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

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.

18 pages

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Cited by in corpus (19)

• Classical and Quantum Superintegrability with Applications
• Invariant classification of second-order conformally flat superintegrable systems
• General $N^{th}$-order superintegrable systems separating in polar coordinates
• Supersymmetric partners of the harmonic oscillator with an infinite potential barrier
• Quantum Integrals from Coalgebra Structure
• SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations
• Recurrence approach and higher rank cubic algebras for the $N$-dimensional superintegrable systems
• On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields
• Two-dimensional superintegrable systems from operator algebras in one dimension
• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• Coherent states for the supersymmetric partners of the truncated oscillator
• Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator
• New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
• Higher Order Supersymmetric Truncated Oscillators
• Dynamical symmetries of the anisotropic oscillator
• Superintegrable systems, polynomial algebra structures and exact derivations of spectra
• General Nth order integrals of the motion
• Examples of Complete Solvability of 2D Classical Superintegrable Systems
• Jacobi last multiplier and two-dimensional superintegrable oscillators