Spectral asymptotics for generalized Schrödinger operators

Let d is an element of {3, 4, 5, ...}. Consider L = - 1/omega div(A del u) + mu over its maximal domain in L-w(2)(R-d). Under certain conditions on the weight w, the coefficient matrix A and the positive Radon measure mu we obtain upper and lower bounds on N(lambda, L)-the number of eigenvalues of L that are at most lambda >= 1. Furthermore we show that the eigenfunctions of L corresponding to those eigenvalues are exponentially decaying. In the course of proofs, we develop generalized Poincare and weighted Young convolution inequalities as the main tools for the analysis.