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Some exactly solvable \(\mathcal {PT}\)-invariant potentials with real spectra via the (extended) Nikiforov–Uvarov method

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Abstract

The (extended) Nikiforov–Uvarov method is employed to find exact solutions of the Schrödinger operator for three \(\mathcal {PT}\)-invariant potentials (periodic exponential, cotangent and \(\mathcal {PT}\)-symmetric harmonic plus centrifugal). It is shown that their corresponding Schrödinger operators can exhibit real energy eigenvalues. The results are compared with similar works but with different methods. The comparisons led to Rodrigues formulas of some functions of interest. The eigenfunctions of these examples are expressed in terms of Hankel functions, Romanovski polynomials and Heun functions. The method is proved to be felicitous and leads to closed energy formulas for the potentials under study.

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References

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

    Article  ADS  MathSciNet  Google Scholar 

  2. C M Bender, Contemp. Phys. 46, 277 (2005)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  4. B Midya, B Roy and R Roychoudhury, Phys. Lett. A 374, 2605 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  5. F Cannata, G Junker and J Trost, Phys. Lett. A 246, 219 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  6. B Rath, arXiv:1802.01981v1 [quant-ph]

  7. N Beblano and J da Providencia, J. Math. Phys. 60, 012104 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  8. S Mitra and S Chakrabarty, Eur. Phys. J. Plus 132, 114 (2017)

    Article  Google Scholar 

  9. B Margolis and E S Troubetzkoy, Phys. Rev. 106, 105 (1957)

    Article  ADS  Google Scholar 

  10. A J Sous, Turk. J. Phys. 32, 123 (2008)

    ADS  Google Scholar 

  11. V K Srivastava and S K Bose, Indian J. Pure Appl. Phys. 47, 547 (2009)

    Google Scholar 

  12. B G Márkus and F Márkus, Indian J. Phys. 90(4), 441 (2016)

    Article  ADS  Google Scholar 

  13. Z Ahmed, D Ghosh and J A Nathan, Phys. Lett. A  379, 1639 (2015)

    Article  ADS  Google Scholar 

  14. H Ciftci, R L Hall and N Saad, Cent. Eur. J. Phys. 11, 37 (2013)

    Google Scholar 

  15. B Bagchi, C Quesne and M Znojil, Mod. Phys. Lett. A 16, 2047 (2001)

    Article  ADS  Google Scholar 

  16. A F Nikiforov and V B Uvarov, Special functions of mathematical physics (Birkhauser, Basel, 1988)

    Book  Google Scholar 

  17. H Karayer, D Demirhan and F Büyükkilic, J. Math. Phys. 59, 053501 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  18. H Karayer, D Demirhan and F Büyükkilic, J. Math. Phys. 56, 063504 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  19. H Karayer, D Demirhan and F Büyükkilic, Rep. Math. Phys.76(3), 271 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  20. A Al-Jamel, Int. J. Mod. Phys. A 34, 1950054 (2019)

    Article  ADS  Google Scholar 

  21. W Magnus, F Oberhettinger and R Soni, Formulas and theorems for the special functions of mathematical physics, 3rd edn (Springer, Berlin, 1966)

    Book  Google Scholar 

  22. C B Compean and M Kirchbach, J. Phys. A 39, 547 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  23. R De, R Dutt and U Sukhatme, J. Phys. A 25, L843 (1992)

    Article  ADS  Google Scholar 

  24. A Al-Jamel, Mod. Phys. Lett. A 33, 1850185 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  25. A P Raposo, H J Weber, D E Alvarez-Castillo and M Kirchbach, CEJP 5(3), 253 (2007)

    ADS  Google Scholar 

  26. B Bagchi, C Quesne and M Znojil, Mod. Phys. Lett. A 16, 2047 (2001)

    Article  ADS  Google Scholar 

  27. G Levai, P Siegl and M Znojil, J. Phys. A 42, 295201 (2009)

    Article  MathSciNet  Google Scholar 

  28. S Longhi, Europhys. Lett. 115, 61001 (2016)

    Article  ADS  Google Scholar 

  29. A Singh and S R Jain, Pramana – J. Phys. 92: 47(2019)

    Article  ADS  Google Scholar 

  30. A Ronveaux, Heun’s differential equations (Oxford University Press, 1995).

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thanks the reviewers for their efforts, which helped in improving the quality of the manuscript.

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Authors and Affiliations

  1. Physics Department, Faculty of Science, Al al-Bayt University, P.O. Box 130040, Mafraq, 25113, Jordan

    M Abusini, M Serhan, Ahmed Al-Jamel & Eqab M Rabei

  2. Mathematics Department, Yarmouk University, P.O. Box 566, Irbid, 21163, Jordan

    Mohammad F Al-Jamal

Authors
  1. M Abusini
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  2. M Serhan
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  3. Mohammad F Al-Jamal
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  4. Ahmed Al-Jamel
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  5. Eqab M Rabei
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Correspondence to Ahmed Al-Jamel.

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Abusini, M., Serhan, M., Al-Jamal, M.F. et al. Some exactly solvable \(\mathcal {PT}\)-invariant potentials with real spectra via the (extended) Nikiforov–Uvarov method. Pramana - J Phys 93, 93 (2019). https://doi.org/10.1007/s12043-019-1860-x

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  • DOI: https://doi.org/10.1007/s12043-019-1860-x

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