On the Perturbation Theory of Self-Adjoint Operators

Abstract

We show that all types of self-adjoint perturbations of a semi-bounded operator $A$ (purely singular, mixed singular, and regular) can be described and studied from a unique point of view in the framework of the extension theory as well as in the framework of the additive perturbation theory. We also show that any singular finite rank perturbation $\widetilde{A}$ can be approximated in the norm resolvent sense by regular finite rank perturbations of $A$. An application is given to the study of Schr\"{o}dinger operators with point interactions.