The Courant‐Herrmann conjecture

The Courant‐Herrmann Conjecture (CHC) concerns the sign properties of combinations of the Dirichlet eigenfunctions of elliptic pde's, the most important of which is the Helmholtz equation $\Delta u + \lambda \rho u = 0$ for \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$D \in \mathbb{R}^N$\end{document} . If the eigenvalues are ordered increasingly, CHC states that the nodal set of a combination $v = \sum_{i=1}^nc_iu_i$ of the first $n$ eigenfunctions, divides $D$ into no more than $n$ sign domains in which $v$ has one sign. The conjecture is classically known to hold for $N=1$ , we conjecture that it is true for rectangular boxes in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathbb{R}^N (N\geq2)$\end{document} , but show that it is false in general.