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The question of determining for which eigenvalues there exists aneigenfunction which has the same number of nodal domains as the label of theassociated eigenvalue (Courant-sharp property) was motivated by the analysis ofminimal spectral partitions. In previous works, many examples have beenanalyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots .A natural toy model for further investigations is the M\"obius strip, anon-orientable surface with Euler characteristic $0$, and particularly the"square" M\"obius strip whose eigenvalues have higher multiplicities. In thiscase, we prove that the only Courant-sharp Dirichlet eigenvalues are the firstand the second, and we exhibit peculiar nodal patterns.

Courant-sharp property for Dirichlet eigenfunctions on the M\"obius strip