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

Symmetry Integrability and Geometry-Methods and Applications

article

Bjorn K. Berntson¹  Ernest G. Kalnins²  Willard Miller³ 

[1] Department of Mathematics, KTH Royal Institute of Technology;Department of Mathematics, University of Waikato;School of Mathematics, University of Minnesota

关键词: superintegrable systems;    Calogero 3 body system;    functional linear dependence 2020 Mathematics Subject Classification 20C35;    35B06;    70H20;    81Q80;    81R12;   

DOI  :  10.3842/SIGMA.2020.135

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown