### Abstract

Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators, but their symmetry algebras do not close under commutation and not enough is known about their structure to give a complete classification. Some examples are known for which the 3-parameter system can be extended to a 4th order superintegrable system with a 4-parameter potential and 6 linearly independent symmetry generators. In this paper we use Bocher contractions of the conformal Lie algebra so(5,C) to itself to generate a large family of 3-parameter systems with 4th order extensions, on a variety of manifolds, all from Bocher contractions of a single generic system on the 3-sphere. We give a contraction scheme relating these systems. The results have myriad applications for finding explicit solutions for both quantum and classical systems.

Original language | English (US) |
---|---|

Article number | 095203 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 50 |

Issue number | 9 |

DOIs | |

State | Published - Jan 27 2017 |

### Fingerprint

### Keywords

- conformal superintegrability
- contractions
- quadratic algebras
- superintegrable systems

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*50*(9), [095203]. https://doi.org/10.1088/1751-8121/aa5843

**Toward a classification of semidegenerate 3D superintegrable systems.** / Escobar-Ruiz, M. A.; Miller, Willard.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 50, no. 9, 095203. https://doi.org/10.1088/1751-8121/aa5843

}

TY - JOUR

T1 - Toward a classification of semidegenerate 3D superintegrable systems

AU - Escobar-Ruiz, M. A.

AU - Miller, Willard

PY - 2017/1/27

Y1 - 2017/1/27

N2 - Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators, but their symmetry algebras do not close under commutation and not enough is known about their structure to give a complete classification. Some examples are known for which the 3-parameter system can be extended to a 4th order superintegrable system with a 4-parameter potential and 6 linearly independent symmetry generators. In this paper we use Bocher contractions of the conformal Lie algebra so(5,C) to itself to generate a large family of 3-parameter systems with 4th order extensions, on a variety of manifolds, all from Bocher contractions of a single generic system on the 3-sphere. We give a contraction scheme relating these systems. The results have myriad applications for finding explicit solutions for both quantum and classical systems.

AB - Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators, but their symmetry algebras do not close under commutation and not enough is known about their structure to give a complete classification. Some examples are known for which the 3-parameter system can be extended to a 4th order superintegrable system with a 4-parameter potential and 6 linearly independent symmetry generators. In this paper we use Bocher contractions of the conformal Lie algebra so(5,C) to itself to generate a large family of 3-parameter systems with 4th order extensions, on a variety of manifolds, all from Bocher contractions of a single generic system on the 3-sphere. We give a contraction scheme relating these systems. The results have myriad applications for finding explicit solutions for both quantum and classical systems.

KW - conformal superintegrability

KW - contractions

KW - quadratic algebras

KW - superintegrable systems

UR - http://www.scopus.com/inward/record.url?scp=85013356897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85013356897&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aa5843

DO - 10.1088/1751-8121/aa5843

M3 - Article

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 9

M1 - 095203

ER -