Second Order Derivative Supersymmetry and Scattering Problem

TL;DR

This paper extends supersymmetric quantum mechanics to include higher order derivatives, explores scattering amplitudes, and introduces a q-deformed algebra, providing explicit models with reflectionless potentials and infinite bound states.

Contribution

It introduces higher order derivative supercharges and a polynomial algebra in supersymmetric quantum mechanics, along with explicit scattering models and a q-deformed algebraic structure.

Findings

Derived a higher order polynomial supersymmetry algebra.

Constructed explicit scattering amplitude models using hypergeometric functions.

Identified reflectionless potentials with infinitely many bound states.

Abstract

Extensions of standard one-dimensional supersymmetric quantum mechanics are discussed. Supercharges involving higher order derivatives are introduced leading to an algebra which incorporates a higher order polynomial in the Hamiltonian. We study scattering amplitudes for that problem. We also study the role of a dilatation of the spatial coordinate leading to a q-deformed supersymmetric algebra. An explicit model for the scattering amplitude is constructed in terms of a hypergeometric function which corresponds to a reflectionless potential with infinitely many bound states.