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

Abstract

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-\alpha }$ for $\alpha >0$. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $\alpha >\frac{1}{2}$; a transition from pure point to singular continuous spectrum in the critical region $\alpha =\frac{1}{2}$; and pure point spectrum in the sub-critical region $\alpha <\frac{1}{2}$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.