Mathematical Sciences: Inverse Scattering Theory

本次颁奖期间将重点开展工作 关于高维逆散射问题和相关的 一维问题。 这项工作的基本思想是， 自然是从测量的数据中构建障碍， 粒子通过障碍物。 从未来的过渡到 解，比如说，波动方程，到过去被称为 散射变换 关于变换或其 频谱被认为是理解任何波或 经历散射的广义波。 这项工作将考虑延长结果的时间依赖 具有缓慢衰减势的薛定谔方程，其中 逆方程允许简单极点。 高维 因此，不能排除多极化的出现。 但其结构尚不清楚。 问题 至于会否累积，亦会研究。 一种方法是考虑时间的解决方案 不一定有界的非独立薛定谔方程 及时 该方法已被用于实现完整的 一维nxn逆的解（对所有势） 散射问题以及一维n阶 标量方程 几个经典的方程和系统， 预计将使用通过此开发的结果进行分析 调查 其中包括扎哈罗夫-沙巴特系统和 KPI方程。

Work to be done during the period of this award will focus on higher dimensional inverse scattering problems and on related one dimensional problems. The underlying idea for work of this nature is that of constructing obstacles from data measured as particles pass the obstacle. The transition from the future of a solution of, say, the wave equation, to the past is called the scattering transform. Knowledge of either the transform or its spectrum is considered to be central to understanding any wave or generalized wave which experiences scattering. This work will consider extending results on time dependent Schrodinger equations with slowly decaying potentials in which the inverse equation admits simple poles. In higher dimensional settings the advent of multiple poles cannot be ruled out. However, their structure is not understood. The question of whether or not they may accumulate will also be studied. One method to be used will consider solutions for time dependent Schrodinger equations which are not necessarily bounded in time. This method has been used to achieve the complete solution (for all potentials) for the one dimensional nxn inverse scattering problem as well as for the one dimensional n-th order scalar equation. Several classical equations and systems are expected to be analyzed using results developed through this investigation. They include the Zakharov-Shabat system and the KPI equation.