{
  "id": "1302.5044",
  "version": "v1",
  "published": "2013-02-20T17:47:07.000Z",
  "updated": "2013-02-20T17:47:07.000Z",
  "title": "On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set",
  "authors": [
    "Raffaele Carlone",
    "Mark Malamud",
    "Andrea Posilicano"
  ],
  "comment": "accepted for publication in Journal of Differential Equations",
  "doi": "10.1016/j.jde.2013.01.026",
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We investigate spectral properties of Gesztesy-\\v{S}eba realizations D_{X,\\alpha} and D_{X,\\beta} of the 1-D Dirac differential expression D with point interactions on a discrete set $X=\\{x_n\\}_{n=1}^\\infty\\subset \\mathbb{R}.$ Here $\\alpha := \\{\\alpha_{n}\\}_{n=1}^\\infty$ and \\beta :=\\{\\beta_{n}\\}_{n=1}^\\infty \\subset\\mathbb{R}. The Gesztesy-\\v{S}eba realizations $D_{X,\\alpha}$ and $D_{X,\\beta}$ are the relativistic counterparts of the corresponding Schr\\\"odinger operators $H_{X,\\alpha}$ and $H_{X,\\beta}$ with $\\delta$- and $\\delta'$-interactions, respectively. We define the minimal operator D_X as the direct sum of the minimal Dirac operators on the intervals $(x_{n-1}, x_n)$. Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator $D_X^*$ in the case $d_*(X):=\\inf\\{|x_i-x_j| \\,, i\\not=j\\} = 0$. It turns out that the boundary operators $B_{X,\\alpha}$ and $B_{X,\\beta}$ parameterizing the realizations D_{X,\\alpha} and D_{X,\\beta} are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schr\\\"odinger operators with point interactions. We show that certain spectral properties of the operators $D_{X,\\alpha}$ and $D_{X,\\beta}$ correlate with the corresponding spectral properties of the Jacobi matrices $B_{X,\\alpha}$ and $B_{X,\\beta}$, respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy--{\\vS}eba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light $c\\to\\infty$. Most of our results are new even in the case $d_*(X)> 0.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-02-20T17:47:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34L05",
      "34L40",
      "47E05",
      "47B25",
      "47B36",
      "81Q10"
    ],
    "keywords": [
      "point interactions",
      "spectral theory",
      "discrete set",
      "spectral properties",
      "gesztesy-šeba realizations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Differential Equations",
      "year": 2013,
      "volume": 254,
      "number": 9,
      "pages": 3835
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013JDE...254.3835C"
    }
  }
}