 
Summary: THE RELAXATIONTIME WIGNER EQUATION
Anton Arnold
Abstract. The relaxation{time (RT) Wigner equation models the quantum{mechanical mo
tion of electrons in an electrostatic eld, including their interaction with phonons. We discuss
the conditions on a Wigner distribution function for being `physical', and show that they will
stay `physical' under temporal evolution. Particular attention is paid to the proper de nition
of the particle density for Wigner functions w =2 L1. For the 1D{periodic, self{consistent
RT{Wigner{Poisson equation we give a local convergence result towards the steady state.
1. Introduction. This paper is concerned with the analysis of the relaxation{time Wigner
equation and the physical propertiesof its solution. The Wigner formalism, which represents
a phase{space description of quantum mechanics, has in recent years attracted considerable
attention of solid state physicists for including quantum e ects into the simulation of ultra{
integrated semiconductor devices, like resonant tunneling diodes, e.g. ( 7], 10], 5]). Also,
the Wigner ({Poisson) equation has recently been the objective of a detailed mathematical
analysis. For a physical derivation and the discussion of many of its analytical properties
we refer the reader to 15], 12], 6] (and references therein).
The real{valued Wigner (quasi) distribution function w = w(x v t) describes the state
of an electron ensemble in the 2d{dimensional position{velocity (x v){phase space. In
the absence of collision and scattering, and in the e ective{mass approximation, its time
evolution under the action of the (real{valued) electrostatic potential V(x t) is governed by
