The Calogero Model: Integrable Structures and Orthogonal Basis

TL;DR

This paper explores the algebraic and integrable structures of the Calogero model, demonstrating that Hi-Jack polynomials form an orthogonal basis through quantum Lax and Dunkl operator formulations.

Contribution

It introduces an algebraic construction of eigenfunctions, revealing the W-algebra structure and establishing Hi-Jack polynomials as the orthogonal basis for the model.

Findings

W-algebra structure among operators

Hi-Jack polynomials form an orthogonal basis

Algebraic construction of eigenfunctions

Abstract

Integrability, algebraic structures and orthogonal basis of the Calogero model are studied by the quantum Lax and Dunkl operator formulations. The commutator algebra among operators including conserved operators and creation-annihilation operators has the structure of the W-algebra. Through an algebraic construction of the simultaneous eigenfunctions of all the commuting conserved operators, we show that the Hi-Jack (hidden-Jack) polynomials, which are an multi-variable generalization of the Hermite polynomials, form the orthogonal basis.