Direct and inverse spectral problems for rank-one perturbations of self-adjoint operators

TL;DR

This paper characterizes the eigenvalues of rank-one perturbations of self-adjoint operators with discrete spectra and explores reconstructing the perturbation from spectral data.

Contribution

It provides a complete characterization of eigenvalues for such perturbations and addresses the inverse spectral problem for reconstructing the perturbation.

Findings

Eigenvalues of rank-one perturbations are fully characterized.

The inverse problem of reconstructing the perturbation from spectrum is discussed.

Results apply to self-adjoint operators with discrete spectra.

Abstract

For a given self-adjoint operator $A$ with discrete spectrum, we completely characterize possible eigenvalues of its rank-one perturbations~$B$ and discuss the inverse problem of reconstructing $B$ from its spectrum.