One-dimensional Lieb–Thirring conjecture

Let $L_{\gamma,1}$ be the optimal Lieb–Thirring constant in dimension $1$, and let $L_{\gamma,1}^{\mathrm{cl}}$ denote the semiclassical constant. Lieb–Thirring conjecture in dimension one. The optimal constant is

$$L_{\gamma,1}=\begin{cases} 2\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2}L_{\gamma,1}^{\mathrm{cl}},&1/2\leq\gamma\leq3/2,\\ L_{\gamma,1}^{\mathrm{cl}},&3/2\leq\gamma. \end{cases}$$

This is the one-dimensional specialization of the Lieb–Thirring conjecture. It has been proved for $\gamma=1/2$ and for $\gamma\geq3/2$, but remains open for $1/2<\gamma<3/2$.