A Poisson relation for conic manifolds

Abstract

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 $\Delta$ 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 $\partial X$ in any direction, or as ``geometric'' geodesics which must enter and leave $\partial X$ at points which are connected by a geodesic of length $\pi$ in $\partial X.$ Let $\DIFF=\{0\} \cup \{\pm lengths of closed diffractive geodesics\}$ and $\GEOM=\{0\} \cup \{\pm lengths of closed geometric geodesics\}.$ We show that $$ \Tr \cos t \sqrt\Delta \in C^{-n-0}(\RR) \cap C^{-1-0}(\RR\backslash \GEOM) \cap C^\infty(\RR\backslash \DIFF).$$ This generalizes a classical result of Chazarain and Duistermaat-Guillemin on boundaryless manifolds, which in turn follows from Poisson summation in the case $X=S^1.$