Quantum diffusion of the random Schrodinger evolution in the scaling limit

Abstract

We consider random Schrödinger equations on $\bR^d$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential. Denote by $λ$ the coupling constant and $ψ_t$ the solution with initial data $ψ_0$. The space and time variables scale as $x\sim λ^{-2 -κ/2}, t \sim λ^{-2 -κ}$ with $0< κ< κ_0(d)$. We prove that, in the limit $λ\to 0$, the expectation of the Wigner distribution of $ψ_t$ 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 analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the {\it non-ladder} diagrams is smaller than their &#34;naive size&#34; by an extra $λ^c$ factor {\em per non-(anti)ladder vertex} for some $c > 0$. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.