Some Properties of the Calogero-Sutherland Model with Reflections

TL;DR

This paper explores the Calogero-Sutherland Model with reflections, establishing duality properties, deriving generating functions for eigenfunctions, and analyzing wave-function symmetries for classical Lie group root systems.

Contribution

It proves duality relations in the BC_N model, derives generating functions for generalized Jacobi polynomial eigenfunctions, and discusses wave-function symmetries for classical Lie groups.

Findings

Duality relates eigenfunctions with different coupling constants.

Generated polynomial eigenfunctions are generalized Jacobi polynomials.

Wave-function symmetries are characterized for classical Lie group root systems.

Abstract

We prove that the Calogero-Sutherland Model with reflections (the BC_N model) possesses a property of duality relating the eigenfunctions of two Hamiltonians with different coupling constants. We obtain a generating function for their polynomial eigenfunctions, the generalized Jacobi polynomials. The symmetry of the wave-functions for certain particular cases (associated to the root systems of the classical Lie groups B_N, C_N and D_N) is also discussed.