On Courant’s Nodal Domain Property for Linear Combinations of Eigenfunctions Part II

According to Courant's theorem, an eigenfunction as\-sociated 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 simple and explicit examples for which the extended Courant property is false: convex domains in $\R^n$ (hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on the round sphere $\mathbb{S}^2$, and on a flat torus $\mathbb{T}^2$.