  Recent perturbative studies have shown the existence of long-lived,
quasi-stationary configurations of scalar fields around black holes. In
particular, such configurations have been found to survive for cosmological
timescales, which is a requirement for viable dark matter halo models in
galaxies based on such type of structures. In this paper we perform a series of
numerical relativity simulations of dynamical non-rotating black holes
surrounded by self-gravitating scalar fields. We solve numerically the coupled
system of equations formed by the Einstein and the Klein-Gordon equations under
the assumption of spherical symmetry using spherical coordinates. Our results
confirm the existence of oscillating, long-lived, self-gravitating scalar
fields configurations around non-rotating black holes in highly dynamical
spacetimes with a rich scalar field environment. Our numerical simulations are
long-term stable and allow for the extraction of the resonant frequencies to
make a direct comparison with results obtained in the linearized regime. A
byproduct of our simulations is the existence of a degeneracy in plausible
long-lived solutions of Einstein equations that would induce the same motion of
test particles, either with or without the existence of quasi-bound states.
