Spectral Theory

谱理论

• 批准号: 1665526
• 负责人: Barry Simon
• 金额: $ 16.8万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665526&HistoricalAwards=false
• 关键词: Spectral; Theory

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.

PI将继续他对光谱理论的研究，尤其是正交多项式的光谱理论。光谱理论涉及数学模型与其“光谱特征”的关系，并包括计算机断层扫描，声纳分析和亚原子颗粒的散射等领域。正交多项式是最简单的范式，尤其有用，因为逆问题 - 通常很难显示具有解决方案 - 是如此明确。 PI将研究与正交多项式有关的几个问题及其近亲Chebyshev多项式。将研究这三个主要区域。 第一个涉及较高的总和规则，这是PI是先驱的主题。 它旨在使用大偏差的方法来跟进最近的进展，这是概率理论的技术。 第二个涉及研究Chebyshev多项式的问题。 最近，PI解决了一个45年历史的猜想，涉及与真实线的子集相关的多项式，并将研究新赠款期在复杂平面的一般子集的更微妙案例。 第三个区域涉及施罗宾格经营者的周期性潜力。该范式是真实行的定期施罗宾格运营商，但是有一些新问题与对称人群现在是非亚伯利亚人有关的事实。

项目成果

Tosio Kato’s work on non-relativistic quantum mechanics, Part 2

加藤 Tosio Kato 的非相对论量子力学著作，第 2 部分

• DOI: 10.1142/s166436071950005x
• 发表时间: 2019
• 期刊: Bulletin of Mathematical Sciences
• 影响因子: 1.2
• 作者: Simon, Barry

Poncelet's theorem, paraorthogonal polynomials and the numerical range of compressed multiplication operators

Poncelet 定理、平行正交多项式和压缩乘法算子的数值范围

• DOI: 10.1016/j.aim.2019.04.027
• 发表时间: 2019
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Martínez–Finkelshtein, Andrei;Simanek, Brian;Simon, Barry
• 通讯作者: Simon, Barry

Periodic Jacobi matrices on trees

• DOI: 10.1016/j.aim.2020.107241
• 发表时间: 2019-11
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Nir Avni;Jonathan Breuer;B. Simon

Similarity Between Two Projections

• DOI: 10.1007/s00020-017-2414-6
• 发表时间: 2017-05
• 期刊: Integral Equations and Operator Theory
• 影响因子: 0.8
• 作者: A. Böttcher;B. Simon;I. Spitkovsky

Asymptotics of Chebyshev polynomials, II: DCT subsets of ${\mathbb{R}}$

切比雪夫多项式的渐近，II：${mathbb{R}}$ 的 DCT 子集

• DOI: 10.1215/00127094-2018-0045
• 发表时间: 2019
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Christiansen, Jacob S.;Simon, Barry;Yuditskii, Peter;Zinchenko, Maxim
• 通讯作者: Zinchenko, Maxim