We consider a non-trapping n-dimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the "northern cap" of the compactification. For small perturbations of Minkowski space that fit into our framework, our asymptotic expansions yield a rate of decay that improves on the Klainerman-Sobolev estimates.

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For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, nondirect, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave.

We investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities. This class of manifolds includes, and is modelled on, the product of a smooth manifold and a cone over a compact fiber. Our main results are a general 'diffractive' theorem showing that the spreading of singularities at the edge only occurs along the fibers and a more refined 'geometric' theorem showing that for appropriately regular (nonfocusing) solutions, the main singularities can only propagate along geometrically determined rays. Thus, for the fundamental solution with initial pole sufficiently close to the edge, we are able to show that the regularity of the diffracted front is greater than that of the incident wave.

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M • , g). (Euclidean R n , with a compactly supported metric perturbation, is an example of such a space.) Let ∆ be the positive Laplacian on (M, g), and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator (h 2 ∆ + V − (λ 0 ± i0) 2 ) −1 , at a nontrapping energy λ 0 > 0, uniformly for h ∈ (0, h 0 ), h 0 > 0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, e −it(∆/2+V ) , t ∈ (0, t 0 ) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.