Rational solutions for the Riccati-Schr\"odinger equations associated to translationally shape invariant potentials

TL;DR

This paper introduces a novel method to construct eigenfunctions of translationally shape-invariant potentials using continued fractions, providing explicit formulas tailored to different potential classes.

Contribution

It presents a new approach to derive eigenfunctions via continued fractions, enhancing understanding of shape-invariant potentials and their eigenstates.

Findings

Eigenfunctions expressed as terminating continued fractions.

Explicit formulas for eigenstates depending on potential class.

Applicable to all potentials within the Barclay-Maxwell class.

Abstract

We develop a new approach to build the eigenfunctions of a translationally shape-invariant potential. For this we show that their logarithmic derivatives can be expressed as terminating continued fractions in an appropriate variable. We give explicit formulas for all the eigenstates, their specific form depending on the Barclay-Maxwell class to which the considered potential belongs.