Anomalous diffusion in quantum systems with absolutely continuous spectra

For a power-law dispersion relation $\omega (k)\propto k^{\alpha}$, corresponding to an absolutely continuous spectrum of plane waves in spatial dimension $d_{\rm e}$, it is shown that the condition $\alpha\ge 2d_{\rm e}$ leads to the divergence of the time of permanence in the initial region of localization. We define this regime as quasi-absence of diffusion. Another interesting case, defined as quasi-diffusion, turns out to be $d_{\rm e}\le\alpha