Courant-sharp eigenvalues of Neumann 2-rep-tiles
Courant-sharp eigenvalues of Neumann 2-rep-tiles
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Neumann 2-rep-tiles 的 Courant-sharp 特征值
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
David Fajman
中科院分区：
文献类型：
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作者：
R. Band;Michael Bersudsky;David Fajman
我们求出了某些2-rep-tile域上Laplacian的Courant-sharp Neumann特征值。在$$mathbb {R}^{2}$$R2中，我们考虑的域是等腰直角三角形和边比为$$sqrt{2}$$2的矩形（也称为A4纸）。在$$mathbb {R}^{n}$$Rn中，域是推广上述平面矩形的盒子。这些区域的对称性揭示了它们本征函数的一种特殊结构，我们称之为折叠展开。这种结构会影响本征函数的节点集，这反过来又允许导出Courant锐度的必要条件。此外，这些域的本征值被布置为允许节点计数和谱位置之间的比较的晶格。使用这些方法排除了大多数特征值的Courant锐度。此外，这种分析允许估计节点的缺陷，光谱位置和节点计数之间的差异。
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 foldingunfolding. 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.