Spectral and Scattering Theory for Schrodinger Operators

这个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中的新问题。本项目已取得了相当多的研究成果，本文仅介绍了本项目主要研究者和合作者的部分成果.利用相空间隧穿理论，证明了在恒磁场存在下，薛定谔算符本征函数的指数衰减率在半经典极限下较大.研究了散射的半经典渐近性。特别是，它表明，光谱位移函数有一个快速跳跃（大小为2π倍整数）附近的每个量子共振.结果表明，对应于两个不相交的能量面之间的相互作用的散射矩阵的系数在半经典极限下呈指数衰减。一种新的方法来分析相空间隧穿的开发和使用（与A.马丁内斯，V.Sordoni的联合工作）。证明了在Anderson型随机磁场作用下，二维离散Schrodinger算符和连续Schrodinger算符（任意维数）积分态密度的Lifshitz尾.基于谱移函数理论，给出了Wegner估计的一个新的证明。Wegner估计在随机薛定谔算子的安德森局部化的证明中起着至关重要的作用（与J.M.Combes，P.D.Hislop的联合工作）。

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).