论文标题
特殊的Hermite微分方程及其最小表面表示的一般解决方案
General solution of the exceptional Hermite differential equation and its minimal surface representation
论文作者
论文摘要
本文的主要目的是研究具有固定分区$λ=（1）$的特殊Hermite微分方程的一般解决方案，以及与该溶液相关的最小表面的构建。我们得出了一个线性二阶的普通微分方程，与Codimension二的特定多项式特定家族相关。我们表明，这些多项式可以用经典的Hermite多项式表示。基于这一事实，我们证明了存在于构造该多项式的分区引起的特殊Hermite多项式的规范与差距序列之间存在联系。我们发现频谱没有差距的特殊Hermite微分方程的一般分析解。我们表明，频谱与非多项式溶液相辅相成。我们介绍了根据一般解决方案来表达的表面所获得的结果的实现，用于利用经典的Enneper-WeierStrass公式将沉浸在欧几里得空间$ \ Mathbb {e}^3 $中，导致最小的表面。提出了这些表面的三维显示。
The main aim of this paper is the study of the general solution of the exceptional Hermite differential equation with fixed partition $λ= (1)$ and the construction of minimal surfaces associated with this solution. We derive a linear second-order ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can be expressed in terms of classical Hermite polynomials. Based on this fact, we demonstrate that there exists a link between the norm of an exceptional Hermite polynomial and the gap sequence arising from the partition used to construct this polynomial. We find the general analytic solution of the exceptional Hermite differential equation which has no gap in its spectrum. We show that the spectrum is complemented by non-polynomial solutions. We present an implementation of the obtained results for the surfaces expressed in terms of the general solution making use of the classical Enneper-Weierstrass formula for the immersion in the Euclidean space $\mathbb{E}^3$, leading to minimal surfaces. Three-dimensional displays of these surfaces are presented.
