{
  "id": "2409.03027",
  "version": "v1",
  "published": "2024-09-04T18:34:39.000Z",
  "updated": "2024-09-04T18:34:39.000Z",
  "title": "Non-harmonic analysis of the wave equation for Schrödinger operators with complex potential",
  "authors": [
    "Aparajita Dasgupta",
    "Lalit Mohan",
    "Shyam Swarup Mondal"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This article investigates the wave equation for the Schr\\\"{o}dinger operator on $\\mathbb{R}^{n}$, denoted as $\\mathcal{H}_0:=-\\Delta+V$, where $\\Delta$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove that the operator $\\mathcal{H}_0$, with $\\operatorname{Re}(V)\\geq 0$ and $\\operatorname{Re}(V)(x)\\to\\infty$ as $|x|\\to\\infty$, has a purely discrete spectrum under certain conditions. In the spirit of Colombini, De Giorgi, and Spagnolo, we also prove that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev spaces, and when the propagation speed is H\\\"{o}lder continuous (or more regular), it is well-posed in Gevrey spaces. Furthermore, we prove that it is very weakly well-posed when the coefficients possess a distributional singularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-04T18:34:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46F05",
      "58J40",
      "22E30"
    ],
    "keywords": [
      "wave equation",
      "non-harmonic analysis",
      "schrödinger operators",
      "complex potential",
      "complex-valued multiplication operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}