  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.
