Inverse problems for Hankel operators

该研究属于谱理论领域。一般来说，谱理论的目的是研究严格的数学术语之间的联系，一方面结构和几何性质的刚体，频率和模式的振动（振荡），这些机构的另一方面。所讨论的振动可能是机械的、声学的、弹性的、电磁的，或者可能涉及微观粒子。在数学上，这通常简化为对线性算子的离散谱（即本征值）的研究。谱理论的数学研究通常简化为对大量系统共有的非常一般的性质的分析。特别是，模型系统扮演着至关重要的角色;这些是理想化的数学对象，旨在模拟真实的物理系统的一些重要特征，但又足够简单，以便它们能够进行严格的数学分析。建议的研究集中在这样一个模型系统：汉克尔算子。汉克尔算子是理想化的数学模型，揭示了谱理论与其他数学领域（函数论和复分析）之间的深层结构联系。广义地说，谱理论中的问题可以分为正问题和逆问题。直接问题：“给定一个线性算子的某类，如何确定它的谱？反问题：给定某类线性算子的谱，如何重构该算子？反问题已经由M.Kac通过他著名的问题“人们能听到鼓的形状吗？”“该项目的目的是研究汉克尔算子的逆问题。更准确地说：每个汉克尔算子都有一组本征值;这些本征值（连同一些附加参数）被称为谱数据。我们的目标是研究Hankel算子和它们的谱数据之间的一一对应关系。

The proposed research belongs to the area of Spectral Theory. Generally speaking, the aim of Spectral Theory is to study in rigorous mathematical terms the connection between structural and geometric properties of rigid bodies on the one hand, and frequencies and patterns of vibrations (oscillations) of these bodies on the other hand. Oscillations in question may be mechanical, acoustic, elastic, electromagnetic, or may involve microscopic particles. Mathematically, this usually reduces to the study of the discrete spectrum (i.e. eigenvalues) of linear operators. The mathematical study of Spectral Theory usually reduces to the analysis of very general properties that are common to a large number of systems. In particular, a crucial role is played by model systems; these are idealized mathematical objects designed to emulate some important features of the real physical systems, yet being sufficiently simple so that they are amenable to rigorous mathematical analysis. The proposed research focuses on one such model system: Hankel operators. Hankel operators are idealized mathematical models that reveal deep structural links between Spectral Theory and other areas of mathematics: Function Theory and Complex Analysis. Broadly speaking, the problems in Spectral Theory can be classified into direct and inverse problems. Direct problem: "Given a linear operator of a certain class, how to determine its spectrum?" Inverse problem: "Given a the spectrum of a linear operator of a certain class, how to reconstruct the operator?" Inverse problems have been popularised by M.Kac through his famous question "Can one hear the shape of a drum?"The aim of the project is to study the inverse problem for Hankel operators. More precisely: every Hankel operator has a set of eigenvalues; these eigenvalues (together with some additional parameters) are called the spectral data. We aim to study the one-to-one correspondence between the Hankel operators and their spectral data.