Survey of Nodeless Regular Almost-Everywhere Holomorphic Solutions for Exactly Solvable Gauss-Reference Liouville Potentials on the Line I.Subsets of Nodeless Jacobi-Seed Solutions Co-Existent with Discrete Energy Spectrum

TL;DR

This survey compiles nodeless, holomorphic solutions for specific Sturm-Liouville problems that enable the construction of isospectral rational SUSY partner potentials via multi-step transformations, expanding the toolkit for exactly solvable quantum models.

Contribution

It provides a comprehensive list of nodeless AEH solutions for rational Sturm-Liouville equations suitable for generating isospectral SUSY partner potentials.

Findings

Identified all nodeless AEH solutions compatible with the Schrödinger equation on the line.

Showed these solutions can serve as seed functions for multi-step Liouville-Darboux transformations.

Demonstrated the resulting potentials are conditionally exactly quantized by Jacobi-Seed Heine polynomials.

Abstract

The paper collates a complete list of nodeless regular almost-everywhere holomorphic (AEH) solutions for a subset of rational canonical Sturm-Liouville equations (RCSLEs) exactly quantized on a finite interval by classical Jacobi polynomials. The subset was constrained by the requirement that the appropriate Liouville transformation results in the Schrodinger equation on the line. The common remarkable feature of the selected nodeless solutions co-existent with the discrete energy spectrum is that they can be used as seed functions for multi-step 'canonical Liouville-Darboux transformations' (CLDTs) to convert the Gauss-Reference (GRef) potential (appearing in the resultant Schrodinger equation) into its isospectral rational SUSY partners conditionally exactly quantized by the so-called 'Jacobi-Seed' Heine polynomials.