Jack polynomials and the multi-component Calogero-Sutherland model

TL;DR

This paper constructs orthogonal polynomials related to a multicomponent Calogero-Sutherland model and conjectures their identity with Jack polynomials, supported by analytic and numerical evidence.

Contribution

It introduces a new approach to connect multicomponent Calogero-Sutherland models with Jack polynomials, providing conjectures and evaluations of related integrals.

Findings

Polynomials constructed as orthogonal in multicomponent model coordinates

Conjecture that these are Jack polynomials with specific parameters

Evaluation of normalization and related integrals

Abstract

Using the ground state $\psi_0$ of a multicomponent generalization of the Calogero-Sutherland model as a weight function, orthogonal polynomials in the coordinates of one of the species are constructed. Using evidence from exact analytic and numerical calculations, it is conjectured that these polynomials are the Jack polynomials $J_\kappa^{(1+1/\lambda)}$, where $\lambda$ is the coupling constant. The value of the normalization integral for $\psi_0 J_\kappa^{(1+1/\lambda)}$ is conjectured, and some further related integrals are evaluated.