Superintegrable 3-body systems on the line

TL;DR

This paper studies classical three-body systems on a line with specific integrals, revealing their superintegrability and superseparability, and explores their quantum symmetries, including well-known models like Calogero.

Contribution

It introduces a class of superintegrable three-body systems on a line with quadratic first integrals and analyzes their quantum symmetry operators and conformal superseparability.

Findings

Systems are superseparable and superintegrable.

Quantum symmetry operators are constructed for these systems.

Includes well-known models like Calogero and Wolfes.

Abstract

We consider classical three-body interactions on a Euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momenta first integrals. These systems are superseparable (i.e. multiseparable), superintegrable and equivalent (up to rescalings) to a one-particle system in the three-dimensional Euclidean space. Common features of the dynamics are discussed. We show how to determine the quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal superseparability is proved and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other three-body interactions widely studied in mathematical physics.