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

TL;DR

This paper derives detailed asymptotic expansions for the energies of the ground state of the Gross-Pitaevskii equation in the Thomas-Fermi limit across different dimensions, confirming and extending previous results.

Contribution

It provides a rigorous proof of the kinetic energy expansion in 3D and extends the asymptotic analysis to 1D and 2D cases, adding an extra term to the expansion.

Findings

Rigorous proof of kinetic energy expansion in 3D

Extension of energy expansions to 1D and 2D

Calculation of an additional term in the asymptotic expansion

Abstract

From the asymptotic expansion of the ground state of the Gross-Pitaevskii equation in the Thomas--Fermi limit given by Gallo and Pelinovsky in a previous work, 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 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.