Spectral asymptotics for generalized Schrödinger operators

Abstract

Letd∈ {3,4,5, . . .}. ConsiderL=−1wdiv(A∇u) +μover its maximal domaininL2w(Rd). Under certain conditions on the weightw, the coefficient matrixAand the positiveRadon measureμwe obtain upper and lower bounds onN(λ, L)—the number of eigenvalues ofLthat are at mostλ≥1. Furthermore we show that the eigenfunctions ofLcorresponding to thoseeigenvalues are exponentially decaying. In the course of proofs, we develop generalized Poincaréand weighted Young convolution inequalities as the main tools for the analysis.