Bôcher Contractions of Conformally Superintegrable Laplace Equations
The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often ''hidden''. The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebra...
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Інститут математики НАН України
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| Zitieren: | Bôcher Contractions of Conformally Superintegrable Laplace Equations / E.G. Kalnins, Willard Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 38 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1477372025-02-23T19:24:38Z Bôcher Contractions of Conformally Superintegrable Laplace Equations Kalnins, E.G. Miller Jr., Willard Subag, E. The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often ''hidden''. The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems on constant curvature spaces are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group SO(4,C), and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that Bôcher's prescription for coalescing roots of these forms induces contractions of the conformal algebra so(4,C) to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html. This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller Jr). 2016 Article Bôcher Contractions of Conformally Superintegrable Laplace Equations / E.G. Kalnins, Willard Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 38 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R05; 81R12; 33C45 DOI:10.3842/SIGMA.2016.038 https://nasplib.isofts.kiev.ua/handle/123456789/147737 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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English |
| description |
The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often ''hidden''. The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems on constant curvature spaces are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group SO(4,C), and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that Bôcher's prescription for coalescing roots of these forms induces contractions of the conformal algebra so(4,C) to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details. |
| format |
Article |
| author |
Kalnins, E.G. Miller Jr., Willard Subag, E. |
| spellingShingle |
Kalnins, E.G. Miller Jr., Willard Subag, E. Bôcher Contractions of Conformally Superintegrable Laplace Equations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Kalnins, E.G. Miller Jr., Willard Subag, E. |
| author_sort |
Kalnins, E.G. |
| title |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_short |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_full |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_fullStr |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_full_unstemmed |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_sort |
bôcher contractions of conformally superintegrable laplace equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147737 |
| citation_txt |
Bôcher Contractions of Conformally Superintegrable Laplace Equations / E.G. Kalnins, Willard Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 38 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2025-11-24T15:49:13Z |
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2025-11-24T15:49:13Z |
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