Localization at all energies in the weakly disordered 2D Anderson model · revision history

Revision 2

Accepted · core change · record

Changes against revision 1:

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 · changed · core
Revision 1: 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.
Revision 2: 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 $\exp(c/\lambda^2)$. 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

Decisions