LJKDeterministic Models and Algorithms: EDPMOISE Seminar

On Thursday March 16 2006 at 11h00 in Salle 1  Tour IRMA

Seminary of Grégoire ALLAIRE (CMAP, Ecole Polytechnique)

Homogénéisation de l'équation de Schrödinger : théorème de masse effective.

Summary

We study the homogenization of a Schrödinger equation with a large periodic potential. The chosen scaling is not that of semiclassical analysis but instead corresponds to longer time scales. In other words we look for a limit after the usual effective hamiltonian regime.
We obtain a rigorous derivation of socalled effective mass theorems in solid state physics. More precisely, for wellprepared initial data concentrating on a Bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating Bloch eigenfunction and of a slowly varying solution of an homogenized Schrödinger equation. The homogenized coefficients depend on the chosen Bloch eigenvalue, and the homogenized solution may experience a large drift. The homogenized limit may be a system of equations having dimension equal to the multiplicity of the Bloch eigenvalue. Our method is based on a combination of classical homogenization techniques (twoscale convergence and suitable oscillating test functions) and of Bloch waves decomposition.
