{
  "id": "1803.01828",
  "version": "v1",
  "published": "2018-03-05T18:46:05.000Z",
  "updated": "2018-03-05T18:46:05.000Z",
  "title": "Scattering approach to Anderson localisation",
  "authors": [
    "A. Ossipov"
  ],
  "comment": "4+2 pages",
  "categories": [
    "cond-mat.mes-hall",
    "cond-mat.dis-nn",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We develop a novel approach to the Anderson localisation problem in a $d$-dimensional disordered sample of dimension $L\\times M^{d-1}$. Attaching a perfect lead with the cross-section $M^{d-1}$ to one side of the sample, we derive evolution equations for the scattering matrix and the Wigner-Smith time delay matrix as a function of $L$. Using them one obtains the Fokker-Plank equation for the distribution of the proper delay times and the evolution equation for their density at weak disorder. The latter can be mapped onto a non-linear partial differential equation of the Burgers type, for which a complete analytical solution for arbitrary $L$ is constructed. Analysing the solution for a cubic sample with $M=L$ in the limit $L\\to \\infty$, we find that for $d<2$ the solution tends to the localised fixed point, while for $d>2$ to the metallic fixed point and provide explicit results for the density of the delay times in these two limits.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-05T18:46:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scattering approach",
      "non-linear partial differential equation",
      "evolution equation",
      "wigner-smith time delay matrix",
      "fixed point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}