On the Spectrum of an Operator Pencil with Applications to Wave Propagation in Periodic and Frequency Dependent Materials

We study wave propagation in periodic and frequency dependent materials. The approach in this paper leads to spectral analysis of a quadratic operator pencil where the spectral parameter relates to the quasimomentum and the frequency is a parameter. We show that the underlying operator has a discrete spectrum, where the eigenvalues are symmetrically placed with respect to the real and imaginary axis. Moreover, we discretize the operator pencil with finite elements and use a Krylov space method to compute eigenvalues of the resulting large sparse matrix pencil.