The Second Order 2D Behaviors of a 3D Bose Gases in the Gross-Pitaevskii Regime

Abstract

We consider a system of $N$ bosons interacting in a three-dimensional box endowed with periodic boundary condition that is strongly confined in one direction such that the normalized thickness of the box $d\ll 1$. We assume particles to interact through a repulsive, radially symmetric and short-range interaction potential with scattering length scale $a\ll d$. We present a comprehensive study of such system in the Gross-Pitaevskii regime, up to the second order ground state energy, starting from proving optimal Bose-Einstein condensation results which were not previously available. The fine interplay between the parameters $N$, $a$ and $d$ generates three regions. Our result in one region on the one hand, is compatible with the classical three-dimensional Lee-Huang-Yang formula. On the other hand, it reveals a new mechanism exhibiting how the second order correction compensates and modifies the first order energy, which was previously thought of as containing a jump, and thus explains how a three-dimensional Bose gas system smoothly transits into two-dimensional system. Moreover, delving into the analysis of this new mechanism exclusive to the second order, we discover a dimensional coupling correlation effect, deeply buried away from the expected 3D and quasi-2D renormalizations, and calculate a new second order correction to the ground state energy.

145 pages, 2 figures. accomodating comments from A.Triay

References in corpus (6)

• Atomic Bose-Einstein condensate in a twisted-bilayer optical lattice
• The Second Order Upper Bound for the Ground Energy of a Bose Gas
• Atomic Bose-Einstein condensate to molecular Bose-Einstein condensate transition
• Supersymmetric field theory of local light diffusion in semi-infinite media
• The excitation spectrum of two dimensional Bose gases in the Gross-Pitaevskii regime
• The rigorous derivation of the ${\mathbb{T}}^2$ focusing cubic NLS from 3D