  An ultra-light axion field with mass $\sim 10^{-22}\ {\rm eV}$, also known as
wave or fuzzy dark matter, has been proposed as a component of the dark matter
in the Universe. We study the evolution of the axion dark matter distribution
in the central region of a halo, assuming the mass is dominated by this axion
field, and that gravity is the only important interaction. We calculate the
excited axion states in the spherical gravitational potential generated by the
self-gravitating ground-state, also known as soliton. These excited states are
similar to the states of the hydrogen atom with quantum numbers $(n,l,m)$, here
designating oscillation modes of a classical wave. At fixed $n$, the modes with
highest $l$ have the lowest energy because of the extended mass distribution
generating the potential. We use an approximate analytical treatment to derive
the distribution of mass in these states when a steady-state is reached by
dynamical relaxation, and find that a corona with a mass density profile $\rho
\propto r^{-5/3}$ should be set up around the central soliton, analogous to the
Bahcall-Wolf cusp predicted for the stellar distribution around a central black
hole. The central soliton accretes dark matter from the corona as dynamical
relaxation proceeds and negative orbital energy flows out. This density profile
should remain valid out to the radius where the mass in the corona is
comparable to the mass of the central soliton; further than that, the
gravitational potential depends on the initial distribution of dark matter and
the relaxation time increases rapidly with radius.
