Complex spectral dimensionality on fractal structures

Abstract

Fractal structures have been associated with scaling properties of many physical sys- tems. On the basis of a solvable model, it is asserted that the above analogy leads in a natural way to complex indices, for instance, in the power law describing the density of states. The corresponding oscillations are, in fact, necessary to reproduce the singular part of the spectrum, and are governed by the nearest complex singularities of the Mellin transform of the spectral density. Observation of such oscillations in actual systems might allow the effective fractal dimension of the involved struc- ture to be determined.