Research on Problems Related to Random Schrodinger Operators

随机薛定谔算子相关问题的研究

• 批准号: 15540166
• 负责人: UEKI Naomasa
• 金额: $ 2.18万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540166/
• 关键词: Random Schrodinger Operators; Stochastic Analysis; Operator Theory; Spectrum; Anderson Localization; Random Field; Differential Equations; Wegner Estimates; Wenger評価; 状態密度関数; random Shrodinger作用素

The aim of this project is to study problems related to random Schrodinger operators from various viewpoints. The main problem is related to the Anderson transition. For this, we gave a Wegner type estimate for a class of random Schrodinger operators with electromagnetic potentials given by Gaussian random fields, and proved the Anderson localization at sufficiently low energy for these operators. This is the first proof of the strong dynamical localization for Gaussian random potentials. For the proof, we extend Klopp's vector field method for models with lattice structures to our models without lattice structures and extend Germinet-Klein theory on the Anderson localization for operators bounded below to our models unbounded below. On the other hand, the other attractive problem is to study the asymptotic behavior of the integrated density of states. For this, we study that of the Pauli Hamiltonian with a random magnetic field whose mean is zero, and we gave a lower bound on a neighborhood of zero in terms of logarithmic functions in a basic case where the magnetic field is uniform in one direction. This shows that our integrated density of states increases rapidly at zero and our Pauli Hamiltonian has many states whose energies are close to zero. This phenomenon is contrary to the phenomenon known as Lifschitz tail for other random Schrodinger operators where the integrated density of states increases slowly. The speed of the increment we showed is the fast one in the known results except for the case that the integrated density of states has a jump. We showed this by applying the Aharonov-Casher theory rigorously.

该项目的目的是从不同的角度研究与随机薛定谔算子相关的问题。主要问题与安德森转变有关。为此，我们对一类具有由高斯随机场给出的电磁势的随机薛定谔算子进行了韦格纳型估计，并证明了这些算子在足够低能量下的安德森定域性。这是高斯随机势强动态局域化的第一个证明。为了证明，我们将具有晶格结构的模型的 Klopp 向量场方法扩展到没有晶格结构的模型，并将下界算子的安德森定位的 Germinet-Klein 理论扩展到我们的无界模型。另一方面，另一个有吸引力的问题是研究积分态密度的渐近行为。为此，我们研究了均值为零的随机磁场的保利哈密顿量，并且在磁场在一个方向上均匀的基本情况下，我们以对数函数的形式给出了零邻域的下界。这表明我们的积分态密度在零时迅速增加，并且我们的泡利哈密顿量有许多能量接近于零的态。这种现象与其他随机薛定谔算子的利夫希茨尾部现象相反，其中状态的积分密度缓慢增加。我们展示的增量速度是已知结果中除了积分态密度有跳跃的情况之外最快的。我们通过严格应用阿哈罗诺夫-卡舍尔理论来证明这一点。

项目成果

A viewpoint on positivity of pseudodifferential operators from the Wick calculus

从Wick演算看赝微分算子的正性

• 发表时间: 2005
• 期刊: 京都大学数理解析研究所講究録 1412
• 作者: Nicolas Lerner;Yoshinori Morimoto
• 通讯作者: Yoshinori Morimoto

Regularity of weak solutions for a class of infinitely degenerate elliptic semilinear equations

一类无限简并椭圆半线性方程弱解的正则性

• 发表时间: 2004
• 期刊: Seminaire Equations aux Derivees Partielles 2003-2004
• 作者: Yoshinori Morimoto;Chao-Jiang Xu
• 通讯作者: Chao-Jiang Xu

ガウスのTheorema elegantissimum

高斯优雅定理

• 发表时间: 2004
• 期刊: 津田塾大学 数学・計算機科学研究所報 25
• 作者: Yoshinori Morimoto;Chao-Jiang Xu;西和田公正
• 通讯作者: 西和田公正

H.Tsuiki: "A Domain Theoretic Semantics of LAX generic functions"Theoretical Computer Science. 307-311 (2003)

H.Tsuiki：“LAX 通用函数的领域理论语义”理论计算机科学。

A thermodynamic limit of relative Fredholm determinant of Green operators for random Schrodinger operators

随机薛定谔算子的格林算子相对 Fredholm 行列式的热力学极限

• 发表时间: 2004
• 期刊: Markov Processes and Related Fields 9・4
• 作者: Yoshinori Morimoto;Chao-Jiang Xu;西和田公正;Shin-ichi Kotani;Naomasa Ueki;Naomasa Ueki;Kimimasa Nishiwada;Shin-ichi Kotani;N.Ueki;N.Ueki;H.Tsuiki;S.Kotani
• 通讯作者: S.Kotani