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

TitreFredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation
Type de publicationArticle de revue
AuteurBertola, Marco, Cafasso, Mattia
VilleBerlin ; Heildeberg
TypeArticle scientifique dans une revue à comité de lecture
Pagination793 - 833
Titre de la revueCommunications in Mathematical Physics
Mots-clésClassical and Quantum Gravitation, Relativity Theory, quantum, Statistical Physics, Dynamical Systems and Complexity, Theoretical, Mathematical and Computational Physics
Résumé en anglais

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.

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