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We derive a sufficient condition for a Hermitian $N \times N$ matrix $A$ tohave at least $m$ eigenvalues (counting multiplicities) in the interval$(-\epsilon, \epsilon)$. This condition is expressed in terms of the existenceof a principal $(N-2m) \times (N-2m)$ submatrix of $A$ whose Schur complementin $A$ 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 acriterion that allows us to control the probability of having $m$ closely lyingeigenvalues for $H_\omega$-a result known as an $m$-level Wegner estimate. Wedemonstrate its usefulness by verifying the input condition of our criterionfor some physical models. These include the Anderson model and random blockoperators that arise in the Bogoliubov-de Gennes theory of dirtysuperconductors.

Eigenvalue counting inequalities, with applications to Schrodinger operators