Revisiting (quasi-)exactly solvable rational extensions of the Morse potential

TL;DR

This paper explores new rational extensions of the Morse potential within supersymmetric quantum mechanics, revealing a novel isospectral family with an 'enlarged' shape invariance property and connecting extended radial oscillator potentials to Morse potentials.

Contribution

It identifies a new family of rationally-extended Morse potentials that are isospectral and exhibit an enlarged shape invariance, expanding the understanding of solvable quantum potentials.

Findings

Re-derivation of known extended Morse potentials.

Discovery of a new isospectral extended potential family.

Application of point canonical transformation to connect radial oscillator and Morse potentials.

Abstract

The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials $V_{A,B,{\rm ext}}(x)$, obtained from a conventional Morse potential $V_{A-1,B}(x)$ by the addition of a bound state below the spectrum of the latter, is re-obtained. More importantly, the existence of another family of extended potentials, strictly isospectral to $V_{A+1,B}(x)$, 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.