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：我们针对修正亥姆霍兹方程的广义Steklov问题进行了数值研究，重点考察其谱与区域几何结构之间的关系。本研究探讨了三个不同方面：（i）多边形区域特征值的渐近行为；（ii）特征函数积分对区域对称性的依赖性；（iii）光滑形状及含角点情形下Steklov特征函数远离边界时的局域化与指数衰减性质。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区域上相关Dirichlet-to-Neumann算子的特征值与特征函数。此外，我们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。