Diffractive Propagation on Conic Manifolds

TL;DR

This survey reviews advances in understanding wave equation solutions on conic manifolds, focusing on energy decay, wave trace singularities, and scattering resonances related to diffractive geodesics.

Contribution

It consolidates recent results on singularity propagation, energy decay, and resonance distribution on manifolds with conical singularities, extending previous work with multiple collaborators.

Findings

Energy decay on noncompact manifolds with diffractive trapped orbits

Singularities of the wave trace from diffractive closed geodesics

Distribution patterns of scattering resonances linked to closed geodesics

Abstract

In this survey, we review some applications and extensions of the author's results with Richard Melrose on propagation of singularities for solutions to the wave equation on manifolds with conical singularities. These results mainly concern: the local decay of energy on noncompact manifolds with diffractive trapped orbits (joint work with Dean Baskin); singularities of the wave trace created by diffractive closed geodesics (joint work with G. Austin Ford); and the distribution of scattering resonances associated to such closed geodesics (joint work with Luc Hillairet).