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

在这个项目中，我们对声学方程、Klein-Gordon方程、Schro“dinger方程的各种波传播现象感兴趣。研究的主要部分是波在磁场中传播现象的渐近性。我们进一步研究了在应用物理中重要的散射正问题和反问题。为了更具体地解释这些结果，我们在这里列出了文献(1)、(2)和(3)的摘要。(1)在一般散射理论的框架下，我们综述了薛定谔方程、Klein-Gordon方程和波动方程的一些基本问题。下列主题是在适当的衰变和/或小的条件下处理的微扰项。广义特征函数的增长估计，预解估计，散射正问题和反问题，光滑性和Strichartz估计。由于我们提出的加权能量法，一些主题自然地扩展到含时和/或非自伴扰动。(2)我们处理无限射线和一点相连的环的图上的逆散射问题。我们的问题相当于在算子散射数据的基础上重建势。(3)本文综述了具有强奇异性的具有外势的磁薛定谔算子的一些基本结果。在适当的磁场和外势衰减条件下，讨论了下列问题：算子的自伴性，广义特征函数的增长性估计，极限吸收原理，一致预解估计，以及相应发展方程的光滑性。

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.