Propagation of Singularities in Many-body Scattering in the Presence of Bound States

In this paper we describe the propagation of singularities of tempered distributional solutions u ∈ S of (H−λ)u = 0, where H is a many-body Hamiltonian H = ∆+ V , ∆ ≥ 0, V = P a Va, and λ is not a threshold of H, under the assumption that the inter-particle (e.g. two-body) interactions Va are real-valued polyhomogeneous symbols of order −1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term ‘singularity’ refers to a microlocal description of the lack of decay at infinity. Thus, we prove that the set of singularities of u is a union of maximally extended broken bicharacteristics of H. These are curves in the characteristic variety of H−λ, which can be quite complicated due to the existence of bound states. We use this result to describe the wave front relation of the S-matrices. We also analyze Lagrangian properties of this relation, which shows that the relation is not ‘too large’ in terms of its dimension.