Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras

TL;DR

This paper introduces a general quartic Poisson algebra for 2D superintegrable systems, extends algebraic realization methods, and applies them to derive energy spectra using deformed oscillator representations.

Contribution

It develops the first comprehensive construction of quartic Poisson and associative algebras and their realizations as deformed oscillators for superintegrable systems.

Findings

Derived the Casimir operator for quartic algebras.

Constructed finite-dimensional unitary irreducible representations.

Applied the framework to a superintegrable system related to Laguerre exceptional polynomials.

Abstract

We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog and we extend Daskaloyannis' construction in obtained in context of quadratic algebras and we obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite dimensional unitary irreductible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptionnal orthogonal polynomials introduced recently.