The Spectral Position of Neumann domains on the torus

TL;DR

This paper investigates the spectral position of Neumann domains of Laplacian eigenfunctions on the torus, providing a complete solution for separable eigenfunctions and numerical insights for random waves, advancing understanding of Neumann domain spectra.

Contribution

It offers the first comprehensive analysis of the spectral position of Neumann domains on the torus, combining analytic solutions for separable eigenfunctions with numerical results for random waves.

Findings

Complete solution for separable eigenfunctions on the torus.

Numerical analysis of random wave Neumann domains.

Answers to existing spectral position questions and new research questions.

Abstract

Neumann domains of Laplacian eigenfunctions form a natural counterpart of nodal domains. The restriction of an eigenfunction to one of its nodal domains is the first Dirichlet eigenfunction of that domain. This simple observation is fundamental in many works on nodal domains. We consider a similar property for Neumann domains. Namely, given a Laplacian eigenfunction $f$ and its Neumann domain $\Omega$, what is the position of $\ .f\ |_{\Omega}$ in the Neumann spectrum of $\Omega$? The current paper treats this spectral position problem on the two-dimensional torus. We fully solve it for separable eigenfunctions on the torus and complement our analytic solution with numerics for random waves on the torus. These results answer questions from [37, 8] and arouse new ones.