Dynamical and invariance algebras of the $d$-dimensional Dunkl-Coulomb problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914071975034880 |
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| author | Quesne, Christiane |
| author_facet | Quesne, Christiane |
| contents | It is shown that the rich algebraic structure of the standard $d$-dimensional Coulomb problem can be extended to its Dunkl counterpart. Replacing standard derivatives by Dunkl ones in the so($d+1$,2) dynamical algebra generators of the former gives rise to a deformed algebra with similar commutation relations, except that the metric tensor becomes dependent on the reflection operators and that there are some additional commutation or anticommutation relations involving the latter. It is then shown that from some of the dynamical algebra generators it is straightforward to derive the integrals of motion of the Dunkl-Coulomb problem in Sturm representation. Finally, from the latter, the components of a deformed Laplace-Runge-Lenz vector are built. Together with the Dunkl angular momentum components, such operators insure the superintegrability of the Dunkl-Coulomb problem in Schrödinger representation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_07862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamical and invariance algebras of the $d$-dimensional Dunkl-Coulomb problem Quesne, Christiane Mathematical Physics Quantum Physics It is shown that the rich algebraic structure of the standard $d$-dimensional Coulomb problem can be extended to its Dunkl counterpart. Replacing standard derivatives by Dunkl ones in the so($d+1$,2) dynamical algebra generators of the former gives rise to a deformed algebra with similar commutation relations, except that the metric tensor becomes dependent on the reflection operators and that there are some additional commutation or anticommutation relations involving the latter. It is then shown that from some of the dynamical algebra generators it is straightforward to derive the integrals of motion of the Dunkl-Coulomb problem in Sturm representation. Finally, from the latter, the components of a deformed Laplace-Runge-Lenz vector are built. Together with the Dunkl angular momentum components, such operators insure the superintegrability of the Dunkl-Coulomb problem in Schrödinger representation. |
| title | Dynamical and invariance algebras of the $d$-dimensional Dunkl-Coulomb problem |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2410.07862 |