{
  "id": "2411.10401",
  "version": "v1",
  "published": "2024-11-15T18:11:13.000Z",
  "updated": "2024-11-15T18:11:13.000Z",
  "title": "Pointwise Weyl Laws for Quantum Completely Integrable Systems",
  "authors": [
    "Suresh Eswarathasan",
    "Allan Greenleaf",
    "Blake Keeler"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of H\\\"ormander, following important prior contributions by G\\\"arding, Levitan, Avakumovi\\'c, and Agmon-Kannai (to name only some), obtained pointwise asymptotics (or a \"pointwise Weyl law\") for a single elliptic, self-adjoint operator. Here, we establish a microlocalized pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, $\\overline{P}=(P_1,P_2,\\dots, P_n)$, where $P_i$ are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold $M^n$, with $\\sum P_i^2$ elliptic and $[P_i,P_j]=0$ for $1\\leq i,j\\leq n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-\\Delta)^\\frac12$. We illustrate our result with several examples, including surfaces of revolution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-15T18:11:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pointwise weyl law",
      "integrable systems",
      "compact manifold",
      "important prior contributions",
      "self-adjoint"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}