# Conductance and absolutely continuous spectrum of 1D samples

5 CPT - E5 Physique statistique et systèmes complexes
CPT - Centre de Physique Théorique - UMR 7332
Abstract : 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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Cited literature [34 references]

https://hal.archives-ouvertes.fr/hal-01146220
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Submitted on : Monday, April 27, 2015 - 11:22:04 PM
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### Citation

Laurent Bruneau, Vojkan Jaksic, Yoram Last, Claude-Alain Pillet. Conductance and absolutely continuous spectrum of 1D samples. Communications in Mathematical Physics, Springer Verlag, 2016, 344 (3), pp.959-981. ⟨10.1007/s00220-015-2501-y⟩. ⟨hal-01146220⟩

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