8 problems

Let $E(N,\lambda v)$ and $E(N-1,\lambda v)$ denote the ground-state energies of the trapped Bose gas with interaction scaling $\lambda$, and let $T$ and $v$ satisfy appropriate condit…

Let $\rho_Z$ be the one-body density of a ground state of the many-body atomic Hamiltonian with nuclear charge $Z$ and $N=N_c(Z)$ electrons. Universality conjecture. Along subseque…

Let $H_N^{\rm Comb}$ be the Hamiltonian with two-body potential $W$ at scaling parameter $\beta=1$ and three-body potential $V$. Let $e^{\rm Comb}_{\rm GP}(b_1,b_2)$ be the minimum…

Let $H_N^{\rm GP}$ be the Gross–Pitaevskii Hamiltonian, let $E_{\rm GP}(N)$ denote its ground-state energy, and let $e_{\rm GP}$ be the minimum of the corresponding Gross–Pitaevski…

Nonvanishing-minimizer conjecture. Any minimizer $\psi$ of $F^{\boldsymbol{\alpha},(0)}(r)$ satisfies

Let a system with initial wave function $\psi_0$ evolve under interaction potential $w$ and external potential $v$. For a different interaction potential $w'$, there should be a un…

Let $H=T+V_{\mathrm{ee}}+V_{\mathrm{ext}}$ be the Hamiltonian of a bounded many-electron system, with external potential $v$ and one-particle density $n$ in a non-degenerate ground…

Let $u(t)$ be the unique solution to the Hartree equation for some $u_0\in H^1(\mathbb{R}^3)$, let $\mathscr{A}_Z$ be the global attractor, and let $\gamma_c$ be the constant appea…