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...
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2010 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2010
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146503 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862543151491710976 |
|---|---|
| author | Kalnins, E.G. Kress, J.M. Willard Miller, Jr. |
| author_facet | Kalnins, E.G. Kress, J.M. Willard Miller, Jr. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
|
| first_indexed | 2025-11-24T21:43:24Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146503 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T21:43:24Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kalnins, E.G. Kress, J.M. Willard Miller, Jr. 2019-02-09T19:30:20Z 2019-02-09T19:30:20Z 2010 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 https://nasplib.isofts.kiev.ua/handle/123456789/146503 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tools for Verifying Classical and Quantum Superintegrability Article published earlier |
| spellingShingle | Tools for Verifying Classical and Quantum Superintegrability Kalnins, E.G. Kress, J.M. Willard Miller, Jr. |
| title | 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_short | Tools for Verifying Classical and Quantum Superintegrability |
| title_sort | tools for verifying classical and quantum superintegrability |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146503 |
| work_keys_str_mv | AT kalninseg toolsforverifyingclassicalandquantumsuperintegrability AT kressjm toolsforverifyingclassicalandquantumsuperintegrability AT willardmillerjr toolsforverifyingclassicalandquantumsuperintegrability |