标题： 小扰动对低能拉普拉斯特征函数的影响 显示英文标题

标题： On the effects of small perturbation on low energy Laplace eigenfunctions

摘要： 我们研究拉普拉斯特征函数的节点几何和拓扑的多个方面，特别强调低频区域。 这包括对佩恩性质、节点域的开角估计、(基本)谱隙的饱和等的调查，以及上述所有内容在小尺度扰动下的行为。 我们的目标是突出与基态/低能特征函数相关的谱理论和节点现象的有趣方面，而不是渐近结果。 显示英文摘要

摘要： We investigate several aspects of the nodal geometry and topology of Laplace eigenfunctions, with particular emphasis on the low frequency regime. This includes investigations in and around the Payne property, opening angle estimates of nodal domains, saturation of (fundamental) spectral gaps etc., and behaviour of all of the above under small scale perturbations. We aim to highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions, as opposed to asymptotic results.