Propagation of Wigner functions for the Schrödinger equation with a perturbed periodic potential
S. Teufel, G. Panati
In P. Blanchard, G. Dell'Antonio, (Hrsg.), Multiscale Methods in Quantum Mechanics. Birkhäuser, 2004.
[BlochRomArch.ps (347.9 kB)]
Zusammenfassung
Let VΓ be a lattice periodic potential and A and φ 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=½(-i∇x - A(εx))2 + VΓ(x) + φ(εx) propagates along the flow of the semiclassical model of solid states physics up an error of order ε. If ε-dependent corrections to the flow are taken into account, the error is improved to order ε2. 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