Geodesic
The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip
Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 809-823.

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The Schrödinger operator in a thin infinite strip with $\mathcal{PT}$-symmetric boundary conditions and a localized potential is studied. The case of a virtual level on the threshold of the essential spectrum of an efficient one-dimensional operator is considered. Sufficient conditions for the transformation of this level into an isolated eigenvalue are obtained and the first terms of the asymptotic expansion are calculated for this eigenvalue. Sufficient conditions for the absence of such an eigenvalue are also obtained.
Keywords: elliptic operator, thin infinite strip, $\mathcal{PT}$-symmetric boundary condition, localized potential, isolated eigenvalue, asymptotics. 
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D.I. Borisov. The Emergence of Eigenvalues of a $\mathcal{PT}$-Symmetric Operator in a Thin Strip. Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 809-823. http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a1/

[1] D. Borisov, D. Krejčiřík, “The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions”, Asymptot. Anal., 76:1 (2012), 49–59 | MR | Zbl

[2] D. I. Borisov, “Diskretnyi spektr tonkogo $\mathcal{PT}$-simmetrichnogo volnovoda”, Ufimsk. matem. zhurn., 6:1 (2014), 30–58

[3] C. M. Bender, S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry”, Phys. Rev. Lett., 80:24 (1998), 5243–5246 | DOI | MR | Zbl

[4] A. Mostafazadeh, “Pseudo-Hermiticity versus $PT$-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian”, J. Math. Phys., 43:1 (2002), 205–214 | DOI | MR | Zbl

[5] A. Mostafazadeh, “Pseudo-Hermiticity versus $PT$-Symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum”, J. Math. Phys., 43:5 (2002), 2814–2816 | DOI | MR | Zbl

[6] A. Mostafazadeh, “Pseudo-Hermiticity versus $PT$-Symmetry. III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries”, J. Math. Phys., 43:8 (2002), 3944–3951 | DOI | MR | Zbl

[7] A. Mostafazadeh, “On the Pseudo-Hermiticity of a class of PT-symmetric Hamiltonians in one dimension”, Mod. Phys. Lett. A, 17:30 (2002), 1973–1977 | DOI | MR | Zbl

[8] M. Znojil, “Exact solution for Morse oscillator in PT-symmetric quantum mechanics”, Phys. Lett. A, 264:2-3 (1999), 108–111 | DOI | MR | Zbl

[9] M. Znojil, “Non-Hermitian matrix description of the $\mathcal{PT}$-symmetric anharmonic oscillators”, J. Phys. A: Math. Gen., 32:42 (1999), 7419–7428 | DOI | MR | Zbl

[10] M. Znojil, “$\mathcal{PT}$-symmetric harmonic oscillators”, Phys. Lett. A, 259:3-4 (1999), 220–223 | DOI | MR | Zbl

[11] G. Lévai, M. Znojil, “Systematic search for $\mathcal{PT}$-symmetric potentials with real energy spectra”, J. Phys. A: Math. Gen., 33:40 (2000), 7165–7180 | DOI | MR | Zbl

[12] C. M. Bender, “Making sense of non-Hermitian Hamiltionians”, Rep. Prog. Phys., 70:6 (2007), 947–1018 | DOI | MR

[13] E. Caliceti, F. Cannata, S. Graffi, “Perturbation theory of $\mathcal{PT}$-symmetric Hamiltonians”, J. Phys. A, 39:32 (2006), 10019–10027 | DOI | MR | Zbl

[14] P. Dorey, C. Dunning, R. Tateo, “Spectral equivalences, Bethe ansatz equations, and reality properties in $\mathcal{PT}$-symmetric quantum mechanics”, J. Phys. A, 34:28 (2001), 5679–5704 | DOI | MR | Zbl

[15] H. Langer, Ch. Tretter, “A Krein space approach to PT-symmetry”, Czechoslovak J. Phys., 54:10 (2004), 1113–1120 | DOI | MR | Zbl

[16] K. C. Shin, “On the reality of the eigenvalues for a class of $\mathcal{PT}$-symmetric oscillators”, Commun. Math. Phys., 229:3 (2002), 543–564 | DOI | MR | Zbl

[17] M. Znojil, “$\mathcal{PT}$-symmetric square well”, Phys. Lett. A, 285:1-2 (2001), 7–10 | DOI | MR | Zbl

