Compactness of Schr\"odinger semigroups

TL;DR

This paper investigates conditions under which Schr"odinger semigroups are compact, linking spectral properties to potential and measure perturbations, with applications to operators with thin sublevel sets at infinity.

Contribution

It provides a simple criterion for semigroup compactness based on relative compactness of multiplication operators and characterizes compactness for measure perturbations in Dirichlet form contexts.

Findings

Criterion for compactness via sublevel set properties

Characterization of measure perturbation compactness

Application to Schr"odinger operators with thin potentials

Abstract

This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of selfadjoint operators that are bounded below (on an L^2-space). For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of the semigroup in terms of relative compactness of the operators of multiplication with characteristic functions of sublevel sets. In the context of Dirichlet forms, we can even characterize compactness of the semigroup for measure perturbations. Here, certain 'averages' of the measure outside of compact sets play a role. As an application we obtain compactness of semigroups for Schr\"odinger operators with potentials whose sublevel sets are thin at infinity.