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=Δ+V, Δ⩾0, V=∑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.