8 problems

Let $L_+$ be the linearized operator in the Hartree problem, and let $gKer[L_+]$ denote its generalized kernel. At the mass-critical exponent … the linearized operator has an extra…

Dynamical Hartree conjecture. For every initial state $u_0\in H^1(\mathbb{R}^3)$ and every $R>0$,

The Hartree equation is studied with initial data $Q_0(x,y)$ satisfying … where $H_x$ and $\bar{H}_y$ are the magnetic Hamiltonians acting in the $x$- and $y$-variables, respective…

For each $N$, let the many-fermion dynamics with Coulomb interaction and its effective Hartree dynamics be compared in the setting of the paper, where the convergence estimate has…

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…

Let $u(t)$) be the unique solution to the Hartree equation for some $u_0\in H^1(\mathbb{R}^3)$, and let $\mathscr{A}_Z$ denote the global attractor. Take any sequence of times…

Minimal-mass conjecture. The value of $m_0$ should be $+\infty$ in the defocusing case and $M(Q)$ in the focusing case.

Consider the focusing mass-critical Hartree equation in $\mathbb{R}^4$, and let $\delta_0$ be the supremum of the $L^2$-mass thresholds below which its Cauchy problem has a unique…