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

Spherical Schr\"odinger Hamiltonians: Spectral Analysis and Time Decay