Superintegrability and higher order polynomial algebras I

TL;DR

This paper introduces a method to derive higher order integrals and polynomial algebras for 2D superintegrable systems using creation and annihilation operators, expanding understanding of their algebraic structures and spectra.

Contribution

It presents a novel approach to construct higher order polynomial algebras for superintegrable systems, including cases with non-cubic algebras, and provides realizations for spectral analysis.

Findings

Constructed quintic and seventh order polynomial algebras.

Derived realizations in terms of deformed oscillator algebras.

Obtained energy spectra from algebraic representations.

Abstract

We present a method to obtain higher order integrals and polynomial algebras for two-dimensional superintegrable systems from creation and annihilation operators. All potentials with a second and a third order integrals of motion separable in Cartesian coordinates were studied. The integrals of motion of two of them do not generate a cubic algebra. We construct for these Hamiltonians a higher order polynomial algebra from the creation and annihilation operators. We obtain quintic and seventh order polynomial algebras. We give also for polynomial algebras of order 7 realizations in terms of deformed oscillator algebras. These realizations and finite dimensional unitary representations allow us to obtain the energy spectrum.