Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type. III. Factorized Asymptotics

In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions JN(a+,a−,b;x,y) of the Hamiltonians arising in the integrable N-particle systems of hyperbolic relativistic Calogero–Moser type. We focused on the 1st steps of the scheme in Part I and on the cases N=2 and N=3 in Part II. In this paper, we determine the dominant asymptotics of a similarity-transformed function EN(b;x,y) for yj−yj+1→∞⁠, j=1,…,N−1 and thereby confirm the long-standing conjecture that the particles in the hyperbolic relativistic Calogero–Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers N>3⁠.