{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $q_1, q_2 \\colon [0, \\infty) \\to \\R$ be integrable on each interval $[0, R]$ with $R \u003e 0$.\nAssume that both [[Schrödinger operators|89AA6XJY5K]] $-y'' + q_j y$ are in the [[limit-point case|6J9FXT6HAV]] at $\\infty$.\nLet $m_1$ and $m_2$ be their [[Weyl–Titchmarsh functions|XT8PA31VAX]].\nIf $m_1(z) = m_2(z)$ for all $z \\in \\C \\setminus \\R$, then $q_1 = q_2$ almost everywhere on $[0, \\infty)$.\n\n## Remarks\n\nThe function $m$ determines the spectral measure through its Herglotz representation, and the spectral measure determines $q$ by Gelfand–Levitan theory; this gives one proof when $q$ is regular enough.\nExample: if $m(z) = i\\sqrt{z}$ for all non-real $z$, then $q = 0$ almost everywhere, since this is the $m$-function of $q = 0$.\nThe hypothesis cannot be weakened to equality of the spectra: in general the spectrum alone does not determine $q$.\nThere is also a local version: if $m_1(z) - m_2(z)$ decays fast enough as $z \\to \\infty$ along suitable rays, then $q_1 = q_2$ on an initial interval $[0, a]$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that one complex function, the $m$-function, pins down the potential completely.\nTwo different potentials on a half-line cannot have the same $m$-function.\n\n### L2 human\n\nThis is the basic uniqueness result of inverse spectral theory for one-dimensional Schrödinger operators.\nIt sits beside the [[Gelfand–Levitan|R4MRNP9EQT]] reconstruction, which turns the same data into an explicit formula for $q$.\nThe local version shows that the asymptotic behaviour of $m(z)$ for large $\\abs{z}$ already determines $q$ on an initial interval.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "XT8PA31VAX"
    ],
    "kind": "statement",
    "msc": [
      "34A55",
      "34B20",
      "34L40"
    ],
    "name": "Borg–Marchenko uniqueness theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Inverse_scattering_transform"
      }
    ],
    "tag": "Y2GYM93C1W",
    "uses": [
      "89AA6XJY5K",
      "6J9FXT6HAV",
      "XT8PA31VAX"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/Y2GYM93C1W/",
  "links": [
    {
      "field": "about",
      "target": "XT8PA31VAX"
    },
    {
      "field": "uses",
      "target": "89AA6XJY5K"
    },
    {
      "field": "uses",
      "target": "6J9FXT6HAV"
    },
    {
      "field": "uses",
      "target": "XT8PA31VAX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y2GYM93C1W.md",
  "msc": [
    "34A55",
    "34B20",
    "34L40"
  ],
  "name": "Borg–Marchenko uniqueness theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y2G/Y2GYM93C1W.md",
  "status": null,
  "tag": "Y2GYM93C1W",
  "uri": "m2://Y2GYM93C1W",
  "url": "https://math2.tamnd.com/wiki/borg-marchenko-uniqueness-theorem/",
  "writing": {
    "tag": "Y2GYM93C1W",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}