Courant-sharp eigenvalues of Neumann 2-rep-tiles

Abstract

We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In ${\mathbb{R}}^2$ the domains we consider are the isosceles right triangle and the rectangle with edge ratio $\sqrt{2}$ (also known as the A4 paper). In ${\mathbb{R}}^n$ 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\textbackslash{}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.

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Cited by in corpus (2)

• Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
• Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders