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.

中文翻译：

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莫比乌斯带上狄利克雷本征函数的 Courant-sharp 性质

确定哪些特征值存在一个特征函数的问题，该特征函数具有与相关特征值（Courant-sharp 特性）的标签相同数量的节点域，其动机是对最小谱分区的分析。在以前的工作中，已经分析了许多对应于正方形、矩形、圆盘、三角形、圆环等的例子。 进一步研究的自然玩具模型是莫比乌斯带，一种具有欧拉特征为 0 的不可定向表面，特别是''square'' 莫比乌斯带，其特征值具有更高的多重性。在这种情况下，我们证明唯一的 Courant-sharp Dirichlet 特征值是第一个和第二个，并且我们表现出特殊的节点模式。