Exact Solvability of the Calogero and Sutherland Models

TL;DR

This paper demonstrates that the Calogero and Sutherland models are exactly solvable by expressing their Hamiltonians as quadratic polynomials in $sl_N$ algebra generators, leveraging symmetric polynomial coordinates and representation theory.

Contribution

It introduces a new coordinate system using symmetric polynomials and shows how the Hamiltonians can be represented as quadratic $sl_N$ algebra elements, revealing their exact solvability.

Findings

Hamiltonians expressed as quadratic polynomials in $sl_N$ generators

Existence of an infinite flag of representation spaces preserved by the Hamiltonians

Connection established between models and Jack polynomials

Abstract

Translationally invariant symmetric polynomials as coordinates for $N$-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland $N$-body Hamiltonians, after appropriate gauge transformations, can be presented as a {\it quadratic} polynomial in the generators of the algebra $sl_N$ in finite-dimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed.