Discrete Schrödinger operators with random alloy-type potential

Alexander Elgart, Helge Krüger, Martin Tautenhahn, Ivan Veselić

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

7 Scopus citations

Abstract

We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schrödinger operators Hω = -Δ+Vω on l2(ℤd) where Δ is the discrete Laplacian and Vω the multiplication by the function Vω(x) = Σkεℤd ωku(x - k). Here ωk, k ε ℤd, are i.i.d. random variables and u ε l1(ℤd; ℝ) is a so-called single-site potential. Since u may change sign, certain properties of Hω depend in a non-monotone way on the random parameters ωk. This requires new methods at certain stages of the localization proof.

Original languageEnglish
Title of host publicationSpectral Analysis of Quantum Hamiltonians
Subtitle of host publicationSpectral Days 2010
PublisherSpringer Basel
Pages107-131
Number of pages25
ISBN (Electronic)9783034804141
ISBN (Print)9783034804134
DOIs
StatePublished - 1 Jan 2012
Externally publishedYes

Keywords

  • Cartan's theorem
  • Discrete alloy-type model
  • Fractional moment method
  • Localization
  • Multiscale analysis
  • Non-monotone
  • Sign-indefinite
  • Singlesite potential

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