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Sbornik: Mathematics, 2017, Volume 208, Issue 2, Pages 173–199
DOI: https://doi.org/10.1070/SM8657 (Mi sm8657) 		 
This article is cited in 10 scientific papers (total in 10 papers)

On eigenvalues of a $\mathscr{P\!T}$-symmetric operator in a thin layer

D. I. Borisovabc, M. Znojild

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa
b Bashkir State Pedagogical University, Ufa
c University of Hradec Králové, Czech Republic
d Nuclear Physics Institute of the Czech Academy of Sciences, Řež, Czech Republic
English version PDF (605 kB) Citations (10)
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DOI: https://doi.org/10.1070/SM8657
Abstract: We consider an elliptic operator with variable coefficients in a thin three-dimensional layer with $\mathscr{P\!T}$-symmetric boundary conditions. We study the effect of the appearance of isolated eigenvalues at the edges of the gaps in the essential spectrum. We obtain sufficient conditions that guarantee that such eigenvalues either exist or are absent near a given edge of a gap. In the case of existence, the first terms in the asymptotic expansion of these emerging eigenvalues are calculated.
Bibliography: 34 titles.
Keywords: thin domain, $\mathscr{P\!T}$-symmetric operator, edge of a gap, asymptotics, periodic operator.
Funding agency Grant number
Russian Foundation for Basic Research 14-01-97009-р_поволжье_а
Akademie Věd České Republiky RVO61389005
Grantová Agentura České Republiky 16-22945S
D. I. Borisov's research was supported by the Russian Foundation for Basic Research (grant no. 14-01-97009-{\selectlanguage{russian}р_поволжье_а}). M. Znojil's research was supported by the Nuclear Physics Institute of the Czech Academy of Sciences (research plan RVO61389005) and the Czech Science Foundation GAČR (standard grant no. 16-22945S).
Received: 05.01.2016 and 21.05.2016
Russian version:
Matematicheskii Sbornik, 2017, Volume 208, Number 2, Pages 3–30
DOI: https://doi.org/10.4213/sm8657
Bibliographic databases:
UDC: 517.956+517.958
MSC: Primary 35P15; Secondary 35P20, 47F05
Language: English
Original paper language: Russian
Citation: D. I. Borisov, M. Znojil, “On eigenvalues of a $\mathscr{P\!T}$-symmetric operator in a thin layer”, Mat. Sb., 208:2 (2017), 3–30; Sb. Math., 208:2 (2017), 173–199
Citation in format AMSBIB
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    • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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