{
  "id": "1202.2915",
  "version": "v1",
  "published": "2012-02-14T02:58:45.000Z",
  "updated": "2012-02-14T02:58:45.000Z",
  "title": "An Estimate on the Number of Eigenvalues of a Quasiperiodic Jacobi Matrix of Size $n$ Contained in an Interval of Size $n^{-C}$",
  "authors": [
    "Ilia Binder",
    "Mircea Voda"
  ],
  "comment": "37 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift $x\\rightarrow x+\\omega$. We assume that the Lyapunov exponent $L(E_{0})$ of the corresponding Jacobi cocycle satisfies $L(E_{0})\\ge\\gamma>0$. In this setting we prove that the number of eigenvalues $E_{j}^{(n)}(x)$ of a submatrix of size $n$ contained in an interval $I$ centered at $E_{0}$ with $|I|=n^{-C_{1}}$ does not exceed $(\\log n)^{C_{0}}$ for any $x$. Here $n\\ge n_{0}$, and $n_{0}$, $C_{0}$, $C_{1}$ are constants depending on $\\gamma$ (and the other parameters of the problem).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-14T02:58:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "47B36",
      "82B44"
    ],
    "keywords": [
      "quasiperiodic jacobi matrix",
      "eigenvalues",
      "infinite quasi-periodic jacobi self-adjoint matrices",
      "corresponding jacobi cocycle satisfies",
      "real analytic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.2915B"
    }
  }
}