2013/06/14 by Alexander Elgart, Daniel Schmidt, Elgart, Alexander +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Graph theory and applications #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1306.3459
revised version, 26 pages, no figures
openalex publication_date 2013/06/14 · arxiv created 2014/03/11 · arxiv updated 2014/03/12 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We derive a sufficient condition for a Hermitian N × N matrix A to have at least m eigenvalues (counting multiplicities) in the interval (-ε, ε). 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 (-Kε, Kε), with an explicit constant K. We apply this result to a random Schrodinger operator Hω, obtaining a criterion that allows us to control the probability of having m closely lying eigenvalues for Hω-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.