Wigner measures and effective mass theorems

Chabu, Victor; Fermanian Kammerer, Clotilde; Macià, Fabricio

Annales Henri Lebesgue, Tome 3 (2020), pp. 1049-1089.

Nous étudions une équation de Schrödinger qui décrit la dynamique d’un électron dans un crystal en présence d’impuretés et nous considérons des longueurs d’onde de la taille des cellules du crystal. Lorsque la donnée initiale satisfait à des hypothèses ad-hoc, il est bien connu que l’on peut rendre compte des propriétés de la fonction d’onde en considérant la solution d’une équation de Schrödinger plus simple, appelée équation de masse effective. En utilisant la décomposition de Floquet–Bloch, comme il est classique dans ce domaine, nous exhibons des équations de masse effective dans un cadre plus général que dans les travaux antérieurs, en autorisant notamment des dégénérescences des points critiques des bandes de Bloch (ce qui ne peut arriver qu’en dimension plus grande que 1). Notre analyse repose sur l’utilisation des mesures de Wigner et leur application à l’analyse de la dispersion dans des edp-s et aboutit à l’introduction d’équations de masse effective de type Heisenberg.

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 wave-lengths 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.

Classification : 35B27, 58J40, 81S30, 81Q20
Mots clés : Bloch modes, semi-classical analysis on manifolds, Wigner measures, two-microlocal measures, Effective mass theory

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