{
  "id": "1905.05136",
  "version": "v1",
  "published": "2019-05-13T16:42:10.000Z",
  "updated": "2019-05-13T16:42:10.000Z",
  "title": "A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points",
  "authors": [
    "Blake Keeler"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold $M$ with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, $E_\\lambda(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum of eigenspaces with eigenvalue smaller than $\\lambda^2$ as $\\lambda \\to\\infty$. We obtain a uniform logarithmic improvement in the remainder of this asymptotic expansion when the points $x,y$ are close together. This result is a generalization of a work by B\\'erard, which treated the on-diagonal case, $E_\\lambda(x,x)$. The results in this paper allow us to conclude that the rescaled covariance kernel of a monochromatic random wave on a manifold without conjugate points locally converges to a universal scaling limit at an inverse logarithmic rate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-13T16:42:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P20"
    ],
    "keywords": [
      "two-point weyl law",
      "compact riemannian manifold",
      "uniform logarithmic improvement",
      "monochromatic random wave",
      "conjugate points locally converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}