Continuity of eigenvalues for Schrödinger operators, $L^p$-properties of Kato type integral operators

Abstract.

Given an arbitrary relatively compact (finely) open subset of ${\mathbb R}^d, \mu$-eigenvalues of $-A(D)+\nu$ are studied where $A(D)$ is the Dirichlet Laplacian on D and $\mu,\nu$ are measures on ${\mathbb R}^d$ such that $G_X^\mu$ is continuous and $G_X^\nu$ is bounded for every ball X in ${\mathbb R}^d (G_X$ being Green's function for X). Moreover, it is shown that these eigenvalues depend continuously on D and $\nu$. The results are based on very general compactness and convergence properties of integral operators of Kato type which are developed before.