{
  "id": "1411.4078",
  "version": "v1",
  "published": "2014-11-14T23:03:20.000Z",
  "updated": "2014-11-14T23:03:20.000Z",
  "title": "$L^p$ norms, nodal sets, and quantum ergodicity",
  "authors": [
    "Hamid Hezari",
    "Gabriel Riviere"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.DS",
    "math.MP",
    "math.SP"
  ],
  "abstract": "For small range of $p>2$, we improve the $L^p$ bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence of eigenfunctions. We also derive improvements on the size of the nodal sets. Our proof is based on a quantum ergodicity property of independent interest, which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate. In Appendix B, we show that, in the case of a rational torus, this quantum ergodicity property still holds for symbols supported in balls with a radius shrinking at a polynomial rate. We also obtain a polynomial rate of convergence for the analogue of the quantum ergodicity theorem in the case of the rational torus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-14T23:03:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nodal sets",
      "quantum ergodicity property",
      "rational torus",
      "polynomial rate",
      "quantum ergodicity theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.4078H"
    }
  }
}