{
  "id": "1711.05029",
  "version": "v1",
  "published": "2017-11-14T09:51:49.000Z",
  "updated": "2017-11-14T09:51:49.000Z",
  "title": "Analytic scattering theory for Jacobi operators and Bernstein-Szegö asymptotics of orthogonal polynomials",
  "authors": [
    "D. R. Yafaev"
  ],
  "comment": "Dedicated to the memory of Lyudvig Dmitrievich Faddeev",
  "categories": [
    "math.CA",
    "math.FA",
    "math.SP"
  ],
  "abstract": "We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the \"free\" discrete Schr\\\"odinger operator $H_{0}$. Our goal is to construct various spectral quantities of the operator $H$, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair $H_{0}$, $H$, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials $P_{n}(z)$ associated to the Jacobi matrix $H $ as $n\\to\\infty$. In particular, we consider the case of $z$ inside the spectrum $[-1,1]$ of $H_{0}$ when this asymptotics has an oscillating character of the Bernstein-Szeg\\\"o type and the case of $z$ at the end points $\\pm 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-14T09:51:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "39A70",
      "47A40",
      "47B39"
    ],
    "keywords": [
      "analytic scattering theory",
      "jacobi operators",
      "orthogonal polynomials",
      "study semi-infinite jacobi matrices",
      "jacobi matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}