3维Anderson模型小无序的定域化极限和Boltzmann极限
Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3
摘要：
我们证明了三维Anderson模型在弱无序下本征函数的局部化长度的下界。我们的结果类似于Schlag，Shubin和Wolff对一维和二维的结果。我们证明，概率为1，大多数本征函数的本地化长度有界从下面的$O(\frac{λ^{-2}}{\log\frac1λ})$，其中$λ$是无序强度。这是通过依赖于时间的方法来实现的，该方法将Erdös和Yau开发的方法推广到晶格和非高斯情况。此外，我们表明，相应的晶格随机薛定谔动力学的宏观极限是由线性玻尔兹曼方程。
英文摘要：
We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by $O(\frac{λ^{-2}}{\log\frac1λ})$, where $λ$ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erdös and Yau to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lattice random Schrödinger dynamics is governed by the linear Boltzmann equations.
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