On Fredholm property of an integro-differential operator in the problem of electromagnetic wave diffraction on a volumetric body, partially screened by a system of flat screens

Background. The aim of this work is to study a new vector problem of electromagnetic wave scattering on a partially shielded volumetric inhomogeneous anisotropic body. Material and methods . The problem is considered in the quasiclassical formulation; the original boundary value problem is reduced to a system of integro-differential equations; the properties of the system are studied using pseudodifferential calculus in Sobolev spaces on manifolds with a boundary. Results. The quasiclassical formulation of the diffraction problem is proposed; the boundary value problem for Maxwell's equations is reduced to a system of integro-differential equations; the operator of this system is treated as a pseudodifferential operator ($\psi$DO) in Sobolev spaces on manifolds with a boundary; the quadratic form of the matrix $\psi$DO is studied and is shown to be coercive; the Fredholm property of the $\psi$DO is proved. Conclusions. The matrix $\psi$DO is proved to be a Fredholm operator of zero index; this results can be used for further theoretical study of the diffraction problem as well as for validation of numerical methods.