Abstract
The self-similar structure of the dynamical matrix of hierarchical chains of harmonic oscillators is explored to obtain perturbative derivation of the eigensolutions. We focus in particular on the eigenvalue spectrum, determining explicit expressions for the eigenfrequencies, and finding a Cantor set with zero Lebesgue measure. We then bring the calculation to the particular case of a linear chain with equal masses connected by hierarchically distributed springs of power-law type. The spectral dimension, explicitly derived for this system, shows a nonuniversal behavior, being strongly dependent on the details of the coupling. The spectral dimension is then used for computing the lattice specific heat while singularities in the spectrum are analyzed both within the local and global point of view. All results are compared with the numerically determined exact eigenvalues.
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