Expansion of the energy of the ground state of the Gross–Pitaevskii equation in the Thomas–Fermi limit

Abstract

From the asymptotic expansion of the ground state of the Gross–Pitaevskii equation in the Thomas–Fermi limit given by Gallo and Pelinovsky [“On the Thomas-Fermi ground state in a harmonic potential,” Asymptot. Anal. 73(1–2), 53–96 (2011)]10.3233/ASY-2011-1034, we infer an asymptotic expansion of the kinetic, potential, and total energy of the ground state. In particular, we give a rigorous proof of the expansion of the kinetic energy calculated by Dalfovo, Pitaevskii, and Stringari [“Order parameter at the boundary of a trapped Bose gas,” Phys. Rev. A 54, 4213–4217 (1996)]10.1103/PhysRevA.54.4213 in the case where the space dimension is 3. Moreover, we calculate one more term in this expansion, and we generalize the result to space dimensions 1 and 2.