Classical n-body scattering with long-range potentials
Résumé
We consider the scattering of n classical particles interacting via pair potentials which are assumed to be 'long-range', i.e. of order $\mathcal{O}({r}^{-\alpha })$ as r tends to infinity, for some α > 0. We define and focus on the 'free region', the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an explicit, global surface of section for the free region. This surface of section allows us to prove the smoothness of the map sending a point to its final state and to establish a forward conjugacy between the n-body dynamics and free dynamics.