Quantum Cosmological Scenarios of Brans-Dicke Gravity in Einstein and Jordan Frames

The Hamiltonian of any quantum cosmological model must obey some conditions in order to be self-adjoint or to admit self-adjoint extensions. We consider 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 the 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.