{
  "id": "2407.20191",
  "version": "v1",
  "published": "2024-07-29T17:19:11.000Z",
  "updated": "2024-07-29T17:19:11.000Z",
  "title": "Three-dimensional inverse acoustic scattering problem by the BC-method",
  "authors": [
    "M. I. Belishev",
    "A. F. Vakulenko"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The forward acoustic scattering problem that we deal with, is to find $u=u^f(x,t)$ satisfying \\begin{align*} &u_{tt}-\\Delta u+qu=0, && (x,t) \\in {\\mathbb R}^3 \\times (-\\infty,\\infty); \\\\ &u \\mid_{|x|<-t} =0 , && t<0;\\\\ &\\lim_{s \\to -\\infty} s\\,u((-s+\\tau)\\,\\omega,s)=f(\\tau,\\omega), && (\\tau,\\omega) \\in \\Sigma:=[0,\\infty)\\times S^2; \\end{align*} for a real valued compactly supported potential $q=q(x)$ and a control $f \\in L_2(\\Sigma)$. The map $R: L_2(\\Sigma)\\to L_2(\\Sigma)$, \\begin{align*} & (Rf)(\\tau ,\\omega )\\,:= \\lim_{s \\to +\\infty} s\\, u^f((s+\\tau )\\,\\omega ,s), \\quad (\\tau ,\\omega ) \\in \\Sigma \\end{align*} is a response operator. The inverse problem is to determine $q$ from $R$. It is solved by a relevant version of the boundary control method. The procedure that recovers the potential is local: for any fixed $\\xi>0$, given $R\\upharpoonright\\{f\\in L_2(\\Sigma)\\,|\\,\\,f\\mid_{0\\leqslant \\tau<\\xi}=0\\}$ it determines $q\\big|_{|x|\\geqslant\\xi}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-29T17:19:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q93",
      "35L05",
      "78A45"
    ],
    "keywords": [
      "three-dimensional inverse acoustic scattering problem",
      "boundary control method",
      "response operator",
      "forward acoustic scattering problem",
      "real valued compactly supported potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}