The second order 2D behaviors of a 3D Bose gas in the Gross-Pitaevskii regime

We consider a system of N 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 ≪ 1 d\ll 1 . We assume particles to interact through a repulsive, radially symmetric and short-range interaction potential with scattering length scale a ≪ d 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 N , a a and d 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 is consistent with Schnee and Yngvason [Comm. Math. Phys. 269 (2007), pp. 659–691], and thus explains how a three-dimensional Bose gas system smoothly transits into two-dimensional system. From the analysis of this new mechanism exclusive to the second order, we find 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. As a result, there are 2nd order terms not yet seen in the 3D and 2D cases.