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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
9/2006

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

László Erdös, Manfred Salmhofer and Horng-Tzer Yau

Abstract

We consider random Schrödinger equations on $\mathbb{R}^d$ 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 (L. Erdős, M. Salmhofer and H.-T. Yau, Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams. Submitted to Ann. Math. (2005)) 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.

Received:
Jan 23, 2006
Published:
Jan 23, 2006
MSC Codes:
60J65, 81T18, 82C10, 82C44

Related publications

inJournal
2007 Repository Open Access
Laszlo Erdös, Manfred Salmhofer and Horng-Tzer Yau

Quantum diffusion of the random Schrödinger evolution in the scaling limit. Pt. 2 : the recollision diagrams

In: Communications in mathematical physics, 271 (2007) 1, pp. 1-53