  The Hamiltonian of any quantum cosmological model must obey some conditions
in order to be self-adjoint, or to admit self-adjoint extensions. We will
consider in this paper the self-adjoint character of the Hamiltonian resulting
from the Brans-Dicke theory, both in Einstein and Jordan frames. In both cases
we find the wave-function solutions in order to obtain concrete predictions for
the evolution of the Universe. We show that the specific cosmological scenarios
obtained depend very weakly on the conditions of having a self-adjoint
operator. The problem of equivalence between Einstein and Jordan frames is
considered, and it is shown that this equivalence implies a specific ordering
parameter, as well as a particular choice of the physical variables.
