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

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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2012, Volume 18, Number 2, Pages 22–37 (Mi timm805)

This article is cited in 12 scientific papers (total in 12 papers)

On a

P T

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

D. I. Borisov^ab

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

Full-text PDF (259 kB) Citations (12)

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Abstract: A planar

P T

$\mathcal{PT}$ -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

P T

$\mathcal{PT}$ -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:

P T

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

Received: 12.09.2011

English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2013, Volume 281, Issue 1, Pages 5–21
DOI: https://doi.org/10.1134/S0081543813050027

Bibliographic databases:

Document Type: Article

UDC: 517.984.5+517.955.8

Language: Russian

Citation: D. I. Borisov, “On a

P T

$\mathcal{PT}$ -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

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\jour Proc. Steklov Inst. Math. (Suppl.)

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Linking options:

https://www.mathnet.ru/eng/timm805

https://www.mathnet.ru/eng/timm/v18/i2/p22

This publication is cited in the following 12 articles:

D. I. Borisov, P. Exner, “Approximation of point interactions by geometric perturbations in two-dimensional domains”, Bull. Math. Sci., 13:02 (2023)
Denis Ivanovich Borisov, “Geometric Approximation of Point Interactions in Two-Dimensional Domains for Non-Self-Adjoint Operators”, Mathematics, 11:4 (2023), 947
D. I. Borisov, “Norm Resolvent Convergence of Elliptic Operators in Domains with Thin Spikes”, J Math Sci, 261:3 (2022), 366
D. B. Davletov, O. B. Davletov, R. R. Davletova, A. A. Ershov, “Skhodimost sobstvennykh elementov kraevoi zadachi tipa Steklova dlya operatora Lame”, Tr. IMM UrO RAN, 27, no. 1, 2021, 37–47
D. I. Borisov, D. A. Zezyulin, “Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole. I. Meromorphic Continuations”, Russ. J. Math. Phys., 28:4 (2021), 416
D. I. Borisov, G. Cardone, G. A. Chechkin, Yu. O. Koroleva, “On elliptic operators with Steklov condition perturbed by Dirichlet condition on a small part of boundary”, Calc. Var., 60:1 (2021)
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
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
D. I. Borisov, A. I. Mukhametrakhimova, “The Norm Resolvent Convergence for Elliptic Operators in Multi-Dimensional Domains with Small Holes”, J Math Sci, 232:3 (2018), 283
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
D.I. Borisov, “The Emergence of Eigenvalues of a $\mathcal{PT}$ -Symmetric Operator in a Thin Strip”, Math. Notes, 98:6 (2015), 872–883
D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$ -symmetric waveguide”, Ufa Math. J., 6:1 (2014), 29–55

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

Trudy Instituta Matematiki i Mekhaniki UrO RAN

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