Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials

TL;DR

This paper develops the structure theory for quadratic algebras in 2D second order superintegrable systems with 1-parameter potentials, showing they close at order three and are Stäckel equivalent to constant curvature systems.

Contribution

It completes the structure theory for 1-parameter superintegrable systems, proving algebra closure at order three and establishing their Stäckel equivalence to constant curvature spaces.

Findings

Quadratic algebra closes at order three for these systems.

Functional relationship between four generators is of order four.

All 1-parameter systems are Stäckel equivalent to constant curvature systems.

Abstract

The structure theory for the quadratic algebra generated by first and second order constants of the motion for 2D second order superintegrable systems with nondegenerate (3-parameter) and or 2-parameter potentials is well understood, but the results for the strictly 1-parameter case have been incomplete. Here we work out this structure theory and prove that the quadratic algebra generated by first and second order constants of the motion for systems with 4 second order constants of the motion must close at order three with the functional relationship between the 4 generators of order four. We also show that every 1-parameter superintegrable system is St\"ackel equivalent to a system on a constant curvature space.