论文标题
一维离散的安德森模型具有衰减的随机潜力：从A.C.频谱到动态定位
One-dimensional Discrete Anderson Model in a Decaying Random Potential: from a.c. Spectrum to Dynamical Localization
论文作者
论文摘要
我们考虑了一个一维的安德森模型，该模型平均衰减，例如$ n^{ - α} $，$α> 0 $。已知这种简单的模型显示出具有不同类型的频谱的丰富相图，随着衰减率$α$的变化。 我们回顾了Kiselev，Last和Simon的一篇文章，作者在其中显示了A.C.在超临界情况下的频谱$α> \ frac12 $，在关键情况下从单数连续到纯点光谱的过渡$α= \ frac12 $，以及在亚临界情况下$α<\ frac12 $中的密集的纯点光谱。我们提供了$α\ ge \ frac12 $的案例的完整证明，并简化了一些论点。我们通过讨论模型的动态方面来补充上述结果。我们给出了一个简单的论点，表明尽管光谱过渡，但对于$α= \ frac12 $的所有能量都会发生运输。最后，我们讨论了Simon的定理，内容涉及亚临界区域$α<\ frac12 $的动态定位。这尤其意味着频谱是该制度的纯粹点。
We consider a one-dimensional Anderson model where the potential decays in average like $n^{-α}$, $α>0$. This simple model is known to display a rich phase diagram with different kinds of spectrum arising as the decay rate $α$ varies. We review an article of Kiselev, Last and Simon where the authors show a.c. spectrum in the super-critical case $α>\frac12$, a transition from singular continuous to pure point spectrum in the critical case $α=\frac12$, and dense pure point spectrum in the sub-critical case $α<\frac12$. We present complete proofs of the cases $α\ge\frac12$ and simplify some arguments along the way. We complement the above result by discussing the dynamical aspects of the model. We give a simple argument showing that, despite of the spectral transition, transport occurs for all energies for $α=\frac12$. Finally, we discuss a theorem of Simon on dynamical localization in the sub-critical region $α<\frac12$. This implies, in particular, that the spectrum is pure point in this regime.
