Quantum Depletion for Fixed Bounded Repulsive Potentials

We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is $\frac{8}{3\sqrt{\pi }}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$ as $\rho \downarrow 0$. The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.