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

TL;DR

This paper demonstrates that the Wigner function for the Schrödinger equation with a perturbed periodic potential propagates approximately along a semiclassical flow, with errors controlled by the small parameter epsilon, extending previous Egorov-type results.

Contribution

It establishes the propagation of Wigner functions in perturbed periodic potentials with precise error estimates, improving understanding of semiclassical limits in solid state physics.

Findings

Wigner function propagates along semiclassical flow with order epsilon error

Including epsilon-dependent corrections reduces error to order epsilon^2

Results extend previous Egorov theorem to perturbed periodic potentials

Abstract

Let $V_\Gamma$ be a lattice periodic potential and $A$ and $\phi$ 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 $H = {1/2} (-\I\nabla_x - A(\epsilon x))^2 + V_\Gamma (x) + \phi(\epsilon x)$ 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).