Isospectral orthogonal polynomials from the Darboux transforms

Orthogonal polynomials (OP) are used in many branches of the mathematical and physical sciences; in particular they are part of the eigenfunctions of quantum chemical (QC) potential models. Recently, in the search for new solvable potentials to be useful in QC applications, the use of supersymmetry (SUSY) and Hamiltonian intertwining methods have shown the existence of isospectral potential partners. Also, it has been proved that SUSY is equivalent to the standard Darboux transform (DT) applied to Sturm-Liouville-type problems. Consequently, because of the similarity between the differential equations of OP and of SL, we consider the application of the standard and generalized DT to find the isospectral OP that are partners of the standard one. To attain this purpose, we use a pair of transformations, a point canonical and a gauge, to convert the SL differential equation into a Schrodinger-like equation allowing OP solutions. As a useful application of our proposal, we consider some of the most important OP by obtaining the generalized isospectral OP partners. Also, we shown the associated isospectral QC potentials for each of the orthogonal polynomials considered in this work. The proposed procedure can be used to obtain new orthogonal polynomials as well as to find new solvable QC potential models.