标题： 黎曼-希尔伯特方法在汉克尔乘积算子的弗雷德霍姆行列式中的应用：标量值核 显示英文标题

标题： A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

摘要： 我们通过矩阵值黎曼-希尔伯特问题来表征一类汉克尔复合算子的弗雷德霍姆行列式，针对加法和乘法复合情况。 底层积分算子的标量值核不假设表现出从Its、Izergin、Korepin和Slavnov的开创性工作中已知的可积结构\cite{IIKS}。 然而，我们能够仅通过利用核的汉克尔复合结构，通过一个自然相关的Zakharov-Shabat型黎曼-希尔伯特问题来描述相应的弗雷德霍姆行列式。 我们通过一系列例子展示这种方法的效率，然后以黎曼-希尔伯特数据的形式计算几个秩一扰动行列式，最后推导出适用于适当核类的Akhiezer-Kac渐近定理。 显示英文摘要

摘要： We characterize Fredholm determinants of a class of Hankel composition operators via matrix-valued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are not assumed to display the integrable structure known from the seminal work of Its, Izergin, Korepin and Slavnov \cite{IIKS}. Yet we are able to describe the corresponding Fredholm determinants through a naturally associated Riemann-Hilbert problem of Zakharov-Shabat type by solely exploiting the kernels' Hankel composition structures. We showcase the efficiency of this approach through a series of examples, we then compute several rank one perturbed determinants in terms of Riemann-Hilbert data and finally derive Akhiezer-Kac asymptotic theorems for suitable kernel classes.