Mathematical Scattering Theory in Quantum Waveguides

Abstract

A waveguide occupies a domain G with several cylindrical ends. The waveguide is described by a nonstationary equation of the form \(i{{\partial }_{t}}f = \mathcal{A}f\), where \(\mathcal{A}\) is a selfadjoint second order elliptic operator with variable coefficients (in particular, for \(\mathcal{A} = - \Delta \), where Δ stands for the Laplace operator, the equation coincides with the Schrödinger 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.