Spectral statistics for random Schr"odinger operators in the localized regime
Spectral statistics for random Schr"odinger operators in the localized regime
复制标题
局部区域随机薛定格算子的谱统计
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
F. Klopp
中科院分区：
文献类型：
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作者：
F. Germinet;F. Klopp
研究了局域状态下随机哈密顿量的特征值和特征函数的各种统计量。考虑局域相中能量为E的随机哈密顿量。假设状态密度函数在$E$附近不是太平坦。将其限制为某个大的立方体$Lambda$。现在考虑$I_Lambda$，一个以$E$为中心的小能量区间，当立方体$Lambda$的体积增长到无穷大时，它渐近地包含了无限多个特征值。我们证明了，在大体积极限下，随机哈密顿函数的特征值是由独立的同分布随机变量给出的，概率为1，误差可达立方体体积的任意次幂。因此，我们导出了局部展开特征值的均匀泊松行为，以及大尺度范围内展开能量和展开局域中心联合分布的泊松行为。*展开的水平间距的分布，局部和全局，*展开的定位中心的分布，局部和全局。
We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$. Restrict it to some large cube $Lambda$. Consider now $I_Lambda$, a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $Lambda$ grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive * uniform Poisson behavior of the locally unfolded eigenvalues, * a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales. * the distribution of the unfolded level spacings, locally and globally, * the distribution of the unfolded localization centers, locally and globally.