Hilbert space theory for relativistic dynamics with reflection. Special cases

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 and r satisfying the Yang–Baxter equations.