Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

Pierre Bérard (1) , Bernard Helffer (2, 3)

1 IF - Institut Fourier
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We address the question of determining the eigenvalues $\lambda_n$ (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 $n$ 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 the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.

Mots clés: Nodal lines; Courant theorem.; Nodal domains

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]

Pierre Bérard, Bernard Helffer. Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle. Letters in Mathematical Physics, 2016, 106 (12), pp.1729--1789. ⟨hal-01120958v3⟩