Schrodinger-Newton solitons with axial symmetry

TL;DR

This paper finds axially symmetric solutions to the Schrödinger-Newton system, allowing for non-spherical gravitational potentials and analyzing the partial-wave structure of these self-consistent solutions.

Contribution

It introduces solutions with axial symmetry in the Schrödinger-Newton system without assuming spherical symmetry, expanding understanding of gravitationally bound quantum states.

Findings

Solutions exhibit axial symmetry in gravitational potential and wavefunction.

Partial-wave analysis reveals contributions from multiple angular momentum states.

Provides a framework for studying non-spherical quantum gravitational systems.

Abstract

We solve the Schr\"odinger-Newton problem of Newtonian gravity coupled to a nonrelativistic scalar particle for solutions with axial symmetry. The gravitational potential is driven by a mass density assumed to be proportional to the probability density of the scalar. Unlike related calculations for condensates of ultralight dark matter or boson stars, no assumption of spherical symmetry is made for the effective gravitational potential. Instead, the potential has only axial symmetry, consistent with the axial symmetry of the particle's probability density for eigenstates of $L_z$. With total angular momentum no longer a good quantum number, there are in general contributions from a range of partial waves. This permits us to study the partial-wave content of self-consistent solutions of the Schr\"odinger-Newton system.