论文标题
杰出的Hermite方程的复发关系和一般解决方案
Recurrence relations and general solution of the exceptional Hermite equation
论文作者
论文摘要
异常的正交赫米特和拉瓜多项式与谐波和奇异振荡器的K-Step扩展有关。特殊的多项式允许从超对称量子力学的Darboux-Crum和Krein-Adler构建体中存在不同的增压。它们还允许存在不同类型的阶梯关系及其相关的复发关系。这种关系的存在是这些多项式的独特特性。这些关系已被用来构建可整合的2D模型，并显示有趣的频谱，变性和有限维统一表示。在以前的作品中，仅讨论了光谱的物理或多项式部分。众所周知，通用解决方案与其他类型的复发/梯子关系有关。我们计划详细讨论特殊的Hermite多项式$ x_2^{（1）} $的情况，并明确介绍通过与不同类型的梯子操作员一起表现而获得的新链。我们将利用其中一位作者[32]的最新结果，其中构建了一般的分析解决方案并与非分类汇合HEUN方程相连。构建了用于通用溶液的模拟Rodrigues公式。可以从2链表示的图中获取其他状态的有限状态集并非唯一，但是从2链表示的图中获得的消失箭头和对角线箭头可以获得最小的集合。然后，利用这些Rodrigues公式，不仅以纯粹的代数方式构建了多项式和非多项式的状态，而且还以基于代数方式的阶梯运算符的作用从阶梯运算符的作用中获取系数，也基于发电机之间的进一步的换向关系。
Exceptional orthogonal Hermite and Laguerre polynomials have been linked to the k-step extension of harmonic and singular oscillators. The exceptional polynomials allow the existence of different supercharges from the Darboux-Crum and Krein-Adler constructions of supersymmetric quantum mechanics. They also allow the existence of different types of ladder relations and their associated recurrence relations. The existence of such relations is a unique property of these polynomials. Those relations have been used to construct 2D models which are superintegrable, and display an interesting spectrum, degeneracies and finite-dimensional unitary representations. In previous works, only the physical or polynomial part of the spectrum is discussed. It is known that the general solutions are associated with other types of recurrence/ladder relations. We plan to discuss in detail the case of the exceptional Hermite polynomials $X_2^{(1)}$ and to explicitly present new chains obtained by acting with different types of ladder operators. We will exploit a recent result by one of the authors [32], where the general analytic solution was constructed and connected with the non-degenerate confluent Heun equation. The analogue Rodrigues formulas for the general solution are constructed. The set of finite states from which the other states can be obtained algebraically is not unique but the vanishing arrow and diagonal arrow from the diagram of the 2-chain representations can be used to obtain minimal sets. These Rodrigues formulas are then exploited, not only to construct the states, polynomial and non-polynomial, in a purely algebraic way, but also to obtain coefficients from the action of ladder operators also in an algebraic manner based on further commutation relations between monomials of the generators.
