Exceptional orthogonal polynomials: theory and applications

我目前的研究重点是例外正交理论 多项式。经典的正交多项式是一种基本的 数学工具的应用范围从原子物理和 电路设计中对基本算法的运用 在科学计算中。 例外正交多项式是一种新的、令人着迷的 经典概念的泛化。它们也是作为解决方案而出现的 二阶微分方程式，但考虑到这种可能性 由此得到的多项式族不包含每个 学位。这种对理论假设的温和放松导致了 更丰富的示例集，随之而来的是灵活性的增加 在超对称量子力学等应用中。 早在2008-2009年间，我就与合作者一起引入并开发了这些新技术。 从那时起，该领域经历了快速增长，在应用和理论方面都取得了许多令人振奋的发展。 我的研究目标是集中于特殊的正交多项式的理论基础，以期 探索与现有数学理论分支的新联系。

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.