Investigation of the Spectrum of the Problem of Normal Waves in a Closed Regular Inhomogeneous Dielectric Waveguide of Arbitrary Cross Section

Abstract—The problem of normal waves in a closed (screened) regular waveguide of arbitrary cross section is considered. It is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The solutions are defined using the variational formulation of the problem. The problem is reduced to the study of an operator function. The properties of the operators involved in the operator function (which are necessary for analyzing its spectral properties) are examined. Theorems are proved concerning the discrete character of the spectrum and the distribution of characteristic numbers of the operator function on the complex plane. The completeness of the system of eigen- and associated vectors of the operator function is investigated. It is proved that the system of eigen- and associated vectors of the operator function is doubly complete with a finite defect.