Wave propagation on Euclidean surfaces with conical singularities. I: Geometric diffraction

TL;DR

This paper analyzes how wave singularities behave on Euclidean surfaces with conical points, extending classical results to include diffraction effects and identifying the wave propagator as a special Fourier integral operator.

Contribution

It extends the understanding of wave trace singularities to conical surfaces, incorporating diffraction and characterizing the wave propagator as an intersecting Lagrangian Fourier integral operator.

Findings

Computed leading-order singularity for periodic orbits with degenerate diffractions.

Extended classical wave trace results to surfaces with conical singularities.

Identified wave propagators as singular Fourier integral operators associated with intersecting Lagrangians.

Abstract

We investigate the singularities of the trace of the half-wave group, $\mathrm{Tr} \, e^{-it\sqrt\Delta}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author \cite{Hil} and the two-dimensional case of the work of the first author and Wunsch \cite{ForWun} as well as the seminal result of Duistermaat and Guillemin \cite{DuiGui} in the smooth setting. As an intermediate step, we identify the wave propagators on $X$ as singular Fourier integral operators associated to intersecting Lagrangian submanifolds, originally developed by Melrose and Uhlmann \cite{MelUhl}.