Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip
Pierre Bérard
Universite´ Grenoble Alpes and CNRS, FranceBernard Helffer
Université de Nantes, FranceRola Kiwan
American University in Dubai, United Arab Emirates
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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, .... 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.
Cite this article
Pierre Bérard, Bernard Helffer, Rola Kiwan, Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip. Port. Math. 78 (2021), no. 1, pp. 1–41
DOI 10.4171/PM/2059