Gravitational atoms: general framework for the construction of multistate axially symmetric solutions of the Schr\"odinger-Poisson system

We present a general strategy to solve the stationary Schr\"odinger-Poisson (SP) system of equations for multistates with axial symmetry. The approach allows us to obtain the well known single and multistate solutions with spherical symmetry, Newtonian multistate $\ell-$boson stars and axially symmetric multistate configurations. For each case we construct particular examples that illustrate the method, whose stability properties are studied by numerically solving the time-dependent SP system. Among the stable configurations there are the mixed-two-state configurations including spherical and dipolar components, which might have an important value as potential anisotropic dark matter halos in the context of ultralight bosonic dark matter scenarios. This is the reason why we also present a possible process of formation of these mixed-two-state configurations that could open the door to the exploration of more general multistate structure formation scenarios.