  We consider the question of scattering for the boson star equation in three
space dimensions. This is a semi-relativistic Klein-Gordon equation with a
cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by
exploiting a special null structure present in the equation, and a refined
asymptotic analysis performed in Fourier space, to obtain global solutions
evolving from small and localized Cauchy data. We describe the behavior at
infinity of such solutions by identifying a suitable nonlinear asymptotic
correction to scattering. As a byproduct of the weighted energy estimates
alone, we also obtain global existence and (linear) scattering for solutions of
semi-relativistic Hartree equations with potentials decaying faster than
Coulomb.
