Classical and Quantum Superintegrability with Applications

TL;DR

This review explores classical and quantum superintegrable systems, focusing on their classification, algebraic structures, and connections to orthogonal polynomials, with new insights into higher-order integrals and algebraic relations.

Contribution

It provides a comprehensive classification of second-order superintegrable systems and analyzes the algebraic structures of integrals of motion, including higher-order cases and their relation to orthogonal polynomials.

Findings

Classification of 2D second-order superintegrable systems

Partial solutions for third-order integrals in Euclidean space

Connection between superintegrability and orthogonal polynomial classification

Abstract

A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta. We present a classification of second-order superintegrable systems in two-dimensional Riemannian and pseudo-Riemannian spaces. It is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis. The determining equations for the existence of integrals of motion of arbitrary order in real Euclidean space $E_2$ are presented and partially solved for the case of third-order integrals. A systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order. The algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally. The relation between superintegrability and the classification of orthogonal polynomials is analyzed.