Conductance and absolutely continuous spectrum of 1D samples

TL;DR

This paper links the absolutely continuous spectrum of 1D Schrödinger operators to the limiting behavior of conductances in finite samples, providing a spectral characterization via quantum transport measures.

Contribution

It establishes a novel characterization of the absolutely continuous spectrum using Landauer-Büttiker and Thouless conductances in one-dimensional quantum systems.

Findings

Conductances remain non-zero in the limit iff the energy lies in the absolutely continuous spectrum.

The main result connects spectral properties with quantum transport measures.

Discussion of the relationship with the Schrödinger Conjecture.

Abstract

We characterize the absolutely continuous spectrum of the one-dimensional Schr\"odinger operators $h=-\Delta+v$ acting on $\ell^2(\mathbb{Z}_+)$ in terms of the limiting behavior of the Landauer-B\"uttiker 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 ${\rm sp}_{\rm ac}(h)\cap I=\emptyset$. We also discuss the relationship between this result and the Schr\"odinger Conjecture.