Problems in Mathematical Physics

派：巴里·西蒙，加州理工学院DMS-0140592摘要西蒙教授将继续他的研究，重点是薛定谔算子及其离散类似物的正、逆谱问题。他计划探索和规则对谱理论的影响，包括雅可比矩阵(他和Killip最近使用和规则解决了来自正交多项式社区的几个长时间开放猜想)和薛定谔算子。他预计要研究的其他问题包括离散谱问题的等谱流形结构、高维共振计算以及混合奇异连续谱和点谱的研究。在广泛的科学领域中，一个基本问题是确定某些只能间接观察的对象与你间接观察到的信息之间的关系。问题有两个方面：确定对于系统的给定状态间接观察到的东西，称为正问题；以及试图从间接观测中归纳出系统的性质，称为逆问题。西蒙教授将研究量子力学正反问题的各个方面。虽然重点将放在量子理论的方程式上，但层析成像和雷达/声纳的正反问题也有潜在的衍生问题。

PI: Barry Simon, California Institute of TechnologyDMS-0140592AbstractProfessor Simon will continue his research focusing on direct and inversespectral problems for Schrodinger operators and their discrete analogs. Heplans to explore the impact of sum rules for spectral theory both for Jacobimatrices (where he and Killip recently solved several long open conjecturesfrom the Orthogonal Polynomial community using sum rules) and Schrodingeroperators. Other problems he expects to look at include the structure ofthe isospectral manifold for problems with discrete spectrum, resonancecounting in higher dimensions, and the study of mixed singular continuousand point spectrum.A fundamental problem in wide swaths of science is determining the relationbetween some object that can only be observed indirectly and the informationyou can observe indirectly. There are two sides of issue: determine whatis indirectly observed for a given state of the system, called the directproblem, and trying to induce properties of the system from the indirectobservations, called the inverse problem. Professor Simon will studyvarious aspects of direct and inverse problems in quantum mechanics. Whilethe focus will be on the equations of quantum theory, there are potentialspinoffs to tomography and radar/sonar direct and inverse problems.