Equivalence of the Calogero-Sutherland Model to Free Harmonic Oscillators

TL;DR

This paper demonstrates that the Calogero-Sutherland model with inverse-square and harmonic interactions can be transformed into free harmonic oscillators, simplifying the analysis of its eigenfunctions, constants of motion, and algebraic structure.

Contribution

It introduces a similarity transformation that maps complex interacting particle systems to free oscillators, revealing their underlying simplicity and algebraic properties.

Findings

Complete eigenfunctions can be derived from the transformation.

Constants of motion are explicitly constructed.

The method applies to a broad class of long-range interacting models.

Abstract

A similarity transformation is constructed through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension, can be mapped identically, to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion and a linear $W_{1+\infty}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.