  Self-gravitating quantum matter may exist in a wide range of cosmological and
astrophysical settings from the very early universe through to present-day
boson stars. Such quantum matter arises in a number of different theories,
including the Peccei-Quinn axion and UltraLight (ULDM) or Fuzzy (FDM) dark
matter scenarios. We consider the dynamical evolution of perturbations to the
spherically symmetric soliton, the ground state solution to the
Schr\"{o}dinger-Poisson system common to all these scenarios. We construct the
eigenstates of the Schr\"{o}dinger equation, holding the gravitational
potential fixed to its ground state value. We see that the eigenstates
qualitatively capture the properties seen in full ULDM simulations, including
the soliton "breathing" mode, the random walk of the soliton center, and
quadrupolar distortions of the soliton. We then show that the time-evolution of
the gravitational potential and its impact on the perturbations can be well
described within the framework of time-dependent perturbation theory. Applying
our formalism to a synthetic ULDM halo reveals considerable mixing of
eigenstates, even though the overall density profile is relatively stable. Our
results provide a new analytic approach to understanding the evolution of these
systems as well as possibilities for faster approximate simulations.
