Research on Problems Related to Random Schrodinger Operators

本课题的目的是从多个角度研究与随机薛定谔算子有关的问题。主要问题与安德森转变有关。为此，我们给出了一类电磁势为高斯随机场的随机Schrodinger算子的Wegner型估计，并证明了这类算子在足够低能量下的安德森局部化.这是高斯随机势的强动力学局域性的第一个证明。对于证明，我们扩展Klopp的向量场方法的模型与晶格结构，我们的模型没有晶格结构和扩展Germinet-Klein理论的安德森本地化的运营商有界以下我们的模型无界以下。另一方面，另一个吸引人的问题是研究积分态密度的渐近行为。为此，我们研究的泡利哈密顿与随机磁场的平均值为零，我们给出了一个下界的邻域为零的对数函数在一个基本的情况下，磁场是均匀的在一个方向上。这表明我们的积分态密度在零处迅速增加，我们的泡利哈密顿量有许多能量接近零的态。这种现象与其他随机薛定谔算符的Lifschitz尾现象相反，在Lifschitz尾中，积分态密度缓慢增加。除了积分态密度有跳跃的情况外，我们给出的增量速度是已知结果中最快的。我们通过严格应用Aharonov-Casher理论来证明这一点。

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.