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.

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关于传输特征函数的局部和全局结构及其他

（内部）传输特征值问题是波散射理论中出现的一类非椭圆、非自伴和非线性光谱问题。它们以微妙的方式连接到许多方面的直接和逆向散射问题。多年来，传输特征值的性质得到了广泛而深入的研究，而传输特征函数的内在性质却鲜有研究。最近，在一系列论文中，发现了传输特征函数的几个有趣的局部和全局几何结构。此外，那些长期揭晓的几何特性产生了一些有趣的应用，对直接和逆散射问题具有理论和实际意义。本文通过总结迄今为止获得的结果并讨论其背后的基本原理，回顾了文献中的这些发展。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，对传输特征值问题与几个具有挑战性的逆散射问题之间联系的一些有趣观察，以及对传输特征值和特征函数的光谱特性的几个猜想，其中大多数是文学新手。