The diffractive wave trace on manifolds with conic singularities

TL;DR

This paper analyzes the singularities in the wave trace on manifolds with conic singularities, generalizing classical results to include diffraction effects from cone points.

Contribution

It computes the principal amplitude of wave trace singularities caused by diffractive geodesics on conic manifolds, extending previous theorems to singular settings.

Findings

Derived explicit formulas for diffraction-induced singularities.

Extended Duistermaat-Guillemin theorem to conic singularities.

Connected wave trace singularities to geometric invariants of geodesics.

Abstract

Let $(X,g)$ be a compact manifold with conic singularities. Taking $\Delta_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t \sqrt{ \smash[b]{\Delta_g}}}$ 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.