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.
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