Mathematical scattering theory in electromagnetic waveguides

A waveguide occupying a 3D domain $G$ with several cylindrical outlets to infinity is described by the non-stationary Maxwell system with conductive boundary conditions. Dielectric permittivity and magnetic permeability are assumed to be positive definite matrices $\varepsilon(x)$ and $\mu(x)$ depending on a point $x$ in $G$. At infinity, in each cylindrical outlet, the matrix-valued functions converge with exponential rate to matrix-valued functions that do not depend on the axial coordinate of the cylinder. For the corresponding stationary problem with spectral parameter, we define continuous spectrum eigenfunctions and the scattering matrix. The non-stationary Maxwell system is extended up to an equation of the form $i\,\partial_t \mathcal{U}(x,t)=\mathcal{A}(x,D_x)\mathcal{U}(x,t)$ with elliptic operator $\mathcal{A}(x,D_x)$. We associate with the equation a boundary value problem and, for an appropriate couple of such problems, construct the scattering theory. We calculate the wave operators, define the scattering operator, and describe its relation to the scattering matrix. From the obtained results we extract information about the original Maxwell system.