm{ext}}(\varepsilon t,x)u^\varepsilon (t,x) =0. \end{aligned}$$

Hence, the asymptotic limit we are interested in can be viewed as performing simultaneously the semi-classical and long-time limits. This approach was pursued in [14, 15] to deal with the case where no crossings between Bloch bands are present, and this point of view is also adopted in references [2, 54]. Here the situation is more complicated, as interactions between projections on different Bloch bands may occur. This regime involving performing simultaneously the semi-classical and long time limit has been useful in other contexts, we refer the reader to the survey articles [5, 43, 44].

In Section 3.1 below, we recall elements of the theory of semi-classical measures that we apply to $(\psi ^\varepsilon )_{\varepsilon >0}$ in the next two sections. We first discuss in Section 3.2 the relations between the semi-classical measures of $(\psi ^\varepsilon )_{\varepsilon >0}$ and those of the families $(\psi ^\varepsilon _n)_{\varepsilon >0}$ that have been introduced in (2.5) and (2.6). Then, we analyze the localisation properties of these semi-classical measures in Section 3.3, which motivates a two-microlocal approach.

3.1 Semi-classical Measures and Energy Densities

Let us recall briefly some basic facts of the theory of semi-classical measures [34, 35, 40] that will be needed in the sequel. From now on, for every $s\in {{\mathbb {R}}},N\in {{\mathbb {N}}}^*$, $H^s_\varepsilon ({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$ will denote the space $H^s_\varepsilon ({{\mathbb {R}}}^d)^N$ equipped with the norm

$$\begin{aligned} \Vert \Psi \Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}=\left( \sum _{j=1}^N\Vert \Psi _j\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d)}^2\right) ^{1/2},\;\;\Psi =(\Psi _1,\cdots , \Psi _N). \end{aligned}$$

We associate to every $\Psi \in L^2({{\mathbb {R}}}^d,{{{\mathbb {C}}}}^N):=H^0_\varepsilon ({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$ a microlocal version $W^\varepsilon _{\Psi }$ of the (matrix-valued) energy density

$$\begin{aligned} \Psi \otimes {\overline{\Psi }}=(\Psi _i{{\overline{\Psi }}}_j)_{1\le i,j\le N}\in {{\mathbb {C}}}^{N\times N}. \end{aligned}$$

The matrix-valued function $W^\varepsilon _{\Psi }\in L^2({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$ is defined by

$$\begin{aligned} W^\varepsilon _{\Psi }(x,\xi )=\int _{{{\mathbb {R}}}^d} \mathrm{e}^{i\xi \cdot v}\Psi \left( x-\frac{\varepsilon v}{2}\right) \otimes {\overline{\Psi }}\left( x+\frac{\varepsilon v}{2}\right) \frac{dv}{(2\pi )^d}, \end{aligned}$$

(3.1)

and its action on symbols $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$ is related with semi-classical pseudodifferential calculus according to

$$\begin{aligned} \int _{{{\mathbb {R}}}^{2d}} {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(a(x,\xi )W^\varepsilon _{\Psi }(x,\xi ))\mathrm{d}x\,\mathrm{d}\xi = \left( \mathrm{op}_\varepsilon (a) \Psi , \Psi \right) _{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)} \end{aligned}$$

where $\mathrm{op}_\varepsilon (a)$ denotes the matrix-valued semi-classical pseudodifferential operator of symbol a. The Wigner function satisfies the following bounds for every $a\in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$ and $\varepsilon >0$:

$$\begin{aligned} \left| \left( \mathrm{op}_\varepsilon (a) \Psi , \Psi \right) _{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)} \right| \le C_d \Vert \Psi \Vert ^2_{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N})}\Vert a\Vert _{{\mathcal {C}}^{d+2}({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})}; \end{aligned}$$

(3.2)

for $C_d>0$ depending only on d. If in addition $a\ge 0$ (meaning that a takes values in the set of non-negative Hermitian matrices),

$$\begin{aligned} \int _{{{\mathbb {R}}}^{2d}} {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(a(x,\xi )W^\varepsilon _{\Psi }(x,\xi ))\mathrm{d}x\,\mathrm{d}\xi \ge -C_a\varepsilon \Vert \Psi \Vert ^2_{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N})}, \end{aligned}$$

(3.3)

for some $C_a>0$ that can be computed in terms of a and its derivatives. Estimate (3.2) is a consequence of the Calderón-Vaillancourt theorem (C.1), whereas (3.3) is a reformulation of Gårding’s inequality (C.3). A direct computation also shows that $W^\varepsilon _\Psi$ is actually a lift of $\Psi \otimes {\overline{\Psi }}$:

$$\begin{aligned} \int _{{{\mathbb {R}}}^d}W^\varepsilon _\Psi (x,\xi )\mathrm{d}\xi =\Psi \otimes {\overline{\Psi }}(x). \end{aligned}$$

(3.4)

Suppose now that $(\Psi ^\varepsilon )_{\varepsilon >0}$ is a bounded sequence in $L^2({{\mathbb {R}}}^d,{{{\mathbb {C}}}}^N)$; then (3.2) ensures that $(W^\varepsilon _{\Psi ^\varepsilon })$ is a bounded sequence of distributions. In addition, (3.3) implies that all its accumulation points are non negative Radon matrix-valued measures, that is, measures valued on the set of complex $N\times N$ Hermitian positive-semidefinite matrices. Moreover, any measure $\mu$ obtained from $(W^\varepsilon _{\Psi ^\varepsilon })$ along some subsequence $(\varepsilon _\ell )$ satisfies (see (C.4), (C.5))

$$\begin{aligned} \mu ({{\mathbb {R}}}^{2d})\le \liminf _{\ell \rightarrow \infty }\Vert \Psi ^{\varepsilon _\ell }\Vert _{L^2({{\mathbb {R}}}^d)}^2. \end{aligned}$$

These measures are called semi-classical or Wigner measures of the family $(\Psi ^\varepsilon )_{\varepsilon >0}$.

Remark 3.1

If $(W^{\varepsilon _\ell }_{\Psi ^{\varepsilon _\ell }})$ converges in ${{\mathcal {S}}}'({{\mathbb {R}}}^{2d})$ to the semiclassical measure $\mu$ then, for every $a\in C_0({{\mathbb {R}}}^{d}_{x}\times {{\mathbb {R}}}^d_{\xi })$ that is $d+2$ times continuously differentiable in x the following holds:

$$\begin{aligned} \int _{{{\mathbb {R}}}^{2d}} {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(a(x,\xi )W^{\varepsilon _\ell }_{\Psi ^{\varepsilon _\ell }}(x,\xi ))dx\,d\xi \mathop {\longrightarrow }\limits _{\ell \rightarrow \infty }\int _{{{\mathbb {R}}}^{2d}} {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(a(x,\xi )\mu (dx,d\xi )). \end{aligned}$$

This follows from Remark C.3 and assertion (1) of Lemma C.4.

Finally, the lift property (3.4) is transferred to an accumulation point $\mu$ generated from a subsequence $(\Psi ^{\varepsilon _\ell })$,i.e.,

$$\begin{aligned} \forall \phi \in {\mathcal {C}}_0({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N\times N}),\quad \lim _{\ell \rightarrow \infty }&\int _{{{\mathbb {R}}}^d}{{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}} (\phi (x) (\Psi ^{\varepsilon _\ell }\otimes \overline{\Psi ^{\varepsilon _\ell }})(x))\mathrm{d}x\\ =&\int _{{{\mathbb {R}}}^d}{{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(\phi (x)\mu (dx,d\xi )), \end{aligned}$$

provided that no mass of $(\Psi ^{\varepsilon _\ell })$ is lost at infinity in Fourier space:

$$\begin{aligned} \limsup _{\ell \rightarrow \infty }\int _{|\varepsilon _\ell \xi |>R}|\widehat{\Psi ^{\varepsilon _\ell }}(\xi )|^2 \mathrm{d}\xi \mathop {\longrightarrow }\limits _{R\rightarrow \infty } 0. \end{aligned}$$

(3.5)

This condition, referred sometimes to as $\varepsilon$-oscillation, is fulfilled as soon as the sequence $(\Psi ^\varepsilon )_{\varepsilon >0}$ is bounded in $H^s_\varepsilon ({{\mathbb {R}}}^d)^N$ for some $s>0$.

Let us conclude this concise review of semi-classical measures by recalling how the matrix-valued semi-classical measure $\mu =(\mu _{i,j})_{1\le i,j,\le N}$ is related to the semi-classical measures of the families of components $(\Psi ^\varepsilon _j)$, for $j=1,\ldots ,N$. Suppose that the subsequence $(\Psi ^{\varepsilon _\ell })$ gives the semi-classical measure $\mu$. Then, for every $1\le i,j, \le N$,

$$\begin{aligned} \forall a\in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{2d}),\quad \lim _{\ell \rightarrow \infty }\left( \mathrm{op}_\varepsilon (a) \Psi ^{\varepsilon _\ell }_i, \Psi ^{\varepsilon _\ell }_j \right) _{L^2({{\mathbb {R}}}^d)}=\int _{{{\mathbb {R}}}^{2d}}a(x,\xi )\mu _{i,j}(dx,d\xi ).\nonumber \\ \end{aligned}$$

(3.6)

Moreover, since $\mu$ takes values on the set of Hermitian positive-semidefinite matrices, one also has that the $\mu _{i,i}$ are non-negative (scalar) Radon measures and that $\mu _{i,j}$ is absolutely continuous with respect to both $\mu _{i,i}$ and $\mu _{j,j}$. The latter condition implies that $\mu _{i,j}=0$ as soon as $\mu _{i,i}$ and $\mu _{j,j}$ are mutually singular. In particular,

$$\begin{aligned} \mu _{i,i}\,\bot \,\mu _{j,j}\implies \forall a\in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{2d}),\quad \lim _{\ell \rightarrow \infty }\left( \mathrm{op}_\varepsilon (a) \Psi ^{\varepsilon _\ell }_i, \Psi ^{\varepsilon _\ell }_j \right) _{L^2({{\mathbb {R}}}^d)}=0. \end{aligned}$$

(3.7)

In this article, we are mainly interested in time-dependent versions of these objects. The modifications required in order to adapt the theory to this context are rather straightforward. Suppose now that $(\Psi ^\varepsilon )_{\varepsilon >0}$ is bounded in $L^\infty ({{\mathbb {R}}}_t; L^2({{\mathbb {R}}}^d_x,{{\mathbb {C}}}^N))$. Define $W^\varepsilon _{\Psi ^\varepsilon }$ as

$$\begin{aligned} W^\varepsilon _{\Psi ^\varepsilon }(t,x,\xi ):= & {} W^\varepsilon _{\Psi ^\varepsilon (t,\cdot )}(x,\xi )\nonumber \\= & {} \int _{{{\mathbb {R}}}^d} \mathrm{e}^{i\xi \cdot v}\Psi ^\varepsilon \left( t,x-\frac{\varepsilon v}{2}\right) \otimes \overline{\Psi ^\varepsilon }\left( t,x+\frac{\varepsilon v}{2}\right) \frac{dv}{(2\pi )^d}. \end{aligned}$$

(3.8)

Then (3.2) again implies that, for every $\theta \in L^1({{\mathbb {R}}})$ and every $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$,

$$\begin{aligned}&\left| \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t) {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}( a(x,\xi ) W^\varepsilon _{\Psi ^\varepsilon }(t,x,\xi ))\mathrm{d}x\,\mathrm{d}\xi \,\mathrm{d}t\right| \nonumber \\&\quad \le C_d \Vert \Psi ^\varepsilon \Vert _{L^\infty ({{\mathbb {R}}}_t; L^2({{\mathbb {R}}}^d_x,{{\mathbb {C}}}^N))}^2 \Vert \theta \Vert _{ L^1({{\mathbb {R}}})} \Vert a\Vert _{{\mathcal {C}}^{d+2}({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})}. \end{aligned}$$

(3.9)

This ensures that $(W^\varepsilon _{\Psi ^\varepsilon })$ is bounded in ${{\mathcal {S}}}'({{\mathbb {R}}}\times {{\mathbb {R}}}^{2d})$. Moreover, any accumulation point $\mu$ of this family is a non negative Radon measure on ${{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}$, because of (3.3). It follows from (3.9) that the projection of $\mu$ onto the t-variable is absolutely continuous with respect to the Lebesgue measure on ${{\mathbb {R}}}$. Therefore, we conclude using the disintegration theorem the existence of a measurable map from $t\in {{\mathbb {R}}}$ to non negative, finite, matrix-valued Radon measures $\mu ^t$ on ${{\mathbb {R}}}^{2d}$ such that $\mu (dt,dx,d\xi )=\mu ^t(dx,d\xi ) dt$.

Summing up, for every sequence $(\varepsilon _\ell )_{\ell \in {{\mathbb {N}}}}$ going to 0 as $\ell$ goes to $+\infty$ such that $(W^{\varepsilon _\ell }_{\Psi ^{\varepsilon _\ell }})$ converges in the sense of distributions the following holds: for all $\theta \in L^1({{\mathbb {R}}})$ and $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$,

$$\begin{aligned}&\int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t) {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(a(x,\xi )W^{\varepsilon _\ell }_{\Psi ^{\varepsilon _\ell }}(t,x,\xi ))\mathrm{d}x\,\mathrm{d}\xi \, \mathrm{d}t\mathop {\longrightarrow }\limits _{\ell \rightarrow \infty }\nonumber \\&\quad \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}}\theta (t) {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(a(x,\xi )\mu ^t(dx,d\xi )) \mathrm{d}t. \end{aligned}$$

(3.10)

If the sequence $(\Psi ^{\varepsilon _\ell }(t,\cdot ))$ is in addition $\varepsilon$-oscillating (3.5) for almost every $t\in {{\mathbb {R}}}$, the projections of the measures $\mu ^t$ on the $\xi$-variable are the limits of the energy densities: for every $\theta \in L^1({{\mathbb {R}}})$, $\phi \in {\mathcal {C}}_0({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N\times N})$,

$$\begin{aligned}&\int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^d} \theta (t) {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}} (\phi (x) (\Psi ^{\varepsilon _\ell }\otimes \overline{\Psi ^{\varepsilon _\ell }})(t,x))\mathrm{d}x \mathop {\longrightarrow }\limits _{\ell \rightarrow \infty } \nonumber \\&\quad \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^d} \theta (t) {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}(\phi (x)\mu ^t(dx,d\xi ))\,\mathrm{d}t. \end{aligned}$$

(3.11)

Remark 3.2

Time-dependent analogues of (3.6), (3.7) also hold after replacing $\mu _{i,j}$ by $\mu ^t_{i,j}$ and averaging in the t-variable. So does the analogue of Remark 3.1.

3.2 The Semi-classical Measure of $(\psi ^\varepsilon )_{\varepsilon >0}$ in Terms of Those of the Sequences $(\psi ^\varepsilon _n)_{\varepsilon >0}$, $n\in {{\mathbb {N}}}$

We now focus on the basic properties of semi-classical measures associated to a sequence $(\psi ^\varepsilon )_{\varepsilon >0}$ of solutions to (1.1), issued from initial data $(\psi ^\varepsilon _0)_{\varepsilon >0}$ that are bounded in $H^s_\varepsilon ({{\mathbb {R}}}^d)$ for some $s>d/2$, and onto clarifying how they are related to those of the families of projections $(\psi ^\varepsilon _n)_{\varepsilon >0}$ defined in (2.6).

First note that the highly oscillating character of the Schrödinger propagator prevents in general to be able to extract a subsequence along which $W^\varepsilon _{\psi ^\varepsilon (t,\cdot )}$ will converge for every $t\in {{\mathbb {R}}}$. Following [3, 6, 41] we consider time averages of the Wigner functions we just described.

Identity (3.10) applied to this context states that whenever $(W^{\varepsilon _\ell }_{\psi ^{\varepsilon _\ell }})$ converges in the sense of distributions for some sequence $(\varepsilon _\ell )_{\ell \in {{\mathbb {N}}}}$ going to 0 as $\ell$ goes to $+\infty$ the following holds: for all $\theta \in L^1({{\mathbb {R}}})$ and $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d})$,

$$\begin{aligned} \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t) a(x,\xi )W^{\varepsilon _\ell }_{\psi ^{\varepsilon _\ell }}(t,x,\xi )\mathrm{d}x\,\mathrm{d}\xi \, \mathrm{d}t\mathop {\longrightarrow }\limits _{\ell \rightarrow \infty } \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t) a(x,\xi )\varsigma ^t(dx,d\xi ) \mathrm{d}t,\nonumber \\ \end{aligned}$$

(3.12)

where, for a.e. $t\in {{\mathbb {R}}}$, $\varsigma ^t$ is a non negative Radon measure on ${{\mathbb {R}}}^{2d}$.

In addition, $\varsigma ^t$ can be related to the weak limits of the energy densities since the family $(\psi ^\varepsilon )_{\varepsilon >0}$ is $\varepsilon$-oscillating.

Remark 3.3

If $(\psi ^\varepsilon _0)_{\varepsilon >0}$ is bounded in $H^s_\varepsilon ({{\mathbb {R}}}^d)$ for some $s>d/2$ then $(\psi ^\varepsilon (t,\cdot ))_{\varepsilon >0}$, the corresponding family of solutions to (1.1), is $\varepsilon$-oscillating for every $t\in {{\mathbb {R}}}$. This follows from [15, Lemma 6.2] applied to the family $(U^\varepsilon )_{\varepsilon >0}$ of solutions to (2.2) once one notices that for $r\in ({d\over 2},s)$, $R>0$ and $t\in {{\mathbb {R}}}$,

$$\begin{aligned} \int _{|\varepsilon \xi |>R}\Vert \widehat{U^\varepsilon }(t,\xi ,\cdot )\Vert _{H^r({{\mathbb {T}}}^d )}^2\mathrm{d}\xi \le R^{-2(s-r)}(\Vert \psi ^\varepsilon _0\Vert _{H^s_\varepsilon ({{\mathbb {R}}}^d)}+C_s\varepsilon |t|)^2, \end{aligned}$$

as follows from estimate (2.3).

As a consequence of this, (3.11) implies one has for the subsequence $(\varepsilon _\ell )$ of (3.12) and for $\theta \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}})$, $\phi \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^d)$,

$$\begin{aligned} \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^d}\theta (t) \phi (x)|\psi ^{\varepsilon _\ell }(t,x)|^2 \mathrm{d}x\, \mathrm{d}t \mathop {\longrightarrow }\limits _{\ell \rightarrow +\infty } \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t) \phi (x) \varsigma ^t( dx,d\xi )\mathrm{d}t.\nonumber \\ \end{aligned}$$

(3.13)

For $n,n'\in {{\mathbb {N}}}^*$, we use the notation $W^\varepsilon _{n,n'}$ to refer to the Wigner function of the pair $\psi ^\varepsilon _n,\psi ^\varepsilon _{n'}$. In other words, for every $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d})$,

$$\begin{aligned} \int _{{{\mathbb {R}}}^{2d}} a(x,\xi ) W^\varepsilon _{n,n'}(t,x,\xi )\mathrm{d}x\,\mathrm{d}\xi = \left( \mathrm{op}_\varepsilon (a) \psi ^\varepsilon _n (t,\cdot ) , \psi ^\varepsilon _{n'}(t,\cdot ) \right) _{L^2({{\mathbb {R}}}^d)}. \end{aligned}$$

The same argument presented before shows that for $n,n'\in {{\mathbb {N}}}$, there exists a sequence $(\varepsilon _\ell )_{\ell \in {{\mathbb {N}}}}$ going to 0 as $\ell$ goes to $+\infty$ such that, for all $\theta \in L^1({{\mathbb {R}}})$, $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d})$,

$$\begin{aligned}&\int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t) a(x,\xi ) W^\varepsilon _{n,n'}(t,x,\xi )\mathrm{d}x\,\mathrm{d}\xi \, \mathrm{d}t\mathop {\longrightarrow }\limits _{\ell \rightarrow \infty } \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t) a(x,\xi )\mu ^t_{n,n'} (dx,d\xi )) \mathrm{d}t,\nonumber \\ \end{aligned}$$

(3.14)

where, for a.e. $t\in {{\mathbb {R}}}$, $\mu ^t_{n,n'}$ is a (signed) Radon measure on ${{\mathbb {R}}}^{2d}$.

Proposition 3.4

There exist a subsequence $(\varepsilon _\ell )_{\ell \in {{\mathbb {N}}}}$ going to 0 as $\ell$ goes to $+\infty$ such that (3.12) and (3.14) hold simultaneously for all $n,n'\in {{\mathbb {N}}}^*$. In addition, for a.e. $t\in {{\mathbb {R}}}$,

$$\begin{aligned} \varsigma ^t= \sum _{ n,n'\in {{\mathbb {N}}}^*} \mu ^t_{n,n'}, \end{aligned}$$

the convergence of the series being understood in the weak-$*$ topology of the space of Radon measures on ${{\mathbb {R}}}^{2d}$.

Proof

We proceed to a first extraction to have (3.12) and we keep denoting by $\varepsilon$ the resulting subsequence. We put

$$\begin{aligned} \Psi ^\varepsilon _N:=(\psi ^\varepsilon _1,\ldots ,\psi ^\varepsilon _N)\in {\mathcal {C}}({{\mathbb {R}}}_t; L^2({{\mathbb {R}}}^d_x,{{\mathbb {C}}}^N)). \end{aligned}$$

We know that $(W^\varepsilon _{\Psi ^\varepsilon _N})$, defined by (3.8), are uniformly bounded in ${\mathcal {C}}({{\mathbb {R}}}_t;{{\mathcal {S}}}'({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N}))$, both in $\varepsilon >0$ and $N\in {{\mathbb {N}}}^*$.

By (3.10), any accumulation point of $(W^\varepsilon _{\Psi ^\varepsilon _N})$ obtained along some subsequence $(\varepsilon _\ell )_{\ell \in {{\mathbb {N}}}}$ is a time-dependent family of non negative matrix-valued Radon measures $\mu _N^t$. By diagonal extraction, we can find a sequence $(\varepsilon _\ell )_{\ell \in {{\mathbb {N}}}}$ such that $(W^{\varepsilon _\ell }_{\Psi ^{\varepsilon _\ell }_N})_{\varepsilon >0}$ converge for every $N\in {{\mathbb {N}}}^*$. We denote by $(\mu ^t_N)_{N\in {{\mathbb {N}}}^*}$ their respective limits. By (3.6) we know that, for every $n,n'\le N\le N'$ one has

$$\begin{aligned} (\mu ^t_N)_{n,n'}=(\mu ^t_{N'})_{n,n'}=\mu ^t_{n,n'}, \end{aligned}$$

where $\mu ^t_{n,n'}$ is obtained through (3.14). This shows that we can find a sequence $(\varepsilon _\ell )_{\ell \in {{\mathbb {N}}}}$ as claimed.

