  This paper studies the scattering matrix $\Sigma(E;\hbar)$ of the problem \[
  -\hbar^2 \psi''(x) + V(x) \psi(x) = E\psi(x) \] for positive potentials $V\in
C^\infty(\R)$ with inverse square behavior as $x\to\pm\infty$. It is shown that
each entry takes the form
$\Sigma_{ij}(E;\hbar)=\Sigma_{ij}^{(0)}(E;\hbar)(1+\hbar \sigma_{ij}(E;\hbar))$
where $\Sigma_{ij}^{(0)}(E;\hbar)$ is the WKB approximation relative to the
{\em modified potential} $V(x)+\frac{\hbar^2}{4} \la x\ra^{-2}$ and the
correction terms $\sigma_{ij}$ satisfy $|\partial_E^k \sigma_{ij}(E;\hbar)| \le
C_k E^{-k}$ for all $k\ge0$ and uniformly in $(E,\hbar)\in (0,E_0)\times
(0,\hbar_0)$ where $E_0,\hbar_0$ are small constants. This asymptotic behavior
is not universal: if $-\hbar^2\partial_x^2 + V$ has a {\em zero energy
resonance}, then $\Sigma(E;\hbar)$ exhibits different asymptotic behavior as
$E\to0$. The resonant case is excluded here due to $V>0$.
