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 article, 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 |${\rm E}_2(b;x,y)$| and |${\rm E}_3(b;x,y)$|⁠. In particular, we determine the dominant asymptotics for |$y_1-y_2\to\infty$| and |$y_1-y_2,y_2-y_3\to\infty$|⁠, respectively, from which the conjectured factorized scattering can be read off.