Eigenvalues in spectral gaps of differential operators

Abstract

Spectral problems with band-gap spectral structure arise in numerous applications, including the study of crystalline structure and the determination of transmitted frequencies in photonic waveguides. Numerical discretization of these problems can result in spurious results, a phenomenon known as spectral pollution. We present a method for calculating eigenvalues in the gaps of self-adjoint operators which avoids spectral pollution. The method perturbs the problem into a dissipative problem in which the eigenvalues to be calculated are lifted out of the convex hull of the essential spectrum, away from the spectral pollution. The method is analysed here in the context of one-dimensional Schrodinger equations on the half line, but is applicable in a much wider variety of contexts, including PDEs, block operator matrices, multiplication operators, and others.