{
  "id": "math-ph/0102028",
  "version": "v3",
  "published": "2001-02-22T20:22:55.000Z",
  "updated": "2001-03-31T02:21:50.000Z",
  "title": "Reconstruction of the potential from I-function",
  "authors": [
    "A. G. Ramm"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "If $f(x,k)$ is the Jost solution and $f(x) = f(0,k)$, then the $I$-function is $I(k) := \\frac{f^\\prime(0,k)}{f(k)}$. It is proved that $I(k)$ is in one-to-one correspondence with the scattering triple ${\\mathcal S} :=\\{S(k), k_j, s_j, \\quad 1 \\leq j \\leq J\\}$ and with the spectral function $\\rho(\\lambda)$ of the Sturm-Liouville operator $l= -\\frac{d^2}{dx^2} + q(x)$ on $(0, \\infty)$ with the Dirichlet condition at $x=0$ and $q(x) \\in L_{1,1} := \\{q: q= \\bar q, \\int^\\infty_0 (1+x) |q(x) dx < \\infty\\}$. Analytical methods are given for finding $\\mathcal S$ from $I(k)$ and $I(k)$ from $\\mathcal S$, and $\\rho(\\lambda)$ from $I(k)$ and $I(k)$ from $\\rho(\\lambda)$. Since the methods for finding $q(x)$ from $\\mathcal S$ or from $\\rho(\\lambda)$ are known, this yields the methods for finding $q(x)$ from $I(k)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2001-03-31T02:21:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34R30"
    ],
    "keywords": [
      "reconstruction",
      "i-function",
      "jost solution",
      "one-to-one correspondence",
      "spectral function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.ph...2028R"
    }
  }
}