Solvable rational extensions of the isotonic oscillator

TL;DR

This paper presents a straightforward method to generate infinite families of regular rational solvable extensions of the isotonic oscillator potential using Riccati-Schrödinger solutions, Bäcklund transformations, and symmetries.

Contribution

It introduces a direct approach combining Riccati-Schrödinger equations, Bäcklund algorithms, and symmetries to construct infinite sets of solvable extensions of the isotonic oscillator.

Findings

Generated three infinite sets of rational extensions (L1, L2, L3)

Provided a transparent method for constructing these extensions

Enhanced understanding of shape invariant potentials

Abstract

Combining recent results on rational solutions of the Riccati-Schr\"odinger equations for shape invariant potentials to the finite difference B\"acklund algorithm and specific symmetries of the isotonic potential, we show that it is possible to generate the three infinite sets (L1, L2 and L3 families) of regular rational solvable extensions of this potential in a very direct and transparent way.