On the discreteness of the spectrum of integrodifferential operator-functions in the problem of oscillations in open volume resonators

Background. The purpose of this work is to investigate the properties of the resonance frequency spectrum in the problem of oscillations of volumetric magneto-dielectric resonators.
Materials and methods. The study is carried out by reducing the problem to the analysis of the system of 3D integro-differential equations in the domain of heterogeneity that determines the operator-function of the spectral parameter.
Results. We prove the theorem on the discreteness of the resonance frequency spectrum in the problem of oscillations in volume resonators that are bounded 3D anisotropic magneto-dielectric bodies whose permittivity and permeability functions are piecewise smooth. The problem is reduced to the analysis of the system of volume singular integral equations that defines a holomorphic Fredholm operator-function of the spectral parameter.
Conclusions. The method of volume integro-differential equations is an effective tool for analyzing the properties of the problem of electromagnetic oscillations of volume magneto-dielectric resonators.