  In this paper we show that spherically symmetric BEC dark matter halos, with
the $\sin r/r$ density profile, that accurately fit galactic rotation curves
and represent a potential solution to the cusp-core problem are unstable. We do
this by introducing back the density profiles into the fully time-dependent
Gross-Pitaevskii-Poisson system of equations. Using numerical methods to track
the evolution of the system, we found that these galactic halos lose mass at an
approximate rate of half of its mass in a time scale of dozens of Myr. We
consider this time scale is enough as to consider these halos are unstable and
unlikely to be formed. We provide some arguments to show that this behavior is
general and discuss some other drawbacks of the model that restrict its
viability.
