{
  "id": "1008.4396",
  "version": "v4",
  "published": "2010-08-25T21:49:54.000Z",
  "updated": "2011-09-25T14:36:44.000Z",
  "title": "Non-concentration of quasimodes for integrable systems",
  "authors": [
    "Jared Wunsch"
  ],
  "comment": "One addition to acknowledgments; to appear in Comm. PDE",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set $WF_h$ of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of $WF_h$ along positive codimension sub-tori of a Liouville torus $\\mathcal{L}$ if there exist rational relations among the frequencies of the flow on $\\mathcal{L}.$ We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. The main tools are the spreading of Lagrangian regularity on $\\mathcal{L}$ previously shown by Vasy and the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-09-25T14:36:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integrable systems",
      "hamilton flow",
      "principal symbol",
      "non-concentration",
      "higher order transport equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.4396W"
    }
  }
}