The method of operator beams and operator functions in the problem of normal waves of a closed regular inhomogeneous dielectric waveguide of arbitrary cross section

Background. The aim of this work is to study the spectrum of the problem of normal waves of a closed regular inhomogeneous dielectric waveguide of arbitrary cross section. Material and methods. To determine the solution, a variational formulation of the problem was used. The variational problem is reduced to the study of an operator pencil that depends nonlinearly on the spectral parameter. Results. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of an operator function on the complex plane are proved. The question of completeness of the system of eigenvectors and associated vectors of the operator-function is considered. A theorem on the double completeness of the system of eigenvectors and associated vectors of an operator function with a finite defect is proved. Conclusions. The proposed analytical method makes it possible to prove the discreteness of the spectrum in the problem of a closed inhomogeneous waveguide of an arbitrary cross section. In addition, this method can be used to study the spectral properties of more complex waveguide structures.