On the spectrum of the linear transport operator in a semi-infinite medium

The linear mono-energetic Boltzmann equation with isotropic scattering is considered for a semi-infinite medium in plane geometry and the spectrum of the corresponding operator under perfectly reflecting, vacuum, generalized or diffusely reflecting boundary conditions is explored in the frame of the 'initial-value problem'. By the Hille-Yosida theorem, the existence and uniqueness of the solutions of these problems are assured. As a common feature, one observes the absence of a true isolated asymptotic eigenmode, the solution displaying, due to the infinite extent of the medium, only 'transient' modes.