We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on the domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for various simply-connected planar domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate several conjectures with relevance in spectral geometry.

翻译：本文对修正亥姆霍兹方程的广义Steklov问题进行了数值研究，重点关注其谱与域几何结构之间的关系。我们探讨了三个不同方面：（i）多边形域特征值的渐近行为；（ii）特征函数积分对域对称性的依赖关系；（iii）光滑形状及含尖角情形下Steklov特征函数远离边界处的局域化与指数衰减。为此，我们实现了两种互补的数值方法，用于计算各类单连通平面域的相关Dirichlet-to-Neumann算子的特征值与特征函数。此外，我们还讨论了所得结果在扩散控制反应理论中的应用，并提出了若干与谱几何相关的猜想。