Toward a classification of semidegenerate 3D superintegrable systems

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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• Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups
• Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
• Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras
• Dynamical symmetry algebra of two superintegrable two-dimensional systems
• Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 2. Systems with dilatation and shift symmetries
• Torsion-free connections of second-order maximally superintegrable systems