Nodal sets of Laplace eigenfunctions under small perturbations

Abstract

We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if such perturbations are relatively small, more formally, supported at a sub-wavelength scale. We do not need any generic assumption on the topology of the nodal sets or the simplicity of the Laplace spectrum. As an indirect application, we are able to show that a certain “Payne property” concerning the second nodal set remains stable under controlled perturbations of the domain.