19 problems

Let $L_{\kappa,1}^{(N)}$ denote the rank-$N$ Lieb–Thirring constant in dimension $1$, and let $L_{\kappa,1}^{(1)}$ be its one-bound-state value. One-bound-state conjecture. … If tr…

For a Schrödinger operator in dimension $n$, let $L_{0,n}$ be the sharp constant in the Lieb–Thirring inequality at exponent $\gamma=0$, and let $L_{0,n}^{1}$ be the sharp constant…

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…

Glaser–Grosse–Martin conjecture. The map $N\in\mathbb{N}\mapsto \ell^{(N)}_{0,d}$ attains its maximum for some finite $N_c$ in every dimension, with $N_c=1$ when $d\in\{3,4,5,6\}$.

Lundholm–Nam–Portmann conjecture. In the strong-coupling limit,

Levy–Lieb upper-bound conjecture. The functional satisfies

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…

Let $d\geq1$ and let $\gamma$ be in the range for which the Lieb–Thirring inequality is defined. Denote by $L_{\gamma,d}$ its optimal constant, by $L_{\gamma,d}^{(1)}$ the best con…

Let $d\geq 3$ and set $\gamma=0$. Define … and let $L_{0,d}$ be the optimal CLR constant and $L_{0,d}^{\mathrm{cl}}$ its semiclassical value. Glaser–Grosse–Martin conjecture. … Thi…

Hansmann–Katriel conjecture. The inequality above is no longer true when the assumption that $J$ is self-adjoint is dropped. The conjecture concerns the extension of Lieb–Thirring…

Let $\kappa_{\rm sc}(d)$ denote the threshold separating the semiclassical regime from the non-semiclassical regime, and interpret the limiting potential $V_\infty$ of the cluster…

Let $V_N$ be an optimal potential for the rank-$N$ Lieb–Thirring constant $L_{\kappa,d}^{(N)}$, and let $N_k$ be a subsequence along which the corresponding $N$ fermions attract ea…

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

Let $N$ spinless fermions in $\operatorname{\mathbb{R}}^3$ have normalized wave function $\psi$ and one-particle density $\rho$. Their kinetic energy is … The sharp semiclassical L…

Let $n_\gamma$ denote the number of bound states of a maximizer of the Lieb–Thirring variational functional in dimension three, and let $\gamma$ approach $1$. Bound-state prolifera…

Let $V_{\gamma,2,1}$ denote the radial potential with exactly one bound state in the two-dimensional Lieb–Thirring variational problem, and let $\gamma_{c,2}$ be the critical expon…

For the Lieb–Thirring variational problem, let $V_{\gamma,1,1}$ denote the potential with exactly one bound state that appears as the one-bound-state critical point, and let the fu…

Frank et al.'s conjecture. The restriction $\kappa\geq 1$ is superfluous, and the displayed estimate should hold for $\kappa$ satisfying the selfadjoint restrictions:…

Failure conjecture. The displayed inequality is not true for non-selfadjoint Jacobi operators.