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

Abstract

We study wave diffraction on Euclidean surfaces with conic singularities $(X,g)$. We determine, for the first time, the precise microlocal structure of the wave at the intersection of the direct (or geometric) and diffracted fronts. Namely, we show that the wave kernel is a singular Fourier integral operator in a calculus associated to two intersecting Lagrangian submanifolds (corresponding to the two fronts), introduced originally byMelrose and Uhlmann [23].

We investigate the singularities of the trace of the half-wave group, Tr$e^{-it\sqrt{\Delta }}$, on $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author [12] and the two-dimensional case of the work of the first author and Wunsch [10] as well as the seminal result of Duistermaat and Guillemin [7] in the smooth setting.

In future work, we shall use these results to obtain inverse spectral results on Euclidean surfaces with conic singularities.