{
  "id": "math-ph/0606023",
  "version": "v6",
  "published": "2006-06-07T17:58:04.000Z",
  "updated": "2006-10-03T19:17:17.000Z",
  "title": "Distribution of particles which produces a \"smart\" material",
  "authors": [
    "A. G. Ramm"
  ],
  "comment": "corrected typos",
  "doi": "10.1007/s10955-007-9303-3",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "If $A_q(\\beta, \\alpha, k)$ is the scattering amplitude, corresponding to a potential $q\\in L^2(D)$, where $D\\subset\\R^3$ is a bounded domain, and $e^{ik\\alpha \\cdot x}$ is the incident plane wave, then we call the radiation pattern the function $A(\\beta):=A_q(\\beta, \\alpha, k)$, where the unit vector $\\alpha$, the incident direction, is fixed, and $k>0$, the wavenumber, is fixed. It is shown that any function $f(\\beta)\\in L^2(S^2)$, where $S^2$ is the unit sphere in $\\R^3$, can be approximated with any desired accuracy by a radiation pattern: $||f(\\beta)-A(\\beta)||_{L^2(S^2)}<\\epsilon$, where $\\epsilon>0$ is an arbitrary small fixed number. The potential $q$, corresponding to $A(\\beta)$, depends on $f$ and $\\epsilon$, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles $D_m\\subset D$, $1\\leq m\\leq M$, distributed in an a priori given bounded domain $D\\subset\\R^3$. The geometrical shape of a small particle $D_m$ is arbitrary, the boundary $S_m$ of $D_m$ is Lipschitz uniformly with respect to $m$. The wave number $k$ and the direction $\\alpha$ of the incident upon $D$ plane wave are fixed.It is shown that a suitable distribution of the above particles in $D$ can produce the scattering amplitude $A(\\alpha',\\alpha)$, $\\alpha',\\alpha\\in S^2$, at a fixed $k>0$, arbitrarily close in the norm of $L^2(S^2\\times S^2)$ to an arbitrary given scattering amplitude $f(\\alpha',\\alpha)$, corresponding to a real-valued potential $q\\in L^2(D)$.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2006-10-03T19:17:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J05",
      "35J10",
      "70F10",
      "74J25",
      "81U40",
      "81V05",
      "35R30"
    ],
    "keywords": [
      "distribution",
      "scattering amplitude",
      "incident plane wave",
      "arbitrary small fixed number",
      "small acoustically soft particles"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}