{
  "id": "1105.0873",
  "version": "v2",
  "published": "2011-05-04T16:59:29.000Z",
  "updated": "2011-07-06T00:59:26.000Z",
  "title": "Effective limiting absorption principles, and applications",
  "authors": [
    "Igor Rodnianski",
    "Terence Tao"
  ],
  "comment": "91 pages, no figures, submitted, CMP. Some references fixed, other minor corrections",
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate quantitative (or effective) versions of the limiting absorption principle, for the Schr\\\"odinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form $$ \\| R(\\lambda+i\\eps) f \\|_{H^{0,-1/2-\\sigma}} \\leq C(\\lambda, H) \\| f \\|_{H^{0,1/2+\\sigma}}.$$ We are particularly interested in the exact nature of the dependence of the constants $C(\\lambda,H)$ on both $\\lambda$ and $H$. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies $\\lambda \\ll 1$, medium energies $\\lambda \\sim 1$, and large energies $\\lambda \\gg 1$, and there is also a non-trivial distinction between \"qualitative\" estimates on a single operator $H$ (possibly obeying some spectral condition such as non-resonance, or a geometric condition such as non-trapping), and \"quantitative\" estimates (which hold uniformly for all operators $H$ in a certain class). Using elementary methods (integration by parts and ODE techniques), we give some sharp answers to these questions. As applications of these estimates, we present a global-in-time local smoothing estimate and pointwise decay estimates for the associated time-dependent Schr\\\"odinger equation, as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation, under some additional assumptions on the operator $H$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-07-06T00:59:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "35L05",
      "35Q41"
    ],
    "keywords": [
      "effective limiting absorption principles",
      "applications",
      "pointwise decay estimates",
      "integrated local energy decay estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 91,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.0873R"
    }
  }
}