Intertwining Symmetry Algebras of Quantum Superintegrable Systems

TL;DR

This paper explores the algebraic structure of quantum superintegrable systems using shape-invariant intertwiners that form Lie algebra pairs, aiding in solving for eigenvalues and eigenfunctions, with classical analogs also examined.

Contribution

It introduces a novel algebraic framework using shape-invariant intertwiners to analyze superintegrable quantum systems and their classical counterparts.

Findings

Intertwining operators form Lie algebra pairs like (su(n), so(2n)).

Eigenstates belong to unitary representations of these algebras.

Intertwining operators facilitate eigenvalue and eigenfunction determination.

Abstract

We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like $(su(n),so(2n))$ or $(su(p,q),so(2p,2q))$. The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness.