基于传输特征值的具有不连续折射率内部传输问题逆谱分析

内部传输问题是非均匀声学及电磁介质的逆散射理论中特征值问题，其中最重要的是逆传输特征值问题，近年来倍受研究者关注。本项目基于传输特征值数据研究具有不连续折射率声波方程的逆谱问题。研究内容主要包括：(1)通过研究大波数对应初值问题解特性和特征行列式的渐近特征，刻画传输特征值分布，并基于Galerkin方案给出其数值计算；(2)探究是否传输特征值能唯一确定折射率及其不连续性，否则为确保其唯一性还需要怎样的其它谱数据？运用指数系闭性等理论探究确定唯一性的谱数据的最优性；(3)运用Borg方法研究逆传输特征值问题的局部可解性及稳定性，并理论上给出折射率重构程序；(4)基于离散化方法与Newton型算法，我们拟提供由传输特征值数据重构折射率的数值计算及重构算法性能分析。该项目研究及其成果将广泛应用于逆散射理论中，为层状材料无损检测提供基于传输特征值的有效方法，对研究散射体内部结构性质具有重要意义。

The interior transmission problem is an eigenvalue problem that arises in the inverse scattering theory for inhomogeneous acoustic and electromagnetic media, and one of the most significant issues related to the interior transmission problem is the inverse transmission eigenvalue problem, which has attracted much attention of researchers in recent years. Based on transmission eigenvalues, this project intends to investigate inverse spectral problems related to the sound wave equation with a discontinuous refractive index. The main contents of the study include the following aspects. (1) By investigating properties of the solution of the initial value problems and the asymptotic behavior of the characteristic determinants for large wave numbers, we characterize the distribution of transmission eigenvalues and give the numerical computation of transmission eigenvalues by adopting the Galerkin scheme.(2) We explore whether transmission eigenvalues can uniquely determine the refractive index and its discontinuity, if not, what other spectral data do one need to ensure the uniqueness？By using some closedness and basis tests for exponential systems, we shall explore the optimality of the spectral data to determine the uniqueness. (3) By applying the Borg method, we study a local solvability and stability of the inverse transmission eigenvalue problem and theoretically give a reconstructive procedure for the refractive index. (4) Based on the discretization method and Newton-type algorithms, we propose to provide a numerical computation for the reconstruction of the refractive index by using transmission eigenvalues as input data, and present the performance analysis of the reconstruction algorithm. The research and results of the project will be widely used in the inverse scattering theory, and provide effective methods for the nondestructive testing of layered materials based on transmission eigenvalues. It is of great significance to the study of the internal structure properties of the scatterers.