Courant-sharp eigenvalues of Neumann 2-rep-tiles
arXiv:1507.03410 · doi:10.1007/s11005-016-0926-7
Abstract
We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In the domains we consider are the isosceles right triangle and the rectangle with edge ratio (also known as the A4 paper). In 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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