Scattering and Inverse Scattering for Linear and Nonlinear Wave Propagations

线性和非线性波传播的散射和逆散射

基本信息

批准号：
16540204
负责人：
MOCHIZUKI Kiyoshi
金额：
$ 2.3万
依托单位：
Chuo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540204/
关键词：
Wave equation Schr"odinger equation scattering theory Problem of inverse scattering Inverse scattering on graphs time dependent potential scattering theory time dependent perturbation wave equation Schro" dinger equation inverse scatte

项目摘要

In this project we have been studying several kind of wave propagation phenomena for the fundamental equations of physics like Maxwell equations and Schr"odinger equations. Our main subjects are in the scattering theory and inverse scattering theory, and these problems are important not only in the fields of mathematical analysis but also in those of applied sciences.We shall summarize the results of the head investigator.In [1] is developed the scattering theory for wave equations in exterior domain with coefficients depending on both space and time. The space-time weighted energy estimates play the crucial role in this problem. In [5] we studied similar problems for Schr"odinger equations with time dependent potentials. The space time LAp-LAq estimates becomes important in this problem. In [2] is obtained a Rellich type asymptotic estimates for generalized eigenfunctions for the Schr"odinger operator with oscillating longrange potentials. The results are applied to show the principle of limiting absorption for this operator. [3] is the joint work with Prof. M. Nakao, and a standard decay condition of energy is shown for wave equations in exterior domain with dissipation which is effective only near infinity. In [4] is studied the inverse scattering for Schr"odinger equations on graphs. These problems are recently recognized to be important not only in the classical field of nano-scale technology (like solid state physics and electrionics) but also in the filed of information stechnology. [4] is obtained as a joint work with Profs V. Marchenko and I. Trooshin, and is restricted to the simplest graph including a compact part. Our joint works will proceed into more general graphs including compact parts. Another work is done on the scattering inverse problem for non-selfadjoint wave equation as a continuation of the former work of the same title (Kluwer Academic Publishers, ISAAK 10 (2003), 303-316).

在这个项目中，我们一直在研究麦克斯韦方程和薛定谔方程等物理基本方程的几种波传播现象。我们的主要研究对象是散射理论和逆散射理论，这些问题不仅在数学分析领域以及应用科学领域。我们将总结首席研究员的结果。在[1]中发展了外域波动方程的散射理论，其系数取决于空间和时间。加权能量估计发挥着在这个问题中，我们研究了具有时间相关势的薛定谔方程的类似问题。时空 LAp-LAq 估计在此问题中变得很重要。在[2]中获得了具有振荡长程势的薛定格算子的广义本征函数的Rellich型渐近估计。结果用于显示该算子的限制吸收原理。[3]是与Prof.的共同工作。 M. Nakao 给出了具有耗散的外部域波动方程的标准能量衰减条件，该条件仅在无穷大附近有效。在[4]中研究了逆散射。图上的薛定谔方程。这些问题最近被认为不仅在纳米级技术的经典领域（如固态物理和电子学）而且在信息技术领域都很重要。 [4] 是与 V. Marchenko 教授和 I. Trooshin 教授的联合工作获得的，并且仅限于包括紧凑部分的最简单的图。我们的联合工作将继续研究更通用的图表，包括紧凑的零件。另一项工作是关于非自共轭波动方程的散射反问题，作为同名前一工作的延续（Kluwer 学术出版社，ISAAK 10 (2003), 303-316）。