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

D.I. Borisov

Ufa mathematical journal, Tome 6 (2014) no. 1, pp. 29-55

Résumé

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.

Détail

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

Bibliographie

[1] C. M. Bender, S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry”, Phys. Rev. Lett., 80:24 (1998), 5243–5246 | DOI | MR | Zbl

[2] 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

[3] 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

[4] 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

[5] 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

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

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

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

[9] G. Levai, 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

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

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

[12] E. Caliceti, S. Graffi, J. Sjöstrand, “Spectra of $\mathcal{PT}$-symmetric operators and perturbation theory”, J. Phys. A, 38:1 (2005), 185–193 | DOI | MR | Zbl

[13] E. Caliceti, F. Cannata, S. Graffi, “$\mathcal{PT}$-symmetric Schrödinger operators: reality of the perturbed eigenvalues”, SIGMA, 6 (2010), id 009, 8 pp. | DOI | MR

[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] D. Krejčiřík, H. Bíla, M. Znojil, “Closed formula for the metric in the Hilbert space of a $\mathcal{PT}$-symmetric model”, J. Phys. A, 39:32 (2006), 10143–10153 | DOI | MR | Zbl

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

[17] K. C. Shin, “On the reality of the eigenvalues for a class of $\mathcal{PT}$-symmetric oscillators”, Commun. Math. Phys., 220:3 (2002), 543–564 | DOI | MR

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

[19] D. Borisov, D. Krejcirik, “$\mathcal{PT}$-symmetric waveguide”, Integr. Equat. Oper. Th., 62:4 (2008), 489–515 | DOI | MR | Zbl

[20] D. Krejčiřík, M. Tater, “Non-Hermitian spectral effects in a PT-symmetric waveguide”, J. Phys. A, 41:24 (2008), id 244013 | MR | Zbl

[21] D. Krejčiřík, P. Siegl, “$\mathcal{PT}$-symmetric models in curved manifolds”, J. Phys. A, 43:48 (2010), id 485204 | MR | Zbl

[22] Borisov D. I., “O $\mathcal{PT}$-simmetrichnom volnovode s paroi malykh otverstii”, Trudy Instituta matematiki i mekhaniki UrO RAN, 18, no. 2, 2012, 22–37

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

[24] Nazarov S. A., “Variatsionnyi i asimptoticheskii metody poiska sobstvennykh chisel pod porogom nepreryvnogo spektra”, Sib. matem. zhurn., 51:5 (2010), 1086–1101 | MR | Zbl

[25] Gadylshin R. R., “O lokalnykh vozmuscheniyakh operatora Shrëdingera na osi”, Teor. i matem. fizika, 132:1 (2002), 97–104 | DOI | MR | Zbl

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

[27] Nazarov S. A., Asimptoticheskii analiz tonkikh plastin i sterzhnei, v. 1, Ponizhenie razmernosti i integralnye otsenki, Nauchnaya kniga (IDMI), Novosibirsk, 2002, 406 pp.

[28] 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

[29] D. Borisov, P. Freitas, “Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in Rd”, J. Funct. Anal., 258:3 (2010), 893–912 | DOI | MR | Zbl

[30] D. Borisov, G. Cardone, “Complete asymptotic expansions for the eigenvalues of the Dirichlet Laplacian in thin three-dimensional rods”, ESAIM. Contr. Op. Ca. Va., 17:3 (2011), 887–908 | DOI | MR | Zbl

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

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

[33] Borisov D. I., “Asimptotiki i otsenki sobstvennykh elementov Laplasiana s chastoi neperiodicheskoi smenoi granichnykh uslovii”, Izv. RAN. Ser. matem., 67:6 (2003), 23–70 | DOI | MR | Zbl

[34] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974, 408 pp. | MR

[35] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1982, 740 pp. | MR

[36] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976, 392 pp. | MR

[37] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973, 577 pp. | MR

[38] D. Borisov, “Asymptotic behaviour of the spectrum of a waveguide with distant perturbation”, Math. Phys. Anal. Geom., 10:2 (2007), 155–196 | DOI | MR | Zbl