Spectral properties of finite-dimensional waveguide systems

This is a largely expository paper in which a finite dimensional model for gyroscopic/waveguiding systems is studied. Properties of the spectrum that play an important role when computing with such models are studied. The notion of “waveguide-type” is explored in this context. The main theorem 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.