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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 [A. Arnold, H. Lange, and P.F. Zweifel, J. Math. Phys., 41 (2000), pp. 7167--7180] we prove without any additional prerequisite conditions that the solution of the Wigner equation with inflow boundary conditions will be symmetric only if the potential is symmetric. This improves the result in [D. Taj, L. Genovese, and F. Rossi, Europhys. Lett., 74 (2006), pp. 1060--1066], which depends on the convergence of the 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, contrary to the argument given in [D. Taj, L. Genovese, and F. Rossi, Europhys. Lett., 74 (2006), pp. 1060--1066].