Propagation of Wigner functions for the Schroedinger equation with a perturbed periodic potential

Abstract

Let $V_{\Gamma }$ be a lattice periodic potential and $A$ and $\varphi$ external electromagnetic potentials which vary slowly on the scale set by the lattice spacing. It is shown that the Wigner function of a solution of the Schroedinger equation with Hamiltonian operator $H = {1/2} (-\I\nabla_x - A(εx))^2 + V_Γ(x) + ϕ(εx)$ propagates along the flow of the semiclassical model of solid states physics up an error of order $\epsilon$. If $\epsilon$-dependent corrections to the flow are taken into account, the error is improved to order ${\epsilon }^2$. We also discuss the propagation of the Wigner measure. The results are obtained as corollaries of an Egorov type theorem proved in a previous paper (math-ph/0212041).

arXiv comment14 pages; to appear in the proceedings of the conference "Multiscale methods in Quantum Mechanics", Accademia dei Lincei, Roma (Italy), December 16-20, 2002