Fractional Lieb-Thirring coupling conjecture for the optimal constant

Let $d\geq 1$, $s>0$, and $\lambda\geq 0$. For normalized many-body wave functions $\Psi\in\mathcal{H}^s_{d,N}$, define

$$C_{\rm BLT}(\lambda):= \inf_{N \ge 2} \ \inf_{\substack{\Psi \in \mathcal{H}^s_{d,N} \\ \left\lVert \Psi \right\rVert_2=1}} \frac{ \left\langle \Psi, \left( \sum_{i=1}^N (-\Delta_{i})^s + \lambda W_s \right) \Psi \right\rangle }{ \int_{\mathbb{R}^d} \rho_{\Psi}^{1+2s/d} },$$

where

$$\mathcal{H}^s_{d,N}:= \left\{ \Psi \in H^{s}(\mathbb{R}^{dN}): \int_{\mathbb{R}^{dN}} W_s|\Psi|^2 < \infty \right\}, \qquad W_s(x):= \sum_{1\le i<j \le N} \frac{1}{|x_i-x_j|^{2s}}.$$

Let $C_{\mathrm{GN}}$ be the optimal constant in the one-body fractional Gagliardo–Nirenberg inequality,

$$C_{\mathrm{GN}}:= \inf_{\substack{u \in H^s(\mathbb{R}^d)\\ \left\lVert u \right\rVert_2 = 1}} \frac{\langle u, (-\Delta)^s u \rangle}{\int_{\mathbb{R}^d} |u|^{2 (1+2s/d)}}.$$

Fractional Lieb-Thirring coupling conjecture. The optimal constant also satisfies

$$C_{\mathrm{BLT}}(0)>0 \quad\text{for }2s>d,$$

and, for all $d\geq 1$ and $s>0$,

$$\lim_{\lambda\to\infty}C_{\mathrm{BLT}}(\lambda)=C_{\mathrm{GN}}.$$

These assertions describe the limiting behavior of the optimal interacting Lieb-Thirring constant: positivity at zero coupling in the regime $2s>d$, and convergence to the one-body optimal constant at strong coupling. The source presents them as beliefs, and gives no proof or resolution.