Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
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DOI：
10.1090/tran/7832
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发表时间：
2017-06
影响因子：
1.3
通讯作者：
Valmir Bucaj;D. Damanik;J. Fillman;Vitaly Gerbuz;Tom VandenBoom;Fengpeng Wang;Zhenghe Zhang
中科院分区：
文献类型：
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作者：
Valmir Bucaj;D. Damanik;J. Fillman;Vitaly Gerbuz;Tom VandenBoom;Fengpeng Wang;Zhenghe Zhang
从F urstenberg定理给出的李雅普诺夫指数的正性出发，给出了一维安德森模型的谱和动力学局部化的完备和完备的证明.也就是说，在$\ell^2（\mathbb{Z}）$中的Schr\“odinger算子，其势由独立同分布（i.i.d.）随机变量几乎必然具有纯点谱，其特征函数呈指数衰减，其酉群在时间上均匀地呈指数非对角衰减。这是通过一个新的结果：对于安德森模型，一个典型的所有广义本征函数的李雅普诺夫行为。我们还解释了如何获得扩展CMV矩阵的Verblunsky系数是i.i.d.的类似声明，以及这些模型的半线类似物。
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by F\"urstenberg's theorem. That is, a Schr\"odinger operator in $\ell^2(\mathbb{Z})$ whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typically has Lyapunov behavior for all generalized eigenfunctions. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogs of these models.