{
  "id": "2202.02087",
  "version": "v1",
  "published": "2022-02-04T11:40:21.000Z",
  "updated": "2022-02-04T11:40:21.000Z",
  "title": "Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials",
  "authors": [
    "D. R. Yafaev"
  ],
  "comment": "86 pages",
  "categories": [
    "math.CA",
    "math.FA",
    "math.SP"
  ],
  "abstract": "We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\\to\\infty$ as $n\\to\\infty$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_{n}$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_{n}, b_{n}$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_{n}(z)$, $n\\geq -1$, of such equations by a condition for $n\\to\\infty$ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schr\\\"odinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions $P_{n}(z)$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $P_{n}(z)$ as $n \\to\\infty$ in terms of the Wronskian of the solutions $ P_{n} (z) $ and $ f_{n} (z)$. The formulas obtained for $P_{n}(z)$ generalize the asymptotic formulas for the classical Hermite polynomials where $a_{n}=\\sqrt{(n+1)/2}$ and $b_{n}=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-04T11:40:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "39A70",
      "47A40",
      "47B39"
    ],
    "keywords": [
      "jacobi operators",
      "asymptotic behavior",
      "orthogonal polynomials",
      "spectral analysis",
      "asymptotic formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 86,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}