Persistence of solvability in quantum systems deformed by Dunkl operators
DOI:
https://doi.org/10.14311/AP.2025.65.0222Keywords:
Schrödinger equation, Dunkl operator, bound states, position-dependent massAbstract
We study persistence of solvability in nonrelativistic quantum systems with positiondependent mass upon introduction of a deformation by Dunkl operators. Conditions are derived for the governing Schrödinger equation of the conventional system to admit the same solutions as in the deformed case, up to a reparametrisation of coupling constants. These conditions require the positiondependent mass or the potential of the system to have a specific form. If this is the case for a particular system, then the Schrödinger equations for its conventional version and for the Dunkl-deformed partner share solutions in the same functional form.
Downloads
References
E. P. Wigner. Do the equations of motion determine the quantum mechanical commutation relations? Physical Review 77:711–712, 1950. https://doi.org/10.1103/PhysRev.77.711
L. M. Yang. A note on the quantum rule of the harmonic oscillator. Physical Review 84:788–790, 1951. https://doi.org/10.1103/PhysRev.84.788
H. S. Green. A generalized method of field quantization. Physical Review 90:270–273, 1953. https://doi.org/10.1103/PhysRev.90.270
O. W. Greenberg, A. M. L. Messiah. Selection rules for parafields and the absence of para particles in nature. Physical Review 138:B1155–B1167, 1965. https://doi.org/10.1103/PhysRev.138.B1155
M. S. Plyushchay. Deformed Heisenberg algebra and fractional spin field in 2 + 1 dimensions. Physics Letters B 320(1–2):91–95, 1994. https://doi.org/10.1016/0370-2693(94)90828-1
M. S. Plyushchay. Deformed Heisenberg algebra with reflection. Nuclear Physics B 491(3):619–634, 1997. https://doi.org/10.1016/S0550-3213(97)00065-5
M. Plyushchay. Hidden nonlinear supersymmetries in pure parabosonic systems. International Journal of Modern Physics A 15(23):3679–3698, 2000. https://doi.org/10.1142/S0217751X00001981
C. F. Dunkl. Differential-difference operators associated to reflection groups. Transactions of the American Mathematical Society 311:167–183, 1989. https://doi.org/10.1090/S0002-9947-1989-0951883-8
J. F. van Diejen, L. Vinet. Calogero-Moser-Sutherland models. CRM Series in Mathematical Physics. Springer-Verlag, New York, 2000. https://doi.org/10.1007/978-1-4612-1206-5
B. Hamil, B. C. Lütfüoğlu, M. Merad. Dunkl-Schrödinger equation in higher dimensions. Physica Scripta 100(3):035301, 2025. https://doi.org/10.1088/1402-4896/ada9b2
A. Benchikha, B. Hamil, B. C. Lütfüoğlu, B. Khantoul. Dunkl-Schrödinger equation with time-dependent harmonic oscillator potential. International Journal of Theoretical Physics 63(10):248, 2024. https://doi.org/10.1007/s10773-024-05786-6
A. Benchikha, B. Hamil, B. C. Lütfüoğlu. A path integral treatment of time-dependent Dunkl quantum mechanics. International Journal of Geometric Methods in Modern Physics p. 2550113. Preprint. https://doi.org/10.1142/S0219887825501130
H. Bouguerne, B. Hamil, B. C. Lütfüoğlu, M. Merad. Dunkl algebra and vacuum pair creation: Exact analytical results via Bogoliubov method. Nuclear Physics B 1007:116684, 2024. https://doi.org/10.1016/j.nuclphysb.2024.116684
A. Hocine, B. Hamil, F. Merabtine, et al. On Dunkl-Bose-Einstein condensation in harmonic traps. Revista Mexicana de Física 70(5):051701, 2024. https://doi.org/10.31349/RevMexFis.70.051701
M. Rosenblum. Generalized Hermite polynomials and the Bose-like oscillator calculus. In A. Feintuch, I. Gohberg (eds.), Nonselfadjoint operators and related topics, vol. 73, pp. 369–396. Birkhäuser, Basel, 1994. https://doi.org/10.1007/978-3-0348-8522-5_15
B. Hamil, B. C. Lütfüoğlu. Dunkl graphene in constant magnetic field. The European Physical Journal Plus 137(11):1241, 2022. https://doi.org/10.1140/epjp/s13360-022-03463-3
C. Quesne. Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials. Modern Physics Letters A 38(22–23):2350108, 2023. https://doi.org/10.1142/S0217732323501080
