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

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.

💡 Research Summary

The paper investigates the role of higher‑order ladder operators in one‑dimensional classical mechanics and uses the resulting polynomial Heisenberg algebras (PHAs) to classify all possible potentials that admit third‑ and fourth‑order ladder operators. Starting from the well‑known first‑order ladder operators of the harmonic oscillator, the authors recall that second‑order ladder operators lead to the isotropic and anisotropic oscillators, while third‑ and fourth‑order operators have been studied in quantum mechanics in connection with supersymmetric quantum mechanics (SUSYQM) and Painlevé transcendents.

In the classical setting the commutator is replaced by the Poisson bracket, and a ladder operator of order (n) is written as a polynomial in the momentum (P) with coefficient functions (f_k(x)). The defining relations of a PHA are
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