Comprehensive analysis of conditionally exactly solvable models

TL;DR

This paper thoroughly examines a conditionally exactly solvable quantum potential, providing alternative constructions, analyzing bound states, and establishing its classification as an exactly solvable Natanzon-class potential.

Contribution

It introduces a supersymmetric approach to a CES model and clarifies its status as a Natanzon-class exactly solvable potential.

Findings

Only one root of the cubic equation yields physical bound states.

The CES potential is classified as a Natanzon-class exactly solvable potential.

Alternative supersymmetric construction of the potential is provided.

Abstract

We study a quantum mechanical potential introduced previously as a conditionally exactly solvable (CES) model. Besides an analysis following its original introduction in terms of the point canonical transformation, we also present an alternative supersymmetric construction of it. We demonstrate that from the three roots of the implicit cubic equation defining the bound-state energy eigenvalues, there is always only one that leads to a meaningful physical state. Finally we demonstrate that the present CES interaction is, in fact, an exactly solvable Natanzon-class potential.