Conductance and absolutely continuous spectrum of 1D samples

We characterize the absolutely continuous spectrum of the one-dimensional Schr"odinger operators $h=-\mathrm{\Delta }+v$ acting on ${\mathrm{\ell }}^2({\mathbb{Z}}_{+})$ in terms of the limiting behavior of the Landauer-B"uttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting $h$ to a finite interval $[1,L]\cap {\mathbb{Z}}_{+}$ and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval $I$ are non-vanishing in the limit $L\to \mathrm{\infty }$ iff ${\mathrm{s}\mathrm{p}}_{\mathrm{a}\mathrm{c}}(h)\cap I=\mathrm{\varnothing }$. We also discuss the relationship between this result and the Schr"odinger Conjecture.