Operator Function Method in the Problem of Normal Waves in an Inhomogeneous Waveguide

The problem of normal waves in a closed (screened) regular waveguiding structure of arbitrary cross-section is considered by reducing it to a boundary value problem for the longitudinal electromagnetic field components in Sobolev spaces. The variational statements of the problem is used to determine the solution. The problem is reduced to studying an operator function. The properties of the operators contained in the operator function necessary to analyze its spectral properties are studied. Theorems on the spectrum discreteness and the distribution of characteristic numbers of the operator function on the complex plane are proved. The problem of completeness of the system of root vectors of the operator function is considered. The theorem on the double completeness of the system of root vectors of the operator function with finite deficiency is proved.