Free harmonic oscillators, Jack polynomials and Calogero-Sutherland systems

TL;DR

This paper explores the algebraic structure of Calogero-Sutherland and Sutherland models, revealing their connections to free harmonic oscillators and particles, and introduces operators that link their eigenspaces.

Contribution

It establishes an exact connection between Cherednik eigenfunctions and monomials, and constructs a harmonic oscillator algebra for these models.

Findings

Connection between Cherednik eigenfunctions and monomials

Harmonic oscillator algebra for CSM and SM

Differences in excitations of CSM and SM

Abstract

The algebraic structure and the relationships between the eigenspaces of the Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators. We find an exact connection between the simultaneous non-symmetric eigenfunctions of the $A_{N-1}$ Cherednik operators, from which the eigenfunctions of the CSM and SM are constructed, and the monomials. This construction, not only, allows one to write down a harmonic oscillator algebra involving the Cherednik operators, which yields the raising and lowering operators for both of these models, but also shows the connection of the CSM with free oscillators and the SM with free particles on a circle. We also point out the subtle differences between the excitations of the CSM and the SM.