Mathematical Sciences: Well-Posed Inverse Problems

这项工作的对象，反问题的研究， 数学分析，是恢复物理性质之一 从远程获取的数据中对物体或现象进行分析 意见。 问题是适定的，因为 数据集并不是那么繁重，以至于排除了任何解决方案，但 足以重建。确定a的问题 量子力学势从它的散射振幅 回到量子力学的开端。 但很少有人关注 是用来寻找数据集的， 适定的意思是从势到 散射数据是连续可逆的。 一个可能 这样的数据集的候选者是后向散射数据， of this research研究. 已经完成了许多成功的工作， 这个问题在三个方面，虽然主要限于小 潜力 本研究试图消除这一假设， 寻找更一般的结果，可能放弃一些程度 良好的姿态。 工作将继续进行 二维反向散射，以及，其中一种类型的 参数中的奇异性使得后向散射更加 难以处理的问题，即使假设紧凑的支持， 潜在的。在这个项目的背景下，反问题是 用经典方程表示，例如 薛定谔方程这些方程可以表示波 涉及磁势和电势的现象，声学 和其他类型的波在障碍物的存在。 虽然 大部分的研究集中在数学分析的 逆映射，它与物理世界的联系是 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.