Dynamical self-averaging for a lattice Schrödinger equation with weak random potential

Abstract

We study the kinetic, weak coupling limit of the dynamics governed by a discrete random Schrödinger operator on ${\mathbb{Z}}^3$. For sequences of ${\ell }^2\left({\mathbb{Z}}^3\right)$-bounded initial states and convergent initial Wigner transform, we prove that the scaled Wigner transform converges to the solution of a linear Boltzmann equation in $L^r\left(\mathbb{P}\right)$for all $r>0$, thus considerably strengthening a previous result by Chen. The key ingredients for the proof are a finer classification of graphs in the expansion of the perturbed dynamics as well as a novel resolvent estimate for the unperturbed Schrödinger operator. Under some additional assumption on the sequence of initial states we even prove almost sure convergence.

38 pages, 7 figures Added proof for almost sure convergence