Define now $\psi ^{N,\varepsilon }:=\sum _{n=1}^N\psi ^\varepsilon _n$. One has that for $a\in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{2d})$ and $t\in {{\mathbb {R}}}$,

$$\begin{aligned} \int _{{{\mathbb {R}}}^{2d}} a(x,\xi )W^{\varepsilon _\ell }_{\psi ^{N,\varepsilon _\ell }}(t,x,\xi )\mathrm{d}x\,\mathrm{d}\xi =\int _{{{\mathbb {R}}}^{2d}}a(x,\xi ) {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}\left( Q\, W^{\varepsilon _\ell }_{\Psi ^{\varepsilon _\ell }_N}(t,x,\xi ) \right) \mathrm{d}x\,\mathrm{d}\xi , \end{aligned}$$

where Q is the $N\times N$ matrix whose all entries are equal to one. Therefore, $(W^{\varepsilon _\ell }_{\psi ^{N,\varepsilon _\ell }})_{\ell \in {{\mathbb {N}}}}$ converges to the semi-classical measure given, for a.e. $t\in {{\mathbb {R}}}$, by

$$\begin{aligned} \varsigma ^t_N=\sum _{1\le n,n'\le N}\mu ^t_{n,n'}. \end{aligned}$$

Finally, (2.5) and Lemma 2.1 imply that for every $\theta \in L^1({{\mathbb {R}}})$,

$$\begin{aligned} \limsup _{\ell \rightarrow \infty } \int _{{\mathbb {R}}}\theta (t) \Vert \psi ^{\varepsilon _\ell }(t,\cdot )-\psi ^{N,\varepsilon _\ell }(t,\cdot )\Vert _{L^2({{\mathbb {R}}}^d)}^2\mathrm{d}t\mathop {\longrightarrow }\limits _{N\rightarrow \infty } 0; \end{aligned}$$

which in turn guarantees that, for every $\theta \in L^1({{\mathbb {R}}}), a\in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{2d})$,

$$\begin{aligned} \int _{{{\mathbb {R}}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t)a(x,\xi )\varsigma ^t_N(dx,d\xi )dt\mathop {\longrightarrow }\limits _{N\rightarrow \infty } \int _{{{\mathbb {R}}}}\int _{{{\mathbb {R}}}^{2d}} \theta (t)a(x,\xi )\varsigma ^t(dx,d\xi )\mathrm{d}t. \end{aligned}$$

$\square$

The rest of this article is devoted to computing the measures $\mu ^t_{n,n'}$.

3.3 Localization of Semi-classical Measures

If the operator $L^\varepsilon P^\varepsilon _{\varphi _n}$ is applied to problem (2.2), one deduces that $\psi ^\varepsilon _n$ (which is defined by (2.6)) satisfies the pseudo-differential equation

$$\begin{aligned} \left\{ \begin{array}{l} i\varepsilon ^2 \partial _t \psi _n^\varepsilon (t,x)) = \varrho _n(\varepsilon D_x) \psi _n^\varepsilon (t,x)+ \varepsilon ^2 f^\varepsilon _n(t,x),\quad (t,x)\in {{\mathbb {R}}}\times {{\mathbb {R}}}^d,\quad \\ \psi _n^\varepsilon (0,x)=\varphi _n\left( {x\over \varepsilon },\varepsilon D_x\right) \int _{{{\mathbb {T}}}^d} \overline{\varphi _n}(y,\varepsilon D_x) \psi ^\varepsilon _0(x)\mathrm{d}y, \end{array}\right. \end{aligned}$$

(3.15)

with

$$\begin{aligned} f^\varepsilon _n(t,x):= \varphi _n\left( {x\over \varepsilon },\varepsilon D_x\right) \int _{{{\mathbb {T}}}^d} \overline{\varphi _n}(y,\varepsilon D_x) (V_{\mathrm{ext}} (t,x) U^\varepsilon (t,x,y))\mathrm{d}y. \end{aligned}$$

This fact will be used to obtain all the information on the measures $\mu ^t_{n,n'}$ defined in (3.14) that is relevant to our purposes. In this section we gather some basic facts; in following sections we will introduce a more precise machinery that will allow us to obtain a complete picture.

Proposition 3.5

Let $(\psi ^\varepsilon _0)$ be bounded in $H^s_\varepsilon ({{\mathbb {R}}}^d)$ for some $s>d/2$. For any $n,n'\in {{\mathbb {N}}}^*$, let $(\psi ^\varepsilon _n)_{\varepsilon >0}$ and $(\psi ^\varepsilon _{n'})$ be defined by (2.6) and let $\mu ^t_{n,n'}$ be given by (3.14). Let $\Omega \subseteq {{\mathbb {R}}}^d$ be open and invariant by translations by $2\pi {{\mathbb {Z}}}^d$. Then the following hold:

• (1) If $\nabla _\xi \varrho _n\in \mathrm{Lip}({{\mathbb {R}}}^d)$ on $\Omega$ and $\nabla _\xi \varrho _n|_\Omega \not =0$, then

  $$\begin{aligned} \mu _{n,n} ^t({{\mathbb {R}}}^d\times \Omega ) =0,\quad \text { for a.e. }t\in {{\mathbb {R}}}. \end{aligned}$$

• (2) Let $\delta >0$ and suppose that

  $$\begin{aligned} \Omega \subset \{\xi \in {{\mathbb {R}}}^d\,:\, |\varrho _n(\xi )-\varrho _{n'}(\xi )|\ge \delta \}. \end{aligned}$$

  Then $\displaystyle {|\mu _{n,n'} ^t|({{\mathbb {R}}}^d\times \Omega ) =0,\quad \text { for a.e. }t\in {{\mathbb {R}}}.}$

Proof

Point 1 is proved in an analogous manner than Proposition 3.4 in [15]. Using the calculus of semi-classical pseudo-differential operators with low regularity of Lemma C.4 it is possible to prove that for every $\theta \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}})$ and $a\in {\mathcal {C}}^\infty _0 ({{\mathbb {R}}}^d\times \Omega )$,

$$\begin{aligned} \int _{{\mathbb {R}}}\theta (t) (\mathrm{op}_\varepsilon (\nabla _\xi \varrho _n \cdot \nabla _x a) \psi ^\varepsilon _n(t,\cdot ) ,\psi ^\varepsilon _{n}(t,\cdot ))_{L^2({{\mathbb {R}}}^d)}\mathrm{d}t \mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0} 0. \end{aligned}$$

By (3.14), this implies that, for almost every $t\in {{\mathbb {R}}}$,

$$\begin{aligned} \int _{{{\mathbb {R}}}^{d}\times \Omega }\nabla _\xi \varrho _n (\xi )\cdot \nabla _x a(x,\xi )\mu _{n,n}^t(dx,d\xi )=0. \end{aligned}$$

This implies that the measure $\mu _{n,n}^t \mathbf{1 }_{{{\mathbb {R}}}^d\times \Omega }$ is invariant by the flow $(x,\xi ) \mapsto (x+s\nabla \varrho _n(\xi ), \xi )$. Since $\mu _{n,n}^t$ is non negative and finite, necessarily it is identically 0.

For proving Point 2, it is enough to obtain, for every $\theta \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}})$ and $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{d}\times \Omega )$:

$$\begin{aligned} \int _{{\mathbb {R}}}\theta (t) \left( \mathrm{op}_\varepsilon (a) \psi ^\varepsilon _n(t,\cdot ),\psi ^\varepsilon _{n'}(t,\cdot )\right) _{L^2({{\mathbb {R}}}^d)} \mathrm{d}t\mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0}0. \end{aligned}$$

We have

$$\begin{aligned}&i\varepsilon ^2 \frac{d}{dt} \left( \mathrm{op}_\varepsilon (a) \psi ^\varepsilon _n(t,\cdot ),\psi ^\varepsilon _{n'}(t,\cdot )\right) _{L^2({{\mathbb {R}}}^d)} \nonumber \\&\quad = \left( \left( \varrho _{n'}(\varepsilon D_x) \mathrm{op}_\varepsilon (a) - \mathrm{op}_\varepsilon (a)\varrho _n(\varepsilon D_x)\right) \psi ^\varepsilon _n(t,\cdot ),\psi ^\varepsilon _{n'}(t,\cdot )\right) _{L^2({{\mathbb {R}}}^d)}+\varepsilon ^2 R^\varepsilon (t),\nonumber \\ \end{aligned}$$

(3.16)

where $|R^\varepsilon (t)|\le C\Vert f^\varepsilon _n(t,\cdot )\Vert _{L^2({{\mathbb {R}}}^d)}^2$ is locally uniformly bounded in $t\in {{\mathbb {R}}}$ for every $\varepsilon >0$.

By Lemma C.4 (2), the following holds with respect to the ${{\mathcal {L}}}(L^2({{\mathbb {R}}}^d))$ norm:

$$\begin{aligned} \varrho _{n'}(\varepsilon D_x) \mathrm{op}_\varepsilon (a) - \mathrm{op}_\varepsilon (a)\varrho _n(\varepsilon D_x)= \mathrm{op}_\varepsilon \left( (\varrho _{n'}-\varrho _n) a\right) +O(\varepsilon ). \end{aligned}$$

This identity together with integration by parts transforms (3.16) into

$$\begin{aligned}&\int _{{\mathbb {R}}}\theta (t) \left( \mathrm{op}_\varepsilon \left( (\varrho _{n'}-\varrho _n) a\right) \psi ^\varepsilon _n(t,\cdot ),\psi ^\varepsilon _{n'}(t,\cdot )\right) _{L^2({{\mathbb {R}}}^d)} \mathrm{d}t\\&\quad = \frac{\varepsilon ^2}{i} \int _{{\mathbb {R}}}\theta '(t) \left( \mathrm{op}_\varepsilon (a) \psi ^\varepsilon _n(t,\cdot ),\psi ^\varepsilon _{n'}(t,\cdot )\right) _{L^2({{\mathbb {R}}}^d)} \mathrm{d}t + O(\varepsilon ). \end{aligned}$$

Taking limits $\varepsilon \rightarrow 0$, which is possible by Remarks 3.1 and 3.2, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}}\theta (t)(\varrho _{n'}(\xi )-\varrho _n(\xi )) a(x,\xi )\mu _{n,n'} ^t(dx,d\xi ) \mathrm{d}t=0. \end{aligned}$$

By density, this relation holds for all $a\in {{\mathcal {C}}}_0 ({{\mathbb {R}}}^{d}\times \Omega )$, in particular for ${\tilde{a}} = (\varrho _n-\varrho _{n'})^{-1} a$. This shows that, as we wanted to prove

$$\begin{aligned} \forall \theta \in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}),\; \forall a\in {{\mathcal {C}}}_0 ({{\mathbb {R}}}^{d}\times \Omega ),\quad \int _{{\mathbb {R}}}\int _{{{\mathbb {R}}}^{2d}}\theta (t) a(x,\xi )\mu _{n,n'} ^t(dx,d\xi ) \mathrm{d}t=0. \end{aligned}$$

$\square$

Proposition 3.5 shows that $\mu ^t_{n,n}$ can only charge the sets $\Lambda _n$ of critical points of $\varrho _n$ or the set where $\varrho _n$ has a conical crossing with another Bloch energy (i.e. where $\varrho _n$ ceases to be ${\mathcal {C}}^{1,1}({{\mathbb {R}}}^d)$). It also shows that $\Sigma _{n,n'}$ is the only region where the measures $\mu ^t_{n,n'}$ can be non-zero. Since, assuming H1, the crossing sets reduce to $\cup _{n\in {{\mathbb {N}}}_0} \Sigma _n$, the analysis of the measures $\mu ^t_{n,n'}$ will be performed in the following sections by means of a second microlocalisation above the sets $\Lambda _n$ and $\Sigma _n$, under the assumption H1, H2 and H3.

4 Two Microlocal Analysis

The analysis of the concentration of a family on a submanifold of the phase space turned out to be an important element of the analysis of its behavior. Two-microlocal semi-classical measures gives a quantitative overview on these concentration phenomena. They were first introduced simultaneously and independently in [50] and by one of the author in her thesis (see the articles [21, 22, 24]), and they have found applications in different fields, as for example [26, 42] and articles connected to these ones. We recall in Section 4.1 the definition of two-microlocal semi-classical measures that is useful in our context and apply the theory to families $(\psi ^\varepsilon )_{\varepsilon >0}$ of solution to (1.1) in Section 4.1 in the frameworks of Theorem 1.5 and of Theorem 1.8. This leads to the statement of two results (Theorems 4.5 and 4.6) that will be proved in the next Sections 5 and 6.

4.1 Two-Scale Semi-classical Measures

We study here the concentration of a bounded family $(\Psi ^\varepsilon )_{\varepsilon >0}$ of $L^\infty ({{\mathbb {R}}}, L^2( {{\mathbb {R}}}^d,{{\mathbb {C}}}^N))$ on a set ${{\mathbb {R}}}^d\times X$ where X is assumed to be a connected, closed embedded submanifold of $({{\mathbb {R}}}^d)^*$ of codimension p. Following [15], we achieve a second microlocalization above ${{\mathbb {R}}}^d\times X$ and we crucially use that the geometric properties of X imply that there exists a tubular neighbourhood U of $\{(\sigma ,0)\,:\,\sigma \in X\}\subseteq N X$ such that the tubular coordinate map

$$\begin{aligned} U\ni (\sigma ,v)\longmapsto \sigma +v\in ({{\mathbb {R}}}^d)^* \end{aligned}$$

is a diffeomorphism onto its image V. In that case, there exists a smooth map $\sigma _X:V\longrightarrow X$ such that, for every $\xi \in V$,

$$\begin{aligned} \begin{array}{ccl} \xi =\sigma +v,\quad (\sigma ,v)=(\sigma _X(\xi ),\xi -\sigma _X(\xi ))\in U. \end{array} \end{aligned}$$

(4.1)

We extend the phase space $T^*{{\mathbb {R}}}^d:={{\mathbb {R}}}^d_x\times ({{\mathbb {R}}}^d)^*_\xi$ with a new variable $\eta \in \overline{ {{\mathbb {R}}}^d}$, where $\overline{{{\mathbb {R}}}^d}$ is the compactification of $({{\mathbb {R}}}^d)^*$ obtained by adding a sphere $\mathbf{S}^{d-1}$ at infinity. The space ${\mathcal {A}}^{(2)}$ of test functions associated with this extended phase space is formed by those functions

$$\begin{aligned} a\in {{\mathcal {C}}}^\infty (T^*{{\mathbb {R}}}^d_{x,\xi }\times {{\mathbb {R}}}^d_\eta ,{{\mathbb {C}}}^{N\times N}) \end{aligned}$$

which satisfy the two following properties:

• (1) there exists a compact $K \subset T^*{{\mathbb {R}}}^d$ such that, for all $\eta \in {{\mathbb {R}}}^d$, the map $(x,\xi )\longmapsto a(x,\xi ,\eta )$ is a smooth matrix-valued function compactly supported in K;

• (2) there exists a smooth matrix-valued function $a_\infty$ defined on $T^*{{\mathbb {R}}}^d\times \mathbf{S}^{d-1}$ and $R_0>0$ such that, if $|\eta |>R_0$, then $a(x,\xi ,\eta )=a_\infty (x,\xi ,\eta /|\eta |)$.

For $a\in {\mathcal {A}}^{(2)}$ supported in ${{\mathbb {R}}}^d\times V\times {{\mathbb {R}}}^d$, we write

$$\begin{aligned} a_\varepsilon (x,\xi ):=a\left( x,\xi ,\frac{\xi -\sigma _X(\xi )}{\varepsilon }\right) . \end{aligned}$$

We associate to $\Psi ^\varepsilon (t)$ a two-microlocal Wigner distribution

$$\begin{aligned} W^{X,\varepsilon }(t)\in {\mathcal {D}}'({{\mathbb {R}}}^d\times V\times \overline{{{\mathbb {R}}}^d}), \;\;W^{X,\varepsilon }_{\Psi ^\varepsilon }(t)=(W_{j,k}^{X,\varepsilon })_{1\le j,k\le N}; \end{aligned}$$

its action on test functions $a\in {{\mathcal {A}}}^{(2)}$ supported in ${{\mathbb {R}}}^d\times V\times {{\mathbb {R}}}^d$ is defined by

$$\begin{aligned} \left\langle W^{X,\varepsilon }_{\Psi ^\varepsilon }(t),a\right\rangle :=\left( \mathrm{op}_\varepsilon (a_\varepsilon ) \Psi ^\varepsilon (t),\,\Psi ^\varepsilon (t)\right) _{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}. \end{aligned}$$

(4.2)

Since the family of operators $(\mathrm{op}_\varepsilon (a_\varepsilon ))_{\varepsilon >0}$ is uniformly bounded in $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N\times N})$ (as a consequence of the Calderón-Vaillancourt theorem, see “Appendix C”), it follows that $(W^{X,\varepsilon }_{\Psi ^\varepsilon }(t))$ is a bounded sequence of distributions. In addition, any smooth, compactly supported test function $a\in {{\mathcal {C}}}^\infty _0({{\mathbb {R}}}^d\times V,{{\mathbb {C}}}^{N\times N})$ can be naturally identified to an element of ${{\mathcal {A}}}^{(2)}$ which does not depend on the last variable. For such a, one clearly has

$$\begin{aligned} \left\langle W^{X,\varepsilon }_{\Psi ^\varepsilon }(t) ,a\right\rangle = \left\langle W^{\varepsilon }_{\Psi ^\varepsilon }(t) ,a\right\rangle ; \end{aligned}$$

hence $W^{X,\varepsilon }_{\Psi ^\varepsilon }(t)$ is a lift of $W^{\varepsilon }_{\psi ^\varepsilon }(t)$ to the extended phase-space. We thus focus on the asymptotic description of the quantities

$$\begin{aligned} \int _{{\mathbb {R}}}\theta (t) \langle W^{X,\varepsilon }_{\Psi ^\varepsilon }(t),a\rangle \mathrm{d}t,\;\;\theta \in L^1({{\mathbb {R}}}),\;\;a\in {{\mathcal {A}}}^{(2)}. \end{aligned}$$

(4.3)

In order to describe the limits of these quantities, we must introduce some notations. We consider an open subset W of V where there exists $\varphi : W \longrightarrow {{\mathbb {R}}}^p$ a smooth function such that the $\xi \in W$ for which $\varphi (\xi )=0$ are precisely those which are in $W\cap X$. We also assume that $d\varphi (\sigma )$ for $\sigma \in W\cap X$ is of maximal rank. These coordinates functions give parametrization of the manifolds under consideration and for every $\sigma \in W\cap X$, we can write

$$\begin{aligned} N_\sigma X= \{ \, ^td\varphi (\sigma ) z\;:\; z\in {{\mathbb {R}}}^p\}. \end{aligned}$$

This parametrization allows to define a measure on $N_\sigma X$ and the space $L^2(N_\sigma X,{{\mathbb {C}}}^2)$. Different $\varphi$ will give equivalent norms. The function $\varphi$ also induces a smooth map B from the neighbourhood W of $\sigma$ into the set of $d\times p$ matrices such that

$$\begin{aligned} \xi -\sigma _X(\xi )=B(\xi )\varphi (\xi ),\;\;\xi \in W. \end{aligned}$$

(4.4)

Therefore, given a function $a\in {{\mathcal {C}}}^\infty _0({{\mathbb {R}}}^d\times W\times {{\mathbb {R}}}^d,{{\mathbb {C}}}^{N\times N})$ and a point $(\sigma ,v)\in T^*_\sigma X$, we can use $\varphi$ to define an operator acting on $f\in L^2(N_\sigma X,{{\mathbb {C}}}^N)$ by

$$\begin{aligned} Q_a^\varphi (\sigma ,v)f(z)=\int _{{{\mathbb {R}}}^p\times {{\mathbb {R}}}^p}a\left( v+ \, ^td\varphi (\sigma ) \frac{z+y}{2},\sigma , B(\sigma ) \eta \right) f(y)\mathrm{e}^{i\eta \cdot (z-y)}\frac{d\eta \,dy}{(2\pi )^p}. \end{aligned}$$

In other words, $Q_a^\varphi (\sigma ,v)$ is obtained from a by applying the non-semi-classical Weyl quantization to the symbol $a\left( v+ \, ^td\varphi (\sigma ) \, \cdot \,,\sigma , B(\sigma ) \, \cdot \,\right) \in {{\mathcal {C}}}^\infty _0({{\mathbb {R}}}^p\times {{\mathbb {R}}}^p,{{\mathbb {C}}}^{N\times N})$,

$$\begin{aligned} Q_a^\varphi (\sigma ,v)= a^W\left( v+ \, ^td\varphi (\sigma ) z,\sigma , B(\sigma ) D_z\right) . \end{aligned}$$

(4.5)

Using invariance properties with respect to changes of coordinate systems that are precisely described in [15], Section 4, one can conclude that a induces an operator $Q_a^X$ on $L^2(N_\sigma X,{{\mathbb {C}}}^N)$. Clearly, $Q_a^X(\sigma ,v)$ is smooth and compactly supported in $(\sigma ,v)$; moreover, $Q_a^X(\sigma ,v)$ is a compact operator on $L^2(N_\sigma X,{{\mathbb {C}}}^N)$ for every $(\sigma ,v)\in T^*X$.

Proposition 4.1

(Proposition 4.2 and 4.4 of [15]) There exist a sequence $(\varepsilon _\ell )$, a measurable map $t\mapsto \gamma ^t$ valued in the set of non negative (matrix-valued) measures on $T^*{{\mathbb {R}}}^d\times \mathbf{S}^{d-1}$, a measurable family of (scalar) non negative measures $\nu ^t$ on $T^*X$ and a measurable map $t\mapsto M^t$, where

$$\begin{aligned}&M^t:T^*X\ni (\sigma ,v)\longmapsto M^t(\sigma ,v)\in {\mathcal {L}}^1(L^2(N_\sigma X,{{\mathbb {C}}}^N)) \\&\quad \text{ and }\qquad {{\,\mathrm{Tr}\,}}_{L^2(N_\sigma X,{{\mathbb {C}}}^N)} M^t(\sigma ,v)=1,\quad \nu ^t\text {-a.e. }(\sigma ,v)\in T^*X \end{aligned}$$

such that, for every $\theta \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}})$ and $a\in {{\mathcal {A}}}^{(2)}$ supported in ${{\mathbb {R}}}^d\times V\times {{\mathbb {R}}}^d$, one has:

$$\begin{aligned}&\int _{{\mathbb {R}}}\theta (t) \left\langle W^{X,\varepsilon _\ell }(t),a\right\rangle \mathrm{d}t \mathop {\longrightarrow }\limits _{\varepsilon _\ell \rightarrow 0} \nonumber \\&\quad \int _{{\mathbb {R}}}\theta (t) \int _{T^*X}{{\,\mathrm{Tr}\,}}_{L^2(N_\sigma X,{{\mathbb {C}}}^N)}(Q_a^X(\sigma ,v)M^t(\sigma ,v))\nu ^t(d\sigma ,dv)\mathrm{d}t \nonumber \\&\quad + \int _{{\mathbb {R}}}\theta (t) \int _{ T^*{{\mathbb {R}}}^d\times \mathbf{S}^{d-1}} {{\,\mathrm{Tr}\,}}_{{{\mathbb {C}}}^{N\times N}}\left( a_\infty (x,\sigma ,\omega ) \gamma ^t(dx,d\sigma ,d\omega ) \right) \mathrm{d}t .\end{aligned}$$

(4.6)

The family of operators $M^t(\sigma ,v)$ describes the part of the concentration that comes from finite distance while the measure $\gamma ^t(dx,d\sigma ,d\omega )$ is often called the part at infinity of the two-scale semi-classical measure. In particular

$$\begin{aligned} \gamma ^t(x,\xi ,\omega ) \mathbf{1}_{\xi \notin X}= \mu ^t (x,\xi ) \otimes \delta \left( \omega -\frac{\xi -\sigma _X(\xi )}{|\xi -\sigma _X(\xi )|}\right) , \end{aligned}$$

where $\mu ^t$ is a semi-classical measure of $(\Psi ^\varepsilon (t))_{\varepsilon >0}$.

