Heun-Polynomial Representation of Regular-at-Infinity Solutions for the Basic SUSY Ladder of Hyperbolic P\"oschl-Teller Potentials Starting from the Reflectionless Symmetric Potential Well

TL;DR

This paper demonstrates that solutions to the hyperbolic Pöschl-Teller potential can be expressed using Heun polynomials, enabling the construction of new exactly solvable rational potentials through canonical transformations.

Contribution

It introduces a novel representation of regular-at-infinity solutions using Heun polynomials and applies this to generate new shape-invariant rational potentials via Liouville-Darboux transformations.

Findings

Solutions are expressible as Heun polynomials at negative energies.

Heun polynomials form a subset of Lambe-Ward polynomials with zero accessory parameter.

New rational potentials are generated using canonical transformations.

Abstract

It is shown that the regular-at-infinity solution of the 1D Schrodinger equation with the hyperbolic Poschl-Teller (h-PT) potential with integer parameters is expressible in terms of a n-order Heun polynomial in y=thr at an arbitrary negative energy. It was proven that the Heun polynomials in question form a subset of generally complex Lambe-Ward polynomials corresponding to zero value of the accessory parameter. Since the mentioned solution expressed in the new variable y has an almost-everywhere holomorphic (AEH) form it can be used as the factorization function (FF) for canonical Liouville-Darboux transformations (CLDTs) to construct a continuous family of shape-invariant rational potentials exactly-solvable by the Hp-seed (HpS) Heine polynomials. There are also two sequences of infinitely many rational potentials generated using CLDTs with nodeless regular-at-origin AEH FFs.