Mathematical scattering theory in electromagnetic waveguides

The waveguide occupies a domain $G\subset \bR^3$ having several cylindrical outlets to infinity and is described by the non-stationary Maxwell system with perfectly conductive boundary conditions. For the corresponding stationary problem with spectral parameter, the continuous spectrum eigenfunctions and the scattering matrix are defined. We calculate the wave operators, introduce the scattering operator and describe its connection with the scattering matrix. The proof is based on extending the Maxwell system to an equation of the form $i\partial_t \Psi(x,t)=\cA(x,D_x)\Psi(x,t)$ with elliptic operator $\cA(x,D_x)$. We associate to the equation an initial boundary value problem and develop the scattering theory for the problem. The information on the Maxwell system is derived from the results obtained for the extended problem.
Based on a joint research with B.A. Plamenevskii and O.V. Sarafanov.