Stationary solutions of the Schr\"{o}dinger-Newton model - An ODE approach

Abstract

We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schr\"{o}dinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second order differential equations. It turns out that $d=6$ is critical for the existence of finite energy solutions and the equations for positive spherically symmetric solutions reduce to a Lane-Emden equation for all $d\geq 6$. Our result implies in particular the existence of stationary solutions for two-dimensional self-gravitating particles and closes the gap between the variational proofs in $d=1$ and $d=3$.