Remark 4.2

• (1) In a stationary setting, similar objects can be associated with (non time dependent) bounded families in $L^2({{\mathbb {R}}}^d)$. More precisely, if $(f^\varepsilon )_{\varepsilon >0}$ is a bounded family in $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$, one can associate with $(f^\varepsilon )_{\varepsilon >0}$ a pair $M_0d\nu _0$ defined by the existence of a subsequence $(\varepsilon _\ell )$ such that for all $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{3d},{{\mathbb {C}}}^{N\times N})$,

  $$\begin{aligned} \left( \mathrm{op}_{\varepsilon _\ell }(a_{\varepsilon _\ell }) f^{\varepsilon _\ell },f^{\varepsilon _\ell }\right) \mathop {\longrightarrow }\limits _{\varepsilon _\ell \rightarrow 0} \int _{T^*X}{{\,\mathrm{Tr}\,}}_{L^2(N_\sigma X,{{\mathbb {C}}}^N)}(Q_a^X(\sigma ,v)M_0(\sigma ,v))\nu _0(d\sigma ,dv). \end{aligned}$$

  The initial data in the the von Neumann Equation (1.15) (Theorem 1.5) and in the von Neumann Equations (1.19) and (1.20) (Theorems 1.8) are constructed in that manner, with $N=1$, $X=\Lambda _n$ and $f^\varepsilon =\psi ^\varepsilon _n(0)$ for Equation (1.15) and with $N=2$, $X=\Sigma _n$ and $f^\varepsilon =\,^t( \psi ^\varepsilon _n(0),\psi ^\varepsilon _{n+1}(0))\in {{\mathbb {C}}}^2$ for Equations (1.19) and (1.20).

• (2) When $X=\{\xi _0\}$, then $\sigma _X(\xi )=\xi -\xi _0$ and one has for $a\in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{3d})$

  $$\begin{aligned} \left( \mathrm{op}_\varepsilon (a_\varepsilon ) f^\varepsilon ,f^\varepsilon \right) = \left( \mathrm{op}_1(a(x,\xi _0+\varepsilon \xi , \xi ) \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot \xi _0} f^\varepsilon , \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot \xi _0} f^\varepsilon \right) \end{aligned}$$

  and the operator $\mathrm{op}_1(a(x,\xi _0, \xi )$ is compact As a consequence, the part at finite distance of any two-microlocal measure associated with the concentration of $(f^\varepsilon )$ on X is a projector $|f^{\xi _0}\rangle \langle f^{\xi _0}|$ where $f^{\xi _0}$ is a weak limit in $L^2$ of the family $(\mathrm{e}^{-\frac{i}{\varepsilon }x\cdot \xi _0} f^\varepsilon )$.

• (3) Note that for determining the part of the concentration that comes from finite distance, it is enough to consider symbols a that are compactly supported in all the variables.

4.2 Two Microlocal Semi-classical Measures for the Families $(\psi ^\varepsilon _n(t))_{n\in N}$

These objects allow to determine the semi-classical measure $\varsigma ^t$. Indeed, in [15], we have proved that they allow to describe $\varsigma ^t$ above critical points of $\varrho _n$ for which the hessian of $\varrho _n$ is of maximal rank on the set of critical points $\Lambda _n$ (see assumption H2 and H2’). We will use them to prove that $\varsigma ^t=0$ above all crossing sets satisfying H3 and to show that $\varsigma ^t$ can be non zero because of modes interactions above degenerate crossing points satisfying H3’.

4.2.1 Critical Points

We recall here results from [14, 15], mainly Theorem 2.2 in [15] which gives a precise description of the measures $\mu ^t_{n,n}$ above the set $\Lambda _n$ of critical points of $\varrho _n$ (see (1.4)). Let $\Omega$ be an open set of ${{\mathbb {R}}}^d$ such that $\Lambda _n\cap \Omega$ is a submanifold.

Theorem 4.3

[15] Let $(M^t_{n}d\nu ^t_{n}, \gamma ^t_{n})$ be a pair of two-microlocal semi-classical measures associated with the concentration of $(\psi ^\varepsilon _n(t))$ above $\Lambda _n\cap \Omega$. Then, there exists $M_n d\nu _{n}$, a two-microlocal measure associated to the concentration at finite distance of $(\psi ^\varepsilon _n(0))$ on $\Lambda _n\cap \Omega$ such that $\nu ^t_{n}=\nu ^0_{n}=\nu _n$, $t\mapsto M^t_{n}(\xi ,v)$ belongs to the space ${\mathcal {C}}({{\mathbb {R}}};{\mathcal {L}}_+^1(L^2(N_\xi \Lambda _n))$ and solves the von Neumann equation (1.15) with initial data $M^0_{n}=M_n$. Moreover, if the Hessian of $\varrho _n$ is of maximal rank on $\Lambda _n\cap \Omega$, then $\gamma ^t_{n}=0$.

Remark 4.4

• (1) The maximal rank assumption consists in saying that

  $$\begin{aligned} \mathrm{Rank}\, \mathrm{Hess}\,\varrho _n(\sigma )= \mathrm{codim}\, \Lambda _n,\;\;\sigma \in \Lambda _n, \end{aligned}$$

  or equivalently: $\displaystyle {\mathrm{Ker} \, \mathrm{Hess}\,\varrho _n(\sigma )= T_\sigma \Lambda _n,\;\;\sigma \in \Lambda _n.}$

• (2) It is important to notice that the families $(M^t_{n})$ are completely determined by the initial data: up to a subsequence for which one has

  $$\begin{aligned} \left( \mathrm{op}_{\varepsilon _\ell }(a) \psi ^{\varepsilon _\ell }_n(0), \psi ^{\varepsilon _\ell }_n(0) \right) \mathop {\longrightarrow }\limits _{\varepsilon _\ell \rightarrow 0} \int _{T^*\Lambda _n}{{\,\mathrm{Tr}\,}}_{L^2(N_\xi \Lambda )}\left[ Q_{a}^{\Lambda _n}(\xi ,v)M^0_{n}(\xi ,v)\right] \nu ^0_{n}(d\xi ,dv). \end{aligned}$$

4.2.2 Conical Crossing Points

When H1, H2 and H3 for all $n\in {{\mathbb {N}}}^*$, the crossing sets $\Sigma _n$ are manifolds. Besides, because of the periodicity of the Bloch energies, $\Sigma _n$ thus is the union of connected, closed embedded submanifold of $({{\mathbb {R}}}^d)^*$ and we can focus on each of these connected components by considering the two-microlocal setting of Section 4.1 with $N=1$ and the family $(\psi ^\varepsilon _n)_{\varepsilon >0}$ for this submanifold.

Theorem 4.5

Assume H1, H2 and H3 holds for some $n\in {{\mathbb {N}}}^*$. Let $\Sigma$ be a connected component of $\Sigma _n$. Then any pair $(M^t_{n} d\nu _{n}^t, d\gamma _{n}^t)$ of two-microlocal semi-classical measures associated with the concentration of $(\psi ^\varepsilon _n(t))$ on $\Sigma$ satisfy $\nu ^t_{n}=0$ and $\gamma ^t_{n}=0$. Therefore $\mu ^t_{n,n} \mathbf{1} _\Sigma =0$.

The proof of this result is performed in Section 5.

4.2.3 Degenerate Crossing Points

We now suppose that n is fixed and we consider the concentration of $\psi ^\varepsilon _n(t)$ and $\psi ^\varepsilon _{n+1}(t)$ when the crossing set $\Sigma _n$ involving the two Bloch energies $\varrho _n$ and $\varrho _{n+1}$ satisfies H3’. We consider a connected component Y of $\Sigma _n$ which is assumed to be included into $\Lambda _n$ and $\Lambda _{n+1}$, the sets of critical points of $\varrho _n$ and $\varrho _{n+1}$ respectively. We consider the two-microlocal setting of Section 4.1 for $N=2$, the submanifold Y and the family

$$\begin{aligned} \Psi ^\varepsilon (t)=(\psi ^\varepsilon _n(t),\psi ^\varepsilon _{n+1}(t))\in {{\mathbb {C}}}^2. \end{aligned}$$

In view of Lemma B.1, the equation satisfied by $\Psi ^\varepsilon$ is

$$\begin{aligned} i\varepsilon ^2 \partial _t \Psi ^\varepsilon = \Theta (\varepsilon D) \Psi ^\varepsilon +\varepsilon ^2 V_{\mathrm{ext}} (t,x) \Psi ^\varepsilon +\varepsilon ^3 F^\varepsilon (t,x) \end{aligned}$$

(4.7)

with $(F^\varepsilon (t))$ uniformly bounded in $L^2({{\mathbb {R}}}^d)$ and

$$\begin{aligned} \Theta (\xi )&= \mathrm{Diag} (\varrho _n(\xi ), \varrho _{n+1}(\xi ))=\lambda _n(\xi ) \mathrm{Id} - g_n\left( \xi , \xi -\sigma _{Y}(\xi )\right) J,\;\;\;\; \nonumber \\ J&=\begin{pmatrix} 1 &{} 0\\ 0 &{} -1\end{pmatrix}, \end{aligned}$$

(4.8)

where $g_n\in {\mathcal {C}}^\infty \left( \sqcup _{\xi \in \Omega } \left( \{\xi \} \times N_{\sigma _{\Sigma _n}(\xi )}\Sigma _n\right) \right)$ (see also (1.11) where the restriction of $g_n$ to points of $\Sigma _n$ is introduced), and the function $\lambda _n$ is defined in (1.12). Note that by assumption H3’, there exists $c>0$ such that we have $g_n(\sigma ,\eta )\le c |\eta |^q$ for all $\sigma \in \Sigma _n$ and $\eta \in N_\sigma \Sigma _n$, and $g_n(\xi ,\eta )=|\eta |^2 \theta _n(\xi )$ by (2) of Lemma B.1.

Theorem 4.6

We suppose that H1’, H2’ and H3’ hold. Consider a connected component Y of $\Sigma _n$ that is included in $\Lambda _n\cap \Lambda _{n+1}$. Let $(M^td\nu ^t,d\gamma ^t)$ be a pair associated with the concentration of the family $(\Psi ^\varepsilon (t))_{\varepsilon >0}$ on Y. Then, $\gamma ^t=0$ and there exists $Md\nu ^0$ associated with the concentration at finite distance of $(\Psi ^\varepsilon (0))_{\varepsilon >0}$ on Y such that $\nu ^t=\nu ^0$ and the following holds:

• (1) If $q=2$, $M^t$ satisfies (1.20) with initial data M.

• (2) If $q>2$, $M^t$ satisfies (1.19) with initial data M.

Remark 4.7

Note that, even in Case (2), it can happen that the modes interact above the crossing, if they were doing so at time $t=0$. Corollary E.3 provides examples of such initial data.

The proof of Theorem  4.6 is the subject of Section 6.

5 Proof of Theorem 4.5

We prove Theorem 4.5 in two steps: first we focus on the part of the two-scale semi-classical measure that comes from infinity in Section 5.1, then we concentrate on the part at finite distance in Section 5.2. We use the characterization of Lemma B.1 and write

$$\begin{aligned} \varrho _n(\xi ) =\lambda _n(\xi )- g_n(\xi ,\xi -\sigma _{\Sigma }(\xi )),\;\; \varrho _{n+1}(\xi ) = \lambda _n(\xi )+ g_n(\xi ,\xi -\sigma _{\Sigma }(\xi ))\nonumber \\ \end{aligned}$$

(5.1)

with $\lambda _n$ smooth and $g_n\in {\mathcal {C}}^\infty \left( \sqcup _{\xi \in \Omega } \left( \{\xi \} \times N_{\sigma _{\Sigma _n}(\xi )}\Sigma _n\right) \right)$ and $\eta \mapsto g_n(\xi ,\eta )$ homogeneous of order 1 in $\eta$ (see (1) in Lemma B.1). Note that the function introduced in the introduction in (1.11) is the restriction of $g_n$ to $N\Sigma _n$ (and thus have been denoted similarly).

5.1 The Two-Scale Semiclassical Measures at Infinity

Let $a\in {{\mathcal {A}}}^{(2)}$ supported in ${{\mathbb {R}}}^d\times W\times {{\mathbb {R}}}^d$ where W is an open subset of ${{\mathbb {R}}}^d$ where we have tubular coordinates for $\Sigma$. Let $\chi \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^d)$ such that $\chi =1$ on B(0, 1) and $\chi =0$ on $B(0,2)^c$ with $0\le \chi \le 1$. We set, for $R,\delta >0$

$$\begin{aligned} a^{R,\delta }(x,\xi ,\eta )= a(x,\xi ,\eta ) ((1-\chi (\eta /R)) \chi ((\xi -\sigma _{\Sigma }(\xi ))/\delta ). \end{aligned}$$

Then, in view of equation (3.15),

$$\begin{aligned} i\varepsilon {d\over dt} (\mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) \psi ^\varepsilon _n(t),\psi ^\varepsilon _n(t))=\varepsilon ^{-1} \left( [\mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) ,\varrho _n(\varepsilon D)] \psi ^\varepsilon _n(t),\psi ^\varepsilon _n(t)\right) +O(\varepsilon ).\nonumber \\ \end{aligned}$$

(5.2)

Using (5.1), the homogeneity of $g_n$, and the notation introduced in (D.1), we write

$$\begin{aligned} \varrho _n(\varepsilon D)= \lambda _n(\varepsilon D)-\varepsilon g_n(\varepsilon D, D -\varepsilon ^{-1} \sigma _\Sigma (\varepsilon D)) = \lambda _n(\varepsilon D) -\varepsilon (g_n)_\varepsilon (\varepsilon D). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \varepsilon ^{-1} \left[ \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) ,\varrho _n(\varepsilon D)\right] = \mathrm{op}_\varepsilon (\nabla _x a^{R,\delta }_\varepsilon \cdot \nabla \lambda _n) - \left[ \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) , (g_n)_\varepsilon (\varepsilon D)\right] +O(\varepsilon ). \end{aligned}$$

We can now apply Lemma D.1 with $k=0$, and we obtain

$$\begin{aligned} \varepsilon ^{-1} \left[ \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) ,\varrho _n(\varepsilon D)\right] = \mathrm{op}_\varepsilon ( b_\varepsilon )+O(\varepsilon ) + O(R^{-1})+O(\delta ) \end{aligned}$$

with $b = \nabla _x a^{R,\delta }\cdot \nabla \lambda _n -\nabla _x a^{R,\delta }\cdot \nabla _\eta g_n$. Passing to the limits $\varepsilon \rightarrow 0$, then $R\rightarrow +\infty$ after time integration against $\theta \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}})$, we obtain by (5.2),

$$\begin{aligned} \int _{{{\mathbb {R}}}}\theta (t) \left( \mathrm{op}_\varepsilon (b_\varepsilon ) \psi ^\varepsilon _n(t),\psi ^\varepsilon _n(t) \right) \mathrm{d}t= O(\varepsilon ) + O(R^{-1})+O(\delta ). \end{aligned}$$

We deduce that

$$\begin{aligned} \int _{{{\mathbb {R}}}\times {{\mathbb {R}}}^d\times \Sigma \times \mathbf{S}^{d-1}} \theta (t) (\nabla \lambda _n(\sigma ) -\nabla _\eta g_n(\sigma ,\omega ) )\cdot \nabla _xa_\infty (x,\sigma ,\omega ) d\gamma ^t_{n}(x,\sigma ,\omega )=0. \end{aligned}$$

This implies that the measure $\gamma ^t _{n}(x,\sigma ,\omega )$ is invariant by the flow

$$\begin{aligned} (x,\sigma ,\omega )\mapsto (x+s(\nabla \lambda _n(\sigma )-\nabla _\eta g_n(\sigma ,\omega ) ,\sigma ,\omega ),\;\;s\in {{\mathbb {R}}}. \end{aligned}$$

As a consequence, $\gamma ^t_{n}$ is supported on $\{\nabla \lambda _n(\sigma )-\nabla _\eta g_n(\sigma ,\omega )=0\}$, and by H 3, $\gamma _{n}^t=0$.

5.2 The Two-Scaled Semiclassical Measures Coming from Finite Distance

In view of Remark 4.2 (3), we now choose $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{d}\times W\times {{\mathbb {R}}}^d)$ where W is as above. Let $\theta \in L^1({{\mathbb {R}}})$. Arguing as in (5.2), we observe

$$\begin{aligned} \int _{{\mathbb {R}}}\theta (t) \left( [\mathrm{op}_\varepsilon (a_\varepsilon ), \varepsilon ^{-1} \varrho _n(\varepsilon D_x) ] \psi ^\varepsilon _n(t) ,\psi ^\varepsilon _n(t)\right) =O(\varepsilon ). \end{aligned}$$

Using that a is compactly supported in the variable $\eta$ and the homogeneity of g, we obtain in ${\mathcal {L}}(L^2({{\mathbb {R}}}^d))$,

$$\begin{aligned} \frac{1}{\varepsilon }[\mathrm{op}_\varepsilon (a_\varepsilon ), \varrho _n(\varepsilon D_x) ] =i \mathrm{op}_\varepsilon (\nabla \lambda _n(\xi )\cdot \nabla _x a_\varepsilon ) -[\mathrm{op}_\varepsilon (a_\varepsilon ), (g_n)_\varepsilon (\varepsilon D) ] +O(\varepsilon ) . \end{aligned}$$

Passing to the limit $\varepsilon \rightarrow 0$ thanks to Lemma D.2, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}}\theta (t) \mathrm{Tr}_{L^2(N_\sigma \Sigma )} ( i Q^{\Sigma }_{\nabla \lambda _n \cdot \nabla _x a}-[Q_a^{\Sigma }(\sigma ,v), Q_g^{\Sigma }(\sigma )] M^t_n(\sigma ,v))\nu ^t_n(\sigma ,v) \mathrm{d}t=0.\nonumber \\ \end{aligned}$$

(5.3)

This relation has important consequences on the structure of $M^t_{n,n}$ and $\nu ^t_n$. For stating them, we write

$$\begin{aligned} \nabla \lambda _n(\sigma )=\nabla ^\perp \lambda _n(\sigma )+\nabla ^\sharp \lambda _n(\sigma ),\;\;\nabla ^\perp \lambda _n(\sigma )\in N_\sigma \Sigma \;\; \text{ and }\;\; \nabla ^\sharp \lambda _n(\sigma )\in T_\sigma \Sigma . \end{aligned}$$

Lemma 5.1

Equation (5.3) implies

$$\begin{aligned} \mathrm{supp} (\nu ^t_{n}) \subset \{ (\sigma ,v)\in T{\Sigma },\;\; \nabla ^\sharp \lambda _n(\sigma )=0\} \end{aligned}$$

and

$$\begin{aligned}{}[Q_F^\Sigma ( \sigma ), M^t_{n}(\sigma ,v) ]=0\;\;d\nu ^t_{n} \, a.e. \, (\sigma ,v)\in N\Sigma , \end{aligned}$$

where $F(\sigma ,\eta )=\nabla ^\perp \lambda _n(\sigma ) \cdot \, \eta + g_n(\sigma , \eta ).$

Proof of Lemma 5.1

We use a system of equations $\varphi (\xi )=0$ of $\Sigma$ and the matrix B defined in (4.4). For $\sigma \in X$ and $\zeta \in T_\sigma {{\mathbb {R}}}^d$, we have

$$\begin{aligned} (\mathrm{Id} -d\sigma _{\Sigma }(\sigma ))\zeta = B(\sigma ) d\varphi (\sigma )\zeta , \end{aligned}$$

which allows to decompose $\zeta$ as

$$\begin{aligned} \zeta = d\sigma _{\Sigma }(\sigma )\zeta + B(\sigma ) d\varphi (\sigma )\zeta ,\;\; d\sigma _{\Sigma }(\sigma )\zeta \in T_\sigma {\Sigma }\;\;\text{ and } \;\;B(\sigma ) d\varphi (\sigma )\zeta \in N_\sigma {\Sigma }. \end{aligned}$$

In particular, $B(\sigma ) d\varphi (\sigma )=\mathrm{Id}$ on $N_\sigma {\Sigma }$. In view of this observation, we write for $(\sigma ,v)\in T\Sigma$, and $(z,\zeta ) \in (N_\sigma {\Sigma })^*$

$$\begin{aligned}&\nabla \lambda _n(\sigma )\cdot \nabla _x a(v\!+ \! ^t d\varphi (\sigma ) z,\sigma , B(\sigma ) \zeta )\\&=\nabla ^\sharp \lambda _n(\sigma ) \cdot \nabla _v a(v\!+ \! ^t d\varphi (\sigma ) z,\sigma , B(\sigma ) \zeta )+\nabla ^\perp \lambda _n(\sigma ) \cdot (B( \sigma )\nabla _z)\\&\quad \cdot \left( a(v+ \, ^t d\varphi (\sigma ) z,\sigma , B(\sigma ) \zeta ) \right) \end{aligned}$$

We obtain

$$\begin{aligned} Q^\varphi _{\nabla \lambda _n(\xi )\cdot \nabla _x a}(\sigma ,v) = \nabla ^\sharp \lambda _n(\sigma ) \cdot \nabla _v Q_a^\varphi (\sigma , v ) -i [ Q_F^\varphi (\sigma ), Q_a ^\varphi (\sigma ,v)], \end{aligned}$$

with $Q_F^\varphi (\sigma )= F(\sigma , B(\sigma )D_z)$. Therefore, equation (5.3) can be written as

$$\begin{aligned}&\int _{{\mathbb {R}}}\theta (t) \mathrm{Tr}_{L^2(N_\sigma \Sigma )} ( i \nabla ^\sharp \lambda _n(\sigma ) \cdot \nabla _v Q^\Sigma _a (\sigma , v )\\&\quad +[Q_a^{\Sigma }(\sigma ,v), Q_F^\Sigma (\sigma )] M^t_n(\sigma ,v))\nu ^t_n(\sigma ,v) \mathrm{d}t=0, \end{aligned}$$

We deduce that

$$\begin{aligned} i \nabla ^\sharp \lambda _n (\sigma ) \cdot \nabla _v (M^t_{n} d\nu _{n}^t) + [Q^\Sigma _F(\sigma ) , M^t_{n} d\nu ^t_{n}] =0 . \end{aligned}$$

Taking the trace gives

$$\begin{aligned} \nabla ^\sharp \lambda _n(\sigma ) \cdot \nabla _v \nu ^t_{n}=0, \end{aligned}$$

whence the invariance of $\nu ^t_n$ by the flow defined on $T\Sigma$ by

$$\begin{aligned} (\sigma ,v)\mapsto (\sigma , v+s\nabla ^\sharp \lambda _n(\sigma )),\;\;s\in {{\mathbb {R}}}, \end{aligned}$$

which implies the results. $\quad \square$

We conclude the analysis of the two-scaled Wigner measures at finite distance $M^t$ by using Lemma 5.2 below. For this, we need to check that its assumptions are satisfied. Hypothesis H 3 implies that if $\nabla ^\sharp \lambda _n(\sigma )=0$, then for all $\eta \in N_\sigma {\Sigma }\setminus \{0\}$,

$$\begin{aligned} \nabla ^\perp \lambda _n(\sigma )-\nabla _\eta g_n(\sigma ,\eta ) \not =0. \end{aligned}$$

Considering $\nabla _\zeta (F(\sigma ,B(\sigma )\zeta ))$, we have

$$\begin{aligned} \nabla _\zeta (F(\sigma ,B(\sigma )\zeta )) =\, ^t B(\sigma )\left( \nabla ^\perp \lambda _n(\sigma )-\partial _\eta g_n(\sigma ,\eta ) \right) \not =0, \end{aligned}$$

because $B(\sigma )$ is invertible on $N_\sigma \Sigma$, and the assumptions of the next lemma are satisfied.

