Dynamical self-averaging for a lattice Schr\"odinger equation with weak random potential

Abstract

We study the kinetic, weak coupling limit of the dynamics governed by a discrete random Schr\"odinger 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 Schr\"odinger operator. Under some additional assumption on the sequence of initial states we even prove almost sure convergence.