Dynamics of quantum hyperbolic Ruijsenaars-Schneider particles via diagonalization of analytic difference operators

We present a new generalised eigenfunction of the reduced two-particle, mixed-charge, hyperbolic Ruijsenaars-Schneider (or, relativistic A_1-Calogero-Moser) Hamiltonian. The asymptotics of this function displays transmission and reflection in a way that generalizes the familiar non-relativistic picture. Using this function we construct integral transforms diagonalizing the Hamiltonian (an analytic difference operator, or A\DeltaO). When the three parametric dependences of the Hamiltonian are restricted to a certain polytope, these transforms can be used for a functional-analytic Hilbert space theory with all the desired quantum mechanical features (self-adjointness, spectrum, S-matrix etc.). As a final consideration we see how such a theory can be constructed in a different way for a special choice of the coupling parameter, with accompanying special features.