Abstract A set of continuum and discrete eigenfunctions, derived previously for half-space transport problems with a scattering ratio that is a bilinear function of position, is examined. For a Holder continuous angular flux incident on the half-space, this eigenfunction set is proved to be half-range complete and the expansion coefficient for the continuum eigenfunctions is proved to be a continuous function of the spectral variable Y. Also, equations for the exiting flux, which are derived using this eigenfunction set, are proved to possess a unique, analytic solution.