  We rewrite the Klein-Gordon (KG) equation in an arbitrary space-time
transforming it into a generalized Schr\"odinger equation. Then we take the
weak field limit and show that this equation has some differences with the
traditional Schr\"odinger equation plus a gravitational field. Thus, this
procedure shows that the Schr\"odinger equation derived in a covariant manner
is different from the traditional one. With this new Schr\"odinger equation, we
study the KG equation in a Newtonian space-time to describe the behavior of a
scalar particle in an inertial system. We give some examples where it is
possible to study the energy levels, effective potential and the wave function
of the systems, these results contain the gravitational effects due to the
curvature of space-time. We show that it is possible to verify experimentally
these effects in a laboratory using non-inertial reference frames.
