Quantum diffusion of the random Schrodinger evolution in the scaling limit II. The recollision diagrams

Erdos, Laszlo; Salmhofer, Manfred; Yau, Horng-Tzer

Quantum diffusion of the random Schrodinger evolution in the scaling limit II. The recollision diagrams

Authors

Erdos, Laszlo

Salmhofer, Manfred

Yau, Horng-Tzer

Description

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 a rigorous analysis of Feynman diagrams. In the companion paper 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.
57 pages, 15 figures Extended version on Jun 10: some more definitions from the companion paper are added to make the paper self-contained. Literature enlarged