Field work and dispersion relations of excitations on fractals

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Abstract
An approach to finding the dispersion law of excitations on fractals for an arbitrary interaction is developed. It is based on the scale invariance of the probe-field work and allows one to express exponents of the dispersion law in terms of the spectral and Hausdorff dimensions. The expressions obtained for the dispersion-law exponent are different for vibrational (Goldstone-type) and dipolar (non-Goldstone-type) excitations, in agreement with previous results of Alexander and Orbach and those of the authors of the present paper.