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Arnold, Anton The Relaxation-Time von Neumann-Poisson Equation

Summary: Arnold, Anton
The Relaxation-Time von Neumann-Poisson Equation
This paper is concerned with the relaxation-time von Neumann-Poisson (or quantum Liouville-Poisson) equation in
three spatial dimensions which describes the self-consistent time evolution of an open quantum mechanical system
that include some relaxation mechanism. This model and the equivalent relaxation-time Wigner-Poisson system play
an important role in the simulation of quantum semiconductor devices.
We prove that the evolution is a positivity preserving map, satisfying the Lindblad condition. The nonlinear
evolution problem is formulated as an abstract Cauchy problem in the space of Hermitian trace class operators.
For initial density matrices with nite kinetic energy we prove the global-in-time existence and uniqueness of mild
solutions. A key ingredient for our analysis is a new generalization of the Lieb-Thirring inequality for density matrix
operators. We also present a local convergence result towards the steady state of the system.
1. Introduction
In this paper we shall discuss the relaxation-time von Neumann-Poisson (RT ;vNP ) equation, which is the simp-
lest physically relevant quantum mechanical model to account for electron{phonon scattering. Together with the
equivalent RT -Wigner-Poisson equation, it is an important model for the numerical simulation of ultra-integrated
semiconductor devices, like resonant tunneling diodes ( 11], 6], 7]). Here, we will mainly focus on existence and
uniqueness results for this problem in three spatial dimensions, and on the large-time behavior of its solution. For
results on the (technically much simpler) RT -Wigner-Poisson system in one dimension with space-periodic boundary
conditions we refer to 1].
With a constant RT > 0 the RT ; vNP system is the following time evolution equation for the density


Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien


Collections: Mathematics