Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials

TL;DR

This paper proves the shape invariance of two families of exactly solvable quantum potentials by establishing new polynomial identities involving cubic products of Laguerre and Jacobi polynomials, extending the understanding of quantum solvability.

Contribution

It provides elementary proofs for shape invariance of deformed potentials using novel polynomial identities, advancing the mathematical framework of quantum mechanics.

Findings

Established shape invariance for new classes of potentials.

Derived and proved cubic polynomial identities for Laguerre and Jacobi polynomials.

Connected polynomial identities to quantum potential solvability.

Abstract

We provide analytic proofs for the shape invariance of the recently discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux-P\"oschl-Teller potential by a degree \ell (\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3\ell involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.