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Zbl 0987.81114
Vasy, András
Propagation of singularities in many-body scattering. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 34, No. 3, 313-402 (2001). ISSN 0012-9593

Author's summary: We describe the propagation of singularities of tempered distributional solutions $u\in {\cal S}'$ of $(H-\lambda) u=0$, $\lambda>0$, where $H$ is a many-body Hamiltonian $H=\Delta +V$, $\Delta\ge 0$, $V=\sum_a V_a$, 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 $\bbfS^{n-1}$ at time $\pi$. We also present a natural geometric generalization to asymptotically Euclidean spaces.
[Roger G.Newton (Bloomington)]

MSC 2000:: *81U10 n-body potential scattering theory

Keywords: scattering Schrödinger equation; propagation of singularities; tempered distributional solutions; many-body Hamiltonian

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