Riemann-Hilbert approach to a generalized sine kernel and applications

N. Kitanine (1) , Karol K. Kozlowski (2) , Jean Michel Maillet (2) , N. A. Slavnov (3) , Véronique Terras (2, 4)

1 LPTM - UMR 8089 - Laboratoire de Physique Théorique et Modélisation
2 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
3 SMI - Steklov Mathematical Institute [Moscow]
4 LPTA - Laboratoire de Physique Théorique et Astroparticules

We investigate the asymptotic behavior of a generalized sine kernel acting on a finite size interval [-q,q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener--Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.

Keywords: Fredholm determinant; Generalized sine kernel; Wiener-Hopf operators; Integrable models

Domains: Physics [physics]; Mathematical Physics [math-ph]; Mathematics [math]; Mathematical Physics [math-ph]

N. Kitanine, Karol K. Kozlowski, Jean Michel Maillet, N. A. Slavnov, Véronique Terras. Riemann-Hilbert approach to a generalized sine kernel and applications. Communications in Mathematical Physics, 2009, 291 (3), pp.691-761. ⟨10.1007/s00220-009-0878-1⟩. ⟨ensl-00283404v3⟩