Superintegrable quantum u(3)--systems and higher rank factorizations

TL;DR

This paper explores a class of two-dimensional superintegrable quantum systems on constant curvature surfaces, revealing an $so(6)$ dynamical algebra and novel hierarchical structures through shape-invariance and higher-rank factorizations.

Contribution

It introduces a new framework for superintegrable systems using higher-rank factorizations and shape-invariance, connecting to $so(6)$ algebra and polyhedral lattice representations.

Findings

Identification of $so(6)$ dynamical algebra in superintegrable systems

Development of Hamiltonian hierarchies via shape-invariance

Representation of systems through three-dimensional polyhedral lattices

Abstract

A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant intertwining operators we arrive at a $so(6)$ dynamical algebra and its Hamiltonian hierarchies. We pay attention to those associated to certain unitary irreducible representations that can be displayed by means of three-dimensional polyhedral lattices. We also discuss the role of superpotentials in this new context.