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Title: Absolutely continuous spectrum of the 3D Anderson model at weak disorder
Plain: 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: 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.
Settled by: A proof of absolutely continuous spectrum, or of delocalized eigenfunctions in finite volume uniformly in the volume, for $d = 3$.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Absolutely continuous spectrum at weak disorder is proven on tree graphs (Klein, 1998, and later work of Aizenman and Warzel), not on $Z^d$.
Posed since: 1958