A Poisson Relation for Conic Manifolds

Let X be a compact Riemannian manifold with conic singularities, i.e. a Riemannian manifold whose metric has a conic degeneracy at the boundary. Let ∆ be the Friedrichs extension of the Laplace-Beltrami operator on X. There are two natural ways to define geodesics passing through the boundary: as “diffractive” geodesics which may emanate from ∂X in any direction, or as “geometric” geodesics which must enter and leave ∂X at points which are connected by a geodesic of length π in ∂X. Let DIFF = {0} ∪ {±lengths of closed diffractive geodesics} and GEOM = {0} ∪ {±lengths of closed geometric geodesics}. We show that Tr cos t √ ∆ ∈ C(R) ∩ C(R\GEOM) ∩ C(R\DIFF). This generalizes a classical result of Chazarain and Duistermaat-Guillemin on boundaryless manifolds, which in turn follows from Poisson summation in the case X = S.