Torsion and semi-degeneracy of second-order maximally superintegrable systems

TL;DR

This paper investigates second-order maximally superintegrable systems with multiple parameters, revealing geometric conditions involving torsion and conformal invariance that determine their potential extensions.

Contribution

It introduces a geometric framework based on statistical manifolds with torsion to classify and extend superintegrable systems with multiple parameters.

Findings

A necessary and sufficient condition for potential extension involving trace-free tensor fields.

Extension condition is conformally invariant, applicable to conformally superintegrable systems.

Identification of affine connection torsion as key to potential compatibility.

Abstract

The isotropic harmonic oscillator and the Kepler-Coulomb system are pivotal models in the Sciences. They are two examples of second-order (maximally) superintegrable (Hamiltonian) systems. These systems are classified in dimension two. A partial classification exists in dimension three. In this paper, our focus is on second-order superintegrable systems with a $(n+1)$-parameter potential with $n\geq3$. We find that these systems are underpinned by an information-geometric structure, namely the structure of a statistical manifold with torsion. We obtain a necessary and sufficient condition for such systems to extend to non-degenerate systems, i.e. to admit a maximal family of compatible potentials. The condition is geometric: we show that a $(n+1)$-parameter potential is the restriction of a non-degenerate potential if and only if a certain trace-free tensor field vanishes. We interpret this condition as the requirement that a certain affine connection has vectorial torsion. We also show that the condition for a system to be extendable is conformally invariant, allowing us to extend our results to second-order conformally superintegrable systems with a $(n+1)$-parameter potential.