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

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.

💡 Research Summary

The paper investigates two infinite families of exactly solvable one‑dimensional quantum mechanical potentials that were recently introduced by Odake and Sasaki. These families are generated by deforming the well‑known radial oscillator potential and the Darboux‑Pöschl‑Teller (DPT) potential with an eigenpolynomial of degree ℓ (ℓ = 1, 2, …). The deformation is performed at the level of the supersymmetric pre‑potential W(x): for the radial oscillator one adds the logarithmic derivative of a Laguerre polynomial Lℓ^(g‑½)(x²), while for the DPT one adds the logarithmic derivative of a Jacobi polynomial Pℓ^(A‑½,B‑½)(cos 2x). The resulting potentials Vℓ(x)=Wℓ²(x)−Wℓ′(x) contain rational terms but retain the supersymmetric structure.

The central claim is that despite the deformation, the new potentials remain shape‑invariant. Shape invariance means that the partner Hamiltonians H⁻(a) and H⁺(a) satisfy H⁺(a)=H⁻(a′)+R(a) with a simple shift of the parameter set a→a′ and an additive constant R(a). This property guarantees that the entire spectrum and eigenfunctions can be obtained algebraically by repeated application of the supersymmetric ladder operators.

To prove shape invariance the authors translate the condition into a polynomial identity. For the Laguerre‑deformed potentials the condition reduces to a cubic identity of degree 3ℓ involving products of three Laguerre polynomials with shifted parameters: \