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}
}
• Published 28 February 2015
• Mathematics, Physics
• Letters in Mathematical Physics
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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