Scattering Theory for Manifolds with Asymptotically Cylindrical Ends

Let X be compact manifold with boundary and exact b-metric, so X has asymptotically cylindrical ends. Let Δ be the Laplacian on X. In this paper it is shown that [formula] where F− 1 is the inverse Fourier transform, b-Tr is a regularization of the integral of the restriction of the Schwartz kernel to the diagonal, and Ψ is the scattering matrix. The first term is analogous to the case of a compact manifold without boundary, and the rest are similar to terms in a trace formula for one-dimensional scattering.