Vibration modes of
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n
-gaskets and other fractals
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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