Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 2. P. 119 – 141. ISSN 2079-6641

PHYSICS
https://doi.org/10.26117/2079-6641-2026-54-2-119-141
Research Article
Full text in English
MSC 81V99, 82B30

Contents of this issue

Read Russian Version

Thermodynamic Properties of a Quantum Isotonic Oscillator with Position-Dependent Effective Mass

E. S. Bokpe^1, F. A. Dossa^{2,3,4,\ast}, G. Y. Avossevou^1

^1Institut de Math´ematiques et de Sciences Physiques (IMSP), Universit´e d’Abomey-Calavi (UAC), Porto-Novo, Benin
^2´Ecole Doctorale des Sciences, Technologies, Ing´enierie et Math´ematiques (ED-STIM), UNSTIM, Abomey, Benin
^3Laboratoire de Physique et Applications (LPA), UNSTIM, Abomey, Benin
^4D´epartement de Physique, Facult´e des Sciences et Techniques (FAST), Natitingou, Benin

Abstract. We examine the thermodynamic properties of a quantum isotonic oscillator with positiondependent effective mass, described by a strictly negative deformation parameter \alpha. By solving the corresponding Schr¨odinger equation using the Nikiforov-Uvarov method, we obtain the energy spectrum and wave functions of the system. From these results we construct the partition function, from which we derive the main thermodynamic quantities: internal energy, specific heat, free energy, and entropy. The analysis shows that the deformation of the effective mass significantly modifies the energy levels and the thermal response of the system. More precisely, the parameter \alpha exerts a decisive influence on thermal transitions, while the isotonic parameter \beta becomes determinant when the mass is no longer constant. We further show, through explicit analytical and numerical analysis, that the condition \alpha<0 is not a simple arbitrary choice but the necessary and sufficient condition for the thermodynamic stability of the model. These observations underscore the need to account for mass deformation in quantum models, in particular to faithfully describe the thermal behavior of nanostructured materials and semiconductor devices.

Key words: Isotonic oscillator; Position-dependent mass; Nikiforov-Uvarov method; Thermodynamics
properties.

Received: 28.05.2026; Revised: 09.07.2026; Accepted: 10.07.2026; First online: 14.09.2026

For citation. Bokpe E. S., Dossa F. A., Avossevou G.Y. Thermodynamic properties of a quantum isotonic oscillator with position-dependent effective mass. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 2, 119-141. EDN: FQFUER. https://doi.org/10.26117/2079-6641-2026-54-2-119-141.

Funding. The work was carried out without external funding.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to the writing of the article and are responsible for the final version.

^{\ast}Correspondence: E-mail: dossafanselme@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Bokpe E. S., Dossa F. A., Avossevou G. Y., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

