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

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We study the kinetic, weak coupling limit of the dynamics governed by a discrete random Schrödinger operator on Z³. For sequences of ℓ²(Z³)-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 Lʳ(P)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.