# On the Properties of Phononic Eigenvalue Problems

## TL;DR

This paper investigates the fundamental properties of phononic eigenvalue problems, providing a unified theoretical framework, extending the Plane Wave Expansion method, and supporting findings with numerical calculations.

## Contribution

It offers a unified understanding of phononic eigenvalues and eigenvectors, and extends the PWE method to broader classes of eigenvalue problems.

## Key findings

- Eigenvalues can be real or complex, with eigenvectors forming complete bases under certain conditions.
- Phononic problems can be reformulated into linear eigenvalue problems for direct computation of physical quantities.
- The extended PWE method applies to new eigenvalue forms, broadening its applicability.

## Abstract

In this paper, we consider the operator properties of various phononic eigenvalue problems. We aim to answer some fundamental questions about the eigenvalues and eigenvectors of phononic operators. These include questions about the potential real and complex nature of the eigenvalues, whether the eigenvectors form a complete basis, what are the right orthogonality relationships, and how to create a complete basis when none may exist at the outset. In doing so we present a unified understanding of the properties of the phononic eigenvalues and eigenvectors which would emerge from any numerical method employed to compute such quantities. We show that the phononic problem can be cast into linear eigenvalue forms from which such quantities as frequencies, wavenumbers, and desired components of wavevectors can be directly ascertained without resorting to searches or quadratic eigenvalue problems and that the relevant properties of such quantities can be determined apriori through the analysis of the associated operators. We further show how the Plane Wave Expansion (PWE) method may be extended to solve each of these eigenvalue forms, thus extending the applicability of the PWE method to cases beyond those which have been considered till now. The theoretical discussions are supplemented with supporting numerical calculations. The techniques and results presented here directly apply to wave propagation in other periodic systems such as photonics.