Article
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Use of Normalizable Seed Solutions for Constructing
Version 1
: Received: 15 August 2023 / Approved: 16 August 2023 / Online: 17 August 2023 (13:02:11 CEST)
How to cite: Natanson, G. Use of Normalizable Seed Solutions for Constructing. Preprints 2023, 2023081263. https://doi.org/10.20944/preprints202308.1263.v1 Natanson, G. Use of Normalizable Seed Solutions for Constructing. Preprints 2023, 2023081263. https://doi.org/10.20944/preprints202308.1263.v1
Abstract
The paper examines rational Darboux-Crum transforms (RDCTs) of the Jacobi-reference (JRef) potential on the line, with an emphasis on the transformations using only normalizable seed solutions. The Cooper-Ginocchio-Khare (CGK) potential constitutes the simplest example of this rich family of exactly solvable one-dimensional (1D) potentials. It was explicitly confirmed that that an arbitrary Darboux-Crum transformation (DCT) of the given Sturm-Liouville equation (SLE) can be represented as a chain of sequential Darboux deformations of the associated Liouville potentials. If the rational SLE (RSLE) under consideration is exactly solvable and the transformation function (TF) used at each step is nodeless then each RSLE constructed in such a way was proven being exactly solvable itself. The exact solvability of RDCTs of the JRef canonical SLE (CSLE) was also confirmed for the DCTs using pairs of juxtaposed eigenfunctions (an extension of the renowned Adler theorem to the RCSLEs of our interest). By re-expressing the reference polynomial fraction (RefPF) of the transformed CSLE in terms of the Krein determinant (KD), instead of Crum Wronskian (CW), it was shown that the eigenfunctions under consideration are expressible in terms of polynomial solutions of a generalized Heun equation with positions of the most poles dependent on the exponent parameters. The outlined technique was extended to two isospectral SUSY partners of the JRef potential (as well as to its third sibling with the inserted new lowest-energy level) which, by analogy with the CGK potential, are quantized by polynomial solutions of the 2-Heun equation with all four poles at some fixed positions on the real axis.
Keywords
Cooper-Ginocchio-Khare potential; rational Sturm-Liouville equation; Liouville transformation; Darboux-Crum transformation; Crum Wronskian; Krein determinant; Adler theorem; generalized Heun equation; differential equation of Heine type.
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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