Résumé
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 ́-II equation. In the space of one dimension, these results allow us to characterize e the distribution of eigenvalues in the point spectrum of the Schr ̈dinger operator associated with the nonlinear ground state.