{
  "id": "2005.10323",
  "version": "v1",
  "published": "2020-05-20T19:17:43.000Z",
  "updated": "2020-05-20T19:17:43.000Z",
  "title": "Weyl formulae for Schrödinger operators with critically singular potentials",
  "authors": [
    "Xiaoqi Huang",
    "Christopher D. Sogge"
  ],
  "categories": [
    "math.AP",
    "math.CA",
    "math.DG",
    "math.SP"
  ],
  "abstract": "We obtain generalizations of classical versions of the Weyl formula involving Schr\\\"odinger operators $H_V=-\\Delta_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials $V$. In particular, we extend the classical results of Avakumovi\\'{c} , Levitan and H\\\"ormander by obtaining $O(\\lambda^{n-1})$ bounds for the error term in the Weyl formula in the universal case when we merely assume that $V$ belongs to the Kato class, ${\\mathcal K}(M)$, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. In this case, we can also obtain extensions of the Duistermaat-Guillemin theorem yielding $o(\\lambda^{n-1})$ bounds for the error term under generic conditions on the geodesic flow, and we can also extend B\\'erard's theorem yielding $O(\\lambda^{n-1}/\\log \\lambda)$ error bounds under the assumption that the principal curvatures are non-positive everywhere. We can obtain further improvements for tori, which are essentially optimal, if we strengthen the assumption on the potential to $V\\in L^p(M)\\cap {\\mathcal K}(M)$ for appropriate exponents $p=p_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-20T19:17:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J50",
      "35P15"
    ],
    "keywords": [
      "critically singular potentials",
      "weyl formula",
      "schrödinger operators",
      "error term",
      "compact boundaryless riemannian manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}