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 first 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 \rE_N(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.Comment: 21 page