{
  "id": "2405.16448",
  "version": "v1",
  "published": "2024-05-26T06:29:47.000Z",
  "updated": "2024-05-26T06:29:47.000Z",
  "title": "Understanding of linear operators through Wigner analysis",
  "authors": [
    "Elena Cordero",
    "Gianluca Giacchi",
    "Edoardo Pucci"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators reside in (weighted) modulation spaces, particularly in Sj\\\"ostrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes. Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schr{\\\"o}dinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations $S \\in Sp(d, \\mathbb{R})$. The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator $ T $ on $ \\mathbb{R}^d $ into a pseudodifferential operator $ K$ on $ \\mathbb{R}^{2d}$. This transformation involves a symbol $\\sigma$ well-localized around the manifold defined by $ z = S w $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-26T06:29:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S05",
      "35S30",
      "47G30"
    ],
    "keywords": [
      "linear operators",
      "fourier integral operator",
      "wigner analysis",
      "extend wigners original framework",
      "pseudodifferential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}