On Courant's nodal domain property for linear combinations of eigenfunctions

Pierre Bérard (1) , Bernard Helffer (2)

1 IF - Institut Fourier
2 LMJL - Laboratoire de Mathématiques Jean Leray

We revisit Courant's nodal domain property for linear combinations of eigenfunctions, and propose new, simple and explicit counterexamples for domains in $\mathbb R^2$, $\mathbb S^2$, $\mathbb T^2$, and $\mathbb R^3$, including convex domains.

Mots clés: Eigenfunction; Nodal domain; Courant nodal domain theorem

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

Pierre Bérard, Bernard Helffer. On Courant's nodal domain property for linear combinations of eigenfunctions. 2017. ⟨hal-01519629v3⟩