Discrete spectrum of cranked, branching, and periodic waveguides
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S. A. Nazarov
Translated by: A. Plotkin - St. Petersburg Math. J. 23 (2012), 351-379
- DOI: https://doi.org/10.1090/S1061-0022-2012-01200-8
- Published electronically: January 24, 2012
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Abstract:
The variational method is applied to the study of the spectrum of the Laplace operator with mixed boundary conditions and with Dirichlet conditions in planar or multidimensional domains (waveguides) with cylindrical or periodic exits to infinity. The planar waveguides of constant width are discussed completely, such as cranked, broken, smoothly bent, or branching waveguides. For them, the existence of eigenvalues below the continuous spectrum threshold is established. A similar result is obtained for the multidimensional cranked and branching waveguides, and also for some periodic ones. Several open questions are stated; in particular, they concern problems with Neumann boundary conditions, full multiplicity of the discrete spectrum, and planar waveguides with piecewise constant boundary.References
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Bibliographic Information
- S. A. Nazarov
- Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, St. Petersburg 199178, Russia
- MR Author ID: 196508
- Email: srgnazarov@yahoo.co.uk
- Received by editor(s): October 10, 2009
- Published electronically: January 24, 2012
- Additional Notes: Supported by RFBR (grant no. 09-01-00759)
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 351-379
- MSC (2010): Primary 35Q60
- DOI: https://doi.org/10.1090/S1061-0022-2012-01200-8
- MathSciNet review: 2841676