Isospectral potentials: Special functions partners and factorization operators

In supersymmetric quantum mechanics (SUSY QM), the isospectral potentials are those involved in a Schrödinger equation, which remains invariant under the standard Darboux transform. In this work, we propose a reciprocal and a generalized Darboux transform to find the isospectral potentials having eigenfunctions related to special functions (SF) partners. To that, the differential equation for SF is converted into a Schrödinger-like equation allowing SF solutions for the corresponding potential-like involved. This in turn is related with the equivalent of the Witten superpotential, occurring in SUSY QM. The equivalent of the generalized Witten superpotential consequently leads to the generalized SF as well as to their factorization operators. As a useful application of the proposal, we explicitly consider the SF partners, factorization operators and isospectral potentials for some classical SF: Hermite, Laguerre and Legendre. However, the method is general and can be applied to obtain new isospectral potentials, which could be useful in quantum chemical applications.