In this paper, 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.

在本文中，我们提出了关于拉普拉斯特征函数的几何结构及其在二维中与特征函数的定量行为的深层关系的一些新颖而有趣的发现。我们引入了拉普拉斯特征函数的广义奇异线的新概念，并仔细研究了这些奇异线和节点线。研究表明，其中两条线之间的相交角与本征函数在相交点的消失次序密切相关。我们建立关系的准确和全面的定量表征。粗略地讲，如果相交角不合理，则消失顺序通常是无限的，如果相交角是有理的，则消失序是有限的。实际上，在后一种情况下，消失的顺序就是合理程度。理论发现是原始的，在光谱理论中具有重大意义。而且，它们直接应用于一些非常重要的物理问题，包括反障碍物散射问题和反衍射光栅问题。在某种多边形设置中显示，可以通过有限的许多远场图案来恢复未知散射体的支撑以及表面阻抗参数。实际上，对于某些重要应用，最多两个远场模式就足够了。在逆散射理论中，有限的许多远场模式的唯一可识别性仍然是一个极富挑战性的基本数学问题。