Lemma 5.2

Let $p\in {{\mathbb {N}}}$ and M be a non negative trace-class operator on $L^2({{\mathbb {R}}}^p)$, and $F\in {\mathcal {C}}^\infty ({{\mathbb {R}}}^p\setminus \{0\})$ such that $\nabla _\zeta F(\zeta )\not =0$ for all $\zeta \in {{\mathbb {R}}}^p\setminus \{0\}$. Assume $\left[ F(D_z) , {M} \, \right] =0.$ Then ${ M}=0$.

Proof

Let $\phi \in L^2({{\mathbb {R}}}^p)$ be an eigenvector of M for an eigenvalue $\ell \not =0$. Then, for all $j\in {{\mathbb {N}}}$,

$$\begin{aligned} \phi _j:=(F( D_z))^j \phi \end{aligned}$$

is also an eigenvector for $\ell$. Since $\ell$ is of finite multiplicity because M is trace-class, we deduce that the set $\{ \phi _j, j\in {{\mathbb {N}}}\}$ is of finite dimension. Let $k\in {{\mathbb {N}}}^*$ the first index such that the family $(\phi _j)_{0\le j\le k}$ is not a family of independent vectors. Then, there exist $\alpha _0,\cdots \alpha _k\in {{\mathbb {R}}}$ non all equal to 0, and such that $\displaystyle {\sum _{j=0}^k \alpha _j \phi _j=0.}$ In Fourier variables, we obtain $\displaystyle {\left( \sum _{j=0}^k \alpha _j F( \zeta )^j\right) {{\widehat{\phi }}}(\zeta )=0.}$ The set $\displaystyle {{\mathcal {C}}} =\left\{ \zeta \in {{\mathbb {R}}}^p,\;\; \sum _{j=0}^k \alpha _j F(\zeta )^j =0\right\}$ is the union of a finite number of sets ${{\mathcal {C}}}_\beta$,

$$\begin{aligned} {\mathcal {C}}_\beta = \{ F(\zeta )=\beta \} \end{aligned}$$

for $\beta$ a real-valued root of the polynomial $\sum _{0\le j\le k} \alpha _j X^j$. Since $\nabla _\zeta F(\zeta )\not =0$ for all $\zeta \not =0$, these sets ${\mathcal {C}}_\beta$ are hypersurfaces of ${{\mathbb {R}}}^p$ and thus of Lebesgues measure 0. So it is for ${{\mathcal {C}}}$ and we deduce that $\phi =0$. $\quad \square$

6 Proof of Theorem 4.6

Theorem 4.6 contains two statements. First, it states that the two-scale semi-classical measures at infinity is 0, what we prove in Section 6.1 below, by showing invariance properties of its diagonal elements. Secondly, it gives transport equations that allow to compute the two-scale semi-classical measures coming from finite distance from the knowledge of the initial data. We focus on this latter point in Section 6.2.

6.1 Analysis at Infinity

We perform the proof for $q=2$, the proof for $q>2$ is similar. Let $\Psi ^\varepsilon$ be a family of solutions to equation (4.7). Let $a\in {{\mathcal {A}}}^{(2)}$ supported in ${{\mathbb {R}}}^d\times W\times {{\mathbb {R}}}^d$ where W is an open subset of ${{\mathbb {R}}}^d$ where we have tubular coordinates for the manifold Y. Let $\chi \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^d)$ such that $\chi =1$ on B(0, 1) and $\chi =0$ on $B(0,2)^c$ with $0\le \chi \le 1$. We set for $R,\delta >0$

$$\begin{aligned} a^{R,\delta }(x,\xi ,\eta )= a(x,\xi ,\eta ) ((1-\chi (\eta /R)) \chi ((\xi -\sigma _{Y}(\xi ))/\delta ) \end{aligned}$$

and we consider the symbol

$$\begin{aligned} {{\tilde{a}}}^{R,\delta }(x,\xi ,\eta )=|\xi -\sigma _Y(\xi )|^{-1} a^{R,\delta }(x,\xi ,\eta ). \end{aligned}$$

By Lemma C.4 (1) (see also “Appendix D”), there exists a constant $C>0$ such that

$$\begin{aligned} \Vert \mathrm{op} _\varepsilon ({{\tilde{a}}}^{R,\delta }_\varepsilon )\Vert _{{\mathcal {L}}(L^2({{\mathbb {R}}}^d))}\le C (\varepsilon R)^{-1}. \end{aligned}$$

In view of (4.8) and of (2) in Lemma B.1, we write with the notations introduced in (D.1)

$$\begin{aligned} \Theta (\varepsilon D)= \lambda _n(\varepsilon D)\mathrm{Id} - \varepsilon ^2 (g_n)_\varepsilon (\varepsilon D). \end{aligned}$$

Therefore, if E is a constant diagonal matrix of ${{\mathbb {C}}}^{2\times 2}$, we obtain

$$\begin{aligned} \left[ \mathrm{op}_\varepsilon ({{\tilde{a}}}^{R,\delta }_{\varepsilon } E ) \;,\; \Theta (\varepsilon D) \right]= & {} \left[ \mathrm{op}_\varepsilon ({{\tilde{a}}}^{R,\delta }_{\varepsilon }) \;,\; \lambda _n(\varepsilon D)\right] E\\&- \varepsilon ^2 \left[ \mathrm{op}_\varepsilon ({{\tilde{a}}}^{R,\delta }_{\varepsilon }) \;,\; (g_n)_\varepsilon (\varepsilon D)\right] EJ, \end{aligned}$$

where we have used that $EJ=JE$. We observe that setting

$$\begin{aligned} b(x,\xi ,\eta )= |\eta |^{-1} a^{R,\delta } (x,\xi ,\eta ), \end{aligned}$$

we have

$$\begin{aligned} \varepsilon \, \mathrm{op}_\varepsilon ({{\tilde{a}}}^{R,\delta }_{\varepsilon }) =\mathrm{op}_\varepsilon (b_{\varepsilon }) \end{aligned}$$

and we can apply Lemma D.1 because $b\in {\mathcal {A}}^{(2)}_{-1}$ and $g_n\in {\mathcal {H}}_2$. We deduce that

$$\begin{aligned} {\varepsilon } \left[ \mathrm{op}_\varepsilon ({{\tilde{a}}}^{R,\delta }_{\varepsilon }) \;,\; (g_n)_\varepsilon (\varepsilon D)\right] = \left[ \mathrm{op}_\varepsilon (b_{\varepsilon }) \;,\; (g_n)_\varepsilon (\varepsilon D)\right] = \mathrm{op}_\varepsilon ((\nabla _x b \cdot \nabla _\eta g)_\varepsilon ) \end{aligned}$$

with $\nabla _x b(x,\xi ,\eta )= |\eta |^{-1} \nabla _x a^{R,\delta } (x,\xi ,\eta ).$ Therefore, we are left with

$$\begin{aligned} \frac{1}{\varepsilon } \left[ \mathrm{op}_\varepsilon ({{\tilde{a}}}^{R,\delta }_{\varepsilon }E) \;,\; \Theta (\varepsilon D) \right]&= \mathrm{op}_\varepsilon (\nabla _x {{\tilde{a}}}^{R,\delta }_{\varepsilon }\cdot \nabla _\xi \lambda _n(\xi ) ) E - \mathrm{op}_\varepsilon ((\nabla _x b \cdot \nabla _\eta g_n)_\varepsilon ) EJ \nonumber \\&\quad + O(\varepsilon )+O(R^{-1})+O(\delta ). \end{aligned}$$

We use $\nabla \lambda _n(\xi )=\mathrm{Hess}\, \lambda _n(\sigma _Y(\xi ) )(\xi -\sigma _Y(\xi )) +O((\xi -\sigma _Y(\xi ))^2)$ and we set

$$\begin{aligned} c(x,\xi ,\eta ):= & {} \nabla _x a^{R,\delta }(x,\xi ,\eta )\cdot \mathrm{Hess} \, \lambda _n(\sigma _Y(\xi )) \frac{\eta }{|\eta |} E\\&- \nabla _x a^{R,\delta }(x,\xi ,\eta ) \cdot \frac{1}{|\eta |} \nabla _\eta g_n\left( \xi ,\eta \right) EJ. \end{aligned}$$

Note that $c\in {{\mathcal {A}}}^{(2)}$ and

$$\begin{aligned} \frac{1}{\varepsilon }\left[ \mathrm{op}_\varepsilon ({{\tilde{a}}}^{R,\delta }_{\varepsilon } E) \;,\; \Theta (\varepsilon D) \right] = \mathrm{op}_\varepsilon (c_\varepsilon ) + O(\varepsilon )+O(R^{-1})+O(\delta ). \end{aligned}$$

Therefore, passing to the limit $\varepsilon$ to 0, then R to $+\infty$ and finally $\delta$ to 0, we obtain for all $\theta \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}})$,

$$\begin{aligned}&\int _{{{\mathbb {R}}}} \theta (t)\int _{{{\mathbb {R}}}^d\times Y\times \mathbf{S}^{d-1}} \mathrm{Tr}_{{{\mathbb {C}}}^{2\times 2}}\left( \nabla _xa_\infty (x, \sigma ,\omega )\cdot (\mathrm{Hess}\, \lambda _n(\sigma ) \omega \, E - \nabla _\eta g_n(\sigma , \omega ) EJ)\right) \\&\quad \gamma ^t (dx,d\sigma , d\omega )\mathrm{d}t =0. \end{aligned}$$

Let us denote by $\gamma _{n,n}^t$ and $\gamma _{n+1,n+1}^t$ the diagonal coefficients of the matrix-valued measure $\gamma ^t$. Choosing the $2\times 2$ diagonal matrix E such that $EJ=E$, we deduce that the measure $\gamma _{n,n}^t$ is invariant by the flow

$$\begin{aligned} (x,\sigma ,\omega ) \mapsto (x+s(\mathrm{Hess}\,\lambda _n(\sigma ) \omega -\nabla _\eta g_n(\sigma ,\omega ) ) ,\sigma ,\omega ),\;\; s\in {{\mathbb {R}}}. \end{aligned}$$

Then, choosing E such that $EJ=-E$, we obtain that the measure $\gamma _{n+1,n+1}^t$ is invariant by the flow

$$\begin{aligned} (x,\sigma ,\omega ) \mapsto \left( x+s(\mathrm{Hess}\,\lambda _n(\sigma ) \omega +\nabla _\eta g_n(\sigma , \omega )),\sigma , \omega \right) ,\;\; s\in {{\mathbb {R}}}. \end{aligned}$$

From assumption H3’, we deduce $\gamma ^t_{n,n}=0$ and $\gamma ^t_{n+1,n+1}=0$, and the positivity of $\gamma ^t$ implies that $\gamma ^t=0$. One argues similarly when $q>2$, and proves that the term in $g_n$ does not contribute to the limit.

6.2 The Two-Scale Semiclassical Measures Coming from Finite Distance

Here again, we write the proof for $q=2$. We choose $\theta \in L^1({{\mathbb {R}}}^d)$, $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{d}\times W\times {{\mathbb {R}}}^d,{{\mathbb {C}}}^{2\times 2} )$ where W is a tubular neighbrohood of Y where the function $\sigma _Y$ is defined. Using the homogeneity of the function $g(\xi ,\eta )$, we have

$$\begin{aligned} i\frac{d}{dt} \left( \mathrm{op}_\varepsilon (a_\varepsilon ) \Psi ^\varepsilon (t),\Psi ^\varepsilon (t)\right) = I^\varepsilon _1(t)+I^\varepsilon _2(t) \end{aligned}$$

(6.1)

with

$$\begin{aligned} I^\varepsilon _1(t)&= \left( [ \mathrm{op}_\varepsilon (a_\varepsilon ),\varepsilon ^{-2} \lambda _n(\varepsilon D_x) + V_{\mathrm{ext}}(t,x) ] \Psi ^\varepsilon (t) ,\Psi ^\varepsilon (t)\right) \\ I^\varepsilon _2 (t)&=- \left( \left( \mathrm{op}_\varepsilon (a_\varepsilon ) J (g_n)_\varepsilon (\varepsilon D)- (g_n)_\varepsilon (\varepsilon D)J \mathrm{op}_\varepsilon (a_\varepsilon ) \right) \Psi ^\varepsilon (t) ,\Psi ^\varepsilon (t)\right) . \end{aligned}$$

Note that if $q>2$, the homogeneity implies $I^\varepsilon _2 (t)= O(\varepsilon ^{q-2})$. Section 5.1 in [15] gives the uniform boundedness of the family of time dependent functions $t\mapsto I^\varepsilon _1(t)$ and Lemma D.2 yields the uniform boundedness of the family of time dependent functions $t\mapsto I^\varepsilon _2(t)$. Therefore, the left-hand side of (6.1) is uniformly bounded with respect to $\varepsilon$. Therefore, the maps $t\mapsto M^t(\sigma , v) d\nu ^t(\sigma ,v)$ defined on TY will be continuous in time.

Remark 6.1

At that level of the proof, one sees that by Ascoli theorem, one can find for each $T>0$ a sequence $\varepsilon _\ell$ for which the limit of $\left( \mathrm{op}_{\varepsilon _\ell }(a_{\varepsilon _\ell }) \Psi ^{\varepsilon _\ell }(t),\Psi ^{\varepsilon _\ell }(t)\right)$ exists for all $t\in {{\mathbb {R}}}$. One then deduces the convergence t by t of these quantities.

We now integrate equation (6.1) against a function $\theta$ and pass to the limit $\varepsilon \rightarrow 0$. By Section 5.1 in [15], we have for $\theta \in {\mathcal {C}}_0^\infty ({{\mathbb {R}}})$, up to the subsequence defining $M^td\nu ^t$

$$\begin{aligned}&\int \theta (t) I^\varepsilon _1(t) \mathrm{d}t \mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0} \\&\;\int _{{\mathbb {R}}}\theta (t) \int _{TY} \mathrm{Tr}_{L^2(N_\sigma Y,{{\mathbb {C}}}^2)} \Biggl ( \left[ Q^Y_a(\sigma , v) , \frac{1}{2} \mathrm{Hess} \lambda _n (\sigma )\,D_z\cdot D_z\right. \left. +m_{V_{\mathrm{ext}} (t,\cdot )}(x,v)\right] M^t(\sigma ,v) \Biggr )\\&\quad \nu ^t(\sigma ,v)\mathrm{d}t. \end{aligned}$$

By Lemma D.2 for studying the term $I^\varepsilon _2$,

$$\begin{aligned}&\int \theta (t) I^\varepsilon _2(t) dt \mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0}\\&\; \int _{{\mathbb {R}}}\theta (t) \int _{TY} \mathrm{Tr}_{L^2(N_\sigma Y,{{\mathbb {C}}}^2)} \left( Q^Y_a(\sigma ,v) J Q^Y_{g_n}(\sigma )-Q^Y_{g_n}(\sigma )JQ_a^Y(\sigma , v)\right) M^t(\sigma ,v) \Biggr )\nu ^t(\sigma ,v)\mathrm{d}t.\\&\qquad = \int _{{\mathbb {R}}}\theta (t) \int _{TY} \mathrm{Tr}_{L^2(N_\sigma Y,{{\mathbb {C}}}^2)} \left( \left[ Q^Y_a(\sigma ,v) \;,\; J Q^Y_{g_n}(\sigma )\right] M^t(\sigma ,v) \right) \nu ^t(\sigma ,v)\mathrm{d}t \end{aligned}$$

Reporting the result in (6.1), we obtain

$$\begin{aligned}&-i\int _{{\mathbb {R}}}\theta '(t) \int _{TY} \mathrm{Tr}_{L^2(N_\sigma Y,{{\mathbb {C}}}^2)} (Q^Y_a(\sigma ,v)M^t(\sigma ,v)) \mathrm{d}\nu ^t(\sigma ,v) \mathrm{d}t \\&\quad = \int _{{\mathbb {R}}}\theta (t) \int _{TY} \mathrm{Tr}_{L^2(N_\sigma Y,{{\mathbb {C}}}^2)} \Biggl (\Bigg [ Q^Y_a(\sigma , v) ,\Bigg ( \frac{1}{2} \mathrm{Hess} \lambda _n (\sigma )\,D_z\cdot D_z+m_{V_{\mathrm{ext}} (t,\cdot )}(x,v)\biggr )\mathrm{Id} \\&\qquad + J Q^Y_{g_n}(\sigma ) \Bigg ] M^t(\sigma ,v) \Biggr )\nu ^t(\sigma ,v)\mathrm{d}t. \end{aligned}$$

We deduce that

$$\begin{aligned} \partial _t (M^t d\nu ^t)= \left[ \left( \frac{1}{2} \mathrm{Hess} \lambda _n (\sigma )\,D_z\cdot D_z+m_{V_{\mathrm{ext}} (t,\cdot )}(x,v)\right) \,\mathrm{Id} + Q^Y_{g_n}(\sigma ) J \;,\; M^t d\nu ^t\right] d\nu ^t. \end{aligned}$$

Taking the trace of this expression gives $\partial _t\nu ^t=0$, whence $\nu ^t=\nu ^0$ (because of the continuity of $t\mapsto \nu ^t$), and the equation satisfied by $M^t$.

7 Proof of the Main Theorems

7.1 Proofs of Theorem 1.5 and Proposition 1.6

We prove here the results obtained under H1, H2 and H3, which corresponds to a general setting without too much assumptions on the initial data and with generic hypothesis on the Bloch energies.

Proof of Theorem 1.5

Let $(\varepsilon _\ell )$ be a sequence given by Proposition 3.4 and $\varsigma ^t$ and $\mu ^t_{n,n'}$ the corresponding semi-classical measures along that sequence. Because of the assumption H1 and Part (2) of Proposition 3.5, $\mu ^t_{n,n'}=0$ for a.e. $t\in {{\mathbb {R}}}$ as soon as $|n-n'|>1$. Besides, by H2, we can use Theorem 4.3 to determine $\mu ^t_{n,n}\mathbf{1}_{\Lambda _n}$. Finally, by H3 and Theorem 4.5, $\mu _{n,n}^t= \mu ^t_{n,n}{} \mathbf{1}_{\Lambda _n}$ and the result follows. We obtain that for a subsequence $\varepsilon _\ell$ for $a,b\in {{\mathbb {R}}}$ and $\varphi \in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{d})$,

$$\begin{aligned}&\int _a^b\int _{{{\mathbb {R}}}^d} \phi (x) | \psi ^{\varepsilon _\ell }(t,x)|^2dx \mathrm{d}t\mathop {\longrightarrow }\limits _{\varepsilon _\ell \rightarrow 0}\\&\qquad \sum _{n\in {{\mathbb {N}}}_0} \int _a^b\int _{T^*X}{{\,\mathrm{Tr}\,}}_{L^2(N_\sigma \Lambda _n}(m_\phi ^{\Lambda _n}(\sigma )M_n^t(\sigma ,v))\nu _n(d\sigma ,dv)\mathrm{d}t, \end{aligned}$$

once if; observed that, for $a(x,\xi ):= \phi (x)$, the operator $Q_a(\sigma ,v)$ coincides with $m_\phi ^{\Lambda _n}(\sigma )$. $\quad \square$

Proof of Proposition 1.6

For the data considered in that statement, one has $U^\varepsilon _0= \varphi _{n_0}(y,\varepsilon D_x) u^\varepsilon _{n_0}$. Therefore, $M_n=0$ for $n\not =n_0$ and $M_n^t$ too. We then focus on calculating $M_{n_0}$ above any $\xi \in \Lambda _{n_0}$. By Corollary E.3 (1), since $\xi$ is an isolated point of $\Lambda _{n_0}$, the measure $\nu _{n_0}$ is given by

$$\begin{aligned} \nu _{n_0}(d\xi )= \Vert v_{n_0}\Vert _{L^2}^2 \sum _{j\in 2\pi {{\mathbb {Z}}}^d} |c_{n_0} (\xi _{n_0}+ j)|^2 \delta (\xi - \xi _{n_0}-j), \end{aligned}$$

and, for $j\in 2\pi {{\mathbb {Z}}}^d$, the operator $M_{n_0}(\xi _{n_0} +j)$ is the projector on ${{\mathbb {C}}}v_{n_0}$. As a consequence, the solution $M^t(\xi _{n_0} +j)$ of (1.15) is the orthogonal projection on ${{\mathbb {C}}}\psi ^{\xi _{n_0}}(t)$ where $\psi ^{\xi _{n_0}}$ satisfies (1.16). We obtain that for any $\phi \in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^d)$,

$$\begin{aligned} \int _{\Lambda _{n_0}} \mathrm{Tr}_{L^2({{\mathbb {R}}}^d)}m_\phi ^{\Lambda _{n_0}} (M^t_{n_0}(\xi ))\nu _{n_0} (d\xi )&= \left( \int _{{{\mathbb {R}}}^d} \phi (x) | \psi ^{\xi _{n_0}}(t,x)|^2 dx \right) \sum _{j\in {{\mathbb {Z}}}^d} |c_{n_0} (\xi _{n_0}+ j) |^2\\&=\int _{{{\mathbb {R}}}^d} \phi (x)| \psi ^{\xi _{n_0}}(t,x)|^2dx, \end{aligned}$$

where we have used that $N\Lambda _{n_0}={{\mathbb {R}}}^d$, $m^{\Lambda _{n_0}}_\phi$ is the operator of multiplication by $\phi$, and the conservation of $L^2$ norms for the equation (1.16) ($\Vert v_{n_0}\Vert _{L^2}=\Vert \psi ^{\xi _{n_0}}(t)\Vert _{L^2}$ for all $t\in {{\mathbb {R}}}$). $\quad \square$

7.2 Proof of Theorem 1.8 and Proposition 1.9

We focus here on degenerate crossings involving two energies isolated from the remainder of the spectrum and well-prepared data that concentrate on these modes. We will indeed prove a more general result than Theorem 1.8, assuming that $\Sigma _n$ is included in $\Lambda _n\cup \Lambda _{n+1}$ but not necessarily equal to $\Lambda _n$ and $\Lambda _{n+1}$ (the latter being non necessarily equal). Thus, one has to take into account the additional contributions to the energy densities generated by the points of $(\Lambda _n\cup \Lambda _{n+1})\setminus \Sigma _n$ and Theorem 1.8 is a straightforward corollary of the next result.

