基于混合谱数据的Krein弦方程的逆问题研究

Krein弦方程作为描述弦振动而形成的广义微分方程，其自伴生成算子的谱与逆谱问题一直是应用数学领域中最为活跃的研究课题之一。本项目以Krein弦方程为主要研究对象,基于混合谱数据，考察其自伴生成算子的谱与逆谱问题。通过对混合谱数据进行整合和规范，以Simon的A函数和Krein的变换函数为主要工具，拟形成和得到Krein弦方程逆问题研究的新方法和新结果。具体内容有：建立质量分布函数与算子谱的相互制约关系，得到特征值和特征函数的渐近性；规范和探索混合谱数据的格式和特性，将其纳入到适当的空间中，建立其与谱测度空间的同胚映射，以实现质量分布函数的唯一确定性；建立混合谱数据与A函数的一一对应关系，解决质量分布函数的重构问题；刻画Krein弦方程“三区间”问题的非唯一性特征，并解决其唯一确定性问题。该研究将进一步丰富和拓展微分算子理论，为解决相关物理问题提供理论基础。

As the generalized differential equation for describing the vibration of strings, the spectral and inverse spectral problem for Krein's strings have always been one of the most active research topics in the field of applied mathematics. In this project, we will take the Krein's string as the main research object, and study the spectral and inverse spectral problems in terms of the mixed spectral data. By normalizing and integrating the mixed spectral data, and by taking the Simon's A function and Krein's transformation function as the main tool, the main goal of this project is to establish and obtain some new methods and new results which are associated with the inverse problems of the Krein's strings. The main contents are as follows. Establish the inter restricted relationship between the mass distribution function and the spectral data, characterize the asymptotic formulas for the eigenvalues and eigenfunctions. Explore the format and characteristics for the mixed spectral data, search for appropriate space, establish the homeomorphic mapping between it and spectral measure space, so as to realize the uniqueness of the mass distribution function. Establish the relationship between the mixed spectral data and the A function, solve the reconstruction of the mass distribution function. Characterize the nonuniqueness property of the three-interval problem for Krein's strings, and further solve its uniqueness problem. This study will further enrich and expand the theory of differential operators, and provide a theoretical basis for solving related physical problems.