Some applications of heat flow to Laplace eigenfunctions

TL;DR

This paper explores how Laplace eigenfunctions concentrate mass, their localization near nodal sets, and their decay properties on complex domains, using heat diffusion techniques to reveal new geometric and spectral insights.

Contribution

It introduces heat diffusion methods to analyze eigenfunction localization, nodal set behavior, and decay on complex geometries, providing new results on eigenfunction structure.

Findings

Eigenfunctions concentrate near nodal sets.

Existence and structure of thin nodal domains are characterized.

Eigenfunction decay rates are established on complex Euclidean domains.

Abstract

We consider mass concentration properties of Laplace eigenfunctions $\varphi_\lambda$, that is, smooth functions satisfying the equation $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$, on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" part and "thin" elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets.