Scattering Metrics and Geodesic Flow at Infinity

Any compact # manifold with boundary admits a Riemann metric on its interior taking the form x -4 dx + x -2 h # near the boundary, where x is a boundary defining function and h # is a smooth symmetric 2-cotensor restricting to be positive-definite, and hence a metric, h, on the boundary. The scattering theory associated to the Laplacian for such a `scattering metric' was discussed by the first author and here it is shown, as conjectured, that the scattering matrix is a Fourier integral operator which quantizes the geodesic flow on the boundary, for the metric h, at time #. To prove this the Poisson operator, of the associated generalized boundary problem, is constructed as a Fourier integral operator associated to a singular Legendre manifold