Vibration Modes of 3n-gaskets and other fractals

Vibration Modes of 3n-gaskets and other fractals -...

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Vibration modes of 3 n -gaskets and other fractals N Bajorin, T Chen, A Dagan, C Emmons, M Hussein, M Khalil, P Mody, B Steinhurst, A Teplyaev E-mail: teplyaev@math.uconn.edu Department of Mathematics, University of Connecticut, Storrs CT 06269 USA Abstract. We rigorously study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric Fnitely ramiFed fractals, which include the Sierpinski gasket and other 3 n -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. AMS classiFcation scheme numbers: 28A80, 31C25, 34B45, 60J45, 94C99 PACS numbers: 02.30.Sa, 02.20.Bb, 02.50.Ga, 02.60.Lj, 02.70.Hm Submitted to: J. Phys. A: Math. Gen. Confidential: not for distribution. Submitted to IOP Publishing for peer review 6 November 2007
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Vibration modes of 3 n -gaskets and other fractals 2 1. Introduction There is a large body of physics and mathematics literature devoted to analysis on fractals. A small sample of it, containing many more references, is [3, 8, 18, 22, 53] and [1, 31, 32, 30, 33, 34, 43, 42, 52, 54, 55, 56, 57, 59, 61, 62]. For example, tools for the numerical analysis of the Sierpi´nski gasket were developed in [12, 24], and fractal antennae were considered in [20, 29, 46, 48]. One of the most recent papers where the random walks on the Sierpi´nski gasket play a role is [11]. In most of these works fractals provide examples of irregular or scale-invariant media. In this paper we rigorously study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar fully symmetric ±nitely rami±ed fractals. Such studies originated in [49, 50], where it was observed that on the Sierpi´nski lattice there are highly localized eigenfunctions corresponding to eigenvalues of very high multiplicity. Later the spectrum of the Laplacian on the Sierpi´nski gasket was studied in detail in [21]. The main purpose of our paper is to develop a theoretical matrix analysis, including analysis of singularities, which allows the exact computation of eigenvalues, eigenfunctions and their multiplicities for a large class of complex fractals. We consider the classical Laplacian on fractals which generalizes the usual one dimensional second derivative, is the generator of the self-similar diffusion process (see [5, 6]), and has possible applications as the quantum Hamiltonian. The latter is especially relevant because it was the original motivation of [49, 50], and because
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Vibration Modes of 3n-gaskets and other fractals -...

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