A simple 2nd order lower bound to the energy of dilute Bose gases

Abstract: 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 $e(\rho) \geq 4\pi a \rho² (1- C \sqrt{\rho a^3} \,)$.