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Anderson localization for Bernoulli and other singular potentials

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Abstract

We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Hölder continuous distributions and for bounded potentials whose distribution is a convex combination of a Hölder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions.

We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.

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Author notes

  1. Fabio Martinelli

    Present address: Dipartimento di Matematica, Universita di Roma “La Sapienza”, Piazzale A. Moro, 2, I-00185, Rome, Italy

Authors and Affiliations

  1. Department of Mathematics, University of California, 92717, Irvine, CA, USA

    Rene Carmona, Abel Klein & Fabio Martinelli

Authors
  1. Rene Carmona
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  2. Abel Klein
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  3. Fabio Martinelli
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Additional information

Communicated by B. Simon

Partially supported by NSF grant DMS 85-03695

Partially supported by NSF grant DMS 83-01889

Partially supported by G.N.F.M. C.N.R.

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Carmona, R., Klein, A. & Martinelli, F. Anderson localization for Bernoulli and other singular potentials. Commun.Math. Phys. 108, 41–66 (1987). https://doi.org/10.1007/BF01210702

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  • DOI: https://doi.org/10.1007/BF01210702

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