  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.
