Vibration modes of 3n-gaskets and other fractals
Vibration modes of 3n-gaskets and other fractals
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DOI：
10.1088/1751-8113/41/1/015101
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发表时间：
2008-01-11
影响因子：
2.1
通讯作者：
Teplyaev, A.
中科院分区：
文献类型：
--
作者：
Bajorin, N.;Chen, T.;Teplyaev, A.
我们严格地研究了自相似的对称分形的特征值和特征函数（振动模式），其中包括Sierpinski垫片和其他3 n-垫片。我们认为经典的拉普拉斯分形推广通常的一维二阶导数，是自相似扩散过程的生成器，并有可能作为量子哈密顿的应用。我们开发了一个理论矩阵分析，包括奇异性分析，这使我们能够准确地计算本征值，本征函数及其多重性。我们支持我们的理论分析的符号和数值计算。特别是，我们的分析允许计算分形上的谱zeta函数和本征值的极限分布（即，积分态密度）。我们认为这样的例子作为3级Sierpinski垫片，分形3-树，和钻石分形。
We rigorously study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals, which include the Sierpinski gasket and other 3n-gaskets. We consider the classical Laplacian on fractals which generalizes the usual one-dimensional second derivative, is the generator of the self-similar diffusion process, and has possible applications as the quantum Hamiltonian. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations. Our analysis, in particular, allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues (i.e., integrated density of states). We consider such examples as the level-3 Sierpinski gasket, a fractal 3-tree, and the diamond fractal.