{
  "id": "math/0512431",
  "version": "v2",
  "published": "2005-12-18T20:15:57.000Z",
  "updated": "2006-01-16T10:25:55.000Z",
  "title": "A Liouville-type theorem for Schrödinger operators",
  "authors": [
    "Yehuda Pinchover"
  ],
  "comment": "14 pages, the main result was improved, and a few more applications were added",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator $P_1$, such that a nonzero subsolution of a symmetric nonnegative operator $P_0$ is a ground state. Particularly, if $P_j:=-\\Delta+V_j$, for $j=0,1$, are two nonnegative Schr\\\"odinger operators defined on $\\Omega\\subseteq \\mathbb{R}^d$ such that $P_1$ is critical in $\\Omega$ with a ground state $\\phi$, the function $\\psi\\nleq 0$ is a subsolution of the equation $P_0u=0$ in $\\Omega$ and satisfies $|\\psi|\\leq C\\phi$ in $\\Omega$, then $P_0$ is critical in $\\Omega$ and $\\psi$ is its ground state. In particular, $\\psi$ is (up to a multiplicative constant) the unique positive supersolution of the equation $P_0u=0$ in $\\Omega$. Similar results hold for general symmetric operators, and also on Riemannian manifolds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-01-16T10:25:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B05"
    ],
    "keywords": [
      "schrödinger operators",
      "ground state",
      "liouville-type theorem",
      "general symmetric operators",
      "similar results hold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12431P"
    }
  }
}