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April 2007 Persistence of Anderson localization in Schrödinger operators with decaying random potentials
Alexander Figotin, François Germinet, Abel Klein, Peter Müller
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Ark. Mat. 45(1): 15-30 (April 2007). DOI: 10.1007/s11512-006-0039-0

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| 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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Alexander Figotin. François Germinet. Abel Klein. Peter Müller. "Persistence of Anderson localization in Schrödinger operators with decaying random potentials." Ark. Mat. 45 (1) 15 - 30, April 2007. https://doi.org/10.1007/s11512-006-0039-0

Information

Received: 18 April 2006; Published: April 2007
First available in Project Euclid: 31 January 2017

zbMATH: 1159.47059
MathSciNet: MR2312950
Digital Object Identifier: 10.1007/s11512-006-0039-0

Rights: 2007 © Institut Mittag-Leffler

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Vol.45 • No. 1 • April 2007
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