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
Email:
[email protected]
Department of Mathematics, University of Connecticut, Storrs CT 06269 USA
Abstract.
We
rigorously
study
eigenvalues
and
eigenfunctions
(vibration
modes) on the class of selfsimilar symmetric finitely ramified 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
selfsimilar
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 level3 Sierpinski gasket, a fractal 3tree, and the
diamond fractal.
AMS classification 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 scaleinvariant media.
In this paper we rigorously study eigenvalues and eigenfunctions (vibration
modes) on the class of selfsimilar fully symmetric finitely ramified 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 selfsimilar 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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 Spring '08
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 Eigenvalue, eigenvector and eigenspace, ... ..., Vn, Fractal, M0, Selfsimilarity

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