A simple 2nd order lower bound to the energy of dilute bose gases - Brietzke, Birger; Fournais, Søren; Solovej, Jan Philip (Communications in mathematical physics, 2020)
For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound $$e(ρ ) \ge 4π a ρ ^2 (1- C \sqrt{ρ a^3} \,)$$e(ρ)≥4πaρ2(1-Cρa3).