# Analytical solutions for the radial Scarf II potential

## TL;DR

This paper derives analytical bound and resonance solutions for the radial Scarf II potential, connecting it to its one-dimensional form and exploring its spectral properties and potential applications.

## Contribution

It provides explicit analytical solutions for the radial Scarf II potential and links these to the one-dimensional case, expanding understanding of their spectral characteristics.

## Key findings

- Derived bound and resonance solutions for the radial Scarf II potential.
- Connected radial solutions to the one-dimensional Scarf II potential.
- Analyzed the spectral properties and potential applications of the radial Scarf II potential.

## Abstract

The real Scarf II potential is discussed as a radial problem. This potential has been studied extensively as a one-dimensional problem, and now these results are used to construct its bound and resonance solutions for $l=0$ by setting the origin at some arbitrary value of the coordinate. The solutions with appropriate boundary conditions are composed as the linear combination of the two independent solutions of the Schr\"odinger equation. The asymptotic expression of these solutions is used to construct the $S_0(k)$ s-wave $S$-matrix, the poles of which supply the $k$ values corresponding to the bound, resonance and anti-bound solutions. The location of the discrete energy eigenvalues is analyzed, and the relation of the solutions of the radial and one-dimensional Scarf II potentials is discussed. It is shown that the generalized Woods--Saxon potential can be generated from the Rosen--Morse II potential in the same way as the radial Scarf II potential is obtained from its one-dimensional correspondent. Based on this analogy, possible applications are also pointed out.