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

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.

36 pages. In v2 we have revised section 3.3, made some minor corrections and added three references