Eigenvalue counting inequalities, with applications to Schrödinger operators

  • Alexander Elgart

    Virginia Tech, Blacksburg, USA
  • Daniel Schmidt

    Virginia Tech, Blacksburg, USA

Abstract

We derive a sufficient condition for a Hermitian N×NN \times N matrix AA to have at least mm eigenvalues (counting multiplicities) in the interval (ϵ,ϵ)(-\epsilon, \epsilon). This condition is expressed in terms of the existence of a principal (N2m)×(N2m)(N-2m) \times (N-2m) submatrix of AA whose Schur complement in AA has at least mm eigenvalues in the interval (Kϵ,Kϵ)(-K\epsilon, K\epsilon), with an explicit constant KK. We apply this result to a random Schrödinger operator HωH_{\omega}, obtaining a criterion that allows us to control the probability of having mm closely lying eigenvalues for HωH_{\omega} – a result known as an mm-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.

Cite this article

Alexander Elgart, Daniel Schmidt, Eigenvalue counting inequalities, with applications to Schrödinger operators. J. Spectr. Theory 5 (2015), no. 2, pp. 251–278

DOI 10.4171/JST/97