{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] of functions on $\\R^n$, and let $\\phi \\in X$ be an incoming wave.\nLet $V$ be multiplication by a potential.\nLet $G_0$ be the outgoing free resolvent, the convolution with the outgoing Green's function of $\\Delta + k^2$.\nAssume that $T = G_0 V$ is a bounded linear operator on $X$.\nConsider the scattering equation $\\psi = \\phi + T \\psi$, an [[integral equation of the second kind|8KJCN49V64]].\nThe *Born series* of $\\psi$ is the formal series\n$$\\psi = \\sum_{n = 0}^{\\infty} T^n \\phi = \\phi + G_0 V \\phi + G_0 V G_0 V \\phi + \\cdots.$$\nIts partial sum $\\phi + G_0 V \\phi$ is the *(first) Born approximation*.\n\n## Remarks\n\nIf the [[operator norm|06EKW1AKXF]] satisfies $\\norm{T} \u003c 1$, the series converges in $X$ and its sum is the unique solution of $\\psi = \\phi + T \\psi$, by the [[Neumann series|8QJ91709V2]].\nReplacing $V$ by $g V$ with a coupling constant $g$, this holds for all small enough $\\abs{g}$.\n\nExample: in one dimension, take the point interaction $V = g \\delta_0$ with $g \\in \\R$; then the outgoing Green's function is $\\frac{e^{ik\\abs{x}}}{2ik}$, and $G_0 V \\psi(x) = \\frac{g}{2ik} e^{ik\\abs{x}} \\psi(0)$.\nFor $\\phi(x) = e^{ikx}$ each term is a multiple of $e^{ik\\abs{x}}$, and the series sums to $\\psi(x) = e^{ikx} + \\frac{g/(2ik)}{1 - g/(2ik)} e^{ik\\abs{x}}$ when $\\abs{g} \u003c 2k$.\nAt $x = 0$ this gives $\\psi(0) = \\frac{1}{1 - g/(2ik)}$, which solves $\\psi(0) = 1 + \\frac{g}{2ik} \\psi(0)$, as it must.\n\nNon-example: for $\\abs{g} \u003e 2k$ in the same model the geometric series diverges, although the equation still has the solution given by the closed formula.\nSo the Born series may fail to converge even when the scattering problem is well posed.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "2PCVRG9NBN",
    "kind": "definition",
    "msc": [
      "81U05",
      "45B05"
    ],
    "name": "Born series",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Born_series"
      }
    ],
    "tag": "EYA944RVSY",
    "uses": [
      "8KJCN49V64",
      "06GZ956657",
      "06EKW1AKXF"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/EYA944RVSY/",
  "links": [
    {
      "field": "defines",
      "target": "2PCVRG9NBN"
    },
    {
      "field": "uses",
      "target": "8KJCN49V64"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "06EKW1AKXF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/EYA944RVSY.md",
  "msc": [
    "81U05",
    "45B05"
  ],
  "name": "Born series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EYA/EYA944RVSY.md",
  "status": null,
  "tag": "EYA944RVSY",
  "uri": "m2://EYA944RVSY",
  "url": "https://tamnd.github.io/math2/wiki/born-series-definition/",
  "writing": {
    "tag": "EYA944RVSY",
    "kind": "definition",
    "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
      }
    ]
  }
}