Local fluctuation of the spectrum of a multidimensional Anderson tight binding model
Local fluctuation of the spectrum of a multidimensional Anderson tight binding model
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多维安德森紧束缚模型光谱的局部涨落
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发表时间：
1996
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通讯作者：
Nariyuki Minami
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作者：
Nariyuki Minami
考虑Anderson紧束缚模型H = − Δ + V作用于l~2（Zd）及其对有限超立方体H~Λ <$Zd的限制.其中V ={Vx; x ∈ Zd}是由独立同分布的随机变量组成的随机势。设{Ej（Λ）} j是H Λ的特征值，且设Ej（Λ，E）=| Λ|（Ej（Λ）-E），j <$1，是它的重标度特征值。然后，假设绿色函数的分数阶矩的指数衰减在复能量E附近成立，态密度n（E）在E处存在，我们将证明随机序列{n_j（Λ，E）}_j，作为R_1上的点过程，当Λ变大时，弱收敛到具有强度测度（E）dx的平稳Poisson点过程，从而推广了Molchanov对一维连续随机Schr dinger算子证明的结果。另一方面，最近Aizenman，Molchanov和Graf建立了绿色函数分数阶矩的指数衰减，作为证明大无序或极端能量下安德森局域化的技术引理。因此，本文的结果可以概括如下：在期望安德森局域化的能量E附近，如果Λ很大，则H Λ的本征值之间没有相关性。
We consider the Anderson tight binding modelH=−Δ+V acting inl2(Zd) and its restrictionHΛ to finite hypercubes Λ⊂Zd. HereV={Vx;x∈Zd} is a random potential consisting of independent identically distributed random variables. Let {Ej(Λ)}j be the eigenvalues ofHΛ, and let ξj(Λ,E)=|Λ|(Ej(Λ)−E),j≧1, be its rescaled eigenvalues. Then assuming that the exponential decay of the fractional moment of the Green function holds for complex energies nearE and that the density of statesn(E) exists atE, we shall prove that the random sequence {ξj(Λ,E)}j, considered as a point process onR1, converges weakly to the stationary Poisson point process with intensity measuren(E)dx as Λ gets large, thus extending the result of Molchanov proved for a one-dimensional continuum random Schrödinger operator. On the other hand, the exponential decay of the fractional moment of the Green function was established recently by Aizenman, Molchanov and Graf as a technical lemma for proving Anderson localization at large disorder or at extreme energy. Thus our result in this paper can be summarized as follows: near the energyE where Anderson localization is expected, there is no correlation between eigenvalues ofHΛ if Λ is large.