Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties

TL;DR

This paper classifies all nondegenerate second order superintegrable systems in complex Euclidean 3-space, revealing their structure as points on an algebraic variety constrained by quadratic relations and identifying exactly 10 such potentials.

Contribution

It provides a complete classification of nondegenerate potentials for superintegrable systems in 3D Euclidean space using algebraic geometry methods.

Findings

Exactly 10 nondegenerate potentials identified.

Superintegrable systems correspond to points on an algebraic variety.

The variety is characterized by six quadratic polynomial constraints.

Abstract

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n-1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schroedinger operator, deep connections with special functions and with QES systems. Here we announce a complete classification of nondegenerate (i.e., 4-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in 10 variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly 10 nondegenerate potentials.