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Persistence of Anderson localization in Schrödinger operators with decaying random potentials

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Arkiv för Matematik

Abstract

We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than |x|-2 at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as |x|-α at infinity, we determine the number of bound states below a given energy E<0, asymptotically as α↓0. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent α; (b) dynamical localization holds uniformly in α.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, Irvine, CA, 92697-3875, USA

    Alexander Figotin & Abel Klein

  2. Département de Mathématiques, Université de Cergy-Pontoise, Laboratoire AGM, UMR 8088 CNRS, 2 avenue Adolphe Chauvin, FR-95302, Cergy-Pontoise cedex, France

    François Germinet

  3. Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, DE-37077, Göttingen, Germany

    Peter Müller

Authors
  1. Alexander Figotin
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  2. François Germinet
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  3. Abel Klein
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  4. Peter Müller
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Correspondence to Abel Klein.

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Figotin, A., Germinet, F., Klein, A. et al. Persistence of Anderson localization in Schrödinger operators with decaying random potentials. Ark Mat 45, 15–30 (2007). https://doi.org/10.1007/s11512-006-0039-0

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  • DOI: https://doi.org/10.1007/s11512-006-0039-0

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