Gauss-Seed Nets of Sturm-Liouville Problems With Energy-Independent Characteristic Exponents and Related Sequences of Exceptional Orthogonal Polynomials I. Canonical Darboux Transformations Using AEH Functions

Abstract

The paper applies the so-called 'Canonical-Darboux-Transformation' (CDT) method to reproduce general expressions for rational potentials (RPs) quantized in terms of exceptional orthogonal polynomial systems (X-OPSs). The benchmark of the new method recently developed by the author for implicit potentials solvable via hypergeometric functions is that rationally-extended SUSY partners of the original potential are quantized in terms of sequences of the so-called 'Gauss-seed' (GS) Heine polynomials starting from a polynomial of non-zero order. The common mark of the Darboux-Poschl-Teller (DPT) potential and isotonic oscillator discussed in this paper is that the appropriate rational Sturm-Liouville (RSL) equations have energy-independent characteristic exponents at both singular end points and as a result the appropriate sequences of GS Heine polynomials turn into X-OPSs with infinitely many members.