Method for computing waveguide scattering matrices in the vicinity of thresholds
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B. A. Plamenevskiǐ, A. S. Poretskiǐ and O. V. Sarafanov
Translated by: B. A. Plamenevskiǐ - St. Petersburg Math. J. 26 (2015), 91-116
- DOI: https://doi.org/10.1090/S1061-0022-2014-01332-5
- Published electronically: November 21, 2014
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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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Bibliographic 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
- 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)
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 26 (2015), 91-116
- MSC (2010): Primary 47A40
- DOI: https://doi.org/10.1090/S1061-0022-2014-01332-5
- MathSciNet review: 3234806