Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Abstract

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the gl(n) Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

“…In the more interesting superintegrable case, the presence of additional constants of motion implies the existence of non-abelian algebras spanned by the integrals [1]. Usually, these non-abelian structures are Lie, quadratic, cubic or higher order polynomial algebras [2][3][4][5][6][7][8][9][10][11][12][13][14]. The standard example that is often recalled in the literature is the 3D Kepler-Coulomb system.…”