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Persistence of Anderson localization in Schrödinger operators with decaying random potentials

Alexander Figotin Department of Mathematics, University of California, Irvine François Germinet Département de Mathématiques, Université de Cergy-Pontoise Abel Klein Department of Mathematics, University of California, Irvine Peter Müller Institut für Theoretische Physik, Georg-August-Universität Göttingen

TBD mathscidoc:1701.333102

Arkiv for Matematik, 45, (1), 15-30, 2006.4
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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@inproceedings{alexander2006persistence,
  title={Persistence of Anderson localization in Schrödinger operators with decaying random potentials},
  author={Alexander Figotin, François Germinet, Abel Klein, and Peter Müller},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203619498795911},
  booktitle={Arkiv for Matematik},
  volume={45},
  number={1},
  pages={15-30},
  year={2006},
}
Alexander Figotin, François Germinet, Abel Klein, and Peter Müller. Persistence of Anderson localization in Schrödinger operators with decaying random potentials. 2006. Vol. 45. In Arkiv for Matematik. pp.15-30. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203619498795911.
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