报 告 人： 刁怀安 副教授
报告题目：On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems
报告时间：2019年6月15日（周六）上午11:00-12:00
报告地点：静远楼204报告厅
主办单位：数学与统计学院、科学技术研究院
报告人简介：
刁怀安，东北师范大学数学与统计副教授，研究方向数值代数与反散射问题，发表科研论文三十余篇；出版学术专著一本；现为国际线性代数系会会员；曾多次访问普渡大学、麦克马斯特大学、汉堡工业大学以及香港浸会大学等高校。
报告摘要：
In this talk, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisation of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is irrational, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field patterns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory.
联 系 人： 贾志刚