Conductance and absolutely continuous spectrum of 1D samples

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 behavior 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 ${\mathrm{sp}}_{\mathrm{ac}}(h)\cap I=\varnothing$. We also discuss the relationship between this result and the Schrödinger Conjecture.

References in corpus (1)

• Landauer-Büttiker and Thouless conductance

Cited by in corpus (4)

• Ballistic Transport for Limit-Periodic Jacobi Matrices with Applications to Quantum Many-Body Problems
• Crystalline conductance and absolutely continuous spectrum of 1D samples
• Mean--field stability for the junction of quasi 1D systems with Coulomb interactions
• Stochastic process emerged from lattice fermion systems by repeated measurements and large-time limit