theory as our guide, wherever possible. We will also exploit certain special features of the scattering problem not found in generic bound state systems, such as the asymptotic potential condition $V'(x) \ra 0$ as $x \ra \pm \infty$ (where primes denote spatial differentiation), and its usual implications for scattering theory and applications.[@taylor]

The basic WKB solutions are given by \^(x) = r\_(x) e\^[i s\_(x)/]{}, \[scsoln\] where r\_(x) = s’\_(x) = \[scrs\] The corresponding positive and negative momentum functions, specifying the semiclassical LMs, are given by $p^\sc_\pm(x) = \pm
s'_\sc(x)$. Equations (\[scsoln\]) and (\[scrs\]) apply to both bound and scattering cases; note that for both, $\Ppm^\sc(x)$ are complex conjugates of each other. The asymptotic potential condition ensures that these approach exact quantum plane waves asymptotically, with the usual scattering interpretations, i.e. $\Psi_+(x)$ in the $x\ra-\infty$ asymptotic region is the incoming wave from the left (usually taken to be the incident wave), $\Psi_+(x)$ as $x\ra\infty$ is the outgoing wave from the left (the usual transmitted wave), etc.

Insofar as determining the corresponding exact quantum solutions $\Ppm(x)$, the procedure described in paper I is still appropriate for bound and semi-bound (i.e. on one side only) states, in that the results satisfy the correspondence 