A Poisson relation for conic manifolds

الأصل · EN

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 Δ 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 ={0} ∪ {± lengths of closed diffractive geodesics} and ={0} ∪ {± lengths of closed geometric geodesics}. We show that t √Δ∈ C⁻ⁿ⁻⁰() ∩ C⁻¹⁻⁰() ∩ C∞(). This generalizes a classical result of Chazarain and Duistermaat-Guillemin on boundaryless manifolds, which in turn follows from Poisson summation in the case X=S¹.