On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I

Abstract

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 Extended Courant Property. 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.