Quantum superintegrability and exact solvability in N dimensions

TL;DR

This paper constructs a family of maximally superintegrable and exactly solvable quantum systems in N-dimensional space, revealing their algebraic structure and solutions in terms of orthogonal polynomials.

Contribution

It introduces a new class of superintegrable systems in N dimensions with explicit algebraic and analytical solutions, extending the understanding of quantum integrability.

Findings

Constructed N-dimensional superintegrable systems including Coulomb as a special case

Identified two sets of commuting second order operators

Solved the systems using classical orthogonal polynomials

Abstract

A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in N-dimensional Euclidean space. Two different sets of N commuting second order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and N further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.