Spectral and Scattering Theory for Schrodinger Operators

本项目的目的是研究一般薛定谔算子的谱和散射理论。此外，它还意在探索量子物理中的新问题领域和相关课题。在该项目中已取得了不少研究成果，仅给出了一些由首席研究员和合作者精选的结果。利用相空间隧穿理论，证明了在恒定磁场作用下，薛定谔算子本征函数在半经典极限下的指数衰减率较大。研究了散射的半经典渐近性。特别地，研究表明，在每个量子共振附近，光谱位移函数都有一个快速跳跃(大小为2π×整数)。结果表明，与两个不相交的能面相互作用对应的散射矩阵的系数在半经典极限下呈指数衰减。发展并采用了一种分析相空间隧穿的新方法(与A.Martinez，V.Sordoni共同工作)。证明了具有Anderson型随机磁场的二维离散薛定谔算符和连续薛定谔算符(任意维算符)的积分态密度的Lifshitz尾。基于谱移位函数理论，给出了Wegner估计的一种新证明。Wegner估计在随机薛定谔算子的Anderson局部化证明中起着至关重要的作用(与J.M.Combes，P.D.Hislop共同工作)。

The purpose of this project is to investigate the spectral and scattering theory for Schrodinger operators in general. Moreover, it is also intended to explore new area of problems in quantum physics and related topics. Quite a few reserch results has been obtained in the project, and only a selected results by the head investigator and collaborators are presented here.1. By employing the theory of phase space tunneling, it is proved that the exponential decay rate of eigenfunctions for Schrodinger operator is larger in the semiclassical limit in the presence of constant magnetic field.2. Semiclassical asymptotics of the scattrering is investigated. In particular, it is shown that the spectral shift function has a rapid jump (of the size 2π times integer) near each quantum resonance.3. It is shown that the coefficients of the scattering matrix corresponding to the interaction between two nonintersecting energy surfaces decay exponentially in the semiclassical limit. A new method to analyze the phase space tunneling is developed and employed (joint work with A.Martinez, V.Sordoni).4. The Lifshitz tail for the integrated density of states is proved for 2 dimensional discrete Schrodinger operators and continuous Schrodinger operators (arbitrary dimension) with Anderson-type random magnetic fields.5. A new proof of the Wegner estimate based on the theory of the spectral shift function is developed. The Wegner estimate plays crucial role in the proof of Anderson localization for random Schrodinger operators (joint work with J.M.Combes, P.D.Hislop).