Courant-sharp eigenvalues of Neumann 2-rep-tiles
| Autor: | Ram Band, David Fajman, Michael Bersudsky |
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| Rok vydání: | 2015 |
| Předmět: |
Physics
010102 general mathematics Lattice (group) Statistical and Nonlinear Physics Eigenfunction Mathematics::Spectral Theory 01 natural sciences 35B05 58C40 58J50 Combinatorics Mathematics - Spectral Theory Position (vector) 0103 physical sciences Isosceles triangle Homogeneous space FOS: Mathematics 010307 mathematical physics Rectangle 0101 mathematics Laplace operator Spectral Theory (math.SP) Mathematical Physics Eigenvalues and eigenvectors |
| DOI: | 10.48550/arxiv.1507.03410 |
| Popis: | 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\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. |
| Databáze: | OpenAIRE |
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