Quantum Super-Integrable Systems as Exactly Solvable Models

TL;DR

This paper explores quantum super-integrable systems, demonstrating their exact solvability through algebraic methods, and constructing eigenfunctions and representations of associated quadratic algebras.

Contribution

It introduces a framework linking super-integrability with exact solvability via nonlinear Lie algebra extensions and eigenfunction construction methods.

Findings

Quantum super-integrable systems can be exactly solved using algebraic techniques.

Eigenfunctions are constructed through commuting operators.

Finite-dimensional representations of quadratic algebras are developed.

Abstract

We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between super-integrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Finite dimensional representations of the quadratic algebras are thus constructed in a way analogous to that of the highest weight representations of Lie algebras.