Intertwining symmetry algebras of quantum superintegrable systems on the hyperboloid

TL;DR

This paper explores the algebraic structure of quantum superintegrable systems on a hyperboloid, revealing how intertwining operators form su(2,1) and so(4,2) Lie algebras and characterizing their spectra via unitary representations.

Contribution

It demonstrates the algebraic closure of intertwining operators into su(2,1) and so(4,2) algebras and links these to the spectral properties of the systems, a novel insight in hyperboloid quantum systems.

Findings

Intertwining operators form su(2,1) Lie algebra.

Broader operators close the so(4,2) algebra.

Spectral states characterized by unitary representations.

Abstract

A class of quantum superintegrable Hamiltonians defined on a two-dimensional hyperboloid is considered together with a set of intertwining operators connecting them. It is shown that such intertwining operators close a su(2,1) Lie algebra and determine the Hamiltonians through the Casimir operators. By means of discrete symmetries a broader set of operators is obtained closing a so(4,2) algebra. The physical states corresponding to the discrete spectrum of bound states as well as the degeneration are characterized in terms of unitary representations of su(2,1) and so(4,2).