Theorem 7.1

Then, there exists a subsequence $\varepsilon _\ell \mathop {\longrightarrow }\limits _{\ell \rightarrow +\infty } 0$, three non negative measures $\nu _n\in {\mathcal {M}}^+(T^*\Lambda _n)$, $\nu _{n+1}\in {\mathcal {M}}^+(T^*\Lambda _{n+1})$ and $\nu ^0\in {\mathcal {M}}^+(\Sigma _n)$ depending on $(\psi ^{\varepsilon _\ell }_0)$, three measurable trace-class operators $M_n$, $M_{n+1}$ and M

$$\begin{aligned}&M_n:\,T^*_\xi \Lambda _n\ni (\xi ,v) \mapsto M_n (\xi ,v) \in {\mathcal {L}}^1_+(L^2(N_\xi \Lambda _n)),\\&\qquad \mathrm{Tr} _{L^2(N_\xi \Lambda _n)}M_n (\xi ,v) =1\; d\nu _n \; a.e.\\&M_{n+1}:\,T^*_\xi \Lambda _{n+1}\ni (\xi ,v) \mapsto M_{n+1}(\xi ,v)\in {\mathcal {L}}^1_+(L^2(N_\xi \Lambda _{n+1})),\\&\qquad \mathrm{Tr} _{L^2(N_\xi \Lambda _{n+1})}M_{n+1} (\xi ,v) =1\; d\nu _{n+1} \; a.e. \\&M:\,T^*_\xi \Sigma _n\ni (\xi ,v) \mapsto M(\xi ,v) \in {\mathcal {L}}^1_+(L^2(N_\xi \Sigma _n,{{\mathbb {C}}}^2)),\\&\qquad \mathrm{Tr}_{L^2(N_\xi \Sigma _n,{{\mathbb {C}}}^2)}M (\xi ,v) =1\; d\nu ^0\; a.e. \end{aligned}$$

such that for every $a<b$ and every $\phi \in {{\mathcal {C}}}_0({{\mathbb {R}}}^d)$ one has

$$\begin{aligned} \lim _{\ell \rightarrow +\infty }&\int _a^b\int _{{{\mathbb {R}}}^d} \phi (x) |\psi ^{\varepsilon _\ell } (t,x)|^2 dx \mathrm{d}t \\&\quad =\sum _{j=n,n+1} \int _a^b \int _{T^*(\Lambda _j\setminus \Sigma _n)}\mathrm{Tr} _{L^2(N_\xi \Lambda _j)}[m^{\Sigma _n}_\phi (\xi ,v)M^t_j (\xi ,v) ] \nu _j(d\xi ,dv)\mathrm{d}t\\&\qquad + \int _a^b \int _{T^*\Sigma _n}\mathrm{Tr} _{L^2(N_\xi \Sigma _n, {{\mathbb {C}}})} [m^{\Sigma _n}_\phi (\xi ,v) (m_n^t + m_{n+1}^t \\&\qquad +2 \mathrm{Re} (m_{n,n+1}^t ))(\xi ,v) ] \nu ^0(d\xi , dv)\mathrm{d}t , \end{aligned}$$

where

$$\begin{aligned} M^t(\xi ,v) = \begin{pmatrix} m_n^t(\xi ,v) &{} m_{n,n+1}^t(\xi ,v) \\ m_{n,n+1}^t(\xi ,v)^* &{} m_{n+1}^t(\xi ,v)\end{pmatrix} \end{aligned}$$

is a non negative trace class operators on $L^2( N_\xi \Sigma _n,{{\mathbb {C}}}^2)$.

Besides, the map $t\mapsto M^t_n(x,\xi )\in {\mathcal {C}}({{\mathbb {R}}}, {\mathcal {L}}^1_+(L^2(N_\xi \Lambda _n))$ solves the von Neumann equation (1.15) and similarly for $M^t_{n+1}$ and $\varrho _{n+1}$, and the map $t\mapsto M^t (\xi ,v)\in {\mathcal {C}}({{\mathbb {R}}}, {\mathcal {L}}^1_+(L^2(N_\xi \Sigma _n,{{\mathbb {C}}}^2))$ solves (1.19) if $q>2$ and (1.20) if $q=2$. All the initial data depend on $(\psi ^{\varepsilon _\ell }_0)$ as in Remark 4.2.

Proof

We have

$$\begin{aligned} U^\varepsilon _0(x,y)=\varphi _n(y,\varepsilon D_x) u^\varepsilon _{n}(x)+ \varphi _{n+1}(y,\varepsilon D_x) u^\varepsilon _{n+1}(x) \end{aligned}$$

and we are going to take advantage of the fact that $U^\varepsilon _0\in \mathrm{Ran } \,\Pi (\xi )$, the spectral projector on

$$\begin{aligned} \mathrm{Ker}(P(\xi )-\varrho _n(\xi )) \oplus \mathrm{Ker}(P(\xi )-\varrho _{n+1}(\xi )). \end{aligned}$$

By assumption H1’, the band of the spectrum of $P(\xi )$ consisting of the pair $\{ \varrho _n(\xi ), \varrho _{n+1}(\xi )\}$ is separated from the remainder of the spectrum by a gap, which implies that $\xi \mapsto \Pi (\xi )$ is analytic. We claim that a consequence of this is that if $(\varepsilon _\ell )$ is a sequence given by Proposition 3.4, then

$$\begin{aligned} \varsigma ^t = \mu ^t_{n,n}+\mu ^t_{n+1,n+1} + \mu ^t_{n,n+1} +\mu ^t_{n+1,n}. \end{aligned}$$

(7.1)

By Proposition 3.5, $\varsigma ^t$ has only support above $\Lambda _n$ (because of $\mu ^t_{n,n}$), $\Lambda _{n+1}$ (because of $\mu ^t_{n,n+1}$) and $\Sigma _n$ (because of the crossed terms). Then, the result of Theorem 1.8 comes from two observations:

• (1) assumption H2’ allows to use Theorem 4.3 to determine $\mu ^t_{n,n}$ above $\Lambda _n\setminus \Sigma _n$ and $\mu ^t_{n+1,n+1}$ above $\Lambda _{n+1}\setminus \Sigma _{n+1}$,

• (2) assumption H3’ allows to use Theorem 4.6 to compute $\mu ^t_{n,n}$, $\mu ^t_{n+1,n+1}$, $\mu ^t_{n,n+1}$ and $\mu ^t_{n+1,n}$ above $\Sigma _n$ in terms of the coefficients of the matrix-valued measure $M^t d\nu ^0$. In view of (7.1), we obtain that for all $\phi \in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^d)$,

  $$\begin{aligned} \int _{{{\mathbb {R}}}^d\times \Sigma _n} \phi (x) \varsigma ^t(dx,d\xi )= & {} \int _{T^*\Sigma _n} \mathrm{Tr} \left[ m_\phi ^{\Sigma _n} (\xi ,v) E M^t(\xi , v)\right] \nu ^0 (d\xi ,dv),\;\; \nonumber \\ E= & {} \begin{pmatrix} 1 &{} 1 \\ 1 &{} 1 \end{pmatrix} \end{aligned}$$

  (7.2)

It remains to discuss Equation (7.1), which comes from (1.8) and the estimate

$$\begin{aligned} \left\| (1-\Pi (\varepsilon D_x)) U^\varepsilon (t)\right\| _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\le \varepsilon \, C_s (1+|t|) \end{aligned}$$

for $s>d/2$. We observe that the family

$$\begin{aligned} W^\varepsilon (t,x)= (1-\Pi )(\varepsilon D_x) U^\varepsilon (t,x). \end{aligned}$$

satisfies the system

$$\begin{aligned} i\varepsilon ^2\partial _t W^\varepsilon =P(\varepsilon D_x) W^\varepsilon +\varepsilon ^2 V_{\mathrm{ext}} W^\varepsilon +\varepsilon ^3 G^\varepsilon ,\;\; W^\varepsilon (0)=0 \end{aligned}$$

with $G^\varepsilon (t) =- \varepsilon ^{-1} \left[ \Pi (\varepsilon D_x), V_{\mathrm{ext}}(t)\right] U^\varepsilon (t)$ uniformly bounded in $L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$. Therefore, when $s=0$, the estimate comes from an energy argument. We then proceed as in Lemma 6.7 in [15] by induction in $s\in {{\mathbb {N}}}$ and interpolation between s and $s+1$, observing that, in view of Remark 2.2, it is enough to prove that $P(\varepsilon D_x)^{s/2}W^\varepsilon$ and $\langle \varepsilon D_x \rangle ^s W^\varepsilon$ go to 0 in $L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$. $\quad \square$

As a by-product of the proofs, we have the following remark:

Remark 7.2

In view of Remark 6.1, for all $T>0$, there exists a sequence $\varepsilon _\ell$ for which one has for any $\phi \in {\mathcal {C}}^\infty ({{\mathbb {R}}}^d)$,

$$\begin{aligned} \int _{{{\mathbb {R}}}^d}\phi (x) |\psi ^\varepsilon (t,x)|^2 dx\ge & {} \int _{T^*\Sigma _n} \mathrm{Tr}_{L^2(N_\xi \Sigma _n)} \left( m^{\Sigma _n}_\phi (\xi ,v) (m_n^t+m^t_{n+1} \right. \nonumber \\&\left. +2\mathrm{Re}\, m^t_{n,n+1} )(\sigma ,v)\right) \nu ^0(d\xi ,dv). \end{aligned}$$

(7.3)

Let us assume

$$\begin{aligned} \int _{T^*\Sigma _n} \mathrm{Tr}_{L^2(N_\xi \Sigma _n)} \left( m_n^0+m^0_{n+1} +2\mathrm{Re}\, m^0_{n,n+1} \right) d\nu ^0=\Vert \psi ^\varepsilon _0\Vert _{L^2}. \end{aligned}$$

Then $\displaystyle { \int _{T^*\Sigma _n} \mathrm{Tr}_{L^2(N_\xi \Sigma _n)} \left( m_n^t+m^t_{n+1} +2\mathrm{Re}\, m^t_{n,n+1} \right) d\nu = |\psi ^\varepsilon (t,x)|^2 dx}$, and the inequality (7.3) becomes an equality. One has obtained a t by t description of the limit of the energy density. The same observation holds in the frame of Theorem 4.3.

It remains to prove Proposition 1.9.

Proof of Proposition 1.9

According to the assumptions, we have $\Lambda _n=\Lambda _{n+1}=\Sigma _n$. Therefore, we only have to compute $M d\nu ^0$ and solve the von Neumann equations defining $M^t$ in both studied cases. For the data considered in that statement, one has $\psi ^\varepsilon _0=\psi ^\varepsilon _{0,n}+\psi ^\varepsilon _{0,n+1}$ (where the latter families are defined in (1.16)) with $\xi _n=\xi _{n+1}$. Therefore, by Corollary E.3, $\nu ^0$ is given by

$$\begin{aligned} \nu ^0(d\xi )= (\Vert v_{n}\Vert _{L^2}^2+ v_{n+1}\Vert _{L^2}^2) \sum _{j\in 2\pi {{\mathbb {Z}}}^d} |c_{n} (\xi _{n}+ j)|^2 \delta (\xi - \xi _{n}-j) \end{aligned}$$

and, for $j\in 2\pi {{\mathbb {Z}}}^d$, the operator $M^0(\xi _{n} +j)$ is the projector on ${{\mathbb {C}}}\, ^t(v_{n},v_{n+1})$. The solution of the Heisenberg equations (1.19) and (1.20) then are orthogonal projectors on ${{\mathbb {C}}}\, ^t(\psi ^{\xi _n}_{n},\psi ^{\xi _n}_{n+1})$ as defined in the statement of Proposition 1.9. In view of (7.2) and of $m_\phi ^{\Sigma _n} = \phi (x)$, we conclude for $\phi \in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^d)$,

$$\begin{aligned}&\int _{\Sigma _{n}} \mathrm{Tr}_{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^2)} (m_\phi ^{\Sigma _{n}} E M^t(\xi ))\nu ^0 (d\xi )\\&\quad = \left( \int _{{{\mathbb {R}}}^d} \phi (x) | \psi ^{\xi _{n}}_n(t,x) +\psi ^{\xi _n}_{n+1} (t,x)|^2 dx \right) \sum _{j\in {{\mathbb {Z}}}^d} |c_{n_0} (\xi _{n_0}+ j) |^2\\&\quad =\int _{{{\mathbb {R}}}^d} \phi (x)| \psi ^{\xi _{n}}_n(t,x) +\psi ^{\xi _n}_{n+1} (t,x)|^2dx \end{aligned}$$

$\square$

7.3 The 1-d Case: Proof of Theorem 1.1 and Discussion of Proposition 1.6

We now focus on the results devoted to the 1-dimensional case. By Lemma A.1, the Bloch energies $\varrho _n$ have only non-degenerate critical points and $\Lambda _n\subset \pi {{\mathbb {Z}}}$. Besides, they are smooth outside the set of crossing points $\Sigma _n=\pi {{\mathbb {Z}}}\setminus \Lambda _n$, that are all conical. Therefore, the assumptions of Theorem 1.5 are satisfied and

$$\begin{aligned} \varsigma ^t=\sum _{n\in I_n}\mu ^t_{n,n}. \end{aligned}$$

with $\mu ^t_{n,n}$ determined by the pairs $M^t_n) \nu ^t_n$. It remains to characterize the pairs $(M^t_n,\nu _n)$ that are associated with the discrete sets $\Lambda _n$. For this reason, $T^*\Lambda _n=\Lambda _n\times \{0\}$ and $N\Lambda _n = {{\mathbb {R}}}^d$, the measure $\nu ^t_n$ is a sum of Dirac masses and the operator $M^t_n$ is constant and an orthogonal projector on a function $\psi _\xi ^{(n)}$ that has to satisfy (1.9) since $M^t_n$ satisfies (1.15) (see also Corollary 1.4 in [15]).

It can be illuminating to see how Proposition 1.6 can be deduced from Theorem 1.1 when $d=1$. The assumptions of Proposition 1.6 correspond to the choice of

$$\begin{aligned} \psi ^\varepsilon _0(x)= \varphi _{n_0}\left( \frac{x}{\varepsilon },\varepsilon D_x\right) \mathrm{e}^{\frac{i}{\varepsilon }x\xi _{n_0}}v_{n_0}^\varepsilon (x) \end{aligned}$$

with $\xi _{n_0}\in \Lambda _{n_0}$ and $(v^\varepsilon _{n_0})_{\varepsilon >0}$ bounded in $H^s({{\mathbb {R}}})$ with $s>1$. This in turn, corresponds to setting

$$\begin{aligned} U^\varepsilon _{n_0} (x,y) = \varphi _{n_0}\left( y,\varepsilon D_x\right) \mathrm{e}^{\frac{i}{\varepsilon }x\xi _{n_0}}v_{n_0}^\varepsilon (x). \end{aligned}$$

As a consequence, we have $\Pi _n(\varepsilon D_x) U^\varepsilon _0=0$ for $n\not =n_0$ and $\Pi _{n_0}(\varepsilon D_x) U^\varepsilon _0=U^\varepsilon _0$, whence

$$\begin{aligned} L^\varepsilon \Pi _n(\varepsilon D_x) U^\varepsilon _0 =\left\{ \begin{array}{rc} 0 &{} \text{ for }\; n\not =n_0 , \\ \psi ^\varepsilon _0 &{} \text{ for }\; n=n_0. \end{array}\right. \end{aligned}$$

We deduce that the weak limits $\psi _\xi ^{(n)}$ ($n\in {{\mathbb {N}}}^*$ and $\xi \in \Lambda _n$) that we have to consider in Theorem 1.1 satisfy

$$\begin{aligned} \psi ^{(n)}_\xi (0) = \left\{ \begin{array}{cc} 0 &{} \text{ for }\; n\not =n_0 , \\ \mathop {{\mathrm{w-lim}}}\limits _{\varepsilon \rightarrow 0} \mathrm{e}^{-\frac{i}{\varepsilon }x\xi }\psi ^\varepsilon _0 &{} \text{ for }\; n=n_0 . \end{array}\right. \end{aligned}$$

Lemma E.2 yields

$$\begin{aligned} \psi ^{(n_0)}_\xi (0) = \mathop {{\mathrm{w-lim}}}\limits _{\varepsilon \rightarrow 0} \mathrm{e}^{-\frac{i}{\varepsilon }x\xi }\psi ^\varepsilon _0= \left\{ \begin{array}{cl} 0 &{} \text{ if }\; \xi \notin \xi _{n_0}+2\pi {{\mathbb {Z}}}, \\ c_{n_0}(\xi _{n_0}+j) v_{n_0} &{} \text{ if }\; \xi =\xi _{n_0}+j,\;j\in 2\pi {{\mathbb {Z}}}. \end{array}\right. \end{aligned}$$

We recall that the sequence $(c_n(\xi +j))_{j\in {{\mathbb {Z}}}}$ are the Fourier coefficients of $y\mapsto \varphi _{n}(y,\xi )$. As a consequence, for $t\in {{\mathbb {R}}}$, we have

$$\begin{aligned} \psi ^{(n_0)}_\xi (t) = \left\{ \begin{array}{cl} 0 &{} \text{ if }\; \xi \notin \xi _{n_0}+2\pi {{\mathbb {Z}}}, \\ c_{n_0}(\xi _{n_0}+j) \psi ^{\xi _{n_0}}(t) &{} \text{ if }\; \xi =\xi _{n_0}+j,\;j\in 2\pi {{\mathbb {Z}}}\end{array}\right. \end{aligned}$$

where $\psi ^{\xi _{n_0}}(t)$ solves (1.9) with $n=n_0$ and initial data $\psi ^{\xi _{n_0}}=v_{n_0}$. Applying Theorem 1.1, we obtain

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _a^b \int _{{\mathbb {R}}}\phi (x) |\psi ^\varepsilon (t,x)|^2 dxdt&= \sum _{j\in 2\pi {{\mathbb {Z}}}} \int _a^b \int _{{{\mathbb {R}}}}\phi (x) |c_{n_0}(\xi _{n_0}+j)|^2 | \psi ^{\xi _{n_0}}(t,x)|^2 dxdt \\&= \int _a^b \int _{{{\mathbb {R}}}}\phi (x) | \psi ^{\xi _{n_0}}(t,x)|^2 \mathrm{d}x\mathrm{d}t, \end{aligned}$$

where we have used that

$$\begin{aligned} \sum _{j\in 2\i {{\mathbb {Z}}}} |c_{n_0}(\xi _{n_0}+j)|^2=\Vert \varphi (\cdot ,\xi _{n_0}\Vert ^2_{L^2({{\mathbb {T}}})} =1. \end{aligned}$$

7.4 Extension of the Setting to More General Situations

Our results could be formulated differently by assuming that the initial data is localised in Fourier variables on a set $\Omega$ in which the assumptions H 1, H 2 and H 3 of Theorem 1.5 are satisfied. More precisely, we assume that $\Omega$ is a ${{\mathbb {Z}}}^d$-periodic open subset of ${{\mathbb {R}}}^d$ such that there exists a unit cell ${{\mathcal {B}}}$ of ${{\mathbb {Z}}}^d$ for which $\Omega \cap {\mathcal {B}}$ is strictly included in ${\mathcal {B}}$. We prove here that the analysis of the semi-classical measure $\varsigma ^t$ of $(\psi ^\varepsilon )_{\varepsilon >0}$ in ${{\mathbb {R}}}\times {{\mathbb {R}}}^d\times \Omega$ can be performed by localizing the initial data $\psi ^\varepsilon _0$, which allows to extend the results of Theorem 1.5 to data with less strict assumptions on the Bloch energies.

Lemma 7.3

Let $\Omega$ as above and $\chi \in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^d)$ be $2\pi {{\mathbb {Z}}}^d$-periodic, supported in the interior of ${\mathcal {B}}+2\pi {{\mathbb {Z}}}^d$ and equal to 1 on $\Omega$. Let $U_\chi ^\varepsilon (t)$ be the solution of equation (2.2) with initial data $\chi (\varepsilon D) U^\varepsilon _0$. Then, for every $s\ge 0$ there exists a constant $C_s>0$ such that, for all $t\in {{\mathbb {R}}}$,

$$\begin{aligned} \left\| U^\varepsilon _\chi (t)-\chi (\varepsilon D_x) U^\varepsilon (t)\right\| _{H^s_\varepsilon ({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)}\le \varepsilon \, C_s (1+|t|). \end{aligned}$$

Moreover, if $\psi ^\varepsilon _0=L^\varepsilon U^\varepsilon _0$, then there exist $C>0$ such that

$$\begin{aligned} \left\| L^\varepsilon U_\chi ^\varepsilon (t)-\chi (\varepsilon D_x) \psi ^\varepsilon (t)\right\| _{L^2 ({{\mathbb {R}}}^d)}\le \varepsilon \, C (1+|t|). \end{aligned}$$

Proof of Lemma 7.3

Note that we have $U^\varepsilon _\chi (0)=\chi (\varepsilon D_x) U^\varepsilon _0$. We observe that ${{\widetilde{U}}}^\varepsilon = \chi (\varepsilon D_x) U^\varepsilon$ satisfies the system

$$\begin{aligned} i\varepsilon ^2\partial _t {\widetilde{U}}^\varepsilon =P(\varepsilon D_x){\widetilde{U}}^\varepsilon +\varepsilon ^2 V_{\mathrm{ext}}{\widetilde{U}}^\varepsilon +\varepsilon ^3 F^\varepsilon , \end{aligned}$$

with $F^\varepsilon (t) = \varepsilon ^{-1} \left[ \chi (\varepsilon D_x), V_{\mathrm{ext}}(t)\right] U^\varepsilon (t)$ is uniformly bounded in $L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$. A standard energy estimate then gives the result for $s=0$. Then, in view of Remark 2.2, it is enough to prove that $P(\varepsilon D_x)^{s/2}({{\widetilde{U}}}^\varepsilon -U^\varepsilon _\chi )$ and $\langle \varepsilon D_x \rangle ^s ({{\widetilde{U}}}^\varepsilon -U^\varepsilon _\chi )$ go to 0 in $L^2({{\mathbb {R}}}^d\times {{\mathbb {T}}}^d)$. We proceed by induction in $s\in {{\mathbb {N}}}$ and interpolation between s and $s+1$, following the arguments of the proof of Lemma 6.7 in [15]. This proves the first estimate of the lemma.

To prove the second estimate, note that whenever $\chi$ is $2\pi {{\mathbb {Z}}}^d$-periodic, we have

$$\begin{aligned} \chi (\varepsilon D) \left( \mathrm{e}^{\frac{i}{\varepsilon }k\cdot x}\cdot \right) = \mathrm{e}^{\frac{i}{\varepsilon }k\cdot x} \chi (\varepsilon D+k)= \mathrm{e}^{\frac{i}{\varepsilon }k\cdot x}\chi (\varepsilon D) \end{aligned}$$

for $k\in 2\pi {{\mathbb {Z}}}^d$. Thus $[\chi (\varepsilon D), L^\varepsilon ]=0$ where $L^\varepsilon$ is the operator defined in (1.8). We deduce that

$$\begin{aligned} \chi (\varepsilon D) \psi ^\varepsilon (t)= \chi (\varepsilon D) L^\varepsilon U^\varepsilon (t) = L^\varepsilon \chi (\varepsilon D) U^\varepsilon (t). \end{aligned}$$

Therefore, combining (1.8) and the previous estimate, finishes the proof of the lemma. $\quad \square$

Appendices

Appendix A. One Dimensional Bloch Modes

We review the main aspects of the theory that are needed here; the reader can refer to the books [48, 53] or the articles [32, 39, 47] among others for additional details.

