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Quantum diffusion of the random Schrödinger evolution in the scaling limit

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Acta Mathematica

Abstract

We consider random Schrödinger equations on R d for d ≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and \(\psi_t\) the solution with initial data \(\psi_0\). The space and time variables scale as \( x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)} \). We prove that, in the limit λ → 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 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 λc 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.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany

    László Erdős

  2. Max Planck Institute for Mathematics and Theoretical Physics, University of Leipzig, Inselstrasse 22, D-04103, Leipzig, Germany

    Manfred Salmhofer

  3. Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

    Horng-Tzer Yau

Authors
  1. László Erdős
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  2. Manfred Salmhofer
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  3. Horng-Tzer Yau
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Correspondence to László Erdős.

Additional information

The first auhor was partially supported by NSF grant DMS-0307295 and MacArthur Fellowship. The third author was partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRN-CT-2002-0027.

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Erdős, L., Salmhofer, M. & Yau, HT. Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math 200, 211–277 (2008). https://doi.org/10.1007/s11511-008-0027-2

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  • DOI: https://doi.org/10.1007/s11511-008-0027-2

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