An infinite family of solvable and integrable quantum systems on a plane

TL;DR

This paper introduces an infinite family of exactly-solvable and integrable quantum potentials on a plane, unifying known rational potentials and revealing their algebraic structures, with conjectures on superintegrability and a related quasi-exactly-solvable generalization.

Contribution

It presents a new infinite family of solvable and integrable potentials, unifies known cases, and explores their algebraic and superintegrability properties.

Findings

All known rational potentials are special cases of this family.

The algebraic structure and hidden algebra of the potentials are characterized.

A quasi-exactly-solvable generalization is also constructed.

Abstract

An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.