论文标题
关于传输本征及以后的本地和全球结构
On local and global structures of transmission eigenfunctions and beyond
论文作者
论文摘要
（内部）透射特征值问题是在波散射理论中出现的一种非涡旋，非频谱和非线性光谱问题。它们以微妙的方式与许多方面的直接散射问题联系在一起。多年来，透射特征值的特性已经进行了广泛而深入的研究，而透射特征功能的内在特性的研究却少得多。最近，在一系列论文中，发现了几种有趣的局部和全局几何结构。此外，那些久远的几何特性在直接和反向散射问题中产生了一些有趣的应用和实际重要性的有趣应用。本文通过总结到目前为止获得的结果并讨论其背后的原理来回顾文献中的这些发展。本文有一些附带结果，包括几种类型的传输特征值问题的一般公式，关于传输特征值问题之间的联系的一些有趣的观察结果以及几个具有挑战性的反向散射问题，以及关于传输特征值和特征功能的频谱特性的几种猜想，其中大多数是文献的新知识。
The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature.
