Muller boundary integral equations in the spectral theory of dielectric waveguides

Background. Muller boundary integral equations are widely used for theoretical and numerical analysis of various spectral problems of the mathematical theory of diffraction. They are also used for calculation of superficial own waves of homogeneous weakly guiding dielectric waveguides without losses. The aim of the work is to develop a method of using the latter for searching of both superficial and leaky waves of such waveguides, and also to research qualitative spectral proprties. Materials and methods. The research of qualitative spectral properties was carried out using the methods of the theory of regularization of problems of open waveguides' own waves. Reduction of the initial problem to a spectral problem for the system of integral equations was carried out by the potential theory methods. The further analysis is based on the known results on isolation of characteristic values of the Fredholm holomorphic operator-function with the presence of at least one regular point in the area of its holomorphy, and on behavior of characteristic values of such operator-function as a function of non-spectral parameters. Results. It is proved that the initial problem for the surface Helmholtz equation is equivalent to a nonlinear spectral problem for Muller boundary integral equations with a quite continuous operator. It is proved that the characteristic set of the built operator-value function may consist of only isolated points on the corresponding Riemann surface. Each characteristic value continuously depends on nonspectral parameters and may occur and disappear on the boundary of this surface. Cocnlusions. The developed technique of Muller boundary integral equations application may be successfully implemented for solving spectral problems of the theory of dielectric waveguides, namely for searching superficial leaky own waves, as well as for researching qualitative spectral properties.