Localization at all energies in the weakly disordered 2D Anderson model
In plain words
Scaling theory predicts that in two dimensions every electron state is trapped by any amount of disorder, however weak. A proof exists only near the edges of the energy band or at strong disorder.
Precise statement
For $H = \Delta + \lambda V$ on $\ell^2(Z^2)$, $\Delta$ the nearest-neighbour hopping operator (spectrum $[-4, 4]$ at $\lambda = 0$), $V$ i.i.d. with bounded density (or Bernoulli), prove that for every $\lambda > 0$ the spectrum is almost surely pure point with exponentially decaying eigenfunctions at all energies, with localization length expected to grow as $\operatorname{exp}(c/\lambda^2)$. Answer: a proof.
What would settle it
A proof of complete localization for small $\lambda$ including the band center.
Status in the literature
Localization is known at large disorder and near band edges, including for Bernoulli disorder near the edge (Ding and Smart, 2020).