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 Zap. Nauchn. Sem. POMI, 2012, Volume 409, Pages 130–150 (Mi znsl5516)

Structure of the spectrum of the periodic family of identical cells connected through apertures of reducing sizes

a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia
b St. Petersburg State University, St. Petersburg, Russia
c University of Helsinki, Department of Mathematics and Statistics, Helsinki, Finland

Abstract: A waveguide is constructed such that the Dirichlet problem for the Laplace operator gets the essential spectrum implying a countable set of points in the real positive semi-axis. The waveguide is obtained by joining identical cells through apertures in their common walls. Size of the apertures decreases at distance from the “central” cell. It is shown that the first point of the essential spectrum is a limit of an infinite sequence of eigenvalues of the problem from its discrete spectrum. A hypothesis on the structure of the discrete spectrum inside spectral gaps is formulated and other open questions are mentioned.

Key words and phrases: Dirichlet problem, Helmholtz operator, waveguide, essential spectrum, infinite family of spectral gaps.

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English version:
Journal of Mathematical Sciences (New York), 2013, 194:1, 72–82

Bibliographic databases:

UDC: 517.956.227

Citation: S. A. Nazarov, J. Taskinen, “Structure of the spectrum of the periodic family of identical cells connected through apertures of reducing sizes”, Mathematical problems in the theory of wave propagation. Part 42, Zap. Nauchn. Sem. POMI, 409, POMI, St. Petersburg, 2012, 130–150; J. Math. Sci. (N. Y.), 194:1 (2013), 72–82

Citation in format AMSBIB
\Bibitem{NazTas12} \by S.~A.~Nazarov, J.~Taskinen \paper Structure of the spectrum of the periodic family of identical cells connected through apertures of reducing sizes \inbook Mathematical problems in the theory of wave propagation. Part~42 \serial Zap. Nauchn. Sem. POMI \yr 2012 \vol 409 \pages 130--150 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl5516} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3032233} \transl \jour J. Math. Sci. (N. Y.) \yr 2013 \vol 194 \issue 1 \pages 72--82 \crossref{https://doi.org/10.1007/s10958-013-1508-1} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84898972295}