The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials VA, B, ext (x), obtained from a conventional Morse potential VA-1, B(x) by the addition of a bound state below the spectrum of the latter, is reobtained. More importantly, the existence of another family of extended potentials, strictly isospectral to VA+1, B(x), is pointed out for a well-chosen range of parameter values. Although not shape invariant, such extended potentials exhibit a kind of "enlarged" shape invariance property, in the sense that their partner, obtained by translating both the parameter A and the degree m of the polynomial arising in the denominator, belongs to the same family of extended potentials. The point canonical transformation connecting the radial oscillator to the Morse potential is also applied to exactly solvable rationally-extended radial oscillator potentials to build quasi-exactly solvable rationally-extended Morse ones.
Novel Enlarged Shape Invariance Property and Exactly Solvable Rational Extensions of the Rosen-Morse II and Eckart Potentials