{
  "id": "1606.00297",
  "version": "v1",
  "published": "2016-06-01T14:16:51.000Z",
  "updated": "2016-06-01T14:16:51.000Z",
  "title": "Transport and large deviations for Schrodinger operators and Mather measures",
  "authors": [
    "Artur O. Lopes",
    "J. Mohr"
  ],
  "categories": [
    "math.DS",
    "math-ph",
    "math.CA",
    "math.MP",
    "math.PR",
    "quant-ph"
  ],
  "abstract": "In this mainly survey paper we consider the Lagrangian $ L(x,v) = \\frac{1}{2} \\, |v|^2 - V(x) $, and a closed form $w$ on the torus $ \\mathbb{T}^n $. For the associated Hamiltonian we consider the the Schrodinger operator ${\\bf H}_\\beta=\\, -\\,\\frac{1}{2 \\beta^2} \\, \\Delta +V$ where $\\beta$ is large real parameter. Moreover, for the given form $\\beta\\, w$ we consider the associated twist operator ${\\bf H}_\\beta^w$. We denote by $({\\bf H}_\\beta^w)^*$ the corresponding backward operator. We are interested in the positive eigenfunction $ \\psi_\\beta$ associated to the the eigenvalue $ E_\\beta$ for the operator ${\\bf H}_\\beta^{w} $. We denote $ \\psi_\\beta^*$ the positive eigenfunction associated to the the eigenvalue $ E_\\beta$ for the operator $({\\bf H}_\\beta^{w})^* $. Finally, we analyze the asymptotic limit of the probability $\\nu_\\beta= \\psi_\\beta\\, \\psi_\\beta^*$ on the torus when $\\beta \\to \\infty$. The limit probability is a Mather measure. We consider Large deviations properties and we derive a result on Transport Theory. We denote $L^{-}(x,v) = \\frac{1}{2} \\, |v|^2 - V(x) - w_x(v) $ and $L^{+}(x,v) = \\frac{1}{2} \\, |v|^2 - V(x) + w_x(v) $. We are interest in the transport problem from $\\mu_{-}$ (the Mather measure for $L^{-}$) to $\\mu_{+}$ (the Mather measure for $L^{+}$) for some natural cost function. In the case the maximizing probability is unique we use a Large Deviation Principle due to N. Anantharaman in order to show that the conjugated sub-solutions $u$ and $u^*$ define an admissible pair which is optimal for the dual Kantorovich problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-01T14:16:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A50",
      "37A60",
      "37N20",
      "60F10",
      "81Q20"
    ],
    "keywords": [
      "mather measure",
      "schrodinger operator",
      "dual kantorovich problem",
      "large real parameter",
      "large deviation principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}