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

TL;DR

This paper establishes the existence and uniqueness of stationary spherically symmetric solutions for the Schrödinger-Newton model across various dimensions using an ODE approach, highlighting the critical dimension d=6.

Contribution

It provides a rigorous proof of stationary solutions' existence and uniqueness for the Schrödinger-Newton model in any dimension, connecting previous results and identifying critical dimensions.

Findings

Existence and uniqueness of solutions in all dimensions

d=6 is a critical dimension for finite energy solutions

Reduction to Lane-Emden equation for d≥6

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 $d=3$.