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

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

2005-12-06

arxiv.org/abs/math-ph/0512015v2

Physics

Mathematical Physics

57 pages, 15 figures Extended version on Jun 10: some more definitions from the companion paper are added to make the paper

Scientific paper

We consider random Schr\"odinger equations on $\bR^d$ 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 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.

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