含δ杂质朗道哈密顿量谱的特征
Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities
摘要：
考虑外磁场中的随机薛定谔算子。随机势由随机强度的δ函数组成，δ函数位于规则二维晶格的位置上。当磁场足够强时，我们描述了这个随机哈密顿量的最低N朗道带中的谱，这取决于N。我们表明，在这些频带的频谱是完全纯点，符合朗道能级的能量是无限退化的，对应的能量在频谱的其余部分的本征函数是本地化的一致有界的本地化长度。通过将哈密顿量与格算子相关联，我们能够使用Aizenman-Molchanov方法来证明局部化。
英文摘要：
We consider a random Schrodinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman-Molchanov method to prove localization.
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