Eigenvalue counting inequalities, with applications to Schrodinger operators

TL;DR

This paper establishes a new eigenvalue counting inequality for Hermitian matrices and applies it to control the probability of closely spaced eigenvalues in random Schrödinger operators, with implications for physical models.

Contribution

It introduces a novel eigenvalue counting inequality involving Schur complements and applies it to derive probabilistic eigenvalue estimates for random Schrödinger operators.

Findings

Derived a sufficient eigenvalue counting condition using principal submatrices.

Applied the criterion to random Schrödinger operators, including the Anderson model.

Verified the condition for physical models like superconductors.

Abstract

We derive a sufficient condition for a Hermitian $N \times N$ matrix $A$ to have at least $m$ eigenvalues (counting multiplicities) in the interval $(-\epsilon, \epsilon)$. This condition is expressed in terms of the existence of a principal $(N-2m) \times (N-2m)$ submatrix of $A$ whose Schur complement in $A$ has at least $m$ eigenvalues in the interval $(-K\epsilon, K\epsilon)$, with an explicit constant $K$. We apply this result to a random Schrodinger operator $H_\omega$, obtaining a criterion that allows us to control the probability of having $m$ closely lying eigenvalues for $H_\omega$-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.