A New Estimate on the Two-Dimensional Indirect Coulomb Energy

We prove a new lower bound on the indirect Coulomb energy in two-dimensional quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the Lieb–Solovej–Yngvason bound with a smaller constant, ${C = (4/3)^{3/2} \sqrt{5 \pi -1} \approx 5.90 < C_{\rm LSY} = 192 \sqrt{2 \pi} \approx 481.27}$ , which also involves an additive gradient energy term of the single particle density.