Spectral asymptotics for generalized Schrödinger operators

Abstract

Let $d\in \{3,4,5,\dots \}$. Consider $L=-\frac{1}{w}\mathrm{div}(A\mathrm{\nabla }u)+\mu$ over its maximal domain in $L_w^2({\mathbb{R}}^d)$. 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.