{
  "id": "math/0202264",
  "version": "v2",
  "published": "2002-02-25T18:49:55.000Z",
  "updated": "2002-09-24T17:06:41.000Z",
  "title": "A Poisson relation for conic manifolds",
  "authors": [
    "Jared Wunsch"
  ],
  "comment": "Exposition substantially improved. 1 figure added. Title changed",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "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 $X=S^1.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-09-24T17:06:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J47",
      "58J50",
      "35L05"
    ],
    "keywords": [
      "conic manifolds",
      "poisson relation",
      "compact riemannian manifold",
      "conic degeneracy",
      "friedrichs extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......2264W"
    }
  }
}