Novel enlarged shape invariance property and exactly solvable rational extensions of the Rosen–Morse II and Eckart potentials

The existence of a novel enlarged shape invariance property valid for some rational extensions of shape-invariant conventional potentials, first pointed out in the case of the Morse potential, is confirmed by deriving all rational extensions of the Rosen–Morse II and Eckart potentials that can be obtained in first-order supersymmetric quantum mechanics. Such extensions are shown to belong to three different types, the first two strictly isospectral to some starting conventional potential with different parameters and the third with an extra bound state below the spectrum of the latter. In the isospectral cases, the partner of the rational extensions resulting from the deletion of their ground state can be obtained by translating both the potential parameter $A$ (as in the conventional case) and the degree $m$ of the polynomial arising in the denominator. It therefore belongs to the same family of extensions, which turns out to be closed.