{
  "id": "1009.3084",
  "version": "v2",
  "published": "2010-09-16T03:24:27.000Z",
  "updated": "2012-05-01T00:41:38.000Z",
  "title": "Resolvent at low energy III: the spectral measure",
  "authors": [
    "Colin Guillarmou",
    "Andrew Hassell",
    "Adam Sikora"
  ],
  "comment": "42 pages, 4 figures",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "Let $M^\\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\\circ$, in the sense that $M^\\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $\\Delta$ be the positive Laplacian associated to $g$, and $P = \\Delta + V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(\\lambda) = (\\lambda/\\pi i) \\big(R(\\lambda+i0) - R(\\lambda - i0) \\big)$ of $P_+^{1/2}$, where $R(\\lambda) = (P - \\lambda^2)^{-1}$, as $\\lambda \\to 0$, in a manner similar to that done previously by the second author and Vasy, and by the first two authors. The main result is that the spectral measure has a simple, `conormal-Legendrian' singularity structure on a space which is obtained from $M^2 \\times [0, \\lambda_0)$ by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators $\\cos(t \\sqrt{P_+})$ and $\\sin(t \\sqrt{P_+})/\\sqrt{P_+}$, and the Schr\\\"odinger propagator $e^{itP}$, as $t \\to \\infty$. In particular, we prove the analogue of Price's law for odd-dimensional asymptotically conic manifolds. This result on the spectral measure has been used in a follow-up work by the authors (arXiv:1012.3780) to prove sharp restriction and spectral multiplier theorems on asymptotically conic manifolds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-01T00:41:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P25",
      "47A40",
      "58J50"
    ],
    "keywords": [
      "spectral measure",
      "low energy",
      "complete noncompact manifold",
      "asymptotically conic riemaniann metric",
      "wave solution operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.3084G"
    }
  }
}