On the effects of small perturbation on low energy Laplace eigenfunctions

TL;DR

This paper explores how small perturbations affect the geometry and topology of low energy Laplace eigenfunctions, focusing on spectral properties, nodal domain shapes, and stability of spectral gaps.

Contribution

It provides new insights into the stability and geometric features of low energy eigenfunctions under small perturbations, emphasizing spectral and nodal phenomena.

Findings

Analysis of nodal domain opening angles

Saturation of spectral gaps under perturbations

Behavior of eigenfunctions near the ground state

Abstract

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.