Recurrence approach and higher rank cubic algebras for the $N$-dimensional superintegrable systems

TL;DR

This paper develops a recurrence approach to construct higher order integrals of motion for N-dimensional superintegrable systems, revealing their algebraic structures and enabling spectrum derivation.

Contribution

It introduces a novel recurrence method combined with coupling constant metamorphosis to analyze superintegrable systems and their higher rank cubic algebras.

Findings

Higher order integrals of motion are constructed for N-dimensional systems.

The algebraic structure is identified as a higher rank cubic algebra involving Casimir operators.

Finite dimensional unitary representations are used to derive energy spectra.

Abstract

By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the $N$-dimensional superintegrable Kepler-Coulomb model with non-central terms and the double singular oscillators of type ($n, N-n$). We show how the integrals of motion generate higher rank cubic algebra $C(3)\oplus L_1\oplus L_2$ with structure constants involving Casimir operators of the Lie algebras $L_1$ and $L_2$. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.