Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 111, 19 pages      arXiv:1107.5911      https://doi.org/10.3842/SIGMA.2011.111
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum

Alexander A. Andrianov a, b and Andrey V. Sokolov a
a) V.A. Fock Department of Theoretical Physics, Sankt-Petersburg State University, 198504 St. Petersburg, Russia
b) ICCUB, Universitat de Barcelona, 08028 Barcelona, Spain

Received August 06, 2011, in final form November 25, 2011; Published online December 05, 2011

Abstract
Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined.

Key words: non-Hermitian quantum mechanics; supersymmetry; exceptional points; resolution of identity.

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References

  1. Berry M.V., Physics of non-Hermitian degeneracies, Czechoslovak J. Phys. 54 (2004), 1039-1047.
    Heiss W.D., Exceptional points of non-Hermitian operators, J. Phys. A: Math. Gen. 37 (2004), 2455-2464, quant-ph/0304152.
    Dembowski C., Dietz B., Gräf H.-D., Harney H.L., Heine A., Heiss W.D., Richter A., Encircling an exceptional point, Phys. Rev. E 69 (2004), 056216, 7 pages, nlin.CD/0402015.
    Stehmann T., Heiss W.D., Scholtz F.G., Observation of exceptional points in electronic circuits, J. Phys. A: Math. Gen. 37 (2004), 7813-7819, quant-ph/0312182.
    Mailybaev A.A., Kirillov O.N., Seyranian A.P., Geometric phase around exceptional points, Phys. Rev. A 72 (2005), 014104, 4 pages, quant-ph/0501040.
    Müller M., Rotter I., Exceptional points in open quantum systems, J. Phys. A: Math. Theor. 41 (2008), 244018, 15 pages.
  2. Sokolov A.V., Andrianov A.A., Cannata F., Non-Hermitian quantum mechanics of non-diagonalizable Hamiltonians: puzzles with self-orthogonal states, J. Phys. A: Math. Gen. 39 (2006), 10207-10227, quant-ph/0602207.
  3. Andrianov A.A., Cannata F., Sokolov A.V., Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians. I. General properties, Nuclear Phys. B 773 (2007), 107-136, math-ph/0610024.
  4. Sokolov A.V., Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians. II. Rigorous results, Nuclear Phys. B 773 (2007), 137-171, math-ph/0610022.
  5. Klaiman S., Günther U., Moiseyev N., Visualization of branch points in PT-symmetric waveguides, Phys. Rev. Lett. 101 (2008), 080402, 4 pages, arXiv:0802.2457.
    Günther U., Samsonov B.F., The Naimark dilated PT-symmetric brachistochrone, Phys. Rev. Lett. 101 (2008), 230404, 4 pages, arXiv:0807.3643.
    Günther U., Samsonov B.F., The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes, Phys. Rev. A 78 (2008), 042115, 9 pages, arXiv:0709.0483.
  6. Pavlov B.S., On the spectral theory of non-selfadjoint differential operators, Dokl. Akad. Nauk SSSR 146 (1962), 1267-1270 (English transl.: Sov. Math. Dokl. 3 (1963), 1483-1487).
    Pavlov B.S., On a non-self-adjoint Schrödinger operator, Probl. Math. Phys., Vol. 1, Spectral Theory and Wave Processes, Izdat. Leningrad Univ., Leningrad, 1966, 102-132 (in Russian).
  7. Naimark M.A., Linear differential operators, Frederick Ungar Publishing Co., New York, 1967.
  8. Heiss W.D., Phase transitions of finite Fermi systems and quantum chaos, Phys. Rep. 242 (1994), 443-451.
    Heiss W.D., Müller M., Rotter I., Collectivity, phase transitions and exceptional points in open quantum systems, Phys. Rev. E 58 (1998), 2894-2901, quant-ph/9805038.
    Narevicius E., Moiseyev N., Fingerprints of broad overlapping resonances in the e+H2 cross section, Phys. Rev. Lett. 81 (1998), 2221-2224.
    Narevicius E., Moiseyev N., Trapping of an electron due to molecular vibrations, Phys. Rev. Lett. 84 (2000), 1681-1684.
  9. Schomerus H., Frahm K.M., Patra M., Beenakker C.W.J., Quantum limit of the laser linewidth in chaotic cavities and statistics of residues of scattering matrix poles, Phys. A 278 (2000), 469-496, chao-dyn/9911004.
  10. Hernández E., Jáuregui A., Mondragón A., Degeneracy of resonances in a double barrier potential, J. Phys. A: Math. Gen. 33 (2000), 4507-4523.
    Dembowski C., Gräf H.-D., Harney H.L., Heine A., Heiss W.D., Rehfeld H., Richter A., Experimental observation of the topological structure of exceptional points, Phys. Rev. Lett. 86 (2001), 787-790.
  11. Bender C.M., Wu T.T., Analytic structure of energy levels in a field-theory model, Phys. Rev. Lett. 21 (1968), 406-409.
