Mathematical Research Letters

Volume 9, Issue 6, November 2002 pp. 813-828.

A Poisson relation for conic manifolds

Authors: Jared Wunsch
Author institution: Northwestern University

Summary: 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 $\Lap$ 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 $\dif=\{0\} \cup \{{\Psi^{m}_\infty (\bbR^n)} \text{lengths of closed diffractive geodesics}\}$ and $\geo=\{0\} \cup \{{\Psi^{m}_\infty (\bbR^n)} \text{lengths of }$ $\text{closed geometric geodesics}\}.$ We show that $$ \Tr \cos t \sqrt\Lap \in \mathcal{C}^{-n-0}(\mathbb{R}) \cap \mathcal{C}^{-1-0}(\RR\backslash \geo) \cap {{\mathcal{C}}^{\infty}}(\mathbb{R}\backslash \dif).$$ This generalizes a classical result of Chazarain and Duistermaat-Guillemin on boundaryless manifolds, which in turn follows from Poisson summation in the case $X=S^1.$

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