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

✍ Scribed by András Vasy

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 597 KB
• Volume: 184
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3763

✦ Synopsis

In this paper we describe the propagation of singularities of tempered distributional solutions u # S$ of (H&*) u=0, where H is a many-body Hamiltonian H=2+V, 2 0, V= a V a , and * is not a threshold of H, under the assumption that the inter-particle (e.g., 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'' 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&\*, 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.