Supersymmetry and Algebraic Darboux Transformations

We describe a class of algebraically solvable SUSY models by considering the deformation of invariant polynomial flags by means of the Darboux transformation. The algebraic deformations corresponding to the addition of a bound state to a shape-invariant potential are particularly interesting. The polynomial flags in question are indexed by a deformation parameter m = 1, 2, dots, and lead to new algebraically solvable models. We illustrate these ideas by considering deformations of the hyperbolic Pöschl–Teller potential.