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arXiv · 1611.02977

Toward a classification of semidegenerate 3D superintegrable systems

M. A. Escobar-Ruiz·W. Miller Jr
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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 don't 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 Bôcher 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, and all from Bôcher 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.

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M. A. Escobar-Ruiz, W. Miller Jr. 2016-11-09. Toward a classification of semidegenerate 3D superintegrable systems. https://doi.org/10.1088/1751-8121%2Faa5843

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