Gravitationally induced inhibitions of dispersion according to a modified Schr\"odinger-Newton equation for a homogeneous-sphere potential

We modify the time dependent Schr"odinger-Newton equation by using a potential for a solid sphere suggested by J"a"askel"ainen (J"a"askel"ainen 2012 Phys. Rev. A 86 052105) as well as a hollow-sphere potential. Compared to our recent paper (Giulini and Gro{\ss}ardt 2011 Class. Quantum Grav. 28 195026) where a single point-particle, i.e. a Coulomb potential, was considered this has been suggested to be a more realistic model for a molecule. Surprisingly, compared to our previous results, inhibitions of dispersion of a Gaussian wave packet occur at even smaller masses for the solid-sphere potential, given that the width of the wave packet is not exceeded by the radius of the sphere.