Communications in Mathematical Physics

, Volume 344, Issue 3, pp 959–981 | Cite as

Conductance and Absolutely Continuous Spectrum of 1D Samples

  • L. Bruneau
  • V. JakšićEmail author
  • Y. Last
  • C.-A. Pillet


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).


Continuous Spectrum Transfer Matrix Jacobi Matrice Open Quantum System Bloch Wave 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • L. Bruneau
    • 1
  • V. Jakšić
    • 2
    Email author
  • Y. Last
    • 3
  • C.-A. Pillet
    • 4
    • 5
    • 6
    • 7
  1. 1.Département de Mathématiques, UMR 8088, CNRSUniversité de Cergy-PontoiseCergy-PontoiseFrance
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Institute of MathematicsThe Hebrew UniversityJerusalemIsrael
  4. 4.CPTAix-Marseille UniversitéMarseille Cedex 9France
  5. 5.UMR 7332CNRSMarseille Cedex 9France
  6. 6.CPTUniversité de ToulonLa Garde CedexFrance
  7. 7.FRUMAMMarseilleFrance

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