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

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Submission date: 23. Jan. 2006
Pages: 53
published in: Communications in mathematical physics, 271 (2007) 1, p. 1-53 
DOI number (of the published article): 10.1007/s00220-006-0158-2
MSC-Numbers: 60J65, 81T18, 82C10, 82C44
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We consider random Schrödinger equations on formula9 for formula11 with a homogeneous Anderson-Poisson type random potential. Denote by formula13 the coupling constant and formula15 the solution with initial data formula17. The space and time variables scale as formula19 with formula21. We prove that, in the limit formula23, the expectation of the Wigner distribution of formula15 converges weakly to the solution of a heat equation in the space variable x for arbitrary formula29 initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper (L. Erdos, 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.

18.10.2019, 02:13