One-dimensional perturbations of unbounded selfadjoint operators with empty spectrum

TL;DR

This paper investigates how one-dimensional singular perturbations affect the spectrum of unbounded selfadjoint operators, providing criteria for spectrum removal and characterizing rank-one perturbations leading to Volterra operators.

Contribution

It offers new criteria for spectrum elimination via singular perturbations and characterizes rank-one perturbations resulting in Volterra operators.

Findings

Criteria for spectrum removal by singular perturbations

Complete description of rank-one perturbations leading to Volterra operators

Spectral properties of unbounded selfadjoint operators under perturbations

Abstract

We study spectral properties of one-dimensional singular perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one perturbation is a Volterra operator.