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 − α for α > 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 α > 1 2 $\alpha >\frac 12$ ; a transition from pure point to singular continuous spectrum in the critical region α = 1 2 $\alpha =\frac 12$ ; and pure point spectrum in the sub-critical region α < 1 2 $\alpha <\frac 12$ . 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.

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衰减随机势中的一维离散狄拉克算子 I：谱和动力学

我们研究了随机势中的一维离散狄拉克算子的频谱和动力学，该随机势是通过使用 n - α 类型的包络对 α > 0 的 iid 环境进行阻尼而获得的。我们恢复了先前为模拟安德森模型获得的所有频谱范围在随机衰减势中，即：超临界区绝对连续谱 α > 1 2 $\alpha >\frac 12$ ；临界区从纯点到奇异连续谱的跃迁 α = 1 2 $\alpha =\frac 12$ ；和亚临界区的纯点谱 α < 1 2 $\alpha <\frac 12$ 。从动力学的角度来看，超临界区域的离域化遵循 RAGE 定理。在临界区，我们展示了一个基于特征函数下界的简单论证，表明即使存在点谱也不会发生动态定位。最后，我们通过分数矩方法显示了亚临界区域的动态定位，并提供了对本征函数的控制。