Derivation of Hartree theory for two-dimensional attractive Bose gases in almost Gross-Pitaevskii regime

TL;DR

This paper rigorously derives Hartree theory for two-dimensional attractive Bose gases in regimes close to the Gross-Pitaevskii limit, establishing stability, energy convergence, and Bose-Einstein condensation.

Contribution

It extends the derivation of Hartree and NLS limits to more general scalings, including exponential, for attractive interactions in 2D Bose gases.

Findings

Proves stability of the many-body system with attractive interactions.

Shows convergence of ground state energy to a nonlinear Schrödinger functional.

Establishes Bose-Einstein condensation in a broader diluteness regime.

Abstract

We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schr\"odinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross--Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose-Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.