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Mat. Sb., 2017, Volume 208, Number 2, Pages 3–30 (Mi msb8657)  

This article is cited in 5 scientific papers (total in 5 papers)

On eigenvalues of a $\mathscr{PT}$-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

Abstract: We consider an elliptic operator with variable coefficients in a thin three-dimensional layer with $\mathscr{PT}$-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, $\mathscr{PT}$-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).

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English version:
Sbornik: Mathematics, 2017, 208:2, 173–199

Bibliographic databases:

UDC: 517.956+517.958
MSC: Primary 35P15; Secondary 35P20, 47F05
Received: 05.01.2016 and 21.05.2016

Citation: D. I. Borisov, M. Znojil, “On eigenvalues of a $\mathscr{PT}$-symmetric operator in a thin layer”, Mat. Sb., 208:2 (2017), 3–30; Sb. Math., 208:2 (2017), 173–199

Citation in format AMSBIB
\by D.~I.~Borisov, M.~Znojil
\paper On eigenvalues of a~$\mathscr{P\!T}$-symmetric operator in a~thin layer
\jour Mat. Sb.
\yr 2017
\vol 208
\issue 2
\pages 3--30
\jour Sb. Math.
\yr 2017
\vol 208
\issue 2
\pages 173--199

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    This publication is cited in the following articles:
    1. M. Znojil, “Bound states emerging from below the continuum in a solvable $\mathscr{PT}$-symmetric discrete Schrödinger equation”, Phys. Rev. A, 96:1 (2017), 012127  crossref  mathscinet  isi  scopus
    2. J. Behrndt, M. Langer, V. Lotoreichik, J. Rohleder, “Spectral enclosures for non-self-adjoint extensions of symmetric operators”, J. Funct. Anal., 275:7 (2018), 1808–1888  crossref  mathscinet  zmath  isi  scopus
    3. A. Yu. Trynin, “Skhodimost protsessov Lagranzha–Shturma–Liuvillya dlya nepreryvnykh funktsii ogranichennoi variatsii”, Vladikavk. matem. zhurn., 20:4 (2018), 76–91  mathnet  crossref
    4. Borisov D. Cardone G., “Spectra of Operator Pencils With Small P & Xdcab;& X1D4Af;& Xdcaf;-Symmetric Periodic Perturbation”, ESAIM-Control OPtim. Calc. Var., 26 (2020), UNSP 21  crossref  mathscinet  isi
    5. A. Yu. Trynin, “O ravnomernom priblizhenii interpolyatsionnymi mnogochlenami Lagranzha po matritse uzlov Yakobi ${\mathcal L}_n^{(\alpha_n,\beta_n)}$ funktsii ogranichennoi variatsii”, Izv. RAN. Ser. matem., 84:6 (2020), 197–222  mathnet  crossref
  • Математический сборник Sbornik: Mathematics (from 1967)
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