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Propagation of singularities in many-body scattering. (English) Zbl 0987.81114

Author’s summary: We describe the propagation of singularities of tempered distributional solutions \(u\in {\mathcal S}'\) of \((H-\lambda) u=0\), \(\lambda>0\), where \(H\) is a many-body Hamiltonian \(H=\Delta +V\), \(\Delta\geq 0\), \(V=\sum_a V_a\), under the assumption that no subsystem has a bound state and that the 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. We use this result to prove that the wave front relation of the free-to-free \(S\)-matrix (which, under our assumptions, is all of the \(S\)-matrix) is given by the broken geodesic flow, broken at the ‘singular directions’, on \(\mathbb{S}^{n-1}\) at time \(\pi\). We also present a natural geometric generalization to asymptotically Euclidean spaces.

MSC:

81U10 \(n\)-body potential quantum scattering theory
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