B. Hamil, B. C. Lütfüoğlu. Dunkl-Klein-Gordon equation in three-dimensions: The Klein-Gordon oscillator and Coulomb potential. Few-Body Systems 63(4):74, 2022. https://doi.org/10.1007/s00601-022-01776-8
G. Junker. On the path integral formulation of Wigner-Dunkl quantum mechanics. Journal of Physics A: Mathematical and Theoretical 57(7):075201, 2024. https://doi.org/10.1088/1751-8121/ad213d
R. D. Mota, D. Ojeda-Guillén, M. A. Xicoténcatl. The generalized Fokker-Planck equation in terms of Dunkl-type derivatives. Physica A: Statistical Mechanics and its Applications 635:129525, 2024. https://doi.org/10.1016/j.physa.2024.129525
H. Bouguerne, B. Hamil, B. C. Lütfüoğlu, M. Merad. Dunkl-Pauli equation in the presence of a magnetic field. Indian Journal of Physics 98(12):4093–4105, 2024. https://doi.org/10.1007/s12648-024-03170-y
R. D. Mota, D. Ojeda-Guillén. Effect of the two-parameter generalized Dunkl derivative on the two-dimensional Schrödinger equation. Modern Physics Letters A 37(33–34):2250224, 2022. https://doi.org/10.1142/S0217732322502248
R. D. Mota, D. Ojeda-Guillén. Exact solutions of the Schrödinger equation with Dunkl derivative for the free-particle spherical waves, the pseudo-harmonic oscillator and the Mie-type potential. Modern Physics Letters A 37(1):2250006, 2022. https://doi.org/10.1142/S0217732322500067
A. Schulze-Halberg. Closed-form bound states for two Dunkl-Liénard oscillator systems. International Journal of Modern Physics A 39(2–3):2450013, 2024. https://doi.org/10.1142/S0217751X24500131
T. Gora, F. Williams. Electronic states of homogeneous and inhomogeneous mixed semiconductors. In D. G. Thomas (ed.), II–VI Semiconducting Compounds. Benjamin, New York, 1967.
T. Gora, F. Williams. Theory of electronic states and transport in graded mixed semiconductors. Physical Review 177:1179–1182, 1969. https://doi.org/10.1103/PhysRev.177.1179
P. T. Landsberg. Solid state theory: methods and applications. Wiley-Interscience, London, 1969.
O. von Roos, H. Mavromatis. Position-dependent effective masses in semiconductor theory. II. Physical Review B 31:2294–2298, 1985. https://doi.org/10.1103/PhysRevB.31.2294
O. von Roos. Position-dependent effective masses in semiconductor theory. Physical Review B 27:7547–7552, 1983. https://doi.org/10.1103/PhysRevB.27.7547
R. M. Lima, H. R. Christiansen. The kinetic Hamiltonian with position-dependent mass. Physica E: Low-dimensional Systems and Nanostructures 150:115688, 2023. https://doi.org/10.1016/j.physe.2023.115688
O. Rosas-Ortiz. Position-dependent mass systems: Classical and quantum pictures. In P. Kielanowski, A. Odzijewicz, E. Previato (eds.), Geometric Methods in Physics XXXVIII, pp. 351–361. Springer International Publishing, Cham, 2020. https://doi.org/10.1007/978-3-030-53305-2_24
M. F. Rañada. A quantum quasi-harmonic nonlinear oscillator with an isotonic term. Journal of Mathematical Physics 55(8):082108, 2014. https://doi.org/10.1063/1.4892084
P. M. Mathews, M. Lakshmanan. On a unique nonlinear oscillator. Quarterly of Applied Mathematics 32(2):215–218, 1974.
P. M. Mathews, M. Lakshmanan. A quantum-mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction. Il Nuovo Cimento A (1965–1970) 26(3):299–316, 1975. https://doi.org/10.1007/BF02769015
S. Karthiga, V. Chithiika Ruby, M. Senthilvelan, M. Lakshmanan. Quantum solvability of a general ordered position dependent mass system: Mathews-Lakshmanan oscillator. Journal of Mathematical Physics 58(10):102110, 2017. https://doi.org/10.1063/1.5008993
M. Abramowitz, I. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New York, 1964.
Wolfram Research. Gauss hypergeometric function 2F1 identities (formula 07.23.17.0056), 2001. [2025-05-01]. https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/17/02/07/0004/
V. Chithiika Ruby, V. K. Chandrasekar, M. Lakshmanan. Quantum solvability of a nonlinear δ-type mass profile system: coupling constant quantization. Journal of Physics Communications 6(8):085006, 2022. https://doi.org/10.1088/2399-6528/ac8522
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Axel Schulze-Halberg
This work is licensed under a Creative Commons Attribution 4.0 International License.
