Abstract
The vibrational properties of one-dimensional hierarchical systems are investigated and results are obtained for both their eigenvalues and eigenvectors. Two cases are considered, the first one with a hierarchy of spring constants and the latter with a hierarchy in the masses. In both cases the eigenspectrum is found to be a zero-measure, two-scale Cantor set with a fractal dimension between 0 and 1. The scaling properties of the spectra are calculated using renormalization group techniques and are verified by extensive numerical work. The low-frequency density of states and low-temperature specific heat are calculated and a singularity is found in the scaling behavior. The eigenvectors are found to be either extended or critical and self-similar. A transfer matrix formalism is introduced to calculate the scaling properties of the envelope of the critical eigenvectors. Furthermore, a connection is established between the hierarchical vibration and diffusion problems, as well as to the same problems in random systems, thus showing the universality of the observed features.
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Keirstead, W.P., Ceccatto, H.A. & Huberman, B.A. Vibrational properties of hierarchical systems. J Stat Phys 53, 733–757 (1988). https://doi.org/10.1007/BF01014223
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DOI: https://doi.org/10.1007/BF01014223