发表机构
La Trobe University; University of Hawaii; The University of Queensland（拉筹伯大学; 夏威夷大学; 昆士兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于Darboux-Crum变换，构造了与III型Hermite例外正交多项式相关的量子系统新准精确可解变形，给出Bethe ansatz方程并应用于多能隙模型。
AI 中文摘要
准精确可解（QES）的谐振子和奇异振子变形已通过多种方法被广泛研究。在本文中，我们获得了与例外正交多项式（EOPs）相关的可解量子系统的新QES变形族。该构造基于Darboux-Crum变换理论以及与Hermite型EOPs相关的精确可解量子系统的分类。研究表明，这些变形打破了未变形系统的精确可解性，并在变形系统中引入了模型参数，使得存在有限数量的多项式解，其根满足代数方程组。所提出并研究的新族包括与任意余维数的III型Hermite EOPs相关的量子系统的变形。我们给出了多项式和有理变形，并在每种情况下，通过Bethe ansatz方程和参数约束分析了准精确可解性的条件。一般而言，底层多项式解的结构不再与众所周知的经典正交多项式相关联。因此，对于一般情况，我们主要侧重于提出新方法以及Bethe ansatz方程和模型参数的约束。作为应用，我们构造了与允许一个、两个和最多四个能隙的可解模型相关的特定QES变形族，并获得了其波函数和谱的闭式表达式。我们分析了模型参数空间中QES解的存在性，提供了给定参数下解数量的信息。
英文摘要
Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.
