Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials

TL;DR

This paper explores superintegrable quantum systems with third order integrals, constructing cubic algebras, deriving energy spectra, and analyzing rational potentials within supersymmetric and PT-symmetric frameworks.

Contribution

It develops a general formalism for cubic algebras in superintegrable systems with rational potentials, including new realizations and spectral calculations.

Findings

Constructed the most general cubic algebra for these systems

Derived energy spectra for rational potentials

Connected superintegrability with supersymmetric and PT-symmetric quantum mechanics

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 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 quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.