C. Gallo and D. Pelinovsky
On the Thomas-Fermi ground state in a harmonic potential
Asymptotic Analysis 73, 53-96 (2011)
Abstract:
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.
Keywords:
Gross-Pitaevskii equation, ground states, Painleve-II equation,
existence and stability, semiclassical limit