Some results on the eigenfunctions of the quantum trigonometric Calogero-Sutherland model related to the Lie Algebra $D_4$

TL;DR

This paper expresses the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to Lie algebra D4 using Weyl-invariant variables, enabling solutions for eigenfunctions and analysis of associated orthogonal polynomials.

Contribution

It introduces a new parametrization of the Hamiltonian in terms of Lie algebra characters, facilitating the explicit solution of eigenfunctions and polynomial properties.

Findings

Explicit eigenfunctions derived for the D4 model

Recurrence relations for associated orthogonal polynomials

Generating functions for the polynomial system

Abstract

We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra $D_4$ in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra. This parametrization allows us to solve for the energy eigenfunctions of the theory and to study properties of the system of orthogonal polynomials associated to them such as recurrence relations and generating functions.