{
  "id": "2411.14824",
  "version": "v1",
  "published": "2024-11-22T09:50:03.000Z",
  "updated": "2024-11-22T09:50:03.000Z",
  "title": "Spectral regularity with respect to dilations for a class of pseudodifferential operators",
  "authors": [
    "Horia D. Cornean",
    "Radu Purice"
  ],
  "comment": "10 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We continue the study of the perturbation problem discussed in \\cite{CP3} and get rid of the 'slow variation' assumption by considering symbols of the form $a\\big(x+\\delta\\,F(x),\\xi\\big)$ with $a$ a real H\\\"{o}rmander symbol of class $S^0_{0,0}(\\mathbb{R}^d\\times\\mathbb{R}^d)$ and $F$ a smooth function with all its derivatives globally bounded, with $|\\delta|\\leq1$. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of $\\delta=0$ is still of the order $\\sqrt{|\\delta|}$, the distance between their spectral edges behaves like $|\\delta|^\\nu$ with $\\nu\\in[1/2,1)$ depending on the rate of decay of the second derivatives of $F$ at infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-22T09:50:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "81Q15",
      "35S05"
    ],
    "keywords": [
      "pseudodifferential operators",
      "spectral regularity",
      "spectral edges behaves",
      "low variation",
      "smooth function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}