Scattering metrics and geodesic flow at infinity. (English) Zbl 0855.58058

Melrose, Richard; Zworski, Maciej

Invent. Math. 124, No. 1-3, 389-436 (1996).

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.

Reviewer: P.Godin (Bruxelles)

MSC:

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

Keywords:

scattering metric; Fourier integral operator; geodesic flow