  Dark matter (DM) self-interactions have important implications for the
formation and evolution of structure, from dwarf galaxies to clusters of
galaxies. We study the dynamics of self-interacting DM via a light mediator,
focusing on the quantum resonant regime where the scattering cross section has
a non-trivial velocity dependence. While there are long-standing indications
that observations of small scale structure in the Universe are not in accord
with the predictions of collisionless DM, theoretical study and simulations of
DM self-interactions have focused on parameter regimes with simple analytic
solutions for the scattering cross section, with constant or classical velocity
(and no angular) dependence. We devise a method that allows us to explore the
velocity and angular dependence of self-scattering more broadly, in the
strongly-coupled resonant and classical regimes where many partial modes are
necessary for the achieving the result. We map out the entire parameter space
of DM self-interactions --- and implications for structure observations --- as
a function of the coupling and the DM and mediator masses. We derive a new
analytic formula for describing resonant s-wave scattering. Finally, we show
that DM self-interactions can be correlated with observations of Sommerfeld
enhancements in DM annihilation through indirect detection experiments.
