Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

TL;DR

This paper characterizes all Courant-sharp eigenvalues for the equilateral torus and triangle, showing only the first few eigenvalues are Courant-sharp, and extends some results to related triangles.

Contribution

It provides a complete classification of Courant-sharp eigenvalues for the equilateral torus and triangle, extending previous partial results.

Findings

Only the first and second eigenvalues are Courant-sharp for the equilateral torus.

Only the first, second, and fourth eigenvalues are Courant-sharp for the equilateral triangle.

Sketches similar results for related triangles.

Abstract

We address the question of determining the eigenvalues $\lambda\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.