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

点击翻译按钮获取中文摘要

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.