Quantum diffusion for all times in the weakly disordered 3D lattice
In plain words
An electron in a weakly disordered 3D crystal should wander like a random walker, with its mean squared distance growing linearly in time forever. This is proven only up to a long but finite time.
Precise statement
For $H = \Delta + \lambda V$ on $Z^3$, $\Delta$ the nearest-neighbour hopping operator, show that E sum_x |x|^2 |<x|exp(-itH)|0>|^2 ~ D(lambda) t as $t \to \infty$ with a diffusion constant $D(\lambda) > 0$ for small $\lambda$ (or the analogous statement for an energy-localized initial state). Erdos, Salmhofer and Yau proved diffusion up to times $t \sim \lambda^{-2-\kappa}$ for some $\kappa > 0$. Answer: a proof for $t \to \infty$ at fixed small $\lambda$.
What would settle it
A proof of diffusive spreading uniformly in time at fixed small $\lambda$.
Status in the literature
Diffusion is controlled only up to times $\lambda^{-2-\kappa}$ (Erdos, Salmhofer and Yau, around 2007-2008).