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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Method for computing waveguide scattering matrices in the vicinity of thresholds


Authors: B. A. Plamenevskiǐ, A. S. Poretskiǐ and O. V. Sarafanov
Translated by: B. A. Plamenevskiǐ
Original publication: Algebra i Analiz, tom 26 (2014), nomer 1.
Journal: St. Petersburg Math. J. 26 (2015), 91-116
MSC (2010): Primary 47A40
DOI: https://doi.org/10.1090/S1061-0022-2014-01332-5
Published electronically: November 21, 2014
MathSciNet review: 3234806
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Abstract: A waveguide occupies a domain $ G$ in $ \mathbb{R}^{n+1}$, $ n\geq 1$, having several cylindrical outlets to infinity. The waveguide is described by the Dirichlet problem for the Helmholtz equation. The scattering matrix $ S(\mu )$ with spectral parameter $ \mu $ changes its size when $ \mu $ crosses a threshold. To calculate $ S(\mu )$ in a neighborhood of a threshold, an ``augmented'' scattering matrix $ \mathcal {S} (\mu )$ is introduced, which keeps its size near the threshold and is analytic in $ \mu $ there. A minimizer of a quadratic functional $ J^R(\,\cdot \,, \mu )$ serves as an approximation to a row of the matrix $ \mathcal {S}(\mu )$. To construct such a functional, an auxiliary boundary-value problem is solved in the bounded domain obtained by cutting off the waveguide outlets to infinity at a distance $ R$. As $ R\to \infty $, the minimizer $ a (R, \mu )$ tends exponentially to the corresponding row of $ \mathcal {S}(\mu )$ uniformly with respect to $ \mu $ in a neighborhood of the threshold. The neighborhood may contain some waveguide eigenvalues corresponding to eigenfunctions exponentially decaying at infinity. Finally, the elements of the ``ordinary'' scattering matrix $ S(\mu )$ are expressed in terms of those of the augmented matrix $ \mathcal {S}(\mu )$.

If an interval $ [\mu _1, \mu _2]$ of the continuous spectrum contains no thresholds, the corresponding functional $ J^R(\,\cdot \,, \mu )$ should be defined for the usual matrix $ S(\mu )$ and, as $ R\to \infty $, its minimizer $ a (R, \mu )$ tends to the row of the scattering matrix at exponential rate uniformly with respect to $ \mu \in [\mu _1, \mu _2]$.


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Additional Information

B. A. Plamenevskiǐ
Affiliation: Division of Mathematical Physics, Physics Department, St. Petersburg State University, Russia
Email: boris.plamen@gmail.com

A. S. Poretskiǐ
Affiliation: Division of Mathematical Physics, Physics Department, St. Petersburg State University, Russia
Email: poras1990@list.ru

O. V. Sarafanov
Affiliation: Division of Mathematical Physics, Physics Department, St. Petersburg State University, Russia; Division of Mathematical Information Technology, University of Jyväskylä, Finland
Email: saraf@math.nw.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01332-5
Keywords: Augmented scattering matrix, threshold limits, minimizer for a functional, exponential convergence
Published electronically: November 21, 2014
Additional Notes: The authors were supported by RFBR (grant no. 12-01-00247a) and by Scientific Schools (grant no. 357.2012.1)
Article copyright: © Copyright 2014 American Mathematical Society

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