Spectral Theory

PI将继续他对谱理论的研究，特别是正交多项式的谱理论。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，特别有用，因为逆问题--通常甚至很难证明有解--是如此明确。PI将研究与正交多项式及其近亲切比雪夫多项式有关的几个问题。主要研究三个方面。第一个涉及更高和规则，这是PI是先驱的一个主题。它的目的是利用概率论中的一种技术--大偏差方法来跟踪最近的进展。第二部分研究了切比雪夫多项式。最近，PI解决了一个已有45年历史的猜想，该猜想涉及与实线的子集相关的多项式，并将在新的授权期内研究复平面的一般子集的更微妙的情况。第三个领域涉及具有周期位势的树上的薛定谔算子。这个范例是实线上的周期薛定谔算符，但有一些新的问题与对称群现在是非阿贝尔的事实有关。

The PI will continue his research into spectral theory, especially the spectral theory of orthogonal polynomials. 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, especially useful because the inverse problem --often difficult to even show has solutions -- is so explicit. The PI will study several problems connected with orthogonal polynomials and its close relative, the Chebyshev polynomials.Three main areas will be studied. The first involves higher sum rules, a subject where the PI was a pioneer. It is intended to follow up on recent progress using the method of large deviations, a technique from probability theory. The second concerns studying Chebyshev polynomials. Recently, the PI settled a 45 year old conjecture involving polynomials associated to subsets of the real line and will study the more subtle case of general subsets of the complex plane in the new grant period. The third area concerns Schrodinger operators on trees with periodic potential. The paradigm is periodic Schrodinger operators on the real line but there are new issues connected with the fact that the symmetry group is now non-abelian.