  We study the stationary Wigner equation on a bounded, one-dimensional spatial
domain with inflow boundary conditions by using the parity decomposition in
(Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). The
decomposition reduces the half-range, two-point boundary value problem into two
decoupled initial value problems of the even part and the odd part. Without
using a cutoff approximation around zero velocity, we prove that the initial
value problem for the even part is well-posed. For the odd part, we prove the
uniqueness of the solution in the odd $L^2$-space by analyzing the moment
system. An example is provided to show that how to use the analysis to obtain
the solution of the stationary Wigner equation with inflow boundary conditions.
