论文标题
一维离散的狄拉克运算符，具有衰减的随机电势I：频谱和动力学
One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics
论文作者
论文摘要
我们研究了通过抑制I.I.D的随机电势，研究一维离散dirac运算符的频谱和动力学。带有$ n^{ - α} $的环境，$α> 0 $。我们以随机的衰减潜力恢复了以前为模拟模型获得的所有光谱制度，即：超临界区域的绝对连续光谱$α> \ frac12 $;从纯点到关键区域中的单数连续频谱的过渡$α= \ frac12 $;和亚临界区域$α<\ frac12 $中的纯点光谱。从动力学的角度来看，超临界区域中的离域遵循愤怒定理。在关键区域，我们基于征征的下限表现出一个简单的论点，表明即使在存在点频谱的情况下，也不会发生动态定位。最后，我们通过分数矩方法显示在亚临界区域中的动态定位，并提供对本征函数的控制。
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 $α>\frac12$; a transition from pure point to singular continuous spectrum in the critical region $α=\frac12$; and pure point spectrum in the sub-critical region $α<\frac12$. 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.
