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 $d\ll1$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2024
- DOI:
- 10.48550/arXiv.2401.15540
- arXiv:
- arXiv:2401.15540
- Bibcode:
- 2024arXiv240115540C
- Keywords:
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- Mathematical Physics;
- Mathematics - Analysis of PDEs
- E-Print:
- 142 pages, 2 figures