4 problems

Let $d\geq 2$, let $q>1$, and let $\gamma$ be a density matrix satisfying $0\leq\gamma\leq 1$. Write … for a rank-$N$ projection, and let $\rho_\gamma$ denote its density. The asso…

Let $d\geq2$ and $q\in(d/2,d]$. Let $V$ be a bounded complex-valued potential on $\mathbb{R}^d$, and let $z$ be an eigenvalue of $-\Delta-V$ with…

Let $d\geq1$ and let $\gamma$ satisfy the constraint defining the admissible Lieb–Thirring regime. Let $L_{\gamma,d}$ be the optimal Lieb–Thirring constant, let…

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