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 $V>0$.
Categories: math-ph math.MP