Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

Request Permissions   Purchase Content 


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
Published electronically: November 21, 2014
MathSciNet review: 3234806
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]$.

References [Enhancements On Off] (What's this?)

  • 1. S. A. Nazarov and B. A. Plamenevskiĭ, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expos. Math., vol. 13, Walter de Gruyter, Berlin, 1994. MR 1283387 (95h:35001)
  • 2. B. A. Plamenevskiĭ and O. V. Sarafanov, On a method for computing the scattering matrices of waveguides, Algebra i Analiz 23 (2011), no. 1, 200-231; Englisgh transl., St. Petersburg Math. J. 23 (2012), no. 1, 139-160. MR 2760152 (2012d:35259)
  • 3. -, On a method for computing waveguide scattering matrices in the presence of point spectrum, Functsional. Anal. i Prilozen. 48 (2014), no. 1, 61-72. (Russian) MR 3204678
  • 4. M. Costabel and M. Dauge, Stable asymptotics for elliptic systems on plane domains with corners, Comm. Partial Differential Equations 19 (1994), no. 9-10, 1677-1726. MR 1294475 (95g:35051)
  • 5. V. G. Maz'ya and J. Rossmann, On a problem of Babuška (stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points), Math. Nachr. 155 (1992), 199-220. MR 1231265 (94i:35059)
  • 6. I. V. Kamotskiĭ and S. A. Nazarov, Wood's anomalies and surface waves in the problem of scattering by a periodic boundary. I-II, Mat. Sb. 190 (1999), no. 1, 109-138; no. 2, 43-47; English transl., Sb. Math. 190 (1999), no. 1, 111-141; no. 2, 205-231. MR 11700697 (2000j:35047); MR 1701000 (2000j:35048)
  • 7. S. A. Nazarov and B. A. Plamenevskiĭ, Selfadjoint elliptic problems with radiation conditions on the edges of the boundary, Algebra i Analiz 4 (1992), no. 3, 196-225; English transl., St. Petersburg Math. J. 4 (1993), no. 3, 569-594. MR 1190778 (93k:35080)
  • 8. I. V. Kamotskiĭ and S. A. Nazarov, An augmented scattering matrix and exponentially decreasing solutions of an elliptic problem in a cylindrical domain, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 264 (2000), 66-82; English transl., J. Math. Sci. (N.Y.) 111 (2002), no. 4, 36-3666. MR 1796996 (2001m:35246)
  • 9. I. C. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of linear operators. I, Oper. Theory: Adv. Appl., vol. 49, Birkhäuser, Basel-Boston-Berlin, 1990. MR 1130394 (93d:47002)
  • 10. I. C. Gohberg and E. I. Sigal, An operator generalization of the logarithmic residue theorem and Rouche's theorem, Math. Sb. 84 (1971), no. 4, 607-629. (Russian) MR 0313856 (47:2409)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 47A40

Retrieve articles in all journals with MSC (2010): 47A40

Additional Information

B. A. Plamenevskiǐ
Affiliation: Division of Mathematical Physics, Physics Department, St. Petersburg State University, Russia

A. S. Poretskiǐ
Affiliation: Division of Mathematical Physics, Physics Department, St. Petersburg State University, Russia

O. V. Sarafanov
Affiliation: Division of Mathematical Physics, Physics Department, St. Petersburg State University, Russia; Division of Mathematical Information Technology, University of Jyväskylä, Finland

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

American Mathematical Society