The density in a three-dimensional radial potential (Q1804657)

In [Bull. Am. Math. Soc., New Ser. 23, No. 2, 525-530 (1990; Zbl 0722.35072)], we announced the proof of an asymptotic formula for the ground-state energy of a non-relativistic atom. A crucial step in the proof of that formula is the estimate \[ \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} (\rho(x)- \rho_{\text{sc}}(x)) (\rho(y)- \rho_{\text{sc}}(y)) {dx dy\over |x- y|}\leq CZ^{5/3- a}\qquad (a> 0),\tag{1} \] where \(\rho\) and \(\rho_{\text{sc}}\) arise from \(V_{\text{TF}}\) for large \(Z\). The main result of this paper is that (1) holds, modulo an assumption which will be proven in two later papers.