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Title: Localization at all energies in the weakly disordered 2D Anderson model
Plain: 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: 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.
Settled by: A proof of complete localization for small $\lambda$ including the band center.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Localization is known at large disorder and near band edges, including for Bernoulli disorder near the edge (Ding and Smart, 2020).
Posed since: 1979