Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics

TL;DR

This paper explores how coupling constant metamorphosis and Stäckel transforms map integrable and superintegrable systems between different manifolds, preserving polynomial symmetries and algebraic structures in classical and quantum mechanics.

Contribution

It provides a detailed analysis of special cases where CCM preserves polynomial symmetries and demonstrates new superintegrable systems derived via CCM in two-dimensional spaces.

Findings

Examples of 3rd and 4th order superintegrable systems obtained through CCM.

Identification of conditions under which CCM preserves polynomial symmetries.

Insights into the structure of symmetry algebras in classical and quantum cases.

Abstract

We review the fundamentals of coupling constant metamorphosis (CCM) and the St\"ackel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the momenta to polynomials in the momenta. We study specializations of these actions which do preserve polynomials and also the structure of the symmetry algebras in both the classical and quantum cases. We give several examples of non-constant curvature 3rd and 4th order superintegrable systems in 2 space dimensions obtained via CCM, with some details on the structure of the symmetry algebras preserved by the transform action.