C. Gallo and D. Pelinovsky

On the Thomas-Fermi ground state in a harmonic potential

Asymptotic Analysis 73, 53-96 (2011)

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

Gross-Pitaevskii equation, ground states, Painleve-II equation, existence and stability, semiclassical limit