{
  "id": "1704.06744",
  "version": "v1",
  "published": "2017-04-22T04:15:22.000Z",
  "updated": "2017-04-22T04:15:22.000Z",
  "title": "Reducibility of quantum harmonic oscillator on $ R^d$ with differential and quasi-periodic in time potential",
  "authors": [
    "Zhenguo Liang",
    "Zhiguo Wang"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We improve the results by Gr\\'ebert and Paturel in \\cite{GP} and prove that a linear Schr\\\"odinger equation on $R^d$ with harmonic potential $|x|^2$ and small $t$-quasiperiodic potential as $$ {\\rm i}u_t - \\Delta u+|x|^2u+\\varepsilon V(\\omega t,x)u=0, \\ (t,x)\\in R\\times R^d $$ reduces to an autonomous system for most values of the frequency vector $\\omega\\in R^n$. The new point is that the potential $V(\\theta,\\cdot )$ is only in ${\\mathcal{C}^{\\beta}}(T^n, \\mathcal{H}^{s}(R^d))$ with $\\beta$ large enough. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in some suitable Sobolev norms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-22T04:15:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum harmonic oscillator",
      "time potential",
      "differential",
      "reducibility",
      "quasi-periodic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}