Quantum diffusion of the random Schrödinger evolution in the scaling limit

Abstract

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 $\psi_t$ the solution with initial data $\psi_0$. The space and time variables scale as $ x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)} $. We prove that, in the limit λ → 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 L2 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 non-ladder diagrams is smaller than their “naive size” by an extra λc factor 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.