Courant-sharp eigenvalues of Neumann 2-rep-tiles

We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In $$\mathbb {R}^{2}$$R2, the domains we consider are the isosceles right triangle and the rectangle with edge ratio $$\sqrt{2}$$2 (also known as the A4 paper). In $$\mathbb {R}^{n}$$Rn, the domains are boxes which generalize the mentioned planar rectangle. The symmetries of those domains reveal a special structure of their eigenfunctions, which we call folding\unfolding. This structure affects the nodal set of the eigenfunctions, which, in turn, allows to derive necessary conditions for Courant-sharpness. In addition, the eigenvalues of these domains are arranged as a lattice which allows for a comparison between the nodal count and the spectral position. The Courant-sharpness of most eigenvalues is ruled out using those methods. In addition, this analysis allows to estimate the nodal deficiency—the difference between the spectral position and the nodal count.