Translational Shape-Invariance of Radial Jacobi-Reference Potential Under Two Sequential Darboux Transformations

The paper represents a further development of the mathematical formalism relating the conventional quantum-mechanical theory of the exactly-solvable translationally shape-invariant (TSI) potentials to the preservation of the form of the corresponding Rational Canonical Sturm-Liouville Equation (RCSLE) under the Darboux deformation of its Liouville potential using the so-called ’basic’ solution as the Transformation Function (TF). Since these ‘Liouville-Darboux transformations’ (as we term them) simply shift one of the parameters by 1 we refer to this feature of the mentioned RCSLE as its ‘translational form-invariance’. The main purpose of this paper is to demonstrate that, by analogy with its TSI limit represented by the hyperbolic Pöschl-Teller potential, the radial potential exactly solvable via hypergeometric functions (termed ‘radial Jacobi-reference potential in the paper) has two pairs of quasi-rational solutions such that their analytical continuations do not have zeros at any regular point in the complex plane. It is essential that the Characteristic Exponents (ChExps) of these four ‘basic’ solutions for the pole at the origin differ only by sign and can be then grouped into the pairs via the requirement that the paired solutions share the same characteristic exponent for the mentioned singularity. Each pair of the basic solutions is then used as seed functions for the second-order Darboux-Crum Transformation (DCT) of the radial potential in question. It is proven that both transformations simply shift by 2 the exponent difference for the pole of the RCSLE at the origin while keeping two other parameters unchanged. In other words, the DCT in question brings us back to the initial radial potential by either deleting the ground-energy state or inserting a new one. The important novel element of our approach is the conversion of the Crum Wronskian (CW) to the Krein Determinant (KD) which makes it possible to express the eigenfunctions of the transformed RCSLE in terms of polynomial solutions of a Heine-type differential equation with degree-dependent exponential parameters. As an illustration of the Darboux-Crum-Krein theory of CSLEs put forward by the author we make use of the Krein representation to explicitly confirm that the state-deleting DCT of the Jost solution simply shifts the third parameter of the corresponding hypergeometric series by 2 as the direct consequence of the translational form-invariance of the RCSLE under the DCT in question.