Construction of angular-dependent potentials from trigonometric Pöschl-Teller systems within the Dunkl formalism

Axel Schulze-Halberg

doi:10.14311/AP.2023.63.0273

Authors

Axel Schulze-Halberg Indiana University Northwest, Department of Mathematics and Acturial Science and Department of Physics, 3400 Broadway, Gary IN 46408, United States of America

DOI:

https://doi.org/10.14311/AP.2023.63.0273

Keywords:

Dunkl operator, Schrodinger equation, trigonometric Poschl-Teller potential, angular equation, Darboux-Crum transformation

Abstract

We generate solvable cases of the two angular equations resulting from variable separation in the three-dimensional Dunkl-Schrödinger equation expressed in spherical coordinates. It is shown that the Dunkl formalism interrelates these angular equations with trigonometric Pöschl-Teller systems. Based on this interrelation, we use point transformations and Darboux-Crum transformations to construct new solvable cases of the angular equations. Instead of the stationary energy, we use the constants due to the separation of variables as transformation parameters for our Darboux-Crum transformations.

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References

C. F. Dunkl. Differential-difference operators associated to reflection groups. Transactions of the American Mathematical Society 311(1):167–183, 1989. https://doi.org/10.2307/2001022

M. Rösler. Orthogonal polynomials and special functions. In Dunkl operators: Theory and applications, vol. 1817, pp. 93–135. Springer, Berlin, D, 2003. https://doi.org/10.1007/3-540-44945-0_3

J. F. Diejen, L. Vinet. Calogero-Moser-Sutherland models. Springer, New York, NY, 1st edn., 2000. https://doi.org/10.1007/978-1-4612-1206-5

P. Etingof. Calogero-Moser systems and representation theory. European Mathematical Society, Zurich, CH, 2007. https://doi.org/10.4171/034

M. Feigin, T. Hakobyan. On Dunkl angular momenta algebra. Journal of High Energy Physics 2015:107, 2015. https://doi.org/10.1007/JHEP11(2015)107

M. Faigin, M. Vrabec. Intertwining operator for AG2 Calogero-Moser-Sutherland system. Journal of Mathematical Physics 60:073503, 2019. https://doi.org/10.1063/1.5090274

J. P. Anker. An introduction to Dunkl theory and its analytic aspects. In Analytic, Algebraic and Geometric Aspects of Differential Equations. Trends in Mathematics, pp. 3–58. Birkhäuser, Basel, CH, 2017. https://doi.org/10.1007/978-3-319-52842-7_1

H. Mejjaoli. Nonlinear generalized Dunkl-wave equations and applications. Journal of Mathematical Analysis and Applications 375(1):118–138, 2011. https://doi.org/10.1016/j.jmaa.2010.08.058

C. F. Dunkl, Y. Xu. Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge, UK, 2nd edn., 2014. https://doi.org/10.1017/CBO9781107786134

W. S. Chung, H. Hassanabadi. One-dimensional quantum mechanics with Dunkl derivative. Modern Physics Letters A 34(24):1950190, 2019. https://doi.org/10.1142/S0217732319501906

V. X. Genest, M. E. H. Ismail, L. Vinet, A. Zhedanov. The Dunkl oscillator in the plane II: Representations of symmetry algebra. Communication in Mathematical Physics 329:999–1029, 2014. https://doi.org/10.1007/s00220-014-1915-2

V. X. Genest, M. E. H. Ismail, L. Vinet, A. Zhedanov. The Dunkl oscillator in the plane I: Superintegrability, separated wavefunctions and overlap coefficients. Journal of Physics A: Mathematical and Theoretical 46(14):145201, 2013. https://doi.org/10.1088/1751-8113/46/14/145201

V. X. Genest, A. Lapointe, L. Vinet. The Dunkl-Coulomb problem in the plane. Physics Letters A 379(12–13):923–927, 2015. https://doi.org/10.1016/j.physleta.2015.01.023

R. D. Mota, D. Ojeda-Guillén, M. Salazar-Ramírez, V. D. Granados. Exact solutions of the 2D Dunkl-Klein-Gordon equation: The Coulomb potential and the Klein-Gordon oscillator. Modern Physics Letters A 36(23):2150171, 2021. https://doi.org/10.1142/S0217732321501716

C. Quesne. Rationally-extended Dunkl oscillator on the line. Journal oh Physics A: Mathematical and Theoretical 56(26):265203, 2023. https://doi.org/10.1088/1751-8121/acd736

D. Ojeda-Guillén, R. D. Mota, M. Salazar-Ramírez, V. D. Granados. Algebraic approach for the one-dimensional Dirac-Dunkl oscillator. Modern Physics Letters A 35(31):2050255, 2020. https://doi.org/10.1142/S0217732320502557

S. Sargolzaeipor, H. Hassanabadi, W. S. Chung. Effect of the Wigner-Dunkl algebra on the Dirac equation and Dirac harmonic oscillator. Modern Physics Letters A 33(25):1850146, 2018. https://doi.org/10.1142/S0217732318501468

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. Contreras-Astorga, D. J. Fernández. Supersymmetric partners of the trigonometric Pöschl-Teller potentials. Journal of Physics A: Mathematical and Theoretical 41(47):475303, 2008. https://doi.org/10.1088/1751-8113/41/47/475303

A. Arabasghari, H. Hassanabadi, A. Schulze-Halberg, W. S. Chung. Bound state solutions of the Dunkl-Schrödinger equation for a class of angular-dependent potentials, 2023. Preprint.

D. J. Fernández. Trends in supersymmetric quantum mechanics. In Integrability, Supersymmetry and Coherent States, pp. 37–68. Springer, New York, NY, 2019. https://doi.org/10.1007/978-3-030-20087-9_2

M. Abramowitz, I. A. Stegun. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, NY, 9th edn., 1964.

A. D. Alhaidari, I. A. Assi, A. Mebirouk. Bound-states for generalized trigonometric and hyperbolic Pöschl-Teller potentials. International Journal of Modern Physics A 37(3):2250012, 2022. https://doi.org/10.1142/S0217751X22500129

A. Schulze-Halberg, P. Roy. Construction of zero-energy states in graphene through the supersymmetry formalism. Journal of Physics A: Mathematical and Theoretical 50(36):365205, 2017. https://doi.org/10.1088/1751-8121/aa8249

A. Schulze-Halberg, P. Roy. Darboux transformations for Dunkl-Schrödinger equations with energy-dependent potential and position-dependent mass. [2023-01-27]. arXiv:2301.11622

D. Bermudez, D. J. Fernández, N. Fernández-García. Wronskian differential formula for confluent supersymmetric quantum mechanics. Physics Letters A 376(5):692–696, 2012. https://doi.org/10.1016/j.physleta.2011.12.020