Convergency of the Monte Carlo algorithm for the solution of the Wigner quantum-transport equation

The Wigner function provides a convenient description for single-particle quantum transport in space dependent systems, such as modern nanoelectronic devices. A Monte Carlo algorithm has been recently introduced for the solution of this integro-differential equation. However, when the potential applied to the system has different limits at + and −∞, a convergence problem arises for the kernel of the integral part of the equation. In this paper, we discuss the rigorous mathematical aspects of the convergency of the solution of the Wigner equation and of the Neumann expansion on which the Monte Carlo algorithm is based.