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

TL;DR

This paper analyzes wave propagation in periodic, frequency-dependent materials through spectral analysis of a quadratic operator pencil, revealing a discrete, symmetric spectrum and employing finite element discretization with Krylov methods for eigenvalue computation.

Contribution

It introduces a spectral analysis framework for quadratic operator pencils in wave propagation, including discretization and numerical eigenvalue computation methods.

Findings

The operator has a discrete spectrum with symmetric eigenvalues.

Finite element discretization enables large sparse matrix eigenvalue problems.

Krylov space methods effectively compute eigenvalues of the discretized operator.

Abstract

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.