Effective fractal dimension of energy levels

Abstract

Energy-level systems can be seen as sets of a series of discrete points that exhibit fractal properties. To characterize these fractals, we calculated the effective fractal dimension, which behaves as a function of the measuring scale. The computerized box-counting method was employed to obtain the fractal dimension that is irrelevant to the level number. This feature allows a physically meaningful connection that can be established between the fractal and statistics. The procedure was applied to the Poisson- and Wigner-distributed numbers and to the vibrational levels of SO2.