Large-\(x\) analysis of an operator-valued Riemann-Hilbert problem (Q2801964)

The authors investigate an operator of the form \(I+V\) acting on the space \(L^{2}(a,b)\), where \(V\) is the integral operator with the kernel NEWLINE\[NEWLINE V(\lambda,\mu) = \frac{icF(\lambda)}{2i\pi(\lambda-\mu)} \left\{ \frac { \exp\left(\frac{ix}{2}[p(\lambda)-p(\mu)]\right) } {(\lambda-\mu)+ic} + \frac { \exp\left(\frac{ix}{2}[p(\mu)-p(\lambda)]\right) } {(\lambda-\mu)-ic} \right\}, NEWLINE\]NEWLINE assuming that the function \(p\) and \(F\) satisfy some additional conditions. In the main result of the paper, they find the first term of \(x\)-asymptotic expansion of its Fredholm determinant. The approach uses the operator-valued Riemann-Hilbert problem, including results of the authors' paper [ibid. 2014, No. 24, 6826--6838 (2014; Zbl 1302.47074)] and [\textit{V. E. Korepin} and \textit{N. A. Slavnov}, J. Phys. A, Math. Gen. 30, No. 23, 8241--8255 (1997; Zbl 0928.35165)].