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

Pierre Bérard (1) , Bernard Helffer (2) , Rola Kiwan (3)

1 IF - Institut Fourier
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 American University in Dubai

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, Möbius 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.

Mots clés: Spectral theory; Courant theorem; Laplacian; Nodal sets; Klein bottle; Cylinder

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Géométrie différentielle [math.DG]

Pierre Bérard, Bernard Helffer, Rola Kiwan. Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders. 2020. ⟨hal-02937338v3⟩