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

Abstract

We study wave diffraction on Euclidean surfaces with conic singularities (X, g)(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 by Melrose and Uhlmann [23]. We investigate the singularities of the trace of the half-wave group, Tre^{-it\sqrt\Delta}e−it Δ, on (X, g)(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.