Scattering and Inverse Scattering for Linear and Nonlinear Wave Propagations

在这个项目中，我们研究了物理学基本方程如麦克斯韦方程和薛定谔方程的几种波传播现象。本文的主要研究内容是散射理论和逆散射理论，这些问题不仅在数学分析领域而且在应用科学领域都是重要的，本文将总结主要研究者的结果，文[1]发展了系数既依赖于空间又依赖于时间的外区域波动方程的散射理论。时空加权能量估计在这个问题中起着至关重要的作用。文[5]中我们研究了含时势的Schr“odinger方程的类似问题。空时LAp-LAq估计在这个问题中变得重要。文[2]给出了具有振荡长程势的Schr“odinger算子的广义本征函数的Rellich型渐近估计。结果表明了该算子的限制吸收原理。[3]是与M.教授的联合工作。Nakao等提出的方法，给出了具有耗散的外区域波动方程的能量衰减的一个标准条件，该条件仅在无穷远处有效。文[4]研究了图上Schr“odinger方程的逆散射。这些问题最近被认为是重要的，不仅在经典领域的纳米尺度技术（如固体物理学和电子学），而且在信息技术领域。[4]是与V. Marchenko教授和I. Trooshin，并限于最简单的图，包括一个紧凑的部分。我们的联合工作将进行到更一般的图，包括紧凑的部分。本文在Kluwer Academic Publishers，ISAAK 10（2003），303-316的基础上，进一步研究了非自伴波动方程的散射反问题.

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).