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Almost standing waves in a periodic waveguide with resonator, and near-threshold eigenvalues


Author: S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 28 (2016), nomer 3.
Journal: St. Petersburg Math. J. 28 (2017), 377-410
MSC (2010): Primary 81Q10
DOI: https://doi.org/10.1090/spmj/1455
Published electronically: March 29, 2017
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Abstract: The definition and an existence criterion are given for the standing waves at the threshold of the continuous spectrum for a periodic quantum waveguide with a resonator (the Dirichlet problem for the Laplace operator). Such waves and their linear combinations do not transfer energy to infinity, and they only differ from the standing waves with the zero Floquet parameter by an exponentially decaying term. It is shown that the almost standing and trapped waves at the threshold generate eigenvalues in the discrete spectrum of a waveguide with a regular sloping local perturbation of the wall.


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Additional Information

S. A. Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvoretz, 198504 St. Petersburg, Russia;; Peter the Great St. Petersburg Polytechnical University, Polytechnicheskaya ul. 29, 195251 St. Petersburg, Russia;; Institute of Mechanical Engineering Problems, Bol′shoǐ pr. V. O. 61, 199178 St. Petersburg, Russia
Email: srgnazarov@yahoo.co.uk, s.nazarov@spbu.ru

DOI: https://doi.org/10.1090/spmj/1455
Keywords: Periodic waveguide, resonator, discrete spectrum, almost standing waves, threshold scattering matrix, asymptotics
Received by editor(s): November 20, 2015
Published electronically: March 29, 2017
Additional Notes: Supported by RFBR (project no. 15-01-02175)
Article copyright: © Copyright 2017 American Mathematical Society