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Quantum diffusion of the random Schrödinger evolution in the scaling limit. II: The recollision diagrams. (English) Zbl 1205.82123

Summary: We consider random Schrödinger equations on \(\mathbb{R}^d\) for \(d \geq 3\) 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 \(\lambda \rightarrow 0\), 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 [Acta Math. 200, No. 2, 211–277 (2008; Zbl 1155.82015)] 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.

MSC:

82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
47B80 Random linear operators
47N50 Applications of operator theory in the physical sciences
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Citations:

Zbl 1155.82015
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References:

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