Spectral Properties of Finite-dimensional Waveguide Systems

Abstract

This is a largely expository paper in which we study a finite dimensional model for gyroscopic/waveguiding systems. We study properties of the spectrum that play an important role when computing with such models. The notion of "waveguide type" is defined and explored in this context and Theorem 3.1 provides a form of the central result (due to Abramov) concerning the existence of real spectrum for such systems. The roles of semisimple/defective eigenvalues are discussed, as well as the roles played by eigenvalue "types" (or "Krein signatures"). The theory is illustrated with examples.