Multistep DBT and regular rational extensions of the isotonic oscillator

In some recent articles, we developed a new systematic approach to generate solvable rational extensions of primary translationally shape invariant potentials. In this generalized SUSY QM partnership, the DBTs are built on the excited states Riccati–Schrödinger (RS) functions regularized via specific discrete symmetries of the considered potential. In the present paper, we prove that this scheme can be extended in a multistep formulation. Applying this scheme to the isotonic oscillator, we obtain new towers of regular rational extensions of this potential which are strictly isospectral to it. We give explicit expressions for their eigenstates which are associated to the recently discovered exceptional Laguerre polynomials and show explicitly that these extensions inherit the shape invariance properties of the original potential.