Skip to main content
Log in

Approximate Solutions of the Dunkl–Schrödinger Equation for the Hyperbolic Pöschl–Teller Potential

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

We construct an approximate solution to the Schrödinger equation for the hyperbolic Pöschl–Teller potential within the Dunkl formalism. Our approximation is based on a series expansion of the inverse quadratic term generated by the Dunkl operator. Two methods of establishing correct parity in the approximate solutions are discussed, and an example is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availability

No datasets were generated or analysed during the current study.

References

  1. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1964)

    Google Scholar 

  2. F. Ahmed, S.K. Nayek, Impact of topological defects and Yukawa potential combined with inverse square on eigenvalue spectra of diatomic molecules \(O_2\), \(NO\), \(LiH\), \(HCl\). Phys. Scr. 99, 025401 (2024)

    Article  ADS  Google Scholar 

  3. A.O. Barut, A. Inomata, R. Wilson, Algebraic treatment of second Pöschl–Teller Morse–Rosen and Eckart equations. J. Phys. A 20, 4083 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  4. O. Bayrak, E. Aciksoz, Corrected analytical solution of the generalized Woods–Saxon potential for arbitrary \(\ell \) states. Phys. Scr. 90, 015302 (2015)

    Article  ADS  Google Scholar 

  5. C.F. Dunkl, Y. Xu, “Orthogonal Polynomials of Several Variables’’, Encyclopedia of Mathematics and Its Applications, 2nd edn. (Cambridge University Press, Cambridge, 2014)

    Book  Google Scholar 

  6. C.F. Dunkl, Differential–difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167 (1989)

    Article  MathSciNet  Google Scholar 

  7. P. Etingof, Calogero–Moser Systems and Representation Theory, Zurich Lect. Adv. Math., vol. 4. (European Mathematical Society, Zurich, 2007)

    Book  Google Scholar 

  8. F.J.S. Ferreira, F.V. Prudente, Pekeris approximation: another perspective. Phys. Lett. A 377, 3027 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  9. I.S. Gomez, E.S. Santos, O. Abla, Morse potential in relativistic contexts from generalized momentum operator: Schottky anomalies, Pekeris approximation and mapping. Mod. Phys. Lett. A 36, 2150140 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  10. R.L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential. Phys. Rev. A 14, 2363 (1976)

    Article  ADS  Google Scholar 

  11. T. Hakobian, O. Lechtenfeld, A. Nersessian, Superintegrability of generalized Calogero models with oscillator or Coulomb potential. Phys. Rev. D 90, 101701 (2014)

    Article  ADS  Google Scholar 

  12. G. Junker, On the Path Integral Formulation of Wigner–Dunkl Quantum Mechanics, preprint (2024). arXiv:2312.12895

  13. G.D. Mahan, Exactly solvable models. in Many-Particle Physics. Physics of Solids and Liquids (Springer, Boston)

  14. S. Miraboutalebi, L. Rajaei, Solutions of N-dimensional Schrödinger equation with Morse potential via Laplace transforms. J. Math. Chem. 52, 1119 (2014)

    Article  MathSciNet  Google Scholar 

  15. R.D. Mota, D. Ojeda-Guillen, M.A. Xicotencatl, The generalized Fokker–Planck equation in terms of Dunkl-type derivatives. Phys. A 635, 129525 (2024)

    Article  MathSciNet  Google Scholar 

  16. D. Ojeda-Guillen, R.D. Mota, M. Salazar-Ramirez, V.D. Granados, Algebraic approach for the one-dimensional Dirac–Dunkl oscillator. Mod. Phys. Lett. A 35, 2050255 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  17. C.L. Pekeris, The rotation-vibration coupling in diatomic molecules. Phys. Rev. 45, 98 (1934)

    Article  ADS  Google Scholar 

  18. C. Quesne, Quasi-Exactly Solvable Potentials in Wigner–Dunkl Quantum Mechanics. preprint (2024). arXiv:2401.04586

  19. C. Quesne, Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials. Mod. Phys. Lett. A 38, 2350108 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  20. M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics. Operator Theory: Advances and Applications, vol. 73, ed. by A. Feintuch, I. Gohberg (Birkhauser, Basel, 1994)

    Google Scholar 

  21. R. Sasaki, Exactly-solvable quantum mechanics. The Universe 2, 2 (2014)

    Google Scholar 

  22. A. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics (Taylor and Francis, New York, 1994)

    Google Scholar 

  23. J.F. van Diejen, L. Vinet, Calogero–Sutherland–Moser Models, CRM Series in Mathematical Physics (Springer-Verlag, 2000)

  24. E.P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations? Phys. Rev. 77, 711 (1950)

    Article  ADS  MathSciNet  Google Scholar 

  25. H. Yanar, A. Tas, M. Salti, O. Aydoglu, Ro-vibrational energies of CO molecule via improved generalized Pöschl–Teller potential and Pekeris-type approximation. Eur. Phys. J. Plus 135, 292 (2020)

    Article  Google Scholar 

  26. L.M. Yang, A note on the quantum rule of the harmonic oscillator. Phys. Rev. 84, 788 (1951)

    Article  ADS  MathSciNet  Google Scholar 

  27. Y. You, F.-L. Lu, D.-S. Sun, C.-Y. Chen, S.-H. Dong, Solutions of the second Pöschl–Teller potential solved by an improved scheme to the centrifugal term. Few-Body Syst. 54, 2125 (2013)

    Article  ADS  Google Scholar 

  28. https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/17/02/07/0004/

Download references

Author information

Authors and Affiliations

Authors

Contributions

A.S.H. carried out all work related to this manuscript.

Corresponding author

Correspondence to Axel Schulze-Halberg.

Ethics declarations

Conflict of interest

The authors declare no Conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schulze-Halberg, A. Approximate Solutions of the Dunkl–Schrödinger Equation for the Hyperbolic Pöschl–Teller Potential. Few-Body Syst 65, 58 (2024). https://doi.org/10.1007/s00601-024-01931-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00601-024-01931-3

Navigation