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On Pseudo-Hermitian Operators with Generalized C-symmetries

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Modern Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 190))

Abstract

The concept of C-symmetries for pseudo-Hermitian Hamiltonians is studied in the Krein space framework. A generalization of C-symmetries is suggested.

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References

  1. Z. Ahmed, P, T, PT, and CPT-invariance of Hermitian Hamiltonians. Phys. Lett. A. 310 (2003), 139–142.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Albeverio, S.M. Fei, and P. Kurasov, Point interactions: PT-Hermiticity and reality of the spectrum. Lett. Math. Phys. 59 (2002), 227–242.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Albeverio and S. Kuzhel, Pseudo-Hermiticity and theory of singular perturbations. Lett. Math. Phys. 67 (2004), 223–238.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Albeverio and S. Kuzhel, One-dimensional Schrödinger operators with P-symmetric zero-range potentials. J. Phys. A. 38 (2005) 4975–4988.

    Article  MATH  MathSciNet  Google Scholar 

  5. T.Ya. Azizov and I.S. Iokhvidov Linear Operators in Spaces with Indefinite Metric. Wiley, Chichester, 1989.

    Google Scholar 

  6. A. Batal and A. Mostafazadeh, Physical aspects of pseudo-Hermitian and PT-symmetric quantum mechanics. J. Phys. A. 37 (2004), 11645–11679.

    Article  MATH  MathSciNet  Google Scholar 

  7. C.M. Bender, Calculating the C operator in PT-quantum mechanics. Czechoslovak J. Phys. 54 (2004), 1027–1038.

    Article  MathSciNet  Google Scholar 

  8. C.M. Bender, Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70 (2007), 947–1018.

    Article  MathSciNet  Google Scholar 

  9. C.M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT-symmetry. Phys. Rev. Lett. 80 (1998), 5243–5246.

    Article  MATH  MathSciNet  Google Scholar 

  10. C.M. Bender, D.C. Brody, and H.F. Jones, Complex extension of quantum mechanics. Phys. Rev. Lett. 89 (2002), 401–405.

    Article  MathSciNet  Google Scholar 

  11. C.M. Bender, D.C. Brody, and H.F. Jones, Must a Hamiltonian be Hermitian? Amer. J. Phys., 71 (2003), 1095–1102.

    Article  MathSciNet  Google Scholar 

  12. C.M. Bender and H.F. Jones, Semiclassical calculation of the C operator in PT-quantum mechanics. Phys. Lett. A. 328 (2004), 102–109.

    Article  MATH  MathSciNet  Google Scholar 

  13. C.M. Bender and B. Tan, Calculation of the hidden symmetry operator for a PT-symmetric square well. J. Phys. A. 39 (2006), 1945–1953.

    Article  MATH  MathSciNet  Google Scholar 

  14. P.A.M. Dirac, The physical interpretation of quantum mechanics. Proc. Roy. Soc. London A. 180 (1942), 1–40.

    Article  MathSciNet  Google Scholar 

  15. P. Dorey, C. Dunning, and R. Tateo, Spectral equivalence, Bethe ansatz, and reality properties in PT-symmetric quantum mechanics. J. Phys. A: Math. Gen. 34 (2001), 5679–5704.

    Article  MATH  MathSciNet  Google Scholar 

  16. H.B. Geyer, W.D. Heiss, and F.G. Scholtz, Non-Hermitian Hamiltonians, metric, other observables and physical implications. ArXiv:0710.5593v1 [quant-ph] 30 Oct. 2007.

    Google Scholar 

  17. U. Gunther, F. Stefani, and M. Znojil, MHD α 2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator. J. Math. Phys. 46 (2005), 063504–063526.

    Article  MathSciNet  Google Scholar 

  18. S. Gupta, Theory of longitudinal photons in quantum electrodynamics. Proc. Phys. Soc. 63 (1950), 681–691.

    Article  Google Scholar 

  19. H.F. Jones, On pseudo-Hermitian Hamiltonians and their Hermitian counterparts. J. Phys. A. 38 (2005) 1741–1746.

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Kretschmer and L. Szymanowski, Quasi-Hermiticity in infinite-dimensional Hilbert spaces. Phys. Lett. A. 325 (2004) 112–122.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Kuzhel and S. Kuzhel, Regular Extensions of Hermitian Operators. VSP, Utrecht, 1998.

    Google Scholar 

  22. H. Langer, Zur Spektraltheorie J-selbstadjungierter Operatoren. Math. Ann. 146 (1962), 60–85.

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Langer, Maximal dual pairs of invariant subspaces of J-self-adjoint operators. Mat. Zametki 7 (1970), 443–447. (Russian)

    MATH  MathSciNet  Google Scholar 

  24. H. Langer, Invariant subspaces for a class of operators in spaces with indefinite metric. J. Funct. Anal. 19 (1975), 232–241.

    Article  MATH  Google Scholar 

  25. H. Langer and C. Tretter, A Krein space approach to PT-symmetry. Czechoslovak J. Phys. 54 (2004), 1113–1120.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Lévai and M. Znojil, Systematic search for PT-symmetric potentials with real energy spectra. J. Phys. A. 33 (2000), 7165–7180.

    Article  MATH  MathSciNet  Google Scholar 

  27. 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 (2002), 205–214.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Mostafazadeh, Pseudo-Hermiticity and generalized PT — and CPT-symmetries. J. Math. Phys. 44 (2003), 979–989.

    Google Scholar 

  29. A. Mostafazadeh, Krein-space formulation of PT-symmetry, CPT-inner products, and pseudo-Hermiticity. J. Math. Phys. 44 (2003), 979–989.

    Google Scholar 

  30. W. Pauli, On Dirac’s new method of field quantization. Rev. Mod. Phys. 15 (1943), 175–207.

    Article  MATH  MathSciNet  Google Scholar 

  31. F.G. Scholtz, H.B. Geyer, and F.J.W. Hahne, Quasi-Hermitian operators in quantum mechanics and the variational principle. Ann. Phys. (N.Y.) 213 (1992).

    Google Scholar 

  32. T. Tanaka, General aspects of PT-symmetric and P-self-adjoint quantum theory in a Krein space. J. Phys. A. 39 (2006), 14175–14203.

    Article  MATH  MathSciNet  Google Scholar 

  33. B. Thaller, The Dirac Equation. Springer-Verlag, Berlin 1992.

    Google Scholar 

  34. M. Znojil, PT-symmetric harmonic oscillators. Phys. Lett. A. 259 (1999), 220–223.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str, 01601, Kiev, Ukraine

    S. Kuzhel

Authors
  1. S. Kuzhel
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Vadim M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    Israel Gohberg

  3. Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivska st., 01601, Kyiv-4, Ukraine

    Anatoly Kochubei , Yurij Berezansky , Myroslav Gorbachuk  & Valentyna Gorbachuk , ,  & 

  4. Department of Mathematical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Gennadiy Popov

  5. Institute of Analysis and Scientific Computing, Technical University of Vienna, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria

    Heinz Langer

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Kuzhel, S. (2009). On Pseudo-Hermitian Operators with Generalized C-symmetries. In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 190. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9919-1_23

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