We look at the long-time behavior of solutions to a semi-classical Schrödinger equation on the torus. We consider time scales which go to infinity when the semi-classical parameter goes to zero and we associate with each time-scale the set of semi-classical measures associated with all possible choices of initial data. On each classical invariant torus, the structure of semi-classical measures is described in terms of two-microlocal measures, obeying explicit propagation laws. We apply this construction in two directions. We first analyze the regularity of semi-classical measures, and we emphasize the existence of a threshold: for time-scales below this threshold, the set of semi-classical measures contains measures which are singular with respect to Lebesgue measure in the "position" variable, while at (and beyond) the threshold, all the semi-classical measures are absolutely continuous in the "position" variable, reflecting the dispersive properties of the equation. Second, the techniques of two-microlocal analysis introduced in the paper are used to prove semiclassical observability estimates. The results apply as well to general quantum completely integrable systems.

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We study a Schrödinger equation which describes the dynamics of an electron in a crystal in the presence of impurities. We consider the regime of small wavelengths comparable to the characteristic scale of the crystal. It is well-known that under suitable assumptions on the initial data and for highly oscillating potential, the wave function can be approximated by the solution of a simpler equation, the effective mass equation. Using Floquet-Bloch decomposition, as it is classical in this subject, we establish effective mass equations in a rather general setting. In particular, Bloch bands are allowed to have degenerate critical points, as may occur in dimension strictly larger than one. Our analysis leads to a new type of effective mass equations which are operator-valued and of Heisenberg form and relies on Wigner measure theory and, more precisely, to its applications to the analysis of dispersion effects.

Résumé. -L'étude de la dynamique semi-classique d'électrons dans un cristal débouche naturellement sur le problème de l'évolution des mesures semi-classiques en présence d'un croisement de modes. Dans ce travail, nousétudions un système 2 × 2 qui présente un tel croisement.À cet effet, nous introduisons des mesures semiclassiquesà deuxéchelles qui décrivent comment la transformée de Wigner usuelle se concentre sur l'ensemble des trajectoires rencontrant ce croisement. Puis nouś etablissons des formules explicites de type Landau-Zener reliant les traces de ces mesures de part et d'autre du croisement.Abstract (Semi-classical measures and eigenvalue crossings). -Semiclassical study of multidimensional crystals leads naturally to the following question: how do semiclassical measures propagate through energy level crossings ? In this contribution, we discuss a simple 2 × 2 system which displays such a crossing. For that purpose, we introduce two-scaled semi-classical measures, which describe how the usual Wigner transforms are concentrating on trajectories passing through the crossing points. Then we derive explicit formulae for the branching of such measures. These formulae are generalizations of the so-called Landau-Zener formulae. Texte reçu le 6 avril 2001, accepté le 3 juillet 2001