Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. I. Asymptotic Expansion Formulas
Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. I. Asymptotic Expansion Formulas
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随机离散薛定谔算子谱的极值理论 I. 渐近展开公式。
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发表时间：
2008
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通讯作者：
A. Astrauskas
中科院分区：
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作者：
A. Astrauskas
本文研究了多维格环面上随机薛定谔算子的谱问题，其中随机薛定谔算子的独立同分布是从格环面增加到整个格环面。势（安德森哈密顿）。在中秩情形下，只要势的上分布尾在无穷远处的衰减慢于双指数函数，我们得到了极值本征值和本征函数的显式几乎处处渐近展开式.对于分数指数尾（包括威布尔分布和高斯分布），证明了特征值的极值型极限定理，并描述了模型参数对规范化常数指定的强烈影响.在证明中，我们使用了基于簇展开的有限秩扰动参数作为预解式，我们的结果说明了谱的极值理论与独立同分布的极值性质之间的密切联系。潜力另一方面，相应的本征函数的局部化性质给出了抛物型安德森模型的长时间不稳定性的基本信息。
We consider the spectral problem for the random Schrödinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We obtain the explicit almost sure asymptotic expansion formulas for the extreme eigenvalues and eigenfunctions in the intermediate rank case, provided the upper distributional tails of potential decay at infinity slower than the double exponential function. For the fractional-exponential tails (including Weibull’s and Gaussian distributions), extremal type limit theorems for eigenvalues are proved, and the strong influence of parameters of the model on a specification of normalizing constants is described. In the proof we use the finite-rank perturbation arguments based on the cluster expansion for resolvents.The results of our paper illustrate a close connection between extreme value theory for spectrum and extremal properties of i.i.d. potential. On the other hand, localization properties of the corresponding eigenfunctions give an essential information on long-time intermittency for the parabolic Anderson model.