  Based on the well-posedness of the stationary Wigner equation with inflow
boundary conditions given in (A. Arnold, H et al. J. Math. Phys., 41, 2000), we
prove without any additional prerequisite conditions that the solution of the
Wigner equation with symmetric potential and inflow boundary conditions will be
symmetric. 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 to
the symmetric profile for three different numerical schemes. This implies that
the upwind schemes can also yield a symmetric numerical solution, on the
contrary to the argument given in (D. Taj et al. Europhys. Lett., 74, 2006).
