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

DOI:
https://doi.org/10.1090/S1061-0022-2011-01190-2

Published electronically:
November 10, 2011

MathSciNet review:
2760152

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A waveguide lies in 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 , where is the spectral parameter, the minimizer of a quadratic functional 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 . As , the minimizer 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.

**1.**V. Grikurov, E. Heikkola, P. Neittaanmäki, and B. Plamenevskii,*On computation of scattering matrices and on surface waves for diffraction gratings*, Numer. Math.**94**(2003), no. 2, 269-288. MR**1974556 (2004c:78018)****2.**V. O. Kal'vin, P. Neittaanmäki, and B. A. Plamenevskiĭ,*Method for computing scattering matrices for general dissipative and selfadjoint elliptic problems in domains with cylindrical ends*, Probl. Mat. Anal., No. 28, Tamara Rozhkovskaya, Novosibirsk, 2004, pp. 37-65; English transl., J. Math. Sci. (N. Y.)**122**(2004), no. 3, 3212-3245. MR**2084178 (2005h:35072)****3.**S. A. Nazarov and B. A. Plamenevskiĭ,*Elliptic problems in domains with piecewise smooth boundaries*, Nauka, Moscow, 1991; English version, de Gruyter Exp. in Math., vol. 13, Walter de Gruyter and Co., Berlin, 1994. MR**1283387 (95h:35001)****4.**V. Maz'ya, S. Nazarov, and B. Plamenevskii,*Asymptotic theory of elliptic boundary value problems in singularly perturbed domains*. Vol. I, Oper. Theory Adv. Appl., vol. 111, Birkhäuser, Basel, 2000. MR**1779977 (2001e:35044a)****5.**I. Gohberg, S. Goldberg, and M. A. Kaashoek,*Classes of linear operators*. Vol. I, Oper. Theory Adv. Appl., vol. 49, Birkhäuser, Basel, 1990. MR**1130394 (93d:47002)****6.**V. A. Kondrat'ev,*Boundary value problems for elliptic equations in domains with conical or angular points*, Trudy Moskov. Mat. Obshch.**16**(1967), 209-292; English transl. in Trans. Moscow Math. Soc.**1967**(1968). MR**0226187 (37:1777)****7.**L. Bers, F. John, and M. Schechter,*Partial differential equations*, Lectures in Appl. Math., vol. 3A, Amer. Math. Soc., Prodidence, RI, 1979. MR**0598466 (82c:35001)**

Retrieve articles in *St. Petersburg Mathematical Journal*
with MSC (2010):
35P25

Retrieve articles in all journals with MSC (2010): 35P25

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:
https://doi.org/10.1090/S1061-0022-2011-01190-2

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