发表机构
Université de Montréal; Centre de recherches mathématiques, Université de Montréal; Laboratoire d’Annecy de Physique Théorique（蒙特利尔大学; 蒙特利尔大学数学研究中心; 安纳西理论物理实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带反射的 Scarf I 超对称哈密顿量及其离散 para-Bannai--Ito 版本，确定其连续极限，证明本征向量收敛性，并揭示完美传递与复兴性质。
AI 中文摘要
para-Bannai--Ito 多项式在 Bannai--Ito 双格点上正交，该双格点是双侧的——它从中心值 $c_0$ 向两侧延伸——并且除了两个网格参数外，还依赖于一个等谱形变参数。其 Jacobi 矩阵 $J$ 仅在该参数的一个值（persymmetric 点）处对位点反转不变，我们在此处确定其连续极限。由于网格是双侧的，自然的研究对象是平移算子 $Q=J-c_0$，极限在能带中心附近取：$Q$ 收缩为一阶 Dunkl 算子（超电荷），$H=Q^2$ 收缩为广义 Pöschl--Teller（Scarf I）哈密顿量，具有小 $-1$ Jacobi 本征函数——带反射的超对称量子力学，反射进入哈密顿量的平方根。第二个网格参数为哈密顿量本身添加反射项，双格点（在平移意义下）是极限超电荷的谱。本征向量的收敛性直接在多项式上证明。在此网格上，persymmetric 模型在参数的算术条件下能完美传递状态，但从不分数复兴，复兴是形变的作用；这两个性质都传递到极限。这些结果适用于偶数个位点；对于奇数个位点，极限则是 Pöschl--Teller 哈密顿量的矩阵超对称对，反射作为 Dunkl 势出现在哈密顿量中。离散模型的双谱对过渡到小 $-1$ Jacobi 多项式的反对易子代数的 Dunkl 实现。
英文摘要
The para-Bannai--Ito polynomials are orthogonal on a Bannai--Ito bi-lattice that is \emph{two-sided} --- it extends on both sides of a central value $c_0$ --- and depend, besides the two grid parameters, on an isospectral deformation parameter. Their Jacobi matrix $J$ is invariant under the reversal of the sites at one value of that parameter only, the persymmetric point, and we determine its continuum limit there. Because the grid is two-sided, the natural object is the shifted operator $Q=J-c_0$, and the limit is taken about the centre of the band: $Q$ contracts to a first--order Dunkl operator, a supercharge, and $H=Q^2$ to the generalized Pöschl--Teller (Scarf I) Hamiltonian, with little $-1$ Jacobi eigenfunctions --- supersymmetric quantum mechanics with reflections, the reflection entering the square root of the Hamiltonian. The second grid parameter adds a reflection term to the Hamiltonian itself, and the bi-lattice is, up to a shift, the spectrum of the limiting supercharge. The convergence of the eigenvectors is proved directly on the polynomials. On this grid the persymmetric model transfers states perfectly, under an arithmetic condition on the parameters, but never revives fractionally, revival being the work of the deformation; both properties pass to the limit. These results are for an even number of sites; for an odd number of sites the limit is instead the matrix supersymmetric pair of Pöschl--Teller Hamiltonians, with the reflection in the Hamiltonian as a Dunkl potential. The bispectral pair of the discrete model goes over to the Dunkl realization of the anticommutator algebra of the little $-1$ Jacobi polynomials.
Comments34 pages
