On Spectral Stability for Rank One Perturbations

TL;DR

This paper investigates the spectral stability of rank one singular perturbations of self-adjoint operators, showing that most such perturbations lack eigenvalues on a dense set and that eigenvalues are typically isolated.

Contribution

It introduces a unified approach to analyze spectral stability under rank one perturbations, extending previous methods and results.

Findings

Most rank one singular perturbations have no eigenvalues on a dense Gδ set.

For a dense Gδ set of perturbations, eigenvalues are isolated.

The spectrum's structure under perturbations is characterized using self-adjoint extension theory.

Abstract

Embedded point spectra of rank one singular perturbations of an arbitrary self-adjoint operator A on a Hilbert space H is studied. It is shown that these perturbations can be regarded as self-adjoint extensions of a densely defined closed symmetric operator B with deficiency indices (1; 1). Assuming the deficiency vector of B is cyclic for its self-adjoint extensions, we prove that the spectrum of A contains a dense G{\delta} subset where it is not possible to have eigenvalues for any rank one singular perturbation. Moreover, for a dense G{\delta} set of rank one singular perturbations of A their eigenvalues are isolated. The approach presented here unifies points of view taken by different authors.