On the Thomas-Fermi ground state in a harmonic potential

Details On the Thomas-Fermi ground state in a harmonic potential On the Thomas-Fermi ground state in a harmonic potential

2009-11-19

arxiv.org/abs/0911.3913v1

Physics

Mathematical Physics

38 pages, no figures

Scientific paper

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.

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