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

2014/10/07 by Gregory Natanson, Natanson, Gregory · 2 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1410.1515

54 pages

arxiv created 2014/10/07 · openalex publication_date 2014/10/07 · arxiv updated 2014/10/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• The generating functions of Lame equation in Weierstrass's form

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