  There is increasing numerical evidence that scalar fields can form long-lived
quasi-bound states around black holes. Recent perturbative and numerical
relativity calculations have provided further confirmation in a variety of
physical systems, including both static and accreting black holes, and
collapsing fermionic stars. In this work we investigate this issue yet again in
the context of gravitationally unstable boson stars leading to black hole
formation. We build a large sample of spherically symmetric initial models,
both stable and unstable, incorporating a self-interaction potential with a
quartic term. The three different outcomes of unstable models, namely migration
to the stable branch, total dispersion, and collapse to a black hole, are also
present for self-interacting boson stars. Our simulations show that for
black-hole-forming models, a scalar-field remnant is found outside the
black-hole horizon, oscillating at a different frequency than that of the
original boson star. This result is in good agreement with recent spherically
symmetric simulations of unstable Proca stars collapsing to black holes.
