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Communications in Mathematical Physics

, Volume 309, Issue 3, pp 793–833 | Cite as

Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

  • M. Bertola
  • M. Cafasso
Article

Abstract

We extend the formalism of integrable operators à 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 \({\frac{E_1^T(\lambda) E_2(\mu)}{\lambda+\mu}}\), 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 Painlevé II (recently introduced by Retakh and Rubtsov) and a related noncommutative equation of Painlevé type. We construct a particular family of solutions of the noncommutative Painlevé II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painlevé 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.

Keywords

Convolution Operator Trace Class Hilbert Problem Resolvent Operator Fredholm Determinant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada
  2. 2.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada

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