On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori

TL;DR

This paper investigates the parity of the number of nodal domains of Laplacian eigenfunctions on flat tori, proving evenness in one case and constructing an example with three domains in another.

Contribution

It proves that eigenfunctions on the standard 2D torus have an even number of nodal domains and constructs a specific eigenfunction with three nodal domains on a scaled torus.

Findings

Eigenfunctions on the standard torus have an even number of nodal domains.

Constructed an eigenfunction with three nodal domains on a scaled torus.

Provides insight into the nodal domain structure of Laplacian eigenfunctions.

Abstract

In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus $(\mathbb{R}/2\pi\mathbb{Z})^{2}$, a non-constant eigenfunction has an even number of nodal domains. We then consider the torus $(\mathbb{R}/2\pi\mathbb{Z})\times(\mathbb{R}/2\rho\pi\mathbb{Z})\,$, with $\rho=\frac{1}{\sqrt{3}}\,$, and construct on it an eigenfunction with three nodal domains.