References

1. Von Ross O., Mavromatis H.Position-dependent effective masses in semiconductor theory. II, Phys. Rev. B., 1985. vol. 31, pp. 2294.
2. De Saavedra F.A., Boronat J., Polls A., Fabrocini A. Effective mass of one He 4 atom in liquid He 3, Phys. Rev. B., 1994. vol. 50, pp. 4248.
3. Geller M.R., Kohn W. Quantum mechanics of electrons in crystals with graded composition, Phys. Rev. Lett., 1993. vol. 70, pp. 3103.
4. Harrison P., Valavanis A. Quantum wells, wires and dots: theoretical and computational physics of semiconductor nanostructures. Chichester: John Wiley and Sons, 2016.
5. BenDaniel D.J., Duke C.B. Space-charge effects on electron tunneling, Phys. Rev., 1966. vol. 152, pp. 683.
6. Dossa F.A. Thermodynamic properties and algebraic solution of the N-dimensional harmonic oscillator with minimal length uncertainty relations, Phys. Scr., 2021. vol. 96, pp. 105703.
7. Amir N., Iqbal S. Ladder operators and associated algebra for position-dependent effective mass systems, Europhys. Lett., 2015. vol. 111, pp. 20005.
8. Tchoffo M., Migueu F.B., Vubangsi M., Fai L.C. Supersymmetric approach to coherent states for nonlinear oscillator with spatially dependent effective mass, Heliyon, 2019. vol. 5.
9. Gendenshtein L. Derivation of exact spectra of the Schrödinger equation by means of supersymmetry, JETP Lett., 1983. vol. 38, pp. 356.
10. Ikhdair S.M., Sever R. Relativistic and nonrelativistic bound states of the isotonic oscillator by Nikiforov-Uvarov method, J. Math. Phys., 2011. vol. 52.
11. Cariñena J.F., Perelomov A.M., Rañada M.F., Santander M.A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator, J. Phys. A: Math. Theor., 2008. vol. 41, pp. 085301.
12. Cariñena J.F., Inzunza L., Plyushchay M.S. Rational deformations of conformal mechanics, Phys. Rev. D, 2018. vol. 98, pp. 026017.
13. Zhu D.A new potential with the spectrum of an isotonic oscillator, J. Phys. A: Math. Gen., 1987. vol. 20, pp. 4331.
14. Wang J.S., Liu T.K., Zhan M.S. Nonclassical properties of even and odd generalized coherent states for an isotonic oscillator, J. Opt. B: Quantum Semiclass. Opt., 2000. vol. 2, pp. 758.
15. Thirulogasanthar K., Saad N. Coherent states associated with the wavefunctions and the spectrum of the isotonic oscillator, J. Phys. A: Math. Gen., 2004. vol. 37, pp. 4567.
16. Cariñena J.F., Rañada M.F., Santander M., Senthilvelan M.A non-linear oscillator with quasiharmonic behaviour: two- and n-dimensional oscillators, Nonlinearity, 2004. vol. 17, pp. 1941.
17. Cariñena J.F., Rañada M.F., Santander M.A superintegrable two-dimensional non-linear oscillator with an exactly solvable quantum analog, SIGMA, 2007. vol. 3, pp. 030.
18. Cariñena J.F., Rañada M.F., Santander M.A nonlinear deformation of the isotonic oscillator and the Smorodinski-Winternitz system: integrability and superintegrability, Regular and Chaotic Dynamics, 2005. vol. 10, pp. 423.
19. Sakiroglu S., Sari H. Contribution of the impurity to the thermodynamic properties of a quantum dot with a Gaussian confinement potential, Eur. Phys. J. Plus, 2025. vol. 140, pp. 1.
20. Onate C.A. et al. Thermodynamic properties and the confinement frequency for Second Pöschl-Teller potential, Results in Physics, 2025. vol. 70, pp. 108144.
21. Nikiforov A.F., Uvarov V.B. Special functions of mathematical physics. Basel: Birkhäuser, 1988. 205 pp.
22. Nikiforov A.F., Uvarov V.B., Suslov S.K. Classical orthogonal polynomials of a discrete variable. Berlin: Springer, 1991.
23. Hassanabadi H., Maghsoodi E., Ikot A.N., Zarrinkamar S. Minimal Length Schrödinger Equation with Harmonic Potential in the Presence of a Magnetic Field, Advances in High Energy Physics, 2013. vol. 2013, pp. 923686.
24. Dimirci M., Sever R. Arbitrary l-state solutions of the Klein-Gordon equation with the Eckart plus a class of Yukawa potential and its non-relativistic thermal properties, Eur. Phys. J. Plus, 2023. vol. 138, pp. 1.
25. Berkdemir C. Theoretical Concepts of Quantum Mechanics, 2012.
26. Falaye B.J., Oyewumi K.J., Abbas M. Exact solution of Schrödinger equation with q-deformed quantum potentials using Nikiforov-Uvarov method, Chin. Phys. B, 2013. vol. 22, pp. 110301.
27. Onate C.A., Okon I.B., Jude G.O., Onyeaju M.C., Anita Z.D. Uncertainty Relation and the Thermal Properties of an Isotropic Harmonic oscillator (IHO) with the Inverse Quadratic (IQ) Potentials and the Pseudo-Harmonic (PH) with the Inverse Quadratic (IQ) Potentials, Quantum Reports, 2023. vol. 5, pp. 38.
28. Gora T., Williams F. Theory of electronic states and transport in graded mixed semiconductors, Phys. Rev., 1969. vol. 177, pp. 1179.
29. Dekar L., Chetouani L., Hammann T.F. An exactly soluble Schrödinger equation with smooth position-dependent mass, J. Math. Phys., 1998. vol. 39, pp. 2551.
30. Myers N.M., McCready J., Deffner S. Quantum heat engines with singular interactions, Symmetry, 2021. vol. 13, pp. 978.
31. Tezcan C., Sever R.A general approach for the exact solution of the Schrödinger equation, Int. J. Theor. Phys., 2009. vol. 48, pp. 337.
32. Abramowitz M., Stegun I.A. Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables, vol. 55, 1965.
33. Morse P.M., Feshbach H. Methods of Theoretical Physics. New York: Technology Press, 1946.
34. Strekalov M.L. Thermodynamic Functions of Triatomic Molecules: An Analytical View, Russ. J. Phys. Chem. A, 2024. vol. 98, pp. 1671.
35. Song X.Q., Wang C.W., Jia C.S. Thermodynamic properties for the sodium dimer, Chem. Phys. Lett., 2017. vol. 673, pp. 50.
36. Bastard G. Wave Mechanics Applied to Semiconductor Heterostructures. New York: John Wiley and Sons, 1989.
37. Serra L., Lipparini E. Spin response of unpolarized quantum dots, Europhys. Lett., 1997. vol. 40, pp. 667.
38. Dong S.H. Factorization method in quantum mechanics, vol. 150. Berlin: Springer, 2007.
39. Plastino A.R., Rigo A., Casas M., Garcias F., Plastino A. Supersymmetric approach to quantum systems with position-dependent effective mass, Phys. Rev. A, 1999. vol. 60, pp. 4318.
40. Cruz C.S., Negro J., Nieto L.M. Classical and quantum position-dependent mass harmonic oscillators, Phys. Lett. A, 2007. vol. 369, pp. 400.

---

Information about the authors

---

---

---