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

TL;DR

The paper surveys two-dimensional quantum superintegrable systems in flat space, showing that six models are exactly solvable and possess algebraic forms for their Hamiltonians and integrals. Each model exhibits a hidden algebra $g^{(s)}$ and a finite, 4-generator polynomial algebra of integrals, with the integrals acting on infinite flags of finite-dimensional invariant subspaces. The TTW system with integer $k$ reveals a family of hidden algebras $g^{(k)}$ and a high-order polynomial algebra of order $(k+1)$ for the integrals, unifying the exact-solvability and superintegrability framework across several classic models. The Montreal conjecture is reinforced by demonstrating exact solvability and algebraic structure for these planar systems, including reductions to the Calogero and Wolfes models in singular limits and connections to dihedral symmetry invariants.

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.