Study of asymptotic distribution of the Schroedinger operators with strong magnetic fields

Iwasuka证明了具有泊松随机势的薛定谔算子的谱是整个实线势有负部分。本文利用正则换易关系的摄动方法，对恒定磁场被多个螺线线（点）磁场摄动时朗道能级间特征值的个数进行了详细估计。Doi通过分析Hamilton流的渐近行为，研究了带扰动的二次势薛定谔方程解的奇异性结构。Ito和Tamura证明了具有多个类delta磁场的薛定谔算子的自伴随性、波算子的渐近完备性和特征值的不存在性，并给出了它们的散射振幅在磁场中心距离趋于无穷大时的渐近行为的前导项。Tamura还证明了二维Dirac算子在光滑场近似时解的范数收敛性。他还利用他最近关于无界算子的Trotter积公式在算子范数意义上收敛的结果，得到了奇异势情况下薛定谔半群核的渐近公式。

Iwasuka showed that the spectrum of the Schroedinger operators with Poisson random potential is the whole real line potential has some negative part.Mine obtained some detailed estimate of the number of the eigenvalues lying between the Landau levels when constant magnetic field is perturbed by multiple solenoidal (point) magnetic field, by using method of the perturbation of the canonical commutation relation.Doi studied the structure of the singularity of the solution to the Schroedinger equations with quadratic potential with perturbation by using the analysis of the asymptotic behavior of the Hamilton flow.Ito and Tamura proved the selfadjointness, the asymptotic completeness of the wave operator and the nonexistence of the eigenvalue for the Schrodinger operators with multiple delta like magnetic fields, and gave the leading term of the asymptotic behavior of their scattering amplitude when the distance between the centers of the magnetic fields tends to infinity.Tamura also proved the convergence in norm of the resolvent of the two dimensional Dirac operators when a delta like magnetic field is approximated by smooth fields. He also obtained an asymptotic formula for the kernel of the Schroedinger semigroup in the case of singular potentials by applying his recent result that the Trotter product formula converges in the sense of operator norm for unbounded operators.