Exceptional orthogonal polynomials: theory and applications

例外正交多项式：理论与应用

• 批准号: RGPIN-2014-04031
• 负责人: Milson, Robert
• 金额: $ 1.02万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=659839
• 关键词: Exceptional; orthogonal; polynomials; theory; applications

My current research focus is the theory of exceptional orthogonal*polynomials. Classical orthogonal polynomials are a fundamental*mathematical tool with applications ranging from atomic physics and*the design of electrical circuits to fundamental algorithms utilized*in scientific computation.**Exceptional orthogonal polynomials are a new and fascinating*generalization of the classical concept. They also arise as solutions*of second-order differential equations, but allow for the possibility*that the resulting family of polynomials does not contain every*degree. This mild relaxation of the theoretical assumption leads to a*much richer set of examples with an attendant increase of flexibility*in applications such as super-symmetric quantum mechanics.**Working with collaborators, I introduced and developed these novel techniques back in 2008-2009.*Since then the field has seen rapid growth with many exciting developments both in applications and theory.*My research goals is to focus on the theoretical underpinnings of exceptional orthogonal polynomials with a view*towards exploring novel connections with existing branches of mathematical theory.

我目前的研究重点是异常正交*多项式理论。 经典正交多项式是一种基本的*数学工具，其应用范围从原子物理学和*电路设计到*科学计算中使用的基本算法。**特殊的正交多项式是经典概念的一种新的、令人着迷的*概括。 它们也作为二阶微分方程的解*出现，但允许产生的多项式族不包含每个*度的可能性*。 这种对理论假设的温和放松导致了*更丰富的示例集，并随之增加了超对称量子力学等应用中的灵活性*。**与合作者合作，我早在 2008 年就引入并开发了这些新技术 - 2009 年。*从那时起，该领域迅速发展，在应用和理论方面都有许多令人兴奋的发展。*我的研究目标是专注于特殊正交多项式的理论基础，以期*探索与数学理论现有分支的新颖联系。