Tools for Verifying Classical and Quantum Superintegrability
Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n−1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are al...
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| Дата: | 2010 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2010
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146503 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Tools for Verifying Classical and Quantum Superintegrability / E.G. Kalnins, J.M. Kress, Jr. Willard Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1465032019-02-10T01:24:41Z Tools for Verifying Classical and Quantum Superintegrability Kalnins, E.G. Kress, J.M. Willard Miller, Jr. Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n−1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the constructions to date are for n=2 but cases where n>2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stäckel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential. 2010 Article Tools for Verifying Classical and Quantum Superintegrability / E.G. Kalnins, J.M. Kress, Jr. Willard Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C99; 20C35; 22E70 DOI:10.3842/SIGMA.2010.066 http://dspace.nbuv.gov.ua/handle/123456789/146503 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 |
Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n−1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the constructions to date are for n=2 but cases where n>2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stäckel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential. |
| format |
Article |
| author |
Kalnins, E.G. Kress, J.M. Willard Miller, Jr. |
| spellingShingle |
Kalnins, E.G. Kress, J.M. Willard Miller, Jr. Tools for Verifying Classical and Quantum Superintegrability Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Kalnins, E.G. Kress, J.M. Willard Miller, Jr. |
| author_sort |
Kalnins, E.G. |
| title |
Tools for Verifying Classical and Quantum Superintegrability |
| title_short |
Tools for Verifying Classical and Quantum Superintegrability |
| title_full |
Tools for Verifying Classical and Quantum Superintegrability |
| title_fullStr |
Tools for Verifying Classical and Quantum Superintegrability |
| title_full_unstemmed |
Tools for Verifying Classical and Quantum Superintegrability |
| title_sort |
tools for verifying classical and quantum superintegrability |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146503 |
| citation_txt |
Tools for Verifying Classical and Quantum Superintegrability / E.G. Kalnins, J.M. Kress, Jr. Willard Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:24:41Z |
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