On Well-Posedness of a Boltzmann-Like Semiconductor Model

Abstract

The paper deals with an analysis of well-posedness of the Boltzmann-like semiconductor equation with unbounded collision frequency, introduced recently by Majorana and Milazzo.17The equation is derived by writing the balance of the electrons lost and gained at each energy level due to scattering on the crystalline lattice of the semiconductor. As the total amount of electrons is expected to be constant, the process can be viewed as a Markov process, and from the functional analytic point of view it fits into the general theory of substochastic semigroups.5,26In this paper we present two methods of solving the evolution equation describing this process: one is a generalization of the approach of Reuter and Ledermann23to solving differential equations governing Markov processes with denumerably many states, while the other is based on the Kato–Voigt perturbation technique for substochastic semigroups.15,26,2,3,5The combination of these two techniques is a powerful tool yielding strong results on the existence and uniqueness of conservative solutions. It is also shown how the solution method employed in Ref. 17 fits into the theory developed in this paper.