Microlocal properties of scattering matrices for Schrödinger operators on manifolds∗

We consider a d-dimensional C∞-manifold M such that M = Mc ∪ M∞, where Mc is relatively compact open submanifold, and M∞ is the noncompact part which is diffeomorphic to (0,∞)×∂M . Here ∂M is a compact manifold without boundary. In the following, we designate a point in M∞ by (r, θ) with r ∈ (0,∞) and θ ∈ ∂M . Often we also denote by θ ∈ ∂M a variables in a local coordinate of ∂M . Let H(θ)dθ be a positive density on ∂M , and (hjk(θ)) d−1 j,k=1 be a positive (2, 0)-tensor on ∂M . Typically (hjk(θ)) is a Riemannian (co)metric, and H(θ) is the corresponding density. Then we set