  We study the unitary propagation of a two-particle one-dimensional
Schr\"odinger equation by means of the Split-Step Fourier method, to study the
coherent evolution of a spatially indirect exciton (IX) in semiconductor
heterostructures. The mutual Coulomb interaction of the electron-hole pair and
the electrostatic potentials generated by external gates and acting on the two
particles separately are taken into account exactly in the two-particle
dynamics. As relevant examples, step/downhill and barrier/well potential
profiles are considered. The space- and time-dependent evolution during the
scattering event as well as the asymptotic time behavior are analyzed. For
typical parameters of GaAs-based devices the transmission or reflection of the
pair turns out to be a complex two-particle process, due to comparable and
competing Coulomb, electrostatic and kinetic energy scales. Depending on the
intensity and anisotropy of the scattering potentials, the quantum evolution
may result in excitation of the IX internal degrees of freedom, dissociation of
the pair, or transmission in small periodic IX wavepackets due to dwelling of
one particle in the barrier region. We discuss the occurrence of each process
in the full parameter space of the scattering potentials and the relevance of
our results for current excitronic technologies.
