Spectral and Scattering Theory for Schrodinger Operators

本项目的目的是研究一般薛定谔算子的光谱和散射理论。此外，它还旨在探索量子物理和相关主题的新问题领域。在这个项目中已经取得了相当多的研究成果，这里只介绍了一些主要研究者和合作者的研究成果。利用相空间隧穿理论，证明了恒定磁场存在下薛定谔算子的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。特别是，谱移函数在每个量子共振附近有一个快速的跳跃（大小为2π乘以整数）。结果表明，在半经典极限下，两个不相交的能量面相互作用对应的散射矩阵的系数呈指数衰减。与A. martinez， V.Sordoni合作，提出并应用了一种分析相空间隧道的新方法。对于具有anderson型随机磁场的二维离散薛定谔算子和连续薛定谔算子（任意维），证明了态密度积分的Lifshitz尾。基于谱移函数理论，提出了一种新的Wegner估计的证明方法。Wegner估计在随机薛定谔算子的Anderson定位证明中起着至关重要的作用（与J.M.Combes， P.D.Hislop联合工作）。

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).