Well-Posed Inverse Problems

从散射幅度确定量子力学势的问题可以追溯到量子力学的起源。问题的一个方面是找到问题适定的数据集，即从势到散射数据集的映射是连续可逆的。这项工作主要集中在一个这样的集合上，即所谓的后向散射数据。它是将散射幅度限制在与入射平面波方向相反的方向上而产生的数据。在三维和更高的维度中，背向散射和势的关系被很好地记录下来。在二维情况下或在数据限于半空间的情况下，在本项目过程中将分析一些障碍。关于逆散射的其他工作将集中在具有可变声速的波动方程和具有电势和磁势的薛定谔方程。第二个研究方向是定义在具有楔形边界的区域上的双曲型偏微分方程解的奇性传播，以及存在边界楔形区域时二阶方程初边值问题解的奇性传播。偏微分方程式是物理科学中数学建模的基础。人们知道，涉及运动、材料和能量等连续变化的现象遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。本提案中描述的反问题在这些研究中起着核心作用。它们的目的不是陈述关系，而是从这些关系中提取定性和定量的意义。

The problem of determining a quantum mechanical potential from its scattering amplitude goes back to the beginning of quantum mechanics. One aspect of the problem which has received little attention is finding data sets for which the problem is well-posed, i.e., that the mapping from the potential to the scattering data set is continuously invertible. This work focuses on one such set, the so-called backscattering data. It is the data which occurs from the restriction of the scattering amplitude to the direction opposite to the direction of the incident plane wave. In dimensions three and higher, the relationship of the backscattering and potential is well documented. In two dimensions or in cases where the data is restricted to half-spaces, there are obstacles which will be analyzed during the course of this project. Other work on inverse scattering will focus on the wave equation with variable speed of sound and the Schrodinger equation with both electric and magnetic potential. A second line of investigation concerns hyperbolic partial differential equations defined in domains with wedgelike boundaries and the propagation of singularities of solutions of initial-boundary value problems for second order equations in the presence of boundary wedges. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The inverse problems described in this proposal play a central role in these studies. Their object is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.