{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:15:26Z","timestamp":1760116526196,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,10]],"date-time":"2024-11-10T00:00:00Z","timestamp":1731196800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2024R974"],"award-info":[{"award-number":["RSPD2024R974"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Quantum deformations offer valuable perspectives into quantum mechanics, particularly by advancing our understanding of symmetry and algebraic structures.The Dunkl oscillator, which integrates Dunkl operators into the harmonic oscillator framework, advances the system\u2019s algebraic properties and opens new approaches for exploring quantum phenomena. Supersymmetric quantum mechanics (SSQM) unifies bosonic and fermionic aspects and facilitates the construction of solvable models using generalized Dunkl operators. This paper introduces a new approach to the Dunkl oscillator, employing a complex reflection operator to deepen the understanding of its connection to Hermite polynomials on radial lines. The results offer new perspectives on the Dunkl oscillator\u2019s algebraic structure and its relevance to SSQM and quantum deformation theory, expanding the potential for discovering solvable quantum models.<\/jats:p>","DOI":"10.3390\/sym16111508","type":"journal-article","created":{"date-parts":[[2024,11,11]],"date-time":"2024-11-11T03:52:07Z","timestamp":1731297127000},"page":"1508","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Supersymmetric Quesne-Dunkl Quantum Mechanics on Radial Lines"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2743-2036","authenticated-orcid":false,"given":"Fethi","family":"Bouzeffour","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1103\/PhysRev.77.711","article-title":"Do the equations of motion determine the quantum mechanical commutation relations?","volume":"77","author":"Wigner","year":"1950","journal-title":"Phys. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1103\/PhysRev.84.788","article-title":"A note on the quantum rule of the harmonic oscillator","volume":"84","author":"Yang","year":"1951","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1090\/S0002-9947-1989-0951883-8","article-title":"Differential-difference operators associated to reflection groups","volume":"311","author":"Dunkl","year":"1989","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dunkl, C.F., and Xu, Y. (2014). Orthogonal Polynomials of Several Variables, Cambridge University Press. No. 155.","DOI":"10.1017\/CBO9781107786134"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1006\/aphy.1996.0012","article-title":"Deformed Heisenberg algebra, fractional spin fields, and supersymmetry without fermions","volume":"245","author":"Plyushchay","year":"1996","journal-title":"Ann. Phys. (N. Y.)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0375-9601(98)00046-2","article-title":"C\u03bb-extended harmonic oscillator and (para)supersymmetric quantum mechanics","volume":"240","author":"Quesne","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"145201","DOI":"10.1088\/1751-8113\/46\/14\/145201","article-title":"The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients","volume":"46","author":"Genest","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1007\/s10773-020-04667-y","article-title":"C\u03bb-Extended Oscillator Algebra and d-Orthogonal Polynomials","volume":"60","author":"Bouzeffour","year":"2021","journal-title":"Int. J. Theor. Phys."},{"key":"ref_9","unstructured":"Nersessian, H.B. (1995). Generalized Hermite polynomials on the radial rays in the complex plane. Theory of Functions and Applications, Louys Publishing House."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1006\/jmaa.1997.5199","article-title":"A class of orthogonal polynomials on the radial rays in the complex plane","volume":"206","author":"Milovanovic","year":"1995","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"L749","DOI":"10.1088\/0305-4470\/25\/12\/008","article-title":"Supersymmetry and fractional statistics in two-dimensional field theory","volume":"25","author":"Khare","year":"1992","journal-title":"J. Phys. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/0550-3213(81)90006-7","article-title":"Dynamical breaking of supersymmetry","volume":"188","author":"Witten","year":"1981","journal-title":"Nucl. Phys. B"},{"key":"ref_13","first-page":"356","article-title":"Supersymmetry in the problem of the harmonic oscillator","volume":"38","author":"Gendenshtein","year":"1983","journal-title":"JETP Lett."},{"key":"ref_14","unstructured":"Plyushchay, M.S. (1994). Supersymmetry without fermions. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"435301","DOI":"10.1088\/1751-8113\/44\/43\/435301","article-title":"Supersymmetric quantum mechanics with reflections","volume":"44","author":"Post","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1007\/s44198-024-00224-x","article-title":"The Extended Dunkl Oscillator and the Generalized Hermite Polynomials on the Radial Lines","volume":"31","author":"Bouzeffour","year":"2024","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_17","unstructured":"Chihara, T.S. (2011). An Introduction to Orthogonal Polynomials, Courier Corporation."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Koekoek, R., Lesky, P.A., and Swarttouw, R.F. (2010). Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-05014-5"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1508\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:29:46Z","timestamp":1760113786000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1508"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,10]]},"references-count":18,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["sym16111508"],"URL":"https:\/\/doi.org\/10.3390\/sym16111508","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,11,10]]}}}