  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.
