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The Hamiltonian of any quantum cosmological model must obey some conditionsin order to be self-adjoint, or to admit self-adjoint extensions. We willconsider in this paper the self-adjoint character of the Hamiltonian resultingfrom the Brans-Dicke theory, both in Einstein and Jordan frames. In both caseswe find the wave-function solutions in order to obtain concrete predictions forthe evolution of the Universe. We show that the specific cosmological scenariosobtained depend very weakly on the conditions of having a self-adjointoperator. The problem of equivalence between Einstein and Jordan frames isconsidered, and it is shown that this equivalence implies a specific orderingparameter, as well as a particular choice of the physical variables.

Cosmological scenarios in quantum cosmology of Brans-Dicke theory of gravity on Einstein and Jordan frames