{
  "id": "0909.0523",
  "version": "v1",
  "published": "2009-09-02T20:23:57.000Z",
  "updated": "2009-09-02T20:23:57.000Z",
  "title": "Property $C$ and applications to inverse problems",
  "authors": [
    "A. G. Ramm"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\ell_j:=-\\frac{d^2}{dx^2}+k^2q_j(x),$ $k=const>0, j=1,2,$ $0<c_0\\leq q_j(x)\\leq c_1,$ %$q\\in BV([0,1])$, $q$ has finitely many discontinuity points $x_m\\in [0,1],$ and is real-analytic on the intervals $[x_m,x_{m+1}]$ between these points. The set of such functions $q$ is denoted by $M.$ Only the following property of $M$ is used: if $q_j\\in M$, $j=1,2,$ then the function $p(x):=q_2(x)-q_1(x)$ changes sign on the interval $[0, 1]$ at most finitely many times. Suppose that $(*)\\quad \\int_0^1p(x)u_1(x,k)u_2(x,k)dx=0,\\quad \\forall k>0,$ where $p\\in M$ is an arbitrary fixed function, and $u_j$ solves the problem $\\ell_ju_j=0,\\quad 0\\leq x\\leq 1,\\quad u'_j(0,k)=0,\\quad u_j(0,k)=1.$ If $(*)$ implies $h=0$, then the pair $\\{\\ell_1,\\ell_2\\}$ is said to have property $C$ on the set $M$. This property is proved for the pair $\\{\\ell_1,\\ell_2\\}$. Applications to some inverse problems for a heat equation are given. the set $M$. This property is proved for the pair",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-02T20:23:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "74J25",
      "34E05"
    ],
    "keywords": [
      "inverse problems",
      "applications",
      "heat equation",
      "arbitrary fixed function",
      "discontinuity points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.0523R"
    }
  }
}