0 Ju n 20 06 Quantum diffusion of the random Schrödinger evolution in the scaling limit

We consider random Schrödinger equations on Rd for d ≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ−2−κ/2, t ∼ λ−2−κ with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to the solution of a heat equation in the space variable x for arbitrary L2 initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the nonrepetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation. AMS 2000 Subject Classification: 60J65, 81T18, 82C10, 82C44 Partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRNCT-2002-0027. Partially supported by NSF grant DMS-0602038.