Abstract

In this talk I shall describe the characterization of the absolutely continuous spectrum of the one-dimensional Schr ̈odinger operators h = −∆ v acting on 2 (Z ) in terms of the limiting behavior of the Landauer-B ̈ uttiker and Thouless conductances of the associated finite samples. The finite samples are 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 → ∞ (physical characterization of the metallic regime) iff sp ac (h) ∩ I = ∅ (mathematical characterization of the metallic regime). This result is of importance for the foundations of quantum mechanics since it provides the first complete dynamical character- ization of the absolutely continuous spectrum of Schr ̈odinger operators. I shall also discuss its relation with Avila’s counterexample to the Schr ̈odinger Conjecture. This talk is based on a joint work with L. Bruneau, Y. Last, and C-A. Pillet.