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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
##### Keywords:
wave front relation of the S-matrices
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