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

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions JN (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 J₂(b; x, y) and J₃(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 E₂(b; x, y) and E₃(b; x, y). In particular, we

determine the dominant asymptotics for y₁ − y₂ → ∞ and y₁ − y₂, y₂ − y₃ → ∞, resp., from which the conjectured factorized scattering can be read off.