An ultra-light axion field with mass \sim 10 ^ { -22 } { 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 .