A Recurrence Relation Approach to Higher Order Quantum Superintegrability
We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of h...
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| Дата: | 2011 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2011
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146806 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Recurrence Relation Approach to Higher Order Quantum Superintegrability / E.G Kalnins, J.M. Kress, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 30 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1468062019-02-12T01:24:22Z A Recurrence Relation Approach to Higher Order Quantum Superintegrability Kalnins, E.G. Kress, J.M. Miller Jr., W. We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of k, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each k. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the Stäckel transform acts and show that it preserves the structure equations. 2011 Article A Recurrence Relation Approach to Higher Order Quantum Superintegrability / E.G Kalnins, J.M. Kress, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 30 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C99; 20C35; 22E70 DOI:10.3842/SIGMA.2011.031 http://dspace.nbuv.gov.ua/handle/123456789/146806 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of k, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each k. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the Stäckel transform acts and show that it preserves the structure equations. |
| format |
Article |
| author |
Kalnins, E.G. Kress, J.M. Miller Jr., W. |
| spellingShingle |
Kalnins, E.G. Kress, J.M. Miller Jr., W. A Recurrence Relation Approach to Higher Order Quantum Superintegrability Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Kalnins, E.G. Kress, J.M. Miller Jr., W. |
| author_sort |
Kalnins, E.G. |
| title |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability |
| title_short |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability |
| title_full |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability |
| title_fullStr |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability |
| title_full_unstemmed |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability |
| title_sort |
recurrence relation approach to higher order quantum superintegrability |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146806 |
| citation_txt |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability / E.G Kalnins, J.M. Kress, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 30 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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