Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems

TL;DR

This paper explores classical and quantum systems with ladder operators, deriving general solutions involving polynomial equations, and introduces new superintegrable systems with unique trajectory patterns.

Contribution

It presents the most general one-dimensional classical systems with third and fourth order ladder operators satisfying polynomial Heisenberg algebras, and introduces new superintegrable systems.

Findings

Systems described by solutions of quartic and quintic equations

Two new families of superintegrable systems

Examples of trajectories as deformed Lissajous figures

Abstract

We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems respectively with a third and a fourth order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformed Lissajous's figures.