Comment on fractality of quantum mechanical energy spectra

The fractal properties of the energy spectra of quantum systems are discussed in connection with the paper by S'aiz and Mart'inez [Phys. Rev. E 54, 2431 (1996)]. It is shown that for discrete energy levels the Hausdorff–Basicovitch dimension is zero and differs from the Renyi scaling exponents computed by the standard box counting algorithm. The Renyi exponents for the inverse power series data sets (x_n = 1/n^a, n=1,2,…) are computed analytically and they are shown to be d_0 = 1/(1+a) and, as a consequence, d_0 = 1/3 for the Balmer formula.