{
  "id": "math-ph/0702021",
  "version": "v1",
  "published": "2007-02-08T09:48:24.000Z",
  "updated": "2007-02-08T09:48:24.000Z",
  "title": "A uniform quantum version of the Cherry theorem",
  "authors": [
    "Carlos Villegas Blas",
    "Sandro Graffi"
  ],
  "comment": "17 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider in $L^2(\\R^2)$ the operator family $H(\\epsilon):=P_0(\\hbar,\\omega)+\\epsilon F_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $\\om$, $F_0$ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and $\\ep\\in\\C$. Then there exist $\\ep^\\ast >0$ independent of $\\hbar$ and an open set $\\Omega\\subset\\C^2\\setminus\\R^2$ such that if $|\\ep|<\\ep^\\ast$ and $\\om\\in\\Om$ the quantum normal form near $P_0$ converges uniformly with respect to $\\hbar$. This yields an exact quantization formula for the eigenvalues, and for $\\hbar=0$ the classical Cherry theorem on convergence of Birkhoff's normal form for complex frequencies is recovered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-02-08T09:48:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81S30",
      "70H08"
    ],
    "keywords": [
      "uniform quantum version",
      "quantum harmonic oscillator",
      "birkhoffs normal form",
      "diophantine frequency vector",
      "exact quantization formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math.ph...2021V"
    }
  }
}