Spherically-symmetric solutions of the Schrödinger-Newton equations

As part of a programme in which quantum state reduction is understood as a gravitational phenomenon, we consider the Schrödinger-Newton equations. For a single particle, this is a coupled system consisting of the Schrödinger equation for the particle moving in its own gravitational field, where this is generated by its own probability density via the Poisson equation. Restricting to the spherically-symmetric case, we find numerical evidence for a discrete family of solutions, everywhere regular, and with normalizable wavefunctions. The solutions are labelled by the non-negative integers, the nth solution having n zeros in the wavefunction. Furthermore, these are the only globally defined solutions. Analytical support is provided for some of the features found numerically.