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 Trudy Inst. Mat. i Mekh. UrO RAN, 2012, Volume 18, Number 2, Pages 22–37 (Mi timm805)

On a $\mathcal{PT}$-symmetric waveguide with a pair of small holes

D. I. Borisovab

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
b Bashkir State Pedagogical University

Abstract: A planar $\mathcal{PT}$-symmetric waveguide with a pair of small holes is considered. The waveguide is modeled by a planar infinite strip in which a pair of symmetric small holes is cut out. The operator is the Laplacian with $\mathcal{PT}$-symmetric boundary condition at the edges of the strip and Neumann condition at the boundaries of the holes. For this operator, the uniform resolvent convergence is established and the convergence rate is estimated. The effect of the generation by the holes of new eigenvalues from the boundary of the continuous spectrum is studied. Sufficient conditions for the existence and absence of such eigenvalues are obtained and the first terms of their asymptotic expansions are found.

Keywords: $\mathcal{PT}$-symmetric waveguide, small hole, uniform resolvent convergence, asymptotics.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2013, 281, suppl. 1, 5–21

Bibliographic databases:

UDC: 517.984.5+517.955.8

Citation: D. I. Borisov, “On a $\mathcal{PT}$-symmetric waveguide with a pair of small holes”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 2, 2012, 22–37; Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 5–21

Citation in format AMSBIB
\Bibitem{Bor12} \by D.~I.~Borisov \paper On a~$\mathcal{PT}$-symmetric waveguide with a~pair of small holes \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2012 \vol 18 \issue 2 \pages 22--37 \mathnet{http://mi.mathnet.ru/timm805} \elib{http://elibrary.ru/item.asp?id=17736183} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2013 \vol 281 \issue , suppl. 1 \pages 5--21 \crossref{https://doi.org/10.1134/S0081543813050027} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000320460300002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84879143282} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide”, Ufa Math. J., 6:1 (2014), 29–55
2. D.I. Borisov, “The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883
3. D. B. Davletov, D. V. Kozhevnikov, “The problem of Steklov type in a half-cylinder with a small cavity”, Ufa Math. J., 8:4 (2016), 62–87
4. D. I. Borisov, M. N. Konyrkulzhaeva, “Simplest graphs with small edges: asymptotics for resolvents and holomorphic dependence of spectrum”, Ufa Math. J., 11:2 (2019), 56–70
5. Paul B., Dhar H., Chowdhury M., Saha B., “Treating Ostrogradski Instability For Galilean Invariant Chern-Simon'S Model Via Pt Symmetry”, Phys. Rev. D, 99:6 (2019), 065018
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