On approximations of the Schr\"odinger-Newton equation by harmonic potentials

The evolution of the, initially Gaussian, centre-of-mass wave-function for a homogeneous, spherical particle according to the Schr\odinger-Newton equation can be approximated by a harmonic potential, if the wave-function is narrow compared to the size of the particle. Here, the validity of a previously proposed approximation of the Schr\odinger-Newton equation is studied, where this is extended beyond the regime of narrow wave-functions, replacing the coupling constant of the harmonic potential by a function of the wave-function width. It turns out that such an extension beyond the narrow wave-function regime is not a good approximation for the self-gravitational evolution according to the Schr\odinger-Newton equation.