Toward a classification of semidegenerate 3D superintegrable systems

M. A. Escobar-Ruiz, Willard Miller

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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 languageEnglish (US)
Article number095203
JournalJournal of Physics A: Mathematical and Theoretical
Issue number9
StatePublished - Jan 27 2017

Bibliographical note

Funding Information:
This work was partially supported by a grant from the Simons Foundation (#412351, Willard Miller, Jr) and by CONACYT grant (# 250881 to M A Escobar). The author M A Escobar is grateful to ICN UNAM for the kind hospitality during his visit, where a part of the research was done, he was supported in part by DGAPA grant IN108815 (Mexico).


  • conformal superintegrability
  • contractions
  • quadratic algebras
  • superintegrable systems

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