International Journal of Theoretical Physics

, Volume 50, Issue 4, pp 1012–1018 | Cite as

Spectrum of the Metric Operator of a Simple Open image in new window -Symmetric Model

  • Jakub Železný


In papers (Krejčiřík D. et al.: J. Phys. A: Math. Gen.: 39(32), 10143–10153 (2006); Krejčiřík D., Tater M.: J. Phys. A: Math. Theor. 41(24), 244 (2008)) a new very simple Open image in new window -symmetric model was introduced and closed formula for the metric operator was found. We use an alternative integral form of this metric operator to study the spectrum of the metric.


Open image in new window-symmetry Non-Hermiticity Quasi-Hermiticity Robin boundary conditions Metric operator 


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  1. 1.
    Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70(6), 947–1018 (2007) MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian hamiltonians having \(\mathcal{PT}\)-symmetry. Phys. Rev. Lett. 80(24), 5243–5246 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Bender, C.M., Boettcher, S., Meisinger, P.N.: \(\mathcal{PT}\)-symmetric quantum mechanics. J. Math. Phys. 40(5), 2201–2229 (1999) MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Borisov, D., Krejčiřík, D.: \(\mathcal{PT}\)-symmetric waveguides. Integral Equ. Oper. Theory 62(4), 489–515 (2008) zbMATHCrossRefGoogle Scholar
  5. 5.
    Conway, B.J.: A Course in Functional Analysis. Springer, New York (1985) zbMATHGoogle Scholar
  6. 6.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1995) zbMATHGoogle Scholar
  7. 7.
    Krejčiřík, D.: Calculation of the metric in the Hilbert space of a \(\mathcal{PT}\)-symmetric model via the spectral theorem. J. Phys. A, Math. Theor. 41(24), 244 (2008) 012 Google Scholar
  8. 8.
    Krejčiřík, D., Siegl, P.: \(\mathcal{PT}\)-symmetric models in curved manifolds. J Phys A: Math Theor (2010), to appear Google Scholar
  9. 9.
    Krejčiřík, D., Tater, M.: Non-Hermitian spectral effects in a \(\mathcal{ PT}\)-symmetric waveguide. J. Phys. A, Math. Theor. 41(24), 244 (2008) 013 Google Scholar
  10. 10.
    Krejčiřík, D., Bíla, H., Znojil, M.: Closed formula for the metric in the Hilbert space of a \(\mathcal{PT}\)-symmetric model. J. Phys. A, Math. Gen. 39(32), 10143–10153 (2006) ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Kretschmer, R., Szymanowski, L.: Quasi-Hermiticity in infinite-dimensional Hilbert spaces. Phys. Lett. A 325(2), 112–117 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Mostafazadeh, A.: Pseudo-hermitian representation of quantum mechanics. Int. J. Geom. Methods Mod. Phys. (2010), to appear Google Scholar
  13. 13.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, INC (1995) Google Scholar
  14. 14.
    Scholtz, F.G., Geyer, H.B., Hahne, F.J.W.: Quasi-Hermitian operators in quantum mechanics and the variational principle. Ann. Phys. 213(1), 74–101 (1992) MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Scholtz, F.G., Geyer, H.B., Heiss, W.D.: Non-Hermitian Hamiltonians, metric, other observables and physical implications, arXiv:0710.5593
  16. 16.
    Siegl, P.: Supersymmetric quasi-Hermitian Hamiltonians with point interactions on a loop. J. Phys. A, Math. Theor. 41(24), 244 (2008) 025 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Stone, M., Goldbart, P.: Methods of mathematical physics. (2005)

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePragueCzech Republic
  2. 2.Department of Theoretical PhysicsNuclear Physics Institute ASCRRez near PragueCzech Republic

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