论文标题
关于一般的三阶形状不变的汉密尔顿人的家族
On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials
论文作者
论文摘要
这项工作报告并根据广义的Hermite多项式对最一般的理性量子电位进行了分类。这是通过利用三阶形状不变的汉密尔顿人和第四个painlevé方程之间的固有关系来实现的，从而使广义的Hermite多项式从$ -1/x $和-2x $ -2X $ -2X $ $ -2X $ nierations中出现。这种关系明确地建立了离散的频谱结构，通常，该结构是由间隙分隔的等距特征值的有限和无限二维序列组成的。广义的Hermite多项式的两个指标决定了有限序列和间隙的维度。同样，可以将完整的本征粒子分解为两个不相交的子集。在这种形式中，每个组中的征粒子都写为在真实线时定义的重量函数的乘积。这些多项式实现了二阶差分方程，并根据三项复发关系（二阶差异方程）确定，其初始条件也是根据广义的Hermite多项式固定的。
This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlevé equation, such that the generalized Hermite polynomials emerge from the $-1/x$ and $-2x$ hierarchies of rational solutions. Such a relation unequivocally establishes the discrete spectrum structure, which, in general, is composed as the union of a finite- and infinite-dimensional sequence of equidistant eigenvalues separated by a gap. The two indices of the generalized Hermite polynomials determine the dimension of the finite sequence and the gap. Likewise, the complete set of eigensolutions can be decomposed into two disjoint subsets. In this form, the eigensolutions within each set are written as the product of a weight function defined on the real line times a polynomial. These polynomials fulfill a second-order differential equation and are alternatively determined from a three-term recurrence relation (second-order difference equation), the initial conditions of which are also fixed in terms of generalized Hermite polynomials.
