The essential numerical range and a theorem of Simon on the absorption of eigenvalues

TL;DR

This paper extends Simon’s theorem using the essential numerical range to analyze when eigenvalues become part of the essential spectrum in holomorphic families of self-adjoint operators.

Contribution

It generalizes Simon’s theorem by providing a sufficient condition for the essential spectrum minimum to be an eigenvalue at threshold points, based on eigenvalue approach rates.

Findings

Eigenvalue approach rates relate to eigenvalues of a bounded operator.

The essential numerical range is key to understanding eigenvalue absorption.

Provides a criterion for eigenvalue inclusion at the essential spectrum threshold.

Abstract

Let $A(t)$ be a holomorphic family of self-adjoint operators of type (B) on a complex Hilbert space $\mathcal{H}$. Kato-Rellich perturbation theory says that isolated eigenvalues of $A(t)$ will be analytic functions of $t$ as long as they remain below the minimum of the essential spectrum of $A(t)$. At a threshold value $t_0$ where one of these eigenvalue functions hits the essential spectrum, the corresponding point in the essential spectrum might or might not be an eigenvalue of $A(t_0)$. Our results generalize a theorem of Simon to give a sufficient condition for the minimum of the essential spectrum to be an eigenvalue of $A(t_0)$ based on the rate at which eigenvalues approach the essential spectrum. We also show that the rates at which the eigenvalues of $A(t)$ can approach the essential spectrum from below correspond to eigenvalues of a bounded self-adjoint operator. The key insight behind these results is the essential numerical range which was recently extended to unbounded operators by B\"{o}gli, Marletta, and Tretter.