  We study the viability of a complex scalar field $\chi$ with self-interacting
potential $ V = m^\chi_0/2 \, |\chi|^2 + h \, |\chi|^4$ as dark matter. The
scalar field is produced at reheating through the decay of the inflaton field
and then, due to the self-interaction, a Bose-Einstein condensate of $\chi$
particles forms. The condensate represents dark matter in that model. We
analyze the cosmological evolution of the model, stressing how, due to the
presence of the self-interaction, the model naturally admits dark matter
domination at late times, thus avoiding any fine tuning on the energy density
of the scalar field at early times. Finally we give a lower bound for the size
of dark matter halos at present time and we show that our model is compatible
with dark matter halos greater than $0.1 \, Kpc$ and with BBN and CMB bounds on
the effective number of extra neutrinos $\Delta_\nu^{eff}$. Therefore, the
model is viable and for $h\simeq 10^{-4}-10^{-12}$ one obtains a mass $m^\chi
\simeq m^{\chi}_0 \simeq 1-10^{-2} \, eV$ for dark matter particles from
radiation-matter equality epoch to present time, but at temperatures $T_\gamma
\gg 10 \, eV$, where $T_\gamma$ is the photons temperature, thermal corrections
to $m^\chi_0$ due to the self-coupling $h$ are dominant.
