From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model

TL;DR

This paper reviews known integrable and quasi-exactly-solvable quantum models with rational potentials, emphasizing their symmetries, polynomial solutions, and algebraic structures, and discusses models related to quantum $A_N$ and $H_4$ systems.

Contribution

It provides a concise overview of the algebraic and symmetry properties of rational quantum models, highlighting connections between different integrable systems.

Findings

Models characterized by discrete symmetry and polynomial eigenfunctions.

Existence of separation of variables and second-order integrals.

Algebraic form in invariants of symmetry groups.

Abstract

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate and, hence, the existence of the 2nd order integral, (v) an algebraic form in invariants of a discrete symmetry group (in space of orbits).