Eigenvalue counting inequalities, with applications to Schrödinger operators

Alexander Elgart, Daniel Schmidt

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We derive a sufficient condition for a Hermitian N×N matrix A to have at least m eigenvalues (counting multiplicities) in the interval (-∈,∈). This condition is expressed in terms of the existence of a principal (N - 2m) × (N-2m) submatrix of A whose Schur complement in A has at least m eigenvalues in the interval (-K∈;K∈), with an explicit constant K. We apply this result to a random Schrödinger operator Hω, obtaining a criterion that allows us to control the probability of having m closely lying eigenvalues for Hω-a result known as an m-level Wegner estimate. We demonstrate its usefulness by verifying the input condition of our criterion for some physical models. These include the Anderson model and random block operators that arise in the Bogoliubov-de Gennes theory of dirty superconductors.

Original languageEnglish
Pages (from-to)251-278
Number of pages28
JournalJournal of Spectral Theory
Volume5
Issue number2
DOIs
StatePublished - 1 Jan 2015
Externally publishedYes

Keywords

  • Anderson models
  • Eigenvalue counting
  • Minami estimate
  • Random block operators
  • Wegner estimate

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