7 problems

Hohenberg–Kohn conjecture. For every $\rho\in\mathcal{D}^N$, either

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

Continuity conjecture. If

Let $d$ be the spatial dimension, let $v(x)=|x|^{-\alpha}$ with $0<\alpha\leq 1$, and consider … with $0<\beta\leq 1/d$. Let $I^1$ denote the trace-class space used for the density…

Consider the time-dependent density-matrix equation with self-interaction satisfying condition (SI2), and let $d$ denote the spatial dimension and $\beta$ the exponent in that cond…

Semiclassical mean-field limit conjecture. For every function $f$ satisfying $f\geqslant 0$, $\sqrt{f}\in H^1(\mathbb{R}^d)$ and $\int_{\mathbb{R}^d}f=1$, one has

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…