Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators
We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dep...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211084 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators. Bjorn K. Berntson, Ernest G. Kalnins and Willard Miller Jr. SIGMA 16 (2020), 135, 33 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862577031095517184 |
|---|---|
| author | Berntson, Bjorn K. Kalnins, Ernest G. Miller, Willard Jr. |
| author_facet | Berntson, Bjorn K. Kalnins, Ernest G. Miller, Willard Jr. |
| citation_txt | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators. Bjorn K. Berntson, Ernest G. Kalnins and Willard Miller Jr. SIGMA 16 (2020), 135, 33 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and a 2-parameter rational potential is the best-known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces.
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| first_indexed | 2026-03-13T15:15:17Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211084 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T15:15:17Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Berntson, Bjorn K. Kalnins, Ernest G. Miller, Willard Jr. 2025-12-23T13:11:58Z 2020 Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators. Bjorn K. Berntson, Ernest G. Kalnins and Willard Miller Jr. SIGMA 16 (2020), 135, 33 pages 1815-0659 2020 Mathematics Subject Classification: 20C35; 35B06; 70H20; 81Q80; 81R12 arXiv:2004.00933 https://nasplib.isofts.kiev.ua/handle/123456789/211084 https://doi.org/10.3842/SIGMA.2020.135 We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and a 2-parameter rational potential is the best-known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces. We thank the referees for suggestions that improved our presentation, in particular, one referee for pointing out a gap in our original classification. B.K.B. acknowledges support from the Goran Gustafsson Foundation. W.M. was partially supported by a grant from the Simons Foundation (# 412351 to Willard Miller, Jr.). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators Article published earlier |
| spellingShingle | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators Berntson, Bjorn K. Kalnins, Ernest G. Miller, Willard Jr. |
| title | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators |
| title_full | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators |
| title_fullStr | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators |
| title_full_unstemmed | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators |
| title_short | Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators |
| title_sort | toward classification of 2nd order superintegrable systems in 3-dimensional conformally flat spaces with functionally linearly dependent symmetry operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211084 |
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