Quantum two-dimensional superintegrable systems in flat space: exact-solvability, hidden algebra, polynomial algebra of integrals

TL;DR

This review demonstrates that six two-dimensional quantum superintegrable systems in flat space are exactly solvable, possess hidden algebraic structures, polynomial integrals, and invariant subspaces, confirming the Montreal conjecture.

Contribution

It provides a detailed analysis confirming the exact solvability and algebraic structures of six specific superintegrable quantum systems, expanding understanding of their polynomial and hidden symmetries.

Findings

All six systems are exactly solvable with polynomial eigenfunctions.

They admit algebraic forms for Hamiltonian and integrals, with hidden Lie algebra structures.

Each system has an infinite flag of finite-dimensional invariant subspaces.

Abstract

In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, $G_2$ rational, or $I_6$) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index $k$. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure $g^{(k)}$ with various $k$, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra $g^{(k)}$ for a certain $k$. In all presented cases the algebra of integrals is a 4-generated $(H, I_1, I_2, I_{12}\equiv[I_1, I_2])$ infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.