{
  "id": "2209.09672",
  "version": "v1",
  "published": "2022-09-20T12:17:13.000Z",
  "updated": "2022-09-20T12:17:13.000Z",
  "title": "Algebraic delocalization for the Schrödinger equation on large tori",
  "authors": [
    "Henrik Ueberschaer"
  ],
  "comment": "10 pages, no figures",
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP",
    "math.SP"
  ],
  "abstract": "Let $\\mathcal{L}$ be a fixed $d$-dimensional lattice. We study the localization properties of solutions of the stationary Schr\\\"odinger equation with a positive $L^\\infty$ potential on tori $\\mathbb{R}^d/L\\mathcal{L}$ in the limit, as $L\\to\\infty$, for dimension $d \\leq 3$. We show that the probability measures associated with $L^2$-normalized solutions, with eigenvalue $E$ near the bottom of the spectrum, satisfy an algebraic delocalization theorem which states that these probability measures cannot be localized inside a ball of radius $r = o(E^{-1/4+\\epsilon})$, unless localization occurs with a sufficiently slow algebraic decay. In particular, we apply our result to Schr\\\"odinger operators modeling disordered systems, such as the d-dimensional continuous Anderson- Bernoulli model, where almost sure exponential localization of eigenfunctions, in the limit as $E \\to 0$, was proved by Bourgain-Kenig in dimension $d \\geq 2$, and show that our theorem implies an algebraic blow-up of localization length in this limit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-20T12:17:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P99",
      "81Q99"
    ],
    "keywords": [
      "large tori",
      "schrödinger equation",
      "probability measures",
      "sure exponential localization",
      "algebraic delocalization theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}