Infinitely many shape invariant potentials and new orthogonal polynomials

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic P"oschl-Teller potentials in terms of their degree \ell polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (\ell=1,2,…) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and G'omez-Ullate et al’s are the first members of these infinitely many potentials.

💡 Research Summary

The paper presents a systematic construction of infinitely many exactly solvable one‑dimensional quantum‑mechanical potentials that retain the property of shape invariance. The authors start from two well‑known families of shape‑invariant potentials: the radial oscillator (the radial part of the three‑dimensional harmonic oscillator) and the trigonometric/hyperbolic Pöschl‑Teller potentials. In their original form the eigenfunctions of these systems are expressed through classical orthogonal polynomials – Laguerre polynomials for the radial oscillator and Jacobi polynomials for the Pöschl‑Teller family.

The central idea is to deform each original potential by a polynomial factor of degree ℓ (ℓ = 1, 2, …). This deformation is implemented via Darboux‑Crum transformations that either delete or insert the ℓ‑th excited state of the original Hamiltonian. Concretely, the new superpotential Wℓ(x) is defined as the logarithmic derivative of the ℓ‑deformed ground‑state wavefunction, and the corresponding partner potentials Vℓ±(x) satisfy the shape‑invariance condition

 Wℓ²(x) + Wℓ′(x) = Wℓ+1²(x) − Wℓ+1′(x) + R(ℓ),

where R(ℓ) is a constant depending only on ℓ. This relation guarantees that the hierarchy of Hamiltonians {Hℓ} remains shape invariant: each step in ℓ changes only a set of parameters while preserving the functional form of the potential. Consequently the entire energy spectrum can be generated recursively, giving the same linear spacing as in the undeformed case, while the eigenfunctions acquire a non‑trivial polynomial factor.

The eigenfunctions of the deformed Hamiltonians are no longer expressed by the standard Laguerre or Jacobi polynomials. Instead they involve new families of orthogonal polynomials, denoted Xℓ‑Laguerre and Xℓ‑Jacobi. These “exceptional” polynomials differ from the classical families in that they miss a finite number of low‑degree terms (the ℓ lowest degrees are absent), yet they still satisfy a Sturm‑Liouville differential equation, a three‑term recurrence relation, and a Rodrigues‑type formula with respect to a modified weight function wℓ(x) = w(x)/