Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras

TL;DR

This paper introduces a new family of superintegrable quantum systems using extended Laplace-Runge-Lentz vectors, providing their integrals of motion, exact solutions, and quadratic algebra relations involving higher rank Lie algebra Casimirs.

Contribution

It presents a novel method for discovering superintegrable systems and constructs their integrals of motion with algebraic structures involving higher rank Lie algebras.

Findings

New superintegrable systems with explicit solutions

Quadratic algebra relations involving higher rank Lie algebra Casimirs

Exact energy eigenvalues and eigenfunctions obtained

Abstract

We present a useful proposition for discovering extended Laplace-Runge-Lentz vectors of certain quantum mechanical systems. We propose a new family of superintegrable systems and construct their integrals of motion. We solve these systems via separation of variables in spherical coordinates and obtain their exact energy eigenvalues and the corresponding eigenfunctions. We give the quadratic algebra relations satisfied by the integrals of motion. Remarkably these algebra relations involve the Casimir operators of certain higher rank Lie algebras in the structure constants.