New Shape Invariant Potentials in Supersymmetric Quantum Mechanics

TL;DR

This paper introduces a broad class of new shape invariant potentials in supersymmetric quantum mechanics, characterized by reflectionlessness and infinite bound states, extending known solutions through a scaling ansatz and q-deformations.

Contribution

The authors develop a scaling ansatz to generate new shape invariant potentials, including reflectionless and q-deformed solutions, expanding the set of exactly solvable quantum potentials.

Findings

New shape invariant potentials with infinite bound states

Potentials are reflectionless and include q-deformations

Explicit energy eigenvalues and transmission coefficients provided

Abstract

Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, we obtain a large class of new shape invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in our potentials as a special case is the self-similar potential recently discussed by Shabat and Spiridonov.