Research on spectrum of random magnetic Schr {"o} dinger operators

我们考虑$mathbf{R}^2$上的磁薛定谔算子。磁场是晶格上均匀磁场和周期性变化的点状磁场的总和。我们将给出每个Landau能级是无穷退化本征值的一个充分条件。这一条件对于最低的朗道能级也是必要的。在阈值情况下，我们看到在最低朗道能级附近的光谱是纯绝对连续的。此外，我们还将给出朗道能级及其能隙的态密度的估计。证明是基于Geyler和v Sv‘iv cek的方法、磁布洛赫理论和正则对易关系。

We consider the magnetic Schr"odinger operator on $mathbf{R}^2$. The magnetic field is the sum of a homogeneous magnetic field and periodically varying pointlike magnetic fields on a lattice. We shall give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. This condition is also necessary for the lowest Landau level. In the threshold case, we see that the spectrum near the lowest Landau level is purely absolutely continuous. Moreover, we shall give an estimate for the density of states for Landau levels and their gaps. The proof is based on the method of Geyler and v Sv tov'iv cek, the magnetic Bloch theory, and canonical commutation relations.