{
  "id": "1412.6420",
  "version": "v1",
  "published": "2014-12-19T16:31:11.000Z",
  "updated": "2014-12-19T16:31:11.000Z",
  "title": "Bound States for Nano-Tubes with a Dislocation",
  "authors": [
    "Rainer Hempel",
    "Martin Kohlmann",
    "Marko Stautz",
    "Jürgen Voigt"
  ],
  "comment": "28 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "As a model for an interface in solid state physics, we consider two real-valued potentials $V^{(1)}$ and $V^{(2)}$ on the cylinder or tube $S=\\mathbb R \\times (\\mathbb R/\\mathbb Z)$ where we assume that there exists an interval $(a_0,b_0)$ which is free of spectrum of $-\\Delta+V^{(k)}$ for $k=1,2$. We are then interested in the spectrum of $H_t = -\\Delta + V_t$, for $t \\in \\mathbb R$, where $V_t(x,y) = V^{(1)}(x,y)$, for $x > 0$, and $V_t(x,y) = V^{(2)}(x+t,y)$, for $x < 0$. While the essential spectrum of $H_t$ is independent of $t$, we show that discrete spectrum, related to the interface at $x = 0$, is created in the interval $(a_0, b_0)$ at suitable values of the parameter $t$, provided $-\\Delta + V^{(2)}$ has some essential spectrum in $(-\\infty, a_0]$. We do not require $V^{(1)}$ or $V^{(2)}$ to be periodic. We furthermore show that the discrete eigenvalues of $H_t$ are Lipschitz continuous functions of $t$ if the potential $V^{(2)}$ is locally of bounded variation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-19T16:31:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "35P20",
      "81Q10"
    ],
    "keywords": [
      "bound states",
      "essential spectrum",
      "dislocation",
      "nano-tubes",
      "solid state physics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.6420H"
    }
  }
}