线I上代数量子化反射Liouville势的单源网。具有二阶极点的有理正则Sturm-Liouville方程的几乎处处全纯解
Single-Source Nets of Algebraically-Quantized Reflective Liouville Potentials on the Line I. Almost-Everywhere Holomorphic Solutions of Rational Canonical Sturm-Liouville Equations with Second-Order Poles
摘要：
本文提出了一种构造有理Liouville势（RLP）超对称阶梯的统一方法，该方法从所谓的“高斯参考”（GRef）势出发，通过经典Jacobi、经典（广义）Laguerre或Romanovski-Routh多项式（指数依赖于能量）在线精确量子化.每个RLP是通过适当的有理规范Sturm-Liouville方程（RCSLE）与二阶极点的刘维尔变换（LT）。所提出的分析利用了典型的Sturm-Liouville方程（CSLE）的一般因式分解的交织“广义”达布算子。我们将后一种算子称为规范刘维-达布变换（CLDTs），以强调它们等价于三步操作：i）从CSLE到薛定谔方程的LT; ii）适当LP的达布变换（DT）; iii）从薛定谔方程到新CSLE的逆LT。证明了CLDT保持RCSLE的有理形式，如果它的分解函数（FF）是RCSLE的几乎处处全纯（AEH）解（或者，换句话说，具有有理对数导数的解）。如本文所解释的，有多达四个规范变换将我们感兴趣的每个RCSLE转换为具有能量依赖多项式系数的二阶微分方程。最重要的结果是，这些方程的多项式解属于Heine多项式序列，它是通过在奇点的固定值处改变自由项和适当的特征指数而得到的。这使我们能够构建网络的多项式解决方案-所谓的“r-，c-或i-Gauss种子”（r-，c-或i-GS）海涅多项式-从雅可比，（广义）拉盖尔或劳斯多项式，分别。
英文摘要：
The paper presents the unified technique for constructing SUSY ladders of rational Liouville potentials (RLPs) starting from the so-called "Gauss-reference" (GRef) potentials exactly quantized on the line via classical Jacobi, classical (generalized) Laguerre, or Romanovski-Routh polynomials with energy-dependent indexes. Each RLP is obtained by means of the Liouville transformation (LT) of the appropriate rational canonical Sturm-Liouville equation (RCSLE) with second-order poles. The presented analysis takes advantage of the generic factorization of canonical Sturm-Liouville equations (CSLEs) in terms of intertwining "generalized" Darboux operators. We refer to the latter operators as the canonical Liouville-Darboux transformations (CLDTs) to stress that they are equivalent to three-step operations: i) the LT from the CSLE to the Schrodinger equation; ii) the Darboux transformation (DT) of the appropriate LP; and iii) the inverse LT from the Schrodinger equation to the new CSLE. It is proven that the CLDT preserves the rational form of the RCSLE if its factorization function (FF) is an almost-everywhere holomorphic (AEH) solution of the RCSLE (or, in other words, a solution with a rational logarithmic derivative). As explained in the paper there are up to four gauge transformations which convert each RCSLE of our interest into the second-order differential equations with energy-dependent polynomial coefficients. The most important result of the paper is that polynomial solutions of these equations belong to sequences of Heine polynomials obtained by varying free terms at fixed values of singular points and the appropriate characteristic exponents. This allows us to construct networks of polynomial solutions -- the so-called "r-, c-, or i-Gauss-seed" (r-, c-, or i-GS) Heine polynomials -- starting from Jacobi, (generalized) Laguerre or Routh polynomials, respectively.
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