  We study the process of dispersion of low-regularity solutions to the
Schr\"odinger equation using fractional weights (observables). We give another
proof of the uncertainty principle for fractional weights and use it to get a
lower bound for the concentration of mass. We consider also the evolution when
the initial datum is the Dirac comb in $\mathbb{R}$. In this case we find
fluctuations that concentrate at rational times and that resemble a realization
of a L\'evy process. Furthermore, the evolution exhibits multifractality.
