**Abstract: ** We will present some recent results concerning with time decay
properties of scaling-invariant electromagnetic Schrödinger flows. A
representation formula determines the relation between the spectral
properties of the spherical hamiltonian and solutions to the flow. We prove
sharp time decay estimates for a large family of operators; moreover, we
prove some polynomially improved time-decay in weighted spaces, once the
previous estimates hold. This last phenomenon is quite interesting, because
show new estimates which do not hold for the free flow.
The results are obtained in collaboration with V. Felli (Milano - Bicocca),
M. Fontelos (ICMAT - Madrid), G. Grillo (Milano - Politecnico), H. Kovarik
(Univ. Brescia), and A. Primo (UAM - Madrid).