Density profile of multi-state fuzzy dark matter

Equations of motion for excited states of weakly self-interacting bosons forming fuzzy dark matter are solved using the WKB approximation. The contribution of self-interactions are neglected in the equations of motion. Wave functions of excited states are expressed in terms of a yet undetermined gravitational potential. At equilibrium, the contributions of states to the density distribution are summed using Bose-Einstein statistics. Combined with the Poisson equation, a differential equation is obtained for the gravitational potential, which has physically acceptable solutions only if the energy spectrum of excited states has a finite gap, corresponding to a finite virial radius. Such a gap could be created by decay processes, in first order perturbation of the self-interaction potential. The obtained density profile is found to be similar to the Burkert profile.