{
  "id": "2302.07519",
  "version": "v1",
  "published": "2023-02-15T08:29:10.000Z",
  "updated": "2023-02-15T08:29:10.000Z",
  "title": "Scattering theory for some non-self-adjoint operators",
  "authors": [
    "Nicolas Frantz"
  ],
  "comment": "33 pages, 1 figure, the results proven rely on the material introduced in arxiv:2203.12406 which is recalled in this paper",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider a non-self-adjoint $H$ given as the perturbation of a self-adjoint operator $H_0$. We suppose that $H$ is of the form $H=H_0+CWC$ where $C$ is a bounded metric operator, relatively compact with respect to $H_0$, and $W$ is bounded. We suppose that $C(H_0-z)^{-1}C$ is uniformly bounded in $z\\in\\mathbb{C}\\setminus\\mathbb{R}$. We define the regularized wave operators associated to $H$ and $H_0$ by $W_\\pm(H,H_0) := \\displaystyle\\mathrm{slim}_{t\\rightarrow\\infty} e^{\\pm itH}r_\\mp(H)\\Pi_\\mathrm{p}(H^\\star)^\\perp e^{\\mp itH_0}$ where $\\Pi_\\mathrm{p}(H^\\star)$ is the projection onto the direct sum of all the generalized eigenspace associated to eigenvalue of $H^\\star$ and $r_\\mp$ is a rational function that regularizes the `incoming/outgoing spectral singularities' of $H$. We prove the existence and study the properties of the regularized wave operators. In particular we show that they are asymptotically complete if $H$ does not have any spectral singularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-15T08:29:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-self-adjoint operators",
      "scattering theory",
      "regularized wave operators",
      "spectral singularity",
      "incoming/outgoing spectral singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}