Spectral asymptotics for generalized Schrödinger operators

Abstract

Let \(d \in \{3,4,5,\ldots\}\). Consider \(L = -\frac{1}{w} \, \operatorname{div}(A \, \nabla u) + \mu\) over its maximal domain in \(L^2_w(\mathbb{R}^d)\). 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 \(\lambda \ge 1\). Furthermore we show that the eigenfunctions of \(L\) corresponding to those eigenvalues are exponentially decaying. In the course of proofs, we develop generalized Poincaré and weighted Young convolution inequalities as the main tools for the analysis.