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A proof exists only near the edges of the energy band or at strong disorder.","posed_since":"1979","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.","problem_ref":null,"references":"","settled_by":"A proof of complete localization for small $\\lambda$ including the band center.","status_note":"Localization is known at large disorder and near band edges, including for Bernoulli disorder near the edge (Ding and Smart, 2020).","title":"Localization at all energies in the weakly disordered 2D Anderson 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