The Bogoliubov quantum-depletion law for the dilute Bose gas in the thermodynamic limit

For hard spheres (October 5): for every bounded continuous $f$ and $\varepsilon>0$, at all sufficiently small fixed $\rho$, every thermodynamic accumulation value of $\frac{1}{N\sqrt{\rho a^3}}\sum_{k\ne0}f(k/\sqrt{8\pi\rho a})\,n(k)$ is within $\varepsilon$ of $\int f\,d\nu_{\mathrm{Bog}}$, uniformly over ground states; taking $f=1$ gives depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$. A companion proves the same depletion asymptotic for fixed bounded, nonnegative, radial, finite-range potentials with positive scattering length. Not covered: positive temperature, the next-order correction, or unbounded non-hard-core potentials.

For hard spheres (October 5): for every bounded continuous $f$ and $\varepsilon>0$, at all sufficiently small fixed $\rho$, every thermodynamic accumulation value of $\frac{1}{N\sqrt{\rho a^3}}\sum_{k\ne0}f(k/\sqrt{8\pi\rho a})\,n(k)$ is within $\varepsilon$ of $\int f\,d\nu_{\mathrm{Bog}}$, uniformly over ground states; taking $f=1$ gives depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$. A companion proves the same depletion asymptotic for fixed bounded, nonnegative, radial, finite-range potentials with positive scattering length. Not covered: positive temperature, the next-order correction, or unbounded non-hard-core potentials.