Preprint 8/2006

Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams.

László Erdös, Manfred Salmhofer, and Horng-Tzer Yau

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Submission date: 23. Jan. 2006
Pages: 59
published in: Acta mathematica, 200 (2008) 2, p. 211-277 
DOI number (of the published article): 10.1007/s11511-008-0027-2
MSC-Numbers: 60J65, 81T18, 82C10, 82C44
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We consider random Schrödinger equations on formula11 for formula13 with a homogeneous Anderson-Poisson type random potential. Denote by formula15 the coupling constant and formula17 the solution with initial data formula19. The space and time variables scale as formula21 with formula23. We prove that, in the limit formula25, the expectation of the Wigner distribution of formula17 converges weakly to the solution of a heat equation in the space variable x for arbitrary formula31 initial data.

The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the non-ladder diagrams is smaller than their ``naive size" by an extra formula33 factor per non-(anti)ladder vertex for some c > 0. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series. The analysis of the recollision graphs is given in the companion paper (L. Erdos, M. Salmhofer and H.-T. Yau, Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Submitted to Commun. Math. Phys.)

23.06.2018, 02:11