principle globally, as desired (for semi-bound examples, consult the Appendix). For true scattering states, however, this procedure fails, in the sense that if $\Psi_+(x)$ is chosen to match the normalization and flux of $\Psi^\sc_+(x)$ in the $x\ra\infty$ asymptote, then it will necessarily approach a nontrivial linear superposition of $\Psi^\sc_+(x)$ and $\Psi^\sc_-(x)$ in the $x \ra -\infty$ asymptote, and vice-versa. There is therefore an ambiguity as to how the corresponding quantum $\Ppm(x)$’s should be defined, i.e. which asymptotic region should be used to effect the correspondence. More significantly though, [*either*]{} choice will result in component functions $\Ppm(x)$ with substantial interference in one of the two asymptotic regions. This is due to partial reflection of the exact quantum scattering states, which is not predicted by the basic WKB approximation. Thus, in the large action limit, the exact quantum solutions manifest large-magnitude quantum potentials, $q_\pm(x)$, and rapidly oscillating field functions $q_\pm(x)$, $r_\pm(x)$, and $p_\pm(x)$—exactly the undesirable behavior that the CPWM was introduced to avoid—whereas the corresponding basic WKB functions are smooth, and asymptotically uniform.

The lack of any partial reflection is a well-understood shortcoming of the WKB approximation[@berry72; @froman; @heading; @poirier03capI]—i.e., the basic $\Ppm^\sc(x)$ components, though elegantly constructed from smooth classical functions $r_\sc(x)$ 