  We investigate the oscillations of neutron stars using a purely Newtonian
approach and three other pseudo-Newtonian formulations. Our work is motivated
by the fact that pseudo-Newtonian formulations are commonly used in
core-collapse supernova (CCSN) simulations. We derive and solve numerically the
radial and nonradial perturbation equations for neutron star oscillations using
different combinations of modified Newtonian hydrodynamics equations and
gravitational potentials. We pay special attention to the formulation proposed
recently by Zha et al. [Phys. Rev. Lett. 125, 051102 (2020)] that implements
the standard Case A effective potential in CCSN simulations with an additional
lapse-function correction to the hydrodynamics equations. We find that this
"Case A+lapse" formulation can typically approximate the frequency of the
fundamental radial mode of a $1.4 M_\odot$ neutron star computed in general
relativity to about a few tens of percent for our chosen EOS models. For the
nonradial quadrupolar $f$ mode, which is expected to contribute strongly to the
gravitational waves emitted from a protoneutron star, the Case A+lapse
formulation performs much better and can approximate the $f$ mode frequency to
within about a few percent even for the maximum-mass configuration in general
relativity.
