Superintegrability in a two-dimensional space of non-constant curvature

TL;DR

This paper investigates superintegrability in a two-dimensional curved space, identifying specific potentials that preserve superintegrability and analyzing the associated classical and quantum algebraic structures.

Contribution

It extends the study of superintegrability to arbitrary curved manifolds, specifically analyzing a space of revolution and classifying potentials that maintain superintegrability.

Findings

Identified three distinct potentials for superintegrability on the space of revolution.

Analyzed separation of variables in Hamilton-Jacobi and Schrödinger equations.

Determined classical and quantum quadratic algebras for these potentials.

Abstract

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton-Jacobi and Schroedinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.