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 Ufimsk. Mat. Zh., 2014, Volume 6, Issue 1, Pages 30–58 (Mi ufa231)

Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide

D.I. Borisovab

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

Abstract: In a thin multidimensional layer we consider a differential second order $\mathcal{PT}$-symmetric operator. The operator is of rather general form and its coefficients are arbitrary functions depending both on slow and fast variables. The $\mathcal{PT}$-symmetry of the operator is ensured by the boundary conditions of Robin type with pure imaginary coefficient. In the work we determine the limiting operator, prove the uniform resolvent convergence of the perturbed operator to the limiting one, and derive the estimates for the rates of convergence. We establish the convergence of the spectrum of perturbed operator to that of the limiting one. For the perturbed eigenvalues converging to the limiting discrete ones we prove that they are real and construct their complete asymptotic expansions. We also obtain the complete asymptotic expansions for the associated eigenfunctions.

Keywords: $\mathcal{PT}$-symmetric operator, thin domain, uniform resolvent convergence, estimates for the rate of convergence, spectrum, asymptotic expansions.

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English version:
Ufa Mathematical Journal, 2014, 6:1, 29–55 (PDF, 671 kB); https://doi.org/10.13108/2014-6-1-29

Bibliographic databases:

UDC: 517.9
MSC: 35P05, 35B25, 35C20

Citation: D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide”, Ufimsk. Mat. Zh., 6:1 (2014), 30–58; Ufa Math. J., 6:1 (2014), 29–55

Citation in format AMSBIB
\Bibitem{Bor14} \by D.I.~Borisov \paper Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide \jour Ufimsk. Mat. Zh. \yr 2014 \vol 6 \issue 1 \pages 30--58 \mathnet{http://mi.mathnet.ru/ufa231} \elib{http://elibrary.ru/item.asp?id=21290425} \transl \jour Ufa Math. J. \yr 2014 \vol 6 \issue 1 \pages 29--55 \crossref{https://doi.org/10.13108/2014-6-1-29} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000371149500003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84899697311} 

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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, “The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883
2. R. Novak, “Bound States in Waveguides With Complex Robin Boundary Conditions”, Asymptotic Anal., 96:3-4 (2016), 251–281
3. D. I. Borisov, M. Znojil, “On eigenvalues of a $\mathscr{PT}$-symmetric operator in a thin layer”, Sb. Math., 208:2 (2017), 173–199
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