Mathematical Sciences: Inverse Scattering Theory

在此奖项期间所做的工作将集中于高维逆散射问题和相关的一维问题。这种性质的工作的基本思想是根据粒子通过障碍物时测量的数据来构建障碍物。从波动方程的未来解到过去解的过渡称为散射变换。对变换或其谱的了解被认为是理解任何经历散射的波或广义波的核心。这项工作将考虑扩展具有缓慢衰减电位的时间相关薛定谔方程的结果，其中逆方程允许简单极点。在更高维度的设定中，多极的出现不能被排除。然而，它们的结构尚不清楚。它们是否会积累的问题也将被研究。一种方法是考虑不一定在时间上有界的时变薛定谔方程的解。该方法已被用于实现一维nxn逆散射问题和一维n阶标量方程的完全解（所有势）。几个经典方程和系统预计将被分析使用的结果，通过这项调查发展。它们包括Zakharov-Shabat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.