Conductance and Absolutely Continuous Spectrum of 1D Samples

L. Bruneau, V. Jakšić*, Y. Last, C. A. Pillet

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators h= - Δ + v acting on ℓ2(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 languageAmerican English
Pages (from-to)959-981
Number of pages23
JournalCommunications in Mathematical Physics
Volume344
Issue number3
DOIs
StatePublished - 1 Jun 2016

Bibliographical note

Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

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