On Solutions of the Killingbeck Potential and Clarifying Comments on a Related Analytical Approach

  • Fatma Zohra Khaled LPRIM, Department of Physics, University of Batna I, Batna, Algeria https://orcid.org/0009-0005-2853-2265
  • Mustafa Moumni LPRIM, Department of Physics, University of Batna I, Batna, Algeria; LPPNNM, Department of Matter Sciences, University of Biskra, Biskra, Algeria https://orcid.org/0000-0002-8096-6280
  • Mokhtar Falek LPPNNM, Department of Matter Sciences, University of Biskra, Biskra, Algeria; Faculty of Technology, University of Khenchela, Khenchela, Algeria https://orcid.org/0000-0002-0466-9559
DOI: https://doi.org/10.26565/2312-4334-2025-4-14
Keywords: Schrödinger equation, Killingbeck potential, Yukawa potential, Serie expansion method, Heun equation

Abstract

The work presents analytical solutions to the Schrödinger equation for the Killingbeck potential, a hybrid model combining harmonic, linear, and Coulombic terms, as well as an approximate model of Yukawa-type potentials. The radial Schrödinger equation is solved by means of the series expansion method, thus yielding the exact expressions of both bound-state solutions and eigen-functions for systems such as quarkonium and confined hydrogen-like atoms in plasma environments. Furthermore, we offer a constructive commentary on the work of Obu et al. (East Eur. J. Phys. 3, 146–157, 2023), with the aim of clarifying a mathematical misstatement utilised in their analytical treatment of analogous systems.

Downloads

Download data is not yet available.

References

J.P. Killingbeck, J. Phys. A: Math. Gen. 14, 1005 (1981). https://doi.org/10.1088/0305-4470/14/5/020

H. Hamzavi, and A.A. Rajabi, Ann. Phys. 334, 316 (2013). https://doi.org/10.1016/j.aop.2013.04.007

M. Chabab, A. Lahbas, and M. Oulne, Eur. Phys. J. A, 51, 131 (2015). https://doi.org/10.1140/epja/i2015-15131-y

O.J. Oluwadare, and K.J. Oyewumi, Chinese Phys. Lett. 34, 110301 (2017). https://doi.org/10.1088/0256-307X/34/11/110301

M. Hamzavi, A.A. Rajabi, and H. Hassanabadi, Few-Body Syst. 48, 171 (2010). https://doi.org/10.1007/s00601-010-0095-7

C.Y. Chen, and S.H. Dong, Phys. Lett. A, 335, 374 (2005). https://doi.org/10.1016/j.physleta.2004.12.062

N. Brambilla, et al., Rev. Mod. Phys. 77, 1423 (2005). https://doi.org/10.1103/RevModPhys.77.1423

E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, and T.-M.Yan, Phys. Rev. D, 17, 3090 (1978). https://doi.org/10.1103/PhysRevD.17.3090

F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995). https://doi.org/10.1016/0370-1573(94)00080-M

H. Yukawa, Proc. Phys. Math. Soc. Jpn. 17, 48 (1935). https://doi.org/10.11429/ppmsj1919.17.0 48

S. Fl¨ugge, Practical Quantum Mechanics, (Springer, Berlin, 1999). https://doi.org/10.1007/978-3-642-61995-3

M. Sreelakshmi, and R. Akhilesh, J. Phys. G: Nucl. Part. Phys. 50, 073001 (2023). https://doi.org/10.1088/1361-6471/acd1a3

A. Kievsky, E. Garrido, M. Viviani, et al. Few-Body Syst. 65, 23 (2024). https://doi.org/10.1007/s00601-024-01893-6

M. Sreelakshmi, and R. Akhilesh, Int. J. Theor. Phys. 64, 58 (2025). https://doi.org/10.1007/s10773-025-05924-8

N. Mukherjee, C.N. Patra, and A.K. Roy, Phys. Rev. A, 104, 012803 (2021). https://doi.org/10.1103/PhysRevA.104.012803

Zhan-Bin Chen, Phys. Plasmas, 30, 032103 (2023). https://doi.org/10.1063/5.0140534

Tong Yan, et al., Phys. Rev. Plasmas, 31, 042110 (2024). https://doi.org/10.1063/5.0185339

B. G¨on¨ul, K. K¨oksal, and E. Bakir, Phys. Scr. 73, 279 (2006). https://doi.org/10.1088/0031-8949/73/3/007

A. Arda, and R. Sever, Zeitschrift f¨ur Naturforschung A, 69, 163 (2014). https://doi.org/10.5560/zna.2014-0007

J.A. Obu, E.P. Inyang, E.S. William, D.E. Bassey, and E.P. Inyang, East Eur. J. Phys. (3), 146 (2023). https://doi.org/10.26565/2312-4334-2023-3-11

A. Guvendi, and O. Mustafa, Eur. Phys. J. C, 84, 866 (2024). https://doi.org/10.1140/epjc/s10052-024-13192-9

O. Mustafa, and A. Guvendi, Int. J. Geom. Methods Mod. Phys. 2550091 (2024). https://doi.org/10.1142/S0219887825500914

O. Mustafa, and A. Guvendi, Eur. Phys. J. C, 85, 34 (2025). https://doi.org/10.1140/epjc/s10052-025-13779-w

A. Guvendi, and O. Mustafa, Nucl. Phys. B, 1014, 116874 (2025). https://doi.org/10.1016/j.nuclphysb.2025.116874

A. Guvendi, and O. Mustafa, Ann. Phys. 473, 169897 (2025). https://doi.org/10.1016/j.aop.2024.169897

F.M. Fernandez, arXiv:2205.07884 https://doi.org/10.48550/arXiv.2205.07884

A. Ronveaux (Ed.), Heun’s Differential Equations, (Oxford University Press, Oxford, 1995). https://doi.org/10.1093/oso/9780198596950.001.0001

M.S. Child, S-H. Dong, and X-G. Wang, J. Phys. A, 33, 5653 (2000). https://doi.org/10.1088/0305-4470/33/32/303

P. Amore, and F.M. Fernandez, Phys. Scr. 95, 105201 (2020). https://doi.org/10.1088/1402-4896/abb252

Published
2025-12-08
Cited
How to Cite
Khaled, F. Z., Moumni, M., & Falek, M. (2025). On Solutions of the Killingbeck Potential and Clarifying Comments on a Related Analytical Approach. East European Journal of Physics, (4), 164-171. https://doi.org/10.26565/2312-4334-2025-4-14
Issue
No. 4 (2025): East European Journal of Physics
Section
Original Papers

Copyright (c) 2025 Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

    • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
    • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
    • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).