## Abstract

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators h= - Δ + 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) ∩ 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).

Original language | American English |
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Pages (from-to) | 959-981 |

Number of pages | 23 |

Journal | Communications in Mathematical Physics |

Volume | 344 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jun 2016 |

### Bibliographical note

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