Well-Posed Inverse Problems

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。该问题的一个很少受到关注的方面是寻找问题是适定的数据集，即从势到散射数据集的映射连续可逆。这项工作的重点是一组这样的数据，即所谓的反向散射数据。它是由于散射振幅被限制到与入射平面波方向相反的方向而产生的数据。在三维及更高的维度中，后向散射与势的关系是有充分文献记载的。在二维或在情况下，数据被限制在半空间，有障碍，将在这个项目的过程中进行分析。反散射的其他工作将集中在变声速波动方程和同时具有电势和磁势的薛定谔方程上。第二个研究方向是关于在楔形边界域中定义的双曲型偏微分方程，以及在边界楔存在下二阶方程初边值问题解的奇异性传播。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。本提案中描述的逆问题在这些研究中起着核心作用。他们的目的不是陈述关系，而是从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.