{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "3X4ZQ41PXG"
    },
    {
      "field": "uses",
      "target": "3X4ZQ41PXG"
    }
  ],
  "body": "\n## Example\n\nLet $\\lambda, \\alpha, \\theta$ be real numbers.\nThe *almost Mathieu operator* $H_{\\lambda,\\alpha,\\theta}$ acts on the Hilbert space $\\ell^2(\\Z)$ of square-summable sequences $u = (u(n))_{n \\in \\Z}$ by\n$$(H_{\\lambda,\\alpha,\\theta}u)(n) = u(n + 1) + u(n - 1) + 2\\lambda\\cos\\big(2\\pi(n\\alpha + \\theta)\\big)u(n).$$\nIt is a bounded [[self-adjoint operator|0NEGN5CRSC]].\nCheck: each of the three terms has norm at most $1$, $1$ and $2\\abs{\\lambda}$, so $\\norm{H_{\\lambda,\\alpha,\\theta}} \\le 2 + 2\\abs{\\lambda}$.\nThe shifts $u(n) \\mapsto u(n + 1)$ and $u(n) \\mapsto u(n - 1)$ are adjoint to each other, so their sum is self-adjoint, and multiplication by a real sequence is self-adjoint.\n\n## Remarks\n\nThe operator is a discrete [[Schrödinger operator|89AA6XJY5K]] with a quasi-periodic potential when $\\alpha$ is irrational.\nFor irrational $\\alpha$, its [[spectrum|BMKATJ802A]] does not depend on $\\theta$.\nAubry duality relates the operators with coupling $\\lambda$ and $1/\\lambda$, and $\\lambda = 1$ is the critical coupling.\nFor irrational $\\alpha$, the Lebesgue measure of the spectrum is $\\abs{4 - 4\\abs{\\lambda}}$; in particular it is zero at $\\lambda = 1$.\nThe [[ten martini problem|3X4ZQ41PXG]] asked whether the spectrum is a Cantor set for all irrational $\\alpha$ and all $\\lambda \\ne 0$; Avila and Jitomirskaya proved that it is.\nPlotting the spectrum against $\\alpha$ for $\\lambda = 1$ gives Hofstadter's butterfly.\n\n## Explanations\n\n### L1 human\n\nThe operator models an electron hopping along a chain of atoms, one site to the next, with an energy at each site that varies like a cosine.\nWhen $\\alpha$ is irrational, the pattern of site energies never repeats exactly, but it comes close to repeating again and again.\nThis in-between situation, neither periodic nor random, gives a spectrum of remarkable fractal structure.\nThe operator is a test case for the spectral theory of quasi-periodic systems and also models electrons in a crystal under a magnetic field.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "instance_of": [
      "0NEGN5CRSC"
    ],
    "kind": "example",
    "msc": [
      "47B36",
      "81Q10",
      "47A10"
    ],
    "name": "Almost Mathieu operator",
    "same_as": {
      "wikidata": "Q4733975",
      "wikipedia": "Almost Mathieu operator"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Almost_Mathieu_operator"
      }
    ],
    "tag": "D08SKTH2T2"
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/D08SKTH2T2/",
  "links": [
    {
      "field": "instance_of",
      "target": "0NEGN5CRSC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/D08SKTH2T2.md",
  "msc": [
    "47B36",
    "81Q10",
    "47A10"
  ],
  "name": "Almost Mathieu operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D08/D08SKTH2T2.md",
  "status": null,
  "tag": "D08SKTH2T2",
  "uri": "m2://D08SKTH2T2",
  "url": "https://math2.tamnd.com/wiki/almost-mathieu-operator/",
  "writing": {
    "tag": "D08SKTH2T2",
    "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
      }
    ]
  }
}