Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals (Q469580)

The authors investigate nonlinear eigenvalue problems for differential operators \(F(\lambda)v=0\), \(\lambda\in \Omega\), where \(F(\lambda)\) are Fredholm operators of index \(0\) that depend analytically on \(\lambda\in \Omega\). They propose a numerical method using contour integrals of solutions for resolvent equations, which is based on the Keldysh' theorem. This method extends a recent method for nonlinear eigenvalue problems with matrices. All eigenvalues inside a given contour \(\Gamma\subset \mathbb{C}\) are determined, and the authors prove that the approximate equations are well-conditioned as the original eigenvalue problem. The errors are well-controlled when the resolvent equations are solved via boundary value problems on finite domains. They also discuss suitable normalizations of the Evans function, relate their contour method to such a normalized Evans function, and apply this method to first order differential systems with \(\lambda\)-dependent matrices. Some applications to Schrödinger operators on the real line and on bounded intervals, and to the FitzHugh-Nagumo system are also presented.