First of all, note that $\phi \in L^2({{\mathbb {T}}})$ solves $P(\xi )\phi =\lambda \phi$ for some $\xi ,\lambda \in {{\mathbb {R}}}$ if and only if $f(y,\lambda ):=\mathrm{e}^{i\xi y}\phi (y)$ is a solution to:

$$\begin{aligned} -{1\over 2} \partial _y^2 f(y,\lambda ) +V_{\mathrm{per}}(y) f(y, \lambda )=\lambda f(y, \lambda ), \quad y\in {{\mathbb {R}}}, \end{aligned}$$

(A.1)

satisfying the periodicity condition

$$\begin{aligned} f(1,\lambda )=\mathrm{e}^{i\xi }f(0,\lambda ). \end{aligned}$$

(A.2)

Given $\lambda \in {{\mathbb {R}}}$, the solutions of (A.1) are linear combinations of two solutions $f_1(y,\lambda )$ and $f_2(y,\lambda )$ satisfying

$$\begin{aligned} f_1(0,\lambda )=\partial _y f_2(0,\lambda )=1,\;\; f_2(0,\lambda )= \partial _y f_1(0,\lambda )=0. \end{aligned}$$

Define:

$$\begin{aligned} M_\lambda (y):=\left( \begin{array}{cc} f_1(y,\lambda ) &{} f_2(y,\lambda ) \\ \partial _y f_1(y,\lambda ) &{} \partial _y f_2(y,\lambda ) \end{array} \right) ; \end{aligned}$$

then the existence of a solution to (A.1) satisfying (A.2) is equivalent to the fact that $\mathrm{e}^{i\xi }$ is an eigenvalue of $M_\lambda (1)$. One can check that $\det M_\lambda (y)=1$ for every $y, \lambda \in {{\mathbb {R}}}$; therefore, letting $\Delta (\lambda ):={{\,\mathrm{Tr}\,}}M_\lambda (1)$, we find that $\mathrm{e}^{i\xi }\in {{\,\mathrm{Sp}\,}}M_\lambda (1)$ if and only if:

$$\begin{aligned} \Delta (\lambda )=2\cos \xi . \end{aligned}$$

(A.3)

It can be shown that solutions to (A.1) depend analytically on $\lambda$, and that moreover, $\Delta$ extends to an entire function of order 1/2. The real solutions to equations $\Delta (\lambda )=\pm 2$ form infinite increasing sequences $(a_i^\pm )$ that tend to infinity.

The following facts (the reader may find helpful to consult [47, Figure 1, p. 145] or [53, Section XIII.16]) are illustrated in Fig. 1:

• The sequences $(a_i^\pm )$ are intertwined. More precisely, one has:

  $$\begin{aligned} a_1^+<a_1^-\le a_2^-< a_2^+\le a_3^+< a_3^-\cdots , \end{aligned}$$

  (A.4)

• Let be $I_{2i-1} = (a_{2i-1}^+,a_{2i-1}^-)$ and $I_{2i} = (a_{2i}^-,a_{2i}^+)$. Then $I_i$ has non-empty interior and is strictly decreasing for i odd and strictly increasing for i even.

• If $a_i^\sigma =a_{i+1}^\sigma$ for some $i\in {{\mathbb {N}}},\,\sigma \in \{+,-\}$ then $\Delta ^\prime (a_i^\sigma )=0.$

These properties have important implications on the behavior of Bloch energies. For every $n\in {{\mathbb {N}}}$ the following hold.

• Fact 1 The $n^{\mathrm{th}}$ Bloch energy is the solution to .

• Fact 2 $\varrho _n$ is $2\pi {{\mathbb {Z}}}$-periodic (we knew this already), and moreover

  $$\begin{aligned} \varrho _n(\xi )=\varrho _n(2\pi -\xi ),\quad \forall \xi \in {{\mathbb {R}}}. \end{aligned}$$

• Fact 3 is strictly increasing if n is odd (resp. strictly decreasing if n is even) and analytic in the interior of the interval. If it is differentiable at $\xi =0,\pi$ then necessarily $\varrho _n^\prime (\xi )=0$ and $\varrho _n$ is analytic around that point.

• Fact 4 A crossing can happen only at two consecutive Bloch energies. Let $n\in {{\mathbb {N}}}$ be such that

  $$\begin{aligned} \Sigma _n:=\{\xi \in {{\mathbb {R}}}\,:\,\varrho _n(\xi )=\varrho _{n+1}(\xi )\}\ne \emptyset ; \end{aligned}$$

  then $\Sigma _n=\pi {{\mathbb {Z}}}\setminus 2\pi {{\mathbb {Z}}}$ if n is odd, $\Sigma _n=2\pi {{\mathbb {Z}}}$ if n is even. Moreover

  $$\begin{aligned} \Delta ^\prime (\varrho _n(\xi ))=0,\quad \forall \xi \in \Sigma _n. \end{aligned}$$

  (A.5)

Fig. 1

Hill’s discriminant for $V_{\mathrm{per}}(y) = 5\cos (2y)$ numerically calculated. All graphics show the same set of data plotted with different ranges. In the first two it is possible to observe both divergence of $\Delta (\lambda )$ in the region $\lambda \approx \inf _y V_{\mathrm{per}}(y)$ and non-decaying oscillations of period $\sim \sqrt{\lambda }$ for $\lambda \longrightarrow \infty$. The third one illustrates how the intervals $I_i$, wherein solutions to (A.1) are stable, are never empty, and why their borders $a_i^\pm$ must satisfy (A.4). Roots $a_i^+$ admit $\pi$-periodic solutions, and $a_i^-$ $2\pi$-periodic solutions; in each case, there are two linearly independent such solutions if and only if $a_i^\pm$ coincides with $a^\pm _{i+1}$ or $a^\pm _{i-1}$. If this is not the case, as for roots $a^\pm _1$, $a_2^\pm$ and $a^+_3$ seen in the last image, then the equation also admits a unstable solution. A complete study of $\Delta (\lambda )$ in one dimension is found in [46]

In addition, critical points of Bloch energies in the one dimensional case are never degenerate nor can occur at a crossing point.

Lemma A.1

The set of critical points of any Bloch energy $\varrho _n$ is contained in $\pi {{\mathbb {Z}}}$ and all the critical points are non-degenerate. Moreover, the crossing set $\Sigma _n$ associated with two consecutive Bloch energies $\varrho _n$ and $\varrho _{n+1}$ does not contain any critical points of the Bloch energies $\varrho _n$ and $\varrho _{n+1}$.

Proof

The first assertion on the critical points is property (3) above, whereas the second follows from differentiating twice equation (A.3) and evaluating at a critical point $\xi =k\pi$, $k\in {{\mathbb {Z}}}$ to get:

$$\begin{aligned} \Delta ^\prime (\varrho _n(k\pi ))\varrho _n^{\prime \prime }(k\pi )=(-1)^{k+1}2. \end{aligned}$$

This relation also shows that $\Delta ^\prime (\lambda )$ cannot vanish at $\lambda =\varrho _n(k\pi )$. Together with (A.5) this shows that a critical point cannot be a crossing point. $\quad \square$

Remark A.1

In the free case ($V_{\mathrm{per}}=0$) there is only a Bloch band of infinite multiplicity. More generally, it has been proved in [10] that the absence of spectral gap is equivalent to the periodic potential $V_{\mathrm{per}}$ being constant.

Appendix B. The Properties of the Bloch Energies at Crossing Points

Here we present a normal form for the expression of two Bloch energies $\varrho _n(\xi )$ and $\varrho _{n+1}(\xi )$ close to the crossing set $\Sigma _n$ (defined as $\Sigma _n=\Sigma _{n,n+1}$ in (1.5)).

Lemma B.1

Let $\sigma _0$ be a point in the crossing set $\Sigma _n$ of two consecutive Bloch energies $\varrho _n$ and $\varrho _{n+1}$ having neighborhood U with the following properties:

• (i) $\Sigma _n\cap U$ is a smooth manifold.

• (ii) The multiplicities of $\varrho _n(\xi ),\varrho _{n+1}(\xi )$ are constant on each connected component of $U\setminus \Sigma _n$.

• (iii) There exists $\delta _0>0$ such that for all $\xi \in U$,

  $$\begin{aligned} d\left( \{ \varrho _n(\xi ),\varrho _{n+1}(\xi ) \}, \mathrm{Sp} \, P(\xi ) \setminus \{ \varrho _j(\xi ),\quad \varrho _j(\xi )=\varrho _n(\xi ) \; or\; \varrho _j(\xi )=\varrho _{n+1}(\xi ) \} \right) \ge \delta _0. \end{aligned}$$

Then, there exist $\Omega \subseteq U$, a neighborhood of $\sigma _0$ that is $2\pi {{\mathbb {Z}}}^d$-invariant, two functions $\lambda _n\in {\mathcal {C}}^\infty (\Omega )$, $g_n\in {\mathcal {C}}^\infty \left( \sqcup _{\xi \in \Omega } \left( \{\xi \} \times N_{\sigma _{\Sigma _n}(\xi )}\Sigma _n\right) \right)$, and a function $m\in L^\infty (U)$ which is constant on each connected component of U such that

$$\begin{aligned}&\forall \xi \in \Omega \setminus \Sigma _n, \;\;\varrho _n(\xi ) =\lambda _n(\xi )- g_n(\xi ,\xi -\sigma _{\Sigma _n}(\xi )),\;\; \nonumber \\&\qquad \qquad \qquad \qquad \quad \varrho _{n+1}(\xi )= \lambda _n(\xi )+m(\xi ) g_n(\xi ,\xi -\sigma _{\Sigma _n}(\xi )). \end{aligned}$$

Moreover,

• (1) If the crossing set $\Sigma _n$ is conical in U, then for all $\xi \in U$, the map $N_{\sigma (\xi )} \Sigma \ni \eta \mapsto g_n(\xi ,\eta )$ is homogeneous of degree 1 and $g_n(\sigma ,\eta )\not =0$ when $(\sigma ,\eta )\in N\Sigma _n$ with $\eta \not =0$,

• (2) If none of the points of $\Sigma _n$ are conical crossings in U, then there exists $\theta _n\in {\mathcal {C}}^\infty ({{\mathbb {R}}}^d )$ such that $g_n(\xi ,\eta )= |\eta |^2 \theta _n (\xi )$, which implies that $\varrho _n$, $\varrho _{n+1}\in {\mathcal {C}}^{1,1}({{\mathbb {R}}}^d)$,

• (3) If the multiplicities of $\varrho _n,\varrho _{n+1}$ are equal on $U\setminus \Sigma _n$ then $m=1$.

Remark B.2

Note that in case (2), the function $\theta _n$ can be zero on $\Sigma _n$.

Proof

We denote by $j_-(\xi ),j_+(\xi )$ the functions valued in ${{\mathbb {N}}}$ and constant on connected component of $U\setminus \Sigma _n$ such that for all $\xi \in U\setminus \Sigma _n$ $\varrho _{n-j+1}(\xi )=\varrho _n(\xi )$ for $1\le j\le j_-(\xi )$ and $\varrho _{n+j}(\xi )=\varrho _{n+1}(\xi )$ for $1\le j\le j_+(\xi )$. We denote by $\Pi (\xi )$ the projector on

$$\begin{aligned} F_\xi =\mathrm{Ker}(P(\xi )-\varrho _n(\xi )) \oplus \mathrm{Ker}(P(\xi )-\varrho _{n+1}(\xi )). \end{aligned}$$

By the assumptions on U, the pair $\{\varrho _n(\xi ),\varrho _{n+1}(\xi )\}$ is isolated from the remainder of the spectrum of $P(\xi )$ when $\xi \in U$, this implies that the map $U\ni \xi \mapsto \Pi (\xi )\in {\mathcal {L}}(L^2({{\mathbb {T}}}^d))$ is analytic and the function $\dim F_\xi$ is constant for $\xi \in U$. We denote by $\ell _0$ this constant and we have $\ell _0=j_-(\xi )+j_+(\xi )$ for all $\xi \in U\setminus \Sigma _n$. Moreover, $\varrho _n(\xi )$ and $\varrho _{n+1}(\xi )$ are the two only eigenvalues of the operator $\Pi (\xi )P(\xi )\Pi (\xi )$ which maps $F_\xi$ onto $F_\xi$ for any $\xi \in {{\mathbb {R}}}^d$.

Let us first show that it is possible to find $\Omega \subseteq U$, with $\sigma _0\in \Omega$ and construct, for every $\xi \in \Omega$, an orthonormal basis $(\phi _j(\xi ,\cdot ))_{1\le j\le \ell _0}$ of $F_\xi$ such that the maps $\xi \mapsto \phi _j(\xi ,\cdot )$ are analytic for all $j\in \{1,\cdots \ell _0\}$. To see this, consider $(\varphi _i(\sigma _0,\cdot ))_{1\le i\le \ell _0}$, a basis of $F_{\sigma _0}$. Chose a neighborhood $\Omega$ of $\sigma _0$ small enough to ensure that the vectors

$$\begin{aligned} \Pi (\xi ) \varphi _{j}(\sigma _0,\cdot ),\;\; j\in \{1,\ldots ,\ell _0\} \end{aligned}$$

form a rank $\ell _0$ family. Then apply the standard Schmidt orthonormalization process to this family.

Let $A(\xi )$, $\xi \in \Omega$, be the matrix of the operator $\Pi (\xi )P(\xi )\Pi (\xi )$ in the basis we just constructed. This is a $\ell _0\times \ell _0$ analytic matrix that we can write

$$\begin{aligned} A(\xi )=\lambda _n(\xi ) \mathrm{Id} + A_0(\xi ) \end{aligned}$$

with $\lambda _n(\xi ):= \frac{1}{\ell _0} \mathrm{Tr}_{{{\mathbb {C}}}^{\ell _0}} A(\xi )$ and $A_0(\xi )$ analytic and trace-free. Moreover, $A(\xi )$ is diagonalizable and has only two eigenvalues $\varrho _n(\xi )$ and $\varrho _{n+1}(\xi )$ that we write

$$\begin{aligned} \varrho _n(\xi ) = \lambda _n(\xi ) - g(\xi ) ,\quad \varrho _{n+1}(\xi ) = \lambda _n(\xi ) + m(\xi )g(\xi ) , \end{aligned}$$

with $g(\xi )>0$ and where, for $\xi \in \Omega \setminus \Sigma _n$, $m(\xi )$ is the ratio between the multiplicities of $\varrho _n(\xi )$ and $\varrho _{n+1}(\xi )$,

$$\begin{aligned} m(\xi )= \frac{j_-(\xi )}{j_+(\xi )} \end{aligned}$$

and m is constant in the connected componnent of $U\setminus \Sigma _n$.

The functions $-g(\xi )$ and $m(\xi )g(\xi )$ are the two eigenvalues of $A_0(\xi )$. Therefore, they are homogeneous function of degree 1 of the coefficients of $A_0(\xi )=(a_{i,j}(\xi ))_{1\le i,j\le \ell _0}$: we write $g(\xi )= G(A_0(\xi ))$ where G is a homogeneous function on ${{\mathbb {R}}}^{\frac{\ell _0^2-1}{2}}$. Here, we have considered that a $\ell _0\times \ell _0$ trace-free Hermitian matrix is a function of $\ell _0-1$ real-valued diagonal coefficients and of $\frac{\ell _0(\ell _0-1)}{2}$ complex-valued coefficients (those under the diagonal being the conjugate of those above the diagonal), and we have observed that $(\ell _0-1) +\frac{\ell _0(\ell _0-1)}{2}= \frac{\ell _0^2-1}{2}$.

By the definition of the crossing set, $A_0(\xi )=0$ if and only if $\xi \in \Sigma _n$. Since the map $\xi \mapsto A_0(\xi )$ is analytic, it vanishes on $\Sigma _n$ at finite order $q\in {{\mathbb {N}}}$ and the crossing set is conical if and only if $q=1$ for all points of $\Sigma _n$. Therefore, in case (1), there exists a smooth tensor $T^{\ell _0,1}(\xi )$ such that

$$\begin{aligned} A_0(\xi ) = T^{\ell _0,1}(\xi ) [\xi -\sigma _{\Sigma _n}(\xi )], \end{aligned}$$

with

$$\begin{aligned} \forall \sigma \in \Sigma _n\cap \Omega , \;\; \forall \eta \in N_{\sigma }\Sigma _n \setminus \{0\},\;\; T^{\ell _0,1}(\sigma ) \eta \not =0_{{{\mathbb {C}}}^{\ell _0\times \ell _0}}. \end{aligned}$$

We deduce that

$$\begin{aligned} g(\xi )= g_n(\xi ,\xi -\sigma _{\Sigma _n}(\xi )),\;\;\text{ with } \;\;g_n(\xi ,\eta ):=G\left( T^{\ell _0,1}(\xi )\left[ \eta \right] ^q\right) \end{aligned}$$

where $g_n$ is homogeneous of degree 1 in the variable $\eta$. Besides, if none of the crossing points are conical, we write $A_0(\xi ) = T^{\ell _0,2}(\xi ) [\xi -\sigma _{\Sigma _n}(\xi )]^2$ with $T^{\ell _0,2}(\xi )$ a smooth tensor, which allows to prove Point (2) with

$$\begin{aligned} \theta _n(\xi )= |\xi -\sigma _{\Sigma _n}(\xi )|^{-2}G(T^{\ell _0,2}(\xi ) [\xi -\sigma _{\Sigma _n}(\xi )]^2). \end{aligned}$$

That concludes the proof since Point (3) is obvious. $\quad \square$

Appendix C. Semi-classical Pseudo-differential Calculus

We recall here results about matrix-valued semi-classical pseudo-differential operators. We denote by $S_{N\times N}$ the set of functions $a=(a_{i,j})\in {\mathcal {C}}^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$ which are bounded together with their derivatives in matrix-norm. Then, for $a\in S_{N\times N}$, one defines the Weyl semi-classical pseudo-differential operator of symbol a as

$$\begin{aligned} \mathrm{op}_\varepsilon (a) f(x)=\int _{{{\mathbb {R}}}^{2d}} \mathrm{e}^{{i\over \varepsilon } \xi \cdot (x-y)} a\left( {x+y\over 2},\xi \right) f(y)\mathrm{d}y\, \frac{d\xi }{(2\pi \varepsilon )^{d}},\quad \forall f\in {{\mathcal {S}}}({{\mathbb {R}}}^d,{{\mathbb {C}}}^N). \end{aligned}$$

Properties of these matrix-valued pseudo-differential operators follow from the well-understood scalar theory, once the definition of the product of matrices and its non-commutativity is taken into account. Unless stated otherwise, the reader may found proofs of the scalar versions of the results presented here in [18, 25, 57], for instance.

The Calderón-Vaillancourt theorem [11, 12] extends to the matrix-valued case and ensures the existence of a constant $C_d>0$ such that for every $a\in S_{N\times N}$ one has

$$\begin{aligned} \Vert \mathrm{op}_\varepsilon (a)\Vert _{{{\mathcal {L}}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N}))}\le C_d\, N_d^\varepsilon (a), \end{aligned}$$

(C.1)

where

$$\begin{aligned} N_d^\varepsilon (a):=\sum _{\alpha \in {{\mathbb {N}}}^{2d},|\alpha |\le d+2} \varepsilon ^{|\alpha |/2}\sup _{{{\mathbb {R}}}^d\times {{\mathbb {R}}}^d}|\partial _{x,\xi }^\alpha a|_{{{\mathbb {C}}}^{N\times N}}. \end{aligned}$$

This estimate shows that semi-classical pseudo-differential operators are uniformly bounded in $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$ for $\varepsilon \in (0,1]$. Moreover, for every $a\in S_{N\times N}$,

$$\begin{aligned} \mathrm{op}_\varepsilon (a)^*=\mathrm{op}_\varepsilon (a^*). \end{aligned}$$

(C.2)

In particular, semi-classical pseudo-differential operators whose symbols take values in the space of Hermitian matrices are self-adjoint on $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$.

In addition, the symbolic calculus for matrix-valued pseudodifferential operators goes as follows:

Proposition C.1

Let $a,b\in S_{N\times N}$, then

$$\begin{aligned} \mathrm{op}_\varepsilon (a)\mathrm{op}_\varepsilon (b) = \mathrm{op}_\varepsilon (ab)+\frac{\varepsilon }{2i} \mathrm{op}_\varepsilon (\{a,b\})+\varepsilon ^2 R^{(1)}_\varepsilon , \end{aligned}$$

with $\{a,b\}=\sum _{j=1}^d\partial _{\xi _j} a\, \partial _{x_j} b-\partial _{x_j}a\, \partial _{\xi _j} b$ and

$$\begin{aligned} \left[ \mathrm{op}_\varepsilon (a),\mathrm{op}_\varepsilon (b)\right] =\mathrm{op}_\varepsilon ([a,b])+ {\varepsilon \over 2i}(\mathrm{op}_\varepsilon (\{a,b\}) -\mathrm{op}_\varepsilon (\{b,a\}))+\varepsilon ^2 R_\varepsilon ^{(2)}, \\ \Vert R^{(j)}_\varepsilon \Vert _{{{\mathcal {L}}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))}\le C \,\sup _{|\alpha |+|\beta |=2} N_d^\varepsilon (\partial _\xi ^\alpha \partial _x^{\beta } a) N_d^\varepsilon ( \partial _\xi ^\beta \partial _x^{\alpha } b),\quad j\in \{1,2\}, \end{aligned}$$

for some constant $C>0$ independent of a, b and $\varepsilon$.

