One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics

Abstract

In this paper, we develop the radial transfer matrix formalism for unitary one-channel operators. This generalizes previous formalisms for CMV matrices and scattering zippers. We establish an analog of Carmona's formula and deduce criteria for absolutely continuous spectrum which we apply to random Hilbert Schmidt perturbations of periodic scattering zippers. Contents 1. Setup and result 1 1.1. One-channel unitary operators 3 1.2. Transfer matrices 6 1.3. Spectral average formula and criteria for a.c. spectrum 11 1.4. Absolutely continuous spectrum for periodic one-channel scattering zippers with random decaying perturbation 12 2. Examples 14 2.1. One-dimensional quantum walks 14 2.2. Generalized one-channel quantum walks 16 2.3. Stroboscopic unitary dynamics on Z 2 17 3. Transfer matrix and Resolvent 18 4. Spectral averaging and weak convergence 21 5. Perturbation of periodic scattering zipper 24 Appendix A. Facts of linear algebra 29 References 33

One-dimensional Discrete Anderson Model in a Decaying Random Potential: from a.c. Spectrum to Dynamical Localization

“…for all ϕ 0 ∈ l 2 (Z) with bounded support. A proof of these simple facts can be found in [9] in a related discrete model or in [8] in the continuum.…”

“…The proof of Theorem 2.3 in [38] uses the Kunz-Souillard method (KSM) [13,35]. In [9], we prove localization for the one-dimensional Dirac operator in a sub-critical potential by means of the fractional moments method instead. This approach allowed us to greatly generalize the hypothesis required in [38].…”

“…Using the Taylor expansion log The following lemma is the key to estimate the martingale terms. It corresponds to [33,Lemma 8.4] but this shorter proof is taken from [9].…”

“…The next lemma actually provides a lower bound on any non-trivial generalized solution of H ω,λ x = Ex for α = 1 2 , λ > 0, uniformly in E ranging over compact intervals of (−2, 2). The proof is taken from [9] where it was developed in the context of the discrete Dirac model. Lemma 5.6.…”

Dynamical Localization for the One-dimensional Continuum Anderson Model in a Decaying Random Potential

“…From the dynamical point of view, it is standard to show that the system propagates for α > 1 2 . For the critical case α = 1 2 , no transition occurs at the dynamical level, despite of the spectral transition: there are non-trivial transport exponents for all values of the coupling constant [28] for both the discrete and continuum model (see also [8,9] for elementary arguments showing delocalization). This provides yet another example of a model where spectral localization and transport coexist.…”