Mathematical scattering theory in quantum waveguides

A waveguide occupies a domain G with several cylindrical ends. The waveguide is described by a nonstationary equation of the form i∂_t f = Af , where A is a selfadjoint second order elliptic operator with variable coefficients (in particular, for A = -Δ, where Δ stands for the Laplace operator, the equation coincides with the Schrodinger equation). For the corresponding stationary problem with spectral parameter, we define continuous spectrum eigenfunctions and a scattering matrix. The limiting absorption principle provides expansion in the continuous spectrum eigenfunctions. We also calculate wave operators and prove their completeness. Then we define a scattering operator and describe its connections with the scattering matrix.