Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher

For each $d\ge 3$ there is $\lambda_d>0$ such that for every $0<\lambda<\lambda_d$, almost surely a fixed open interval $I_d$ near the bottom band edge (for $d\ge 4$, $-2d+(1/200,1/100)$) lies in the spectrum and the spectral restriction to $I_d$ is purely absolutely continuous and nonzero. $I_d$ does not depend on $\lambda$. Not shown: a.c. spectrum in the bulk of the band, the location of a mobility edge, quantum diffusion, uniformity in $d$, or other laws.

For each $d\ge 3$ there is $\lambda_d>0$ such that for every $0<\lambda<\lambda_d$, almost surely a fixed open interval $I_d$ near the bottom band edge (for $d\ge 4$, $-2d+(1/200,1/100)$) lies in the spectrum and the spectral restriction to $I_d$ is purely absolutely continuous and nonzero. $I_d$ does not depend on $\lambda$. Not shown: a.c. spectrum in the bulk of the band, the location of a mobility edge, quantum diffusion, uniformity in $d$, or other laws.