What is the absolutely continuous spectrum?

TL;DR

The paper surveys a program to characterize the absolutely continuous spectrum $sp_{ m ac}(H)$ of a self-adjoint operator through transport in associated open quantum systems. It develops a Jacobi-matrix representation of spectral triples and analyzes electronic transport via Electronic Black Box models under Landauer–Büttiker, Thouless, and crystalline transport notions, establishing transport-based criteria that characterize $sp_{ m ac}(H)$; it also clarifies the limitations of linear response through Avila’s counterexample and presents weaker but robust equivalences. The work connects deep spectral questions (including Schrödinger Conjectures and their ergodic variants) to concrete transport observables, providing a rigorous foundation for physical heuristics linking ac spectrum to conducting behavior and offering scalable criteria that extend beyond purely self-adjoint operators. The results yield a unifying framework that ties resolvent/transfer-matrix analysis to large-scale transport, with potential extensions to interacting many-body systems and deeper investigations into the rate of convergence of currents and the role of periodic/crystalline structures in inducing or suppressing transport.

Abstract

We summarize (and comment on) the research program carried out in (CMP 319, 501 (2013)), (CMP 338, 347 (2015)), (CMP 344, 959 (2016), (LMP 106, 787 (2016)). This program is devoted to the characterization of the absolutely continuous spectrum of a self-adjoint operator H in terms of the transport properties of a suitable class of open quantum systems canonically associated to H.

Figures (3)

• Figure 1: A finite sample of length $L$ coupled to two electronic reservoirs
• Figure 2: The periodic EBB model associated to the Hamiltonian $J_{L,\mathrm{per}}$.
• Figure 3: The EBB model associated to the sample hamiltonian $J_L^{(N)}$ for $N=7$.

Theorems & Definitions (4)

• Theorem 4.1
• Theorem 7.1
• Theorem 7.2
• Theorem 7.3