2011/01/20 by M. Bertola, M. Cafasso · 2 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Commutative property #Formalism (music) #Fredholm determinant #Fredholm integral equation #Fredholm theory #Holomorphic and Operator Theory #Integrable system #Noncommutative geometry #Operator theory #Resolvent #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1007/s00220-011-1383-x
published as Comm Math Phys, February 2012, Volume 309, Issue 3, pp 793-833 · 46 pages, no figures (oddly)
arxiv created 2011/01/20 · openalex publication_date 2011/12/02 · arxiv updated 2013/06/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 E1T(x) E2(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.