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 | References | Similar Articles | Additional Information

Abstract: A waveguide occupies a domain in , , having several cylindrical outlets to infinity. The waveguide is described by the Dirichlet problem for the Helmholtz equation. The scattering matrix with spectral parameter changes its size when crosses a threshold. To calculate in a neighborhood of a threshold, an ``augmented'' scattering matrix is introduced, which keeps its size near the threshold and is analytic in there. A minimizer of a quadratic functional serves as an approximation to a row of the matrix . 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 . As , the minimizer tends exponentially to the corresponding row of uniformly with respect to 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 are expressed in terms of those of the augmented matrix .

If an interval of the continuous spectrum contains no thresholds, the corresponding functional should be defined for the usual matrix and, as , its minimizer tends to the row of the scattering matrix at exponential rate uniformly with respect to .

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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