Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. II. The two- and three-variable cases

TL;DR

This paper analyzes the joint eigenfunctions of the hyperbolic relativistic Calogero-Moser Hamiltonians for two and three particles, proving their meromorphic extension, invariance, duality, and asymptotic behavior, confirming conjectured scattering properties.

Contribution

It establishes the meromorphic extension, invariance, duality, and asymptotic properties of joint eigenfunctions for the 2- and 3-particle cases, confirming conjectured scattering behavior.

Findings

Eigenfunctions extend to globally meromorphic functions.

Invariance and duality properties are proven.

Asymptotic behaviors match conjectured factorized scattering.

Abstract

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this paper we focus on the cases $N=2$ and $N=3$, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing $a_+,a_-$ positive, we prove that $J_2(b;x,y)$ and $J_3(b;x,y)$ extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions E$_2(b;x,y)$ and E$_3(b;x,y)$. In particular, we determine the dominant asymptotics for $y_1-y_2\to\infty$ and $y_1-y_2,y_2-y_3\to\infty$, resp., from which the conjectured factorized scattering can be read off.