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Discrete spectrum of cranked, branching, and periodic waveguides


Author: S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 23 (2011), nomer 2.
Journal: St. Petersburg Math. J. 23 (2012), 351-379
MSC (2010): Primary 35Q60
DOI: https://doi.org/10.1090/S1061-0022-2012-01200-8
Published electronically: January 24, 2012
MathSciNet review: 2841676
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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.


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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, St. Petersburg 199178, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/S1061-0022-2012-01200-8
Keywords: Eigenvalues, discrete spectrum, Laplace operator, cranked, branching, and periodic waveguides
Received by editor(s): October 10, 2009
Published electronically: January 24, 2012
Additional Notes: Supported by RFBR (grant no. 09-01-00759)
Article copyright: © Copyright 2012 American Mathematical Society