Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip

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,. .. . A natural toy model for further investigations is the Möbius strip, a non-orientable surface with Euler characteristic 0, and particularly the "square" Möbius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns.

Mots clés: Laplacian; Nodal sets; Courant theorem; Spectral theory; Möbius strip.

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

Pierre Bérard, Bernard Helffer, Rola Kiwan. Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip. Portugaliae Mathematica, 2021, 78 (1), pp.1--41. ⟨10.4171/pm/2059⟩. ⟨hal-02569100v3⟩