On the existence of propagators in stationary Wigner equation without velocity cutoff

In a previous work by Barletti and Zweifek [2], a parity decomposition method was applied to recast the one-dimensional, stationary Wigner equation with inflow boundary conditions into two decoupled evolution equations but with coupling remaining in the initial conditions. The singularity introduced by the division by the velocity v forced the authors to perform the analysis in a simplified situation with the velocity cut-off close to zero. In this note we shall show that the operators introduced in [2] generate evolution families in suitably weighted L^2 spaces, without introducing the velocity cut-off. These spaces are different for each equation, though there is a common space in which both evolutions take place provided the initial conditions are appropriately selected. This allows to solve the two evolution equations but falls short of providing the solution to the original problem with complete inflow boundary conditions.