{
  "id": "2210.10941",
  "version": "v1",
  "published": "2022-10-20T00:42:00.000Z",
  "updated": "2022-10-20T00:42:00.000Z",
  "title": "Spectral theory of $p-adic$ Hermite operator",
  "authors": [
    "Tianhong Zhao"
  ],
  "categories": [
    "math-ph",
    "math.FA",
    "math.MP",
    "math.NT",
    "math.SP",
    "quant-ph"
  ],
  "abstract": "We give the definition of $p-adic$ Hermite operator and set up the $p-adic$ spectral measure. We compare the Archimedean case with non-Archimedean case. The structure of Hermite conjugate in $C^{*}$-Algebra corresponds to three canonical structures of $p-adic$ ultrametric Banach algebra: 1. mod $p$ reduction 2. Frobenius map 3. Teichm\\\"uller lift. There is a nature connection between Galois theory and Hermite operator spectral decomposition. The Galois group $\\mathrm{Gal}(\\bar{\\mathbb{F}}_p|\\mathbb{F}_p)$ generate the $p-adic$ spectral measure. We point out some relationships with $p-adic$ quantum mechanics: 1. creation operator and annihilation operator 2. $p-adic$ uncertainty principle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-20T00:42:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral theory",
      "spectral measure",
      "hermite operator spectral decomposition",
      "ultrametric banach algebra",
      "hermite conjugate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}