Sharp one-dimensional Lieb–Thirring constants

We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every $\frac{1}{2}<\gamma <\frac{3}{2}$, the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in $L^{\gamma +1/2}(\mathbb{R})$, with all negative eigenvalues included.