Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals

摘要

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