    Bender C.M., Wu T.T., Anharmonic oscillator, Phys. Rev. 184 (1969), 1231-1260.
  12. Curtright T., Mezincescu L., Biorthogonal quantum systems, J. Math. Phys. 48 (2007), 092106, 35 pages, quant-ph/0507015.
    Mostafazadeh A., Non-Hermitian Hamiltonians with a real spectrum and their physical applications, Pramana J. Phys. 73 (2009), 269-277, arXiv:0909.1654.
  13. Sokolov A.V., Resolutions of identity for some non-Hermitian Hamiltonians. II. Proofs, SIGMA 7 (2011), 112, 16 pages, arXiv:1107.5916.
  14. Cooper F., Freedman B., Aspects of supersymmetric quantum mechanics, Ann. Physics 146 (1983), 262-288.
  15. Fernández C. D.J., New hydrogen-like potentials, Lett. Math. Phys. 8 (1984), 337-343.
  16. Andrianov A.A., Borisov N.V., Ioffe M.V., Quantum systems with equivalent energy spectra, JETP Lett. 39 (1984), 93-97.
    Andrianov A.A., Borisov N.V., Ioffe M.V., The factorization method and quantum systems with equivalent energy spectra, Phys. Lett. A 105 (1984), 19-22.
    Andrianov A.A., Borisov N.V., Ioffe M.V., Factorization method and Darboux transformation for multidimensional Hamiltonians, Theoret. and Math. Phys. 61 (1985), 1078-1088.
  17. Sukumar C.V., Supersymmetry, factorisation of the Schrödinger equation and a Hamiltonian hierarchy, J. Phys. A: Math. Gen. 18 (1985), L57-L61.
    Sukumar C.V., Supersymmetric quantum mechanics of one-dimensional systems, J. Phys. A: Math. Gen. 18 (1985), 2917-2936.
    Sukumar C.V., Supersymmetric quantum mechanics and the inverse scattering method, J. Phys. A: Math. Gen. 18 (1985), 2937-2955.
  18. Bagrov V.G., Samsonov B.F., Darboux transformation of Schrödinger equation, Phys. Particles Nuclei 28 (1997), 374-397.
  19. Andrianov A.A., Cannata F., Nonlinear supersymmetry for spectral design in quantum mechanics, J. Phys. A: Math. Gen. 37 (2004), 10297-10321, hep-th/0407077.
  20. Aoyama H., Sato M., Tanaka T., N-fold supersymmetry in quantum mechanics: general formalism, Nuclear Phys. B 619 (2001), 105-127, quant-ph/0106037.
  21. Samsonov B.F., Roy P., Is the CPT norm always positive?, J. Phys. A: Math. Gen. 38 (2005), L249-L255, quant-ph/0503040.
    Samsonov B.F., SUSY transformations between diagonalizable and non-diagonalizable Hamiltonians, J. Phys. A: Math. Gen. 38 (2005), L397-L403, quant-ph/0503075.
  22. Fernández C. D.J., Fernández-García N., Higher-order supersymmetric quantum mechanics, AIP Conf. Proc. 744 (2005), 236-273, quant-ph/0502098.
  23. Gel'fand I.M., Vilenkin N.J., Generalized functions, Vol. 4, Some applications of harmonic analysis, Academic Press, New York, 1964.
  24. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
    Bender C.M., Boettcher S., Meisinger P., PT-symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201-2229, quant-ph/9809072.
    Bender C.M., Brody D.C., Jones H.F., Complex extension of quantum mechanics, Phys. Rev. Lett. 89 (2002), 270401, 4 pages, Erratum, Phys. Rev. Lett. 92 (2004), 119902, 1 page, quant-ph/0208076.
    Bender C.M., Chen J.H., Milton K.A., PT-symmetric versus Hermitian formulations of quantum mechanics, J. Phys. A: Math. Gen. 39 (2006), 1657-1668, hep-th/0511229.
    Bender C.M., Making sense of non-Hermitian Hamiltonians, Rept. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
  25. Schrödinger E., The factorization of the hypergeometric equation, Proc. Roy. Irish Acad. Sect. A 46 (1941), 53-54, physics/9910003.
  26. Infeld L., Hull T.E., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
  27. Andrianov A.A., Borisov N.V., Ioffe M.V., Eides M.I., Supersymmetric origin of equivalent quantum systems, Phys. Lett. A 109 (1985), 143-148.
    Andrianov A.A., Borisov N.V., Ioffe M.V., Eides M.I., Supersymmetric mechanics: a new look at the equivalence of quantum systems, Theoret. and Math. Phys. 61 (1985), 965-972.
  28. Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integrals and series. Elementary functions, Nauka, Moscow, 1981 (in Russian).
  29. Andrianov A.A., Cannata F., Sokolov A.V., Spectral singularities for non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity, J. Math. Phys. 51 (2010), 052104, 22 pages, arXiv:1002.0742.

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