  We study nonlinear ground states of the Gross-Pitaevskii equation in the
space of one, two and three dimensions with a radially symmetric harmonic
potential. The Thomas-Fermi approximation of ground states on various spatial
scales was recently justified using variational methods. We justify here the
Thomas-Fermi approximation on an uniform spatial scale using the
Painlev\'{e}-II equation. In the space of one dimension, these results allow us
to characterize the distribution of eigenvalues in the point spectrum of the
Schr\"{o}dinger operator associated with the nonlinear ground state.
