A new numerical approach to Anderson (de)localization
A new numerical approach to Anderson (de)localization
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安德森（去）定位的新数值方法
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发表时间：
2012
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通讯作者：
C. Liaw
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作者：
C. Liaw
我们开发了一种新的方法来解决安德森本地化问题。这种方法的实现产生了强有力的数值证据，导致一个（令人惊讶的许多）猜想：二维离散随机薛定谔算子与小的混乱允许状态，动态离域与正概率。这种方法基于Abakumov-Liaw-Poltoratski最近的一个结果，该结果植根于秩1扰动下的谱行为的研究，并指出每个非零向量对于算子的奇异部分几乎必然是循环的。与该领域的其他数值方法相比，所提出的数值工作是相当简单的。此外，该方法消除了边界条件的影响。虽然我们进行的数值实验几乎完全在二维离散随机薛定谔算子的情况下，我们包括设置为一般类的安德森模型称为安德森型哈密顿。我们跟踪的位置时，最初位于原点的波包的能量是根据离散随机薛定谔算子的演变。这种方法并没有提供新的洞察力的能量制度的扩散发生。1 ar X iv：1 20 7. 28 43 v1 [ m at hph ] 1 2 Jul l 2 01 2 A new numerical approach to安德森（de）localization Constanze Liaw Department of Mathematics Texas A&M University Mailstop 3368学院站，TX 77843 USA电子邮件：conni@math.tamu.edu
We develop a new approach for the Anderson localization problem. The implementation of this method yields strong numerical evidence leading to a (surprising to many) conjecture: The two dimensional discrete random Schrödinger operator with small disorder allows states that are dynamically delocalized with positive probability. This approach is based on a recent result by Abakumov–Liaw–Poltoratski which is rooted in the study of spectral behavior under rank-one perturbations, and states that every non-zero vector is almost surely cyclic for the singular part of the operator. The numerical work presented is rather simplistic compared to other numerical approaches in the field. Further, this method eliminates effects due to boundary conditions. While we carried out the numerical experiment almost exclusively in the case of the two dimensional discrete random Schrödinger operator, we include the setup for the general class of Anderson models called Anderson-type Hamiltonians. We track the location of the energy when a wave packet initially located at the origin is evolved according to the discrete random Schrödinger operator. This method does not provide new insight on the energy regimes for which diffusion occurs. 1 ar X iv :1 20 7. 28 43 v1 [ m at hph ] 1 2 Ju l 2 01 2 A new numerical approach to Anderson (de)localization Constanze Liaw Department of Mathematics Texas A&M University Mailstop 3368 College Station, TX 77843 USA email: conni@math.tamu.edu