Accumulation of Complex Eigenvalues of a Class of Analytic Operator Functions

TL;DR

This paper investigates how complex eigenvalues of certain analytic operator functions accumulate near the essential spectrum, establishing conditions for eigenvalue accumulation and demonstrating applications in electromagnetic field theory.

Contribution

It provides new theoretical results on eigenvalue accumulation, minimality, and completeness for analytic operator functions, with practical implications for rational operator functions and electromagnetic applications.

Findings

Eigenvalues accumulate near the essential spectrum.

Minimality and completeness of eigenvector systems are established.

Conditions for eigenvalue accumulation at poles and infinity are derived.

Abstract

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given.