{
  "id": "1811.09254",
  "version": "v1",
  "published": "2018-11-22T17:45:55.000Z",
  "updated": "2018-11-22T17:45:55.000Z",
  "title": "Semiclassical asymptotic behavior of orthogonal polynomials",
  "authors": [
    "D. R. Yafaev"
  ],
  "categories": [
    "math.CA",
    "math.FA",
    "math.SP"
  ],
  "abstract": "Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\\to\\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and study the corresponding second order difference equation. We suggest an Ansatz for its solutions playing the role of the semiclassical Green-Liouville Ansatz for solutions of the Schr\\\"odinger equation. The formulas obtained for $P_{n}(z)$ as $n\\to\\infty$ generalize the classical Bernstein-Szeg\\\"o asymptotic formulas.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-22T17:45:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "39A70",
      "47A40",
      "47B39"
    ],
    "keywords": [
      "semiclassical asymptotic behavior",
      "orthogonal polynomials",
      "asymptotic formulas",
      "corresponding second order difference equation",
      "spectral theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}