  We investigate the way in which the Gribov problem is manifested in the BRST
quantization of simple quantum mechanical models by comparing models with and
without a Gribov problem. We show that the hermiticity and nilpotency of the
BRST charge together with the Batalin-Vilkovisky theorem yield non-trivial
supplementary conditions on gauge fixing fermions. If the gauge fixing fermion
satisfies the supplementary conditions, the BRST physical states form a space
isomorphic to the Dirac space, and the BRST formal path integral does not
suffer from the Gribov problem. The conventional gauge fixing fermion, that
gives rise to the Faddeev-Popov integral, fails to satisfy the supplementary
conditions due to the Gribov problem. Alternatively, enforcing the conventional
gauge fixing fermion, these supplementary conditions imply restrictions on the
BRST physical states for which the Batalin-Vilkovisky theorem holds. We find
that these BRST physical states are not isomorphic the Dirac states. This can
be interpreted as a violation of the Batalin-Vilkovisky theorem on the space of
Dirac states and implies a breakdown of unitarity and a general dependence of
physical quantities on the gauge condition.
