Conductance and absolutely continuous spectrum of 1D samples
Résumé
We characterize the absolutely continuous spectrum of the one-dimensional
Schrödinger operators h = −∆ + v acting on $l^2 (Z _+ )$ 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
$sp_{ac} (h)\cap I = \emptyset$. We also discuss the relationship between this result and the Schrödinger
Conjecture.
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