MATHPH In the literature: open

Absolutely continuous spectrum of the 3D Anderson model at weak disorder

In plain words

Prove that at least some electron states in a weakly disordered 3D lattice spread over the whole sample. This is the half of the mobility-edge problem that has no proof at all.

Precise statement

For $H = \Delta + \lambda V$ on $\ell^2(Z^3)$, with $\Delta$ the nearest-neighbour hopping operator (spectrum $[-6, 6]$ at $\lambda = 0$), i.i.d. bounded $V$ and $\lambda$ small, prove that the almost-sure spectrum has a nonzero absolutely continuous component in some energy interval inside $(-6, 6)$, or at least that eigenfunctions in that interval are not exponentially localized. Answer: a proof.

What would settle it

A proof of absolutely continuous spectrum, or of delocalized eigenfunctions in finite volume uniformly in the volume, for $d = 3$.

Status in the literature

Absolutely continuous spectrum at weak disorder is proven on tree graphs (Klein, 1998, and later work of Aizenman and Warzel), not on $Z^d$.