Conductance and Absolutely Continuous Spectrum of 1D Samples

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators h= - Δ + v acting on ℓ²(Z₊) in terms of the limiting behaviour of the Landauer–Büttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting h to a finite interval [ 1 , L] ∩ 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→ ∞ iff sp _ac(h) ∩ I≠ ∅. We also discuss the relationship between this result and the Schrödinger Conjecture (Avila, J Am Math Soc 28:579–616, 2015; Bruneau et al., Commun Math Phys 319:501–513, 2013).