  We present and study a novel class of one-dimensional Hilbert space
eigenfunction transforms that diagonalize analytic difference operators
encoding the (reduced) two-particle relativistic hyperbolic Calogero-Moser
dynamics. The scattering is described by reflection and transmission amplitudes
$t$ and $r$ with function-theoretic features that are quite different from
nonrelativistic amplitudes. The axiomatic Hilbert space analysis in the
appendices is inspired by and applied to the attractive two-particle
relativistic Calogero-Moser dynamics for a sequence of special couplings.
Together with the scattering function $u$ of the repulsive case, this leads to
a triple of amplitudes $u, t, r$ satisfying the Yang-Baxter equations.
