Dual Lieb–Thirring conjecture for the kinetic-energy constant

Let $K_d$ be the optimal constant in the kinetic-energy inequality referred to as equation (kinetic), and let $K_d^{\mathrm{one}}$ and $K_d^{\mathrm{cl}}$ be the corresponding one-particle and semiclassical constants. Dual Lieb–Thirring conjecture for $\gamma=1$. The optimal constant is

$$K_d=\begin{cases} K_d^{\mathrm{one}},&d\leq2,\\ K_d^{\mathrm{cl}},&d\geq3. \end{cases}$$

By the stated duality, the same assertion also gives the optimal constants in the associated alternative kinetic-energy formulations. The conjecture is presented as the dual formulation of the Lieb–Thirring conjecture at $\gamma=1$; the supplied text does not state a resolution of this formulation.