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Scattering metrics and geodesic flow at infinity. (English) Zbl 0855.58058
Any compact \(C^\infty\) manifold with boundary admits a so-called scattering metric on its interior. In a previous work, the first author discussed the scattering theory of the corresponding Laplacian. In the present work, it is proved, as conjectured, that the scattering matrix is a Fourier integral operator which quantizes the geodesic flow on the boundary (for a metric defined canonically by the scattering metric) at time \(\pi\). This is proved by showing that the Poisson operator of the associated generalized boundary problem is a Fourier integral operator associated to a singular Legendre manifold.

58J40 Pseudodifferential and Fourier integral operators on manifolds
35P25 Scattering theory for PDEs
35S30 Fourier integral operators applied to PDEs
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