The Diffractive Wave Trace on Manifolds with Conic Singularities

Let (X, g) be a compact manifold with conic singularities. Taking ∆g to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group e−it √ ∆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. 0. Introduction In this paper, we consider the trace of the half-wave group U(t) def = e−it √ ∆g on a compact manifold with conic singularities (X, g). Our main result is a description of the singularities of this trace at the lengths of closed geodesics undergoing diffractive interaction with the cone points. Under the generic assumption that the cone points of X are pairwise nonconjugate along the geodesic flow, the resulting singularity at such a length t = L has the oscillatory integral representation ∫ Rξ e−i(t−L)·ξ a(t, ξ) dξ, where the amplitude a is to leading order a(t, ξ) ∼ L · (2π) kn 2 e ikπ(n−3) 4 · χ(ξ) ξ− k(n−1) 2 ×  k ∏ j=1 i−mγj ·Dαj (qj , q′ j) · dist γj g (Yαj+1 , Yαj ) −n−1 2 ·Θ− 12 (Yαj → Yαj+1)  as |ξ| → ∞ and the index j is cyclic in {1, . . . , k}. Here, n is the dimension of X and k the number of diffractions along the geodesic, and χ is a smooth function supported in [1,∞) and equal to 1 on [2,∞). The product is over the diffractions undergone by the geodesic, with Dαj a quantity determined by the functional calculus of the Laplacian on the link of the j-th cone point Yαj , the factor Θ − 12 (Yαj → Yαj+1) is (at least on a formal level) the determinant of the differential of the flow between the j-th and (j+ 1)-st cone points, and mγj is the Morse index of the geodesic segment γj from the j-th to (j + 1)-st cone points. All of these factors are described in more detail below. To give this result some context, we recall the known results for the LaplaceBeltrami operator ∆g = d ∗ g ◦ d on a smooth (C∞) compact Riemannian manifold (X, g). In this setting, there is a countable orthonormal basis for L(X) comprised