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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On a method for computing waveguide scattering matrices


Authors: B. A. Plamenevskiĭ and O. V. Sarafanov
Translated by: B. A. Plamenevskiĭ
Original publication: Algebra i Analiz, tom 23 (2011), nomer 1.
Journal: St. Petersburg Math. J. 23 (2012), 139-160
MSC (2010): Primary 35P25
Published electronically: November 10, 2011
MathSciNet review: 2760152
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Abstract: A waveguide lies in $ \mathbb{R}^2$ and, outside a large disk, coincides with the union of finitely many nonoverlapping semistrips (``cylindrical ends''). It is described by a Dirichlet problem for the Helmholtz equation. As an approximation for a row of the scattering matrix $ S(\mu )$, where $ \mu $ is the spectral parameter, the minimizer of a quadratic functional $ J^R(\cdot , \mu )$ is used. To construct this functional, an auxiliary boundary value problem is solved in the bounded domain obtained by truncating the cylindrical ends of the waveguide at a distance $ R$. As $ R\to \infty $, the minimizer $ a (R, \mu )$ tends with exponential rate to the corresponding row of the scattering matrix uniformly on every finite closed interval of the continuous spectrum containing no thresholds. Such an interval may contain eigenvalues of the waveguide (with eigenfunctions exponentially decaying at infinity). The applicability of this method goes far beyond the simplest model considered in the paper.


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Additional Information

B. A. Plamenevskiĭ
Affiliation: Department of Mathematical Physics, St. Petersburg State University, Ul′yanovskaya 1, St. Petersburg 198504, Russia
Email: boris.plamen@gmail.com

O. V. Sarafanov
Affiliation: Department of Mathematical Physics, St. Petersburg State University, Ul′yanovskaya 1, St. Petersburg 198504, Russia
Email: saraf@math.nw.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01190-2
PII: S 1061-0022(2011)01190-2
Keywords: Waveguide, scattering matrix, approximation, minimizer
Received by editor(s): September 1, 2010
Published electronically: November 10, 2011
Additional Notes: Supported by grants NSh-816.2008.1 and RFBR-09-01-00191-a
Dedicated: To the memory of M. Sh. Birman
Article copyright: © Copyright 2011 American Mathematical Society