Spectra of rank-one perturbations of self-adjoint operators

TL;DR

This paper characterizes the spectra of rank-one perturbations of self-adjoint operators, showing they can include any number of eigenvalues with arbitrary algebraic multiplicities, both real and non-real.

Contribution

It provides a complete description of the possible spectra resulting from rank-one perturbations of self-adjoint operators with discrete spectra.

Findings

The spectrum of a rank-one perturbation can include any number of eigenvalues.

Eigenvalues can be real or non-real with arbitrary algebraic multiplicities.

The characterization applies specifically to operators with discrete spectra.

Abstract

We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic multiplicity