Preprint of the project:
12-099 Wolf-Jürgen Beyn, Yuri Latushkin, Jens Rottmann-Matthes.
Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals
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.