Remark C.2

The term of order $\varepsilon ^2$ above has a particularly simple expression when $b\in S_{N\times N}$ does not depend on x. The following hold in ${{\mathcal {L}}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))$:

$$\begin{aligned} \mathrm{op}_\varepsilon (b)\mathrm{op}_\varepsilon (a)= & {} \mathrm{op}_\varepsilon (ba)+\frac{\varepsilon }{2i}\sum _{j=1}^d \mathrm{op}_\varepsilon ( \partial _{\xi _j} b\, \partial _{x_j}a)\\&+{\varepsilon ^2\over 8} \sum _{1\le \ell ,p\le d} \mathrm{op}_\varepsilon ( \partial ^2_{\xi _\ell \xi _p}b\, \partial ^2_{x_\ell x_p} a )+ O(\varepsilon ^3),\\ \mathrm{op}_\varepsilon (a)\mathrm{op}_\varepsilon (b)= & {} \mathrm{op}_\varepsilon (ab)-\frac{\varepsilon }{2i} \sum _{j=1}^d\mathrm{op}_\varepsilon (\partial _{x_j}a\, \partial _{\xi _j} b)\\&+{\varepsilon ^2\over 8} \sum _{1\le \ell ,p\le d} \mathrm{op}_\varepsilon (\partial ^2_{x_\ell x_p} a \, \partial ^2_{\xi _\ell \xi _p}b)+ O(\varepsilon ^3). \end{aligned}$$

There are also analogues of Gårding’s inequality for elliptic differential operators [33] in this context. If $a\in S_{N\times N}$ takes values in the set of non-negative Hermitian matrices then there exist $C_a>0$, which depends on a finite number of derivatives of a such that

$$\begin{aligned} \mathrm{op}_\varepsilon (a)+C_a\varepsilon {{\,\mathrm{Id}\,}}\ge 0. \end{aligned}$$

(C.3)

This follows by considering $a*\rho _\varepsilon$, with $\rho _\varepsilon (x,\xi ):=(\pi \varepsilon )^{-d} e^{-\frac{|x|^2+|\xi |^2}{\varepsilon }}$. A direct computation shows that $\mathrm{op}_\varepsilon (a*\rho _\varepsilon )$ is a non-negative operator (see for instance [35]). One concludes by noticing that, due to (C.1),

$$\begin{aligned} \mathrm{op}_\varepsilon (a)-\mathrm{op}_\varepsilon (a*\rho _\varepsilon )=O(\varepsilon ), \end{aligned}$$

(C.4)

in ${{\mathcal {L}}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))$. The operator $\mathrm{op}_\varepsilon (a*\rho _\varepsilon )$ is called the anti-Wick quantization of the symbol $a\in S_{N\times N}$, besides the fact that non-negative symbols correspond to non-negative operators, the operator norm satisfies a simpler bound than that satisfied by their Weyl counterparts, namely:

$$\begin{aligned} \Vert \mathrm{op}_\varepsilon (a*\rho _\varepsilon )\Vert _{{{\mathcal {L}}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N}))}\le \Vert a\Vert _{L^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})}. \end{aligned}$$

(C.5)

Through the article, it is necessary to understand the boundedness and symbolic calculus properties of operators with symbols of limited regularity. Denote by ${\mathcal {A}}$ the completion of ${\mathcal {C}}^\infty _0({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$ with respect to the norm:

$$\begin{aligned} \Vert a\Vert _{\mathcal {A}}:= \sup _{\xi \in {{\mathbb {R}}}^d} \sup _{|\alpha |\le d+2} \int _{{{\mathbb {R}}}^d} | \partial _x^\alpha a(x,\xi ) |_{{{\mathbb {C}}}^{N\times N}} \mathrm{d}x . \end{aligned}$$

(C.6)

Remark C.3

By a mollification argument, one can show that ${\mathcal {A}}$ contains all the functions $a\in {\mathcal {C}}_0({{\mathbb {R}}}^{d}_x\times {{\mathbb {R}}}^d_\xi ,{{\mathbb {C}}}^{N\times N})$ that are $d+2$ times continuously differentiable with respect to the first variable.

This regularity assumption is sufficient for our purposes (see [34, Lemma 3.7] and [28, Section 3] for related results).

Lemma C.4

The space ${\mathcal {A}}$ enjoys the following properties:

• (1) There exists a universal constant $C_d>0$ only depending on d such that, for every $a\in {\mathcal {A}}$,

  $$\begin{aligned} \Vert \mathrm{op}_\varepsilon (a)\Vert _{{\mathcal {L}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))} \le C_d \Vert a\Vert _{\mathcal {A}}. \end{aligned}$$

• (2) Suppose $\varrho \in \text {Lip}({{\mathbb {R}}}^d_\xi ,{{\mathbb {C}}}^{N\times N})$ and $a\in {\mathcal {A}}$. The following hold in ${{\mathcal {L}}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))$:

  $$\begin{aligned} \mathrm{op}_\varepsilon (a\,\varrho )= \mathrm{op}_\varepsilon (a) \varrho (\varepsilon D_x)+O(\varepsilon ), \\ \mathrm{op}_\varepsilon (\varrho \, a)=\varrho (\varepsilon D_x)\mathrm{op}_\varepsilon (a) +O(\varepsilon ). \end{aligned}$$

• (3) Suppose $\varrho \in {\mathcal {C}}^{1,1}({{\mathbb {R}}}^d_\xi ,{{\mathbb {R}}})$ (that is $\nabla \varrho \in \text {Lip}({{\mathbb {R}}}^d,{{\mathbb {R}}}^d)$) and $a\in {\mathcal {A}}$. The following hold in ${{\mathcal {L}}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))$:

  $$\begin{aligned} \frac{i}{\varepsilon }[\mathrm{op}_\varepsilon (a),\varrho (\varepsilon D_x)\mathrm{Id} ] =\mathrm{op}_\varepsilon (\nabla _x a\cdot \nabla \varrho (\xi ))+O(\varepsilon ). \end{aligned}$$

Proof

Suppose that $A\in {\mathcal {L}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))$ is of the form

$$\begin{aligned} Af(x)=\frac{1}{\varepsilon ^d}\int _{{{\mathbb {R}}}^d}k\left( \frac{x+y}{2},\frac{x-y}{\varepsilon }\right) f(y)\mathrm{d}y, \quad \forall f\in {{\mathcal {S}}}({{\mathbb {R}}}^d,{{\mathbb {C}}}^N); \end{aligned}$$

(C.7)

then one deduces, after change of variables and using Hölder’s inequality, that

$$\begin{aligned}&|(A\,f,g)_{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}|\le \Vert f\Vert _{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}\Vert g\Vert _{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}\int _{{{\mathbb {R}}}^d}\sup _{x\in {{\mathbb {R}}}^d}|k(x,v)|_{{{\mathbb {C}}}^{N\times N}}\mathrm{d}v.\nonumber \\&\forall f,g\in {{\mathcal {S}}}({{\mathbb {R}}}^d,{{\mathbb {C}}}^N). \end{aligned}$$

(C.8)

In order to prove item (1), notice that the following identity holds,

$$\begin{aligned} \mathrm{op}_\varepsilon (a)=({{\mathcal {F}}}^\varepsilon )^*\mathrm{op}_\varepsilon ({\underline{a}}){{\mathcal {F}}}^\varepsilon ,\quad {\underline{a}}(x,\xi ):=a(-\xi ,x), \end{aligned}$$

(C.9)

where ${\mathcal {F}}^\varepsilon$ stands for the semi-classical Fourier transform:

$$\begin{aligned} {{\mathcal {F}}}^\varepsilon (f)(\xi )= \int _{{{\mathbb {R}}}^d} \mathrm{e}^{- i \frac{\xi }{\varepsilon }\cdot x} f(x) \frac{dx}{(2\pi \varepsilon )^{d/2}}. \end{aligned}$$

The operator $\mathrm{op}_\varepsilon ({\underline{a}})$ is of the form (C.7) with $k=(2\pi )^{-d}{\widehat{a}}$, where ${\widehat{a}}$ denotes the Fourier transform of a with respect to the first variable. Using (C.8), (C.9) and Plancherel’s formula we conclude

$$\begin{aligned}&|(\mathrm{op}_\varepsilon (a)f,g)_{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}|\\&\quad \le \Vert f\Vert _{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}\Vert g\Vert _{L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}\int _{{{\mathbb {R}}}^d}\sup _{\xi \in {{\mathbb {R}}}^d}|{\widehat{a}}(v,\xi )|_{{{\mathbb {C}}}^{N\times N}}\frac{dv}{(2\pi )^d}\\&\forall f,g\in {{\mathcal {S}}}({{\mathbb {R}}}^d,{{\mathbb {C}}}^N). \end{aligned}$$

The constant $C_d>0$ is then chosen such that:

$$\begin{aligned} \int _{{{\mathbb {R}}}^d} \sup _{\xi \in {{\mathbb {R}}}^d} |{\widehat{a}}(v,\xi )|_{{{\mathbb {C}}}^{N\times N}}\frac{dv}{(2\pi )^d}\le C_d \Vert a\Vert _{\mathcal {A}}. \end{aligned}$$

This concludes the proof of the first assertion.

In order to prove the second assertion, we show that

$$\begin{aligned} R^\varepsilon :=({\mathcal {F}}^\varepsilon )^*(\mathrm{op}_\varepsilon (a\,\varrho )- \mathrm{op}_\varepsilon (a) \varrho (\varepsilon D_x)){\mathcal {F}}^\varepsilon , \end{aligned}$$

satisfies $\Vert R^\varepsilon \Vert _{{\mathcal {L}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))}=O(\varepsilon )$; the proof of the other identity is analogous. From (C.9), we deduce that

$$\begin{aligned} R^\epsilon f(x)=\int _{{{\mathbb {R}}}^d}r^\varepsilon \left( \frac{x+y}{2},\frac{x-y}{\varepsilon }\right) \,f(y)\frac{dy}{(2\pi \varepsilon )^d},\quad \forall f\in {{\mathcal {S}}}({{\mathbb {R}}}^d,{{\mathbb {C}}}^N), \end{aligned}$$

where $r^\varepsilon (x,v):={\widehat{a}}(v,x)(\varrho (x)-\varrho (x-\varepsilon v))$. By (C.8) we can estimate:

$$\begin{aligned} \Vert R^\varepsilon \Vert _{{\mathcal {L}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))} \le \int _{{{\mathbb {R}}}^d}\sup _{x\in {{\mathbb {R}}}^d} |r^\varepsilon (x,v)|_{{{\mathbb {C}}}^{N\times N}}\frac{dv}{(2\pi )^d}. \end{aligned}$$

By hypothesis, we can find $L_\varrho >0$ such that

$$\begin{aligned} |\varrho (x)-\varrho (x-\varepsilon v)|_{{{\mathbb {C}}}^{N\times N}}\le L_\varrho \varepsilon |v|,\quad \forall (v,x)\in {{\,\mathrm{supp}\,}}{\widehat{a}}. \end{aligned}$$

Therefore, using

$$\begin{aligned} |v||{{\widehat{a}}}(v,x) |\le (1+|v|^2) | {{\widehat{a}}}(x,v)| = | {{\widehat{a}}}(x,v) |+ |\widehat{(-\Delta a)} (x,v)|, \end{aligned}$$

we deduce

$$\begin{aligned} \Vert R^\varepsilon \Vert _{{\mathcal {L}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))} \le \varepsilon \, C_d L_\varrho (\Vert a\Vert _{{\mathcal {A}}} + \Vert \Delta a\Vert _{{\mathcal {A}}}). \end{aligned}$$

For the third assertion, we show that

$$\begin{aligned} {{\tilde{R}}}^\varepsilon :=({\mathcal {F}}^\varepsilon )^*(\frac{1}{\varepsilon }[\mathrm{op}_\varepsilon (a),\varrho (\varepsilon D)]- \mathrm{op}_\varepsilon (\nabla _x \cdot \nabla \varrho )){\mathcal {F}}^\varepsilon , \end{aligned}$$

satisfies $\Vert {{\tilde{R}}}^\varepsilon \Vert _{{\mathcal {L}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))}=O(\varepsilon )$. Indeed, the kernel of ${{\tilde{R}}}^\varepsilon$ is of the form (C.7) with

$$\begin{aligned} {{\tilde{r}}}^\varepsilon (x,v)&=\frac{i}{\varepsilon }{{\widehat{a}}}(v,x)\left( \varrho (x)-\varrho (x-\varepsilon v)\right) -\widehat{\nabla _x a} (v,x) \cdot \nabla \varrho (x)\\&= i{{\widehat{a}}}(v,x) v\cdot \nabla \varrho (x) - \widehat{\nabla _x a}(v,x)\cdot \nabla \varrho (x) + \varepsilon \theta (x,v){{\widehat{a}}}(x,v) \end{aligned}$$

and there exists $L_{\nabla \varrho }>0$ such that

$$\begin{aligned} |\theta (x,v) |\le L_{\nabla \varrho } |v|^2. \end{aligned}$$

Using $iv{{\widehat{a}}}(v,x)= \widehat{\nabla _x a}(x,v)$ and $|v|^2 | {{\widehat{a}}}(x,v)| = |\widehat{(-\Delta a)} (x,v)|$, we deduce

$$\begin{aligned} \Vert {{\tilde{R}}}^\varepsilon \Vert _{{\mathcal {L}}(L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N))}\le \varepsilon C_d L_{\nabla \varrho } \Vert \Delta a\Vert _{{\mathcal {A}}} . \end{aligned}$$

$\square$

Appendix D. Two-Scale Pseudodifferential Operators

We prove here technical lemma concerning the pseudodifferential operators considered in Section 4.1, the formalism of which we follow. Due to the properties of Bloch energies, we are going to consider more general classes of symbols than those of ${{\mathcal {A}}}^{(2)}$ (as defined in Section 4.1). For $k\in {{\mathbb {Z}}}$, we introduce the class ${{\mathcal {A}}}^{(2)}_k$ of smooths functions on ${{\mathbb {R}}}^{3d}$ that are compactly supported in the variables $(x,\xi )$ uniformly with respect to $\eta$ and coincide with an homogeneous function of degree k in $\eta$ as soon as $|\eta |>R_0$ for some $R_0>0$. With these notations, ${\mathcal {A}}^{(2)}={\mathcal {A}}^{(2)}_0$.

Of particular interest for us are functions $g(\xi ,\eta )$, independent of the variable x, that are smooth in $\xi$, and satisfy a symbol estimate of order k in $\eta$. We denote by ${\mathcal {H}}_k$ the set of these functions. For X a connected, closed embedded submanifold of $({{\mathbb {R}}}^d)^*$ and $g\in {\mathcal {H}}_k$, the operator

$$\begin{aligned} g_\varepsilon (\varepsilon D)= g(\varepsilon D, D-\varepsilon ^{-1} \sigma _X(\varepsilon D)) \end{aligned}$$

(D.1)

is then well defined as an operator mapping $H^s({{\mathbb {R}}}^d)$ into $H^{s-k}({{\mathbb {R}}}^d)$ uniformly in $\varepsilon$ when $s\in {{\mathbb {R}}}$. We are interested in the pseudodifferential calculus involving two-scale pseudodifferential operator of the form $\mathrm{op}_\varepsilon (a)$ for a in ${{\mathcal {A}}}_k^{(2)}$ for some $k\in {{\mathbb {Z}}}$ and Fourier multipliers $g_\varepsilon (\varepsilon D)$ with $g\in {{\mathcal {H}}}_p$ for some $p\in {{\mathbb {Z}}}$.

The first type of pseudodifferential calculus results that we shall use concerns the large values of $\eta$. We consider $\chi \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^d)$ such that $\chi =1$ on B(0, 1) and $\chi =0$ on $B(0,2)^c$ with $0\le \chi \le 1$. We set for $R,\delta >0$

$$\begin{aligned} a^{R,\delta }(x,\xi ,\eta )= a(x,\xi ,\eta ) ((1-\chi (\eta /R)) \chi ((\xi -\sigma _{\Sigma }(\xi ))/\delta ). \end{aligned}$$

On the support of $a^{R,\delta }$, $\varepsilon R \le |\xi -\sigma _{\Sigma }(\xi )|\le 2\delta$. Note that if $k>0$ and $a\in {{\mathcal {A}}}^{(2)}_{-k}$, the estimate (C.6) yields that in ${\mathcal {L}}(L^2({{\mathbb {R}}}^d))$,

$$\begin{aligned} \mathrm{op}_\varepsilon (a_\varepsilon ^{R,\delta })= O((\varepsilon R)^{-k}). \end{aligned}$$

Lemma D.1

Let $k\in {{\mathbb {N}}}$, $a\in {{\mathcal {A}}}^{(2)}_{-k}$, $g=g(\xi ,\eta )$ in ${{\mathcal {H}}}_{k+1}$ and $\delta ,R>0$. Then, in ${\mathcal {L}}(L^2({{\mathbb {R}}}^d))$,

$$\begin{aligned} \left[ \mathrm{op}_\varepsilon (a_\varepsilon ^{R,\delta }), g_\varepsilon (\varepsilon D)\right] = i \,\mathrm{op}_\varepsilon \left( (\nabla _x a^{R,\delta } \cdot \nabla _\eta g)_\varepsilon \right) +O(\varepsilon )+O(\delta )+O(1/R). \end{aligned}$$

Proof

We want to take advantage from the fact that we have $|\xi -\sigma _{\Sigma } (\xi )|>R\varepsilon$ on the support of $a^{R,\delta }_\varepsilon$ to avoid the singularity of the function g. Indeed, $a^{R,\delta }_\varepsilon$ is supported outside the singularity of the function g. Let $\kappa \in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^d)$ supported outside 0 and such that $(1-\chi )\kappa =(1-\chi )$ and denote by $\kappa ^R$ the function defined by $\kappa ^R(\eta )=\kappa (\eta /R)$ so that

$$\begin{aligned} \kappa _\varepsilon ^R(\xi )= \kappa \left( \frac{\xi -\sigma _{X} (\xi )}{R\varepsilon }\right) . \end{aligned}$$

Standard symbolic calculus gives that for all $N\in {{\mathbb {N}}}$, we have

$$\begin{aligned} \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon )&= \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon )\kappa _\varepsilon ^R(\varepsilon D)+R^{-N} \mathrm{op}_\varepsilon ( r^{1,N} _\varepsilon (x,\xi ))\\&=\kappa _\varepsilon ^R(\varepsilon D) \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon )+R^{-N} \mathrm{op}_\varepsilon ( r^{2,N} _\varepsilon (x,\xi )) \end{aligned}$$

where the symbols $r^{j,N}_\varepsilon$ have symbol norms that are uniformly bounded and are supported in the set $\{c_0 \varepsilon R< \xi -\sigma _{X} (\xi )<C_0\varepsilon R\}$ for some $0<c_0<C_0$. In particular, by Proposition C.1 and because $g\in {\mathcal {H}}_{k+1}$, $a\in {{\mathcal {A}}}^{(2)}_{-k}$,

$$\begin{aligned} \mathrm{op}_\varepsilon ( r^{1,N}_\varepsilon ) g_\varepsilon (\varepsilon D)= \mathrm{op}_\varepsilon ( r^{1,N}_\varepsilon g_\varepsilon ) +O(R^{k+1-k})= \mathrm{op}_\varepsilon (r^{1,N}_\varepsilon g_\varepsilon ) +O(R), \end{aligned}$$

whence

$$\begin{aligned} \mathrm{op}_\varepsilon ( r^{1,N}_\varepsilon ) g_\varepsilon (\varepsilon D)= O(R). \end{aligned}$$

Similarly, we have

$$\begin{aligned} g_\varepsilon (\varepsilon D) \mathrm{op}_\varepsilon ( r^{2,N}_\varepsilon ) = O(R). \end{aligned}$$

We deduce that

$$\begin{aligned} \left[ \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) ,g_\varepsilon (\varepsilon D)\right] = \left[ \mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) ,(\kappa ^Rg)_\varepsilon (\varepsilon D)\right] +O( R^{-N+1}). \end{aligned}$$

The function $\xi \mapsto \kappa ^R g$ is now smooth, which allows to use standard results of symbolic calculus, what we shall do in local coordinates.

We consider a system of local coordinates $\varphi (\xi )=0$ of X and the $d\times p$ smooth matrix $B(\xi )$ such that

$$\begin{aligned} \xi -\sigma _X(\xi )= B(\xi ) \varphi (\xi ) \end{aligned}$$

where $\varphi (\xi )\in {{\mathbb {C}}}^{p\times 1}$ is a column. We associate with $\varphi$ the diffeomorphism

$$\begin{aligned} \Phi : (\,^t\varphi (\xi ),\xi '')\mapsto \xi \end{aligned}$$

and, according to Lemma 4.3 in [15], there exists an isometry ${{\mathcal {U}}}_\varepsilon$ of $L^2({{\mathbb {R}}}^d)$ such that for all $b\in {{\mathcal {A}}}^{(2)}$ and $f\in L^2({{\mathbb {R}}}^d)$

$$\begin{aligned} (\mathrm{op}_\varepsilon (b_\varepsilon ) f,f)= \left( \mathrm{op}_\varepsilon \left( b\left( \, ^t d\Phi (\xi )^{-1}x, \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) {{\mathcal {U}}}_\varepsilon f,{{\mathcal {U}}}_\varepsilon f\right) +O(\varepsilon ). \end{aligned}$$

Note that if $\xi ,\zeta ,x\in {{\mathbb {R}}}^d$,

$$\begin{aligned} d\Phi (\xi )^{-1} \zeta = (d\varphi (\Phi (\xi )) \zeta ,\zeta ''),\;\; \, ^td\Phi (\xi )^{-1} x= \, ^td\varphi (\Phi (\xi )) x'+(0,x''), \end{aligned}$$

where $d\varphi (\xi )$ is the $p\times d$ matrix with lines the gradient of each of the component of $\varphi$. Therefore, focusing on the commutator

$$\begin{aligned} L^\varepsilon = \left[ \mathrm{op}_\varepsilon \left( a^{R,\delta }\left( \, ^t d\Phi (\xi )^{-1}x, \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) ,(\kappa ^R g)\left( \Phi (\varepsilon D), B(\Phi (\varepsilon D) )D_{x'} \right) \right] , \end{aligned}$$

we obtain, in ${\mathcal {L}}(L^2({{\mathbb {R}}}^d))$, that

$$\begin{aligned} L^\varepsilon&= \mathrm{op}_\varepsilon \left( d\varphi (\Phi (\xi )) \nabla _{x} a^{R,\delta } \left( \, ^t d\Phi (\xi )^{-1}x, \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \cdot \,^tB(\Phi (\xi ) )\nabla _\eta (\kappa ^Rg)\right. \\&\qquad \left. \left( \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) + O(R^{-1})+O(\varepsilon ). \end{aligned}$$

We observe that if $\sigma \in X$, $B(\sigma ) d\varphi (\sigma )=\mathrm{Id} -d\sigma _X(\sigma )$ and $d\sigma _X(\sigma ) \nabla _\eta g(\sigma ,\eta )=0$ because $\nabla _\eta g(\sigma ,\eta )\in N_\sigma X$ and $d\sigma _X(\sigma )\zeta =0$ if $\zeta \in N_\sigma X$. We deduce

$$\begin{aligned}{}[\mathrm{op}_\varepsilon (a^{R,\delta }_\varepsilon ) ,g_\varepsilon (\varepsilon D)]= i \mathrm{op}_\varepsilon (b_\varepsilon ) +O(\varepsilon ) +O(R^{-1})+O(\delta ), \end{aligned}$$

with

$$\begin{aligned} b(x,\xi ,\eta )= \kappa ^R (\eta )\nabla _x a^R(x,\xi ,\eta ) \cdot \nabla _\eta g\left( \xi ,\eta \right) \in {{\mathcal {A}}}^{(2)}_{0}. \end{aligned}$$

Using $\kappa (1-\chi )=1-\chi$, we obtain

$$\begin{aligned} b(x,\xi ,\eta )= \nabla _x a^R(x,\xi ,\eta ) \cdot \nabla _\eta g\left( \xi ,\eta \right) . \end{aligned}$$

$\square$

We shall also need properties of two-scale symbolic calculus at finite distance, i.e. for symbols that are compactly supported in all the variables, including the variable $\eta$. Here again, the use of local coordinates and Lemma 4.3 in [15] are a crucial argument.

