Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams

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 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 Partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRNCT-2002-0027. Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.