Integro-Differential Equations in the Problem of Scattering of Electromagnetic Waves by a Graphene-Coated Dielectric Body

We consider the problem of determining the resonance frequencies of dielectric bodiescoated with graphene. In the considered formulation of the problem, the nonlinearity of grapheneis not taken into account. The initial–boundary value problem for Maxwell’s equations is reducedto a system of integro-differential equations on the graphene surface. We prove the Fredholmproperty of this system under certain sufficient conditions and establish the discreteness of thespectrum of an operator-valued function corresponding to this system in a certain domain of thecomplex plane of the spectral parameter (circular frequency).