Courant-sharp eigenvalues of Neumann 2-rep-tiles

TL;DR

This paper characterizes Courant-sharp Neumann eigenvalues for specific 2D and higher-dimensional domains with symmetries, revealing their structure and providing criteria to identify which eigenvalues are Courant-sharp.

Contribution

It introduces a novel folding/unfolding eigenfunction structure and uses lattice arrangements to analyze Courant-sharpness in certain symmetric domains.

Findings

Most eigenvalues are not Courant-sharp due to symmetry constraints.

Eigenvalues form a lattice structure enabling spectral and nodal count comparison.

The method estimates the nodal deficiency for these eigenvalues.

Abstract

We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In $\R^{2}$ the domains we consider are the isosceles right triangle and the rectangle with edge ratio $\sqrt{2}$ (also known as the A4 paper). In $\R^{n}$ the domains are boxes which generalize the mentioned planar rectangle. The symmetries of those domains reveal a special structure of their eigenfunctions, which we call folding\textbackslash{}unfolding. This structure affects the nodal set of the eigenfunctions, which in turn allows to derive necessary conditions for Courant-sharpness. In addition, the eigenvalues of these domains are arranged as a lattice which allows for a comparison between the nodal count and the spectral position. The Courant-sharpness of most eigenvalues is ruled out using those methods. In addition, this analysis allows to estimate the nodal deficiency - the difference between the spectral position and the nodal count.