Eigenvalue Estimates for the Perturbed Andersson Model

Consider the operator H = −∆ + pω − V, where pω is the potential of the Andersson type and V ≥ 0 is a function that decays slowly at the infinity. We study the rate of accumulation of eigenvalues of H to the bottom of the essential spectrum. 1. STATEMENT OF THE MAIN RESULT The question we study is rooted in two different areas of mathematics: the spectral theory of differential operators and the theory of percolations. We begin with a discussion of the topics that are more important for the CwikelLieb-Rozenblum and Lieb-Thirring estimates. Let Ej be the negative eigenvalues of the operator −∆− V (x), V ≥ 0. Then ∑