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Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation

  • Strong Field Phenomena
  • Published:
Laser Physics

Abstract

We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.

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References

  1. H. Friedrich and D. Wintgen, Phys. Rev. A 32, 3231 (1985).

    Article  ADS  Google Scholar 

  2. H. Geyer, D. Heiss, and M. Znojil (guest editor), Special Issue Dedicated to the Physics of Non-Hermitian Operators (PHHQP IV) (Univ. of Stellenbosch, South Africa, 2005), J. Phys. A 39, 9965 (2006).

  3. C. Figueira de Morisson Faria and A. Fring, J. Phys. A 39, 9269 (2006).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (MacGraw-Hill, New York, 1978).

    MATH  Google Scholar 

  6. A. Mostafazadeh, J. Phys. A 38, 3213–3234 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  7. H. F. Jones and J. Mateo, Czech. J. Phys. 55, 1117 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  8. H. F. Jones and J. Mateo, Phys. Rev. D 73, 085002(4) (2006).

  9. J. A. C. Weideman, J. Phys. A 39, 10229 (2006).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401(4) (2002).

  11. P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. K. C. Shin, J. Phys. A 38, 6147 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. E. Wigner, J. Math. Phys. 1, 409 (1960).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. S. Weigert, J. Opt. B: Quantum Semiclass. Opt. 5, S416 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  15. S. Weigert, J. Phys. A 39, 10239 (2006).

    Google Scholar 

  16. Z. Ahmed, Phys. Lett. A 290, 19 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. A. Mostafazadeh, J. Math. Phys. 43, 2814 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. F. G. Scholtz, H. B. Geyer, and F. Hahne, Ann. Phys. 213, 74 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. W. Heisenberg, Rev. Mod. Phys. 29, 269 (1957).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. A. Mostafazadeh, J. Math. Phys. 43, 205 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. A. Mostafazadeh, J. Math. Phys. 43, 3944 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. A. Mostafazadeh, J. Phys. A 36, 7081 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. A. Mostafazadeh, J. Math. Phys. 44, 974 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. S. Weigert, Phys. Rev. A 68, 062111(4) (2003).

  25. I. Rotter and A. F. Sadreev, Phys. Rev. E 71, 0362271(14) (2005).

  26. E. Persson, T. Gorin, and I. Rotter, Phys. Rev. E 54, 3339 (1996).

    Article  ADS  Google Scholar 

  27. M. Leone, A. Paoletti, and N. Robotti, Phys. Perspect. 6, 271 (2004).

    Article  ADS  Google Scholar 

  28. H. Silverstone and C.N. Nicolaides (Ed.) (Plenum Press, New York, 1990), pp. 295.

  29. A. Fring, V. Kostrykin, and R. Schrader, J. Phys. B 29, 5651 (1996).

    Article  ADS  Google Scholar 

  30. C. Figueira de Morisson Faria, A. Fring, and R. Schrader, Laser Phys. 9, 379 (1999).

    Google Scholar 

  31. C. Figueira de Morisson Faria, A. Fring, and R. Schrader, J. Phys. B 33, 1675 (2000).

    Article  ADS  Google Scholar 

  32. M. Born and V. Fock, Z. Phys. 51, 165 (1928).

    Article  ADS  Google Scholar 

  33. A. Fleischer and N. Moiseyev, Phys. Rev. A 72, 032103(11) (2005).

  34. M. Lewenstein, P. Salieres, A. L’Huillier, and P. Antoine, Adv. At. Mol. Opt. Phys. 41, 83 (1999).

    Article  Google Scholar 

  35. W. Becker, F. Grasbon, R. Kopold, et al., Adv. At. Mol. Opt. Phys. 48, 35 (2002).

    Google Scholar 

  36. C. Joachain, M. Dörr, and N. Kylstra, Adv. At. Mol. Opt. Phys. 42, 225 (2000).

    Article  Google Scholar 

  37. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. D 70, 025001(19) (2004).

  38. A. Mostafazadeh, J. Phys. A 38, 6557 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. C. Figueira de Morisson Faria and A. Fring, Czech. J. Phys. 56, 899 (2006).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. E. Caliceti, F. Cannata, and S. Graffi, J. Phys. A 39, 10019 (2006).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. F. G. Scholtz and H. B. Geyer, Phys. Lett. B 634, 84 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  42. F. G. Scholtz and H. B. Geyer, J. Phys. A 39, 10189 (2006).

    Google Scholar 

  43. D. B. Fairlie, J. Chaos, Solitons Fractals 10, 365 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  44. A. Mostafazadeh, J. Math. Phys. 47, 072103(11) (2006).

  45. C. M. Bender, D. C. Brody, J.-H. Chen, et al., Phys. Rev. D 74, 025016(10) (2006).

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  48. E. Witten, Nucl. Phys. B 202, 253 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  49. F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).

    Article  MathSciNet  Google Scholar 

  50. M. Znojil, Phys. Lett. A 259, 220 (1999).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  51. C. Figueira de Morisson Faria, A. Fring, and H. Jones (unpublished notes).

  52. C. M. Bender, J.-H. Chen, and K. A. Milton, J. Phys. A 39, 1657 (2006).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, quanth-ph/0609032 (2006).

  54. K. A. Milton, Czech. J. Phys. 54, 85 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  55. C. M. Bender, I. Cavero-Pelaez, K. A. Milton, and K. V. Shajesh, Phys. Lett. B 613, 97 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  56. C. Figueira de Morisson Faria and A. Fring (in preparation).

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Authors and Affiliations

  1. Centre for Mathematical Science, City University, Northampton Square, London, EC1V 0HB, UK

    C. F. M. Faria & A. Fring

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  1. C. F. M. Faria
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  2. A. Fring
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Original Text © Astro, Ltd., 2007.

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Faria, C.F.M., Fring, A. Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation. Laser Phys. 17, 424–437 (2007). https://doi.org/10.1134/S1054660X07040196

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  • DOI: https://doi.org/10.1134/S1054660X07040196

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