Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals

TL;DR

This paper introduces a contour integral method for detecting and approximating eigenvalues of holomorphic Fredholm operator pencils, with applications to Schrödinger operators and the FitzHugh-Nagumo system, offering an alternative to Evans function analysis.

Contribution

It extends contour integral techniques to boundary value problems for operator pencils, providing a controlled, general approach for eigenvalue computation in unbounded domain problems.

Findings

Effective eigenvalue detection via contour integrals with controlled errors.

Application to Schrödinger operators on real line and bounded intervals.

Numerical analysis of FitzHugh-Nagumo system demonstrating method's practicality.

Abstract

Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh' theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are well-controlled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schr\"odinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHugh-Nagumo system. We also relate the contour method to the well-known Evans function and show that our approach provides an alternative to evaluating and computing its zeroes.