Action-angle maps and scattering theory for some finite-dimensional integrable systems. Pt. 1

We construct an action-angle transformation for the Calogero-Moser systems with repulsive potentials, and for relativistic generalizations thereof. This map is shown to be closely related to the wave transformations for a large class C of Hamiltonians, and is shown to have remarkable duality properties. All dynamics in C lead to the same scattering transformation, which is obtained explicitly and exhibits a soliton structure. An auxiliary result concerns the spectral asymptotics of matrices of the form M exp(tD) as t → ∞. It pertains to diagonal matrices D whose diagonal elements have pairwise different real parts and to matrices M for which certain principal minors are non-zero. (orig.)