A second order upper bound on the ground state energy of a Bose gas beyond the Gross-Pitaevskii regime

TL;DR

This paper extends the upper bound on the ground state energy of a Bose gas beyond the Gross-Pitaevskii regime, approaching the Lee-Huang-Yang formula for a broader range of interaction scaling.

Contribution

It provides a more precise second order upper bound on the ground state energy for Bose gases with interaction scaling parameter <7/12, improving previous results.

Findings

Extended the upper bound to <7/12.

Refined error analysis for better bounds.

Supports the Lee-Huang-Yang formula in a broader regime.

Abstract

We consider a system of $N$ bosons in a unitary box in the grand-canonical setting interacting through a potential with scattering length scaling as $N^{-1+\kappa},$ $\kappa\in (0,2/3).$ This regimes interpolate between the Gross-Pitaevskii regime ($\kappa=0$) and the thermodynamic limit ($\kappa=2/3$). In Forum Math. Sigma 9 (2021), e74, as an intermediate step to prove an upper bound in agreement with the Lee-Huang-Yang formula in the thermodynamic limit, it is obtained a second order upper bound on the ground state energy for $\kappa<5/9.$ In this paper, thanks to a more careful analysis of the error terms, we extend the above mentioned result to $\kappa<7/12$.