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Based on the well-posedness of the stationary Wigner equation with inflowboundary conditions given in (A. Arnold, H et al. J. Math. Phys., 41, 2000), weprove without any additional prerequisite conditions that the solution of theWigner equation with symmetric potential and inflow boundary conditions will besymmetric. This improve the result in (D. Taj et al. Europhys. Lett., 74, 2006)which depends on the convergence of solution formulated in the Neumann series.By numerical studies, we present the convergence of the numerical solution tothe symmetric profile for three different numerical schemes. This implies thatthe upwind schemes can also yield a symmetric numerical solution, on thecontrary to the argument given in (D. Taj et al. Europhys. Lett., 74, 2006).

Stationary Wigner Equation with Inflow Boundary Conditions: Will a Symmetric Potential Yield a Symmetric Solution?