{
  "id": "1807.00219",
  "version": "v1",
  "published": "2018-06-30T19:33:02.000Z",
  "updated": "2018-06-30T19:33:02.000Z",
  "title": "The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates",
  "authors": [
    "Burak Erdogan",
    "Michael Goldberg",
    "William R. Green"
  ],
  "comment": "37 pages, submitted",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We investigate $L^1\\to L^\\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\\frac12}$ decay rate, which may be improved to $t^{-\\frac12-\\gamma}$ for any $0\\leq \\gamma<\\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-30T19:33:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "massless dirac equation",
      "dispersive estimates",
      "zero-energy obstructions",
      "natural decay rate",
      "dirac evolution satisfies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}