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Let ( X , g ) be a compact manifold with conic singularities. Taking Δ g to be the Friedrichs extension of the Laplace–Beltrami operator, we examine the singularities of the trace of the half-wave group e − i t Δ g arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat–Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points. [ABSTRACT FROM AUTHOR]

Let P h be a self-adjoint semiclassical pseudodifferential operator on a manifold M such that the bicharacteristic flow of the principal symbol on T* M is completely integrable and the subprincipal symbol of P h vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes u h of P h . We show that on a nondegenerate rational invariant torus, Lagrangian regularity of u h (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional “diffractive” manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics. [ABSTRACT FROM AUTHOR]

We obtain the Strichartz inequalities ∥u∥LtqLxr([0,1]×M) ≤ C∥u(0)∥L²(M) for any smooth n-dimensional Riemannian manifold M which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and nontrapping, where u is a solution to the Schrödinger equation iut + ½ΔMu = 0, and 2 < q, r ≤ ∞ are admissible Strichartz exponents (2⁄2 + n⁄r = n⁄2). This corresponds with the estimates available for Euclidean space (except for the endpoint (q, r) = (2, (Multiple line equation(s) cannot be represented in ASCII text)) when n > 2). These estimates imply existence theorems for semi-linear Schrödinger equations on M. by adapting arguments from Cazenave and Weissler and Kato. This result improves on our previous result, which was an Lt,x4 Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically fiat manifolds. [ABSTRACT FROM AUTHOR]

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, non-direct, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave. [ABSTRACT FROM AUTHOR]