Physics ©Springer-Verlagl988 Action-Angle Maps and Scattering Theory for Some Finite-Dimensional Integrable Systems

Abstract. 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 ^ of Hamiltonians, and is shown to have remarkable duality properties. All dynamics in ̂ 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(ίD) as ί-> oo. 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