Mathematical Sciences: Mathematical Physics

小行星9401491 西蒙教授将继续研究 薛定谔算子的谱理论以及有关逆谱和逆散射问题的研究。 这两组研究都利用了对克莱因光谱位移的详细研究。 将探讨的领域包括：扩展秩一理论，包括迹类扰动，以及无界电位的逆问题，这应该以某种方式确定的特征值和Dirichlet特征值。 特别是，将试图描述这个问题的等谱流形。 西蒙教授将研究一些与量子力学有关的数学理论问题。 他工作的一个方面涉及固体和随机介质中电子模型的长期行为研究，其性质处于创新材料设计的最前沿。 西蒙还希望研究逆光谱和散射理论中的问题：从可以容易观察到的数据中确定微观结构的一般问题。 逆理论在雷达探测潜艇和解释医学断层成像等领域都是核心。 ***

9401491 Simon Professor Simon will continue his research looking into the spectral theory of Schrodinger operators and research concerning inverse spectral and inverse scattering problems. Both sets of research exploit detailed studies of the Krein spectral shift. Among areas that will be explored are: extending the rank one theory to include trace class perturbations, and the inverse problem for unbounded potentials, which should be determined in some way by the eigenvalues and Dirichlet eigenvalues. In particular, an attempt will be made to describe the isospectral manifold for this problem. Professor Simon will study a number of problems in the mathematical theory connected with quantum mechanics. One aspect of his work involves the study of long-time behavior in the model of electrons in solid and random media, whose properties are at the forefront of innovative materials design. Simon also expects to study problems in inverse spectral and scattering theory: the general issue of determining a microscopic structure from data that can be readily observed. Inverse theory is central to areas as varied as radar detection of submarines and the interpretation of medical tomography. ***