Comprehensive analysis of conditionally exactly solvable models

TL;DR

The paper reexamines the Dutt–Khare–Varshni conditionally exactly solvable potential through both the point canonical transformation and a supersymmetric quantum mechanics lens, clarifying how the bound-state spectrum emerges from a cubic energy equation. It proves that only the middle root of the cubic yields physical bound states under proper normalizability, correcting previous ambiguities. By embedding the CES potential within the Natanzon class, it shows the model is an exactly solvable system and harmonizes the PCT and SUSY perspectives within a unified framework. This strengthens the classification of CES potentials and clarifies their connections to Natanzon-type solvable models and Jacobi-polynomial solutions.

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.

Figures (1)

• Figure 1: Graphical solution of Eq. (\ref{['eu']}).