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 2 +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 π. 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.