{
  "id": "0710.5903",
  "version": "v2",
  "published": "2007-10-31T16:45:38.000Z",
  "updated": "2008-01-07T15:46:27.000Z",
  "title": "Leaky Quantum Graphs: A Review",
  "authors": [
    "Pavel Exner"
  ],
  "comment": "42 pages, to appear in the proceedings of the Isaac Newton Institute programme \"Analysis on Graphs and Applications\"; the final version with minor improvements",
  "categories": [
    "math-ph",
    "cond-mat.mes-hall",
    "math.AP",
    "math.MP",
    "math.SP",
    "quant-ph"
  ],
  "abstract": "The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-\\Delta -\\alpha \\delta (x-\\Gamma)$ with a singular attractive interaction supported by a graph-like set in $\\mathbb{R}^\\nu,\\: \\nu=2,3$. We will explain how such singular Schr\\\"odinger operators can be properly defined for different codimensions of $\\Gamma$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $\\Gamma$ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leaky-graph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-01-07T15:46:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q05",
      "35J10",
      "35P15",
      "58J05"
    ],
    "keywords": [
      "leaky quantum graphs",
      "open problems",
      "strong-coupling asymptotic behavior",
      "discrete counterpart",
      "leaky-graph hamiltonians"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.5903E"
    }
  }
}