  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.