[18] D. Borisov, D. Krejčiřík, “$\mathcal{PT}$-symmetric waveguide”, Integral Equations Operator Theory, 62:4 (2008), 489–515 | DOI | MR | Zbl

[19] D. Borisov, “On a quantum waveguide with a small PT-symmetric perturbation”, Acta Polytechnica, 47:2-3 (2007), 57–59

[20] D. Borisov, “O $\mathcal{PT}$-simmetrichnom volnovode s paroi malykh otverstii”, Tr. IMM UrO RAN, 18, no. 2, 2012, 22–37

[21] S. A. Nazarov, Asimptoticheskii analiz tonkikh plastin i sterzhnei T. 1. Ponizhenie razmernosti i integralnye otsenki, Nauchnaya kniga, Novosibirsk, 2002

[22] G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005 | MR

[23] D. Borisov, P. Freitas, “Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26:2 (2009), 547–560 | DOI | MR | Zbl

[24] D. Borisov, G. Cardone, “Complete asymptotic expansions for the eigenvalues of the Dirichlet Laplacian in thin three-dimensional rods”, ESAIM Control Optim. Calc. Var., 17:3 (2011), 887–908 | DOI | MR | Zbl

[25] G. Cardone, A. Corbo Esposito, G. P. Panasenko, “Asymptotic partial decomposition for diffusion with sorption in thin structures”, Nonlinear Anal., Theory Methods Appl., 65:1 (2006), 79–106 | DOI | MR | Zbl

[26] G. P. Panasenko, E. Pérez, “Asymptotic partial decomposition of domain for spectral problems in rod structures”, J. Math. Pures Appl. (9), 87:1 (2007), 1–36 | DOI | MR | Zbl

[27] V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Asimptoticheskie resheniya nerelyativistskikh uravnenii kvantovoi mekhaniki v iskrivlennykh nanotrubkakh. I. Reduktsiya k prostranstvenno-odnomernym uravneniyam”, TMF, 141:2 (2004), 267–303 | DOI | MR | Zbl

[28] V. V. Grushin, “Asimptoticheskoe povedenie sobstvennykh znachenii operatora Laplasa v tonkikh beskonechnykh trubkakh”, Matem. zametki, 85:5 (2009), 687–701 | DOI | MR | Zbl

[29] V. V. Belov, S. Yu. Dobrokhotov, S. O. Sinitsyn, “Asimptoticheskie resheniya uravneniya Shredingera v tonkikh trubkakh”, Asimptoticheskie razlozheniya. Teoriya priblizhenii. Topologiya, Sbornik statei, Tr. IMM UrO RAN, 9, no. 1, 2003, 15–25 | MR | Zbl

[30] V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Quantum and classical dynamics of electron in thin curved tubes with spin and external electromagnetic fields taken into account”, Russ. J. Math. Phys., 11:1 (2004), 109–119 | MR | Zbl

[31] V. V. Grushin, “Asimptoticheskoe povedenie sobstvennykh znachenii operatora Shredingera v tonkikh zamknutykh trubkakh”, Matem. zametki, 83:4 (2008), 503–519 | DOI | MR | Zbl

[32] I. M. Glazman, Pryamye metody spektralnogo kachestvennogo analiza singulyarnykh differentsialnykh operatorov, Fizmatlit, M., 1963 | MR | Zbl

[33] R. R. Gadylshin, “O lokalnykh vozmuscheniyakh operatora Shredingera na osi”, TMF, 132:1 (2002), 97–104 | DOI | MR | Zbl

[34] D. I. Borisov, “Diskretnyi spektr pary nesimmetrichnykh volnovodov, soedinennykh oknom”, Matem. sb., 197:4 (2006), 3–32 | DOI | MR | Zbl

[35] D. I. Borisov, “O spektre dvumernogo periodicheskogo operatora s malym lokalizovannym vozmuscheniem”, Izv. RAN. Ser. matem., 75:3 (2011), 29–64 | DOI | MR | Zbl

[36] A. Majda, “Outgoing solutions for perturbation of $-\Delta$ with applications to spectral and scattering theory”, J. Differential Equations, 16:3 (1974), 515–547 | DOI | MR | Zbl

[37] E. Sanches-Palensiya, Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR | Zbl

[38] D. Borisov, P. Exner, R. Gadyl'shin, “Geometric coupling thresholds in a two-dimensional strip”, J. Math. Phys., 43:12 (2002), 6265–6278 | DOI | MR | Zbl