{
  "id": "1711.07583",
  "version": "v1",
  "published": "2017-11-21T00:14:40.000Z",
  "updated": "2017-11-21T00:14:40.000Z",
  "title": "Adiabatic evolution and shape resonances",
  "authors": [
    "Michael Hitrik",
    "Andrea Mantile",
    "Johannes Sjoestrand"
  ],
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schr\\\"odinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter $\\varepsilon $ with $\\ln\\varepsilon \\asymp -1/h$, where $h$ denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length $\\varepsilon ^{-N}$ with an error ${\\cal O}(\\varepsilon ^N)$. Here $N>0$ is arbitrary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-21T00:14:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "35P20",
      "35B34",
      "35S05"
    ],
    "keywords": [
      "shape resonances",
      "general linear adiabatic evolution problem",
      "time dependent potential",
      "potential wells",
      "non-linear evolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}