General elliptic boundary value problems with spectral parameter are studied in a domain \( G \subset {\mathbb{R}^{n + 1}} \) with several “quasicylindrical exits” at infinity. The problems are selfadjoint relative to the Green formula. The coefficients of the boundary value problems are smooth and are slowly stabilized at each exit at infinity to functions that are periodic with respect to the axis t-coordinate. We describe the asymptotics of solutions at infinity. We formulate a well-posed problem with radiation conditions, introduce a unitary scattering matrix, and justify a method for computing this matrix. The results are new even in the particular case where the role of “quasicylinders” are played by usual cylinders (with a constant cross-section) and the coefficients are stabilized at infinity to functions independent of the axis coordinate. Bibliography: 12 titles.
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Translated from Problems in Mathematical Analysis 64, 2012, p. 93–117.
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Neittaanmäki, P., Plamenevskii, B.A. & Sarafanov, O.V. Radiation and scattering in domains with periodic waveguides under slow stabilization of characteristics of a medium. J Math Sci 184, 331–361 (2012). https://doi.org/10.1007/s10958-012-0871-7
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DOI: https://doi.org/10.1007/s10958-012-0871-7