Dissipative dynamics in semiconductors at low temperature

A mathematical model is introduced which describes the dissipation of electrons in lightly doped semi-conductors. The dissipation operator is proved to be densely defined and positive and to generate a Markov semigroup of operators. The spectrum of the dissipation operator is studied and it is shown that zero is a simple eigenvalue, which makes the equilibrium state unique. Also it is shown that there is a gap between zero and the rest of its spectrum which makes the return to equilibrium exponentially fast in time.