The diffractive wave trace on manifolds with conic singularities

Abstract

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 $e^{- i t \sqrt{ \smash[b]{Δ_g}}}$ 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.

46 pages; 3 figures. New version reflects correction to trace formula (factor of primitive length rather than length), as well as some corrections to discussions of Jacobi fields in section 1.3