Localization theorems for nonlinear eigenvalue problems

TL;DR

This paper introduces new localization theorems for nonlinear eigenvalue problems that extend classical results like Gershgorin's theorem and the Bauer-Fike theorem, with applications to delay differential equations, Hadeler's problem, and quantum resonance.

Contribution

The paper generalizes classical eigenvalue localization theorems to nonlinear problems, providing new tools for analyzing complex eigenvalue problems in various applications.

Findings

New localization theorems for nonlinear eigenvalue problems

Applications to delay differential equations and quantum resonance

Extended classical eigenvalue bounds to nonlinear settings

Abstract

Let $T : \Omega \rightarrow \bbC^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset \bbC$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.