On the equivalence of differential and integral formulations of the problem for eigenwaves of weakly guiding dielectric waveguides

The use of the method of Muller boundary integral equations for solving the problem of eigenwaves of weakly guiding dielectric waveguides was justified. A theorem was proved about the spectral equivalence between the original differential problem and the problem for the system of Muller boundary integral equations on the physical sheet of the Riemann surface, where the eigenvalues, the propagation constants of the eigenwaves, are sought. With this aim, the localization regions of the spectra of the original problem and a so-called “turned inside out” problem generating spurious eigenvalues were analyzed. A sufficient condition of equivalence was obtained: the problems are equivalent if the problem turned inside out has only a trivial solution. Consequently, as confirmed by the results of the numerical experiments, only true eigenvalues on the physical sheet of the Riemann surface can be found by using the method of Muller boundary integral equations.