We study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals which includes 3n-gaskets. We consider such examples as the Sierpinski gasket, a non-p.c.f. analog of the Sierpinski gasket, the level-3 Sierpinski gasket, a fractal 3-tree, the hexagasket, and one dimensional fractals. 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.

We show how to calculate the spectrum of the Laplacian operator on fully symmetric, finitely ramified fractals. We consider well-known examples, such as the unit interval and the Sierpiński gasket, and much more complicated ones, such as the hexagasket and a non-post critically finite self-similar fractal. We emphasize the low computational demands of our method. As a conclusion, we give exact formulas for the limiting distribution of eigenvalues (the integrated density of states), which is a purely atomic measure (except in the classical case of the interval), with atoms accumulating to the Julia set of a rational function. This paper is the continuation of the work published by the same authors in Ref. 1.

We analyse the spectrum of a self-adjoint operator on a Laakso space using the projective limit construction originally given by Barlow and Evans. We will use the hierarchical cell structure induced by the choice of approximating quantum graphs to calculate the spectrum with multiplicities. We also extend the method for using the hierarchical cell structure to more general projective limits beyond Laakso spaces.

It is known that Laplacian operators on many fractals have gaps in their spectra. This fact precludes the possibility that a Weyl-type ratio can have a limit and is also a key ingredient in proving that the Fourier series on such fractals can have better convergence results than in the classical setting. In this paper we prove that the existence of gaps is equivalent to the total disconnectedness of the Julia set of the spectral decimation function for the class of fully symmetric p.c.f. fractals, and for self-similar fully symmetric finitely ramified fractals with regular harmonic structure. We also formulate conjectures related to geometry of finitely ramified fractals with spectral gaps, to complex spectral dimensions, and to convergence of Fourier series on such fractals. 2000 Mathematics Subject Classification. Primary 28A80; Secondary 35P05, 35J05. Theorem 2. Under the conditions of the theorem above, there exist gaps in σ(∆) if and only if J R is totally disconnected.This result can be generalized for a larger class of finitely ramified symmetric fractals with weights, and with more elaborate combinatorial structure, but doing so would require dealing with many technical details and auxiliary results which are not available in the existing literature. The main ideas and techniques behind these results come from [3,32,36,46]. One of the newest examples of fractals we consider can be found in [16]. We also give results about the location of gaps which generalize the results in [46].

We consider the spaces introduced by Laakso in 2000 and, building on the work of Barlow, Bass, Kumagai, and Teplyaev, prove the existence and uniqueness of a local symmetry invariant diffusion via heat kernel estimates.