Metric versus observable operator representation, higher spin models

Open Access
Regular Article
  • 32 Downloads

Abstract.

We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

References

  1. 1.
    A. Fring, T. Frith, Phys. Rev. A 95, 010102(R) (2017)ADSCrossRefGoogle Scholar
  2. 2.
    C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    C.M. Bender, Rep. Prog. Phys. 70, 947 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    A. Mostafazadeh, Int. J. Geom. Methods Mod. Phys. 7, 1191 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Cannata, M. Ioffe, G. Junker, D.J. Nishnianidze, J. Phys. A 32, 3583 (1999)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    A.A. Andrianov, F. Cannata, J. Phys. A 37, 10297 (2004)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    C. Figueira de Morisson Faria, A. Fring, J. Phys. A 39, 9269 (2006)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    C. Figueira de Morisson Faria, A. Fring, Laser Phys. 17, 424 (2007)ADSCrossRefGoogle Scholar
  9. 9.
    A. Mostafazadeh, Phys. Lett. B 650, 208 (2007)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Znojil, Phys. Rev. D 78, 085003 (2008)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Fring, M.H.Y. Moussa, Phys. Rev. A 93, 042114 (2016)ADSCrossRefGoogle Scholar
  12. 12.
    F.S. Luiz, M.A. Pontes, M.H.Y. Moussa, Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps, arXiv:1611.08286 (2016)Google Scholar
  13. 13.
    A. Fring, M.M.H.Y. Moussa, Phys. Rev. A 94, 042128 (2016)ADSCrossRefGoogle Scholar
  14. 14.
    G. von Gehlen, J. Phys. A 24, 5371 (1991)ADSCrossRefGoogle Scholar
  15. 15.
    O.A. Castro-Alvaredo, A. Fring, J. Phys. A 42, 465211 (2009)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    V. Ermakov, Univ. Izv. Kiev. 20, 1 (1880)Google Scholar
  17. 17.
    E. Pinney, Proc. Am. Math. Soc. 1, 681 (1950)MathSciNetGoogle Scholar
  18. 18.
    A. Hone, Phys. Lett. A 263, 347 (1999)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    R.M. Hawkins, J.E. Lidsey, Phys. Rev. D 66, 023523 (2002)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    J.R. Choi, B.H. Kweon, Int. J. Mod. Phys. B 16, 4733 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    J.R. Choi, Int. J. Theor. Phys. 42, 853 (2003)CrossRefGoogle Scholar
  22. 22.
    N. Ferkous, A. Bounames, M. Maamache, Phys. Scr. 88, 35001 (2013)CrossRefGoogle Scholar
  23. 23.
    S. Dey, A. Fring, Phys. Rev. D 90, 084005 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    M.V. Ioffe, H. Korsch, Phys. Lett. A 311, 200 (2003)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    S. Dey, A. Fring, L. Gouba, J. Phys. A 48, 40FT01 (2015)CrossRefGoogle Scholar
  26. 26.
    C. Eliezer, A. Gray, SIAM J. Appl. Math. 30, 463 (1976)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics, CityUniversity of LondonLondonUK

Personalised recommendations