  We consider scattering by general compactly supported semi-classical
perturbations of the Euclidean Laplace-Beltrami operator. We show that if the
suitably cut-off resolvent of the Hamiltonian quantizes a Lagrangian relation
on the product cotangent bundle, the scattering amplitude quantizes the natural
scattering relation. In the case when the resolvent is tempered, which is true
under some non-resonance assumptions, and when we work microlocally near a
non-trapped ray, our result implies that the scattering amplitude defines a
semiclassical Fourier integral operator associated to the scattering relation
in a neighborhood of that ray. Compared to previous work we allow this relation
to have more general geometric structure.
