On the exact solubility in momentum space of the trigonometric Rosen-Morse potential

TL;DR

This paper demonstrates the exact transformability of the trigonometric Rosen-Morse potential's solutions from position to momentum space in Euclidean space, highlighting the algebraic nature of the resulting equations and the potential for numerical wavefunction solutions.

Contribution

It shows that the Rosen-Morse potential's solutions can be exactly transformed to momentum space, revealing algebraic equations and discussing the potential for numerical solutions.

Findings

Exact solutions in position space can be transformed to momentum space.

The resulting momentum space equation is algebraic, not integral.

Numerical methods are needed for wavefunctions in some cases.

Abstract

The Schrodinger equation with the trigonometric Rosen-Morse potential in flat three dimensional Euclidean space, E3, and its exact solutions are shown to be also exactly transformable to momentum space, though the resulting equation is purely algebraic and can not be cast into the canonical form of an integral Lippmann-Schwinger equation. This is because the cotangent function does not allow for an exact Fourier transform in E3. In addition we recall, that the above potential can be also viewed as an angular function of the second polar angle parametrizing the three dimensional spherical surface, S3, of a constant radius, in which case the cotangent function would allow for an exact integral transform to momentum space. On that basis, we obtain a momentum space Lippmann-Schwinger-type equation, though the corresponding wavefunctions have to be obtained numerically.