{
  "id": "1606.08066",
  "version": "v1",
  "published": "2016-06-26T18:43:32.000Z",
  "updated": "2016-06-26T18:43:32.000Z",
  "title": "Quantum ergodicity and $L^p$ norms of restrictions of eigenfunctions",
  "authors": [
    "Hamid Hezari"
  ],
  "categories": [
    "math.AP",
    "math.CA",
    "math.DG",
    "math.SP"
  ],
  "abstract": "We prove an analogue of Sogge's local $L^p$ estimates for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-G\\'erard-Tzvetkov, Hu, and Chen-Sogge. The improvements are logarithmic on negatively curved manifolds (without boundary) and by $o(1)$ for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get $o(1)$ improvements on $L^\\infty$ estimates of Cauchy data away from a shrinking neighborhood of the corners, and as a result using the methods of Ghosh-Reznikov-Sarnak and Jung-Zelditch, we get that the number of nodal domains of two dimensional ergodic billiards tends to infinity as $\\lambda \\to \\infty$. These results work only for a full density subsequence of any given orthonormal basis of eigenfunctions. We also present an extension of the $L^p$ estimates of Burq-G\\'erard-Tzvetkov, Hu, and Chen-Sogge, for the restrictions of Dirichlet and Neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary. This part does not assume ergodicity on the manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-26T18:43:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum ergodicity",
      "restrictions",
      "piecewise smooth boundary",
      "dimensional ergodic billiards tends",
      "ergodic geodesic flows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}