  The direct and inverse scattering problems on the full line are analyzed for
a first-order system of ordinary linear differential equations associated with
the derivative nonlinear Schr\"odinger equation and related equations. The
system contains a spectral parameter and two potentials, where the potentials
are proportional to the spectral parameter and hence are called
energy-dependent potentials. Using the two potentials as input, the direct
problem is solved by determining the scattering coefficients and the
bound-state information consisting of bound-state energies, their
multiplicities, and the corresponding norming constants. By using two different
methods, the corresponding inverse problem is solved by determining the two
potentials when the scattering data set is used as input. The first method
involves the transformation of the energy-dependent system into two distinct
energy-independent systems. The second method involves the establishment of the
so-called alternate Marchenko system of linear integral equations and the
recovery of the energy-dependent potentials from the solution to the alternate
Marchenko system.
