Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

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Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

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Title: Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams
Author: Erdos, Laszlo; Salmhofer, Manfred; Yau, Horng-Tzer

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Citation: Erdős, László, Manfred Salmhofer, and Horng-Tzer Yau. 2007. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams.” Communications in Mathematical Physics 271 (1) (January 31): 1–53. doi:10.1007/s00220-006-0158-2.
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Abstract: We consider random Schrödinger equations on {mathbb{R}d} for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ t the solution with initial data ψ0. The space and time variables scale as {x˜ λ^{-2 -kappa/2}, t ˜ λ^{-2 -kappa}} with 0 < κ < κ0( d). We prove that, in the limit λ → 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 [10] 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.
Published Version: doi:10.1007/s00220-006-0158-2
Other Sources: https://arxiv.org/abs/math-ph/0512015
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:32706724
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