{
  "id": "math-ph/0511032",
  "version": "v1",
  "published": "2005-11-09T20:22:33.000Z",
  "updated": "2005-11-09T20:22:33.000Z",
  "title": "A second eigenvalue bound for the Dirichlet Schroedinger operator",
  "authors": [
    "Rafael D. Benguria",
    "Helmut Linde"
  ],
  "doi": "10.1007/s00220-006-0041-1",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\lambda_i(\\Omega,V)$ be the $i$th eigenvalue of the Schr\\\"odinger operator with Dirichlet boundary conditions on a bounded domain $\\Omega \\subset \\R^n$ and with the positive potential $V$. Following the spirit of the Payne-P\\'olya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\\star$, we prove that $\\lambda_2(\\Omega,V) \\le \\lambda_2(S_1,V_\\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $\\lambda_1(\\Omega,V) = \\lambda_1(S_1,V_\\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $\\lambda_2(B_R, V) / \\lambda_1(B_R, V)$ decreases when the radius $R$ of the ball $B_R$ increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-09T20:22:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "49Rxx",
      "81Q10"
    ],
    "keywords": [
      "second eigenvalue bound",
      "dirichlet schroedinger operator",
      "convexity assumptions",
      "dirichlet boundary conditions",
      "th eigenvalue"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Commun. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}