Propagation of singularities in many-body scattering

Abstract

In this paper we describe the propagation of singularities of tempered distributional solutions u∈ S ′ of ( H− λ) u=0, λ>0, where H is a many-body Hamiltonian H= Δ+ V, Δ⩾0, V=∑ a V a , under the assumption that no subsystem has a bound state and that the two-body interactions V a are real-valued polyhomogeneous symbols of order −1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term ‘singularity’ provides a microlocal description of the lack of decay at infinity. We use this result to prove that the wave front relation of the free-to-free S-matrix (which, under our assumptions, is all of the S-matrix) is given by the broken geodesic flow, broken at the ‘singular directions’, on S n−1 at time π. We also present a natural geometric generalization to asymptotically Euclidean spaces.