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Propagation of singularities in many-body scattering in the presence of bound states. (English) Zbl 1003.35012
Journées “Équations aux dérivées partielles”, Saint-Jean-de-Monts, France, 31 mai au 4 juin 1999. Exposés Nos. I–XIX. Nantes: Université de Nantes. Exp. No. XVI, 20 p. (1999).
Summary: In these lecture notes we describe the propagation of singularities of tempered distributional solutions \(u\in{\mathcal S}'\) of \((H-\lambda)u = 0\), where \(H\) is a many-body Hamiltonian \(H= \Delta+ V\), \(\Delta\geq 0\), \(V= \sum_a V_a\), and \(\lambda\) 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’ provides a microlocal description of the lack of decay at infinity. Our result is then 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. Here we only present the statement of the results and sketch some of the ideas in proving them, the complete details have been published in J. Funct. Anal. 184, 177-272 (2001; Zbl 1085.35010).
For the entire collection see [Zbl 0990.00047].

MSC:
35A21 Singularity in context of PDEs
81U10 \(n\)-body potential quantum scattering theory
35A20 Analyticity in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J47 Propagation of singularities; initial value problems on manifolds
35P25 Scattering theory for PDEs
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