{
  "id": "1404.0739",
  "version": "v2",
  "published": "2014-04-03T00:36:44.000Z",
  "updated": "2014-04-22T08:34:40.000Z",
  "title": "A Jost-Pais-type reduction of Fredholm determinants and some applications",
  "authors": [
    "Alan Carey",
    "Fritz Gesztesy",
    "Denis Potapov",
    "Fedor Sukochev",
    "Yuri Tomilov"
  ],
  "comment": "50 pages; some typos removed",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We study the analog of semi-separable integral kernels in $\\cH$ of the type {equation*} K(x,x')={cases} F_1(x)G_1(x'), & a<x'< x< b, \\\\ F_2(x)G_2(x'), & a<x<x'<b, {cases} {equation*} where $-\\infty\\leq a<b\\leq \\infty$, and for a.e.\\ $x \\in (a,b)$, $F_j (x) \\in \\cB_2(\\cH_j,\\cH)$ and $G_j(x) \\in \\cB_2(\\cH,\\cH_j)$ such that $F_j(\\cdot)$ and $G_j(\\cdot)$ are uniformly measurable, and {equation*} \\|F_j(\\cdot)\\|_{\\cB_2(\\cH_j,\\cH)} \\in L^2((a,b)), \\; \\|G_j (\\cdot)\\|_{\\cB_2(\\cH,\\cH_j)} \\in L^2((a,b)), \\quad j=1,2, {equation*} with $\\cH$ and $\\cH_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\\cdot, \\cdot)$ generates a trace class operator $\\bsK$ in $L^2((a,b);\\cH)$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant ${\\det}_{L^2((a,b);\\cH)}(\\bsI - \\alpha \\bsK)$, $\\alpha \\in \\bbC$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\\cH$ (and $\\cH_1 \\oplus \\cH_2$). Explicit applications of this reduction theory are made to Schr\\\"odinger operators with suitable bounded operator-valued potentials. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-22T08:34:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B10",
      "47G10",
      "34B27",
      "34L40"
    ],
    "keywords": [
      "jost-pais-type reduction",
      "hilbert spaces",
      "trace class operator",
      "jost-pais reduction theory",
      "appropriate fredholm determinants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.0739C"
    }
  }
}