Integro-differential equations in the problem of electromagnetic wave scattering on a dielectric body covered with graphene

We consider the determination of resonance frequencies of dielectric bodies coated with graphene. In the addressed problem statement, the graphene nonlinearity is not taken into account. The initial boundary-value problem for Maxwell’s equations is reduced to a system of integro-differential equations on the graphene surface. We prove the Fredholm property of this system under certain sufficient conditions and establish the discreteness of the spectrum of an operator-valued function corresponding to this system in a certain region of the complex plane of the circular frequency spectral parameter.