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Published online by Cambridge University Press: 26 June 2025
This chapter 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 noncom pact symmetric spaces.
1. Introduction
This chapter is an effort to explain in relatively nontechnical terms recent results in many-body scattering and related topics. Thus, 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, i.e. wave phenomena. Motivated by this, I first briefly describe propagation of singularities for the wave equation. This is a remarkable relationship between geometric optics (the particle view of light) and the solutions of the wave equation (the wave view).
Next, in Section 3, I explain the geometry of many-body scattering, which includes both that of the configuration space and phase space. This geometry is closely related to classical mechanics, playing the role of geometric optics, but even at this point quantum phenomena emerge.
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