{
  "id": "1101.3997",
  "version": "v1",
  "published": "2011-01-20T19:24:32.000Z",
  "updated": "2011-01-20T19:24:32.000Z",
  "title": "Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation",
  "authors": [
    "M. Bertola",
    "M. Cafasso"
  ],
  "comment": "46 pages, no figures (oddly)",
  "journal": "Comm Math Phys, February 2012, Volume 309, Issue 3, pp 793-833",
  "doi": "10.1007/s00220-011-1383-x",
  "categories": [
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We extend the formalism of integrable operators a' la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi-infinite interval and to matrix integral operators with a kernel of the form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painleve' II (recently introduced by Retakh and Rubtsov), a related noncommutative equation of Painleve' type. We construct a particular family of solutions of the noncommutative Painleve' II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painleve' II. Such a solution plays the same role as its commutative counterpart relative to the Tracy-Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-20T19:24:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fredholm determinant",
      "noncommutative painleve",
      "pole-free solutions",
      "matrix integral operators",
      "specific riemann-hilbert problems"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2012,
      "month": "Feb",
      "volume": 309,
      "number": 3,
      "pages": 793
    },
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012CMaPh.309..793B"
    }
  }
}