MiS Preprint Repository

We consider random Schrödinger equations on ${\mathbb{R}}^d$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential. Denote by $\lambda$ the coupling constant and ${\psi }_t$ the solution with initial data ${\psi }_0$. The space and time variables scale as $x\sim {\lambda }^{-2-\kappa /2},t\sim {\lambda }^{-2-\kappa }$ with $0<\kappa <{\kappa }_0(d)$. We prove that, in the limit $\lambda \to 0$, the expectation of the Wigner distribution of ${\psi }_t$ converges weakly to the solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data.

The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper (L. Erdős, M. Salmhofer and H.-T. Yau, Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams. Submitted to Ann. Math. (2005)) the analysis of the non-repetition 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.