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

Pierre Bérard (1) , Bernard Helffer (2)

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

According to Courant's theorem, an eigenfunction associated with the $n$-th eigenvalue $\lambda_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to $\lambda_n$. We call this assertion the \emph{Extended Courant Property}.\smallskip In this paper, we propose new, simple and explicit examples for which the extended Courant property is false: convex domains in $\mathbb{R}^n$ (hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on the round sphere $\mathbb{S}^2$, and on a flat torus $\mathbb{T}^2$. We also give numerical evidence that the extended Courant property is false for the equilateral triangle with rounded corners, and for the regular hexagon.

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

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, Part I. 2018. ⟨hal-01519629v4⟩