Semiclassical limit for the Schroedinger equation with a short scale periodic potential

Abstract

We consider the dynamics generated by the Schroedinger operator $H=-{1/2}Δ+ V(x) + W(\epsi x)$, where $V$ is a lattice periodic potential and $W$ an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit $\epsi \to 0$ the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics.