10 problems

Let $W$ denote the band width and let $\ell$ denote the localization length of a two-dimensional random band matrix; let $L$ be the linear size of the system and $W_c$ its critical…

Consider the Anderson Hamiltonian … on $\ell^2(\mathbb Z^d)$, where $\lambda>0$ is the coupling constant and $V_{\omega}$ is an i.i.d. random potential. For sufficiently large coup…

Consider the random block Schrödinger operators of the source, with coupling parameter $\lambda$ and model-dependent scale $\Lambda_{\Psi}$. Critical-threshold conjecture. The crit…

For any $u\in\{1,\ldots,N\}$, let $S^{u}_{xy}=\delta_{xy}S_{ux}$ and $S^{1/2,u}_{xy}=\delta_{xy}S^{1/2}_{ux}$. Let $G_s(z)$ be the resolvent, let $w_s$ be the associated spectral p…

Let $H=H_\beta=-\beta\Delta+V$ be the Anderson Hamiltonian on a discrete torus $\Lambda$ approximating $\mathbb{Z}^d$, and let … In the same setting, the more precise quantum diffu…

Let $u$ be one of the weak solutions to the sixth-order quantum-diffusion equation constructed in the proof of the existence theorem, let $u^0$ be its initial datum, and let…

Localization-delocalization transition conjecture. There exists $W_c\in[1,L]$ such that if $W\ll W_c$, the bulk eigenvectors of $H$ are localized, whereas if $W\gg W_c$, they are d…

Let $W$ denote the band width and let $\ell$ denote the localization length of a one-dimensional random band matrix; let $L$ be the linear size of the system and $W_c$ its critical…

Almost Mathieu diffusion conjecture. For almost every $\theta,\alpha\in[0,1]$, the equation has a diffusion constant $D(g,\lambda)\in(0,\infty)$ that is smooth for all…

Diffusive propagation and disorder-scaling conjecture. For every bounded potential $u$ and every $\lambda>0$, these constants are positive and finite, with…