  In this survey, we review recent results concerning the canonical dispersive
flow $e^{itH}$ led by a Schr\"odinger Hamiltonian $H$. We study, in particular,
how the time decay of space $L^p$-norms depends on the frequency localization
of the initial datum with respect to the some suitable spherical expansion. A
quite complete description of the phenomenon is given in terms of the
eigenvalues and eigenfunctions of the restriction of $H$ to the unit sphere,
and a comparison with some uncertainty inequality is presented.
