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

TL;DR

This paper analyzes the semiclassical limit of the Schrödinger equation with a short-scale periodic potential and an external potential, showing convergence of quantum observables to classical dynamics as the scale parameter approaches zero.

Contribution

It provides a rigorous proof of the convergence of quantum observables to semiclassical dynamics in the presence of a short-scale periodic potential and a slowly varying external potential.

Findings

Strong convergence of the position operator in the semiclassical limit

Semiclassical observables follow classical dynamics as epsilon approaches zero

Results applicable to quantum systems with short-scale periodic structures

Abstract

We consider the dynamics generated by the Schroedinger operator $H=-{1/2}\Delta + V(x) + W(\epsi x)$, 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.