Well-Posed Inverse Problems

确定量子力学势的问题 从它的散射振幅可以追溯到 量子力学 问题的一个方面， 很少有人注意到的是， 适定的，即，从势能到 散射数据集是连续可逆的。 这项工作 集中在一个这样的集合，所谓的后向散射数据。 它 是从散射的限制中产生的数据 的方向相反的方向上的振幅 入射平面波 在三维和更高的维度中， 后向散射与电势有良好的关系 记录在案。 在二维空间中，或者在数据 限制在半空间，有障碍，这将是 在这个项目的过程中进行了分析。 上的其他工作 逆散射将集中在波动方程与变量 声速和薛定谔方程， 磁势。第二条调查线涉及 双曲型偏微分方程 楔形边界和奇点的传播 二阶初边值问题的解 方程中存在边界楔。 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 中描述的逆问题 这一建议在这些研究中发挥了核心作用。其对象 并不是陈述它们之间的关系，而是提取 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.