Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip

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