Eigenvalue counting inequalities, with applications to Schrödinger operators

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.