Analysis of wave propagation phenomena in the magnetic fields and inverse scattering problems

在这个项目中，我们对声学，Klein-Gordon, Schro“dinger方程的各种波传播现象感兴趣。研究的主要部分是磁场中波传播现象的渐近性。进一步研究了应用物理中重要的散射正逆问题。为了更具体地解释结果，我们在这里列出了(1)、(2)和(3)论文的摘要。(1)在一般散射理论的框架叉中，我们研究了Schro " dinger、Klein-Gordon和波动方程的一些基本问题。以下主题是在适当的衰减和/或摄动项的小条件下处理的。广义特征函数的增长估计，解决估计，散射正反问题，平滑性质和Strichartz估计。由于我们的加权能量方法的公式，一些主题自然地被扩展到时间相关和/或非自伴随扰动。(2)研究了具有无限射线和在一点连接的环的图上的逆散射问题。我们的问题就是在算子散射数据的基础上进行势的重建。(3)讨论了具有强奇异性的外势磁Schro " dinger算子的一些基本结果。在适当的磁场和外部势的衰减条件下，讨论了以下主题：算子的自伴随性、广义特征函数的生长估计、极限吸收原理、均匀分解估计和相应演化方程的平滑性质。

In this project we are interested in various wave propagation phenomena of Acoustic, Klein-Gordon, Schro”dinger equations. Main part of the study is the asymptotics of the waves propagation phenomena in the magnetic fields. We further study scattering direct and inverse problems which are important in applied physics. In order to explain the results more specifically, we list here the abstracts of the papers (1), (2) and (3).(1) We survey some basic problems of Schro”dinger, Klein-Gordon and wave equations in the frame fork of general scattering theory. The following topics are treated under suitable decay and/or smallness conditions on the perturbationterms. Growth estimates of generalized eigenfunctions, Resolvent estimates, Scattering direct and inverse problems, Smoothing properties and Strichartz estimates. Due to our formulation of the weighted energy method, some topics are naturally extended to time-dependent and/or non-selfadjoint perturbations.(2) We treat an inverse scattering problem on a graph with an infinite ray and a loop joined at one point. Our problem amounts to the reconstruction of potential on the basis of scattering data of operator.(3) In this article we survey some basic resulta for the magnetic Schro”dinger operator with external potential which has a strong singularity. The following topics are treated under suitable decay conditions on the magnetic field and external potential: Selfadjointness of the operator, Growth estimates of generalized eigenfunctions, Principle og limiting absorption, Uniform resolvent estimates, and Smoothing properties for corresponding evolution equations.