On the existence of propagators in stationary Wigner equation without velocity cut-off

If Ref. [2], a parity decomposition method was applied to recast the one-di mensional, stationary Wigner equation with inflow boundary conditions into two decoupled evolution equations but with coupling remaining in the initia l conditions. The singularity introduced by the division by the velocity nu 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 ope rators introduced in [2] generate evolution families in suitably weighted L -2 spaces, without introducing the velocity cut-off. These spaces are diffe rent for each equation, though there is a common space in which both evolut ions take place provided the initial conditions are appropriately selected. This allows to solve the two evolution equations but falls short of provid ing the solution to the original problem with complete inflow boundary cond itions.