Explicit Computations for the Classical and Quantum Integrability of the 3-Dimensional Rational Calogero-Moser System

TL;DR

This paper explicitly verifies the classical and quantum integrability of the 3D rational Calogero-Moser system using Lax pairs and constructs associated operators for general root systems, providing detailed computational and theoretical insights.

Contribution

It offers explicit computations for integrability in the 3D case and constructs Olshanetsky-Perelomov operators for general root systems using Dunkl operators.

Findings

Explicit verification of integrability for n=3 case

Construction of Olshanetsky-Perelomov operators for general root systems

Presentation of classical analogues of Dunkl and Olshanetsky-Perelomov operators

Abstract

The integrability of the classical and quantum rational Calogero-Moser systems is verified explicitly via the Lax pair method for the case $n=3$. We provide an extensive survey of reflection groups and root systems. The Olshanetsky-Perelomov operators are constructed for a general root system via Dunkl operators, associated to root systems. The integrability of the quantum rational Calogero-Moser system is discussed via the Olshanetsky-Perelomov operators, which provide a set of commuting integrals of motion. The classical analogues of both the Dunkl and the Olshanetsky-Perelomov operators are also presented.