Exact Solvability of Two-Dimensional Real Singular Morse Potential

TL;DR

This paper demonstrates the exact solvability of a two-dimensional Morse potential using supersymmetric methods, providing explicit bound state energies and wave functions, and extending results through shape invariance.

Contribution

It introduces a supersymmetric approach to solve a non-separable two-dimensional Morse potential and extends the solution to a hierarchy of models.

Findings

Explicit bound state energies are derived.

All wave functions are constructed analytically.

The model is completely integrable despite non-separability.

Abstract

The supersymmetric approach in the form of second order intertwining relations is used to prove the exact solvability of two-dimensional Schrodinger equation with generalized two-dimensional Morse potential for $a_0=-1/2$. This two-parametric model is not amenable to conventional separation of variables, but it is completely integrable: the symmetry operator of fourth order in momenta exists. All bound state energies are found explicitly, and all corresponding wave functions are built analytically. By means of shape invariance property, the result is extended to the hierarchy of Morse models with arbitrary integer and half-integer values $a_k=-(k+1)/2.$