The Maxwell system in waveguides with several cylindrical ends

Abstract:

A waveguide coincides with a domain $G$ in $\mathbb {R}^3$ having finitely many cylindrical outlets to infinity; the boundary $\partial G$ is smooth. In $G$, the stationary Maxwell system is considered with spectral parameter $k \in \mathbb {R}$ and the identity matrices of dielectric permittivity and magnetic permeability. The boundary $\partial G$ is assumed to be perfectly conductive. In the presence of charges and currents, the solvability is studied of the corresponding boundary value problem supplemented with “intrinsic” radiation conditions at infinity. For all $k$ in the continuous spectrum of the problem (including the thresholds and eigenvalues), a basis in the space of continuous spectrum eigenfunctions is described and the scattering matrix is defined and is shown to be unitary. For this, the Maxwell system is extended to an elliptic one, and the latter is studied in detail. The information on the Maxwell boundary value problem comes from that obtained for the elliptic problem.