MATHPH In the literature: open

Existence of a mobility edge in the 3D Anderson model

In plain words

Show that in a weakly disordered 3D crystal, electrons at energies near the band edges are trapped while those in the middle of the band move freely. The boundary energy between the two regimes is the mobility edge.

Precise statement

$H = \Delta + \lambda V$ on $\ell^2(Z^3)$, with $(\Delta \psi)(x) = \sum_{\left|y-x\right|=1} \psi(y)$ the hopping operator (spectrum $[-6, 6]$ at $\lambda = 0$), $V(x)$ i.i.d. (e.g. uniform on $[-1,1]$), $\lambda > 0$ small. Prove that there is $E_c(\lambda)$ such that, almost surely, the spectrum in $\left|E\right| > E_c$ is pure point with exponentially decaying eigenfunctions and the spectrum in a nonempty interval inside $\left|E\right| < E_c$ is continuous. Answer: a proof.

What would settle it

A proof of coexistence of localized and delocalized spectrum for one 3D disorder distribution.

Status in the literature

Localization near band edges is proven; no delocalization result exists on $Z^d$.