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- Title
Courant-sharp eigenvalues of Neumann 2-rep-tiles.
- Authors
Band, Ram; Bersudsky, Michael; Fajman, David
- 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\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.
- Subjects
VON Neumann algebras; EIGENVALUES; PLANAR graphs; MATHEMATICAL models; EIGENFUNCTIONS
- Publication
Letters in Mathematical Physics, 2017, Vol 107, Issue 5, p821
- ISSN
0377-9017
- Publication type
Article
- DOI
10.1007/s11005-016-0926-7