Geometry and analysis in many-body scattering. (English) Zbl 1086.35508

Vasy, András

Uhlmann, Gunther (ed.), Inside out: Inverse problems and applications. Cambridge: Cambridge University Press (ISBN 0-521-82469-9/hbk). Math. Sci. Res. Inst. Publ. 47, 333-379 (2003).

Summary: This paper explains in relatively nontechnical terms recent results in many-body scattering and related topics. Many results in the many-body setting should be understood as new results on the propagation of singularities, here understood as lack of decay of wave functions at infinity, with much in common with real principal type propagation (wave phenomena). Classical mechanics plays the role that geometric optics has in the study of the wave equation, but even at this point quantum phenomena emerge. Propagation of singularities has immediate applications to the structure of scattering matrices and to inverse scattering; these topics are addressed here. The final section studies a problem very closely related to many-body scattering, namely scattering on higher rank noncompact symmetric spaces.
For the entire collection see [Zbl 1034.78003].

MSC:

35P25	Scattering theory for PDEs
81U10	\(n\)-body potential quantum scattering theory
81U20	\(S\)-matrix theory, etc. in quantum theory
78A05	Geometric optics

Keywords:

propagation of singularities; wave functions; geometric optics; wave equation; scattering matrices