{
  "id": "1204.1305",
  "version": "v2",
  "published": "2012-04-05T18:50:55.000Z",
  "updated": "2013-05-08T15:23:35.000Z",
  "title": "Microlocal limits of plane waves and Eisenstein functions",
  "authors": [
    "Semyon Dyatlov",
    "Colin Guillarmou"
  ],
  "comment": "78 pages, 5 figures; to appear in Annales de l'ENS",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "We study microlocal limits of plane waves on noncompact Riemannian manifolds (M,g) which are either Euclidean or asymptotically hyperbolic with curvature -1 near infinity. The plane waves E(z,\\xi) are functions on M parametrized by the square root of energy z and the direction of the wave, \\xi, interpreted as a point at infinity. If the trapped set K for the geodesic flow has Liouville measure zero, we show that, as z\\to +\\infty, E(z,\\xi) microlocally converges to a measure \\mu_\\xi, in average on energy intervals of fixed size, [z,z+1], and in \\xi. We express the rate of convergence to the limit in terms of the classical escape rate of the geodesic flow and its maximal expansion rate - when the flow is Axiom A on the trapped set, this yields a negative power of z. As an application, we obtain Weyl type asymptotic expansions for local traces of spectral projectors with a remainder controlled in terms of the classical escape rate.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-08T15:23:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "plane waves",
      "eisenstein functions",
      "classical escape rate",
      "geodesic flow",
      "weyl type asymptotic expansions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 78,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.1305D"
    }
  }
}