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 \({x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}}\) 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.
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Communicated by M. Aizenman
Partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRN-CT-2002-0027.
Partially supported by NSF grant DMS-0602038.
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Erdős, L., Salmhofer, M. & Yau, HT. Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams. Commun. Math. Phys. 271, 1–53 (2007). https://doi.org/10.1007/s00220-006-0158-2
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DOI: https://doi.org/10.1007/s00220-006-0158-2