{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nConsider the Lieb–Liniger model of bosons on a line with delta interaction of strength $c \u003e 0$, in units with $\\hbar = 2m = 1$.\nIn the thermodynamic limit its ground state is described by a density $\\rho$ of quasi-momenta on an interval $[-K, K]$, with $K \u003e 0$.\nThis density solves\n$$\\rho(k) = \\frac{1}{2\\pi} + \\frac{1}{\\pi} \\int_{-K}^{K} \\frac{c}{c^2 + (k - q)^2} \\, \\rho(q) \\, dq, \\quad -K \\le k \\le K.$$\nThe particle density and the ground state energy per length are $n = \\int_{-K}^{K} \\rho(k) \\, dk$ and $e = \\int_{-K}^{K} k^2 \\rho(k) \\, dk$.\n\nThe equation is a [[Fredholm equation of the second kind|DEYYNVCV9J]] on $[-K, K]$ with the continuous [[difference kernel|B5AB451TB7]] $\\frac{1}{\\pi} \\frac{c}{c^2 + (k - q)^2}$.\nThis kernel is positive and $\\frac{1}{\\pi} \\int_{-\\infty}^{\\infty} \\frac{c}{c^2 + u^2} \\, du = 1$.\nSo for each $k$ the integral of the kernel over the bounded interval $[-K, K]$ is strictly less than $1$.\nBy continuity and compactness, its maximum over $k$ is some $\\theta \u003c 1$.\nHence the integral operator has norm $\\theta \u003c 1$ on $C[-K, K]$ with the supremum norm.\nThen the [[Liouville–Neumann series|H1HWSWME6B]] converges to the unique continuous solution, which is positive.\n\n## Remarks\n\nIn the limit $c \\to \\infty$ the kernel tends to $0$ uniformly on $[-K, K]^2$, so $\\rho \\to \\frac{1}{2\\pi}$.\nThen $n = \\frac{K}{\\pi}$ and $e = \\frac{K^3}{3\\pi} = \\frac{\\pi^2 n^3}{3}$, the energy of free fermions with the same density.\n\n## Explanations\n\n### L1 human\n\nBosons that repel each other on a line can be solved exactly with the Bethe ansatz.\nFor many particles the allowed momenta fill an interval with some density, and this density solves a linear integral equation.\nThe kernel is a Lorentzian bump of width $c$, so for strong repulsion the integral term is small and the density is almost constant.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "DEYYNVCV9J"
    ],
    "kind": "example",
    "msc": [
      "82B23",
      "45B05"
    ],
    "name": "Lieb–Liniger integral equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lieb%E2%80%93Liniger_model"
      }
    ],
    "tag": "Z219Y8PJHD"
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/Z219Y8PJHD/",
  "links": [
    {
      "field": "instance_of",
      "target": "DEYYNVCV9J"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Z219Y8PJHD.md",
  "msc": [
    "82B23",
    "45B05"
  ],
  "name": "Lieb–Liniger integral equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z21/Z219Y8PJHD.md",
  "status": null,
  "tag": "Z219Y8PJHD",
  "uri": "m2://Z219Y8PJHD",
  "url": "https://tamnd.github.io/math2/wiki/lieb-liniger-integral-equation/",
  "writing": {
    "tag": "Z219Y8PJHD",
    "kind": "example",
    "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
      }
    ]
  }
}