Affiliations: Institut de Mathématiques et de Modélisation de Montpellier, Université Montpellier II, 34095 Montpellier, France | Department of Mathematics, McMaster University, Hamilton, ON L8S 4K1, Canada
Note: [] Supported in part by the project ANR-07-BLAN-0250 of the Agence Nationale de la Recherche.
Note: [] Supported in part by the NSERC.
Abstract: We study non-linear 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 Painlevé-II equation. In the space of one dimension, these results allow us to characterize the distribution of eigenvalues in the point spectrum of the Schrödinger operator associated with the non-linear ground state.