Lemma D.2

Let $a\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{3d})$ and $g\in {\mathcal {H}}_k$ for $k\in {{\mathbb {N}}}$. Let $(f^\varepsilon )_{\varepsilon >0}$ a bounded family in $L^2({{\mathbb {R}}}^d)$ and $Md\nu$ the two-scale Wigner measure at finite distance associated with its concentration on X. Then, there exists a constant $C>0$ such that for all $\varepsilon >0$,

$$\begin{aligned} \left( \mathrm{op}_\varepsilon (a) g_\varepsilon (\varepsilon D) f^\varepsilon ,f^\varepsilon \right) \le C\Vert f^\varepsilon \Vert _{ L^2({{\mathbb {R}}}^d)}. \end{aligned}$$

In addition, up to the subsequence defining $Md\nu$,

$$\begin{aligned} \left( \mathrm{op}_\varepsilon (a) g_\varepsilon (\varepsilon D) f^\varepsilon ,f^\varepsilon \right) \mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0} \int _{TX^*} \mathrm{Tr} _{L^2(N_\sigma X)}(Q^X_a(v,\sigma ) Q^X_g(\sigma ) M(\sigma ,v) )d\nu (\sigma ,v). \end{aligned}$$

We recall that the notations of this section have been introduced in Section 4.1. Of course, this lemma has standard generalizations to vector-valued families and to time dependent families which are bounded in $L^\infty ({{\mathbb {R}}}, L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$.

Proof

Here again, we work in local coordinates $\varphi (\xi )=0$ of X and we consider the $d\times p$ smooth matrix $B(\xi )$ such that

$$\begin{aligned} \xi -\sigma _X(\xi )= B(\xi ) \varphi (\xi ) \end{aligned}$$

where $\varphi (\xi )\in {{\mathbb {C}}}^{p\times 1}$ is a column. We associate with $\varphi$ the diffeomorphism

$$\begin{aligned} \Phi : (\,^t\varphi (\xi ),\xi '')\mapsto \xi \end{aligned}$$

and, we consider the isometry ${{\mathcal {U}}}_\varepsilon$ of $L^2({{\mathbb {R}}}^d)$ given by Lemma 4.3 in [15] such that for all $b\in {{\mathcal {A}}}^{(2)}$ and $f\in L^2({{\mathbb {R}}}^d)$

$$\begin{aligned} (\mathrm{op}_\varepsilon (b_\varepsilon ) f,f)= \left( \mathrm{op}_\varepsilon \left( b\left( \, ^t d\Phi (\xi )^{-1}x, \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) {{\mathcal {U}}}_\varepsilon f,{{\mathcal {U}}}_\varepsilon f\right) +O(\varepsilon ). \end{aligned}$$

We then concentrate on the operator

$$\begin{aligned} \mathrm{op}_\varepsilon \left( a\left( \, ^t d\Phi (\xi )^{-1}x, \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) \mathrm{op}_\varepsilon \left( g\left( \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) . \end{aligned}$$

Since a is compactly supported in all variables, we obtain, in ${\mathcal {L}}(L^2({{\mathbb {R}}}^d))$, that

$$\begin{aligned}&\mathrm{op}_\varepsilon \left( a\left( \, ^t d\Phi (\xi )^{-1}x, \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) \\&\qquad = \mathrm{op}_\varepsilon \left( a\left( \, ^t d\Phi (0,\xi '')^{-1}x, \Phi (0,\xi ''), B(\Phi (0,\xi '') )\frac{\xi '}{\varepsilon }\right) \right) + O(\varepsilon )\\&\qquad = \mathrm{op}_\varepsilon \left( a^W\left( \, ^t d\Phi (0,\xi '')^{-1}x, \Phi (0,\xi ''), B(\Phi (0,\xi '') )D_{x'} \right) \right) + O(\varepsilon ). \end{aligned}$$

One can then interpret this operator as an operator acting on $L^2({{\mathbb {R}}}^d_{x''}, L^2({{\mathbb {R}}}^d_{x'}))$ where, as explained in Section 4.1 of [15], for any $(\sigma ,v)=((0,\xi ''), (0,x''))\in TX^*$, the map

$$\begin{aligned} (z,\zeta )\mapsto a\left( \, ^t d\Phi (0,\xi '')^{-1}(z,x''), \Phi (0,\xi ''), B(\Phi (0,\xi '') )\zeta \right) \end{aligned}$$

defines a function on $T^*(N_\sigma X)$, which implies that the operator

$$\begin{aligned} a^W\left( \, ^t d\Phi (0,\xi '')^{-1}x, \Phi (0,\xi ''), B(\Phi (0,\xi '') )D_{x'} \right) \end{aligned}$$

acts on $L^2(N_\sigma X)$. It is the expression of the operator $Q_a(\sigma , v)$ in the local coordinates induced by the choice of equations $\varphi (\xi )=0$ of X.

The difficulty with $g_\varepsilon (\varepsilon D)$ is that the map $(\sigma ,\eta ) \mapsto g(\sigma ,\eta )$ is not bounded in $\eta$. Therefore, we decompose $g(\sigma ,\eta )$ into two parts thanks to a function $\chi \in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^d)$ such that $\chi =1$ on B(0, 1) and $\chi =0$ on $B(0,2)^c$ with $0\le \chi \le 1$. Writing like before $\chi ^R(\eta )=\chi (\eta /R)$ for $R>0$, we set

$$\begin{aligned} g=g\chi ^R + g(1-\chi ^R) \end{aligned}$$

and we first focus on $\mathrm{op}_\varepsilon (a_\varepsilon ) (g(1-\chi ^R))_\varepsilon (\varepsilon D)$. Since now $g(1-\chi _R)$ is smooth, we can use standard symbolic calculus, and we have

$$\begin{aligned} \mathrm{op}_\varepsilon (a_\varepsilon ) (g(1-\chi ^R))_\varepsilon (\varepsilon D)= \mathrm{op}_\varepsilon ((ag(1-\chi ^R))_\varepsilon ) + O(R^{-1}) \end{aligned}$$

because $\nabla _\xi ( (g(1-\chi ^R))_\varepsilon )= O(R^{-1})$. Moreover, as soon as R is large enough, we have $ag(1-\chi ^R)=0$. We conclude

$$\begin{aligned} g_\varepsilon (\varepsilon D)= (g\chi ^R)_\varepsilon (\varepsilon D) +O(R^{-1}) \end{aligned}$$

and

$$\begin{aligned}&\mathrm{op}_\varepsilon \left( g\left( \Phi (\xi ), B(\Phi (\xi ) )\frac{\xi '}{\varepsilon }\right) \right) = \mathrm{op}_\varepsilon \left( (g\chi ^R)(\Phi (0,\xi ''), B(\Phi (0,\xi '') ) D_{x'})\right) \\&\qquad +O_R(\varepsilon ) +O(R^{-1}) \end{aligned}$$

in ${\mathcal {L}}(L^2({{\mathbb {R}}}^d))$. Note that in the following we will let first $\varepsilon$ go to 0, and then R to $+\infty$, so that $O_R(\varepsilon )$ is negligible. Besides, when R goes to $+\infty$, we are left with the operator $g(\Phi (0,\xi ''), B(\Phi (0,\xi '') D_{x'})$ (with strong convergence), which is the expression in local coordinates of the operator $Q_g(\sigma )$.

At this stage of the proof, we are left with the quantity

$$\begin{aligned}&\Big ( \mathrm{op}_\varepsilon \left( a^W\left( \, ^t d\Phi (0,\xi '')^{-1}x, \Phi (0,\xi ''), B(\Phi (0,\xi '') )D_{x'} \right) \right) \mathrm{op}_\varepsilon (g(\Phi (0,\xi ''), \nonumber \\&B(\Phi (0,\xi '') ) D_{x'})) {{\mathcal {U}}}_\varepsilon f,{{\mathcal {U}}}_\varepsilon f\Big ). \end{aligned}$$

It turns out that the pair $M d\nu$ has been defined in [15] (see Proposition 4.2) as a semi-classical measure of the family $({{\mathcal {U}}}_\varepsilon f)$, which is a bounded family in $L^2({{\mathbb {R}}}^{d-p}_{x''}, L^2({{\mathbb {R}}}^p_{x'}))$. Therefore, in coordinates

$$\begin{aligned}&\left( \mathrm{op}_\varepsilon \left( a^W\left( \, ^t d\Phi (0,\xi '')^{-1}x, \Phi (0,\xi ''), B(\Phi (0,\xi '') )D_{x'} \right) \right) \right. \\&\qquad \left. \mathrm{op}_\varepsilon (g(\Phi (0,\xi ''), B(\Phi (0,\xi '') ) D_{x'})) {{\mathcal {U}}}_\varepsilon f,{{\mathcal {U}}}_\varepsilon f\right) \\&\mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0} \int _{ {{\mathbb {R}}}^{2(d-p)}} \mathrm{Tr}_{L^2({{\mathbb {R}}}^p)} \\&\qquad \left( \mathrm{op}_\varepsilon \left( a^W\left( \, ^t d\Phi (0,\xi '')^{-1}x, \Phi (0,\xi ''), B(\Phi (0,\xi '') )D_{x'} \right) \right) M(\xi '',x'') \right) \mathrm{d}\nu (\xi '',x'')\\&\qquad \qquad = \int _{TX^*} \mathrm{Tr} _{L^2(N_\sigma X)}(Q^X_a(v,\sigma ) Q^X_g(\sigma ) M(\sigma ,v) )\mathrm{d}\nu (\sigma ,v). \end{aligned}$$

$\square$

Appendix E. Well-Prepared Data

We prove here properties of well-prepared initial (1.17). We shall use the Fourier coefficients of the Bloch waves $\varphi _n(y,\xi )$ for $n\in {{\mathbb {N}}}^*$. Recall that they satisfy the Bloch periodicity condition:

$$\begin{aligned} \varphi _n(y,\xi +j) = \mathrm{e}^{-i j\cdot y} \varphi _n(y,\xi ),\;\;\forall j\in 2\pi {{\mathbb {Z}}}^d. \end{aligned}$$

(E.1)

Integrating (E.1) with respect to y on ${{\mathbb {T}}}^d$ gives an expression for the Fourier coefficients of $\varphi _n(\cdot ,\xi )$: for all $y\in {{\mathbb {T}}}^d$ and $\xi \in {{\mathbb {R}}}^d$

$$\begin{aligned} \varphi _n(y,\xi )=\sum _{j\in 2\pi {{\mathbb {Z}}}^d} c_n(\xi + j)\mathrm{e}^{i j\cdot y},\;\; c_n(\xi )=\int _{{{\mathbb {T}}}^d} \varphi _n(y,\xi ) dy. \end{aligned}$$

(E.2)

For $s>0$, there exists $C_{n,s}>0$ such that

$$\begin{aligned} \sum _{j\in 2\pi {{\mathbb {Z}}}^d} \langle j\rangle ^{2s} |c_n(\xi +j)|^2 \le C_{n,s} \langle \xi \rangle ^{2s} \end{aligned}$$

(E.3)

It turns out that these class of data are closely related to those studied in [1, 2].

Lemma E.1

Let $( \psi ^\varepsilon _{0,n})_{\varepsilon >0}$ as in (1.17) with $u^\varepsilon _n= \mathrm{e}^{\frac{i}{\varepsilon }\xi _n\cdot x}v^\varepsilon _n(x)$, $v_n^\varepsilon$ uniformly bounded in $H^s({{\mathbb {R}}}^d)$, $s>d/2$ (or $s>1$ when $d=1$). Assume that $\xi \mapsto \varphi _n(\cdot , \xi )$ is Lipschitz. Then, we have, in $L^2({{\mathbb {R}}}^d)$,

$$\begin{aligned} \psi ^\varepsilon _{0,n}= \mathrm{e}^{\frac{i}{\varepsilon }x\cdot \xi _n}\varphi _n\left( \frac{x}{\varepsilon },\xi _n\right) v^\varepsilon _n+O(\varepsilon ). \end{aligned}$$

Besides, if $v^\varepsilon _n \rightarrow v_n$ in $L^2$, then $\Vert \psi ^\varepsilon _{0,n}\Vert _{L^2}= \Vert v_n\Vert _{L^2}+o(1)$ as $\varepsilon$ goes to 0.

Proof

We set ${{\widetilde{\psi }}}^\varepsilon _{0,n}= \mathrm{e}^{\frac{i}{\varepsilon }x\cdot \xi _n}\varphi _n\left( \frac{x}{\varepsilon },\xi _n\right) v^\varepsilon _n$. We write for $x\in {{\mathbb {R}}}^d$

$$\begin{aligned} \psi ^\varepsilon _{0,n}(x)=\sum _{j\in 2\pi {{\mathbb {Z}}}^d} \mathrm{e}^{\frac{i}{\varepsilon }j\cdot x}c_n(j+\varepsilon D_x) u^\varepsilon _n(x). \end{aligned}$$

Let $\theta \in L^2({{\mathbb {R}}}^d)$, we write $(\theta ,\psi ^\varepsilon _{0,n})_{L^2({{\mathbb {R}}}^d)} = \sum _{j\in 2\pi {{\mathbb {Z}}}^d} \alpha ^\varepsilon _j$ with

$$\begin{aligned} \alpha ^\varepsilon _j= \left( \theta ,\mathrm{e}^{\frac{i}{\varepsilon }j\cdot x}c_n(j+\varepsilon D_x) u^\varepsilon _n \right) _{L^2} =\left( \theta ,\mathrm{e}^{\frac{i}{\varepsilon }(j+\xi _n)\cdot x}c_n(j+\xi _n+\varepsilon D_x) v^\varepsilon _n \right) _{L^2}. \end{aligned}$$

By (E.3), there exists $C>0$ such that

$$\begin{aligned} |\alpha ^\varepsilon _j| \le C \langle j\rangle ^{-s} \Vert \langle \varepsilon D_x\rangle ^s u^\varepsilon _n\Vert _{L^2}\Vert \theta \Vert _{L^2}. \end{aligned}$$

Therefore, the sum enters into the frame of Lebesgue dominated convergence. The same holds for the series

$$\begin{aligned} (\theta ,{{\widetilde{\psi }}}^\varepsilon _{0,n})_{L^2({{\mathbb {R}}}^d)} = \sum _{j\in 2\pi {{\mathbb {Z}}}^d} {{\widetilde{\alpha }}}^\varepsilon _j,\;\; {{\widetilde{\alpha }}}^\varepsilon _j= \left( \theta ,\mathrm{e}^{\frac{i}{\varepsilon }(j+\xi _n)\cdot x}c_n(j+\xi _n) v^\varepsilon _n \right) _{L^2}. \end{aligned}$$

Besides

$$\begin{aligned} \alpha ^\varepsilon _j- {{\widetilde{\alpha }}} ^\varepsilon _j= \left( \theta , \mathrm{e}^{\frac{i}{\varepsilon }(j+\xi _n)\cdot x}(c_n(j+\xi _n+\varepsilon D_x)-c_n(j+\xi _n)) v^\varepsilon _n \right) _{L^2}. \end{aligned}$$

The conclusion then comes from the observation that $\xi \mapsto c_n(\xi )$ is uniformly Lipschitz and thus, one has

$$\begin{aligned} \left| \alpha ^\varepsilon _j- {{\widetilde{\alpha }}} ^\varepsilon _j\right| \le \Vert \theta \Vert _{L^2} \Vert (c_n(j+\xi _n+\varepsilon D_x)-c_n(j+\xi _n)) v^\varepsilon _n\Vert _{L^2} \le C\varepsilon \Vert \theta \Vert _{L^2} \Vert D_x v^\varepsilon _n\Vert _{L^2} \end{aligned}$$

which gives the result by the boundedness of the family $(v^\varepsilon _n)_{\varepsilon >0}$ in $H^s({{\mathbb {R}}}^d)$ with $s\ge 1$.

Let us now compute the norm of $\psi ^\varepsilon _{0,n}$. One has $\Vert \psi ^\varepsilon _{0,n}\Vert _{L^2}^2=\sum _{j,j'\in 2\pi {{\mathbb {Z}}}^d} \beta ^\varepsilon _{j,j'}$ with

$$\begin{aligned} \beta ^\varepsilon _{j,j'}&= \left( \mathrm{e}^{\frac{i}{\varepsilon }\cdot x (j-j')} c_n(j+\xi _n+\varepsilon D_x) v^\varepsilon _n(x) ,c_n(j'+\xi _n+\varepsilon D_x ) v^\varepsilon _n(x)\right) _{L^2}. \end{aligned}$$

Additionally, by (E.3), there exists $C>0$ such that

$$\begin{aligned} |\beta ^\varepsilon _{j,j'}|\le C \langle j\rangle ^{-s}\langle j'\rangle ^{-s} \Vert \langle \varepsilon D_x\rangle ^s v^\varepsilon _n\Vert ^2_{L^2}. \end{aligned}$$

Therefore, the sum enters into the frame of Lebesgue dominated convergence. If $|j-j'|\ge 1$ with $j_\ell -j'_\ell \not =0$, $0\le \ell \le d$, an integration by parts give

$$\begin{aligned} \beta ^\varepsilon _{j,j'}= & {} i \varepsilon (j_\ell -j'_\ell )^{-1} \int _{{{\mathbb {R}}}^d} \mathrm{e}^{\frac{i}{\varepsilon }\cdot x (j-j')}\\&\partial _{x_\ell }\left( c_n(j+\xi _n+\varepsilon D_x) v^\varepsilon _n(x) \overline{c_n(j'+\xi _n+\varepsilon D_x ) v^\varepsilon _n(x)}\right) \mathrm{d}x, \end{aligned}$$

whence $\beta ^\varepsilon _{j,j'}\rightarrow 0$ as $\varepsilon \rightarrow 0$ since $(v^\varepsilon _n)_{\varepsilon >0}$ is uniformly bounded in $H^1({{\mathbb {R}}}^d)$. Moreover

$$\begin{aligned} \sum _{j\in 2\pi {{\mathbb {Z}}}^d}\beta ^\varepsilon _{j,j}= \sum _{j\in 2\pi {{\mathbb {Z}}}^d} (2\pi )^{-d}\int _{{{\mathbb {R}}}^d} |c_n(j+\xi _n+\varepsilon \xi )|^2 |{{\widehat{v}}}^\varepsilon _n (\xi ) |^2 \mathrm{d}\xi . \end{aligned}$$

Using $\sum _{j\in 2\pi {{\mathbb {Z}}}^d} |c_n(\xi )|^2=\Vert \varphi _n(\cdot ,\xi )\Vert _{L^2({{\mathbb {T}}}^d)}^2=1$, we deduce

$$\begin{aligned} \sum _{j\in 2\pi {{\mathbb {Z}}}^d}\beta ^\varepsilon _{j,j}= (2\pi )^{-d}\int _{{{\mathbb {R}}}^d}|v^\varepsilon _n (\xi ) |^2 \mathrm{d}\xi =\Vert v^\varepsilon _n\Vert ^2_{L^2}\mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0} \Vert v_n\Vert ^2 \end{aligned}$$

and the conclusion follows. $\quad \square$

Let us now examine weak limits of such families.

Lemma E.2

Let $( \psi ^\varepsilon _{0,n})_{\varepsilon >0}$ as in (1.17) with $u^\varepsilon _n= \mathrm{e}^{\frac{i}{\varepsilon }\xi _n\cdot x}v_n(x)$, $(v_n^\varepsilon )_{\varepsilon >0}$ bounded in $H^s({{\mathbb {R}}}^d)$ with $s>d/2$ (or $s>1$ when $d=1$). Assume one has as $\varepsilon$ goes to 0

$$\begin{aligned} v_n^\varepsilon \rightharpoonup v_n \;\; \text{ in }\;\; L^2. \end{aligned}$$

Then, the weak limits of $\left( \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot \xi }\psi ^\varepsilon _{0,n}\right) _{\varepsilon >0}$ are non zero if and only if $\xi \in \xi _{n}+2 \pi {{\mathbb {Z}}}^d$. Besides, for any $j\in 2\pi {{\mathbb {Z}}}^d$, one has as $\varepsilon$ goes to 0

$$\begin{aligned} \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot (\xi _n+j)}\psi ^\varepsilon _{0,n} \rightharpoonup c_n(\xi _n+j) v_{n} \;\; \text{ in }\;\; L^2. \end{aligned}$$

Proof

One writes

$$\begin{aligned} \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot \xi }\psi ^\varepsilon _{0,n}&= \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot \xi }\sum _{j\in 2\pi {{\mathbb {Z}}}^d} \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot j} c_n(\varepsilon D_x+j) \mathrm{e}^{\frac{i}{\varepsilon }x\cdot \xi _n} v^\varepsilon _n \\&= \sum _{j\in 2\pi {{\mathbb {Z}}}^d} \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot (j+\xi -\xi _n)} c_n(\xi _n+j +\varepsilon D_x) v^\varepsilon _n \end{aligned}$$

and the result follows. $\quad \square$

As a corollary, we obtain a description of the concentration of these families on subsets of interest.

Corollary E.3

Let X be a $2\pi {{\mathbb {Z}}}^d$ periodic subset X of ${{\mathbb {R}}}^d$ consisting of isolated points and containing $\xi _n$.

• (1) Let $M^Xd\nu ^X$ be the two-microlocal measures at finite distance associated with the concentration of $(\psi ^\varepsilon _{0,n})_{\varepsilon >0}$ on X. Then,

  $$\begin{aligned} \nu ^X=\Vert v_n\Vert ^2_{L^2}\sum _{j\in 2\pi {{\mathbb {Z}}}^d}|c_n(\xi _n+j)|^2 \delta (\xi -\xi _n-j) \end{aligned}$$

  and, for $j\in 2\pi {{\mathbb {Z}}}^d$, the operator $M^X(\xi _{n} +j)$ is the projector of $L^2({{\mathbb {R}}}^d)$ on ${{\mathbb {C}}}v_{n}$.

• (2) Consider the family of $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^2)$, $\Psi _{n,n'}^\varepsilon =\, ^t\left( \psi ^\varepsilon _{0,n},\psi ^\varepsilon _{0,n'}\right)$ with $n\not =n'$ and $\xi _{n'}=\xi _n$. Let $M^Xd\nu ^X$ be the two-microlocal measures at finite distance associated with the concentration of $\Psi _{n,n'} ^\varepsilon$ on X. Then

  $$\begin{aligned} \nu ^X=\left( \Vert v_n\Vert ^2_{L^2} + \Vert v_{n+1}\Vert ^2_{L^2}\right) \sum _{j\in 2\pi {{\mathbb {Z}}}^d}|c_n(\xi _n+j)|^2 \delta (\xi -\xi _n-j) \end{aligned}$$

  and, for $j\in 2\pi {{\mathbb {Z}}}^d$, the operator $M^X(\xi _{n} +j)$ is the projector of $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^2)$ on ${{\mathbb {C}}}\,^t(v_{n},v_{n+1})$.

Proof

Since $\xi _n$ is an isolated point of X, by Remark 4.2, $M^X$ is the orthogonal projector on a weak limit in $L^2$ of

$$\begin{aligned} x\mapsto \mathrm{e}^{-\frac{i}{\varepsilon }x\cdot \xi }\varphi _{n}\left( \frac{x}{\varepsilon }, \varepsilon D_x\right) u^\varepsilon _{n}. \end{aligned}$$

By Lemma E.2, any of this weak limit is 0 if $\xi \notin \xi _{n}+2\pi {{\mathbb {Z}}}^d$ and if $\xi =\xi _{n}+j$ with $j\in 2\pi {{\mathbb {Z}}}^d$, then the limit is $c_{n_0}(\xi _{n_0}+j)v_{n_0}$ whence we get the Part 1 of the result. The proof of Part 2 follows the same lines. $\quad \square$