  We construct spherically symmetric equilibrium solutions of the
Schr\"odinger-Poisson (SP) system of equations with a core-tail structure that
could serve as models of Fuzzy Dark Matter (FDM) halos. The core is assumed to
be a solitonic ground state equilibrium configuration of the SP equations, and
the tail is integrated from a transition radius onwards. The total mass of the
system parametrizes the family of solutions and constrains the tail density
profile. The tail has a radial velocity profile, whereas the core is
stationary. We investigate the evolution of these equilibrium configurations
and find that the tail initially perturbs the core, and consequently, the whole
solution oscillates around a virialized solution that we call 'relaxed', whose
average also has a core-tail structure. We measure the departure of the relaxed
configuration from the equilibrium solution in order to estimate the utility of
the latter. We also find that the core-halo scaling relation of equilibrium
configurations has an exponent $\alpha=1/3$, whereas relaxed configurations
exhibit a scaling with $\alpha=0.54$.
