{
  "id": "2110.07154",
  "version": "v3",
  "published": "2021-10-14T05:03:58.000Z",
  "updated": "2023-07-31T06:03:36.000Z",
  "title": "Decay estimates for fourth-order Schrödinger operators in dimension two",
  "authors": [
    "Ping Li",
    "Avy Soffer",
    "Xiaohua Yao"
  ],
  "comment": "62 Pages. This is a final version which was published in JFA 2023",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper we study the decay estimates of the fourth order Schr\\\"{o}dinger operator $H=\\Delta^{2}+V(x)$ on $\\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the zero threshold in the presence of resonances or eigenvalue, and then use them to establish the $L^1-L^\\infty$ decay estimates of $e^{-itH}$generated by the fourth order Schr\\\"{o}dinger operator $H$. Our methods used in the decay estimates depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, we classify these zero resonances as the distributional solutions of $H\\phi=0$ in suitable weighted spaces. Due to the degeneracy of $\\Delta^{2}$ at zero threshold and the lower even dimension (i.e. $n=2$), we remark that the asymptotic expansions of resolvent $R_V(\\lambda^4)$ and the classifications of resonances are more involved than Schr\\\"odinger operator $-\\Delta+V$ in dimension two.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2023-07-31T06:03:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "decay estimates",
      "fourth-order schrödinger operators",
      "asymptotic expansions",
      "zero threshold",
      "fourth order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}