Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible representations of the quantum quadratic algebras though the constr...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2008 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2008
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148996 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D / E.G. Kalnins, W.Jr. Miller, S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 47 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862547561055780864 |
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| author | Kalnins, E.G. Post, S. Miller Jr., W. |
| author_facet | Kalnins, E.G. Post, S. Miller Jr., W. |
| citation_txt | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D / E.G. Kalnins, W.Jr. Miller, S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 47 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible representations of the quantum quadratic algebras though the construction of models in which the symmetries act on spaces of functions of a single complex variable via either differential operators or difference operators. In another paper we have already carried out parts of this analysis for the generic nondegenerate superintegrable system on the complex 2-sphere. Here we carry it out for a degenerate superintegrable system on the 2-sphere. We point out the connection between our results and a position dependent mass Hamiltonian studied by Quesne. We also show how to derive simple models of the classical quadratic algebras for superintegrable systems and then obtain the quantum models from the classical models, even though the classical and quantum quadratic algebras are distinct.
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| first_indexed | 2025-11-25T16:04:12Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148996 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T16:04:12Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kalnins, E.G. Post, S. Miller Jr., W. 2019-02-19T12:50:44Z 2019-02-19T12:50:44Z 2008 Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D / E.G. Kalnins, W.Jr. Miller, S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 47 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 20C99; 20C35; 22E70 https://nasplib.isofts.kiev.ua/handle/123456789/148996 There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible representations of the quantum quadratic algebras though the construction of models in which the symmetries act on spaces of functions of a single complex variable via either differential operators or difference operators. In another paper we have already carried out parts of this analysis for the generic nondegenerate superintegrable system on the complex 2-sphere. Here we carry it out for a degenerate superintegrable system on the 2-sphere. We point out the connection between our results and a position dependent mass Hamiltonian studied by Quesne. We also show how to derive simple models of the classical quadratic algebras for superintegrable systems and then obtain the quantum models from the classical models, even though the classical and quantum quadratic algebras are distinct. This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D Article published earlier |
| spellingShingle | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D Kalnins, E.G. Post, S. Miller Jr., W. |
| title | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D |
| title_full | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D |
| title_fullStr | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D |
| title_full_unstemmed | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D |
| title_short | Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D |
| title_sort | models for quadratic algebras associated with second order superintegrable systems in 2d |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148996 |
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