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Conductance and Absolutely Continuous Spectrum of 1D Samples

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Abstract

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators \({h = -\Delta + v}\) acting on \({\ell^2(\mathbb{Z}_+)}\) 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 \({[1, L] \cap \mathbb{Z}_+}\) 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 \({L \to \infty}\) iff \({{\rm sp}_{\rm ac}(h) \cap I \neq \emptyset}\). 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).

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Authors and Affiliations

  1. Département de Mathématiques, UMR 8088, CNRS, Université de Cergy-Pontoise, 95000, Cergy-Pontoise, France

    L. Bruneau

  2. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    V. Jakšić

  3. Institute of Mathematics, The Hebrew University, 91904, Jerusalem, Israel

    Y. Last

  4. CPT, Aix-Marseille Université, 13288, Marseille Cedex 9, France

    C.-A. Pillet

  5. UMR 7332, CNRS, 13288, Marseille Cedex 9, France

    C.-A. Pillet

  6. CPT, Université de Toulon, B.P. 20132, 83957, La Garde Cedex, France

    C.-A. Pillet

  7. FRUMAM, Marseille, France

    C.-A. Pillet

Authors
  1. L. Bruneau
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  2. V. Jakšić
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  3. Y. Last
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  4. C.-A. Pillet
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Correspondence to V. Jakšić.

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Communicated by Y. Kawahigashi

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Bruneau, L., Jakšić, V., Last, Y. et al. Conductance and Absolutely Continuous Spectrum of 1D Samples. Commun. Math. Phys. 344, 959–981 (2016). https://doi.org/10.1007/s00220-015-2501-y

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