Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 135, 33 pages      arXiv:2004.00933      https://doi.org/10.3842/SIGMA.2020.135

Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators

Bjorn K. Berntson a, Ernest G. Kalnins b and Willard Miller Jr. c
a) Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
b) Department of Mathematics, University of Waikato, Hamilton, New Zealand
c) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

Received April 07, 2020, in final form December 09, 2020; Published online December 16, 2020

Abstract
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 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.

Key words: superintegrable systems; Calogero 3 body system; functional linear dependence.

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