quantum mechanics - Can we apply Schrödinger equation in Newton gravitational potential and derive the deterministic Newton's gravitation as a special case of it

We know the solutions for wave functions of a an atom of hydrogen, and the energy values as given by spectral analysis of radiation emitted by hydrogen confirms the possible energy states as predicted by the Schrödinger equation.

My question: in the above case the potential used is Coulomb potential which is mathematically the same as gravitational potential. Hence can it be created a similar hydrogen equation but with point masses (instead of point charges) in Newton's gravitational field (instead of charged particles like electrons in Coulomb field)?

And in this equation, making some appropriate parameter going to 0 or ∞, can we show that the deterministic Newton's law of gravitation emerges out of it?

Last but not the least, if not for the classical gravitation, can we do this for relativistic gravitation (GR)?

I guess this is the main theme of the research in quantum gravity. Please correct me if I am wrong.