Exact Solvability of Superintegrable Systems

TL;DR

This paper demonstrates that all four superintegrable quantum systems on the Euclidean plane share a hidden $sl(3)$ algebra structure, which explains their exact solvability through polynomial-preserving Hamiltonians and integrals of motion.

Contribution

It reveals the common underlying $sl(3)$ algebra structure in all four superintegrable systems on the Euclidean plane, providing a unifying algebraic framework for their exact solutions.

Findings

All systems possess the same hidden $sl(3)$ algebra.

Hamiltonians preserve a polynomial flag related to $sl(3)$ representations.

Integrals of motion form finite-dimensional $sl(3)$ modules.

Abstract

It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra $sl(3)$. The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the $sl(3)$-algebra, realized by first order differential operators.