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 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.
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