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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

The Maxwell system in waveguides with several cylindrical ends


Authors: B. A. Plamenevskiĭ and A. S. Poretskiĭ
Translated by: B. A. Plamenevskiĭ
Original publication: Algebra i Analiz, tom 25 (2013), nomer 1.
Journal: St. Petersburg Math. J. 25 (2014), 63-104
MSC (2010): Primary 35N25, 35Q61, 35P25, 78A50
Published electronically: November 20, 2013
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Abstract: A waveguide coincides with a domain $ G$ in $ \mathbb{R}^3$ having finitely many cylindrical outlets to infinity; the boundary $ \partial G$ is smooth. In $ G$, the stationary Maxwell system is considered with spectral parameter $ k \in \mathbb{R}$ and the identity matrices of dielectric permittivity and magnetic permeability. The boundary $ \partial G$ is assumed to be perfectly conductive. In the presence of charges and currents, the solvability is studied of the corresponding boundary value problem supplemented with ``intrinsic'' radiation conditions at infinity. For all $ k$ in the continuous spectrum of the problem (including the thresholds and eigenvalues), a basis in the space of continuous spectrum eigenfunctions is described and the scattering matrix is defined and is shown to be unitary. For this, the Maxwell system is extended to an elliptic one, and the latter is studied in detail. The information on the Maxwell boundary value problem comes from that obtained for the elliptic problem.


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Additional Information

B. A. Plamenevskiĭ
Affiliation: Mathematical Physics Division, Physics Department, St. Petersburg State University, Ulianovskaya str. 3, Petrodvoretz, St. Petersburg 198504, Russia
Email: boris.plamen@gmail.com

A. S. Poretskiĭ
Affiliation: Mathematical Physics Division, Physics Department, St. Petersburg State University, Ulianovskaya str. 3, Petrodvoretz, St. Petersburg 198504, Russia
Email: poras1990@list.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01280-5
PII: S 1061-0022(2013)01280-5
Keywords: Elliptic extension, radiation principle, eigenfunctions of continuous spectrum, unitary scattering matrix
Received by editor(s): May 28, 2012
Published electronically: November 20, 2013
Additional Notes: Supported by the Chebyshev Laboratory, St. Petersburg State University (RF Government grant 11.G34.31.0026) and by RFBR (grant no. 12-01-00247a)
Article copyright: © Copyright 2013 American Mathematical Society