Spectral Theory

Simon教授计划研究正交多项式及其相关系统的谱理论，并在过去十年中继续这一学科的实质性进展。他打算研究的一个有趣的问题是与Cantor测度相关的正交多项式的谱理论，目前有强有力的数值证据表明递归系数是渐近几乎周期的，Simon教授希望证明定理。第二个问题涉及有限间隙Jacobi矩阵的指数环面元素的指数衰减扰动，其中需要使用Riemann曲面理论的技巧，这给推广Damanik和Simon早先获得的单带结果带来了一些困难。第三个问题涉及关于单位圆上的正交多项式的零点的猜想二分法，其中数值证据表明，圆上的每个点都是零点的极限，否则所有的零点都远离单位圆。最后，Simon教授计划将过去关于实直线和单位圆上的正交多项式的结果推广到复平面上更一般的测量的背景下。逆光谱理论是通过测量的光谱特性来确定物体的特性的理论。实际的例子包括使用x射线来确定晶体物质的结构，以及医学成像技术，如CAT扫描。基本的数学理论是这些重要应用领域进一步发展的种子。正交多项式的谱理论意义重大，因为它是一种有解反问题的一般显式公式集的理论。在他的职业生涯中，Simon教授为发展我们的科学基础设施做出了重要贡献，他指导了30多名研究生博士，并指导了近50名博士后和其他科学家。他们中的许多人都有了自己的杰出职业生涯。他预计，在新的赠款期间，这种情况将继续下去。

Professor Simon plans to study the spectral theory of orthogonal polynomials and related systems, following up on the substantial progress in the subject within the past ten years. One intriguing problem he intends to study is the spectral theory of the orthogonal polynomials associated to the Cantor measure where there is currently strong numerical evidence that the recursion coefficients are asymptotically almost periodic and Prof. Simon hopes to prove theorems. A second problem concerns exponential decaying perturbations for elements of the exponential torus of finite gap Jacobi matrices where the need to employ techniques from the theory of Riemann surfaces presents some difficulties in extending the one band results obtained earlier by Damanik and Simon. A third problem concerns a conjectured dichotomy about the zeros of orthogonal polynomials on the unit circle where numeric evidence suggests that either every point on the circle is a limit of zeros or else all the zeros stay away from the unit circle. Finally, Professor Simon is planning to explore extensions of the results obtained in the past for orthogonal polynomials on the real line and unit circle to the context of more general measures in the complex plane.The proposed research concerns the spectral and inverse spectral theory of orthogonal polynomials. Inverse spectral theory is connected with determining the properties of an object from its measured spectral properties. Practical examples include using x-rays to determine the structure of crystalline matter and medical imaging technology such as CAT scans. The underlying mathematical theory is the seed corn for further developments in these important applied areas. The spectral theory of orthogonal polynomials is significant because it is a theory where there is a general explicit set of formulae for solving the inverse problem. During his career, Professor Simon has made important contributions to developing our scientific infrastructure with over 30 graduate student PhD supervisions and mentoring of almost 50 postdocs and other scientists. Many of these have gone on to distinguished careers of their own. He expects this to continue during the new grant period.