Spectrum of harmonic excitations on fractals

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Abstract
The density of states and the nature of the eigenmodes of the vibrating d-dimensional Sierpinski gasket are investigated. The results hold for the tight-binding or any general quadratic Hamiltonian. For d > 1, the spectral measure is shown to be a superposition of two distinct pure point measures of relative weights d/(d + 1) and 1/(d + 1). The eigenmodes associated with each part are explicitly calculated The first part of the spectrum is associated with localized modes, with non zero amplitudes only on a finite number of sites (molecular modes), whereas the second part is associated with a new kind of states : the hierarchical modes. The influence of the boundary conditions is also elucidated as well as the importance of this kind of spectrum in quantum percolation and incommensurate potentials