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Inverse problems for Hankel operators

Hankel 算子的逆问题

基本信息

批准号：
EP/N022408/1
负责人：
Alexander Pushnitski
金额：
$ 9.11万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN022408%2F1
关键词：
Inverse problems Hankel operators

项目摘要

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.

本研究属于光谱学的范畴。一般而言，谱理论的目的是用严格的数学术语研究刚体的结构和几何性质与这些刚体的振动(振荡)的频率和模式之间的联系。所讨论的振荡可能是机械的、声学的、弹性的、电磁的，也可能涉及微观颗粒。从数学上讲，这通常归结为研究线性算子的离散谱(即本征值)。频谱理论的数学研究通常归结为对大量系统所共有的非常一般的性质的分析。特别是，模型系统扮演着关键的角色；它们是理想化的数学对象，旨在模拟真实物理系统的一些重要特征，但又足够简单，以至于它们可以接受严格的数学分析。所提出的研究集中在一个这样的模型系统：Hankel算子。Hankel算子是理想化的数学模型，揭示了谱理论和其他数学领域：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：“给定某类线性算子，如何确定其谱？”反问题：“给定某一类线性算子的谱，如何重构该算子？”逆问题由M.Kac通过他的著名问题“人们能听到鼓的形状吗？”得到普及。该项目的目的是研究Hankel算子的逆问题。更准确地说：每个Hankel算子都有一组特征值；这些特征值(与一些附加参数一起)称为光谱数据。我们的目标是研究Hankel算子与它们的谱数据之间的一一对应关系。