Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion

TL;DR

This paper develops a polynomial associative algebra framework for quantum superintegrable systems with third-order integrals, enabling the calculation of energy spectra for such systems.

Contribution

It introduces the most general associative cubic algebra for these systems and provides specific realizations to compute energy spectra.

Findings

Constructed the general associative cubic algebra for the system

Derived specific algebra realizations for energy spectrum calculations

Applied the formalism to a quantum superintegrable potential

Abstract

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in cartesian coordinates with a third order integral are known. The general formalism is applied to one of the quantum potentials.