Mathematical Sciences: Well-Posed Inverse Problems

这项工作的目的是研究数学分析中的逆问题，从远程观测获得的数据中恢复物体或现象的物理性质。从数据集没有累赘到排除任何解决方案的意义上说，这些问题是很好的，但对于重建来说是足够的。从散射振幅确定量子力学势的问题可以追溯到量子力学的开始。但是，很少有人注意到寻找问题是适定的数据集，即从势到散射数据的映射是连续可逆的。这种数据集的一个可能的候选者是本研究的主题——后向散射数据。在三维空间的问题上已经完成了许多成功的工作，尽管主要局限于小的势能。目前的研究试图消除这一假设，寻找更普遍的结果，可能会放弃某种程度的适切性。二维后向散射的研究也将继续进行，其中参数中的一种奇点使得后向散射成为一个更加难以处理的问题，即使假设有紧凑的势能支持。在这个项目的背景下，逆问题是根据经典方程，如薛定谔方程来表述的。这些方程可以表示在存在障碍物的情况下，涉及磁和电势、声学和其他类型的波的波动现象。虽然大部分研究都集中在逆映射的数学分析上，但它与物理世界的联系很容易追踪。

The object of this work, the study of inverse problems in mathematical analysis, is one of recovering physical properties of an object or phenomenon from data obtained by remote observations. The problems are well-posed in the sense that the data sets are not so burdensome as to rule out any solutions, but sufficient for the reconstruction. The problem of determining a quantum mechanical potential from its scattering amplitude goes back to the beginning of quantum mechanics. But little attention is given to finding data sets for which the problem is well-posed in the sense that the mapping from the potential to the scattering data is continuously invertible. One likely candidate for such a data set is backscattering data, the subject of this research. Much successful work has been accomplished on the question in three dimensions, though mainly confined to small potentials. The present work seeks to remove this assumption and look for more general results, possibly giving up some degree of well-posedness. Work will continue on two-dimensional backscattering as well, where a type of singularity in a parameter makes backscattering a much more untractable problem, even assuming compact support of the potential.Inverse problems in the context of this project are formulated in terms of classical equations, such as the Schrodinger equation. These equations may represent wave phenomena involving magnetic and electric potentials, acoustics and other type of waves in the presence of obstacles. Although much of the research focuses on the mathematical analysis of the inverse mapping, its connections with the physical world are easily traced.