Some exact results on quantum Newtonian cosmology

We obtain the wave functions associated to the quantum Newtonian universe with a cosmological constant which is described by the Schr"{o}dinger equation and discuss some aspects of its dynamics for all forms of energy density, namely, matter, radiation, vacuum, dark energy, and quintessence. These wave functions of the quantum Newtonian universe are obtained in terms of the Heun’s functions and the respective energy levels are shown. We use these solutions to investigate the expansion of the universe and found that the asymptotic behavior for the scale factor is $R\sim e^t$ for whatever the form of energy density is. We also analyze the behavior of the universe at early stages.