发表机构
Pabna University of Science and Technology; La Trobe University（帕布纳科技大学; 拉筹伯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究两个三维量子超可积磁系统，通过二次代数、分离变量和Sturm代数化获得三种互补的代数描述，揭示其隐藏结构与可解性。
AI 中文摘要
我们研究了两个在非零轴对称磁场中的三维量子超可积系统。对于每个模型，积分生成一个带有Casimir算子的二次代数，有限维形变振子表示给出代数谱。相同的谱通过变量分离也能恢复。在圆抛物坐标中，分离后的方程是六次双合流Heun方程，具有有限多项式扇区。我们将它们表述为Sturm特征值问题，以物理能量为参数，分离常数被对角化。经过规范变换后，每个方程都允许$sl(2)$代数化，并约化为有限三对角矩阵。我们还构造了每个完整哈密顿量的预分离Sturm表示，以逆平方耦合作为Sturm特征值。变换后的积分保持对易，在固定轴向积分、改变变量和规范旋转后，双变量Sturm哈密顿量及其非中心积分属于普遍包络代数$U(sl(3))$。因此，我们获得了三种互补的代数描述：原始哈密顿量的二次代数/形变振子表示，抛物分离后的$sl(2)$代数化，以及分离前完整Sturm问题的$sl(3)$代数化。尽管两个磁哈密顿量在原始分类中不同，固定轴向积分将它们约化为相同的奇异2:1振子正规形，但具有不同的有效参数嵌入。这解释了共同的$sl(3)$结构，而不意味着原始磁系统的等价性。这些结果表明，不同的隐藏代数结构和精确或准精确可解性如何在多重可分离超可积系统的不同阶段出现。
英文摘要
We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an $sl(2)$ algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra $U(sl(3))$. We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the $sl(2)$ algebraization after parabolic separation, and the $sl(3)$ algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common $sl(3)$ structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.
