Abstract
Abstract We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb {R}^3)$ , gives better rates and, in our opinion, is substantially simpler.
Highlights
Introduction and main resultWe considerN bosons in a finite box ΛL = [− L 2 ]3 ⊂R3, interacting via a two-body nonnegative, radial, compactly supported potential V with scattering length a
It has been applied to establish the validity of Bogoliubov theory in the Gross–Pitaevskii regime in [3, 2]; while our approach is inspired by these papers, we need new tools to deal with the large boxes considered in Proposition 1.3
To show Proposition 1.3, we find it convenient to work with rescaled variables
Summary
R3, interacting via a two-body nonnegative, radial, compactly supported potential V with scattering length a. The bulk of the article contains the proof of the following proposition, establishing the existence of a trial state with the correct energy per unit volume and the correct expected number of particles on boxes of size L = ρ−γ. It has been applied to establish the validity of Bogoliubov theory in the Gross–Pitaevskii regime in [3, 2]; while our approach is inspired by these papers, we need new tools to deal with the large boxes considered in Proposition 1.3 (a simple computation shows that the Gross– Pitaevskii regime corresponds to the exponent γ = 1/2; to control localisation errors, we need instead to choose γ > 1).
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