Finiteness of the lower spectrum of Schrödinger operators with singular potentials

Abstract:

We assume that $q:{\mathbb {R}^m} \to \mathbb {R}(m \geq 3)$ is a measurable function with the property that its negative and positive parts, respectively, belong to the Kato class $K({\mathbb {R}^m})$ and ${K_{loc}}({\mathbb {R}^m})$. We prove a conjecture by B. Simon concerning the finiteness of the lower spectrum of an s.a. realization of the Schrödinger expression $- \Delta + q$ in ${L^2}({\mathbb {R}^m})$ bounded from below.