Continuum energy eigenstates via the factorization method

TL;DR

This paper extends the factorization method to continuum energy eigenstates in quantum mechanics, introducing a single-shot approach that leverages confluent hypergeometric functions for solving the Ricatti equation.

Contribution

It generalizes the factorization method to continuum states using a novel single-shot approach involving confluent hypergeometric functions.

Findings

Enables solving continuum energy eigenstates with a new factorization technique

Provides an alternative to differential equation methods for quantum solutions

Requires knowledge of confluent hypergeometric functions

Abstract

The factorization method was introduced by Schroedinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of auxiliary Hamiltonians that share common eigenvalues with their adjacent Hamiltonians in the chain, except for the lowest eigenvalue. In this work, we generalize the factorization method to continuum energy eigenstates. Here, one does not generically have a factorization chain -- instead all energies are solved using a "single-shot factorization," enabled by writing the superpotential in a form that includes the logarithmic derivative of a confluent hypergeometric function. The single-shot factorization approach is an alternative to the conventional method of "deriving a differential equation and looking up its solution," but it does require some working knowledge of confluent hypergeometric functions. This can also be viewed as a method for solving the Ricatti equation needed to construct the superpotential.