Orthogonal Symmetric Polynomials Associated with the Calogero Model

TL;DR

This paper develops a Rodrigues formula for Hi-Jack polynomials, providing an orthogonal basis for the Calogero model, a key quantum integrable system with inverse-square interactions.

Contribution

It introduces a novel Rodrigues formula for Hi-Jack polynomials, extending the algebraic framework of the Calogero model.

Findings

Constructed Rodrigues formula for Hi-Jack polynomials

Established orthogonal basis for Calogero model

Linked algebraic structures of Calogero and Sutherland models

Abstract

The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a one-dimensional quantum integrable system with inverse-sine-square interactions. Inspired by the Rodrigues formula for the Jack polynomials, which form the orthogonal basis of the Sutherland model, recently found by Lapointe and Vinet, we construct the Rodrigues formula for the Hi-Jack (hidden-Jack) polynomials that form the orthogonal basis of the Calogero model.