Arkiv för Matematik

Persistence of Anderson localization in Schrödinger operators with decaying random potentials

Alexander Figotin, François Germinet, Abel Klein, and Peter Müller

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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.

Article information

Source
Ark. Mat., Volume 45, Number 1 (2007), 15-30.

Dates
Received: 18 April 2006
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898978

Digital Object Identifier
doi:10.1007/s11512-006-0039-0

Mathematical Reviews number (MathSciNet)
MR2312950

Zentralblatt MATH identifier
1159.47059

Rights
2007 © Institut Mittag-Leffler

Citation

Figotin, Alexander; Germinet, François; Klein, Abel; Müller, Peter. Persistence of Anderson localization in Schrödinger operators with decaying random potentials. Ark. Mat. 45 (2007), no. 1, 15--30. doi:10.1007/s11512-006-0039-0. https://projecteuclid.org/euclid.afm/1485898978


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