We consider a waveguide that occupies a domain G with several cylindrical ends and is descried by the nonstationary equation 
where 
is a selfadjoint second order elliptic operator with variable coefficients. For the boundary condition we consider the Dirichlet, Neumann, or Robin ones. For the stationary problem with parameter we describe eigenfunctions of the continuous spectrum and a scattering matrix. Based on the limiting absorption principle, we obtain an expansion in eigenfunctions of the continuous spectrum. We compute wave operators and prove their completeness. We define a scattering operator and describe its connection with the scattering matrix. As a consequence, we construct scattering theory for the wave equation 
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Translated from Problemy Matematicheskogo Analiza 115, 2022, pp. 87-110.
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Plamenevskii, B.A., Poretskii, A.S. & Sarafanov, O.V. Mathematical Scattering Theory in Quantum and Acoustic Waveguides. J Math Sci 262, 329–357 (2022). https://doi.org/10.1007/s10958-022-05820-0
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DOI: https://doi.org/10.1007/s10958-022-05820-0