Behavior of waveguide scattering matrices in a neighborhood of thresholds
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B. A. Plamenevskii and A. S. Poretskii
Translated by: B. A. Plamenevskii - St. Petersburg Math. J. 30 (2019), 285-319
- DOI: https://doi.org/10.1090/spmj/1543
- Published electronically: February 14, 2019
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Abstract:
A waveguide occupies a $d+1$-dimensional domain with several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem with spectral parameter $\mu$, selfadjoint with respect to the Green formula. At infinity, the coefficients of the problem stabilize at an exponential rate to functions independent of the axial variable in the corresponding cylinder. On every interval of the continuous spectrum between neighboring “thresholds”, a unitary scattering matrix $S(\mu )$ is defined; the size of $S(\mu )$ is finite for any $\mu$, remains to be constant on any such interval, and varies from an interval to an interval. The basic result claims the existence of finite one-sided limits of the scattering matrix $S(\mu )$ at every threshold.References
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Bibliographic Information
- B. A. Plamenevskii
- Affiliation: St. Petersburg State University, Universitetskaya Emb. 7/9, 199034 St. Petersburg, Russia
- Email: b.plamenevskii@spbu.ru
- A. S. Poretskii
- Affiliation: St. Petersburg State University, Universitetskaya Emb. 7/9, 199034 St. Petersburg, Russia
- Email: st036768@student.spbu.ru
- Received by editor(s): July 12, 2018
- Published electronically: February 14, 2019
- Additional Notes: The study was supported by the project of Russian Science Foundation no. 17-11-01126
- © Copyright 2019 American Mathematical Society
- Journal: St. Petersburg Math. J. 30 (2019), 285-319
- MSC (2010): Primary 35P25; Secondary 47A70
- DOI: https://doi.org/10.1090/spmj/1543
- MathSciNet review: 3790737
Dedicated: Dedicated to the memory of Vladimir Ivanovich Smirnov on the occasion of the 130th anniversary of his birth