On the fredholm property of integral equations system in the problem of electromagnetic waves propagation in a graphene-coated rod

Background. The problem of electromagnetic waves propagation in a dielectric rod of arbitrary cross-section covered with a layer of graphene, which is considered infinitely thin, is considered. The main problem in describing the process of wave propagation in the waveguiding structure is to obtain and analyze the system of integral equations to determine propagation constants. Materials and methods. Maxwell's equations are solved in the frequency domain. The coupling conditions contain the conductivity of graphene. In this article, we neglect the nonlinearity of graphene. The method of Green's functions is applied. Results and conclusions. The system of integral equations for determining the propagation constants is obtained. The Fredholm property of the system and the discreteness of the spectrum of the problem are proved.