Conservative and non-conservative Boltzmann-type models of semiconductor theory

In this paper we analyze the linear Boltzmann equation of semiconductor theory with unbounded collision term modelling both elastic and inelastic scattering of electrons on the crystalline lattice (corresponding to scattering on impurities and optical phonons), in both bounded and unbounded domains. We prove the existence of a substochastic semigroup solving this problem and, for a large class of scattering cross-sections, we also characterize the generator of this semigroup as the closure of the formal right-hand side operator showing thus that the semigroup is conservative (stochastic) in this case. On the other hand, we provide an example of a cross-section growing at an exponential rate for which the semigroup is not conservative.