{
  "id": "1712.03431",
  "version": "v1",
  "published": "2017-12-09T20:12:55.000Z",
  "updated": "2017-12-09T20:12:55.000Z",
  "title": "Local Weak Limits of Laplace Eigenfunctions",
  "authors": [
    "Maxime Ingremeau"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's random wave conjecture. Using recent results of Bourgain, Buckley and Wigman, we will prove that some deterministic families of eigenfunctions on $\\mathbb{T}^2$ satisfy the conclusions of the random wave conjecture. We also show that on an arbitrary domain, a sequence of Laplace eigenfunctions always admits local weak limits. We explain why these local weak limits can be a powerful tool to study the asymptotic number of nodal domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-09T20:12:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplace eigenfunctions",
      "admits local weak limits",
      "berrys random wave conjecture",
      "local weak convergence",
      "arbitrary domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}