Spectral Theory

西蒙教授将继续他对谱理论的研究，特别是正交多项式的谱理论。这个项目将集中在频谱有有限数量的间隙的理论上。它还将研究周期薛定谔算子的微扰，这是一种固体中杂质的模型。光谱理论涉及数学模型与其“光谱特性”的关系，包括计算机层析成像、声纳分析和亚原子粒子散射等领域。正交多项式是最简单的范例；它们特别有用，因为逆问题--通常甚至很难证明有解--是如此明确。此外，单位圆(Simon教授在其上写了一本书)上的正交多项式是电子电路滤波器设计的核心工具。

Professor Simon will continue his research into spectral theory, especially the spectral theory of orthogonal polynomials. This project will focus on the theory where the spectrum has a finite number of gaps. It will also study perturbations of periodic Schrodinger operators, a model of impurities in solids.Spectral theory concerns the relation of mathematical models to their "spectral characteristics" and includes such areas as computer tomography, sonar analysis, and scattering of subatomic particles. Orthogonal polynomials are the simplest paradigm; they are especially useful because the inverse problem--often difficult to even show has solutions--is so explicit. In addition, orthogonal polynomials on the unit circle (on which Professor Simon has written a book) are a central tool in electronic circuit filter design.