{
  "id": "1301.1282",
  "version": "v1",
  "published": "2013-01-07T17:43:25.000Z",
  "updated": "2013-01-07T17:43:25.000Z",
  "title": "Control for Schrödinger equations on 2-tori: rough potentials",
  "authors": [
    "Jean Bourgain",
    "Nicolas Burq",
    "Maciej Zworski"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "For the Schr\\\"odinger equation, $ (i \\partial_t + \\Delta) u = 0 $ on a torus, an arbitrary non-empty open set $ \\Omega $ provides control and observability of the solution: $ \\| u |_{t = 0} \\|_{L^2 (\\T^2)} \\leq K_T \\| u \\|_{L^2 ([0,T] \\times \\Omega)} $. We show that the same result remains true for $ (i \\partial_t + \\Delta - V) u = 0 $ where $ V \\in L^2 (\\T^2) $, and $ \\T^2 $ is a (rational or irrational) torus. That extends the results of \\cite{AM}, and \\cite{BZ4} where the observability was proved for $ V \\in C (\\T^2) $ and conjectured for $ V \\in L^\\infty (\\T^2) $. The higher dimensional generalization remains open for $ V \\in L^\\infty (\\T^n) $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-07T17:43:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schrödinger equations",
      "rough potentials",
      "higher dimensional generalization remains open",
      "arbitrary non-empty open set",
      "result remains true"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}