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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2011 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2011
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| Онлайн доступ: | https://nasplib.isofts.kiev.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. |
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Kalnins, E.G. Kress, J.M. Miller Jr., W. 2019-02-11T15:45:22Z 2019-02-11T15:45:22Z 2011 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 https://nasplib.isofts.kiev.ua/handle/123456789/146806 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. This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Recurrence Relation Approach to Higher Order Quantum Superintegrability Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability |
| spellingShingle |
A Recurrence Relation Approach to Higher Order Quantum Superintegrability Kalnins, E.G. Kress, J.M. Miller Jr., W. |
| 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 |
| author |
Kalnins, E.G. Kress, J.M. Miller Jr., W. |
| author_facet |
Kalnins, E.G. Kress, J.M. Miller Jr., W. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.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. |
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