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

磁场中的波传播现象和逆散射问题分析

基本信息

批准号：
22540204
负责人：
MOCHIZUKI Kiyoshi
金额：
$ 1.91万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2010
资助国家：
日本
起止时间：
2010 至 2012
项目状态：
已结题

项目摘要

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.

在这个项目中，我们感兴趣的是声学，克莱因-戈登，薛定谔方程的各种波传播现象。本文主要研究电磁波在磁场中传播的渐近性。进一步研究了应用物理中重要的散射正问题和散射反问题。为了更具体地解释结果，我们在这里列出论文（1），（2）和（3）的摘要。(1)本文在广义散射理论的框架下，综述了薛定谔方程、克莱因-戈登方程和波动方程的一些基本问题。在适当的衰变和/或小性条件下，对扰动项进行了讨论。广义本征函数的增长估计，预解估计，散射正问题和反问题，光滑性质和齐次估计。由于我们制定的加权能量方法，一些主题自然扩展到时间相关和/或非自伴扰动。(2)本文讨论了一个图上的逆散射问题，该图中有一条无限长的射线和一个在一点上连接的环。我们的问题实质上是在算符散射数据的基础上重建位势。(3)本文综述了具有强奇异性的带外势的磁Schro“dinger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，广义本征函数的增长估计，极限吸收原理，一致预解式估计，以及相应演化方程的光滑性质。