{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2FE4P23NVC"
    },
    {
      "field": "uses",
      "target": "CSMF0WVH67"
    },
    {
      "field": "uses",
      "target": "KH7EPKDCH3"
    },
    {
      "field": "defines",
      "target": "Z8CW6Z2K7N"
    }
  ],
  "body": "\n## Summary\n\nAn essentially self-adjoint operator is a symmetric operator on a Hilbert space whose closure is self-adjoint.\nSuch an operator has exactly one self-adjoint extension, so it determines a unique quantum observable and a unique spectral theory.\nIn practice, differential operators are first defined on small spaces of smooth functions, and essential self-adjointness is what makes this choice harmless.\n\n## Explanations\n\n### L1 human\n\nUnbounded operators, such as derivatives, can only be defined on part of a Hilbert space, and the choice of this part matters.\nA symmetric operator on a small domain may admit many self-adjoint extensions, which correspond to different boundary conditions and different physics.\nAn essentially self-adjoint operator leaves no choice: there is exactly one way to extend it to a self-adjoint operator.\nFor example, $-i \\, d/dx$ on smooth compactly supported functions on the whole line $\\R$ is essentially self-adjoint, because the line has no boundary where a condition could be imposed.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47B25"
    ],
    "name": "Essentially self-adjoint operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Self-adjoint_operator"
      }
    ],
    "specializes": [
      "H971WGMV6C"
    ],
    "tag": "DT49GKR9AS"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/DT49GKR9AS/",
  "links": [
    {
      "field": "specializes",
      "target": "H971WGMV6C"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/DT49GKR9AS.md",
  "msc": [
    "47B25"
  ],
  "name": "Essentially self-adjoint operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DT4/DT49GKR9AS.md",
  "status": null,
  "tag": "DT49GKR9AS",
  "uri": "m2://DT49GKR9AS",
  "url": "https://math2.tamnd.com/wiki/essentially-self-adjoint-operator/",
  "writing": {
    "tag": "DT49GKR9AS",
    "kind": "concept",
    "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
      }
    ]
  }
}