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

Based on the well-posedness of the stationary Wigner equation with inflow boundary conditions given in [1], 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 [6] 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 [6].