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Pseudo-invariants theory and real phases for systems with non-Hermitian time-dependent Hamiltonians

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Abstract.

In this paper, the Lewis-Riesenfeld invariant theory is generalized for the study of systems with non-Hermitian time-dependent Hamiltonians. Explicitly time-dependent pseudo-Hermitian invariants theory, with a time-dependent metric, is developed. We derive a simple relation between the eigenstates of this pseudo-Hermitian invariant and the solutions of the Schrödinger equation. A physical system is treated in detail: the time-dependent Swanson model, where an explicitly time-dependent pseudo-Hermitian invariant is derived as well as their eigenvalues and eigenstates.

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References

  1. C.M. Bender, Rep. Prog. Phys. 70, 947 (2007)

    Article  ADS  Google Scholar 

  2. A. Mostafazadeh, Int. J. Geom. Methods Mod. Phys. 07, 1191 (2010)

    Article  Google Scholar 

  3. F.G. Scholz, H.B. Geyer, F.J. Hahne, Ann. Phys. 213, 74 (1992)

    Article  ADS  Google Scholar 

  4. C. Figueira de Morisson Faria, A. Fring, J. Phys. A: Math. Theor. 39, 9269 (2006)

    Article  ADS  Google Scholar 

  5. C. Figueira de Morisson Faria, A. Fring, Laser Phys. 17, 424 (2007)

    Article  ADS  Google Scholar 

  6. F. Bagarello, J.P. Gazeau, F.H. Szafraniec, M. Znojil, Non-selfadjoint Operators in Quantum Physics: Mathematical Aspects (Wiley, 2015)

  7. M. Znojil, Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution, arXiv:0710.5653

  8. Reply to Comment on “Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution”, arXiv:0711.0514

  9. Which operator generates time evolution in Quantum Mechanics?, arXiv:0711.0535

  10. A. Mostafazadeh, Phys. Lett. B 650, 208 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  11. A. Mostafazadeh, Comment on “Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution”, arXiv:0711.0137

  12. Comment on “Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution”, arXiv:0711.1078

  13. M. Znojil, Phys. Rev. D 78, 085003 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  14. M. Znojil, SIGMA 5, 001 (2009) arXiv:0901.0700

    Google Scholar 

  15. H. Bíla, Adiabatic time-dependent metrics in PT-symmetric quantum theories, arXiv:0902.0474

  16. J. Gong, Q.H. Wang, Phys. Rev. A 82, 012103 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  17. J. Gong, Q.H. Wang, J. Phys. A 46, 485302 (2013)

    Article  MathSciNet  Google Scholar 

  18. M. Maamache, Phys. Rev. A 92, 032106 (2015)

    Article  ADS  Google Scholar 

  19. A. Fring, M.H.Y. Moussa, Phys. Rev. A 93, 042114 (2016)

    Article  ADS  Google Scholar 

  20. A. Fring, M.H.Y. Moussa, Phys. Rev. A 94, 042128 (2016)

    Article  ADS  Google Scholar 

  21. B. Khantoul, A. Bounames, M. Maamache, Eur. Phys. J. Plus 132, 258 (2017)

    Article  Google Scholar 

  22. A. Fring, T. Frith, Phys. Rev. A 95, 010102(R) (2017)

    Article  ADS  Google Scholar 

  23. 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

  24. F.S. Luiz, M.A. Pontes, M.H.Y. Moussa, Gauge linked time-dependent non-Hermitian Hamiltonians, arXiv:1703.01451

  25. H.R. Lewis, W.B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)

    Article  ADS  Google Scholar 

  26. L.D. Landau, E.M. Lifchitz, Mécanique quantique, Tome 3 (Théorie non relativiste), 3e édition (MIR, 1975)

  27. H.R. Lewis, J. Math. Phys. 9, 1976 (1968)

    Article  ADS  Google Scholar 

  28. Z. Ahmed, Phys. Lett. A 294, 287 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  29. M.S. Swanson, J. Math. Phys. 45, 585 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  30. H.F. Jones, J. Phys. A 38, 1741 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  31. B. Bagchi, C. Quesne, R. Roychoudhury, J. Phys. A 38, L647 (2005)

    Article  ADS  Google Scholar 

  32. D.P. Musumbu, H.B. Geyer, W.D. Heiss, J. Phys. A 40, F75 (2007)

    Article  Google Scholar 

  33. C. Quesne, J. Phys. A 40, F745 (2007)

    Article  ADS  Google Scholar 

  34. A. Sinha, P. Roy, J. Phys. A 40, 10599 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  35. Eva-Maria Graefe, Hans Jurgen Korsch, Alexander Rush, Roman Schubert, J. Phys. A 48, 055301 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  36. S.S. Mizrahi, M.H.Y. Moussa, B. Baseia, Int. J. Mod. Phys. B 8, 1563 (1994)

    Article  ADS  Google Scholar 

  37. M. Maamache, J. Phys. A 31, 6849 (1996)

    Article  ADS  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Laboratoire de Physique Quantique et Systèmes Dynamiques, Faculté des Sciences, Université Ferhat Abbas Sétif 1, 19000, Sétif, Algeria

    Mustapha Maamache, Oum Kaltoum Djeghiour, Naima Mana & Walid Koussa

  2. Département de Physique, Université de Jijel, BP 98 Ouled Aissa, 18000, Jijel, Algeria

    Oum Kaltoum Djeghiour

Authors
  1. Mustapha Maamache
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  2. Oum Kaltoum Djeghiour
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  3. Naima Mana
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  4. Walid Koussa
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Correspondence to Mustapha Maamache.

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Maamache, M., Kaltoum Djeghiour, O., Mana, N. et al. Pseudo-invariants theory and real phases for systems with non-Hermitian time-dependent Hamiltonians. Eur. Phys. J. Plus 132, 383 (2017). https://doi.org/10.1140/epjp/i2017-11678-2

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