Spherical Schrödinger Hamiltonians: Spectral Analysis and Time Decay

In this survey, we review recent results concerning the canonical dispersive flow $e^{itH}$ led by a Schrödinger 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.

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