Two-body Coulomb problem and hidden $g^{(2)}$ algebra: superintegrability and cubic polynomial algebra

TL;DR

This paper introduces a new exactly-solvable superintegrable quantum system derived from the two-body Coulomb problem, revealing a hidden $g^{(2)}$ algebra and a cubic polynomial algebra of integrals in curved space.

Contribution

It uncovers a novel superintegrable system with a hidden $g^{(2)}$ algebra and cubic polynomial algebra, expanding understanding of algebraic structures in quantum superintegrability.

Findings

Identification of a new superintegrable system in curved space.

Discovery of a cubic polynomial algebra of integrals.

Demonstration that the algebra is a subalgebra of the universal enveloping algebra.

Abstract

It is shown that the two-body Coulomb problem in the Sturm representation leads to a new two-dimensional, exactly-solvable, superintegrable quantum system in curved space with a $g^{(2)}$ hidden algebra and a cubic polynomial algebra of integrals. The two integrals are of orders two and four, they are made from two components of the angular momentum and from the modified Laplace-Runge-Lenz vector, respectively. It is demonstrated that the cubic polynomial algebra is an infinite-dimensional subalgebra of the universal enveloping algebra $U_{g^{(2)}}$.