Comment on fractality of quantum mechanical energy spectra

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The fractal properties of the energy spectra of quantum systems are discussed in connection with the paper by Sáiz and Martínez [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 (xₙ = 1/nᵃ, n=1,2,...) are computed analytically and they are shown to be d₀ = 1/(1+a) and, as a consequence, d₀ = 1/3 for the Balmer formula.