Generation of a Novel Exactly Solvable Potential

TL;DR

This paper introduces a new exactly solvable shape invariant extension of the Morse potential with unique properties, expanding the class of known isospectral potentials in quantum mechanics.

Contribution

It presents a novel shape invariant Morse potential extension that differs in asymptotic behavior and does not follow standard isospectral deformation structures.

Findings

Same eigenenergies as Morse potential

Different asymptotic limits from Morse

Does not conform to standard isospectral deformation structure

Abstract

We report a new shape invariant (SI) isospectral extension of the Morse potential. Previous investigations have shown that the list of "conventional" SI superpotentials that do not depend explicitly on Planck's constant $\hbar$ is complete. Additionally, a set of "extended" superpotentials has been identified, each containing a conventional superpotential as a kernel and additional $\hbar$-dependent terms. We use the partial differential equations satisfied by all SI superpotentials to find a SI extension of Morse with novel properties. It has the same eigenenergies as Morse but different asymptotic limits, and does not conform to the standard generating structure for isospectral deformations.