Summary: Operator Splitting Methods Applied to Spectral Discretizations
of Quantum Transport Equations
Anton ARNOLD and Christian RINGHOFER
Fachbereich Mathematik, MA 6-2, TU-Berlin, Stra e des 17. Juni 136, D-10623 Berlin, Germany.
Department of Mathematics, Arizona State University, Tempe, AZ 85287
Typeset by AMS-TEX
Abstract: The Wigner-Poisson equation is the most successful basis for transient simulations of
quantum-e ect semiconductor devices so far. We present a full discretization of this nonlinear
pseudo-di erential system using a mixed spectral-collocation and operator splitting method. Con-
vergence and nonlinear stability of the scheme are proven.
Keywords: operator splitting methods, collocation methods, Wigner functions AMS (MOS)
subject classi cation: 65M70, 81S30
In this paper we present and analyze an operator splitting method for the solution of the Wigner-
Poisson system. This system consists of the Wigner (or quantum Liouville-) equation coupled to the
Poisson equation. The Wigner equation is a pseudo di erential equation of the form
@tw + v rxw + V]w = 0: (1.1)