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An ultra-light axion field with mass $\sim 10^{-22}\ {\rm eV}$, also known aswave or fuzzy dark matter, has been proposed as a component of the dark matterin the Universe. We study the evolution of the axion dark matter distributionin the central region of a halo, assuming the mass is dominated by this axionfield, and that gravity is the only important interaction. We calculate theexcited axion states in the spherical gravitational potential generated by theself-gravitating ground-state, also known as soliton. These excited states aresimilar to the states of the hydrogen atom with quantum numbers $(n,l,m)$, heredesignating oscillation modes of a classical wave. At fixed $n$, the modes withhighest $l$ have the lowest energy because of the extended mass distributiongenerating the potential. We use an approximate analytical treatment to derivethe distribution of mass in these states when a steady-state is reached bydynamical 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 theBahcall-Wolf cusp predicted for the stellar distribution around a central blackhole. The central soliton accretes dark matter from the corona as dynamicalrelaxation proceeds and negative orbital energy flows out. This density profileshould remain valid out to the radius where the mass in the corona iscomparable to the mass of the central soliton; further than that, thegravitational potential depends on the initial distribution of dark matter andthe relaxation time increases rapidly with radius.

Bosonic dark matter halos: excited states and relaxation in the potential of the ground state