9 problems

Let $E^c_{\kappa}$ and $e^c_{\kappa}$ be the energy levels defined above, and let ${\cal E}^c_{\kappa}$ be the Dirac–Fock energy on $G_N(P^+_{\kappa,\widetilde{W}} H^{1/2})$. A sub…

Mean-field TDHF conjecture. When $N$ is large, the single-body density operator $D_{N:1}(t)$ should nearly obey the time-dependent Hartree-Fock equation, with $F(0)=D_{N:1}(0)$. Th…

Let $\mu \in (0,4d)$ be the chemical potential, and let $U$ be the Hubbard interaction strength. Denote by $E_{\mu,U}^{(\mathrm{hf})}$ and $E_{\mu,U}^{(\mathrm{hfz})}$ the minimal…

Let a frustration-free FBI Hamiltonian be a many-body Hamiltonian whose ground-state energy is attained when each positive-semidefinite local term is minimized, and let its Hartree…

Weak Mittleman's conjecture. The Dirac–Fock model is an approximation of the electron-positron Hartree–Fock model in the precise sense of the displayed inequality. This weaker stat…

Let $e_q$ denote the physical ground state energy of the electron-positron Hartree–Fock model, obtained by maximizing the model's ground state energy over all allowed one-particle…

Long-range extension conjecture. All the results of Theorems corresponding to uniqueness of critical points and uniqueness of paramagnetic minimisers are valid throughout a region…

Let $w$ be a positive radial non-increasing function. For density $\rho\geq 0$ and temperature $T\geq 0$, let $E_{\textnormal{no-spin}}^{\rm HF}(\rho,T)$ denote the no-spin Hartree…

Let $L$ be the side length of a box, and let $\gamma_{r_s,T,L}$ denote the paramagnetic Hartree–Fock fluid state of density $\rho$ at temperature $T>0$. Uniqueness conjecture. The…