Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

TL;DR

This paper reviews the invariant classification of 3D superintegrable systems and demonstrates how all such systems on conformally flat manifolds can be derived from singular limits of a generic system on the sphere, using geometric root transformations.

Contribution

It introduces a method to construct all 3D superintegrable systems from singular limits of a generic system, based on invariant classification and geometric root transformations.

Findings

All superintegrable systems on conformally flat 3D manifolds can be obtained from sphere systems.

Singular limits are used to connect different superintegrable systems.

Invariant classification guides the construction of these limits.

Abstract

The invariant classification of superintegrable systems is reviewed and utilized to construct singular limits between the systems. It is shown, by construction, that all superintegrable systems on conformally flat, 3D complex Riemannian manifolds can be obtained from singular limits of a generic system on the sphere. By using the invariant classification, the limits are geometrically motivated in terms of transformations of roots of the classifying polynomials.