{
  "id": "1204.0728",
  "version": "v1",
  "published": "2012-04-03T16:31:03.000Z",
  "updated": "2012-04-03T16:31:03.000Z",
  "title": "On Self-Adjointness Of 1-D Schrödinger Operators With $δ$-Interactions",
  "authors": [
    "I. I. Karpenko",
    "D. L. Tyshkevich"
  ],
  "comment": "To be published in Methods of Functional Analysis and Topology (2012)",
  "categories": [
    "math.FA"
  ],
  "abstract": "In the present work we consider in $L^2(\\mathbb{R}_+)$ the Schr\\\"odinger operator $\\mathrm{H_{X,\\alpha}}=-\\mathrm{\\frac{d^2}{dx^2}}+\\sum_{n=1}^{\\infty}\\alpha_n\\delta(x-x_n)$. We investigate and complete the conditions of self-adjointness and nontriviality of deficiency indices for $\\mathrm{H_{X,\\alpha}}$ obtained in \\cite{karpiiKost}. We generalize the conditions found earlier in the special case $d_n:=x_{n}-x_{n-1}=1/n$, $n\\in \\mathbb{N}$, to a wider class of sequences $\\{x_n\\}_{n=1}^\\infty$. Namely, for $x_n=\\frac{1}{n^{\\gamma}\\ln^\\eta n}$ with $<\\gamma,\\eta>\\in(1/2, 1)\\times(-\\infty,+\\infty)\\:\\cup\\:\\{1\\}\\times(-\\infty,1]$, the description of asymptotic behavior of the sequence $\\{\\alpha_n\\}_{n=1}^{\\infty}$ is obtained for $\\mathrm{H_{X,\\alpha}}$ either to be self-adjoint or to have nontrivial deficiency indices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-03T16:31:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34L40",
      "47E05",
      "47B25",
      "47B36",
      "81Q10"
    ],
    "keywords": [
      "schrödinger operators",
      "self-adjointness",
      "interactions",
      "nontrivial deficiency indices",
      "wider class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.0728K"
    }
  }
}