Schrödinger-Newton solitons with axial symmetry

We solve the Schrödinger-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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>L</a:mi><a:mi>z</a:mi></a:msub></a:math>. 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 Schrödinger-Newton system.