Eigenvalue counting inequalities, with applications to Schrödinger operators

Abstract

ABSTRACT. We derive a sufficient condition for a Hermitian N × N matrix A to have at least m eigenvalues (counting multiplicities) in the interval (−ǫ, ǫ). 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 (−Kǫ, Kǫ), with an explicit constant K.We apply this result to a random Schrödinger operator H ω , obtaining a criterion that allows us to control the probability of having m closely lying eigenvalues for H ω -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.

Localization for Random Block Operators Related to the XY Spin Chain

“…However, random block operators have recently found considerable interest, see [22,14,15,13], with physical motivation also provided by the Bogoliubov-de Gennes equation in the mean-field approximation of BCS theory. We also mention that the model given by the anisotropic XY chain has been considered in the quantum information literature under the name Majorana chain, see [23] and [6].…”