Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle

@article{Brard2015CourantSharpEF,
  title={Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle},
  author={Pierre H. B{\'e}rard and Bernard Helffer},
  journal={Letters in Mathematical Physics},
  year={2015},
  volume={106},
  pages={1729-1789}
}
We address the question of determining the eigenvalues $${\lambda_{n}}$$λn (listed in nondecreasing order, with multiplicities) for which Courant’s nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $${n}$$n nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of… 
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