Lieb–Thirring conjecture for optimal spectral constants

Let $L_{\gamma,d}$ be the optimal constant in the Lieb–Thirring inequality, let $L_{\gamma,d}^{\mathrm{cl}}$ be the semiclassical constant, and let $L_{\gamma,d}^{\mathrm{one}}$ be the optimal constant in the corresponding one-particle bound. Lieb–Thirring conjecture. The optimal constant is

$$L_{\gamma,d}=\max\left(L_{\gamma,d}^{\mathrm{cl}},L_{\gamma,d}^{\mathrm{one}}\right).$$

Equivalently, in the notation of the source, $R_{\gamma,d}=\max\left(1,L_{\gamma,d}^{\mathrm{one}}/L_{\gamma,d}^{\mathrm{cl}}\right)$. The conjecture is known to hold in some cases and to fail in others; the paper surveys the state of the problem.