作为逆问题研究的典型代表，Sturm-Liouville逆谱问题已发展成为应用数学领域中最为活跃的研究课题之一。本项目以Sturm-Liouville微分算子为主要研究对象，以特征函数的结点为谱数据，从非线性映射的新观点出发研究其逆问题的存在性、唯一重构以及稳定性。具体内容有：探明使得唯一性成立的非超定结点数据的特性，拟给出两组数列为该结点数据的条件并建立其与势函数之间的同胚映射以实现逆结点问题的存在性；建立该结点数据与特征值、规范常数集之间的对应关系，以GLM积分方程为主要工具实现势函数的唯一重构；通过分析以上同胚映射的连续性和解析性，实现逆结点问题的稳定性；应用已建立的方法，以特征函数组成的Wronskians行列式的结点作为谱数据研究一类新逆结点适定性问题。该项目的研究旨在形成和得到处理逆结点问题的新方法和新结果，进一步丰富和拓展微分算子理论。

As a typical representative of inverse problems, inverse Sturm-Liouville problems have gradually developed into one of the most active research topics in applied mathematics. From the nonlinear mapping point of new view, in this project we will take the Sturm-Liouville differential operators as the main research object, and study the existence, unique reconstruction, and stability of the inverse problems in terms of the spectral data consisting of the subset of nodal points (zeros) of eigenfunctions. The main contents are as follows. By analyzing the properties of the non-overdetermined nodal points which ensure the uniqueness, we will provide the conditions that two sequences can be used as the nodal points, and establish the homeomorphism mapping between the two sequences and the potential functions so as to achieve the existence. By establishing the relationship between the subset of nodal points and the set of eigenvalues and norming constants, and taking the GLM integral equation as the main tool, we will realize the unique reconstruction of potential. By analyzing the continuity and analyticity of the above homeomorphism mapping, we will achieve the stability of the inverse nodal problem. Basing on the methods established above, we will study the well-posedness of a new inverse nodal problem by using the subset of nodal points of Wronskians determinant formed by eigenfunctions. The study of this project intends to form and obtain some new methods and new results of dealing with the inverse nodal problems, and further enrich and develop the theory of differential operators.