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Accepted · first revision · record

Title: Existence of a mobility edge in the 3D Anderson model
Plain: 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: $H = \Delta + \lambda V$ 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.
Settled by: A proof of coexistence of localized and delocalized spectrum for one 3D disorder distribution.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Localization near band edges is proven; no delocalization result exists on $Z^d$.
Posed since: 1958