Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

TL;DR

This paper characterizes the Courant-sharp eigenvalues for flat Klein bottles and cylinders, showing only the first two eigenvalues are Courant-sharp in these geometries, extending previous spectral analysis results.

Contribution

It identifies the only Courant-sharp eigenvalues for flat Klein bottles associated with the square torus and for certain flat cylinders, providing new insights into spectral properties of these surfaces.

Findings

Only the first and second eigenvalues are Courant-sharp for the flat Klein bottle with square torus.

Only the first and second eigenvalues are Courant-sharp for the flat cylinders considered.

Extends the classification of Courant-sharp eigenvalues to non-orientable surfaces and cylinders.

Abstract

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, M\"obius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,\pi) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues.